Photoemission from Optoelectronic Materials and their Nanostructures (Nanostructure Science and Technology)

Nanostructure Science and Technology

A volume in the Nanostructure Science and Technology series. Further titles in the series can be found at: http://www.springer.com/series/6331

Kamakhya Prasad Ghatak · Debashis De · Sitangshu Bhattacharya

Photoemission from Optoelectronic Materials and their Nanostructures

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Kamakhya Prasad Ghatak Department of Electronic Science The University of Calcutta Kolkata, West Bengal 700009 India [email protected]

Debashis De Department of Computer Science and Engineering West Bengal University of Technology Salt Lake City Kolkata 700064 India [email protected]

Sitangshu Bhattacharya Nano Scale Device Research Laboratory Centre for Electronics Design and Technology Indian Institute of Science Bangalore 560012 India [email protected]

ISBN 978-0-387-78605-6 e-ISBN 978-0-387-78606-3 DOI 10.1007/978-0-387-78606-3 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009928695 © Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Professor Somenath Sarkar for his invaluable inspiration, selfless advice and research-proactive leadership

Preface

The creation of quantized structures like quantum wells (QWs) in ultrathin films (UFs), quantum well wires (QWWs), and quantum dots (QDs) is due to two factors: • reduction of the symmetry of the wave-vector space of the charge carriers in electronic materials having various band structures, and • emergence of modern fabrication techniques like molecular beam epitaxy (MBE), metal organic chemical vapor deposition (MOCVD), fine line lithography (FLL), etc. Quantized structures have garnered much interest in nanoscience because of their promise for unearthing both new scientific revelations and new technological applications. In QWs in UFs, the quantization of the motion of the carriers in the direction perpendicular to the film exhibits the two-dimensional behavior of the charge carriers. Another new structure known as a QWW has been proposed to investigate the physical properties in these materials where the carrier gas is quantized in two transverse directions and they can move only in the longitudinal direction. As the concept of quantization increases from 1D to 3D, the degree of freedom of the free carriers annihilates totally, and the density-of-states (DOS) function changes from Heaviside step function to the Dirac’s delta function forming QDs. An enormous range of important applications of such nanostructures for nanoscience in the quantum regime together with a rapid increase in computing power, have generated considerable interest in the study of the optical properties of quantum effect devices. Examples of such new applications include quantum resistors, resonant tunneling diodes, quantum switches, quantum sensors, quantum logic gates, quantum transistors, optical switching systems, etc. Although many new effects in quantized structures have already been reported, the interest in further research of other aspects of such quantum-confined materials is becoming increasingly important. One such significant property is photoemission, which is a physical phenomenon, and its importance has already been established since the inception of Einstein’s photoelectric effect (for which Einstein won Nobel Prize in 1921), which in recent years finds extensive applications in modern optoelectronics, photoemission spectroscopy and related aspects in connection with nanostructures. It is well known that the photo-emitted current density depends on the DOS function which, in turn, affects significantly the different physical properties of various vii

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materials. Photoemission from different materials having degenerate carrier concentration is determined by their respective energy band structures. It exhibits different values in different materials and varies with the incident photon energy, the carrier concentration, the quantizing electric field, the magnitude of the reciprocal quantizing magnetic field under magnetic quantization, the nano-thickness as in QWs in UFs, QWWs and QDs, superlattice period as in the quantum confined superlattices having various carrier energy spectra, etc. It is noteworthy in this regard that the available books, review articles and monographs on optoelectronics and related topics in general cannot even afford to devote an entire chapter to photoemission from different materials and their nanostructures. This monograph is based on our investigations of photoemission for the last twenty years and an attempt has been made to present the same from a plethora of materials and their nanostructures having different energy-wave vector dispersion relations of the carriers under various physical conditions. In Chapter 1, the fundamentals of photoemission have been investigated from bulk samples of wide gap materials having parabolic energy bands and also under magnetic quantization, cross fields configuration, QWs in UFs, QWWs, QDs, and magneto size quantizations, respectively. Since Iijima’s discovery (S. Iijima, Nature 354, 56 [1991]), carbon nanotubes (CNs) have been recognized as an important quantum material, which has generated new research avenues in the areas of nanoscience and technology in general. In Chapter 2, the elementary theory of photoemission from QWs in UFs and QWWs of nonlinear optical, III–V, II–VI, nGallium Phosphide, n-Germanium, Platinum Antimonide, stressed materials, Bismuth and carbon nanotubes has been presented. Chapter 3 contains an analysis of photoemission from QDs of nonlinear optical, III–V, II–VI, n-Gallium Phosphide, n-Germanium, Tellurium, Graphite, Platinum Antimonide, zero-gap, Lead Germanium Telluride, Gallium Antimonide, stressed materials, Bismuth, IV–VI, II–V, Zinc and Cadmium diphosphides, Bismuth Telluride and Antimony on the basis of their respective carrier energy spectra. Semiconductor superlattices (SLs) enjoy extensive applications in optoelectronics, and in Chapter 4 photoemission has been investigated from III–V, II–VI, IV–VI, and HgTe/CdTe quantum well superlattices (QWSL) with graded interfaces under magnetic quantization. The same chapter also includes photoemission from III–V, II–VI, IV–VI, and HgTe/CdTe quantum well effective mass superlattices under magnetic quantization, together with the quantum dots of the aforementioned microstructures. It is worth remarking that, in the methods as given in the literature, the physics of photoemission has been incorporated in the lower limit of the photoemission integral, and assumes that the band structure of the bulk materials becomes an invariant quantity in the presence of the photo-excitation necessary for photoelectric effect. The basic band structure of optoelectronic materials changes in the presence of external light waves in a fundamental way, which has been incorporated mathematically through the expressions of the DOS function and the velocity along the direction of photoemission respectively, in addition to the appropriate fixation of the lower limit of the photo-emission integral for the purpose of investigating the photoemission from bulk specimens of optoelectronic materials in Chapter 5.

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An important concept highly relevant to the measurement of bandgap in electronic materials in the presence of external photo-excitation has also been discussed in this perspective. The effects of quantizing magnetic field on the band structures of optoelectronic materials are easily observed in experiments. Various important physical features originate from the significant changes in the carrier energy spectra of the materials under magnetic quantization. In Chapter 6, photoemission has been investigated under magnetic quantization from optoelectronic materials on the basis of the concept presented in Chapter 5. Chapter 7 covers the study of the photoemission from QWs in UFs, QWWs, and QDs of optoelectronic materials as an extension of the new dispersion relations of the bulk optoelectronic materials investigated in Chapter 5. In chapters 8 and 9, the photoemission from quantum confined effective mass superlattices of optoelectronic materials and quantum confined superlattices of optoelectronic materials with graded interfaces have respectively been presented in the presence of external photo-excitation. Chapter 10 briefly discusses experimental results, in which we have proposed a single multidimensional open research problem for experimentalists regarding multi-photon photoemission from quantized materials having various band structures under different physical conditions. The book ends with Chapter 11, which contains the conclusion and outlines the scope for future research. To the best of our knowledge, there is no other book devoted totally on the photoemission from optoelectronic compounds and their nanostructures together with the other technologically important macro and micro materials. We earnestly hope that the present book will be a useful reference not only for the present and the next generation of readers but also for researchers in materials science, optoelectronics, and related fields. It is needless to say that the production of an error-free first edition of any book from every point of view is practically impossible, in spite of our joint efforts. We will be grateful to our readers for their constructive criticisms, and will incorporate them in future editions. From Chapter 5 through the end, we have presented 125 open research problems in this important topic, which will be useful not only for alert readers but also for PhD aspirants who wish to contribute to different aspects of photoemission from quantized structures. These open research problems form an integral part of this book, and it will also be useful in graduate courses on modern optoelectronic devices in many universities and institutes. We are confident that our esteemed readers will enjoy the extensive investigations of photoemission from low-dimensional nonlinear optical, III–V, II–VI, Gallium Phosphide, Germanium, Platinum Antimonide, zero-gap, stressed, Bismuth, carbon nanotubes, Gallium Antimonide, IV–VI, Lead Germanium Telluride, Graphite, Tellurium, II–V, Zinc and Cadmium diphosphides, Bismuth Telluride, Antimony, III–V, II–VI, IV–VI, and HgTe/CdTe quantum well superlattices with graded interfaces under magnetic quantization, III–V, II–VI, IV–VI, and HgTe/CdTe effective mass superlattices under magnetic quantization, the quantum dots of the aforementioned superlattices, quantum confined effective mass superlattices and superlattices of optoelectronic materials with graded interfaces on the basis of appropriate respective dispersion relations. The experimental investigations of photoemission from the

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aforementioned nanostructures are relatively less common in the literature, although such studies will provide the key to unlock the band structures of quantized materials which, in turn, control the transport phenomena in such quantum effect devices in the presence of external photo-excitation. Various mathematical analyses and a few chapters of this book are appearing here for the first time. Finally, we hope that our joint efforts will fire the imagination to initiate frontier line researches on this fascinating subject by anyone engaged either in research or in industries connected with nanostructured optoelectronics in general.

Acknowledgments Acknowledgment by Kamakhya Prasad Ghatak I am grateful to G. P. Agarwal for triggering my interest in the active field of nanophotonics while delivering an invited lecture at the University of Calcutta. I express my gratitude to D. Bimberg, V. S. Letokhov, W. L. Freeman, and H. L. Hartnagel for various kinds of academic advice during the last two decades. I am indebted to S. Chatterjee, A. Mallick, and A. Karmakar for numerous academic discussions. I am grateful to S. C. Dasgupta for teaching an engineering student the methods of theoretical physics and inspiring me to solve independently the problems from the two-volume classics of Morse-Feshbach 35 years ago. I simultaneously offer my special thanks to S. S. Baral for creating the passion for theoretical acoustics from the fundamental works of Morse and Ingard. The late P. N. Butcher encouraged me repeatedly to write a research trilogy on the band structure-dependent properties of nanostructured materials. To honor him, we present The Einstein Relation in Compound Semiconductors and Their Nanostructures in the Springer Series in Materials Science, Vol. 116, as the first one; the present book as the second one; and the thermoelectric power in nanostructured materials under strong magnetic field as the third one, which will be published in the Springer Series in Materials Science in 2010, to complete the trilogy as our tribute to Late P. N. Butcher, a thorough scientist in the real sense of the term. I must acknowledge the hidden contribution of my numerous research students who at present hold positions of repute in various academic institutions. My family members deserve a very special mention for really forming the backbone of my long unperturbed research career. I am grateful to my young PhD students not only for confining me in the infinitely deep quantum wells of Ramanujan and Rumi, but also for inspiring me to teach quantum mechanics and related topics from the eightvolume classics of Greiner et al. P. K. Sarkar and S. Bania of my department always tranquilize me at rather difficult moments. In addition, this book has been completed under grant 8023/B0R/RID/RPS-95/2007-08 as sanctioned by the All India Council for Technical Education in their research promotion scheme 2008.

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Acknowledgment by Sitangshu Bhattacharya It is virtually impossible to express my gratitude to all the admirable persons who have influenced my academic and social life from every point of view; nevertheless, a short memento to my teachers S. Mahapatra and H. S. Jamadagni, at the Centre for Electronics Design and Technology, A. K. Sood, at the Department of Physics and M. S. Hegde, at the Solid State and Structural Chemistry Unit and presently the Dean of Science at Indian Institute of Science, Bangalore, for their fruitful academic advices and guidance remains everlasting. I am indebted to my sister Ms. S. Bhattacharya for her belief in me and my work, which still stimulates and amplifies my efficiency for performing in-depth research. I am grateful in the real sense of the term to my teacher K. P. Ghatak for instilling in me the monochromatic idea that the performing good research containing fundamental and innovative concepts is the keystone to excel in creative research activity. The author S. B. is grateful to the DST, India for sanctioning the research grant under the proposal “Fasttrack scheme for young scientist” 2008–2009 having the reference number SR/FTP/ETA-37/08 under which the simulations have been completed. Finally and eternally, I believe our Mother Nature has propelled this joint collaboration in her own unseen way in spite of several insurmountable obstacles.

Acknowledgment by Debashis De I express my gratitude to K. P. Ghatak, S. N. Sarkar, S. Sengupta, P. K. Roy, and A. K. Sen for their constructive academic support. I am highly indebted to my brother S. De for his constant inspiration and mental support. I am grateful to the All India Council for Technical Education for granting me the aforementioned project jointly in their research promotion scheme 2008, under which this book has been completed.

Joint Acknowledgments We are indebted to Dr. K. Howell, Senior Editor, Springer; Dr. D. J. Lockwood, Series Editor, Nanostructure Science and Technology; Dr. A. Greene, Editorial Director, Springer; and C. Balmes, Editorial Assistant, Springer, for their priceless technical assistance. We are grateful to Ms. S. Roy and A. Saha, the two prominent young members of our research team, for their overall help from every point of view without which the writing of this book would be a mere dream. Naturally, the authors are responsible for nonimaginative shortcomings. Kolkata, India Bangalore, India Kolkata, India November 2008

K. P. Ghatak D. De S. Bhattacharya

Contents

1 Fundamentals of Photoemission from Wide Gap Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Theoretical Background . . . . . . . . . . . . . . . . . . 1.2.1 Photoemission from Bulk Semiconductors . . . . 1.2.2 Photoemission Under Magnetic Quantization . . . 1.2.3 Photoemission in the Presence of Cross Fields . . 1.2.4 Photoemission from Quantum Wells in Ultrathin Films of Wide Gap Materials . . . . . . . . . . . 1.2.5 Photoemission from Quantum Well Wires of Wide Gap Materials . . . . . . . . . . . . . . . . 1.2.6 Photoemission from Quantum Dots of Wide Gap Materials . . . . . . . . . . . . . . . . . . . 1.2.7 Photoemission Under Magneto-Size Quantization (MSQ) . . . . . . . . . . . . . . . . . . . . . . . 1.3 Results and Discussions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Fundamentals of Photoemission from Quantum Wells in Ultrathin Films and Quantum Well Wires of Various Nonparabolic Materials . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . 2.2.1 Photoemission from Nonlinear Optical Materials . . . . 2.2.2 Photoemission from III–V Materials . . . . . . . . . . 2.2.3 Photoemission from II–VI Compounds . . . . . . . . . 2.2.4 Photoemission from n-Gallium Phosphide . . . . . . . 2.2.5 Photoemission from n-Germanium . . . . . . . . . . . 2.2.6 Photoemission from Platinum Antimonide . . . . . . . 2.2.7 Photoemission from Stressed Materials . . . . . . . . . 2.2.8 Photoemission from Bismuth . . . . . . . . . . . . . . 2.2.9 Photoemission from (n, n) and (n, 0) Carbon Nanotubes 2.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Fundamentals of Photoemission from Quantum Dots of Various Nonparabolic Materials . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . 3.2.1 Photoemission from Nonlinear Optical Materials . . . 3.2.2 Photoemission from III–V Materials . . . . . . . . . 3.2.3 Photoemission from II–VI Materials . . . . . . . . . 3.2.4 Photoemission from n-Gallium Phosphide . . . . . . 3.2.5 Photoemission from n-Germanium . . . . . . . . . . 3.2.6 Photoemission from Tellurium . . . . . . . . . . . . 3.2.7 Photoemission from Graphite . . . . . . . . . . . . . 3.2.8 Photoemission from Platinum Antimonide . . . . . . 3.2.9 Photoemission from Zero-Gap Materials . . . . . . . 3.2.10 Photoemission from Lead Germanium Telluride . . . 3.2.11 Photoemission from Gallium Antimonide . . . . . . . 3.2.12 Photoemission from Stressed Materials . . . . . . . . 3.2.13 Photoemission from Bismuth . . . . . . . . . . . . . 3.2.14 Photoemission from IV–VI Materials . . . . . . . . . 3.2.15 Photoemission from II–V Materials . . . . . . . . . . 3.2.16 Photoemission from Zinc and Cadmium Diphosphides 3.2.17 Photoemission from Bismuth Telluride . . . . . . . . 3.2.18 Photoemission from Quantum Dots of Antimony . . . 3.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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107 107 109 109 110 120 121 122 124 126 128 129 131 132 137 138 142 146 147 149 150 152 170

4 Photoemission from Quantum Confined Semiconductor Superlattices 173 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . 174 4.2.1 Magneto-photoemission from III–V Quantum Well Superlattices with Graded Interfaces . . . . . . . . . 174 4.2.2 Magneto-Photoemission from II–VI Quantum Well Superlattices with Graded Interfaces . . . . . . . . . 179 4.2.3 Magneto-Photoemission from IV–VI Quantum Well Superlattices with Graded Interfaces . . . . . . . . . 181 4.2.4 Magneto-Photoemission from HgTe/CdTe Quantum Well Superlattices with Graded Interfaces . . . 185 4.2.5 Magneto-Photoemission from III–V Quantum Well Effective Mass Superlattices . . . . . . . . . . . . . 186 4.2.6 Magneto-Photoemission from II–VI Quantum Well Effective Mass Superlattices . . . . . . . . . . . . . 188 4.2.7 Magneto-Photoemission from IV–VI Quantum Well Effective Mass Superlattices . . . . . . . . . . . . . 191 4.2.8 Magneto-Photoemission from HgTe/CdTe Quantum Well Effective Mass Superlattices . . . . . . . . 193

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4.2.9

Photoemission from III–V Quantum Dot Superlattices with Graded Interfaces . . . . . . . 4.2.10 Photoemission from II–VI Quantum Dot Superlattices with Graded Interfaces . . . . . . . 4.2.11 Photoemission from IV–VI Quantum Dot Superlattices with Graded Interfaces . . . . . . . 4.2.12 Photoemission from HgTe/CdTe Quantum Dot Superlattices with Graded Interfaces . . . . . . . 4.2.13 Photoemission from III–V Quantum Dot Effective Mass Superlattices . . . . . . . . . . . . . . . . . 4.2.14 Photoemission from II–VI Quantum Dot Effective Mass Superlattices . . . . . . . . . . . . . . . . . 4.2.15 Photoemission from IV–VI Quantum Dot Effective Mass Superlattices . . . . . . . . . . . . 4.2.16 Photoemission from HgTe/CdTe Quantum Dot Effective Mass Superlattices . . . . . . . . . . . . 4.3 Results and Discussions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Photoemission from Bulk Optoelectronic Materials 5.1 Introduction . . . . . . . . . . . . . . . . . . . 5.2 Theoretical Background . . . . . . . . . . . . . 5.3 Results and Discussions . . . . . . . . . . . . . 5.4 Open Research Problems . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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6 Photoemission under Quantizing Magnetic Field from Optoelectronic Materials . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.2 Theoretical Background . . . . . . . . . . . . . . . 6.3 Results and Discussions . . . . . . . . . . . . . . . 6.4 Open Research Problems . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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Photoemission from Quantum Wells in Ultrathin Films, Quantum Wires, and Dots of Optoelectronic Materials . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 7.2 Theoretical Background . . . . . . . . . . . . . . . . . 7.2.1 Photoemission from Quantum Wells in Ultrathin Films of Optoelectronic Materials . . . . . . . . 7.2.2 Photoemission from Quantum Well Wires of Optoelectronic Materials . . . . . . . . . . . 7.2.3 Photoemission from Quantum Dots of Optoelectronic Materials . . . . . . . . . . . 7.3 Results and Discussions . . . . . . . . . . . . . . . . .

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7.4 Open Research Problems . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Photoemission from Quantum Confined Effective Mass Superlattices of Optoelectronic Materials . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 8.2 Theoretical Background . . . . . . . . . . . . . . . . . 8.2.1 Magneto-Photoemission from Quantum Well Effective Mass Superlattices . . . . . . . . . . . 8.2.2 Photoemission from Effective Mass Quantum Well Wire Superlattices . . . . . . . . . . . . . 8.2.3 Photoemission from Quantum Dots of Effective Mass Superlattices . . . . . . . . . . . . . . . . 8.2.4 Magneto-Photoemission from Effective Mass Superlattices . . . . . . . . . . . . . . . . 8.3 Results and Discussions . . . . . . . . . . . . . . . . . 8.4 Open Research Problems . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 Photoemission from Quantum Confined Superlattices of Optoelectronic Materials with Graded Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . 9.2.1 Magneto Photoemission from Quantum Well Superlattices . . . . . . . . . . . . . . . . . . . 9.2.2 Photoemission from Quantum Well Wire Superlattices 9.2.3 Photoemission from Quantum Dot Superlattices . . . 9.2.4 Magneto-Photoemission from Superlattices of III-V Optoelectronic Materials . . . . . . . . . . . . 9.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . 9.4 Open Research Problems . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Review of Experimental Results 10.1 Experimental Works . . . . 10.2 Open Research Problem . . References . . . . . . . . . . . .

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Conclusion and Future Research . . . . . . . . . . . . . . . . . . . 11.1 Open Research Problems . . . . . . . . . . . . . . . . . . . . .

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Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Materials Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Symbols

α α0 δ δ || ⊥  o λ ε εsc ε0 ζ (2r)  (j + 1) α0 φ ω0 υ a ac a0 ,b0 − → A B B2 c C0 C1 C2 dx , dy , dz e E EF EFB Eo

Band nonparabolicity parameter Optical absorption coefficient Crystal field splitting constant Dirac’s delta function Spin-orbit splitting constants parallel Spin-orbit splitting constants perpendicular to the C-axis Isotropic spin-orbit splitting constant Interface width in superlattices Wavelength Trace of the strain tensor/energy as measured from the center of the band gap Semiconductor permittivity Permittivity of vacuum Zeta function of order 2r Complete Gamma function Probability of photoemission Work function Cyclotron resonance frequency Frequency The lattice constant Nearest neighbor C–C bonding distance The widths of the barrier and the well for superlattice structures Vector potential Quantizing magnetic field Momentum matrix element Velocity of light Splitting of the two-spin states by the spin orbit coupling and the crystalline field Conduction band deformation potential Strain interaction between the conduction and valance bands Nanothickness along the x, y, and z directions Magnitude of electron charge Total energy of the carrier Fermi energy Fermi energy in the presence of magnetic field Electric field

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xviii E0 Enz EF0D Eg 0 EB EQD E FL Eg EFBL EF2DL Eg EF1D Fj (η) f (E) f0 G0 gv h 

H I ˆi, ˆj and kˆ J k0 I0 J J2D kB k lx ¯l, m, ¯ n¯ L0 m0 m∗ m∗|| m∗⊥ m1 m2 m3 m2 m∗t m∗l m∗⊥,1 , m∗||,1 mr mv

List of Symbols Ground state energy of the electron in the presence of crossed electric and magnetic field Energy of the nth subband 0D Fermi energy Unperturbed band gap Bohr electron energy Totally quantized energy Fermi energy in the presence of light waves Perturbed band gap Fermi energy under quantizing magnetic field in the presence of light waves Fermi energy in QWs in UFs in the presence of light waves Increased band gap Fermi energy in the presence of 2D quantization One parameter Fermi-Dirac integral of order j Fermi Dirac occupation probability factor Optical frequency Nonlinear response from the optical excitation of the free carriers Valley degeneracy Planck’s constant Dirac’s constant Heaviside step function Photocurrent Orthogonal triads photoelectric current density Inverse Bohr radius Light intensity Bulk photoemission current density 2D photocurrent density Boltzmann’s constant Electron wave vector Sample length along x direction Matrix elements of the strain perturbation operator Superlattices period length Free electron mass Isotropic effective electron masses at the edge of the conduction band Longitudinal effective electron masses at the edge of the conduction band Transverse effective electron masses at the edge of the conduction band Effective carrier masses at the band-edge along x direction Effective carrier masses at the band-edge along y direction Effective carrier masses at the band-edge along z direction Effective mass tensor component at the top of the valence band (for electrons) or at the bottom of the conduction band (for holes) The transverse effective masses at k = 0 The longitudinal effective masses at k = 0 Transverse and longitudinal effective electron masses at the edge of the conduction band for the first material in superlattice Reduced mass Effective mass of the heavy hole at the top of the valance band in the absence of any field

List of Symbols N (E) Nc nx , ny , nz n0 N2DT (E) N1DT (E) n1D n2D N0DT (E) n0D n P0 T tc ν (E) v, i υ0 V0 W x, y

xix Density of states in bulk specimens Effective number of states in the conduction band Size quantum numbers along the x, y, and z directions Total bulk electron concentration Total 2D density of states function Total 1D density of states function Electron concentration per unit length Electron concentration per unit area Total 0D density of states function 0D electron concentration per subband Landau quantum number/chiral indices Momentum matrix element Temperature Tight binding parameter Velocity of the emitted electron Integer Threshold frequency Potential barrier Electron affinity Alloy compositions

Chapter 1

Fundamentals of Photoemission from Wide Gap Materials

1.1 Introduction It is well known that the photoelectric effect occupies a singular position in the whole arena of materials science and related disciplines in general together with the fact that the photoemission from the electronic materials is also a vital physical phenomenon from the viewpoint of modern optoelectronics and photoemission spectroscopy [1]. The classical  equation of the photo-emitted current density is [2]  J = 4πem∗ gv (kB T)2 / h3 exp [(hν − φ) / (kB T)], where e, m∗ , gv , kB , T, h, hυ and φ are the electron charge, effective electron mass at the edge of the conduction band, valley degeneracy, the Boltzmann constant, temperature, the Planck’s constant, incident photon energy along z-axis and work function, respectively. The aforementioned equation is valid for both the charge carriers, and in this conventional form it appears that the photoemission changes with the effective mass, temperature, work function, and the incident photon energy, respectively. This relation holds only under the condition of carrier nondegeneracy. The photoemission has different values for different materials and varies with doping and with external fields, which creates quantization of the wave-vector space of the carriers, leading to various types of quantized structures. The nature of these variations has been studied [2–35], and some of the significant features are as follows: 1. The photoemission from bulk materials increases with the increase in doping. 2. The photoemission exhibits oscillatory dependence with inverse quantizing magnetic field because of the Shubnikov–de Haas (SdH) effect. 3. The photoemission changes significantly with the magnitude of the externally applied quantizing electric field in electronic materials. 4. The photoemission from quantum confined Bismuth, nonlinear optical, III–V, II– VI, and IV–VI materials oscillate with nanothickness in various manners which are totally band structure–dependent. 5. The nature of the variations is significantly influenced by the energy band constants of various materials having different band structures. 6. The photoemission has significantly different values in quantum confined semiconductor superlattices and various other quantized structures. K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_1,  C Springer Science+Business Media, LLC 2009

1

2

1 Fundamentals of Photoemission

In Section 1.2.1 of the theoretical background, an elementary analysis of the photoemission from bulk specimens of wide gap semiconductors having parabolic energy bands is presented. It is well known that the band structure of any electronic material changes fundamentally in the presence of external fields [36]. The effects of the quantizing magnetic field on the energy wave vector dispersion relation of the charge carriers of small gap semiconductors can be observed easily in experiments. It is well known that under magnetic quantization, the motion of the charge carrier parallel to the magnetic field remains unaltered, while the area of the wave vector space normal to the direction of the magnetic field gets quantized in accordance with the Landau rule of area quantization in the wave-vector space of the charge carriers [37]. The physics of the Landau levels is the signature of the concept of singularity of complex variables and the quantized energies are known as Landau sub-bands. The quantizing magnetic field tends to remove the degeneracy and increases the band gap. A semiconductor, placed in a magnetic field B, can absorb the radiative energy with the frequency ω0 (≡ eB / m∗ ). The effect of energy quantization is experimentally noticeable when the separation between any two consecutive Landau levels is greater than kB T. A number of interesting transport phenomena originate from the change in the basic band structure of the semiconductor under magnetic quantization, and these have been widely investigated and have also served as diagnostic tools for characterizing the different materials having different carrier dispersion laws. The discreteness in the Landau levels leads to a whole crop of magneto-oscillatory phenomena, important among which are (i) Shubnikov–de Haas oscillations in magneto-resistance; (ii) de Haas–Van Alphen oscillations in magnetic susceptibility; (iii) magneto-phonon oscillations in thermoelectric power, etc. In Section 1.2.2, the photoemission from the semiconductors having parabolic energy bands under magnetic quantization is explored. The effect of crossed electric and quantizing magnetic fields on the transport properties of different semiconducting compounds has been relatively less investigated compared to the corresponding magnetic quantization, although the cross fields studies are very important with respect to the experimental findings [38– 42]. It is well known that in the presence of electric field (E0 ) along the x-axis and the quantizing magnetic field B along the z-axis, the dispersion relations of the conduction electrons in semiconductors become modified in a fundamental way and the electron moves in both the z and y directions. The motion along the y direction is purely due to the presence of E0 along the x-axis, in the absence of which the effective electron mass along the y-axis becomes infinite, indicating the fact that the electron motion along the y-axis is prohibited. The effective electron mass of the isotropic, bulk semiconductors having parabolic energy bands under cross fields configuration exhibits mass anisotropy which, in turn, depends on the electron energy, the magnetic quantum number, the electric and the magnetic fields, respectively. In Section 1.2.3, the photoemission under cross field configuration from semiconductors having parabolic energy bands is investigated. In recent years, with the advent of fine lithographical methods [43], molecular beam epitaxy [44], metal-organic chemical vapor deposition [45], and other experimental techniques, the restrictions of the motion of the carriers of bulk materials in one (QWs in UFs, NIPI structures, inversion and accumulation layers), two

1.1

Introduction

3

(QWWs), and three (QDs, magneto-size quantized systems, magneto inversion layers, magneto accumulation layers, quantum dot superlattices, magneto quantum well superlattices, and magneto NIPI structures) dimensions have in the last few years attracted much attention not only for their potential in uncovering new phenomena in nanostructured science and technology but also for their interesting quantum device applications [46–48]. In QWs in UFs, the restriction of the motion of the carriers in the direction normal to the film (say, the z direction) may be viewed as carrier confinement in an infinitely deep 1D rectangular potential well, leading to quantization (known as quantum size effect [QSE]) of the wave vector along the direction of the potential well, allowing 2D electron transport parallel to the surface of the film representing new physical features not exhibited in bulk semiconductors [49]. The low-dimensional hetero-structures based on various materials are widely investigated because of the enhancement of carrier mobility [50]. These properties make such structures suitable for applications in quantum well lasers [51], hetero-junction FETs [52], tunneling hot-electron transfer amplifier (THETA) devices [53], high-frequency microwave circuits [54], optical modulators [55], optical switching systems [56], and other devices. The constant energy 3D wave-vector space of bulk semiconductors becomes a 2D wave-vector surface in QWs in UFs because of dimensional quantization. Thus, the concept of reduction of symmetry of the wave-vector space is needed to create low-dimensional structures. In Section 1.2.4, the detailed formulation of photoemission from QWs in UFs of wide gap materials is presented. In QWWs, the restriction of the motion of the carriers along two directions may be viewed as carrier confinement by two infinitely deep 1D rectangular potential wells, along any two orthogonal directions leading to quantization of the wave vectors along the said directions, allowing 1D electron transport. With the help of modern fabricational techniques, such one-dimensional quantized structures have been experimentally realized and enjoy an enormous range of important applications in the realm of nanoscience in the quantum regime. They have generated much interest in the analysis of nanostructured devices for investigating their electronic, optical, and allied properties [57–60]. Examples of such new applications are based on the different transport properties of ballistic charge carriers which include quantum resistors [61–66], resonant tunneling diodes and band filters [67, 68], quantum switches [69], quantum sensors [70–72], quantum logic gates [73–74], quantum transistors and subtuners [75–77], hetero-junction FETs [78], high-speed digital networks [79], high-frequency microwave circuits, optical modulators, optical switching systems, and other devices [80]. In Section 1.2.5, the photoemission from QWWs of wide gap materials is studied. As the dimension of the QWs in UFs increases from 1D to 3D, the degree of freedom of the free carriers decreases drastically and the density-of-states function is changed from a stepped cumulative one to the Dirac’s delta function [81–82], forming QDs. The importance of QDs is already well known in the whole field of nanostructured science and technology. In Section 1.2.6, the photoemission from QDs of wide band gap materials has been formulated. The densityof-states function is also quantized in the presence of magneto-size quantization,

4

1 Fundamentals of Photoemission

and in Section 1.2.7, the photoemission is studied under this condition. Section 1.3 contains the result, and discussions pertinent to this chapter.

1.2 Theoretical Background 1.2.1 Photoemission from Bulk Semiconductors The consequence of the photoelectric effect is the creation of the concept of photoelectric current density (J) which, can, in turn, be written through the photoemission integral (PI ) as [2]: α0 e J= (1.1) (PI ) 4 where α0 is the probability of photoemission, ∞ PI =

    N E νz E f (E) dE

(1.2)

E0

in which, E0 ≡ W − hυ, W is the electron affinity, E ≡ E − E0 , E is the total energy of the electron as measured from the edge of the conduction    band in the vertically upward direction in the absence of any quantization, N E is the DOS   function at E = E ,νz E is the velocity of the emitted electron along the z-axis when E = E , f (E) is the Fermi-Dirac occupation probability factor and can be written as f (E) = [1 + exp ((E − EF )/kB T)]−1 , EF is the Fermi energy as measured from the edge of the conduction band in a vertically upward direction in the absence of any quantization. It appears then that the formulation of J needs an expression of DOS function which can in turn be derived as follows: The DOS function in three dimensions is defined as the number of carrier states per unit volume of wave-vector space per unit energy interval. The generalized formula of the DOS function for bulk specimens is given by: N (E) =

∂ {V (E)} (2π ) ∂E 2gv

3

(1.3)

where V (E) is the volume of k-space which should be determined from the energywave vector dispersion relation of the carriers. For n-type wide gap semiconductors having parabolic energy bands, the energy spectrum of the conduction electrons assumes the form: 2m∗ E (1.4) k2 = 2 where k is the electron wave vector and  (≡ h/ (2π )) is known as Dirac’s constant. Equation (1.4) is basically the equation of an electron parabola in a two-dimensional E-k diagram which can be written as: 2m∗ E . (1.5) kx2 + ky2 + kz2 = 2

1.2

Theoretical Background

5

Therefore, the constant energy surface is a sphere in k-space whose volume can be expressed as: V (E) =

4π 3



2m∗ E 2

3/2 .

(1.6)

Using (1.3) and (1.6), the DOS function assumes the form, N (E) = 4πgv

2m∗ h2

3/2

√ E.

(1.7)

Equation (1.7) is known as the inverted-parabolic dependence of the electronic three-dimensional density-of-states (3D DOS) function for parabolic n-type semiconductors. The expression of the DOS function is very important in semiconductor science in general and is used in investigating the carrier density, the photoelectric current density, Hall coefficient, thermoelectric power, mobility, and almost all transport properties of semiconductor devices. The increasing importance of the DOS function becomes apparent with the advent of nanoscience and technology. In two dimensions, the unit of the DOS function is (1/(eVm2 )), and in one dimension, the unit of the same function is (1/eVm). The plots (a), (b), and (c) of Fig. 1.1 exhibit the plot of the normalized 3D DOS function for bulk specimens of n-type InSb, InAs, and GaAs, respectively, as a function of electron energy, where the real band structure has been neglected for the purpose of simplified presentation, and the numerical values of the effective electron masses at the edge of the conduction band have been taken from Appendix A. Since m∗ of n-InSb is lowest as compared with the other two materials, the 3D DOS function exhibits the lowest value for bulk specimens of n-InSb, indicating the fact that the greatest electron energy states are available for the purpose of electronic conduction in n-InSb as compared with n-InAs and n-GaAs respectively. We observe that the DOS function can be written from (1.7) for E = E as:   N E = 4πgv



2m∗ h2

3/2 √ E .

(1.8)

  The term νz E is the velocity of the emitted electron along the z-axis and can be written as:    1 ∂E  . (1.9) νz E =  ∂kz E=E , kx =ky =0 The use of (1.4) and (1.9) yields the following equation:   νz E =



2 m∗

1/2 √ E .

(1.10)

6

1 Fundamentals of Photoemission

Fig. 1.1 Plot of the normalized 3D density-of-states function in bulk specimens of (a) InSb, (b) InAs, and (c) GaAs, versus electron energy

Combining (1.1), (1.8), and (1.10), it can be written as: 4π α0 em∗ gv J= h3

∞ E0

E dE

. F 1 + exp E−E kB T

(1.11)

Substituting ((E − E0 )/(kB T)) ≡ y and η0 ≡ ((hν − φ)/(kB T)) from (1.11), the expression of the photo-emitted current density from the bulk specimens having parabolic energy bands can be expressed as: J=

4π α0 em∗ gv (kB T)2 F1 (η0 ) h3

(1.12)

where F1 (η0 ) is the special case of the one-parameter Fermi-Dirac integral of order j which assumes the form [83]: Fj (η) =

1  (j + 1)



∞ 0

xj dx , 1 + exp (x − η)

j > −1

(1.13)

1.2

Theoretical Background

7

or for all j, analytically continued as a complex contour integral around the negative x-axis:

 +0 xj dx  (−j) , (1.14) Fj (η) = √ 2π −1 −∞ 1 + exp (−x − η) √ where  (j + 1) = j (j), (1/2) = π, and  (0) = 1. A few important properties of the Fermi-Dirac integral are listed in this context without proof [84] since these are very useful in obtaining direct results without cumbersome mathematics, and are needed for further progress.  d  (1.15a) Fj (η) = Fj−1 (η). 1. dη  (1.15b) 2. Fj (η)dη = Fj+1 (η).

4 3/2 π2 1 + 2 , η > 1.25. (1.15c) 3. F1/2 (η) ≈ √ η 8η 3 π 4. F0 (η) = ln |1 + eη |. 5. Fj (η) ≈ eη ,

η < 0 for all j.

(1.15d) (1.15e)

It appears that the evaluation of J as a function of electron concentration needs an expression of electron statistics, which can be formulated as follows: It is well known that N (E) f (E) dE is the number of electrons in the conduction band per unit volume lying in the E to E + dE energy range. Thus the total electron concentration is given by:  n0 =

ρTop

ρ0

N (E) f (E) dE,

(1.16)

where the lower limit ρ0 can be determined from the equation N (ρ0 ) = 0; and from the nature of variation of f (E), the upper limit ρTop can be replaced by infinity without introducing appreciable error in the subsequent investigations. Therefore, using the expressions of N (E) from (1.7), (1.16) can be written as: n0 = 4πgv

2m∗ h2

3/2 

∞

0

√ E

dE. F 1 + exp E−E kB T

(1.17)

The lower limit of the integral in (1.17) starts with zero of DOS function. Substituting x ≡ (E/kB T) (x is a new variable of normalized energy) and η ≡ (EF /kB T) (normalized Fermi energy) in (1.17) results in the emergence of the following equation: n0 = 4πgv

2m∗ kB T h2

3/2 

∞ 0

x1/2 dx. 1 + exp (x − η)

(1.18)

8

1 Fundamentals of Photoemission

Using (1.13) and (1.18), the electron concentration can be expressed as: n0 = Nc F1/2 (η) ,

(1.19)

where Nc ≡ 2gv

2π m∗ kB T h2

3/2

is known as the effective number of states in the conduction band. A Few Special Cases Case I:: When the Fermi level lies within the band gap, the semiconductor becomes nondegenerate and η<0. Under this condition, the use of (1.15e) and (1.19) leads to the expression: n0 = Nc exp (η) .

(1.20)

Case II: When EF touches the edge of the conduction band, then η → 0, and the semiconductor becomes critically degenerate. The electron concentration assumes the form: n0 = Nc F1/2 (0) .

(1.21)

Case III: Under highly degenerate electron concentration, using (1.15c), (1.19) can be written as:

  π2 4 η3/2 1 + 2 . (1.22) n0 = Nc √ 8η 3 π From (1.22), using the Binomial Theorem and the theory of approximation of algebraic equations, it can easily be proved that:  EF (T) = EF (0) 1 −

π 2 kB2 T 2 12 (EF (0))2

 ,

(1.23)

where EF (T) is the Fermi energy at temperature T and EF (0)is the Fermi energy when T → 0. Thus, EF (T) changes with temperature in a parabolic manner for semiconductors having parabolic energy bands. Case IV: Under the condition of extreme carrier degeneracy, the contribution of the second term of (1.22) is much less than the first term, so that the same can be expressed as: 8gv n0 = √ 3 π



2π m∗ h2

3/2 (EF (0))3/2 .

(1.24)

1.2

Theoretical Background

9

From (1.24) it is observed that under the condition of extreme degeneracy, the electron concentration is independent of temperature. In general, any electronic property of any semiconductor under the condition of extreme degeneracy will be temperature independent. This is logical, since extreme degeneracy is obtained when T→0 and f (E) →1, together with the fact that all states are filled up to the Fermi level, and above the Fermi level all states are vacant. Therefore (1.24) assumes the form: EF (0) = b1 (n0 )2/3 , where

b1 ≡

h2 2πm∗

(1.25)

√ 2/3 3 π . 8gv

Equation (1.25) reflects the fact that the Fermi energy for semiconductors having parabolic energy bands under the condition of extreme carrier degeneracy is proportional to the (2/3)rd power of the electron concentration. Under the condition of extreme carrier degeneracy F1 (η0 ) = (η02 /2), and (1.12) assumes the form: J=

2π α0 em∗ gv (hυ − φ0 )2 . h3

(1.26)

Let φ0 ≡ hυ0 , in which υ0 is the threshold frequency. Therefore: J=

2π α0 em∗ gv (υ − υ0 )2 . h

(1.27)

Thus, when the energy of the light quantum is much greater than the work function of the material, the condition of extreme degeneracy is reached and the current density is independent of temperature. For the opposite inequality, the system becomes nondegenerate and (1.12) is transformed to the well known classical photoelectric current density equation as given in the introduction. Under the condition hυ ≡ φ,η0 = 0, and using the well-known relation F1 (0) = (π 2 /2), from (1.12) we get: J=

π 2 α0 em∗ gv kB 2 T 2 . 3 h3

(1.28)

Thus, for the said condition, J again exhibits parabolic dependence with temperature.

1.2.2 Photoemission Under Magnetic Quantization In the presence of a quantizing magnetic field B along the z direction, the area of the wave vector space of the charge carriers perpendicular to the direction of magnetic

10

1 Fundamentals of Photoemission

quantization becomes quantized and the corresponding area quantization rule of Landau [37] can be expressed as:

2πeB 1 n+ , An =  2

(1.29)

where n = 0, and 1, 2, . . . is known as the Landau quantum number. Since the cross-section of a sphere is a circle, in this case we can write: π ks

2



2πeB 1 = n+ ,  2

(1.30)

in which ks2 ≡ kx2 + ky2 . Equation (1.5) can be written as: E=

2 kz2 2 ks2 + . 2m∗ 2m∗

(1.31)

Using (1.30) and (1.31), the dispersion relation of the conduction electrons in semiconductors having parabolic energy bands in the presence of a quantizing magnetic field B along the z direction can be expressed as:

2 kz2 1 . ω0 + E = n+ 2 2m∗

(1.32)

It is interesting to note that although the Landau area quantization rule is valid for large values of n, in the absence of electron spin, the operator method, the Schrödinger differential equation technique, and the method of the area quantization rule of the wave vector space lead to the identical result. From (1.32), it appears that the electron energy spectra for constant B is a set of displaced parabolas for various values of n, since the said equation is of the form y = an + bx2 , where y ≡ E, an ≡ (n + (1/2)) ω0 , b ≡ (2 /2m∗ ), and x ≡ kz . Thus for constant B, E vs kz curves constitute the well known Landau parabolas as shown in Fig. 1.2. The Landau energy levels can be obtained from (1.32) by substituting E = En and kz = 0, thus getting En = (n + (1/2)) ω0 , which shows that the consecutive Landau levels are equidistant in energy scale, since En+1 − En = ω0 . Besides, the surface enclosed by (1.32) is a set of concentric symmetric cylinders whose axes are parallel to the magnetic field (where n = 0 is known as the magnetic quantum limit [MQL]), as shown in Fig. 1.3. With increase in the magnetic field, the electrons will reside in the surface of the lowermost cylinder specified by n = 0. Using (1.3) and (1.30), the total DOS function under magnetic quantization (NB (E)) , excluding electron spin, can be written as: NB (E) =

nmax ∂kz eBgv  H (E − En ), ∂E π 2 n=0

(1.33)

1.2

Theoretical Background

11

Fig. 1.2 The energy spectrum of the electron in a quantizing magnetic field

Fig. 1.3 Magnetic quantization converts a spherical constant energy surface in k-space into discrete concentric cylinders whose axes are parallel to the direction of the quantizing magnetic field

where H is called the Heaviside step function. Using (1.33) and (1.32), we get: √ nmax H (E − En ) eBgv 2m∗  . NB (E) = √ 2 2 2π  E − En n=0

(1.34)

For fixed n, the DOS function per subband will be small for E > (n + (1/2)) ω0 . For E → (n + (1/2)) ω0 , NB (E) → ∞. Thus the DOS function under magnetic quantization tends to infinity with respect to electron energy at the Landau singularity. Converting the summation over n to the integration over n, (1.34) converts to the well-known (1.7). It is worth remarking that the Landau singularity is the basic

12

1 Fundamentals of Photoemission

Fig. 1.4 The plot of the normalized density-of-states function under magnetic quantization as a function of normalized electron energy for n-InSb

cause of oscillations of the electronic properties of electronic materials under magnetic quantization with inverse quantizing magnetic field. Figure 1.4 exhibits the plot of the normalized DOS function under magnetic quantization versus normalized electron energy. The velocity of the photoemitted electron per subband is: √ 

2 1 E− n+ ω0 . vzB (E) = √ 2 m∗

(1.35)

The photoelectric current density JBP per subband is given by:

JBP

α0 gv e = 2

∞

NB (E)νzB (E) f (E) dE,

(1.36)

En

where En ≡ (En + W − hν) and NB (E) is the DOS function per subband. Using (1.34), (1.35), and (1.36), the total photocurrent density can be written as: nmax  nmax α0 e2 Bgv  α0 e2 Bgv kB T  f dE = F0 (η1 ), JB = (E) 2π 2 2 2π 2 2 ∞

n=0 E

n

n=0

(1.37)

1.2

Theoretical Background

13

where η1 ≡

EFB − En . kB T

EFB is the Fermi energy in the presence of a magnetic field as measured from the edge of the conduction band in the absence of any quantization in the vertically upward direction. It is important to note that the velocity of the photo-emitted electron in the nth sub-band and the DOS function of the nth sub-band in this case cancels each other, leaving a constant prefactor. This cancellation effect is very important in QWWs and determines the limit of the maximum resistance offered by one-dimensional systems. Under the condition of extreme degeneracy from (1.37), we can write: nmax   α0 e2 Bgv  EFB − En . (1.38) JB = 2π 2 2 n=0

From (1.38), we observe that JB is independent of temperature under the condition of extreme carrier degeneracy. Under the condition of nondegeneracy from (1.37), we get: JB =

nmax α0 e2 Bgv kB T  exp (η1 ). 2π 2 2

(1.39)

n=0

For bulk materials, in the absence of a magnetic field, converting the summation over n to the corresponding integration over n from (1.39), we can write: ⎤ ⎡



∞

2 Bg k T α e hν − φ −nω −ω 0 v B 0 0 0 exp exp exp dn⎦ . Lim (JB ) = Lim ⎣ B→0 B→0 kB T 2kB T kB T 2π 2 2 0

(1.40) By carrying out elementary integration, we can get the classical photoemission equation as given in the introduction. It appears that the computation of JB requires an expression of electron concentration, which can be written as: √ nmax  ∞ f (E) dE eBgv 2m∗  . (1.41) n0 = √ 2 2 2π  E − En n=0 En Substituting,

EFB − En E − En ≡ x0 and ≡ η2 , from (1.41), one can obtain, kB T kB T n0 = Nc θ

n max n=0

F 1 (η2 ), θ ≡ 2

ω0 . kB T

(1.42)

The decrease of electron concentration with increase in magnetic field under magnetic quantum limit together with the condition of carrier non-degeneracy is known as the magnetic freeze out of the carriers. Under the condition of carrier

14

1 Fundamentals of Photoemission

non-degeneracy, converting the summation over n to the integration over n, (1.42) converts into the well-known (1.20). From (1.42) one can infer that the application of a magnetic field always tends to lower the Fermi energy level with respect to the edge of the conduction band, i.e., it always tends to make the semiconductor less degenerate.

1.2.3 Photoemission in the Presence of Cross Fields From classical electromagnetic theory, one can write: − → − → B =∇× A,

(1.43)

− → where A is the vector potential. In the presence of quantizing magnetic field B along the z direction, (1.43) assumes the form:     ˆ   ∂i ∂j k∂  0 i + 0 j + Bkˆ =  ∂x ∂y ∂z  , A A A  x y z where  i,  j and  k are unit vectors along the x-, y-, and z-axes, respectively. From the above equation, we can write: ∂Az ∂y ∂Ax ∂z ∂Ay ∂x

− − −

∂Ay ∂z ∂Az ∂x ∂Ax ∂y

=0 =0. =B

(1.44)

This particular set is satisfied by Ax ≡ 0, Ay ≡ Bx, and Az ≡ 0. Therefore in the presence of the electric field E0 along the x-axis and the quantizing magnetic field B along the z-axis for the present case we can write: 2

E + eE0 xˆ =

pˆ 2z (ˆpy − eBˆx) pˆ 2x + + , 2m∗ 2m∗ 2m∗

(1.45)

where  x, pˆ x , pˆ y , and pˆ z are operators. We introduce a new operator  θ , which can be expressed as: θˆ = −ˆpy + eBˆx −

m∗ E0 . B

(1.46)

Eliminating the operator  x, between (1.45) and (1.46) the electron energy spectrum in semiconductors having parabolic energy bands in the presence of cross fields configuration can be written as:

E0 ky [kz ]2 m∗ Eo2 1 − . ω0 + − E = n+ ∗ 2 2m B 2B2

(1.47)

1.2

Theoretical Background

15

The photo-emitted current density in this case is thus given by: J (E0 , B) = gv

nmax  ∞ ∂E ∂f (E) −α0 e  I0 dE, ∂kz ∂E 4lx π 2  E¯

(1.48)

n=0

where lx is the sample length along the x direction: E¯ ≡ xl ≡



 xh m∗ E02 1 + W − hν , I ≡ kz dky , n+ ω0 − 0 2 2B2 xl



−m∗ E0 eBlx m∗ E0 , and xh ≡ − . B  B

Using (1.47) and (1.48), we get: J (E0 , B) ≡  where φ¯ ≡ 2B Therefore,



eBlx

−



J (E0 , B) =

−gv α0 e eBlx 2π 2 lx 

2m∗ E0 B

n max



E − E¯ + E0 φ¯

n=0

 ∂f (E) dE, ∂E

α0 e2 Bgv 2π 2 2

.

n max  n=0 0

∞

 eG−η2 dG GkB T + E0 φ¯  2 , 1 + eG−η2

(1.49)

where   EFE0 B − E¯ E − E¯ G≡ , η2 ≡ , and EFE0 B kB T kB T is the Fermi energy in the present case. Differentiating (1.13) with respect to η, one can write: 

∞ 0



  xj ex−η dx 2 =  (j + 1) Fj−1 (η) 1 + ex−η

(1.50)

Using (1.50) and (1.49), the photoemission current density in this case is given by: J (E0 ,B) =

nmax   α0 e2 BkB Tgv  ¯ −1 (η2 ) , F + ψF (η ) 0 2 2π 2 2

(1.51)

n=0

¯

where, ψ¯ ≡ EkB0Tφ . In the absence of electric field (E0 → 0), (1.51) is simplified into the form as given by (1.37).

16

1 Fundamentals of Photoemission

The DOS function in this case is given by: √ nmax  2m∗ m∗ ω0  [E − E3n ]1/2 H (E − E3n ) N (E,E0 ,B) = gv lx π 2 2 eE0 n=0  1/2 − [E − E4n ] H (E − E4n ) ,

(1.52)

  where E3n ≡ (n + (1/2)) ω0 + e2 E02 /2m∗ ω02 − eE0 lx and E4n ≡ E3n + eE0 lx . In Fig. 1.5, we have plotted the normalized DOS function versus the normalized electron energy in the presence of cross fields configuration for three values of electric field. It appears from (1.52) and Fig. 1.5 that the N (E, E0 , B) oscillates with the electron energy.   From (1.47) one can easily observe that the ground state energy E0 of the electron in the  presence of crossed electric and magnetic fields is given by E0 = (1/2)ω0 + e2 E02 /2m∗ ω02 , which, in the absence of an electric field, leads to the ground state energy as ((1/2)ω0 ). The N (E, E0 , B) inthe presence of  cross fields shifts right to the ground state energy, by an amount of e2 E02 /2m∗ ω02 . Figure 1.6 exhibits the plot of the normalized DOS function versus the electric field in this case, and it appears that the DOS function oscillates with the electric field.

Fig. 1.5 Plot of the normalized density-of-states function versus normalized electron energy in the presence of crossed electric and quantizing magnetic fields for n-InSb, where for curve (a) E0 = 105 Vm–1 , for curve (b) E0 = 1.52 × 105 Vm–1 , and for curve (c) E0 = 1.72 × 105 Vm–1

1.2

Theoretical Background

17

Fig. 1.6 Plot of the normalized DOS function versus electric field in the presence of crossed electric and quantizing magnetic fields for n-InSb, where for curve (a) B = 2 tesla and for curve (b) B = 1.25 tesla

Combining (1.52) with the Fermi-Dirac occupation probability factor, the electron concentration can be written as: n0 = C0

n max 

 F 1 (η¯ 1 ) − F 1 (η¯ 2 ) ,

n=0



2

2

(1.53)

e2 E02 kB T eE0 lx , and , η¯ 2 ≡ η¯ q − Nc θ , η¯ 1 ≡ η¯ 2 + C0 ≡ eE0 lx kB T 2m∗ kB Tω02



 1 1 EFE0 B − n + ω0 . η¯ q ≡ kB T 2 In the absence of an electric field, E0 → 0, and applying L’Hospital’s rule, one can show that (1.53) is simplified into the form given by (1.42).

1.2.4 Photoemission from Quantum Wells in Ultrathin Films of Wide Gap Materials The photoelectric current density in QWs in UFs can be written as:

J2D

nzmax ∞ αo egv  = N2D (E)f (E)vz (Enz )dE, 2dz n zmin E nz

(1.54)

18

1 Fundamentals of Photoemission

where dz is the nanothickness along the z-direction, Enz is the energy of the nth subband, N2D (E) is the density-of-states function per subband, vz (Enz ) is the velocity of the electron in the nthz subband, and the factor (1/2) originates because only half of the electron will migrate towards the surface and escape [35]. Therefore, it appears that the formulation of J2D requires an expression of N2D (E) which can in turn be written per subband as: N2D (E) =

2gv ∂A , (2π)2 ∂E

(1.55)

  where A ≡ πks2 is the 2D area of the wave vector space perpendicular to the direction of size quantization and should be obtained from the 2D electron dispersion law of the QWs in UFs. In the presence of size quantization, the wave vector of the electron [85] along the direction of an infinitely deep potential well gets quantized in accordance with the wave vector quantization rule as: ki =

ni π , di

(1.56)

where i ≡ x,y,z,ni and di are the size quantum number and the film thickness in the ith direction, respectively. Thus, combining (1.31) with (1.56), the 2D electron energy spectrum in this case can be written as: 2 ks2 + Enz , E= 2m∗

2 En z ≡ 2m∗



nz π dz

2 .

(1.57)

The vz (Enz ) in this case can be expressed as: 





vz Enz =

2 m∗





Enz .

(1.58)

Using (1.55) and (1.57), the DOS function per subband can be written as: N2D (E) =

m∗ gv . π2

(1.59)

The total DOS function (N2DT (E)) in QWs in UFs of wide gap semiconductors, the dispersion relation of whose bulk electron is defined by the parabolic energy bands can be expressed as: N2DT (E) =

nzmax   m∗ gv  H E − Enz . 2 π nz =1

(1.60)

1.2

Theoretical Background

19

Fig. 1.7 Plot of the normalized 2D DOS function for the quantum wells in ultrathin films of (a) InAs and (b) GaAs as a function of normalized electron energy

Figure 1.7 exhibits the plot of the normalized 2D DOS function versus normalized electron energy for the quantum wells in ultrathin films of n-type InAs and GaAs, respectively, as shown in plots (a) and (b), where the real band structure has been neglected for the purpose of simplified presentation. It appears that the N2DT (E) exhibits step functional dependence with the electron energy. For E < E1 where E1 = (2 /2m∗ )(π/dz )2 , N2DT (E) becomes zero, whereas the 3D DOS function exists which indicates that for E < E1 the surface electron concentration vanishes for 2D systems, and whereas for the corresponding bulk semiconductors, the electron concentration is finite. When E = E1 , the 2D DOS function exhibits a quantum jump having the value (m∗ gv /π 2 ). The 2D DOS function becomes constant for E < 4E1 and when E = 4E1 the said function again jumps with a constant step height (m∗ gv /π 2 ). It is important to note that the upper points of all the steps for the 2D systems must lie on the curve of the corresponding 3D DOS function for bulk semiconductors. The quantization of the 2D DOS function is the basic reason behind the step behavior of many electronic properties of the two-dimensional systems. Thus, combining (1.54), (1.58), and (1.59), we get [25]:

J2D

nzmax αo ekB Tgv  = nz [F0 (η4 )], 2dz2 n zmin

(1.61)

20

1 Fundamentals of Photoemission

where nzmin ≥

dz π

√

2m∗ 



 (W − hν)1/2 , η4 ≡ (EF2D − Enz ) kB T, and EF2D

is the Fermi energy in the presence of size quantization as measured from the edge of the conduction band in the vertically upward direction in the absence of any quantization. The evaluation of J2D in this case requires the expression of electron statistics, which can be expressed as:

n2D

nzmax ∞ 2gv  ∂A = f (E) dE. ∂E (2π)2

(1.62)

nz =1E

nz

Thus, one obtains:

n2D

nzmax m∗ kB Tgv  = F0 (η4 ). π2

(1.63)

nz =1

For bulk semiconductors having parabolic energy bands, converting the summation over nz to the corresponding integration over nz , under the condition of nondegeneracy, we can write from (1.61) that:

J2D

α0 ekB Tgv = e 2dz2

EF kB T

∞

−an2z

nz e

2 dnz , where a ≡ 2m∗

nzmin



π dz

2

(kB T)−1 . (1.64)

Performing the integration, (1.64) converts into the well-known form of the photoemission as given in the introduction of this chapter.

1.2.5 Photoemission from Quantum Well Wires of Wide Gap Materials The expression of 1D dispersion relation for QWWs of semiconducting materials whose bulk energy band structures are defined by the parabolic energy bands can be written as: 2 2 kx2 + Eny nz , Eny nz ≡ E= ∗ 2m 2m∗



ny π dy



2 +

nz π dz

2  .

(1.65)

1.2

Theoretical Background

21

The expression of the total DOS function can be written for the present case as: nymax nzmax  ∂kx   2gv  H E − En y n z N1DT (E) = π ∂E ny =1 nz =1

  nymax nzmax  H E − Eny nz 2gv √ ∗   2m . = h E − Eny nz

(1.66)

ny =1 nz =1

In Fig. 1.8, the normalized 1D DOS function versus normalized electron energy for the QWW of (a) InAs, (b) InSb, and (c) GaAs has been plotted. It appears that the 1D DOS function exhibits oscillatory dependence with energy, and the singularity points are determined by E = Eny nz . The expressions for the vz (Enz ), Enz , Enzmin , and nzmin , respectively, in this case are given by: 

2  Enz , m∗

2 nz π 2 Enz = , 2m∗ dz

vz (Enz ) =

(1.67) (1.68)

Fig. 1.8 Plot of the normalized 1D DOS function for the QWWs of (a) InAs, (b) InSb, and (c) GaAs as a function of normalized electron energy

22

1 Fundamentals of Photoemission



2 nzmin π 2 Enzmin = , 2m∗ dz √ dz 2m∗  (W − hv). nzmin ≥ π 

(1.69) (1.70)

Thus, using the appropriate equations, the 1D photoemission current from QWWs of wide gap materials can be written as [26]:   nymax nzmax ∞   2  nz π gv αo e  2 √ ∗ f (E) dE 2m  J1D = √ 2 m∗ 2m∗ dz h E − Eny nz ny =1 nzmin E ny nz (1.71) √ nymax nzmax  gv α0 e π kB T  = nz F− 1 (η5 ), √ 2 dz 2m∗ n =1 nz y

min

where η5 ≡

EF1D − Eny nz kB T

,

in which EF1D is the Fermi energy in the presence of 2D quantization of the wave vector space as measured from the edge of the conduction band in the vertically upward direction in the absence of any quantization. The electron concentration per unit length can be expressed as: ∞   

nymax nzmax

n1D =

N1D (E)f (E)dE.

