Infrared Ellipsometry on Semiconductor Layer Structures: Phonons, Plasmons, and Polaritons (Springer Tracts in Modern Physics)

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Springer Tracts in Modern Physics Volume 209 Managing Editor: G. H¨ohler, Karlsruhe Editors: J. K¨uhn, Karlsruhe Th. M¨uller, Karlsruhe A. Ruckenstein, New Jersey F. Steiner, Ulm J. Tr¨umper, Garching P. W¨olﬂe, Karlsruhe

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Mathias Schubert

Infrared Ellipsometry on Semiconductor Layer Structures Phonons, Plasmons, and Polaritons

With 77 Figures

123

Mathias Schubert Institut f¨ur Experimentelle Physik II Universit¨at Leipzig Linn´estr. 5 04013 Leipzig, Germany E-mail: [email protected]

Library of Congress Control Number: 2004113130

Physics and Astronomy Classiﬁcation Scheme (PACS): 71.36.+c, 78.20.-e, 78.20.Ls

ISSN print edition: 0081-3869 ISSN electronic edition: 1615-0430 ISBN 3-540-23249-4 Springer Berlin Heidelberg New York 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 springeronline.com © Springer-Verlag Berlin Heidelberg 2004 Printed in Germany 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. Typesetting: by the author and TechBooks using a Springer LATEX macro package Cover concept: eStudio Calamar Steinen Cover production: design &production GmbH, Heidelberg Printed on acid-free paper

SPIN: 10930953

56/3141/jl

543210

Dedicated to my parents

Preface

The scope of this book is introduction and application of ellipsometry as solid state spectroscopy technique at long wavelengths for analysis of phonon and plasmon properties in semiconductor layer structures. Fundamental issues about infrared ellipsometry, its mathematical and technical implementation as well as data analysis procedures are covered. The book is intended for students as well as for experienced researchers. A new-comer in ellipsometry may ﬁnd basic introductory sections, and likewise the expert may ﬁnd inspiration for further applications upon adopting schemes developed and pursued in this book. Particular emphasis is given to issues of infrared light propagation in layered structures, including situations where low symmetry materials are involved. Special attention is paid to the “generalized ellipsometry” concept and its mathematical implementation for treatment of arbitrary anisotropic materials. Use of the “Jones matrix” and the “Mueller matrix” approaches in ellipsometry are explained. The set up of instrumentation is brieﬂy reviewed. The long-wavelength dielectric function of materials with polar lattice modes and free charge carriers is discussed, with special attention paid to the Drude model. Phonon, plasmon, bulk and interface polariton eﬀects in semiconductor layer structures are explained, highlighted by numerous examples of contemporary interest. It is demonstrated why and how surface mode excitations provide extreme sensitivity to free-charge-carrier properties and phonon mode parameters in ﬁlms of thickness much less than the probing wavelength. Phosphide-, arsenide-, nitride- and oxide-based semiconductors are examined, which comprise the building blocks of present and future electronic and opto-electronic devices such as light emitting diodes, or laser diodes, etc. Examples demonstrate analysis of free-charge-carrier parameters, chemical composition, ordering, strain and structural knowledge from multinary alloys materials. Information retrieved from buried layers without performing electrical contacts or destructive sample preparation is demonstrated for simple as well as for complex-layered heterostructures. Unprecedented, infrared ellipsometry is capable of “weighing” the inertial mass of free charge carriers in layered structures according to their moment of inertia when brought into an external magnetic ﬁeld.

VIII

Preface

Many colleagues have contributed to the results described in this book. The author thanks Dr. G. Wagner (Institut f¨ ur Nichtklassische Chemie e.V. Leipzig) for providing Figs. 6.13, 7.4, 7.24, Dr. E. Schubert (Institut f¨ ur Oberﬂ¨ achenmodiﬁzierung Leipzig, e.V.) for Figs. 2.6, 6.6, 8.3, V. Darakchieva (University Linkoping) for Fig. 7.5, Prof. Dr. M. Grundmann for Fig. 7.3, and C. Bundesmann (Universit¨ at Leipzig) for Fig. 7.8. I acknowledge the assistance of providing high-quality samples by Dr. V. Gottschalch (Universit¨at Leipzig), Dr. I. Pietzonka, Dr. J. Oﬀ, Dr. K. Streubel (Osram Opto Semiconductors GmbH, Regensburg), Prof. Dr. F. Scholz (Universit¨at Ulm), and Dr. B. Kuhn (Bosch), Prof. Dr. D. Hommel, Dr. S. Einfeldt, S. Figge, Dr. T. B¨ ottcher, M. Diesselberg (Universit¨at Bremen), Prof. Dr. A. Krost, Dr. A. Dadgar (Universit¨ at Magdeburg), Prof. Dr. D. J. As (Gesamthochschule Paderborn), Prof. Dr. T. Paskova, and Prof. Dr. B. Monemar (University Linkoping), and Prof. Dr. W. Dollase (University of Calˇ (On Semiconifornia Los Angeles). I acknowledge my coworkers Dr. J. Sik ductors), Dr. S. Zangooie, Dr. A. Kasic (Inﬁneon technologies), T. Hofmann, Dr. G. Leibiger, C. Bundesmann, N. Ashkenov, and C. v. Middendorf (Universit¨at Leipzig) for their help. I wish to express my sincere gratitude to Prof. Dr. J. A. Woollam (University Nebraska-Lincoln) for continued support and encouragement, and to Prof. Dr. M. Grundmann, Prof. Dr. W. Grill and Dr. V. Riede for providing excellent working conditions at the Institut f¨ ur Experimentelle Physik II. Many stimulating ideas came upon helpful discussions with Prof. Dr. B. Rheinl¨ ander (Universit¨ at Leipzig). Dr. A. R¨oseer (ISAS), Dr. T. E. Tiwald, Dr. C. M. Herzinger (J. A. Woollam Co., Inc.) and Dr. C. Bernhard (MPI Stuttgart) were pioneering in this ﬁeld, and I am grateful for many enlightening conversations. I thank the colleagues from the Mechanische Werkstatt at Faculty of Physics and Geoscience. I acknowledge collaboration with Prof. Dr. H. Arwin (University Linkoping) and Prof. Dr. P. Esquinazi (Universit¨ at Leipzig). Ing. U. Teschner and D. W. Thompson are acknowledged for their technical help. I thank M. Grundmann, B. Rheinl¨ ander, C. M. Herzinger, and T. Hofmann for critical reading of the manuscript. I thank M. Sousa for her kind advice, and Dr. A. Lahee (Springer Verlag) for editorial support. Finally I thank my family for their patience. The research was supported in part by the National Science Foundation, the University of Nebraska-Lincoln, the Bundesministerium f¨ ur Bildung und Forschung, the Deutsche Forschungsgemeinschaft, the Universit¨ at Leipzig, the Swedish Foundation for International Cooperation in Research and Higher Education, and the University of Linkoping. The author greatly appreciates support given by the Joanneum Research Forschungsgesellschaft mbH Graz, the L.O.T.-Oriel GmbH Darmstadt, the Solarion GmbH Leipzig, and the OSRAM Opto Semiconductors GmbH, Regensburg.

Leipzig, September 2004

Mathias Schubert

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 4

2

Ellipsometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Jones Matrix Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mueller Matrix Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Standard Ellipsometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Generalized Ellipsometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Brewster and Interface-Mode Conditions . . . . . . . . . . . . . . . . . . 2.5.1 Generalized Brewster Conditions . . . . . . . . . . . . . . . . . . . 2.5.2 Generalized Interface-Mode Conditions . . . . . . . . . . . . . . 2.6 Anisotropic Dielectric Function Tensor . . . . . . . . . . . . . . . . . . . . 2.6.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Orthogonal Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Bond Polarizability Model . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Light Propagation in Layered Anisotropic Media . . . . . . . . . . . 2.8 Ellipsometry Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Thickness and Dielectric Function . . . . . . . . . . . . . . . . . . 2.8.2 Wavelength-by-Wavelength Fit . . . . . . . . . . . . . . . . . . . . . 2.8.3 Parameterized Model Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Ellipsometry Test Functions . . . . . . . . . . . . . . . . . . . . . . . 2.8.5 Parameter Accessibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Infrared Ellipsometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 8 9 11 12 13 13 14 14 14 15 16 18 20 20 21 21 22 23 23 27

3

Infrared Model Dielectric Functions . . . . . . . . . . . . . . . . . . . . . . 3.1 Polar Lattice Resonance Contributions . . . . . . . . . . . . . . . . . . . . 3.2 Free-Charge-Carrier Contributions . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Low-Polarity Mode Contributions . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Free-Charge-Carrier Magneto-Optic Contributions . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 35 38 39 42

X

Contents

4

Polaritons in Semiconductor Layer Structures . . . . . . . . . . . . 4.1 Bulk Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Example: i -GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Surface Polaritons and Surface Guided Waves . . . . . . . . . . . . . . 4.2.1 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Example: i-GaAs Film on n-GaAs Substrate . . . . . . . . . 4.2.3 The Berreman Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 45 47 48 50 52 62 64

5

Anisotropic Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Bands of Total Reﬂection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Sapphire (Hexagonal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Stibnite (Orthorhombic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 68 70 76 79

6

Zincblende-Structure Materials (III–V) . . . . . . . . . . . . . . . . . . . 6.1 Free Charge Carriers (p-Type GaAs) . . . . . . . . . . . . . . . . . . . . . . 6.2 Alloying (AlGaInP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Ordering (AlGaInP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Strain and Composition (GaNP, GaInAsN) . . . . . . . . . . . . . . . . 6.4.1 GaNP/GaP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 InGaNAs/GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 85 90 96 96 100 104

7

Wurtzite-Structure Materials (Group-III Nitrides, ZnO) . . 7.1 Crystal Quality: Bulk and Epilayer (ZnO) . . . . . . . . . . . . . . . . . 7.2 Total-Reﬂection-Bands and Dielectric Anisotropy (ZnO) . . . . . 7.3 Epitaxial Relationship (ZnO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Free Charge Carriers (p- and n-Type GaN) . . . . . . . . . . . . . . . . 7.5 Strain and Alloying (InAlN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Device Heterostructures (Group-III Nitrides) . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 112 116 117 123 128 133 143

8

Magneto-Optic Ellipsometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Experimental Setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Polar MO Setup: Example . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Oblique MO Setup: Example . . . . . . . . . . . . . . . . . . . . . . 8.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 i-GaAs(d)/n-GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 n-B0.03 In0.06 Ga0.91 As(d)/i-GaAs . . . . . . . . . . . . . . . . . . . 8.2.3 n-GaAs(d)/n-Al0.19 Ga0.33 In0.48 P(d)/i-GaAs . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147 149 150 152 153 154 159 160 163

Contents

XI

A

4 × 4 Transfer Matrix Tp for Isotropic Films . . . . . . . . . . . . . . 165 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

B

4 × 4 Transfer Matrix Tp for Dielectric Homogeneous Films . . . . . . . . . . . . . . . . . . . . . . . . 169 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

C

4 × 4 Transfer Matrix Tp for Magneto-optic Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

D

4 × 4 Transfer Matrix Tp for Dielectric Helical Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

E

TM Waves at the Interface Between Two Half-Inﬁnite Media . . . . . . . . . . . . . . . . . . . . . . . . . 181 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

F

TM Waves at Two Stratiﬁed Interfaces . . . . . . . . . . . . . . . . . . . 185 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

1 Introduction

Semiconductor device structures consist of stacked layer systems with layers of diﬀerent alloy compositions, individual dopant incorporation, free-chargecarrier properties, and possibly with hundreds of interfaces. Such structures, as envisioned by Zhores I. Alferov and Herbert Kroemer, are the foundation for today’s world wide high-speed communication system, high-power light sources, and numerical processing abilities.1 Novel metastable materials, unavailable from natural sources, can be grown atomic layer by atomic layer using modern non-equilibrium growth techniques. Determination of their physical properties is mandatory for the appropriate design of heterostructure devices. Likewise, demands for control and characterization of individual layer properties during and after deposition reach out for adequate experimental techniques. Optical tools are non-invasive and non-destructive, and provide immediate access to fundamental physical parameters. This book is devoted to applications of infrared ellipsometry for exploration of lattice vibration (phonon) and free-charge-carrier (plasmon) modes in complex semiconductor layer structures. The infrared dielectric function of a multinary semiconductor alloy is a fundamental material property and can be obtained by ellipsometry. Lattice and free-charge-carrier contributions to the dielectric function can be diﬀerentiated and quantiﬁed upon model lineshape analysis, and are sensitive to strain, composition, and the state-of-order. Concentration, mobility, and effective mass parameters can be derived from the dielectric function, either in combination with electrical Hall-eﬀect or magneto-optic (generalized) ellipsometry measurements. For the latter, the sign of the free-charge-carrier species can be diﬀerentiated. Polarized Fano-, Brewster- or surface-guided waves enrich the variety of optical eﬀects in semiconductor heterostructures, and provide increased sensitivity to free-charge-carrier and optical phonon mode parameters. 1

The Nobel Prize in Physics for 2000 was awarded to Alferov, Kroemer and Kilby whose work laid the foundation of modern information technology, particularly through their invention of rapid transistors, laser diodes, and integrated circuits (chips).

Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 1–6 (2004) c Springer-Verlag Berlin Heidelberg 2004

2

1 Introduction

An experimental technique for precise determination of the complexvalued dielectric function is ellipsometry.2 Based upon Drude’s work more than hundred years ago,3 ellipsometry measures the polarization state of a parallel electromagnetic radiation ﬁeld after interaction with a layered sample of plane-parallel interfaces. Two independent parameters are obtained, which suﬃce to evaluate both real and imaginary parts of the dielectric function, or alternatively, to determine accurately the thickness of a single ﬁlm. Despite providing information about numerous materials physics parameters (Fig. 1.1), the importance of the infrared dielectric function was not studied for semiconductors, likely because precise infrared dielectric function spectra of semiconductors in layered structures were inaccessible. In the mideighties when David Aspnes and coworkers at Bell Labs solved the problem of automated instrumentation for spectroscopic ellipsometry in the visible to near ultra violet spectral region, precise dielectric function spectra for semiconductor materials became available [1, 7, 10, 11, 12, 13]. The ground work was laid for exact analysis of band-to-band transition energies from the dielectric function leading to direct comparison with theoretical band structure calculations [14, 15]. Eﬀects of surface overlayers, such as natural oxides, and the ability for in-situ monitoring paved the way for much improved epitaxial growth techniques [16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Inﬂuence of strain, alloying, and ordering are still subject of intense ellipsometric investigations [26, 27, 28]. In 1990 Arnulf R¨ oseler published a pioneering book merging the viable Fourier-transform infrared spectroscopy technique with the ellipsometry approach. A homemade prototype ellipsometer proved successful [29]. A main experimental problem in the infrared and far-infrared spectral range is the absence of intense and compact light sources. In the extreme cases this has been circumvented by making use of synchrotron light, which provides by about three orders of magnitude more brilliant light in the farinfrared as compared to conventionally available sources, like mercury arc lamps [30, 31, 32, 33, 34, 35]. The need for extension of ellipsometry to long wavelengths has led to diﬀerent designs of desk-top based systems using the less intense thermal sources (globars). With the commercial availability of laboratory-type infrared ellipsometers, using black-body radiator light 2

3

In principle, one can also use the intensity change of the electromagnetic radiation ﬁeld for reconstruction of the dielectric function. This is because of the physically required Kramers-Kronig consistency of the response function. The necessary information on the second part of the response function can be obtained upon Kramers-Kronig analysis of the measured intensity spectrum. This approach, extensively used before spectroscopic ellipsometers became available, delivers fundamentally erroneous results due to the unavoidable extension of the Kramers-Kronig integral into spectral regions were data are unavailable [1]. Paul Karl Ludwig Drude (12 July 1863–5 July 1906) worked in G¨ ottingen, Leipzig (1894–1898), Giessen and Berlin. Fundamental papers derived the equations of ellipsometry [2], and reported on ﬁrst experiments [3, 4, 5, 6, 7].

1 Introduction

3

ω [cm-1]

403

150

565

726

8066

40328

72590

10

ε2 E1TO

100

GaN

5

AlN

E1TO

50 0 0

E2 E1

ε1

0

E0 8

-50 4 -100 0.06

0.08

1 3 5 7 9

0

hω [eV]

Fig. 1.1. Complex dielectric function exemplarily for zincblende Al0.12 Ga0.88 N, determined by spectroscopic ellipsometry from the mid-infrared to the vacuum ultraviolet spectral region from a semiconductor layer structure [8, 9]. From ≈ 1 eV to ≈ 9 eV, the dielectric function reveals the electronic band-to-band transition energies (E0 , E1 , E2 ), exciton properties and the below-band-gap index of refrac√ tion ( ε), for example. At smaller wavenumbers, the dielectric function is governed by polar lattice mode excitations, caused here by the Ga-N and Al-N sublattices, and the free-charge-carrier excitations, which are reﬂected by the asymptotic increase (decrease) of the imaginary (real) parts of the dielectric function towards longer wavelengths according to the Drude model of the free-charge-carrier dielectric polarizability. Note the increase of magnitude of the polarizability at infrared wavelengths, and the amount of information contained within the long wavelength part

sources, since the last ﬁfteen years new light is being shed on the signiﬁcance of the long wavelength dielectric function in materials research [36, 37, 38]. As demonstrated here, phonon- and plasmon-related polariton excitations expose the spectral behavior of the dielectric function. Ellipsometry provides a precise means for their measurement. Phonon mode excitations cause distinct structures within the dielectric function similar to those observed near direct electronic band-to-band transitions at shorter wavelengths. These infrared phonon features are inﬂuenced by strain, composition, and ordering, like their electronic transition critical point counterparts, but often with diﬀerent and

4

1 Introduction

improved sensitivities. In-situ infrared ellipsometry techniques will be an exciting development for exploitation of the extreme interface mode sensitivity to surface overlayers [36]. A large potential for applications of the infrared magneto-optic ellipsometry approach for exploration of valence and conduction band structure parameters for semiconductor materials is at hand. The concept for understanding of the long-wavelength response of thinﬁlm samples and its characterization using spectroscopic ellipsometry is not restricted to inorganic semiconductors. Organic polymer and biologic materials are currently being intensively investigated. Determination of individual bond strengths, and bond orientations involve the generalized infrared ellipsometry concept developed here, addressing interesting research applications [39, 40, 41, 42, 43, 44]. Studies of free-charge-carrier properties in organic conducting or semiconducting materials for use in all-organic heterostructures establish another exciting research ﬁeld [45, 46]. Determination of the thin-ﬁlm metal-oxides’ dielectric functions in diﬀerent electrochromic states using infrared ellipsometry was demonstrated recently for design of all-solidstate (thin-ﬁlm metal-oxide) electrochromic devices for control of the 300 K low-orbit satellite surface-heat-emittance modulation [47, 48, 49, 50, 51]. The eﬀects of phase and microstructure on the infrared dielectric function of polymorph-polycrystalline boron nitride thin ﬁlms enabled measurement of phase contents and crystallite orientations and indicated moisture within thin-ﬁlm micropores – the potential cause for the problematic adhesion failure of the superhard coatings [52, 53, 54, 55, 56]. Nanostructure design and optical properties of electrochemically etched porous Bragg reﬂector layer structures can be controlled and optimized by use of infrared ellipsometry [57, 58].

References 1. D. E. Aspnes, in E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic Press, New York, 1998), vol. 1, 2nd ed. 2 2. P. Drude, Ann. Phys. 32, 584 (1887). 2 3. P. Drude, Ann. Phys. 34, 489 (1888). 2 4. P. Drude, Ann. Phys. 36, 532 (1888). 2 5. P. Drude, Ann. Phys. 36, 865 (1889). 2 6. P. Drude, Ann. Phys. 39, 481 (1890). 2 7. K. Vedam, Thin Solid Films 313-314, 1 (1998). 2 8. A. Kasic, M. Schubert, T. Frey, U. K¨ ohler, D. J. As, and C. M. Herzinger, Phys. Rev. B 65, 184302 1 (2002). 3 9. A. Kasic, Phonon, Free-Carrier Properties, and Electronic Interband Transitions of Binary, Ternary, and Quaternary Group-III Nitride Layers Measurend by Spectroscopic Ellipsometry (Shaker Verlag, Aachen, 2003). 3 10. D. E. Aspnes and A. Studna, Appl. Opt. 14, 220 (1975). 2 11. D. E. Aspnes and A. Studna, Rev. Sci. Instrum. 49, 291 (1978). 2 12. D. E. Aspnes, J. B. Theeten, and F. Hottier, Phys. Rev. B 20, 3292 (1979). 2

References

5

13. D. E. Aspnes, R. F. Potter, and S. S. So, eds., Optical Characterization Techniques for Semiconductor Technology, vol. 0276, Bellingham, WA (1981), SPIE. 14. P. Yu and M. Cardona, Fundamentals of Semiconductors (Springer-Verlag, Berlin, 1999). 2 15. U. Rossow and W. Richter, in G. Bauer and W. Richter, eds., Optical Characterization of Epitaxial Semiconductor Layers (Springer, Berlin, 1996), pp. 68–128. 16. D. E. Aspnes, W. E. Quinn, and S. Gregory, Appl. Phys. Lett. 56, 2569 (1990). 17. D. E. Aspnes, W. E. Quinn, M. C. Tamargo, M. A. A. Pudensi, S. A. Schwarz, M. J. S. P. Brasil, R. E. Nahory, and S. Gregory, Appl. Phys. Lett. 60, 1244 (1992). 2 18. J. A. Roth, W. S. Williamson, D. H. Chow, and G. L. Olson, J. Vac. Sci. Technol. B 18, 1439 (2000). 2 19. J. Phillips, D. Edwall, D. Lee, and J. Arias, J. Vac. Sci. Technol. 19, 1580 (2001). 2 20. S. Peters, T. Schmidtling, T. Trepk, U. W. Pool, J.-T. Zettler, and W. Richter, J. App. Phys. 88, 4085 (2000). 2 21. C. M. Herzinger, B. Johs, P. Chow, D. Reich, G. Carpenter, D. Croswell, and J. V. Hove, Mat. Res. Soc. Symp. Proc. 406, 347 (1996). 2 22. E. A. Irene and J. A. Woollam, MRS Bull. 20, 24 (1995). 2 23. B. Johs, C. M. Herzinger, J. H. Dinan, A. Cornfeld, J. D. Benson, D. Doctor, G. Olson, I. Ferguson, M. Pelczynski, P. Chow, et al., Thin Solid FIlms 313314, 490 (1998). 2 24. A. Bonanni, D. Stifter, A. Montaigne-Ramil, K. Schmidegg, K. Hingerl, and H. Sitter, J. Cryst. Growth 248, 211 (2003). 2 25. C. Chen, I. An, and R. W. Collins, Phys. Rev. Lett. 90, 217402 (2003). 2 26. J. Rivory and F. Abeles, Spectroscopic Ellipsometry: Proceedings of the First International Conference (Elsevier, Lausanne, 1993). 2 27. R. W. Collins, D. E. Aspnes, and E. A. Irene, eds., Spectroscopic Ellipsometry: Proceedings of the Second International Conference (Elsevier, Lausanne, 1998). 28. M. Fried, J. Huml´ıˇcek, and K. Hingerl, eds., Spectroscopic Ellipsometry: Proceedings of the Third International Conference (Elsevier, Lausanne, 2004). 2 29. A. R¨ oseler, Infrared Spectroscopic Ellipsometry (Akademie-Verlag, Berlin, 1990). 2 30. R. Henn, Untersuchung Infrarotaktiver Phononen und Freier Ladunsgtr¨ ager in Hochtemperatursupraleitern Mittels Ferninfrarot-Ellipsometrie, Ph.D. thesis, Max-Planck-Institut f¨ ur Festk¨ orperforschung Stuttgart (1997). 2 31. J. Kircher, R. Henn, M. Cardona, P. L. Richards, and G. P. Williams, J. Opt. Soc. Am. B 14, 705712 (1997). 2 32. R. Henn, C. Bernhard, A. Wittlin, M. Cardona, and S. Uchida, Thin Solid Films 313-314, 643648 (1998). 2 33. C. Bernhard, R. Henn, A. Wittlin, M. Kl¨ aser, G. M¨ uller-Vogt, C. Lin, and M. Cardona, Phys. Rev. Lett. 80, 1762 (1998). 2 34. A. A. Sirenko, C. Bernhard, A. Golnik, A. Clark, J. Hao, W. Si, and X. Xi, Nature 404, 373 (2000). 2 35. M. Gensch, K. Hinrichs, A. R¨ oseler, E. H. Korte, and U. Schade, Analytical and Bioanalytical Chemistry 376, 626 (2003). 2 36. E. H. Korte, K. Hinrichs, and A. R¨ oseler, Spectrochimica Acta B - Atomic Spectroscopy 57, 1625 (2002). 3, 4

2

2 2

2

6

1 Introduction

37. J. Huml`ıˇcek and T. Zettler, in E. A. Irene and H. W. Tompkins, eds., Handbook of Ellipsometry (William Andrew Publishing, 2004). 3 38. A. Kasic, M. Schubert, J. Oﬀ, B. Kuhn, F. Scholz, S. Einfeldt, T. B¨ ottcher, D. Hommel, D. J. As, U. Koehler, et al., Phonons and Free-Carrier Properties of Binary, Ternary, and Quaternary Group-III Nitride Layers Measured by Infrared Spectroscopic Ellipsometry (Wiley-VCH, 2003), pp. 1750 – 1769, Physica Status Solidi (C) - Conferences and Critical Reviews. 3 39. A. R¨ oseler and K.-H. Korte, Thin Solid Films 313-314, 708 (1998). 4 40. K. Hinrichs, M. Gensch, A. R¨ oseler, E. H. Korte, K. Sahre, K. J. Eichhorn, N. Esser, and U. Schade, Appl. Spectroscopy 57, 1250 (2003). 4 41. B. Jordanov, D. Tsankov, K. Hinrichs, R. Dietel, and E. H. Korte, Analytical and Bioanalytical Chemistry 375, 1246 (2003). 4 42. D. Tsankov, K. Hinrichs, E. H. Korte, R. Dietel, and A. R¨ oseler, Langmuir 18, 6559 (2002). 4 43. K. Hinrichs, D. Tsankov, E. H. Korte, A. R¨ oseler, K. Sahre, and K. J. Eichhorn, Applied Spectroscopy 56, 737 (2002). 4 44. D. Tsankov, K. Hinrichs, A. R¨ oseler, and E. H. Korte, phys. stat. sol. (a) 188, 1319 (2001). 4 45. M. Schubert, C. Bundesmann, G. Jakopic, and H. Arwin, Appl. Phys. Lett. 84, 200 (2003). 4 46. M. Schubert, C. Bundesmann, H. V. Weckstern, G. Jakopic, A. Haase, N.K. Persson, F. Zhang, H. Arwin, and O. Ingan¨ as, Appl. Phys. Lett. 84, 1311 (2004). 4 47. E. Franke, M. Schubert, C. L. Trimble, M. J. DeVries, J. A. Woollam, and F. Frost, J. Appl. Phys. 88, 5166 (2000). 4 48. E. Franke, M. Schubert, C. L. Trimble, and J. A. Woollam, Thin Solid Films 388, 283 (2001). 4 49. E. Franke, C. L. Trimble, J. S. Hale, M. Schubert, and J. A. Woollam, J. Appl. Phys. 88, 5777 (2000). 4 50. E. Franke, H. Neumann, M. Schubert, C. L. Trimble, and J. A. Woollam, Surf. Coat. Techn. 151-152, 285 (2002). 4 51. E. Franke, C. L. Trimble, J. S. Hale, M. Schubert, and J. A. Woollam, Appl. Phys. Lett. 77, 930 (2000). 4 52. M. Schubert, B. Rheinl¨ ander, E. Franke, H. Neumann, T. E. Tiwald, J. A. Woollam, J. Hahn, and F. Richter, Phys. Rev. B 56, 13306 (1997). 4 53. E. Franke, H. Neumann, M. Schubert, T. E. Tiwald, J. A. Woollam, and J. Hahn, Appl. Phys. Lett. 70, 1668 (1997). 4 54. E. Franke, M. Schubert, H. Neumann, T. E. Tiwald, D. W. Thompson, J. A. Woollam, J. Hahn, and F. Richter, J. Appl. Phys. 82, 2906 (1997). 4 55. M. Schubert, E. Franke, H. Neumann, T. E. Tiwald, D. W. Thompson, J. A. Woollam, and J. Hahn, Thin Solid Films 313-314, 692 (1998). 4 56. E. Franke, M. Schubert, J.-D. Hecht, H. Neumann, T. E. Tiwald, H. Yao, J. A. Woollam, and J. Hahn, J. Appl. Phys. 84, 526 (1998). 4 57. S. Zangooie, T. E. Tiwald, M. Schubert, and J. A. Woollam, J. Mat. Res. 16, 1241 (2001). 4 58. S. Zangooie, M. Schubert, C. Trimble, D. W. Thompson, and J. A. Woollam, Appl. Opt. 88, 906 (2001). 4

2 Ellipsometry

Ellipsometry, in general, determines the complex ratio ρ of linearly independent electric ﬁeld components of polarized electromagnetic plane waves. More speciﬁc, the change of the polarization state of an electromagnetic plane wave upon interaction with a sample is addressed thereby. Explicitly, for a given mode description in terms of, e.g., p and s polarized ﬁelds (amplitudes A stand for incident, B for exiting waves. Figure 2.1)1 Bp Ap ρ= / . (2.1) Bs As

P

As a

sample normal

Ap

E

Bp

E

A

Bs

Fig. 2.1. Deﬁnition of the plane of incidence (p plane) and the incidence angle Φa through the wave vectors of the incident and emerging (reﬂection set up) plane waves. Ap , As , Bp , and Bs , denote the complex amplitudes of the p and s modes before and after reﬂection, respectively. P and A are the azimuth angles of the linear polarizer used, e.g., in the standard arrangement of rotating-analyzer (polarizer) ellipsometer. P , or A are equal to zero if their preference directions are parallel to the p plane. (Both P and A rotate clockwise with respect to the light propagation.)

Traditionally, this ratio is measured in reﬂection or transmission for light polarized parallel (p), and perpendicular (s) to the plane of incidence [1, 2, 3, 4, 5, 6] (Fig. 2.1: Reﬂection set up). The result of an ellipsometry 1

Considerations are given for a reﬂection set up, but hold for the complex ratio τ of polarized plane wave components in the transmission arrangement as well.

Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 7–29 (2004) c Springer-Verlag Berlin Heidelberg 2004

8

2 Ellipsometry

measurement is usually presented by real-valued parameters Ψ and ∆, where tan Ψ is deﬁned as the absolute value of the complex ratio, and ∆ denotes the relative phase change of the p and s components of the electric ﬁeld vector [1]2 ρ = tan Ψ exp (i∆) . (2.2) The complex ratio ρ is often presented within the so-called pseudodielectric function. The pseudodielectric function is a direct inversion of the ellipsometric Ψ , ∆ parameters in (2.1), measured at angle of incidence Φa , assuming bulk behavior of the sample surface. The two-phase (ambient-substrate) model relates the (pseudo) substrate dielectric function with the ellipsometric parameters [1] 2 1− 2 2 sin Φa + cos Φa tan2 Φa . (2.3) ε = εa 1+ The ε presentation is commonly used if the surface overlayer eﬀects are negligible. The (true) substrate ε is then comparable with ε [7]. The complex ratio ρ can be addressed within diﬀerent presentations of the electromagnetic plane wave response. Also, depending on the sample properties, i.e., whether a surface reﬂects (or transmits) light upon mode conversion or not, the ellipsometric parameter set must be further expanded in to the so-called generalized ellipsometry parameter set. In such cases the Jones matrix presentation provides a convenient and suﬃcient frame. When samples depolarize the incident light the ellipsometric parameters must include the degree of polarization. Likewise, when the experimental equipment produces already partially depolarized light the Mueller matrix presentation is more appropriate.

2.1 Jones Matrix Presentation For nondepolarizing samples, the so-called Jones matrix, containing the complex-valued polarized reﬂection (transmission) coeﬃcients, provides a complete description for ellipsometric analysis [1, 8, 9, 10, 11]. In Fig. 2.2 a set of multiple beams, characterized by plane parallel wave fronts, may emerge from a nondepolarizing sample. Tracing the mode response of individual input (A) and output (X) channels, the Jones matrix elements jkl are obtained in generic form as follows3 deﬁning the Jones matrix j 2

3

The Ψ , ∆ notation is a traditional but not a necessary presentation. Unless used unambiguously as running index, the symbol “i” addresses the imaginary unit √ −1. In this notation the ﬁrst index denotes the incident polarization mode, and the second index refers to the outgoing polarization mode.

2.2 Mueller Matrix Presentation

Xi Xj

=j

Ai Aj

=

jii jji jij jjj

Ai Aj

9

.

(2.4)

p’

k’

s’

p’’

p k s

nondepolarizing sample

k’’

s’’ p ’’’ k’’’

s’’’’

p’’’’

s ’’’

k’’’’ Fig. 2.2. Incident and emergent electromagnetic plane waves at a nondepolarizing optical system. Jones matrices connect the incident (A) modes (p,s) with emergent (X) plane wave components (modes (p,s)’, (p,s)”, . . . ) with respect to a chosen coordinate system, e.g., the p and s planes in Fig. 2.1

If a sample with plane parallel boundaries is considered, a reﬂected and a transmitted beam may emerge, characterized by the Jones reﬂection and transmission matrices, r and t, respectively: rpp rsp tpp tsp r= , t= . (2.5) rps rss tps tss The oﬀ-diagonal elements are nonzero for optical systems that convert p into s waves and vice versa.4

2.2 Mueller Matrix Presentation An alternative description of the polarized sample response is the Mueller matrix and the Stokes vector formalism. This approach can furthermore completely account for depolarization. The four real-valued Stokes parameters of 4

The coordinate systems presenting the two mode polarizations for both the input and output channels must not necessarily coincide. For example, the input channel can be described in the p-s-system, and the output channel can be cast into left- and right-handed elliptically polarized modes. Oﬀ-diagonal Jones matrix elements then possibly appear diﬀerent from zero, although the sample may be isotropic.

10

2 Ellipsometry

an electromagnetic plane wave are deﬁned in terms of the p- and s-polarized coordinate system5 S0 = Ip + Is ,

(2.6)

S1 = Ip − Is , S2 = I45 − I−45 , S3 = Iσ+ − Iσ− ,

(2.7) (2.8) (2.9)

where Ip , Is , I45 , I−45 , Iσ+ , and Iσ− denote the intensities for the p-, s-, +45◦ , −45◦ , right-, and left-handed circularly polarized light components, respectively [1, 12]. S0 is proportional to the total intensity of the light beam, where S0 ≥ S12 + S22 + S32 . (2.10) The equality sign in (2.10) holds only if the light beam is totally polarized. Arranging the Stokes parameters into a column vector, the Mueller matrix then describes the changes of each quantity upon interaction of the electromagnetic plane wave with an optical system.6 One has      M11 M12 M13 M14 S0 S0  M12 M22 M23 M24   S1   S1      = . (2.11)  M13 M32 M33 M34   S2   S2  S3 output M14 M42 M43 M44 S3 input The advantage of this concept is the ability to handle situations with partial polarization of the electromagnetic plane wave, particularly when caused by device components’ imperfections. Details, use in ellipsometry, and relation to the Jones concept have been outlined previously [1, 12, 13, 14, 15, 16]. For example, in a rotating-analyzer-system the Mueller matrix elements of row 4 and column 4 are not measured, which does not impair the accessibility of the normalized Jones matrix elements, deﬁned in Sect. 2.1, except for its relative phase, and which can only be obtained by including compensator(s). Compensators are needed, in general, to access the additional Mueller matrix elements [12]. For a nondepolarizing system [1], a one-to-one relation exists between matrices r and M [17] (2.12)

M12

(2.13)

M13 5 6

1 2 2 2 2 |rpp | + |rss | + |rsp | + |rps | 2 1 2 2 2 2 |rpp | − |rss | − |rsp | + |rps | = 2 = Re (rpp r¯sp + r¯ss rps )

M11 =

(2.14)

The Stokes parameters have dimensions of intensities. Sample, mirrors, rotators, optical devices within the light path, and any combinations thereof.

2.3 Standard Ellipsometry

M14 = Im (rpp r¯sp + r¯ss rps ) 1 2 2 2 2 M21 = |rpp | − |rss | + |rsp | − |rps | 2 1 2 2 2 2 |rpp | + |rss | − |rsp | − |rps | M22 = 2 M23 = Re (rpp r¯sp − r¯ss rps ) M24 = Im (rpp r¯sp − r¯ss rps )

11

(2.15) (2.16) (2.17) (2.18) (2.19)

M31 = Re (rpp r¯ps + r¯ss rsp ) M32 = Re (rpp r¯ps − r¯ss rsp ) M33 = Re (rpp r¯ss + r¯ps rsp )

(2.20) (2.21) (2.22)

M34 = Im (rpp r¯ss − r¯ps rsp ) M41 = −Im (rpp r¯ps + r¯ss rsp )

(2.23) (2.24)

M42 = −Im (rpp r¯ps − r¯ss rsp ) M43 = −Im (rpp r¯ss + r¯ps rsp )

(2.25) (2.26)

M44 = Re (rpp r¯ss − r¯ps rsp )

(2.27)

where the vertical bar means the complex conjugate. The Mueller matrix for an isotropic sample is given by [15]   1 −N 0 0  −N 1 0 0   M = (2.28) 0 0 C S  . 0 0 −S C The quantities N , S, and C provide access to the ellipsometric parameters N = cos 2Ψ, S = sin 2Ψ sin ∆,

(2.29) (2.30)

C = sin 2Ψ cos ∆ .

(2.31)

N , S, and C are not independent, and are constrained for non-depolarizing samples by the relation N 2 + S2 + C2 = 1 . (2.32) The complex ratio ρ is obtained from N , S and C ρ=

C + iS . 1+N

(2.33)

2.3 Standard Ellipsometry By deﬁnition, in the standard ellipsometry situation Ψ and ∆ do not depend on the polarization state of the incident plane wave. Within the Jones presentation the generic expression is

12

2 Ellipsometry

ρ=

jp = tan Ψ exp (i∆) . js

(2.34)

Here jp and js denote the p- and s-polarized complex reﬂection (“jp,s ” = “rp,s ”) or transmission coeﬃcients (“jp,s ” = “tp,s ”).

2.4 Generalized Ellipsometry By deﬁnition, in the generalized ellipsometry situation Ψ and ∆ depend on the polarization state of the incident plane wave. This concept is valid within both, the Mueller matrix as well as within the Jones matrix formalism. Within the Jones presentation generic expressions for the generalized ellipsometry parameters are Ψij , ∆ij (“J, j” = “T , t” or “R, r”)7 jpp ≡ Jpp = tan Ψpp exp(i∆pp ) , jss

(2.35)

jps ≡ Jps = tan Ψps exp(i∆ps ) , jpp

(2.36)

jsp ≡ Jsp = tan Ψsp exp(i∆sp ) . jss

(2.37)

Note that the diagonal elements from the same column of the Jones matrix are used to normalize the matrix elements. This choice is convenient for Rotating-Analyzer ellipsometry [8]. For a Rotating-Polarizer ellipsometry setup a more intuitive choice would be to normalize the oﬀ-diagonal elements by the on-diagonal elements from the same row. The real-valued quantities Ψpp , Ψps , Ψsp , ∆pp , ∆ps , ∆sp now comprise the generalized ellipsometry data format. This deﬁnition may be changed by other users. However, the three complex quantities chosen here are independent, and all other choices may be expressed as a linear combination of this basic set. In this notation the complex oﬀ-diagonal ratios Rps (Tps ) and Rsp (Tsp ) behave similarly if the polarization transfer functions for the s and p polarized light are similar. This situation is observed for, e.g., chiral liquid crystal cells in normal incidence transmission [19], or in certain conﬁgurations for a uniaxial bulk or ﬁlm [20]. Depending on the sample properties and its orientation during the generalized ellipsometry experiment the oﬀ-diagonal normalized Jones matrix elements may be symmetrical or antisymmetrical, Hermitean or anti-Hermitean, completely diﬀerent or simply zero [21, 22, 23]. For small oﬀ-diagonal ratios the on-diagonal ratio Rpp (Tpp ) is very similar to the complex ratio determined by standard ellipsometry. In particular, if Rps (Tps ) and Rsp (Tsp ) vanish, Rpp (Tpp ) is identical to the standard ellipsometric ratio ρ (τ ) deﬁned in (2.34). 7

This set lacks the light beam’s absolute intensity and the light beam’s absolute phase information contained within the Jones matrix. For acquisition of generalized ellipsometry parameters from anisotropic samples see [1, 8, 18].

2.5 Brewster and Interface-Mode Conditions

13

Combining (2.1), (2.4), and (2.35)–(2.37) it can be seen that the complex ratio ρ represents a bilinear transformation of the incident polarization state χ ρ=

rpp /rss + (rsp /rss ) χ−1 , 1 + (rps /rpp ) (rpp /rss ) χ

χ≡

Ap , As

(2.38)

where the generalized ellipsometry parameters constitute the transformation coeﬃcients Rpp + Rsp χ−1 . (2.39) ρ= 1 + Rpp Rps χ Therefore, one can always express ρ in terms of tan Ψ and cos ∆ for any set of generalized ellipsometry parameters, which is then valid for a speciﬁc value of χ only. This implies that anisotropic samples can be measured in the “standard ellipsometry” mode, but for subsequent data model analysis the value of χ used must be stored in addition to Ψ and ∆. It can be seen that Ψ and ∆ will depend on χ exactly when the ratios containing oﬀ-diagonal Jones matrix elements do not vanish.8

2.5 Brewster and Interface-Mode Conditions The Jones matrix contains important information about polarization conditions of reﬂected or transmitted electromagnetic waves. These are the socalled generalized Brewster- and interface-mode-conditions [24, 25]. 2.5.1 Generalized Brewster Conditions In similarity with the situation under which Brewster observed the polarizing properties of surfaces when light is reﬂected under a distinct angle, a generalization can be formulated for an arbitrary non-depolarizing sample, described 8

Some commercial ellipsometer devices perform automatically so-called zoneaveraged measurements where ρ is determined as arithmetic mean from multiple settings of χ. Such positions include settings of the linear input polarizer mirrored by the plane of incidence to cancel out systematic polarizer errors. Because (2.39) is not symmetrical in χ, such zone-averaged data will be useless for further model analysis. For anisotropic samples zone-averaging cannot be used to account for slight analyzer or polarizer calibration oﬀsets because such oﬀsets virtually rotate the sample Jones matrix producing virtual oﬀ-diagonal elements. See also corresponding footnote in Sect. 2.1. Precise calibration of a rotating-element ellipsometer set up is mandatory to obtain correct generalized ellipsometry data. Care must also be taken in Rotating-Analyzer or -Polarizer ellipsometry, where commonly the polarizer position is tracked when operating in the standard ellipsometry mode. This is done because most sensitivity to the actual Ψ value is obtained when the input polarizer azimuth is close to the sample Ψ . Appropriate consideration of this “polarizer tracking” will also entangle subsequent data analysis.

14

2 Ellipsometry

properly by its Jones matrix r9 This is identical with the formulation of the polarizing condition, i.e., when completely depolarized light is reﬂected with one distinct polarization state. Then the polarization state described upon (2.4), Xi /Xj , is independent on the choice of Ai , Aj . It is easy to show that this is fullﬁlled when the determinant of r vanishes rii rji det =0. (2.40) rij rjj In general, two ﬁelds with diﬀerent polarizations emerge upon illumination with depolarized light rii Xi = , Xj rij

Xi rji = . Xj rjj

(2.41)

In the classical (isotropic) Brewster situation, only s-polarized light is reﬂected (“i” = “p”, “j” = “s”), and Xp /Xs = 0, regardless of Ap , As . 2.5.2 Generalized Interface-Mode Conditions The inversion of (2.40) can be used to ﬁnd situations where mode propagation parallel to the surface occurs. (rii rjj − rij rji )

−1

=0.

(2.42)

For isotropic surfaces, a motivation for (2.42) is given in Appendices E and F for electromagnetic ﬁeld propagation parallel to the interface with electric ﬁeld polarization perpendicular to the surface (transverse magnetic wave). In this case, the denominator of rpp must vanish, concordant with (2.42).

2.6 Anisotropic Dielectric Function Tensor 2.6.1 General Description In non-cubic solids, the phonon mode frequencies of the polar lattice vibrations depend, in general, on the phonon mode propagation direction. Likewise, directionally dependent free-charge-carrier scattering rates and the anisotropic inverse eﬀective free-charge-carrier mass tensor will produce nonscalar free-charge-carrier contributions. The infrared dielectric function is then represented by a complex-valued second-rank tensor ε, which can be 9

The equivalent consideration can be made for the transmission case.

2.6 Anisotropic Dielectric Function Tensor

15

expressed in Cartesian coordinates (x,y,z)10 

 εxx εxy εxz D = ε˜0 (P + E) = ε˜0 εE = ε˜0  εyx εyy εyz  E , εzx εzy εzz

(2.43)

where the ﬁeld-phasors displacement D, polarization ﬁeld P , and electric ﬁeld E are given along the unit directions x, y, z (˜ ε0 is the vacuum permittivity) D = xDx + yDy + zDz ,

(2.44)

E = xEx + yEy + zEz , P = xPx + yPy + zPz .

(2.45) (2.46)

2.6.2 Orthogonal Rotations Orthogonal rotations relate interior and exterior Cartesian coordinate axes of a sample under consideration. In order to address the ε tensor appropriately, let the plane of incidence (x, z) and the sample surface (x, y) set a righthanded Cartesian system (x, y, z) with origin at the sample surface. The (real-valued) Euler angles ϕ, ψ, and θ can be used to rotate between the laboratory (x, y, z) and the Cartesian crystal coordinate system (ξ, η, ζ) in Fig. 2.3. ε(x, y, z) = Aε(ξ, η, ζ)A−1 ,

(2.47) 11

where the unitary matrix A is the orthogonal rotation matrix [26] 



cos ψ cos ϕ − cos θ sin ϕ sin ψ − sin ψ cos ϕ − cos θ sin ϕ cos ψ sin θ sin ϕ A =  cos ψ sin ϕ + cos θ cos ϕ sin ψ − sin ψ sin ϕ + cos θ cos ϕ cos ψ − sin θ cos ϕ  . sin θ sin ψ sin θ cos ψ cos θ (2.48)

For orthorhombic, tetragonal, hexagonal, and trigonal symmetry, a real-valued rotation matrix A can be found such that ε is diagonal in (ξ , η , ζ ) 10

11

A nonsymmetric tensor decomposes into its symmetric and its antisymmetric part, and both can have three major complex elements and diﬀerent center-ofgravity systems. In general, the permittivity tensor is a function of the photon energy ¯ hω due to non-local response within the time domain (frequency dispersion). For mediums without spatial mirror (chiral) symmetry (optically active mediums), ε depends on the photon wave vector ¯ hk. This eﬀect, due to nonlocal response within the space domain (spatial dispersion), however, will not be addressed in this book. Note that A−1 = AT , where “T ” denotes the transpose of a matrix. The Euler angles describe the rotation from the laboratory system into the major crystal system.

16

2 Ellipsometry

z

ζ

η

θ

x

ϕ

ψ

ξ

y

Fig. 2.3. Deﬁnition of the Euler angles ϕ, ψ, and θ, and the orthogonal rotations as provided by A. (ξ, η, ζ) and (x, y, z) refer to the crystal and the laboratory coordinate systems, respectively





εx 0 0 ε = A  0 εy 0  A−1 . 0 0 εz

(2.49)

2.6.3 Bond Polarizability Model Restricting to linear polarization, the second-rank polarizability tensor χ is deﬁned by   χxx χxy χxz P = χE =  χyx χyy χyz  E . χzx χzy χzz

(2.50)

Intrinsic bond polarizations (eigenvectors) set up a spatial non-Cartesian (monoclinic, triclinic ), or Cartesian (orthorhombic, tetragonal, hexagonal, trigonal, and cubic ) center-of-gravity system, with axes described by vectors a = xax + yay + zaz , b = xbx + yby + zbz , and c = xcx + ycy + zcz . The linear polarization response is additive, and may be split into P = Pa +Pb +Pc ,

(2.51)

P a = ρa (aE) ,

(2.52)

P b = ρb (bE) ,

(2.53)

P c = ρc (cE) .

(2.54)

where

The complex-valued scalar polarizabilities ρa , ρb , ρc must obey KramersKronig consistency, and correspond to the intrinsic center-of-gravity bond

2.6 Anisotropic Dielectric Function Tensor

17

polarization system. Accordingly, the corresponding part of the (symmetric) dielectric function tensor ε is easily deconvoluted by expanding (2.50) and (2.52)–(2.54) [27] εxx = 1 + ax ax ρa + bx bx ρb + cx cx ρc ,

(2.55)

εxy = ax ay ρa + bx by ρb + cx cy ρc ,

(2.56)

εxz = ax az ρa + bx bz ρb + cx cz ρc ,

(2.57)

εyy = 1 + ay ay ρa + by by ρb + cy cy ρc ,

(2.58)

εyz = ay az ρa + by bz ρb + cy cz ρc ,

(2.59)

εzz = 1 + az az ρa + bz bz ρb + cz cz ρc .

(2.60)

Note that naturally εij = εji (i = j ). Functions ρa , ρb , ρc can be described, in principle, by model functions f (¯hω , X1 , X2 , X3 , . . .), which depend on parameters Xi and photon energy ¯hω . Thereby, appropriate physical lineshape models must be invoked, as shown exemplarily for phonon and plasmon modes in Chap. 3. For anisotropic materials, besides the coordinates of the unit axes a, b, and c, three polarizability functions need to be diﬀerentiated, which can be identiﬁed by (major-axes) dielectric function spectra εj (ω ) for certain symmetries only. For biaxial materials with orthorhombic symmetry, with their coordinate system coincident with a given laboratory system (ay = az = bx = bz = cx = cy = 0, ax = by = cz = 1) one has expectedly D = x(1 + ρa )Ex + y(1 + ρb )Ey + z(1 + ρc )Ez ,

(2.61)

D = xεa Ex + yεb Ey + zεc Ez ,

(2.62)

where the polarization vector in a given direction exclusively depends upon the electric ﬁeld component in that same direction.12 For uniaxial materials (hexagonal, tetragonal, trigonal ), such as for the wurtzite group-III nitrides, or for sapphire, titanium dioxide, or the noncubic polytypes of silicon carbide for example, the index j corresponds to electric ﬁeld polarization parallel (j = “||”) or perpendicular (j = “⊥”) to the high-symmetry lattice c-axis. The dielectric tensor is symmetric in all indices, regardless to the strength of the anisotropy, and regardless to the actual orientation of the dielectric tensor’s center-of-gravity (major tensor axis orientations) with respect to the laboratory frame of reference. For uniaxial and orthorhombic systems real and imaginary parts of the dielectric tensor belong to the same Cartesian center-of-gravity coordinate system. Monoclinic or triclinic systems are not discussed in this book, but are of potential interest. Such systems virtually possess diﬀerent center-of-gravity 12

Note that for a purely dielectric material, an optical axis must be deﬁned by a line, not by a (one-directional) vector [28]. Indeed, changing the sign of a, b, and/or c does not change the result for the polarization ﬁeld!

18

2 Ellipsometry

systems for real and imaginary parts when presented by a dielectric function tensor within Cartesian coordinates. As a result a wavelength-dependent rotation of the (apparent) center-of-gravity systems will be observed, an eﬀect which has become known as angular dispersion.13

2.7 Light Propagation in Layered Anisotropic Media A 4 × 4 matrix approach, often cited as the Berreman-formalism, can be used to calculate the electromagnetic response of stratiﬁed anisotropic materials with plane parallel boundaries conveniently in terms of the Jones matrix elements [29].14 This approach is devoted to details of the internal sample polarization-modifying process, whereas the Jones matrices describe the external polarizing sample properties. A global transfer matrix T describes the response of the entire multiple-layered structure (Fig. 2.4)     As

 Bs     Ap  Bp

frontside

Cs

 Ds   = T  Cp  Dp

.

(2.63)

backside

A

C

B

D

Fig. 2.4. Schematic presentation of incident (A), reﬂected (B), and transmitted (C) plane waves across a sample with plane parallel interfaces, and multiple layer stacks at the front side of the supporting substrate. (Indices for eigenpolarizations, e.g., p,s, are suppressed.) D modes (if present) are incident from the right. The substrate may totally absorb C and/or D. The wave vectors for modes C and D split if the substrate is anisotropic

A characteristic transfer matrix, Tp , accounts for the optical properties of a single homogeneous layer within a stratiﬁed sample.15 Analytical solutions for Tp are given within Appendix A if ε is a complex scalar (isotropic materials), Appendix B if ε is a complex symmetric anisotropic 13

14

15

Such systems can be analyzed using the generalized ellipsometry technique that includes materials with arbitrary dielectric tensor properties [8, 18]. Details of the Berreman-formalism have been outlined on numerous occasions, and derivation is omitted here for brevity. See, for instance [1, 18, 30, 31]. The permeability is assumed unity throughout (µ = 1).

2.7 Light Propagation in Layered Anisotropic Media

19

tensor (uniaxial and biaxial materials), Appendix C if ε is a complex nonsymmetric anisotropic tensor (magneto-optic materials), and Appendix D if ε is a complex symmetric anisotropic tensor with rotational twist perpendicular to the layer interfaces.16 Matrices for incident (ambient) (La ) and exit (Lf ) mediums embed the layer stack according to the layer stack surrounding [8, 38]. The transfer matrix T follows from the ordered product of all n layers’ matrices Tp , embraced by the incident and exit matrices [9, 38, 39] −1 −1 T = L−1 a Tp1 . . . Tpn Lf .

