Nanomaterials: Volume 2: Quantization and Entropy 9783110661194, 9783110659726

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Kamakhya Prasad Ghatak, Madhuchhanda Mitra Nanomaterials

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Kamakhya Prasad Ghatak, Madhuchhanda Mitra

Nanomaterials

Volume 2: Quantization and Entropy

Authors Prof. Dr. Engg Kamakhya Prasad Ghatak Institute of Engineering and Management University of Engineering and Management Department of Computer Science and Engineering Kolkata and Jaipur India [email protected] Prof. Dr. Madhuchhanda Mitra University of Calcutta Department of Applied Physics 92 A.P.C. Road Kolkata-700009 India [email protected]

ISBN 978-3-11-065972-6 e-ISBN (PDF) 978-3-11-066119-4 e-ISBN (EPUB) 978-3-11-065999-3 Library of Congress Control Number: 2020934280 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Cover image: Science Photo Library/University of Sydney/Ammrf Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck www.degruyter.com

After spending forty years in Nanomaterials, I have finally realized that my education is nothing but heducation and my real I, is not at all equal to the sum of the body I wear and the mind which drives the body. From the desk of the first author: The author dedicates this book to the following beautiful trio whose combined effort is just like a Decadron to him to revive the last phase of his academic life. 1. Professor Dr. S. Chakrabarti, the Hon’ble Director of the Institute of Engineering and Management, his present mentor for kindly picking him from National Institute of Technology at Agartala at rather academic turbulent moments and transforming myself as a member of IEM and UEM family. 2. Professor Dr. B. Chatterjee, the Hon’ble Vice-Chancellor of the University of Engineering and Management, Jaipur for his warm personality which generates in the mind the confidence that “Yes, I can”. 3. Professor R. Paul Dean Research (Associate) of the University of Engineering and Management, Kolkata for behaving like the academic daughter of the author to feel that at least in odd times she will be besides him. From the desk of the Second author: The second author dedicates this book to Mr. B. Nag, her colleague and lifelong time tested friend for making her life always pleasant in the Department of Applied Physics of the University of Calcutta.

Preface Calmness and tolerance act like air conditioning in a hot room and increase everyone’s efficiency.

The applications and the importance of nanomaterials in many dimensions are already well known since its creation in the domain of low dimensional science and technology. An enormous range of powerful applications of such low dimensional structures in the quantum regime together with a rapid increase in computing power, have generated considerable interest in the study of the quantum effect devices based on various new materials of reduced dimensionality. Examples of such new applications include quantum registers, quantum switches, quantum sensors, quantum logic gates, quantum well (QWs), nanowires (NWs), quantum box (QBs), quantum wire transistors, quantum cascade lasers, high-speed digital networks, high-resolution terahertz spectroscopy, advanced integrated circuits, superlattice photo-oscillator, superlattice photo-cathodes, resonant tunneling diodes and transistors, superlattice coolers, thermoelectric devices, thin film transistors, microoptical systems, high performance infrared imaging systems, single electron/molecule electronics, nano-tube based diodes, and other nano-electronic devices [1–14]. In volume 1 of the series on nanomaterials, we have investigated few Electronic Properties of Opto electronic nanomaterials having various band structures under different physical conditions in the presence of intense photon field with the use of the Heisenberg’s Uncertainty Principle (HUP). In this context, it may be written that the available reports on the said areas cannot afford to cover even an entire chapter regarding the ENTROPY in heavily doped (HD) nanomaterials and after thirty years of continuous effort, we see that the complete investigations of the entropy comprising of the whole set of materials and allied sciences is really a sea and is a permanent member of the domain of impossibility theorems. It may be noted that the entropy is a significant concept and a physical phenomenon which occupies a singular position in the whole arena of science and technology in general and whose importance has already been established since the inception of second law of thermodynamics which in recent years finds extensive applications in modern thermodynamics of nanomaterials, characterization and investigation of condensed matter systems, thermal properties of thermal semiconducting devices and related aspects in connection with the investigations of the thermal properties of nanomaterials [15–19]. It is well known that the entropy is the measure of disorder or uncertainty about a system [15–19]. The equilibrium state of a system maximizes the entropy as all the information about the initial conditions except that the conserved variables are lost. According to the second law of thermodynamics the total entropy of any system will not decrease other than by increasing the entropy of some other system. A reduction in the increase of entropy in a specified process, such as a chemical https://doi.org/10.1515/9783110661194-202

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reaction, means that it is energetically more efficient. In accordance to the second law of thermodynamics the entropy of a system that is not isolated may decrease. In mechanics, the second law in conjunction with the fundamental thermodynamic relation places limits on a system’s ability to do useful work. The entropy change of a system at temperature T absorbing an infinitesimal amount of heat δq in a reversible way is given by δq=T. Statistical mechanics demonstrates that entropy is governed by probability, thus allowing for a decrease in disorder even in an isolated system. According to Boltzmann’s the entropy is a measure of the number of possible microscopic states (or microstates) of a system in thermodynamic equilibrium. Entropy is the only quantity in the physical sciences that seems to imply a particular direction of progress, sometimes called an arrow of time. The significant work of Zawadzki [20] reflects the fact that the entropy for materials having degenerate electron concentration is independent of scattering mechanisms and is exclusively determined by the dispersion laws of the respective carriers. It will, therefore, assume different values for different systems and varies with the doping, the magnitude of the reciprocal quantizing magnetic field under magnetic quantization, the nanothickness in ultrathin films, quantum wires and dots, the quantizing electric field as in inversion layers, the carrier statistics in various types of quantum confined superlattices having different carrier energy spectra and other types of low-dimensional field assisted systems. It is well known that heavy doping and carrier degeneracy are the keys to unlock the important properties of Materials and they are especially instrumental in dictating the characteristics of Ohomic contacts and Schottky contacts, respectively [21–31]. It is an amazing fact that although the heavily doped materials (HDS) have been investigated in the literature but the study of the corresponding entropies of HDS is still one of the open research problems. This first monograph solely investigates the entropy in HD non-linear optical, III-V, II-VI, Gallium Phosphide, Germanium, Platinum Antimonide, stressed, IV-VI, Lead Germanium Telluride, Tellurium, II-V, Zinc and Cadmium di-phosphides, Bismuth Telluride, III-V, II-VI, IVVI and HgTe=CdTequantum well HD superlattices with graded interfaces under magnetic quantization, III-V, II-VI, IV-VI and HgTe=CdTe HD effective mass superlattices under magnetic quantization, quantum confined effective mass superlattices and superlattices of HD optoelectronic materials with graded interfaces respectively. Our method is not at all related with the Density-of-States (DOS) technique as used in the literature. From the electron energy spectrum, one can obtain the DOS but the DOS technique, as used in the literature cannot provide the E-k dispersion relation. Therefore, our study is more fundamental than those in the existing literature, because the Boltzmann transport equation, which controls the study of the charge transport properties of the semiconductor devices, can be solved if and only if the E-k dispersion relation is known. This book is divided into two parts each containing 5 and 4 chapters and 5 appendices is partially based on our on-going researches on the entropy in HDS and

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an attempt has been made to present a cross section of the entropy for wide range of HDS and their quantized-structures with varying carrier energy spectra under various physical conditions. It is well known that the band tails are being formed in the forbidden zone of the HDS and can be explained by the overlapping of the impurity band with the conduction and valence bands [32]. Kane [33] and Bonch Bruevich [34] have independently derived the theory of band tailing for materials having unperturbed parabolic energy bands. Kane’s model [33] was used to explain the experimental results on tunneling [35] and the optical absorption edges [36, 37] in this context. Halperin and Lax [38] developed a model for band tailing applicable only to the deep tailing states. Although Kane’s concept is often used in the literature for the investigation of band tailing [39, 40], it may be noted that this model [33, 41] suffers from serious assumptions in the sense that the local impurity potential is assumed to be small and slowly varying in space coordinates [40]. In this respect, the local impurity potential may be assumed to be a constant. In order to avoid these approximations, we have developed in this book, the electron energy spectra for HDS for studying the entropy based on the concept of the variation of the kinetic energy [32, 40] of the electron with the local point in space coordinates. This kinetic energy is then averaged over the entire region of variation using a Gaussian type potential energy. On the basis of the E–k dispersion relation, we have obtained the electron statistics for different HDS for the purpose of numerical computation of the respective entropy. It may be noted that, a more general treatment of many-body theory for the DOS of HDS merges with one-electron theory under macroscopic conditions [32]. Also, the experimental results for the Fermi energy and others are the average effect of this macroscopic case. So, the present treatment of the one-electron system is more applicable to the experimental point of view and it is also easy to understand the overall effect in such a case [31]. In a HDS, each impurity atom is surrounded by the electrons, assuming a regular distribution of atoms, and it is screened independently [39, 42, 43]. The interaction energy between electrons and impurities is known as the impurity screening potential. This energy is determined by the interimpurity distance and the screening radius (popularly known as the Debye screening length). The screening length changes with the band structure. Furthermore, these entities are important for HDS in characterizing the semiconductor properties [44–47] and the modern electronic devices [39, 46]. The works on Fermi energy and the screening length in an n-type GaAs have already been initiated in the literature [47], based on Kane’s model. Incidentally, the limitations of Kane’s model [33, 40], as mentioned above, are also present in their studies. The part one deals with the influence of quantum confinement on the entropy of non-parabolic HDS and in chapter one we study the entropy in QWs of HD nonlinear optical materials on the basis of a generalized electron dispersion law introducing the anisotropies of the effective masses and the spin orbit splitting constants respectively together with the inclusion of the crystal field splitting within the framework of

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the ~ k.~ p formalism. We will observe that the complex electron dispersion law in HDS instead of real one occurs from the existence of the essential poles in the corresponding electron energy spectrum in the absence of band tails. The physical picture behind the existence of the complex energy spectrum in HD non-linear optical Materials is the interaction of the impurity atoms in the tails with the splitting constants of the valance bands. The more is the interaction, the more the prominence of the complex part than the other case. In the absence of band tails, there is no interaction of the impurity atoms in the tails with the spin orbit constants and consequently, the complex part vanishes. One important consequence of the HDS forming band tails is that the effective electron mass (EEM) exists in the forbidden zone, which is impossible without the effect of band tailing. In the absence of band tails, the effective mass in the band gap of Materials is infinity. Besides, depending on the type of the unperturbed carrier energy spectrum, the new forbidden zone will appear within the normal energy band gap for HDS. The results of HD III-V (e.g. InAs, InSb, GaAs etc.), ternary (e.g. Hg1 − x Cdx Te), quaternary (e.g. In1 − x Gax As1 − y Py lattice matched to InP) compounds form a special case of our generalized analysis under certain limiting conditions. The entropy in HD QWs of II-VI, IV-VI, stressed Kane type materials, Te, GaP, PtSb2 , Bi2 Te3 , Ge, GaSb, II-V, Lead Germanium Telluride, Zinc and Cadmium Diphosphides has also been investigated in the appropriate sections. The importance of the aforementioned materials has also been described in the same chapter. In the absence of band tails and under the condition of extreme carrier degeneracy together with certain limiting conditions, all the results for all the entropies for all the HD QWs of chapter one get simplified into the form of isotropic parabolic energy bands exhibiting the necessary mathematical compatibility test. In the second and third chapters, the entropy for HD nanowires (NWs) and quantum boxes (QBs) of all the materials of chapter 1 have respectively been investigated. As a collateral study we shall observe that the EEM in such QWs and NWs becomes a function of size quantum number, the Fermi energy, the scattering potential and other constants of the system which is the intrinsic property of such 2D and 1D electronic materials. In this context, it may be noted that the effects of quantizing magnetic field (B) on the band structures of compound materials are most striking than that of the parabolic one and are easily observed in experiments. A number of interesting physical features originate from the significant changes in the basic energy wave vector relation of the carriers caused by the magnetic field. The valuable information could also be obtained from experiments under magnetic quantization regarding the important physical properties such as Fermi energy and effective masses of the carriers, which affect almost all the transport properties of the electron devices [48] of various materials having different carrier dispersion relations [49]. Specifically in chapter four we study the entropy in HD non-linear optical materials in the presence of strong magnetic field leading to the magnetic quantization of the energy band states of the corresponding bulk HD materials. The results

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of HD III-V (e.g. InAs, InSb, GaAs etc.), ternary (e.g. Hg1 − x Cdx Te), quaternary (e.g. In1 − x Gax As1 − y Py lattice matched to InP) compounds form a special case of our generalized analysis under certain limiting conditions. The entropy for HD II-VI, IV-VI, stressed Kane type materials, Te, GaP, PtSb2 , Bi2 Te3 , Ge, and GaSb has also been investigated by formulating the respective appropriate HD energy band structure. In the absence of band tails and under the condition of extreme carrier degeneracy together with certain limiting conditions, all the results for all the entropies for all the HD materials of this chapter one get simplified into the wellknown parabolic energy bands under strong magnetic quantization exhibiting the necessary mathematical compatibility test. In chapter five we have studied the entropy for all the HD materials of chapter four in the presence of magneto size quantization. In part two we have studied the entropies in HD quantum confined superlattices (SLs). It is well known that Keldysh [50] first suggested the fundamental concept of a SL, although it was successfully experimental realized by Esaki and Tsu [51]. The importance of SLs in the field of nano-electronics has already been described in [52–53]. The most extensively studied III-V SL is the one consisting of alternate layers of GaAs and Ga1-xAlxAs owing to the relative ease of fabrication. The GaAslayer forms quantum wells and Ga1 − x Alx As form potential barriers. The III-V SL’s are attractive for the realization of high speed electronic and optoelectronic devices [54]. In addition to SLs with usual structure, SLs with more complex structures such as II-VI [55], IV-VI [56] and HgTe=CdTe [57] SL’s have also been proposed. The IV-VI SLs exhibit quite different properties as compared to the III-V SL due to the peculiar band structure of the constituent materials [58]. The epitaxial growth of IIVI SL is a relatively recent development and the primary motivation for studying the mentioned SLs made of materials with the large band gap is in their potential for optoelectronic operation in the blue [59]. HgTe=CdTeSL’s have raised a great deal of attention since 1979, when as a promising new materials for long wavelength infrared detectors and other electro-optical applications [60]. Interest in Hgbased SL’s has been further increased as new properties with potential device applications were revealed [61]. These features arise from the unique zero band gap material HgTe [62] and the direct band gap materials CdTe which can be described by the three band mode of Kane [63]. The combination of the aforementioned materials with specified dispersion relation makes HgTe=CdTe SL very attractive, especially because of the possibility to tailor the material properties for various applications by varying the energy band constants of the SLs. In addition to it, for effective mass SLs, the electronic sub-bands appear continually in real space [64, 65, 66]. We note that all the aforementioned SLs have been proposed with the assumption that the interfaces between the layers are sharply defined, of zero thickness, i.e., devoid of any interface effects. The SL potential distribution may be then considered as a one dimensional array of rectangular potential wells. The aforementioned advanced experimental techniques may produce SLs with physical interfaces between

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the two materials 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 exists for the electrons. The influence of finite thickness of the interfaces on the electron dispersion law is very important, since the electron energy spectrum governs the electron transport in SLs. The chapter six explores the entropy in III-V, II-VI, IV-VI, HgTe=CdTe and strained layer heavily doped Quantum wire superlattices (QWHDSLs) with graded interfaces and heavily doped quantum wire effective mass super lattices respectively. The chapter seven investigates the entropy in quantum dot HDSLs for all cases of chapter six. The chapter eight of part two of this book contains the study of the entropy in HD SLs under magnetic quantization for all the cases of chapter six. The chapter nine contains the conclusions and future research in this context. With the advent of nano-devices, the built-in electric field becomes so large that the electron energy spectrum changes fundamentally instead of being invariant and the chapter 10 (Appendix A) of this book investigates the entropy under intense electric field in bulk specimens of HD III-V, ternary and quaternary materials. The same chapter explores the influence of electric field on the entropy in the presence of magnetic quantization, cross-fields configuration, QWs, NWs, QBs, magneto size quantum effect, inversion and accumulation layers, magneto inversion and magneto accumulation layers, doping superlattices, magneto doping superlattices, QWHD, NWHD and QBHD effective mass superlattices, magneto QWHD effective mass superlattices, magneto HD effective mass superlattices, QWHD, NWHD and QBHD superlattices with graded interfaces, magneto QWHD superlattices with graded interfaces and magneto HD superlattices with graded interfaces and respectively magnetic quantization, size quantization, accumulation layers, HD doping superlattices and effective mass HD superlattices under magnetic quantization respectively. It is interesting to note that the EEM depends on the strong electric field (which is not observed elsewhere) together with the fact that the EEM in the said systems depends on the respective quantum numbers in addition to the Fermi energy, the scattering potential and others system constants which are the characteristics features of such hetero-structures. The Chapter 10 (Appendix A) investigates the entropy in bulk specimens HD Kane type materials under intense electric field in the presence of strong magnetic quantization after formulating the electron dispersion law in the present case. The same appendix studies the entropy under cross-fields configuration, QWs, NWs, QBs, magneto size quantum effect, inversion and accumulation layers, magneto inversion and magneto accumulation layers, doping superlattices, magneto doping superlattices, QWHD, NWHD and QBHD effective mass superlattices, magneto QWHD effective mass superlattices, magneto HD effective mass superlattices, QWHD, NWHD and QBHD superlattices with graded interfaces, magneto QWHD superlattices with graded interfaces and magneto HD superlattices with graded interfaces and respectively.

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In this context we have already noted that the semiconductor superlattices (SLs) composed of alternative layers of two different degenerate layers with controlled thickness [67] have found wide applications in many new devices such as photodiodes, photo-resistors [67, 68], transistors [69], light emitters [70], tunneling devices [71], etc. [72–84]. The investigations of the physical properties of narrow gap SLs have increased extensively; since they are important for optoelectronic devices and because of the quality of hereto-structures, involving narrow gap materials have been improved. It may be noted in this context that the doping superlattices, are crystals with a periodic sequence of ultrathin film layers [85, 86] of the same semiconductor with the intrinsic layer in between together with the opposite sign of doping. All the donors will be positively charged and all the acceptors negatively. This periodic space charge causes a periodic space charge potential which quantizes the motions of the carriers in the z-direction together with the formation of the sub-band energies. In Chapter 11(Appendix B), the entropy in doping superlattices of HD nonlinear optical, III-V, II-VI, IV-VI, and stressed Kane type Materials has been investigated. In this case we will note that the EEM in such doping supper lattices becomes a function of nipi sub-band index, surface electron concentration, Fermi energy, the scattering potential and other constants of the system which is the intrinsic property of such 2D quantized systems. In Chapter 12 (Appendix C) the entropy in QWHDSLs under magnetic quantization have been studied. It is well known that the electrons in bulk materials in general, have three dimensional freedom of motion. When, these electrons are confined in a one dimensional potential well whose width is of the order of the carrier wavelength, the motion in that particular direction gets quantized while that along the other two directions remains as free. Thus, the energy spectrum appears in the shape of discrete levels for the one dimensional quantization, each of which has a continuum for the two dimensional free motion. The transport phenomena of such one dimensional confined carriers have recently studied [87] with great interest. For the metal-oxide-semiconductor (MOS) structures, the work functions of the metal and the semiconductor substrate are different and the application of an external voltage at the metal-gate causes the change in the charge density at the oxide semiconductor interface leading to a bending of the energy bands of the semiconductor near the surface. As a result, a one dimensional potential well is formed at the semiconductor interface. The spatial variation of the potential profile is so sharp that for considerable large values of the electric field, the width of the potential well becomes of the order of the de Broglie wavelength of the carriers. The Fermi energy, which is near the edge of the conduction band in the bulk, becomes nearer to the edge of the valance band at the surface creating inversion layers. The energy levels of the carriers bound within the potential well get quantized and form electric sub bands. Each of the sub-band corresponds to a quantized level in a plane perpendicular to the surface leading to a quasi two dimensional electron gas. Thus, the extreme band bending at low temperature allows us to observe the quantum effects at

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the surface. Although, considerable work has already been done regarding the various physical properties of different types of inversion layers having various band structures, nevertheless it appears from the literature that there lies scopes in the investigations made while the interest for studying different other features of accumulation layers is becoming increasingly important. In chapter 13 (Appendix D), the Entropy in accumulation layers of HD nonlinear optical, III-V, II-VI, IV-VI, stressed Kane type Materials and Ge have been investigated. For the purpose of relative comparisons, we have also studied the entropy in inversion layers of the afore-mentioned materials. It is interesting to note that the EEM in such layers is a function of electric sub-band index, surface electric field, Fermi energy, the scattering potential and other constants of the system which is the intrinsic property of such 2D electrons. It is worth remarking that the influence of crossed electric and quantizing magnetic fields on the transport properties of materials having various band structures has relatively less investigated as compared with the corresponding magnetic quantization, although, the cross-fields are fundamental with respect to the addition of new physics and the related experimental findings in modern quantum effect devices. It is well known that in the presence of electric field ðE0 Þ along x-axis and the quantizing magnetic field ðBÞ along z-axis, the mass of the carriers in materials become modified and for which the carrier moves in both the z and y directions. The motion along y-direction is purely due to the presence of E0 along x-axis and in the absence of electric field, the effective electron mass along y-axis tends to infinity which indicates the fact that the electron motion along y-axis is forbidden. The effective electron mass of the isotropic, bulk materials having parabolic energy bands exhibits mass anisotropy in the presence of cross fields and this anisotropy depends on the electron energy, the magnetic quantum number, the electric and the magnetic fields respectively, although, the effective electron mass along z- axis is a constant quantity. In 1966, Zawadzki and Lax [88] formulated the entropy for III-V materials in accordance with the two band model of Kane under cross fields configuration which generates the interest to study this particular topic of solid state science in general [89]. The chapter 14 (Appendix E) investigates the entropy under cross-field configuration in HD nonlinear optical, III-V, II-VI, IV-VI and stressed Kane type materials respectively. This chapter also tells us that the EEM in all the cases is a function of the finite scattering potential, the magnetic quantum number and the Fermi energy even for HD materials whose bulk electrons in the absence of band tails are defined by the parabolic energy bands. The last chapter 15 (Appendix F) contains the numerical values of the energy band constants of few materials. It is needless to say that this monograph is based on the ‘iceberg principle’ [90] and the rest of which will be explored by the researchers of different appropriate fields. Since, there is no existing report devoted solely to the study of entropy for HD quantized structures to the best of our knowledge, we hope that the present book will a useful reference source for the present and the next generation of the

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readers and the researchers of nano-materials and allied sciences in general. We have discussed enough regarding the Entropies in different quantized HD materials although lots of new computer oriented numerical analysis are being left for the purpose of being computed by the readers, to generate the new graphs and the inferences from them which all together is a sea in itself. The production of error free first edition of any book from every point of view is a permanent member in the domain of impossibility theorems, the same stands very true for this monograph also. Various expressions and a few chapters of this book have been appearing for the first time in printed form. The suggestions from the readers for the development of the book will be highly appreciated for the purpose of inclusion in the future edition, if any. In this book, from chapter one to till chapter fourteen, we have presented circa 200 open research problems for the graduate students, PhD aspirants, researchers, engineers in this pinpointed research topic. We strongly hope that alert readers of this monograph will not only solve the said problems by removing all the mathematical approximations and establishing the appropriate uniqueness conditions, but also will generate new research problems both theoretical and experimental and, thereby, transforming this monograph into a solid book. Incidentally, our readers after reading this book will easily understand that how little is presented and how much more is yet to be investigated in this exciting topic which is the signature of coexistence of new physics, advanced mathematics combined with the inner fire for performing creative researches in this context from the young scientists since like Kikoin [91] we feel that “A young scientist is no good if his teacher learns nothing from him and gives his teacher nothing to be proud of”. We emphatically write that the problems presented here form the integral part of this book and will be useful for the readers to initiate their own contributions on the entropy for HDS and their quantized counter parts since like Sakurai [92] we firmly believe “The reader who has read the book but cannot do the exercise has learned nothing”. In this monograph, we have formulated the expressions of effective electron mass and the sub-band energy throughout this monograph as a collateral study, for the purpose of in-depth investigations of the said important pinpointed research topics. Thus, in this book, the readers will get much information regarding the influence of quantization in HD low dimensional materials having different band structures. Although the name of the book is extremely specific, from the content, one can easily infer that it should be useful in graduate courses on materials science, condensed matter physics, solid states electronics, nano-science and technology and solid-state sciences and devices in many Universities and the Institutions in addition to both Ph.D. students and researchers in the aforementioned fields. Last but not the least, we do hope that our humble effort will kindle the desire to delve deeper into this fascinating and deep topic by any one engaged in materials research and device development either in academics or in industries.

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References [1]

[2]

[3]

[4]

[5]

[6]

Rieth M., Schommers W., (edited by), Handbook of computational and theoretical nanoscience (American Scientific Publishers, Los Angeles, 2006); Vols. 1–10; Rieth M., and Schommers W., J. Nanosci. Nanotechnol. 2, 679 (2002); Rieth M., Nano-Engineering in Science and TechnologySeries on the Foundations of Natural Science and Technology Vol. 6 (2003). Das P.K., Ghatak K.P., Journal of Nanoscience and Nanotechnology 19, 2909 (2019); Ghatak K.P., Chakrabarti S., Chatterjee B., Materials Focus, 7, 361 (2018); Ghatak K.P., Chakrabarti S., Chatterjee B., Das P.K., Dutta P., Halder A., Materials Focus, 7, 390 (2018); Chatterjee B., Debbarma N., Mitra M., Datta T., Ghatak K.P., Journal of Nanoscience and Nanotechnology, 17, 3352 (2017); Ghatak K.P., De D., Materials Focus, 6, 114 (2017); Das P.K., Dutta P., Halder A., Pal J., Debbarma N., Debbarma S., Ghatak K.P., Materials Focus, 6, 167 (2017); Das P.K., Dutta P., Halder A., Bhattacharjee R., Ghatak K.P., Materials Focus, 6, 133 (2017); Bhattacharjee R., Ghatak K.P., Journal of Nanoscience and Nanotechnology, 17, 640 (2017); Mitra M., Sen T.N., Datta T., Bhattacharjee R., Singh L.S., Ghatak K.P., Journal of Nanoscience and Nanotechnology, 17, 256 (2017); Sen T.N., Ghatak K.P., Quantum Matter, 5, 732 (2016); Sen T.N., Ghatak K.P., Quantum Matter, 5, 721 (2016); Ghatak K.P., Sarkar K., Chakrabarti S., Kumar M., Debbarma M., Sen T.N., Chakraborty M., Reviews in Theoretical Science, 4, 199 (2016); Bhattacharjee R., Sarkar K., Kumar M., Chatterjee B., Ghatak K.P., Quantum Matter, 5, 557 (2016); Ghatak K.P., Sarkar K., Debbarma N., Singh L.S., Quantum Matter, 5, 427 (2016); Ghatak K.P., De D., Journal of Nanoengineering and Nanomanufacturing, 6, 1 (2016); Biswas S.K., Mitra M., Ghatak K.P., Journal of Nanoengineering and Nanomanufacturing, 6, 63 (2016); Paitya N., Ghatak K.P., Quantum Matter, 5, 191 (2016); Chatterjee B., Chakrabarti S., Sen S.K., Mitra M., Ghatak K.P., Quantum Matter, 5, 85 (2016). Sen T.N., Ghatak K.P., Journal of Nanoscience and Nanotechnology 16, 1229 (2016); Ghatak K. P., Einstein’s Photoemission, Springer Tracts in Modern Physics, Vol. 262, Springer International Publishing, Switzerland (2015). Sarkar K., Chakraborty M., Chakravarti S., Chatterjee B., Ghatak K.P., Journal of Nano-engineering and Nano-manufacturing 5, 43 (2015); Chakrabarti S., Singh L.S., Ghatak K.P., Advances in Optical Science and Engineering, 17 (2015); Chatterjee B., Sarkar K., Ghatak K.P., Advances in Optical Science and Engineering, 621 (2015); Adhikari S.M., Ghatak K.P., Journal of Advanced Physics, 2, 130 (2013); Bhattacharya S., De D., Ghosh S., Ghatak K.P., Journal of Computational and Theoretical Nano-science, 10, 664 (2013); Paitya N., Bhattacharya S., De D., Ghosh S., Ghatak K.P., Journal of Nano-engineering and Nano-manufacturing, 2, 211 (2012). Ghatak K.P., Fizika A., 1, 197 (2006); Singh L.J., Choudhury S., Singha S., Roy S., Ghatak K.P., Electrical Engineering, 87, 19 (2005); Chakraborty P.K., Datta G.C., Ghatak K.P., Physica Scripta, 68, 368 (2003); Ghatak K.P., Mukhopadhyay J., Banerjee J.P., SPIE Proceedings Series, 4746, 1292 (2002); Ghatak K.P., Mukhopadhyay J., Banerjee J.P., SPIE Proceedings Series, 4746, 1296 (2002); Ghatak K.P., Mukhopadhyay J., Banerjee J.P., Proceedings-SPIE The International Society for Optical Engineering, 2, 1296 (2002); Ghatak K.P., Mukhopadhyay J., Banerjee J.P., SPIE proceedings series, 4746, 347 (2002); Ghatak K.P., Bose P.K., Banerjee J.P., SPIE proceedings series, 4746, 351 (2002); Ghatak K.P., SPIE proceedings series, 4746, 292 (2002); Ghatak K.P., Journal of Wave Material Interaction, 14, 157 (1999); Ghatak K.P., Bose P.K., Journal of Wave Material Interaction, 12, 53 (1997). Ghatak K.P., Nag B., Journal of Wave Material Interaction 12, 85 (1997); Mitra B., Basu D.K., Nag B., Ghatak K.P., Nonlinear Optics-Reading, 17, 171 (1997); Ghatak K.P., Bose P.K., Journal of Wave Material Interaction, 12, 60 (1997); Bose P.K., Ghatak K.P., Journal of Wave Material

Preface

[7] [8]

[9]

[10]

[11]

[12] [13] [14]

[15] [16] [17] [18] [19]

XVII

Interaction, 12, 67 (1997), Ghatak K.P., Bose P.K., Majumder G., MRS Proceedings, 494, 157 (1997). Ghatak K.P., Basu D.K., Basu D., Nag B., Il Nuovo Cimento D 18, 947 (1996); Ghatak K.P., Dutta S., Ali A., Banerjee S., Nag B., Journal of Wave Material Interaction, 11, 127 (1996). Ghatak K.P., Chakrabarty P.K., Nag B , Journal of Wave Material Interaction, 11, 159 (1996); Nag B., Chakrabarty P.K., Ghatak K.P., Journal of Wave Material Interaction, 11, 211 (1996); Ghatak K.P., Chakrabarty P.K., Nag B., Journal of Wave Material Interaction, 11, 159 (1996); Ghatak K.P., Banik S.N., Fizika A-ZAGREB-, 5, 31 (1996); Ghatak K.P., Banerjee J.P., Chakrabarty P.K., Nag B., Journal of Wave Material Interaction, 11, 166 (1996); Chakrabarty P.K., Nag B., Dutta S., Ghatak K.P., Journal of Wave Material Interaction, 11, 111 (1996). Ghatak K., Beta S., Ali A., Nag B., A-ZAGREB- F., 5, 111 (1996); Ghatak K.P., Banerjee J.P., Goswami B., Nag B., Nonlinear Optics-reading, 16, 241 (1996); Ghatak K.P., Mitra M., Goswami B., Nag B., Nonlinear Optics-reading, 16, 167 (1996); Ghatak K.P., Bhattacharya D., Basu D., Nag B., Physica Status Solidi (b), 191, 141 (1995); Ghatak K.P., Nag B., Mitra M., Bannerjee J.P., Journal of Wave Material Interaction, 10, 11 (1995); Ghatak K.P., Basu D.K., Nag B., Journal of Wave Material Interaction, 10, 29 (1995); Ghatak K.P., Dutta S., Basu D., Nag B., Journal of Wave Material Interaction, 10, 1 (1995); Ghatak K.P., Nag B., Mazumder G., MRS Proceedings, 379, 85 (1995); Ghatak K.P., Nag B., Mazumder G., MRS Proceedings, 379, 109 (1995), Ghatak K.P., Banik S.N., Fizika A., 3, 155 (1994); Banik S.N., Ghatak K.P., Biswas S., Fizika A., 3, 77 (1994); Ghatak K.P., Fizika A., 2, 133 (1993); Ghatak K.P., Biswas S.N., MRS Proceedings, 313, 375 (1993). Ghatak K.P., Biswas S.N., MRS Proceedings, 308, 445 (1993); Ghatak K.P., Bhattacharyya D., Journal of Wave Material Interaction, 8, 233 (1993); Ghatak K.P., Biswas S.N., Acta Physica Slovaca, 43, 425 (1993); Ghatak K.P., Biswas S.N., Nanostructured Materials, 2, 91 (1993). Ghatak K.P., Mondal M., Fizika A., 1, 113 (1992); Ghatak K.P., De B., MRS Proceedings, 262, 911 (1992); Ghatak K.P., De B., MRS Proceedings, 242, 373 (1992); Ghatak K.P., De B., MRS Online Proceedings Library Archive, 228, (1992); Ghatak K.P., Bhattacharyya S., Biswas S.N., Acta Physica Hungarica, 70, 83 (1991); Ghatak K.P., Acta Physica Hungarica, 69, 121 (1991); Ghatak K.P., De B., MRS Proceedings, 234, 59 (1991). Ghatak K.P., De B., MRS Proceedings, 234, 55 (1991); Ghatak K.P., Acta Physica Hungarica, 68, 253 (1990); Ghatak K.P., Acta Physica Hungarica, 67, 407 (1990). Ghatak K.P., MRS Proceedings, 216, 469 (1990); Ghatak K.P., MRS Proceedings, 216, 465 (1990). Ghatak K.P., Ghoshal A., Bhattacharyya S., Mondal M., Ultrahigh-and High-speed Photography, Videography, Photonics, and Velocimetry’90; Eighth in a Series: 10–13 July 1990, San Diego, California, 8, 471 (1990); Ghatak K.P., De B., Mondal M., Biswas S.N., MRS Proceedings, 184, 261 (1990); Ghatak K.P., De B., Mondal M., Biswas S.N., MRS Proceedings, 198, 333 (1990); Mitra B., Ghatak K.P., Solid-State Electronics, 32, 810 (1989); Ghatak K.P., Mondal M., Journal of Applied Physics, 62, 922 (1987); Biswas S.N., Ghatak K.P., Megagauss Technology and Pulsed Power Applications, 8, 219 (1987); Mondal M., Ghatak K.P., Czechoslovak Journal of Physics, 36, 1396 (1986); Chakravarti A.N., Ghatak K.P., Ghosh K.K., Ghosh S., Dhar A., Physica Status Solidi B-Basic Research, 103, K55 (1981). Landsberg P.T., Phys. Lett 102A, 171 (1984). Ben-Naim A., Entropy 9, 132 (2007); Ben-Naim A., Entropy Demystified. World Scientific. Singapore, (2007); Srednicki M., Phys. Rev. Lett. 71, 666 (1993). Landsberg P.T., J. Stat. Phys 35, 159 (1984). Das P.K., Dutta P., Halder A., Bhattacharjee R., Ghatak K.P., Materials Focus 6, 133 (2017). Bhattacharjee R., Ghatak K.P., Journal of Nanoscience and Nanotechnology 17, 640 (2017).

XVIII

References

[20] Zawadzki W., Two-Dimensional Systems, Hetero structures and Superlattices; Bauer G., Kuchar F., Heinrich H. (ed.) Springer Ser. Solid-State Sci., Vol. 53 (Springer-Verlag, Germany, 1984) Zawadzki. W., 11th Internet Conf. Phys. Of Semicond. Vol. 1 (Elsevier Publishing Company, Netherlands, 1972); Zelenin S.P., Kondrat’ev A.S., Kuchma A.E., Sov. Phys. Semicond. 16, 355, (1982). [21] Pan H.J., Wang W.C., Thai K.B., Cheng C.C., Yu K.H., Lin K.W., Wu C.Z., Liu W.C., Semicond. Sci. Technol 15, 1101 (2000). [22] Mohammad S.N., J. Appl. Phys 95, 4856 (2004). [23] Arora V.K., Appl. Phys. Lett 80, 3763 (2002). [24] Mohammad S.N., J. Appl. Phys 95, 7940 (2004). [25] Mohammad S.N., Philos. Mag 84, 2559 (2004). [26] Mohammad S.N., J. Appl. Phys 97, 063703 (2005). [27] Suzue K., Mohammad S.N., Fan Z.F., Kim W., Aktas O., Botchkarev A.E., Morkoç H., J. Appl. Phys 80, 4467 (1996). [28] Mohammad S.N., Fan Z.F., Kim W., Aktas O., Botchkarev A.E., Salvador A., Morkoç H., Electron. Lett 32, 598 (1996). [29] Fan Z., Mohammad S.N., Kim W., Aktas O., Botchkarev A.E., Suzue K., Morkoç H., J. Electron. Mater 25, 1703 (1996). [30] Lu C., Chen H., Lv X., Xia X., Mohammad S.N., J. Appl. Phys 91, 9216 (2002). [31] Dmitriev S.G., Markin Yu V., Materials 34, 931 (2000). [32] Willardson R.K., Beer A.C. eds., Semiconductors and Semimetals, Vol. 1 102 (Academic, New York, 1966). [33] Kane E.O., Phys. Rev 131, 79 (1963); Phys. Rev. B., 139, 343 (1965). [34] Bonch Bruevich V.L., Sov. Phys.Sol. Stat 4, 53 (1963). [35] Logan R.A., Chynoweth A.G., Phys. Rev 131, 89 (1963). [36] Hwang C.J., J. Appl. Phys 40, 3731 (1969). [37] Pankove J.I., Phys. Rev. A 130, 2059 (1965). [38] Halperin B.I., Lax M., Phys. Rev 148, 722 (1966). [39] Abram R.A., Rees G.J., Wilson B.L.H., Adv. Phys 27, 799 (1978). [40] Shklovskii B.I., Efros A.L., Electronic Properties of Doped Semiconductors, Springer series in Solid State Sciences, Vol. 45 (Springer, Berlin, 1984). [41] Kane E.O., Solid State Electron 28, 3 (1985). [42] Chakraborty P.K., Biswas J.C., J. Appl. Phys 82, 3328 (1997). [43] Nag B.R., Electron Transport in Compound Materials, Springer Series in Solid state Sciences, Vol. 11 (Springer, Heidelberg, 1980). [44] Schmid P.E., Phys. Rev. B 23, 5531 (1981). [45] Jellison G.E. Jr., Modine F.A., White C.W., Wood R.F., Young R.T., Phys. Rev. Lett 46, 1414 (1981). [46] Fistul V.I., HD Materials (New York, Plenum, 1969); ch 7. [47] Hwang C.J., J. Appl. Phys 41, 2668 (1970); Sritrakool W., Glyde H. R., Yakanit. V. Sa, Can. J. Phys., 60, 373 (1982); Ikoma. H., J. Phys. Soc. Japan, 27, 514 (1969). [48] Debbarma S., Ghatak K.P., Rev.Theo. Sci 3, 16 (2015). [49] Modern Perspective on Thermoelectrics and related materials, MRS Symposium Proceedings, Spring Meeting 234, 59, (1991); Ghatak K.P., Chowdhury A.K., Ghosh S., Chakravarti A.N., Czecho. Jour of Phys. B 30, 925 (1980); Ghatak K.P., Chowdhury A.K., Ghosh S., Chakravarti A.N., Acta Phys. Polon. A 58, 389 (1980); Chakravarti A.N., Ghatak K.P., Ghosh K.K., Ghosh. S., Dhar A., Phys. Stat. Sol. (b) 103, K55 (1981); Chakravarti A.N., Ghatak K.P., Ghosh S., Chowdhury A.K., Phys. Stat. Sol. (b) 109, 705 (1982); Ghatak K.P., Ghosh K.K., Mukherjee H. M., Chakravarti A.N., Phys. Stat. Sol. (b) 110, 323 (1982); Ghatak K.P., Bhattacharya S.,

Preface

[50] [51] [52] [53] [54]

[55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66]

[67]

XIX

Bhowmik S., Benedictus R., Chowdhury S., J. Appl. Phys. 103, 094314 (2008); Chakravarti A. N., Ghatak K.P., Rao G.B., Ghosh K.K., Phys. Stat. Sol. (b) 112 75 (1982); Mondal M., Ghatak K. P., Phys. Stat. Sol. (b) 120, K63 (1983); Mondal M., Ghatak K.P., Acta Phys. Polon. A 66, 539 (1984); Mondal M., Ghatak K.P., Acta Phys. Polon. A 66, 47 (1984); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 126, K41 (1984); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 126, K47 (1984); Mondal M., Banik S., Ghatak K.P., Jour. of low Temp. Phys. 74, (423) (1989); Mitra B., Ghatak K.P.,) Phys. Scr. 40, 776 (1989); Ghatak K.P., Mondal M., Phys. Stat. Sol. (b) 160, 673 (1990); Ghatak K.P., Mondal M., Jour. of Appl. Phys. 69, 1666 (1991); Ghatak K.P., Il Nuovo Cimento D 14, 1187 (1992); Nag B., Ghatak K.P., Jour. of Phys. and Chem. of Sol. 59, 713 (1998); Chakraborty P.K., Ghatak K.P., Jour. of Phys. D. Appl. Phys. 32, 2438 (1999); Chakraborty P.K., Biswas S.K., Ghatak K.P., (Physica B: Condensed Mattr, 352, 111 (2004). Keldysh L.V., Soviet Phys. Solid State 4 (1962). Esaki L., Tsu R., IBM J. Research and Develop. 14, 61 (1970). Tsu R., Superlattices to nanoelectronics (Elsevier, 2005); Ivchenko E.L., Pikus. G., (1995) Superlattices and other heterostructures, Springer-Berlin. Bastard G., Wave mechanics applied to heterostructures (Editions de Physique, Les Ulis, France, 1990). Ghatak K.P., De B., MRS Proceedings, Vol. 299, 65 (Cambridge University Press, 1994); Ghatak K.P., Mondal M., Physica Status Solidi (b) 175, 113 (1993); Ghatak K.P., De B., MRS Proceedings, Cambridge University Press, 242, 377 (1992); Ghatak K.P., San Diego,’91, San Diego, CA, Intern. Soc. Opt. Photon. 282 (1991); Mitra B., Ghatak K. P. physica status solidi (b) 164, K13 (1991); Ghatak K.P., Properties of II-VI Materials: Bulk Crystals, Epitaxial Films, Quantum Well Structures, and Dilute Magnetic Systems, Materials Research Society, 161, 371 (1990); Ghatak K.P., Biswas S.N., MRS Proceedings, Cambridge University Press, 161 (1989); Bhattacharyya S., Ghatak K.P., Biswas S., OE/Fibers’ 87, 73 (1987); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 122, K93 (1984). Vaidyanathan K.V., Jullens R.A., Anderson C.L., Dunlap H.L., Solid State Electron 26, 717 (1983). Wilson B.A., IEEE, J. Quantum Electron. 24, 1763 (1988). Krichbaum M., Kocevar P., Pascher H., Bauer G., IEEE, J. Quantum Electron. 24, 717 (1988). Schulman J.N., McGill T.C., Appl. Phys. Letts 34, 663 (1979). Kinoshita H., Sakashita T., Fajiyasu H., J. Appl. Phys. 52, 2869 (1981). Ghenin L., Mani R.G., Anderson J.R., Cheung J.T., Phys. Rev. B 39, 1419 (1989). Hoffman C.A., Mayer J.R., Bartoli F.J., Han J.W., Cook J.W., Schetzina J.F., Schubman J.M., Phys. Rev. B. 39, 5208 (1989). Yakovlev V.A., Sov. Phys. Semicon 13, 692 (1979). Kane E.O., J. Phys. Chem. Solids 1, 249 (1957). Sasaki H., Phys. Rev. B 30, 7016 (1984). Ghatak K.P., Bhattacharya S., Pahari S., De D., Ghosh S., Mitra M., Annalen dER Physik 17, 195 (2008). Bose P.K., Paitya N., Bhattacharya S., De D., Saha S., Chatterjee K.M., Pahari S., Ghatak K.P., Quantum Matter 1, 89 (2012); Bhattarchya S., De D., Ghosh S., Banerjee P., Saha S., Mitra M., Nag B., Pal M., Biswas S.K., Ghatak K.P., Jour. of Comput. and Theo. Nanosci. 7, 1066 (2010); Ghatak K. P., Bhattacharya S., Pahari S., Mitra S.N., Bose P.K., De D., Jour. of Phys. and Chem. of Solids. 70, 122 (2009); Bhattacharya S., Paul N.C., De D., Ghatak K.P., Phys. B: Condensed Matter. 403, 4139 (2008). Anderson N.G., Laidig W.D., Kolbas R.M., Lo Y.C., J. Appl. Phys 60, 2361 (1986); Paitya N., Ghatak K.P., J. Adv. Phys. 1, 161 (2012); Bhattacharya S., De D., Adhikari S.M., Ghatak K.P., Superlatt. Microst. 51, 203 (2012); De D., Bhattacharya S., Adhikari S.M., Kumar A., Bose P.K.,

XX

[68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87]

References

Ghatak K.P., Beilstein J. Nanotech. 2, 339 (2012); De D., Kumar A., Adhikari S.M., Pahari S., Islam N., Banerjee P., Biswas S.K., Bhattacharya S., Ghatak K.P., Superlatt. and Microstruct. 47, 377 (2010); Pahari S., Bhattacharya S., Roy S., Saha A., De. D., Ghatak K.P., Superlatt. and Microstruct. 46, 760 (2009); Chowdhary S., Singh L.J., Ghatak K. P., Phys. B: Conden. Matter, 365, 5 (2005); Ghatak K.P., Mukhopadhyay J., Banerjee J.P., SPIE proceedings series, 4746 1292 (2002); Ghatak K.P., Dutta S., Basu D.K., Nag B., Il Nuovo Cimento. D 20, 227 (1998); Ghatak K.P., De B., Mat. Resc. Soc. Proc. 300, 513 (1993); Ghatak K.P., Inter. Soci.Opt. and Photon. Proc. Soc. Photo Opt. Instru. Engg., 1626, 115 (1992); Ghatak K.P., Ghoshal A., Biswas S.N., Mondal M., Proc. Soc. Photo Opt. Instru. Engg. 1308, 356 (1990); Biswas S.N., Ghatak K.P., Internat. Jour. Electronics Theo. Exp. 70, 125 (1991); Mitra B., Ghatak K.P., Phys. Lett. A. 146, 357 (1990); Mitra B., Ghatak K.P., Phys. Lett. A. 142, 401 (1989); Ghatak K.P., Mitra B., Ghoshal A., Phy. Stat. Sol. (b) 154, K121 (1989); Mitra B., Ghatak K.P., Phys. Stat. Sol. (b). 149, K117 (1988); Bhattacharyya S., Ghatak K.P., Biswas S., OE/Fibers’ 87, Inter. Soc. Opt. Photon. 73 (1987); Ghatak K.P., Chakravarti A.N., Phys. Stat. Sol. (b). 117, 707 (1983). Capasso F., Semiconductors and Semimetals 22, 2 (1985). Capasso F., Mohammed K., Cho A.Y., Hull R., Hutchinson A.L., Appl. Phys. Letts 47, 420 (1985). Capasso F., Kiehl R.A., J. Appl. Phys 58, 1366 (1985). Ploog K., Doheler G.H., Adv. Phys 32, 285 (1983). Capasso F., Mohammed K., Cho A.Y., Appl. Phys. Lett 478 (1986). Grill R., Metzger C., Döhler G.H., Phys. Rev. B 63, 235316 (2001). Kost A.R., Jupina M.H., Hasenberg T.C., Garmire E.M., J. Appl. Phys. 99, 023501 (2006). Smirnov A.G., Ushakov D.V., Kononenko V.K., Proc. SPIE 4706, 70 (2002). Ushakov D.V., Kononenko V.K., Manak I.S., Proc. SPIE 4358, 171 (2001). Wang J.Z., Wang Z.G., Wang Z.M., Feng S.L., Yang Z., Phys. Rev. B 62, 6956 (2000). Kost A.R., West L., Hasenberg T.C., White J.O., Matloubian M., Valley G.C., Appl. Phys. Lett. 63, 3494 (1993). Bastola S., Chua S.J., Xu S.J., J. Appl. Phys 83, 1476 (1998). Yang Z.J., Garmire E.M., Doctor D., J. Appl. Phys 82, 3874 (1997). Avetisyan G.H., Kulikov V.B., Zalevsky I.D., Bulaev P.V., Proc. SPIE 2694, 216 (1996). Pfeiffer U., Kneissl M., Knüpfer B., Müller N., Kiesel P., Döhler G.H., Smith J.S., Appl. Phys. Lett 68, 1838 (1996). Vaghjiani H.L., Johnson E.A., Kane M.J., Grey R., Phillips C.C., J. Appl. Phys 76, 4407 (1994). Kiesel P., Gulden K.H., Hoefler A., Kneissl M., Knuepfer B., Dankowski S.U., Riel P., Wu X.X., Smith J.S., Doehler G.H., Proc. SPIE 85, 278 (1993). Doheler G.H., Phys. Script 24, 430 (1981). Mukherjee S., Mitra S.N., Bose P.K., Ghatak A.R., Neoigi A., Banerjee J.P., Sinha A., Pal M., Bhattacharya S., Ghatak K.P., J. Compu. Theor Nanosc 4, 550 (2007). Quinn J.J., Styles P. J. (ed.), Electronic Properties of Quasi Two Dimensional Systems (North Holland, Amsterdam, 1976); Antcliffe G.A., Bate R.T., Reynolds R.A., (1971) Proceedings of the International Conference, Physics of Semi-metals and Narrow-Gap Materials (ed.) Carter D.L., Bate R.T., Pergamon Press, Oxford, (499); Paasch G., Fiedler T., Kolar M., Bartos I., Phys. Stat. Sol. (b) 118, 641 (1983); Lindner Th., Paasch G., J. Appl. Phys., 102, 054514 (2007); Lamari S., J. Appl. Phys. 91, 1698 (2002); Ghatak K.P., Biswas S.N., J. Vac. Sc. and Tech., 7B, 104) (1989); Mitra B., Ghatak K.P., Sol. State Electron. 32, 177 (1989); Ghatak K.P., Mondal M., J. Appl. Phys., 62, 922 (1987); Mondal M., Ghatak K.P., Phys. Script., 31, 613 (1985); Ghatak K.P., Mondal M., Z F, Physik B, 64, 223 (1986); Ghatak K.P., Biswas S.N., Sol. State Electron., 37, 1437 (1994); Choudhury D.R., Chowdhury A.K., Ghatak K.P., Chakravarti A.N.,

Preface

[88] [89]

[90] [91] [92]

XXI

Phys. Stat. Sol. (b) 98, K141 (1980); Chakravarti A.N., Chowdhury A.K., Ghatak K.P., Phys. Stat. Sol. (a) 63, (K97) (1981); Mondal M., Ghatak K.P., Acta Phys. Polon. A 67, 983 (1985); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 128, K21 (1985); Mondal M., Ghatak K.P., Phys. Stat. Sol. (a) 93, 377 (1986); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 139, 185 (1987); Ghatak K.P., Chatterjee N., Mondal M., Phys. Stat. Sol. (b) 139, K25 (1987); Ghatak K.P., Mondal M., Phys. Stat. Sol. (b) 148, 645 (1988); Ghatak K.P., Ghosal A., Phys. Stat. Sol. (b) 151, K135 (1989); Ghatak K.P., Chattopadhyay N., Mondal M., Appl. Phys. A 48, 365 (1989). Zawadzki W., Lax B., Phys. Rev. Lett 16, 1001 (1966). Ghatak K.P., Banerjee J.P., Goswami B., Nag B., Non. Opt. and Quan. Opt 16, 241 (1996); Mitra B., Ghoshal A., Ghatak K.P., Phys. Stat. Sol. (b), 154, K147 (1989); Ghatak K.P., Goswami B., Mitra M., Nag B., Non. Opt. and Quan. Opt., 16, 9 (1996); Mondal M., Ghatak K.P., Ann. der Physik. 46, 502 (1989); Ghatak K.P., OE Fiber-DL, 435 (1991); Ghatak K.P., Mitra B., Internat.Jour. Electron. Theo. Exp., 70, 343 (1991); Ghatak K.P., Proc. Soc. Photo-Optical Instru. Engg., USA, 1280, 53 (1990); Mitra B., Ghatak K.P., Phys. Lett. A 137, 413 (1989); Mondal M., Ghatak K.P., Annalen der Physik, 501, (502) (1989); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 147, K179 (1988); Biswas S., Chattopadhyay N., Ghatak K.P., International Society for Optics and Photonics, Proc. Soc. Photo-Optical Instru. Engg., USA, 836, 175 (1987). Pais A., Robert J., Oppenheimer, xviii (Oxford University Press, U.K, Vol. 2006). Kikoin I.K., Science for Everyone: Encounters with Physicists and Physics, 154 (Mir Publishers, Russia, 1989). Sakurai J.J., Advanced Quantum Mechanics (Addision –Wesley Publishing Company, USA, 1967).