(1.72)

ny =1 nz =1E

ny nz

From (1.72), one can write that: ∞

nymax n zmax  f (E) dE 2√ ∗   2m n1D = gv h E − Eny nz ny =1 nz =1E

ny nz

√

(1.73)

nymax nzmax

2gv 2m∗ πkB T   = F− 1 (η5 ). 2 h ny =1 nz =1

1.2.6 Photoemission from Quantum Dots of Wide Gap Materials The electron dispersion relation for the QDs of wide gap materials having parabolic energy bands can be written as: EQD

2 π 2 = 2m∗



nx dx



2 +

ny dy



2 +

nz dz

2  ,

(1.74)

1.2

Theoretical Background

23

where EQD is the totally quantized energy in this case. The total DOS function in this case is given by: nxmax nymax nzmax     2gv  δ  E − EQD , N0DT (E) = dx dy dz

(1.75)

nx =1 ny =1 nz =1

  where δ  E − EQD is the Dirac’s delta function. Figure 1.9 exhibits the plot of the normalized 0D DOS function versus normalized electron energy for QDs of n-GaAs. It appears that the density-of-states becomes a delta function, which is physically correct, since all components of the electron wave vector are quantized in QDs. The expressions of the 0D electron concentration per subband (n0D ), Enz ,nzmin , J0D , and total electron concentration (n0D ), for QDs of wide-gap materials can, respectively, be written as: 2gv F−1 (η) , dx dy dz

2 nz π 2 , E nz = 2m∗ dz

n0D =

(1.76)

(1.77)

Fig. 1.9 Plot of the normalized 0D DOS function for the QDs of GaAs versus normalized electron energy

24

1 Fundamentals of Photoemission

nzmin ≥ J0D

dz π

√

2m∗  (W − hv), 

(1.78)

nxmax nymax nzmax    αo eπgv  nz F−1 (η6 ) , = ∗ 2 m dx dy dz n

(1.79)

nxmax nymax nzmax   2gv  = F−1 (η6 ), dx dy dz

(1.80)

nx =1 ny =1

n0D

zmin

nx =1 ny =1 nz =1

where η6 ≡

EF0D − EQD , kB T

in which EF0D is the Fermi energy in the present case. For bulk specimens of nondegenerate wide-gap materials, converting the summations over nx , ny , and nz to the corresponding integrations over the said variables, (1.79) gets simplified to the well-known expression of the photoemission as given in the introduction.

1.2.7 Photoemission Under Magneto-Size Quantization (MSQ) The dispersion relation of the conduction electrons in QWs in ultrathin semiconducting films in the presence of a quantizing magnetic field B along the z-axis can be written as:



1 2 nz π 2 , ω0 + EMSQE = n + 2 2m∗ dz

(1.81)

where EMSQE is the totally quantized energy. The total DOS function in this case is given by: 

N0DE EMSQE



nnmax nzmax    eBgv  = δ  E − EMSQE . π

(1.82)

n=0 nz =1

It appears that the DOS becomes a delta function, which is physically correct, since all components of the electron wave vector are quantized in MQSE in a different way as compared with the corresponding QDs. In the case of MQSE, the unit of −1 −1   instead of eVm3 as in the case of QDs. Besides, the DOS function is eVm2 quantized energy points on the energy axis in QDs is determined by the equation E ≡ EQD , whereas the same for MQSE is determined by the equation E ≡ EMQSE . The velocity of the electron in the nthz subband is given by: 





2Enz vz Enz = √ , m∗

(1.83)

1.3

Results and Discussions

25

where

2 nz π 2 . 2m∗ dz The electron concentration per unit area is given by: Enz ≡

nMSQE =

nnmax nzmax    eBgv  F−1 ηMSQE , π n =n n=0

z

(1.84)

zmin

where  EFMSQE − EMSQE ≡ and EFMSQE kB T 

ηMSQE

is the Fermi energy in this case. Therefore, the current density can be written as [28]: JMSQE =

nnmax nzmax    e2 Bgv α0  nz F−1 ηMSQE , ∗ 2 m dz

(1.85)

n=0 nz =1

where 

nmax

nzmax

  √

EFMSQE − (W − hν) 2m∗ dz  1 ≤ − ,nzmin ≥ (W − hν) and ω0 2  π √ 

2m∗ dz 1 ≤ EFMSQE − n + ω0 .  π 2

1.3 Results and Discussions Using (1.12) and (1.19), and the table in Appendix A, we have plotted the normalized photoemitted current density as a function of the normalized incident photon energy from bulk samples of n-InSb, assuming parabolic energy bands in Fig. 1.10 by neglecting the nonparabolic behavior for the purpose of elementary presentation. For the fixed value of the electron concentration, it appears that the photoemitted current density increases with increasing incident photon energy for bulk materials. Using (1.37) and (1.42), we have plotted the normalized photoemitted current density in the presence of a quantizing magnetic field as a function of normalized incident photon energy for n-InSb as shown in Fig. 1.11. From Fig. 1.11 it is observed that the photoemission is an increasing function of the incident photon energy.

26

1 Fundamentals of Photoemission

Fig. 1.10 Plot of the normalized photocurrent density from bulk specimens of n-InSb as a function of normalized incident photon energy

Figure 1.12 exhibits the dependence of normalized magneto photocurrent density from n-InSb on the inverse quantizing magnetic field. It appears from Fig. 1.12 that the photoemission current density is an oscillatory function of an inverse quantizing magnetic field. The oscillatory dependence is due to the crossing over of the Fermi level by the Landau subbands in steps, resulting in a successive reduction of the number of occupied Landau levels with the increase in the magnetic field. For each coincidence of a Landau level with the Fermi level, there would be a discontinuity in the DOS function resulting in a peak of oscillation. The peaks should occur whenever the Fermi energy is a multiple of the energy separation between the two consecutive Landau levels. It may be noted that the origin of oscillations in the magneto-photoemission is the same as that of the SdH oscillations. The variations of the photoemission current density are periodic with the quantizing magnetic field. With an increase in the magnetic field, the amplitude of the oscillation will increase; and, ultimately, at very large values of the magnetic field, the conditions for the quantum limit will be reached when the photoemission current density will exhibit a monotonic dependence with an increase in the magnetic field.

1.3

Results and Discussions

27

Fig. 1.11 Plot of the normalized photocurrent density from n-InSb as a function of normalized incident photon energy in the presence of a quantizing magnetic field

Fig. 1.12 Plot of the normalized photocurrent density from n-InSb as a function of an inverse quantizing magnetic field

28

1 Fundamentals of Photoemission

Using (1.51) and (1.53), we have plotted the normalized J (E0 ,B) as a function of the normalized incident photon energy, as shown in Fig. 1.13 for n-InSb; and it appears that the J (E0 , B) increases with the increasing incident photon energy in the presence of crossed electric and quantizing magnetic fields. Using (1.61) and (1.63), in Figs. 1.14, 1.15, and 1.16, we have plotted the normalized photoemission current density as a function of dz , the normalized incident photon energy, and normalized electron degeneracy for the quantum wells in ultrathin films of n-GaAs, n-InAs, and n-InSb. It appears that photoemitted current density increases with decreasing film thickness, increasing photon energy, and increasing electron statistics in a step-like manner exhibiting the signature of 1D quantization of the wave vector space of the conduction electron. Using (1.71) and (1.73), the normalized photocurrent density from the QWWs of the said materials has been plotted in Figs. 1.17, 1.18, and 1.19 for all the cases of Figs. 1.14, 1.15, and 1.16, respectively. The variations of the said variable in QWWs with respect to film thickness, normalized incident photon energy, and the normalized electron degeneracy are more or less similar to the corresponding plots for QWs in UFs, as given in Figs. 1.14, 1.15, and 1.16; only the numerical magnitudes and the dimensions of the appropriate x- and y-axis are different. This is a consequence of 2D quantization of the wave vector space of the conduction electrons. Using (1.79) and (1.80), in Figs. 1.20, 1.21, and 1.22, we have plotted the normalized photoemission from the QDs of the aforementioned materials for all the cases mentioned in Figs. 1.14, 1.15, and 1.16, respectively. For QDs, the numerical values of the photoemitted current

Fig. 1.13 Plot of the normalized photocurrent density from n-InSb as a function of normalized incident photon energy under cross fields configuration

1.3

Results and Discussions

29

Fig. 1.14 Plot of the normalized photocurrent density from quantum wells in ultrathin films of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of film thickness

Fig. 1.15 Plot of the normalized photocurrent density from quantum wells in ultrathin films of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of normalized incident photon energy

30

1 Fundamentals of Photoemission

Fig. 1.16 Plot of the normalized photocurrent density from quantum wells in ultrathin films of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of normalized electron degeneracy

Fig. 1.17 Plot of the normalized photocurrent density from QWWs of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of film thickness

1.3

Results and Discussions

31

Fig. 1.18 Plot of the normalized photocurrent density from QWWs of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of normalized incident photon energy

Fig. 1.19 Plot of the normalized photocurrent density from QWWs of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of normalized electron degeneracy

32

1 Fundamentals of Photoemission

Fig. 1.20 Plot of the normalized photocurrent density from QDs of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of film thickness

Fig. 1.21 Plot of the normalized photocurrent density from QDs of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of normalized incident photon energy

1.3

Results and Discussions

33

Fig. 1.22 Plot of the normalized photocurrent density from QDs of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of normalized electron degeneracy

Fig. 1.23 Plot of the normalized photoemission current density from n-InSb in the presence of magneto-size quantization as a function of normalized incident photon energy

34

1 Fundamentals of Photoemission

densities change due to 3D quantization of the wave vector space of the conduction electrons for all the plots, as given in Figs. 1.14, 1.15, and 1.16. Using (1.84) and (1.85), we have plotted the oscillatory normalized photoemitted current density as a function of the normalized incident photon energy for n-InSb under magneto-size quantized conditions, as shown in Fig. 1.23. Moreover, the photoemission from quantum-confined compounds can become several orders of magnitude larger than that of bulk specimens of the same materials, which is also a direct signature of quantum confinement. The photoemitted current density under magneto-size quantized conditions increases with increasing photon energy in an oscillatory manner. Finally, we note that this oscillatory dependence will be less and less prominent with increasing film thickness; and ultimately, for bulk specimens of the same material, the photoemission will be found to increase continuously with increasing electron degeneracy in a nonoscillatory manner.

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References

35

18. K. P. Ghatak, Long Wave Length Semiconductor Devices, Materials and Processes Symposium Proceedings, MRS Symposium Proceedings, MRS Spring Meeting, vol. 216, p. 469 (1990). 19. K. P. Ghatak, A. Ghoshal, S. Bhattacharyya, SPIE, Nonlinear Optical Materials and Devices for Photonic Switching, USA, vol. 1216, p. 282 (1990). 20. K. P. Ghatak, SPIE, Nonlinear Optics III, USA, vol. 1626, p. 115 (1992). 21. K. P. Ghatak, A. Ghoshal, B. De, SPIE, Optoelectronic Devices and Applications, USA, vol. 1338, p. 111 (1990). 22. R. Houdré, C. Hermann, G. Lampel, P. M. Frijlink, Surface Sci. 168, 538 (1986). 23. T. C. Chiang, R. Ludeke, D. E. Eastman, Phys. Rev. B. 25, 6518 (1982). 24. S. P. Svensson, J. Kanski, T. G. Andersson, P. O. Nilsson, J. Vacuum Sci. Technol. B 2, 235 (1984); S. F. Alvarado, F. Ciccacci, M. Campagna, Appl. Phys. Letts. 39, 615 (1981). 25. C. Majumdar, A. B. Maity, A. N. Chakravarti, Phys. Stat. Sol. (b) 140, K7 (1987). 26. C. Majumdar, A. B. Maity, A. N. Chakravarti, Phys. Stat. Sol. (b) 141, K35 (1987). 27. N. R. Das, K. K. Ghosh, D. Ghoshal, Phys. Stat. Sol. (b) 197, 97 (1996). 28. C. Majumdar, A. B. Maity, A. N. Chakravarti, Phys. Stat. Sol. (b), 144, K13, (1987). 29. N. R. Das, A. N. Chakravarti, Phys. Stat. Sol. (b) 176, 335 (1993). 30. S. Sen, N. R. Das and A. N. Chakravarti, J. Phys: Conden. Mat. 19, 186205 (2007); N. R. Das, S. Ghosh, A. N. Chakravarti, Phys. Stat. Sol. (b) 174, 45 (1992). 31. A. B. Maity, C. Majumdar, A. N. Chakravarti, Phys. Stat. Sol. (b) 144, K93, (1987). 32. A. B. Maity, C. Majumdar, A. N. Chakravarti, Phys. Stat. Sol. (b) 149, 565 (1988). 33. N. R. Das, A. N. Chakravarti, Phys. Stat. Sol. (b) 169, 97 (1992). 34. A. Modinos, Field, Thermionic and Secondary Electron Emission Spectroscopy (Plenum Press, USA, 1984). 35. A. V. D. Ziel, Solid State Physical Electronics, (Prentice Hall, Inc. USA, 1957). 36. B. R. Nag, Electron Transport in Compound Semiconductors, Springer Series in Soild-State Science, Vol. II (Springer Verlag, Germany, 1980). 37. L. Landau, E. M. Liftshitz, Statistical Physics, Part-II, (Pergamon Press, UK, 1980). 38. W. Zawadzki, B. Lax, Phys. Rev. Lett. 16, 1001 (1966). 39. K. P. Ghatak, M. Mondal, Zeit. fur Phys. B 69, 471 (1988); M. Mondal, N. Chattopadhyay, K. P. Ghatak, J. Low Temp. Phys. 66, 131 (1987). 40. M. Mondal, K. P. Ghatak, Phys. Letts. A 131, 529 (1988); B. Mitra, K. P. Ghatak, Phys. Letts. A 137, 413 (1989). 41. M. J. Harrison, Phys. Rev. A 29, 2272 (1984). 42. J. Zak, W. Zawadzki, Phys. Rev. 145, 536 (1966); W. Zawadzki, Q. H. F. Vrehen, B. Lax, Phys. Rev. 148, 849 (1966); Q. H. F. Vrehen, W. Zawadzki, M. Reine, Phys. Rev. 158, 702 (1967); M. H. Weiler, W. Zawadzki, B. Lax, Phys. Rev. 163, 733 (1967). 43. P. M. Petroff, A. C. Gossard, W. Wiegmann, Appl. Phys. Letts. 45, 620 (1984); J. M. Gaines, P. M. Petroff, H. Kroemar, R. J. Simes, R. S. Geels, J. H. English, J. Vac. Sci. Tech. B 6, 1378 (1988) 44. J. Cibert, P. M. Petroff, G. J. Dolan, S. J. Pearton, A. C. Gossard, J. H. English, Appl. Phys. Letts. 49, 1275 (1986). 45. T. Fukui, H. Saito, Appl. Phys. Letts. 50, 824 (1987). 46. H. Sasaki, Jpn. J. Appl. Phys. 19, 94 (1980). 47. P. M. Petroff, A. C. Gossard, R. A. Logan, W. Wiegmann, Appl. Phys. Lett. 41, 635 (1982). 48. H. Temkin, G. J. Dolan, M. B. Panish, S. N. G. Chu, Appl. Phys. Lett. 50, 413 (1988); B. I. Miller, A. Shahar, U. Koren, P. J. Corvini, Appl. Phys. Lett. 54, 188 (1989). 49. L. L. Chang, H. Sakaki, C. A. Chang, L. Esaki, Phys. Rev. Letts. 38, 1489 (1977); K. Lee, M. S. Shur, J. J. Drummond, H. Morkoc, IEEE Trans. Electron. Dev. 30, 207 (1983). 50. N. T. Linch, Festkorperprobleme 23, 227 (1985). 51. D. R. Scifres, C. Lindstrom, R. D. Burnham, W. Streifer, T. L. Paoli, Electron. Letts. 19, 169 (1983).

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52. P. M. Solomon, Proc. IEEE, 70, 489 (1982); T. E. Schlesinger, T. Kuech, Appl. Phys. Lett. 49, 519 (1986). 53. H. Heiblum, D. C. Thomas, C. M. Knoedler, M. I. Nathan, Appl. Phys. Letts. 47, 1105 (1985). 54. O. Aina, M. Mattingly, F. Y. Juan, P. K. Bhattacharya, Appl. Phys. Letts. 50, 43 (1987). 55. I. Suemune, L. A. Coldren, IEEE J. Quant. Electron. 24, 1778 (1988). 56. D. Miller, D. Chemla, T. Damen, T. Wood, C. Burrus, A. Gossard, W. Weigmann, IEEE J. Quant. Electron. 21, 1462 (1985). 57. F. Sols, M. Macucci, U. Ravaioli, K. Hess, Appl. Phys. Lett. 54, 350 (1980). 58. C. S. Lent, D. J. Kirkner, J. Appl. Phys. 67, 6353 (1990). 59. C. S. Kim, A. M. Satanin, Y. S. Joe, R. M. Cosby, Phys. Rev. B, 60, 10962 (1999). 60. S. Midgley, J. B. Wang, Phys. Rev. B 64, 153304 (2001). 61. T. Sugaya, J. P. Bird, M. Ogura, Y. Sugiyama, D. K. Ferry, K. Y. Jang, Appl. Phys. Lett. 80, 434 (2002). 62. B. E. Kane, G. R. Facer, A. S. Dzurak, N. E. Lumpkin, R. G. Clark, L. N. Pfeiffer, K. N. West, Appl. Phys. Lett. 72, 3506 (1998). 63. C. Dekker, Physics Today, 52, 22 (1999). 64. A. Yacoby, H. L. Stormer, N. S. Wingreen, L. N. Pfeiffer, K. W. Baldwin, K. W. West, Phys. Rev. Lett. 77, 4612 (1996). 65. Y. Hayamizu, M. Yoshita, S. Watanabe, H. Akiyama, L. N. Pfeiffer, K. W. West, Appl. Phys. Lett. 81, 4937 (2002). 66. S. Frank, P. Poncharal, Z. L. Wang, W. A. de Heer, Science 280, 1744 (1998). 67. I. Kamiya, I. Tanaka, K. Tanaka, F. Yamada, Y. Shinozuka, H. Sakaki, Physica E 13, 131 (2002). 68. A. K. Geim, P. C. Main, N. La Scala, Jr., L. Eaves, T. J. Foster, P. H. Beton, J. W. Sakai, F. W. Sheard, M. Henini, G. Hill, M. A. Pate, Phys. Rev. Lett. 72, 2061 (1994). 69. A. S. Melnikov, V. M. Vinokur, Nature 415, 60 (2002). 70. K. Schwab, E. A. Henriksen, J. M. Worlock, M. L. Roukes, Nature 404, 974 (2000). 71. L. Kouwenhoven, Nature 403, 374 (2000). 72. S. Komiyama, O. Astafiev, V. Antonov, T. Kutsuwa, H. Hirai, Nature 403, 405 (2000). 73. E. Paspalakis, Z. Kis, E. Voutsinas, A. F. Terzis, Phys. Rev. B 69, 155316 (2004). 74. J. H. Jefferson, M. Fearn, D. L. J. Tipton, T. P. Spiller, Phys. Rev. A 66, 042328 (2002). 75. J. Appenzeller, Ch. Schroer, Th. Schapers, A. v. d. Hart, A. Fröster, B. Lengeler, H. Lüth, Phys. Rev. B 53, 9959 (1996). 76. J. Appenzeller, C. Schroer, J. Appl. Phys. 87, 3165 (2000). 77. P. Debray, O. E. Raichev, M. Rahman, R. Akis, W. C. Mitchel, Appl. Phys. Lett. 74, 768 (1999). 78. P. M. Solomon, Proc. IEEE 70, 489 (1982); T. E. Schlesinger, T. Kuech, Appl. Phys. Lett. 49, 519 (1986). 79. D. Kasemset, C. S. Hong, N. B. Patel, P. D. Dapkus, Appl. Phys. Letts. 41, 912 (1982). 80. K. Woodbridge, P. Blood, E. D. Pletcher, P. J. Hulyer, Appl. Phys. Lett. 45, 16 (1984). 81. D. Bimberg, M. Grundmann, N. N. Ledentsov, Quantum Dot Heterostructures (John Wiley and Sons, USA, 1999) 82. T. Tsuboi, Phys. Stat. Sol. (b), 146, K11 (1988) (and the references cited therein). 83. K. P. Ghatak, S. Bhattacharya, S. Pahari, D. De, S. Ghosh, M. Mitra, Annalen der Physik, 17, 195 (2008). 84. J. S. Blakemore, Semiconductor Statistics (Dover Publications, USA, 1987). 85. W. Zawadzki, In: Two Dimensional Systems, Hetrostructures and Superlattices, Edited by G. Bauer, F. Kuchar, H. Heinrich (Springer-Verlag, Germany, 1984).

Chapter 2

Fundamentals of Photoemission from Quantum Wells in Ultrathin Films and Quantum Well Wires of Various Nonparabolic Materials

2.1 Introduction In chapter 1, the photoemission from wide-gap materials having parabolic energy bands under different physical conditions has been studied. For the purpose of indepth study, in this chapter, the same has been investigated from QWs in UFs and QWWs of non-parabolic materials having different band structures. The journey towards the knowledge temple known as the photoelectric effect begins with the non-linear optical compounds which find applications in non-linear optics and light emitting diodes [1]. The quasi-cubic model can be used to investigate the symmetric properties of both the bands at the zone center of wave vector space of the same compound [2]. Including the anisotropic crystal potential in the Hamiltonian, and special features of the nonlinear optical compounds, Kildal [3] formulated the electron dispersion law under the assumptions of the isotropic momentum matrix and the isotropic spin orbit splitting constant, respectively, although the anisotropies in the two aforementioned band constants are the significant physical features of the said materials [4]. In Section 2.2.1, the photoemission from QWs in UFs and QWWs of nonlinear optical materials is investigated by considering the combined influence of the anisotropies of the said energy band constants together with the inclusion of the crystal field splitting. The III–V compounds finds extensive usage in infrared detectors [5], quantum dot light-emitting diodes [6], quantum cascade lasers [7], quantum well wires [8], optoelectronic sensors [9], high electron mobility transistors [10], etc. The III–V, ternary, and quaternary materials are called Kane-type compounds since their electron energy spectra are defined by the three-band model of Kane [11]. In Section 2.2.2, the photoemission from QWs in UFs and QWWs of III–V materials has been studied and the simplified results for the two-band model of Kane and that of wide gap materials have further been demonstrated as special cases. The II–VI compounds are being extensively used in nanoribbons, blue green diode lasers, photosensitive thin films, infrared detectors, ultra high-speed bipolar transistors, fiber optic communications, microwave devices, photovoltaic and solar cells, semiconductor gamma-ray detector arrays, and semiconductor detector gamma camera; and they allow for a greater density of data storage on optically addressed compact discs [12–14]. The carrier energy spectra in II–VI materials 37 K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_2,  C Springer Science+Business Media, LLC 2009

38

2 Fundamentals of Photoemission from Quantum Wells

are defined by the Hopfield model [15], where the splitting of the two-spin states by the spin-orbit coupling and the crystalline field has been taken into account. Section 2.2.3 describes the investigation of the photoemission from QWs in UFs and QWWs of II–VI compounds. The n-Gallium Phosphide (n-GaP) is being used in quantum dots, light-emitting diodes [16], high efficiency yellow solid state lamps, light sources, and high peak current pulse for high gain tubes. The green and yellow light-emitting diodes made of nitrogen-doped n-GaP possess a longer device life at high drive currents [17]. In Section 2.2.4, the photoemission from QWs in UFs and QWWs of n-GaP has been studied. The importance of Germanium has been well known since the inception of transistor technology, and in recent years it has been used in memory circuits, single photon detectors, single photon avalanche diodes, ultrafast all-optical switches, THz lasers, and THz spectrometers [18–19]. The investigation of photoemission from QWs in UFs and QWWs of Ge has been presented in Section 2.2.5. The Platinum Antimonide (PtSb2 ) finds application in device miniaturization, colloidal nanoparticle synthesis, sensors and detector materials, and thermophotovoltaic devices [20]. It may also be noted that stressed materials are being widely investigated for strained silicon transistors, quantum cascade lasers, semiconductor strain gauges, thermal detectors, and strained-layer structures [21]. The photoemission from QWs in UFs and QWWs of PtSb2 and stressed materials (taking stressed n-InSb as an example) has, respectively, been investigated in Sections 2.2.6 and 2.2.7. In recent years, Bismuth (Bi) nanolines have been fabricated, and Bi also finds use in arrays of antennas, which leads to the interaction of electromagnetic waves with such Bi-nanowires [22]. Several dispersion relations of the carriers have been proposed for Bi. Shoenberg [23] experimentally verified that the de Haas–Van Alphen and cyclotron resonance experiments supported the ellipsoidal parabolic model of Bi, although the magnetic field dependence of many physical properties of Bi supports the two-band model [24]. The experimental investigations on the magneto-optical [25] and the ultrasonic quantum oscillations [26] support the Lax ellipsoidal nonparabolic model [24]. Kao [27], Dinger and Lawson [28], and Koch and Jensen [29] demonstrated that the Cohen model [30] is in conformity with the experimental results in a better way. Additionally, the hybrid model of bismuth developed by Takoka et al. also finds use in the literature [31]. McClure and Choi [32] devised a new model of Bi and they showed that it can explain the data for a large number of magneto-oscillatory and resonance experiments. In Section 2.2.8, we formulate the photoemission from QWs in UFs and QWWs of Bi in accordance with the aforementioned energy band models for the purpose of relative assessment. It has further been demonstrated that under certain limiting conditions all the results for all the models of 2D and 1D system reduce to the well-known result of the photoemission from bulk samples of non-degenerate wide gap materials. This above statement stands for the compatibility test of the theoretical analysis. With the discovery of carbon nanotubes (CNs) in 1991 [33], novel electronic properties pertaining to their particular tubular structures have found wide applications in modern quantum effect devices [34–39]. Such devices include nanooscillators, ultra-fast optical filters, nano bearings, nano probes, nano cantilevers, field

2.2

Theoretical Background

39

emission displays, nano gears, nano motors, and other nanoelectronic devices [40]. The CNs can be tailored into a metal or a semiconductor depending on the diameter and the chiral index numbers (m,n) [41], where the integers m and n denote the number of unit vectors along two directions in the honeycomb crystal lattice of graphene. For armchair and zigzag nanotubes, the chiral indices are given as m = n [41]. Another class of CN called as chiral CN has distinct integers m and n. Additionally, a CN can be a metallic if m − n = 3q, where q = 1,2,3 . . .; otherwise it is a semiconductor. Single-walled nanotubes (SWN) are a very important class of CNs and can be used as excellent conductors [42], thus miniaturizing electronics beyond the micro-electromechanical scale. One useful application of SWNs is in the development of the first intramolecular field effect transistors (FETs) [43]. Modern industrial technology predicts that traditional interconnects between devices will be a major performance and reliability bottleneck as size reduces. Metallic single-wall CNs (SWCNs) have received considerable attention as potential substitutions for traditional interconnect materials like Cu due to their excellent inherent electrical and thermal properties. Since the carriers are confined in a metallic SWCN [44], the inclusion of the subband energy owing to Born–Von Karman (BVK) boundary conditions [45] for their unique band structure becomes prominent. The quantization of function due to van Hove singularity (VHS) [2.46] of the wave vectors. The investigation of photoemission from carbon nanotubes has been presented in Section 2.2.9. The last section 2.3 contains the results and discussions pertaining to this chapter.

2.2 Theoretical Background 2.2.1 Photoemission from Nonlinear Optical Materials Considering the anisotropies of the effective electron mass and the spin orbit splitting constant, and including the influence of crystal field splitting, the electron dispersion law in bulk specimens of nonlinear optical materials within the framework of k.p formalism can be written as [47]: γ (E) = f1 (E) ks2 + f2 (E) kz2 ,

(2.1)

where 

       1 E + Eg0 E + Eg0 + || + δ E + Eg0 + || γ (E) ≡ E E + Eg0 3 

  2 + E E + Eg0 2|| − 2⊥ , 9 E is the electron energy as measured from the edge of the conduction band in the vertically upward direction in the absence of any quantization, Eg0 is the band gap

40

2 Fundamentals of Photoemission from Quantum Wells

in the absence of any external field, || and ⊥ are the spin orbit splitting constants parallel and perpendicular to the crystal axis, respectively, δ is the crystal field splitting constant,

−1 ! 

2 1 Eg0 + ⊥ δ E + Eg0 + || f1 (E) ≡  Eg0 Eg0 + ⊥ 3 3 



  2 2 + E + Eg0 E + Eg0 + || + 2|| − 2⊥ , ks2 ≡ kx2 + ky2 , 3 9 

−1 !   2 2 ∗ f2 (E) ≡  Eg0 Eg0 + || 2m|| Eg0 + || 3 

   2 E + Eg0 E + Eg0 + || , 3 2







2m∗⊥



and m∗|| and m∗⊥ are the effective masses of the electron at the edge of the conduction band parallel and perpendicular to the direction of the c axis, respectively. The 2D electron energy spectrum for QWs in UFs of nonlinear optical materials in the presence of size-quantization along the x-direction can be expressed as:

πnx γ (E) = f1 (E) dx

2 + f1 (E) ky2 + f2 (E) kz2 .

(2.2)

The subband energy (Enx ) is given by:     γ Enx = f1 Enx (π nx / dx )2 .

(2.3)

The condition for photoemission in this case assumes the form Enxmin + hv ≥ W,

(2.4)

where Enxmin can be expressed through (2.3) as:



 2 γ Enxmin = f1 Enxmin πnxmin / dx .

(2.5)

The total photoelectric current density from QWs in UFs of nonlinear optical materials in this case can, in general, be written as:

J2D

nxmax ∞   αo egv  = N2D (E) f (E) vx Enx dE, 2dx n xmin E nx

(2.6)

2.2

Theoretical Background

41

where nxmin is the nearest integer of the following inequality: nxmin

dx ≥ π



γ (W − hv) f1 (W − hv)

1/ 2 .

(2.7)

It appears that the formulation of J2D requires an expression of N2D (E) which can in turn be written in this case using (2.2) as: N2D (E) =

gv ∂ {φ1 (E,nx )} , (2π) ∂E



−1/2

(2.8)

where φ1 (E,nx ) ≡ f1 (E) f2 (E)





π nx γ (E) − f1 (E) dx

2  .

  The vx Enx in this case can be expressed as:   1   vx Enx = 1 Enx ,  ⎡ "   #  $ #  $ "   ⎤−1 f1 Enx γ Enx γ Enx f1 Enx     ⎦ f1 Enx . "   "   1 Enx ≡ ⎣ − 2 γ Enx 2 f1 Enx

(2.9)

Thus combining the appropriate equations, we get: J2D =

nxmax   αo egv  1 Enx [φ1 (EF2D ,nx ) + φ2 (EF2D ,nx )] , 2hdx n

(2.10)

xmin

where φ2 (EF2D ,nx ) ≡ mation,

%so

r=1 Zr2 [φ1 (EF2D ,nx )]

 Zr,Y ≡ 2 (kB T)

2r



1−2

1−2r

, s0 is the upper limit of the sum-

 ∂ 2r ζ (2r) , ∂EFYD 2r

Y = 2 and ζ (2r) is the Zeta function of order 2r [48]. The evaluation of J2D in this case requires the expression of electron statistics, which can be expressed using (2.8) as: n2D =

nxmax gv  [φ1 (EF2D ,nx ) + φ2 (EF2D ,nx )]. 2π nx =1

(2.11)

42

2 Fundamentals of Photoemission from Quantum Wells

In this context, the photocurrent from QWWs of nonlinear optical materials can be formulated in the following way: For electron motion along the x-direction only, the 1D electron dispersion law in this case can be written, following (2.1), as: 2  γ (E) = f1 (E) kx2 + f1 (E) πny / dy + f2 (E) (π nz / dz )2 .

(2.12)

  The subband energy E is given by the equation:   2     γ E = f1 E π ny / dy + f2 E (π nz / dz )2 .

(2.13)

The 1D DOS function per subband is given by: N1D (E) =

2gv ∂kx . π ∂E

(2.14)

The velocity of the emitted electrons along the x-direction can be written as: vx (E) =

1 ∂E .  ∂kx

(2.15)

Therefore the photocurrent is given by:

nymax nzmax ∞

 1 ∂E 2 ∂kx αo egv  f (E) dE, I= 2 π ∂E  ∂kx

(2.16)

ny =1 nz =1

1

where 1 ≡ E + W − hν. Using (2.16), one can write: nymax nzmax  αo egv kB T  F0 (η6 ) ,where η6 ≡ I= π ny =1 nz =1



  EF1D − E + W − hν . (2.17) kB T

This is the general expression of the photocurrent from QWWs, where EF1D and E are the two-band structure–dependent quantities. Thus, it appears that the evaluation of J1D requires an expression of carrier statistics which can, in turn, be written combining (2.12), (2.14), and the Fermi-Dirac occupation probability factor as: n1D =

2gv π

n ymax nzmax  ny =1 nz =1

     t1 EF1D ,ny ,nz + t2 EF1D ,ny ,nz ,

(2.18)

2.2

Theoretical Background

43

where 1/ 2    2  t1 EF1D ,ny ,nz ≡ γ (EF1D ) − f1 (EF1D ) π ny / dy − f2 (EF1D ) (π nz / dz )2  −1 2 f1 (EF1D ) / 



t2 EF1D ,ny ,nz ≡

S0 

   Zr,Y t1 EF1D ,ny ,nz , and Y = 1.

r=1

2.2.2 Photoemission from III–V Materials (a) Under the conditions || = ⊥ =  (the isotropic spin splitting constant), δ = 0, m∗|| = m∗⊥ = m∗ , (2.1) assumes the form: 2 k 2 2m∗

   E E + Eg0 E + Eg0 +  Eg0 + 13 

. = I (E) , I (E) ≡   Eg0 Eg0 +  E + Eg0 + 23 

(2.19)

Equation (2.19) describes the dispersion relation of the conduction electrons in III–V materials and is well known in the literature as the three-band model of Kane [11]. The dispersion relation of the 2D electrons in this case is given by: 2 ky2 2m∗

+

2 kz2 2 + (πnx / dx )2 = I (E) . ∗ 2m 2m∗

(2.20)

The subband energy can be written as:   2 I Enx = (πnx / dx )2. 2m∗

(2.21)

The velocity of the electron in the nth x band can be determined from the equation: 





vx Enx =

    2 2 Enx , 2 Enx ≡ ∗ m

"  &   I Enx I  Enx .

(2.22)

The photoemission in this case can be expressed as:

J2D

egv αo = π 2 dx



m∗ 2

1 n xmax 2 nxmin

  2 Enx [T3 (EF2D ,nx ) + T4 (EF2D ,nx )] ,

(2.23)

44

2 Fundamentals of Photoemission from Quantum Wells

where √ nxmin ≥

2m∗ 



dx π



T4 (EF2D ,nx ) ≡



2 I (W − hv),T3 (EF2D ,nx ) ≡ I (EF2D ) − 2m∗

sv 



π nx dx

2 

Zr,Y T3 (EF2D ,nx ) , and Y = 2.

r=1

The investigation of J2D requires an expression of electron statistics, which can be written as: n2D =

nxmax m∗ gv  [T3 (EF2D ,nx ) + T4 (EF2D ,nx )] . π2

(2.24)

nx =1

The photocurrent from QWWs of III–V materials has been discussed in accordance with the three-band model of Kane as follows: The one-dimensional electron dispersion law is given by:   2 kx2 + G ,n n = I (E) , 2 y z 2m∗       2 where G2 ny ,nz ≡ 2 π 2 /2m∗ ny / dy + (nz / dz )2 .

(2.25)

The subband energyE can be written as:     G2 ny ,nz = I E .

(2.26)

The photocurrent is given by (2.17), in which E is given by (2.26) and EF1D should be determined from the following equation:

n1D

√ nymax nzmax      2gv 2m∗  = t3 EF1D ,ny ,nz + t4 EF1D ,ny ,nz , π

(2.27)

ny =1 nz =1

where     1 2 t3 EF1D ,ny ,nz ≡ I(EF1D ) − G2 ny ,nz / , 



t4 EF1D ,ny ,nz ≡

S0 

   Zr,Y t3 EF1D ,ny ,nz , and Y = 1.

r=1

(b) Under the inequalities  >> Eg0 or  << Eg0 , (2.19) assumes the form E(1 + αE) = (2 k2 / 2m∗ ), α ≡ 1 / Eg0 .

(2.28)

2.2

Theoretical Background

45

Equation (2.28) is known as the two-band model of Kane and should be used for studying the electronic properties of the materials whose band structures obey the above inequalities [11]. For QWs in UFs of III–V materials whose bulk energy band structures obey the two-band model of Kane, the 2D electron dispersion law, the density-of-states per subband, and vx (Enz ) assume the forms: E(1 + αE) =

2 ky2 2m∗

+

2 kz2 2 + ∗ 2m 2m∗



nx π dx

2 ,

m∗ gv (1 + 2αE) , π2 "     Enx 1 + αEnx 2  . vx (Enx ) = 3 (Enx ), 3 (Enx ) ≡  m∗ 1 + 2αEnx N2D (E) =

(2.29) (2.30)

(2.31)

Therefore, J2D is given by:

J2D

√ nxmax   αo ekB Tgv 2m∗  = 3 (Enz ) (1 + 2αEnx )F0 (ηn ) + 2αkB TF1 (ηn ) , 2π 2 dx n xmin

(2.32) where nxmin

dx [(W − hν) {1 + α (W − hν)}]1/2 ≥ π

 √  &  2m∗ , ηn ≡ (EF2D − Enx ) kB T , 

in which Enx is given by:   "   Enx = [2α]−1 −1 + 1 + 2α2 /m∗ (π nx /dx )2 .

(2.33)

Combining (2.30) with the Fermi-Dirac occupation probability factor and using (1.13), the electron statistics in this case can be written as:

n2D

nxmax   m∗ gv kB T  = (1 + 2αEnx )F0 (ηn ) + 2αkB TF1 (ηn ) . 2 π

(2.34)

nx =1

The expression of the 1D dispersion relation, for QWWs of III–V materials whose energy band structures are defined by the two-band model of Kane, assumes the form: E(1 + αE) =

  2 kx2 + G2 ny ,nz . 2m∗

(2.35)

46

2 Fundamentals of Photoemission from Quantum Wells

In this case, the quantized energy E is given by: 

E = (2α)

−1

 "   −1 + 1 + 4αG2 ny ,nz .



(2.36)

The photocurrent is given by (2.17), in which E is given by (2.36), and EF1D should be determined from the following equation:

n1D

2gv = π

√

nymax nzmax   2m∗  t5 (EF1D ,ny ,nz ) + t6 (EF1D ,ny ,nz ) , 

(2.37)

ny =1 nz =1

where     1 2 t5 EF1D ,ny ,nz ≡ EF1D (1 + αEF1D ) − G2 ny ,nz / , S0       Zr,Y t5 EF1D ,ny ,nz , and Y = 1. t6 EF1D ,ny ,nz ≡ r=1

In the absence of band nonparabolicity, the photocurrent assumes the form:

I=

nymax nzmax   $   αo gv ekB T  1 # . F0 EF1D − G2 ny ,nz + W − hν π kB T

(2.38)

ny =1 nz =1

Converting the summation over quantum numbers to the corresponding integrations in (2.38), the photocurrent density from 1D semiconductors having isotropic parabolic energy bands with nondegenerate electron concentration is transformed into the well-known form given in the introduction of Chapter 1.

2.2.3 Photoemission from II–VI Compounds The dispersion relation of the carriers in bulk specimens of II–VI compounds in accordance with the Hopfield model can be expressed as [15]: E = Ao ks2 + Bo kz2 ± Co ks ,

(2.39)

whereA0 ≡ 2 / 2m∗⊥ , B0 ≡ 2 / 2m∗ , and C0 represents the splitting of the two-spin states by the spin orbit coupling and the crystalline field. In the presence of the size quantization along the z-direction, (2.39) assumes the form

2.2

Theoretical Background

47

E = A0 ks2 + B0

π nz dz

2 ± C0 ks .

(2.40)

The expressions of the subband energy Enz , the 2D density-of-states function per subband, N2DT (E), the velocity of the carriers in the nthz subband, nzmin , J2D , and n2D can, respectively, be given by: Enz = B0 (πnz / dz )2 ,   √   C0 / 2 A0 gv m∗⊥ , 1− √ N2D (E) = π2 E + δ51 (nz )  √   nzmax    C0 / 2 A0 gv m∗⊥  N2DT (E) = H E − En z , 1− √ 2 π E + δ51 (nz )

(2.41)

(2.42)

(2.43)

nz =1

J2D

  2  Bo Enz , vz Enz =  √ dz W − hν , nzmin ≥ √ π Bo  1/2 nz  max

C f (E ,n )   Teαo gv kB m∗⊥ 0 s F2D z = n η F , √ z 0 nz3 − m∗|| 2dz2 2 A 0 kB T nz

(2.44) (2.45)

(2.46)

min



C f (E ,n )  gv m∗⊥ kB T  0 s F2D z η F , √ 0 nz3 − π2 2 A 0 kB T n =1 nzmax

and n2D =

z

where 

 π nz 2 1 2 δ51 (nz ) ≡ , (C0 ) − 4A0 B0 4A0 dz   ηnz3 ≡ (kB T)−1 EF2D − B0 (πnz / dz )2 , ⎡ "

 ⎢ fs (EF2D ,nz ) ≡ ⎣2 ηnz3 + δ52 (nz ) − δ52 (nz ) ⎡ ⎤⎤ s

2r−1  (−1) (2r − 1)! ⎥⎥ ⎢ 1−2r ζ (2r) + ⎣2 1 − 2

2r ⎦⎦ , r=1 ηnz3 + δ52 (nz )   B0 (πnz / dz )2 + δ51 (nz ) and δ52 (nz ) ≡ . kB T

(2.47)

48

2 Fundamentals of Photoemission from Quantum Wells

The 1D dispersion relation for QWWs of II–VI materials can be written as:   E = B0 kz2 + G3,± nx ,ny , where 



⎡

G3,± nx ,ny ≡ ⎣A0



πnx dx



2 +

π ny dy

2 !

± C0

(2.48)

π nx dx



2 +

π ny dy

2 !1/2

⎤ ⎦.

The photocurrent from QWWs of II–VI materials is given by: nxmax nymax ) )     *  α0 egv kB T  I= F0 (kB T)−1 EF1D − G3,+ nx ,ny + W − hν 2π  nx =1 ny =1 )    **  +F0 (kB T)−1 EF1D − G3,− nx ,ny + W − hν . (2.49)

The 1D electron statistics can be written as: n1D =

nxmax nymax      gv  t7 EF1D ,nx ,ny + t8 EF1D ,nx ,ny , √ π B0 n =1 n =1 x

(2.50)

y

where     1/ 2   1/ 2   + EF1D − G3,− nx ,ny , t7 EF1D ,nx ,ny ≡ EF1D − G3,+ nx ,ny S0       Zr,Y t7 EF1D ,nx ,ny , and Y = 1. t8 EF1D ,nx ,ny ≡ r=1

2.2.4 Photoemission from n-Gallium Phosphide The dispersion relation of the conduction electrons in bulk specimens of n-GaP is given by [49]:  1/ 2 

4 k02 2 2  2 2 ks2 2 2 2 + + |VG | k + kz − ks + kz + |VG | E= 2m∗⊥ 2m∗|| s m∗2 ||

(2.51)

where k0 and |VG | are constants of the energy spectrum. The 2D electron dispersion relation in size quantized n-GaP can be expressed as: 1/2  , E = aks2 + C (nz π / dz )2 + |VG | − Dks2 + |VG |2 + D (nz π / dz )2

(2.52)

2.2

Theoretical Background

49

in which a≡

2 2 2 + , C≡ , ∗ ∗ 2m⊥ 2m|| 2m∗||

and



D ≡ 4 k02 / m2|| .

  The subband energy Enz is given by: 1/ 2  Enz = C (πnz / dz )2 + |VG | − |VG |2 + D (π nz / dz )2 .

(2.53)

The nzmin should be the nearest integer of the following equation:

π nzmin (W − hν) = C dz

2





π nzmin + |VG | − |VG |2 + D dz

2 1/ 2 .

(2.54)

  The velocity vz Enz is given as: 



vz Enz = C

√

2   θ4 Enz , 

(2.55)

where θ4

 ) *1/ 2 −1/ 2  2 2 2 Enz ≡ C − 2CD 4C |VG | − 4CDEnZ + D − 4CD |VG |



 

*1/ 2  ) D − 2C |VG |+2CEnz − D2 +4CDEnz +4C2 |VG |2 −4CD |VG |

−1/ 2 .

The total DOS function is given by: nzmax    ∂  gv  t9 (E,nz ) H E − Enz , N2DT (E) = ∂E 4πa2

(2.56)

nz =1

in which  )  *1/ 2  t9 (E,nz ) ≡ {2a (E − t1 ) + D} − [2a (E − t1 ) + D]2 − 4a2 (E − t1 )2 − t2 , t1 ≡ |VG | + C (π nz / dz )2 , and t2 ≡ |VG |2 + D (πnz / dz )2.

The electron statistics in QWs in UFs in n-GaP assumes the form: n2D =

nzmax   gv  t9 (EF2D ,nz ) + t10 (EF2D ,nz ) , 2 4πa nz =1

Where t10 (EF2D ,nz ) ≡

So % r=1

  Zr,Y t9 (EF2D ,nz ) in which Y = 2.

(2.57)

50

2 Fundamentals of Photoemission from Quantum Wells

The expression of the photo-emitted current density is given by: J2D =

nzmax    eCα0 gv  θ4 Enz t9 (EF2D ,nz ) + t10 (EF2D ,nz ) . √ 2 2a dz h 2 nz

(2.58)

min

In this case, the 1D dispersion relation can be written as:  kx2 =



2a2

−1

t9 (E,nz ) −

ny π dy

2  .

(2.59)

The use of (2.59) leads to the expression of the 1D carrier concentration as: n1D =

nymax nzmax      2gv  t11 EF1D ,ny ,nz + t12 EF1D ,ny ,nz , π

(2.60)

ny =1 nz =1

where 

t11 EF1D ,ny ,nz





−1 π ny 2 ≡ 2a2 . t9 (EF1D ,nz ) − dy

1/ 2 ,

So       Zr,Y t11 EF1D ,ny ,nz , and Y = 1. t12 EF1D ,ny ,nz ≡ r=1

The photocurrent in this case is given by (2.17), where EF1D should be determined from (2.60) and the subband energy E is given by: ⎡



πny E = ⎣a dy

2



πnz +C dz

2



π ny +|VG |− D dy

2



π nz + |VG |2 + D dz

2 !1/ 2

⎤ ⎦.

(2.61)

2.2.5 Photoemission from n-Germanium It is well known that the conduction electrons of n-Ge obey two different types of dispersion laws, since band nonparabolicity has been included in two different ways as given in the literature [50–51]. (a) The energy spectrum of the conduction electrons in bulk specimens of n-Ge can be expressed in accordance with Cardona et al. [50] as:

2.2

Theoretical Background

51

 2 1/ 2 Eg20 2 kz2 Eg0  2 E=− + , + + Eg0 ks ∗ 2 2m|| 4 2m∗⊥

(2.62)

where in this case m∗|| and m∗⊥ are the longitudinal and transverse effective masses along the <111> direction at the edge of the conduction band, respectively. Equation (2.62) can be written as: 2 ks2 = E (1 + αE) + α 2m∗⊥



2 kz2 2m∗||

2



2 kz2 − (1 + 2αE) 2m∗||

 .

(2.63)

In the presence of size quantization along the kz direction, the 2D dispersion relation of the conduction relations in QWs in UFs of n-Ge can be written by extending the method as given in [52] as: 2 ky2 2 kx2 + = γ (E,nz ) , 2m∗1 2m∗2

(2.64)

where m∗1 ≡ m∗⊥ , m∗2 = ⎡

m∗⊥ + 2m∗|| 3

,

γ (E,nz ) ≡ ⎣E (1 + αE) − (1 + 2αE)

2



2m∗3

nz π dz



2 +α

2

2m∗3



nz π dz

2 2

⎤ ⎦,

and m∗3 ≡

3m∗|| m∗⊥

2m∗|| + m∗⊥

.

The area of ellipse of the 2D surface as given by (2.64) can be written as:  2π m∗1 m∗2 A (E,nz ) = γ (E,nz ) . 2

(2.65)

Therefore the DOS function per subband can be expressed as:   

 4 m∗1 m∗2 2 π nz 2 N2D (E) = 1 + 2αE − 2α . 2m∗3 dz π2

(2.66)

52

2 Fundamentals of Photoemission from Quantum Wells

The total DOS function is given by: 

 nzmax    4 " ∗ ∗ 2 π nz 2 m1 m2 N2DT (E) = 1 + 2αE − 2α H E − Enz , ∗ 2 2m3 dz π nz =1

(2.67) where Enz is the positive root of the following equation: 





Enz 1 + αEnz − 1 + 2αEnz





2 2m∗3



π nz dz

2 

 +α

2 2m∗3



π nz dz

2 2 = 0. (2.68)

The velocity of the electron in the nth z subband is given by: 



vz Enz =



2 m∗||

1/2



Enz .

(2.69)

Thus, combining (2.67) with the Fermi-Dirac occupation probability factor and using (1.13), the electron statistics in this case can be written as:

n2D

1/2 nzmax         16kB T m∗1 m∗2 A1 (nz ) + 2αEnz F0 Enz + 2αkB TF1 Enz , = 2 π nz =1

(2.70)       where A1 (nz ) ≡ 1 + 2α 2 / 2m∗3 (π nz / dz )2 and ηnz ≡ (1 / kB T) EF2D −Enz . Using the appropriate equations, the photo-emitted current density assumes the form: nzmax

J2D = C1 kB T

        Enz A1 (nz ) + 2αEnz F0 Enz + 2αkB TF1 Enz ,

(2.71)

nzmin

where



  1/ 2 " m∗1 m∗2 1 2 C1 ≡ ,nzmin ≥ , −A3 + A3 − 4A2 A4 2m∗|| 2A4  2  2 π A3 ≡ (1 + 2α (W − hν)) , ∗ 2m3 dz 16π α0 e dz h2

A2 ≡ (W − hν) (1 + α (W − hν)) , and  2 2 2 π . A4 ≡ α ∗ 2m3 dz

2.2

Theoretical Background

53

Following (2.64), the expression of the 1D dispersion relation of the electrons in QWWs of Ge can be written as: 2 kx2 2 = γ − (E,n ) z 2m∗1 2m∗2



π ny dy

2 .

(2.72)

The use of (2.72) leads to the expression of the 1D carrier statistics in QWWs of Ge in the present case as: n1D

 nymax nzmax      2gv 2m∗1  t13 EF1D ,ny ,nz + t14 EF1D ,ny ,nz , = π

(2.73)

ny =1 nz =1

in which 

t13 EF1D ,ny ,nz





2 ≡ γ (EF1D ,nz ) − 2m∗2

t14 (EF1D ,nz ) ≡

So 



π ny dy

2 1/ 2 ,

   Zr,Y t13 EF1D ,ny ,nz , and Y = 1.

r=1

The photocurrent is given by (2.17) in which the quantity E is the positive root of the following equation:   2 γ E ,nz = 2m∗2



π ny dy

2 .

(2.74)

(b) The dispersion relation of the conduction electron in bulk specimens of n-Ge can be expressed in accordance with the model of Wang and Ressler [51] and can be written as: 2 kz2 2 ks2 + − c1 E= 2m∗|| 2m∗⊥



2 ks2 2m∗⊥

2

− d1

2 ks2 2m∗⊥



2 kz2 2m∗||



 − e1

where

2 c1 ≡ C 2m∗⊥ / 2 ,

C ≡ 1.4A,

m∗⊥ 2 1 4 ∗2 A≡ ,  / Eg0 m⊥ 1− 4 m0  ∗ ∗ 4m⊥ m d1 ≡ d , d ≡ 0.8A, 4

2 e1 ≡ e0 2m∗ / 2 , and e0 ≡ 0.005A.

2 kz2 2m∗||

2 , (2.75)

54

2 Fundamentals of Photoemission from Quantum Wells

Therefore the 2D dispersion law can be expressed as: E = A5 (nz ) + A6 (nz )β − c1 β 2 where 2 A5 (nz ) ≡ 2m∗3 



A6 (nz ) ≡ 1 − d1 β≡



 2 π nz 2 1 − e1 , 2m∗3 dz



 2 nz π 2 , and 2m∗3 dz

π nz dz

2 

(2.76)



2 ky2 2 kx2 + . 2m∗1 2m∗2

Equation (2.76) can be written as: 2 ky2 2 kx2 + = I1 (E,nz ) , (2.77) 2m∗1 2m∗2   1/ 2  where I1 (E,nz ) ≡ (2c1 )−1 A6 (nz ) − A26 (nz ) − 4c1 E + 4c1 A5 (nz ) . From (2.77), the area of the 2D ks -space is given by:  2π m∗1 m∗2 A (E,nz ) = I1 (E,nz ) . 2

(2.78)

The DOS function per subband can be written as:  4 m∗1 m∗2 {I1 (E,nz )} , N2D (E) = π 2   where {I1 (E,nz )} ≡ (∂ / ∂E) I1 (E,nz ) . The total DOS function assumes the form:  nzmax   4 m∗1 m∗2  {I1 (E,nz )} H E − Enz , N2DT (E) = 2 π

(2.79)

(2.80)

nz =1

  where the subband energy Enz is given by: 2 Enz = 2m∗3



π nz dz



2 − e1

2 2m∗3



π nz dz

2 2 .

(2.81)

The nzmin is the nearest integer of the following inequality: nzmin ≥ (2A7 e1 )

−1/ 2

2 1/ 2  2 π , A7 ≡ . (2.82) 1 − 1 − 4e1 (W − hν) ∗ 2m3 dz



2.2

Theoretical Background

55

The velocity of the electron in the nz th subband is given by:   1 vz Enz = 



2B0 e1

1/ 2

  θ5 Enz ,

(2.83)

where B0 ≡

" 1/ 2    1/ 2  2 , θ E 1 − 4e E . E ≡ 1 − 4e 1 − 5 nz 1 nz 1 nz 2m∗||

The electron statistics can be written as: n2D

 nzmax   4 m∗1 m∗2  t15 (EF2D ,nz ) + t16 (EF2D ,nz ) , = 2 π

(2.84)

nz =1

where t15 (EF2D ,nz ) ≡ I1 (EF2D ,nz ) , t16 (EF2D ,nz ) ≡

So 

Zr,Y (t15 (EF2D ,nz )) , and Y = 2.

r=1

The photo-emitted current density is given by: J2D

α0 e = 2dz 



2B0 e1

1/ 2   ∗ ∗  n  4 m1 m2 zmax     θ5 Enz t15 (EF2D ,nz ) + t16 (EF2D ,nz ) . 2 dz π n zmin

(2.85) In QWWs of n-Ge, following the model of Wang and Ressler, the 1D electron dispersion relation, extending (2.77), can be written as: 

 2 π ny 2 2 kx2 = I1 (E,nz ) − . 2m∗1 2m∗2 dy

(2.86)

The use of (2.86) leads to the expression of the 1D carrier statistics in QWWs of Ge in the present case as: n1D

 nymax nzmax      2gv 2m∗1  t17 EF1D ,ny ,nz + t18 EF1D ,ny ,nz , = π ny =1 nz =1

in which 

t17 EF1D ,ny ,nz





2 ≡ I (EF1D ,nz ) − 2m∗2



π ny dy

2 1/ 2 ,

(2.87)

56

2 Fundamentals of Photoemission from Quantum Wells So       t18 EF1D ,ny ,nz ≡ Zr,Y t17 EF1D ,ny ,nz , and Y = 1. r=1

The expression of the photocurrent is given by (2.17), where E is given by: 



I1 E ,nz



2 = 2m∗2



π ny dy

2 .

(2.88)

2.2.6 Photoemission from Platinum Antimonide The dispersion relation for the n-type PtSb2 can be written as [53]:  2  2   4  2  2   a 2 a 2 a 2 a 4 2 a k − lkS E + δ0 − υ k −n kS = I k , E + λ0 4 4 4 4 16 (2.89)



where 

  2  2   2   2  2 a a a a a ω1 ≡ λ0 −l , ω2 ≡ λ0 , ω3 ≡ n +υ , 4 4 4 4 4   2 2  2 a a , , I1 ≡ I ω4 ≡ υ 4 4 λ0 , l, δ0 , ν, and n are the band constants, and a is the lattice constant. Equation (2.89) can be expressed as:   

2 E + ω1 ks2 + ω2 kz2 E + δ0 − ω3 ks2 − ω4 kz2 = I1 kz2 + ks2 .

(2.90)

The use of (2.90) leads to the expression of the 2D dispersion law in ultra thin films of n-PtSb2 as:   " ks2 = [2A9 ]−1 −A10 (E,nz ) + A210 (E,nz ) + 4A9 A11 (E,nz ) , (2.91) where A9 ≡ [I1 + ω1 ω3 ] ,  A10 (E,nz ) ≡ ω3 E + ω1 E + δ0 − ω4



πnz dz

2 !

+ ω 2 ω3

π nz dz

2



 π nz 2 , +2I1 dz

2.2

Theoretical Background

57

 



A11 (E,nz ) ≡ E E + δ0 − ω4 + ω2

π nz dz

2 

π nz dz

2 

E + δ0 − ω4

π nz dz

2 

− I1

π nz dz

4  .

The area of ks space can be expressed as: A (E,nz ) =

π t19 (E,nz ) , 2A9

(2.92)

  " where t19 (E,nz ) ≡ −A10 (E,nz ) + A210 (E,nz ) + 4A9 A11 (E,nz ) . The DOS function per subband is given by: N2D (E) =

 gυ ∂  t19 (E,nz ) . 4πA9 ∂E

(2.93)

The total DOS function assumes the form: nzmax    ∂  gυ  t19 (E,nz ) H E − Enz , N2DT (E) = 4πA9 ∂E

(2.94)

nz =1

where the quantized levels Enz can be expressed through the equation:  

 π nz 2 Enz = (2) + δ0 − ω4 − ω2 dz ⎧ 



2

⎨ π nz 2 π nz 2 π nz 4 + + δ0 − ω4 + 4 I1 ω2 ⎩ dz dz dz ⎤



!1/ 2 π nz 4 π nz 2 ⎦. + ω2 ω4 − ω2 δ0 dz dz −1



π nz dz

2



(2.95)

  The vz Enz is given by:   1   νz Enz = θ6 Enz ,  in which 

θ6 Enz



  A13 Enz  , ≡ A12 Enz

(2.96)

58

2 Fundamentals of Photoemission from Quantum Wells

    A13 Enz ≡ 2 [2 (ω2 ω4 + I1 )]1/ 2 ω2 (E + δ0 ) − Eω4 − {Eω4 − ω2 (E + δ0 )}2 1/ 2 + 4 (ω2 ω4 + I1 ) E (E + δ0 )]1/ 2 ,    1 A12 Enz ≡ (ω2 − ω4 ) − 2 (Eω4 − ω2 (E + δ0 )) (ω4 − ω2 ) 2  + 4 (ω2 ω4 + I1 ) (2E + δ0 ) , −1/ 2   {Eω4 − ω2 (E + δ0 )}2 + 4 (ω2 ω4 + I1 ) E (E + δ0 ) . The electron statistics can be written as: n2D

nzmax ∞  π  ∂  = t19 (E,nz ) f (E) dE, 2 2A ∂E (2π) 9

2gv

nz =1E

nz

n2D =

gv 4πA9

nzmax



 t19 (EF2D ,nz ) + t20 (EF2D ,nz ) ,

(2.97)

nz =1

where t20 (EF2D ,nz ) ≡

S 

  Zr,Y t19 (EF2D ,nz ) , and Y = 2.

r=1

The photo-emitted current density assumes the form: J2D

nzmax    α0 gv e  = θ6 Enz t19 (EF2D ,nz ) + t20 (EF2D ,nz ) , 4dz hA9 n

(2.98)

zmin

where ⎡

nzmin

2 2 2 ! 1 ⎢ π π π ≥ √ ⎣ ω2 (W − hν) + ω2 δ0 − ω4 (W − hν) dz dz dz 2 ⎡ 2 2 2 !2 π π π + ω2 δ0 − ω4 (W − hν) − ⎣ ω2 (W − hν) dz dz dz  + 4 ω2 ω4



π dz

4

+ I1

π dz

4  

⎤1/ 2 ⎤1/ 2  ⎥ (W − hν)2 + δ0 (W − hν) ⎦ ⎦ .

Extending (2.91), the electron dispersion relation in QWWs of PtSb2 can be written as:

2.2

Theoretical Background

 kx2

= [2A9 ]

−1

59

2   " π n y −A10 (E,nz ) + A210 (E,nz ) + 4A9 A11 (E,nz ) − . dy



(2.99) The use of (2.99) leads to the expression for 1D electron statistics: n1D =

nymax nzmax      2gv  t21 EF1D ,ny ,nz + t22 EF1D ,ny ,nz , π

(2.100)

ny =1 nz =1

where   )  t21 EF1D ,ny ,nz ≡ [2A9 ]−1 −A10 (EF1D ,nz ) *1/ 2  π n 2 ) y 2 − + A10 (EF1D ,nz ) + 4A9 A11 (EF1D ,nz ) dy

!1/ 2 ,

S       Zr,Y t21 EF1D ,ny ,nz , and Y = 1. t22 EF1D ,ny ,nz ≡ r=1

The photoemission in this case is given by (2.17), in which E is given by: 

−1

E = (2)



)

*1/ 2 

− (l1 + l2 ) + (l1 + l2 ) − 4 (l1 l2 − l3 ) 2

,

(2.101)

where l1 ≡ ω1

π ny dy

2

+ ω2

π nz dz

l3 ≡ I 1

2 !

, l2 ≡ δ0 − ω3

πny dy



2 +

π nz dz

2 !2

π ny dy

2

− ω4

π nz dz

2 ! , and

.