The ambient matrix  L−1 a

0 0 1 =  2  (cos Φa )−1 −(cos Φa )−1

1 1 0 0

−(na cos Φa )−1 (na cos Φa )−1 0 0

(2.64)



0  0  , 1/na  1/na

(2.65)

depends on the angle of incidence Φa and the index of refraction na of the (isotropic) ambient material. If the substrate is birefringent the C modes may couple. The exit matrix follows from the assumption that the total transmitted ﬁeld within the substrate consists of a linear combination of the Eigenvectors Ξi of the characteristic coeﬃcient matrix ∆ (A.2). The coeﬃcients are the amplitudes Cp and Cs . The exit matrix Lf is then obtained as follows (Lf )jk = Ξjk , j, k = 1 . . . 4 ,

(2.66)

and explicit expressions of Ξjk for materials with ε being a complex nonsymmetric anisotropic tensor are given in [39]. If the major axes of a material with symmetric tensor ε coincide with the laboratory axes (x, y , z ), modes C √ decouple, and Lf reads (nl = εl , “l ” = “x ”, “y ”, “z ”)   0

0

cos Φz − cos Φz  0  .  0 nx nx

1 1 0 Lf =   −ny cos Φy ny cos Φy 0 0

0

(2.67)

The angle Φl is calculated from Snell’s law cos Φl =

1 − ([na /nl ] sin Φa )2 .

(2.68)

The Jones matrices r and t follow then from T, solving (2.63).17 For example, the reﬂection mode generalized ellipsometry parameters read (“Jij ” = “Rij ”, “ij ” = “pp ”, “ps ”, “sp ” ) [9] 16

17

Chiral nematic liquid crystals [32] or helically-shaped sculptured solid-state thin ﬁlms comprise such materials [33, 34, 35, 36, 37]. For example, when deriving elements of r, then modes D in Fig. 2.4 must be set to zero.

20

2 Ellipsometry Rpp =

T11 T43 − T41 T13 , T21 T33 − T23 T31

(2.69)

Rps =

T11 T12 − T21 T13 , T11 T43 − T41 T13

(2.70)

Rsp =

T41 T33 − T43 T31 . T21 T33 − T23 T31

(2.71)

2.8 Ellipsometry Data Analysis The ellipsometric parameters depend on the photon energy ¯hω ,18 the layer sequence within the sample, each layer’s (major-axes) dielectric function spectra εj (ω ),19 each layer’s ε tensor Euler orientation angles ψ , ϕ, and θ, each layer’s thickness d, the dielectric function of the substrate material that supports the layer stack, the ambient material’s dielectric function, and the angle of incidence Φa . The standard model for analyzing ellipsometry data is based on a sequence of homogeneous layers with smooth and parallel interfaces. For anisotropic materials, the ellipsometric measurements depend on the principal-axis dielectric functions εx , εy , and εz , the orientation of the crystal principal axes with respect to the plane of incidence, and the polarization state of the incident light beam [1, 8, 20, 38, 40]. For a general situation of optical anisotropy and sample orientation the simple deﬁnition of ρ in (2.2) is no longer a complete description, and the generalized ellipsometry approach outlined in Sect. 2.4 is needed. 2.8.1 Thickness and Dielectric Function The dielectric function of a bulk material with no overlayer can be directly inverted from the measured ellipsometry spectra by using (2.3). For bulk materials with orthorhombic symmetry, similar inversions can be found when combining standard ellipsometry data taken at high symmetry orientations in multiple model analysis procedures. In principle, thickness and/or ε of each layer from a complex layer structure can be determined from an ellipsometry experiment by comparing calculated data to measured data, and by using numerical regression analysis techniques. For layers with isotropic material, the thickness and the complex-valued dielectric function parameters are correlated, and cannot be diﬀerentiated from a single-wavelength ellipsometry 18

19

This monograph is explicitly concerned with spectroscopic ellipsometry measurements. For materials with monoclinic and triclinic symmetry the ellipsometric parameters depend on the polarization functions ρa , ρb , ρc and their non-Cartesian axes a, b, and c described in Sect. 2.6. Casting these into eﬀective major-axes dielectric function spectra εj (ω) diagonalizing the dielectric function tensor may not always be possible.

2.8 Ellipsometry Data Analysis

21

experiment, regardless of the number of angles of incidence involved.20 In order to avoid this general correlation problem one can either acquire spectroscopic ellipsometry data over regions where the dielectric function undergoes suﬃcient changes, or one can invoke the so-called multiple-sample analysis technique. For the latter, layers of identical material but with suﬃciently diﬀerent thickness values must be available. A combined analysis including data from both (or more) layers simultaneously enables to break up the correlation between the individual thickness values and the dielectric function. This situation naturally occurs during the in-situ observation of a ﬁlm growth by using spectroscopic ellipsometry, and this is why such techniques enable on-line determination of growth rate and material properties. For the case where only one layer is available, the spectroscopic ellipsometry parameters provide suﬃcient sensitivity to the thickness and the dielectric function if part of the data cover a spectral region where the layer is not too much absorbing, and where the real part of the dielectric function changes according to some known physical behavior. Such situations arise when transparent layers are measured below the fundamental electronic band-to-band transitions, where the real part of the dielectric function can be well described by the well-known Cauchy formulae. A similar situation is that of a polar material at infrared wavelengths, where below and above the restrahlen band the layer is transparent but highly absorbing within, and where well-known lineshape functions (e.g. Lorentzian broadened harmonic oscillator lineshapes) are suﬃcient to describe the dielectric function. 2.8.2 Wavelength-by-Wavelength Fit Traditionally, “Wavelength-by-wavelength ﬁts” (also referred to as “pointby-point ﬁts”) are performed where the dielectric function values of interest are extracted from the experimental data for each wavelength, and independent of all other spectra data points. For this procedure, thickness of the particular layer as well as dielectric functions and thickness of all other sample constituents have to be known. The dielectric function obtained from the point-by-point ﬁt needs then further comparison with model assumptions in order to obtain values of physically relevant parameters such as phonon mode frequencies and broadening parameters. 2.8.3 Parameterized Model Fit Instead, ﬁtting parameterized model dielectric functions to experimental data, simultaneously for all spectral data points, provides direct connection between measured data and physical parameters of interest. Parametric models further prevent wavelength-by-wavelength random measurement errors 20

The correlation problem between thickness and dielectric function is of less concern when studying arbitrarily-oriented anisotropic samples [41]. See Sect. 7.3.

22

2 Ellipsometry

from becoming part of the extracted dielectric functions. Parametric model assumptions greatly reduce the number of free parameters. Use of parametric models involves a certain risk for subtle spectral features to be subsumed by the lineshape of the model function. Nevertheless, parameterization of ε based on a physical model is the best choice for ellipsometry data analysis, especially when the wavelength-by-wavelength ﬁt method is inapplicable. Kramers-Kronig consistent model functions can impose additional physicality on the modeling procedure, which potentially can provide resistance to small systematic errors.21 2.8.4 Ellipsometry Test Functions In addition to Ψ and ∆ (the standard ellipsometry situation), or Ψpp , ∆pp , Ψps , ∆ps , and Ψsp , ∆sp (the generalized ellipsometry situation), their standard deviations σ Ψ and σ ∆ are measured [42, 43]. The Levenberg-Marquardt algorithm can be used, for instance, in order to ﬁt the model parameters by minimizing the following weighted test function (maximum likelihood approach) [44] 2 2 S Ψi − Ψ c ∆ i − ∆c 1 2 ξSE =

i

2S − K + 1

i

+

σiΨ

i=1

σi∆

,

(2.72)

where S denotes the number of measured data pairs (Ψi , ∆i ), K is the number of real-valued ﬁt parameters, and Ψic and ∆ci are the calculated ellipsometric parameters at photon energy E = ¯hωi [43, 45]. For the generalized ellipsometry situation, the test function is set up accordingly [20] 2 Ψ −Ψ c 2 Ψ −Ψ c 2 S c Ψpp,i −Ψpp,i ps,i sp,i 2 ps,i sp,i 1 ξGE = +

1 6S−K+1

i=1 ∆ S

i=1

+

Ψ σpp,i

6S−K+1

c pp,i −∆pp,i ∆ σpp,i

2 +

∆

Ψ σps,i c ps,i −∆ps,i ∆ σps,i

+

2 +

∆

Ψ σsp,i c sp,i −∆sp,i ∆ σsp,i

2

. (2.73)

Likewise, if Mueller matrix elements Mij are involved, a test function may be constructed in the following manner 2 ξM E

S J 1 = JS − K + 1 i=1 l=1

21

Mil − Milc M

σil il

2 ,

(2.74)

For example, in wurtzite GaN two of its principal dielectric functions, the extraordinary ε|| (parallel to the c-axis), and the ordinary ε⊥ (perpendicular to the c-axis), diﬀer from each other. For c-plane oriented ﬁlms the standard ellipsometry situation applies and point-by-point ﬁts are inapplicable. This is because two independent values are provided by Ψ and ∆, but four independent pieces of information are needed for ε|| and ε⊥ .

2.9 Infrared Ellipsometry

23

where J denotes the number of individual Mueller matrix elements, and σilMil are their standard deviations. Combinations of test functions may be obtained when data sets measured in diﬀerent ellipsometry modes are to be included into multiple-data-type best-ﬁt regression procedures. This situation arises, for instance, during analysis of magneto-optic generalized ellipsometry data taken from isotropic semiconductor layer structures. There the ﬁeld-dependent data eventually must be taken in the Mueller matrix ellipsometry mode, alternatively in the generalized ellipsometry mode, and ﬁeldfree measurements are performed in the standard ellipsometry mode (see Sect. 8.2). 2.8.5 Parameter Accessibility Issues of parameter accessibility from spectroscopic ellipsometry data have been thoroughly discussed, for instance by Jellison ([42] and therein). The random experimental errors (σiΨ , σi∆ ) must be propagated appropriately into the error bars on the ﬁt parameters. These error bars represent existing and ﬁnite correlation values between the ﬁt parameters. The light propagation within the sample is calculated using standard matrix formalism for anisotropic multilayered systems with plane parallel interfaces, as outlined in Sect. 2.7.22 Parameter correlation as well as parameter accessibility must always be checked upon numerical evaluation of the parameter correlation matrix elements provided by the numerical regression procedure.

2.9 Infrared Ellipsometry A schematic drawing of an infrared ellipsometry set up is shown in Fig. 2.5. Brieﬂy, the light source illuminates the entrance slot of a Michelson interferometer (Fourier-transform infrared spectrometer), which in principle acts as the wavelength-diﬀerentiating device.23 The output slot of the interferometer illuminates the beam-folding optics, which redirects the beam onto the sample. The light reﬂected or transmitted from the sample is collected by another beam-folding optics, which sends the remaining infrared light to the detector. 22

23

Data acquisition and analysis for anisotropic materials in generalized ellipsometry was discussed in detail previously [8, 19, 20, 37, 38, 40, 46, 47, 48, 49, 50, 51, 52, 53]. A forthcoming handbook publication on ellipsometry further contains substantial information [6]. A detailed mathematical and technical description of the Fourier-transformbased infrared ellipsometry schemes is given in the textbook by A. R¨ oseler [12], where the interested reader is referred to. Special attention must be paid to the polarization characteristics of the Fourier-transform light source, which must be properly accounted for during the calibration and measurement procedures. Likewise, the polarization characteristics (including partial depolarization eﬀects) of all optical elements must be determined prior to the ellipsometric measurement.

24

2 Ellipsometry

A visible-wavelength pre-aligned laser beam, which can be brought into the system by a switchable mirror, is used for sample adjustment. Computerized stepper-motor driven rotation stages hold polarizer, analyzer, and eventually compensator, if available.

Computer control

Alignment beam coupler Motor control

FTIR source

20:1

HeNe Laser

Focusing optics

Rotating polarizer

Theta stage (sample)

2-Theta stage (analyzer, detector) Rotating compensator Rotating analyzer Focusing optics Detector

Fig. 2.5. Schematic drawing of a (far) infrared rotating-polarizer sample rotatingcompensator rotating-analyzer ellipsometry setup

At ﬁxed angle of incidence Φa , polarizer (P ), analyzer (A), and compensator azimuth (C ), a complete interferogram is monitored and stored, eventually repeated by a certain number of mirror scans and averaged for noise reduction. The next step is to rotate, for example, the compensator by a certain amount, and repeat interferogram monitoring. This is done until one full turn of the compensator is stored. This cycle may be repeated after either polarizer or analyzer, or both, are rotated to another position. Finally, the data set is Fourier-transformed into intensity spectra versus polarizer, analyzer and compensator azimuths. Then, for each wavenumber, these intensities are

2.9 Infrared Ellipsometry

25

analyzed by another Fourier analysis with respect to rotations P , A, and C , and the resulting coeﬃcients provide access to the Mueller matrix element information of the sample [12]. A fundamental advantage of rotating-polarizer, rotating-compensator ellipsometry over traditional infrared intensity spectroscopy techniques is that the atmospheric absorption, in general, does not strongly aﬀect the ellipsometric parameters as long as the atmospheric conditions remain reasonably stable during one full revolution of the polarizer and analyzer. Slowly time-dependent changes are cancelled out by the acquisition procedure. However, rapid changes of the atmospheric conditions will aﬀect the data, and should be avoided. Background measurements, which are necessary to suppress the spurious atmospheric absorption structures from reﬂectance or transmittance data for even stable atmospheric conditions, are dispensed with the ellipsometry technique. The diﬃculties for setting up precise ellipsometric systems operating at long wavelengths are mainly (i) the imperfections of available polarizers, and (ii) the low intensity of feasible light sources. The latter point requires to work with large apertures in front of the source, which is commonly a silicon carbide rod (globar) held at elevated temperatures. Consequently, the infrared light beam transfers through the optical set up as a diverging ray bundle, rather than a highly collimated beam. Focussing mirrors are needed to redirect and recollect the beam, introducing geometrical abberations. Always, the light beam that interacts with the sample contains wave fronts entering under a range for angles of incidence. The exit beam contains a mixture of polarization states belonging to wave fronts reﬂected or transmitted within this angular range. The exit beam is partially depolarized. For a 10 mm aperture and a symmetric arrangement of mirrors with 300 mm focal lengths this angular beam deviation range is within 3◦ . . . 5◦ . The ﬁrst point is of more concern and more diﬃcult to handle. Polarizer imperfections cause depolarization. As discussed in Sect. 2.2 the Stokes and Mueller matrix formalism is appropriate for description of depolarization. Before ellipsometric measurements, determination of the Mueller matrices of all optical components must be performed. Once these matrices are known, the sample Mueller matrix can be extracted from measurement of the ellipsometric data from the sample. A concise description of depolarization treatment in infrared ellipsometry was given in the textbook by Arnulf R¨ oseler. Many details are discussed there, and instructions for possible set ups are provided. Of particular importance is also the eﬀect of the Michelson interferometer that is used as the monochromatic light source. Such interferometers possess characteristic polarization dependence, which must be taken into account properly. Meanwhile, infrared ellipsometers are available commercially, largely based on R¨ oseler’s concepts. Most examples discussed in this

26

2 Ellipsometry

(1)

(2)

(3) (5) (4) (6)

(9)

(7)

(8) 5 mm Fig. 2.6. Schematic drawing (to scale) of the light path through a far infrared ellipsometer. A plane mirror (1) folds the (nearly parallel) source beam onto an oﬀ-axis paraboloidal mirror (2) (f = 295 mm), which focuses the beam through the polarizer (3) onto the sample surface (9). After reﬂection through the analyzer (4), and compensator (5), a second oﬀ-axis paraboloidal mirror (6), f = 295 mm reﬂects a parallel bundle onto another oﬀ-axis paraboloidal mirror (not shown, f = 190.5 mm), whose focal point coincides with the detector area of a liquidhelium cooled bolometer (7). The sample stage is mounted on the θ arm, analyzer, compensator, and bolometer are mounted on the 2θ arm of the θ − 2θ goniometer (8)

book were measured with a rotating-polarizer-, rotating-compensator-type, Fourier-transform-based variable angle-of-incidence conﬁguration. A prototype far infrared ellipsometer was built upon similar schemes, replacing the detector unit by a liquid helium cooled bolometer (Fig. 2.6). For details of data acquisition procedures, alignment, and treatment of systematic and random error sources the reader is referred to [6, 12, 54, 55, 56, 57].

References

27

References 1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (NorthHolland Publ. Co., Amsterdam, 1984). 7, 8, 10, 12, 18, 20 2. D. Kliger, J. Lewis, and C. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, New York, 1990). 7 3. H. Tompkins, A User’s Guide to Ellipsometry (Academic Press, New York, 1993). 7 4. R. A. Chipman, in Handbook of Optics (McGraw-Hill, New York, 1995), vol. 2, chap. 22, 2nd ed. 7 5. R. M. A. Azzam, in Handbook of Optics (McGraw-Hill, New York, 1995), vol. 2, chap. 27, 2nd ed. 7 6. H. Thompkins and E. A. Irene, eds., Handbook of Ellipsometry (William Andrew Publishing, Highland Mills, 2004). 7, 23, 26 7. D. Aspnes, J. Vac. Sci. Technol. 17, 1057 (1980). 8 8. M. Schubert, in E. A. Irene and H. W. Tompkins, eds., Handbook of Ellipsometry (William Andrew Publishing, 2004). 8, 12, 18, 19, 20, 23 9. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988). 8, 19 10. M. Mansuripur, The physical principles of magneto-optical recording (University Press, Cambridge, 1995). 8 11. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984). 8 12. A. R¨ oseler, Infrared Spectroscopic Ellipsometry (Akademie-Verlag, Berlin, 1990). 10, 23, 25, 26 13. G. E. Jellison and F. Modine, Appl. Opt. 36, 8184 (1998). 10 14. G. E. Jellison and F. A. Modine, Appl. Opt. 36, 8190 (1998). 10 15. G. E. Jellison, in E. A. Irene and H. W. Tompkins, eds., Handbook of Ellipsometry (William Andrew Publishing, 2004). 10, 11 16. C. Chen, I. An, and R. W. Collins, Phys. Rev. Lett. 90, 217402 (2003). 10 17. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998). 10 18. M. Schubert, in W. S. Weiglhofer and A. Lakhtakia, eds., Introduction to Complex Mediums for Optics and Electromagnetics (SPIE, Bellingham, WA, 2004), pp. 677 – 710. 12, 18 19. M. Schubert, B. Rheinl¨ ander, C. Cramer, H. Schmiedel, B. Johs, C. M. Herzinger, and J. A. Woollam, J. Opt. Soc. Am. A 13, 1930 (1996). 12, 23 20. M. Schubert, B. Rheinl¨ ander, J. A. Woollam, B. Johs, and C. M. Herzinger, J. Opt. Soc. Am. A 13, 875 (1996). 12, 20, 22, 23 21. M. Elshazly-Zaghloul, R. M. A. Azzam, and N. M. Bashara, Surf. Sci. 96, 41 (1980). 12 22. H. W¨ ohler, M. Fritsch, G. Haas, and D. A. Mlynski, J. Opt. Soc. Am. A 8, 536 (1991). 12 23. K. Eidner, J. Opt. Soc. Am. A 6, 1657 (1989). 12 24. D. B. Brewster, Philos. Trans. R. Soc. Lond. 105, 105 (1815). 13 25. A. Lakhtakia, Optik 90, 184 (1992). 13 26. H. Goldstein, Classical Mechanics (Addison-Wesley, 1965). 15 27. C. M. Herzinger (2002), private communication. 17 28. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1980), 6th ed. 17 29. D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972). 18

28

2 Ellipsometry

30. I. Abdulhalim, L. Benguigui, and R. Weil, J. Phys. 46, 815 (1985). 18 31. M. Garriga, Ph.D. thesis, Universit¨ at Stuttgart (198?). 18 32. S. Chandrasekhar, Liquid Crystals (Cambridge University Press, Cambridge, 1992). 19 33. R. Messier and A. Lakthakia, Mat. Res. Innovat. 2, 217 (1998). 19 34. Q. Wu, I. Hodgkinson, and A. Lakhtakia, Opt. Eng. 39, 1863 (2000). 19 35. I. Hodgkinson, Q. Wu, B. Knight, A. Lakhtakia, and K. Robbi, Appl. Opt. 39, 642 (2000). 19 36. I. Hodgkinson and Q.Wu, Appl. Opt. 38, 3621 (1999). 19 37. M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Oﬀ, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, et al., in A. Lakhtakia, G. Dewar, and M. W. McCall, eds., Complex Mediums III: Beyond Linear Isotropic Dielectrics (SPIE, Bellingham, WA, 2002), vol. 4806, pp. 264 – 276. 19, 23 38. M. Schubert, Phys. Rev. B 53, 4265 (1996). 19, 20, 23 39. W. Xu, L. Wood, and T. Golding, Phys. Rev. B 61, 1740 (2000). 19 40. M. Schubert, Thin Solid Films 313-314, 323 (1998). 20, 23 41. C. Bundesmann, N. Ashkenov, M. Schubert, A. Rahm, H. v. Wenckstern, E. M. Kaidashev, M. Lorenz, and M. Grundmann, Thin Solid Films 455-456, 161 (2004). 21 42. G. E. Jellison, Thin Solid Films 313-314, 33 (1998). 22, 23 43. C. M. Herzinger, P. G. Snyder, B. Johs, and J. A. Woollam, J. Appl. Phys. 77, 1715 (1994). 22 44. W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes: The Art of Scientiﬁc Computing (Cambridge University Press, Cambridge, 1988). 22 45. C. M. Herzinger, H. Yao, P. G. Snyder, F. G. Celii, Y.-C. Kao, B. Ohs, and J. A. Woollam, J. Appl. Phys. 77, 4677 (1995). 22 46. T. E. Tiwald and M. Schubert, in L. M. Hanssen, ed., Optical Diagnostics Methods for Inorganic Materials II (SPIE, Bellingham, WA, 2000), vol. 4103, pp. 19–29. 23 47. M. Schubert, T. E. Tiwald, and J. A. Woollam, Appl. Opt. 38, 177 (1999). 23 48. J.-D. Hecht, A. Eiﬂer, V. Riede, M. Schubert, G. Krauß, and V. Kr¨ amer, Phys. Rev. B 57, 7037 (1998). 23 49. E. Franke, H. Neumann, M. Schubert, T. E. Tiwald, J. A. Woollam, and J. Hahn, Appl. Phys. Lett. 70, 1819 (1997). 23 50. M. Schubert, T. Hofmann, B. Rheinl¨ ander, I. Pietzonka, T. Saß, V. Gottschalch, and J. A. Woollam, Phys. Rev. B 60, 16618 (1999). 23 51. A. Kreuter, K. Otte, G. Lippold, G. Wagner, A. Schindler, and M. Schubert, Appl. Phys. Lett. 78, 195 (2001). 23 52. T. Hofmann, M. Schubert, and C. M. Herzinger, in A. Duparr´e and B. Singh, eds., Advanced Characterization Techniques for Optical, Semiconductor, and Data Storage Components (SPIE, Bellingham, WA, 2002), vol. 4779, pp. 90– 97. 23 53. M. Schubert, T. Hofmann, and C. M. Herzinger, J. Opt. Soc. Am. A 20, 347 (2003). 23 54. T. E. Tiwald, D. W. Thompson, J. A. Woollam, W. Paulson, and R. Hance, Thin Solid Films 313-314, 661 (1998). 26 55. D. W. Thompson, M. J. DeVries, T. E. Tiwald, and J. A. Woollam, Thin Solid Films 313-314, 341 (1998). 26

References

29

56. T. Hofmann, Infrarot-Spektralellipsometrie und Raman Spektroskopie an III-VHalbleiter¨ ubergittern, Master’s thesis, Universit¨ at Leipzig (2000). 26 57. T. Hofmann, Far-Infrared Spectroscopic Ellipsometry on A-III B-V Semiconductor Heterostructures, Ph.D. thesis, Universit¨ at Leipzig (2004). 26

3 Infrared Model Dielectric Functions

This chapter reviews a selection of dielectric function models progressed for lineshape analysis of (far) infrared ellipsometry data. Contributions to the dielectric function due to polar lattice vibrations (phonons), impurity (lowpolarity) vibration modes, and free-charge-carrier plasma excitations are discussed. A condensed derivation for the classical anisotropic magneto-optic free-charge-carrier contribution is given.

3.1 Polar Lattice Resonance Contributions Figure 3.1 depicts transverse optical (TO) phonon and longitudinal optical (LO) phonon modes of selected binary zincblende III-V compound semiconductors as a function of the equilibrium lattice constants.1 Here, the optical equivalent of the phonon mode frequency is given in units of µm, and the classiﬁcation of their location within the mid and far infrared spectral regions is obvious (vertical arrows).2 The lattice mode frequencies of the group-III nitrides correspond to infrared wavelengths, with some overlap to the far infrared, whereas the group-III-(As,P)-compounds reveal phonon mode frequencies within the far infrared. The lighter (heavier) elements generally cause higher (lower) frequencies. The “strictness” of a particular phonon mode, i.e., the amount of the energy equivalent of the mode frequency, is a measure for the strength of the bond, the masses attached by the bond, and the distance of the separated masses (length of the bond). The lattice modes possess polar momentums due to the large ionicity of the bonds between the group-III and group-V elements. Hence, the optical phonon modes split into transverse and longitudinal branches. The amount of the TO-LO splitting is a measure for the ionicity of the respective bond. The polar momentum also aﬀects the dielectric polarizability of the materials. The dielectric function exhibits signiﬁcant dispersion near the phonon mode frequencies. The lattice 1

2

For individual sources see, e.g., Landolt-B¨ ornstein, Numerical Data and Functional relationshsip in Science and Technology, edited by O. Madelung, M. Schultz, H. Weiss, New Series, vol. 17 (1982); reprinted in Semiconductors-Basic Data, 2nd edn, ed by O. Madelung (Springer, New York 1996). Unless speciﬁed otherwise, it is generally referred to the infrared spectral range.

Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 31–43 (2004) c Springer-Verlag Berlin Heidelberg 2004

32

3 Infrared Model Dielectric Functions

constant increases with increasing atomic radius of the group-V-component, and the bonds “soften” and the phonon mode wavelength equivalents extend to longer wavelengths.

1000

BN

BP

-1

ωTO & ωLO [cm ]

AlN

BAs AlP

GaN

100

4

InN

AlAs

InP GaP GaAs InAs

5 a0 [Å]

6

Fig. 3.1. Γ -point TO and LO phonon mode frequencies ωTO (solid symbols) and ωLO (open symbols) of the binary zincblende III-V compound semiconductor materials as a function of their equilibrium lattice constant a0

Model Dielectric Functions are employed for the lineshape analysis of ellipsometry data. The infrared dielectric response ε at photon energy E = ¯hω of a solid crystalline media with l (multiple) polar optical phonon modes is commonly described by harmonic oscillator functions with Lorentzian lineshapebroadening for each infrared-active phonon branch (HOA approach) ε = ε∞ +

l i

2 2 4πFTOi ωTOi , 2 − ω − iωΓTOi

2 ωTOi

(3.1)

where FTOi , ΓTOi and ωTOi are strength, broadening, and resonance energy parameters of the i-th TO phonon, respectively, and ε∞ is the high-frequency dielectric constant. Resonant excitation of TO phonons by electromagnetic light waves causes large dielectric displacement D , and corresponds to the poles of ε. Propagation of polar LO phonons requires zero divergence of the dielectric displacement (∇D = 0), and therefore zeros in ε [1]. The poles of the dielectric loss function, ε−1 , are at ωLOi , and their broadening parameters ΓLOi , and strengths FLOi are related to the large dielectric loss at LO phonon

3.1 Polar Lattice Resonance Contributions

33

frequencies.3 The dielectric loss function ε−1 within the HOA approach is form invariant to ε, and yields ε−1 = ε−1 ∞ +

l i

2 2 4πFLOi ωLOi , 2 ωLOi − ω 2 − iωΓLOi

(3.2)

and ωLOi , ΓLOi , FLOi are functions of ωTOj , ΓTOj , FTOi , and ε∞ . Gervais and Piriou [2] pointed out that diﬀerent phonon decay times should be considered for LO and TO phonons if an LO frequency equals several times that of its corresponding TO frequency. Huml´ıˇcek commented on the HOA approach that the broadening constants of the same phonon branch, ΓLOi and ΓTOi , diﬀer already from each other in multiple-phonon-mode crystals with large TO-LO splitting values [3]. However, within the HOA model the parameters ωLOi , ΓLOi , and FLOi depend only on ωTOj , ΓTOj , and FTOi , and the decay mechanism of each LO phonon is tied to that of all TO phonons. Equation (3.1) can also be considered as partial fraction decomposition ε = ε∞

l 2 ωLOi − ω 2 − iωγLOi i

2 ωTOi − ω 2 − iωγTOi

.

(3.3)

In this symmetric form, the dielectric function ε and the dielectric loss function ε−1 can easily be interchanged [4, 5]. Within the HOA approach the free parameters are ε∞ , ωTOi , ωTOi , and γTOi . The parameters γLOi could be derivable from the other parameters and the deﬁnition that, e.g., (∂ /∂ω )[ω Im{ε−1 (ω )}] = 0 at ω = ωLOi [5]. Lowndes [5], and later Gervais and Piriou [2] extended that model for l = 1, and l > 1, respectively, by identifying γLOi as adjustable parameters.4 In this factorized form of the dielectric function, independent broadening constants γLOi and γTOi for the LO- and TO-phonons of all i phonon branches are considered, respectively. Allowing for diﬀerent broadening constants includes the eﬀects of anharmonic phonon mode coupling, whereas the harmonic approximation fails to describe the typically asymmetric dielectric response function of multiple phonon mode materials especially near phonon mode frequencies.5 In both model equations (3.1) and (3.3) the static dielectric constant ε0 is related to the high-frequency dielectric constant ε∞ through ωLOi and ωTOi 3

4

5

The actual peak positions in Im{ε} and Im{ε−1 } occur at slightly diﬀerent frequencies than the TO and LO modes, respectively. This, however, is not of further importance because both parameters are accessed through line shape analysis of the ellipsometry data, and not from peak position assignment within dielectric function spectra. Note, however, that (3.1) and (3.3) describe the same lineshape if Γi = γTO,i ≈ γLO,i ωTO,i . A plausible picture for anharmonic phonon mode coupling eﬀects using classical oscillators is given by Barker and Hopﬁeld [6].

34

3 Infrared Model Dielectric Functions

[ε or ε−1 at ω = 0 in (3.1), (3.2), or (3.3), and γLOi , γTOi ωLOi , ωTOi ], which is known as the Lyddane-Sachs-Teller relation [7, 8, 9].6 ε0 = ε∞

l 2 ωLOi i

2 ωTOi

.

(3.4)

Gervais and Piriou pointed out that the conditions found by Lowndes γLOi −γTOi > 0 and γLOi /γTOi ≤ (ωLOi /ωTOi )2 [5], must be satisﬁed to keep the physical meaning of ε, i.e., to obtain positive extinction coeﬃcients. However, these conditions were derived by Lowndes for l = 1, and are not valid for l >

1, as can be easily proved. In fact, the data obtained by Gervais and Piriou demonstrated that the condition γLO > γTO is not fulﬁlled for the second (Eu ) TO-LO pair in sapphire. A more general condition for multiple mode materials, i.e., for l > 1 can be obtained from the high-frequency limit in (3.3). Setting ζ = 1/ω , and expanding the denominator in (3.3) for ζ → 0, one obtains l 2 2 ε∼ 1 + iζγLOi + O ζ 1 − iζγTOi + O ζ . (3.5) = ε∞ i ζ→0

From there one has Im {ε} > 0|ζ→0 ⇔

l

(γLOi − γTOi ) > 0 ,

(3.6)

i

which is a generalization of the ﬁrst of Lowndes’ conditions.7 For multiplemode-materials the condition γLO > γTO has to be replaced by (3.6), i.e., the 6

7

Confusion often arises concerning the use of ε∞ within (3.1), (3.2), and (3.3). This parameter accounts for the electronic contributions to the dielectric polarizability whose transition energies fall outside the upper limit of the actual spectral range studied. In contrast to ε0 , which is a well-deﬁned physical quantity, ε∞ depends on that part of the electronic polarization, which has been excluded upon data acquisition. For well-separated absorption bands, such as due to the infrared-active phonon modes, or the electronic band-to-band transitions, ε∞ obtained as best-ﬁt parameter from an infrared ellipsometry data analysis may well correspond to the index of refraction within the band gap, i.e., for photon energies well above the phonon modes but also well below the fundamental bandto-band transition energy. However, in a strict sense, the electronic transitions at higher energies always induce dispersion. The index of refraction within the band gap is always superimposed by a small fraction of that dispersion, which renders a precise assessment of ε∞ from infrared data analysis diﬃcult. In the same sense, care must be taken when comparing ε∞ parameters determined from infrared data, and “ε∞ ” values derived from below-band-gap index of refraction measurements. This condition still does not guarantee that Im{ε} ≥ 0 at frequencies near ωTOi , ωLOi . There should exist further constraints on the choice of γTOi , γLOi , which must depend on ωTOi , ωLOi , and all other parameters ωTOj , ωLOj , γTOj , γLOj , and ε∞ .

3.2 Free-Charge-Carrier Contributions

35

sum over the diﬀerences between the LO and TO broadening parameters of all infrared-active branches. For some of those, γTOi may exceed γLOi , as, e.g., for sapphire and rutile [10, 11].

3.2 Free-Charge-Carrier Contributions Drude’s contribution to physical optics – the discovery of the duality between electrical and optical properties of matter – has spurred many opticians and physicists since more than a century. Atoms contain electrons, some of which are free to move, governed by the Fermi energy and allowed momentum states, within correlated systems when literally countless atoms compose to liquid or solid state materials. In semiconductors, valence band electrons, captured from their host sites into bound acceptor states, leave positively charged lattice sites behind, which serve for a “free-hole” transport upon spatial rearrangement of other valence band electrons with similar momentum and energy. Likewise, electrons from bound ionized donator states populate unoccupied conduction band states increasing the number of freely moveable electrons. Attracted by electric ﬁelds the free charge carriers drift, upon acceleration, with their peculiar mobility between regions of diﬀerent potentials, being dressed with an eﬀective free-charge-carrier mass according to the law of inertia. Coulomb attraction forces staggering allegiance of the free charge carriers to electromagnetic radiation, with “inertial”-mass and mobility depending on frequency and momentum. Energy can be transferred from an electromagnetic wave to the system matter upon carriers suﬀering collisions with other carriers, phonons, impurities, defects, and so on. Thereby, valuable band structure and lattice quality information can be obtained.8 Free charge carriers in semiconductors form collective excitation modes, the so-called plasma modes (plasmons). The plasma modes will couple to the LO lattice modes and form the so-called coupled LO plasmonphonon (LPP) modes. Depending on the strength of the coupling the free carriers thereby inﬂuence the dielectric function. The excitation of polar 8

Although not invoked within this monograph, the polaron-mass picture must also be mentioned here. In this concept, the free-charge-carrier is subject to Coulomb attraction by opposite charge centers in lattices with strong polar moment. Upon movement, the free-charge-carrier causes local distortion of the bond charge distribution, impeding its intertial moment. The system free-charge-carrier and local bond distortion forms the so-called polaron. From Drude’s point of view, however, it is indistinguishable whether a free charge carrier is moving through a (completely) unpolar lattice or if instead a polaron has formed due to strongly polarized local bond distributions. In a strict sense, the simultaneous occurrence of infrared-active polar lattice resonances is indicative for the existence of polarons, and simple approximation schemes can be used to subtract the polaron eﬀect from the eﬀective mass parameter determined by an infrared optical experiment [12].

36

3 Infrared Model Dielectric Functions

lattice vibrations in III-V semiconductor compound materials with infrared or far infrared wavelength-equivalents occurs for incident electromagnetic waves with small wavevectors q : |q| ≈ 100 cm−1 . . . ≈ 600 cm−1 (far infrared) or |q| ≈ 600 cm−1 . . . ≈ 3000 cm−1 (infrared), respectively. The wavevectors amount to a small fraction of the Brillouin zone only (≈108 cm−1 ), and the q -dependence of the phonon and plasmon mode frequencies near the Γ -point can be neglected. The latter is important for the determination of the freecharge-carrier plasma properties from their fundamental plasma excitation frequency, ωp (q), which can obey considerable dispersion for light scattering at visible wavelengths [13]. The classical Drude equation holds suﬃciently for description of the contribution to ε(FC) due to k free-charge-carrier species [9, 14] (FC)

ε

(ω) = −

k s=1

with ∗ ωp,s

=e

∗ (ωp,s )2 , ω(ω + iγp,s )

(3.7)

Ns . ε˜0 ms

(3.8)

The unscreened plasma frequencies ω *p,s depend on the sth species’ freecharge-carrier concentration Ns , and eﬀective mass ms (ε˜0 is the vacuum permittivity, and e the amount of the electrical unit charge). The plasmon broadening parameters γp,s are related to the inverse of the energy-averaged carrier-momentum relaxation time τm s . In a constant free-charge-carrier scattering regime (the time derivative of the electron momentum distribution function is constant), the complex average momentum lifetime −1 τm = τm (1 − iωτm )

can be written as

,

τm

= τm (1 − iω τm )−1 ,

(3.9)

(3.10)

and τm can be substituted by the optical free-charge-carrier mobility µ [1, 15, 16]9 −1 e γp,s ≡ τm s

=

ms µs

.

(3.11)

Plasmons strongly interact with LO phonons when the screened plasma frequency ∗ ωp,s ωp,s = √ ε∞,s

9

(3.12)

Free-charge-carrier species, which diﬀer in their eﬀective mass and mobility parameters, could potentially reveal similar plasma broadening parameters, virtually merging their contributions to the dielectric polarizability into that of a singlecomponent plasma term. In this case, a magneto-optical infrared-ellipsometry experiment, as described in Sect. 8.2, potentially yields further information on the multiple-component plasma properties.

3.2 Free-Charge-Carrier Contributions

37

is in the range of the lattice TO and LO modes. Coupling of LPP modes is often studied by Raman scattering [16], e.g., in n-type wurtzite GaN [17, 18, 19, 20]. However, the LPP modes can only be observed in Raman spectra when the plasmon is not too much damped. Especially in the case of p-doping, such as in wurtzite p-type GaN or p-type GaAs, overdamping of plasmons (ωp γp ) occurs. This is because of small free-hole densities due to higher acceptor activation energies, larger hole eﬀective masses, and usually lower mobility values of holes compared to electrons [21, 22, 23]. However, as discussed in Sect. 7.4, overdamped plasmons still contribute to the dielectric function, from where their properties can be obtained upon lineshape analysis. Kukharskii [24] suggested that the dielectric function for semiconductors with free charge carriers can be transformed into a factorized expression similar to that described above allowing for anharmonic coupling eﬀects between plasmon excitations and LO phonons in multiple polar phonon mode materials (k species of free charge carriers, l lattice phonon bands):

k+l

2 ω 2 + i˜ γLO,j ω − ω ˜ LO,j

j=1

ε(L+FC) (ω) = ε∞ ωk

k

(ω + iγp,s )

s=1

l

2 ω 2 + iγTO,j ω − ωTO,j

,

(3.13)

j=1

where ω ˜ LO,ij and γ˜LO,ij are the eigenfrequencies and the broadening values of the (k + l) LPP modes, respectively. The γp,s values are treated as the plasmon broadening parameters in the long-wavelength limit. Equation (3.13) was successfully used to describe the anharmonic character of LPP modes within reﬂection intensity, and ellipsometry data of low to moderately and heavily doped n- and p-type GaAs [24], and wurtzite n-type GaN, respectively [25, 26]. For example, in n- and p-type wurtzite GaN (k = l = 1), the LPP mode frequencies ω ˜ LO,i in (3.13) are the well-known roots of ε = ε(L) + ε(FC) , if damping is neglected (γTO = γLO = γp = 0; j = “1”, “2”): ω ˜ LO,j =

1 2 ωLO + ωp2 + (−1)j 2

2 2 + ωp2 )2 − 4ωp2 ωTO , (ωLO

(3.14)

where ω ˜ LO,1 (ω ˜ LO,2 ) is the screened frequency of the lower (upper) LPP mode branch. The equations above connect N , m, and µ with ε(L+FC) . Note that only the quantities ζ≡

N , m

ξ≡

N , µ

(3.15)

can be determined from the plasma frequency and broadening parameters.10 Hence, the eﬀective mass and mobility parameters can only be obtained when 10

These quantities should not be confused with the Cartesian coordinates in Sect. 2.6.

38

3 Infrared Model Dielectric Functions

the free-charge-carrier concentration N is known. Likewise, N can be determined when the m and/or µ values are known. Note further that ε∞ aﬀects the plasma frequency obtained from lineshape analysis of the infrared dielectric function, and incorrect ε∞ values potentially distort the free-charge-carrier parameter results.

3.3 Low-Polarity Mode Contributions Equation (3.14) is no longer valid for materials with more than one infraredactive phonon band (l > 1). Infrared-active modes (ν = 1 . . . n) with small 2 2 − ωTO,ν , i.e., modes with low polarity LO-TO splitting values δων2 = ωLO,ν modes such as lattice resonances caused by impurities (impurity modes IM, ωIM,ν ≡ ωTO,ν ), contribute to ε as a small perturbation only11 ε(L+FC+IM) (ω) = ε(L+FC) (ω)

n

1+

ν=1

iδγν ω − δων2 2 ω 2 + iγIM,ν ω − ωIM,ν

,

(3.16)

where a new broadening parameter was introduced: δγν ≡ γLO,ν − γTO,ν . For (L+FC) δγν ≈ 0, and small Im{εj } values for ω ≈ ωIM,ν , it follows from (3.14 that δων2 < 0 [δων2 > 0] if the frequency of the IM is within the spectral range ωTO,i < ωIM,ν < ωLO,i [ωLO,i < ωIM,ν < ωTO,i+1 ].12 If contributions due to IM’s are present within experimental spectra, but neglected during data analysis, then it can be read from (3.16) that a new “high-frequency” limit value ε˜∞ will result from the line shape ﬁt n 2 ε˜∞ (ω = 0) = ε∞

1+

ν=1

δων 2 ωIM,ν

.

(3.17)

Accordingly, ε˜∞ will be too small (too large) for IM’s not considered, but located within ωTO,i < ωIM,ν < ωLO,i (ωLO,i < ωIM , ν < ωTO,i+1 ). Appropriate consideration of low polar IM’s allows to derive the screened plasma frequencies ωp,j , and one obtains for l = 1 lattice phonon band and n impurity modes 11

12

The use of the term “impurity mode” is largely historical. Lattice imperfections (elementary impurities, voids, interstitial atoms) may cause vibration modes conﬁned to respective lattice sites, and with frequencies, which commonly diﬀer from the host lattice modes. Alloying activates modes with small polarity, supposedly caused by diﬀerent states of local order and composition “roughness”, and which can be well described by the “impurity mode” model. Such order-induced modes are denoted by “alloy-induced” modes or “additional” modes (AM). This condition follows from the natural “TO-LO” sequence rule, a consequence of the linear response theorem. Other than the generalized “Lowndes” rule, which ensures Im{ε} ≥ 0 for ω → ∞, the “TO-LO” rule here ensures Im{ε} ≥ 0 in general, i.e., for all ω, provided that δγ ≈ 0.

3.4 Free-Charge-Carrier Magneto-Optic Contributions

ωp = e

N ε∞ ε˜0 m

n

1+

ν=1

δων2 2 ωIM,ν

39

0.5 .

(3.18)

From the requirement Im{ε(L+FC+IM) }|ω→∞ ≥ 0, one can derive yet another constraint for δγν , regardless of δων2 k+l

(˜ γLO,i − γTO,i ) +

i=1

n

δγν ≥ 0 .

(3.19)

ν=1

In summary, for parameterization of the long-wavelength dielectric function of a multiple (l) polar mode semiconductor material with species (k) of free charge carriers and impurity (n) band absorptions either the harmonic sum presentation (suppressing anharmonic coupling eﬀects) ε (ω) = ε∞ +

l j

k

−

s=1

2 2 4πFTO,j ωTO,j 4πAIM,j ωIM,j + 2 2 ωTO,j − ω 2 − iωΓTO,j ωIM,j − ω 2 − iωΓTO,j n

j

ω∗2p,s ω (ω + iγp,s )

,

(3.20)

or the anharmonic factorized form may be used allowing for independent broadening of TO- and LO-type excitations ε (ω) = ε(L+FC+IM) .

(3.21)

3.4 Free-Charge-Carrier Magneto-Optic Contributions In this section, the Drude infrared dielectric function tensor model for freecharge-carrier magneto-optic eﬀects in semiconductors will be shortly reviewed. This tensor has to be added to the bond polarizability model in Sect. 2.6.3 if polar lattice vibrations are present. Chapter 8 contains examples for application of the free-charge-carrier magneto-optic plasma tensor and bond polarizability model. The free-charge-carrier magneto-optic contribution to the dielectric tensor ε is obtained through the conductivity tensor σ 13   εxx εxy εxz i ε(FC−MO) =  εyx εyy εyz  = − σ. ε˜0 ω εzx εzy εzz

(3.22)

The components of σ in SI units and Cartesian coordinates can be derived from the equation of motion for a free charge carrier with charge q (q = +|e| for free holes, and q = −|e| for free electrons) 13

See, e.g., Pidgeon, [14].

40

3 Infrared Model Dielectric Functions

m

me q

(γ + ∂t ) v = [E + µ0 H (v × h)] ,

(3.23)

and with anisotropic inverse eﬀective mass tensor (m0 m )−1 , and spatially dispersive but energy-independent inverse relaxation time tensor (scattering rate) γ when moving in a constant magnetic ﬁeld µ0 H = µ0 H(hx , hy , hz ) = µ0 Hh, with velocity where 14

is the driving ﬁeld. using Ohm’s law

v = (∂t x, ∂t y, ∂t z) ,

(3.24)

E = (Ex , Ey , Ez )e−iωt ,

(3.25)

Assuming a harmonic plane wave solution for v , and j = σE = qN v ,

(3.26)

results in   0 ε(FC−MO) (ω) = − ωp∗2  ω 2 + iωγ I − iω ωc  hz

−1

−hz hy 0 −hx  −hy hx 0

, (3.27)

where the following tensor may be deﬁned as the (unscreened) “plasma frequency” tensor ∗2 e2 −1 ωp

≡N

ε˜0 m0

m

,

(3.28)

and which depends on the free-charge-carrier concentration N (free-chargecarrier density), and the free-charge-carrier inverse eﬀective mass tensor. The tensor leading the antisymmetric magnetic ﬁeld contribution in (3.27) may be termed “cyclotron frequency” tensor ωc ≡ q

µ0 H m0

m −1 ,

(3.29)

which depends on the magnitude H of the magnetic ﬁeld, the inverse eﬀective free-charge-carrier mass tensor, and the free-charge-carrier type. As can be seen from ε(FC−MO) , the magnetic ﬁeld causes symmetry breaking in ε. For H = 0, (3.27) is identical with (3.7). In (3.27) the magnetic ﬁeld contribution scales with the inverse eﬀective mass tensor m −1 . Therefore, if the magneticﬁeld induced changes in the dielectric function tensor can be determined, information on the eﬀective mass tensor elements can be obtained. With B = µ0 H(0, 0, −1), and with isotropic scattering and inverse eﬀective mass tensors (scalars), the MO plasma tensor in (3.22) simpliﬁes15 14

15

Note that µij = µji and γij = γji is assumed here. For isotropic scatter rates γ = γp is the commonly used plasma broadening parameter. Within the coordinate system used here for the ellipsometric data analysis, the z-axis points towards the bottom of the sample. Refer to Chap. 2, and Fig. 2.4. The magnetic ﬁeld orientation addresses the PMO setup deﬁned in Sect. 7.1.

3.4 Free-Charge-Carrier Magneto-Optic Contributions

(FC−MO)

ε

(ω) = −

ωp ω

2

with ε˜xx (ω) = (1 + i ε˜xy (ω) = (1 +



ε˜xx

i˜ εxy 0

0

0

 −i˜ εxy ε˜xx 0  ,

1+i

γp ω

γp 2 ) ω

−

ωc ω γp 2 iω) −

(3.30)

ε˜zz

ω 2 ,

(3.31)

c

ω

ωc 2 ,

(3.32)

ω

1 . γ 1 + i ωp

ε˜zz (ω) =

41



(3.33)

For ω ωc , εxy is proportional to ωc = µ0 H(q /m). Its wavenumber dependence provides q /m. The in-plane component εxx provides ωp∗ and γp . The out-of-plane component εzz does not depend on the magnetic ﬁeld, and is best measured prior to the MO experiment. Resonance in εxx and εxy occurs for ω ≈ ωc , provided that ωc γp . Typical frequencies of ωc are within the ω ωc , εxx ≈ εzz and εxy ≈ ω −3 . microwave region. For √ √ For B = µ0 H(1/ 2, 0, −1/ 2), and with isotropic scattering and inverse eﬀective mass tensors (scalars), the MO plasma tensor reads16     2 ε˜xx 0 ε˜xz 0 ε˜xy 0 ωp  0 ε˜yy 0  + i  −˜ ε(FC−MO) (ω) = (3.34) εxy  , εxy 0 −˜ ω

ε˜xz 0

ε˜xx

with (1 + i

ε˜xx (ω) = (1 +

γp 2 ) ω

γ i ωp )((1 1 2

ε˜xz (ω) = (1 + i

0

γp )((1 ω

ε˜yy (ω) = (1 + i

+

−

1 2

(1 + i

c

ω

−

ω γp 2 ) ω

−

ωc 2

1+i

γp ω

γp 2 ) ω

−

1 ωc √ 2 ω

ε˜xy (ω) =

ω 2

γ i ωp )2

+i

γp 2 ) ω

−

ε˜xy 0

ω 2 , c

ω

ω 2 , c

ω

(3.35)

)

(3.36)

)

ωc 2 ,

(3.37)

ω

ω 2 .

(3.38)

c

ω

The tensor ε(FC−MO) (ω) is composed of a symmetric and an antisymmetric part. For ω ωc , εxx ≈ εyy , εxy ≈ ω −3 , and εxz ≈ ω −4 . The symmetric contribution to the MO birefringence εxz depends on the square of the cyclotron frequency, and is not sensitive to sgn(q). 16

This addresses a possible OMO setup deﬁned in Sect. 8.1.

42

3 Infrared Model Dielectric Functions

Provided the magnetic ﬁeld magnitude is known (h can actually be obtained from the data analysis as well), an infrared magneto-optic generalized ellipsometry experiment can provide suﬃcient information to determine N , m, and µ of a free-charge-carrier species independently, and avoiding to make contacts as needed for electrical Hall-eﬀect measurements. Experiments on layer structures constructed of layered mediums with scalar eﬀective mass tensors are discussed in Chap. 8.

References 1. C. M. Wolfe, N. Holonyak, and G. E. Stillmann, Physical Properties of Semicoductors (Prentice Hall, New Jersey, 1989). 32, 36 2. F. Gervais and B. Piriou, J. Phys. C 7, 2374 (1974). 33 3. J. Huml´ıˇcek, Phil. Mag. B 70, 699 (1994). 33 4. D. Berreman and F. Unterwald, Phys. Rev. 174, 791 (1968). 33 5. R. P. Lowndes, Phys. Rev. B 1, 2754 (1970). 33, 34 6. A. S. B. Jr. and J. J. Hopﬁeld, Phys. Rev. 135, A 1732 (1964). 33 7. R. H. Lyddane, R. G. Sachs, and E. Teller, Phys. Rev. 59, 673 (1941). 34 8. W. Cochran and R. A. Cowley, J. Phys. Chem. Solids 23, 4471 (1962). 34 9. C. Kittel, Introduction to Solid States Physics (John Wiley and Sons, New York, 1986). 34, 36 10. M. Schubert, T. E. Tiwald, and C. M. Herzinger, Phys. Rev. B 61, 8187 (2000). 35 11. T. E. Tiwald and M. Schubert, in L. M. Hanssen, ed., Optical Diagnostics Methods for Inorganic Materials II (SPIE, Bellingham, WA, 2000), vol. 4103, pp. 19–29. 35 12. C. Klingshirn, Semiconductor Optics (Springer-Verlag, Berlin, 1995). 35 13. G. Abstreiter, M. Cardona, and A. Pinczuk, in M. Cardona and G. G¨ untherodt, eds., Light Scattering in Solids (Springer, Berlin, 1984), vol. 4. 36 14. C. Pidgeon, in M. Balkanski, ed., Handbook on Semiconductors (North-Holland, Amsterdam, 1980), pp. 223 – 328. 36, 39 15. B. Rheinl¨ ander, Phys. Lett. 29A, 420 (1969). 36 16. P. Yu and M. Cardona, Fundamentals of Semiconductors (Springer-Verlag, Berlin, 1999). 36, 37 17. T. Kozawa, T. Kachi, H. Kano, Y. Taga, M. Hashimoto, N. Koide, and K. Manabe, J. Appl. Phys. 75, 1098 (1994). 37 18. P. Perlin, J. Camassel, W. Knap, T. Taliercio, J. C. Chervin, T. Suski, I. Grzegory, and S. Porowski, Appl. Phys. Lett. 67, 2524 (1995). 37 19. D. Kirillov, H. Lee, and J. S. Harris, J. Appl. Phys. 80, 4058 (1996). 37 20. F. Demangeot, J. Frandon, M. Renucci, C. Meny, O. Briot, and R. Aulombard, J. Appl. Phys. 82, 1305 (1997). 37 21. G. Popovici, G. Xu, A. Botchkarev, W. Kim, H. Tang, A. Salvador, H. Morko¸c, R. Strange, and J. White, J. Appl. Phys. 82, 4020 (1997). 37 22. F. Demangeot, J. Frandon, M. Renucci, N. Grandjean, B. Beaumont, J. Massies, and P. Gibart, Solid State Commun. 106, 491 (1998). 37 23. H. Harima, T. Inoue, S. Nakashima, K. Furukawa, and M. Taneya, Appl. Phys. Lett. 73, 2000 (1998). 37 24. A. A. Kukharskii, Sol. State Com. 13, 1761 (1973). 37

References

43

25. A. Kasic, M. Schubert, S. Einfeld, D. Hommel, and T. Tiwald, Phys. Rev. B 62, 7365 (2000). 37 26. A. Kasic, M. Schubert, B. Kuhn, F. Scholz, S. Einfeldt, and D. Hommel, J. Appl. Phys. 87, 3720 (2001). 37

4 Polaritons in Semiconductor Layer Structures

The infrared optical response of layered structures composed of polar semiconductor materials is strongly inﬂuenced by resonant excitation of phononand plasmon-supported bulk polaritons. In addition, subtle eﬀects are caused by interface modes (surface polaritons , surface guided waves ) at which electromagnetic radiation is mediated parallel to the interfaces of thin-ﬁlm structures. Precise information on phonon and plasmon mode parameters can be obtained from measurement and model analysis of the frequency-dependent sample response, e.g., in terms of the complex ratio ρ in (2.1).1 A rich variety of plasmon-phonon supported bulk and interface modes can occur in thinﬁlm structures composed of polar semiconductors, as will be shown here. A brief outline of the concept of bulk and interface polaritons is given in the following section. The occurrence of surface polaritons and surface guided waves in polar semiconductor thin-ﬁlm structures is explained and demonstrated.