Acknowledgments Real gentleness in a person is the power that sees, understands, yet never interferes.

Acknowledgment by Kamakhya Prasad Ghatak I am grateful to A.N. Chakravarti, formerly Head of the Department of the Institute of Radio Physics and Electronics of the University of Calcutta, my mentor and an ardent advocator of the concept of theoretical minimum of Landau, who convinced a 21 years old circuit theorist that Condensed Matter Physics, in general, is the hidden dual dance of quantum mechanics, statistical mechanics together with advanced mathematical techniques, and even to appreciate the incredible beauty, he not only placed a stiff note for me to derive all the equations in the Monumental Course of Theoretical Physics, the Classics of Landau Lifshitz together with the two-volume classics of Morse-Feshbach 45 years ago but also forced me to stay in the creative critical zone of research till date. I express my gratitude to Late B.R. Nag, formerly Head of the Departments of Radio Physics and Electronics and Electronic Science of the University of Calcutta, to whom I am ever grateful as a student and research worker, the Semiconductor Grandmaster of India for his three books on semiconductor science in general and more than 200 research papers (many of them are absolutely in honor’s class), which still fire my imagination. I consider myself to be rather fortunate to study Mathematics under the direct influence of Late S.C. Dasgupta, formerly Head of the Department of Mathematics of the then Bengal Engineering College, Shibpur (presently Indian Institute of Engineering Science and Technology) and M. Mitra (both of them were formidable Applied Mathematicians with deep physical insight and could solve any problem from the two-volume classics of MorseFeshbach instantly on the blackboard) of the said Department of Mathematics for teaching me deeply the various methods of different branches of Applied Mathematics with special emphasis to analytic number theory when I was pursuing the bachelor degree in the branch of Electronics and Telecommunication Engineering 45 years ago. I offer my thanks to Late S.S. Boral, formerly Founding Head of the Department of Electronics and Telecommunication Engineering of the then Bengal Engineering College, Shibpur, for teaching me the Theoretical Acoustics from the classics of Morse and Ingard by urging me to solve the problems. I am grateful to S. K. Sen, formerly Head of the Department of Electrical Engineering of Bengal Engineering College, Shibpur, Ex-Vice-Chancellor of Jadavpur University and Ex-Power Minister of West Bengal State for teaching me deeply non-linear network analysis and synthesis and non-linear control systems in general. I am indebted to Late C.K. Majumdar, Head of the Department of Physics of the University of Calcutta for lighting the fire for Theoretical Physics within me. https://doi.org/10.1515/9783110661194-203

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I am grateful to all my friends and colleagues for the last 40 years from my school days till date for forming my inner fire to solve independently the problems from five volumes of Berkley Physics Course, three volumes of Sakurai, ten volumes of Addison-Wesley Modular Series on Solid State Devices, two volumes of CohenTannoudji et al., three volumes of Edwards, three volumes of Carslaw, six volumes of Guillemin, three volumes of Tuttle, two volumes of Scott, two volumes of Budak, five volumes of Chen. It is curious to note that they are insisting me in the real sense of the term to verify all the entries of Gradshteyn-Ryzhik and three volumes of Bateman manuscript project for the last 40 years. It may be noted that academic output of a nice scholar is the response of a good quality RC coupled amplifier in the mid-band zone, whereas the same for a fine research worker is the Dirac’s delta function. Incidentally, I can safely say that I belong to neither as a consequence of the academic surroundings which this life presents to me. I express my warm gratitude to H. L. Hartnagel, D. Bimberg, W. L. Freeman, and W. Schommers for various academic interactions spanning over the last three decades. I remember the sweet memories of P.N. Robson, I.M. Stephenson, V.S. Letokhov, J. Bodnar, and K.F. Brennan with true reverence for inspiring me with the ebullient idea that the publications of the research monographs from internationally reputed publishers containing innovative concepts is of prime importance to excel in creative research activity. On 30 December 2006, I wrote a letter to Late P.T. Landsberg (popularly known as P.T.L among his scientific friends) for writing a book on Electron Statistics of nanostructured materials which can at least transform me in to a better scientist but P.T.L in turn requested me repeatedly to complete the proposed book at the earliest and often expressed his desire to write a foreword for this book. Incidentally, due to previous other heavy academic and related commitments I was unable to finish this present monograph and his wife Mrs. Sophia Landsberg in her letter of desperate sadness dated January 12, 2009, informed me that from 2008 onward due to Alzheimer’s disease her husband had to give up scientific works. The disappearance of P.T.L. from my research scenario is a big shock to me, which I will have to bear till my own physical disappearance. I still remember his power-packed words that “The definition of Kamakhya Prasad Ghatak (K.P.G.) = M.Tech. + Ph.D. + D. Engg. + more than 300 research papers in nanoscience and technology in SCI Journals + more than 60 research papers and International Conferences of SPIE and MRS of USA + Ph.D. guide of more than three dozens of Ph.D. students + many research monographs form Springer-Verlag, Germany + HOD + Dean+ Senior Professor is a big zero to P.T.L. if K.P.G. cannot live in a world full of new creative concepts and not merely repeating the past successes, since past success is a dead concept at least to P.T.L. On 21 December 1974, A. N. Chakravarti (an internationally recognized expert on Einstein Relation in general), my mentor in my first interaction with him, emphatically

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energized me by making myself acquainted with the famous Fermi-Dirac Integrals and telling me that I must master Semiconductor Statistics (Pergamon Press, London, 1962) by J. S. Blakemore for my initiation in semiconductor physics. Later on, I got in touch with K. Seeger, the well-known author of the book Semiconductor Physics (Springer Series in Solid State Sciences, vol. 40, Springer-Verlag, Germany, 1982). The solid mathematical physicist Late P. N. Butcher has been a steady hidden force since 1983 before his sad demise with respect to our scripting the series in band structure dependent properties of nanostructured materials. Both P. T. L. and P. N. Butcher visited the Institute of Radio Physics and Electronics of the University of Calcutta, my ALMA MATER where I started my research as a M. Tech. student and later on as a faculty member. I formed permanent covalent bonds with J. S. Blakemore, K. Seeger, P. N. Butcher, and P.T. L through letters (pre-email era), and these four first class semiconductor physicist, in turn, shared with pleasure their vast creative knowledge of materials and related sciences with a novice like me. I offer special thanks to Late N. Guhachoudhury of Jadavpur University for instilling in me the thought that the academic output = ((desire × determination × dedication) – (false enhanced self ego pretending like a true friend although a real unrecognizable foe)). I am grateful to all the members of my research group (from 1983 till date) for not only quantum confining me in the infinitely deep quantum wells of Ramanujan and Rabindranath but also inspiring me to teach quantum mechanics and related topics from the eight volumes classics of Greiner et al. In this context, from the flashback of my memory I wish to offer my indebtedness to M. Mondal, the first member of my research team who in 1983 joined with me to complete his Ph.D. work under my guidance when R. K. Poddar, the then Vice-chancellor of the University of Calcutta selected me as a Lecturer (presently Assistant Professor) of the famous Department of Radio Physics and Electronics. In 1987, S. K. Sen, the then Vice-chancellor of Jadavpur University accepted me as the Reader (presently Associate Professor) in the Department of Electronics and Telecommunication Engineering and since then many young researchers (more than 12 in numbers consisting of B. Mitra, A. Ghoshal, D. Bhattacharya, A. R. Ghatak, and so on) around me created my research team and insisted me to work with them at the rate of 16 hours per day including holidays in different directionsof research for the purpose of my creative conversion from an ordinary engineer to a 360° research scientist, and consequently I enjoyed the high rate of research publications in different reputed international journals in various aspects of band structuredependent properties of quantized structures. It is nice to note that these young talented researchers obtained their respective Ph. D degree under my direct supervision. Incidentally, in 1994, R. K. Basu, the then Vice-chancellor of the University of Calcutta selected me as a Professor in the Department of Electronic Science and another flood of research overwhelmed me in a new direction of materials science. The persons responsible for this change include S. Datta, S. Sengupta, A. Ali, and so on. The 11th and 12th names of this golden series are S. Bhattacharya and D. De,

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respectively, who, in turn, formed permanent covalent bonds with me not only with respect to research (S. Bhattacharya and D. De are respectively the coauthors of seven and two monographs in different series of Springer) but also in all aspects of life in general. It is interesting to note that after serving 18 years as a Professor in the Department of Electronic Science, in 2012, P. K. Bose, and the then Director of the National Institute of Technology, Agartala, requested me to join as a Professor and Departmental Head in Electronics and Communication Engineering. He being my life-long friend, I accepted his offer (and later on as a Dean), and more than 10 young scholars around me again directed my research in an altogether new direction. In 2015 the respected Director Professor S. Chakrabarti of Institute of Engineering and Management in Salt Lake City, Kolkata, has kindly offered me the position of Research Director and Senior Professor in his famous institute in the academic fag end of my life to complete my last run toward the creative knowledge temple with my new young research workers. I am grateful to him for his creative gesture. In my 40+ years of teaching life (I have the wide experience of teaching Physics, Mathematics, Applied Mechanics – from engineering statics up to nonlinear mechanics including indeterminate structures) and 70% of the courses of Electronics and Telecommunication and Electrical Engineering, respectively) and 40+ years of research life (mostly in Materials Science, Nano-science and Number Theory), I have finally realized that teaching without research is body without brain and research without teaching is body without blood although my alltime hero, creatively prolific number-theorist Godfrey Harold Hardy in his classic book entitled A Mathematician’s Apology (Cambridge University Press, 1990) tells us “I hate teaching.” Incidentally, one young theoretician friend of mine often tells me that many works in theoretical semiconductor science are based on the following seven principles: – Principles of placing the necessary and sufficient conditions of a proof in the band-gap regime. – Principles of interchange of the summation and the integration processes and unconditioned convergences of the series and the integrals. – Principles of random applications of one electron theory and superposition theorem in studying the properties of materials, although the many body effects are very important together with the fact that the nature is fundamentally nonlinear in nature. – Principles of using the invariant band structure concept of materials even in the presence of strong external fields (light, electric, heavy doping, etc.) and the random applications of perturbation theory, which is in a sense quantum mechanical Taylor series without considering the related consequences. – Principle of random applications of the binomial theorem without considering the important concept of branch cut. – Principle of little discussion regarding the whole set of materials science comprising of different compounds having various band structures under different

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physical conditions as compared with the simplified two-band model of Kane for III–V materials. – Principle of using the Fermi’s golden rule, the band structure, and the related features, which are valid for nondegenerate materials to materials having degenerate carrier concentrations directly. Although my young friend is a purist in his conjecture, there are no doubt certain elements of truth inside his beautiful comments. I hope that our readers will present their intricate and advanced theories after paying due weightage of his aforementioned seven principles. I offer special thanks to the members of my research team for placing their combined effort toward the development of this book in the DO-LOOP of a computer and critically reading the manuscript in its last phase before sending it to C. Ascheron, Ex-Executive Editor Physics, Springer-Verlag. Last but not the least; I am grateful forever to our lifelong time-tested friend S. Sanyal, Principal, Lake School for Girls, Kolkata, for not only motivating me at rather turbulent moments of my academic carrier but also instilling in me the concept that the ratio of total accumulated knowledge in my chosen field of research to my own contribution in my topic of research tends to infinity at any time and is the definition of nonremovable pole in the transfer function of my life. As always, myself with the members of my research team are grateful forever to Dr. C. Ascheron, Ex-Executive Editor Physics, Springer Verlag, Germany, in the real sense of the term for his inspiration and priceless technical assistance from the very start of our first monograph and that too from Springer. C. Ascheron is the hidden force behind the publications of nine monographs, the collective output of myself and my research group for the last 40 years, from Springer. Dr. Ascheron has proposed my name for being a future author in the STEM program of De Gruyter and forwarded the manuscript of this book to Dr. K. Berber-Nerlinger, Acquisitions Editor, Physics, of De Gruyter for accepting our bookproposal like the first volume of the series on Nanomaterials. We are extremely grateful to Dr. V. Schubert. Dr. R. Fritz, and Dr. S. Manogarane because without their concentrated combined joined effects this volume could not be produced in printed form. Naturally, the authors are responsible for nonimaginative shortcomings. We firmly believe that Mother Nature has propelled this project in her own unseen way in spite of several insurmountable obstacles.

Acknowledgment by Madhuchanda Mitra It is a great pleasure to express my gratitude to Professor K.P. Ghatak for instigating me to carry out extensive researches in the fields of materials science. I am thankful to the faculty members and staffs of Department of Applied Physics, University of Calcutta, for support and cooperation. My family members especially my husband

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and son deserve a very special mention for forming the backbone of my long unperturbed research carrier. Kolkata, India 9 January 2019

K. P. Ghatak, M. Mitra

Contents Preface

VII

Acknowledgments

XXIII

About the Authors

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Symbols

XXXIX

Part I: Entropy in heavily doped quantum confined nonparabolic materials 1 1.1 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7 1.2.8 1.2.9 1.2.10 1.2.11 1.2.12 1.2.13 1.2.14 1.3 1.4

The entropy in quantum wells of heavily doped materials 3 Introduction 3 Theoretical background 6 Entropy in quantum wells (QWs) of HD nonlinear optical materials 6 Entropy in quantum wells (QWs) of HD III–V materials 20 The entropy in quantum wells (QWs) of HD II–VI materials 36 The entropy in quantum wells (QWs) of HD IV–VI materials 40 The entropy in quantum wells (QWs) of HD stressed Kane type materials 51 The entropy in quantum wells (QWs) of HD Te 59 The entropy in quantum wells (QWs) of HD gallium phosphide 63 The entropy in quantum wells (QWs) of HD platinum antimonide 67 The entropy in quantum wells (QWs) of HD bismuth telluride 72 The entropy in quantum wells (QWs) of HD germanium 74 The entropy in quantum wells (QWs) of HD gallium antimonide 82 The entropy in quantum wells (QWs) of HD II–V materials 86 The entropy in quantum wells (QWs) of HD lead germanium telluride 87 The entropy in quantum wells (QWs) of HD Zinc and Cadmium diphosphides 88 Result and discussions 89 Open research problems 97 References 104

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2 The entropy in nanowires of heavily doped materials 111 2.1 Introduction 111 2.2 Theoretical background 111 2.2.1 The entropy in NWs of HD nonlinear optical materials 111 2.2.2 The entropy in NWsof HD III–V materials 113 2.2.3 The entropy in nanowires of HD II–VI materials 120 2.2.4 The entropy in nanwires of HD IV–VI materials 121 2.2.5 The entropy in nanowiresof HD stressed kane-type materials 2.2.6 The entropy in nanowires of HD Te 130 2.2.7 The entropy in nanowires of HD gallium phosphide 131 2.2.8 The entropy in nanowires of HD platinum antimonide 133 2.2.9 The entropy in nanowires of HD bismuth telluride 134 2.2.10 The entropy in nanowires of HD germanium 136 2.2.11 The entropy in nanowires of HD galium antimonide 139 2.2.12 The entropy in nanowires of HD II–V materials 140 2.2.13 Entropy in nanowires of HD lead germanium telluride 143 2.2.14 Entropy in nanowires of HD zinc and cadmium diphosphides 2.3 Results and discussion 147 2.4 Open research problems 152 References 154

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3 Entropy in quantum box of heavily doped materials 157 3.1 Introduction 157 3.2 Theoretical background 158 3.2.1 Entropy in QB of HD nonlinear optical materials 158 3.2.2 Entropy in QB of HD III–V materials 159 3.2.3 Entropy in QB of HD II–VI materials 164 3.2.4 Entropy in QBof HD IV–VI materials 165 3.2.5 Entropy in QB of HD stressed Kane-type materials 167 3.2.6 Entropy in QB of HD Te 167 3.2.7 Entropy in QB of HD gallium phosphide 168 3.2.8 Entropy in QB of HD platinum antimonide 169 3.2.9 Entropy in QB of HD bismuth telluride 170 3.2.10 Entropy in QB of HD germanium 171 3.2.11 The entropy in quantum box (QB) of HD gallium antimonide 173 3.2.12 The entropy in quantum box (QB) of HD II–V materials 174 3.2.13 The entropy in quantum box (QB) of HD lead germanium telluride 175 3.2.14 The entropy in quantum box (QB) of HD zinc and cadmium diphosphides 177

Contents

3.3 3.4

4 4.1 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.2.6 4.2.7 4.2.8 4.2.9 4.2.10 4.2.11 4.2.12 4.2.13 4.3 4.4

5 5.1 5.2 5.2.1 5.2.2 5.2.3 5.2.4

Results and discussion Open research problems References 197

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179 196

Entropy in heavily doped materials under magnetic quantization 201 Introduction 201 Theoretical background 202 Entropy in HD nonlinear optical materials under magnetic quantization 202 Entropy in QWs of HD III–V materials under magnetic quantization 205 Entropy in HD II–VI materials under magnetic quantization 211 Entropy in HD IV–VI materials under magnetic quantization 212 Entropy in HD stressed Kane-type materials under magnetic quantization 218 Entropy in HD Te under magnetic quantization 219 Entropy in HD Gallium Phosphide under magnetic quantization 220 Entropy in HD platinum antimonide under magnetic quantization 221 Entropy in HD Bismuth Telluride under magnetic quantization 221 Entropy in HD Germanium under magnetic quantization 222 Entropy in HD Gallium antimonide under magnetic quantization 223 The entropy in HD II–V materials under magnetic quantization 224 Entropy in HD lead germanium telluride under magnetic quantization 226 Results and discussion 227 Open research problems 237 References 239 The entropy in heavily doped nanomaterials under magneto-size quantization 241 Introduction 241 Theoretical background 241 Electron energy spectrum in HD nonlinear optical materials under magneto-size quantization 241 Entropy in QWs of HD III–V materials under magneto-size quantization 242 Entropy in HD II–VI materials under magneto-size quantization 245 Entropy in HD IV–VI materials under magneto size-quantization 245

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5.2.5

Entropy in HD stressed Kane type materials under magneto-size quantization 248 Entropy in HD Te under magneto size-quantization 248 Entropy in HD gallium phosphide under magneto size quantization 249 Entropy in HD platinum antimonide under magneto size quantization 250 Entropy in HD bismuth telluride under magneto size quantization 250 Entropy in HD germanium under magneto size quantization 251 Entropy in HD gallium antimonide under magneto-size quantization 252 Entropy in HD II–V materials under magneto size quantization 252 Entropy in HD lead germanium telluride under magneto size quantization 253 Results and discussion 253 Open research problems 259 References 260

5.2.6 5.2.7 5.2.8 5.2.9 5.2.10 5.2.11 5.2.12 5.2.13 5.3 5.4

Part II: Entropy in heavily doped quantum confined superlattices 6 6.1 6.2 6.2.1 6.2.2 6.2.3 6.2.4

Entropy in quantum wires of heavily doped superlattices 265 Introduction 265 Theoretical background 266 Entropy in III–V quantum wire HD SLs with graded interfaces 266 Entropy in II–VI quantum wire HD SLs with graded interfaces 270 Entropy in IV–VI quantum wire HD SLs with graded interfaces 273 Entropy in HgTe/CdTe quantum wire HD SLs with graded interfaces 275 6.2.5 Entropy in strained layer quantum wire HD SLs with graded interfaces 277 6.2.6 Entropy in III–V quantum wire HD effective mass SLs 280 6.2.7 Entropy in II–VI quantum wire HD effective mass SLs 282 6.2.8 Entropy in IV–VI quantum wire HD effective mass SLs 284 6.2.9 Entropy in HgTe/CdTe quantum wire HD effective mass SLs 287 6.2.10 Entropy in strained layer quantum wire HD effective mass SLs 288 6.3 Results and discussion 290 6.4 Open research problem 293 References 294

Contents

7 7.1 7.2 7.2.1 7.2.2 7.2.3 7.2.4

XXXIII

Entropy in quantum dot HDSLs 297 Introduction 297 Theoretical background 297 Entropy in III–V quantum dot HD SLs with graded interfaces 297 Entropy in II–VI quantum dot HD SLs with graded interfaces 298 Entropy in IV–VI quantum dot HD SLs with graded interfaces 298 Entropy in HgTe/CdTe quantum dot HD SLs with graded interfaces 299 7.2.5 Entropy in strained layer quantum dot HD SLs with graded interfaces 299 7.2.6 Entropy in III–V quantum dot HD effective mass SLs 300 7.2.7 Entropy in II–VI quantum dot HD effective mass SLs 300 7.2.8 Entropy in IV–VI quantum dot HD effective mass SLs 301 7.2.9 Entropy in HgTe/CdTe quantum dot HD effective mass SLs 301 7.2.10 Entropy in strained layer quantum dot HD effective mass SLs 302 7.3 Results and discussion 302 7.4 Open research problems 305 References 306 8 8.1 8.2 8.2.1

Entropy in HDSLs under magnetic quantization 309 Introduction 309 Theoretical background 309 Entropy in III–VHD SLs with graded interfaces under magnetic quantization 309 8.2.2 Entropy in II–VI HD SLs with graded interfaces under magnetic quantization 312 8.2.3 Entropy in IV–VI HD SLs with graded interfaces under magnetic quantization 314 8.2.4 Entropy in HgTe/CdTe HD SLs with graded interfaces under magnetic quantization 316 8.2.5 Entropy in strained layer HD SLs with graded interfaces under magnetic quantization 319 8.2.6 Entropy in III–V HD effective mass SLs under magnetic quantization 321 8.2.7 Entropy in II–VI HD effective mass SLs under magnetic quantization 322 8.2.8 Entropy in IV–VI HD effective mass SLs under magnetic quantization 323 8.2.9 Entropy in HgTe/CdTe HD effective mass SLs under magnetic quantization 325 8.2.10 Entropy in strained layer HD effective mass SLs under magnetic quantization 326

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8.3 8.4

9

10

Contents

Results and discussion Open research problems References 333

327 331

Conclusion and scope for future research References 338

335

Appendix A: The entropy under intense electric field in HD kane type materials 339 10.1 Introduction 339 10.2 Theoretical background 340 10.2.1 Entropy in the presence of intense electric field in HD III–V, ternary, and quaternary materials 340 10.2.2 The entropy under magnetic quantization in HD Kane-type materials in the presence of intense electric field 350 10.2.3 Entropy in QWs in HD Kane-type materials in the presence of intense electric field 351 10.2.4 Entropy in NWs in HD Kane-type materials in the presence of intense electric field 352 10.2.5 Magneto entropy in QWs of HD Kane-type materials in the presence of intense electric field 353 10.2.6 Entropy in accumulation and inversion layers of Kane-type materials in the presence of intense electric field 354 10.2.7 Entropy in doping superlattices of HD Kane-type materials in the presence of intense electric field 356 10.2.8 Entropy in QWHD effective mass superlattices of Kane-type materials in the presence of intense electric field 357 10.2.9 Entropy in NWHD effective mass superlattices of Kane-type materials in the presence of intense electric field 359 10.2.10 Magneto entropy in QWHD effective mass superlattices of Kanetype materials in the presence of intense electric field 360 10.2.11 Entropy in QWHD superlattices of Kane-type materials with graded interfaces in the presence of intense electric field 361 10.2.12 Entropy in NWHD superlattices of Kane-type materials with graded interfaces in the presence of intense electric field 365 10.2.13 Entropy in QDHD superlattices of Kane-type materials with graded interfaces in the presence of intense electric field 368 10.2.14 Magneto entropy in HD superlattices of Kane-type materials with graded interfaces in the presence of intense electric field 369 10.2.15 Magneto entropy in QWHD superlattices of Kane-type materials with graded interfaces in the presence of intense electric field 372

Contents

10.3

11 11.1 11.2 11.2.1 11.2.2 11.2.3 11.2.4 11.2.5 11.3

Open research problems References 373

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Appendix B: Entropy in doping superlattices of HD nanomaterials Introduction 375 Theoretical background 375 Entropy in doping superlattices of HD nonlinear optical materials 375 Entropy in doping superlattices of HD III–V, ternary, and quaternary materials 377 Entropy in doping superlattices of HD II–VI materials 382 Entropy in doping superlattices of HD IV–VI materials 384 Entropy in doping superlattices of HD stressed Kane-type materials 387 Open research problems 389 References 389

375

12 Appendix C: Entropy in QWHDSLs under magnetic quantization 391 12.1 Introduction 391 12.2 Theoretical background 391 12.2.1 Entropy in III–V QWHDSLs with graded interfaces under magnetic quantization 391 12.2.2 Entropy in II–VI quantum well HD superlattices with graded interfaces under magnetic quantization 392 12.2.3 Entropy in IV–VI quantum well HD superlattices with graded interfaces under magnetic quantization 392 12.2.4 Entropy in HgTe/CdTe quantum well HD superlattices with graded interfaces under magnetic quantization 393 12.2.5 Entropy in strained layer quantum well HD superlattices with graded interfaces under magnetic quantization 394 12.2.6 Entropy in III–V quantum well HD effective mass super lattices under magnetic quantization 394 12.2.7 Entropy in II–VI quantum well HD effective mass super lattices under magnetic quantization 395 12.2.8 Entropy in IV–VI quantum well HD effective mass super lattices under magnetic quantization 396 12.2.9 Entropy in HgTe/CdTe quantum well HD effective mass super lattices under magnetic quantization 396 12.2.10 Entropy in strained layer quantum well HD effective mass super lattices under magnetic quantization 397 12.3 Open research problems 398 References 398

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13 13.1 13.2 13.2.1 13.2.2 13.2.3 13.2.4 13.2.5 13.2.6 13.3

Contents

Appendix D: Entropy in accumulation and inversion layers of non-parabolic materials 399 Introduction 399 Theoretical background 400 The entropy in accumulation and inversion layers of non-linear optical materials 400 Entropy in accumulation and inversion layers of III–V, ternary, and quaternary materials 403 Entropy in accumulation and inversion layers of II–VI materials 410 Entropy in accumulation and inversion layers of IV–VI materials 413 Entropy in accumulation and inversion layers of stressed III–V materials 415 Entropy in accumulation and inversion layers of germanium 418 Open research problems 419 References 420

14 Appendix E: Entropy in HDs under cross-fields configuration 423 14.1 Introduction 423 14.2 Theoretical background 423 14.2.1 Entropy in HD nonlinear optical materials under cross-fields configuration 423 14.2.2 Entropy in HD Kane-type III–V materials under cross-fields configuration 427 14.2.3 Entropy in HD II–VI materials under cross-fields configuration 431 14.2.4 Entropy in HD IV–VI materials under cross-fields configuration 433 14.2.5 Entropy in HD stressed materials under cross-fields configuration 435 14.3 Open research problems 438 References 439 15

Appendix F: The numerical values of the energy band constants of few materials 441 References 443

Material index Subject index

447 449

About the Authors When Love is real, people don’t need to impress, for Love in itself leaves a solid positive impression.

Professor Kamakhya Prasad Ghatak (h-index34, i10 index-168, maximum citation of a research paper 359 and total citations5295) is the first recipient of the Doctor of Engineering Degree of Jadavpur University, Kolkata, in 1991 since the inception of the university in 1955, and in the same year he received the Indian National Science Academy visiting fellowship award for IITKharagpur. He is the principal coauthor of more than 400 research papers in SCI journals, seven research monographs on various topics of nanotechnology from Springer-Verlag, Germany (Vol.170 in Solid State Science, Vols. 116, 137, and 167 in Materials Science, Vols. 255 and 260 in Tracts in Modern Physics (TMP), 1 in Nanostructured Science and Technology series) and vol. 1 in Nanomaterials series of DeGruyter, Germany. He is the solo author of Vols. 262 and 265 in TMP of the Springer-Verlag, Germany, and Vols. 7 and 8 in the Series on the Foundations of Natural Science and Technology of World Scientific. He is the Principal Editor of the two edited monographs in series in Nanotechnology of NOVA, USA. The score in Vidwan Portal Unit, Government of India, of Prof. Ghatak is 8.9 out of 10. Prof. Ghatak has successfully supervised more than 30 PhD students. AICTE, India, has selected the first R & D project in his life for the best project award in electronics and second best research project award considering all the branches of engineering for the year 2006, and he is the first recipient of the academic excellence award of the University of Engineering & Management, Kolkata, from the present Governor of West Bengal in its first convocation in 2017. He was Assistant Professor in the Department of Radiophysics and Electronics of the University of Calcutta from 1983 to 1987, Associate Professor in the Department of ETCE of Jadavpur University from 1987 to 1994, Professor and HOD of Electronic Science of the University of Calcutta from 1994 to 2012, HOD and Dean of the National Institute of Technology, Agartala from 2012 to 2014 and from January 2015 he has joined in the IEM UEM group, Kolkata, as Research Director and Senior Professor. He is the referee, Editor and Guest Editor of different reputed international journals. His present research interests are nanomaterials and genetic electronics, respectively. His brief CV has been enlisted in many biographical references of USA and UK. Professor Madhuchhanda Mitra received her B.Tech, M.Tech, and Ph.D. (Tech) degrees in 1987, 1989, and 1998, respectively, from the University of Calcutta, Calcutta, India. She is a recipient of “Griffith Memorial Award” of the University of Calcutta. She is the principal co-author of 150 scientific research papers in international peer-reviewed journals. Her present research interests are nano-science and nanotechnology, identification of different biomarkers and biomedical signal processing which include feature extraction, compression, encryption and classification of electrocardiography (ECG) and photoplethysmography (PPG) signals. Presently, she is professor in the Department of Applied Physics, University of Calcutta, India, where she has been actively engaged in both teaching and research.

https://doi.org/10.1515/9783110661194-205

Symbols α a a0 ,b0 A0 ~ A AðE, nz Þ B B2 b c C1 C2 ΔC44 ΔC456 δ Δ
0  1 Δ B d0 D0 ðEÞ DB ðEÞ DB ðE, λÞ dx , dy , dz Δk Δ? Δ d3 k 2 ε ε0 ε∞ εsc ΔEg jej E E0 ,ζ 0 Eg Ei Eki EF EFB En EFs E
Fn EFSL ~ εs EFQWSL

Band non-parabolicity parameter The lattice constant The widths of the barrier and the well for superlattice structures The amplitude of the light wave The vector potential The area of the constant energy 2D wave vector space for ultrathin films Quantizing magnetic field The momentum matrix element Bandwidth Velocity of light Conduction band deformation potential A constant which describes the strain interaction between the conduction and valance bands Second-order elastic constant Third-order elastic constant Crystal field splitting constant Interface width Period of SdH oscillation Superlattice period Density-of-states (DOS) function DOS function in magnetic quantization DOS function under the presence of light waves Nano thickness along the x, y and zdirections Spin-orbit splitting constants parallel Spin-orbit splitting constants perpendicular to the C-axis Isotropic spin-orbit splitting constant Differential volume of the k space Energy as measured from the center of the band gap Trace of the strain tensor Permittivity of free space Semiconductor permittivity in the high frequency limit Semiconductor permittivity Increased band gap Magnitude of electron charge Total energy of the carrier Electric field Band gap Energy of the carrier in the ith band Kinetic energy of the carrier in the ithband Fermi energy Fermi energy in the presence of magnetic quantization Landau sub band energy Fermi energy in the presence of size quantization Fermi energy for nipis Fermi energy in superlattices Polarization vector Fermi energy in quantum wire superlattices with graded interfaces

https://doi.org/10.1515/9783110661194-206

XL

EFL EFBL EF 2DL EF 1DL Eg0 Erfc Erf EFh E
hd Fs F ðVÞ Fj ðηÞ f0 f0i gv G G0 g* h ^ H ^ H′ HðE − En Þ ^i, ^j and k^ i I jci k kB λ
0 λ
l, m,
n
Lx , Lz L0 LD m1 m2 m3 m′2 m*i m*k m*? mc m*?, 1 , m*k, 1 mr mv n

Symbols

Fermi energy in the presence of light waves Fermi energy under quantizing magnetic field in the presence of light waves 2D Fermi energy in the presence of light waves 1D Fermi energy in the presence of light waves Un-perturbed band-gap Complementary error function Error function Fermi energy of HD materials Electron energy within the band gap Surface electric field Gaussian distribution of the impurity potential One parameter Fermi-Dirac integral of order j Equilibrium Fermi-Dirac distribution function of the total carriers Equilibrium Fermi-Dirac distribution function of the carriers in the ith band Valley degeneracy Thermoelectric power under classically large magnetic field Deformation potential constant Magnitude of the band edge g-factor Planck’s constant Hamiltonian Perturbed Hamiltonian Heaviside step function Orthogonal triads Imaginary unit Light intensity Conduction current contributed by the carriers of the ith band Magnitude of the wave vector of the carrier Boltzmann’s constant Wavelength of the light Splitting of the two spin-states by the spin-orbit coupling and the crystalline field Matrix elements of the strain perturbation operator Sample length along x and z directions Superlattices period length Debyescreening length Effective carrier masses at the band-edge along x direction Effective carrier masses at the band-edge along y direction The 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) Effective mass of the ithcharge carrier in the ith band Longitudinal effective electron masses at the edge of the conduction band Transverse effective electron masses at the edge of the conduction band Isotropic effective electron masses at the edge of the conduction band 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 Landau quantum number

Symbols

nx ,ny , nz n1D ,n2D n2Ds , n2Dw
2Dw
2Ds , n n ni Nnipi ðEÞ N2DT ðEÞ N2D ðE, λÞ N1D ðE, λÞ n0
0 n ni P Pn Pk P? ~ r Si ~ s0 t tc T τ i ðEÞ k,~ rÞ, u2 ð~ k,~ rÞ u1 ð~ VðEÞ V0 Vð~ rÞ x, y zt μi μ ζð2rÞ Γðj + 1Þ η ηg ω0 θ μ0 ω " ′,# ′

XLI

Size quantum numbers along the x, y and z directions 1D and 2D carrier concentration 2D surface electron concentration under strong and weak electric field Surface electron concentration under the strong and weak electric field quantum limit Mini-band index for nipi structures DOS function for nipi structures 2D DOS function 2D DOS function in the presence of light waves 1D DOS function in the presence of light waves Total electron concentration Electron concentration in the electric quantum limit Carrier concentration in the ith band Isotropic momentum matrix element Available noise power Momentum matrix elements parallel to the direction of crystal axis Momentum matrix elements perpendicular to the direction of crystal axis Position vector Zeros of the Airy function Momentum vector of the incident photon Time scale Tight binding parameter Absolute temperature Relaxation time of the carriers in the ith band Doubly degenerate wave functions Volume of κ space Potential barrier encountered by the electron Crystal potential Alloy compositions Classical turning point Mobility of the carriers in the ith band Average mobility of the carriers Zeta function of order 2r Complete Gamma function Normalized Fermi energy Impurity scattering potential Cyclotron resonance frequency Angle Bohr magnetron Angular frequency of light wave Spin up and down function

Part I: Entropy in heavily doped quantum confined nonparabolic materials Knowledge is proud, he knows too much, but the wise is humble, he knows no more.

1 The entropy in quantum wells of heavily doped materials If my only desire is to be desireless, then my consciousness is reversed.

1.1 Introduction In recent years, with the advent of fine lithographical methods [1, 2], molecular beam epitaxy [3], organ metallic vapor-phase epitaxy [4], and other experimental techniques, the restriction of the motion of the carriers of bulk materials in one [quantum wells (QWs), doping superlattices, accumulation, and inversion layers], two (nanowires), and three (quantum dots, magneto-size quantized systems, magneto inversion layers, magneto accumulation layers, quantum dot superlattices, magneto QW superlattices, and magneto doping superlattices) dimensions has in the past few years attracted much attention not only for their potential in uncovering new phenomena in nanoscience but also for their interesting quantum device applications [5–8]. In QWs, 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 of the carriers along the direction of the potential well, allowing 2D carrier transport parallel to the surface of the film representing new physical features not exhibited in bulk materials [9–13]. The lowdimensional heretostructures based on various materials are widely investigated because of the enhancement of carrier mobility [14].These properties make such structures suitable for applications in QWs lasers [15], hereto-junction field-effect transistors (FETs) [16, 17], high-speed digital networks [18–21], high-frequency microwave circuits [22], optical modulators [23], optical switching systems [24], and other devices. The constant energy 3D wave-vector space of bulk materials becomes 2D wave-vector surface in QWs due to dimensional quantization. Thus, the concept of reduction of symmetry of the wave-vector space and its consequence can unlock the physics of low-dimensional structures. In this chapter, we study the entropy in QWs of HD nonparabolic materials having different band structures in the presence of Gaussian band tails. At first, we shall investigate the entropy in QWs of HD nonlinear optical compounds, which are being used in nonlinear optics and light emitting diodes [25]. 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. Including the anisotropic crystal potential in the Hamiltonian, and special features of the nonlinear optical compounds, Kildal [26] formulated the electron dispersion law under the assumptions of isotropic momentum matrix element and the isotropic spin-orbit splitting constant, respectively, although the anisotropies in https://doi.org/10.1515/9783110661194-001

4

1 The entropy in quantum wells of heavily doped materials

the two aforementioned band constants are the significant physical features of the said materials [27–29]. In Section 1.2.1, the entropy in QWs of HD nonlinear optical materials has been investigated on the basis of newly formulated HD dispersion relation of the said compound by considering the combined influence of the anisotropies of the said energy band constants together with the inclusion of the crystal field splitting, respectively, within the framework of ~ k ·~ p formalism. The III–V compounds find applications in infrared detectors [30], quantum dot light emitting diodes [31], quantum cascade lasers [32], QWs wires [33], optoelectronic sensors [34], high electron mobility transistors [35], etc. The electron energy spectrum of III–V materials can be described by the three- and two-band models of Kane [36, 37], together with the models of Stillman et al. [38], Newson and Kurobe [39] and, Palik et al. [40], respectively. In this context it may be noted that the ternary and quaternary compounds enjoy the singular position in the entire spectrum of optoelectronic materials. The ternary alloy Hg1–xCdxTe 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 30μm [41] by adjusting the alloy composition. Hg1–xCdxTe finds extensive applications in infrared detector materials and photovoltaic detector arrays in the 8–12μm wave bands [42]. The above uses have generated the Hg1–xCdxTe technology for the experimental realization of high-mobility single crystal 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 [43]. Besides, the quaternary alloy In1−x Gax Asy P1−y lattice matched to InP, also finds wide use in the fabrication of avalanche photo-detectors [44], hereto-junction lasers [45], light emitting diodes [46] and avalanche photodiodes [47], field effect transistors, detectors, switches, modulators, solar cells, filters, and new types of integrated optical devices are made from the quaternary systems [48]. It may be noted that all types of band models as discussed for III–V materials are also applicable for ternary and quaternary compounds. In Section 1.2.2, the Entropy in QWs of HD III–V, ternary and quaternary materials has been studied in accordance with the corresponding HD formulation of the band structure and the simplified results for wide gap materials having parabolic energy bands under certain limiting conditions have further been demonstrated as a special case in the absence of heavy doping and thus confirming the compatibility test. The II–VI materials are being used in nano-ribbons, blue green diode lasers, photosensitive thin films, infrared detectors, ultra-high-speed bipolar transistors, fiber optic communications, microwave devices, solar cells, semiconductor gamma-ray detector arrays, semiconductor detector gamma camera and allow for a greater density of data storage on optically addressed compact discs [49–56]. The carrier energy spectra in II–VI compounds are defined by the Hopfield model [57] where the splitting of the two-spin states by the spin-orbit coupling and the crystalline field has been taken into account. Section 1.2.3 contains the investigation of the Entropy in QWs of HD II–VI compounds.

1.1 Introduction

5

Lead Chalcogenides (PbTe, PbSe, and PbS) are IV–VI nonparabolic materials whose studies over several decades have been motivated by their importance in infrared IR detectors, lasers, light-emitting devices, photo-voltaic, and high temperature thermo-electrics [58–62]. PbTe, in particular, is the end compound of several ternary and quaternary high performance high temperature thermoelectric materials [63–67]. It has been used not only as bulk but also as films [68–71], QWs [72] superlattices [73, 74] nanowires [75] and colloidal and embedded nano-crystals [76, 77, 78, 79], and PbTe films doped with various impurities have also been investigated [80–87] These studies revealed some of the interesting features that had been seen in bulk PbTe, such as Fermi-level pinning and, in the case of superconductivity [88]. In Section 1.2.4, the 2D Entropy in QWs of HD IV–VI materials has been studied taking PbTe, PbSe, and PbS as examples. The stressed materials are being investigated for strained silicon transistors, quantum cascade lasers, semiconductor strain gages, thermal detectors, and strained-layer structures [89–92]. The Entropy in QWs of HD stressed compounds (taking stressed n-InSb as an example) has been investigated in Section 1.2.5 The vacuum deposited Tellurium (Te) has been used as the semiconductor layer in thin-film transistors (TFT) [93] which is being used in CO2 laser detectors [94], electronic imaging, strain sensitive devices [95, 96], and multichannel Bragg cell [97]. Section 1.2.6 contains the investigation of Entropy in QWs of HD Tellurium. The n-Gallium Phosphide (n-GaP) is being used in quantum dot light emitting diode [98], high efficiency yellow solid state lamps, light sources, 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 [99–101]. In Section 1.2.7, the Entropy in QWs of HD n-GaP has been studied. The Platinum Antimonide (PtSb2) finds application in device miniaturization, colloidal nanoparticle synthesis, sensors and detector materials and thermo-photovoltaic devices [102–104]. Section 1.2.8 explores the Entropy in QWs of HD PtSb2. Bismuth telluride (Bi2Te3) was first identified as a material for thermoelectric refrigeration in 1954 [105] and its physical properties were later improved by the addition of bismuth selenide and antimony telluride to form solid solutions. The alloys of Bi2Te3are useful compounds for the thermoelectric industry and have been investigated in the literature [106–110]. In Section 1.2.9, the Entropy in QWs of HD Bi2Te3 has been considered. The usefulness of elemental semiconductor Germanium is already well known since the inception of transistor technology and, it is also being used in memory circuits, single photon detectors, single photon avalanche diode, ultrafast optical switch, THz lasers and THz spectrometers [111–114]. In Section 1.2.10, the Entropy has been studied in QWs of HD Ge. Gallium Antimonide (GaSb) finds applications in the fiber optic transmission window, heretojunctions, and QWs. A complementary hereto-junction field effect transistor in which the channels for the p-FET device and the n-FET device forming the complementary FET are formed from GaSb. The band gap energy of GaSb makes it suitable for low power operation [115–120]. In Section 1.2.11, the Entropy in QWs of HD GaSb has

6

1 The entropy in quantum wells of heavily doped materials

been 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 0.8V and power conservation coefficients near 6 percent for 720-nm light [121]. They are also used in ultrasonic amplification [122]. The development of an evaporated thin film transistor using cadmium selenide as the semiconductor has been reported by Weimer [123, 124]. The Entropy in HD QWs of II–V materials has been presented in Section 1.2.12. In Section 1.2.13, the Entropy in HDQWs of Pb1–xGaxTe has been investigated [125]. The diphosphides finds prominent role in biochemistry where the folding and structural stabilization of many important extra-cellular peptide and protein molecules, including hormones, enzymes, growth factors, toxins, and immunoglobulin are concerned [126]. Besides, artificial introduction of extra diphosphides into peptides or proteins can improve biological activity [127] or confer thermal stability [128]. 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 spectro-photometric titration of the released 3-nitro-2-pyridinethiol [129–134]. In Section 1.2.14, the Entropy in HD QWs of zinc and cadmium diphosphides has been investigated. Section 1.3 contains the result and discussions pertaining to this chapter. The last Section 1.4 contains 25 open research problems.