2.2.7 Photoemission from Stressed Materials The dispersion relation of the conduction electrons in bulk specimens of stressed materials can be written as [47]: kx2 [a∗ (E)]2 where

+

ky2 [b∗ (E)]2

+

kz2 [c∗ (E)]2

= 1,

(2.102)

60

2 Fundamentals of Photoemission from Quantum Wells



∗

2

a (E)

=



K 0 (E) A0 (E) + 12 D0 (E)

, K 0 (E) = E − C1 ε −

2 2C22 εxy



3E g0

3E g0

2B22

 ,

C1 is the conduction band deformation potential, ε is the trace of the strain tensor εˆ which can be written as ⎡

⎤ εxx εxy 0 εˆ = ⎣ εxy εyy 0 ⎦ , 0 0 εzz C2 is a constant which describes the strain interaction between the conduction and valance bands, E g0 = Eg0 + E − C1 ε, Eg0 is the band gap in the absence of any field, B2 is the momentum matrix element, 

b0 ε (a0 + C1 ) 3b0 εxx A0 (E) = 1 − + −  Eg0 2Eg0 2E g0 a0 = −

 ,

  1 1 2n l + 2m , b0 = l − m , d0 = √ , 3 3 3

l,m,n are the matrix elements of the strain perturbation operator, D0 (E) = √ εxy d0 3 Eg , 0

 ∗ 2 b (E) =

2  K0 (E) , and , c∗ (E) = L0 (E) A0 (E) −   b0 ε (a0 + C1 ) 3b0 εzz L0 (E) = 1 − +  −  E g0 E g0 2E g0 K0 (E)

1 2 D0 (E)

In the presence of size quantization along the x-direction, the 2D electron energy spectrum assumes the form: ky2 [b∗ (E)]2

+

kz2 [c∗ (E)]2

=1−

1 [a∗ (E)]2

(π nx / dx )2 .

(2.103)

The subband energy (Enx ) is given by:   a∗ Enx = nx π / dx .

(2.104)

The area of 2D wave vector space enclosed by (2.103) can be written as:  A = πb∗ (E) c∗ (E) 1 −



π nx dx a∗ (E)

2  .

(2.105)

2.2

Theoretical Background

61

The velocity of the electron in the subband characterized by the energy Enx is given by:   1 1  , vx Enx =  θ7 Enx

(2.106)

where 



2 

 2x1 En3x + (x2 + 3x1 x5 ) En2x + 2x2 x5 Enx − x5 x3 + x4 ,     3εC1 3 x1 ≡ , x2 ≡ 2x1 Eg0 − C1 ε − , 2B22 2B22   2 C22 εxy  2 3εC1  3  x3 ≡ + 2 Eg0 − C1 ε − 2 Eg0 − C1 ε , B22 B2 2B2   2  2 C22 εxy  3εC1  x4 ≡ Eg0 − C1 ε + Eg0 − C1 ε , and B22 B22 √ ⎤ ⎡ d 3 εxy 0 b0 ε 3b0 εxx ⎦. x5 ≡ ⎣Eg0 − C1 ε − (a0 + C1 ) + − + 2 2 2

θ7 Enx



≡

1 Enx + x5

The emission condition is: Enzmin + hv ≥ W,

(2.107)



  whereEnzmin should be determined by the equation a∗ Enxmin = nxmin (π / dx ), in which nxmin satisfies the following inequality: nxmin ≥

 dx  ∗ a (W − hν) . π

(2.108)

The electron statistics can be written as: n2D

nxmax   gv  = t23 (EF2D ,nx ) + t24 (EF2D ,nx ) , 2π

(2.109)

nx =1

where  ∗

∗

t23 (EF2D ,nx ) ≡ b (EF2D ) c (EF2D ) 1 − t24 (EF2D ,nx ) ≡

so  r=1



π nx ∗ dx a (EF2D )

Zr,Y t23 (EF2D ,nx ), where Y = 2.

2 ! and

62

2 Fundamentals of Photoemission from Quantum Wells

The photoemission current density is given by:

J2D

nxmax 1 α0 gv e  [t23 (EF2D ,nx ) + t24 (EF2D ,nx )]. = 2hdx n θ7 (Enx )

(2.110)

xmin

  In the absence of stress and under the substitution B22 ≡ 32 Eg0 / 4m∗ ,, equation (2.109) assumes the same form as given by (2.34). The 1D dispersion relation in stressed Kane-type materials can be written extending (2.103) as: kx2

∗

= [a (E)]

2

1 1− ∗ [b (E)]2



π ny dy

2

1 − ∗ [c (E)]2



π nz dz

2 ! .

(2.111)

Using (2.111), the 1D electron statistics can be written as: n1D =

nymax nzmax   2gv  p1 (EF1D ,ny ,nz ) + p2 (EF1D ,ny ,nz ) , π

(2.112)

ny =1 nz =1

where  p1 (EF1D ,ny ,nz ) ≡ [a∗ (EF1D )]2 1 −

2 π ny 1 ∗ 2 dy [b (EF1D )]

2 !1/ 2 1 π nz − ∗ , 2 dz [c (EF1D )]

p2 (EF1D ,ny ,nz ) ≡

so 

  Zr,Y p1 (EF1D ,ny ,nz ) , and Y = 1.

r=1

The expression of the photocurrent in this case is given by (2.17), where E is defined through the following equation: 1 [b∗ (E )]2



π ny dy

2

1 + ∗  2 [c (E )]



π nz dz

2 = 1.

(2.113)

2.2.8 Photoemission from Bismuth 2.2.8.1 The McClure and Choi Model The dispersion relation of the carriers in Bi can be written, following McClure and Choi [32], as:

2.2

Theoretical Background

E (1 + αE) =

63

  p2y p2y p2 p2x m2 + + z + αE 1 − 2m1 2m2 2m3 2m2 m2

+

p4y α 4m2 m2

−

αp2x p2y 4m1 m2

−

αp2y p2z 4m2 m3

(2.114)

,

where pi ≡ ki , i = x,y,z, m1 ,m2 and m3 are the effective carrier masses at the band-edge along the x, y, and z directions, respectively, and m2 is the effective-mass tensor component at the top of the valence band (for electrons) or at the bottom of the conduction band (for holes). In the presence of size quantization along the y-direction, the 2D dispersion relation is given by:  kx2



!

 π ny 2 π ny 2 α2 2 2 − + kz dy 2m3 4m2 m3 dy (2.115)

4 !

  π ny π ny 2 α m2 . − αE 1 − 4m2 m2 dy m2 dy

α2 4m1 m2

2 − 2m1

= E (1 + αE) −



The 2D area assumes the form: √  2π m1 m3  t25 E,ny , A (E) = 2 

(2.116)

where 

t25 E,ny







−1 

π ny 4 α2 π ny 2 α ≡ 1− E (1 + αE) − 2m2 dy 4m2 m2 dy   

2 πny m2 −αE 1 − . m2 dy

The subband energy Eny can be expressed as:   Eny 1 + αEny −

α 4m2 m2



π ny dy

4

 

π ny 2 m2 −αEny 1 − = 0. (2.117) m2 dy

The DOS function per subband is given by: N (E) =

√ $ gv m1 m3 #  t25 E,ny . 2 π

(2.118)

The total 2D DOS function can be written as: N2DT (E) =

nymax √ #  $   gv m1 m3  t25 E,ny H E − Eny . 2 π ny =1

(2.119)

64

2 Fundamentals of Photoemission from Quantum Wells

The velocity of the photoemitted electron in the nth y subband is given by:   vy Eny = where 

θ10 Eny



πny  , dy θ10 Eny

(2.120)





2 αEny α 1 m2 1 m2 ≡ + − 1−  + 1−  m2 m2 2 m2 m2 m2   !−1/2 

 4αEny 1 + αEny αEny m2 1 + + 1 − m2 m2 m2 m2 m2   

2αEny 1 + 2αEny α m2 1−  + . m2 m2 m2 m2

m2 m2 2α

The nymin can be determined from the following equation:

π nymin 4 α (W − hν) (1 + α (W − hν)) − 4m2 m2 dy  

2 π nymin m2 − α (W − hν) 1 − = 0.  m2 dy

(2.121)

Using (2.119) and the Fermi-Dirac occupation probability factor, the electron statistics can be written as: n2D

nymax √      gv m1 m3  t25 EF2D ,ny + t26 EF2D ,ny , = π2

(2.122)

ny =1

so   %    where t26 EF2D ,ny ≡ Zr,Y t25 EF2D ,ny and Y = 2. r=1

The photo-emitted current density is given by: J2D =

nymax √   −1      α0 egv m1 m3  ny θ10 Eny t25 EF2D ,ny + t26 EF2D ,ny . 2 2dy n ymin

(2.123) The 1D dispersion relation of the carriers in Bi in this case can be written as:    2!

 π ny 2 m2 2 kx2 α2 π ny 2 αE 1−  E (1 + αE) = 1− +G12 + , 2m1 2m2 dy 2m2 m2 dy (2.124)

2.2

Theoretical Background

65

where ⎧   ⎫ ⎨ 2 π n 2 2 π n 2 2n n π 2 2⎬ 4 π n 4  α α y y y z z G12 ≡ + + − . ⎩2m2 dy ⎭ 2m3 dz 4m2 m2 dy 4m2 m3 dy dz Using (2.124), the 1D electron statistics can be expressed as: √

nymax nzmax      2m1  n1D t27 EF1D ,ny ,nz + t28 EF1D ,ny ,nz ,  ny =1 nz =1 ⎧

−1/ 2   ⎨ α2 π ny 2 t27 EF1D ,ny ,nz ≡ 1− ⎩ 2m2 dy

2gv = π

 EF1D (1 + αE1DF ) − G12 −

2 2m2

 αE 1 −



m2 m2



π ny dy

(2.125)

⎫

2 1/2 ⎬ ⎭

,

so       Zr,Y t27 EF1D ,ny ,nz , and Y = 1. t28 EF1D ,ny ,nz ≡ r=1

The expression of the photocurrent is given by (2.17), where E is given by: 

E = (2α)  in which l4 ≡ 1 −

2 α 2m2



−1



 " 2 −l4 + l4 + 4αG12 ,

(2.126)





πny 2 m2 1 −  dy m2

.

2.2.8.2 The Hybrid Model The dispersion relation of the carriers in bulk specimens of Bi in accordance with the Hybrid model can be represented as [31]:  2 αγ0 4 ky4 θ0 (E) ky 2 kz2 2 kx2 + + + , E (1 + αE) = 2M2 2m1 2m3 4M22

(2.127)

in which   M2 M2 , δ0 ≡  , θ0 (E) ≡ 1 + αE (1 − γ0 ) + δ 0 , γ0 ≡ m2 M2 and the other notations are defined in [31]. In the presence of size quantization along the y-direction, the 2D electron dispersion relation can be written as:

66

2 Fundamentals of Photoemission from Quantum Wells

2 kz2 2 kx2 θ0 (E) 2 + = E (1 + αE) − 2m1 2m3 2M2



π ny dy

2 −

αγ0 4 4M22



π ny dy

4 .

(2.128)

The 2D area is given by: √  2π m1 m3  (2.129) t29 E,ny A E,ny = 2   



  θ0 (E) 2 π ny 2 αγ0 4 π ny 4 − . t29 E,ny = E (1 + αE) − 2M2 dy dy 4M24 



The subband energy are given as: 

Eny 1 + αEny



 



θ0 Eny 2 π ny 2 αγ0 4 π ny 4 − = 0. − 2M2 dy dy 4M22

(2.130)

The total DOS function in this case can be written as: nymax √ #  $   gv m1 m3  N2DT (E) = t29 E,ny H E − Eny . π2

(2.131)

ny =1

The electron velocity in the nth y band is given by:   vy Eny =

πny  , dy θ11 Eny

(2.132)

in which 

θ11 Eny



⎡    ⎤  θ0 Eny α (1 − γ0 ) + αγ0 1 + 2αEny M2 ⎣ ⎦. "  = −α (1 − γ0 ) +    αγ0 θ02 Enuy + 2αγ0 Eny 1 + αEny

The use of (2.129) leads to the 2D electron statistics in QWs in UFs of Bi in this case as: n2D

nymax √      gv m1 m3  t29 EF2D ,ny + t30 EF2D ,ny , = 2 π

(2.133)

ny =1

so   %    Zr,Y t29 EF2D ,ny , and Y = 2. in which t30 EF2D ,ny = r=1

The photo-emitted current density in this case is expressed as: J2D

nxmax √      αo gv e m1 m3  1   t29 EF2D ,ny + t30 EF2D ,ny , = ny 2 2dy  θ11 Eny n xmin

(2.134)

2.2

Theoretical Background

67

where nymin can be determined from: θ0 (W − hυ) 2 (W − hν) (1 + α (W − hν)) 2M2



π nymin dy

2

αγ0 4 − 4M22



π nymin dy

4 = 0. (2.135)

The 1D dispersion relation in this case assumes the form: E (1 + αE) =

2 kx2 2 + G14 + 2m1 2M2



π ny dy

2 αE (1 − γ0 ) ,

(2.136)

where  G14 =

2 2m3



π nz dz

2

2 + 2M2



π ny dy

2



1 + δ0



αγ0 4 + 4M22



π ny dy

4  .

The use of (2.136) leads to the expression for the electron concentration per unit length as:

n1D

2gv = π

where 



√

nymax nzmax      2m1  t31 EF1D ,ny ,nz + t32 EF1D ,ny ,nz , 

(2.137)

ny =1 nz =1



t31 EF1D ,ny ,nz ≡ EF1D (1 + αEF1D ) − G14 −

2

2M2



π ny dy

1/ 2

2 αEF1D (1 − γ0 )

,

so       Zr,Y t31 EF1D ,ny ,nz , and Y = 1. t32 EF1D ,ny ,nz ≡ r=1

The photocurrent is given by (2.17) where E is given by:   " E = (2α)−1 −l6 + l62 + 4αG14 , where

 l6 = 1 −

2 α 2M2



π ny dy

(2.138)

!

2 (1 − γ0 )

.

2.2.8.3 The Cohen Model In accordance with the Cohen model [30], the dispersion law of the carriers in Bi is given by:

68

2 Fundamentals of Photoemission from Quantum Wells

E (1 + αE) =

αEp2y p2y (1 + αE) αp4y p2 p2x + z − + + . 2m1 2m3 2m2 2m2 4m2 m2

(2.139)

In the presence of size quantization along the y-direction, the 2D electron dispersion law in this case is given by:





αE2 π ny 2 (1 + αE)2 π ny 2 α4 π ny 4 2 kx2 2 kz2 − − = + . 2m2 dy 2m2 dy 4m2 m2 dy 2m1 2m3 (2.140) The subband energy can be written as:

E (1 + αE)+

  αEny 2 Eny 1 + αEny + 2m2



πny dy

2 −

(1 + αEny )2 2m2



πny dy

2 −

α 4 4m2 m2



π ny dy

4 = 0. (2.141)

The 2D area can be expressed as: √   2π m1 m3   A E,ny = t33 E,ny , 2 

(2.142)

where   αE2 t33 E,ny = E (1 + αE) + 2m2



πny dy

2 −

(1 + αE)2 2m2



π ny dy

2 −

α 4 4m2 m2



π ny dy

4 .

The total DOS function in this case assumes the form: nymax √ #  $   gv m1 m3  t33 E,ny H E − Eny . N2DT (E) = 2 π

(2.143)

ny =1

The electron velocity in the ny th subband for the Cohen model is same as that of the McClure and Choi models, respectively, and therefore the photo-emitted current density is given by: J2D =

nxmax √      αo gv e m1 m3  1   n ,n ,n E + t E , t y 33 F2D y 34 F2D y 2dy2  θ10 Eny n

(2.144)

xmin

where nymin is defined through the equation:

α (W − hν) 2 π nymin 2 (W − hν) (1 + α (W − hν)) + 2m2 dy 

2  

 2 π nymin π nymin 4 (1 + α (W − hν) ) α4 − − = 0, 2m2 dy 4m2 m2 dy (2.145) and

2.2

Theoretical Background

69

so       t34 EF2D ,ny = Zr,Y t33 EF2D ,ny , and r=1

Y = 2. The electron statistics assumes the form: n2D =

nymax √      gv m1 m3  t E + t E . ,n ,n 33 F2D y 34 F2D y π2

(2.146)

ny =1

The 1D carrier dispersion law in this case can be written as: αE2 + El7 − G15 =

2 kx2 , 2m1

(2.147)

where 





 α2 π ny 2 α2 π ny 2 + and l7 = 1 − 2m2 dy 2m2 dy 





 π ny 4 2 π nz 2 2 π ny 2 α4 G15 = + + . 2m3 dz 2m2 dy 4m2 m2 dy The 1D electron concentration per unit length is given by: n1D =

2gv π

√

nymax nzmax      2m1  t35 EF1D ,ny ,nz + t36 EF1D ,ny ,nz , 

(2.148)

ny =1 nz =1

where 1/2    2 + EF1D l7 − G15 , t35 EF1D ,ny ,nz = αEF1D so       Zr,Y t35 EF1D ,ny ,nz , and t36 EF1D ,ny ,nz = r=1

Y = 1. The photocurrent is given by (2.17), where E is given by: 

E = (2α)

−1



 " 2 −l7 + l7 + 4αG15 .

(2.149)

70

2 Fundamentals of Photoemission from Quantum Wells

2.2.8.4 The Lax Model The electron energy spectra in bulk specimens of Bi in accordance with the Lax model can be written as [24]: p2y p2 p2x + + z . 2m1 2m2 2m3

E (1 + αE) =

(2.150)

The 2D electron dispersion relation can be written as: 2 E (1 + αE) − 2m2



π ny dy

2 =

2 kz2 2 kx2 + . 2m1 2m3

(2.151)

The subband energy is given by:   2 2  πny / dy . Eny 1 + αEny = 2m2

(2.152)

The 2D area can be written as: √   2π m1 m3   A E,ny = E , t , n 37 F2D y 2

(2.153)

where 

t37 EF2D ,ny





2 = E (1 + αE) − 2m2



π ny dy

2  .

The total DOS function in this case assumes the form: N2DT (E) =

nymax √ #  $   gv m1 m3  t37 E, ny H E − Eny . 2 π

(2.154)

ny =1

The electron velocity in the ny th subband is given by:   vy Eny =

π ny  . dy m3 1 + 2αEny

(2.155)

The photoemission current density in this case can thus be given as: J2D =

α0 gv ekB T 2dy2 



m1 m3

+2αkB TF1 ηny

1/2 n xmax 

nxmin

ny 

    1  1 + 2αEny F0 ηny 1 + 2αEny

,

where nymin satisfies the following inequality:

(2.156)

2.2

Theoretical Background

nymin and ηny

71

dy ≥ π

√ 2m2 [(W − hν) (1 + α (W − hν))]1/2 

(2.157)

⎡ 1/2 ⎤  2 π n 2  1  2α y ⎦. = EF2D − Eny and Eny = (2α)−1 ⎣−1 + 1 + kB T m2 dy

The use of (2.154) and Fermi-Dirac probability function leads to the expression of the electron statistics as: n2D =

nymax √       (kB Tgv ) m1 m3  1 + 2αEny F0 ηny + 2αkB TF1 ηny . (2.158) 2 π ny =1

The 1D dispersion relation in this case can be written as: E (1 + αE) =

2 kx2 + G16 , 2m1

(2.159)

where G16 =

2 2m2



π ny dy

2 +

2 2m3



π nz dz

2 .

The 1D electron statistics are given by: n1D where

2gv = π

√

nymax nzmax      2m1  t37 EF1D , ny , nz + t38 EF1D , ny , nz , 

(2.160)

ny =1 nz =1

  t37 EF1D , ny , nz = [EF1D (1 + αEF1D ) − G16 ]1/2 , so       Zr,Y t37 EF1D , ny , nz , and Y = 1. t38 EF1D , ny , nz = r=1

The photocurrent is given by (2.17), where E is given by:    E = (2α)−1 −1 + 1 + 4αG16 .

(2.161)

It may be noted that under the conditions α → 0, M2 → ∞ and isotropic effective electron mass at the edge of the conduction band, all models of Bismuth convert into isotropic parabolic energy bands. Thus under the aforementioned conditions and the conversion of the summation over the quantum numbers to the corresponding integrations, lead to the well known expression of the photo-emitted current density for non-degenerate wide-gap materials as given in the introduction.

72

2 Fundamentals of Photoemission from Quantum Wells

2.2.9 Photoemission from (n, n) and (n, 0) Carbon Nanotubes For armchair and zigzag carbon nanotubes, the energy dispersion relations are given by [47]:  √  √ 1/2 ky ac 3 2 ky ac 3 , cos + 4 cos E = tc 1 + 4 cos n 2 2 √ √ −π/ 3ac < ky < π/ 3ac

 vπ 1/2 3ky ac 2 vπ , cos + 4 cos E = tc 1 + 4 cos 2 n n   −π/3ac < ky < π/3ac , 



vπ

(2.162)

(2.163)

where tc is the tight binding parameter, v = 1, 2, . . . , 2n, and ac is the nearest neighbor C–C bonding distance. Using (2.162) and (2.163), the electron statistics for both cases can, respectively, be written as [47] : max       8 Ac1 EF1 , i + Bc1 EF1 , i √ πac 3 i=1

i

n1D =

and n1D =

(2.164)

imax      8  Ac2 EF1 , i + Bc2 EF1 , i , 3πac

(2.165)

i=1

where ⎡ ⎡

Ac 1

⎤⎤  ⎡ 2  2 ⎤1/2 2 2 E E E ⎢1 ⎢ ⎥⎥ F1 − 1 ⎦ ⎦⎦ , EF1 ,i = cos−1 ⎣ ⎣− 2i − 5 + ⎣ 2i − 5 + 16 8 tc tc tc2





Ei =



|3i − m + n| ac |tc | , 2 r0

r0 is the radius of the nanotube, EF1 is the Fermi energy as measured from the  middle  E , B , i = of the band gap in the vertically upward direction, i = 1,2,3 . . . i max c F 1 1    %s Z , i , Y = 1, A E c1 F1 r=1 rY Ac2



 EF1 , i = cos−1



EF2 1 tc2

−1−

s       Bc2 EF1 , i = ZrY Bc1 EF1 , i . r=1

2  

Ei −1 tc

2Ei −1 tc

−1  , and

2.3

Results and Discussions

73

The generalized expression of photoemission from both types of CNs assumes the form:

imax 4α0 ekB T  [F0 (ξ1 )], (2.166) I1Dn,n = 2π i=1

  where ξ1 ≡ (kB T)−1 EF1 − Ei .

2.3 Results and Discussions Using (2.10) and (2.11), and taking the energy band constants as given in Appendix A, we have plotted the normalized photoemitted current density from QWs in UFs of CdGeAs2 (an example of nonlinear optical materials) as a function of dx , as shown in plot (a) of Fig. 2.1, in which plot (b) corresponds to δ = 0. Plot (c) has been drawn in accordance with the three-band model of Kane, and plot (d) refers to the two-band model of Kane. Plot (e) exhibits variation in accordance with the parabolic energy bands for the overall assessments of the energy band constants on photoemission in this case. Figure 2.2 exhibits the plots of the normalized photoemitted current density from QWs in UFs of CdGeAs2 as a function of the normalized incident photon energy for all cases of Fig. 2.1 and 2.3 show the dependence of the said variable on the normalized electron degeneracy for all cases of Fig. 2.1.

Fig. 2.1 Plot of the normalized photocurrent density from QWs in UFs of CdGeAs2 as a function of dz in accordance with (a) generalized band model, (b) δ = 0, (c) the three-band model of Kane, (d) the two-band model of Kane, and (e) the parabolic energy bands

74

2 Fundamentals of Photoemission from Quantum Wells

Fig. 2.2 Plot of the normalized photocurrent density from QWs in UFs of CdGeAs2 as a function of normalized incident photon energy for all cases of Fig. 2.1

Fig. 2.3 Plot of the normalized photocurrent density from QWs in UFs of CdGeAs2 as a function of normalized electron degeneracy for all cases of Fig. 2.1

2.3

Results and Discussions

75

The normalized photoemitted current density from QWs in UFs of n-InAs (an example of III–V materials) in accordance with the three- (using (2.23) and (2.24)) and two- (using (2.32) and (2.34)) band models of Kane as functions of film thickness, normalized incident photon energy, and the normalized electron degeneracy have been presented in Figs. 2.4, 2.5, and 2.6 respectively. Figures 2.7, 2.8, and 2.9 exhibit the variations of normalized photoemitted current density for n-InSb as functions of film thickness, normalized incident photon energy, and the normalized electron degeneracy, respectively. Using (2.46) and (2.47), the variations of the normalized photocurrent density from QWs in UFs of CdS (an example of II–VI materials) as functions of thickness, normalized incident photon energy, and normalized electron degeneracy have respectively been drawn in Figs. 2.10, 2.11, and 2.12, where the plots for Co = 0 have further been drawn for the purpose of assessing the influence of the splitting of the two-spin states by the spin orbit coupling and the crystalline field. The thickness, normalized photon energy, and normalized electron degeneracy dependences of normalized photoemitted current density from GaP have been shown in Figs. 2.13, 2.14, and 2.15, respectively, by using (2.58) and (2.57). The dependence of normalized photoemitted current density with reference to the aforementioned variables from QWs in UFs of n-Ge (using (2.70), (2.71) and (2.84), (2.85)) and PtSb2 (using (2.97) and (2.98)) has been shown in Figs. 2.16, 2.17, 2.18, 2.19, 2.20, and 2.21 in accordance with the models of Cardona et al.

Fig. 2.4 Plot of the normalized photocurrent density from QWs in UFs of n-InAs as a function of dz in accordance with (a) the three-band model of Kane and (b) the two-band model of Kane

76

2 Fundamentals of Photoemission from Quantum Wells

Fig. 2.5 Plot of the normalized photocurrent density from QWs in UFs of n-InAs as a function of normalized incident photon energy in accordance with (a) the three-band model of Kane and (b) the two-band model of Kane

Fig. 2.6 Plot of the normalized photocurrent density from QWs in UFs of n-InAs as a function of normalized electron degeneracy in accordance with (a) the three-band model of Kane and (b) the two-band model of Kane

2.3

Results and Discussions

77

Fig. 2.7 Plot of the normalized photocurrent density from QWs in UFs of n-InSb as a function of dz in accordance with (a) the three-band model of Kane and (b) the two-band model of Kane

Fig. 2.8 Plot of the normalized photocurrent density from QWs in UFs of n-InSb as a function of normalized incident photon energy in accordance with (a) the three-band model of Kane and (b) the two-band model of Kane

78

2 Fundamentals of Photoemission from Quantum Wells

Fig. 2.9 Plot of the normalized photocurrent density from QWs in UFs of n-InSb as a function of normalized electron degeneracy in accordance with (a) the three-band model of Kane and (b) the two-band model of Kane

Fig. 2.10 Plot of the normalized photocurrent density from QWs in UFs of CdS as a function of dz with (a) C0  = 0 and (b) C0 = 0

2.3

Results and Discussions

79

Fig. 2.11 Plot of the normalized photocurrent density from QWs in UFs of CdS as a function of normalized incident photon energy with (a) C0 = 0 and (b) C0 = 0

Fig. 2.12 Plot of the normalized photocurrent density from QWs in UFs of CdS as a function of normalized electron degeneracy with (a) C0 = 0 and (b) C0 = 0

80

2 Fundamentals of Photoemission from Quantum Wells

Fig. 2.13 Plot of the normalized photocurrent density from QWs in UFs of n-GaP as a function of dz

Fig. 2.14 Plot of the normalized photocurrent density from QWs in UFs of n-GaP as a function of normalized incident photon energy

2.3

Results and Discussions

81

Fig. 2.15 Plot of the normalized photocurrent density from QWs in UFs of n-GaP as a function of normalized electron degeneracy

Fig. 2.16 Plot of the normalized photocurrent density from QWs in UFs of n-Ge as a function of thickness in accordance with the models of (a) Cardona et al. and (b) Wang et al.

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Fig. 2.17 Plot of the normalized photocurrent density from QWs in UFs of n-Ge as a function of normalized incident photon energy for both the cases of Fig. 2.16

Fig. 2.18 Plot of the normalized photocurrent density from QWs in UFs of n-Ge as a function of normalized electron degeneracy for both the cases of Fig. 2.16

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83

Fig. 2.19 Plot of the normalized photocurrent density from QWs in UFs of n-PtSb2 as a function of thickness

[50], Wang and Ressler [51], and Emtage [53], respectively. Figures 2.22, 2.23, and 2.24 manifest the variations of the normalized photoemitted current density from QWs in UFs of stressed n-InSb (using (2.109) and (2.110)) as functions of the film thickness, normalized incident photon energy, and the normalized electron degeneracy respectively. Figures 2.25, 2.26, and 2.27 exhibit the normalized photoemitted current density from QWs in UFs of bismuth as functions of film thickness, normalized incident photon energy, and normalized electron degeneracy, in accordance with the models of (a) McClure and Choi (using (2.122) and (2.123)), (b) Cohen (using (2.144) and (2.146)), (c) Hybrid (using (2.133) and (2.134)), and (d) Lax (using (2.156) and (2.158)), respectively. The normalized photocurrent from QWWs for all the aforementioned materials as a function of film thickness, normalized incident photon energy, and normalized electron degeneracy has been plotted in Figs. 2.28, 2.29, 2.30, 2.31, 2.32, 2.33, 2.34, 2.35, 2.36, 2.37, 2.38, 2.39, 2.40, 2.41, 2.42, 2.43, 2.44, 2.45, 2.46, 2.47, 2.48, 2.49, 2.50, 2.51, 2.52, 2.53, and 2.54, respectively. Using (2.164), (2.165), and (2.166), we have plotted in Figs. 2.55 and 2.56 the variations of the normalized photocurrent as functions of normalized electron degeneracy and normalized incident photon energy for (13, 6), (16, 0), (10, 10), and (22, 19) SWCNs having diameter of 1.36, 1.28, 1.38, and 2.82 nm, respectively. It should be noted that (16, 0) and (10, 10) are semiconductor zigzag and metallic armchair CNs, while (13, 6) and (22, 19) are chiral semiconductor and metallic CNs, respectively.

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Fig. 2.20 Plot of the normalized photocurrent density from QWs in UFs of n-PtSb2 as a function of normalized incident photon energy

Fig. 2.21 Plot of the normalized photocurrent density from QWs in UFs of n-PtSb2 as a function of normalized electron degeneracy

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85

Fig. 2.22 Plot of the normalized photocurrent density from QWs in UFs of stressed n-InSb as a function of film thickness

Fig. 2.23 Plot of the normalized photocurrent density from QWs in UFs of stressed n-InSb as a function of normalized incident photon energy

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2 Fundamentals of Photoemission from Quantum Wells

Fig. 2.24 Plot of the normalized photocurrent density from QWs in UFs of stressed n-InSb as a function of normalized electron degeneracy

Fig. 2.25 Plot of the normalized photocurrent density from QWs in UFs of Bismuth as a function of film thickness in accordance with the models of (a) McClure and Choi, (b) Cohen, (c) Hybrid, and (d) Lax, respectively

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87

Fig. 2.26 Plot of the normalized photocurrent density from QWs in UFs of Bismuth as a function of normalized incident photon energy for all the cases of Fig. 2.25

Fig. 2.27 Plot of the normalized photocurrent density from QWs in UFs of Bismuth as a function of normalized electron degeneracy for all the cases of Fig. 2.25

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Fig. 2.28 Plot of the normalized photocurrent from QWWs of CdGeAs2 as a function of dz in accordance with (a) generalized band model, (b) δ = 0 , (c) the three-band model of Kane, (d) the two-band model of Kane, and (e) the parabolic energy bands

Fig. 2.29 Plot of the normalized photocurrent from QWWs of CdGeAs2 as a function of normalized incident photon energy for all cases of Fig. 2.28

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89

Fig. 2.30 Plot of the normalized photocurrent from QWWs of CdGeAs2 as a function of normalized electron degeneracy for all cases of Fig. 2.28

Fig. 2.31 Plot of the normalized photocurrent from QWWs of n-InAs as a function of dz in accordance with (a) the three-band model of Kane and (b) the two-band model of Kane

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Fig. 2.32 Plot of the normalized photocurrent from QWWs of n-InAs as a function of normalized incident photon energy in accordance with (a) the three-band model of Kane and (b) the two-band model of Kane

Fig. 2.33 Plot of the normalized photocurrent from QWWs of n-InAs as a function of normalized electron degeneracy in accordance with (a) the three-band model of Kane and (b) the two-band model of Kane

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91

Fig. 2.34 Plot of the normalized photocurrent from QWWs of n-InSb as a function of dz in accordance with (a) the three-band model of Kane and (b) the two-band model of Kane

Fig. 2.35 Plot of the normalized photocurrent from QWWs of n-InSb as a function of normalized incident photon energy in accordance with (a) the three-band model of Kane and (b) the two-band model of Kane

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Fig. 2.36 Plot of the normalized photocurrent from QWWs of n-InSb as a function of normalized electron degeneracy in accordance with (a) the three-band model of Kane and (b) the two-band model of Kane

Fig. 2.37 Plot of the normalized photocurrent from QWWs of CdS as a function of dz with (a) C0  = 0 and (b) C0 = 0

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93

Fig. 2.38 Plot of the normalized photocurrent from QWWs of CdS as a function of normalized incident photon energy with (a) C0 = 0 and (b) C0 = 0

Fig. 2.39 Plot of the normalized photocurrent from QWWs of CdS as a function of normalized electron degeneracy with (a) C0 = 0 and (b) C0 = 0

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Fig. 2.40 Plot of the normalized photocurrent from QWWs of n-GaP as a function of dz

Fig. 2.41 Plot of the normalized photocurrent from QWWs of n-GaP as a function of normalized incident photon energy

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95

Fig. 2.42 Plot of the normalized photocurrent from QWWs of n-GaP as a function of normalized electron degeneracy

Fig. 2.43 Plot of the normalized photocurrent from QWWs of n-Ge as a function of thickness in accordance with the models of (a) Cardona et al. and (b) Wang et al.

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Fig. 2.44 Plot of the normalized photocurrent from QWWs of n-Ge as a function of normalized incident photon energy for all the cases of Fig. 2.43

Fig. 2.45 Plot of the normalized photocurrent from QWWs of n-Ge as a function of normalized electron degeneracy for all the cases of Fig. 2.43

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Results and Discussions

97

Fig. 2.46 Plot of the normalized photocurrent from QWWs of n-PtSb2 as a function of film thickness

Fig. 2.47 Plot of the normalized photoemission from QWWs of n-PtSb2 as a function of normalized incident photon energy

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2 Fundamentals of Photoemission from Quantum Wells

Fig. 2.48 Plot of the normalized photocurrent from QWWs of n-PtSb2 as a function of normalized electron degeneracy

Fig. 2.49 Plot of the normalized photocurrent from QWWs of stressed n-InSb as a function of film thickness

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Results and Discussions

99

Fig. 2.50 Plot of the normalized photocurrent from QWWs of stressed n-InSb as a function of normalized incident photon energy

Fig. 2.51 Plot of the normalized photocurrent from QWWs of stressed n-InSb as a function of normalized electron degeneracy

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2 Fundamentals of Photoemission from Quantum Wells

Fig. 2.52 Plot of the normalized photocurrent from QWWs of Bismuth as a function of film thickness in accordance with the models of (a) McClure and Choi, (b) Cohen, (c) Hybrid, and (d) Lax, respectively

Fig. 2.53 Plot of the normalized photocurrent from QWWs of Bismuth as a function of normalized incident photon energy for all cases of Fig. 2.52

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Results and Discussions

101

Fig. 2.54 Plot of the normalized photocurrent from QWWs of Bismuth as a function of normalized electron degeneracy for all cases of Fig. 2.52

Fig. 2.55 Plot of the normalized photocurrent in (a) (13, 6) chiral semiconductor carbon nanotube, (b) (16, 0) zigzag semiconductor carbon nanotubes, (c) (10, 10) metallic armchair carbon nanotube, and (d) (22, 19) chiral metallic carbon nanotube as function of normalized electron degeneracy

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Fig. 2.56 Plot of the normalized photocurrent as a function of normalized incident photon energy for all the cases of Fig. 2.55

The influence of quantum confinement is immediately apparent from Figs. 2.1, 2.4, 2.7, 2.10, 2.13, 2.16, 2.19, 2.22, 2.25, 2.28, 2.31, 2.34, 2.37, 2.40, 2.43, 2.46, 2.49, and 2.52, since the photoemission depends strongly on the thickness of the quantum-confined materials in contrast with the corresponding bulk specimens. The photoemission decreases with increasing film thickness in an oscillatory way, with different numerical magnitudes for QWs in UFs and QWWs, respectively. It appears from the aforementioned figures that photoemission exhibits spikes for particular values of film thickness which, in turn, depend on the particular band structure of the specific material. Moreover, photoemission from QWs in UFs and QWWs of different compounds can become several orders of magnitude larger than that of bulk specimens of the same materials, which is also a direct signature of quantum confinement. This oscillatory dependence will be less and less prominent with increasing film thickness. It appears from Figs. 2.3, 2.6, 2.9, 2.12, 2.15, 2.18, 2.21, 2.24, 2.27, 2.30, 2.33, 2.36, 2.39, 2.42, 2.45, 2.48, 2.51, and 2.54 that the photoemission increases with increasing degeneracy and also exhibits spikes for all types of quantum confinement as considered in this chapter. For bulk specimens of the same material, the photoemission will be found to increase continuously with increasing

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103

electron degeneracy in a nonoscillatory manner. Figures 2.2, 2.5, 2.8, 2.11, 2.14, 2.17, 2.20, 2.23, 2.26, 2.29, 2.32, 2.35, 2.38, 2.41, 2.44, 2.47, 2.50, and 2.53 illustrate the dependence of photoemission from quantum-confined materials on the normalized incident photon energy. The photoemission increases with increasing photon energy in a step-like manner for all the figures. The appearance of the discrete jumps in all the figures is due to the redistribution of the electrons among the quantized energy levels, when the size quantum number corresponding to the highest occupied level changes from one fixed value to another. With varying electron degeneracy, a change is reflected in the photoemission through the redistribution of the electrons among the size-quantized levels. It may be noted that at the transition zone from one subband to another, the height of the peaks between any two subbands decreases with the increase in the degree of quantum confinement, and is clearly shown in all the curves. It should be noted that although the photoemission varies in various manners with all the variables as evident from all the figures, the rates of variations are totally band-structure dependent. It appears from Fig. 2.55 that the photoemission from CNs exhibits oscillatory dependence with increasing carrier degeneracy in a completely different manner as compared with that from QWs in UFs and QWWs, respectively. The numerical values of the photocurrent in all (m, n) cases vary widely, and are determined thoroughly by the chiral indices and the diameter of the CNs. From Figs. 2.55 and 2.56, we can assess the influence of chiral index numbers on the photoemission from CNs, and it further appears that the numerical values of the photocurrent from CNs are the greatest, together with the fact that the oscillatory dependence results from the crossing over of the Fermi level by the quantized level due to van Hove singularities. It should be noted that the rate of increment is totally dependent on the band structure and the spectrum constants of the SWCNs. This oscillatory dependence will be less and less prominent with increasing nanotube radius and carrier degeneracy, respectively. Ultimately, for larger diameters, the photocurrent will be found to be less prominent, resulting in monotonic increasing variation. The smoothing of the spikes region of all the curves is due to the broadening of the energy levels as a result of higher lattice temperature. It may be noted that with the advent of MBE and other experimental techniques, it is possible to fabricate quantum-confined structures with an almost defect-free surface, and this fact has made it possible to study the volume photoelectric effects for QWs in UFs [47]. The numerical computations have been performed using the fact that the probability of photon absorption in direct band-gap compounds is close to unity [47]. If the direction normal to the film was taken differently from that as assumed in this work, the expressions for photoemission from QWs in UFs and QWWs would be different analytically, since the basic dispersion relations for many materials are anisotropic. The influence of the energy band models on the photoemission from various types of quantum-confined materials can also be assessed from the plots. With different sets of energy band parameters, we shall get different numerical values of the photoemission, although the nature of the variations of the same as shown here would be similar for the other types of materials; and the simplified analysis of

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this chapter exhibits the basic qualitative features of the photoemission phenomena from such compounds. It must be mentioned that a direct research application of the quantum confinement of materials is in the area of band structure [54]. Finally, it may be noted that the basic aim of this chapter is not only to solely to demonstrate the influence of quantum confinement on the photoemission from QWs in UFs and QWWs of nonlinear optical, III–V, II–VI, n-GaP, n-Ge, PtSb2 , stressed compounds, Bi, and carbon nanotubes, respectively, but also to formulate the appropriate electron statistics in the most generalized form, since the transport and other phenomena in quantized structures having different band structures and the derivation of the expressions of many important electronic properties are based on the temperature-dependent electron statistics in such materials [55].

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Chapter 3

Fundamentals of Photoemission from Quantum Dots of Various Nonparabolic Materials

3.1 Introduction It is well known that in quantum dots (QDs), all directions of motion of the electron in its wave vector space are quantized and the DOS function changes from the Heaviside step function to the Dirac’s delta function, as discussed in Chapter 1. For QDs of different shapes, the potential is nonseparable and the Schrodinger differential equation should in general be solved through numerical analysis [1]. In this chapter, in Section 3.2.1 on theoretical background, the photoemission from QDs of nonlinear optical materials is investigated. In Sections 3.2.2 through 3.2.5 the photoemission from QDs of III–V, II–VI, n-GaP, and n-Ge is studied, respectively. It may be noted that Tellurium (Te) finds applications in thin-film transistors (TFT) [2], CO2 laser detectors [3], electronic imaging, strain sensitive devices [4] and multichannel Bragg cells [5]. The Section 3.2.6 contains the investigation of photoemission from QDs of Tellurium. The Section 3.2.7 presents the investigation of photoemission in QDs of graphite which are often used as suitable model for investigation of low dimensional systems and in particular for investigation of phase transition in such systems [6]. The Section 3.2.8 contains the study of photoemission from QDs of PtSb2 . It recent years, gapless materials are used in optical wave-guide switches or modulators that can be fabricated by the application of the electro-optic and thermo-optic effects for facilitating optical communications and signal processing. There are extensive applications in infrared detectors and night vision cameras [7]. Section 3.2.9 contains the investigation of photoemission from QDs of gapless materials, taking n-HgTe as an example. Lead chalcogenides (PbTe, PbSe, and PbS) are IV–VI nonparabolic semiconductors whose studies over several decades have been motivated by their importance in infrared IR detectors, lasers, light-emitting devices, photovoltaics, and high temperature thermoelectrics [8]. PbTe in particular is the end compound of several ternary and quaternary high–performance,

107 K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_3,  C Springer Science+Business Media, LLC 2009

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high-temperature thermoelectric materials [9]. It has been used not only as bulk but also as films [10], quantum wells [11], superlattices [12], nanowires [13], and colloidal and embedded nanocrystals [14]. PbTe films doped with various impurities have been studied recently [15]. These studies revealed some of the interesting features that have been seen in bulk PbTe, such as Fermi level pinning and superconductivity [16]. In Section 3.2.10, the photoemission from QDs of Pb1–x Gax Te has been presented. Gallium antimonide (GaSb) finds applications in the fiber-optic transmission window, heterojunctions, and quantum wells. A complementary heterojunction field effect transistor (CHFET), in which the channels for the p-FET device and the n-FET device form the complementary FET, are formed from GaSb. The band gap energy of GaSb makes it suitable for low power operation [17]. In Section 3.2.11, the photoemission from QDs of GaSb is investigated. In Sections 3.2.12, 3.2.13, and 3.2.14, the photoemission from QDs of stressed materials, Bismuth, and IV–VI compounds, respectively, are studied. The II–V materials have been studied in photovoltaic cells constructed of single crystal semiconductor materials in contact with electrolyte solutions. Cadmium selenide shows an open-circuit voltage of 0.8 V and power conservation coefficients near 6 percent for 720-nm light [18]. They are also used in ultrasonic amplification [19]. The development of an evaporated thin film transistor using cadmium selenide as the semiconductor has been reported by Weimer [20–21]. The photoemission from QDs of II–V materials has been presented in Section 3.2.15. The diphosphides find a prominent role in biochemistry, where the folding and structural stabilization of many important extracellular peptide and protein molecules, including hormones, enzymes, growth factors, toxins, and immunoglobulin, are concerned [22]. Additionally, artificial introduction of extra diphosphides into peptides or proteins can improve biological activity [23] or confer thermostability [24]. The asymmetric diphosphide bond formation in peptides containing a free thiol group takes place over a wide pH range in aqueous buffers and can be crucially monitored by spectrophotometric titration of the released 3-nitro-2-pyridinethiol [25, 26]. In Section 3.2.16, photoemission from QDs of zinc and cadmium diphosphides is investigated. Bismuth telluride (Bi2 Te3 ) was first identified as a material for thermoelectric refrigeration in 1954 [27], and its properties were later improved by the addition of bismuth selenide and antimony telluride to form solid solutions [28]. The alloys of Bi2 Te3 are very important compounds for the thermoelectric industry. In Section 3.2.17, the photoemission from QDs of Bi2 Te3 is considered. It may be noted that glasses made from antimony (Sb) are very promising in the near infrared spectral range for third- or second-order nonlinear processes. The chalcogenide glasses are in general associated with high nonlinear properties for their infrared transmission from 0.5–1 to 12–18 μm [29]. Alloys of Sb are used as ultra high frequency indicators and in thin film thermocouples [29]. In Section 3.2.18, the photoemission from QDs of Sb is studied. Section 3.3 contains the result and discussion pertinent to this chapter.

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Theoretical Background

109

3.2 Theoretical Background 3.2.1 Photoemission from Nonlinear Optical Materials Let Eni (i = x,y and z) be the quantized energy levels due to infinitely deep potential well along ith -axis with ni (= 1,2,3, . . .) as the size quantum numbers. Therefore, from (2.1), one can write:

  πnx 2   , (3.1) γ Enx = f1 Enx dx

  π ny 2   , (3.2) γ Eny = f1 Eny dy

  πnz 2   . (3.3) γ Enz = f2 Enz dz   From (2.1), the totally quantized energy EQD1 , can be expressed as:     γ EQD1 = f1 EQD1



πnx dx



2 +

π ny dy

2 



+ f2 EQD1





π nz dz

2  .

(3.4)

The total density-of-states function in this case is given by: n

n

n

xmax ymax zmax     2gv  δ  E − EQD1 . N0DT (E) = dx dy dz n n n x=1

y=1

(3.5)

z=1

Using (3.5) and the Fermi-Dirac occupation probability factor, the total electron concentration can be written as: n

n0D

n

n

xmax ymax zmax     2gv  = F−1 (η31 ), where η31 ≡ (kB T)−1 EF0D − EQD1 . dx dy dz n n n x=1

y=1

z=1

(3.6) Therefore the electron concentration per subband is given by: n0D =

2gv dx dy dz

F−1 (η31 ) .

(3.7)

The expression of the total photoemitted current density in this case is: n

J0D

n

n

xmax ymax zmax     eα0 gv  = (n0D ) vz Enz , 2 n n n x=1

y=1

zmin

(3.8)

110

3 Fundamentals of Photoemission from Quantum Dots

where the nzmin should be the nearest integer of the following inequality: nzmin ≥

√ dz γ (W − hν) , √ π f2 (W − hν)

(3.9)

and the velocity of the photoemitted electrons in the nth z subband can be written as:   1   vz Enz = Q1 Enz , 

(3.10)

in which 

Q1 Enz



   1/2 3/2  Enz γ Enz 2f2     . ≡      f2 Enz γ Enz − γ Enz f2 Enz

Using (3.7), (3.8), and (3.10), we get: J0D =

eα0 gv dx dy dz

n xmax nymax nzmax  

  F−1 (η31 ) Q1 Enz .

(3.11)

nx=1 ny=1 nzmin

3.2.2 Photoemission from III–V Materials The dispersion relation of the conduction electrons of III–V semiconductors are described by the models of Kane (both three and two bands) [30], Stillman et al. [31], Newson and Kurobe [32], Rossler [33], Palik et al. [34], Johnson and Dickey [35], and Agafonov et al. [36], respectively. For the purpose of complete and coherent presentation, the photoelectric effect in densities QDs of III–V compounds have also been investigated in accordance with the aforementioned different dispersion relations for relative comparison as follows: 3.2.2.1 The Three-Band Model of Kane Under the conditions δ = 0,  = ⊥ = , and m∗ = m⊥ ∗ = m∗ , (2.1) gets simplified to the three-band Kane model as given by (2.19). The quantized energy levels in QDs of III–V semiconductors in accordance with the three-band model of Kane can be expressed as: 

I Enx



2 = 2m∗

  2 I Eny = 2m∗



π nx dx π ny dy

2 ,

(3.12)

,

(3.13)

2

3.2

Theoretical Background

111

and 

I En z



2 = 2m∗



π nz dz

2 .

(3.14)

  The totally quantized energy EQD2 in this case assumes the form: 

I EQD2



2 π 2 = 2m∗



nx dx



2 +

ny dy



2 +

nz dz

2  .

(3.15)

The velocity of the photoelectron in the nth z quantized level is given by: 





vz Enz =

  2 Q2 Enz , ∗ m

(3.16)

where 

Q2 Enz

   −1 "   I Enz . ≡ I Enz



Thus the photoemitted current density can be expressed as:  J0D =

α0 egv   dx dy dz



2 m∗

 nx ny nz max  max  max 

  Q2 Enz F−1 (η32 ),

(3.17)

nx=1 ny=1 nzmin

in which nzmin is the nearest integer of the following inequality:  nzmin ≥

dz π

 √ 2m∗ 1/2 [I (W − hν)] , 

(3.18)

  and η32 = (kB T)−1 EF0D − EQD2 , where EQD2 has to be determined from (3.15). The form of n0D as given by (3.6) remains unchanged, in which η31 should be replaced by η32 . 3.2.2.2 The Two-Band Model of Kane Under the constraints  >> Eg0 or  << Eg0 , the three-band model of Kane gets simplified to the two-band model of Kane and the photoemitted current density assumes the form: J0D

α0 gv e = dx dy dz



2 m∗

 nx ny nz max  max  max  nx=1 ny=1 nzmin

  Q3 Enz F−1 (η33 ),

(3.19a)

112

3 Fundamentals of Photoemission from Quantum Dots

where 

 √ dz 2m∗ 1/2 [(W − hν) [1 + α (W − hν)]] , nzmin ≥ π  "        Q3 Enz ≡ Enz 1 + αEnz / 1 + 2αEnz , and η33 ≡

 1  EF0D − EQD3 . kB T

Enz obeys the equation:   2 π 2 Enz 1 + αEnz = 2m∗



nz dz

2 .

(3.19b)

The totally quantized energy (EQD3 ) in this case is given by 

EQD3 1 + αEQD3



2 π 2 = 2m∗



nx dx



2 +

ny dy



2 +

nz dz

2  .

(3.20)

The form of n0D as given by (3.6) remains unchanged, in which η31 should be replaced by η33 , where EQD3 has to be determined from (3.20). For α → 0, the photoemission as given by (3.19a) gets simplified to (1.79). 3.2.2.3 The Model of Stillman et al. In accordance with the model of Stillman et al. [31], the electron dispersion law of III–V materials assumes the form: E = t11 k2 − t12 k4 ,

(3.21)

where 2 and 2m∗



2 

 $−1  m∗ 2 2 22 # ≡ 1− 3Eg0 + 4 + . Eg0 +  2 + 3Eg0 . m0 2m∗ Eg0

t11 ≡ t12

The (3.21) can be expressed as: 2 k 2 = I11 (E) , 2m∗   where I11 (E) ≡ a11 1 − (1 − a12 E)1/2 ,

(3.22)

3.2

Theoretical Background

113

a11 ≡

2 t11 4m∗ t12

, a12 ≡

4t12 . 2 t11

Therefore, proceeding as above, the photoelectric current density in this case can be written as: J0D =

α0 gv e dx dy dz



2 m∗

1/2 n xmax nymax nzmax  

  Q5 Enz F−1 (η34 ),

(3.23a)

nx =1 ny =1 nzmin

where nzmin η34

√     1/2    −1  dz 2m∗ [I11 (W − hν)]1/2 , Q5 Enz ≡ I11 Enz ≥ I11 Enz , and π    ≡ (kB T)−1 EF0D − EQD4 .

Enz obeys the equation:

2 2m∗



πnz dz

2

  ≡ I11 Enz .

(3.23b)

The EQD4 in this case can be defined as 

I11 EQD4



2 π 2 = 2m∗



nx dx



2 +

ny dy



2 +

nz dz

2  .

(3.24)

The form of n0D as given by (3.6) remains unchanged, in which η31 should be replaced by η34 , where EQD4 has to be determined from (3.24). 3.2.2.4 The Model of Newson and Kurobe In accordance with the model of Newson and Kurobe, the electron dispersion law in this case can be expressed as [32]: 

E=

a13 kz4



2 2 2 2 2 2 2 4 4 + + a k + k + a k k + a + k k , k 14 14 13 s z s x y x y 2m∗ 2m∗

(3.25)

where a13 is the non-parabolicity constant, a14 (≡ 2a13 + a15 ), and a15 is known as the warping parameter.  vz Enz and Enz are given by:      3/4 vz Enz = 4 (a13 )1/4 · Enz · ()−1 , Enz = a13

π nz dz

(3.26)

4 .

(3.27)

114

3 Fundamentals of Photoemission from Quantum Dots

The totally quantized energy EQD5 in this case can be written as:  





π ny 2 2 π nz 4 π nx 2 π nz 2 = a13 + + a + . 14 dz 2m∗ dx dy dz  





2 π ny 2 2 πnx 2 4 nx ny + + π + a (3.28) 14 2m∗ dx dy dx dy 

4  ny nx 4 4 + a13 π + . dx dy

EQD5

The photoemitted current density in this case is given by:

J0D

nxmax nymax nzmax    3/4 4α0 egv 1/4  Enz = F−1 (η35 ), a13 dx dy dz n nx =1 ny =1

(3.29)

zmin

where nzmin ≥

dz EF0D − EQD5 . (a13 )−1/4 (W − hν)1/4 and η35 ≡ π kB T

The form of n0D as given by (3.6) remains unchanged, in which η31 should be replaced by η35 , where EQD5 has to be determined from (3.28). 3.2.2.5 The Model of Rossler The dispersion relation of the conduction electrons in this case in accordance with the model of Rossler can be written as [33]: E=

  2 k 2 4 2 2 2 2 2 2 [α + + α k] k + + β k) k k + k k + k k (β 11 12 11 12 x y y z z x 2m∗

1/2   2 2 2 ± γ11 + γ12 k k kx ky + ky2 kz2 + kz2 kx2 − 9kx2 ky2 kz2 ,

(3.30)

where α11,α12 , β11 , β12 , γ11 and γ12 are energy-band constants. The νz Enz and Enz in this case are given by:   1   νz Enz = Q6 Enz , 

(3.31a)

where 



Q6 Enz ≡



2

m∗



πnz dz



+ α11

π nz dz

4



π nz +4 dz

3 

π nz α11 + α12 dz

−1 , and

En z

3.2

Theoretical Background

115

obeys the equation  Enz =

2 2m∗



π nz dz

2



πnz + α11 + α12 dz



π nz dz

4  .

(3.31b)

The photoemitted current density can be expressed as:

J0D

α0 egv = 2



1 dx dy dz

n xmax nymax nzmax  

    Q6 Enz F−1 η36,± ,

(3.32)

nx =1 ny =1 nzmin

  where η36,± ≡ EF0D − EQD6,± (kB T)−1 , and EQD6,± in this case assumes the form:

EQD6,±

2 π 2 ≡ 2m∗ ⎡



nx dx

+

+ ⎣α11 + α12



π nx dx



2

ny dy

+

2 +

nz dz

2 

π ny dy



2 +

π nz dz

2 1/2

⎤ ⎦.



2 π ny 2 πnz 2 + + dy dz ⎤ 

2

2

2 1/2 πn π n πn y x z ⎦. + ⎣β11 + β12 + + dx dy dz 

2

2

2  4 nx ny 4 ny nz 4 nz nx +π +π π dx dy dy dz dz dx ⎡ ⎤ 

2

2

2 1/2 π ny π nz π nx ⎦. ± ⎣γ11 + γ12 + + dx dy dz 





2

2 π ny 2 π nz 2 πnx 2 4 nx ny 4 ny nz + + +π π dx dy dz dx dy dy dz   1/2

2  6 4 nz nx 6 +π . − 9π nx ny nz /dx dy dz dz dx (3.33) is the nearest integer of the equation: 

πnx dx ⎡

The nzmin



2

(W − hν) =

2

2 2m∗





πnzmin dz

2





π nzmin π nzmin 4 + α11 + α12 . (3.34a) × dz dz

116

3 Fundamentals of Photoemission from Quantum Dots

The electron concentration can be written as: n

n0D =

n

n

xmax ymax zmax     gv  F−1 η36,± . dx dy dz n n n x=1

y=1

(3.34b)

z=1

3.2.2.6 The Model of Palik et al. In accordance with the model of Palik et al. [34], the energy spectrum of the conduction electrons in III–V materials up to the fourth order in effective mass theory, taking into account the interactions with the heavy hole, the light hole, and the splitoff bands, can be written as [34]: E=

2 k 2 − b11 k4 , 2m∗

(3.35)

where  b11 ≡

 ⎡ x2 ⎤ 

−1 1+ 11 ⎣ x2 ⎦ . (1−y1 )2 , x11 ≡ 1+  , and y1 ≡ m∗ /m0 . 1+ 211 Eg0 4Eg0 (m∗ )2 4

From (3.35) we get: 2 k2 = I12 (E) , 2m∗

(3.36)

where  I12 (E) ≡ b12 a12 −

"

 a212

− 4Eb11 , b12 ≡

a12 2 , and a12 ≡ . 2b11 2m∗

The photoemission current density in this case is given by: J0D =

α0 egv dx dy dz



2 m∗

1/2 n xmax nymax nzmax  

  Q7 Enz F−1 (η37 ) ,

nx =1 ny =1 nzmin

where √ 

2m∗ dz [I12 (W − hν)]1/2 , nzmin ≥  π     1/2    /I12 Enz , and Q7 Enz ≡ I12 Enz   EF0D − EQD7 η37 ≡ . kB T

(3.37)

3.2

Theoretical Background

117

The Enz and EQD7 are defined by the following equations:

  2 π nz 2 , I12 Enz = 2m∗ dz 



   πny 2 2 π nz 2 π nx 2 I12 EQD7 = + + . 2m∗ dx dy dz

(3.38)

(3.39)

The form of n0D as given by (3.6) remains unchanged, in which η31 should be replaced by η37 , where EQD7 has to be determined from (3.39). 3.2.2.7 The Model of Johnson and Dickey In accordance with the model of Johnson and Dickey [35], the electron dispersion law in III–V materials assumes the form: Eg 2 k 2 E=− 0 + 2 2



 1/2 Eg0  1 1 2 − , 1 + a + k (E) 15 m∗ m0 2

(3.40)

where

  Eg0 +  E + Eg0 + 23  42

 a14 (E) , a14 (E) ≡ a15 (E) ≡ , 2m∗ Eg0 Eg0 + 23  E + Eg0 + 

⎤ ⎡ 2 E +  g0    2 3 m0  ⎦ , and P = P ⎣   ∗ m Eg0 Eg0 +  is the energy band constant. The EQD8 in this case is given by:

EQD8

  



 π ny 2 Eg0 1 π nz 2 2 1 πnx 2 =− − + + + 2 2 m∗ m0 dx dy dz  



1/2 (3.41)   π ny 2 Eg0 π nz 2 πnx 2 + + + . 1 + a15 EQD 2 dx dy dz

  The νz Enz can be written as:   1   νz Enz = Q8 Enz , 

(3.42)

118

3 Fundamentals of Photoemission from Quantum Dots

where 

Q8 Enz



⎡⎡

⎤  

2 

2 −1/2 −1    E a E π n πn g n z z ⎢ z ⎦ , ≡ ⎣⎣1 − 0 15 1 + a15 Enz 4 dz dz 

 

Eg0 a15 Enz π nz +  4 dz ⎤ ⎤ 

−1/2   πnz 2 ⎦⎥ 1 + a15 Enz ⎦, dz

2

1 1 − ∗ m m0



nz π dz



and Eg 2 E nz = − 0 + 2 2



nz π dz

2

1 1 − ∗ m m0



Eg + 0 2

 1 + a15



1/2  π nz 2 Enz . dz (3.43)



The photo-emitted current density can be written as: J0D =

nxmax nymax nzmax     α0 gv e  Q8 Enz F−1 (η38 ), dx dy dz n nx =1 ny =1

(3.44)

zmin

where nzmin is the nearest integer of the following equation:



Eg0 1 2 π nzmin 2 1 − + (W − hν) = − 2 2 dz m∗ m0 

1/2 Eg0 π nzmin 2 + , 1 + a15 (W − hν) 2 dz

(3.45)

and η38 ≡ (EF0D − EQD8 /kB T). The form of n0D as given by (3.6) remains unchanged, in which η31 should be replaced by η38 , where EQD8 has to be determined from (3.41). 3.2.2.8 The Model of Agafonov et al. In accordance with the model of Agafonov et al. [36], the electron dispersion law can be expressed as: y − Eg0 E= 2



 1 − T5

kx4 + ky4 + kz4 yk2

 ,

(3.46)

3.2

Theoretical Background

119

where √  2

 D 3 − 3B 8  , B = −21 , (y)2 = Eg20 + P20 k2 , T5 = 3 2 2m0 2

 D = −40 , and P0 2m0 is the momentum   matrix element. The νz Enz in this case can be written as:   1   νz Enz = Q9 Enz , 

(3.47)

where 

Q9 Enz



⎡⎡ ⎡ ⎤

2

  1 8 π n π nz z ≡ ⎣⎣ ⎣ Eg20 + P20 − Eg0 ⎦ −2T5 . 2 3 dz dz Eg20

8 + P20 3



π nz dz

2 −1/2





 8 2 π nz 3 + P T5 . 3 0 dz ⎤⎤ 

2 −3/2

 2 −1/2 8 8 8 π n π n π nz z z 2 2 2 2 2 ⎦⎦ + P0 Eg0 + P0 Eg0 + P0 3 dz 3 dz 3 dz ⎡ ⎤ ⎤ ⎡ ⎤ 



1/2 −1 8 2 π nz 2 π nz 2 ⎣ 2 ⎢ ⎥ ⎦ ⎥ . Eg0 + P0 ⎣1 − T5 ⎦⎦ , dz 3 dz 

and Enz is defined by the following equation: ⎡ ⎤

1/2 1⎣ 2 8 2 π nz 2 Enz = − Eg0 ⎦ Eg0 + P0 2 3 dz ⎡ ⎤ 

2 −1/2

2 8 π n π n z z ⎣1 − T5 . Eg2 + P20 ⎦. 0 3 dz dz

(3.48)

The totally quantized energy can be written as:  





 ψ30 − Eg0 π ny 4 π nz 4 π nx 4 = + + 1 − T5 . 2 dx dy dz ⎤   (3.49)



−1 π ny 2 πnz 2 π nx 2 ⎦, + + ψ30 dx dy dz

EQD9

120

3 Fundamentals of Photoemission from Quantum Dots

where  ψ30 = Eg20

8 + P20 3



π nx dx



2 +

π ny dy



2 +

π nz dz

2 1/2 .