4.1 Bulk Polaritons 4.1.1 Dispersion Relation Polaritons are electromagnetic eigenmodes of matter, which obey Maxwell’s postulates.2 More precisely, polaritons are the eigenstates of the (linear) system matter and radiation ﬁeld, and are seen as composite particles of the 1

2

Most features within infrared ellipsometry spectra are caused by bulk and interface modes. These features depend crucially on the behavior of the dielectric functions of the semiconductor materials involved. For example, the resonance lineshapes depend on the absorption loss near the interface mode frequencies. Because ε depends on phonon and plasmon mode parameters, as discussed in Chap. 3, infrared ellipsometry is extremely sensitive to lattice-vibration and free-charge-carrier properties of heterostructures constituents, eventually with ﬁlm (layer) thickness less than a fraction of the probing wavelength. The term “polariton” was originally introduced by Hopﬁeld [1] to describe the “uncoupled polarization ﬁeld particle” of a medium, such as excitons, phonons etc. Later on the term was designated to the coupled polarization excitation – photon modes [2]. Mills and Burstein have generalized the term polariton to include the coupled magnetic dipole excitation modes in magnetic media as well [3]. In frequency regions where the electromagnetic waves in a medium

Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 45–65 (2004) c Springer-Verlag Berlin Heidelberg 2004

46

4 Polaritons in Semiconductor Layer Structures

photons and the elementary matter excitations. Polaritons involve electromagnetic radiation and charged particle displacement, and propagate along certain directions.3 When the electromagnetic wave propagates in a material, the electric and magnetic ﬁelds associated with the wave excite the internal degrees of freedom within the medium. In a dielectric, the electric ﬁeld will couple with infrared active phonons with the same wave vector as the incident radiation.4 As a consequence the dielectric permittivity of the medium will diﬀer from unity, and the phase velocity of the wave diﬀers from the velocity of light in vacuum [3]. Constitutive relations are required to connect the source ﬁelds in vacuum (electric ﬁeld E and magnetic induction H ) with the dielectric displacement D and magnetic ﬁeld B within the medium, caused by the electromagnetic polarization response.5 The description of the propagation of electromagnetic radiation in matter follows upon Maxwell’s postulates, written here in the space-time domain rotH = ε

1 ∂ E, c ∂t

1 ∂ H, c ∂t divD = 0 ,

rotE = −

divH = 0 .

(4.1) (4.2) (4.3) (4.4)

If the medium responds in a linear manner (the polarization responses follows the source ﬁeld excitation linearly), D and E can be related within the frequency-time domain by the dielectric function tensor ε(ω), thereby accounting for temporal dispersion D = ε(ω)E ,

(4.5)

Assuming solutions with wave-like character of wave frequency ω and wave vector k , one obtains a wave vector equation that involves only the electric ﬁeld E [3]

3

4 5

are predominantly dipole excitation in character (such as optical phonons, excitons, magnons), the modes are designated as optical phonon-polaritons, excitonpolaritons, or magnon-polaritons, etc. For a detailed introduction see, e.g., Burstein and De Martini [4], or Mills and Burstein [3]. It is typical for coupled systems (the photons couple with the elementary matter excitations) that almost always neither side carries the total amount of energy. Near resonance, however, the full electromagnetic energy may be transferred to the elementary matter excitation, provided there is no or only little scattering of the polariton mode energy into diﬀerent, radiant or non-radiant (polariton) modes. It is assumed that the magnetic permeability tensor µ is unity. A detailed discussion of constitutive characterization of simple and complex mediums is given in [5]. Because magnetic response is omitted in this work, magnetic induction and magnetic ﬁeld are equal, except for the vacuum permeability µ0 = 4π × 10−7 NA−2 .

4.1 Bulk Polaritons k × [k × E] +

47

2 ω c

ε (ω) E = 0 .

(4.6)

This equation has the form of an eigenvalue equation, from which k may be determined once ω is speciﬁed. A special case is that of an insulating solid for which the dielectric function tensor is isotropic. Then the solutions have E in the plane transverse to k , and the dispersion relation becomes

ck ω

2 = ε (ω) .

(4.7)

For the matter discussed in this work ε is assumed to be independent of k , i.e., spatial dispersion eﬀects are neglected. Note, however, that spatial dispersion was already neglected upon setting (4.1)–(4.4).6 In frequency regions where ε is independent on ω , the dispersion relation takes the wellknown form where the phase and group velocities are equal, and diﬀer from the velocity of light c in vacuum ck ω= √ . ε

(4.8)

4.1.2 Example: i -GaAs A simple example for bulk polaritons in polar semiconductors shall be given for better clarity of the subsequent discussion of the surface polariton and surface guided wave eﬀects in layered semiconductor structures. The case of non-doped (intrinsic: “i”) GaAs with a single infrared active phonon resonance is considered. Equation (4.7) can be solved as a function of k !   " 2 " 2 2 2 " 1 (ck) (ck) (ck) 2  2 2 + ωLO ± + ωLO −4 ωTO . (4.9) ω± = #  2

6

ε∞

ε∞

ε∞

For instance, in a free-electron plasma, the free-charge-carrier contribution to the dielectric function is not anymore determined by the local electric ﬁeld but inﬂuenced by the ﬁeld along the path, which the electrons travel when their mean free path or cyclotron radius exceeds the wavelength of the excitation. Such non-local eﬀects can be observed in thin foils of alkali metals, and require a non-local k-dependent form of the dielectric function as suggested by Lindhard [6]. Non-local eﬀects, however, can be omitted when vF k ω p , where vF is the Fermi velocity. In general, the wave vector dependence of the dielectric function arises from the wave vector dependence of the physical quantities that aﬀect the dielectric function, such as ε∞ , ω TO , ω LO , and so on, and which is neglected here. A discussion of spatial dispersion eﬀects on bulk and interface polaritons is given by D.L. Mills in “Spatial dispersion and its eﬀects on the properties of dielectrics” [4] pp 147. One may also refer to the interesting paper by A.R. Melnyk, “Optical excitation of longitudinal plasmons: A Fresnel equation theory” [4] pp 221, and therein. See also Chap. 6 in [7].

48

4 Polaritons in Semiconductor Layer Structures

For each k, two frequencies, ω + (k) and ω − (k), are allowed (bulk polaritons). A plot of the polariton dispersion ω + , ω − is shown in Fig. 4.1a. For √ ω ω TO the polariton propagates with the velocity c/ ε∞ because the frequency is too high for the lattice to follow the rapid electromagnetic ﬁeld oscillations.7 For ω ω TO the lower branch polariton propagates with the √ velocity c/ ε0 , which is slower than for ω ω TO , because for GaAs ε∞ = 10.6 < ε0 = 12.685. When ε∞ and ε0 diﬀer appreciably, the wave that propagates through the lattice for ω ω TO has partly optical phonon character, and part of the ﬁeld energy is stored within the lattice motion [4]. In the frequency region between ω TO and ω LO Re{ε(ω)} is negative, and no propagating normal mode exists. In an optical experiment, where the incident √ radiation is coupled in through a medium with na = εa , εa independent of ω and k, only one polariton mode can be studied at the same time. The intersections of the polariton branches with the “light line” are shown in Fig. 4.1b. Total reﬂection will occur between ωTO and ω+ , and ω = 0 and ω− . With light incident at oblique angle Φa at a plane parallel boundary, Φa can be used as experimental degree of freedom, and one can “swap” the “light √ √ line” from left to right in Fig. 4.1b for 0 ≤ qx ≤ εa , where qx = εa sin Φa is the dimensionless x-component of the wave vector, that is parallel to the interface (Fig. 4.2).

4.2 Surface Polaritons and Surface Guided Waves Interface modes are tied to the existence of a surface, or more speciﬁc, to the existence of an interface between two media of certain dielectric properties.8 The term surface mode addresses phenomena, which involve resonant propagation of electromagnetic waves and energy transport parallel to the surface. Surface guided waves (SGW) denote those with transverse electric (TE; electric ﬁeld vector parallel to interface) radiation ﬁelds. Such eﬀects comprise Brewster modes (total internal reﬂection) and guided slab modes. Surface polaritons (SP) are interface-bound electromagnetic waves of transverse magnetic (TM; magnetic ﬁeld vector parallel to interface) character.9 SP modes at a single interface cannot be easily excited within a standard reﬂection or transmission experiment because their allowed dispersion range is 7

8

9

The high-frequency dielectric constant ε∞ diﬀers from unity and is a measure of the polarizability due to higher-frequency electronic transitions. The surface of a sample may also be treated as an interface. The terms “surface” and “interface” have the same meaning here. The role of surface (interface) polaritons in propagation of radio-frequency electromagnetic wave propagation along the earth’s surface and in the skin eﬀect of metals was recognized at the begin of the 20th century by Cohn [8], Uller [9], Zenneck [10], and Sommerfeld [11], who showed that surface waves occur at the boundary of two media when one of the media is either “lossy” dielectric or a metal, and the other is a loss-free dielectric.

4.2 Surface Polaritons and Surface Guided Waves

(b)

(a) ck

400

-1

ω [cm ]

ck/√ε∞

200

0

0

500

400

ck/√ε0

ω+ ωLO ωTO

49

ω+ √ε∞

ω-

1000

1500 -1

ck [cm ]

√ε0

2000 0

1

2

3

ω-

4

200

5

0

√εasinΦa

Fig. 4.1. Bulk polariton dispersion for the case where the electromagnetic ﬁeld interacts with a single infrared active phonon in a crystal with cubic symmetry. (a) The two polariton branches ω+ , and ω− for k → 0 (∞) approach zero (ωTO ) and √ ωLO (inﬁnity), respectively. For light incident from vacuum ( εa = 1, dotted line) only the upper branch polariton can be excited, which will occur slightly above ωLO . √ (b) Intersections of the polariton branches with the “light line” qx = εa sin Φa when the ambient index of refraction is varied, e.g., upon a high-index-prism, where qx = (c/ω)kx is the dimensionless wave vector parallel to the interface. No polariton √ √ √ can propagate for ε∞ ≤ qx ≤ ε0 = ε∞ (ωLO /ωTO ). Note that in a linear reﬂection or transmission experiment only one polariton branch is active at the same time. (Parameters for GaAs are given in Table 4.1.)

to the right of the “light line” ω = ck, and high-index-of-refraction prisms are necessary to increase the momentum of the incident light wave.10 However, as will be shown and discussed in detail below, dielectric properties of polar semiconductors at infrared wavelengths, bound within a thin-ﬁlm structure, produce boundary conditions for certain wavelengths, where resonant excitation of SP’s can be observed without use of a high-index prism. For simplicity, only the double-interface situation will be addressed here. Multiple-interface (multilayer) systems are tractable by the same approach. 10

Such arrangements include the well-known “Otto”- and “Kretschmann”conﬁgurations [12, 13, 14, 15]. See also Fig. 4.9.

50

4 Polaritons in Semiconductor Layer Structures

Ψ [deg] na/i-GaAs

ω [cm-1]

400

45˚ 200

0˚

30

60

90

Φa[deg]

Fig. 4.2. Calculated contour-plot of the reﬂection ellipsometric parameter Ψ for a √ 300 µm thick slab of i-GaAs immersed into a medium with εa = 5.5, versus Φa and ω, for demonstration of the bulk polariton dispersion. The bounds of the total reﬂection regions (Rp = Rs = 1) where Ψ drops oﬀ 45◦ render the bulk polariton branches ω+ , ω− in Fig. 4.1b. (Parameters for GaAs are given in Table 4.1.)

4.2.1 Dispersion Relation The denominators of the Fresnel reﬂection coeﬃcient for p- and s-polarized light provide a convenient form of the dispersion for the interface-bound TM or TE modes in thin-ﬁlm heterostructures [16], and shall be studied here. Refer to Appendices E and F. The coordinate system is depicted in Figs. 2.1, E.1, F.1. Single Interface SP Modes (Bound TM Waves) For the single interface the p-polarized Fresnel reﬂection coeﬃcient can be rearranged into 0=

kz,1 kz,2 + , ε1 ε2

(4.10)

where ε1 (kz,1 ) and ε2 (kz,2 ) are the complex dielectric functions of the materials (wave vector z -components) on both sides of the interface. The associated x-component of the wave vector can be expressed as kx0 =

ω c

ε1 ε2 ε1 + ε2

1/2 .

The associated z -component within the j th medium is (j = “1,”, “2”)

(4.11)

4.2 Surface Polaritons and Surface Guided Waves

2 kz,j =

ω c

2

εj − kx0

51

1/2 .

(4.12)

For a bound TM wave, the associated electric ﬁelds must be evanescent on both sides of the interface, and the kz,j ’s must possess imaginary parts. In case of no or only little absorption, i.e., Im(ε1 ) ≈ Im(ε2 ) ≈ 0, (4.10)–(4.12) are fulﬁlled in two diﬀerent situations, termed here (SP1) and (SP2): Fano modes: (SP1) kz,1 , and kz,2 are imaginary and sign(ε1 ) = −sign(ε2 ); Brewster modes: (SP2) kz,1 , and kz,2 , are real, sign(kz,1 ) = −sign(kz,2 ), and ε1 > 0 and ε2 > 0. The situation (SP1) occurs if ε1 and ε2 have opposite signs. These modes correspond to surface TM waves, which propagate along the interface and decay exponentially to zero as z → ±∞. Fano was ﬁrst who recognized the occurrence of bound surface TM waves when ε1 and ε2 of adjacent media have opposite signs [17]. Modes of situation (SP1) are therefore called Fano modes. The situation (SP2) refers to SP modes, which occur when both ε1 and ε2 are positive. Such modes correspond to (and have been termed) Brewster modes, for which the intensity of the reﬂected wave is zero or minimal (see, e.g., [4], Sect. 2.1.1.). The Brewster modes have smaller kx0 magnitudes than the magnitudes of the bulk polaritons in either medium at the same frequency as opposed to the Fano modes, which have larger kx0 magnitudes. The Fano modes cannot linearly couple to the bulk modes, and are non-radiant [3, 4].11 For the half-inﬁnite media, the Brewster modes are not bound to the interface, i.e., they do not attenuate with distance from the interface, because kz,1 , and kz,2 are real-valued. Accordingly, no bound SP mode occurs for situation (SP2) either. Double Interface SP Modes (Bound TM Waves) The existence of a second interface (Fig. F.1), separated from the ﬁrst interface by a fraction of the probing wavelength only, causes a subtle shift ∆kx of the x-component of the wave vector kx0 , where kx0 may refer to a solution of (4.11) for either interface. The new SP resonance occurs at kx0 +∆kx , where the shift ∆kx can be expressed through the substrate (εb ) and ﬁlm (εf ) dielectric functions, the ﬁlm thickness d,12 kx0 , and the p-reﬂectivity of the ambient – 0 at kx0 [16] ﬁlm interface rp01 0 ∆kx = 2rp01 kx0 11

12

εf εb ei2α0 . (εf + εb ) (εf − εb )

(4.13)

The case in which medium 1 is lossy and has a complex dielectric function, and medium 2 is loss-free and has a positive real part of the dielectric function (lossy-dielectric-air interface) are known as Zenneck modes. As shown in Appendix F the ﬁlm thickness d enters (4.13) upon the exponential √ argument α0 = (ω/c)d [εf (εf + εb )−1 ].

52

4 Polaritons in Semiconductor Layer Structures

Due to this shift, the individual interface SP wave vector dispersions are displaced, and may cross the individual bulk polariton branches, and the new SP wave vector x-component is complex-valued. Hence, the associated z -components are complex-valued as well, and light, mediated through one of the media into the interface, can excite the SP mode, which will decay exponentially away from the interface. Unfortunately, the ω -dependence of kx0 + ∆kx cannot be solved explicitly, and has to be evaluated numerically. It is, however, more eﬃcient and accurate to use the denominator of the p-polarized reﬂection coeﬃcient for this purpose FTM = 1 + rp01 rp12 ei2α0 ,

(4.14)

which is identical with the TM mode condition derived in Appendix F, and which should vanish near SP resonance.13 Double Interface SGW Modes (TE Waves) Similar to the case above, the relation for surface guided TE waves (SGW) at a thin ﬁlm follows from the denominator of the s-polarized reﬂection coeﬃcient FTE = 1 + rs01 rs12 ei2α0 ,

(4.15)

which must also vanish near SGW resonance. 4.2.2 Example: i-GaAs Film on n-GaAs Substrate The occurrence of SP and SGW modes in polar semiconductor heterostructures is demonstrated here by calculations as well as by experiment. Samples with a single i-GaAs ﬁlm deposited on n-GaAs substrates are studied. Table 4.1 contains typical GaAs far infrared model dielectric functions parameters, which are used for calculation of εf (ω ) and εb (ω).14 To begin with, 13

14

The alternative for observation of SP modes is the use of a high-index-prism, where the evanescent electromagnetic ﬁeld at the base of the prism is brought close to the interface of interest (see Fig. 4.9). Thereby, the wave vector of the ﬁeld coupled into the system is larger than the allowed bulk modes and larger than within normal ambient (εa = 1), and resonant excitation of SP modes can be observed upon loss of the p-polarized light intensity transmitted through the symmetric prism [12, 14]. Note the gap in the “Otto”-conﬁguration (Fig. 4.9) is actually a “ﬁlm” of index εa = 1. The subsequent discussion holds for polar heterostructures as well (e.g., AlGaInP-GaAs), except for diﬀerent material-related SP and SGW resonance frequencies. The plasma broadening values chosen here correspond to very high free-charge-carrier mobility parameters (≈ 5 × 104 cm2 /(Vs)), for better visibility of the SP and SGW eﬀects within the calculated spectra.

4.2 Surface Polaritons and Surface Guided Waves

53

the solution of (4.11) is discussed for the i-GaAs/n-GaAs interface.15 For simplicity, phonon (γ = 0) and plasmon broadening (γp = 0) is omitted for solving (4.11). The dielectric functions are then εb = ε∞

2 ωp2 ωLO − ω2 − 2, 2 2 ω ωTO − ω

εf = ε∞

2 ωLO − ω2 , 2 ωTO − ω 2

(4.16)

with ε∞ , ωTO , ωLO , and ωp as high-frequency dielectric constant, TO, LO, and plasmon frequency parameters, respectively. Equation (4.11) has four physical roots $ 1 ωs1,2 =

ωs3,4 =

2 1 2

s+ ∓

s− ∓

$

2 s2+ − 8ωTO [s+ − 2ωLO ] ,

(4.17)

2 s2− − 8ωTO [s− − 2ωLO ] ,

(4.18)

with ωp2 2(ckx )2 2 s± = + 2ωLO + ± ε∞ ε∞

4(ckx )4 + ωp4 . ε2∞

(4.19)

Table 4.1. GaAs lattice-mode and free-charge-carrier parameters used for the examples discussed in this chapter. ε∞ εf εb

ωTO ωLO [cm−1 ] [cm−1 ]

10.6 268 10.6 268

292 292

γ [cm−1 ]

N [cm3 ]

ωp [cm−1 ]

µ [cm2 /(V s)]

m [m0 ]

1.5 1

0 1017.5

0 670

– 104.7

– 0.063b)

a)

(a)

For the calculation of the SP and SGW dispersions no plasma broadening was assumed. (b) Electron eﬀective mass from [18].

It is worth to consider the small and large wave vector behavior of the four solutions. Branches ωs1 , ωs2 , and ωs4 approach ωLPP− , ωLPP+ , and ωLO √ when k → 0, respectively, whereas ωs3 ≈ ck / ε∞ . Note that ωs3 would also remain ﬁnite at k = 0 when εf would contain a plasma term with parameters √ diﬀerent from that in εb . For k → ∞, ωs1 → ωTO , and ωs2 ≈ ck/ (2/ε∞ ), whereas ! " 2 " ωp2 ωp2 ωp2 1# 2 2 ω = 2ω + − 2ω + − 8ω 2 . (4.20) s3,k→∞

15

2

LO

ε∞

LO

ε∞

TO

ε∞

This is for demonstration purpose only. One may also study (4.11) for the air/iGaAs interface. The actual modes observable at the double-interface structure follow through numerical analysis of (4.14) and (4.15) anyway, and diﬀer from those of the individual interfaces.

54

4 Polaritons in Semiconductor Layer Structures

(a)

-1

Φa=70°

b f

ck

400 ω [cm ]

ωs2

(b)

√2ck/√ε∞

ωs4

ωLO ωTO

400

ω+

ωLPP+

ωs1 ωLPP-

200

200

ω-

ωs3 0

0

500

1000

1500 -1

ck [cm ]

2000 0

500

1000

1500

0 2000

-1

ck [cm ]

Fig. 4.3. (a) SP dispersions for the i-GaAs/n-GaAs interface (solid lines). The hatched areas indicate ω-regions where either n-GaAs (gray; 0 ≤ ω ≤ ωLPP− and ωTO ≤ ω ≤ ωLPP+ ), or i-GaAs (dark-gray: ωTO ≤ ω ≤ ωLO ) have negative dielectric function values. Branches ωs1 , and ωs2 (ωs3 , and ωs4 ) follow closely those of the bulk polariton dispersion branches ω− and ω+ , label b and the dashed lines, for the nGaAs (label f , dash-dotted lines, for the i-GaAs) for small k, and remain always above (below ) the bulk polariton dispersion, respectively (Fig. 4.3b). Branches ωs1 , ωs2 , and ωs3 , ωs4 correspond to situations (SP2) and (SP1), Brewster and Fano modes, respectively. The mode associated with ωs4 causes the Berreman-eﬀect. Note that this mode has only very little dispersion. (b) A second interface, bound by a medium with εa , aﬀects the SP dispersion slightly, and causes crossing with the bulk dispersions near the light line, e.g., at Φa = 70◦ angle of incidence (dotted circles). Accordingly, modes ωs1 , ωs2 , and ωs4 can be observed just as in a high-index-prism experiment, but already with index εa = 1, i.e., illuminating the structure through air. (Parameters for GaAs are given in Table 4.1.)

Accordingly, modes ωs1 , ωs2 obey strong dispersion, whereas ωss is bound within a very small frequency region. Figure 4.3a depicts the SP dispersions and their boundaries. The particular type of the SP mode is determined by the nature of the dielectric functions of the two media. The hatched areas in Fig. 4.3a indicate the frequency regions in which either εb or εf are negative. From there it is obvious that ωs1 , ωs2 , and ωs3 , ωs4 belong to situations (SP2) and (SP1), Brewster (BSP) and Fano (FSP) modes, respectively. Figure 4.3b shows the bulk dispersion branches for both n-GaAs (dashed lines) and iGaAs (dash-dotted lines) for comparison. Due to the presence of the free charge carriers, both bulk polariton n-GaAs branches are now active for the

4.2 Surface Polaritons and Surface Guided Waves

55

“light line”. The SP branches ωs1 and ωs2 remain above the bulk polariton branches of the n-GaAs, whereas ωs3 and ωs4 are found below the i-GaAs bulk polariton branches (Fig. 4.3b). Accordingly, the wave vector x-components are smaller (larger) than the associated bulk modes at the same frequency, in accord with the deﬁnition of the Brewster and Fano modes mentioned above. No intersection exists between SP and bulk polariton branches (Fig. 4.3b). The SP modes are non-radiant. However, if one medium is terminated by a second interface (thin ﬁlm), and bound by a third medium (εa ) through which the interface between εb and εf can be illuminated, both SP branches are modiﬁed by a small wave vector shift ∆kx . As a result, the branches ω s1 , ω s2 , and ω s4 become active and can be observed upon illuminating the structure with polarized infrared electromagnetic radiation. Figure 4.4 depicts numerical solutions of the minima in functions FTE (4.14) and FTM (4.15) versus the ﬁlm thickness d. Three SP branches emerge at frequencies ωs1 , ωs2 , and ωs4 shown in Fig. 4.3a. According to the above deﬁnition, those bound by ωs1 , ωs2 for d → 0 are termed here BSP− and BSP+ , and the mode bound by ωs1 is termed FSP.16 The minimum in FTM for mode BSP− is weak for small thickness d, and drawn here as dotted line. All SP modes undergo slight changes with increasing d, and the inset enlarges that for FSP. Two SGW modes occur with strong thickness dependence. Both SGW branches merge with modes BSP+,− for d → 0, and are therefore termed SGW+,− accordingly. Because ωs1 and ωs2 depend on ωp , modes BSP+,− and SGW+,− are extremely sensitive to the free-charge-carrier properties. Figure 4.5 depicts calculated p- and s-polarized normalized reﬂection coeﬃcient diﬀerences for Rp,s = ||rp,s ||2 for the thin-ﬁlm example, and Fig. 4.6 presents the ellipsometric parameters Ψ and ∆ for diﬀerent values of the ﬁlm thickness d, and at Φa = 70◦ . The reﬂection coeﬃcients for the bare substrate (d = 0, pure reﬂectivity spectra not divided by d) are shown in the insets, and for the ellipsometric parameters as dotted lines in Fig. 4.5b. The parameters γ and µ are now as shown in Table 4.1. The modes in Fig. 4.4 can be identiﬁed within the data: (FSP) The changes in Rp,s , Ψ , and ∆ imposed by the thin i-GaAs layer, are extremely small, except for the strong resonance, denoted by FSP, which occurs near ωLO .17 This SP mode is actually a lossy Fano mode, because both εb and εa are complex-valued. The lineshape is Lorentzian, and can be 16

17

Note that ωs1 , ωs2 are bound by the lower and upper LPP modes within the n-GaAs, from where the “− ” and “+ ” in BSP+,− is adopted. The mode FSP is the origin of the Berreman eﬀect. Note that the changes scale almost linearly with d as long as d is small enough compared with λ = 2 πc/ω. This is why p-reﬂectivity or ellipsometry data near this SP resonance own superior sensitivity to the thickness of a surface chargecarrier depletion layer despite their very small thickness compared with the probing wavelength [19, 20].

56

4 Polaritons in Semiconductor Layer Structures

air/i-GaAs(d)/n-GaAs

+

BSP

300

FSP

293 +

-1

ω [cm ]

SGW 225

SGW

FSP

-

292 0

2

4 -

BSP

150

Φa = 70˚

ωp = 670 cm

-1

75 0

1

2

3

4

5

d [µm]

Fig. 4.4. SP (FSP, BSP+,− ) and SGW (SGW+,− ) mode frequencies for the air/iGaAs(d)/n-GaAs thin-ﬁlm structure as a function of the ﬁlm thickness d. The frequencies for the SP and SGW modes follow from numerical evaluation of (4.14) and (4.15), respectively

fully described by (F.37).18 The width of the lineshape is directly related to Im{εf }, and presents an extremely sensitive measure for small absorption within the ﬁlm.19 (BSP) The p-reﬂectivity is further aﬀected by subtle changes, upon increase in d, at frequencies near ωs1 (k = 0) = ωLPP− (BSP− ) and ωs2 (k = 0) = ωLPP+ (BSP+ ). The subtle modes with Lorentzian lineshape in Rp refer to situation (SP2), which is also contained in (F.37). It is likely that such small resonance eﬀects will fall within the experimental error bars. Their experimental observation was not yet reported. 18

19

This situation, obtained here for a polar semiconductor ﬁlm structure near ω ≈ ωLO is identical with that, under which Berreman reported the loss of the preﬂectivity of a LiF ﬁlm near the LiF LO mode frequency, where the ﬁlm was backed by silver [21]. This eﬀect has been termed “Berreman eﬀect”, and which is frequently interpreted as due to a resonant excitation of LO lattice modes within the ﬁlm (see, for instance, Chap. 10 in [22]). It is, however, evident from the above discussion that the SP situation is not related to a LO lattice mode vibration, because the associated electromagnetic displacement pattern across the ﬁlm is that of a TM mode with propagation direction parallel to the ﬁlm interfaces. If both components of the GaAs-GaAs ﬁlm structure would be doped, but with diﬀerent free-charge-carrier parameters, for the situation discussed here, the mode FSP would split into modes FSP−,+ occurring near the ﬁlm modes ωLPP− , ωLPP+ , respectively.

4.2 Surface Polaritons and Surface Guided Waves

57

-1

{(R012-R02)/d}[nm ] 5.0x10

p

-4

air/i-GaAs(d)/n-GaAs BSP

-

TO

+

FSP BSP

0.0 1.0

d [nm] 1 10 100 500

0.5

-4

-5.0x10

0.0 200 1.0x10

300

-3

Φa = 70˚

s 0.0 1.0

0.5

-3

-1.0x10

SGW

-

0.0

SGW 200

150

200

+

300 250

300

350

Wavenumber [cm-1] Fig. 4.5. Calculated normalized reﬂection coeﬃcient diﬀerences for p- (upper panel ) and s-polarized plane waves (lower panel ) for an air/i-GaAs(d)/n-GaAs structure with diﬀerent thickness d. The insets display reﬂectivity spectra Rp,02 and Rs,02 of the bare n-GaAs substrate. The reﬂectivity diﬀerences are divided by d in units of nm. Vertical arrows (FSP, BSP+,− , SGW+,− ) indicate the SP and SGW mode frequencies from Fig. 4.4. The mode FSP occurs in Rp,012 and Ψ , ∆, where a bound TM wave, described by (F.37) for situation (SP1), propagates parallel to the interfaces of the homostructure. The modes labelled by BSP+,− are identical with those in (F.37) for situation (SP2), and are hardly to observe experimentally. (Note the small resonance compared with the mode FSP.) The s-polarization reveals a resonant loss of reﬂectivity at SGW+,− . (The parameters for calculation of εf and εb are given in Table 4.1.)

58

4 Polaritons in Semiconductor Layer Structures

Rp/Rs = tanΨexp{i∆} 90

air/i-GaAs(d)/n-GaAs SGW

-

+

SGW

60

Ψ [˚]

d=0 TO d = 100 nm d = 1000 nm

FSP

30

0 270

∆ [˚]

180 90 0 -90

Φa = 70˚

150

200

250

300

350

Wavenumber [cm-1] Fig. 4.6. Ψ (upper panel ) and ∆ (lower panel ) spectra for an air/i-GaAs(d)/nGaAs structure with diﬀerent thickness d addressed in Fig. 4.5, for d = 0, 100 nm, and 1000 nm. The SGW mode shift with d is obvious, also the occurrence of the FSP mode. Resonance due to modes BSP are hardly to detect because their eﬀects are subsumed by the strong changes in the s-polarization due to the SGW modes and due to the bulk polariton mode excitation

(SGW) Strong changes occur within the s-polarized reﬂectivity upon increase in d. The frequency shift of the modes termed SGW−,+ agrees excellently with their dispersion shown in Fig. 4.4. The upward-pointed peaks in Ψ reveal this shift most clearly.20 20

Note that the frequencies of these modes separate from the bulk polariton mode frequencies as d increases. The bulk polariton excitation causes the sudden drop in p- and s-reﬂectivity above the frequencies where the “light line” in Fig. 4.3b crosses the branches ω −,+ .

4.2 Surface Polaritons and Surface Guided Waves

air/i-GaAs(d)/n-GaAs

SGW

+

59

Φa = 45˚

FSP

90 C

Ψ [˚]

d = 602 nm

B

60

d = 776 nm

A

d = 1819 nm

30 200

250

300

350

-1

Wavenumber [cm ] Fig. 4.7. Observation of Fano (FSP; vertical dotted line) and SGW+ modes (vertical arrows) in semiconductor homostructures: Experimental (dashed lines) and best-ﬁt (solid lines) Ψ spectra from three diﬀerent i-GaAs/n-GaAs(Te-doped) samples with thickness d, and slightly varying free-charge-carrier concentration N , and mobility µ (Data are given in Table 4.2). Spectra for sample B and C are shifted by 20◦ and 40◦ for convenience, respectively. The FSP mode occurs slightly above the ﬁlm LO frequency, and is not (very little) aﬀected by the n-GaAs bulk dispersion, as expected. The upper SGW+ modes shift with increasing N because ωs2 shifts to higher frequencies. The modes SGW− are overdamped and do not occur, because of the relatively low free-charge-carrier mobility, which leads to stronger absorption in εb near the frequencies ωLPP− compared to that near ωLPP+ . The SGW+ mode is red-shifted due to the decreasing ﬁlm thickness d, as shown in Fig. 4.8 Table 4.2. Best-ﬁt parameters free-charge-carrier concentration N , mobility µ, and thickness d for the i-GaAs(d)/n-GaAs(Te-doped: N , µ) samples obtained from analysis of the ellipsometric data shown in Fig. 4.7. The electron eﬀective mass parameter was taken from [18]

Sample

N [cm3 ]

µ [cm2 /(Vs)]

d [nm]

A B C

3.39 × 1017 3.50 × 1017 4.47 × 1017

2750 2450 2290

2145 761 602

60

4 Polaritons in Semiconductor Layer Structures 325

air/i-GaAs(d)/n-GaAs

ωp = 746 cm

-1

ωp = 663 cm

+

320

-1

SGW

ωp = 645 cm -1

ω [cm ]

-1

315

310

305

300 0

1

2

3

d [µm] Fig. 4.8. Calculated thickness dependence of the SGW+ mode for the plasma frequency values in the i-GaAs(d)/n-GaAs samples studied in Fig. 4.7. The arrows indicate the observed frequencies, which provide exactly the thickness values obtained for the i-GaAs ﬁlm in samples A, B, and C from the best-ﬁt shown in Fig. 4.7.

Figure 4.7 presents experimental Ψ spectra for three i-GaAs(d)/n-GaAs ﬁlm structures, with diﬀerent ﬁlm thickness d, plasma frequency, and plasma damping parameters (slightly diﬀerent free-charge-carrier concentrations and mobility parameters within the Te-doped n-GaAs substrates). Resonant excitation of the FSP and the SGW+ modes occurs, according to the explanation given above.21 The FSP mode is virtually unaﬀected by the thickness and the freecharge-carrier concentration, as expected. The SGW+ modes shift to higher wavenumbers with increasing plasma frequency (free-charge-carrier concentration), and to lower wavenumbers with increasing thickness d. The actual SGW+ mode frequencies are the net eﬀect of both shifts. Fig. 4.8 contains the d-dependence of the SGW+ mode for the plasma frequency values obtained from the best-ﬁt analysis of the experimental data in Fig. 4.7. The branches are sifted to higher frequencies with increasing ωp . The arrows indicate the peak positions of the features labeled by SGW+ in Fig. 4.7. The intersections with the according function SGW+ (d, ωp ) provide the thickness of the i-GaAs 21

The infrared ellipsometry study of non-doped epitaxial 4H silicon carbide on heavily doped 4H silicon carbide homostructures reported by Zollner et al. [23] also reveal this SGW+ mode, which causes a strong resonance in Ψ for ω above the LO frequency of the non-doped SiC epilayer in Fig. 4.1 of [23].

4.2 Surface Polaritons and Surface Guided Waves

εp

εp

εgap < εp εmetal

εgap < εp

(a)

(b)

εa=1

εa=1

εmetal

εf

i-GaAs

εf

i-GaAs

εb

n-GaAs

εb

n-GaAs

(c)

61

(d)

Fig. 4.9. a and b: Total-internal-reﬂection symmetric prism conﬁgurations used for observation of SP modes at (bulk-like) metal surfaces (Otto-conﬁguration, Fig. 4.9a), or at thin metal ﬁlms (Kretschmann-conﬁguration, Fig. 4.9b). In both cases a “gap” medium, usually air or vacuum, with index of refraction smaller than the prism material is involved. Analogous situations occur within a polar semiconductor layer structure above the LO (or likewise the LPP) mode frequencies, where either the ﬁlm (Fig. 4.9c, the analogue of the Otto-conﬁguration) or the half-inﬁnite substrate (Fig. 4.9d – the analogue of the Kretschmann-conﬁguration) possess indexof-refraction values less than the ambient index na . The optical properties of the opposite material (shaded ) can be metal-like (large negative dielectric function between a pair of related TO-LO (LPP) frequencies) or dielectric (outside a pair of related TO-LO frequencies). In both cases, the ambient air acts as a virtual prism of index 1, and the heterostructure component, for which the frequency is slightly above an LO- or LPP-frequency, acts as the “gap” material

layer for all three samples in excellent agreement with the best-ﬁt thickness obtained from the line shape ﬁt of the ellipsometry data. Figure 4.9 depicts total-internal-reﬂection prism arrangements known as Otto- and Kretschmann conﬁgurations, together with the analogous situations, which occur within the air/i-GaAs(d)/n-GaAs system. Inspecting εb and εf near SP mode frequencies, either the real part of εb or εf is positive but small compared with εa . This component acts as the “gap” material, and thereby the ambient virtually forms a “high-index” prism, because the index of the “gap” material is less than εa , and which may serve as a solely but nonetheless incisive explanation for the occurrence of the SP modes in polar semiconductor layer structures when εa = 1. However, care must be taken because this simpliﬁed view does not account for the d-dependence of the SP modes. Also, it does not serve for explanation of the SGW mode eﬀects.

62

4 Polaritons in Semiconductor Layer Structures

4.2.3 The Berreman Eﬀect The Berreman eﬀect has gained new attention since the availability of Fouriertransform-based infrared ellipsometers. This eﬀect occurs in almost all infrared ellipsometry data taken from layer structures made from materials with polar lattice mode and/or free-charge-carrier resonances [19, 20, 23, 24, 25, 26, 27, 28]. Berreman observed the resonant loss of p-reﬂectivity near the LO frequency of a LiF ﬁlm backed by a metal. This eﬀect was originally explained by the resonant absorption of infrared waves within the ﬁlm at wavelengths equivalent to the LiF-LO phonon mode frequency [29]. The interpretation of this observation reads in contemporary textbooks as if light is absorbed by resonant excitation of the thin-ﬁlm LO modes [22]. Fundamental problems exist with the interpretation in [22], because transverse electromagnetic waves should not excite lattice modes with LO displacement pattern in a linear optical experiment [30]. R¨ oseler analyzed the surface polariton dispersion relation for the double-interface situation FTM (4.14) when the ﬁlm is a polar medium (the dielectric function is described by a harmonic oscillator) and the half-inﬁnite medium is a metal. It was demonstrated that the Berreman feature is not due to resonant LO mode excitation (implementing that there is no electromagnetic ﬁeld absorption in the ﬁlm), but caused by an interference optical eﬀect with energy transported along the interface. The fact that both layer and substrate must reveal small absorption coeﬃcients for the eﬀect to become observable was also pointed out. Multiple local minimums versus d were found in FTM . The one with smallest thickness was assigned to the so-called Berreman-thickness, the value d for which the reﬂectance loss in Rp is strongest. For much smaller thickness, the frequency of the minimum in FTM is shifted to shorter wavenumbers. This frequency was interpreted as that of a true SP mode, which was thought to be diﬀerent than the Berreman mode. Infrared ellipsometry experiments for LiF ﬁlms deposited on silver, silicon [25], and for non-doped 4H-SiC ﬁlms deposited on heavily-doped 4H-SiC substrates [23] highlighted the importance of this optical phenomena. Further extension of the physical origin was suggested by the “intensiﬁed-absorption-loss” which occurs within the ﬁlm at wavenumbers near the Berreman resonance. The intensiﬁed electric ﬁeld was explained as an interference optical phenomenon at a wavelength where the index of refraction is less than one, and where the ﬁeld inside the ﬁlm is composed of back and forward travelling waves. However, interference should be ineﬀective across interfaces separated by very small fraction of the probing wavelength λ only. Likewise, absorption must exist within the ﬁlm in order to excite the Berreman resonance. For example, the FSP mode in Fig. 4.5 occurs regardless to the actual thickness d, as long as d is small. An appreciable change in reﬂectivity should be already measurable for d = 1 nm. As discussed in Chap. 4, and Appendices E, F, the cause of the observed resonance loss is the excitation of a Fano-type surface polariton mode. This mode is accompanied by energy transport along the interface, and accommodates

4.2 Surface Polaritons and Surface Guided Waves

63

air/i-GaAs(d)/n-GaAs 297.5

FSP

εa = 32

16

-1

ω [cm ]

8 4

295.0

2 εa = 1

292.5 -1

-1

d = 2000nm, ωp = 670cm , γp = 0.1cm 0

30

60

90

Φa[˚]

Fig. 4.10. Angle-of-incidence and square of the ambient index-of-refraction (n2a = εa ) dependences of the FSP mode for the ambient (εa )/i-GaAs(d)/n-GaAs structure discussed in Sect. 4.2.2. The frequency shift is small for εa = 1 (vacuum), and increases for εa > 1, but remains bound between ωLO = ωs4 (k = 0) and ωs4 (k → ∞) for εa → ∞, as expected from the SP mode dispersion shown in Fig. 4.3

absorption loss within the layer structures by homogeneous and inhomogeneous broadening of polar and non-polar lattice modes. Naturally, a loss of reﬂected p-polarized light is detected. As shown in Fig. 4.4, for semiconductor layer structures the FSP mode frequency (solutions of type (SP1) for equation FTM ) is a continuous function versus d.22 Figure 4.10 depicts the FSP mode frequency when the ambient index-of-refraction is changed hypothetically. The FSP mode remains bound between ωs4 (k = 0) and ωs4 (k → ∞) for εa → ∞, as expected from the SP mode dispersion discussed above. Finally, Fig. 4.11 depicts far infrared ellipsometry data from a semiconductor layer structure with multiple layer interfaces. This example contains a superlattice sequence consisting of n-type AlGaAs and i-type GaAs, deposited on the standard buﬀer/substrate situation discussed above. Two decoupled bands of FSP and SGW+ modes occur, which can be clearly identiﬁed. The occurrence of the FSPAlGaAs mode near ω ≈ 282 cm−1 reveals that the AlGaAs sublayers are insulating, and indicates charge transfer of electrons into the GaAs sublayers. The second mode FSPGaAs near ω ≈ 292 cm−1 originates from the i-GaAs (d ≈ 135 nm)/n-GaAs undoped-buﬀer/doped-substrate situation discussed above. As will be shown somewhere else, the interface mode resonances are extremely sensitive to magnetic ﬁeld induced changes of the free-charge-carrier polarizabilites, and hence allow for detection of their properties upon magneto-optic ellipsometry measurements. The magneto-optic 22

The two diﬀerent modes identiﬁed in [30] actually belong to the same SP mode branch.

64

4 Polaritons in Semiconductor Layer Structures

1.0

+,SL

+,GaAs

SGW

SGW 0.5

FSP

GaAs

Mij /M11

AlGaAs

FSP

0.0

M12 -0.5

M33 -1.0 100

200

300

400

500

600

ω [cm ] -1

Fig. 4.11. Low-temperature (10 K) experimental (dotted lines) and best-ﬁt calculated (solid lines) ellipsometry data from a 20×[AlGaAs(15 nm)/GaAs(20 nm)-SL deposited on i-GaAs (d ≈ 135 nm)/n-GaAs. The AlGaAs barrier layers are n-type doped, whereas the GaAs well layers are undoped. The free electrons move into the GaAs wells, where the conduction band states are lower than in the barriers. The electron concentration within the substrate is N ≈ 8 × 1017 cm−3 . Two FSP and SGW+ mode pairs occur, one caused by the interface i-GaAs/n-GaAs, another one caused by the AlGaAs barrier layers. The GaAs-like LO mode within the AlGaAs layers amounts to 281.5 cm−1 . The FSPAlGaAs mode now occurs closely above this frequency, which is a direct indication for the free charge carrier depletion within the AlGaAs barriers. [31]

measurements then clearly prove the free-charge-carrier transfer from the barrier into the well regions [31].

References 1. J. J. Hopﬁeld, Phys. Rev. 112, 1555 (1958). 45 2. V. M. Agranovich and D. L. Mills, Surface Polaritons (1982). 45 3. D. L. Mills and E. Burnstein, in J. M. Ziman, ed., Reports on Progress in Physics (The Institute of Physics, London, 1974). 45, 46, 51 4. E. Burstein and F. D. Martini, eds., Polaritons (Pergamon Press, New York, 1974). 46, 47, 48, 51 5. W. S. Weiglhofer and A. Lakhtakia, Introduction to Complex Mediums for Optics and Electromagnetics (SPIE, Bellingham, 2003). 46 6. J. Lindhard, Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd. 28(8) (1954). 47 7. D. L. Mills and E. Burstein, Rep. Prog. Phys. 37, 817 (1974). 47

References

65

8. E. Cohn, Das Elektromagnetische Feld (Julius Springer, Berlin, 1927), revised version of ”Das elektromagnetische Feld. Vorlesungen u ¨ ber die Maxwellsche Theorie”, published in 1900 (Leipzig). 48 9. K. Uller, Beitr¨ age Zur Theorie der Elektromagnetischen Strahlung, Ph.D. thesis, Universit¨ at Rostock (1903). 48 10. J. Zenneck, Ann. Phys. 23, 846 (1907). 48 11. A. Sommerfeld, Ann. Phys. 28, 665 (1909). 48 12. A. Otto, Zeit. f¨ ur Physik 216, 398 (1968). 49, 52 13. A. Otto, Z. Angew. Phys. 27, 207 (1969). 49 14. E. Kretschmann and H. Raether, Z. Naturforsch. 23a, 2135 (1968). 49, 52 15. E. Kretschmann, Z. Phys. 241, 313 (1971). 49 16. H. Raether, Surface Polaritons (Springer, Berlin, 1988). 50, 51 17. U. Fano, J. Opt. Soc. Am. 31, 213 (1941). 51 18. P. Yu and M. Cardona, Fundamentals of Semiconductors (Springer-Verlag, Berlin, 1999). 53, 59 19. J. Huml´ıˇcek, R. Henn, and M. Cardona, Appl. Phys. Lett. 69, 2581 (1996). 55, 62 20. A. Kasic, M. Schubert, S. Einfeld, D. Hommel, and T. Tiwald, Phys. Rev. B 62, 7365 (2000). 55, 62 21. D. W. Berreman, Phys. Rev. 130, 2193 (1963). 56 22. C. Kittel, Introduction to Solid States Physics (John Wiley and Sons, New York, 1986). 56, 62 23. S. Zollner, J. P. Carrejo, T. E. Tiwald, and J. A. Woollam, phys. stat. sol. 208, R3 (1998). 60, 62 24. E. Woldt, J. Bremer, and O. Hunderi, J. Appl. Phys. 75, 1739 (1994). 62 25. J. Huml´ıˇcek, phys. stat.sol. (b) 215, 155 (1999). 62 ˇ M. Schubert, T. Hofmann, and V. Gottschalch, MRS Internet J. Nitride 26. J. Sik, Semicond. Res. 5, 3 (2000). 62 27. M. Schubert, B. Rheinl¨ ander, E. Franke, H. Neumann, T. E. Tiwald, J. A. Woollam, J. Hahn, and F. Richter, Phys. Rev. B 56, 13306 (1997). 62 28. G. Leibiger, V. Gottschalch, and M. Schubert, J. Appl. Phys 90, 5951 (2001). 62 29. D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972). 62 30. A. R¨ oseler, Infrared Spectroscopic Ellipsometry (Akademie-Verlag, Berlin, 1990). 62, 63 31. C. V. Middendorf, T. Hofmann, G. Leibiger, and M. Schubert. 64

5 Anisotropic Substrates

A prerequisite for infrared ellipsometry investigations of semiconductor layer structures is an accurate knowledge of the substrate materials’ dielectric functions. Appropriate consideration of the anisotropic optical properties of noncubic substrate materials is the subject of the present chapter. The goal must be to determine the intrinsic dielectric functions to the most accurate level, because these functions will be incorporated during the best-ﬁt data analysis of the unknown heteroepitaxial layers.1 Errors on the substrate dielectric function data base will propagate into those for the materials of interest. The required information – the major polarizability functions a , b , c , and their center-of-gravity system by vectors a, b, c described in Sect. 2.6 – cannot be simply obtained from a single surface alone. Measurements should be done on samples with surfaces cut under diﬀerent crystallographic angles from the same bulk crystal. For low-symmetry surfaces, use of the generalized ellipsometry concept is mandatory. For systems with orthogonal vectors a, b, c, however, the standard ellipsometry approach can be maintained if the sample can be oriented such that all of vectors a, b, c are parallel to the laboratory coordinate axes. For many bulk crystals this can be achieved to a suﬃcient level of accuracy. Identiﬁcation of good starting values (phonon mode parameters) for the best-ﬁt model lineshape analysis presents another diﬃculty, even if the ellipsometry data were acquired correctly. A procedure for identiﬁcation of phonon modes and their symmetry for trigonal, tetragonal, hexagonal, and orthorhombic symmetry materials is therefore described in this chapter. This procedure makes use of the characteristic reﬂectivity signatures due to the so-called bands of total reﬂection. In order to enlighten the reﬂectivity signatures and there connection with the intrinsic phonon mode parameters the analytical behavior of the p- and s-polarized reﬂection coeﬃcients for high-symmetry orientations of the anisotropic samples will be inspected. Once the understanding of the reﬂection coeﬃcients have been obtained, it is straightforward to “read” these signatures from the ellipsometric parameters. The goal is to identify the number of TO-LO pairs for each lattice direction and to guess reasonable starting values for the numerical data 1

Existing examples include silicon [1], silicon carbide [2, 3], rutile [4], LaSrGaO4 and LaSrAlO4 [5], sapphire [6], and ZnO [7].

Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 67–79 (2004) c Springer-Verlag Berlin Heidelberg 2004

68

5 Anisotropic Substrates

regression procedures. Sapphire (tetragonal) and stibnite (orthorhombic) are discussed exemplarily.

5.1 Bands of Total Reﬂection For isotropic, trigonal, tetragonal, hexagonal, and orthorhombic symmetry materials – when the major polarizability axes coincide with the laboratory coordinate system – the p- and s-polarized reﬂectance coeﬃcients can be written as [8]2 rp =

Nxa − Naz , Nxa + Naz

(5.1)

rs =

Naa − Nyy , Naa + Nyy

(5.2)

with Nij = ni cos Φj =

√

εi

1−

εa sin2 Φa . εj

(5.3)

At total reﬂection, rp rp = ||rp ||2 and/or rs rs = ||rs ||2 equal unity.3 Total reﬂection occurs for phonon energies near optical lattice resonance frequencies; these regions are called the reststrahlen bands. In isotropic materials, both p- and s-polarized components of the incident light beam are totally reﬂected for photon energies where ε is less than εa sin2 Φa . (For simplicity, consideration of absorption is omitted during this section.) This situation is diﬀerent in anisotropic materials, and somewhat more complex in the presence of multiple-phonon-modes. However, the frequency as well as the type of the phonon mode can be identiﬁed from the signature of experimentally observed bands of total reﬂection at oblique incidence for p- and s-polarized light. A simple rule to identify bands of total reﬂection from high-symmetry surfaces cut from bulk materials can be obtained by inspecting the p- and s-polarized reﬂection coeﬃcients.4 p-Polarization It follows from ||rp ||2 = 1 in (5.1) that total reﬂection occurs for p-polarized light when 2

3 4

Refer to Fig. 2.1 for deﬁnition of the laboratory system, and (2.49) with matrix A set to unity. For anisotropic sample measurements the Ψ and ∆ notation is still used to deﬁne a complex ratio as in (2.2), however, direct inversions of the ellipsometric parameters into εx = 1 + a , εy = 1 + b , εz = 1 + c , as in (2.3) for an isotropic two-phase model, do not exist, and numerical regression procedures are needed for data analysis. √ εa denotes the ambient dielectric function, εa = na . A similar approach was described by Lang and Pashabekova for low-symmetry cuts of uniaxial single-mode materials at normal incidence [9].

5.1 Bands of Total Reﬂection

%

√

Re

ε∗x

1−

sin2 Φa εz

69

& =0,

(5.4)

where Re{} denotes the real part of a complex quantity, and the star indicates the complex conjugation. This condition is fulﬁlled if (εa is set to 1) Case (a)εx < 0 and (sin2 Φa /εz ) < 1, or if Case (b)εx > 0 and (sin2 Φa /εz ) > 1. For convenience, the quantities ωLO∗ are introduced, and deﬁned as the frequencies, at which the dielectric functions equal the square of the xcomponent of the dimensionless wavevector q qx2 = (na sin Φa )2 ,

(5.5)

εj (ωLO∗j ) = (na sin Φa )2 .

(5.6)

for j = “x”, “y ”, “z ” Note that ωLO∗j ≥ ωLOj at Φa ≥ 0 (ωLO∗j = ωLOj at Φa = 0). For case (a) the incident photon energy ω must be within one of the spectral regions where εx is negative, that is ω ∈ {ωTO , ωLO }x,i .

(5.7)

However, total reﬂection will occur only if ω is not within the frequency ranges ω ∈ {ωLO , ωLO∗ }z,j ,

(5.8)

and the indices “i” and “j ” indicate the i-th and j -th phonon band for polarization along x and z , respectively. The conclusive intersection of both frequency sets provides spectral regions where the condition of total reﬂection is abandoned due to dielectric loss along z . Case (b) is fulﬁlled when ω falls within ω ∈ {ωLO , ωLO∗ }z,i ,

(5.9)

ω ∈ {ωTO , ωLO }x,i .