1.2 Theoretical background 1.2.1 Entropy in quantum wells (QWs) of HD nonlinear optical materials The form of k.p matrix for nonlinear optical compounds can be expressed extending Bodnar [27] as " #
2
1 H H
(1:1) H=
+ H
1 H 2 where, 2
Eg0 6 6
1 = 6 0 H 6

4 Pjj kz 0

ð − 2Δjj =3Þ pffiffiffi ð 2Δ? =3Þ


jj
kz P pffiffiffi ð 2Δ? =3Þ   − δ + 31Δjj

0

0

0

0

3

7 07 7, 7 05 0

2

0 6
6 f ,+
=6 H 6 0 4
f ,+

−
f ,+

0
f ,−

0

0

0

0

0

0

3

7 0 7 7 0 7 5 0


jj and P
? are the momen
g is the band gap in the absence of any field, P in which E 0 tum matrix elements parallel and perpendicular to the direction of crystal axis axis, respectively, δ is the crystal field splitting constant, Δjj and Δ? are the spinorbit splitting constants parallel and perpendicular to the C-axis, respectively,

1.2 Theoretical background

7

pffiffiffi pffiffiffiffiffiffiffi  
f,± ≡ P
? = 2
kx ± i
ky and i = − 1. Thus, neglecting the contribution of the higher bands and the free electron term, the diagonalization of the above matrix leads to the dispersion relation of the conduction electrons in bulk specimens of nonlinear optical materials as

k2 +
f2 ðEÞ

k2
=
f1 ðEÞ γðEÞ s z

(1:2)

where
+E
g ÞðE
+E
g + Δjj Þ + δðE
+E
g + 2 Δjj Þ + 2 ðΔ2 − Δ2 Þ
= Eð
E
+E
g Þ½ðE γðEÞ ? 0 0 0 0 3 3 jj
is the total energy of the electron as measured from the edge of the conduction E band in the vertically upward direction in the absence of any quantization,
k2 =
k2 +
k2 , s x y 

g + Δ? Þ 

g ðE
h2 E 1 2 0 0






f1 ðEÞ =
g + 2 Δ? Þ δ E + Eg0 + 3 Δjj + ðE + Eg0 Þ E + Eg0 + 3 Δjj
*? ðE ½2m 3 0  1 2 2 + ðΔjj − Δ? Þ , 9
= f2 ðEÞ



g + Δjj Þ 
2 Eg0 ðE h 0
+E
g + 2 Δjj ,
h = h
+E
g Þ E ð E 0 0
g + 2 Δ? Þ
*jj ðE 3 2π ½2m 3 0


* are the longitudinal and transverse effective
* jj and m h is Planck’s constant and m ? electron masses at the edge of the conduction band, respectively. Thus the generalized unperturbed electron energy spectrum for the bulk specimens of the nonlinear optical materials in the absence of band tails can be expressed following (1.2) as ! ( 
jj
c
jj 
jj E
E
+ 1Þðb
+ 1Þ αb b Eðα
h2
kz2 h2
ks2 2 2 ?
2
δE + ðΔjj − Δ? Þ +
= +
+ 1Þ
cjj
*jj
*? 9 2m ð
cjj E b?
cjj 2m " !( ! ! )

2 2 2 2
jj
c b 2 αbjj ðΔjj − Δ? Þ
h2
ks2 δ Δjj − Δ? α ? − +

+ 1Þ − 2m


*? 6Δ c 9
cjj ð
cjj E 2 αE + 1 b? jj jj ( 2 )! #) 2 Δjj − Δ?
cjj δ + (1:3) −

6Δjj 2 cjj E + 1 where

−1

jj ≡ ðE
? ≡ ðE
g + Δjj Þ − 1 ,
c? ≡ E
g + Δ? Þ − 1 ,
g + 2 Δ? b ,b 0 0 0 3 −1

g Þ − 1 .
g + 2 Δjj
cjj = E and α = ðE 0 0 3

8

1 The entropy in quantum wells of heavily doped materials


VÞ
of the impurity potential is given by [131] The Gaussian distribution Fð
2 =η2 Þ
VÞ
= ðπη2 Þ1=2 expð − V Fð g g

(1:4)

where ηg is the impurity scattering potential. It appears from (1.4) that the variance parameter ηg is not equal to zero, but the mean value is zero. Further, the impurities are assumed to be uncorrelated and the band mixing effect has been neglected in this simplified theoretical formalism. Using the (1.3) and (1.4), we get ! " # " # ð
ð

jj
c b
h2
kz2 E



h2
ks2 E



?
FðVÞdV + FðVÞdV =
?
cjj 2m
*jj − ∞
*? − ∞ 2m b (ð
E
jj ðE

− VÞ½αð

− VÞ
+ 1½b
− VÞ
+1 ðE E
V
VÞ
d
+ αbjj Fð
− VÞ
+ 1
cjj ½
cjj ðE −∞ " ð
#

ð E
E 2 2 2 αbjj 2 2









δ ðE − VÞFðVÞdV + ðΔjj − Δ? Þ ðΔjj − Δ2? Þ FðVÞdV −
cjj 9 9 −∞ −∞ " ) ð E

jj ð E

V
VÞ
d
αb Fð
V
− VÞ
Fð
VÞ
d
+ 2 ðΔ2 − Δ2 Þ δ ðE −+ ?



c 9 jj jj − ∞ ½cjj ðE − VÞ + 1 −∞ ) ! #

ð E
ð E

2 k
2
V
VÞ
d

h 2 αbjj 2 Fð 2 s



ðΔjj − Δ? Þ FðVÞdV −
− VÞ
+ 1 − 2m
*? 9
cjj cjj ðE −∞ − ∞ ½
( !" ! ð E
2 2
jj
c
V
VÞ
d
b δ Δjj − Δ? Fð ? α +


6Δjj 2 bjj ?
cjj − ∞ ½αðE − VÞ + 1 #) ! ð
2 2 E
V
VÞ
d
δ Δjj − Δ? Fð
cjj + (1:5) +
− VÞ
+ 1 6Δjj 2 cjj ðE − ∞ ½
The (1.5) can be written as !
jj
c
h2
k2 b
h2
kz2
1 s
Ið1Þ +
Ið1Þ =
*jj
*? 2m b?
cjj 2m ( ) 


jj  αb 2 2 2 αbjj 2 2 2



= I 3 ð
cjj Þ + δIð4Þ + ðΔjj − Δ? ÞIð1Þ − ðΔjj − Δ? ÞI 6 ð
cjj Þ
cjj 9 9
cjj #) !( !" ! ( 2 )! 2 2 2
jj
c b
h2
ks2 δ Δjj − Δ?
δ Δjj − Δ? 1

cjj Ið
cjj Þ − αIðαÞ + +
?
cjj
*? 6Δjj 6Δjj 2 2 2m b

ð1:6Þ

Let us substitute


Ið1Þ =

ðE
−∞


V
VÞ
d
Fð

(1:7)

1.2 Theoretical background

ð E



I 3 ð
cjj Þ =

−∞

ð E



I 3 ð
cjj Þ =

−∞

ð E



I 3 ð
cjj Þ =
Ið4Þ =
IðαÞ =

−∞

ð E
−∞

ðE
−∞


jj ðE
− VÞ½αð

− VÞ
+ 1½b
− VÞ
+ 1 ðE E
V
VÞ
d
Fð

½
cjj ðE − VÞ + 1
jj ðE
− VÞ½αð

− VÞ
+ 1½b
− VÞ
+ 1 ðE E
V
VÞ
d
Fð ½cjj ðE − VÞ + 1

(1:8)


jj ðE
− VÞ½αð

− VÞ
+ 1½b
− VÞ
+ 1 ðE E
V
VÞ
d
Fð

½
cjj ðE − VÞ + 1


V
− VÞ
Fð
VÞ
d
ðE

(1:9)


V
VÞ
d
Fð

½αðE − VÞ + 1

(1:10)


−V
≡
x and,
x ≡
t0 we get from (1.7) Substituting E ηg 
h

i Ð∞ 2E t 0 2
Ið1Þ = ðexp − E
2 =pffiffiffi
π Þ exp − t + d
t0 0 η η2 g


Ið1Þ =

9

0

h1 + Erf ðE=η i
Þ

g

(1:11)

g

2


Þ).
Þs the error function of (ðE=η where Erf ðE=η g g From (1.9), we can write ð
pffiffiffi E


E ½1 + Erf ðE=η
− VÞ
expð − V
2 =η2 Þd
Ið4Þ = 1=η π
Þ ðE g g g V= 2 −∞ ( ) ð E
1


expð − V
2 =η2 Þd − qffiffiffiffiffiffiffiffi V g V 2 πηg − ∞

(1:12)

After computing this simple integration, we obtain Thus,
pffiffiffi
2 =η2 Þð2 πÞ + E ð1 + Erf ðE=η
η Þ
Ið4Þ = η expð − E
ÞÞ = γ ðE, g g g g 0 2

(1:13)

From (1.10), we can write 1
IðαÞ = qffiffiffiffiffiffiffi ffi πη2g

ð E
−∞




2 =η2 Þd expð − V g V
− VÞ
+ 1 ½αðE

(1:14)


=η2 Þ ! 0
! ± ∞,
1
! 0 and; expð − V When, V g ½αðE − VÞ + 1 Thus (1.14) can be expressed as 2

10

1 The entropy in quantum wells of heavily doped materials

pffiffiffi
IðαÞ = ð1=αη πÞ g

ð∞ −∞

2

t
−
tÞ − 1 d expð −
t Þðu

(1:15)

where,

1 + αE V

≡ = t and u αη ηg It is well known that [123, 124] ð∞

t
−
tÞ − 1 expð −
t2 Þd
ZÞ
= ði=πÞ ðZ Wð

(1:16)

−∞

pffiffiffiffiffiffiffi In which i = − 1 and Z, in general, is a complex number. We also know [123, 124],

ZÞ
= ði=πÞ expð − Z
2 ÞErfcð −
iZÞ Wð

(1:17)

where
≡ 1 − Erf ðZÞ.
ErfcðZÞ
Þ = 1 − Erf ð − iu
Þ Thus, Erfcð − iu
Þ = − Erf ðiu
Þ Since, Erf ð − iu Thus, pffiffiffi
IðαÞ = ½ − i π=αη  expð − u
2 Þ½1 + Erf ðiu
Þ g

(1:18)

We also know that [123, 124] 2  e−x ð1 − cosð2xyÞÞ + i sinð2xyÞ Erf ð
x + i
yÞ = Erf ð
xÞ + 2π
x  ∞
2 =4Þ
2 2 X expð − p + e−x ½f p ð
x,
yÞ + ig
p ð
x,
yÞ + εð
x,
yÞ 2

2 π p = 1 ðp + 4x Þ




(1:19)

where
fp ð
x,
yÞ = ½2x − 2x coshðp

yÞ cosð2
x
yÞ + p
sinhðp

yÞ sinð2
x
yÞ,  

yÞ sinð2
x
yÞ + p sinhðp

yÞ cosð2
x
yÞεð
x,
yÞÞj ≈ 10 − 15 Erf ð
x + i
yÞj g
p ð
x,
yÞ ≡ ½2
x coshðp
in (1.19), we obtain, Substituting
x = 0 and
y = u ( )
X
2 =4Þ 2i ∞ expð − p
u
Þ
Þ = sinhðp Erf ðiu
π p
= 1 p

(1:20)

1.2 Theoretical background

11

Therefore, we can write
21 ðα, E,
η Þ − iD
21 ðα, E,
η Þ
IðαÞ = C g g

(1:21)

where, "

# ( " )# ∞ 2 X
2 expð − p =4Þ
21 ðα, E,
η Þ=
u
2 Þ
Þ pffiffiffi expð − u sinhðp and C g
p αηg π
=1 p " pffiffiffi # π 2


Þ expð − u D21 ðα, E, ηg Þ = αηg The (1.21) consists of both real and imaginary parts and therefore,
I3 ðαÞ is complex, which can also be proved by using the method of analytic continuation of the subject Complex Analysis. The integral
I3 ð
cjj Þ in (1.8) can be written as ! !





cjj
I3 ð
cjj Þ = αbjj
Ið5Þ + αcjj + bjj cjj − αbjj
Ið4Þ + 1 1 − α
Ið1Þ 1 −
cjj
cjj
cjj
cjj
c2jj ) ! (

jj b 1 α
Ið
cjj Þ 1− 1− (1:22) −
cjj
cjj
cjj where
Ið5Þ =

ð E
−∞


V
− VÞ
2 Fð
VÞ
d
ðE

(1:23)

From (1.23) we can write " ð
! ! ð E
E
2
2 1 − V − V 2
V
V
− 2E



exp
Ið5Þ = qffiffiffiffiffiffiffiffi E d d exp V η2g η2g −∞ −∞ πη2g ! # ð E

2 −V 2
V

d + V exp η2g −∞ The evaluations of the component integrals lead us to write ! " !#

2
ηg E −E 1 2 E 2
Ið5Þ = p

η Þ ffiffiffi exp + ðηg + 2E Þ 1 + Erf = θ0 ðE, g η2g 4 ηg 2 π

(1:24)

Thus combining the aforementioned equations,
I3 ð
cjj Þ can be expressed as
21 ðE,
η Þ +
iB
21 ðE,
η Þ
I3 ð
cjj Þ = A g g where,

(1:25)

12

1 The entropy in quantum wells of heavily doped materials

( " ! !)#

jj η E
2 1
α b − E E 2 g 2
η Þ=
Þ 1 + Erf
21 ðE, pffiffiffi exp A ðη + 2E g
cjj 2 π η2g 4 g ηg " #( )
2 =η2 Þ
jj
cjj − αb
jj
η expð − E α
cjj + b E g
pffiffiffi + ½1 + Erf ðE=ηÞ +
c2jj 2 2 π !

jj 1 b 1 α
1− + 1− ½1 + Erf ðE=ηÞ
cjj 2
cjj
cjj ) ! (

jj b 2 α 2
1 Þ pffiffiffi 1 − 1− expð − u −
cjj
cjjjj
c2jj ηg π )# " ( " # ∞
X 1 +
cjj E expð − p2 =4Þ
u
1 ≡
1 Þ u sinhðp
cjj ηg p p=1 "

and
η Þ≡
21 ðE, B g

! pffiffiffi

jj b α π
21 Þ 1− 1− expð − u
cjj
cjj
c2jj ηg

Therefore, the combination of all the appropriate integrals together with algebraic manipulations leads to the expression of the dispersion relation of the conduction electrons of HD nonlinear optical materials forming Gaussian band tails as h2
kz2 h2
ks2 +
hg Þ 2m
hg Þ = 1
21 ðE,
22 ðE,
*jj T
*? T 2m

(1:26)


hg Þ and T
22 ðE,
hg Þ have both real and complex parts and are given by
21 ðE, where, T 


hg Þ = ½T
27 ðE,
hg Þ + iT
28 ðE,
hg Þ, T
27 ðE,
hg Þ = T 23 ðE, hg Þ
21 ðE, T
hg Þ
5 ðE, T " #
jj  a b 1 2 2
hg Þ ≡ A
21 ðE,
hg Þ +
hg Þ + ðD − D Þ½1 + Erf ðE=h
gÞ
23 ðE, dg0 ðE, T ?
cjj 9 jj )# ! (
jj 2 ab 2 2

hg Þ , ðDjj − D? ÞG21 ð
cjj , E, − 9
cjj ∞ o n expð − p X
2 =4Þ 2
u
1 Þ , sinhðp pffiffiffi expð − u21 Þ
cjj hg p
p
=1 p 1
hg Þ ≡ ½1 + Erf ðE=h
5 ðE,
g Þ, T 2 " # 


jj 2 ab T 24 ðE, hg Þ

2 2





ðDjj − D? ÞH21 ð
cjj , E, hg Þ , T24 ðE, hg Þ ≡ B21 ðE, hg Þ + T28 ðE, hg Þ ≡

9
cjj T 5 ðE, hg Þ


21 ðE,
hg Þ ≡ G

1.2 Theoretical background


hg Þ ≡
21 ð
cjj , E, H

13

 pffiffiffi  p 2
22 ðE,
hg Þ ≡ ½T
29 ðE,
hg Þ + iT
30 ðE,
hg Þ,
1 Þ , T expð − u hg
cjj










hg Þ ≡ T 3 ðE, hg ÞT 25 ðE, hg Þ − T 24 ðE, hg ÞT 26 ðE, hg Þ
29 ðE, T 2
25 ðE,
hg Þ + T
26 ðE,
hg Þ2  ½T " !  ! " 2 #!


jj
c
jj
cjj Djj − D2? b b 1 d E ?
21 ðajj , E,
hg Þ ≡
hg Þ
25 ðE, ajj C T +
?
cjj 2 1 + Erf hg + b
?
c? 6Djj 2 b ! " 2 #!
jj
c? Djj − D2? b d
21 ðajj , E,
hg Þ, +
G − 6Djj 2 b? "
21 ðajj , E,
hg Þ ≡ C
hg Þ ≡ T26 ðE,

" ## ∞ X 2 expð − p2 =4Þ 2
u
Þ
Þ , sinhðp pffiffiffi expð − u a phg p p=1
jj
c b ?
?
cjj b

!


jj
c? b +
b?

" 2 #! Djj − D2? d
hg Þ
21 ðajj , E, aD − 6Djj 2 " 2 #! Djj − D2? d
hg Þ,
21 ð
cjj , E, H − 6Djj 2

and ηg ¼ hg ; Djj ¼ Δ11 ; D? ¼ Δ?








hg Þ ≡ T 24 ðE, hg ÞT 25 ðE, hg Þ + T 23 ðE, hg ÞT 26 ðE, hg Þ
30 ðE, T 2 2



½ðT 25 ðE, hg ÞÞ + ðT 26 ðE, hg ÞÞ  From (1.26), it appears that the energy spectrum in HD nonlinear optical materials is complex. The complex nature of the electron dispersion law in HD materials occurs from the existence of the essential poles in the corresponding electron energy spectrum in the absence of band tails. It may be noted that the complex band structures have already been studied for bulk materials and superlattices without heavy doping [135] and bear no relationship with the complex electron dispersion law as indicated by (1.26). The physical picture behind the formulation of the complex energy spectrum in HDS is the interaction of the impurity atoms in the tails with the splitting constants of the valance bands. More is the interaction; more is the prominence of the complex part than the other case. In the absence of band tails, ηg ! 0, and there is no interaction of the impurity atoms in the tails with the spin orbit constants. As a result, there exist no complex energy spectrum and (1.26) gets converted into (1.2) when ηg ! 0. Besides, the complex spectra are not related to same evanescent modes in the band tails and the conduction bands. It is interesting to note that the single important concept in the whole spectra of materials and allied sciences is the effective electron mass which is in disguise in the apparently simple (1.26), and can, briefly be described as follows:

14

1 The entropy in quantum wells of heavily doped materials

Effective Electron Mass (EEM): The effective mass of the carriers in materials, being connected with the mobility, is known to be one of the most important physical quantities, used for the analysis of electron devices under different operating conditions [136]. The carrier degeneracy in materials influences the effective mass when it is energy dependent. Under degenerate conditions, only the electrons at the Fermi surface of n-type materials participate in the conduction process and hence, the effective mass of the electrons corresponding to the Fermi level (EEM) would be of interest in electron transport under such conditions. The Fermi energy is again determined by the electron energy spectrum and the carrier statistics and therefore, these two features would determine the dependence of the effective electron mass in degenerate n-type materials under the degree of carrier degeneracy. In recent years, various energy wave vector dispersion relations have been proposed [137–139] which have created the interest in studying the effective mass in such materials under external conditions. It has, therefore, different values in different materials and varies with electron concentration, with the magnitude of the reciprocal quantizing magnetic field undermagnetic quantization, with the quantizing electric field as in inversion layers, with the nano-thickness as in UFs and nano wires and with superlattice period as in the quantum confined superlattices of small gap materials with graded interfaces having various carrier energy spectra [140–156].
F of HDS in the The transverse and the longitudinal EEMs at the Fermi energy E n presence of band tails as measured from the edge of the conduction band in the vertically upward direction in the absence of band tails of HD nonlinear optical materials can, respectively, be expressed as 
29 ðE,
η Þ g′


*? = m*? fT (1:27) m E = EF g n

and 
27 ðE,
η Þg′

m*jj = m*jj fT E = EF g

n

(1:28)

where the primes denote the differentiations of the differentiable functions with respect to Fermi energy in the appropriate case. In the absence of band tails ηg ! 0 and we get # " 2

′ − ψ ðEÞfψ

′ h
ψ ð EÞfψ ð EÞg ð EÞg * 2 1 1 2
=
F , OÞ
? ðE (1:29) m
2 fψ2 ðEÞg

E = EF

and
=
F , OÞ m*jj ðE

# "

′

′
h2 ψ3 ðEÞfψ 1 ðEÞg − fψ1 ðEÞgfψ3 ðEÞg
2 fψ3 ðEÞg



E=E F

(1:30)

1.2 Theoretical background

15


F is the Fermi energy as measured from the edge of the conduction band in where E
= γðEÞ,
ψ ðEÞ
= the vertically upward direction in the absence of band tails ψ1 ðEÞ 2
f1 ðEÞ,
and ψ ðEÞ
=
f2 ðEÞ,
3

Comparing the aforementioned equations, one can infer that the effective masses exist in the forbidden zone, which is impossible without the effect of band tailing. For materials, in the absence of band tails the effective mass in the band gap is infinity. The DOS function is given by qffiffiffiffiffiffiffiffiffi
*? 2m*jj 2g
ν m
η Þ=
η Þ cos½ψ ðE,
η Þ
11 ðE,
HD ðE, (1:31a) R N g g 11 g 3π2
h3 where, g
ν is the valley degeneracy, "" qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T
η Þf
xðE,
η Þg′
ðE,
η Þ ≡ fT
29 ðE,
η Þg′
xðE,
η Þ + 29 qgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g
11 ðE, R g g g
η Þ 2
xðE, g # qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T
η ÞfyðE,
η Þg′ 2
30 ðE, g g
η Þg′
yðE,
η Þ
30 ðE, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − fT g g
η Þ 2
yðE, g " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T
η Þf
yðE,
η Þg′
ðE,
29 ðE,
η Þg′
yðE,
η Þ + 29 qgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g + fT g g
η Þ 2
yðE, g ## qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T
η Þf
xðE,
η Þg′ 2 1=2
30 ðE, g g
30 ðE,
η Þg′ xðE,
η Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + fT , g g
η Þ 2
xðE, g  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1


27 ðE,
η Þg2 + fT
28 ðE,
η Þg2 , T27 ðE, ηg Þ + fT g g 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
27 ðE,
η Þg2 + fT
28 ðE,
η Þg2 − T
η Þ
η Þ≡ 1
27 ðE,
yðE, f T g g g g 2
η Þ≡
xðE, g

and ""
η Þ ≡ tan ψ11 ðE, g

−1


η Þg′
29 ðE, fT g #"

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
η Þ
29 ðE, g
η Þ+ q
η Þg′
xðE,
η Þ
30 ðE, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + fT yðE, g g g
η Þ 2
yðE, g

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T
η Þf
xðE,
η Þg′
ðE,
η Þg′g
xðE,
η Þ + 29 qgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g
29 ðE, ffT g g
η Þ 2
xðE, g # − 1# qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T
η Þg′
f
tðE, g
η Þg′
yðE,
η Þ + 30qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
30 ðE, ffi − fT g g
2
yðE, ηg Þ


η Þg′
30 f
xðE, T g + qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
η Þ 2
xðE, g

16

1 The entropy in quantum wells of heavily doped materials

The oscillatory nature of the DOS for HD nonlinear optical materials is apparent
η Þ ≥ π, the cosine function becomes negative leading to the from (1.31a). For, ψ11 ðE, g negative values of the DOS. The electrons cannot exist for the negative values of the DOS and therefore, this region is forbidden for electrons, which indicates that in the band tail, there appears a new forbidden zone in addition to the normal band gap of the semiconductor. The use of (1.31a) leads to the expression of the electron concentration as pffiffiffiffiffiffiffiffi  
s X
*
* 2m 2g
m
F , η Þ +
F , η Þ
I11 ðE

rÞ½
I 11 ðE
0 = v ? 3 (1:31b) Lð n g g h h 3π2
h
r = 1 where, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F , η Þ ≡ ½T
29 ðE
F , η Þ
xðE
F , η Þ − T
30 ðE
F , η Þ yðE
F , η Þ
I11 ðE g g g g g h h h h h 2r

rÞ = 2ð
kB TÞ
2r ð1 − 21 − 2r Þξð2
rÞ ∂ Lð
2r ∂E

(1:31c)

Fs


r is the set of real positive integers whose upper s and ξð2
rÞ is the Zeta function of order 2
r [133, 134]. The entropy per unit volume which can be written as ∂Ω S0 = −
 ∂T E
= E
F

(1:31d)

in which Ω is the thermodynamic potential which, in turn, can be expressed in accordance with the Fermi–Dirac statistics as  

 X  E F − E δ0 

  Ω = − kB T ln1 + exp
(1:31e)
 kB T where the summation is carried out over all the possible δ0 states and
kB is Boltzmann constant. Thus, combining 1.31d and1.31e, the magnitude of the entropy for HD systems can be written in a simplified form as

0 ∂n
S0 = ðπ2
k2 T=3Þ
(1:31f) B
F − E
hd ∂ðE
hd is the electron energy within the band gap, as measured from
k = 0 and where E should be obtained from the dispersion relation of the HD materials under the con
−E
hd when
k = 0. It should be noted that being a thermodynamic relation ditions E and temperature induced phenomena, the entropy as expressed by (1.31f), in general, is valid for electronic materials having arbitrary dispersion relations and their nanostructures. In addition to bulk materials in the presence of strong magnetic

1.2 Theoretical background

17

field, (1.31f) is valid under one-, two- and three-dimensional quantum confinement of the charge carriers (such as quantum wells in ultrathin films, nipi structures, inversion and accumulation layers, quantum well superlattices, carbon nanotubes, quantum wires, quantum wire superlattices, quantum dots, magneto inversion and accumulation layers, quantum dot superlattices, magneto nipis, quantum well superlattices under magnetic quantization, ultrathin films under magnetic quantization, etc.). The formulation of
S0 requires the relation between electron statistics and the corresponding Fermi energy, which is basically the band structure-dependent quantity and changes under different physical conditions. It is worth remarking to  note that the number π2 3Þ has occurred as a consequence of mathematical analysis and is not connected with the well- known Lorenz number. For quantum wells in ultrathin films, nipi structures, inversion and accumulation layers, quantum well superlattices, magneto inversion and accumulation layers, magneto nipis, quantum well superlattices under magnetic quantization and magneto size quantization, the carrier concentration is measured per unit area whereas, for quantum wires, quantum wires under magnetic field, quantum wire superlattices and such allied systems, the same can be measured per unit length. Besides, for bulk materials under strong magnetic field, quantum dots, quantum dots under magnetic field, quantum dot superlattices and quantum dot superlattices under magnetic field, the carrier concentration is expressed per unit volume.
hd is the smallest negative root of the equation For HD nonlinear materials, E
27 ðE
hd , η ÞT





½T g 29 ðEhd , ηg Þ − T28 ðEhd , ηg ÞT30 ðEhd , ηg Þ = 0

(1:31g)

Therefore, the entropy can be numerically evaluated by using (1.31b), (1.31f), (1.31g) and the allied definitions. For dimensional quantization along z-direction, the dispersion relation of the 2D electrons in this case can be written following (1.26) as
z Þ2
z π=d
2 ðn h
h2
ks2 + * *

η Þ =1


jj T 21 ðE, ηg Þ 2m
? T 22 ðE, 2m g

(1:32)


z are the size quantum number and the nano-thickness
z ð = 1, 2, 3Þ and d where, n along the z-direction, respectively.
can, in general, be ex
2DT ðEÞÞ The general expression of the total 2D DOS ðN pressed as nzmax


X
z Þ


∂AðE, n
= 2g v
2DT ðEÞ HðE − Enz Þ N 2
∂E ð2πÞ nz = 1

(1:33)


E,
n
z Þ is the area of the constant energy 2D Wave vector space and in this case where Að
n is the corresponding
E
−E
n Þ is the Heaviside step function and E it is for QWs Hð z z

18

1 The entropy in quantum wells of heavily doped materials


2DT ðEÞ
for QWs of sub-band energy. Using (1.32) and (1.33), the expression of the N HD nonlinear optical materials can be written as
zmax n
*
X
= m? g v



η ,n
1D ′ðE,
2DT ðEÞ T N g
z ÞHðE − Enz D1 Þ π
h2 n
z = 1

(1:34)

where, " #
z Þ2
z π=d
2 ðn h




z Þ = 1 − T1D ðE, ηg , n
η Þ T22 ðE, ηg Þ,
21 ðE,
*jj T 2m g
n D is the corresponding sub-band
E
−E
n Þ is the Heaviside step function and E Hð z z 1 energy which in this case is given by the following equation
z Þ2
z π=d
2 ðn h
n D , η Þ = 1
21 ðE
*jj T 2m z 1 g

(1:35)

Thus we observe that both the total DOS and sub-band energies of QWs of HD nonlinear optical materials are complex due to the presence of the pole in energy axis of the corresponding materials in the absence of band tails. EEM in this case is given by
F1HD , η , n
′1D ðE
F1HD , η , n
*? ½Real part of T
* ðE m g
z Þ = m g
z Þ

(1:36)


F1HD is the Fermi energy in the presence of size quantization of the QWs of where E HD nonlinear optical materials as measured from the edge of the conduction band in the vertically upward direction in the absence of any perturbation. Thus, we observe that EEM is the function of size quantum number and the Fermi energy due to the combined influence of the crystal filed splitting constant and the anisotropic spin-orbit splitting constants, respectively. Besides it is a function of ηg due to which EEM exists in the band gap, which is otherwise impossible. Combining (1.34) with the Fermi–Dirac occupation probability factor, integrat
n D1 to infinity and applying the generalized Summerfield’s lemma ing betweenE z [152–153], the 2D carrier statistics in this case assumes the form
2D = n

max
*? g
v n
X m
1D ðE
F1HD , η n


z Þ ½Real part of ½T g,
z Þ + T2D ðEF1HD , ηg , n 2 π
h n
z = 1

(1:37)

where,
F1HD , η , n
2D ðE T g
z Þ =


s X
r = 1


F1HD , η , n

rÞ½T
1D ðE Lð g
z Þ,

Therefore combining (1.37) and (1.31f) we can study the entropy in this case.

(1:38)

1.2 Theoretical background

19

In the absence of heavy doping, the 2D dispersion relation EEM in the x–y plane at the Fermi level, the total 2D DOS and the electron concentration for QWs of nonlinear optical materials in the absence of band tails can, respectively, be written as
z Þ2
= ψ ðEÞ

k2 + ψ ðEÞð
n
z π=d ψ1 ðEÞ 2 3 s ( " !
2 )
z π
h2 n 2 *




′ ′
z Þ =
ðEFs , n ½ψ2 ðEFs Þ ψ2 ðEFs Þ fψ1 ðEFs Þg − fψ3 ðEFs Þg
m 2 dz

(1:39)

(

#
2 )
π n z
Fs Þ − ψ ðE

Fs Þg′ − ψ1 ðE fψ2 ðE 3 Fs Þ
dz



= g
v
2DT ðEÞ N 2π

 n
X zmax
z = 1 n

"

(

#
z π 2 n



E
−E
n Þ ′ − fψ1 ðEÞ − ψ3 ðEÞð
Þ gfψ2 ðEÞg Hð z1 dz
2
z π
n Þ = ψ ðE
n Þ n ψ1 ðE z1 z1 2
z d
2D = n





z π

fψ ðE

′ n ′ ½ψ2 ðEÞ ψ2 ðEÞ 1 Fs Þg − fψ3 ðEÞg
z d 2


zmax n g
v X
51 ðE
Fs , n
52 ðE
Fs , n
z Þ + T
z Þ ½T 2π n
z = 1

2 )

(1:40)

(1:41)

(1:42a)

(1:42b)


= γðEÞ,
ψ ðEÞ
=
f1 ðEÞ,
ψ ðEÞ
=
f2 ðEÞ,
E
n are the sub-band energies, E
Fs is where ψ1 ðEÞ z1 2 3 the Fermi energy in the 2D sized quantized material in thepresence of size quantization and in the absence of heavy doping as measured from the edgeof the conduction band in the vertically upward direction in the absence of any quantization, " #
z Þ2
Fs Þ − ψ ðE
Fs Þðn
z π=d ψ1 ð E 3


z Þ ≡ T51 ðEFs , n
Fs Þ ψ2 ðE and
Fs , n
52 ðE
z Þ ≡ T

s X


Fs , n

rÞ½ T
51 ðE
z Þ Lð

(1:43)

r=1

In the absence of band tails, the entropy can be written as

0 ∂n
S0 = ðπ2
k2 T=3Þ
B
F ∂E

(1:44)

Thus, using (1.43) and (1.44), we can study the entropy in this case. In the absence of heavy doping, the DOS for bulk specimens of nonlinear optical materials is given by

20

1 The entropy in quantum wells of heavily doped materials


= g
v ð3π2 Þ − 1 ψ ðEÞ

0 ðEÞ D 4

(1:45)

where, 2 qffiffiffiffiffiffiffiffiffiffiffiffi 3 3=2 3=2
½ψ ðEÞ
′ ð EÞ ψ



′ ′ 1 1 ½ψ ð EÞ ½ψ ð EÞ 3 ½ψ ð EÞ ½ψ ð EÞ 1 7 3 1
≡6 qffiffiffiffiffiffiffiffiffiffiffiffi − 2 q1 ffiffiffiffiffiffiffiffiffiffiffiffi − ψ4 ðEÞ 4 5, 3=2 2 2 ψ ðEÞ 2

ψ ðEÞ


ψ ðEÞ½ψ ð EÞ ½ψ2 ðEÞ ψ3 ðEÞ 2 3 2 3
′ ≡ ½ð2E
+E
g Þψ ðEÞ½
Eð
E
+E
g Þ − 1 + Eð
E
+E
g Þð2E
+ 2E
g + δ + Δjj Þ, ½ψ1 ðEÞ 1 
 − 1  
′ ≡ 2m
g + Δ? Þ δ + 2E
+ 2E
g + 2 Δjj
g + 2 Δ?
g ðE
*? E ½
h2 E ½ψ2 ðEÞ 3 3 and 
 − 1  
′ ≡ 2m
g + Δjj Þ 2E
+ 2E
g + 2 Δjj
g + 2 Δjj
g ðE
*? E ½ψ3 ðEÞ ½
h2 E 3 3 Combining (1.45) with the Fermi–Dirac occupation probability factor and using the generalized Summerfield’s lemma[153], the electron concentration can be written as
E
F Þ + Nð
E
F Þ
0 = g
v ð3π2 Þ − 1 ½Mð n

(1:46)

where, 2
E
F Þ ≡ 6 Mð 4

3 Þ2

3


F ½ψ1 ðE 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi5,
F Þ ψ ðE
F Þ ψ2 ðE 3


F is the Fermi energy of the bulk specimen in the absence of band tails as meaE sured from the edge of the conduction band in the vertically upward direction and
E
F Þ ≡ Nð


s X



rÞMð
E
F Þ Lð

(1:47)


r = 1

Thus, using (1.46a) and (1.44), we can study the entropy in this case. 1.2.2 Entropy in quantum wells (QWs) of HD III–V materials The dispersion relation of the conduction electrons of III–V compounds are described by the models of Kane (both three and two bands) [36, 37], Stillman et al. [38, 39] and Palik et al. [40], respectively. For the purpose of complete and coherent presentation and relative comparison, the entropy in QWs of HD III–V materials have also been investigated in accordance with the aforementioned different dispersion relations as follows:

21

1.2 Theoretical background

(a) The Three-Band Model of Kane Under the conditions δ = 0, Δjj = Δ? = Δ (isotropic spin orbit splitting constant) and
*? = m
c (isotropic effective electron mass at the edge of the conduction band),
*jj = m m (1.2) gets simplified as 2
+E
g + ΔÞðE
g + 2 ΔÞ
E
+E
g ÞðE Eð
2
k



h 3 0 0 0 = I11 ðEÞ, I11 ðEÞ ≡
g + ΔÞðE
+E
g + 2 ΔÞ
g ðE
c 2m E 3 0 0 0

(1:48)

which is known as the three-band model of Kane [36] and is often used to investigate the physical properties of III–V materials. Under the said conditions, the HD electron dispersion law in this case can be written from (1.26) as
2
k h
31 ðE,
η Þ + iT
32 ðE,
η Þ =T g g
c 2m 2

(1:49)

where, !"

b 2 a
η Þ θ0 ðE, g

c 1 + Erf ðE=ηg Þ " !#

 1

c − αb

b α
c + b 1 α

E
γ0 ðE, ηg Þ + + 1− 1− 1 + Erf
c 2
c
c
c2 ηg " ##
 ∞
X expð − p
2 =4Þ b 1 α

2 2
u
2 Þ
2 Þ , pffiffiffi expð − u − 1− 1− sinhðp
c cηg π
c
c
p
=1 p


η Þ≡
31 ðE, T g


−1
≡ ðE
g + 2 Δ
g + ΔÞ − 1 ,
c ≡ E b 3 ! pffiffiffi

 pffiffiffi b 1 +
cE 2 1 α

π π


22 Þ
2 ≡ and T32 ðE, ηg Þ ≡ expð − u u 1 − 1 −
Þ
c
c cηg cηg
cηg
c 1 + Erf ðE=η g
32 ðE,
η Þ and this imagiThus, the complex energy spectrum occurs due to the term T g nary band is quite different from the forbidden energy band. EEM at the Fermi level is given by 
F , η Þ = m
31 ðE,
η Þg′


c fT
* ðE ð1:50Þ m E = EF s g g s

Thus, EEM in HD III–V, ternary and quaternary materials exists in the band gap, which is the new attribute of the theory of band tailing. In the absence of band tails, ηg ! 0 and EEM assumes the form 
F Þ = m
′


c f
I11 ðEÞg
* ðE (1:51) m E = EF

22

1 The entropy in quantum wells of heavily doped materials

The DOS function in this case can be written as


c 3=2


η Þ = g v 2m
η Þ
HD ðE, N R21 ðE, ηg Þ cos½ϑ21 ðE, g g 3π2
h2

(1:52)

where, "
η Þ≡
21 ðE, R g


η Þg′2 ½fβ ðE,
η Þg′2 ½fα11 ðE, g 11 g
η Þ + 4β ðE,
η Þ 4α11 ðE, g 11 g

#1=2

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1



33 ðE,
η Þg2 + fT
34 ðE,
η Þg2 , α11 ðE, ηg Þ ≡ T33 ðE, ηg Þ + fT g g 2 h i
η Þ ≡ fT
31 ðE,
η Þg3 − 3T
31 ðE,
η ÞfT
32 ðE,
η Þg2 ,
33 ðE, T g g g g h i
η Þ ≡ 3T
32 ðE,
η ÞfT
31 ðE,
η Þg2 − fT
32 ðE,
η Þg3 ,
34 ðE, T g g g g
η Þ≡ 1 β11 ðE, g 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2





fT 33 ðE, ηg Þg + fT 34 ðE, ηg Þg − T33 ðE, ηg Þ

and "
η Þ ≡ tan − 1 ϑ21 ðE, g


η Þg′ fβ11 ðE, g +
fα11 ðE, η Þg′ g

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#
η Þ α11 ðE, g
η Þ β ðE, 11

g


η Þ ≥ π and a Thus, the oscillatory DOS function becomes negative for ϑ21 ðE, g new forbidden zone will appear in addition to the normal band gap. The electron concentration can be expressed as
0 = n


  
s X
c 3=2 g
v 2m
F , η Þ +
I111e ðE
F , η Þ
I111e ðE lðrÞ½ s s h h 3π2
h2
r = 1

(1:53)

where 3=2
F , η Þ = fγ ðE

111e ðE T g 2 Fh , ηg Þg h


hd is given by In this case, E
hd , η Þ = 0
31 ðE T g

(1:54)

The numerical evaluation of the entropy has been done by using (1.53), (1.31f), (1.54) and the allied definitions. For dimensional quantization along z-direction, the dispersion relation of the 2D electrons in this case can be written following (1.49) as

1.2 Theoretical background


z Þ2
h2 ð
ks Þ2
z π=d
2 ðn h
31 ðE,
η Þ + iT
32 ðE,
η Þ + =T g g
c
c 2m 2m

23

(1:55)


in this case assumes the form
2DT ðEÞ The expression of the N
zmax n
c g
v X m

η ,n



′5D ðE,
T N2DT ðEÞ = g
z ÞHðE − Enz D5 Þ π
h2 n
z = 1

(1:56)

where
z Þ2 ð2m
5D ðE,
η ,n




z π=d
cÞ − 1 T h2 ð n g
z Þ = ½T31 ðE, ηg Þ + iT32 ðE, ηg Þ −

n D5 in this case given by and the sub-band energies E z
z Þ2 gð2m
31 ðE
n D5 , η Þ
z =d
cÞ − 1 = T f
h2 ðn z g

(1:57)

Thus, we observe that both the total DOS in QWs of HD III–V compounds and the sub-band energies are complex due to the presence of the pole in energy axis of the corresponding materials in the absence of band tails. EEM in this case is given by
F1HD , η , n
′31 ðE
F1HD , η , n
c ½T
* ðE m g
z Þ = m g
z Þ

(1:58)

Therefore, under the same conditions as used in obtaining (1.48) from (1.2), the 2D carrier statistics in this case can be written by using the same conditions from (1.37) as
2D = n


max n X
c g
v zz m
5D ðE
F1HD , η , n


z Þ ½Real part of ½T g
z Þ + T 6D ðEF1HD , ηg , n 2 π
h n
z = 1

(1:59)

where
F1HD , η , n
6D ðE T g
z Þ =

s X r=1


F1HD , η , n

rÞ½T
5D ðE Lð g
z Þ

Therefore, combining (1.31f) and (1.59) we can study the entropy in this case. In the absence of band tails, the 2D dispersion relation, EEM in the x–y plane at the Fermi level, the total 2D DOS, the sub-band energy and the electron concentration for QWs of III–V materials assume the following forms
2
ks2 h
h2
x Þ2 =
I11 ðEÞ

π=d + ðn
c 2m
c z 2m

(1:60)


Fs Þ = m
Fs Þg′
c f
I11 ðE
* ðE m

(1:61)

24

1 The entropy in quantum wells of heavily doped materials


zmax n
c g
v X
= ðm
′Hð
E
−E
n Þg
2DT ðEÞ Þ f½
I 11 ðEÞ N z2 π
h2 n
z = 1

(1:62)


n can be expressed as where the sub-band energies E z2
2
z Þ2
N Þ = h
I11 ðE
π=d ðn Z2
c z 2m
 n
zmax
c g
v X m
53 ðE
Fs , n
54 ðE
Fs , n
z Þ + T
z Þ
2D = ½T n π
h2 n
z = 1

(1:63)

(1:64)

where " #
2 h 2






ðn π=dz Þ T55 ðEFs , nz Þ ≡ I11 ðEFs Þ −
c z 2m and
Fs , n
54 ðE
z Þ ≡ T


s X


Fs , n

rÞT
53 ðE
z Þ Lð

(1:65)


r = 1

It is worth noting that the EEM in this case is a function of Fermi energy alone and is independent of size quantum number. Thus, using (1.64) and (1.44), we can study the entropy in this case. In the absence of band tails, the DOS function and the electron concentration, in bulk III–V, ternary, and quaternary materials in accordance with the unperturbed three-band model of Kane assume the following forms.
 ffi
c 3=2 q
ffiffiffiffiffiffiffiffiffiffiffi
= 4πgv 2m


0 ðEÞ ð EÞ ½
I ′11 ðEÞ I D 11
h2

c 3=2

2m g

1 ðE
F Þ
0 = v2 ½M1 ðEF Þ + N n 3π
h2

(1:66)

(1:67)

where "

# 1 1 1 1
≡
I11 ðEÞ


3=2


I ′11 ðEÞ
+E
+E
g + E
+E
g + Δ − E
+E
g + 2 Δ , M1 ðEF Þ ≡ ½I11 ðEÞ , E 3 and
1 ðE
F Þ ≡ N


s X


F Þ

rÞM
1 ðE Lð


r = 1

Thus using (1.44) and (1.67) we can study the entropy in this case

(1:68)

1.2 Theoretical background

25


g or Δ < < E
g , (1.48) can be expressed as Under the inequalities Δ > > E 0 0
2
k
+ αEÞ
=h Eð1
c 2m

2

(1:69)

It may be noted that (1.69) is the well-known two band model of Kane and is used in the literature to study the physical properties of those III–V and opto-electronic materials whose energy band structures obey the aforementioned inequalities. The dispersion relation in HD III–V, ternary and quaternary materials whose energy spectrum in the absence of band tails obeys the two band model of Kane as defined by (1.69), can be written as
2
k h
η Þ = γ2 ðE, g
c 2m 2

(1:70)

where "
η Þ≡ γ2 ðE, g

# h i 2
η Þ + αθ0 ðE,
η Þ , ð E, γ g g 0
Þ 1 + Erf ðE=η g

The EEM in this case can be written as
F , η Þ = m
η Þg′j


* ðE
c fγ2 ðE, m g g E = EF h

(1:71)

h

Thus, one again observes that the EEM in this case exists in the band gap. In the absence of band tails, ηg ! 0 and the EEM assumes the well-known form
F Þ = m



c f1 + 2αEgj
* ðE m E = EF

(1:72)

The DOS function in this case can be written as

3=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

η Þ = g v 2 mc
η Þfγ ðE,
η Þg′
HD ðE, γ2 ðE, N g g g 2 2π2
h2

(1:73)

Since, the original two band Kane model is an all zero and no pole function in the finite energy plane with respect to energy, therefore the HD counterpart will be totally real and the complex band vanishes. The electron concentration is given by
0 = n where


  
s X
c 3=2

g
v 2m
Fs , η Þ

rÞ½
I111 ðE ð E , η Þ + I Lð 111 Fs g g 3π2
h2
r = 1

(1:74)

26

1 The entropy in quantum wells of heavily doped materials

3=2
Fs , η Þ = fγ ðE

I111 ðE g 2 Fs , ηg Þg


hd is given by In this case, E
hd , η Þ = 0 γ2 ðE g

(1:75)

One can numerically compute the entropy by using (1.74), (1.75), (1.31f) and the allied definitions in this case. For dimensional quantization along z-direction, the dispersion relation of the 2D electrons in this case can be written following (1.70) as
z Þ
z π=d
2 ðn h
h2 ð
ks Þ
η Þ + = γ2 ðE, g
c
c 2m 2m 2

2

(1:76)


2DT ðEÞ
in this case can be written The expression of the N nzmax
c g
v X
=m



η ,n
2DT ðEÞ
7D ′ðE, N T g
z ÞHðE − Enz D7 Þ π
h2 nz = 1

(1:77)

where
z Þ2 ð2m
η ,n

7D ðE,
z π=d
c Þ − 1 , h2 ð n T g
z Þ = ½γ2 ðE, ηg Þ −

n D7 in this case given by The sub-band energies E z
z Þ2 gð2m
n D7 , η Þ
z π=d
c Þ − 1 = γ2 ðE f
h2 ðn z g

(1:78)

Thus, we observe that both the total DOS and sub-band energies of QWs of HD III–V compounds in accordance with two band model of Kane are not at all complex since the dispersion relation in accordance with the said model is an all zero function with no pole in the finite complex plane. The EEM in this case is given by
F1HD , η , n
F1HD , η , n
c ½γ2 ′ðE
* ðE m g
z Þ = m g
z Þ

(1:79)

Therefore under the same conditions as used in obtaining (1.48) from (1.2), the 2D carrier statistics in this case can be written by using the same conditions from (1.77) as
2D = n where

z max
c g
v n
X m
7D ðE
FLHD , η , n


z Þ ½T g
z Þ + T8D ðEF1HD , ηg , n 2 π
h n
z = 1

(1:80)

1.2 Theoretical background


F1HD , η , n
8D ðE T g
z Þ =


s X
r = 1

27


F1HD , η , n

rÞT
7D ðE Lð g
z Þ

Therefore, combining (1.31f) and (1.80) we can get the entropyin this case.
g , (1.60) assumes the form
g or Δ < < E Under the inequalities Δ > > E 0 0
2
z π
2
ks2
h2 n
+ αEÞ
=h + Eð1
z

2mc 2mc d

(1:81a)

The EEM can be written from (1.81a) as
Fs Þ = m
Fs Þ
c ð1 + 2αE
* ðE m

(1:81b)

The total 2D DOS function assumes the form
zmax n
c g
v X
=m
Hð
E
−E
n Þ
2DT ðEÞ ð1 + 2αEÞ N z3 π
h2 n
z = 1

(1:82)


n Þ can be expressed as where the sub-band energy ðE z3
2 h
z Þ2 = E
n ð1 + αE
n Þ
π=d ðn z3 z3
c z 2m

(1:83)

The 2D electron statistics can be written as
2D = n


zmax ð ∞ n
E

c g
v X ð1 + 2αEÞd m



2 E − E π
h n
z = 1 E
nz3 1 + exp

Fs k T B

zmax
g
v n
X
c
kB T m
n ÞF
0 ðη Þ + 2α
kB T
F
1 ðη Þ ½ð1 + 2αE = z3 n1 n1 π
h2 n
z = 1

(1:84)


Fs − E
n Þ=
kB T
and F
j ðηÞ is the one parameter Fermi–Dirac integral of where ηn1 ≡ ðE z3 order j which can be written [154] as
 ð∞
xj d
x 1

,
j > − 1 (1:85) Fj ð nÞ = Γð
j + 1Þ 0 1 + expð
x − ηÞ or for all j, analytically continued as a complex contour integral around the negative x-axis

jÞ  ð + 0
xj d
x
j ðηÞ = Γðp−ffiffiffiffiffiffi ffi (1:86) F 2π − 1 − ∞ 1 + expð −
x − ηÞ where η is the dimensionless parameter and
x is independent variable,

28

1 The entropy in quantum wells of heavily doped materials

Therefore in this case the entropy can be investigated by using (1.44) and (1.84). The forms of the DOS and the electron statistics for bulk specimens of III–V materials in the absence of band tails whose energy band structures are defined by the two-band model of Kane can, respectively, be written as

c 3=2 q
ffiffiffiffiffiffiffiffiffiffiffiffiffi
= 4πg
v 2m
½
I ′11e ðEÞ

0 ðEÞ , I 11e ðEÞ D
h2

c 3=2

g
v 2m
2 ðE
F Þ
0 = 2 ½M2 ðEF Þ + N n 3π
h2

(1:87)

(1:88)

where
I11e ðEÞ
≡ Eð1
+ αEÞ,

I ′11e ðEÞ
≡ ð1 + 2αEÞ,

F Þ ≡ ½
I11e ðE
F Þ3=2
2 ðE M

(1:89)

and
2 ðE
F Þ ≡ N


s X


F Þ

rÞM
2 ðE Lð

(1:90)


r = 1


g or Δ < < E
g together with the inequality αE
F < < 1, (c) Under the constraints Δ > > E 0 0 the (1.87) and (1.88) assumes the forms as  



c F
1=2 ðηÞ + 15αkB T F
3=2 ðηÞ
0 = g
v N (1:91) n 4 where

3=2
*
kB T
c ≡ 2 2πm N and η ≡ h2


F E
kB T


and
F E η≡

kB T The entropy can be written as
2
 



S0 = g
v N
− 1=2 ðηÞ + 15αkB T F
c π kB F
1=2 ðηÞ 3 4

(1:92)

1.2 Theoretical background

29

The dispersion relation in HDS whose energy spectrum in the absence of band tails obeys the parabolic energy bands (1.69) is given by
2
k h
η Þ = γ3 ðE, g
c 2m 2

(1:93)

where "
η Þ≡ γ3 ðE, g

# 2

ÞÞ γ0 ðE, ηg Þ ð1 + Erf ðE=η g

Since the dispersion relation in accordance with the said model is an all zero function with no pole in the finite complex plane, therefore the HD counterpart will be totally real, which is also apparent form the expression (1.93). The EEM in this case can be written as
Fh , η Þ = m
Fh , η Þg′
c fγ3 ðE
* ðE m g g

(1:94)

In the absence of band tails, ηg ! 0 and the EEM assumes the form
F Þ = m
c
* ðE m

(1:95)

It is well known that the EEM in unperturbed parabolic energy bands is a constant quantity in general excluding cross-fields configuration. However, the same mass in the corresponding HD bulk counterpart becomes a complicated function of Fermi energy and the impurity potential together with the fact that the EEM also exists in the band gap solely due to the presence of finite ηg . The DOS function in this case can be written as
 ffi
c 3=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g
v 2m


η Þfγ ðE,
η Þg′ NHD ðE, ηg Þ = 2 ð E, γ g g 3 3 2π
h2 The electron concentration is given by #
 "
s X
c 3=2

g
v 2m
F , η Þ

rÞ½
I 113 ðE
0 = 2 I113 ðEFh , ηg Þ + Lð n g h 3π
h2
r = 1

(1:96)

(1:97)

where 3=2
F , η Þ = fγ ðE

I113 ðE g 3 Fh , ηg Þg h


hd is given by In this case, E
hd , η Þ = 0 γ3 ðE g

(1:98)

30

1 The entropy in quantum wells of heavily doped materials

One can numerically compute the entropy by using (1.97), (1.98), (1.31f) and the allied definitions in this case. For dimensional quantization along z-direction, the dispersion relation of the 2D electrons in this case can be written following (1.93) as
z Þ
z z=d
2 ðn h
h2 ð
ks Þ
η Þ + = γ3 ðE, g
c
c 2m 2m 2

2

(1:99)


2DT ðEÞ
in this case can be written as The expression of the N
zmax n
c g
v X
=m



η ,n
9D ′ðE,
2DT ðEÞ T N g
z ÞHðE − Enz D9 Þ π
h2 n
z = 1

(1:100)

where
z Þ2 ð2m
9D ðE,
η ,n

z π=d
c Þ − 1 . T h2 ðn g
z Þ = ½γ3 ðE, ηg Þ −

n D9 in this case given by The sub-band energies E z
z Þ2 gð2m
n D9 , η Þ
z π=d
c Þ − 1 = γ3 ðE f
h2 ðn z g

(1:101)

The EEM in this case can be written as
F1HD , η , n
F1HD , η Þ
* ðE
c ½γ′3 ðE m g
z Þ = m g

(1:102)

Therefore under the same conditions as used in obtaining (1.48) from (1.2), the 2D carrier statistics in this case can be written by using the same conditions from (1.77) as
2D = n


zmax n
c g
v X m
9D ðE
F1HD , η , n


z Þ ½T g
z Þ + T 10D ðEF1HD , ηg , n π
h2 n
z = 1

(1:103)

where
10D ðE
F1HD , η , n T g
z Þ


s X
r = 1


F1HD , η , n

rÞ½T
9D ðE Lð g
z Þ,

Therefore combining (1.31f) and (1.80) we can get the entropy in this case. Under the condition α ! 0, the expressions of total 2D DOS, for materials without forming band tails whose bulk electrons are defined by the isotropic parabolic energy bands can, be written from (1.82) as
zmax n


c g
v X
=m
E
−E
n
2DT ðEÞ H N z p π
h2 n
z = 1

(1:104)


n Þ the n
2D and the entropy can, respectively, be expressed as The sub-band energy ðE zp

1.2 Theoretical background


2
z π
2 n
n = h E zp
z
c d 2m
2D = n

31

(1:105)

z max


g
v n
X
c
kB T m
0 η F n2 2 π
h
z = 1 n

(1:106a)


z max

2
2


n X c gv
S0 = π kB T m
−1 η F n2 2 3π
h
z = 1 n

(1:106b)

(b) The Model of Stillman et al. In accordance with the model of Stillman et al. [38], the electron dispersion law of III–V materials assumes the form
=
t11 k2 −
t12
k4 E

(1:107)

where !2 " ! #
 2 2
c 2
h2
h 2Δ m − 1

g + 4Δ +

t11 ≡ ;
t = 1 − 3E 0
g .fðEg0 + ΔÞð2Δ + 3Eg0 Þg
c 12
c
0 2m 2m m E 0
0 is the free electron mass and m In the presence of band tails, (1.107) gets transformed as
2
k

h = I12 ðE, ηg Þ
c 2m 2

(1:108)

where
η Þ=a
η ÞÞ1=2 , a
I12 ðE,
11 ½1 − ð1 − a
12 γ3 ðE,
11 ≡ g g

!
2
t11 h , 4mc
t12

and 4
t
12 = 212 a
t11 The EEM can be written as
F , η Þ = m
F , η Þg′
c f
I12 ðE
* ðE m g g h h

(1:109)

The DOS function in this case can be written as

c 3=2 q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g
v 2m

η Þf
I12 ðE,
η Þg′
NHD ðE, ηg Þ = 2 I 12 ðE, g g 2π
h2 The electron concentration is given by

(1:110)

32

1 The entropy in quantum wells of heavily doped materials


0 = n


  
s X
c 3=2

g
v 2m



ð E , η Þ + ð E , η Þ I Lð r Þ½ I 121 Fh g 121 Fh g 3π2
h2
r = 1

(1:111)

where
F , η Þ = f
I12 ðE
F , η Þg3=2
I121 ðE g g h h
hd is expressed through the equation In this case, E
hd , η Þ = 0 γ3 ðE g

(1:112)

One can numerically compute the entropy by using (1.111), (1.112), (1.31f) and the allied definitions in this case. For dimensional quantization along z- direction, the dispersion relation of the 2D electrons in this case can be written following (1.108) as
z Þ2
h2 ð
ks Þ2
z π=d
2 ðn h
η Þ + =
I12 ðE, g
c
c 2m 2m

(1:113)


2DT ðEÞ
in this case can be written as the expression of the N
zmax n
c g
v X
=m
−E
n D11 Þ
η , nz ÞHðE
11D ′ðE,
2DT ðEÞ T N z g 2 π
h n
z = 1

(1:114)

where h i
z Þ2 ð2m
11D ððE,
η ,n


z π=d
cÞ − 1 T h2 ðn g
z Þ = I12 ðE, ηg Þ −

n D11 in this case given by The sub-band energies E z
z Þ2 gð2m
n D11 , η Þ
z π=d
c Þ − 1 =
I12 ðE f
h2 ðn z g

(1:115)

The EEM in this case assumes the form
F1HD , η , n
F1HD , η , n
E
c f
I12 ′ðE m*ð g
z Þ = m g
z Þ

(1:116)

The 2-D electron statistics in this case can be written as
2D = n

i z max h
c g
v n
X m
F1HD , η , n
12D ðE
F1HD , η , n
11D ðE

Þ + T Þ T z z g g π
h2 n
z = 1

(1:117)

where
F1HD , η , n
12D ðE T g
z Þ =


s X
r = 1

h i
F1HD , η , n

rÞ T
11D ðE
Þ , Lð z g

Therefore combining (1.117) and (1.31f) we can get the entropyin this case.