The photoemitted current density is given by: J0D =

nxmax nymax nzmax     α0 gv e  Q9 Enz F−1 (η39 ), dx dy dz n nx =1 ny =1

(3.50)

zmin

  in which, η39 = (1/kB T) EF0D − EQD9 , and nzmin should be determined from the following equation: ⎡1 ⎤  2

2  8 π n 1 ⎣2 z min 3E2 + P2 − Eg0 ⎦ (W − hν) = g0 2 3 0 dz ⎤ (3.51) ⎡ 

2 

2 −1/2 8 π nzmin ⎦. ⎣1 − T5 π nzmin Eg20 + P20 dz 3 dz The form of η0D is given by (3.6), where η31 should be replaced by η39 .

3.2.3 Photoemission from II–VI Materials Using (2.39), the totally quantized energy in this case can be expressed as:  EQD10,± = A0







1/2 π ny 2 π ny 2 π nx 2 π nx 2 π nz 2 + B0 + ± C0 + . dx dy dz dx dy (3.52)

  The νz Enz and Enz are given by:



 π nz vz Enz = ∗ , m|| dz

2 π nz 2 Enz = . 2m∗|| dz 



(3.53)

(3.54)

The electron concentration can be written as: n0D

nxmax nymax nzmax     gv  F−1 η40,± , = dx dy dz nx =1 ny =1 nz =1

(3.55)

3.2

Theoretical Background

121

  where η40,± = (1/kB T) EF0D − EQD10,± . The photoemitted current density is given by: J0D

nxmax nymax nzmax     α0 gv πe  = nz F−1 η40,± , ∗ 2dx dy dz2 m|| n nx =1 ny =1

(3.56)

zmin

where

nzmin

" ∗ dz 2m|| ≥ (W − hν)1/2 . π 

(3.57)

3.2.4 Photoemission from n-Gallium Phosphide Using (2.51), the totally quantized energy EQD11 in this case is given by:  





 πny 2 π ny 2 2 π nz 2 πnx 2 2 π nx 2 EQD11 = + + + + 2m∗⊥ dx dy 2m∗|| dx dy dz    1/ 2



4 k02 π ny 2 π nz 2 π nx 2 − + + + |VG | . + |VG |2 d d d m∗2 x y z || (3.58)   The νz Enz and Enz can be respectively written as: 

vz Enz



 = ∗ m||



π nz dz



  Q10 Enz ,

(3.59)

where ⎡ 



⎢ Q10 Enz = ⎣1 −



2 k02 m∗||

⎧ ⎫−1/2 ⎤ ⎪

2 ⎪ ⎨ 2 4  k π nz ⎬ ⎥ |VG |2 + 0 2 ⎦, ⎪ ⎪ d z ∗ ⎩ ⎭ m||

and 2 Enz = 2m∗||



π nz dz



2 −

4 k02 m∗2 ||



π nz dz

1/ 2

2 + |VG |2

+ |VG | .

(3.60)

The electron concentration assumes the form: n0D

nxmax nymax nzmax    2gv  F−1 (η41 ) , = dx dy dz nx =1 ny =1 nz =1

(3.61)

122

3 Fundamentals of Photoemission from Quantum Dots

  where η41 = (1/kB T) EF0D − EQD , in which EQD is determined from (3.58). The photoemitted current density is given by: nxmax nymax nzmax     α0 πgv e  Q10 Enz F−1 (η41 ), ∗ 2 dx dy dz m|| n

J0D =

nx =1 ny =1

(3.62)

zmin

where nzmin is the nearest integer of the following equation:

(W − hν) =

2

2m∗||



π nzmin dz

2

⎡ 4 k02

⎢ + |VG | − ⎣|VG |2 +

m∗||

2



π nzmin dz

2

⎤1/2 ⎥ ⎦

. (3.63)

3.2.5 Photoemission from n-Germanium (a) Using the model of Cardona et al. [37], as discussed in (2.62), the totally quantized energy EQD12 in this case is given by:

EQD12

Eg 2 =− 0+ ∗ 2 2m



π nz dz



2  2 2

!1/ 2 Eg0 π ny 2  π nx 2 + + . + Eg0 4 2m∗⊥ dx dy (3.64)

  The νz Enz and Enz can be respectively written as:



   π nz , vz Enz = ∗ m|| dz

2 π nz 2 Enz = . 2m∗|| dz

(3.65) (3.66)

The electron concentration assumes the form: n0D

nxmax nymax nzmax    2gv  F−1 (η42 ) , = dx dy dz

(3.67)

nx =1 ny =1 nz =1

  where η42 = (1/kB T) EF0D − EQD12 , in which EQD is determined from (3.64). The photoemitted current density is given by:

J0D

nxmax nymax nzmax   α0 πgv e  = nz F−1 (η42 ), dx dy dz2 m∗|| n nx =1 ny =1

zmin

(3.68)

3.2

Theoretical Background

123

where nzmin is the nearest integer of the following inequality: " dz ≥ π

nzmin

2m∗ 

(W − hν)1/2 .

(3.69)

(b) Using the model of Wang and Ressler [38], as discussed in (2.75), the totally quantized energy EQD in this case is given by: EQD13

2 = 2m∗||



2

π nz dz

2 + 2m∗⊥



πnx dx



2 +

π ny dy

2 !



!2 π ny 2 πnx 2 − c1 + dx dy 2

2

! 2 

π ny 2  πnx  π nz 2 − d1 + 2m∗⊥ dx dy 2m∗|| dz  2

π nz 4 2 − e1 . 2m∗|| dz

2 2m∗⊥

2

(3.70)

  The νz Enz and Enz can be respectively written as: 

vz Enz



 = ∗ m||



nz π dz



  Q11 Enz ,

(3.71)

where ⎡   ⎢ Q11 Enz = ⎣1 − e1 2 Enz = 2m∗||



π nz dz

2 m∗|| 

2

− e1



2

2 2m∗||

π nz dz

2

2

⎤ ⎥ ⎦,

π nz dz

4 .

(3.72)

The electron concentration assumes the form: n0D =

nxmax nymax nzmax    2gv  F−1 (η43 ) , dx dy dz

(3.73)

nx =1 ny =1 nz =1

  where η43 = (1/kB T) EF0D − EQD13 , in which EQD is determined from (3.70). The photoemitted current density is given by: J0D =

nxmax nymax nzmax     α0 πgv e  Q11 Enz F−1 (η43 ), ∗ 2 dx dy dz m n nx =1 ny =1

zmin

(3.74)

124

3 Fundamentals of Photoemission from Quantum Dots

where nzmin is the nearest integer of the following inequality: 2 (W − hν) = 2m∗||



π nzmin dz



2 − e1

2 2m∗||

2

π nzmin dz

4 .

(3.75)

3.2.6 Photoemission from Tellurium The carriers of Tellurium find various descriptions for the energy-wave vector dispersion relations in the literature. Among them we shall use the E-k dispersion relations as given by Bouat et al. [39] and Ortenberg and Button [40], respectively. (i) The dispersion relation of the conduction electrons of Tellurium can be written in accordance with Bouat et al. as [39]:  1/2 , E = A6 kz2 + A7 ks2 ± A8 kz2 + A9 ks2

(3.76)

where A 6 , A7 , A8 and A9 are the energy band constants. The vz Enz,± in this case is given by:

1

vz Enz,± = Q12 Enz,± , 

(3.77)





nz π Q12 Enz,± = 2A6 ± A8 , dz

where

in which Enz,± = A6

π nz dz



2 ±

πnz dz



A8 .

The totally quantized energy can be written as: EQD14,± = A6 

π nz dz

± A8



2 + A7 π nz dz

2

πnx dx 

+ A9



2 + πnx dx

π ny dy

2 +

2 

π ny dy

2 1/2

(3.78) .

The electron concentration is given by: n0 =

nxmax nymax nzmax     gv  F−1 η44,± , dx dy dz nx =1 ny =1 nz =1

(3.79)

3.2

Theoretical Background

125

  where η44,± = EF0D − EQD14,± /kB T. The photoemitted current density is given by: nxmax nymax nzmax

    α0 egv  Q12 Enz,± F−1 η44,± , 2dx dy dz

J0D =

(3.80)

nx =1 ny =1 nz =1

where nzmin can be determined from W − hν = A6

π nzmin dz

2 ±



A8 .

π .nz . dz min

(3.81)

(ii) The dispersion relation of the conduction electrons of Tellurium can be written in accordance with the model of Ortenberg and Button as [40]: 

E = t1 + t2 kz2 + t3 ks2 + t4 ks4 + t5 ks2 kz2 ±

t1 + t6 ks2

2

1/2 + t7 kz2

,

(3.82)

where t1 , t 2 , t3 , t 4 , t5 , t6 , and t7 are the energy band constants. The vz Enz,± in this case is given by:

1

vz Enz,± = Q13 Enz,± , 

(3.83)

where

⎡

Q13 Enz,± = ⎣2t2



π nz dz



± t7

⎤



2 −1/2 π nz π nz ⎦, . t12 + t7 dz dz

in which Enz,± = t1 + t2

π nz dz



2 ±

t12

+ t7

π nz dz

2 1/2 .

The totally quantized energy can be written as:  







2 π ny 2 π ny 2 π nz 2 π nx 2 π nx 2 = t1+ t2 + t3 + + + t4 dz dx dy dx dy 

2

2 

2 πny πnx π nz + t5 + dx dy dz

EQD15,±

126

3 Fundamentals of Photoemission from Quantum Dots

⎡ ± ⎣ t1 + t6



πnx dx



2 +

π ny dy

2 !2

+ t7

π nz dz

2

⎤1/2 ⎦

(3.84)

.

The electron concentration is given by: nxmax nymax nzmax     gv  F−1 η45,± , n0 = dx dy dz

(3.85)

nx =1 ny =1 nz =1

  where η45,± = EF0D − EQD15,± /kB T. The photoemitted current density can be written as: J0D

nxmax nymax nzmax

    α0 egv  = Q13 Enz,± F−1 η45,± , 2dx dy dz

(3.86)

nx =1 ny =1 nz =1

where nzmin should be determined from the following equation: W − hν = t1 + t2

π nzmin dz



2 ±

t12

+ t7

π nzmin dz

2 1/2 .

(3.87)

3.2.7 Photoemission from Graphite The carrier dispersion law in graphite can be written as [41]: 1/2  1 1 2 2 (3.88) k , (E2 + E3 ) ± (E2 − E3 )2 + η46 2 4   where E2 = 1 − 2γ1 cos φ0 + 2γ5 cos2 φ0 ,1 ,γ1 , and γ5 are energy band constants. √ 

3 2 a (γ0 + 2γ4 cos φ0 ) , c and a φ0 = (ckz /2) , E3 = 2γ2 cos φ0 , η46 = 2

are constants of the spectrum. The vz Enz,± in this case is given by: E=



1

vz Enz,± = Q14 Enz,± ,  where





1 c (π nz ) c (π nz ) Q14 Enz,± = γ1 c sin − γ5 c sin 2 2dz 2dz

(3.89)

3.2

Theoretical Background

127







c (πnz ) c (π nz ) 1 1 − γ2 c sin 1 − 2γ1 cos ± 2dz 2 4 2dz



2 c (πnz ) c (π nz ) − 2γ2 cos2 + 2γ5 cos2 2dz 2dz



−1/2 2 c (π nz ) 2 3 (a)2 nz π + . γ0 + 2γ4 cos . 4 dz 2dz  





 c (πnz ) 1 2 c (π nz ) 2 c (π nz ) 1 − 2γ1 cos + 2γ5 cos − 2γ2 cos 2 2dz 2dz 2dz



  c (π nz ) c (πnz ) c (π nz ) cγ1 sin − cγ5 sin + γ2 c sin 2dz 2dz 2dz



2 3 πnz c (πnz ) + (a)2 γ0 + 2γ4 cos 4 dz 2dz

2 

 c (πnz ) π nz 2 3 − (a) cγ4 γ0 + 2γ4 cos , 2 dz 2dz in which

Enz,± =







 1 c (π nz ) c (π nz ) c (π nz ) 1 − 2γ1 cos + 2γ5 cos2 + 2γ2 cos2 2 2dz 2dz 2dz





2  c (π nz ) c n c (π nz ) 1 (π ) z 1 − 2γ1 cos + 2γ5 cos2 − 2γ2 cos2 ± 4 2dz 2dz 2dz

2

2 1/2 2 3 (a) c (π nz ) π nz γ0 + 2γ4 cos + . 4 2dz dz

The totally quantized energy can be written as:

EQD16,±





c (π nz ) 1 2 c (π nz ) = 1 − 2γ1 cos + 2γ5 cos 2 2dz 2dz

 

c n 1 (π ) z 2 2 c (π nz ) + 2γ2 cos 1 − 2γ1 cos ± 2dz 4 2dz



2 c (πnz ) c (π nz ) 3 + (a)2 − 2γ2 cos2 + 2γ5 cos2 2dz 2dz 4





1/2 π ny 2 nz π c 2 π nz 2 πnx 2 + + . γ0 + 2γ4 cos 2dz dx dy dz (3.90)

128

3 Fundamentals of Photoemission from Quantum Dots

The electron concentration is given by: nxmax nymax nzmax       gv  n0 = F−1 η46,± , where η46,± = EF0D − EQD16,± /kB T. dx dy dz nx =1 ny =1 nz =1

(3.91) The photoemitted current density can be written as: J0D

nxmax nymax nzmax

    α0 gv e  = F−1 η46,± Q14 Enz,± , 2dx dy dz  n nx =1 ny =1

(3.92)

zmin

where nzmin should be determined from the following equation:        c πnzmin 1 2 c π nzmin 1 − 2γ1 cos + 2γ5 cos (W − hν) = 2 2dz 2dz          c π nzmin 1 2 c πnzmin +2γ2 cos 1 − 2γ1 cos ± 2dz 4 2dz       2 c πnzmin c π nzmin + 2γ5 cos2 − 2γ2 cos2 2dz 2dz ⎤     2

1/2 c πnzmin 3 (a)2 π nzmin 2 ⎦ + . γ0 + 2γ4 cos 4 2dz dz (3.93)

3.2.8 Photoemission from Platinum Antimonide The vz (Enz ) in this case is given by: vz (Enz ) =

  1 Q15 Enz , 

where 

Q15 Enz







−1 π nz 2 π nz 2 = 2Enz + δ0 − ω4 + ω2 . dz dz 





 π nz 3 π nz π nz 2 + 2ω4 Enz + ω2 4I1 dz dz dz



2  π nz π nz − 2ω2 E + δ0 − ω4 , dz dz

(3.94)

3.2

Theoretical Background

129

in which Enz should be determined from the following equation: 

Enz + ω2

πnz dz

2  



π nz 2 π nz 4 . Enz + δ0 − ω4 = I1 dz dz

(3.95)

The totally quantized energy EQD17 is given by: 







 π ny 2 π nz 2 πnx 2 + EQD17 + ω1 + ω2 . dx dy dz  



 πny 2 π nz 2 π nx 2 + − ω4 EQD17 + δ0 + ω3 dx dy dz 

2

2

2 2 π ny πnz π nx = I1 + + . dx dy dz

(3.96)

The carrier concentration is given by: p0D

nxmax nymax nzmax   2gv  = F−1 (η47 ), where η47 = (EF0D − EQD17 )/(KB T). dx dy dz nx =1 ny =1 nz =1

(3.97) The photoemitted current density can be written as: J0D =

nxmax nymax nzmax     α0 egv  Q15 Enz F−1 (η47 ) , dx dy dz  n nx =1 ny =1

(3.98)

zmin

where nzmin should be determined from the following equation: 

(W − hν) + ω2



   π nzmin 2 π nzmin 2 π nzmin 4 = I1 . (W − hν) + δ0 − ω4 dz dz dz (3.99)

3.2.9 Photoemission from Zero-Gap Materials The dispersion relation of the conduction electrons in gapless materials [42] is given by:    kz  3e2 k 2 2 k 2 + − EB ln   , E= ∗ 2m 128εsc π k0

(3.100)

where εsc is the semiconductor permittivity, EB is the Bohr electron energy, and k0 is the inverse Bohr radius.

130

3 Fundamentals of Photoemission from Quantum Dots

The vz (Enz ) in this case is given by: vz (Enz ) =

  1 Q16 Enz , 

(3.101)

where 

Q16 Enz





 πnz 2EB dz  3e2 1 . − 2 = + , m∗ dz  128εsc π nz 

in which Enz should be determined from the following equation:  Enz =

 2m∗



π nz dz

2

 

 π nz  3e2 πnz 2  , + . − EB ln  128εsc dz π dz k0 

(3.102)

The totally quantized energy EQD18 is given by:

EQD18

2 2 

ny nz nx 2 + + dx dy dz 



1/2 π ny 2 3e2 π nz 2 π nx 2 − + + 128εsc dx dy dz     2 EB  (π nx /dx )2 + πny /dy + (π nz /dz )2  − ln  .   π k02

2 π 2 = 2m∗



(3.103)

The electron concentration is given by: nxmax nymax nzmax     2gv  F−1 (η48 ), where η48 = EF0D − EQD18 /kB T. n0 = dx dy dz nx =1 ny =1 nz =1

(3.104) The photoemitted current density can be written as:

J0D

nxmax nymax nzmax     α0 gv e  = Q16 Enz F−1 (η48 ) , dx dy dz

(3.105)

nx =1 ny =1 nz =1

where nzmin should be determined from the following equation:  (W − hν) = 2m∗



πnzmin dz

2

3e2 + 128εsc



π nzmin dz



   π nzmin  2  . − EB ln  π dz 

(3.106)

3.2

Theoretical Background

131

3.2.10 Photoemission from Lead Germanium Telluride The dispersion law of n-type Pb1–x Gex Te with x = 0.01 can be expressed following Vassilev [43] as:   E − 0.606ks2 − 0.722kz2 E + Eg0 + 0.411ks2 + 0.377kz2   = 0.23ks2 + 0.02kz2 ± 0.06Eg0 + 0.061ks2 + 0.0066kz2 ks ,



(3.107a)

where Eg0 = 0.21 eV, kx , ky , and kz are in the units of 109 m−1 The vz (Enz ) in this case is given by:   1   vz Enz = Q17 Enz , 

(3.107b)

where 

Q17 Enz





−1 πnz 2 = 2Enz − 0.0345 . dz 





 π nz 2 πnz πnz + 0.1444 Enz + 0.0377 0.04 dz dz dz



2  π nz π nz − 0.0754 Enz − 0.0722 , dz dz

in which Enz should be determined from the following equation: 

 π nz 2 − Enz −Enz − Eg0 − 0.0377 dz

2 

2

2 1/2 

 πny πnz π nx π nz 2 = 0.02 ± + 0.06Eg0 + 0.0066 . dz dx dy dz (3.108)   The totally quantized energy EQD19,± is given by:

π nz 0.0722 dz

2





 EQD19,± − 0.606

 EQD19,± + 0.471



πnx dx πnx dx





2 +



2 +

πny dy πny dy

2 

2 



π nz − 0.0722 dz

2 

+ Eg0

π nz + 0.0377 dz

2 

132

3 Fundamentals of Photoemission from Quantum Dots









1/2 π ny 2 π ny 2 π nx 2 π nz 2 = 0.23 + ± + + 0.02 dy dz dx dy  

2

2 

2  π ny π nx π nz + 0.06Eg0 + 0.061 + 0.0066 . dx dy dz (3.109) The electron concentration is given by: n0 =

π nx dx

2



nxmax nymax nzmax   EF0D − EQD,± gv  F−1 (η49 ), where η49 = . dx dy dz kB T

(3.110)

nx =1 ny =1 nz =1

The photoemitted current density can be written as: J0D =

nxmax nymax nzmax     α0 gv e  Q17 Enz F−1 (η49 ) , 2dx dy dz

(3.111)

nx =1 ny =1 nz =1

where nzmin should be determined from the following equation:

 πnzmin 2 (W − hν) − 0.0722 dz 



πnzmin 2 π nzmin 2 . = 0.02 (W − hν) + Eg0 + 0.0377 dz dz 



(3.112)

3.2.11 Photoemission from Gallium Antimonide The conduction electrons of n-GaSb obey the three dispersion relations as provided by Seiler et al. [44], Mathur et al. [45], and Zhang [46], respectively. The photoemission from QDs of GaSb is presented in accordance with the aforementioned models for the joint purpose of coherent presentation and relative assessment. (i) In accordance with the model of Seiler et al. [44], the dispersion relation of the conduction electrons in n-GaSb assumes the form:   1/2 ζ 2 k2 Eg0  Eg 0 v0 θ1 (k) 2 w0 θ2 (k) 2 0 2 + + ± + 1 + α4 k , E= − 2 2 2m0 2m0 2m0 (3.113) where



 −1 2 , θ1 (k) = k−2 kx2 ky2 + ky2 kz2 + kz2 kx2 α4 = 4P20 Eg0 +  Eg20 Eg0 +  3

3.2

Theoretical Background

133

) represents the warping of the Fermi surface, θ2 (k) = k2 kx2 ky2 + ky2 kz2 + kz2 kx2  *1/2 .k−1 represents the inversion asymmetry splitting of the conduction −9kx2 ky2 kz2 band, ζ0 , v0 and w0 represent the constants of the spectrum. The vz (Enz ) in this case is given by:

1

vz Enz,± = Q18 Enz,± , 

(3.114)

where

⎡

Q18 Enz,± = ⎣α4 Eg0



π nz dz

⎤ 



−1/2 

ζ0  2 π nz 2 π nz ⎦ + 1 + α4 , dz m0 dz

in which Enz,± should be determined from the following equation:

Enz,±

Eg Eg =− 0 + 0 2 2

 1 + α4



π nz dz

2 1/2 +

ζ0  2 2m0



π nz dz

2 .

(3.115)

The totally quantized energy EQD,± is given by: 

EQD,±

 



1/2 π ny 2 Eg0 Eg0 π nz 2 πnx 2 = − + + + 1 + α4 2 2 dx dy dz   





 πny 2 ζ0  2 v0 2 w0 S7 2 πnz 2 π nx 2 + + + S6 ± , + 2m0 dx dy dz 2m0 2m0 (3.116)

where 





−2 πny 2 πnx 2 πnz 2 + + S6 = dx dy dz ⎡ ⎤ 2  2 2 n n 2 π 2 ny nz π 2 nx ny π z x ⎣ ⎦, + + dx dy dy dz dx dz  



 πny 2 π nx 2 π nz 2 + + S7 = dx dy dz

134

3 Fundamentals of Photoemission from Quantum Dots



n2x n2y π 4 dx2 dy2



πnx dx

+

n2y n2z π 4 dy2 dz2

2 +

π ny dy

n2 n2 π 4 + z 2x 2 dz dx

2 +

π nz dz

 −

9π 6 n2x n2y n2z

!1/2

dx2 dy2 dz2 ⎤

2 −1/2

⎦.

The electron concentration is given by: n0D =

nxmax nymax nzmax       gv  F−1 η50,± , where η50,± = EF0D − EQD20,± /kB T. dx dy dz nx =1 ny =1 nz =1

(3.117) The photoemitted current density can be written as: J0D =

nxmax nymax nzmax

    α0 egv  Q18 Enz,± F−1 η50,± , 2dx dy dz n nx =1 ny =1

(3.118)

zmin

where nzmin should be determined from the following equation: Eg Eg (W − hν) = − 0 + 0 2 2



1 + α4

π nzmin dz

2 1/2

+ ζ0

2 2m0



π nzmin 2 . dz (3.119)

(ii) In accordance with the model of Mathur et al [45], the electron dispersion law in n-GaSb can be written as: E = α9 k2 +

 Eg1  1 + α10 k2 − 1 , 2

(3.120)

where       22 2 5 × 10−S T 2 1 1   ,Eg1 = Eg0 + − , α10 = . α9 = 2m0 2 (112 + T) m∗ m0 Eg1 From (3.120), we get: k2 = where

E + α11 − [α12 E + α13 ]1/2 , α9

(3.121)

3.2

Theoretical Background

α11 =

2 Eg1



8α92

135

 2     4 2 Eg1 Eg1 10α α 4α9 8α 9 10 2 + 29 − α10 + , α12 = , α13 = α10 . Eg1 Eg1 64α94 Eg1 α93

The vz (Enz ) in this case is given by:   1   vz Enz = Q19 Enz , 

(3.122)

where 

⎡



Q19 Enz = ⎣2α9



π nz dz



⎡ ⎤⎤



2 −1/2 Eg1 π n π n z z ⎣α10 ⎦⎦ , + 1 + α10 2 dz dz

in which Enz should be determined from the following equation: Enz =

2nz π dz



⎡



⎣α9 +

α10 Eg1 1 + α10 4



π nz dz

2 −1/2

⎤ ⎦.

(3.123)

The totally quantized energy EQD21 assumes the form: 

EQD21





 π ny 2 πnz 2 π nx 2 = α9 + + dx dy dz ⎡1 ⎤ (3.124)  2

2

2

2  π n Eg1 2 π n π n y x z ⎣31 + α10 + + + − 1⎦ . 2 dx dy dz

The electron concentration is given by:

n0D

nxmax nymax nzmax     2gv  = F−1 (η51 ), where η51 = EF0D − EQD21 / (kB T) . dx dy dz n nx =1 ny =1

zmin

(3.125) The photoemitted current density can be written as: J0D =

nxmax nymax nzmax     α0 egv  Q19 Enz F−1 (η51 ) , dx dy dz n nx =1 ny =1

(3.126)

zmin

where nzmin should be determined from the following equation: W − hν = α9

π nzmin dz

2

⎡ ⎤ 

2 Eg1 π n zmin ⎣−1 + 1 + α10 ⎦. + 2 dz

(3.127)

136

3 Fundamentals of Photoemission from Quantum Dots

(iii) The dispersion relation of the conduction electrons in n-GaSb can be expressed in accordance with Zhang [46] as:       E = E1 + E2 Z1 (k) k2 + E3 + E4 Z1 (k) k4 + k6 E5 + E6 Z1 (k) + E7 Z2 (k) , (3.128) where  √  5 21 kx4 + ky4 + kz4 3 − , Z1 (k) = 4 5 k4       639639 1/2 kx2 ky2 kz2 1 kx4 + ky4 + kz4 3 1 Z2 (k) = + − − , 32 12 5 105 k6 k4 the coefficients are in eV, the values of k are in 10(a/2π) times k in atomic units (a is lattice constant), and E1 , E2 , E3 , E4 , E5 , E6 , and E7 are energy-band constants. The vz (Enz ) in this case is given by:   1   vz Enz = Q20 Enz , 

(3.129)

where 

Q20 Enz



  √ √



 21 21 π nz π nz 3 E3 + =2 E1 + .E2 + 4 .E4 dz 2 dz 2 √



 π nz 5 21 E7 639639 1/2 +6 . E5 + .E6 + , dz 2 42 32

in which Enz should be determined from the following equation: 

  √   √ 



21 21 π nz 6 πnz 2 π nz 4 + E3 + E4 + Enz = E1 + E2 2 dz 2 dz dz   √ 

  21 639639 1/2 1 E5 + E6 + E7 . 2 32 42 (3.130) The totally quantized energy EQD22 is given by: EQD22

  = E1 + E2 Z3  +  +

πnx dx πnx dx



πnx dx



2 +



2 +



2

πny dy πny dy

+

πny dy



2 +



2 +



2

π nz dz π nz dz

+

2 2

2 3

π nz dz

2 

  . E3 + E4 Z3   . E5 + E6 Z3 + E7 Z4 ,

(3.131)

3.2

Theoretical Background

137

where √ 



 ny π 4 5 21 nz π 4 nx π 4 + + Z3 = . 4 dx dy dz ⎤ ⎡

2

2 −2 ny π 2 π π n n 3 x z ⎣ + + − ⎦ and dx dy dz 5    

2 2 ny 639639 1/2 nx ny nz 2 nx Z4 = + 32 dx dy dz dx dy ⎤  2 −3 nz 1 1 ⎦ + + . √ Z3 − dz 105 (15) 21 The electron concentration is given by: n0D =

nxmax nymax nzmax     gv  F−1 (η52 ), where η52 = EF0D − EQD22 /kB T. dx dy dz nx =1 ny =1 nz =1

(3.132) The photoemitted current density can be written as: J0D =

nxmax nymax nzmax     α0 egv  Q20 F−1 Enz F−1 (η52 ), 2dx dy dz n nx =1 ny =1

(3.133)

zmin

where nzmin should be determined from the following equation: 

  √ 

 21 21 π nzmin 4 πnzmin 2 W − hν = E1 + E2 + E3 + E4 2 dz dz 2 √ 



 πnzmin 6 21 1 639639 1/2 + E5 + E6 + E7 . . dz 2 42 32 (3.134) √

3.2.12 Photoemission from Stressed Materials   The vz Enz for QDs of stressed materials in this case is given by: 

vz Enz



    Q21 Enz 1 = , where Q21 Enz =    . ∗  c Enz

(3.135)

138

3 Fundamentals of Photoemission from Quantum Dots

The Enz can be expressed through the equation:   nz π c∗ Enz = . dz

(3.136)

The totally quantized energy EQD23 in this case assumes the form:



 −2 −2 −2 π ny 2  ∗  π nz 2  ∗  a∗ EQD23 b EQD23 c EQD23 + + = 1. dy dz (3.137) The electron concentration is given by:



πnx dx

2

n0D =



nxmax nymax nzmax       2gv  F−1 η53 , where η53 = (kB T)−1 EF0D − EQD23 . dx dy dz nx =1 ny =1 nz =1

(3.138a) The photoemitted current density can be written as:

J0D

nxmax nymax nzmax       α0 egv  = Q21 Enz F−1 η53 , dx dy dz 

(3.138b)

nx =1 ny =1 nz =1

where nzmin is given by (3.139); where nzmin satisfies the following inequality: nzmin ≥

dz π





 c∗ (W − hν) .

(3.139)

3.2.13 Photoemission from Bismuth 3.2.13.1 The McClure and Choi Model   The vz Enz of the conduction electrons in QDs of Bi in accordance with McClure and Choi [47] can be expressed as:   vz Enz =

πnz  . dz m3 1 + 2αEnz

(3.140)

Enz should be determined from the following equation: 

Enz 1 + αEnz



2 = 2m3

The totally quantized energy EQD24 is given by:



π nz dz

2 .

(3.141)

3.2

Theoretical Background

139





2 π ny 2 2 π nz 2 π nx 2 + + dx 2m2 dy 2m3 dz

2



2 π ny 4  αEQD24 π ny m2 α4 + 1−  + 2m2 dy m2 4m2 m2 dy  

α4 π 2 n2x π 2 n2y α4 π 4 ny nz 2 − . 2 . − 4m1 m2 4m2 m3 dy dz dx2 dy (3.142) The electron concentration is given by:   2 EQD24 1 + αEQD24 = 2m1

n0 =



nxmax nymax nzmax     2gv  F−1 (η54 ), where η54 = (kB T)−1 EF0D − EQD24 . dx dy dz nx =1 ny =1 nz =1

(3.143) The photoemitted current density can be written as: J0D =

nxmax nymax nzmax   nz F−1 (η54 ) α0 egv π   , 2 m3 .dx dy dz 1 + 2αE n z n n =1 n =1 x

y

(3.144)

zmin

where nzmin should be determined from the following equation: 2 π 2 (W − hv) [1 + α (W − hv)] = 2m3



nzmin dz

2 .

(3.145)

3.2.13.2 The Hybrid Model The vz (Enz ) for QDs of Bi in accordance with the Hybrid model [48] can be written as:   vz Enz =

πnz  , dz m3 1 + 2αEnz

(3.146)

in which Enz should be determined from the following equation:   2 Enz 1 + αEnz = (πnz /dz )2 . 2m3

(3.147)

The totally quantized energy EQD25 is given by:   2 EQD25 1 + αEQD25 = 2m1 



π nz dz

2 +

2 2m3



π nz dz

2 +

2 2M2

  γ 0 4 1 + δ 0 + αEQD25 1 − γ 0 + 4M22 Eg0



π ny 2 dy

π ny 4 . dy (3.148)

140

3 Fundamentals of Photoemission from Quantum Dots

The electron concentration is given by: n0 =

nxmax nymax nzmax     2gv  F−1 (η55 ), where η55 = (kB T)−1 EF0D − EQD25 . dx dy dz nx =1 ny =1 nz =1

(3.149) The photoemitted current density can be written as:

J0D

nxmax nymax nzmax   nz F−1 (η55 ) α0 egv π   , = 2 m3 .dx dy dz 1 + 2αEnz n =1 n =1 n x

y

(3.150)

zmin

where nzmin should be determined from the following equation: (W − hv) [1 + α (W − hv)] =

2 2  π nzmin /dz . 2m3

(3.151)

3.2.13.3 The Cohen Model The vz (Enz ) for QDs of Bi in accordance with the model of Cohen [49] can be written as:   vz Enz = 

πnz  , 1 + 2αEnz dz m3

(3.152)

in which Enz should be determined from the following equation:   2 Enz 1 + αEnz = 2m3



π nz dz

2 .

(3.153)

The totally quantized energy EQD26 is given by:





π ny 4 2 π nz 2 α4 π nx 2 + + dx 2m3 dz 4m2 m2 dy



 2 π ny 2 m2 + 1 + αEQD26 1 −  . 2m2 dy m2 (3.154) The electron concentration is given by:   2 EQD26 1 + αEQD26 = 2m1

n0 =



nxmax nymax nzmax     2gv  F−1 (η56 ), where η56 = EF0D − EQD26 /kB T. dx dy dz nx =1 ny =1 nz =1

(3.155)

3.2

Theoretical Background

141

The photoemitted current density can be written as:

J0D

nxmax nymax nzmax   nz F−1 (η56 ) π α0 egv   , = 2 m3 .dx dy dz 1 + 2αEnz n =1 n =1 n x

y

(3.156)

zmin

where nzmin should be determined from the following equation: 2 (W − hv) [1 + α (W − hv)] = 2m3



π nzmin dz

2 .

(3.157)

3.2.13.4 The Lax Model The vz (Enz ) for QDs of Bi in accordance with the model of Lax [50] is given by:   vz Enz =

π nz  , dz .m3 1 + 2αEnz

(3.158)

in which Enz should be determined from the following equation:   2 Enz 1 + αEnz = (πnz /dz )2 . 2m3

(3.159)

The totally quantized energy EQD27 is given by:   2 EQD27 1 + αEQD27 = 2m1



π nx dx

2 +

2 2m2



π ny dy

2 +

2 2m3



π nz dz

2 . (3.160)

The electron concentration is given by: n0 =

nxmax nymax nzmax     2gv  F−1 (η57 ) , where η57 = EF0D − EQD27 /kB T. dx dy dz nx =1 ny =1 nz =1

(3.161) The photoemitted current density can be written as:

J0D

nxmax nymax nzmax   nz F−1 (η57 ) π α0 egv   , = m3 .dx dy dz2 1 + 2αEnz n =1 n =1 n x

y

(3.162)

zmin

where nzmin should be determined from the following equation: (W − hv) [1 + α (W − hv)] =

2 2  π nzmin /dz . 2m3

(3.163)

142

3 Fundamentals of Photoemission from Quantum Dots

3.2.14 Photoemission from IV–VI Materials In addition to the Cohen and Lax models, the carriers of IV–VI compounds are described by three more types of dispersion relations as provided by Dimmock [51], Bangert and Kastner [52], and Foley et al. [53], respectively. In this section, the photoemission from QDs of IV–VI materials has been presented in accordance with the models of Dimmock [51], Bangert and Kastner [52] and Foley et al. [53] respectively, although the carrier dispersion laws in this compound also follows Cohen and Lax models. (i) The dispersion relation of the conduction electrons in IV–VI semiconductors can be expressed in accordance with Dimmock [51] as:    2 kz2 2 kz2 Eg0 Eg0 2 ks2 2 ks2 2 2 2 2 − − + ε+ ε− − − + − = P⊥ ks + P kz , (3.164) 2 2 2m− 2m 2m 2m t t l l where ε is the energy as measured from the center of the band gap Eg0 , and m± t and represent the contributions to the transverse and longitudinal effective masses m± l − →→ p perturbations with the other of the external L+ and L− bands arising from the k .− 6

6

bands taken to the second order.

Eg Substituting ε = E + 20 , P2⊥ = 2 Eg0 /2m∗t and P2 = 2 Eg0 /2m∗l (m∗t and m∗l are the transverse and longitudinal effective masses at k = 0), in (3.164), the following equation is obtained:     2 2

2 kz2 2 kz2 2 kz2  ks 2 ks2 2 ks2 + . 1 + αE + α = E− ∗ − − − + +α + 2mt 2m∗l 2mt 2ml 2mt 2ml (3.165)   The vz Enz is given by:   πnz   Q21 Enz , vz Enz = ∗ dz ml where 





Q21 Enz = 1 + 

α2 m∗l − m+ l ml



π nz dz

2

α2 × 1 + 2αEnz + 2m+ l

+

m∗l 

m− l π nz dz

(3.166) 

1 + αEnz

2  1−



m+ l

m− l

−1 ,

in which Enz should be determined from the following equation: 

Enz 1 + αEnz









 π nz 2 π nz 2  2 2 + αEnz . 1 + αEnz − + − dz dz 2ml 2ml (3.167)



4 2 2 π nz π nz α4 = . − − dz 2m∗l dz 4m+ l ml

3.2

Theoretical Background

143

The totally quantized energy EQD28 assumes the form:

 π nz 2 2 + − dz 2m− l !



 π ny 2 α2 π nz 2 πnx 2 × 1 + αEQD28 + + +α + (3.168) dx dy dz 2m+ 2ml t 



πny 2 2 π nx 2 2 π nz 2 = + . + 2m∗t dx dy 2m∗l dz



2 EQD28 − 2m− t 



πnx dx

2



πny dy

2

2 !

The electron concentration is given by:

n0D

nxmax nymax nzmax     2gv  = F−1 (η57 ), where η57 = EF0D − EQD28 /kB T. dx dy dz nx =1 ny =1 nz =1

(3.169) The photoemitted current density can be written as:

J0D =

nxmax nymax nzmax   πα0 egv 

dx dy dz m∗l n =1 n =1 nz x y min 2

  Q21 Enz F−1 (η57 ),

(3.170)

where nzmin should be the nearest integer of the following equation:



2 2 π nzmin 2 π nzmin 2 − (W − hv) [1 + α (W − hv)] + α (W − hv) dz dz 2m+ 2m− l l



  4 2 4 2  π nzmin π nzmin α = . (1 + α (W − hν)) − ∗ + − dz 2ml dz 4ml ml (3.171) (ii) The electron dispersion law in IV–VI compounds can be written in accordance with Bangert and Kastner [52] as:  (E) = F 1 (E) ks 2 + F 2 (E) kz 2 , where   2  2  R¯ Q (¯s)2 + + ,  (E) ≡ 2E, F 1 (E) ≡ E + Eg0 E + c E + c   2  2  s¯ + Q 2 A¯ F 2 (E) ≡ + , R, s, E + Eg0 E + c c  , Q, c  , and A¯

(3.172)

144

3 Fundamentals of Photoemission from Quantum Dots

are the spectrum   constants. The vz Enz is given by: 







vz Enz

  Q22 Enz = , 

(3.173)

where 

Q22 Enz





πnz =2 dz



F 2 Enz

 







 Enz −

π nz dz

2

   F 2 Enz

−1 ,

in which Enz should be determined from the following equation:

  π nz 2   .  Enz = F 2 Enz dz

(3.174)

The totally quantized energy EQD29 can be expressed as: 





 EQD29 = F1 EQD29





πnx dx



2 +

π ny dy

2 



  π nz 2 . (3.175) +F2 EQD29 dz

The electron concentration is given by: n0D =

nxmax nymax nzmax     2gv  F−1 (η58 ), where η58 = (kB T)−1 EF0D − EQD29 . dx dy dz nx =1 ny =1 nz =1

(3.176) The photoemitted current density can be written as: J0D =

nxmax nymax nzmax     α0 egv  Q22 Enz F−1 (η58 ), dx dy dz n nx =1 ny =1

(3.177)

zmin

where nzmin should be the nearest integer of the following equation:

π nzmin  (W − hv) = F 2 (W − hv) dz

2 .

(3.178)

  2  2  It may be noted that under the condition S = Q = 0, R = A = 2 Eg0 /m∗ , equation (3.169) reduces to the well-known two-band model of Kane. (iii) In accordance with Foley et al [53], the electron dispersion relation assumes the form:

3.2

Theoretical Background

145

⎡ ⎤1/2 2 Eg0 Eg0 2 kz 2 2 ks 2 2 kz 2 ⎣ 2 ks 2 E+ + + + + + P 2 kz 2 + P⊥ 2 ks 2 ⎦ , = 2 2 2m⊥ − 2m − 2m⊥ + 2m + (3.179) where     1 1 1 1 1 1 1 1 = ± = ± , , mtc and mlc 2 mtc mtv 2 mlc mlv m± m± ⊥  are the transverse and longitudinal effective electron masses of the conduction electrons at the edge of the conduction band, and mtv and mlv are the transverse and longitudinal hole masses of the holes at the edge of the valence band.  effective  The vz Enz is given by: 

vz Enz



  Q23 Enz = , 

(3.180)

where 

Q23 Enz









2 −1/2  π nz  1 Eg0 π nz 2 2 π nz = − + P + + dz  2 dz dz m 2m+    



 Eg0 2 2 π nz π nz 2 π nz × + , + P 2 dz dz 2m+ m+   dz

in which Enz should be determined from the following equation: Eg 2 Enz = − 0 + 2 2m− 



π nz dz

2

⎡

Eg0 2 +⎣ + 2 2m+ 



π nz dz

2 2

+ P2

π nz dz

2

⎤1/2 ⎦

.

(3.181) The totally quantized energy EQD30 for this model is given by:

EQD30





ny π 2 Eg0 2 nz π 2 nx π 2 2 =− + + + 2 dx dy dz 2m− 2m− ⊥  ⎡

2

2

!2 2 Eg0 ny π 2 π π   n n z x +⎣ + + + 2 dz dx dy 2m+ 2m+  ⊥ +

P2

nz π dz

2

+ P2⊥

nx π dx



2 +

ny π dy

2 !1/2

. (3.182)

146

3 Fundamentals of Photoemission from Quantum Dots

The electron concentration assumes the form: n0D =

nxmax nymax nzmax     2gv  F−1 (η59 ), where η59 = EF0D − EQD30 / (kB T) . dx dy dz nx =1 ny =1 nz =1

(3.183) The photoemitted current density can be written as: J0D =

nxmax nymax nzmax     α0 egv  Q23 Enz F−1 (η59 ) , dx dy dz  n nx =1 ny =1

(3.184)

zmin

where nzmin should be the nearest integer of the following equation: 2

Eg (W − hv) = − 0 + 2 2m−  +

P2

π nzmin dz



π nzmin dz

2

2 1/2

⎡

Eg0 2 +⎣ + 2 2m+ 



π nzmin dz

2 2

. (3.185)

3.2.15 Photoemission from II–V Materials The dispersion relation of the holes in II–V compounds in accordance with Yamada [54] can be expressed as: E = A10 kx2 + A11 ky2 + A12 kz2 + A13 kx  1/2

2 ± A14 kx2 + A15 ky2 + A16 kz2 + A17 kx + A18 ky2 + A219 ,

(3.186)

where A10 , A11 , A12 , A13 , A14 , A15 , A16 , A17 , A18 , and A19 are energy band constants. The vz (nz ) in this case is given by:   1   vz Enz = Q24,± Enz ,  where

⎡

Q24,± (nz ) = ⎣2A12



π nz dz



± 2A216

π nz dz

3 

(3.187)

A216

π nz dz

−1/2 ⎤

4 + A219

⎦,

in which Enz should be determined from the following equation: Enz = A12

π nz dz



2 ±

A216

π nz dz

1/2

4 + A219

.

(3.188)

3.2

Theoretical Background

147

The totally quantized energy EQD31,± is given by:

EQD31,±







π ny 2 πnx 2 π nz 2 π nx = A10 + A11 + A12 + A13 dx dy dz dx ⎡

2



2

2 π ny 2 π n π n π n x z x ± ⎣ A14 + A15 + A16 + A17 dx dy dz dx + A18

π ny dy

1/2

2 + A219

. (3.189)

The hole concentration is given by: nxmax nymax nzmax    −1 gv  1 + exp η60,± , dx dy dz nx =1 ny =1 nz =1   = EF0D − EQD31,± /kB T.

p0D = where η60,±

(3.190)

The photoemitted current density can be written as: J0D =

nxmax nymax nzmax      −1 α0 egv  Q24,± Enz 1 + exp η60,± , 2dx dy dz n nx =1 ny =1

(3.191)

zmin

where nzmin should be the nearest integer of the following equation: W − hν = A12

π nzmin dz



2 ±

A216

π nzmin dz

1/2

4 + A219

.

(3.192)

3.2.16 Photoemission from Zinc and Cadmium Diphosphides The dispersion relation of the holes of Cadmium and Zinc diphosphides can approximately be written following Chuiko [55] as:  

  β2 β3 (k) 2 β2 β3 (k) 2 k ± β4 β3 (k) β5 − k E = β1 + 8β4 8β4   1/2   β 2 (k) β 2 (k) 2 +8β42 1 − 3 , − β2 1 − 3 k 4 4 where β1 ,β2 ,β4 and β5 are system constants and β3 (k) =



(3.193)



kx2 + ky2 − 2kz2 .k−2 .

148

3 Fundamentals of Photoemission from Quantum Dots

The vz (nz ) in this case is given by:   1   vz Enz = Q25 Enz , 

(3.194)

where







  β2 β2 π nz 3 πnz Q25 (nz ) = 2 β1 − ∓ −2β4 β5 + − β2 dz 4β4 dz 4β4 2 ⎤  



2 −1/2 β2 3 π nz ⎦, 12β42 + +2β4 β5 + + β2 4β4 2 dz in which Enz should be determined from the following equation: !1/2



β2 3 π nz 2 2 ± −2β4 β5 + + 12β4 . − β2 4β4 2 dz (3.195) The totally quantized energy EQD32,± is given by:

β2 Enz = β1 − 4β4



πnz dz

2

  

 β2 β6 β2 β6 EQD32,± = β1 + β72 ± β4 β6 β5 − β72 8β4 8β4   !1/2   2 2 β β , +8β42 1 − 6 − β2 1 − 6 β72 4 4

(3.196)

where 



 πny 2 1 π nz 2 πnx 2 + −2 and β6 = 2 dx dy dz β7 



 π ny 2 π nx 2 πnz 2 2 β7 = + + . dx dy dz The hole concentration assumes the form: nxmax nymax nzmax    −1 gv  1 + exp η61,± ,where dx dy dz nx =1 ny =1 nz =1   = EF0D − EQD32,± / (kB T) .

p0D = η61,±

(3.197)

The photoemitted current density can be written as: J0D

nxmax nymax nzmax      −1 α0 egv  = Q25 Enz 1 + exp η61,± , 2dx dy dz n nx =1 ny =1

zmin

(3.198)

3.2

Theoretical Background

149

where nzmin should be the nearest integer of the following equation:



β2 πnzmin 2 W − hν = β1 − 4β4 dz !1/2 (3.199)



β2 3 π nzmin 2 2 ± −2β4 β5 + + 12β4 . − β2 4β4 2 dz

3.2.17 Photoemission from Bismuth Telluride The dispersion relation of the holes in Bi2 Te3 can be expressed as [56]: E (1 + αE) =

2 α11 kx2 + α22 ky2 + α33 kz2 + 2α23 ky kz , 2m0

(3.200)

where α11, α22, α33 and α23 are spectrum constants. The vz Enz is given by:   vz Enz =

πα33 nz  , dz m0 1 + 2αEnz

(3.201)

in which Enz should be determined from the following equation: 

Enz 1 + αEnz



2 = 2m0



α33

π nz dz

2  .

(3.202)

The totally quantized energy EQD33 can be written as: 

EQD33 1 + αEQD33







ny π 2 2 nx π 2 = + α22 α11 2m0 dx dy  

2 π 2 ny nz nz π + α33 + 2α23 . dz dy dz

(3.203)

The hole concentration is given by:

p0D =

nxmax nymax nzmax    −1 2gv  1  1 + exp (η62 ) , where η62 = EF0D − EQD33 . dx dy dz kB T nx =1 ny =1 nz =1

(3.204)

150

3 Fundamentals of Photoemission from Quantum Dots

The photoemitted current density assumes the form: J0D

  nxmax nymax nzmax   nz 1 + exp (η62 ) −1 π α0 egv α33    = , m0 dx dy dz2 1 + 2αEnz n =1 n =1 n x

y

(3.205)

zmin

where nzmin should be the nearest integer of the following equation: (W − hv) [1 + α (W − hv)] =

2 2 α33  π nzmin /dz . 2m0

(3.206)

3.2.18 Photoemission from Quantum Dots of Antimony The dispersion relation of the conduction electrons in Antimony (Sb) can be written following Ketterson [57] as: 2m0 E = α11 kx2 + α22 ky2 + α33 kz2 + 2α23 ky kz , 2

(3.207)

2m0 E = a1 kx2 + a2 ky2 + a3 kz2 + a4 ky kz + a5 kz kx + a6 kx ky , 2 2m0 E = a1 kx2 + a2 ky2 + a3 kz2 + a4 ky kz − a5 kz kx − a6 kx ky , 2

(3.208) (3.209)

where 1 1 (α11 + 3α22 ) , a2 = (α22 + 3α11 ) , a3 = α33 , a4 = α33 , 4 4 √ √ a5 = 3, a6 = 3 (α22 − α11 ) , α11 , α22 , α33 , and α23 a1 =

are the system constants. For  the  QDs whose bulk carrier dispersion law is described by (3.207), the vz1 Enz1 can be expressed as:    πnz1 vz1 Enz1 = α33 . , m0 dz

(3.210)



2 πnz1 2 . . 2m0 dz

(3.211)

where Enz1 = α 33

The totally quantized energy EQD1 can be written as: 2m0 EQD1 = α 11 2



π nx1 dx

2

+ α 22

π ny1 dy

2

+ α 33

π nz1 dy



2 + 2α 23

 π 2 ny1 .nz1 . dy dz (3.212)

3.2

Theoretical Background

151

The electron concentration for the QDs whose bulk carrier dispersion law is described by (3.207) can be written as:

n0D1

nx1max ny1max nz1max     2 1  = F−1 (η63 ),where η63 = EF0D − EQD1 . dx dy dz kB T nx1 =1 ny1 =1 nz1 =1

(3.213) For the QDs whose bulk carrier dispersion law is described by (3.207), the photoemitted current density can be expressed as: nx1max ny1max nz1max   α0 eπ α¯ 33  nz1 F−1 (η63 ), m0 .dx dy dz2 n

J0D1 =

nx1 =1 ny1 =1

(3.214)

z1min

where nz1min should be the nearest integer of the following equation: W − hv = α¯ 33



2 πnz1min 2 . . 2m0 dz

(3.215)

Similarly, for QDs whose bulk carrier dispersion law is given by (3.208), the corresponding photocurrent density and the electron concentration can, respectively, be expressed as: J0D2 =

nx2max ny2max nz2max   π α0 ea3  nz2 F−1 (η64 ) 2 m0 .dx dy dz n

(3.216)

nx2max ny2max nz2max    2 = F−1 (η64 ), dx dy dz

(3.217)

nx2 =1 ny2 =1

z2min

and n0D2

nx2 =1 ny2 =1 nz2 =1

  where η64 = (1/kB T) EF0D − EQD2 , EQD2 is given by: EQD2







ny2 π 2 2 nx2 π 2 nz2 π 2 = + a2 + a3 a1 2m0 dx dy dz  ny2 nz2 π 2 π 2 nx2 ny2 nx2 nz2 π 2 + a4 + a5 + a6 . dy dz dx dz dx dy

(3.218)

The nz2min should be the nearest integer of the following equation: W − hν = a3

2 2m0



πnz2min dz

2 .

(3.219)

152

3 Fundamentals of Photoemission from Quantum Dots

Similarly, for QDs whose bulk carrier dispersion law is given by (3.209), the corresponding photocurrent density and the electron concentration can, respectively, be written as: J0D3

nx3max ny3max nz3max   π α0 ea3  = nz3 F−1 (η65 ) 2 m0 .dx dy dz n nx3 =1 ny3 =1

(3.220)

z3min

and n0D3 =

nx3max ny3max nz3max    2 F−1 (η65 ), dx dy dz

(3.221)

nx3 =1 ny3 =1 nz3 =1

  where η65 = (1/kB T) EF0D − EQD3 , EQD3 is given by: EQD3







ny3 π 2 2 nx3 π 2 nz3 π 2 = + a2 + a3 a1 2m0 dx dy dz  ny3 nz3 π 2 π 2 nx3 ny3 nx3 nz3 π 2 + a4 − a5 − a6 . dy dz dx dz dx dy

(3.222)

The nz2min should be the nearest integer of the following equation: W − hv = a3

2 2m0



πnz3min dz

2 .

(3.223)

Therefore, the respective simplified expressions of the total electron concentration and the current density from QDs of antimony can be written as: J0D = J0D1 + J0D2 + J0D3

(3.224)

n0D = n0D1 + n0D2 + n0D3 .

(3.225)

and

3.3 Results and Discussions Using (3.6) and (3.11) and the band constants from Appendix A, we have plotted the normalized photoemitted current density from QDs of CdGeAs2 as a function of film thickness as shown in plot (a) of Fig. 3.1, where plot (b) indicates the case for δ = 0. Plots (c) and (d) represent the photoemission in this case in accordance with the three- (using (3.17)) and two- (using (3.19a)) band models of Kane. In Figs. 3.2 and 3.3, the aforementioned variables have been plotted as functions of normalized incident photon energy and normalized electron degeneracy, respectively, for

3.3

Results and Discussions

153

Fig. 3.1 Plot of the normalized photocurrent density from QDs of CdGeAs2 as a function of dz in accordance with (a) generalized band model, (b) δ = 0, (c) the three-band model of Kane, and (d) the two-band model of Kane

Fig. 3.2 Plot of the normalized photocurrent density from QDs of CdGeAs2 as a function of normalized incident photon energy for all the cases of Fig. 3.1

154

3 Fundamentals of Photoemission from Quantum Dots

Fig. 3.3 Plot of the normalized photocurrent density from QDs of CdGeAs2 as a function of normalized electron degeneracy for all the cases of Fig. 3.1

Fig. 3.4 Plot of the normalized photocurrent density as a function of dz from QDs of n-InSb in accordance with the (a) three-band model of Kane, (b) two-band model of Kane, (c) model of Stillman et al., (d) model of Palik et al., (e) model of Johnson et al., and (f) model of Agafonov et al. The plots (g) and (h) refer to QDs of GaAs in accordance with the models of Newson et al. and Rossler, respectively

3.3

Results and Discussions

155

all cases of Fig. 3.1. In Fig. 3.4, the normalized photoemitted current density from QDs of n-InSb has been plotted as a function of dz in accordance with the (a) threeband model of Kane, (b) two-band model of Kane, the models of (c) Stillman et al. (using (3.23a), (d) Palik et al. (using (3.37)), (e) Johnson et al. (using (3.44)), and (f) Agafonov et al (using (3.50)), respectively. Plots (g) and (h) refer to QDs of GaAs in accordance with the models of Newson et al. (using (3.29)) and Rossler (using (3.32)), respectively, in the same figure. In Figs. 3.5 and 3.6, the normalized photoemissions for all cases of Fig. 3.4 have been plotted as functions of normalized incident photon energy and normalized electron degeneracy, respectively. Figure 3.7 exhibits the normalized photoemitted current density from QDs of CdS (using (3.56)) as a function of dz as shown in plot (a), in which plot (b) is valid for C0 = 0. Plot (c) in the same figure is valid for QDs of GaP and has been drawn by using (3.61) and (3.62), respectively. In Figs. 3.8 and 3.9, the aforementioned variable has been plotted as functions of normalized incident photon energy and normalized electron degeneracy, respectively, for all cases of Fig. 3.7. Figure 3.10 exhibits the plot of the normalized photoemitted current density as a function of film thickness from QDs of n-Ge in accordance with the models of (a) Cardona et al. (using (3.67) and (3.68)) and (b) Wang et al. (using (3.73) and (3.74)), respectively. The curves (c) and (d) in the same figure refer to the photoemission

Fig. 3.5 Plot of the normalized photocurrent density from QDs of n-InSb and n-GaAs as a function of normalized incident photon energy for all the cases of Fig. 3.4

156

3 Fundamentals of Photoemission from Quantum Dots

Fig. 3.6 Plot of the normalized photocurrent density from QDs of n-InSb and n-GaAs as a function of normalized electron degeneracy for all the cases of Fig. 3.4

Fig. 3.7 Plot of the normalized photocurrent density from QDs of (a) CdS with C0  = 0, (b) C0 = 0, respectively, and (c) GaP as a function of dz

3.3

Results and Discussions

157

Fig. 3.8 Plot of the normalized photocurrent density from QDs of (a) CdS with C0  = 0, (b) C0 = 0, respectively, and (c) GaP as a function of normalized incident photon energy

Fig. 3.9 Plot of the normalized photocurrent density from QDs of (a) CdS with C0  = 0, (b) C0 = 0, respectively, and (c) GaP as a function of normalized electron degeneracy

158

3 Fundamentals of Photoemission from Quantum Dots

Fig. 3.10 Plot of the normalized photocurrent density from QDs of (a) n-Ge in accordance with the model of Cardona et al., (b) n-Ge in accordance with the models of Wang et al. as a function of film thickness. The curves (c) and (d) refer to the same plot for QDs of Tellurium in accordance with the models of Bouat et al. and Ortenberg et al., respectively. The curve (e) stands for the QDs of graphite

from QDs of Tellurium in accordance with the models of Bouat et al. (using (3.79) and (3.80)) and Ortenberg et al. (using (3.85) and (3.86)), respectively. Additionally, the curve (e) of the same figure has been plotted for QDs of Graphite (using (3.91) and (3.92)). In Figs. 3.11 and 3.12, the normalized photoemitted current density has been plotted as a function of normalized incident photon energy and normalized electron degeneracy, respectively, for all cases of Fig. 3.10. In Fig. 3.13, we have drawn the plots of the normalized photoemission as a function of film thickness from QDs of (a) n-PtSb2 (using (3.97) and (3.98)), (b) zerogap materials, taking HgTe as an example (using (3.104) and (3.105)), and (c) Pb1–x Gex Te (using (3.110) and (3.111)). In Figs. 3.14 and 3.15, the normalized photoemission in this case has been plotted as a function of normalized incident photon energy and normalized carrier degeneracy, respectively, for all cases of Fig. 3.13. In Fig. 3.16, we have drawn the plots of the normalized photoemission as a function of film thickness from QDs of GaSb in accordance with the models of (a) Seiler et al. (using (3.117) and (3.118)), (b) Mathur et al. (using (3.125) and (3.126)), and (c) Zhang et al. (using (3.132) and (3.133)), respectively. The curve (d) in the same figure refers to the QDs of stressed materials, where stressed n-InSb has been considered as an example (using (3.138a) and (3.139b)). In Figs. 3.17 and 3.18, the normalized photocurrent density in this

3.3

Results and Discussions

159

Fig. 3.11 Plot of the normalized photocurrent density as a function of normalized incident photon energy for all the cases of Fig. 3.10

Fig. 3.12 Plot of the normalized photocurrent density as a function of normalized electron degeneracy for all the cases of Fig. 3.10

160

3 Fundamentals of Photoemission from Quantum Dots

Fig. 3.13 Plot of the normalized photocurrent density from QDs of (a) PtSb2 , (b) HgTe, and (c) Pb1–x Gex Te as a function of film thickness

Fig. 3.14 Plot of the normalized photocurrent density from QDs of (a) PtSb2 , (b) HgTe, and (c) Pb1–x Gex Te as a function of normalized incident photon energy

3.3

Results and Discussions

161

Fig. 3.15 Plot of the normalized photocurrent density from QDs of (a) PtSb2 , (b) HgTe, and (c) Pb1–x Gex Te as a function of normalized carrier degeneracy

Fig. 3.16 Plot of the normalized photocurrent density from QDs of GaSb in accordance with the models of (a) Seiler et al., (b) Mathur et al., and (c) Zhang et al., respectively, as a function of film thickness. The plot (d) refers to QDs of stressed InSb

162

3 Fundamentals of Photoemission from Quantum Dots

Fig. 3.17 Plot of the normalized photocurrent density as a function of normalized incident photon energy for all the cases of Fig. 3.16

Fig. 3.18 Plot of the normalized photocurrent density as a function of normalized electron degeneracy for all the cases of Fig. 3.16

3.3

Results and Discussions

163

Fig. 3.19 Plot of the normalized photocurrent density as a function of film thickness from QDs of Bismuth in accordance with the models of (a) McClure and Choi, (b) Cohen, (c) Hybrid, and (d) Lax, respectively

case has been plotted as a function of normalized incident photon energy and the normalized electron degeneracy, respectively, for all cases of Fig. 3.16. Figure 3.19 exhibits the plot of the normalized photocurrent density as a function of film thickness from QDs of Bismuth in accordance with the models of (a) McClure and Choi (using (3.143) and (3.144)), (b) Cohen (using (3.155) and (3.156)), (c) Hybrid (using (3.149) and (3.150)), and (d) Lax (using (3.161) and (3.162)), respectively. In Figs. 3.20 and 3.21, the same variable has been plotted as a function of normalized incident photon energy and normalized electron degeneracy, respectively, for all cases of Fig. 3.19. Figure 3.22 depicts the plot of the normalized photoemission current density as a function of film thickness from QDs of IV–VI materials taking n-PbTe as an example in accordance with the models of (a) Dimmock (using (3.169) and (3.170)), (b) Cohen (using (3.155) and (3.156)), (c) Bangert et al. (using (3.177) and (3.176)), and (d) Foley et al. (using (3.184) and (3.183)), respectively. Additionally, plot (e) of Fig. 3.22 exhibits the photoemission current density from QDs of II–V materials taking CdSb as an example (using (3.191) and (3.190)). In Figs. 3.23 and 3.24, the normalized photoemission in this case has been plotted as a function of normalized incident photon energy and normalized carrier degeneracy, respectively, for all cases of Fig. 3.22. Figure 3.25 exhibits the plot of the normalized photoemission current density as a function of film thickness from QDs of (a) cadmium and (b) zinc diphosphides,

164

3 Fundamentals of Photoemission from Quantum Dots

Fig. 3.20 Plot of the normalized photocurrent density as a function of normalized incident photon energy for all the cases of Fig. 3.19

Fig. 3.21 Plot of the normalized photocurrent density as a function of normalized electron degeneracy for all the cases of Fig. 3.19

3.3

Results and Discussions

165

Fig. 3.22 Plot of the normalized photocurrent density from QDs of PbTe in accordance with the models of (a) Dimmock, (b) Cohen, (c) Bangert et al., and (d) Foley et al., respectively, as a function of film thickness. The plot (e) refers to QDs of CdSb

Fig. 3.23 Plot of the normalized photocurrent density as a function of normalized incident photon energy for all the cases of Fig. 3.22

166

3 Fundamentals of Photoemission from Quantum Dots

Fig. 3.24 Plot of the normalized photocurrent density as a function of normalized carrier degeneracy for all the cases of Fig. 3.22

Fig. 3.25 Plot of the normalized photocurrent density from QDs of (a) cadmium and (b) zinc diphosphides as a function of film thickness

3.3

Results and Discussions

167

Fig. 3.26 Plot of the normalized photocurrent density from QDs of (a) cadmium and (b) zinc diphosphides as a function of normalized incident photon energy

Fig. 3.27 Plot of the normalized photocurrent density from QDs of (a) Cadmium and (b) zinc diphosphides as a function of normalized carrier degeneracy

168

3 Fundamentals of Photoemission from Quantum Dots

Fig. 3.28 Plot of the normalized photocurrent density from QDs of (a) Bi2 Te3 and (b) Sb as a function of film thickness

respectively, which were obtained by using (3.198) and (3.197). In Figs. 3.26 and 3.27, the normalized photoemissions in this case have been plotted as functions of normalized photon energy and normalized carrier degeneracy, respectively, for all cases of Fig. 3.25. Figure 3.28 shows the plot of the normalized photoemission as a function of film thickness from QDs of (a) Bi2 Te3 (using (3.205) and (3.204)) and (b) Sb (using (3.225) and (3.224)), respectively. In Fig. 3.29 and 3.30, the normalized photoemissions in this case have been plotted as functions of normalized photon energy and normalized carrier degeneracy, respectively, for all cases of Fig. 3.28. It appears from Figs. 3.1, 3.4, 3.7, 3.10, 3.13, 3.16, 3.19, 3.22, 3.25, and 3.28 that the normalized photoemitted current density increases with decreasing film thickness and exhibits spikes for various values of dz which are totally band structure– dependent. Figures 3.2, 3.5, 3.8, 3.11, 3.14, 3.17, 3.20, 3.23, 3.26, and 3.29 exhibit the step-functional dependence of the normalized photoemitted current density from the QDs of different materials with the incident photon energy. It is apparent from Figs. 3.3, 3.6, 3.9, 3.12, 3.15, 3.18, 3.21, 3.24, 3.27, and 3.30 that the normalized photoemitted current density from the QDs of various materials increases with increasing normalized carrier degeneracy and exhibits spikes for different values of carrier concentration which are again band structure–dependent. It may be noted that the QDs lead to discrete energy levels, somewhat like atomic energy levels, which produce very large changes. This follows from the inherent nature of the quantum confinement of the carrier gas dealt with here. In QDs, there remain no free

3.3

Results and Discussions

169

Fig. 3.29 Plot of the normalized photocurrent density from QDs of (a) Bi2 Te3 and (b) Sb as a function of normalized incident photon energy

Fig. 3.30 Plot of the normalized photocurrent density from QDs of (a) Bi2 Te3 and (b) Sb as a function of normalized carrier degeneracy

170

3 Fundamentals of Photoemission from Quantum Dots

carrier states in between any two allowed sets of size-quantized levels, unlike those found for QWs in UFs and QWWs where the quantum confinements are 1D and 2D, respectively. Consequently, the crossing of the Fermi level by the size-quantized levels in QDs would have much greater impact on the redistribution of the carriers among the allowed levels, as compared to that found for QWs in UFs and QWWs, respectively. Although photoemission varies in different manners with all the variables in all the limiting cases, as evident from all the figures, the rates of variation are totally band structure–dependent. The quantum signature of QDs for photo-electric effect is rather prominent as compared to the same from QWs in UFs and QWWs. The photoemission from QWs in UFs, QWWs, and QDs in the presence of external photoexcitation will be further investigated in detail in Chapter 7, with the realization that it is the band structure which changes in a fundamental way and consequently alters the photoemission, together with the fact that in general all the physical properties of all the electronic materials change radically, leading to new physical concepts.

