Lateral Alignment of Epitaxial Quantum Dots

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NanoScience and Technology

NanoScience and Technology Series Editors: P. Avouris B. Bhushan D. Bimberg K. von Klitzing H. Sakaki R. Wiesendanger The series NanoScience and Technology is focused on the fascinating nano-world, mesoscopic physics, analysis with atomic resolution, nano and quantum-effect devices, nanomechanics and atomic-scale processes. All the basic aspects and technologyoriented developments in this emerging discipline are covered by comprehensive and timely books. The series constitutes a survey of the relevant special topics, which are presented by leading experts in the ﬁeld. These books will appeal to researchers, engineers, and advanced students. Applied Scanning Probe Methods I Editors: B. Bhushan, H. Fuchs, and S. Hosaka Nanostructures Theory and Modeling By C. Delerue and M. Lannoo Nanoscale Characterisation of Ferroelectric Materials Scanning Probe Microscopy Approach Editors: M. Alexe and A. Gruverman Magnetic Microscopy of Nanostructures Editors: H. Hopster and H.P. Oepen Silicon Quantum Integrated Circuits Silicon-Germanium Heterostructure Devices: Basics and Realisations By E. Kasper, D.J. Paul The Physics of Nanotubes Fundamentals of Theory, Optics and Transport Devices Editors: S.V. Rotkin and S. Subramoney Single Molecule Chemistry and Physics An Introduction By C. Wang, C. Bai Atomic Force Microscopy, Scanning Nearﬁeld Optical Microscopy and Nanoscratching Application to Rough and Natural Surfaces By G. Kaupp

Applied Scanning Probe Methods II Scanning Probe Microscopy Techniques Editors: B. Bhushan, H. Fuchs Applied Scanning Probe Methods III Characterization Editors: B. Bhushan, H. Fuchs Applied Scanning Probe Methods IV Industrial Application Editors: B. Bhushan, H. Fuchs Nanocatalysis Editors: U. Heiz, U. Landman Roadmap of Scanning Probe Microscopy Editors: S. Morita Nanostructures – Fabrication and Analysis Editor: H. Nejo Applied Scanning Probe Methods V Scanning Probe Microscopy Techniques Editors: B. Bhushan, H. Fuchs, S. Kawata Applied Scanning Probe Methods VI Characterization Editors: B. Bhushan, S. Kawata Applied Scanning Probe Methods VII Biomimetics and Industrial Applications Editors: B. Bhushan, H. Fuchs Fundamentals of Friction and Wear Editors: E. Gnecco, E. Meyer

Oliver G. Schmidt

Lateral Alignment of Epitaxial Quantum Dots With 446 Figures

123

Editor: Prof. Dr. Oliver G. Schmidt Prof. Dr. Oliver G. Schmidt Institute for Integrative Nanosciences IFW Dresden Helmholtzstrasse 20 01069 Dresden, Germany e-mail: [email protected]

Series Editors: Professor Dr. Phaedon Avouris IBM Research Division Nanometer Scale Science & Technology Thomas J. Watson Research Center, P.O. Box 218 Yorktown Heights, NY 10598, USA

Professor Bharat Bhushan Nanotribology Laboratory for Information Storage and MEMS/NEMS (NLIM) W 390 Scott Laboratory, 201 W. 19th Avenue The Ohio State University, Columbus Ohio 43210-1142, USA

Professor Dr. Dieter Bimberg

Professor Dr., Dres. h. c. Klaus von Klitzing Max-Planck-Institut für Festkörperforschung Heisenbergstrasse 1, 70569 Stuttgart, Germany

Professor Hiroyuki Sakaki University of Tokyo Institute of Industrial Science, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan

Professor Dr. Roland Wiesendanger Institut für Angewandte Physik Universität Hamburg Jungiusstrasse 11, 20355 Hamburg, Germany

TU Berlin, Fakutät Mathematik, Naturwissenschaften, Institut für Festkörperphysik Hardenbergstr. 36, 10623 Berlin, Germany

ISBN 978-3-540-46935-3 Springer Berlin Heidelberg New York Library of Congress Control Number: 2007923281 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Product liability: The publishers cannot guarantee the accuracy of any information about dosage and application contained in this book. In every individual case the user must check such information by consulting the relevant literature. Typesetting: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover: eStudioCalamar S.L., F. Steinen-Bro, Pau/Girona, Spain SPIN 11373360

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Introduction

The unique success story of semiconductor physics and technology relies on the ability to highly integrate micro- and nanometer sized functional units on a single chip. Within the last years epitaxial quantum dots have become such functional units and moved to the forefront of cutting edge research to study the exciting physics of single quantum structures and to fathom their tremendous potential for device applications. Quantum dots constitute a natural template to construct reﬁned artiﬁcial matter, such as artiﬁcial atoms, molecules and possibly artiﬁcial crystals with entirely new electronic and optical properties. However, the full advantage of their unique properties can be exploited, only, if a controlled positioning or growth of the quantum dots inside a more complex device structure or a precise coupling between the quantum dots and a macroscopic periphery can be achieved. The prime task of this book is to review recent techniques, which allow the controlled positioning and lateral alignment of quantum dots on standard substrate surfaces. The alignment techniques range from pure self-ordering mechanisms to advanced quantum dot growth on patterned substrates. In the former case, growth conditions, substrate orientations and layer sequences are optimized to achieve a high degree of lateral ordering. In the latter case, the nucleation centers of the quantum dots are deﬁned by appropriate pre-patterning of the substrate surfaces. This approach allows for an absolute positioning of the quantum dots relative to marker structures, which are necessary to deﬁne a device at the position of the quantum dot in subsequent processing steps. While this book clearly documents the great advance made in controlling the spatial position of quantum dots, there remain huge challenges that need rigorous tackling in future years. One of the biggest problems is the nonresonant energy spectrum of quantum dot ensembles, even if they are located in an apparently perfectly ordered array. The reason is that each quantum dot is slightly diﬀerent in size, shape and composition and therefore emits a photon with a diﬀerent energy. The question of “How identical are nanostructures and can we create identical nanostructures?” addresses many ﬁelds of today’s integrative nanotechnologies and is not inherent to quantum dots. For quantum dots, a solution might be a self-limiting growth mechanism or the manipulation of individual quantum dots after growth.

VI

Introduction

Part I of this book concentrates on the lateral self-alignment of epitaxial quantum dots. This self-alignment is realized by choosing appropriate growth conditions and special substrate surfaces. The self-alignment on a short range scale is exploited to create compact lateral quantum dot molecules. By stacking multiple quantum dot layers pronounced lateral ordering on a medium range scale is accomplished. The underlying growth mechanisms governing these phenomena are described and reviewed in detail in the ﬁrst part of the book. In the second Part the aim is to control the absolute lateral position of quantum dots on a long-range scale. Such long-range ordered quantum dot arrays might be useful for a high integration of single quantum dot devices, or to realize one, two, and three dimensional quantum dot crystals. Part II demonstrates that such artiﬁcial crystals can be created with high structural integrity and excellent optical quality. However, at the present stage the distances between quantum dots are too large as to observe new electronic band structures. My gratitude goes to all authors having composed the 26 chapters of this book. It took more than two years to put together this work, but I am sure the eﬀort was worthwhile and the book will serve as a helpful platform to understand the many fundamental questions of quantum dot growth as well as to further our eﬀorts to eventually integrate single quantum dots on a single chip. Oliver G. Schmidt

Contents

Part I Lateral Self-Alignment I.1 General 1 Physical Mechanisms of Self-Organized Formation of Quantum Dots V. Shchukin, D. Bimberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2 Routes Toward Lateral Self-Organization of Quantum Dots: the Model System SiGe on Si(001) C. Teichert, M.G. Lagally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 I.2 Compact Lateral Quantum Dot Conﬁgurations 3 Short-Range Lateral Ordering of GeSi Quantum Dots Due to Elastic Interactions J.A. Floro, R. Hull, J.L. Gray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4 Hierarchical Self-Assembly of Lateral Quantum-Dot Molecules Around Nanoholes A. Rastelli, R. Songmuang, S. Kiravittaya, O.G. Schmidt . . . . . . . . . . . . 103 I.3 Ordering in Single Layers 5 Energetics and Kinetics of Self-Organized Structure Formation in Solution Growth – the SiGe/Si System S.H. Christiansen, M. Schmidbauer, H. Wawra, R. Schneider, W. Neumann, H.P. Strunk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6 Ge Quantum Dot Self-Alignment on Vicinal Substrates I. Berbezier, A. Ronda, A. Karmous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7 Lateral Arrangement of Ge Self-Assembled Quantum Dots on a Partially Relaxed Six Ge1−x Buﬀer Layer H.-j. Kim,Y.-H. Xie, K.L. Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

VIII

Contents

8 Ordering of Wires and Self-Assembled Dots on Vicinal Si and GaAs (110) Cleavage Planes G. Abstreiter, D. Schuh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 I.4 Ordering by Layer Stacking 9 Stacking and Ordering in Self-Organized Quantum Dot Multilayer Structures G. Springholz, V. Holy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 10 Self-Organized Anisotropic Strain Engineering for Lateral Quantum Dot Ordering R. N¨ otzel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 11 Towards Quantum Dot Crystals via Multilayer Stacking on Diﬀerent Indexed Surfaces Z.M. Wang, G.J. Salamo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Part II Forced Alignment II.5 Growth on Shallow Modulated Surfaces II.5.1 SiGe Islands 12 One-, Two-, and Three-Dimensionally Ordered GeSi Islands Grown on Prepatterned Si (001) Substrates Z. Zhong, G. Bauer, O.G. Schmidt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 13 Ordered SiGe Island Arrays: Long Range Material Distribution and Possible Device Applications G.S. Kar, S. Kiravittaya, M. Stoﬀel, O.G. Schmidt . . . . . . . . . . . . . . . . . . 373 14 Nanoscale Lateral Control of Ge Quantum Dot Nucleation Sites on Si(001) Using Focused Ion Beam Implantation A. Portavoce, R. Hull, F.M. Ross . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 15 Ge Nanodroplets Self-Assembly on Focused Ion Beam Patterned Substrates I. Berbezier, A. Karmous, A. Ronda. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 16 Metallization and Oxidation Templating of Surfaces for Directed Island Assembly O.D. Dubon, J.T. Robinson, K.M. Itoh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

Contents

IX

II.5.2 InGaAs/GaAs Quantum Dots 17 Site Control and Selective-Area Growth Techniques of InAs Quantum Dots with High Density and High Uniformity K. Asakawa, S. Kohmoto, S. Ohkouchi, Y. Nakamura . . . . . . . . . . . . . . . 463 18 In(Ga)As Quantum Dot Crystals on Patterned GaAs(001) Substrates S. Kiravittaya, H. Heidemeyer, O.G. Schmidt . . . . . . . . . . . . . . . . . . . . . . . 489 II.6 Growth on Surface Modulations of High Amplitude II.6.1 SiGe Islands 19 Directed Arrangement of Ge Quantum Dots on Si Mesas by Selective Epitaxial Growth K.L. Wang, H.-j. Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 20 Directed Self-Assembly of Quantum Dots by Local-Chemical-Potential Control via Strain Engineering on Patterned Substrates H. Wang, F. Liu, M. Lagally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 21 Structural and Luminescence Properties of Ordered Ge Islands on Patterned Substrates L. Vescan, T. Stoica, E. Sutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 22 Formation of Si and Ge Nanostructures at Given Positions by Using Surface Microscopy and Ultrathin SiO2 Film Technology M. Ichikawa, A. Shklyaev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 II.6.2 III-V Quantum Dots 23 Pyramidal Quantum Dots Grown by Organometallic Chemical Vapor Deposition on Patterned Substrates E. Kapon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 24 Large-Scale Integration of Quantum Dot Devices on MBE-Based Quantum Wire Networks H. Hasegawa, T. Sato, S. Kasai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

X

Contents

25 GaAs and InGaAs Position-Controlled Quantum Dots Fabricated by Selective-Area Metalloorganic Vapor Phase Epitaxy T. Fukui, J. Motohisa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 26 Spatial InAs Quantum Dot Positioning in GaAs Microdisk and Posts G.S. Solomon, Z. Xie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691

List of Contributors

Gerhard Abstreiter Walter Schottky Institut TU M¨ unchen, 85748 Garching Germany Kiyoshi Asakawa TARA Center University of Tsukuba 1-1-1, Ten-noudai Tsukuba 305-8577, Japan G¨ unther Bauer Institute for Semiconductor and Solid State Physics Johannes Kepler University Altenbergerstr. 69, 4040 Linz Austria I. Berbezier L2MP UMR CNRS 6137 Polytech’Marseille - Technopole de Chˆateau Gombert 13451 Marseille Cedex 20, France Dieter Bimberg Technische Universit¨ at Berlin Institut f¨ ur Festk¨ orperphysik Hardenbergstr. 36, 10623 Berlin Germany Silke H. Christiansen Max-Planck-Institute of Microstructure Physics Weinberg 2, 06120 Halle/Saale Germany

Oscar D. Dubon, Jr. Department of Materials Science and Engineering University of California, Berkeley CA 94720, USA Jerrold A. Floro Sandia National Laboratories Surface and Interface Sciences Department, Albuquerque NM 87185-1415, USA Takashi Fukui Graduate School of Information Science and Technology, and Research Center for Integrated Quantum Electronics, Hokkaido University N14, W9, Sapporo, 060-0814 Japan Jennifer L. Gray University of Virginia Department of Materials Science and Engineering 116 Engineers Way, Charlottesville VA 22904-4745, USA Hideki Hasegawa Research Center for Integrated Quantum Electronics (RCIQE) and Graduate School of Information Science and Technology Hokkaido University N 13, W 8, Sapporo 060-8628, Japan

XII

List of Contributors

H. Heidemeyer Max-Planck-Institut f¨ ur Festk¨ orperforschung Heisenbergstrasse 1, 70569 Stuttgart Germany V. Holy Department of Electronic Structures Faculty of Mathematics and Physics Charles University 12116 Prague, Czech Republic Robert Hull University of Virginia Department of Materials Science and Engineering, 116 Engineers Way Charlottesville, VA 22904, USA Masakazu Ichikawa Quantum-Phase Electronic Center Department of Applied Physics Graduate School of Engineering The University of Tokyo 7-3-1 Hongo Bunkyo-ku Tokyo 113-8656, Japan Kohei M. Itoh Keio University, 3-14-1 Hiyoshi Kouhoku-Ku, Yokohama 223-8522, Japan Eli Kapon Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Laboratory of Physics of Nanostructures Lausanne CH-1015, Switzerland Gouri S. Kar Max-Planck-Institut f¨ ur Festk¨ orperforschung Heisenbergstrasse 1, 70569 Stuttgart Germany

A. Karmous L2MP UMR CNRS 6137 Polytech’Marseille - Technopole de Chˆateau Gombert 13451 Marseille Cedex 20 France Seiya Kasai Research Center for Integrated Quantum Electronics (RCIQE) and Graduate School of Information Science and Technology Hokkaido University N 13, W 8, Sapporo 060-8628, Japan Hyung-jun Kim Department of Electrical Engineering University of California Los Angeles USA Suwit Kiravittaya Max-Planck-Institut f¨ ur Festk¨ orperforschung Heisenbergstrasse 1, 70569 Stuttgart Germany Shigeru Kohmoto System Devices Research Labs NEC Corporation 2-9-1 Seiran, Ohtsu Shiga 520-0833, Japan Max Lagally Department of Materials Science and Engineering University of Wisconsin-Madison 1509 University Avenue, Madison WI 53706, USA Feng Liu Department of Materials Science and Engineering, University of Utah Salt Lake City, UT 84112, USA

List of Contributors

Junichi Motohisa Graduate School of Information Science and Technology, and Research Center for Integrated Quantum Electronics, Hokkaido University N14, W9, Sapporo, 060-0814 Japan Yusui Nakamura Department of Electrical and Computer Engineering Kumamoto University, 2-39-1, Kurokami Kumamoto 300-2635, Japan W. Neumann Humboldt-University of Berlin Institute of Physics Newtonstraße 15 12489 Berlin, Germany Richard N¨ otzel eiTT/COBRA Inter-University Research Institute Eindhoven University of Technology 5600 MB Eindhoven The Netherlands Shunsuke Ohkouchi Fundamental and Environmental Research Laboratories NEC Corporation 34, Miyukigaoka, Tsukuba Ibaraki 305-8501, Japan Alain Portavoce University of Virginia Department of Materials Science and Engineering, 116 Engineers Way Charlottesville, VA 22904, USA Armando Rastelli Max-Planck-Institut f¨ ur Festk¨ orperforschung Heisenbergstr. 1, 70569 Stuttgart Germany

XIII

Jeremy T. Robinson Department of Materials Science and Engineering University of California Berkeley, CA 94720, USA A. Ronda L2MP UMR CNRS 6137 Polytech’Marseille - Technopole de Chˆateau Gombert 13451 Marseille Cedex 20 France Frances M. Ross IBM Research Division T.J. Watson Research Center 1101 Kitchawan Road, Yorktown Heights, NY 10598, USA Gregory J. Salamo Department of Physics University of Arkansas Fayetteville, Arkansas 72701 USA Taketomo Sato Research Center for Integrated Quantum Electronics (RCIQE) and Graduate School of Information Science and Technology Hokkaido University N 13, W 8, Sapporo 060-8628, Japan Oliver G. Schmidt Max-Planck-Institut f¨ ur Festk¨ orperforschung Heisenbergstr. 1, 70569 Stuttgart Germany M. Schmidbauer Institute of Crystal Growth Max-Born-Straße 2 12489 Berlin, Germany

XIV

List of Contributors

R. Schneider Humboldt-University of Berlin Institute of Physics Newtonstraße 15 12489 Berlin, Germany Dieter Schuh Walter Schottky Institut TU M¨ unchen, 85748 Garching Germany Vitaly Shchukin Technische Universit¨ at Berlin Institut f¨ ur Festk¨ orperphysik Hardenbergstr. 36, 10623 Berlin Germany A. Shklyaev Quantum-Phase Electronic Center Department of Applied Physics Graduate School of Engineering The University of Tokyo and Japan Science and Technology Agency CREST, 7-3-1 Hongo, Bunkyo-ku Tokyo 113-8656, Japan Glenn S. Solomon Solid-State Photonics Laboratory Stanford University, Stanford CA 94305, USA Rudeesun Songmuang Max-Planck-Institut f¨ ur Festk¨ orperforschung Heisenbergstr. 1, 70569 Stuttgart Germany G¨ unther Springholz Institut f¨ ur Halbleiterund Festk¨ orperphysik, Johannes Kepler Universit¨ at Linz 4040 Linz, Austria Mathieu Stoﬀel Max-Planck-Institut f¨ ur Festk¨ orperforschung Heisenbergstrasse 1, 70569 Stuttgart Germany

Toma Stoica ISG, Forschungszentrum J¨ ulich 52425 J¨ ulich, Germany and INCDFM, Magurele, POB Mg7 Bucharest, Romania H.P. Strunk Institute for Materials Science and Engineering VII Cauerstraße 6 91058 Erlangen Germany Eli Sutter Center for Functional Nanomaterials Brookhaven National Laboratory P.O. Box 5000, Bldg. 480, Upton NY 11973-5000, USA Christian Teichert Institute of Physics Montanuniversit¨ at Leoben Franz Josef Str. 18, 8700 Leoben Austria Lili Vescan Richtericher Str. 86, 52072 Aachen Germany Hangyao Wang Department of Materials Science and Engineering University of Utah Salt Lake City, UT 84112 USA Kang L. Wang Department of Electrical Engineering, University of California Los Angeles USA

List of Contributors

Zhiming M. Wang Department of Physics University of Arkansas Fayetteville, Arkansas 72701 USA Herbert Wawra Institute of Crystal Growth Max-Born-Straße 2 12489 Berlin, Germany Ya-Hong Xie Department of Electrical Engineering

XV

University of California Los Angeles USA Zhigang Xie Solid-State Photonics Laboratory Stanford University, Stanford CA 94305, USA Zhenyang Zhong Institute for Semiconductor and Solid State Physics Johannes Kepler University Linz 4040 Linz, Austria

Part I

Lateral Self-Alignment

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots Vitaly Shchukin and Dieter Bimberg Technische Universit¨ at Berlin, Hardenbergstr. 36, 10623 Berlin, Germany

1.1 Introduction A move towards smaller dimensions is the general trend in modern solid state physics and technology. The size of modern devices is approaching the nanometer scale, for both vertical and lateral dimensions. Applications of ultrathin layers, or quantum wells, for micro- and optoelectronics had gained broad acceptance in the 1980s. The development of heterostructures with still lower dimensionality [quantum wires, where carriers are conﬁned in two directions and move freely in only one direction, and quantum dots (QDs), where carriers are completely conﬁned] took much longer. It became clear that defect-free quantum wire- and especially dot-structures constitute the utmost technological challenge but provide enormous advantages for devices. The largest class of QDs, which has attracted most attention in basic research and is today ﬁnding numerous applications, is that of semiconductor QDs. QDs are ultrasmall insertions of a narrow band gap semiconductor material in a wide band gap semiconductor matrix. At the beginning of the 1990s, a few outstanding discoveries of self-organization phenomena at crystal surfaces for direct fabrication of nanostructures led to a major change of paradigms in semiconductor physics and technology. First, the main focus has been shifted from growing and studying layers (quantum wells) to QDs. The latter have a discrete electronic spectrum and manifest themselves in this respect like artiﬁcial atoms, though they typically consist of 103 to 104 atoms. Second, to obtain nanometer-scale structures on surfaces, selforganization must be employed instead, or at least in addition to, lithography. Here, the uniform deposition of a foreign material on a substrate, gives rise to a nanostructure like those shown in Fig. 1.1a, c, instead of forming a ﬂat homogeneous ﬁlm. Third, using lattice-mismatched heteroepitaxy has not only allowed the growing of defect-free nanostructures, but is a necessary prerequisite for self-organization eﬀects to occur. The overgrowth of a surface nanostructure, typically by the substrate material (Fig. 1.1b, d), may lead to coherent (defect-free) inclusions in a semiconductor matrix with zero-dimensional electronic properties persistent up to room temperature. This new approach in epitaxy enables fast parallel fabrication of large densities of QDs for almost unlimited material

6

V. Shchukin, D. Bimberg

Fig. 1.1. Examples of spontaneously formed nanostructures. a Array of twodimensional (2D) islands after submonolayer (SML) heteroepitaxy. b Capped structure a forming 2D quantum dots (QDs) embedded in a matrix. c Array of 3D coherently strained islands on a wetting layer on a substrate surface. d Capped structure c forming an array of zero-dimensional QDs embedded in a matrix

combinations and has become the basis of a powerful new branch in nanotechnology. Intense research into QDs by a large number of leading laboratories, undertaken during the last decade, has clearly demonstrated that understanding the eﬀects of self-organization alone is far from being suﬃcient to bring novel objects to practical applications. A profound understanding of the physics of nanostructures and the development of tools for controlled tuning of geometrical parameters and thus electronic spectra is a must for any signiﬁcant success in nanotechnology. To achieve this goal requires complementary studies, including specially designed growth experiments, theoretical modeling of growth, structural and optical characterization on a nanoscale and theory of electronic properties. Combining eﬀects generously provided by nature with further engineering allows signiﬁcant progress in fabricating nanostructures suitable for devices. Certain aspects of self-organization in epitaxy have been addressed by us and others in books and reviews [1–10]. Here, the main focus is on the basic mechanisms of the self-organized formation of QDs. In Sect. 1.2 twodimensional (2D) islands are discussed, the role of which in QD formation is often underestimated. These 2D islands form, e.g., during submonolayer (SML) heteroepitaxy, and have unambiguously demonstrated the electronic properties of QDs. An array of 2D islands is a good model system allowing profound study of many self-organization phenomena. The results of such

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

7

a study can be extended to arrays of nanoislands extending in all three dimensions. In Sect. 1.3. arrays of three-dimensional (3D) strained nanoislands are dealt with. For arrays of both 2D and 3D islands we focus on the question of the relative roles of thermodynamics and kinetics, which is often debated in the literature. Decisive experiments are discussed that allow distinguishing between close-to-equilibrium QD arrays, the formation of which is thermodynamically dominated, and far-from-equilibrium arrays, the formation of which is mostly kinetically controlled. In Sect. 1.4 we cover a large number of techniques and results of engineering of nanostructures: engineering exciton wavefunctions and control of the photoluminescence (PL) polarization via the stacking of dots, seeding of QDs allowing independent control of dot size and density, nanoengineering using a transition between diﬀerent vertical arrangements of the dots in multistacks, shifting the QD optical spectra towards longer wavelengths via activated alloy phase separation in the cap layer, defect reduction techniques using a multiple cycle of partial overgrowth and thermal etching of the dots. For the In(Ga)As/GaAs model system, a combination of these techniques is demonstrated to allow defect-free nanostructures on GaAs substrates emitting in the technologically important wavelength of 1.3 μm and probably beyond. The summary in Sect. 1.5 presents a prospective view on the future development of semiconductor nanotechnology.

1.2 Arrays of 2D Strained Islands Heteroepitaxial semiconductor systems where a material 2 is deposited on a substrate of a material 1 sustain a large variety of self-organized nanostructures. Equilibrium theory of heteroepitaxial growth, traditionally distinguishes [11] among three growth modes. These are the Frank–van der Merwe (FM), Volmer–Weber (VW), and Stranski–Krastanow (SK) modes. They can be described as layer-by-layer growth (2D), island growth (3D), and wetting layer plus islands (Fig. 1.2). The particular growth mode for a given system depends on the interface energies and on the size of the lattice mismatch. In lattice-matched systems the growth mode is governed by interface and surface energies only. If the sum of the epilayer surface energy γ2 and of the interface energy γ12 is lower than or equal to the energy of the substrate surface, γ2 + γ12 ≤ γ1 , i.e., if the deposited material wets the substrate, the FM mode occurs. A change in γ2 + γ12 alone can drive a transition from FM to VW growth. For a strained epilayer with small interface energy, initial growth may occur layer by layer, but a thicker layer has a large strain energy and can lower its energy by forming isolated islands in which strain is relaxed. Than SK growth occurs.

8

V. Shchukin, D. Bimberg

Fig. 1.2. The three growth modes for heteroepitaxial systems: Frank–van der Merwe (FM), Volmer–Weber (VW), and Stranski–Krastanow (SK)

Fig. 1.3. Possible surface conﬁgurations after SML deposition. a Random distribution of atoms of the deposit across the surface. b Surface array of ordered 2D islands. c Large random islands formed via ripening

If a given heteroepitaxial systems grows according to the FM or the SK mode, and the amount of the deposited material is below one monolayer (ML), one may ﬁnd the following possible arrangements of the deposit (Fig. 1.3). Atoms of the deposit may be distributed randomly across the surface (Fig. 1.3a). Atoms may order into small islands (Fig. 1.3b) depending on the intrinsic properties of the system. The small islands may undergo ripening and form large islands limited in size only by surface inhomogeneities, extended defects or slow kinetics (Fig. 1.3c). Atoms of the deposit will form some surface reconstruction on an atomic scale. Strictly speaking, a transition from a 2D morphology to a 3D one can occur even below 1 ML average coverage. However, in the present section we focus on systems in which 2D morphology persists at least up to 1 ML. In semiconductor systems, narrow gap SML insertion can be overgrown by the substrate material, thus eventually forming an attractive potential for electrons and holes. In the case of a random distribution of atoms different from those of the substrate (Fig. 1.3a), the “layer” is a quantum well characterized by an alloy composition. In the case of large islands (Fig. 1.3c) with a characteristic extension exceeding the exciton Bohr radius, the layer can be regarded as consisting of fragments of a quantum well.

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

9

Finally, in the case of islands (Fig. 1.3b) of size less than the exciton Bohr radius, the islands are quantum dots localizing excitons also in the lateral plane. The existence of an optimum size of islands follows from the theory of capillarity eﬀects on crystal surfaces. The formation of equilibrium domain structures has been predicted theoretically by Andreev [12], Marchenko [13], Alerhand et al. [14] and Vanderbilt [15]. The ﬁrst observations are likely to be those in the metallic systems O/Cu(110) by Kern et al. [16]. The ﬁrst experimental studies of a SML semiconductor system were done by Wang et al. [17]. Optical reﬂectance spectroscopy of a SML (0.3 ML) insertion of InAs in a GaAs matrix revealed a large optical anisotropy. Such an anisotropy is not consistent with the picture of a homogeneous quantum well or with large fragments of a quantum well and can be explained only by the formation of anisotropic islands where an anisotropy of the island shape and/or of the strain pattern leads to optical anisotropy. Crosssectional high resolution transmission electron microscopy (HRTEM) studies of 1 ML CdSe insertions in a ZnSe matrix [19] revealed the formation of ﬂat nanoscale islands. A more detailed overview of experimental data is found in [9]. 1.2.1 Energetics of a SML Array of Islands Let N adatoms on a surface form an island. Then the change of the Helmholtz free energy equals to √ √ √ (1.1) F˜ (N ) = −W N − μad N + C1 N − C2 N ln( N ) . Here the ﬁrst term is the binding energy of the islands due to formation of the chemical bonds between neighboring atoms. The second term represents the decrease of the free energy due to the transition of N atoms from a dilute gas of adatoms to a compact surface phase. The third term is the energy of the island boundary. The last term is due to the elastic relaxation energy caused by surface stress relaxation at the island boundaries. Stress relaxation at the island boundaries is based on capillarity phenomena on solid surfaces. The theory, developed by Andreev [12, 20] and Marchenko [13, 21], demonstrates that every solid surface is characterized by, besides a scalar quantity, the surface energy γ, a 2D surface stress tensor ταβ . A microscopic interpretation of this surface stress is given in [9]. The nature of the surface stress is related to the fact that the surface atoms of a solid are in a diﬀerent environment than the bulk atoms. Then, when a surface layer is matched to a bulk crystal, the surface layer is intrinsically strained or compressed, even if it is the surface of a homogeneous crystal. Furthermore, at every boundary between two surface phases, the intrinsic surface stress has a discontinuity. This occurs at the edge between two

10

V. Shchukin, D. Bimberg

surfaces having diﬀerent crystallographic orientations [20, 21] or at a boundary between planar surface domains [12, 13]. The latter can be a boundary between surface domains diﬀering in surface reconstruction, like (2 × 1) and (1 × 2) domains on Si(001) [14], or a boundary between two chemically diﬀerent phases of the surface, like a boundary of a monolayer-high island to a substrate. At every boundary the discontinuity in the tensor ταβ results in an eﬀective elastic force monopole applied to the boundary. This creates an inhomogeneous strain ﬁeld and results in an elastic relaxation. It should be noted that no elastic force monopole occurs at an island boundary in a homoepitaxial system. The boundary manifests itself only as a source of an elastic dipole. This makes a crucial diﬀerence in the energetics of islands in homo- and heteroepitaxial systems. By dividing Eq. (1.1) by N , one obtains the change of the free energy per atom in the islands: √ C2 C1 F (N ) = −W − μad + √ − √ ln( N ) . N N

(1.2)

The energy per atom Eq. (1.2) always has a minimum for an optimum number of atoms in the island (the optimum volume of the island) Nopt = exp [2 (C1 /C2 + 1)] .

(1.3)

The optimum volume of an island (Eq. 1.3) refers to a dilute array when the coverage q → 0. Upon a coverage increase, an elastic interaction between the islands mediated by the substrate becomes important and governs the preferred structure. At coverage 0 ≤ q ≤ 0.286 the preferred structure is a hexagonal array of circular disks, or Droplets; at coverage 0.286 ≤ q ≤ 0.714 the system forms a one-dimensional periodic structure of stripes; and at coverage 0.714 ≤ q ≤ 1.0 the system favors a hexagonal array of antidisks, or inverse droplets [15, 22]. Kinetic theories developed for SML islands in homoepitaxial systems demonstrate that an optimum island size is obtained for a relatively slow steady-state growth which follows the nucleation stage. Kinetic theory yields a scaling behavior of the island volume distribution [23–25] that appears to ﬁt experiments on homoepitaxy for Si/Si(001) [26], etc. well Kinetic theories developed for homoepitaxial systems are applicable to heteroepitaxial systems if the elastic relaxation at island boundaries is a weak eﬀect, the optimum number of atoms in the island Nopt is large, and the actual islands contain considerably fewer atoms than Nopt . Several experiments have revealed scaling to be valid for some heteroepitaxial systems, e.g., Ag/Si(111) [27]. In this regard, it becomes important to develop experimental tools that allow distinction between thermodynamically controlled and kinetically con-

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trolled arrays of islands. Such tools, developed in [28], refer to the temperature dependence of the SML arrays. 1.2.2 Characteristic Energies in SML Arrays of Disks To address the behavior of an array of SML islands at ﬁnite temperatures, it is worth considering ﬁrst characteristic energies for an array of islands at T = 0. The energy normalized per atom in an optimum island is equal to C2 −E0 = − . Nopt

(1.4)

The value E0 is the energy gain per atom in an optimum island with respect to the energy per atom in a large (N → ∞) island. The constant C2 related to the surface stress discontinuity at the island boundary was estimated for a model system of InAs/GaAs islands as C2 ≈ 120 meV. For islands having Nopt = 1, 000, Eq. (1.4) yields E0 ≈ 3.7 meV. A characteristic temperature T = E0 /kB ≈ 43 K. This temperature is smaller by an order of magnitude than typical growth temperatures 300 – 600 celsius, or 600 – 900 K. Consequently such a small energy E0 does not give the correct energy scale of ﬁnite temperature eﬀects at realistic temperatures. The same characteristic energy, normalized per island, equals to Θ = E0 Nopt = C2 Nopt . (1.5) The energy normalized per island is about 3.7 eV, giving a characteristic temperature ≈ 43,000 K that again can not be directly related to the temperature dependence at realistic experimental temperatures. To obtain a characteristic temperature for an array of 2D islands it is necessary to consider ﬂuctuations in the system. 1.2.3 Conﬁguration Entropy in Arrays of SML Islands At ﬁnite temperatures, three types of ﬂuctuations occur in an equilibrium array of islands: ﬂuctuations of island position, volume, and shape. If one approximates them to be independent, the conﬁguration entropy is then a sum of three contributions. The relative roles of the diﬀerent types of ﬂuctuation are related to the corresponding stiﬀness of the array of islands. A detailed analysis [9] has shown that for nanometer-scale islands having a typical number of atoms N > 100, the stiﬀness of the array of islands against shape ﬂuctuations is larger than the stiﬀness against the other two types of ﬂuctuation. Therefore shape ﬂuctuations are not important and their contribution to the total entropy can be neglected.

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V. Shchukin, D. Bimberg

Fluctuations of the number of atoms in the islands contribute essentially to the entropy Svol , if the variation of the island energy does not exceed kB T . The entropy of volume ﬂuctuations per island is 2πk T B Svol = kB ln 2N . (1.6) Θ To evaluate the entropy of island positions for a dilute array, it is possible to neglect elastic interactions between islands. The system reduces then to an array of identical disks each of which has N atoms and can occupy an arbitrary position on the surface under the constraint that disks are not allowed to overlap. The entropy per island is then: αN Spos = kB ln , (1.7) q where α is a numerical factor of the order of unity. By adding up the contributions Eq. (1.6) and Eq. (1.7) to the entropy, and dividing the sum by the number of atoms in the island N , one obtains the entropy per atom. Then, adding up the energy E and entropy terms (−T S), one gets the Helmholtz free energy per atom, √ C2 C1 (0) F = −W + Eelast + √ − √ ln( N ) N N

αN kB T 2πkB T − ln . (1.8) + ln 2N N q Θ The entropy term in the free energy (Eq. 1.8) is negative and its absolute value decreases monotonously with N . Therefore it shifts the free energy minimum towards smaller values of N . This implies an entropy-driven shrinkage of islands with temperature. To estimate the characteristic temperature at which entropy eﬀects become essential, we equate the entropy term in Eq. (1.8) to the energy per atom in the optimum island, E0 . This yields the value of the characteristic temperature: Tchar ∼

Θ . 2 kB ln Nq

(1.9)

The value of Tchar agrees with our earlier results [28]. Remarkably, the characteristic temperature has an intermediate value between the energy per atom in an optimum island and the energy per island, E0 kB Tchar Θ .

(1.10)

A typical value of the characteristic temperature is about 800 – 1000 K which lies in the range of typical growth temperatures. This comparison conﬁrms the decisive role of entropy eﬀects for the formation of arrays of strained SML islands in real experimental systems.

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1.2.4 Equilibrium Distribution of the Islands in Volumes For a dilute array of noninteracting islands, the equilibrium distribution is described by the Boltzmann–Gibbs distribution function,

) μN − E(N [μ − E(N )] N P (N ) = exp = exp . (1.11) kB T kB T The chemical potential μ can be found from the constraint that the total number of atoms in the adatom gas and in the islands is ﬁxed by the total coverage q, q = qad + N P (N ) , (1.12) N

where the summation is taken over macroscopic volumes of the islands, N 1. Single adatoms are taken into account separately, and islands containing a small number of atoms N = 2, 3, . . . etc. are neglected. We assume the elastic interaction energy of a given island with the other islands to be equal to such an energy in a perfect hexagonal array with the coverage q−qad . Here, we neglect the contribution of adatoms to the elastic interaction energy, since adatoms are strongly relaxed. The equilibrium distribution function can then be written in a way similar to Eq. (1.11), where the interaction energy of every given island with the others is taken in a mean ﬁeld approximation. It follows

[μ − E(N ; q − qad )] N . (1.13) P (N ) = exp kB T Equations (1.12) and (1.13) form a set of two equations with two unknowns, the chemical potential μ and the density of adatoms qad . The set of equations Eq. (1.12) and Eq. (1.13) were solved in [29] by choosing the value of the binding energy W = 369 meV and of the coeﬃcient related to the energy of the island boundary C1 = 392 meV. With these parameters, the distribution function of the number of atoms in the island was obtained, and can also be rewritten in terms of the distribution function of the island sizes, P (N )dN = 2P (L)π(4a2 )−1 LdL where L is the diameter of a circular island. Figure 1.4a depicts the distribution function of the island sizes at a given coverage q = 0.15 and diﬀerent temperatures. At T = 0 K the distribution function is an inﬁnitely sharp peak at some optimum size Lopt . With increasing temperature the distribution of island sizes broadens and its maximum shifts to smaller sizes. The second maximum corresponding to individual adatoms evolves. The islands have a bimodal size distribution where one local maximum corresponds to single adatoms, and the other one refers to nanometer-scale islands. With decreasing temperature the local maximum

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V. Shchukin, D. Bimberg

Fig. 1.4. Equilibrium distribution of the sizes of 2D islands. a Distribution at a given surface coverage q = 0.15 and diﬀerent temperatures. 1: T = 0, 2: T = 250 K, 3: T = 500 K, 4: T = 750 K, 5: T = 1, 000 K. b Distribution at a given temperature T = 750 K and diﬀerent coverage. 1: q = 0.05, 2: q = 0.10, 3: q = 0.15, 4: q = 0.20, 5: q = 0.25

corresponding to nanoscale islands disappears and the bimodal size distribution is transformed to a unimodal one. Dashed lines refer to a relative small island size, say L < 2 nm, where the continuum approach does not apply quantitatively but may give a correct qualitative description. It should be noted that a bimodal size distribution of strained 2D islands was obtained earlier from kinetic Monte Carlo (MC) simulations in Ref. [30]. However, no discussion of the temperature dependence of such distribution was given there. Figure 1.4b demonstrates the change of the island size distribution at a given temperature as a function of island coverage. The eﬀect of coverage is similar to the eﬀect of temperature. At larger coverages islands have a bimodal size distribution having some optimum size whereas at lower coverages the size distribution becomes unimodal and an optimum size can no longer be resolved. The main result of our thermodynamic consideration is the shrinkage of the island volume and, hence, of the island size with temperature. This eﬀect provides an eﬃcient experimental tool allowing us to distinguish between equilibrium, or thermodynamically controlled arrays of islands and kinetically controlled ones. In homoepitaxial systems where equilibrium corresponds to the ripening of the islands, dense arrays of islands having some ﬁnite size are kinetically controlled arrays representing some intermediate state on the path to ripening. In such arrays the average size of the islands increases with the substrate temperature (see, e.g., experimental results for Fe/Fe islands [31]). The same is expected for heteroepitaxial systems in the case where the size of kinetically controlled islands is far below the equilibrium value. On the other hand, the decrease of the island size with temperature is an indication

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

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of the equilibrium nature of an array of islands in a given heteroepitaxial system. 1.2.5 Crossover from Kinetically Controlled to Thermodynamically Limited Growth of 2D Strained Islands In order to develop a uniform description of a heteroepitaxial system, which would cover both kinetically controlled and thermodynamically controlled growth stages, Meixner et al. [32] carried out kinetic MC simulations of the formation of 2D strained islands upon growth interruption (GI). These simulations used an event-based algorithm applied to a solid-onsolid model with deposition and diﬀusion as the relevant processes. Diﬀusion of adatoms occurs on a square lattice by nearest-neighbor hopping. Atoms can cross island edges by surmounting a Schw¨ obel barrier. The relevant energies in our simulations are the binding energy to the surface Es = 0.7 eV and the strength of the n ≤ 4 nearest-neighbor bonds Eb = 0.3 eV that inﬂuence the time scale for diﬀusion and island formation, respectively. Existing islands generate an elastic strain ﬁeld caused by the lattice mismatch. This strain ﬁeld inﬂuences detachment from island boundaries and the motion of adatoms in the vicinity of islands through a position-dependent energy correction term Estr . The hopping rate for a single atom is then given by an Arrhenius law Es + n Eb − Estr p = ν exp − , (1.14) kB T with the attempt frequency ν = 1013 s−1 . The strain energy density was calculated by using a Green’s function approach and normalized per atomic bond. The simulations were performed on a lattice of 250 × 250 atomic sites. As an initial step a coverage of 4% was deposited randomly on the surface at a ﬂux of 1 ML s−1 . Every 0.01 s a histogram of the island size distribution is recorded. To reduce the noise, ten simulations with diﬀerent initial conditions have been used to calculate an average. Figure 1.5 √ displays the results for the temporal evolution of an average island size N for temperatures of T = 675 K, 700 K, and 725 K. From Fig. 1.5, it is clear that in the initial stages of island growth the size distribution is kinetically controlled. At lower temperatures, many small islands are formed whereas at higher temperatures, fewer and larger islands emerge. On short time scales of a few seconds the islands do not grow by a considerable amount and the scaling of the island size with temperature is still kinetically controlled. At lower temperatures the nucleation of islands is the dominant process. Since the adatom mobility is low, the density of single adatoms increases

16

V. Shchukin, D. Bimberg

rapidly during the deposition and pairs of atoms are formed randomly. Those act as nuclei for islands. Consequently, one observes many small islands for low temperatures. With increasing temperature, the adatoms become more and more mobile. A single adatom in a hot system can travel a long distance until it ﬁnds an existing island to which it will attach. The adatom density therefore decreases and nucleation of new islands is suppressed. The ﬁnal spatial conﬁguration in the kinetically controlled regime exhibits few and large islands. Immediately after deposition, however, the islands begin to equilibrate. The system is now in an intermediate state between kinetically and thermodynamically controlled growth conditions. The slow increase of island sizes and a crossover of the average island size for systems of diﬀerent temperatures is characteristic for this regime. At low temperatures, the growth process is slowest. The higher the Temperature, the faster the islands approach their average equilibrium size. Once the equilibrium size distribution is reached, the average island diameter remains constant. In the course of equilibration the islands in the low temperature system continue to grow until they reach their equilibrium size at an average diameter above that of the islands of the hotter systems, as is expected for islands grown under equilibrium conditions. From the results of the thermodynamic theory and of the kinetic simulations, an experimental tool emerges that allows distinguishing between kinetically and thermodynamically controlled islands. If, with increasing substrate temperature, the average number of atoms in the islands, or the average island volume increases, the island formation is controlled predominantly by the growth kinetics. If, with increasing substrate temperature, the average

Fig. 1.5. Temporal evolution of the average island size for T = 675 K, T = 700 K, and T = 725 K. Monte Carlo simulations have been performed on a 250 × 250 grid and averaged over ten runs with the same set of parameters

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island volume decreases, the island formation is controlled predominantly by thermodynamics. For SML islands, the height is ﬁxed, and the island volume is proportional to the square of the island lateral size, thus the above arguments apply to the dependence of the lateral size on temperature. 1.2.6 Practical Example: SML InAs/GaAs Islands Figure 1.6 demonstrates experimental results for the temperature dependence of SML arrays of InAs/GaAs islands [33]. The heteroepitaxial InAs/GaAs (001) system containing SML (0.3 ML) insertions of InAs in a GaAs(001) matrix has been grown at diﬀerent temperatures by molecular beam epitaxy (MBE). GI of 10 s has been introduced, and the system has been overgrown by GaAs. Figure 1.6a displays cross-sectional high resolution transmission electron microscopy (HRTEM) images processed by the digital analysis of lattice images (DALI) evaluation program which yields a map of local lattice parameters in the vertical direction, az . Brighter regions in Fig. 1.6a correspond to a larger az , or, in other words, to a larger content of indium. For a sample grown at 350 ◦ C, rather large islands (> 8 nm) are revealed by HRTEM. For a sample grown at 480 ◦ C, the islands shrink, and the typical size is less than 4 nm. Figure 1.6b displays photoluminescence (PL) spectra

Fig. 1.6. Dependence of a SML [0.3 monolayer (ML)] array of InAs/GaAs islands on growth temperatures. a Digital analysis of lattice images (DALI)-processed cross section high resolution transmission electron microscopy (HRTEM) images of the SML array of InAs/GaAs islands grown at 350 ◦ C. Brighter regions correspond to higher Indium content. b DALI-processed cross section HRTEM image of the system grown at 480 ◦ C. White arrows point to the island boundaries. c Photoluminescence (PL) spectra of SML arrays of InAs/GaAs islands grown at diﬀerent temperatures

18

V. Shchukin, D. Bimberg

of structures grown at 350, 450, and 480 ◦ C. With increasing growth temperature, the PL peak shifts towards higher energies, which corresponds to smaller sizes of SML islands localizing excitons. The shrinkage of the size of SML islands is in agreement with our theory and conﬁrms the thermodynamic equilibrium nature of arrays of SML InAs/GaAs islands. 1.2.7 SML Islands: Broad Field of Applications SML islands have a large number of advantages enabling a large variety of device applications. 1. A major advantage of SML islands in heteroepitaxial systems is their relatively small dimensions. Typical height of the islands may be strictly 1 ML, or weakly smeared up to 2 – 3 ML; the lateral dimensions may be about 5 nm, and the lateral spacing between centers of neighboring islands may be about 10 nm. Thus, an area density of the islands may be extremely high, up to 1012 cm−2 and more. 2. Further, SML islands provide a possibility of growing a multilayered structure with a large number of layers while keeping the whole structure coherent, i.e., dislocation-free. Since the volume of every single SML island is signiﬁcantly smaller than that of a typical 3D island, the onset of misﬁt dislocations will occur later. 3. The exciton oscillator strength deduced from the optical reﬂection measurements of SML InAs/GaAs structures [18] is of the order 0.5 × 10−2 – 10−1 , and the lifetime of excitons is about 50 ns. So, large oscillator strengths are of the same order as in GaAs/GaAlAs quantum wells of the width of 50 – 100 ˚ A and considerably larger than that expected for ultrathin quantum wells of 1-ML thickness. Such an ultrastrong optical response from a single-sheet SML array suggests growing of multisheet systems. Repeating a sheet of SML islands will create a medium with a high volume density of oscillators, each of high oscillator strength. Thus, a 3D medium of ﬁnite thickness exhibiting an ultrastrong optical response near the exciton resonance will be obtained. 4. A medium with a strong optical response may be employed to achieve the resonant waveguiding eﬀect proposed by Alferov et al. [34] and realized by Ledentsov et al. [35] for a CdSe/ZnSe superlattice. A strong refractive index modulation near the optical absorption peak may effectively conﬁne an optical wave without the need for a conventional waveguide. 5. The high exciton absorption coeﬃcients in QDs convert to ultrahigh exciton/biexciton gain coeﬃcients at high excitation densities, since no screening of excitons occurs. This allows lasing for very short cavity lengths in edge geometry, or enables surface lasing in vertical geometry even if no highly reﬂecting Bragg mirrors are used. For example,

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

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ZnMgSSe/GaAs and ZnMgSSe/air interfaces could lead to 30% reﬂectivity, which allows surface lasing in structures with 20 sheets of CdSe SML insertions in a ZnMgSSe matrix [36]. 6. Optical studies of superlattices formed by SML Ge insertions in a Si matrix [37] have suggested that the islands form quasi-type-I QDs versus type-II QDs, which might be expected from the zone diagrams of the constituent bulk materials. This results in a bright PL and shows broad prospectives for Si-based optoelectronics, e.g., Si-based light emitting diodes, etc.

1.3 Arrays of 3D Strained Islands If a heteroepitaxial system grows according to the Stranski–Krastanow (SK) mode and the amount of the deposited material exceeds a certain critical thickness, a transition from 2D to 3D morphology occurs and islands form with a larger extension in growth direction. Alternatively, in Volmer–Weber growth mode, 3D islands form before the ﬁrst monolayer is complete. The formation of 3D islands was observed in the early 1980s, ﬁrst for In(Ga)As/GaAs and Ge(Si)/Si, and later for a great variety of other material systems (a review of experiments is found in [9]). The experiments have revealed two surprising phenomena. First, under certain growth parameters, 3D islands in a lattice-mismatched system form in a coherent way, i.e., dislocation-free. Second, if the deposition is stopped and a heteroepitaxial system is subjected to annealing or GI, size-limited growth has been observed: the 3D islands reach a certain size and apparently show no Ostwald ripening. Coherent formation of islands was explained by Vanderbilt and Wickham [38], who showed that 3D growth and the onset of dislocations are two competing mechanisms for strain energy relaxation, and that in a certain range of material parameters depending on the island volume, the lattice mismatch, the surface energies and the dislocation formation energies, coherent islands are energetically preferred. Basically, islands can be coherent until a certain volume, and become dislocated as they grow larger. The physical nature of size-limited growth has long been and still is a controversial issue. A few alternative explanations have been proposed. The thermodynamic theory developed by Shchukin et al. [39] and extended further by Daruka and Barab´ asi [40] and Shchukin and Bimberg [4] shows the following: surface energies in a strained system are a function of the local strain. In a certain range of material parameters, the formation of a 3D island from an initially ﬂat surface, which is accompanied by a decrease in the elastic strain energy, may also be accompanied by a decrease in the total surface energy despite an increase in the overall surface area. In this case, no thermodynamic driving force to ripening exists, and an array of islands of an optimum size corresponds to a stable or metastable state of the system. Kinetic theories focus basically on various mechanisms of the slowing down of the ripening

20

V. Shchukin, D. Bimberg

process which may eventually lead to size-limited growth even if there is a global thermodynamic driving force to ripening. Priester and Lannoo [41] considered the formation of 2D platelets of optimum size which transform to 3D islands, ensuring a narrow size distribution of the latter. Madhukar et al. [42] and Kobayashi et al. [43] focused on the dependence of the barriers on attachment of adatoms to the islands and their detachment from the islands due to island-induced strain, which eventually results in a size limited growth. Besides that, they hypothesized that the strain ﬁeld between the islands gives rise to preferred migration of adatoms from larger islands to smaller islands, thus favoring equalization of island sizes [42, 43]. Chen et al. [44] and Jesson et al. [45] showed that the growth of any new atomic layer on ﬂat side facets of a 3D strained faceted island occurs via nucleation, and the corresponding barriers become progressively larger with increasing island size, eventually resulting in size-limited island growth. Wang et al. [46] considered the situation where the islands ﬁrst nucleate from an initial 2D ﬁlm. This process governs their density. Then a local equilibrium between every island and the surrounding wetting layer is established which determines the island volume. It should be emphasized that the debates on the thermodynamicallydominated versus kinetically-dominated nature of an array of strained islands refer to particular experiments. A system may be subject, after a deposition of Q monolayers of a material 2 on a substrate of material 1, to annealing or GI. Such a system may be with good accuracy considered as a closed system. If one neglects the possibility of evaporation of the deposited material or its alloying with the substrate, the number of atoms of the material 2 is conserved. In this chapter, the thermodynamic theory of the size-limited growth will be discussed in more detail. An overview of kinetic models may be found in [9]. Experiments particularly focused the relative roles of thermodynamics and kinetics in the formation of arrays of InAs/GaAs islands will be discussed. 1.3.1 Ordering of Islands in Size versus Ripening The classical picture of island formation in mass-conserving systems usually considers the process as consisting of three stages. At the nucleation stage, islands form and dissolve due to thermal ﬂuctuations in the system. The change of the total Helmholtz free energy of a system, ΔF , due to the formation of an island can be written as a sum of the surface and volume contributions, ΔF = ΔFsurf + ΔFV .

(1.15)

The surface contribution is usually positive, ΔFsurf > 0 due to the energy cost of a new surface, while the volume contribution (in the case of a supersaturated gas of adatoms) is negative, ΔFV < 0. As a function of the island volume, ΔF ﬁrst increases as the volume is below a certain value, V < Vc , and then decreases as V > Vc . Vc is the volume of a critical nucleus. An

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

21

Fig. 1.7. The formation of coherent islands on the substrate surface. Q and (Q−Q ) are the nominal coverage and wetting layer thickness, respectively. L is the island base length

island having a volume V < Vc will with probably dissolve. If an island has a volume exceeding that of the critical nuclei, V > Vc , it will probably grow further in volume. At the growth stage a suﬃcient density of islands exceeding the critical volume is present. The islands grow due to attachment of adatoms, the gas of adatoms is being depleted, and no new islands nucleate. At the latest stage of coarsening, or Ostwald ripening, the gas of adatoms is depleted to a large extent and supersaturation is decreased to a very low value. Correspondingly, the volume of a critical nucleus Vc has increased to a large value, and islands that initially had volumes above Vc turn out to have volumes below the new value of Vc and start to dissolve. At each moment, larger islands are growing at the expense of smaller islands. Ostwald ripening is well known in the general picture of ﬁrst-order phase transitions [47–49]. For a particular growth of 3D islands on a surface, the latter has been extended by Chakraverty [50]. Its detailed kinetics depend on the slowest process, either attachment/detachment or diﬀusion of adatoms between the islands. In both cases, ripening yields a rather large size distribution of islands that evolves with time in a similar way, leading to an increase in the average volume of an island and to a corresponding decrease in island density. Numerous experiments revealing size-limited island growth have indicated an apparent absence of Ostwald ripening. The thermodynamic theory addressing this issue focuses on an equilibrium state of a lattice-mismatched heteroepitaxial system. If, under certain conditions, an equilibrium corresponds to an array of islands of ﬁnite size, this will mean that no thermodynamic driving force to Ostwald ripening exists for such a system, and the ripening will not occur. Consider Q monolayers of the material 2 deposited on a substrate of material 1. Let (Q − Q ) ML remain in a ﬂat ﬁlm, and the rest Q ML form an array of equal-shape and equal-size islands (see Fig. 1.7). Then the total energy of the system equals E = felast (Q − Q )a + (1 − q)W (Q − Q ) +

inter E 1 Eisland + . (1.16) A0 2

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V. Shchukin, D. Bimberg

Here, the ﬁrst term is the elastic strain energy of a nonrelaxed planar uniform ﬁlm of the thickness (Q − Q )a, the second term is the energy of the wetting layer of a thickness (Q − Q ) ML, the third term is the energy of a single island, and the fourth term is the interaction energy of a single island with all other islands, A0 denoting the unit cell area of the superlattice comprised of islands. By assuming all islands having equal volume and shape are forming a perfect superlattice on the surface, the total energy of the system (Eq. 1.16) becomes a function of two parameters: the amount of material in all islands (Q ML) and the volume of each island. Let us ﬁrst ﬁx the value Q and seek the energy minimum of an array of equally sized islands. Then, instead of minimizing the total energy, one can seek the minimum energy per atom in the islands. Take, for simplicity, the islands having a shape of a square-based pyramid with the tilt angle of side facets ϑ0 and focus on the energy per atom as the function of the island base length L, E(L) [39] (ΔΓ ) − (6 cot θ0 )2/3 E(L) = Ω −f1 (θ0 )λε20 + (6 cot θ0 )1/3 L

L f2 (θ0 )τ 2 2/3 f3 (θ0 )η × ln . (1.17) + (6 cot θ0 ) λL2 2πa L2 The ﬁrst term in Eq. (1.17) is the energy of the volume elastic relaxation V elast . It is always negative. The second term is the change of the renorΔE malized surface energy of the system due to the island formation. For concreteness, we will write (ΔΓ ) for an island having the shape of a pyramid with a square L × L base and a tilt angle of side facets θ0 . Then renorm surf = (ΔΓ )(1/6 tan θ0 )2/3 L2 , and ΔE interface − W (Q ) − g1 (θ0 )τ ε0 (ΔΓ ) = (6 cot θ0 )2/3 γ2 (θ0 ) sec θ0 + γ12 −g2 (θ0 )Sε20 . (1.18) Here the change of the surface energy includes the contributions due to the appearance of tilted facets of the island, due to the appearance of the interface between the deposited material and the substrate underneath the island, due to the disappearance of the planar surface area, and due to renormalization terms, both linear and quadratic in ε0 . The key point of the further analysis is that the quantity (ΔΓ ) can be of either sign. V It is worth noting that the volume elastic relaxation energy ΔEelast (the ﬁrst term in Eq. (1.17)) does not depend on the island size L. To seek the minima of E(L) from Eq. (1.17) we introduce the characteristic length f3 (θ0 )ηλ 1 + L0 = 2πa exp , (1.19) f2 (θ0 )τ 2 2

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

23

and the characteristic energy per atom, E0 =

Ωf2 (θ0 )(6 cot θ0 )2/3 τ 2 . 2λL20

(1.20)

Then we can write the sum of all L-dependent terms in E(L) as follows [51] 1/2

2 e L L0 2α L0 E (L) = E0 −2 ln + 1/2 . (1.21) L L0 L e The function E (L) is governed by the control parameter α=

e1/2 λL0 (ΔΓ ) , f2 (θ0 )(6 cot θ0 )1/3 τ 2

(1.22)

which is the ratio of the change of surface energy due to island formation and edges |. of the contribution of the edges to the elastic relaxation energy, |ΔEelast The energy of a dilute array of islands per atom versus the size of the island L is displayed in Fig. 1.8 for diﬀerent values of α. If α ≤ 1, there exists an optimum size of islands Lopt , corresponding to the absolute minimum of the energy, minE (L) ≡ E(Lopt ) < 0. On the other hand, the ripening of islands would correspond to L → ∞ where the energy E (L) → 0. It means that an array of identical islands of optimum size Lopt is a stable array. Islands will show size-limited growth up to this value, and will not undergo further ripening. If 1 < α < 2 e−1/2 ≈ 1.2, there exists only a local minimum of the

Fig. 1.8. The energy of a dilute array of 3D coherently strained islands per atom versus. the size of the island. The parameter α is the ratio of the change of the renorm surface energy due to the formation of islands, ˛ ΔEsurf˛ , and of the contribution ˛ edges ˛ of the edges to the elastic relaxation energy, ˛ΔEelast ˛

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energy, corresponding to a metastable array where E (L ) > 0. If α ≥ 2 e−1/2 , the local minimum in the energy E (L) disappears. For both latter cases, where α > 1, there exists the thermodynamic tendency to ripening. The energy minimum corresponds then to a single huge island where all deposited material is collected. If (ΔΓ ) < 0 (and α < 0), the formation of a 3D island, besides a decrease of the strain energy due to the elastic relaxation, leads also to a decrease of the renormalized surface energy. 1.3.2 Relative Role of Thermodynamic and Kinetic Eﬀects: Experimental Focus One of the key issues of size-limiting island growth is the relative role of thermodynamic and kinetic eﬀects. Focusing on this debate, criteria were formulated [4] to allow testing for any particular system, whether an observed array of islands is controlled predominantly by thermodynamics or by kinetics. To resolve this question, it is important to verify experimentally whether the following properties of an equilibrium systems are present in a given system. 1. Upon GI or annealing, the system evolves towards equilibrium. When the equilibrium is reached, no further changes occur. 2. The equilibrium state of the system depends only on thermodynamic parameters, and not on prehistory. For heteroepitaxial systems in question, these are the amount of the deposited material and temperature. For III–V and II–VI compound semiconductors, there exists one more thermodynamic parameter. The GI is provided by switching oﬀ the supply of cations whatever the anion (e.g., arsenic) vapor pressure. The anion vapor pressure is the third thermodynamic parameter which can aﬀect the morphology of the system. 3. For a system in equilibrium it is possible, by varying thermodynamic parameters of the system, to cause reversible changes of the morphology. To resolve this issue, particular experiments have been designed and undertaken in the In(Ga)As/GaAs systems, aimed at distinguishing thermodynamically controlled arrays of islands from kinetically controlled arrays. Thorough and comprehensive experimental studies have revealed the following: 1. An evolution of the dot size up to a limiting value upon growth interruption for MBE-grown InAs QDs, is observed only in a certain range of growth conditions. 2. A reversible phase transition in the InAs/GaAs system from 3D to 2D morphology is driven by a reduction of As pressure. 3. A reversible phase transition from 3D to 2D morphology in the GaInAs(P)/GaAs system is driven by the switching oﬀ/on of As and P.

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4. An irreversible phase transition from coherently strained islands to dislocated islands, i.e., the “switching on” of Ostwald ripening is driven by an increase in As pressure. 5. The formation of coherent InAs QDs in metalorganic chemical vapor deposition (MOCVD) is possible only in MBE-like conditions, at very low pressure of As. 6. The general tendency towards Ostwald ripening of InAs islands follows upon an increase in arsenic pressure in both MBE and MOCVD. 7. The preferred alignment of nearest neighboring dots is that along elastically soft directions 100. 8. A decrease of the average volume of InAs islands upon an increase in the substrate temperature. 9. Reversible changes in island shape, volume, and density upon cyclic variations of the substrate temperature are observed for InAs/GaAs islands. First, the dependence of the QD density, shape, size and PL spectra on the formation temperature have been studied [55, 56]. For each growth temperature, the deposition of 3 ML of InAs was followed by 10 s of GI, and the system was capped by GaAs. Capped structures were studied by the plan-view transmission electron microscopy (TEM) and by PL spectroscopy. Figure 1.9 demonstrates the increase of the lateral size (Fig. 1.9b) and the decrease of the island density (Fig. 1.9c) with the temperature of island formation. The plan-view TEM images of Fig. 1.9d, e illustrate the increase of the lateral size with temperature. Cross section TEM images (Fig. 1.9g, h) reveal, at the same time, the decrease of the island height with temperature. Since lateral size and height of the islands show opposite changes with temperature, an additional experimental approach is needed to address the dependence of the island volume on temperature. If a QD changes its volume while preserving its shape, a shift of the PL spectrum is obvious. Larger QDs have larger exciton localization energy and therefore, the PL peaks should be shifted towards lower photon energy (red shift). The dependence of the PL peak position on the QD shape has been analyzed theoretically by means of eight-band k · p theory [1, 52] including strain, piezoelectric, and excitonic eﬀects. The calculations carried out for truncated pyramids having the same volume and diﬀerent levels of truncation have revealed the following. As the QD volume remains constant, and the dots become ﬂatter, the PL peak must shift towards lower photon energy (red shift). Thus, with increasing temperature, islands become ﬂatter, but experiments reveal a blue shift in the PL. Since the shape change from steep to ﬂat contributes to the red shift, the total blue shift can only be explained by a large decrease in the island volume: the blue shift due to the volume decrease overcomes the red shift due to the ﬂattening of the islands. Flattening of 3D strained islands and simultaneous decrease of the volume is in agreement with the general behavior of the equilibrium shape of

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a strained island as a function of the volume, where islands of a larger volume are steeper, and islands of a smaller volume are ﬂatter [53, 54]. The decrease of the island volume simultaneous with the decrease of the island density indicates an excess amount of In atoms present on the surface in the form of adatoms and small 2D islands. After overgrowth these adatoms and 2D islands contribute to the eﬀective thickness of the wetting layer. There has been additional focus on the reversibility of the changes of the QD upon cooling. The deposition of 3 ML InAs at 500 ◦ C was followed by 10 s GI at the same temperature, further cooling down to 450 ◦ C during 120 s, and capping at 450 celsius. The comparison of the array of islands after 120 s cooling (Fig. 1.9f, i) with the array formed and capped at 500 ◦ C (Fig. 1.9e,h) indicates that cooling results in a decrease of the lateral size of the islands.

Fig. 1.9. Eﬀects of temperature and temperature ramping to lower values on the lateral size and the height of InAs/GaAs(001) islands. Eﬀect of the formation temperature on parameters of the array of 3D coherently strained islands of InAs on GaAs(001). (a) The position of the PL peak from QDs. (b) The average lateral size of the islands. (c) The island density. (d), (e), (f ) Plan-view symmetrical [001] zone-axis transmission electron microscopy (TEM) images of the arrays of InAs/GaAs islands. (d) Islands are grown at 450 celsius, the growth interruption (GI) is 10 s. The density of the islands is 7.1 × 1010 cm−2 , the average lateral size 14.9 ± 0.5 nm. (e) Islands are grown at 500 ◦ C, the GI is 10 s. The island density is 2.8×1010 cm−2 , the average lateral size is 19.5±0.5 nm. (f ) The islands are grown at 500 ◦ C and cooled down to 450 ◦ C over 120 s. The island density is 4.5 × 1010 cm−2 , the average lateral size equals 17.5 ± 0.5 nm. g, h, i Cross-sectional (010) dark-ﬁeld TEM images with g = 002 of the arrays of InAs/GaAs islands. Growth conditions are the same as in (d), (e), and (f ), respectively

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

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Thus, the lateral size of the islands after cooling is intermediate between the value of the array deposited and capped at 450 ◦ C (Fig. 1.9d, g), and the value of the array deposited and capped at 450 ◦ C (Fig. 1.9e, h). The same is true for the island density and the height, which are signiﬁcantly increased after cooling (Fig. 1.9f, i) from 2.8 × 1010 cm−2 to 4.5 × 1010 cm−2 . An even more dramatic increase in QD density was manifested for fast cooling of QDs formed at 520 ◦ C (from 1.7 × 1010 cm−2 to 5.0 × 1010 cm−2 ) accompanied by a strong reduction of the lateral size and increase in height. This indicates a partial reversibility of changes for arrays of InAs strained islands upon temperature variations. More about the eﬀects of cooling on an array of strained islands has been revealed by the PL spectra. The PL spectra of two samples have been compared. In the ﬁrst sample, the QDs were formed at 500 ◦ C, subjected to 10 s GI, and capped by GaAs at the same temperature. In the sample 2, the array was formed at 500 ◦ C, cooled down to 450 ◦ C over 120 s, and capped by GaAs at 450 ◦ C. The comparison shows two eﬀects. First, cooling leads to a red shift of the QD PL peak to 1270 nm (with only a 30% drop in intensity at 1300 nm) indicating an increase in the QD volume. This again shows a reversibility of the volume change upon the change of temperature. Second, the cooling leads to a blue shift of the PL peak from the wetting layer. This is related to a strong reduction of adatom density, as adatoms condense, increasing the volume of existing 3D islands and forming new islands. The reversibility is partial because it is diﬃcult for the system to create or eliminate enough islands in a limited time. Other aspects of the eﬀect of reversible change of temperature and vapor pressure of V-group element on an array of InAs/GaAs islands are discussed in detail in [9]. To conclude, these results present strong evidence in favor of a close-to-equilibrium nature of the formation of 3D coherently strained islands of InAs on GaAs(001) substrates. 1.3.3 Dense Arrays of 3D Islands: Lateral Arrangement The larger is the island density, the more important becomes the elastic interaction between the islands. The elastic interaction mediated by the substrate may result in speciﬁc lateral arrangements of the islands. Bimberg et al. [57] observed the square-like spatial correlation in nearest neighboring dots arrangements. Figure 1.10a shows the plan-view transmission electron microscopy (TEM) micrograph of a single-sheet array of MBE-grown InAs QDs [57]. Preferential alignment of dots in rows parallel to elastically soft 100 directions is visible. Figure 1.10b displays the histogram of the direction between a given dot and a nearest neighboring dot. This histogram reveals a well-pronounced maximum for 100 directions. The observed arrangement was explained theoretically by Shchukin et al. [39] who showed that the square lattice arrangement is energetically preferred with respect to other types of lateral arrays. Two factors favor the square

28

V. Shchukin, D. Bimberg

Fig. 1.10. a Plan-view transmission electron microscopy (TEM) micrograph of a single sheet of InAs dots grown in molecular beam epitaxy (MBE) by 4-ML deposition of InAs. Dots are preferentially aligned in rows parallel to 100. b Histogram of the direction of the nearest neighboring dots

lattice: the cubic anisotropy of elastic moduli of the medium, and the square shape of the base of a single island. 1.3.4 Dense Arrays of 3D Islands: Stabilizing Role of Near-Field Elastic Interaction The role of the elastic interaction between the islands, when considered on average, is destabilizing. It was shown in [39] that the elastic interaction energy is on average the positive energy of elastic repulsion. Therefore it reduces the domain of the phase diagram corresponding to a stable array of equally sized islands and favors ripening, or coarsening.

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However, elastic interaction plays a completely diﬀerent role due to local variations in the strain ﬁeld in dense arrays of islands. Shchukin et al. [58] and Jesson et al. [59] considered an array of strained islands of conical shape as a model example. The total energy of the island array is then given by, ρa ρb 3 w Va Vb 2/3 , , (1.23) Etotal = βVa − wJ tan ϑVa + 3 F 2 π Rab Rab Rab b=a

where the ﬁrst term represents the additional surface energy associated with the island formation. The second term is the elastic self-relaxation energy of the island, and the third term represents the elastic interaction energy between the ath island and all other islands. Here Va and ρa are the respective volume and the base radius of the ath cone and Rab is the distance between the basal centers of islands a and b. The coeﬃcient w = (1 + ν)(1 − ν)−1 Y ε20 , where ε0 is the lattice mismatch between the deposit and the substrate, Y and ν are Young’s modulus and Poisson’s ratio, respectively, assumed to be equal in both materials, and the numerical factor J = 1.059. The coeﬃcient β = 2π 1/3 3−1/3 (cot ϑ)2/3 (ΔΓ ), where ΔΓ = γ(ϑ) sec ϑ − γ(0), and γ(ϑ) and γ(0) being the surface energies of the tilted surface of the island and of the ﬂat surface of the wetting layer, respectively. The role of local elastic interactions is more pronounced in the case where ΔΓ > 0 so that, even without elastic interactions, islands would tend to ripen to reduce the overall surface energy. The evolution of an array of islands in the regime of attachment-limited kinetics is controlled by islands that can attach atoms from the adatom sea and detach atoms which go to the adatom sea. The local ﬂux of atoms to/from each island is governed by the local diﬀerence between the chemical potential ¯, of an atom, μa and the adatom sea, μ dVa = Va1/3 [¯ μ − μa ] , dt

(1.24)

where the chemical potential of an island is deﬁned as μa = ∂Etotal /∂Va . The elastic interaction energy between the two conical islands has been calculated exactly in [60]. The key feature of this energy is that its contribution to the chemical potential diverges as two islands nearly contact each other. To emphasize the impact of the elastic interaction on the evolution of a dense array of islands, an initially hexagonal array of identical conical islands was considered, and an initial perturbation in island volume and position was introduced. Figure 1.11a, b compares the evolution of island radii without and with strain. The evolution of a dense array of islands without strain is dominated by the coalescence events, when two islands touch each other and form a single island by adding up their volumes. The coalescence events manifest themselves as abrupt jumps in island radii. When the strain is included, no abrupt jumps occur. The latter means that no coalescence on

30

V. Shchukin, D. Bimberg

Fig. 1.11. Time evolution for the scaled radii ρ of the initial array. a For zero strain. b With strain included

impact occurs in an array of strained islands, and the coarsening proceeds via the Ostwald ripening mechanism. Thus, a dramatic increase of the chemical potential of a strained island in the vicinity of another strained island due to local strain ﬁelds suppresses the coalescence. In addition, the elastic interaction alters the temporal behavior of the average density and radius of the islands, which is in agreement with experimental data on dense arrays of GeSi/Si islands. In ultradense arrays, where islands nearly touch each other, a metastable state may occur, which is stable against small perturbations in island volumes and position. Thus, surprisingly, a positive elastic energy of elastic repulsion between islands can stabilize ultradense arrays.

1.4 Nanoengineering The phenomenon of spontaneous formation of nanostructures on crystal surfaces has given rise to a large branch of nanotechnology employing selforganized growth of QDs and wires. It has been realized, however, that simply exploiting natural phenomena is far from being suﬃcient for both physics and application. Just to mention a few limitations: –

Although geometrical and electronic parameters of quantum wire- and QD-structures can be tuned by varying the amount of the deposited material, substrate temperature and vapor pressure (in the case of compound semiconductors, e. g., III–V, II–VI, or III–N), a much higher degree of ﬂexibility and tunability is required.

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

31

Fig. 1.12. Schematics of the growth of multisheet arrays of QDs. a Single-sheet array of SML islands. b Multisheet array of SML islands. c Single-sheet array of 3D (3D) coherently strained islands. d Multisheet array of 3D coherently strained islands

–

–

–

–

–

The volume density of QDs is rather low. Thus, for laser applications, the overlap of the active medium with the optical mode is rather small, which deteriorates major device characteristics. The maximum obtained wavelength of the PL peak from 3D InAs/GaAs QDs is 1.24 μm which is too small as compared to the practically important spectral region of 1.3–1.55 μm. The InAs/GaAs structure showing the PL maximum at 1.24 μm contains, along with coherent islands, a signiﬁcant density of dislocated islands which makes the structure not suitable for laser applications. Thus, the wavelength of lasing is limited to even smaller values. Localization of electrons and holes by 3D InAs/GaAs QDs is not suﬃciently strong to maintain the performance of QD structures up to room temperatures. SML InAs/GaAs islands provide even weaker localization of carriers. Thus, lasers based on these structures would not operate at room temperature.

A way to surmount these problems has been recognized. The way is to work together with nature and thus to combine phenomena of spontaneous formation of nanostructures with engineering of complex systems. Figure 1.12 shows a basic approach to this kind of engineering. Essential is the growth of multilayered structures, where layers of a narrow band gap material forming quantum wires or QDs alternate with layers of a wide band gap material forming barriers for electrons and holes. Thus multisheet arrays of islands form in a matrix. Arrays based on a proper combination of materials represent multisheet arrays of quantum wires or dots. These complex nanostructures demonstrate signiﬁcant advantages with respect to single-sheet arrays of wires or dots:

32

V. Shchukin, D. Bimberg

1. Multisheet arrays of islands show a larger degree of tunability. The material and thickness of every sheet of wires or dots as well as material and thickness of every spacer layer can be adjusted independently. 2. Multisheet arrays of QDs have a larger overlap with the optical wave in optical devices like lasers thus improving laser characteristics. 3. In multisheet arrays the dots of the neighboring sheets can be electronically coupled. This coupling will shift electron and hole levels to lower energies, and the PL spectra should shift towards lower photon energies, or to longer wavelengths (red shift). For InAs/GaAs QDs this will mean a shift towards the highly requested spectral region of 1.3–1.55 μm. 4. A shift of electronic levels to lower energies will mean a larger localization energy of electrons and holes and thermal stability of the lasers. 1.4.1 Wavefunction Engineering in Multilayered Arrays of QDs Additional engineering of complex nanostructures has created novel unique properties that had hardly been expected before: 1. The possibility of independent control of density and volume of the QDs by using a concept of “seeding”. 2. The possibility to control the PL polarization via exciton wavefunction engineering. 3. The possibility to tune between diﬀerent types of vertical correlation between the islands. 4. The overgrowth of initial by strained islands by an alloy followed by activated alloy phase separation. The formation of geometrically coupled vertical columns of islands has already been observed in the ﬁrst stages of QD research [61]. It was easily acepted that if the spacers between the QD layers are rather thin, coupling of electronic states occurs, which may lower the exciton transition energy and results in a corresponding shift of the optical absorption and PL spectra towards lower photon energies (“red” shift). The growth of multisheet arrays of QDs, along with a shift in the energy of electronic states, allows control of the wavefunctions of electrons and holes, and, therefore, the wavefunctions of excitons that determine such optical characteristics as the oscillator strength of the optical transition and polarization of absorption and PL. Figure 1.13a, b shows schematically multisheet arrays of 3D islands with thick (a) and a thin (b) spacers. In the case of a thick spacer, QDs of neighboring sheets are electronically uncoupled, and the wave functions of electrons and holes are localized in separate QDs. For InAs QDs in a GaAs matrix with a square-based pyramid shape with {101}-side facets, Stier et al. [52] calculated electron and hole states by using an eight-band (k · p) model taking into account the inhomogeneous strain distribution and piezoelectric eﬀects assuming a homogeneous

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

33

Fig. 1.13. Schematics of the exciton wavefunction engineering by stacking QDs. a Sheets of QDs are separated by thick spacers, so that the dots are electronically not coupled. Wavefunction of an exciton localized by QDs roughly follows the shape of the QD and is extended in the lateral plane. Edge photoluminescence (PL) is transverse electric or TE-polarized. b Sheets of QDs are separated by thin spacers, QDs are electronically coupled, and edge PL is transverse magnetic or TM-polarized. Dashed lines show schematically exciton wavefunctions

In distribution. It was shown that the hole wavefunctions are conﬁned to the pyramid base and, therefore, strongly ﬂattened compared with the shape of the pyramid itself. As a consequence, both the absorption and the emission of light are polarized in the (001) plane. Thus, the edge PL will be transverse or TE-polarized. For InGaAs QDs of a ﬂatter shape, this eﬀect will be even stronger. In contrast, in multisheet arrays of electronically coupled QDs, the wavefunctions of electrons and holes are extended over the entire vertical columns of QDs. Then PL will be transverse magnetic or TM-polarized. Transition from TE-polarized PL to TM-polarized PL can be driven by varying the spacer thickness, or by increasing the number of QD sheets in a structure. Yu et al. [62] studied PL from multisheet arrays of InGaAs/GaAs QDs. Structures were grown by MBE and contained 1, 3, 10, and 20 layers of In0.5 Ga0.5 As QDs deposited at 485 celsius and separated by 5-nm-thick GaAs spacers. The QDs are pyramidal with a base length of about 18 nm, a height of about 5 nm, and an average lateral separation of 55 nm. The vertical alignment of the QDs was identiﬁed in X-ray studies [63]. The edge PL from the structures with 1, 3, and 10 sheets demonstrated TE-polarization, while the edge PL from the 20-sheet structure was TM-polarized, in accordance with Fig. 1.13. 1.4.2 Seeding of QDs A concept of independent control of density and volume of QDs was proposed independently by Maximov et al. [64] Mukhametzhanov et al. [65] and Hein-

34

V. Shchukin, D. Bimberg

Fig. 1.14. Seeding of InGaAs QDs on a layer of InAlAs stressors. a The concept of growth of columns of vertically correlated islands on small stressors. b Energy band gaps of the multisheet structure showing a shallow localization potential for carriers in the ﬁrst layer and a deeper one in the subsequent layers. c Cross-sectional TEM image of the multisheet structure revealing smaller islands in the ﬁrst sheet and larger islands in the subsequent sheets. d Plan-view TEM image of the multisheet structure revealing a highly uniform array of QDs with a high density. e PL spectrum of the composite QDs

richsdorﬀ et al. [66] Both approaches include diﬀerent deposition schemes in the ﬁrst sheet and in the subsequent sheets. Maximov et al. [64] employed the fact, that InAlAs QDs have an area density of 2 × 1011 cm−2 which is 4 times larger than that of InGaAs QDs deposited under the same growth conditions, as shown in [67]. InAlAs QDs have a much larger band gap energy and, thus, much smaller localization energy for the same matrix and, therefore can hardly be used to improve laser characteristics. The large band gap of InAlAs QDs causes their depopulation already at low temperatures. It is advantageous if these QDs are used as passive prelayers for seeding of QDs having larger localization energy (InAs or InGaAs QDs).

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

35

Maximov et al. [64] proposed and performed seeding of InGaAs islands on top of a sheet of InAlAs islands used as stressors via the concept of vertically coupled QDs [68]. The idea is illustrated in Fig. 1.14a, b. In the ﬁrst sheet, InAlAs is deposited on a GaAlAs substrate resulting in the formation of InAlAs small islands. These islands serve as stressors providing the growth of columns of vertically correlated islands (Fig. 1.14a), where islands in the subsequent sheet have a larger volume than those of the ﬁrst sheet. Figure 1.14b depicts energy band gaps in all the sheets of QDs showing a shallow localization potential for electrons and holes in the ﬁrst sheet, and a deep potential in the subsequent sheets. The experimental data of Fig. 1.14c–e conﬁrm the anticipated eﬀect. Cross-sectional transmission electron microscopy (TEM) (Fig. 1.14c) reveals smaller islands in the ﬁrst sheet, and larger islands in the subsequent sheets. The plan-view TEM image of Fig. 1.14d shows a large density of QDs with a high uniformity in sizes. PL spectra of composite QDs (Fig. 1.14e) reveal a single maximum around 1.15 eV. It is shifted with respect to the PL maximum from a single sheet of InGaAs QDs (Fig. 1.14c) at the value of 1.26 eV due to electronic coupling in a vertical column of QDs. At the same time, PL from composite QDs reveals no contribution from high-energy states in the InAlAs QDs of the ﬁrst layer. Thus, the novel growth regime of seeding of a high density of small QDs in the ﬁrst sheet and growth of vertical columns of vertically correlated and electronically coupled large QDs has indeed provided a high density of large QDs with a large localization energy. 1.4.3 Vertical Correlation and Anticorrelation: Role of Elastic Anisotropy Vertical correlations and the formation of vertical columns of strained islands have been well understood, observed in various material systems and widely employed to fabricate arrays of vertically coupled QDs (QDs). Surprisingly, Straßburg et al. [69] observed a diﬀerent type of relative arrangement of islands.in a multilayered array of CdSe islands in a ZnSe matrix. Figure 1.15a shows an anticorrelation between islands in the neighboring sheets: islands of the subsequent sheet form above the spacings between islands of the previous sheet. The observations of Straßburg et al. [69] had been a real puzzle contradicting previous experimental and theoretical work. This puzzle has been solved theoretically by Shchukin et al. [70]. The decisive point is the elastic anisotropy of the semiconductor material. Semiconductors having a diamond structure, like Si, Ge, or zinc-blende structure, like III–V and II–VI binary materials, are characterized by a large elastic anisotropy parameter, ξ=

c11 − c12 − 2c44 , c44

(1.25)

36

V. Shchukin, D. Bimberg

Fig. 1.15. Anticorrelation and correlation in multisheet arrays of 2D CdSe islands in ZnSe matrix. Cross-sectional high resolution transmission electron microscopy (HRTEM) image processed by the DALI evaluation program. Gray scale corresponds to the local lattice parameter in the vertical direction. Dark areas refer to smaller values of the lattice parameter in the vertical direction, or to lower content of Cd, whereas bright areas refer to larger values of the lattice parameter in the vertical direction, or to higher content of Cd. Thus, Cd-rich islands are shown as bright areas revealing a vertically anticorrelated arrangement for thicker spacer a and a vertically correlated arrangement for thinner spacer b

where c11 , c12 and c44 are elastic moduli in the Voigt notation. In diamondlike and zinc-blende-like materials, the parameter ξ < 0, which implies that {100} axes are elastically soft axes, whereas {111} are elastically hard ones. A key feature of elastically anisotropic materials is the existence of generalized Rayleigh waves, which means that an acoustic wave generated in some plane, exhibits an oscillatory decay away from the source. The same is valid for static strained ﬁelds. As a consequence, the elastic strain created by the buried islands exhibits an oscillatory decay with distance from the source (Fig. 1.16). Therefore, the interaction between successive sheets of islands exhibits this oscillatory decay with the separation between the sheets, i. e., with the spacer thickness. The transition between vertical correlation and vertical anticorrelation opens a possibility of eﬃcient wavefunction engineering and control of the PL polarization. Results of optical studies are discussed in detail in [36, 71]; also see the review [6]. After anticorrelation had been discovered experimentally in CdSe/ZnSe system by Straßburg et al. [70] and explained theoretically by Shchukin

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

37

Fig. 1.16. Strain ﬁeld at the surface created by the sheet of buried islands. Schematic plot demonstrates an oscillatory decay of the strain with the spacer thickness. Dependent on the spacer thickness, the two sheets of islands exhibit a correlated (left) or an anticorrelated (right) arrangement

et al. [70], similar anticorrelated arrangement of QDs were observed by Springholz et al. [72] in PbSe/PbEuTe system. IV–VI semiconductors have a rock salt crystal structure, having a positive value of the elastic anisotropy parameter ξ; thus the axes {111} are soft axes, and {100} are hard ones. Theoretical studies by Hol´ y et al. [73] showed a good agreement with experiment explaining the observed transition between correlated and anticorrelated arrangement on the basis of continuum elasticity theory. Recently, an anticorrelated arrangement of QDs was observed in III–V systems, including InAs/InAlAs quantum wires on InP substrates [74] as well as InGaAs/GaAs QDs [75]. This conﬁrms once again the universal character of the transition between vertical correlation and anticorrelation which is governed by the elastic anisotropy of the system. 1.4.4 Alloy Growth on Stressors: Activated Alloy Phase Separation The concept of the overgrowth of InAs islands by GaInAs alloy proposed by Maximov et al. [76] is illustrated in Fig. 1.17. Coherent InAs/GaAs islands overgrown by an InGa(Al)As alloy layer serve as a model system. The laterally varying strain at the surface created by the InAs islands aﬀects the overgrowth with an InGa(Al)As alloy by strain-driven surface migration. One expects the following qualitative picture of alloy phase separation activated by stressors. For the conventional overgrowth of InAs QDs by pure GaAs it had earlier been found [77] that Ga atoms prefer to migrate away from the QDs towards pseudomorphically strained regions, having an in-plane lattice parameter equal to that of unstrained GaAs. A similar eﬀect should occur during the overgrowth of InAs islands by an InGa(Al)As alloy: In atoms will accumulate at the InAs islands increasing their lateral size. When the islands are completely covered, the tensile strain on top of the QDs will favor In accumulation from the growing alloy. The latter could eventually increase the height of the

38

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Fig. 1.17. Schematics of the overgrowth of InAs islands by GaInAs alloy. a Coherently strained 3D InAs islands over InAs wetting layer on GaAs substrate. b Initial stage of the overgrowth of InAs islands by GaInAs alloy. Adatoms of In incorporate preferably at the facets of initial InAs islands forming In-rich domains in the capping layer around initial InAs islands. c Latest stages of the overgrowth of InAs islands by the alloy. When InAs islands are overgrown completely, Inrich domains form in the capping layer over the islands. d Completely overgrown structure. In-rich domains are formed in the vicinity of initial InAs islands enlarging an eﬀective size of the QDs in both lateral and vertical directions

islands. Thus, we expect that the activated phase separation of the alloy increases the In content in the vicinity of the InAs islands. In other words, the eﬀective lateral size and height of the QDs will increase providing enhanced localization of electron and holes and a red shift in the PL spectrum of the QDs. The driving force for phase separation in the growing alloy will additionally contribute to the formation of In-rich domains close to and on top of the InAs islands. This eﬀect will be pronounced within a certain temperature range, whereas it will be hindered at high temperatures due to entropy eﬀects and at low temperatures due to a decrease in the surface diﬀusivity of adatoms.

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

39

Fig. 1.18. Cross-sectional TEM images of InAs/GaAs islands overgrown by a Ga0.85 In0.15 As, and b Ga0.70 Al0.15 In0.15 As. Addition of Al leads to an increase of the eﬀective island height

Plan-view TEM studies of InAs QDs overgrown by InGaAs (given in [76]) conﬁrm an increase in the lateral size of QDs. PL spectra reveal a red shift of the PL maxima when the dots are overgrown by an InGaAs alloy. However, there are two eﬀects that may contribute to the observed red shift: the lowering of the average energy band gap of the surrounding InGaAs matrix, and the formation of In-rich domains in the vicinity and above initial InAs islands. These two eﬀects become clearly distinguishable if initial InAs islands are overgrown by a quaternary InGaAlAs alloy. A cross-sectional TEM image of the alloy-capped sample is shown in Fig. 1.18a, b. The overgrowth by a quaternary Ga0.70 Al0.15 In0.15 As alloy (Fig. 1.18b) leads to a signiﬁcant increase in the island height compared to the overgrowth by a ternary Ga0.85 In0.15 As alloy (Fig. 1.18a). These data indicate that addition of Al enhances the phase separation whereas an increase in height is less pronounced when islands are overgrown by InGaAs. The eﬀect of Al on the PL spectra of the QDs is shown in Fig. 1.19. Despite an increase in the energy band gap in the alloy layer of Ga0.7 Al0.15 In0.15 As compared to that of Ga0.85 In0.15 As, the PL maximum from the QDs exhibits a red shift. This eﬀect can be unambiguously attributed to the enhanced phase separation due to Al, in agreement with the cross-section TEM data of Fig. 1.18. Thus, by varying the alloy composition, the thickness of the alloy cap layer, and the amount of initially deposited InAs, it is possible to tune the ground state transition energy of the QDs. For QDs overgrown by Ga0.7 Al0.15 In0.15 As, the ground state transition energy is signiﬁcantly (up to 200 meV) red shifted as compared to the original InAs/GaAs QDs. This allows the technologically important 1.3 μm spectral region to be reached, maintaining the high PL eﬃciency and low defect density. Besides that, it is possible to combine two complex growth modes described in this paper

40

V. Shchukin, D. Bimberg

Fig. 1.19. PL spectra of InAs/GaAs QDs overgrown by Ga0.85 In0.15 As (solid lines) and Ga0.7 Al0.15 In0.15 As (dashed lines)

and to fabricate a multisheet array of QD layers each capped by an alloy material. A detailed description of this growth mode and characteristics of the corresponding QD injection laser are given in [76]. Thus, the overgrowth of coherently strained InAs islands by InGa(Al)As alloy leads to activated alloy phase separation and the formation of In-rich domains in the vicinity of the initial islands. This results in an increase of the eﬀective lateral size and height of the QDs and in a signiﬁcant red shift of the PL spectrum. 1.4.5 Defect Reduction Techniques In the epitaxy of In(Ga)As QDs, much of the eﬀort has been focused and is still focused on fabricating QD lasers emitting in the practically impor-

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

Fig. 1.20. procedure

41

Evaporation of defect-containing islands using a two-step annealing

tant spectral range of 1, ,300 nm. This implies rather big QDs. The islands formed always have some distribution in sizes. Shifting this distribution towards larger sizes implies the presence of some very big islands, which might be dislocated. Thus, obtaining QD structures emitting at still longer wavelengths is connected with a need to reduce or eliminate defects. The defect reduction technique proposed by Ledentsov [78] is shown schematically in Fig. 1.20. This technique includes the overgrowth of InAs islands with GaAs such that high and presumably dislocated islands remain only partially covered (Fig. 1.20b). Annealing (Fig. 1.20c) results in the evaporation of uncovered islands (Fig. 1.20d). There still remain covered islands containing defects. However, the strain ﬁeld on top of capped coherently strained islands and plastically relaxed dislocated islands is diﬀerent. Capping by AlAs leaves areas on top of defected islands uncovered (Fig. 1.20e). As AlAs is stable against heating, the high-temperature annealing (Fig. 1.20f) results in the elimination of the buried defects, whereas areas covered by AlAs remain untouched. The subsequent capping by GaAs (Fig. 1.20g) leads to a capped defect-free structure. Employing defect reduction techniques in MBE growth of QDs has been discussed in detail in [9]. Metalorganic chemical vapor deposition (MOCVD) growth has clearly shown the advantages of such technique as well. An application of the defect reduction technique in MOCVD growth of QDs [79]

42

V. Shchukin, D. Bimberg

Fig. 1.21. Realization of the defect-reduction technique in metalorganic chemical vapor deposition (MOCVD) QD growth. a, b Plan-view (TEM) images of a control sample A (2.7-ML In0.65 Ga0.35 As QDs) overgrown with 5 nm In0.2 Ga0.8 As, and sample B subsequently in situ annealed at 600 ◦ C. Each plan-view image displays an area of about 1 × 1 μm2 . c Cross-sectional dark ﬁeld TEM image of sample B

is shown in Fig. 1.21. The sample with no defect reduction technique (DRT) technique (Fig. 1.21a) shows defects marked by white triangles. An annealing step results in the elimination of defects and formation of craters at their locations. The craters are marked by white triangles in Fig. 1.21b. The crosssectional TEM shows the local thinning of the deposited In(Ga)As layer at the places where defects have been eliminated. Another branch of defect reduction techniques refers to QD growth on metamorphic buﬀers [80]. Thus, a thick GaInAs layer is being grown on a GaAs substrate and a network of dislocations is formed in the deposited layer. Then a similar technique as above involving a two-step overgrowth and annealing is being introduced, after which the dislocations are blocked within the initial GaInAs layer. Further overgrowth of the structure results in a nearly dislocation-free GaInAs which serves eﬀectively as a new ‘substrate’ for further growth of InAs QDs. Lasers grown by employing this technique have shown lasing at 1515 nm, continuous wave (CW) operation with output power Pout = 220 mW, and diﬀerential eﬃciency up to 50% [81–83]. These results indicate that GaAs-based QD lasers might take over after the 1.3 μm spectral region also the next important one, around 1.5 μm. To conclude, it has been demonstrated in this section that combining self-organization phenomena and subsequent nanoengineering allows the fabrication of nanostructures to meet almost any requirement on geometrical parameters and electronic spectra. Improvement of the size uniformity and spatial ordering by multilayered growth of QDs, seeding of QDs, enables an

1 Physical Mechanisms of Self-Organized Formation of Quantum Dots

43

independent control of the island volume and density, engineering of the exciton wavefunction, allowing tuning of the spectral position and polarization of the emitted light. Activated alloy phase separation and defect reduction techniques, are powerful experimental tools that considerably expand possible device applications of epitaxial nanostructures.

1.5 Conclusion About a decade after QD research became the mainstream in semiconductor physics and technology, it is worth summing up the state of the art and overview the perspectives. Employing eﬀects of self-organization at crystal surfaces allows fabrication of semiconductor heterostructures with a high density of coherent inclusions in a wide band gap matrix displaying a discrete electronic spectrum up to room temperature and above. These structures have a low density of defects, can be fabricated in a production-friendly massive parallel way, can store or transfer electrons, reveal bright PL and allow their use as active media for ultrahigh quality semiconductor diode lasers and other optoelectronic devices. The substantial progress achieved in the understanding of the basic physics of self-organization has led to the invention of numerous approaches in nanoengineering allowing construction of various nanoworlds inside semiconductor wafers. For example, growing multisheet arrays, it is possible to improve the uniformity of sizes and arrangement of nanoinsertions. By choosing diﬀerent materials for QDs in the ﬁrst and in the subsequent sheets, one can control the density and the size of QDs independently and grow a large density of large dots. By varying the spacer thickness in multisheets, one can choose between vertical columns and a checkerboard arrangement of dots in the cross section plane. By manipulating the geometry, it is possible to tune between unpolarized and highly polarized surface emission from the dots as well as to choose between diﬀerent polarizations of the edge emission. By capping the dots with an alloy, the phase separation in the cap layer can be activated, eﬀectively increasing the size of the QDs and shifting the PL spectrum towards ultimately longer wavelengths. It is possible to grow large islands, among which a substantial fraction are dislocated, and to apply a multicycle overgrowth and thermal etching, selectively eliminating dislocated islands and keeping coherent ones. By applying a set of sophisticated techniques of nanogrowth and nanoengineering, QD lasers have been developed that outperform conventional quantum well lasers in major parameters: threshold current density, temperature stability of the threshold current density, and diﬀerential eﬃciency. Lasers have been grown on GaAs-substrates emitting at 1.3 μm with key parameters far exceeding those of the state-of-the-art quantum well lasers on InP substrate. A relatively low modal gain in QD lasers reduces undesirable nonlinear eﬀects, particularly suppressing ﬁlamentation. A symmetric gain

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spectrum in QD lasers enables very low chirp operation (a detailed state-ofthe-art of QD lasers is discussed in [84]). Self-organized formation of QDs in completely new material combinations, e. g., InAs/Si create the possibility of forming optoelectronic devices directly combined with a silicon integrated circuit which is particularly promising in view of the foreseen use of optical interconnects in microelectronics [85]. The success in the development of QD lasers which have overcome the conventional quantum well lasers in all major parameters is just only the ﬁrst example of a major breakthrough of nanotechnology. QD arrays may serve as a new media for data storage by controlling the eﬀects of the Coulomb blockade which may path the way to a vast application of QD structures in microelectronics. QDs are particularly exciting for single-photon emitters and thus for the use in quantum cryptography [86]. Last, but not least, the long spin relaxation time in QDs make them particularly promising for spintronics, and, thus for quantum computing. Acknowledgement. Parts of this work were supported by the Deutsche Forschungsgemeinschaft (Sfb 296) and by the SANDiE Network of Excellence of the European Commission, contract number NMP4-CT-2004-500101.

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77. Q. Xie, P. Chen, A. Madhukar, Appl. Phys. Lett. 65, 2051 (1994) 78. N.N. Ledentsov, US Patent 6,653,166 79. I.N. Kaiander, R.L. Sellin, T. Kettler, N.N. Ledentsov, D. Bimberg, N.D. Zakharov, P. Werner, Appl. Phys. Lett. 84, 2992 (2004) 80. V.A. Shchukin, N.N. Ledentsov, US Patent 6,784,074 81. N.N. Ledentsov, A.R. Kovsh, A.E. Zhukov, N.A. Maleev, S.S. Mikhrin, A.P. Vasilev, E.S. Semenova, M.V. Maximov, Yu.M. Shemyakov, N.V. Kryzhanovskaya, V.M. Ustinov, D. Bimberg, Electron. Lett. 39, 1126 (2003) 82. M.V. Maximov, Yu.M. Shernyakov, N.V. Kryzhanovskaya, A.G. Gladyshev, Yu.G. Musikhin, N.N. Ledentsov, A.E. Zhukov, A.P. Vasil´ıev, A.R. Kovsh, S.S. Mikhrin, E.S. Semenova, N.A. Maleev, E.V. Nikitina, V.M. Ustinov, Zh.I. Alferov, Fiz. Tekhn. Poluprovodn. 38, 766 (2004) [Semiconductors 38, 732 (2004)] 83. L.Ya. Karachinsky, T. Kettler, N.Yu. Gordeev, I.I. Novikov, M.V. Maximov, Yu.M. Shernyakov, N.V. Kryzhanovskaya, A.E. Zhukov, E.S. Semenova, A.P. Vasilev, V.M. Ustinov, N.N. Ledentsov, A.R. Kovsh, V.A. Shchukin, S.S. Mikhrin, A. Lochmann, O. Schulz, L. Reissmann, D. Bimberg, to be published 84. D. Bimberg, J. Phys. D, Appl. Phys. 38, 2055 (2005) 85. N.N. Ledentsov, in Future Trends in Microelectronic: The Nano, the Giga, and the Ultra, ed. by S. Luryi, J. Xu, A. Zaslavsky (Wiley, New York, 2004) pp. 358–384 86. N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Mod. Phys. 74, 145 (2002)

2 Routes toward Lateral Self-Organization of Quantum Dots: the Model System SiGe on Si(001) Christian Teichert1 and Max G. Lagally2 1

2

Institute of Physics, Montanuniversit¨ at Leoben, Franz-Josef-Str. 18, 8700 Leoben, Austria Department of Materials Science and Engineering, University of Wisconsin-Madison, 1509 University Avenue, Madison, WI 53706, USA

2.1 Introduction Because its lattice constant is 4.2% larger, Ge grows on Si in the Stranski– Krastanov (SK) growth mode, that is, with formation of coherent threedimensional (3D) nanocrystals after completion of a two-dimensional (2D) wetting layer of one or several monolayers (MLs) [1]. The formation of such 3D nanocrystals in SK growth was convincingly demonstrated for the ﬁrst time in 1990, using scanning tunneling microscopy (STM) of Ge grown on the technologically relevant Si(001) surface [2]. At the same time it was shown, using cross-sectional transmission electron microscopy (TEM), that these nanocrystals are coherent with the substrate and therefore free of interfacial (misﬁt) dislocations [3]. Some years earlier, 3D morphologies arising in strained-layer growth had been observed using TEM [4,5], but with questionable interpretations. Mo et al. were able to image the facet structure of Ge nanocrystals growing on Si(001) with atomic resolution and proposed a model for the atomic structure of the facets [2]. The corresponding STM image is reproduced in Fig. 2.1a. The nanocrystal has a rectangular base with edges in 110 directions and four {105} facets, two of which are trapezoidal and two triangular, giving a prism-like shape with canted ends. The shape is best described as a hut conformation, resulting in the appellation “hut cluster” [2]. Hut shapes can vary from square to rectangular, with the elongation, if it exists, along either the [100] or the [010] direction, and with aspect ratios of up to about 4. The huts, when they grow, arrange themselves in an interlocked, labyrinth-like array, as vividly shown in the atomic force microscopy (AFM) image shown in Fig. 2.1b. The nonuniformity in size, shape, and position (see Sect. 2.3) a priori hampers the use of densely packed arrays of SiGe nanocrystals for quantum dot (QD) applications. It becomes relevant to ask whether there are simple ways to improve the uniformity of crystallite size and shape. Since the discovery of strain-driven formation of QDs, it has been found that strain can also be used to improve lateral ordering, resulting in a signiﬁcant increase in size uniformity, a process that is particularly eﬀective in Ge- or SiGe-grown

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Fig. 2.1. {105}-faceted Ge “hut” nanocrystals on Si(001). a 3D STM image and schematic top and side views. The height of the nanocrystal is 2.8 nm while the base dimensions are of the order of 20 and 40 nm. b AFM image of a Si0.25 Ge0.75 ﬁlm with an interlocked array of these huts

Si(001). Because this system is so amenable to STM and AFM studies, the ordering process can also be quantiﬁed, making Ge/Si(001) an ideal model system to explore routes toward lateral self-organization of QDs. In this chapter a brief overview of experimental observations of straininduced self-organization of nanostructures in SiGe/Si(001) heteroepitaxy is provided, based on a recent comprehensive review [6] and organized as follows. In Sect. 2.2 we present a brief introduction to AFM-based quantitative characterization of surface morphologies. We will discuss how to deduce essential information about nanocrystal facet orientations and describe power spectral density (PSD) analysis that allows quantiﬁcation of the degree of self-organization at the growth front. In Sect. 2.3 the overall scenario of strain-relief morphologies in semiconductor S–K growth is introduced, using the SiGe/Si(001) system as the basis, with the main emphasis being on the coherently strained faceted hut nanocrystal and its “magic” {105} facet. Building on Sects. 2.1–2.3, we will discuss, in the following sections, the three main routes to lateral ordering of {105}-faceted SiGe huts. In Sect. 2.4, we will explain the very eﬃcient lateral ordering of huts by using successive alloy layers in SiGe/Si superlattices. In Sect. 2.5 the interplay of nanocrystal arrangement and the dislocation network in partially strain-relaxed SiGe ﬁlms, leading to checkerboard-like crystallite arrays are described. We will illustrate the inﬂuence of the substrate vicinality on the lateral ordering of the huts in single SiGe ﬁlms in Sect. 2.6. In Sect. 2.7, we will, by comparing Ge/Si(001) with III–V, II–VI, and IV–VI heteroepitaxial-growth systems, demonstrate that self-organization routes in Ge/Si(001) are widely applicable to semiconductor heteroepitaxy. In Sect. 2.8 we provide a summary as well as a perspective on potential applications of laterally ordered heteroepitaxial nanocrystal arrays other than QDs, focusing again primarily on SiGe/Si(001).