1.2 Theoretical background

33

For unperturbed material, the 2-D EEM can be expressed as
Fs Þg′
Fs Þ = m
c f
I12 ðE
E m*ð

(1:118)

where h i
=a
1=2
I12 ðEÞ
11 1 − ð1 − a
12 ðEÞÞ It appears that the EEM in this case is a function of Fermi energy alone and is independent of size quantum number. The total 2D DOS function in the absence of band tails in this case can be written as
 n
zmax 


c g
v X
= m
′

−E
n Þ
2DT ðEÞ EðE (1:119) ½
I12 ðEÞ N z3 2 π
h
z = 1 n
n can be expressed as where the sub-band energies E z3
2
z Þ2
n Þ = h
I12 ðE
π=d ðn z3
c z 2m The 2D electron concentration assumes the form
 n
zmax  
c g
v X
Fs , n
56 ðE
Fs , n
55 ðE
2D = m
z Þ + T
z Þ T N 2 π
h
z = 1 n

(1:120)

(1:121)

where " 2 # 2


h π n z
Fs , n
Fs Þ −
55 ðE
Z Þ ≡
I12 ðE T
z
c d 2m and
Fs , n
56 ðE
Z Þ ≡ T


s X


Fs , n

rÞT
55 ðE
Z Þ Lð


r = 1

Thus using (1.44) and (1.121) we can study the Entropy in this case. The expression of electron concentration for bulk specimens of III–V materials (in the absence of band tails) can be written in accordance with the model of Stillman et al. as
0 = n where


 
c 3=2 
g
v 2m
F Þ + N
A ðE
F Þ MA10 ðE 10 2 2 3π
h

(1:122)

34

1 The entropy in quantum wells of heavily doped materials


F Þ = ½
I12 ðE
F Þ3=2 and N
A ðE
F Þ =
A ðE M 10 10


s X
r = 1


F Þ

rÞ½M
A ðE Lð 10

Thus using (1.44) and (1.122) we can study the entropy in this case. (c) Model of Palik et al. 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 of heavy hole, light hole and the split-off holes can be expressed in accordance with the model of Palik et al. [40] as k
2

11
k4
= h −B E
c 2m 2

(1:123)

where #2 4 1+
h
11 = 4 B 2

cÞ 4Eg0 ðm 1+ "

3

"
x2 11 2 2 5


x11 ð1 − y11 Þ , x11 1 + 2

Δ
E g0

!# − 1 and
y11 =


c m
0 m

The (1.123) gets simplified as
2
k

h = I13 ðEÞ
c 2m 2

(1:124)

where h i
12 a
=b
B
11 Þ1=2 , a
I13 ðEÞ
12 Þ2 − 4E
12 =
12 − ðða


2 h
c 2m

!

 
12
12 = a and b
11 2B

Under the condition of heavy doping forming Gaussian band tails, (1.124) assumes the form
2
k

h = I13 ðE, ηg Þ
c 2m 2

(1:125)

where h i 2
12 a
I13 ðE,
η Þ=b
11 γ ðE,
η ÞÞ1=2

− ðð a Þ − 4 B 12 12 g g 3 The EEM can be written as
F , η Þ = m
F , η Þg′
c f
I13 ðE
* ðE m g g h h The DOS function in this case can be expressed as

(1:126)

1.2 Theoretical background




c 3=2 q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
η Þ = g v 2m
η Þf
I13 ðE,
η Þg′
HD ðE, I 13 ðE, N g g g 2π2
h2

35

(1:127)

Since, the original band model in this case is a no pole function, in the finite complex plane therefore, the HD counterpart will be totally real and the complex band vanishes. The electron concentration is given by
0 = n


  
s X
c 3=2

g
v 2m



I123 ðEFh , ηg Þ + LðrÞ½I 123 ðEFh , ηg Þ 3π2
h2
r = 1

(1:128)

where
Fh , η Þ = f
I123 ðE
Fh , η Þg3=2
I123 ðE g g

(1:129)


hd is given by In this case, E
hd , η Þ γ3 ðE g

(1:130)

One can numerically compute the entropy by using (1.128), (1.129), (1.31f) and the allied definitions in this case. For dimensional quantization along z-direction, the dispersion relation of the 2D electrons in this case can be written following (1.108) as
in this case can be written as
2DT ðEÞ the expression of the N
zmax n
c g
v X
=m
η ,n



′13D ðE,
2DT ðEÞ T N g
z ÞHðE − Enz D13 Þ π
h2 n
z = 1

(1:131)

where h i
z Þ2 ð2m
13D ðE,
η ,n


z π=d
cÞ − 1 T h2 ðn g
z Þ = I13 ðE, ηg Þ −

n D13 in this case given by The sub-band energies E z
z Þ2 gð2m
n D13 , η Þ
c Þ − 1 =
I13 ðE
z π=d f
hðn z g

(1:132)

The EEM in this case can be expressed as h i
F1HD , η , n
F1HD , η , n
c
I ′13 ðE
* ðE m g
z Þ = m g
z Þ

(1:133)

The 2-D electron statistics in this case can be written as
2D = n

i max h
c g
v n
X m
13D ðE
F1HD , η , n


z Þ T g
z Þ + T14D ðEF1HD , ηg , n 2 π
h n
z = 1

(1:134)

36

1 The entropy in quantum wells of heavily doped materials

where
F1HD , η , n
14D ðE T g
z Þ =


s X
r = 1


13D ðE
F1HD , η , n LðrÞ½T g
z Þ

Therefore combining (1.134) and (1.31f) we can get the entropy in this case. The 2D electron dispersion relation in the absence of band tails this case assumes the form
2
ks2 h
h2
z Þ2 =
I13 ðEÞ

π=d + ðn
c 2m
c z 2m

(1:135a)

The EEM in this case can be written from (1.135a) as
F Þ = m
F Þ′
c ½
I13 ðE
* ðE m S S The total 2D DOS function can be written as
 n
zmax

X
= mc g v
′Hð
E
−E
n Þg
2DT ðEÞ f½
I13 ðEÞ N z4 π
h2 n
z = 1

(1:135b)

(1:136)


n can be expressed as where the sub-band energies E z4
2
z Þ2
n Þ = h
I13 ðE
π=d ðn z4
c z 2m

(1:137)

The 2D electron concentration assumes the form
2D = n

z max
c g
v n
X m
57 ðE
Fs , n
58 ðE
Fs , n
z Þ + T
z Þ ½T 2 π
h n
z = 1

(1:138)

where " 2 #
s 2
X

h π n z
Fs , n
Fs Þ −
Fs , n
Fs , n
58 ðE

rÞT
57 ðE
57 ðE
z Þ ≡
I13 ðE
z Þ ≡
z Þ and T Lð T
z
c d 2m
r = 1 (1:139) Thus by using (1.138) and (1.44) we can study the entropy in this case.

1.2.3 The entropy in quantum wells (QWs) of HD II–VI materials The carrier energy spectra in bulk specimens of II–VI compounds in accordance with Hopfield model [57] can be written as

1.2 Theoretical background


′0
k2 +
λ0
ks
=a
′0
ks2 + b E z

37

(1:140)


′0 ≡
h2 =2m
′0 ≡
h2 =2m
*? , b
*jj , and
λ0 represents the splitting of the two-spin where a states by the spin orbit coupling and the crystalline field. Therefore the dispersion relation of the carriers in HD II–VI materials in the presence of Gaussian band tails can be expressed as
′0
k2 ±
λ0
ks
η Þ=a
′0
ks2 + b γ3 ðE, g z

(1:141)

Thus, the energy spectrum in this case is real since the corresponding E-k relation in the absence of band tails as given by (1.141) is a no pole function in the finite complex plane. The transverse and the longitudinal EEMs masses are, respectively, given by 2 0 13  
λ0 6 B C 7 * *

′
? fγ3 ðE, ηg Þg 41 + @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA5E
− E

? ðEFn , ηg Þ = m (1:142) m 2  Fn
η Þ
0 ′γ3 ðE, ð
λ0 Þ + 4a g and
F , η Þ = m
η Þg′j


*k fγ3 ðE,
*k ðE m n g g E − EFn

(1:143)

Thus the transverse EEM in HD II–VI materials is a function of electron energy and is double valued due to the presence of
λ0 and due to heavy doping the same mass exists in the band gap. In the absence of band tails, ηg ! 0, we get 13 2 0  
λ0 B 6 C 7 * *

? 41 + @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA5E
= E

? ðEF Þ = m (1:144) m  F 2
0 ′
EÞ ð
λ0 Þ + 4a and
F Þ = m
*jj
*jj ðE m

(1:145)

The volume in k- space as enclosed (1.141) can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " 2
η Þ ð
λ0 Þ γ3 ðE, g 4π 3
η Þg3=2 +
E,
η Þ= p ffiffiffiffiffi ffi ð E, fγ Vð g g 3 ′
8
0 a
′0 b′0 3a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ## "


2
η Þ γ3 ðE,
g 3 λ0 ð λ Þ 0 − 1
η Þ+ pffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ3 ðE, sin ± g 2 4 a
′0 4a
′0
η Þ + ð
λ0
Þ γ3 ðE, g 4d 0

(1:146)

38

1 The entropy in quantum wells of heavily doped materials

Therefore, the electron concentration can be written as  
s X g
v
F , η Þ +
F , η Þ

rÞ½
I 124 ðE
0 = pffiffiffiffiffiffiffi
I124 ðE n Lð g g h h
′0
′0 b 3π2 a
r = 1

(1:147)

where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2
η Þ ð
λ0 Þ γ3 ðE, g 3
F , η Þ = 4fγ ðE
F , η Þg3=2 +
I124 ðE 5 g g 3 h h ′ 8
0 a 2


hd is given by In this case, E
hd , η Þg = 0 fγ3 ðE g

(1:148)

Thus, one can numerically evaluate the entropy by using (1.147), (1.31f), (1.148) and the allied definitions in this case. The dispersion relation of the conduction electrons of QWs of HD II–VI materials for dimensional quantization along z- direction can be written following (1.141) as
2
z
k2 + b
′0 πn
η Þ=a ′
±
λ0
ks γ3 ðE, 0 g s
dz

(1:149)

The EEM can be expressed following (1.149) as 2

3

6 7 ð
λ0 Þ 7
F1HD , n
z , ηg Þ = m
*? 6
* ðE m 61 ∓  7  1=2 5

4 2 2
′0 n
z π + 4a
′
′0 b
γ ð E , η Þ ð
λ0 Þ − 4a 0 3 F1HD g
d

(1:150)

z

Thus we observe that the doubled valued effective mass in 2-D QWs of HD II–VI materials is a function of Fermi energy, size quantum number and the screening potential, respectively, together with the fact that the same mass exists in the band gap due to the sole presence of the splitting of the two-spin states by the spin orbit coupling and the crystalline field. The sub-band energy in this case is given by
2
z
′0 πn
n D14 , η Þ = b γ3 ðE z g
dz

(1:151)

The surface electron concentration at low temperatures assumes the form
2D = n

z max
*? n
X g
v m
F1HD , η Þ − E
n D + ð
λÞ2 m
*?
h − 2 γ3 ðE g 2 Z 14 π
h n
Z = 1

Therefore combining (1.152) and (1.31f) we can get the entropy in this case.

(1:152)

1.2 Theoretical background

39

The dispersion relation of the conduction electrons of QWs of II–VI materials for dimensional quantization along z- direction in the absence of band tails can be written following (1.140) as
2
z π
′0 n
0
k2 + b
=d ±
λ0
ks E s
z d

(1:153)

Using (1.153), the EEM in this case can be written as 2

3

6 7 ð
λ0 Þ 7
Fs , n
z Þ = m
*? 6
* ðE m 61 ∓  7  1=2 5 2 4 2
z π n


′ ′ ′
0b 0

0 EFs + 4a ðλ0 Þ − 4a d

(1:154)

z


n assumes the form The sub-band energy E z4
2
z π
′0 n
n = b E z5
z d The area of constant energy 2D quantized surface in this case is given by " # π 2 2 1=2









z Þ =
0 ′ðE − Enzs Þ ± λ0 ½ðλ0 Þ + 4a
0 ′ðE − EnzS Þ  ½ðλ0 Þ + 2a A ± ðE, n 2
0 ′Þ 2ða The surface electron concentration can be expressed in this case as
z max ð ∞ − 2g
v nX ∂
+ ðE
Fs, n
− ðE
Fs, n
E

2D =
z Þ + A
z Þ f
f0 ðEÞgd n ½A 2
∂E 2ð2πÞ n
z = 1 E
nzS

(1:155a)

(1:155b)

(1:156)


is the Fermi–Dirac occupation probability factor. where
f0 ðEÞ From (1.156) we get
2D = n

z max
n
X
*?
kB T g
v m
0 ðη Þ F nzS 2 π
h
z = 1 n

(1:157)

where
Fs − E
n + ð
λÞm
−1
*?
h − 2 Þð
kB TÞ ηnzS = ðE zS Therefore the entropy is given by
X zmax 2
2

*
n ? gv
S0 = π kB T m
− 1 ðη Þ F nz8 3π
h2
z = 1 n

(1:158)

40

1 The entropy in quantum wells of heavily doped materials

1.2.4 The entropy in quantum wells (QWs) of HD IV–VI materials The dispersion relation of the conduction electrons in IV–VI materials can be expressed in accordance with Dimmock [156] as " #" #
g
g E E
h2
ks2
h2
kz2
h2
ks2
h2
kz2 0 0
2
k2
2
k2 + P ε− − + (1:159) − + ε+ =P ? s jj z
t− 2m
t−
t− 2m
t− 2 2m 2 2m
g , m
t± and m
l± whereε is the energy as measured from the center of the band gap E 0 represent the contributions to the transverse and longitudinal effective masses of
− bands arising from the ~
± and L k ·~ p perturbations with the other the external L 6 6 bands taken to the second order. h
i   2 
h2 E
g0 
g =2m
≡
+ Eg0
2 ≡
h2 E
*t , P
*t and (where m and ε = E Substituting, P ? jj 2 0
* 2m l
* are the transverse and the longitudinal effective masses at k
= 0), (1.159) gets transm l

formed as "

2
2 2
2
−
h ks −
h kz E
t− 2m
l− 2m

#"

# 2
2 2
2 k k
h h

h2
ks2
h2
kz2 s z
+α + α + 1 + αE =
t+
l+
*t
*l 2m 2m 2m 2m

From (1.160), we can write "
# 
 α
h4
ks4 1 1 1 1 α
h2
kz2 2
2
+ αE + +
h ks − −
t+ m
l− m
t−
t− 2m
t+
t−
t+
*t 2m 4m 4m 2m 2m " +


2
kz2
h2
kz2 h +
l−
*l 2m 2m

!

(1:160)

ð1:161Þ

# 
2
1 αE 1 α
h4
kz4 2


− + + − Eð1 + αEÞ = 0 +
h kz
l+ m
t−
l− m
l 4m 2 m

Using (1.161), the dispersion relation of the conduction electrons in HD IV–VI materials can be expressed as h i α
h4
ks4


η Þ
k2 +
λ72 ðE,
η Þ Z 0 ðE, ηg Þ +
h2
ks2
λ11 ðE, g g z + −
t m
l 4m h i
η Þ
k2 +
λ74 ðE,
η Þ
k4 −
λ75 ðE,
η Þ =0 +
λ73 ðE, g z g z g where " 1
η Þ≡
0 ðE, Z 1 + Erf g 2 " α
h2
η Þ≡ λ71 ðE, g
t− m
l+ 4m


E ηg

!#
η Þ≡ ,
λ70 ðE, g


η Þ+
0 ðE, Z g

α
h2
l− m
t+ 4m

α
η Þ,
ðE, Z g
t+ m
t− 0 4m #
η Þ ,
0 ðE, Z g

(1:162)

41

1.2 Theoretical background


  1 1

1 1
γ ðE, ηg Þ , −
− Z0 ðE, ηg Þ + α −
t− 2m
t+ 0
*t 2E 2m 2m t " # !

h2
h2

α
h2 1 1

γ ðE, ηg Þ , + − Z0 ðE, ηg Þ + λ73 ðE, ηg Þ ≡
l−
l− 2m
l+ 0
*l 2m 2 m 2m


η Þ≡ λ72 ðE, g


η Þ= λ74 ðE, g





η Þ
0 ðE, α
h4 Z g
η Þ ≡ ½γ ðE,
η Þ + αθ0 ðE,
η Þ and λ75 ðE, g g g 0 +

l− 4ml m

Thus, the energy spectrum in this case is real since the corresponding dispersion relation in the absence of band tails as given by (1.162) is a pole-less function with respect to energy axis in the finite complex plane. The respective transverse and the longitudinal EEMs’ in this case can be written as # " "
λ78 ðE,
η Þg′ f g − 2
F , η Þ = f2Z0 ðE,
η Þg
η Þ − f
λ72 ðE,
η Þg′ + qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
*? ðE
z0 ðE, m g g g g h
η Þ 2
λ78 ðE, g  #  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
0 ðE,
η Þg′ −
λ72 ðE,
η Þ +
λ78 ðE,
η Þ  − fZ (1:163) g g g




EF

h

where h i
λ78 ðE,
η Þ ≡ 4
λ70 ðE,
η Þ
λ75 ðE,
η Þ g g g and

3

2


F , η Þ =
*k ðE m g h




η Þg′f
λ84 ðE,
η Þg′ + 2ff
λ85 ðE,
η Þg′  f
λ84 ðE,
h 6

7 g g g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 − fλ84 ðE, ηg Þg′ + 5 2 4 



ðλ84 ðE, ηg ÞÞ + 4λ85 ðE, ηg Þ
2


E−E Fh

(1:164)
η Þ≡ in which,
λ84 ðE, g


λ ðE,
η Þ 73 g
λ ðE,
η Þ 74 g


η Þ≡ and
λ85 ðE, g


λ ðE,
η Þ 75 g
λ ðE,
η Þ 74 g

Thus, we can see that the both the EEMs’ in this case exist in the band gap. In the absence of band tails, ηg ! 0, we get " #
311 ðEÞg
′ 
511 fT
h2 a *

′
11 ðEÞg + pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
? ðEF Þ = (1:165) − fa m
311 ðEÞ


2 2 T E−E F

where

42

1 The entropy in quantum wells of heavily doped materials




t+ m
t−
≡ 2m
α211 ðEÞ ≡ 1 + αE + 1 + αE , α11 ðEÞ α ð EÞ, 211
t−
*t 2m
*t 2m 2m α
h2  2  + −
t m
t 2m α 1 α511 ≡ ω11 ðω11 Þ ≡ +m − 2
16 m α
h t
t "  #1=2 2
α2 1 1 α2
≡ ω311 ðEÞ + − + − T311 ðEÞ ðω11 Þ ≡ + + + − − −
t m
t m
t m
t
t
t m
t m
t 16 m 4m m ðω11 Þ2 " # 


+ αEÞ

2 αEð1 1 αE ð1 + αEÞ
+ ω311 ðEÞ ≡ + −
t−
t+ m
t+
t−
*t m 2m 2m 2m and

"
+ −


F Þ = mt ml
*k ðE m α

(

α α − +
l
l− 2m 2m

 2

1 + 2




1
* 2m l


1 + αE
− 2m

+ "

l

1
* 2m l

−

+


αE
+ 2m




l

l


1 + αE
− 2m l

 α
− 2m

−


αE
+ 2m l

−

α
− 2m

+


+ αEÞ
αEð1
+
−m m

2

+

l

l


αð1 + 2αEÞ
+
−m m

#

l

l

)#

l

(1:166)
−E
E F

The volume in k- space as enclosed by (1.162) can be written through the integral as ð λ86 ðE,
η Þh i g
E,
η Þ = 2π
η Þ
k2 +
λ80 ðE,
η Þ Vð − ½λ ð E, 79 g g z g 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
η Þ
k4 +
λ82 ðE,
η Þ
k2 +
λ83 ðE,
η Þd
kz +
λ81 ðE, g z g z g

(1:167)

where ffi 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=2
η Þ2 + 4
λ85 ðE,
η Þ −
λ84 ðE,
η Þ
λ ðE,
η Þ ½
λ84 ðE, g g g g
λ86 ðE,
η Þ≡4
η Þ ≡ 71 5 ,
λ79 ðE, g g 2
η Þ
0 ðE, 2 2
h Z g
λ81 ðE,
η Þ≡ g


λ76 ðE,
η Þ g
η Þ ≡ ½
λ71 ðE,
η Þ2 ,
λ76 ðE,
η Þ ≡ ½
λ71 ðE,
η Þ2 ,
λ76 ðE, g g g g 2 4

4
h ½Z 0 ðE, η Þ g

h i
λ77 ≡ 2
λ71 ðE,
η Þ
λ72 ðE,
η Þ − 4
λ70 ðE,
η Þ
λ73 ðE,
η Þ − 4
λ70 ðE,
η Þ
λ74 ðE,
η Þ g g g g g g
λ83 ðE,
η Þ≡ g


λ78 ðE,
η Þ g 2 4

9
h ½Z 0 ðE, η Þ g

43

1.2 Theoretical background

and
λ78 ðE,
η Þ ≡ ½4
λ70 ðE,
η Þ
λ75 ðE,
η Þ g g g Thus, 
η Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i ð λ86 ðE, g
k4 +
λ88 ðE,
E,
η Þ =
λ87 ðE,
η Þ
η Þ
k2 +
λ89 ðE,
η Þ −
λ90 ðE,
η Þ d
kz Vð g g g z g g z 0

(1:168) Where
λ87 ðE,
η Þ ≡ 2π g

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
λ ðE,
λ ðE,
η Þ
η Þ g
g
η Þ,
λ88 ðE,
η Þ ≡ 82
η Þ ≡ 83 , λ ðE, λ81 ðE, g g g
λ81 ðE,
λ81 ðE,
η Þ 89
η Þ g g

and " #
λ79 ðE,
η Þf
λ86 ðE,
η Þg3 g g
λ90 ðE,
η Þ ≡ 2π
η Þ
λ89 ðE,
η Þ +
λ80 ðE, g g g 3 The (1.168) can be written as h i
η Þ
λ95 ðE,
η Þ −
λ90 ðE,
η Þ
E,
η Þ =
λ87 ðE, Vð g g g g

(1:169)

in which, " #
λ91 ðE,
η Þ g
λ95 ðE,
η Þ≡
η Þ,
λ94 ðE,
η Þ
t ½
λ93 ðE, ½−E g g g 3
η Þg2 + f
λ92 ðE,
η Þg2 + 2f
λ92 ðE,
η Þg2 F
η Þ,
λ94 ðE,
η Þ
t ½
λ93 ðE, ½f
λ91 ðE, g g g g g !

fλ86 ðE, ηg Þg
η Þg2 + f
λ91 ðE,
η Þg2 + 2f
λ92 ðE,
η Þg2  + ½f
λ86 ðE, g g g 3
η Þg2 + f
λ86 ðE,
η Þg2 1=2 ½f
λ92 ðE,
η Þg2 + f
λ86 ðE,
η Þg2  − 1=2 , ½½f
λ91 ðE, g g g g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
λ88 ðE,
η Þg2 ≡ 1
η Þg2 − 4λ89 ðE,
η Þ +
λ88 ðE,
η Þ, E
i ½
λ93 ðE,
η Þ,
λ94 ðE,
η Þ , f
λ91 ðE, f g g g g g g 2

44

1 The entropy in quantum wells of heavily doped materials

is the incomplete elliptic integral of the 2nd kind and is given by [133, 134],
η Þ,
λ94 ðE,
η Þ ≡
i ½
λ93 ðE, E g g

 ð
λ93 ðE,
η Þ g 1=2
η Þg2 sin2 ξg dξ; f1 − f
λ94 ðE, g 0

ξ is the variable of integration in this case, " #  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
λ ðE,η
Þ g 2 1

λ93 ðE,η

Þ≡tan −1 86


Þg2 −4
λ89 ðE,η
Þ , fλ ðE,ηg Þg ≡ λ88 ðE,ηg Þ− f
λ88 ðE,η g g g
λ92 ðE,η
Þ 92 2 g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Þg2 −f
λ92 ðE,η
Þg2 f
λ91 ðE,η g g
λ94 ðE,η
i ½
λ93 ðE,η
Þ≡
Þ,
λ94 ðE,η
Þ ,F g g g
λ91 ðE,η
Þ g

is the incomplete elliptic integral of the 1st kind and is given by [133, 134],
η Þ,
λ94 ðE,
η Þ ≡
i ½
λ93 ðE, F g g

ð
λ93 ðE,
η Þ g 0


η Þg2 sin2 ξg f1 − f
λ94 ðE, g

− 1=2

 dξ.

The DOS function in this case is given by i
h
η Þ = gv f
λ87 ðE,
η Þg′
λ95 ðE,
η Þ + f
λ95 ðE,
η Þg′
λ87 ðE,
η Þ − f
λ90 ðE,
η Þg′
HD ðE, N g g g g g g 4π3 (1:170) Therefore the electron concentration can be expressed as   
s X g

F , η Þ +
F , η Þ

rÞ
I 125 ðE
0 = v3
I125 ðE n Lð g g h h 4π
r = 1

(1:171)

where h i
F , η Þ = f
λ87 ðE
F , η Þg
λ95 ðE
F , η Þ − f
λ90 ðE
F , η Þg
I125 ðE g g g g h h h h
hd is given by In this case, E
λ95 ðE
hd , η Þ = 0 g

(1:172)

Thus, one can numerically evaluate the entropy by using (1.31f), (1.171) and (1.172) and the allied definitions in this case. The 2D dispersion relation of the conduction electrons in QWs of IV–VI materials in the absence of band tails for the dimensional quantization along z direction can be expressed as

1.2 Theoretical background

!  2
2 2

h2
ky2
z π k
h x


h n
+ αEÞ
+ αE
+ + αE Eð1
z − ð1 + αEÞ 2
x4 2
x5 2
x6 d ! ! 2 2 2 2
h2
kx2
h
ky
h2
kx2
h
ky
h2
kx2 −α + + + −α 2
x1 2
x2 2
x4 2
x5 2
x1

45

! 2 2
2
kx2
h
ky h + 2
x1 2
x2 !
2
h2
ky2
h2 n
z π
z 2
x2 2
x6 d

! 2  2 2
2 2
2

h2
ky2

k
h π h
π
h n n z z x
− ð1 + αEÞ +
z . − α 2
x3 d
z 2
x3 d 2
x4 2
x5 −α


2 2
2
 2 2
z π
z π
z π
2 n h
h n
2
kx2
h
ky h
h2 n = + +
z
z
z
1 2m
2 2m
3 d 2
x3 d 2
x6 d 2m

ð1:173Þ

where
t+ ,
x5 =
x4 = m


l+
t+ m
l−
t− m
t+ + 2m
l+
t− + 2m
l− 3m 3m m m −


, x = m , x = = , x6 = , x 1 2 3 t
l+ + m
t+
l− + m
t− 3 2m 3 2m


*t , m
2 =
1 =m m


*l
*t + 2m m 3

and
3 = m


*l m
*t 3m . *
*l
t + 2m m

Therefore, the HD 2-D dispersion relation In this case assumes the form !  2
2 2

h2
ky2
z π 2 k h
x
η Þ + αγ ðE,
η Þ
η Þ
h n γ2 ðE, + ð E, + αγ g g g 3 3
z 2
x4 2
x5 2
x6 d ! ! ! 2
2 2
2 2
2
h2
ky2
h2
ky2
h2
ky2 k k k h
h
h
x x x
η ÞÞ − ð1 + αγ3 ðE, + + + −α g 2
x1 2
x2 2
x1 2
x2 2
x4 2
x5 !
2  2 2

z π
z π 2
h2
kx2
h
ky2
h2 n
η ÞÞ
h n −α + − ð1 + αγ ð E, g 3
z
z 2
x1 2
x2 2
x6 d 2
x3 d !
2 2
2 2 2
z π
h2 n
h kx
h
ky −α +
z 2
x3 d 2
x4 2
x5 −α


2 2
2
2 h2
ky2
z π
z π
z π
2 n h
h n
2
kx2
h
h2 n = + +
z
z
z
1 2m
2 2m
3 d 2
x3 d 2
x6 d 2m

ð1:174Þ

Substituting,
kx =
rCosθ and
ky =
rSinθ (where
r and θ are 2D polar coordinates in 2D wave vector space) in (1.174), we can write

46

1 The entropy in quantum wells of heavily doped materials

"

!# ! "
2 cos2 θ
h2 sin2 θ h 1
h2 cos2 θ
h2 sin2 θ 2 + + +
r r
1
2
x4
x5 2 m m ! !
2 2 2
2
z π
z π
h2 n
h cos θ
h2 sin2 θ
h2 cos2 θ
h2 sin2 θ
h2 n +α + + +α
x4
x5
x1
x2 2
x3 dz 2
x6 dz #
2
2 2  2  cos θ sin θ cos θ sin θ 2 2
η ÞÞ
η Þ −
h αγ3 ðE, +
h ð1 + αγ3 ðE, + + g g
x1
x4
x5 x2 "  2 2
2

z π 2
η Þ + αγ ðE,
η Þ
h n
η ÞÞ
h nz π − ð1 + αγ ð E, − γ2 ðE, g g g 3 3 2
x6 dz 2
x3 dz !#

z π 4
h4 n −α =0 (1:175) 4
x3
x6 dz


4

1
h2 cos2 θ h
2 sin2 θ α +
x1
x2 4

!


E,
n
z Þ of the 2D wave vector space can be expressed as The area Að
E,
n
z Þ =
J1 −
J2 Að

(1:176)

where
J1 ≡ 2

ð π=2 0


c1
1 dθ b

(1:177)



c21 a
3 dθ b 1

(1:178)

and
J2 ≡ 2

ð π=2 0

in which 
4 
2 

h cos θ sin2 θ cos2 θ sin2 θ
≡ α , + + a
x1
x2
x4
x5 4 !
 "
!"
 
z π 2 cos2 θ sin2 θ
h2 cos2 θ sin2 θ
h2 n
+α b1 ≡ + +
z
1
2
x4
x5 2 2
x3 m m d !
 "

z π 2 cos2 θ sin2 θ
h2 n + +α
z
1
2 2
x6 m m d
2 
2 ### cos θ sin2 θ cos θ sin2 θ
− αγ3 ðE, ηg Þ + + + ð1 + αγ3 ðE, ηg ÞÞ
x1
x2
x4
x5 and

47

1.2 Theoretical background

"

!
 2
z π 2
h n
η Þ + αγ ðE,
η Þ
c1 ≡ γ2 ðE, g g 3
z 2
x6 d !

4 
4 # 2 2

z π h
π
h n n z
η ÞÞ − ð1 + αγ3 ðE, −α g
z
z 2
x3 4
x3
x6 d d The (1.177) can be expressed as
J1 = 2

ð π=2 0


n
t31 ðE,
z Þdθ 2


11 ðE,
n

z Þsin2 θ A11 ðE, nz Þcos θ + B

where
h2


n
11 ðE,
n
t31 ðE,
z Þ ≡
c1 , A
z Þ ≡
z Þ, t11 ðE, n 2m1 " " ##

2
n
n
z Þ
z Þ αγ3 ðE,
z π 1 α
h2 nz π 2 α
h2 n 1 + αγ2 ðE,
n
t11 ðE,
z Þ ≡ 1 + m
1 + + −
z
x4 2
x3 dz
x1
x4 2
x1
x6 d
2 h
n
n
11 ðE,
z Þ ≡
z Þ t21 ðE, B 2m2 and "

"

## 2 2 2
2

n
n



1 + αγ ð E, Þ αγ ð E, Þ α
h π α
h π n n z z z z 3
n
t21 ðE,
z Þ ≡ 1 + m2 + + − 3
z
z
x2
x5 2
x3
x5 d 2
x2
x6 d Performing the integration, we get
n
11 ðE,
n
11 ðE,
n
J1 = π
t31 ðE,
z Þ½A
z ÞB
z Þ − 1=2

(1:179)

From (1.178) we can write
2

4
J2 = αt31 ðE, nz Þ
h
I
3 ðE,
n
z Þ 2B 11

(1:180)

where
I ≡

ð∞ 0


1 + a
2
z2 Þða
3 + a
4
z2 Þdz ða 3


Þ2 +
z2  ½ða


≡
≡ 1, a in which a
1
x1 2

n
z Þ A1 ðE, 2
Þ ≡

ða .
z Þ B1 ðE, n

1
x2





z Þ A11 ðE, n 2
, ðaÞ =

,
z Þ B11 ðE, n


3 ≡ ,
z = tan θ, θ is a new variable, a

(1:181) 1
x4


4 ≡ , a

1
x5

and

The use of the Residue theorem leads to the evaluation of the integral in (1.181) as

48

1 The entropy in quantum wells of heavily doped materials


I = π ½a
2 a
+ 3a
4 
a
1 4 4a

(1:182)

Therefore, the 2D area of the 2D wave vector space can be written as
  
n
n
z Þ
z Þ
h4 π
t31 ðE, 1 1 3 α
t31 ðE,
n
HD ðE,
z Þ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1− + A
2 ðE,
n
11 ðE,
n
11 ðE,
n
x5
x1
x2 8B
z Þ
z ÞB
z Þ A 11 The EEM for the HD QWs of IV–VI materials can thus be written as  
h2 *


m ðE, nz Þ = ½θ5HD ðE, nz Þ 2
=E
E

(1:183)

(1:184)

F1HD

Thus, the EEM is a function of Fermi energy and the quantum number due to the band nonparabolicity. The total DOS function can be written as
 n
X zmax g


nz ÞHð
E
−E
n
θ5HD ðE, (1:185) N2DT ðEÞ = v z7HD Þ 2π n
z = 1
n where the sub-band energy ðE z7HD Þ in this case can be written as
n γ2 ðE 2

7HD


2
2
z π
z π
2 n h
h2 n
n
γ3 ðE
− ð1 + α , η ÞÞ g 27HD
z
z 7HD 2
x6 d 2
x3 d
2 2
2 " 2
2 #
z π
z π
z π
h2 = 0 n
h n
h n −α − ð1:186Þ


z 2x3 2x 2m dz dz d 6 3


n
γ3 ðE
g Þ + α ,η 2


g Þ ,η

The use (1.185) leads to the expression of 2D electron statistics as
n


2D = n

max g
v X
55HD ðE
F1HD , nz Þ + T
56HD ðE
F1HD , nz Þ ½T 2π n
z = 1

(1:187)

where s X


F1HD , nz Þ ≡ AHD ðEF1HD , nz Þ and T
F1HD , nz Þ ≡
F1HD , nz Þ
55HD ðE

rÞT55HD ðE
56HD ðE T Lð π r=1

Using (1.187) and (1.31f) we can numerically study the entropy in this case. In the absence of heavy doping the EEM in QWs of IV–VI materials can be written as  2 
nz Þ =
h ½θ5 ðE,
nz Þ
* ðE, (1:188) m 

2 E=E Fs

where

1.2 Theoretical background

49

"

#

nz Þ
h4

t30 ðE, 1 1 3 α
nz Þ ≡ 1 −
nz Þ B
10 ðE,
nz Þ − 1
10 ðE, θ5 ðE, + ½A
x5
x1
x2 8½B
10 ðE,
nz Þ2 "qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( 

1=2 1

B10 ðE, nz Þ






′
′ ′ fA10 ðE, nz Þg

A10 ðE, nz ÞB10 ðE, nz Þft30 ðE, nz Þg − ft30 ðE, nz Þg 2 A10 ðE, nz Þ " #)#

1

′ A10 ðE, nz Þ + fB 10 ðE, nz Þg

nz Þ 2 B10 ðE, −



nz Þα
h4
t30 ðE, 1 1 1 3


10 ðE,
nz Þg2 f
t30 ðE,
nz Þg′ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½B10 ðE, nz Þ − 4 ½fB +
10 ðE,
nz Þ B
10 ðE,
nz Þ
x5
x1
x2 8 A


nz ÞfB
10 ðE,
nz Þg′
t30 ðE,
nz Þ,
t30 ðE,
nz Þ =
c0 ,
10 ðE, − 2B "
+α
+α

EÞ
E
c0 ≡ Eð1


2 h 2
x6

!
 !
 !
 #
z π 2
z π 2
z π 4
2 h
h2 n n n

EÞ
− ð1 + α − α ,
z
z
z 2
x3 4
x3
x6 d d d


2


nz Þ ≡ h
nz Þ
10 ðE, t ðE, nz Þ,
t10 ðE, A
1 10 2m " " ##
2
2
αE


h2 n

h2 n
z π
z π 1 α 1 + αE α
1 + + − ≡ 1+m
z
z
x4 2
x3 d
x1
x4 2
x1
x6 d
2


nz Þ ≡ h
10 ðE, t ðE, nz Þ B
2 20 2m and " " ##
2
2
α

E

h2 n
z π

h2 n
z π
E 1+α α α
nz Þ ≡ 1 + m
t20 ðE,
2 + + −
z
z
x2
x5 2
x3
x5 d 2
x2
x6 d Thus, the EEM is a function of Fermi energy and the quantum number due to the band nonparabolicity. The total DOS function can be written as
 n
X zmax g


nz ÞHð
E
−E
n Þ
θ5 ðE, (1:189) N2DT ðEÞ = v z7 2π n
z = 1
n Þ in this case can be written as where the sub-band energy ðE z7

50

1 The entropy in quantum wells of heavily doped materials


2  2

z π
z π 2
2 n
n Þ + α
n h
n Þ
h n
n ð1 + α


E E − ð1 + α E E z7 z7 z7 z7
z
z 2x6 d 2
x3 d "
2 2
2
2 #
z π
z π
z π
h2 n
h n
h2 n −α − =0
z
z
z
3 d 2
x3 d 2
x6 d 2m

(1:190)

In the absence of heavy doping, the expression of 2D electron statistics can be written as
n


2D = n

max g
v X
550 ðE
Fs , nz Þ + T
560 ðE
Fs , nz Þ ½T 2π n
z = 1

(1:191)

where


Fs , nz Þ ≡ A0 ðEFs , nz Þ , A
nz Þ
550 ðE
0 ðE, T π
 
nz Þ
nz Þ
h4  π
t30 ðE, 1 1 3 α
t30 ðE, = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + 1−
2 ðE,
nz Þ ,
10 ðE,
nz Þ B
10 ðE,
nz Þ
x5
x1
x2 8B A 10 and
Fs , nz Þ =
560 ðE T


s X


Fs , nz Þ

rÞT
550 ðE Lð

(1:192)


r = 1

Thus using (1.44) and (1.192) we can study the entropy in this case. For bulk specimens of IV–VI materials, the expressions of electron concentration and the ENTROPY assume the forms
 g
v


A ðE
F Þ
0 = ½MA4 ðEFb Þ + N (1:193) n 4 b 2π2 " 2 #


S0 = g
v kB T ½M
F Þ + N
′A ðE
F Þ
′A ð E (1:194) 4 4 b b 6 where    + − 
t m
t α4 2m 3








MA4 ðEFb Þ = αJA1 ðEFb Þ − α3 ðEFb ÞτA1 ðEFb Þ − ½τA1 ðEFb Þ , α5 = ωA 1 , 3 α
h2
J A ðE
F Þ = 1 b


F Þ
F Þ
4 ðE
τA1 ðE A b b

2


2



2 ðE − ½ − ðA A Fb Þ + BA ðEFb ÞÞEðλ, qÞ + 2BA ðEFb ÞFðλ, qÞ + 3 3

1=2
2
− 1=2
F ÞÞ2 + A
2 ðE

2

2

2

2
½ð
τA1 ðE A Fb Þ + 2BA ðEFb Þ½AA ðEFb Þ + τA1 ðEFb Þ ½BA ðEFb Þ + τA1 ðEFb Þ b

2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2

2 ðE
A
ð E Þ τ A Fb Þ − BA ðEFb Þ A F 5,
=4 λ = tan − 1

b , q
F Þ
A ðE BA ðE F Þ A b

b

51

1.2 Theoretical background

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i1=2 .pffiffiffi h
F Þ = τA ð E
F Þ + τ2 ðE
F Þ − 4τA ðE
F Þ
A ðE 2, A A2 2 3 b b b b  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 .pffiffiffi


F Þ − 4τA ðE
F Þ
2, BA ðEFb Þ = τA2 ðEFb Þ − τ2A2 ðE 3 b b
F Þ = τA 2 ð E b


F Þ
Þ ωA2 ðE ω ðE b
F Þ = A3 Fb , , τ ð E A 3 b ω2A1 ω2A1

" " # 
F
F  1 α.E 1 + αE α 1 1 b b
− + + . ωA2 ðEFb Þ =
t+
t−
t− ml+ m
l− m
t+
*t 2m 2m 2 2m m " ##
F
F αE 1 + αE α 1 b b + + − + −
l+
l−
t m
t m
*l 2m 2m m "
F Þ = ωA3 ðE b

" #
F 1 + αE
F
F
F αE
F 1 + αE
F αE αE 1 + αE 1 b b b b b b + − + + − + − + − *
t
t
t m
t
t m
t
t 2m 2m m m 2m " " ## #
F
F
F
F αE 1 + αE αE 1 + αE 1 1 b b b b
+ − + − + , α2 ðEFb Þ = ,
t+
t−
t+
t−
*t 2m
*t 2m 2m 2m 2m 2m

  α
h2 1 1 + − + , α3 =
l+ m
t
t− m
l m 4 m "
F Þ = τA1 ðE b


l+ m
l− 2m α
h2

"" +

#1=2 " "


F
F 1 + αE αE 1 b b + − −  −
l+
l
l 2m m 2m


F
F 1 + αE αE 1 b b + −
l+
l
l− 2m 2m m

#2

#


F ð1 + αE
F Þ αE b b +
l+
l− m m

#1=2 #1=2 ;

is the in complete Elliptic integral of second kind, Fðλ, qÞ is the incomplete Elliptic
Fb Þ = P

s Lð
A ðE



integral of first kind N r = 1 rÞ½MA4 ðEFb Þ 4

1.2.5 The entropy in quantum wells (QWs) of HD stressed Kane type materials The electron energy spectrum in stressed Kane type materials can be written [152–157] as

2

2

2 ky kx kz +
+ =1



0 ðEÞ
c0 ðEÞ a b0 ðEÞ

(1:195)

52

1 The entropy in quantum wells of heavily doped materials

where " #
2
ε2
3E

′g 
0 ðEÞ 2C K 2 xy
1
ε −
≡
≡ E
−C
0 ðEÞ
0 ðEÞ ½a , K
+ 1D
2


0 ðEÞ
′g 2B A 3E 2 2 0 ðEÞ 2


1 is the conduction band deformation potential, ε is the trace of the strain tensor ^ε C which can be written as 2 3 εxx εxy 0 7 ^ε = 6 4 εxy εyy 0 5, 0 0
εzz
2 is a constant which describes the strain interaction between the conduction and C
1
ε, B
g + E
−C
2 , is the momentum matrix element,
′g ≡ E valance bands, E 0 0 " #
0
ε
0
εxx
1 Þ 3b
b ð a + C 0
≡ 1−
0 ðEÞ + − , A
′g
′g
′g E 2E 2E 0

0

0


1
2n
0 ≡ 1 ð
l − mÞ,
0 ≡ p
b
0 ≡ − ðb
d ffiffiffi , a 0 + 2mÞ, 3 3 3 pffiffiffi
l, m,
0 3Þ
εxy
0 ðEÞ
≡ ðd
n
are the matrix elements of the strain perturbation operator, D
′g E 0

0 ðEÞ K
0 ðEÞ
2≡

2 ½b

, ½c0 ðEÞ ≡
0 ðEÞ
A0 ðEÞ − 21 D



0
c0 ðEÞ K

L0
c0 ðEÞ

and "


0
ε
0
εzz
1 Þ 3b
0 + C b
= 1 − ða
0 ðEÞ + − L


′g E′g0 E′g0 2E 0

#

The use of (1.195) can be written as
−α
−α
−α
3 −
t2 E
2 +
t3 E
+ t4
1 Þ
kx2 + ðE
2 Þ
kY2 + ðE
3 Þ
kz2 =
t1 E ðE where 
0
εxx −
1
ε − ða
1 Þ
ε + 3 b
g − C
1 ≡ E
0 + C α 0 2 
0
εxx −
1
ε − ða
1 Þ
ε + 3 b
g − C
2 ≡ E
0 + C α 0 2


pffiffiffi 
0 b 3

εxy d0 ,
ε + 2 2
pffiffiffi 
0 b 3
0
εxy d ε− 2 2

(1:196a)

53

1.2 Theoretical background



0
εzz − b0
ε ,
1
ε − ða
1 Þ
ε + 3 b
g − C
3 ≡ E
+ C α 0 0 2 2

  





t1 ≡ 3 ,
t2 ≡ 1
2
2 6ðEg0 − C1 εÞ + 3C1 ε , 2B 2 B 2 2
h i
1
εÞ2 + 6C
1
εðE
1
εÞ − 2C
2
ε2
g − C
g − C
t3 ≡ 1 3ð E 2 xy 0 0
2 2B 2 and
t4 ≡




 i 1 h

1
εÞ2 + 2C
2
ε2 .
g − C
εððE 3 − C 1 2 xy 0 2
2B 2

The (1.196a) can be written as  
17
k2 − T
27
k2 − T
37
k2 = q
67 E
67 E

k2 − T
3 − R
2 + V
+ρ
67 E
67 E x y z

(1:196b)

where
27 = α
37 = α
67 ,
t3 = V
67
17 = α
1 , T
2 , T
3 ,
t1 = q
67 ,
t2 = R T and
t4 =
ρ67 Under the condition of heavy doping, (1.196b) can be written as  
Ið4Þ
k2 − T
17
Ið1Þ

27
Ið1Þ
k2 − T
37
k2
Ið1Þ = q
67
Ið5Þ + V
67
Ið4Þ + ρ
67
Ið6Þ − R
67
Ið1Þ kx2 − T y z (1:196c) where
Ið6Þ =

ð E
−∞


V
− VÞ
3 Fð
VÞ
d
ðE

(1:197)

The (1.197) can be written as
2
Ið7Þ + 3E

Ið8Þ −
Ið9Þ
Ið6Þ = E
3
Ið1Þ − 3E

(1:198)

In which,
Ið7Þ =

ð E
−∞


V
Fð
VÞ
d
V

(1:199)

54

1 The entropy in quantum wells of heavily doped materials


Ið8Þ =
Ið9Þ =

ð E



V
2 Fð
VÞ
d
V

(1:200)


V
3 Fð
VÞ
d
V

(1:201)

−∞

ð E
−∞

Using (1.4), together with simple algebraic manipulations, one obtains !
2
−η
Ið7Þ = pffiffiffig exp − E
2g η 2 π " !#

2g η E
Ið8Þ = 1 + Erf
g 4 η

(1:202)

(1:203)

and
2
3 −η
Ið9Þ = pffiffiffig exp − E
2g η 2 π

!"