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3 Fundamentals of Photoemission from Quantum Dots of Millimeter and Sub-Millimeter Waves, 2001. The Fourth International Kharkov 1, 331 (2001); Z. Helin, D. M. Rowe, S. G. K. Williams, 2001. Proc. XX International Conference on Thermoelectrics, 314 (2001). B. R. Nag, Electron Transport in Compound Semiconductors, (Springer Verlag, Germany, 1980); E. O. Kane, In Semiconductors and Semimetals, Vol. 1, Ed. By R. K. Willardson, A. C. Beer (Academic Press, USA, 1966) p. 75. G. E. Stillman, C. M. Wolfe, J. O. Dimmock, in Semiconductors and Semimetals, Ed. R. K. Willardon, A. C. Beer 12, p. 169 (Academic Press, IV,V, USA, 1977). D. J. Newson, A. Karobe, Semiconductor. Sci. Tech. 3, 786 (1988). U. Rossler, Solid State Commun. 49, 943 (1984). E. D. Palik, G. S. Picus, S. Teither, R. E. Wallis, Phys. Rev. 122, 475 (1961). E. J. Johnson, D. H. Dickey, Phys. Rev. 1, 2676 (1970). V. G. Agafonov, P. M. Valov, B. S. Ryvkin, I. D. Yarashetskin, Sov. Phys. Semiconduct. 12, 1182 (1978). M. Cardona, W. Paul, H. Brooks, Helv, Acta Phys. 33, 329 (1960); A. F. Gibson, In: Proceeding of International School of Physics “ENRICO FERMI” course XIII, Ed. R. A. Smith, p. 171 (Academic Press, USA, 1963). C. C. Wang, N. W. Ressler, Phys. Rev. 2, 1827 (1970). J. Bouat, J. C. Thuillier, Surface Sci. 73, 528 (1978). M. V. Ortenberg, K. J. Button, Phys. Rev. B 16, 2618 (1977). H. Ushio, T. Dau, Y. Uemura, J. Phys. Soc. Jap. 33, 1551 (1972). V. I. Ivanov-Omskii, A. Sh. Mekhtisev, S. A. Rustambekova, E. N. Ukraintsev, Phys. Stat. Sol. (b) 119, 159 (1983). L. V. Vassilev, Phys. Stat. Sol. (b), 121, 203 (1984). D. G. Seiler, W. M. Beeker, K. M. Roth, Phys. Rev. 1, 764 (1970). P. C. Mathur, S. Jain, Phys. Rev. 19, 1359 (1979). H. I. Zhang, Phys. Rev. B 1, 3450 (1970). J. W. McClure, K. H. Choi, Solid State Comm. 21, 1015 (1977). S. Takaoka, H. Kawamura, K. Murase, S. Takano, Phys. Rev. B, 13, 1928 (1976). M. H. Cohen, Phys. Rev. 121, 387 (1961). B. Lax, J. G. Mavroides, H. J. Ziegeer, R. Keyes, J. Phys. Rev. Lett. 5, 241 (1960). J. O. Dimmock, In: The Physics of Semimetals and Narrowgap Semiconductors, Ed. D. L. Carter, R. T. Bates (Pergamon Press, UK, 1971). E. Bangert, P. Kastner, Phys. Stat. Sol. (b), 61, 503 (1974). G. M. T. Foley, P. N. Langenberg, Phys. Rev. B, 15, 4850 (1977). Y. Yamada, J. Phys. Jpn. 35, 1600 (1973); M. Singh, P. R. Wallace, S. D. Jog, E. Arushanov, J. Phys. Chem. Solids 45, 409 (1984). G. P. Chuiko, Sov. Phys. Semi. 19, 1381 (1985). M. Stordeur, W. Kuhnberger, Phys. Stat. Sol. (b), 69, 377 (1975); D. R. Lovett, Semimetals and Narrow-Bandgap Semiconductor, (Pion Limited, UK, 1977); H. Köhler, Phys. Stat. Sol. (b), 74, 591 (1976). J. B. Ketterson, Phys. Rev. 129, 18 (1963).

Chapter 4

Photoemission from Quantum Confined Semiconductor Superlattices

4.1 Introduction In recent years with the advent of MBE, MOCVD, FLL, and other modern fabricational techniques, semiconductors with superlattice structures (SLs), in which alternate layers of two different degenerate materials set up a periodic potential with a periodicity many times the crystal dimensions, resulting in energy minibands, have been experimentally realized [1–4]. The SL has found wide applications in many new device structures, such as avalanche photodiodes [5], photo detectors [6], transistors [7], light emitters [8], electro-optical modulators [9], etc. Among the III–V SLs, the GaAs/Ga1–x Alx As SL has been extensively studied, in which the GaAs layers form the quantum wells and Ga1–x Alx As layers form the potential barriers. The III–V SLs find extensive application in the realization of high speed optoelectronic devices [10]. Additionally, II–VI [11], IV–VI [12], and HgTe/CdTe [13] SLs have also been experimentally realized. The IV–VI SLs exhibit new physical properties in comparison with the III–V SL because of the important band structure of the constituent materials [14]. The II–VI SLs are being used for optoelectronic operation in the blue wavelength [14]. HgTe/CdTe SLs also find applications for long wavelength infrared detectors and other electro-optical applications [15]. These features arise from the direct band gap compound CdTe whose conduction electrons obey the three-band model of Kane and gapless material HgTe [15–17]. Besides, in the effective mass SLs, the subbands of the electrons exist in real space [18]. It is worth remarking that said SLs have been proposed with the idealistic case that the interfaces between the layers are sharply defined, of zero thickness. Incidentally, the distribution of the potential of the SL may be assumed as a one dimensional array of rectangular potential wells. The aforementioned advanced experimental techniques may produce SLs with physical interfaces between the two materials that are crystallographically abrupt; adjoining their interface will change at least on an atomic scale. As the potential form changes from a well (barrier) to a barrier (well), an intermediate potential region comes into play to interact with the electrons. The influence of finite thickness of the interfaces on the electron dispersion law is very significant, since the carrier energy spectra in turn control the carrier transport in various types of SLs. 173 K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_4,  C Springer Science+Business Media, LLC 2009

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4 Quantum Confined Semiconductor Superlattices

In this chapter, we shall study the magneto-photoemission from III–V, II–VI, IV–VI, and HgTe/CdTe quantum well superlattices (QWSLs) with graded interfaces in Sections 4.2.1, 4.2.2, 4.2.3, and 4.2.4. In Sections 4.2.5, 4.2.6, 4.2.7, and 4.2.8, we shall present the photoemitted current density under magnetic quantization from III–V, II–VI, IV–VI, and HgTe/CdTe quantum well effective mass SLs respectively. In Sections 4.2.9, 4.2.10, 4.2.11, 4.2.12, 4.2.13, 4.2.14, 4.2.15, and 4.2.16, we shall investigate the same from the quantum dots of the aforementioned SLs. Section 4.3 contains results and discussions pertinent to this chapter.

4.2 Theoretical Background 4.2.1 Magneto-photoemission from III–V Quantum Well Superlattices with Graded Interfaces The electron dispersion law in bulk specimens of III–V semiconductors whose energy band structures are defined by the three-band model of Kane, and which are the constituent materials of III–V SLs, can be expressed as: 2 k 2 = EG(E, Eg0i , i ), 2m∗i

(4.1)

where, i = 1, 2, . . . ,

   Eg0i + 23 i E + Eg0i + i E + Eg0i

 , G(E, Eg0i , i ) ≡    Eg0i Eg0i + i E + Eg0i + 23 i

and the other notations have already been defined in Chapter 2. The electron energy spectrum of III–V SLs with graded interfaces can be expressed [19] as: 1  (E, ks ) , (4.2) 2 where L0 (≡ a0 + b0 ) is the SL period, a0 and b0 are the widths of the barrier and the well respectively, cos (L0 k) =

  (E, ks) ≡ 2 cosh {β (E, ks )} cos {γ (E, ks )}+ε (E, ks ) sinh {β (E, ks )} sin {γ (E, ks )}   K12 (E, ks ) − 3K2 (E, ks ) cosh {β (E, ks )} sin {γ (E, ks )} + 0 K2 (E, ks )    {K2 (E, ks )}2 + 3K1 (E, ks ) − sinh {β (E, ks )} cos {γ (E, ks )} K1 (E, ks )

4.2

Theoretical Background

175



+ 0 2 {K1 (E, ks )}2 − {K2 (E, ks )}2 cosh {β (E, ks )} cos {γ (E, ks )}   1 5 {K2 (E, ks )}3 5 {K1 (E, ks )}3 + + − 34K2 (E, ks ) K1 (E, ks ) 12 K1 (E, ks ) K2 (E, ks )  sinh {β (E, ks )} sin {γ (E, ks )}

, 

β (E, ks ) ≡ K1 (E, ks ) [a0 − 0 ] , K1 (E, ks ) ≡ E ≡ (V0 − E) ,V0

2m∗2 E G(E − Vo ,α2 , 2 ) + ks2 2

is the potential barrier encountered by the electron αi ≡ 1/Eg0i , ks2 = kx2 + ky2 , o is the interface width, 

1/ 2 ,

   V0 = Eg02 − Eg01  ,

2m∗1 E G(E,α ,  ) − ks2 γ (E, ks ) ≡ K2 (E, ks ) [b0 − 0 ] ,K2 (E, ks ) ≡ 1 1 2   K1 (E, ks ) K2 (E, ks ) ε (E, ks ) ≡ − . K2 (E, ks ) K1 (E, ks )

1/2 , and

Therefore, the simplified electron dispersion law in III–V QWSL with graded interfaces can be expressed under magnetic quantization as: 



     2 2 |e| B 1 1 πnz 2 −1 1 = 2 cos − f4 E4, 1 , n n+ , (4.3) dz 2  2 L0 where

     , n = f f4 E4, n) (E, E=E4, 1 , 5 1

 and E4, 1 is the totally quantized energy in this case.

f5 (E, n) = [2 cosh [t1 (E, n)] cos [t2 (E, n)] + ε4 (E, n) sinh [t1 (E, n)] sin [t2 (E, n)]   t42 (E, n) + 0 − 3t5 (E, n) cosh (t1 (E, n)) sin (t2 (E, n)) t5 (E, n)    t52 (E, n) + 3t4 (E, n) − sinh (t1 (E, n)) cos (t2 (E, n)) t4 (E, n) 

+ 0 2 t42 (E, n) − t52 (E, n) cosh (t1 (E,n)) cos (t2 (E,n))   1 5t53 (E, n) 5t43 (E, n) + + − 34t4 (E, n) t5 (E, n) 12 t4 (E, n) t5 (E, n)

176

4 Quantum Confined Semiconductor Superlattices

 sinh (t1 (E, n)) sin (t2 (E, n)) , t1 (E, n) = t4 (E, n) (a0 − 0 ) , 

  1/ 2 2m∗2 (V0 − E)  1 2 |e| B G E − V α ,  , n+ + t4 (E, n) = 0, 2 2  2 2 t2 (E, n) = t5 (E, n)(bo − 0 ),

  ∗  2 |e| B  2m1 1 1/ 2 t5 (E, n) = − E.G E α ,  n + , 1 1  2 2   t4 (E, n) t5 (E, n) and ε4 (E, n) ≡ − . t5 (E, n) t4 (E, n) The electron concentration in this case (n0 ) is given by: nzmax nmax    eBgv  n0 = F−1 η4, 1 , π

(4.4)

nz =1 n=0



 where η4, 1 = (kB T)−1 EFBQWSLGI − E4, 1 and EFBQWSLGI is the Fermi energy in the present case. The velocity of the photoemitted electron in the nz th subband assumes the form:

   sin L0 πnz     dz 2L0     = R4 Enz , nz , vz (Enz ) =    Enz 

(4.5a)

where the term φ  (Enz ) is given by:  φ  (Enz ) = φ  (E)E=E , nz

(4.5b)

in which φ  (E) =



 2β  (E, 0) sinh (β (E, 0)) cos (γ (E, 0)) − (2γ  (E, 0)) cosh (β (E, 0))

sin (γ (E, 0)) + [ζ1 (E, 0)]   2K1 (E, 0) K1 (E, 0) K12 (E, 0) K2 (E, 0)  + 0 − − 3 K2 (E, 0) . K2 (E, 0) K22 (E, 0) cosh (β (E, 0)) sin (γ (E, 0))   #   K12 (E, 0) + β (E, 0) sinh (β (E, 0)) sin (γ (E, 0)) − 3K2 (E, 0) K2 (E, 0) $ + γ  (E, 0) cos (γ (E, 0)) cosh (β (E, 0)) + sinh (β (E, 0)) cos {γ (E, 0)}

4.2

Theoretical Background

177

! 2K2 (E, 0) K2 (E, 0) K22 (E, 0) K1 (E, 0) 3 + K1 (E, 0) K12 (E, 0)   K22 (E, 0) #  + 3K1 (E, 0) − β (E, 0) cosh {β (E, 0)} cos {γ (E, 0)} K1 (E, 0)  $  −γ (E, 0) sin {γ (E, 0)} sinh {β (E, 0)} K1 (E, 0) −

 # $ + 0 4 K1 (E, 0) K1 (E, 0) − K2 (E, 0) K2 (E, 0) cosh {β (E, 0)} cos {γ (E, 0)}

+ 2 K12 (E, 0) − K22 (E, 0) #  $ β (E, 0) sinh (E, 0) cos {γ (E, 0)} − γ  (E, 0) sin {γ (E, 0)} cosh {β (E, 0)}  5K23 (E, 0) K1 (E, 0) 15K12 (E, 0) K1 (E, 0) 1 15K22 (E, 0) K2 (E, 0) + − + 12 K1 (E, 0) K2 (E, 0) K12 (E, 0) −

5K13 (E, 0) K2 (E, 0) K22 (E, 0)



− 34 K2 (E, 0) K1 (E, 0) −34K2 (E, 0) K1 (E, 0)

   1 5K23 (E, 0) 5K13 (E, 0) + sinh (β (E, 0)) sin (γ (E, 0)) + 12 K1 (E, 0) K2 (E, 0)   −34K1 (E, 0) K2 (E, 0) β  (E, 0) cosh {β (E, 0)} sin {γ (E, 0)} 

+ γ  (E, 0) cos {γ (E, 0)} sinh {β (E, 0)}



,

where β  (E, 0) = K1 (E, 0) (a0 − 0 ) ,

−1

[−G (E − V0 ,α2 , 2 )] K1 (E, 0) = m∗2 K1 (E, 0) 2 



1 1 1 1 + (E − V0 ) + − E + Eg02 E + Eg02 + 2 E + Eg02 + 23 2  ∗ 1/2 2m2 (V0 − E)G(E − V0 ,α2 , 2 ) , K1 (E, 0) = 2 β(E, 0) = K1 (E, 0)(a0 − 0 ), 

1/2

2m∗1 E γ (E, 0) = K2 (E, 0)(b0 − 0 ),K2 (E, 0) = G(E,α1 , 1 ) 2

,

 ,

178

4 Quantum Confined Semiconductor Superlattices

γ  (E, 0) = K2 (E, 0) (b0 − 0 ) ,   m∗1  [G (E,α1 , 1 )] K2 (E, 0) = K2 (E, 0) .2 



1+E

1 1 1 + − E + Eg01 E + Eg01 + 1 E + Eg01 + 23 1

 ,

 ζ1 (E, 0) = ε (E, 0) sinh {β (E, 0)} sin {γ (E, 0)} + ε (E, 0) β  (E, 0) cosh {β (E, 0)}  sin {γ (E, 0)} + γ  (E,0) ε (E,0) sinh (β (E,0)) cos {γ (E,0)} , and   K1 (E, 0) K1 (E, 0) K2 (E, 0) K2 (E, 0) K2 (E, 0) K1 (E, 0)  ε (E, 0) = − + − . K2 (E, 0) K1 (E, 0) K22 (E, 0) K12 (E, 0) The Enz can be determined from the equation:  φ(Enz ) = 2 cos

 πL0 nz , dz

(4.6)

  where φ Enz is given by the equation: φ(Enz ) = φ(E)|E=Enz , in which  φ(E) = 2 cosh{β(E, 0)} cos{γ (E, 0)} + ε(E, 0) sinh{β(E, 0)} sin{γ (E, 0)} 

K12 (E, 0) − 3K2 (E, 0)) cosh (β(E, 0)) sin (γ (E, 0)) K2 (E, 0)   K22 (E, 0) sinh (β(E, 0)) cos (γ (E, 0)) +(3K1 (E, 0) − K1 (E, 0)

+ 0

+ 0 [2{K12 (E, 0) − K22 (E, 0)} cosh{β(E, 0) cos{γ (E, 0)}   1 5K23 (E, 0) 5K13 (E, 0) + −34K1 (E, 0)K2 (E, 0) + 12 K1 (E, 0) K2 (E, 0)   sinh{β(E, 0)} sin{γ (E, 0)} , and

(4.7)

4.2

Theoretical Background

179



 K1 (E, 0) K2 (E, 0) ε(E, 0) ≡ − . K2 (E, 0) K1 (E, 0) Thus the photoemitted current density from III–V quantum well superlattices with graded interfaces under magnetic quantization (J) can be written as: nmax n zmax α0 e2 BL0 gv  J= F−1 (η4, 1 )R1 (Enz , nz ), π2 dz n n=0

where

(4.8)

zmin

   sin L0 πnz    dz    and nzmin R1 (Enz , nz ) =    Enz 

is the nearest integer of the following inequality: nzmin ≥

dz πL0



cos−1



 1 φ(W − hv) . 2

(4.9)

4.2.2 Magneto-Photoemission from II–VI Quantum Well Superlattices with Graded Interfaces The energy spectrum of the conduction electrons of the constituent materials of II–VI SLs are given by: [20] E=

2 kz2 2 ks2 + ± C0 ks 2m∗⊥, 1 2m∗, 1

(4.10)

and   2 k 2 = EG E, Eg02 , 2 , ∗ 2m2

(4.11)

where m∗⊥, 1 and m∗, 1 are the transverse and longitudinal effective electron masses, respectively, at the edge of the conduction band for the first material. The electron dispersion law in II–VI SLs with graded interfaces can be expressed [19] as: cos (Lo k) = where

1 1 (E, ks ) , 2

 1 (E, ks ) ≡ 2 cosh {β1 (E, ks )} cos {γ1 (E, ks )}

(4.12)

180

4 Quantum Confined Semiconductor Superlattices

+ ε1 (E, ks ) sinh {β1 (E, ks )} sin {γ1 (E, ks )}   {K3 (E, ks )}2 + 0 − 3K4 (E, ks ) cosh {β1 (E, ks )} sin {γ1 (E, ks )} K4 (E, ks )    {K4 (E, ks )}2 + 3K3 (E, ks ) − sinh {β1 (E, ks )} cos {γ1 (E, ks )} K3 (E, ks ) 

+ 0 2 {K3 (E, ks )}2 − {K4 (E, ks )}2 cosh {β1 (E, ks )} cos {γ1 (E, ks )} 1 + 12



 5 {K3 (E, ks )}3 5 {K4 (E, ks )}3 + − 34K4 (E, ks ) K3 (E, ks ) K4 (E, ks ) K3 (E, ks ) 

sinh {β1 (E, ks )} sin {γ1 (E, ks )}

,

1/2 2m∗2 E 2 β1 (E, ks ) ≡ K3 (E, ks ) [a0 − 0 ] ,K3 (E, ks ) ≡ G(E − Vo ,α2 , 2 ) + ks , 2 1/2  ∗  2m, 1 2 ks2 ∓ C0 ks , γ1 (E, ks ) ≡ K4 (E, ks ) [b0 − 0 ] ,K4 (E, ks ) ≡ E− 2m∗⊥, 1 2 

and 

 K3 (E, ks ) K4 (E, ks ) − . ε1 (E, ks ) ≡ K4 (E, ks ) K3 (E, ks ) The expressions of n0 and J are, respectively, given by: nzmax nmax    gv eB  n0 = F−1 η4, 2 h n

(4.13)

nzmax nmax      α0 e2 gv BL0  J= F−1 η4, 2 R1 Enz , nz 2 2π dz n

(4.14)

z=1

n=0

and

zmin

n=0



  where η4, 2 ≡ (1/kB T) EFBQWSLGI − E4, 2 , E4, 2 is the root of the equation

π nz dz

2

 1 = 2 cos−1 L0

!2 

 2 |e| B 1 1 − f6 (E, n) n+ , 2  2 E=E 4, 2

(4.15)

4.2

Theoretical Background

181

  and f6 (E, n) ≡ f5 (E, n) E=E4, 2 ,

in which the terms K2 (E, 0) , K2 (E, 0) and t5 (E, n) of III–V SL are respectively replaced in this case by:  

2m∗, 1 2

1/ 2 ,E

  , [K2 (E, 0)]1/ 2 m∗, 1 / 2 and



2m∗, 1

1 E− n+ 2

2





eB ∓ C0 m∗⊥, 1



!1/ 2 1 1/ 2 2 |e| B . n+  2

  The term R1 Enz , nz in (4.14) is given by:

   sin L0 πnz      dz R1 Enz , nz =      ,  φ1 Enz    where φ1 Enz = φ  (E) of (4.5b) at E = Enz . Besides, all the definitions of symbols of III–V SL remain as they are for II–VI SL.

4.2.3 Magneto-Photoemission from IV–VI Quantum Well Superlattices with Graded Interfaces The E-k dispersion relation of the conduction electrons of the constituent materials of the IV–VI SLs can be expressed [21] as:  E=

ai ks2

+ bi kz2

+



ci ks2

+ di kz2



+

ei ks2

+ fi kz2

Eg + 0i 2

2 1/2 −

Eg0i , (4.16) 2

where  ai ≡  fi ≡

2 2m− ⊥, i 2 2m+ , i



 , bi ≡

 , and

2 2m− , i



 ,

ci ≡

P2⊥, i ,

di ≡

P2, i ,

ei ≡

 2 , 2m+ ⊥, i

i = 1, 2.

The electron dispersion law in IV–VI SLs with graded interfaces can be expressed as: cos (Lo k) =

1 2 (E, ks ) , 2

(4.17)

182

4 Quantum Confined Semiconductor Superlattices

where  2 (E, ks ) ≡ 2 cosh {β2 (E, ks )} cos {γ2 (E, ks )} + ε2 (E, ks ) sinh {β2 (E, ks )} sin {γ2 (E, ks )}   {K5 (E, ks )}2 − 3K6 (E, ks ) cosh {β2 (E, ks )} sin {γ2 (E, ks )} + 0 K6 (E, ks )    {K6 (E, ks )}2 sinh {β2 (E, ks )} cos {γ2 (E, ks )} + 3K5 (E, ks ) − K5 (E, ks ) 

+ 0 2 {K5 (E, ks )}2 − {K6 (E, ks )}2 cosh {β2 (E, ks )} cos {γ2 (E, ks )} 1 + 12





5 {K6 (E, ks )}3 5 {K5 (E, ks )}3 + − 34K6 (E, ks ) K5 (E, ks ) K6 (E, ks ) K5 (E, ks ) 

sinh {β2 (E, ks )} sin {γ2 (E, ks )}

,

β2 (E, ks ) ≡ K5 (E, ks ) [a0 − 0 ] ,       1/2 K5 E, kx , ky ≡ (E − V0 )2 T32 + (E − V0 ) T42 kx , ky + T52 kx , ky   1/2  fi2 − (E − V0 ) T12 + T22 kx , ky , T3i ≡  2 , b2i − fi2

     2 −1  T4i kx , ky ≡ 4T1i 4bi di + 4bi fi Eg0i + 4fi2 Eg0i + 8 kx2 + ky2 ei fi bi − ai fi2 , T5i (kx , ky ) ≡



2 4T1i

−1 

kx2 + ky2

2 

−8ai bi ei fi + 4b2i e2i + 4fi2 a2i





 + kx2 + ky2 4ei fi Eg0i bi + 4ei fi di + 4ei fi2 Eg0i − 4ai b2i Eg0i − 4ai bi di − 4ai bi fi Eg0i + 4b2i ei Eg0i + 4b2i ci + 4b2i Eg0i ai  −4fi2 ei Eg0i − 4fi2 ci − 4fi2 Eg0i ai   + Eg20i b2i + di2 + fi2 g2i + 2Eg0i bi di + 2Eg20i bi fi + 2di fi Eg0i ,

−1 , T1i ≡ b2i − fi2 

   T2i kx , ky ≡ [2T1i ]−1 Eg0 i bi + di + fi Eg0i + 2 (ei fi − ai bi ) kx2 + ky2 ,

4.2

Theoretical Background

183

γ2 (E, ks ) ≡ K6 (E, ks ) [b0 − 0 ] ,      K6 E, kx , ky ≡ ET11 + T21 kx , ky      1/2 1/2 2 − T31 E + ET41 kx , ky + T51 kx , ky and  ε2 (E, ks ) ≡

 K5 (E, ks ) K6 (E, ks ) − . K6 (E, ks ) K5 (E, ks )

The expressions of n0 and J for III–V SL as given by (4.4) and (4.8) remain as they are for IV–VI SL, where the expressions for K1 (E, 0), K1 (E, 0), t4 (E, n), K2 (E, 0), K2 (E, 0), and t5 (E, n) of III–V SL should respectively be replaced by K5 (E, 0), K5 (E, 0), t4 (E, n), K6 (E, 0), K6 (E, 0), and t5 (E, n) as follows: K5 (E, 0) ≡



 1/2 T32 (E − V0 )2 + T42 (0, 0)(E − V0 ) + T52 (0, 0)

− [T12 (E, V0 ) + T22 (0, 0)]]1/2 ,   T42 (0, 0) ≡ [2T12 ]−1 4b2 d2 + 4b2 f2 Eg02 + 4f22 Eg02 ,   T52 (0, 0) ≡ [2T12 ]−2 Eg202 b22 + d22 + f22 g22 + 2Eg02 b2 d2 + 2Eg202 b2 f2 + 2d2 f2 Eg02 ,    T22 (0, 0) ≡ [2T12 ]−2 Eg02 e2 + d2 + f2 Eg02 , K5 (E, 0) ≡ [2K5 (E, 0)]−1   (2T32 (E − V0 ) + T42 (0, 0)) −T12 +  1/ 2 , 2. T32 (E − V0 )2 + T42 (0, 0) (E − V0 ) + T52 (0, 0)  1/ 2 t4 (n) ≡ T32 (E − V0 )2 + T42 (n) (E − V0 ) + T52 (n) 

− T12 (E − V0 ) + T22 (n)

 1/ 2 ,

 T42 (n) ≡ (2T12 )−2 4b2 d2 + 4b2 f2 Eg02 + 4f22 Eg02

 8 |e| B 1  + n+ e2 f2 b2 − f22 a2 ,  2 

 2 2 1 2 −2 4 |e| B 2 2 2 2 b e f + 4b e + 4f a −8a n + T52 (n) ≡ (2T12 ) 2 2 2 2 2 2 2 2 2 2

 2 |e| B 1 + n+ 4e2 f2 Eg02 b2 + 4e2 f2 d2 + 4e2 f22 Eg02  2

184

4 Quantum Confined Semiconductor Superlattices

− 4a2 b22 Eg02 − 4a2 b2 d2 − 4a2 b2 f2 Eg02 + 4b22 e2 Eg02 + 4b22 c2  + 4b22 Eg02 a2 − 4f22 e2 Eg02 −4f22 c2 − 4f22 Eg02 a2   + Eg202 b22 + d22 + f22 g22 + 2Eg02 b2 d2 + 2Eg202 b2 f2 + 2d2 f2 Eg02 ,

  4 |e| B 1 n+ T22 (n) ≡ [2T12 ]−1 Eg02 b2 + d2 + f2 Eg02 + (e2 f2 − a2 b2 ) ,  2   1/ 2 1/ 2 2 , K6 (E, 0) ≡ [T11 E + T21 (0, 0)] − T31 E + T41 (0, 0) E + T51 (0, 0)   T21 (0, 0) ≡ [2T11 ]−1 Eg01 b1 + d1 + f1 Eg01 , T41 (0, 0) ≡ [2T11 ]−2   4b1 d1 + 4b1 f1 Eg01 + 4f12 Eg201 ,   T51 (0, 0) ≡ (2T11 )−2 Eg201 b21 +d12 + f12 g21 + 2Eg01 b1 d1 + 2Eg201 b1 f1 + 2d1 f1 Eg01 ,   (2T31 E + T41 (0, 0))  −1 T11 −  K6 (E, 0) ≡ [2K2 (E, 0)] 1 2 , 2. T31 E2 + T41 (0, 0) E + T51 (0, 0) /   1/ 2 1/ 2 t5 (E, n) ≡ [T11 E + T21 (n)] − T31 E2 + T41 (n) E + T51 (n) ,

   1 4 |e| B n+ , T21 (n) ≡ (2T11 )−1 Eg01 b1 + d1 + f1 Eg01 + (e1 f1 − a1 b1 )  2  T41 (n) ≡ [2T11 ]−2 4b1 d1 + 4b1 f1 Eg01 + 4f12 Eg01

 8 |e| B 1  2 + n+ e1 f1 b1 − f1 a1 , and  2 

 2 2 1 2 2 2 −2 4 |e| B 2 2 e + 4f a − 8a b e f 4b n + T51 (n) ≡ [2T11 ] 1 1 1 1 1 1 1 1 2 2

1  2 |e| B n+ 4e1 f1 Eg01 b1 + 4e1 f1 d1 + 4e1 f12 Eg01 − 4a1 b21 Eg01 +  2 − 4a1 b1 d1 − 4a1 b1 f1 Eg01 + 4b21 e1 Eg01 + 4b21 c1 + 4b21 Eg01 a1  −4f12 e1 Eg01 − 4f12 c1 − 4f12 Eg01 a1   + Eg201 b21 + d12 + f12 g21 +2Eg01 b1 d1 + 2Eg201 b1 f1 + 2d1 f1 Eg01 .

4.2

Theoretical Background

185

4.2.4 Magneto-Photoemission from HgTe/CdTe Quantum Well Superlattices with Graded Interfaces The dispersion relation of the conduction electrons of the constituent materials of HgTe/CdTe SLs can be expressed [15] as: 3 |e|2 k 2 k 2 + , 2m∗1 128εsc   = EG E1 , Eg02 , 2 .

E= 2 k2 2m∗2

(4.18) (4.19)

The electron energy dispersion law in HgTe/CdTe SL is given by: cos (Lo k) =

1 3 (E, ks ) , 2

(4.20)

where  3 (E, ks ) ≡ 2 cosh {β3 (E, ks )} cos {γ3 (E, ks )} + ε3 (E, ks ) sinh {β3 (E, ks )} sin {γ3 (E, ks )}   {K7 (E, ks )}2 − 3K8 (E, ks ) cosh {β3 (E, ks )} sin {γ3 (E, ks )} + 0 K8 (E, ks )    {K8 (E, ks )}2 + 3K7 (E, ks ) − sinh {β3 (E, ks )} cos {γ3 (E, ks )} K7 (E, ks ) 

+ 0 2 {K7 (E, ks )}2 − {K8 (E, ks )}2 cosh {β3 (E, ks )} cos {γ3 (E, ks )} 

5 {K8 (E, ks )}3 5 {K7 (E, ks )}3 + − 34K7 (E, ks ) K8 (E, ks ) K7 (E, ks ) K8 (E, ks )   sinh {β3 (E, ks )} sin {γ3 (E, ks )} ,

1 + 12



β3 (E, ks ) ≡ K7 (E, ks ) [a0 − 0 ] ,  ∗  1/2 2m2 E 2 G(E − V , E ,  ) + k , K7 (E, ks ) ≡ 0 g02 2 s 2 γ3 (E, ks ) ≡ K8 (E, ks ) [b0 − 0 ] , " ⎡ ⎤1/2 B20 + 2AE − B0 B20 + 4AE 2 3 |e|2 K8 (E, ks ) ≡ ⎣ − ks2 ⎦ , B0 ≡ ,A≡ and 2 128εsc 2m∗1 2A

186

4 Quantum Confined Semiconductor Superlattices



 K7 (E, ks ) K8 (E, ks ) ε3 (E, ks ) ≡ − . K8 (E, ks ) K7 (E, ks )

The expressions of n0 and J for III–V SL as given by (4.4) and (4.8) remain as they are for HgTe/CdTe, where the expressions for K2 (E, 0) , K2 (E, 0) and t5 (E, n) of III–V SL are respectively replaced by K8 (E, 0) , K8 (E, 0) and t6 (E, n) as follows: 

1/ 2 2m∗1 ω1 (E) , K8 (E, 0) ≡ 2 ⎡    1/ 2 ⎤

2 ∗ ∗ 4 2 2 2 3e m1 9m1 e 2 E 3e ⎦, +E− + ω1 (E) ≡ ⎣ 128εsc m∗1 128εsc 2 (128εsc )2 

m∗1 ω1 (E)

K8 (E, 0) ≡ 





 , and t6 (E, n) ≡

2 K8 (E, 0)



 2m∗1 ω1 (E) 2 |e| B 1 1/ 2 − . n +  2 2

4.2.5 Magneto-Photoemission from III–V Quantum Well Effective Mass Superlattices Following Sasaki [18], the electron dispersion law in III–V effective mass superlattices (EMSLs) can be written as:  kx2

=

 *2 1 ) −1   2 f E, ky , kz − k⊥ , cos L02

(4.21)

  in which f E, ky , kz ≡ a1 cos [a0 C1 (E, k⊥ ) + b0 D1 (E, k⊥ )] 2 ≡ k2 + k2 , a2 cos [a0 C1 (E, k⊥ ) − b0 D1 (E, k⊥ )], k⊥ y z  a1 ≡

2  −1 m∗2 1/2 , 4 m∗1

m∗2 +1 m∗1

1/2   2m∗1 E 2 − k ,  , G E, E g01 1 ⊥ 2 1/2  ∗

  2m2 E 2 D1 (E, k⊥ ) ≡ , and G E, Eg02 , 2 − k⊥ 2  2  −1  m∗2 1/2 m∗2 a2 ≡ −1 + . 4 m∗1 m∗1 

C1 (E, k⊥ ) ≡

The electron concentration in this case is given by:

−

4.2

Theoretical Background

187 nmax n xmax eBgv  n0 = F−1 (η5 ), π

(4.22)

n=0 nx =1

where η5 ≡

EFBQWSLEM − E5 , EFBQWSLEM kB T

is the Fermi energy in the present case, and E5 is the root of the equation

π nx dx



2 =

  2 2 |e| B 1  −1 1  n+ , cos f 5 (E, n) E=E5 −  2 L02

(4.23)

in which      f 5 (E, n) ≡ a1 cos a0 g1 (E, n) + b0 h1 (E, n) − a2 cos a0 g1 (E, n) − b0 h1 (E, n) ,  2 |e| B 2m∗1 E  G E, Eg01 , 1 − n+ g1 (E, n) ≡  2  ∗  2 |e| B 2m2 E  h1 (E, n) ≡ G E, Eg02 , 2 − n+  2 

1 2 1 2

1/ 2 , and

1/ 2 .

The photoemitted current density is given by: J=

nmax n xmax   α0 gv e2 B  F−1 (η5 ) L5 Enx , 0 , 2 2π L0 dx n n=0

(4.24)

xmin

where       #  $2 −1/ 2 L5 Enx , 0 ≡ I5 Enx , 0 1 − f 5 Enx , 0         I5 Enx , 0 ≡ a2 a0 g1 Enx , 0 − b0 h1 Enx , 0       sin a0 g1 Enx , 0 − b0 h1 Enx , 0      − a1 a0 g1 Enx , 0 + b0 h1 Enx , 0       sin a0 g1 Enx , 0 + b0 h1 Enx , 0          m∗1    G Enx , Eg01 , 1 + Enx G Enx , Eg01 , 1 , g1 Enx , 0 ≡ 2 g1 Enx , 0       −1  −1 Enx + Eg0i + i + Enx + Eg0i G Enx , Eg0i , i ≡ G Enx , Eg0i , i

188

4 Quantum Confined Semiconductor Superlattices

− Enx + Eg0i

2 + i 3

−1  ,



      2 G Enx , Eg0i , i ≡ Enx + Eg0i Enx + Eg0i + i Eg0i + i 3

−1    2 , Eg0i Eg0i + i Enx + Eg0i + i 3          m∗2    G Enx , Eg02 , 2 + Enx G Enx , Eg02 , 2 , [h1 Enx , 0 ≡ 2 h1 Enx , 0 

  /12 2m∗1 Enx  g1 Enx , 0 ≡ G Enx , Eg01 , 1 , 2   ∗   /12  2m2 Enx  h1 Enx , 0 ≡ G E , E ,  , and nx g02 2 2 



        f 5 Enx , 0 ≡ a1 cos a0 g1 Enx , 0 + b0 h1 Enx , 0      −a2 cos a0 g1 Enx , 0 − b0 h1 Enx , 0

(4.25)

Enx is the root of the equation     L0 πnx f 5 Enx = cos . dx

(4.26)

The nzmin should be the nearest integer of the following inequality: nxmin ≥

dx πL0



  cos−1 f 5 (W − hv, 0) ,

(4.27)

where f 5 (W − hv, 0)is obtained by replacing Enx by W − hv  and the same  in (4.25), replacement should be done in the definitions of g1 Enx , 0 and h1 Enx , 0 , together with the related definitions.

4.2.6 Magneto-Photoemission from II–VI Quantum Well Effective Mass Superlattices The electron concentration in this case is given by: n0 =

nmax n zmax eBgv  F−1 (η6 ), h n=0 nz =1

(4.28)

4.2

Theoretical Background

189

where

η6 =

EFBQWSLEM − E7 , E7 kB T

is the root of the equation

πnz dz



2 =

 2 2 |e| B  1  −1 1  n+ cos f7 (E, n) E=E7 −  2 L02

(4.29)

     f7 (E, n) ≡ a3 cos a0 g2 (E, n) + b0 h2 (E, n) − a4 cos a0 g2 (E, n) − b0 h2 (E, n) , (4.30) in which

 a3 ≡

2 ⎡  1/ 2 ⎤−1  ∗ 1/ 2 2m m∗2 1, 1 ⎣4 ⎦ , g2 (E, n) ≡ m∗, 1 2

m∗2 +1 m∗, 1 



 1/ 2  1 1/ 2 1 2 |e| B  |e| B , ∓ C0 n+ n+ E− ∗ m, 1 2  2

 ∗ 1/ 2    2m2 1 eB 1/ 2 h2 (E, n) ≡ , and EG E, Eg02 ,2 − n + 2 m∗2 2 ⎡  1/ 2 ⎤2 ⎡  1/ 2 ⎤−1 m∗2 m∗2 ⎦ ⎣4 ⎦ a4 ≡ ⎣−1 + m∗, 1 m∗, 1

In this case the photoemitted current density is given by:

n

nmax zmax    α0 gv e2 B  J= L6 Enz , 0 F−1 (η6 ), 2 4π L0 dz n n=0

where

zmin

(4.31)

190

4 Quantum Confined Semiconductor Superlattices       $2 −1/ 2 #  L6 Enz , 0 ≡ I6 Enz , 0 1 − f7 Enz , 0 ,               I6 Enz , 0 ≡ a4 a0 g2 Enz , 0 − b0 h2 Enz , 0 sin a0 g2 Enz , 0 − b0 h2 Enz , 0            sin a0 g2 Enz , 0 + b0 h2 Enz , 0 , − a3 a0 g2 Enz , 0 + b0 h2 Enz , 0   g2 Enz , 0 ≡



2m∗, 1

1/ 2

" 4Enz

2

−1

,

      m∗2   Enz , 0 ≡ G Enz , Eg02 , 2 + Enz G Enz , Eg02 , 2 , 2 h2 Enz , 0 

      2 G Enz , Eg0i , i ≡ Enz + Eg0i Enz + Eg0i + i Eg0i + i 3 

−1   2 Eg0i Eg0i + i Enz + Eg0i + i , 3       −1  −1 Enz + Eg02 + 2 + Enz + Eg02 G Enz , Eg02 , 2 ≡ G Enz , Eg02 , 2 h2







− Enz + Eg02   g2 Enz , 0 ≡



2m∗, 1

1/ 2



2

2 + 2 3

Enz

1/ 2

−1 

, and i = 2 in this case.

,

  ∗   1/ 2   2m2 Enz G Enz , Eg02 , 2 , h2 Enz , 0 ≡ 2          f7 Enz , 0 ≡ a3 cos a0 g2 Enz , 0 + b0 h2 Enz , 0      − a4 cos a0 g2 Enz , 0 − b0 h2 Enz , 0 .

(4.32)

The Enz is the root of the equation:

    π nz = f7 Enz , 0 . cos L0 dz

(4.33)

The nzmin should be the nearest integer of the following inequality: nzmin ≥

dz πL0

   cos−1 f7 (W − hv, 0) ,

(4.34)

where f6 (W − hv, 0)is obtained by replacing Enz byW−hv in (4.32),  and the replacement should be done in the definitions of g2 Enz , 0 and h2 Enz , 0 , together with the related definitions.

4.2

Theoretical Background

191

4.2.7 Magneto-Photoemission from IV–VI Quantum Well Effective Mass Superlattices The electron concentration in this case is given by: n

n0 =

nmax zmax  2eBgv  F−1 (η7 ), h

(4.35)

n=0 nz =1

where

η7 =

EFBQWSLEM − E8 kB T

, E8

is the root of the equation 



2 2 |e| B  1  −1 1 πnz 2 = n+ (4.36) cos f8 (E, n) E=E8 − dz  2 L02      f8 (E, n) ≡ a5 cos a0 g3 (E, n) + b0 h3 (E, n) − a6 cos a0 g3 (E, n) − b0 h3 (E, n) (4.37)



2 

−1 M2∗ 1/ 2 M2∗ +1 , 4 a5 ≡ M1∗ M1∗      2 2 2  + a c + a e E − e E a Mi∗ ≡  2 i i i i i g g i 0i 0i ai − e2i 

Eg20i a2i + c2i + e2i Eg0i + 2ci Eg0i ei − 2Eg0i ai ci − 2ai ei Eg20i

1/ 2 

,

 1/ 2 1/ 2 2 , g3 (E, n) ≡ [T11 E + T21 (n)] − T31 E + T41 (n) E + T51 (n) 

  1/ 2 1/ 2 h3 (E, n) ≡ [T12 E + T22 (n)] − T32 E2 + T42 (n) E + T52 (n) ,  a6 ≡ −1 +



M2 ∗ M1 ∗

2 

−1 M2 ∗ 1/ 2 , 4 M1 ∗

and the rest of the symbols have already been defined in Section 4.2.3. The photoemitted current density is given by: n

nmax zmax    α0 gv e2 B  J= L7 Enz , 0 F−1 (η7 ), 2 2π L0 dz n n=0

zmin

(4.38)

192

4 Quantum Confined Semiconductor Superlattices

where       #  $2 −1/ 2 L7 Enz , 0 ≡ I7 Enz , 0 1 − f8 Enz , 0 ,               I7 Enz , 0 ≡ a6 a0 g3 Enz , 0 − b0 h3 Enz , 0 sin a0 g3 Enz , 0 − b0 h3 Enz , 0            −a5 a0 g3 Enz , 0 + b0 h3 Enz , 0 sin a0 g3 Enz , 0 + b0 h3 Enz , 0 ,       −1 1 g3 Enz , 0 ≡ 2g3 Enz , 0 2Enz T31 + T41 (0, 0) T11 − 2  −1/ 2  2 T31 Enz + T41 (0, 0) Enz + T51 (0, 0) , 1/ 2 1/ 2      , g3 Enz , 0 ≡ T11 Enz + T21 (0, 0) − T31 En2z + T41 (0, 0) Enz + T51 (0, 0) h3

    −1 1 Enz , 0 ≡ 2h3 Enz , 0 2Enz .T32 + T42 (0, 0) T12 − 2





 −1/ 2  2 T32 Enz + T42 (0, 0) Enz + T52 (0, 0) , 1/ 2      , h3 Enz , 0 ≡ T12 Enz + T22 (0, 0) − T32 En2z + T42 (0, 0) Enz +T52 (0, 0)]1/ 2

and the rest of the symbols have been defined in Section 4.2.3. Additionally,         f8 Enz , 0 ≡ a5 cos a0 g3 Enz , 0 + b0 h3 Enz , 0      −a6 cos a0 g3 Enz , 0 − b0 h3 Enz , 0 . The Enz can be determined from the equation: 

   π nz cos L0 = f8 Enz , 0 . dz

(4.39)

The nzmin should be the nearest integer of the following inequality: nzmin ≥

dz πL0

   cos−1 f8 (W − hv, 0) ;

(4.40)

f8 (W − hv, 0) should be formulated in the same way as written in Section 4.2.6.

4.2

Theoretical Background

193

4.2.8 Magneto-Photoemission from HgTe/CdTe Quantum Well Effective Mass Superlattices The electron concentration in this case is given by: n

zmax nmax  eBgv  n0 = F−1 (η8 ), π

(4.41)

n=0 nz =1

where η8 ≡

EFBQWSLEM − E9 kB T

, E9 and f9 (E, n)

are defined by the following equations:

πnz dz



2 =

 2 2 |e| B  1  −1 1 n+ , cos f9 (E, n) E=E9 −  2 L02

(4.42)

     f9 (E, n) = a3 cos a0 g4 (E, n) + b0 h4 (E, n) − a4 cos a0 g4 (E, n) − b0 h4 (E, n) , (4.43) where  g4 (E, n) ≡ ⎡ ω1 (E) ≡ ⎣



−



1/ 2 1 2 |e| B , n+  2 

9m∗1 e4 (128εsc )2

 h4 (E, n) ≡

2m∗1 ω1 (E) 2

+E−

3e2 m∗1



128εsc 2

  

3e2 128εsc

2

22 E + m∗1

1/ 2 ⎤ ⎦ , and



1/ 2  2m∗2  2 |e| B 1 G E, Eg02 , 2 − . n+  2 2

The photoemitted current density is given by: n

J=

zmax nmax    α0 gv e2 B  L8 Enz , 0 .F−1 (η8 ), 2 2π L0 dz n

n=0

where

zmin

(4.44a)

194

4 Quantum Confined Semiconductor Superlattices

    #  $2 −1/ 2       a4 a0 g4 Enz , 0 − b0 h4 Enz , 0 L8 Enz , 0 ≡ 1 − f9 Enz , 0       sin a0 g4 Enz , 0 − b0 h4 Enz , 0      −a3 a0 g4 Enz , 0 + b0 h4 Enz , 0       sin a0 g4 Enz , 0 + b0 h4 Enz , 0 , in which         f9 Enz , 0 ≡ a3 cos a0 g4 Enz , 0 + b0 h4 Enz , 0      − a4 cos a0 g4 Enz , 0 − b0 h4 Enz , 0 ,   ∗    1/ 2   ∗   2 1/ 2  2m2  g4 Enz , 0 ≡ 2m1 ω1 Enz /  , h4 Enz , 0 ≡ G Enz , Eg02 , 2 , 2 ⎡  −1/ 2 ⎤ ∗

2 2 2 2      m 2 Enz 3e 3e −1 1 ⎣ ⎦ 1− + g4 Enz , 0 ≡ g4 Enz , 0 128εsc 128εsc m∗1 2 and        −1 m∗2   E . G , E ,  h4 Enz , 0 ≡ h4 Enz , 0 nz g02 2 2 The Enz should be determined from the equation

    π nz = f9 Enz , 0 . cos L0 dz

(4.44b)

The nzmin should be the nearest integer of the following inequality: nzmin ≥

dz πL0

   cos−1 f9 (W − hv, 0) .

(4.45)

The process of finding f9 (W − hυ, 0) is same as f7 (W − hv, 0).

4.2.9 Photoemission from III–V Quantum Dot Superlattices with Graded Interfaces The electron concentration in this case is given by: N0 =

nxmax nymax nzmax   2gv  F−1 (η9 ), dx dy dz nx =1 ny =1 nz =1

where

(4.46)

4.2

Theoretical Background

195

η9 =

EFQDSLGI − EQD10 , kB T

in which EFQDSLGI is the Fermi energy in this case, the EQD10 is the root of the equation



π nz dz



2 =

   2

    π ny 2 1 π nx 2 −1 1  − f10 E, nx , ny E=EQD10 − cos , 2 dx dy L02 (4.47)

and    #  $ #  $ f10 E, nx , ny = 2 cosh β0 E, nx , ny cos γ0 E, nx , ny   #  $ #  $ + ε0 E, nx , ny sinh β0 E, nx , ny sin γ0 E, nx , ny     2 E, n , n   #  $ K10 x y   − 3K20 E, nx , ny cosh β0 E, nx , ny + 0 K20 E, nx , ny    2 E, n , n #  $   K20 x y   sin γ0 E, nx , ny + 3K10 E, nx , ny − K10 E, nx , ny $ #  $ #  sinh β0 E, nx , ny cos γ0 E, nx , ny



 )    * 2 2 + 0 2 K10 E, nx , ny − K20 E, nx , ny $ #  $ #  cosh β0 E, nx , ny cos γ0 E, nx , ny      3 E, n , n 3 E, n , n 5K10 1 5K20 x y x y   +   + 12 K10 E, nx , ny K20 E, nx , ny (4.48)

    −34K10 E, nx , ny K20 E, nx , ny #



sinh β0 E, nx , ny

in which

$

#  $  sin γ0 E, nx , ny



196

4 Quantum Confined Semiconductor Superlattices

    β0 E, nx , ny ≡ K10 E, nx , ny (a0 − 0 ) , 

1/ 2   2m∗2 (V0 − E) G (E − V0 ,α2 , 2 ) π ny 2 π nx 2 + + , K10 E, nx , ny ≡ dx dy 2       γ0 E, nx , ny ≡ K20 E, nx , ny (b0 − 0 ) , K20 E, nx , ny 

1/ 2 2m∗1 EG (E,α1 , 1 ) π ny 2 π nx 2 − − , and ≡ dx dy 2        K10 E, nx , ny K20 E, nx , ny  −   . ε0 E, nx , ny ≡ K20 E, nx , ny K10 E, nx , ny The photoemitted current density is given by:

   sin L0 nz π       dz 2α0 eL0 gv   , F−1 (η9 )    J1 = dx dy dz   φ Enz , 0  n nxmax nymax nzmax nx =1 ny =1

(4.49)

zmin

where Enz is determined from     L0 πnz , φ Enz = 2 cos dz

(4.50)

and the nzmin should be the nearest integer of the following inequality: nzmin ≥

dz πL0

cos

−1



 1 φ (W − hv, 0) , 2

(4.51)

      where φ Enz ≡ φ Enz , 0 and φ  Enz are the  same  as the functions in (4.7) and (4.5b), respectively, To obtain φ Enz , 0 and φ  Enz for the present case, one should  (E, 0) by replace K10 {K10 (E, 0)}−1



m∗2 2



  . −G (E − V0 ,α2 , 2 ) + (V0 − E) G (E − V0 ,α2 , 2 ) ,

2 K10 (E, 0)

by 2m∗2 (V0 − E) G (E − V0 ,α2 , 2 )  , K20 (E, 0) by 2

 m∗1  {K20 (E, 0)}−1 . G (E,α1 , 1 ) + EG (E,α1 ,1 ) , and 2

4.2

Theoretical Background

197

1/2  K20 (E, 0) by 2m∗1 EG (E,α1 , 1 ) .−2 .

4.2.10 Photoemission from II–VI Quantum Dot Superlattices with Graded Interfaces The electron concentration is given by:

N0 =

nxmax nymax nzmax   gv  F−1 (η10 ), dx dy dz

(4.52)

nx =1 ny =1 nz =1

where

η10 =

EFQDSLGI − EQD11 and EQD11 kB T

is the root of the equation

π nz dz



2 =

   2

    π ny 2 1 π nx 2 −1 1  − f11 E, nx , ny E=EQD11 − cos . 2 dx dy L02 (4.53)

  To get f11 E, nx , ny E=EQD11 , we have to change  K20 (E, 0) by 



K20 E, nx , ny by

⎣E −

2

2m∗, 1

 1/ 2



2m∗, 1

m, 1

1/ 2 ,

2

1/ 2 ×

2 π nx dx

  , K20 (E, 0) by [K20 (E, 0)]1/ 2

E

2



⎡

2m∗,1



2 +

πny dy

2 !

∓ C0

π nx dx



2 +

π ny dy

2 !1/ 2

⎤1/ 2 ⎦

,

198

4 Quantum Confined Semiconductor Superlattices





πny 2 dy 1/ 2 2m∗2 (V0 − E) + G (E − V0 ,α2 , 2 ) , 2  ∗

1/ 2 2m2  K10 (E, 0) by − E) G − V ,α ,  and K10 (E (V ) (E, 0) by 0 0 2 2 2   m∗2   [−G (E − V0 ,α2 , 2 )] K10 (E, 0) ,2 



K10 E, nx , ny by



π nx dx

2

+



1 1 1 + − 1 + (E − V0 ) E + Eg02 E + Eg02 + 2 E + Eg02 + 23 2



in (4.48), (4.7), and (4.5b) respectively. The other equations of Section 4.2.5 remain invariant in this case. The photoemitted current density is given by:

   sin L0 nz π  nxmax nymax nzmax     dz α0 eL0 gv    , F−1 (η10 )    J1 = dx dy dz   φ Enz , 0  n nx =1 ny =1

(4.54)

zmin

where     L0 πnz φ Enz = 2 cos dz

(4.55)

and the nzmin should be the nearest integer of the following inequality: nzmin ≥

dz πL0

cos

−1



 1 φ (W − hv, 0) , 2

(4.56)

      where φ Enz ≡ φ Enz , 0 and φ  Enz , 0 are the same as the functions in (4.5b) and (4.7), respectively.

4.2.11 Photoemission from IV–VI Quantum Dot Superlattices with Graded Interfaces The electron concentration is given by: nxmax nymax nzmax   2gv  F−1 (η11 ), N0 = dx dy dz n nx =1 ny =1

zmin

(4.57)

4.2

Theoretical Background

199

where η11 =

EFQDSLGI − EQD12 , and EQD12 kB T

is the root of the equation 

  2

   π ny 2 1 π nx 2 −1 1  = − − f12 E, nx , ny E=EQD12 cos , 2 dx dy L02 (4.58)     where to get f12 E, nx , ny , we have to change in (4.48) K10 E, nx , ny by

π nz dz

2



   1/ 2 T32 (E − V0 )2 + T42 nx , ny (E − V0 ) + T52 nx , ny 



− T12 (E − V0 ) + T22 nx , ny

  1/ 2

    T2i nx , ny ≡ [2T1i ]−1 Eg0 i bi + di + fi Eg0 i  + 2 (ei fi − ai bi )

πnx dx



2 +

πny dy

2  ,i = 1, 2,...,

   T4i nx , ny ≡ (2T1i )−2 4bi di + 4bi fi Eg0 i + 4fi2 Eg0 i  +8 

πnx dx ⎡



T5i nx , ny ≡ (2T1i )−2 ⎣ 



2 +

πnx dx

π ny dy



2 +

−8ai bi ei fi + 4b2i e2i



2  

ei fi bi − fi2 ai

π ny dy



 ,

2 !2

+ 4fi2 a2i

. 