(5.10)

but not within Here, the p-component of the light beam is totally reﬂected when the dielectric drops from its positive maximum until it reaches (qx2 )−1 loss function ε−1 z , and if the electromagnetic waves with polarization along x at the same time experience real refractive indices. Hence, the p-polarized reﬂection conditions for multiple-phonon-mode crystals are obtained by the sum of the conclusive intersections between the frequency sets {ωTO , ωLO }x,i , and {ωLO , ωLO∗ }z,j

ω ||rp ||2 = 1 ∈

'

l,m

i,j

{ωTO , ωLO }x,i \ {ωLO , ωLO∗ }z,j ,

(5.11)

70

5 Anisotropic Substrates

where i and j run over all l, m (far) infrared-active phonon branches, which are active in εx and εz , respectively, and l + m reﬂection bands occur in rp . It can be directly read from (5.11) that no signature indicative for ωTO,z will occur.5 Unless ωLO,x equals ωLO,z , the p-polarized reﬂectivity contains sharp spectral cutoﬀs or insets at frequencies of LO-phonons polarized along both, the x-and z -axes, and hence allows for immediate experimental access to the LO-phonon energies. This situation is diﬀerent from the isotropic case, where the reststrahlen spectra of bulk materials have no speciﬁc features at the LO-phonon frequency, as for example in Fig. 6.1 [10, 11]. s-Polarization The s-polarized reﬂection bands are due to εy only. The condition for total reﬂection of s-polarized light, which has the same form as (5.4), reads % & $ sin2 Φa Re

ε∗y

1−

εy

=0,

(5.12)

and is satisﬁed if (a) εy < 0 and (sin2 Φa /εy ) < 1, or if (b) εy > 0 and (sin2 Φa /εy ) > 1. Both cases are similar to those above, except that the frequency sets {ωTO , ωLO }y,i and {ωLO , ωLO∗ }y,i do not overlap. Hence, the s-polarized reﬂection conditions for multiple-phonon-mode crystals are the uniﬁcation of frequency sets {ωTO , ωLO }y,i and {ωLO , ωLO∗ }y,i ,

ω ||rs ||2 = 1 ∈

n '

{ωTO , ωLO∗ }y,i ,

(5.13)

i

and the sum has to be taken over all n (far) infrared-active phonon branches for polarization parallel y , and n bands of total reﬂectivity occur in rs . This also implies that no signatures directly related to ωLO,y can be identiﬁed in rs !

5.2 Sapphire (Hexagonal) High-quality synthetic sapphire is used as substrate material for solid-state device applications. Compound materials such as Alx Ga1−x N or Inx Ga1−x N can be grown on sapphire using metal-organic vapor phase epitaxy [12]. 5

Following similar lines of derivation, for c-plane oriented wurtzite-structure thin ﬁlms, the A1 (TO) mode cannot be determined from an infrared optical experiment.

5.2 Sapphire (Hexagonal)

71

Recent development permitted growth of heterostructures, multilayers, and superlattices from group III-nitride materials on sapphire, and progress in semiconductor research enabled fabrication of GaN-based blue-light emitting diodes and laser devices [13, 14]. Sapphire is an excellent insulator with high thermal conductivity [15, 16]. Due to its wide band gap the material is highly transparent for photons with energies from the far infrared to the deep-ultraviolet, i.e., from the lattice reststrahlen band (≈0.12 eV) to the onset of the electronic band-to-band transitions (≈9 eV) [10, 16, 17]. Sapphire (α-Al2 O3 ) consists of a hexagonal closely packed (hcp ) lattice of aluminum atoms with oxygen atoms at octahedral sites; Al atoms at 12(c) sites [point symmetry C2 ], O atoms at 18(e) sites [C3 ])[15]. Because of its rombohedral structure, sapphire exhibits uniaxial optical anisotropy. Group theory considerations predict four infrared-active modes for αAl2 O3 with dipole-moment oscillation perpendicular (electric ﬁeld vector E⊥c; Eu ), and 2 modes with dipole-moment oscillation parallel to the c-axis (E||c, A2u ) [18] (Table 5.2). Coulomb interaction splits the lattice vibrations into LO and TO modes. Two of three sapphire principal dielectric polarizability values diﬀer, the ordinary (perpendicular c; ε⊥ ), and extraordinary (parallel c; ε|| ) dielectric functions.6 Brieﬂy, response from a c-plane sample is obtained by orienting the sample in the {x, y , z} laboratory coordinates such that {εx , εy , εz } = {ε⊥ , ε⊥ , ε|| }. The a-plane situations with c parallel, or perpendicular to the plane of incidence are {ε|| , ε⊥ , ε⊥ }, or {ε⊥ , ε|| , ε⊥ }), respectively. The corresponding conditions for the bands of total reﬂection are summarized in Table 5.1. Experimental data (dashed lines) of the ellipsometric parameter Ψ from high-symmetry orientations of α-Al2 O3 are depicted in Fig. 5.2. The solid lines refer to the best-ﬁt calculation of the experimental data under variation of the dielectric function model parameters in (3.3) for ε|| and ε⊥ . The transversal and longitudinal A2u and Eu phonons of sapphire are indicated by brackets (solid brackets: TO phonons, dotted brackets: LO phonons). Bands of total reﬂection between active phonon frequencies at the high-symmetry orientations of sapphire can be recognized upon comparison with Fig. 5.1 and Table 5.1. For the sample orientations in Fig. 5.2 the standard ellipsometry approach is valid, because no mode conversion between p- and s-polarized light occurs upon reﬂection. Analysis of the diﬀerent data sets, acquired from commercially available c- and a-plane cut sapphire, is done by employing a multiple-sample modelling approach. Diﬀerent models are set up according to the respective surface orientation. The thereby calculated ellipsometric data 6

The use of the terms ordinary and extraordinary here is to diﬀerentiate between the two intrinsic polarizability functions, and descends from the terms ordinary and extraordinary indices of refraction. This connection is strictly speaking not correct, because the indices of refraction stand for the propagation speeds of ordinary and extraordinary waves within uniaxial anisotropic materials, whereas the dielectric functions relate electric and displacement phasor ﬁelds.

72

5 Anisotropic Substrates c-plane

(a)

a-plane c || x

(c)

2

2

|rp| =1

1000

ω [cm ]

|rp| =1

-1

-1

ω [cm ]

1000

500

500

a-plane c ⊥ x

(b)

c-plane

(d)

2

2

1000

|rs| =1

|rs| =1

-1

ω [cm ]

-1

ω [cm ]

1000

500

500

0.0

0.5 2

sin Φa

1.0

0.0

0.5

1.0

2

sin Φ a

Fig. 5.1. Bands of total reﬂection (hatched areas) for high-symmetry orientations of α-Al2 O3 (uniaxial) as a function of qx2 = sin2 Φa , and wavenumber ω. Refer to (5.11) and (5.13), and Tables 5.1, 5.2. Broadening is omitted here for clarity. The restrahlen bands for rp (c⊥x) and rs (c||x) are identical with those for rs (c||z). See also [6]

are compared with the corresponding experimental spectra. The evaluation is done simultaneously for all data sets. All models share a common set of dielectric functions ε|| and ε⊥ , which may be either parameterized by use of (3.3), or which can as well be obtained by the wavelength-by-wavelength inversion procedure. For the latter, suﬃcient information is contained within the multiple-sample data sets. For the model dielectric function approach, good starting values are required for the numerical data regression. One can read the corresponding phonon mode frequencies from the experimental Ψ spectra in Fig. 5.2 with the help of Table 5.1. For the c-plane sample at low wavenumbers, total p-reﬂection occurs between 385 cm−1 (ωTO1 ) . . . and ≈387.8 cm−1 , giving rise to the sharp resonance feature in Ψ . The ﬁrst frequency must be a TO mode with Eu symmetry, the second is supposedly the frequency ωLO∗1 , which belongs to the (Eu ) LO mode associated with ωTLO1 . At the next (Eu ) branch, a similar but broader band is seen between ωTO2 [439.1 cm−1 ] . . . ωLO∗2 [≈483 cm−1 ]. The sharp dip in Ψ within a few wavenumbers at frequencies above (Eu )ωLO∗2 is a subtle feature due to almost zero reﬂectivity in rs above (Eu )ωLO∗2 when ε⊥ approaches

5.2 Sapphire (Hexagonal)

73

Table 5.1. Bands of total reﬂection in rp and rs for high-symmetry orientations of uniaxial bulk crystals. The frequencies ωLO∗ are deﬁned by ε(ωLO∗ )||,⊥ = εa sin2 Φa . Total reﬂection at a given frequency ω occurs if ω matches within one of the conclusive sets of frequencies. Conclusive means: If ω matches both sets of frequencies, total reﬂection does not occur. For sapphire, l = m = 4, and n = 2 (Table 5.2) {εx , εy , εz }

p-polarization

s-polarization

c-plane: j=n i= (m

{ε⊥ , ε⊥ , ε|| }

{ωTO , ωLO }⊥,i \ {ωLO , ωLO∗ }||,j

i=1 j=1

m (

{ωTO , ωLO∗ }⊥,i

i=1

a-plane: j=n i= (m

{ε|| , ε⊥ , ε⊥ }

{ωTO , ωLO }||,j \ {ωLO , ωLO∗ }⊥,i

i=1 j=1

m (

{ωTO , ωLO∗ }⊥,i

i=1

a-plane: m (

{ε⊥ , ε|| , ε⊥ }

m (

{ωTO , ωLO∗ }⊥,i

i=1

Eu

{ωTO , ωLO∗ }||,i

i=1

α-Al2O3

A2u

Ψ [deg]

60

30

c-plane 0

500

750

1000 -1

ω [cm ]

1250

500

750

1000 -1

ω [cm ]

c⊥x

c || x

a-plane

a-plane

1250

500

750

1000

1250

-1

ω [cm ]

Fig. 5.2. Ellipsometric parameter Ψ from high-symmetry orientations of α-Al2 O3 (Φa = 70◦ )

74

5 Anisotropic Substrates

unity, and a non-zero reﬂectivity in rp , which begins to raise because of the next (Eu ) TO-LO band. However, prior to this band at (Eu )ωTO3 [569 cm−1 ] . . . (Eu )ωLO∗3 [≈ 629.7 cm−1 ], the additional condition due to ε|| causes a very sharp band in rp only, and a pronounced structure appears in Ψ between 510.8 cm−1 . . . ≈ 515.5 cm−1 , which is indicative of an ωLO . . . ωLO∗ sequence for polarization along c (A2u ). A fourth phonon band with Eu symmetry extends from ωTO4 [633.6 cm−1 ] . . . ωLO∗4 [≈ 1021 cm−1 ]. This band is split between ω ≈ 880 cm−1 and ω ≈ 907 cm−1 caused by a loss in p-reﬂectivity due to an overlap between the fourth Eu -symmetry band with the second A2u symmetry band: (A2u ) ωLO2 [881.1 cm−1 ] . . . (Eu ) ωLO4 [906.6 cm−1 ]. The assignment of the latter follows, however, from simultaneous consideration of the remaining spectra. For the a-plane situation when c||x, a narrow rs -reﬂectivity band between (Eu )ωTO1 . . . (Eu ) ωLO∗1 merges with the steep rise in rp at the low energy side of the reﬂection band in Ψ at (A2u )ωTO1 . . . (A2u ) ωLO1 . The total reﬂectivity band in rp due to (Eu ) ωLO1 . . . (Eu ) ωLO∗1 is too narrow to be detected at the spectral resolution used here. Within (A2u ) ωTO1 , and (Eu ) ωTO2 ||rp ||2 is 1, whereas ||rs ||2 is less than 1, and hence Ψ extends above 45◦ . The start of the plateau in Ψ ≈ 45◦ marks (Eu ) ωTO2 . Between (Eu ) ωLO2 . . . (Eu ) ωLO∗2 a sharp cut-oﬀ occurs in rp , leading to the notch in Ψ , and marking (Eu ) ωLO2 . Above (Eu ) ωLO∗2 , rp recovers its total reﬂectivity condition, but rs does not. Hence, Ψ extends again beyond 45◦ until rp drops oﬀ at (A2u ) ωLO1 . The p-reﬂectivity reaches unity again at (A2u ) ωTO2 , where rs is already total reﬂecting because of (Eu ) ωTO3 < (A2u ) ωTO2 . The peak in Ψ denotes the spectral range where rp rp is still unity, but rs rs is not, i.e., between (Eu ) ωLO3 . . . (Eu ) ωTO4 . Note that rp should not actually be total reﬂecting between (Eu ) ωLO3 . . . (Eu ) ωLO∗3 , and a sharp notch should appear in Ψ . However, because (Eu ) ωLO3 and (Eu ) ωLO∗3 are so close in their wavenumbers, this structure cannot be seen at the resolution used here (2 cm−1 ). The dip in Ψ is due to the end of the rp reﬂection band at (A2u ) ωLO2 , and the inset of the additional band at (Eu ) ωLO4 , throughout which rs continues its total reﬂectivity. Both reﬂection bands in rp and rs end at (Eu ) ωLO∗4 . For the a-plane situation when c⊥x total reﬂection in rp occurs between (Eu ) ωTOi . . . (Eu ) ωLOi , and in rs between (A2u ) ωTOj . . . (A2u ) ωLO∗j . The p-bands can be easily recognized within the Ψ spectra. The small peak in Ψ at (A2u ) ωLO∗1 is caused by total loss of s-reﬂectivity, whereas rp is non-zero (but not totally reﬂective). An inversion in Ψ occurs between (Eu ) ωTO3 , where rp enters the third band of total reﬂection whereas ||rs ||2 < 1, and (A2u ) ωTO2 , where the second band of total reﬂection in rs begins. The corresponding phonon mode frequencies, after achieving the best-ﬁt calculations for the experimental spectra, are summarized in Table 5.2. One must also include the phonon mode broadening parameters (Table 5.3). As can be seen, the conditions for the LO and TO lattice mode broadening parameters derived

5.2 Sapphire (Hexagonal)

75

Table 5.2. Room temperature transverse and longitudinal optical phonon frequencies of α-Al2 O3 in units of cm−1 (ε0,⊥ = 9.385, ε∞,⊥ = 3.077, ε0,|| = 11.614, ε∞,|| = 3.072) [6] l ωTO ωLO ωTO ωLO

(Eu ) (Eu ) (A2u ) (A2u )

1

2

3

4

384.99 387.60 397.52 510.87

439.10 481.68 582.41 881.1

569.00 629.50

633.63 906.6

Table 5.3. Room temperature transverse and longitudinal optical phonon broadening parameters of α-Al2 O3 in units of cm−1 [6] l γTO γLO γTO γLO

(Eu ) (Eu ) (A2u ) (A2u )

1

2

3

4

3.3 3.1 5.3 1.1

3.1 1.9 3.0 15.4

4.7 5.9

5.0 14.7

in Sect. 3 (3.6) is fullﬁlled for sapphire. A more detailed discussion can be found in [6]. The high-wavenumber frequencies ωLO of the ordinary and extraordinary dielectric functions determine the shape of the high-energy edge of the material’s reststrahlen region, which depends on the angle of incidence. For sapphire at Φa = 70◦ , both p- and s-polarized reﬂectivities undergo loss near ω ≈ 1000 cm−1 . If ωLO⊥ diﬀers from ωLO|| , a distinct ﬁne structure of this reststrahlen edge appears in the ellipsometric parameters.7 According to (5.11) and (5.13), loss of p- and s-polarized reﬂectivity is determined by ωLOx and ωLOz . For sapphire, (Eu ) ωLO4 > (A2u ) ωLO2 and ε|| > ε⊥ near ω ≈ 1000 cm−1 . Hence, depending on the surface orientation and the wavenumber sequence of ωLOx and ωLOz , the reststrahlen edge resembles a positive (Ψ increases and then decreases) or negative (Ψ decreases and then increases) derivative-like feature, or just that of an isotropic material. For c-plane sapphire (refer to Fig. 5.2), the s-reﬂectivity drops ﬁrst at (A2u ) ωLO2 , and the p-reﬂectivity follows later at (Eu ) ωLO4 , leading to the negative derivative-like structure. For a-plane when c⊥x, this sequence is reversed, and so is the derivative-like structure. For a-plane when c||x, both reﬂectivities drop at (Eu ) ωLO4 so as if the material would be isotropic. The sequence (Eu ) ωLO4 > (A2u ) ωLO2 can be directly read from the Ψ data, provided that the surface orientations are known. This sequence directly stems from the fact, that ε|| > ε⊥ . Hence, the character (positive or negative) of the 7

This eﬀect is easily missed in unpolarized reﬂection intensity measurements.

76

5 Anisotropic Substrates

optical anisotropy of a uniaxial material above the reststrahlen edge can be recognized from the experimental data without explicit numerical evaluation.

5.3 Stibnite (Orthorhombic) Materials with orthogonal center-of-gravity systems can be treated by similar procedures as discussed above for the uniaxial situation. Similar conditions as in Table 5.1 can be set for each high-symmetry plane. Although rarely in use as substrate materials for semiconductor layer structures, investigations of fundamental phonon mode properties is still a task to be fulﬁlled for most of the naturally occurring minerals. Stibnite is such a material, which is chosen here for example.8 The ore mineral discussed here originates from Iyo, Japan. Samples were cut from the same single crystal approximately parallel to (100) (“a-plane”), (010) (“b-plane”), and (001) (“c-plane”). After initial polishing, orientations of the a, b, c axes, relative to the sample axes, were measured on a conventional back reﬂection Laue x-ray diﬀraction instrument. Then all samples were given a ﬁnal polishing [22, 23]. Figure 5.3 contains Ψ spectra from the approximate-oriented a-plane, bplane, and c-plane surfaces. The best-ﬁt calculated data were obtained from the wavenmuber-by-wavenumber approach and a multiple-sample-, multipleorientation analysis. During this analysis, measurements from all surfaces as well as from multiple sample orientations (multiple polar sample angles ϕ) taken at multiple angles of incidence were combined into one best-ﬁt parameter regression procedure. Figure 5.3 shows such oﬀ-axis oriented data for the b- and c-plane surfaces, where the angle between the stibnite a-axis and the laboratory x-axis (plane of incidence) is approximately 45◦ . a-Plane The high-symmetry a-plane situation is depicted in Fig. 5.3. The bands of total reﬂection are indicated by horizontal bars, labelled by p or s, for c || x, and c || y . In each case, however, the TO frequencies for polarization parallel a cannot be determined, similar to the TO wavenumbers the for z -polarization in uniaxial c-plane samples. Stars in Fig. 5.3 indicate the wavenumbers ωLO for polarization parallel the a-axis. The bands-of-reﬂection interchange upon 8

The metal chalgogenide stibnite, Sb2 S3 , is orthorhombic, Pnma, a = 11.302 ˚ A, b = 3.834 ˚ A, c = 11.222 ˚ A, with excellent cleavage on (100), and crystals commonly elongated along the b-axis [19]. (Some older literature interchange axial labels leading to Pbnm, a = 11.2 ˚ A, b = 11.3 ˚ A, c = 3.8 ˚ A, cleavage on (010), and crystals elongated along c). Sb2 S3 has become of interest in recent years because of its use in television cameras, microwave, switching, and optoelectronic devices [20]. Potentials for use in optical data storage media [21], and as solar-cell material was identiﬁed.

5.3 Stibnite (Orthorhombic) a*

* *

* s p

b c

60

c-plane

b c *

a *

*

60

*

ps

Ψij [deg]

Ψ [deg]

a-plane c||x

77

30

30 pp sp

a-plane c||y

Ψ [deg]

60

a*

* *

* p s

b c

0

b-plane

a b *

* 60

c ps

30

30

pp

Sb2S3 0

100

150

Ψij [deg]

0

sp

200

250 -1

ω [cm ]

300

350

100

150

200

250

300

350

0

-1

ω [cm ]

Fig. 5.3. Experimental (dotted lines) and best-ﬁt (wavenmuber-by-wavenumber multiple-sample-, multiple-orientation analysis) calculated ellipsometry data (solid lines), obtained from nearly a-plane, b-plane, and c-plane oriented surfaces of stibnite (Φa = 70◦ ). Data shown for the b- and c-plane surfaces were acquired with the a-axis rotated ≈45◦ away the laboratory x-axis. The horizontal bars indicate the spectral regions of total reﬂection for p and s polarization. The labels “a”, “b”, “c” address polarizations parallel crystal axes a, b, c, respectively. The vertical bars denote TO and LO frequencies. The stars indicate frequencies ωLO∗

switching between the two high-symmetry orientations, and help to identify the TO and LO frequencies for polarization along axes b and c. b-Plane The oﬀ-axis data in Fig. 5.3 can only be shown in the generalized ellipsometry mode, where the Jones matrix presentation is used here. Strong p-s-mode coupling occurs across the stibnite reststrahlen band. For the b-plane, the TO wavenumbers for polarization parallel to axis b do not occur, and cannot be obtained. The bands of total reﬂection are governed by axes a and c, leading to mixtures (mode coupling) of p and s reﬂectivities. c-Plane Data reading is similar to b-plane, except that now the TO wavenumbers for polarization parallel to axis c do not occur. The bands of total reﬂection are governed by axes a and b.

78

5 Anisotropic Substrates

313-Plane Oﬀ-axis cut (skew cut) surfaces present ellipsometric challenges because of the intricate sample response. However, such surfaces also bear increased information, and in principle one can obtain all functions εa , εb , εc , in addition to the Euler angle parameters of the surface orientation [22]. One out of the six generalized ellipsometric data sets in the Jones matrix presentation is depicted in Fig. 5.4. The three-dimensional surface plot contains calculated spectra versus sample rotation. For each polar angle a diﬀerent spectrum is obtained, providing sensitivity the actual sample orientation.

Ψ ps [˚]

50

Sb2S3

25

0 350

360 300

W av

270

250

e nu

180

200

mbe r[

90

150

cm -1 ]

100 0

ϕ 0 [˚

]

Fig. 5.4. Calculated three-dimensional surface plot of the far infrared generalized ellipsometry parameter Ψps for an approximate (313)-plane surface of stibnite versus relative polar sample rotation ϕ0 at Φa = 45◦ . The model dielectric functions used here are given in [23] (At ϕ0 = 0◦ in Fig. 2.3: ϕ = 45◦ , ψ = 45◦ , θ = 45◦ in (2.49).)

Analysis of such data sets is done by use of the wavenumber-bywavenumber procedure. Initially, if no good estimate for TO and LO frequencies exists, a narrow spectral region may be selected, and Re{εa }, Re{εb }, Re{εc }, Im{εa }, Im{εb }, Im{εc }, must be varied together with sets {ϕ, ψ, θ} for each (!) sample orientation. Parameters ψ and θ are common to all sets from the same sample. If a precise goniometer for polar sample rotation was used, then the polar angle steps δϕ between individual measurements can be implemented as parameter constraints. If a good ﬁt is achieved, where several starting values should be tested, the procedure is extended into neighboring wavenumber regions. The Euler angles obtained thereby should be wavenumber-independent. This implies that the Euler angles can also be determined.

References

79

References 1. T. E. Tiwald, D. W. Thompson, J. A. Woollam, W. Paulson, and R. Hance, Thin Solid Films 313-314, 661 (1998). 67 2. T. E. Tiwald, J. A. Woollam, S. Zollner, J. Christiansen, B. R. Gregory, T. Wetteroth, and R. S. Wilson, Phys. Rev. B 60, 11464 (1999). 67 3. S. Zollner, J. P. Carrejo, T. E. Tiwald, and J. A. Woollam, phys. stat. sol. 208, R3 (1998). 67 4. T. E. Tiwald and M. Schubert, in L. M. Hanssen, ed., Optical Diagnostics Methods for Inorganic Materials II (SPIE, Bellingham, WA, 2000), vol. 4103, pp. 19–29. 67 5. J. Huml´ıˇcek, R. Henn, and M. Cardona, Phys. Rev. B 61, 14554 (2000). 67 6. M. Schubert, T. E. Tiwald, and C. M. Herzinger, Phys. Rev. B 61, 8187 (2000). 67, 72, 75 7. N. Ashkenov, B. N. Mbenkum, C. Bundesmann, V. Riede, M. Lorenz, E. M. Kaidashev, A. Kasic, M. Schubert, M. Grundmann, G. Wagner, et al., J. Appl. Phys. 93, 126 (2003). 67 8. M. Schubert, Phys. Rev. B 53, 4265 (1996). 68 9. I. Lang and U. Pashabekowa, Sov. Phys. - Solid State 6, 2913 (1965). 68 10. E. D. Palik, in E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic Press, Boston, 1991), vol. 2, p. 313. 70, 71 11. C. Kittel, Introduction to Solid States Physics (John Wiley and Sons, New York, 1986). 70 12. R. Davis, J. Cryst. Growth 137, 161 (1994). 70 13. J. Orton and C. Foxon, Rep. Prog. Phys. 61, 1 (1998). 71 14. S. Nakamura, M. Senoh, N. Iwasa, S. Nagahama, T. Yamada, and M. Mukai, Jpn. J. Appl. Phys. 34, L1332 (1995). 71 15. W. J. Tropf and M. E. Thomas, in E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic Press, Boston, 1998), vol. 3, p. 653. 71 16. F. Gervais, in E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic Press, Orlando, 1985), vol. 1, p. 177. 71 17. E. T. Arakawa and M. W. Williams, J. Phys. Chem. Solids 29, 735 (1968). 71 18. A. S. Barker, Phys. Rev. 132, 1474 (1963). 71 19. R. Poerschke, Semiconductors Other Than Group IV and III-V Compounds (Springer, Berlin, 1992). 76 20. A. G. Vesdeshwar, J. Phys. III France 5, 1161 (1995). 76 21. P. Arun and A. G. Vedeshwar, J. Appl. Phys. 79, 4029 (1996). 76 22. M. Schubert and W. Dollase, Opt. Lett. 27, 2073 (2002). 76, 78 23. M. Schubert, T. Hofmann, C. M. Herzinger, and W. Dollase, Thin Solid Films 455-456, 619 (2004). 76, 78

6 Zincblende-Structure Materials (III–V)

The following chapters will provide diﬀerent examples including various semiconductor materials of contemporary interest and with diverse properties, where diﬀerent layer structure situations will be addressed.

6.1 Free Charge Carriers (p-Type GaAs) In zincblende p-type group-III group-V materials heavy- (“hh ”) and light-hole (“lh ”) contributions to the infrared dielectric response is expected because of the heavy- and light-hole valence band states at the Γ -point of the Brillouin zone [1]. Multiple-component free-charge-carrier excitations can be detected, for instance, in highly Zn-doped p-type GaAs substrates at far-infrared wavelengths. Physical model assumptions must be invoked for derivation of quantitative information about the physical material parameters, such as mobility, concentration or eﬀective mass parameters from the infrared ellipsometry experiment. The Fermi-level-position (EF ) depends on the energy-dependent curvature of the valence bands (Ev (k )) submerged by the Fermi-energy-level, the freecharge-carrier density (N ), and the temperature (T ). Determination of EF requires knowledge of the band curvatures. For arbitrary energy surfaces and ﬁnite T the optical mass, which represents the average local curvature of the band, m∗op , must be evaluated from the following equation [2] ) 2 ∇k Ef (E − EF ) dk 1 ) , (6.1) = ∗ 2 mop

3¯ h

f (E − EF ) dk

and which must be compared with the eﬀective mass in (3.8) [2]. The density N is obtained from * N=

1 4π 3

f (E − EF ) dk ,

(6.2)

where E = E(k) is the band energy, and f is the Fermi function1 1

k = 1.38 × 10−23 J/K and h = 6.623 × 10−34 Js denote Boltzmann’s and Planck’s constants, respectively. The semiconductor is said to be nondegenerate if Ev + 4kT < EF

Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 81–107 (2004) c Springer-Verlag Berlin Heidelberg 2004

(6.3)

82

6 Zincblende-Structure Materials (III–V) 1

f (E, EF , T ) =

1 + exp

E−EF .

(6.4)

kT

For ellipsoidal energy band extrema the eﬀective-density-of-states concept can be addressed. For a single valence band extremum at Ev with spherical symmetry, the number of free holes per unit volume depends on the temperature and the Fermi energy EF [3] Nh = Nv F1/2 (η) ,

(6.5)

with the eﬀective density of states Nv = 2

2πkT mdh h2

.

(6.6)

The Fermi-Dirac integral of order 1/2 is deﬁned as [3] * ∞ 1/2 2 F1/2 (η) ≡ √ π

0

x dx , 1 + exp (x − η)

(6.7)

Ev − EF . kT

(6.8)

with the dimensionless variables x=

Ev − E , kT

η=

For single extrema with spherical symmetry the density-of-states eﬀective mass mdh equals that of the hole eﬀectice mass m∗h at the top of the valence band.2 For nonequivalent extrema at the same energy, such as light hole and heavy hole bands in zincblende materials, the total number of free holes Nh is obtained by adding the contributions of each band Nlh = Nv,lh F1/2 (η) ,

Nhh = Nv,hh F1/2 (η) .

(6.10)

Accordingly, for ellipsoidal bands one should obtain Nhh = Nlh

mhh mlh

32

,

(6.11)

or multiplying both sides with mlh /mhh , mlh Nhh = mhh Nlh 2

mhh mlh

12

,

(6.12)

or degenerate otherwise. For minima with nonspherical symmetry the density-of-states eﬀective mass is obtained by (6.9) mdh = (m1 m2 m3 )1/3 , where the mi s are the eﬀective mass tensor values diagonalized by a suitable choice of axes.

6.1 Free Charge Carriers (p-Type GaAs)

83

regardless of the Fermi energy and the sample temperature. The left side is equivalent to the ratio ζhh /ζlh (3.15), and the validity of (6.11) can be checked against the experimentally determined parameters ζhh and ζlh . Lowney and Kahn used k·p theory and obtained the valence band shapes in GaAs near the Γ -point [4]. According to their calculations, the curvatures of the three topmost valence bands near the Γ -point are not constant and increase diﬀerently for the three bands when the Fermi level merges deeper into the valence bands. Accordingly, one should expect deviations from the above considerations, which were made under the assumptions of parabolic bands, i.e., one should expect to obtain experimental ratios ζhh /ζlh , which deviate from (6.12) when GaAs with diﬀerent free-charge-carrier concentration parameters is studied. Figure 6.1 depicts far infrared ellipsometry spectra in the pseudodielectric function presentation measured on a GaAs single crystal. The supplier’s speciﬁcation of the sum concentration of free holes amounted to Nh ≈ 1.4 × 1019 cm−3 with a room-temperature DC free-hole-mobility value of 100

p-GaAs

80

(a)

ε2

60 40 20 0 30

(b)

20

ε1

10 0

-10 -20 -30

100

200

300

ω [cm ] -1

400

500

Fig. 6.1. Imaginary (a) and real part (b) of the pseudodielectric function ε (small dots) and the best-ﬁt model dielectric function ε for p-GaAs with Nh ≈ 1.4 × 1019 cm−3 (solid lines). The individual model lineshape contributions due to the GaAs TO mode (dashed lines), and due to free heavy holes (dotted lines) and free light holes (dash-dotted lines) are also shown

84

6 Zincblende-Structure Materials (III–V)

µp ≈ 70 cm2 /(Vs).3 Assuming a parabolic shape of the GaAs valence bands, then the Fermi-level EF at the Γ -point should be situated above the spin-

split valence band states [1, 5]. A potentially existing surface charge-carrierdepletion layer should be extremely thin (≈ 1 nm in Schottky approximation) due to the large bulk free-charge-carrier concentration. This layer is therefore omitted in the following analysis. Figure 6.1 further contains best-ﬁt ε data of p-GaAs (solid lines) in the far-infrared spectral range. Because of no overlayer was considered the model dielectric function ε equals ε and contains one lattice mode term (3.1), and two Drude terms (3.7) allowing for the Γ -point free-charge-carrier contributions due to light-hole and heavy-hole states.4 Two sets of quantities ζs = Ns /ms , and ξs = Ns µs (s = “hh”, “lh”) follow from the two Drude terms (Sect. 3.2). If the directionally (isotropic) averaged Γ -point (k = 0) light- and heavy-hole mass parameters are employed for data reduction (mlh = 0.08 m0 , mhh = 0.56 m0 ) one obtains the following concentration and mobility parameters [5]: Nhh = (1.26 ± 0.03)×1019 cm−3 , µhh = (85 ± 1) cm2 /(Vs), Nlh = (5.89 ± 0.02)×1018 cm−3 , µlh = (57 ± 1) cm2 /(Vs).

The sum of Nlh + Nhh is in good agreement with the supplier’s Nh value. Assuming decoupled bands, one can compare the free-charge-carrier mobilities obtained by the optical experiment with the electrical Hall-eﬀect result. The inverse of the energy-averaged momentum relaxation times may be added 1 1 1 = + , τp τlh τhh

(6.13)

and the eﬀective mobility value may be derived as follows µp,eﬀ =

µlh Nlh + µhh Nhh , Nlh + Nhh

(6.14)

which amounts here to 76 cm2 /(Vs). The electrical (DC) Hall mobility µp is found in good agreement with the optical result. This is not surprising because of the high crystal quality of the sample studied here. DC and optical mobility values in general address diﬀerent macroscopic quantities, averaged over ensembles of electronic states, and energy-dependent interaction processes [3, 6]. In essence, the optical excitation does not involve charge transport, whereas the electrical (DC) Hall-eﬀect incorporates scattering events with free-charge-carriers travelling over large distances. However, one must bear in mind that the optical mobility parameters derived upon (3.11) depend on the eﬀective mass parameters, and the assumption of spherical, parabolic 3

4

Both parameters were obtained by the supplier from electrical Hall-eﬀect measurements, presumably from diﬀerent specimens of the same lot. The best-ﬁt lattice mode parameters are ε∞ = 10.52 ± 0.02, ωTO = (267.4 ± 0.1) cm−1 , ωLO = (291.5 ± 0.2) cm−1 , and γ = (3.1 ± 0.1) cm−1 .

6.2 Alloying (AlGaInP)

85

energy bands. For GaAs, assuming parabolic bands, ζhh /ζlh = 0.684, regardless of Nh (6.12). For the p-type sample studied here one obtains ζhh /ζlh = 0.305 [5], which is indicative for the non-parabolic energy-dependence of the valence bands in GaAs near the Γ -point. In order to reconstruct the actual shape of the energetic states versus momentum one can determine ζhh /ζlh as a function of the total free-charge-carrier concentration, and solve iteratively for the Fermi energy and the individual distribution of holes within the valence bands.

6.2 Alloying (AlGaInP) Alloying of binary constituents into ternary, quaternary, or multinary groupIII group-V semiconductor solid solutions allows for tailoring of physical material properties, such as the energies and curvatures of valence and conduction bands, and the equilibrium lattice constants. Alloying provides another degree of freedom, unknown from the binary constituents: Ordering. Group-III or group-V elements can be placed by certain pattern onto their respective lattice sites (ordering), or may be randomly distributed.5 The way how this arrangement is done deﬁnes a certain state of ordering. One type of atomic ordering is that with CuPt-type structure, which is shown exemplarily for GaInP in Sect. 6.3. In the perfectly disordered alloy the elements are substituted by each other randomly. The atomic ordering occurs spontaneously during growth, and depends on the growth parameters. In real samples the order is not perfect, and the samples are partially ordered. Likewise, no complete disorder may be achieved. Order-induced properties can be used to evaluate the degree of ordering [7, 8]. Certain diﬀraction spots disappear in electron beam diﬀraction patterns when the ordered state diminishes (Fig. 6.6). It depends on the precision with which a speciﬁc technique can trace such properties (phonon modes, band-to-band transitions, birefringence, and so on). If a sample is found disordered, a more precise technique may still reveal ﬁnite ordering. Therefore, disordered samples are termed “highly-disordered”, indicating that none of the existing tools have revealed ordering-induced properties. The state of ordering inﬂuences the material properties, which becomes important for heterostructures design. The infrared dielectric function of multinary compounds contains valuable information about the physical state of the material: Strain, composition, and ordering, for instance. The inﬂuence of composition can be monitored by ellipsometry, discussed here exemplarily for randomly distributed mixed-anion (group-III) elements, common cation (group-V) element compound alloys.6 Speciﬁcally, unstrained, undoped, and highly-disordered Alx Ga1−x InP epilayers, grown on n-type Te-doped (001)-oriented GaAs by metal organic vapor 5

6

In alloys diﬀerent elements are allowed to occupy similar sites on common sublattices, e.g., Ga and In in GaInP. Cation mixing is expected to induce characteristic infrared ﬁngerprints as well.

86

6 Zincblende-Structure Materials (III–V) GaAs (buffer)

(Al0.33Ga0.67)0.52In0.48P

60

Ψ [deg]

InP GaP

40

350 400

SGW

AlP

+

FSP

20

GaAs (substrate) AM1 AM2 AM3

100

200

300

400

500

600

ω [cm-1]

Fig. 6.2. Experimental (symbols) and calculated (solid line) far-infrared Ψ spectra (Φa = 50◦ ) for a (Al0.33 Ga0.67 )0.52 In0.48 P ﬁlm grown on Te-doped n-type (001)GaAs substrate and undoped GaAs buﬀer layer (metal organic vapor phase growth). The lattice modes from the binary constituents are indicated by brackets (TO: solid ; LO: dotted vertical lines). The inset enlarges the spectral region of AM1 –AM3 . The dotted line presents the best-ﬁt lineshape without AM1 –AM3 . Note the occurrence of the FSP (very small dip) and SGW+ (at smaller wavenumbers than FSP) modes, bound to the interface n-GaAs-i-GaAs, where the position of the SGW+ mode is strongly modiﬁed by the quaternary epilayer

phase epitaxy, are discussed for compositions 0 ≤ x ≤ 1. Prior to deposition of the multinary layer, an undoped GaAs buﬀer layer was grown, which reveals its existence upon interface polariton mode propagation eﬀects.7 Figure 6.2 depicts experimental and calculated far-infrared Ψ spectra (Φa = 50◦ ) for a (Al0.33 Ga0.67 )0.52 In0.48 P epilayer sample. The reststrahlen band of the GaAs substrate extends between ωTO (GaAs) and LPP+ (GaAs), and causes distinct spectral features within the sample response at long wavelengths. The sharp peak near ω ≈ 284 cm−1 is caused by resonant excitation of the SGW+ mode within the n-GaAs-i-GaAs-AlGaInP heterostructure. This mode was discussed in Chap. 4. The GaAs buﬀer layer reveals its existence upon the subtle FSP resonance, which causes a notch in Ψ near ωLO (GaAs). The bracket “GaAs(buﬀer)” indicates ωTO (GaAs) and ωLO (GaAs), and points to the small dip caused by the FSP mode near ωLO (GaAs). The model dielectric function used for calculation of ε for the quaternary material contains multiple TO-LO-pairs to account for far-infrared-active 7

Transmission electron microscopy images and selected-area diﬀraction pattern revealed the highly disordered state of the group-III elements in these ﬁlms. Further details on electronic band-to-band transitions, optical constants, strain and structure evaluation of the samples discussed here are available [9, 10]. The perpendicular lattice mismatch |∆a/a|⊥ of all epilayers with respect to the GaAs substrate is less than 1 × 10−3 .

6.2 Alloying (AlGaInP)

87

60 (AlxGa1-x)0.52In0.48P 40

Al0.52In0.48P

60 40

x = 0.82

60 40

x = 0.7

Ψ [deg]

20 40

x = 0.48

20 40

x = 0.33

20 40

Ga0.52In0.48P

20 0 100

200

300

400

500

600

ω[cm-1]

Fig. 6.3. Experimental (dashed lines) and calculated (solid lines) Ψ spectra (Φa = 50◦ ) from various (Alx Ga1−x )0.52 In0.48 P/GaAs samples. Vertical lines indicate TO, LO, and AM mode frequencies from the respective model dielectric functions. The diﬀerent GaAs TO-LPP+ bands reﬂect the diﬀerent doping and free-charge-carrier mobility levels within the Te-doped substrates [10]

(i) binary lattice modes (GaP, InP, AlP), and (ii) alloy-induced low polarity modes (AM) with small TO-LO splitting. Brackets in Fig. 6.2 denote the InP-like, the GaP-like, the AlP-like TO and LO modes, and three low-polarity modes, AM1 –AM3 . The inset in Fig. 6.2 enlarges the spectral region where AM1 –AM3 occur, and shows the best-ﬁt lineshape of Ψ without considering contributions of AM1 –AM3 to ε of the quaternary ﬁlm. The low-polarity modes clearly improve the match between the model lineshape and the experimental data. The physical origin of modes AM1 –AM3 is related to the state of ordering within the group-III sublattice sites, addressed within Sect. 6.3. Figure 6.3 contains spectra from samples over the entire composition range. The vertical lines indicate TO, LO, and AM mode frequency parameters determined from the ellipsometry model analysis.8 8

The Te-doped GaAs substrates used for the individual samples were taken from diﬀerent lots, and diﬀer in their free-charge-carrier concentration and mobility values. This is reﬂected in Fig. 6.3 by the diﬀerent GaAs TO-LPP+ bands seen within the individual spectra, which are not further indexed here. Detailed parameter values are available in [10].

6 Zincblende-Structure Materials (III–V) 10

x=1

10

x = 0.82

1 0.1 0.01

x = 0.82

1 10

x = 0.7

Im(ε)

1 10

x = 0.48

1 0.1 0.01

x = 0.7

1 0.1 0.01

x = 0.48

1 10

x = 0.33

1 0.1 0.01

x = 0.33

1 10

x=0

300

1 0.1 0.01

x=1

1

x=0

1 0.1 0.01

1

350

400 -1

ω [cm ]

450

-Im(1/ε)

88

350

400-1

ω [cm ]

450

500

Fig. 6.4. ε of unstrained, highly-disordered (Alx Ga1−x )0.52 In0.48 P alloy epilayers. Dotted lines: direct inversion (wavelength-by-wavelength ﬁt) of the ellipsometry data by holding the GaAs-parameters and layer thicknesses constant; solid lines: best-ﬁt model dielectric functions. Vertical lines indicate frequencies of TO, LO, and AM modes. The dashed lines (x = 0) indicate the noise limit for the determination of the dielectric function values, where the average experimental error bars on Ψ and ∆ are propagated into error bars for ε. The short-dashed lines (x = 0) represent calculated ε spectra assuming the linear average between the dielectric functions of GaP- and InP [10]

Real and imaginary parts of the model dielectric functions together with the wavelength-by-wavelength extracted (point-by-point ﬁt) ε spectra of Alx Ga1−x InP are displayed in Fig. 6.4. The excellent match between the point-by-point extracted dielectric function data, and the lineshape produced by the model dielectric function approach is remarkable.9 The spectra demonstrate the complexity of the far-infrared dielectric functions of multinary compound materials. A simple approach is often found, where summation of harmonic oscillator line shapes weighted in a linear scheme by the volume fractions is done for approximation of ε, thereby accounting for the volume fractions of the binary constituents. Figure 6.4 depicts such spectra, exemplarily for Ga0.52 In0.48 P, assuming a linearly weighted average of the GaP and 9

The point-by-point ﬁt procedure does not impose constraints on the spectral dependence of ε.

6.2 Alloying (AlGaInP)

89

(AlxGa1-x)0.52In0.48P 440

AlP-like

ω [cm-1]

AM3

400

GaP-like

AM2

360 InP-like

AM4

320 0.0

AM1

0.2

0.4

0.6

0.8

1.0

Al Composition x Fig. 6.5. TO (solid symbols) and LO modes (open symbols) of unstrained, highlydisordered AlGaInP lattice matched to GaAs. Circles and triangles indicate parameters obtained from ellipsometry, squares denote values obtained from Raman scattering. The shaded areas emphasize the InP-, GaP- and AlP-like TO-LO bands. The dash-dotted lines follow the compositional dependencies of the alloy-induced modes (AM1 , AM2 , AM3 ), which are unknown from the binaries. Mode AM4 is observed for Ga0.52 In0.48 P only [10, 12]

InP dielectric functions. The result is strikingly diﬀerent from the measured dielectric function, and such interpolation schemes should be avoided.10 Figure 6.5 summarizes the compositional dependencies of all modes obtained from the model dielectric functions shown in Fig. 6.4. In addition, results from Raman scattering investigations are included for comparison [10].11 Excellent agreement is found between modes detected within the Raman scattering spectra, and those identiﬁed upon model analysis of the ε spectra. The shaded areas highlight phonon bands, which descend from those of the binary constituents. In addition, modes AM1 –AM3 occur consistently within all samples, except for mode AM3 , which seems to be related to the existence of Al within the group-III sublattice. These modes form bands, which obey slight dispersion upon x, except for mode AM1 . A fourth mode AM4 is observed for x = 0 only. The GaP-like band disappears for Al content x > 0.33. Likewise, the AlP-like mode is estimated to vanish for x < 0.15, if the linear dependencies of the AlP-like TO and LO modes are extrapolated. The frequencies of all modes may be used to monitor the composition of a quaternary (Alx Ga1−x )0.52 In0.48 P ﬁlm on GaAs. Attempts to 10

11

Adding harmonic oscillators according to the number of binary constituents somehow mimics the infrared reﬂectivity spectra of multinary alloys. This approach was previously used, e.g. for AlGaAs, or InGaAs [11]. In zincblende semiconductors the Γ -point Raman-active and infrared-active phonon modes are identical [1].

90

6 Zincblende-Structure Materials (III–V)

calculate the dielectric function at far-infrared wavelengths using ﬁrst-principles density-functional linear-response theory have been reported for random-alloy Ga0.5 In0.5 P [13, 14]. Ga0.5 In0.5 P should reveal GaP- and InP-like phonon bands, in accordance with the above results. Figure 6.5 contains results for the unstrained compounds, i.e., for certain but ﬁxed compositions for which the alloy adopts the same lattice constants as the substrate. For lattice matched growth on other substrates, or for strained growth, diﬀerent mode frequencies will be observed. For the biaxial strain situation, reduction of lattice symmetry and mode splitting is expected furthermore. The strain dependencies of the zone-center optical phonon modes can be approximated through the linear phonon deformation potential values, which are well known for most binary III-V compounds [1]. For example, for z = 0.48 in (Alx Ga1−x )z In1−z P grown pseudomorphic to GaAs, another scale must be added to Fig. 6.5. (see also Sect. 6.4.)

6.3 Ordering (AlGaInP) Long-range chemical ordering is widely observed in ternary or quaternary III-V semiconductor alloys grown by metal organic vapor phase epitaxy or molecular beam epitaxy [8]. The spontaneous arrangement of group-III elements during the growth process in alternating {111}B planes constitutes ordering of the CuPt(B) type in (Ay B1−y )III C III D2V zincblende compounds such as Aly Ga1−y InP2 , Aly Ga1−y InAs2 , or Inx Ga1−x Asy P1−y . The stacking direction of the superlattice planes is usually found according to ordering within either one of the {111}B planes, i.e., parallel to the [¯111] or the [1¯11] direction (Fig. 6.6, hereafter referred to as CuPt-type).12 The ordered phase consists of a 1,1-superlattice of monolayer planes (Ay B1−y )x+η/2 Cx−η/2 and (Ay B1−y )x−η/2 Cx+η/2 , where x and y are the composition parameters of the random substitutional alloy, and η is the diﬀerence between the compositions x of two subsequent sublattice planes within the CuPt superlattice. Spontaneously CuPt-type ordered (Al)GaInP(001) alloys usually consist of two domain types: Within domains, the sublattice stacking direction is constant, but ordering is incomplete (partial ordering, according to the degree of ordering η ). The domains diﬀer by the two possible ordering variants. Boundaries between domains of the same ordering variant consist of an atomic sequence change (antiphase boundaries). It was observed that all domains obey a common degree of ordering. There is no disordered material between diﬀerent domains (see, e.g. [15, 16, 17]). A perfect alternate stacking of the superlattice planes was not reported yet. Only partially ordered epilayers are commonly formed during growth [7]. Correlation between the surface structure during growth and the CuPt-type 12

Two other possible but rarely observed variants are due to ordering within the {111}A planes (CuPtA ).

6.3 Ordering (AlGaInP)

(a)

91

(b)

Fig. 6.6. Crystal structure of perfectly CuPt(B) ordered GaInP2 for ordering along the [1¯ 11] direction (Fig. 6.6a). Typical selected area electron diﬀraction pattern from an ordered A1+η B1−η C/A1−η B1+η C compound with CuPtB structure ([110]pole) The spots at 1/2 [¯ 111] (arrow a) and 1/2 [1¯ 11] (arrow b) reveal the CuPt-type ¯ ordering along the [111] and [1¯ 11] directions. A favorite [¯ 111] ordering is found in this sample [18], where the spots at 1/2 [¯ 111] appear with higher intensities than these at 1/2 [1¯ 11] (Fig. 6.6b)

ordering was observed in GaInP [19]. Ordering was found to develop during the vapor phase growth if it takes place in the phosphorous-rich (2 × 1) mode, whereas it is suppressed in the less-phosphorous-rich (2 × 4) regime, and surface strain was proposed to be responsible for the surface-induced ordering [20, 21]. An ordering parameter of η ≈ 0.45 was observed for the best achievable (2 × 1) regime [19]. Theoretical results relate diﬀerent surface structures of binary III-As(001) and III-P(001) to surface strain caused by diﬀerent atomic radii of the respective constituents.13 For partially ordered material the parameter η can be used to quantify the degree of ordering. η varies between 0 (random) and 1 (perfect ordering). For non-stoichiometric alloys (x = 0.5), 0 ≤ η/2 ≤ min(x, 1 − x). For lattice matched growth on GaAs x = 0.48 for Gax In1−x P. Lattice match is required to avoid further eﬀects due to biaxial strain.14 As a result of CuPt-type ordering the symmetry of the ordered structure is lowered from cubic (Td ) to rhombohedral (C3v ) by the superperiodicity along the ordering direction. In a strict sense, this symmetry change can only be assigned for a transition from completely random-substitutional to perfectly-ordered solid solutions. However, as discussed below, the symmetry properties of physical quantities of partially ordered compounds are very similar to those of materials with C3v symmetry. 13

14

It is thought that surface strain and its local reduction plays a key role in ordering formation [22]. For derivation of η, however, the ideal composition x = 0.5 (GaInP2 ) is assumed here. One may consider the remaining 0.02 Ga as being located at a Group III site randomly.

92

6 Zincblende-Structure Materials (III–V)

This (partial) symmetry change aﬀects the physical properties of the ordered material, and has caused strong research interest [7, 8, 23, 24, 25, 26]. The most commonly observed consequences of CuPt ordering are the reduction of the Γ -point band gap energy, and the splitting of the Γ -point valence bands [18, 23, 27, 28, 29, 30, 31, 32, 33]. Determination of optical properties is the most prominent method for giving an estimate of the relative degree of ordering within alloys of the same composition [23]. Ordering-aﬀected sample properties P scale with η according to the simple square law dependence: P (η) = P (0) + [P (1) − P (0)] η 2 .

(6.15)

For example, Γ -point band-gap energies E0 , crystal-ﬁeld splitting ∆cf and spin-orbit splitting parameters ∆so were calculated for perfectly-ordered AlInP2 , AlInAs2 , GaInP2 , and GaInAs2 [26]. Together with the random-alloy values, experimentally determined parameters from spontaneously ordered samples have been used to derive η , e.g., in GaInP2 [7, 32, 34, 35], AlInP2 [18], GaInAs2 [36, 37], and AlGaInP2 [27]. The symmetry change from Td to C3v lifts the degeneracy of the TO and LO phonon modes in the zincblende lattice, which diﬀer now for propagation along diﬀerent crystal directions. The Γ -point zincblende TO and LO modes split into modes with A1 (displacement pattern with polarization parallel to the ordering direction) and E (perpendicular) character. From one TO-LO pair with Td symmetry, one threefold- (E (TO)) and one singledegenerate (A1 (LO)) mode with phonon wave vector parallel to the ordering direction [1¯11], (q || [1¯11]), and three threefold-degenerate (A1 (TO), E (TO), E (LO)) modes with phonon wave vector perpendicular to the ordering direction [1¯11], (q ⊥ [1¯11]) will emerge. The number of modes with Td symmetry is doubled within the C3v structure. The far-infrared dielectric function obeys tensor character   ε⊥ 0 0 ε =  0 ε⊥ 0  , 0 0 ε||

(6.16)

where ε⊥ and ε|| denote the major dielectric functions for wave propagation with electric ﬁeld polarization perpendicular or parallel to the ordering direction, respectively.15 According to the above notation, TO and LO phonon modes with E (A1 ) symmetry represent poles and zeros within ε⊥ (ε|| ), respectively. For epilayers with exact or nearly-approximate (001) surface orientation the coordinate system of the CuPt-type ordered structure does not coincide with the laboratory coordinate system (x, y, z) because the optical axis of the uniaxial material is tilted from the sample normal (z -axis) by √ cos θ = 1/ 3, i.e., by θ ≈ 54.7◦ . The result of an optical measurement depends on the sample orientation, i.e., the position of the optical axis with 15

Here the ε tensor is represented within the coordinate system (ξ, η, ζ) (Fig. 2.3) of the CuPt-type ordered structure (Fig. 6.6), where the ζ-direction is taken parallel to the ordering direction.

6.3 Ordering (AlGaInP)

93

respect to the plane of incidence as well as with respect to the incident polarization. As discussed in Chap. 2, the standard ellipsometry situation is then inapplicable, and the generalized ellipsometry approach is needed. However, under certain conditions the standard ellipsometry approach can be maintained for anisotropic materials. The CuPt-type ordered epilayers are uniaxial. In general, for uniaxial ﬁlms, embedded within isotropic ambient and substrate materials, the oﬀ-diagonal Jones matrix elements rps and rsp are very small or zero, respectively, regardless of the axis inclination θ from the sample normal z if (Fig. 2.1) [38] (O|| ) the optic axis is within the (x, z) plane, or when (O⊥ ) the optic axis is within the (y, z) plane. These conditions can be used to maintain the standard ellipsometry approach, and acquire Ψ and ∆ spectra as if the sample would be isotropic. The spectra measured at orientation O|| will diﬀer from those taken at orientation O⊥ . The CuPt-type ordered ﬁlm is modelled as uniaxial anisotropic material during ellipsometry data analysis. Suﬃcient information for the two dielectric functions ε|| and ε⊥ are available thereby. Figure 6.7 depicts experimental and calculated Ψ and ∆ spectra for a nominally-undoped CuPt-type ordered Ga0.52 In0.48 P ﬁlm lattice matched to (001)GaAs,16 and with high degree of ordering (η ≈ 0.47).17 The calculated spectra were obtained upon modeling the GaInP2 ﬁlm as a uniaxial layer, where the Euler angle coordinates are known for both orientations.18 The unknown parameters are the model dielectric function parameters for the set of ε|| and ε⊥ , the thicknesses of the GaAs buﬀer layer and the GaInP2 layer, and the free-charge-carrier parameters of the n-GaAs substrate. The reststrahlen band of the n-GaAs substrate extends between ωTO(GaAs) ≈ 268 cm−1 (solid vertical line) and LPP+ ≈ 307 cm−1 (dotted vertical line). The small dip very close to the GaAs LO mode near ωLO(GaAs) ≈ 292 cm−1 (dashed vertical line, and arrow with double head denoted by FSP) is caused by resonant excitation of the Fano-type SP mode (“Berreman”eﬀect). The vertical arrows denoted by SGW+,− indicate the s-polarized surface guided modes bound to the n-GaAs/i-GaAs-GaInP heterostructure. Note the occurrence of both SGW+,− modes. InP-, and GaP-like TO (solid vertical lines) and LO (dotted vertical lines) modes are indicated by brackets. The inset enlarges the spectral region of AM1 , AM2 , and AM4 . A strong sample orientation dependence of the ellipsometry spectra can be seen within the spectral region where the modes AM1 , AM2 , and AM4 occur. These modes were already seen in randomly ordered (Alx Ga1−x )0.52 In0.48 P, as shown in the previous section, but modes AM1 and AM2 now reveal strong anisotropy upon 16 17

18

The substrate surface orientation was 2◦ oﬀ towards the nearest (111)Ga plane. The order parameter η was determined from generalized ellipsometry near-bandgap birefringence measurements [27]. In (6.16) ψ = 0◦ and θ = 54.7◦ ; φ = 90◦ (O|| ) or 0◦ (O⊥ ).