2 E 1+ 2
g η

# (1:204)

Thus (1.197) can be written as " " # ! # !  i
2 h

ηg 3 2 −E 2 2 2


I ð6Þ = E 1 + Erf E E + ηg + pffiffiffi exp 4E + ηg η2g 2 2 ηg 2 π

(1:205)

Thus, combining the appropriate equations, the dispersion relations of the conduction electrons in HD stressed materials can be expressed as
11 ðE,
η Þ
k2 + Q
η Þ
k2 + S
11 ðE,
η Þ
k2 = 1
11 ðE, P g x g y g z

(1:206)

where "

#
η Þ − ðT
17 =2Þ½1 + Erf ðE,
η Þ γ0 ðE, g g
η Þ≡
11 ðE, , P g
η Þ
14 ðE, Δ g

"
η Þ≡ q
67 Δ14 ðE, g

( "
E


E 1 + Erf 2 ηg

!#"

# ηg 3 2 2
E + ηg + pffiffiffi exp 2 2 π


−E η2g #



67 θ0 ðE,
η Þ+V
67 γ ðE,
η Þ + ρ67 ½1 + Erf ðE=η
Þ , −R g g g 0 2 "


η Þ − ðT
27 =2Þ½1 + Erf ðE=η
Þ γ ðE, g g
11 ðE,
η Þ≡ 0 Q g
Δ14 ðE, η Þ g

and

#

2

)

!
2 + η2  ½4E g

55

1.2 Theoretical background

"


η Þ − ðT
27 =2Þ½1 + Erf ðE=η
Þ γ ðE, g g
S11 ðE,
η Þ ≡≡ 0 g
η Þ Δ14 ðE,

#

g

Thus, the energy spectrum in this case is real since the dispersion relation of the corresponding materials in the absence of band tails as given by (1.195) is a poleless function in the finite complex plane. The EEMs along
x,
y and
z directions in this case can be written as "
h2 

F ; η Þ
xx ðEFh ; ηg Þ = ½γ0 ðE m g h 2
F ; η Þ − 2 ½Δ14 ðE
F ; η Þg0 ½γ ðE

17 =2Þ½1 + Erf ðE − ðT g g 0 Fh ; η g Þ h h
F ; η Þ
27 =2Þ½1 + Erf ðE − ðT g h " " !# (
F
17 E 1 T h
pffiffiffi exp − − Δ14 ðEFh ; ηg Þ 1 + Erf ηg 2 ηg
x


2F E h η2g

!)##

(1:207) "


2
F ; η Þ = h
F ; η Þ
yy ðE ½γ0 ðE m g g h h 2
F ; η Þ − 2 ½Δ14 ðE
F ; η Þg0 ½γ ðE

27 =2Þ½1 + Erf ðE − ðT g g 0 Fh ; ηg Þ h h
F ; η Þ
27 =2Þ½1 + Erf ðE − ðT g h " " !# (
F
27 E 1 T h
pffiffiffi exp − Δ14 ðEFh ; ηg Þ − 1 + Erf ηg 2 ηg
x


c η2g

!)##

(1:208) and " 2
h
F ; η Þ =
F ; η Þ
zz ðE ½γ0 ðE m g g h h 2
F ; η Þ − 2 ½Δ14 ðE
F ; η Þg0 ½γ ðE

37 =2Þ½1 + Erf ðE − ðT g g 0 Fh ; ηg Þ h h
F ; η Þ
37 =2Þ½1 + Erf ðE − ðT g h " " !# (
F
37 E 1 T h
pffiffiffi exp − − Δ14 ðEFh ; ηg Þ 1 + Erf ηg 2 ηg x


2F E h η2g

!)##

(1:209) Thus, we can see that the EEMs in this case exist within the band gap. In the absence of band tails, ηg ! 0 we get

56

1 The entropy in quantum wells of heavily doped materials


F Þ =
h2 a
F Þfa
F Þg′
0 ðE
*xx ðE
0 ðE m

(1:210)


0 ðE
0 ðE
F Þ =
h2 b
F Þfb
F Þg′
*xx ðE m

(1:211)


F Þ =
h2
c0 ðE
F Þf
c0 ðE
F Þg′
*xx ðE m

(1:212)

and

The DOS function in this case can be written as " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g
v −2 3


η Þg Δ14 ðE,
η ÞfΔ14 ðE,
η Þg′
NHD ðE, ηg Þ = 2 fΔ15 ðE=ηg Þg fΔ15 ðE, g g g 3π 2 # 3=2

′ − fΔ14 ðE, η Þg fΔ15 ðE, η Þg g

(1:213)

g

where
η Þ ≡ ½½γ ðE,
η Þ − ðT
17 =2Þ½1 + Erf ðE,
η Þ½γ ðE,
η Þ − ðT
27 =2Þ½1 + Erf ðE,
η Þ Δ15 ðE, g g g g g 0 0
η Þ − ðT
37 =2Þ½1 + Erf ðE,
η Þ1=2 ½γ0 ðE, g g Using (1.213), the electron concentration at can be written as  
s X g

F , η Þ +
F , η Þ

rÞ½
I 126 ðE
0 = v2
I126 ðE n Lð g g h h 3π
r = 1

(1:214)

where "


F , η Þg3=2 fΔ14 ðE g h
F , η Þ =
I126 ðE g h
F , η Þ Δ15 ðE h

#

g


hd is given by In this case, E
hd , η Þg = 0 fΔ14 ðE g

(1:215)

Thus, one can numerically evaluate the entropy by using (1.214), (1.215), (1.31f) and the allied definitions in this case. The dispersion relation of the conduction electrons in HD QWs of Kane type materials can be written as
2
z
11 ðE,
η Þ
k2 + Q
η Þ
k2 + S
11 ðE,
η Þ πn
11 ðE, ≡1 P g x g y g
z d The EEM can be expressed as

(1:216)

1.2 Theoretical background


2
′
h
F1HD , η , n

z Þ
* ðE Þ = A 56 ðEF1HD , ηg , n m z g 2

57

(1:217)

where i h
η Þðη

z πÞ2 π 1 −
S11 ðE, g dz
56 ðE
F1HD , η , n A g
z Þ = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


η Þ
P11 ðE, ηg ÞQ11 ðE, g From (1.217), it appears that the EEM is a function of Fermi energy, and size quantum number and the same mass exists in the band gap. Thus, the total 2D DOS function can be expressed
zmax n X
= g
v
F1HD , η , nz Þ
′56 ðE
2DT ðEÞ A N g 2π n
z = 1

(1:218)


n The sub-band energies (E z8HD ) are given by
S11 ðE
n

2 z8HD , ηg Þðπ nz =dz Þ = 1

(1:219)

The 2D surface electron concentration per unit area for QWs of stressed HD Kane type compounds can be written as
n


2D = n

max g
v X
57HD ðE
F1HD , η , η


z Þ ½T g
z Þ + T58HD ðEF1HD , ηg , η 2π n
z = 1

(1:220)

where
F1HD , η , η


57HD ðE
z Þ T g
z Þ ≡ A56HD ðEF1HD , ηg , η and
F1HD , η , η
58HD ðE T g
z Þ ≡

s X


F1HD , η , η

rÞT
57HD ðE Lð g
z Þ

r=l

Using (1.31f) and (1.220) we can study the entropy in this case. In the absence of band tails, the 2D electron energy spectrum in QWs of stressed materials assumes the form
k2
k2 1 y x
z Þ2 = 1
z π=d + + ðn 2 2 2





½c0 ðEÞ ½a0 ðEÞ ½b0 ðEÞ The area of 2D wave vector space enclosed by (1.221) can be written as
0 ðEÞ
E,
n
2 ðE,
n
b

z Þ = πP
z Þa
0 ðEÞ Að

(1:221)

58

1 The entropy in quantum wells of heavily doped materials

where
z
c0 ðEÞ
n
2
2 ðE,
z Þ = ½1 − ½n
z π=d P From (1.221), the EEM can be written as
F , n
z Þ =
* ðE m s


2
2
h
0 ðE
F Þb
F Þ′
z Þa
0 ðE ½P ðEFs , n s s 2

Thus, the total 2D DOS function can be expressed as
 n
X zmax

= gv
n
E
−E
n Þ
2DT ðEÞ
z ÞHð θ6 ðE, N z11 2π n
z = 1

(1:222)

(1:223)

in which,
θ6 ðE,
0 ðEÞ
0 ðEÞ
n
E,
n
E,
n
b
+ fPð
E,
n
′b

z Þ = ½2Pð
z ÞfPð
z Þg′a
0 ðEÞ
z Þg2 fa
0 ðEÞg
0 ðEÞg
′a

E,
n
0 ðEÞ
z Þg2 fb + fPð
n Þ are given by The sub-band energies ðE z11
z
n Þ = n
z π=d
c0 ðE z11

(1:224)

The 2D surface electron concentration per unit area for QWs of stressed Kane type compounds can be written as
2D = n


zmax n g
v X
61 ðE
Fs , n
62 ðE
Fs , n
z Þ + T
z Þ ½T 2π n
z = 1

(1:225)

where
0 ðE
61 ðE
Fs , n
Fs , n
Fs Þb
Fs Þ
z Þ ≡ ½p
z Þa
0 ðE
2 ðE T and
Fs , n
62 ðE
z Þ ≡ T


s X


Fs , n

rÞT
61 ðE
z Þ Lð


r = 1

The entropy in this case assumes the form " # zmax h i
2
n
X
S0 = g
v πkB T
Fs , n
Fs , n
z Þ + T ′62 ðE
z Þ T ′61 ðE 6
z = 1 n

(1:226)

The DOS function for bulk specimens of stressed Kane type materials in the absence of band tail can be written as

59

1.2 Theoretical background

 
0 ðEÞ
0 ðEÞ½
0 ðEÞ
= g
v ð3π2 Þ − 1 a
b

c0 ðEÞ
′+a
b
′
c0 ðEÞ
+ ½a
′b

c0 ðEÞ

0 ðEÞ
0 ðEÞ½
0 ðEÞ
0 ðEÞ D (1:227) Combining (1.227) with the Fermi–Dirac occupation probability factor and using the generalized Summerfield lemma the electron concentration in this case can be expressed as
4 ðE
F Þ + N
4 ðE
F Þ
0 = g
v ð3π2 Þ − 1 ½M n

(1:228)

where
0 ðE
4 ðE
F Þ ≡ ½a
F Þb
F Þ
c0 ðE
F Þ
0 ðE M and
F Þ ≡
4 ðE N


s X


F Þ

rÞM4 ðE Lð


r = 1

The entropy in this case is given by " 2 #
kB T
g
v  
0 =
F Þ + N
′4 ð E
F Þ
′4 ð E S M 9

(1:229)

1.2.6 The entropy in quantum wells (QWs) of HD Te The dispersion relation of the conduction electrons in Te can be expressed as [158]  
= ψ
k2 + ψ
k2 ± ψ2
k2 + ψ2
k2 1=2 E 1 z 2 s 3 z 4 s

(1:230)

where the values of the system constants are given in table as given annexure (15). The carrier energy spectrum in HD Te can be written as  
η Þ = ψ
k2 + ψ
k2 ± ψ2
k2 + ψ2
k2 1=2 γ3 ðE, g 1 z 2 s 3 z 4 s

(1:231)

The EEMs along
kz and
ks directions assume the forms 2 3
2 6 ψ3 7

F , η Þ = h
*z ðE ffi5γ3 ðE m 41 − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fh , η g Þ g h 2 2ψ1
ψ3 + 4ψ1 γ3 ðEFh , ηg Þ

(1:232)

and 2

3

2

ψ3
6 7

F , η Þ = h
*s ðE ffi5γ3 ðE m 41 − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fh , η g Þ g h 2 2ψ1
F , η Þ ψ4 + 4ψ1 γ3 ðE g h

(1:233)

60

1 The entropy in quantum wells of heavily doped materials

The investigations of EEMs require the expression of electron concentration, which can be written from (1.231) as
0 = n

i g
v h

F , η Þ +
t2HD ðE
F , η Þ t1HD ðE g g h h 2 3π

where 3
F , η Þ = ½3ψ ðE



t1HD ðE g 5HD Fh , ηg ÞΓ3HD ðEFh , ηg Þ − ψ6 Γ3HD ðEFh , ηg Þ h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #
F , η Þ
F , η Þ ψ2 ψ23 + 4ψ1 γ3 ðE γ3 ðE g h g h 4
F , η Þ =
F , η Þ = ψ5HD ðE + ð E , Γ 3HD g g 2 h h 2ψ1 ψ2 2ψ2

ψ6 =

ψ1 ψ2

and
F , η Þ =
t2HD ðE g h


s X
r = 1


F , η Þ

rÞ
t1HD ðE Lð g h

(1:234a)


Fhd is given by In this case E
F , η Þg = 0 fγ3 ðE g hd

(1:234b)

Therefore by using (1.31f), (1.234a) and (1.234b) we can study the entropy in this case The 2D electron energy spectrum in HD QW of Te can be written using (1.230) as "
2
2 #1=2

z π n z 2
k2 = ψ ðE,
η Þ−ψ
η Þ − πn ± ψ7 ψ8HD ðE, 5 g 6 g s

dz dz where ψ7 =

ψ4

pffiffiffiffiffi ψ1 3=2

ψ2

and "
η Þ= ψ8HD ðE, g


η Þψ ψ2 + 4ψ2 ψ2 ψ44 + 4γ3 ðE, g 2 4 2 2 4ψ1 ψ2 ψ24

The EEM in this case is given by
2 h
F , n
F , n
z Þ =
t′40 ðE
z Þ
* ðE m s s 2

#

(1:235)

1.2 Theoretical background

61

2

3

′ ψ8HD ðEF1HD , ηg Þψ 8HD ðEF1HD , ηg Þ7
6 h
F1HD , η , n
F1HD , η Þ + q
* ðE ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 m 4ψ5HD ðE g
z Þ = g 2
z Þ2
F1HD , η Þ − ðπn
z =d ψ8HD ðE g 2

(1:236) The total DOS function in this case can be expressed as
zmax n
X
= gv
η ÞHð
E
−E
n
2DT ðEÞ ψ′5HD ðE, Þ N g z59HD π n
z = 1

(1:237)


n where E is the lowest positive root of the equation z59HD
n ψ′5HD ðE

"

2 #1=2
z
z πn πn 2
± ψ7 ψ8HD ðEnz59HD , ηg Þ −
, ηg Þ − ψ6
=0 z59HD dz dz

(1:238)

The surface electron concentration is given by
2D = n


max n g
v X
F1HD , η , n


z Þ ½
t1HDTe ðE g
z Þ + t 2HDTe ðEF1HD , ηg , n π n
z = 1

(1:239)

where "
2 #
z πn
F1HD , η , n



t1HDTe ðE
z Þ ψ5HD ðEF1HD , ηg , n
z Þ − ψ6 g
z Þ = t1HDTe ðEF1HD , ηg , n
dz and
F1HD , η , n
t2HDTe ðE g
z Þ =


s X
r = 1


F1HD , η , n

rÞ½
t1HDTe ðE Lð g
z Þ

Thus using (1.239) and (1.31f) we can study the entropy in this case. The 2D electron energy spectrum in QWs of Te in the absence of band tails assumes the form "

2 #1=2
z
z πn 2

k2 = ψ ðEÞ
− ψ πn 5 6 s
z ± ψ7 ψ8 ðEÞ − d
z d where "

2

E + ψ4 ψ5 ðEÞ ψ2 2ψ22

#

(1:240)

62

1 The entropy in quantum wells of heavily doped materials

and 4 2 2 2

= ψ4 + 4Eψ2 ψ4 + 4ψ2 ψ3 ψ28 ðEÞ 4ψ1 ψ2 ψ24

Thus, the total 2D DOS function can be expressed as
=
2DT ðEÞ n

g n
X max v
n
E
−E
n Þ
t′40 ðE,
z ÞHð z12 π n
z = 1

(1:241)

where 2

3 "

2 #1=2 1=2
z
π n π n z 2

n
−ψ 5
t40 ðE,
z Þ = 4ψ5 ðEÞ 6
z ± ψ7 ψ8 ðEÞ − d
z d
n ) are given by The sub-band energies (E z12
2

z
z πn
n = ψ πn E ± ψ 1 3 z12

dz dz

(1:242a)

Using (1.240) the EEM can be expressed as
F , n
z Þ =
* ðE m s


2
′
h
z Þ t 40 ðEFs , n 2

(1:242b)

The 2D surface electron concentration per unit area for QWs of Te can be written as
2D = n




nz max g
v X
Fs , n
E
Fs , n
z ÞHð
z Þ ½
t′40 ðE π n
z = 1

(1:243)

where
Fs , n
t41 ðE
z Þ ≡


s X


Fs , n

rÞt40 ðE
z Þ Lð


r¼1

The entropyin this case is given by " #  zmax
kB 2 T
n
X
π g v
0 =
Fs , n
Fs , n
z Þ +
t′41 ðE
z Þ ½
t′40 ðE S 3
z = 1 n

(1:244)

The electron concentration and the entropy for bulk specimens of Te in the absence of band tails can, respectively, be expressed as
0 = n

g
v


9 ðE
F Þ ½M9 ðEF Þ + N 3π2

(1:245)

63

1.2 Theoretical background

and "

#
kB 2 T


g v
S0 =
F Þ + N
′9 ðE
F Þ ½M′9 ðE 9

(1:246)

where

"

# 2
F ψ E 3

F Þ = ½3ψ ðE



9 ðE + 4 M 5 F ÞΓ3 ðEF Þ − ψ6 Γ3 ðEF Þ, ψ5 ðEF Þ = ψ5 2ψ22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
s X 2
9 ðE
F Þ = ½2ψ  − 1
F Þ ≡
F Þ
F − ψ and N

rÞM
9 ðE ψ + 4ψ Γ3 ðE E Lð 1 1 3 3
r = 1

1.2.7 The entropy in quantum wells (QWs) of HD gallium phosphide The energy spectrum of the conduction electrons in n-GaP can be written as [159] " #1=2  
2
ks2 h
h2
′
2
2
4
k02
2
2 h 2


G E= + ½A ks + kz  − ðks + kz Þ + jVG j + V * * *2


2m? 2mjj 2mjj

(1:247)


′=1. where
k0 and jVG j are constants of the energy spectrum and A The dispersion relation of the conduction electrons in HD n- GaP can be expressed as " #1=2
2
ks2 h
h2
′
2
2
4
k02
2
2 h 2
+ ½ A k s + kz  − ðks + kz Þ + jVG j − jVG j γ3 ðE, ηg Þ =
*? 2m
*jj
*2 2m m jj

(1:248)

The EEMs assume the forms as
F ; η Þ =
*z ðE m g h

"
F ; η Þ
2 γ′3 ðE h g h
D
Cγ
DÞ½
ðE
+b
2 + 4b
2 + 4b

C 1 ± ðC 3 Fh ; ηg Þ b # 2 − 1=2
C
γ ðE
D
F ; η ÞD

+ 4b − 4b 3

h

(1:249)

g

And
2
′

F , η Þ = h
F , η Þ
*s ðE m ½t11 γ 3 ðEFh , ηg Þ −
t41 t
′5 ðE g g h h 2 where ! pffiffiffiffiffi 2 2
2 g
3 k
h 1
h 1
h2 0






′
, C= , D = jVG j, t11 = , a = + A b, t41 = b=

*jj
*?
*jj
2 2 m a 2m 2m 2a

(1:250)

64

1 The entropy in quantum wells of heavily doped materials


+ 4a
ðE
F , η Þ = ½g
2 − 4a

−1

bc
2
cÞ,
t52 ðE
Cγ
2
2
2


g
3 = ð4a g 3 Fh , ηg Þðg3 Þ , g2 = ð4a b + C + 4aCDÞ h The electron concentration can be expressed as
0 = n


s X g
v


F , η Þ

rÞ½
I 127 ðE ½ I ð E , η Þ + Lð 127 F g g h h 4π2
r = 1

(1:251)

where
F , η Þ = ½M
1HD ðE
F , η Þ
I127 ðE g g h h

"

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F , η Þ = 2ð
t11 γ ðE
F , η Þ +
t21 Þ t81 + t91 γ ðE

1HD ðE M g g 3 3 Fh , η g Þ + h h

+


t41 2

!"


t31 3

!
F , η Þ θ3, − ðE g h

# qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2



θ, − ðEFh , ηg Þ θ − ðEFh , ηg Þ + t5 ðEFh , ηg Þ − t5 ðEFh , ηg Þ

 # θ ðE
, η Þ + qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2


θ ð E , η Þ + t ð E , η Þ Fh g 5 Fh g   , − Fh g −


qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + ðt41 t5 ðEFh , ηg Þ=2Þ ln    
F , η Þ
t5 ðE g h




+ 2a

t81 = ½
t4 + 4
t2
t21
t31 + ð4
t2
t2 g
2 Þðg
3 Þ − 1 ,
t31 = b ,
t21 = g1 , g
1 = − ðC
DÞ, 41 41 31 41 2

a 2a 2 2 2
2


− 1
t91 = ½4
t11
t31
t41 + 8
t11
t21
t31 − ð16
t31 t41 aCÞðg3 Þ , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i pffiffiffi − 1 h
F , η Þ = ð
t31 2Þ
t61 +
t71 γ ðE


t81 +
t91 γ ðE

2

θ − ðE g 3 Fh , η g Þ − 3 Fh , ηg Þ , t61 = ðt41 + 2t21 t31 Þ h

and
t71 = ð2
t11
t31 Þ
hd in this case is given by the equation The E
hd , η Þ = 0 γ3 ðE g

(1:252)

Therefore using (1.251), (1.252), and (1.31f) we can study the entropyin this case. The 2D dispersion relation in QW of HD GaP can be expressed following (1.248) as
k2 s

"
 #1=2
2 2

π n π n z z 2
η Þ +
t21 −
t31
η Þ =
t11 γ3 ðE, −
t41 +
t5 ðE, g g
z
z d d

The EEM in this case can be written following (1.253) as

(1:253)

1.2 Theoretical background

65

" 2
h
F1HD ; η ; n



′

t11 γ′ ðE m ðE g
z Þ = 3 F1HD ; ηg Þ − t 41 t5 ðEF1HD ; ηg Þt 5 ðEF1HD ; ηg Þ 2
*

"


z πn
dz

# − 1=2 #

!2


F1HD ; η Þ +
t52 ðE g

The total DOS function assumes the form "
zmax n g
v X






′

t11 γ′ ðE N 2DT ðE, ηg Þ = 3 F1HD , ηg Þ − t 41 t5 ðEF1HD , ηg Þt 5 ðEF1HD , ηg Þ 2π n
z = 1 "


z πn
z d

# − 1=2 #

!2
F1HD , η Þ +
t52 ðE g

(1:254)

(1:255)


E
−E
n Þ Hð z8tHD


n is given by the equation where E z8THD "
 #1=2

z
z 2 2
πn πn
n

t11 y3 ðE


− t41 Þ + t21 − t31
+ t5 ðEnz8THD , ηg Þ =0 z8THD
z dz d

(1:256)

The surface electron concentration in QW of HD n-GaP can be written as
s = n


max n g
v X
F1HD , η , n


z Þ ½
t3HDGaP ðE g
z Þ + t 4HDGaP ðEF1HD , ηg , n π n
z = 1

(1:257)

where
2
2
z
z πn πn
F1HD , η , n

t1HDGap ðE



− t41
g z Þ = t11 γ3 ðEF1HD , ηg , nz Þ + t21
dz dz 1=2
F1HD , η , n +
t52 ðE g
z Þ


F1HD , η , n
t4HDGap ðE g
z Þ =


s X
r = 1


F1HD , η , n

rÞ½
t3HDGap ðE Lð g
z Þ

Thus using (1.257) and (1.31f) we can study the entropy in this case. The 2D electron dispersion relation in size-quantized n-GaP in the absence of band tails assumes the form #1=2
2 "
2
z
z n 2 2
n 2





− c
+ cks + jVG j + jVG j E = aks + b
dz dz
n Þ are given by The sub-band energy ðE z13

(1:258)

66

1 The entropy in quantum wells of heavily doped materials

h i1=2
z Þ2 + jVG j − jVG j2 + Dðπ
z Þ2
n
n =
cðπn
z =d
z =d E z13

(1:259)

The (1.258) can be expressed as
k2 =
t42 ðE,
nz Þ s

(1:260)

in which,
nz Þ ≡ ½f2a
−
t1 Þ + Dg
− f½2a
−
t1 Þ + D
2 − 4a
−
t1 Þ2 −
t2 g1=2 
t42 ðE,
ðE
ðE
2 ½ðE
z Þ2
n
t1 ≡ jVG j + Cðπ
z =d and
z Þ2 ,
n
t2 ≡ jVG j2 + Cðπ
z =d The total DOS function is given by N2DT ðEÞ =


zmax n g
v X
E
−E
n 13 Þ
n
z ÞHð ½
t42 ′ðE, z 4πa2 n
z = 1

(1:261a)

Using (1.260) the EEM can be expressed as
F , n
z Þ =
* ðE m s


2
′
h
z Þ t 42 ðEFs , n 2

(1:261b)

The electron statistics in QWs in n-GaP assumes the form  n
X 
 zmax g
v






½t42 ðEFs , nz Þ + t43 ðEFs , nz Þ  n2D = 4πa2 n
z = 1
F , n
t43 ðE
z Þ = s


s X
r = 1


F , n
z Þ  ½
t42 ðE s

(1:262)

where
F , n
t43 ðE
z Þ = s


s X
r = 1


F , n
z Þ. ½
t42 ðE s

The entropy in this case is given by # " #" zmax
2
n
X
S0 = g
v πkB T


′
′
z Þ + t 43 ðEFs , n
z Þ ½t 42 ðEFs , n
2 12a
=1 n

ð1:263Þ

z

The EEMs in bulk specimens of n-GaP in the absence of band tails can be written as

67

1.2 Theoretical background

2
F Þ =
h ½
t′11 −
t41 ðE
F Þ
t′5 ðE
F Þ
*s ðE m 2

(1:264)


2
2 D
V
C
2 − 4b
D
b
E
F Þ = h
C
− 1=2 
F + 4b
2 + C m*z ðE ½1 − C½4 b

(1:265)

where 
E
1=2
C
F Þ = g
2 − 4a
t5 ðE g
3 The electron concentration and the entropy in this case assume the forms g
v


1 ðE
F Þ ½M1 ðEF Þ + N 4π2 " #
kB 2 T


g v
S0 =
F Þ + N
′1 ðE
F Þ ½M′1 ðE 12


0 = n

where

(1:266)

(1:267)

"

" qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

t41 t 31 3


F Þ +
t5 ðE
F Þ

F Þ ϕ2 ðE

M1 ðEF Þ = 2ð
t11 EF +
t21 Þ
t91 EF +
t81 + ϕ ðEF Þ + ϕðE 3 2

"  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi##
F Þ + ϕ2 ðE
F Þ +
t5 ðE
F Þ ϕðE
F Þ
t41
t5 ðE   pffiffiffiffiffiffiffiffiffiffiffiffiffi ln  +  ,
F Þ  
t5 ðE 2 pffiffiffi  
F
t71 − ½
t81 +
t91 E
F Þ = ð
t31 2Þ − 1
t61 + E
F  1=2 ϕðE
F Þ =
1 ðE N


s X



rÞM
1 ðE
F Þ ½Lð


r = 1

1.2.8 The entropy in quantum wells (QWs) of HD platinum antimonide The dispersion relation for the n-type PtSb2 can be written as [160]   
4
2
2

2 a
2

2
2
a

4 a a
+ λ0 a
′
ks2 k − lks =I k E + δ0 −
v
k2 − n E 4 4 4 4 16 The (1.268) assumes the form

(1:268)

68

1 The entropy in quantum wells of heavily doped materials


+ ω1
k2 + ω2
k2 ½E
+ δ0 + ω3
k2 − ω4
k2  =
I1 ð
k2 +
k2 Þ2 ½E s z s z s z

(1:269)

where    
2 2
2
a
2
2
2
2
2

a
a a a a a ′


, ω2 = λ0 , ω3 = n , ω4 = ν , I1 = I , −ν ω1 = λ0 + l 4 4 4 4 4 4 4
′ and a
are the band constants. λ0 ,
l, δ0 ,
v, n The carrier dispersion law in HD PtSb2 can be written as
2


11
k4 −
k2 ½T



2
4

T s s 21 ðE, ηg Þ − T31 kz  + ½T41 kz − T51 ðE, ηg Þkz − T61 ðE, ηg Þ = 0

(1:270)

Where
21 ðE,
η Þ = ½ω1 δ0 + ω1 γ ðE,
η Þ + ω3 γ ðE,
η Þ,
11 = ð
I1 − ω2 ω3 Þ, T T g g g 3 3
41 = ½2
I 1 + ω2 ω4 , T
51 ðE,
η Þ = ½ω2 γ − ω4 γ ðE,
η Þ
31 = ½2
I 1 + ω2 ω4 − ω2 ω3 , T T g g 0 3
η Þ, T
61 ðE,
η Þ = ½γ ðE,
η Þ + γ γ ðE,
η Þ + ω2 γ3 ðE, g g g g 8 0 3
η Þ = 2θ0 ðE,
η Þ½1 + Erf ðE,
η Þ − 1 and γ8 ðE, g g g The EEMs are given by 2

3






′ ′ ð T ð E , η Þ T , η Þ + 2 T , η Þ ð E ð E T 21 Fh g 21 Fh g 11 61 Fh g 7
F , η Þ =
h 6
′

*s ðE qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi m 5 g h
11 4T21 ðEFh , ηg Þ + 2T 2




′ T21 ðEFh , ηg Þ + 4T 11 T 61 ðEFh , ηg Þ 2

(1:271)
F , η Þ =
*z ðE m g h


2 h
11 2T

!"
51 ′ðE
51 ′ðE
51 ′ðE
F , η Þ + ½T
F , η ÞT
F , η Þ + 2T
F , η Þ
51 ′ðE T g g g g h h h h #


2 ðE


′
½T 51 Fh , ηg Þ + 4T 41 T 51 ðEFh , ηg Þ

− 1=2

(1:272) The electron concentration assumes the form " #
s X g
v





0 = 2 I128 ðEFh , ηg Þ + n Lð
rÞ½I 128 ðEFh , ηg Þ 3π
r = 1 where
F , η Þ = ½M
6HD ðE
F , η Þ
I128 ðE g g h h

(1:273)

69

1.2 Theoretical background

"

#
F , η Þ ρ′2HD ðE g h
F , η Þ = T
91HD ðE
F , η Þρ ðE


F , η Þ
11
J3 ðE
6HD ðE −T M g g 2HD Fh , ηg Þ − T101 g h h h 3
F , η Þ =
91HD ðE T g h


F , η Þ
21 ðE T g h
11 2T

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiii1=2 h h
F , η Þ = ð2T
41 Þ − 1 T
F , η Þ + T
2 ðE




51 ðE ρ2HD ðE , g g 51 Fh , ηg Þ + 4T 41 T 61 ðEFh , ηg Þ h h
31 =2T
11 
101 = ½T T
F , η Þ =
J 3 ðE g h


F , η Þ ρ2HD ðE g h

2




2 ðE

½½A 3HD Fh , ηg Þ + B3HD ðEFh , ηg ÞE0 ðηðEFh , ηg Þ, tðEFh , ηg ÞÞ 3
2 ðE
F , η Þ − B
2 ðE
F , η ÞF
0 ðηðE
F , η Þ,
tðE
F , η ÞÞ − ½A 3HD

g

h

3HD

h

g

h

g

h

g


F , η Þ ρ2HD ðE g h 2


2

2 ðE ½ðA + 3HD Fh , ηg Þ − ρ2HD ðEFh , ηg ÞÞðB3HD ðEFh , ηg Þ 3
F , η ÞÞ1=2 , − ρ22HD ðE g h
F , η ÞÞ,
tðE
F , η Þ
0 ðηðE E g g h h and
F , η ÞÞ,
tðE
F , η Þ
0 ðηðE
0 E F g g h h are the incomplete elliptic integrals of second and first, respectively. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiii 1 hh



2 ðE



2 ðE A T12 ðEFh , ηg Þ + T 3HD Fh , ηg Þ = 12 Fh , ηg Þ − 4T 13 ðEFh , ηg Þ , 2
F , η Þ = ½T
7 ðE
F , η Þ=T
61 
12 ðE T h


2
61 = ½T T 31

g

h

g


11 T
7 ðE
F , η Þ = ½2T
31 T
F , η Þ − 4T
11 T
F , η Þ
41 , T
21 ðE
51 ðE − 4T g g g h h h


F , η Þ = ðT
8 ðE
F , η Þ=T
8 Þ, T
8 ðE
F , η Þ = ½T
2 ðE




13 ðE T g g g 21 Fh , ηg Þ + 4T11 T61 ðEFh , ηg Þ, h h h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiii pffiffiffiffiffiffiffi 1 hh



2 ðE






2 ðE T12 ðEFh , ηg Þ − T B 3HD Fh , ηg Þ = 12 Fh , ηg Þ − 4T 13 ðEFh , ηg Þ , T11 = ½ T 61 =2T11  2 " #
F , η Þ ρ2 ð E g − 1 h
3 ðE
F , η Þ=A
3 ðE
F , η Þ, ηðE
F , η Þ = sin
F , η Þ = ½B
tðE g g g g h h h h
F , η Þ
3 ðE B g h
hd in this case is given by the equation The E
hd , η Þ = 0
61 ðE T g

(1:273b)

Using (1.273a), (1.273b), and (1.31f), we can study the entropy in this case. From (1.270) the dispersion relation in QWs of HD PtSb2can be expressed as

70

1 The entropy in quantum wells of heavily doped materials


2

1HD ðE,
η ,n

11
k4 − P
z Þ = 0 T g
z Þks + P2HD ðE, ηg , n s

(1:274)

where
z Þ
1HD ðE,
η ,n


31 ðπn
z Þ − T
z =d P g
z Þ = ½T21 ðE, ηg , n
z Þ4 − T
z Þ2 − T
η ,n

51 ðE
F , η Þðπn
61 ðE
F , η Þ
2HD ðE,
z =d
z =d P g
z Þ = ½T41 ðπ n g g h h (1.274) can be written as
k2 = A
60 ðE
F , η , n g
z Þ s h

(1:275)

where "

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F , η , n


2 ðE




60 ðE
z Þ − P

A g
z Þ = P1HD ðEFh , ηg , n 1HD Fh , ηg , nz Þ − 4T 11 P2HD ðEFh , ηg , nz Þ h

#

The EEM assumes the form
F1HD , η , n
* ðE m g
z Þ =


2
′
h
z Þ A 60 ðEF1HD , ηg , n 2

(1:276)

The surface electron concentration is given by
2D = n


zmax n g
v X
F1HD , η , n


z Þ ðE g
z Þ + B60 ðEF1HD , ηg , n 2π n
z = 1

(1:277)

where
60 ðE
F1HD , η , n B g
z Þ =


s X
r = 1


F1HD , η , n

rÞ½A
60 ðE Lð g
z Þ

Thus by using (1.277) and (1.31f) we can study the entropy in this case. From (1.269), we can write the expression of the 2D dispersion law in QWs of n-PtSb2 in the absence of band tails as
k2 =
t44 ðE,
n
z Þ s

(1:278)

where
n
9 − 1½ − A
10 ðE,
n
t44 ðE,
z Þ = ½2A
z Þ +

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
9A
n
11 ðE,
2 ðE,
z Þ + 4A
z Þ, A 10

1.2 Theoretical background

"

(


10 ðE,
n
9 ≡ ½
I1 + ω1 ω3 , A
+ ω1
z Þ ≡ ω3 E A

71


2 )

+ δ 0 − ω4 π nz E
z d


2
2 #
z
z πn πn + ω2 ω3
+ 2
I1
, dz dz and " "
2 #
2 "
2 #
4 #
z
z
z
z πn πn πn πn






+ ω2
− I1
E + δ0 − ω
A11 ðE, nz Þ ≡ E E + δ0 − ω4
dz dz dz dz The area of ks space can be expressed as
n
E,
n
z Þ
z Þ = π
t44 ðE, Að

(1:279)

The total DOS function assumes the form
zmax n
X
= gv
n
−E
n Þ
2DT ðEÞ
z ÞHðE ½
t′44 ðE, N z14 2π n
z = 1

(1:280)


n Þ can be expressed through the equation where the quantized levels ðE z14 " " −1

2 − ω Enz14 = ð2Þ (" +

ω2


z πn
dz


z πn
dz

!2 + δ0 − ω4

!2 + δ0 − ω4

" !4
z πn
+ 4 I1
+ ω2 ω4 dz


z πn
dz


z πn
dz


z πn
dz

!2 #

!2 #2 (1:281a)

!4 − ω2 δ0


z πn
dz

!2 #)1=2 ##

Using (1.278), the EEM in this case can be written as
2 ′
h
F , n

z Þ
* ðE Þ = t 44 ðEFs , n m z s 2

(1:281b)

The electron statistics can be written as
2D = n


zmax n g
v X
Fs , n
Fs , n
z Þ +
t45 ðE
z Þ ½
t44 ðE 2π n
z = 1

where
Fs , n
t45 ðE
z Þ ≡


s X
r = 1


Fs , n
Fs , n

rÞ½
t44 ðE
z Þ +
t45 ðE
z Þ Lð

(1:282)

72

1 The entropy in quantum wells of heavily doped materials

The entropyin this case is given by # " #" zmax
kB 2 T
n
X
π g v
0 =
Fs , n
Fs , n
z Þ +
t′45 ðE
z Þ ½
t′44 ðE S 6
z = 1 n

(1:283)

1.2.9 The entropy in quantum wells (QWs) of HD bismuth telluride The dispersion relation of the conduction electrons in Bi2Te3 can be written as [161]
+ αEÞ
= ω1
k2 + ω2
k2 + ω3
k2 + 2ω4
kz
ky Eð1 x y z

(1:284)

where ω1 =


2 h
2 h
2 h
h2 α11 , ω2 = α22 , ω3 = α33 , ω4 = α ,
0
0
0
0 23 2m 2m 2m 2m


22 , α
33 and α
23 are system constants.
11 , α in which α The dispersion relation in HD Bi2Te3 assumes the form
η Þ = ω1
k2 + ω2
k2 + ω3
k2 + 2ω4
k
ky γ2 ðE, g x y z z

(1:285)

The EEMs can, respectively, be expressed as
2 h
,η Þ γ ðE
1 2 Fh g 2w

(1:286)


2
F , η Þ = h
,η Þ
*y ðE γ ðE m g h
2 2 Fh g 2w

(1:287)

2
F , η Þ =
h γðE
F , η Þ
*z ðE m g g h h
3 2w

(1:288)


F , η Þ =
*x ðE m g h

The DOS function in this case is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3=2 γ ðE,
η Þγ′ ðE,
η Þ
g g 2 2 2 m 0
EÞ
= 4πg
v q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð h2 α α α − 4α α2 11 22 33

(1:289)

11 23

Thus combining (1.289) with the Fermi–Dirac occupation probability factor, the electron concentration can be written as

0 3=2 g
v 2m
1HD ðE
F , η Þ + U
2HD ðE
F , η Þ
0 = 2 ðα11 α22 α33 − 4α11 α223 Þ − 1=2 ½U n g g h h 3π
h2 (1:290a)

1.2 Theoretical background

73

where 3=2

F , η Þ = ½γ ðE

F , η Þ =
1HD ðE , U2HD ðE U g g 2 Fh , ηg Þ h h

s X r=1


F , η Þ

rÞ½U
1HD ðE Lð g h


hd in this case is given by the equation The E
hd , η Þ = 0 γ2 ðE g

(1:290b)

Using (1.290a), (1.290b), and (1.31f), we can study the entropy in this case. The dispersion relation in QWs of HD Bi2Te3 can be expressed as

x 2 πn
+ ω2
ky2 + ω3
kz2 + 2ω4
kz
ky γ2 ðE, ηg Þ = ω1
dx

(1:291)

The EEM can be expressed as
0 m
F1HD , η Þ = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F1HD , η Þ
* ðE ffi γ′2 ðE m g g α1 α33 − 4α223

(1:292)

The surface electron concentration can be written as
zmax n g
v X
60 ðE
F1HD , η , n


x Þ ½R g
x Þ + R61 ðEF1HD , ηg , n 2π n
z = 1 " #

F1HD , η Þ 2m
0 γ′2 ðE 2m
0 πn
x 2 1 g



− 2 R60 ðEF1HD , ηg , nx Þ = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x α11 d
h2
h
11 α
2
33 − 4α a


2D = n

(1:293)

23

and
61 ðE
F1HD , η , n R g
z Þ =


s X
r = 1


F1HD , η , n

rÞ½R
60 ðE Lð g
z Þ

Using (1.293) and (1.31f) we can study the entropy in this case. The 2D electron dispersion law in QWs of Bi2Te3 in the absence of band tails assumes the form

x π 2 n

+ ω2
ky2 + ω3
kz2 + 2ω4
kz
ky Eð1 + αEÞ = ω1
dx The area of the ellipse is given by "
2 #
+α



π 2 m π Eð1 EÞ n 0 x
n
n ðE,
x Þ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 −ω A
x
22 α
23
23 − 4α α d
h2 The total DOS function assumes the form

(1:294)

(1:295)

74

1 The entropy in quantum wells of heavily doped materials


=
2DT ðEÞ N


xmax n X
0 g
v m
Hð
E
−E
n Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 + 2αEÞ z15 2 2

23 nx = 1
22 α
33 − 4α π
h α

(1:296)


n Þ can be expressed through the equation where ðE z15

x π 2
n Þ = ω1 n
n ð1 + αE E z15 z15
x d

(1:297a)

The EEM in this case assumes the form as
F ÞÞ
0 ð1 + 2αðE m s
F Þ = p
* ðE ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m s α22 α33 − 4α2 23

(1:297b)

The electron concentration can be written as !n zmax X
0 g
v m
n ÞF
o ðη Þ + 2α
kB TF
1 ðη Þ
2D = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð1 + 2αE n z15 n15 n15 α22 α33 − 4α223 nz = 1

(1:298)

where ηn = 15


n
Fs − E E z15
kB T


Using (1.298) the entropy in this case is given by # !" #" 2 zmax
n
X
0 m g
v π2
k B T

S0 = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
~
ffi ½ð1 + 2αEnz15 ÞF − 1 ðηnz15 Þ + 2αkB TF0 ðηn Þ  z15 3
z = 1 n α22 α33 − 4α223 (1:299)

1.2.10 The entropy in quantum wells (QWs) of HD 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 [162, 163]. (a) The energy spectrum of the conduction electrons in bulk specimens of n-Ge can be expressed in accordance with Cardona et al. [162] as 2
2

= Eg0 +
h kz + E
*jj 2 2m

"


2 E g0 2

!
g
k2 +E 0 z


2 h
*? 2m

!#1=2 (1:300)

1.2 Theoretical background

75

where in this case m* and m*? are the longitudinal and transverse effective masses k along direction at the edge of the conduction band, respectively The (1.300) can be written as ! !
h2
ks2

h2
kz2
h2
kz2

= Eð1 + αEÞ + α − ð1 + 2αEÞ (1:301)
*jj
*jj
*jj 2m 2m 2m The dispersion relation under the condition of heavy doping can be expressed from (1.301) as ! 2
2 2
2 k
h2
ks2 h
z
η Þ+α
η ÞÞ
h kz
ðE, = γ2 ðE, (1:302) − ð1 + 2αEγ g g 3 * *
?
jj
*jj 2m 2m 2m The EEMs can be written as
Fh , η Þ = m
Fh , η Þ
* ðE
*? γ′2 ðE m g g

(1:303)

and
F , η Þ = m
F , η Þ −
*jj γ′3 ðE
* ðE m g g


F , η Þ ½1 + 2αγ ðE
′
γ ′3 ð E g 3 F , ηg Þ − γ 2 ðEF , ηg Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F , η Þ2 − 4αγ ðE
½1 + 2αγ3 ðE g 2 F , ηg Þ

The electron concentration can be written as qffiffiffiffiffiffiffiffiffiffi 
s
k* 
*? 2m 8πg
v m X
F , η Þ +
F , η Þ
I129 ðE

rÞ½
I129 ðE
0 = Lð N g g h h
3 h
r = 1

(1:304)

(1:305a)

where
F , η Þ = ½M
8HD ðE
F , η Þ,
I129 ðE g" g h h # i γ ðE h
F ,η Þ α g 3 1=2 2 h
8HD ðE
F , η Þ = γ ðE

F , η Þ + γ ðE
F , η Þ −
F ,η Þ M γ2 ðE ½1+2αγ3 ðE 3 Fh , ηg Þ h g h g h g 3 5 3 h g
hd in this case is given by the equation The E
hd , η Þ = 0 γ2 ðE g

(1:305b)

Thus by using (1.305a), (1.305b) and (1.31f), we can study the entropy in this case. In the presence of size quantization, the dispersion law in QW of HD Ge can be written following (1.302) as
z Þ2
z π=d
2
ks2 h
2 ðn
η Þ+α h = γ ð E, g 2
*?
*jj 2m 2m

!2
η ÞÞ − ð1 + 2αγ3 ðE, g


z Þ2
z π=d
2 ðn h
*jj 2m

(1:306a)

76

1 The entropy in quantum wells of heavily doped materials

The EEM assumes the form " # 2 2

α
h π n z * *
F1HD , η , n
F1HD , η Þ −

? γ′2 ðE
ðE m g
z Þ = m g
z γ3 ðEFh , ηg Þ
*jj d m

(1:306b)

The surface electron concentration per unit area is given by
xmax n
*? X


1 ðE
F1HD , η , n
2D = g
v m
z Þ ½R N g
z Þ + S1 ðEF1HD , ηg , n π
h2 n
x = 1

(1:307)

where " !
z Þ2 2
z π=d
2 ðn h



z Þ = γ2 ðEF1HD , ηg Þ + α
R1 ðEF1HD , ηg , n
*jj 2m 2
z Þ2
z π=d
F1HD , η ÞÞ
h ðn
γ3 ðE − ð1 + 2α g
*jj 2m

#

and
S1 ðE
F1HD , η , n g
z Þ =


s X
r = 1


F1HD , η , n

rÞ½R
1 ðE Lð g
z Þ

Thus using (1.307) and (1.31f) we can study the entropyin this case. In the presence of size quantization along kz direction, the 2D dispersion relation of the conduction relations in QWs of n-Ge in the absence of band tails can be written by extending the method as given in [158] as 2 2
2
kx2
h
ky h
n
z Þ + = γðE, *
1 2m
*2 2m

where, m*1 = m*?

2 3 " 2  2 #2 2
2
*


h
π
h π m n n z z
+ αEÞ
− ð1 + 2αEÞ

n 5
z Þ ≡ 4Eð1
*2 = ? , γðE, +α m
z
z
*3 d
*3 d 3 2m 2m

and
*3 = m


*jj m
*? 3m
*jj + m
*? 2m

The area of ellipse of the 2D surface as given by (1.308) can be written as

(1:308)

77

1.2 Theoretical background


E,
n
z Þ = Að

2π

pffiffiffiffiffiffiffiffiffiffiffi
*2

*1 m m
z Þ γðE, n 2
h

(1:309a)

The EEM in this case can be written as
qffiffiffiffiffiffiffiffiffiffiffi
F , n
F , n

z Þ′
* ðE
*2 ½γðE
*1 m Þ = m m z s s

(1:309b)

The DOS function per sub-band can be expressed as pffiffiffiffiffiffiffiffiffiffiffi "
2 !#
z
*2
*1 m 4 m
h2 πn

2D = 1 + 2αE − 2α n 2 *

2 m3 d z π
h The total DOS function is given by "  2 !# qffiffiffiffiffiffiffiffiffiffiffi 2

4 h
π n z
− 2α
=
E
−E
n Þ
2D ðEÞ
*2 1 + 2αE
*1 m Hð m N z16
z
*3 d 2m π
h2

(1:310)

(1:311)


n is the positive root of the following equation where E z16
n Þ − ð1 + 2αE
n Þ
n ð1 + E E z16 z16 z16


2 !
 2 !2
z
z
2 πn h
h2 πn =0 +α
z
z
*3 d
*3 d 2m 2m

(1:312)

Thus combining (1.311) with the Fermi–Dirac occupation probability factor, the 2D electron statistics in this case can be written as pffiffiffiffiffiffiffiffiffiffiffi zmax
n
X
*2

*1 m kB T 4 m
1 ðn
0 ðη Þ2α
kB T
F
1 ðη Þ
z Þ + 1 + 2αηn ÞF
2D = × ½ðA (1:313) n n16 nz16 2 z16 π
h
z = 1 n where "

2 # 2


h π n z
1 ðn
z Þ ≡ 1 + 2α
A
z
*3 d 2m and 1

n  −E ηnz16 =
½E z16
F2D kB T The entropy in this case is given by " pffiffiffiffiffiffiffiffiffiffiffi 2 #n
 zmax
X
*2
k B T
*1 m
S0 = 4π m
1 ðn
n ÞF
− 1 ðη Þ + 2α
kB T
F
0 ðη Þ
½ð A Þ + 2α E z nz16 nz16 z16 3
h2
z = 1 n (1:314)

78

1 The entropy in quantum wells of heavily doped materials

The expressions of EEMs’ in bulk specimens of Ge in the absence of band tails can be written following (1.301)
F Þ = m
*jj
* ðE m

(1:315)


F Þ = m
F Þ
*? ð1 + 2αE
* ðE m

(1:316)

The DOS function for bulk specimens of Ge in the absence of band tails can be written following (1.301) as
* 3=2 "
* 2 # 1
D
m
11
1 2m 5
3=2 18α



*,?2 m
D = ðm
*k Þ1=3 (1:317) E2 − αE + E2 ; m NðEÞ = 4πg
v 2 h 6 5
h2 Using (1.317), the electron concentration in bulk specimens of Ge can be written as " #
*

2

3=2
*D
kB T
11 kB T
5


189

m 2πm



0 = Nc1 F1 ðηÞ − αkB T F3 ðηÞ + n F7 ðηÞ ; Nc1 = 2g
v αkB T 2 2 2
2 4 4
h2 h (1:318) The use of (1.318) leads to the expression of entropy in this case as 2 3 ! 
2

2
*k
kB T m k π 5 189 N c1 B
S0 =
1 ðηÞ − α
kB T
F
3 ðηÞ +

7 ðηÞ5 4F F α
kB T 2 2 2 3 4 4
h2

(1:319)

(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 [163] can be written as ! ! ! ! kz2
h2
ks2
h2

h2
ks2
h2
ks2
h2
kz2
h2
kz2
+ − α4 − α5 − α6 (1:320) E=
*k 2m
*?
*?
*?
*k
*k 2m 2m 2m 2m 2m where
*
?

2m
, , β4 = 1.4β α4 = β 4 5
h2
* *  4
?m
k 4m −1 2 * −1
= α
h ½ðm



β , α7 = 0.8β Þ − ð m Þ  , α = α 0 5 7 5 5 ? 4 4
h and
Þ α = ð0.005β 5


*k 2m
2 h

!2

79

1.2 Theoretical background

The energy spectrum under the condition of heavy doping can be written as
2
kz2
h2
ks2
2
ks2 h
η Þ= h γ3 ðE, + − α4 g * *
k 2m
?
*? 2m 2m

!


2
ks2 h
*? 2m


5 −α

!