 +

π nx dx



2 +

π ny dy

2 

4ei fi bi Eg0i + 4ei fi di + 4ei fi2 Eg0i − 4ai bi Eg0i − 4ai bi di − 4ai bi fi Eg0i +4b2i ei Eg0i + 4b2i ci + 4b2i Eg0i ai − 4fi2 ei Eg0i − 4fi2 ci − 4fi2 Eg0i ai   + Eg20 b2i + di2 + fi2 g2i +2Eg0i bi di + 2Eg20i bi fi + 2di fi Eg0i



200

4 Quantum Confined Semiconductor Superlattices

  K20 E, nx , ny by         1/ 2 1/ 2  , T11 E + T21 nx , ny − T31 E2 + T41 nx , ny E + T51 nx , ny K10 (E, 0) by 

1/ 2

T32 (E − V0 )2 + T42 (0, 0) (E − V0 ) + T52 (0, 0)

 − [T12 (E − V0 ) + T22 (0, 0)]]1/ 2 , K10 (E, 0) by   − V + T 0)) (2T (E ) (0, 32 0 42 [2K10 (E, 0)]−1 −T12 +  1/ 2 , 2 T32 (E − V0 )2 + T42 (0, 0) (E − V0 ) + T52 (0, 0)

K20 (E, 0) by 



1/ 2 1/ 2

[T11 E + T21 (0, 0)] − T31 E + T41 (0, 0) E + T51 (0, 0) 2

 [2K20 (E, 0)]

−1

(2T31 E + T41 (0, 0))

T11 −

 , K20 (E, 0) by



 1 2 , where 2. T31 E2 + T41 (0, 0) E + T51 (0, 0) /   T2i (0, 0) ≡ [2T1i ]−1 Eg0i bi + di + fi Eg0i , T4i (0, 0)   ≡ [2T1i ]−2 4bi di + 4bi fi gi + 4fi2 Eg0i and   T5i (0, 0) ≡ [2T1i ]−2 Eg20i b2i + di2 + fi2 g2i + 2Eg0i bi di + 2Eg20i bi fi + 2di fi Eg0i . These changes we should make in (4.48), (4.7), and (4.5b), respectively. The photoemitted current density is given by:

   sin L0 πnz  nxmax nymax nzmax     dz 2α0 eL0 gv    , F−1 (η11 )    J1 = dx dy dz   φ Enz , 0  n nx =1 ny =1

(4.59)

zmin

where     L0 πnz , φ Enz = 2 cos dz and the nzmin should be the nearest integer of the following inequality: nzmin ≥

dz πL0



cos−1



 1 φ (W − hv, 0) . 2

(4.60)

4.2

Theoretical Background

201

4.2.12 Photoemission from HgTe/CdTe Quantum Dot Superlattices with Graded Interfaces The electron concentration is given by: N0 =

nxmax nymax nzmax   2gv  F−1 (η12 ), dx dy dz

(4.61)

nx =1 ny =1 nz =1

where η12 =

EFQDSLGI − EQD13 , and EQD13 kB T

is the root of the equation 

  2

   π ny 2 1 π nx 2 −1 1  = − − f13 E, nx , ny E=EQD13 cos . 2 dx dy L02 (4.62)   To get f13 E, nx , ny E=EQD13 , one should make the following substitutions in (4.48):

π nz dz

2





1/ 2 2m∗2 π ny 2 π nx 2 + , K10 E, nx , ny ≡ (V0 − E) G (E − V0 ,α2 , 2 ) + dx dy 2 

1/ 2   2m∗1 π ny 2 π nx 2 ω1 (E) − − , K20 E, nx , ny ≡ dx dy 2  ∗ 1/ 2 2m2 K10 (E, 0) ≡ , (V0 − E) G (E − V0 ,α2 , 2 ) 2 ∗

m2  [−G (E − V0 ,α2 , 2 )] K10 (E, 0) ≡ [K10 (E, 0)]−1 2    1 1 1 1 + (E − V0 ) + − , E + Eg02 E + Eg02 + 2 E + Eg02 + 23 2  ∗ 1/ 2 2m1 ω1 (E) , K20 (E, 0) ≡ 2  ∗   −1 2m1  ω (E) , K20 (E, 0) ≡ [2K20 (E, 0)] and 2 1 ⎡  −1/ 2 ⎤

2 2 2 2 2 E 3e 3e ⎦. ω1 (E) ≡ ⎣1 − + 128εsc 128εsc m∗1 



The photoemitted current density is given by:

202

4 Quantum Confined Semiconductor Superlattices

⎤⎫ ⎧ ⎡  sin L0 πnz ⎬ ⎨      dz 2α0 eL0 gv   , ⎦ ⎣   J1 ≡ F−1 (η11 )  ⎭  ⎩ dx dy dz  φ Enz   nxmax nymax nzmax

(4.63)

nx =1 ny =1 nz =1

where     L0 πnz , φ Enz ≡ 2 cos dz

(4.64)

and nzmax ≥

dz πL0



cos−1





1 φ (W − hv, 0) 2

(4.65)

4.2.13 Photoemission from III–V Quantum Dot Effective Mass Superlattices The electron concentration is given by: N0 =

nxmax nymax nzmax   2gv  F−1 (η13 ), dx dy dz

(4.66)

nx =1 ny =1 nz =1

where η13 =

EFQDSLEM − EQD14 , EFQDSLEM kB T

is the Fermi energy in this case, andEQD14 is the root of the equation

πnx dx



2 =



2

2   2 π n 1  −1   π n y z f14 E, ny , nz E=EQD14 − − cos , dy dz L02 (4.67)

in which         f14 E, ny , nz E=EQD14 = a1 cos a0 g5 E, ny , nz + b0 h5 E, ny , nz       −a2 cos a0 g5 E, ny , nz − b0 h5 E, ny , nz E=EQD14 (4.68) 

 

1/ 2 2m∗1 EG E, Eg01 , 1 π ny 2 π nz 2 g5 E, ny , nz ≡ − − , dy dz 2   

1/ 2   2m∗2 EG E, Eg02 , 2 π ny 2 π nz 2 − − . h5 E, ny , nz ≡ dy dz 2 



4.2

Theoretical Background

203

The photoemitted current density is given by:

J1 =

nymax nzmax nxmax     α0 egv  L5 Enx , 0 F−1 (η13 ), dx dy dz  n ny =1 nz =1

(4.69)

xmin

  where L5 Enx , 0 is given by (4.24) in Section 4.2.5. Equations (4.25), (4.26), and (4.27) are perfectly valid in this case, where f 5 (W − hv, 0) is obtained by replacing Enx by W − hv in (4.25); and the same replacement should be done in the definition of g1 (Enx , 0) and h1 (Enx , 0), together with the related definitions.

4.2.14 Photoemission from II–VI Quantum Dot Effective Mass Superlattices The electron concentration is given by: nxmax nymax nzmax   gv  F−1 (η14 ), N0 = dx dy dz

(4.70)

nx =1 ny =1 nz =1

where

η14 =

EFQDSLEM − EQD15 , kB T

and EQD15 is the root of the equation

π nz dz



2 =



2   2 1  −1   π n x f15 E, nx , ny E=EQD15 − cos , dx L02

(4.71)

in which         f15 E, nx , ny E=EQD15 ≡ a3 cos a0 g6 E, nx , ny + b0 h6 E, nx , ny       −a4 cos a0 g6 E, nx , ny − b0 h6 E, nx , ny E=EQD15 (4.72)

204

4 Quantum Confined Semiconductor Superlattices









g6 E, nx , ny





! π ny 2 π nx 2 ≡ + dx dy 2 ⎤ 1/2

!1/2 πny 2 πnx 2 ⎦ and ∓C0 + dx dy

h6 E, nx , ny ≡



2m∗, 1

1/2 



2 E− 2m∗⊥, 1

 2m∗2 E  G E, Eg02 , 2 − 2



π nx dx



2 +

π ny dy

2 !1/2 .

The photoemitted current density is given by:

J1 =

nxmax nymax nzmax     α0 egv  L6 Enz , 0 F−1 (η14 ), 2dx dy dz  n nx =1 ny =1

(4.73)

zmin

  where L6 Enz , 0 is given by the definition of (4.31) in Section 4.2.6. Equations (4.32), (4.33), and (4.34) are perfectly valid in this case, where f7 (W − hv, 0) is obtained by replacing W − hv in (4.32), and the same replacement should be made in the definitions of g2 (Enx , 0) and h2 (Enx , 0), together with the related definitions.

4.2.15 Photoemission from IV–VI Quantum Dot Effective Mass Superlattices The electron concentration is given by:

N0 =

nxmax nymax nzmax   2gv  F−1 (η15 ), dx dy dz n nx =1 ny =1

(4.74)

zmin

where η15 =

EFQDSLEM − EQD16 , kB T

and EQD16 is the root of the equation

π nz dz

where



2 =



  2 π ny 2 1  −1   π nx 2  f16 E, nx , ny E=EQD16 − − cos , dx dy L02 (4.75)

4.2

Theoretical Background

205

        f16 E, nx , ny E=EQD16 ≡ a5 cos a0 g7 E, nx , ny + b0 h7 E, nx , ny       −a6 cos a0 g7 E, nx , ny − b0 h7 E, nx , ny E=EQD16 (4.76)       g7 E, nx , ny ≡ T11 E + T21 nx , ny −    1/2 T31 E2 + T41 nx , ny E + T51 nx , ny       h7 E, nx , ny ≡ T12 E + T22 nx , ny −

1/2

1/2

1/2











T32 E + T42 nx , ny E + T52 nx , ny 2

,

in which the other symbols have been defined in Section 4.2.11. The photoemitted current density is given by: nxmax nymax nzmax     α0 egv  L7 Enz , 0 F−1 (η15 ), J1 = dx dy dz  n nx =1 ny =1

(4.77)

zmin

  where L7 Enz , 0 is given in the definition of (4.38) in Section 4.2.7. Equations (4.39) and (4.40) are also valid in this case, where f8 (W −hv, 0) should be formulated in the same way as in Section 4.2.6.

4.2.16 Photoemission from HgTe/CdTe Quantum Dot Effective Mass Superlattices The electron concentration is given by: nxmax nymax nzmax   2gv  F−1 (η16 ), N0 = dx dy dz

(4.78)

nx =1 ny =1 nz =1

where η16 =

EFQDSLEM − EQD17 , kB T

and EQD17 is the root of

π nz dz

where



2 =



  2 π ny 2 1  −1   π nx 2  f17 E, nx , ny E=EQD17 − − cos , dy dx L02 (4.79)

206

4 Quantum Confined Semiconductor Superlattices

        f17 E, nx , ny E=EQD17 ≡ a3 cos a0 g8 E, nx , ny + b0 h8 E, nx , ny       −a4 cos a0 g8 E, nx , ny − b0 h8 E, nx , ny E=EQD17 , (4.80) in which 





g8 E, nx , ny ≡

2m∗1 ω1 (E) − 2



π nx dx



2 +

π ny dy

2 1/ 2

and 



h8 E, nx , ny ≡



 2m∗2  G E, Eg02 , 2 − 2 



nx π dx



2 +

ny π dy

2 1/ 2 .

The photoemitted current density is given by: nxmax nymax nzmax     α0 egv  L8 Enz , 0 F−1 (η16 ), J1 ≡ dx dy dz  n nx =1 ny =1

(4.81)

zmin

  where L8 Enz , 0 is given by the definition of (4.44a) in Section 4.2.8. Equations (4.44b) and (4.45) are also valid in this case. The process of finding f9 (W − hv, 0)is same as\ f7 (W − hv, 0) in Section 4.2.6.

4.3 Results and Discussions Using (4.8) and (4.4) and the energy band constants from Appendix A in Figs. 4.1, 4.2, 4.3, and 4.4, the normalized photocurrent density from QW III–V SLs (taking GaAs/Ga1–x Alx As, and Inx Ga1−x As/InP QW SLs) with graded interfaces under the magnetic quantization has been plotted as functions of the inverse quantizing magnetic field, normalized electron degeneracy, film thickness, and the normalized incident photon energy, respectively. It appears from Fig. 4.1 that photoemission in this case oscillates with the inverse quantizing magnetic field due to the SdH effect. Figure 4.2 exhibits the fact that the photocurrent density increases with increasing carrier degeneracy in an oscillatory way, and the nature of the oscillations is different compared with Fig. 4.1. From Fig. 4.3, it can be inferred that photoemission oscillates with film thickness, and for certain values of film thickness photoemission exhibits very large values. From Fig. 4.4, it appears that photoemission increases with increasing photon energy in quantum steps. The plot of the normalized magneto-photoemission from II–VI QW SLs (taking CdS/ZnSe QWSL as an example) with graded interfaces as a function of inverse quantizing magnetic field has been obtained by using (4.14) and (4.13), as shown in the curve (b) of Fig. 4.5, where the plot (a) has been drawn with C0 = 0 for the purpose of assessing the splitting of the two spin states by the spin orbit

4.3

Results and Discussions

207

Fig. 4.1 Plot of the normalized photocurrent density from (a) GaAs/Ga1–x Alx As and (b) Inx Ga1−x As/InP quantum well superlattices with graded interfaces as a function of inverse quantizing magnetic field

Fig. 4.2 Plot of the normalized photocurrent density from (a) GaAs/Ga1–x Alx As and (b) Inx Ga1−x As/InP quantum well superlattices with graded interfaces under quantizing magnetic field as a function of normalized electron degeneracy

208

4 Quantum Confined Semiconductor Superlattices

Fig. 4.3 Plot of the normalized photocurrent density from (a) GaAs/Ga1–x Alx As and (b) Inx Ga1−x As/InP quantum well superlattices with graded interfaces under quantizing magnetic field as a function of film thickness

Fig. 4.4 Plot of the normalized photocurrent density from (a) GaAs/Ga1–x Alx As and (b) Inx Ga1−x As/InP quantum well superlattices with graded interfaces under quantizing magnetic field as a function of normalized incident photon energy

4.3

Results and Discussions

209

Fig. 4.5 Plot of the normalized photocurrent density from (a) CdS/ZnSe with Co = 0, (b) CdS/ZnSe with Co  = 0, (c) HgTe/CdTe, and (d) PbSe/PbTe quantum well superlattices with graded interfaces as a function of inverse magnetic field

coupling and the crystalline field on the magneto-photoemission in this case. The plot (c) of Fig. 4.5 has been drawn for HgTe/CdTe QWSL, whereas the plot (d) is valid for IV–VI QWSL (using PbSe/PbTe as an example). Figures 4.6, 4.7, and 4.8 demonstrate the plots of the photoemission as functions of normalized electron degeneracy, film thickness, and normalized incident photon energy, respectively, for all the cases of Fig. 4.5. The plot of the normalized magnetophotoemission for III–V QW effective mass SLs (taking GaAs/Ga1–x Alx As as an example) as a function of inverse quantizing magnetic field has been obtained by using (4.24) and (4.22), as shown in the curve (a) of Fig. 4.9. The plots (b), (c), and (d) in the same figure have been drawn for II–VI QW effective mass SL (taking CdS/ZnSe as an example and using (4.28) and (4.31)), IV–VI QW effective mass SL (taking PbSe/PbTe as an example and using (4.35) and (4.38)), and HgTe/CdTe QW effective mass SL (using (4.41) and (4.44a)), respectively. The plots for normalized photoemission as functions of normalized electron degeneracy, film thickness, and normalized incident photon energy for all the cases of Fig. 4.9 have been drawn in Figs. 4.10, 4.11, and 4.12, respectively. The plot (a) of Fig. 4.13 exhibits the variation of normalized photoemission as a function of film thickness for QD SLs with graded interfaces of (a) HgTe/CdTe (using (4.61) and (4.63)), (b) HgTe/Hg1–x Cdx Te (an example ofIII–V QD SLs and using (4.46) and

210

4 Quantum Confined Semiconductor Superlattices

Fig. 4.6 Plot of the normalized photocurrent density from (a) CdS/ZnSe with Co = 0, (b) CdS/ZnSe with Co  = 0, (c) HgTe/CdTe, and (d) PbSe/PbTe quantum well superlattices with graded interfaces under quantizing magnetic field as a function of normalized electron degeneracy

Fig. 4.7 Plot of the normalized photocurrent density from (a) CdS/ZnSe with Co = 0, (b) CdS/ZnSe with Co  = 0, (c) HgTe/CdTe, and (d) PbSe/PbTe quantum well superlattices with graded interfaces under quantizing magnetic field as a function of film thickness

4.3

Results and Discussions

211

Fig. 4.8 Plot of the normalized photocurrent density from (a) CdS/ZnSe with Co = 0, (b) CdS/ZnSe with Co  = 0, (c) HgTe/CdTe, and (d) PbSe/PbTe quantum well superlattices with graded interfaces under quantizing magnetic field as a function of normalized incident photon energy

Fig. 4.9 Plot of the normalized photocurrent density from (a) GaAs/Ga1–x Alx As, (b) CdS/ZnSe with Co  = 0, (c) PbSe/PbTe, and (d) HgTe/CdTe quantum well effective mass superlattices as a function of inverse quantizing magnetic field

212

4 Quantum Confined Semiconductor Superlattices

Fig. 4.10 Plot of the normalized photocurrent density from (a) GaAs/Ga1–x Alx As, (b) CdS/ZnSe with Co  = 0, (c) PbSe/PbTe, and (d) HgTe/CdTe quantum well effective mass superlattices under quantizing magnetic field as a function of normalized electron degeneracy

Fig. 4.11 Plot of the normalized photocurrent density from (a) GaAs/Ga1–x Alx As, (b) CdS/ZnSe with Co  = 0, (c) PbSe/PbTe, and (d) HgTe/CdTe quantum well effective mass superlattices under quantizing magnetic field as a function of film thickness

4.3

Results and Discussions

213

Fig. 4.12 Plot of the normalized photocurrent density from (a) GaAs/Ga1–x Alx As, (b) CdS/ZnSe with Co  = 0, (c) PbSe/PbTe, and (d) HgTe/CdTe quantum well effective mass superlattices under quantizing magnetic field as a function of normalized incident photon energy

Fig. 4.13 Plot of the normalized photocurrent density from (a) HgTe/CdTe, (b) HgTe/Hg1–x Cdx Te, (c) CdS/ZnSe with Co = 0, and (d) PbSe/PbTe quantum dots of superlattices with graded interfaces as a function of film thickness

214

4 Quantum Confined Semiconductor Superlattices

Fig. 4.14 Plot of the normalized photoemitted current density from (a) HgTe/CdTe, (b) HgTe/Hg1–x Cdx Te, (c) CdS/ZnSe with Co = 0, and (d) PbSe/PbTe quantum dots of superlattices with graded interfaces as a function of normalized electron degeneracy

Fig. 4.15 Plot of the normalized photocurrent density from (a) HgTe/CdTe, (b) HgTe/Hg1–x Cdx Te, (c) CdS/ZnSe with Co = 0, and (d) PbSe/PbTe quantum dots of superlattices with graded interfaces as a function of normalized incident photon energy

4.3

Results and Discussions

215

Fig. 4.16 Plot of the normalized photocurrent density from (a) HgTe/CdTe, (b) HgTe/Hg1–x Cdx Te, (c) CdS/ZnSe with Co = 0, and (d) PbSe/PbTe quantum dots effective mass superlattices as a function of film thickness

Fig. 4.17 Plot of the normalized photocurrent density from (a) HgTe/CdTe, (b) HgTe/Hg1–x Cdx Te, (c) CdS/ZnSe with Co = 0, and (d) PbSe/PbTe quantum dots effective mass superlattices as a function of normalized electron degeneracy

216

4 Quantum Confined Semiconductor Superlattices

Fig. 4.18 Plot of the normalized photocurrent density from (a) HgTe/CdTe, (b) HgTe/Hg1–x Cdx Te, (c) CdS/ZnSe with Co = 0, and (d) PbSe/PbTe quantum dots effective mass superlattices as a function of normalized incident photon energy

(4.49)), (c) CdS/ZnSe (an example of II–VI QD SLs and using (4.52) and (4.54) with C0  = 0), and (d) PbSe/PbTe (an example of IV–VI QD SLs and using (4.57) and (4.59)), respectively. Figures 4.14 and 4.15 demonstrate the plots for normalized photoemission as functions of normalized electron degeneracy and normalized incident photon energy, respectively, for all the cases of Fig. 4.13. The plot (a) of Fig. 4.16 exhibits the variation of normalized photoemission as a function of film thickness for QD effective SLs of (a) HgTe/CdTe (using (4.78) and (4.81)), (b) HgTe/Hg1–x Cdx Te (an example of III–V SLs and using (4.66) and (4.69)), (c) CdS/ZnSe (an example of II–VI QD SLs and using (4.70) and (4.73) with C0  = 0), and (d) PbSe/PbTe (an example of IV–VI QD SLs and using (4.74) and (4.77)), respectively. Figures 4.17 and 4.18 exhibit the plots for photoemission as functions of normalized electron degeneracy and the normalized incident photon energy, respectively, for all the cases of Fig. 4.16. The nature of the variation of the plots in the different types of SLs under different physical conditions as shown in Figs. 4.5–4.18 have already been discussed in describing the plots of Figs. 4.1, 4.2, 4.3, and 4.4. Finally, it is logical to conclude that the numerical values of the photoemitted current density are totally different in all cases which exhibit the signature of the respective band structure under different physical conditions, and that the rates of variation are again totally energy spectrum–dependent.

References

217

References 1. L. L. Chang, L. Esaki, Progr. Crystal Growth Charact. 2, 3 (1979). 2. R. Dingle, H. L. Stormer, A. C. Gossard, W. Weigman, Appl. Phys. Letts. 33, 665 (1978). 3. G. A. Sai-Halasz, L. L. Chang, J. M. Welter, C. A. Chang, L. Esaki, Solid State Commun. 27, 935 (1978). 4. G. A. Sai-Halasz, L. Esaki, W. Harrison, Phys. Rev. B18, 2812 (1978). 5. F. Capasso, In: Semiconductors and Semimetal, Ed. R. W. Willardson, A. C. Beer, 22, P.2 (Academic Press, USA, 1985). 6. J. S. Smith, L. C. Chiu, S. Magralit, A. Yariv, J. Vac. Sci. Tech. 31, 376 (1983). 7. F. Capasso, R. A. Keihl, J. Appl. Phys. 58, 1366 (1985). 8. K. Ploog, G. H. Dohler, Adv. Phys. 32, 285 (1983). 9. D. R. Suifres, C. Kindstrom, R. D. Burnhaum, W. Streifer, T. L. Paoli, Electron. Letts. 19, 170 (1983). 10. K. V. Vaidyanathan, R. A. Jullens, C. L. Anderson, H. L. Dunlap, Solid State Electron. 26, 717 (1983). 11. B. A. Wilson, IEEE, J. Quant. Electron. 24, 1763 (1988). 12. M. Krichbaum, P. Kocevar, H. Pascher, G. Bauer, IEEE, J. Quant. Electron. 24, 717 (1988). 13. L. Esaki, IEEE J. Quant. Electron. 22, 1611 (1986). 14. M. Kinoshita, T. Sakashita, H. Fajiyasu, J. Appl. Phys, 52, 2869 (1981). 15. V. A. Yakovlev, Sov. Phys. Semicon. 13, 692 (1979). 16. A. N. Chakravarti, K. P. Ghatak, G. B. Rao, K. K. Ghosh, Phys. Stat. Solids. (b), 112, 75 (1982) 17. E. O. Kane, J. Phys. Chem. Solids, 1, 249 (1957). 18. H. Sasaki, Phys. Rev. B, 30, 7016 (1984). 19. K. P. Ghatak, S. Bhattacharya, D. De, Einstein Relation in Compound Semiconductors and Their Nano Structures (Springer Series in Materials Science, Vol. 116, Springer Verlag, Germany, 2008) 20. J. J. Hopfield, J. Phys. Chem. Solids, 15, 97 (1960). 21. G. M. T. Foley, P. N. Langenberg, Phys. Rev. B,15, 4850 (1977).

Chapter 5

Photoemission from Bulk Optoelectronic Materials

5.1 Introduction The ternary and quaternary compounds enjoy a singular position in the entire spectrum of optoelectronic materials. It is well known that the ternary alloy Hg1–x Cdx Te is a classic narrow gap compound. The band gap of this ternary alloy can be varied to cover the spectral range from 0.8 to over 30 μm [1] by adjusting the alloy composition. Hg1–x Cdx Te finds extensive applications in infrared detector materials and photovoltaic detector arrays in the 8–12 μm wave bands [2]. The above uses have generated the Hg1–x Cdx Te technology for the experimental realization of high mobility single crystals with specially prepared surfaces. The same compound has emerged to be the optimum choice for illuminating the narrow sub-band physics because the relevant material constants can easily be experimentally measured [3]. It may be mentioned that the quaternary alloy In1–x Gax Asy P1–y lattice matched to InP also finds wide use in the fabrication of avalanche photodetectors [4], heterojunction lasers [5], light-emitting diodes [6], and avalanche photodiodes [7]—and in addition, the field effect transistors, detectors, switches, modulators, solar cells, filters, and new types of integrated optical devices are made from quaternary systems [8]. In Section 5.2, the photoemission from the said materials has been investigated in the presence of external photo-excitation whose unperturbed electron energy spectra are respectively defined by the three and two band models of Kane together with parabolic energy bands. In the same section, the carrier statistics and the photoemitted current density for unperturbed three- and two-band models of Kane together with parabolic energy bands have been shown as special case. The same section contains two different applications of our analysis and the photoemission has also been numerically investigated by taking the said materials as examples of ternary and quaternary compounds in Section 5.3. The measurement of the band gap of semiconducting materials in the presence of light waves has also been discussed in this context. The Section 5.4 presents open research problems pertinent to this chapter.

219 K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_5,  C Springer Science+Business Media, LLC 2009

220

5

Photoemission from Bulk Optoelectronic Materials

5.2 Theoretical Background The simplified electron energy spectra in optoelectronic materials up to the second order in the presence of external photoexcitation whose unperturbed dispersion relations of the conduction electrons are defined by the three- and two-band models of Kane together with parabolic energy bands, can, respectively, be expressed as [9]: 2 k 2 = β0 (E, λ) 2m∗

(5.1)

2 k 2 = ω0 (E, λ) 2m∗

(5.2)

2 k 2 = ρ0 (E, λ) , 2m∗

(5.3)

where β0 (E, λ) ≡ [I (E) − θ0 (E, λ)] ,

   E E + Eg0 E + Eg0 +  Eg0 + 13 

I(E) ≡ ,   Eg0 Eg0 +  E + Eg0 + 23   

I0 λ2 Eg0 Eg0 +  β 2 ρ 2 1 e2

θ0 (E, λ) ≡ t + √ √ φ0 (E) 96mr πc3 εsc ε0 E + 2  4 2 g0



− δ

Eg0 1+ φ0 (E) + δ 

+ (Eg0

3



1 1 −δ ) −  φ0 (E) + δ Eg0 + δ  



1/2

Eg0 + δ  1 −  2 φ0 (E) + δ  Eg0 − δ 

1/2 ⎫2 ⎬ ⎭

,

∗ −1 + m−1 , m is the effective mr is the reduced mass and is given by m−1 v r = (m ) v mass of the heavy hole at the top of the valence band in the absence of any field, I0 is the light intensity of wavelength λ, c is the velocity of light, ε0 is the permittivity of vacuum, and εsc is the permittivity of the material,



 1/2 6(Eg0 + 2/3)(Eg0 + ) /χ , χ ≡ 6Eg20 + 9Eg0  + 42 , 1/2

1/2  , ρ ≡ 42 /3χ , t ≡ 6(Eg0 + 2/3)2 /χ



m∗ I (E) 1/2 , φ0 (E) ≡ Eg0 1 + 2 1 + mv Eg0

δ  ≡ Eg20  (χ )−1 , ω0 (E, λ) ≡ E (1 + αE) − B0 (E, λ) , β≡



5.2

Theoretical Background

221

 2 

e2 I0 λ2 Eg0 Eg0 1 1 1 , 1 + + E − √ g0 φ1 (E) φ1 (E) Eg0 384π c3 mr εsc ε0 φ1 (E)   2m∗ E(1 + αE) 1/2 , and φ1 (E) ≡ Eg0 1 + mr Eg0  ∗

 2m E −3/2 e2 I0 λ 2 . ρ0 (E,λ) ≡ E − 1+ √ mr Eg0 96πc3 mr εsc ε0

B0 (E, λ) ≡

Thus, under the limiting condition k → 0, from (5.1), (5.2), and (5.3), we observe that E  = 0 and is positive. Therefore, in the presence of external light waves, the energy of the electron does not tend to zero when k → 0, whereas the unperturbed three- and two-band models of Kane together with parabolic energy bands reflect the fact that for k → 0, E → 0. As the conduction band is taken as the reference  level of energy, therefore  the lowest positive value of E for k → 0 provides the  increased band gap Eg of the materials, due to photon excitation. The values of the increased band gap can be obtained by computer iteration processes for various values of I0 and λ, respectively.    Using (5.1), vz E and N E for optoelectronic materials in the presence of light waves whose unperturbed conduction electrons obey the three-band model of Kane can be written as: √    1/2   2 β0 E , λ vz E = √ m∗ β0 (E , λ)

(5.4)

and  

     2m∗ 3/2 .gv β0 (E , λ)β0 E , λ , N E = 4π 2 h

(5.5)

∂ [β0 (E, λ)]. where β0 (E, λ) ≡ ∂E Using (1.1), (1.2), (5.4), and (5.5), the photoemitted current density in this case can be written using the generalized Sommerfield’s lemma as: [10]  JL =

4πem∗ (kB T)2 α0 gv h3

 ⎡ ⎣

1 + 23 α 

(1 + α)

2αkB TF2 (ηL )



(5.6) E0 + αE02 + 13 αE0 1 + 1 + 2αE0 + α F1 (ηL ) + F−1 (ηL ) 3 kB T         a0 + ηL  B51  + φ5 (ηL ) − + a0 ln  I2 a0  (kB T)2   ηL ≡ EFL − E0 (kB T)−1 , EFL is the Fermi energy in the presence of light waves as measured from the edge of the conduction band in the absence of any field,

222

5

Photoemission from Bulk Optoelectronic Materials



2 a0 ≡ E0 + Eg0 +  (kB T)−1 , 3 s0

 (−1)2r−1 (2r − 1)! φ (ηL ) ≡ 2 1 − 21−2r ξ (2r) , 2r + η (a ) 0 L r=1  

I0 λ2 Eg0 Eg0 +  β 2 e2 ρ 2

, t+ √ B51 ≡ √ 48mr π c3 εsc ε0 Eg + 2  4 2 0 3  ∞   I2 ≡ f1 E f (E) dE , E0

and   f1 E  ≡

1 φ0 (E ) 

1+

Eg 0 − δ  φ0 (E ) + δ  + δ



Eg 1 − 0 2 φ0 (E ) + δ  Eg0 − δ 

+ (Eg0 1/2 ⎫2 ⎬ ⎭



1 1 −δ ) − φ0 (E ) + δ  Eg0 + δ  

1/2

.

It appears then that the determination of JL as a function of electron concentration requires an expression of electron statistics which can in turn be written as:

−1 2m∗ 3/2      gv M51 EFL , λ + N51 EFL , λ , n0L = 3π 2 2 

(5.7)

where 

M51 (EFL , λ) ≡ βL

s0   3/2  EFL , λ , N51 (EFL , λ) ≡ ZrY M51 (EFL , λ) , and Y = L.



r=1

The expressions of JL and n0L for optoelectronic materials in the presence of light waves whose unperturbed conduction electrons obey the two-band model of Kane can be written following (5.2) as: 



B51 α 4πem∗ (kB T)2 α0 gv TF F + 2αk − JL = (η ) (η ) 1 L B 2 L kB T h3    3C51 F0 (ηL ) − kB TF1 (ηL ) +2C52 (kB T)2 F2 (ηL ) 2

(5.8)

and

−1 2m∗ 3/2      M E E g , λ + N , λ , n0L = 3π 2 v 52 F 52 F L L 2

(5.9)

5.2

Theoretical Background

223

where I0 λ2 Eg0 e2 , √ 192mr πc3 εsc ε0   2

 15 C51 2m∗ α 3   , C51 ≡ , C52 ≡ − α C51 mv 8 2   3/2 , M52 (EFL , λ) ≡ ω0 EFL , λ B51 ≡

N52 (EFL , λ) ≡

S0 

ZrY M52 (EFL , λ), and Y = L.

r=1

The expressions of JL and n0L for optoelectronic materials in the presence of light waves whose unperturbed conduction electrons obey the parabolic energy bands can be expressed as: 

 4π em∗ (kB T)2 α0 gv JL = h3 



 m∗ B53 F1 (ηL ) − F0 (ηL ) − 3αkB T 1 + F1 (ηL ) kB T mv and



n0L = 3π

2

where

−1 2m∗ 3/2 2 e2

B53 ≡

(m∗ )2 c3

     .gv M53 EFL , λ + N53 EFL , λ ,

I0 λ2 , √ εsc ε0

8mr π s0  N53 (EFL , λ) ≡ ZrY M53 (EFL , λ).

  3/2 M53 (EFL , λ) ≡ ρ0 EFL , λ ,

(5.10)

(5.11)

and

r=1

SpecialCase: Formulation of electron concentration and current density for unperturbed three- and two-band models of Kane for optoelectronic materials (i) The expressions of J and n0 in accordance with the unperturbed three-band model of Kane assume the forms:

 ⎡  2  4π em∗ (kB T)2 α0 gv ⎣ 1 + 3 α 2αkB TF2 (η0 ) J= h3 (1 + α)

E0 + αE02 + 13 αE0 1 + 1 + 2αE0 + α F1 (η0 ) + F−1 (η0 ) 3 kB T ⎤      a0 + η0   + φ (η0 ) ⎦ +a0 ln  a0 

(5.12)

224

5

and

Photoemission from Bulk Optoelectronic Materials



−1 2m∗ 3/2 n0 = 3π 2 gv [M54 (EF ) + N54 (EF )] , 2

where a0 =

2 2 9 (kB T)2

(5.13)

2 1 + α , 3



S0

 (−1)2r−1 (2r − 1)! 2 1 − 21−2r ζ (2r) , M54 (EF ) ≡ [I (EF )]3/2 , 2r + η (a ) 0 0 r=1  S 0 

 ∂ 2r 2r 1−2r N54 (EF ) ≡ Zr M54 (EF ), and Zr ≡ 2 (kB T) 1 − 2 ζ (2r) 2r . ∂EF r=1

φ (η0 ) =

(ii) In accordance with the unperturbed two-band model of Kane, the corresponding expressions of J and n0 are given by:  J= and

 4πem∗ (kB T)2 α0 gv [F1 (η0 ) + 2αkB TF2 (η0 )] h3



−1 2m∗ 3/2 n0 = 3π 2 gv [M55 (EF ) + N55 (EF )] , 2

where M55 (EF ) ≡ [EF (1 + αEF )]3/2 and N55 (EF ) ≡

So 

(5.14)

(5.15)

Zr M55 (EF ).

r=1

For α → 0, (5.14) and (5.15) are simplified into the well-known form as given by (1.12) and (1.19), respectively. The two different applications of the results of this chapter in the field of materials science in general are: 1. The nonlinear response from the optical excitation of the free carriers is given by [11]: G0 =

−e2 (2πf0 )2 2

∞

 0

∂kx kx ∂E

−1

N (E) f (E) dE,

(5.16)

where f0 is the optical frequency. From the band structure we can derive the term x (kx ( ∂k ∂E )) as a function of E, and by using the DOS function as formulated, we can study the G0 for all types of optoelectronic and III–V materials. 2. The effective mass of the charge carriers in electronic materials, being inversely related with the mobility, is known to be one of the central concepts in the whole field of materials science and related disciplines [12]. It is important to note that

5.2

Theoretical Background

225

various definitions exist for the effective carrier mass [13]. Among them, it is the effective momentum mass (EMM) that should be regarded as the basic quantity [14]. This is because the EMM appears in the analysis of transport phenomena and all other properties of the carriers in a band with arbitrary band nonparabolicity [15]. It can be proved that it is this mass which enters into various transport coefficients and plays the most dominant role in explaining the experimental results of different mechanisms of scattering [16–17]. The doping in degenerate materials influences the effective mass when it is energy-dependent [17]. Under the condition of carrier degeneracy, only the carriers at the Fermi surface of electronic materials participate in the conduction process, and hence the EMM of the electrons corresponding to the Fermi level would be of interest in carrier transport under such conditions. The Fermi energy is again determined by the carrier dispersion laws and the carrier concentration; therefore, these two important features would determine the dependence of the EMM in degenerate materials under the degree of carrier degeneracy. The expression of the EMM is given by: ∗

m







5

EFL = (k)

1 ∂E  ∂k

   

E=EFL

 ∂k  = k . ∂E E=EF 2

(5.17)

L

Using (5.1), (5.2), (5.3), and (5.17), the expressions of the EMMs in the presence of light waves for III–V, ternary, and quaternary materials whose unperturbed conduction electrons obey the three- and two-band models of Kane together with parabolic energy bands can be written as:        m∗ EFL = m∗ I  EFL − θ0 EFL , λ ,

(5.18)

      m∗ EFL = m∗ 1 + 2αEFL − B0 EFL , λ ,

(5.19)

     m∗ EFL = m∗ ρ0 EFL , λ ,

(5.20)

where the primes indicate the differentiation of the differentiable functions with respect to Fermi energy. In the absence of light waves, the corresponding wellknown expressions [18] are, respectively, given by:     m∗ EFL = m∗ I  (EF ) ,

(5.21)

  m∗ EFL = m∗ [(1 + 2αEF )] ,

(5.22)

m∗ (EF ) = m∗ .

(5.23)

Comparing (5.18) and (5.21), and again (5.19) and (5.22), we observe that the EMM, in addition to doping, changes with light intensity and wavelength when the material is being exposed to photoexcitation; and it is important to note that in

226

5

Photoemission from Bulk Optoelectronic Materials

(5.21), (5.22), and (5.23), the outcome of conventional analysis cannot explain this important phenomenon. Equation (5.23) exhibits the fact that the EMM is a constant quantity for electronic materials having unperturbed parabolic energy bands; whereas (5.20) reflects the idea that the EMM of the same material in the presence of light waves changes with concentration, light intensity, and wavelength, respectively, in various manners.

5.3 Results and Discussions Using the appropriate equations, the normalized incremental band gap (Eg ) has been plotted as a function of normalized I0 (for a given wavelength and considering red light for which λ = 660 nm) at T= 4.2 K in Figs. 5.1 and 5.2 for n-Hg1–x Cdx Te and n-In1–x Gax Asy P1–y lattice matched to InP in accordance with the perturbed three- and two-band models of Kane and that of perturbed parabolic energy bands, respectively. In Figs. 5.3 and 5.4, the normalized incremental band gap has been plotted for the aforementioned optoelectronic compounds as a function of λ. Using the appropriate equations, the normalized photoemission from n-Hg1–x Cdx Te has been plotted as functions of normalized I0 (for a given wavelength and considering red light for which λ is about 640 nm), λ (assuming I0 = 10 nWm–2 ), and the

  Fig. 5.1 Plots of the normalized incremental band gap Eg for n-Hg1–x Cdx Te as a function of normalized light intensity in which the curves (a) and (b) represent the perturbed three- and twoband models of Kane, respectively. The curve (c) represents the same variation in n-Hg1–x Cdx Te, in accordance with the perturbed parabolic energy bands

5.3

Results and Discussions

227

  Fig. 5.2 Plots of the normalized incremental band gap Eg for In1–x Gax Asy P1–y lattice matched to InP as a function of normalized light intensity for all cases of Fig. 5.1

  Fig. 5.3 Plots of the normalized incremental band gap Eg for Hg1–x Cdx Te as a function of wavelength for all cases of Fig. 5.1

228

5

Photoemission from Bulk Optoelectronic Materials

  Fig. 5.4 Plots of the normalized incremental band gap Eg for In1–x Gsx Asy P1–y lattice matched to InP as a function of wavelength for all cases of Fig. 5.2

normalized electron degeneracy at T= 4.2 K in accordance with the perturbed threeand two-band models of Kane and that of perturbed parabolic energy bands in Figs. 5.5, 5.6, and 5.7, respectively. Figures 5.8, 5.9, and 5.10 exhibit all the aforementioned cases for n-In1–x Gax Asy P1–y lattice matched to InP, respectively. It appears that J increases with the increasing electron degeneracy in accordance with all the band models. The combined influence of the energy band constants on the photoemission from ternary and quaternary materials can easily be assessed from all the figures. It appears that the photoemission decreases with increasing light intensity for all the materials and also decreases as the wavelength shifts from violet to red. The influence of light is immediately apparent from all the plots, since the photoemission depends strongly on the light intensity for all types of perturbed band models. This is in direct contrast with the bulk specimens of the said compounds, whose formulations depend on the general idea that the band structure is an invariant quantity in the presence of external photo-excitation, together with the fact that the physics of photoemission is being converted mathematically by using the lower limit of integration as E0 , as often used in the literature. The dependence of JL on light intensity and wavelength reflects the direct signature of the light wave on the band structure–dependent physical properties of electronic materials in general, in the presence of external photo-excitation and photon-assisted transport for the corresponding optoelectronic semiconductor devices. Although JL tends to decrease with increasing intensity and wavelength, the rate of increase is totally band structure–dependent.

5.3

Results and Discussions

229

Fig. 5.5 Plot of the normalized photoemitted current density from n-Hg1–x Cdx Te as a function of normalized light intensity in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane together with parabolic energy bands, respectively

It is worth remarking that our basic equation (5.1) covers various materials having different energy band structures. Under certain limiting conditions, all the results of the photoemission for different materials having various band structures lead to the well-known expression of the same for wide-gap materials having simplified parabolic energy bands. This indirect test not only exhibits the mathematical compatibility of the formulation but also shows the fact that the presented simple analysis is a more generalized one, since well-known results can be obtained under certain limiting conditions of the generalized expressions. It is worth remarking that the influence of external photo-excitation radically changes the original band structure of the material. Because of this change, the photon field causes the band gap of semiconductors to increase. We propose the following two experiments for the measurement of band gap of semiconductors under photo-excitation. (A) A white light with color filter is allowed to fall on a semiconductor, and the optical absorption coefficient (α 0 ) is measured experimentally. For different colors of light, α 0 is measured, and α 0 versus ω (the incident photon energy) is plotted, and we extrapolate the curve such that α 0 → 0 at a particular value ω1 . During this process, we vary the wavelength with fixed I0 . From our present study, we have observed that the band gap of the semiconductor increases for

230

5

Photoemission from Bulk Optoelectronic Materials

Fig. 5.6 Plot of the normalized photoemitted current density from n-Hg1–x Cdx Te as a function of wavelength for all cases of Fig. 5.5

Fig. 5.7 Plot of the normalized photoemitted current density from n-Hg1–x Cdx Te as a function of normalized electron degeneracy for all cases of Fig. 5.5

5.3

Results and Discussions

231

Fig. 5.8 Plot of the normalized photoemitted current density from In1–x Gax Asy P1–y lattice matched to InP as a function of normalized light intensity in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane together with parabolic energy bands, respectively

various values of λ when I0 is fixed (from Figs. 5.3 and 5.4). This implies that the band gap of the semiconductor measured (i.e., ω1 = Eg ) is not the unperturbed band gap Eg0 but the perturbed band gap Eg ; where Eg = Eg0 + Eg , Eg is the increased band gap at ω1 . Conventionally, we consider this Eg as the unperturbed band gap of the semiconductor, and this particular concept needs modification. Furthermore, if we vary I0 for a monochromatic light (when λ is fixed), the band gap of the semiconductor will also change (Figs. 5.1 and 5.2). Consequently, the absorption coefficient will change with the intensity of light [19]. For an overall understanding, the detailed theoretical and experimental investigations are needed in this context for various materials having different band structures. (B) The conventional idea for the measurement of the band gap of the semiconductors is the fact that the minimum photon energy hν (ν is the frequency of the monochromatic light) should be equal to the band gap Eg0 (unperturbed) of the semiconductor, i.e., hν = Eg0 .

(5.24)

232

5

Photoemission from Bulk Optoelectronic Materials

Fig. 5.9 Plot of the normalized photoemitted current density from In1–x Gax Asy P1–y lattice matched to InP as a function of wavelength for all cases of Fig. 5.8

In this case, λ is fixed for a given monochromatic light and the semiconductor is exposed to a light of wavelength λ. Also the intensity of the light is fixed. From Figs. 5.3 and 5.4, we observe that the band gap of the semiconductor is not Eg0 (for a minimum value of hν), but Eg , the perturbed band gap. Thus, we can rewrite the above equality as: hν = Eg .

(5.25)

Furthermore, if we vary the intensity of light (Figs. 5.1 and 5.2) for the study of photoemission, the minimum photon energy should be: hν1 = Eg1 ,

(5.26)

where Eg1 is the perturbed band gap of the semiconductor due to the various intensity of light when ν and ν1 are different. Thus, we arrive at the following conclusions: (a) Under different intensity of light, keeping λ fixed, the condition of band gap measurement is given by: hν1 = Eg1 = Eg0 + Eg1 .

(5.27)

5.3

Results and Discussions

233

Fig. 5.10 Plot of the normalized photoemitted current density from In1–x Gax Asy P1–y lattice matched to InP as a function of normalized electron degeneracy for all cases of Fig. 5.8

(b) Under different color of light, keeping the intensity fixed, the condition of band gap measurement assumes the form: hν = Eg = Eg0 + Eg ,

(5.28)

and not the conventional result as given by (5.24).

We have not considered other types of optoelectronic and III–V materials and other external variables for the purpose of concise presentation. Besides, the influence of energy band models and the various band constants on the photoemission for different materials can also be studied from all the figures of this chapter. The numerical results presented in this chapter would be different for other materials but the nature of the variation would be unaltered. The theoretical results as given here would be useful in analyzing various other experimental data related to this phenomenon. Finally, it appears that this theory can be used to investigate the thermoelectric power, the Debye screening length, the magnetic susceptibilities, the Burstien Moss shift, the plasma frequency, the Hall coefficient, the specific heat, and other different transport coefficients of modern optoelectronic devices operated in the presence of light waves.

234

5

Photoemission from Bulk Optoelectronic Materials

5.4 Open Research Problems The open research problems in this section of the book are by far most important for alert readers and PhD aspirants for their own contributions in this fascinating topic of semiconductor optoelectronics in general. All the following problems should be investigated in the presence of external photo-excitation which changes the band structure in a fundamental way, together with the proper inclusion of the variations of work function in appropriate cases.

(R5.1) Investigate multiphoton photoemission for bulk specimens of all the materials whose unperturbed carrier energy spectra are defined in Chapter 3 in the presence of arbitrarily oriented photo-excitation by incorporating the appropriate changes. (R5.2) Investigate multiphoton photoemission in the presence of an arbitrarily oriented nonquantizing nonuniform electric fields and photo-excitation, respectively, for all the cases of R5.1. (R5.3) Investigate multiphoton photoemission in the presence of arbitrarily oriented nonquantizing alternating electric fields and photo-excitation, respectively, for all the cases of R5.1. (R5.4) Investigate multiphoton photoemission for arbitrarily oriented photoexcitation from the heavily-doped materials in the presence of Gaussian, exponential, Kane, Halperin, Lax, and Bonch-Bruevich types of band tails for all materials whose unperturbed carrier energy spectra are defined in Chapter 3. (R5.5) Investigate multiphoton photoemission from all the materials in the presence of arbitrarily oriented nonquantizing nonuniform electric fields and photoexcitation for all the appropriate cases of problem R5.4. (R5.6) Investigate multiphoton photoemission from all the materials in the presence of arbitrarily oriented nonquantizing alternating electric field and photoexcitation for all the appropriate cases of problem R5.4. (R5.7) Investigate multiphoton photoemission from negative refractive index, organic, magnetic, disordered and other advanced materials in the presence of arbitrarily oriented photo-excitation. (R5.8) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and alternating nonquantizing electric fields for all the problems of R5.7. (R5.9) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and nonquantizing nonuniform electric fields for all the problems of R5.7. (R5.10) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and alternating nonquantizing electric fields for all the problems of R5.7. (R5.11) Investigate all the problems from R5.1 to R5.10 by removing all the mathematical approximations and establishing the respective appropriate uniqueness conditions.

References

235

References 1. P. Y. Lu, C. H. Wung, C. M. Williams, S. N. G. Chu, C. M. Stiles, Appl. Phys. Letts. 49, 1372 (1986). 2. N. R. Taskar, I. B. Bhat, K. K. Prat, D. Terry, H. Ehasani, S. K. Ghandhi, J. Vac. Sci. Tech. 7A, 281 (1989). 3. F. Koch, In: Two-Dimensional Systems, Hetrostructures, and Superlattices, Ed. G. Bauer, F. Kuchar, H. Heinrich, Springer Series in Solid States Sciences, Vol. 53, p. 20 (SpringerVerlag, Germany, 1984). 4. L. R. Tomasetta, H. D. Law, R. C. Eden, I. Reyhimy, K. Nakano, IEEE J. Quant. Electron.14, 800 (1978). 5. T. Yamato, K. Sakai, S. Akiba, Y. Suematsu, IEEE J. Quant. Electron. 14, 95 (1978). 6. T. P. Pearsall, B. I. Miller, R. J. Capik, Appl. Phys. Letts. 28, 499 (1976). 7. M. A. Washington, R. E. Nahory, M. A. Pollack, E. D. Beeke, Appl. Phys. Letts. 33, 854 (1978). 8. M. I. Timmons, S. M. Bedair, R. J. Markunas, J. A. Hutchby, Proceedings of the 16th IEEE Photovoltaic Specialist Conference (IEEE, San Diego, California 666, 1982). 9. K. P. Ghatak, S. Bhattacharya, D. De, Einstein Relation in Compound Semiconductors and their nanostructures, Vol. 116, (Springer Series in Materials Science, Springer-Verlag, Germany, 2008). 10. R. K. Pathria, Statistical Mechanics, 2nd Edition, (Butterworth-Heinmann, UK, 1996). 11. A. S. Filipchenko, I. G. Lang, D. N. Nasledov, S. T. Pavlov, L. N. Radaikine, Phys. Stat. Solids (b) 66, 417 (1974). 12. S. Adachi, J. Appl. Phys. 58, R11 (1985). 13. R. Dornhaus, G. Nimtz, Springer Tracts in Modern Physics, Vol. 78, p. 1. (Springer, Germany, 1976). 14. W. Zawadzki, Handbook of Semiconductor Physics, vol. 1, Ed. W. Paul, p. 719 (North Holland, The Netherlands, 1982). 15. I. M. Tsidilkovski, Cand. Thesis Leningrad University USSR (1955). 16. F. G. Bass, I. M. Tsidilkovski, Ivz. Acad. Nauk Azerb USSR 10, 3 (1966). 17. K. P. Ghatak, M. Mondal, Z. fur Naturforschung A 41a, 881 (1986); K. P. Ghatak, M. Mondal, Z. fur Physik B, Cond. Matter 69, 471 (1988); B. Mitra, K. P. Ghatak, Solid State Electron. 32, 515 (1989); B. Mitra, K. P. Ghatak, Solid State Electron. 32, 177 (1989); A. N. Chakravarti, K. P. Ghatak, K. K. Ghosh, S. Ghosh, A. Dhar, Z. fur Phys. B. Cond. Matter 47, 149 (1982); M. Mondal, N. Chattopadhyay, K. P. Ghatak, J. Low Temp. Phys. 66, 131 (1987); K. P. Ghatak, A. Ghoshal, B. Mitra, Nouvo Cimento D 14D, 903 (1992); P. K. Chakraborty, G. C. Dutta, K. P. Ghatak, Phys. Scripta 68, 368 (2003); A. N. Chakravarti, A. K. Choudhury, K. P. Ghatak, S. Ghosh, A. Dhar, Appl. Phys. 25, 105 (1981); M. Mondal, K. P. Ghatak, Phys. Letts. A 131A, 529 (1988); K. P. Ghatak, S. N. Biswas, Nonlin. Opt. Quant. Opt. 12, 83 (1995); K. P. Ghatak, S. N. Biswas, Nonlin. Opt. Quant. Opt. 4, 347 (1993); K. P. Ghatak, A. Ghoshal, B. Mitra, Nouvo Cimento D 13D, 867 (1991); B. Mitra, A. Ghoshal, K. P. Ghatak Nouvo Cimento D 12D, 891 (1990). 18. B. R. Nag, Electron Transport in Compound Semiconductors, Springer Series in Solid-State Science, Vol. 11 (Springer-Verlag, Germany, 1980). 19. P. K. Chakraborty, L. J. Singh, K. P. Ghatak, J. Appl. Phys. 95, 5311 (2004).

Chapter 6

Photoemission under Quantizing Magnetic Field from Optoelectronic Materials

6.1 Introduction In this chapter, we shall study photoemission under magnetic quantization in optoelectronic materials in the presence of external photo-excitation whose conduction electrons obey the energy wave-vector dispersion relations as given by (5.1), (5.2), and (5.3), respectively. Section 6.2 contains the theoretical background. The dependence of magneto photoemission from n-Hg1–x Cdx Te and n-In1–x Gax Asy P1–y lattice matched to InP on the inverse quantizing magnetic field, the carrier concentration, the intensity of light, and the wavelength have been discussed in Section 6.3. The Section 6.4 presents open research problems pertinent to this chapter.

6.2 Theoretical Background Using (1.29), (5.1), (5.2), and (5.3), the magneto-dispersion relations, in the absence of electron spin, for optoelectronic materials in the presence of photo-excitation, whose unperturbed conduction electrons obey the three- and two-band models of Kane, together with parabolic energy bands, are given by [1]:

2 kz2 1 ,  ω0 + β0 (E, λ) = n + 2 2m∗

2 kz2 1 ,  ω0 + τ0 (E, λ) = n + 2 2m∗

2 kz2 1 .  ω0 + ρ0 (E, λ) = n + 2 2m∗

(6.1) (6.2) (6.3)

The velocity of the emitted electrons along the z-direction for magneto dispersion relations (6.1), (6.2), and (6.3) can, respectively, be expressed as:  vzB (E) =

√ 2 β0 (E, λ) − (n + (1/2))  ω0 . , ∗ {β0 (E, λ)} m

(6.4)

237 K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_6,  C Springer Science+Business Media, LLC 2009

238

6

 vzB (E) =  vzB (E) =

Photoemission under Quantizing Magnetic Field

√ 2 τ0 (E, λ) − (n + (1/2))  ω0 . , {τ0 (E, λ)} m∗

(6.5)

√ 2 ρ0 (E, λ) − (n + (1/2))  ω0 . . ∗ {ρ0 (E, λ)} m

(6.6)

The DOS function per subband for (6.1), (6.2), and (6.3) can, respectively, be written as: NB (E,

 √

 −1/2  gv |e| B 2m∗ 1  {β0 (E, λ)} β0 (E, λ)− n +  ω0 , (6.7) λ) = 2 2π 2 2

 √

 −1/2  ∗ |e| 2m B g 1 v {τ0 (E, λ)} τ0 (E, λ)− n + NB (E, λ) =  ω0 , (6.8) 2 2π 2 2  √

 −1/2  ∗ |e| 2m B g 1 v {ρ0 (E, λ)} ρ0 (E, λ)− n + NB (E, λ) =  ω0 . (6.9) 2 2π 2 2 Using (1.36), (6.4), (6.5), (6.6), (6.7), (6.8), and (6.9), the magneto photoemission from optoelectronic materials, whose unperturbed bulk conduction electrons obey (5.1), (5.2), and (5.3) can, respectively, be expressed as:

JBL

nmax α0 e2 Bgv kB T  = F0 (η61 ), 2π 2 2

(6.10)

nmax α0 e2 Bgv kB T  F0 (η62 ), 2π 2 2

(6.11)

nmax α0 e2 Bgv kB T  F0 (η63 ), 2π 2 2

(6.12)

n=0

JBL =

n=0

JBL =

n=0

where EFBL − (EnL1 + (W − hv)) EFBL − (EnL2 + (W − hv)) , η62 ≡ , kB T kB T EFBL − (EnL3 + (W − hv)) ≡ , kB T

η61 ≡ η63

EFBL is the Fermi energy under quantizing magnetic fields in the presence of light waves as measured from the edge of the conduction band in the vertically upward direction in the absence of any quantization, EnL1 , EnL2 , and EnL3 are, respectively, the Landau subband energy for (6.1), (6.2), and (6.3), which in turn can be expressed as:

6.3

Results and Discussions

239



1 β0 (EnL1 , λ) = n +  ω0 , 2

1 τ0 (EnL2 , λ) = n +  ω0 , 2

1 ρ0 (EnL3 , λ) = n +  ω0 . 2

(6.13) (6.14) (6.15)

It appears then that the evolution of the magneto photo-current density requires the expression of electron statistics which can, respectively, be written as: n0L

n0L

n0L

√ nmax gv |e| B 2m∗  [M61 (EFBL , B, λ) + N61 (EFBL , B, λ)], = π 2 2 n=0 √ nmax gv |e| B 2m∗  [M62 (EFBL , B, λ) + N62 (EFBL , B, λ)], = π 2 2 n=0 √ nmax gv |e| B 2m∗  [M63 (EFBL , B, λ) + N63 (EFBL , B, λ)], = π 2 2

(6.16)

(6.17)

(6.18)

n=0

where 

1/2 1 M61 (EFBL , B, λ) ≡ β0 (EFBL , λ) − n + , ω0 2 s  Zr, Y M61 (EFBL , B, λ) N61 (EFBL , B, λ) ≡ r=1



1/2 1 Y = BL, M62 (EFBL , B, λ) ≡ τ0 (EFBL , λ) − n + , ω0 2 s  Zr,Y M62 (EFBL , B, λ),M63 (EFBL , B, λ) N62 (EFBL , B, λ) ≡ r=1



1/2 1 ≡ ρ0 (EFBL , λ) − n + ,  ω0 2 s  Zy,Y M63 (EFBL , B, λ). and N63 (EFBL , B, λ) ≡ r=1

6.3 Results and Discussions Using (6.10), (6.11), and (6.12); and (6.16), (6.17), and (6.18), we have plotted the normalized magneto-photoemission current density from n-Hg1–x Cdx Te versus inverse quantizing magnetic field in accordance with the perturbed three- and

240

6

Photoemission under Quantizing Magnetic Field

Fig. 6.1 Plot of the normalized photocurrent density as a function of inverse magnetic field from n-Hg1–x Cdx Te, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane together with parabolic energy bands, respectively

two-band models of Kane and that of perturbed parabolic energy bands as shown in Fig. 6.1. Figures 6.2, 6.3, and 6.4 exhibit the variation of the aforementioned quantity from n-Hg1–x Cdx Te as functions of the normalized electron degeneracy, the normalized intensity of light and wavelength at T= 4.2 K, respectively. Figures 6.5, 6.6, 6.7, and 6.8 represent the said variations of photoemitted current density under magnetic quantization from n-In1–x Gax Asy P1–y lattice matched to InP. It appears from Figs. 6.1 and 6.5 that the photoemitted current density under magnetic quantization oscillates with inverse quantizing magnetic field, and the numerical values are different in various cases, which is the direct signature of the band structure. It may be noted that the origin of the oscillation is the same as that of SdH oscillations as discussed in detail in Chapter 1. From Figs. 6.2 and 6.6, we observe that the said physical quantity oscillates with electron degeneracy, although the numerical values of the oscillatory photoemission are different in different cases. Figures 6.3 and 6.7 exhibit the fact that the normalized magneto-photoemitted current density decreases with increasing intensity and the slopes directly reflect the influence of the energy band constants. Figures 6.4 and 6.8 reflect the fact that the magneto-photoemitted current density decreases with increasing wavelength. Finally, we note that the form of the expression of the said physical quantity in this case as given by (6.10), (6.11), and (6.12) is

6.3

Results and Discussions

241

Fig. 6.2 Plot of the normalized photocurrent density as a function of normalized carrier degeneracy from n-Hg1–x Cdx Te, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane together with parabolic energy bands, respectively

Fig. 6.3 Plot of the normalized photocurrent density as a function of normalized light intensity from n-Hg1–x Cdx Te, in which the curves (a), (b), and (c) represent the perturbed three- and twoband models of Kane together with parabolic energy bands, respectively

242

6

Photoemission under Quantizing Magnetic Field

Fig. 6.4 Plot of the normalized photocurrent density as a function of wavelength from nHg1–x Cdx Te, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane together with parabolic energy bands, respectively

Fig. 6.5 Plot of the normalized photocurrent density as a function of inverse magnetic field from In1–x Gax Asy P1–y lattice matched to InP, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane together with parabolic energy bands, respectively

6.3

Results and Discussions

243

Fig. 6.6 Plot of the normalized photocurrent density as a function of normalized carrier degeneracy from In1–x Gax Asy P1–y lattice matched to InP, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane together with parabolic energy bands, respectively

Fig. 6.7 Plot of the normalized photocurrent density as a function of normalized light intensity from In1–x Gax Asy P1–y lattice matched to InP, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane together with parabolic energy bands, respectively

244

6

Photoemission under Quantizing Magnetic Field

Fig. 6.8 Plot of the normalized photocurrent density as a function of wavelength from In1–x Gax Asy P1–y lattice matched to InP, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane together with parabolic energy bands, respectively

generalized where E nLi (i=1, 2, and 3) and the Fermi energy under magnetic fields are two-band structure–dependent quantities.

6.4 Open Research Problems Investigate the following open research problems in the presence of external photoexcitation which changes the band structure in a fundamental way, together with the proper inclusion of the electron spin, the variation of work function, and the broadening of Landau levels, respectively, for appropriate problems. (R6.1) Investigate multiphoton photoemission from all the materials as discussed in Chapter 3 in the presence of arbitrarily oriented photo-excitation and quantizing magnetic fields, respectively. (R6.2) Investigate multiphoton photoemission from all the materials as discussed in Chapter 3 in the presence of an arbitrarily oriented nonquantizing nonuniform electric field, photo-excitation, and quantizing magnetic fields, respectively. (R6.3) Investigate multiphoton photoemission from all the materials as discussed in Chapter 3 in the presence of an arbitrarily oriented nonquantizing alternating electric field, photo-excitation, and quantizing magnetic fields, respectively.