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2.2 Quantitative Morphological Analysis of Laterally Self-Organized QD Arrays To analyze pattern formation on the nanometer scale, methods are required that allow two things: ﬁrst, the clear identiﬁcation of individual surface structures and second, the determination of distribution functions of size, shape, and separation of the structures to quantify the structure uniformity, including the degree of lateral ordering. The former is, of course, accomplished by direct imaging, whereas for the latter, scattering techniques are best suited. Provided suﬃciently large image sizes covering a large ensemble of individual structures are available, applying scattering analysis to real-space images can also supply the desired information on structure uniformity. Modern scanning probe microscopy (SPM), i. e., STM and AFM, are very well suited to imaging surface structures on the nanometer scale, even down to atomic resolution. Because AFM does not require conducting or semiconducting surfaces it allows ex-situ imaging of semiconductor surfaces that frequently form an insulating oxide layer after coming into contact with air. AFM is therefore the favored imaging technique to investigate nanostructures on semiconductor surfaces. In AFM, the resulting “image”, a recorded matrix zij = z(xi ,yj ) = z(r ) typically containing N × N equidistant pixels, represents a 3D topography. Assuming a proper calibration of the piezoelectric scanner (or the real-time measurement of the scanner’s movement in closed loop feedback) and an inﬁnitely sharp AFM tip, the measured topography corresponds to the true 3D topography of the investigated surface. Because of the necessarily ﬁnite dimensions of the AFM tip, its “convolution” with the probe geometry is measured [7]. For lateral structure sizes that exceed considerably the tip radius (for virgin, conventional tips this value is typically 10–20 nm) and structure slopes less than half the opening angle of the AFM tip, this eﬀect can be neglected. Figure 2.2 demonstrates how AFM images from a given QD array can be analyzed with respect to individual structure sizes and shapes, as well as to obtaining quantitative information on size and shape distributions. All AFM images are shown in color or gray-scale presentation with light corresponding to higher z values. The color/gray-scale range (hereafter called the z scale) is given with each image. The sample chosen for illustration in Fig. 2.2, a homoepitaxial Si ﬁlm grown on a Si(001) substrate that is tilted by 4◦ with respect to the [110] direction, exhibits nanostructures of a certain size and shape uniformity. The growth conditions resulted in the kinetically induced formation of 3D nanocrystals. As one can see from the high-resolution AFM image (Fig. 2.2a) the crystallites have preferential edges along the close-packed 110 direction. The size and 3D shape of the individual structures are obtained by analyzing one-dimensional (1D) cross sections in [110] and [¯110], as is demonstrated in Fig. 2.2b, c. In addition to the lateral and vertical dimensions of the individual nanocrystals, information on their 3D shape is also accessible from the

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Fig. 2.2. Morphology analysis of an epitaxial QD array. a 1 μm × 1 μm AFM image, z scale: 40 nm. b One-dimensional (1D) cross section perpendicular to the miscut direction along the solid line indicated in a. c 1D cross section in miscut direction along the dashed line in a. The z scales in b and c are exaggerated by a factor of 12.5. d Two-dimensional (2D) histogram of the orientations of local surface normals calculated from the AFM image in a after correction for the substrate miscut. The histogram is displayed in polar-plot presentation, with the white circle marking a polar angle of 90◦ . Crosses denote speciﬁc crystallographic orientations. e 10 μm × 10 μm AFM image, z scale: 30 nm. f 2D power spectrum calculated from the large-area scan presented in e, lateral frequency range from −25 μm−1 to 25 μm−1 . Light means high density. g Radial power spectrum calculated from the image presented in e. The arrows mark the full width at half-maximum (FWHM) of the ring structure visible in f

cross sections. The trapezoidal cross sections in Fig. 2.2b, c indicate that the crystallites have the shape of a fourfold, truncated pyramid that is terminated by a (001) oriented terrace. The asymmetric cross section in Fig. 2.2c is due to substrate miscut. The slopes of the four side walls of the mesa are about 25◦ with respect to (001), indicating that the sides are {113} facets. In order to characterize an ensemble of self-organized nanostructures with regard to faceting, easy access to quantitative information on the existence and orientation of facets is obtained by analyzing a 2D histogram of the local surface orientations [8]. For this purpose, the local surface normals n are calculated for each image point. For vicinal substrates, a transformation of coordinates must ﬁrst be applied to the image z(x,y) to correct for the substrate miscut [6]. The polar angle nθ and the azimuthal angle nφ of all surface

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normal vectors will be displayed as a polar plot, yielding the histogram of the orientations of all local surface normals. In this plot, the origin corresponds to the orientation of the chosen low-index plane, e. g., (001). The azimuthal angle in the plot equals the azimuthal angle of n and the distance of a point from the origin is proportional to the polar angle of n. The gray-scale in the polar plot is given by the number of image points with the corresponding n orientation. Because of uncertainties in the performance of the piezoelectric scanner, the inﬂuence of the tip shape and noise, the accuracy of the determination of nθ and nφ is about ± 4◦ , which is suﬃcient to distinguish between diﬀerent low-index (11n) and (10n) facets. Figure 2.2d represents the facet orientation analysis for the image shown in Fig. 2.2a. It shows the existence of the four {113} facets and of the (001) terraces of the truncated pyramids. The stronger peak at (¯ 1¯ 13) compared to that for (113) clearly mirrors the asymmetric shape of the mesas caused by the substrate miscut. Thus, one can even semiquantitatively distinguish between the area fractions of certain facets. Further, it can be seen that the transition from the {113} facets towards the (001) face occurs neither abruptly nor continuously but rather by insertion of shallower facets, namely {117} facets having 11.4◦ tilt angle (with respect to the (001) plane). Back transformation of the individual peaks in the histogram of local surface normals into a topographic image [8] reveals their spatial origin and therefore yields information on size, shape, and the arrangement of the individual nanofacets having the corresponding orientation. Figure 2.2e–g demonstrates how the analysis of the PSD yields easy access to distribution functions of dot sizes and dot separations. The PSD or power spectrum of the surface roughness [9, 10] is the product of the Fourier transform of the recorded image zij with its complex conjugate. Detailed formalisms to calculate the power spectrum from a digitized SPM image are given in [11]. In order to improve the statistics, a larger image size must be chosen for the PSD analysis, simultaneously assuring that the structures of interest are covered by a suﬃcient number of image pixels. Figure 2.2e–f presents a 10 × 10 μm image together with the corresponding 2D PSD. In general, for a periodic surface pattern the 2D power spectrum would consist of a periodic array of sharp peaks. From orientation and symmetry of the peak pattern, the real-space orientation and symmetry of the pattern can be deduced. The location of the ﬁrst-order peak in reciprocal space yields in real space the lateral periodicity of the pattern in the corresponding direction. For a nonideal ordering, the peaks broaden and the number of detectable higher-order peaks decreases. The full width at half-maximum (FWHM) of the ﬁrst-order peak is a quantitative measure of the width of the distribution function of the structure separations. In other words, the narrower the peaks in the power spectrum and the more high-order peaks are visible, the higher is the uniformity of the surface pattern. Here, the 2D PSD shows a ring structure in the center that is surrounded by four weaker peaks. The ring is caused by the random arrangement of the mesas. Its circular shape indicates that

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the arrangement of the mesa-like crystallites is laterally isotropic, i. e., there is no preferential orientation of the dot centers. From the ring radius (dashed line, marked A) the preferential separation of the crystallites of about 250 nm is deduced. The FWHM of the ring is most reliably determined by analyzing the radial PSD, where the 1D PSDs are averaged over all lateral directions (see Fig. 2.2g). From this value we derive that the width of the separation function distribution ranges from 175 to 300 nm. The outer structure in the power spectrum shows a fourfold symmetry, resulting from the square shape of the (001) terraces with edges along [110] on top of the mesas. From the peak position (marked B), the preferential size, i. e., the edge length, of these terraces is found to be 135 nm. In order to obtain information on the uniformity of the structures, the width of the corresponding peaks in the PSD must be determined from 1D cuts through the 2D power spectra, as will be shown in Sect. 2.4.

2.3 Overall Scenario of S–K Growth in Si1−xGex/Si(001) Heteroepitaxy Growth of pure Ge or Si1−x Gex alloy layers on Si(001) proceeds in the S–K growth mode [1], in which pseudomorphic growth of a tetragonally distorted, smooth wetting layer is followed by the formation of 3D nanocrystals, but the process of strain relief via morphological changes is complex. The ﬁrst stage of strain relief in Si1x Gex /Si(001) is the formation of a (2xn) surface reconstruction that has already begun at submonolayer coverages for pure Ge [12, 13] and is delayed for reduced Ge content in the ﬁlm [14]. The reconstruction evolves from the Si(001)-(2x1) reconstruction, in which rows of dimers are formed. Dimers also form in the Ge ﬁlm; however, on average every nth dimer is missing in each dimer row (the stress is compressive along the rows even in clean Si(001) and more so for the Ge ﬁlm). For submonolayer Ge ﬁlms, n is 9 and decreases slightly with thickness [15]. The rebonding of the second layer atoms at these missing dimers provides strain relief in the ﬁlm [16]. Interactions between the dimer vacancies cause them to form increasingly well-ordered rows [17] that appear as trenches in the surface morphology. This strain-driven alignment of the dimer vacancies results in the observed (2xn) reconstruction. Thus, the (2xn) reconstruction of the Si1x Gex wetting layer during growth on Si(001) can be regarded as the selforganization of eight-dimer-wide Ge or SiGe stripes that arrange themselves in two domains. If surface reconstruction by insertion of dimer vacancies is no longer sufﬁcient to relieve strain in the wetting layer, 3D nanocrystals form: the initial ones are the {105}-faceted huts. Their elongation may be related to strain relaxation [18]; however, more recent in-situ STM measurements have shown that it is rather due to a kinetically self-limiting growth [19]. This elongation causes the crystallites to arrange themselves in an interlocked array exhibiting

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Fig. 2.3. Surface morphology of a 2.5 nm Si0.25 Ge0.75 single layer grown on vicinal Si(001) substrate (θ = 0.25◦ , φ = 25◦ ). a 0.8 × 0.8 μm AFM image. b Corresponding polar plot of the orientations of local surface normals. c 5 μm × 5 μm AFM image showing that the crystallites are superimposed on a step-bunched ripple pattern. The z scale in a and c is 5 nm. d 2D power spectrum, ranging from −50 to 50 μm−1 calculated from the image presented in c. The split peak in the center reﬂects the azimuthal orientation (25◦ ) and the periodicity of the ripples (600 nm) and the broad frame-like feature is due to the arrangement of the {105}-faceted nanocrystals

a broad distribution of sizes and shapes. Figure 2.3 shows the surface morphology of a 2.5 nm Si0.25 Ge0.75 alloy ﬁlm grown by molecular beam epitaxy (MBE) at 550 ◦ C on a vicinal Si(001) substrate. The AFM image (Fig. 2.3a) exhibits a labyrinth-like array of rectangular huts and square-based pyramids. The analysis of the local surface normals presented in Fig. 2.3b reveals four clear peaks of equal probability at a polar angle of 11◦ ± 3◦ and azimuthal orientations along 100 corresponding to four {105} facets. Thus we can conclude ﬁrst that the rectangular and square-based structures are all {105}-faceted. Second, the equal area fraction of all four facets indicates

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that the elongation of the hut crystallites occur in both directions, [100] and [010] with the same probability. Third, from the vanishing peak in the center of the polar plot (corresponding to the (001) orientation) we conclude that the surface is completely faceted under these particular growth conditions. In other words, the hut-like and pyramid-like crystallites touch each other, or at least approach to within ∼ 4 nm, the resolution owing to the size and shape of the AFM tips used. Thus, the {105}-faceted crystallites form a close-packed and interlocked 2D array. In Fig. 2.3c, d, a 5 × 5 μm AFM image and the corresponding 2D power spectrum are presented. The fourfold broad feature that is dominating the power spectrum reﬂects the interlocked crystallite array. By analyzing 1D cuts through the 2D power spectrum in

100 directions, information on the hut size distribution is obtained [20]. From the peak location, the predominant spacing of the hut centers (and equivalently the length of the base edges, because the huts touch each other) is calculated to be 35 nm. From the FWHM, the width of the crystallite size distribution is found to range from 20 to 60 nm. Thus, the ratio of the FWHM of the crystallite size distribution to the predominant crystallite size is greater than 1, indicating a very broad distribution. Further, from a manual analysis of the island bases an average ratio of the long edge to the short one of 1.45 has been determined again with a broad distribution function [20]. The large-scale real-space image (Fig. 2.3c) also reveals that the interlocked dot array is superimposed on a ripple pattern. From the split peak in the center of the power spectrum, the ripple period is determined to be 600 nm. The ripple pattern is due to rearrangement of the steps preexisting on the vicinal substrate [21,22]. Originally considered to be a strain-induced eﬀect [23], the step bunching already occurs in the Si buﬀer layers commonly grown prior to Ge deposition [24, 25] and is therefore a kinetic growth instability [26]. In a rather narrow growth temperature window, the resulting ripple morphology is maintained upon subsequent growth of SiGe alloy layers [27, 28]. In Sect. 2.6 it will be shown how this phenomenon can be utilized to align {105}-faceted crystallites. X-ray diﬀraction measurements [29–31] can determine the inhomogeneous strain ﬁeld within the hut crystallites: They are fully strained at the bottom and completely relaxed at the apex (i. e., with the Ge lattice constant) [32]. A particular element of the strain relief are the {105} facets. The (105) surface consists of two-atom-wide (001) terraces separated by 100 steps; the dangling bonds of the surface atoms can eﬃciently be removed by dimer formation. Thus, the driving force for the formation of the {105} facets is an eﬀective strain relief along this crystallographic direction in conjunction with the full dimerization of the {105} face that overcompensates the increase in surface energy [2, 33]. More recently it has been found by STM investigations of Ge growth on Si(105) that the characteristic feature of the Ge(105) face is a strained rebonded step structure [34]. For III–V systems, a hybrid approach – combining calculations of the surface energies by densityfunctional theory and the bulk deformation energies by continuum elasticity

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theory – has been used to determine the shape and stability of coherently strained crystallites [35, 36]. The mechanisms governing the formation of hut crystallites are not completely clear. It has been argued that {105}-faceted Si1−x Gex crystallites form via 3D nucleation [37]. However, it has been shown by low-energy electron microscopy (LEEM) that for small Ge concentration x (0.2 < x < 0.6), their formation is not via 3D nucleation but starts from shallow-stepped mounds [38,39] in agreement with the Asaro–Tiller–Grinfeld instability [40–42]. In this context, the diﬀerences in surface diﬀusivity of the atom species in strained-alloy epitaxy play a crucial role [43, 44]. Only very recently, the combination of ﬁrst-principles calculation with continuum modeling yielded more insight into the formation and stability of {105}-faceted hut islands [45]. At a later stage of strain relief, 3D nanocrystals with facets steeper than {105} arise, allowing the crystallites to grow larger, laterally and in height [2]. The facets that have been observed are {113} (tilt angle 25.2◦ with respect to (001)) [8] and {15 3 23} (tilt angle 33.6◦ ) [46,47]. If they are large enough, the nanocrystals are terminated with (001) terraces on top [2]. In a diﬀerent study, crystallites with {113} and {102} facets (25.6◦ tilt angle) were observed and labeled “domes” [48], a term now used in general for multifaceted nanocrystals. The shape transition from {105}-faceted crystallite to domes has been studied in detail during growth and annealing. Depending on the experimental conditions, coarsening [46, 49] or elastic repulsion [50] has been identiﬁed as the driving forces for the shape transition. Daruka et al. have suggested that the shape transition is typically ﬁrst-order with crystallite size, with the discontinuous introduction of the steeper facets at the crystallite edges [51]. The principal evolution of base shape and cross section of 3D faceted nanocrystals in the SiGe/Si(0010 system – found for growth by both chemical vapor deposition (CVD) and MBE – is sketched in Fig. 2.4. The side views reveal that the nanocrystals are very shallow structures at all stages. This situation changes only if plastically relaxed “superdomes” with facets up to {111} appear [52]. It is worthwhile mentioning that slightly diﬀerent facet orientations are observed if a surfactant like Sb or a diﬀerent growth technique, such as liquid phase epitaxy, is used (see also Chap. 5). Also, upon Si overgrowth, a shape change toward even shorter nanocrystals occurs, due to intermixing of the Si being deposited [53, 54]. The strain-relief mechanisms discussed above, i. e., surface reconstruction in the wetting layer as well as formation of coherent 3D huts and domes, are elastic. Beyond a critical layer thickness hc , these mechanisms are no longer suﬃcient to relieve strain. The formation of so-called misﬁt dislocations resulting in plastic (irreversible) strain relief sets in. For a recent review on strain relaxation by a single misﬁt dislocation and by an array of dislocations see [55]. The critical layer thickness, at which dislocation formation for Si1x Gex growth on Si(001) starts, depends on the Ge concentration x. According to mechanical-equilibrium theory, hc is about 1 nm for pure Ge

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Fig. 2.4. Illustration of the shape transition of faceted Si1−x Gex nanocrystals during molecular-beam epitaxy (MBE) growth on Si(001) with increasing strain (from left to right). The cross section in the bottom are cutting the crystallites through their centers. For simplicity, the starting structure is assumed to be a square hut [a {105}-faceted pyramid a] rather than an elongated hut. With increasing strain, steeper facets such as {113} b and pairs of {15 3 23} facets c evolve

and increases up to 100 nm for x = 0.05 [56]. The experimentally determined critical thicknesses (the “kinetic critical thickness”) can be larger by more than an order of magnitude [5]. Dislocations in Ge and SiGe are 60◦ dislocations that glide on a {11¯ 1} plane. Their nucleation and the eﬀect of substrate miscut on it has been discussed in detail by Mooney et al. [57]. The dislocations that are generated at the substrate/ﬁlm interface manifest themselves at the surface by ridges and troughs [58, 59]. These are extended in either [110] or [¯ 110] directions. Thus, at a suﬃciently high number density of dislocations, the development of a misﬁt dislocation network occurs at the substrate/ﬁlm interface. Such a network, consisting of two arrays of single dislocations with alternating glide planes, is sketched in Fig. 2.5a. Frequently, single dislocations pile up into multiple dislocations by a modiﬁed Frank–Read mechanism [60], resulting in a rougher ridge–trough structure on the surface [58]. The dislocation network manifests itself at the surface as a so-called “cross-hatch” pattern [8, 58], which is shown in the AFM image presented in Fig. 2.5b. The pattern is characterized by a network of very straight ridges and troughs along 110 that may extend over several tens of micrometers. Further, it is seen that – under the particular growth conditions applied here [61] – the cross-hatch pattern coexists with the crystallites, oﬀering the possibility of utilizing the interplay of the two strain-relief mechanisms for QD self-organization, as will be demonstrated in Sect. 2.5. The scenario presented above is based on morphological changes. It does not involve a classiﬁcation with respect to Ge segregation at the surface during SiGe alloy growth [62] or strain-driven intermixing during pure Ge growth [63–66]. Such chemical eﬀects will modify surface and nanocrystal free energies, but will not change the sequence of ordered structures or their self-organization.

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Fig. 2.5. Dislocation network and its manifestation at the ﬁlm surface. a Schematic diagram of two orthogonal arrays of alternating single misﬁt dislocations forming at the ﬁlm/substrate interface (after [55]). b 10 × 10 μm AFM image of an 80 nm Si0.7 Ge0.3 ﬁlm (ﬁrst 30 nm grown at a low temperature) showing the cross-hatch pattern with nanocrystals and step bunched ripples superimposed. The z scale is 10 nm

A dense array of coherently strained SiGe hut crystallites (Fig. 2.3) is useful in some applications, such as photodetectors, but the range of applications increases signiﬁcantly if the size uniformity can be improved. In the following three sections we present successful routes to fabricating dense arrays of {105}-faceted SiGe nanocrystals that are much more uniform than those shown in Fig. 2.3: the strain-mediated lateral and vertical ordering in SiGe/Si superlattices [20, 67] (Sect. 2.4), the formation of a checkerboard array of {105}-faceted pyramids induced by a dense dislocation network [61] (Sect. 2.5), and nanocrystal ordering guided by step-bunched ripples on vicinal Si(001) [22, 27] (Sect. 2.6).

2.4 Strain Induced Lateral Self-Organization of {105}-faceted Nanocrystals in SiGe/Si Multilayer Films The regularity of the nanocrystal array shown in Fig. 2.3 does not improve at all if the ﬁlm thickness is increased [68]. However, a dramatic morphological change occurs upon growing an N × (2.5 nm Si0.25 Ge0.75 /10 nm Si) multilayer stack. Figure 2.6 vividly illustrates this point by comparing the morphology of the surface of an alloy-terminated 20-bilayer ﬁlm to the single alloy layer. The 20th alloy layer also consists of huts that touch each other. However, the huts become larger and, most surprising, exhibit a degree of self-organization. They are more uniform in size, they adopt more or less square bases, and they arrange preferentially on chains along 100 or even on a square pattern.

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Fig. 2.6. Lateral self-organization in a SiGe/Si superlattice. 3D AFM images of a single SiGe alloy layer (a) and the 20th alloy layer (b). Note the diﬀerent vertical scales. The insets illustrate the sample structure

To quantify this self-organization phenomenon, detailed PSD analysis of the AFM images as a function of the bilayer number, N, has been performed [20]. The results are summarized in Fig. 2.7. The morphological evolution is clearly manifested in the change of the 2D power spectrum from a diﬀuse and frame-like pattern for a single layer into four relatively sharp peaks with correspondingly higher orders. The predominant crystallite spacing along 100 directions (and equivalently the base size, as the dots touch each other) and the crystallite size distribution are determined from cuts through the 2D power spectrum along the [010] direction (Fig. 2.7c, d). The location of the peaks (arrows) yields the mean crystallite size in [010] and the peak’s FWHM (horizontal lines) gives the width of the size distribution Δ. Figure 2.7e shows the evolution of and the relative width of the size distribution Δ/ . starts at about 35 nm for a single layer and approaches a stationary value of roughly 100 nm. The width of the size distribution relative to the average crystallite size decreases from 1.1 for the ﬁrst layer to 0.3 for 20 bilayers. This narrowing of the size distribution – mainly occurring within the ﬁrst ten layers – is clear evidence for the self-organization process. The increase in dot uniformity is accompanied by a change in dot shape from hut-shape to a four-sided pyramid [20]. The dot height changes from ∼3 nm to ∼10 nm. With increasing thickness, the vertical roughness of the alloy layers increases. However, upon overgrowth of the SiGe dots with the 10 nm Si spacer layer, the morphology always smoothens to nearly that of the Si substrate, with no indications of the underlying 3D structures [53, 68]. If the Si spacer thickness is increased to 30 nm, the self-organization eﬀect is less pronounced [68] and for 70 nm spacer thickness it is completely gone [69].

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Fig. 2.7. a–c AFM images (1 × 1 μm) of SiGe alloy-terminated surfaces of 2.5 nm Si0.25 Ge0.75 /10 nm Si multilayer ﬁlms and the corresponding 2D power spectra as a function of the number of alloy layers N. The 2D power spectra, ranging from −50 to 50 μm−1 , have been calculated from 5 × 5 μm images. Cuts along [010] through the 2D power spectra are shown in c, the ﬁrst alloy layer and d, the 20th alloy layer. e Evolution of the mean dot spacing along 100 (solid line) and relative width of the size distribution, Δ/ (dashed line). The error bars indicate the width of the distribution of dot spacings. The lines are guides to the eye

Thus we can conclude that the Si spacer layers act as a control on the selforganization of the SiGe crystallites in the multilayer stack. How can we explain the spontaneous lateral ordering of the QDs in the superlattices? X-ray diﬀraction (XRD) and TEM measurements of the superlattices reveal a tendency of the huts to pile up in vertical columns [6,21,53]. We therefore assume that the Si spacer layer acts to create a strain distribution that is favorable for the formation of regularly spaced and sized huts. In the following, this eﬀect will be examined in the framework of continuum elasticity theory. The basic features of the self-organization are already revealed within a simple and relatively generic model in 1+1 dimensions, in which the coherently strained huts are treated as spherical inclusions in an

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Fig. 2.8. a Normalized surface strain ε versus lateral displacement (scaled by the spacer layer thickness ho ) due to a strained, point-like dot buried under an isotropic elastic spacer layer at a depth ho and at lateral position x = 0. b Same curve with enhanced ε scaleu The minimum in strain causes the preferential nucleation of a dot in a subsequent alloy layer directly on top of a buried dot as is sketched by the dotted circle in c. d Surface strain ε due to four buried dots (arrows at bottom). Arrows on top indicate minima in ε, i. e., favored positions for subsequent dot formation. e The self-organization ﬁnally leads to the evolution of a 3D QD lattice, as sketched

isotropic elastic matrix [67] (Fig. 2.8). The dots are located on a ﬂat surface and are then buried under a thickness ho of additional matrix material (e. g., Si). The new surface will also be ﬂat (Fig. 2.8c), and thus the strain is the only factor biasing nucleation at this surface. Following Maradudin and Wallis [70], the surface strain ε due to a dot buried at a depth L and lateral position x = 0 is ε(x) = C(x2 + h2o )−3/2 [1 − 3h2o /(x2 + h2o )] .

(2.1)

The coeﬃcient C is proportional to the volume of the buried dot and the misﬁt, and also involves the elastic constants. The resulting strain is shown in Fig. 2.8a, b. For ε a sign convention is chosen so that ε is negative in the tensile region. There is a deep minimum just on top of the buried dot as well as an area with a positive strain (visible better in Fig. 2.8b at an enhanced ε scale). The preferential nucleation site is at the minimum of ε, in other words, directly above the buried dot. The minimum of ε is proportional to 3 the dot volume and indirectly √ proportional to ho . Nucleation is disfavored at lateral oﬀsets larger than 2ho (as can be seen in Fig. 2.8b), with the strain

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dropping smoothly to zero at large distances. The continuous repetition of the preferential formation of a new dot in the multilayer stack just above the buried one results in the piling up of dots (found in numerous TEM studies for the SiGe/Si system, e. g., [53, 71, 72]) and for III–V systems even two decades ago [4]. However, the strain-mediated preferential nucleation on top of a buried dot alone cannot explain the increased lateral ordering in the superlattice because that would just result in copying the crystallite arrangement from alloy layer to alloy layer. Self-organization comes into play if we consider the lateral interaction of the strain ﬁelds of adjacent buried dots [67] as is demonstrated in Fig. 2.8d for an illustrative arrangement of buried dots. The strain ε at the surface of the Si spacer layer is calculated simply by summing Eq. (2.1) over all the buried dots. In this deterministic model, dots are assumed to nucleate at each minimum of ε, with volumes proportional to the areas of the corresponding Vorono¨ı polygons (the region closer to that dot than to any other). Besides replication of the lateral dot positions (labeled A in Fig. 2.8d), two other mechanisms occur. Extremely closely spaced buried dots cause only one minimum, resulting in a “thinning out” of closely spaced dots (labeled B). The shallow minimum of ε between widely spaced dots or random nucleation in these areas cause a ﬁlling in of gaps in the dot arrangement (labeled C). As a result of processes B and C – both of which are clearly observable in TEM [53] – the new positions in Fig. 2.8d are much more uniformly spaced than the initial dot locations. Thus, the observed self-organization is due to the interplay of the strain ﬁelds of adjacent buried dots, i. e., it is mediated by the spacer layer. Once a uniform size and arrangement of the dots is adopted by this process, the vertical replication of the laterally ordered arrangement [6,53] will ultimately result in the self-organized formation of a spatially ordered tetragonal QD lattice, as is sketched in Fig. 2.8e. Although it is simple, the model qualitatively explains the main experimental results: the narrowing of the dot size distribution and the increase of the average dot size with increasing layer number. The experimentally observed change in crystallite base shape cannot be accounted for by the model because of the assumption of point-like spherical dots. To study shape transitions a ﬁnite crystallite size and an anisotropic shape must be introduced. Considering a 1+1-dimensional dot of height H (H ho ) and width W , Eq. 2.1 changes to: ε(x) = −CH/h2o [u(1 + u2 )−3/2 (2 + u2 ) − v(1 + v 2 )−3/2 (2 + v 2 )] ,

(2.2)

with u = (x + W /2)/ho and v = (x – W /2)/ho . C again includes the misﬁt and elastic constants [73]. This reﬁned model yields the result that the width of the tensile region (where ε is negative) is proportional to W 1/2 . This dependence provides a gradual shape transition from rectangular to square bases [73]. The preferential alignment of the huts normal to the growth front is an inherent consequence in the models presented above because of the assumption

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of an isotropic elastic spacer layer. To account for possible elastic anisotropies in the spacer layers, Holy et al. solved the equilibrium stress equations for a point-like dot within a semi-inﬁnite, anisotropic matrix [74,75]. They found that the elastic energy distributions on the surface above a buried dot are determined by two essential parameters, the elastic anisotropy of the matrix material on the one hand and the surface orientation on the other (for details see Chap. 9). The elastic anisotropy is particularly large for the IV–VI semiconductors SnTe and PbTe, producing for PbSe/Pb1−xEux Te superlattices on PbTe(111) an fcc(111)-like ABC QD stacking at proper layer spacings [76]. For Si(001), the elastic anisotropy causes a splitting into side minima, which are, however, very shallow and the angle with respect to the normal under which they appear is small. Thus, in practice only one dot is formed on top of the buried one as has been experimentally observed. Nevertheless, the elastic anisotropy in the case of Si(001) is shown to promote the lateral ordering on a square lattice [77]. For our close-packed dot array, in which the crystallite bases touch each other, the most signiﬁcant contribution to the observed strong lateral ordering is expected to originate in the faceted crystallite shape [20, 73]. To estimate this contribution, an extension of the anisotropic model with point-like stress sources [74] to ﬁnite islands with well deﬁned shapes will be necessary. The experimentally observed improvement of the relative width of the dot size distribution Δ/ from 1.1 for the interlocked hut cluster array to 0.3 (i. e., ± 15% size uniformity) for the 20th alloy layer is quite promising. However, a continuation of the superlattice does not result in further improvement of lateral ordering, as can be seen for the data of the 40-bilayer sample (Fig. 2.7e). Instead, the island uniformity decreases because of the onset of dome formation [6], a direct consequence of the accumulation of elastic strain in the superlattice. This drawback can be overcome by reducing the alloy layer thickness for increasing layer number [78]. The most elegant idea to avoid completely the accumulating strain in the multilayer stack is the growth of strain-symmetrized superlattices which lead to an astonishingly small value of ± 6% for the size distribution in a 100-bilayer PbSe/Pb1−xEux Te superlattice [76] (see Chap. 9).

2.5 Self-Organization of Nanocrystals via Interaction with Misﬁt Dislocations Under certain preparation conditions, spontaneous lateral ordering of 3D SiGe dots on Si(001) is already observed in a single alloy layer, for example, when the nanocrystal formation rests on an existing dislocation network. First observations of this interplay were reported in 1995 [79]. An initially dislocation-free and smooth Si0.5 Ge0.5 ﬁlm that was MBE-grown at 400 ◦ C on Si(001) showed after annealing at 560 ◦ C nanocrystals decorating the signatures on the growth surface, the misﬁt dislocations occurring

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at the ﬁlm/substrate interface. This preferential nucleation of pyramid-like nanocrystals at the surface at positions where the misﬁt dislocations exit (leaving a step and a strain ﬁeld) has been utilized to create regular arrays that are guided by an underlying rectangular network of misﬁt dislocations. On a compositionally graded Si1−x Gex layer that exhibits a regular cross-hatch pattern, Ge dots of about 40 nm height and 280 nm base width almost exclusively nucleate along the ridge–trough structures of the crosshatch pattern, thus forming a highly ordered but diluted dot ensemble [80]. On a slightly diﬀerent substrate, well-deﬁned {105}-faceted pyramids form mainly on top of the intersections of the dislocation network [81] since the misﬁt is smallest above the intersections. However, in both cases, the dot density is too small for potential applications. The dot density can be signiﬁcantly increased under conditions of progressive relaxation that result in a dense dislocation network [61]. Such conditions are provided by a two-temperature growth procedure originally designed to fabricate virtual substrates for vertical metal oxide semiconductor transistors [82]. In the ﬁrst stage, 30 nm Si0.7 Ge0.3 ﬁlm is deposited at low temperatures (LTs) (150 ◦ C – 200 ◦ C) and the subsequent 50 nm alloy layer is grown after the sample temperature has been increased to 550 ◦ C. Figure 2.9 shows the AFM results for LT growth at 150 ◦ C. The morphology is characterized by a superposition of close-packed crystallite arrays and a cross-hatch pattern. The cross-hatch pattern is decorated by uniform four-sided pyramids that stick out of the surrounding crystallite arrays (Fig. 2.9a). The degree of relaxation, measured by X-ray diﬀraction, is about 20% and about 90% of these dislocation lines extend to more than 30 μm length [61]. Along these straight dislocation lines, uniform {105}-faceted pyramids arrange themselves like beads. Their height is about 15 nm, with slightly higher crystallites (17 nm) at the intersections of the dislocation network. The pyramid-decorated dislocation lines running along 110 result in a sharp cross in the 2D power spectrum that is similar to the diﬀraction spots from needle-like structures. This cross shows bright spots, indicating a preferential separation of the pyramid chains. The brightest spot corresponds to the pyramid spacing, which is about 190 nm. Analysis of the surface normals of the AFM image reveals that the areas between the pyramid chains are indeed ﬁlled by irregular arrays of hut crystallites. From the frame-like halo in the power spectrum, their average base size has been determined to be 160 ± 40 nm. As is seen in the center of the AFM image presented in Fig. 2.9a, two or three dislocation lines are occasionally spaced so closely that the pyramids touch each other and form locally a uniform square array. One can expect, for a suﬃciently high density of dislocation lines, that the {105}faceted nanocrystals will all arrange themselves on the intersections of the dislocation network. Figure. 2.10 shows the results for an 80 nm Si0.7 Ge0.3 alloy ﬁlm grown under identical conditions as the one presented in Fig. 2.9, with the only diﬀerence being that the ﬁrst 30 nm were grown under simultaneous 1-keV Si+ ion bombardment [83]. The morphology is now clearly dominated

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Fig. 2.9. Chains of {105}-faceted pyramids guided by a cross-hatch pattern. a 5 μm × 5 μm image. b 2D power spectrum calculated from a 30 μm × 30 μm image ranging from −8.5 to 8.5 μm−1 . The cross originates from the cross-hatch pattern and the frame-like halo from the interlocked hut crystallites

by a dense cross-hatch pattern that is decorated by uniform pyramid-like nanocrystals [61]. In the areas of dense dislocations we observe a surprisingly uniform surface pattern (see the high-resolution image presented in Fig. 2.10b). It looks like a man-made, lithographically patterned “top-down” structure rather than one that spontaneously self-organized. The pattern consists of four-sided pyramids and pits that form a checkerboard array, as is demonstrated by the 3D view (Fig. 2.10c). The periodicity in 110 directions is about 190 nm as obtained from the 2D PSD (inset in Fig. 2.10a). Analysis of the orientations of local surface normals reveal that the entire surface in these areas consists of {105} facets. In other words, the checkerboard array of {105}-faceted pyramids and pits is nothing else than a close-packed array of uniform {105} nanofacets. Figure 2.10d shows a 3D model. For negligible substrate miscut, the nanofacets have a rhombic shape that deviates only slightly from a square (acute angle of 88.9◦ ). The edge length of the rhombs is about 95 nm. The driving force for the checkerboard evolution originates in the special elastic properties of the Ge{105} facet [6]. The crystallites nucleate almost exclusively on intersection sites. The geometry of the dislocation network and the base shape of the {105}-faceted pyramids lead to a c(2×2) crystallite ordering. Driven by the low elastic energy of the {105} facets [2, 34], the areas between the pyramids transform into {105}-faceted pits supplying SiGe material to the pyramids. The magnitude of the lateral period in the checkerboard pattern and how it can be controlled are not yet completely understood. With this knowledge one can expect to extend the long-range ordering within the checkerboard array from the present ∼1 μm to several tens of micrometers and to reduce the pattern periodicity. The latter has not yet been achieved with the technique described here. However, twist wafer bonding

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Fig. 2.10. Checkerboard array of self-organized SiGe pyramids. a 10 × 10 μm image Inset 2D power spectrum calculated from a 20 × 20 μm image ranging from −12.8 to 12.8 μm−1 . b 1 × 1 μm image of the area framed in a. c 3D presentation of the image shown in b. d 3D model of the surface (to scale). T denotes the tip of a pyramid and B points to the bottom of a pyramidal pit

results in a very dense dislocation network that can be used, after strain selected etching, as a template for Ge growth. Crystallite sizes of 20 nm have been observed [84].

2.6 Lateral Alignment on Vicinal Substrates The second approach to self-organized dot ordering in a single SiGe alloy layer utilizes step bunching of pre-existing steps on vicinal Si(001) substrates. As mentioned in Sect. 2.3, the step bunching occurs during buﬀer layer growth but can be maintained upon subsequent growth of SiGe alloy layers. It has been shown that ripple distance and orientation can be tailored by varying polar and azimuthal miscut of the substrate [22, 27]. Because the ripples evolve prior to the {105}-faceted crystallites, one might expect an inﬂuence of the underlying ripple structure on the crystallite formation in cases where the ripple direction is parallel to one pair of crystallite edges, i. e., along [100] or

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Fig. 2.11. Step bunched ripples to guide {105}-faceted hut nanocrystals. a 2.5 × 1.5 μm image of a 2.5 nm Si0.55 Ge0.45 ﬁlm grown on vicinal Si(001) with polar miscut θ = 2◦ along [100]. b Schematic 3D diagram of the ripples. c 1 × 1 μm AFM image of a 2.5 nm Si0.25 Ge0.75 ﬁlm grown on the same substrate. d 2D power spectrum, calculated from a 5 × 5 μm image and ranging from −50 to 50 μm−1

[010], and the ripple separation is in the range of the crystallite size. A ripple structure that fulﬁlls the conditions mentioned above is realized on a Si(001) substrate with a polar miscut angle of 2◦ towards [100]. The situation is illustrated in Fig. 2.11. For a 2.5 nm Si0.55 Ge0.45 ﬁlm grown on this substrate, we observe ripples running along [010] (Fig. 2.11a). From the corresponding power spectrum a mean ripple separation of 70 nm is determined. The ripples have a symmetric triangular cross section with both side slopes in the range of the polar miscut angle θ. They consist of an extended (001) terrace on the one side and a step bunch (having a local miscut that is 2θ) with doubled step density compared to that of the substrate on the other side [22] as is sketched in Fig. 2.11b. The ripple morphology changes to an ordered array of hut nanocrystals after growing an additional 2.5 nm Si0.25 Ge0.75 ﬁlm (Fig. 2.11c) [22,27]. The hut crystallites – now exclusively elongated in [010] – are arranged in straight chains parallel to the [010] direction. The 2D power spectrum (Fig. 2.11d) shows clear peaks even with higher orders. From the peak positions the mean width (+) and length (×) of the guided huts can be derived. They have a very uniform base width of 35 ± 3.5 nm. The lengths of the crystallites are more broadly distributed, ranging from 35 to 65 nm

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and reﬂecting less ordering within the chains. From the situation presented in Fig. 2.11, it becomes evident that on each side of a ripple, [both the (001) terraces and the step bunches], chains of nanocrystals form. One can conclude that the step-bunched ripples guide the subsequently forming {105}-faceted crystallites. Similar eﬀects, resulting in diﬀerent dot patterns, are observed when growing on vicinal Si(001) substrates with higher polar miscut, i. e., on low index Si(11n) and (10n) substrates. For growth on Si(113) ordered arrays of rather elongated Ge crystallites terminated by {159} facets are observed [85]. On a step-bunched Si(113) substrate, close-packed rows of compact crystallites with a diameter of 80 nm form along large step bunches [86]. On Si(001) with a 10◦ polar miscut, alignment of rather large domes has been observed [87]; using Sb as a surfactant helped to reduce their lateral size to 35 nm [88]. A further case of directed nanocrystal self-assembly on vicinal Si(001) is based on a step bunch to facet transition [27] observed for SiGe growth on Si(001) substrates with a polar miscut angle of θ = 4◦ towards 100. Figure 2.12 shows an AFM image of the growth morphology of a single 2.5 nm Si0.55 Ge0.45 layer grown on such a substrate. Here, the ripples are no longer straight but consist of quite uniform triangular structures. The well oriented triangles form chains along [¯ 110] appearing in the height representation as a zigzag pattern. Analysis of the orientations of local surface normals (Fig. 2.12b) reveals a peak in the center, indicating the presence of (001) oriented areas, as is the case for the straight ripples at lower miscuts. 1¯ 10], two peaks, indicating (¯105) and Instead of a further peak at 2θ = 8◦ in [¯ ¯ (015) facets, appear. Back transformation of the individual facet peaks shows that the (001) terraces are predominantly square shaped (35 nm × 35 nm) and the two {105} facets are parallelograms (25 nm × 35 nm, acute angle 47◦ ). The ideal arrangement of these nanofacets, illustrated in Fig. 2.12c [22], is reminiscent of a tiling frequently used by M.C. Escher to pretend threedimensionality [89]. A similar pattern, consisting of three types of {113} facets, is observed for 10 nm Si0.7 Ge0.3 on Si(118) grown at 520 ◦ C and annealed at this temperature for 70 h [90]. The observed step bunch to facet transition can be considered as a collective meandering of straight step bunch segments [6] that does not require any new steps to form. Only the length √ of the step edges has to be increased by a factor of 2. This meandering results in an increased surface area that is accompanied by an increase of the root mean square roughness compared to the ripple pattern [27]. However, the increase in surface energy is overcompensated by the reduction in strain energy eﬀected by the formation of the {105} facets. The observed lateral self-organization is therefore the result of a strain-driven collective transition from bunches of straight steps into nanofacets. As has been shown recently, this transition is kinetically hindered for growth temperatures below 400 ◦ C [91]. The nanofaceted morphology shown in Fig. 2.12 is a precursor to an ordered array of fully {105}-faceted nanocrystals. Figure. 2.13a shows the mor-

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Fig. 2.12. Step bunch to facet transition in a Si0.55 Ge0.45 ﬁlm grown on vicinal 1¯ 10]. a 1 × 1 μm AFM image. b CorSi(001) substrate with a 4◦ miscut towards [¯ responding polar plot of the orientations of local surface normals. c 3D model of the ideal facet arrangement (to scale). T points to the top and B to the bottom

phology of a 2.5 nm Si0.25 Ge0.75 ﬁlm that is grown on a substrate with the same vicinality as was used in Fig. 2.12. It exhibits {105}-faceted crystallites with a uniform elongation of their bases along [110], in contrast to the orientation of the hut nanocrystals. The unexpected shape of the resulting dots becomes evident when one assumes that the step-bunch faceting triggers the nanocrystal formation (Fig. 2.13b): guided by the already existing pair of {105} facets, four-sided pyramids will form on top of the (001) facets. The resulting nanocrystal (Fig. 2.13c) is terminated by trapezoidal (¯105) and (0¯ 15) facets and triangular (015) and (105) facets. As a consequence of this shape, the nanocrystals order in a distorted hexagonal array that is described by a centered rectangular unit mesh. Improved quasi-hexagonal ordering has been obtained by growing a Si0.55 Ge0.45 ﬁlm on the same substrate orientation at 600 ◦ C [91] and by depositing Ge on a rippled SiGe ﬁlm [92] (see also Chap. 8). The examples presented in this section clearly demonstrate the possibility of obtaining lateral ordering of coherently strained QDs by growing on vicinal surfaces with a suﬃciently high step density. The self-organization results from an interplay of kinetically induced bunching of the preexisting substrate steps and strain-driven nanocrystal formation. The step bunching, already

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¯10]. ¯ a 1 × 1 μm Fig. 2.13. Dot ordering on a substrate with a 4◦ miscut towards [1 AFM image of a Si0.25 Ge0.75 ﬁlm. b Model of the ideal crystallite arrangement. c Resulting crystallite shape. d Top view of the ideal dot array with a centered rectangular unit mesh

occurring during growth of the Si buﬀer layer, results in ripple formation consisting of step bunches and extended step-free terraces. For suﬃciently high miscut, when the resulting step bunch and terrace dimensions are below 100 nm, the ripple pattern acts as a template for crystallite growth. Also, a strain-induced step bunch to facet transition resulting in alternating facet segments can induce dot ordering. By exploring various high-miscut substrates, a variety of self-organized dot patterns has been obtained [87] (see also Chap. 6). The approach clearly demonstrates that morphological surface patterns can trigger QD ordering. In recent years, instead of using templates spontaneously formed during buﬀer layer growth on vicinal substrates, man-made templates have been used, yielding much better ordering (see Part II of this book).

2.7 Comparison to III–V, II–VI, and IV–VI Heteroepitaxial Growth In III–V, II–VI and IV–VI semiconductors systems, strain-driven self-organization of nanocrystals also results in poor order in a single layer, although the nanocrystals tend to be more uniform in size and more compact. The routes

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discussed above for improved ordering also work in most of these systems. Strain-mediated self-organization has been early found in InAs/GaAs superlattices [93–95]. Vertical alignment was also found for GaN dots in GaN/AlN multilayers [96]. For IV–VI semiconductors, the already mentioned strain symmetrized superlattices of lead salts revealed the most superior ordering obtained so far [76]. The strong elastic anisotropy in the spacer layer leads, in this system, to an fcc(111)-like ABC dot stacking. An ABAB stacking was observed for CdSe/ZnSe [97]. Lateral self-organization due to an interplay with the dislocation network has been found, in addition to the Group IV systems we describe, so far only – to our knowledge – for the III–V system [98]. Lateral alignment on vicinal substrates has been observed for the III–V system in numerous studies. Highly packed InGaAs dots form, e. g., when growing on GaAs(311)B [99] whereas steps or step bunches have been used to align InAs or InGaAs dots in low-density arrays [100, 101]. Recently, the latter eﬀect has also been successfully applied in the II–VI system [102]. The above list is far from complete. For a detailed discussion of the peculiarities of nanostructure self-organization in the various semiconductor materials, the reader is referred to a comprehensive review by Stangl et al. [30].

2.8 Summary and Outlook After a brief introduction to SPM-based quantitative analysis of quasiperiodic QD arrays, we reviewed the strain-driven self-organization of 3D nanostructures for S–K growth of SiGe on Si(001). The following routes for self-alignment of coherently strained crystallites have been discussed and analyzed in this model system: 1. Lateral and spatial ordering of {105}-faceted nanopyramids in SiGe/Si multilayer ﬁlms that is driven by the strain-mediation in the Si spacer layers. 2. Interplay of crystallite arrangement and cross-hatched dislocation networks, yielding a self-organized checkerboard array of {105}-faceted pyramids. 3. Crystallite ordering on vicinal Si(001) guided by spontaneously forming step-bunched ripple patterns on vicinal Si(001). These routes – which are not restricted to group IV semiconductors – allow control of the size, shape, and arrangement of the QDs by tuning substrate miscut, layer composition, individual layer thickness, and growth conditions, resulting in a diversity of possible surface patterns. Because of the large parameter set (which can even be increased by dynamic variation of growth conditions) we expect further improvement of ordering for the already-known patterns, as well as a plethora of novel self-organized nanostructure arrays that await discovery. Besides their natural potential for optoelectronic applications, these laterally ordered arrays of nanofaceted crystallites oﬀer the

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possibility of use as large-area templates for the growth and organization of other materials, such as magnetic thin ﬁlms [103] or organic ﬁlms [104]. Finally, self-organized nanopatterns may serve as a stamp to transfer (via nanoimprint lithography) the self-organized surface patterns onto other materials [105]. Acknowledgement. We would like to thank our coworkers and collaborators M. Bauer, J.C. Bean, C. Hofer, E. Kasper, F. Liu, K. Lyutovich, Y.-M. Mo, L.J. Peticolas, Y.H. Phang, and J. Tersoﬀ for their contributions as well as F.S. Flack and C. Hofer for technical support preparing the manuscript.

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3 Short-Range Lateral Ordering of GeSi Quantum Dots Due to Elastic Interactions Jerrold A. Floro1 , Robert Hull2 , and Jennifer L. Gray2 1

2

Sandia National Laboratories, Surface and Interface Sciences Department, Albuquerque, NM 87185-1415, USA University of Virginia, Department of Materials Science and Engineering, 116 Engineers Way, Charlottesville, VA 22904-4745, USA

3.1 Introduction GeSi alloy ﬁlm growth on Si (001) represents perhaps the most intensively studied of all heteroepitaxial materials systems. This is truly a model system for exploring fundamental scientiﬁc issues, oﬀering complete chemical miscibility of the constituents, relatively low volatility, a simple diamond cubic lattice, a range of lattice misﬁt up to 4% for pure Ge on Si, and a nearly linear dependence of lattice parameter on alloy composition [1]. Furthermore, the atomic structure and the nature of surface diﬀusion processes on the Si (001) surface have been extensively characterized. Finally, the obvious importance of Si to microelectronics, and the rapidly increasing technological importance of Ge and GeSi, lends additional motivation to studies of this material system. In this chapter we will examine ordering processes occurring during molecular beam epitaxial growth (MBE) of dilute Gex Si1−x alloys, x = 0.2 − 0.3, on Si (001). We will distinguish between the formation of long-range order within dense quantum dot (QD) arrays, which we will refer to as extended order, and local ordering processes that can produce symmetric quantum dot molecules (QDMs). These strain-induced mechanisms take place under diﬀerent deposition conditions; ideal QD arrays form “near-to-equilibrium”, whereas QDMs form under “kinetically limited” conditions. Operational deﬁnitions of these terms shall be provided later. Nonetheless, ordering processes in both cases result from elastic interactions mediated by the underlying material. Our investigations of ordering have been aimed at developing fundamental understanding of the mechanisms that govern strain-driven morphological evolution during growth. However, it is also useful to consider whether ordering would be technologically important. For optoelectronic applications, e.g., where quantum dots would serve as the active medium for light emission or detection, ordering is not explicitly required. However, highly ordered arrays of quantum dots are likely to possess related characteristics that are useful for optoelectronics, including narrow size distributions, compositional homogeneity, and high areal density. For photonics applications such as the self-assembly of a photonic lattice (in 2D or 3D), extended ordering is nec-

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essary by deﬁnition. For applications such as nanologic, quantum dots must be precisely located for circuit functionality and addressability, clearly an even more demanding requirement than simple ordering since it requires patterns that need not be periodic, and which occur over a wide range of length scales. In this chapter, we will ﬁrst brieﬂy review the experimental techniques, including a novel light scattering approach for real-time monitoring of QD formation. Then we examine extended ordering in dense arrays of QDs. We follow by describing the origins of local ordering to form QDMs, and how these ordering processes can be modiﬁed and controlled.

3.2 Experimental Techniques 3.2.1 MBE Growth GeSi alloy ﬁlms were grown using electron beam co-evaporation in a customdesign molecular beam epitaxy (MBE) chamber. Evaporation sources were 40 cc monolithic starter sources with 99.999% initial purity. The base pressure of the chamber was below 1 × 10−10 Torr, but the pressure rose to 1 – 2 × 10−8 Torr during deposition, primarily due to hydrogen, but also with some (less than 1 × 10−9 Torr partial pressure) CO, CO2 , and CH4 from the hot ﬁlaments. Partial pressures of O2 and H2 O remained below 1 × 10−10 Torr. Deposition rates were controlled using calibrated quartz crystal oscillators. All heating was by radiative transfer from a nude W ﬁlament and the sample temperature was monitored using a pyrometer, with an absolute temperature accuracy of about 25 ◦ C. The substrates were diced from undoped Si (001) wafers, with a miscut no greater than 0.1◦ , to dimensions 0.5 × 1.5 × 0.012 thick. Cleaning for epitaxy involved chemical formation of a nonstoichiometric oxide that was ultimately removed by in-situ desorption just before buﬀer growth. All chemicals were clean-room electronic grade and the rinse water was ﬂowing, ultraﬁltered 18 MΩ deionized water. After dicing, the substrates were ﬁrst subjected to solvent degreasing, both in ultrasonic (2-propanol and acetone) and at elevated temperatures (trichloroethylene at 80 ◦ C). Next, residual hydrocarbons were removed using an exothermically heated 4:1 H2 SO4 :H2 O2 mixture. Trace transition metals were then removed using the sequence: etch in 1:1:4 HCl:H2 O2 :H2 O at 80 ◦ C, rinse, oxide removal in 7:1 buﬀered oxide etch, which was repeated three times. The ﬁnal chemical oxide was formed using a 3:1:1 HCl:H2 O2 :H2 O solution at 80 ◦ C, followed by an extensive rinse, N2 blow dry, after which the sample was immediately mounted to a Mo platen and pumped down in the MBE load lock. After transfer into the growth chamber, samples were degassed by ramping from room temperature to 630 ◦ C over 14 – 20 h. Oxide desorption occurred during 820 ◦ C annealing for 15 min, with continuous monitoring of the

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surface structure using reﬂection high-energy electron diﬀraction (RHEED). A Si buﬀer layer was grown 100 nm thick at 750 ◦ C, using a low/high/low sequence for the deposition rate. The RHEED pattern after buﬀer growth, monitored along the 110 azimuth, typically consisted of a Laue circle of intense spots at both integral and half-order positions, characteristic of a smooth, 2×1 reconstructed surface. During growth, the GeSi ﬁlms were also monitored using the multibeam optical stress sensor (MOSS) and light-scattering spectroscopy (LiSSp). MOSS is a curvature-based technique for sensitive real-time stress measurement that is optimized for use in vibrationally noisy vacuum environments. However, as this technique plays only a secondary role in our discussion of ordering process, interested readers are referred elsewhere [2, 3]. LiSSp is described in detail below, as it was central to our measurements of simultaneous extended ordering and coarsening associated with elastic repulsion. After growth, ex-situ measurements of morphology were performed using contact atomic force microscopy (AFM), scanning electron microscopy (SEM), and transmission electron microscopy (TEM). 3.2.2 Light Scattering Spectroscopy (LiSSp) Nonspecular light scattering is a relatively simple and inexpensive method to obtain information on the surface spatial period, the areal density of islands, and the degree of lateral ordering [4]. With some additional knowledge, it is also possible to obtain the mean island volume and the areal coverage. The minimum spatial resolution is 100 nm, so the QD spacing must be larger than this for light scattering to be useful. For light scattering to be viable as an in-situ, real-time mapping of the surface power spectral density during MBE growth, we developed a novel approach using broadband illumination and spectroscopic detection, which we now describe. The scattered optical power spectrum from surface roughness is proportional to the average power spectral density (PSD) [5]: 2 1 Δz (ρ) ei(k0 −k)•ρ d2 ρ , (3.1) gr (k0 − k) = A where k0 is the incident wavevector, k is the scattered wavevector, ρ is a vector in the plane of the surface, Δz(ρ) is the surface height proﬁle, and A is the area of the illuminating optical beam. Note that the integral in Eq. 1 depends only on the component of the scattering vector K = k0 − k parallel to the sample surface. For the geometry shown in Fig. 3.1, the parallel component is given by K = (2π/λ) [sin (θ0 ) + sin (θf )] ,

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Fig. 3.1. The light-scattering spectroscopy (LiSSp) setup and the scattering geometry

Fig. 3.2. LiSSp spectral evolution during molecular beam epitaxial (MBE) growth of Ge0.2 Si0.8 /Si (001) at 750 ◦ C and 0.01 nm/s. Note the increasing intensity and progressive red shift (towards smaller frequencies) as the ﬁlm thickens (arrow ) and islands become coarser. The inset shows the peak position, used to obtain the mean island spacing, and the full width at half maximum (ΔK , used to determine the correlation length of the island array

where λ is the wavelength of light. In conventional light scattering using a monochromatic laser source, K = (2π/λ) [sin (θ0 ) + sin (θf )] is varied using angle-resolved measurements by changing the sample and detector angles. However, this is slow and inconvenient for in-situ, real-time measurements. An alternative is to use a ﬁxed scattering geometry, and to vary the wave-

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length of the incident light. We employ a broadband Xe arc lamp as the source, and a commercial spectroscopic detector consisting of a blazed grating fronting a CCD array, as shown in Fig. 3.1. The only other components needed for LiSSp are a pair of UV transmissive lenses and optical ﬁbers, a lambertian scattering source used to precharacterize the system response, and computer-control software/hardware. In our experimental setup, θ0 = 52◦ , θf =48◦ , and the spectral response of the source and detector covers the spectral range of 200 – 900 nm. This enables us to access a range of surface spatial periods from 130 – 590 nm. Figure 3.2 shows the evolution of light scattering spectra during growth of Ge20 Si80 ﬁlms on Si (001) at 755 ◦ C and 0.01 nm/s. In order to obtain these plots from the raw scattering data, we (1) subtract the thermal background emission from the sample (acquired in the early stages of ﬁlm growth when the ﬁlm is known to be smooth); (2) normalize the data using the response proﬁle from the lambertian scatterer; and (3) apply a λ4 correction to account for the Rayleigh scattering eﬃciency. The mean lateral spacing, λ = 2πK||,p , is obtained from the peak of the power spectral density, as shown in the inset of Fig. 3.2, by ﬁtting the upper portion of the PSD to a Gaussian. The correlation length, Λ, is obtained as Λ =2π/ΔK|| , where ΔK|| is the full width at half maximum of the ﬁrst peak in the PSD. The areal density of islands (number per unit area) is simply NA = λ−2 .

3.3 Extended Ordering During Near-to-Equilibrium Growth Much of our research on extended ordering has been on alloy ﬁlms with low Ge content: a Ge fraction of 0.2, and misﬁt strain of 0.8% [2,6]. Since there is an inverse relationship between the strain and the natural length scale associated with 3D roughening [7], low strain alloy growth eﬀectively enlarges all the relevant dimensions associated with the 3D roughening process. This has the advantage of reducing the resolution requirements on imaging and diﬀraction, making morphological characterization signiﬁcantly easier. The range of ﬁlm thickness over which morphological transitions occur is also extended, so that diﬀerent regimes of roughening, and the transitions between regimes, can be studied in greater detail [2, 8]. This magniﬁcation of length scales using low strain growth is what enables our use of light scattering as a characterization method for “quantum” dot arrays. Of course, it is important to show that the fundamental growth behavior of QDs does not depend on the length scale. 3.3.1 Generic Behavior, Near-to-Equilibrium GeSi/Si (001) alloys pass through a well-characterized sequence of morphological transitions as a function of the mass equivalent deposited ﬁlm thick-

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ness. Epitaxial growth conditions that permit this sequence to occur will be referred to here as “near-to-equilibrium”. Initially, the alloy ﬁlm grows as a planar wetting layer to some critical thickness that depends on the Ge fraction. Below this thickness, the wetting layer is stable against 3D roughening. Above this thickness, additionally deposited material leads to 3D roughening that typically occurs in the form of a nonlocal instability [9–12]. There is evidence that for very high strains, e.g., for pure Ge on Si with 4% strain, roughening can occur via nucleation [13, 14]. However, in either case, with further deposition the surface roughness evolves into discrete{105}-faceted pyramidal islands [15]. Under near-to-equilibrium conditions, the pyramids have a symmetric, compact shape [16], as shown in Fig. 3.3a whereas extended “huts”, asymmetric, {105}-faceted ridge-like structures, result from deposition conditions that partially limit surface mobility [2, 15, 17]. Lower strain alloys tend to exhibit denser QD arrays compared with pure Ge on Si, presumably a byproduct of island formation via the collective instability at lower strain. With further growth, pyramids can exceed a critical volume and transform to so-called “dome clusters” that exhibit steeper facets and relieve more stress, as shown in Fig. 3.3b [16, 18–20]. Eventually, domes become large enough that misﬁt dislocations form at the island/Si interface that eﬃciently relieve strain and aﬀect island size and shape [2, 21, 22]. This sequence of transitions is observed across essentially the full range of alloy compositions from pure Ge down to 20% Ge. The latter does not represent a special minimum composition for islanding; we are simply unaware of any work examining islanding at lower Ge fractions using MBE or chemical vapor deposition (however, see the work using liquid phase epitaxy by Christiansen et al., Chap. 5). It is important to realize that, since the length scale for islanding depends inversely on the strain, observation of the generic morphological transition sequence requires that the adatom diﬀusion length matches or exceeds the intrinsic length scale for roughening. Operationally, this is achieved using a combination of elevated deposition temperature and low deposition rate. For example, Ge ﬁlm growth at 550 – 600 ◦ C and 0.1 monolayer/s leads to formation of compact pyramids, with lateral dimensions

Fig. 3.3. atomic force microscopy (AFM) micrographs, 2 × 2 μm, of a {105}faceted pyramidal islands and b dome clusters

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of order 40 nm, containing of order 105 atoms [16]. For Ge0.2 Si0.8 , compact pyramids have lateral dimensions about 200 nm, and contain of order 107 atoms [6]. In this case, growth of compact pyramids requires 750 ◦ C, 0.1 ml/s deposition conditions. At even lower Ge contents, island formation should be possible, but the temperature required will eventually become large enough that Ge re-evaporation will become signiﬁcant relative to the Ge deposition rate. At 900 ◦ C, the Ge vapor pressure (above pure Ge) is about 2 × 10−7 Torr, corresponding to an evaporation rate of about 0.05 ml/s. 3.3.2 The Pyramid Regime in Si0.8 Ge0.2 Using a combination of in-situ stress sensing and ex-situ imaging, we have shown that {105} pyramids in Ge0.2 Si0.8 ﬁlms on Si (001) exist over an extended ﬁlm thickness regime from 6 – 13 nm, for deposition at 750 ◦ C and 0.01 nm/s [6]. Within this regime, the pyramid arrays are observed to simultaneously increase their degree of spatial ordering (onto a square lattice), and to undergo coarsening. Since the two processes are coupled, both will be discussed here. LiSSp directly measures the mean lateral spacing, areal density, and correlation length of the island array, as discussed above. However, in the pyramid regime, where the morphology behavior is well characterized, additional information can be extracted from the LiSSp data. The mean island volume is determined if we know the deposited thickness. We will assume that the wetting layer thickness is both known and ﬁxed. In our experiments, the wetting layer thickness can be determined directly from our concurrent real-time

Fig. 3.4. The areal coverage, Q, during MBE growth of Ge0.2 Si0.8 /Si (001) at 750 ◦ C and 0.01 nm/s, from LiSSp. This data is from the pyramid regime

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stress measurements [2], and XTEM suggests that the wetting layer thickness remains ﬁxed within the pyramid regime. Then the mean volume is given by

V = (hf − hwl )λ2 , where hf is the mass equivalent ﬁlm thickness and hwl is the wetting layer thickness. If we also know the island shape, then the coverage is obtained as Q = A/λ2 , where A is the mean basal area of the island array. For compact {105} pyramids, A = (6V /tanθ)2/3 , where θ is the angle between the facet and the (001) plane, equal to 11.3◦ for {105}. Figure 3.4 shows how Q evolves with ﬁlm thickness in the pyramid regime. Extended Lateral Ordering By measuring the light scattering spectrum in real-time during GeSi alloy ﬁlm deposition, we can track the lateral spacing, the areal coverage, and the correlation length of the island array. The latter is used to characterize the degree of extended spatial ordering. The light scattering spectrum was measured along a 100 in-plane azimuth, in anticipation that ordering would occur along these elastically softer directions. As a check, after completion of ﬁlm growth we rotated the sample about its 001 normal and veriﬁed that the LiSSp spectra exhibited fourfold symmetry, with the minimum spatial period occurring along the 100 directions as expected. Figure 3.5 shows the normalized correlation length as a function of island areal coverage during continuous MBE growth of Ge0.2 Si0.8 /Si (001) at 750 ◦ C and 0.01 nm/s, and during an identical growth to 10 nm, followed by annealing at the growth temperature [23]. At each point, we have normalized Λ by the mean lateral spacing of the island array, λ. The normalized correlation length essentially represents the number of near neighbor islands that are well-correlated. Corresponding SEM images are shown in Fig. 3.6. As deposition proceeds, the areal coverage of the islands increases (see Fig. 3.4), and there is a corresponding increase in the degree of extended ordering. With continuous deposition, the ordering reaches a maximum (Fig. 3.6a), and then begins to decrease. In this regime, SEM indicates that pyramids are undergoing copious impingement and coalescence, as shown in Fig. 3.6b, hence the loss of order. The maximum degree of ordering is fairly limited, reaching a peak value of about 3.5, i.e., the array is well-correlated over a range of 3 – 4 islands, then there is a mistake in the ordering. Unfortunately, Fig. 3.5 also demonstrates that ordering cannot be improved by annealing. Ordering is rapidly lost during annealing at the deposition temperature as the areal coverage decreases due to coarsening. Fig. 3.6c clearly shows that annealing improves the symmetry and faceted form of individual pyramids, but both the areal coverage and order decrease signiﬁcantly. The key conclusion is that ordering is a dynamic phenomenon occurring during growth of dense, interacting island arrays. Several between islands might play an important role in biasing the evolution of the island array [24, 25]. Shchukin performed an approximate analytical calculation of the elastic energy change associated with diﬀerent ordering

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Fig. 3.5. The normalized correlation length, Λ/λ, during MBE growth of Ge0.2 Si0.8 /Si (001) at 750 ◦ C and 0.01 nm/s, as a function of areal coverage, Q. The data points with open boxes about them correspond to scanning electron microscopy (SEM) images in Fig. 3.6. Squares Continuous deposition to 13 nm ﬁlm thickness, circles deposition to 10.4 nm ﬁlm thickness, followed by annealing at the growth temperature. The circle data has been oﬀset downwards by Λvλ = 0.5 for clarity. Note the rapid loss of ordering with annealing

Fig. 3.6. SEM images of islands a before the ordering peak, b at the peak of the ordering curve in Fig. 3.5, c at the end of the ordering curve in Fig. 3.5, where copious coalescence is taking place, and d after annealing. The marker bars are all 1 μm

schemes (a simple square lattice, a face-centered lattice, and a hexagonal lattice), and showed that the square lattice is energy minimizing, in agreement with our observations. We have performed quasi-2D ﬁnite element calculations of the excess elastic energy as a function of the areal coverage of islands. In this analysis, islands are treated as cones of rotation, and the coverage is varied by changing the lateral cell size relative to the island base dimension. Figure 3.7 shows the result of the calculations. In Fig. 3.7 the excess elastic interaction energy increases slowly at low coverage, but then rises rapidly for areal coverage exceeding about 0.7. This

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Fig. 3.7. Quasi-3D ﬁnite element analysis of excess elastic interaction energy as a function of areal coverage of islands. The line is an algebraic ﬁt to the data

behavior is characteristic of a relatively short-range interaction. As a result, signiﬁcant lateral ordering only occurs within high-density arrays of quantum dots. Unfortunately, disordering processes such as island coalescence, and eventually the dome cluster transition, become increasingly severe at very high areal coverage, limiting the attainable ordering. There is a growth scheme that could potentially improve lateral correlations. This would be to continuously adjust deposition rate (downwards) and/or temperature (upwards) during MBE to maintain constant areal coverage value around 0.7–0.8. In this manner, ordering could continue on the upward trend shown in Fig. 3.5 while limiting deleterious impingement eﬀects. Since the thermodynamic critical transition volume to domes is reduced by increasing areal coverage [20], Q, the dome transition would also be retarded during constant Q growth. Clearly, this is a diﬃcult approach to carry out, requiring real-time feedback control based on the coverage. Coarsening The elastic interaction simultaneously drives lateral ordering and Ostwald ripening of the island array. These are coupled processes: it is via the coarsening process that dynamic ordering actually takes place (as opposed to, for example, translational motion of islands). Figure 3.8 shows the behavior of the mean island volume and the areal density of islands, from LiSSp. The island volume evolves as t3 , while the areal density decreases with a time dependence that exhibits a negative second derivative. Taken together this data represents an example of accelerating coarsening kinetics: as the system length scale gets coarser, the rate of coarsening increases. Using stan-

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Fig. 3.8. LiSSp measurements of the evolution of the a mean island volume and b the areal density of islands during MBE growth of Ge0.2 Si0.8 /Si (001) at 750 ◦ C and 0.01 nm/s. The circles are measured data, while the lines are results of mean ﬁeld simulations incorporating elastic interactions and a deposition ﬂux

dard hydrodynamic mean ﬁeld analysis, we have shown that this unusual behavior cannot be explained simply by capillarity-driven Ostwald ripening in an “open” system where an external ﬂux is present [8]. However, by adding a coverage-dependent elastic interaction term to the system energetics (based on an analytical ﬁt to the FEM results of Fig. 3.7), the mean ﬁeld analysis can reasonably reproduce the coarsening kinetics we observe. The lines in Fig. 3.8 show the modeling results. More details can be found in [8], but the essential physics are that, as the areal coverage of QDs increases during deposition, the increasing elastic interaction drives up the chemical potential, resulting in faster coarsening. In this kinetic regime, however, coarsening is not actually fast enough to keep up with deposition, and the coverage continues to increase, thereby continuously accelerating the coarsening rate. Summary In the {105} pyramidal island regime associated with GeSi heteroepitaxy under near-to-equilibrium deposition conditions, elastic interactions associated with dense island arrays drives coupled coarsening and extended lateral ordering amongst the islands. At low Ge content, the lateral length scale of the island array is suﬃciently large that the kinetics of ordering and coarsening can be measured in real-time using light scattering spectroscopy. Because of the short-range nature of the interaction, ordering occurs only in dense arrays, and is quite limited in extent. Unfortunately, since this process is dynamically associated with growth, it cannot be improved by simple annealing. In order to obtain extended ordering in two (or three) dimensions over macroscopic distances, alternative schemes will be needed, likely requiring the use

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of preimposed patterns (which can be patterned topography, chemistry, or strain) in order to direct the self-assembly process.

3.4 Nonequilibrium Quantum Dot Microstructures We now turn our focus to special, locally ordered structures that can occur when GeSi/Si (001) growth is performed further away from equilibrium than for the experiments described previously. At suﬃciently reduced growth temperatures (whose magnitudes depend strongly upon lattice mismatch strain, but are typically below about 350 – 550 ◦ C, the higher ends of this range being for lower lattice mismatch), surface migration lengths and islanding/roughening are largely suppressed. Under these conditions the ﬁlm necessarily grows in a quasi-planar fashion, with interfacial misﬁt dislocations serving as the primary strain relief mechanism as the critical thickness for dislocation introduction is exceeded. Such structures contain relatively high strain energy densities. 3.4.1 Fourfold Quantum Dot Molecule Microstructures A more complex and more interesting regime is the region of intermediate adatom migration lengths, i.e. where adatoms can diﬀuse over signiﬁcant distances, but not over suﬃcient lengths to attain the equilibrium set of microstructures. In this regime, periodic surface roughening is frequently observed [17, 26, 27]. In addition, under speciﬁc ranges of kinetically limited growth conditions, other well-deﬁned but more complex microstructures may exist. We have recently described a new metastable morphological transition state that occurs during growth of Gex Si1−x /Si (001) with 0.2 < × < 0.4 within speciﬁc ranges of epilayer growth temperature and growth rate [28–30] the “quantum dot molecule”. The QDM structure comprises a shallow {105}faceted pit bounded by four {105}-faceted islands (Fig. 3.9). It forms in regimes of intermediate adatom/ad-dimer mobility, as controlled by epitaxial growth temperature and growth rate. Thus for Ge0.3 Si0.7 /Si (001) heterostructures, QDMs were initially discovered for deposition at 550 ◦ C, 0.09 nm/s (relatively standard conditions for MBE growth of GeSi ﬁlms [28]). Maintaining temperature at 550 ◦ C, but reducing the rate to 0.015 nm/s results in the loss of the QDM microstructure (extended 105 surface ripple microstructures form), whereas increasing the rate to 0.3 nm/s retains the QDMs. Increasing deposition temperature to 750 ◦ C causes loss of the QDM microstructure for rates of 0.015 nm/s or 0.09 nm/s (equilibrium structures form in both cases), while deposition at 350 ◦ C and 0.09 nm/s reduces adatom mobility such that no substantial 3D surface morphology forms. We note that the QDM microstructure is entirely reproducible in the range of kinetic regimes in which it forms [28, 30].

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Fig. 3.9. a Quantum dot molecules in 20 nm Ge0.3 Si0.7 /Si(001), 550 ◦ C, 0.09 nm/s. AFM image, 1 × 1 mm ﬁeld of view. b Schematic cross-section along a 100 inplane direction, illustrating the shape of a QDM

The complete microstructural evolution of the alloy ﬁlm under the QDM growth conditions is shown in Fig. 3.10. After growth of an initial Ge0.3 Si0.7 wetting layer with thickness c. 5 nm, shallow square pits start to form with angles of a few degrees to the surface, and penetrating just a nanometer or so into the ﬁlm. With subsequent GeSi growth, the pits broaden and deepen, and become bounded by small islands on each of the four sides of the pit (Fig. 3.10b), akin to the cooperative nucleation process described by [31]. The islands then elongate along the pit edges to form a quasi-continuous wall (Fig. 3.10c). The sidewall facets on both islands and pits have now steepened to {105}. As dislocations start to appear (Fig. 3.10d–f), the ﬁlm strain is relaxed and the QDMs destabilize. Figure 3.11 shows a series of 3D perspective views detailing how an individual QDM evolves from the initial shallow pit to its ﬁnal conﬁguration as a faceted pit surrounded by a continuous island wall. While it has been established previously that pits, as well as islands, can morphologically relieve strain in growing heteroepitaxial ﬁlms e.g., [32], the reason for their prevalence (as opposed to the more generally observed islands) under these speciﬁc growth conditions remains an open question. Previous reports of similar growth morphologies [33] arose from deliberate C contamination. However we believe this does not apply here as the pits do not nucleate at the obvious locations for concentration of C, such as growth interfaces (we note that none of the QDMs extend down to the Si/GeSi interface, and that the original Si buﬀer layer is highly planar with RMS roughness < one monolayer). There is also a possibility, not yet adequately explored, that surface hydrogen plays a role in the pit nucleation. 3.4.2 Size Stability of Quantum Dot Molecules As well as providing a reproducible complex nanostructure with signiﬁcant potential applications (Sect. 3.4.6), nature also self-limits the evolution of this structure: the QDM structure exhibits an extraordinary size and shape

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Fig. 3.10. AFM images of growth of Ge0.3 Si0.7 /Si(100) at 550 ◦ C, 0.09 nm/s, with GeSi layer thicknesses of a 0 nm, b 15 nm, c 30 nm, d 53 nm, e 100 nm, f 200 nm. Fields of view are 5 × 5 μm with edges align roughly along [110]. Insets to b and c show evolution of the quantum dot molecule (QDM) microstructure

Fig. 3.11. AFM images of Ge0.3 Si0.7 /Si(100) ﬁlms grown at 550 ◦ C, 0.09 nm/s with mean ﬁlm thicknesses of a 5 nm, b 25 nm, c 20 nm, d 30 nm. Fields of view are 400 × 400 nm

stability [34]. Once the “mature” QDM structure is attained, i.e., with continuous island walls around the pit (Figure 3.11d), both the shape and size are stable with further epitaxial growth, until misﬁt dislocations enter the structure. Both pits and walls are deﬁned by {105} facets, while the dimension (measured between exterior wall edges) is ﬁxed at ∼ 220 nm for

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Ge0.3 Si0.7 ﬁlms (Fig. 3.12). This is true also for annealed ﬁlms, although additional QDMs and {105}-faceted ridge structures do form. The reasons for this size stability are not obvious. Finite element calculations demonstrate that the system strain energy decreases continuously as the QDM dimensions increase [34], so that the observed maximum in QDM size is not due to an energetic minimum. Rather, we attribute the size stability to a kinetic barrier to adatom diﬀusion. As shown in Fig. 3.13, a continuous energy barrier exists to motion of adatoms out of the pit when bounded with the completed island wall, The ridge represents a local minimum in elastic energy, and it is particularly favorable for the next new facet to nucleate on the upper portion of the interior {105} facet, which can then create an effective barrier for adatom escape from the pit. This results in self-replicating (conformal) growth of the mature QDMs [34]. However, “escape paths” do exist in the earlier stages of formation when the bounding wall is not continuous and in this regime the QDM size and shape do evolve. The stable size of the mature QDMs varies inversely with the system strain, Fig. 3.14, as observed more generally for morphological microstructures in this system [10, 11].

Fig. 3.12. a A percentile plot summary of the size distributions of QDMs for Ge0.3 Si0.7 /Si(100) grown at 550 ◦ C and 0.09 nm/s for diﬀerent ﬁlm thicknesses, with and without 1 h anneals at growth temperature. Each open box encloses the inner 90% of the data and each ﬁlled box encloses the inner 50% of the data. The white bar represents the mean. Each distribution represents 39–45 individual measurements. b Measured pit angles (from AFM) vs. QDM lateral dimensions

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Fig. 3.13. Finite element modeling of local strain energy density in a a mature QDM, showing the entire model of one-quarter of the QDM structure and surrounding area, b the corresponding calculation for the QDM before wall coalescence, with possible adatom diﬀusion pathways out of the pit arrowed. Units in the strain energy density scale are in 109 J/m3

Fig. 3.14. Scaling of mature QDM size with system strain (Gex Si1−x composition). QDM sizes vary from 16 nm at x = 0.4, to 22 nm at x = 0.3, to 38 nm at x = 0.2

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3.4.3 1D Instability of Precursor Pits While the QDM microstructure is essentially stable with respect to moderate (1 h at the growth temperature) thermal annealing, see boxes labeled “annealed” in Fig. 3.12a, the precursor pits are not. Rather, they exhibit uniaxial instability and elongate along one of the in-plane 001 edges if they are annealed before the bounding islands form [35]. Figure 3.15 shows the result of annealing a pit array in a 5 nm thick ﬁlm for 1 h at 550 ◦ C. The pits have elongated into {105}-faceted grooves that have length-to-width aspect ratios as large as 15. Again, the grooves, being 2 – 3 nm deep, do not penetrate through the thickness of the GeSi ﬁlm. Note this morphological instability is bistable with respect to the two in-plane 001 directions, with no obvious mechanism governing the local choice between the two elongation directions. Further, long, quasi-1D islands form along the extended edges of the grooves; these islands exhibit a range of shapes from {105}-faceted huts to irregular lenticular morphologies. The reasons for this 1D instability are not clear, but are likely to be associated with the facet nucleation barrier originally proposed by [36]. We also observe that the cross-sectional shape of the bounding islands follows a progression as a function of the island cross-sectional area that is in qualitative agreement with recent predictions of the evolution of island shape as a function of volume assuming that {001} and {105} are surface energy minima of the system [12]. As the observed length-to-width aspect ratios in this conﬁguration can signiﬁcantly exceed those obtainable from growth of {105} Ge hut clusters on Si(100), this mechanism might also be exploited for the formation of quantum wire structures. 3.4.4 Directed Assembly of Quantum Dot Molecule Arrays On unmodiﬁed Si (001) surfaces, QDMs apparently nucleate randomly. However, we have succeeded in controlling QDM nucleation through prepatterning of the growth surface with pits created ex situ to the MBE chamber by focused ion beam (FIB) sputtering, enabling hierarchical assembly of semiconductor nanostructures over many orders of magnitude [37]. Dimensions range from the tens of nanometers of individual QDM components, to the few hundred nanometers of the overall QDM dimension, to microscopic, even macroscopic dimensions of the QDM array. Figure 3.16 illustrates this capability. A 1 pA (nominal diameter c. 10 nm) 30 keV Ga+ focused ion beam is used to create an array of small depressions on a Si (001) surface, with individual pits about 5 nm deep and 30 nm in diameter. Some component of this pit array apparently survives the subsequent wafer cleaning process described earlier. A relatively thin Si buﬀer layer of 7 nm (to avoid the pit broadening observed in this conﬁguration, as discussed later in this section) is then grown at 750 ◦ C, followed by “standard” QDM growth conditions of 20 nm Ge0.3 Si0.7 at 550 ◦ C, 0.9 A/s.

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Fig. 3.15. 1D instability of unbounded pits. a 5 × 5 μm AFM image showing a 5 nm thick Ge0.3 Si0.7 ﬁlm, annealed at 550 ◦ C for 1 h. b Enlargement of a 1.4 × 1.4 μm area in a

Fig. 3.16. SEM images of ordered arrays of QDMs formed by focused ion beam (FIB) prepatterning of holes 10 nm deep, 30 nm in diameter, followed by 7 nm Si buﬀer layer at 750 ◦ C, 20 nm Ge0.3 Si0.7 at 550 ◦ C, 0.9 A/s. Original hole spacings are a 1.0 μm, b 0.5 μm, c 0.25 μm, and d 0.15 μm. Each image shows an area approximately 1.92 × 1.26 μm

It is observed that QDMs do nucleate around the original FIB holes, as intended. The degree of ﬁdelity of the QDM array to the original pattern (i.e., the realization of the goal of one QDM to each FIB feature, with no QDMs forming between pattern features) depends upon the array feature spacing, s. For large hole spacings, s = 1.0 μm (Fig. 3.16a) large numbers of QDMs also nucleate between the “seeded” QDM lattice that forms on the original hole array. At s =0.5 μm (Fig. 3.16b), very few QDMs nucleate between the seeded array sites. At s = 0.25 and 0.15 μm (Fig. 3.16c,d), lattices of highly interacting QDMs are formed. We note also that the aspect ratios of the

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Fig. 3.17. Growth of patterned QDM arrays following growth of a 36 nm Si(100) buﬀer layer. As-fabricated FIB features had diameter 30 nm, depth 5 nm, spacing 250 nm. Following a 36 nm Si buﬀer layer at 750 ◦ C, 20 nm Ge0.3 Si0.7 at 550 ◦ C, 0.09 nm/s, arrays of QDMs form on interstices between holes. a AFM image where circles highlight a QDM in unpatterned region, a QDM in patterned region (light gray circle), and an FIB-initiated hole (dark gray circle). b AFM perspective view of ordered QDM arrays. c Line scan across one FIB-initiated pit and surrounding walls

QDMs on the seeded array are greater (i.e., higher facet angles with respect to the (001) surface than {105}) than for QDMs that form away from the seeding sites, or on unpatterned surfaces. If the Si buﬀer layer is grown to a larger thickness, we observe that the original FIB-fabricated pits expel material during buﬀer layer growth, resulting in deeper and broader pits with increasing buﬀer thickness. This is illustrated in Fig. 3.17, for deposition of a 36 nm Si buﬀer layer at 750 ◦ C, followed by 20 nm of Ge0.3 Si0.7 at 550 ◦ C, 0.09 nm/s onto a Si (001) surface patterned by an array of FIB-created pits about 5 nm deep, 30 nm in diameter, and spaced by 250 nm. As observed in the cross-sectional AFM scan, the pits substantially enlarge both in diameter and depth during Si buﬀer layer growth, implying that Si did not want to attach in the original pit region. The reason for this material expulsion is unclear, but presumably is associated with the implanted Ga from the original FIB fabrication. Subsequent

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GeSi growth then leads to formation of {105} facetted islands next to the enlarged pits, such that QDMs formed on the interstices between the holes when the size of the interstices matched the QDM dimensions. To date, all our topographic patterning experiments aimed at forming ordered QDM arrays have employed FIB sputtering to create the initial surface topography. Of course, other – and ultimately more eﬃcient – methods exist for creating such topography such as electron beam lithography. While there is no doubt that such methods will successfully create the initial topography, it appears likely that the locally implanted Ga from FIB patterning is aﬀecting surface evolution during the Si buﬀer layer homoepitaxy, both preserving and enlarging the initial pits. It may thus be that such local chemical modiﬁcation is a necessary component of enabling such topographical patterning of the QDM arrays. Further experiments utilizing electron beam lithography and a focused Si ion beam (from a mass-ﬁltered AuSi liquid metal ion source) will elucidate these mechanisms. 3.4.5 Summary and Potential Applications The quantum dot molecule comprises a new complex nanoscale morphology that forms reproducibly under a relatively narrow range of conditions of intermediate adatom mobility in Gex Si1−x /Si(100). In this regime, nature assembles a complex local morphology comprising a fourfold set of islands bounding a square pit in the epilayer surface, with both pit and wall facets close to {105}. The diameter of the overall molecule is of the order a few hundred nanometers, with the component islands having widths of a few tens of nanometers. These dimensions scale inversely with the system strain. Further, the QDM dimensions self-limit at the point when the four bounding islands elongate along the pit edges to form a quasi-continuous wall. Thus the QDM structure is also highly self size-selecting. Finally, the ability to organize these naturally self-assembling objects into ordered patterns and arrays through surface topographic patterning has substantial ramiﬁcations for practical applications. One application to which the QDM geometry naturally lends itself is the quantum cellular automata (QCA) architecture [38, 39]. Extra charge (electrons or holes) in four closely spaced quantum dots will adopt one of two bistable conﬁgurations due to Coulomb repulsion, representing the two states of digital logic. Switching between states requires electron or hole tunneling between adjacent dots in the cell. Further, “cells” can be spatially aligned so that the state of one cell controls the state of the next, ultimately enabling logic gates and digital circuits to be assembled. The potential advantages of this architecture with respect to conventional Si CMOS are much lower power-delay products, and much simpler interconnect schemes (as signals in two intersecting QCA cell chains can essentially propagate across each other). The primary disadvantage is the extremely challenging lithographic requirements for practical realization of this architecture, as the dot dimensions,

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and even more critically the dot separations, have to be tens of nanometers or less for operation at anything other than cryogenic temperatures (this is because the energy required to switch logic states is very small at anything other than very small dot dimensions/spacings). To date, the experimental state of the art in Al/Al2 O3 structures fabricated by electron beam lithography correspond to a handful of logic gates operating at less than 1 Kelvin. The conﬁguration of the semiconductor QDM lends itself directly to the QCA architecture, provided suitable carrier doping; localization and surface passivation techniques can be realized at this scale, oﬀering substantial promise for development of this concept. More generally, the QDM and related structures form a broad set of lowdimensional structures that could provide the “building blocks” for more complex architectures. In addition to the QDM itself, it is understood how to modify growth conditions to form standard singular quantum dot structures (clusters or domes). The precursor pit in QDM formation could be regarded as an “antidot”, and the one dimensional pits and islands created on annealing the precursor pits as “antiwires” and “wires”. Substantially improved understanding and control would be required, however, if these diﬀerent structures were to be controllably generated and interlinked. With respect to the fundamental mechanisms of QDM assembly and stability, several open questions remain. The ﬁrst is the reason for the initial evolution of strain relieving pits, as opposed to islands. While we believe the predominance of evidence is against direct nucleation by impurities such as carbon, other more subtle eﬀects, such as the possible role of hydrogen, cannot be ruled out yet. Also, while the presence of an energy barrier at the bounding island ridges is a plausible mechanism for prohibiting adatom diﬀusion out of the QDM pit when the bounding walls are complete, a more deﬁnitive description of the observed size-selection of mature QDMs is needed. The factors controlling the 1D instability of the precursor pits are not understood, nor are the local conﬁgurations that determine which in-plane 001 direction the instability chooses. Most fundamentally, what determines why, of all possible conﬁgurations, 107 atoms evolve towards this speciﬁc conﬁguration under conditions of limited adatom mobility? Perhaps the most fascinating question of all is whether the QDM is the only conﬁguration that forms under conditions of intermediate adatom mobility in this system (or others), or is merely the ﬁrst of a set waiting to be discovered. Already we have discovered variants of the QDM microstructure, one in which four {105}-faceted islands surround not a central pit, but a {103}-faceted dome island, and one where double walls form around a QDM pit [30]. Many other structures may be waiting to be discovered. Acknowledgement. Work at Sandia was funded by the DOE Oﬃce of Basic Energy Sciences. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under

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Contract No. DE-AC04-94AL85000. Work at UVa was funded by NSF-DMR (Grant #0075116) under the UVA-UIUC-IBM-Sandia Focused Research Group “Nanoscale Morphological Evolution of Semiconductor Surfaces” and by the NSF Materials Research Science and Engineering Center at UVa.

References 1. H.-J. Herzog, in Properties of Strained and Relaxed Silicon Germanium, ed. by E. Kasper, INSPEC, London (1995), p. 49 2. J.A. Floro, E. Chason, L.B. Freund, R.D. Twesten, R.Q. Hwang, Phys. Rev. B 59, 1990(1999) 3. J.A. Floro, E. Chason, in In situ and Real Time Characterization of Thin Films, ed. by O. Auciello, A.R. Krauss, Wiley, New York (2001), p. 191 4. E. Chason, M.B. Sinclair, J.A. Floro, J.A. Hunter, R.Q. Hwang, Appl. Phys. Lett. 72, 3276 (1998) 5. J.M. Elson, Phys.Rev. B 30, 5460 (1984) 6. J.A. Floro, E. Chason, R.D. Twesten, R.Q. Hwang, L.B. Freund, Phys. Rev. Lett. 79, 3946 (1997) 7. D.J. Srolovitz, Acta Metall. 37, 621 (1989) 8. J.A. Floro, M.B. Sinclair, E. Chason, L.B. Freund, R.D. Twesten, R.Q. Hwang, G.A. Lucadamo, Phys. Rev. Lett. 84, 701 (2000) 9. M.A. Grinfeld, D.J. Srolovitz, in Properies of Strained and Relaxed Silicon Germanium, ed. by E. Kasper, INSPEC, London, (1995), p. 3 10. P. Sutter, M.G. Lagally, Phys. Rev. Lett. 84, 4637 (2000) 11. R.M. Tromp, F.M. Ross, M.C. Reuter, Phys. Rev. Lett. 84, 4641 (2000) 12. J. Tersoﬀ, B.J. Spencer, A. Rastelli, H. von Kanel, Phys. Rev. Lett. 89, 196104 (2002) 13. I. Goldfarb, P.T. Hayden, J.H.G. Owen, G.A.D. Briggs, Phys. Rev. Lett. 78, 3959 (1997) 14. D.E. Jesson, M. Kastner, B. Voigtlaender, Phys. Rev. Lett. 84, 330 (2000) 15. Y.W. Mo, D.E. Savage, B.S. Swartzentruber, M.G. Lagally, Phys. Rev. Lett. 65, 1020 (1990) 16. G. Medeiros-Ribeiro, A.M. Bratkovski, T.I. Kamins, D.A.A. Ohlberg, R.S. Williams, Science 279, 353 (1998) 17. A.J. Pidduck, D.J. Robbins, A.G. Cullis, W.Y. Leong, A.M. Pitt, Thin Solid Films 222, 78 (1992) 18. M. Tomitori, K. Watanabe, M. Kobayashi, O. Nishikawa, Appl. Surf. Sci. 76/77, 322 (1994) 19. R.M. Ross, J. Tersoﬀ, R.M. Tromp, Phys. Rev. Lett. 80, 984 (1998) 20. J.A. Floro, G.A. Lucadamo, E. Chason, L.B. Freund, M. Sinclair, R.D. Twesten, R.Q. Hwang, Phys.Rev. Lett. 80, 4717 (1998) 21. M. Krishnamurthy, J.S. Drucker, J.A. Venables, J. Appl. Phys.69, 6461 (1991) 22. M. Hammar, F.K. LeGoues, J. Tersoﬀ, M.C. Reuter, M. Tromp, Surf. Sci. 349, 129 (1996) 23. J.A. Floro, E. Chason, M.B. Sinclair, L.B. Freund, G.A. Lucadamo, Appl. Phys. Lett. 73, 951 (1998) 24. V.A. Shchukin, N.N. Ledentsov, P.S. Kop’ev, D. Bimberg, Phys. Rev. Lett. 75, 2968(1995)

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25. N.P. Kobayashi, T.R. Ramachandran, P. Chen, A. Madhukar, Appl. Phys. Lett. 68, 3299 (1996) 26. N.-E. Lee, D.G. Cahill, J.E. Greene, J. Appl. Phys. 80, 2199 (1996) 27. D.E. Jesson, S.J. Pennycook, J.-M. Baribeau, D.C. Houghton, Phys. Rev. Lett. 71, 1744(1993) 28. J.L. Gray, R. Hull, J.A. Floro, Appl. Phys. Lett. 81, 2445 (2002) 29. R. Hull, J.L. Gray, M. Kammler, S. Atha, P. Kumar, T. Vandervelde, J.C. Bean, J.A. Floro, F.M. Ross, Mater. Sci. Eng. B101, 1 (2003) 30. T.E. Vandervelde, P. Kumar, T. Kobayashi, J.L. Gray, T. Pernell, J.A. Floro, R. Hull, J.C. Bean, Appl. Phys. Lett. 83, 5205 (2003) 31. D.E. Jesson, K.M. Chen, S.J. Pennycook, T. Thundat, R.J. Warmack, Phys. Rev. Lett. 77, 1330 (1996) 32. J. Tersoﬀ, F. LeGoues, Phys. Rev. Lett. 72, 3570 (1994) 33. X. Deng, M. Krishnamurthy, Phys. Rev. Lett. 81, 1473 (1998) 34. J.L. Gray, N. Singh, D.M. Elzey, R. Hull, J.A. Floro, Phys. Rev. Lett. 92, 135504 (2004) 35. J.L. Gray, R. Hull, J.A. Floro, Appl. Phys. Lett. 85, 3253 (2004) 36. D. Jesson, G. Chen, K.M. Chen, S.J. Pennycook, Phys. Rev. Lett. 80, 5156 (1998) 37. J.L. Gray, S. Atha, R. Hull, J.A. Floro, Nanoletters 4, 2447 (2004) 38. C.S. Lent, P.D. Tougaw, Proc. IEEE 85, 541 (1997) 39. I. Amlani, A.O. Orlov, G. Toth, G.H. Bernstein, C.S. Lent, G.L. Snider, Science 284, 289 (1999)

4 Hierarchical Self-Assembly of Lateral Quantum-Dot Molecules Around Nanoholes A. Rastelli, R. Songmuang, S. Kiravittaya, and O.G. Schmidt Max-Planck-Institut f¨ ur Festk¨ orperforschung, 70569, Stuttgart, Germany

4.1 Introduction Quantum dots (QDs) can be fabricated as inclusions of a narrow energyband-gap semiconductor in a matrix of larger band gap material. In QDs, charge carriers are conﬁned in all directions, and they are hence referred to as artiﬁcial atoms. One of the most elegant and convenient approaches to produce QDs is to exploit the Stranski–Krastanow (SK) growth mode during lattice-mismatched heteroepitaxial growth. The availability of such a simple fabrication method has rendered it possible to study the properties of selfassembled QDs in great detail [1–3]. In contrast to single QDs, relatively little work has been done up to now on coupled QD systems, i.e., on artiﬁcial molecules [4–7]. QD molecules are interesting, both as a new playground for studying interacting electronic systems and for their potential application as building blocks of quantum information processing devices [8]. In fact, single QDs can be used as one [9, 10] or two “qubit” [11] systems, but cannot be scaled to perform complex operations. For this purpose, chains or groups of QDs are required. However, the fabrication of QD molecules by self-assembly is more complex than the growth of single QDs, since a single self-assembly step rarely leads to welldeﬁned groups of spatially close QDs, which possibly behave as QD molecules. A relatively simple way to fabricate vertical QD molecules is to grow stacks of QDs. It is known that the surface strain ﬁeld modulation from a buried island layer controls the island nucleation in the next layer and this leads to a spontaneous vertical alignment (see, e.g., [12, 13]). The electronic coupling between vertically aligned QDs has been demonstrated [5–7]. The main disadvantage of this approach is that the composition and strain state of the diﬀerent layers are usually diﬀerent and, most importantly, it is hard to envision a controlled tuning of the QD potential proﬁles, especially of the barrier between them. This would require electrodes attached to the side of an etched mesa that contains the QDs [14]. Therefore, a lateral geometry is desirable. The most promising method for achieving long-range laterally ordered self-assembled QDs consists in combining substrate prepatterning and self-assembled growth (See [15–17] and Part II). The fabrication of laterally ordered SK QDs is still a time-consuming process, and not much is known on the optical quality of the grown QDs and QD molecules. Moreover, for fun-

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damental studies, spatial ordering is not always required. For these reasons, fully self-assembled structures are still very attractive. In this chapter we present a simple route to the fabrication of lateral QD molecules, based on the use of hierarchical self-assembly. In hierarchically self-assembled structures the result of a self-assembly step is used as the starting point for the subsequent step. Here, the starting point is represented by SK-grown InAs islands on GaAs(001) substrate. Islands are buried with a thin GaAs layer and then an in-situ etching step is applied. The strain modulation from the buried InAs QDs increases the etching rate of GaAs [18, 19], leading to the spontaneous formation of holes on the GaAs cap surface. By depositing InAs on the surface with self-assembled nanoholes, groups of closely spaced InAs islands naturally form at the hole edges. In the following, such groups will be referred to as lateral QD molecules, as has been done in the literature [19–22]. The number of QDs per QD molecule can be tuned by changing the InAs growth conditions. The combination of growth and etching in the same molecular beam epitaxy (MBE) system can be used for in situ processing, since it allows a quick switching between epitaxial growth and atomic layer precise etching in a contamination-free environment. The nanohole template created with this technique is therefore clean, requires no prepatterning, and has nanometer size [18, 19, 21, 22]. Here, the QD molecule growth process will be described along with the optical properties of ensembles of QD molecules. In order to demonstrate the lateral coupling between QDs belonging to the same molecule, detailed single QD molecule investigations are desirable. The most convenient way to address single QDs [1, 3] or individual QD molecules [4–6] is by microphotoluminescence spectroscopy, in which the laser beam used for the excitation is focused through a microscope objective to obtain a micrometric size spot. For this kind of study, it is useful to have QD molecules with low surface density (less than about 108 cm−2 ) and with emission energy in the working range of sensitive Si detectors. This chapter is organized as follows: in Sect. 2 we will present the fabrication method and the properties of self-assembled nanoholes. Then we will discuss how QD molecules with diﬀerent multiplicity can be obtained by growing InAs on top of the GaAs surface with nanoholes (Sect. 3). Eventually, we will show how QD bimolecules with low surface density and emission in the sensitive range of Si detectors can be obtained, opening the route to the investigation of individual, laterally coupled QD systems.