! !2
2
kz2 h
2
kz2 h − α6
*k
*k 2m 2m

(1:321a)

The (1.321) can be expressed as
2
ks2 h
η Þ kz2 − α10 ½
kz4 + α11
kz2 + α12 ðE, = α8 − α9
g
*? 2m

(1:321b)

where 1 α5 , α9 = α8 =
2α4 2α4

!
2 h 1 , α10 = *
k 2α4 2m


h2
*k 2m

!

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
*k  4α 2m
4 − 2α
5 α25 − 4α4 α6 , α11 = 2
h α25 − 4α4 α6

and
η Þ= α12 ðE, g


*k 2m

!"

#
η Þ ð1 − 4α4 γ3 ðE, g α25 − 4α4 α6


2 h

The EEMs’ can be written as # !"
Fh , η Þ
*k
*k γ3 ðE 2m m g 2
mz ðEFh , ηg Þ =
Fh , η Þ 1 − 4α4 γ3 ðE
h2 g
Fh , η Þ = m2? ðE g


*k 2m
h2

!"


Fh , η Þ m2? γ3 ðE g
Fh , η Þ 1 − 4α4 γ3 ðE g

(1:322)

# (1:323)

The electron concentration in HD Ge in accordance with the model of Wang and Ressler can be expressed as
0 = n


2? g
v 

m
Fh , η Þ I3 ðEFh , ηs Þ +
I4 ðE s 2 π2
h

where α9 3

Fh , η Þ½α8 ρ ðE

Fh , η Þ
I3 ðE ρ ðEFh , ηs Þα10
J10 ðE 10 Fh , ηs Þ − s s 3 10 " 2 #1  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi21
k 2 1 m

Fh , η Þ ρ10 ðEFh , ηs Þ = 1 − 1 − 4α6 γ3 ðE s
h α6

(1:324)

80

1 The entropy in quantum wells of heavily doped materials



2 ðE A 1 Fh , ηg Þ
Fh , η Þ, q

0 ðλðE ½−E g
ðEFh , ηg ÞÞ 3
Fh , η Þ + B
2 ðE

2



2 ðE

½A g 1 1 Fh , ηg Þ + 2B1 ðEFh , ηg ÞF 0 ðλðEFh , ηg Þ, qðEFh , ηg ÞÞ " #

Fh , η Þ + ρ2 ðE

2 ðE
2 ðE A A g 1 Fh , ηg Þ 1 10 Fh , ηs Þ 2
2



+ ½ρ10 ðEFh , ηs Þ + A1 ðEFh , ηg Þ + 2B1 ðEFh , ηg Þ
2

Fh , η Þ 3 B1 ðEFh , ηg Þ + ρ210 ðE s


Fh , η Þ =
J 10 ðE g

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1h 1h

Fh , η Þ , B
2 ðE

Fh , η Þ
2 ðE A α11 + α211 − 4α212 ðE α11 + α211 − 4α212 ðE g g 1 Fh , ηg Þ = 1 Fh , ηg Þ = 2 2 " # "" #
,η Þ

2

2 ðE ρ ðE A 1 Fh , ηg Þ − B1 ðEFh , ηg Þ
Fh , η Þ = tan − 1 10 Fh g , q
Fh , η Þ =
ðE λðE g g 2


ðE
ðE B A 1 Fh , ηg Þ 1 Fh , ηg Þ and
I4 ðE
Fh , η Þ = g


s X
r = 1


Fh , η Þ

rÞ½
I 3 ðE Lð g


hd in this case is given by the equation The E
hd , η Þ γ2 ðE g

(1:324b)

Thus using (1.324a), (1.324b), and (1.31f) we can study the entropy in this case. The dispersion relation in QW of HD Ge can be written as "
 #1=2
2
2
z π
z π 4
z π
2
ks2 h n n n
η Þ = α8 − α9
− α10 + α11
+ α12 ðE, g
z
*? 2m dz d dz

(1:325)

The (1.325) can be expressed as
2
ks2

h
z Þ = A15 ðE, ηg , n
*? 2m

(1:326)

where 2

"
 #1=2 3
2
2 4


π n π n π n z z z
15 ðE,
η ,n
η Þ 4 5 A − α10 + α11
+ α12 ðE, g
z Þ = α8 − α9 g
z
z d d dz The EEM is given by
F1HD , η , n
′75 ðE
F1HD , η , n
*s ðE
*? = A m g
z Þm g
z Þ The electron concentration per unit area assumes the form

(1:327)

81

1.2 Theoretical background


2D = n


zmax n
*? gv X m
75 ðE
F1HD , η , n


z Þ ½A g
z Þ + A76 ðEF1HD , ηg , n π
h2 n
z = 1

(1:328)

where
F1HD , η , n
76 ðE A g
z Þ =


s X
r = 1


F1HD , η , n

rÞ½A
75 ðE Lð g
z Þ

Using (1.328) and (1.31f) we can study the entropy in this case The 2D dispersion law in the absence of band tails can be expressed as
6 ðn
=A
5 ðn
z Þ + A
z Þβ − α4 β2 E

(1:329)

where !
 # " !
 # 2 " 2 2
2 2


z 2
h π n
h π n h
πn z z
6 ðn
5 ðn
z Þ =
z Þ = 1 − α5 1 − α , A A 6 * * *
z
z
z
3 d
3
3 2m 2m 2m d d and β=

2 2
2
kx2
h
ky h +
*1 2m
*2 2m

The (1.329) can be written as 2 2
2
kx2
h
ky

h
z Þ + = I1 ðE, n
*1 2m
*2 2m

(1:330)

where h i 1=2
n
2 ðn
I1 ðE,
6 ðn
+ 4α4 A
5 ðn
z Þ = ð2α4 Þ − 1 A
z Þ − A

Þ − 4α Þ E z 4 z 6 From (1.330), the area of the 2D ks -space is given by pffiffiffiffiffiffiffiffiffiffiffi
*2
*1 m 2π m
6 ðE,
n
n
I1 ðE,
z Þ =
z Þ A 2
h Using (1.331a) in this case can be expressed as
qffiffiffiffiffiffiffiffiffiffiffi  
F , n
F , n

z Þ ′
* ðE
*2
I1 ðE
*1 m Þ = m m z s s The DOS function per sub-band can be written as pffiffiffiffiffiffiffiffiffiffiffi
*2


*1 m 4 m


z Þg′ N2D ðEÞ = fI1 ðE, n π
h2

(1:331a)

(1:331b)

(1:332)

82

1 The entropy in quantum wells of heavily doped materials

∂


n
z Þg′ =
where, f
I1 ðE,
½I1 ðE, nz Þ ∂E The total DOS function assumes the form pffiffiffiffiffiffiffiffiffiffiffi n
zmax
*2 X


*1 m 4 m


z Þg′ HðE − Enz17 Þ fI1 ðE, n N2DT ðEÞ = 2 π
h
z = 1 n where the sub-band energy ðEnz17 Þ are given by !
 " !
 #
z 2
z 2
h2 πn
h2 πn
1 − α Enz17 = 6
z
z
*3
*3 2m 2m d d The electron statistics can be written as pffiffiffiffiffiffiffiffiffiffiffi n
zmax
*2 X
*1 m 4 m
Fs , n
Fs , n
z Þ +
t47 ðE
z Þ
2D = ½
t46 ðE n 2 π
h
z = 1 n

(1:333)

(1:334)

(1:335)

where
Fs , n
Fs , n
Fs , n
t46 ðE
z Þ =
I1 ðE
z Þ,
t47 ðE
z Þ =


s X


Fs , n

rÞð½
t46 ðE
z ÞÞ Lð


r = 1

Using (1.335), the entropy in this case is given by " #  qffiffiffiffiffiffiffiffiffiffiffi 2
2
n
X zmax k π T B * *
0 = 4 m
Fs , n

t′47 ðE
′


2
1m ½ Þ + t ð E , n Þ S z 48 Fs z 3π
h2
z = 1 n

(1:336)

1.2.11 The entropy in quantum wells (QWs) of HD gallium antimonide The dispersion relation of the conduction electrons in n-GaSb can be written as [164] " #1=2 2 2

′g E
′g E k
2
h 2
h2
k 1 1 0 0
E= − − 1+
′g
0
c m
0 2m 2 2 m E 0 where "

# − 5
2 5.10 T
g +
′g = E
E 0 0
eV 2ð112 + TÞ The (1.337) can be expressed as

(1:337)

1.2 Theoretical background


2
k

h = I36 ðEÞ
c 2m

83

2

(1:338)

where
= ½E
+E
′ − ðm
′ =2Þ − ½ðE
′ =2Þ2 ½ððE
′ Þ2 =2Þð1 − ðm
I 36 ðEÞ
c =m
0 ÞðE
c =m
0 ÞÞ g0 g0 g0 g0
E
′ ð1 − ðm
′ =2Þð1 − ðm
c =m
0 ÞÞ2 + E
c =m
0 ÞÞ + ½ðE g0 g0

1=2



Under the condition of heavy doping (1.338) assumes the form
2
k

h = I36 ðE, ηg Þ
c 2m 2

(1:339)

where
I 36 ðE,
η Þ = ½γ ðE,
η Þ+E
′ − ðm
′ =2Þ
c =m
0 ÞðE g g g0 g0 3
′ =2Þ2 ½ððE
′ Þ2 =2Þð1 − ðm
c =m
0 ÞÞ − ½ðE g0 g0 ′

′


=2Þð1 − ðm
ð1 − ðm
c =m
0 ÞÞ2 + E
c =m
0 ÞÞ1=2  + ½ðE g0 g0 where ""
η Þ=
I 36 ðE; g

+

′



c E g
η Þ+E
′ − m γ3 ðE; : − g g
0 2 m
′ E g0

!2

2


c m 1−
0 m

"


′ E g 2

!2 "

!
η ÞE
′ + γ3 ðE; g g0


′ E g + 2


c m 1−
0 m


c m 1−
0 m

!##2

!#1=2 #

The EEM can be written as
F , η Þ = m
F , η Þg′
c f
I36 ðE
* ðE m g g h h

(1:340)

The DOS function in this case can be written as
 ffi

c 3=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F , η Þ = gv 2m
F , η Þf
I36 ðE
F , η Þg′
HD ðE N I36 ðE g g g 2 h h h 2 2π
h

(1:341)

Since, the original band model in this case is a no pole function, therefore, the HD counterpart will be totally real, and the complex band vanishes. The electron concentration is given by

84

1 The entropy in quantum wells of heavily doped materials


0 = n



c 3=2

g
v 2m ½fI36 ðEFh , ηg Þg3=2  3π2
h2

(1:342)


hd is given by In this case, E
F , η Þ = 0
I36 ðE g h

(1:343)

One can numerically compute the entropy by using (1.342), (1.343), (1.31f) and the allied definitions in this case. For dimensional quantization along z- direction, the dispersion relation of the 2D electrons in QWs of HD GaSb can be written following (1.339) as
z Þ
z π=d
2 ðn h
h2 ð
ks Þ

+ = I36 ðE, ηg Þ
c
c 2m 2m 2

2

(1:344)


2DT ðEÞ
in this case can be written as The expression of the N
zmax n

X
= mc g v
F , η , n



2DT ðEÞ T ′119D ðE N g
z ÞHðE − EnzD119 Þ h π
h2 n
z = 1

(1:345)

where
z Þ2 ð2m
119D ðE
F , η , n


z π=d
c Þ − 1 , T h2 ðn g
z Þ = ½I36 ðEFh , ηg Þ −
h
n The sub-band energies E in this case given by zD119 n o
z Þ2 ð2m
n
z π=d
c Þ − 1 =
I36 ðE , ηg Þ
h2 ðn zD119

(1:346)

The EEM in this case assumes the form
F1HD , η , n
F1HD , η , n
c ½I ′36 ðE
* ðE m g
z Þ = m g
z Þ The 2-D electron statistics in this case can be written as
 n
zmax
c g
v X m
119D ðE
F1HD , η , n



z Þ ½T N2DT ðEÞ = g
z Þ + T129D ðEF1HD , ηg , n π
h2 n
z = 1

(1:347)

(1:348)

where
129D ðE
F1HD , η , n T g
z Þ =


s X
r = 1

nX  zmax
119D ðE
F1HD , η , n

rÞ
½T Þ Lð z g nz =

Therefore combining (1.348) and (1.31f) we can get the entropyin this case. The total 2D DOS function in the absence of band tails in this case can be written as

1.2 Theoretical background


 n
zmax
c g
v X
= m
′Hð
E
−E
n Þg
2DT ðEÞ f½
I 36 ðEÞ N z44 π
h2 n
z = 1

85

(1:349)


n can be expressed as where the sub-band energies E z3 2
z Þ2
n Þ =
h ðπn
I36 ðE
z =d z44
c 2m

(1:350a)

The EEM in this case can be written as
F Þ = ðm
F Þ′
c Þ½
I36 ðE
* ðE m s s The 2D carrier concentration assumes the form
 n
zmax
c g
v X m
55 ðE
Fs , n
56 ðE
Fs , n

z Þ + T
z Þ ½T N2DT = π
h2 n
z = 1

(1:350b)

(1:351)

where
2
z Þ2 
Fs , n
Fs Þ − h
55 ðE
z Þ = ½
I36 ðE
z =d ðπn T
c 2m and
Fs , n
56 ðE
z Þ = T


s X


Fs , n

rÞ½T
55 ðE
z Þ Lð


r = 1

Using (1.351), the entropy in this case is given by # " #" max
kB 2 T
n
X π
0 = m
55 ðE
Fs , n
56 ðE
Fs , n
c g
v
z ÞÞ′ + ðT
z ÞÞ′ ½ðT S 3
h2
z = 1 n

(1:352)

The expression of electron concentration for bulk specimens of GaSb (in the absence of band tails) can be expressed as
0 = n



c 3=2
g
v 2m
F Þ + N
A ðE
F Þ ½MA10 ðE 10 3π2
h2

where
F Þ½I36 ðE
F Þ3=2
A ðE M 10 and
F Þ =
A ðE N 10


s X
r = 1


F Þ

rÞ½M
A ðE Lð 10

(1:353)

86

1 The entropy in quantum wells of heavily doped materials

The entropy in this case can be expressed as
 

c 3=2 g
v
kB2 T
S0 = 2m
F ÞÞ′ + ðN A ðE
F ÞÞ′ ½ðMA10 ðE 10 9
h2

(1:354)

1.2.12 The entropy in quantum wells (QWs) of HD II–V materials The dispersion relation (DR) of the holes in II–V compounds in accordance with Yamada [165] can be expressed as
11
k2 + A
12
k2 + A
13
kx ± ½ðA
14

15
k2 + A
16
k2 + A
17
kx Þ2 + A
18
k2 + A
2 
=A
10
k2 + A kx2 + A E x y z y z y 19 (1:355)
11 , A
12 , A
13 , A
14 , A
15 , A
16 , A
17 , A
18 and A
19 are energy band con
10 , A where A stants. The DR under the condition of formation of band tails can be written in this case as
η Þ=A
10
k2 + A
11
k2 + A
12
k2 + A
13
kx γ3 ðE, g x y z
14
k2 + A
15
k2 + A
16
k2 + A
17
kx Þ2 + A
18
k2 + A
2 1=2 ± ½ðA x y z y 19

(1:356)

The whole energy spectrum in this case assumes the form


x π 2

2

2
n
x π n

γ3 ðE, ηg Þ = A10
+ A11 ky + A12 kz + A13
dx dx "
2
!
x π
x π 2 n n 2

2




2 1=2
18
k2 + A ± ðA14
+ A15 ky + A16 kz + A17
+A y 19 dx dx

(1:357)


n HD401 Þ is the lowest positive root of the following equation The subband energy ðE z
n HD401 , η Þ = A
10 γ3 ðE z g





x π n
x d

2

2 31=2

2
!2


π π π n n n
14 x
13 x
17 x
2 5 4 A +A +A +A 19
x ±
x
x d d d (1:358)

The EEM and the DOS function for both the cases should be calculated numerically. Using (1.31f) and (1.358) we can study the entropy numerically.

1.2 Theoretical background

87

1.2.13 The entropy in quantum wells (QWs) of HD lead germanium telluride The dispersion law of n-type Pb1–xGexTe with x = 0.01 can be expressed following Vassilev [125] as   
− 0.606
k2 − 0.722
k2 E
+E
g + 0.411
k2 + 0.377
k2 E s z s z 0   (1:359)
g + 0.061
k2 + 0.0066
k2
ks = 0.23
ks2 + 0.02
kz2 ± 0.06E s z 0
g = 0.21 eV,
kx ,
ky and
kz are the units of 109 m
−1 where E 0 2 The electron energy spectrum n − type
kz Pb1−x Gex Te under the condition of formation of band tails can be written as 2 3  2
η Þ + γ ðE,
η Þ½E
g − 0.195k
2 − 0.345
k2  = 0.23
k2 + 0.02
k2 4 5θ0 ðE, g g s z s z 3
0 1 + Erf ðηE Þ g      
g + 0.061
k2 + 0.0066
k2
ks + E
g + 0.411
k2 + 0.377
k2 0.606k2 + 0.722
k2 ± 0.06E s z s z s z 0 0 (1:360) The E − ks relation in HD QWs of n-type Pb1–xGexTe assumes the form 2 3 "
2 #
z π 2 2
η Þ + γ ðE,
η Þ E
g − 0.195
k − 0.345 n 4 5θ ðE, g g s 3 0
z

r
f ð E
Þ 0 d 1+E ηg "
2 "
2 #
z π
z π n n 2 2
ks


± 0.06Eg0 + 0.061ks + 0.0066
= 0.23ks + 0.02
dz dz "
2 #"
2 ##

z π π n n z 2 2


g + 0.411k + 0.377 + E 0.606ks + 0.722
s 0
dz dz

(1:361)

The subband energy (EnzHD400 ) is the lowest positive root of the following equation 2 3 "
2 #
2 6 7

n
g − 0.345 nz π E


5θ0 ðEnzHD400 , ηg Þ + γ3 ðE , ηg Þ E 4 0 zHD400
z nzHD400 d 1 + Erf ηg "
2 "
 2 #"
2 ##
z π
z π
z π n n n
= 0.02
± Eg0 + 0.377
0.722
(1:362) dz dz dz The EEM and the DOS function for both the cases should be calculated numerically. Using (1.31f) and (1.362) we study the entropy numerically.

88

1 The entropy in quantum wells of heavily doped materials

1.2.14 The entropy in quantum wells (QWs) of HD Zinc and Cadmium diphosphides The DR of the holes of Cadmium and Zinc diphosphides can approximately be written following Chuiko [166] as  


 
= β + β2 β3 ðkÞ
k2 ± β β ð
kÞ β − β2 β3 ðkÞ k2 E 1 4 3 5 8β4 8β ! ! 4 2
2
β ðkÞ β ðkÞ
2 1=2 + 8β24 1 − 3 k (1:363) − β2 1 − 3 4 4
2


2


2


= kx + ky − 2kz where β1 , β2 , β4 and β5 are system constants and β3 ðkÞ 2
k Under the condition of formation of band tail, the above equation assumes the form ( 

kÞ
  β β ð 2 2 3
kÞ β − β2 β3 ðkÞ
k2
η Þ= β + k ± β β ð γ3 ðE, g 1 4 3 5 8β4 8β4 ! ! )1=2 β2 ð
kÞ β2 ð
kÞ
2 + 8β24 1 − 3 k (1:364) − β2 1 − 3 4 4 The DR in HD QWs of Zinc and Cadmium diphosphides can be written as " 
2 # (

kÞ
z π β β ð β2 β31 ð
kÞ n 2 3 2
2


kx + ky +
γ3 ðE, ηg Þ = β1 + ± β4 β31 ðkÞ β5 − 8β4 8β4 dz " ! !
2 ##
z π β231 ð
kÞ β231 ð
kÞ 2
k2 +
k2 + n 1 − 1 − + 8β − β 2 x y 4
z 4 4 d " #)
2 1=2
z π n k
x2 + k
y2 +
(1:365) dz where 2

2 3
k2 +
k2 − 2 n
z π 6 x y 7
z d β31 ð
kÞ = 6 2 7 4 5
k2 +
k2 + n
z π
x y d z


n The subband energy ðE Þ is the lowest positive root of the following equation zHD402 "
2 # ("
"
2 ##)1=2


z π
z π β2 β2 n n
γ3 EnzHD401 , ηg = β1 − ± − 2β4 β5 −
z
z 4β4 4β4 d d (1:366)

1.3 Result and discussions

89

The EEM and the DOS function for both the cases should be calculated numerically. Using (1.31f) and (1.366) we can study entropy numerically. Thus, we can summarize the whole mathematical background in the following way. In this chapter, we have investigated the 3D and 2D entropies in HD bulk and QWs of nonlinear optical materials on the basis of a newly formulated electron dispersion law considering the anisotropies of the effective electron masses, the spin orbit splitting constants and the influence of crystal field splitting within the framework of ~ k ·~ p formalism. The results for 3D and 2D entropy’s for HD bulk and QWs of III–V, ternary and quaternary compounds in accordance with the three and two band models of Kane form a special case of our generalized analysis. We have also studied the entropy in accordance with the models of Stillman et al. and Palik et al., respectively, since these models find use to describe the electron energy spectrum of the aforesaid materials. The 3D and 2D entropy’s has also been derived for HD bulk and QWs of II–VI, IV–VI, stressed materials, Te, n − GaP, p − PtSb2 , Bi2 Te3 , n − Ge, n − GaSb, II–V, Lead Germanium Telluride and Zinc and Cadmium Diphosphides compounds by using the models of Hopfield, Dimmock, Seiler, Bouat et al., Rees, Emtage, Kohler, Cardona, Wang et al. Mathur et al., Yamada, Vassilev and Chuiko, respectively, on the basis of the appropriate carrier energy spectra. The well-known expressions of the entropies in the absence of band tails for wide gap materials have been obtained as special cases of our generalized analysis under certain limiting conditions. This indirect test not only exhibits the mathematical compatibility of our formulation but also shows the fact that our simple analysis is a more generalized one, since one can obtain the corresponding results for relatively wide gap materials having parabolic energy bands under certain limiting conditions from our present derivation.

1.3 Result and discussions Using the appropriate equations and taking the energy band constants as given in Table of appendix (15), the normalized entropy in QWs of HD CdGeAs2 (an example of nonlinear optical materials) have been plotted as a function of film thickness as shown in Figure 1.1 in accordance with the generalized band model (δ≠0), three and two band HD models of Kane together with parabolic HD energy bands as shown by curves (a), (c), (d) and (e), respectively. The special case for δ = 0 has also been shown in plot (b) in the same figure to assess the influence of crystal field splitting. The Figure 1.2 exhibits the plots of the normalized entropy in QWs of HD CdGeAs2 as a function of the surface electron concentration per unit area for all cases of Figure 1.1.The Figures 1.3 and 1.5 exhibit the normalized entropy for QWs of HD InAs and InSb as a function of film thickness for three and two HD band models of Kane together with HD parabolic energy bands as shown by curves (a), (b) and (c), respectively, in both the figures. The Figures 1.4 and 1.6 show the corresponding

90

1 The entropy in quantum wells of heavily doped materials

Film thickness (in nm) Normalized entropy

0.04

0

20 40 60 80 100 120 140 160 180 200

0.01 (e) (d) (c) (b) (a)

0.001

Figure 1.1: Plot of the normalized entropy in UFs of HD CdGeAs2 as a function of film thickness in accordance with (a) the generalized band model (δ ≠ 0), (b) δ = 0, (c) the three and (d) the two band models of Kane together with (e) the parabolic energy bands.

Carrier concentration (in x1018 m–2) 0.1

0.01

0.001

Normalized entropy

0.1 1

10

(c)

100 (e)

(b) (a) (d)

Figure 1.2: Plot of the normalized entropy in UFs of HD CdGeAs2 as a function of carrier concentration for all cases of Figure 1.1.

Film thickness (in nm) 0.04 0 20 40 60 80 100 120 140 160 180 200

Normalized entropy

0.01

(c) (b) 0.001

(a)

0.0001 Figure 1.3: Plot of the normalized entropy in UFs of HD InAs as a function of film thickness in accordance with the (a) three and (b) two band models of Kane together with (c) parabolic energy bands.

1.3 Result and discussions

91

Carrier concentration (in x1018 m–2) 10

1 1 0.1

0.01

0.001

Normalized entropy

0.1

100

(b) (c) (a)

0.0001 Figure 1.4: Plot of the normalized entropy in UFs of HD InAs as a function of carrier concentration for all the cases of Figure 1.3.

Film thickness (in nm)

Normalized entropy

0.01

0 20 40 60 80 100 120 140 160 180 200

0.001 (C) (b) (a)

0.0001 Figure 1.5: Plot of the normalized entropy of UFs of HD InSbas a function of film thickness for all the cases of Figure 1.3.

dependences on the surface electron concentration for QWs of HD InAs and InSb. In Figure 1.7, the normalized entropy has been plotted in QWs of HD CdS as a function of film thickness for both
λ0 = 0 and
λ0 ≠0 as shown by curves (b) and (a), respectively, for the purpose of assessing the splitting of the two spin states by the spin orbit coupling and the crystalline field. The Figure 1.8 shows the corresponding carrier statistics dependence of the entropy for all the cases of Figure 1.7. In Figure 1.9, the normalized entropy has been plotted for HD QWs of PbTe, PbSnTe and HD stressed InSb as a function of film thickness in accordance with the appropriate band models as shown by curves (a), (b) and (c), respectively. The Figure 1.10 exhibits the corresponding dependence on the surface electron concentration per

92

1 The entropy in quantum wells of heavily doped materials

Carrier concentration (in x 1018 m–2) 1

0.1

0.01

10

1

Normalized entropy

0.1

100

(c)

(a) (b)

0.001 0.0001 Figure 1.6: Plot of the normalized entropy in UFs of HD InSbas a function of carrier concentration for all the cases of Figure 1.3.

Film thickness (in nm)

Normalized entropy

0.06

0 20 40 60 80 100 120 140 160 180 200 (a)

(b)

0.01 Figure 1.7: Plot of the normalized entropy in UFs of HD CdS as a function of film thickness for (a)
0 = 0 in accordance with the model of Hopfield.
0 ≠ 0 and (b) λ λ

unit area. Figure 1.11 demonstrates the plots of the normalized entropy in QWs of HD GaP, PtSb2, Bi2 Te3 and Cadmium Antimonide, respectively, as a function of film thickness. In Figure 1.12, the normalized entropy has been plotted as a function of carrier concentration for all the cases of Figure 1.11. The influence of 1D quantum confinement is immediately apparent from Figure (1.1), (1.3), (1.5), (1.7), (1.9) and (1.11) since the entropy depends strongly on the thickness of the quantum-confined materials which is in direct contrast with bulk specimens. The entropy increases with increasing film thickness in an oscillatory way with different numerical magnitudes for HD QWs, respectively. It appears from the aforementioned figures that the entropy in HD QWs exhibits spikes for

1.3 Result and discussions

93

Carrier concentration (in x 1018 m–2) 0.06 1

10

Normalized entropy

0.1

100 (a) (b)

0.01 Figure 1.8: Plot of the normalized entropy of UFs of HD CdSas a function of carrier concentration for the cases of Figure 1.7.

0.18

0.0155

Normalized entropy

0.0145 0.014

0.14

0.0135

(a)

0.12

(b)

0.013

(c)

0.0125

0.10

0.013 0.08

Normalized entropy

0.015

0.16

0.0115

0.06 0

20

40

60

80

100

120

0.011 140 160 180 200

Film thickness (in nm)

Figure 1.9: Plot of the normalized entropy for UFs of HDs (a) PbTe, (b) PbSnTe and (c) stressed InSbas a function of film thickness.

Carrier concentration (in x 1018 m–2) 0.4 0.1

1 Normalized entropy

0.1

10

100 (a) (b)

(c)

0.01

Figure 1.10: Plot of the normalized entropy for UFs of HDs (a) PbTe, (b) PbSnTe and (c) stressed InSbas a function of carrier concentration.

94

1 The entropy in quantum wells of heavily doped materials

Film thickness (in nm)

Normalized entropy

0.1

0

10

20

30

40

50

60

70

80

90

100

(d) (c) (b)

(a) 0.01

Figure 1.11: Plot of the normalized entropy in HDs QWs of (a) GaP, (b) PtSb2 (c) Bi2Te3 and (d) Cadmium Antimonide, respectively, as a function of film thickness. Carrier concentration (in x 1018 m–2) 0.5

0.1

0.01

Normalized entropy

0.1

10 (b)

100 (c) (d)

(a)

Figure 1.12: Plot of the normalized entropy for HDs QWs of (a) Gap, (b) PtSb2, (c) Bi2Te3 and (d) Cadmium Antimonide, respectively, as a function of carrier concentration.

particular values of film thickness which, in turn, is not only the signature of the asymmetry of the wave vector space but also the particular band structure of the specific material. Moreover, the entropy in HD QWs of different compounds can become several orders of magnitude larger than that of the bulk specimens of the same HD 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 Figure (1.2), (1.4), (1.6), (1.8), (1.10) and (1.12) that the entropy decreases with increasing carrier degeneracy for 1D quantum confinement as considered for the said figures. For relatively high values of carrier degeneracy, the influence of band structure of a specific HD material is large and the plots of entropy differ widely from one another whereas for low values of the carrier degeneracy, they exhibit the converging tendency. For bulk specimens of the same material, the entropy will be found to increase continuously with increasing electron degeneracy in a nonoscillatory manner in an altogether different way. The appearance of the humps of the respective curves is due to the redistribution of the electrons among the quantized energy levels when the quantum number corresponding to the highest occupied level changes from one fixed value to the

1.3 Result and discussions

95

others. With varying electron concentration, a change is reflected in the 2D entropy through the redistribution of the electrons among the quantized levels. Although the 2D entropy varies in various manners with all the variables in all the limiting cases as evident from all the curves of Figures. 1.1 and 1.2, the rates of variations are totally band-structure dependent.The influence of the energy band constants on the entropy in both the cases is apparent for all the materials as considered here. The normalized 2D entropy for QWs of HD stressed Kane type n − InSb has been plotted in Figures 1.9 and 1.10 as functions of nano-thickness and surface electron concentration, respectively, as shown in plot (a) in the presence of stress while the plot (b) exhibits the same in the absence of stress for the purpose of assessing the influence of stress on the 2D entropy in QWs of HD of stressed n − InSb. In the presence of stress, the magnitude of the 2D entropy is being increased as compared with the same under stress free condition. It may be noted that with the advent of modern experimental techniques, it is possible to fabricate quantum-confined structures with an almost defect-free surface. If the direction normal to the film was taken differently from that as assumed in this work, the expressions for the 2D entropy in quasi two-dimensional structures would be different analytically, since the basic dispersion laws of many important materials are anisotropic. It may be noted that under certain limiting conditions, all the results for all the models as derived here get simplified to have transformed into the well-known expressions of 3D and 2D entropy’s. This indirect test not only exhibits the mathematical compatibility of the present formulation but also shows the fact that our simple analysis is a more generalized one, since one can obtain the corresponding results for relatively wide gap 2D materials having parabolic energy bands under certain limiting conditions from the present generalized analysis. Thus, the present investigations cover the study of 2D entropy for QWs of HD nonlinear optical, III–V, ternaries, quaternaries, II–VI, IV–VI, stressed compounds, Te, GaP, PtSb2 , Bi2 Te3 , Ge and GaSb having different band structures. One striking understanding as a collateral study as considered here is that, the EEM becomes a function of the size quantum number the Fermi energy and other energy band constants depending on the respective HD 2D dispersion laws as formulated already in the respective theoretical background of this chapter together with the fact that the EEMs exists in the band gap, a phenomena which is impossible without the concept of band tailing. It must be mentioned that a direct research application of the quantized materials is in the area of band structure. The theoretical results as derived in this chapter exhibit the basic qualitative features of 2D entropy for different quantum confined HD materials. One important concept of this chapter is the presence of poles in the finite complex plane in the dispersion relation of the materials in the absence of band tails creates the complex energy spectrum in the corresponding HD samples. Besides, from the DOS function in this case, it appears that a new forbidden zone has been created in addition to the normal band gap of the semiconductor. If the basic dispersion relation in the absence of band tails contains no poles in the finite complex

96

1 The entropy in quantum wells of heavily doped materials

plane, the corresponding HD energy band spectrum will be real, although it may be the complicated functions of exponential and error functions and deviate considerably from that in the absence of band tailing.Another important point in this context is the existence of the effective mass within the forbidden zone, which is impossible without the formation of band tails. It is an amazing fact that the study of the carrier transport in HD quantized materials through proper formulation of the Boltzmann transport equationwhich needs in turn, the corresponding HD carrier energy spectra is still one of the open research problems. 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 defectfree surface. In formulating the generalized electron energy spectrum for nonlinear optical materials, we have considered the crystal-field splitting parameter, the anisotropies in the momentum-matrix elements, and the spin-orbit splitting parameters, respectively. In the absence of heavy doping, the crystal field splitting parameter together with the assumptions of isotropic effective electron mass and isotropic spin orbit splitting, our basic relation as given by eq (1.2) converts into eq (1.48). The eq (1.48) is the well-known three-band Kane model and is valid for III–V compounds, in general. It should be used as such for studying the electronic properties of n-InAs where the spin-orbit splitting parameter (Δ) is of the order of band
g and under this inequality, eq
g ). For many important materials Δ > > E gap (E 0 0 2
2 −1



c which is the well-known two-band (1.48) assumes the form Eð1 + EE g0 Þ =
h k =2m
g ! ∞, the above equation gets simplified Kane model. Also under the condition, E 0
=
h2
k2 =2m
c . It is important to to the well-known form of parabolic energy bands as E note that under certain limiting conditions, all the results for all the models as derived here have transformed into the well-known expression of the 2D entropy for size quantized materials having parabolic bands. We have not considered other types of compounds or external physical variables for numerical computations in order to keep the presentation brief. With different sets of energy band constants, we shall get different numerical values of the HD 2D entropy though the nature of variations of the HD 2D entropy as shown here would be similar for the other types of materials and the simplified analysis of this chapter exhibits the basic qualitative features of the HD 2D entropy for such compounds. We must note that the study of transport phenomena and the formulation of the electronic properties of HD nano-compounds are based on the dispersion relations in such materials. The theoretical results of our chapter can be used to determine the 2D entropy and the constituent heavily-doped bulk materials in the absence of size effects It is worth remarking that this simplified formulation exhibits the basic qualitative features of HD 2D entropy for nano-materials. The basic objective of this chapter is not solely to demonstrate the influence of quantum confinement on the 2D entropy for HD QWs of nonparabolic materials but also to formulate the appropriate electron statistics in the most generalized form, since the transport and other phenomena in HD nano-materials having different band structures and the

1.4 Open research problems

97

derivation of the expressions of many important electronic properties are based on the temperature-dependent electron statistics in such compounds. Our method is not at all related to the DOS technique as used in the literature. From the E-k dispersion relation, we can obtain the DOS, but the DOS technique as used in the literature cannot provide the E-k dispersion relation. Therefore, our study is more fundamental than those of the existing literature because the Boltzmann transport equation, which controls the study of the charge transport properties of semiconductor devices, can be solved if and only if the E-k dispersion relation is known. We wish to note that we have not considered the many body effects in this simplified theoretical formalism due to the lack of availability in the literature of proper analytical techniques for including them for the generalized systems as considered in this chapter. Our simplified approach will be useful for the purpose of comparison when methods of tackling the formidable problem after inclusion of the many body effects for the present generalized systems appear. It is worth remarking in this context that from our simple theory under certain limiting conditions we get the wellknown result of the entropy for wide gap materials having parabolic energy bands. The inclusion of the said effects would certainly increase the accuracy of the results, although the qualitative features of the 2D entropy in QWs of HD materials discussed in this chapter would not change in the presence of the aforementioned effects. The influence of energy band models and the various band constants on the entropy 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 variation would be unaltered. The theoretical results as given here would be useful in analyzing various other experimental data related to this phenomenon. Finally, we can write that the analysis as presented in this chapter can be used to investigate, the Burstein Moss shift, the carrier contribution to the elastic constants, the specific heat, screening length, activity coefficient, reflection coefficient, Hall coefficient, plasma frequency, various scattering mechanisms and other different transport coefficients of modern HD nonparabolic quantum confined HD devices operated under different external conditions having varying band structures.

1.4 Open research problems The problems under these sections of this monograph are by far the most important part for the readers and few open research problems are presented from this chapter till appendix 14. The numerical values of the energy band constants for various materials are given in Table of appendix 15 for the related computer simulations. (R.1.1) Investigate the entropy for the HD bulk materials whose respective dispersion relations of the carriers in the absence of band tails and any externally applied field are given below:

98

1 The entropy in quantum wells of heavily doped materials

(a) The electron dispersion law in n-GaP can be written as [167] "
2 #1=2

ks2
h2
ks2 Δ
2
h Δ 2 2
2




+ ∓ ± + P? kz + D1 kx ky E= 2 2m*k 2m*? 2

(R1:1)


1 a1 and a
m,
1 = P
m
P
1 = 2 × 10 − 10
eV
= 335m
1 = 2 × 10 − 10
eV
D


eV, where Δ (b) The dispersion relation for the conduction electrons for IV–VI materials can also be described by the models of Cohen [168], McClure and Choi [169], Bangert et al. [170] and Foley et al. [171], respectively. (i) In accordance with Cohen [168], the dispersion law of the carriers is given by ! 2 4
p 2 2


2y α E α p p

p p y y

+ αEÞ
= x + z − + ð1 + αEÞ + Eð1
1 2m
3 2m
2 2m 2m
′2
2m
′ 4m

(R1:2)

where m1 , m2 and m3 are the effective carrier masses at the band-edge along x, y and z directions, respectively, and m′2 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). (ii) The carrier energy spectra can be written, following McClure!and Choi [169], as
p
2y
4y
2y
2y αE αp p p
2x
2 p
−

+ αEÞ
= p + + z + αE + ð1 + αEÞ (R1:3a) Eð1
1 2m
2 2m
3 2m
2 2m
′2
2m
′ 2m 4m 

2x p
2y p
2y
2y
4y α
2y
2z αp αp p p p
2x
2 p
+ αEÞ
= p
1 − m2 Eð1 + + z + αE − − +
1 2m
2 2m
3 2m
2
1m
2m
2 4m
3 2m
′2
′2 4m
2m m 4m (R1:3b)

(iii) In accordance with Bangert and Kastner [170], the dispersion relation is given by
2+F
2 ðEÞk
2
=F
1 ðEÞk ΓðEÞ s z
≡ 2E,
F
1 ðEÞ
≡ where ΓðEÞ


≡
2 ðEÞ F

2C52
+E
g E 0

+


2 R 1
+E
g E 0

(R1:4) +

S
21
E + Δ′c

+


2 Q 1
+E
g , E 0


Þ2 ðS
1 + Q 1
+ Δ′′c E


2 = 0.83 × 10 − 19 ðeVmÞ2 , Q
2 = 1.3R
2 ,
S2 = 4.6R
2 , Δ′c = 3.07eV
2 = 2.3 × 10 − 19 ðeVmÞ2 , C R 1 1 1 1 1 1
1 = 0, ′′ Δ c = 3.028eV and gv = 4. It may be noted that under the
S1 = 0, Q 2E 2E 2
2 2
2
g
g h
h
k h
k h
0 0 , (R1.4) assumes the form Eð1
+ αEÞ
= s + z which is the
2 ≡
2 ≡ , C R * * 1


* m ?

5


* 2m

jj

simplified Lax model.


2m ?


2m

jj

99

1.4 Open research problems

(iv) The carrier energy spectrum of IV–VI materials in accordance with Foley et al. [171] can be written as " #1=2
g
g 2 E E 2 2 2 2 0




+ ðkÞ +
k +P
k
− ðkÞ + E
+ +P =E E ? s jj z 2 2
+ = ð
kÞ = where E

ks2 h2


+ 2m ?

+

kz2 h2


+ 2m jj

, E − = ðkÞ =

h2 ks2
− 2m?

+

h2 kz2
2m − k

(R1:5)

represents the contribution

from the interaction of the conduction and the valance band hedge states i with 1 1 1 1 1 the more distant bands and the free electron term, = ± = h i
tc
tv , m 2 m m
±
± m ? k 1 1 1
± m
2 m 1c

1v


k = 4.61 × 10 − 10 eVm,
? = 4.61 × 10 − 10 eVm, P For n-PbTe P
0 m
tc m

= 11.36,


0 m
1c m


0 m
tv m

= 10.36,


0 m
tv m

= 0.75,

= 1.20 and g
v = 4 The hole energy spectrum of p-type zero-gap

Materials (e.g. HgTe) is given by [172]

 
 2 k
h2
k 3e
2EB
ln
 k− + E=
v 128ε∞ 2m π k0

(R1:6)


*v is the effective mass of the hole at the top of the valence band, Where-m
0 e2
0 e2 m
B ≡ m and
ko ≡ 2 . E 2 2 2
h ε∞ 2
h ε2∞ (c) The conduction electrons of n-GaSb obey the following two dispersion relations: (i) In accordance with the model of Seiler et al. [173] " # " 2
g
g
g
E E E
h2
k
f 1 ðEÞ 2 0 0 0

+ ½1 + α4 k  + 1+4 E= −
2 2 2 2m′c Eg0

(R1:7)


E
g + 2 ΔÞ½E
g ðE
g + ΔÞ − 1 , P is the isotropic momentum matrix where α4 ≡ 4Pð 3 0 0 0 element,
f1 ð
kÞ ≡
k − 2 ½
kx2
ky2 +
ky 2
kz2 +
kz2
kx2  represents the warping of the Fermi ky2
kz2 g1=2
k − 1  represents the inversurface,
f2 ð
kÞ ≡ ½f
k − 2 ð
kx2
ky2 +
ky2
kz2 +
kz2
kx2 Þ − 9
kx2

0 represent sion asymmetry splitting of the conduction bandand
ς0 ,
ν0 and ω the constants of the electron spectrum in this case. (ii) In accordance with the model of Zhang et al. [174]
ð2Þ K
ð1Þ + E
ð2Þ K
ð1Þ + E
ð2Þ K
ð3Þ
k6, 1
4, 1 
k2 + ½E
4, 1 
k4 +
k6 ½E
4, 1 + E
= ½E
ð1Þ + E E 4 4 2 2 6 6 6

(R1:8)

100

1 The entropy in quantum wells of heavily doped materials

where

" # ! # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"
2
2
2 4 4 4 2 2 2 5 pffiffiffiffiffi
kx +
ky +
kz
639639 kx ky kz 1
kx
ky
kz 3 1
+ − K4, 1 ≡ , K6, 1 ≡ 21 − k4 k6 k6 4 32 22 5 105

 a
 times those of k in atomic the coefficients are in eV, the values of k are 10 2π ð1Þ units (a is the lattice constant), E
ð2Þ = 0, E
ð1Þ = − 1.1320772,
= 1.0239620, E 2

2

4


ð2Þ = 0.05658, E
ð1Þ = 1.1072073, E
ð2Þ = − 0.1134024 and E
ð3Þ = − 0.0072275. E 4 6 6 6

(d) In addition to the well-known band models of III–V materials as discussed in this monograph, the conduction electrons of such compounds obey the following three dispersion relations: (i) In accordance with the model of Rossler [175] 2 h i
2
k
= h E kx2  ± γ10 ½
ky2 +
kz2 +
kx2 Þ − 9
ky2
kz2 g 1=2 α10
k4 + β10 ½
kx2
ky2 +
ky2
kz2 +
kz2
k − 2 ð
kx2
ky2
kz2
kx2
*
2m (R1:9)


β
=β
+β
k
and
γ =
γ +
γ
k in which, α
11 = − 2132 ×
10 = α
11 + α
12 k, where α 10 11 12 10 11 12 −40 4 − 50 5 −40 4

= 12594 ×
12 = 9030 × 10 eVm , β11 = − 2493 × 10 eVm , β 10 eVm , α 12 10−50 eVm5 ,
γ11 = 30 × 10 − 30 eVm3 and
γ12 = 154 × 10 − 42 eVm4 . (ii) In accordance with Johnson and Dickey [176], the electron energy spectrum assumes the form 
2
2 
Eg0
= − Eg0 +
h k 1 + 1 E
v
o m
γb 2m 2 m 2

"

#1=2 2

2
k
f 1 ðEÞ h 1+4

2m′c Eg0

where 2 3
g + ΔÞðE + E
g + 2ΔÞ
g + 2Δ ðE E 0 0 0
mo 6 7 3 3 , m
≡

′c = 0.139m
0 and ≡ P2 4

5, f1 ðEÞ 2Δ Eg0 ðEg0 + ΔÞ m′c
g + ΔÞ
g + ðE + E E 0 0 3   1 2
γb = − m
′c m 0 m (iii) In accordance with Agafonov et al. [177], the electron energy spectrum can be written as 8 9" #1=2 " pffiffiffi 2
2 <

g
=
kx4 +
ky4 +
kz4 η − E k h
D 3 − 3 B 0


(R1:10) 1− E= −
k4
m
* : 2
h2 ; 2 2η c
* 2m

1.4 Open research problems

101

where
1=2 2
≡ − 21
h
2 + 8 P
2
k2
≡ E , B η g0
0 2m 3 and
≡ − 40 D


2 h
0 2m

!

(e) The energy spectrum of the carriers in the two higher valance bands and the single lower valance band of Te can, respectively, be expressed as [179]
=A
10
k2 + B
10
k2 ± ½Δ2 + ðβ
kz Þ2 1=2 and E
= Δjj + A
10
k2 + B
10 k2 ± β
kz E 10 10 z s 10 z s

(R1:11)


is the energy of the hole as measured from the top of the valance and where E within it,
10 = 3.57 × 10−19 eVm2 , Δ10 = 0.628eV, ðB
10 Þ2 = 6 × 10−20
10 = 3.77 × 10−19 eVm2 , B A 2 −5 ðeVmÞ and Δk = 1004 × 10 eV are the spectrum constants. (f) The dispersion relation in graphite can be written following Brandt [180] as  1=2 1

1

2 2 2
E = ½E2 + E3  ± ðE2 − E3 Þ + η2 k 2 4

(R1:12)

where



2 ≡ Δ
− 2
γ cos ϕ , ϕ ≡ c6 kz , E γ2 cos2 ϕ0 E 3 ≡ 2
0 0 1 2 pffiffi



6 ð
γ0 + 2
γ4 cos ϕ0 Þ in which the band constants are Δ, and η2 ≡ 23 a
6 and
c6 , respectively.
γ0 ,
γ1 ,
γ2 ,
γ4 ,
γ5 , a (g) The dispersion relation of the holes in p- InSb can be written in accordance with Cunningham [181–185] as   pffiffiffipffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffi
4 g
4
k
=
c4 ð1 + γ
f4 Þ
k2 ± 1 2 2
c4 16 + 5γ E E 4 4 3 where
c4 ≡

2
4 b
2 h
h2
5 + 2θ4 , b
4 ≡ 3 b
5 ≡ 2.4
h , + θ4 , θ4 ≡ 4.7 , γ4 ≡ , b
0
0
0
c4 2m 2m 2m 2


f4 ≡ 1 ½sin2 2θ + sin4 θsin2 2ϕ, 4

(R1:13)

102

1 The entropy in quantum wells of heavily doped materials

θ is measured from the positive z- axis, ϕ is measured from positive x-axis,
4 = 5 × 10 − 4 eV g
4 ≡ sin θ½cos2 θ + 41 sin4 θsin2 2ϕ and E (h) The energy spectrum of the valance bands of CuClin accordance with Yekimovet. Al. [183] can be written as 2
2
h = ðγ − 2γ Þ
h k E 6 7
0 2m

and

(R1:14)

2 !2 31=2 2
2 2
2 2
2 Δ21 k γ h
k k
h Δ h
1 7
l, s = ðγ + γ Þ 5 − +9 ± 4 + γ7 Δ1 E 6 7
0
0
0 2m 2 4 2m 2m

(R1:15)

where γ6 = 0.53, γ7 = 0.07, Δ1 = 70meV. (i) In the presence of stress, χ6 along and directions, the energy spectra of the holes in Materials having diamond structure valance bands can be, respectively, expressed following Roman [184] et al. as

and


2 + δ2 + B
7 δ6 ð2
k2 −
k2 Þ1=2
=A
6
k2 ± ½B E 7 z s 6

(R1:16)

 1=2
6 D 2 2 2
4 2
2




E = A6 k ± B7 k + δ7 + pffiffiffi δ7 ð2kz − ks Þ 3

(R1:17)


6 are inverse mass band parameters in which δ6 ≡
l7
7, D
6 and C
6, B where A 2



2 ≡ ðB
2 +
c6 Þ and ðS11 − S12 Þχ6 , , Sij are the usual elastic compliance constants, B 7 7 5

d S
8 = − 4.1eV,
l7 = − 2.3eV, A
6 = 19.2
h2 , B
7 = 26.3
h2 , ffiffi Þχ6 . For gray tin, d δ7 ≡ ð 2gp44
0
0 2m 2m 3
6 = 31
h2 and
c2 = − 1112
h2 . D 6

2m 2m 0

0

(R. 1.2) Investigate the entropy for bulk specimens of the heavily–doped materials in the presences of Gaussian, exponential, Kane, Halperian, Lax and Bonch-Burevich types of band tails [37] for all systems whose unperturbed carrier energy spectra are defined in R1.1. (R. 1.3) Investigate the entropy for QWs of all the HD materials as considered in R1.2. (R. 1.4) Investigate the entropy for HD bulk specimens of the negative refractive index, organic, magnetic and other advanced optical materials in the presence of an arbitrarily oriented alternating electric field.