References

245

(R6.4) Investigate multiphoton photoemission from all the materials as discussed in Chapter 3 in the presence of an arbitrarily oriented nonquantizing alternating electric field, photo-excitation, and quantizing alternating magnetic fields, respectively. (R6.5) Investigate multiphoton photoemission from all the materials as discussed in Chapter 3 in the presence of an arbitrarily oriented photo-excitation and crossed electric and quantizing magnetic fields, respectively. (R6.6) Investigate multiphoton photoemission for arbitrarily oriented photoexcitation and quantizing magnetic fields from the heavily-doped materials in the presence of Gaussian, exponential, Kane, Halperin, Lax, and BonchBruevich types of band-tails for all materials whose unperturbed carrier energy spectra are defined in Chapter 3. (R6.7) Investigate multiphoton photoemission for arbitrarily oriented photoexcitation and quantizing alternating magnetic fields for all the cases of R6.6. (R6.8) Investigate multiphoton photoemission for arbitrarily oriented photoexcitation and nonquantizing alternating electric fields and quantizing magnetic fields for all the cases of R6.6. (R6.9) Investigate multiphoton photoemission for arbitrarily oriented photoexcitation and nonuniform alternating electric fields and quantizing magnetic fields for all the cases of R6.6. (R6.10) Investigate multiphoton photoemission for arbitrarily oriented photoexcitation and crossed electric and quantizing magnetic fields for all the cases of R6.6. (R6.11) Investigate multiphoton photoemission from negative refractive index, organic, magnetic, heavily doped, disordered, and other advanced optical materials in the presence of arbitrary oriented photo-excitation and quantizing magnetic fields. (R6.12) Investigate multiphoton photoemission in the presence of arbitrary oriented photo-excitation, quantizing magnetic fields, and alternating nonquantizing electric fields for all the problems of R6.11. (R6.13) Investigate multiphoton photoemission in the presence of arbitrary oriented photo-excitation, quantizing magnetic fields, and nonquantizing nonuniform electric fields for all the problems of R6.11. (R6.14) Investigate multiphoton photoemission in the presence of arbitrary oriented photo-excitation, alternating quantizing magnetic fields, and crossed alternating nonquantizing electric fields for all the problems of R6.11. (R6.15) Investigate all the problems from R6.1 to R6.14 by removing all the mathematical approximations and establishing the respective appropriate uniqueness conditions.

References 1. K. P. Ghatak, S. Bhattacharya, D. De, Einstein Relation in Compound Semiconductors and Their Nanostructures, Vol. 116, Springer Series in Materials Science (Springer-Verlag, Germany, 2008) and the references cited therein.

Chapter 7

Photoemission from Quantum Wells in Ultrathin Films, Quantum Wires, and Dots of Optoelectronic Materials

7.1 Introduction In this chapter, in 7.2.1, 7.2.2, and 7.2.3 on theoretical background, we shall study photoemission from QWs in UFs, QWWs, and QDs of optoelectronic materials, whose bulk conduction electrons are defined by the dispersion relations as given by (5.1), (5.2), and (5.3), respectively. In Section 7.2, we investigate photoemission from the aforementioned quantum confined materials with respect to various external variables; and Section 7.3 includes results and discussions. The Section 7.4 presents open research problems pertinent to this chapter.

7.2 Theoretical Background 7.2.1 Photoemission from Quantum Wells in Ultrathin Films of Optoelectronic Materials The dispersion relation of the 2D electrons in QWs in UFs of optoelectronic materials, the conduction electrons of whose bulk samples are defined by the dispersion relations as given by (5.1), (5.2), and (5.3) can, respectively, be expressed following [1] as: kx2

+ ky2

2m∗ β0 (E, λ) = − 2

kx2

+ ky2

2m∗ τ0 (E, λ) = − 2

kx2

+ ky2

2m∗ ρ0 (E, λ) = − 2



π nz71 dz π nz72 dz π nz73 dz

2 ,

(7.1)

,

(7.2)

,

(7.3)

2

2

where nz7J (J = 1,2,3)is the size quantum number. The 2D electron statistics assume the form: 247 K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_7,  C Springer Science+Business Media, LLC 2009

248

7

n2DL = n2DL = n2DL =

m∗ gv π2 m∗ gv π2 m∗ gv π2

Quantum Wells in Ultrathin Films, Quantum Wires, and Dots

nz71 max 



 φ71 (EF2DL , nz71 ) + φ72 (EF2DL , nz71 ) ,

(7.4)

nz71 =1

nz72 max 



 φ73 (EF2DL , nz72 ) + φ74 (EF2DL , nz72 ) ,

(7.5)

nz72 =1

nz73 max 



 φ75 (EF2DL , nz73 ) + φ76 (EF2DL , nz73 ) ,

(7.6)

nz73 =1

where EF2DL is the Fermi energy in QWs in UFs in the presence of light waves as measured from the edge of the conduction band in the vertically upward direction in the absence of any quantization,  φ71 (E2DF , nz71 ) =

φ72 (EF2DL , nz71 ) =

2m∗ β0 (EF2DL , λ) − 2

s0 



πnz71 dz

2  ,

  Zr,Y φ71 (EF2DL , nz71 ) ,Y = 2DL,

r=1

 φ73 (EF2DL , nz72 ) =

φ74 (EF2DL , nz72 ) =

2m∗ τ (EF2DL , λ) − 2

s0 



πnz72 dz

2  ,

  Zr,Y φ73 (EF2DL , nz72 ) ,

r=1

 φ75 (EF2DL , nz73 ) =

φ76 (EF2DL , nz73 ) =

2m∗ ρ0 (EF2DL , λ) − 2

s0 



π nz73 dz

2  ,

and

  Zr,Y φ75 (EF2DL , nz73 )

r=1

The velocity of the electron in the nz71 th, nz72 th, and nz73 th subbands for the 2D electron energy spectra as given by (7.1), (7.2), and (7.3) can, respectively, be written as: 



υz Enz71 =



m∗

& ⎡"

−1

2

2

 ⎤ β0 Enz71 , λ ⎣   ⎦, β0 Enz71 , λ

(7.7)

7.2

Theoretical Background

249

  υz Enz72 =





& ⎡"  ⎤ τ0 Enz72 , λ 2 ⎣  ⎦,  τ0 Enz72 , λ

−1

2



m∗

υz Enz73 =

(7.8)

& ⎡"  ⎤ ρ0 Enz73 , λ 2 ⎣   ⎦, ρ0 Enz73 , λ

m∗

−1

2

(7.9)

where the subband energies Enz71 , Enz72 , and Enz73 are respectively defined through the following equations: β0

 2 Enz71 , λ = 2m∗



 Enz72 , λ =

τ0







2 2m∗

π nz71 dz



2

π nz72 dz

,

(7.10)

2 ,

(7.11)

.

(7.12)

and   ρ0 Enz73 , λ =



2 2m∗



π nz73 dz

2

The respective expressions of the photoemission are given by:

J2DL =

α0 gv e π 2 dz



m∗ 2

&

−1

2

nz71max



nz71min

⎡"  ⎤ β0 Enz71 , λ   ⎣  ⎦ φ71 (EF2DL , nz71 ) + φ72 (EF2DL , nz71 ) ,   β0 Enz71 , λ (7.13)

where nz71min ≥

J2DL

α0 gv e = π 2 dz



m∗ 2

dz π

√

2m∗ 

 [β0 (W − hυ, λ)]

&

−1

2

,

⎡"  & ⎤

−1 nz72 max τ0 Enz72 , λ  2   ⎣  ⎦ φ73 (EF2DL , nz72 ) + φ74 (EF2DL , nz72 ) ,   E τ , λ n z72 0 n z72min

(7.14)

where nz72min ≥ and

dz π

√

2m∗ 





τ0 (W − hυ, λ)

250

J2DL

7

α0 gv e = π 2 dz



m∗

&

−1

2

2

Quantum Wells in Ultrathin Films, Quantum Wires, and Dots

nz73max



nz73min

⎡"

⎤  ρ0 Enz73 , λ   ⎣  ⎦ φ75 (EF2DL , nz73 ) + φ72 (EF2DL , nz73 ) ,   ρ0 Enz73 , λ (7.15)

where nz73min ≥

dz π

√

2m∗ 

 [ρ0 (W − hυ, λ)]1/2 .

7.2.2 Photoemission from Quantum Well Wires of Optoelectronic Materials The dispersion relations of the 1D electrons in QWWs of optoelectronic materials in the presence of light waves can be expressed from (7.1), (7.2), and (7.3) as: ky2 =

2m∗ β0 (E, λ) − 2

ky2 =

2m∗ τ0 (E, λ) − 2

ky2

2m∗ ρ0 (E, λ) = − 2



π nz71 dz πnz72 dz πnz73 dz



2 −



2 −



2 −

π nx71 dx π nx72 dx π nx73 dx

2 ,

(7.16)

,

(7.17)

,

(7.18)

2

2

where nx7J (J = 1,2,3)is the size quantum number. The electron concentrations per unit length are, respectively, given by: n1DL =

√ nx71max nz71max    2gv 2m∗  φ77 (EF1DL , nx71 , nz71 ) + φ78 (EF1DL , nx71 , nz71 ) , π nx71 =1 nz71 =1

(7.19)

n1DL =

√ nx72max nz72max    2gv 2m∗  φ79 (EF1DL , nx72 , nz72 ) + φ80 (EF1DL , nx72 , nz72 ) , π nx72 =1 nz72 =1

(7.20)

n1DL

√ nx73max nz73max    2gv 2m∗  = φ81 (EF1DL , nx73 , nz73 ) + φ82 (EF1DL , nx73 , nz73 ) , π nx73 =1 nz73 =1

(7.21) where EF1DL is the Fermi energy in QWWs in the presence of light waves, as measured from the edge of the conduction band in the vertically upward direction in the absence of any quantization,

7.2

Theoretical Background

251

 1/2 φ77 (EF1DL , nx71 , nz71 ) = β0 (EF1DL , λ) − G71 (nx71 , nz71 ) , 2 G7i (nx7i , nz7i ) = ∗ 2m φ78 (EF1DL , nx71 , nz71 ) =

s0 



πnx7i dx



2 +

π nz7i dz

2  ,

  Zr,Y φ77 (EF1DL , nx71 , nz71 ) ,

r=1

Y =1DL , φ79 (EF1DL , nx72 , nz72 )  1/2 , = τ0 (EF1DL , λ) − G72 (nx72 , nz72 ) φ80 (EF1DL , nx72 , nz72 ) =

s 

  Zr,Y φ79 (EF1DL , nx72 , nz72 ) ,

r=1



1/2

φ81 (EF1DL , nx73 , nz73 ) = ρ0 (EF1DL , λ) − G73 (nx73 , nz73 ) φ82 (EF1DL , nx73 , nz73 ) =

s 

, and

  Zr,Y φ81 (EF1DL , nx73 , nz73 ) .

r=1

The generalized expression of photo current in this case is given by: IL =

nx7imax nz7imax  α0 egv kB T  F0 (η7i ), π

(7.22a)

nx7i =1 nz7i =1

where    + W − hυ EF1DL − E7i η7i = kB T  are the subband energies in this case and are defined through the following and E7i equations:

   β0 E71 , λ = G71 (nx71 , nz71 )    τ0 E72 , λ = G72 (nx72 , nz72 )    ρ0 E73 , λ = G73 (nx73 , nz73 ) .

(7.22b)

7.2.3 Photoemission from Quantum Dots of Optoelectronic Materials The dispersion relations of the electrons in QDs of optoelectronic materials in the presence of light waves can respectively be expressed from (7.16), (7.17),

252

7

and (7.18) as:

Quantum Wells in Ultrathin Films, Quantum Wires, and Dots

    2m∗ β0 EQ1 , λ = H71 nx71 , ny71 , nz71 , 2

(7.23)

    2m∗ τ0 EQ2 , λ = H72 nx72 , ny72 , nz72 , 2

(7.24)

    2m∗ ρ0 EQ3 , λ = H73 nx73 , ny73 , nz73 , 2 

(7.25)

where EQi is the totally quantized energy and   H7i nx7i , ny7i , nz7i =



π nx7i dx

2

π ny7i dy



2 +

π nz7i dz

2 .

The electron concentration can, in general, be written as: n0DL =

2gv dx dy dz

nx7i max nz7i max ny7i max   

F−1 (η7i0D ),

(7.26)

nx7i =1 ny7i =1 nz7i =1

where η7i0D = (EF0DL − EQi )/kB T and EF0DL is the Fermi energy in QDs in the presence of light waves as measured from the edge of the conduction band in the vertically upward direction in the absence of any quantization. The photoemitted current densities in this case are given by the following equations:

J0DL

(α0 egv ) = dx dy dz



m∗ 2

⎡"  & ⎤

−1 nx71 max nz71 max max ny71 β0 Enz71 , λ   2  ⎣   ⎦F−1 (η710D ) , β0 Enz71 , λ n =1 n =1 n x71

y71

z71 min

(7.27)

J0DL =

(α0 egv ) dx dy dz



m∗

&

−1

n max nz72 max max ny72   2 x72

2

nx72 =1 ny72 =1 nz72 min

⎡"  ⎤ τ0 Enz72 , λ ⎣  ⎦F−1 (η720D ) ,  τ0 Enz72 , λ (7.28)

and

J0DL =

(α0 egv ) dx dy dz



m∗

&

−1

2

2

nx73 max nz73 max max ny73    nx73 =1 ny73 =1 nz73 min

⎡"  ⎤ ρ0 Enz73 , λ ⎣   ⎦F−1 (η730D ) . ρ0 Enz73 , λ (7.29)

7.3

Results and Discussions

253

7.3 Results and Discussions Using the numerical values of the energy band constants from Appendix A, we have plotted normalized photoemitted current density from QWs in UFs of nHg1–x Cdx Te, under external photo-excitation whose band structure follows the perturbed three- (using (7.13) and (7.4)) and two- (using (7.14) and (7.5)) band models of Kane, and that of the perturbed parabolic (using (7.15) and (7.6)) energy bands, as shown by curves (a), (b), and (c) of Fig. 7.1 as functions of film thickness. The plots of Figs. 7.2, 7.3, and 7.4 exhibit the dependence of the normalized photoemitted current density on the normalized electron degeneracy, normalized intensity, and wavelength, respectively, for all cases of Fig. 7.1. The variations of the normalized photoemitted current density from QWs in UFs of n-In1–x Gax Asy P1–y lattice matched to InP as functions of film thickness, normalized carrier degeneracy, normalized incident light intensity, and wavelength, respectively, have been drawn in Figs. 7.5, 7.6, 7.7, and 7.8 for all cases of Fig. 7.1. The dependences of the normalized photocurrent from QWWs of n-Hg1–x Cdx Te with respect to film thickness, normalized carrier degeneracy, normalized light intensity, and wavelength have been drawn in Figs. 7.9, 7.10, 7.11, and 7.12 in accordance with perturbed three- (using (7.22a) and (7.19)) and two- (using (7.22a) and (7.20)) band models of Kane, together with parabolic (using (7.22a) and (7.21)) energy bands as shown by curves (a), (b), and (c), respectively. The variations of normalized photocurrent for QWWs of n-In1–x Gax Asy P1–y lattice matched to InP, have been drawn in Figs. 7.13, 7.14, 7.15, and 7.16 as functions of film thickness, normalized carrier degeneracy, normalized incident light intensity, and wavelengths, respectively. 0.06 Normalized Photocurrent Density

λ = 610 nm 0.05 0.04 0.03 0.02 (c) (b)

0.01

(a)

0 10

15

20

25 30 35 Thickness (in nm)

40

45

50

Fig. 7.1 Plot of the normalized photocurrent density from quantum wells in ultrathin films of n-Hg1–x Cdx Te as a function of film thickness, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively

254

7

Quantum Wells in Ultrathin Films, Quantum Wires, and Dots

1

0.8

0.6

0.4

0.2

0.01

0.1

Normalized Photocurrent Density

1.2

dx = 10 nm λ = 610 nm

01 Normalized Electron Degeneracy

(a)

(b)

(c)

100

10

Fig. 7.2 Plot of the normalized photocurrent density from quantum wells in ultrathin films of n-Hg1–x Cdx Te as a function of normalized electron degeneracy for all cases of Fig. 7.1

Normalized Photocurrent Density

0.06

0.05

λ = 610 nm dx = 10 nm

(c)

0.04

0.03

(b)

0.02 (a) 0.01

0 0.01

0.02

0.03

0.04

0.05 0.06 0.07 Normalized Intensity

0.08

0.09

0.1

Fig. 7.3 Plot of the normalized photocurrent density from quantum wells in ultrathin films of n-Hg1–x Cdx Te as a function of normalized light intensity for all cases of Fig. 7.1

The dependences of the normalized photoemitted current density from QDs of n-Hg1–x Cdx Te on the film thickness, normalized carrier degeneracy, normalized light intensity, and wavelength have been drawn in Figs. 7.17, 7.18, 7.19, and 7.20 in accordance with perturbed three- (using (7.27) and (7.26)) and two- (using (7.28)

Results and Discussions

255

0.011405

0.03422

dx = 10 nm 0.05705

0.0342 0.057

0.011395 0.01139

0.05695

(a)

0.011385 0.01138

0.0569

(c) 0.011375 0.05685

0.01137

(b)

0.011365

0.0568

Normalized Photocurrent Density

Normalized Photocurrent Density

0.0114

0.01136 0.05675

0.011355

410

460

510

560

610

0.03418 0.03416 0.03414 0.03412 0.0341 0.03408

Normalized Photocurrent Density

7.3

0.03406

660

Wavelength (in nm)

Fig. 7.4 Plot of the normalized photocurrent density from quantum wells in ultrathin films of n-Hg1–x Cdx Te as a function of light wavelength for all cases of Fig. 7.1

0.05 Normalized Photocurrent Density

0.045

λ = 610 nm

0.04 0.035 0.03 0.025

(b)

0.02 0.015

(a)

0.01 (c)

0.005 0

10

15

20

25 30 35 Thickness (in nm)

40

45

50

Fig. 7.5 Plot of the normalized photocurrent density from quantum wells in ultrathin films of nIn1–x Gax Asy P1–y lattice matched to InP as a function of film thickness, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively

and (7.26)) band models of Kane, together with parabolic (using (7.29) and (7.26)) energy bands as shown by curves (a), (b) and (c), respectively. The variations of normalized photocurrent density from QDs of n-In1–x Gax Asy P1–y lattice matched to InP have been drawn in Figs. 7.21, 7.22, 7.23, and 7.24 as functions of film

256

7

Quantum Wells in Ultrathin Films, Quantum Wires, and Dots

Normalized Photocurrent Density

0.35

dx = 15 nm λ = 610 nm

0.3 0.25 0.2 0.15 0.1 0.05

(c)

(b) (a)

0 0.01

0.1

1

100

10

Normalized Electron Degeneracy

Fig. 7.6 Plot of the normalized photocurrent density from quantum wells in ultrathin films of n-In1–x Gax Asy P1–y lattice matched to InP as a function of normalized electron degeneracy for all cases of Fig. 7.5

Normalized Photocurrent Density

0.022

λ = 610 nm dx = 15 nm

0.02 (c) 0.018

0.016

(b)

0.014

0.012

0.01 0.01

(a)

0.02

0.03

0.04

0.05 0.06 0.07 Normalized Intensity

0.08

0.09

0.1

Fig. 7.7 Plot of the normalized photocurrent density from quantum wells in ultrathin films of n-In1–x Gax Asy P1–y lattice matched to InP as a function of normalized light intensity for all cases of Fig. 7.5

thickness, normalized carrier degeneracy, normalized incident light intensity, and wavelengths, respectively, for all the cases of Fig. 7.17. From Figs. 7.1 and 7.5, it appears that photoemitted current density from QWs in UFs of optoelectronic

Results and Discussions

257

Normalized Photocurrent Density

dx = 15 nm 0.01339

0.0228 0.02279 0.02278

0.01338

0.02277 0.02276

0.01337 (b)

(a) 0.01336

0.02275 0.02274 0.02273

0.01335 (c) 0.01334 410

460

510 560 Wavelength (in nm)

610

0.01568

0.02281

0.02272 0.02271 660

0.01567 0.01566 0.01565 0.01564 0.01563 0.01562

Normalized Photocurrent Density

0.01334

Normalized Photocurrent Density

7.3

0.01561

Fig. 7.8 Plot of the normalized photocurrent density from quantum wells in ultrathin films of nIn1–x Gax Asy P1–y lattice matched to InP as a function of light wavelength for all cases of Fig. 7.5

0.05

Normalized Photocurrent

0.045

dy = 15 nm λ = 610 nm

0.04 0.035 (c)

0.03 0.025 0.02 0.015 0.01

(b) (a)

0.005 0 10

15

20 25 30 Film Thickness (d z) (in nm)

35

40

Fig. 7.9 Plot of the normalized photocurrent from quantum well wires of n-Hg1−x Cdx Te as a function of film thickness, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively

materials decreases with increasing film thickness in an oscillatory manner. From Figs. 7.9 and 7.13, it appears that the photocurrent from QWWs of optoelectronic materials increases with decreasing film thickness, exhibiting trapezoidal variation

258

7 0.4

Normalized Photocurrent

0.35

Quantum Wells in Ultrathin Films, Quantum Wires, and Dots

dz = 10 nm λ = 610 nm

0.3 0.25 0.2 (c) 0.15 (b) 0.1 (a)

0.05 0 0.001

0.101

0.201

0.301 0.401 0.501 0.601 0.701 Normalized Electron Degeneracy

0.801

0.901

1.001

Fig. 7.10 Plot of the normalized photocurrent from quantum wells wires of n-Hg1–x Cdx Te as a function of normalized electron degeneracy for all cases of Fig. 7.9

0.05 dy = 15 nm dx = 10 nm λ = 610 nm

Normalized Photocurrent

0.045 0.04

(c)

0.035 0.03 0.025 0.02

(b)

0.015

(a)

0.01 0.01

0.02

0.03

0.04

0.05 0.06 0.07 Normalized Intensity

0.08

0.09

0.1

Fig. 7.11 Plot of the normalized photocurrent from quantum well wires of n-Hg1−x Cdx Te as a function of normalized light intensity for all cases of Fig. 7.9

of varying shapes for a very small thickness bandwidth for the whole range of thicknesses considered. The widths of the trapezoids depend on the energy band constants of n-Hg1–x Cdx Te and n-In1–x Gax Asy P1–y lattice matched to InP, respectively.

Results and Discussions

259

0.02215

0.01582

0.0443

0.0158195 0.044295

(b)

0.02214 0.022135

0.04429

0.02213 0.022125

0.044285

(c)

0.02212

0.04428

0.022115 0.02211 0.022105

dy = 15 nm dx = 10 nm

0.0221 410 430

(a)

0.044276

Normalized Photocurrent

Normalized Photocurrent

0.022145

470

490 510 530 550 570 Wavelength (in nm)

0.0158185 0.015818 0.0158175 0.015817 0.0158165 0.015816 0.0158155

0.04427

450

0.015819

Normalized Photocurrent

7.3

0.015815

590 610

Fig. 7.12 Plot of the normalized photocurrent from quantum well wires of n-Hg1−x Cdx Te as a function of light wavelength for all cases of Fig. 7.9

0.025 dy = 15 nm λ = 610 nm

Normalized Photocurrent

0.02

0.015 (c) 0.01 (b) 0.005

(a)

0 10

15

25 20 30 Film Thickness (d z) (In nm)

35

40

Fig. 7.13 Plot of the normalized photocurrent from quantum well wires of n-In1–x Gax Asy P1–y lattice matched to InP as a function of film thickness, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively

From Figs. 7.17 and 7.21, we observe that the photoemission from QDs of optoelectronic materials decreases with increasing film thickness, exhibiting prominent trapezoidal variation for relatively large thickness bandwidth. These three types of variations are the special signatures of 1D confinement in QWs in UFs, 2D

260

7

Normalized Photocurrent

0.2501

0.2001

Quantum Wells in Ultrathin Films, Quantum Wires, and Dots

dz = 10 nm dy = 15 nm λ = 610 nm

(c)

0.1501

0.1001 (b) 0.0501

0.0001 0.01

(a)

0.11

0.21

0.61 0.71 0.31 0.41 0.51 Normalized Electron Degeneracy

0.81

0.91

1.01

Fig. 7.14 Plot of the normalized photocurrent from quantum well wires of n-In1–x Gax Asy P1–y lattice matched to InP as a function of normalized electron degeneracy for all cases of Fig. 7.13

0.02

Normalized Photocurrent

0.018

dy = 15 nm dx = 10 nm λ = 610 nm

(c)

0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.01

(b)

(a) 0.02

0.03

0.04

0.05 0.06 0.07 Normalized Intensity

0.08

0.09

0.1

Fig. 7.15 Plot of the normalized photocurrent from quantum well wires of n-In1–x Gax Asy P1–y lattice matched to InP as a function of normalized light intensity for all cases of Fig. 7.13

confinement in QWWs, and 3D confinement in QDs of optoelectronic materials, respectively, in the presence of light. From Figs. 7.2 and 7.6, it appears that the normalized photoemitted current density from QWs in UFs increases with increasing carrier degeneracy and for relatively large values of the same variable; it exhibits quantum jumps for all types of band models when the size quantum number changes

Results and Discussions

261

0.009492 0.0094915 Normalized Photocurrent

0.0056955

0.018984

dy = 15 nm dx = 10 nm

0.018983

0.005695

0.018982

0.009491

0.0056945 0.018981

0.0094905 (b)

0.00949

(a)

0.0094895

(c)

0.01898

0.005694

0.018979

0.0056935

0.018978

0.005693

0.009489

0.018977

0.0094885

0.018976

0.009488 410

0.0056925

0.018975

430

450

470

490

510 530 550 570 Wavelength (in nm)

590

Normalized Photocurrent

7.3

0.005692

610

Fig. 7.16 Plot of the normalized photocurrent from quantum well wires of n-In1–x Gax Asy P1–y lattice matched to InP as a function of light wavelength for all cases of Fig. 7.13

0.4 dy = 15 nm dx = 10 nm λ = 610 nm

Normalized Photocurrent Density

0.35 0.3 0.25 (c) 0.2 (b)

0.15 0.1

(a)

0.05 0 10

15

20

25 30 Film thickness dz (nm)

35

40

Fig. 7.17 Plot of the normalized photocurrent density from QDs of n-Hg1–x Cdx Te as a function of film thickness, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively

from one fixed value to another. Figures 7.10 and 7.14 show respectively that the normalized photocurrent in QWWs of optoelectronic materials increases with increasing normalized electron degeneracy. Figures 7.18 and 7.22 demonstrate that the photoemitted current density from QDs of optoelectronic materials increases with increasing electron degeneracy,

262

7

Quantum Wells in Ultrathin Films, Quantum Wires, and Dots

Normalized Photocurrent Density

10

Normalized Electron Degeneracy

1 0

0.1 (c)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a)

0.1

(b)

0.01

dy = 15 nm dx = 10 nm dz = 10 nm λ = 610 nm

0.001

0.0001

Fig. 7.18 Plot of the normalized photocurrent density from QDs of n-Hg1–x Cdx Te as a function of normalized electron degeneracy for all cases of Fig. 7.17

Normalized Photocurrent Density

0.4 dy = 15 nm dx = 10 nm dz = 10 nm λ = 610 nm

0.35 (c) 0.3

0.25

(b)

0.2

(a)

0.15

0.1 0.01

0.02

0.03

0.04

0.05 0.06 Normalized Intensity

0.07

0.08

0.09

0.1

Fig. 7.19 Plot of the normalized photocurrent density from QDs of n-Hg1–x Cdx Te as a function of normalized light intensity for all cases of Fig. 7.17

again in a different oscillatory manner. From Figs. 7.3, 7.7, 7.11, 7.15, 7.19, and 7.23, it appears that the photoemission increases with decreasing intensity for all types of quantum confinement. From Figs. 7.4, 7.8, 7.12, 7.16, 7.20, and 7.24, we can conclude that the normalized photoemission decreases with increasing wavelength for QWs in UFs, QWWs, and QDs of optoelectronic materials. Finally, we

Results and Discussions

263

Normalized Photocurrent Density

0.22145 0.2214 (b)

0.348

0.3479

0.22135 0.3478

0.2213 (a) 0.22125

0.3477

(c)

0.2212

0.3476

0.22115 0.3475

0.2211 0.22105 410

0.1614

0.3481

dy = 15 nm dx = 10 nm dz = 10 nm

0.3474

430

450

450

490

510 530 550 Wavelength (in nm)

570

590

0.16135 0.1613 0.16125 0.1612 0.16115 0.1611

Normalized Photocurrent Density

0.2215

Normalized Photocurrent Density

7.3

0.16105

610

Fig. 7.20 Plot of the normalized photocurrent density from QDs of n-Hg1–x Cdx Te as a function of light wavelength for all cases of Fig. 7.17

Normalized Photocurrent Density

0.3 dy = 15 nm dx = 10 nm λ = 610 nm

0.25

0.2

(c)

0.15

0.1 (b) 0.05 (a) 0 10

15

20

25

30

35

40

Film thickness dz (nm)

Fig. 7.21 Plot of the normalized photocurrent density from QDs of n-In1–x Gax Asy P1–y lattice matched to InP as a function of film thickness, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively

can write that it is apparent from all the figures that the photoemission from quantum confined ternary materials is larger as compared with the quantum confined quaternary compounds for all types of quantum confinement.

264

7

Quantum Wells in Ultrathin Films, Quantum Wires, and Dots

10

Normalized Photocurrent Density

Normalized Electron Degeneracy 1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c) 0.1 (b) (a)

.01

dy = 15 nm dx = 10 nm dz = 10 nm λ = 610 nm

001

0.0001

Fig. 7.22 Plot of the normalized photocurrent density from QDs of n-In1–x Gax Asy P1–y lattice matched to InP as a function of normalized electron degeneracy for all cases of Fig. 7.21

0.06

dy = 15 nm dx = 10 nm dz = 10 nm λ = 610 nm

Normalized Photocurrent Density

0.25

0.2

(c) 0.15

0.1

(b) 0.05

(a) 0 0.01

0.11

0.21

0.31

0.41 0.51 0.61 Normalized Intensity

0.71

0.81

0.91

1.01

Fig. 7.23 Plot of the normalized photocurrent density from QDs of n-In1–x Gax Asy P1–y lattice matched to InP as a function of normalized light intensity for all cases of Fig. 7.21

Open Research Problems

265

dy = 15 nm dx = 10 nm 0.2531 dz = 10 nm

0.04113

0.041125

0.25308

(a)

(c)

0.25306

0.04112 0.25304

0.041115 (b) 0.04110 410

0.12656

0.25312

0.25302

0.253

430

450

470

490

510 530 550 570 Wavelength (in nm)

590

610

0.12655

0.12654

0.12653

0.12652

0.12651

Normalized Photocurrent Density

Normalized Photocurrent Density

0.041135

Normalized Photocurrent Density

7.4

0.1265

Fig. 7.24 Plot of the normalized photoemitted current density from QDs of n-In1–x Gax Asy P1–y lattice matched to InP as a function of light wavelength for all cases of Fig. 7.21

7.4 Open Research Problems Investigate the following open research problems in the presence of external photoexcitation which changes the band structure in a fundamental way, together with the proper inclusion of the electron spin, the variation of work function, and the broadening of Landau levels, respectively, for the appropriate problems.

(R7.1) Investigate multiphoton photoemission from all the quantum confined materials (i.e., multiple quantum wells, wires, and dots) whose unperturbed carrier energy spectra are defined in Chapter 3, in the presence of arbitrarily oriented photo-excitation and quantizing magnetic field, respectively. (R7.2) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and alternating quantizing magnetic field, respectively, for all the various cases of R7.1. (R7.3) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation, alternating quantizing magnetic fields, and an additional arbitrarily oriented nonquantizing nonuniform electric field, respectively, all the various cases of R7.1. (R7.4) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation, alternating quantizing magnetic fields, and an additional arbitrarily oriented nonquantizing alternating electric field, respectively, all the various cases of R7.1. (R7.5) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation, and crossed quantizing magnetic and electric fields, respectively, for all the various cases of R7.1.

266

7

Quantum Wells in Ultrathin Films, Quantum Wires, and Dots

(R7.6) Investigate multiphoton photoemission for arbitrarily oriented photoexcitation and quantizing magnetic fields from the entire quantum confined heavily-doped materials in the presence of exponential, Kane, Halperin, Lax, and Bonch-Bruevich types of band-tails for all materials whose unperturbed carrier energy spectra are defined in Chapter 3. (R7.7) Investigate multiphoton photoemission for arbitrarily oriented photoexcitation and alternating quantizing magnetic fields for all the cases of R7.6. (R7.8) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation, alternating quantizing magnetic fields, and an additional arbitrarily oriented nonquantizing nonuniform electric field for all the cases of R7.6. (R7.9) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation, alternating quantizing magnetic fields, and an additional arbitrarily oriented nonquantizing alternating electric field, respectively, for all the cases of R7.6. (R7.10) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation, and crossed quantizing magnetic and electric fields, respectively, for all the cases of R7.6. (R7.11) Investigate multiphoton photoemission for all the appropriate problems from R7.1 to R7.10 in the presence of finite potential wells. (R7.12) Investigate multiphoton photoemission for all the appropriate problems from R.7.1 to R.7.10 in the presence of parabolic potential wells. (R7.13) Investigate multiphoton photoemission for all the above appropriate problems for quantum rings. (R7.14) Investigate multiphoton photoemission for all the above appropriate problems in the presence of elliptical Hill and quantum square rings respectively. (R7.15) Investigate multiphoton photoemission from carbon nanotubes in the presence of arbitrary photo-excitation. (R7.16) Investigate multiphoton photoemission from carbon nanotubes in the presence of arbitrary photo-excitation and nonquantizing alternating electric fields. (R7.17) Investigate multiphoton photoemission from carbon nanotubes in the presence of arbitrary photo-excitation and nonquantizing alternating magnetic fields. (R7.18) Investigate multiphoton photoemission from carbon nanotubes in the presence of arbitrary photo-excitation and crossed electric and quantizing magnetic fields. (R7.19) Investigate multiphoton photoemission from heavily doped semiconductor nanotubes in the presence of arbitrary photo-excitation for all the materials whose unperturbed carrier dispersion laws are defined in Chapter 3. (R7.20) Investigate multiphoton photoemission from heavily doped semiconductor nanotubes in the presence of nonquantizing alternating electric fields and arbitrary photo-excitation for all the materials whose unperturbed carrier dispersion laws are defined in Chapter 3.

7.4

Open Research Problems

267

(R7.21) Investigate multiphoton photoemission from heavily doped semiconductor nanotubes in the presence of nonquantizing alternating magnetic fields and arbitrary photo-excitation for all the materials whose unperturbed carrier dispersion laws are defined in Chapter 3. (R7.22) Investigate multiphoton photoemission from heavily doped semiconductor nanotubes in the presence of arbitrary photo-excitation and nonuniform electric fields for all the materials whose unperturbed carrier dispersion laws are defined in Chapter 3. (R7.23) Investigate multiphoton photoemission from heavily doped semiconductor nanotubes in the presence of arbitrary photo-excitation and alternating quantizing magnetic fields for all the materials whose unperturbed carrier dispersion laws are defined in Chapter 3. (R7.24) Investigate multiphoton photoemission from heavily doped semiconductor nanotubes in the presence of arbitrary photo-excitation and crossed electric and quantizing magnetic fields for all the materials whose unperturbed carrier dispersion laws are defined in Chapter 3. (R7.25) Investigate multiphoton photoemission in the presence of arbitrary photo-excitation for all the appropriate nipi structures of the materials whose unperturbed carrier energy spectra are defined in Chapter 3. (R7.26) Investigate multiphoton photoemission in the presence of arbitrary photo-excitation for all the appropriate nipi structures of the materials whose unperturbed carrier energy spectra are defined in Chapter 3 in the presence of an arbitrarily oriented nonquantizing nonuniform additional electric field. (R7.27) Investigate multiphoton photoemission for all the appropriate nipi structures of the materials whose unperturbed carrier energy spectra are defined in Chapter 3 in the presence of an arbitrarily oriented photoexcitation and nonquantizing alternating magnetic field. (R7.28) Investigate multiphoton photoemission for all the appropriate nipi structures of the materials whose unperturbed carrier energy spectra are defined in Chapter 3 in the presence of an arbitrarily oriented photoexcitation and quantizing alternating magnetic field. (R7.29) Investigate multiphoton photoemission for all the appropriate nipi structures of the materials whose unperturbed carrier energy spectra are defined in Chapter 3 in the presence of an arbitrarily oriented photoexcitation and crossed electric and quantizing magnetic fields. (R7.30) Investigate multiphoton photoemission from heavily doped nipi structures for all the appropriate cases of all the above problems. (R7.31) Investigate multiphoton photoemission in the presence of arbitrary photo-excitation for the appropriate inversion layers of all the materials whose unperturbed carrier energy spectra are defined in Chapter 3. (R7.32) Investigate multiphoton photoemission in the presence of arbitrary photo-excitation for the appropriate inversion layers of all the materials whose unperturbed carrier energy spectra are defined in Chapter 3 in the

268

7

Quantum Wells in Ultrathin Films, Quantum Wires, and Dots

presence of an arbitrarily oriented nonquantizing nonuniform additional electric field. (R7.33) Investigate multiphoton photoemission for the appropriate inversion layers of all the materials whose unperturbed carrier energy spectra are defined in Chapter 3 in the presence of an arbitrarily oriented photoexcitation and nonquantizing alternating magnetic field. (R7.34) Investigate multiphoton photoemission for the appropriate inversion layers of all the materials whose unperturbed carrier energy spectra are defined in Chapter 3 in the presence of an arbitrarily oriented photoexcitation and quantizing alternating magnetic field. (R7.35) Investigate multiphoton photoemission for the appropriate inversion layers of all the materials whose unperturbed carrier energy spectra are defined in Chapter 3 in the presence of an arbitrarily oriented photoexcitation and crossed electric and quantizing magnetic fields, by considering electron spin and broadening of Landau levels. (R7.36) Investigate multiphoton photoemission in the presence of arbitrary photo-excitation for the appropriate accumulation layers of all the materials whose unperturbed carrier energy spectra are defined in Chapter 3, by modifying the above appropriate problems. (R7.37) Investigate multiphoton photoemission in the presence of arbitrary photo-excitation from wedge shaped, cylindrical, ellipsoidal, conical, triangular, circular, parabolic rotational and parabolic cylindrical QDs of all the materials whose unperturbed carrier energy spectra are defined in Chapter 3. (R7.38) Investigate multiphoton photoemission in the presence of arbitrary photo-excitation from wedge shaped, cylindrical, ellipsoidal, conical, triangular, circular, parabolic rotational and parabolic cylindrical QDs of all the materials whose unperturbed carrier energy spectra are defined in Chapter 3 in the presence of an arbitrarily oriented nonquantizing nonuniform additional electric field. (R7.39) Investigate multiphoton photoemission from wedge shaped, cylindrical, ellipsoidal, conical, triangular, circular, parabolic rotational and parabolic cylindrical QDs of all the materials whose unperturbed carrier energy spectra are defined in Chapter 3 in the presence of an arbitrarily oriented photo-excitation and nonquantizing alternating magnetic field. (R7.40) Investigate multiphoton photoemission from wedge shaped, cylindrical, ellipsoidal, conical, triangular, circular, parabolic rotational and parabolic cylindrical QDs of all the materials whose unperturbed carrier energy spectra are defined in Chapter 3 in the presence of an arbitrarily oriented photo-excitation and quantizing alternating magnetic field. (R7.41) Investigate multiphoton photoemission from wedge shaped, cylindrical, ellipsoidal, conical, triangular, circular, parabolic rotational and parabolic cylindrical QDs of all the materials whose unperturbed carrier energy spectra are defined in Chapter 3 in the presence of an arbitrarily oriented photo-excitation and crossed electric and quantizing magnetic fields.

Reference

269

(R7.42) Investigate multiphoton photoemission from wedge shaped, cylindrical, ellipsoidal, conical, triangular, circular, parabolic rotational and parabolic cylindrical QDs in the presence of an arbitrarily oriented photoexcitation and arbitrarily oriented alternating nonquantizing electric field together with arbitrarily oriented alternating quantizing magnetic fields. (R7.43) Investigate all the problems from R7.1 to R7.42 by removing all the mathematical approximations and establishing the respective appropriate uniqueness conditions.

Reference 1. K. P. Ghatak, S. Bhattacharya, D. De, Einstein Relation in Compound Semiconductors and their nanostructures, Springer Series in Materials Science, Vol. 116 (Springer-Verlag, Germany, 2008) and the references cited therein.

Chapter 8

Photoemission from Quantum Confined Effective Mass Superlattices of Optoelectronic Materials

8.1 Introduction In Chapter 4, photoemission was studied from SLs having various band structures, assuming that the band structures of the constituent materials are invariant quantities in the presence of external photo-excitation. In this chapter, this assumption has been removed, and in Section 8.2.1, an attempt is made to study the magneto-photoemission from QW effective mass SL of optoelectronic materials. In Section 8.2.2, photoemission from effective mass QWW SLs of optoelectronic materials is investigated; and in Section 8.2.3, the magneto-photoemission from effective mass QD SLs of optoelectronic materials is studied. Section 8.2.4 explores magneto-photoemission from effective mass SLs of optoelectronic materials. The Sections 8.3 and 8.4 contain respectively the result and discussions and open research problems pertinent to this chapter.

8.2 Theoretical Background 8.2.1 Magneto-Photoemission from Quantum Well Effective Mass Superlattices The electron energy spectrum in this case can be expressed following [1] as:  kx2

=

 2 1  −1  2 − k⊥ , cos f30 E, ky , kz L02

(8.1)

where      f30 E, ky , kz = a1 cos a0 g30 (E, k⊥ ) + b0 h30 (E, k⊥ )   −a2 cos a0 g30 (E, k⊥ ) − b0 h30 (E, k⊥ ) , 1/2  ∗  2m1  2 β0 E, λ, Eg01 , 1 − k⊥ , g30 (E, k⊥ ) = 2 271 K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_8,  C Springer Science+Business Media, LLC 2009

272

8

Quantum Confined Effective Mass Superlattices

       β0 E, λ, Eg0i , i = γi E, Eg0i , i − θi E, Eg0i , i , i = 1, 2,

   2   E E + Eg0i E + Eg0i + i Eg0i + 3 i

γi E, Eg0i , i = ,   Eg0i Eg0i + i E + Eg0i + 23 i ⎡ ⎤  

2   I0 λ2 .Eg0i Eg0i + i βi2 |e|2 ρ 1 i ⎦

× . θi E, Eg0i ,i =⎣ ti + √ 4 φ0i (E) 96mri πc3 √ε0 εsci Eg + 2 i 2 0i 3    1/2     Eg0i − δi 1 1   + Eg0i − δi 1+  − φ0i (E) + δi Eg0i + δi φ0i (E)+δi ⎫ 1/2 2  

−1 ⎬ Eg0i + δi 1 1 1 × − , m = , + ri  2 ⎭ φ0i (E) + δi m∗i mυi Eg − δ  0i i

 ⎡  ⎤1/2 2

6 Eg0i + 3 i Eg0i + i ⎦ , χi = 6Eg2 +9Eg0i i +42i , βi = ⎣ 0i χi

⎡

2⎤1/2  1/2 6 Eg0i + 23 i 42i ⎢ ⎥ ti = ⎣ , ⎦ , ρi ≡ χi 3χi     1/2 2  E γ0 E, Eg0i , i i g 0i φ0i (E) = Eg0i , δi = , and Eg0i χi 1/2  ∗  2m2  2 β0 E, λ, Eg02 ,2 − k⊥ . h30 (E, k⊥ ) = 2 



m∗ 1+2 1+ i mυi



When the unperturbed bulk dispersion law of the constituent materials is definedby  E, λ, Eg0i , i is the two-band model of Kane, (8.1) remains as it is, except that β 0   replaced by τ0 E, λ, Eg0i , where   

  I0 λ2 .Eg0i Eg0i |e|2 1 τ0 E, λ, Eg0i = E (1 + α0i E) − . 1 + √ ε0 εsci Ui (E) Ui (E) 384mri πc3  2 1 1 1 +Eg0i , α0i = − , and Ui (E) Ui (E) Eg0i Eg0i 1/2  ∗

2mi . = Eg0i 1+ α0i E [1+α0i E] mr When the unperturbed bulk dispersion law of the constituent materials is defined by  E, λ, E should ,  parabolic energy bands, (8.1) remains as it is except that β 0 g i 0i   be replaced by ρ0 E, λ, Eg0i , where   −3/2 ∗

  2mi |e|2 I0 λ2 E . α ρ0 E, λ, Eg0i = E − 1 + √ 0i mri 96mri πc3 ε0 εsci

8.2

Theoretical Background

273

In the presence of a quantizing magnetic field along the x-direction, the electron dispersion relation in quantum well effective mass superlattices is given by:

π nx dx

2 =

2 2 |e| B  1  −1 1 E = E f − n + , cos n (E, ) 30 x 30  2 L02

(8.2)

where E30 is the totally quantized energy,      f30 (E, n) = a1 cos a0 g30 (E, n) + b0 h30 (E, n) − a2 cos a0 g30 (E, n) − b0 h30 (E, n)  g30 (E, n) =  h30 (E, n) =

   2 |e| B 1 1/2 E, λ, E − β , , and n + 0 g01 1  2 2

2m∗1

   2 |e| B 1 1/2 E, λ, E − β , . n + 0 g 2 02  2 2

2m∗2

The electron concentration is given by: n0L =

nmax n xmax   gv eB  F−1 η30,1 , π

(8.3)

n=0 nx =1

where η30,1 =

EFBQWSLEM − E30 and EFBQWSLEM kB T

is the Fermi energy in this case. The photoelectric current density is given by: JL =

gv e2 Bα0 2π2 L0 dx

n xmax max n

    F−1 η30,1 L30 Enx ,0 ,

n=0 nxmin

where      −1/2 2 Enx , 0 , L30 Enx , 0 = I30 Enx , 0 1 − f30             sin a0 g30 Enx , 0 I30 Enx , 0 = a2 a0 g30 Enx , 0 − b0 h30 Enx , 0      −b0 h30 Enx , 0 − a1 a0 g30 Enx , 0 +b0 h30 g30











Enx , 0

Enx , 0 = g30 Enx , 0

 

−1







sin a0 g30 Enx , 0 + b0 h30



m∗1 2



β0



  Enx , 0 ,





Enx , λ, Eg01 , 1



,

(8.4)

274

8

Quantum Confined Effective Mass Superlattices

  ∗   1/2     2m1  E g30 Enx , 0 = β , λ, E ,  , β0 Enx , λ, Eg0i , i 0 nx g01 1 2       = β0 E, λ, Eg0i , i E = Enx , β0 E, λ, Eg0i , i   1   1 + = γi E, Eg0i , i E E + Eg0i    1 1 + − + θi E, Eg0i , i 2 E + Eg0i + i E + Eg0i + 3 i      (E) 2ψi E, Eg0i , i φ0i   + , − φ0i (E) ψi E,Eg0i ,i ψi

   −1/2   (E) δ  − E φ0i 1 1 g0i i E, Eg0i , i =  − 2 +    2 φ0i (E) + δi Eg0i + δi 2 φ0i (E) + δi  1/2   Eg0i + δi 1  ×  −  2 φ0i (E) + δi Eg0i − δi   −1/2  Eg0i + δi 1 − +  2 φ0i (E) + δi Eg0i − δi  1/2 ⎤⎤   1 1 ⎦⎦ , ψi E,Eg0i ,i  −  φ0i (E) + δi Eg0i + δi     1/2     Eg0i − δi 1 1   = 1+  − δ − + E g i 0i φ0i (E)+δi Eg0i +δi φ0i (E) + δi ⎫   1/2 ⎬   Eg0i + δi 1 , −   2 ⎭ φ0i (E) + δi Eg0i − δi  ∗

       −1 m2  E h30 Enx ,0 = h30 Enx , 0 , λ, E ,  β , nx g02 2 0 2







h30

 ∗   1/2  2m2  Enx ,0 = E β , λ, E ,  , 0 n g 2 x 02 2





      f30 (E, 0) = a1 cos a0 g30 (E, 0) + b0 h30 (E, 0) − a2 cos a0 g30 (E, 0) − b0 h30 (E, 0) g230 (E, 0) =

2m∗1 2

   2m∗  β0 E, λ, Eg01 , 1 and h230 (E, 0) = 22 β0 E, λ, Eg02 , 2 . 

Enx should be determined from the equation f30



 L0 π nx Enx , 0 = cos . dx





The nxmin in (8.4) must be determined from the inequality

(8.5)

8.2

Theoretical Background

275

nxmin ≥

dx πL0



  cos−1 f30 (W − hυ, 0) .

(8.6)

When the unperturbed bulk dispersion relation of the constituent materials is defined by the two-band model of Kane, all the pertinent above equations remain    E, λ, E is to be replaced by τ E, λ, E ,  unchanged, except that β 0 g0i i 0 g0i and     β0 E, λ, Eg0i , i should be replaced by τ0 E, λ, Eg0i , where τ0



 |e|2 I0 λ2 Eg30i Ui (E)  E, λ, Eg0i = 1 + 2α0i E +  , in which √ 32mri πc3 ε0 εsci Ui4 (E)  ∗

  −1 mi Eg0i Ui (E) ≡ [Ui (E)] (1 + 2α0i E) . mυi





For the perturbed parabolic bulk dispersion relation materials in this    of constituent    E, λ, E ,  be replaced by ρ E, λ, E and β case, β0 E, λ, Eg0i , i should 0 g0i g0i i 0   should be replaced by ρ0 E, λ, Eg0i , where 

ρ0 E, λ, Eg0i







|e|2 I0 λ2 m∗i α0i = 1+ √ 32m2ri πc3 ε0 εsci



1+

2m∗i mri

5/2 

α0i E

.

8.2.2 Photoemission from Effective Mass Quantum Well Wire Superlattices The electron dispersion relation in this case is given by:  kx2

=



 2 πny 2 1  −1  π nz 2 − − cos f31 E, ny , nz , dy dz L02

(8.7)

where         − a2 cos f31 E, ny , nz =  a1 cos a0g31 E, n y , nz + b0 h31 E, ny , nz a0 g31 E, ny , nz − b0 h31 E, ny , nz ,  

1/2    2m∗1  π ny 2 π nz 2 g31 E, ny , nz = β0 E, λ, Eg01 , 1 − + , and dy dz 2 



h31 E, ny , nz =



 2m∗2  β0 E, λ, Eg02 , 2 − 2 



π ny dy



2 +

π nz dz

2 1/2 .

The subband energy E31 is given by: 

2

2  2  π n 1  −1  π n y z + cos f31 E, ny , nz E = E31 = . dy dz L02

(8.8)

276

8

Quantum Confined Effective Mass Superlattices

The photo-emitted current is given by:

I1L

nymax nzmax ∞  e 2 1  ∂kx ∂E = α0 gv . . . f (E) dE, where E¯ 31 = E31 + W − hυ. 2 π  ∂E ∂x ny =1 nz =1 ¯ E31

Therefore, I1L =

α0 gv ekB T π

n ymax nzmax 

F0 (η31 ) ,

(8.9)

ny =1 nz =1

where η31 =

EFQWWSLEM − (E31 + W − hυ) , kB T

in which EFQWWSLEM is the Fermi energy in the present case. The electron concentration per unit length is given by: n01L

nymax nzmax      2gv  K31 EFQWWSLEM , ny , nz + K32 EFQWWSLEM , ny , nz , = π ny =1 nz =1

(8.10) where 



K31 EFQWWSLEM , ny , nz =



2 1  −1  cos f31 EFQWWSLEM , ny , nz − 2 L0

1/2 πnz 2 − , dz



πny dy

2

s       Zr, Y K31 EFQWWSLEM , ny , nz , and Y = QWWSLEM. K32 EFQWWSLEM , ny , nz = r=1

energy For the perturbed  two-band model of Kane and that of  the parabolic   band, β0 E, λ, Eg0i , i should be replaced by τ0 E, λ, Eg0i and ρ0 E, λ, Eg0i , respectively. The basic forms of (8.9) and (8.10) remain unchanged.

8.2.3 Photoemission from Quantum Dots of Effective Mass Superlattices The electron energy spectrum in this case is given by:

π nx dx



2 =

 2 π n 2 π n 2  1  −1  y z − cos f31 E, ny , nz E = E32 − , (8.11) dy dz L02

where E32 is the totally quantized electron energy in this case.

8.2

Theoretical Background

277

The electron concentration is given by: N0L

nxmax nymax nzmax   2gv  = F−1 (η32 ) , dx dy dz

(8.12)

nx =1 ny =1 nz =1

where η32 =

EFQDSLEM − E32 , kB T

in which EFQDSLEM is the Fermi energy in the present case. The photoemitted current density is given by: J1L =

2L0 egv α0 dx dy dz 

n xmax nymax nzmax  

  L30 Enx , 0 F−1 (η32 ) ,

(8.13)

nxmin ny =1 nz =1

          2 E , 0 −1/2 , I where L30 Enx , 0 = I30 Enx , 0 1 − f30 nx 30 Enx , 0 , and f30 Enx , 0 are defined in connection with (8.4) and nxmin ≥

dx πL0

  cos−1 {f30 (W − hυ,0)} .

(8.14)

Enx should be determined from the equation

    L0 nx π f30 Enx ,0 = cos . dx

(8.15)

8.2.4 Magneto-Photoemission from Effective Mass Superlattices The electron dispersion law in this case assumes the form:  kx2

=

 2 2 |e| B 1  −1 1 n+ . cos {f30 (E, n)} −  2 L02

(8.16)

The Landau energy level E33 can be expressed as:

 $2 2 |e| B 1 1  −1 #  f = n + . cos n) (E, 30 E = E33  2 L02

(8.17)

The photoemitted current density is given by: JML =

α0 e2 BkB T 2π 2 2

n max n=0

F0 (η33 ),

(8.18)

278

8

Quantum Confined Effective Mass Superlattices

when η33 =

EFBSLEM − (E33 + W − hυ) , kB T

in which EFBSLEM is the Fermi energy in the present case. The electron concentration is given by: n0ML =

nmax gv eB  [K33 (EFBSLEM , n) + K34 (EFBSLEM , n)], π 2

(8.19)

n=0

where K33 (EFBSLEM , n) = K34 (EFBSLEM , n) =



1/2 2  2 |e| B 1  −1 1 , n+ cos f30 (EFBSLEM , n) −  2 L02

s 

Zr,Y [K33 (EFBSLEM , n)], and Y = BSLEM.

r=1

8.3 Results and Discussions Using (8.4) and (8.3), the normalized photoemitted current density from QW HgTe/Hg1–x Cdx Te effective mass SL has been plotted as a function of the inverse quantizing magnetic field as shown in plot (a) of Fig. 8.1, whose constituent materials obey the perturbed three-band model of Kane in the presence of external photo-excitation. The curves (b) and (c) of the same figure have been drawn for the perturbed two-band model of Kane and that of the perturbed parabolic energy bands, respectively. The curves (d), (e), and (f) in the same figure exhibit the corresponding plots of QW Inx Ga1–x As/InP effective mass SL. Figures 8.2, 8.3, 8.4, and 8.5 show the variations of the normalized photoemitted current density from the said SLs as functions of normalized electron degeneracy, normalized intensity, wavelength, and thickness, respectively, for all the cases of Fig. 8.1. Using (8.10) and (8.9), the normalized photocurrent from QWW effective mass HgTe/Hg1–x Cdx Te SL as a function of film thickness has been depicted in plot (a) of Fig. 8.6, whose constituent materials obey the perturbed three-band model of Kane in the presence of external light waves. The curves (b) and (c) of the same figure have been drawn for the perturbed two-band model of Kane and the perturbed parabolic energy bands, respectively. The curves (d), (e), and (f) in the same figure exhibit the corresponding plots of Inx Ga1–x As/InP QWW effective mass SL. Figures 8.7, 8.8, 8.9, and 8.10 exhibit the plots of the normalized photoemitted current as functions of normalized carrier concentration, normalized intensity, wavelength, and normalized incident photon energy, respectively, for all the cases of Fig. 8.6. Using (8.13) and (8.12), the normalized photoemitted current density from HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP effective mass QD SLs, respectively, has

8.3

Results and Discussions

279

Fig. 8.1 Plot of the normalized photoemission current density from QW effective mass superlattices of HgTe/Hg1–x Cdx Te as a function of an inverse magnetic field, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively. The curves (d), (e), and (f) exhibit the corresponding plots of Inx Ga1–x As/InP

Fig. 8.2 Plot of the normalized photoemission from QW effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as a function of normalized electron degeneracy for all cases of Fig. 8.1

280

8

Quantum Confined Effective Mass Superlattices

Fig. 8.3 Plot of the normalized photoemission current density from QW effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as a function of normalized light intensity for all cases of Fig. 8.1

Fig. 8.4 Plot of the normalized photoemission current density from QW effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as a function of light wavelength for all cases of Fig. 8.1

8.3

Results and Discussions

281

Fig. 8.5 Plot of the normalized photoemission current density from QW effective mass superlattices of HgTe/Hg1–x Cdx Te as a function of film thickness, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively. The curves (d), (e), and (f) exhibit the corresponding plots of Inx Ga1–x As/InP

Fig. 8.6 Plot of the normalized photocurrent from quantum well wire effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as function of film thickness for all cases of Fig. 8.5

282

8

Quantum Confined Effective Mass Superlattices

Fig. 8.7 Plot of the normalized photocurrent from quantum well wire effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as function of normalized electron degeneracy for all cases of Fig. 8.5

Fig. 8.8 Plot of the normalized photocurrent from quantum well wire effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as function of normalized light intensity for all cases of Fig. 8.5

8.3

Results and Discussions

283

Fig. 8.9 Plot of the normalized photocurrent from quantum well wire effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as function of light wavelength for all cases of Fig. 8.5

Fig. 8.10 Plot of the normalized photocurrent as function of normalized incident photon energy from quantum well wire effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP for all cases of Fig. 8.5

284

8

Quantum Confined Effective Mass Superlattices

been plotted for all types of band models as a function of film thickness, as shown in Fig. 8.11. Figures 8.12, 8.13, 8.14, and 8.15 exhibit the plots of normalized photoemitted current density from the said SLs as functions of normalized electron degeneracy, normalized intensity, wavelength, and normalized incident photon energy, respectively, for all cases of Fig. 8.11. Using (8.19) and (8.18), the normalized photoemitted current density from effective mass HgTe/Hg1–x Cdx Te SL under magnetic quantization has been plotted as a function of the quantizing inverse magnetic field, as shown in plot (a) of Fig. 8.16, whose constituent materials obey the perturbed three-band model of Kane in the presence of external photoexcitation. The curves (b) and (c) of the same figure have been drawn for the perturbed two-band model of Kane and the perturbed parabolic energy bands, respectively. The curves (d), (e), and (f) in the same figure exhibit the corresponding plots of Inx Ga1–x As/InP SL. Figures 8.17, 8.18, 8.19, and 8.20 exhibit the said variation in this case as functions of normalized electron degeneracy, normalized intensity, wavelength, and normalized incident photon energy, respectively, for all the cases of Fig. 8.16. It appears from Fig. 8.1 that the normalized photoemitted current density from QW effective mass HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP SLs oscillates with the inverse quantizing magnetic field due to the SdH effect, where the oscillatory amplitudes and the numerical values are determined by the respective energy band constants. From Fig. 8.2, it appears that the photoemitted current density increases with increasing carrier concentration in an oscillatory way. Figures 8.3 and 8.4 show that the photoemitted current density decreases with increasing intensity and wavelength in different manners. From Fig. 8.5, it appears that the normalized photoemitted current density from QW effective mass HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP SLs decreases with increasing film thickness in an oscillatory manner with different numerical values as specified by the energy band constants of the aforementioned SLs. From Fig. 8.6, it appears that the normalized photoemission from QWW effective mass HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP SLs increases with decreasing thickness and exhibits large oscillations. From Fig. 8.7, it appears that the normalized photocurrent for the said system increases with increasing carrier concentration, exhibiting a quantum jump for a particular value of the said variable for all the models of both the SLs. From Figs. 8.8 and 8.9, it can be inferred that the normalized photocurrent in this case increases with decreasing intensity and wavelength in different manners. From Fig. 8.10, we can write that the normalized photocurrent from QWW effective mass HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP SLs increases with increasing normalized incident photon energy and exhibits quantum steps for specific values of the said variable. From Fig. 8.11, it appears that photoemitted current density from QD effective mass HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP SLs exhibits the same type of variations as given in Figs. 8.5 and 8.6, respectively, although the physics of QD effective mass SLs is completely different as compared with the magneto quantum well effective mass SLs and quantum wire effective mass SLs, respectively. The different physical phenomena in the former as compared with the latter two cases yield different numerical values of photoemission and different thicknesses, respectively,

8.3

Results and Discussions

285

Fig. 8.11 Plot of the normalized photoemission current density from quantum dot effective mass superlattices of HgTe/Hg1–x Cdx Te as function of film thickness, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively. The curves (d), (e), and (f) exhibit the corresponding plots of Inx Ga1–x As/InP

Fig. 8.12 Plot of the normalized photoemission current density from quantum dot effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as function of normalized electron degeneracy for all cases of Fig. 8.11

286

8

Quantum Confined Effective Mass Superlattices

Fig. 8.13 Plot of the normalized photoemission current density from quantum dot effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as function of normalized light intensity for all cases of Fig. 8.11

Fig. 8.14 Plot of the normalized photoemission current density from quantum dot effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as function of light wavelength for all cases of Fig. 8.11

8.3

Results and Discussions

287

Fig. 8.15 Plot of the normalized photoemission current density from quantum dot effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as function of normalized incident photon energy for all cases of Fig. 8.11

Fig. 8.16 Plot of the normalized photoemission current density from effective mass superlattices of HgTe/Hg1–x Cdx Te as function of inverse magnetic field and in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively. The curves (d), (e), and (f) exhibit the corresponding plots of Inx Ga1–x As/InP

288

8

Quantum Confined Effective Mass Superlattices

Fig. 8.17 Plot of the normalized magneto photoemission current density from effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as function of normalized electron degeneracy for all cases of Fig. 8.16

Fig. 8.18 Plot of the normalized magneto photoemission current density from effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as function of normalized light intensity for all cases of Fig. 8.16

8.3

Results and Discussions

289

Fig. 8.19 Plot of the normalized magneto photoemission current density from effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as function of light wavelength for all cases of Fig. 8.16

Fig. 8.20 Plot of the normalized magneto photoemitted current density from effective mass superlattices of HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP as function of normalized incident photon energy for all cases of Fig. 8.16

290

8

Quantum Confined Effective Mass Superlattices

for exhibiting quantum jump. From Figs. 8.12, 8.13, and 8.14, it appears that the photoemission current density from QD effective mass HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP SLs increases with increasing carrier concentration, decreasing intensity, and decreasing wavelength, respectively, in various manners. Figure 8.15 demonstrates the fact that the photoemission current density from QD effective mass HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP SLs exhibits quantum steps with increasing photon energy for both the cases. Figure 8.16 exhibits the fact that the normalized photoemission current density from effective mass HgTe/Hg1–x Cdx Te and Inx Ga1–x As/InP SLs oscillates with an inverse quantizing magnetic field. Figure 8.17 exhibits the fact that the photoemission in this case increases with increasing carrier concentration. Figures 8.18 and 8.19 demonstrate that photoemission current density decreases with increasing intensity and wavelength in different manners. From Fig. 8.20, we can infer that photoemission exhibits step functional dependence with increasing photon energy for both the SLs in this case with different numerical magnitudes.