4.2 Self-Assembled Nanoholes from a Template of InAs/GaAs QDs 4.2.1 AsBr3 Atomic Layer Precise In-Situ Etching There are several demonstrations of chemical etching of GaAs or InP by using group V halides such as AsCl3 , PCl3 , and AsBr3 under ultra high vacuum

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conditions [23,24]. In this work, AsBr3 was selected as etching gas due to the lower electron aﬃnity of Br compared to F and Cl, and hence to the lower chemical reactivity with the parts composing the MBE system [25]. Moreover, the melting point of AsBr3 is 32.8 ◦ C, which is above room temperature, rendering this material relatively easy to handle. The vapor pressure of AsBr3 at 20 ◦ C is suﬃcient to introduce the etching gas into the growth chamber, with no need of a precracking process or of a carrier gas. In addition, the use of AsBr3 is advantageous because, during the etching process, AsBr3 can provide the As overpressure needed to stabilize the GaAs surface [25]. In our setup an AsBr3 in-situ etching gas system is integrated into a modiﬁed RIBER 32P solid source MBE machine. The gas-line is evacuated by a turbomolecular pump and is heated at about 80 ◦ C to avoid the condensation of the etching gas on the tube walls. The ﬂow of the etching gas is controlled by means of a mass ﬂow controller. In order to start the etching of a GaAs substrate, AsBr3 is supplied into the growth chamber. When AsBr3 molecules impinge and migrate on the GaAs surface, GaBrx molecules form due to their higher binding energy compared to AsBrx . The etching process occurs by desorption of GaBrx molecules from the GaAs surface. Zhang et al. [24] found out that the factor limiting the etching rate is the formation or the desorption of GaBrx and not the thermal decomposition of AsBr3 molecules. The etching process occurs on a layer-by-layer fashion, as demonstrated by the observation of clear intensity oscillations in the reﬂection high-energy electron diﬀraction (RHEED) pattern during etching (see inset of Fig. 4.1). Therefore, the etching rate can be calibrated by RHEED oscillations, analogous to the growth rates. In Fig. 4.1 the etching rate determined in this way is plotted as a function of substrate temperature. In this plot we can distinguish two diﬀerent regions. At low substrate temperatures, the etching rate increases with increasing temperature (see inset of Fig. 4.1). This regime can be referred to as reaction rate-limited region, since the etching rate is limited by the rate of formation and desorption of GaBrx molecules. At higher substrate temperatures, the etching rate is practically independent from the substrate temperature. This regime is indicated as supply rate-limited region [24,25]. In this regime, the etching rate increases linearly with the ﬂow of the etching gas [23] rendering it possible to tune the etching rate by means of a mass ﬂow controller. An interesting result is that the supply of an As ﬂux to maintain As-rich atmosphere during etching has little eﬀect on the etching rate, since AsBr3 can provide a large enough As overpressure to stabilize the GaAs surface. In addition, Kaneko et al. [26] observed the decaying of RHEED intensity and the disappearing of the oscillation at intermediate substrate temperatures and their reappearing at lower temperatures. AsBr3 etches GaAs as well as other III–V compounds, such as AlAs, and InAs. The etching process shows two diﬀerent temperature regimes for each material, but the transition temperatures between the reaction rate-

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Fig. 4.1. Temperature dependent etching rate for GaAs (001) measured by reﬂection high-energy electron diﬀraction (RHEED) intensity oscillations. The inset shows the RHEED intensity oscillations during the etching process at diﬀerent substrate temperatures

limited and the supply rate-limited regimes are diﬀerent. The etching gas shows material selectivity, since it etches InAs at the highest etching rate and AlAs at the lowest rate (see [25] for details). 4.2.2 Fabrication of Self-Assembled Nanoholes All the samples discussed here are grown on epi-ready semi-insulating GaAs(001) wafers. Prior to buﬀer growth, the surface oxide layer is removed at 630 ◦ C for 10 min, under an As4 beam equivalent pressure of 6 – 8 × 10−6 mbar. Subsequently, a 400-nm thick GaAs buﬀer is grown. Nanoholes are then fabricated as illustrated in Fig. 4.2. Self-assembled InAs islands are grown by depositing nominal 1.8 monolayers (ML) of InAs at a substrate temperature of 500 ◦ C and a very low InAs growth rate of 0.01 ML/s. A 30-s growth interruption is introduced in order to improve the island size homogeneity [27]. The substrate temperature is subsequently decreased to 470 ◦ C

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Fig. 4.2. Schematic illustration of the nanohole fabrication process. a Initial InAs QDs. b Overgrowth of InAs QDs with GaAs. c Nanohole formation by supply of AsBr3 etching gas onto the GaAs cap layer

Fig. 4.3. 1 × 1 μm2 Atomic force microscopy (AFM) images of the surface evolution during the nanohole fabrication process: InAs QDs obtained by deposition of 1.8 ML InAs (a), rhombus structures obtained by 10 nm GaAs overgrowth (b), nanoholes obtained by nominal etching depth of 1 nm (c), 3 nm (d), 5 nm (e), 8 nm (f ) and 10 nm (g)

and a thin GaAs layer is deposited to cover the InAs QDs. The cap layer deposition is followed by an in-situ etching step with AsBr3 gas. The gas ﬂow is ﬁxed at 80 milli standard cubic centimeter per minute (msccm), corresponding to an etching rate of 0.23 ML/s for planar GaAs(001). The substrate temperature is kept at 500 ◦ C, which is in the supply rate-limited regime. The etching is performed while supplying additional As4 ﬂux. The thickness of the GaAs cap layer and the amount of AsBr3 are systematically varied. The nominal etching depth is deﬁned as the etching depth of an unstrained GaAs (001) surface at 500 ◦ C. Figure 4.3 shows selected atomic force microscopy (AFM) images illustrating the fabrication of self-assembled nanoholes. After covering the InAs

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QDs shown in Fig. 4.3a with 10 nm of GaAs, rhombus-shaped structures (Fig. 4.3b) with a tiny hole at the center are obtained [28]. The surface morphology of the GaAs cap layer changes when the AsBr3 etching gas is supplied. By increasing the nominal etching depth, the holes at the center of the rhombus structures show an increase both in depth and diameter as shown in Fig. 4.3c, d. Subsequently, the holes assume a “bow-tie” shape (Fig. 4.3e). This morphology is attributed to the anisotropic shape of the initial rhombus structures and to anisotropic etching rate [25]. Figure 4.3f, g reveals that by increasing the nominal etching depth beyond 5 nm the hole diameter signiﬁcantly increases. The detailed analysis of the hole structure evolution are presented in Fig. 4.4, where the average hole depth and width in the [110] direction are plotted as a function of nominal etching depth. According to Fig. 4.4, the etching mechanism can be divided into two distinct regimes. In the ﬁrst regime, where the nominal etching depth is equal or less than 1 nm, the diameter and the depth increase with increasing nominal etching depth. If additional etching gas is supplied, we enter the second regime, where the hole diameter increases while the depth slightly decreases. In this regime, the holes transform from rhombus-like structures with a small hole in the middle to bow-tie-shaped structures [19, 22]. 4.2.3 Nanohole Formation Mechanism We use photoluminescence (PL) spectroscopy as a tool to gather a deeper insight into the etching mechanism. After a 400-nm thick GaAs buﬀer layer, a 20-nm thick Al0.4 Ga0.6 As cladding layer is grown at 610 ◦ C. Nanoholes are created with the conditions described in Sect. 4.2.2 and overgrown with 100-nm GaAs, 20-nm Al0.4 Ga0.6 As and 20-nm GaAs layers. The samples are excited by the 488-nm line of an Ar+ laser and the signal is detected by a liquid-nitrogen-cooled Ge photodetector. Representative PL spectra and a summary of the PL peak energy at 8 K versus the nominal etching depth are shown in Fig. 4.5. A systematic blue shift of the QD related peak is observed. The peak of the nonetched InAs QDs

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Fig. 4.5. a Low temperature photoluminescence (PL) spectra of samples containing nanoholes. The peaks at lower energy come from the remainders of partially etched InAs QDs. b Summary of the PL peak energies as a function of nominal etching depth. At 5 nm nominal etching depth, the peak from the buried InAs QDs disappears

is located at 1.022 eV. After 0.33 nm nominal etching depth, the peak shifts to 1.081 eV and then to 1.320 eV when the etching depth reaches 3 nm. The blue shift of the QD-related peak is the evidence that the etching gas removes material from the QDs, causing a decrease of the dot size. The changes in QD size and shape occurring during etching were recently investigated by selective removal of the GaAs cap layer [29]. The peak disappears after 5-nm nominal etching depth, implying that the InAs material is completely removed. The position of the wetting layer peak at 1.412 eV remains unchanged during the whole etching process. Remarkably, the linewidth of the PL peak associated with the buried QDs remains in the range of 24 – 40 meV during etching, suggesting that diﬀerent QDs are etched by AsBr3 in a rather uniform way. We qualitatively explain the in-situ etching mechanism by three possible parameters shown in Fig. 4.6: the material selectivity of the etching gas [25], the strain state of the GaAs surface, produced by the buried InAs QDs, and the atomic bonding strength of Ga atoms at the hole edges. Schuler et al. [25] reported that the AsBr3 etching rate for InAs is 1.24 times higher than that for GaAs in the supply-rate limited regime. The etching rate of an InGaAs alloy can be linearly interpolated from the values for InAs and GaAs. For the ﬁrst regime, where the hole depth and diameter rapidly increase, the presence of an In rich region above the QDs cannot explain the observed features of the nanoholes. For example, for the 1-nm etched sample with 10-nm GaAs cap layer, the average hole depth is 6 nm. This result indicates

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that the etching rate above the QDs is 6 times higher than the etching rate on the ﬂat surface, which is not consistent with the material selectivity of the etching gas. Since the InAs material in the QDs is more relaxed compared to InAs in the wetting layer, the GaAs cap layer above the QDs is under tensile stress. Therefore, the AsBr3 etching gas can remove preferentially the material from the regions above the buried QDs and create the nanoholes. Since the strain energy decreases with increasing distance from the stressor [12], the importance of the strain-enhanced mechanism decreases if the cap layer thickness is increased. The gradual increase of the hole depth in the case of a 15 nm thick cap is a clear evidence of the reduced strain with larger cap thickness [19]. At an early stage of the etching process, the etched surface is rather smooth and the sample surface is approximately ﬂat. Therefore, the strain from the buried InAs is probably the major parameter determining the etching mechanism. However, we observe a systematic blue shift of the QD PL peak (Fig. 4.5) with increasing etching depth, which indicates that the material from the buried InAs QDs is removed by the etching gas. Consequently, the strain from the buried islands decreases during etching. From the AFM results in the second regime, where the nominal etching depth is larger than 1 nm, we observe a slight decrease of the hole depth and a continuous increase of the hole diameter, at a slower rate compared to the ﬁrst regime (see Fig. 4.4). The AFM data suggest that during etching the surface tends to ﬂatten, resulting in the increase of the hole diameter during the etching process. We can argue that the atomic bonds of the material at the hole edges are weaker compared to other locations, thus the AsBr3 etching gas can preferentially remove atoms from those positions.

Fig. 4.6. Schematic illustration of the possible parameters which determine the nanohole formation, i.e., the material selectivity of the etching gas due to In segregation, the strain ﬁeld from the buried InAs and the atomic bonds of Ga atoms at the hole edges

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In conclusion, in the ﬁrst region, where the depth and diameter increase rapidly, the strain-enhanced etching rate is the dominant mechanism responsible for the nanohole formation. In the second region, where the depth shows saturation, while the diameter still increases, the etching gas preferably removes atoms loosely bound at the hole edges. 4.2.4 Stability Of The Nanohole Template The thermal stability of the holes was investigated by annealing the samples at 450 ◦ C and 500 ◦ C in As4 atmosphere for annealing times up to 300 s. The average hole depth as a function of annealing time at diﬀerent temperatures, shown in Fig. 4.7a, reveals that the depth decreases with annealing temperature. The error bars in Fig 7a correspond to the standard deviations. In Fig. 4.7b–e are AFM images of the nanoholes during the annealing process at 450 ◦ C. The initial holes are bow-tie shaped and are obtained by 5-nm nominal etching of a 10-nm GaAs cap layer grown at 500 ◦ C. The hole depth decreases with annealing time and the hole periphery changes from bow-tie shape to an elliptical structure elongated in the [110] direction. The observed results can be qualitatively explained by the gradient of the chemical potential on the surface. Since the buried InAs QDs are completely removed during the etching step used to produce the nanoholes for this investigation, the strain contribution is negligible. The gradient of the chemical potential due to the surface curvature term drives the Ga adatoms downhill into the holes and causes the decrease of the hole depth during annealing. By increasing the annealing temperature, the hole depth decreases more rapidly, because of the higher adatom density on the surface and larger diﬀusivity of Ga adatoms. We indicate the depth decrease of the nanoholes during annealing as self-collapsing process [21].

Fig. 4.7. Evolution of the hole depth as a function of annealing time at 450 ◦ C and 500 ◦ C (a) and 1×1 μm2 AFM images of nanoholes prior to annealing (b) and after annealing at 450 ◦ C for 60 s (c), 120 s (d), 300 s (e)

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4.3 Self-Assembly of Lateral InAs/GaAs QD Molecules 4.3.1 Growth of QD Bimolecules The surface with nanoholes is used as a template for the fabrication of lateral QD molecules. Nanoholes are obtained as described in Sect. 2.2, using a 10-nm thick GaAs cap layer above the InAs QDs and a nominal etching depth of 5 nm. Bow-tie shaped nanoholes, with a depth of 5 – 6 nm (Fig. 4.8a) are then overgrown with InAs at 500 ◦ C. The amount of InAs deposition is systematically varied up to 2.5 ML to understand the molecule formation mechanism. The growth rate of InAs is kept at 0.008 – 0.01 ML/s throughout this experiment. Figure 4.8b–d illustrates that the holes become shallower and narrower along the [1-10] direction during 0 – 0.6 ML InAs deposition. An atomically ﬂat surface is recovered after 0.8 – 1.4 ML InAs deposition (Fig. 4.8e–f). The initial stages of the QD “bimolecule” formation is observed at an InAs coverage of about 1.6 ML InAs (not shown). The bimolecules are composed of two QDs, with slightly diﬀerent sizes, aligned along the [1-10] direction. Their density increases when 1.8 ML InAs are deposited, as shown in Fig. 4.8g. Further InAs deposition (2.5 ML) leads to fully developed bimolecule structures, as shown in Fig. 4.8h. The center-to-center distance distribution of the QD bimolecule ensemble, obtained from 2×2 μm2 AFM images, is shown as a histogram in Fig. 4.9a. By using a Gaussian ﬁt, we obtain a mean center-to-center distance of 45±4 nm, compatible with the distance between the QDs in the cross-sectional transmission electron microscopy (TEM) image shown in Fig. 4.9c. From the di-

Fig. 4.8. 1×1 μm2 AFM images showing the surface evolution during overgrowth of the nanoholes (a) with increasing amounts of InAs at 500 ◦ C: 0.2 ML (b), 0.4 ML (c), 0.6 ML (d), 0.8 ML (e), 1.4 ML (f ), 1.8 ML (g) and 2.5 ML (h)

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mension of the initial hole in Fig. 4.9b, we infer that the QDs nucleate at the hole edges. 4.3.2 Growth of QD Multimolecules From the nanohole annealing experiments, we found out that the nanoholes rapidly collapse at a substrate temperature of 500 ◦ C, due to the negative surface curvature of the nanoholes, which drives Ga adatoms into the holes. In order to prevent the self-collapsing process during the InAs growth, the substrate temperature is decreased to 450 ◦ C, where the collapsing rate is low (see Fig. 4.7a). Figure 4.10 shows the comparison between selected AFM images during the ﬁlling process at 450 ◦ C and 500 ◦ C with diﬀerent amounts of InAs. At a substrate temperature of 450 ◦ C, we observe the nucleation of InAs QDs at the edges of the nanoholes (Fig. 4.10a) already at a coverage of 1.2 ML of InAs (by far less than the critical value for QD formation on ﬂat surface). With further deposition of InAs (1.4 ML), the QD density increases and nucleation keeps occurring preferentially at the nanohole edges (Fig. 4.10b). With the growth of 1.6 ML of InAs, QD “multimolecule” structures are obtained (Fig. 4.10c). Islands are also observed on the ﬂat surface between the QD molecules and their density increases by increasing the amount of deposited InAs. By changing the substrate temperature during InAs deposition from 450 to 500 ◦ C and the amount of InAs from 1.8 to 2.5 ML, the dominant number of QDs per molecules (n) can be increased up to 6 [22]. Figure 4.11a–d shows selected three-dimensional (3D) AFM images of a bimolecule, a trimolecule, a quadmolecule, and a hexamolecule, which are obtained by the deposition of 2.5 ML InAs at 500 ◦ C, 1.8 ML InAs at 470 ◦ C, 2.0 ML InAs at 470 ◦ C, and 1.8 ML at 450 ◦ C, respectively.

Fig. 4.9. a The center-to-center distance between the quantum dots (QDs) in QD bimolecules. The inset shows a 125 ×125 nm2 AFM image of a bimolecule structure. b and c show cross-sectional transmission electron microscopy (TEM) images of an in-situ etched nanohole and lateral InAs QD bimolecule, respectively

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Fig. 4.10. 1×1 μm2 AFM images of multimolecule structures fabricated by deposition of InAs onto nanoholes at 450 ◦ C (top panels) and 500 ◦ C (lower panels). The amount of deposited InAs is 1.2 ML (a, d); 1.4 ML (b, e); 1.6 ML (c, f )

Fig. 4.11. 3D view of 200×200 nm2 AFM images of multimolecule structures fabricated by deposition of diﬀerent amounts of InAs onto nanoholes at diﬀerent substrate temperatures: 2.5 ML at 500 ◦ C (a), 1.8 ML InAs at 470 ◦ C (b), 2.0 ML InAs at 470 ◦ C (c), and 1.8 ML InAs at 450 ◦ C (d)

For a statistical analysis we select diﬀerent samples grown under growth conditions, where the percentage of a certain n-fold QD molecule is particularly high. For 2.5 ML InAs deposition at 500 ◦ C, we obtain 59% bimolecules and 40% isolated dots, while for 2 ML InAs deposition at 470 ◦ C, we obtain 52% quadmolecules, 28% trimolecules, 4% bimolecules, and 16% others. In the case of 1.8-ML InAs deposition at 450 ◦ C, we obtained 32% hexamolecules, 22% pentamolecules, 8% heptamolecules and 38% others. We ob-

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Fig. 4.12. Maximum percentage of dominant QD molecules on the sample surface as a function of the dominant multiplicity (n)

serve that the maximum percentage of n-fold QD molecules decreases with increasing n. Figure 4.12 provides information about the maximum percentage on a sample surface as a function of n. n-fold QD molecules with large n tend to form when the InAs growth is performed at lower substrate temperature, because In adatoms have a higher probability to nucleate new islands before possibly being incorporated into existing QD molecules. We note that the formation of QD molecules with even multiplicity tends to have a higher probability than that of QD molecules with odd n. We attribute this eﬀect to the twofold symmetry of the hole structure. 4.3.3 Formation Mechanism of Lateral QD Molecules The surface morphology after overgrowing the holes with InAs at 450 ◦ C shows that the edge of the nanoholes is the dominant factor leading to the formation of the QD molecule structures. Moreover, the QD nucleation can be observed before the InAs thickness reaches the critical value for the dot formation. A simple mechanism, which may explain the formation of the QD molecules, is schematically depicted in Fig. 4.13. The negative curvature of the initial nanohole in Fig. 4.13a is likely to be the driving force for the migration of Ga (from the self-collapsing mechanism) and In (from deposition) into the holes. At the initial stage of the deposition, both In and Ga adatoms probably grow at the bottom of the holes (Fig. 4.13b). The topmost layer is

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Fig. 4.13. Schematic illustration of the QD molecule formation. a Initial Nanohole. b Early stage of ﬁlling process. c Preferential InAs growth at the hole edge. d QDs molecule formation

expected to be InAs due to the segregation of In [30]. Since the InAs layer is compressed at the bottom of the hole, the In atoms preferentially attach at the hole edges, where the InAs layer can relax more easily (Fig. 4.13c). This growth model is supported by the AFM images in Fig. 4.8a–d and by the observation that the hole diameter drastically narrows with increasing amount of deposited InAs, while annealing shows a much smaller eﬀect on the hole diameter. Eventually, the hole edges with the partially relaxed InAs induce a current of In adatoms towards the hole, leading to the QD molecule nucleation (Fig. 4.13d). This picture is supported by the AFM images shown in Fig. 4.10a–c where nanoholes are overgrown with InAs at 450 ◦ C. At this temperature, where the self-collapsing process is not pronounced, we clearly observe the QD molecule formation around the remainders of the holes. We expect that the same mechanism is responsible for QD multimolecule and bimolecule formation, even though we do not observe the remainders of the holes when the QD bimolecules form (Fig. 4.10f). Most probably this is due to the larger diﬀusivity of Ga, causing a higher collapsing rate and producing a ﬁlling-up of the holes with GaAs as discussed in Sect. 2.4. 4.3.4 Photoluminescence of QD Bimolecule Ensembles In order to better understand the process of QD molecule formation and investigate the optical quality of the samples with QD molecules, we use PL spectroscopy. After InAs deposition onto the nanoholes, a GaAs cap layer is deposited. Figure 4.14a shows the result of a systematic PL study (at 8 K) of the bimolecule formation process. For 5 nm nominal etching depth, the wetting layer (WL) signal at 1.414 eV is the dominant peak, indicating that the underlying QDs are completely removed and only the WL remains (see also Fig. 4.5). At 0.2 ML InAs deposition, we observe another peak (labeled with FH) at 1.396 eV, which we attribute to the second InAs layer that partially ﬁlls the etched holes. For 1.6 ML InAs deposition, two peaks coexist in the spectrum, (apart from the third peak at 1.412 eV, which originates from the WL of the initial layer). The peak at 1.392 eV is appointed to some holes ﬁlled with InGaAs and the peak at 1.328 eV is attributed to the initial stage of the QD bimolecule (labeled with QDBM) formation. This peak red-shifts to 1.046 eV with further InAs deposition and can therefore be attributed to fully developed QD bimolecules. The linewidth of the peak is 29 meV, indicating

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Fig. 4.14. a PL spectra at 8 K of the structures containing “ﬁlled nanoholes” (FH ) during the QD bimolecule (QDBM ) fabrication process. WL: wetting layer. b PL peak energy at room temperature versus amount of deposited InAs

a good size uniformity of the QD bimolecules. Figure 4.14b contains a summary of the PL peak position at room temperature as a function of deposited amount of InAs. The WL signal is the dominant peak up to 1.6 ML InAs deposition and then the peak from the QDs in the second layer can be observed. It is noteworthy that for 2.5 ML InAs deposition, the QD bimolecules emit at 0.972 eV, have a linewidth of 30 meV, and the PL intensity is comparable to the original QD layer, which underlines a good size uniformity of the structure and the high crystal quality of the samples, respectively.

4.4 Lateral QD Molecules for Single QD Molecule Optical Investigations As discussed in Sect. 2.1, the main interest towards QD molecules relies in the opportunity they oﬀer of studying interacting electronic systems. Since ensembles of self-assembled objects are always aﬀected by inhomogeneities, it is important to have access to individual QD molecules to study their peculiar properties and to possibly apply them in new devices. The most common approach to investigate individual QDs or QD molecules is by optical spectroscopy [1,3–6,9–11]. For this kind of study, the structures should satisfy the following requirements (see, e.g., [31]): (1) The structure should contain a homogeneous distribution of objects, i.e., only QD molecules and as few as possible isolated QDs, to facilitate the interpretation of the optical spectra;

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(2) QD molecules should have low surface density (less than about 108 cm−2 ), so that single QD molecules can be accessed with a focused laser with typical spot diameter of 1 – 2 μm; (3) the transition energies of QD molecules should occur in the range 1.3 – 1.7 eV, where the most advanced Si detectors and tunable Ti:Sa lasers operate. In the following we describe a route to the fabrication of homogeneous self-assembled QD bimolecules with low density and light emission in the desired spectral range. 4.4.1 Improvement of the Homogeneity of QD Bimolecules The procedure used to grow InAs QD molecules is based on several selfassembly steps: growth of InAs islands, overgrowth and creation of selfassembled nanoholes by etching, overgrowth with InAs. Since the result of a self-assembly step is used as the starting point for the following step, the control of the ﬁnal structure is quite diﬃcult. As shown in Fig. 4.12, the surface with QD molecules usually contains QD molecules with diﬀerent multiplicity and an appreciable number of isolated QDs (see AFM images in Figs. 4.8h and 4.15a). Since the best homogeneity is obtained for the smallest multiplicity, i.e., for bimolecules, where the yield reaches about 60%, we optimized the growth procedure of QD bimolecules [32]. As shown in Figs. 4.8h and 4.15a, QD bimolecules align along the [1-10] direction, most probably because of a higher adatom diﬀusivity in this direction compared to the [110] and because of In enrichment a the hole edges along [1-10] during the hole-ﬁlling process (see Fig. 4.8a–d and Fig. 4.12).

Fig. 4.15. 1.8 × 1.8 μm2 AFM images of lateral InAs QD bimolecule. In a the nanohole template is grown on a surface with relatively large (001) terraces, while in b growth is performed on a surface with stepped mounds aligned along the [1-10] directions. Stepped mounds are obtained by overgrowing a seeding layer of InAs islands with a thick GaAs layer and lead to improved yield of QD bimolecules

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It is also known that islands tend to nucleate at surface step edges [33], when available. Therefore, we attempted the growth of QD bimolecules on a surface containing a large number of steps aligned along [1-10]. A typical result is shown in Fig. 4.15b, where we see that the percentage of QD bimolecules has drastically increased, compared to isolated QDs. A qualitative explanation for this behavior is the following. The InAs islands used to produce the nanoholes are found to nucleate preferentially at the long-side walls (parallel to the [1-10] direction) of the mounds. After GaAs overgrowth and etching, nanoholes form thus on the same side of the mounds. Therefore the nanohole edges, which are preferential QD nucleation sites, become even more favorable because of their location with respect to the stepped mounds. The combination of the two factors leads to the improved homogeneity of the QD bimolecule ensemble, as also demonstrated on large scale by X-ray investigations [34]. The stepped mounds on the GaAs surface are easily created by burying a “seeding layer” of InAs islands below a 100-nm thick GaAs layer grown at 500 ◦ C. Because of the relatively low substrate temperature, a ﬂat GaAs surface is not recovered and mounds appear on the surface. This rough surface is used as a template for the growth of QD bimolecules as described in Sect. 3.1. Other methods to produce a surface with stepped mounds are being investigated. In conclusion, the addition of a self-assembly step (creation of a rough surface) to the hierarchically self-assembled structure allows us to improve considerably the homogeneity of the ensemble. 4.4.2 Optimization of QD Bimolecules Surface Density and Emission Wavelength The typical surface density of the QD bimolecules grown as described in Sect. 3.1 is 4×109 cm−2 , which is rather high for single QD bimolecule investigations. The ﬁrst step to reduce such a density is to produce a surface with lower density of nanoholes, which in turn requires the growth of a lowdensity ensemble of InAs islands. This can be obtained by properly reducing the amount of deposited InAs [31]. Also the amount of InAs deposited on the surface with nanoholes has to be reduced accordingly. Figure 4.16a shows the surface of a sample in which about 92% of the objects are QD bimolecules, with a low surface density of 6×107 cm−2 . InAs QD bimolecules have a typical emission energy of 1.05 eV at 8 K (see Sect. 3.4). This spectral range is not particularly suited for coherent optical studies, and it is desirable to blue-shift the emission energy. A possible method to achieve this goal is to apply post-growth rapid thermal annealing (RTA, see [31] and references therein), promoting bulk In-Ga intermixing and consequent emission blue-shift. This technique has the disadvantage of producing a similar blue-shift of the emission of the InAs seeding layer (see Sect. 4.1), possibly generating an unwanted signal during PL measurements.

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Fig. 4.16. 3.8 × 3.8 μm2 AFM image of low-density InAs/GaAs (QDBMs) grown on stepped surface (a) and PL spectrum of low-density QD bimolecules with blueshifted emission obtained by partial capping with 2 nm GaAs and annealing for 4 min at 500 ◦ C (b)

In order to decouple the emission of the QD bimolecules from that of the seeding layer, we blue-shift the emission by partially capping the QD bimolecules with a thin (2-nm thick) layer of GaAs and annealing the structure for different times at 500 ◦ C before continuing the capping process with additional 100 nm of GaAs (see [35]). This method does not require any post-growth annealing and allow us to achieve the desired blue-shift, as shown in the PL spectrum of Fig. 4.15b, where the annealing time is 4 min. The laser spot used for this measurement has a diameter of about 180 μm. The emission of the high-density QDs used to produce a rough surface (seeding layer) is visible on the low-energy side of the spectrum. The QD bimolecules generate a signal peaked at 1.325 eV, with intensity lower than the wetting layer (WL) peak because of their low surface density. Single-QD bimolecule measurements show resolution-limited sharp lines, witnessing the high quality of the structure [36]. 4.4.3 Hierarchical Self-Assembly of GaAs/AlGaAs QD Bimolecules In Sect. 4.2 we have described a method to blue-shift the emission of QD bimolecules by partial capping and annealing [35]. The main disadvantage of this approach is that the shape and composition proﬁles of the resulting QD bimolecules are hard to determine, and this renders it diﬃcult to predict or interpret their electronic properties. This is generally the case for strained QD structures, where pronounced intermixing usually occurs. In order to solve this problem, we have recently demonstrated a method, allowing

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Fig. 4.17. Schematic representation of the hierarchical self-assembly of GaAs/ AlGaAs QDs (a–d) and GaAs/AlGaAs QD bimolecules (e–h)

Fig. 4.18. 1 × 1 μm2 AFM images of a GaAs nanoholes and b GaAs biholes overgrown with 10 nm of Al0.44 Ga0.56 As at 500 ◦ C. The biholes are obtained by 10-nm GaAs overgrowth of InAs QD bimolecules followed by 5-nm nominal etching. The AlGaAs surface can be used to fabricate unstrained hierarchically self-assembled QDs (a) and QD bimolecules (b)

self-assembled unstrained QDs with predictable properties to be grown in the GaAs/AlGaAs system [37]. The procedure is schematically illustrated in Fig. 4.17: a surface with self-assembled nanoholes (Fig. 4.17a) is overgrown with a thin AlGaAs layer at a substrate temperature of 500 ◦ C, where the reduced diﬀusivity of Al allows the holes to be transferred to the AlGaAs surface (Fig. 4.17b); the nanoholes are ﬁlled with GaAs by overgrowing them with GaAs and annealing the surface for 2 min (Fig. 4.17c); the GaAs-ﬁlled nanoholes are eventually overgrown with an upper AlGaAs barrier to form GaAs QDs in AlGaAs matrix. Since intermixing between GaAs and AlGaAs is reduced at 500 ◦ C, the morphology of the GaAs QDs is essentially that of the AlGaAs nanoholes (see, e.g., Fig. 4.18a) and the composition is known. This allows the QD electronic properties of the structure to be calculated with no adjustable structural parameters [37].

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A similar method can be used to create GaAs/AlGaAs QD bimolecules with predictable properties and optimum emission energy, as schematically illustrated in Fig. 4.17e–h: InAs QD bimolecules (Fig. 4.17e) are overgrown with 10-nm GaAs and etched for nominal 5 nm by AsBr3 . This results in the formation of “biholes” (Fig. 4.17f), which can be overgrown with AlGaAs (Fig. 4.17g) to create GaAs/AlGaAs QD bimolecules (Fig. 4.17h). An AlGaAs surface with biholes is shown in Fig. 4.18b. Since GaAs QDs and QD bimolecules are obtained by hierarchical selfassembly starting from InAs QDs and QD bimolecules, their density can be tuned as in the case of self-assembled InAs structures. The shape and size of the nanoholes can be widely tuned by varying the growth parameters of the AlGaAs layer [37] or the geometry of the GaAs nanoholes. GaAs/AlGaAs heterostructures have light emission in the optimum range of Si detectors, and no further optimization is needed for single-QD and QD bimolecule optical investigation.

4.5 Summary In this chapter, a method has been presented to fabricate fully self-assembled structures consisting of laterally close groups of quantum dots. The fabrication process, based on the combination of MBE growth and in situ etching, has been discussed together with the morphological and optical properties of the obtained quantum dot molecules. The growth procedure has been optimized for obtaining quantum dot bimolecules with properties suitable for coherent optical investigations. Diﬀerent approaches to fabricate self-assembled lateral QD molecules are currently being investigated [38–40]. Acknowledgement. The authors thank B. Krause for fruitful discussion and K.V. Klitzing for continuous support and interest. This work was ﬁnancially supported by the BMBF (01BM906/4 and 03N8711).

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5 Energetics and Kinetics of Self-Organized Structure Formation in Solution Growth: the SiGe/Si System S.H. Christiansen1 , M. Schmidbauer2 , H. Wawra2 , R. Schneider3 , W. Neumann3 , and H.P. Strunk4 1

2 3

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Max-Planck-Institute of Microstructure Physics, Weinberg 2, 06120 Halle/Saale, Germany Institute of Crystal Growth, Max-Born-Straße 2, 12489 Berlin, Germany Humboldt-University of Berlin, Institute of Physics, Newtonstraße 15, 12489 Berlin, Germany Institute for Materials Science and Engineering VII, Cauerstraße 6, 91058 Erlangen, Germany

Abstract When analyzing and modeling the self-organized structures and their evolution in misﬁt epitaxial growth, one encounters the problem of ﬁnding out whether the formation process is kinetically or energetically controlled. Generally the applied growth methods operate with rather high supersaturations just to ensure the appropriately high growth rate. The controlling step is in most cases the surface diﬀusion, for example in the physical or chemical vapor deposition methods. Thus, the resulting structure formation, though inﬂuenced by elastic energy reduction (elastic relaxation), is generally kinetically controlled. On the other hand, solution growth, i.e., liquid-phase epitaxy, while maintaining high deposition rates, deposits the epitaxial layer from a solvent that contains the respective solute in a small supersaturation generally maintained by a small undercooling. Such a small deviation from equilibrium and the high diﬀusivity of the solute in the liquid justiﬁes solution growth to be considered energetic. One has to keep in mind, however, that the growth system, at any moment of growth, realizes the lowest energy conﬁguration. The investigation of the evolution of the various self-organized structures in this case provides insight into the low-energy structures and helps to identify, by comparison to other growth experiments, kinetically inﬂuenced growth processes. Our work reports on the self-organized growth of Si1−x Gex alloys on Si(001) substrates by liquid phase epitaxy, mostly from Bi solution (composition range from zero to 100% Ge, corresponding to a lattice misﬁt of up to 4.2%). We analyze the evolution of ripples and islands, the nature of their formation (as a result, rippling is an instability, islanding occurs thermodynamically activated), their interactions and interdependencies leading to a variety of regular geometrical arrangements, especially due to ripples templating the distribution of islands. Having the unique option in solution

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growth to keep the system after growth under the liquid for many hours we can study the stability aspects of the self-organized arrangements, of which the stability of the regular island arrangement (no Ostwald ripening) is the most important result. In order to identify the limits of the energetically determined growth we present an in-depth analysis of island growth inclusive of kinetically caused features. The ﬁeld of energetically determined growth can be described by means of a master curve. Deviations from this relationship “coverage with islands versus cooling rate” indicate a kinetic inﬂuence. A brief glance to diﬀusional and plastic relaxation serves to complete the text. This report is based on many experimental and theoretical/simulational approaches. We use various microscopies (optical, atomic force, imaging and analytical transmission electron microscopies) and X-ray diﬀraction techniques (laboratory and synchrotron radiation, high-resolution and glancingangle diﬀraction) to access structures in all length scales. We calculate the three-dimensional (3D) strain ﬁelds and interactions between self-organized structures with the ﬁnite-element method and model the observations with these energetic aspects, but also with linear and nonlinear stability analyses. Monte Carlo and molecular dynamics simulations complement our tool pool.

5.1 Introduction Today, many semiconductor devices require epitaxial processes. In the case of Si, extremely thin layers are utilized, e.g., for SiGe, strained Si, dielectrics, and other applications, especially when band gap and mobility engineered structures are based on quantum wells, wires, and dots. In the world of compound semiconductors epitaxy has always been required due to the use of extremely high quality alloy layers in devices. A variety of epitaxial growth techniques have been developed for semiconductors over the past 50 years. Solution growth by liquid-phase epitaxy (LPE) is one of the oldest techniques that proved to be particularly useful to produce high quality thin layers. However, these techniques have largely been displaced by more ﬂexible techniques, such as metal organic chemical vapor deposition (MOCVD) or molecular beam epitaxy (MBE), for the growth of a large range of materials and special structures. Each of the epitaxial growth processes is extremely complex when viewed in detail at the atomic level and thus, even after thousands of man years of eﬀort, we have not yet reached a complete understanding. For each growth process, the fundamentals are often broken down into the separate categories of thermodynamics, kinetics and mass transport, which are, however, tightly coupled. As a result of the above-mentioned complexity of epitaxial growth, early studies were largely empirical, giving epitaxy the appearance of a magic art. Persuading a better understanding of the epitaxial processes, sophisticated in-situ diagnostic tools were developed, which just showed that the processes were even more complex than we thought.

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Thus, a complete understanding of epitaxial processes remains elusive, however, since these epitaxial processes are essentially controlled phase transitions, i.e., thermodynamics controls many of the resulting characteristics of the processes and the resulting epitaxial layers. Having said that, it must also be acknowledged, that the very process of growing an epitaxial layer demonstrates that the system cannot be completely at equilibrium. It is the intentional deviation from equilibrium that gives the driving force for epitaxy. Nevertheless, the extremely slow growth rates and the relatively high temperatures, typically used in high quality epitaxy, give conditions approximating equilibrium between the growing solid and the nutrient phase right at the interface. This guarantees a dominant role of thermodynamics in most cases. Especially, these conditions are essentially valid for liquid-phase epitaxy due to the fact that this method operates at comparably low driving forces compared to methods such as MBE and CVD (cf. Fig. 5.1). Thus, it seems entirely natural for LPE to describe the growth process by thermodynamics and this method proves to be the model method to experimentally realize near thermodynamic growth conditions. It is the aim of this paper to brieﬂy review selected near thermodynamic, pseudomorphic LPE growth experiments in the comparably easy model system (Si,Ge)/Si (with a lattice mismatch between 0 and 4.2%). It shows a number of cases related to nucleation and growth of self-assembled SiGe

Fig. 5.1. Estimated thermodynamic driving force for several epitaxial growth processes. All calculations are for the growth of GaAs at 1000 K (after Stringfellow [1]). MBE Molecular beam epitaxy, LPE liquid-phase epitaxy, MOVPE metal-organic vapour-phase epitaxy

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ripples and islands on Si(001) surfaces, where a deep and partly even quantitative insight into the growth process can be obtained by considering the thermodynamics and nucleation kinetics involved. Thus, we will establish a balanced understanding of growth phenomena, where kinetic and thermodynamic growth regimes are two extremes in a continuum, which can be scanned by the selected speciﬁcs of each experiment. The system SiGe is comparably easy due to the fact that it is an essentially isotropic cubic material with Si and Ge being entirely miscible in the full range of alloy compositions at usual growth temperatures. Ge has a larger lattice constant than Si. Growth of SiGe on Si substrates initially takes place pseudomorphic, i.e., free of extended defects but strained in compression. Eventually, the stored strain energy reaches such a level that the system reduces it by initiating relaxation. These processes can be of diﬀerent nature, plastically by dislocation formation, or elastically by the formation of a rippled surface or free-standing nanoscopic islands. In this review we will restrict ourselves to elastic relaxation experiments, i.e., to growth conditions where self-organized ripple and island formation takes place. Speciﬁcally, we show results of experiments that are unique to LPE growth. Among those are so-called equilibration experiments where the sample remains under the saturated solution at elevated growth temperatures. Thus, the system obtains time to assume its equilibrium conﬁguration. Among those are also dissolution experiments where the layer is ﬁrst deposited by cooling the system down by ΔT1 and then dissolved partly or entirely again by heating the system by a temperature interval ΔT2 . With this type of growth experiments reversibility of growth stages as well as true energy minimum conﬁgurations, as given by island sizes, shapes, and assemblies can be determined. Further we use experiments, which yield growth stages for diﬀerent growth times (and thus ΔT ) on one substrate (temperature-resolved growth experiment). Such samples ensure otherwise identical deposition conditions and simplify the analysis of the time evolution of growth. Performing such experiments at rather low temperatures and thus low growth rates we could monitor very initial stages of growth. With this low temperature LPE we are able to describe an energetic pathway of heteroepitaxial SiGe/Si growth and can moreover distinguish between energetically and kinetically determined growth morphologies.

5.2 Fundamentals of LPE 5.2.1 The Technique Growth from the solution takes place near thermodynamic equilibrium. Since adatoms diﬀuse in the solution with diﬀusion constants that are four orders of magnitude higher than on a free surface, we have generally no ki-

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netic limitations due to diﬀusion (the border line to the kinetic growth regime will be considered later). Thus, growth from a solution can best be described in terms of equilibrium thermodynamics. In the case of homoepitaxial growth (i.e., growth of a semiconductor on a substrate of the same material) the growth can be described starting from the phase diagram of the solvent and the growing material (semiconductor). In the present case of heteroepitaxial growth, additional energy terms, mainly due to phase boundaries and lattice strain, have to be considered. They essentially become evident in the growth topology as we show in detail in this work. The LPE experiments in the high-misﬁt regime are performed in a sliding graphite boat as sketched in Fig. 5.2. Hydrogen gas (puriﬁed by a Pd cell) served as a protective atmosphere. The Si substrate source wafers, the appropriate amounts of the metal (Bi, In) solvent (typically 10 g of a purity of 99.9999%) together with the appropriate amount of the solute Ge are loaded to diﬀerent boat chambers. The substrates for epitaxy with a size of 4 × 4 cm2 are cut from polished Si(001) wafers (Siltronic). The native oxide of the substrates and of the Si pieces that are used for saturation of the solution are removed by an RCA treatment (Kern and Puotinen 1970 [2]), followed by a (2.5%) HF dip. After loading to the boat, in-situ oxide desorption at 940 ◦ C is performed prior to adjustment of the growth temperature Ts . The growth

Fig. 5.2. Schematic of solution growth by LPE shows how the specimen in a sliding boat is moved below the saturated solution and out again. With this outward movement, the solution is stripped away from the specimen surface. In the temperature T vs. time t plot, lines 1 and 2 indicate growth experiments with high and low cooling rates ΔT /Δt, resp., for identical cooling interval ΔT . Line 3 denotes a so-called “equilibrium experiment”, line 4 denotes a so-called dissolution experiment

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process starts when the substrate is brought into contact with a liquid–metal solution containing Ge and Si by sliding the graphite boat as sketched in Fig. 5.2. The solution consisting of the metal (Bi, In) solvent and the corresponding solute Ge is prepared and moved to the solution chamber. A Si wafer there guarantees Si saturation of the solution whichever growth temperature is selected. The cooling interval ΔT , given by the diﬀerence between growth temperature Ts and end temperature Te determines the volume of deposited material at equilibrium. ΔT is realized via a cooling rate ΔT /Δt and a cooling time Δt. To be able to monitor diﬀerent growth stages and the development of the pseudomorphic growth structures on one sample, we utilize low growth temperatures that result in considerably longer growth times. We generally worked with growth temperatures between 450 ◦ C and 850 ◦ C. The use of these remarkably low growth temperatures required to solve the problem of oxidation of the substrate that usually comes along with low growth temperatures. We slide the specimen in 2 or 3 steps under the solution (it can also be a slow continuous slide) thus realizing diﬀerent growth times and thus ΔT under otherwise identical growth conditions on a single specimen (temperatureresolved growth experiment). The growth times strongly diﬀer for diﬀerent cooling rates, growth temperatures and misﬁts. The temperature change with time throughout diﬀerent types of LPE experiments are shown in Fig. 5.2. In this ﬁgure lines in diﬀerent colors show diﬀerent experimental conditions that are partly unique to LPE: the black and the blue line mark identical cooling intervals ΔT , but diﬀerent cooling rates ΔT /Δt. The faster the cooling the further the experiments deviates from thermodynamic equilibrium conditions. The red and green lines show growth experiments that are unique to LPE growth. With LPE we can perform so-called equilibration experiments. In these experiments we realize a given ΔT thus depositing material on the substrate and then leave the specimen under the saturated solution at deposition temperature for quite some time (in this work we show specimens that were equilibrated even for 3 days). With these equilibration experiments it is possible to provide the time the system needs to assume its true energy equilibrium conﬁguration. We can moreover realize dissolution experiments (green line in Fig. 5.2). With these experiments we, in a ﬁrst step deposit material, given by ΔT , and in a second step heat substrate and solution up again by a ΔT0 , so that deposited material dissolves again. These types of experiments make it possible to check for reversibility of growth stages. 5.2.2 The Concept of Driving Forces Layer growth is usually discussed in terms of nucleation and growth which in turn are usually discussed in terms adsorption of atoms from the gas

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phase or a solution onto a surface, diﬀusion of these atoms on that surface, and attachment of the atoms to a growing nucleus, island, or layer. All these process steps require time, which is given by the respective rates of the various steps. In cases the growth experiment limits the available time for the series of all these steps to occur and the system gets, in consequence, into a state far from equilibrium with rather high driving forces, the growth result does generally not correspond to the energetically most favorable state, i.e., growth is kinetically controlled. Such a situation occurs generally at low temperatures, where diﬀusion is slow, and typical deposition rates result in adatom concentrations that exceed the equilibrium concentration of adatoms on the surface by far so that growth is indeed controlled by kinetics. At high temperatures, for deposition from the gas phase or for solution growth, however, the equilibrium concentration of adatoms may be already so high that an appreciable concentration of additional adatoms due to an external ﬂux may still represent only slight relative increase. Diﬀusion can be so fast that spatially separated regions on the surface interact on a time scale that is not slow relative to the growth process. In such cases driving forces are extremely low and collective phenomena are more important than individual atomic events, and thermodynamics is more important than kinetics. As will be seen, the thermodynamic aspect holds in solution growth experiments in most cases very well and the gained insights are most useful. Thermodynamics deﬁnes the maximum growth rate, often determines the composition of the solid being grown including alloy composition, stoichiometry and dopant incorporation. Thermodynamics also deﬁnes the equilibrium microstructure of the solid and so can be used to predict whether the solid alloy will be uniform (random), clustered or ordered. In the following, we will describe solution growth experiments in the SiGe/Si(001) system and relate them to thermodynamics of the growth process and use this example to explain the concept of driving forces during LPE. In order to put LPE into perspective to high driving force growth methods, we brieﬂy review observations from LPE and in addition from MBE and CVD experiments. As a matter of fact, this review will essentially sketch the present stage of knowledge on ripple and island formation and growth. We regard here ripples as the undulatory deviation of the growing epitaxial surface from the planar shape. The special case, which is realized where the troughs of the ripple structure reach the substrate will be signiﬁcant in the context of island formation. Islands are all those structures that grow pseudomorphically onto the wetting layer and form isolated and laterally limited entities (the Stranski–Krastanow growth mode). Ripples are reported essentially for comparatively low misﬁts f (say f < 1% corresponding to 15 at.% Ge) to form during growth [3–5]. Their

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formation was also reported during post-growth annealing [5, 6]. The discussion, however, in the last decade was governed by the observation of island structures that are divided into “hut clusters” (after Mo et al. [7]), i.e., small islands at low coverage, and “domes” (after Tomitori et al. [8])([6,9–15]), i.e., larger islands at a higher coverage. In the course of these eﬀorts, based on a larger body of MBE and CVD experiments, the attempt was undertaken to deﬁne equilibrium growth stages and classify huts and domes as equilibrium conﬁgurations for certain deposition ranges. However, during our LPE experiments that have, due to the low driving forces, the best access to true equilibrium conﬁgurations, neither huts nor domes appeared, but we observed at low misﬁts ripples followed by islands, at high misﬁts islands with speciﬁc equilibrium shape, size and distribution and faint or no ripples followed by the same islands. In the following we will summarize the conﬁgurations as obtained in experiments as conducted far from equilibrium and compare these to our LPE results. High Driving Forces: Kinetically Inﬂuenced Growth Huts are {105}-faceted square-based pyramids with generally a narrow size distribution [7, 14]. These “huts” are stable in width for some time during growth and then transform into domes. These domes appear isotropic in shape, and can often be square-based or form octagonal pyramids bound by several diﬀerent facets, primarily {102}, {110}, and {113}. They have a decreased aspect ratio (width/height) relative to the huts. The hut-todome transition occurs statistically and reproduces the hut cluster arrangement. Some intermediate sizes and shapes of huts and domes may occur. A diversity of attempts exist to explain the limited lateral sizes and shapes of huts and domes, the instant of the conversion of huts into domes, and the stability of hut and dome conﬁgurations and arrangements. For example the hut-to-dome transition is explained by Medeiros-Ribeiro et al. [14] by energy diagrams as a function of the deposited volume that have each a minimum at a diﬀerent volume. Thus, they assign “stability ranges” to huts and domes [14] such as do other authors [6, 9, 11]. Ross et al. [9], on grounds of in situ experiments in the electron microscope, conclude this transition to be a coarsening process (kind of Ostwald ripening [16]). Thus, they state implicitly that huts are not necessarily a stable conﬁguration. On the other hand, Kamins and Williams [10] performed annealing experiments and came to the conclusion that the hut-to-dome transition is characterized by a barrier that decreases with increasing hut density, i.e., huts are (kinetically) stabilized at least at low densities. The experiments further indicate that dome arrays partly coarsen, partly shrink to huts that further decrease in number. K¨astner and Voigtl¨ ander’s experiments [17] (and also experiments by other authors [13, 18]) even show that square-based huts

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are unstable against the formation of a rectangular base which in consequence – as also observed – leads to a considerably broad size distribution of the huts [13–15, 17]. Low Driving Forces: Energetics Of Growth The morphological development of SiGe layers on Si(001) substrates appears rather transparent in LPE experiments: essentially at low misﬁts ripples ﬁrst appear and, during extended growth, islands form [19–23]. The ripples, when forming, cover almost instantly the entire surface and align, at the beginning with larger scatter, later very prominently, along the elastically soft 100 directions (cf. Fig. 5.3). The islands have the form of truncated four-sided pyramids with four {111} side facets and an {001} top facet as shown in the cross-sectional TEM micrograph in Fig. 5.4 and the plane-view micrograph in Fig. 5.5 (see also sketch in Fig. 5.8). All forming islands have this shape independent of their width, i.e., this shape forms self-assembled. The base widths λ∗ of the islands follow a 1/f 2 rule [24] and thus behave as does the wavelength λ of ripples. Figure 5.6 shows the measured scaling of island widths with the misﬁt. This observation conﬁrms that islands grow self-limited. While the island width is determined by the misﬁt, the island height A∗ is correlated with the island width λ∗ : in all our experiments we observe a range of aspect ratios of λ∗ /A∗ = 2 − 2.5 [25, 26] for those islands that are at the transition to plastic relaxation.

Fig. 5.3. Developing layer morphology: (left) from irregular ripples, which ﬁrst gain order (middle) and eventually transform into a sort of regular template (right) for island nucleation. Low misﬁtting heteroepitaxial Si0.95 Ge0.05 /Si(001) growth at 600 ◦ C, growth interval ΔT = 3 K. Atomic force microscopy (AFM) micrographs [19]

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Fig. 5.4. Equilibrium island shape, Si0.40 Ge0.60 as an example. a Truncated squarebased pyramid with four {111} side facets and an {001} top facet. The island is cut along a {110} plane. b Zoomed out view in high resolution around the re-entrant island corner. For both pictures, multibeam conditions with the incident electron beam parallel to the 110-zone axis

We consider next the implications of the low and high misﬁt ranges. The boundary between low and high misﬁts lies at about 15% Ge in the alloy. This line between the high and low misﬁt regime may appear somewhat arbitrary, however, experimental results show that roughly at this composition a change in the interplay of relaxation mechanisms occurs. At low misﬁts, the amplitude of the ripples grows initially exponentially as shown in

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Fig. 5.5. Equilibrium island assembly, Si0.40 Ge0.60 as an example. Same islands as in Fig. 5.4 but in plane view. Islands are preferably arranged along the elastically [100] and [010] directions. Experimental conditions: electron beam parallel to the [001] zone axis

Fig. 5.6. Island width or ripple wavelength λ dependence on the misﬁt f : λ ∼ 1/f 2 . A larger body of experimental data is shown in the inset diagram. For direct comparison, cross-sectional micrographs of two diﬀerent SiGe alloy compositions show islands of diﬀerent size according to the misﬁt but identical shape (truncated, square-based {111}-faceted pyramid)

Fig. 5.7 [23] and the transition to islands occurs in a speciﬁc way shown in Fig. 5.3 (middle, right). Due to the enhancement of the strain in the ripple troughs, as quantiﬁed by 3D ﬁnite element calculations (cf. Fig. 5.12) [21,27], at a certain instant the chemical potential at the growing surface compensates the one of the growth-driving force. As a consequence local dissolution within the troughs sets in (e.g., Fig. 5.3, middle) and islands form from the ripple structure. As a matter of fact, the regular ripple array serves as a template for a regular arrangement of islands (e.g., Fig. 5.3, right). At the

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Fig. 5.8. Model of a truncated pyramid with base width λ*, top width b, height A∗ and side angle α

early stage of this transformation patches of island form with the islands preferentially arranged along the elastically soft 100 directions (as do the ripples), at an average island spacing of roughly the island width. In the later stage the islands form a periodic lattice with their base lines aligned along

110 directions and covering the whole growth surface, e.g., Fig. 5.8. For details of the self-assembled island conﬁgurations, why and how they form, see Sect. 5.3.4. Though ripples and islands are found at higher misﬁts also, most recent results indicate that there exists a signiﬁcant diﬀerence to the phenomena ob-

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Fig. 5.9. AFM micrographs (8 μm × 8 μm) of growth stages of Si0.40 Ge0.60 /Si(001) as an example. a A faint ripple template that covers the entire surface; bright features are due to statistically nucleated islands. b At a later growth stage the density of islands has increased [19]

Fig. 5.10. Growth state of Si0.67 Ge0.33 /Si(001) as an example for high misﬁts with slightly increased island density compared to that shown in Fig. 5.9a: islands are distributed on a faint ripple template. Plane view presentation. The black line in a indicates the distance sampled for the height proﬁle in b, which shows large amplitudes of islands and small amplitudes of ripples. This ripple amplitude stays constant during growth. Growth temperature 600 ◦ C, growth interval ΔT = 3 K [19]

served at low misﬁts. Figure 5.9 shows the main topological aspects of growth in the high misﬁt regime in atomic force micrographs in a 3D view. Figure 5.9a shows a state where a faint ripple template covers the whole surface with scattered islands that obviously have statistically nucleated. Figure 5.9b shows an increased density of islands. Figure 5.10 quantiﬁes these statements. Figure 5.10a shows a growth stage slightly more advanced (for a smaller Ge content) than that shown in Fig. 5.9a (in a plane-view representation). The black line in Fig. 5.10a indicates where the height proﬁle depicted in Fig. 5.10b originates. This line scan shows large amplitudes that belong to the islands and small amplitudes belonging to the ripple template. The wavelength for ripples and islands is roughly identical and follows the λ (or λ∗ for islands) ∼ 1/f 2 rule (cf. Fig. 5.6). Thus the wavelength dependence on the

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misﬁt of islands and ripples in the high and in the low misﬁt regime coincides. Moreover, the islands for both misﬁt regimes align along the elastically soft

100-directions as does the underlying ripple structures (cf. Fig. 5.3 and Fig. 5.9a). Contrary to the ﬁndings at low misﬁts (cf. Fig. 5.3) the amplitude A of the ripples is very small at high misﬁts. At the given wavelength λ ripple aspect ratios of λ/A = 100 occur. Careful analysis revealed that this amplitude A of the ripples remains even constant during growth (in our experiments up to two hours). This interesting growth stage will be discussed further in Sect. 5.3.2, Sect. 5.3.2 explains the need for nonlinear instability analysis.

5.3 SiGe/Si(001) Heteroepitaxy This chapter will discuss aspects of layer growth in the heteroepitaxial, misﬁt SiGe/Si(001) system. An outline of the growth process close to thermodynamic equilibrium, i.e., with low driving forces operational, has already been given in Sect. 5.2.2. In the following we will analyze the growth process in terms of relaxation behavior. Moreover, we will discuss kinetically determined deviations from the described energetic growth process. 5.3.1 Classiﬁcation of Relaxation Phenomena In the heteroepitaxial (Si)Ge/Si system the maximum lattice misﬁt between pure Si and pure Ge is 4.2% and the related elastic strain increases with increasing layer thickness. As soon as it exceeds a critical thickness for roughening, Hcr , the layer forms undulations [28–31] or even islands or cusps [32] dependent on the actual process parameters and the amount of strain. (This critical thickness has to be discriminated from that of Matthews and Blakeslee, hc , for dislocation formation [33].) This roughening process has intensively been investigated experimentally and theoretically. The main results, important to this work, will brieﬂy be summarized: the physical reason for roughening, as pointed out by diﬀerent authors [28–31] is the inherent instability of strained thin layers, known as the Asaro–Tiller –Grinfeld instability. The criterion for roughening to occur is that the additional surface energy Esurf needed can be nourished by a reduction in strain energy Estrain . In other words, the driving force for roughening is the potential of rough surfaces to relax strain elastically [25,34–36] at their free-standing parts. Srolovitz [30] found, using a simple 2D model of periodic square shaped surface undulations that only wavelengths λ are stable with: λ

8 · E · Esurf , σ2

(5.1)

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with E comprises the Young modulus and σ the stress in the layer. Thus, λ depends on the misﬁt f according to: λ

Esurf , f2

(5.2)

since misﬁt f and stress σ are directly proportional. Srolovitz further found that Eq. 5.2 is a very general functional dependence for strained thin undulated layers independent of the shape of the undulation and the growth limiting step, i.e., independent of the special kinetics valid let it be either surface diﬀusion or condensation. Eq. 5.2 relates the wavelength of the undulation that forms due to the instability to one over the misﬁt square. Gao [31] also proposes conditions for roughening. His numerical analysis (perturbation analysis) derived three entities in addition to the stable ﬂat layer surface that are prominent in strained layer epitaxy: sinusoidal undulations, cusping and 3D islanding. He states that below a certain critical wavelength λcr undulations are unstable independent of the layer thickness. Stable sinusoidal undulations form for λ < λcr . When exceeding the critical thickness Hcr for roughening, sinusoidal undulations form that transform into a complete cusp network that leaves separated islands behind. After Gao, islands are always energetically favored compared to a ﬂat continuous strained layer. The layer thickness however beyond which island formation starts, depends on the misﬁt. This numerically described growth process coincides very well with our observations outlined in Sect. 5.2.2. We also observe layers that are unstable against roughening by forming sinusoidal undulations. Depending on the misﬁt the undulated layer thickness varies with the misﬁt. The lower the misﬁt the thicker the undulated layer. At Ge contents in the SiGe alloy beyond 15%, the ripple template becomes very faint and at even higher misﬁts, roughly above 25% the ripple template becomes nondiscernible. It is our interpretation that in this misﬁt range the template layer is so thin that it cannot hold even the faintest undulation. How eﬀective elastic relaxation through undulation formation is depends essentially on the aspect ratio of the undulation as given by the wavelength or island width λ and the amplitude A or island height A∗ . The smaller the aspect ratio λ/A, the larger is the amount of elastically relaxed strain [25,36]. To obtain a measure for elastic strain relaxation due to the undulations we follow here an approach ﬁrst proposed by Luryi and Suhir [37] and later adopted by our group for ripples and islands [36]. These authors introduced a numerical function Φ (we shall call it “elastic relaxation parameter”) that reduces the nominal misﬁt f in undulated layers to an eﬀective reduced misﬁt, fr according to: (5.3) fr = f · φ. Physically interpreted, fr is the misﬁt of a pseudomorphic planar layer that has the same strain energy density as the undulated system. Thus, Φ can be calculated when the undulated layer is periodic (which we assume through-

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out) and the periodicity elements describe the unit volume. Since the strain energy density is proportional to f 2 , we obtain: Φ2 =

(E/V )strain,3D fr2 = , f2 (E/V )strain,2D

(5.4)

where (E/V )strain,3D is the average strain energy density stored in a 3D laterally limited structure (consisting of layer and substrate) and (E/V )strain,2D is the strain energy density stored in a continuous, ﬂat, strained layer, with the same volume and misﬁt as the 3D arrangement (in the following referred to as “2D- reference”). Eq. 5.5 gives the functional dependence (in the framework of isotropic continuum elasticity theory) that deﬁnes the strain energy per considered volume: E/Vstrain = 2 · G

1+ν · (f · Φ)2 , 1−ν

(5.5)

with Φ is unequal 1 for the undulated “3D layers” and equal 1 for the “2D reference”. In Eq. 5.5 “G” represents the shear modulus, ν the Poisson ratio and “f” the misﬁt between layer and substrate. The average strain energy density (E/V )strain,2D can be calculated from Eq. 5.5, with Φ = x and f is the nominal misﬁt in the layer. For the calculation of the average strain energy density (E/V )strain,3D based on Eq. 5.5, the reduced eﬀective misﬁt fr , however, has to be determined. These calculations have been carried out on the basis of 3D ﬁnite element (3D FE) calculations and yield the average strain energy density (E/V )strain,3D at any site of the geometry according to: 1 E/Vstrain,3D = σij · εij , (5.6) 2 with i,j ∈ 1,2,3. All stress and strain components of the respective tensors (σij and εij , respectively) are given for each cuboid element (e) of the ﬁnite element meshing of the volume Ve . Thus, the total average strain energy density distribution (E/V )strain,3D of the mismatched, undulated layer including the underlying substrate divided by the total strain energy density of the 2D reference, (E/V )strain,2D , can be derived from 3D FE calculations and the elastic relaxation parameter Φ can be determined according to:

(E/V )strain,3D

(e)

· V (e)

(e)

· V (e)

(e)

Φ2 =

(E/V )strain,2D

.

(5.7)

(e)

Figure 5.11 shows an example of 3D FE calculations of the strained state, here of a sinusoidally undulated (rippled) SiGe layer on a Si(001) substrate. Due to symmetry, only half a wavelength λ is modeled with the ripple ridges

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Fig. 5.11. Model representation with 3D ﬁnite elements of the ripple stage of a SiGe-layer (xGe = 0.25) on Si(001). The strain energy density distribution is displayed by grey scale coding. In this example the ripple has an aspect ratio of λ/A = 5. Ridges are elastically relaxed towards the equilibrium lattice constant of SiGe (very dark in the grey scale), troughs are strained in compression (bright areas). For the original colour coded image see [21]. Owing to symmetry, only half the wavelength λ is shown and periodic pursuit of the structure in the x = [110], −x, y = [1,-1,0], −y directions is accounted for by the boundary conditions

lying aligned along the elastically soft 100 directions. Periodic pursuit is accounted for by periodic boundary conditions. The simulations show that the ridges of the sinusoidal undulations are elastically strain relaxed towards the bulk equilibrium lattice constant of the SiGe alloy while the lattice in the troughs is compressed. Figure 5.12 shows an example of 3D FE calculations of a strained pseudomorphic, {111}-faceted Si0.67 Ge0.33 island on an Si(001) substrate. Due to symmetry, one half of the wavelength λ plus the surrounding substrate in x and y direction is shown only. In our example the island has an aspect ratio of λ∗ /A∗ = 2.5. The strain energy density distribution is color coded. Regions with a reduced elastic strain energy density are shown dark in this ﬁgure, regions with enhanced elastic strain energy density are bright. The function Φ as a measure for elastic strain relaxation by such type of ripples or islands is given in Fig. 5.13 as dependent on the aspect ratio of the laterally limited structures. The smaller the aspect ratio is the more eﬀective is the elastic relaxation and thus the smaller Φ. The solid lines in Fig. 5.13 are ﬁts to the calculated

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Fig. 5.12. Model representation with 3D ﬁnite elements of an island in the SiGe layer (xGe = 0.33) on Si(001) The strain energy density distribution is displayed. In this example the island has an aspect ratio of λ∗ /A∗ = 2.5. Due to symmetry, only one quarter of the square based island is shown and periodic pursuit of the structure in the x = [110], -x, y=[1,-1,0], -y directions is accounted for by the boundary conditions. The island’s top region with a reduced elastic strain energy density appears dark. The bottom edges of the islands show an enhanced strain energy density which is coded bright. For the original colour coded image see [21]

data values and yield the very helpful analytical dependency: Φ = 1 − exp(−p ·

λ ), A

(5.8)

with p representing an adjustable constant. For the truncated, pyramidal islands as they are used in our ﬁnite element calculations, p = 0.27, for the sinusoidal undulations (pure sinus nearly without a 2D homogeneous crystal base), p = 0.16. Once the strained state at the growing surface is known at every site, one can envisage qualitatively the atomic incorporation process during growth. Germanium has a larger atomic radius than silicon and thus prefers to attach at sites where the lattice constant is widened. Such sites are the relaxed ones, i.e., the protruding ones in ripples and islands (cf. Figs. 5.11 and 5.12). There the material with the intended composition will grow. In consequence

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Fig. 5.13. Elastic strain relaxation parameter Φ(λ/A) as calculated from 3D ﬁnite element (FE) models with aspect ratios in the range of 0 < λ/A < 40 for faceted square-based islands (triangles, xGe = 0.85) and sinusoidal undulations (circles, xGe = 0.03)

the incorporation rate of germanium is reduced in areas of comparably small lattice constant, i.e., in areas of compression, especially in the troughs and bottom edges. Thus these incorporation processes lead to an eﬀective alloy segregation as will be quantiﬁed in Sect. 5.3.3. The quantiﬁcation of the local incorporation rates has to consider also that the locally varying stress state modiﬁes the local driving force for growth as given by the supersaturation. This modiﬁcation enters essentially into the energetic term in the rate equation: ΔG ), (5.9) R(r) ∝ Nad · ν · exp(− kB · T with R(r) is the adatom incorporation rate, Nad is the adatom density at the surface, ν is the attempt frequency for incorporation, ΔG(r) is the activation energy for adatom incorporation. 5.3.2 Topology Formation Essentially two in principle diﬀerent models can be distinguished that describe the formation of laterally limited structures such as ripples and islands. These models are based either on instability theories [28–31, 38] or thermodynamic equilibrium theories [6, 9, 11, 12]. Both types of models start from the fact that the formation of undulations and/or islands reduces the total energy of the strained system where its ingredients “strain energy” (decreases) and “surface energy” (increases) act as counterparts. For the instability approach a small arbitrary perturbation

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is applied to the system and the reaction of the system to this perturbation is considered. As a result, in strained layers an undulated or, under certain conditions (strain, layer thickness), even cusped or islanded layers may arise ( [31]). For the thermodynamic approach, in contrast, the start and end situations are compared and the transitions from start to end states are inferred to be associated with activation energies (in case we assume one step transitions). Evaluation of characteristic activation energies requires the identiﬁcation of the rate determining process and the many inﬂuences like cooling interval, surface diﬀusion, desorption, and adsorption processes etc. This multiparameter situation is probably the reason for the many attempts of interpretation of self-assembled growth phenomena for which the equilibrium stage is often obscured by kinetically determined conﬁgurations in MBE or CVD growth, as mentioned before. In discussing now the nature of ripples and islands as observed by LPE growth we need to ﬁnd only their signatures. Ripples: Linear Instability Analysis The instability theory as put forward by a few authors [30,31,38] analyzes the development of the system after a disturbance in form of say a ﬂuctuation. The result of these approaches is that, provided the disturbance has a wavelength longer than a critical one, the system changes into the energetically favorable state without a thermally activated step. Of course the velocity of this change is controlled by the respective mechanism, which itself may be thermally activated. Srolovitz [30] has calculated the kinetics when surface diﬀusion or condensation is involved. The characteristics of these two types of instabilities are: (i) the structure formation occurs simultaneously on the whole surface, (ii) the amplitude increases exponentially in time (at least initially), (iii) a characteristic wavelength forms, which scales with λ ∼ 1/f 2 (cf. also Sect. 5.2.2). Comparison with the LPE experiments shows that the ripples, observed at low misﬁts, exhibit this λ ∼ 1/f 2 characteristics, and the exponential increase of the ripple amplitude with time (see Fig. 5.7) as typical for the linear instability. Ripples thus form barrierless. We have to mention here that such a linear assessment cannot reproduce the ripple amplitude that is constant with growth time as observed at higher misﬁts during LPE growth (c.f. Fig. 5.9 and Fig. 5.10). The Need for Nonlinear Instability Analysis A stable amplitude is described by the weakly nonlinear instability theory [39–42]. The basic equation for development of the amplitude A of a disturbance is: dA = sA − a1 A3 . (5.10) dt

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The linear approach is included for a1 = 0. Here a1 is the so-called Landau parameter (it contains a combination of growth and material parameters), s is the growth rate. Putting the time derivative to zero for the stable amplitude, we obtain for the amplitude A0 : A20 =

s . a1

(5.11)

For s > 0 ( i.e., growth), amplitudes exist that are constant with growth time, when a1 > 0. In this case after the initial ﬂuctuation grew exponentially in amplitude, the amplitude stays constant with the value given by Eq. 5.12. The physical reason for this stable amplitude is not known yet. As an accompanying eﬀect, the characteristic wavelength of the developing structure is the same as in the linear approach. In this way the ripple structure can be used as a very regular template for island growth as demonstrated in Fig. 5.3. Islands: a Nucleation Phenomenon In the following we will describe some growth experiments taken from an ongoing comprehensive eﬀort that shows that the island formation is a thermally activated process. Figure 5.14 contains a collection of AFM micrographs showing the dependence of island formation on growth temperature TS (600 ◦ C–750 ◦ C in steps of 50 ◦ C) and growth time (20, 40, and 60 min) for Si0.75 Ge0.25 /Si(001) as an example. These experiments belong to the type of the time-resolved experiments (see Sect. 5.2.1). The cooling rate was chosen slow (0.17 K/min for Fig. 5.14) to assure low growth rates and thus conditions as close as possible to thermodynamic equilibrium. The growth process is in all experiments conducted in a way that the temperature decreases linearly from an initial value (the growth temperature Ts ) to a maximum of ΔT = 10 K at the end of the growth experiment after 60 min (right column of AFM micrographs). A close inspection shows that after 20 min at all temperatures the aforementioned ripple structures exist. These ripples are superimposed by obviously statistically distributed islands. After 40 min of growth, the island density is increased with a certain regularity in the island arrangement as given by the underlying ripple template. After 60 min of growth the island number density is still further increased. Independent of the necessity for further statistically reliable data in a broad parameter range (growth velocity, Ge content to check for misﬁt dependence of nucleation) we can show here that the observations corroborate thermally activated nucleation of islands. Focusing on the left column in Fig. 5.14 we count the numbers of islands and plot these in an Arrhenius diagram (Fig. 5.15). We can deduce the activation energy of island nucleation from Fig. 5.15 and obtain 0.84 ± 0.13 eV.

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Fig. 5.14. Time–temperature matrix (AFM micrographs) of island nucleation and growth at high misﬁt f (Si0.75 Ge0.25 ). Growth temperatures and times are indicated. In all examples the growth interval after 60 min of growth is ΔT = 10 K [19, 43]

In the evaluation, the island density at 600 ◦ C is neglected since the time of 20 min contains a large contribution due to the incubation time, which is a characteristic feature at the beginning of nucleation processes in general. This contribution can be neglected at higher temperatures. In view of the preliminary nature of our evaluation we cannot assign a speciﬁc process to the island nucleation. The activation energy value is reasonable and is

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Fig. 5.15. Arrhenius plot of the island number density N after 20 min of growth (data taken from Fig. 5.14). Ts is the temperature at start of growth. The activation energy for island nucleation is 0.84 ± 0.13 eV [20, 43]

consistent with ﬁrst results of the analysis based on Avrami–Johnson–Mehl type [44, 45]. Identifying Kinetic Growth Conditions The fact derived from experiment that islands nucleate statistically with a characteristic activation energy strongly suggest that it should be possible to describe island nucleation and growth in time the same way as other nucleation processes such as e.g., nucleation of crystallites in an amorphous matrix [46], i.e., in terms of statistical thermodynamics. In the following we show that this is indeed the case. As an additional result we shall be able to identify the borderline to the kinetic growth regime. We refer in the following to growth experiments that establish an evolution matrix of structural data [43]. In addition to the growth time, cooling rate and Ge content (or misﬁt) are the parameters. The growth temperature Ts has just been considered. The following paragraphs in this chapter show how the island coverage β (and thus nucleation) evolves with time and changes with the selected parameters (one parameter only at a time while the others are kept constant). Nucleation Theory Transferred to Island Nucleation on a Substrate The crystallization processes are usually described in terms of the crystallized volume fraction. In our case of heteroepitaxy the SiGe islands nucleate on a substrate surface and rapidly grow to a ﬁnite size. It is thus easier to evaluate the area fraction covered with islands, i.e., to evaluate the coverage

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as a measure for the phase transformation. The coverage β, as e.g., measured from AFM micrographs, is given by: β = F · N,

(5.12)

where F represents the average area covered by a single island, and N represents the island number density on the surface. We note that this approach reduces the 3D problem of island growth on a surface to a 2D problem. We can use this approach since our LPE-grown islands follow the λ ∼ 1/f 2 [24,47] rule, i.e., all islands on one specimen have nearly the same misﬁt dependent width λs thus F ∼ 1/f 4. Moreover, all islands have the same aspect ratio λ/A ∼ 2.5 and thus all islands have roughly the same volume, Vpyramid , at a given width, i.e., Vpyramid ∼ λz ∼ 1/f 6 . This 2D-approach is in general not suitable in growth experiments for example by CVD or MBE. In these cases huts and domes have been reported to form that have diﬀerent base areas and thus diﬀerent volumes [9,13]. Here the volumes have to be determined directly. In order to determine the coverage β we need to measure for each growth experiment and growth time the island density and the average base area F . These entities can be evaluated rather easily from AFM micrographs, but have speciﬁc errors that should be considered [43]. The island base area F is evaluated from the AFM micrographs by means of the integrated AFM software and we deduce a relative error of 10% after correction due to the ﬁnite size of the used tip. Island counting on different areas of the same specimen sector results in the variation of island density of up to 10%. The relative error of the area coverage β is then ∼ 14%. Modeling: Change of Coverage β with Time For the modeling of the time evolution of coverage two experimental aspects have to be considered. First, our observations show that island formation is dominated by nucleation with island growth playing a negligible role: as soon as an island has nucleated it assumes its ﬁnal 111-faceted truncated pyramidal shape with its volume Vpyramid ∼ 1/f 6 (at least when growing close to equilibrium, i.e., at low cooling rates). Thus the time needed for the formation of an island is small compared to the usual growth times. The nucleation time is much larger than the growth time for an island, which was implicitly assumed already in Sect. 5.3.2. Second, the islands start to form after an incubation period t0 . This incubation time t0 is rather insensitive to the process parameters (especially to all used cooling rates ΔT /Δt) except for the growth temperature Ts (between 500 ◦ C and 900 ◦ C), which causes a small variation around t0 = 8.5 min. Thus, t0 behaves in analogy to the crystallization of amorphous Si in thin layers [48, 49].

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We start the mathematical description from the fact that the change of the coverage with time, β(t), is at any time proportional to the still uncovered area (1 − β(t)). We can write: dβ(t) = κ · [1 − β(t)] , dt

(5.13)

where κ is a constant which represents the initial slope of the β(t) dependence. Integration leads to: β(t) = 1 − exp[−N˙ (t0 ) · F¯ · (t − t0 )] .

(5.14)

N˙ (t0 ) is the island nucleation rate at the beginning of growth (at incubation time t0 ) In the following, we simply refer to N˙ (t0 ) as the initial island nucleation rate. N˙ (t0 ) depends on the cooling rate ΔT /Δt, the misﬁt f and the growth temperature Ts . In the following we will show how island nucleation is aﬀected by the various process parameters and especially how kinetic eﬀects show up as limits to the energetically controlled growth domain. The Coverage β Versus Cooling Rate The change in the cooling rate ΔT /Δt aﬀects especially the initial island nucleation rate N˙ (t0 ). An example for the nucleation behavior as dependent of the cooling rate is given in Fig. 5.16 where the coverage is plotted versus growth time. Here, three diﬀerent cooling rates are compared: ΔT /Δt = 0.056 K/min, ΔT /Δt = 0.111 K/min and ΔT /Δt = 0.333 K/min. It is visible that the initial nucleation rate is larger for larger cooling rates under otherwise identical conditions. Quantitatively we determine N˙ (t0 ) from the ﬁt of Eq. 5.16 to our experimental data. The parameters extracted from Fig. 5.16 are summarized in Table 5.1. We can use the data presented in Fig. 5.16 for a check whether kinetic limitations come into play at high cooling rates. The analysis is based on the fact that in the energetically determined growth domain the deposited volume depends only on the cooling interval ΔT . Figure 5.17 shows the replotted data from Fig. 5.16. In fact the curves superimpose for the two low cooling Table 5.1. Experimental LPE growth data for diﬀerent cooling rates derived from AFM measurements and Fig. 5.16 [19] cooling rate cooling island ΔT /Δt interval width λ [K/min] ΔT30min [K] [μm]

κ [min−1 ]

N (t0 ) N (Δt (min · μm2 )−1 = 30 min) [μm−2 ]

0.056 0.111 0.333

0.90×10−3 2.02×10−3 6.97×10−3

0.34×10−3 0.85×10−3 3.56×10−3

1.666 3.333 10

1.64 1.54 1.40

1.01×10−2 2.56×10−2 10.7×10−2

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Fig. 5.16. Coverage β vs. time t. Ge content 0.10. Experimental data are marked with squares, pentagons and circles, ﬁts to the data are based on Eq. 5.16. Lines of equal cooling interval ΔT are indicated (ΔT = 11.2 K, 8.4 K, 5.6 K, 2.8 K [43]). Ts = 700 ◦ C [43]

Fig. 5.17. Coverage β vs. cooling interval ΔT . Experimental data are marked with squares, pentagons and circles. While the curves for low cooling rates are superimposed, the curve for high cooling rate lies lower indicating a kinetic inﬂuence on the growth. Ts = 700 ◦ C [43]

rates (circle, square). For the largest cooling rate ΔT /Δt = 0.333 K/min (pentagon), however, the curve lies below the two others. Thus less material as compared to the other two growth experiments is deposited island conﬁgurations. In contrast to equilibrium conditions where equilibrium shaped {111}-faceted truncated pyramids form, kinetically determined growth conditions at high cooling rates cause also smaller islands with deviating shapes in our cases essentially ﬂat {115}-faceted pyramids (cf. Fig. 5.18). The type of kinetic limitation that is present here cannot undoubtedly be determined. The diﬀusion processes to the liquid/solid interface may be too slow or there might be a kinetic limitation to adatom incorporation into the crystal at too high cooling rates. Thus, further experiments are needed to clarify the type of kinetic limitation. A certain hint may be given by

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Fig. 5.18. Si0.75 Ge0.25 deposition. a Low cooling rate ΔT /Δt = 0.11 K/min. b High cooling rate ΔT /Δt = 0.48 K/min. Only equilibrium shaped islands ({111}-faceted truncated pyramids) are observed in a, whereas in b equilibrium shaped islands and kinetically controlled smaller islands with essentially {115} facets are seen. AFM micrographs [43]

Fig. 5.19. Islands and ripples in a coexistence stage, deposited at high cooling rate ΔT /Δt = 0.333 K/min. Dissolution of ripples around the islands is observed. AFM micrograph [43]

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Fig. 5.19. This ﬁgure shows islands, nucleated on a ripple template during deposition at high cooling rates ΔT /Δt = 0.333 K/min). The ripple template is dissolved close to the islands. An interpretation may be that the solution in a certain volume fraction around the islands is impoverished of incorporation-ready material. For the case that the transport of material to the islands in the solution is too slow, material from the ripples will be dissolved [43]. This ripple template dissolution follows its strain distribution, which is shown in the 3D ﬁnite element (3D FE) model and plane-view TEM micrographs in Fig. 5.20. The areas around the islands are strained in compression with some ten MPa for the given misﬁt. These highly strained areas around the growing islands are dissolved. The driving force is the accompanying reduction of system strain energy. A more detailed description of the

Fig. 5.20. Comparison of strain-energy density values in the interface between a an islanded Si0.75 Ge0.25 layer and the Si substrate. The strain energy density at the substrate/island interface is displayed as calculated by 3D FE (a, b) and is compared to plan-view TEM diﬀraction-contrast images (c, d) with the electron beam parallel to the 001 direction for a single island (a, c) and a close arrangement of two islands (b, d) [50]

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calculation of the strained state of island and ripple assemblies is given in Sect. 5.3.3. Another nonequilibrium island shape obtained under conditions of extremely fast cooling is shown in the scanning electron micrograph in Fig. 5.21. This multifaceted island, terminated by alternating {113} and {110} facets and an {001} top facet has formed under a droplet of Bi solvent that was unintentionally not drawn away under the sliding boat of our LPE machine. Under this metallic solvent we expect very high cooling rates that indeed turn into a faceted island however with facets that deviate from the minimum surface energy {111} facets. It is interesting to note that this island is very similar in shape to some of the “domes” reported in MBE and CVD experiments [6, 8, 9, 11]. The Coverage β Versus Ge Content Figure 5.22a shows the island structure of two samples diﬀering in the Ge concentrations that have been grown with otherwise identical parameters, especially the growth time (t = 180 min). It is obvious that island number density and also the coverage are higher at higher Ge concentration. This observation holds for the whole growth process, see Fig. 5.22b, signaling that the island nucleation rate is higher. Such an observation is unexpected since nucleation models point to the same rates or even a smaller one at higher Ge concentration when the increased strain energy density due to the higher misﬁt is taken into account. The lack of further growth data makes attempts to clarify the reasons for this behavior premature. Instead we can show for

Fig. 5.21. Si0.75 Ge0.25 island at extremely high cooling rates. It was found under a solution droplet (Bi) that has been etched away with HNO3 solution from the grown layer: the nonequilibrium shaped island exhibits {113} and {110} side facets and an {001} top facet. The large diameter of the island is 730 nm. Scanning electron micrograph [19]

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Fig. 5.22. a Islands deposited at Ts = 700 ◦ C, ΔT /Δt = 0.056 K/min and for two diﬀerent Ge compositions in the SiGe alloy: xGe = 0.25, xGe = 0.4. AFM micrographs 10μm × 10μm. b Evaluation of the data in a plotted as coverage β vs. deposition time t. c Plot of the same data shown in b but now for: coverage β vs. “misﬁt modiﬁed time” tf . Experimental data in b and c is marked with squares and circles

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the available data set that the deposited volume of epitaxial material is the same at any time of growth for the two Ge concentrations. We start from Eq. 5.16 according to which the coverage β for a certain time t (t0 is in our case practically constant) is controlled by the term N˙ (t0 ) · F¯ , which is the product of the area covered by a single island, F , with the initial island nucleation number density per unit area, N˙ (t0 ). The larger this product, g, the higher the coverage and the steeper the slope of the β(t) curve for a certain growth time t. While N˙ (t0 ) depends in a complex way on thermodynamic parameters, F just depends on the misﬁt f [24], i.e., on the Ge content, according to F ∼ λy g1/f 4. The island volume is Vpyramid ∼ λz ∼ 1/f 6 (see Sect. 5.3.2). The deposited total volume Vtot consists then in the volume of a single island Vpyramid multiplied with the number of islands N (t) at a time t. The evaluation of Vtot for diﬀerent Ge contents as exempliﬁed in Fig. 5.22a for xGe = 0.25 and xGe = 0.40, but identical cooling interval ΔT , identical growth temperature Ts and identical growth time t yields 0.66 μm3 /μm2 and 0.63 μm3 /μm2 , i.e., roughly the same amount of epitaxial material is deposited. We thus can write: N (t, f1 ) · Vpyramid (f1 ) ≈ N (t, f2 ) · Vpyramid (f2 ) ≈ const

(5.15)

This equation states that N (t, f ) depends inversely proportional to Vpyramid and also N˙ (t, f ) (at least in the linear region of Eq. 5.16). Thus we can insert N˙ (t, f ) ∝ f 6 in Eq. 5.16. This results in: β(t) ≈ 1 − exp(−N˙ (t0 , f ) · F · (t − t0 ) = 1 − exp(−const · f 2 · (t − t0 )). (5.16) This result suggests to deﬁne a generalized time coordinate tf = f 2 ·t, a “misﬁt modiﬁed time”. In the respective replot of the data from Fig. 5.22b in Fig. 5.22c the two curves coincide, which supports the presented approach. Further work to clarify the physical meaning of this result is under way. 5.3.3 Strain and Composition Measurements and Simulations of Strain We get access to the strained state of the diﬀerent growth stages by locally probing and averaging methods such as convergent-beam electron diﬀraction (CBED) [25, 27] in the TEM, lattice displacement analysis in the highresolution (HR)TEM using the digital analysis of lattice images (DALI) software [51, 52], by micro-Raman spectroscopy [27, 53, 54] and by XRD methods [55]. All the experimental methods probe only certain components of the strained state and are best complemented by strain simulations with 3D ﬁnite elements. Details of the ﬁnite element simulation are shown in Sect. 5.3.1. In the following we present aspects of local strain probing by HRTEM and

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DALI simulations, and of average strain analysis over usually several thousand and even more islands by the XRD method. For CBED and Raman measurements of strain we refer to publications [25, 27, 53, 54]. In transmission electron microscopy, the classical diﬀraction-contrast method can be applied to visualize the strain ﬁeld in the surroundings of the islands, which can also support the analysis of the geometry of small islands based on image interpretation and appropriate theoretical structure models as calculated by means of molecular dynamics [56]. In order to analyze structure and composition of the islands on an atomic scale high-resolution HRTEM has to be applied. The interpretation of HRTEM images does also require image simulations owing to the complexity of both the scattering and the imaging process. New possibilities of structural investigations have been opened by spherical-aberration correction in TEM [57,58]. Additionally to the improvement of the resolution limit a drastic reduction of artifacts due to delocalization eﬀects can be attained. Another way of structure retrieval is the determination of the scattered wave function at the exit surface of the crystalline specimen by electron holography [59] or focus series reconstruction techniques [60]. Furthermore, various methods of quantitative HRTEM (qHRTEM) exist to measure the local strain and chemical composition on atomic scale [61]. The strain state can also be analyzed by large-angle convergent beam electron diﬀraction (LACBED). When (Si,Ge) islands are imaged in a TEM under diﬀraction-contrast conditions the general contrast behavior is essentially inﬂuenced by the strain ﬁeld in the island regions. It is mainly the strain-induced local distances between the particular lattice planes including their bending that leads to diﬀerences in the position and intensity of diﬀracted beams and thus to contrast pattern variations in the image. An example for such an analysis is shown in Fig. 5.23. The FE calculations concerning the strain tensor component εxx (z) in the (1-10) plane through the middle of a Si0.6 Ge0.4 island pyramid (130 nm basal width, 65 nm height) are displayed in Fig. 5.23a, a TEM cross section image in adequate projection in Fig. 5.23b. The calculated pattern shows mirror symmetry relative to the middle plane of the pyramid, where the color-coded contour levels of εxx (z) vary from approximately −8.7×10−3 (dark) to 1.3×10−2 (bright) on a linear scale. There is a rather good agreement between the simulated strain-ﬁeld distribution and the shapes of the thickness contours in the experimental image (caused by local changes in the excitation error). From Fig. 5.23a the following characteristic features can brieﬂy be concluded: –

The inner zone of the island is tetragonally distorted, i.e., the lattice is dilated in [001] direction and compressed along both 110 directions, where the degree of distortion depends on the position in the {110} planes.

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Fig. 5.23. Comparison of results of FE calculations and plan-view TEM diﬀractioncontrast images for a single Si1−x Gex island in cross section. a Strain tensor component εxx in the (-110) plane through the middle of the pyramid b Bright-ﬁeld TEM image of a single Si0.6 Ge0.4 island

– –

The (Si,Ge) lattice is fully relaxed in the region of the island apex and partly near the {111} side facets. At the edges of the island, where εxx (z) is negative, the crystal lattice is strongly compressed such that the local lattice constant is even smaller than that of silicon.

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Fig. 5.24. Structure model of an Si0.5 Ge0.5 island on Si (001) relaxed by molecular dynamics [64]

–

The strain ﬁeld caused by the island reaches into the substrate, hence below the middle of the island and in a certain distance to its edges the Si lattice is widened.

In the same manner, of course, plane-view TEM images of (Si,Ge)/Si islands can be compared to calculated strain-ﬁeld distributions. We would like to consider now an example, which utilizes structure modeling by the molecular dynamics (MD) method (see, e.g., [62, 63]). In these cases relaxation of an initially stiﬀ model of a lattice-mismatched (Si,Ge)/Si structure with respect to the intrinsic strain leads to results which permit to calculate TEM contrasts of islands with rather good similarity to the experimental ones. In Fig. 5.24 an example is given of a MD-relaxed structure model representing a truncated Si0.5 Ge0.5 pyramid of homogeneous composition directly placed on Si (001). The island has a square basis with a width of about 11 nm and its shape is given by {111} side facets and an (001) top layer, as observed. Owing to the relaxation process small deviations from the perfect crystal structure appear, e.g., in the interfacial region between pyramid bottom and Si substrate (see region marked by an arrow). Likewise, at the {111} side planes of the Si0.5 Ge0.5 lattice atomic rows are slightly bent due to the relaxation, too. Based on this structure model of a Si0.5 Ge0.5 pyramid diﬀraction-contrast image series were calculated by means of the EMS software package [65] for a thickness range of the complete (Si,Ge)/Si layer from 9.8 nm to 134.7 nm. The accelerating voltage is 200 kV, defocus value is Δf = 25 nm lying between the Gauss and the Scherzer focus. In principle, both the simulated as well as the experimental micrographs of (Si,Ge) islands exhibit a wide contrast variability due to diﬀerences in island size, the corresponding strain ﬁeld, sample thickness and orientation. Typical results are shown in Fig. 5.25. Despite the diﬀerences in the island size chosen in the model (11 nm basal width) and really present (about 130 nm in width) a relatively good agreement is found of the image-contrast simulations with the experimental ﬁndings. In each case, contrast features with fourfold symmetries are

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Fig. 5.25. Comparison of simulated diﬀraction-contrast TEM images (a1, b1, c1 ) of (Si,Ge) islands on Si with experimental images (a2, b2, c2 ) in plan-view for diﬀerent specimen thicknesses. The thickness values used for simulation are a1: 9.8 nm, b1: 48.9 nm, and c1: 87.9 nm. Basal width of the simulated islands 11 nm, of the imaged islands about 130 nm

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obtained, which additionally have more or less pronounced ﬁne structures depending on the speciﬁc specimen thickness. As it can be deduced from Fig. 5.25a, b, these dynamical contrasts do solely appear if the thickness exceeds a certain minimum value. Because of the partly rather good match of experimental and simulated diﬀraction-contrast TEM images it is obvious that the contrast behavior depends only little on the size, but strongly on the general shape of the 3D object including its faceting and the strain-ﬁeld distribution. A detailed analysis of HRTEM images using the DALI routine [51, 52] is carried out. DALI permits the determination of lattice strains. This routine comprises noise-ﬁltering, determination of the maxima of image intensity, generation of a net given by these maxima, deﬁnition of a reference lattice in an undisturbed region with known lattice constant, and calculation of the local displacements from the real lattice with respect to the reference

Fig. 5.26. Displacement analysis by digital analysis of lattice images (DALI) of the (001) lattice planes at the basal region of a (Si,Ge) island. The viewing direction in the high-resolution TEM (HRTEM) micrograph is parallel to the [110] edge of the islands base. This image is a gray-scale modiﬁcation of an originally color-coded one, which can be found in the online-version of [67]

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lattice. There are data available that demonstrate the potential of DALI studies to obtain information on structural and compositional changes as a result of thermal annealing [66] (will not be discussed here). Successful DALI analyses require a certain degree of homogeneity over the selected area, especially as is concerned homogeneous specimen thickness and composition. Too strong variations would lead to contrast artifacts like reversals or changes to completely diﬀerent black/white contrast features. Thus generally only a small fraction of highly resolved structural micrographs are accessible to such an analysis. The evaluation of the strain ﬁelds measured with DALI in terms of sources is straightforward when there is one physical source only. This holds for the strain ﬁeld caused by the misﬁt in the substrate below a pseudomorphic island. We shall discuss one example here. On the other hand such an analysis becomes complicated when additional sources for the strain come into play, as is the case for the islands themselves where concentration gradients can modify the misﬁt-induced strain ﬁelds (in addition to relaxation eﬀects due to free surfaces). One example will be discussed in the next chapter. Figure 5.26 shows an HRTEM image of the crystal volume adjacent to the interface between island and substrate near to one edge of the island [67]. The color-coded map of displacement u [001] parallel to [001] (i.e., parallel to the normal of the horizontal (001) lattice planes) within a selected area is superimposed to the image. The component u [001] increases. towards the region underneath the island. Hence, the (001) lattice planes are bent upwards in [001] direction. The lateral strain along [¯ 110], u [¯ 110], is analyzed. in the same area in Fig. 5.27. Close to the island edge the (220) lattice planes are compressed while expansion is found when getting closer to the center of the base of the island. This result is compatible with the general strain distribution as calculated by FEM, e.g., Fig. 5.23. A rather instructive example is presented in Fig. 5.28, where an island is seen along [010] direction, i.e., along the diagonal line in the base plane, cf. the sketch. The color coded map represents the displacement u [¯100] of the (100) lattice planes. This sketch of the island as seen from top also indicates schematically isostrain contours in the substrate, the blue square gives the base of island. Finite element calculations e.g., in Figs. 5.12 and 5.23 show that the maximum strains exist close to the edges of the basal planes [68]. This fact is nicely corroborated by the analysis in Fig. 5.28. Composition Analysis The analysis of a possible chemical composition gradient might serve to clarify the growth processes in more detail. Diﬀerent approaches are available which essentially vary in the spatial resolution and, of course, in the sensitivity. We describe selected examples in the following.

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The preceding chapter has demonstrated the DALI method that permits to analyze local lattice parameters. Local changes in the lattice parameter can also be due to variations in the chemical composition, which may happen in the GeSi islands where Si and Ge have diﬀerent atomic radii. A representative image of small Si0.5 Ge0.5 islands taken under HRTEM conditions is given in Fig. 5.29a. The individual island in Fig. 5.29b is free of dislocations and stacking faults, thus the strain distribution is solely due to elastic relaxation of the misﬁt. The coverage of the islands by an amorphous ﬁlm is due to glue used in the course of TEM sample preparation. The amorphous glue is partially removed during ﬁnal ion milling. The rectangular ﬁeld in Fig. 5.29b marks the area of analysis the results of which are shown in Fig. 5.30. Since the Si substrate could not be used because of its thickness the bottom of the island itself was used to measure any displacement of atomic columns. The ﬁeld of view selected in Fig. 5.29b is reproduced in Fig. 5.30a, the arrow diagram in Fig. 5.30b summarizes the

Fig. 5.27. Strain analysis of the (220) planes. This image is a gray-scale modiﬁcation of an originally color-coded one, which can be found in the online-version of [67]

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deviations of the positions of the atomic columns as measured with respect to the bottom reference lattice. The diagram Fig. 5.30d shows the displacement values averaged parallel to the substrate/island interface, i.e., along the traces of the individual (001) planes, plotted as a function of the number of lattice plane in growth direction. Four zones, I, II, III and IV, can be discriminated in growth direction, a ﬁrst thin one (zone I) that is the pyramid’s bottom followed by two transition regions II and III reaching to a fourth zone (IV) which begins at about 4/5 of the height of the rectangle. In general, both strain and composition contribute to the displacements. For a ﬁrst assessment, we can neglect the strain, since the highest strain energy density in an island is restricted to a thin layer near to the interface to the substrate, e.g., Fig. 5.12. Beyond, the strain energy density is rather small. In this part of the volume we can obtain information on the chemical composition, i.e., the Ge concentration. Under these conditions the Ge content is proportional to the slope of the dependency displayed in Fig. 5.30d. Consequently, in the ﬁrst zone (the reference lattice) with an extension of about

Fig. 5.28. Strain distribution within the set of (100) planes at the bottom region of a (Si,Ge) island. Projection is along the diagonal [010] of the island, see inserted sketch. For the color-coded original ﬁgure see the online version of [67]

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one tenth in growth direction, slope zero indicates a constant, but unknown Ge concentration. The regions II and III exhibit two diﬀerent slopes. They translate into respective Ge concentrations that increase stepwise from the bottom layer. The slope of about zero in zone IV can be judged to indicate that here the Ge concentration is the same as in the reference area (zone I). Absolute values for the Ge concentration can thus not be obtained. However, measurements with reference to the substrate lattice permit a quantiﬁcation of strain, with the error of the strained substrate below the island (e.g., Fig. 5.23a). We will now consider the evaluation of the Ge composition by X-ray diﬀuse scattering. Small changes in the Ge concentration x inside a SiGe island lead to very small eﬀects. Therefore, the main impact on the X-ray diﬀuse intensity is the strain ﬁeld which is changed by a chemical composition gradient and not the structure amplitude which is, in principle, also sensitive via the diﬀerent atomic form amplitudes of the diﬀerent atomic species. The strain distribution of an island of ﬁxed shape, size, and chemical composition coherently grown on a substrate is numerically determined by the ﬁnite element method. This is demonstrated in Fig. 5.31 for the example

Fig. 5.29. HRTEM imaging of Si0.5 Ge0.5 islands in cross section. a Overview. b Detailed image of an individual island. The marked rectangular ﬁeld is used for DALI analysis. Island width w ≈ 65 nm, island height h ≈ 30 nm

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of an Si1−x Gex truncated pyramid (λ* = 130 nm, A* = 65 nm) with constant Ge composition of x = 0.30. We integrate the calculated strain ﬁeld in the island into X-ray diﬀuse scattering simulations based on the kinematical scattering theory. The procedure is described in more detail in [69]. The results of these simulations are presented in Fig. 5.32a–c, which should be compared with the corresponding experimental data shown in Fig. 5.32d. Various Ge concentration proﬁles have been used. Let us ﬁrst focus on the diﬀuse scattering from a homogeneous SiGe island, i.e., a constant Ge content of x = 0.30 (Fig. 5.32a). A more detailed examination of these calculations shows that

Fig. 5.30. Analysis of displacement of atomic columns in the inner zone of a Si0.5 Ge0.5 island by DALI. a HRTEM image of a selected region of Fig. 5.29b. b Drawing of the evaluated displacement vectors in the (110) plane, c Corresponding two-dimensional plot of displacement values, d Displacement values along [001] averaged along each (001) plane as seen in part a

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Fig. 5.31. FE calculation for a Si0.70 Ge0.30 pyramid grown coherently on Si(001). a Elastic strain energy density in the (110) plane through the middle (1-10) plane of the pyramid. The contour levels vary from 0 (bright) to 2.5 × 107 Jm−3 (dark ) on a linear scale. The thin white lines represent the used FEM mesh. b Total strain tensor component xx in this (110) plane. The contour levels vary from −4.2 × 10−3 (dark ) to 1.4 × 10−2 (bright) on a linear scale [69]

the central peak (P1) arises from scattering from the upper half of the island, whereas the butterﬂy-shaped diﬀuse intensity P2 around P1 is due to diﬀuse scattering from sections of the island that are close to the island– substrate interface. One of the most important results is that the upper half of the island is strongly relaxed and exhibits near-cubic symmetry, whereas the lower half of the island is strongly distorted and thus gives rise to broad diﬀuse scattering. The two main features P1 and P2 are accompanied by weak thickness fringes that are due to the ﬁnite island height (h = 65 nm). The bending of the fringes indicates that they are sensitive to strain. A similar strain-induced deformation of characteristic features in reciprocal space has been found for quantum wire structures [70–72].