94

6 Zincblende-Structure Materials (III–V)

100

Ga0.52In0.48P

+ E(AM4) E(AM2) SGW E(AM ) 1

120

(η = 0.47)

100

80 -

A1(AM1) A1(AM4) A1(AM2)

Ψ⊥ [˚]

325 E(InP)

FSP

40

350

80

375

Ψ|| [˚]

SGW

60

60

E(GaP)

40

20

20

0 GaAs

(a)

A1(InP) A1(GaP)

300 250

E(AM1)

(b) GaAs

0 E(AM2)

350

E(AM2)

E(InP)

300

SGW

+

-

FSP

100

250 A1(AM2)

325 350 E(GaP)

200

∆|| [˚]

SGW

150

A1(AM1) A1(AM4)

∆⊥ [˚]

200

150

50 100 0 50

A1(InP)

-50

A1(GaP)

100

200

300 -1 ω [cm ]

400

0 500

Fig. 6.7. Experimental (symbols) and calculated (solid lines) Ψ (a) and ∆ (b) spectra (Φa = 70◦ ) for a Ga0.52 In0.48 P ﬁlm lattice matched to (001)GaAs. Spectra 11] direction parallel were measured in orientation O|| (||) and O⊥ (⊥), with the [0¯ and perpendicular to the plane of incidence, respectively. For O⊥ , Ψ (∆) spectra are shifted by 15◦ (90◦ ) for convenience

the change of sample orientation, and are shifted with respect to modes AM1 and AM2 depicted in Fig. 6.5. Figure 6.8 presents the best-ﬁt Ga0.52 In0.48 P model dielectric functions for ε|| and ε⊥ . The anisotropy caused upon the partial CuPt-type ordering is obvious.19 19

The free-charge-carrier concentration Ne = (3.5 ± 0.03) × 1017 cm−3 and mobility parameter µe = (2.76 ± 0.06) × 103 cm2 /(Vs) within the Te-doped GaAs substrate was determined from the ellipsometry data analysis. The buﬀer layer thickness is (130 ± 5) nm, the thickness of the GaInP2 layer is (1463 ± 5) nm.

6.3 Ordering (AlGaInP)

95

Ga0.52In0.48P η = 0.47

10 100 1

Im(ε⊥)

10

Im(ε||)

1 1 0.1

Im(-1/ε⊥)

0.01 1 0.1

Im(-1/ε||)

250

0.01

300

350

400

-1

ω [cm ]

Fig. 6.8. Model dielectric functions ε|| and ε⊥ of Ga0.52 In0.48 P obtained from data in Fig. 6.7. The local maxima in Im(ε) and Im(−1/ε) locate the TO and LO mode frequencies with E (⊥) and A1 symmetry (||)

A1 (ε||)

GaP-like

InP-like

AM2

360 II

AM2 III

AM4

-1

InP-like II

340

380

I

ω [cm ]

360

E (ε⊥)

GaP-like

I

-1

ω [cm ]

380

340

III

AM4

IV

IV 320

320 V

AM1 0.0

0.1

0.2 η 0.3

V

AM1 0.4

1.0

0.0

0.1

0.2

η

0.3

0.4

1.0

Fig. 6.9. Ga0.52 In0.48 P model dielectric function TO (solid ) and LO (open symbols) mode frequencies as a function of the order parameter η. Vertical dashed brackets are phonon mode frequencies calculated by Ozolinˇs and Zunger [13, 14] for η = 0 and 1. The random-alloy phonon bands evolve into ﬁve bands, bound by their respective TO and LO modes, which are highlighted by the hatched areas [12]

96

6 Zincblende-Structure Materials (III–V)

Figure 6.9 depicts the results in terms of the phonon mode frequency parameters from the best-ﬁt MDF’s for E and A1 symmetry as a function of η , respectively. Both ﬁgures also include theoretical predictions (dashed vertical brackets) [13, 14]. The agreement for η = 0, except for the observation of the additional modes AM1 (forming band I), AM2 , and AM4 , is quite fair considering the complexity of the theoretical problem. Modes AM2 , and AM4 split the InP-like band into three sub-bands at η = 0 (bands II, III, IV in Fig. 6.9), as already seen in Fig. 6.5. Evolution of the Ga0.48 In0.52 P phonon modes for η > 0.47 remains subject of speculation. For the superlattice with exact C3v symmetry, the A1 -TO modes in Fig. 6.9 are threefold degenerate, whereas the A1 -LO modes are singlets. As the perfect order begins to deteriorate, the degenerate TO modes may split into their three submodes with diﬀerent frequencies, and to which diﬀerently ordered parts of the sublattices may contribute. As the degree of ordering continues to diminish, the split modes continue to evolve and may form the ﬁve bands observed for η < 0.47. The alloy-induced modes, observed throughout the AlGaInP system, seem to have their origin in residual ordering of the group-III sublattice sites. The phonon modes of perfectly ordered material need to be studied experimentally before deﬁnite conclusions can be drawn regarding the exact origin of modes AM1 –AM4 . Likewise, evolution of the lattice and alloy-induced modes shall be studied for diﬀerent ordering types such as with ordering direction along [001] (CuAu-I-type ordering [7]). For CuPt-type ordering the phonon mode dispersion depicted in Fig. 6.9 can be used to monitor the degree of ordering.

6.4 Strain and Composition (GaNP, GaInAsN) Composition and strain are important parameters for group-III group-V heterostructure design due to the large variation of the electronic (band energies, eﬀective mass, mobility, etc.) and lattice properties (equilibrium lattice constant, expansion coeﬃcients,. . . ) within the constituent alloy materials. In combination with x-ray diﬀraction techniques ellipsometry at longwavelengths can be used for simultaneous determination of strain and composition in multinary alloy layers. The following two examples demonstrate the ellipsometry use for ternary and quaternary structures. 6.4.1 GaNP/GaP Nitrogen-doped GaP (GaNy P1−y , yN < 0.001) has been used for decades as active material in green light emitting diodes. Nitrogen inclusion causes tensile strain. With the availability of modern non-equilibrium growth techniques, metastable GaNP alloys can be grown successfully using metal-organic vapor phase epitaxy [39], or gas-source molecular beam epitaxy [40]. Similar to GaNy As1−y , incorporation of only a few percent of nitrogen in GaNy P1−y leads to a strong redshift of the fundamental band-gap energy [39, 40, 41, 42],

6.4 Strain and Composition (GaNP, GaInAsN)

97

and GaNy P1−y alloys emerge as candidates for III-V-compound materials engineering. Figure 6.10 summarizes Ψ spectra of various GaNP layers grown on GaP, together with the spectrum of a bare (001) i-GaP substrate with i-GaP buﬀer layer deposited only.20 High-resolution x-ray scattering spectra are depicted in Fig. 6.11, which were recorded using diﬀraction from the (004) lattice planes. The GaNy P1−y diﬀraction peak is shifted towards higher diﬀraction angles, which reﬂects the increasing nitrogen incorporation upon the increased biaxial tensile strain under pseudomorphic growth conditions. The full-width at half-maximum values indicate a good structural quality of the epitaxial layers, and increase slightly with y . Similar to GaNAs, two lattice resonance terms are required in (3.3) for calculation of ε for GaNy P1−y : one for the GaP-like resonance (ωTO1 , ωLO1 , γLO1 ), and a second for the GaN-like resonance (ωTO2 , ωLO2 , γLO2 ). The infrared active nitrogen-related mode within the GaP host lattice occurs in the impurity limit (y < 0.001) at ωGaP:N ≈ 492 cm−1 . . . 496 cm−1 [46, 47, 48, 49, 50]. For y > 0.001 the GaN-like phonon mode emerges from ωGaP:N , and splits into a TO and LO branch with increasing y . The phonon mode dependence on y is inﬂuenced by the biaxial tensile layer strain as well as by alloying, i.e., the replacement of P by N atoms. In order to diﬀerentiate between eﬀects of strain and alloying, the composition dependencies of the phonon deformation potentials and the elastic coeﬃcients of GaNy P1−y would be required. The observed redshift of ωLO1 seems reasonable because the local phosphor mode in cubic GaN is smaller than ωLO1 of pure GaP, and a biaxial tensile strain should also result in a redshift of ωLO1 . The frequency ωTO2 seems independent on y , and is found within (494.4 ± 0.6) cm−1 . This mode coincides with ωGaP:N , the well-known local mode of nitrogen in GaP. The extended linear diatomic-chain model suggests that ωGaP:N = 495 cm−1 [51], and ωGaP:N was observed between 492 cm−1 . . . 496 cm−1 for yN < 0.001 [46, 47, 48, 49, 50]. The TO frequency of cubic GaN, ωTO,β−GaN = 553 cm−1 (y = 1) [137] is larger than ωGaP:N , the local N-mode in GaP, and alloying should cause a blue shift of ωTO2 , the GaN-like TO-mode in GaNy P1−y . No shift of this mode is observed, and one may conclude that the eﬀects of biaxial strain and alloying must compensate, at least for the compositions y ≤ 0.0285 discussed here. Figure 6.12 summarizes the lattice mode parameters in terms of the normalized amplitude fj = (ωLOj − ωTOj )/ωTOj , and lattice broadening 20

The GaNy P1−y layers with thickness ≈350 – 380 nm were grown on undoped (i) (001) GaP at growth temperature TG = 650◦ C using low-pressure metal-organic vapor phase epitaxy, and standard precursors. The N -concentration was determined by high-resolution x-ray diﬀraction, shown in Fig. 6.11 [43, 44, 45]. Detection of the GaN-like phonon modes with small polarity by infrared reﬂection or transmission intensity measurements failed. Nitrogen-induced non-resonant Raman scattered light, collected from the samples shown in Fig. 6.10, was found too weak for quantitative analysis.

98

6 Zincblende-Structure Materials (III–V)

ωLO1

120

GaNyP1-y on GaP y = 0.0285 y = 0.023

80

Ψ (˚)

y = 0.0135 y = 0.009 y = 0.006 40

ωTO1

y=0

ωLO1

ωTO2

400

500

600 -1

Wavenumber (cm )

y = 0.0285

y = 0.023

4

10

y = 0.009

intensity (a. u.)

10

GaNyP1-y on GaP y = 0.006

5

GaP (004)

Fig. 6.10. Experimental (dotted lines) and best-ﬁt (solid lines) Ψ spectra of GaNP thin ﬁlms on (001) i-GaP (buﬀer)-i-GaP substrates. The subtle notch at the highfrequency end of the GaP restrahlen band indicates the occurrence of the FSP mode guided by the GaNP layer near their GaP-like LO mode. The GaN-like sublattice modes emerge near ωTO2 . Spectra for y > 0 are shifted for convenience by 15◦ with respect to each other

3

10

2

10

1

10

0

10

0

200

400

600

800 1000 1200 1400

∆Ω(arcsec) Fig. 6.11. High-resolution x-ray data from GaNy P1−y layers with y = 0.006 (Full width at half maximum = 58 arcsec), 0.009 (60 arcsec), 0.023 (71 arcsec), and 0.0285 (72 arcsec) grown on GaP by metal-organic vapor phase epitaxy [43]

6.4 Strain and Composition (GaNP, GaInAsN)

99

8

10

6

GaP-like phonon

GaNyP1-y

0.8

0.4

-1

2 8

γ (cm )

4

-2

f (10 )

9

20 15

GaN-like phonon

10 0.0 5 0.000

0.005

0.010

0.015

0.020

0.025

y Fig. 6.12. GaN-like sublattice mode strength parameter fj = (ωLOj − ωTOj )/ωTOj (squares) and broadening parameters γj (triangles) of the GaP-like (j = 1) and the GaN-like (j = 2) phonon band in GaNy P1−y grown on GaP. The decrease (increase) of f1 (f2 ) reﬂects the decrease (increase) of the polar momentum of the GaP (GaN) sublattice mode. For example, the parameter f2 increases linearly upon y with ∆f /∆y = 0.27 ± 0.02, which can be used to determine y in strained GaNy P1−y layers grown pseudomorphically on GaP [43]

parameters γj , reﬂecting the change of polarity (strength) of the GaP- and GaN-like band, and lattice disorder upon changes in γj . The strength of the bands can be used for testing the nitrogen content in the biaxial strained GaNP layer grown pseudomorphic on GaP. If the inﬂuence of strain can be separated from the eﬀects of alloying, one could diﬀerentiate between the state of strain and the composition of a GaNP layer using ωTO1 , ωLO1 , ωTO2 , and ωLO2 , as demonstrated below for InGaNAs.21 The broadening parameters γ1 and γ2 increase slightly with y , in correspondence with the line width values of the x-ray diﬀraction curves. The increase in broadening reﬂects nonideal lattice properties, such as formation of nitrogen clusters leading to local strain ﬁelds, or strain-induced dislocations, but also reﬂects inhomogeneous broadening due to alloying. 21

Identiﬁcation of pseudomorphic, or partially relaxed layer growth on a particular substrate should be possible thereby.

100

6 Zincblende-Structure Materials (III–V)

6.4.2 InGaNAs/GaAs (In)GaNAs alloys have attracted increasing interest in the past few years. In contrast to the well-known chemical trends of usual III-V alloys the incorporation of nitrogen into InGaAs and GaAs reduces drastically the band gap [52, 53, 54, 55]. Several models have been applied to explain this phenomenon theoretically [56, 57, 58, 59]. As shown experimentally and theoretically, the bowing of the energy gap depends strongly on the N content, due to the large chemical and size diﬀerences of the As and N atoms [40, 60]. Alloying of indium with GaNAs results in quaternary compounds with lattice match to GaAs or InP [61, 62]. In1−x Gax N1−y Asy is of high interest for optoelectronic devices operating at wavelengths of 1.3 . . . 1.55 µm [55, 63, 64, 65, 66, 67, 68, 69, 70, 71], required for high-speed optical-ﬁbre communication networking. Applications include laser diodes [72], bipolar transistors [73], resonant-cavity enhanced photodiodes [74], and high-eﬃciency solar cells [75]. Diﬀerent epitaxial methods have been used to grow GaNx As1−x on GaAs including metal-organic molecular beam and vapor phase epitaxy. Nitrogen fractions up to x ≈ 10% and higher were reported [76, 77]. Coupling of nitrogen-induced states causes red-shift of the Γ -point conduction band states, and blue-shift of the L- and X-point states, which can be measured by ellipsometry at near-infrared to ultra-violet wavelengths [43, 44, 45, 78, 79, 80, 81, 82, 83, 84, 85]. An interesting alternative is the incorporation of boron instead of nitrogen. Boron is less aﬀective on the host electronic states, yet reduces lattice mismatch between GaAs and (B) InGaAs [86]. This route is attractive for solar cell absorber materials, eventually incorporated in tandem structures. The nitrogen-induced sublattice phonon mode frequency can be measured by ellipsometry. Important for heterostructure device design the lattice mode parameters extracted from the dielectric function of thin undoped GaNy As1−y and In1−x Gax N1−y Asy layers can be used to determine the composition parameters x and y , together with the state of strain [81, 82]. Furthermore, strain- or dislocation-induced free carriers can be sensed upon detection of their contribution to the dielectric function [78, 80]. Due to the strong lattice mismatch between GaAs and GaNx As1−x for increasing nitrogen incorporation the critical thickness for the pseudomorphic layer growth of the tensile strained GaNx As1−x decreases rapidly. Alloys of In1−x Gax N1−y Asy can be grown lattice matched to GaAs [53]. Growth of strained superlattice (SL) structures can stabilize the GaNx As1−x sublayers evading the critical-thickness limitation for single epilayers. GaNx As1−x / GaAs SL structures with few monolayers of InAs constitute another alternative to alloying of In1−x Gax N1−y Asy on GaAs. Depending on the InAs fraction, the average strain within the InAs/GaNx As1−x /GaAs SL can be compensated, tensile or compressive.

6.4 Strain and Composition (GaNP, GaInAsN)

101

3-4 nm InAs

oxide

2-3 nm

GaAs

30 nm

GaAs

18 nm

GaN xAs 1-x

8 nm

InAs

GaN xAs 1-x

3 nm

GaAs

9 nm

InAs

InAs GaAs buffer

20x

10x

5x

InAs oxide

0.3 nm ~ 500 nm

GaAs buffer

InAs

~ 300 nm

InAs (001)GaAs

~ 1 mm

(a)

(001)GaAs

(b)

~1 mm

(c)

Fig. 6.13. Schematic presentation of the GaAs/(InAs)1 /(GaNx As1−x )12 (a) and GaAs/GaNx As1−x (b) superlattice structures. Figure 6.13c depicts a highresolution transmission electron microscopy image taken from the 1-monolayerInAs/12-monolayer-GaNx As1−x superlattice sequence in Fig. 6.13a, where the individual monolayers and lattice sites are resolved

InAs/GaNAs-SL’s The set of InAs/GaNx As1−x /GaAs SL samples shown in Fig. 6.13 can be used to deﬁne the strain and alloy dependence of the GaN-like phonon modes. The InAs monolayer fractions where chosen such that the overall SL’s are compressively strained, whereas those without the InAs monolayer are tensile strained. Figure 6.14 depicts infrared ellipsometry data, from which the strength f = (ωLO2 -ωTO2 )/ωTO2 and frequency of the GaN-like sublattice TO mode ωTO2 is obtained. Besides the occurrence of FSP and SGW modes, bound to the i-GaAs(buﬀer layer)-n-GaAs(substrate) interface, a resonance band evolves with increasing nitrogen concentration near ωTO2 . The frequency shift of ωTO2 versus x is found negligible. On the other hand, its relative magnitude f , related to the polar strength of the Ga-N phonon branch, increases linearly with x. This dependence is depicted in Fig. 6.15 versus the average SL in-plane strain εxx = (a − a0 [x])/a0 [x] and versus x.22 Two diﬀerent linear f -dependencies on x, one for the tensile (∆f /∆x ≈ 0.43), and one for the compressively strained SL’s (∆f /∆x ≈ 0.26) are obtained here. Accordingly, f vs. x reveals diﬀerent slopes. The latter can be used to diﬀerentiate between the strain and the composition induced phonon shifts of the Ga-N mode for small x23 22

23

The in-plane lattice constant a is equal to that of GaAs, and a0 [x] is the lattice constant of the unstrained In1−x Gax N1−y Asy alloy. According to x-ray investigations, all SL’s without InAs are under tensile strain, which increases with increasing x. The SL’s with InAs are under average compressive strain, which decreases with increasing x. It is implied that ωTO2 does not depend on x, then f can be decomposed into fractions linear in x and εxx .

6 Zincblende-Structure Materials (III–V) SGW

x =3.3 % TO2

30 FSP

TO1

SGW

90

8

x = 0.9 %

x =1.3 %

6

x =1.3 %

x = 0.9 %

4

x = 3.3 %

GaAs1-xNx /GaAs-SL

2

+

InAs/GaAs1-xNx /GaAs-SL

x =2.4 %

8

x =2.0 %

60

x = 2.4 %

6 x =1.5 %

x =2.0 %

30

4

x = 1.5 %

0

10

2LO1

TO2

GaAs1-xNx /GaAs-SL

60

0

Ψ [deg]

+

300

Ψ [deg]

Ψ [deg]

90

400 500 -1 ω ( cm )

600

700

Ψ [deg]

102

InAs/GaAs1-xNx /GaAs-SL

400

500

600

700

-1

ω ( cm )

Fig. 6.14. Left side: Experimental (symbols), and best-ﬁt calculation (solid lines) of Ψ for the GaNx As1−x /GaAs-SL’s (upper panels), and for the InAs/GaNx As1−x /GaAs-SL’s (lower pannels) shown in Fig. 6.13 (Φa = 70◦ ). Vertical lines indicate the GaAs-like (ωTO1 ), and the GaN-like TO (ωTO2 ) lattice mode frequencies. The FSP and SGW+ modes are excited near ω ≈ 268 cm−1 (ωLO(GaAs) ) and 310 cm−1 (ωLPP+ (GaAs) ), respectively, bound by the i-GaAs-n-GaAs interface. Spectra are to scale, but shifted for convenience by 20◦ each. Right side: Same spectra for the GaN-like resonance spectral region. The forbidden second harmonic of the GaAs LO frequency LO1 is detected for the tensile strained sample with x = 3.3%. This observation is explained by the breakdown of selection rules due to the increase of the strain-induced lattice disorder, in agreement with the observation that the InAs monolayers improve the morphology of the GaNx As1−x sublayers [80]

f = αx + βεxx ,

(6.17)

with α = (3.3 ± 0.1) × 10−1 and β = (5.1 ± 0.5) × 10−1 . The tensile strained GaNAs/GaAs SL’s reveal eﬀects of free charge carriers upon the best-ﬁt model calculations [78]. No detectable free-chargecarrier concentration was obtained for the compressively strained InAs/ GaNAs/GaAs SL’s [80], and for undoped GaNAs single layers discussed in [84]. The origin of the free charge carriers within the tensile strained GaNAs/GaAs SL’s is attributed to strain-induced acceptor or donator states. The lack of carriers within the compressively strained SL’s may indicate a better sample morphology because of less overall SL strain. It is also likely that the isolated InAs monolayers bind free holes and free electrons thus forming localized bound excitons [87, 88]. Figure 6.15b depicts the free-charge-carrier

6.4 Strain and Composition (GaNP, GaInAsN) 160

19.0

InAs/GaNxAs1-x/GaAs

3.3

-4

1.3 0.9

1.5

tensile

-6

-4

-2

0

2

4

6

εxx = 0

17.5

GaNxAs1-x/GaAs 0

100 18.0

-4

2.4 2.0

40

tensile

18.5

50

InAs/GaNAs/GaAs compressive 0

8

f [10 ]

80

150

GaNAs/GaAs

compressive

log n

120 f [10 ]

103

1

2

εxx [10 ]

0

3

x [%]

-3

(a)

(b)

Fig. 6.15. (a): GaN-like lattice mode strength parameter f versus in-plane strain εxx . The numbers denote the nitrogen fractions x of the GaNx As1−x sublayers (in %) [80]. (b): GaN-like lattice mode strength parameter f versus x. Lines are drawn to guide the eye. The free-charge-carrier concentration parameter N was derived from the best-ﬁt lineshape analysis for the GaNAs/GaAs SL samples. No free charge carriers are detected within the samples containing the InAs monolayers [80]

i-InxGa1-xAs1-yNy/i-GaAs/n-GaAs ωTO(GaN)

ωTO(GaAs) SGW

+

(∆a/a)⊥ -3 1.9x10

y = 0.024

x = 0.124

y = 0.023

x = 0.110

y = 0.022

x = 0.112

y = 0.019

-3 -3

x = 0.111

y = 0.013

-3

Ψ [a.u.]

y = 0.029 x = 0.094

7.8x10

-3

6.2x10

-3

7.4x10 8.1x10 7.3x10

Ψ [a.u.]

FSP{ωLO(GaAs)}

x = 0.091 100

200

300

400 -1

ω [cm ]

(a)

500

600

400

450

500

550

-1

ω [cm ]

(b)

Fig. 6.16. (a): Experimental (symbols) and best-ﬁt model calculation (solid lines) Ψ spectra of single InGaNAs layers grown on i-GaAs-n-GaAs buﬀer-substrate structures. The perpendicular lattice mismatch {∆a/a}⊥ was determined from highresolution x-ray diﬀraction measurements. The FSP mode bound to the i-GaAs-nGaAs structure is present, also the SGW+ mode. (b): Same as Fig. 6.16a for the GaN-like sublattice resonance spectral region. The magnitude of the resonance in Ψ is related to the strength of the lattice resonance (polarity), which can be used together with the perpendicular lattice mismatch to quantify the In and N contents within the quaternary layers [82]

104

6 Zincblende-Structure Materials (III–V)

concentration N of the GaNx As1−x sublayers, assuming m = 0.063 m0 , obtained from the infrared dielectric functions of the tensile strained SL’s. The parameter N increases considerably with x. The eﬀect of the carriers, which exhibit much lower mobility values than in GaAs, can be seen in the Ψ spectra in Fig. 6.14. The SGW+ resonance broadens with increasing x. The surface guided modes are extremely sensitive to absorption, which is here provided by the plasma damping mechanism within the GaNAs SL sublayers. On the contrary, the SGW+ resonance remains sharp within the SL’s containing InAs monolayers. InGaNAs The dependence of f on εxx and the nitrogen content can be used to determine the composition of quaternary Inx Ga1−x Ny As1−y layers, if the assumption holds that the Ga-N sublattice resonance is not inﬂuenced by the presence of In, except for the related strain. Infrared ellipsometry data from singlelayer Inx Ga1−x Ny As1−y are shown in Fig. 6.16a. The perpendicular lattice mismatch ∆a/a⊥ parameters are given as well, together with the resulting x and y contents (Fig. 6.16b). The nitrogen composition y follows from rearrangement of f with respect to y , which now includes three known (f , α, β ) and one unknown (εxx ) parameter. The biaxial strain εxx is related to the lattice mismatch perpendicular to the growth surface (∆a/a)⊥ via the elastic constants C11 and C12 [89] εxx ≡

C11 aGaAs − aInGaNAs aGaAs =− aInGaNAs aInGaNAs C11 + 2C12

∆a a

⊥

,

(6.18)

with aInGaNAs being the lattice constant of the unstrained (relaxed) InGaNAs layer. C11 , C12 , and aInGaNAs also depend on x and y . An iterative scheme, developed in [82], employs Vegard’s law, and starts with the C11 /C22 values of GaAs, and provides after few iteration cycles a convergent solution for the In and N contents [90].

References 1. P. Yu and M. Cardona, Fundamentals of Semiconductors (Springer-Verlag, Berlin, 1999). 81, 84, 89, 90 2. M. Cardona, Phys. Rev. 121, 752 (1961). 81 3. C. M. Wolfe, N. Holonyak, and G. E. Stillmann, Physical Properties of Semicoductors (Prentice Hall, New Jersey, 1989). 82, 84 4. J. R. Lowney and A. H. Kahn, J. Appl. Phys. 64, 447 (1988). 83 5. S. Zangooie, M. Schubert, D. W. Thompson, and J. A. Woollam, Appl. Phys. Lett. 78, 937 (2001). 84, 85 6. S. M. Sze, Physics of Semiconductor Devices (John Wiley and Sons, New York, 1981), 2nd ed. 84

References

105

7. A. Zunger, MRS Bull. 22, 20 (1997). 85, 90, 92, 96 8. A. Zunger and S. Mahajan, Atomic Ordering and Phase Separation in Epitaxial III-V Alloys (Elsevier, Amsterdam, 1994), vol. 3, chap. 19, 2nd ed. 85, 90, 92 9. M. Schubert, J. A. Woollam, G. Leibiger, B. Rheinl¨ ander, I. Pietzonka, T. Saß, and V. Gottschalch, J. Appl. Phys. 86, 2025 (1999). 86 10. T. Hofmann, G. Leibiger, V. Gottschalch, I. Pietzonka, and M. Schubert, Phys. Rev. B 64, 155206 (2001). 86, 87, 88, 89 11. S. Adachi, GaAs and Related Materials (World Scientiﬁc, New Jersey, 1994). 89 12. T. Hofmann, V. Gottschalch, and M. Schubert, Phys. Rev. B 66, 195204 1 (2002). 89, 95 13. V. Ozoli¸ nˇs and A. Zunger, Phys. Rev. B 57, R9404 (1998). 90, 95, 96 14. V. Ozoli¸ nˇs and A. Zunger, Phys. Rev. B 63, 087202 (2001). 90, 95, 96 15. I. Pietzonka, Wachstumsuntersuchungen an Geordnetem (GaIn)P unter Verwendung Verschiedener Phosphorquellen, Ph.D. thesis, Universit¨ at Leipzig (1999). 90 16. T. Sass, Zusammenhang Zwischen MOVPE-Z¨ uchtungsparametern und der ¨ Uberstrukturbildung in Epitaktischen (GaIn)P - Schichten, Ph.D. thesis, Universit¨ at Leipzig (1999). 90 17. J. Dakin, Handbook of Optoelectronics (Inst. of Physics Pub. Inc., 2004). 90 ˇ 18. M. Schubert, B. Rheinl¨ ander, E. Franke, I. Pietzonka, J. Skriniarov´ a, and V. Gottschalch, Phys. Rev. B 54, 17616 (1996). 91, 92 19. M. Zorn, P. Kurpas, A. Shkrebtii, B. Junno, A. Bhattacharya, K. Knorr, M. Weyers, L. Samuelson, J. Zettler, and W. Richter, Phys. Rev. B 60, 8185 (1999). 91 20. B. Philips, A. Norman, T. Seong, S. Mahajan, G. Booker, M. Skrowronski, J. Harbison, and V. Keramidas, J. Cryst. Growth 140, 249 (1994). 91 21. S. Zhang, S. Froyen, and A. Zunger, Appl. Phys. Lett. 67, 3141 (1995). 91 22. W. Schmidt, F. Bechstedt, N. Esser, M. Pristovsek, C. Schultz, and W. Richter, Phys. Rev. B 57, 14596 (1998). 91 23. S.-H. Wei and A. Zunger, Phys. Rev. B 49, 14337 (1994). 92 24. S.-H. Wei, A. Franceschetti, and A. Zunger, Phys. Rev. B 51, 13097 (1995). 92 25. S.-H. Wei, D. B. Laks, and A. Zunger, Appl. Phys. Lett. 62, 1937 (1993). 92 26. S.-H. Wei and A. Zunger, Phys. Rev. B 57, 8983 (1998). 92 27. M. Schubert, T. Hofmann, B. Rheinl¨ ander, I. Pietzonka, T. Saß, V. Gottschalch, and J. A. Woollam, Phys. Rev. B 60, 16618 (1999). 92, 93 28. A. Gomyo, T. Suzuki, and S. Iijima, Phys. Rev. Lett. 60, 2645 (1988). 92 29. T. Kanata, M. Nishimoto, H. Nakayama, and T. Nishino, Phys. Rev. B 45, 6637 (1992). 92 30. A. Mascarenhas, S. Kurtz, A. Kibbler, and J. M. Olson, Phys. Rev. Lett. 63, 2108 (1989). 92 31. P. Ernst, C. Geng, F. Scholz, H. Schweizer, Z. Y., and A. Mascarenhas, Appl. Phys. Lett. 67, 2347 (1995). 92 32. M. Schubert, B. Rheinl¨ ander, and V. Gottschalch, Solid State Commun. 95, 723 (1995). 92 33. B. Rheinl¨ ander, M. Schubert, and V. Gottschalch, phys. stat. sol. (a) 152, 287 (1995). 92 34. A. Mascarenhas, B. Fluegel, Y. Zhang, J. Greisz, and J. Olson, phys. status solidi (a) 64, 477 (1997). 92

106

6 Zincblende-Structure Materials (III–V)

35. P. Kiesel, T. Kippenberg, E. Greger, M. Moser, U. Hilburger, J. Krauss, G. Schmiedel, and G. D¨ ohler, Pysica E 2, 599 (1998). 92 36. R. Wirth, H. Seitz, M. Geiger, J. Porsche, F. Scholz, and A. Hangleiter, J. Appl. Phys. 83, 6196 (1998). 92 37. R. Wirth, J. Porsche, F. Scholz, and A. Hangleiter, Phys. Rev. B 59, 8592 (1999). 92 38. M. Schubert, B. Rheinl¨ ander, J. A. Woollam, B. Johs, and C. M. Herzinger, J. Opt. Soc. Am. A 13, 875 (1996). 93 39. J. Baillargeon, K. Cheng, G. Hoﬂer, P. Pearah, and K. Hsieh, Appl. Phys. Lett. 60, 2540 (1992). 96 40. W. Bi and C. Tu, Appl. Phys. Lett. 69, 3710 (1996). 96, 100 41. S. Miyoshi, H. Yaguchi, K. Onabe, R. Ito, and Y. Shiraki, Appl. Phys. Lett. 63, 3506 (1993). 96 42. X. Liu, S. Bishop, J. Baillargeon, and K. Cheng, Appl. Phys. Lett. 63, 208 (1993). 96 43. G. Leibiger, V. Gottschalch, A. Kasic, and M. Schubert, Appl. Phys. Lett. 79, 3407 (2001). 97, 98, 99, 100 44. G. Leibiger, M. Schubert, V. Gottschalch, G. Benndorf, and R. Schwabe, Phys. Rev. B 65, 245207 1 (2002). 97, 100 45. G. Leibiger, V. Gottschalch, G. Benndorf, R. Schwabe, and M. Schubert, phys. stat. sol. (a) 228, 279 (2001). 97, 100 46. D. Thomas, J. Hopﬁeld, and C. Frosch, Phys. Rev. Lett. 15, 857 (1965). 97 47. F. Thompson and R. Newman, J. Phys. C 4, 3249 (1971). 97 48. V. Riede, H. Sobotta, and H. Neumann, phys. stat. sol. (a) 89, K147 (1985). 97 49. A. Kachare, J. Cherlow, T. Yang, W. Spitzer, and F. Euler, J. Appl. Phys. 47, 161 (1976). 97 50. M. D. Santos, C. Hirlimann, and M. Balkanski, phys. stat. sol. (b) 126, K113 (1984). 97 51. G. Lucovsky, M. Brodsky, and E. Burstein, Phys. Rev. B 2, 3295 (1970). 97 52. M. Weyers, M. Sato, and H. Ando, Jpn. J. Appl. Phys. 31(2), L853 (1992). 100 53. M. Kondow, K. Uomi, K. Hosomi, and T. Mozume, Jpn. J. Appl. Phys. 33(2), L1056 (1994). 100 54. F. H¨ ohnsdorf, J. Koch, C. Agert, and W. Stolz, J. Cryst. Growth 195, 391 (1998). 100 55. S. Sato and S. Satoh, J. Cryst. Growth 192, 381 (1998). 100 56. S. Sakai, Y. Ueta, and Y. Terauchi, Jpn. J. Appl. Phys. 32, 4413 (1993). 100 57. L. Bellaiche, Appl. Phys. Lett. 75, 2578 (1999). 100 58. W. Shan, W. Walukiewicz, J. A. III, E. Haller, J. Geisz, D. Friedmann, J. Olson, and S. Kurtz, Phys. Rev. Lett. 82, 1221 (1999). 100 59. T. Mattila, S.-H. Wei, and A. Zunger, Phys. Rev. B 60, R11245 (1999). 100 60. S.-H. Wei and A. Zunger, Phys. Rev. Lett. 76, 664 (1997). 100 61. M. Kondow, K. Uomi, A. Niwa, T. Kitatani, S. Watahiki, and Y. Yazawa, Jpn. J. Appl. Phys. 35(1), 1273 (1996). 100 62. M. Gokhale, J. Wei, H. Wang, and S. Forrest, Appl. Phys. Lett. 74, 1287 (1999). 100 63. M. Kondow, S. Nakatsuka, T. Kitatani, Y. Yazawa, and M. Okai, Jpn. J. Appl. Phys. 35(1), 1273 (1996). 100 64. M. Gokhale, J. Wie, P. Studenkov, H. Wang, and S. Forest, IEEE Photonics. Technol. Lett. 11, 952 (1999). 100

References

107

65. F. H¨ ohnsdorf, J. Koch, S. Leu, W. Stolz, B. Borvhert, and M. Druminski, Electron. Lett. 35, 571 (1999). 100 66. C. Ellmers, C. H¨ ohnsdorf, J. Koch, C. Agert, S. Leu, D. Karaiskaj, M. Hofmann, W. Stolz, and W. R¨ uhle, Appl. Phys. Lett. 74, 2271 (1999). 100 67. D. Mars, D. Babic, Y. Kaneko, Y.-L. Chang, S. Subramanya, J. Kruger, P. Perlin, and E. Weber, J. Vac. Sci. Technol. B 17, 1272 (1999). 100 68. X. Yang, M. Jurkovic, J. Heroux, and W. Wang, Appl. Phys. Lett. 75, 178 (1999). 100 69. X. Yang, J. B. Heroux, M. J. Jurkovic, and W. I. Wang, Appl. Phys. Lett. 76, 795 (2000). 100 70. D. Gollub, M. Fischer, M. Kamp, and A. Forchel, Appl. Phys. Lett. 81, 4330 (2002). 100 71. M. Fischer, M. Reinhardt, and A. Forchel, Electron. Lett. 36, 1208 (2000). 100 72. C. Coldren, S. Spruytte, J. Harris, and M. Larson, J. Vac. Sci. Technol. B 18, 1480 (2000). 100 73. P. Chang, A. Baca, N. Li, P. Sharps, H. Hou, J. Laroche, and R. Ren, Appl. Phys. Lett. 76, 2788 (2000). 100 74. J. Heroux, X. Yang, and W. Wang, Appl. Phys. Lett. 75, 2716 (1999). 100 75. S. Kurtz, A. Allermann, E. Jones, J. Gee, J. Banas, and B. Hammons, Appl. Phys. Lett. 74, 729 (1999). 100 76. Y. Qiu, S. Nikishin, H. Temkin, N. Faleev, and Y. Kudriavtsev, Appl. Phys. Lett. 70, 3242 (1997). 100 77. C. Jin, Y. Qiu, S. Nikishin, and H. Temkin, Appl. Phys. Lett. 74, 3516 (1999). 100 ˇ M. Schubert, T. Hofmann, and V. Gottschalch, MRS Internet J. Nitride 78. J. Sik, Semicond. Res. 5, 3 (2000). 100, 102 79. J. Sik, M. Schubert, G. Leibiger, G. Kirpal, V. Gottschalch, and J. Huml´ıˇcek, Appl. Phys. Lett. 76, 2859 (2000). 100 80. J. Sik, M. Schubert, G. Leibiger, and V. Gottschalch, J. Appl. Phys. 89, 294 (2001). 100, 102, 103 81. G. Leibiger, V. Gottschalch, and M. Schubert, phys. stat. sol. (a) 228, 259 (2001). 100 82. G. Leibiger, V. Gottschalch, and M. Schubert, J. Appl. Phys 90, 5951 (2001). 100, 103, 104 83. G. Leibiger, V. Gottschalch, B. Rheinl¨ ander, J. Sik, and M. Schubert, Appl. Phys. Lett. 77, 1650 (2000). 100 84. G. Leibiger, V. Gottschalch, B. Rheinl¨ ander, J. Sik, and M. Schubert, J. Appl. Phys. 89, 4927 (2001). 100, 102 85. G. Leibiger, V. Gottschalch, A. Kasic, B. Rheinl¨ ander, J. Sik, and M. Schubert, Mat. Res. Soc. Symp. 639, G6.35 (2001). 100 86. G. Leibiger, V. Gottschalch, V. Riede, M. Schubert, J. N. Hilﬁker, and T. E. Tiwald, Phys. Rev. B 67, 195205 (2003). 100 87. R. Iotti, L. Andreani, and M. Ventra, Phys. Rev. B 57, R15072 (1998). 102 88. K. M¨ ader and A. Baldereschi, in A. D’Andrea, R. Girlanda, and A. Quattropani, eds., Optics of Excitons in Conﬁned Systems: Proceedings of the International Meeting, Italy, 24-27 September 1991 (Instutite of Physics, Bristol, 1992). 102 89. J. Hornstra and W. L. Bartels, J. Cryst. Growth 44, 513 (1978). 104 90. G. Leibiger, A-III B-V-Mischkristallbildung mit Stickstoﬀ und Bor (Shaker Verlag, Aachen, 2004). 104

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

The success in growing high-quality wide-band-gap group-III-nitride alloys over the past several years has led to immense eﬀort in nitride-based semiconductor device research. In particular, the achievement of controlled p- and n-type doping of respective device constituents resulted in the development of short-wavelength light-emitting diodes, laser diodes, and high-temperature, high-power, high-frequency electronic devices [1, 2, 3, 4, 5, 6, 7]. Studies of fundamental physical properties of novel materials is essential for designing new device structures. Device heterostructures may consist of numerous layers of compound materials with diﬀerent compositions. Metrology and control of individual layer properties – such as the free-charge-carrier concentration and mobility, as well as strain and composition – represent a challenge. The previous sections have demonstrated the applicability of (generalized) infrared ellipsometry for characterization of free-charge-carrier properties, ordering, strain, and alloying eﬀects in arsenide- and phosphide-based semiconductor compound materials. This section focuses on the intricate infraredpolarization response of the optically anisotropic wurtzite III-nitrides and their complex device heterostructures. The infrared ellipsometry approach tackles this phenomenon, and its future perspective may be envisioned as nondestructive and noninvasive optical metrology tool for semiconductor research and production environments. 4 describes the crystalline structure of the binary The space group C6v wurtzite III-nitrides with two formula units in the primitive cell (Fig. 7.1). The optical phonons at the Γ -point of the Brillouin zone belong to the following irreducible representation [8, 9, 10] Γopt = 1A1 + 2B1 + 1E1 + 2E2 .

(7.1)

Both A1 and E1 modes are polar and split into TO and LO phonons with diﬀerent frequencies due to the macroscopic electric ﬁelds associated with the LO phonons. The short-range inter-atomic forces cause anisotropy, and A1 and E1 modes possess diﬀerent frequencies. Because the electrostatic forces dominate the anisotropy in the short-range forces, the TO-LO splitting is larger than the A1 -E1 splitting. For the lattice vibrations with A1 and E1 symmetry, the atoms move parallel and perpendicular to the wurtzite c-axis, respectively (Fig. 7.2). Both A1 and E1 modes are Raman and infrared active. Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 109–145 (2004) c Springer-Verlag Berlin Heidelberg 2004

110

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

Fig. 7.1. Crystal structure of the binary wurtzite-structure group-III nitrides. A wurtzite crystal consists of two interpenetrating hexagonal closely packed sublattices, displaced by 5/8 along the c [0001] axis

[0001]

A1

E1

E2(1)

E2(2)

B1(1)

B1(2)

[2110]

Fig. 7.2. Displacement pattern for the six optical Γ -point phonon modes in the wurtzite structure

The two non-polar infrared inactive E2 modes (E2(1) , E2(2) ) are Raman active. The B1 modes are infrared and Raman inactive (silent modes).1 The behavior is that of an optically uniaxial material. The infrared dielectric function is a tensor with two major dielectric functions   ε⊥ 0 0 ε =  0 ε⊥ 0  , 0 0 ε||

(7.2)

where ε⊥ and ε|| stand for wave propagation with electric ﬁeld polarization perpendicular or parallel to the wurtzite c-axis, respectively.2 In formal agreement with the CuPt-type ordered materials (Sect. 6.3), TO and LO phonon modes with E1 (A1 ) symmetry represent poles and zeros within ε⊥ (ε|| ), respectively. For epilayers with exact (0001) surface orientation the coordinate 1

2

Lattices with no center of inversion are piezoelectric, and some of the phonon modes can be both Raman and infrared active. Here the ε tensor is represented within the coordinate system (ξ, η, ζ) (Fig. 2.3) of the wurtzite structure (Fig. 7.1), where the ζ-direction is taken parallel to the wurtzite c-axis.

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

111

system of the wurtzite structure coincides with the laboratory system (x, y, z) and the optical axis of the uniaxial material is parallel to the sample normal (z -axis). This is an often observed situation because many wurtzite-structure layers adopt the basal plane growth orientation with (0001) surfaces. However, speciﬁcally when grown on low-symmetry cuts of anisotropic substrates, e.g., r-plane sapphire, wurtzite-structure layers likely adopt a diﬀerent growth orientation, e.g., a-plane. For the latter case, the optical axis is parallel to the growth surface, and the result of an optical measurement will depend on the orientation of the optical axis with respect to the plane of incidence as well as with respect to the incident polarization. The standard ellipsometry situation is then inapplicable, and the generalized ellipsometry approach is needed. Prior to the usage of infrared ellipsometry for the characterization of complex heterostructures, databases with typical model dielectric function parameters for the individual layer materials are needed. A database should comprise TO and LO phonon frequencies, broadening values, high-frequency dielectric constants, and free-charge-carrier eﬀective mass quantities. Relevant physical parameters are then allowed to vary when parameter sets are ﬁtted to an actual experiment upon the best-ﬁt layer model calculation. Such parameters could be the free-charge-carrier concentration and mobility to take into account diﬀerent doping levels, the phonon mode frequencies because of strain- or alloy-induced mode shifts, and the lattice broadening values to reﬂect diﬀerent crystalline quality of the individual layer constituents. Free-charge-carrier eﬀective mass parameter determination is discussed for n- and p-type GaN [11],3 and n-type InN [12]. Disorder-related phonon modes and eﬀects of free-charge-carrier depleted surface layers are shown exemplarily for highly Si-doped n-type GaN [13]. Inﬂuence of strain and composition in ternary and quaternary alloys will be addressed here.4 Lateral free-charge-carrier conﬁnement and strain-induced phonon mode shifts in strained AlGaN/GaN superlattices are discussed, leading to understanding and successful modelling of complex-device heterostructure samples for lightemitting and laser diodes. In the course of the initial sections, assessment of crystal quality, evaluation of anisotropic phonon modes, and determination of the epitaxial relationship for oﬀ-axis layer growth is discussed exemplarily for another wurtzite structure material of contemporary interest: ZnO. 3

4

The wurtzite modiﬁcation of group-III nitride materials is commonly indicated by “α”, whereas the cubic modiﬁcation is termed “β”, such as for “α”-GaN or “β”-GaN. Unless speciﬁed otherwise, only wurtzite materials are discussed in this chapter, and the preﬁx “α” is suppressed throughout. Further available data include AlN [14, 15], InN [16], AlGaN [17, 18, 19, 20], InGaN [21], AlInGaN [22], cubic AlGaN [23], and Alx Ga1−x N/Aly Ga1−y N superlattices [24, 25, 26]. Reviews are in [27], and [28].

112

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

7.1 Crystal Quality: Bulk and Epilayer (ZnO) ZnO possesses the same wurtzite-type structure as GaN, and thus belongs to the space group C64 v . ZnO is a direct-band-gap material (Eg ≈ 3.37 eV) with large exciton binding energy Eex ≈ 60 meV [30, 31]. Alloying with Mg and Cd allows for band gap engineering over large spectra regions. Addition of Ga and Al, for example, results in highly electrically conductive transparent materials. Such transparent conductive oxides (TCO) possess technological importance, for instance in display applications. ZnO-based compounds have gained renewed interest for use in electronic and optoelectronic device structures for applications in the ultra-violet to the deep-ultra-violet spectral region. Its optical properties render that of a uniaxial material, with optical axis parallel to the hexagonal lattice c-axis (Fig. 7.8). Previous studies, mostly performed on single crystal bulk samples have revealed many of its fundamental properties. In particular, detailed reports exist on the optical phonon modes and static dielectric constants [8, 32, 33, 34, 35]. However, recent availability of high-quality heteroepitaxial ZnO thin ﬁlms (Fig. 7.3), intended for incorporation into semiconductor device heterostructures, has provided new grounds for materials studies. Many of the growth peculiarities are not well understood. While n-type conductivity is fairly good controlled [29], attempts are currently focused on suﬃcient and reliable p-type conductivity, the prerequisite for bipolar device layer structures. Phonon modes and dielectric functions may be compared for singlecrystalline materials, present either in bulk form (crystal) or as epitaxial layer (thin ﬁlm). The fundamental lattice modes as well as their broadening parameters provide information about built-in strain and crystal quality. Infrared ellipsometry can thereby deliver quantitative and comparative evaluation of the epitaxial layer crystal structure properties. Wurtzite ZnO is chosen here as an example.

Fig. 7.3. A 10 × 10 × 0.3 mm3 sapphire crystal coated with a transparent, electrically conductive ZnO layer (thickness ≈ 1 µm). The epitaxial layer was deposited by the Pulsed Laser Deposition technique [29]. In the corners Au-pad are sputtered for Hall-eﬀect measurements (M. Grundmann, private communication)

7.1 Crystal Quality: Bulk and Epilayer (ZnO)

113

Fig. 7.4. Cross-sectional high-resolution transmission electron microscopy image and selected area diﬀraction pattern (inset) of a ZnO ﬁlm deposited by Pulsed Laser Deposition on a [0001]-oriented sapphire crystal surface [36]. The ZnO ﬁlm adopts the [0001] as the growth direction, indicated by an arrow. The incident electron beam direction is parallel to the [2¯ 1¯ 10] direction of the ﬁlm and parallel to the [1¯ 100] direction of the substrate. See also Fig. 7.8

A ZnO thin ﬁlm was deposited by the Pulsed Laser Deposition technique on a 10 × 10 × 0.5 mm3 sized c-plane sapphire (0001) crystal (Fig. 7.4). A KrF excimer laser (λ = 248 nm) was used for ablation of a polycrystalline ZnO target [29]. The undoped, naturally n-type bulk ZnO single crystal with cplane cut and double-side polished surface (Eagle Picher Technologies, LLC) was grown by the seeded chemical vapor transport method. The free-chargecarrier concentration in both samples was below the estimated infrared ellipsometry detection limit (Ne < 8 × 1016 cm−3 , Sect. 7.4). High-resolution x-ray diﬀraction scans of the ZnO thin ﬁlm and the bulk crystal are shown in Fig. 7.5. Figure 7.4 depicts a cross-sectional high-resolution transmission electron microscopy image and a selected area diﬀraction pattern from the interfacial region between the sapphire crystal and the ZnO ﬁlm. The selected area diﬀraction indicates the monocrystallinity of the ﬁlm, and reveals the epitaxial relationship [0001]sapphire ||[0001]ZnO and [1¯100]sapphire ||[2¯1¯10]ZnO . The lattice parameters are a = (0.325 ± 0.007) nm and c = (0.521 ± 0.01) nm, and identical with those of the bulk ZnO crystal (a = 0.3240 nm, c = 0.5206 nm). A very thin nucleation zone of about 2–3 monolayers on top of the substrate can be seen [36]. The (0002) ZnO diﬀraction peaks of the radial (2θ − ω ) scans possess diﬀerent full width at half maximum (FWHM) values, and the ZnO layer peak (0.047◦ ) is two times larger than that of the single crystal (0.023◦ ). This can be related to the existence of grain boundaries within the ﬁlm in the direction perpendicular to the sample surface. The (0002) peak of

114

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

ZnO/Al2O3 FWHM: 0.095˚ ZnO bulk FWHM: 0.24˚

ZnO/Al2O3 FWHM: 0.047˚ ZnO bulk FWHM: 0.023˚

1

1

0.1

0.1

0.01

0.01

34.2

34.4

2θ−ω (deg)

34.6

-0.4

-0.2

0.0

0.2

Normalized intensity (a.u.)

Normalized intensity (a.u.)

the ZnO layer appears at slightly smaller scattering angle (34.40◦ ) compared to the ZnO bulk peak (34.41◦ ), indicating a very small degree of residual compressive strain in the layers. Some contribution of a strain gradient to the broadening of the radial scan peak of the layer is also possible. On the other hand, the FWHM of the ZnO ﬁlm (0002) rocking curve was measured to be 0.095◦ (Fig. 7.5), indicating a low degree of mosaicity and a good overall crystal quality. The comparison with the bulk ZnO shows that its (0002) rocking curve is broader (FWHM: 0.24◦ ) most probably due to the surface polishing processing, which causes lattice damage in the near surface region.

0.4

∆ω (deg)

Fig. 7.5. ZnO (0002) high-resolution x-ray diﬀraction peak of the radial (2θ − ω) scan, and high-resolution x-ray diﬀraction rocking (∆ω) curve of the ZnO (0002) peak. (Solid line: ZnO epilayer ≈ 1 µm on sapphire, dotted line: ZnO bulk ≈ 10×10 × 0.3 mm3 crystal

Figure 7.6 contains the infrared ellipsometry spectra of the two samples. The calculated data were obtained using a two-phase model (ambient / (0001)-ZnO) for the bulk sample, and a three-phase model (ambient/(0001)ZnO/(0001)-Al2 O3 ) for the ZnO ﬁlm sample. The dielectric functions were parameterized using (3.3), and are contained in Fig. 7.7. The spectral positions of the infrared-active ZnO phonon modes are indicated by vertical lines in Fig. 7.6. The sapphire A2u and Eu symmetry modes are indicated for comparison. The bands of total reﬂection (reststrahlen bands) can be recognized by the plateaus in Ψ , where Ψ ≈ 45◦ . In Fig. 7.7 the TO phonons give rise to peaks in Im[ε], while the LO phonons peak in Im[ε−1 ] (Table 7.1). The

7.1 Crystal Quality: Bulk and Epilayer (ZnO) 300

60

40

115

A1 E1

A1 E1

ZnO

200

20

0

∆ [˚]

Ψ [˚]

100

-100

0 60

A1 E1

ZnO/α-Al2O3

A1

300

E1

200

40

100 20

0

A2u 0 400

600

Eu

Φa = 70˚

800

1000

-1

ω [cm ]

1200

A2u Eu 400

600

800

1000

-100 1200

-1

ω [cm ]

Fig. 7.6. Ψ and ∆ spectra of a ZnO crystal with (0001)-surface cut, and an epitaxial ﬁlm deposited by Pulsed Laser Deposition on a sapphire crystal with (0001) surface. (Dotted lines: experimental data; solid lines: best-ﬁt spectra.) The best-ﬁt ZnO ﬁlm thickness is d = (1401 ± 5) nm. The ZnO phonon modes of A1 and E1 symmetry, as well as the A2u and Eu phonon modes of sapphire are indicated by vertical lines (solid lines: TO phonons; dotted lines: LO phonons) [36, 37]

dielectric functions for both the ﬁlm and the bulk samples are almost indistinguishable, except for small deviations near the lattice mode resonances. The lattice broadening parameter γ of the bulk sample is slightly larger than for the ﬁlm sample, which reﬂects the residual lattice defects within the near surface region caused by the polishing process. This observation is concordant with the rocking curve data. The E1 (TO) mode of the ﬁlm sample is blueshifted by about one wavenumber (Table 7.1), indicating the same residual compressive strain in the layer as obtained from the radial x-ray scans (Fig. 7.5). The high-frequency dielectric constants of the bulk crystal are slightly larger than the ﬁlm values, which may be indicative for defects within the ﬁlm, likely located at grain boundaries. Hence, crystal quality assessments, similar to those obtained from x-ray diﬀraction investigations, can be obtained from analysis of the infrared ellipsometry data.