8.4 Open Research Problems Investigate the following open research problems in the presence of external photoexcitation which changes the band structure in a fundamental way, together with the proper inclusion of the electron spin, the variation of work function, and the broadening of Landau levels, respectively, for the appropriate problems. (R8.1) Investigate multiphoton photoemission from quantum confined III–V, II– VI, IV–VI, HgTe/CdTe effective mass superlattices together with short period, strained layer, random, Fibonacci, polytype, and sawtooth superlattices in the presence of arbitrarily oriented photo-excitation. (R8.2) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and quantizing magnetic field respectively for all the cases of R8.1. (R8.3) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and nonquantizing nonuniform electric fields, respectively, for all the cases of R8.1. (R8.4) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and nonquantizing alternating electric fields, respectively, for all the cases of R8.1. (R8.5) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and crossed electric and quantizing magnetic fields, respectively, for all the cases of R8.1. (R8.6) Investigate multiphoton photoemission from heavily doped quantum confined superlattices for all the cases of R8.1. (R8.7) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and quantizing magnetic fields, respectively, for all the cases of R8.6.

Reference

291

(R8.8) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and nonquantizing nonuniform electric fields, respectively, for all the cases of R8.6. (R8.9) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and nonquantizing alternating electric fields, respectively, for all the cases of R8.6. (R8.10) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and crossed electric and quantizing magnetic fields, respectively, for all the cases of R8.6. (R8.11) Investigate all the problems from R8.1 to R8.10 by removing all the mathematical approximations and establishing the respective appropriate uniqueness conditions.

Reference 1. K. P. Ghatak, S. Bhattacharya, D. De, Einstein Relation in Compound Semiconductors and their nanostructures, Springer Series in Materials Science, Vol. 116 (Springer-Verlag, Germany, 2008) and the references cited therein.

Chapter 9

Photoemission from Quantum Confined Superlattices of Optoelectronic Materials with Graded Interfaces

9.1 Introduction In Section 9.2.1, magneto-photoemission from QW SL of optoelectronic materials with graded interfaces, the dispersion relation of whose constituent materials obeys (5.1), (5.2), and (5.3), is investigated. In Section 9.2.2, photoemission from QWW SLs of optoelectronic materials with graded interfaces is studied, and in Section 9.2.3, magneto-photoemission from QDs of optoelectronic materials with graded interfaces is studied. Section 9.2.4 explores the magneto-photoemission from SLs of optoelectronic materials with graded interfaces. It is worth remarking that all the said investigations are made in the presence of external photo-excitation. The Sections 9.3 and 9.4 contain, respectively, the result and discussions and open research problems pertinent to this chapter.

9.2 Theoretical Background 9.2.1 Magneto Photoemission from Quantum Well Superlattices The electron energy spectrum in this case can be expressed following [1] as: kz2

   2 1 −1 1 = 2 cos − ks2 , φ91 (E, ks ) 2 L0

(9.1)

where

 φ91 (E, ks ) = 2 cosh {β91 (E, ks )} cos {γ91 (E, ks )} + ε91 (E, ks ) sinh {β91 (E, ks )} sin {γ91 (E, ks )}   2 (E, k ) K91 s + 0 −3K92 (E, ks ) cosh {β91 (E, ks )} sin {γ91 (E, ks )}. K92 (E, ks )   2 (E, k ) K92 s + 3K91 (E, ks ) − K91 (E, ks )

293 K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_9,  C Springer Science+Business Media, LLC 2009

294

9 Quantum Confined Superlattices of Optoelectronic Materials

 sinh {β91 (E, ks )} cos {γ91 (E, ks )}  ) * 2 2 + 0 2 K91 (E, ks ) − K92 (E, ks )

1 + 12



cosh {β91 (E, ks )} cos {γ91 (E, ks )}



2 (E, k ) 2 (E, k ) 5K92 5K91 s s + − 34K91 (E, ks ) K92 (E, ks ) K91 (E, ks ) K92 (E, ks )  sinh {β91 (E, ks )} sin {γ91 (E, ks )} ,

β91 (E, ks ) =K91 (E, ks ) (a0 − 0 ) , K91 (E, ks )    1/ 2 2m∗2  2 , = ks − 2 β0 E − V0 , λ, Eg02 , 2 

.

ks2 = kx2 + ky2 ,   V0 = Eg02 − Eg01  ,γ91 (E, ks ) = K92 (E, ks ) (b0 − 0 ) , K92 (E, ks )  ∗ 1/ 2  2m1  2 β0 E, λ, Eg01 ,1 − ks . = 2 In the presence of a quantizing magnetic field B along the z direction the magneto dispersion relation assumes the form: kz2



  2 1 2 |e| B 1 −1 1 = 2 cos − φ91 (E, n) n+ , 2  2 L0

(9.2)

where φ91 (E, n) = [2 cosh {β91 (E, n)} cos {γ91 (E, n)} + ε91 (E, n) sinh {β91 (E, n)} sin {γ91 (E, n)}   2 (E, n) K91 + 0 − 3K92 (E, n) cosh {β91 (E, n)} sin {γ91 (E, n)} K92 (E, n)    2 (E, n) K92 + 3K91 (E, n) − sinh {β91 (E, n)} cos {γ91 (E, n)} K91 (E, n)  ) * 2 2 + 0 2 K91 (E, n) − K92 (E, n) cosh {β91 (E, n)} cos {γ91 (E, n)}   2 (E, n) 2 (E, n) 5K91 1 5K92 + + − 34K91 (E, n) K92 (E, n) 12 K91 (E, n) K92 (E, n)  sinh {β91 (E, n)} sin {γ91 (E, n)} ,

9.2

Theoretical Background

295

β91 (E, n) =K91 (E, n) (a0 − 0 ) , 

  1/ 2 2m∗  1 2 |e| B , n+ − 22 β0 E − V0 ,λ,Eg02 ,2 K91 (E, n) =  2  γ91 (E, n) =K92 (E, n) (b0 − 0 ) ,

  ∗  2 |e| B 2m1  1 1/ 2 β0 E,λ, Eg01 , 1 − and n+ K92 (E, n) =  2 2   K91 (E, n) K92 (E, n) − . ε91 (E, n) = K92 (E, n) K91 (E, n) In quantum well superlattices with graded interfaces the magneto dispersion law assumes the form:

π nz dz

2



  2   1 2 |e| B 1 −1 1  = 2 cos φ91 (E, n) E=E91 − n+ , 2  2 L0

(9.3)

where E91 is the quantized energy in this case. The electron concentration is given by: n0L =

nzmax nmax  gv eB  F−1 (η91 ), π

(9.4)

nz =1 n=0

where η91 =

EFQWSLGI − E91 , kB T

in which EFQWSLGI is the Fermi energy in this case. The photoemission current density is given by: JL =

nzmax nmax    α0 e2 gv BL0  F−1 (η91 ) ρ91 Enz , 2 π dz n zmin

n=0

where 

ρ91 Enz



   sin L0 πnz       dz     , in which φ91 =    Enz ,0 = φ91 (E,0) E=Enz ,  φ91 Enz ,0 

where    φ91 (E, 0) = 2β91 (E, 0) sinh {β91 (E, 0)} cos {γ91 (E, 0)}  $   − 2γ91 (E, 0) cosh {β91 (E, 0)} sin γ91 (E, 0)

(9.5)

296

9 Quantum Confined Superlattices of Optoelectronic Materials

 + ω91 (E, 0) + 0

 (E, 0) K 2 (E, 0)K  (E, 0) 2K91 (E, 0)K91 − 91 2 92 K92 (E, 0) K92 (E, 0)

cosh {β91 (E,0)} sinh {γ91 (E, 0)}   2 (E, 0) #   K91 + β91 (E, 0) sinh {β91 (E, 0)} − 3K92 (E, 0) K92 (E, 0)  − 3K92 (E, 0)

 sin {γ91 (E, 0)} + γ91 (E, 0) cos {γ91 (E, 0)}  × cosh {β91 (E, 0)} + sinh {β91 (E, 0)} cos {γ91 (E, 0)} 3 K91 (E, 0)

− +

 (E, 0) 2K92 (E, 0) K92 K91 (E, 0) ! 2  (E, 0) K92 (E, 0) K91



2 (E, 0) K91

 2 (E, 0) #  K92  + 3K91 (E, 0) − β91 (E, 0) cosh {β91 (E, 0)} cos {γ91 (E, 0)} K91 (E, 0)   − γ91 (E, 0) sin {γ91 (E, 0)} sinh {β91 (E, 0)}} # $   + 0 [4 K91 (E, 0) K91 (E, 0) − K92 (E, 0) K92 (E, 0) cosh {β91 (E, 0)} cos {γ91 (E, 0)}

2 2  + 2 K91 (E, 0) − K92 (E, 0) {β91 (E, 0) sinh {β91 (E, 0)} cos {γ91 (E, 0)}  − γ91 (E, 0) sin {γ91 (E, 0)} cosh {β91 (E, 0)}}   (E, 0) 2 (E, 0) K  (E, 0) 5K92 1 10K92 (E, 0) K92 91 + − 2 (E, 0) 12 K91 (E, 0) K91

+

 (E, 0) 2 (E, 0) K  (E, 0) 10K91 (E, 0) K91 5K91 92 − 2 (E, 0) K92 (E, 0) K92

  − 34 K91 (E, 0) K92 (E, 0) − 34K91 (E, 0) K92 (E, 0)  2 (E, 0) 2 (E, 0)   5K91 1 5K92 + sinh {β91 (E, 0)} sin {γ91 (E, 0)} + 12 K91 (E, 0) K92 (E, 0)  −34K91 (E, 0) K92 (E, 0)  [β91 (E, 0) cosh {β91 (E, 0)} sin {γ91 (E, 0)}  +γ91 (E, 0) cos {γ91 (E, 0)} sinh {β91 (E, 0)}]]],

   β91 (E, 0) = K91 (E, 0) (a0 − 0 ) , K91 (E, 0) ∗

  m2  −γ2 E − V0, Eg02 , 2 = [K91 (E, 0)]−1 2 

9.2

Theoretical Background

297

     + (V0 − E) γ2 E − V0, Eg02 ,2 + ψ2 E − V0, Eg02 ,2 ,   γ2 E − V0, Eg02 ,2     =γ2 E,Eg02 ,2 | E=E−V0 ,γ2 E, Eg02 ,2     1 1 1 =γ2 E, Eg02 ,2 + − , E + Eg02 E + Eg02 + 2 E + Eg02 + 23 2   ψ2 E − V0, Eg02 ,2      =ψ  E, Eg ,2 E=E−V ψ  E, Eg ,i 2

02

0

i

0i

has been defined in connection with (8.4),    γ91 (E,0) =K92 (E,0) (b0 − 0 ) ,K92 (E, 0) ∗

  m1 β0 E,λ,Eg01 ,1 , = [K92 (E, 0)]−1 2  ω91 (E, 0) = [91 (E, 0) {β91 (E, 0)} sin {γ91 (E, 0)}    + ε91 (E, 0) β91 (E, 0) cosh {β91 (E, 0)} sin {γ91 (E, 0)}  + γ91 (E, 0)

 =

 ε91 (E, 0) sinh {β91 (E, 0)} cos {γ91 (E, 0)} ,91 (E, 0)

 (E, 0)  (E, 0) K91 K91 (E, 0) K92 − 2 (E, 0) K92 (E, 0) K92

 (E, 0) K  (E, 0) K92 (E, 0) K91 − 92 + 2 (E, 0) K91 (E, 0) K91



nzmin and Enz are determined from:   dz 1 cos−1 φ91 (W − hυ,0) L0 π 2     L0 π nz φ91 Enz ,0 = 2 cos , dz

nzmin ≥

in which

where

(9.6) (9.7)

   φ91 Enz ,0 = φ91 (E,0) E=Enz ,

 φ91 (E, 0) = 2 cosh {β91 (E, 0)} cos {γ91 (E, 0)} + ε91 (E, 0) sinh {β91 (E, 0)} sin {γ91 (E, 0)}   2 (E, 0) K91 + 0 − 3K92 (E, 0) cosh {β91 (E, 0)} sin {γ91 (E, 0)} K92 (E, 0)

298

9 Quantum Confined Superlattices of Optoelectronic Materials



K 2 (E, 0) + 3K91 (E,0) − 92 K91 (E, 0)



 × sinh {β91 (E, 0)} cos {γ91 (E, 0)}  ) * 2 2 + 0 2 K91 (E, 0) − K92 (E, 0) cosh {β91 (E, 0)} cos {γ91 (E, 0)}   2 (E, 0) 2 (E, 0) 5K91 1 5K92 + + − 34K91 (E, 0) K92 (E, 0) 12 K91 (E, 0) K92 (E, 0)   sinh {β91 (E, 0)} sin {γ91 (E, 0)} , β91 (E, 0) =K91 (E, 0) (a0 − 0 ) , K91 (E, 0)  ∗   2m2 # = (V0 − E) γ2 E − V0 , Eg02 , 2 2   $ 1/ 2 +ψ2 E − V0 , Eg02 , 2 , γ91 (E, 0) =K92 (E, 0) (b0 − 0 ) ,   ∗  1/ 2 2m1  β0 E, λ, Eg01 , 1 , and K92 (E, 0) = 2   K91 (E, 0) K92 (E, 0) − . ε91 (E,0) = K92 (E, 0) K91 (E, 0)

9.2.2 Photoemission from Quantum Well Wire Superlattices The electron energy spectrum in this case is given by:  kz2 =

2  

2

2    π n 1 π n 1 y x − − φ91 E,nx ,ny cos−1 , 2 dx dy L02

where    #  $ #  $ φ91 E, nx ,ny = 2 cosh β91 E, nx ,ny cos γ91 E, nx ,ny   #  $ #  $ + ε91 E, nx ,ny sinh β91 E, nx ,ny sin γ91 E, nx ,ny     2 E, n ,n   K91 x y   − 3K92 E, nx ,ny + 0 K92 E, nx ,ny #  $ #  $ cosh β91 E, nx ,ny sin γ91 E, nx ,ny

(9.8)

9.2

Theoretical Background

299



  2 E, n ,n  $ # K92 x y   sinh β91 E, nx ,ny + 3K91 E, nx ,ny − K91 E, nx ,ny  #  $ cos γ91 E, nx ,ny  )   2 + 0 2 K91 E, nx ,ny  * #  $ #  $ 2 E, nx ,ny cosh β91 E, nx ,ny cos γ91 E, nx ,ny − K92    2 E,n ,n 1 5K92 x y   + 12 K91 E, nx ,ny    2 E, n ,n     5K91 x y   + − 34K91 E, nx ,ny K92 E, nx ,ny K92 E, nx ,ny  #  $ #  $ sinh β91 E, nx ,ny sin γ91 E, nx ,ny ,       β91 E, nx ,ny =K91 E, nx ,ny (a0 − 0 ) , K91 E, nx ,ny 1/ 2 



 πny 2 2m∗2  πnx 2 + − 2 β0 E − V0 ,λ,Eg02 ,2 , = dx dy        γ91 E, nx ,ny =K92 E, nx ,ny (b0 − 0 ) ,K92 E, nx ,ny 1/ 2 



 πny 2 2m∗1  πnx 2 − + 2 β0 E,λ,Eg01 ,1 , and = − dx dy         K91 E, nx ,ny K92 E, nx ,ny  −   . ε91 E, nx ,ny = K92 E, nx ,ny K91 E, nx ,ny 



The subband energy E92 is given by:   2



  π ny 2 1 π nx 2 −1 1  E, n ,n = + . φ cos 91 x y E=E92 2 dx dy L02

(9.9)

The photoelectric current is given by: I1L

nxmax nymax  gv ekB Tα0  = F0 ( η92 ), π nx =1 ny =1

where 

η92

 EFQWWSLGI − (E92 + W − hυ) = , kB T

(9.10)

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9 Quantum Confined Superlattices of Optoelectronic Materials

in which EFQWWSLGI is the Fermi energy in this case. The electron concentration per unit length is given by: n01L =

nxmax nymax      2gv  K¯ 91 EFQWWSLGI ,nx ,ny + K¯ 92 EFQWWSLGI ,nx ,ny , (9.11) π nx =1 ny =1

where 



K¯ 91 EFQWWSLGI ,nx ,ny =

and



     2 1 −1 1 E ,n ,n φ cos 91 FQWWSLGI x y 2 L02

1/ 2 π ny 2 π nx 2 − − dx dy

s0       K¯ 92 EFQWWSLGI ,nx ,ny = Zr K¯ 91 EFQWWSLGI ,nx ,ny . r=1

9.2.3 Photoemission from Quantum Dot Superlattices The electron energy spectrum in this case is given by:   

 2

2

2     π n π nz 2 1 π n y x −1 1 = − φ91 E, nx ,ny E=E93 − cos , dz 2 dx dy L02 (9.12) where E93 is the totally quantized energy in this case. The electron concentration is given by: N0L

nxmax nymax nzmax   2gv  = F−1 (η93 ), dx dy dz

(9.13)

nx =1 ny =1 nz =1

where η93 =

EFQDSLGI − E93 , kB T

in which EFQDSLGI is the Fermi energy in this case. The photoelectric current density is given by: J1L

nxmax nymax nzmax     2eL0 gv α0  = F−1 (η93 ) ρ91 Enz , dx dy dz  n nx =1 ny =1

(9.14)

zmin

  where ρ91 Enz is derived in the definition of (9.5), and all the discussions after (9.5) is also totally valid as it is in this very particular case.

9.3

Results and Discussions

301

9.2.4 Magneto-Photoemission from Superlattices of III-V Optoelectronic Materials The dispersion relation in this case is given by (9.2), and the Landau level energy E94 is given by:

  2   2 |e| B 1 1 −1 1  φ91 (E,n) E=E94 = n+ . cos 2  2 L02

(9.15)

The photoemission current density is given by: JML =

α0 e2 BkB T 2π 2 2

n max

F0 (η94 ),

(9.16)

n=0

where η94 =

EFBSLGI − (E94 + W − hυ) , kB T

in which EFBSLGI is the Fermi energy in this case. The electron concentration is given by: n0ML =

nmax   gv eB  K¯ 93 (EFBSLGI ,n) + K¯ 94 (EFBSLGI ,n) , 2 π 

(9.17)

n=0

where  K¯ 93 (EFBSLGI ,n) = K¯ 94 (EFBSLGI ,n) =

2

1/ 2   1 2 |e| B 1 −1 1 − and φ91 (EFBSLGI ,n) n+ cos 2  2 L02

s0 

  Zr K¯ 93 (EFBSLGI ,n) .

r=1

9.3 Results and Discussions Using (9.4) and (9.5), the plot of the normalized magneto photoemitted current density from QW HgTe/Hg1–x Cdx Te SL with graded interfaces as a function of inverse quantizing magnetic field is shown in plot (a) of Fig. 9.1, whose constituent materials obey the perturbed three-band model of Kane in the presence of external light waves. The curves (b) and (c) of the same figure refer to the perturbed two-band model of Kane and that of the perturbed parabolic energy bands, respectively. The plots (d), (e), and (f) of the same figure exhibit the said dependences for Inx Ga1−x

302

9 Quantum Confined Superlattices of Optoelectronic Materials

Fig. 9.1 Plot of the normalized photoemission current density from QW superlattices of graded interface of HgTe/Hg1−x Cdx Te as a function of an inverse magnetic field, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively. The curves (d), (e), and (f) exhibit the corresponding plots of Inx Ga1−x As/InP

As/InP QW SL with graded interfaces. Figures 9.2, 9.3, 9.4, and 9.5 show the dependences of the normalized magneto photoemitted current density from the said SLs as functions of normalized electron degeneracy, normalized intensity, wavelength, and normalized incident photon energy,respectively, for all the cases of Fig. 9.1. Using (9.11) and (9.10), the normalized photocurrent from QWW HgTe/Hg1–x Cdx Te SL with graded interfaces as a function of film thickness is shown in plot (a) of Fig. 9.6, whose constituent materials obey the perturbed three-band model of Kane in the presence of photo-excitation. The curves (b) and (c) of the same figure exhibit the same dependence for the perturbed two-band model of Kane and the perturbed parabolic energy bands, respectively. The curves (d), (e), and (f) in the same figure are valid for Inx Ga1−x As/InP QWW SL with graded interfaces in the present case. The Figs. 9.7, 9.8, 9.9, and 9.10 refer to the plots of the normalized photoemitted current as functions of normalized electron degeneracy, normalized intensity, wavelength, and normalized incident photon energy respectively, for all the cases of Fig. 9.6. Using (9.14) and (9.13), the normalized photoemitted current density from HgTe/Hg1−x Cdx Te and Inx Ga1–x As/InP QD SLs with graded interfaces, respectively, is shown in Fig. 9.11 for all types of band models as a function of film thickness. Additionally, Figs. 9.12, 9.13, 9.14, and 9.15 further exhibit the plots of

9.3

Results and Discussions

303

Fig. 9.2 Plot of the normalized photoemission current density from QW superlattices of graded interface of HgTe/Hg1−x Cdx Te and Inx Ga1−x As/InP as a function of normalized electron degeneracy for all cases of Fig. 9.1

Fig. 9.3 Plot of the normalized photoemission current density from QW superlattices of graded interface of HgTe/Hg1−x Cdx Te and Inx Ga1−x As/InP as a function of normalized light intensity for all cases of Fig. 9.1

304

9 Quantum Confined Superlattices of Optoelectronic Materials

Fig. 9.4 Plot of the normalized photoemission current density from QW superlattices of graded interface of HgTe/Hg1−x Cdx Te and Inx Ga1−x As/InP as a function of light wavelength for all cases of Fig. 9.1

Fig. 9.5 Plot of the normalized photocurrent from QW superlattices of graded interface of HgTe/Hg1−x Cdx Te and Inx Ga1−x As/InP as a function of normalized incident photon energy for all the cases of Fig. 9.1

9.3

Results and Discussions

305

Fig. 9.6 Plot of the normalized photocurrent from QWW superlattices of graded interface of HgTe/Hg1–x Cdx Te as a function of film thickness, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively. The curves (d), (e), and (f) exhibit the corresponding plots of Inx Ga1−x As/InP

Fig. 9.7 Plot of the normalized photocurrent from QWW superlattices of graded interface of HgTe/Hg1−x Cdx Te and Inx Ga1−x As/InP as a function of normalized electron degeneracy for all cases of Fig. 9.6

306

9 Quantum Confined Superlattices of Optoelectronic Materials

Fig. 9.8 Plot of the normalized photocurrent from QWW superlattices of graded interface of HgTe/Hg1−x Cdx Te and Inx Ga1− x As/InP as a function of normalized light intensity for all cases of Fig. 9.6

Fig. 9.9 Plot of the normalized photocurrent from QWW superlattices of graded interface of HgTe/Hg1−x Cdx Te and Inx Ga1−x As/InP as a function of light wavelength for all cases of Fig. 9.6

9.3

Results and Discussions

307

Fig. 9.10 Plot of the normalized photocurrent from QWW superlattices of graded interface of HgTe/Hg1−x Cdx Te and Inx Ga1−x As/InP as a function of normalized incident photon energy for all cases of Fig. 9.6

Fig. 9.11 Plot of the normalized photocurrent density from quantum dot superlattices of graded interface of HgTe/Hg1–x Cdx Te as a function of film thickness, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively. The curves (d), (e), and (f) exhibit the corresponding plots of Inx Ga1−x As/InP

308

9 Quantum Confined Superlattices of Optoelectronic Materials

Fig. 9.12 Plot of the normalized photocurrent density from quantum dot superlattices of graded interface of HgTe/Hg1–x Cdx Te and Inx Ga1−x As/InP as a function of normalized electron degeneracy for all cases of Fig. 9.11

Fig. 9.13 Plot of the normalized photocurrent density from quantum dot superlattices of graded interface of HgTe/Hg1–x Cdx Te and Inx Ga1−x As/InP as a function of normalized light intensity for all cases of Fig. 9.11

9.3

Results and Discussions

309

Fig. 9.14 Plot of the normalized photocurrent density from quantum dot superlattices of graded interface of HgTe/Hg1–x Cdx Te and Inx Ga1−x As/InP as a function of light wavelength for all cases of Fig. 9.11

Fig. 9.15 Plot of the normalized photocurrent density from quantum dot superlattices of graded interface of HgTe/Hg1–x Cdx Te and Inx Ga1−x As/InP as a function of normalized incident photon energy for all cases of Fig. 9.11

310

9 Quantum Confined Superlattices of Optoelectronic Materials

Fig. 9.16 Plot of the normalized photoemission current density from superlattices of graded interface of HgTe/Hg1–x Cdx Te as function of inverse magnetic field, in which the curves (a), (b), and (c) represent the perturbed three- and two-band models of Kane, together with parabolic energy bands, respectively. The curves (d), (e), and (f) exhibit the corresponding plots of Inx Ga1−x As/InP

normalized photoemitted current density from the said SLs as functions of normalized carrier concentration, normalized intensity, wavelength, and normalized incident photon energy, respectively, for all cases of Fig. 9.11. Using (9.17) and (9.16), the normalized magneto photoemitted current density from HgTe/Hg1–x Cdx Te SL with graded interfaces has been shown as a function of an inverse quantizing magnetic field, as seen in plot (a) of Fig. 9.16 in the present case. The curves (b) and (c) of the same figure have been drawn for the perturbed two-band model of Kane and the perturbed parabolic energy bands, respectively. The curves (d), (e), and (f) of Fig. 9.16 exhibit the corresponding plots of Inx Ga1−x As/InP SL. Figures 9.17, 9.18, 9.19, and 9.20 show the said variation in this case as functions of normalized carrier concentration, normalized intensity, wavelength, and normalized incident photon energy, respectively, for all the cases of Fig. 9.16. From Fig. 9.1, it appears that the normalized photoemitted current density from QW HgTe/Hg1–x Cd x Te and In x Ga1−x As/InP SLs with graded interfaces oscillates with the inverse quantizing magnetic field due to the SdH effect, where the oscillatory amplitudes and the numerical values are determined by the respective constants of the electron energy spectrum. From Fig. 9.2, it appears that photoemission increases with increasing carrier degeneracy; and for particular values of the same, the quantum jump is observed in Fig. 9.2, where the quantum numbers changes from one fixed set to another allowable set which is the direct signature of the quantization effect. From Figs. 9.3 and 9.4, it is observed that the

9.3

Results and Discussions

311

Fig. 9.17 Plot of the normalized photoemission current density from superlattices of graded interface of HgTe/Hg1–x Cdx Te and Inx Ga1−x As/InP as a function of normalized electron degeneracy for all cases of Fig. 9.16

Fig. 9.18 Plot of the normalized photoemission current density from superlattices of graded interface of HgTe/Hg1–x Cdx Te and Inx Ga1−x As/InP as a function of normalized light intensity for all cases of Fig. 9.16

312

9 Quantum Confined Superlattices of Optoelectronic Materials

Fig. 9.19 Plot of the normalized photoemission current density from superlattices of graded interface of HgTe/Hg1–x Cdx Te and Inx Ga1−x As/InP as a function of light wavelength for all cases of Fig. 9.16

Fig. 9.20 Plot of the normalized photocurrent density from superlattices of graded interface of HgTe/Hg1–x Cdx Te and Inx Ga1−x As/InP as a function of normalized incident photon energy for all cases of Fig. 9.16

9.4

Open Research Problems

313

photoemitted current density decreases with increasing intensity and wavelength in altogether different manners. Figure 9.5 exhibits the fact that the photoemission from QW HgTe/Hg1–x Cd x Te and In x Ga1−x As/InP effective mass SLs with graded interfaces increases with increasing photon energy, showing the step functional dependence. Figure 9.6 depicts the fact that the photocurrent from QWW HgTe/Hg1–x Cd x Te and In x Ga1−x As/InP SLs with graded interfaces increases with decreasing thickness and exhibits large oscillations. From Fig. 9.7, it has been observed that the photocurrent for the said systems increases with increasing electron degeneracy, exhibiting a quantum jump for a particular value of the same for all the models of both the SLs. From Figs. 9.8 and 9.9, it appears that the photocurrent for the present system increases with decreasing intensity and wavelength in different manners. From Fig. 9.10, it is observed that the photocurrent from QWW HgTe/Hg1–x Cd x Te and In x Ga1−x As/InP SLs with graded interfaces increases with increasing photon energy in step-like fashion. From Fig. 9.11 it appears that the photoemission from QD HgTe/Hg1–x Cd x Te and In x Ga1−x As/InP SLs with graded interfaces increases with decreasing film thickness, with larger oscillations again exhibiting the quantization effect. From Fig. 9.12, it is observed that the photoemission for the present system increases with increasing carrier degeneracy exhibiting quantum jumps for a particular value depending on the energy band constants. From Figs. 9.13 and 9.14, it appears that the photoemission for the said systems increases with decreasing intensity and decreasing wavelength, respectively, in various manners. Figure 9.15 demonstrates the fact that the photocurrent density for the present system exhibits quantum steps with increasing photon energy for both the cases. Figure 9.16 exhibits the fact that the photoemission from HgTe/Hg 1–x Cd x Te and In x Ga1−x As/InP SLs with graded interfaces oscillates with an inverse quantizing magnetic field. Figure 9.17 exhibits the fact that the photoemission in this case increases with increasing carrier concentration. Figures 9.18 and 9.19 demonstrate that photoemission decreases with increasing intensity and wavelength in different manners. From Fig. 9.20, we can infer that photoemission exhibits step functional dependence with increasing photon energy for both the SLs in this case with different numerical magnitudes. Finally, we can write that although the photoemission from SLs has been investigated in Chapters 4, 8, and 9, still one can easily infer how little is presented and how much more there is yet to be investigated in the research field of photoemission from quantum confined SLs having different band structures, which is a sign of the coexistence of new physics, related mathematics, and the passion for research in this context.

9.4 Open Research Problems Investigate the following open research problems in the presence of external photoexcitation which changes the band structure in a fundamental way, together with the proper inclusion of the electron spin, the variation of work function, and the broadening of Landau levels respectively for appropriate problems.

314

9 Quantum Confined Superlattices of Optoelectronic Materials

(R9.1) Investigate multiphoton photoemission from quantum confined III-V, II-VI, IV-VI, HgTe/CdTe superlattices with graded interfaces together with short period, strained layer, random, Fibonacci, polytype, and sawtooth superlattices in this context, in the presence of arbitrarily oriented photoexcitation. (R9.2) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and quantizing magnetic fields, respectively, for all the cases of R9.1. (R9.3) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and nonquantizing nonuniform electric fields, respectively, for all the cases of R9.1. (R9.4) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and nonquantizing alternating electric fields, respectively, for all the cases of R9.1. (R9.5) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and crossed electric and quantizing magnetic fields, respectively, for all the cases of R9.1. (R9.6) Investigate multiphoton photoemission from heavily doped quantum confined superlattices for all the cases of R9.1. (R9.7) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and quantizing magnetic fields, respectively, for all the cases of R9.6. (R9.8) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and nonquantizing nonuniform electric fields, respectively, for all the cases of R9.6. (R9.9) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and nonquantizing alternating electric fields, respectively, for all the cases of R9.6. (R9.10) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and crossed electric and quantizing magnetic fields, respectively, for all the cases of R9.6. (R9.11) Investigate all the problems from R9.1 to R9.10 by removing all the mathematical approximations and establishing the respective appropriate uniqueness conditions.

Reference 1 K. P. Ghatak, S. Bhattacharya, D. De, Einstein Relation in Compound Semiconductors and their Nanostructures, Springer Series in Materials Science, Vol. 116 (Springer-Verlag, Germany, 2008) and the references cited therein.

Chapter 10

Review of Experimental Results

10.1 Experimental Works In recent years, there has been considerable interest in investigating photoemission from solids and metal surfaces [1], photoelectron spectroscopy [2], internal photoemission spectroscopy [3], solid state photoemission [4], photoemission spectroscopy measured on Ge nanodots [5], photoemission from self-assembled silver nanocrystals [6], UV photoemission spectroscopy of single-walled carbon nanotubes on Si substrates [7], multiphoton photoemission of self-assembled silver nanocrystals [8], nonlinear photoemission effects in III–V semiconductors [9], Xray photoemission electron microscopy investigation of magnetic thin film antidote arrays [10], optical properties of CdS nanocrystalline semiconductors [11], etc. It appears from the literature that photoemission from quantized structures have been relatively less investigated. In fact, R. Houdré et al. reported for the first time regarding photoemission experiments from superlattice and a single quantum well as functions of excitation energy [12]. In this book, we have discussed in detail photoemission from quantum confined nonlinear optical, III–V, II–VI, GaP, Ge, PtSb2 , zero-gap, stressed, Bismuth, GaSb, IV–VI, Pb1–x Gex Te, Graphite, Tellurium, II–V, Cadmium and Zinc diphosphides, Bi2 Te3 , Antimony, III–V, II–VI, IV–VI, and HgTe/CdTe quantum well superlattices with graded interfaces under magnetic quantization, III–V, II–VI, IV–VI, and HgTe/CdTe effective mass superlattices under magnetic quantization, the QDs of the aforementioned superlattices, quantum confined effective mass superlattices, and superlattices of optoelectronic materials with graded interfaces, on the basis of appropriate carrier energy spectra. It is also interesting to note that although we have considered a plethora of materials having different band structures and photoemission from such quantized structures theoretically, the corresponding experimental studies have been relatively less investigated, as noted already. Thus, detailed experimental works are needed for an in-depth study of photoemission from such low-dimensional structures as functions of incident photon energy, film thickness, carrier concentration, and other externally controllable quantities, which in turn will add new physical phenomena in the regime of photoemission spectroscopy of nanostructures and related materials. In this context, we believe that the identification of 315 K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_10,  C Springer Science+Business Media, LLC 2009

316

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Review of Experimental Results

open research problems is one of the biggest problems in research. We hope that physicists will carry out new experimental researches, not only in the said directions, but also in many other interdisciplinary topics, to add new knowledge and concepts in the experimental aspect of this particularly important area of solid state optoelectronics.

10.2 Open Research Problem The single open research problem of this chapter is for skilled experimentalists and requires a huge amount of funding and concentrated effort amalgamated with indepth experimental insight in this context. (R10.1) Investigate experimentally multiphoton photoemission for all the appropriate open research problems of this monograph.

References 1. M. Cardona, L. Ley, Photoemission in Solids I and II, Topics in Applied Physics, Vol. 26, 27 (Springer-Verlag, Germany, 1978); M. L. Glasser, A. Bagchi, Theories of Photoemission from Metal Surfaces, In: Progress in Surface Science Vol. 7, p. 113 (Pergamon Press, USA, 1975). 2. S. Hüfner, Photoelectron Spectroscopy: Principles and Applications (Springer-Verlag, Germany, 2003). 3. V. V. Afanas’ev, Internal Photoemission Spectroscopy: Principles and Applications (Elsevier, The Netherlands, 2007). 4. W. Schattke, M. A. Van Hove, Solid-State Photoemission and Related Methods: Theory and Experiment (Wiley, USA, 2007). 5. I. Matsuda, S. Hasegawa, A. Konchenko, Y. Nakayama, Y. Nakamura, M. Ichikawa, Phys. Rev. B, 73,113311 (2006); V. L. Colvin, A. P. Alivisatos, J. G. Tobin, Phys. Rev. Lett. 66, 2786 (1991). 6. B. Schroeter, K. Komlev, W. Richter, Mat. Sci. Eng. B88, 259 (2002); G. F. Bertsch, N. Van Giai, N. Vinh Mau, Phys. Rev. A 61, 033202 (2000). 7. L. Fleming, M. D. Ulrich, K. Efimenko, J. Genzer, A. S. Y. Chan, T. E. Madey, S. -J. Oh, O. Zhou, J. E. Rowe, Vac. Sci. Technol. B, 22, 2000 (2004); H. Shimoda, B. Gao, X. -P. Tang, A. Kleinhammes, L. Fleming, Y. Wu, O. Zhou, Phys. Rev. Lett. 88, 015502 (2002). 8. M. Maillard, P. Monchicourt, M. P. Pileni, Chem. Phys. Lett. 380,704 (2003). 9. H. Tang, R. K. Alley, H. Aoyagit, J. E. Clendenin, J. C. Frisch, C. L. Garden, E. W. Hoyt, R. E. Kirby, L. A. Klaisner, A. V. Kulikov, C. Y. Prescott, P. J. Saez, D. C. Schultz, J. L. Turner, M. Woods, M. S. Zolotorev, Proc. 1993 IEEE Particle Accelerator Conference. 17–20 May 1993, Washington, DC. 15th IEEE Particle Accelerator Conference, 3036. 10. L. J. Heyderman, F. Nolting, Quitmann, Appl. Phys. Letts. 83, 1797 (2003); R. P. Cowburn, A. O. Adeyeye, J. A. C. Bland, Appl. Phys. Lett. 70, 2309 (1997); L. Torres, L. Lopez-Diaz, J. Iniguez, Appl. Phys. Lett. 73, 3766 (1996); A. Y. Toporov, R. M. Langford, A. K. PetfordLong, Appl. Phys. Lett. 77, 3063 (2000); L. Torres, L. Lopez-Diaz, O. Alejos, J. Iniguez, J. Appl. Phys. 85, 6208 (1999). 11. S. Tiwari, S. Tiwari, Cryst. Res. Technol. 41, 78 (2006). 12. R. Houdré, C. Hermann, G. Lampel, P. M. Frijlink, A. C. Gossard, Phys. Rev. Lett. 55, 734 (1985).

Chapter 11

Conclusion and Future Research

In this book, the photoemission has been investigated from various types of quantum wells in ultrathin films, quantum wires, quantum dots, effective mass superlattices, and superlattices with graded interfaces, under different physical conditions which generate useful information regarding photoemission from various materials and their nanostructures having different band structures. Our investigations are based on the simplified k.p formalism of semiconductor science and do not incorporate the advanced techniques of photon electron interaction mechanisms in this context.

11.1 Open Research Problems Open research problems have been presented to our esteemed readers and we enjoy the pleasure of further presenting the following challenging research problems in this particular area of optoelectronics, which is a sea in itself. (R11.1) Investigate multiphoton photoemission from inversion and accumulation layers for the research problems of Chapter 7 in the presence of surface states and hot electron effects. (R11.2) Investigate multiphoton photoemission from finite multiple quantum wells and wires of negative refractive index, organic, magnetic, heavily doped, disordered, and other advanced optical materials in the presence of arbitrarily oriented photo-excitation. (R11.3) Investigate multiphoton photoemission from wedge shaped, cylindrical, ellipsoidal, conical, triangular, circular, parabolic rotational and parabolic cylindrical QDs of all the appropriate problems of R11.2 in the presence of arbitrary photo-excitation. (R11.4) Investigate multiphoton photoemission in the presence of an arbitrarily oriented photo-excitation and quantizing magnetic field for all the appropriate cases of R11.3. (R11.5) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and alternating nonquantizing electric fields for all the appropriate cases of R11.3. 317 K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_11,  C Springer Science+Business Media, LLC 2009

318

11

Conclusion and Future Research

(R11.6) Investigate multiphoton photoemission in the presence of an arbitrarily oriented photo-excitation and nonuniform nonquantizing electric fields for all the appropriate cases of R11.3. (R11.7) Investigate multiphoton photoemission in the presence of an arbitrarily oriented photo-excitation and crossed electric and quantizing magnetic fields for all the appropriate cases of R11.3. (R11.8) Investigate multiphoton photoemission in the presence of an arbitrarily oriented photo-excitation and quantizing magnetic field for all the appropriate cases of R11.2. (R11.9) Investigate multiphoton photoemission in the presence of arbitrarily oriented photo-excitation and alternating nonquantizing electric fields for all the appropriate cases of R11.2. (R11.10) Investigate multiphoton photoemission in the presence of an arbitrarily oriented photo-excitation and nonuniform nonquantizing electric field for all the cases of R11.2. (R11.11) Investigate multiphoton photoemission in the presence of an arbitrarily oriented photo-excitation and crossed electric and quantizing magnetic field for all the appropriate cases of R11.2. (R11.12) Investigate multiphoton photoemission from all the systems and the open research problems of Chapters 5, 6, 7, 8, 9, and 11 in the presence of many-body effects. (R11.13) (a) Investigate the static photoelectric effect for all the appropriate problems of this monograph. (b) Investigate all the appropriate problems for open quantum dots of different materials whose unperturbed carrier energy spectra are defined in Chapter 3 of this monograph. (c) Investigate all the problems from R11.1 to R11.13 (b) by removing all the mathematical approximations and establishing the respective appropriate uniqueness conditions. Dear readers, we sincerely believe that you not only will solve these open and challenging research problems but also will generate new concepts, both theoretical and experimental. This particular approach will, in turn, transform you into mature scientists bubbling with creativity much more original than ours, although there is no hidebound prescription for creativity. Let us recall our hero Hardy, who in his classic of desperate sadness entitled “A Mathematician’s Apology” (Cambridge University Press, 1990, pp. 37), tells us “in his roll-call of mathematicians: ‘Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty . . . I do not know an instance of a major mathematical advance initiated by a man past fifty’.” Therefore, we greet your appearance in the research screen pertaining to this particular topic in lieu of us. This elementary book is the signature of our combined and continuous effort over the last twenty years of research and we wish to induce the passion for research activity in you, since we eternally like to hope that you are the right person to carry forward the lineage of this subject to escalate greater heights and supersede us. It may be noted that Albert Einstein’s description of the cause of photoelectric effect by absorption of quanta of light (photons)

11.1

Open Research Problems

319

appears in his classic research paper entitled “On a Heuristic Viewpoint Concerning the Production and Transformation of Light” (Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt) (Volume 322 Issue 6, Pages 132–148, Annalen der Physik). His explanation of the features of the photoelectric effect and the characteristic frequency won him the Nobel Prize in 1921. We consider ourselves to be really fortunate to present this monograph as a small token of our contribution as a dedication on the occasion of the completion of the eighty-ninth year of Einstein’s winning the Nobel Prize for the explanation of the aforementioned physical phenomena which in recent years finds extensive applications in modern nanoelectronics together with the fact that it is really amazing to note that the gigantic contributions of the majestic person is like a garden of different but enigmatic scientific flowers such as the theory of relativity, photoelectric effect, etc. to name a few. It has always occurred to us that whenever we were academically very fatigued, we turned our attention to the Einstein’s garden and were always gifted with unexplored scientific flower readily convertible into a sea of new knowledge with tremendous potential in modern science. So it is only natural that we would like to suggest to our young readers to visit the Einstein’s garden whenever exhausted to pluck an unexplored scientific flower. In the mean time our research interest has been shifted and finally, we are moving towards our obsession, Fowler-Nordheim field emission from nanostructured materials, leaving the pedigree of Einstein’s photoemission on your able shoulders with the hope that you are in tune with the fact that “Exposition, criticism, appreciation is the work for second-rate minds.” (G. H. Hardy, A mathematician’s Apology, Cambridge University Press, 1990, pp. 61).

Appendix A The Numerical Values of the Energy Band Constants of a Few Materials

Materials

Numerical values of the energy band constants

1

The conduction electrons of n-Cadmium Germanium Arsenide can be described by three types of band models

2

n-Indium Arsenide

3

n-Gallium Arsenide

4

n-Gallium Aluminium Arsenide

1. The values of the energy band constants in accordance with the generalized electron dispersion relation of nonlinear optical materials (as given by (2.1)) are as follows: Eg0 = 0.57 eV,  = 0.30 eV, ⊥ = 0.36 eV, m∗ = 0.034m0 , m∗⊥ = 0.039m0 , T = 4 K, δ = −0.21 eV, gv = 1 [1, 2], εsc = 18.4ε0 [3] (εsc and ε0 are the permittivity of the semiconductor material and free space, respectively) and the work function (W) = 4 eV [4]. 2. In accordance with the three-band model of Kane (as given by (2.19)), thespectrum constants are given by:  = || + ⊥ /2 = 0.33 eV, Eg0 = 0.57 eV,

m∗ = m∗|| + m∗⊥ /2 = 0.0365m0 and δ = 0 eV. 3. In accordance with two-band model of Kane, Eg0 = 0.57 eV and m∗ = 0.0365m0 . The values Eg0 = 0.36 eV,  = 0.43 eV, m∗ = 0.026m0 , gv = 1, εsc = 12.25ε0 [5] and W = 5.06 eV [6] are valid for the three-band model of Kane as given by (2.19). The values Eg0 = 1.55 eV,  = 0.35 eV, m∗ = 0.07m0 , gv = 1, εsc = 12.9ε0 [5] and W = 4.07 eV [7] are valid for the three-band model of Kane as given by (2.19). values a13 = −1.97 × 10−37 eVm4 and a15 = −2.3 × 10−34 eVm4 [8] are valid for (3.25). The values α11 = −2132 × 10−40 eVm4 , α12 = 9030 × 10−50 eVm5 , β11 = −2493 × 10−40 eVm4 , β12 = 12594 × 10−50 eVm5 , γ11 = 30 × 10−30 eVm3 , γ12= −154 × 10−42 eVm4 [9]  are valid for (3.30). Eg0 = 1.424 + 1.266x + 0.26x2 eV,  = (0.34 − 0.5x) eV, m∗ = [0.066 + 0.088x] m0 , gv = 1, εsc = [13.18 − 3.12x] ε0 [10] and W = (3.64 − 0.14x) eV [11].

321 K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_BM2,  C Springer Science+Business Media, LLC 2009

322

5

Appendix A The Numerical Values of the Energy Band Constants of a Few Materials

Materials

Numerical values of the energy band constants

n-Mercury Cadmium Telluride

Eg0 = ( − 0.302 + 1.93x + 5.35 × 10−4

6

n-Indium Gallium Arsenide Phosphide lattice matched to Indium Phosphide

7

n-Indium Antimonide n-Gallium Antimonide

8

9

10

n-Cadmium Sulphide

n-Lead Telluride

(1 − 2x)T − 0.810x2 + 0.832x3 ) eV,    = 0.63 + 0.24x − 0.27x2 eV, m∗ = 0.1m0 Eg0 (eV)−1 , gv = 1,  εsc = 20.262 − 14.812x + 5.22795x2 ε0 [12] and    eV [13]. W=   4.23 − 0.813 Eg0 − 0.083 Eg0 = 1.337 − 0.73y + 0.13y2 eV,    = 0.114 + 0.26y − 0.22y2 eV, m∗ = (0.08 − 0.039y) m& 0, y = (0.1896 − 0.4052x) (0.1896 − 0.0123x),   gv = 1, εsc = 10.65 + 0.1320y ε0 and W (x,y) = [5.06 (1 − x) y + 4.38(1 − x)(1 − y) + 3.64xy + 3.75 {x(1 − y)} ] eV [14]. Eg0 = 0.2352 eV,  = 0.81 eV, m∗ = 0.01359m0 , gv = 1, εsc = 15.56ε0 [5] and W = 4.72 eV [6]. The values of Eg0 = 0.81 eV,  = 0.80 eV, P0 = 9.48 × 10−10 eVm, ς¯0 = −2.1, v¯ 0 = −1.49, ω¯ 0 = 0.42, gv = 1 [15] and εsc = 15.85ε0 [15, 16] are valid for the model of Seiler et al. [15] as given by (3.113). The values E1 = 1.024 eV, E2 = 0 eV, E3 = −1.132 eV, E4 = 0.05 eV, E5 = 1.107 eV, E6 = −0.113 eV and E7 = −0.0072 eV [16] are valid for the model of Zhang [16] as given by (3.128). m∗ = 0.7m0 , m∗⊥ = 1.5m0 , C0 = 1.4 × 10−8 eVm, gv = 1 [5], εsc = 15.5ε0 [17] and W = 4.5 eV [6] are valid for the Hopfield model [5] as given by (2.39) − The values m− t = 0.070m0 , ml = 0.54m0 , + + mt = 0.010m0 , ml = 1.4m0 , P|| = 141 meVnm, P⊥ = 486 meVnm, Eg0 = 190 meV, gv = 4 [5], εsc = 33ε0 [5, 18] and W = 4.6 eV [19] are valid for the Dimmock model [20] as given by (3.165). The  2 values R = 2.3 × 10−19 (eVm)2 , Eg0 = 0.16 eV,  2  2  2 (s)2 = 4.6 R , c = 3.07 eV, Q = 1.3 R ,   2 c = 3.28 eV, A = 0.83 × 10−19 (eVm)2 [21] and W = 4.21 eV [6] are valid for the model of Bangert and Kastner [21] as given by (3.172). The values mtv = 0.0965m0 , mlv = 1.33m0 , mtc = 0.088m0 , mlc = 0.83m0 [19] are valid for the model of Foley et al. [19] as given by (3.179). The values m1 = 0.0239m0 , m2 = 0.024m0 , m2 = 0.31m0 , m3 = 0.24m0 [22] are valid for the Cohen model [23] as given by (2.139).

Appendix A The Numerical Values of the Energy Band Constants of a Few Materials

323

Materials

Numerical values of the energy band constants

11

Stressed n-Indium Antimonide

12

Bismuth

13

Mercury Telluride

14

Platinum Antimonide

15

n-Gallium Phosphide

16

Germanium

17

Tellurium

18

Graphite

19

Lead Germanium Telluride

The values m∗ = 0.048mo , Eg0 = 0.081 eV, B2 = 9 × 10−10 eVm, C1 = 3 eV, C2 = 2 eV, a0 = −10 eV, b0 = −1.7 eV, d0 = −4.4 eV, Sxx = 0.6 × 10−3 (kbar)−1 , Syy = 0.42 × 10−3 (kbar)−1 , Szz = 0.39 × 10−3 (kbar)−1 , Sxy = 0.5 × 10−3 (kbar)−1 , εxx = σ Sxx , εyy = σ Syy , εzz = σ Szz , εxy = σ Sxy , σ is the stress in kilobar, gv = 1[24] are valid for the model of Seiler et al. [24] as given by (2.102). Eg0 = 0.0153 eV, m1 = 0.00194m0 , m2 = 0.313m0 , m3 = 0.00246m0 , m2 = 0.36m0 , gv = 3, gs = 2 [25], M2 = 0.128m0 , M2 = 0.80m0 [26] and W = 4.34 eV. m∗v = 0.028m0 , gv = 1, εsc = 15.2ε0 [27] and W = 5.5 eV [28]. For valence bands, along <111> direction, λ1 = 0.33 eV, l1 = 1.09 eV, ν1 = 0.17 eV, n = 0.22 eV, a = 0.643 nm, I0 = 0.30 (eV)2 , δ0 = 0.33 eV, gv = 8 [29], εsc = 30ε0 [30] and work function (φw ) ≈ 3.0 eV [31]. m∗|| = 0.92m0 , m∗⊥ = 0.25m0 , k0 = 1.7 × 1019 m−1 , |VG | = 0.21 eV, gv = 6, gs = 2 [32] and W = 3.75 eV [6]. Eg0 = 0.785 eV, m∗|| = 1.57m0 , m∗⊥ = 0.0807m0 [7] and W = 4.14 eV [6] The values A6 = 6.7 × 10−16 meVm2 , A7 = 4.2 × 10−16 meVm2 ,  2 A8 = 6 × 10−8 meVm and  2 A9 = 3.6 × 10−8 meVm [33] are valid for the model of Bouat et al. [33] as given by (3.76). The values t1 = 0.06315 eV, t2 = −10.02 /2m0 , t3 = −5.552 /2m0 , t4 = 0.3 × 10−36 eVm4 , t5 = 0.3 × 10−36 eVm4 , t6 = −5.552 /2m0 , t7 = 6.18 × 10−20 (eVm)2 [34] and W = 1.9708 eV [35] are valid for the model of Ortenberg and Button [34] as given by (3.82). The values 1 = −0.0002 eV, γ1 = 0.392 eV, γ5 = 0.194 eV, c¯ = 0.674 nm, γ2 = −0.019 eV, a¯ = 0.246 nm, γ0 = 3 eV, γ4 = 0.193 eV [36] and W = 4.6 eV [37] are valid for the model of Brandt et al. [36] as given by (3.88). The valuesEg0 = 0.21 eV, gv = 4 [38] and φw ≈ 6 eV [39] are valid for the model of Vassilev [38] as given by (3.107a).

324

Appendix A The Numerical Values of the Energy Band Constants of a Few Materials

Materials

Numerical values of the energy band constants

20

Cadmium Antimonide

21

Cadmium Diphosphide

22

Zinc Diphosphide

23

Bismuth Telluride

24

Carbon Nanotube

25

Antimony

26

Zinc Selenide

27

Lead Selenide

The values A10 = −4.65 × 10−19 eVm2 , A11 = −2.035 × 10−19 eVm2 , A12 = −5.12 × 10−19 eVm2 , A13 = −0.25 × 10−10 eVm, A14 = 1.42 × 10−19 eVm2 , A15 = 0.405 × 10−19 eVm2 , A16 = −4.07 × 10−19 eVm2 , A17 = 3.22 × 10−10 eVm, A18 = 1.69 × 10−20 (eVm)2 , A19 = 0.070 eV [40] and φ ≈ 2 eV [41] are valid for the model of Yamada [40] as given by (3.186). The values β1 = 8.6 × 10−21 eVm2 , β2 = 1.8 × 10−21 (eVm)2 , β4 = 0.0825 eV, β5 = −1.9 × 10−19 eVm2 [42] and φ ≈ 5 eV [43] are valid for the model of Chuiko [42] and is given by (3.193). The values β1 = 8.7 × 10−21 eVm2 , β2 = 1.9 × 10−21 (eVm)2 , β4 = −0.0875 eV, β5 = −1.9 × 10−19 eVm2 [42] and W ≈ 3.9 eV [43] are valid for the model of Chuiko [42] and is given by (3.193). The values Eg0 = 0.145 eV, α11 = 3.25, α22 = 4.81, α33 = 9.02, α23 = 4.15, gs = 2, gv = 6 [44] and φ = 5.3 eV [45] are valid for the model of Stordeur et al. [44] using (3.200). The values ac = 0.144 nm [46], tc = 2.7 eV [47], r0 = 0.7 nm [48] and W = 3.2 eV [49] are valid for the graphene band structure realization of carbon nanotube [48] and are given by (2.162) and (2.163). The values α¯ 11 = 16.7, α¯ 22 = 5.98, α¯ 33 = 11.61, α¯ 23 = 7.54 [50] and W = 4.63 eV [6] are valid for the model of Ketterson [50] and are given by (3.207), (3.208) and (3.209) respectively. m∗2 = 0.16m0 , 2 = 0.42 eV, Eg02 = 2.82 eV[7] and W = 3.2 eV [51]. − + m− t = 0.23m0 , ml = 0.32m0 , mt = 0.115m0 , + ml = 0.303m0 , P|| ≈ 138 meVnm, P⊥ = 471 meVnm, Eg0 = 0.28 eV [52], εsc = 21.0ε0 [32] and W = 4.2 eV [53].

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Subject Index

A Area quantization, 2, 10 B Band structure Agafonov, 118 antimony, 150 bismuth telluride, 149 bismuth, 62 carbon nanotube, 72 II-V compounds, 146 II-VI compound, Hopfield model, 46 IV-VI compounds, 142 diphosphides, 147 gallium antimonide, 132 gallium phosphide, 48 germanium, 50 graphite, 126 Johnson, 117 Newson, 113 nonlinear optical, 39 Palik, 116 parabolic model, 4 Pb1−x Gex Te, 131 platinum antimonide, 56 Rossler, 114 Stillman model, 112 stressed materials, 59 tellurium, 124 three band Kane, 43 two band Kane, 44 zero-gap, 129 Bismuth nanowire, 38 Bulk, 4, 219 C Carbon nanotubes (CNs), 39 Cyclotron resonance, 38

D de Haas-Van Alphen oscillations, 2 Delta function, 23, 24 Density-of-states (DOS) bulk, 4, 5 cross field, 16 magnetic field, 12 MSQ, 24 quantum dot, 22 quantum well, 18 quantum well wire, 20 Effective mass (EMM), 225 Effective number of states, 8 Extreme degeneracy, 9 F Fermi energy, 4 Fermi-Dirac integral, 6–7 Fermi-Dirac probability, 4 G Graded interfaces, 179, 194 H Heavy hole, 220 Heaviside step function, 11 K k.p, 39 L Landau subbands/levels, 2, 10, 11, 26 Landau singularities, 12 Light waves, 219, 221 M Magnetic freeze out, 13–14 Magnetic field, 2, 11–14, 28 Magnetic quantum limit, 10

327

328 O Optical absorption coefficient, 229 Optical excitation, 224 P Photoemission bulk, 4, 6, 219, 226 classical, 1, 9 cross fields, 14, 16 degenerate, 8 nondegenerate, 9 magnetic field, 11, 14, 240 MSQE, 24–25 Photoexcitation, 220 Q Quaternary, 219 Quantum well wires, 20, 39, 250 Quantum dots, 22, 110, 251 S Shubnikov de Hass (SdH), 1–2 Superlattice (SL), 173 II-VI, 179 III-V, 174

Subject Index III-V effective mass, 186 IV-VI, 181 effective mass, 275 HgTe/CdTe, 185 T Ternary, 219 U Ultrathin film, 19, 43, 46, 48, 50, 56, 59, 62, 247, 253–255 effective mass superlattices, 186, 188, 191, 193, 202, 203, 204, 206, 277, 279, 281 graded superlattices, 181, 185, 194, 197, 198, 201, 298, 300–301 quantum dots, 22–24, 110–120, 121–170, 251 quantum well wire, 20–22, 50, 250 Ultrathin films/Quantum wells, 19, 39, 247 V Vector potential, 14 Z Zeta function, 41

Materials Index

A Antimony, 108, 168–169 B Bismuth, 1, 38, 83, 86–87, 100–101, 108, 138, 163, 315, 323 Bi2 Te3 , 108, 149, 168–169, 315, 324 C Cadminum diphosphide, 108, 147, 166–167 Carbon nanotubes, 38–39, 72, 101, 104–105, 263, 315 CdGeAs2 , 73–74, 88–89, 152–154 CdS, 75, 78–79, 92–93, 155–157, 315 CdS/ZnSe, 206–216 G GaAs, 5, 6, 19, 21, 23, 28–33, 154–156, 173, 322 GaAs/Ga1−x Alx As, 173, 206–209, 211–213 GaSb, 108, 132, 134, 136, 158, 161, 315 GaP, 38, 48–49, 80–81, 94–95, 104, 107, 121, 155–157, 315 Germanium, 38, 50, 81–82, 95–96, 122, 158, 321, 323 Graphite, 107, 126, 158, 315, 323 H Hg1−x Cdx Te, 219, 226–230, 237, 239–242, 254–255, 257–258, 259, 261–263 HgTe, 107, 158–161, 173

HgTe/CdTe, 173–174, 185–186, 193, 201, 205, 209–216 HgTe/Hg1−x Cdx Te, 213–216, 278–290, 301–313 I InAs, 5, 6, 19, 21, 28–33, 75–76, 89–90 InSb, 5, 6, 12, 17, 21, 25–34, 38, 75, 77–78, 83, 85–86, 91–92, 98–99, 154–156, 158, 161 Inx Ga1−x As/InP, 206–208, 278–290, 310 In1−x Gax Asy P1−y lattice matched to InP, 219, 226–228, 231–233, 237, 240, 242, 244, 255–261, 263–264, 265 P Pb1−x Gex Te, 131, 158, 160–161, 315 PbSe/PbTe, 209–216 PbTe, 107–108, 163, 165 PtSb2 , 38, 56, 58, 75, 83–84, 97–98, 104, 107, 128, 158–159, 315 S Stressed n-InSb, 38, 83, 85–86, 98–99, 158 T Tellurium, 107, 124–125, 158, 315, 323 Z Zinc diphosphide, 147, 163, 166–167, 315, 324

329