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Fig. 5.32. a–c X-ray diﬀuse scattering simulations in the vicinity of the 004 reciprocal-lattice point (q100 − q001 -plane) of an Si1−x Gex island with an island base width of λ* = 130 nm and height A* = 65 nm. The corresponding island models diﬀering in the Ge concentration proﬁle are shown on the left. Scattering A−1 ) has been omitted in the simulations. d from the Si substrate (q001 = 4.628 ˚ Experiment [55, 73]

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With regard to the results from a homogenous island, the agreement with the experiment can be improved by allowing a variation of the Ge composition inside the island, which is the only free ﬁtting parameter, as the island size and shape are well known from SEM and AFM. The simulations clearly prove that the Ge content at the top of the island is signiﬁcantly higher than in regions at the island base. Surprisingly, an abrupt change of the Ge content at one-third of the island height from x = 0.25 to x = 0.30 yields the best result (Model b, Fig. 5.32b), which is at least in qualitative agreement with the DALI analysis in Fig. 5.30 (where a diﬀerent nominal Ge concentration is used). Other simulations based either on a linear gradient, an abrupt change at one-half of the island height, or an inclusion with shallow {115} facets clearly fail [50, 73]. However, a simulation with a combination of a linear gradient and a constant Ge content (model c; Fig. 5.32c) yields a result that is very close to the model b. This result is important, since it resembles the result from DALI (see Fig. 5.30) and since – as we will discuss later – model c is supported by investigations based on energy-ﬁltered TEM and energydispersive X-ray spectroscopy (EDXS). How can we explain the observed inhomogeneous Ge distribution? First, it is energetically favorable that Ge is deposited at the top of the island where the lattice can relax more eﬃciently than at the island base (see Fig. 5.31). On the other hand, the strain energy density jumps to a comparatively low level at about one third of the island height, which supports the close relationship between the observed Ge distribution and the elastic strain energy density. The abrupt change of the Ge content can be further substantiated by early stages of growth [73], which can be observed directly by a special growth procedure. We have grown a layer at T = 600 ◦ C with a gradient of 0.1 K min−1 and a total growth time of 60 min. After growth, the supersaturation was stopped and the sample was kept at a constant temperature of T = 594 ◦ C for 18 hours. A variety of islands diﬀering in size and shape could be observed that represent diﬀerent stages of island growth. Figure 5.33a shows the initial stages of growth where, as indicated by the height proﬁles, the islands develop from the ripple pattern (proﬁle 1) to ﬂat islands (proﬁles 2 and 3) with a ﬁxed square base whose size is of the same order as the ripple pattern wavelength. The side faces of these islands exhibit discrete angles and the largest angles (around 16◦ ) that are observed correspond to {115} facets. At that stage of growth top facets have not yet formed. As soon as {115} facets have developed, the islands quickly change to ﬂat truncated pyramids with {111} side facets, see Fig. 5.33b. However, the island height is still close to the values observed for the {115} islands (Fig. 5.33b). After the {111}-side facets have evolved, the growth proceeds mainly in the vertical [001] direction while the island base width is constant or at least increases only slightly (Fig. 5.33c). It is just this growth stage, which occurs with an increased Ge concentration, as modeled before. (It may be interesting to mention that also decreas-

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Fig. 5.33. A variety of SiGe islands diﬀering in size and shape can be observed by AFM under special growth conditions (for details see text). a Early stages of growth: constant island base width, evolution of side facets. b Intermediate stage: at a critical height {111} side facets have developed. c Final stages of growth: vertical growth of islands with slightly increasing base widths [73]

ing facet angles that could be called “inverse” have been observed during growth [74, 75]. Here, the shape evolution is determined by strain-driven alloying.) In this context, it is noteworthy to mention that it is still not clear, why the SiGe islands exhibit a limited and constant size, i.e., why there is no Ostwald ripening [9, 76–79]. This question does not arise only for SiGe islands but is important for any strained system that is grown in the Stranski–Krastanow growth mode. Kinetic limitations could be responsible for that behavior. For example, recent studies [13, 17] have shown that a kinetic energy barrier for the MBE growth on the island facets can lead to self-limiting growth, i.e., the growth rate is decreasing with increasing island size. It is also very likely that the depletion of the wetting layer plays a central role in these kinetic considerations [80]. However, in LPE the island growth is governed mainly by vertical mass transport from the solvent towards the layer and not by surface

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diﬀusion. From that point of view, a substantial inﬂuence of the wetting layer depletion on growth kinetics is not expected. Also energetics might potentially govern the island size distribution. Shchukin et al. [81] have tried to explain the limited size of the islands by taking into account not only the elastic strain relaxation and the surface tension but also the elastic interaction of the facet edges of the island. According to this model, there will be a minimum in the energy of the island as a function of the island volume, and a narrow size distribution is favored [82]. The chemical composition of the islands can be directly determined by means of energy-dispersive X-ray spectroscopy (EDXS), for details see [83] and energy-ﬁltered TEM (EFTEM). (Electron energy loss spectrometry is an alternative, however respective results will not be discussed here. For details see [84]). Figure 5.34 shows typical results of EDXS line-proﬁle analyses obtained from Si0.6 Ge0.4 islands. The X-ray counts of the Si-Kα and Ge-Kα lines were recorded at 35 points along a straight line through the middle plane of the pyramid (cf. scanning transmission electron microscopy (STEM) dark-ﬁeld image in Fig. 5.34a). The resulting qualitative curves are mainly aﬀected by the specimen thickness including the speciﬁc geometry of the pyramid. Correction for the thickness gradient yields spatially resolved quantitative Ge and Si concentration proﬁles as given in Fig. 5.34c. Because of this thickness inﬂuence, the quantitative element distribution is quite different from the pure intensity proﬁles in Fig. 5.34b. Beginning at the substrate/pyramid interface, which is according to the ﬁnite width of the sudden intensity changes somewhat smeared-out, it exhibits that the Ge content shows a steady increase to an almost constant value at about two third of the pyramid’s height, where the absolute concentration of about 60 atom% Ge is reached. This value appears too high in view of the nominal Ge concentration of 40%. One reason might be that quantiﬁcation was performed by use of theoretical Cliﬀ –Lorimer factors since no reliable standard materials of known Si/Ge composition were available. In any case, the measured gradient of the Ge concentration indicates that strain-driven Ge segregation occurs in growth direction during island growth owing to the diﬀerence in the lattice parameters between Si substrate and Si1−x Gex layer. In addition, both the qualitative and the quantitative Ge line-proﬁles show a small jump in intensity or concentration, respectively, at approximately one third of the height of the island as inferred from the X-ray diﬀraction experiments (see Fig. 5.32). These EDXS analyses have been complemented with energy-ﬁltered TEM where the element distribution in the island region is mapped two-dimensionally in images taken in the respective characteristic loss energy window. For a specimen of homogeneous thickness the local brightness in an electronspectroscopic image, for example taken with the characteristic ionization edges Ge-L23 or Si-K, is a direct measure of the element concentration. Unfortunately, because of the above-mentioned thickness variation this state-

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ment does not hold for the (Si,Ge)/Si samples under investigation. Nevertheless, EFTEM imaging can give information about the local content of an element when means are applied to correct for the thickness inﬂuence. The procedure is demonstrated in Fig. 5.35 which shows the Ge distribution in the region of a (Si,Ge) island with nominally 40 atom% Ge. In the

Fig. 5.34. Element distribution in the region of a single Si0.6 Ge0.4 island as revealed by energy-dispersive X-ray spectroscopy (EDXS) line-proﬁle analysis. a STEM dark-ﬁeld image. b Qualitative line proﬁle. c Corresponding quantitative proﬁle (from top to bottom)

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Ge-L23 map (ionization energy of about 1217 eV) the approximately 175 nm high island is clearly observable with bright contrast. In this image the signal intensity is modulated by both the Ge concentration and the local thickness (cf. Fig. 5.35a). This behavior can also be seen in the corresponding intensity proﬁle of Fig. 5.35c (taken from the rectangular ﬁeld marked in the Ge-L23 map). This proﬁle of the net Ge-L23 counts is very similar to the

Fig. 5.35. Element distribution in the region of a single Si0.6 Ge0.4 island as revealed by energy-ﬁltered TEM. a Ge-L23 map. b Pre-edge image. c, d Corresponding line proﬁles. e Thickness-corrected Ge proﬁle

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qualitative EDXS line proﬁle given in Fig. 5.35b. Elimination of the thickness modulation is possible, when this image intensity is divided by a Ge-L23 pre-edge image whose signal intensity represents predominantly the specimen thickness. The resulting thickness-corrected proﬁle (Fig. 5.35e) exhibits a continuous rise of the Ge content from the bottom of the pyramid along the growth direction. In the topmost region the composition seems to be constant. In summary, it may be said that both analytical TEM techniques, i.e., EDXS and EFTEM, reveal a compositional gradient from the substrate/island interface to middle regions in form of an increase of the Ge/Si ratio. Therefore, the widening of the crystal lattice found by diﬀerent kinds of structure analysis can be attributed to this chemical inhomogeneity. However, the DALI result of identical Ge concentrations in the regions near the interface to the Si substrate and in the topmost regions are identical (see Fig. 5.30) is obviously at variance. Hence, the same value of the evaluated (001) plane distances might be accidental and the topmost layer might be elastically completely relaxed. 5.3.4 Positional Ordering Template-Mediated Ordering Ripples as Template We have already considered in Sect. 5.2.2 the role of the ripples as a template for the islands that form in a later stage of growth on their crests. In this way the potential sites of islands is predetermined and the island arrangement is dominated by the regular ripple structure. In consequence the islands form a rather regular lattice, in which the islands align along the elastically soft 100 directions of the substrate, i.e., at 45◦ to their 110 base edges. Thus, an impression of a face-centered lattice is generated. The analysis of these relationships was facilitated by the use of time resolved specimen that exhibit diﬀerent growth stages under identical growth conditions. One observation important for this template mediated ordering is that the amplitude of the ripples decreases with the misﬁt and that at high misﬁt (well beyond 15%) they are nondiscernible. This result can indicate that a diﬀerent mechanism for the regular island arrangement exists and one might be the formation of linear island arrays in the initial stage of islanding. This aspect is considered below, eg. Fig. 5.42. Diﬀerent Classes of Islands During ongoing growth there will be an instance where the surface is covered by equally sized islands, the base width λ∗ and interdistance w∗ of

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which are given by the 1/f 2 dependency (Eq. 5.2, f : misﬁt parameter). Further growth requires nucleation of further islands, which can only occur between the existing islands. Large-ﬁeld-of-view micrographs of such samples are shown in Fig. 5.36a, b that in fact display a bimodal island size distribution. Figure 5.36b shows a magniﬁcation of this bimodal island size distribution. The islands are appropriately framed. Now the face centered arrangement has changed into a simple cubic arrangement, when the two diﬀerent island

Fig. 5.36. a Islands at an earlier (low coverage) and later (high coverage) growth stage on one sample. At the low coverage larger {111}-faceted islands reside periodically arranged along elastically soft 100 directions. The island edges lie parallel to the 110 directions. At higher coverage a bimodal island size distribution occurs. A second set of smaller {111}-faceted islands forms that ﬁll the empty places between four of the larger islands. b A magniﬁcation of this “bimodal size” growth stage with appropriately framed islands [19, 85]

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sizes are neglected. The larger islands belong to the initial self-assembled arrangement of islands (stage of shorter growth time t), the smaller islands have formed at the later stage. Their smaller size corresponds to the fact that they sit in between the larger ones at which site due to the compressive strain ﬁelds by the initial islands the strain is increased beyond the one described 2 law the respective by the (geometrical) misﬁt. According to the λ ∼ 1/feﬀ islands become smaller in width. A model of this arrangement has been used to calculate the strain ﬁelds and the energy densities [85], see Fig. 5.37. The composition Si0.75 Ge0.25 is used, the aspect ratio is λ∗ /A = 2.5. The diﬀerences in system energies are as follows: for the face centered lattice Fig. 5.37a (left), the average strain energy density in the layer is 7 × 107 J/m3 and in the substrate 3 × 106 J/m3 . For the square lattice (Fig. 5.37a at right) it is 1 × 108 J/m3 and in the substrate 3.5 × 106 J/m3 thus appreciably higher in the islands. Island–Island Correlations Quantitative Analysis of Island Ordering To quantify spatial island correlation eﬀects we perform Fourier transformations of AFM micrographs (power density spectrum). Owing to its insensitivity to strain, grazing incidence small angle X-ray scattering (GISAXS) represents another tool that is particularly well suited for the quantitative analysis of spatial correlation eﬀects. With GISAXS we realize probe sizes of about 0.1 mm × 10 mm. Depending on the island coverage and island size we thus get correlation information from about 104 –107 islands, i.e., the data exhibit high statistical relevance. In the following AFM micrographs and corresponding GISAXS intensity pattern are displayed for varying island area coverage β.

Fig. 5.37. 3D FE calculation of the average strain energy density (between zero and 2 × 108 J/m3 ) in island assemblies. Left Face-centered island lattice, right quadratic island lattice [85]

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The GISAXS patterns show distinct peaks, which are caused by positional ordering (correlation). From the positions q of these correlation peaks we are able to calculate the mean island spacings, d, via: d=

2π , q

(5.17)

where q denotes the scattering vector. The line widths δq yield information about the corresponding correlation lengths L via the Scherrer equation: L=

2π . δq

(5.18)

This quantitative analysis is performed for a series of samples exhibiting approximately the same island size, diﬀering only in the island area coverage β. At very low island density (Fig. 5.38) the islands seem to be randomly distributed, despite the occurrence of island dimers and linear trimers. However, a ring-shaped intensity pattern appears in the GISAXS distribution, which indicates that there is a constant, isotropic nearest-neighbor distance. The ensemble of islands behaves like a two-dimensional (2D) ﬂuid or glass [86] rather than a 2D gas. Evaluation of the peak spacings leads to a mean island–island distance of about d = 1000 nm, which is deﬁnitely larger than the interisland distance within a dimer. Consequently, GISAXS is sensitive to the mean distance between clusters rather than to the distances between the islands within a dimer.

Fig. 5.38. a AFM micrograph of a sample with low island area coverage (β= 0.02, x = 0.30,λ* = 150 nm, A* = 75 nm). b Corresponding grazing incidence small angle X-ray scattering (GISAXS) in-plane intensity distribution. The grayscale varies linearly from bright (low intensity) to dark (high intensity) [87]

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When the island coverage is increased (see Fig. 5.39), the island dimers and trimers develop into chains (rows) oriented along 100. However, at suﬃciently low coverage, as in this case, the island chains themselves are short and still not correlated to each other, i.e., they are independent of each other. Therefore, each island can be unambiguously assigned to a single chain. This aspect can be checked in Fig. 5.39. The corresponding diﬀuse intensity now exhibits correlation peaks along the 100 directions; however, the correlation peaks are smeared out with respect to the 100 directions. This result can be understood if we account for the coexistence of two characteristic length scales: the island distances within a row seem to be well deﬁned (d = 195 nm), whereas the mean chain–chain distance is deﬁnitely larger and shows a rather broad statistical distribution (d = 250 . . . 500 nm). Surprisingly, although they are very weak, correlation peaks along the 110 directions are also observed in the intensity pattern. When the island coverage is further increased the interisland distances are now so small that the gaps are partly ﬁlled by slightly smaller islands

Fig. 5.39. a AFM micrograph of a sample with medium island area coverage (β = 0.20, x = 0.30, λ* = 130 nm, A* = 65 nm). b Corresponding AFM in-plane power spectrum. c GISAXS in-plane intensity distribution. Correlation peaks are observed in both the 100 and 110 directions

Fig. 5.40. a AFM micrograph of a sample with high island area coverage (β = 0.48, x = 0.30, λ* = 150 nm, A* = 75 nm). b Corresponding AFM in-plane power spectrum. c GISAXS in-plane intensity distribution. Correlation peaks are observed in both the 100 and 110 directions [87]

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(Fig. 5.40). This process locally changes the morphology from a face-centered square symmetry to a primitive square symmetry as discussed before. However, owing to the reduced island size homogeneity, the GISAXS intensity pattern does not exhibit exact square symmetry anymore (d100 : d110 = 1.21 < 21/2 ), unlike that observed for medium island coverage. To summarize, both experiments, Fourier analysis from AFM data and GISAXS, yield correlation lengths of islands. These measurements permit to assess the degree of self-assembly, i.e., the degree of periodicity on the substrate surface depending on the direction on the sample surface. While for the low coverage the correlation length L is larger in the 100 direction (L100 = 780 nm) and smaller in the 110 direction (L110 = 540 nm) it is reversed for the high coverage sample. There L100 = 360 nm in the 100 direction and L110 = 510 nm in the 110 direction. The Autocorrelation Function It is interesting to mention that the correlation length L as deﬁned in Eq. (5.18) does not represent an adequate measure of the correlation properties. These properties are fully represented in the convolution product, C(r), of the island positions. We have direct experimental access to this correlation function by simple Fourier transformation of the measured diﬀuse intensity distribution [88]. In the following we will discuss this procedure and subsequent conclusions. In addition to the X-ray diﬀuse scattering data in GISAXS geometry and the respective atomic force micrographs, the island-island correlation is also visible in the X-ray diﬀuse scattering in high-resolution X-ray diﬀraction (HRXRD) geometry in the form of satellite peaks. These peaks are clearly resolved in the vicinity of the 00λ truncation rod (see Fig. 5.32d). A respecA−1 is shown tive section of Fig. 5.32d in 100 direction at q001 = 4.560 ˚ in Fig. 5.41a. It is striking that many orders of correlation peaks are visi-

Fig. 5.41. HRXRD from Si0.70 Ge0.30 islands with medium island coverage (β = 0.20) in the vicinity of the 004 reciprocal lattice point. a Linear section at q001 = 4.560˚ A−1 through experimental intensity distribution shown in Fig. 5.35d. b Corresponding autocorrelation function C(r) in real space [88]

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ble which disperse slightly outwards with increasing horizontal momentum transfer q100 . The occurrence of high order satellite peaks is not necessarily a consequence of very good spatial correlation, i.e., of large correlation lengths. Rather, they may be generated by an asymmetric line proﬁle in the real-space autocorrelation function C(r), which is the Fourier transform of the X-ray diﬀuse intensity. This is indeed the case as can be seen from Fig. 5.41b. The correlation function C(r) exhibits only ﬁrst order peaks (labeled as C−1 and C1 in Fig. 5.41b), indicating the presence of short-range ordering. The peak positions of C±1 can be evaluated as r = ±266 nm, indicating a somewhat smaller mean island distance as evaluated from the 1st order satellite peak positions in Fig. 5.44a which give r = ±292 nm. This deviation is caused by the asymmetric line proﬁles of C±1 : while the peak positions are given by r = ±266 nm, the respective “centers of mass” of C±1 are located slightly more outwards and their positions agree well with r = ±292 nm. Both values for r are marked in Fig. 5.41b as dashed lines. Island chain does not fall below values of about r ≈ 220 nm, although the island base diagonal width would allow values down to r = 180 nm. On the other hand much larger distances than d100 = 266 nm are possible: in AFM images, distances up to about 400 nm are observed, that correspond to the distance between adjacent island chains. As a consequence of this behavior, an asymmetric distribution of the island–island distances is induced, and the proﬁle of the ﬁrst order peak in the autocorrelation function becomes asymmetric. The values r = 220 nm (minimum island distance) and r = 400 nm (maximum island distance) observed in the atomic force micrographs is also found in the experimental autocorrelation function (Fig. 5.41b). At these r values the correlation vanishes. There is thus direct agreement between AFM and corresponding X-ray results. Evolution of Island Rows At comparatively high Ge concentrations (x > 15%), i.e., at high lattice misﬁts f , very regular arrays of SiGe islands can be observed (e.g., Fig. 5.14), although surface rippling cannot be detected. A closer inspection of the surface morphology reveals extended island chains (rows) oriented along the 100 directions. It is very interesting to track the evolution of these island rows as a function of island coverage [89]. At a very low island density there is already a large fraction of islands that are clustered. These clusters consist mainly of island dimers, with a small fraction of linear three-island trimers that are oriented along the 100 directions (Fig. 5.42, curve a). When the island coverage increases (Fig. 5.42, curves b–d) the dimers and trimers develop into extended rows of islands aligned along 100. The observed behavior clearly proves that the island rows are independently formed: there are no crossing points between the rows, and each island

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Fig. 5.42. Statistical distribution of rows (along 100 directions) of self-organized Si0.70 Ge0.30 islands ( λ∗ = 130 nm, A = 65 nm). The growth times were varied between diﬀerent areas of the same sample (“time resolved growth”), resulting in diﬀerent island area coverages of β = 0.04 (a), β = 0.07 (b), β = 0.11 (c), and β = 0.18 (d ) [89]

can be unambiguously assigned to a single row. It is also striking that the mean distance between the islands within a single row is very well deﬁned, while there is a remarkable ﬂuctuation of the distances between neighboring rows. A statistical evaluation of the observed length of the island (Fig. 5.42) allows drawing three important conclusions: 1. The number of isolated islands normalized to the total number of islands decreases with increasing coverage. 2. With higher island coverage β the maximum length of the rows increases. 3. The mean length of the rows increases with higher island coverage. The observed clustering is a clear indication of short-range ordering and suggests that during the growth the nucleation of an island is strongly inﬂuenced by already existing islands. One possible explanation is that the extended strain ﬁeld related to a single island induces additional centers of nucleation, which are preferentially located along the elastically soft direction of the island base diagonal whereas along the hard 110 directions no such centers are induced. Indeed, the elastic strain energy density (Fig. 5.14a, b) favors island nucleation along the 100 directions. However, the evolution of clusters can ultimately not be explained in this way, since the elastic strain energy density around a single island does not exhibit any energy minimum along

100, nor any local maximum along the 110 direction, which can lead to an averaged ﬂux of adatoms into the elastically soft direction. However, several papers [90, 91] report on the phenomenon that the strain en-

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Fig. 5.43. a AFM micrograph of Si0.75 Ge0.25 /Si(001) islands at low island overage (area coverage β = 0.02, excluding the wetting layer). A high percentage of islands is arranged in dimers and trimers oriented along 100 (some of which are marked by circles). b Results of a kinetic Monte Carlo simulation (T = 650 K, growth rate 0.01 monolayers/s, coverage c = 0.05, after 200 s growth interruption) [92]

ergy around a Stranski–Krastanow island can overcompensate the driving forces of growth and even can cause a removal of deposited material around an island. Indeed, also in our case of LPE SiGe islands, wetting layer depletion has been found [26, 47], where the depleted area follows exactly the symmetry of strain energy density in the wetting layer. The depletion of the wetting layer impedes nucleation in the immediate vicinity of the island. The evolution of island dimers and longer rows can be simulated by kinetic Monte Carlo (KMC) growth simulations as shown by Meixner et al. [92] for the case of MBE growth. A direct KMC growth simulation is not yet feasible in case of LPE, where the mass transport is given by diﬀusion in the solvent and not by surface diﬀusion as in the case of MBE. In fact, even for MBE, KMC methods cannot treat the 3D island growth of arrays of fully developed islands. However, with appropriately chosen growth parameters and a selfconsistently included anisotropic strain ﬁeld, KMC methods can simulate the initial stages of growth close to equilibrium conditions until platelets of islands arise [93]. Figure 5.43b displays results of a respective KMC calculation at low island coverage [92]. The simulation was performed close to equilibrium conditions, leading to a qualitative agreement with experimental observations on LPE grown SiGe islands (see Fig. 5.43a) in that island dimers and trimers are also present in the simulations. However, at high island coverage the experimental island chains exhibit a higher degree of ordering. A poorer spatial ordering, which is obtained by KMC analysis can be attributed to large ﬂuctuations that occur at a ﬁnite simulation time. The approach of equilibrium is driven by local variations of the strain ﬁeld. The equilibrium state is reached by small

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strain-induced diﬀerences in the diﬀusion constant. The closer the system is to the equilibrium state, the smaller is the driving forces and thus approach to equilibrium. Thus, perfect ordering can also be expected as a result from KMC simulations, however, calculations within reasonable times require huge computational power. Reversibility and Stability Of Growth Stages Thus far we have considered self-organized structures as they form during growth under low driving forces, i.e., near thermodynamic equilibrium, to a certain stage of growth. Experimentally we could show that, independent of the growth conditions and of the misﬁt, at the beginning of growth ripples form due to an instability [19, 20], which can be described in terms of a linear instability phenomenon (characteristics: exponential increase of amplitude) [28–31], see Sects. 5.3.2 and 5.3.2. These ripples represent a template that imposes its regular structure onto the islands that form later, however, by thermal activation [19], Sect. 5.3.2. In this stage of coexistence of ripples and islands, the ripple amplitude stays constant, which can in principle be understood in terms of a weakly nonlinear stability analysis [19, 20, 39], see also Fig. 5.10. In the discussion below of the stability of the island structure we have also to resort to the nonlinear instability. The question remains open whether at very high misﬁt ripples still exist as a template or whether a diﬀerent mechanism ensures regularity of the island arrangement, such as preferential formation of island rows, as discussed before (Sect. 5.3.4). The near equilibrium growth now oﬀers the possibility to further conﬁrm the equilibrium nature of the growth structures by appropriate experiments made possible by the liquid-phase epitaxy (LPE). We shall brieﬂy consider results obtained by dissolution, equilibration and annealing experiments. Dissolution Experiments Figure 5.2 shows the temperature proﬁle of a dissolution experiment. The amount of dissolved material is proportional to the heating interval ΔT . The negative ΔT reduces the eﬀective cooling interval ΔT . The corresponding removal of already crystallized material makes it is possible to see whether earlier growth stages are recovered which is a signature of equilibrium structures. Especially the transition from ripples to islands is interesting in this respect. In fact, as demonstrated in Fig. 5.44 we recover the ripples (Fig. 5.44b) when we dissolve the islands (Fig. 5.44a), which indicates reversibility of this transition and the ripple template stage. The TEM plane view micrographs in Fig. 5.45 show that the dissolution process of the islands is a statistical process, island dissolution occurs thermally activated. In Fig. 5.45a still a few islands are visible, in Fig. 5.45b only faint remnants of islands are present at

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this stage of dissolution. Figure 5.46 shows the recovery of the ripple template for the low misﬁt regime. As a result, independent of the misﬁt, the system traces the growth sequence “ﬁrst form ripples and then islands” in the reverse direction thus proving the process to be reversible and part of an energetic pathway of growth. Equilibration Experiments Figure 5.2 shows the temperature proﬁle of an equilibration experiment, in which the growth is stopped and the system is kept at the respective end temperature with the melt still on the epitaxial layer. Such experiments have been conducted for the very early ripple stage. Since the growth times in this case are rather short we used very low growth temperatures (600 ◦ C). Such experiments permit to assess the equilibrium ripple conﬁguration independently of cooling rates ΔT /Δt. This way very regular ripples form with a wavelength strictly following the λ ∼ 1/f 2 rule (Eq. 5.2). We show here the

Fig. 5.44. Si0.75 Ge0.25 . a Deposited at 550 ◦ C at a cooling interval of ΔT = 10 K. b Dissolution at a heating interval of ΔT 1 = 1 K. AFM micrographs

Fig. 5.45. TEM micrographs of a sample after dissolution (as described in Fig. 5.43): islands at low density are visible (square shaped features) in addition to wavy bumps that remain from dissolved islands. a Lower magniﬁcation. b Higher magniﬁcation. The incident electron beam is parallel to the 100 direction

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eﬀect of such an experiment on the island arrangement, see Fig. 5.47. It is evident that a very regular arrangement of equally sized islands has formed. This arrangement is stable in time (see the extraordinarily extended equilibration time of two days) and can only be understood in terms of a nonlinear stability phenomenon, which prevents Ostwald ripening. It is interesting to note that a face centered lattice formed. As stated in Sect. 5.3.4, further growth in such a lattice would lead to smaller islands between the existing islands, the structure then turns simple cubic (e.g., Fig. 5.36). Equilibration of such a structure results in dissolution of this second set of islands. This dissolution is an experimental proof that this particular island square lattice is not an equilibrium conﬁguration. Annealing Experiments An annealing experiment in the present context is an equilibration experiment with the melt removed. Such experiments have been carried out at 900 ◦ C and 980 ◦ C (30, 60, 90 min.) in N2 atmosphere for specimens with high island coverages β. We will summarize the major ﬁndings and show the AFM micrographs in Fig. 5.48 as an example. Annealing reduces the island density drastically, the more the higher the annealing temperatures and annealing time. In Fig. 5.48 (left) a high bimodal island density exists after growth with islands arranged in roughly a square lattice. After anneal the smaller island population has disappeared leaving behind a face-centered lattice, admittedly with a certain degree of disorder. In any case, these annealing experiments yield results that are compatible with those from dissolution and equilibration experiments. Especially the unimodal face centered island lattice is a stable low energy self-assembled structure.

Fig. 5.46. Si0.95 Ge0.05 island dissolution at a heating interval of ΔT 1 = 1 K. AFM micrograph

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Fig. 5.47. a AFM micrograph of Si0.95 Ge0.05 /Si(001), deposited at a cooling rate of 0.166 K/min with subsequent equilibration under the saturated solution at growth temperature for two days. b Fourier transform of the micrograph shown in a

Fig. 5.48. Atomic force micrographs (2 × 2μm) of Si0.67 Ge0.33 /Si deposited at a cooling rate of 0.166 K/min for 30 min at 550 ◦ C. Left Bimodal island distribution showing a square lattice as explained in Fig. 5.19 right after annealing at 980 ◦ C in N2 atmosphere for 30 min; the smaller islands have disappeared resulting in a unimodal island distribution of face centered type

5.3.5 Diﬀusive and Plastic Strain Relaxation Although beyond the present text that treats elastic relaxation and resulting self-assembling phenomena, we would like, for completion, to brieﬂy comment on diﬀusive and plastic relaxation. The elastic relaxation during the epitaxial pseudomorphic growth leads immediately to a growth surface with locally varying strain, i.e., varying lattice parameter. Examples are displayed in Figs. 5.11 and 5.12. This varying strain modiﬁes locally the equation generally used for the incorporation rate: R(r) ∝ Nad · ν · exp( −ΔG kT ),

(5.19)

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where R(r) is the adatom incorporation rate, Nad is the adatom density at the surface, ν is the attempt frequency for incorporation, ΔG is the activation energy for adatom incorporation. The modiﬁcation for strain appears essentially in the activation energy ΔG, which thus varies locally due to an appropriate correction term. For the present purpose we can use the mechanistic notion that the incorporation rate of Si and Ge is maximum when a local composition is formed that has the same lattice parameter as the strained surface. In consequence an inhomogeneous Ge concentration develops which causes corresponding diﬀusional ﬂows within the grown crystal (also across the interface between substrate and overgrowth) driven in addition to the concentration gradients also by the inhomogeneous strain ﬁeld. We call the resulting change in the strain ﬁeld diﬀusive relaxation. This diﬀusive relaxation has been predicted [94], however, it is diﬃcult to measure since segregation of a few % Ge in nanosized islands under conditions of inhomogeneous strain has to be determined (cf. Sect. 5.3.3). A respective ﬁrst analysis by transmission electron microscopy can be found in [95]. Plastic relaxation of strain by formation of misﬁt dislocations at the interface between substrate and overgrowth is a very eﬃcient process, however, also a very deleterious one to optoelectronic properties. Thus in most cases of semiconductor epitaxy the critical parameters at which dislocations form have to be avoided in growth and processing and a whole wealth of respective work on their determination is available. For continuous epitaxial layer growth we have to consider two critical thicknesses. The ﬁrst one is the critical thickness for dislocation glide, hc , which is usually formulated for a planar heteroepitaxial layer [33]. Since such layers undergo elastic relaxation with ripple and island formation, dislocations may glide already at a considerably smaller (eﬀective) thickness due to the stress enhancement in these structures. SiGe on Si is a fortuitous case since Si is initially free of dislocations and before any glide of misﬁt dislocations can occur dislocations have to be nucleated. This nucleation process has a high activation energy and is thus kinetically impeded so that dislocation-free layers can be grown with a thickness well above the critical thickness hc . Such layers, however, are intrinsically unstable against misﬁt dislocation formation, especially during processing, and thus of limited use for devices. The second critical thickness, Hcr , ([28–31]) is that for roughening due to ripple formation since such surfaces are in most cases unwanted in electronics. The larger the misﬁt, the smaller is the thickness of a continuous layer that can be grown with a planar surface. A practical limit is a Ge content of 25% (corresponding to a misﬁt of 1%). Higher Ge contents result in rough surfaces at layer thicknesses smaller than the usually used ones in heteroepitaxial device manufacturing. In consequence both critical thicknesses have generally to be considered with regard to potential applications of the layers. The SiGe/Si system is rather benign since a well-deﬁned pseudomorphic growth regime exists due to the comparably low misﬁt.

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5.4 Conclusions We have assembled a summarizing view on experimental results that have been obtained in the last decade by studies of the self-organized structure evolutions in lattice-misﬁtted epitaxy. The special aspect arises from the fact that the utilized method, solution growth or LPE, inherently operates with a very low driving force for growth and thus can be considered to operate near thermodynamic equilibrium. The analysis of the pseudomorphic growth in the system Si1−x Gex /Si(001) shows especially that two modes of elastic relaxation of the misﬁt strain occur, a ﬁrst stage consisting in a criss-cross of undulations of the surface which is a true thickness modulation of the epitaxial layer. A second mode is the formation of islands that in the present case are characterized by a shape of a truncated four-sided pyramid with very low energy {111} side faces. The evaluation of time and temperature dependencies of the evolution of these structures indicates that rippling behaves according to linear stability theory, and that islands form by thermal activation. In this context one has also to mention the new insight that the regular rippling structure serves as template for the forming islands thus assuring a very regular arrangement of islands at later stages of growth. We have presented direct evidence for this result for the lower range of Ge composition (xGe < 0.15). Although regular island arrangements form also at high Ge composition, this templating eﬀect could not be supported by experiment in this range, very probably because the rippling stage is too short to be seen: islanding starts already at an epitaxial layer thickness of a few atomic layers. In this growth stage observations have been put forward that indicate correlated island nucleation in form of linear island rows along the elastically soft

100 directions in the (001) growth surface. The unique possibility oﬀered by LPE of equilibration (keeping the liquid on the epitaxial layer at constant temperature) and of dissolution (increasing the temperature) of a grown islanded layer has shed new light onto the physical nature of such arrangements. These experiments have shown that the island arrangement is inherently stable against Ostwald ripening, which was completely unexpected. Moreover, the arrangement even tends to assume a very equidistant island distribution. This geometric arrangement can be understood based on the stress distribution in islands and substrate as calculated by the ﬁnite-element method in the framework of linear elasticity theory. In addition, we have shown that the stability of the island arrangement against Ostwald ripening can in principle be understood in terms of a nonlinear stability analysis, where, however, the responsible physical interaction processes still remain to be identiﬁed. In this article we have presented extensive investigations substantiating the claim that LPE is a near-thermodynamic-equilibrium growth technique due to its low driving forces. These investigations permit to formulate a master equation that describes the borderline between energetically and kinetically controlled growth. The crossing of this border is manifested by the

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fact that less material (in volume) is deposited in a given temperature interval than is possible in the energetically controlled range. The kinetically controlled growth regime may cause more complicated island shapes to appear many of which exhibit facets diﬀerent from {111}, like {115}, {113}, or {110}. Looking this way onto the energetic–kinetic border it becomes clear that it is deﬁned by the time the system has available during the complete growth experiment to reach the lowest energy state given by strain and surface energies. In view of the results presented in this article, we have to presume that during growth of misﬁtting layers by growth methods with generally rather high driving forces and/or limited diﬀusion and incorporation rates, i.e., different from LPE, the arising self-organized structures, their reactions, interactions and shape changes are kinetically controlled and stabilized, though contributing to the reduction in Gibbs energy. Evaluation of such structures can thus be supported by a comparison with the results from LPE. Acknowledgement. The authors are indebted to many colleagues. M. Albrecht, M. Becker, A.-K. Gerlitzke, M. Hanke, J. Michler, and A.-T. Tham, who accompanied our work at least for certain periods in the past, have been of indispensable help in collecting, analyzing, and interpreting experimental data. We also owe thanks to those colleagues who always have been prepared to share their knowledge with us in discussions and who supported our work in various other ways: S. Besedine, U. G¨ osele, D. Grigoriev, R. K¨ ohler, R. Otto, H. Raidt, P. Sch¨ afer, K. Scheerschmidt, and Th. Wiebach. Moreover we thank A. Rosenauer for providing the DALI software package and we kindly acknowledge technical support from O. Seeck and W. Drube (HASYLAB/DESY Hamburg). The presented work has been supported within the framework of SFB 296 and the DFG projects KO 1510/2, STR 277/14, and WA 1453/1.

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6 Ge Quantum Dot Self-Alignment on Vicinal Substrates I. Berbezier, A. Ronda, and A. Karmous L2MP UMR CNRS 6137, Polytech’Marseille - Technopole de Chˆ ateau Gombert 13451 Marseille Cedex 20, France

6.1 Introduction In this chapter we propose a two-step growth process consisting of speciﬁc self-nanopatterning and subsequent self-assembly of Ge dots. During the ﬁrst step, a nanostructured template layer is obtained by growth, which serves to order the Ge islands during the second step. Growth instabilities that develop during epitaxy of Si and Si1−x Gex on silicon substrates produce large-scale, highly corrugated, periodic and reproducible morphologies [1–8]. Such instabilities could be eﬃciently used as templates for subsequent ordering of Ge islands [9–13] without the need of sophisticated lithographic tools. To address this issue, perfect control and characterization of the morphological evolution of surface instabilities is needed. At the least, it is necessary to determine the optimized experimental conditions in order to develop speciﬁc patterns. Theoretical models used to describe stress-driven instability [14–21] present large discrepancies with experimental results. For instance, they neither explain the morphological evolution of the instability with the substrate orientation [7] nor the opposite eﬀects of stresses of opposite sign are explained by the models. Also, morphological evolutions of the instabilities are still under debate, probably because kinetics (surface diﬀusion) [22, 23] and atomistic parameters (step edge energy), should also play a major role in these evolutions [24]. Moreover, the driving force of Ge island nucleation is not fully determined. Based on scanning tunneling microscopy (STM) observations, random nucleation [25] and preferential nucleation at step edges [26] or in between dimer vacancy lines [27] have been suggested. So far, the relative roles of stress, surface energy, surface diﬀusion . . . have not been accurately quantiﬁed. Nevertheless, it has been shown that 3D Ge islands preferentially nucleate along the edges of mesas [28] and on dislocation lines [29]. In these two cases, it was suggested that stress gradients were mainly responsible for 3D island nucleation, but Eggleston and Voorhees et al [30] have reported other driving forces too, such as surface energy anisotropy. The aim of this section is to determine the role of thermodynamic and kinetic parameters (stress gradients, chemistry, step edges and surface energy . . .) on the ordering of Ge island.

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In the ﬁrst part, we fabricate well-controlled patterns with sizable features. Some examples of spontaneous substrate nanopatterning created by periodic instabilities that develop during Si or Si1−x Gex growth are presented. Particular attention is paid to the morphological evolution of instabilities as a function of experimental conditions (growth temperature, substrate, misorientation and misﬁt). In the second part, subsequent ordering of Ge islands on self-nanopatterned template layers is analyzed and the eﬃciency of various driving forces deduced. The results highlight the dominant role of stress, chemistry and surface energy and a surprisingly remarkable ordering obtained on a Si1−x Gex undulated template layer demo. In this case, Ge dots align perpendicularly to the step edges, on the top of the undulations.

6.2 Experimental The samples were grown in a Riber molecular beam epitaxy (MBE) chamber with a base pressure in the 10 × 10−11 torr range. Si and Ge were evaporated from an electron beam evaporator and an eﬀusion cell, respectively. The deposition rate of Si was maintained at a constant of about 0.03 nm/s. Si1−x Gex alloyed layers with x between 0 and 0.3 were deposited either on nominal or on vicinal (001) and (111) surfaces. Vicinal (001) and (111) surfaces were misoriented in the [110] and the [-1-12] or [11-2] directions, respectively, with a miscut angle, θ, varying between 0◦ and 10◦ . In-situ cleaning of the substrate consisted of a two-temperature process (830 ◦ C/30 min and 1230 ◦ C/2 min), which resulted in a sharp 2D reﬂection high-energy electron diﬀraction (RHEED) pattern, indicative of a clean surface. Prior to the growth of the Si1−x Gex layers, a 50-nm thick Si buﬀer layer was systematically grown at 750 ◦ C to achieve a reproducible ﬂat surface. Morphological characterization of the samples was performed by atomic force microscopy (AFM) operating in air and by transmission electron microscopy (TEM) cross-sectional observations.

6.3 Results 6.3.1 Si/Si Instabilities First, we consider the epitaxy of Si/Si (001) vicinal substrates (i.e., growth without stress). It has already been observed that step-bunching instability develops on Si substrates at a critical temperature [2, 31–33]. Nice periodic undulations perpendicular to the miscut direction (i.e., parallel to the step edges) were observed for the miscut angles investigated (0.5◦ < θ < 10◦ ). The periodicity (∼250 nm on the example presented) of the patterns

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created can be appreciated in the AFM image of Fig. 6.1a. Systematic investigation of amplitude (A) and correlation function (λ) as a function of growth temperature (T g) and deposited thickness (h) was performed. The results evidence a very narrow temperature regime of the instability, which coincides with the growth mode transition between 2D islands nucleation and step ﬂow. Kinetic Monte Carlo simulations [34], have shown that the instability originates from the anisotropy of diﬀusion barrier-induced by the (2 × 1 + 1 × 2) surface reconstruction. Time-dependent evolution of the undulation was ﬁtted by power laws (λα t α and A α t β ) with values of the exponents α ∼ 0.3 and β ∼ 0.53. (Fig. 6.1c, d). TEM crosssectional observation (not shown) of the layer revealed an asymmetric periodic corrugation, which consists of a succession of ﬂat terraces and step bunches. It can be anticipated that step bunches could be favorable nucleation sites for 3D islands, because ﬁrst, adatoms attachment is easier at a step edge and second, the elastic energy of the system (sub-

Fig. 6.1. Some characteristics of the kinetic growth instability which develops during epitaxy of Si on Si (001) 1.5◦ oﬀ: a Atomic force microscopy (AFM) image (scan size 5 × 5 μm2 ) of 500 nm Si deposited at T g = 450 ◦ C, b morphological evolution as a function of T g, c and d give the time dependent evolution of the correlation length and of the root mean square roughness (RMS ), respectively. Data points are linked by dashed lines while the continuous line is the ﬁt

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strate/wetting layer/island) is reduced when islands are set on a train of steps [34]. We turn now to the epitaxial growth of Si/Si (111) vicinal with misorientations around [1-10] towards the [11-2] or [-1-12] directions. In the ﬁrst case (we take the example of a misorientation of 10◦ towards [11-2]), the starting surface consists of regular train of triple-layer steps (Fig. 6.2a). During Si growth (or high temperature annealing) the surface breaks down into large {111} facets and {112} sawtooth facets (Fig. 6.2b). This behavior is attributed to a faceting instability which has been well described in the literature [35]. It is reasonable to assume that {112} nanofaceted surfaces [36] are favorable nucleation sites because of the presence of multiple stepped subunits which form this orientation. The sawtooth shape of the periodic instability can be well appreciated in the 3D AFM view of Fig. 6.2c. In the second case (we take the example of a misoriention of 10◦ towards the [-1-12] direction), the starting surface consists of a regular train of single-layer steps. During Si growth, a periodic step-bunching instability develops, as can be seen on Fig. 6.3a. In this case, alternation of areas with high step- and low step-densities can be observed. Such a morphology should not be confused with the faceting instability morphology described above (absence of facets). The temperature range of the instability regime

Fig. 6.2. Transmission electron microscopy (TEM) cross section images of Si(111) misoriented 10◦ towards [11-2] a before growth and b after growth (magniﬁcation is ×106 ). c is a 3D AFM view of the instability (scan size 5 × 5 μm2 ) after deposition of 500 nm

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is between 650 and 800 ◦ C (Fig. 6.3b) with a maximum amplitude at about 750 ◦ C. At lower and at higher temperatures, the instability vanishes and growth recovers the normal step ﬂow mode. The same behavior was also observed in the case of a substrate misoriented 1.5◦ towards [-1-12]. The temperature of instability onset coincides with the gradual destabilization of 7 × 7 surface reconstruction at the expense of the 1 × 1, which leads to the coexistence of the two reconstructions. We suggest that the step-bunching instability observed here could be attributed to the diﬀerence of surface diffusion barriers on 7 × 7 and 1 × 1 reconstructed areas. The time-dependent evolution of the surface morphology has been investigated. Variations of λ and A with the deposited thickness (h) were ﬁtted by power laws with critical exponents α ∼ 0.23 and β ∼ 0.5 respectively (Fig. 6.3c and d respectively). Kinetic Monte Carlo simulations of this instability regime are in progress.

Fig. 6.3. Typical characteristics of the kinetic growth instability which develops during epitaxy of Si on Si (111) 10◦ oﬀ to [-1-12]: a AFM image (scan size 5 × 5 μm2 ) of 500 nm Si deposited at T g =700 ◦ C; b morphological evolution as a function of T g; c and d give the time-dependent evolution of correlation length and amplitude respectively. Data points are linked by dashed lines while the continuous line is the ﬁt

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6.3.2 SiGe/Si Instabilities In this part we focus on the instability that develops during the growth of Si1−x Gex (with low Ge concentrations) on nominal and misoriented Si(001) substrates. Although it is outside the scope of this chapter to discuss the origin of the instability (see [32] for more details), it is necessary to quantitatively determine its morphological evolution as a function of experimental parameters in order to control and optimize the pattern features created for the subsequent ordering of Ge dots. First of all, it should be recalled that the Si1−x Gex instability is not a step-bunching instability as already shown in [37]. Indeed, it was found that the periodic undulations formed during the growth of Si1−x Gex on Si misoriented substrate lie on the train of single-layer steps on vicinal Si(001) substrates [37] perpendicularly to the step edges, with their sides consisting of (105) facets [33]. Examples of patterns developed after the growth of 10 nm Si1−x Gex (x = 0.35) on vicinal 1.5◦ oﬀ and 10◦ oﬀ Si(001) are presented in Fig. 6.4. At low miscut angles (< 2◦ ), undulations have rectangular bases aligned along the two {010} directions (Fig. 6.4a), while at higher miscut angles undulations transform into long wires (Fig. 6.4b) perpendicular to the step edges. This eﬀect is explained by the stabilization of (105) facets along the undulation sides which become parallel to the misorientation direction for a miscut angle of 8◦ . However, even if the miscut angle of the substrate has a large eﬀect on the undulation shape, it inﬂuences neither the correlation length (λ) nor the root mean square roughness (RMS ) of the instability. Time-dependent evolution of surface morphology has been measured during the deposition of Si1−x Gex (x = 0.25) on 1.5◦ oﬀ vicinal substrate. The results show that λ is almost constant during the deposition (Fig. 6.5a) while RMS increases linearly with h up to a deposited thickness h ∼ 100 nm (Fig. 6.5b). For h > 100 nm, the RMS has abruptly changed due to the formation of misﬁt dislocations, which are evidenced by the presence of cross-hatch patterns. We have also measured the morphological evolution of Si1−x Gex surfaces as a function of the Ge concentration x (or misﬁt stress m) for h ∼ 50 nm (Fig. 6.5c, d). The results show that λ is inversely proportional to m while the RMS increases proportionally with m. An abrupt transition of the RMS is observed when dislocations are formed (at m ≥ 1.15%) due to the appearance of the cross-hatch pattern (not investigated here). It should be added that the misﬁt range tested was very narrow because of experimental limitations: absence of instability at low misﬁt and formation of dislocations at large misﬁt. However, by adjusting the deposited thickness for larger Ge concentrations of the deposited layer, a periodicity of λ ∼ 25 nm was obtained for x = 0.5 and h = 5 nm. This result is in good agreement with the value extrapolated from the ﬁt of the data given in Fig. 6.5c. In conclusion, the morphological features of the kinetic stress-driven instability that develops during the growth of Si1−x Gex can be controlled by

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Fig. 6.4. Surface morphology of 10 nm Si1−x Gex layers (x ∼ 0.35) deposited on vicinal Si(001): a 1.5◦ oﬀ and b 10◦ oﬀ. Scan size is 5 × 5 μm2

Fig. 6.5. Morphological evolution of Si1−x Gex layers deposited on Si (001) 1.5◦ oﬀ as a function of the deposited thickness (h) in a and b and of the Ge concentration (x) in c and d. a, c give the evolution of the correlation length (λ). b, d give the evolution of the RMS

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the Ge concentration x, the deposited thickness and the misorientation angle of the substrate: λ decreases with x, RMS increases with h and x and the shape of undulations is determined by the miscut angle. 6.3.3 Self-Assembling We now investigate at the self-assembling of Ge dots on self-nanopatterned template layers with the morphology described above. Eight monolayers MLs of Ge were systematically deposited on the patterned surfaces at a growth temperature Tg ∼ 600 ◦ C. In a ﬁrst example, we used the periodic undulation of the kinetic stepbunching instability described in Fig. 6.1a. Patterns were created during the growth of 500 nm Si on Si (001) 1.5◦ oﬀ substrate at 450 ◦ C. They consisted of periodic undulations with a low slope (periodicity ∼ 250 nm and amplitude ∼ 2 nm). In this case, a preferential nucleation of 3D Ge islands along the step bunches could be expected. Figure 6.6a shows that the dome-like Ge islands, are not perfectly aligned along the surface undulations, even if preferential formation of some islands along the undulation sides, on the step bunches can be observed (Fig. 6.6b). Under the same experimental conditions, a very nice alignment of Ge islands is obtained on the faceting instability which develops on Si(111) misoriented 1.5◦ towards [11-2] as described in Fig. 6.2 and also on the kinetic instability which develops on Si(111) misoriented 1.5◦ towards [-1-12] as described in Fig. 6.3. In the latter case, one can see that most of the step bunches are decorated by long chains of Ge islands (Fig. 6.6c). The perfect alignment of triangular shaped Ge islands along step edges can be better appreciated at higher magniﬁcation (Fig. 6.6d). These series of experiments show that (111) step edges have a considerable eﬀect on 3D island nucleation while, in contrast, (001) steps have a very small eﬀect. These diﬀerences can be attributed to the diﬀerent step height and step edge energy of these two surfaces. In a second example, we used the kinetic stress-driven instability-induced patterning obtained during the deposition of Si1−x Gex (x = 0.35, h= 10 nm) on (001) Si 1.5◦ oﬀ (as presented in Fig. 6.4a). In that case, the patterns consist of a 2D square lattice of undulations. By deposition of Ge on such patterns we obtain 2D dense square arrays of homogeneous dome-like Ge islands (Fig. 6.7a). One can see that Ge islands align perfectly on the top of the rectangular base patterns created by the instability. Since stress relaxation is assumed to be larger on the top of the undulation than in the bottom, we then deduce that ordering of Ge islands is induced by stress relaxation. In order to conﬁrm this result we have used the same kinetic-stress-driven instability but on a (001)Si substrate misoriented by 10◦ . In that case, unidirectional periodic undulations are formed (see Fig. 6.4b). It is clearly visible, that Ge islands are again perfectly aligned along the undulations (Fig. 6.7b); careful observation of AFM images at lower scale reveals that Ge dots, of

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Fig. 6.6. AFM images of 8 ML Ge (T g 600 ◦ C) deposited on instability-induced patterns developed during the growth of 500 nm Si on: a Si(001) misoriented 10◦ and c Si(111) misoriented 10◦ towards [-1-12]. Scan size is 5 × 5 μm2 . b, d Enlarged 3D views of a and c respectively. Scan size is 3 × 3 μm2

both dome-like and hut islands types, are grown on the top of the undulations (Fig. 6.7c, d), which are perpendicular to the regular train of monolayer steps of the substrate, separated by 0.8-nm-wide terraces (due to the miscut). It can then be deduced that the ML steps of the substrate have no eﬀect on the ordering of the Ge dots. On the contrary, it is reasonable to suggest that stress relaxation induced by the presence of underlying ML steps and by the undulation, is at the origin of Ge dots’ ordering. The eﬀect of preferential nucleation of Ge islands due to stress relaxation (along the cross-hatch pattern [29] or on the edges of narrow holes [38]) has already been evidenced in the literature. If we concentrate now on the formation of islands on these patterned substrates, it is clear that they ﬁrst form by ampliﬁcation of the undulation and then evolve by shape transition into islands. A systematic description

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Fig. 6.7. AFM images of 8 ML Ge (T g 600 ◦ C) deposited on instability-induced patterns developed during the growth of 10 nm SiGe (x = 0.35) on vicinal Si(001): a 1.5◦ oﬀ and b 10◦ oﬀ. c Enlarged 3D view of b. Scan size is 3 × 3 μm2 . d is a 3D AFM image of Ge deposited using the same growth conditions than the deposit exhibited in b and c, but with a lower Ge thickness (h ∼ 6 ML). Scan size is 1.5 × 1.5 μm2

of the islands’ morphology and microstructure is given elsewhere [37]. For both huts and domes islands, the size is not aﬀected by the template layer wavelength but by energetic (balance between surface energy and elastic energy) and kinetic eﬀects (surface diﬀusion of adatoms). The combination of these eﬀects imposes the formation of (105) facets that leads to elongated ﬂat islands with large lateral size (on misoriented substrates). In order to change the island size, kinetic and energetic parameters have to be modiﬁed. For this purpose we make use of surfactant mediated growth (SMG) by Sb. In Fig. 6.8 we show the eﬀect of surfactant coverage on the morphology of Ge deposits. In this set of samples the template layer was obtained by depositing 15 nm of Ge0.3 Si0.7 . This results in an average wavelength of ∼ 200 nm.

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From Fig. 6.8a to Fig. 6.8c the Sb coverage corresponds to θ = 0, θ = 0.5, θ = 1. Diﬀerent observations can be deduced from this set of experiments. Ordering of Ge islands is achieved when the size of the islands matches the half wavelength of the patterned layer (Fig. 6.8a). In this sample, without Sb deposition, large islands (130 nm diameter) are observed. A dramatic reduction in island size is evidenced due to the predeposition of Sb. For the sample covered by 0.5 ML (Fig. 6.8b) an island size of ∼50 nm is achieved, but poor ordering is observed. No islands nucleated on the sample covered by 1 ML Sb (Fig. 6.8c). The inhibition of the 2D–3D transition induced by Sb surfactant mediated growth is fully discussed in [37]. In summary, partial Sb coverage and reduction of the wavelength are both necessary to obtain ordered ultrasmall Ge islands. In the rest of this section we will consider the morphology of Ge islands grown on a template layer Ge0.5 Si0.5 (h ∼ 3 nm) with an average wavelength λ ∼ 90 nm. In Fig. 6.9 we have investigated the eﬀect of Sb coverage on the small wavelength patterns. In Fig. 6.9a, with small Sb predeposition (θ ∼ 0.1) large Ge islands are visible. Their diameter (∼ 120 nm) is much larger than the ripple wavelength which prevents their ordering. In Fig. 6.9b no islands are present, since the 1 ML Sb coverage fully inhibits the 2D–3D transition. In Fig. 6.9c, at intermediate coverage (θ ∼ 0.5), islands align along the ripples, forming closely packed chains. The average island size is 35 nm. These islands are fully strained and they do not present any visible facet, in contrast to hut islands that exhibit (105) facets when obtained during Ge/Si(001) heteroepitaxy (see [39]). However the SMG does not improve the ordering of Ge islands. Indeed, island chains obtained here have a lower ordering degree (Fig. 6.9e) than those in Fig. 6.7. We interpret this result as a consequence of the nucleation of Ge islands on the Sb-free areas, which are distributed randomly below 1 ML coverage. In order to organize the Ge nucleation sites during SMG we make use of preferential desorption of Sb from the step edges.

Fig. 6.8. AFM images of 8 ML Ge deposited on the θ ML Sb/15 nm Si0.7 Ge0.3 /Si(100), 10◦ oﬀ at 600 ◦ C. a θ = 0 b θ = 0.5 ML. c θ = 1 ML. Note: during Sb deposition, the sample is kept below 400 ◦ C, to prevent Sb desorption. Scan sizes are 1.5 × 1.5 μm2

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Fig. 6.9. 5 μm × 3.2 μm2 AFM images of Ge deposited on θ Mono Layer (ML) Sb/Ge0.5 Si0.5 3 nm thick. a θ ∼ 0.1: large Ge islands (120 nm diameter) are visible. b θ ∼ 1: no islands are present, since the 1 ML Sb coverage inhibits the 2D–3D transition c, d θ ∼ 0.5: diﬀerent procedures have been used to get this surfactant thickness. In c the 0.5 ML coverage was obtained by direct deposition, while in d it was obtained by controlled partial desorption of 1 ML. e, f Zooms of c and d respectively with scan size (1.5 × 0.9 μm2 )

The sample in Fig. 6.9d was obtained by controlled partial desorption from 1 ML Sb resulting in Sb coverage ∼ 0.5. In these experimental conditions, island chains align much better either on the top of the ripples or in the channels.

6.4 Conclusion We have shown that by understanding and controlling the morphological evolution of Si/Si and (Si1−x Gex )/Si growth instabilities as a function of ex-

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perimental parameters, it is possible to create instability-induced patterns of scalable periodicity and amplitude. In particular, kinetic step-bunching instability and faceting instability, which both develop during the growth of Si on vicinal (111) and (001) Si substrates, were investigated. Typical patterns developed on such instability were presented. Patterns which develop on kinetic-stress-driven instability, during the growth of Si1−x Gex on vicinal Si(001), were also evidenced. We have shown that the degree of ordering of Ge dots depend on the characteristics of substrate prepatterning. In particular, we have shown that (001) steps (MLs as well as bunches) have a small inﬂuence on 3D islands nucleation. In contrast, almost perfect ordering on kinetic-stress-driven instability is attributed to the stress relaxation eﬀect. By combining the use of prepatterned Si1−x Gex template layers deposited on misoriented Si(001) surfaces and Sb SMG of Ge we have successfully ordered ultrasmall Ge islands. In this process, which is based only on growth steps, the relevant experimental parameters are the Ge concentration of the Si1−x Gex alloy, which controls the ripple wavelength, and the Sb coverage, which controls the island size.

References 1. A.G. Cullis, MRS Bull. 21, 21 (1996) 2. C. Schelling, G. Springholz, F. Sch¨ aﬄer, Phys. Rev. Lett. 83, 995 (1999) 3. D.E. Jesson, K.M. Chen, S.J. Pennycook, T. Thundat, R.J. Warmack, Phys. Rev. Lett. 77, 1330 (1996) 4. P. Sutter, M.G. Lagally, Phys. Rev. Lett. 84, 4637 (2000) 5. R.M. Tromp, F.M. Ross, M.C. Reuter, Phys. Rev. Lett. 84, 4641 (2000) 6. I. Berbezier, B. Gallas, L. Lapena, J. Fernez, J. Derrien, B. Joyce, J. Vac. Sci. & Technol. B 16, 1582 (1998) 7. I. Berbezier, B. Gallas, A. Ronda, J. Derrien, Surf. Sci. 412–413, 415 (1998) 8. A. Ronda, M. Abdallah, J.M. Gay, J. Stettner, I. Berbezier, Appl. Surf. Sci. 162–163, 576 (2000) 9. K. Sakamoto, H. Matsuhata, M.O. Tanner, D. Wang, K.L. Wang, Thin Solid Films 321, 55 (1998) 10. J. Zhu, K. Brunner, G. Abstreiter, Appl. Phys. Lett. 73, 620 (1998) 11. M. Abdallah, I. Berbezier, P. Dawson, M. Serpentini, G. Bremond, B. Joyce, Thin Solid Films 336, 256 (1998); I. Berbezier, M. Abdallah, A. Ronda, G. Bremond, Mater. Sci. Eng. B 69–70, 367 (2000) 12. H. Omi, T. Ogino, Thin Solid Films 369, 88 (2000) 13. C. Teichert, J.C. Bean, M.G. Lagally, Appl. Phys. A 67, 675 (1998) 14. H. Gao, J. Mech. Phys. Solids 42, 741 (1994) 15. R.J. Asaro, W.A. Tiller, Metall. Trans. 3, 1789 (1972) 16. M. Grinfeld, Soviet Phys. Dokl. 31, 831 (1986) 17. D.J. Srolovitz, Acta Metall. 37, 621 (1989) 18. A. Pidduck, D. Robbins, A. Cullis, W.Y. Leong, A. Pitt, Thin Solid Films 222, 78 (1992) 19. A. Oral, R. Ellialtioglu, Surf. Sci. 323, 295 (1995)

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20. C. Duport, P. Nozi`eres, J. Villain, Phys. Rev. Lett. 74, 134 (1995) 21. F. L´eonard, J. Tersoﬀ, Appl. Phys. Lett. 83, 72 (2003) 22. W. Barvosa-Carter, M.J. Aziz, L.J. Gray, T. Kaplan, Phys. Rev. Lett. 81, 1445 (1998) 23. B.J. Spencer, P.W. Voorhees, J. Tersf, Phys. Rev. B 64 235318 (2001) 24. V.B. Shenoy, L.B. Freund, J. Mech. Phys. Solids 50, 1817 (2002) 25. B. Voigtler, T. Weber, P. Smilauer, D.E. Wolf, Phys. Rev. Lett. 78, 2164 (1997) 26. N. Motta, J. Phys. Cond. Matter 14, 8353 (2002) 27. P. Sutter, I. Schick, W. Ernst, E. Sutter, Phys. Rev. Lett. 91, 176102 (2003) 28. L. Vescan, T. Stoica, J. Appl. Phys. 91, 10119 (2002) 29. C. Teichert, C. Her, K. Lyutovich, M. Bauer, E. Kasper, Thin Solid Films 380, 25 (2000) 30. J.J. Eggleston, P.W. Voorhees, Appl. Phys. Lett. 80, 306 (2002) 31. M. Ladeveze, I. Berbezier, Arnaud F. D’avitaya, Surf. Sci. 352–354, 797 (1996) 32. I. Berbezier, A. Ronda, A. Portavoce, J. Phys. Cond. Matter 14, 8283 (2002) 33. A. Ronda, I. Berbezier, Physica E 23, 370 (2004) 34. A. Pascale, I. Berbezier, A. Ronda, P. Kelires, submitted (2007) 35. F.K. Men, F. Liu, P.J. Wang, C.H. Chen, D.L. Cheng, J.L. Lin, F.J. Himpsel, Phys. Rev. Lett. 88, 096105 (2002) 36. A. Baski, S.C. Erwin, L.J. Whitman, Surf. Sci. 392, 69 (1997) 37. I. Berbezier, A. Ronda, F. Volpi, A. Portavoce, Surf. Sci. 531, 231 (2003) 38. A. Karmous, A. Cuenat, A. Ronda, I. Berbezier, S. Atha, R. Hull, Appl. Phys. Lett. 85, 6401 (2004) 39. A. Portavoce, I. Berbezier, A. Ronda Phys. Rev. B 69, 155416 (2004)

7 Lateral Arrangement of Ge Self-Assembled Quantum Dots on a Partially Relaxed Six Ge1−x Buﬀer Layer Hyung-jun Kim, Ya-Hong Xie, and Kang L. Wang Department of Electrical Engineering, University of California Los Angeles, USA

7.1 Introduction Since the observation of pyramidal Ge islands formed on Si (001) by epitaxy in the early 1990s, there has been substantial interest in fabricating dense and uniform arrays of self-assembled Ge quantum dots. This chapter is dedicated to the review of one of several methods used to fabricate laterally ordered arrays of semiconductor structures with 3D quantum conﬁnement, namely, epitaxial growth on partially relaxed SiGe buﬀer layers. A convenient technique for creating self-assembled quantum dots is the growth of a strained epitaxial layer to create small islands in a wide range of lattice-mismatched material systems, where the substrate material has a larger band gap than the epitaxial layer. The epitaxy of quantum dots uses the transition from the two-dimensional (2D) layer to three-dimensional (3D) island growth that takes place during the deposition of a pseudomorphically strained epitaxial layer. Heteroepitaxial growth of highly strained structures has attracted interest lately because it oﬀers the possibility of fabricating islands with very narrow size distributions without the need for any substrate patterning or surface treatment. Quantum dot formation has been observed for a wide range of material/substrate combinations, including InAs/GaAs, InGaAs/GaAs, InP/GaAs, SiGe/Si, and Ge/Si. Among all these diﬀerent material combinations, Ge/Si represents the simplest system. There have been a great number of approaches to fabricating semiconductor quantum dot arrays of uniform size and shape with regular spatial distribution. In order to achieve this goal, it is essential to advance understanding of the fundamental formation mechanisms of Ge quantum dots. To date, several formation mechanisms still remain unclear. The issue of thermodynamics (see Chap. 1) is complicated by the following eﬀects: interdiﬀusion between quantum dots and substrate resulting in nonuniform composition of individual quantum dots, nonuniform distribution of quantum dots across the substrate surface, local strain ﬁelds at subsurfaces, surface energies associated with the facets of diﬀerently shaped quantum dots, and the possible dot-size-dependent energy conﬁguration. On the other hand, the prevailing morphology from a particular fabrication process could be limited extensively by kinetics. The key elements inﬂuencing the kinetic process are the surface diﬀusion of adatoms during the growth of the wetting layer (2D) and quantum

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dots (3D), adatom density, dot nucleation density, and Ostwald ripening (also known as coarsening). All these parameters are determined by experimental variables, such as substrate temperature, growth rate, and cleanliness of the substrate surface. The ability to decouple the eﬀects of each of these experimental variables will allow us to clearly understand the formation mechanism of Ge quantum dots. For example, a high substrate temperature during dot growth not only enhances the surface diﬀusion of Ge adatoms, but also increases the rate of interdiﬀusion, leading to a higher Si concentration in the Ge quantum dots.

7.2 Ge Self-Assembled Quantum Dots on a Partially Relaxed SixGe1−x Buﬀer Layer We describe the lateral arrangement of Ge dots when a partially relaxed Six Ge1−x buﬀer layer is used as a template for nucleation. The buried misﬁt dislocations located at the interface between the buﬀer layer and the Si (001) substrate are eﬀective for preferential nucleation. Figure 7.1 shows a schematic illustration of the buﬀer layer structure, consisting of a thick Six Ge1−x and a thin Si capping layer, for the preferential nucleation of Ge A-thick Si cap layer dots. A 600 – 800 ˚ A-thick Six Ge1−x buﬀer layer and a 100 ˚

Fig. 7.1. Schematic drawing of a relaxed template consisting of Six Ge1−x and Si capping layers. The resulting dislocation network forms at the interface between the relaxed Six Ge1−x buﬀer layer and Si substrate. Thicker and thinner lines indicate misﬁt dislocations and the resulting undulated strain ﬁelds, respectively

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were grown at 550 ◦ C and 600 ◦ C, respectively. The Six Ge1−x buﬀer layer was almost completely strained as grown. Subsequently, the samples underwent a post-growth anneal at 700 ◦ C for 30 min that led to the partial strain relaxation of the Six Ge1−x buﬀer layer via dislocation generation. The purpose of the Si cap layer under tensile strain is to preserve a ﬂat surface due to the smaller lattice constants than that of the Six Ge1−x layer. The resulting samples consist of an undulating strain ﬁeld with a relatively ﬂat top surface of Si atoms. The surface of the relaxed buﬀer layer has typical roughness of ∼9.9 ˚ A root-mean square roughness (RMS )[according to atomic force microscopy (AFM)]. The buried dislocation density can be controlled by the Ge concentration of the relaxed Six Ge1−x buﬀer layer. Figure 7.2 represents 2D AFM image topography of Ge dots grown on a partially relaxed Si0.75 Ge0.25 buﬀer layer. All Ge dots exclusively nucleate over the buried dislocations and form orthogonal arrays along 110 directions because the average dislocation spacing is as small as the size of the dot. Assuming that the buried dislocations are single ones, the strain of the Si0.75 Ge0.25 buﬀer layer is relaxed by approximately 10%. The measured average dislocation spacing is ∼5000 ˚ A while a 100% relaxed Si0.75 Ge0.25 buﬀer layer corresponds to the spacing of ∼500 ˚ A. Most of the literature, in contrast, reports randomly nucleated Ge dots when they are grown directly on bulk Si (001) substrates. Surface diffusion is a dominant factor determining the size and spatial distributions of Ge dots. Figure 7.3 shows the corresponding plan-view TEM image representing the ordering of Ge dots along two sets of the buried dislocations in the relaxed Si0.85 Ge0.15 buﬀer layer. Each array of dots is not only clearly associated with a single buried dislocation, as opposed to dislocation pile-ups, but is also observed on one side of the buried misﬁt dislocations. In spite of the higher dot density in the sample, the TEM image of low magniﬁcation exhibits only relatively big dots such as 200 ˚ A-tall dots. Projections of Burgers vectors onto the interface plane are marked with arrows. There are two possible

Fig. 7.2. Two-dimensional (2D) atomic force microscopy (AFM) image topography of Ge self-assembled quantum dots (SAQDs) grown on a partially relaxed Si0.75 Ge0.25 buﬀer layer

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Fig. 7.3. A plan-view transmission electron microscopy (TEM) micrograph of the sample with Ge quantum dots grown on the relaxed Si0.85 Ge0.15 buﬀer layer is shown [16]

slip planes possessing two of each possible Burgers vectors for a 60◦ misﬁt dislocation line. The 60◦ dislocation lying in the [110] direction can have either (111) or (111) slip planes. Two possible Burgers vector directions of [011] and [101] and another two of [111] and [011] correspond to the (111) and (111) planes, respectively. Therefore the Burgers vector analysis allows us to ﬁnd the corresponding slip plane of the dislocation. Ross showed the strain ﬁeld variation of the top surface due to buried dislocations using 2D linear strain analysis [1]. The lowest energy sites occur at the intersection between the slip plane and the top surface where islands preferentially nucleate. The average distance (650 ± 13 ˚ A) between the dislocation line and the array of Ge dots, in Fig. 7.3, approximately agrees with the calculated value (640 ˚ A) that is the lateral distance from the buried dislocation to the position where the {111} slip plane intersects the top surface of the 900-˚ A-thick underneath buﬀer layers. A solid line interlinks the centers of the dots that belong to a horizontal dislocation below. It is evident from the experimental observations that the preferential nucleation of dots along the dislocation lines is the direct result of the lower misﬁt strain in dots. The clear contrast in the critical sizes of dots further illustrates the fact that a larger critical size is associated with lower misﬁt strain. The relaxed Si0.85 Ge0.15 buﬀer layer in Fig. 7.3 causes two clearly diﬀerent regions distinguished by the spatial distribution of Ge dots because it provides considerably larger dislocation spacing than the average surface dif-

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fusion length of Ge adatoms. As a result, there coexist the preferential and random distributions over the dislocation lines and in the region between dislocations, respectively. What drives the preferential and random nucleation of Ge dots coexisting in a sample? Two diﬀerent types of spatial distribution of Ge dots are observed when the relaxed Si0.85 Ge0.15 buﬀer layer is used for the Ge dots, while only the preferential nucleation existed on the relaxed Si0.75 Ge0.25 buﬀer layer. The clue is presumably closely linked to the mobility of Ge adatoms that randomly hop on a top surface.

7.3 The Observation of Three-Stage Nucleation Now we will discuss the nucleation and growth of Ge quantum dots at three distinct surface sites on the relaxed Six Ge1−x buﬀer layer, followed by the introduction of a new method for the quantitative observation of surface diﬀusion processes as a function of the substrate temperature and the growth rate of Ge. When the average spacing of the underlying stressor is much larger than the surface diﬀusion length of adatoms, three types of surface sites can be distinguished with various Ge coverages. The three types of sites include those over the dislocation intersections, those over single dislocation lines, and those far away from dislocations. We designate them site A (points), B (lines), and C (surface area), respectively. From the viewpoint of adatom density, the Stranski–Krastanow (SK) growth mode itself is an interesting contrast to homoepitaxial growths of Si on Si that proceeds in Frank–van der Merwe (FV) mode. The latter has been studied in detail by Mo et al. and has been shown to grow via either step propagation or 2D island nucleation and coalescence [2]. As a result, the adatom density reaches a steady state value for a given incident molecular ﬂux and remains at that value throughout the rest of the growth process. SK growths such as Ge on Si or InAs on GaAs, on the other hand, go through FV mode during the ﬁrst couple of monolayers’ (ML’s) coverage, i. e., during the formation of the wetting layer. At the completion of the wetting layer formation, surface steps cease functioning as sinks for adatoms and 2D island nucleation and growth are prohibited energetically. As a result, the adatom density starts to increase linearly with time until it reaches the supersaturation value for the nucleation of dots. The three types of surface sites of the partially relaxed buﬀer layer have diﬀerences in their in-plane lattice constants, with the value at site A being the closest to that of unstrained Ge. Intuitively, such diﬀerences can translate into either a diﬀerence in the surface diﬀusion coeﬃcient of the adatoms, in the wetting layer thickness, in the critical size of pyramid-to-dome transition, or all of the above.

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Through ﬁne control of the Ge coverage to a fraction of an angstrom, prenucleation ridges over single dislocation lines are observed immediately before the formation of Ge dots. This is the more direct evidence of the preferential nucleation of Ge dots over the buried dislocations compared to the larger critical size. Subsequent Ge dots have the characteristic pyramid shape and undergo pyramid-to-dome transition with increasing Ge coverage. The critical size of pyramid-to-dome transition is diﬀerent among the dots located over the three diﬀerent types of surface sites. Under the assumption that the transition represents a switch in the minimum free energy shape as proposed by Ross et al. [3], the diﬀerence in the critical island size can be taken as an indicator of the diﬀerence in misﬁt strain energy among the three types of sites. AFM topographies of samples with Ge coverage of 4.0, 4.5, 5.0 and 6.0 ˚ A are shown in Fig. 7.4a–d. The ﬁrst three images clearly illustrate the existence of three types of surface sites and the resulting three-stage nucleation over dislocation intersections (site A), single dislocation lines (site B), and in regions far away from dislocations (site C), respectively. Three-stage nucleation is clearly shown as the Ge coverage increases. At 4.0 ˚ A, Ge dots of pyrami-

Fig. 7.4. 2D AFM topographic images of Ge quantum dots with 4 – 6 ˚ A coverage A Ge coverage with Ge dots only on a partially relaxed Si0.9 Ge0.1 buﬀer layer. a 4.0 ˚ at site A. b 4.5 ˚ A Ge coverage with Ge dots at site A and B. c 5.0 ˚ A Ge coverage with Ge dots at site A, B, and C. d 6.0 ˚ A Ge coverage with Ge dots at site A, B, and C [17]

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dal shape nucleate exclusively at A sites. These pyramids form a rectangular array with perfect correspondence to the network of buried dislocations. An additional 0.5 ˚ A of Ge growth causes the preferential nucleation of Ge dots over dislocation lines. Figure 7.4b shows a large rectangle bordered by dislocations consisting of dots formed at A and B sites. Although the Ge coverage at this point is approximately the wetting-layer thickness (established in the literature to be ∼3 ML ≈ 4.2 ˚ A in the case of Ge grown on bulk Si (001) substrate [4]), it is important to notice that there is no dot in the region between dislocations. The dislocation spacing of ∼9 μm is much larger than the surface diﬀusion length of Ge adatoms reported in literature under similar conditions. The appearance of dots at sites A and B indicates that either the Ge adatom density is higher or the nucleation barrier is lower at these sites.

7.4 The Prenucleation Stage, Simultaneous Self-Assembled Quantum Dots (SAQDs) Growth and Nucleation In an eﬀort to diﬀerentiate between the two possibilities, surface topography immediately prior to the nucleation of any islands is examined in detail. The location of the buried dislocations is revealed by lines of steps such as the one shown in Fig. 7.5a. Arrows indicate the step over a buried single dislocation line prior to Ge growth. Evidence of a higher density of Ge adatoms is observed at site B in the form of a ridge (not islands) of very low aspect ratio as shown in Fig. 7.5b. The ridge height in samples with 3.0 ˚ A Ge coverage is determined to be 7.5 ± 1.5 ˚ A, a signiﬁcant increase from the 2.9 ± 0.7 ˚ A value in the sample with no Ge coverage. The original ridge height from samples with no Ge coverage agrees well with the observation by Lutz et al. who claimed a ridge height of 2.5 ± 0.3 ˚ A for individual buried dislocations [5]. Although there is deﬁnite nonplanarity at these sites, (001) remains to be the prevailing facets. The average terrace width of ∼240 ˚ A calculated from the aspect ratio of the ridges is comparable to that on typical Si (001) surfaces. The only diﬀerence is the polarities of the “force dipole” at these steps are more aligned. We call these ridges “pile-ups” of Ge in order to diﬀerentiate them from dots. The lack of 3D Ge islands at this Ge coverage indicates that the wetting-layer thickness (>∼4.5 ˚ A) at these sites is much thicker than at site C, presumably due to the much-reduced misﬁt strain at these sites. This observation favors the higher Ge adatom density possibility over that of a lower nucleation barrier at B sites. Similar “pile-up” of Ge is also observed at site A. There, due to the fourfold symmetry, the “pile-ups” are in pyramidal shape with extremely low (∼1:145) aspect ratio. Furthermore, the aspect ratio seems to vary continuously with time, instead of staying at a constant value such as in the case of typical Ge pyramids on Si. The fact that no single facet is preferred indicates

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Fig. 7.5. 2D AFM topographic images of the sample; before Ge growth (a) and with 3.0 ˚ A Ge (b). Arrows indicate the ridge on the top surface resulting from buried single dislocation. Line scan of a–b and c–d illustrates that ridge height is 2.9 ± 0.7 ˚ A and 7.5 ± 1.5 ˚ A, respectively [17]

that such “pile-ups” of Ge are not dots. In other words, they do not represent energetically stable islands. Instead, they could be explained by the existence of a chemical potential gradient for Ge adatoms toward site A and B caused by the undulating strain ﬁeld. Supporting this believe is the fact that such pile-ups are never observed at site C, where there is no directional diﬀusion of Ge adatoms.

7.5 The Observation of a Rare Event of SAQDs Forming After a Single Propagating Dislocation In Fig. 7.6 the nucleation of Ge dots proceeds from point A to B along a buried single dislocation. Dot size becomes smaller and the interdot spacing becomes larger as we move from the center towards the end of the dislocation half loop. This trend is symmetric towards the other end of the dislocation half loop. Three line scans reveal that ridge heights of a–b, c–d, and e–f are 7.2 ˚ A, 4.8 ˚ A ˚ and 2.9 A, respectively. This trend is a direct consequence of the fact that the accumulation of Ge adatoms by surface diﬀusion happens at a ﬁnite rate. The time it takes is comparable to the time it takes for the dislocation to expand by several micrometers. As a result, the time period during which directional surface diﬀusion happens is noticeably shorter near the end of the buried dislocation half loop than near the center. The ridge height of

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Fig. 7.6. 3D AFM topographic images of the sample in which the nucleation of Ge quantum dots proceeds along an underlying dislocation. Ridge heights across single dislocation of a–b, c–d, and e–f are ∼ 7.2 ˚ A, ∼ 4.8 ˚ A and ∼2.9 ˚ A, respectively [17]

e–f is comparable to the one in samples without Ge growth corresponding to Fig. 7.5a, whereas that of a–b is practically the value measured across any buried dislocation line in Fig. 7.4a,b. The total length of this propagating buried dislocation is found to be ∼24 μm by following the ridge. Apparently, the dislocation half loop is introduced near the beginning of the Ge growth that took place at 700 ◦ C. Such a substrate temperature is suﬃcient to allow a dislocation half loop to propagate at a rate that is on the order of a few micrometers per minute. Ge dots formed along the line are practically trailing the expanding dislocation, allowing us to observe the unfolding of such a dynamic process.

7.6 Diﬀerent Determining Factors for the InterDot Spacing A partially relaxed buﬀer layer is an excellent experimental vehicle for the study of the various stages of nucleation and growth of dots following the SK mode. There is a complex interplay of the directional surface diﬀusion of Ge adatoms with the diﬀerent dot nucleation barriers at the various surface sites. The experimental results provide insight into the formation process of Ge dots on Si. At the same time, these results also raise more unknowns. Following the evolution of Ge adatom density near the three types of sites on the surface, we can “trace” the formation of Ge dots on Si (001). At the beginning of the growth, Ge adatom density increases all across the sample surface. Together with this increase, directional diﬀusion takes place, presumably as a result of the diﬀerence in Ge adatom diﬀusivity near the three

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Fig. 7.7. Schematic drawing of adatom density (ns ) as a function of growth time (t) at a growth rate of R. Wetting-layer completes at τo and 3D quantum dots nucleation occurs at τr ; nv is the equilibrium adatom density [17]

types of surface sites. Ge adatoms dwell longer at sites A and B because the lattice constants there are closer to that of unstrained Ge. Consequently, there is a net ﬂux of Ge adatoms from the surrounding region towards sites A and B, and the adatom density near these sites increases as a result. The wetting-layer thickness at these sites is apparently much larger than on bulk Si surfaces. There is no Ge dot nucleation at site B for Ge layer thickness values below ∼6.0 ˚ A. Such a large wetting layer thickness can again be explained using the much lowered misﬁt strain between the dots and the substrate at sites A and B compared to that at site C. Intuitively, a zero misﬁt strain should lead to a wetting-layer thickness approaching inﬁnity. Directional diﬀusion leads to a partial depletion of Ge adatoms near sites A and B that is later manifested as a “denuded zone”, free of Ge dots. The extent of the directional diﬀusion is approximately the diﬀusion length during the experimental process and is on the order of micrometers. As a result, Ge adatom density in regions far from sites A and B is largely unaﬀected. It is expected to stay constant at ﬁrst, until the completion of the wetting layer. At that point, step edges stop functioning as sinks for adatoms and 2D nucleation becomes energetically unfavorable, and Ge adatom density starts to increase with deposition time in a linear fashion. Figure 7.7 depicts the expected evolution of Ge adatom density with time. Under a growth rate of R the wetting layer completes at τo . The adatom density abruptly drops at τr with the nucleation of 3D quantum dots. After this the evolution of the adatom density is relevant to the consecutive 3D nucleation at sites A, B, and C.

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Upon increaseing Ge coverage, dot nucleation begins ﬁrst at site A, and then at site B. All dots begin with the pyramidal shape that is followed by an abrupt transition to dome shape. The reason that dots form at sites A and B is interpreted as being a result of the abundant supply of Ge adatoms due to the strain-induced chemical potential gradient. Although no appropriate experiments have yet been done, it is expected that the length of time that the substrate temperature is suﬃciently high is the determining factor for the preferential nucleation of dots at sites A and B. The longer the duration, the more Ge adatoms gather at these sites, and therefore there are high degrees of supersaturation. Further increase in Ge coverage leads eventually to the nucleation of dots at site C. This occurs when Ge supersaturation reaches the critical value for dot nucleation. The combined knowledge of the Ge ﬂux and the deposition time allows a rough estimate of the critical value of Ge supersaturation necessary for the nucleation of dots. The only uncertainty in such an estimate is the precise wetting-layer thickness that cannot be determined by the absence of dots, but only by monitoring the adatom density in real time. The majority of the wetting layer information quoted in the literature uses the former deﬁnition and is therefore, strictly speaking, inaccurate. Immediately after the onset of dot nucleation at site C, directional diﬀusion takes place across the entire sample surface. At this point, all existing dots function as sinks for Ge adatoms. Figure 7.8 shows a schematic drawing of the Ge adatom density near dots at site C. Directional diﬀusion contribute to the decrease Ge adatom density between dots. The rate of decrease is inversely proportional to the interdot spacing. Depending on the relative magnitude of the incoming Ge ﬂux to the directional diﬀusion rate, the Ge adatom density between dots may increase further, leading to more dots being nucleated, which is associated with a decrease in the spacing between dots as well as an increase in Ge adatom loss due to directional diﬀusion. During this process, experimental observations show continued nucleation of dots that corresponds to Ge coverage from 4.5 to 6.0 ˚ A in our experiments. Eventually, the spacing between dots becomes small enough that the directional diﬀusion ﬂux overtakes the incoming Ge ﬂux and the Ge adatom density drops below the critical supersaturation value (nc ). The dot nucleation process comes to a halt. Many researchers have reported the surface diﬀusion of heteroepitaxial growths because of practical demands as well as interest in the understanding of surface processes. Formation of Ge dots on Si (001) depends strongly on the surface diﬀusion of Ge adatoms [6, 7]. The simultaneous processes of surface diﬀusion of adatoms and SK mode island nucleation and growth have greatly hindered our ability to correctly interpret experimental results. The large number of modeling attempts using the molecular-dynamic (MD) technique and ab initio total energy calculations have contributed to the understanding of adsorption and diﬀusion of Ge adatoms [8,9]. Such theoretical studies suﬀer

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Fig. 7.8. Schematic drawing of Ge adatom density between two existing Ge quantum dots at site C. When the distance between two dots is larger than the equilibrium interdot spacing, another nucleation occurs at the position Ge supersaturation reaches the critical value (nc ) with the increase of time (t1 to t5 ) [17]

from the lack of an appropriate potential for surface processes in the case of MD simulation, and limited computational power of modern computers in the case of ab initio calculation. The rate equation approach has mathematically described the detailed nucleation and growth behavior [10, 11]. The correct values of the constants of the rate equation, however, could come only from experiments. Mo et al. observed anisotropic diﬀusion of Si adatoms on Si (001) using the denuded zone free of 2D Si islands in the vicinity of surface steps [12]. They used extremely low Si coverage (0.07 ML) and derived the relationship between the island density, the denuded zone width, and the surface diﬀusion coeﬃcient. Their result is not applicable to heteroepitaxy following the SK mode, since ﬁlm growth is no longer via the attachment of adatoms to step edges as in the case of FV (layer-by-layer) growth mode. It can be stated that a clear understanding of the diﬀusion of adatoms is lacking, especially in the case of heteroepitaxy such as Ge dots on Si (001). Now we discuss how the denuded zone width (WDZ ) and the interdot spacing (di ) in the region far away from dislocations deﬁne the surface diﬀusion coeﬃcient (Ds ) and activation energy (Es ). As the growth temperature increases or the growth rate decreases, the surface diﬀusion length of Ge adatoms increases, resulting in both larger WDZ and di . The activation energy of surface diﬀusion is obtained from Arrhenius plot of the areal dot density (N ) in the region far from dislocations as a function of the growth temperature. Figure 7.7 shows the schematic drawing of the adatom density (ns ) as a function of the growth time. It is generally accepted that the equilibrium saturated adatom density is much less than 1 ML (∼ = 0.1 ML) [13]. The incorporation of Ge adatoms during the wetting layer growth is either via 2D island nucleation or via step ﬂow. The Ge adatom density is slightly above the equilibrium value in view of a combination of the abundance of surface steps as adatom sinks and the low incoming ﬂux of Ge. Upon the completion of the wetting layer, Ge adatoms can no longer incorporate into the 2D ﬁlm

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due to the high strain energy in the ﬁlm. Then the adatom density increases linearly with growth time until it reaches the supersaturation value for the nucleation of 3D Ge dots. At the onset of Ge dots nucleation, the Ge adatom density drops abruptly. The adatom density between dots initially nucleated increases again for another nucleation event. This process continues until the interdot spacing is reduced to such a value the incoming Ge ﬂux equals to the surface diﬀusion ﬂux in between dots. Based on the above postulations the mean surface diﬀusion length (ds ) of Ge adatoms can be expressed by, 1 (7.1) ds = Ds τ = N where τ is the mean lifetime of Ge adatom. Using Eq. (7.1) and Fig. 7.1 4Ds

Δns,max ∼R ads /2

(7.2)

where, a is a hopping distance and R is a growth rate [cm−2 sec−1 ]. Δns,max = ns,max − nv , ns and nv are an adatom density and an equilibrium adatom density, respectively. Thus ns,max ≤ R(τr − τo ), and τo and τr are time of 2D wetting layer completion and onset of 3D island nucleation, respectively. Finally we have (7.3) N ∝ exp(2Es/kB T ) where, kB is the Boltzmann constant. Since N is equal to 1/(di )2 , the measurement of the average dot density at various substrate temperatures provides the activation energy (Es ) in an Arrhenius plot. Assuming that the volume increase of dots over the buried dislocations is contributed by the incorporation of Ge adatoms impinging within the corresponding denuded zones, the volume changes of dots over dislocations are carefully measured. Fick’s ﬁrst law is used to obtain the preexponential factor based on the fact that there is an adatom concentration gradient. Since the denuded zone width and interdot spacing are directly associated with surface diﬀusion, the experimental results presented in this paper will provide fundamental understanding of surface processes for the Ge dot formation. Figure 7.9 shows AFM topographic images corresponding to 6.0 ˚ A Ge coverage grown at growth temperatures ranging from 650 to 750 ◦ C. Figure 7.9a–e are samples with a growth rate of 0.05 ˚ A/s, while Fig. 7.9f–j are samples with a higher growth rate of 0.1 ˚ A/s. Dot-free denuded zones exist on both sides of buried dislocations along the two 110 directions. Furthermore, the denuded zone is wider as the growth temperature of Ge dots increases, indicating the increase in the diﬀusion coeﬃcient. It is important to note the necessity of using properly designed relaxed SiGe buﬀer layers with the dislocation spacing larger than the surface diﬀusion length of Ge adatoms. Failure to do so will lead to the absence of random dots, and with that, plenty

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of useful information [14]. Figure 7.9f shows no preferential nucleation of Ge dots over dislocations, but the random nucleation of Ge dots in the region away from dislocations. The resulting quantum dot morphology reveals that the surface diﬀusion length of Ge adatoms is too short to be inﬂuenced by the chemical potential gradient over buried dislocations. In particular, incident Ge ﬂux is expected to be drastically higher than the diﬀusing Ge adatoms on the surface before the formation of dots. Supersaturation abruptly decreases with the nucleation of large number of dots. The average surface diﬀusion length is longest for the sample of Fig. 7.9e grown at the highest growth temA/s). Therefore, more perature (750 ◦ C) with the slowest growth rate (0.05 ˚ Ge adatoms incorporate into the region over buried dislocations and the probability of the nucleation of random dots decreases due to the lack of Ge adatoms. Eventually, larger denuded zone widths form compared to the other samples grown at lower growth temperature or higher growth rate. The AFM images of Fig. 7.9e, j shows the relatively nonuniform size distribution of dots grown over dislocations comparing other samples. Coarsening is believed to cause the broader size distribution as well as the larger average size of dots at the expense of the decrease of dot density. The interdot spacing over dislocations is much smaller than that in the region between dislocations in all samples. Thus, coarsening occurs in dots over dislocations at an earlier stage. Moreover, dots over dislocations in Fig. 7.9e show an additional event leading to the broad size distribution, i. e., coalescence among two or more dots. The perimeters of two adjacent dots overlap, leading to a two-hump conﬁguration to be observed prior to becoming one bigger dot. The small spacing among

˚ Ge coverage grown at ﬁve diﬀerent Fig. 7.9. 2D AFM topographic images of 6.0 A growth temperatures on partially relaxed Si0.9 Ge0.1 buﬀer layers in which buried dislocations provide the preferential nucleation sites. The growth rate of the lowtemperature Ge wetting-layer (3.4 ˚ A) is 0.05 ˚ A/s. Subsequent 2.6 ˚ A Ge are grown at 0.05 ˚ A/sec for a–e and at 0.1 ˚ A/s for f –j. Particularly, e and f show two extremes caused by high and low surface diﬀusion coeﬃcients, respectively [18]

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dots over dislocations causes the rapid increase in average size. In contrast, coalescence is not observed in Fig. 7.9j, grown at same growth temperature with a higher growth of 0.1 ˚ A/s. The density of random dots in the region far from dislocations is strongly dependent on the surface diﬀusion of Ge adatoms. As a result, it clearly decreases with increasing growth temperature and the decreasing growth rate. Coarsening is manifested in this region as deviation from the linear relationship in Arrehenius plot of the dot density (N ) as a function of the growth temperature. It will be discussed below. From AFM images in Fig. 7.9, we further notice that denuded zone widths are symmetric with respect to the two perpendicular 110 directions. In other words, the anisotropic diﬀusion induced by the two types of terraces on a Si (001) surface [15] is not observed presumably due to the fact that the denuded zone widths are much larger than typical terrace widths and the anisotropy in diﬀusion is averaged out as a result.

7.7 Activation Energy and Pre-exponential Factor of Surface Diﬀusion of Ge Adatom on Si (001) Figure 7.10 shows Arrehenius plot of the areal density of random dots in the region away from dislocations as a function of the growth temperature between 650 and 750 ◦ C. The areal dot densities are obtained from the saturated stage. The ﬁgure shows near-perfect Arrehenius behavior from all samples grown at 650 as well as 750 ◦ C with one exception. The only data point that deviates signiﬁcantly is grown at 750 ◦ C with the low growth rate of 0.05 ˚ A/s. We believe the deviation is a result of the extremely long surface diﬀusion length of Ge adatoms on the surface leading to non-negligible coarsening. The apparent spacing between dots increases when smaller dots disappear as a direct consequence of coarsening. Based on the equation of N ∝ exp(2Es /kB T ), the activation energies obtained from two diﬀerent growth rates of 0.05 ˚ A/s and 0.1 ˚ A/s are 0.676 ± 0.03 eV and 0.671 ± 0.03 eV, respectively. Excluding the sample undergoing coarsening, similar activation energies are obtained from two growth rates. The experimental activation energies are comparable to the ones calculated in the literature. Now we obtain the pre-exponential factor from the volume increase of dots over dislocations as shown in Fig. 7.11. An additional two samples with 5.0 ˚ A Ge coverage were grown at 700 ◦ C with growth rates of ˚ 0.05 A/s and 0.1 ˚ A/s, respectively. The average volumes of dots were carefully measured to compare with two other samples with 6.0 ˚ A Ge coverage. At this growth temperature the dot size distributions are found to be relatively uniform at positions over buried dislocations and the region away from dislocations. Moreover, no coarsening eﬀect is observed at two different Ge coverages. The average volume of dots over buried dislocations is apparently larger than that in the region away from dislocations. It is

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Fig. 7.10. Arrehenius plots of the areal density of randomly nucleated dots in the region between dislocations versus the growth temperature [18]

known that many more Ge adatoms are incorporated into the region over buried dislocations due to the chemical potential gradient. The volume increase of dots over dislocations is approximately proportional to the denuded zone width and larger than that of those in the region away from dislocations. We found that the total number of Ge atoms impinging onto the denuded zones is slightly larger than that for the average volume increase of dots over dislocations at both growth rates. This is apparent evidence that the majority of Ge adatoms incident within the denuded zone incorporates into the existing dots over dislocations between 5.0 and 6.0 ˚ A Ge coverage. Furthermore, the line density of dots over dislocations is identical at two Ge coverages with complete domes. This observation implies the absence of additional dot nucleation as well as pyramid-to-dome shape transition. Based on the simple assumption that there is a concentration gradient of Ge adatom in the denuded zones caused by the chemical potential gradient, surface diﬀusion coeﬃcient (Ds ) is calculated using Fick’s ﬁrst law Δn J = −Ds . Δx Flux J (atoms per second) is given by the average volume increase of dots over dislocation between two diﬀerent Ge coverages of 5.0 and 6.0 ˚ A. The number of Ge atoms impinging directly onto the existing dots is negligible because the area of denuded zone on both sides is signiﬁcantly larger than that of dots. Δn/Δx (atoms per centimeter squared) is the number of Ge

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Fig. 7.11. Schematic drawing showing preferentially nucleated quantum dots over a buried dislocation and randomly nucleated ones in the region far away from a dislocation. The increase of dot volume is equivalent to the number of adatoms incorporated from the denuded zone. Grey and black dots indicate the average sizes of 5.0 ˚ A and 6.0 ˚ A Ge coverages, respectively. WDZ denuded zone width [18]

atoms arriving onto the unit area of denuded zone. In the samples grown with a growth rate of 0.05 ˚ A/s, the volume measurement reveals that ∼7.1 × 105 atoms are used for the volume increase of an individual dot over buried dislocations for 20 s (2000 nm3 ≈ 88,400 Ge atoms). Based on the fact that 1.0 ˚ A Ge coverage corresponds to ∼4.48 × 1014 atoms/cm2 , it is found that ∼8.0 × 105 atoms arrive onto the square region in the denuded zone as shown in Fig. 7.11. As a result, Ds =∼7.97 × 10−11 cm2 /s. Using the well-known equation of Ds = Do exp(−Es /kT ) and the activation energy (0.676 eV), a pre-exponential factor (Do ) is ∼2.53 × 10−7 cm2 /s. At the higher growth rate of 0.1 ˚ A/s much less number of Ge atoms (∼1.1 × 105 atoms) incorporated for the volume increase, presumably due to the smaller denuded zones. Consequently, the volume increase of dots over dislocations is considerably smaller than that of the sample with 0.05 ˚ A/s. Using the activation energy (0.671 eV) obtained from the samples of 0.1 ˚ A/s we obtain a slightly smaller pre-exponential factor (∼1.0 × 10−7 cm2 /s), implying a slower surface diﬀusion rate.

7.8 Simulation for Adatom Proﬁle The incorporation of Ge adatoms during the wetting-layer growth is either via 2D island nucleation or via step ﬂow. The Ge adatom density is only slightly above the equilibrium value in view of a combination of the abun-

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dance of surface steps as adatom sinks and the low incoming ﬂux of Ge. Upon the completion of the wetting layer (t > τo ), Ge adatoms can no longer incorporate into the 2D ﬁlm, presumably due to the high strain energy in the ﬁlm. The critical adatom density for dot nucleation is ﬁrst achieved in the regions of slow diﬀusion, i. e., near the intersections of dislocation lines and then above the dislocation lines themselves. These dots start acting as adatom sinks, leading to strips of decreased adatom density near the dislocation lines. If the spacing between dislocations is larger than the adatom diﬀusion length, then the adatom density in the region away from the dislocations continues to increase approximately linearly with time, as shown for τo < t < τr in Fig. 7.7, until it reaches the supersaturation value for the nucleation of 3D Ge dots. At the onset of the Ge quantum dot nucleation, the density of Ge adatoms drops abruptly. The adatom density between the initially nucleated dots increases again with incoming Ge ﬂux until another dot nucleates between two initially widely spaced dots. This process continues until the interdot spacing is reduced a value for which the incoming Ge ﬂux equals to the surface diﬀusion ﬂux between dots. Qualitatively, the spatial distribution of dots and the existence of a denuded zone without dots near the dislocation lines can be traced back to the adatom density proﬁle at times preceding the formation of randomly nucleated dots (t < τr in Fig. 7.7). The adatom density n(x, y, t) obeys the Fick’s diﬀusion equation ∂n = ∇(Ds ∇n) + F ∂t

(7.4)

where Ds is the surface diﬀusion coeﬃcient and F is the deposition rate. Equation (7.1) can be used to describe the adatom density in a rectangular area Lx × Ly between the dislocations. We consider the time period when dots have already nucleated both on the intersections and dislocation lines, i. e., along the edges of this rectangular region. Dots will grow by incorporating adatoms from the surrounding areas, in other words, dots over the dislocation lines act as adatom sinks, which can be modeled in the framework of Eq. (7.1) by the following boundary conditions: n|x=0 = n|x=Lx = n|y=0 = n|y=Ly = nv

(7.5)

where nv is the equilibrium adatom density. To keep things simple, we assume that nv is not aﬀected by the strain due to the dislocation and growing dots. Furthermore, we neglect possible spatial variation of the diﬀusion constant nv in the strained regions. Finally, we set the initial density at t = 0 as uniform, n = nv . These assumptions are justiﬁed if, as in our case, one’s aim is to arrive at qualitative conclusions regarding the time evolution of the density proﬁle. Our methods for obtaining quantitative estimates of nv , described below, do not rely on these approximations. Equation (7.1) can be solved analytically using Fourier series expansion in spatial coordinates. The calculated density proﬁle n(τ, t) is shown in Fig. 7.12 at consecutive times. The critical density

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Fig. 7.12. The simulation result of Ge adatom density proﬁle in the region between two dislocations [18]

for nucleating dots is shown schematically as a dashed line. It is seen that the adatom density increases linearly in time over an extended region between the dislocation lines, where n(τ, t) stays practically ﬂat. At a certain time, the critical density is reached simultaneously everywhere in this central region, leading to a sudden appearance of randomly nucleated dots and growth. In contrast, near the dislocation lines the density remains below the critical value, leading to the formation of denuded zones. The width of these zones depends on the ratio n(τ, t)/n(τ, t), and increases with n(τ, t)/n(τ, t) if n∗ is unchanged. Increase in growth temperature should increase both n∗ and n∗ , but the eﬀect of increase in n∗ should dominate and lead to a widening of the denuded zone. This approach represents a new method to obtain the surface diﬀusion coeﬃcient of heteroepitaxial growth. The behavior of Ge dots grown on the relaxed SiGe buﬀer layer is considerably dependent on the surface diﬀusion of Ge adatoms. All experimental observations are based on the fact that the denuded zone width and the interdot spacing of random dots are strongly dominated by the surface diﬀusion process. Two diﬀerent values carefully measured at diﬀerent growth temperatures and growth rates provide insight into surface diﬀusion process and the subsequent formation of Ge dots on Si substrate. Alternative approaches for measuring the surface diﬀusion coeﬃcient have used elaborate equipments consisting both of evaporation sources and in-situ surface analytical tools such as scanning Auger microscopy or ion scattering. In contrast, the new approach has practically employed 3D Ge dots formation on Si surface following the SK mode. Surface diﬀusion process in heteroepitaxial growth has been characterized by AFM observations of the morphology changes as functions of growth temperature and rate. Moreover, the buried dislocation network used in this study is an excellent tool to investigate the

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surface diﬀusion of adatoms. The approach is not limited to the Si/Ge system but can be used for other heteroepitaxial systems in SK growth mode and in the presence of surfactants such as Sb, As, H, or C.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

F.M. Ross, IBM J. Res. Dev. 44, 489 (2000) Y.-W. Mo, J. Kleiner, M.B. Webb, M.G. Lagally, Surf. Sci., 268, 275 (1992) F.M. Ross, J. Tersoﬀ, R.M. Tromp, Phys. Rev. Lett. 80, 984 (1998) H.-J. Gossmann, L.C. Feldman, W.M. Gibson, Surf. Sci. 155, 413 (1985) M.A. Lutz, R.M. Feenstra, F.K. LeGoues, P.M. Mooney, J.O. Chu, Appl. Phys. Lett. 66, 724 (1995) T.I. Kamins, R.S. Williams, Appl. Phys. Lett. 71, 1201 (1997) Y.-H. Wu, C.-Y. Wang, A. Elfving, G.V. Hansson, W.-X. Ni, Mater. Sci. Eng. B 89, 151 (2002) D. Srivastava, B.J. Garrison, Phys. Rev. B 46, 1472 (1992) C. Roland, G.H. Gilmer, Phys. Rev. B 47, 16286 (1993) R. Kariotis, M.G. Lagally, J. Vac. Sci. Technol. B 7, 269 (1989) M. Tomellini, M. Fanfoni, Curr. Opin. Solid State Mater. Sci. 5, 91 (2001) Y.-W. Mo, J. Kleiner, M.B. Webb, M.G. Lagally, Surf. Sci. 268, 275 (1992) M. Zinke-Allmang, L.C. Feldman, M.H. Grabow, Surf. Sci. 200, L427 (1988) Y.H Xie, S.B. Samavedam, M. Bulsara, T.A. Langdo, E.A.Fitzgerald, Appl. Phys. Lett. 71, 3567 (1997) Y.-W. Mo, M.G. Lagally, Surf. Sci. 248, 313 (1991) H.J. Kim, J.Y. Chang and Y.H. Xie, J. Crystal Growth 247(3–4), 251–254 (2003) H.J. Kim, Z.M. Zhao, and Y.H. Xie, Phys. Rev. B 68, 205312–205317 (2003) H. J. Kim, Z. M. Zhao, J. Liu, V. Ozolins, J. Y. Chang, and Y. H. Xie, J. Appl. Phys. 95(11), 6065–6071 (2004)

8 Ordering of Wires and Self-Assembled Dots on Vicinal Si and GaAs (110) Cleavage Planes Gerhard Abstreiter1 and Dieter Schuh1,2 1 2

Walter Schottky Institut, TU M¨ unchen, 85748 Garching, Germany Present address: Institut f¨ ur Experimentelle und Angewandte Physik, Universit¨ at Regensburg, 93040 Regensburg, Germany

8.1 Introduction The ability to precisely control the growth of semiconductor quantum dots and quantum wires is a topic that has recently attracted much interest worldwide. Such nanostructures can be fabricated by e. g., self-assembly, induced by elastic strain relaxation in lattice-mismatched systems like GaAs/InAs or Si/Ge. In addition to the recent progress made in growing quantum dots of well-deﬁned size and shape the controlled positioning of self-assembled dot and wire systems is one of the major challenges of today’s heteroepitaxy. The main methods for preparing state of the art quantum dots are molecular beam epitaxy (MBE) (e. g., [1]) or epitaxy by chemical vapor deposition (CVD) (e. g., [2]). Diﬀerent methods have also been applied to achieve a spatial ordering or well-deﬁned positioning of such self-assembled nanostructures. One possible approach is the exploitation of modiﬁed growth kinetics which occurs e. g., on high index vicinal surfaces with regularly ordered atomic steps. These growth techniques have been used, for example, to grow In(Ga)As quantum dots and wires on (311)A or B GaAs (e. g., [3, 4]), on miscut (100)-oriented GaAs [5] or at crystal defects [6]. Other methods are based on modifying the surface morphology and subsequent growth or regrowth of the quantum dot structures (e. g., [7]). One of the most common approaches is the use of lithographically patterned substrates as templates in order to force controlled dot nucleation (e. g., [8–10]). In this chapter we discuss selected examples of growth of Ge or SiGe quantum wires and quantum dots on high-index vicinal Si surfaces with regularly ordered atomic steps, as well as the ability to controllably position InAs quantum dots of a well-deﬁned size on (110)-oriented cleavage planes of GaAs. Misﬁt lattice strain of SiGe material deposited on Si substrates can relax via the bunching of atomic surface steps with SiGe agglomeration at the step edges, but also by nucleation of Ge-rich quantum dots in the Stranski–Krastanow growth mode. Size, density and composition of such Si/Ge nanostructures can be tuned in a wide range by the growth parameters. Local strain ﬁelds inﬂuence the nucleation and the lateral arrangement and thus can be applied for self-ordering in the vertical as well as in the lat-

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eral direction. An overview of the fabrication of Si/Ge-based nanostructures using these techniques is presented in Sect. 8.2 (see also [11]). The use of cleaved edge overgrowth (CEO) allows for the fabrication of wire-like templates with the atomic precision given by MBE. This method has been applied recently to realize ordered arrays of InAs quantum dots on (110) cleavage planes of GaAs/AlAs multilayer structures. Here, we make use of self-assembly of InAs on thin epitaxial AlAs layers due to diﬀerent diﬀusion lengths, sticking coeﬃcients and desorption rates of In on AlAs with respect to GaAs. The results achieved so far with this new method are reviewed in Sect. 8.3 (see also [12, 13]).