1.4 Open research problems

103

(R. 1.5) Investigate the entropy for the QWs of HD negative refractive index, organic, magnetic and other advanced optical materials in the presence of an arbitrarily oriented alternating electric field. (R. 1.6) Investigate the entropy for the multiple QWs of HD materials whose unperturbed carrier energy spectra are defined in R1.1 (R. 1.7) Investigate the entropy for all the appropriate HD low dimensional systems of this chapter in the presence of finite potential wells. (R. 1.8) Investigate the entropy for all the appropriate HD low dimensional systems of this chapter in the presence of parabolic potential wells. (R. 1.9) Investigate the entropy for all the appropriate HD systems of this chapter forming quantum rings. (R 1.10) Investigate the entropy for all the above appropriate problems in the presence of elliptical Hill and quantum square rings. (R. 1.11) Investigate the entropy for parabolic cylindrical HD low dimensional systems in the presence of an arbitrarily oriented alternating electric field for all the HD materials whose unperturbed carrier energy spectra are defined in R1.1. (R. 1.12) Investigate the entropy for HD low dimensional systems of the negative refractive index and otheradvanced optical materials in the presence of an arbitrarily oriented alternating electric field and nonuniform light waves. (R. 1.13) Investigate the entropy for triangular HD low dimensional systems of the negative refractive index, organic, magnetic and other advanced optical materials in the presence of an arbitrarily oriented alternating electric field in the presence of strain. (R. 1.14) Investigate the entropy in HD QWs of nonparabolic materials as discussed in this chapter in the presence of nonuniform magnetic field. (R. 1.15) Investigate the entropy for all the problems of (R1.14) in the presence of arbitrarily oriented magnetic field. (R. 1.16) Investigate the entropy for all the problems of (R1.14) in the presence of alternating electric field. (R. 1.17) Investigate the entropy for all the problems of (R1.14) in the presence of alternating magnetic field. (R. 1.18) Investigate the entropy for all the problems of (R1.14) in the presence of crossed electric field and quantizing magnetic fields. (R. 1.19) Investigate the entropy for all the problems of (R1.14) in the presence of crossed alternating electric field and alternating quantizing magnetic fields. (R. 1.20) Investigate the entropy for HD QWs of the negative refractive index, organic and magnetic materials in the presence of nonuniform electric field. (R. 1.21) Investigate the entropy for HD QWs of the negative refractive index, organic and magnetic materials in the presence of alternating time dependent nonuniform magnetic field.

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(R. 1.22) Investigate the entropy for HD QWs of the negative refractive index, organic and magnetic materials in the presence of in the presence of crossed alternating electric field and alternating quantizing nonuniform magnetic fields. (R. 1.23) a) Investigate the entropy for HD low dimensional systems of the negative refractive index, organic, magnetic and other advanced optical materials in the presence of an arbitrarily oriented alternating electric field considering many body effects. b) Investigate all the appropriate problems of this chapter for a Dirac electron. (R. 1.24) investigate all the appropriate problems of this chapter by including the many body, image force, broadening and hot carrier effects, respectively. (R. 1.25) Investigate all the appropriate problems of this chapter by removing all the mathematical approximations and establishing the respective appropriate uniqueness conditions.

References [1] [2]

Petroff P.M., Gossard A.C., Wiegmann W., Appl. Phys. Lett. 45, 620 (1984). Gaines J.M., Petroff P.M., Kroemar H., Simes R.J., Geels R.S., English J.H., J. Vac. Sci. Technol. B6, 1378 (1988). [3] Cilbert J., Petroff P.M., Dolan G.J., Pearton S.J., Gossard A.C., English J.H., Appl. Phys. Lett 49, 1275 (1986). [4] Fujui T., Saito H., Appl. Phys. Lett 50, 824 (1987). [5] Sasaki H., Jpn. J. Appl. Phys 19, 94 (1980). [6] Petroff P.M., Gossard A.C., Logan R.A., Weigmann W., Appl. Phys. Lett 41, 635 (1982). [7] Temkin H., Dolan G.J., Panish M.B., Chu S.N.G., Appl. Phys. Lett 50, 413 (1988). [8] Miller I., Miller A., Shahar A., Koren U., Corvini P.J., Appl. Phys. Lett 54, 188 (1989). [9] Chang L.L., Esaki H., Chang C.A., Esaki L., Phys. Rev. Lett 38, 14811 (1977). [10] Hess K., Shur M.S., Drunnond J.J., Morkoc H., IEEE Trans. Electron. Devices ED-30, 07 (1983). [11] Bastard G., Wave Mechanics Applied to Semiconductor Hetero-structures (Halsted, Les Ulis, Les Editions de Physique, New York, 1988). [12] Kelly M.J., Low dimensional materials: materials, physics, technology, devices (Oxford University Press, Oxford, 1995). [13] Weisbuch C., Vinter B., Quantum Semiconductor Structures (Boston Academic Press, Boston, 1991). [14] Linch N.T., Festkorperprobleme 23, 27 (1985). [15] Sciferes D.R., Lindstrom C., Burnham R.D., Streifer W., Paoli T.L., Electron. Lett 19, 170 (1983). [16] Solomon P.M., Proc. IEEE 70, 4811 (1982). [17] Schlesinger T.E., Kuech T., Appl. Phys. Lett 49, 5111 (1986). [18] Kasemet D., Hong C.S., Patel N.B., Dapkus P.D., Appl. Phys. Lett 41, 912 (1982). [19] Woodbridge K., Blood P., Pletcher E.D., Hulyer P.J., Appl. Phys. Lett 45, 16 (1984). [20] Tarucha S., Okamoto H.O., Appl. Phys. Lett 45, 16 (1984). [21] Heiblum H., Thomas D.C., Knoedler C.M., Nathan M.I., Appl. Phys. Lett 47, 1105 (1985). [22] Aina O., Mattingly M., Juan F.Y., Bhattacharyya P.K., Appl. Phys. Lett 50, 43 (1987).

References

105

[23] Suemune I., Coldren L.A., IEEE J. Quant. Electron 24, 1178 (1988). [24] Miller D.A.B., Chemla D.S., Damen T.C., Wood J.H., Burrus A.C., Gossard A.C., Weigmann W., IEEE J. Quant. Electron 21, 1462 (1985). [25] Rowe J.W., Shay J.L., Phys. Rev. B 3, 451 (1973). [26] Kildal H., Phys. Rev B10, 5082 (1974). [27] Bodnar J., in Proceedings of the International Conference on Physics of Narrow-gap materials (Polish Science Publishers, Warsaw, 1978) [28] Chuiko G.P., Chuiko N.N., Sov. Phys. Semicond 15, 7311 (1981). [29] Ghatak K.P., Biswas S.N., Proc. SPIE 1484, 1411 (1991). [30] Rogalski A., J. Alloys Comp 371, 53 (2004). [31] Baumgartner A., Chaggar A., Patanè A., Eaves L., Henini M., Appl. Phys. Lett 92, 091121 (2008). [32] Devenson J., Teissier R., Cathabard O., Baranov A.N., Proc. SPIE 6909, 69090U (2008). [33] Passmore B.S., Wu J., Manasreh M.O., Salamo G.J., Appl. Phys. Lett 91, 233508 (2007). [34] Mikhailova M., Stoyanov N., Andreev I., Zhurtanov B., Kizhaev S., Kunitsyna E., Salikhov K., Yakovlev Y., Proc. SPIE 6585, 658526 (2007). [35] Kruppa W., Boos J.B., Bennett B.R., Papanicolaou N.A., Park D., Bass R., Electron. Lett 42, 688 (2006). [36] Kane E.O., Semiconductors and Semimetals, Vol. 1 R.K. Willardsonand A.C. Beer eds (Academic Press, New York, 1966). [37] Nag B.R., Electron Transport in Compound Semiconductors (Springer, Heidelberg, 1980). [38] Stillman G.E., Wolfe C.M., Dimmock J.O., Semiconductors and Semimetals, Vol. 12 R.K. Willardonand A.C. Beer eds (Academic Press, New York, 1977). [39] Newson D.J., Karobe A., Semicond. Sci. Tech 3, 786 (1988). [40] Palik E.D., Picus G.S., Teither S., Wallis R.E., Phys. Rev 122, 475 (1961). [41] Lu P.Y., Wung C.H., Williams C.M., Chu S.N.G., Stiles C.M., Appl. Phys. Lett 49, 1372 (1986). [42] Taskar N.R., Bhat I.B., Prat K.K., Terry D., Ehasani H., Ghandhi S.K., J. Vac. Sci. Tech 7A, 281 (1989). [43] Koch F., Springer Series in Solid States Sciences (Springer, Germany, 1984). [44] Tomasetta L.R., Law H.D., Eden R.C., Reyhimy I., Nakano K., IEEE J. Quant. Electron 14, 800 (1978). [45] Yamato T., Sakai K., Akiba S., Suematsu Y., IEEE J. Quantum Electron 14, 95 (1978). [46] Pearsall T.P., Miller B.I., Capik R.J., Appl. Phys. Lett 28, 4911 (1976). [47] Washington M.A., Nahory R.E., Pollack M.A., Beeke E.D., Appl. Phys. Lett 33, 854 (1978). [48] Timmons M.I., Bedair S.M., Markunas R.J., Hutchby J.A. (1982), “in Proceedings of the 16th IEEE Photovoltaic Specialist Conference” (IEEE, San Diego, California 666,) [49] Zapien J.A., Liu Y.K., Shan Y.Y., Tang H., Lee C.S., Lee S.T., Appl. Phys. Lett 90, 213114 (2007). [50] Park M., Proc. SPIE 2524, 142 (1995). [51] Hur S.-G., Kim E.T., Lee J.H., Kim G.H., Yoon S.G., Electrochem. Solid-State Lett 11, H176 (2008). [52] Kroemer H., Rev. Mod. Phys 73, 783 (2001). [53] Nguyen Duy T., Meslage J., Pichard G., J. Crys. Growth 72, 490 (1985). [54] Aramoto T., Adurodija F., Nishiyama Y., Arita T., Hanafusa A., Omura K., Morita A., Solar Energy Mater. Solar Cells 75, 211 (2003). [55] Barber H.B., J. Electron. Mater 25, 1232 (1996). [56] Taniguchi S., Hino T., Itoh S., Nakano K., Nakayama N., Ishibashi A., Ikeda M., Electron. Lett 32, 552 (1996). [57] Hopfield J.J., J. Appl. Phys 32, 2277 (1961). [58] Agrawal G.P., Dutta N.K., Semicond. Lasers (Van Nostrand Reinhold, New York, 1993).

106

[59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92]

1 The entropy in quantum wells of heavily doped materials

Chatterjee S., Pal U., Opt. Eng. (Bellingham) 32, 2923 (1993). Chaudhuri T.K., Int. J. Energy Res 16, 481 (1992). Dughaish J.H., Phys. B 322, 205 (2002). Wood C., Rep. Prog. Phys 51, 4511 (1988). Hsu K.-F., Loo S., Guo F., Chen W., Dyck J.S., UhER C., Hogan T., Polychroniadis E.K., Kanatzidis M.G., Science 303, 811 (2004). Androulakis J., Hsu K.F., Pcionek R., Kong H., Uher C., D’Angelo J.J., Downey A., Hogan T., Kanatzidis M.G., Adv. Mater 18, 1170 (2006). Poudeu P.F.P., Angelo J.D., Downey A.D., Short J.L., Hogan T.P., Kanatzidis M.G., Angew. Chem. Int. Ed 45, 3835 (2006). Poudeu P.F., D’ A.J., Kong H., Downey A., Short J.L., Pcionek R., Hogan T.P., Uher C., Kanatzidis M.G., J. Am. Chem. Soc 128, 14347 (2006). Sootsman J.R., Pcionek R.J., Kong H., Uher C., Kanatzidis M.G., Chem. Mater 18, 4993 (2006). Mountvala A.J., Abowitz G., J. Am. Ceram. Soc 48, 651 (1965). Rogacheva E.I., Krivulkin I.M., Nashchekina O.N., AYu S., Volobuev V.A., Dresselhaus M.S., Appl. Phys. Lett 78, 3238 (2001). Lee H.S., Cheong B., Lee T.S., Lee K.S., Kim W.M., Lee J.W., Cho S.H., Huh J.Y., Appl. Phys. Lett 85, 2782 (2004). Kishimoto K., Tsukamoto M., Koyanagi T., J. Appl. Phys 92, 5331 (2002). Rogacheva E.I., Nashchekina O.N., Grigorov S.N., Us M.A., Dresselhaus M.S., Cronin S.B., Nanotechnology 14, 53 (2003). Rogacheva E.I., Nashchekina O.N., Meriuts A.V., Lyubchenko S.G., Dresselhaus M.S., Dresselhaus G., Appl. Phys. Lett 86, 063103 (2005). Rogacheva E.I., Grigorov S.N., Nashchekina O.N., Tavrina T.V., Lyubchenko S.G., AYu S., Volobuev V.V., Fedorov A.G., Dresselhaus M.S., Thin Solid Films 493, 41 (2005). Qiu X., Lou Y., Samia A.C.S., Devadoss A., Burgess J.D., Dayal S., Burda C., Angew. Chem. Int. Ed 44, 5855 (2005). Wang C., Zhang G.F., Li S.Y., J. Phys. Chem. Solids 62, 1957 (2001). Poudel B., Wang W.Z., Wang D.Z., Huang J.Y., Ren Z.F., J. Nanosci. Nanotechnol 6, 1050 (2006). Zhang B., He J., Tritt T.M., Appl. Phys. Lett 88, 043119 (2006). Heiss W., Groiss H., KaQWmann E., Hesser G., Böberl M., Springholz G., Schäffler F., Koike K., Harada H., Yano M., Appl. Phys. Lett 88, 192109 (2006). Akimov B., Bogoyavlenskiy V.A., Ryabova L.I., Vasil’kov V.N., Phys. Rev. B 61, 16045 (2000). Ya A U., Samoilov A.M., Sharov M.K., Yatsenko O.B., Akimov B.A., Inorg. Mater 38, 12 (2002). Ya A U., Samoilov A.M., Buchnev S.A., Yu V.S., Sharov M.K., Inorg. Mater 38, 450 (2002). Samoilov A.M., Buchnev S.A., Synorov Y., Agapov B.L., Khoviv A.M., Inorg. Mater 39, 1132 (2003). Samoilov A.M., Buchnev S.A., Dolgopolova E.A., Synorov Y., Khoviv A.M., Inorg. Mater 40, 349 (2004). Murakami H., Hattori W., Aoki R., Phys. C 269, 83 (1996). Murakami H., Hattori W., Mizomata Y., Aoki R., Phys. C 273, 41 (1996). Murakami H., Aoki R., Sakai K., Thin Solid Films 27, 343 (1999). Volkov B.A., Ryabova L.I., Khokhlov D.R., Phys. Usp 45, 819 (2002), and references therein. Hüe F., Hÿtch M., Bender H., Houdellier F., Claverie A., Phys. Rev. Lett 100, 156602 (2008). Banerjee S., Shore K.A., Mitchell C.J., Sly J.L., Missous M., IEE Proc. Circuits Devices Syst 152, 497 (2005). Razeghi M., Evans A., Slivken S., Yu J.S., Zheng J.G., Dravid V.P., Proc. SPIE 5840, 54 (2005). Stradling R.A., Semicond. Sci. Technol 6, C52 (1991).

References

[93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114]

[115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128]

107

Weimer P.K., Proc. IEEE 52, 608 (1964). Ribakovs G., Gundjian A.A., IEEE J. Quant. Electron QE-14, 42 (1978). Dey S.K., J. Vac. Sci. Technol 10, 227 (1973). Lynch S.J., Thin Solid Films 102, 47 (1983). Kudzin V.V., Kulakov V.S., Pape’ D.R., Kulakov S.V., Molotok V.V., IEEE. Ultrason. Symp 1, 749 (1997). Hatami F., Lordi V., Harris J.S., Kostial H., Masselink W.T., J. Appl. Phys 97, 096106 (2005). Wessels B.W., J. Electrochem. Soc 722, 402 (1975). Tolfree D.W.L., J. Sci. Instrum 41, 788 (1964). Hart P.B., Proc. IEEE 61, 880 (1973). Hines M.A., Scholes G.D., Adv. Mater 15, 1844 (2003). Wang C.A., Huang R.K., Shiau D.A., Connors M.K., Murphy P.G., O’Brien P.W., Anderson A.C., DePoy D.M., Nichols G., Palmisiano M.N., Appl. Phys. Lett 83, 1286 (2003). Hitchcock C.W., Gutmann R.J., Borrego J.M., Bhat I.B., Charache G.W., IEEE Trans. Electron. Devices 46, 2154 (1999). Goldsmid H.J., Douglas R.W., Br. J. Appl. Phys 5, 386 (1954). Rosi F.D., Abeles B., Jensen R.V., J. Phys. Chem. Sol 10, 191 (1959). T.M. Tritt (ed.), Materials and Semimetals, vol. 69, 70 and 71: Recent Trends in Thermoelectric Materials Research I, II and III (Academic Press, New York, 2000). Rowe D.M (ed.), CRC Handbook of Thermoelectric (CRC Press, Boca Raton, 1995). Rowe D.M., Bhandari C.M., Modern Thermoelectrics (Reston Publishing Company, Virginia, 1983). Rowe D.M (ed.), Thermoelectrics Handbook: Macro to Nano (CRC Press, Boca Raton, 2006). Choi H., Chang M., Jo M., Jung S.J., Hwang H., Electrochem (2008); Solid-State Lett. 11, H154. Cova S., Ghioni M., Lacaita A., Samori C., Zappa F., Appl. Opt 35, 1956 (1996). Lee H.W.H., Taylor B.R., Kauzlarich S.M., Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 12 (Technical Digest, 2000). Brundermann E., Heugen U., Bergner A., Schiwon R., Schwaab G.W., Ebbinghaus S., Chamberlin D.R., Haller E.E., Havenith M.., IN 29th International Conference on Infrared and MillimeterWaves and 12th International Conference on Terahertz, Electronics, vol 283 (2004). Baranov A.N., Voronina T.I., Zimogorova N.S., Kauskaya L.M., Yakoviev Y.P., Sov. Phys. Semicond 19, 1676 (1985). Yano M., Suzuki Y., Ishii T., Matsushima Y., Kimata M., Jpn. J. Appl. Phys 17, 2091 (1978). Yuang F.S., Su Y.K., Li N.Y., Jpn. J. Appl. Phys 30, 207 (1991). Yuang F.S., Su Y.K., Li N.Y., Gan K.J., J. Appl. Phys 68, 6383 (1990). Su Y.K., Chen S.M., J. Appl. Phys 73, 8349 (1993). Haywood S.K., Henriques A.B., Mason N.J., Nicholas R.J., Walker P.J., Semicond. Sci. Technol 3, 315 (1988). Young G.C., Anderson W.W., Anderson L.B., Trans. Electron Dev. IEEE 24, 492 (1977). Gordon R.L., Neeley V.I., Curtin H.R., Proc. IEEE 54, 2014 (1966). Weimer P.K., Proc. IRE 50, 1462 (1962). Lee M.J., Wright S.W., Judge C.P., Cheung P.Y., (1991) Display Research Conference, International Conference Record, 211 Vassilev L.A., Phys. Stat. Sol. (b) 121, 203 (1984). Thornton J.M., Mol J., Biol 151, 261 (1981), Creighton T.E., (1986) Methods Enzymol. 131, 83; Creighton T.E., (1988) Bio Essays 8, 57. Hruby V.J., Life Sciences 31, 189 (1982), Hruby V.J., Al-Obeidi F., Kazmierski W., (1990) Biochem J., 268, 249. Wetzel R., Trends Biochem. Sci 12, 478 (1987).

108

1 The entropy in quantum wells of heavily doped materials

[129] Kimura T., Matsueda R., Nakagawa Y., Kaiser E.T., Anal Biochem 122, 274 (1982). [130] Andreu D., Albericio F., Sole N.A., Munson M.C., Ferrer M., Barnay G., Methods in Molecular Biology, 35 Peptide Synthesis Protocols in Ed M. W. Pennington, B. M. Dunna, (Humana Press Inc., USA, 1994). [131] Chakrabarti S., Chatterjee B., Debbarma S., Ghatak K.P., Jour. of Nanoscience and Nanotech 15 (2015). [132] Debbarma S., Ghatak K.P., Reviews in Theoretical Science 3, 16 (2015). [133] Abramowitz M., Stegun I.A., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (New York, Wiley, 1964). [134] Gradshteyn I.S., Ryzhik I.M., Tables of Integrals, Series and Products (New York, Academic Press, 1965). [135] Heine V., Proc. Phys. Soc 81, 300 (1963), Schulman J.N., Chang Y.C., (1981), Phys. Rev B, 24, 4445. [136] Adachi S., J. Appl. Phys 58, R11 (1985). [137] Ghatak K.P., Banerjee J.P., Bhattacharya D., Nanotechnology 7, 110 (1996), Bhattacharya S., Choudhury S., Ghatak K.P., (2010), Superlatt. and Microstruct., 48, 257; Ghatak K.P., Bhattacharya S., Pahari S., De D., Ghosh S., Mitra M., (2008), Ann. Phys., 17, 195; Pahari S., Bhattacharya S., Ghatak K.P., (2009), (2088) J. Comput. Theor. Nanosci., (Invited Paper), 6. [138] Krieehbaum M., Kocevar P., Pascher H., Bauer G., IEEE QE 24, 1727 (1988), Lundstrom M.S., Guo J., Nanoscale Transistors, Device Physics, Modeling and Simulation (Springer, 2006, USA); Saito R., Dresselhaus G., Dresselhaus M.S., Physical Properties of Carbon Nanotubes (Imperial College Press, London, 1998); Yang X., Ni J., (2005) Phys. Rev. B72, 195426; Mintmire W., White C.T., (1998), Phys. Rev. Letts. 81, 2506. [139] Bir G.L., Pikus G.E., Symmetry and Strain – Induced effects in Semiconductors (Nauka, Russia, 1972); (in Russian); Mondal M., Ghatak K.P., (1986), Phys. Stat. Sol. (b) 135, K21; Wu C.C., Lin C.J., (1984), J. Low. Temp. Phys. 57, 469; Yamada Y., (1973), J. Phys. Soc. Jpn. 35, 1600. [140] Chakravarti A.N., Ghatak K.P., Ghosh K.K., Ghosh S., Mukherjee H.M., Phys. Stat. Sol. b 118, 843 (1983), Bhattacharya S., De D., Adhikari S.M., Ghatak K.P. (2012), Superlatt. and Microstruc. 51, 203; Ghatak K.P., Mondal M., (1986), Z.F. Naturforschung 41A, 821. [141] Chakravarti A.N., Choudhury A.K., Ghatak K.P., Ghosh S., Dhar A., Appl. Phys 25, 105 (1981), Chakraborty P.K., Datta G.C., Ghatak K.P., (2003), Physica Scripta, 68, 368. [142] Chakravarti A.N., Ghatak K.P., Dhar A., Ghosh K.K., Ghosh S., Appl. Phys A26, 165 (1981); Ghatak K.P., Bhattacharya S., Biswas S.K., Dey A., Dasgupta A.K., (2007), Phys. Scrip, 75, 820. [143] Ghatak K.P., Mondal M., Z.F. Physik B B69, 471 (1988), Chakravarti A.N., Ghatak K.P., Ghosh K.K., Ghosh S., Dhar A., (1982), Z.F, Physik B. 47, 149. [144] Lyden H.A., Phys. Rev 135, A514 (1964), Palik E.D., Wright G.B., in Semiconductors and Semimetals, edited by Willardson R.K., BeER A.C., 3, p-421 (1967, Academic Press, New York, USA). [145] Mondal M., Ghatak K.P., Phys. Letts 131 A, 529 (1988), Ghatak K.P., Mitra B., (1992), Int. J. Electron. 72, 541; Mitra B., Ghoshal A., Ghatak K.P., (1990), Nouvo Cimento D 12D, 891; Ghatak K.P., Biswas S.N., (1993), Nonlinear Optics and Quantum Optics 4, 347. [146] Ghatak K.P., Ghoshal A., Mitra B., Nouvo Cimento 14D, 903 (1992), Ghatak K.P., Ghoshal A., Mitra B., (1991), Nouvo Cimento. 13D, 867; Mitra B., Ghatak K.P. (1989), Solid State Electron. 32, 177; Mondal M., Chattapadhyay N., Ghatak K.P., (1987), J. Low Temp. Phys. 66 131. [147] Hai P.N., Chen W.M., Buyanova I.A., Xin H.P., Tu C.W., Appl. Phys. Lett 77, 1843 (2000), DiVincenzo D.P., Mele E.J., (1984), Phys. Rev B 29, 1685; Perlin P., Litwin-Staszewska E., Suchanek B., Knap W., Camassel J., Suski T., Piotrzkowski R., Grzegory I., Porowski S., Kaminska E., Chervin J.C., (1996), Appl. Phys. Lett. 68, 1114; Smith G.E. (1962), Phys. Rev.

References

[148] [149] [150] [151] [152]

[153] [154] [155] [156] [157]

[158] [159] [160] [161]

[162]

[163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174]

109

Lett., 9, 487; Schneider D., Rurup D., Plichta A., Grubert H.-U., Schlachetzki A., Hansen K., (1994), Z. Phys. B, 95, 281; Masia F., Pettinari G., Polimeni A., Felici M., Miriametro A., Capizzi M., Lindsay A., Healy S.B., O’Reilly E.P., Cristofoli A., Bais G., Piccin M., Rubini S., Martelli F., Franciosi A., Klar P.J., Volz K., Stolz W., (2006), Phys. Rev. B, 73, 073201. Arora V.K., Jeafarian H., Phys. Rev. B 13, 4457 (1976); Ostapov S.E., Zhikharevich V.V., Deibuk V.G., (2006), Semicond. Phys., Quan. Electron. and Optoelectron, 9, 29. Aubin M.J., Caron L.G., Jay G.J.P., Phys. Rev. B 15, 3872 (1977); Sewall S.L., Cooney R.R., Kambhampati P., (2009), Appl. Phys. Lett, 94, 243116. Tanaka K., Kotera N.; in 20th Internat. Conf. on Indium Phosphide and Related Materials, (2008), 25–29 May 2008, p 1–4, Versailles, France. Singh M., Wallace P.R., Jog S.D., Erushanov J., J. Phys. Chem. Solids 45, 409 (1984). Zawadzki W., Adv. Phys 23, 435 (1974); Zawadzki W., in Two-Dimensional Systems, Heterostructures and Superlattices, Bauer G., Kuchar F., Heinrich H., (ed.) Springer Ser. Solid-State Sci., Vol. 53 (Springer-Verlag, Germany, 1984); Zawadzki W., 11th Internet Conf. Phys. Of Semicond. Vol. 1 (Elsevier Publishing Company, Netherlands, 1972); Zelenin S.P., Kondrat’ev A.S., Kuchma A.E., (1982), Sov. Phys. Semicond. 16, 355. Pathria R.K., Statistical Mechanics, 2nd (Butterworth-Heinmann, Oxford, 1996). Ghatak K.P., Mitra B., Basu D.K., Nag B., Nolinear Optics 17, 171 (1997). Nag B.R., Chakravarti A.N., Solid State Electron 18, 109 (1975), (1974), Phys. Stal Sol. (a) 22, (K153). Dimmock J.O., The Physics of Semimetals and Narrowgap Materials ed. by D.L. Carter, R.T. Bates (Pergamon Press, Oxford, 1971). Seiler D.G., Bajaj B.D., Stephens A.E., Phys. Rev. B 16, 2822 (1977); Germaneko A.V., Minkov G.M., (1994), Phys. Stat. Sol. (b) 184, (9); Bir G.L., Pikus G.E., (1972), Symmetry and Strain – Induced effects in Materials Nauka, Russia, (in Russian). Bouat J., Thuillier J.C., Surf. Sci 73, 528 (1978). Rees G.J., Physics of Compounds, in Proceedings of the 13th International Conference ed. by F.G. Fumi (North Holland Company, 1976), p. 1166 Entage P.R., Phys. Rev 138, A246 (1965). Stordeur M., Kuhnberger W., Phys. Stat. Sol 69, 377 (1975); Lovett D.R., Semimetals and Narrow-Bandgap Semiconductor (Pion Limited, UK, 1977); Kohler H., (1976), Phys. Stat. Sol. 74, 591. Cardona M., Paul W., Brooks H., Helv Acta Phys 33, 329 (1960); A.F. Gibson in Proceeding of International School of Physics, Enrico fermi, course XIII, ed. by R.A. Smith (Academic Press, New York, 1963), p. 171. Wang C.C., Ressler N.W., Phys. Rev 2, 1827 (1970). Mathur P.C., Jain S., Phys. Rev 19, 1359 ( 1979). Yamada Y., J. Phys. Japan 35, 1600 (1973); Singh M., Wallace P.R., Jog S.D., Arushanov E., (1984) J. Phys. Chem. Solids, 45, 409. Chuiko G.P., Sov. Phys. Semi 19 (12), 1381 (1985). Ivchenko E.L., Pikus G.E., Sov, Phys. Semicond 13, 579 (1979). Cohen M.H., Phys. Rev 121, 387 (1961). McClure J.W., Choi K.H., Solid State Comm 21, 1015 (1977). Bangert E., Kastner P., Phys. Stat. Sol (b) 61, 503 (1974). Foley G.M.T., Langenberg P.N., Phys. Rev. B 15B, 4850 (1977). Ivanov-Omskii V.I., Mekhtisev A., Rustambekova S.A., Ukraintsev E.N., Phys. Stat. Sol. (b) 119, 159 (1983). Seiler D.G., Beeker W.M., Roth L.M., Phys. Rev 1, 764 (1970). Zhang H.I., Phys. Rev. B 1, 3450 (1970).

110

1 The entropy in quantum wells of heavily doped materials

[175] Rossler U., Solid State Commun 49, 943 (1984). [176] Johnson J., Dickey D.H., Phys. Rev 1, 2676 (1970). [177] Agafonov V.G., Valov P.M., Ryvkin B.S., Yarashetskin I.D., Sov. Phys. Semiconduct 12, 1182 (1978). [178] Vassilev L.A., Phys. State sol (b) 121, 203 (1984). [179] Averkiev N.S., Asnin V.M., Bakun A.A., Danishevskii A.M., Ivchenko E.L., Pikus G.E., Rogachev A.A., Sov. Phys. Semicond 18, 379 (1984). [180] Brandt N.B., Davydov V.N., Kulbachinskii V.A., Nikitina O.M., Sov. Phys. Sol. Stat 29, 1014 (1987). [181] Ketterson J.B., Phys. Rev 129, 18 (1963). [182] Cunningham R.W., Phys. Rev 167, 761 (1968). [183] Yekimov A.I., Onushchenko A.A., Plyukhin A.G., Efros A.I.L., J. Expt. Theor. Phys 88, 1490 (1985). [184] Roman B.J., Ewald A.W., Phys. Rev B5, 3914 (1972). [185] Chuiko G.P., Sov. Phys. Semiconduct 19 (12), 1381 (1985).

2 The entropy in nanowires of heavily doped materials To be the best, I must extract the best from myself.

2.1 Introduction It is well known that in nanowires (NWs), the restriction of the motion of the carriers along two directions may be viewed as carrier confinement by two infinitely deep one-dimensional (1D) rectangular potential wells, along any two orthogonal directions leading to quantization of the wave vectors along the said directions, allowing 1D carrier transport [1]. With the help of modern experimental techniques, such1D 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 nano-structured devices for investigating their electronic, optical and allied properties [2–4]. Examples of such new applications are based on the different transport properties of ballistic charge carriers which include quantum resistors [5–10], resonant tunneling diodes and band filters [11, 12], quantum switches [13], quantum sensors [14–16], quantum logic gates [17–18], quantum transistors and sub tuners [19–21], heterojunction FETs [22], high-speed digital networks [23], high-frequency microwave circuits [24], optical modulators [25], optical switching systems [26, 27], and other devices. In this chapter through Sections 2.2.1–2.2.14 we have investigated the entropy’s in NWs of heavily doped (HD) nonlinear optical, III–V, II–VI, stressed Kane type, Te, GaP, PtSb2, Bi2Te3, Ge, GaAs, II–V, lead germanium telluride and zinc and cadmium phosphides respectively. Section 2.3 contains the summary and conclusions pertaining to this chapter. Section 2.4 presents 19 open research problems.

2.2 Theoretical background 2.2.1 The entropy in NWs of HD nonlinear optical materials The dispersion relation of 1D electrons in this case can be written following (1.32) as 2
y Þ2
z Þ2
y π=d
z π=d
h2 ðn
h2 ðn
h2
kx + +
η Þ 2m
η Þ 2m
η Þ =1
21 ðE,
22 ðE,
21 ðE,
*jj T
*jj T
*jj T 2m g g g

https://doi.org/10.1515/9783110661194-002

(2:1)

112

2 The entropy in nanowires of heavily doped materials


z are the size quantum number and the nanothickness
z ð = 1, 2, 3, .....Þ, d where n
y are the size quantum num
y ð = 1, 2, 3, .....Þ and d along the z-direction respectively, n ber and the nanothickness along the y-direction respectively The 1D DOS function per sub-band is given by

= 2g
v ∂kx
1D ðEÞ N
π ∂E

(2:2)


′1HDNW ) can be expressed in this case as The complex sub-band energy (E
y Þ2
z Þ2
y π=d
z π=d
h2 ðn
2 ðn h + =1
′1HDNW , η Þ 2m
′1HDNW , η Þ
22 ðE
*jj T21 ðE
*? T 2m g g

(2:3)

The EEM in this case in given by
F1HDNW , n
y , n
z , ηg Þ =
* ðE m

 
2 h ∂
n
1HDNW ðE,
y , n
z , ηg Þ2 ½T Real part of
2 ∂ðEF1HDNW Þ (2:4)

where, ""
1HDNW ðE,
n
y , n
z , ηg Þ = T

# #
η Þ 1=2
22 ðE,
y Þ 2 2m
z Þ2
*? T
y π=d
z π=d
h2 ðn
2 ðn h g 1−
η Þ − 2m
η Þ
21 ðE,
22 ðE,
*jj T
*? T 2m
h2 g g


F1HDNW is the Fermic energy in this case and E Thus, we observe that the EEM is the function of size quantum numbers in both the directions and the Fermi energy due to the combined influence of the crystal filed splitting constant and the anisotropic spin-orbit splitting constants respectively. Besides it is a function of ηg due to which the EEM exists in the band gap, which is otherwise impossible. The carrier statistics can be written as
 2g
v
1D = Real part of n π (2:5)
ymax n n zmax X
X







½T 1HDNW ðEF1HDNW ; ny ; nz ; ηg Þ + T 2HDNW ðEF1HDNW ; ny ; nz ; ηg Þ
y¼1 n
z¼1 n

and
2HDNW ðE
F1HDNW , n
y , n
z , ηg Þ = T


s X
r = 1


F1HDNW , n

rÞ½T
1HDNW ðE
y , n
z , ηg Þ Lð

Using (2.5) and (1.31f) we can find the entropy in this case. In the absence of bandtails, for electron motion along x-direction only, the 1D electron dispersion law in this case can be written following (1.2) as

113

2.2 Theoretical background


y Þ2 +
f2 ðEÞðπ
z Þ2
=
f1 ðEÞk
2 +
f1 ðEÞðπ
n
n
y =d
z =d γðEÞ x

(2:6)


′1 Þ are given by the equation The sub-band energyðE
y Þ2 + f2 ðE
z Þ2
1 ′Þðπn
1 ′Þðπn
1 ′Þ = f1 ðE
y =d
z =d γðE

(2:7)

The electron concentration per unit length can be written as
 n
X
zmax ymax n X 2g
v
F1d , n
F1d , n
1D =
y, n
z Þ +
t2 ðE
y, n
z Þ n ½
t1 ðE π n
y = 1 n
z = 1

(2:8)


F1d is the Fermic energy in this case, where E
y Þ2 −
f2 ðE
z Þ2 1=2 ½
f1 ðE
F1d , n
F1d Þ −
f1 ðE
F1d Þðπn
F1d Þðπn
F1d Þ − 1=2
y , n
z Þ ≡ ½γðE
y =d
z =d t1 ðE and
F1d , n
t2 ðE
y , n
z Þ ≡


s X


F1d , n
y , n
z Þ LðrÞ½
t1 ðE


r = 1

using (1.44) and (2.8), we can find entropy in this case.

2.2.2 The entropy in NWsof HD III–V materials (i) Three-band model of Kane The dispersion relation of 1D electrons in this case can be written following (1.55) as
y Þ2
h2
kx 2
z Þ2 h
y π=d
z π=d
2 ðn
2 ðn h
31 ðE,
η Þ + iT
31 ðE,
η Þ + + =T g g
c
c
c 2m 2m 2m

(2:9)


′2HDNW ) in this case can be expressed as The sub-band energy (E
y Þ2
z Þ2 h
y π=d
z π=d
2 ðn
2 ðn h
31 ðE
′2HDNW , η Þ + iT
31 ðE
′2HDNW , η Þ + =T g g
c
c 2m 2m

(2:10)

The EEM in this case is given by
F1HDNW , η Þ = m
31 ′ðE
F1HDNW , η Þ
* ðE
c ½T m g g

(2:11)

The carrier statistics can be written as
 2g
v
1D = Real Part of n π
ymax n n zmax X
X
y = 1 n
z = 1 n

(2:12)
3HDNW ðE
F1HDNW , n
4HDNW ðE
F1HDNW , n
y , n
z , ηg Þ + T
y , n
z , ηg Þ ½T

114

2 The entropy in nanowires of heavily doped materials

where ""
F1HDNW ; n
3HDNW ðE
y ; n
z ; ηg Þ = T


31 ðE
F1HDNW ; η Þ +
iT
31 ðE
F1HDNW ; η Þ T g g

# #1=2
y Þ 2 2m
z Þ2 h
y π=d
z π=d
c
2 ðn
2 ðn h − −
c
c 2m 2m
h2 where
F1HDNW , n
4HDNW ðE
y , n
z , ηg Þ = T


s X
r = 1


F1HDNW , n

rÞ½T
3HDNW ðE
y , n
z , ηg Þ Lð

Using (1.31f) and (2.12) we can investigate the entropy in this case. In the absence of doping, 1D electron dispersion law is given by
2
kx2
h

y , n
z Þ =
I11 ðEÞ + G2 ðn
c 2m

(2:13)

where
y Þ2 + ðn
z Þ2 
2 ðn
y , n
z Þ ≡ ð
h2 π2 =2m
c Þ½ðn
y =d
z =d G
′2 can be written as The sub-band energy E
2 ðn
′2 Þ
y , n
z Þ =
I11 ðE G

(2:14)

The electron statistics in this case can be written as pffiffiffiffiffiffiffiffi n
X
zmax n
c ymax X 2g
v 2m
F1d , n
F1d , n
y , n
z Þ +
t4 ðE
y , n
z Þ
1D = ½
t3 ðE n π
h ny = 1 nz = 1

(2:15)

where
2 ðn
F1d , n
F1d Þ − G
t3 ðE
y , n
z Þ ≡ ½
I11 ðE
y , n
z Þ1=2 and
F1d , n
t4 ðE
y , n
z Þ ≡


s X


F1d , n

rÞ½
t3 ðE
y , n
z Þ Lð


r = 1

Thus using (1.44) and (2.15), we can study the entropy in this case. (ii) Two band model of Kane The dispersion relation under heavy doping of 1D electrons in this case can be written as

2.2 Theoretical background


y Þ2
h2
kx 2
z Þ2 h
y π=d
z π=d
2 ðn
2 ðn h
η Þ + + = γ2 ðE, g
c
c
c 2m 2m 2m

115

(2:16)


′3HDNW ) in this case can be expressed as The sub-band energy (E
y Þ
z Þ
y π=d
z π=d
2 ðn h
2 ðn h
′3HDNW , η Þ + = γ2 ðE g

2 mc 2 mc 2

2

(2:17)

The EEM in this case is given by
F1HDNW , η Þ = m
F1HDNW , η Þ
* ðE
c ½γ2 ′ðE m g g

(2:18a)

The carrier statistics can be written as
1D = n


 n
X
zmax ymax n X 2g
v
7HDNW ðE
F1HDNW , n
8HDNW ðE
F1HDNW , n
y , n
z , ηg Þ + T
y , n
z , ηg Þ ½T π n
y = 1 n
z = 1 (2:18b)

where ""
F1HDNW , n
7HDNW ðE
y , n
z ,ηg Þ= T

# #1=2 2
y Þ 2 2m
z Þ2
h2 ðn
y π=d
z π=d
c
h ð n
F1HDNW ,η Þ− γ2 ðE − g
c
c 2m 2m
h2

and
F1HDNW , n
8HDNW ðE
y , n
z , ηg Þ = T


s X
r = 1


F1HDNW , n

rÞ½T
7HDNW ðE
y , n
z , ηg Þ Lð

Thus using (1.44) and (2.18b) we can study the entropy in this case. The expression of 1D dispersion relation, for NWs of III-V materials whose energy band structures are defined by the two-band model of Kane in the absence of band tailing assumes the form
2
kx2

+ αEÞ
=h
y , n
z Þ + G2 ðn Eð1
c 2m
′3 is given by In this case, the quantized energy E  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ðn
′3 = ð2αÞ − 1 − 1 + 1 + 4αG
y , n
z Þ E The carrier statistics in the case can be expressed as pffiffiffiffiffiffiffiffi n
X
zmax n
c ymax X 2g
2m
F1d , n
F1d , n
y , n
z Þ +
t6 ðE
y , n
z Þ
1D = v ½
t5 ðE n π
h n
y = 1 n
z = 1

(2:19)

(2:20)

(2:21)

116

2 The entropy in nanowires of heavily doped materials

where,
2 ðn
F1d , n
F1d ð1 + αE
F1d Þ − G
t5 ðE
y , n
z Þ ≡ ½E
y , n
z Þ1=2 and
F1d , n
t6 ðE
y , n
z Þ ≡


s X


F1d , n

rÞ½
t5 ðE
y , n
z Þ Lð


r = 1

Thus using (1.44) and (2.21) we can study the entropy in this case. (iii) Parabolic energy bands The dispersion relation of 1D electrons under the condition of heavy doping this case can be written as
y Þ2
h2
kx 2
z Þ2 h
y π=d
z π=d
2 ðn
2 ðn h
η Þ + + = γ3 ðE, g
c
c
c 2m 2m 2m

(2:22)


′5HDNW ) in this case can be expressed as The sub-band energy (E
y Þ2
z Þ2 h
y π=d
z π=d
2 ðn
2 ðn h
′5HDNW , η Þ + = γ3 ðE g
c
c 2m 2m

(2:23)

The EEM in this case is given by
F1HDNW , η Þ = m
F1HDNW , η Þ
* ðE
c ½γ3 ′ðE m g g

(2:24a)

Thus the carrier statistics can be written as
1D = n


 n
X
zmax h ymax n i X 2g
v
F1HDNW , n
10HDNW ðE
F1HDNW , n
9HDNW ðE
y , n
z , ηg Þ + T
y , n
z , ηg Þ T π n
y = 1 n
z = 1 (2:24b)

where, ""
9HDNW ðE
F1HDNW , n
y , n
z ,ηg Þ= T

# #1=2
y Þ 2 2m
z Þ2
h2 ðn
y π=d
z π=d
c
h2 ðn
γ3 ðEF1HDNW ,ηg Þ− − ,
c
c 2m 2m
h2

and
F1HDNW , n
y , n
z , ηg Þ = T10HDNW ðE


s X
r = 1


F1HDNW , n

rÞ½T
9HDNW ðE
y , n
z , ηg Þ Lð

Thus using (1.44) and (2.24b) we can study the entropy in this case.

2.2 Theoretical background

117

The expression of 1D dispersion relation, for NWs of III-V materials whose energy band structures are defined by parabolic isotropic energy bands in the absence of band tailing assumes that
2
kx2

= h
y , n
z Þ + G2 ðn E
c 2m

(2:25)


′7 is given by In this case, the quantized energy E
2 ðn
′7 = G
y , n
z Þ E

(2:26)

The carrier statistics in the case can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n
y n
z max X max
X
c π
kB T 2g
v 2m
− 1 ðη Þ ,
F n1D = 67 2 h
y = 1 n
z = 1 n

(2:27)

where η67 =




νÞ
′7 + W
−h EF1d − ðE
kB T


Using(1.44) and (2.27), we can write # − 1 " n
#
2
" n
X
zmax
zmax ymax n ymax n X X X π kB


. F − 1 ðη67 Þ F − 3 ðη67 Þ G= 2 2 3
e
y = 1 n
z = 1
y = 1 n
z = 1 n n

(2:28)

Under the condition of non degeneracy, (2.28) get transformed into the well-known expression as given in the preface. (iv) The Model of Stillman et al. The dispersion relation of 1D electrons under heavy doping in this case can be written as
y Þ
z Þ
y π=d
z π=d
2 ðn h
2 ðn h
h2
kx
η Þ + + = θ4 ðE, g
c
c
c 2m 2m 2m 2

2

2

(2:29)

where,
η Þ =
I12 ðE,
η Þ θ4 ðE, g g
′9HDNW ) in this case can be expressed as The sub-band energy (E
y Þ
z Þ
y π=d
z π=d
2 ðn h
2 ðn h
′9HDNW , η Þ + = θ4 ðE g
c
c 2m 2m 2

2

The EEM in this case is given by

(2:30)

118

2 The entropy in nanowires of heavily doped materials


F1HDNW , η Þ = m
F1HDNW , η Þ
c ½θ4 ′ðE
* ðE m g g

(2:31a)

The carrier statistics canbe written as

1D = n

2g
v π

 n
X
zmax ymax n X
y = 1 n
z = 1 n


11HDNW ðE
F1HDNW , n
12HDNW ðE
F1HDNW , n
y , n
z , ηg Þ + T
y , n
z , ηg Þ ½T (2:31b)

where, ""
F1HDNW , n
11HDNW ðE
y , n
z , ηg Þ = T

2
z Þ2
z π=d
F1HDNW , η Þ −
h ðn θ4 ðE g
c 2m # # 1=2
y Þ2 2m
y π=d
c
h2 ðn −
c 2m
h2

Thus using (1.31f) and (2.31b) we can study the entropy in this case. The expression of 1D dispersion relation for NWs of III–V materials whose energy band structures are defined by the model of Stillman et al in the absence of band tailing assumes the form
2
kx2

=h
I12 ðEÞ
y , n
z Þ + G2 ðn
c 2m

(2:32)


′9 is given by In this case, the quantized energy E
2 ðn
′9 Þ = G
I12 ðE
y , n
z Þ

(2:33)

The carrier statistics in the case can be expressed as pffiffiffiffiffiffiffiffi n
X
zmax n
c ymax X 2g
2m
9 ðE
9 ðE
F1d , n
F1d , n
1D = v
y , n
z Þ + Q
y , n
z Þ n ½P π
h n
y = 1 n
z = 1 where
2 ðn
F1d , n
F1d Þ − G
9 ðE
y , n
z Þ ≡ ½
I12 ðE
y , n
z Þ1=2 P and
9 ðE
F1d , n
y , n
z Þ ≡ Q


s X


F1d , n

rÞ½P
9 ðE
y , n
z Þ Lð


r = 1

Thus using (1.44) and (2.34), we can study the entropy in this case. (v) The Model of Palik et al. The dispersion relation of 1D electrons in this case can be written as

(2:34)

2.2 Theoretical background


y Þ2
h2
kx 2
z Þ2 h
y π=d
z π=d
2 ðn
2 ðn h
η Þ + + = θ5 ðE, g
c
c
c 2m 2m 2m

119

(2:35)

where
η Þ =
I13 ðE,
η Þ θ5 ðE, g g
′10HDNW ) in this case can be expressed as The sub-band energy (E
y Þ2
z Þ2 h
y π=d
z π=d
2 ðn
2 ðn h
′10HDNW , η Þ + = θ5 ðE g
c
c 2m 2m

(2:36)

The EEM in this case is given by
F1HDNW , η Þ = m
F1HDNW , η Þ
* ðE
c ½θ5 ′ðE m g g

(2:37a)

The carrier statistics can be written as
1D = n


 n
X
zmax ymax n X 2g
v
13HDNW ðE
F1HDNW , n
14HDNW ðE
F1HDNW , n
y , n
z , ηg Þ + T
y , n
z , ηg Þ ½T π n
y = 1 n
z = 1 (2:37b)

where ""
13HDNW ðE
F1HDNW ,n
y ,n
z ,ηg Þ= T

# #1=2
y Þ 2 2m
z Þ2
h2 ðn
y π=d
z π=d
c
h2 ðn
θ5 ðEF1HDNW ,ηg Þ− +
c
c 2m 2m
h2

and
F1HDNW , n
14HDNW ðE
y , n
z , ηg Þ = T


s X
r = 1


F1HDNW , n

rÞ½T
13HDNW ðE
y , n
z , ηg Þ Lð

Thus, using (1.31f) and (2.37b), we can investigate the entropy in this case. The expression of 1D dispersion relation for NWs of III–V materials whose energy band structures are defined by the model of Palik et al in the absence of band tailing assumes the form
2
kx2

=h
I13 ðEÞ
y , n
z Þ + G2 ðn
c 2m

(2:38)


′10 is given by In this case, the quantized energy E
2 ðn
′10 Þ = G
I13 ðE
y , n
z Þ The carrier statistics in the case can be expressed as

(2:39)

120

2 The entropy in nanowires of heavily doped materials


1D = n

2g
v π

pffiffiffiffiffiffiffiffi n
X
zmax n
c ymax X 2m
12 ðE
11 ðE
F1d , n
F1d , n
y , n
z Þ + Q
y , n
z Þ ½P
h n
y = 1 n
z = 1

(2:40)

where
2 ðn
F1d , n
F1d Þ − G
11 ðE
y , n
z Þ ≡ ½
I13 ðE
y , n
z Þ1=2 P and
12 ðE
F1d , n
y , n
z Þ ≡ Q


s X


F1d , n

rÞ½P
11 ðE
y , n
z Þ Lð


r = 1

Using (1.44) and (2.40), we can study the entropy in this case.

2.2.3 The entropy in nanowires of HD II–VI materials The 1D electron dispersion law in NW of HD II–VI materials can be written following (1.141) as
η Þ=a
′0 γ3 ðE, g

"



x π n
x d

"
2
# 2
2 #1=2
y π 2

π n n π
h2
kz2 n y x +
+ + ±
λ0
x
y
*jj 2m d dy d

(2:41)


′13HDNW ) in this case can be expressed as The sub-band energy (E
′13HDNW , η Þ = a
′0 γ3 ðE g

"



x π n
x d

"
2
# 
 #1=2
y π 2
y π 2
x π 2 n n n
+
+
± λ0
x d dy dy

(2:42)

The EEM in this case is given by
F1HDNW , η Þ = m
F1HDNW , η Þ
*k γ′3 ðE
* ðE m g g

(2:43)

The carrier statistics can be written as
1D = n


ymax
xmax n n X g
v X
17HDNW ðE
F1HDNW , n
18HDNW ðE
F1HDNW , n
x , n
y , ηg Þ + T
x , n
y , ηg Þ ½T π n
x = 1 n
y = 1

(2:44) Where, ""
F1HDNW ; n
17HDNW ðE
x ; n
y ; ηg Þ = T


F1HDNW ; η Þ − a
0 0 γ3 ðE g

"



x π n
x d

2
#
y π 2 n +
dy

121

2.2 Theoretical background

∓
λ0 and
F1HDNW , n
18HDNW ðE
x , n
y , ηg Þ = T


s X
r = 1

"



x π n
x d

2
 #1=2 #

 #1=2 2mjj
y π 2 n +
dy
h2



rÞ½T17HDNW ðEF1HDNW , nx , ny , η Þ Lð g

Using (1.31f) and (2.44), we can study the entropy in this case. The 1D dispersion relation for NWs of II–VI materials in the absence of bandtails can be written as
′0
k2 + G
3, ± ðn
=b
x , n
y Þ E z

(2:45)

where, 2
3, ± ðn
x , n
y Þ ≡ 4a
′0 G

(



x πn
dx

3 (
2
) 2
2 )1=2
y 2

πn π n π n y x 5 +
+
±
λ0
x dy d dy

The 1D electron statistics can be written as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
zmax
n
X ymax n
*jj π
kB T g
v 2m X

−1


− 1 ðη

x , n
y Þ n1D = F 68, ± Þ, η68, ± = ðkB TÞ ½EF1d − ½G3, ± ðn
2 h
y = 1 n
z = 1 n

(2:46) Thus using (1.44) and (2.46) we can study the entropy in this case.