116

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

ZnO Re{ε}

75 0 -75

Im{ε}

A1(TO) E1(TO) 150 75

A1(LO) E1(LO)

-1

Im{-ε }

0 15.0

7.5

0.0 300

400

500

600

-1

ω [cm ] Fig. 7.7. Best-ﬁt calculated model dielectric function spectra (real and imaginary part), and dielectric loss function (Im[ε−1 ]) for ZnO in the reststrahlen band region. Solid lines: ﬁlm sample, dotted lines: bulk sample [36, 37] Table 7.1. ZnO phonon mode frequencies and broadening parameters in units of cm−1 obtained from model dielectric function lineshape analysis of the ellipsometry data in Fig. 7.6. Error bars in parenthesis represent the 90% conﬁdence limits. The high-frequency dielectric constants (ε∞ ) were calculated from the static limits (ε0 ) using the Lyddane-Sachs-Teller relation (3.4) [38, 39, 40]. For the two examples discussed here the A1 (TO) mode cannot be determined. See also Sect. 7.2

Film Bulk

ω[E1 (TO)]

ω[E1 (LO)]

ω[A1 (LO)]

γ

ε∞,⊥

ε∞,||

409.1(0.9) 408.2(0.3)

588.3(0.7) 592.1(0.2)

574.5(0.3) 577.1(0.4)

10 ± 1 13 ± 1

3.61(0.01) 3.70(0.01)

3.76(0.10) 3.78(0.05)

7.2 Total-Reﬂection-Bands and Dielectric Anisotropy (ZnO) As discussed in Sect. 5.2, the phonon mode frequencies and their symmetry of uniaxial bulk materials can be read from the infrared ellipsometry data by identiﬁcation of the bands of total reﬂection at oblique incidence for p- and s-polarized light. Furthermore, the sign of the dielectric anisotropy, i.e., the

7.3 Epitaxial Relationship (ZnO)

117

√

diﬀerence between ordinary (n⊥ = ε⊥ ) and extraordinary index of refraction √ (n|| = ε|| ) at wavenumbers closely above the restrahlen range can be seen just by inspecting the raw ellipsometry data. In Fig. 7.6, for ZnO, total reﬂection occurs for wavenumbers between ω[E1 (TO)] and ω [E1 (LO), Φa ],5 giving rise to a smooth plateau in Ψ for the ZnO bulk sample. The small dip within the plateau between A1 (LO) and E1 (LO) is caused by the loss in p reﬂectivity, and localizes both frequencies. Note that for the bulk sample, due to the uniaxial anisotropy, the E1 (LO) mode frequency can be directly obtained from the experiment, whereas for the ﬁlm sample this mode must be obtained through a lineshape analysis. A similar plateau, which originates from the ZnO reststrahlen band is present in the ﬁlm Ψ spectra. The sapphire substrate phonon modes introduce then additional features. For both samples, all materials are c-plane oriented, which means that the optical axes are oriented perpendicular to the surface. For this situation, the TO phonon mode parameter for polarization parallel to the optical axis cannot be obtained.6 The derivative-like structure in Ψ for bulk ZnO around 650 cm−1 in Fig. 7.6 is driven by a subtle anisotropy eﬀect. This eﬀect was explained in Sect. 5.2, with sapphire used as example. The derivative-like structure occurs just above the reststrahlen band of the ZnO bulk sample. The ZnO dielectric function spectra ε|| and ε⊥ approach unity at diﬀerent wavenumbers above the reststrahlen band (Fig. 7.7). Because rp (rs ) reaches its loss when |ε|| | ≈ 1 (|ε⊥ | ≈ 1), Ψ undergoes a minimum (maximum). It can be directly read from the Ψ spectra in Fig. 7.6 that ZnO is uniaxial positive, i.e., √ √ |ε|| | > |ε⊥ |, at wavenumbers immediately above the reststrahlen band. ZnO remains uniaxial positive throughout the entire band gap region [30].

7.3 Epitaxial Relationship (ZnO) The surface orientation of epitaxial layers adopted during heteroepitaxial growth depends on type and crystallographic surface orientation of the growth template (substrate). The epitaxial relationship between template 5 6

See Sect. 5.2 for deﬁnition of ω [E1 (LO), Φa ]. For the lineshape analysis, however, one must include this value into the model dielectric function. Additional experiments, for example Raman scattering, must be performed to determine the A1 (TO) mode(s). Oﬀ-axis oriented thin ﬁlm materials, if available, present a very useful alternative. As discussed in the following section, one can then obtain all symmetry modes from an infrared ellipsometry experiment. If no data is available, one must at least include a good estimate into the model dielectric function analysis for the c-plane oriented ﬁlms. However, the fact that a reﬂection or transmission experiment cannot provide the A1 (TO) mode(s) for c-plane oriented uniaxial ﬁlms also implies that these values can be widely varied without aﬀecting the calculated reﬂection or transmission coeﬃcients.

118

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

a3

c

c-plane (0001) a2

a1

a-plane (1120)

r-plane (1102)

Fig. 7.8. Schematic presentation of the hexagonal unit cell with coordinates a1 , 20) (“a”), and (1¯ 102) a2 , a3 , and c. Hatched areas indicate the (0001) (“c”), (11¯ (“r”) lattice planes

and epitaxial layer is then a crucial information, which is commonly obtained from x-ray diﬀraction measurements. Analysis of reciprocal space maps is required in order to obtain the indices of collinear lattice planes and directions. Generalized infrared ellipsometry experiments can provide the epitaxial relationship for non-c-plane oriented samples, as will be discussed in this section. ZnO and GaN growth can be performed on diﬀerent templates, such as silicon, silicon carbide, or sapphire. In addition, diﬀerent surface orientations can be used thereby varying surface reconstruction mechanisms and lattice spacings [14, 15, 41].7 Figure 7.8 depicts the crystallographic indices and orientations of a, r, and c planes exemplarily within the hexagonal unit cell. For growth on r-plane sapphire, the epitaxial layer adopts a-plane orientation, thereby minimizing the lattice mismatch between the template and the epitaxial layer. The oﬀ-axis orientations of both the substrate and the thin ﬁlm require appropriate anisotropy consideration in optical experiments. Often, such samples are considered useless for reﬂection or transmission studies due to the complicated polarization mode coupling. However, the generalized ellipsometry approach can easily handle this situation, and valuable information can be obtained. Figure 7.9 demonstrates generalized infrared ellipsometry data for an a-plane ZnO ﬁlm grown on r-plane sapphire. This three-dimensional surface plot for Ψsp (one out of six!) illuminates the capability of ellipsometry to determine the optical axis orientations for both the substrate and the epitaxial layer. 7

ZnO grown on r-plane sapphire has been demonstrated for acousto-optic, photochromic, and piezoelectric device applications [42].

7.3 Epitaxial Relationship (ZnO)

119

Ψ sp [˚]

a-ZnO/r-Al2O3 18 15 12 9 6 3 0 700

650

W av

360 600

e nu

550

270 500

mbe r[

180 450

cm -1 ]

400

90 350 0

ϕ 0 [˚

]

Fig. 7.9. Calculated three-dimensional surface plot of the infrared generalized ellipsometry parameter Ψsp for an a-plane ZnO ﬁlm on c-plane sapphire versus polar angle ϕ0 = ϕZnO = ϕsapphire at Φa = 70◦ . The model dielectric functions contained in Fig. 7.12 are used. Refer to Fig. 2.3 for Euler angle deﬁnition, and to (7.2) for the dielectric function tensor description

For the uniaxial materials, assignment for the lattice orientation can be made from the knowledge of the optical axis orientation. Besides the structural information, from such a sample the complete dielectric function tensor can be obtained, even without implementation of physical lineshape functions. The point-by-point data regression procedure can be applied, and the resulting dielectric function spectra can be directly compared to model dielectric functions. The infrared dielectric functions provide access to the complete set of A1 -and E -symmetry phonon-mode, the anisotropic free-chargecarrier mobility parameters, and their density.8 In addition to the anisotropic dielectric functions, the ﬁlm thickness can be determined. In principle, alternative methods, such as Raman scattering, x-ray diﬀraction, and electrical Hall-eﬀect measurements can provide similar information. However, the ellipsometry experiment can deliver the information from a single optical experiment, and in principle allows for analysis of complex layer structures. The latter is diﬃcult for correct assignment of Raman scattering and Hall-eﬀect data. Electrical contacts are not necessary for the ellipsometry technique. An a-plane oriented ZnO layer was grown by Pulsed Laser Deposition on r-plane oriented sapphire [29]. Ellipsometry data were acquired for three diﬀerent sample orientations (polar angle ϕ in Fig. 2.3) by rotating the sample manually about the sample normal in steps of δϕ ≈ 45◦ , and at multiple 8

Infrared ellipsometry data measured on c-plane oriented wurtzite-structure thin ﬁlms, e.g., ZnO, MgZnO, GaN, InN, . . . cannot provide the A1 (TO) mode frequency and broadening parameters (see also Sect. 7.4). For such studies, additional Raman scattering measurements are required [11, 12, 36, 43, 44].

120

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

Ψpp [˚]

(a) (b)

30

(c)

0

Ψps [˚]

(d) (e) 15 (f) 0

Ψsp [˚]

(g) (h)

10 (i) 0

500

750

1000 -1

wave number [cm ]

Fig. 7.10. Experimental (symbols) and best-ﬁt (solid lines) generalized infrared ellipsometry spectra Ψij of an a-plane ZnO thin ﬁlm grown on r-plane sapphire. The spectra are shifted for clarity. Vertical solid lines indicate the ZnO phonon mode frequencies in units of wavenumbers. Dotted lines denote the Eu and A2u symmetry sapphire modes. The vertical arrow indicates the F SP mode bound to the ZnO/sapphire interface. Spectra were measured at ϕ = (−43.9 ± 0.5)◦ [(a), (d), (g)], (0.4 ± 0.5)◦ [(b), (e), (h)], and (46.5 ± 0.5)◦ [(c), (f), (i)]. The angle of incidence for the data depicted here was Φa = 70◦ . Note the diﬀerent scales for Ψps , Ψsp , and Ψpp

angles of incidence. For comparison, structural analysis was performed by xray diﬀraction and conductivity evaluation was done by Hall-eﬀect measurements. The epitaxial relation was found as follows: [0001]ZnO ||[0¯111]sapphire and (11¯ 20)ZnO ||(01¯ 12)sapphire .9 Figures 7.10 and 7.11 show measured and best-ﬁt calculated spectra of the generalized ellipsometry parameters Ψij and ∆ij . Figure 7.12 depicts the 9

¯ ZnO, and (0224) ¯ and (1235) ¯ sapphire reﬂections were The (11¯ 20) and (2130) used for the determination of the epitaxial relationship. The ZnO (11¯ 20) rocking curve peak FWHM value was ≈0.22◦ , the value of the peak within the 2θ-ω-scan was ≈ 0.04◦ , reﬂecting very good crystal properties and coherent growth. The a lattice constant was (3.255 ± 0.001)˚ A, in good agreement with the ZnO bulk value [43]. The electrical Hall-eﬀect free-charge-carrier concentration at room temperature was found as Ne = 6.7(0.1) × 1017 cm−3 .

7.3 Epitaxial Relationship (ZnO)

121

∆pp [˚]

(a) (b) 180 90 0

(c)

∆ps [˚]

(d) (e) 360 180 0

(f)

∆sp [˚]

(g) (h) 360 180 0

(i) 500

750

1000

wave number [cm-1] Fig. 7.11. Same as Fig. 7.10 for ∆ij . The spectra are shifted for clarity

resulting dielectric functions extracted from the point-by-point (wavelenghtby-wavelength) ﬁt of the ellipsometry data, together with the result from a subsequent best-ﬁt lineshape analysis using (3.3, 3.7–3.11). Table 7.2 lists the model parameters for polarization parallel and perpendicular to the ZnO c-axis. The phonon mode frequencies agree well with those from Sect. 7.1. Parameters ε∞,⊥ and ε∞,|| were found as 3.60(0.01) and 3.53(0.01), respectively. These values are slightly smaller than for the c-plane ﬁlms, indicative for increased number of lattice vacancies. The free-charge-carrier concentration parameter is in very good agreement with the Hall-eﬀect measurement. The small diﬀerence might be due to uncertainties in the eﬀective electron mass, which was assumed to be m = 0.29 me . The best-ﬁt value of the c-axis azimuth angle (θ in Fig. 2.3) for the sapphire substrate was θsapphire = 54.9(0.8)◦ . The nominal value for a perfectly r-plane sapphire crystal is θsapphire = 57.6◦ , with manufacturing tolerance of ±1◦ . For the ZnO ﬁlm θZnO = 89.0(1.0)◦ , which means that the ZnO thin ﬁlm on r-plane sapphire indeed exhibited an a-plane orientation with the caxis lying in the growth plane. The diﬀerence between the ﬁlm and substrate in-plane azimuth angles (ϕ in Fig. 2.3) must be set close to zero in order to

122

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

Table 7.2. Best-ﬁt model dielectric function parameters in (3.3, 3.7–3.11) for aplane ZnO on r-plane sapphire. Compare also with Table 7.1. The free-electronconcentration parameter was Ne = 5.8(0.2) × 10 17 cm−3 , in very good agreement with electrical Hall-eﬀect measurements. Error bars in parenthesis represent the 90% conﬁdence interval for the last digit(s). Note that for single-mode situations no anharmonic coupling can occur, and γTO = γLO = γ

|| ⊥

ε∞

ωTO [cm−1 ]

ωLO [cm−1 ]

γ [cm−1 ]

µ [cm2 /(V s)]

3.60(2) 3.53(2)

376.5(1) 410.7(2)

574.1(4) 590.4(6)

15.1(2) 10.7(5)

32(5) 55(5)

Re(ε)

50

(a)

E⊥c

E||c

0

-50

150

(b)

E||c E⊥c

Im(ε)

100

50

0 400

450

500

ω [cm-1]

Fig. 7.12. Real (a) and imaginary part (b) of the anisotropic dielectric functions for polarization E||c and E⊥c determined from the a-plane ZnO thin ﬁlm in the reststrahlen region (dotted lines: point-by-point analysis, solid lines: model dielectric function approach with parameters given in Table 7.2)

7.4 Free Charge Carriers (p- and n-Type GaN)

123

match the complex data sets depicted in Figs. 7.10 and 7.11. This reﬂects the same epitaxial relationship obtained from x-ray analysies [42, 43, 45, 46], i.e., the ZnO [0001] direction is parallel to the sapphire [0¯111] direction, and the ZnO (11¯ 20) plane is parallel to the sapphire (01¯12) plane. For the azimuth angles one has ϕZnO = ϕsapphire .

7.4 Free Charge Carriers (p- and n-Type GaN) Figure 7.13 reviews typical ellipsometry spectra for undoped, Mg- (p-type) and Si-doped (n-type) GaN layers deposited by molecular beam epitaxy on c-plane sapphire [47, 48]. Most structures within the Ψ and ∆ spectra are due to excitation of bulk and interface polaritons caused by the sapphire substrate, and shall not be further discussed here. Anisotropy The wurtzite GaN ﬁlms are optically uniaxial, and two ε spectra, ε|| and ε⊥ are necessary to model the infrared response. Because of the c-plane orientation (the GaN epilayers adopt the sapphire surface orientation), not all of the model dielectric function parameters can be accessed [18, 37]. In particular, the A1 (TO) mode parameters (frequency and broadening) are inaccessible by the infrared ellipsometry technique for c-plane oriented wurtzite-structure layers, and thus are inserted into the equations as ﬁxed parameters. The phonon mode frequency can be determined from Raman scattering analysis of the same layers, or may be taken from theoretical predictions [49]. The broadening value is diﬃcult to access independently, and are often coupled to the broadening value of the E1 (TO) mode.10 Free-Charge-Carrier Parameters Analysis of the spectra shown in Fig. 7.13 provided, together with parameters Ne and Nh obtained from electrical Hall-eﬀect measurements, precise values for the n- and p-type eﬀective mass parameters of GaN. Because the ε spectra are anisotropic, mobility and eﬀective mass parameters can be accessed for polarization parallel and perpendicular to the GaN c-axis. For some wurtzite GaN layers grown on c-plane sapphire it was observed that 10

The broadening values γ⊥ , γ|| may diﬀer from each other, in general. However, analysis of a-plane oriented layers where all model parameters are accessible revealed no distinct diﬀerences between γ⊥ and γ|| in ZnO and GaN. The approximation made for c-plane layer data analysis, γ⊥ = γ|| , is suﬃciently accurate. Note further that setting γ|| to a ﬁxed value over a large parameter range does not inﬂuence the best-ﬁt results on the remaining parameters, which is just a diﬀerent way of expressing the nonexistence of sensitivity to this parameter.

124

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO) A1(TO)

50

E1(TO) E1(LO)

A1(LO)

180

D

D 0

0 50

180 C

B

B

Ψ [˚]

0 50

180

*

0 50

+

180

*

0 50

Eu

A2u

600

800 -1

ω [cm ]

1000

180 S

S 400

0

A 0

A

0

0

∆ [˚]

C

400

600

800

0

1000

-1

ω [cm ]

Fig. 7.13. (a) Experimental (symbols) and calculated (solid lines) Ψ -spectra of unintentionally doped (A: Ne = 7.8 × 1016 cm−3 , B: Ne = 3.2 × 10 17 cm−3 ), Mgdoped (C: Nh = 8 × 10 17 cm−3 ), and Si-doped (D: Ne = 1 × 1019 cm−3 ) [0001]oriented GaN layers on [0001] (c-plane) sapphire substrates [11]. The spectra of bare c-plane sapphire are included for comparison (S). The sapphire TO and LO A2u phonon modes are indicated by brackets (solid brackets: TO; dotted brackets: LO). The dip marked by an asterisk is caused by the (anisotropic) FSP mode near the GaN high-frequency A1 -LPP mode (Φa = 72◦ ). (b) Same as Fig. 7.13a for ∆, and with the Eu phonon mode frequencies of sapphire for comparison. The small peak marked by an asterisk is caused by the GaN E1 (TO) mode excitation, the structure marked by a cross is due to the (anisotropic) FSP mode near the GaN A1 (LO) mode. The vertical lines indicate the positions of GaN phonon mode frequencies, but only the E1 (TO) mode causes a resonance structure. The A1 (LO) (or LPP+ ) mode is responsible for the occurrence of the FSP mode. Mode E1 (LO) follows from the lineshape ﬁt only. Mode A1 (TO) is “invisible” for this c-axis orientation of the wurtzite-structure layers

the mobility parallel to c is higher by about one order of magnitude than the mobility perpendicular to the GaN c-axis. This is commonly attributed to the columnar growth modus. Accordingly, scattering across domain boundaries for movement parallel to the ﬁlm interface aﬀects the free-charge-carrier mobility more than that perpendicular to the growth interface. The anisotropy of the eﬀective mass parameters was found to be smaller than the error limit, in agreement with theoretical predictions [11]. A comparison between mobility parameters obtained by infrared ellipsometry and Hall resistivity measurements is given in [50].

7.4 Free Charge Carriers (p- and n-Type GaN)

125

Impurity Modes Prior to the determination of the exact free-charge-carrier parameters from ε|| and ε⊥ for the Si-doped samples, subtle features, too small to be seen in Fig. 7.13, need to be considered. Two eﬀects are superimposed on the ellipsometry spectra of these samples: (i) subtle distortions of the GaN-lattice band lineshape, caused by low-polar mode contributions, and (ii ) the existence of a carrier-depleted surface region on the top of the Si-doped GaN layers.11 Figure 7.14 enlarges the spectra for sample D shown in Fig. 7.13, together with two model calculations. The upper graphs present those considering (i) and (ii ), and the lower spectra were obtained including the GaN lattice modes and the free-charge-carrier contributions only. Three additional infrared active modes with low polarity can be identiﬁed from the ellipsometry data of sample D. The model dielectric function in (3.13) with k = 1 and l = 1 is no longer appropriate for description of the experimental data for GaN. Figure 7.14 compares model calculations using diﬀerent model dielectric functions [k = l = 1 (lower spectra), and isotropic low-polar impurity modes (3.16): Iν , ν = 1 . . . 3; δων2 = 0; upper spectra)] with the experiment. Excellent match is obtained only upon implementing the modes I1 , I2 , and I3 , with best-ﬁt parameters I1 : ωIM,1 = (574 ± 5)cm−1 , γIM,1 = (59 ± 17)cm−1 , −δγ1 = 8cm−1 , I2 : ωIM,2 = (746 ± 5)cm−1 , γIM,2 = (173 ± 17)cm−1 , −δγ2 = 59cm−1 , I3 : ωIM,3 = (851 ± 8)cm−1 , γIM,3 = (125 ± 42)cm−1 , −δγ3 = 16cm−1 . The origins of these modes were tentatively assigned to the disorder-activated high E2 GaN mode (I1 ), and to an acoustic-optical combination band (I3 ), whereas the origin of the second mode is still unclear. These modes were found within both, MBE and MOVPE grown highly Si-doped GaN layers, and seem to be related to the existence of Si within the wurtzite lattice [13, 51, 52]. Depletion Layer The FSP mode, which occurs in undoped (doped) GaN near the A1 (LO) (A1 LPP+ ) mode, is driven by the “gap” condition provided by ε|| (Fig. 4.9). In Fig. 7.13, this mode is slightly shifted to higher wavenumbers for sample B compared with sample A, because of the higher free-electron concentration in sample B. For sample D the GaN-layer FSP mode has broadened because of the increased absorption induced by the relatively low-mobile free charge carriers, and occurs near ω ≈ 1047 cm−1 . However, a subtle dip remains in sample D near the unscreened A1 (LO) mode. This dip is the FSP mode caused 11

It was mentioned in Sect. 3.3 that omission of low-polar mode contributions to ε rescales the apparent ε∞ best-ﬁt parameter. Then care must be taken because ε∞ aﬀects the screened plasma frequency from which the free-charge-carrier parameters have to be evaluated.

126

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

*

40 30

I1

I2

I3

20

Ψ [˚]

10

*

0 650

40

700

750

30 20 10 0

GaN/[0001]Al2O3 400

600

800

1000

-1

ω [cm ] Fig. 7.14. Experimental (dotted lines) and calculated (solid lines) Ψ -spectra of the highly Si-doped GaN/sapphire sample D (Ne = 1 × 1019 cm−3 ) within the spectral region of the GaN phonon modes. The calculated spectrum in the upper panel was obtained by including contributions from three additional isotropic low-polar modes (I1 , I2 , I3 ) to ε|| and ε⊥ of the doped GaN layer, and adding an undoped thin GaN layer on top of the doped GaN layer (surface carrier depletion layer). The undoped GaN layer causes a subtle FSP mode resonance near the A1 (LO) frequency, indicated by the star (inset). The lower panel demonstrates the best-ﬁt calculation without considering the surface depletion layer and without the lowpolarity modes, i.e., using the same model as for the unintentionally doped and Mg-doped GaN layers in Fig. 7.13

by a very thin free-charge-carrier surface depletion layer (inset in Fig. 7.14). This eﬀect can be explained by Fermi-level pinning at surface states, which are localized within the band gap [53]. The resulting band bending causes a surface depletion layer. Accordingly, the A1 (LO) mode is not screened by free charge carriers within this very thin surface layer. Assuming this depletion layer to be homogeneous and without free-charge-carrier contribution to the GaN ε|| and ε⊥ spectra, a thickness of about 5 nm results from the

7.4 Free Charge Carriers (p- and n-Type GaN)

127

best-ﬁt model calculation including an undoped (screened) surface layer into the layer model.12 Free-Charge-Carrier Detection Limits Finally, the question about when and under what conditions can free-chargecarriers be detected in an infrared ellipsometry experiment is addressed. In general, there are no overall valid numbers to be given because the answer to this question will strongly depend on the actual sample structure and the material under study. A single-layer situation is taken here for the sake of simplicity in order to shed some light on the lower detection limits. The effective mass and mobility parameters are used as input values together with the ﬁlm thickness d. For example, consider a n- or p-type GaN layer on cplane sapphire, with isotropic eﬀective mass. The dielectric functions ε|| and ε⊥ were calculated for GaN through (3.14), and for sapphire through (3.3) and parameters given in [37].13 The free-charge-carrier eﬀects were judged as detectable when the test function in (2.72) (omitting data for ∆) exceeds a 2 ≈ 130), that is estimated from experience [27]. For the stancertain value (ξSE dard deviations it was assumed σiΨ = 0.1◦ , and Ψi and Ψic refer to calculated data for the situation without and with free-charge-carriers, respectively. The index i runs over all data included for wavenumbers from 300 . . . 1200 cm−1 with 1 cm−1 step width, and angles of incidence Φa = 55◦ . . . 75◦ with 5◦ step width. Figure 7.15 depicts the thereby obtained free-charge-carrier detection limits as a function of concentration and mobility for n- and p-type GaN ﬁlms with d = 100 nm and 1000 nm. Below the lines, for any given pair (N , µ), the infrared ellipsometry data will not provide suﬃcient sensitivity in order to quantify the parameters N and µ. The detection limits depend on d, and smaller values N may be detectable for larger values µ. The limits are more favorable for electrons (m = 0.2 m0 ) than for holes (m = 0.8 m0 ). For realistic free electron mobility values µ = 100 . . . 500 cm2 /(Vs) in GaN, the detection limits for N is in the range between 1 . . . 3 × 1017 cm−3 for d = 100 . . . 1000 nm. For p-type GaN with currently typical mobility values µ = 5 . . . 30 cm2 /(Vs) the detection limit depends strongly on µ and d, and estimations are for N in the range between 2 . . . 20 × 1017 cm−3 for d = 100 . . . 1000 nm. Note that the detection limits will also depend on the individual layers within complex heterostructures, if such samples are under consideration. 12

13

Huml`ıˇcek et al. observed the same eﬀect in heavily-doped n-type GaAs by farinfrared ellipsometry [54]. As outlined in Sect. 4.2.2 (Fig. 4.6) the FSP mode resonance provides extreme sensitivity to the thickness of this very thin layer compared with the wavelength of the probing light beam. The other parameters used in this estimation were: ωTO,|| = 535 cm−1 , ωTO,⊥ = 560 cm−1 , ωLO,|| = 735 cm−1 , ωLO,⊥ = 742 cm−1 , γTO,|| = γTO,⊥ = 4 cm−1 , γ˜LO,1,|| = γ˜LO,1,⊥ = 500 cm−1 , γ˜LO,2,|| = γ˜LO,2,⊥ = 6 cm−1 , ε∞|| = ε∞⊥ = 5.2, m|| = m⊥ , µ|| = µ⊥ .

128

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

18

-3

N (cm )

10

17

10

1

10

100

µ (cm /Vs) 2

Fig. 7.15. Calculated detection limits in infrared ellipsometry for the eﬀects of free-charge-carriers in a GaN ﬁlm deposited on c-plane sapphire. For (µ, N ) pairs above the respective lines, the ellipsometry spectra provide suﬃcient sensitivity to the parameters µ, N . The detection limits depend on the eﬀective free-chargecarrier mass parameter and the thickness d of the layer. Solid line: d = 1 µm, m = 0.2 m0 ; dotted line: d = 0.1 µm, m = 0.2 m0 ; dashed line: d = 1 µm, m = 0.8 m0 ; dash-dotted line: d = 0.1 µm, m = 0.8 m0 [27]

7.5 Strain and Alloying (InAlN) The ternary compound Inx Al1−x N exhibits large variation in band gap energy from Eg ≈ 6.2 eV (AlN) to Eg ≈ 1.9 eV (InN), and is a promising candidate for application in optoelectronic group-III-nitride device heterostructures. Inx Al1−x N can be used for lattice matched conﬁnement layers in GaN-based devices, and may also serve as active material for nitride-based light-emitting diode and laser diode devices operating in the ultra violet to near-infrared spectral range. Knowledge about phonon mode frequencies of wide-band gap alloys is useful, but diﬃcult to receive from non-resonant Raman scattering, expecially when the ﬁlms are very thin. Thereby, the inﬂuence of strain and composition can be studied, as demonstrated here for the E1 (TO) phonon of undoped Inx Al1−x N (0.117 ≤ x ≤ 0.212) thin ﬁlms. For this alloy a twomode behavior was predicted [49], but a one-mode behavior is observed. From the E1 (TO) frequency shift vs. x conclusions can be drawn concerning the strain state of the ≈120 nm thick Inx Al1−x N layers deposited either on GaN/AlN/Al2 O3 , or directly on sapphire [19] .

7.5 Strain and Alloying (InAlN)

129

Figure 7.16 presents experimental and calculated Ψ data from an Inx Al1−x N ﬁlm deposited on top of a sequence of GaN and AlN layers grwon initially on c-plane sapphire. From analysis of the ellipsometry data using a three-layer model, and optical dielectric function spectra for sapphire, GaN, and AlN determined individually and prior to this experiment, relevant model dielectric function parameters of the Inx Al1−x N ﬁlm can be obtained. The vertical arrow indicates the position of the E1 (TO) mode. The inset in Fig. 7.16 highlights the shift of the Inx Al1−x N E1 (TO) mode versus the indium content x. This shift is caused by two eﬀects, the composition variation and the ﬁlm in-plane strain xx . Biaxial Strain In the case of biaxial strain in the plane normal to the c-axis, the space-group 4 is conserved, and strain and stress symmetry of the wurtzite structure C6v tensors take diagonal form. The in-plane (xx , yy ) and normal (zz ) strain components are then given by xx = yy =

and zz =

a − a0 , a0

c − c0 . c0

(7.3)

(7.4)

Here, a0 (a) and c0 (c) denote the a and c lattice constants of the unstrained (strained) crystal, and the convention is used that xx < 0 (xx > 0) for layers under compressive (tensile) in-plane strain. The corresponding stress tensor components follow according to Hooke’s law σxx = (C11 + C12 ) xx + C13 zz ,

(7.5)

σzz = 2C13 xx + C33 zz ,

(7.6)

and where the parameters Cij denote the elastic constants. For strain normal to the c-axis, the group-III nitride crystals retain their hexagonal symmetry (σzz = 0), and there is only a shift but no splitting of the phonon frequencies. In the range of the validity of Hooke’s law, the magnitude of the frequency shift ∆ωλ for each phonon mode λ is characterized by two phonon deformation potential constants aλ and bλ [55] ∆ωλ = 2aλ xx + bλ zz .

(7.7)

In order to determine aλ and bλ from an observed mode shift, another quantity – the bulk Gr¨ uneisen parameter – must be known [56]. The Gr¨ uneisen parameter is related to the elastic constants and the phonon

130

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

InxAl1-xN/GaN/AlN/[0001]Al2O3 * A B C D E F

80

60 E1(TO) Ψ [˚]

600

640

680

40 * C x = 0.156

20

0

400

600

800

1000 1200 1400 -1

ω [cm ] Fig. 7.16. Experimental (symbols) and calculated (solid line) Ψ data from an In0.156 Al0.844 N/GaN/AlN/[0001]Al2 O3 sample (C) at 70◦ angle of incidence. The asterisk marks the small resonance due to the E1 (TO) mode of the AlN layer. The inset shows data from series of samples within the spectral region of the Inx Al1−x N E1 (TO) mode. The Inx Al1−x N layers were grown on GaN (A-E), or directly on sapphire (F), with compositions x = 0.117 (A), 0.150 (B), 0.187 (D), 0.212 (E), and 0.128 (F). The vertical arrows depict the E1 (TO) mode, which shifts to smaller wavenumbers for increasing In-content. (The spectra in the inset are shifted vertically.) The observed E1 (TO) mode is neither purely “AlN”-like nor “InN”-like. [19]

deformation potential constants, and characterizes the phonon mode frequency shift at hydrostatic compression.14 The phonon mode shift for biaxially strained layers follows from (7.6) and (7.7) 14

For hydrostatic pressure the diagonal components of the stress tensor are equal, σxx = σyy = σzz = σ ,

(7.8)

and the phonon mode shift is given by ∆ωλ =

∂ω σ, ∂P

(7.9)

7.5 Strain and Alloying (InAlN) ∆ωλ = βλ xx ,

with

β λ ≡ 2 aλ −

C13 bλ C33

131 (7.12)

.

(7.13)

Composition In the case of a ternary Ax B1−x C layer, phonon mode shifts with respect to the unstrained binary constituents AC , BC can be further induced by the composition x. It is often necessary to include a bowing term, α1 x(1 − x), for description of the composition-induced mode shift in order to describe experimental observations appropriately ∆ωλ (x) = α0,λ x + α1,λ x(1 − x) ,

(7.14)

where α0,λ represents the diﬀerences between the phonon mode frequencies from the binary end compounds α0,λ = ωBC − ωAC .

(7.15)

Pseudomorphic Growth Pseudomorphic growth results in coherent in-plane lattice constants between template and epilayer. Biaxial strain results when mismatch exists between the equilibrium in-plane lattice constants of the ﬁlm and the underlying material. The combined eﬀects of strain and composition onto the phonon mode frequencies is then represented by ∆ωλ = ∆ωλ (x) + ∆ωλ (xx ) = α0,λ x + α1,λ x(1 − x) + βλ xx .

(7.16)

For the Inx Al1−x N samples in Fig. 7.16, xx = (aInAlN − aGaN )/aGaN , where 0 0 InN AlN aInAlN = xa0 + (1 − x)a0 is the in-plane lattice constant of the strained ∂ω where the linear pressure coeﬃcient ∂P can be expressed by the compliance constants Sij and deformation potentials aλ , bλ

−

∂ω = 2aλ (S11 + S12 + S13 ) + bλ (2S13 + S33 ) . ∂P

(7.10)

The Gr¨ uneisen parameter γ is related to the linear pressure coeﬃcient, the strain free phonon mode ω0 , and the bulk modulus B via γ=

∂ω B . ∂P ω0

(7.11)

Normally, high-pressure experiments on single bulk crystals using diamond-anvil cells are required for determination of the Gr¨ uneisen parameter.

132

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO) 640

InN-like

ω [cm-1]

ω [cm-1]

600

AlN-like

550 500

on GaN on Al2O 3 composition only composition + strain

E1(TO)

650

InxAl1-xN 0.0

0.2

0.4

x

(a)

0.6

0.8

1.0

630

compressive tensile

620

0.12

0.16

x

0.20

(b)

Fig. 7.17. E1 (TO) frequency vs. composition for ≈120 nm thick Inx Al1−x N layers determined from ellipsometry (squares: Inx Al1−x N grown on GaN/AlN/c-plane Al2 O3 , triangles: Inx Al1−x N grown on c-plane Al2 O3 ). Figure 7.17a: Dotted lines are calculations for the “InN”-like and “AlN”-like E1 (TO) mode frequencies [49]. The experiment (symbols) reveals a one-mode behavior. The dashed line depicts the unstrained dependence of the E1 (TO) frequency. The solid line reﬂects in-plane strain and composition dependencies. The intersection between both lines indicates the transformation between compressive and tensile strain for growth on GaN (Fig. 7.17b). A sample grown on sapphire (up triangle) reveals its E1 (TO) mode apart from those grown on GaN, and falls on the dashed line according to its (apparently complete) relaxed growth

Inx Al1−x N epilayer obtained from high-resolution x-ray diﬀraction measurements.15 For biaxial strain in the plane normal to the Inx Al1−x N c-axis the alloy In0.18 Al0.82 N is expected to be lattice-matched to GaN, which is nearly given for sample D.16 The E1 (TO) mode frequency is depicted vs. x in Fig. 7.17, and which can be well described by using α0,λ = -188 cm−1 , α1,λ = (−25 ± 1) cm−1 , and βλ = (987 ± 25) cm−1 .17 The intersection of the solid line ∆ωλ (x) + ∆ωλ (xx ) with the dashed line ∆ωλ (x) at x = 0.180 denotes the transition between tensile (xx < 0) and compressive (xx > 0) in-plane ﬁlm strain, and is in good agreement with the lattice match condition expected from Vegard’s law. 15

16

17

Data used for the in-plane lattice constants of strain-free GaN, InN and AlN are AlN ˚ = 3.189 ˚ A, aInN = 3.111 ˚ A, respectively [57]. aGaN 0 0 = 3.544 A, and a0 Associated with pseudomorphic growth is the critical ﬁlm thickness until which no partial relaxation occurs. Above the critical thickness the epilayer begins to relax upon dislocation and defect formation. For Inx Al1−x N on GaN the critical thickness was reported ≈400 nm, interestingly without signiﬁcant dependence on the composition x of the epilayers [58]. E1 (TO) = 476 cm−1 and 664.0 cm−1 for unstrained (xx = 0) InN (x = 1) and AlN (x = 0), respectively [17, 19, 59].

7.6 Device Heterostructures (Group-III Nitrides)

133

Relaxation Dislocations and defects may partially destroy the lattice coherency, and result in partial pseudomorphic conditions relaxing the pseudomorphic strain ﬁelds. Likewise, complete lattice relaxation may occur, and the growing ﬁlm can adopt its equilibrium lattice constant. The observed E1 (TO) mode shift for the Inx Al1−x N ﬁlm grown directly on sapphire (F) is an example for such a situation. The phonon mode shift can be explained by alloy composition eﬀects only. The strain inside the ﬁlm with nominally 12.8% indium content, √ which arises from the lattice mismatch ( 3aInAlN − asapphire )/asapphire of approximately 15%, was relieved in a region close to the ﬁlm-substrate interface. The E1 (TO) mode observed here follows the compositional shift (dashed line) in Fig. 7.17.

7.6 Device Heterostructures (Group-III Nitrides) The infrared dielectric “ﬁngerprints” due to polar lattice modes and coupling of free-charge-carrier plasmons to the LO phonon modes can be used for contact-less and non-destructive measurement of free-charge-carrier and crystal-structure properties of individual layers in complex semiconductor heterostructures for electronic and optoelectronic device applications. Examples, shown here for wurtzite-structure group-III nitride-based coherent and incoherent light emitter structures, give rise to attractive prospects of infrared ellipsometry as novel metrology tool in academic and industrial environments [28, 60, 61, 62]. For analysis of such complex structures, databases must exist, which contain the infrared dielectric function information in appropriate and suﬃciently detailed form. For the group-III nitrides, and the structures exempliﬁed here, appropriate model dielectric functions for individual and typical constituents were determined on single or simple-layered structures in advance. Such parameter collections comprise phonon mode and broadening parameters, high-frequency dielectric constants, and the free-charge-carrier eﬀective mass values [28]. During best-ﬁt analysis of the ellipsometry data from complex heterostructures, only signiﬁcant quantities are further varied such as the free-charge-carrier concentration and mobility, or the lattice resonance and broadening parameters. Superlattice sequences are often involved for the design of optical conﬁnement layers. A p-type AlGaN/GaN superlattice, designed as typical optical conﬁnement element in laser diodes, is therefore included here. The same superlattice sequence is incorporated within the coherent emitter structure discussed at the end of this section. Important technological quantities such as free-charge-carrier concentration and mobility parameters in n- and p-type regions, the vertical position of the p − n-junction, layer thicknesses of individual constituents, composition of ternary constituents, and assessment of the crystal-quality upon lattice

134

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

p-GaN:Mg 1.5 nm p-Al 0.15 Ga 0.85 N:Mg 3 nm

×115

p-GaN 1.5 nm Al 0.15 Ga 0.85 N 3 nm

i-GaN:Mg - 500 nm GaN 2000 nm

GaN 1495 nm

GaN 25 nm (0001) sapphire

(a)

(0001) sapphire

(b)

Fig. 7.18. (a): Schematic cross-section of a p-type AlGaN/GaN SL structure. The thicknesses refer to the intended growth rates and durations. (b): Layer model used for the best-ﬁt calculations. The corresponding spectra are shown in Fig. 7.19. Thickness values are the best-ﬁt results

broadening and carrier mobility parameters can be evaluated from analysis of the ellipsometry data. The wurtzite III-N-compound materials studied here were grown by MOVPE on c-plane sapphire and annealed subsequently under N2 -atmosphere in order to activate the Mg donators [47].18 The layer sequences of the superlattice (SL) structure, the light-emitting diode (LED) structure, and the laser diode (LD) structures are depicted in Figs. 7.18a, 7.20a, and 7.22a. The SL structure, which is also part of the laser diode structure, consists of 115 periods of Al0.15 Ga0.85 N/GaN layers. The SL was doped with Mg in order to achieve p-type conductivity. A thin GaN nucleation layer, followed by a thick GaN buﬀer layer, was deposited ﬁrst. The SL was then deposited on top of a highly resistive GaN layer, which was slightly doped with Mg in order to compensate residual donors. The LED structure consists of a thick Si-doped n-type GaN layer deposited onto a buﬀer/nucleation layer sequence. The active region consists of a Si-doped In0.09 Ga0.91 N/GaN multiple quantum well (MQW), which is capped by a 30 nm thick Mg-doped Al0.1 Ga0.9 N electronblocking layer. A highly Mg-doped GaN layer is used for the top contact. The LED structure provides emission at λ ≈ 395 nm under electrical excitation. The LD structure constitutes the most complicated sample studied here. Two AlGaN/GaN SL sequences compose the bottom and top cladding (optical conﬁnement) layers. The top SL is Mg-doped, whereas the bottom SL is Si-doped (only the AlGaN sublayers, Fig. 7.22a). The nucleation/buﬀer 18

The Si and Mg doping rates during the MOCVD growth were adjusted to obtain hole concentrations of ≈2 × 1017 cm−3 within the p-type contact layers, and free-electron concentrations of 1 . . . 2 × 1018 cm−3 within all Si doped layers. The free-hole concentration within the p-type AlGaN superlattice was aimed at 1 × 1017 cm−3 .

7.6 Device Heterostructures (Group-III Nitrides)

135

layer zone is similar to that of the LED structure. The active region, which also includes the electron conﬁnement layer, is the same as for the LED structure. Both components are embedded from bottom and top by n- and p-type GaN layers, respectively. A highly Mg-doped p-type contact layer terminates the top of the structure. Superlattice Structure Figure 7.19 presents experiment and best-ﬁt calculation data with excellent match.19 The sapphire lattice modes dominate the spectra for wavenumbers ω < 1200 cm−1 . Above ω ≈ 1200 cm−1 optical thickness interference oscillations occur within the SL structure.20 The brackets in the upper panel indicate the infrared active Eu and A2u modes of sapphire [37]. The sapphire phonon signatures are superimposed by phonon, plasmon, and polariton modes within the GaN and AlGaN sublayers. The GaN, AlGaN, and Al2 O3 infrared dielectric functions ε|| and ε⊥ were parameterized using (3.3)–(3.11). The brackets within the lower panel of Fig. 7.19 index the materials’ phonon modes parallel (A1 ) and perpendicular (E1 ) to the c-axis, where the dashed vertical lines denote LO, and the solid (dotted) lines denote TO modes. The spectral position of the symmetry forbidden (“insensitive”) A1 (TO) mode is marked by dotted lines. Figure 7.18b displays the layer model used for the best-ﬁt calculation. The nucleation/buﬀer/i-GaN-layer sequence is optically not diﬀerentiable, and was treated as a single homogeneous layer. All GaN-sublayers can be well described by a common set of phonon mode parameters [24, 25]. The free-charge-carrier contributions within the GaN-SLsublayers, however, diﬀer from those of the thick underlying GaN ﬁlm. The GaN lattice mode parameters obtained from the best-ﬁt calculations given in Table 7.3 are in good agreement with results for single-layer GaN [11]. The AlGaN response exhibits the known two-mode behavior for polarization perpendicular to the c-axis [18, 49, 63]. The best-ﬁt AlGaN lattice mode parameters are given in Table 7.3 as well. The AlGaN A1 (TO) mode was set to 550 cm−1 , according to theoretical calculations [49]. The comparatively large phonon broadening parameters are due to alloy-induced mode broadening [64, 65], as well as lower crystal quality of the AlGaN sublayers in comparison with the GaN layers. 19

20

The c-plane sapphire orientation causes group-III nitride layer growth with the same crystallographic orientation. Accordingly, the oﬀ-diagonal elements of the Jones reﬂection matrix are zero, regardless of the angle of incidence, the ﬁlm anisotropy, and the sample orientation. Ellipsometry data are presented in the standard (Ψ , ∆) notation. The thickness parameters of the device constituents follow particularly from the interference pattern above the reststrahlen bands of the materials. Note the pattern change upon varying the angle of incidence. This results from diﬀerent optical path lengths within the layer stack, which provides strong constraints for the individual best-ﬁt thickness values.

136

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

p-{AlGaN/GaN}-SL 270

Eu

[deg]

180 90 0 A2u

-90

[deg]

A1 E1

60

= 40deg = 55deg = 70deg a model fit a

Al 0.15 GaN

a

*(p)

30 A1 E1 0

GaN

500

2000

1000

[cm-1] Fig. 7.19. Experimental (broken lines) and best-ﬁt calculated (solid lines) Ψ and ∆ data for diﬀerent angles of incidence Φa , from the p-type AlGaN/GaN-SL shown in Fig. 7.18a. The best-ﬁt calculated spectra were obtained using the model depicted in Fig. 7.18b. The sapphire Eu and A2u modes are indicated [37]. (Note the logarithmic wavenumber scale) Table 7.3. Best-ﬁt infrared model dielectric function parameters (frequencies in units of cm−1 ) for the SL shown in Fig. 7.18.

GaN Al0.15 Ga0.85 N

a)

εj

ε∞

|| ⊥ || ⊥ ⊥

5.17 ± 0.10 5.17 ± 0.10a) 5.0 ± 0.1a) 5.0 ± 0.1a)

ωTO b)

536 560.0 ± 0.2 550b) 563 ± 1 660 ± 5

ωLO 735.8 ± 0.3 742.0 ± 0.3 765 ± 1 650 ± 4 781 ± 2

γTO

γLO b)

4±1 4±1a) 8b) 8±1 60 ± 5

8±2a) 8±2a) 60±12

Isotropically averaged between “||” and “⊥”. Parameter values related to A1 (TO) were assumed, and not varied during the best-ﬁt analysis. b)

7.6 Device Heterostructures (Group-III Nitrides)

137

For the p-type GaN-SL sublayers an isotropic eﬀective hole-mass of 0.8 m0 [11] was assumed resulting in Nh ≈ 6.6 × 1017 cm−3 , and µh ≈ 20 cm2 /(Vs), the latter also being treated isotropic. It was assumed that all free holes are located within the GaN wells. The free holes within the GaN-SL sublayers couple with the LO phonons, and lead to the coupled LPP modes, which split into low (LPP− ) and high frequency (LPP+ ) branches. The LPP+ mode of A1 symmetry (denoted by “*(p)” in Fig. 7.19 causes subtle asymmetric broadening of the dip in Ψ near the GaN A1 (LO) mode. The dip is caused by the FSP mode excitation (Chap. 4). The FSP mode belongs to the low-conductive nucleation/buﬀer/i-GaN layer sequence.21 The best-ﬁt thicknesses are given in Fig. 7.18b, which are in good agreement with the intended growth values. Electrical Hall-eﬀect measurements provided the free-hole concentration and lateral µh,⊥ mobility values of 1×1017 cm−3 and 20 cm2 /(Vs), respectively, assuming that the entire SL structure is homogeneously conductive. Both values compare very well with the optical results if one considers that within the layer model only the p-type GaN-SL sublayers were assumed to be conductive, and the AlGaN sublayers were treated resistive. Light-Emitting Diode Structure Figure 7.21 presents experimental and best-ﬁt calculated Ψ and ∆ spectra from the LED structure delineated in Fig. 7.20a. For this sample, the thick GaN layer, which follows the nucleation (nominal thickness ≈25 mn) – buﬀer layer (nominal thickness ≈300 nm) sequence, is of n-type and diﬀers optically from the former. The infrared ellipsometry layer model does not explicitly account for the InGaN/GaN sequence, and also not for the AlGaN electron-blocking layer. Suﬃcient sensitivity to the AlGaN- or InGaN-related phonon mode properties, which are here subsumed by the strong GaN-related phonon and free-charge-carrier absorption eﬀects, was not contained within the data.22 The GaN E1 (TO) and A1 (LO) modes are indicated by vertical arrows in Fig. 7.21. Three diﬀerent GaN layers can be distinguished upon data analysis. All layers reveal common lattice mode parameters, equal to those obtained from the SL sample, but with diﬀerent free-charge-carrier properties. The nominally undoped nucleation/buﬀer layer sequence is modelled as a single layer without free-charge-carrier contribution. The strong resonance in Ψ near the GaN A1 (LO) mode is caused by the FSP mode associated with the undoped/doped GaN layer sequence (Fig. 7.21). Assuming an isotropic 21

22

It is always implied here that “free of free charge carriers” means the level of the free-charge-carrier concentration is below the free-charge-carrier detection limit, which is typically less than ≈1 × 1017 cm−3 for electrons in wurtzite GaN. See also Sect. 7.4. Note that the thicknesses of both components contribute about 3% to the overall device thickness only.

138

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO) p-GaN:Mg 400 nm p-Al0.1Ga0.9N:Mg 30 nm n-GaN:Si 4 nm In0.09Ga0.91N 2.5 nm

p-GaN 356 nm ×5

n-GaN 1993 nm

n-GaN:Si 2000 nm GaN 300 nm

GaN 247 nm

GaN 25 nm (0001) sapphire

(a)

(0001) sapphire

(b)

Fig. 7.20. Same as Fig. 7.18 for a light-emitting diode structure

electron eﬀective mass parameter of me = 0.24 m0 , the parameters Ne = (2.2 ± 0.2) × 1018 cm−3 and µe = (110 ± 20) cm2 /(Vs) follow for the n-type layer. The A1 -LPP+ mode in this layer is located at ω ≈ 792 cm−1 , and the corresponding FSP mode resonance is indicated by “LPP+ (n)” in Fig. 7.21. The p-type layer provided Nh = (6 ± 2) × 1016 cm−3 and µh ≈ 30 cm2 /(Vs) using mh = 0.8 m0 . Its A1 -LPP+ mode, indicated by “LPP+ (p)” in Fig. 7.21, is located at ω ≈ 733.5 cm−1 , and which is almost indistinguishable from the uncoupled A1 (LO) mode within the nucleation/buﬀer layer sequence. The free-charge-carrier concentration values obtained here agree well with the doping rate intended during the MOCVD growth of the device heterostructure. The vertical location of the p − n junction is estimated ≈356 nm below the top surface of the heterostructure.23 Laser Diode Structure Experimental and best-ﬁt calculated Ψ and ∆ spectra from the LD structure are shown in Fig. 7.23. This sample contains layer sequences from structures discussed above. Two AlGaN/GaN-SLs act as top and bottom optical cladding layers for the GaN waveguide, and diﬀer only in their doping properties. As for the LED structure, the ellipsometry data did not reveal identiﬁable InGaN or Al0.10 Ga0.90 N related phonon absorption signatures, and 23

A uniform free-charge-carrier distribution within layers, and abrupt changes across interfaces was assumed here. A non-uniform free-charge-carrier distribution due to surface or interface states, or spontaneous and strain-induced polarization ﬁelds may cause inhomogeneous free-charge-carrier depth proﬁles [66], which need then appropriate consideration within the model calculation.

7.6 Device Heterostructures (Group-III Nitrides)

139

LED 270

∆ [deg]

180 90 0

Ψ [deg]

60

Φa = 40deg Φa = 55deg Φa = 70deg

A1(LO)

E1(TO)

-90

model fit

30 +

+

LPP (p) LPP (n) 0

500

2000

1000

ω [cm ] -1

Fig. 7.21. Experimental (broken lines) and best-ﬁt calculated (solid lines) Ψ and ∆ data for diﬀerent angles of incidence Φa , from the light-emitting diode structure shown in Fig. 7.20a. The best-ﬁt calculated spectra were obtained using the model depicted in Fig. 7.20b

therefore the layer model excludes the InGaN/GaN layer sequence and the AlGaN electron blocking layers. The resulting model is depicted in Fig. 7.22b. The arrows in Fig. 7.23 indicate the spectral positions of the GaN and AlGaN lattice modes, which are identical with those obtained in Figs. 7.19 and 7.21. Besides the top and bottom SL’s, the ellipsometry data analysis distinguishes between three different GaN layers, which reveal common lattice mode parameters, equivalent to the SL and LED structures, but with diﬀerent free-charge-carrier properties. Assuming me = 0.24 m0 , and common free-charge-carrier concentration and mobility values for all GaN constituents, the best-ﬁt data analysis resulted in Ne = (2.2 ± 0.2) × 1018 cm−3 and µe = (120 ± 20) cm2 /(Vs) for the n-type regions. The corresponding A1 -LPP+ mode is at ω ≈ 792 cm−1 , indicated by “LPP+ (n)” in Fig. 7.23. Similarly, mh = 0.8 m0 was assumed for all p-type GaN layers, including those within the top SL, and Nh = (4 ± 1) × 1017 cm−3 and µh ≈ 30 cm2 /(Vs) followed accordingly. The p-type GaN A1 LPP+ mode (≈737 cm−1 ) is denoted by “LPP+ (p)” in Fig. 7.23. This mode is now clearly distinguished from the uncoupled A1 (LO) mode, the latter being due to the low-conductive nucleation/buﬀer layer sequence. The ellipsometry

140

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO) p-GaN:Mg 50 nm p-GaN:Mg 1.5 nm p-Al0.15 Ga 0.85 N:Mg 3 nm p-GaN:Mg 100 nm p-Al 0.1Ga 0.9N:Mg 30 nm n-GaN:Si 4 nm In0.09 Ga 0.91 N 2.5 nm

×115

n-GaN:Si 2000 nm

g

p-GaN:Mg, 1.5 nm (fix) p-Al0.159Ga0.841N:Mg, 3.05 (0.05) nm

e´

p -GaN:Mg, 111 (8) nm

a, b

×5

c, d n-GaN:Si, 98 (12) nm

n-GaN:Si 100 nm n-GaN 1.5 nm n-Al 0.15Ga 0.85 N:Si 3 nm

p-GaN:Mg, 34 (4) nm

×115

n-GaN:Si, 1.5 nm (fix) n-Al0.159Ga0.841N:Si, 3.05 (0.05) nm n-GaN:Si, 2009 (7) nm

f

GaN 300 nm GaN 25 nm (0001) sapphire

(a)

e

GaN, 290 (4) nm

(0001) sapphire

(b)

Fig. 7.22. Same as Fig. 7.18 for a laser diode structure. See also Fig. 7.24

layer model depicted in Fig. 7.22b also contains the best-ﬁt thicknesses, which agree well with the intended thickness values based on the adjusted growth rates. Figure 7.24 depicts cross-section high-resolution transmission electron microscopy images of the laser diode structure, shown from the sapphire substrate to the bottom of the top-SL. Figure 7.22b labels individual distances between heterostructure constituents, identiﬁed in Fig. 7.24 (Table 7.4). The agreement between results from the microscopic (destructive) TEM and the macroscopic (non-destructive) ellipsometry approach is excellent.24 The freecharge-carrier parameters correspond well with the intended doping properties, adjusted during the growth of the heterostructures. It is interesting to note that optical techniques with probing wavelengths several times larger than the structures to be studied deliver precise information on individual layer thickness and dielectric function spectra. One necessary assumption is of course the requirement of perfect plane parallel interfaces between individual device constituents with suﬃcient lateral 24

Note that the n- and p-type regions between both SL’s were expected to be 25 nm thicker on each side according to the growth rates, in slight disagreement with the ellipsometry results (compare values given in Figs. 7.22a,b). The transmission electron microscopy images, however, demonstrated the correctness of the optical results.