8.2 SiGe Wires and Dots 8.2.1 Substrate Materials Used for Self-Ordering of Wires And Dots Si substrates with nominally (001)-oriented surfaces, as used, for example, in the microelectronics industry, are typically slightly misoriented up to about 0.5◦ in an arbitrary direction. This misorientation gives rise to intrinsic surface steps with a height of one monolayer (ML). The average lateral separation of these steps depends on miscut angle and is typically a few tens of nanometers. The steps can be very smooth for intentionally prepared miscut angles towards certain highly symmetric crystal directions. For miscut angles larger than 1.5◦ , biatomic steps are formed by the bunching of monoatomic steps. Epitaxial growth of Si layers on such vicinal surfaces together with certain annealing procedures result in a regular arrangement of these biatomic steps. Zhu et al. [14] have analyzed the ordering of such steps by in-situ reﬂection high-energy electron diﬀraction (RHEED) analysis. They found, for example, a very regular arrangement of biatomic steps with a separation of 3.9 nm for a 100-nm-thick epitaxial Si layer grown at 700 ◦ C on a vicinal (001) Si wafer with a miscut angle of 4◦ . An annealing step at about 1100 ◦ C for 10 min was required to achieve a very high regularity of the step arrangement. The step separation can be directly analyzed from the splitting of the (00) RHEED pattern as shown in Fig. 8.1. Higher-indexed crystal surfaces such as (118) or (113) are realized by increasing the miscut angle of the (001) surface towards the [110] direction. Such surfaces should no longer be considered as miscut surfaces but as stable surfaces with their own speciﬁc properties. The (113) surface, which is nearly as stable as (001) Si, shows a very strong Step-bunching behavior [15,16]. An example is shown in Fig. 8.2 where atomic force microscopy (AFM) images of Si ﬁlms that were grown on slightly miscut (113) surfaces at high temperatures are depicted. The (113) substrates used for these studies are miscut by 0.37◦ towards a direction which is 36◦ oﬀ the [-110] direction. Quite regular surface steps with an average separation of 100 nm, corresponding to a step

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Fig. 8.1. Reﬂection high-energy electron diﬀraction (RHEED) patterns from a 100nm-thick Si layer grown at 700 ◦ C on a vicinal Si (001) surface with miscut angle of 4◦ and annealed at 1100 ◦ C for 10 min. The (00) RHEED spot is studied for diﬀerent azimuth angles. a Φ = 0◦ . b Φ = 7◦ . c Φ = 14◦ . A step separation of 3.9 nm is deduced from the observed splitting which is consistent with regular biatomic steps for this miscut angle [14]

Fig. 8.2. Atomic force microscopy (AFM) images of a 100-nm-thick Si layer grown at 700 ◦ C on a slightly miscut Si (113) Substrate. a Direct after growth. b After annealing for 10 min at 1100 ◦ C

height of 4 ML, have already formed after the deposition of a 100-nm-thick Si layer at 700 ◦ C. The main orientation of the steps is perpendicular to the miscut direction but certain kinks are observed which tend to bunch along the [1-10] direction. After annealing such a sample at 1100 ◦ C for 10 min, these faceted kinks disappear and very straight steps with length scales in the 10 μm range are observed. The average terrace widths are now 100, 200 and 300 nm, corresponding to step heights of 4, 8, and 12 ML [17]. The bunching of steps in larger multiples of 4-atomic steps proceeds further with deposition of Si at high temperatures. The 4-atomic step is obviously a natural unit for the surface rearrangement of slightly miscut (113) surfaces at high temperatures. The pronounced bunching of long and rather high steps is a peculiar property of the Si (113) surface. The two examples discussed here demonstrate that a speciﬁc high-temperature treatment of vicinal Si surfaces can lead to very regular surface steps which can be used as natural templates for the fabrication of self-organized Si/Ge wire and quantum dot structures. This will be described in the following section.

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8.2.2 Arrays of Si/SiGe Wires A natural way of fabricating wire-like one-dimensional (1D) structures is the deposition of Ge on Si surfaces with regular steps under step-ﬂow growth conditions. The lattice mismatch of about 4% between Ge and Si results in a critical thickness for Ge on Si of a few monolayers. The exact onset of strain relaxation depends on growth conditions [18]. At higher growth temperatures the built-in strain is relaxed via formation and gliding of dislocations. Under certain growth conditions, elastic strain relaxation occurs via formation of three-dimensional (3D) islands after a certain critical layer thickness. Elastic strain relaxation can also occur along steps due to the modiﬁed boundary conditions with additional free bonds with respect to a ﬂat surface. The elastic SiGe relaxation causes a distortion of the underlying Si substrate that reduces the relaxation energy gained by the system. This induces an instability in the strained layers on vicinal surfaces due to accumulation of material at step edges [11, 19–21]. The lateral length scale of step-induced surface undulations can range from intrinsic surface terrace widths to a micrometer scale due to enhanced step bunching. The accumulation of Ge on monoatomic surface steps has been demonstrated, for example, by Sunamura et al. [22] for submonolayer deposition of Ge. The accumulation of Ge along surface steps can be attributed to the step-ﬂow growth conditions, the lower surface energy of Ge compared with Si and an enhanced strain relaxation of Ge wires at Si step edges. Using SiGe alloys instead of pure Ge allows the growth of thicker layers due to the smaller lattice mismatch. Furthermore, the local strain relaxation of wire-like structures enhances wire formation and induces self-ordering in multilayer structures [20, 21]. A cross-sectional TEM image of a 20-period multilayer of 10 nm Si and 2.5 nm Si0.55 Ge0.45 grown on (001) Si substrate with a miscut angle of 2◦ towards the [100] direction is shown in Fig. 8.3 [11]. The dark lines which correspond to the SiGe layers show a variation in thickness which develops during the ﬁrst few periods. These wire-like structures are correlated vertically from layer to layer. The lateral separation of the wires gets regular after about ﬁve multilayer periods with a lateral wire separation of 120 nm. The vicinal Si surfaces appear smooth with a rather homogeneous distribution of monoatomic or biatomic surface steps. The SiGe surfaces are ﬁnally faceted by (001) surface planes and by regions of enhanced density of surface steps. This results in a rather symmetric shape of the wires. Lattice strain in the SiGe layers enhances the bunching of surface steps and results in an undulated SiGe surface. Partial relaxation of lattice strain in such an array of wires causes tensile strain in the Si on top of the SiGe agglomerations and a compensating lateral compression of the Si in between. SiGe tends to accumulate in the regions of increased lattice constant, which results in a vertical correlation of the wires. This stacking and the repulsive interaction of laterally neighboring SiGe wires caused by

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the lattice compression of Si, results also in lateral ordering and formation of a periodic array. The ordering mechanism leads to stacks of wires with a separation which is well deﬁned by the nominal layer widths and the selfassembling process. Using diﬀerent layer and growth parameters may lead to an asymmetric local strain ﬁeld above the stepped SiGe accumulations, resulting in an inclined wire correlation. This has been observed, for example, in Si/Si0.7 Ge0.3 multilayer structures on (001) Si [23]. An inclined wire correlation has also been observed on vicinal (113) Si substrates, and was attributed to additional step bunching of the Si surfaces and a modiﬁed ordering mechanism [24]. The step bunches locate at the strain maxima and SiGe accumulates, laterally shifted at the step edges. Homogeneous arrays of wires over length scales of 15 μm have been achieved in this way on vicinal (113) surfaces (Fig. 8.4).

Fig. 8.3. Transmission electron microscopy (TEM) image with a cross-sectional view of wire-like SiGe structures formed by step bunching within a 10 nm Si/2.5 nm Si0.55 Ge0.45 multilayer on vicinal Si (001) with a miscut of 2◦ towards [100]. The lateral separation of the wires is 120 nm

Fig. 8.4. AFM image of wire-like structures formed by step bunching within Si/SiGe multilayers grown on slightly miscut Si (113) wafers [24]

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8.2.3 Lateral Self-Ordering of Ge Dots The linear strain ﬁelds of an array of wires as shown in Figs. 8.3 and 8.4 can serve as a template for self-organized growth of Ge dots on a Si surface. The larger lattice mismatch of pure Ge deposited on such surfaces may induce the formation of dots with a larger degree of lattice relaxation as compared to the wires. Figure 8.5 shows AFM images of two examples of two-dimensional (2D) arrays of Ge dots [25]. In both cases 5 ML Ge were deposited on vicinal (001) surfaces with underlying wire arrays similar to the ones shown in Fig. 8.3. In both cases, a periodic arrangement of the dots within the plane and good size homogeneity is observed. The lateral separation of the dot rows perpendicular to the wire direction is 120 nm and thus equal to the period of the wire array. The stripe-like strain ﬁelds at the Si surface induce nucleation and alignment of Ge dots along these stripes. The 2D arrangement of the dots is diﬀerent in the two examples. This is due to the diﬀerent orientation of the underlying wire structure. The direction of the wire alignment is determined by the direction of the miscut angle of the used Si wafer. A quadratic ordering of the dots is observed when the SiGe wires are oriented along the [010] direction (Fig. 8.5b), while a face-centered rectangular or nearly hexagonal ordering is realized for wires along [110] (Fig. 8.5a). Both samples show a densely packed 2D array of Ge islands. The 2D lateral ordering of the Ge dots is remarkable. Imperfections are mainly due to inhomogeneities of the underlying wire template. The two diﬀerent symmetries of lateral ordering can be understood in terms of the minimization of the total strain energy. The self-assembled Ge islands are quadratic pyramids with {105} facets and orientation of the base line along the [010] and [100] directions as sketched in Fig. 8.6. The island arrays are interacting systems caused by stress. The

Fig. 8.5. a AFM image of self-organized Ge islands formed by depositing 5 ML of Ge at 500 ◦ C on a one dimensionally rippled Si/SiGe multilayer structure grown on Si (001) with a miscut angle of 1.5◦ towards [1-10]. b AFM image of self-organized Ge islands fabricated as in a, but grown on Si (001) with a miscut angle of 2◦ towards [100]

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theoretical minimum of the strain energy is found for a periodic arrangement of strain domains [26, 27]. A densely packed quadratic alignment along the soft directions [010] and [100] is the favorable arrangement of the islands with the given base orientation on a pure (001) surface with anisotropic elastic properties [28]. Such an alignment is observed for the [010]-oriented wires (Fig. 8.5b), which enhances the lateral ordering. For wires oriented along the [110] direction, however, the underlying strain ﬁelds enforce an alignment of the islands along the [110] direction. The repulsion of the islands with the pyramid shape results in a lower energy for a face-centered arrangement than for the quadratic arrangement in this case. In this way, touching of the corners of the pyramids is avoided. The symmetry of the self-ordered 2D array of islands is controlled by the direction of the substrate miscuts and is transferred to the dot layer by the underlying wire structure. The alignment of islands along rows inherently causes a 2D self-ordering of islands if they interact by their overlapping strain ﬁelds (repulsion). Self-ordering of nanosized islands on strain templates as shown here oﬀers an excellent control of areal island density and prevents ripening or coalescence of islands. It results in improved control and homogeneity of island size compared with statistical ensembles. The growth technique presented here is a truly bottom-up self-organized process for the fabrication of large arrays of ordered quantum dots with high density. Ordering of Ge islands is also observed on the step bunches formed on slightly miscut Si (113). Figure 8.7 shows an AFM image of the Ge islands

Fig. 8.6. Schematics of two possible arrangements of the quadratic Ge pyramids with base orientation along [100] directions aligned along the step bunches along the [110] direction. Partial contours of the repulsive potentials created by the islands are shown schematically by thin solid lines

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grown at 500 ◦ C. Straight rows of dense Ge islands, which can be as long as 10 μm, are formed along some of the step bunches [11]. High-resolution AFM images show that the Ge islands nucleate on the large step bunches present in this sample. The islands align along these steps and show a rather narrow size distribution with an average height of 7 nm and lateral base diameter of about 80 nm. No correlation of the islands in neighboring rows is observed, due to the large separation of the step bunches.

8.3 Arrays of InAs Dots on AlAs/GaAs Cleavage Planes 8.3.1 Introduction The materials system InAs/GaAs has been used extensively to study the formation of the InAs quantum dots that are formed in the Stranski–Krastanow growth mode on GaAs (001) due to the lattice mismatch of about 7% ([29–31]; for an overview see for example [32] and references therein). Lateral orderings of such self-assembled dots on (001) surfaces have been achieved mainly by combining lithographical techniques with MBE (e. g., [8]). In this chapter we present a new approach that allows the controllable positioning of InAs quantum dots of a well-deﬁned size on (110)-oriented GaAs surfaces. Growth of InAs on the GaAs (110) surface, however, does not in general lead to dot formation; Rather, the lattice mismatch is relaxed by formation of diﬀerent types of dislocations [33]. To overcome this problem we used a new kind of (110)-oriented atomically precise template based on the Ga(In,Al)As materials system. The aim was to make use of strain and composition-patterned (110) surfaces in order to optimize conditions for nucleation of 3D islands. The general approach is based on the method of CEO [34]. The growth procedure is shown schematically in Fig. 8.8. In a ﬁrst growth step, a number of epitaxial AlAs/GaAs layers or InGaAs/AlGaAs layers are grown on a (001) GaAs substrate. After ex-situ preparation of these samples they are transferred back into the MBE system for cleavage and regrowth. The natural

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Fig. 8.8. Schematics of the combination of cleaved edge overgrowth (CEO) and self-assembly of InAs quantum dots. a Growth of AlAs/GaAs or InGaAs/AlGaAs multilayers. b After ex-situ preparation, the substrate is cleaved in situ in the MBE. c Deposition of InAs onto the freshly cleaved (110) surface. The nucleation of InAs on stripes with diﬀerent strain and composition may lead to a one-dimensional (1D) alignment of the quantum dots

cleavage plane of GaAs is the (110) plane. InAs is deposited on the freshly cleaved surfaces consisting of stripes of the diﬀerent layers grown in the ﬁrst step. The experimental results obtained in this way showed that the local strain ﬁelds caused by InGaAs layers embedded in GaAs are not suﬃcient to lead to nucleation of InAs dots on (110) cleavage planes, although it was shown that strain patterns are present even after overgrowth of alternating In0.1 Ga0.9 As and Al0.3 Ga0.7 As layers with 10 nm GaAs [35]. The second attempt tries to make use of diﬀerent surface properties for example with respect to surface diﬀusion and desorption. We used AlAs layers embedded in GaAs which are nearly lattice matched but exhibit largely diﬀerent surface properties. It is well known, for example, that the diﬀusion of In adatoms on (001) AlAs is much smaller compared to (001) GaAs [1] and, hence, we expect a similar behavior for the (110) surfaces. Growth of InAs on GaAs cleavage planes containing thin AlAs layers indeed showed nucleation of InAs islands on AlAs stripes [12,13]. This is shown in the AFM images of Fig. 8.9, giving an overview of a cleaved sample which was overgrown with 3.5 ML InAs at 460 ◦ C. On regions of the GaAs substrate without AlAs stripes (Fig. 8.9a), only large and ﬂat triangular shaped InAs islands are observed, which have already been studied on GaAs (110) [36]. In Fig. 8.9b, regions are shown that contain 1, 3, 5, 7, and 9 layers of AlAs with a thickness of 35 nm. It can be clearly seen that InAs dots nucleate and align along these stripes. This demonstrates that (110) cleavage planes with embedded AlAs stripes act as atomically precise templates for the nucleation and ordering of InAs dots in 1D arrays. The mechanisms for this self-ordering are discussed in the following. 8.3.2 Growth Conditions and Sample Preparation In order to understand the nucleation and growth mechanism, we have varied the basic layer sequence, particularly with respect to AlAs layer thickness as

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Fig. 8.9. AFM images of a GaAs/AlAs cleavage plane overgrown with 3.5 ML InAs. A region of the GaAs substrate with no embedded AlAs layers is shown in a. No self-assembled quantum dots are observed in this region. The region shown in b contains diﬀerent numbers of 35-nm-thick AlAs layers. A dense nucleation of InAs dots is observed along these AlAs stripes

well as the thickness of the intermediate GaAs. In the ﬁrst growth step, pure AlAs and GaAs layers were deposited on GaAs (001) wafers with growth rates between 0.4 and 1 ML/s at a substrate temperature of 650 ◦ C and with a V/III ﬂux ratio between 3 and 6. The 2-inch wafers were taken out of the MBE machine and samples were prepared for CEO as described e. g., in [34]. The sample thickness was lapped down to about 100 μm from the backside and cut into 7 × 7 mm pieces. A short scratch deﬁnes the position of the insitu cleave performed later. These small samples are mounted onto a special Ta sample holder in the MBE machine with the (110) surface oriented towards the eﬀusion cells. Between 2.5 and 3.5 ML InAs was deposited immediately after cleaving the samples in the growth chamber. Such samples were then studied ex situ by AFM. Similar samples were also grown with a 50-nm-thick GaAs cap layer for photoluminescence studies. The growth temperature on the (110) surface was chosen between 440 and 500 ◦ C with an InAs growth rate as low as 0.035 ML/s. 8.3.3 AFM Investigations Figure 8.10 shows more detailed AFM images of the cleaved sample with diﬀerent numbers of 35-nm-thick AlAs layers that was overgrown with 3.5 ML InAs. The ﬁve groups of AlAs stripes are separated by a 1 μm GaAs layer and contain one, three, ﬁve, seven, or nine layers of AlAs. The separation within the groups with more than one AlAs layer is 70 nm. It can be clearly seen that InAs quantum-dot-like structures have nucleated above all AlAs stripes. The geometrical properties like lateral base width, average height and linear dot density deduced from these AFM measurements are summarized in Table 8.1. It should be mentioned that the given values are a convolution of the actual size with the size and shape of the AFM tip used. We ﬁnd that the

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Fig. 8.10. AFM picture of InAs quantum dots ordered above AlAs stripes of the same thickness but varying number. The AlAs layers are embedded in GaAs with a thickness of 1 μm between the groups and 70 nm within the groups. The inset shows three-dimensional images of the AFM data

average size of the quantum dots is the same for the diﬀerent sections independent of the number of AlAs stripes, while the linear dot density decreases slightly with increasing number of stripes. This indicates that the formation of dots is not only determined by the amount of InAs deposited on the AlAs stripes but also by the diﬀusion of In from the surrounding GaAs to the AlAs regions. This is also obvious from Fig. 8.10c, where on the outermost of the nine AlAs stripes the InAs density is considerable larger than that of the inner regions, and where the InAs dots already tend to merge into wires. The reservoir for In diﬀusion towards the AlAs stripes is larger for the outermost layers because of the 1 μm GaAs layer which separates the individual groups.

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Apparently the diﬀusion length of In on GaAs (110) is larger than 35 nm, half the separation of the AlAs layers within the groups, at the given growth temperature, and material transport to the AlAs stripes plays a role in the dot density obtained. Actually, diﬀusion lengths of Ga adatoms on GaAs (110) of between 1 and 15 μm have been reported, depending on growth temperature and As4 pressure [37]. Thus, quite a large diﬀusion length for In can also be expected on GaAs. A simple model for the nucleation of InAs on AlAs assumes that the diﬀusion of In on AlAs is strongly reduced compared to the GaAs regions, which results in a net material transport towards the AlAs stripes, leading to an accumulation of InAs on AlAs (see Fig. 8.11). Thus, the critical thickness is achieved earlier on AlAs and nucleation of dots can occur. In this respect, the low surface mobility of adatoms along the AlAs stripes is probably also helpful. A detailed study of the importance of anisotropic diﬀusion and also the inﬂuence of diﬀerent desorption rates has not yet been performed. Nor is it so far obvious that there is a correlation of the dot positions between the rows, as discussed for example in Sect. 8.2 for the Si/Ge system. The lateral size of the dots is similar to the width of the AlAs layers. To study the inﬂuence of the AlAs layer thickness we have prepared a substrate which contains ﬁve groups each consisting of ten AlAs layers, which are separated by 70 nm GaAs. The thickness of the AlAs layers was varied from group to group from 20, 25, 30, 35 to 40 nm. The groups were again separated by 1 μm GaAs spacer layers. AFM images of cleavage planes of this sample overgrown again with InAs are shown in Fig. 8.12. From these Table 8.1. Geometric properties of InAs quantum dots grown on a substrate with diﬀerent numbers of AlAs layers Number of AlAs layers

1

3

5

7

9

Lateral base width (nm) Average height (nm) Lin. dot density (1/μm)

40–46 21.6 14.9

39–45 20.7 14.5

41–47 21.0 13.3

42–46 21.4 13.5

40–46 21.3 12.0

Fig. 8.11. Simple model for the nucleation process of InAs dots on AlAs stripes embedded in GaAs

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images, it is obvious that the size of the dots is correlated with the thickness of the AlAs layers. The geometrical properties deduced from these AFM images are summarized in Table 8.2. The lateral size is determined by the thickness of the AlAs layers and hence is adjustable by the ﬁrst growth step with the precision of MBE. The dot separation is roughly the same for the diﬀerent stripes, however, the stripes with 40 nm layer thickness are covered much less homogeneously with dots compared to the other groups. This group is already quite close to the (001) surface of the GaAs wafer, which may inﬂuence the dot nucleation. Table 8.2. Geometric properties of InAs quantum dots grown on a substrate with AlAs layers of diﬀerent thickness AlAs thickness (nm)

20

25

30

35

40

Lateral base width (nm) Average height (nm) Lin. dot density (1/μm)

27–35 13.0 13.6

30–40 15.9 12.6

35–45 18.8 14.1

35–45 20.9 11.8

40–50 20.1 13.2

Fig. 8.12. AFM images of 2.5 μm × 2.5 μm regions with InAs quantum dots aligned on groups of AlAs stripes of diﬀerent thickness. a 20 nm. b 25 nm. c 30 nm. d 35 nm. e 40 nm. Each group contains ten AlAs layers with the same thickness embedded in GaAs with a thickness of 1 μm between the groups and 70 nm within the groups. The inset shows corresponding 3d images of the AFM data for a selected region of 1 × 1 μm

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8.3.4 Conclusions and Prospects The AFM results presented here demonstrate that the method of CEO can provide atomically precise templates for the realization of ordered arrays of InAs quantum dots on (110) GaAs/AlAs structures. The dot size can be adjusted by the thickness of the AlAs layers. The growth parameters for this new method of self-organized growth of quantum dots are not yet optimized, but the results achieved so far are very promising. This is also supported by recent photoluminescence measurements with high spatial resolution on overgrown InAs dots on such CEO templates, which reveal sharp luminescence lines from individual dots along the stripes [12, 13]. It is expected that very homogeneous 1D arrays of quantum dots can be achieved in this way, which may be of interest for optical as well as transport studies of coupled quantum dots. For smaller separations of the AlAs stripes, a 2D ordering of the dots seems feasible similar to the 2D ordering observed for strain-patterned Si/Ge as discussed in Sect. 8.2. Acknowledgement. This article is based on the work of various diploma and PhD students at the Walter Schottky Institut of Technische Universt¨ at M¨ unchen. Special thanks go to J. Bauer, R. Schulz, E. Uccelli, and J. Zhu for their contributions. We also want to acknowledge fruitful long-term collaborations with K. Brunner, J.J. Finley and W. Wegscheider. The Si/Ge part of this work was supported ﬁnancially by the BMBF, the EU, and the Volkswagenstiftung, the InAs part by the Deutsche Forschungsgemeinschaft via SFB 348 and SFB 631.

References 1. P. Ballet, J.B. Smathers, H. Yang, C.L. Workman, G.J. Salamo, J. Appl. Phys. 90, 481 (2001) 2. T. Yang, J. Tatebayashi, S. Tsukamoto, M. Nishioka, Y. Arakawa, Appl. Phys. Lett. 84, 2817 (2004) 3. R. N¨ otzel, in Lateral Alignment of Epitaxial Quantum Dots ed. by O.G. Schmidt. Springer Series on Nanoscience and Technology. Springer, Berlin Heidelberg New York (2006), p 305 4. R. Noetzel, K.H. Ploog, J. Cryst. Growth 227–228, 8 (2001) 5. H.J. Kim, Y.J. Park, Y.M. Park, E.K. Kim, T.W. Kim, Appl. Phys. Lett. 78, 3253 (2001) 6. R. Leon, S. Chaparro, S.R. Johnson, C. Navarro, X. Jin, Y.H. Zhang, J. Siegert, S. Marcinkeviˇcius, X.Z. Liao, J. Zou, J. Appl. Phys. 91, 5826 (2002) 7. A. Portavoce, R. Hull, in Lateral Alignment of Epitaxial Quantum Dots, ed. by O.G. Schmidt. Springer Series on Nanoscience and Technology. Springer, Berlin Heidelberg New York (2006), p 397 8. S. Kiravittaya, H. Heidemeyer, O.G. Schmidt, in Lateral Alignment of Epitaxial Quantum Dots, ed. by O.G. Schmidt. Springer Series on Nanoscience and Technology. Springer, Berlin Heidelberg New York (2006), p 489

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9 Stacking and Ordering in Self-Organized Quantum Dot Multilayer Structures G¨ unther Springholz1 and V. Holy2 1

2

Institut f¨ ur Halbleiter- und Festk¨ orperphysik, Johannes Kepler Universit¨ at Linz, 4040 Linz, Austria Department of Electronic Structures, Faculty of Mathematics and Physics, Charles University, 12116 Prague, Czech Republic

9.1 Introduction Strained-layer heteroepitaxy has become a powerful tool for the fabrication of self-assembled semiconductor nanostructures [1–4]. It is based on the natural tendency of highly strained layers to spontaneously form coherent, i.e., dislocation-free three-dimensional (3D) nanoislands on the surface of a thin wetting layer [4–7]. This Stranski–Krastanow islanding is driven by the highly eﬃcient elastic strain relaxation within the islands due to lateral elastic expansion or compression in the directions of their free side faces [4–11]. For islands larger than a certain critical size, the relaxed elastic energy thus gained outweighs the corresponding increase in surface energy, leading to an eﬀective lowering of the total free energy of the system [4–10]. Therefore, this Stranski–Krastanow growth transition occurs in most high-misﬁt heteroepitaxial systems. When the 3D surface islands are embedded in a higher energy band gap matrix material, self-assembled quantum dots with sharp, atomiclike electronic transitions are formed [12–14]. Owing to the statistical nature of growth, however, ensembles of self-assembled dots exhibit considerable variations in size and shape. This results in a large inhomogeneous broadening of the energy levels as well as of the related optical transitions [1–3, 12]. In addition, there is little control over the lateral arrangement and position of the nanoislands. Both factors pose considerable limitations to device applications. Three-dimensional stacking of self-assembled quantum dots in multilayer or superlattice (SL) structures provides an eﬀective tool for controlling the vertical and lateral arrangement of the dots [15–21]. In addition, this increases the total volume of the active material and allows tuning of the electronic wave functions due to the quantum mechanical coupling across the spacer layers [22,23]. In this way, quantum dot molecules [23–25] can be obtained, which are of particular interest for solid-state quantum computation devices [25,26]. Interlayer correlations in multilayer structures have been found to yield signiﬁcant improvements in the size uniformity of the dots [17, 18, 27–30] and may even result in the formation of well-ordered quantum dot superstructures [18, 27]. Diﬀerent types of vertical and lateral correlations have been observed in diﬀerent multilayer systems, ranging from a vertical dot

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alignment in columns for Ge/Si [22, 31–39] or InAs/GaAs [15, 16, 30, 40–43] superlattices, to vertical anticorrelations or fcc-like dot stackings for II– VI [44–46] and IV–VI semiconductors [18, 29, 47–55]. The type of correlation actually formed depends on a large variety of parameters such as spacer thickness [15, 33, 34, 43, 47–55], dot size [43, 53, 54], elastic material properties [18, 45, 56, 57], surface orientation [56], growth conditions [43, 54] as well as the chemical composition of the dots and the spacer layers [58–62]. In addition, a particularly eﬀective lateral ordering has been found for multilayers with staggered interlayer dot correlations [18, 29, 48, 55, 56, 58]. In this chapter, the mechanisms for vertical and lateral ordering in multilayer dot structures as well as the diﬀerent resulting dot stackings are reviewed, with particular emphasis on the prototype Si/Ge, InAs/GaAs and PbSe/PbEuTe dot material systems. Section 9.2 presents a brief overview of the diﬀerent interaction mechanisms and the experimentally observed dot stackings. In Sect. 9.3, the elastic interactions between buried and surface dots are described, introducing the far-ﬁeld and near-ﬁeld approximation and treating in detail the eﬀect of the elastic anisotropy and surface orientation. It is shown, in particular, that the elastic anisotropy plays a crucial role in the ordering process and is responsible for the formation of staggered dot stackings. This conclusion is also supported by Monte Carlo growth simulations. Sections 9.4–9.6 give an overview on the experimental results for InAs/GaAs, Si/Ge and PbSe/PbEuTe multilayers, where for the latter, the various ordering transitions and the phase diagram of ordered structures as a function of spacer thickness and growth conditions are presented. In Sect. 9.7, other interaction mechanisms such as surface morphology, surface segregation and alloy decomposition are discussed, and a summary and outlook is given in the ﬁnal section.

9.2 Mechanisms for Interlayer Dot Correlation Formation Interlayer dot correlations in multilayer structures obviously require some kind of mechanism through which the dots in one layer inﬂuence the growth of the dots in the subsequent layers, i.e., by which the information on the dot positions is conveyed from one layer to another. As shown in Fig. 9.1, conceptually, one can think of three kinds of such mechanisms, namely, (1) chemical processes such as surface segregation, (2) nonplanarized corrugated surface morphologies, or (3) long-range elastic interactions due to the strain ﬁelds emerging from the buried dots. Each of these processes causes a signiﬁcant nonuniformity in properties such as strain, surface diﬀusion and surface chemical potential across the spacer layer. If suﬃciently large, this will affect the nucleation of the subsequent dots and under favorable conditions, vertical and lateral correlations in multilayers are formed. Nucleation of

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Fig. 9.1. Mechanisms for formation of interlayer correlations in self-assembled quantum dot multilayer structures. a, b Interlayer correlations caused by the elastic strain ﬁelds of subsurface dots with nucleation of subsequent dots at the minima on the surface strain energy distribution. Depending on the elastic properties of the spacer layer as well as the surface orientation, these minima can be localized above (a) or between (b) the buried dots. c, d Interlayer correlations caused by nonplanar surface morphologies of the spacer layers. Depending on the dominant mechanism of capillary or stress-driven surface mass transport, subsequent dots may nucleate either on top of the mounds (c) or in the troughs in between (d). e, f Correlations induced by nonuniformities in the chemical composition of the spacer layer due to surface segregation (e) or alloy decomposition (f ). All three mechanisms may induce a vertical dot alignment or staggered dot stacking in multilayer structures

Stranski–Krastanow dots is a rather complex process that depends sensitively on many parameters such as surface stress, lattice-mismatch, thickness and composition of the wetting layer, free energies and local curvature of the growth surface, as well as surface step structure and surface kinetics during growth. On a planar and chemically uniform substrate, these parameters are invariant across the surface and thus, homogenous dot nucleation at random surface sites occurs. In multilayers, however, the variations of strain, topography and/or chemical composition induced by the buried dots induce a preferential nucleation at particular surface sites. As a result, the surface dots become spatially correlated to the dots in previous layers and thus, interlayer dot correlations are formed. This preferential nucleation may be driven by local enhancements of the growth rate, changes in surface diﬀusivity, as well as by decreases in the critical wetting layer thickness or energy barrier for island nucleation. In addition the operation of interlayer interactions in multilayer structures is also manifested by changes in dot size [19, 27, 28, 30, 34, 38, 41, 49, 63–66], density [17, 19, 27, 30, 41, 49, 63], shape [33, 49, 63–66], lateral arrangement [18, 19, 27, 29, 30, 49, 67] as well as critical thickness for dot nucleation [37,38,68] observed in many experiments. Three main eﬀects are important for interlayer correlation formation, namely, (1) elastic lattice deformations around the buried dots induced by the lattic-mismatch [15, 17, 18, 28, 56], (2) corrugations in surface topography due to incomplete surface planarization during overgrowth [69], and ﬁnally (3) surface segregation or alloy decomposition within the spacer layer [60–62]

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that often occur in highly strained material systems. As illustrated schematically in Fig. 9.1, all three mechanisms can in principle give rise to a vertical dot alignment as well as to staggered stackings in multilayer structures. In the case of the interaction via the elastic strain ﬁelds (Fig. 9.1a, b), preferential dot nucleation is generally induced at the minima of strain on the wetting layer surface. Depending on the elastic properties of the materials, these minima may be on top (Fig. 9.1a) or on the surface between the buried dots (Fig. 9.1b), resulting in a vertical or staggered dot stacking, respectively. This will be treated in detail in Sects. 9.3–9.6. The same may result also from correlated dot nucleation resulting from surface corrugations of the spacer layer [69] resulting from incomplete surface planarization. Depending on whether subsequent dots nucleate preferentially at convex or concave surface areas, again a vertical or staggered dot stacking may be formed as shown schematically in Fig. 9.1c and d, respectively. Finally, surface segregation of dot material (Fig. 9.1e) or alloy decomposition of the space-layer (Fig. 9.1f) [60–62], can produce a nonuniform chemical composition on the wetting layer above the dots. This may aﬀect the growth of subsequent dot layers due to the resulting variations in chemical potential, strain as well as eﬀective local wetting-layer thicknesses. As will be discussed in more detail in Sect. 9.7, these eﬀects also may cause a vertical alignment or staggered dot stackings in dependence of the experimental conditions. All three mechanisms may generally act simultaneously and amplify or counteract each other. Therefore, it is not always easy to decide which one is the main driving force for correlation formation for a given experimental condition or material system. Experimentally, diﬀerent dot stackings have indeed been found for different multilayer structures. Figure 9.2 shows some representative examples as revealed by cross-sectional transmission electron microscopy (TEM). The most common case of vertical dot alignment is illustrated in Fig. 9.2a for a self-assembled InAs/GaAs dot superlattice with 20 nm GaAs spacer layers [42], for which the vertical dot alignment was found to persist up to GaAs spacer-layer thicknesses of about 50 nm [15, 16, 40–43]. A similar vertical alignment has been also observed for SiGe/Si dot superlattices [31–38] with up to 70 nm spacer thicknesses [33, 34], as well as for InP/GaInP [70] and GaN/AlN [71–73] multilayers. Staggered dot stackings have been found for IV–VI [18, 47, 48, 55] and II–VI semiconductors [44–46] and two examples are shown in Fig. 9.2b and e. Staggered stackings were also observed for self-assembled InAs/AlInAs quantum wire superlattices [58–61] as shown in Fig. 9.2f. For a given material system even transitions between diﬀerent interlayer correlations have been observed, as exempliﬁed in Fig. 9.2c for a Ge/Si dot superlattice [69] in which the initial vertical alignment switches to an oblique dot correlation (see dashed lines) due to an instability in the planarization process. Transitions between diﬀerent dot stackings have also been reported for PbSe/PbEuTe [48, 55] and CdSe/ZnSe [45] dot superlat-

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Fig. 9.2. Examples for diﬀerent types of interlayer dot stackings in self-assembled quantum dot multilayers observed by cross-sectional transmission electron microscopy. a Vertically aligned (001) InAs quantum dot superlattice (SL) with 20 nm GaAs spacer layers. Adopted from Darhuber et al. [36]. b Fcc-like ABCABC . . . dot stacking in a (111) oriented superlattice of 5 ML PbSe dots alternating with 45 nm PbEuTe spacers. Insert: plan-view transmission electron micrography showing the 2D hexagonal dot ordering within the growth plane. After Springholz et al. [18,47,49]. c Inclined dot correlations in a 1.2 nm Ge/40 nm Si (001) dot multilayer. Adopted from Sutter et al. [69]. d Vertically anticorrelated InAs/AlInAs quantum wire superlattice on InP (001) with 3 ML/10nm thicknesses, respectively. Adopted from Brault et al. [60,61]. e Anticorrelated 2 ML CdTe islands intercalated with 15 ML ZnTe spacers. The white contour lines indicate iso lattice-parameters, i.e., the chemical composition extracted from the atomically resolved TEM image. Adopted from Mackowski et al. [46]

tices as a function of spacer-layer thickness. This will be discussed in detail in Sect. 9.6.

9.3 Strain-Field Interactions in Multilayer Structures Strain is the major driving force for self-assembled Stranski–Krastanow quantum dot formation [4–10]. Therefore, the elastic strain ﬁelds produced by buried quantum dots play an essential role for the interlayer correlations formed in multilayer structures. The strain ﬁelds are caused by the strong elastic lattice deformation around the dots due to the large dot/matrix lattice-mismatch and they extend up to the epilayer surface. During subsequent dot layer growth, these strain ﬁelds impose a bias on the diﬀusion current of deposited adatoms due to the concomitant gradients in the surface chemical potential. At the strain minima, therefore, a local enhancement of growth rate and thus, preferential island nucleation will occur. Additionally, preferential dot nucleation may also be enforced by the local reduction of the island nucleation barrier at these minima. The strain ﬁelds created by the buried dots depend on a large number of parameters such as size, shape and chemical composition of the buried dots, the spacer layer thickness, as well as the elastic properties of the materials and the crystallographic orientation of the growth surface. In order to understand and evaluate the diﬀerent interlayer correlations and stacking

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types, obviously, the details of the elastic strain ﬁelds must be taken into account. For this purpose, it is useful to consider two limiting cases, namely, (1) the far-ﬁeld limit, where the dot depth is large as compared to the dot dimensions, and (2) the near-ﬁeld limit, where the buried dots are very close to the growth surface. In the far-ﬁeld limit, the internal structure as well as the actual size and shape of the dots can be ignored, i.e., the dots can be treated as simple point stress sources. Detailed calculations and experimental observations indicate that this applies well when the dot depth and spacer thickness exceeds about two times the size of the dots [18, 48]. Under the far-ﬁeld condition, the surface strain distribution produced by each buried dot is solely determined by the elastic properties of the matrix material and the surface orientation, whereas the internal structure of the dots can be neglected. Thus, the far-ﬁeld limit is particularly instructive in revealing the general trends of the elastic dot interactions. 9.3.1 The Isotropic Point-Source or Far-Field Model For the simplest case of an elastically isotropic matrix material, the far-ﬁeld stresses created by an individual buried dot can be derived analytically [74, 75]. The result is a radially symmetric strain distribution ε|| (r) =1/2(εxx +εyy ) on the surface given by: 2 − r2 /d2 P , (9.1) ε|| (r) = − 3 · d (1 + r2 /d2 )5/2 independent of surface orientation. In this expression, d is the dot depth underneath the surface, r= (x2 + y 2 )1/2 is the radial distance from the center above the dot, and P is the strength of the point stress source. P is given by P = ε0 V0 (1 + ν)/π [74, 75], where ε0 is the dot/matrix lattice-mismatch, V0 is the dot volume and ν the Poisson’s ratio. On the wetting layer, the strain distribution is the sum of the homogeneous lattice-mismatch strain ε0 and the inhomogeneous strain distribution ε|| (x, y) created by the buried dot. Thus, the strain energy variation across the wetting layer surface ΔEs (r) = Es (r)– Es,0 can be approximated as [18, 19]: 2 − r2 /d2 Es,0 · V0 · (9.2) ΔEs (r) = −(1 + υ) π · d3 (1 + r2 /d2 )5/2 where Es,0 = 2 μ (1 + ν)/(1 − ν). ε20 is the constant background strain energy due to the homogenous dot/matrix lattice-mismatch. Here, higher order terms in ε|| have been neglected because ε|| (r) ε0 in the far-ﬁeld limit, i.e., the strains due to buried point-source are much smaller than ε0 . For the isotropic limit, evidently, the strain energy distribution ΔEs (r) is radially symmetric as well and its minimum resides directly above the buried dot with an amplitude of ΔEs,min = −(1 + ν)/π . Es,0 V0 /d3 . Also, the shape of

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the strain distribution is invariant when plotted as a function of the scaled surface coordinates r/d, which means that the width of the strain energy distribution scales as 1/d. This is a general property of the far-ﬁeld point-source solution. 9.3.2 The Eﬀect of The Elastic Anisotropy In contrast to the above assumptions, most materials of practical interest, in particular most semiconductors, exhibit a rather high elastic anisotropy. As a result, the surface strain distribution produced by a buried stress source is signiﬁcantly altered and, in particular, shows a strong dependence on the surface orientation. To obtain the corresponding strain distributions, the equilibrium stress equations must be solved, taking the true elastic properties of the matrix as well as the boundary condition of a free surface with vanishing normal surface stresses into account. This can be done, for example, using a Fourier method [56, 76, 77]. In cubic materials, the main elastic anisotropy axes are the crystallographic 100 and 111 directions, in which the Young’s modulus E hkl reaches its extremal values. This is illustrated in Fig. 9.3, where the variation of Ehkl as a function of crystallographic direction [hkl ] is plotted for several semiconductors such as PbTe and PbSe (Fig. 9.3a) and Si, Ge, GaAs and ZnSe (Fig. 9.3b). Correspondingly, the 100 or 111 directions represent the elastically hard or soft directions of cubic materials. The degree of deviation from the isotropic case is characterized by the dimensionless anisotropy ratio A = 2c44 /(c11 − c12 ), which is essentially the ratio E111 /E100 of the elastic moduli along the main anisotropy axes. For isotropic materials, Ehkl does not depend on the direction of the applied stresses and thus, A = 1. For anisotropic materials, one has to consider two opposite cases, namely, that A is either larger or smaller than one. In the ﬁrst case of A > 1, the 111 directions are the elastically hard directions and the 001 directions the soft directions, i.e., E111 /E100 > 1. As shown in Fig. 9.3b, this essentially applies to all group IV, III–V and II–VI semiconductors with diamond or zinc blende crystal structure because the chemical bonds are along the 111 directions in this case. The anisotropy is largest for the II–VI compounds, with A = 2.04 for ZnTe and 2.53 for ZnS (Fig. 9.3c). For C, Si and Ge, A increases from 1.21, 1.56 to 1.64, respectively, and for the III–V compounds A ranges from 1.83 for GaAs to 2.08 for InAs. In the opposite case of A < 1, now 100 are the elastically hard directions and 111 the soft directions (E111 /E100 < 1, see Fig. 9.3a). This applies, e.g., for materials with rock salt crystal structure in which the nearest neighbors are along the 100 directions. In particular, for the narrow gap IV–VI semiconductors the elastic anisotropy is particularly large, with A = 0.18, 0.27 and 0.51 for SnTe, PbTe and PbS, respectively. With respect to the elastic strain ﬁelds, obviously, the strongest changes will occur for materials with high elastic anisotropy. Thus, when A strongly deviates from one, the strain distributions will become strongly dependent

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Fig. 9.3. a, b Polar plot of the Young’s modulus Ehkl as a function of [hkl ] direction within the (¯ 110) plane: a IV–VI compounds PbTe and PbSe with rock salt structure and elastic anisotropy ratio A < 1, where 001 are the elastically hard directions with maximum Ehkl value. b Group IV, III–V and II–VI semiconductors with zinc-blende structure and 111 as elastically hard axes (A > 1). c Elastic anisotropy ratio A = 2c44 /(c11 –c12 ) plotted for various cubic semiconductors

on surface orientation. Selecting two materials with large anisotropy but opposite directions of the anisotropy axes, the normalized surface strain energy distributions ΔEs (r/d)/Es,0 induced by a buried point-like stress source are shown in Fig. 9.4a–e for PbTe and Fig. 9.4f–j for ZnSe for diﬀerent surface orientations and scaled surface coordinates r/d. It is evident that not only the energy distributions strongly diﬀer as a function of surface orientation, but that they also show the opposite trend when A is larger (ZnSe) or smaller than one (PbTe). In particular, it is found that only when the surface orientation is parallel to the elastically hard direction (i.e., (100) for ZnSe and (111) for PbTe) the strain energy minimum resides exactly above the buried dot, whereas in all other cases, the energy minima are laterally displaced. Furthermore, when the surface is close to an elastically soft direction, the central energy minimum splits into several minima, as shown in Fig. 9.4c–f for PbTe and ZnSe. In the far-ﬁeld or point-source limit, the directions where the surface minima are formed are unique for each surface orientation and elastic anisotropy ratio. This is because in this approximation the elastic strain ﬁelds do not

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Fig. 9.4. a–j Calculated surface strain energy distributions Es (x, y) above a point-like strained buried quantum dot for diﬀerent surface orientations of PbTe (a–e), with A = 0.28 and ZnSe (f–j), with A = 2.52. The energy distributions are shown as iso-strain contour plots as a function of reduced surface coordinates x/d and y/d, where d is the dot depth below the surface. Darker colors indicate areas of lower strain energy. For most surface orientations the strain energy maximum is displaced from the center of the buried dot. When the surface normal is an elastically soft direction, a splitting into several side minima occurs. k Inclination angle α at which the energy minima appear on the surface relative to the (hkl ) surface normal plotted as a function of the angle β between the (hkl ) and the (100) planes. The diﬀerent (hkl ) orientations are indicated in l

depend on the internal structure and shape of the dots, but only on the elastic properties of the matrix material. In addition, the surface strain distributions scale strictly linearly with the dot depth. This means that for each surface orientation there exists a well-deﬁned characteristic correlation angle α arctan(rmin /d) along which the energy minima appear on the surface relative to the surface normal direction. On the other hand, the in-plane displacement direction of the energy minima on the surface is determined by

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the projection of the elastically hard direction γ hard closest to the surface normal ns onto the surface plane. This direction is therefore given by the vector = (ns × γ hard ) × ns where γ hard is the unit vector along 111 or 100 for A larger or smaller than one, respectively. Figure 9.4k shows the systematic variation of the correlation angle α as a function of the (hkl ) surface orientation for three materials with diﬀerent elastic anisotropy (GaAs, ZnSe, and PbTe). In this plot, the (hkl ) surface orientation is parameterized in terms of the angle β between [hkl ] and the [100] direction. Clearly, there is a systematic variation of α as the surface orientation is tilted from (100) through (111) and (101). In particular, the largest interlayer correlation angle appears when the surface is parallel to the elastically soft direction (i.e., (100) for A >1 and (111) for A < 1), whereas the energy minima are almost vertically aligned when the surface is close to the elastically hard direction. Again, it is evident that the behavior is opposite for materials with anisotropy ratio larger or smaller than one. If we now compare the materials with the same hard axis (i.e., ZnSe and GaAs in Fig. 9.4k), one can see that the larger the elastic anisotropy (AZnSe > AGaAs ), the larger the correlation angle and thus, the larger the lateral displacement of the energy minima. A systematic analysis of this trend shows that the displacements and thus, the correlation angles α depend in a linear way on the anisotropy ratio for A > 1, and on its reciprocal value for A < 1 [56]. This is shown in Fig. 9.5, where the energy minima directions α are plotted for the most relevant high symmetry (111) and (100) surface orientations as a function of elastic anisotropy. For the (100) direction and the

Fig. 9.5. Direction α of the surface strain energy minima relative to a buried force nucleus plotted versus elastic anisotropy of the matrix material for a the (001) surface and the group IV and the zinc-blende III–V and II–VI semiconductors, and b for materials with rock salt structure (IV–VI semiconductors, etc.) and (111) surface orientation. In both cases, the surface normal direction is parallel to the elastically soft direction. The elastic anisotropy is determined by the anisotropy ratio A= 2c44 /(c11 -c12 )

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III–V and II–VI semiconductors (Fig. 9.5a), a splitting of the energy minima occurs if the anisotropy exceeds the critical value of Ac > 1.4, and beyond this value α varies linearly according to α100 = 56◦ × (1 − 1.1 · A−1 ). For the (111) surface direction and materials with rock salt structure (Fig. 9.5b), a splitting occurs for Ac < 0.6, and below this value, α varies according α111 = 50◦ × (1 − A) [56]. Thus, α100 increases from 16◦ , to 23◦ and 32◦ for Si, GaAs and ZnSe, respectively, and α111 from 19◦ , to 36◦ and 41◦ for PbS, PbTe and SnTe. A similar behavior also applies for other surface orientations. 9.3.3 Expected Dot Stackings in the Far-Field Limit When the spacer thickness between the dot layers is large compared to the dot size and the lateral dot spacing is wide such that the strain ﬁelds of neighboring subsurface dots do not signiﬁcantly overlap, then the far-ﬁeld strain calculations can be directly used to predict the interlayer dot correlations in multilayer structures. Provided that the surface strain energy minima created by the buried dots are deep enough to enforce a correlated nucleation, the dots will replicate along the energy minima directions. According to Fig. 9.4, it thus follows that for almost all materials and growth orientations inclined interlayer dot correlations should be formed, with corresponding interlayer correlation angles plotted in Fig. 9.4k. Only when the surface normal is parallel to the elastically hard direction (i.e., either (111) or (100), see Fig. 9.4) for which case the strain minima reside directly above the buried dots, an exact vertical dot alignment along the growth direction is expected under all conditions. A particular situation arises when the surface orientation is close to an elastically soft direction. If then the elastic anisotropy is suﬃciently large, the surface strain distribution splits into several side minima (see Figs. 9.4 and 9.5) and when one new dot nucleates at each of these minima, staggered dot stackings will be formed. For (001) growth and an elastic anisotropy A larger than 1.5, four side energy minima occur above each buried dot (Fig. 9.4f). These minima deﬁne a preferred square arrangement of dots in the subsequent growth plane, with the previous dot located in the center of the square below the surface. In the next layer, this dot arrangement is replicated again, which in total results in an ABAB . . . stacking sequence of the dots, as is as shown schematically in Fig. 9.6a. In the ideal case, this will yield an overall body-centered tetragonal 3D dot arrangement with a lateral alignment of the dots along the in-plane 100 surface directions. For the (111) growth orientation and A smaller than 0.6, three side minima appear in the energy distributions (see Fig. 9.4c), which create a triangle with equal sides along the 2¯ 1¯ 1 surface directions. This induces a triangular or hexagonal dot arrangement in the subsequent layer centered above the buried dots below the surface. Each of these new dots will induce the same triangular arrangement in the subsequent layer, and thus it takes in total three of these layers until the dots appear again at the same line along the growth direction

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Fig. 9.6. Staggered three-dimensional dot stackings expected from the pointsource strain calculations for materials with high elastic anisotropy and growth along the elastically soft direction. a ABAB . . . stacking and centered tetragonal dot lattice for the (001) growth and anisotropy ratio A 1.5. b Fcc-like ABCABC . . . dot stacking and resulting trigonal 3D dot lattice for the (111) growth orientation and materials with A < 0.6, as observed for PbSe/PbEuTe superlattices (see Fig. 9.2b)

as the initial dot. Consequently, an ABCABC. . . dot stacking is formed, as shown schematically in Fig. 9.6b. The overall dot arrangement is similar to the atom stacking in face-centered cubic lattices and consists of sheets of 2D hexagonally ordered dots separated by the spacer layers. Because in general, the ratio between the lateral dot spacing and the vertical spacer thickness is not equal to 1.155 as in fcc lattices, the whole dot arrangement represents an overall 3D trigonal lattice of dots. As shown in Fig. 9.4e, two well-deﬁned side energy minima may also occur for the (110) growth orientation. Accordingly, this would give rise to the formation of vertical sheets of 2D rhombohedrally ordered dots in a multilayer structure. 9.3.4 Comparison with Experimental Results The Low-Indexed (111) and (100) Growth Directions Experimentally, interlayer correlations in multilayer structures have been studied in detail only for the main (100) and (111) growth orientations and the prototype InAs/GaAs (100) [15, 16, 40–43], Si/Ge (100) [19, 22, 31, 33–36] and PbSe/PbEuTe (111) [18, 29, 47–55] material systems. In these cases, the growth direction is parallel to the elastically soft direction and therefore, staggered dots stackings are expected in the far-ﬁeld limit. For self-assembled (111) oriented PbSe/PbEuTe dot superlattices, indeed a well-deﬁned ABCABC. . . dot stacking is formed as predicted by the pointsource model for PbEuTe spacer thicknesses ranging from 40 to 55 nm [18]. This is exempliﬁed by the cross-sectional TEM image shown in Fig. 9.2b and will be discussed in detail in Sect. 9.6. The experimentally determined

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interlayer dot correlation angle of α = 39◦ [18, 47, 48] agrees remarkably well with the predicted theoretical value of 36◦ . Moreover, an eﬃcient inplane 2D hexagonal ordering occurs (see inset of Fig. 9.2b) [29], with an in-plane dot separation that scales linearly with spacer thickness [18, 48]. Thus, self-organized fcc-like 3D lattices of dots with tunable lattice constant and trigonal lattice structure are obtained [18]. On the other hand, when the dot layer separation is decreased to below 40 nm, a transition from the fccstacking type to a vertical dot alignment is found [48–55]. As will be discussed in Sect. 9.6, this is due to the ﬁnite size of the dots and the characteristic changes in the strain ﬁelds in the near-ﬁeld limit. The situation is quite diﬀerent for the Ge/Si and InAs/GaAs (100) cases. In spite of the fact that a strain minima splitting is predicted in the farﬁeld limit (Fig. 9.4), experimentally, mostly a vertical dot alignment has been observed, as shown for example in Fig. 9.2a. This is due to the fact that the smaller elastic anisotropy results in a much smaller lateral splitting of the strain minima and that in most experimental studies rather small spacer thicknesses in the range of 10–40 nm were used. As will be shown in Sect. 9.3.6, the far-ﬁeld approximation does not hold in this case, which means that the actual size and shape of the dots has to be taken into account in the strain calculations. Even for larger spacer thickness, the energy minima splitting is not very large, e.g., for a spacer thickness of 50 nm, Lmin is only 20 and 34 nm for the Si/Ge and InAs/GaAs cases, respectively. This is less than the typical lateral island sizes, and thus, it is not possible for the dots to occupy just one single energy minimum. As a result, the expected centered-tetragonal dot stacking shown in Fig. 9.6a cannot be formed, which means that for Si/Ge and III–V structures, the elastic anisotropy seems to be not suﬃciently large for staggered dot stackings in the useful range of spacer thicknesses. An exception is a cross-sectional scanning tunnel microscopy study of Liu et al. [78] that has revealed a staggered dot correlation in InGaAs/GaAs multilayers with large spacer thicknesses. Anticorrelated dot stackings For II–VI semiconductors, the elastic anisotropy is signiﬁcantly larger than for the III–V compounds. Thus, there is a stronger tendency for the formation of staggered dot stackings. Indeed, an “anticorrelated” interlayer dot arrangement has been found for (100) oriented CdSe/ZnSe [44, 45] as well as CdTe/ZnTe [46] multilayers, as is exempliﬁed by the cross-sectional TEM of vertically anticorrelated ZnTe monolayer islands shown in Fig. 9.2e [46]. This anticorrelation is quite similar to the theoretically expected ABAB . . . stacking shown in Fig. 9.6a and results from the fact that a local energy maximum is formed directly above each buried dot (see Fig. 9.4f). Therefore, this point is unfavorable for subsequent island nucleation. On the other hand, the four side minima of the surface strain distributions are separated

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only by a weak saddle point (see Fig. 9.4f). Therefore, there is no strong lateral ordering tendency of the surface islands during growth and the dots remain rather disordered in the lateral direction. As a consequence, no ordered centered-tetragonal dot lattice but only an anticorrelated dot stacking is formed. The experimentally derived interlayer correlation angles of theses structures [44–46] are also found to be signiﬁcantly larger than those expected from the point-source model. For example, α is deduced as 40◦ from the TEM image of Fig. 9.2e as compared to the expected theoretical value of 28◦ of Fig. 9.4k. This may be attributed to the modiﬁcations of the surface strain distributions due to the overlapping strain ﬁelds of neighboring buried dots. The eﬀect of overlapping strain ﬁelds has been modeled, e.g., by Shchukin et al. [57], by considering the elastic interaction energy between sheets of periodic 2D arrays of strained nanoislands using a Green’s function approach. Calculating the interaction energy as a function of the relative vertical and lateral displacements of the 2D island arrays in successive layers, it was found that for a certain spacer thickness range the interaction energy is minimized when the islands are anticorrelated in successive layers. For II–VI superlattices, thus, a transition from a vertical dot alignment to an anticorrelated stacking was predicted for spacer thickness exceeding more than 3 times the lateral array period, in good agreement with the experimental observations [46, 54]. Similar calculations were also performed by Kunert and Sch¨ oll for the InAs/GaAs system [161]. As shown in Sect. 9.3.6, the single-dot strain ﬁeld model also predicts such a transition in the near-ﬁeld limit. In any case, the high elastic anisotropy of the II–VI compounds is the crucial prerequisite for the vertical anticorrelation formation. The limitation of the Shchukin model [57] is that it does not allow a prediction on the lateral ordering of the dots because in this model a perfectly periodic lateral dot array must be assumed as a starting point of the calculations. Anticorrelated stackings have also been observed for InAs/AlInAs quantum wire superlattices on InP (001) [58–60, 79, 80], for which an example is shown in Fig. 9.2d. Although this anticorrelation again complies with the basic trend deduced from the anisotropic strain ﬁeld model, in this case, the observed interlayer correlation angles of around 40◦ are also signiﬁcantly larger than those predicted by the point-source model. Therefore, additional interlayer interaction mechanisms have been invoked as the origin for the anticorrelated stacking in this case [60, 61]. This will be discussed in detail in Sect. 9.7. A particularly interesting mixed interlayer stacking was found for twofold stacked InAs quantum dots on lithographically prepatterned (001) GaAs substrate templates [81]. In this case, two diﬀerent dot types are formed in the second layer, one in the “on top” positions and one in the “staggered” positions. As shown by Heidemeyer et al. [81], this special dot stacking is again caused by elastic anisotropy of the spacer layers, which once more underlines the important role of the elastic anisotropy in the stacking of dots in multilayer structures.

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High-Indexed Growth Orientations Quantum dot multilayers on vicinal or high-indexed surfaces represent a particularly interesting case for interlayer correlation formation. This is because of the highly anisotropic surface strain distributions as well as the inclined interlayer dot correlations predicted by the far-ﬁeld strain interaction (Fig. 9.4). In addition, vicinal and high-indexed surfaces exhibit a high anisotropy of almost all surface properties such as adatom diﬀusion barriers as well as surface and step edge energies. However, in spite of numerous studies of single layer self-assembled quantum dot growth on vicinal and high-indexed surfaces (see, e.g., Refs. [92,93]) only very recently, a systematic study has been performed on the interlayer correlations in high-indexed multilayer structures by Schmidbauer et al. [82]. The investigated structures consisted of 16 period InGaAs/GaAs quantum dot superlattices and 10 ML InGaAs dots alternating with 120 ML GaAs spacer layers grown on high-indexed (n11)B GaAs substrates with n = 3, 4, 5, 7 and 9. Apart from the remarkable lateral ordering attributed to the anisotropic surface diﬀusion [82, 83], the direction of the interlayer dot correlations determined by high-resolution X-ray diﬀraction was found to be inclined to the surface normal towards the in-plane [2nn] directions as predicted by our strain-ﬁeld model. The experimentally determined inclination angles were 8◦ , 17◦ , 19◦ , 36◦ and 21◦ for the (311), (411), (511), (711) and (911) surfaces, respectively [82]. This is remarkably close to the respective angles of 13.5◦ , 16.3◦ , 18.4◦ , 21.5◦ and 22.5◦ predicted by the point-source model (Fig. 9.4). The elastic strain-ﬁelds of the buried dots were also calculated for this case using the ﬁnite element method [82]. For the given lens-shaped dots of 30 nm width and 34 nm GaAs spacer thickness, it was found that the calculated energy minima directions are in quite good agreement with the measured correlation directions. In addition, angles deduced from the ﬁnite element calculations are quite close to those obtained from the point-source approximation, indication that this approximation holds for GaAs spacers as thin as 34 nm. 9.3.5 The Strength of Interlayer Strain Interactions Apart from the modeling of the dot stackings, the point-source model can also be utilized to asses the magnitude of the interaction energies required to induce interlayer correlation formation. Comparing a large number of experimental investigations, it turns out that for various material systems and multilayer structures, interlayer dot correlations are found to persist to spacing layers up to 40–60 nm [15,33,34,43,48] (see also Sects. 9.4–9.6). Since for these spacer thicknesses the point-source approximation already holds, one can readily calculate the corresponding depth of the energy minima ΔEs,min produced by the buried islands at a depth d, below the surface using the simple expression of ∗ · Es,0 V0 /d3 ΔEs,min = −Chkl

(9.3)

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where C*hkl is a constant that depends only on the elastic constants of the materials and the chosen (hkl ) surface orientation, Es,0 is the constant strain energy density on an unperturbed 2D wetting layer induced by the layer/substrate lattice-mismatch, and V0 is the island volume. From the farﬁeld calculations, C*100 is obtained as 0.52 for Si and GaAs (100) and C*111 0.69 for PbEuTe (111). In addition, the misﬁt strain energy Es,0 turns out to be of the same order of magnitude of ∼100 meV/atom pair in all three material systems. To estimate the volume of the buried islands, we take into account that self-assembled quantum dots usually exhibit a pyramidal shape with welldeﬁned side facets. For Ge these can be either {105} facets for hut cluster islands [84] (which we consider here), or higher indexed facets for the larger dome-shaped islands [19, 85–87]. For InAs islands, several diﬀerent facets have been reported [88,89], but for simplicity we approximate them by {113} facets, whereas for (111) PbSe islands only {100} facets have been observed [90]. Using the typical dot base widths of 60 nm for Ge and 30 nm for InAs and PbSe islands, one can then calculate the corresponding island volumes as 7200, 4270 and 1600 nm3 , respectively. If we further assume that the eﬀective island volume is preserved during overgrowth, then for the experimentally observed cut-oﬀ interlayer spacing of around 50 nm, a corresponding depth of the strain energy minima ΔEs,min of only 1 – 2 meV per atom pair is found for all three material systems. This means that surprisingly low surface strain energy variations are suﬃcient to induce a correlated dot nucleation in multilayer structures. Although, the maximal correlation thickness obviously depends on the actual island sizes and growth conditions, the magnitude of these energy variations is evidently not only a factor of 20 smaller than the homogenous mismatch strain energies Es,0 , but also one order of magnitude lower than the typical thermal energies kB T during growth. Therefore, the exact microscopic mechanism by which such small energies modify the nucleation process has not yet been resolved. 9.3.6 The Near-Field Limit of Elastic Interactions For multilayers with small spacer-layer thicknesses, the buried dots can no longer be treated as simple point stress sources but their actual size, shape and compositional variations must be taken into account. These parameters, however, diﬀer strongly from one material to another and strongly depend on the growth conditions, dot layer thickness and utilized growth technique. For example, for (001) SiGe/Si dots, hut cluster islands with {105} facets are formed at low growth temperatures and small coverages [19, 84], whereas for thicker dot layers and higher growth temperatures they are transformed into dome shaped islands with {113} facets [85–87], including the existence of several transitional shapes during the transformation process [86, 87]. Even steeper {111} side facets are formed for SiGe dots grown by liquid phase

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epitaxy [91]. For (001) InAs/GaAs islands, multifaceted islands composed of {317}, {011} and {111} facets have been reported [88, 89]. Other shapes have been found for other growth orientations, such as asymmetric InAs or Ge nano-islands on high-indexed GaAs (113)A [92] or Si (113) substrates [93]. For PbSe/PbTe (111) islands, pyramids with triangular base and {100} side facets were observed [18,90]. A further complication arises from the fact that during overgrowth signiﬁcant changes in dot shape and composition occur due to intermixing with the surrounding matrix material [77,94–98]. This intermixing strongly depends on the growth conditions [96, 97] and changes the chemical composition of the dots as well [77, 97, 98]. As a consequence, no general solution of the near-ﬁeld strain interactions can be given but each particular experimental situation and material system must be considered separately. To calculate the strain ﬁelds of near-surface dots several methods can be used [77]. When the elastic constants between dots and matrix material do not diﬀer too much, the strain ﬁelds can be obtained by convoluting the point-source solution with the given dots shape [18, 77]. Alternatively, one can apply ﬁnite element methods [48,82,97,99–103] or atomistic calculations using semiempirical atom potentials [104–107]. These methods have been employed extensively for InAs/GaAs [102, 104] and Si/Ge [97, 103–106] dots but also for other materials such as InN/AlN [108]. As shown by Pryor et al. [104], all three methods give quite similar results for the strain ﬁelds well outside of the buried dots, as applies for the surface strain ﬁelds relevant for multilayer structures. As a general trend, it turns out that for near-surface dots, in all cases the strain ﬁelds are focused in the surface normal direction, i.e., the surface energy minima are conﬁned more closely to the region directly above the dot. This arises from the fact that the free surface allows a very eﬃcient strain relaxation due to the outward or inward relaxation of the lattice planes. The focusing of the strain ﬁelds of near-surface dots is demonstrated in Fig. 9.7 for (001) InAs/GaAs dots. In this example, the surface energy distributions were calculated for truncated InAs pyramids with ﬁxed size and shape for diﬀerent dot depths below a GaAs spacer layer using the semianalytical approach [77]. The size of the square-based pyramids was ﬁxed to 20 nm in width and 7 nm in height, similar to what has been found in cross-sectional STM studies [66, 98], but possible gradients in the chemical composition of the dots due to intermixing were neglected. Figure 9.7a–d shows the calculated surface strain energy distributions for dot depths of d = 20, 30, 50 and 70 nm, respectively. Whereas for large depths of d ≥ 50 nm (i.e., d larger than twice the dot width) the surface strain energy distributions with four side minima along

110 closely resemble that of the point-source model (compare Fig. 9.7c and d with Fig. 9.4f), with decreasing dot depth the lateral spacing of the energy minima shrinks and eventually they merge into one single minimum located directly above the dot (Fig. 9.7a). As a consequence, for multilayer structures with small spacer thicknesses, InAs dots will be always aligned in the vertical growth direction.

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Fig. 9.7. a–d Iso-strain energy contour plots of the surface strain energy distribution above a buried InAs quantum dot with truncated pyramidal shape, 20 nm base and 7 nm height located in a GaAs matrix at diﬀerent dot depths of d = 20, 30, 50 and 70 nm from a to d, respectively. The darker color corresponds to lower strain energy on the wetting layer, and the separation of the contour lines decreases each by a factor of 4 from 0.2 meV to 0.003 meV from a to d, respectively. The dashed squares indicate the base of the InAs islands. e Direction α of the energy minima relative to the surface normal. f Lateral separation L of the energy minima plotted as a function of the InAs dot depth

Figure 9.7e shows the directional angle α of the surface energy minima relative to the surface normal as a function of the GaAs spacer thickness. For large spacer thicknesses, this angle converges to that of the far-ﬁeld pointsource approximation of α = 25◦ , but at spacer thicknesses below 40 nm, α rapidly drops to zero such that at d ≤ 25 nm only one central minimum is formed. The dependence of the lateral energy minima separation L as a function of dot depth is shown in Fig. 9.7f. For large d, α is constant and thus, L increases linearly with increasing spacer thickness. Below d = 40 nm, again L rapidly drops to zero. For the formation of a staggered ABAB. . . dot stacking as shown in Fig. 9.6a, the spacing of the energy minima L must be larger than the base width b of the dots thatis indicated by the dashed squares in Fig. 9.7a–d. For the chosen dot geometry therefore, d must be larger than 40 nm to meet this stacking condition, as is indicated by the dashed lines in Fig. 9.7f. For such large spacer thicknesses, however, the energy minima and thus, the correlated dot nucleation probability is already so small that it becomes very diﬃcult to sustain any interlayer correlations.

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We have also made similar calculations for the SiGe (100) case. Owing to the smaller elastic anisotropy, the splitting of the surface energy minima is even smaller than in the InAs/GaAs case and, in addition, the dots exhibit a much ﬂatter and wider island shape. Therefore, the spacer thickness must be well above 100 nm in this case in order to obtain a surface energy minima splitting that is suﬃciently large for the formation of ABAB. . . dot stacking. This is clearly well beyond the spacer-layer thicknesses up to which interlayer correlations have been observed in this material system [19, 31, 33, 34]. Thus, there seems to be little chance that such a dot stacking can be experimentally obtained for the Si/Ge system. The same general trend, i.e., the focusing of the strain ﬁelds in the vertical direction applies for all other materials and growth orientations, and a further example of this eﬀect will be given in Sect. 9.6 for (111) PbSe/PbEuTe dots. Recently, ﬁnite element calculations of buried dot strain ﬁelds have also been performed for high-indexed GaAs surfaces for the case of InGaAs/GaAs superlattices with 33 nm GaAs spacers [82]. It was found that for the (n11)B surfaces, the inclined interlayer correlations predicted by the point-source model were in good agreement with those obtained from the ﬁnite element calculations [82], except for surface directions close to (100), where only one strain minimum instead of four side minima was found. The fact that the experimentally measured inclined interlayer correlations for high-indexed GaAs surfaces (see Schmidbauer et al. [82]) are quite close to the angles predicted by the point-source model shows that this approximation indeed works well when the spacer thickness is suﬃciently large. The main general conclusion from such calculations is that even for materials with high elastic anisotropy, strain-induced staggered dot stackings can be formed only for spacer thicknesses above a certain critical value. Detailed experimental investigations that support this conclusion are presented in Sect. 9.6. 9.3.7 Lateral Ordering in Monte Carlo Growth Simulations In multilayer structures, the growth generally starts with a ﬁrst quantum dot layer that is just as disordered as a single-dot layer grown under the same conditions. Owing to the interlayer dot interactions, this lateral dot arrangement is changed in the subsequent layers and the questions arise: (1) how these rearrangements evolve as a function of deposited superlattice periods, (2) what ﬁnal structure is eventually formed and (3) under which conditions a self-organized lateral ordering will take place. Obviously, lateral dot ordering requires a signiﬁcant overlap of the dot strain ﬁelds and/or a splitting of the surface strain energy minima. Otherwise, each dot will merely replicate along the strain minima direction and the disordered dot arrangement of the ﬁrst layer will simply be replicated from one layer to another. This is expected for dilute and low-density dot systems with large lateral dot separations compared to the spacer-layer thickness.

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When the dot density is high, the strain ﬁelds of subsurface dots strongly overlap. Therefore, the locations of the local surface strain minima depend on the relative arrangement of the buried dots and they will be modiﬁed and shifted with respect to the buried dot positions. Even more, for closely spaced dots several minima may merge into one minimum and new minima may be introduced between widely spaced buried dots [17]. In each new layer, therefore, the dot positions will be rearranged until a nearly steady state conﬁguration is reached, which under favorable conditions may exhibit some kind of short-range lateral ordering as well [17,56,109]. A much stronger eﬀect is expected when due to the elastic anisotropy, the surface energy minima above each dot are split into several side minima. Then, the strain energy distribution is drastically altered in each layer and thus, a much faster dot rearrangement and also much more eﬃcient lateral ordering will occur. To assess the lateral ordering tendency in diﬀerent multilayer structures, it is instructive to simulate theoretically the dot rearrangements taking place during multilayer growth. This can be done by introducing a few simple assumptions, namely, that (1) the dot nucleation sites are determined by the local minima of the surface strain ﬁelds of the buried dots due to the attraction of the deposited adatoms, and (2) that the size of the dots created at each energy minimum is proportional to the surface area from which the adatoms are collected [17]. Neglecting the highly complex atomistic processes involved in nucleation and growth of Stranski–Krastanow dots, Tersoﬀ et al. [17] have devised a deterministic continuum growth model that in 1 + 1 dimensions demonstrates a rapid homogenization of the size and lateral dot spacings in a superlattice structure. In our work, we have modiﬁed this model to include explicitly the elastic anisotropy and growth orientation of the materials and capture the 2D arrangement of the dots within the growth plane by extending the simulations to 2 + 1 dimensions [56]. In addition, the dot growth in each layer was simulated using a Monte Carlo approach. The total simulation procedure thus consists of the following steps [56, 109]: 1. Random deposition of atoms. 2. Surface diﬀusion of the adatoms along the gradients in the surface chemical potential determined by the strain ﬁelds of the buried dots until the nearest strain energy minimum is reached. The atoms gathered in a particular minimum create a quantum dot with a size proportional to the number of collected adatoms. The latter is proportional to the Voronoi polygonal area around each island. 3. After deposition of 104 adatoms, the surface is covered by a spacer layer with a given thickness and the new surface energy distribution due to the buried islands is calculated by summation of the strain ﬁelds of all islands in the ﬁrst subsurface dot layer. The small contribution of the lower dot layers in the superlattice stack is neglected. This sequence is repeated N times for the desired number of superlattice periods. The ﬁnite lateral extent of the dots is taken into account by

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excluding nucleation of new dots within a certain radius Rd around already existing dots. This denuded zone also ﬁxes the dot density in the ﬁrst layer and it can be associated with an eﬀective surface diﬀusion length. As a further simpliﬁcation, we consider the buried islands as point-like stress sources, corresponding to the far-ﬁeld limit of strain interactions, and we also ignore the strain ﬁelds induced by the surface dots during the growth of each dot layer. This is because these lateral strain ﬁelds are negligible at the stage of dot nucleation and thus are not expected to inﬂuence much the nucleation sites of the dots. To speed up the simulations and thus allow coverage of very large surface areas (> 1 μm2 ) and large dot ensembles, the hopping of deposited adatoms is performed on a surface mesh of 2.5 × 2.5 nm and whole adatom clusters of this size are moved at the same time. Figure 9.8 shows the simulation results for three representative cases, namely, an elastically isotropic dot multilayer with arbitrary growth orientation (Fig. 9.8a), a (100) oriented SiGe dot superlattice where the Ge dots are separated by 20 nm Si spacer layers (Fig. 9.8b), and a (111) oriented PbSe dot superlattice with 50 nm PbEuTe spacer layers (Fig. 9.8c). In Fig. 9.8, for each case, the dot positions within the 9th, 10th, 11th, and 12th layers are represented by diﬀerent symbols and plotted on top of each other. For the elastically isotropic case (Fig. 9.8a), the dots are vertically stacked on top of each other but there is obviously not any lateral ordering tendency. This is

Fig. 9.8. Monte Carlo growth simulations of self-organized quantum dot superlattices showing the island positions in the last superlattice dot layers. a The 9th (crosses) and 10th (open circles) layers of an elastically isotropic superlattice. b The 9th (crosses) and 10th (open circles) layers of a (001) SiGe/Si dot superlattice with 200 ˚ A Si spacer layers. c The 10th (ﬁlled circles), 11th (open circles) and 12th (ﬁlled triangles) layers for a (111) PbSe/PbEuTe superlattice with 500 ˚ A PbEuTe spacers. The inserts show the 2D Fourier power (FFT) spectrum of the island positions in the last superlattice layer. Lower insert in b: the elastic energy distributions above a buried SiGe island shown as iso-strain energy plot with the darker areas indicating regions of higher strain energy. Thus, there exist four strain energy maxima along the 011 surface directions, which induce a preferential in-plane dot alignment along the 010 surface directions. See Holy et al. [56] and Springholz et al. [19] for details

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underlined by the Fourier transform (FFT) power spectrum of the ﬁnal dot positions shown in the insert. This clear discrepancy to the isotropic 1 + 1D growth simulations by Tersoﬀ et al. [17] is due to the fact that a uniform dot spacing in one dimension is not suﬃcient to produce an ordered structure in two dimensions if there is no additional in-plane anisotropy. For the SiGe/Si (100) growth simulation (Fig. 9.8b), because the assumed denuded zone Rd of 50 nm is signiﬁcantly larger than the lateral separation of the energy minima of L ≤ 12 nm, the dots are nearly perfectly aligned in the vertical growth direction. In addition, however, there is also a weak lateral ordering tendency along the in-plane 010 directions as is indicated by the arrows in the 2D FFT power spectrum of the dot positions shown in the upper insert of Fig. 9.8b. A detailed analysis shows that this preferred dot alignment is due to the fact that above each dot four maxima in strain energy appear on the surface along the 011 directions away from the energy minima [109]. This is shown by the lower insert of Fig. 9.8b, where the strain energy distribution on the 20 nm Si spacer above a buried Ge dot is depicted over an extended surface area and where the strain maxima are indicated by the dark color. These maxima cause a repulsive interlayer dot interaction and therefore, the subsequent surface dots tend to nucleate along the 010 surface directions relative to each other. The same result is also found in simulations for (100) InAs/GaAs dot multilayers because the symmetry and basic shape of the strain distribution is similar to that for the SiGe system. Owing to the higher elastic anisotropy, however, the observed lateral ordering tendency is even enhanced. For the third case of (111)-oriented PbSe/PbEuTe superlattice growth simulations shown in Fig. 9.8c, clearly the lateral ordering is not only much more eﬃcient but also the expected ABCABC . . . dot stacking is well reproduced (compare relative dot positions in the 10th, 11th and 12th layers). Even more, the lateral dot spacing in the ﬁnal layer√of 63 nm is exactly equal to the spacing of the surface energy minima L = 3d tan α (with dspacer = 49 nm and α = 36◦ ) calculated from the point-source model, and thus reproduces very well the experimental results [18, 29, 48] (see also Sect. 9.6.1 for more details). Figure 9.9a–d shows the evolution of the PbSe dot arrangement in the simulations as a function of the superlattice period N = 1, 5, 10 and 15, respectively, with the FFT power spectra of the dot positions depicted as inserts. Whereas the dots in the initial layer are completely disordered, after ﬁve superlattice periods a clear hexagonal lateral ordering has already taken place, as signiﬁed by the appearance of satellite peaks in the FFT power spectra (see inserts). With increasing number of periods, these satellite peaks become increasingly better deﬁned and additional higher order satellites appear. Thus, the hexagonal ordering rapidly improves. Figure 9.9e shows the corresponding evolution of the size distribution as a function of the number of superlattice periods. Obviously, there is a transition from the small dots of the initial layer, for which the dot size is determined by the chosen capture radius and the deposited number adatoms, to

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Fig. 9.9. a–d Monte Carlo simulation of self-organized (111) PbSe/PbEuTe dot superlattice growth, showing the island positions (triangles) on the surface after the N = 1st, 5th, 10th, and 15th superlattice period from a–d, respectively. The inserts show the 2D Fourier power spectra of the island positions indicating the progressing 2D hexagonal in-plane ordering. e shows the evolution of the dot size histograms as a function of number of superlattice periods N , showing a signiﬁcant reduction of the relative width of the distributions after about four superlattice periods. Adapted from Springholz et al. [47]

a by about a factor of 2 larger size in the multilayer stack that is caused by the reduction of the dot density due to the interlayer dot correlations. During the concomitant lateral ordering process, the size distribution at ﬁrst signiﬁcantly broadens for the ﬁrst few superlattice layers but then rapidly narrows, indicating a progressive size homogenization. After about ten layers, the relative width of the size distribution normalized to the average dot size saturates at a value of about half of that of the initial layer. This indicates that the lateral ordering in multilayers can indeed improve the uniformity of self-assembled quantum dots. The observed transient broadening of the size distribution is caused by the extensive rearrangement of the dot positions in the ﬁrst superlattice layers, which gives rise to a larger variation in the dot size (see Fig. 9.9e). Again, this transient behavior is in remarkable agreement with our experimental observations described in Sect. 9.6. Other growth simulations focusing on various other aspects of ordering and correlation formation such as the inﬂuence of the growth temperature can be found in Refs. [110–113]. An important conclusion from the growth simulations is that there actually exist two diﬀerent ordering mechanisms in multilayer structures. The ﬁrst is based on the extended repulsive strain ﬁelds from the subsurface dots, which results in an alignment of the surface dots along certain surface directions in which the mutual dot repulsion is locally reduced. The second mechanism is caused by the attractive strain ﬁelds of buried dots and is operative when the surface strain energy distribution above each dot splits into several side minima. In the ﬁrst case, the dots are still aligned along the ver-

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tical growth direction and the lateral ordering tendency is rather weak and essentially only of the short-range type. In the second case, the interlayer dot correlations are of staggered type and the lateral ordering is more eﬃcient and of longer range. In both cases, the anisotropy of the strain ﬁelds on the surface is a crucial prerequisite for the ordering process; without this anisotropy no lateral ordering is found even for idealized growth simulations. This conclusion also holds for the near-ﬁeld strain interactions when the dots can no longer be approximated by point-sources. Lateral ordering is counteracted by thermal disorder induced by random surface diﬀusion and nucleation at other surface sites. Thus, in general the ordering process in actual growth experiments will be more gradual and less eﬃcient than in the above described growth simulations and it will also depend signiﬁcantly on the growth conditions. In addition, it is noted that in the near-ﬁeld limit, anisotropic surface strain ﬁelds may also be introduced by anisotropies in the island shapes. Thus, for higher-indexed growth orientations and growth conditions that produce highly anisotropic dot shapes, a more eﬃcient lateral dot ordering can be expected. This conclusion is also supported by recent growth experiments [58–61, 82, 83].

9.4 Experimental Results for InAs/GaAs Superlattices The formation of interlayer correlations was ﬁrst observed in InAs/GaAs quantum dot superlattices [40, 41]. In these works, it was found that the InAs dots are vertically aligned along the (100) growth direction forming one dimensional dot columns up to GaAs spacer thicknesses of around 400 ˚ A [15, 16, 43] (see Fig. 9.10). For larger spacer thicknesses, this interlayer correlation is lost, i.e., the dots in each layer nucleate independently from those in the previous layers. This transition has been studied by Xie et al. [15] using cross-sectional TEM (Fig. 9.10a–c) from which the pairing probability of the dots in subsequent layers was determined as a function of spacer thickness, as shown in Fig. 9.10d. For spacer thicknesses of up to about 100 ˚ A the interlayer pairing probability was found to be equal to one, but then gradually decreased until at about 500 ˚ A spacer thickness no interlayer correlations were found any more. Similar results were reported in a cross-sectional STM study by Legrand et al. [43], showing that the pairing probability sensitively depends also on the InAs dot size, i.e., that for larger dots, interlayer correlations persist to larger spacer thicknesses. The spacer thickness d1/2 at which the interlayer pairing probability drops to one half was found in both studies to be in the range of 200–300 ˚ A for growth temperature around 500 ◦ C and InAs dot layer thickness of 2 – 2.3 ML, which yields InAs dots with a typical diameter of 100–200 ˚ A. For multilayers with diﬀerent dot size, Legrand et al. [43] the crossover spacer thickness d1/2 was found to scale linearly with the dot diameter [43].

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Fig. 9.10. a–c: Cross-sectional TEM images of InAs/GaAs multi layers with GaAs spacer thickness of a 42 ML, b 92 ML and c 36 ML. The TEM image of a perfectly correlated InAs/GaAs superlattice with ﬁve periods and 36 ML GaAs spacers is depicted in c. d Pairing probability (open squares) of self-assembled InAs dots deduced from the TEM images as a function of GaAs spacer thickness. The ﬁlled circles show the ﬁt to the experimental data using a model described in Ref. [15]. Adopted from Xie et al. [15]. e, f Atomic force microscopy images of a single InAs dot layer (e) and of a 30-period InAs/GaAs (001) dot superlattice (f ) of 3 ML InAs alternating with 6.5 nm GaAs. The horizontal axis is along the [110] direction. The 2D Fourier Transform (FFT) power spectra of the AFM images, shown as inserts, indicate a lateral ordering of the dots along the 100 directions. Adopted from Solomon et al. [67]. g, h Cross-sectional scanning tunneling microscopy images of ﬁvefold stacks of self-assembled 2.4 ML InAs dots separated by 10 nm GaAs spacer layers showing the strong changes in dot shape and size as a function of the number of bilayers. Image size: 150 × 150 nm2 topography image (g) and enlarged 55 × 55 nm2 current image (h). Adopted from Bruls et al. [66]

The pairing probability in the vertical dot columns was explained by a model in which the lowering of the strain energy above the buried dots leads to an attraction of mobile surface adatoms [15, 43]. The local nucleation probability was then assumed to be proportional to the amount of accumulated InAs atoms at the strain minima. For small spacer thickness and correspondingly deeper energy minima, all deposited adatoms within the surface diﬀusion length are attracted, leading to a unity dot pairing probability. With increasing spacer thickness, the depth of the energy minima (see Sect. 9.3.6) and thus, the pairing probability diminishes until the minima become too weak to induce a correlated dot nucleation. The predictions of this model are represented asfull circles in Fig. 9.10d and were found to agree well with the experimental observations (open squares) [15, 43]. Also, a gradual transition from correlated to uncorrelated

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multilayers was found in several Monte Carlo superlattice growth simulations [111–113]. With respect to the lateral dot ordering, Solomon et al. [67] have shown a clear short-range dot ordering in InAs/GaAs superlattices with small GaAs spacer thicknesses below 100 ˚ A. As shown in Fig. 9.10f the dots are then preferentially aligned along the 100 surface directions, whereas for the singledot reference layer (Fig. 9.10e) no lateral order was found. This is evidenced by the corresponding FFT power spectra of the AFM images shown as an insert. For the superlattice sample, four satellite peaks are seen, indicating a preferred square dot arrangement on the surface. The same type of lateral ordering was also found by Darhuber et al. [36] for InAs dot superlattices with 200 ˚ A GaAs spacers using high-resolution X-ray diﬀraction reciprocal space mapping. Solomon et al. [30] have also reported a signiﬁcant size homogenization of the dots, with a decrease of the FWHM of the dot height distribution from ±20% for single InAs dot reference layers to ±8% in the 20-period dot superlattice. This was also accompanied by a 25% narrowing of the photoluminescence (PL) line widths in the superlattice to 54 meV [16, 30]. A similar narrowing was reported by Nakata et al. [68] for superlattices with 30 ˚ A spacers, observing a line width decrease from 90 meV to 27 meV, and comparable results were obtained by He et al. [114]. Also, for vertically aligned InP/GaInP quantum dot superlattices a signiﬁcant PL narrowing from 41 to 16 meV was found [70]. One general feature of vertically aligned InAs dot superlattices is the signiﬁcant increase in the dot size and broadening of shape along the vertical dot columns. This is illustrated in Fig. 9.10g and h by cross-sectional STM images of ﬁvefold-stacked 2.4 ML InAs dots separated by 100 ˚ A GaAs spacer layers by Bruls et al. [66]. Similar changes were also found in other cross-sectional STM [63–65] and TEM [15, 16, 42] studies. From a statistical evaluation of 20-period InAs/GaAs dot superlattices, Solomon et al. [30] have found a 50% increase in dot height as well as 25% increase in dot width as compared to that of a single InAs dot reference layer. Similar values were reported by Nakata et al. [68], who in addition, found a signiﬁcant decrease of the critical wetting-layer thickness for dot nucleation from hc = 1.6 to 1.1 ML along the multilayer stack. The latter seems to be the main reason for the substantial increase in dot size, because correspondingly more material is available for dot formation. As a result of the increase in dot size, a signiﬁcant red-shift of the PL emission of the vertically aligned InAs dots superlattice by around 100 meV has been found with respect to the PL of single InAs dot layers [16,30,67,68,114]. The size increase should also broaden the overall PL emission in spite of the narrowing of the size distribution in the ﬁnal dot layer. Therefore, contradictory results have been reported by various groups. While on the one hand, some groups have reported a signiﬁcant narrowing of the PL emission of InAs multilayers, as mentioned above [16, 30, 68, 114, 115], others have found little changes [116, 117] or even a broadening of the PL emission [118]. This diﬀer-

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ence may be explained also by taking the electronic coupling between the dots into account. For very small spacer thicknesses (as applies to most experimental studies), there is a strong overlap of the electronic wave functions of the dots in adjacent dot layers and thus, the carriers are actually delocalized over several stacked dots. As a result a certain averaging over the vertical inhomogeneities in the dot sizes occurs. By tunneling, the excited carriers may be transferred to the lowest energy Eigenstates along the dot columns and thus, a narrower PL is obtained. For intermediate spacer thicknesses around 20 nm, the electronic coupling becomes small and thus, the diﬀerently red-shifted PL emission of successive dot layers is superimposed. As a result, the emission may be signiﬁcantly broader than that of a single-dot layer [119], an eﬀect that should be also strongly temperature dependent. For even larger spacer thicknesses, the structural interaction between the dots becomes negligible, and thus the multilayer emission will eventually approach that of a severaltimes-repeated single-dot layer [116, 117]. In spite of these complications, in most studies a notable increase in PL eﬃciency was achieved with stacked InAs dots [16, 30, 67, 68, 114, 116, 117]. Therefore, InAs multilayer dot structures have been successfully utilized to improve the performance of quantum dot lasers (see, e.g., [1, 4, 120, 121] for reviews). A special technique has been developed by Fafard et al. [118] to reduce the variations in dot sizes in vertically aligned InAs multilayers. It is based on a special In ﬂush step after each InAs layer after partial capping of the dots with a thin GaAs layer. This In ﬂush step consist of a short growth interruption during which the excess InAs that sticks out of the GaAs layer is desorbed by a short annealing step at 610 ◦ C before continuing with superlattice growth at the 520 ◦ C growth temperature. This reduces the height of the InAs dots to the thickness of the GaAs capping layer and the layer-tolayer dot height uniformity is signiﬁcantly improved [118]. As a result, very narrow PL emission was obtained from these multilayers, such that even the shell structure of the electronic transitions could be observed [118]. Finally, it should be mentioned that recently, another type of dot ordering has been reported by Wang et al. [122–124] for the case of vertically aligned InGaAs/GaAs dot superlattices with an In concentration of around 50%. From TEM and AFM studies, the dots were found to be aligned in laterally ordered one-dimensional dot chains along the [01¯1] surface directions. This has been attributed to the anisotropy of the surface diﬀusion and of the dot shapes [122–124]. An even more eﬃcient 2D lateral ordering, as well as an inclined interlayer dot correlation, was found by the same group for InGaAs/GaAs superlattices grown on high-indexed (n11)B substrates [82, 83]. The ordering seems to be the best for the (511)B surface, resulting in a nearly square arrangement of the dots in the superlattice structure. Also, a peculiar short-range ordering was recently observed for InGaAs/GaAs dot superlattices on (311)B surfaces by Lippen et al. [125]. In all these cases, the exact ordering mechanisms have not yet been established and remain to be investigated in detail, but there are some indications that

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the ordering could be related to the formation of surface corrugations on the spacer layers.

9.5 Results for SiGe/Si Dot Superlattices For SiGe/Si (100) quantum dot superlattices, the general trends are quite similar to those of the InAs/GaAs system. From cross-sectional TEM [19,31–38] as well as X-ray diﬀraction [34, 36] studies, a predominant vertical SiGe dot alignment was found for Si spacer layers up to 500 – 700 ˚ A. This is demonstrated in Fig. 9.11a by the TEM micrograph of a vertically aligned SiGe/Si dot superlattice with 200 ˚ A Si spacers. The degree of interlayer correlation, i.e., the interlayer pairing probability of Ge/Si multilayers with variable spacer thickness and constant 6.5 ML Ge dot layers has been studied by Kienzle et al. [33] and Stangl et al. [34]. The result is shown in Fig. 9.11b, where the probability of vertical dot alignment, deﬁned as degree of base area overlap of the dots along the growth directions as deﬁned by the insert in Fig. 9.11b is plotted versus spacer-layer thickness. For spacers up to 300 ˚ A, a perfect vertical alignment with 100% pairing probability is found. For thicker spacers the pairing probability decreases until at 1000 ˚ A, no interlayer correlations are found. At 700 ˚ A spacer thickness, a pairing probability of still 50% was deduced. This is at a signiﬁcantly larger spacer thickness than for the InAs/GaAs case, which can be attributed to the fact that the Ge islands in these studies were about a factor of 5–10 larger in diameter (850 ˚ A) [33] than the InAs dots in InAs/GaAs multilayer structures [14, 43]. In the same

Fig. 9.11. a Cross-sectional TEM micrograph of a self-assembled (100) Si0.5 Ge0.5 /Si dot superlattice showing the typical vertical alignment of the dots along the growth direction. The SiGe and Si layer thicknesses are 21 and 200 ˚ A, respectively. b Pairing probability of 6.5 ML Ge dots in ﬁve-stack Ge/Si multilayers determined by cross-sectional TEM as a function of Si spacer thickness. For spacer thicknesses smaller than 400 ˚ A, a nearly perfect dot alignment is obtained. For larger thicknesses the pairing probability decreases such that above 1000 ˚ A, no interlayer correlations are found. Adopted from Kienzle et al. [33]. c AFM image of the last SiGe dot layer of a 20-period SiGe/Si dot superlattice with 5.5 ML SiGe dots alternating with 300 ˚ A Si spacers. Insert AFM image of a single SiGe dot, illustrating the typical pyramidal island shape with {105} side facets

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study, it was also found that along the vertical Ge dot columns the dot size increases linearly with layer index number [33], which is most pronounced for superlattices with very thin spacers. In a subsequent study [37], it was also shown that under these conditions, the Ge wetting-layer thickness decreases from 3.8 ML for the 1st layer to 2.5 ML in the 5th layer. This is again related to the corresponding increase in the dot size. A similar reduction in wetting-layer thickness was also reported by Le Thanh et al. [38], who found this eﬀect to depend strongly on the thickness of the Si spacer layers. Concerning the lateral dot ordering, one has really to distinguish between two diﬀerent cases according to the density and spacing of the dots formed in the ﬁrst superlattice layer, which can be tuned by changes in the SiGe dot layer thickness and composition, as well as by the growth conditions. For the case of high-density SiGe hut cluster islands deposited at comparatively low temperatures around 550 ◦ C [19, 27, 39], Teichert et al. [19, 27] have demonstrated quite an eﬀective lateral ordering along the 100 directions, with the formation of a square nearest-neighbor dot arrangement. This ordering progresses rapidly with the number of deposited superlattice periods and is accompanied by an almost threefold improvement in the dot size uniformity, starting however, from a very broad size dispersion of 110% for the ﬁrst SiGe layer. At the same time, a threefold increase in the dot size also takes place such that the dot uniformity in the stack as a whole does not improve much. In the second case of low-density dots formed by growth at the higher temperatures used in most other studies [32–34,36–38], the dots on the surface are already quite uniform but rather dispersed and widely spaced. This is exempliﬁed in Fig. 9.11c for a 20-period SiGe/Si superlattice. Evidently, the Ge dots in these multilayers show only a very faint lateral ordering tendency (see Fig. 9.11c) even at small spacer thicknesses, although in this case the dots also tend to be aligned along the 100 directions, as has been shown by various high-resolution X-ray diﬀraction studies [31, 34, 36]. These ﬁndings agree quite well with the results of Monte Carlo growth simulations [56, 109] described in Sect. 9.3.7, which have proven that this lateral alignment is due to the elastic anisotropy. The discrepancy between the results of Teichert et al. [19, 27] and those of the other groups [32–34, 36–38] is attributed to diﬀerences in the growth conditions, which lead in the former case to closely spaced dots with a large lateral strain ﬁeld overlap, whereas in the other studies the lateral dot interaction was rather weak.

9.6 Ordering and Stacking in PbSe/PbEuTe Dot Superlattices Self-assembled PbSe/PbEuTe quantum dot superlattices represent a particularly interesting system for investigation of interlayer correlations in multilayer structures. On the one hand, the elastic anisotropy is particularly large and therefore, an exceedingly eﬃcient vertical and lateral ordering takes

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place [18,29]. On the other hand, diﬀerent dot stacking types occur in dependence on spacer thicknesses as well as growth conditions [48–55]. Therefore, this system is very well suited for testing the various theoretical predictions of superlattice growth models. In addition, extensive systematic studies on interlayer correlation formation have been carried out in this material system. Self-assembled PbSe dots are produced by heteroepitaxial growth on PbEuTe (111) at a critical coverage of 2.5 ML [90]. For the (100) growth orientation, this islanding transition is suppressed because strain relaxation by misﬁt dislocations already sets in at a smaller thickness of 1 ML [126]. In the IV–VI compounds, the (111) direction is the elastically soft direction. Therefore, staggered dot stackings are expected in (111) multilayer structures. PbSe dots are under tensile strain because their lattice constant is 5.4% smaller than that of the underlying PbEuTe buﬀer layers. This is contrast to the compressively strained dots present in almost all other studied dot material systems, but from an energetic point of view this does not make any diﬀerence because the sign of the strain is removed in the elastic energy calculations. The growth properties of single PbSe dot layers have been studied in detail by our group [90, 127], showing that the basic behavior is quite similar to that of other self-assembled quantum dot systems. In particular, PbSe dots show a well-deﬁned pyramidal shape with {100} side facets and very narrow size distributions [90]. The size and dot density can be readily controlled by adjusting the growth temperature or dot layer thickness [127], with the dot height being typically in the range 60–200 ˚ A, and the density between 1 × 1010 and 20 × 1010 cm−2 . Because of the narrow energy band gaps, PbSe dots are of interest for fabrication of mid-infrared quantum lasers as demonstrated in our previous work [128]. Multilayer structures are always grown on thick strain-relaxed PbTe buﬀer layers as virtual substrates. In the superlattice structures, the Pb1−x Eux Te spacer-layer composition is typically in the range of x = 4 to 10%. By proper adjustment of the ternary composition, strain-symmetrized superlattices can be produced [29] in which the number of deposited bilayers is not limited by misﬁt dislocation formation. In our studies, the parameters of the superlattice structures were varied over a wide range, i.e., the spacer thickness was changed from 100 to 1000 ˚ A [18, 47–52, 55], the PbSe dot layer thickness from 1 to 8 ML [53, 54] and the substrate temperature from 340 to 400 ◦ C [52, 54]. 9.6.1 3D Trigonal Dot Lattices with fcc Stacking For PbSe/PbEuTe dot superlattices with spacer thicknesses between 420 and 520 ˚ A a remarkably eﬃcient lateral ordering process takes place. This is demonstrated in Fig. 9.12, where the AFM images of the last PbSe dot layer of series of superlattices consisting of N = 1, 10, 30 and 100 periods is depicted. In these samples, the PbSe and Pb1−x Eux Te layer thicknesses were

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Fig. 9.12. a–d AFM surface images (top row ) of PbSe/PbEuTe superlattices with increasing number of superlattice periods of N = 1, 10, 30, and 100 from a to d, respectively. Each bilayer consists of 5 ML PbSe dots alternating with 480 ˚ A PbEuTe spacers (image size: 3 × 3 μm2 ). The center row depicts the FFT power spectra (left) and 1 × 1 μm2 auto correlation (AC) spectra (right) of the AFM images characterizing the hexagonal lateral ordering process. The bottom row shows the corresponding dot height histograms. The solid lines are Gaussian ﬁts of the histograms. See Ref. [29] for details

constant at 5 ML and 470 ˚ A, respectively, and identical growth conditions were used [29]. As shown in Fig. 9.12a, for the single layer N =1 the islands are distributed randomly on the surface without any preferred lateral correlation direction. With increasing number of periods N , a rapidly progressing ordering sets in. Already after ten periods, the dots are aligned in single and double rows along the ¯ 110 directions (Fig. 9.12b). Measurements on samples with fewer than ten bilayers show that this ordering commences ﬁrst with the formation of small patches of hexagonally ordered regions, which subsequently enlarge and join to form row-type structures. With increasing period number (Fig. 9.12c, d), larger and larger ordered regions are formed, such that for samples with large N the perfect hexagonal 2D arrangement is disrupted only by single point defects, such as missing dots, dots at interstitial positions, or occasionally, by additionally inserted dot rows (“dislocations”) (see Fig. 9.12d for N = 100 periods). The development of the lateral ordering was characterized by Fourier transformation (FFT) as well as auto correlation (AC) analysis, as shown in the middle panel of Fig. 9.12. For the single-dot layer, the FFT power spectrum exhibits only a broad and diﬀuse ring that indicates an average inplane dot distance of 800 ˚ A without any preferred nearest-neighbor direction. The relative width (FWHM) of this ring is ±47%, which indicates a substantial variation of the lateral dot separations. Also, the auto correlation

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spectrum does not exhibit any structure outside the central maximum, indicating the lack of any correlations in the lateral dot positions. In contrast, for the ten-bilayer sample, the FFT spectrum (Fig. 9.12b) already shows six well-separated side maxima. This corresponds to a preferred spacing between the dot rows of 488 ˚ A. Six side maxima also appear in the auto correlation spectrum, indicating that the next nearest-neighbors dots are along the ¯110 directions, with a preferred distance of 680 ˚ A. For the 30- and 100-period superlattices, the peaks in the FFT spectra sharpen drastically and many higher order satellite peaks appear (see Fig. 9.12c, d). The FWHM of the satellite peaks, narrows from ±47% for N = 1 to ±6% for the 100-period superlattice, i.e., the dot spacings become extremely well deﬁned. Many higher order peaks also appear in the auto correlation spectra, which shows that large perfectly ordered dot domains are formed, with a nearly perfect registry of the dot position over up to ten nearest neighboring dots. As indicated in the auto correlation images of Fig. 9.12, average domain radii of 1, 2, 5 and 6 hexagonal unit cells are obtained for N =1, 10, 30 and 100 periods, respectively. This underlines the eﬃciency of ordering process. A particular feature of the superlattice structures is that neither the lateral dot spacing nor the average dot size change with increasing number of periods, in contrast to the observations for InAs or SiGe dot superlattices (see Sects. 9.4 and 9.5 and Fig. 9.10). This is proven by the island height histograms of diﬀerent samples depicted in the lower panel of Fig. 9.12. After an initial transition from the disordered to the ordered dot state, a constant average dot height of 120 ˚ A is found for all layers (Fig. 9.12). In addition, no change in the critical wetting-layer thickness [29] and no changes in the faceted pyramidal island shapes were observed. To determine how the lateral ordering of the dots aﬀects the size homogeneity, the width of the size distributions was determined by Gaussian ﬁts as indicated in Fig. 9.12 by the solid lines. For the single PbSe dot layer, the width of the histogram is ±14%. Although the lateral ordering already starts in the ﬁrst few superlattice layers, the height distribution at ﬁrst actually broadens to ±27% after ten superlattice periods, and only thereafter decreases to reach a ﬁnal value of ±10% for N = 100. This transient broadening can be understood in terms of a mismatch of the average lateral dot spacing in the ﬁrst PbSe layer with respect to the preferred spacing in the superlattice stack that is determined by the elastic interlayer interactions. In the ﬁrst superlattice layers, this mismatch is accommodated by the formation of many defects and missing rows in the hexagonally ordered dot regions, and because the size of the dots near these defects deviates from those within the ordered regions, a broadening of the size distribution is induced. Once, however, the ordered dot domains have a larger size, a notable size homogenization occurs. This transient eﬀect has also been found in the Monte Carlo growth simulations (Fig. 9.9). The vertical and lateral correlations in the PbSe dot superlattices were also studied by high-resolution X-ray diﬀraction [18–20, 47, 48, 53–55] and

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Fig. 9.13. a, b X-ray reciprocal space maps of a 60-period PbSe/PbEuTe quantum dot superlattice with 470 ˚ A period recorded around the (222) Bragg reﬂection along the a [¯ 211] and b[¯ 1¯ 12] azimuth direction rotated by 60◦ with respect to each other. The squares indicate the expected intensity maxima for an fcc-like ABCABC . . . dot stacking. c Cross-sectional TEM image of a 30-period superlattice, in which the dashed lines indicate the interlayer correlation direction of the dots. d, e Schematic illustration of the ABCABC . . . dot stacking as seen in cross section d and plan view e. See Springholz et al. [19, 20, 47]

TEM [19,20,48,50,52]. Figure 9.13a and b shows the reciprocal space maps of a 60-period superlattice recorded around the (222) Bragg reﬂection along two diﬀerent in-plane azimuth directions of [¯ 211] and [¯1¯12] respectively. Clearly, a large number of satellite peaks are observed both in the vertical qz as well as the lateral qx direction. This proves that the dots are highly correlated and ordered both laterally and vertically, creating a periodic 3D lattice of dots. A striking feature in the reciprocal space maps is that the lateral satellite peaks are not aligned along qx parallel to the surface but rather along inclined directions as indicated by the dashed lines in Fig. 9.13a and b. Moreover, the arrangement of the satellites is mirror symmetric in the reciprocal space maps along the azimuths rotated by 60◦ with respect to each other. This indicates a 3m symmetry of the 3D dot lattice instead of a hexagonal 6mm symmetry expected if the dots were vertically aligned. Thus, the PbSe dots are correlated along oblique directions in the superlattice stack. A detailed analysis of the X-ray spectra [18,47] shows that the arrangement of the satellite peaks can be precisely explained by a trigonal dot lattice arrangement with corresponding fcc-like interlayer stacking of 2D hexagonally ordered dot layers, as predicted by the strain-interaction model of Sect. 9.3. The corresponding dot arrangement is shown schematically in Fig. 9.6b. From the satellite spacings, the lateral dot separation in the 2D hexagonal dot planes is derived as 680 ˚ A, in excellent agreement not only with the AFM results but also with the calculated strain energy minima separation. The vertical

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separation of the dot planes is d = 470 ˚ A in this case, which in total yields a trigonal lattice constant of a0 = 610 ˚ A and a trigonal angle of α = 39.5◦ . This angle is again in good agreement with the correlation angle predicted by the point-source model (Sect. 9.3.3). Figure 9.13c shows the cross-sectional TEM image of an fcc-stacked 30period PbSe dot superlattice, demonstrating that (i) the dots are indeed aligned in directions inclined to the growth axis (dashed lines in Fig. 9.13c), and (ii) that they form a highly regular periodic lattice both the vertical and lateral direction. The comparison of the TEM image with the schematic dot arrangement is shown in Fig. 9.13d, e showing that indeed an ABCABC . . . interlayer dot stacking sequence is formed. In addition, the TEM cross section also demonstrates that the dot ordering rapidly develops already in the ﬁrst superlattice layers, which again underlines the high eﬃciency of the lateral ordering process, similar to what has been seen in the Monte Carlo growth simulations of Sect. 9.3.7. 9.6.2 Stackings and Correlations as a Function of Spacer Thickness For PbSe/PbEuTe dot superlattices, the dependence of the vertical and lateral dot correlations on spacer thickness was studied. In these samples, the PbSe dot layer thickness of 5 ML as well as the growth conditions were kept constant, while the spacer thickness was varied from 100 to 800 ˚ A. The results are summarized in Fig. 9.14a–d, where representative AFM images of the ﬁnal dot layer after 30 superlattice periods are depicted for spacer thicknesses of 160, 320, 465 and 660 ˚ A, respectively. For intermediate spacer thicknesses of 400 – 550 ˚ A, an fcc-like stacking as well as eﬃcient 2D hexagonal lateral ordering takes place. In addition, the lateral dot spacings were found to scale linearly with the spacer thickness or superlattice period dSL . This is indicated by the data and solid line of region II in Fig. 9.14e, where the measured lateral dot spacings L are plotted versus dSL for the whole range of samples. In this region, also the interlayer correlation angle is essentially constant at about 39◦ [18]. This means that the lattice-constants of the resulting 3D trigonal dot lattices can be tuned over a range from 500–700 ˚ A just by changes in the spacer thickness [18]. With increasing spacer thickness, however, the lateral ordering of the dots becomes weaker and weaker until at about 600 ˚ A, interlayer correlations are no longer observed. This is illustrated in Fig. 9.14g by the cross-sectional TEM image of a superlattice with 680 ˚ A spacers, where obviously, the dots in each layer nucleate at random sites independent of the dots in the previous layers. Thus, these superlattices merely represent a repetition of uncorrelated single-dot layers and no dot rearrangements or lateral ordering occurs (see Fig. 9.14d). As a consequence, the average lateral dot spacing on top of the superlattice stack (ﬁlled circles in region III of Fig. 9.14e) is always equal to the constant value of about 500 ˚ A as found for single-dot reference layers grown under the same conditions.

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Fig. 9.14. a–d AFM surface images of 30-period PbSe/PbEuTe dot superlattices with diﬀerent spacer-layer thickness of a 160, b 320, c 465 and d 660 ˚ A, demonstrating the diﬀerent dot spacings and ordering tendency in the structures as evidenced by the FFT power spectra shown as inserts. e Preferred lateral dot spacing L plotted as a function of vertical superlattice period. Squares indicate superlattices with vertically aligned dots, diamonds those with fcc-like ABCABC . . . stacking and ﬁlled circles those of uncorrelated superlattices. f, g Cross-sectional TEM images of a vertically aligned (f ) and an uncorrelated (g) PbSe/PbEuTe dot superlattice with spacer thicknesses of 320 and 680 ˚ A, respectively. See Springholz et al. [48, 49]

˚ there is again a marked For spacer thicknesses lower than about 400 A change in the preferred lateral dot spacing. As evidenced by the squares of Fig. 9.14e, at this spacer thickness the average dot spacing increases abruptly by more than a factor of 2 to about 1300 ˚ A. With further decreasing spacer thickness it linearly decreases to about 800 ˚ A at a spacer thickness of 200 ˚ A (see solid line in region I of Fig. 9.14e) and thereafter stays essentially constant. As shown by the cross-sectional TEM image of Fig. 9.14f, this transition is caused by a change in the vertical interlayer correlation from the fcc-stacking type of region II to a vertical dot alignment in region I. As shown by the AFM surface images of two such superlattices depicted in Fig. 9.14a, b, in this range of spacer thicknesses, a quite good hexagonal lateral ordering also takes place. The properties of vertically aligned PbSe dot superlattices are summarized in Fig. 9.15, where on the left hand side, the AFM images of three vertically aligned dot superlattices with spacer thickness of 105 – 330 ˚ A are shown, together with the corresponding evolution of the dot size distributions depicted in the center panel. Clearly, there is a very strong tendency for a lateral dot alignment along the in-plane ¯ 110 surface directions. As a result, 2D hexagonally ordered dot arrays are formed, which is evidenced by the sixfold symmetric satellite peaks in the FFT power spectra of the AFM images shown as inserts. In addition, the dot sizes and spacings are not only much larger than that of the single-dot reference layer, but they also systematically increase with increasing spacer thickness. This is proven by the corresponding decrease in the FFT satellite peak separations in Fig. 9.15. The lateral ordering is best for spacer thicknesses around 100 – 220 ˚ A, for which the FFT

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Fig. 9.15. a–c AFM surface images of vertically aligned PbSe/PbEuTe quantum dot superlattices with PbEuTe spacer-layer thicknesses of 105, 215 and 330 ˚ A, from a to c, respectively. The number of 50 periods and the 5 ML thickness of the PbSe dot layer is constant for all samples. The inserts show the 2D FFT power spectra of the AFM images. e Dot height histograms obtained for superlattices with spacer thicknesses of 105, 160, 215, 275 and 330 ˚ A, from d to h, respectively. f, g Dot parameters plotted as a function of the spacer-layer thickness: i lateral dot spacing L obtained from the FFT satellite peaks (squares) as well as the PbSe dot density (circles). Also plotted is the full width at half maximum (FWHM) (triangles) of the FFT satellite peaks. j Average PbSe dot height h (•) and corresponding FWHM (Δh/h) of the height histogram peaks (triangles). k Average dot width b (squares) and corresponding FWHM (Δb/b) of the histograms of the dot widths (diamonds) plotted as a function of spacer thickness. Adapted from Raab et al. [49]

satellite peaks are most narrow in width and for which even weaker second order FFT satellites are visible. For larger spacer thicknesses, the satellites become increasingly smeared out, and they almost disappear for the superlattice with 330 ˚ A spacers (Fig. 9.15c). This is due to the increasing disorder in the dot arrangement and the appearance of a second type of smaller PbSe dots nucleated between the larger dots. Because the 330 ˚ A spacer thickness of this sample is already close to the transition to the fcc-like dot stacking, these interstitial dots obviously represent local fcc-stacked regions in which the lateral dot separation by a factor of 2 is smaller than those for the vertically aligned dots (see Fig. 9.14d). For a quantitative analysis of the lateral ordering process in the vertically aligned structures, the FFT satellite peak widths and separations Δk were deduced. The preferred lateral dot spacing L is then obtained from

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LFFT = √ 1/ (Δk · sin 60◦ ) or, alternatively, from the dot density n by using Ln = 1/ n · sin 60◦ . The results are plotted in Fig. 9.15f as a function of spacer thickness. Clearly, the preferred lateral dot spacing increases strongly with increasing spacer thickness, but does not follow a strict linear dependence as observed for the fcc-stacked superlattices. The relative width of the FFT satellite peaks (triangles in Fig. 9.15f) shows a clear minimum of ±16% A, but for thinner, as well as thicker spacer at a spacer thickness of ds =160 ˚ layers, the FWHM increases to about ±21%. Thus, the best lateral ordering occurs for 160 ˚ A spacers. For larger spacer thicknesses, the mean dot separation obtained from the dot densities also no longer agree with the value deduced from the FFT peak spacing. This is due to the appearance of the smaller dots in the AFM images, which increases the dot density but does not much aﬀect the separation of the larger dots. The dependence of the dot size and shape as a function of spacer thickness was deduced from a statistical analysis that also yields the width of the size distributions. Figure 9.15d–h shows the height histograms for the samples with spacer thicknesses of 105 to 330 ˚ A. Clearly, for spacer thicknesses increasing from 85 to 275 ˚ A, the dot height rapidly increases from h = 85 ˚ A to h = 149 ˚ A, respectively. For thicker spacers, however, a small left hand shoulder (A) starts to emerge at smaller dot heights, and this shoulder becomes even more pronounced for the sample with 330 ˚ A spacers. Also, the average dot size does not increase further, but rather slightly decreases to 138 ˚ A. Both eﬀects are caused by the formation of additional smaller dots on the surface, which reduces the overall amount of PbSe available for the larger dots. Figure 9.15i, k summarizes the dependence of the average dot height h and dot width b as a function of spacer thickness. As indicated in Fig. 9.15i, A, the average dot height increases linearly with increasing up to ds = 275 ˚ spacer thickness, whereas the dot width of about 350 ˚ A remains essentially constant (Fig. 9.15k). This translates into a ﬂattening of the dot shape for small spacers, indicating that the dot growth is enhanced in the lateral direction. AFM images recorded with selected sharp AFM tips show that these dots assume a truncated pyramidal shape with triangular base and {100} side facets, as shown schematically in the insert of Fig. 9.15k. For very thin spacer layers, the PbSe dots are rather ﬂat with an aspect ratio of only about 1:5, whereas for the thicker spacers the aspect ratio increases to about 1:3. This is still below the value of 1:2.2 of the pyramidal dots of single-dot layers that do not show any ﬂattening of the island apex. The modiﬁcations of dot shape are obviously induced by the elastic strain ﬁelds of the buried dots, which are strongest for the thinnest spacer layers but decay rapidly as the spacer thickness increases. Therefore, for spacer thickness larger than 400 ˚ A, the dots exhibit the same pyramidal shape known for unperturbed single-dot layers. Perhaps the most interesting feature is the pronounced narrowing of the size distribution for the well-ordered vertically aligned samples. From the

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dependence of the width of the size distributions plotted in Fig. 9.15j and k as a function of spacer thickness, it is found that the FWHM decreases from ±13% to ±8% when ds increases from 80 to 160 ˚ A, after which it increases A. A similar, but even more pronounced again to above ±15% for ds = 330 ˚ trend is observed for the variation of the lateral dot widths (Fig. 9.15k), which again shows a pronounced minimum for 160 ˚ A spacers. Thus, the highest uniformity is obtained for superlattices with the best hexagonal ordering, demonstrating that the lateral ordering indeed produces a higher uniformity of the dot ensembles. The vertical and lateral ordering was also characterized by anomalous X-ray diﬀraction performed at the ESRF synchrotron light source with an X-ray photon energy tuned to the M shell absorption edge of the Pb atoms. In this way, the structure scattering factor if the PbEuTe matrix is drastically reduced to about 50 times below that of PbSe and therefore, the scattering contrast between the dots and matrix is drastically enhanced. The resulting anomalous reciprocal space maps are depicted in Fig. 9.16a–d for the superlattices with 105, 165, 215 and 465 ˚ A spacer thicknesses, respectively. Clearly, for all samples a large number of satellite peaks is observed in the vertical qz as well as lateral qx direction. However, whereas for the samples with thin spacers the lateral satellite peaks are aligned parallel to the qx direction, for the sample with thick spacers they are aligned along inclined directions (dashed lines in Fig. 9.16). In addition, reciprocal space maps recorded along diﬀerent azimuth directions indicate a sixfold symmetry for the samples with small spacers compared to a threefold symmetry for the other sample. This is further evidence for the diﬀerent interlayer dot stackings in the samples, forming a trigonal fcc-like dot lattice for the latter and a vertically aligned 3D hexagonal dot lattice for the former. The quality of the dot ordering process was assessed from cross-sectional line scans of the reciprocal space maps shown in Fig. 9.16e, f. For the sample with fcc-stacking (Fig. 9.16f) a much larger number of lateral satellites can be resolved than for the vertically aligned dot samples (Fig. 9.16e). In addition, for the latter a signiﬁcant increase of the peak widths with increasing qx scattering vector occurs. For a quantitative analysis, the widths of the satellites were derived by ﬁtting the cross-sectional proﬁles by Gaussians, with the ﬁts represented as solid lines in Fig. 9.16e and f. The resulting FWHM of the lateral satellite peaks are plotted in Fig. 9.16g–j versus in plane scattering vector qx . For the vertically aligned dot superlattices, the lateral peak width rapidly increases, whereas for the superlattice with fcc-stacking the FWHM are almost constant. To deduce the corresponding order parameters of the dot structures, this dependence was ﬁtted using a modiﬁed short-range ordering scattering model, in which the dependence of the satellite peak width Δqx as a function of scattering vector qx is described as [55, 77, 129]: 4 /L2 + M 2 (9.4) Δqx ≈ qx4 σL

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Fig. 9.16. a–d (111) Reciprocal space maps recorded by anomalous synchrotron X-ray diﬀraction at 5.1 ˚ A wavelength of four PbSe quantum dot superlattices with A, from a to d, diﬀerent PbEuTe spacer thicknesses ds = 104, 164, 214 and 454 ˚ respectively. The arrangement of the lateral satellite peaks along qx indicates a 3D hexagonal dot lattice structure and vertical dot alignment for the superlattices with small spacer thickness a to c, as compared to the 3D trigonal dot lattice with fcc-like dot stacking for the sample with 454 ˚ A spacers (d). e–i Cross-sectional qx line scans for the superlattice samples with e 164 ˚ A and f 454 ˚ A spacers recorded along the horizontal, respectively, inclined dashed lines in the reciprocal space maps b and d. g–i FWHM Δqx (open symbols) of the lateral satellite peaks plotted as A. The a function of qx scattering vector for the samples with ds =104, 164, and 454 ˚ solid lines represent the ﬁt of the data with a short-range order scattering model from which the dot order parameters such as variance of the dot spacings σ L as well as the average domain size is obtained. Adopted from Lechner et al. [55]

Here, L is the average lateral dot distance obtained from the satellite peak spacing, σL is the variance in the nearest-neighbor dot distance within the growth plane and M is the relative size of the ordered dot domain in units of L. By ﬁtting the experimental data with this relation (solid lines in Fig. 9.16g–j), σL and M can be deduced. M basically corresponds to the oﬀset of Δqx at qx = 0, and σL is determined by the parabolic increase of Δqx at higher qx values. For the well ordered vertically aligned samples (Fig. 9.16g, h), σL = 117 ˚ A and 86 ˚ A, which corresponds to a relative variance or dispersion of the lateral dot spacings of ±15 and ±10%, respectively, A, corresponding to a much whereas for the fcc-stacked sample σL is only 36 ˚ smaller dispersion of only ±5%. This indicates that the lateral dot ordering is signiﬁcantly better than that of the vertically aligned samples. The same trend also applies for the average domain sizes, with M = 2 for the vertically aligned samples as compared to M = 5 for the superlattice with fcc-stacking. This indicates that the ordering process for superlattices with staggered dot stacking is much more eﬃcient compared to that for vertically

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aligned dots, which agrees well with the results from the Monte Carlo growth simulations (Sect. 9.3.7). From an analogous measurement of the satellite peak widths in the vertical qz direction, it is found that this width is quite small for all four samples and that the widths are nearly independent of qz . This indicates that the layer-to-layer dot correlation along the growth axis is nearly perfect, with a correlation length as large as 25 superlattice periods for the vertically aligned samples and of 7 periods for the superlattice with fcc stacking. This diﬀerence is caused by the decreasing strain ﬁeld interactions with increasing spacer thickness, eventually approaching zero in the far-ﬁeld limit. 9.6.3 Correlations as a Function of Dot Size Apart from the spacer thickness dependence, we have also investigated the inﬂuence of the dot layer thickness and growth temperature on the interlayer correlation formed in self-assembled PbSe dot superlattices [53,54]. As shown by studies of single-dot layers [90, 127], both parameters strongly aﬀect the size and density of the dots and thus, the conﬁguration of the starting layer is strongly changed, which is expected to alter the evolution of the subsequent superlattice growth process. Two series of samples were investigated. In the ﬁrst series, the PbSe dot layer thickness was varied from 1 to 8 ML, while A keeping the substrate temperature of 360 ◦ C, the spacer thickness of 410 ˚ and the number of 100 periods constant [52]. In the second series, the substrate temperature was varied from 320◦ to 400 ◦ C whereas the spacer and dot layer thicknesses were kept constant at 420 ˚ A and 5 ML and the number of superlattice periods was 30 [54]. Figure 9.17a–d shows the AFM surface images of the superlattices with PbSe thicknesses of 3, 4, 5 and 8 ML. At 3 ML PbSe coverage (Fig. 9.17a), the PbSe dots are obviously randomly distributed over the surface without a preferred nearest-neighbor direction (see featureless diﬀuse FFT power spectrum of the AFM image depicted as inset). When the PbSe layer thickness increases to 4 ML, this disordered dot arrangement gives way to a well-ordered hexagonal dot arrangement (see Fig. 9.17b) where the dots are aligned along the

¯ 110 surface directions with a well-deﬁned lateral dot spacing of 580 ˚ A. The good lateral ordering is evidenced by the appearance of sharp, sixfold symmetric satellite peaks in the FFT power spectrum. The same ordering is also observed for PbSe thicknesses of 5 and 6 ML, as shown in Fig. 9.17c. For thicker PbSe layers, however, the dots start to cluster and large “super” dots are formed. This is illustrated in Fig. 9.17d by the AFM image of the 8 ML superlattice sample. Although between the large dots a few smaller dots remain that still show some signs of a preferred hexagonal dot coordination, no distinct satellite peaks appear in the FFT power spectrum. This indicates the existence of a rather disordered overall dot arrangement. To determine the corresponding interlayer dot correlations, high-resolution X-ray diﬀraction reciprocal space maps were recorded along the [¯211] azimuth

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direction. The results are shown in Fig. 9.17e–h for superlattices with 3, 4, 5 and 8 ML PbSe thicknesses. For the 3 ML dot sample (Fig. 9.17e), only a vertically aligned line of satellite peaks along qz appear. Around the 0th-order superlattice peak there is a strongly enhanced diﬀuse scattering caused by the presence of self-assembled quantum dots at the heterointerfaces. Because this scattering component is absent for the higher order satellites, there is no spatial correlation of the dots from one layer to the next layer, i.e., the superlattice merely represents an uncorrelated repetition of disordered single-dot layers, similar as for PbSe dot superlattices with thick spacer layers. For the samples with intermediate PbSe thicknesses (see Fig. 9.17f and g for 4 and 5 ML, respectively), satellite peaks are observed in both the vertical qz and lateral qx direction. Thus, the dots are highly correlated both vertically and laterally, creating a 3D ordered lattice of dots. As discussed in detail in the previous section, because the lateral satellites are aligned along lines inclined by 38◦ degrees to the vertical qz direction, an fcc-like dot stacking is formed in these samples. For the samples with larger PbSe thicknesses, again a striking change occurs, in which the well-deﬁned lateral satellites are replaced by a strong, correlated diﬀuse broadening of all satellites along the lateral qx direction. This indicates that although the dots are disordered in the lateral direction, the interface corrugation is replicated from layer to layer along the growth axis. Thus, the dots are vertically aligned in columns as was found for PbSe superlattices with small spacer thickness.

Fig. 9.17. a–d AFM surface images of 100-period PbSe quantum dot superlattices with constant 410 ˚ A PbEuTe spacers but varying PbSe dot layer thickness of 3, 4, 5 and 8 ML for a to d, respectively. The FFT power spectra shown as insets indicate a diﬀerent lateral ordering for the samples. e–i (222) X-ray reciprocal space maps for the same samples. For the 3 ML superlattice e, no lateral satellites are observed due to the lack of any lateral or vertical dot correlations. For the 4 and 5 ML samples f and g, the lateral satellite indicate an fcc-like dot stacking, whereas for the 8 ML superlattice h the large peak broadening along qx indicates the lack of lateral ordering but a vertical dot alignment along the growth direction. Adopted from Springholz et al. [53]

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From a statistical analysis of the dot size distributions [53,54], the average dot height of the ﬁnal layer was found to increase from 48 ˚ A for the 3 ML sample to 125 ˚ A for the 8 ML sample, and the narrowest size distribution is formed for the well-ordered samples with 4 and 5 ML PbSe thickness, for which the FWHM of the histograms is ±14%, whereas it is more than ±25% for the disordered 3 and 8 ML samples [53, 54]. Thus, an fcc-like dot stacking is formed only for superlattices with an average dot height of 80–120 ˚ A. This demonstrates that the kind of interlayer correlation and dots stacking formed in superlattice structures does not depend only on the spacer thickness but also on the dot size. For the sample series with varied growth temperature, a similar behavior was found [54]. At growth temperatures below 340 ◦ C, the dot density is very high and the average dot size smaller than 70 ˚ A. As a result, no interlayer correlations are formed and no lateral ordering occurs [54]. At temperatures between 360 and 380 ◦ C, in contrast, AFM and X-ray diﬀraction measurements reveal a good hexagonal lateral ordering as well as a well-deﬁned fcclike vertical dot stacking. Again, in this case the average dot height is around a value of 90 ˚ A with a very low ±14% size dispersion. At higher temperatures, very large and disordered dots are formed with a large average height of 160 ˚ A and the X-ray diﬀraction data indicate a preferential vertical dot alignment in the samples [54]. This indicates again that an fcc-type dot stacking is formed only for a very limited range of PbSe dot sizes and that the dot size is an important parameter in the interlayer correlation formation. 9.6.4 Phase Boundaries and Phase Diagram for Vertical and Lateral Dot Ordering To explain the changes in the interlayer dot correlations as a function of spacer thickness and dot size, the dependence of the elastic strain ﬁelds and surface energy distributions on these parameters must be analyzed. For this purpose, we have performed a series of ﬁnite element calculations for pyramidal PbSe dots of diﬀerent size but invariant shape located in a (111)-oriented PbEuTe matrix at various depths d below the surface. In accordance with our AFM studies [90], the shape of the dots was modeled as a √ triangular pyramid with {100} side facets and constant aspect ratio of b/h = 6, where b is the island base and h the island height. A general conclusion of these calculations is that the surface strain energy distributions plotted in terms of dimensionless scaled surface coordinates x/d depend only on the ratio d/b of the island depth to the island size. As a consequence, the directions α of the surface energy minima are determined exclusively by the d/b ratio. Figure 9.18c, d shows two representative scaled strain energy distributions for d/b = 0.5 and 1.5, respectively. Clearly, for spacer thicknesses larger than the island base, i.e., d/b > 1, the strain energy distributions closely resemble that obtained by the point-source model (compare Fig. 9.18d with

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Fig. 9.18. The inﬂuence of the spacer-layer thickness, i.e., vertical superlattice period d as well as the PbSe dot size on the strain energy distributions and interlayer correlations in (111) PbSe/PbEuTe quantum dot superlattices. a Direction α of the surface strain energy minima above pyramidal PbSe islands plotted as a function of the renormalized vertical dot layer separation d/b, where b is the lateral dot base width and d is the superlattice period. The surface strain energy distributions for d/b = 0.5 and 1.5 are plotted in c and d as a function of the reduced surface coordinates x/d. The energy separation between the contour lines is 6 meV and 0.16 meV, respectively. b Calculated lateral separation L of the energy minima (ﬁlled squares) and depth of the minima ΔE (open circles) plotted versus superlattice period for PbSe pyramids with a ﬁxed base width of b = 300 ˚ A and a height of h = 120 ˚ A. e Phase diagram of diﬀerent dot stackings in PbSe dot superlattices as a function of dot layer spacing and PbSe dot size. Squares Vertically aligned dots, diamonds fcc-stacked dots, open circles uncorrelated superlattices. Data obtained from X-ray diﬀraction, TEM and AFM measurements. The phase boundary lines are plotted according to Eqs. (9.5) and (9.6). See Springholz et al. [48]

¯ direcFig. 9.4c), with three well separated energy minima along the 1¯12 tions. Thus, for large d/b values the results converge to that of the far-ﬁeld point-source model. This is also evidenced by Fig. 9.18a, where the minima directions α are plotted versus d/b. Therefore, for d/b > 1, the experimentally observed fcc-dot stacking as well as the measured and calculated interlayer correlation directions are in good agreement with each other. According to Fig. 9.18a, however, when d/b decreases below 1, α rapidly decreases, reaching zero already for d/b ≤ 0.5. This means that the three side energy minima are replaced by one central minimum exactly above the buried dot, as is illustrated by the strain energy distribution plotted in Fig. 9.18c for d/b = 0.5. Thus, at small spacer thicknesses and/or large dot sizes, the fcc dot stacking is replaced by a vertical dot alignment, which is again in good agreement with the experimental observations. Figure 9.18b shows the dependence of the energy minima separation Lmin as well as their depth ΔEmin as a function of the vertical superlattice period d for the case of PbSe dots with 120 ˚ A in height and 300 ˚ A base, as measured

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for the sample series presented in Sect. 9.6.2. The diﬀerently shaded regions indicate the diﬀerent interlayer correlations observed in the experiments. For large spacers, L increases linearly with d, but rapidly drops to zero at d below 200 ˚ A. On the other hand, the depth of the energy minima continuously decreases as the spacer thickness increases due to the decay of the elastic strain ﬁelds. Similar calculations for a constant spacer thickness but varying dot size can be found in Ref. [53]. To deduce the conditions at which the transition between the diﬀerently correlated dot structures occur, one has to compare the minima separation Lmin and the dot base width b as indicated by the dashed triangles in Fig. 9.18c and d. For the case of small dots or thick spacers, the minima separation is much larger than the dot base width (b < L). Therefore, the dots on the surface can easily occupy just one single energy minimum and as the growth proceeds, an fcc-like ABCABC . . . stacking sequence is formed. For very small dots and large spacer thicknesses, however, the energy minima become very shallow because the strain ﬁelds decrease linearly with decreasing dot volume in the far-ﬁeld limit [see Eq. (9.3)]. Experimentally, no interlayer dot correlations were found for vertical superA (see Fig. 9.14e). lattice periods beyond the critical value of d1c = 560 ˚ From Fig. 9.18b, at this point, the depth of the strain energy minima induced by one subsurface dot is as small as ΔE =0.4 meV/atom pair. Because in the multilayers each surface energy minimum is produced by the superposition of the strain ﬁelds of three subsurface dots, the corresponding minimal interaction energy required for interlayer correlation formation is actually 3 times this value, i.e., ΔEcrit = 1.2 meV for the given growth conditions. Surprisingly, this value is more than one order of magnitude smaller than the thermal energies at the given growth temperature. This may be taken as an indication that the nucleation of Stranski–Krastanow islands is mostly governed by the critical nucleus size and nucleation barrier rather than by the hopping and diﬀusion process of the single surface adatoms. For large dots and small spacer thicknesses, the energy minima separation successively decreases (Fig. 9.18b) and they eventually merge into one single minimum. Experimentally, the transition from fcc dot stacking to a vertical alignment already occurs before this point, namely, at a spacer thickness of A (see Fig. 9.14e), i.e., at a d/b value of about 1.3. This is because d2c = 400 ˚ for fcc-stacked dots, not only must the energy minima separation be larger than the dot base widths, but even more a denuded zone around each dot exists must be taken into account where further dot nucleation is kinetically suppressed. According to our experimental data, the size of this denuded zone must be about 1.6 times the base width, which is the eﬀective width of the minimal surface area required for each dot. For a given material system, the two phase boundary conditions can be written in a rather generalized form because under the condition of an invariant island shape there is a ﬁxed relationship between the dot base and

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the dot height and the dot volume is proportional to h3 . The ﬁrst phase boundary corresponds to the cut-oﬀ length of the interlayer correlations that is determined by the minimal interaction energy Ecrit required for correlated dot nucleation. This value is speciﬁc for each material system and growth condition, but according to Eq. (9.3), it is determined only by ratio of the volume over the dot depth V0 /h. As V0 ∼ h3 , this cut-oﬀ condition can be reformulated as: Ecrit (9.5) [h/d]c1 = 3 ∗ ·δ E0 · Chkl where δ = V0 /h3 is constant for a given island shape, and E0 is the surface strain energy density of the uniformly strained 2D reference layer. Using for PbSe the values E0 = 142 meV, C*hkl = 0.69 (see Sect. 9.3.4), δ = 3/2 for {100} faceted pyramids and the experimental value of Ecrit = 1.4 meV, one obtains a critical ratio of [h/d]c1 = 0.22 below which no interlayer correlations should be formed in the superlattice structures. Concerning the second boundary condition, for a ﬁxed island shape the correlation angle α of the energy minima is constant for a given d/b (see Fig. 9.18a) and likewise for a ﬁxed h/d ratio. Since the diameter of the denuded zone weﬀ is proportional to b and thus, weﬀ /h = κ = constant, the condition of weﬀ = Lmin for the transition between the vertical dot alignment and fcc stacking can be written for the (111) case as: √ 3 · tan α111 . [h/d]c2 = (9.6) κ √ For the (100) growth orientation, the 3 factor simply has to be replaced by √ 2 because of the diﬀerent arrangement of the energy minima (Sect. 9.3). Applying the appropriate parameters for the PbSe case (κ 3.9 and α111 = 38◦ ), one obtains a critical ratio of [h/d]c2 = 0.28 above which the PbSe dots should be always vertically aligned. Thus, an fcc-like dot stacking should occur only in the range of 0.22 < [h/d] < 0.28. Compiling in Fig. 9.18 the large body of experimental data obtained by Xray diﬀraction, TEM and AFM for more than 50 diﬀerent PbSe quantum dot superlattice samples with various spacer thicknesses and dot sizes, a complete phase diagram of dot stackings is obtained for this material system in which the phase boundaries follow exactly the dashed boundary lines deﬁned by the stacking conditions of Eqs. 9.5 and 9.6. Moreover, this phase diagram clearly demonstrates that staggered dot stackings can be generally obtained only in a narrow window of parameters conﬁned by the phase boundaries represented by Eqs. (9.5) and (9.6). Its width is determined not only by the elastic properties, the surface orientation and the characteristic island shapes but also by the epitaxial growth conditions that will determine the eﬀective cut-oﬀ energy value Ec . For materials with small elastic anisotropy (e.g., for Si/Ge (100) ) [h/d]c2 will become smaller than [h/d]c1 . This means that no

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set of parameters exist in which a staggered dot stacking may be formed, which poses a strong limitation on the conditions and materials in which a staggered stacking can be obtained.

9.7 Other Mechanisms for Interlayer Correlation Formation Elastic interactions are certainly the most important factor for interlayer correlations formed in quantum dot multilayer structures. As already mentioned in Sect. 9.2, however, there exist at least two other alternative but less obvious mechanisms that may contribute to interlayer correlation formation. These mechanisms are correlated dot nucleation mediated either by (1) nonplanarized surface topographies (Fig. 9.1c,d) or (2) lateral compositional modulations of the spacer layer (Fig. 9.1e,f). Although little work has been carried out to clarify these mechanisms, each of them in principle may give rise to diﬀerent interlayer dot stacking types, depending on the intricate details of the interaction and nucleation process. With respect to the correlations mediated by the growth morphology, one ﬁrst has to realize that each quantum dot layer itself represents a highly corrugated surface structure. Although during overgrowth, this 3D surface will be eventually planarized to minimize the surface energy, a corrugated nonplanar surface structure will be retained when the capping process is incomplete such as for thin spacer layers or when planarization is hindered by sluggish surface kinetics. Obviously, a corrugated surface will strongly inﬂuence the subsequent island nucleation process and, because the surface corrugations are linked to dots below the surface, interlayer dot correlations can be produced as well. That surface morphology plays a profound role in selfassembled dot nucleation has been recently established by investigations of self-assembled quantum dot growth on prepatterned substrates, where it has been shown that dot nucleation can be triggered by lithographically deﬁned surface sites [130–132]. In fact, in this way a near-perfect position control of self-assembled quantum dots has been achieved for Ge on Si [130] as well as InAs on GaAs [131]. For overgrowth of Stranski–Krastanow islands, two diﬀerent growth scenarios may actually occur with quite diﬀerent resulting surface morphologies. On the one hand, when the surface diﬀusivity of spacer adatoms and surface capillarity forces are small, surface planarization is rather slow and moundlike structures are retained above the buried islands when the spacer thickness is not very large. This situation is illustrated schematically in Fig. 9.1c,d. In contrast, when surface mass transport is dominated by the stress ﬁelds of the buried dots, the mobile adatoms during spacer deposition are repelled from the surface above the buried dots due to the opposite sign of strain [15]. As a result, the growth of the spacer layer can be locally suppressed and surface depressions or pits are formed above the dots [15]. This has been reported,

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e.g., for InAs islands overgrown by GaAs [133] or InP [134], as well as for PbSe dots overgrown with PbEuTe [160]. This means that the actual type of spacer morphology strongly depends on the chosen growth conditions. A second complication arises from the fact that the reaction of dot nucleation to the presence of nonplanar corrugated surface morphologies itself depends on the mechanism that dominates the dot layer surface mass transport. If the lateral mass transport during wetting-layer growth is dominated by capillary forces, then dot material will accumulate at the concave surface areas and as a result, preferred dot nucleation at the troughs of the surface morphology will occur as is illustrated in Fig. 9.1d. Experimentally, this behavior has been found for Ge growth on prepatterned Si, where Ge islands were found to nucleate preferentially at the bottom of pits etched into the Si substrate [130, 135]. A similar behavior was also reported for InAs islands grown over GaAs hole patterns [131]. On the other hand, when mass transport is dominated by stress-driven surface diﬀusion, the opposite behavior will take place because adatoms then diﬀuse preferentially towards the convex areas of the surface where part of the misﬁt strain can be elastically relaxed. As a result, the dot material will accumulate at the tops or edges of patterned surface structures where preferential dot nucleation will consequently occur. This behavior has been observed, e.g., for InAs quantum dots deposited on GaAs surface ridge or mesa structures [136–138], as well as for Ge grown over Si mesas [139, 140]. Both eﬀects can be further altered when large diﬀerences in the free surface energies or diﬀusivities exist on diﬀerently oriented portions of the surface morphology. In the light of these complexities, the outcome of an actual growth experiment is diﬃcult to predict but obviously, a nonplanar spacer-layer surface can give rise both to a vertical dot alignment as well as a staggered stacking. An example for the profound eﬀect of the spacer morphology on the interlayer dot correlation is shown in Fig. 9.2c for a Ge/Si dot superlattice [69]. In this case, as a result of the increasing Ge dot size in the superlattice stack, complete planarization of the thin Si spacer layers is no longer achieved and as a result, the vertical dot alignment switches to an oblique dot replication at a certain point of growth [69]. Other examples for nonvertical dot stackings possibly related to nonplanarized spacer-layer morphologies include selfassembled InP/GaInP quantum dot stacks [141] as well as InAs/InP [142] and InAs/InAlAs [60, 143] quantum wire multilayers, in which oblique interlayer dot alignments with varying correlation angles as well as staggered wire stackings have been observed. Also, it is noted that corrugated surface morphologies are quite common in epitaxial growth on vicinal or high-index surfaces due to the action of kinetic [144] or strain-induced [21, 145, 146] step bunching or of spontaneous surface facetation [4]. As shown by previous works [21,147–149], this can also lead to the formation of correlated interface corrugations in superlattice structures, which combined with the growth of self-assembled Stranski–Krastanow dots may also lead to morphology driven correlations in self-assembled quantum dot multilayers [150–153].

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The third mechanism for interlayer correlations is based on the possible formation of lateral variations of the chemical composition of the spacer layer induced by the buried dots. As is indicated in Fig. 9.1e, f, these variations may originate from two diﬀerent processes: (1) from preferential surface segregation of dot material above the buried islands (Fig. 9.1e), or (2) from strain- or morphology-induced alloy decomposition of the spacer material (Fig. 9.1f). The ﬁrst mechanism will be operative for strongly segregating heteroepitaxial materials, which applies to many self-assembled quantum dot systems because they are usually composed of materials with large diﬀerences in lattice constants and binding energies, which are the two major driving forces for surface segregation. As surface segregation will tend to cause an enhanced accumulation of dot material directly above the buried dots, the subsequent wetting-layer growth is locally enhanced and the subsequent dots will therefore nucleate preferentially on top of the previous islands. Practically, this eﬀect is superimposed by the simultaneous action of the elastic strain ﬁelds of the buried dots that for thin spacer layers produce a vertical dot alignment as well. As a result, it is unknown how much surface segregation actually contributes to interlayer correlations in multilayer structures. Several studies have indicated that the critical wetting layer for island nucleation in multilayers is successively reduced with increasing number of deposited layers [37, 38, 68]. This may be taken as an indication that surface segregation could be an important parameter in multilayer growth. Surface segregation strongly depends on the growth conditions and can be altered by the use of surfactants [154]. This may provide a tool for controlling and studying its eﬀect in multilayer structures in more detail. For multilayers with multicomponent alloys as spacer layers, lateral compositional variation can be formed when there exists a tendency for the alloys to decompose into regions of diﬀerent chemical composition. This eﬀect is often driven by strain or surface corrugations during epitaxial growth. Alloy phase separation is quite common in III–V ternary or quaternary alloys for certain chemical compositions and growth conditions (see, e.g., [155] for review). The resulting lateral variations of the chemical composition in the spacer layer not only may cause a composition modulation of the wetting layer, but will also induce additional strain variations due to the resulting variations in the alloy lattice constant. This may amplify or counteract the strain ﬁelds from the buried dots, but in any case, this will modify the dot nucleation and thus contribute to interlayer correlation formation. The most prominent examples for this mechanism are the staggered stackings found for self-assembled InAs/AlInAs (001) quantum wire superlattices [58–62, 79, 80], which are illustrated by the TEM image shown in Fig. 9.2d. In this case, the ternary AlInAs spacer layers show a strong tendency for alloy decomposition due to an immiscibility gap [60, 156]. Under the presence of surface roughness or strain variations, lateral phase separation in In-rich and Al-rich regions occurs [156–159] and as a result, V-shaped In-rich regions emerge from the sides faces of the buried InAs wires. This is

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revealed by the chemical contrast in cross-sectional TEM images and is indicated by the dashed arrows in Fig. 9.2d (see [60,61] for details). Subsequently, InAs quantum wires nucleate preferentially at the intersections of In-rich V arms of neighboring buried wires, which gives rise to a staggered ABAB . . . interlayer stacking clearly visible in Fig. 9.2d. This stacking type has been observed consistently by several groups for varying AlInAs spacer-layer thicknesses from 25–300 ˚ A [61, 79], with interlayer correlations angles around 40◦ . Apart from the chemical contrast visible in the TEM image [60, 61], strong supporting evidence for this mechanism comes from the fact that no such staggered stacking has been found when the AlInAs spacer layers are replaced by GaInAs, GaInP or InP spacers, for which a chemical decomposition does not take place and for which therefore, only the usual vertical alignment was found [59,70,80,160]. This underlines that the staggered correlations could be induced by chemical eﬀects. A model for this process has been recently developed by Priester and Grenet [62]. On the other hand, however, in some TEM images of InAs/AlInAs superlattices it seems that due to the slow surface kinetics, AlInAs spacer layers are not always completely planarized prior to subsequent InAs growth. Therefore, the surface morphology could also play a signiﬁcant role in the formation of the staggered correlations in these material systems. In fact, in Ref. [142], an inclined alignment of InAs quantum wires was also found for pure InP spacer layers in which alloy decomposition does not exist.

9.8 Summary and Outlook In summary, interlayer dot interactions give rise to the formation of various types of correlations and stackings in self-assembled quantum dot multilayers. The correlations may be caused by diﬀerent mechanisms, such as (1) elastic interactions mediated by the strain ﬁelds of the buried dots, (2) morphological interactions due to nonplanarized spacer topographies, or (3) by interactions based on chemical composition variations within the spacer material. All three mechanisms can induce a vertical dot alignment parallel to the growth direction as well as to staggered dot stackings, depending on the material properties, growth orientation, layer thicknesses and growth conditions. Thus, even for a single material system, the interlayer correlations may change from one stacking type to another, as was exempliﬁed for selforganized PbSe quantum dot superlattices. Of key importance for staggered dot stacking is a high elastic anisotropy of the spacer material as well as the chosen orientation. Interlayer stackings have a profound eﬀect on the lateral arrangement of the dots. In particular, an eﬀective lateral ordering as well as size homogenization may be induced, as is desired for device applications. Staggered dot stackings are particularly eﬀective in this respect because in this case, the dot nucleation sites are determined by interactions with several neighboring

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dots below the surface, whereas for vertically aligned dots, the initial lateral arrangement is mainly replicated from layer to layer with only a weak lateral ordering tendency. This diﬀerent behavior is supported by Monte Carlo growth simulations. While the stackings due to elastic interactions are already quite well understood, there are still ample uncertainties and open issues to be resolved for the other interaction mechanisms. This is due to the fact that they strongly depend on the chosen growth conditions and are always superimposed by the simultaneous action of the strain ﬁeld interactions. Therefore, much work is still needed to clarify their actual role in multilayer growth. On the other hand, the interplay between the diﬀerent mechanisms may be utilized to create new and more complex interlayer stacking types and thus, novel quantum dot superstructures. This may be achieved, e.g., by alternating the material compositions, spacer thicknesses as well as growth conditions during multilayer growth, or by combining compressively and tensile-strained dots, as well as by combining interlayer stacking with prepatterning of substrate templates. The latter will allow to tailor the initial dot arrangement and thus, may ultimately lead to the synthesis of fully controlled 3D nanostructures. Acknowledgement. We would like to thank M. Pinczolits, P. Mayer, A. Raab, R.T. Lechner, J. Stangl, P. Simicek, D. Lugovyy, S. Zerlauth, F. Sch¨ aﬄer, L. Salamanca-Riba, H.H. Kang, and G. Bauer for their valuable contributions. This work was supported by the Fonds zur F¨ orderung der Wissenschaftlichen Forschung and the Gesellschaft of Mikroelektronik of Austria, as well as the Ministry of Education (research program MSM 0021620834) and the Grant Agency (project 202/03/0148) of the Czech Republic.

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10 Self-Organized Anisotropic Strain Engineering for Lateral Quantum Dot Ordering Richard N¨ otzel COBRA Inter-University Research Institute, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands

List of Abbreviations AFM FWHM MBE ML MOVPE PL QD QWR RT SK SL XRD

Atomic force microscopy Full-width at half-maximum Molecular beam epitaxy Monolayer Metal organic vapor-phase epitaxy Photoluminescence Quantum dot Quantum wire Room temperature Stranski–Krastanov Superlattice X-ray diﬀraction

10.1 Introduction Lateral ordering of semiconductor quantum dots (QDs) of high quality in well-deﬁned arrangements is essential for the realization of future quantum functional devices with applications in solid state quantum computing and quantum communication [1]. We have developed a new concept for the creation of laterally ordered QD arrays by self-organized epitaxy. The concept is based on self-organized anisotropic strain engineering of (In,Ga)As/GaAs superlattice (SL) templates by molecular beam epitaxy (MBE) and the lateral ordering of (In,Ga)As QDs by local strain recognition. It is demonstrated for one-dimensional (1D) and two-dimensional (2D) QD arrays on planar GaAs (100) [2] and (311)B [3] substrates. Starting from a nanoscale, random (In,Ga)As QD layer, thin GaAs capping, annealing, GaAs overgrowth, and repetition in SL growth produces highly ordered 1D and 2D (In,Ga)As and, thus, strain-ﬁeld modulations on a mesoscopic length scale. SL self-organization is governed by surface reconstruction and strain induced anisotropic surface migration together with

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lateral and vertical strain correlation. When (In,Ga)As QDs are grown on the strain modulated SL templates, they arrange into well-ordered, straight 1D multiple and single QD arrays on GaAs (100), and into a lattice of ordered lateral QD molecules with controlled QD number and single QDs on GaAs (311)B. The combination of self-organized anisotropic strain engineering with step engineering on shallow-patterned GaAs (100) generates complex QD arrays with naturally well-positioned bends and branches. The strain-correlated growth in SL template formation and QD ordering is directly conﬁrmed by atomic force microscopy (AFM) and high-resolution X-ray diﬀraction (XRD). The QD arrays exhibit excellent photoluminescence (PL) emission up to room temperature (RT) with a linewidth that is not increased compared to that at low temperature. The high structural and optical quality of the ordered QD arrays is assigned to the inherent smoothness of the lateral strain ﬁeld modulations generated on the SL template surfaces on the nanometer length scale, highlighting self-organized anisotropic strain engineering for QD ordering.

10.2 1D QD Arrays on GaAs (100) 10.2.1 SL Template: Formation and QD Ordering SL template formation and QD ordering on singular GaAs (100) substrates comprise the following steps: 1. Formation of randomly distributed (In,Ga)As QDs (1.8 – 2.6 nm (In,Ga)As, 540 ◦ C growth temperature, 36 – 41% In composition) in the Stranski– Krastanov (SK) growth mode on a GaAs buﬀer layer. 2. Growth of the thin GaAs cap layer (0.7 – 0.9 nm at 540 ◦ C). 3. Annealing at higher temperature (2 min at 580 ◦ C). The QDs elongate and connect due to preferential anisotropic Ga/In adatom surface migration along [0-11] induced by the (2 × 4) GaAs surface reconstruction. Simultaneous In desorption allows uniform QD connection due to strain reduction and is balanced by the thin GaAs cap layer of step (2). Quantum wires (QWRs) along [0-11] form. 4. Growth of the GaAs spacer layer (to a total thickness of 13 – 16 nm at 580 ◦ C). The lateral strain ﬁeld modulation from the buried QWRs is preserved at the surface. 5. Growth of the subsequent (In,Ga)As QD layer (at 540 ◦ C). The QDs preferentially nucleate above the QWRs where the lateral tensile strain ﬁeld minima reduce the lattice mismatch and induce strain-gradient-driven In adatom surface migration preferentially along [011] towards the minima. This is orthogonal to the surface-reconstruction-induced adatom migration during annealing along [0-11]. Well-separated 1D QD arrays along [0-11] form.

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6. Repetition of steps (1)–(5) in (In,Ga)As/GaAs SL growth. The length of the QWRs and QD arrays increases and the lateral ordering improves due to the vertical, strain-correlated stacking. Figure 10.1 shows the corresponding AFM images taken in tapping mode in air for the 16th period of the In0.41 Ga0.59 As 2.3 nm/GaAs 13.0 nm SL template (Fig. 10.1a) In0.41 Ga0.59 As QDs, (Fig. 10.1b) GaAs cap, (Fig. 10.1c) annealing, and (Fig. 10.1d) GaAs spacer layer. The In0.41 Ga0.59 As growth rate is 0.092 nm/s. 1D QD arrays are observed in Fig. 10.1a. The growth of the GaAs cap layer (Fig. 10.1b) induces only marginal changes in shape and size of the QDs. After annealing (Fig. 10.1c), the surface becomes rather ﬂat with QWR like structures along [0-11], aﬃrming that the QDs have uniformly connected. After growth of the GaAs spacer layer, asymmetric mounds elongated along [0-11] are developed (Fig. 10.1d) due to the growth

Fig. 10.1. Atomic force microscopy (AFM) images of the surfaces during formation of the 16th period of the In0.41 Ga0.59 As 2.3 nm/GaAs 13.0 nm SL template on GaAs (100). a 2.3 nm In0.41 Ga0.59 As grown at 540 ◦ C, b 0.7-nm-GaAs cap deposited at 540 ◦ C. c Annealing for 2 min at 580 ◦ C. d 12.3 nm GaAs grown at 580 ◦ C. The black-to-white height contrast is 20 nm for all images

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Fig. 10.2. AFM images of the 2.1-monolayer (ML)-InAs QDs a on GaAs (100) and b on the 1, c 5, d 10, and e 15-period In0.36 Ga0.64 As 2.6 nm/GaAs 16 nm superlattice (SL) templates. f AFM image of the 1.5-ML-InAs single quantum dot (QD) arrays formed at the low growth rate of 0.0007 nm/s on the 15-period SL template. The black-to-white height contrast is (a–e) 7 nm and f 10 nm

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instability on singular GaAs (100). The much larger length of the QD arrays compared to that of the mounds conﬁrms that the QD ordering is not caused by morphological features like step edges but originates from the uniform lateral strain ﬁeld modulation on the SL template [4]. The evolution of the (In,Ga)As/GaAs SL template is followed by the distinct dependence of the ordering of InAs QDs grown on top on the number of SL periods. The locations of the InAs QDs mark the lateral strain ﬁeld minima on the GaAs surface induced by the underlying (In,Ga)As QWR structure. Figure 10.2 shows the AFM images of the QDs formed by 2.1 monolayers (MLs) InAs at 480 ◦ C at a growth rate of 0.037 nm/s directly on GaAs (100) (Fig. 10.2a) and on the In0.36 Ga0.64 As 2.6 nm/GaAs 16 nm SL template with the number of SL periods of (b) 1, (c) 5, (d) 10, and (e) 15 (Fig. 10.2b–e). The InAs QDs are arranged randomly on the GaAs buﬀer layer (Fig. 10.2a). For one SL period (Fig. 10.2b), QD ordering is hardly observed, indicating incomplete QWR formation. A weak modulation of the QD density along [011] appears when the number of SL periods is increased to 5 (Fig. 10.2c). For 10 and 15 SL periods (Fig. 10.2d, e), a clear ordering in

Fig. 10.3. X-ray diﬀraction (XRD) spectra (theta-two-theta scans) around the asymmetric (311) glancing exit reﬂection a of the 15-period In0.41 Ga0.59 As 2.3 nm/GaAs 13 nm SL template A, b of the 15-period In0.41 Ga0.59 As 1.8 nm/GaAs 13 nm SL template B, and c of the 15-period In0.41 Ga0.59 As 1.8 nm/GaAs 16 nm SL template C. The black (gray) spectra are measured with the X-ray beam parallel to the [011] ([0-11]) direction

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arrays of multiple QDs along [0-11] takes place. The length of the arrays easily exceeds 3 μm with a lateral periodicity of 140 nm. The lateral periodicity agrees with that of the QWRs, i.e., the periodicity of the lateral strain ﬁeld modulation, determined by XRD of 160 nm (see Fig. 10.3), conﬁrming the strain-correlated SL growth and QD ordering. Single InAs QD arrays are formed on the 15-period SL template for reduced InAs thickness of 1.5 ML and a very low growth rate of 0.0007 nm/s (Fig. 10.2f). 10.2.2 SL Template and QD Ordering: Optimization Although fairly long and uniform QD arrays are formed, a signiﬁcant number of bends and branches is still evident. The bends and branches are attributed to excess strain accumulated in the SL template which cannot be relaxed along straight QWRs. Hence, the uniformity of the SL template is improved for reduced supply of (In,Ga)As to a minimum above the critical thickness for island formation and increased GaAs spacer layer thickness while preserving suﬃcient vertical strain correlation between successive (In,Ga)As layers for ordering. The improvement of the SL template is demonstrated by the asymmetric (311) glancing exit XRD spectra shown in Fig. 10.3a of the 15-period In0.41 Ga0.59 As 2.3 nm/GaAs 13.0 nm SL template A, (b) of the 15-period In0.41 Ga0.59 As 1.8 nm/GaAs 13.0 nm SL template B with reduced (In,Ga)As thickness, and (c) of the 15-period In0.41 Ga0.59 As 1.8 nm/GaAs 16.0 nm SL template C with reduced (In,Ga)As thickness and increased GaAs spacer layer thickness. With the X-ray beam along [011] clear satellite peaks are observed due to the periodicity of the lateral strain ﬁeld modulation of the stacked QWRs [5]. As the strain is reduced from template A to C the satellite peaks sharpen and the peak-to-valley ratio increases, indicating the improved uniformity of the QWRs. This is supported by the narrower 0th order peaks with the X-ray beam along [0-11] for samples B and C. The improved uniformity of the SL templates A to C is additionally conﬁrmed by low-temperature PL. The PL full-width at half-maximum (FWHM) decreases from 81 meV for SL template A to 42 meV for SL template C. The improved uniformity of the SL templates A to C directly relates to that of the single (In,Ga)As QD arrays grown on top. Figure 10.4a–c shows the QD arrays formed by 2.3 nm In0.41 Ga0.59 As on template A, and by 1.8 nm In0.41 Ga0.59 As on templates B and C. The number of branches of the QD arrays is signiﬁcantly reduced to the low value of 1.4 μm−2 on SL template C. The single QD arrays are perfectly straight over more than 1 μm and most of the QD arrays are extended over 10 μm length. 10.2.3 1D QD Arrays: Optical Properties The high structural quality of the (In,Ga)As QD arrays on the SL template C is underlined by the optical properties. For the PL measurements the QD

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Fig. 10.4. AFM images a of the 2.3-nm-In0.41 Ga0.59 As QD arrays on the 15-period In0.41 Ga0.59 As 2.3 nm/GaAs 13 nm SL template A, b of the 1.8-nm-In0.41 Ga0.59 As QD arrays on the 15-period In0.41 Ga0.59 As 1.8 nm/GaAs 13 nm SL template B, and c of the 1.8-nm-In0.41 Ga0.59 As QD arrays on the 15-period In0.41 Ga0.59 As 1.8 nm/GaAs 16 nm SL template C. The black-to-white height contrast is 30 nm for all images

arrays are capped with 100 nm GaAs (20 nm at 540 ◦ C plus 80 nm at 580 ◦ C after 10 s growth interruption). The PL is excited by the 532 nm line of a NdYAG (yttrium aluminium garnet) laser with an excitation power density of 0.2 W cm−2 , dispersed by a single monochromator and detected by a cooled InGaAs charge-coupled device. Figure 10.5 shows the temperature dependent PL spectra and Fig. 10.6a, b the corresponding PL intensity and FWHM. At low temperature, the PL emission from the QD arrays, centered at 1.21 eV, exhibits a FWHM of 72 meV. The PL line at 1.37 eV stems from the SL template. The high-energy shift is due to the In desorption during template formation which is independently conﬁrmed by XRD [6]. When the PL of the SL template vanishes around 100 K, the PL intensity of the QD arrays slightly increases (Fig. 10.6a), indicating thermally activated carrier transfer from the QWRs to the QDs. Thermally activated carrier redistribution within the QD arrays is revealed by the PL FWHM which exhibits a minimum around 200 K (Fig. 10.6 b). This minimum reﬂects preferential

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R. N¨ otzel Fig. 10.5. Temperature-dependent photoluminescence (PL) spectra of the capped 1.8-nm-In0.41 Ga0.59 As single QD arrays on the 15-period In0.41 Ga0.59 As 1.8 nm/GaAs 16 nm SL template C

Fig. 10.6. Temperature dependence of a the PL peak intensity and b the full-width at half-maximum (FWHM ) of the capped 1.8-nmIn0.41 Ga0.59 As single QD arrays on the 15period In0.41 Ga0.59 As 1.8 nm/GaAs 16 nm SL template C

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carrier transfer from smaller (higher energy) to larger (lower energy) QDs at intermediate temperatures followed by equilibration of the carrier distribution at higher temperatures when the probability of carrier escape from the large QDs increases [4]. The PL eﬃciency of the QD arrays is strong up to RT. Most notably, the FWHM at RT of 70 meV does not exceed that at low temperature (72 meV), as is expected for QDs with strong carrier conﬁnement. 10.2.4 Complex QD Arrays: Growth on Shallow-Patterned Substrates The relationship of self-organized anisotropic strain engineering with step engineering on shallow mesa-patterned GaAs (100) substrates is established for the realization of advanced, complex QD arrays and networks [7]. Growth on shallow-patterned substrates has been ﬁrst utilized for the formation of highly uniform sidewall QWR and QD arrays on GaAs (311)A [8], related to the unique microscopic surface corrugation [9]. On shallow [0-11] and [011] stripe-patterned GaAs (100) substrates the generated type-A and -B steps diﬀerently aﬀect the adatom surface migration processes during SL template development. While type-A steps along [0-11] have no signiﬁcant eﬀect on the strain-gradient-driven In adatom migration along [011], type-B steps along [011] strongly hinder the surface-reconstruction-induced Ga/In adatom migration along [0-11] to prevent QWR formation and QD ordering. When both type-A and -B steps are present on vicinal substrates which are misoriented towards [101], i.e., the average step edge direction is along [001], 45◦ oﬀ [0-11], the surface-reconstruction-induced Ga/In adatom migration is altered to rotate the QD arrays. On shallow zigzag-patterned substrates, this leads to complex QD arrays and networks with well-positioned bends and branches, which exhibit excellent structural and optical quality. Figure 10.7a shows the 1.8-nm-In0.41 Ga0.59 As QD arrays grown on the 15-period In0.41 Ga0.59 As 1.8 nm/GaAs 13 nm SL template on the shallow zigzag-patterned substrate prepared by optical lithography and wet chemical etching. The 10 μm wide mesas with 30 nm height are separated by 10 μm. The 10 μm long sidewalls are alternately rotated by plus and minus 30◦ oﬀ [0-11]. The magniﬁed image in Fig. 10.7b is taken from the sidewall intersections. For this shallow mesa height the sidewalls after GaAs buﬀer layer growth consist of (100) terraces and ML-height type-A and -B steps rather than of facets, in addition to planar (100) surface areas. The 1D QD arrays on the sidewalls are rotated by plus and minus 16◦ oﬀ [0-11], seen in Fig. 10.7b. The smaller rotation angle of the QD arrays compared to that of the sidewalls indicates that the QD ordering is not associated with the average step-edge direction and related QD nucleation, though step-edge related QD ordering is observed when the QDs are grown directly on the GaAs buﬀer layer [10]. In contrast, the QD ordering on the SL template is due to a step-edge induced rotation of the Ga/In adatom migration during annealing in SL template

314

R. N¨ otzel Fig. 10.7. a AFM image of the 1.8-nm-In0.41 Ga0.59 As QD arrays on the 15-period In0.41 Ga0.59 As 1.8 nm/GaAs 13 nm SL template on shallow zigzag-patterned GaAs (100). b Magniﬁed image of the sidewall intersections. The black-to-white height contrast is a 40 nm and b 20 nm

formation while the QD ordering and strain-gradient-driven In adatom migration are solely governed by the strain ﬁeld and unaﬀected by the presence of steps. This is evidenced by the lateral periodicity of the QD arrays on the sidewalls, which is comparable to that in planar areas. The rotation of the QD arrays on the sidewalls naturally produces the basic building blocks of complex QD networks in a well-deﬁned way. Bends of the QD arrays by ∼32◦ are formed at the sidewall intersections, and periodic arrangements of branches are generated at the intersections of the sidewalls and the planar areas, which contain QD arrays oriented along [0-11], as demonstrated in Fig. 10.7b. The integrated PL intensity of the capped QD arrays on the sidewalls is the same as that of the QD arrays in planar areas. Thus, the optical quality of the complex QD arrays, which are realized on shallow-patterned substrates,

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is not degraded compared to that of QD arrays on planar ones. This is again attributed to the fact that the QD ordering is exclusively governed by the strain ﬁeld while the role of the steps generated by the shallow pattern is to modify (rotate) the adatom surface migration during annealing in SL template formation. This allows the design of strain ﬁelds for advanced lateral ordering of QD arrays and networks of high quality and complexity.

10.3 Lattice of Ordered QD Molecules on GaAs (311)B 10.3.1 SL Template: Formation, Stability, and Pattern Transition Figure 10.8 shows the AFM images of the 3.2-nm-In0.37 Ga0.63 As layers on the (In,Ga)As/GaAs SL templates with (a) one, (b) ﬁve, and (c) ten periods on GaAs (311)B. Each SL period comprises 3.2 nm In0.37 Ga0.63 As grown at 500 ◦ C, thin capping by 0.7 nm GaAs at 500 ◦ C, annealing for 2 min at 580 ◦ C, and growth of a 5.5 nm GaAs spacer layer at 580 ◦ C. The growth rates of GaAs and In0.37 Ga0.63 As are 0.073 and 0.116 nm/s. Upon stacking, the nanoscale 2D QD-like (In,Ga)As surface modulation for the ﬁrst SL period [11] evolves into the distinct mesoscopic mesa-like arrangement when the number of SL periods is increased from ﬁve to ten. This is explained by anisotropic surface migration during annealing, smoothening the mesas to form mesoscopic nodes, see Fig. 10.8d, and strain correlated (In,Ga)As stacking, which is governed by preferential (In,Ga)As accumulation on the nodes due to strain-gradient-driven In adatom migration [12]. The morphology in Fig. 10.8c constitutes the Turing pattern [13] of buried (In,Ga)As quantum disks obtained by metal organic vapor-phase epitaxy (MOVPE) [14]. In contrast to MOVPE, where the strain driven materials reorganization is completed for the ﬁrst (In,Ga)As layer due to the higher growth temperature, the development of the SL template in MBE provides snapshots of the pattern evolution, which have been rarely observed experimentally. In this context it is most important that the mesoscopic surface pattern becomes stable after 10 SL periods. This is revealed in Fig. 10.8e for the 3.2-nm-In0.37 Ga0.63 As layer on the 15-period SL template, which closely resembles that on the 10-period SL template in Fig. 10.8c with mesas of ∼200 nm width and 300 nm lateral periodicity. Within the general description of reaction-diﬀusion systems, in the present case, the reaction term is associated with strain induced island formation during (In,Ga)As growth and the diﬀusion term with adatom surface migration during annealing and (In,Ga)As growth, guided by the lateral strain ﬁeld modulation. This is supported by altering the balance between reaction and diﬀusion when the temperature for (In,Ga)As growth and thin GaAs capping is increased to 550 ◦ C, and the temperature for annealing and GaAs spacer layer growth to 610 ◦ C. This reduces the strain due to enhanced In desorption and increases the adatom surface migration length. As a result,

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Fig. 10.8. AFM images of 3.2 nm In0.37 Ga0.63 As on the a one, b ﬁve, and c ten-period In0.37 Ga0.63 As 3.2 nm/GaAs 6.2 nm SL templates on GaAs (311)B. d The 11-period SL template surface after 3.2-nm-In0.37 Ga0.63 As plus 0.7-nmGaAs cap layer growth at 500 ◦ C, and annealing plus 5.5-nm-GaAs spacer layer growth at 580 ◦ C. e 3.2 nm In0.37 Ga0.63 As on the 15-period SL template. f 3.2 nm In0.37 Ga0.63 As on the 10-period SL template. The growth temperature of the (In,Ga)As and thin GaAs cap layers is increased to 550 ◦ C, and the temperature for annealing and GaAs spacer layer growth to 610 ◦ C. The black-to-white height contrast is a, b, c, e ,f ) 15 nm and d 4 nm

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the mesa-like pattern aligned plus and minus 45◦ oﬀ [0-11] transforms into a zigzag or stripe-like pattern oriented preferentially along the directions plus and minus 70◦ oﬀ [0-11], as shown in Fig. 10.8f. A quantitative analysis of the pattern formation and pattern transition as a function of growth conditions is, however, beyond the scope of the present evaluation. To gain further insight in the SL template evolution, the In composition and thickness of the (In,Ga)As layers in the ten-period SL template are varied, and Al-containing layers are inserted. Regarding the In composition, the highest degree of ordering is observed in a window between 37 and 29%, when adjusting the respective (In,Ga)As layer thickness to 3.2 and 4.0 nm to keep the total amount of In constant. For larger or lower In composition and (In,Ga)As layer thickness the ordering degrades either due to too large strain accumulation or too low strain where the strain correlated stacking is not supported. Inserting Al-containing layers at various locations in the GaAs spacer layers of the SL template signiﬁcantly alters its formation. When 2 MLs AlAs covered by 1.0 nm GaAs are introduced beneath the (In,Ga)As layers or 2 MLs AlAs are inserted on top of the thin GaAs cap layers after annealing, the ordering strongly degrades. Replacing the GaAs spacer layers by Al0.24 Ga0.76 As breaks down the ordering. The diminished lateral ordering of the SL template in the presence of AlAs below and above the (In,Ga)As layers indicates that a signiﬁcant strain-driven lateral mass transport of GaAs below and above is involved in SL template formation, which is hindered due to the lower adatom migration lengths of Ga and In on AlAs. For (Al,Ga)As spacer layers the ordering is entirely suppressed due to the small migration length of Al. 10.3.2 QD Molecules: Formation and QD Number Control InAs QDs grow on top of the SL template, i.e., on the upper GaAs layer, in dense and well-separated ordered groups. Figure 10.9a shows the AFM image of the QD molecules formed by 0.6 nm InAs deposited at 500 ◦ C on the ten-period In0.37 Ga0.63 As 3.2 nm/GaAs 6.2 nm SL template. The InAs growth rate is 0.0013 nm/s. The QDs arrange on the nodes of the SL template where the underlying (In,Ga)As accumulation establishes the tensile strain ﬁeld minima and the related strain-gradient-driven In adatom migration for strain-correlated stacking. The ordering and size uniformity of the InAs QDs within the groups are signiﬁcantly improved by lowering the InAs growth temperature to 470 ◦ C (see the 0.5-nm-InAs QD molecules in Fig. 10.9b). When the temperature is lowered further to 450 ◦ C the QD ordering decreases for too small In adatom migration length. For an InAs growth temperature of 520 ◦ C, large, elongated mounds are observed due to QD coalescence, which is already recognized in Fig. 10.9a in the center of the nodes for the QDs grown at 500 ◦ C. Hence, a growth temperature of 470 ◦ C is identiﬁed for optimum InAs QD ordering within the molecules.

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Fig. 10.9. AFM images of the 0.5-0.6-nm-InAs QD molecules grown at a 500 ◦ C and b 470 ◦ C on the ten-period In0.37 Ga0.63 As 3.2 nm/GaAs 6.2 nm SL template with 5.5 nm upper GaAs separation layer. c 0.5-nm-InAs QD molecules grown at 470 ◦ C on the ten-period SL template with 15 nm upper GaAs separation layer. In d the growth temperature of the (In,Ga)As and thin GaAs cap layers in the tenperiod SL template with 15 nm upper GaAs separation layer is increased to 520 ◦ C. The black-to-white height contrast is 10 nm for all images

To control the average number of InAs QDs within the molecules, a reduction of the amount of InAs rather reduces the QD height, leaving the QD number unchanged. The average number of InAs QDs is controlled by the thickness of the upper GaAs separation layer on the SL template. The number is reduced from 11 for the 5.5 nm thick GaAs separation layer (Fig. 10.9b) to 8.5 for a thickness of 9.5 nm, and to 7.5 for a thickness of 15 nm, depicted in Fig. 10.9c. The growth temperature of the 0.5-nm-InAs QDs is 470 ◦ C for optimized QD ordering within the molecules on the tenperiod In0.37 Ga0.63 As 3.2 nm/GaAs 6.2 nm SL template. The reduction of the average number of QDs per molecule with increasing upper GaAs separation layer thickness on the SL template is attributed to a decrease of the tensile strain ﬁeld minima, shrinking the eﬀective area for preferred QD formation. When, however, the upper GaAs separation layer thickness is

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Fig. 10.10. XRD spectra around the symmetric (311) reﬂection of the ten-period In0.37 Ga0.63 As 3.2 nm/GaAs 6.2 nm SL template of Fig. 10.8c without upper (In,Ga)As layer with the X-ray beam 45◦ oﬀ (solid line) and along (dashed line) [0-11]

increased to 20 nm, the QD ordering degrades for too shallow strain ﬁeld minima. The eﬀective area for QD formation and, thus, the average number of QDs per molecule is further decreased by increasing the (In,Ga)As and thin GaAs cap growth temperature in the SL template, promoting enhanced In migration and In accumulation on the nodes. Figure 10.9d shows the 0.5-nmInAs QD molecules deposited at 470 ◦ C on the ten-period SL template with the In0.37 Ga0.63 As and thin GaAs cap layers grown at 520 ◦ C, and a 15 nm thick upper GaAs separation layer. The average number of QDs per molecule is four. The strain-correlated growth in SL template formation and QD ordering is consistently conﬁrmed by XRD in various scattering geometries. Figure 10.10 shows the symmetric (311) XRD spectra of the ten-period In0.37 Ga0.63 As 3.2 nm/GaAs 6.2 nm SL template of Fig. 10.8c (without the (In,Ga)As layer on top) with the X-ray beam 45◦ oﬀ [0-11] and along [0-11]. The spacings of the satellite peaks close to the substrate reﬂection provide the periodicity of the lateral strain ﬁeld modulation of the SL template in the respective directions, 350 nm along the direction 45◦ oﬀ [0-11] and 420 nm along [0-11]. For completeness, grazing exit asymmetric (004) XRD spectra reveal a lateral periodicity of 400 nm along [0-11]. These values coincide well with the lateral periodicity of the InAs QD molecules on the SL template of 300 nm along the direction 45◦ oﬀ [0-11], which is derived from the 2D fast-Fourier transform analysis of the AFM images in Fig. 10.9b, c. 10.3.3 Single QDs: Formation To realize single InAs QDs on the nodes of the ten-period In0.37 Ga0.63 As 3.2 nm/GaAs 6.2 nm SL template, not only the growth temperature of the In0.37 Ga0.63 As and thin GaAs cap layers is increased but also the temperatures for annealing and GaAs spacer layer growth, and that for InAs QD

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deposition. Figure 10.11a shows the AFM image of the 0.5-nm-InAs QDs grown at 500 ◦ C on the SL template with (In,Ga)As and thin GaAs cap layer growth temperature of 530 ◦ C, and annealing and GaAs spacer layer growth at 610 ◦ C. The upper GaAs separation layer thickness is 15 nm. The number of InAs QDs on the SL template nodes, however, does not change compared to that in Fig. 10.9d where only the growth of the (In,Ga)As layers and thin GaAs caps is at higher temperature. The QD groups are less separated and ordered due to enhanced In desorption, i.e., reduced total strain in SL growth, and a more stripe-like pattern develops with less deﬁned nodes as discussed for the SL template in Fig. 10.8f. Single InAs QDs on the SL template nodes are realized at the elevated growth and annealing temperatures of the ten-period SL template and InAs QDs when an additional 0.5-nm-InAs QD layer is inserted. The additional InAs QD layer is grown at 500 ◦ C on the SL template with 15 nm upper GaAs separation layer, it is thin GaAs capped and annealed as the (In,Ga)As layers in the SL template, and overgrown by 15 nm GaAs. Due to the fact that the InAs QDs in this interlayer solely form on the SL template nodes, the lateral strain ﬁeld modulation most eﬀectively concentrates after thin GaAs capping and annealing. Together with the enhanced In adatom migration length at elevated temperature, supported by the tendency for QD coalescence (discussed for the case of Fig. 10.9a), the resulting shrinkage of the eﬀective area of the tensile strain ﬁeld minima for preferred QD formation produces single InAs QDs in the center of the nodes, as shown in Fig. 10.11b. Only the combination of elevated temperatures for SL template formation and InAs QD growth,

Fig. 10.11. AFM images of the 0.5-nm-InAs QDs grown at 500 ◦ C on the tenperiod In0.37 Ga0.63 As 3.2 nm/GaAs 6.2 nm SL template with growth temperature of 530 ◦ C of the (In,Ga)As and thin GaAs cap layers, and 610 ◦ C for annealing and the GaAs spacer layers. a InAs QDs without and b with additional 0.5-nm-InAs QD insertion layer, thin GaAs capped and annealed, on the SL template with 15 nm upper GaAs separation layer and overgrown by 15 nm GaAs. The black-to-white height contrast is 20 nm for both images

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and the insertion of an additional InAs QD layer realizes single QDs. Applied separately, the average QD number per molecule is not changed signiﬁcantly. 10.3.4 QD Molecules: Optical Properties The high structural and optical quality of the InAs QD molecules manifests itself in the excellent PL properties up to RT. Figure 10.12 shows the PL spectra taken at 5 K and RT of the capped (by 20 nm GaAs at the InAs QD growth temperature plus 180 nm GaAs at 580 ◦ C without annealing) InAs QD molecules of Fig. 10.9c. At 5 K the peak at 1.18 eV with a FWHM of 60 meV originates from the InAs QD molecules and that at 1.32 eV with a FWHM of 44 meV from the ten-period SL template. This assignment follows from the PL peak energy of 1.28 eV of the SL template alone, shown in Fig. 10.12 for reference. The high-energy shift of the PL from the SL template is again due to In desorption and ﬂattening of the (In,Ga)As layers during annealing (unannealed (In,Ga)As layers exhibit a PL peak energy of 1.19 eV) and varies slightly from sample to sample. At RT the PL of the QD molecules is centered at 1.10 eV with a FWHM of 59 meV and a three orders of magnitude reduced peak intensity due to thermally activated escape of carriers from the QDs. The absence of RT PL from the SL template indicates eﬃcient carrier transfer to the QDs. The PL eﬃciency of the QD molecules is not degraded compared to that of InAs QDs directly grown on GaAs with a comparable FWHM. Detailed temperature dependent PL measurements between 5 K and RT reveal a constant value of the FWHM of the QD molecules up to 80 K, shown in Fig. 10.13a. The FWHM then undergoes a distinct minimum at 140 K due to thermally activated carrier redistribution from smaller to larger QDs. This is accompanied by a characteristic enhancement of the low-energy shift of the PL peak position as a function of temperature, depicted in Fig. 10.13b. The FWHM increases steeply to 57 meV at 190 K due to equilibration of the carrier distribution in large and small QDs. Remarkably, the steep increase

Fig. 10.12. PL spectra taken at 5 K (solid line) and room temperature (RT ) (dotted line) of the capped 0.5nm-InAs QD molecules of Fig. 10.9c grown at 470 ◦ C on the ten-period In0.37 Ga0.63 As 3.2 nm/GaAs 6.2 nm SL template with 15 nm upper GaAs separation layer. The PL at 5 K from the ten-period SL template alone (dashed line) is shown for reference

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R. N¨ otzel Fig. 10.13. Temperature dependence of a the full-width at halfmaximum (FWHM ) and b the PL peak position of the capped InAs QD molecules of Fig. 10.9c (solid triangles), the capped (In,Ga)As QDs of Fig. 10.8c (solid circles), and the ten-period SL template of Fig. 10.8c without upper (In,Ga)As layer (solid squares)

of the FWHM is followed by a very weak increase above 190 K to a value of 59 meV at RT. The distinct inﬂection point at 190 K indicates equilibration of the carrier distribution solely within the QD molecules, which is followed by thermal broadening resembling that of isolated QDs. This indicates that the QD molecules are electronically isolated. Such an unusual behavior of the FWHM is not observed for the capped (In,Ga)As QD layer of Fig. 10.8c and for the SL template of Fig. 10.8c without upper (In,Ga)As layer, shown in Fig. 10.13a for reference, nor for the 1D (In,Ga)As QD arrays in Fig. 10.6b. After the minimum of the FWHM and enhanced low-energy shift of the PL peak position (Fig. 10.13b) due to thermally activated carrier redistribution, the FWHM continuously increases with temperature, which is attributed to a continuous increase of the carrier spread over areas fully covered with QDs.

10.4 Conclusions A new concept for the lateral ordering of quantum dots (QDs) by selforganized epitaxy has been introduced. The concept is based on self-organized anisotropic strain engineering of (In,Ga)As/GaAs superlattice (SL) templates by molecular beam epitaxy (MBE) and is demonstrated for the formation of

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highly uniform, straight 1D multiple and single (In,Ga)As QD arrays on GaAs (100), and a lattice of ordered InAs QD molecules with controlled number of QDs and single InAs QDs on GaAs (311)B substrates. During stacking the SL templates develop 1D and 2D lateral strain ﬁeld modulations on a mesoscopic length scale where the QDs on top order due to strain-correlated growth and local strain recognition. The combination of self-organized anisotropic strain engineering with step engineering on shallow-patterned GaAs (100) generates complex QD arrays with naturally well-positioned bends and branches. The ordered QD arrays exhibit excellent structural and optical quality up to room temperature, which is attributed to the inherent smoothness of the lateral strain ﬁeld modulations on the dot-diameter and dot-to-dot distance length scales. Hence, fascinating applications of these ordered QD arrays are foreseen for novel quantum functional devices required for quantum computing and quantum communication in solid state. Acknowledgement. The author thanks T. Mano, T. Lippen, Q. Gong, and J.H. Wolter for the very fruitful collaboration, and G.J. Hamhuis and T.J. Eijkemans for technical support.

References 1. R. N¨ otzel, T. Mano, Q. Gong, J.H. Wolter, Proc. IEEE 91, 1898 (2003). The relevant work performed in other laboratories is covered in the references therein and in this book. 2. T. Mano, R. N¨ otzel, G.J. Hamhuis, T.J. Eijkemans, J.H. Wolter, Appl. Phys. Lett. 81, 1705 (2002) 3. T. Lippen, R. N¨ otzel, G.J. Hamhuis, J.H. Wolter, Appl. Phys. Lett. 85, 118 (2004) 4. T. Mano, R. N¨ otzel, G.J. Hamhuis, T.J. Eijkemans, J.H. Wolter, J. Appl. Phys. 95, 109 (2004) 5. W.Q. Ma, R. N¨ otzel, A. Trampert, M. Ramsteiner, H. Zhu, H.P. Sch¨ onherr, K.H. Ploog, Appl. Phys. Lett. 78, 1297 (2001) 6. T. Mano, R. N¨ otzel, G.J. Hamhuis, T.J. Eijkemans, J.H. Wolter, J. Appl. Phys. 92, 4043 (2002) 7. T. Mano, R. N¨ otzel, D. Zhou, G.J. Hamhuis, T.J. Eijkemans, J.H. Wolter, J. Appl. Phys. 97, 014304 (2005) 8. R. N¨ otzel, Z.C. Niu, M. Ramsteiner, H.P. Sch¨ onherr, A. Trampert, L. D¨ aweritz, K.H. Ploog, Nature (London) 392, 56 (1998) 9. R. N¨ otzel, N.N. Ledentsov, L. D¨ aweritz, M. Hohenstein, K. Ploog, Phys. Rev. Lett. 67, 3812 (1991) 10. R. N¨ otzel, Q. Gong, Nanotechnology 13, 705 (2002) 11. Q. Gong, R. N¨ otzel, G.J. Hamhuis, T.J. Eijkemans, J.H. Wolter, Appl. Phys. Lett. 81, 3254 (2002) 12. T. Lippen, R. N¨ otzel, G.J. Hamhuis, J.H. Wolter, J. Appl. Phys. 97, 044301 (2005) 13. J. Temmyo, R. N¨ otzel, T. Tamamura, Appl. Phys. Lett. 71, 1086 (1997) 14. R. N¨ otzel, J. Temmyo, T. Tamamura, Nature, (London) 369, 131 (1994)

11 Towards Quantum Dot Crystals via Multilayer Stacking on Diﬀerent Indexed Surfaces Zhiming M. Wang and Gregory J. Salamo Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA

List of Abbreviations 1D, 2D, 3D AFM FEM FFT GISAXS MBE ML PL RDS RHEED SK STM TEM THz TRTS WL

One-, two-, three-dimensional Atomic force microscopy Finite element method Fast Fourier transform Grazing-incidence X-ray small angle scattering Molecular beam epitaxy Monolayer Photoluminescence Resonant diﬀuse scattering Reﬂection high-energy electron diﬀraction Stranski–Krastanov Scanning tunneling microscopy Transmission electron microscopy Terahertz Time-resolved terahertz spectroscopy Wetting layers

11.1 Introduction The growth of InGaAs on GaAs with suﬃcient lattice mismatch typically proceeds in the Stranski–Krastanov (SK) mode. In the SK growth mode, the ﬁrst few monolayers (MLs) of InGaAs form a pseudomorphic twodimensional (2D) layer, traditionally called the wetting layer (WL). After a critical thickness, the development of three-dimensional (3D) InGaAs islands, which partially relieve the built-up strain, is more energetically favorable than continuous layers. Such 3D islands standing on the 2D WL, usually referred to as self-assembled quantum dots (QDs), have been commonly used for low-dimensional semiconductor research in the last decade. Generally, the InGaAs QDs are randomly distributed on the 2D WL due to the stochastic nature of the self-assembly process. The resulting selfassembled InGaAs QDs have emerged as an important class of materials

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with potential for modern optoelectronic devices, such as QD-based lasers and detectors. In principle, however, more control over uniformity and spatial organization of the InGaAs QD arrays is desirable for many applications. For example, organized 3D arrays are important for addressing QDs and for a collective behavior uniquely diﬀerent from the individual InGaAs QDs. For the vertical ordering along the growth direction, the SK growth mode has produced excellent results through stacking of multiple InGaAs layers and the corresponding strain interaction through GaAs spacer layers. On the other hand, achieving lateral spatial ordering of QDs has been very challenging and we must question whether the SK growth can by itself achieve the desired lateral control of QD positioning or whether outside help, such as lithography, is needed. For example, hybrid approaches made up from SK self-assembly on substrates that are prepatterned by lithography have demonstrated successful lateral “forced alignment” of the QD position. Another interesting, perhaps more natural approach, is to use substrates that are prestructured by surface orientations. That is, by stacking multiple InGaAs QD layers on diﬀerent indexed substrates, we can explore the opportunities for SK self-assembly alone to produce 3D self-alignment. Stacking of InGaAs multiple layers on GaAs(100) leads to the formation of strings of QDs along the [01-1] direction, so-called QD chains. It is found that the QD chains can be substantially increased in length by the introduction of growth interruptions during the initial stages of growth of the GaAs spacer layer. Amazingly, QD chains that are longer than 5 μm are easily achieved, vertically aligned along the growth direction and lateral stringed by an elongated In-rich band called the 1D WL. In sharp contrast with the 2D WL, which exists before the QD chains form, the 1D WL develops during the GaAs capping of the existing QD chains. The formation of both the QD chains and the 1D WL originates from the anisotropic material distribution and redistribution on the (2 × 4) reconstructed GaAs(100) surface with dimer rows running along [01-1]. The surface anisotropy from the reconstruction can be compensated by creating surface steps along [011] through substrate misorientation from (100). On high index surfaces of GaAs(n11)B with high enough surface misorientations where n is 9, 7, 5, 4 and 3, the QD lateral alignment takes the form of square-like lattices instead of QD chains. Remarkably, due to the elastic anisotropy of the matrix on substrates with diﬀerent indexes, QDs in diﬀerent layers are vertically ordered but aligned with a predictable angle inclined to the growth directions.

11.2 Experimental Setups All the samples investigated were grown in a III–V semiconductor molecular beam epitaxy (MBE) growth chamber (Riber 32P), which is equipped with an optical transmission thermometry system that monitors the sub-

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strate band edge to give accurate growth temperatures. MBE growths under various growth conditions were monitored in situ by reﬂection high-energy electron diﬀraction (RHEED). In addition, the MBE chamber is connected to an Omicron surface analysis chamber which contains a scanning tunneling microscopy (STM) system under high vacuum. Most of surface topographic information is obtained by a Veeco atomic force microscopy (AFM) system, through imaging under ambient conditions. In order to probe the 3D ordering of QD multilayers, transmission electron microscopy (TEM) and X-ray diﬀuse scatting have been performed. TEM analysis was performed using a JEOL JEM-2000FX microscope operated at 200 kV. Typically dark-ﬁeld imaging conditions were used to obtain a chemical composition sensitive contrast. This allowed various InGaAs layers and QDs to be distinguished from the surrounding GaAs matrix. X-ray diﬀuse scattering has been monitored threedimensionally by a special multidetection technique using a CCD detector. The experiments were carried out using highly brilliant synchrotron radiation from beam lines W1 and BW2 at HASYLAB/DESY at a wavelength of 1.55. The optical properties of the samples are characterized by photoluminescence (PL) measurements. PL studies were performed in a variabletemperature helium cryostat (4 – 300 K), under the excitation of the 514.5 nm line of a continuous-wave argon-ion laser. The PL signal was analyzed using a 0.5 m single-grating spectrometer, and detected using a photomultiplier tube.

11.3 Stacking on GaAs(100) 11.3.1 Emerging of Lateral Alignment During the Growth of Multiple InGaAs Layers Using GaAs(100) substrates, a series of samples were grown with single, 2, 7, and 12 layers of InGaAs QDs [1]. Each QD layer was formed by depositing 9.0 ML of In0.5 Ga0.5 As at the substrate temperature of 540 ◦ C. The ﬁrst layer of QDs was observed to form after 4.0 ML of In0.5 Ga0.5 As, whereas the following layers of QDs were each observed to form after only 3.4 ML of In0.5 Ga0.5 As. The observations are based on the transition of the RHEED pattern from a 2D streak pattern to a 3D spot pattern. Except for the outermost In0.5 Ga0.5 As QD layer left exposed for AFM measurements, each of the other QD layers is buried by depositing 60 ML GaAs as a spacer. A typical AFM image of the ﬁrst QD layer is given in Fig. 11.1a. The surface is almost fully covered by closely packed In0.5 Ga0.5 As islands with a density of about 430 μm−2 . They are slightly elongated with the long axis parallel to the [01-1] direction, a common feature of InGaAs islands observed on GaAs(100). Figure 11.1b shows the surface topology of the sample with two layers of In0.5 Ga0.5 As QDs. As mentioned above, the observation of the RHEED pattern indicates that the formation of QDs in the second layer happens earlier

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than in the ﬁrst layer. The strain ﬁeld induced by the buried islands in the ﬁrst layer directs the material in the second layer to the local minimum of lattice mismatch. This gives rise to a change of 0.6 ML for the 2D to 3D transition between the ﬁrst and second layer of QDs. Meanwhile, the surface density of In0.5 Ga0.5 As islands shown in Fig. 11.1b actually reduces to 380 μm−2 . Correspondingly, the islands become wider and taller than that in the ﬁrst layer shown in Fig. 11.1a. In addition, the islands tend to lineup in the [01-1] direction, an observation that becomes clear after the growth of seven layers of In0.5 Ga0.5 As QDs, as shown in Fig. 11.1c. The average island size continues to grow bigger and has a surface density of 350 μm−2 by the seventh layer. While these observations are expected the most surprising characteristic is the development of QD chains running along the [01-1] direction, although there are many isolated islands, especially some relatively small islands that ﬁll in the space between the dots making up the chains. Upon continuing the deposition up to 12 layers of In0.5 Ga0.5 As QDs, nearly all of the small islands between the QD chains disappear, as shown in Fig. 11.1d, leaving remarkably

Fig. 11.1. Atomic force microscopy (AFM) images of 9.0 monolayers (ML) In0.5 Ga0.5 As QDs with a diﬀerent number of stacked layers, illustrating the effect of multilayering on the formation of QD chains. a 1 layer. b 2 layers. c 7 layers. d 12 layers

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long chains of QDs along [01-1]. Many of the QD chains easily exceed 1 μm in length. The measured QD density after 12 layers is reduced even further to 250 μm−2 as the QDs continue to grow. The average lateral distance between QD chains is 83 nm, and the average QD distance within chains is 53 nm. 11.3.2 Vertically Aligned Long Chains of InGaAs QDs The high anisotropic QD alignment in the chainlike structure is a direct consequence of the anisotropy of material diﬀusion on the (2 × 4) reconstructed GaAs(100) surface with dimer rows running along [01-1]. It is important to note that the QD chains are only developed during multilayering In0.5 Ga0.5 As QDs spaced by the GaAs layers. Surprisingly, the growth conditions of the GaAs spacers show a strong impact on the formation of QD chains. It is found that the QD chains can be substantially increased in length by the introduction of growth interruptions during the initial stages of growth of

Fig. 11.2. AFM images of the topmost surface after stacking 17 layers of 6.0 ML In0.5 Ga0.5 As QDs spaced by 60 ML GaAs with six growth interruptions of 10 s for the initial 18 MLGaAs growth

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the GaAs spacer layer [2]. The AFM image in Fig. 11.2 shows the surface morphology after stacking 17 layers of 6 ML In0.5 Ga0.5 As spaced by 60 ML GaAs with introducing growth interruptions, one every 10 s per 3 ML growth for the ﬁrst 18 ML GaAs spacer growth. Most of QD chains are over 5 μm long along [01-1]. As shown in the zoom-in AFM image inset of Fig. 11.2, the average spacing of the QD chains is about 71 nm and the separation between QDs along the chains is about 36 nm. While AFM measurements only show the topmost topography of the InGaAs/GaAs QD multiple layers, TEM analysis through the plan and crosssectional measurements is able to provide a composite view including the buried QD layers underneath. For the TEM images in Fig. 11.3, a darker shade represents increased indium content [3]. The plan-view clearly shows the ordered QD chains. The [011] cross-sectional view (Fig. 11.3b) clearly shows dot stacking within the chains. Likewise, the [01-1] view (Fig. 11.3c) clearly shows the vertical stacking of the QD chains themselves. Since all of the QD layers are imaged, these cross-sectional images also show the development of ordering in the ﬁrst few layers, further conforming the emerging of QD lateral alignment as revealed in Fig. 11.1. For these TEM images we estimate the thickness of the sample imaged to be about 100 nm. This implies that in the plan-view we are imaging through about six layers, while in the [011] ([01-1]) view we are imaging through two chains (four QDs). The fact that the observed contrast for these images is related to the indium concentration integrated over the sample thickness (i.e., a thickness larger than the QD size) can make it diﬃcult to truly reconstruct a 3D structure. However, because we have measured three perpendicular views of the sample, and

Fig. 11.3. Dark-ﬁeld, plan and cross-sectional views of the InGaAs/GaAs QD chains as in Fig. 11.2. Scale bar applies to all images. a Planview image of a 1 × 1 μm region. b A [011] cross-sectional view showing the side view of the quantum dot (QD) chains. c A [01-1] cross-sectional view showing the end view of the QD chains

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there is such a high degree of ordering, we have not had this diﬃculty. Further more, we applied selective chemical etching together with AFM imaging to investigate an InGaAs/GaAs QD multiple layered structure [4]. The preferential etching on GaAs over InGaAs reveals the buried InGaAs layers in a QD layer-by-QD layer manner. AFM measurements reveal every exposed InGaAs layer was characterized with the chainlike lateral ordering. Since two or more QD layers could be exposed at the same time by this approach, we received a unique opportunity to study their vertical alignment by regular plan-view AFM measurements. Consistently, we found that both QD chains and QDs in chains are vertically aligned along the growth direction. Remarkably, the alignment of QD chains is unyielding to a modest deviation of surface orientation, which is ﬁrst deduced by using oval defects as a probing tool to explore the role of surface orientation [5]. Figure 11.4a shows a GaAs oval defect, 45 nm tall over the (100) growth front and 10 μm long along [01-1] and 2 μm wide along [011]. The elongation of the defect is the result of the anisotropic surface diﬀusion and incorporation rate of Ga and In. Based on the analysis of its proﬁles, it is found that this speciﬁc defect provides growth surfaces that are vicinal to (100) in all azimuthal directions with misorientation angles up to 0.7◦ along [01-1] and 8◦ along [011]. As a result, at least two very diﬀerent types of steps are expected due to the substrate misorientation: type-A steps that run along [01-1] while type-B steps run along [011]. Consequently, this variety of steps is expected to introduce additional anisotropic behavior in surface diﬀusion. For example, the type-A steps will enhance the surface anisotropy of GaAs(100) along [01-1]. Since the type of step and the step density depends on the misorientation, diﬀerent spatial regions on the oval defect are expected to have diﬀerent densities of type A and type B steps and therefore inﬂuence the characteristic pattern associated with the QD chain. However, rather amazingly, as shown in Fig. 11.4b, the observed QD chains continuously run over the oval defect, despite the diﬀerent surface kinetics introduced by the curved spatial proﬁle of the oval defect. The survival strength of the QD chains over surface misorientations is further conﬁrmed by the persistence of InGaAs QD chains grown on speciﬁc substrates with misorientations up to 2◦ away from (100) [6] and on shallowpatterned GaAs(100) substrates [7] as well. By using shallow-patterned GaAs(100) substrates (modulation depth of only 35 nm), coupled with the growth of multiple InGaAs layers, we demonstrate the selective formation of QD chains in desired spatial regions. The AFM image in Fig. 11.5 shows the selective formation of QD chains on the sidewalls of stripe-patterned GaAs(100) substrates. The robustness of the QD chain adds to its potential for its future application. To achieve a routine procedure for the fabrication of such QD chains, we have investigated the evolution of the QD chains while varying the thickness of the GaAs spacer layer as well as the content in the InGaAs layer [8]. In Fig. 11.6a–c, STM images emphasize the evolution of lateral spacing between

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Z.M. Wang, G.J. Salamo Fig. 11.4. a 20 × 20 μm AFM image emphasizing one oval defect. b 5 μm × 5 μm AFM image emphasizing the survival strength of QD chains under the inﬂuence of an oval defect. The QD chains are formed after stacking 17 QD layers of 7.6 ML In0.4 Ga0.6 As with 60 ML GaAs spacers

Fig. 11.5. Horizontally gradient AFM image of surface topography after stacking 3.3 ML In0.34 Ga0.66 As for the ﬁrst layer and 2.6 ML In0.34 Ga0.66 As for the following nine layers with GaAs spacers of 90 MLon [01-1] stripe-patterned GaAs(100) substrates

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Fig. 11.6. Scanning tunneling microscopy (STM) images of surface morphologies after stacking 17 layers of 5.8 ML In0.5 Ga0.5 As. a, b and c are for 56 ML, 84 ML and 112 ML GaAs spacer, respectively

QD chains as the function of the spacer thickness. By increasing the GaAs spacer thickness, not only the vertical separation of the QD chains increases, but also the lateral spacing increases. The lateral separation between QD chains is 54 nm, 82 nm and 110 nm, for the GaAs spacer thickness of 56 ML, 84 ML and 112 ML, respectively. Meanwhile AFM images in Fig. 11.7 explore the evolution of QD chains as a function of In content in the InGaAs QD layers. Of course, growth with a diﬀerent In content requires a diﬀerent amount of material, i.e., critical thickness, deposited to form QDs. The critical thickness of the transition was observed to increase as the In content in the InGaAs layer was lowered. To keep the results comparable for all the samples with diﬀerent In content in Fig. 11.7, the same amount of InGaAs was deposited after the critical thickness of the transition was reached as indicated by RHEED. Figure 11.7a shows the typical QD chains we achieved by stacking 17 layers of 6.0 ML of In0.5 Ga0.5 As spaced by 60 ML GaAs. The QD chains along [01-1] are spaced with the separation of about 71 nm and the separation between QDs in the chains is about 36 nm. Figure 11.7b shows the QD chains formed by stacking 17 layers of 7.6 ML of In0.4 Ga0.6 As spaced by 60 ML GaAs. The average spacing of the In0.4 Ga0.6 As QD chains in this case is about 100 nm and the separation between QDs in the chains is about 50 nm, both bigger than in the case of the In0.5 Ga0.5 As QD chains. It is notable that the In0.4 Ga0.6 As

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Z.M. Wang, G.J. Salamo Fig. 11.7. AFM images of surface morphologies after stacking 17 layers of InGaAs spaced by 60 MLGaAs. The InGaAs layer is 6.0 ML In0.5 Ga0.5 As in a, 7.6 ML In0.4 Ga0.6 As in b and 11.5 ML In0.3 Ga0.7 As in c, respectively

QDs are more connected in the chains than the In0.5 Ga0.5 As QDs and often the In0.4 Ga0.6 As QD chains are terminated with a [01-1] elongated plate. Likewise, the resulting morphology after stacking 17 layers of 11.5 ML of In0.3 Ga0.7 As is shown in Fig. 11.7c. The most remarkable feature here is that the height modulation within in the In0.3 Ga0.7 As QD chains is very weak. This is due to the fact that the lower indium content coupled with the transferred strain proﬁle is expected to result in lower strain along the dot chain resulting in the decreased ability to support the formation of QDs. Therefore, one may expect that they will behave more like quantum wires than like QD chains. 11.3.3 1D Wetting Layer to String Quantum Dots The In0.36 Ga0.64 As QDs chains, with surface features lying between Fig. 11.7b and c, appear to sit on a common InGaAs continuous base so called a 1D WL, a wire-like structure connecting the QDs in the chains. Therefore, besides the common QD and 2D WL states, the conduction-band energy diagram for the QD chains uniquely includes the state from the 1D WL, as shown in Fig. 11.8a [9]. Accordingly, the additional peak around 1.39 eV, in the PL spectra shown in Fig. 11.8b, is the manifestation of the transition from ground–electron to ground–hole states in the 1D WL. Meanwhile, the PL peaks around 1.32 eV, 1.45 eV and 1.52 eV can be assigned to the con-

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Fig. 11.8. a The schematic conduction-band energy diagram for the QD chains with 15 layers of 2 nm In0.36 Ga0.64 As spaced by 16 nm GaAs spacers. b Normalized low-temperature photoluminescence PL spectra of this sample. WL Wetting layer

tributions from the QD group states, the 2D WL and the GaAs substrate, respectively. The existence of the 1D WL is further conformed by time-resolved terahertz spectroscopy (TRTS). The terahertz (THz) pulse used in this technique is sensitive to the product of free carrier density and carrier mobility. Once carriers become captured by traps or localized QD states, they are no longer mobile and therefore do not contribute to attenuation of the THz pulse transmitted through the sample. TRTS is therefore an ideal probe of carrier capture dynamics in QD systems, complementing traditional techniques such as photoluminescence. A large anisotropy in the transient photoconductive response is observed depending on the polarization of the THz probe pulse with respect to the orientation of the QD chains [10]. Fast (3.5 – 5.0 ps) and eﬃcient carrier capture into the QDs and 1D WLs is observed below 90 K. Thermal emission out of the QDs and 1D WLs to the 2D WL was found to dominate the photoconductivity for over 90 K. Even though AFM images in Fig. 11.2 show well separated QDs in chains with a higher In content of 0.5, close examination of the QDs in the planview TEM image in Fig. 11.3 shows that between QDs and extending slightly perpendicular from the chain is a region of intermediate gray [3]. This wire-

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like structure, with In content less than that in the QDs, but more than the areas between QD chains, appears to be the 1D WL. Figure 11.9 uses a magniﬁed plan-view TEM image and associated line scans to emphasize this 1D WL. Because the TEM plan-view is actually averaged through about 100 nm, or six layers, this observed 1D structure with uniform In content could be the result of the lack of vertical registry of the QDs in the diﬀerent layers. However, the alignment of the dots in the [011] cross-view Fig. 11.2b argues strongly against this. Moreover such averaging would deﬁnitely not result in the sharp transition from QDs to 1D layer compared with the ﬂat 1D layer itself, shown in the line scan between the QDs in Fig. 11.9b. Figure 11.10 shows magniﬁed cross-sectional TEM images and associated scans that further elucidate this 1D layer and contrast it to the 2D WL and the QDs. The [01-1] view, Fig. 11.10a, unambiguously shows the size and In content in the 2D WL. (The fact that the sample depth probed here is 100 nm is not a problem because the plan-view TEM images show the near-parallel dot chains are well separated.) Similarly, the [011] view in Fig. 11.10b shows the size and In content of the 1D WL plus the 2D WL. (Here the 100 nm is about the separation of the dot chains so that large sections of the [011] view are from one QD chain with contributions from the 2D WL between the chains.) The line scans in Fig. 11.10c are averages over 25 and 21 proﬁles across the spacing between QDs of the 14th layer along [01-1] and [011], respectively. The resulting proﬁles indicate that the 1D WL has less peak indium and is thicker than the 2D WL. Summarizing the TEM results, there is deﬁnite evidence of a 1D WL, distinct from the 2D WL, which connects the QDs in the chains. However, the surface QDs appear to be isolated from each other in the chains as shown

Fig. 11.9. a Dark-ﬁeld, plan-view transmission electron microscopy (TEM) image of In0.5 Ga0.5 As QD chains as in Figs. 11.2 and 11.3. b Intensity proﬁles of lines A–A’ and B–B’. Higher intensity represents higher In content

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by the AFM images in Fig. 11.2. Thus there is no indication of the existence of the 1D WL for the uncapped QD chains with In content of 0.5. By realizing the TEM analysis including capped QD chains, we used AFM to measure the topography of two related In0.5 Ga0.5 As QD chain samples, one that was uncapped, and one that was capped by 3 MLs of GaAs. Figure 11.11 shows the AFM images. Figure 11.11a shows a topographic image of the capped sample. This shows a remarkable similarity to the TEM image except the gray scale is reversed (with AFM QDs are raised and white, while with TEM QDs are Inrich and dark). In the AFM image there appears to be a 1D common base to the QD chain. Of course here it is a topographic feature whereas in the TEM images it represents enhanced In content. Figure 11.11b and c show AFM images of single dot chains of capped and uncapped samples, respectively. In Fig. 11.11c there is no evidence of a 1D common base layer, same as shown in Fig. 11.2, while in Fig. 11.11b there is direct topographic evidence of this 1D common layer. This indicates that the 1D layer is associated with the GaAs capping layers for the InGaAs QD chains with high In contents. For In0.5 Ga0.5 As QD chains, the 1D WL develops either in the spacer layers covering the developing QD chains or the ﬁnal capping layer. This is not the ﬁnal resolution of the issue. The AFM evidence is topographic only, but it directly indicates that the process of capping results in the 1D WL for the QD chains with high In contents. On the other hand, the TEM evidence shows indium content between the QDs in the chains. Based

Fig. 11.10. Dark-ﬁeld, cross-sectional view TEM images of In0.5 Ga0.5 As QD chains as in Figs. 11.2 and 11.3. a Along the [01-1] direction (end view of chain). b Along the [011] direction (side view of chains). c Averaged intensity proﬁles across WLs of the 14th layer in the cross-view along line A in a ([01-1] 2D WL) and line B in b ([011] 2D and 1D WLs). The backgrounds have been aligned and the peak heights normalized

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Fig. 11.11. a AFM image of InGaAs In0.5 Ga0.5 As QD chains, as in Figs. 11.2 and 11.3 but with a 3 ML GaAs capping layer. b A magniﬁed view of a QD chain after a 3 ML GaAs capping. c A magniﬁed view of a QD chain without top GaAs capping, the same sample used in Figs. 11.2 and 11.3. The hatchings along [011] result from poorly tuned feedback of AFM scanning

on the AFM evidence, some of the GaAs capping layer preferentially goes to the gap between the dots, and to be consistent with the TEM, this is accompanied by anisotropic In diﬀusion from the QDs into this gap region adding In. It is the anisotropy of surface diﬀusion of In atoms that gives rise to the indium in the 1D WL.