7.6 Device Heterostructures (Group-III Nitrides)

141

Laser diode 270

∆ [deg]

180 90 0 -90 E1(TO) Al0.15GaN

Ψ [deg]

60

E1(TO) GaN

A1(LO) GaN (733) + A1-LPP (p) GaN (737) A1(LO) AlGaN (765) + A1-LPP (n) GaN (792)

Φa = 40deg Φa = 55deg Φa = 70deg

model fit

30

0

500

2000

1000

ω [cm ] -1

Fig. 7.23. Experimental (broken lines and best-ﬁt calculated (solid lines) Ψ and ∆ data for diﬀerent angles of incidence Φa , from the laser diode structure shown in Fig. 7.22a. The best-ﬁt calculated spectra were obtained using the model depicted in Fig. 7.22b Table 7.4. Selected results from transmission electron microscope images (Fig. 7.24) and infrared ellipsometry data analysis (Fig. 7.23) for the p- (dp = dg + de + da + db ) and n-type (dn = de + df ) sample regions, the Alx Ga1−x N/GaN SL period de/115 , and the Al fraction x in the SL barriers. Further values are given in Fig. 7.22b

dp [nm] dn [nm] de/115 [nm] x Ne [1018 cm−3 ] µe [cm2 /(Vs)] Nh [1017 cm−3 ] µh [cm2 /(Vs)] a)

Infrared ellipsometry

High-resolution TEM

668 ± 18 2920 ± 17 4.55 ± 0.05 0.159 ± 0.002b) 2.2 (±0.1) 184 (±5) 3.9 (±0.2) 22 (±4)

665 ± 7 2879 ± 29 4.54 ± 0.05 a) a) a) a). a)

Not available from TEM. The Al-fraction was obtained from comparison of the A1 (LO) mode frequencies with values obtained from AlGaN single layers [18]. b)

142

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

a e b c 50 nm

f

g

d e

1 µm

50 nm

Fig. 7.24. High-resolution TEM cross-section of parts of the laser diode structure studied here. The image is a photograph of a weak beam diﬀraction pattern {g,3g} with g = (0,0,0,2). The thicknesses refer to the intended growth rates and durations. [e: lower AlGaN/GaN SL; d & a: GaN waveguide; b: AlGaN electron-blocking layer; c: InGaN/GaN SL; f: n-GaN base). The p − n-junction is located between b and c. The thickness parameters for individual constituents are in excellent agreement with the infrared ellipsometry best-ﬁt results (see Table 7.4)

extension.25 The sensitivity to free-charge-carrier parameters in thin layers follows particularly from the subtle eﬀects caused by the resonant excitation of the Fano and surface-guided-wave interface modes discussed in Chap. 4. It was commented and shown there that the resonance lineshape depends crucially on the thickness and the complex-valued dielectric function near the resonance. The polarized light spectra shown in Figs. 7.19, 7.21, and 7.23 remind of an “optical chord” wherein each constituent of the complex layer stack provides “strings” upon allowed phonon, plasmon and polariton modes, and which “vibrate” according to their optical “strength” rendering a fascinating duality between light and polar lattices. 25

The inﬁnite-extension-requirement (plane parallel interfaces) of the sample within the (x,y) plane holds suﬃciently when the lateral size of the surface observed under oblique angle is still large compared to the probing wavelength. Otherwise Fraunhofer-diﬀraction will begin to deteriorate the obliquely reﬂected light.

References

143

References 1. J. Orton and C. Foxon, Rep. Prog. Phys. 61, 1 (1998). 109 2. S. Jain, M. Willander, J. Narayan, and R. V. Overstraeten, J. Appl. Phys. 87, 965 (2000). 109 3. T. H. Myers, R. M. Feenstra, M. S. Shur, and H. Amano, eds., GaN and Related Alloys - 1999, Warrendale, Pennsylvania (2000), Materials Research Society. 109 4. C. Wetzel, M. Shur, U. Mishra, B. Gil, and K. Kishino, eds., GaN and Related Alloys 2000, Warrendale, Pensylvania (2001), Materials Research Society. 109 5. J. E. Northrup, J. Neugebauer, D. C. Look, S. F. Chichibu, and H. Riechert, eds., GaN and Related Alloys–2001, Warrendale, Pennsylvania (2002), Materials Research Society. 109 6. C. Wetzel, E. T. Yu, J. S. Speck, A. Rizzi, and Y. Arakawa, eds., GaN and Related Alloys–2002, Warrendale, Pennsylvania (2003), Materials Research Society. 109 7. H. M. Ng, M. Wraback, K. Hiramatsu, and N. Grandjean, eds., GaN and Related Alloys–2003, Warrendale, Pennsylvania (2004), Materials Research Society. 109 8. C. A. Arguello, D. L. Rousseau, and S. P. S. Porto, Phys. Rev. 181, 1351 (1969). 109, 112 9. I. Gorczyca, N. Christensen, E. P. Y. Blanc´ a, and C. Rodriguez, Phys. Rev. B 51, 11936 (1995). 109 10. R. A. Evarestov and V. P. Smirnov, in M. Cardona, ed., Springer Series in Solid State Sciences (Springer, Heidelberg, 1993), vol. 108. 109 11. A. Kasic, M. Schubert, S. Einfeld, D. Hommel, and T. Tiwald, Phys. Rev. B 62, 7365 (2000). 111, 119, 124, 135, 137 12. A. Kasic, M. Schubert, Y. Soito, Y. Nanishi, and G. Wagner, Phys. Rev. B 65, 115206 1 (2002). 111, 119 13. A. Kasic, M. Schubert, B. Kuhn, F. Scholz, S. Einfeldt, and D. Hommel, J. Appl. Phys. 87, 3720 (2001). 111, 125 14. V. Darakchieva, J. Birch, M. Schubert, A. Kasic, S. Tungasmita, T. Paskova, and B. Monemar, Phys. Rev. B 70, 045511 (2004). 111, 118 15. V. Darakchieva, M. Schubert, J. Birch, A. Kasic, S. Tungasmita, T. Paskova, and B. Monemar, Physica B: Condensed Matter 340 - 342, 416 (2003). 111, 118 16. V. Darakchieva, P. P. Paskov, E. Valcheva, T. Paskova, B. Monemar, M. Schubert, H. Lu, and W. J. Schaﬀ, Appl. Phys. Lett. 84, 3636 (2004). 111 17. M. Schubert, J. A. Woollam, A. Kasic, B. Rheinl¨ ander, J. Oﬀ, B. Kuhn, and F. Scholz, phys. stat. sol. (b) 216, 655 (1999). 111, 132 18. M. Schubert, A. Kasic, T. E. Tiwald, J. Oﬀ, B. Kuhn, and F. Scholz, MRS Internet J. Nitride Semicond. Res. 4, 11 (1999). 111, 123, 135, 141 19. A. Kasic, M. Schubert, J. Oﬀ, and F. Scholz, Appl. Phys. Lett. 78, 1526 (2001). 111, 128, 130, 132 20. A. Kasic, M. Schubert, B. Rheinl¨ ander, J. O. F. Scholz, and C. Herzinger, in Proceedings of the 2000 IEEE 27th International Symposium on Compound Semiconductors (IEEE, Piscataway, NJ, 2001), p. 513. 111 21. A. Kasic, M. Schubert, B. Rheinl¨ ander, J. Oﬀ, F. Scholz, and C. M. Herzinger, Mat. Res. Soc. Symp. 639, G6.13 (2001). 111 22. A. Kasic, M. Schubert, J. Oﬀ, F. Scholz, S. Einfeldt, and D. Hommel, phys. stat. sol. (b) 234, 970 (2002). 111 23. A. Kasic, M. Schubert, T. Frey, U. K¨ ohler, D. J. As, and C. M. Herzinger, Phys. Rev. B 65, 184302 1 (2002). 111

144

7 Wurtzite-Structure Materials (Group-III Nitrides, ZnO)

24. M. Schubert, A. Kasic, T. E. Tiwald, J. A. Woollam, V. H¨ arle, and F. Scholz, MRS Internet J. Nitride Semicond. Res. 5, W11.39 (2000). 111, 135 25. M. Schubert, A. Kasic, J. Sik, S. Einfeldt, D. Hommel, V. Haerle, J. Oﬀ, and F. Scholz, Mat. Sci. Eng. B 82, 178 (2001). 111, 135 26. V. Darakchieva, P. P. Paskov, M. Schubert, E. Valcheva, T. Paskova, H. Arwin, B. Monemar, H. Amano, and I. Akasaki, Mat. Res. Soc. Symp. 798, Y5.60 (2004). 111 27. A. Kasic, Phonon, Free-Carrier Properties, and Electronic Interband Transitions of Binary, Ternary, and Quaternary Group-III Nitride Layers Measurend by Spectroscopic Ellipsometry (Shaker Verlag, Aachen, 2003). 111, 127, 128 28. A. Kasic, M. Schubert, J. Oﬀ, B. Kuhn, F. Scholz, S. Einfeldt, T. B¨ ottcher, D. Hommel, D. J. As, U. Koehler, et al., Phonons and Free-Carrier Properties of Binary, Ternary, and Quaternary Group-III Nitride Layers Measured by Infrared Spectroscopic Ellipsometry (Wiley-VCH, 2003), pp. 1750 – 1769, Physica Status Solidi (C) - Conferences and Critical Reviews. 111, 133 29. E. M. Kaidashev, M. Lorenz, H. von Wenckstern, G. Benndorf, A. Rahm, H.-C. Semmelhack, K.-H. Han, H. Hochmuth, C. Bundesmann, V. Riede, et al., Appl. Phys. Lett. 82, 3901 (2003). 112, 113, 119 30. G. E. Jellison and L. A. Boatner, Phys. Rev. B 58, 3586 (1998). 112, 117 31. R. Schmidt, B. Rheinl¨ ander, M. Schubert, D. Spemann, T. Butz, J. Lenzner, E. M. Kaidashev, M. Lorenz, A. Rahm, H. C. Semmelhack, et al., Appl. Phys. Lett. 82, 2260 (2003). 112 32. T. C. Damen, S. P. S. Porto, and B. Tell, Phys. Rev. 142, 570 (1966). 112 33. R. Callender, S. Sussman, M. Selders, and R. Chang, Phys. Rev. B 7, 3788 (1973). 112 34. J. M. Calleja and M. Cardona, Phys. Rev. B 16, 3753 (1977). 112 35. B. H. Bairamov, A. Heinrich, G. Irmer, V. V. Toporov, and E. Ziegler, phys. stat. sol. (b) 119, 227 (1983). 112 36. N. Ashkenov, B. N. Mbenkum, C. Bundesmann, V. Riede, M. Lorenz, E. M. Kaidashev, A. Kasic, M. Schubert, M. Grundmann, G. Wagner, et al., J. Appl. Phys. 93, 126 (2003). 113, 115, 116, 119 37. M. Schubert, T. E. Tiwald, and C. M. Herzinger, Phys. Rev. B 61, 8187 (2000). 115, 116, 123, 127, 135, 136 38. R. H. Lyddane, R. G. Sachs, and E. Teller, Phys. Rev. 59, 673 (1941). 116 39. W. Cochran and R. A. Cowley, J. Phys. Chem. Solids 23, 4471 (1962). 116 40. C. Kittel, Introduction to Solid States Physics (John Wiley and Sons, New York, 1986). 116 41. S. Tripathy, R. Soni, H. Asahi, K. Iwata, K. Asami, and S. Gonda, J. Appl. Phys. 85, 8386 (1999). 118 42. M. Wraback, H. Shen, S. Liang, C. R. Gorla, and Y. Lu, Appl. Phys. Lett. 74, 507 (1999). 118, 123 43. C. Bundesmann, N. Ashkenov, M. Schubert, A. Rahm, H. v. Wenckstern, E. M. Kaidashev, M. Lorenz, and M. Grundmann, Thin Solid Films 455-456, 161 (2004). 119, 120, 123 44. C. Bundesmann, M. Schubert, D. Spemann, T. Butz, M. Lorenz, E. M. Kaidashev, M. Grundmann, N. Ashkenov, H. Neumann, and G. Wagner, Appl. Phys. Lett. 81, 2376 (2002). 119 45. S. Muthukumar, J. Zhong, Y. Chen, Y. Lu, and T. Siegrist, Appl. Phys. Lett. 82, 742 (2003). 123

References

145

46. B. P. Zhang, Y. Segawa, K. Wakatsuki, Y. Kashiwaba, and K. Haga, Appl. Phys. Lett. 79, 3953 (2001). 123 47. S. Figge, T. B¨ ottcher, S. Einfeldt, and D. Hommel, J. Cryst. Growth. 221, 262 (2000). 123, 134 48. T. B¨ ottcher, S. Figge, S. Einfeldt, R. Chierchia, R. Kr¨ oger, C. Petter, C. Zellweger, H.-J. B¨ uhlmann, M. Dießelberg, D. Rudloﬀ, et al., GaN Based Laser Diodes - Epitaxial Growth and Device Fabrication (Wiley-VCH, 2003), pp. 1846 – 1859, Physica Status Solidi (C) - Conferences and Critical Reviews. 123 49. H. Grille, C. Schnittler, and F. Bechstedt, Phys. Rev. B 61, 6091 (2000). 123, 128, 132, 135 50. A. Kasic, M. Schubert, B. Rheinl¨ ander, V. Riede, S. Einfeldt, D. Hommel, B. Kuhn, J. Oﬀ, and F. Scholz, Mat. Sci. Eng. B 82, 74 (2001). 124 51. W. Sun, K. Chen, Z. Yang, J. Li, Y. Tong, S. Jin, G. Zhang, Q. Zhang, and G. Qin, J. Appl. Phys. 85, 6430 (1999). 125 52. H. Siegle, G. Kaczmarczyk, L. Filippidis, A. Litvinchuk, A. Hoﬀmann, and C. Thomsen, Phys. Rev. B 55, 7000 (1997). 125 53. S. M. Sze, Physics of Semiconductor Devices (John Wiley and Sons, New York, 1981), 2nd ed. 126 54. J. Huml´ıˇcek, R. Henn, and M. Cardona, Appl. Phys. Lett. 69, 2581 (1996). 127 55. V. Y. Davydov, N. S. Averkiev, I. N. Goncharyk, D. K. Nelson, I. P. Nikitina, A. S. Polkovnikov, A. N. Smirnov, M. A. Jacobson, and O. K. Semchinova, J. Appl. Phys. 82, 5097 (1997). 129 56. V. Darakchieva, P. Paskov, E. Valcheva, T. Paskova, M. Schubert, C. Bundesmann, H. Lu, W. J. Schaﬀ, and B. Monemar, Superlattices and Microstructures XXXX, XXXX (2004). 129 57. T. Kachi, K. Tomita, K. Itoh, and H. Tadano, Appl. Phys. Lett. 72, 704 (1998). 132 58. J. Pankove and T. Moustakas, eds., Gallium Nitride, vol. 1 - 2 (Academic Press, New York, 1998). 132 59. V. Davydov, V. Emtsev, I. Goncharuk, A. Smirnov, V. Petrikov, V. Mamutin, V. Vekshin, S. Ivanov, M. Smirnov, and T. Inushima, Appl. Phys. Lett. 75, 3297 (1999). 132 60. M. Schubert, A. Kasic, S. Einfeldt, D. Hommel, U. K¨ ohler, J. D. As, B. Kuhn, J. Oﬀ, F. Scholz, and J. A. Woollam, phys. stat. sol. (a) 228, 437 (2001). 133 61. M. Schubert, A. Kasic, S. Einfeldt, D. Hommel, U. K¨ ohler, J. D. As, B. Kuhn, J. Oﬀ, F. Scholz, J. A. Woollam, et al., in A. Duparr´e and B. Singh, eds., Optical Metrology Roadmap for the Semiconductor, Optical and Data Storage Industries II (SPIE, Bellingham, WA, 2001), vol. 4449, pp. 48 – 58. 133 62. A. Kasic, M. Schubert, S. Einfeldt, and D. Hommel, Vibrational Spectroscopy 179, 121 (2002). 133 63. P. Wisniewski, W. Knapp, J. Malzac, J. Camassel, M. Bremser, R. Davis, and T.Suski, Appl. Phys. Lett. 73, 1760 (1998). 135 64. O. Kim and W. Spitzer, J. Appl. Phys. 50, 4362 (1979). 135 65. M. H. Brodsky, G. Lucovsky, M. Chen, and T. Plaskett, Phys. Rev. B 2, 3303 (1970). 135 66. R. Dimitrov, M. Murphy, J. Smart, W. Schaﬀ, J. Shealy, L. Eastman, O. Ambacher, and M. Stutzmann, J. Appl. Phys. 87, 3375 (2000). 138

8 Magneto-Optic Ellipsometry

The free-charge-carrier properties eﬀective mass m, mobility µ, and carrier concentration N are of fundamental importance for understanding basic material properties of semiconductors. N , m, and µ intriguingly combine concepts of classical (such as the eﬀective “inertial” carrier mass) and quantum physics (e.g., the electron energy-momentum dispersion and the electron band occupation). A long term goal has been the accurate and simultaneous measurement of N , m, and µ by non-contact optical means in materials, which are part of complex layered systems, such as semiconductor heterostructures. The eﬀective mass concept addressed here descends from the similarity with the acceleration of a body with mass m and the acceleration experienced by a free electron (or hole) due to an external force. In general, the thereby obtained eﬀective mass is a tensor and depends on the inverse curvature of the electron energetic states versus electron-momentum dispersion (m)ij ≈ (d2 E/dki dkj )−1 [1]. According to Drude (Sect. 3.2), the dielectric function provides access to the square of the free-charge-carrier plasma frequency ωp and the plasma broadening γp (3.7).1 Under simpliﬁed assumptions about the free-chargecarrier momentum distribution function – a single-species carrier plasma taken as example – ωp and γp translate into the coupled electrical quantities N /m and N µ (3.8), (3.11). No distinction can be made between the density-number of carriers N , their “inertial”-mass m, and mobility µ from measurement of the dielectric function alone, nor can the sign of the charge be diﬀerentiated because ωp2 is proportional to q 2 . For determination of m and µ the quantity N must be known from a diﬀerent experiment.2 In a magnetic ﬁeld, the Lorentz force, an axial vector, diverts the movement of the free charge carriers, which causes magneto-optic (chiral) birefringence. This birefringence depends on the inverse of the free-charge-carrier “inertial” mass (described by a tensor), their spatially-dependent scattering characteristics (described by another tensor), and the sign of the charge q . The dielectric response at optical wavelengths is then a complex tensor (magneto-optic plasma tensor). This tensor was discussed in Sect. 3.4 for a 1

2

Note the diﬀerence between the screened plasma frequency ωp (3.12) and the unscreened plasma frequency ωp (3.8). Often, the electrical Hall eﬀect is measured to access N . See also Sect. 7.4.

Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 147–164 (2004) c Springer-Verlag Berlin Heidelberg 2004

148

8 Magneto-Optic Ellipsometry

classical quasi-free charge carrier plasma characterized by arbitrarily shaped (but energy-independent!) inverse eﬀective (“inertial”) mass and scattering tensors.3 Suﬃcient information is contained within the complex elements of the magneto-optic ε tensor to diﬀerentiate between N , m, and µ, and to obtain sign(q ). Cyclotron resonance occurs for ω ≈ ωc , provided that the plasma broadening parameter is small compared with the cyclotron frequency. Typical frequencies of ωc in semiconductors with free charge carriers are within the microwave region, where cyclotron resonance absorption can be observed. Such experiments are typically performed at low temperatures to meet the condition ωc γp [2]. Measurements at infrared wavelengths make use of the coupling mechanism between the plasmon and cyclotron modes. At long wavelengths, in addition to the plasma response, contributions to the dielectric function arise from the electronic background polarization and the polar bond resonances. The longitudinal lattice modes couple with the plasma modes, and the background polarization is responsible for dielectric screening of both modes. The screening shifts the apparent plasma resonance and longitudinal phonon frequencies (the zeros of ε) to longer wavelengths. The magneto-optic response of semiconductors with multiple polar bond resonances at infrared wavelengths and with high-frequency dielectric constants ε∞ > 1 are best analyzed by dielectric function lineshape models.4 The optical response of semiconductor layer structures with free-chargecarrier plasmas subjected to an external magnetic ﬁeld reveal strong magnetooptic eﬀects near the frequencies of the interface modes discussed in the previous chapters. The physical origin of these eﬀects is the coupling between the longitudinal lattice (ωLO ), plasmon (ωp ) and cyclotron (“magnon”, ωc ) modes, which strongly depends on the direction of the magnetic ﬁeld and the incident light beam direction and polarization. The roots of ε(L+FC+MO) – the frequencies of the LPP modes – split upon the inﬂuence of the Lorentz force into coupled longitudinal plasmon-phonon-magnon (LPPM) modes.5 The LPPM modes support bulk and interface polariton modes and produce clearly observable polarization eﬀects near the frequencies of the LPP modes within the (far)infrared spectral region. Such phenomena will be addressed in this chapter. Examples include exemplarily the i-GaAs(d)/n-GaAs structure 3

4 5

In a strict sense, the polar lattice resonance modes are also aﬀected by the external magnetic ﬁeld. Such eﬀects are ignored here. Instead, it is assumed that the bond strengths are large against the Lorentz force, and that the ionic molecular displacements are small compared to their cyclotron radius. For polar acoustic modes associated with large displacement and strong polarity, one may need to consider the eﬀects due to the Lorentz force on the polar lattice motion. See footnote on (3.4). Eﬀects of coupled plasmon-phonon-magnon bulk and interface polaritons have been studied in theory for surfaces between half-inﬁnite anisotropic polar semiconductor materials [3, 4, 5, 6], and by experiment (attenuated total reﬂection) for n-type InSb [7].

8.1 Experimental Setups

149

discussed in Sect. 4.2.2, and determination of the eﬀective mass parameter in n-type AlGaInP, and n-type BInGaAs, where the quaternary layers are embedded within diﬀerently conductive GaAs.

8.1 Experimental Setups Traditional experimental magneto-optic (MO) conﬁgurations for measurement of normal incidence reﬂection or transmission polarized intensity spectra are with the magnetic ﬁeld parallel (longitudinal case), or perpendicular to the light propagation (transverse case). For the transverse case the incident polarization is either parallel or perpendicular to B = µ0 H. In semiconductors, for small m, but with N and µ large, feasible magnetic ﬁelds cause clearly-measurable eﬀects on the infrared reﬂectivity, e.g., in n-type InAs at B = 3.89 T in transverse MO reﬂection [8]. Smaller magnetic ﬁelds produce measurable eﬀects on the polarization state of a transmitted wave at frequencies above the LPP modes (Faraday rotation), e.g., in n-type GaAs at B = 0.833T [9], or in n-type AlAs at B = 1.63 T [10, 11]. Such techniques were used extensively in the past for determination of eﬀective mass parameters from semiconductor materials, and details shall not be discussed here. An excellent review is given by C. R. Pidgeon in [2]. Fundamental restrictions are (i) strong magnetic ﬁelds required for the MO-reﬂection technique in order to obtain appreciable changes of the infrared reﬂectivity (particularly for the case of thin ﬁlm samples), and (ii) wavelength regions of suﬃcient transparency above the high-frequency LPP mode for the Faraday-rotation technique. The main drawback is (iii) the cumbersome analysis of multiplelayered structures. The MO-reﬂection as well as the Faraday-rotation techniques do not allow for complete measurement of the complex MO plasma tensor. Suﬃcient information for independent determination of m, N , µ, and sign(q ) is not easily obtained.6 For materials with large m, but small N and µ, or for thin ﬁlms, the MO-induced changes of the infrared optical sample properties diminish, and are hard to retrieve experimentally. The alternative is the application of generalized ellipsometry. This requires polarization state measurements and model analysis for oblique angle of incidence, but allows for reconstruction of the dielectric function tensor elements. The increased precision and the additional information (phase) with ellipsometry over intensity techniques permits measurement of much smaller birefringence eﬀects, and hence MO eﬀects in less doped materials with lower mobility can be determined. The latter enables feasible measurements at room temperature, where 6

In the Voigt conﬁguration one can, in principle, measure two independent pieces of information: The MO-induced polarization rotation and ellipticity – also known as magnetic double refraction – which provide ωc = qµ0 H/m, but eﬀects are small [2].

150

8 Magneto-Optic Ellipsometry

semiconductor materials often reveal lower mobility parameters. The generalized ellipsometry approach is easily applicable to situations with arbitrarily oriented magnetic ﬁeld directions as well. In view of generalized ellipsometry applications at oblique incidence MO setups are deﬁned here and addressed accordingly later on as follows:7 The polar magneto-optic setup (PMO): B is oriented parallel to the sample normal. The longitudinal magneto-optic setup (LMO): B is oriented parallel to the sample surface and parallel to the plane of incidence. The transverse magneto-optic setup (TMO): B is oriented perpendicular to the sample normal and perpendicular to the plane of incidence. The oblique magneto-optic setup (OMO): B is oriented otherwise. Examples of PMO and OMO setups for MO generalized ellipsometry measurements are given below. 8.1.1 Polar MO Setup: Example The PMO conﬁguration is easily obtained by attaching a Helmholtz-type electromagnet to a standard ellipsometer (Fig. 2.6) with the θ − 2θ goniometer in straight-through position (θ = 90◦ ). Two plane mirrors are then needed to redirect the incident beam onto the sample and back to the detector arm. Such an arrangement is depicted in Fig. 8.1. Two magnetic coils oppose each other. The gap between the pole caps must be suﬃciently large, here for example ≈4 mm. The gap should leave suﬃcient space to redirect the beam between source and detecting units. The sample is attached with its back side onto one of the poles. The focal length of the mirror, which directed the incidence beam onto the sample here was f ≈ 190 mm, producing a beam divergence of ≈3◦ . A commercial FTIR spectrometer with temperature-controlled globar serves as the light source. The strength of the ﬁeld can be tuned by varying the current, the ﬁeld direction is reversed by switching the current. For the setup shown in Fig. 8.1 the angle of incidence, adjusted by the two Au-coated plane mirrors, can be varied between ≈70◦ . . . ≈80◦ . The optical eﬀects of the multiple plane-of-incidence-rotations and phase changes onto the p- and s-polarized components upon the mirror reﬂections must be corrected numerically. In order to obtain the required model parameters (azimuthal and polar angles of the entrance and exit mirrors) measurement of a given sample must be made prior to the MO experiment. The zero-ﬁeld measurement contains suﬃcient information for precise calibration 7

In the MO literature similar terms are in use: Near-normal-incidence reﬂectiontype Kerr-eﬀect measurements are referred to as transverse, longitudinal, and polar conﬁgurations in conceptual agreement with the above notation. Faradayand Voigt eﬀect measurement address transmission-type linear polarization rotation measurement in the above PMO and mixed LMO-TMO conﬁgurations, respectively [2, 12, 13].

8.1 Experimental Setups

151

H M2

M1

B

P

F

S

F

H (a)

H P M1

A

H

(b) Fig. 8.1. (a) Schematic drawing (not to scale) of the light path (F, dotted line) through a Helmholtz-coil arrangement in the polar magneto-optic (PMO) setup. M1: Entrance mirror, M2: exit mirror, S: sample, H: electromagnetic coils, P: magnet poles, B: magnetic ﬁeld direction. (b) Photographic detail showing the entrance side with one of the Au-coated silicon wafer (M1), which serves as entrance mirror in a PMO ﬁr ellipsometry setup. The magnet consists of two water-cooled coils with poles made from magnetically soft alloy (mumetal). The maximum ﬁeld strength between the poles depends on the distance between the two poles. Feasible ﬁeld strengths achievable with this setup are bound by ≈ ± 2.5 T [14, 15]

152

8 Magneto-Optic Ellipsometry

of the spatial rotations of the polarization states due to both mirror planes. Numerical reduction of the mirror rotation eﬀects is done in the Mueller matrix presentation. The matrices responsible for mirror reﬂection with rotation of the plane of incidence are given in [16]. Some more details about the MO calibration scheme can be found in [14]. Among the beneﬁts of the conﬁguration shown in Fig. 8.1 are avoidance of windows and experimental feasibility. Drawbacks include restriction to room-temperature measurements, and normal-state-conductivity magnetic coils with limited magnetic ﬁeld strength [14, 15]. 8.1.2 Oblique MO Setup: Example Figure 8.2 and 8.3 depict an OMO setup with the magnetic ﬁeld direction parallel to the incident beam, where the angle of incidence is ﬁxed at 45◦ . A superconducting magnetocryostat (Oxford) with optical access ports and temperature-controlled (4 K . . . 300 K) sample holder is attached to the ellipsometer setup shown in Fig. 8.3. In this magnet the strength of the ﬁeld can be varied between µ0 H = ±6 T . The beam must be guided through exterior and interior windows (Fig. 8.2), made for example from polyethylene sheets and/or thin diamond wedges. If a globar is used as the light source, the size of the aperture in front of the source and the focal length of the paraboloidal mirrors P1 and P2 determine the size of the beam at the sample, which is ≈10 mm. However, at the location of the windows, due to the beam divergence, the diameter of the beam expands considerably, which must be taken into account when selecting the apertures for entrance and exit windows.

H W1 F

H

B

W3

S I1

I2

I3

W2 F Fig. 8.2. Schematic drawing (top view ; not to scale) of the light path (F, dotted line) through a cryostatic superconducting (C) Helmholtz-coil (H) arrangement in the oblique-magneto-optic (OMO) setup. The magnetic ﬁeld direction (B) is parallel to the incident beam direction. Exterior (W1, W2, W3: polyethelen) and interior windows (I1, I2, I3: diamond wedges) shield the evacuated coil chamber. The sample (S) temperature can be varied between 4 K and 300 K. In reﬂection arrangement, the angle of incidence is Φa = 45◦ . Windows W3 and I3 allow for transmission-type measurements

8.2 Experiments

153

C W1 F

W2 P2

S

P1 P

A

D

P3

G

Fig. 8.3. The technical drawing (side view ) of the OMO setup depicts the beam path through the optics shown in Fig. 2.6, now with the magnetocryostat attached. Labels are as indexed in the caption of Fig. 8.2. The oﬀ-axis paraboloidal mirror P3 focuses the beam onto the detector area of the bolometer (D)

A third set of windows (I3, W3) allows for transmission-type measurements [15].

8.2 Experiments Magneto-optic infrared ellipsometry systems are not yet available commercially. In order to perform such experiments, systems have to be built in laboratories by users themselves. Likewise, procedures for data acquisition

154

8 Magneto-Optic Ellipsometry

and data analysis are not standardized, and will largely depend on the actual setup employed. Therefore, attention is paid here to diﬀerent procedures, which allow to extract the generalized ellipsometry data and which provide access to the MO birefringence information. Diﬀerent examples are included. Other procedures may be employed in the near future as well, because the application of MO ellipsometry at infrared wavelengths will provide very interesting insights into free-charge-carrier properties of semiconductor layer structures. Further applications may include conducting organic materials or highly-correlated electron systems. 8.2.1 i-GaAs(d)/n-GaAs The homostructure discussed in Sect. 4.2.2 is taken here as ﬁrst example. All measurements were made in the Mueller matrix data acquisition mode. A Tedoped GaAs wafer with nominal free-electron concentration of N = 1.5 × 1018 cm−3 was capped by a non-doped GaAs buﬀer layer with nominal thickness of d = 600 nm using metal-organic vapor phase epitaxy under standard growth regimes. Figures 8.4 and 8.5 present experimental and best-ﬁt calculated MO generalized ellipsometry spectra in the PMO conﬁguration. The calculated data were obtained using the MO Drude model (3.30)–(3.33), with one lattice term (3.3, l = 1, γTO = γLO ) added for parameterization of ε, and the 4 × 4 matrix formalism for treatment of light propagation in stratiﬁed media (Sect. 2.7, Appendices A, C), where the eﬀect of the i-GaAs buﬀer layer was considered appropriately. Excellent match between model lineshape and experiment is achieved. Three diﬀerent approaches are pursued. Approach A consists in matching experimental and calculated Mueller matrix spectra for the mirror-samplemirror system using the model dielectric function for ε described above. Approach B consists in wavelength-by-wavelength extraction of the MO dielectric function tensor elements of the GaAs substrate. The results of both approaches (parameterization and wavelength-by-wavelength derived ε spectra) are then compared. As an alternative (Approach C ), instead of matching Mueller matrix element spectra, one may also extract the generalized ellipsometry parameters in terms of the normalized Jones matrix elements for the sample alone, and compare those to the model calculations from A. In A, the Mueller matrix spectra taken without the magnet attached to the system are used to ﬁnd the best-ﬁt parameters for the model dielectric function parameters and thickness of the i-GaAs layer (ωTO , ωLO , γ , common to both substrate and buﬀer layer; ωp , γp , for the substrate, and d for the ﬁlm). The second measurement (magnet attached, ﬁeld oﬀ) determines the eﬀects of spatial rotations and p-, and s-amplitude changes imposed by the two Au-mirrors (Fig. 8.1). For this one must use the model parameters obtained from the ﬁrst step for calculating the actual sample response, and then by ﬁtting the parameters of the corresponding Mueller matrices for mirror eﬀects

8.2 Experiments

80

155

Φa = 80˚

air/i-GaAs(d)/n-GaAs

Ψpp(H) - Ψ(0) [deg]

Ψps & Ψsp [deg]

Ψpp [deg]

60

FSP

H=0

40 20 15 10 5

0

µ0H = 2.3 T

-1 -2 -3 -4

SGW 200

-

SGW 300

+

400

500

600

ω [cm ] -1

Fig. 8.4. Experimental (dotted lines) and best-ﬁt (solid lines) ﬁr MO generalized ellipsometry spectra (PMO setup) for the i-GaAs(d = 686(5) nm)/n-GaAs homostructure discussed in Sect. 4.2.2. Upper panel : Ψ at zero ﬁeld. Vertical arrows indicate frequencies of the interface modes FSP, SGW+ and SGW− , discussed in Chap. 4. Middle panel : Ψps , Ψsp at µ0 H = 2.3 T. Lower panel : diﬀerence spectra Ψpp (H) − Ψ (0). Coupling into LPPM modes causes splitting of the SGW+,− modes, which occur near the LPP+,− mode frequencies of the n-GaAs substrate. No coupling is seen at the FSP mode frequency for H = 0

156

8 Magneto-Optic Ellipsometry

∆pp [deg]

180

air/i-GaAs(d)/n-GaAs 90

H=0

∆ps [deg]

0 180

90

0 180

∆sp [deg]

µ0H = 2.3 T

SGW

-

+

FSP

SGW

90

0 200

300

400

500

600

ω [cm-1] Fig. 8.5. Same as Fig. 8.4 for ∆-type spectra at H = 0 (upper panel ) and µ0 H = 2.3T (middle and lower panel ) [14]

[14, 16, 17].8 As a result one obtains the spatial rotations of the light beam’s coordinate system upon each mirror reﬂection, the angle of incidence at the sample surface (angle between the ﬁr beam axis and the sample normal Φa = 80.0◦ ± .5◦ ), and at each mirror surface. The Mueller matrix elements spectra from the third step (magnetic ﬁeld on), and the diﬀerences between spectra at µ0 H = 2.3 T and H = 0 can then be used to diﬀerentiate between N , m, and µ. Incorporation of the diﬀerence spectra into the best-ﬁt weighted test function (2.74) can be done for immediate extraction of the magnetic ﬁeld induced eﬀects. Note that the diﬀerence spectra vanish if no free charge carriers are present! 8

The experimental and best-ﬁt calculated Mueller matrix element spectra for this step are shown in [14]. Note that all spectra are normalized to the element M11 .

8.2 Experiments

157

In B, for the wavelength-by-wavelength procedure, one must assume that the mirror eﬀects, the angle of incidence, the i-GaAs-layer thickness and ε obtained by A are suﬃciently correct. This assumption is reasonable because all parameters are gained from ﬁeld free measurements. Then one can ﬁt for the complex-valued elements εxx , εzz , and εxy wavelength-by-wavelength (wavenumber-by-wavenumber) matching all experimental spectra simultaneously (setting εxx = εzz , and εxy = 0 for spectra with H = 0). In C, the generalized ellipsometry spectra (the normalized Jones matrix elements) are extracted from the Mueller matrix by considering the mirror eﬀects. The thereby obtained “experimental” generalized ellipsometry spectra are shown in Figs. 8.4 and 8.5. The upper panels depict Ψ , ∆ without external ﬁeld. The i-GaAs ﬁlm causes excitation of the p- and s-polarized n-GaAs-iGaAs interface-bound modes FSP, and SGW+,− , respectively.9 The middle panels display the oﬀ-diagonal generalized ellipsometry parameters for µ0 H = 2.3 T. Strong p − s polarization conversion occurs near the frequencies of the SGW+,− modes, which split upon the LPPM coupling (ωc ≈ 29.8 cm−1 ), rendering the complex MO birefringence. Similar features can be seen from the lower panels. Figure 8.6 depicts εxx , εzz , and εxy extracted from the experimental data using the point-by-point ﬁt technique (Approach B), and the MO Drude model (Approaches A, C). Note that the point-by-point ﬁt does not include assumptions about the spectral dependence of the ε tensor components, whereas a certain global behavior is implied upon the model lineshape approach. The εzz spectra are equal to ε without the external magnetic ﬁeld. The diﬀerences between εxx and εzz at µ0 H = 2.3 T are negligible. The MO Drude model is in very good agreement with the point-by-point extracted complex functions, particularly for εxy . Note that the complex-valued MO ε tensor elements measured by the generalized ellipsometry approach provide an exciting experiment to support the applicability of the Drude theory in doped semiconductors. The best-ﬁt model lineshape parameters obtained here are N = (1.60 ±0.01) × 1018 cm−3 , µ = (2.22 ± 0.01) × 103 cm2 /(Vs), and m = (0.072 ± 0.001) m0 for the n-GaAs substrate, and ωTO = 267.95 cm−1 , ωLO = 291.0 cm−1 , ε∞ = 10.6, and γTO = 3.9 cm−1 for both substrate and epilayer. The error bars correspond to the 90% conﬁdence interval, and are progressed from the standard deviations measured for each experimental data point.10 The set (N , m) obtained here agrees excellently with that expected for this GaAs 9

10

Refer to Fig. 4.7, where similar Ψ spectra are shown, except for the blue-shift of the SGW+ modes here. The low-frequency mode SGW− can here be recognized most clearly in ∆pp at H = 0) in Fig. 8.5. The strength of the magnetic ﬁeld was measured with a Hall-probe, and its value was taken as input parameter. The error bar on this value also propagates into that of m and N . Alternatively, the ellipsometry approach can also be used for precise measurement of the actual magnetic ﬁeld strength at the sample holder using a material with well-known eﬀective mass, for instance GaAs.

8 Magneto-Optic Ellipsometry µ0H = -2.3 T PMO

30 0

120 90

-30

60

-60

30

Im{ε xx}

Re{ε xx}

158

18

0

µ = 2200 cm /(Vs) ε∞ = 10.6

120 90

2

-30 -60

60 30 0 4

3

Re{ε xy}

-3

2

3

MDF point-by-point fit

1

2 1

0

Im{ε xy}

Re{εzz}

N = 1.6×10 cm

Im{ε zz}

150 m = 0.072 me

30

0 300

400

ω [cm ] -1

500

300

400

ω [cm ] -1

500

Fig. 8.6. Wavelength-by-wavelength inverted (dotted lines) and model-dielectricfunction (MDF) best-ﬁt (solid lines) ε tensor spectra (εxx , εxy , and εzz ) for the n-GaAs within the i-GaAs(d)/n-GaAs homostructure. The calculated data are obtained through model analysis of the experimental data shown in Figs. 8.4 and 8.5. The (wavenumber-by-wavenumber) inverted spectra follow closely those predicted by (3.30)–(3.33) and (3.3). The value of the eﬀective mass for this free-charge-carrier concentration is found in remarkable good agreement with previous data derived from Shubnikov-de-Haas experiments on n-GaAs [14]

sample [18].11 Therewith, the (far) infrared MO generalized ellipsometry approach is demonstrated as an optical “inertial” mass scale for semiconductor materials in layered structures. 11

For GaAs with almost spherical energy bands the conduction band eﬀective mass tensor can be taken as isotropic scalar. The value of the electron eﬀective mass at the bottom of the Γ1c conduction band is well established (m/m0 = 0.067 [18]). For large free electron concentration the Fermi energy is located within the conduction band, which is known to have slight non-parabolicity eﬀects for k = 0 [18]. For N > 1017 cm−3 the eﬀect of the non-parabolicity of the band on the conduction band eﬀective mass is not negligible. The result obtained here agrees very well with the solution of the three-level Kane equation up to the order of k6 given in [18].

8.2 Experiments

GaAs

0.0

Mij /M11

-0.2

SGW AMB

-0.4

M12

-0.6 250

-0.8 100

200

0.0 +

-0.2

AMIn

-0.4 -0.6

FSP

159

300

400 -1

ω [cm ]

M33

300

500

-0.8 600

Fig. 8.7. Room-temperature zero-magnetic-ﬁeld experimental (dotted lines) and best-ﬁt calculation (solid lines) ellipsometry data at Φa = 45◦ . Trivial elements M13 , M22 , M31 , M23 , and M32 are omitted. The GaAs-like phonon (TO, LO), and phonon-plasmon (LPP+ ) modes are indicated by solid, dashed, and dotted vertical lines, respectively. Two additional modes are attributed to InAs- and BAs-like sublattice modes denoted by AMIn and AMB , respectively. Vertical lines within the inset indicate wavenumbers for SGW+ and FSP mode excitations. Spectra are normalized to M11

8.2.2 n-B0.03 In0.06 Ga0.91 As(d)/i-GaAs The strain-free boron-alloyed InGaAs compounds with lattice match to GaAs are materials of contemporary interest for posssible use in novel III-V semiconductor solar cell absorber materials [19]. A non-degenerate, silicon-doped, n-type B0.03 In0.06 Ga0.91 As layer is included here for demonstration of a situation where a single conductive layer is deposited on isolating GaAs. Its band gap energy of 1.35 eV was determined by near-infrared ellipsometry investigations. The ﬁr MO generalized ellipsometry experiments provides a strong increase of the electron eﬀective mass of 44% in B0.03 In0.06 Ga0.91 As compared to In0.06 Ga0.94 As. If this result is taken as the basis of a linear interpolation scheme, for BAs – an experimentally obscure compound – the electron eﬀective mass value would render equal to the free electron mass. Figure 8.7 contains data from experiment and best-ﬁt model calculation for the ﬁeld-free Mueller matrix spectra in the OMO setup (Figs. 8.2, 8.3) at Φa = 45◦ . The ﬁeld-free measurements elucidate the vibrational lattice mode behavior.12 The reststrahlen bands of GaAs and B0.03 In0.06 Ga0.91 As are superimposed, and cause distinct structures in the polarized sample response for wavenumbers from 250 . . . 320 cm−1 . The quaternary layer has its main resonance due to the GaAs-like phonon band. In Fig. 8.7 the high-frequency edge 12

The best-ﬁt phonon-mode parameters are given in [20].

160

8 Magneto-Optic Ellipsometry

of the restrahlen band is shifted above the GaAs-like LO mode due to LPPcoupling, and drops oﬀ near the high-frequency LPP mode at ω ≈ 303 cm−1 . Two additional small bands are identiﬁed near ω = 272 and 355 cm−1 , denoted by AMIn and AMB , respectively. Both bands possess a very small width (≈ 1cm−1 ). Mode AMIn is caused by the InAs-like sublattice vibration. Mode AMB is probably induced by the existence of a boron-containing sublattice [21]. Excitation of a Fano-type interface polariton causes the resonant loss in rp , reﬂected by the sharp notch in M12 . Note that here the substrate is the medium with positive real part of the dielectric function, whereas the ﬁlm has negative values because ωLO,GaAs < ωLPP+ ,BInGaAs . An SGW+ mode, bound to the interface between the substrate and the epilayer, occurs at ≈303 cm−1 . This resonance is very sensitive to the free-charge-carrier mobility parameter, and the broad resonance observed here compared to that in the i-GaAs/nGaAs structure is indicative for smaller mobility values in BInGaAs than in GaAs. Magnetic-ﬁeld induced changes of the Mueller matrix spectra are depicted in Fig. 8.8. In order to match model and experiment in Fig. 8.8 variation of ωc is required only, and the remaining parameters are those, which match already the experimental lineshapes in Fig. 8.7. Note that the magnetic-ﬁeldinduced changes are strongest near the frequencies of the SGW mode, and that all lattice mode contributions are not aﬀected by the magnetic ﬁeld. The model dielectric function parameters for the latter must not be varied for calculation of the Mueller matrix diﬀerence spectra. Data in Figs. 8.7 provided ωp = 753.5 cm−1 , and γp = 113.5 cm−1 . Those in Fig. 8.8 resulted in ωc = 30.1 cm−1 . Hence, N = 5.9 × 1017 cm−3 , µ = 885cm2 /(Vs), and m = 0.093 m0 . Mobility and density compare well with results from Hall-eﬀect measurements (N = 8.9 × 1017 cm−3 , µ = 774 cm2 /(Vs)). For the Hall-eﬀect results an uncertainty limit of 15% is estimated from repeated contact processing. Linear interpolation between the binary constituents provides for strain-free In0.06 Ga0.94 As an eﬀective mass parameter of m = 0.065 m0 . Thus incorporation of 2.7% boron leads to an increase of the eﬀective mass of about 44% [20]. 8.2.3 n-GaAs(d)/n-Al0.19 Ga0.33 In0.48 P(d)/i-GaAs Whereas the two previous examples were focused on the diﬀerent alternatives for obtaining the MO dielectric function of a rather simple sample, the following section is concerned with a heterostructure situation with the quaternary material of interest deposited as a thin ﬁlm embedded within diﬀerently doped GaAs. Often in heteroepitaxial growth, active or passive device constituents (layers) must be covered by diﬀerent epitaxial layers to avoid oxidation, even if the growth is intended for studying fundamental physical material properties. Likewise, buﬀer layer sequences are often required for strain-free or high-quality (crack-free) growth. The heterostructure discussed here as example consists of a nominally 1900 nm thick Te-doped,

8.2 Experiments

161

0.12 0.06

δ M31

0.00

δ M21

-0.06 0.11

δ M12 δ M22

0.05 -0.01

δ M32

-0.07 0.2

δ M33

0.1 -0.1 100

δ M13

δ M23

200

300 400 ω [cm-1]

500

600

Fig. 8.8. Room-temperature experimental (dotted lines) and best-ﬁt (solid lines) Mueller matrix spectra obtained by MO generalized ellipsometry (OMO setup in Figs. 8.2, 8.3) at Φa = 45◦ . The spectra are diﬀerences between those obtained at µ0 H = −3 T and +3 T n-type Al0.19 Ga0.33 In0.48 P layer deposited by metal-organic vapor phase epitaxy on an undoped i-type (001)-oriented GaAs substrate/buﬀer layer sequence. The AlGaInP layer is capped by a nominally 70 nm thick Te-doped n-type GaAs layer. Further information on the materials background of this very interesting quaternary alloy are provided in Sect. 6.2. Prerequisite for determination of the MO-induced changes of the AlGaInP dielectric function is the exact knowledge of the ﬁeld-free ε.13 Figure 8.9 depicts the zero-ﬁeld (standard) ellipsometry data. The lattice term ε(L) contains contributions descending from the binary (GaP, InP, AlP) as well as from disorder-induced (AM1 , AM2 , AM3 ) multiple-phonon-mode excitations discussed in Sect. 6.2 [23]. The MO generalized ellipsometry data shown in Fig. 8.10, here in terms of Mueller matrix element diﬀerence spectra, were measured in the PMO setup (Fig. 8.1). The best-ﬁt model analysis (following the approaches A, B, C discussed in Sect. 8.2.1) provided parameters given in Table 8.1 [22].14 For the GaAs cap layer the eﬀective mass parameter 13 14

Parameter values for the model dielectric function ε(L+FC) are given in [22]. For electron (me ) and hole (mh ) data, a kp-based scheme was suggested in [24], interpolating band structure parameters for the binary and ternary constituents. Using the band-gap-energy Eg = 2.124 eV, determined by near-infrared-visible region generalized ellipsometry for this AlGaInP layer [25], the scheme in [24] provides me = 0.113 m0 . The slightly larger me obtained from the MO

162

8 Magneto-Optic Ellipsometry

Table 8.1. Free-charge-carrier and thickness parameters for the n-GaAs(d)/nAl0.19 Ga0.33 In0.48 P(d)/i-GaAs sample. For the GaAs cap layer the eﬀective mass parameter was taken from [18]

n-GaAs n-Al0.19 Ga0.33 In0.48 P

N [1017 cm−3 ]

m [me ]

µ[cm2 /(Vs)]

d [nm]

18.3 ± 0.5 6.7 ± 0.2

0.073 0.12 ± 0.01

1700 ± 100 339 ± 15

71 ± 5 1868 ± 10

50

70 GaAs

InP AlP GaP

30 Ψ [˚]

Ψ [˚]

50

10

Φa=70˚

30 AM2 AM3

Φa=80˚

10

AM1

100

200

300

400

ω [cm-1]

500

600

Fig. 8.9. Zero-ﬁeld ellipsometry spectra (experiment: dotted lines; model: solid lines) measured at Φa = 70◦ , 80◦ on a n-GaAs(d)/n-Al0.19 Ga0.33 In0.48 P(d)/i-GaAs heterostructure [23]

was tied to the concentration parameter because of its known dependence m = m(N ) [18]. Mobility and concentration obtained from Hall-eﬀect measurements compare very well, after the Hall-coeﬃcients had been reanalyzed considering the two-layer structure with their actual thicknesses, concentrations, and mobilities [22]. The example shown here motivates use of the generalized magneto-optic infrared ellipsometry approach for nondestructive analysis of free-chargecarrier and phonon mode properties of individual constituents in complexlayered heterostructures, such as for high-power light emitting diodes. In-line tools may be developed for production environments, which can test the actual heterostructure properties after deposition prior to the actual device processing and packaging. experiment may be indicative for the non-parabolocity of the conduction band in AlGaInP.