11.4 Stacking on GaAs High Index Surfaces 11.4.1 2D Planar Ordering Both the formation of the QD chains and the 1D WL stringing QDs in chains result from the high anisotropy of material distribution and re-distribution, intrinsic to the (2 × 4) reconstructed GaAs(100) surface. As we suggested earlier, this surface anisotropy can be tuned by the natural surface steps introduced by an intended substrate miscut. As shown in Fig. 11.12, a miscut towards [011] and [01-1] results in type-A steps running along [01-1] and typeB steps running along [011], respectively [6]. As one would expect, the density of steps depends on the degree of misorientation. The InGaAs QD chains could survive through a modest deviation of surface orientation from (100), where the density of steps is not high enough to compensate the intrinsic anisotropy induced by the reconstruction. For the high index (n11)B surfaces under study where n is 9, 7, 5, 4 and 3, the degree of misorientation is able to adjust the nominal surface step density (1/S) from 0.56 nm−1 for (911)B to 1.67 nm−1 for (311)B. The nominal step separation S of a GaAs(n11) nm √ n surface consisting of (100) terraces is given as S = 0.5653 = n × 0.2 nm. 2 2

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Fig. 11.12. Schematic illustration of GaAs(100) surface structures, emphasizing the evolution of surface anisotropy induced by surface reconstruction and steps due to misorientation

Fig. 11.13. AFM images of the surface morphology after stacking 17 layers of 10 MLIn0.4 Ga0.6 As QDs spaced by 120 ML GaAs on GaAs (911)B (a), (711)B (b), (511)B (c), (411)B (d) and (311)B (e) substrates

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As shown in Fig. 11.12, the nominal type-B steps on GaAs(n11)B surfaces create energetic barriers for the surface diﬀusion of the adatoms along [011] on GaAs(100) terraces. Therefore, rather nicely, the anisotropic diﬀusion induced by (2 × 4) reconstruction can be compensated by an appropriate step density on (n11)B high index surfaces. While stacking 17 layers of 10 ML In0.4 Ga0.6 As QDs spaced by 120 ML GaAs results in the formation of QD chains on GaAs(100), 2D square-like QD ordering was observed on GaAs (n11)B surfaces, as shown by AFM images in Fig. 11.13 [11]. The appearance of the 2D lateral ordering on all high index surfaces investigated here suggests the patterns of the 2D surface ordering is surprisingly robust under deviation of GaAs surface indexes. Together with

Fig. 11.14. a The fast Fourier transforms (FFT ) taken from the AFM image of the GaAs(511)B sample as in Fig. 11.13. b Grazing-incidence small-angle X-ray scatting (GISAXS ) from the GaAs(511)B sample. c In-plane deﬂection angle θ of the nearest-neighbor direction from [-2nn] as a function of nominal surface step density (1/S) of the high index surfaces under study. Open circles AFM-FFT, filled circles GISAXS, solid line linear ﬁt to the GISAXS data

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the robustness of the 1D chain-like ordering discussed before, it suggests both ordering patterns corresponding to minimum energy conﬁguration of the system of strained QD multiple layers. As the angle of misorientation increases going from a GaAs(911)B to (311)B surface, a systematic deviation of the nearest-neighbor direction from the [-2nn] direction is observed [12]. The deﬂection angle θ, which is a quantitative measure of this deviation, was evaluated by 2D fast Fourier transforms (FFT) of the AFM images and by grazing-incidence X-ray small angle scattering (GISAXS), as depicted in Fig. 11.14a, b for the sample on GaAs(511)B. The well pronounced intensity pattern of X-ray satellites peaks enables a very accurate analysis of the nearest-neighbor directions and the dot–dot distances. Rather remarkably, the angle θ is experimentally seen to vary linearly with the nominal surface step density (1/S) as shown in Fig. 11.14c. Therefore, the nominal surface steps on high index templates can tune the 2D QD lateral ordering on the surface. This is a new and highly favorable phenomenon for engineering QD patterns with predictable lateral ordering. 11.4.2 Inclined Vertical Inheritance to the Growth Direction As we have already noted, both TEM analysis and plan-view AFM measurement assisted by selective chemical etching reveal vertically alignment of QD chains and QDs in chains grown on GaAs(100), which is further conﬁrmed by X-ray diﬀuse scattering. Interestingly, the 2D laterally ordered QD arrays on high index surfaces are vertically self-aligned, but the vertical alignment deviates from the surface normal by a signiﬁcant inclination angle α [12]. This is demonstrated in Fig. 11.15 which shows two 2D sections of the X-ray diﬀuse scattered intensity from the (511)B sample in the vicinity of the GaAs 5-11 reciprocal lattice point. Figure 11.15a gives the X-ray diﬀuse scattering along the [25-5] direction while Fig. 11.15b gives the X-ray diﬀuse scattering along the [011] direction. Beside the modulation of the crystal truncation rod (located at q25−5 = 0 and q011 = 0) which is caused by the mean superlattice period (tSL = 32.5 nm ± 0.1 nm) along growth direction, considerable X-ray diﬀuse scattering is observed, which is mainly caused by local strain ﬂuctuations inside the sample. The diﬀuse intensity is vertically bunched into so-called resonant diﬀuse scattering (RDS) sheets, indicating that the lateral QD positions are vertically correlated. Horizontally, within the RDS sheets pronounced intensity maxima occur, which are caused by lateral ordering of the QDs. In the direction along [25-5] in Fig. 11.15a, the RDS sheets are tilted from the exact horizontal orientation, and the angle of inclination α exactly corresponds to the deviation of the vertical arrangement of the QD positions from the surface normal towards the [25-5] direction. By contrast, the inclination vanishes in the direction along [011] as shown in Fig. 11.15b. As shown in Fig. 11.16b, the inclination angles α were determined experimentally for all samples under study, with the important result that

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Z.M. Wang, G.J. Salamo Fig. 11.15. X-ray diﬀuse scattering from the (511)B sample in the vicinity of the GaAs 5-11 reciprocal lattice point. In the direction a along [25-5] the X-ray diﬀuse scattering (which is concentrated in RDS sheets) is inclined by the angle α while in b the direction along [011] the Resonant diﬀuse scattering (RDS ) sheets are oriented horizontally. This proves inclined vertical inheritance of the horizontal QD positions towards the [25-5]-direction, i.e., perpendicular to the surface edge direction

Fig. 11.16. a Calculated elastic strain energy density (linear gray scale in arbitrary units) on the (511)B surface above a strained In0.4 Ga0.6 As QD which is embedded into a GaAs matrix at 30 nm below the surface. b Angle of inclined inheritance of the lateral QD positions to the surface normal. Open squares calculations, filled squares experimental

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the inclination points always towards the [2n-n] direction, i.e., the direction perpendicular to the nominal step edge direction. On the other hand the inclination towards [011] always vanishes. In order to understand this behavior, numerical model calculations using elasticity theory were performed. Our calculations are based on the ﬁnite element method (FEM) and take into account the QD size, shape and the full elastic anisotropy of the involved materials. Results of the elastic energy density performed for a lens shaped In0.40 Ga0.60 As dot (base width w = 30 nm, height h = 5 nm) are presented in Fig. 11.16a. The dot is located 30 nm below the (511)B surface of a GaAs matrix. A clear shift of the minimum position towards the [25-5] direction is observed, conﬁrming the experimental result of inclined inheritance towards this direction as shown in Fig. 11.15a. A series of FEM calculations were performed for diﬀerent GaAs surface orientations of (n11)B, where n is equal to 9, 7, 5, 4, and 3. Angles α of inclined inheritance are extracted from these calculations and are compared with corresponding experimental results in Fig. 11.16b. For (111)B and (100) surface orientations the energy density shows pronounced minima just above the underlying QD in the previous layer, thus favoring exact vertical correlation. However, for surface orientations on the path between (100) and (111)B, i.e., (911)B, (711)B, . . . , (311)B, remarkably high values for α are observed, with a maximum for the (711)B surface orientation exactly matching the experimental observation. Overall agreement between the FEM calculations and the experimental values for αgs observed. We would like to emphasize that even a rather small deviation from (100) could lead to a remarkable inclined inheritance, as demonstrated by the data point for (15 11)B in Fig. 11.16b. It is important to note that the achievement of a high degree of laterally and vertically ordering is observed to be accompanied by strong PL emission and a narrow linewidth [11]. A PL linewidth as narrow as 23 meV is observed from the multiple layers of QDs grown on GaAs(511)B with optimum ordering. As the result, spatially self-aligned QD arrays with high optical quality have an excellent opportunity to ﬁnd application as detector and emitter arrays.

11.5 Summary and Outlook We have observed and explained the formation of 3D ordered InGaAs/GaAs QD lattices when the substrate orientations is varied from (100) to (311)B in the direction towards (111)B. Long chains of QDs along [01-1] are formed during stacking multiple layers of InGaAs on highly anisotropic GaAs(100) surfaces. The QD chains and the QDs in the chains are vertically self-aligned along the growth direction. Interestingly, the QDs in chain are stringed by a 1D WL along [01-1], as an In-rich common base for InGaAs QDs with low In content or as an Inrich wetting band formed during GaAs spacing or capping InGaAs QDs with

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high In content. The QD chains and the 1D WL form through the anisotropic diﬀusion of surface In and Ga adatoms that accompanies the change in strain proﬁle during GaAs spacing and capping. Uniquely including the 1D WLs in its steady-state material distribution, the vertically aligned InGaAs/GaAs QD chains become QD networks with the 1D WLs as additional channels for carrier transport. The anisotropy of surface diﬀusion on GaAs(100) can be ﬁne-tuned by introducing B-type nominal steps along [011] through substrate misorientations towards (111)B. As a result, 2D square-like QD lateral ordering is observed on high index surfaces GaAs(n11)B with n of 9, 5, 4 and 3 where the angels of surface misorientations provide appropriate densities of surface steps to compensate the intrinsic surface anisotropy of the nominal GaAs(100) terraces. Remarkably, the deviation of the QD nearest-neighbor direction, as measured by an in-plane deﬂection angle θ observed to vary linearly with the nominal surface step density. Meanwhile, these changes are coupled to changes in the QD alignment along the growth direction. Theoretical investigations based on a linear elasticity theory show a systematic variation of the angle α of vertical inheritance of the lateral QD positions when the surface orientation is changed from (100) towards (111)B. The calculated values for the inclination angle α are mainly determined by the elastic anisotropy of the GaAs matrix and are in excellent agreement with experimental ﬁndings. Thus, we have demonstrated that diﬀusion characteristics of the surface, coupled with the elastic anisotropy of the matrix, provide an excellent opportunity to self-organize QDs towards 3D QD crystals with predictable lateral and vertical ordering. Acknowledgement. The scope of this work is in part a joint endeavor through close and fruitful collaborations with many people, including: Yu.I. Mazur, Sh. Seydmohamadi, K. Holmes, W.Q. Ma, J.H. Lee, J.L. Shultz, X. Wang, and M. Xiao from the University of Arkansas (USA); T.D. Mishima and M.B. Johnson from the University of Oklahoma (USA); M. Schmidbauer, D. Grigoriev, P. Sch¨ afer, M. Hanke, and R. K¨ ohler from Humboldt-Universit¨ at zu Berlin (Germany); D.G. Cooke and F.A. Hegmann from the University of Alberta (Canada); P.M. Lytvin, V.V. Strelchuk, and M.Ya. Valakh from the Institute of Semiconductor Physics (Ukraine).

References 1. Z.M. Wang, H. Churchill, C.E. George, G.J. Salamo, J. Appl. Phys. 96, 6908 (2004) 2. Z.M. Wang, K. Holmes, Y.I. Mazur, G.J. Salamo, Appl. Phys. Lett. 84, 1931 (2004) 3. Z.M. Wang, Y.I. Mazur, J.L. Shultz, G.J. Salamo, T.D. Mishima, M.B. Johnson, J. Appl. Phys. 99, 033705 (2006) 4. Z.M. Wang, L. Zhang, K. Homles, G.J. Salamo, Appl. Phys. Lett. 86, 143106 (2005)

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5. Z.M. Wang, Y.I. Mazur, G.J. Salamo, P.M. Lytvin, V.V. Strelchuk, M.Y. Valakh, Appl. Phys. Lett. 84, 4681 (2004) 6. Z.M. Wang, S. Seydmohamadi, J.H. Lee, G.J. Salamo, Appl. Phys. Lett. 85, 5031 (2004) 7. J.H. Lee, Z.M. Wang, B.L. Liang, W.T. Black, V.P. Kunets, Y.I. Mazur, G.J. Salamo, Nanotechnology 17, 2275 (2006) 8. Z.M. Wang, Y.I. Mazur, K. Homles, G.J. Salamo, J. Vac. Sci. Technol. B 23, 1732 (2005) 9. Y.I. Mazur, W.Q. Ma, X. Wang, Z.M. Wang, G.J. Salamo, M. Xiao, T.D. Mishima, M.B. Johnson, Appl. Phys. Lett. 83, 987 (2003) 10. D.G. Cooke, F.A. Hegmann, Y.I. Mazur, W.Q. Ma, X. Wang, Z.M. Wang, G.J. Salamo, M. Xiao, T.D. Mishima, M.B. Johnson, Appl. Phys. Lett. 85, 3839 (2004) 11. Z.M. Wang, Y.I. Mazur, S. Seydmohamadi, G.J. Salamo, H. Kissel, Appl. Phys. Lett. 87, 213105 (2005) 12. M. Schmidbauer, S. Seydmohamadi, D. Grigoriev, Z.M. Wang, Y.I. Mazur, P. Sch¨ afer, M. Hanke, R. Kohler, G.J. Salamo, Phys. Rev. Lett. 96, 066108 (2006)

Part II

Forced Alignment

12 One-, Two-, and Three-Dimensionally Ordered GeSi Islands Grown on Prepatterned Si (001) Substrates Zhenyang Zhong1 , G¨ unther Bauer1 , and Oliver G. Schmidt2 1

2

Institute for Semiconductor and Solid State Physics, Johannes Kepler University Linz, 4040 Linz, Austria Max-Planck-Institut f¨ ur Festk¨ orperforschung, 70569 Stuttgart, Germany

12.1 Introduction Self-assembled GeSi (or Ge) islands on Si substrates have been extensively investigated not only because they represent simple model systems to understand the fundamental physics of heterostructure growth, but also because they represent promising candidates for devices compatible with sophisticated Si device integration technology [1]. On the Si substrate, GeSi or Ge initially grows in a layer-by-layer mode, and after a critical thickness is reached, islands are spontaneously formed to release misﬁt strain energy at the expense of an increase of surface energy. This so-called Stranski–Krastanow growth mode can occur during mismatched heterostructure growth. The growth of self-assembled GeSi or Ge islands depends on the growth process parameters [2–8], such as growth time (or deposited Ge amount), growth temperature, growth rate, and post-growth annealing. Theoretical studies of growth kinetics [9, 10] or energetics of island formation [11] came to the conclusion that a narrow island size distribution can be expected under critical growth conditions or for extremely long ripening times. In principle, taking into account the elastic interaction between neighboring islands, ordered islands can be realized [12, 13]. However, the self-assembled islands are in general randomly distributed in the growth plane due to the stochastic nucleation of the islands on a ﬂat surface. In addition, the diﬀerence of the number density of the islands in diﬀerent regions on the substrate also deteriorates the size homogeneity of the islands. Such disadvantages hamper both fundamental studies and device applications of single islands. In recent years enormous eﬀorts have been devoted to obtain ordered islands. All those attempts can be categorized into two types. One is based on growing islands on substrates, which are not subjected to any advanced preprocessing. Laterally short range ordered islands can be obtained on top of an island multilayer with appropriate thin Si spacer layers [14, 15]. GeSi islands can also be aligned on the surface above a buried dislocation network [16]. In these two cases, a strain ﬁeld on the surface induced by the buried islands or the dislocations inﬂuences the migration of adatoms or addimers on the surface and the nucleation of the islands. Short-range ordered

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islands can also be realized on vicinal surfaces [17, 18], as a result of an anisotropic migration of adatoms or ad-dimers on the step-bunched surface and the elastic interaction between the neighboring islands. A compact arrangement of GeSi islands may also exhibit a short-range ordering due to the elastic interaction between neighboring islands [19]. The other methods are based on growing islands on preprocessed substrates. A micro-step-network, resulting from a regular arrangement of atomic steps, provides templates for ordered island growth [20]. Strain ﬁelds induced by regularly implanted oxygen ions [21] or electron beam-induced carbon depositions with subsequent Si overgrowth [22] help to improve the lateral ordering of islands. Ge deposition on latex nanospheres masked Si substrates and subsequent annealing can result in well-ordered islands arranged in a hexagonal lattice [23]. Stressselective etching of a surface with buried dislocation networks induced by twist wafer bonding gives rise to a periodic surface morphology ﬂuctuation, which promotes the ordering of subsequently grown islands [24]. A major progress on long-range ordered islands has been made recently by combining the self-assembly with standard lithography techniques [25–29]. A SiO2 mask with one-dimensional (1D) ordered stripes or two-dimensional (2D) ordered pits can be fabricated by lithography on a Si surface covered by a SiO2 layer. Si then selectively grows in the SiO2 windows to form 1D ordered stripes or 2D ordered mesas. Subsequent Ge deposition results in selfassembled islands on the top of the mesas or stripes, which was explained from the point of view of growth kinetics or thermodynamics. Long-range lateral ordering of islands was observed. An alternative route to fabricate long-range ordered islands consists in depositing Si and Ge directly on prepatterned Si substrates without a SiO2 mask [1,30–39]. Due to the periodic morphology change on the patterned surface, the Ge (GeSi) islands prefer to nucleate and to grow at particular sites, resulting in long range ordered islands, rather than as disordered ones. In the following, details about the pattern fabrication and the growth process of the islands will be presented. The 1D, 2D and three-dimensional (3D) ordered GeSi island growth on prepatterned Si (001) substrates will be demonstrated. Structural properties of these ordered islands on modulated surfaces will be addressed and the growth mechanism of self-assembled islands on these surfaces will be investigated.

12.2 Experiments: Pattern Fabrication and Island Growth The prepattern on Si (001) substrates is fabricated by holographic or electron beam lithography and subsequent reactive ion etching (RIE). Figure 12.1 schematically illustrates the setup for holographic lithography. A focused Ar+ laser beam with a wavelength of 458 nm passes through an aperture and a subsequent lens, which results in a parallel laser beam with a diameter of

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Fig. 12.1. Schematic setup for holography

5 cm. Half of this widened laser beam directly projects on a Si wafer covered with photoresist, another half projects on the same wafer after reﬂection from a mirror. The interference of these two laser beams gives rise to a fringe pattern on the photoresist layer. After developing the photoresist and the subsequent RIE step, a 1D stripe-patterned Si template is obtained, as shown in Fig. 12.2a. To obtain 2D pit-patterned Si templates, as shown in Fig. 12.2b, a second exposure after rotating the wafer by an angle ω, (see Fig. 12.2a), is required before developing the exposed photoresist. The period of the pattern can be calculated from d = λ/(2 cos α) (12.1) where λ is the wavelength of the laser, α is the angle as shown in Fig. 12.1. Accordingly, the minimum periodicity of the pattern is λ/2, which in fact cannot be achieved since it would require α = 0◦ . The geometric proﬁles of the pattern depend on exposure and developing time, as well as etching time. It is also aﬀected by the energy and the orientation dispersion of the laser beam, especially for patterns with a small periodicity. Typically, patterns with a period in the submicrometer range and depths of about 50 nm were used. After the RIE process, a rough surface with some structural defects is always left, as shown in Fig. 12.3a. Therefore it is necessary to use processes to remove these defects. In general, cleaning processes, including ex-situ chemical cleaning and in-situ thermal cleaning, are employed before growth. Although several groups reported that a smooth surface can be obtained already with thermal treatment of the patterned substrates alone [31, 38], we found that this takes an extremely long annealing time at an optimized temperature. Therefore, we preferred to deposit a suﬃciently thick Si buﬀer to obtain smooth and defect-free surfaces. For these processes two aspects should be considered: (1) one is to preserve the pattern, (2) the other is to eliminate the surface defects and the roughness induced by the RIE process. The geometrical properties (e.g., the depth and the slope of the sidewalls) of the 1D stripe pattern and the 2D pit pattern are changed during these processes, particularly during buﬀer layer growth, as shown in Fig. 12.3. A higher growth temperature and/or a thicker Si buﬀer layer results in

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Fig. 12.2. Atomic force microscopy (AFM) data of a one-dimensional (1D) stripe pattern and b two-dimensional (2D) pit pattern

Fig. 12.3. Surface morphology of a 2D pit-patterned Si template a after the reactive ion etching (RIE) (depth of the pits: ∼50 nm), and b after the buﬀer layer growth (depth of the inverted truncated pyramids: ∼15 nm) (after [35])

shallower stripes or pits, and even complete ﬂattening the whole pattern may occur. Such a ﬂattening eﬀect of the Si buﬀer layer to the stripe pattern and the pit pattern is also related to the periodicities. A pattern with a smaller periodicity is more readily ﬂattened during buﬀer layer growth. In addition, the ratio of w/d (w, the width of the trenches or the pits, d, the periodicity) inﬂuences the ﬂattening eﬀect. The pattern with a smaller w/d ratio can be ﬂattened faster. These eﬀects become more pronounced at a higher growth temperature (e.g., > 550 ◦ C). These ﬂattening phenomena during buﬀer layer growth on patterned substrates originate from the fact that part of the deposited Si atoms on the mesas migrate to the neighboring trenches or pits. This results in thicker buﬀer layers in the trenches than on the stripe mesas, as conﬁrmed by cross-sectional transmission electron microscopy [33].

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The Si buﬀer layer and the GeSi islands were grown by solid source molecular beam epitaxy. Before inserting them into the load lock, the prepatterned Si templates were cleaned by a standard RCA cleaning processes. For some samples, a subsequent HF dip was used to form a hydrogenated surface. After thermal desorption of the oxide at 900 ◦ C, a suﬃciently thick (in general 100 – 150 nm) Si buﬀer layer was deposited while ramping the substrate temperature from a lower level (e.g., 550 ◦ C or 450 ◦ C) to a higher one (e.g., 650 ◦ C). To study the eﬀect of strain ﬁelds on the nucleation of islands on prepatterned substrates, for some samples a strained Si0.5 Ge0.5 /Si superlattice layer was inserted in the buﬀer layer [32, 33]. A sample of the surface morphology just after the RIE and after the buﬀer layer growth of a 2D pit-patterned substrate is displayed in Fig. 12.3a and b, respectively. This ﬁgure clearly demonstrates that the surface of the patterned substrate becomes much smoother, and that a 2D pit pattern on the surface is preserved after the cleaning processes and the buﬀer layer growth. However, the geometric proﬁle of the pits has changed considerably. In general the pits look like inverted truncated pyramids with a depth of ∼ 5 – 25 nm and sidewall angles of ∼ 5–15◦ . After such processes, a smooth surface of a stripe-patterned substrate has also been observed [32]. Subsequently Ge [410 monolayers (MLs)] is deposited on these smooth and modulated surfaces. To enhance the migration of deposited Ge atoms, a growth q interruption of several seconds after each ML growth is often employed. For multilayer samples grown on the prepatterned substrates, the subsequent Ge island layers were separated by thin Si spacer layers, similarly to those grown on ﬂat substrates.

12.3 Results and Discussion 12.3.1 1D-Ordered Islands 1D ordered islands were realized on stripe-patterned substrates, as shown in Fig. 12.4. It can be seen that the islands prefer to grow in the trenches, whereas no islands are found on the top of the stripes. This result is diﬀerent from previous ones for Ge growth in SiO2 windows [26,12-27,12-28,12-29]. The positions of the islands in the trenches depend on the trench shapes. In shallow V-like trenches, as shown in Fig. 12.5a, the islands tend to grow at the intersections of the sidewalls of the trenches, which is denoted by an arrow. Consequently, exactly linearly arranged islands can be formed in periodic trenches. However, in a deep U-like trench, as shown in Fig. 12.5b, three (or even more) intersections of the sidewalls of the trench exist. Most of the islands tend to grow at these intersections as denoted by arrows in Fig. 12.5b. Such a preferential growth of the islands in the trenches can be simply explained from the point view of growth kinetics. The patterned stripes on

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Z. Zhong, G. Bauer, O.G. Schmidt Fig. 12.4. AFM image of 1D ordered islands on a stripe-patterned Si substrate

Fig. 12.5. AFM image of the islands in shallow V-like trenches (a), and in deep U-like trenches (b) (after [33])

the surface are in general oriented along 110 directions. Based on the surface reconstruction of the terraces and the height of the steps, in general four types of steps might appear on the surface: single layer steps with edges parallel (SA ) or perpendicular (SB ) to the dimer rows of the upper terraces, and double layer steps with the edge parallel (DA ) or perpendicular (DB ) to the dimer row of the upper terraces. It is found that the DB step is energetically favorable and can be readily formed on the vicinal surface due to the faster growth rate of the SB step in comparison to that of the SA step. Thus, we can assume that the sidewalls of the stripes are composed of DB steps [33] (and references therein). The migrating unit on the surface could possible be an ad-dimer [40, 41], considering its lower formation energy. For an ad-dimer on the top terrace of the stripe, two time scales should be taken into account: the mean diﬀusion time td for the ad-dimer to the edge, and the mean nucleation time tn for an ad-dimer forming a static cluster. td can be estimated by the

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Fig. 12.6. Schematic diagram illustrating the growth kinetics of deposited Ge on the 1D stripes. The curve in the top right shows the activation barriers for addimers migrating on the plain surface, Eb , downward over steps, Esd , and upward over steps, Esu . Open circles Ad-dimers, solid circles adatoms

following expression, td = l1 l2 /D,

D = a2 f,

f = γ exp (−Eb /kB T )

(12.2)

where l1 and l2 (l1 + l2 = w, width of the top terrace) are the distances between the original ad-dimer position and the two edges of the top terrace of the stripes, as shown in Fig. 12.6; D is the diﬀusion constant, a is the surface lattice constant (3.84), f is the hopping rate (seconds−1 ), γ(∼1013 ) is a prefactor, kB is the Boltzmann constant, T denotes the growth temperature, and Eb is the diﬀusion barrier, as shown in Fig. 12.6. The longest mean diﬀusion time, tdmax is for ad-dimers at the center of the top terrace. The wider the top terrace, the longer tdmax . tn can be estimated by the following expression, aGe 3w tn = / (12.3) 4g a where aGe is the bulk lattice constant of Ge, g is the growth rate. The factor of 3 appears because not only the adatom deposited on the same row of an atomic site on the surface with the ad-dimer can trigger the nucleation of the small cluster, but also those on both neighboring rows, as shown in Fig. 12.6. The wider the top terrace, the shorter tn . If td is much smaller than tn , the ad-dimers on the top terrace can readily migrate to the edge. Given the high growth temperature (e.g., 650 ◦ C), ad-dimers at the edges can also migrate downwards over steps to the sidewalls of the stripes in spite of an additional Ehrlich–Schwoebel barrier at the step [42,43]. Theoretical studies [42, 43] indicate that the activation energy for ad-dimers downwards over a step, Esd , is generally smaller than that upwards over a step, Esu as schematically shown in Fig. 12.6. As a result, more ad-dimers can migrate downwards than upwards at the step [33]. Such an asymmetrical migration of ad-dimers over steps results in a net ﬂux of ad-dimers downwards at the sidewalls of the stripes. The remaining Ge at the upper part (including the top terraces) of the stripes can only form a layer with a thickness less than the critical one for island formation, although nominally suﬃcient Ge material for island formation is supplied. Thus no islands nucleate there.

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Instead, Ge aggregates near the bottom of the sidewalls, particularly at the concave intersections between the sidewalls, facilitating island formation. For V-like trenches, as shown in Fig. 12.5a, only one intersection exists at the bottom, where the net ﬂuxes of Ge or GeSi ad-dimers downwards from both neighboring sidewalls can accumulate and island nucleation can occur [44]. In the extreme case of very shallow ’V’-like trench, the sidewalls might be composed of a mixture of single and double layer steps [44]. The asymmetrical migrations of ad-dimers at the sidewalls can be not so prominent. The preferential nucleation at the intersection cannot occur. For U-like trenches, the slopes of the sidewalls are diﬀerent, as shown in Fig. 12.5b. One steeper sidewall (SW⊥ ) is composed of steps with smaller width; the other sidewall (SW// ) is composed of wider steps. An intersection exists between these two sidewalls. Considering the step–step interactions at the sidewall [45], the steps of SW⊥ are not the favorable positions for the incorporation of Ge due to the stronger repulsion between the steps. In addition, the narrower steps can contain fewer ad-dimers. Therefore, a large amount of ad-dimers at the SW⊥ will migrate to the SW// . Accordingly, the growth rate of the SW// is larger than that of the SW⊥ . In particular, more of the net ﬂux of Ge (or GeSi) ad-dimers from SW⊥ will be incorporated in the region of the SW// near the intersection with respect to that at the other region of SW// , which facilitates the island nucleation near the intersection. The positions of these intersections at the sidewalls are determined by the local geometry of the pattered stripes and the growth process. These positions can be random at the sidewalls of U-like trenches. Therefore, Ge islands along each trench are arranged rather randomly, as shown in Fig. 11.5b. Such a simple growth kinetic model also explains the fact that the Si buﬀer layer is thicker in the trenches than on the stripe mesas as observed in XTEM [33], since at a high growth temperature the asymmetrical migration of Si adatoms on patterned substrates also takes place. Obviously, the positioning of the islands on patterned substrates can be signiﬁcantly aﬀected by the growth temperature, which essentially determines the migration of the ad-dimers on the smooth surface. We have found that at 600 ◦ C on similar stripe patterned substrates, islands can grow all over the surface, not only in trenches, but also on the top of the stripes [33]. In this case, a lower growth temperature results in a longer time td for ad-dimers migrating to the edge of the top terrace and a lower hopping rate over steps at the sidewalls. Accordingly, most ad-dimers on the top terraces can reside there. In addition, a net ﬂux of Ge (or GeSi) ad-dimers at the sidewalls which occurs at a higher growth temperature will be suppressed and might even not exist at too low growth temperatures. Thus a Ge (or GeSi) layer might essentially grow uniformly all over the surface of the patterned substrate. As a result, no preferential positioning of islands is observed. The positioning of islands on the patterned substrates is also related to the strain ﬁeld on the surface, which aﬀects both the migration of ad-dimers or adatoms and the local critical thickness for the 2D–3D transition. Fig-

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Fig. 12.7. a AFM image of islands grown on a stripe-patterned substrate with a buried Si0.5 Ge0.5 /Si superlattice layer. b Lateral strain distribution (in-plane strain tensor component) at the surface of the stripe (one period) induced by the buried Si/Si0.5 Ge0.5 superlattice buﬀer layer. The schematic buﬀer layer structure of one stripe before Ge deposition is displayed at the bottom. The heights and widths are derived from the AFM and the TEM images ([32, 33], copyright American Institute of Physics, 2003)

ure 12.7a shows an AFM image of islands grown on a patterned substrate with a strained Si0.5 Ge0.5 /Si superlattice layer (ﬁve double layers) as the buﬀer layer. Most of the islands are located on top of the stripes with a 1D ordering perpendicular to the stripes as well. The distribution (one period of the stripe structure) of the lateral strain, ε|| = a|| − aSi /aSi (a|| is the lateral lattice constant of the epilayer, aSi is the lattice constant of the Si bulk) is shown in the upper part of Fig. 12.7b, which is calculated by the ﬁnite element method. The layer structure of the buﬀer layer is schematically shown in the lower part of Fig. 12.7b. Evidently, a strain gradient exists near the corner between the top terrace and the sidewall, which aﬀects the migration of the ad-dimers or adatoms [46]. The strain gradient around the top corner can increase the activation barrier for Ge (or GeSi) ad-dimers to hop down, even resulting in a larger value than the activation energy to hop up over a step at the sidewall above point A in Fig. 12.7b. This means that more ad-dimers can reside on the top terrace, and even an inverted net ﬂux of ad-dimers may migrate from the sidewall above point A to the top terrace, in contrast to the case described above without the strained buﬀer layer. Thus, suﬃcient Ge can remain on the top terrace to form islands. Furthermore, the larger compressive strain in the center of the top terrace, as manifested in Fig. 12.7b, can reduce the critical thickness for the 2D–3D transition with respect to that around the top corner. Accordingly, at ﬁrst the islands tend to nucleate at the center of the top terrace [33]. On the other hand, the strain gradient at the sidewall between point A and point B is nearly zero. Therefore, the ad-dimers in this region can still migrate downwards as in the case where the strain ﬁeld was absent. With suﬃcient Ge deposition, a few GeSi islands can also occur near the bottom of the sidewalls, as denoted by a black arrow in Fig. 12.7a.

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The eﬀect of the external strain ﬁeld from the strained buﬀer layer on the migration of the ad-dimers is sensitive to its magnitude. If only two layers of a Si/Si0.5 Ge0.5 superlattice were grown in the buﬀer layer, we found that most of the islands are still formed in the trench [32]. In addition, the strain ﬁeld induced by a SiO2 layer also aﬀects the migration of ad-dimers or adatoms during the selective epitaxy in SiO2 windows [47, 48]. This might contribute to island formation on the top terrace of stripes or mesas in SiO2 windows [26–29]. 12.3.2 2D-Ordered Islands On 2D pit-patterned Si (001) substrates, 2D ordered islands can be grown as shown in Fig. 12.8. Several features of these islands are clearly visible: 1. The islands grow in the pits, as demonstrated by AFM images, particularly by line proﬁles, as shown in the upper inset of Fig. 12.8a;

Fig. 12.8. AFM images of GeSi islands grown on pit-patterned Si substrates. a 6 ML Ge grown on a pattern with a periodicity of 500 nm; upper inset shows the line proﬁle along the arrows A and B (after [34]). b 6 ML Ge grown on a pattern with periodicity of 400 nm (after [36]). c 7 ML Ge grown on a pattern with periodicity of 360 nm. All samples are grown at 700 ◦ C

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2. Single island occupancy per pit can be achieved under proper growth conditions on appropriately pit-patterned substrates. Based on these two aspects, true 2D long-range ordering (LRO) is realized. 3. The regular arrangement of islands in a square, a parallelogram, or a triangular lattice, as shown in Fig. 12.8a–c, demonstrates that the ordering of the islands can be exclusively determined by the pit pattern. Thus, such a 2D LRO of the islands originates from the periodic pit arrangement rather than from an elastic interactions between them [12, 13]. This unique feature facilitates an arrangement of ordered islands in a desired lattice, promoting also the analysis of individual islands and applications where addressing of particular islands is requested. 4. The homogeneity of the islands on these pit-patterned substrates, manifested both in their size and shape, is considerably improved with respect to their counterparts grown on ﬂat substrates [36]. This fact is clearly demonstrated by the height distribution and the scatter plot of the aspect ratio (height divided by diameter) versus height, as shown in Fig. 12.9. The mean value H and the standard deviation ΔH of the island heights are derived from Gaussian ﬁts. The values of the dispersion δ = ΔH/H and the H are shown in Fig. 12.9. The δ values (≤ 4%) of the islands grown on patterned substrates are considerably smaller than that on ﬂat substrates [36]. The concentration of the island aspect ratios, as shown in the left column of Fig. 12.9, indicates that only islands of one type of shape appear on the patterned substrates.

Fig. 12.9. Size distribution of the islands (height distribution of the islands in the left column, scatter plot of aspect ratio vs. height in the right column) of, a 10 ML Ge on a pattern with a periodicity of 370 nm, grown at 700 ◦ C, b for the sample shown in Fig. 12.8a; c for the sample shown in Fig. 12.8b. δ and H denote the dispersion and the mean value of the island heights ( [36])

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5. The average size of a single island on the patterned substrates depends both on the amount of deposited Ge, as well as on the periodicity of the pit pattern. This is because the amount of Ge possibly available for island formation at the pit bottom is roughly proportional to n/d2 (n is the nominal value of deposited Ge, d is the periodicity of the pit). In general, the average island size on patterned substrates with periods around 400 nm is larger than that on ﬂat substrates due to the smaller island density. Although the islands, which are described by their size distribution in Fig. 12.9a, are rather large, no dislocations are found in the XTEM image in Fig. 12.10. This proves that such islands still grow coherently on patterned substrates. This may be due to a partial strain relaxation in the nonﬂat wetting layer, as well as a more eﬃcient strain relaxation due to the appearance of steeper facets which is associated with the lateral growth limitation of the island by the pit sidewall. For identical growth conditions on ﬂat Si substrates, bimodal distributions (pyramid and dome) of the Ge or GeSi islands are frequently observed; some superdomes are also observed with too much Ge deposition [36]. The facets of the islands grown on patterned substrates can be analyzed in detail from a surface orientation map (SOM). As shown in Fig. 12.11, the normal orientation (h, k, l ) at a surface point of the island is obtained from a local plane, which is determined from its nearest neighboring points on the

Fig. 12.10. Cross-sectional transmission electron microscopy (TEM) image of sample shown in Fig. 12.9a (after [35])

Fig. 12.11. Schematic illustration of the surface orientation map (SOM ) of the islands

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island surface. This orientation (h, k, l ) corresponds to one polar coordinate (ρ, α) in the SOM, ρ represents the polar angle between the surface normal [hkl ] and the [001] growth direction, αrepresents the in-plane azimuth of [hkl ] with respect to the [110] (or [100] or other directions in the plane) direction, as shown in Fig. 12.11. The intensity at each position of (ρ, α) in the SOM represents the number of points on the island surface with the same normal orientation. This means that, the larger the area of the facet, the higher the intensity at the corresponding position in the SOM. The left column of Fig. 12.12 exhibits the SOM of islands of several samples, whose AFM images are shown in the right column of Fig. 12.12. The relative intensity is indicated by a grey scale at the bottom of the SOM. For the smaller islands in Fig. 12.12a, the corresponding SOM indicates that no particular facets appear. The islands look like mounds. With increasing height of the islands, {1,1,3} and {15,3,23} facets are observed in the SOM, as shown in Fig. 12.12b. Some faint {1,0,5} facets are also observable. The islands on this patterned substrate are dome-like, similar to those on the ﬂat substrates [4, 49]. With further increase of the island height, steeper {15,3,20} facets appear at the surface of the islands, as demonstrated in the SOM of the islands, shown in Fig. 12.12c [35]. Such an evolution of the island facets with island size might be attributed to an increasingly eﬃcient strain relaxation, since steeper facets can more eﬃciently reduce the misﬁt strain [50,51]. This is also conﬁrmed by our recent observation of pyramid-like islands with dominant {1,1,1} facets, as demonstrated in Fig. 12.12d, grown at a substrate temperature of 620 ◦ C on a patterned substrate. The average height and aspect ratio of these islands are about 41.9 nm and 0.37, respectively, which are considerably larger than those of previously observed domes. An additional interesting feature of islands grown on patterned substrates is their fast formation. As shown in Fig. 12.12a, islands with an average height of 9.5 nm have been formed in pits with only 4 ML Ge depositions at a growth temperature of 700 ◦ C. However, no island formation is observed on ﬂat substrates under identical growth conditions [35]. The growth of islands on the patterned substrates follows the SK mode. Our observation means that the thickness of the GeSi wetting layer in the pits is already beyond the critical value for the 2D-3D transition with only 4 ML Ge deposition, which is somewhat below the corresponding critical value for island formation on ﬂat substrates. One possibility to explain this phenomenon is that this critical thickness at the bottom of the pit is decreased due to an external compressive strain, which is induced by the geometrical proﬁle of the pit. This possibility is ruled out by the following experimental result. When 4 ML Ge are deposited at 600 ◦ C on a similar patterned substrate as the one used for the sample of Fig. 12.12a, no island formation is observed [35]. If the variation of the critical thickness was the main reason for island formation, as shown in Fig. 12.12a, islands should also be observed for samples grown at 600 ◦ C. This is obviously in contrast to the experimental results. The only remaining explanation for the observation of islanding in Fig. 12.12a is that the GeSi layer at the bottom

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Fig. 12.12. SOM (in the left column) and AFM images (in the right column) of islands grown on patterned substrates. a 4 ML Ge grown on a pattern with periodicity of 370 nm at 700 ◦ C. b Sample in Fig. 12.8a. c Sample in Fig. 12.9a (after [35]). d 7 ML Ge grown on a pattern with periodicity of 360 nm at 620 ◦ C (after [52]). The average height H of the islands is indicated in the ﬁgures

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Fig. 12.13. Surface morphology of a pitpatterned substrate with a periodicity of about 400 nm after Si buﬀer layer growth (height scale: 25 nm). The lines forming a black box indicate a “unit cell” on the patterned substrates

of the pits is thicker than the nominal one on the ﬂat substrates. Such an explanation sheds some light on the growth mechanism of islands on pitpatterned substrates. The accumulation of GeSi at the bottom of the pits due to the asymmetrical migration of ad-dimers at the sidewalls accelerates the island formation rate and determines the island formation. After the Si buﬀer layer growth, the pits look like inverted truncated pyramids, as shown in Fig. 12.13. Their sidewall slopes are in the range of about 4.5–15◦ and the depths are about 10∼25 nm. In analogy to stripe-patterned substrates, the sidewalls of the pits are also composed of steps. As discussed above, Ge (or Ge-Si) ad-dimers on the terraces between the pits can easily migrate to the edges and tend to diﬀuse downwards to the sidewalls. A net ﬂux of Ge (or GeSi) ad-dimers at the sidewalls of the pits can also be caused by lower activation barriers for ad-dimers migrating downstairs than upstairs. Accordingly, Ge atoms tend to aggregate at the bottom of the pits, increasing the growth rate of GeSi layer there. The thickness of the GeSi layer at the pit bottom can quickly reach values beyond the critical thickness for the 2D to 3D transition even with a nominal Ge deposition less than the critical one. For a pit with a sufﬁciently small bottom area, only one island can grow in it. This qualitative model explains the observations of both the 2D LRO and the faster growth rate of islands grown on pit-patterned substrates in comparison to that on ﬂat substrates. It should be pointed out that the preferential formation of the islands at the pit bottom can be changed by external strain ﬁelds, e.g., induced by buried SiC clusters [22]. Such a faster nucleation and growth rate of the islands in the pits, together with the lower number density of the islands on substrates with a long period pattern, results in larger islands compared to those on ﬂat substrates. If the bottom size of the pits is too large, more than one island can form in the same pit, which to some degree will aﬀect the lateral ordering. This novel self-assembly process on patterned substrates contributes to the observed size homogeneity of the islands in the pits. Given the periodic pit structure, the surface of the patterned substrates can be regarded as

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being composed of “unit cells”, represented by a box in Fig. 12.13. Taking into account the asymmetrical migration of ad-dimers at the sidewalls, it is reasonable to argue that only Ge atoms deposited within these unit cells can take part in the islanding in the corresponding pits. This means that the amount of Ge, which can contribute to each island, is about the same. In addition, a self-limiting growth due to a signiﬁcant stress concentration at the periphery of the islands, which increases with the size [9, 10], helps to further improve the size homogeneity of the islands. Recent results indicate that the geometrical proﬁles of the pit can also considerably aﬀect the migration and the incorporation of deposited Ge atoms. Depending on these proﬁles either superlarge pyramid-like islands with dominant {111} facets are observed at the bottom of the pits [52], as shown in Fig. 12.12d, or groups of islands. These groups consist of one at the bottom and essentially four at the top corners of the pits [53], as shown in Fig. 12.14a. The growth conditions for these two cases are quite similar; however, in the former case, the sidewalls of pits are mainly composed of {1, 1, 9} and {1, 0, 5} surfaces [52]. No sharp concaved intersections (SCIs) exist at the sidewalls. In the latter case, the sidewalls of pits are mainly composed of {1, 1, 10} surfaces and SCIs appear between adjacent {1, 1, 10} surfaces [53]. With less Ge deposition on a similar patterned substrate, only one island per pit at the bottom is observed, but the SCIs have become truncated or rounded, as shown in Fig. 12.14b. These observations indicate that the SCIs at the sidewalls can also eﬃciently collect the deposited Ge, facilitating the formation of island embryos and even of fully developed islands [53]. Further studies are needed to investigate the dependence of preferential island nucleation on the geometrical proﬁles of pits. It should be recalled that these proﬁles, before Ge deposition has taken place, depend both on the initial patterning of the Si substrate and on growth conditions. 12.3.3 3D-Ordered Islands It is well known that Ge islands prefer to grow directly above buried ones if the spacer layer thickness is appropriate [14, 15, 46, 54]. This phenomenon

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Fig. 12.15. a Surface morphology after three GeSi island layers are grown on a pit-patterned substrate with a periodicity of 400 nm (after [34]). b The corresponding cross-sectional TEM image (after Ref. 37, copyright Elsevier Science 2004)

Fig. 12.16. a Surface morphology after 13 island layers are grown on a pitpatterned substrate with a periodicity of 400 nm. b The corresponding X-ray diﬀraction map in reciprocal space around Bragg point (0, 0, 4). The arrows (white or black ) and SLn (n: 0, -1) indicate the positions of the vertical satellites, Sub indicates the position of the Si substrate reﬂection (after [37])

is explained in terms of the local minimum energy [14] or the gradient of surface chemical potential [46] for deposited Ge due to the strain ﬁeld of buried islands. By stacking multiple island layers separated by Si spacer layers, 3D ordered island arrays can be realized on patterned substrates [37].

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Figures 12.15a and 12.16a show the surface morphology after the growth of three and thirteen island layers, respectively. Apparently, the lateral ordering and to some extent also the size homogeneity of the islands are preserved. Under appropriate growth conditions, islands in each of the subsequent layers have the identical island in-plane arrangement as in the ﬁrst layer, because of the vertical correlation of the island positions along growth direction. Such a replication is conﬁrmed by the TEM image in Fig. 12.15b. In addition, the size of the islands in the subsequent layers can be adjusted to be nearly the same by changing the growth parameters appropriately, e.g., the growth temperature and the amount of deposited material [37]. The 3D ordering of the islands in the multilayer sample is clearly demonstrated by an X-ray reciprocal space diﬀraction map around the (0, 0, 4) Bragg point, as shown in Fig. 12.16b. Each vertical intensity satellite, indicated by arrows (white or black) and SLn (n: 0, -1) is accompanied by a large number of lateral satellites (Sub indicates the position of the Si substrate reﬂection). The full-width at half-maximum (FWHM) of these peaks does not change with their order, demonstrating their long range ordering.

12.4 Outlook Further studies on the growth of the islands on the patterned substrates are required, for instance to achieve smaller islands (much less than 100 nm in diameter) on patterned substrates with a small 2D periodicity, possibly resulting in improved optoelectronic properties. On the other hand, very sparse arrays of islands are required to facilitate the analysis of electronic properties of single islands. In this case, devices based on a single island could become feasible. Another challenge these ordered islands present is to investigate their properties. Several attempts have been made to study the optical properties of these ordered islands by photoluminescence [29, 34, 37, 55]. However, systematic studies are required to take full advantage of them and their properties. Acknowledgement. This work was partly supported by the FWF, Vienna Austria, project No SFB025. We thank Prof. F. Sch¨ aﬄer for helpful discussions on the MBE island growth and for providing the TEM data.

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13 Ordered SiGe Island Arrays: Long Range Material Distribution and Possible Device Applications G.S. Kar, S. Kiravittaya, M. Stoﬀel, and O.G. Schmidt Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany

13.1 Introduction During the last decade, self-assembled islands, commonly denoted as “quantum dots”, became a central research area in semiconductor physics due to their novel and exciting properties. The reduced lateral size modiﬁes the density of states and allows a charge carrier conﬁnement in three dimensions, which results in strong modiﬁcations of the corresponding bulk material properties. The interest in nanostructures is also driven by their tremendous potential for novel type of devices. Indeed, single electron transistors, single photon sources as well as turnstile devices have already been realized using single self-assembled quantum dots [1–5]. Many diﬀerent lattice mismatched material combinations such as In (Ga)As/GaAs (001) [6–8], In (Ga)P/GaAs (001) [9] or Ge/Si (001) [10, 11] were investigated using a wide variety of fabrication techniques such as solid or gas source molecular beam epitaxy (MBE) or chemical vapor deposition. In most of the growth studies, the position of the quantum dots was more or less random due to the statistical nature of the self-assembling process on ﬂat substrate surfaces. The absence of any precise control of the island positions limits considerably their applications, particularly in electronic devices where individual islands need to be addressed. The full potential of quantum dots can thus only be utilized if their positions can be rigorously controlled. This will open the route towards their integration into more complex device architectures. In order to force quantum dots into lateral alignment and to be able to control their positions, many diﬀerent approaches were put forward during the last few years. We can divide them into two main groups. The ﬁrst exploits the surface properties to self-align quantum dots (see Part I). This includes the growth on high-index or vicinal surfaces, the growth on dislocation networks or the use of stacked island structures. These methods, however, do not allow long-range ordering of quantum dots on planar surfaces, which is mandatory for any large-scale integration of single QD devices. The second group is based on the growth on patterned substrates (see Part II). It can be divided further into two subgroups. The ﬁrst includes the selective epitaxial growth on patterned substrates [12, 13] while the second is based on the growth on modulated surfaces [14–22].

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Among the diﬀerent lattice mismatched material systems, Ge(Si) growth on Si (001) is one of the most studied systems due to its simplicity (only two components are involved) and the compatibility with the well-known complementary metal oxide semiconductor (CMOS) technology. The ﬁrst attempt to order SiGe islands on patterned Si (001) substrates was reported in 1995. Hartmann et al. [17] have grown self-assembled Ge islands on a lithographic patterned Si(001) surface. However, due to the low resolution of the patterning technique, no long-range ordering was achieved. Many eﬀorts were then undertaken to improve the lateral ordering of selfassembled islands. Schmidt et al. [18] succeeded in lateral ordering of Ge islands on planar surfaces by using a combination of substrate patterning and spontaneous vertical ordering along the growth direction. More recently Zhong et al. have demonstrated perfect one-dimensional (1D) and two-dimensional (2D) ordering of Ge islands on holographic patterned surfaces [19–21]. These authors emphasized particularly the importance of the growth conditions for the obtained morphologies. The ability to align self-assembled islands into periodic arrays has opened the route towards novel growth phenomena as well as to fabricate quantum dot arrays with excellent size homogeneity [21]. The understanding of the physics sustaining the size homogeneity over a large area will presumably strongly depend on the diﬀusion mechanism in and around ordered island arrays. This chapter is therefore mainly dedicated to the understanding of long-range material diﬀusion in and around ordered island arrays. Moreover, novel device architectures based on laterally aligned Ge islands are proposed. This chapter is organized in the following way: in the ﬁrst part, we will discuss the growth on ﬁnite sized patterned areas starting with the substrate patterning and the sample growth. We will then study the material distribution across the interface between a planar and a patterned surface in the case of a single island layer deposition. The same study will then be extended to the case of a twofold stack of island layers. In the second part, we will propose a model to explain the observed material distribution. This model takes into account the surface curvature in the case of a single island layer and the strain originating from buried islands in the case of the second island layer. Growth simulations are able to reproduce excellently the experimental material distribution [23]. In the third part, we will present brieﬂy two possible applications of ordered island arrays including the concept of a dot ﬁeld eﬀect transistor (DotFET) [24] and a novel device architecture based on free-standing Si bridges [25].

13.2 Growth on Finite-Sized Patterned Areas In this section, we present the experimental methods used for substrate patterning and subsequent Ge deposition. We further analyze the material dis-

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tribution across the interface between patterned and unpatterned areas for both single and twofold stacked island layers. 13.2.1 Substrate Patterning We fabricate ﬁnite sized patterned areas of 60×50 μm2 on Si (001) substrates by using a combination of standard electron beam lithography and reactive ion etching. The details of each processing step is shown in Fig. 13.1. First, Si (001) substrates were spin coated with 130-nm-thick polymethyl methacrylate (PMMA). Standard electron beam lithography was then used to write the pattern on the PMMA coated Si substrates. In this study the pattern consists of 1D trenches aligned along the [110] direction. The pattern was then transferred to the Si substrate by reactive ion etching using a CHF3 /O2 plasma. Finally the samples were cleaned in an O2 plasma. Scanning electron microscopy (SEM) and atomic force microscopy (AFM) were then used to characterize the patterned surface. Figure 13.2a shows a SEM image of the patterned area after cleaning. We can identify very sharp edges demonstrating the ability of the above-described method for eﬃcient substrate patterning. Similar structures were also observed in the AFM micrograph shown in Fig. 13.2b. A cross-sectional linescan taken along the line as indicated in Fig. 13.2b shows that the pattern consists of trenches aligned along [110] having an average depth of 24 nm, a width of 120 nm and a period of 320 nm. We observe a small height overshoot along the edges of

Fig. 13.1. a–d Schematic of each process step used to pattern the Si (001) surfaces. PMMA Polymethyl methacrylate

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the stripes. During the etching process, the CHF3 /O2 plasma digs Si out from the trench leaving some material along the edges. This additional Si material is removed during the modiﬁed Shiraki cleaning process. The samples were ﬁnally dipped in a HF solution for surface passivation before their transfer into a solid-source MBE system. After in-situ H desorption at 650 ◦ C, a clear (2×1) reﬂection high-energy electron diﬀraction pattern was observed indicating that the surface was deoxidized. A 50-nmthick Si buﬀer layer was subsequently grown at 0.1 nm/s, while the substrate temperature was increased from 460 to 620 ◦ C. During growth, the background pressure was kept below 10−9 mbar. After Si deposition, the surface

Fig. 13.2. a Scanning electron microscopy (SEM) image of the patterned area. b Atomic force microscopy (AFM) image of the same patterned sample. c Cross section height proﬁle along the line deﬁned in b

Fig. 13.3. a AFM image of the patterned area taken after deposition of a 50-nm-thick Si buﬀer layer at 620 ◦ C. b Cross section height proﬁle along the line deﬁned in a

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morphology was characterized using AFM. The morphology of the patterned sample after 50 nm Si buﬀer growth is shown in Fig. 13.3a together with a linescan taken along the [1-10] direction (Fig. 13.3b). The original sharp edge pattern transforms into a smooth sinusoidal modulated surface. We ﬁnd that the periodicity remains unchanged but the depth decreases to 20 nm. This surface is then used as a template for the ﬁrst island layer growth. We deposit 5 or 7 monolayer (ML) Ge at a growth rate of 0.044 ML/s at 620 ◦ C. For the twofold-stacked samples, a 30-nm-thick Si cap was grown on the ﬁrst 7 ML Ge layer. The second Ge island layer was then grown by depositing 4 or 7 ML Ge at 620 ◦ C. 13.2.2 Single Island Layer Surface Morphology Figure 13.4a presents an 80×80 μm2 AFM scan taken after deposition of 5 ML Ge at 620 ◦ C. The patterned area (60×50 μm2 ) can be recognized in the center of the ﬁgure. From the above-mentioned large area AFM scan we can identify neither the size nor the shape of individual islands. However, an island-free region is observed between the patterned area and the randomly distributed islands on the ﬂat surface for the Ge coverage range 5 – 7 ML. To gain a deeper insight into the surface morphology, we took small size (5×5 μm2 ) AFM scans along the path indicated in Fig. 13.4a. Typical AFM images corresponding to the positions of the labeled boxes in Fig. 13.4a are shown in Figs. 13.4b–e. The Ge islands are aligned along

Fig. 13.4. a Large area (80×80 μm2 ) AFM image of a single ﬁve-monolayer (5 ML) Ge island layer sample with the patterned area in the center of the image. b–e AFM images (5×5 μm2 ) of the areas marked in a. An island-free region is clearly observed around the pattern. The insets of b and c are zoomed AFM images (1.2×0.9 μm2 )

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G.S. Kar et al. Fig. 13.5. Distribution of a Ge island density and b total material accommodated in the islands per unit area (Θ) across the interface between patterned and unpatterned area for the ﬁrst layer with 5 (triangles) and 7 (squares) ML of Ge. The simulation results (open symbols) are presented along with experimental results. The experimental data are ﬁtted with an exponential function (solid lines). The diﬀusion lengths are also indicated

the predeﬁned trenches (Fig. 13.4b,c) as already reported [18,20]. In contrast a random island distribution is observed on the ﬂat surface (Fig. 13.4d,e). A more detailed analysis shows that the islands on ﬂat surfaces consist of a bimodal distribution of pyramid and dome islands. We ﬁnd mainly pyramid islands (inset of Fig. 13.4b) in the center of the patterned area, while a higher density of dome islands is observed near the edge of the pattern (inset of Fig. 13.4c). The dome island formation could be either the result of island coarsening or the consequence of material accumulation near the inner edges of the pattern, since during growth at a constant temperature a shape transition from pyramid to dome is generally observed when the Ge coverage increases [26,27]. The former hypothesis can be ruled out, since the island density stays almost constant at 12.6±0.7 μm−2 across the patterned area (Fig. 13.5a). The latter hypothesis is supported by the observation of an island-free region around the patterned area. We therefore conclude that a directional material diﬀusion occurs from the ﬂat, unpatterned area towards the patterned area. Material Distribution In order to quantify the material distribution across the interface, we measure the island density and calculate the total material accommodated in the

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Fig. 13.6. Schematic illustration of the Ge diﬀusion across the interface between the ﬂat unpatterned and the patterned area

islands per unit area Θ, from a series of AFM scans taken along the path as indicated in Fig. 13.4a. The results are summarized in Fig. 13.5b. For both 5 and 7 ML Ge depositions, the island density remains almost constant (12.6±0.7 μm−2 and 13.8±0.7 μm−2 , respectively) in the patterned area, while Θ increases, when moving from the center of the patterned area towards the interface. In contrast, if we consider the surface morphologies along a path going from the unpatterned area towards the interface, both island density and Θ decrease. We ﬁt the evolution of Θ (solid symbols) with the following function that allows us to determine the diﬀusion length Li (i = 1,2) of the Ge adatoms in the ﬂat (i = 1) and patterned (i = 2) areas: A − Bex/L1 x<0 Θ(x) = (13.1) −x/L2 x>0 C + De where A, B, C, D are constants determined by the boundary conditions. We consider x = 0 as the interface (See Fig. 13.6). This ﬁtting function is derived from the steady state diﬀusion equation assuming a constant ﬂux of Ge adatoms across the diﬀerent surfaces [17]. We can thus determine the diﬀusion lengths (Li ) from the above equation if we assume a constant wetting layer thickness and a constant Si ratio in each island. The parameters A, B, C, and D do not inﬂuence the diﬀusion lengths. We obtain almost identical diﬀusion lengths of 5.0±0.5 μm on both patterned and unpatterned areas. Moreover, this length is independent of the Ge coverage (in the range of 5 – 7 ML Ge considered here). 13.2.3 Twofold-Stacked Island Layer Surface Morphology For the twofold stacked layer samples, the initial 7 ML Ge island layer was ﬁrst overgrown with 30 nm Si at 620 ◦ C. The 2×1 μm2 AFM image of the 30nm-capped Si surface is shown in Fig. 13.7a. The root-mean-square roughness

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G.S. Kar et al. Fig. 13.7. a AFM image of an overgrown island layer on the patterned surface. The surface is almost ﬂat after deposition of 30 nm Si at 620 ◦ C. b Cross section height proﬁle along the line deﬁned in a

is about 0.34 nm, which is comparable to the roughness measured on ﬂat Si surfaces (≈ 0.3 nm). This ﬂat but strain-modulated surface acts as a template for the growth of the second island layer. For the second layer, 4 or 7 ML Ge were grown using the same conditions as for the deposition of the ﬁrst island layer. The islands in the second layer will form directly above the buried ones [28–30] due to the strain ﬁeld propagation. The resulting surface morphology is shown in Fig. 13.8. In Fig. 13.8a we show a large AFM image (80×80 μm2 ) obtained after deposition of 7 ML Ge on the ﬂat surface. We can again distinguish well-ordered lines of aligned Ge islands in the center of the image. This directly reﬂects the spontaneous alignment of Ge islands along the growth direction due to the strain originating from buried islands. From this ﬁgure we can also identify an island-free region surrounding the patterned area as for the ﬁrst island layer. This region, which is bordered by randomly distributed islands on the ﬂat surface, is also observed independently of the Ge coverage (4 – 7 ML). In order to obtain a more detailed picture of the surface morphology, we took (5×5 μm2 ) AFM scans along the path indicated in Fig. 13.8a. Typical AFM images corresponding to the positions of the labeled boxes in Fig. 13.8a are shown in Figs. 13.8b–e. In the patterned area, 1D chains of well-aligned Ge islands are clearly observed since the strain ﬁeld from the ordered buried islands control the island nucleation in the second layer. A random island distribution is also evident on the ﬂat surface surrounding the ordered island array. At the center of the ordered array, we observe a bimodal distribution of pyramid and dome islands (inset of Fig. 13.8b), while dome islands (inset of Fig. 13.8c) are located predominantly near the edge of the ordered array. Consequently, we believe that more Ge adatoms accumulate near the edge than in the middle of the ordered island array. The

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Fig. 13.8. a Large area (80×80 μm2 ) AFM image of a twofold stack of 7 ML Ge separated by 30 nm of Si. The ordered island array can be observed in the center of the image. b–e AFM images (5×5 μm2 ) of the areas marked in a. An island-free region is clearly observed around the pattern. The insets of b and c are zoomed AFM images (1.2×0.9 μm2 )

observations of an island-free region suggest that a directional diﬀusion also occurs from the randomly distributed island area towards the ordered island array. Material Distribution: Inﬂuence of Strain In order to evaluate the material distribution across the interface, we measure the island density and calculate the total material accommodated in the islands per unit area Θ from a series of AFM scans, taken along the path indicated in Fig. 13.8a. In contrast with the single layer, the density increases slightly while moving from the center towards the edge of the ordered island array. We expect that this is the result of strain driven material accumulation near the edge of the ordered island array. The randomly distributed island density as well as the material distribution on the whole area (see Fig. 13.9) follows the same trend as observed for the ﬁrst island layer (see Fig. 13.5). A simple exponential function is, however, not able to ﬁt the evolution of Θ across the interface. The situation is complicated by the fact that strain ﬁelds originating from buried islands may signiﬁcantly aﬀect the Ge adatom diﬀusion and nucleation. For the second island layer growth we have to consider three regions (See Fig. 13.10): 1. The surface on top of randomly buried islands 2. The surface on top of the buried-island-free region 3. The surface on top of the buried ordered islands.

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G.S. Kar et al. Fig. 13.9. Distribution of a Ge island density and b total material accommodated in the islands per unit area (Θ) across the interface between randomly and ordered island area with 4 (triangles) and 7 (squares) ML of Ge. The simulation results (open symbols) are presented along with experimental results. The solid lines correspond to ﬁts of the experimental data. The diﬀusion lengths are also indicated

Fig. 13.10. Schematic illustration of the model of Ge diﬀusion across the interface between randomly distributed island (1 ), island-free region (2 ) and ordered island arrays (3 )

The surface on top of the buried-island-free region (Fig. 13.10, region 2) leads to the formation of two interfaces instead of only one in case of the ﬁrst layer (Fig. 13.6). The boundary between regions 1 (Fig. 13.10) and 2 (Fig. 13.10) can not be precisely deﬁned due to the stochastic nature of island formation on ﬂat surfaces. Moreover, the material diﬀusion is complicated by the strain ﬁeld modulation and by the variation of the buried island volume across the interface. We therefore describe the evolution of the total material

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accommodated in the islands per unit area Θ within the framework of the general diﬀusion theory. The ﬁtting is realized by using the following function: 0 A erfc( x−x x<0 2L1 ) Θ(x) = (13.2) x B + C erfc( 2L ) x >0 2 where A, B and C are constants, x0 is the distance from the patterned/unpatterned interface to the boundary between region 1 and 2 (see Fig. 13.10) and Li (i = 1,2 ) is the diﬀusion length for the unpatterned (i = 1) and patterned (i = 2) areas. Here again, the value of A, B and C do not inﬂuence the determination of the diﬀusion lengths. Consequently, A, B and C are taken as ﬁtting parameters. In this case, we ﬁnd a signiﬁcantly shorter diﬀusion length on the unpatterned area (1.2-2.0 μm) than on the patterned area (8.8-10 μm). For 7 ML Ge deposition in the second layer, we observe a slight volume overshoot on the ﬂat surface near the island-free region (marked as “P” in Fig. 13.9b). This eﬀect cannot be explained assuming a simple diﬀusion model, and a more advance theoretical description will be developed in the next section.

13.3 Modeling of the Material Distribution 13.3.1 Growth Model In order to gain a deeper insight into the growth on large-scale areas, we consider the mean-field energy “felt” by the deposited Ge material on the substrate surface. This total energy E is derived from the chemical potential of the adatom/substrate system [31, 32] and is deﬁned as: E = Eb + Ecurv + Estr

(13.3)

where Eb is a binding energy on the planar substrate taken as a constant, Ecurv is the contribution of the surface curvature to the total energy and Estr is the strain energy contribution due to the presence of buried islands. In our model, the energy gradient will essentially aﬀect the directional diﬀusion of the deposited material. “Sink” Model for Single Island Layer First we consider the case of a single island layer. In that case, there is no strain energy contribution to the total energy E (Estr = 0). Instead, the patterned area is exclusively characterized by surface curvature, which is expected to be negative in our case. We can thus assume that the total energy E is lower in the patterned area than on the planar surface. The patterned area can therefore be considered as a “sink” for the deposited material, and

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G.S. Kar et al. Fig. 13.11. Total energy prior to ﬁrst layer growth. On average the surface curvature lowers the total energy on the patterned area

Fig. 13.12. a Schematic diagram of a single buried island for surface strain calculation. b Cross section proﬁle of surface strain energy for deposited Ge material. c Total energy prior to second layer growth. The proﬁle is characterized by shortand long-range strain contributions caused by buried islands in the ﬁrst layer

a directional material diﬀusion is expected to occur from the unpatterned area towards the patterned area. Figure 13.11 shows the energy proﬁle used for the growth simulation. It is simply characterized by an energy diﬀerence between the ﬂat and the patterned area. Strain-Driven Material Distribution For the second layer, the surface is ﬂat prior to island growth (see Fig. 13.7). Consequently, there is no surface curvature contribution to the total energy (Ecurv = 0). We calculate the elastic strain energy proﬁle Estr for Ge material by using an analytical expression for a single island [33]. The buried Ge islands are treated as square-based boxes with 160 nm base length and 20 nm high as schematically shown in Fig. 13.12a. The cross-sectional strain proﬁle calculated for a single buried island is shown in Fig. 13.12b. We deduce the total strain energy by adding the strain contributions from each island in the patterned area. We take into account the size of the patterned area

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(60×50 μm2 ), the mean island distance in the trenches (250 nm) and the trench period (320 nm). For the sake of simplicity, we assume a 2D periodic array. Figure 13.12c shows the energy proﬁle used for the second island layer growth simulation. The patterned area is on average tensile strained (for Si) due to the buried ordered island array while the surrounding area is compressively strained. Since the surface is ﬂat, the tensile strain generated by the buried island array must be compensated by a compressive strain around the pattern [28]. We expect that this compensation will modify the diﬀusion of the Ge material around the patterned area. In addition, the shortrange tensile strain outside the patterned area, originating from the random buried islands, is magniﬁed in the inset of Fig. 13.12c. 13.3.2 Growth Simulation Simulation Procedure On the long-range scale, the Ge material diﬀusion follows the gradient of the total energy E deﬁned in Eq. (13.3). Since an atomistic simulation of the considered growth area (at least 80×80 μm2 ) is obviously well beyond any computational capability, we develop a growth simulation using an analogy with the atomistic case. In this simulation, we choose a smaller simulation domain discretized into an area of 300×300 pixels with periodic boundary conditions. At the center of this domain, an area of 120×100 pixels is considered as the patterned area. In this model, the Ge material will be placed inside the domain (deposition) and can subsequently either nucleate (stop) or diﬀuse further with a diﬀusion probability pi . For material deposited on a single pixel, there are ﬁve possible events: the nucleation at this pixel or the diﬀusion to the four neighboring pixels. For an event i, the diﬀusion probability pi is deﬁned as the ratio between the diﬀusion ﬂux of that event i to the total diﬀusion ﬂux. The latter is deﬁned as the sum of the diﬀerent ﬂuxes from all possible events (diﬀusion or nucleation) for a considered pixel. We can deﬁne the diﬀusion ﬂux Fi by the following equation (the detailed derivation of this equation is given in the Appendix). Fi = F0D,S exp (−ΔE/kB T )

(13.4)

where ΔE is the energy diﬀerence between two neighboring pixels, T is the substrate temperature, kB is the Boltzmann’s constant and the prefactor F0D,S is a constant. During diﬀusion, the Ge material can either nucleate (stop) at a pixel (in that case F0D,S = F0S ) or diﬀuse to a neighboring pixel (in that case F0D,S = F0D ). From Eq. (13.4), one can clearly see that the deposited Ge material will experience no directional diﬀusion on ﬂat, unstrained surfaces (ΔE = 0). Consequently, the simulation will generate random deposition proﬁles on the long-range scale. The principle of our growth simulation is as follows: in each simulation step, the Ge material is randomly deposited onto one of the 300×300 pixels.

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Then, the material will nucleate or diﬀuse to a neighboring pixel according to the probability pi , which depends on the ﬂux deﬁned in Eq. (13.4). The diﬀusion continues until nucleation occurs. This simulation is continued until the deposition of 1 equivalent ML (area of 300×300 pixels). Model Parameters Based on this growth model [Eqs. (13.3) and (13.4)], the ratio F0D /F0S will determine the number of diﬀusion steps before nucleation and thus the diffusion length L. Indeed, a larger ratio (i.e., a larger F0D or a smaller F0S ) will result in an increased number of diﬀusion steps. We can therefore establish a relation between the diﬀusion length of the Ge material on the ﬂat, unstrained surface and the ratio between F0D and F0S by performing a simple simulation. We set ΔE in Eq. (13.4) to zero (ﬂat, unstrained surface) and we consider the material deposition only at the center of a 300×300 pixels area. The obtained proﬁle will then provide the diﬀusion length of the Ge material. Figure 13.13 shows a simulation when we set for example F0D = 400×F0S. The evolution can be ﬁtted with a simple exponential function. By using this approach we obtain a diﬀusion length of 8.1 μm. When the F0D /F0S ratio increases, we obtain a longer diﬀusion length. For the ﬁrst island layer, the simulation is performed using the ratio F0D /F0S as well as the energy diﬀerence ΔEcurv between patterned and unpatterned surface (see the upper part of Fig. 13.14a). The schematic plot of the expected volume proﬁle is shown in the lower part of Fig. 13.14a. For the second island layer, the strain energy originating from the buried islands (shown in Fig. 13.12) is written as a sum of two terms, i.e., Estr = L S L S Estr + Estr , where Estr is the long-range strain energy and Estr is the shortL range tensile strain energy contribution from the buried island array. Estr L,U corresponds to a compressive strain energy outside the pattern (Estr ) while L,P ) (see it corresponds to a tensile strain energy inside the pattern (Estr

Fig. 13.13. Simulation results to determine the diﬀusion length L, when F0D /F0S is set to 400. The number of deposited Ge material is plotted as a function of distance from the deposited position. The inset shows the two-dimensional (2D) proﬁle

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Fig. 13.14. Schematic illustration of the growth model for a the ﬁrst and b the second layer. The upper part presents the assumed energy proﬁle across the patterned area while the lower part shows the expected volume proﬁle of the deposited material S the upper part of Fig. 13.14b). Estr is responsible for the vertical orderS ing in stacked island structures [28–30]. Hence, the Estr term will increase the material accumulation probability on the pixel located directly above S S as δ (x, y)ΔEstr , where δ (x, y) = 1 dia buried island. We deﬁne Estr L,U rectly above a buried island and zero elsewhere. ΔEstr is the amplitude of the long-range compressive strain, which repels Ge material diﬀusing towards the patterned area. Based on our considerations, we schematically plot the energy proﬁles E with the related ﬁtting parameters and the expected volume distributions for the second island layer in the lower part of Fig. 13.14b.

Simulation Results and Comparison Figure 13.15 shows the material distributions obtained from averaging over 100 simulations. In case of the ﬁrst island layer (Fig. 13.15a), a good description of our experiments can be obtained with F0D = 400 × F0S and ΔEcurv =154 meV. The value of F0D /F0S used here corresponds to a diﬀu-

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sion length of 8.1 μm on the ﬂat surface (see Fig. 13.13). However, since the total energy E used in this simulation is a mean-ﬁeld energy, we can not directly establish a relation between ΔEcurv and the surface curvature contribution to the chemical potential. The material distribution along the solid line in Fig. 13.15a is rescaled in the y direction and plotted in Fig. 13.5b (open symbols) together with the experimental data. We obtain a good ﬁt of the experimental results, conﬁrming our assumption that on the long-range scale the patterned area can be considered as a material sink for the deposited Ge material. For the second layer, the simulation result is shown in Fig. 13.15b. We use S =34 meV, the following parameters for this simulation: F0D = 400×F0S, ΔEstr L,U S and ΔEstr = 1.15 eV. The value of ΔEstr is compatible with our calculations of the strain energies on top of the buried islands (18 meV). However, the L,U ﬁtted ΔEstr is much larger than expected (value in the order of millielectronvolts). The reason for this apparent discrepancy simply lies in the lateral scaling of the model. In our simulation the Ge material need less than ten L , while in reality a Ge diﬀusive steps to overcome the long-range barrier ΔEstr adatom hops many thousands times to diﬀuse over this barrier. For a good L ﬁt, the model therefore delivers a value of ΔEstr , which is much larger than what is found in reality. Similar to the ﬁrst island layer, we determine the cross-sectional proﬁle of the material distribution along the line indicated in Fig. 13.15b. The rescaled results are shown as open symbols in Fig. 13.9b. We ﬁnd that our simu-

Fig. 13.15. Material distribution obtained from a ﬁrst layer simulation and b second layer simulation. The simulated volume along the solid lines are rescaled and plotted in Fig. 13.5a and Fig. 13.9b, respectively

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lations can be ﬁtted to the highly asymmetric material distribution across the interface between the patterned and the unpatterned area. Moreover, the simulation is also able to explain the material accumulation at the position “P” in Fig. 13.9b. The long-range compressive strain repels Ge material diffusing towards the patterned area leading thus to a material accumulation in the vicinity of the island-free region. Discussion and Model Limitations For the ﬁrst island layer growth simulation, the total energy on the patterned area is assumed to be lower than on the unpatterned area due to the surface curvature contribution. This lower energy is assumed to be responsible for the directional material diﬀusion from the ﬂat surface towards the patterned area. In order to test the importance of the surface curvature, we perform an additional experiment. We drilled the same area and same depth but without patterned trenches into a Si (001) surface. A schematic of this patterned surface is shown in Fig. 13.16a. This surface was then overgrown using the same growth conditions as described in Sect. 13.2.2. The AFM scan across the interface between the two areas is shown in Fig. 13.16b. We do not observe any island-free region around the etched area, which demonstrates the importance of surface curvature for the directional diﬀusion of Ge adatoms. The observed material distribution could also be the result of a nucleation energy diﬀerence between patterned and unpatterned areas. If the nucleation energy (the nucleation probability) is lower (higher) in the patterned area, islands are expected to form earlier in this region. In our case, the nucleation probability (pnuc ) will be considered by modifying the prefactor term F0D,S in Eq. (13.4) (considered as a constant in the preceding section). According

Fig. 13.16. a Schematic of the drilled surface. b AFM image after deposition of 5 ML Ge on a patterned Si surface at 620 ◦ C

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to our model, the nucleation probability can be written as pnuc = 1 − pdiﬀ

(13.5)

where pdiﬀ is the sum of the diﬀusion probabilities in all directions. In other words, the ratio of F0S /F0D deﬁnes the average number of nucleation events after a single diﬀusion step, i.e., pnuc ∝ F0S /F0D . We have thus performed an additional simulation by taking into account explicitly the nucleation energy diﬀerence between the patterned and the unpatterned areas. If we assume that there is no directional diﬀusion (i.e., ΔE= 0 in Eq. (13.4)) and lower nucleation energy in the patterned area, the value of F0S /F0D on the patterned area will be higher than on the unpatterned area. We took the value of F0S /F0D = 1/400 on the unpatterned area and 1/100 on the patterned area. The proﬁle averaged from 100 simulations is shown in Fig. 13.17a. The corresponding material distribution along the solid line in Fig. 13.17a is plotted in Fig. 13.17b. The simulated results were ﬁtted with a simple exponential function (Eq. (13.1)). We ﬁnd an asymmetric material distribution, which is obviously not observed experimentally (see Fig. 13.5b). We therefore conclude that the nucleation energy diﬀerence can not explain our results but instead, directional diﬀusion must be taken into account to reproduce correctly the observed material distribution. Finally, we critically discuss the limitations of our model. Compared to the considered growth area, we use a simulation domain of extremely low resolution (only 300×300 pixels), which means that an island has a lateral size of a single pixel. The low resolution of our growth simulation implies that we cannot predict any short-range eﬀects such as the island density, their local size homogeneity, nor the properties of the single islands such as their size and shape. Moreover, we neglect the wetting layer and consider a maximum coverage of only one monolayer. These assumptions mean that the calculated

Fig. 13.17. a Material distribution obtained from the simulation, which included the nucleation energy diﬀerence between patterned and unpatterned area. b Cross section height proﬁle along the solid line in a. The simulated results are ﬁtted with exponential function and the diﬀusion lengths are labeled

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material distribution needs to be up-scaled by a free but constant factor to ﬁt our experimental results. However, a careful check of our rescaling procedure reveals that the obtained diﬀusion lengths are not modiﬁed by this procedure.

13.4 Possible Device Applications We present here a concept of a metal oxide semiconductor ﬁeld eﬀect transistor (MOSFET) device that incorporates laterally aligned self-assembled Ge islands, also called DotFET [24]. Figure 13.18 represents a schematic cross section of an n-channel DotFET (Fig. 13.18a) and of a p-channel DotFET (Fig. 13.18b) [24, 30]. In the n-channel DotFET, the electrons ﬂow through the strained Si region located above the Ge islands. It is well known that the tensile strain lifts the sixfold degeneracy of the conduction band and splits the Δ2 − Δ4 valleys, reducing thus the intervalley scattering and the electron mass in the ﬂow direction. The strain induced splitting can be even more improved by increasing the number of closely stacked Ge island layers. Indeed, ﬁnite element calculations [34] showed that the tensile strain reaches about 1% in a center of a ﬁvefold stack of Ge islands, which is almost comparable to the strain in pseudomorphic Si layers grown on a Si0.75 Ge0.25 relaxed buﬀer, commonly used for Si/SiGe MOSFETs [35]. In the p-channel DotFET, the holes ﬂow through the Ge island itself. The source and drain contacts are deﬁned at the edge of the Ge island while the gate contact is located directly above the Ge islands. The charge carrier ﬂow can thus simply be controlled by the voltage applied to the gate. The island height, which depends on the growth temperature, can be chosen in order to ensure suﬃciently strong localization of the hole wavefunction in the channel, where the high mobility of the holes is exploited. Moreover, the Ge island lateral size (typically 30 – 70 nm), which also depends on the island growth temperature, matches the gate lengths required in the next generation of MOSFET technology.

Fig. 13.18. Schematic illustration of a an n-channel and b a p-channel dot ﬁeld eﬀect transistor (DotFET) [24]. In the n-channel DotFET, electrons ﬂow through the strained Si above the Ge dot, while in the p-channel DotFET, holes ﬂow through the Ge dot itself

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The DotFET concept has the advantage of being a buﬀer-free technology, i.e., it avoids the need of thick relaxed SiGe buﬀer layers. Moreover, it was shown that stacked Ge island layers are able to withstand high temperature processing steps [24], which are usually required in CMOS technology. In contrast, such high temperature processing may induce the propagation of dislocations through devices grown on SiGe relaxed buﬀers, which results in a degradation of their performances. By choosing appropriately the growth and capping temperatures, Ge islands can incorporate much higher Ge contents [36] than pseudomorphic SiGe quantum wells and consequently, the p-channel performance can be improved [35, 37]. The continuous device downscaling below 100 nm gate length leads to some limitations, in particular short channel eﬀects and possible tunneling through the gate oxide. A possible solution to circumvent these limitations lies in the ability to create new device architectures. Double gated (DG) MOSFETs [38] such as FinFETs [39] can be considered as promising candidates for ultimate devices but their realization remains still challenging. A novel approach for advanced CMOS, called silicon-on-nothing (SON), was recently proposed as a pathway for getting rid of the drawbacks of the conventional CMOS technology [39, 40]. Here, we propose a novel concept of a SON-MOSFET device based on laterally aligned Ge islands. Figure 13.19a represents a possible layout of such a device. First, the Ge islands are overgrown with Si at suﬃciently low temperatures in order to preserve the initial morphology [41, 42]. Then, a combination of electron beam lithography and reactive ion etching is used to deﬁne trenches. When the newly deﬁned trenches cross the islands in the middle, the SiGe forming the island will become exposed on both sides. The selective etching of the SiGe core will then be possible and only a free-standing Si bridge will be left over on the surface. If we deﬁne a source and drain contact on each side of the bridge and a gate ﬁnger on the top, the free-standing bridge can deﬁne the channel of a transistor. The charge ﬂow between source and drain can thus be simply controlled by the voltage applied on the gate. The control of island growth and overgrowth temperature allows a careful tuning of the bridge shape and size.

Fig. 13.19. a Possible layout of a n-channel FET based on free-standing Si bridges. b SEM image of fabricated Si bridge

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In order to realize the ﬁrst step of such a device, we have realized Si bridges experimentally. Figure 13.19b shows a SEM image of a free-standing Si bridge. The SiGe core is etched using a solution of BPA (consisting of HF (1)/H2 O2 (2)/CH3 COOH (3)) which is known to etch selectively Si1−x Gex alloys over pure Si [43]. The bridge extends over a length of 315 nm and has a height and thickness of about 45 nm and 31 nm, respectively [25].

13.5 Summary In conclusion, we have investigated the distribution of material accommodated in self-assembled islands across the interface between patterned and unpatterned areas. Our measurements give evidence of a material depletion region around the ﬁnite sized patterned area for both single and twofold stacked island layers. For the ﬁrst layer, a simple material “sink” model is able to explain a symmetric material distribution across the interface, which can be interpreted as a directional diﬀusion of Ge adatom towards the patterned area. For the twofold stacked layer, our simulations suggest that the strain modulated surface causes a highly asymmetric diﬀusion behavior in and around the pattern as well as a material accumulation in the vicinity of the island-free region. Our results provide a better understanding of the growth mechanisms on ﬁnite sized patterned areas, which is of fundamental importance for future high integration of novel single-quantum-dot-based devices. We propose that defect-free, ordered arrays of self-assembled Ge islands can be the building blocks of a novel type of device, called dot ﬁeld eﬀect transistor (DotFET). This device uses either the strained Si above Ge islands as an n-channel for electrons or the island itself as a p-channel for holes. Finally, we propose a new device architecture based on free-standing Si bridges, which can provide an alternative way to the novel Si-On-Nothing (SON) technology. Acknowledgement. This work was ﬁnancially supported by the BMBF (03N877) and the EC (012150).

Appendix Derivation of the Equation Used for the Growth Simulation In an atomistic growth simulation, atoms are randomly deposited on the substrate surface at a rate F (monolayers per second). After deposition, they can hop to the nearest neighbor sites with a hopping rate Rn . The latter depends on the bonding energies and the temperature, and is given by the Arrhenius law: ES + nEN 2kB T exp − Rn = (A1) h kB T

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where kB is Boltzmann’s constant, T is the temperature and h is Planck’s constant. ES is the surface binding energy, n is the number of nearest neighbors and EN is the bond energy to the neighboring atoms. The prefactor (2kB T /h) is called vibration frequency. For the growth on ﬂat surfaces, the surface binding energy ES is treated as a constant. Such a growth simulation is called kinetic Monte-Carlo simulation [44, 45]. Let us ﬁrst consider the case of growth on a ﬂat surface. The nucleation of single atoms is implicitly included in Eq. (A1). Indeed, when the number of nearest neighbor atoms n increases, the hopping rate Rn drastically decreases. We can therefore assume that atoms having more than one nearest neighbor will nucleate. By comparing the deposition rate with the hopping rate of single adatom, we can deduce that adatoms hop over million of sites before the deposition of the next atom. Since a realistic atomistic simulation is not possible in our case (domain size of at least 80×80 μm2 ), we use a pixel as a smallest unit for our simulation. One pixel has an area of 0.5×0.5 μm2 and is much larger than a single atomic site (1 atomic site has an area of a2 /2, where a is the lattice constant.). Therefore, the number of hopping events will be reduced by several orders of magnitude in our simulation. Based on this consideration we can thus estimate the ratio between diﬀusion events and nucleation events to several hundreds. In our growth simulation we deﬁne the diﬀusion ﬂux by Fi = F0D,S .

(A2)

The index i accounts for diﬀusion events to the four nearest neighbor sites (Fi = F0D , i = 1–4) and for nucleation events (Fi = F0S , i = 5). The diﬀusion probability is then deﬁned as the ratio of the diﬀusion ﬂux of the event i to the total diﬀusion ﬂux (on the ﬂat surface): pdiﬀ,i =

F0D 4F0D + F0S

(A3)

and the nucleation probability can be deﬁned as: pnuc =

F0S . + F0S

4F0D

(A4)

These two probabilities depend only on the ratio between F0D and F0S . We checked that the absolute values of F0D and F0S do not aﬀect the simulation result. The energy diﬀerence between the patterned and the unpatterned area can be included in the atomistic simulation by changing the surface binding energy ES in Eq. (A1) [45]. In our growth model, the absolute value of the energy is not of fundamental importance since the material diﬀusion depends only on the energy gradient [31, 32]. We consider in our simulation the energy diﬀerence (ΔE) between two neighboring sites where Ge material can

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diﬀuse. The latter quantity is included in the diﬀusion ﬂux that can ﬁnally be written as: Fi = F0D,S exp (−ΔE/kB T ) (A5) where F0D,S is equal to either F0D (diﬀusion) or F0S (nucleation).

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14 Nanoscale Lateral Control of Ge Quantum Dot Nucleation Sites on Si(001) Using Focused Ion Beam Implantation Alain Portavoce1, Robert Hull1 , and Frances M. Ross2 1

2

University of Virginia, Department of Materials Science and Engineering, 116 Engineers Way, Charlottesville, VA 22904 USA IBM Research Division, T. J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, New York 10598 USA

14.1 Introduction This chapter is dedicated to the control of quantum dot (QD) nucleation using focused ion beam (FIB) implantation. After reviewing the existing literature on nanostructure control using FIB, we will describe a new technique that allows lateral control of the positions of individual self-assembled quantum dots on a surface. The materials system we will consider is Ge islands grown on the Si(001) surface. The technique consists of three steps: (1) low dose FIB implantation of Ga+ ions, (2) postimplantation annealing, and (3) Ge deposition using chemical vapor deposition or molecular beam epitaxy. All three steps are performed in situ under ultra high vacuum. Since FIB implantation both sputters the surface and forms Ga clusters beneath it, the implantation and annealing processes permit local modiﬁcation of surface topography, chemistry and strain. The diameter of the modiﬁed area depends on the ion beam spot size, the implantation current and the annealing conditions, and can easily be localized to within 80 – 100 nm. Subsequent Ge deposition then leads to preferential island nucleation on the implanted areas, and under appropriate conditions only one Ge island nucleates on each FIB spot. Consequently, arbitrary patterns of Ge QDs can be formed with dot locations speciﬁed to better than 100 nm laterally. We present an analysis of the pattern characteristics resulting from the implantation and annealing steps, and then we describe Ge growth on the patterned substrate. In particular, we consider the factors aﬀecting island nucleation on pattern features. The patterning technique described here allows the fabrication of arbitrary arrays of QDs, with potential novel device applications, and we believe that it should be applicable to a range of materials systems where QDs form by self-assembly during strained layer epitaxy. 14.1.1 Prior Work on the Fabrication and Organization of Nanostructures by FIB The control of materials at the nanometer scale is expected to greatly impact technologies of the next generation. Particularly in the microelectronics ﬁeld,

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the fabrication and control of the organization of metallic and semiconducting quantum wires and quantum dots are likely to enable new devices based on innovative physical properties, which can improve existing microelectronic technology, and could even support a new system of logic by quantum algorithms. The fabrication of devices such as nanocrystal memories [1–11] and quantum cellular automata (QCA) [12–26] requires rigorous control of the location of the nanostructures in three dimensions. The focused ion beam, with an ion beam lateral size of order ten nanometers, is a recent tool that is playing an increasingly important role in nanoscience, as it allows the control of atom distributions by ion implantation as well as by direct sputtering. FIB patterning of substrates such as GaAs [27, 28], graphite [29, 30], InP [31], Si [32–36], SiO2 [33], and SrTiO3 [38, 39] has been successfully used to produce arrays of nanostructures synthesized by a wide range of deposition techniques such as molecular beam epitaxy (MBE) [31, 38], ultra high vacuum chemical vapor deposition (UHV CVD) [40], metal organic chemical vapor deposition (MOCVD) [32, 33, 36, 41], hybrid vapor phase epitaxy (HVPE) [27, 28], low-energy cluster beam deposition (LECBD) [29, 30], pulsed-laser deposition (PLD) [34] and electrochemical deposition [34, 35]. Nanostructures have been fabricated at desired locations on the surface with three diﬀerent FIB methods: (1) implantation [37, 42], (2) milling [43] and (3) selective etching [44,45]. These techniques have been used to demonstrate interesting electronic properties. For example, Kim et al. [37] formed 6 nm diameter Al nanoparticles in an MgO insulator layer by implanting Ga+ ions into Al deposited on the MgO. These Al nanocrystals were created in the source-drain active layer of a single electron device and were shown to create Coulomb blockades at room temperature. Karl et al. [42] created CdSe nanocrystals in a SiO2 layer using implantation of Cd and Se ions followed by rapid thermal annealing. Using the ability of FIB to sputter holes in metallic and semiconducting layers, nanostructures can be created by FIB at chosen locations after deposition. For example, Nakayama et al. [43] combined conventional lithography and FIB sputtering in order to create single electron transistors. Selective etching can be also used to create nanocrystals at speciﬁc locations. Lugstein et al. [45] created 30 – 120 nm diameter InAs faceted crystallites during normal incidence Ga+ FIB irradiation of (001) InAs substrates. The crystallite formation is due to the selective etching of As atoms. As well as nano-object fabrication, FIB is useful in creating controlled arrays of nanostructures. In this case, FIB is used to create a pattern on the substrate before the deposition of atoms [27, 28, 31–33, 36, 38, 40, 41, 46, 47] or nanoclusters [29, 30]. One technique involves covering the surface with a layer that acts as a mask and then drilling a pattern by FIB in this layer before deposition [32, 34, 36, 41]. For example, Kim et al. [32] controlled the nucleation locations of ZnO islands grown by MOCVD using a SiO2 /Si(111) FIB patterned substrate. After growth of a SiO2 layer on the Si substrate, Si lines 390 nm in width were created on the surface by etching the SiO2 layer

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using a Ga+ FIB. During deposition, Zn atoms diﬀuse into the Si lines leading to the preferential nucleation of ∼15 nm diameter ZnO dots on the Si lines. Lachab et al. [41] developed a technique using a photoassisted wet etching technique after FIB milling and before growth in order to decrease the density of defects due to FIB patterning. They showed that InGaN nanostructures with considerably fewer defects and a better crystal quality can be selectively grown on a Si mask deposited on GaN using this technique. Other studies [27–31, 33, 35, 38] have shown that FIB can also be used to pattern a substrate before growth without the use of a mask. This allows two types of patterned substrates to be created by FIB. First, FIB patterning can be used to form a deep surface topography. For example, Du et al. [38] fabricated holes of diameter ∼ 150 – 200 nm and depth ∼ 50 – 100 nm on a SrTiO3 (100) single crystal, and showed that Cu2 O islands nucleate on the border of the holes. Second, FIB implantation at lower doses can be used to create smaller scale defects on the surface such as kinks, steps and amorphous regions. Sun et al. [28] used FIB in this way to grow InP nanowires by HVPE on planar (001) GaAs substrates. The selective growth of InP on the FIBimplanted lines was attributed to a higher density of surface steps on implanted areas. In both the deep and the shallow patterned substrates discussed above, the topography induced by FIB plays an important role in the mechanism of nanostructure formation. For example, the selective nucleation of dots on implanted zones or around holes is usually attributed to a higher density of kinks and steps on these sites [28, 32, 33]. It has also been shown that atoms and clusters are particularly sensitive to ultrashallow features [29, 30], and that the nature and the morphology of the FIB-induced defects can inﬂuence the morphology of the nanostructures [29]. For electrochemical deposition, it is only the size of the defects created by FIB that controls the lateral size of the selectively grown structures [29]. However, in other cases the selective growth of nanostructures on FIB patterns may be inﬂuenced by the surfactant eﬀect of the implanted ions [33] or by strain caused by implantation. Moreover, kinetic processes will also be important in controlling the surface organization of nanostructures. For example, atom surface diﬀusion needs to be controlled in order to nucleate dots on desired areas [28, 32, 33]. This can be done by adjusting both the growth temperature and the distance between the pattern features. We conclude from this brief survey that the FIB is a highly versatile tool in nanoscale materials development, both allowing nanostructures to be fabricated directly and also enabling surface modiﬁcation that controls the nucleation sites of subsequently grown nanostructures. But although FIB has been applied in a variety of materials systems, it has not been integrated with multistep processing. The presence of a mask layer, or the need to form deep features, are not compatible with continued processing such as encapsulation of self assembled islands by subsequent epitaxial growth. The best prospects for integrated processing are the use of low ion doses to pattern nanostructure

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arrays by creating small perturbations of a planar surface, and this is the approach we focus on in the subsequent discussion.

14.2 Development of a Technique for Control of Single QD Nucleation Sites Using FIB Patterning The ultimate control of the organization of features such as QDs consists of the ability to deﬁne the nucleation site of every dot on the surface, with nanometer scale precision. Several studies have shown that the nucleation of epitaxial islands is inﬂuenced by atomic scale topography [48–59], chemistry, i.e., impurities [15, 60–72], and strain [59, 73–77]. Thus, the goal is to modify at least one of these parameters such that nucleation is controlled to a degree suﬃcient to localize island formation. The use of low ion doses allows this goal to be attained in the case of Ge island growth on Si(001). As mentioned above, the technique used here involves three steps [40]: (1) FIB implantation of Ga+ ions on the Si(001) surface, at a low dose; (2) postimplantation annealing, and (3) Ge deposition. Typically, an ion beam energy of 25 keV and a beam current of 11 pA are used, with an implantation time (tI ) between 0.1 and 0.2 ms. This gives a total of only ∼ 7000 – 14,000 ions per implanted feature, with typical feature diameters of order 100 nm, as shown in Fig. 14.1. At these low doses, the major eﬀects of the implantation are the formation of an amorphous region and the implantation of Ga+ ions. The modiﬁcation of the surface topography is minimal: the sputter yield of Si(001) for normally incident 25 keV Ga+ is ∼ 5, and therefore less than 1 monolayer (ML) of material is sputtered. Fig. 14.2 shows the topography of the surface after implantation using atomic force microscopy

Fig. 14.1. In-situ plan-view transmission electron microscopy (TEM) image of the Si membrane after focused ion beam (FIB) implantation of Ga using an ion beam energy of 25 keV, a beam current of 11 pA and an implantation time of 0.1 ms per spot. Ga has been implanted in a regular array with a distance between implanted features of ∼ 170 nm. The scale bar is 100 nm. All the TEM images presented in this chapter were acquired using a dark ﬁeld (g, 3 g) weak beam condition with g = 022 or 0¯ 2¯ 2

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(AFM) in air. Each implanted feature actually shows up as a swelling but the Si surface steps are not aﬀected. We believe that the swelling results from a higher oxidation rate on the amorphized regions of Si compared to crystalline Si [30, 78–80] prior to the ex-situ AFM observations. The magnitude of the swelling is about 0.6 nm (∼ 4 atomic ML) and the diameter of the aﬀected areas is 90 – 150 nm. The postimplantation annealing step changes the topography, the strain and the chemistry of the implanted areas. During annealing, the implantationinduced damage is removed, and under appropriate conditions speciﬁc types of surface feature can appear. Fig. 14.3 shows the evolution of implanted zones during annealing. These images were obtained by annealing a sample in an ultra high vacuum (UHV) transmission electron microscope (TEM) [81], details of which will be described below. Under the imaging conditions used, the image contrast is due primarily to lattice displacements (strain or crystal damage). During annealing we see that the centers of the damaged areas do not move but the diameters progressively shrink, demonstrating that the

Fig. 14.2. Ex-situ atomic force microscopy (AFM) (in air) image of the surface of the Si membrane after FIB implantation of Ga using a beam energy of 25 keV, a beam current of 11 pA and an implantation time of 0.1 ms per spot. Ga has been implanted in a regular array with a distance between implanted features of ∼ 170 nm

Fig. 14.3. Subsequent in-situ plan view TEM images of the Si membrane during annealing. a After implantation of Ga with an implantation time of 0.1 ms per spot, T = 20 ◦ C. b After 49 s, increasing T from 20 to 115 ◦ C. c After 117 s, increasing T from 115 to 550 ◦ C. d After 8 s at T = 550 ◦ C. The scale bar is 100 nm

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recrystallization is occurring from the edges to the centers of the damaged areas [40]. If the substrate is annealed for long enough and/or at high enough temperature, the contrast due to the damage vanishes completely, and we observe the formation of contrast made up of bright spots randomly distributed in the implanted areas (Fig. 14.4). This contrast is very stable and does not vanish even for high temperature (T ∼ 750 ◦ C) and long time annealing (t > 30 minutes). We believe that this contrast is due to the formation of Ga clusters beneath the surface [46, 80], leading to local strain in the vicinity of implanted zones. Furthermore, if annealing permits some Ga atoms to gather below the surface in clusters, we might expect that it should also allow surface segregation of part of the implanted Ga atoms, which will locally change the surface chemistry. Ga is expected to segregate mainly to locations where the implant damage intersects the surface, as damaged regions constitute a favored channel for diﬀusion. But even though Ga is known to segregate to the Si surface [61, 82, 83], the detection of Ga in these implanted areas is experimentally diﬃcult due to the low number of implanted atoms. In the ﬁnal step, Ge is deposited on the patterned substrate using conventional growth techniques. We show below that both UHV chemical vapor deposition (CVD) and molecular beam epitaxy (MBE) produce arrays of patterned islands. The three steps of implantation, annealing and deposition must be performed in the same interconnected UHV system to preserve the epitaxial relationship between the QDs and the substrate. Given such a UHV system, complex QD arrays are easily created using software-generated patterns, and as we will show below, it is possible to achieve good selectivity with one QD on each feature and none in between. The main limiting factor of the technique is the lateral size of the ion beam, since the minimum distance between two dots is equal to the diameter of the implanted zones. This diameter is of the order of 100 nm in the system used in our studies, although smaller spot diameters down to ∼10 nm are available in commercial FIB systems. 14.2.1 Experimental Details The technique was developed in a UHV system composed of several connected chambers coupled to a modiﬁed 300 kV Hitachi UHV-9000 transmission elec-

Fig. 14.4. In-situ plan-view TEM image of the Si membrane after FIB implantation of Ga (tI = 0.1 ms/spot) and annealing at 550 ◦ C for 1 minute. The scale bar is 100 nm

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tron microscope [40, 81]. The system base pressure, including that within the TEM polepiece, is ∼ 2 × 10−10 Torr. One chamber is equipped with a Ga+ source FIB, while another contains an evaporator for MBE growth of Ge, with a growth rate calibrated using medium energy ion scattering. UHV CVD growth of Ge is actually carried out in situ in the TEM by injecting digermane gas through a leak valve in the column. This allows the formation of the Ge islands to be observed in real time during growth. The sample can be heated (by direct current) in the TEM or in any of the chambers, and can be transferred between locations without breaking vacuum. The samples were made from commercial Si(001) silicon-on-insulator wafers composed of a 220 nm device layer and a 2 μm SiO2 layer on a 750 μm Si substrate. The samples were mechanically thinned and chemically etched from the back in order to create a central area where only the 220 nm thick Si layer remains. This Si membrane is transparent to the electron beam and is used for TEM observations in plan view. After loading into the UHV system, each sample was ﬂashed at 1250 ◦ C to remove the native oxide, and the temperature-current relationship was calibrated using optical and infrared pyrometers. The sample was transferred to the FIB chamber for implantation, then into the TEM column so that the annealing could be followed in real time, and ﬁnally the Ge was deposited, either by CVD in the TEM column (allowing direct observation of island growth), or by MBE in the evaporator chamber (allowing imaging that is postgrowth but still under UHV). 14.2.2 Inﬂuence of the Implantation Dose and the Subsequent Annealing Conditions We have seen that the initial implantation locally degrades the Si crystallinity. The subsequent annealing is a key step because it allows the diﬀusion of Si surface atoms as well as the reorganization (recrystallization) of subsurface atoms in the amorphous region. The diﬀusion of surface atoms is particularly important because, as we show below, it leads to particular types of surface topography that depend on the annealing parameters (temperature and time) and the implantation dose. Speciﬁc surface features are needed for the control of QD nucleation sites. Real time observation of postimplantation annealing in situ in the UHV TEM (Fig. 14.3) showed that the recrystallization velocity in the FIBimplanted spots is suﬃciently rapid at temperatures above 550 ◦ C that it becomes diﬃcult to control the recrystallization process at these very tiny dimensions. Instead, the optimal conditions in our experiments are an anneal at 550 ◦ C for 1 min. This provides a minimum temperature and time budget that results in both useful surface features and good homogeneity and reproducibility of features in the array. The surface topography resulting from this anneal is shown in Fig. 14.5c,d, and consists of small pits a few nanometers deep and tens of nanometers in diameter. More details will be given below. This anneal also forms Ga clusters underneath the surface (Fig. 14.4). Thus

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both strain and topography are modiﬁed at each implanted zone, although it is diﬃcult to determine experimentally the location of Ga atoms that have segregated to the surface during the anneal. Annealing at higher temperatures or for longer times than the optimal values given above results in a complete recovery of the crystallinity and surface ﬂatness. Figure 14.6 shows the surface topography after patterning (as in Figs. 14.1 and 14.2) and annealing at 650 ◦ C for 15 min. The surface is completely ﬂat, the amorphous regions have disappeared and smooth surface steps are visible. Plan-view TEM images of the same sample (Fig. 14.7) show that although the patterning-induced topography has been removed during annealing, Ga clusters are still present below the surface leading to strain

Fig. 14.5. Two-dimensional (2D) topography of the surface of a Si membrane. Data is acquired by exsitu AFM in air after implantation and annealing in ultra high vacuum (UHV). a After Ga implantation, tI = 0.1 ms/spot. b After Ga implantation with tI = 0.2 ms/spot, and annealing 1 min at 500 ◦ C and 11 s at 550 ◦ C. c After Ga implantation with tI = 0.2 ms/spot, and annealing at 550 ◦ C for 1 min. d After Ga implantation with tI = 0.1 ms/spot, and annealing at 550 ◦ C for 1 min

Fig. 14.6. Ex-situ AFM (in air) image of the surface of the Si membrane after FIB implantation of Ga with tI = 0.1 ms/spot and annealing at 650 ◦ C for 15 min. Ga has been implanted in the same array geometry as on Fig. 14.2

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localized at the implanted regions. Furthermore, for this type of annealing, Ga atoms that have segregated to the surface are expected to spread on the surface by diﬀusion, as well as to desorb as the temperature approaches 700 ◦ C [84]. In consequence, high temperature and long time annealing allows engineering of the strain in the vicinity of the substrate surface, while lower temperature or shorter time annealing allows both the surface topography and chemistry to be modiﬁed also. The exact surface topography that forms at each implanted feature following annealing is a complex function of implant dose, annealing temperature and annealing time. Three major classes of topography have so far been obtained. For low temperature annealing (T < 550 ◦ C), where the recrystallization velocity is slow, annealing can be stopped before the formation of the Ga clusters (Fig. 14.3). In this case, which we refer to as a type I pattern, the surface topography on the implanted area is a single irregular pit (Figs. 14.5b and 14.8a) for a range of implant times (0.1 and 0.2 ms are shown). The diameters (L) of these pits are between 30 and 50 nm depending on the temperature and time. The slope of their walls (12 ≤ α ≤ 16◦ ) is observed to be independent of both L and the implantation dose. Their depth (3 ≤ H ≤ 5 nm) is also independent of L, but deeper holes are generally obtained with higher implantation doses. A diﬀerent pattern (type II)

Fig. 14.7. In-situ plan-view TEM image of the Si membrane after FIB implantation of Ga (tI = 0.1 ms/spot) and annealing at 650 ◦ C for 15 min. The scale bar is 100 nm

Fig. 14.8. Topography of the surface of the Si membrane acquired by ex situ AFM in air on implanted areas. a After Ga implantation with tI = 0.2 ms/spot, and annealing 1 min at 500 ◦ C and 11 s at 550 ◦ C: pattern type I. b After Ga implantation with tI = 0.1 ms/spot, and annealing at 550 ◦ C for 1 min: pattern type II. c After Ga implantation with tI = 0.2 ms/spot, and annealing at 550 ◦ C for 1 min: pattern type III. The scale bar is 50 nm

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is obtained by using an implantation time of 0.1 ms followed by an anneal at 550 ◦ C for 1 min. Pits still form but are smaller (Figs. 14.5d and 14.8b), with a diameter of 10 – 30 nm and a depth of ∼ 1 nm. In this case α varies between 3 and 10◦ . Finally, type III patterns are formed with an implantation time of 0.2 ms followed by the same annealing sequence. This pattern is more complex (Figs. 14.5c and 14.8c) and consists of a central plateau surrounded by a trench with a depth of ∼ 0.5 nm. Typically the diameter of the plateau is about 30 nm and the slope of the trench walls is between 3 and 10◦ .

14.3 Ge Deposition on Patterned Si(001) Substrates Having developed the ability to modify surface topography controllably, we now consider the formation of islands on these patterned surfaces. We will discuss the selectivity of nucleation, the number of islands forming on each site, and the shape and size of the islands that form on the patterned surface. 14.3.1 Preferential Ge QD Nucleation on Implanted Sites Ge deposition performed on patterned substrates after high temperature and long time annealing (for example, 650 ◦ C for 15 min) leads to the formation of conventional Ge islands with a random surface distribution. This is unsurprising since, as mentioned above, such annealing conditions result in a complete recovery of the surface ﬂatness. Preferential nucleation on implanted areas is observed only if a surface nanotopography exists after the anneal. Figure 14.9 shows that in this case, the island nucleation is actually very well controlled. In other words, islands are not seen away from the patterned features, even if the features are quite far apart. The reason for this is that the wetting layer thickness, in other words the thickness at which the Ge layer changes from a planar to an islanded form, is actually reduced on patterned surfaces. Thus Ge islands nucleate on patterned features after deposition of a layer that is thinner than is required for island formation on the unpatterned Si surface. For example, for Ge deposition performed by MBE with a growth temperaA/s, islands are seen after only ture (TG ) of 550 ◦ C and a growth rate of 0.02 ˚ 1.5 – 2.0 ML Ge has been deposited on a type II patterned substrate with a feature surface density (df ) of ∼ 4.4 × 109 cm−2 . By comparison, island formation does not occur until 3.5 – 4 ML Ge have been deposited on the unmodiﬁed Si(001) surface. It is interesting to note that the wetting layer thickness decreases when the surface density of pattern features increases. For the same deposition conditions and substrate type as above, but with a lower density pattern of df ∼ 1.1 × 109 cm−2 , the wetting layer thickness was found to have an intermediate value of 2 – 2.5 ML. We believe that these results reﬂect a kinetic component to the wetting layer, but whatever the exact mechanism, this reduction in wetting layer thickness is a key feature of

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Fig. 14.9. Ex-situ AFM (in air) image of the surface of the Si membrane after FIB implantation of Ga with tI = 0.2 ms/spot, annealing at 550 ◦ C for 1 min, and Ge UHV CVD with TG = 480 ◦ C, PG = 3.0 × 10−8 Torr

the FIB patterning method, allowing quite selective nucleation control under optimal conditions. In-situ observations of Ge growth show that island nucleation does not occur simultaneously on all the patterned features. Moreover, as soon as dislocations form in some islands, we ﬁnd that further nucleation is inhibited. Dislocated (strain-relaxed) islands presumably act as strong sinks for the incoming ﬂux [85], so at this point the number of islands on the surface stays constant and instead the average island diameter increases with deposition time. This produces signiﬁcant variations in island sizes and strain states, as well as a low “pattern feature ﬁlling rate” (rF ). For example, Fig. 14.10 shows four adjacent implanted areas, of which two contain dislocated islands, the third has an undislocated island, and the fourth is free of islands. This demonstrates the need to identify the optimum growth conditions that promote the best homogeneity of the Ge QD array, so that we can produce arrays of undislocated islands with similar sizes and achieve complete ﬁlling of all the pattern features. The control of growth kinetics is the key to realizing this goal. Atom surface diﬀusion lengths must be adjusted with respect to the distance between neighboring pattern features in order to improve island homogeneity and feature ﬁlling rate. For example, for a distance between features of 170 nm (pattern type II, df ∼ 3.9 × 109 cm−2 ), Ge UHV CVD at 550 ◦ C with a growth pressure (PG ) of 3 × 10−8 Torr and a growth time (tG ) of 5 min leads to the formation of one large, dislocated island per occupied feature with a lateral island size L ∼ 55 nm and a feature ﬁlling rate of only 16%. The same experiment performed at 500 ◦ C leads to the formation of one small, undislocated island per occupied feature with L ∼ 24 nm and rF ∼ 90%. The growth rate is also an important parameter in controlling island homogeneity and feature ﬁlling rate. For UHV CVD, better results have been obtained for higher growth rate, i.e., higher digermane pressure. If PG = 1.3 × 10−8 Torr and TG = 500 ◦ C, only 10% of the pattern features

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A. Portavoce, R. Hull, F.M. Ross Fig. 14.10. In-situ plan-view TEM image of the Si membrane after FIB implantation of Ga with tI = 0.1 ms/spot, annealing at 550 ◦ C for 1 min, and deposition of 2 monolayers (ML) of Ge by molecular beam epitaxy (MBE) at TG = 550 ◦ C. The scale bar is 100 nm

Fig. 14.11. Ge island nucleation on a pattern of type II. a Number of Ge islands per feature (N ) versus the feature ﬁlling rate of the pattern (rF ). b In-situ plan-view TEM image of the Si membrane after Ge UHV chemical vapor deposition (CVD) (TG = 500 ◦ C, PG = 1.3 × 10−7 Torr and tG = 116 s), for the case of a feature ﬁlling rate of 100%. The scale bar is 100 nm

(pattern type III, df ∼ 3.9 × 109 cm−2 ) are ﬁlled before dislocations appear in some islands. For the same patterned substrate at the same temperature but with PG = 1.3 × 10−7 Torr, 98% of the features can be ﬁlled before dislocations appear. The type of pattern that is created by the implantation and annealing steps has a strong inﬂuence on the way the features are ﬁlled. For type II substrates, we do not see more than one island per feature (Fig. 14.11). But for type III substrates, the nucleation of an additional island on some of the features starts even before a ﬁlling rate of 100% is reached (Fig. 14.12). For both types of substrate, depending on the distance between features, Ge islands can nucleate between the pattern features after reaching a ﬁlling rate of 100% and before dislocations appear in the islands. For exam-

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Fig. 14.12. Ge island nucleation on a pattern of type III. a Average number of Ge islands per feature (N ) versus the feature ﬁlling rate of the pattern (rF ). b Insitu plan-view TEM image of the Si membrane after Ge UHV CVD (TG = 500 ◦ C, PG = 3.6 × 10−7 Torr and tG = 59 s), for the case of a feature ﬁlling rate of 100%. The scale bar is 100 nm

Fig. 14.13. In-situ plan-view TEM image of the Si membrane after deposition of Ge by UHV CVD (TG = 500 ◦ C, PG = 3.6 × 10−7 Torr and tG = 59 s) on a sample where two patterns of type III with diﬀerent surface densities of feature (df ) have been created. a df = 2.3 × 108 cm−2 , arrows show islands nucleated on pattern features. b df = 3.4 × 109 cm−2 . The scale bar is 100 nm. The nucleation of Ge islands between features is observed only for the pattern of lower feature density

ple, Fig. 14.13 presents two areas of a same sample with diﬀerent densities of type III features. For the lower density, after nucleation of several islands on each feature, islands nucleate between features before any dislocations appear. For the higher density, several islands have nucleated on each feature but no island has nucleated in between. Figure 14.14 shows the eﬀects of growth temperature and pressure during UHV CVD on both the feature ﬁlling rate and the nucleation time (tN , which is deﬁned as the time diﬀerence between the beginning of the growth and the beginning of observable island nucleation) versus the substrate type. For these experiments two areas of the same sample were patterned using implan-

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Fig. 14.14. Comparison between Ge island nucleation variations on pattern of type II (tI = 0.1 ms/spot, black histogram) and pattern of type III (tI = 0.2 ms/spot, gray histogram) vs. Ge UHV CVD temperature (TG ) and pressure (PG ). a Variation of the average pattern feature ﬁlling fraction versus growth temperature (PG = 3.0 × 10−8 Torr). b Variations of the island nucleation time (tN ) versus growth temperature (PG = 3.0 × 10−8 Torr). c Variations of the average pattern feature ﬁlling rate versus growth pressure (TG = 500 ◦ C). d Variations of the island nucleation time versus growth pressure (TG = 500 ◦ C). e Proportion of features with diﬀerent numbers of associated islands when TG = 500 ◦ C, PG = 3.0 × 10−8 Torr and tG = 300 s. f Proportion of features with diﬀerent numbers of associated islands when TG = 500 ◦ C, PG = 1.4 × 10−7 Torr and tG = 105 s

tation times of tI = 0.1 ms (producing type II features) and tI = 0.2 ms (type III) with df ∼ 3.9 × 109 cm−2 for both cases. The sample was then annealed at 550 ◦ C for 1 min before Ge UHV CVD. Independent of the growth conditions, the type III substrate always shows a higher feature ﬁlling rate. The nucleation time variation exhibits a complex behavior. Increasing growth pressure leads to a decrease of the nucleation time on both types of substrate, with a greater eﬀect observed on the type III sub-

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strate. Temperature shows an even greater eﬀect on nucleation time. For PG = 3 × 10−8 Torr, an increase of the growth temperature leads to an increase of tN on type II substrates, but a decrease of tN on type III substrates. It is clear from all the results presented in this section that the growth of Ge islands on a patterned substrate depends sensitively on several parameters, most importantly the substrate type, the distance between features, the growth rate or pressure and the temperature. However, it does appear possible to tune these parameters to deﬁne a process window that will lead to a high feature ﬁlling rate, with a single undislocated island on each feature and almost no islands in between. 14.3.2 Characteristics of Ge QDs Nucleated on Implanted Sites We now consider the size and shape of the islands that form on the patterned features. It is well known that Ge deposition on an unmodiﬁed Si(001) surface results in the formation of two types of islands, the hut and the dome clusters, as discussed in Chap. I.1 and shown in Fig. 14.15a,b. Deﬁning the island aspect ratio as the ratio between the height (h) and the lateral size (L) of islands, huts and domes are found to exhibit constant aspect ratios of ∼ 0.1 and 0.2, respectively [52, 57, 85–93]. Dislocations are observed only in dome clusters (Fig. 14.15c) because huts transform into domes (rather than forming dislocations) when they reach a critical elastic energy [91, 94– 99] during growth or annealing. The critical lateral size (LC ) of a dome at which the ﬁrst dislocation nucleates depends on its composition, which in turn depends on the degree of Si/Ge intermixing. Generally LC can vary from ∼ 30 to ∼ 90 nm depending on the temperature during growth and postgrowth annealing [100]. The distance separating two dislocations within a single dome is about 30 nm (Fig. 14.15c). We ﬁnd that islands nucleating on Ga+ FIB-patterned features exhibit morphologies that are diﬀerent from the expected huts and domes. We ﬁrst note that, in contrast to Ge growth on the unmodiﬁed Si(001) surface, only one type of coherent island forms on the pattern features. The shape is different from the hut or dome shape (Fig. 14.16) and the aspect ratio is not constant, but varies with L (Fig. 14.17). Islands nucleated on implanted areas exhibit a smaller critical lateral size LC ∼ 24 – 27 nm at which dislocations nucleate, and this size is independent of growth conditions. Furthermore, the distance between dislocations within an island is about 6 times smaller (3.6 – 4.8 nm) than in a conventional dome. The aspect ratio of these islands also depends on whether they are dislocated or not (Fig. 14.17). Typically, we ﬁnd that the aspect ratio of undislocated islands is lower than the aspect ratio of conventional huts, but once dislocations appear, the aspect ratio increases with L up to a maximum (∼ 0.3) which is higher than the dome aspect ratio. Then, when islands grow very large (L > ∼ 75 nm), the aspect ratio decreases again to a value below 0.2. The fact that islands on pattern features

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Fig. 14.15. In-situ plan-view TEM images of Ge islands on the unaltered Si(001) surface. a Hut clusters. b Dome clusters. c Dislocated dome cluster, image obtained by superimposition of two images obtained in dark ﬁeld (g, 3 g) weak beam condition with g = 220 and g = ¯ 220

Fig. 14.16. In-situ plan-view TEM images of Ge islands nucleated on features of type II. a Nondislocated cluster. b,c Dislocated clusters. The scale bar is 100 nm

Fig. 14.17. Variation of the height (h) of Ge islands nucleated on pattern features of type II and III versus their lateral size (L). h and L were measured by AFM in air from three diﬀerent samples. The two solid lines correspond to the relationship of h and L expected for hut (h/L = 0.1) and dome (h/L = 0.2) islands on unmodiﬁed Si(100) surfaces. Dotted lines corresponding to aspect ratios of 0.06 and 0.3 are also shown for comparison. The shaded region corresponds to lateral sizes for which nucleation of dislocation in the islands has been observed during in-situ TEM deposition

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exhibit both a smaller LC and a smaller distance between dislocations suggests that the elastic strain relaxation is lower in these islands than in dome clusters. This could be explained either by a diﬀerence in Ge content arising from the eﬀects of Ga on interdiﬀusion, or by the observed diﬀerences in shape. It has previously been reported that island strain relaxation depends on shape [101–109], with diﬀerent facet angles leading to diﬀerent levels of relaxation and diﬀerent energy barriers for nucleation of dislocations. In applications of QDs, small, compact three-dimensional (3D) islands rather than larger, shallower islands are advantageous in creating quantum eﬀects in all dimensions. The changes we have described in Ge island shapes associated with FIB patterning may therefore be useful in designing novel devices, and FIB patterning using diﬀerent ion species might be promising in allowing a greater range of island shapes to be formed.

14.4 The Mechanism of Preferential Nucleation on the Implanted Sites So far we have described the FIB patterning technique in detail and have considered the optimal conditions under which a single island may be formed on each patterned feature. By comparing island localization after MBE and CVD, and by examining the eﬀects of diﬀerent surface treatment, we now consider the possible physical processes that lead to the selective nucleation, which are spatially varying surface reactivity, strain, and surface topography. 14.4.1 The Inﬂuence of Surface Reactivity During chemical vapor deposition, molecules in the gas phase undergo multiple processes and reactions which ﬁnally result in the deposition of atoms on the surface of the substrate [110–116]. These chemical reactions are thermally activated and can be divided into two groups: reactions occurring between molecules in the gas phase, and reactions occurring between the gas-phase molecules and the growing surface of the crystal. Usually, gas-phase decomposition is neglected for UHV CVD, as the gas molecule interactions are weak due to the very low growth pressure (< 10−6 Torr) [110, 117]. Instead, two consecutive processes are considered to describe growth: (1) the adsorption of gas molecules (precursors) on the surface, and (2) the reaction of these adsorbed molecules with the surface atoms (surface decomposition). These two processes are greatly aﬀected by growth conditions such as temperature and pressure, as well as by surface properties such as the surface atom species and the nature of their chemical bonds. For example, at higher temperatures, the adsorption rate of molecules on the surface is expected to decrease, but conversely, the decomposition rate of molecules adsorbed on the surface is expected to increase [110]. The reaction rate of precursor decomposition may

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also depend on surface atom density, surface step (or defect) density and surface reconstruction [112, 118]. The notion of diﬀerential reaction rates of molecule decomposition on a surface as a function of its local properties leads to the concept of growth selectivity. Indeed the rate of a surface reaction can be increased, decreased or even extinguished by modifying the properties of the surface, in other words by changing the surface reactivity. For example, the reactivity of the oxidized Si surface being very low, the location of Ge atom deposition by CVD can be controlled by using Si substrates where areas on which Ge deposition is not desired are oxidized [118–128]. Ge is then selectively deposited on the nonoxidized surface areas. It is of course possible that Ga coverage may aﬀect surface reactivity, leading to spatial variations in the deposition of Ge. We examined this potential mechanism by comparing UHV CVD of Ge with solid source MBE on Ga+ FIB patterned substrates. We also deposited Ge onto substrates that had been covered by 1 ML of Ga, to examine the eﬀects of Ge independently of other phenomena associated with implantation [100]. In MBE, atom deposition does not result from chemical reaction, but rather from a condensation process. The concept of surface reactivity is thus irrelevant and is replaced by the concept of atom sticking coeﬃcient, which describes the probability of adsorption of a speciﬁc atom on the surface for particular growth conditions. We found that Ge grown by MBE on patterned substrates exhibits the same characteristics as Ge grown by UHV CVD (Fig. 14.18). Just as for UHV CVD growth described previously, islands nucleate on pattern features after MBE only if a surface topography exists after post-implantation annealing. MBE-grown islands also nucleate after the deposition of a thinner wetting layer than on the conventional Si(001) surface, and they exhibit the same shape and same critical lateral size for dislocation nucleation as in UHV CVD deposition. The same feature ﬁlling rates were observed by UHV CVD and MBE for the same growth conditions on the same patterned substrates. As a result, we conclude that Ge island nucleation on pattern features does not result from an increase in surface reactivity on implanted areas compared to unmodiﬁed surface areas during UHV CVD. Our studies [100] of Ge UHV CVD on Si(001) substrates covered by 1 ML of Ga show, in fact, that surface Ga atoms actually inhibit Ge deposition, presumably by passivating Si surface dangling bonds. Implanted Ga presumably shares the same passivation eﬀect and a decrease in surface reactivity. Ge can thus be deposited on the surface by UHV CVD only for growth temperatures high enough to allow Ga desorption. Consequently, Ga atoms segregated to the surface of implanted areas cannot act as preferential regions for Ge deposition. 14.4.2 The Inﬂuence of Strain Ge growth on Si substrates occurs via the Stranski–Krastanow growth mode (Chap. I.1). The Ge surface energy being lower than that of Si, Ge ﬁrst forms a ﬂat wetting layer. As growth continues, 3D Ge islands form on top

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Fig. 14.18. In-situ plan-view TEM images of the Si membrane after FIB implantation of Ga with tI = 0.1 ms/spot, annealing at 550 ◦ C for 1 min, and: a Ge UHV CVD with TG = 550 ◦ C, PG = 5 × 10−8 Torr, tG = 120 s; b,c deposition of 2 ML Ge by MBE at TG = 550 ◦ C. The scale bar is 100 nm

of this wetting layer [54, 58, 74, 89, 129–132]. The 4.2% lattice mismatch between Si and Ge is usually presented as the driving force for the nucleation of 3D islands on Si(001) [74, 86, 87, 132–140] (although controversy still exists). In this thermodynamic point of view, the elastic strain energy stored in the Ge wetting layer increases with its thickness, up to a critical value for which the formation of 3D islands allows a reduction in strain energy that is greater than the increase in surface energy. Strain thus plays a central role in the mechanism of Ge island nucleation on the unmodiﬁed Si surface. On Si(001) the critical strain energy for 3D Ge island nucleation corresponds to a wetting layer thickness of 3 – 4 ML [52, 87, 141–145]. Local strain ﬁelds are known to inﬂuence the location of Ge island nucleation, especially in the case of 3D stacking of self-organized island layers [146–155] (Chap. I.4). It has also been shown that the strain ﬁelds resulting from the formation of SiO2 precipitates below the Si(001) surface, after oxygen ion implantation and annealing, allows localization of Ge island nucleation on top of the buried clusters [156].

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In the case of FIB patterned surfaces, TEM images (Figs. 14.4 and 14.7) clearly showed that the lattice of type II and III substrates is strained in the vicinity of implanted areas, probably because of the formation of Ga clusters below the surface. These localized strain ﬁelds might be expected to inﬂuence island nucleation on Ga+ FIB implanted areas. However Ge growth performed on FIB patterned substrates after annealing at high temperature (such that none of the nanotopography that localizes nucleation, as described earlier, remains) produces islands with a random surface distribution (Fig. 14.19), even though strain contrast is still observed by TEM in the implanted regions. Furthermore, Ge islands can be localized on nanodepressions of type I, for which no strain contrast is observed in the TEM. These results demonstrate that the strain ﬁelds resulting from the Ga clusters below the surface do not play a dominant role in the nucleation process, and by themselves are not suﬃcient to localize nucleation under the conditions we have studied. The fact that the wetting layer thickness varies with the surface density of the patterned features also suggests that the surface topography of patterned substrates plays a more important role on the wetting layer thickness than the elastic strain energy stored in the wetting layer. Finally, if the islanded Ge on a patterned substrate is annealed after growth, the islands that had nucleated on the features of the pattern actually disappear and a ﬂat surface is recovered. This demonstrates that Ge islands on the features are energetically less stable than a ﬂat layer having a thickness smaller than the usual (i.e., that observed on the unmodiﬁed surface) Ge wetting layer, suggesting that arrays of ordered Ge clusters on the implanted sites are themselves in a metastable state. 14.4.3 The Inﬂuence of Ga Atoms Segregated to the Surface During postimplantation annealing, part of the implanted Ga can diﬀuse to the surface and eventually spread out. Direct detection of these atoms is difﬁcult due to the low number of implanted ions. Even if all the implanted Ga atoms segregated to the surface, the surface density of Ga atoms for a feature surface density of 4 × 109 cm−2 with an implantation time of 0.1 ms is only

Fig. 14.19. In-situ plan-view TEM image of the Si membrane after FIB implantation of Ga with tI = 0.1 ms/spot, annealing at 650 ◦ C for 15 min, and Ge UHV CVD with TG = 550 ◦ C, PG = 5 × 10−8 Torr and tG = 240 s. The scale bar is 100 nm

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about 2.8 × 1013 cm−2 , which would correspond to a uniform surface coverage of ∼ 0.04 ML. However, even though the concentration and the location of Ga on the patterned surface is unknown, the inﬂuence of such Ga atoms on Ge island nucleation can be inferred by comparing such islands with Ge islands (formed by UHV CVD) on Si substrates covered with 1 ML of Ga. Islands forming on such Ga-terminated surfaces exhibit characteristics that are similar to Ge islands nucleated on Ga implanted areas [100]. Their aspect ratio follows a similar behavior (compare Fig. 14.20 to Fig. 14.17) and their critical lateral size for dislocation nucleation (20 – 24 nm) is about the same. Furthermore, the distance between dislocations in an island grown on a Ga-terminated surface is again about 3 – 5 nm. As a ﬁnal point of similarity, we note that dislocated Ge islands grown on Si(001) substrates terminated by a Ga ML have a square basal shape (Fig. 14.21). On FIB patterned substrates, the larger dislocated islands exhibit the same basal shape (Fig. 14.22) (although smaller ones show less regular basal shapes). These similarities between Ge islands on Ga-implanted areas and Ge islands on substrates terminated by a Ga ML suggest that at least some of the implanted ions eﬀectively segregate to the surface and aﬀect the characteristics of the islands. The change in surface energetics arising from surface Ga may be expected to promote the formation of facets on the islands that would not be seen for islands formed on a surface without Ga, and the facets present in turn aﬀect the aspect ratio, basal shape and dislocation introduction. The results also show that variations in surface Ga concentration are not required to localize Ge nucleation during UHV CVD. But this does not exclude the possibility that Ga can play a role in localizing nucleation. For example, it should be noted that Ga reduces the mobility of Ge atoms on

Fig. 14.20. Variation of the height (h) of Ge islands nucleated on the Si(001) surface capped with 1 ML Ga versus their lateral size (L). h and L were measured by AFM in air for two diﬀerent Ge UHV CVD conditions. The two solid lines correspond to the relationship of h and L expected for hut (h/L = 0.1) and dome (h/L = 0.2) islands. Dotted lines corresponding to aspect ratios of 0.05 and 0.3 are also shown for comparison. Compare to Fig. 14.17

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A. Portavoce, R. Hull, F.M. Ross Fig. 14.21. In-situ plan-view TEM image of dislocated Ge islands nucleated on a Si(001) surface caped with 1 ML Ga. Ge has been deposited by UHV CVD. The scale bar is 100 nm

Fig. 14.22. In-situ plan-view TEM image of dislocated Ge islands nucleated on a type II FIB implant pattern. Ge has been deposited by UHV CVD. The scale bar is 100 nm

Si(001) [100], such that atom diﬀusion lengths may be locally reduced on pattern features. 14.4.4 The Inﬂuence of Surface Topography The surface topography that forms at the pattern features after annealing seems to be the main factor controlling island nucleation. If, after postimplantation annealing, the ﬂatness of the surface is completely recovered, conventional huts and domes nucleate with a random distribution. But if a nanotopography remains on implanted areas after annealing, island nucleation is localized to the pattern features. The ﬁlling rate of features then depends upon their density, and the local thickness of the wetting layer depends both on the density and on the topography type (recall that in Fig. 14.14d, under certain growth conditions, islands nucleate on type III features before type II). Furthermore, the topography type inﬂuences the nucleation sites and the number of islands per feature. The details of this are very interesting. On type I nanotopographies, islands have been observed to nucleate on three diﬀerent locations (Fig. 14.23): on the edge of the pit (Fig. 14.23b), on the walls of the pit (Fig. 14.23c) and in the center of the pit (Fig. 14.23d). Two islands can nucleate at two diﬀerent nucleation centers on one feature (Fig. 14.24a), but they quickly coalesce to form a single island straddling the edge of the pit (Fig. 14.24b). For type II nanotopographies only one nucleation site has been observed, the edge of the pit (Fig. 14.25a). Only one island per feature has been observed on this type of topography, but because of the small size

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Fig. 14.23. Ex-situ AFM (in air) measurements after FIB implantation of Ga with tI = 0.2 ms/spot, annealing 1 min at 500 ◦ C and 11 s at 550 ◦ C, and Ge UHV CVD with TG = 500 ◦ C, PG = 2.3 × 10−8 Torr and tG = 780 s. a Surface topography. b Proﬁle of an island located on the border of a hole. c Proﬁle of an island located on the wall of a hole. d Proﬁle of an island located in the middle of a hole

Fig. 14.24. Ex-situ AFM (in air) images of Ge islands nucleated on features of type I. a Two islands nucleated on the same feature, one on the wall (arrow 1 ) and one on the border of the hole (arrow 2 ). b Larger island located half in the hole and half on the border of the hole Fig. 14.25. a,b Exsitu AFM (in air) images of Ge islands nucleated on features of type II

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Fig. 14.26. Ex-situ AFM (in air) images of a Ge island nucleated on a feature of type III. a Plan view. b 2D proﬁle

Fig. 14.27. a,b Ex-situ AFM (in air) images of Ge islands nucleated on features of type III

of the pits compared to the islands, it is diﬃcult to conclude that the single islands result from a single nucleation event or the coalescence of several nuclei (e.g., Fig. 14.25b). On type III substrates, all the islands nucleate on the interior mesa (Fig. 14.26). If island nucleation takes place in the middle of this mesa, it is more probable that the ﬁnal conﬁguration comprises only one island centered on the feature (Fig. 14.27a). If islands nucleate on the edge of the mesa, they are more likely to grow towards the outside of the trench (Fig. 14.27b). Several islands can then nucleate on the same mesa without coalescing, which leads to the formation of multiple islands on the same feature (Fig. 14.27b,c). The diﬀerence in island nucleation between features of types I and II seems to be mainly due to the feature size. Indeed, the side wall angle α is not very diﬀerent for topographies of types I and II, but islands nucleate on the walls and in the middle of the pits only for type I. This suggests that for type II features, islands do not form in the middle of the pits or on the walls because of the smaller pit diameter. The nucleation mechanism on type III substrates seems to be more complex. Each type III feature oﬀers several nucleation sites, and multiple islands that form on the same feature do not necessarily coalesce. The surface of the mesa of type III features is rough and not symmetric. This may lead to a reduced atom mobility on the mesa that

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can explain why one mesa oﬀers several nucleation sites, and that nuclei on this mesa do not coalesce as easily as for features of type I and II.

14.5 Conclusions The patterning technique described in this chapter, which consists of the implantation of low dose patterns of Ga+ ions by a focused ion beam into Si(001), followed by annealing and then Ge deposition, permits a new level of control of Ge island nucleation sites. The location of each island on the surface is controlled at a scale approaching the nanometer and the density of islands can be adjusted as desired. Indeed, islands nucleate on patterned substrates before the usual Ge hut and dome clusters form on unmodiﬁed regions, which allows good control of island density and location without formation of extraneous islands. Figure 14.28 illustrates the degree of control of Ge island positioning that can be attained on Si(001) substrates. Fig-

Fig. 14.28. Ex-situ AFM (in air) image of the surface of the Si membrane after Ge UHV CVD. a Ge deposition on a regular array of type III features, TG = 500 ◦ C, PG = 2.0 × 10−8 Torr and tG = 600 s. b Ge deposition on a sample with two regular array of type II feature with diﬀerent densities, TG = 500 ◦ C, PG = 1.3 × 10−7 Torr and tG = 120 s. c and d Ge deposition on a pattern of type III imitating a (quantum cellular automata) QCA island organization, TG = 480 ◦ C, PG = 3.0 × 10−8 Torr and tG = 600 s

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ure 14.28a,b shows two areas with diﬀerent island densities and Fig. 14.28c,d shows a complex pattern of islands. In terms of the growth mechanism, on such patterned substrates the strain eﬀect on island nucleation appears weak. Instead, a localized nanotopography that forms during post implantation annealing is the principal factor controlling island nucleation. Islands nucleated on the pattern features are not energetically stable and their nucleation can be controlled by adjusting the kinetic parameters of deposition. For these reasons, a similar degree of control of island nucleation may be possible for materials other than Si/Ge using the same patterning technique. The maximum island surface density and the minimum distance between islands seem to be limited only by the lateral size of the implanted zones. The possibility of controlling island density and localization using this technique is expected to ﬁnd application in the realization of future devices based on nanoisland engineering. For example, the control of island density and spacing could be useful in fabricating nanocrystal memories with a good reproducibility. The ability to nucleate an island only on a desired location can be useful in optoelectronic applications, with the possibility of precisely positioning islands in waveguides or resonant cavities of photonic structures. Novel nanoelectronic architectures such as quantum cellular automata form another exciting area for future investigation. Acknowledgement. The authors would like to thank Mark C. Reuter for technical help and M. Kammler for useful discussions. This work was supported by a National Science Foundation Materials Research Science and Engineering Center, the “Center for Nanoscopic Materials Design”, at the University of Virginia.

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15 Ge Nanodroplets Self-Assembly on Focused Ion Beam Patterned Substrates I. Berbezier, A. Karmous, and A. Ronda L2MP UMR CNRS 6137, Polytech’Marseille - Technopole de Chˆ ateau Gombert 13451 Marseille Cedex 20, France

15.1 Introduction In this chapter we describe an original approach to the formation of Ge nanocrystals (NCs) on Si(001) based on a combination of focused ion beam (FIB) nanopatterning and spontaneous formation of NCs on the patterns. Two processes have been developed, the ﬁrst on bare Si substrate and the second on SiO2 . In the ﬁrst case, we investigate the eﬀect of experimental parameters (growth temperature, deposited thickness, holes’ pitch and size) on the NCs’ formation. We show that, depending on the growth temperature, two diﬀerent mechanisms of NCs formation occur. They are explained by kinetically limited nucleation at low temperature and stress-induced nucleation at high temperature. In the second case, we investigate the formation of Ge NCs on an ultrathin tunnel oxide by crystallization during thermal annealing of an amorphous Ge layer. We show that NCs’ size and density can be controlled by simply adjusting the initial deposited thickness. When using FIB-patterned SiO2 substrate, the formation of Ge NCs is forced inside the patterns because of both thermodynamics and kinetics. This is explained by the reduction of total surface energy. Periodic two-dimensional (2D) arrays of perfectly ordered, highly dense (> 1011 /cm2 ) and ultrasmall (∼ 20 nm) Ge NCs were achieved.

15.2 Experimental Two diﬀerent ﬂow charts have been designed, one for a Si substrate (ﬂow chart 1) and the other for SiO2 (ﬂow chart 2), for implementation in a metal oxide semiconductor ﬁeld eﬀect transistor (MOSFET) fabrication process. The main steps of the process ﬂow charts for the fabrication of Ge NCs MOSFETs are presented in Fig. 15.1. In a chronological order, they consist of: (1) FIB nanopatterning into 10 × 10 μm2 areas located by lithographic patterns, (2) restoring process of the patterned sample, (3) formation of Ge NCs, (4) fabrication of gate oxide and (5) of metallic contacts. In ﬂow chart 2 (Fig. 15.1b), a sacriﬁcial oxide is fabricated prior to the FIB milling. This oxide is then removed during the restoration step and a highly clean tunnel oxide is fabricated before the Ge deposition.

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Fig. 15.1. Schematic representation of the main steps of processes a ﬂow chart 1 where patterns are performed directly in the Si substrate and b ﬂow chart 2 where patterns are performed in a sacriﬁcial oxide. FIB Focused ion beam, MBE molecular beam epitaxy, T temperature

A ﬁeld oxide, 500 nm thick, was ﬁrst fabricated on all silicon (001) wafers. Using standard optical lithographic techniques and SiO2 wet chemical etching, 10×10 μm2 windows were opened in this oxide in order to easily locate the arrays of holes produced by FIB. In ﬂow chart 1 (Fig. 15.1a), FIB patterning was carried out directly on the (001) Si substrate. Milling investigations were performed with a FEI FIB XL 200 TEM. The FIB instrument is equipped with a gallium liquid metal primary ion gun. With this type of ﬁeld emission gun the minimum beam spot size is ∼10 nm and the source brightness is very high. These features, combined with the ion mass and the energy used, allow a sharp, fast milling of silicon. The FIB process was performed using a dual-beam system in order to minimize the possible gallium implantation during the image grabs required for target area location. Navigation on the wafer surface was done using the scanning electron microscopy (SEM) imaging mode. FIB was used only for milling. The ion source gallium at normal incidence angle with impact energy of 30 keV and 1 pA primary current was used. After FIB milling, the presence of Ga can be detected in the milled areas. The eﬀect of Ga+ implantation has been described elsewhere (Chap. 14) [1]. Good sample cleanliness is achieved when the density of pits measured after the growth of a thick Si buﬀer layer is < 102 /cm2 . The FIB-patterned substrates were chemically cleaned in an HCl:H2 O solution followed by an-

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Fig. 15.2. Atomic force microscopy (AFM) image of a Si(001) patterned area where holes were created with a 10 pA Ga+ beam after Ga removal and wet chemical cleaning. The diameter of the holes is 45 nm and the pitch is 90 nm. Scan size is 10 × 10 μm2

nealing and another HCl bath to remove Ga contamination induced by the FIB process [2]. The eﬃciency of this restoring process to remove the Ga atoms was checked using SIMS measurements (Ga concentration measured was below the detection limit of Secondary Ion Mass Spectrometer (SIMS) ∼ 1016 /cm2 ). The samples were then chemically cleaned and oxidized before being loaded into the growth chamber. An example of a clean patterned area prior to the introduction into the growth chamber is given in Fig. 15.2. The oxide was desorbed in the molecular beam epitaxy (MBE) chamber at a temperature of 900 ◦ C (2 min) and a thin 4 nm Si buﬀer layer was then grown at 750 ◦ C. At this point atomic force microscopy (AFM) images of the patterned samples do not diﬀer from that obtained after the ex-situ cleaning treatment (see Fig. 15.2). After buﬀer growth, the substrate temperature was lowered to the Ge growth temperature. Growth temperature and Ge thickness were varied to map out the growth kinetic eﬀects on the organization of the dots. In ﬂow chart 2 (Fig. 15.1b), the wafers were oxidized to form an oxide mask for FIB milling (20- or 5-nm-thick SiO2 layer). FIB holes were produced with a larger depth than the mask oxide thickness in order to penetrate into the Si substrate. FIB milling was performed under the same conditions as in ﬂow chart 1 (Fig. 15.1a). A special process was developed in order to remove the sacriﬁcial oxide layer and to restore the substrate. A 3.5-nm-thick tunnel oxide was grown and cleaned again, both chemically and in situ. Amorphous Ge was then deposited at room temperature and crystallized during thermal annealing for 30 min at temperatures between 500 and 700 ◦ C. The temperature ramp of the annealing was 50 ◦ C/min. Reﬂection high energy electron diﬀraction (RHEED) was used to follow the NCs’ crystallization in real time. At the end of the annealing treatment, the samples were rapidly cooled down to room temperature. Ge NCs were obtained by a two-step solid phase epitaxy (SPE) process with the ﬁrst step being a deposition of amorphous Ge at room temperature (RT) and the second, annealing at high temperature. Formation of Ge NCs is

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induced by the combination of crystallization and dewetting of Ge amorphous layer on SiO2 during annealing. Ge deposition was performed at RT to avoid formation of stable polycrystalline Ge layers (with extended defects). For the two ﬂow charts (Fig. 15.1), Ge deposition was carried out in a MBE Riber system with a base pressure in the 10−11 Torr range. Si and Ge were evaporated from an electron beam evaporator and an eﬀusion cell respectively, with deposition rates of 0.03 and 0.017 nm/s. Some of the samples were capped by a thin Si layer for further thermal oxidation in order to embed the Ge NCs in an oxide matrix, while others were dedicated to morphological characterization by AFM operating in air in tapping mode.

15.3 Results 15.3.1 Flow Chart 1 Figures 15.3 (a) and (b) are AFM scans of samples where Ge dots were grown at 700 ◦ C on a FIB-patterned Si wafer. At this temperature, domes with a mean diameter of 120 nm and a density of ∼ 109 cm−2 are produced on

Fig. 15.3. Organization of Ge dots on patterned areas at a growth temperature of 700 ◦ C. Ge deposited thickness is 1,4 nm. The initial holes diameter was 40 nm with a pitch of a 180 nm and b 350 nm. The graphs give the evolution of the mean NCs c size and d density with the holes pitch

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the surface. These domes are organized in between FIB holes with a 180 nm pitch (Fig. 15.3a). For larger hole pitch, (350 nm), the domes remain outside, but nearby the holes (Fig. 15.3b). The density (D) and mean size (φ) of the islands are almost independent of the hole pitch L (Fig. 15.3c, d) and islands formed outside and inside the patterned areas have similar morphological characteristics. FIB patterns, then, have a very small inﬂuence on Ge islands characteristics but they induce preferential island nucleation on the edges of the holes at this temperature. Moreover, as commonly observed on bare Si substrates, a higher density of organized domes (∼ 5.109 /cm2 ) was reached by increasing the thickness of the Ge layer. For a thinner Ge ﬁlm, huts islands formed on the surface as well and were organized in the same way as the domes, i.e., close to FIB-drilled holes. In Fig. 15.4 the results of lower temperature growth are shown; in these cases, Ge islands were grown on a FIB-patterned wafer at 550 ◦ C. The Ge islands ﬁll the FIB holes with a correspondence of one island per hole, independent of the hole pitch, from 350 to 200 and 75 nm (Fig. 15.4a–c). The mean islands’ size and density are directly related to the hole pitch (Fig. 15.4d,e). The evolution of φ as a function of L could not be determined accurately because of the measurement uncertainty of the islands’ mean size from AFM

Fig. 15.4. Organization of Ge dots on patterned areas at a growth temperature of 550 ◦ C. Ge deposited thickness is 1.4 nm. The initial holes’ diameter was 40 nm with a pitch of a 350 nm, b 200 nm, and c 75 nm. The graphs give the evolution of the mean nanocrystal (NC) d size and e density with the holes, pitch

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images. The density of islands clearly follows D ∝ 1/L2 (the evolution expected for a ﬁlling of one dot per hole). The high level of organization reached on FIB patterns is better evidenced in Fig. 15.5 where nicely ordered arrays of islands with a density as large as 2.1010 /cm2 and mean diameter ∼ 45 nm were obtained. These results demonstrate two diﬀerent mechanisms of islands formation: inside the FIB holes at low temperature (550 ◦ C) and on edges of the holes at higher temperature (700 ◦ C). We will ﬁrst discuss the inﬂuence of hole arrays on the local chemical potential at the surface. The diﬀerence in the chemical potential of a patterned surface as compared to a planar surface can be expressed by [3]: Δμ = Ωγ κ(x, y) + ΩEel (x, y) where the ﬁrst term Ωγ κ(x, y)describes the change of the surface energy γ with the surface curvature κ(x, y) and the second term ΩEel (x, y) describes the change of the local strain energy Eel (x, y) induced by the holes. It has been shown [4] that the ﬁrst term is lowered on concave surfaces while the second is lowered on convex surfaces. In the experiments described above, at 700 ◦ C nucleation of dots takes place preferentially on the edges of the holes i.e., on convex areas. The eﬀect of the stress part is then predominant as compared to the surface energy part. Such an eﬀect has been demonstrated for nucleation of Ge dots on mesas [5]. If we now compare the situation at lower temperature, no change of Δμ is expected when the growth temperature is lowered to 550 ◦ C. Consequently, the change in the position of preferential nucleation sites at lower temperature cannot be explained by energetic arguments (lower chemical potential). However, decreasing the temperature dramatically decreases the mobility of surface adatoms. Moreover, the energy barrier of diﬀusion is expected to be larger inside the FIB holes than on a planar (001) terrace because of the presence of steps. The closer to the bottom of the holes the adatoms are, the lower is their surface diﬀusion. Kinetic Monte Carlo simula-

Fig. 15.5. AFM image of Ge NCs with mean diameter ∼ 45 nm and density ∼ 2 × 1010 /cm2 ordered on FIB patterns. Scan size is 3.5 × 3.5 μm2

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tions have been performed [6] to investigate the eﬀect of patterned substrate on the surface diﬀusion rate and on the nucleation of 2D islands. The diﬀusion rate of an adatom was deﬁned by: −E k(E, T ) = k0 exp (15.1) kB T with an activation energy of diﬀusion: E = ED (x, y) + nEN

(15.2)

where ED (x, y) is the substrate term and EN a contribution from each occupied lateral nearest neighbor atom. It was shown that inhomogeneity of ED (x, y) with ED1 inside the hole larger than ED2 on the planar surface (ED1 > ED2 ), produces a net ﬂow of adatoms toward the centers of the holes. Consequently, the holes’ centers act as preferential nucleation sites and strongly inﬂuence both the size and the positioning of the islands. At higher temperature, the change in the total diﬀusion rate becomes smaller. Indeed if we express the rate of surface diﬀusion inside the holes (k1 ) by: ED1 − ED2 k1 (E1 , T ) = k2 (E2 , T ) exp − (15.3) kB T with ED1 − ED2 > 0, then k1 (E1 , T ) and k2 (E2 , T ) tend towards the same value when the temperature increases and so the diﬀerence of diﬀusion rate decreases. This explains why the eﬀect of the inhomogeneity of the diﬀusion barrier, which is predominant at low temperature in the experiments described above, becomes negligible at higher temperature. In addition, if we consider that domes are formed by the merging of small Ge dots at high temperature, even if 2D islands nucleate at the bottom of the FIB holes, it is expected that Ge dots will grow on the most energetically favorable positions determined by the minimization of Δμ [Eq. (15-1)]. Our experiments show that the predominant eﬀect of Δμ minimization is the stress relaxation on the convex areas, i.e., on the edges of the holes. 15.3.2 Flow Chart 2 In a ﬁrst series of experiments on oxidized silicon substrates without FIB patterns, we investigated the eﬀect of annealing temperatures and Ge deposited thickness on the crystallization process. The aim was to determine the experimental growth conditions that control the Ge NCs’ size/density (in order to maximize their density and to reduce their size for memory applications). The RHEED pattern evolution was followed in real time during annealing. After deposition at RT of 2 nm Ge, the RHEED pattern of the surface exhibited a diﬀuse background typical for an amorphous structure (Fig. 15.6a). The appearance of well-deﬁned ring patterns, characteristic for crystalline Ge,

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Fig. 15.6. Reﬂection high energy electron diﬀraction (RHEED) patterns of a 2nm-thick Ge layer deposited at room temperature (RT): a as grown; b during the temperature ramp of annealing (at 400 ◦ C) Table 15.1. Mean density, diameter and height of NCs obtained by annealing (30 min) at diﬀerent temperatures of 2 nm Ge Annealing temperature

500 ◦ C

600 ◦ C

650 ◦ C

700 ◦ C

NCs density D (×1010 cm−2 ) Mean diameter (nm) Average height (nm)

10 28 6.5

12.5 25 6

11.2 27 6

10 27 6.5

occurred during the temperature rise of annealing, at ∼ 400 ◦ C (Fig. 15.6b). Consequently, the onset of Ge crystallization and NC formation occurs at this temperature. In order to understand the mechanism of NC formation and evolution, we annealed 2-nm-thick Ge layers at 500 ◦ C, 600 ◦ C, 650 ◦ C and 700 ◦ C for 30 min. We ﬁrst checked that the volume of NCs was equal to the volume of the deposited layer, which indicates that there is no desorption during the annealing. In all cases, disconnected NCs with large size distribution were formed (size variations were about 50%). Mean NCs’ density, diameter and height measured by AFM are reported in Table 15.1. One can see that in all the samples, the NCs have about the same mean size (diameter φ/height h) and same density (D), (φ ∼ 27 nm/h ∼ 6 nm) and D ∼ 1011 /cm2 respectively. Another series of experiments at diﬀerent annealing times shows that NC features did not vary with the annealing time (from 5 to 60 min). Consequently, in the experimental conditions investigated, NC features are independent of the kinetic parameters (annealing temperature and time). It can then be concluded that a pseudoequilibrium shape characterized by an aspect ratio h/φ ∼ 0.23 and a critical wetting angle θ of 50◦ has been reached. Furthermore, it can also be noted that the NCs’ height is almost 3 times greater than the deposited thickness. This point supports the idea that after reaching the equilibrium shape, NCs continue to grow by capture of a large number of atoms from extended areas around the critical nuclei.

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Fig. 15.7. AFM images of samples with diﬀerent Ge deposited thickness (h): a h = 5 nm, b h = 3 nm, c h = 2 nm and d h = 1.5 nm. All these samples were annealed at 700 ◦ C for 30 min. Evolutions of NC density and mean NC area with h are presented in e and f respectively

In another series of samples, we varied the Ge thickness deposited (h0 ) from 1.5 to 5 nm. All the samples were annealed at 700 ◦ C for 30 min. AFM images of NCs obtained for the diﬀerent thicknesses deposited are presented in Fig. 15.7. All the samples show highly packed NCs randomly distributed and with a large size distribution. The density (D) of NCs is inversely proportional to h0 (D ∝ h −1 0 ) while the NCs’ contact area (SC ) increases linearly with h0 (SC ∝ h 0 ). The opposite evolutions of D and SC are explained by mass conservation (in the case of a constant equilibrium shape). A more detailed analysis of the AFM images shows that NCs are disconnected and that the extension of denuded areas between NCs decreases when their size decreases. Furthermore, the same general remarks as made above seem to legitimate: (1) a constant aspect ratio in all the samples despite the diﬀerent NC sizes, and (2) a height ∼ 3 times larger than h0 . These two last results conﬁrm the conclusions derived above. Two phenomena can be invoked to explain the linear evolution of NCs area with h0 : a kinetically limited coalescence or a saturation nucleus density caused by exclusion zones around the critical nuclei. The ﬁrst argument, being based on limited surface diﬀusion, can be ruled out since we reported above that NC features do not vary with thermal annealing conditions. Let us, then, develop the second argument based on thermodynamics. It is reasonable to assume that nucleation is heterogeneous at the SiO2 /Ge interface. The

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evolution of the growth process cannot be determined from our result, since droplets with pseudoequilibrium shape were obtained whatever the annealing conditions (temperature and time). Nevertheless, diﬀerent evolutions could be considered. If the growth process involves the entire layer volume, each nucleus grows trough the untransformed volume by converting sites neighboring the surface of the growing nuclei to the new phase [7]. Alternatively, growth kinetics could follow a two-step process involving growth of individual nuclei followed by coalescence of neighboring nuclei [8]. During the ﬁrst step, nuclei extend in diameter by capturing a large number of neighboring atoms from the amorphous phase. Gradually the surface covered by crystallites increases, and when the peripheries of droplet neighbors meet, coalescence occurs. As a result, the adatom concentration around the growing crystallites is reduced and the system is locally undersaturated. In these areas (commonly called exclusion areas) nucleation is more or less prohibited. When exclusion zones overlap, a saturation of the nucleus density is reached. New nuclei do not form and growth of the island stops. The system has reached a pseudoequilibrium state. This explains the invariability of the droplets size/density with annealing conditions in this set of experiments. To summarize, with this series of experiments, we have shown that it is possible to form highly dense ultrasmall Ge NCs with a shape imposed by thermodynamic equilibrium and a density/size controlled by the initial deposited thickness. This result is in agreement with experimental results on Si/SiO2 system [9]. We have also used the optimized experimental SPE conditions determined above, in order to form NCs on FIB patterned Ultra Thin Oxide (UTO). Figure 15.8a presents the 2D array of FIB nanopatterns after cleaning just before Ge deposition. Patterns consist of nanometer-scale holes with mean hole diameter and periodicity of ∼ 25 nm and ∼ 50 nm respectively. The same surface after NC formation is presented in Fig. 15.8b. We can see that

Fig. 15.8. AFM images of FIB patterns on Si(001) substrate. a After cleaning. The array period is 50 nm. b After RT deposition of 2 nm Ge and annealing at 600 ◦ C for 30 min. Island density is about 4 × 1010 cm−2

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an eﬃcient ordering has been obtained using FIB-patterned substrates and that Ge NCs’ density is fully controlled by the FIB pattern density (∼ 4 1010 /cm2 in the example presented). Nevertheless, the AFM image after NC formation does not allow a deﬁnite conclusion on the nucleation sites’ location (either inside or in between the FIB holes). Nucleation of Ge NCs on oxide could be driven by one of two phenomena, surface diﬀusion barriers (a lower surface diﬀusion is expected inside the FIB holes) or surface energy reduction. Regarding the ﬁrst argument, even if we have seen above that kinetics (and surface diﬀusion) does not control the evolution of the surface in the Ge/SiO2 system, NC nucleation could be sensitive to the surface diﬀusion inhomogeneity created by the hole patterns. This eﬀect would induce the nucleation of NCs inside the holes. In particular, geometrical conﬁnement in the holes should increase adatoms impingement and nucleation probabilities. Regarding the thermodynamics arguments, NCs’ free energy is lower when they are located on the hole rather than on ﬂat surface [10]. In conclusion, we have successfully used FIB nanopatterning for Ge NC ordering on both Si substrate and ultrathin SiO2 layer. In ﬂow chart 1 (on Si substrate) (Fig. 15.1a), we have described the formation of Ge dots inside the holes at low temperature and on the edges of the holes at higher temperature. We have proposed two diﬀerent mechanisms for the Ge dots formation: a kinetically limited nucleation at low temperature and a stressdriven nucleation at high temperature. Under optimized growth conditions and pattern arrays, the complete ﬁlling of every FIB hole by one Ge NC has been obtained. In that case, density and size of Ge NCs was uniquely controlled by the characteristics of the 2D array of FIB patterns. In ﬂow chart 2, (on ultrathin SiO2 layer) (Fig. 15.1b), we have described a preferential formation of nanocrystals by crystallization of an amorphous Ge layer during thermal annealing on speciﬁc positions deﬁned by the topographic features of FIB-nanopatterned substrates. We have shown that NC shape is determined by thermodynamic equilibrium while density/size are controlled by the initial deposited thickness. The use of FIB-patterned substrate allows ordering of the NCs and homogenizes their size. We show that NCs order inside the FIB holes, which is explained by both thermodynamic and kinetic eﬀects The main advantages of these methods are that they can be scaled down to the 10 nm range (permitting dot density of ∼ 2 × 1011 /cm2 ), and that they allow accurately positioning of the Ge dots on large-scale patterned areas.

References 1. M. Kammler, R. Hull, M.C. Reuter, F.M. Ross, Appl. Phys. Lett. 82, 1093 (2003) 2. A. Karmous, A. Cuenat, A. Ronda, I. Berbezier, S. Atha, R. Hull, Appl. Phys. Lett. 85, 6401 (2004)

440 3. 4. 5. 6. 7. 8. 9. 10.

I. Berbezier, A. Karmous, A. Ronda D.J. Srolovitz, Acta Metall. 37, 621 (1989) M. Borgstr¨ om, V. Zela, W. Seifert, Nanotechnology 14, 264 (2003) B. Yang, F. Liu, M.G. Lagally, Phys. Rev. Lett. 92, 025502 (2004) L. Nurminen, A. Kuronen, K. Kaski, Phys. Rev. B 63, 035407 (2000) M. Weinberg, R. Kapral, J. Chem. Phys. 91, 7146 (1989) B.J. Briscoe, K.P. Galvin, Phys. Rev. A 43, 1906 (1991) Y. Wakayama, T. Tagami, S.I. Tanaka, J. Appl. Phys. 85, 8492 (1999) A. Karmous, I. Berbezier, A. Ronda, Phys. Rev. B 73, 075323 (2006)

16 Metallization and Oxidation Templating of Surfaces for Directed Island Assembly Oscar D. Dubon1,2 , Jeremy T. Robinson1,2 , and Kohei M. Itoh3 1

2

3

Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Keio University, 3-14-1 Hiyoshi, Kouhoku-Ku, Yokohama, 223-8522, Japan

16.1 Introduction Germanium grown on silicon is a model system to study island self-assembly processes in semiconductors. Both materials are elemental, diamond-structure semiconductors and are fully miscible. Germanium growth on Si occurs via the Stranski–Krastanow (S–K) mode due to the 4% lattice mismatch between Ge and Si and the lower surface free energy of Ge. This growth mode is characterized by a transition from initially layer-by-layer growth to three-dimensional (3D) island formation. Extensive investigations have been carried out on the structure and evolution of Ge islands on Si [1–7]. Generally, Ge island ensembles are observed in a random distribution and are given names based upon their shape, e.g., huts (or pyramids), domes, and superdomes. A variety of growth and patterning techniques has been explored in the quest for spatial control of Ge islands, and to varying degrees, one-dimensional, two-dimensional (2D) and 3D ordering has been achieved [8–15]. Here, we describe two strategies for the ordering of Ge islands grown on Si by solid-source molecular beam epitaxy (MBE). In one approach an Au pattern is formed on a Si substrate by Au evaporation through a stencil mask; the resulting pattern induces Ge island ordering across millimeter length scales. In the second approach an atomic force microscope is used to locally oxidize a silicon surface; the locally oxidized regions are etched leaving highly ordered nanometer sized dimples that are collection sites for deposited Ge atoms. This technique provides extremely precise island positions.

16.2 Gold-Mediated Assembly of Ge Islands on Au-Patterned Si (001) Substrate templating by lithographic patterning is an eﬀective means of inducing island ordering. However, patterning by optical or electron-beam lithography typically involves multistep, substrate-to-substrate processing.

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The use of stencil masks to deﬁne patterns on a surface is very attractive compared to other routes; once fabricated a stencil mask may be used many times to extend a pattern across a surface and/or generate a pattern on multiple substrates. The challenge then is to develop a process in which the patterns generated with the stencil mask induce island ordering. Here we describe one such process whereby Ge islands order on a Si substrate that has been patterned simply by the evaporation of Au through a stencil mask [16, 17]. 16.2.1 Substrate Preparation and Ge Deposition Conditions The assembly of Ge islands on Au-patterned Si was achieved by a few relatively simple steps as depicted in Fig 16.1. Silicon (001) wafers were degreased in solvents, HF rinsed to remove the native surface oxide, and then mounted in an electron-beam evaporation chamber with a silicon nitride stencil mask placed directly in contact with the Si surface. One nanometer of Au was deposited through the mask, which contains arrays of square windows ranging from 75 to 400 nm in side length that were produced by electron-beam lithography. The masks are robust and reusable; we have performed over 100 depositions with a single mask without signiﬁcant window blockage or damage. In addition, Au can be removed from the silicon nitride stencil mask by conventional wet processing (e.g., rinsing in a KBr solution). After gold deposition samples were removed from the electron-beam evaporation chamber and transferred to an MBE reactor for Ge deposition by thermal eﬀusion. The substrate was outgassed at 300 ◦ C and 600 ◦ C before Ge deposition, which was usually performed at the latter temperature at a system pressure of 4.5 × 10−10 Torr and a rate of approximately 9 monolayers (ML)/min (1 ML of Ge = 6.27 × 1014 cm−2 ). Surface quality was conﬁrmed via reﬂection high-energy electron diﬀraction (RHEED) by the transformation of the surface from a (1 × 1) to a (2 × 1) reconstruction outside of the Au-patterned regions. Subsequent to deposition samples were cooled from the growth temperature to 300 ◦ C at a rate of 10 ◦ C/min.

Fig. 16.1. Patterning technique used to make arrays of metal dots [17]. a Scanning electron micrograph (SEM) showing a region of the stencil mask. b The mask is placed directly in contact with the Si surface and Au is deposited through the mask by electron-beam deposition. c The mask is removed and the sample is transferred to a molecular beam epitaxy (MBE) system for Ge deposition

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16.2.2 Germanium Island Growth in Unpatterned, Au-Free Regions Two regions of growth present in every sample were studied: (1) a region of Ge deposited on the Au-free Si surface and (2) a region of Ge deposited on the Au-patterned Si surface. The boundary between these two regions is shown in Fig 16.2. By studying island morphologies and distributions on Au-free Si, we can readily compare growth processes to the large body of work available for the Ge/Si system. In this region, we ﬁnd that Ge island evolution is well-described by previous reports [3, 4, 7]. Islanding occurs after approximately 3 ML of Ge with a bimodal distribution of pyramids and domes at lower coverages (< 8 ML). Pyramidal islands have four {105} side facets with approximately a 1:10 (height:base) aspect ratio while domes are multifaceted structures containing {105} and {113} side facets with a 1:5 aspect ratio. At high coverages (> 10 ML), islands grow and coarsen so that only domed and superdomed islands are observed. Superdome islands contain additional {111} facets marking the boundary to the substrate. 16.2.3 Germanium Island Growth in Au-Patterned Regions Island evolution and ordering are uniquely diﬀerent in the Au-patterned regions. We ﬁrst discuss the behavior of Au on Si (001) at the growth temperature. Figure 16.3 shows an atomic force microscopy (AFM) topographic image of the Au pattern on Si after annealing at the growth temperature (600 ◦ C) for 45 min. The original pattern is clearly preserved. The redistribution of gold into solidiﬁed droplets within the deposited Au (square) sites

Fig. 16.2. Atomic force microscopy (AFM) image (20 × 20 μm2 ) taken at the boundary between the Au-patterned and Au-free regions after the deposition of nine monolayers (MLs) of Ge. The Ge islands in the patterned regions appear as white dots. While islands randomly assemble outside of the patterned region, they are highly ordered in the patterned regions. The z-scale in the AFM images is truncated at 50 nm to enhance the Ge island positions relative to the Au-sites

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results from a liquid-to-solid transformation upon cooling. Gold and silicon form a liquid solution above a eutectic temperature of 360 ◦ C; Si from the substrate is dissolved into Au to form a liquid that persists at the growth temperature. Thus, Ge deposition begins in the presence of a pattern of Au-Si droplets. Figure 16.4a shows an AFM scan of a 10 × 10 μm2 section of a Aupatterned Si region on which the equivalent of ∼ 9 ML of Ge was deposited. The tallest features represent Ge islands, which themselves have ordered into a square lattice. The Ge islands assemble at (1/2, 1/2)-type positions of the square lattice formed by the Au pattern as revealed in Fig 16.4b. We shall refer to the position of these Ge islands as center sites. Island arrays extending over 500 × 500 μm2 have been produced; such an array contains on the order

Fig. 16.3. AFM image (5 × 5 μm2 ) of the pattern formed on a Si (100) substrate after the evaporation of 1 nm of Au through a stencil mask and annealing at 600 ◦ C for 45 min. The pattern shown was obtained with a mask region having two window spacings

Fig. 16.4. AFM images of a section in the Au-patterned region (10 × 10 μm2 , height = 150 nm per division), and b a zoom-in section (1.5 × 1.5 μm2 , height = 85 nm/div) showing the relative position of Ge islands in the Au pattern. Germanium islands order into a square lattice in response to the Au pattern. Note that very small islands form in the periphery of the patterned Au squares

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of a half-million islands. The extent of the array is deﬁned by the stencil mask. The location of the Ge islands away from the Au regions is surprising in the context of previous studies. For example, in the well-known vapor–liquid– solid (VLS) process [18], which is widely used in semiconductor nanowire synthesis, growth is catalyzed by Au. Consequently, VLS-based nanowires form selectively where gold droplets are found as molecular (precursor) species interact strongly with the Au droplet relative to the bare substrate surface. An important diﬀerence between VLS growth and the island growth process presented here is that we use an elemental (solid) Ge starting source that does not adsorb exclusively at Au sites; thus, surface processes such as adatom diﬀusion are brought into play. The Ge islands forming the array shown in Fig 16.4 are fundamentally diﬀerent in morphology with respect to islands found in the Au-free region. They have an aspect ratio between 1:2 and 1:3 and a unique truncated pyramidal (TP) shape, which is characterized by {111} side facets and a (001) top facet. The TP shape has been conﬁrmed by transmission electron microscopy (TEM). Truncated pyramidal islands are in addition relaxed with stacking faults lying along {111} planes. The presence of a Ge wetting layer has also been conﬁrmed by TEM. We note that coherent TP islands have been observed in the growth of Si1−x Gex (for x up to 0.3) islands on Si (001) by liquid-phase epitaxy, in which similar island growth dynamics appear to be at play [19, 20]. This suggests that the growth of SiGe islands by Ge and Si coevaporation onto a metal-patterned substrate may result in coherent (dislocation-free) islands. Observations of island shape evolution at low coverages (1.5 – 3 ML) reveal that TP shaped islands originate from lens-like islands that appear at or near center sites (Fig 16.5a) [16, 20]. Several lens-like islands are typically present. The largest ones develop distinct facets marking the boundary with the substrate and evolve into TP islands that continue to grow with further deposition as well as by coarsening presumably through the consumption of the smaller islands, which disappear. At intermediate coverages

Fig. 16.5. AFM images (phase contrast mode) of a lens-like islands, b truncated pyramidal islands, c–d superdome-like island located in the Au-patterned region. The equivalent of ∼ 2.5 ML, ∼ 10 ML, ∼ 15 ML, and ∼ 50 ML of Ge were deposited to obtain images a, b, c and d respectively [17]

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(4 – 10 ML), approximately one to three almost fully evolved truncated pyramidal islands are found at each center site. Above 8 ML a one-TP-islandper-site relationship is widely observed at the optimum Au-pattern spacing (Fig 16.5b). The eﬀect of Au-pattern spacing on island ordering is discussed in Sect. 16.2.4. With further deposition of Ge (10 – 30 ML), a shape transition from a TP to a more superdome-like structure begins (Fig 16.5c,d). This transition includes the introduction of {113} as well as other facets at the top of the islands while the steeper {111} facets continue to mark the boundary with the substrate. At a Ge coverage of 50 ML, islands in the Au-free region have coarsened and grown to domes and superdomes. Thus, TP islands evolve toward the superdome island shape that naturally occurs on the bare Si (001) surface. The trends in island shape evolution can be visualized by comparing island base area (BA) to the corresponding island volume (V ). Figure 16.6 shows a plot of island V versus BA for three diﬀerent growth runs in both the Aupatterned and Au-free regions. During the earliest stages of growth both hut and lens-like islands have similar BA:V ratios. However, as lens-like islands transform into TPs through the growth of {111} side-facets, they diverge to a lower BA:V ratio. At an equivalent Ge coverage of 5.5 ML, the TP islands

Fig. 16.6. Island base area vs. island volume for the Au-patterned (triangles and crosses) and Au-free (circles) regions after 2.5 ML (Au-patterned region only), 5.5 ML and 50 ML of deposited Ge

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formed in the Au-patterned region clearly have a lower BA:V ratio (seen as the downward shift of the data). At a Ge coverage of 50 ML, islands in the Au-free region have coarsened and grown to domes and superdomes. At the same time in the patterned region, a shape transition of TP islands to a more superdome-like structure can be seen by the convergence of the data at larger island sizes. The transition from TP to superdome-like islands appears to be continuous. A similar trend in island evolution is seen in a comparable plot of surface area vs. volume. The remarkable diﬀerences between island shapes in the Au-free and Aupatterned regions raises the possibility that Au may be playing a signiﬁcant role. Chemical analysis by electron dispersive spectroscopy (EDS) was performed on the Ge islands, Au patterned-regions, and the regions in between using a 1 nm probe in a transmission electron microscope. It revealed Si compositions up to 30 at. % in the Ge islands,; however, the presence of Au in the islands was not detected within the instrumental limit. Nevertheless, the role of Au in producing the surprising diﬀerence in island shape cannot be discounted as even a small amount of Au on the surface may modify surface energies and drive the observed faceting. Thus far, we know that Sn can be used instead of Au to direct island organization; when Sn is used, island shapes are markedly diﬀerent from the TP shape observed on Au-patterned Si. 16.2.4 Eﬀect of Au-Pattern Spacing on Island Ordering Formation of island-free regions around each Au site is central to understating the ordering process observed here. By varying the Au-pattern spacing, we can vary the degree to which neighboring Au-site interactions occur. When the Au pattern spacing is larger than the island-free regions, or denuded zones, island ordering is not observed (Fig 16.7a). Germanium islanding occurs completely around each Au-site (deﬁned here as the region including the deposited Au square, the small islands growing in the periphery of the Au square, and the associated denuded zone). When Au spacing is decreased such that neighboring denuded zones begin to interact, the ﬁrst evidence of directed assembly emerges (Fig 16.7b). Nucleation of multiple islands is found at each center site and islanding between neighboring Au sites is not present. As the Au spacing is decreased below the denuded zone widths, a one island or close to one island per site relationship is achieved and islands are ordered over the entire patterned region (Figs. 16.7c and 16.4). It is important to note that in all experiments in which superdome-like islands have not yet formed, there is signiﬁcantly more mass in the islands outside of the denuded zones than at the Au site regardless of Au site spacing. This along with island ordering at center sites suggests the existence of a diﬀusion barrier at the denuded zone boundary. Kinetic Monte Carlo (KMC) simulations have been performed to study the dynamics of nucleation, growth and coarsening on the patterned surface, leaving aside the issue of island morphology [16]. A number of physical

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Fig. 16.7. Schematic representation of island ordering and corresponding AFM images (4 × 4 μm2 ) showing the eﬀect of Au pattern spacing at 500 ◦ C. The height scale is truncated at 20 nm to enhance the Ge island positions relative to the patterned Au sites. In a, the Au spacing is greater than twice the denuded zone radius, Rd , and Ge islanding occurs completely around each patterned Au site. In b the Au spacing is on the slightly greater Rd and islanding is mostly conﬁned at the corner sites. In c the Au spacing is less than Rd and there is essentially a one island per site relationship

processes has been considered as the origin of the denuded zones about the Au sites and the ordering of Ge islands, including the possibility that the patterned Au sites act as strong adatom sinks and conversely as diﬀusion (reﬂecting) barriers. These scenarios fail to produce the observed island ordering behavior. Instead, the only successful model that we have identiﬁed associates a strongly reﬂecting diﬀusion barrier with the outer boundary of the denuded zone. For systems in which the radii of the denuded zones are comparable to their separation, one observes no island ordering at center sites. As diﬀusion barriers are brought into proximity, the islands begin to order. Ordering is pronounced once the diﬀusion barriers are separated by approximately one-third of their radii. As the barrier separation distance decreases, the nucleation allowed areas are localized to the center sites. This trend is clearly observed in experiments (Fig 16.7). The physical origin and actual strength of the diﬀusion barriers and the formation of denuded zones are still not understood. It is very possible that dynamical changes in chemical potential across the surface as a result of factors such as strain evolution lead to conditions favorable for the formation of denuded zones and their associated diﬀusion barriers. These fundamental issues concerning the assembly process are the subject of continuing studies.

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16.2.5 Germanium Islands Formed on Au-Patterned Si (110) We have performed experiments of Ge growth on Au-patterned Si (110). Figure 16.8 shows the boundary between Au-patterned and Au-free Si (110) in which the equivalent of 5 ML Ge was deposited. A remarkable transformation in island shape is seen from dome-like islands in the Au-free region to rodshaped islands in the Au-patterned region. The long axis of the rod-shaped islands is aligned along the in-plane [110] direction regardless of pattern orientation. Using AFM we have indexed the side facets to be of the {111} type and found islands having aspect ratios as low as 1:15 (width:length). For the image shown in Fig 16.8, typical rod island dimensions are a height of 25 nm, width of 75 nm, and length of 360 nm. Just as the case for Si(001) at low coverages, islands in the patterned region have a lens-like shape. The largest lens-like islands nucleate {111} side facets and grow into rods. 16.2.6 Summary We have shown that relatively simple metal patterning of a Si surface can induce the assembly of germanium islands into an extensive ordered array. The diﬀusion-based dynamics of this growth process means that growth temperature can be used to tune the array assembly over a wide range of length scales

Fig. 16.8. AFM images (phase contrast mode) of a the boundary between Aufree and Au-patterned Si (110) after 6 ML Ge [17] and b a close-up showing one island in the Au-patterned region. In c, a cross-sectional proﬁle of the island in b shows the ∼ 36◦ angle between the island side facet and the surface (width is 150 nm; height is 40 nm). d Selected directions from the viewpoint of looking down the length of a rod island. The [111] direction makes a 35.26◦ angle to the surface normal, which is in good agreement with the measured angle in c

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while Au pattern design (i.e., diﬀusion barrier pattering) can be exploited to form a variety of nanostructure arrays. The simplicity and diﬀusion-based nature of the process provides the distinct possibility of applying it to other heteroepitaxial islanding systems where an array of diﬀusion barriers should also lead to island ordering.

16.3 Forced Alignment of Ge Dot Arrays on AFM Templated Substrates 16.3.1 Introduction Precise control of the quantum dot position, size, shape and crystalline structure is required for a number of applications as discussed in many chapters of this book. For certain applications such as all-optical quantum information processing [21], the realization of an array of small diameter (50 nm and below) quantum dots with small spacing (50 nm and below) is important to enhance the quantum eﬀect arising from the charge conﬁnement and interaction between adjacent dots. In the past, selective growth of an array of very small Ge dots (diameter < 20 nm) on Si substrates was realized by forming windows using a scanning tunneling microscope in an ultrathin SiO2 overlayer [22]. However, small dots grown in this manner exhibit a variety of facets leading to a variations in shapes, partly because they were grown on a ﬂat Si substrate [23]. This section concerns with the control of position and size of the small quantum dots (diameter 50 nm and below) with a size homogeneity of ± 5% for the best case. Such quantum dots (InGaAs dots on GaAs substrates [24–26] and Ge dots on Si substrates [27]) have been prepared by MBE on specially prepared substrates templated by scanning probe microscope (SPM) anodic oxidation.

Fig. 16.9. Schematic of the experimental procedure for forced lateral alignment by scanning probe microscope (SPM) anodic oxidation lithography. a Local oxidation. b Dimple formation by etching. c Ge dot preferential growth by MBE

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Figure 16.9 shows the overview of this method. The ﬁrst step is local oxidation of substrate by an SPM tip as was demonstrated successfully for Si [28] and GaAs [28, 29]. Placing the tip very close to (or in contact with) the surface of the substrate followed by biasing the substrate with respect to the electrically grounded tip decomposes H2 O molecules in the air to H+ and OH− ions. When OH− ions arrive at the substrate surface, they oxidize the surface locally to form nanoscale oxide dots (Fig. 16.9a). Two-dimensional scanning of the tip to form the nanooxides in a periodic manner leads to an array of the oxide dots as depicted in Fig. 16.9a. The next step is removal of oxide dots by an etchant, which only attacks the oxide dots. The resulting structure consists of the substrate with a periodic array of nanoscale dimples

Fig. 16.10. The ﬁrst example of forced lateral alignment of InGaAs dots on a GaAs substrate by SPM anodic oxidation lithography. a Nanooxide dots formed by SPM. b Dimples after etching. c Site- and position-controlled InGaAs dots grown preferentially in the sites of dimples. d–f Height contours along the arrays shown in a–c, respectively. After Fig. 16.3 of [24]

Fig. 16.11. The second example of forced lateral alignment of InGaAs dots on a GaAs substrate by SPM anodic oxidation lithography. a Nanooxide dots of diﬀerent sizes positioned by SPM. b Dimples after etching. c Site- and position-controlled InGaAs dots grown preferentially in the sites of dimples. After Fig. 16.6 of [24]

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as shown in Fig. 16.9b. The ﬁnal step is uniform deposition of the element(s), which is (are) known to form self-assembled quantum dots on the given substrate in the Stranski–Krastinow (SK) mode. Deposited elements diﬀuse into dimples and nucleate preferentially to form a periodic array of quantum dots of controlled size and position (Fig. 16.9c). Successful examples are shown in Figs. 16.10, 16.11 and 16.12. Figure 16.10a–c presents AFM images of nanooxides, nanodimples, and InGaAs dots, respectively, formed on a GaAs substrate [24]. Figure 16.10e, f shows corresponding height contours when

Fig. 16.12. Forced lateral alignment of Ge dots on a Si substrate by SPM anodic oxidation lithography. a AFM image of nanooxide dots formed by SPM and a height contour along the dot array. b AFM image of dimples after etching and a height contour along the dimple array. c Site- and position-controlled Ge dots grown preferentially in the sites of dimples and a height contour along the dot array. After Figs. 16.4 and 16.6 of [27]

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the tip is scanned along the oxide dots, nanodimples, and InGaAs dots, respectively. It is seen that uniformly “hamburger” shaped dots of diameter ∼ 30 nm and height ∼ 2.5 nm (∼ 1.25 nm into the substrate and ∼ 1.25 nm out of the substrate) are positioned successfully to realize a two dimensional squared lattice with a lattice constant ∼ 50 nm. The same group achieved arbitrary control of the position and size of InGaAs dots on the GaAs substrate with the size and position accuracy of ∼ 5% as shown in Fig. 16.11 [24, 26]. Figure 16.11a shows position and size controlled oxide dots formed on the GaAs substrate by SPM anodic local oxidation; Fig. 16.11b shows the dimples after etching; and Fig. 16.11c shows InGaAs dots that have been formed preferentially at the sites of dimples, whose positions and diameters reﬂecting the corresponding positions and sizes of dimples [24, 26]. Following this success, the same method was applied to the lateral alignment of Ge dots on silicon substrates [27]. Figure 16.12a shows the position and size of controlled oxide dots formed on a Si substrate by SPM anodic local oxidation; Fig. 16.12b shows the dimples after etching; and Fig. 16.12c shows Ge dots that have been formed preferentially at the dimple sites. The corresponding AFM height contour line scans are also shown. As seen from the above examples, SPM anodic oxidation has been proven eﬀective for the control of size and position of quantum dots, which grow in the SK mode on a given substrate. We will describe in this section some of the key engineering issues involved in this method. However, we do not know exactly why dots grow in dimples at this point, and systemic investigations probing the growth mechanism are underway by a number of groups. Therefore, our speculations of why dots grow in dimples are given at the end of this section. 16.3.2 Local Anodic Oxidation by Scanning Probe Microscope The local anodic oxidation by scanning probe tips allows for the lithography of the smallest semiconductor structures among a variety of top-down processing methods [29]. For example, it is possible to form thin gate oxide tunneling barriers of 10 – 50 nm nanometers [30], and sub-10 nm metal oxide device [31] and single electron transistors [32–34] have been demonstrated successfully using SPM local oxide gates. Once nanoscale dots (lines) of oxides are formed, their etching results in dimples (grooves) of corresponding sizes. Its disadvantage, of course, is the low throughput since drawing of each line via scanning and biasing of the tip takes considerable time. Nevertheless, the fact that one can machine semiconductor surfaces with nanometer precision places SPM anodic oxidation lithography as one of the most attractive tools for basic scientiﬁc research probing formation mechanisms of semiconductor nanostructures. The mechanism of SPM anodic oxidation is shown in Fig. 16.13. Very thin layers of water cover hydrophilic surfaces of semiconductor substrates when exposed in the air or other humid atmospheres. Placing a conductive cantilever tip very close to or in contact the surface water forms

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a “water bridge” because of capillary eﬀects [35,36]. Application of a voltage between the tip as the anode and substrate as the cathode ionizes molecules in the water bridge according to the reaction H2 O → H+ +OH− . (It has been recently proposed that electric ﬁeld plays an important role in the formation of the water bridge itself [37].) Hydroxide (OH− ) ions are then drawn to the cathode substrate to trigger the oxidation reaction at the surface. The reaction of OH− with GaAs substrates is 2GaAs + 12OH− → Ga2 O3 + As2 O3 + 6H2 O + 12e− ,

(16.1)

while that with Si is Si + 4h+ → SiO2 + 2H+ ,

(16.2)

where e− and h+ are electrons and holes, respectively. Oxide dots with higher aspect ratios (height to diameter ratios) lead to nanodimples with higher aspect ratios (depth to diameter ratios) after etching. Dimples with higher aspect ratios are preferred for the site selective growth of dots into dimples. Many factors are important for precise and reproducible SPM local oxidation: tip-surface distance, applied voltage and pulsing sequences, humidity of the atmosphere, tip materials and sharpness, etc. As in the case of atomic force microscopy imaging, SPM local oxidation can be performed in the contact mode (tip touching the surface), tapping mode, and noncontact mode. A recent report has indicated that noncontact mode oxidation can produce smaller oxides at a rate that is faster than that for contact mode [38]. This requires precise control of the distance between the tip and surface. The growth rate of the oxide dots decreases with oxidation time due to the accumulation of the space charge buildup above and below the oxide dot just like the case for parallel metal plate capacitors. The space charge build up is the major limiting factor of the growth rate of the anodic oxidation [39, 40]. For the case of silicon, the increase of H+ concentration in the water bridge according to Eq. (16.2) reduces OH− due to the reverse reaction H+ + OH− → H2 O. This also slows down the oxidation rate signiﬁcantly. An obvious solution to avoid such eﬀects is to modulate the applied voltage to remove the accumulated charge. Modulation pulses have been employed successfully to maintain the growth rate and controllability

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Fig. 16.14. Three bias pulse sequences. a A dc bias for a total of 2 s. b 0.1 s pulses with 0.03 s intervals for a total of 2 s. c 0.05 s pulses with 0.02 s intervals for the total of 2 s. Height contours of resulting oxide dots d–f correspond to the pulse sequences a–c, respectively. Taken from Figs. 16.2 and 16.3 of [27]

of local anodic oxidation of Si [39] and GaAs [41]. Figure 16.14 shows the modulated pulse sequences for the local oxidation of SiO2 : a dc bias for 2 s, b a sequence of 0.1 s bias pulse and 0.03 sinterval for the total of 2 s, and c a sequence of 0.05 s bias pulse and 0.03 s interval for the total of 2 s [27]. Figure 16.14d–f presents the height contour of the resulting dot corresponding to the pulse sequences Fig 16.14a–c, respectively, for a bias −V = 4 V [27]. The dot height in Fig 16.14d produced with the dc bias saturates at ∼ 0.5 nm while the dot produced with the shortest pulse sequence has the height as tall as 2.5 nm (Fig. 16.14f). For the cases of both GaAs and Si shown in Figs. 16.10, 16.11 and 16.12, humidity was not controlled deliberately during SPM anodic oxidation, but the average humidity was about 50%. 16.3.3 Etching Process for Dimple Array Formations The oxide dots on GaAs (Fig. 16.10a and Fig. 16.11a) were selectively etched away by immersing the sample into 1 – 5% diluted HCl for a few minutes at room temperature [24]. It was also shown that ultrasonic cleaning in water can remove the oxide dots from the GaAs surface. With an ultrasonic power of 100 W, it takes about 10 – 30 min to remove these oxide dots completely from the substrate [24]. The resulting dimples are shown in Fig. 16.10b and Fig. 16.11b. The height contour scan along the dimple array (Fig. 16.9e)

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shows successful formation of an array of dimples of about 30 nm in diameter and 1.2 nm in depth. The oxide dots formed on Si by local anodic oxidation can be etched away by immersing the sample into 1 – 5% diluted hydroﬂuoric acid at room temperature [27]. After such treatment, dots are removed completely as shown in Fig. 16.12b. The height contour scan along the dimple array (Fig. 16.14b) shows successful formation of an array of dimples of about 40 nm in diameter and 3 nm in depth. 16.3.4 MBE Growth of Dots Site controlled InGaAs quantum dots shown in Fig. 16.10c and Fig. 16.11c were grown by solid-source molecular beam epitaxy on the dimple-patterned GaAs surface [24–26]. Prebaking of the dimple-patterned GaAs wafer was carried out at 250 ◦ C for 1 hour in vacuum prior to growth. However, a thin oxide layer was formed in the short time required to mount the sample into the MBE chamber. High-temperature thermal cleaning to remove the thin, native oxide layer is not performed because the dimples can smear out at temperatures of 600 ◦ C and above. Therefore, Song et al. have used atomic hydrogen irradiation at temperature below 550 ◦ C for about 15 min [24]. This cleaning process [42] can remove thin oxide layers without changing the shape of dimples. The quality of the surface is increased further by depositing a thin buﬀer layer of GaAs followed by InGaAs dot growth at 480 – 520 ◦ C. The coverage of InGaAs was kept below the transition from 2D to 3D growth. The resulting site-controlled InGaAs dots are shown in Fig. 16.8c and Fig. 16.9c. Site controlled Ge quantum dots shown in Fig. 16.12c were grown also by solid-source MBE on the dimple-patterned Si surface [27]. Prior to the loading to the MBE chamber, the wafer was treated by RCA cleanings followed by immersing into a solution of diluted HF for a minute. The wafer with nanodimples was then placed in the MBE chamber with a vacuum of about 10−10 Torr. The wafer was annealed at 850 ◦ C for 15 min to clean the surface. Figure 16.15a shows the height contour of the nanodimples before and Fig. 16.15b after a 850 ◦ C, 5 min surface cleaning treatment in our ultrahigh vacuum MBE chamber. It is clearly seen that the dimple shapes do not change even after thermal cleaning. Finally, Ge atoms were deposited at the substrate temperature of 650 ◦ C with the growth rate of ∼ 0.1 ML/s. Figure 16.12c shows AFM images of an array of Ge dots and their height contour along the array. It is demonstrated clearly that Ge dots are formed preferentially at the sites of dimples. For this particular growth, the substrate temperature was maintained at for 30 s after the end of Ge deposition. However, the optimum duration of postgrowth annealing varies depending on the dimple diameters, dimple separation, substrate temperature, deposition rate, and surface cleanness. Recently, we found that treatment of dimplepatterned Si substrates to remove surface contamination by carbon prior to loading into the MBE chamber is especially important. Carbon impurities on

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Fig. 16.15. AFM height contours of nanodimples a before and b after 850 ◦ C, 15 min surface cleaning in an MBE chamber. After Fig. 16.5 of [27]

silicon substrates are nucleation centers for Ge dots and therefore interfere with the preferential nucleation at the site of dimples. 16.3.5 Growth Mechanism of Dots in Dimples Dots grow in dimples preferentially when the surface is suﬃciently clean so that there is very little island nucleation on the ﬂat surface in the vicinity of the dimples, which therefore must be within the diﬀusion length of ad-dimers formed from the deposited species. The dots grow in dimples preferentially because of the negative surface curvature of the dimples and the lower activation barrier for ad-dimers to hop downward than to hop upwards. The position dependent chemical potential μ(x,y) at the surface is given by [43] (16.3) μ(x, y) = μ0 + V γc(x, y) + V Es (x, y) where μ0 is the chemical potential of the clean and ﬂat surface, V is the atomic volume of surface constituents, γ is the surface energy, c(x,y) is the position dependent surface curvature, and Es (x,y) is the energy associated with strain arising from stress applied by lattice mismatch and other phenomena. Clearly, the negative curvature lowers μ(x,y) so that the nucleation of dots in dimples are favored. Of course, the third term of Eq. (16.3), the strain energy, competes against the eﬀect of the negative curvature when the lattice constant of the dots is larger than that of the surface. The negative curvature can interfere with the expansion of dots to reduce its strain energy. Therefore, the negative curvature must be in the appropriate range such that it reduces the chemical potential suﬃciently to promote nucleation in dimples but not too much so that it allows for the dots with larger lattice constants

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O.D. Dubon, J.T. Robinson, K.M. Itoh Fig. 16.16. An example of SPM surface lithography using parallel straight line oxidation in x–y directions followed by etching for placing small dimples in a 2D array

to expand. Another important driving force for ad-dimers to hop into dimples preferentially is the smaller hopping barriers for hopping down that that for hopping up [44–46]. Experimentally, preferential nucleation of Ge dots in grooves [47] and pits [48, 49] of Si surfaces have been demonstrated beautifully. Zhong et al. were able to show further that Ge dots grow on top convex ridges when the strain energy term [the third term in Eq. (16.3)] dominates. Understanding of the growth mechanism of dots in dimples requires for experimental investigations in which surface quality, dimple diameter and separation, dimple curvature, substrate temperature, deposition rate, and post annealing temperature and duration are controlled systematically. SPM oxidation lithography discussed in this section is one of the ideal methods to prepare templated substrates for such studies. SPM oxidation lithography allows not only oxide dot formation but also oxide lines and other patterns. For example, parallel straight line oxidation in x–y directions followed by etching leads to substrate shown in Fig. 16.16 with very small dimple to dimple separation, which allows hopefully for positioning of dots with very high density. 16.3.6 Summary InGaAs and Ge quantum dots have been grown preferentially in nanodimples formed by the anodic oxidation of the GaAs and Si surfaces, respectively. It will be of great interest to pursue further studies described in the text to a variety of quantum dot systems which grow via Stranski–Krastinow mode. Acknowledgement. The work on the directed assembly of Ge islands on Aupatterned Si was supported by the Laboratory Directed Research and Development Program of Lawrence Berkeley National Laboratory under the US Department of Energy Contract No. DE-AC03-76SF00098. O.D.D. acknowledges support from the National Science Foundation under contract number DMR-0349257. O.D.D. and J.T.R. thank J.A. Liddle, A. Minor, V.R. Radmilovic, and D.C. Chrzan for their contributions to this work. K.M.I. thanks H.Z. Song, T. Oshima, and Y. Okada for their advice on SPM nanooxidation, A. Hirai, Y. Yoshida, and S. Miyamoto for the Ge on Si work conducted at Keio University. The work on AFM templated substrates is supported in part by the Fujitsu Laboratories and in part by a Grantin-Aid for Scientiﬁc Research in Priority Areas “Semiconductor Nanospintronics (#14076215).”

16 Metallization and Oxidation Templating of Surfaces

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17 Site Control and Selective-Area Growth Techniques of InAs Quantum Dots with High Density and High Uniformity Kiyoshi Asakawa1 , Shigeru Kohmoto2 , Shunsuke Ohkouchi3 , and Yusui Nakamura4 1

2

3

4

TARA Center, University of Tsukuba, 1-1-1, Ten-noudai, Tsukuba 305-8577, Japan System Devices Research Laboratories, NEC Corporation, 2-9-1, Seiran, Ohtsu, Shiga 520-0833, Japan Fundamental and Environmental Research Laboratories, NEC Corporation, 34, Miyukigaoka, Tsukuba, Ibaraki 305-8501, Japan Department of Electrical and Computer Engineering, Kumamoto University, 2-39-1, Kurokami, Kumamoto 860-8555, Japan

17.1 Introduction The internet requires ever more rapid transmission of huge amounts of information in optical communication systems as its use becomes more and more widespread. We have been studying ultrafast devices for optical time-divisionmultiplexing. Ultrafast demultiplexing has been reported with a symmetricMach–Zehnder-type (SMZ) all-optical switch by Tajima et al. [1,2], where the switching speed is not restricted to the carrier lifetime of optical nonlinear materials. Since this device was a 20-mm-long hybrid system, it is important to miniaturize it into monolithic ultrasmall devices of 1 mm square. To approach this target, we have been studying photonic-crystal-based SMZ (PCSMZ) all-optical switches with qu