References

163

δ M21

0.05 -0.01

δ M31

-0.07 normalized Intensity

-0.13 δ M32

0.05 -0.01

δ M12 δ M22

-0.07 -0.13

δM23

0.05 -0.01

δ M33 δ M13

-0.07 -0.13 100

200

300 400 ω [cm-1]

500

600

Fig. 8.10. Diﬀerences between PMO-setup MO generalized ellipsometry data at µ0 H = −2.1 T, and +2.1 T, here in the Mueller matrix presentation [22]

References 1. P. Yu and M. Cardona, Fundamentals of Semiconductors (Springer-Verlag, Berlin, 1999). 147 2. C. Pidgeon, in M. Balkanski, ed., Handbook on Semiconductors (North-Holland, Amsterdam, 1980), pp. 223 – 328. 148, 149, 150 3. J. Brion, R. Wallis, A. Hartstein, and E. Burstein, Phys. Rev. Lett. 28, 1455 (1972). 148 4. J. Brion, R. Wallis, A. Hartstein, and E. Burstein, Surf. Sci. 34, 73 (1973). 148 5. A. Hartstein, E. Burstein, J. Brion, and R. Wallis, Surf. Sci. p. 81 (1973). 148 6. R. Wallis, J. Brion, E. Burstein, and A. Hartstein, Phys. Rev. B 9, 3424 (1974). 148 7. E. D. Palik, S. Teitler, and R. F. Wallis, J. Appl. Phys. 32, 2123 (1961). 148 8. G. Wright and B. Lax, J. Appl. Phys. 32, 2113 (1961). 149 9. M. Cardona, Phys. Rev. 121, 752 (1961). 149 10. B. Rheinl¨ ander and H. Neumann, phys. stat. sol. (b) 45, K9 (1971). 149

164

8 Magneto-Optic Ellipsometry

11. B. Rheinl¨ ander, H. Neumann, P. Fischer, and G. K¨ uhn, phys. stat. sol. (b) 49, K167 (1972). 149 12. B. Rheinl¨ ander, Infrarot-Faraday-Eﬀekt an Halbleitern, Master’s thesis, Universit¨ at Leipzig (1965). 150 13. M. Mansuripur, The physical principles of magneto-optical recording (University Press, Cambridge, 1995). 150 14. M. Schubert, T. Hofmann, and C. M. Herzinger, J. Opt. Soc. Am. A 20, 347 (2003). 151, 152, 156, 158 15. T. Hofmann, Far-Infrared Spectroscopic Ellipsometry on A-III B-V Semiconductor Heterostructures, Ph.D. thesis, Universit¨ at Leipzig (2004). 151, 152, 153 16. A. R¨ oseler, Infrared Spectroscopic Ellipsometry (Akademie-Verlag, Berlin, 1990). 152, 156 17. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (NorthHolland Publ. Co., Amsterdam, 1984). 156 18. A. Raymond, J. Robert, and C. Bernard, J.Phys.C 12, 2289 (1979). 158, 162 19. C. Krahmer, G. Leibiger, H. Herrnberger, and V. Gottschalch, Si- und zndotierung von BGaInAs und InGaNAs mit der MOVPE als m¨ ogliche absorbermaterialien in solarzellen (2004), 18. Workshop des DGKK-Arbeitskreises Epitaxie von III/V-Halbleitern, Bremen, 11.-12. Dezember 2003, Germany. 159 20. T. Hofmann, G. Leibiger, N. Ashkenov, V. Gottschalch, and M. Schubert, Strong increase of the electron eﬀective mass in GaAs incorporating boron and indium (2004), unpublished. 159, 160 21. G. Leibiger, V. Gottschalch, V. Riede, M. Schubert, J. N. Hilﬁker, and T. E. Tiwald, Phys. Rev. B 67, 195205 (2003). 160 22. T. Hofmann, M. Schubert, C. M. Herzinger, and I. Pietzonka, Appl. Phys. Lett. 82, 3463 (2003). 161, 162, 163 23. T. Hofmann, G. Leibiger, V. Gottschalch, I. Pietzonka, and M. Schubert, Phys. Rev. B 64, 155206 (2001). 161, 162 24. I. Vurgaftman, J. Meyer, and L. Ram-Mohan, J. Appl. Phys. 89, 5815 (2001). 161 25. M. Schubert, T. Hofmann, B. Rheinl¨ ander, I. Pietzonka, T. Saß, V. Gottschalch, and J. A. Woollam, Phys. Rev. B 60, 16618 (1999). 161

A 4 × 4 Transfer Matrix Tp for Isotropic Films

Isotropic materials can be treated much easier and by simpler equations than through the 4 × 4 approach. However, the isotropic solution within the 4 × 4 matrix formalism is necessary in order to include isotropic layers into layer models, which also include layers of anisotropic materials. The Berremanformalism describes use of 4 × 4 matrices for calculation of reﬂection and transmission of electromagnetic plane waves by stratiﬁed mediums with plane parallel interfaces. Details of this approach shall not be recalled here, and many excellent reviews exist on this matter. Brieﬂy, a characteristic coeﬃcient matrix, ∆, provides Eigenvalues and self-consistent description of the electromagnetic radiation transport between layer boundaries. At any point z within a homogeneous layer, the x and y components of the electric and magnetic ﬁelds in Gaussian units and Cartesian coordinates (see also Sect. 2.7, and [1, 2, 3, 4]) T (A.1) Ψ (z) ≡ Ex , Ey , Hx , Hy (z) , are related through Maxwell’s postulates in the space-frequency domain presentation (A.2) ∂z Ψ (z) = ik0 ∆(z)Ψ (z) , ω (A.3) k0 ≡ . c Here, ω is the angular frequency, and c is the vacuum velocity of light.1 A solution is wanted such that electromagnetic ﬁelds at parallel interfaces separated by a distance d = z − z0 can be related by a simple matrix multiplication, thereby deﬁning the partial transfer matrix Tp for isotropic ﬁlms ω (A.4) Ψ (z) = exp i ∆d Ψ (z0 ) = Tp Ψ (z0 ) . c The elements of the coeﬃcient matrix ∆ in (A.2) are 1

k0 ≡

ω c

e = 2π hc E [eV ] = 5.067690488 × 106 E [eV ].

Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 165–167 (2004) c Springer-Verlag Berlin Heidelberg 2004

166

A 4 × 4 Transfer Matrix Tp for Isotropic Films

 −kx εεzx zz

    0  ∆=   ε εzx − ε yx  yz εzz   εzx εxx − εxz εzz

ε −kx εzy zz

0

0

-1

kx2 − εyy + εyz εxy − εxz

εzy εzz

0

εzy εzz

0

 kx2 1 − εzz     0   , εyz  kx εzz     ε xz −kx εzz

kx = na sin Φa .

(A.5)

(A.6)

For εij = εδij the Eigenvalues of ∆ are2 q1 = q2 = −q3 = −q4 ≡ q =

ε − kx2 =

n2 − n2a sin2 Φa ,

(A.8)

with ε being the isotropic dielectric function. Appendices B, C, and D continue with solutions for Tp in cases when ε possesses certain tensor properties. For isotropic situations (B.4) are degenerate, and do not provide a solution for Tp .3 However, the expansion of the exponential series in (B.1) and the separation of the associated sums for the Cosine and Sine functions result directly in

2 4 3 (ik0 dq) ∆ k0 dq (k0 dq) (ik0 dq) + + ... + i − + ... , Tp = I 1 + 2! 4! q 1! 3! (A.10) ∆ Tp = I cos(k0 dq) + i sin(k0 dq) , (A.11) q 2

The Kronecker symbol δij deﬁnes the diagonal unit matrix I

I = (δij ) , 3

δij =

1, i = j 0, i = j

.

(A.7)

For very small anisotropy the Eigenvalue approach eventually breaks down for ﬁnding the coeﬃcients βi . For k0 d||q1± − q2± || 1, the attempt to invert (B.4) may fail. On the other hand, the determinant of the right side of (B.4) det (Q) = det (qij ) ,

qij = qij−1 ,

i, j = 1, . . . , 4 .

(A.9)

can be used to check if the optical response of a given layer, to be evaluated at a certain wavelength, can be treated anisotropic or isotropic during a best-ﬁt regression analysis of experimental ellipsometry data.

References



cos(k0 dq) 0 Tp =  0 (iε/q) sin(k0 dq)

0 cos(k0 dq) −iq sin(k0 dq) 0

0 −(i/q) sin(k0 dq) cos(k0 dq) 0

167



(iq/ε) sin(k0 dq)  0  .  0 cos(k0 dq) (A.12) where the identities ∆∆ = q 2 E and ∆∆∆ = q 2 ∆ were used.

References 1. D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972). 165 2. I. Abdulhalim, L. Benguigui, and R. Weil, J. Phys. 46, 815 (1985). 165 3. M. Schubert, in E. A. Irene and H. W. Tompkins, eds., Handbook of Ellipsometry (William Andrew Publishing, 2004). 165 4. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (NorthHolland Publ. Co., Amsterdam, 1984). 165

B 4 × 4 Transfer Matrix Tp for Dielectric Homogeneous Films

The matrix ∆ in (A.5) does not depend on z if the medium is homogeneous, i.e., ε is not a function of z. The solution of (A.2) for a layer with thickness d deﬁnes the partial transfer matrix Tp for dielectric homogeneous ﬁlms ω (B.1) Ψ (z + d) = exp i ∆d Ψ (z) = Tp Ψ (z) , c ω (B.2) Tp ≡ exp i ∆d . c Tp connects the in-plane components of the electric and magnetic ﬁelds at layer interfaces separated by d. Tp includes the eﬀects of all multiple reﬂections if a part of the wave is travelling along a direction with no or weak absorption.1 Tp is computed with ∆ as input. Requiring suﬃciently small thickness d for fast convergence the exponential function in (B.1) can be expanded into series of ∆k0 d = 2π∆d/λ [2]. Reasonable convergence may only be achieved when λ 2π||det{∆}||d. The accurate approach is the calculation of Tp by the theorem of Sylvester-Lagrange, where the latter requirement is abandoned [3, 4] ω Tp ≡ exp i ∆d = β0 E + β1 ∆ + β2 ∆∆ + β3 ∆∆∆ . (B.3) c The scalars βi are obtained from the following linear relations 3 ω βj qkj , k = 1, .., 4 , exp i qk d = c j=0

(B.4)

where qk denote the Eigenvalues of ∆ associated with one of the four electromagnetic Eigenmodes Ξk within the layer (k = 1, . . . , 4). Two solutions have a positive real part and constitute the forward travelling plane waves with respect to the chosen laboratory coordinate system. The other solutions with negative real parts are due to the back-travelling wave components. For mediums with symmetric dielectric properties (εij = εij ) the Eigenvalues are 1

Tp is unitary if the medium is loss less in any direction of propagation as consequence from the conservation law of energy. Otherwise ||Ψ (z + d)|| and ||Ψ (z)|| may not be equal [1].

Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 169–170 (2004) c Springer-Verlag Berlin Heidelberg 2004

170

B 4 × 4 Transfer Matrix Tp for Dielectric Homogeneous Films

εxz εxz 2 εxz 2 1 2 4 − − t1 + (kx ) + Σ ± − t1 + 2(kx ) − Σ− Υ , −kx 2 εzz 3 εzz 3 εzz (B.5) ε ε ε 1 2 4 xz xz 2 xz 2 − q1,2 = −kx + − t1 + (kx ) + Σ ± − t1 + 2(kx ) −Σ+Υ , 2 εzz 3 εzz 3 εzz (B.6) with the abbreviations s3 Υ = , (B.7) 2 2+Σ 4 − 3 t1 + (kx εεxz ) zz

− 13 13 1 1 1 + , Σ= s1 s2 + s22 − 4s31 s2 + s22 − 4s31 2 2 2 (B.8) εxz 2 s1 = t1 + 12 kx t2 + t3 , (B.9) εzz 2 εxz εxz 3 2 s2 = 2t1 + 36kx t1 t2 + 108(t2 + kx t3 ) − 72t1 t3 , (B.10) εzz εzz 2 εxz εxz s3 = −8kx ( kx − t1 ) + 16t2 , (B.11) εzz εzz and 1 εxx ε2xz + ε2yz − εzz εxx + εyy − kx2 1 + , (B.12) t1 = εzz εzz kx t2 = εxz εyy − εxy εyz − εxz kx2 , (B.13) εzz 1 t3 = (t31 + t32 ) , (B.14) εzz (B.15) t31 = 2εxy εxz εyz − εxx ε2yz − εyy ε2xz − εzz ε2xy + εxx εyy εzz , 2 2 2 2 (B.16) t32 = kx εxy + εxz − εxx εyy + εzz − kx .

+ q1,2 =

+ − (q1,2 ) refer to forward- (backward-) travelling waves, respecHere q1,2 tively. (B.1) can now be evaluated using the coeﬃcients βi , which follow from (B.4), by setting q1 = q1+ , q2 = q1− , q3 = q2+ , q4 = q2− , for example. The Eigenvalues of uniaxial materials are contained as special solutions.

References 1. K. Eidner, J. Opt. Soc. Am. A 6, 1657 (1989). 169 2. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (NorthHolland Publ. Co., Amsterdam, 1984). 169 3. H. W¨ ohler, M. Fritsch, G. Haas, and D. A. Mlynski, J. Opt. Soc. Am. A 5, 1554 (1988). 169 4. H. W¨ ohler, M. Fritsch, G. Haas, and D. A. Mlynski, J. Opt. Soc. Am. A 8, 536 (1991). 169

C 4 × 4 Transfer Matrix Tp for Magneto-optic Films

Consider homogeneous mediums with non-symmetric dielectric properties, i.e., materials where εij = εij . Examples are magnetized materials with noncubic symmetry, or free-charge-carriers in semiconductors subjected to an arbitrarily oriented external magnetic ﬁeld (Sect. 2.6). The dielectric function tensor ε is then composed of a symmetric (s) and an anti-symmetric (a) tensor1 (magneto-optic materials) εij = sij + aij , sij = sji , aij = −aji ,

(C.1)

where,

1 1 (εij + εji ) , aij = (εij − εji ) . (C.2) 2 2 The Eigenvalues of the matrix ∆ in (A.5) are [1]   2 2   s s s 1 2 4 xz xz xz + q1,2 = − − t1 + kx +Σ ± − t1 + 2 kx −Σ −Υ , −kx  2 szz 3 szz 3 szz sij =

(C.3)   2 2   sxz sxz sxz 1 2 4 − = + − t1 + kx +Σ ± − t1 + 2 kx −Σ +Υ , q1,2 −kx  2 szz 3 szz 3 szz (C.4) with Σ as deﬁned in (B.8), and S3 , Υ = 2 4 − 3 t1 + (kx ssxz )2 + Σ zz sxz s1 = t21 + 12 kx t2 + t3 , szz 2 sxz sxz s2 = 2t31 + 36kx t1 t2 + 108(t22 + kx t3 ) − 72t1 t3 , szz szz ! 2 sxz sxz s3 = −8kx kx − t1 + 16t2 , szz szz 1

Note that ij must be independent on the photon wavevector k.

Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 171–172 (2004) c Springer-Verlag Berlin Heidelberg 2004

(C.5)

(C.6) (C.7) (C.8)

172

C 4 × 4 Transfer Matrix Tp for Magneto-optic Films

and 1 t1 = szz

t33

sxx 2 2 2 2 2 sxz + syz − szz sxx + syy − kx 1 + − ayz − axz , szz (C.9) kx 2 t2 = sxz syy − sxy syz − sxz kx − ayz axy , (C.10) szz 1 t3 = (t31 + t32 + t33 ) , (C.11) szz

t31 = 2 (sxy sxz syz − sxy axz ayz + sxz ax2 ayz − s23 axz axy ) , (C.12) t32 = kx2 s2xy + s2xz − sxx syy + szz − kx2 − a2xz + a2xy . (C.13) 2 = sxx ayz − s2yz + syy a2xz − s2xz + szz a2xy − s2xy + sxx syy szz . (C.14)

+ − (q1,2 ) again refer to the forward (backward) travelling The eigenvalues q1,2 waves. The transfer matrix Tp for magneto-optic ﬁlms is then obtained in the same way as done for the biaxial ﬁlms in Appendix B. Solutions for magnetooptical ellipsometry in polar, transverse and longitudinal conﬁguration, as described by Viˇsnovsk´ y for bulk and ﬁlms, are retained here as well [2, 3].

References 1. M. Schubert, T. E. Tiwald, and J. A. Woollam, Appl. Opt. 38, 177 (1999). 171 2. S.Viˇsnovsk´ y, Czech. J. Phys. B 36, 625 (1986). 172 3. S. Viˇsnovsk´ y, M. N´ yvlt, V. Prosser, R. U. R. Lopuˇsn´ık, J. Ferr´e, G. P´enissard, D. Renard, and R. Krishnan, Phys. Rev. B 52, 1090 (1995). 172

D 4 × 4 Transfer Matrix Tp for Dielectric Helical Films

Dielectric helical thin ﬁlms (DHTF’s) are described by a homogeneous rotation of an arbitrarily oriented symmetric dielectric function tensor ε assuming that all other linear response dyadics are zero, and that the permeability tensor µ0 µ has its scalar vacuum value µ0 [1, 2, 3, 4, 5, 6, 7]. Such material properties are well-known from chiral liquid crystals [8], and which require ellipsometry for appropriate optical characterization [9, 10]. A new class of designed matter is about to emerge upon sculpturing solid-state materials in thin ﬁlm form. Such designs involve physical deposition techniques in three-dimensional growth regimes, where, depending on growth parameters and appropriate substrate rotation, “zick-zack” pattern, “S”-shapes or helices can be deposited [11, 12, 13, 14]. Design dimensions can be well within the nanometer region, providing interesting grounds for new applications such as micro-ﬁlters, micro-antenna arrays, micro-springboards, or “frozen” chiral liquid crystals for optical ﬁlters. Crucial for design is the choice of the right dimensions, because electrical and optical properties will depend on conﬁnement eﬀects leading to form birefringence. Due to the complexity of such ﬁlms, optical characterization is a challenge. The generalized ellipsometry approach allows for nondestructive characterization of helical dielectric materials [15]. The mathematical description of the transfer matrix Tp for dielectric helical ﬁlms is therefore included here. In order to address the tensor of DHTF appropriately, let the plane of incidence (x, z) and the sample surface (x, y) set a right-handed Cartesian system (x, y, z) with origin at the sample surface. The z-component kz of the incident wave vector ka may be positive. It follows without loss of generality that ky = 0. The angle of incidence is Φa , and na is the ambient index of refraction (Fig. 2.1). At z = z0 , ε may be described by three major index √ of refraction values (nj ), and extinction coeﬃcients (kj ), Nj ≡ (nj +ikj ) = εj , j = “1”, “2”, “3”; Fig. D.1), the real-valued Euler angles (ϕ, ψ, θ) set up matrix A (see (2.48), Fig. 2.3). A transforms ε in (ξ, η, ζ), the tensor’s major axis system at z = z0 , into the laboratory frame of reference (x, y, z) ε = Adiag (ε1 , ε2 ,

ε3 ) A−1 .

(D.1)

Here “diag( ·, ·, ·)” indicates the diagonal matrix. It is implied that ε1 , ε2 , and ε3 do not depend on z0 , whereas εij = εij (z0 ), because ϕ(z0 ), ψ(z0 ), Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 173–179 (2004) c Springer-Verlag Berlin Heidelberg 2004

174

D 4 × 4 Transfer Matrix Tp for Dielectric Helical Films

Nx Nz

Ny Fig. D.1. Schematic presentation of the index-of-refraction ellipsoid of a rod-like birefringent material with intrinsic Cartesian coordinate system (ξ, η, ζ). See also Fig. 2.3

θ(z0 ) depend on the choice of z0 . The helical periodicity of the DHTF along the z-direction is provided by the unitary matrix B, which depends on the number of turns υ per unit length P (Fig. D.2)   cos υ − sin υ 0 2π z. (D.2) B =  sin υ cos υ 0  , υ = P 0 0 1 Thus ε depends on υ,  εxx εxy ε(υ) =  εyx εyy εzx εzy

and υ 0 = (2π/P )z0 .    εxz ε1 0 0 εyz  = B(υ)A  0 ε2 0  A−1 B(υ)−1 . εzz 0 0 ε3

(D.3)

The matrix ∆ in (A.5) depends on z upon homogeneous rotation of ε(ε1 , ε2 , ε3 ) along z.1 The solution of (A.2) for a layer with thickness d cannot be obtained immediately. Therefore, the z-dependence of the coeﬃcient matrix ∆ is substituted by υ, and the electromagnetic ﬁeld amplitudes in (A.2) are transformed into elliptical coordinates H ± = Hx ± iHy ,

Ψ˜ (υ)=

E ± = Ex ± iEy . E+, E−, H +, H −

(D.4) (D.5) T

(υ) ,

(D.6)

through the unitary matrix Γ Ψ˜ (υ) = ΓΨ (υ) ,   1 i 0 0  1 −i 0 0  . Γ≡ 0 0 1 i 0 0 1 −i 1

(D.7)

(D.8)

Tensor ε is written as if (D.1) would address non-symmetric situation as well. Therewith one retains magneto-optic situations for the solutions to be described further below.

D 4 × 4 Transfer Matrix Tp for Dielectric Helical Films

175

υ = 2π

υ=π

z

υ=0 z0

(x, y)

Fig. D.2. Schematic presentation of a DHTF-sequence with 0 < θ <90◦ , projected onto a plane in (x, y, z), which contains the z -axis and the z-axis at z = z0 . The arrows represent an arbitrary but ﬁxed reference direction of the dielectric function tensor, perhaps an optical axis, or the ζ-axis (see Fig. D.1). The axis of the helical twist is the z-axis, which is perpendicular to the sample surface. P is the distance of one complete sequence. The sign of P refers to a right (positive) or left (negative) handed helical twist

The vector Ψ˜ contains the in-plane ﬁeld amplitudes of back- and forward travelling elliptically polarized modes. The matrix F(υ) can be used to separate the helical dependence from Ψ˜ (Oseen transformation [16]) (D.9) F(υ) = diag eiυ , e−iυ , eiυ , e−iυ , introducing a new vector Ψ˜ ˜ Φ(υ) = F(υ)Ψ˜ (υ) .

(D.10)

Substitution of z by υ and replacement of Ψ by Φ˜ in (A.2) yields a new ˜ coeﬃcient matrix ∆ ˜ (υ, υ0 ) Φ ˜ (υ0 ) , ∂υ Φ˜ (υ) = in−1 ∆

(D.11)

with ˜ ∆(υ) = U + V+ eiυ + V− e−iυ + W+ ei2υ + W− e−i2υ .

(D.12)

Matrices U, V+ , V− , W+ , W− depend on υ 0 , and could therefore be written as U = U(υ0 ), V+ = V+ (υ0 ), and so on. These matrices read explicitly

176

D 4 × 4 Transfer Matrix Tp for Dielectric Helical Films



 −n 0 −iκ 0  0 n 0 iκ   U=  f s+ −n 0  , s− −f 0 n   0 0 0 0  iπ+ −iπ− 0 0  , V+ =   0 0 −iπ− 0  0 0 −iπ+ 0   iπ+ −iπ− 0 0  0 0 0 0   , V− =   0 0 0 iπ−  0 0 0 iπ+   0 0 0 0  0 0 ikx2 0    2εzz W+ =   ,  0 0 0 0 2 −ikx 0 0 0 2   −ik2 0 0 0 2εzzx   0 0 0 0  W− =  ik2  , 0 x 0 0  2 0 0 0 0

(D.13)

(D.14)

(D.15)

(D.16)

(D.17)

where the following abbreviations were introduced n= κ=

2π , k0 P

k2 1− x 2εzz

(D.18) ,

kx (εyz ± iεxz ) , 2εzz + ε2yz − εzz εxx + εyy − kx2 ) ,

π± = f =− s± =

i (ε2 2εzz xz

(D.19) (D.20) (D.21)

1 2(εxz εyz − εxy εzz ) ± i ε2yz − ε2xz + εzz {εxx − εyy } . (D.22) 2εzz

Normal Incidence ˜ p at normal incidence, i.e., for kx = 0 (or equivalently for A solution for T ◦ Φa = 0 ) was given in [17, 18], based upon the Sylvester-Lagrange theorem for solution of the diﬀerential (D.11) ˜ p (d) = β0 E + β1 U + β2 UU + β3 UUU , T

(D.23)

D 4 × 4 Transfer Matrix Tp for Dielectric Helical Films

with

# 1 " 2 2 q+ cos κ− − q− cos κ+ , 2χ

(D.24)

# 1 " 2 2 (q+ /q− ) sin κ− − (q− /q+ ) sin κ+ , 2χ

(D.25)

1 {cos κ+ − cos κ− } , 2χ

(D.26)

β0 = β1 = i

177

β2 = β3 = i

1 {(1/q+ ) sin κ+ − (1/q− ) sin κ− } , 2χ q± = n2 − if ± χ , χ = s+ s− − i4n2 f , κ± = k0 dq± .

(D.27) (D.28) (D.29) (D.30)

Oblique Incidence A practical solution for oblique incidence (kx = 0 or Φa = 0◦ ) is obtained by analytical expansion [19, 20]. The reference z-plane at z = z0 may be set to z0 = 0, without loss of generality. The DHTF is bound between z = 0, ˜ is analytical in (υ − and z = d. It follows from (D.11) that the matrix ∆ ˜ is expandable into a series with coeﬃcients υ0 ), 0 < |υ−υ0 | < 2π|P |/d, i.e., ∆ (υ − υ0 )α . The resulting coeﬃcient matrices δα are sorted according to the order α. Let υ0 refer to one arbitrary but ﬁxed z0 -value within the DHTF. Thus ∞ m ˜ δm (υ0 ) (υ − υ0 ) . (D.31) ∆ (υ, υ0 ) = m=0

˜ p (υ, υ0 ), The matrices δm do not depend on υ. Assume further that a matrix T supposedly the solution of (D.11), can be expanded into 4 × 4 coeﬃcient matrices tν (υ0 ), which do not depend on υ 2 ˜ p (υ, υ0 ) = T

∞

ν

tν (υ0 ) (υ − υ0 ) .

(D.32)

ν=0

˜ (υ, υ0 ) is solution to (D.11), it must also fulﬁll the following diﬀerential If ∆ equation, which follows from (D.11) (suppressing the variable υ0 ) ˜ p (υ) = in−1 ∆ ˜ (υ) T ˜ p (υ) . ∂υ T

(D.33)

˜ (υ) and T ˜ p (υ) by the respective series expansions, After substitution of ∆ and carrying out the derivative with respect to υ, one obtains a recurrence ˜ p (υ) [20] relation for the coeﬃcient matrices of the series expansion for T 2

Note that the choice of υ0 will aﬀect tν (υ0 ).

178

D 4 × 4 Transfer Matrix Tp for Dielectric Helical Films

i δm−k tk , n (m + 1) m

tm+1 =

(D.34)

k=0

where ˜ p (υ0 , υ0 ) = E , t0 ≡ T

(D.35)

˜ (υ) follow from its Taylor series is the unit matrix. The coeﬃcients for ∆ expansion δ0 = U (υ0 ) + V+ (υ0 ) + V− (υ0 ) + W+ (υ0 ) + W− (υ0 ) , δm =

1 m m m m [i V+ + (−i) V− + (2i) W+ + (−2i) W− ] , m!

(D.36) (m > 0) .

(D.37) For a uniaxial tensor, the matrices V+ and V− describe the eﬀect of the tilt of the optical axis towards the z-axis. V+ and V− vanish if the optical axis of a uniaxial dielectric tensor rotates within the (x, y) plane. For kx = 0, V+ and V− , as well as W+ and W− vanish, and the recurrence relation (D.32) becomes the series expansion of exp{in −1 U}, which is then calculated more eﬃciently through (D.23) [17]. For convenience, and without loss of generality, υ0 may be set to zero. ˜ p (υ) can be calculated for a DHTF of thickness d by setting υ = 2πd/P . T However, depending on the DHTF’s total rotation per unit length ∆υ = υ(z = d) − υ0 (z = 0), slow convergence of (D.32) may impede eﬃcient calcu˜ p (υ). Faster convergence will be obtained for small rotation steps. lation of T For large ∆υ it is more eﬀective to subdivide the layer into l sublayers with rotations δυ = ∆υ/l. Then, for each sublayer, bound by υs = υ0 + sδυ, and υs+1 (s = 0 . . . l – 1), matrices U(υs ), V± (υs ), W± (υs ) have to be modiﬁed ˜ p (υ, υ0 ) of the entire thin according to the actual ε tensor orientation at υs . T ˜ ps (υs+1 , υs ) from all sublayers ﬁlm follows from the ordered product of T ˜ p (υ, υ0 ) = T ˜ p0 (υ1 , υ0 ) . . . T ˜ p0 (υl , υl−1 ) . T

(D.38)

Finally, the following back transformation needs to be carried out in order to obtain Tp within the laboratory frame of reference after multiplying out the diﬀerent sublayer matrices. For compliance with the 4 × 4 matrix approach in generalized ellipsometry analysis, the inverse of Tp is needed [21, 22, 23].3 2π 2π −1 −1 ˜ (D.39) Tp (d) = Γ Tp υ = − d F(υ = − d)Γ . P P

References 1. V. Venugopal and A. Lakhtakia, Opt. Commun. 145, 171 (1998). 173 2. V. Venugopal and A. Lakhtakia, Proc. Roy. Soc. Lond. A 456, 125 (2000). 173 3

˜ p (υ)]−1 = T ˜ p (−υ) [24]. Note the identity [T

References

179

3. V. Venugopal and A. Lakhtakia, Eur. Phys. J. AP 10, 173 (2000). 173 4. A. Lakhtakia and W. Weiglhofer, IEE Proc. Microw. Antennas Propag. 144, 57 (1997). 173 5. A. Lakhtakia and V. Venugopal, Int. J. Electron. Commun. 53, 287 (1999). 173 6. A. Lakhtakia and V. Venugopal, Microwave and Opt. Techn. Lett. 17, 135 (1998). 173 7. V. Venugopal and A. Lakhtakia, Opt. Commun. 149, 217 (1998). 173 8. S. Chandrasekhar, Liquid Crystals (Cambridge University Press, Cambridge, 1992). 173 9. C. Cramer, H. Binder, M. Schubert, B. Rheinl¨ ander, and H. Schmiedel, Mol. Cryst. Liq. Cryst. 282, 395 (1996). 173 10. C. Benecke, H. Seiberle, and M. Schadt, Jpn. J. Appl. Phys. 39, 525 (2000). 173 11. R. Messier and A. Lakthakia, Mat. Res. Innovat. 2, 217 (1998). 173 12. Q. Wu, I. Hodgkinson, and A. Lakhtakia, Opt. Eng. 39, 1863 (2000). 173 13. I. Hodgkinson and Q.Wu, Appl. Opt. 38, 3621 (1999). 173 14. I. Hodgkinson, Q. Wu, B. Knight, A. Lakhtakia, and K. Robbi, Appl. Opt. 39, 642 (2000). 173 15. M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Oﬀ, F. Scholz, E. Schubert, H. Neumann, I. Hodgkinson, M. Arnold, et al., in A. Lakhtakia, G. Dewar, and M. W. McCall, eds., Complex Mediums III: Beyond Linear Isotropic Dielectrics (SPIE, Bellingham, WA, 2002), vol. 4806, pp. 264 – 276. 173 16. J. W. Oseen, J. Chem. Faraday Trans. II 29, 883 (1933). 175 17. M. Schubert, B. Rheinl¨ ander, C. Cramer, H. Schmiedel, B. Johs, C. M. Herzinger, and J. A. Woollam, J. Opt. Soc. Am. A 13, 1930 (1996). 176, 178 18. W. Weiglhofer and A. Lakhatakia, Optik 96, 179 (1994). 176 19. M. Schubert and C. M. Herzinger, phys. stat. sol. (a) 188, 1563 (2001). 177 20. A. Lakhtakia and W. Weiglhofer, Microwave and Opt. Techn. Lett. 12, 245 (1996). 177 21. M. Schubert, Phys. Rev. B 53, 4265 (1996). 178 22. M. Schubert, in W. S. Weiglhofer and A. Lakhtakia, eds., Introduction to Complex Mediums for Optics and Electromagnetics (SPIE, Bellingham, WA, 2004), pp. 677 – 710. 178 23. M. Schubert, in E. A. Irene and H. W. Tompkins, eds., Handbook of Ellipsometry (William Andrew Publishing, 2004). 178 24. D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972). 178

E TM Waves at the Interface Between Two Half-Inﬁnite Media

From electrodynamic continuum considerations one can derive bound solutions for harmonic plane waves at the interface between two inﬁnite dielectric media (“1”, “2” Fig. E.1). The line of derivation follows from the assumption that a transverse magnetic (TM) wave pattern attached to the interface exists, i.e., for z < 01 H 1 = (0, Hy,1 , 0) exp [i (kx,1 x + kz,1 z − ωt)] ,

(E.1)

E 1 = (Ex,1 , 0, Ez,1 ) exp [i (kx,1 x + kz,1 z − ωt)] ,

(E.2)

H 2 = (0, Hy,2 , 0) exp [i (kx,2 x − kz,2 z − ωt)] ,

(E.3)

E 2 = (Ex,2 , 0, Ez,2 ) exp [i (kx,2 x − kz,2 z − ωt)] .

(E.4)

and for z > 0:

z<0

y

z>0

ε1 ε2

x

z Fig. E.1. Two half-inﬁnite dieletric media with dielectric functions ε1 and ε2 , separated by an interface parallel to the (x, y) plane at z = 0 within a Cartesian coordinate system

The wave vector of the TM mode is k = (kx , 0, kz ). The electric and magnetic ﬁeld components must satisfy Maxwell’s postulates (4.1) – (4.4) within both sides of the interface. The conditions of continuity across the interface follow from (4.3) and (4.4) 1

A similar derivation for bound TM wave solutions at interfaces can be found in [1].

Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 181–183 (2004) c Springer-Verlag Berlin Heidelberg 2004

182

E TM Waves at the Interface Between Two Half-Inﬁnite Media

Ex,1 = Ex,2

(E.5)

Hy,1 = Hy,2 ,

(E.6)

ε1 Ez,1 = ε2 Ez,2 .

(E.7)

(E.5) and (E.6) require that kx,1 = kx,2 = kx . The x-components of the electric ﬁelds follow from (4.1) ω kz,1 Hy,1 = − ε1 Ex,1 , c ω kz,2 Hy,2 = ε2 Ex,2 , c

(E.8) (E.9)

or with (E.5) and (E.6) Hy,1 − Hy,2 = 0 ,

(E.10)

kz,1 kz,2 Hy,1 + Hy,2 = 0 . ε1 ε2

(E.11)

The coeﬃcient determinant of (E.10) and (E.11) must vanish, and the dispersion relation for a TM interface bound wave reads as follows kz,1 kz,2 0= + . (E.12) ε1 ε2 Equations (4.2), (E.1) – (E.4), and (E.8), (E.9), together with the conditions for the z-components for the electric ﬁelds in (4.1) provide the relation between the x- and z-components of the wave vectors on both sides of the $ ω %2 interface 2 + kx2 = εj , j = “1”, “2” . (E.13) kz,j c Using (E.13), the interface TM mode dispersion relation can be rewritten 1/2 ε1 ε2 ω kx = , (E.14) c ε1 + ε2 where kx is the x-component of the TM wave, ω is the wave frequency, and c is the light velocity.2 The solution of (E.14) may be termed kx = kx0 . The associated z-component within medium j follows from (E.13) $ % 0 2 1/2 ω 2 εj − kx , j = “1”, “2” . (E.15) kz,j = c For a TM wave bound to the interface, the associated electric ﬁelds must be evanescent on both sides of the interface. Then, a power ﬂow will occur along the interface, mediated by the electromagnetic ﬁeld of the TM wave. 2

Note that (E.13) and (E.14) leave the sign of kz,j undetermined. Note also that both equations are symmetrical with respect to the indices “1” and “2”. Note further the sign change in the present deﬁnition of the wave vectors with respect to those used by Burstein et al: In “Surface polaritons – electromagnetic waves at interfaces”, in [2] p 89, where ka⊥ = kz,1 , and kb⊥ = −kz,2 .

References

183

The evanescent character of the ﬁelds is represented by the imaginary components of kz,1 , and kz,2 . The existence of the TM wave is related to the dielectric media properties, and must be studied for the particular frequency dependence of ε1 and ε2 . In case of no or only little absorption, i.e., Im(ε1 ) ≈ Im(ε2 ) ≈ 0, (E.14) has two solutions (SP1): kz,1 , and kz,2 , are imaginary, and sign(ε1 ) = − sign(ε2 ), (SP2): kz,1 , and kz,2 , are real, sign(kz,1 ) = − sign (kz,2 ), and ε1 > 0 and ε2 > 0. The solution (SP1) represents a surface polariton (SP), and sign(ε1 ) = sign(ε2 ) follows from (E.12). The wave associated with the SP must propagate parallel to the interface, and kx must be real. It follows then from (E.13) and (E.14) that ε1 < 0, and ε2 < −ε1 . The second solution does not address a physical SP situation yet, because both kz,1 , and kz,2 are real-valued. The second solution, however, will become important in Appendix F when the thin-ﬁlm situation is considered.3

References 1. H. Raether, Surface Polaritons (Springer, Berlin, 1988). 181 2. E. Burstein and F. D. Martini, eds., Polaritons (Pergamon Press, New York, 1974). 182

3

The material with negative ε is often referred to as the active medium. This is largely due to historic assignments within interface polariton studies of metal surfaces, where the metal has a large negative dielectric constant. There, the oscillating charge displacement near the metal surface is seen as the source for the wave pattern associated with the polariton ﬁeld outside the metal. The dielectric function in polar semiconductors is negative for wavelength within the reststrahlen range, extended in doped materials by the coupled LO-phonon-plasmon mode frequencies. For situation (SP1) one material is within the reststrahlen range and negative, but eventually with much smaller absolute values than for metals. For (SP2) the “active” material has a positive dielectric function value. In both cases the electromagnetic waves actually exist within both media. The term “active” medium is somewhat confusing for polar semiconductors, and therefore not in use here. More incisive would be to designate the medium with small ε as the “gap” material. Refer also to Fig. 4.9.

F TM Waves at Two Stratiﬁed Interfaces

Two interfaces, separated by a distance d, embrace a ﬁlm with dielectric function εf between two half-inﬁnite media (ambient: εa , and substrate: εb ). A similar search for TM modes as show in Appendix E for a single interface, directs to the dispersion relation for SP modes, which may now be bound to either side of the ﬁlm. The line of derivation given here can easily be used for dissection of SP resonance conditions in heterostructures with more than two interfaces starting over again with the interlaced Fresnel formulas for the p-polarization of stratiﬁed media [1]. To begin with here, it is helpful to recognize that (E.12) is identical with the denominator of the p-polarized √ Fresnel reﬂection coeﬃcient (Sect. 2.5). With ni = εi and Snell’s law k 2 $ c %2 cos Φi = 1 − x , i = “a”, “f ”, “b” , (F.1) εi ω the single-interface reﬂection coeﬃcient reads rp,ij =

nj cos Φi − ni cos Φj , nj cos Φi + ni cos Φj

(F.2)

Nji − Nij zp,ij ≡ , Nji + Nij np,ij

(F.3)

rp,ij =

where a useful abbreviation was introduced [2] √ Nij = εi cos Φj .

(F.4)

After multiplication ahler ”) and denominator nij (“Nen√ √ of numerator zij (“Z¨ ner ”) with εa εb , suppressing the index “p”, (F.3) is rearranged into zij , nij

(F.5)

rij =

εj Nii − εi Njj , εj Nii + εi Njj

(F.6)

rij =

εj kz,i − εi kz,j , εj kz,i + εi kz,j

(F.7)

rij =

and the new denominator is easily recognized as (E.12), which must vanish at Mathias Schubert: Infrared Ellipsometry on semiconductor layer structures STMP 209, 185–190 (2004) c Springer-Verlag Berlin Heidelberg 2004

186

F TM Waves at Two Stratiﬁed Interfaces

y

εa εf εb

z=0 z=d

x

z Fig. F.1. Two half-inﬁnite media with dielectric functions εa (ambient) and εs (substrate), separated by a dielectric ﬁlm of thickness d with dielectric function εf

the SP resonance.1 For the ambient (“0”)/ﬁlm (“1”)/substrate (“2”) system in Fig. F.1 the p-polarized reﬂection coeﬃcient is [1] r012 =

r01 + r12 ei2α , 1 + r01 r12 ei2α

(F.8)

with α = k0 ωdqz,f = dkz,f .2 The quantity qz,jf = Njj ,

(F.9)

is the dimensionless wave vector. (F.8) is rewritten, for convenience, into −1

r012 = r01

n12 + z12 (r01 ) ei2α , n12 + r01 z12 ei2α

(F.10)

and the SP dispersion relation for the ﬁlm system is obtained through the denominator of (F.10) [3] 0 = n12 + r01 z12 ei2α , or

0=

kz,f kz,a + εf εa

kz,b kz,f + εb εf

+

kz,f kz,a − εf εa

(F.11)

kz,b kz,f − εb εf

ei2α .

(F.12) Even for very small ﬁlm thickness d the solution kx , which must satisfy (F.12), is obviously complex-valued. This solution depends on εa , εf , and εb , as well as on d, due to the electromagnetic ﬁeld coupling between the two interfaces, provided that the interfaces are not too far separated from each other.3 1 2 3

The identity kz,i = (ω/c)Nii was used in this derivation. k0 = 2π × 10−4 , if one counts ω in units of cm−1 , and d in units of µm. For d = 0, (F.12) equals (E.14). For d ω/c, and Im{kz,f } > 0, (F.12) decouples into dispersion relations similar to (E.14) for the two individual interfaces “0” − “1” and “1” − “2”.

F TM Waves at Two Stratiﬁed Interfaces

187

For εa = const. = εf = εb , and without loss of generality, the interface f − b is considered here as that which supports the SP excitation.4 Equation (E.14) provides the wave vector component kx = kx0 of the SP mode for this interface. Due to the second interface separated by d, the right side of (F.12) does not vanish at kx = kx0 . For d ω/c, one can assume that the wave vector of the SP mode may now occur at kx = kx0 + ∆kx . Equation (F.12) can then be used to calculate ∆kx upon a linear expansion of all kz,i near kx0 $ % ω 2 2 kz,i = εi − (kx0 + ∆kx ) , (F.13) c 0 kz,i = kz,i −

with

kx0 0 ∆kx , kz,i

1/2 εb εf ω = , c εb + εf ω εi (εb + εf ) − εb εf 0 , kz,i = c εb + εf kx0 εb εf , 0 = kz,i εi (εb + εf ) − εb εf kx0

(F.14)

(F.15)

(F.16) (F.17)

and, besides the thickness d, ∆kx must also be small. Terms proportional to d∆kx , as well as second- and higher-order terms in ∆kx are suppressed.5 The denominator in (F.12) can be expanded near kx0 0 = n12 kx0 + ∆kx + r01 kx0 + ∆kx z12 kx0 + ∆kx exp i2α kx0 + ∆kx , (F.18) which reads more explicitly, using (F.7) and (F.14) & ' 0 0 kx0 kx0 0 n12 kx + ∆kx = εb kz,f + εf kz,b + ∆kx −εb 0 − εf 0 , (F.19) kz,f kz,b n12 kx0 + ∆kx = n012 + δn012 ∆kx , (F.20) & ' 0 k0 k0 0 + ∆kx −εb 0x + εf 0x . (F.21) − εf kz,b z12 kx0 + ∆kx = εb kz,f kz,f kz,b 4

5

The same line of discussion will hold if one considers the interface “0”−“1” to be supportive for an SP exitation. For the ellipsometry experiments discussed in this work, however, the ambient index of refraction is always 1. Therefore, this interface cannot support bound TM waves. This situation will be diﬀerent if one considers a high-index prism experiment, where the prism produces a larger ambient index of refraction. Note that the exponential α also depends on ∆kx . However, for the experimental situation discussed in this work the thickness d is small compared to the wavelength λ = 2πc/ω. Hence, the exponential does not depend on ∆kx in ﬁrst-order approximation.

188

F TM Waves at Two Stratiﬁed Interfaces

0 0 z12 kx0 + ∆kx = z12 + δz12 ∆kx .

(F.22)

The term n012 vanishes because of (E.12). For polar semiconductor materials, and with εa = 1, it can be shown that6 √ εmax − 1 0 0 0 , (F.23) r01 kx0 + ∆kx = r01 + δr01 ∆kx ≈ r01 ≈√ εmax + 1 where εmax = εf if |εf | > |εb | or εmax = εb if |εb | > |εf | near the resonance wavenumber. The new x-component of the SP wave vector is then obtained as −1 0 0 0 z12 i2α0 r01 0 δz12 i2α0 e , (F.24) ∆kx = 1 + r01 0 e δn001 δn12 with α0 = dk 0z,f . Using (F.15,F.17) one can shown that the second term in the bracket can be neglected at kx = kx0 . ∆kx then reads explicitly ω r0 ei2α0 ∆kx = 2 01 c εf − εb 0 0 ∆kx = 2r01 kx

εf εb εf + εb

3/2 .

εf εb ei2α0 . (εf + εb ) (εf − εb )

(F.25) (F.26)

The eﬀect of the ﬁlm is twofold: First, the new SP dispersion crosses neighbouring bulk polariton branches. Second, the new SP wave vector components are complex-valued, as can be easily assessed from (F.26).7 The occurrence of the SP resonance at kx0 + ∆kx aﬀects the p-polarized reﬂectivity. In order to express (F.10) as a function of kx , (F.18) can be written as follows 0 0 i2α0 r01 e + ∆kx δn012 = A0 + ∆kx B 0 , (F.27) z12 −1 0 0 z12 r01 ei2α0 + ∆kx δn012 = A0 + ∆kx B 0 . (F.28) Near resonance, i.e., for D ≡ kx − (kx0 + ∆kx ), the p-polarized reﬂectivity takes the form 0 0 0 0 A + B ∆kx + B D . (F.29) r012 = r01 A0 + B 0 ∆kx + B 0 D $ 0 −2 % ∆kx 1 − r01 +D 0 . (F.30) r012 = r01 D 6

7

0 Rather [4] assumed that r01 varies only slowly with ∆kx , which is easy to show for metals with large negative dielectric function values. It also holds for polar semiconductors, for wave vectors kx near kx0 obtained through both solutions (SP 1) and (SP 2) of (E.14). However, this derivation is more cumbersome, and shall not be given here. Note that the exponential in (F.26) is not symmetrical with respect to the indices “f ” and “b”. Hence, each situation (SP1) and (SP2) deﬁned in Appendix E possesses two subsets, where (A) |εb | > εa and |εf | is small, and (B) |εf | > εa and |εb | is small.

F TM Waves at Two Stratiﬁed Interfaces

189

0 For both cases (SP1) and (SP2) it follows upon (F.23) that r01 ≈ 1 + iγ, and therefore " 0 # 0 −2 = i2γ , (F.31) ≈ i2Im r01 1 − r01

where γ is small. In (F.26) one may introduce F upon writing ∆kx = Fr 001 , where the exponential in F is approximated by εf . (F.32) eiα0 ≈ 1 + i2α0 = 1 + i2dk0 εf + εb In ﬁrst-order approximation F can be treated as a real-valued quantity, for both subsets A and B in situations (SP 1) and (SP 2) deﬁned in Appendix E. Hence, rad +D 0 2iΓ . (F.33) r012 = r01 D With ∆kx = Re {∆kx } + Im {∆kx } = ∆kx + Γ rad , (F.34) and

" # " # kx0 = Re kx0 + Im kx0 = kx0 + Γ i ,

(F.35)

one can rewrite r012 0 r012 = r01

% $ kx − kx0 + ∆kx − i Γ i − Γ rad [kx − (kx0 + ∆kx )] − i (Γ i + Γ rad )

,

and the intensity |r012 |2 ≡ r012 (r012 )∗ near the SP resonance is ! i rad ( 0 (2 Γ 4Γ 2 . |r012 | = (r01 ( 1 − 2 2 [kx − (kx0 + ∆kx )] + (Γ i + Γ rad )

(F.36)

(F.37)

The resonance line in (F.37) is Lorentzian. Its line width is directly proportional to the imaginary parts introduced upon kx0 and ∆kx . The resonance will disappear when either Γ rad or Γ i or both are zero. From (F.15) one can read that either εf or εb must be complex-valued to have Γ i > 0. Hence, near the resonance, a small fraction of EM wave absorption must exist.8 Otherwise, the resonance will not occur.9 Likewise, ∆kx must receive a small imaginary part, mostly upon the phase shift between wave components introduced by the two interfaces separated by d. One can also read from (F.37) that near resonance, the change in reﬂection is extremely sensitive to small fractions of absorption within the ﬁlm. The resonance itself (for the three-phase thinlayer system) occurs at wave numbers where either εf or εb is close to zero 8

9

This absorption, for example, may be due to lattice mode excitation with ﬁnite momentum lifetime due to scattering at free-charge-carrier plasma excitations, at impurities, or with acoustical phonons. This may be seen as the most striking diﬀerence between the occurrence of a thin-ﬁlm supported SP mode, and a thin-ﬁlm interference.

190

F TM Waves at Two Stratiﬁed Interfaces

(the “gap” material with “εgap ”), in order to fulﬁll either of the condition (SP1) or (SP2).10 The Lorentzian shape in |rp012 |2 locates precisely the roots in εgap . Model calculations of the spectral thin-ﬁlm reﬂectivity dependencies using model dielectric function approaches for the materials dielectric properties provide extreme sensitivity to the LO-type frequency and broadening parameters.11

References 1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (NorthHolland Publ. Co., Amsterdam, 1984). 185, 186 2. M. Schubert, Phys. Rev. B 53, 4265 (1996). 185 3. V. M. Agranovich and D. L. Mills, Surface Polaritons (1982). 186 4. H. Raether, Surface Polaritons (Springer, Berlin, 1988). 188

10 11

√ Note that Γ i is proportional to Im{εgap }. The derivations given above are only intended to demonstrate the character of the SP resonance upon several linear approximations made within the Fresnel reﬂection formulae. Equation (F.37) approximates (F.8) near the SP resonance only. For infrared (ellipsometry) data analysis numerical evaluation is performed using the matrix method discussed in Chap. 2.

Index

4 × 4 matrix approach

18

AlGaAs 63 AlGaInAs 90 AlGaInP 85, 90, 161 AlGaN/GaN superlattices 111 AlInP 92 alloy-induced low polarity modes 87 Alloying 85 AlN 129 ambient 19 ambient matrix 19 angular dispersion 18 anharmonic phonon mode coupling 33 anisotropic materials 17 atmospheric absorption 25 atomic ordering 85 bands of total reﬂection 67, 116 Berreman eﬀect 62 Berreman-formalism 18, 165 biaxial materials 17 Biaxial strain 131 biaxial strain 104, 129 BInGaAs 159 Brewster modes 48, 51 broadening 32 bulk polaritons 47 Cartesian coordinate 15 center-of-gravity system 16, 67 Composition 96 composition 85, 128, 131 concentration 36 conductivity tensor 39 Constitutive relations 46 critical ﬁlm thickness 132 CuPt-type 90

cyclotron frequency 148 cyclotron frequency tensor cyclotron resonance 148

40

deformation potential constants 129 degenerate 82 degree of ordering 85, 90 degree of polarization 8 density-of-states eﬀective 82 depolarization 9 dielectric displacement 46 dielectric function spectra 20 dielectric function tensor 17 Dielectric helical thin ﬁlms 173 dielectric loss function 32, 116 disordered material 90 dispersion relation 47 Drude equation 36 Drude’s 35 eﬀective density of states 82 eﬀective free-charge-carrier mass 35 eﬀective mass 36, 123 elastic constants 104, 129 electric ﬁeld 46 electrical (DC) Hall mobility 84 electron-momentum dispersion 147 ellipsometric parameters 8 Ellipsometry 7 epitaxial relationship 117 Euler angles 15 exit matrix 19 Fano modes 51 far infrared 36 Fermi energy 82 Fermi function 81 Fermi-Dirac integral 82 Fermi-level pinning 126

192

Index

free charge carriers 35 free-charge-carrier detection limit 137 free-charge-carrier detection limits 127 Fresnel reﬂection coeﬃcient 50 GaAs 37, 81, 154 GaInP 91 GaN 37, 111, 123 GaNP 96 generalized ellipsometry 12, 22, 67, 118, 154, 155 generalized ellipsometry parameter 119 generalized ellipsometry parameters 12, 120 global transfer matrix 18 globar 25 Gr¨ uneisen parameter 129 harmonic oscillator 32 high-frequency dielectric constant 33 hydrostatic compression 130 impurity modes 38, 125 InAlN 128 incidence angle 7 inertial-mass 35, 147 infrared 36 infrared ellipsometer 26 infrared ellipsometry 23 InGaAsP 90 InGaNAs 100 InN 111 interface modes 45 inverse eﬀective mass tensor 40 inverse relaxation time tensor 40 isotropic materials 18 Jones matrix 8 Jones matrix elements 8 Jones reﬂection and transmission matrices 9 Kretschmann-conﬁguration

61

laser diode (LD) structures Lattice match 91

134

32,

lattice relaxation 133 LiF 62 light line 49 light-emitting diode (LED) structure 134 longitudinal magneto-optic setup 150 longitudinal optical (LO) phonon 31 longitudinal plasmon-phonon-magnon modes 148 low polarity modes 38 Lyddane-Sachs-Teller relation 34 magnetic ﬁeld 40, 46 magnetic induction 46 magneto-optic 39, 149 magneto-optic birefringence 147 magneto-optic materials 19, 171 magneto-optic plasma tensor 39, 147 Maxwell’s postulates 45, 46 metal organic vapor phase epitaxy 90 mirror eﬀects 154 MO setups 150 mobility 36 mode coupling 77 Model Dielectric Functions 32 molecular beam epitaxy 90, 123 Mueller matrix 9 Mueller matrix for an isotropic sample 11 multiple phonon mode materials 33 Multiple-component free-charge-carrier excitations 81 multiple-sample analysis 21 non-doped intrinsic GaAs 47 non-symmetric dielectric properties 171 nondegenerate 81 nondepolarizing system 10 normalized Jones matrix elements 12 oblique magneto-optic setup 150 oblique-magneto-optic (OMO) setup 152 Ohm’s law 40 optical axis 92 optical mass 81 Ordering 85 orthogonal rotation matrix 15 Otto-conﬁguration 61

Index overdamping

37

phonon branch 32 phonon decay times 33 phonons 31 photon energy 20 plane of incidence 7 plasma broadening 147 plasma excitation frequency 36 plasma frequency 147 plasma frequency tensor 40 plasmon broadening 36 plasmon-phonon-modes 35 plasmons 35 point-by-point ﬁts 21 polar magneto-optic (PMO) setup 151 polar magneto-optic setup 150 Polaritons 45 polarizability functions 67 polarizability tensor 16 polarization state 7 polarizing condition 14 pseudodielectric function 8 Pseudomorphic growth 131 Pulsed Laser Deposition 112 quantum well

134

Raman scattering 89 relaxation time 36 reststrahlen bands 68 sample rotation 78 Sapphire 68 sapphire 17, 70, 112 SGW 58 SiC 62 silent modes 110 silicon carbide 17, 25 single-wavelength ellipsometry 20 skew cut 78 Snell’s law 19 spatial dispersion 47 spectroscopic ellipsometry 20, 21 standard ellipsometry 11, 22, 67 static dielectric constant 33 stibnite 68 Stokes parameters 9

193

Stokes vector 9 Strain 85 strain 96, 128 substrate dielectric function 67 superlattice (SL) structure 134 surface depletion layer 126 Surface guided waves 48 surface guided waves 45 Surface polaritons 48 surface polaritons 45 temporal dispersion 46 test function 22 titanium dioxide 17 total reﬂection 68 total-internal-reﬂection 61 transfer matrix 18 transfer matrix Tp for dielectric helical ﬁlms 173 transfer matrix Tp for isotropic ﬁlms 165 transfer matrix Tp for dielectric homogeneous ﬁlms 169 transfer matrix Tp for magneto-optic ﬁlms 172 transparent conductive oxides 112 transverse electric 48 transverse magnetic 48 transverse magneto-optic setup 150 transverse optical (TO) phonon 31 uniaxial materials valence band

17

81

wave frequency 46 wave vector 46 wave vector equation 46 wavelenght-by-wavelength 121 wavelength-by-wavelength 157 Wavelength-by-wavelength ﬁts 21 wavenmuber-by-wavenumber 76 wurtzite III-nitrides 109 wurtzite structure 110 wurtzite-structure 110 zincblende III-V compound semiconductors 31 ZnO 111

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