2.2.4 The entropy in nanwires of HD IV–VI materials (i) Dimmock Model The 1D electron dispersion law in NW of HD IV–VI materials can be expressed following (1.174) as
x π n
x d

2
η Þ + αγ ðE,
η Þ
h γ2 ðE, g g 3 2
x4


η ÞÞ − ð1 + αγ3 ðE, g
2 h −α 2
x1


x π n
x d

!2


h2 2
x1 +


h2 2
x2

!2


2 h + 2
x5


x π n
x d

!2


y π n
y d


y π n
y d


2 h + 2
x2 !2 !

!2 !
η Þ + αγ3 ðE, g
y π n
y d

!2 !


2 2 h k 2
x6 z

122

2 The entropy in nanowires of heavily doped materials


2 h −α 2
x1


x π n
x d

!2


2 h + 2
x2


y π n
y d

!2 !


2
2
η ÞÞ h k − ð1 + αγ3 ðE, g 2
x3 z !2
x π
h2
2
h2 n
h2 kz −α +
2
x3 2
x4 dx 2
x5 !2
x π
h4
kz4
h2
h2 n −α = +
x
1 d
2 4
x3
x6 2m 2m


2
2 h k 2
x6 z


y π n
y d

!2 !


y π n
y d

!2 +


2
2 h k
3 z 2m

(2:47)

Equation (2.47) can be written as
kz = T
36 ðE,
η ,n
y Þ g
x , n

(2:48)

where
22 Þ − 1 ½ − B
η ;n
HD ðE;
η ;n
36 ðE;
y Þ = ½ð2C
y Þ T g
x ; n g
x ; n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
22 A
2 HD ðE;
η ;n
η ;n
HD ðE;
y Þ + 4C
y Þ 1=2 + B g
x ; n g
x ; n 2 ! !2 4
x π h

h2 n
2 h


4



+ ; BHD ðE; ηg ; nx ; ny Þ = α C22 = α
x 4
x3
x6 2
x1 d 2
x2


y π n
y d

!2 !


2 h 2
x6

2
2
η ÞÞ h
η Þ
h
γ3 ðE;
+ ð1 + α − α γ ð E; g g 3 2
x3 2
x6 !2 !2 !# 2 2 2
y π
x π
h
h
h
h2 n n
+ +α +
x
y
3 2m 2
x3 2
x4 d 2
x5 d

and " " !2 !2 #
y π
x π
h2 n
h2 n

Þ+αγ ðE;η
Þ

x ; n
y Þ= − + γ2 ðE;η AHD ðE;ηg ; n g g 3
x
y 2
x1 d 2
x2 d
x π
h2 n
x 2
x4 d

!2


x π
h2 n −α
x
2x 1 d


y π
h2 n +
y 2
x5 d !2

!2 !


y π
h2 n +
y
2x 2 d

2

ÞÞ
h nx π −ð1+αγ3 ðE;η g
x 2
x1 d

!2 !

!2


x π
h2 n
x
2x 4 d


y π
h2 n +
y 2
x2 d

!2


y π
h2 n +
y
2x 5 d

!2 !#


′14HDNW ) in this case can be expressed as The sub-band energy (E

!2 !

123

2.2 Theoretical background


36 ðE
′14HDNW , η , n
y Þ 0=T g
x , n

(2:49)

The EEM in this case is given by
F1HDNW , η , n
y Þ =
* ðE m g
x , n


2 ∂
2
h


½T36 ðEF1HDNW , ηg , nx , ny Þ 2 ∂E

(2:50)

The carrier statistics can be written as
1D = n


 n
X
ymax xmax n X 2g
v
36HDNW ðE
F1HDNW , n
37HDNW ðE
F1HDNW , n
x , n
y , ηg Þ + T
x , n
y , ηg Þ ½T π n
x = 1 n
y = 1 (2:51)

where
36HDNW ðE
F1HDNW , n
36 ðE
F1HDNW , nx , ny , η Þ
x , n
y , ηg Þ = T T g and
F1HDNW , n
37HDNW ðE
x , n
y , ηg Þ = T


s X
r = 1


F1HDNW , n

rÞ½T
36HDNW ðE
x , n
y , ηg Þ Lð

thus using (1.31f) and (2.51) we can study the entropy in this case. The 1D electron dispersion law in NW of IV–VI materials in the absence of band tails can be expressed as !2 ! 2
y π
h2 n

h
k2 + + αE
2
x5 dy 2
x6 z !2 !2 ! !2
y π
x π
x π
h2 n
h2 n
h2 n
h2
− ð1 + αEÞ + + − α
x
x
y 2
x1 d 2
x2 d 2
x1 d 2
x2 !2 !2 ! !
y π
y π 2
x π
x π 2
h2 n
h2 n
h2 n
h2 n + ð − α
x
x Þ + 2
x2 ð d
y
y Þ 2
x4 d 2
x5 d 2
x1 d !2 !2 !
y π
x π
h2 2
h2
2
h2 2
h2 n
h2 n
k −α k − ð1 + αEÞ k +
x
y 2x6 z 2
x3 z 2
x3 z 2
x4 d 2
x5 d !2 !2
y π
x π
h4
kz4
h2 n
h2 n
h2
2 k −α = + +
x
y
1 d
2 d
3 z 4
x3
x6 2m 2m 2m

2
+ αEÞ
+ αE

h Eð1 2
x4


x π n
x d

!2


y π n
y d

!2 !

(2:52)

Equation (2.52) can be written as
kz = T
40 ðE,
n
x , n
y Þ

(2:53)

124

2 The entropy in nanowires of heavily doped materials

where   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2
22 Þ − 1 − B
22 A
n
0 ðE,
n
2 0 ðE,
n
n
40 ðE,
0 ðE,
x , n
y Þ = ð2C
x , n
y Þ + B
x , n
y Þ + 4C
x , n
y Þ T where

"


x , n
y Þ = α B0 ðE, n


2 h 2
x1


x π n
x d

!2


2 h
h2 + +α
3 2m 2
x3


y π n
y d


2 h + 2
x2


2 h 2
x4


x π n
x d

!2

!2 !

2 2
h2

h − αE

h + ð1 + αEÞ 2
x6 2
x3 2
x6


2 h + 2
x5


y π n
y d

!2 !#

and 2 2


h2
n
0 ðE,
x , n
y Þ = 4 − 4 A 2
x1


x π n
x d


+ αEÞ
+ αE
Eð1
x π n
x d


2 h −α 2
x1
− ð1 + αEÞ

!2


2 h 2
x4

!2


2 h 2
x1


y π n
y d


2 h + 2
x2
x π n
x d

!2

!2 3 5


2 h + 2
x5 !2 !


y π
2 n h +
y 2
x2 d !2
x π
h2 n +
x 2
x2 d


y π n
y d

!2 !


x π
2 n h
x 2
x4 d !2 !#
y π n
y d

!2


2 h + 2
x5


y π n
y d

!2 !


′20 )in this case can be expressed as The sub-band energy (E
′20 , n
40 ðE
x , n
y Þ 0=T

(2:54)

The EEM in this case is given by
F1d , n
x , n
y Þ =
* ðE m


2 ∂
2
h


½T40 ðEF1d , nx , ny Þ 2 ∂E

(2:55)

The carrier statistics assumes the form
1D = n


 n
X
ymax n 2g
v xmax X
40 ðE
F1d , n
41 ðE
F1d , n
x , n
y Þ + T
x , n
y Þ ½T π n
x = 1 n
y = 1

where
41 ðE
F1d , n
x , n
y Þ = T


s X


F1d , n

rÞ½T
40 ðE
x , n
y Þ Lð


r = 1

Thus using (1.44) and (2.56), we can study the entropy in this case.

(2:56)

2.2 Theoretical background

125

(ii) Bangert and Kastner Model The dispersion relation of the conduction electrons in bulk specimens of IV–VI materials in accordance with the model of Bangert and Kastner is given by
2 + ω2 ðEÞk
2=1 ω1 ðEÞk s z

(2:57a)

where " # 2
2
2   ðRÞ ð
SÞ ðQÞ −1

ω1 E = ð2EÞ +
g ð1 + α1 EÞ
+ Δc ′ð1 + α2 EÞ

E Δ′′c ð1 + α3 EÞ 0 and " #
2
2 ðAÞ ð
S + QÞ −1


2
2 = 2.3 × 10 − 10 ð
e
νÞ2 , ð
SÞ2 = 4.6ðRÞ ω2 ðEÞ = ð2EÞ , ðRÞ
g0 ð1 + α1 EÞ
+ Δ′′c ð1 + α3 EÞ
E 1 1 1 ,α = α1 =
, α2 = ,
c′ 3 Δ
′′c Eg0 Δ
2 = 1.3ðRÞ
Δ
ðQÞ
2 , ðAÞ
2 = 0.8 × 10 − 4 ð
eV
m
′c = 3.07
eV,
′′c = 3.28
eV,
2Þ Δ The electron energy spectrum in HD IV–VI materials in accordance with this model can be expressed by using the methods as given in chapter 1 as "
2
2 2

E
g Þ−iD
1 ðα
E
g Þg ðRÞ + f
c2 ðα
E
g Þ −iD
2 ðα
E
g Þg ðSÞ

1 , E,


, E, , E, , E, 2Ið4Þ= ks f
c1 ðα 1 2 2 0 0 0 0
go
′c E Δ # 

2 ðQÞ 2ðAÞ





E
g Þ −iD
1 ðα
E
g Þg
3 , E, Eg Þ− iD3 ðα
3 , E, Eg0 Þg
1 , E,
1 , E, +f
c3 ðα +
kz2
f
c1 ðα 0
′′ Ego Δ c
ðS
+ QÞ
E
g Þ−iD
3 ðα
E
g Þg
3 , E,
3 , E, f
c3 ðα 0 0
′′ Δ c 2

+

(2:57b) where 1 1 1 , α3 = , α1 =
, α2 =

′ E g0 Δc Δ′′c " # " # ∞ X 2 2 2 − 1


1 , E, Eg0 Þ =
=4Þðsinhðp
u

c1 ðα
i ÞÞgp
i Þ × pffiffiffi expð − u fexpð − p αi ηg π
=1 p

126

2 The entropy in nanowires of heavily doped materials

" pffiffiffi # π
E
g Þ =
1 ðα
1 , E,
2i Þ expð − u D 0 α1 ηg Therefore (2.57b) can be written as,
2


2
η Þk
1 ðE, F g s + F2 ðE, ηg Þkz = 1

(2:57c)

where, "
η Þ½2γ ðE,
η Þ
1 ðE, F g g 0 +

−1


ðRÞ
E
g Þ − iD
1 ðα
E
g Þg
1 , E,
1 , E, f
c1 ðα 0 0
Eg


ðSÞ
E
g Þ − iD
2 ðα
E
g Þg
2 , E,
2 , E, f
c2 ðα 0 Δ′c


ðQÞ
E
g Þ − iD
3 ðα
E
g Þg
3 , E,
3 , E, f
c3 ðα + 0 0 Δ′c

#

and " h i − 1 2ðAÞ



E
g Þ − iD
1 ðα
E
g Þg

1 , E,
1 , E, f
c1 ðα F 2 ðE, ηg Þ 2γ0 ðE, ηg Þ 0
Eg0 #
+ QÞ
ðS





3 , E, Eg Þ − iD3 ðα
3 , E, Eg0 Þg f
c3 ðα + Δ′c
1 ðE,
η Þ and F
2 ðE,
η Þ are complex, the energy spectrum is also complex in Since F g g the presence of Gaussian band tails. Following (2.57c), the 1D dispersion relation in NW of IV–VI materials in accordance with the present model can be written as 2 !2 !2 3
y π
x π n n
2 ðE,
η Þ4
η Þk2 = 1
1 ðE, 5+F +
(2:58) F g g z
x d dy The (2.58) can be written as
kz = T
60 ðE,
η ,n
y Þ g
x , n

(2:59)

where ""
η ,n
60 ðE,
y Þ = T g
x , n


1 ðE,
η Þ 1−F g

"



x π n
x d

#1=2 2
 ##
y π 2 n − 1
η Þ
2 ðE, +
½F g dy


′15HDNW ) in this case can be expressed as The sub-band energy (E

2.2 Theoretical background


60 ðE
′15HDNW , η , n
y Þ 0=T g
x , n

127

(2:60)

The EEM in this case is given by
F1HDNW , η , n
y Þ =
* ðE m g
x , n


2 ∂
2
h


½T60 ðEF1HDNW , ηg , nx , ny Þ 2 ∂E

(2:61)

The carrier statistics can be written as ! n
xmax 2g
v X
1D = n π
x = 1 n
ymax n

X


y =1 n


50HDNW ðE
F1HDNW ; n
F1HDNW ; n
51HDNW ðE
F1HDNW ; n
x ; n
y ;ηg ÞðE
x ; n
y ; ηg Þ+ T
x ; n
y ;ηg Þ ½T (2:62)

where,
F1HDNW , n
F1HDNW , n
50HDNW ðE
x , n
y , ηg Þ = T40 ðE
x , n
y , ηg Þ T and
F1HDNW , n
51HDNW ðE
x , n
y , ηg Þ = T


s X
r = 1


F1HDNW , n

rÞ½T50HDNW ðE
x , n
y , ηg Þ Lð

Thus, using (1.31f) and (2.66), we can study the entropy in this case. The 1D dispersion relation in the absence of band tailing can be written in this case following (2.57a) as " !2
x πn
+ ω1 ðEÞ
x d


y πn
y d

!2 #

k2 = 1 + ω2 ðEÞ z

(2:63)

Then (2.63) can be written as
kz = T
61 ðE,
n
x , n
y Þ

(2:64)

where, ""
61 ðE,
n
x , n
y Þ = T


1 − ω1 ðEÞ

"



x π n
x d

#1=2 2
 ##
y π 2 n − 1
+
½ω2 ðEÞ dy


′21 ) in this case can be expressed as The sub-band energy (E
′21 , n
61 ðE
x , n
y Þ 0=T The EEM in this case is given by

(2:65)

128

2 The entropy in nanowires of heavily doped materials


2 ∂
2
h
F1d , n
x , n
y Þ =


* ðE m
½T61 ðEF1d , nx , ny Þ 2 ∂E

(2:66)

The carrier statistics can be written as ! n

ymax xmax n X 2g
v X
61 ðE
F1d , n
62 ðE
F1d , n
x , n
y Þ + T
x , n
y Þ
1D = ½T n π
x = 1 n
y = 1 n

(2:67)

where
62 ðE
F1d , n
x , n
y Þ = T


s X


F1d , n

rÞ½T61 ðE
x , n
y Þ Lð


r = 1

Using (1.44) and (2.67), we can study the entropy in this case.

2.2.5 The entropy in nanowiresof HD stressed Kane-type materials The 1D dispersion relation in this case can be written following (1.206) as
x
η Þ πn
11 ðE, P g
dx

!2



11 ðE,
η Þ π ny +Q g
dy

!2
η Þ
k2 = 1 +
S11 ðE, g z

(2:68)

Then (2.68) can be written as
kz = T
70 ðE,
η ,n
y Þ g
x , n

(2:69)

where 22



η ,n
11 ðE,
η Þ π nx
70 ðE,
y Þ = 44 1 − P T g
x , n g
x d

!2



11 ðE,
η Þ π ny +Q g
y d

31=2 !2 3
η Þ − 1 5 5½
S11 ðE, g


′30HDNW ) in this case can be expressed as The sub-band energy (E
′30HDNW , η , n
70 ðE
y Þ 0=T g
x , n

(2:70)

The EEM in this case is given by
F1HDNW , η , n
y Þ =
* ðE m g
x , n


2 ∂
2
h


½T70 ðEF1HDNW , ηg , nx , ny Þ 2 ∂E

The carrier statistics can be written as

(2:71)

2.2 Theoretical background


1D = n

129


 n
X
ymax xmax n X 2g
v
70HDNW ðE
F1HDNW , n
71HDNW ðE
F1HDNW , n
x , n
y , ηg Þ + T
x, n
y , ηg Þ ½T π n
x = 1 n
y = 1 (2:72)

where
F1HDNW , n
70HDNW ðE
F1HDNW , n
70HDNW ðE
x, n
y , ηg Þ = T
x, n
y , ηg Þ T and
F1HDNW , n
71HDNW ðE
x , n
y , ηg Þ = T


s X
r = 1


F1HDNW , n

rÞ½T
70HDNW ðE
x, n
y , ηg Þ Lð

Thus, using (1.31f) and (2.72), we can study the entropy in this case. In the absence of band tailing,1D dispersion relation in this case assumes the form
kz =
t70 ðE,
n
x , n
y Þ

(2:73)

where " "


t70 ðE, n
x , n
y Þ = ½
c0 ðEÞ 1 −


x πn


0 ðEÞ dx a

!2 −


y πn


dy b0 ðEÞ

!2 ##1=2


′42 ) in this case can be expressed as The sub-band energy (E
′42 , n
x , n
y Þ 0 =
t60 ðE

(2:74)

The EEM in this case is given by
2 ∂
2
h
F1d , n
x , n
y Þ =


* ðE m
½ t60 ðEF1d , nx , ny Þ 2 ∂E The carrier statistics can be written as ! n

ymax xmax n X 2g
v X
F1d , n
F1d , n
x , n
y Þ +
t61 ðE
x , n
y Þ
1D = ½
t60 ðE n π

nx = 1 ny = 1 where
F1d , n
t61 ðE
x , n
y Þ =


s X


F1d , n

rÞ½
t60 ðE
x , n
y Þ Lð


r = 1

Thus, using (1.44) and (2.76), we can study the ENTROPY in this case.

(2:75)

(2:76)

130

2 The entropy in nanowires of heavily doped materials

2.2.6 The entropy in nanowires of HD Te The 1D dispersion relation may be written in this case following (1.235) as
kx =
t72 ðE,
n
y , n
z , ηg Þ

(2:77)

where "
n
t72 ðE,
y , n
z , ηg Þ = −


y π n
y d

!2
η Þ−ψ + ψ5HD ðE, g 6 "


η Þ− ± ψ7 ψ28HD ðE, g


z πn
dz

!2
z πn
z d !2 #1=2 #1=2


′31HDNW ) in this case can be expressed as The sub-band energy (E
′31HDNW , η , n
z Þ 0 =
t72 ðE g
y , n

(2:78)

The EEM in this case is given by
F1HDNW , η , n
z Þ =
* ðE m g
y , n


2 ∂
2
h


½ t72 ðEF1HDNW , ηg , ny , nz Þ 2 ∂E

(2:79)

The carrier statistics can be written as ! n

zmax ymax n X 2g
v X
F1HDNW , n
F1HDNW , n
y , n
z , ηg Þ +
t73HDNW ðE
y , n
z , ηg Þ
1D = ½
t72HDNW ðE n π
y = 1 n
z = 1 n (2:80) where
F1HDNW , n
F1HDNW , n
t72HDNW ðE
y , n
z , ηg Þ =
t72HDNW ðE
y , n
z , ηg Þ and
F1HDNW , n
t73HDNW ðE
y , n
z , ηg Þ =


s X
r = 1


F1HDNW , n

rÞ½
t72HDNW ðE
y , n
z , ηg Þ Lð

Thus, using (1.31f) and (2.80), we can study the entropy in this case. In the absence of band tailing the 1D dispersion relation in this case assumes the form
k = H
n
70 ðE,
y , n
z Þ where

(2:81)

2.2 Theoretical background

"


y , n
z Þ = − H70 ðE, n


y π n
y d

!2


z
− ψ πn + ψ5 ðEÞ 6
dz

!2

"
− ± ψ7 ψ28 ðEÞ


z πn dz

131

!2 #1=2 #1=2


′44 ) in this case can be expressed as The sub-band energy (E
′44 , n
70 ðE
y, n
z Þ 0=H

(2:82)

The EEM in this case is given by
F1d , n
y , n
z Þ =
* ðE m


h2 ∂
2



½H70 ðEF1d , ny , nz Þ 2 ∂E

(2:83)

The carrier statistics can be written as ! n

zmax ymax n X 2gv X
70 ðE
F1d , n
71 ðE
F1d , n
y , n
z Þ + H
y , n
z Þ
1D = ½H n π
y = 1 n
z = 1 n

(2:84)

where
71 ðE
F1d , n
y , n
z Þ = H


s X


70 ðE
F1d , n
y , n
z Þ LðrÞ½H


r = 1

Thus, using (1.44) and (2.84), we can study the entropy in this case.

2.2.7 The entropy in nanowires of HD gallium phosphide The 1D dispersion relation may be written in this case following (1.253) as
kx = u
n
y , n
z , ηg Þ
70 ðE,

(2:85)

where "



70 ðE, ny , nz , ηg Þ = − u

!2
y π n




y + t11 γ3 ðE, ηg Þ + t21 − t31 d #1=2 #1=2 " !2
z π n 2

−
t41
z + t5 ðE, ηg Þ d


z π n
z d

!2


′32HDNW ) in this case can be expressed as The sub-band energy (E
′
y , n
z Þ
70 ðE , ηg , n 0=u 32HDNW The EEM in this case is given by

(2:86)

132

2 The entropy in nanowires of heavily doped materials


2 ∂ 2
h





* ðE
, η , n , n Þ = m y z g 32HDNW
½u70 ðE32HDNW , ηg , ny , nz Þ 2 ∂E

(2:87)

The carrier statistics can be written as ! n

zmax ymax n X 2g
v X
F1HDNW , n
F1HDNW , n
y , n
z , ηg Þ + u
y , n
z , ηg Þ
1D =
70HDNW ðE
71HDNW ðE ½u n π
y = 1 n
z = 1 n (2:88) where,
F1HDNW , n
F1HDNW , n
y , n
z , ηg Þ = u
y , n
z , ηg Þ
70HDNW ðE
70HDNW ðE u and
F1HDNW , n
y , n
z , ηg Þ =
71HDNW ðE u


s X
r = 1


F1HDNW , n

rÞ½u
y , n
z , ηg Þ
70HDNW ðE Lð

By using (1.31f) and (2.88), we can find the entropy in this case. In the absence of band tailing the 1D dispersion relation in this case can be written using (1.260) as
kx = X
71 ðE,
n
y, n
z Þ

(2:89)

where "


y , n
z Þ = − X71 ðE, n


y π n
y d

#1=2

!2
nz Þ +
t42 ðE,


′46 ) in this case can be expressed as The sub-band energy (E
′46 , n
71 ðE
y , n
z Þ 0= X

(2:90)

The EEM in this case is given by 
h2 ∂ 
2

F1d , n
y , n
z Þ =


* ðE m
X71 ðEF1d , ny , nz Þ 2 ∂E The carrier statistics can be written as ! n

zmax ymax n X  2g
v X
F1d , n
72 ðE
F1d , n
71 ðE
y , n
z Þ + X
y , n
z Þ
1D = X n π
y = 1 n
z = 1 n where

(2:91)

(2:92)

2.2 Theoretical background


F1d , n
72 ðE
y , n
z Þ = X


s X

133


71 ðE
F1d , n
y , n
z Þ LðrÞ½X


r = 1

By using (1.44) and (2.92), we can find the entropy in this case.

2.2.8 The entropy in nanowires of HD platinum antimonide The 1D dispersion relation may be written in this case following (1.275) as
kx = V
70 ðE,
n
y , n
z , η
g Þ

(2:93)

where 2
n
70 ðE,
y , n
z , η
g Þ = 4 − V


y π n dy

31=2

!2


60 ðE,
η
g , n
y Þ5 +A

The sub-band energy (E′34HDNW ) in this case can be expressed as
′34HDNW , η
70 ðE
g , n
y , n
z Þ 0=V

(2:94)

The EEM in this case is given by
F1HDNW , η
g , n
y , n
z Þ =
* ðE m


2 ∂
2
h



½V70 ðEF1HDNW , ηg , ny , nz Þ 2 ∂E

(2:95)

The carrier statistics can be written as ! n

zmax h ymax n i X 2g
v X
F1HDNW , n
71HDNW ðE
F1HDNW , n
70HDNW ðE
y , n
z , ηg Þ + V
y , n
z , ηg Þ
1D = V n π
y = 1 n
z = 1 n (2:96) where
F1HDNW , n
70HDNW ðE
F1HDNW , n
70HDNW ðE
y , n
z , ηg Þ = V
y , n
z , ηg Þ V and
F1HDNW , n
71HDNW ðE
y , n
z , ηg Þ = V


s X
r = 1


F1HDNW , n

rÞ½V
70HDNW ðE
y , n
z , ηg Þ Lð

By using (1.31f) and (2.96), we can find the entropy in this case. In the absence of band tailing the 1D dispersion relation in this case can be written using (1.278) as

134

2 The entropy in nanowires of heavily doped materials


kx = D
71 ðE,
n
y , n
z Þ

(2:97)

where

"


y , n
z Þ = − D71 ðE, n


y π n
y d

!2

#1=2

+ t44 ðE, nz Þ

The sub-band energy(ðE′ 48Þ) in this case can be expressed as
′48 , n
71 ðE
y , n
z Þ 0=D

(2:98)

The EEM in this case is given by
F1d , n
y , n
z Þ =
* ðE m


h2 ∂
2



½D71 ðEF1d , ny , nz Þ 2 ∂E

(2:99)

The carrier statistics can be written as ! n

zmax ymax n X 2g
v X
71 ðE
F1d , n
72 ðE
F1d , n
y , n
z Þ + D
y , n
z Þ
1D = ½D n π
y = 1 n
z = 1 n

(2:100a)

where
72 ðE
F1d , n
y , n
z Þ = D


s X


F1d , n

rÞ½D
71 ðE
y , n
z Þ Lð


r = 1

Using (1.44) and (2.100a), we can find the entropy in this case.

2.2.9 The entropy in nanowires of HD bismuth telluride The dispersion relation in this case can be written following (1.285) as
kx =
J70 ðE,
n
y , n
z , ηg Þ

(2:100b)

where ""
J70 ðE,
n
y , n
z , ηg Þ =


η Þ − ω2 γ2 ðE, g


y π n
y d

!2 − ω3


z π n
z d

!2

# #1=2
y n
z π2 n −1 − 2ω4

ðω1 Þ dy dz


′50HDNW ) in this case can be expressed as The sub-band energy (E
′50HDNW , η , n
z Þ 0 =
J70 ðE g
y , n The EEM in this case is given by

(2:101)

135

2.2 Theoretical background


2 ∂
2
h
F1HDNW , η , n




* ðE , n Þ = m y z g
½J70 ðEF1HDNW , ηg , ny , nz Þ 2 ∂E

(2:102)

The carrier statistics which can, in turn, be written as ! n

zmax ymax n X 2g
v X
F1HDNW , η , n
F1HDNW , η , n
z Þ +
J71HDNW ðE
z Þ
1D = ½
J70HDNW ðE n g
y , n g
y , n π
y = 1 n
z = 1 n (2:103) where
J70HDNW ðE
F1HDNW , η , n
F1HDNW , η , n
z Þ =
J70HDNW ðE
z Þ g
y , n g
y , n and
F1HDNW , η , n
J71HDNW ðE
z Þ = g
y , n


s X
r = 1


F1HDNW , η , n

rÞ½
J70HDNW ðE
z Þ Lð g
y , n

Using (1.31f) and (2.103), we can find the entropy in this case. In the absence of band tailing the 1D dispersion relation in this case can be written using (1.278) as
kx = B
71 ðE,
n
y , n
z Þ

(2:104)

where ""
n
71 ðE,
y , n
z Þ = B


+ αEÞ
− ω2 Eð1


y π n
y d

!2 − ω3


z π n
z d

!2

# #1=2
y n
z π2 n −1 − 2ω4

ðω1 Þ dy dz

The sub-band energy (E′50) in this case can be expressed as
′50 , n
71 ðE
y , n
z Þ 0=B

(2:105)

The EEM in this case is given by
h2 ∂
2

F1d , n
y , n
z Þ =


* ðE m
½B 71 ðEF1d , ny , nz Þ 2 ∂E The carrier statistics can be written as ! n

zmax ymax n X 2g
v X
71 ðE
F1d , n
72 ðE
F1d , n
y , n
z Þ + B
y , n
z Þ
1D = ½B n π

ny = 1 nz = 1 where
72 ðE
F1d , n
y , n
z Þ = B


s X
r = 1


F1d , n

rÞ½B
71 ðE
y , n
z Þ Lð

(2:106)

(2:107)

136

2 The entropy in nanowires of heavily doped materials

Using (1.44) and (2.107), we can find the entropy in this case.

2.2.10 The entropy in nanowires of HD germanium (a) Model of Cardona et al The dispersion relation in accordance with this model in the present case can be written following (1.306b) as
kx = L
70 ðE,
n
y , n
z , ηg Þ

(2:108)

where ""
n
70 ðE,
y , n
z , ηg Þ = L

"
η Þ+α γ2 ðE, g


2 h
*jj 2m


z π n
z d


2
η ÞÞ h − ð1 + 2αγ3 ðE, g
*jj 2m

!2 #2


z π n
z d

!2 #


*jj 2m

!#1=2


2 h

The sub-band energy (E′52HDNW ) in this case can be expressed as
′52HDNW , η , n
70 ðE
z Þ 0=L g
y , n

(2:109)

The EEM in this case is given by
2 ∂
2
′ h
′F1HDNW , η , n
z Þ =


* ðE m g
y , n
½L70 ðE F1HDNW , ηg , ny , nz Þ 2 ∂E

(2:110)

The carrier statistics can be written as ! n

zmax ymax n X 2gv X
70HDNW ðE
F1HDNW , n
71HDNW ðE
F1HDNW , n
y , n
z , ηg Þ + L
y , n
z , ηg Þ
1D = ½L n π

ny = 1 nz = 1 (2:111) where
70HDNW ðE
F1HDNW , n
70HDNW ðE
F1HDNW , n
y , n
z , ηg Þ = L
y , n
z , ηg Þ L and
F1HDNW , n
71HDNW ðE
y , n
z , ηg Þ = L


s X
r = 1


F1HDNW , n

rÞ½L
70HDNW ðE
y , n
z , ηg Þ Lð

In the absence of band tailing the 1D dispersion relation in this case can be written using (1.278) as

2.2 Theoretical background


kx = B
77 ðE,
n
y , n
z Þ

137

(2:112)

where ""
77 ðE,
n
y , n
z Þ = B

"

2
+ αEÞ
+ α
h Eð1
*jj 2m


z π n
z d

!2 #2

2

h − ð1 + 2αEÞ 2m*jj

!#1=2 
*jj
z π 2 2m n
z 
h2 d


′ 60) in this case can be expressed as The sub-band energy (E
′60 , n
77 ðE
y , n
z Þ 0=B

(2:113)

The EEM in this case is given by
h2 ∂
2

F1d , n
y , n
z Þ =


* ðE m
½B77 ðEF1d , ny , nz Þ 2 ∂E The carrier statistics can be written as ! n

zmax ymax n X 2g
v X
77 ðE
F1d , n
78 ðE
F1d , n
y , n
z Þ + B
y , n
z Þ
1D = ½B n π
y = 1 n
z = 1 n

(2:114)

(2:115)

where
78 ðE
F1d , n
y , n
z Þ = B


s X


77 ðE
F1d , n
y , n
z Þ LðrÞ½B


r = 1

Using (1.44) and (2.115), we can study the entropy in this case. (b) Model of Wang et al. The dispersion relation in accordance with this model in the present case can be written following (1.326) as
kx = β ðE,
n
y , n
z , ηg Þ 70

(2:116)

where "

" !2 *

2 m π n z
n
y , n
z , ηg Þ = − + 2? α8 − α9
β70 ðE, dz
h #1=2 ##1=2 " !4 !2
z
z πn πn
η Þ − α10 + α11
+ α12 ðE, g
z d dz
y π n
y d

!2

The sub-band energy (E′54HDNW ) in this case can be expressed as
′54HDNW , η , n
z Þ 0 = β70 ðE g
y , n

(2:117)

138

2 The entropy in nanowires of heavily doped materials

The EEM in this case is given by
2 ∂ 2
h
F1HDNW , η , n
z Þ =


* ðE m g
y , n
½β70 ðEF1HDNW , ηg , ny , nz Þ 2 ∂E

(2:118)

The carrier statistics can be written as ! n

zmax ymax n X 2g
v X
F1HDNW , n
F1HDNW , n
y , n
z , ηg Þ + β71HDNW ðE
y , n
z , ηg Þ
1D = ½β70HDNW ðE n π
y = 1 n
z = 1 n (2:119) where
F1HDNW , n
F1HDNW , n
y , n
z , ηg Þ = β70 ðE
y , n
z , ηg Þ β70HDNW ðE and
F1HDNW , n
y , n
z , ηg Þ = β71HDNW ðE


s X
r = 1




rÞ½β
y , n
z , ηg Þ Lð 70HDNW ðEF1HDNW , n

Using (1.31f) and (2.119), we can study the entropy in this case. In the absence of band tailing the 1D dispersion relation in this case can be written using (1.278) as
kx = P
77 ðE,
n
y , n
z Þ

(2:120)

where
n
77 ðE,
y , n
z Þ = P

""
2 h
I1 ðE,
n
z Þ −
*2 2m


y π n
y d

!2 #


*1 2m
h2

!#1=2


′2HDNW ) in this case can be expressed as The sub-band energy (E
′80 , n
77 ðE
y , n
z Þ 0=P

(2:121)

The EEM in this case is given by
F1d , n
y , n
z Þ =
* ðE m


h2 ∂
2



½P77 ðEF1d , ny , nz Þ 2 ∂E

The carrier statistics can be written as ! n

zmax ymax n X 2g
v X
77 ðE
F1d , n
78 ðE
F1d , n
y , n
z Þ + P
y , n
z Þ
1D = ½P n π
y = 1 n
z = 1 n

(2:122)

(2:123)

2.2 Theoretical background

139

where
77 ðE,
n
y , n
z Þ = P

""
2 h
I1 ðE,
n
z Þ −
*2 2m


y π n
y d

!2 #


*1 2m
h2

!#1=2

Using (1.44) and (2.123), we can study the entropy in this case.

2.2.11 The entropy in nanowires of HD galium antimonide The dispersion relation of the 1D electrons in this case can be written as
y Þ2
h2
kx 2
z Þ2 h
y π=d
z π=d
2 ðn
2 ðn h
η Þ + + =
I36 ðE, g
c
c
c 2m 2m 2m

(2:124)

The sub-band energy ðE′100HDNW Þ in this case can be expressed as
y Þ2
z Þ2 h
y π=d
z π=d
2 ðn
2 ðn h + =
I36 ðE′100HDNW , ηg Þ
c 2m 2mc

(2:125)

The EEM in this case is given by
* ðE′F1HDNW , ηg Þ = m
c ½
I36 ′ðE′F1HDNW , ηg Þ m

(2:126)

The carrier statistics can be written as ! n

zmax ymax n X 2g
v X
7HDNW ðE
F1HDNW , n
8HDNW ðE
F1HDNW , n
y , n
z , ηg Þ + R
y , n
z , ηg Þ
1D = ½R n π
y = 1 n
z = 1 n (2:127) where ""
7HDNW ðE
F1HDNW ; n
F1HDNW ; η ÞðE
F1HDNW ; η Þ
y ; n
z ; ηg Þ =
I 36 ðE R g g # #1=2
y Þ 2 2m
z Þ2 h
y π=d
z π=d
c
2 ðn
2 ðn h − −
c
c 2m 2m
h2 and
8HDNW ðE
F1HDNW , n
y , n
z , ηg Þ = R


s X
r = 1


F1HDNW , n

rÞ½R
7HDNW ðE
y , n
z , ηg Þ Lð

Using (1.31f) and (2.127), we can study the entropy in this case.

140

2 The entropy in nanowires of heavily doped materials

The expression of 1D dispersion relation, for NWs of GaSb whose energy band structures in the absence of band tailing assumes the form
2
kx2

=h
I36 ðEÞ
y , n
z Þ + G2 ðn
c 2m

(2:128)

In this case, the quantized energy E′101 is given by
2 ðn
I36 ðE′101 Þ = G
y , n
z Þ

(2:129)

The carrier statistics in the case can be expressed as pffiffiffiffiffiffiffiffi n
X
zmax n
c ymax X 2g
2m
101 ðE
F1d , n
102 ðE
F1d , n
y , n
z Þ + R
y , n
z Þ
1D = v ½R n π
h n
y = 1 n
z = 1

(2:130)

where
2 ðn
F1d , n
F1d Þ − G
101 ðE
y , n
z Þ ≡ ½
I36 ðE
y , n
z Þ1=2 R and
F1d , n
102 ðE
y , n
z Þ ≡ R


s X


F1d , n

rÞ½R
101 ðE
y , n
z Þ Lð


r = 1

Thus, using (1.44) and (2.160), we can study the entropy in this case.

2.2.12 The entropy in nanowires of HD II–V materials The DR of the 1D holes in II-V compounds can be expressed as
η Þ=A
10 γ3 ðE, g 2
14 ±4 A


x π n
x d


x π n
x d

!2

!2
15 +A


11 +A
y π n
y d


y π n
y d

!2
12
k2 + A
13 +A z

!2
16
k2 +A z


17 +A


x π n
x d


x π n
x d

!

!2 !2
18 +A


y π n
y d

!2

31=2
2 5 +A 19 (2:131)

where the numerical values of the energy band constants are given in Appendix A.
n The sub-band energy (E ) is the lowest positive root of the following zHD401 equation

141

2.2 Theoretical background

!2
x π n

x + A13 d


n
10 γ3 ðE , ηg Þ = A zHD401


x π n
x d

! " ±


14 A


x π n
x d

!2
17 +A


x π n
x d

!!2

#1=2
2 +A 19 (2:132)

(2.132) can be expressed as
kx = Δ27 ðE,
η ,n
y Þ g
x , n

(2:133)

where
η ,n
2 − A
2 Þ − 1 ½½Δ21 ðE,
η ,n
12 + A
16 Δ22 ðn
y Þ = ½ðA
y ÞA
x , n
y Þ Δ27 ðE, g
x , n g
x , n 12 16
η ,n
12 + A
16 Δ22 ðn
2 − A
2 ÞΔ25 ðn
η Þ1=2
y ÞA
x , n
y Þ2 − ðA
x , n
y , E, + ½½Δ21 ðE, g
x , n g 12 16 "
η ,n
η Þ−A
10
y Þ = γ3 ðE, Δ21 ðE, g
x , n g "
14
x , n
y Þ = A Δ22 ðn


x π n
x d

!2
15 +A


x π n
x d
y π n
y d

!2


y π n
y d


11 −A

!2
17 +A


x π n
x d

!2
13 −A


x π n
x d

!# ,

!#
η Þ
x , n
y , E, , Δ25 ðn g


η ,n
y Þ − Δ24 ðn
x , n
y Þ, = Δ221 ðE, g
x , n
η ,n
x , n
y Þ = ½Δ222 ðE,
y Þ + Δ23 ðn
y Þ Δ24 ðn g
x , n and "
18
y Þ = A Δ23 ðn


y π n
y d

!2

#
2 +A 19

The DOS function in this case can be written as
ymax
xmax n n X
X
η Þ = gv
n
E
−E
200HDNW Þ
1DHDΓ ðE,
x , n
y , ηg ÞHð Δ′27 ðE, N g π n
x = 1 n
y = 1

(2:134)


′200HDNW is the sub-band energy and in this case can be expressed as In (2.164), E
′200HDNW , n
x , n
y , ηg Þ 0 = Δ27 ðE

(2:135)

The EEM in this case is given by
x , n
y Þ =
* ðE′F1HDNW , ηg , n m


2 ∂ 2 ′ h


½Δ27 ðE F1HDNW , ηg , nx , ny Þ 2 ∂E

The carrier statistics in this case can be written as

(2:136)

142

2 The entropy in nanowires of heavily doped materials

g
v π


1D = n

! n

zmax  ymax n
s h i X X X

rÞ Δ27 ðE′F1HDNW , n
x , n
y , ηg Þ
x , n
y , ηg Þ Δ27 ðE′F1HDNW , n Lð
y = 1 n
z = 1 n


r = 1

(2:137) Thus, using (1.31f) and (2.137), we can study the entropy in this case. In the absence of band-tailing, the 1D hole energy spectrum in this case assumes the form
=A
10 E


x π n
x d

" ±


14 A

!2


x π n
x d


11 +A


y π n
y d

!2
15 +A

!2


y π n
y d


12
k2 + A
13 +A z


x π n
x d

!2
16
k2 +A z

!


x π n
x d


17 +A

!!2
18 +A


y π n
y d

!2

#1=2
2 +A 19 (2:138)


′300 ) is the lowest positive root of the following equation The subband energy (E
′300 = A
10 E


x π n
x d

!2


x π n
x d


13 +A

! " ±


14 A


x π n
x d

!2
17 +A


x π n
x d

!!2

#1=2
2 +A 19

(2:139)

(2.139) can be expressed as
kz = Δ271 ðE,
η ,n g
y Þ

(2:140)

where
n
2 − A
2 Þ − 1 ½½Δ211 ðE,
n
12 + A
16 Δ22 ðn
x , n
y Þ = ½ðA
x , n
y ÞA
x , n
y Þ Δ271 ðE, 12 16
n
12 + A
16 Δ22 ðn
2 − A
2 ÞΔ251 ðn
1=2
x , n
y ÞA
x , n
y Þ2 − ðA
x , n
y , EÞ + ½½Δ211 ðE, 12 16 "


x π n
x d


n
−A
10
x , n
y Þ = E Δ211 ðE, "
14
x , n
y Þ = A Δ22 ðn


x π n
x d

!2

!2
15 +A


y π n
y d


11 −A
y π n
y d

!2

!2
17 +A


13 −A


x π n
x d

!# ,

!#


x π n
x d

,


= Δ2 ðE,
n
x , n
y , EÞ
x , n
y Þ − Δ24 ðn
x , n
y Þ, Δ251 ðn 211 "
n
x , n
y Þ = ½Δ222 ðE,
x , n
y Þ + Δ23 ðn
y ÞandΔ23 ðn
y Þ = Δ24 ðn The DOS function in this case can be written as



18 ny π A
y d

!2

#
2 +A 19

143

2.2 Theoretical background


xmax n n ymax X
X
= g
v
n
E
−E
300HDNW Þ
1DΓ ðEÞ
x , n
y ÞHð Δ′271 ðE, N π n
x = 1 n
y = 1

(2:141)


′300 is the sub-band energy in this case which can be expressed as In (2.171), E
′300 , n
x , n
y Þ 0 = Δ271 ðE

(2:142)

The EEM in this case is given by
2 ∂ 2 h
F1d , η , n ′




* ðE , n Þ = m x y g
½Δ271 ðE F1d , nx , ny Þ 2 ∂E

(2:143)

The DOS function in this case can be written as
1D = n


 n
X  
ymax 
s xmax n X X g
v
F1d , n
F1d , n

rÞ Δ271 ðE
x , n
y Þ
x , n
y Þ Δ271 ðE Lð π n
x = 1 n
y = 1
r = 1

(2:144)

Using (1.44) and (2.144) we can study the entropy in this case.

2.2.13 Entropy in nanowires of HD lead germanium telluride The 1D electron energy spectrum in n-type Pb1–xGexTe under the condition of formation of band tails can be written as "

" " !2
x π n


θ0 ðE, ηg Þ + γ3 ðE, ηg Þ Ego − 0.195

x − d 1 + ðηE Þ #

2

g

"

"

= 0.23


x π n
x d

!2


y π n
y d

−

!2 #

"


x π n
x d

#" + 0.377
kz2

" 0.606

!2


x π n
x d

# − 0.345
kz2

#

" !2
x π n
± 0.06Ego + 0.061
x − d
go + 0.0411 + E

!2 #

− 0.02
kz2

"

"


y π n
y d


y π n
y d


y π n
y d

− !2 −

!2 #

#" − 0.0066
kz2


x π n
x d

!2 −


y π n
y d

!2 #1=2

!2 #


y π n
y d

!2

## + 0.722
kz2 (2:145)


n The subband energy (E ) is the lowest positive root of the following equation zHD500

144

2 The entropy in nanowires of heavily doped materials

2 6 4

3 1 + Erf 2

2 E
2

2


7


4 nx π

5θ0 ðE nzHD500 , ηg Þ+ γ3 ðEnzHD500 , ηg Þ½Ego − 0.195
x nzHD500 d ηg


x π n = 40.234
dx

!2 −


y π n
y d

2


x π
go + 0.0614 n ± ½0.06E
x d 2

2


x π
go + 0.04114 n + 4E
x d

!2 −


y π n
y d

!2 3 5

3 !2 3 5 − 0.02
k2 5 z

!2 −

!2 −


y π n
y d


y π n
y d

!2 32 !2
54 n x π −
x d


y π n
y d

2 !2 332 !2
x π n 5540.6064
x − d

!2 31=2 5


y π n
y d

!2 33# 55 (2:146)

The EEM and the DOS function for both the cases should be calculated numerically. Using (1.31f) and (2.145), we can study the entropy numerically. The 1D dispersion law of n-type Pb1 − x Gex Te with x = 0.01 in the absence of band-tails can be expressed as !2 !2 # #
y π
n π n x 2

− 0.606 − 0.722kz E
x − d
y d " # # "
y π 2
n π n x 2 2

+E
go + 0.411 ð E
x Þ − ð d
y Þ + 0.377kz d " " !2 !2 #
y π
x π n n = 0.23 + 0.02
kz2
x − d
y d " " !2 !2 # #" !2
y π

n π π n n x x
go + 0.061 ± 0.06E + 0.0066
kz2
x − d
x −
y d d

"

"


y π n
y d

!2 #1=2

(2:147)
500 in this can be written as The sub-band energy E "

"


500 − 0.606 E "


x π n
x d

!2 −


y π n
y d

"

!2 #"
go + 0.411
500 + E E


x π n
x d

!2 −


y π n
y d

!2 ##

"

" !2 !2 # " !2
y π
x π
x π n n n
= 0.23 ± 0.06Ego + 0.061
x − d
x
y d d !2 ##" !2 !2 #1=2
y π
y π
x π n n n −

x − d
y d dy

(2:148)

2.2 Theoretical background

145

Using (1.44) and (2.147) we can study the entropy numerically. 2.2.14 Entropy in nanowires of HD zinc and cadmium diphosphides The DR in HD NWs of zinc and cadmium diphosphides can be written as 2 !2 !2 3  
kx , n
y , n
z Þ
2
y π
β β ð n π n z 2 31
η Þ= β + 4k + 5 γ3 ðE, g 1 x
z
y + d 8β4 d

(2 ±

!2 !2 !2 33
kx , n



β β ð , n Þ π n π n y z y z 2 31 2 4β β ð
kx , n 4
k + 55
y , n
z Þ β5 − 4 31 x
z
y + d 8β4 d

+8β4

2

!2 !2 3)1=2 ! !2
y , n
z Þ
y , n
z Þ

β31 2 ð
kx , n β31 2 ð
kx , n π n π n y z 4
k2 + 5 1− −β2 1− x
z
y + d 4 4 d (2:149)

where 2

2 2 3
k2 + n
y π − 2 n
z π
y
z 6 x 7 d d
y , n
z Þ = 6 β31 ð
kx , n 2 2  7 4 5
k2 + n
y π + n
z π

x d d y

z


n in this case assumes the form The sub-band energy E zHD600 2 !2 !2 3  



β β ð0, ny , nz Þ 4 ny π nz π 5
n , ηg Þ = β1 + 2 31 + γ3 ðE zHD600
z
y 8β4 d d

(2 ±

!2 !2


y π β β ð0, n , n Þ n y z 2 31 4β β ð0, n 4
y , n
z Þ β5 − + 4 31
y 8β4 d

+ 8β4 2


y , n
z Þ β 2 ð0, n 1 − 31 4

! − β2

!2 33
z π n 55
z d

!2 !2
y , n
z Þ
y π β31 2 ð0, n n 4 1−
y + 4 d

!2 3)1=2
z π n 5
z d (2:150)

where

146

2 The entropy in nanowires of heavily doped materials

2

 3 2
6 n
y π − 2 n

z π 7 6 7 dz dy 7
y , n
z Þ = 6 β31 ð0, n 6"
2
2 #7 4 n
y π 5 + n

z π
dy

dz

The EEM and the DOS function should be obtained numerically. Using (1.31f) and (2.149) we can study the entropy numerically. The 1D DR in NWs of zinc and cadmium diphosphides in the absence of bandtails can be written as 2 !2 !2 3  
kx , n
y , n
z Þ
2
y π
β β ð n π n z 2 311
= β + 4k + 5 E 1 x
z
y + d 8β4 d 82 !2 !2 <
kx , n


y π β β ð , n Þ n y z 2 311 2

4 4
,n
Þ β5 − kx +
± β β ðk , n + : 4 311 x y z 8β4 dy ! !
y , n
z Þ
y , n
z Þ β2311 ð
kx , n β2311 ð
kx , n 2 + 8β4 1 − − β2 1 − 4 4 2 3  1=2 !2 !2

4
k2 + ny π + nz π 5 x
z
y d d

!2 33
z π n 55
z d

(2:151)


700 ) is the lowest positive root of the following equation The subband energy (E 2 !2 !2 3  



β β ð0, n , n Þ π n π n y z 4 y z
700 = β + 2 311 5 E 1
z
y + d 8β4 d

(2 ±

!2 !2
y , n
z Þ
y π β β ð0, n n 2 311 4β β ð0, n 4
y , n
z Þ β5 − 4 311
y + 8β4 d

+ 8β4

2


z π n
z d

!2 ! !2
y , n
z Þ
y , n
z Þ
y π β2311 ð0, n β2311 ð0, n n 4 1− − β2 1 −
y + 4 4 d

!2 33 55


z π n
z d

!2 3)1=2 5 (2:152)

The EEM and the DOS function for both the cases should be calculated numerically. Using (1.44) and (2.151), we can study the entropy numerically.

2.3 Results and discussion

147

2.3 Results and discussion Figures 2.1–2.12 exhibits the dependences of the normalized entropy in HD NWs of all the materials as considered in Chapter 1 in accordance with all the band models and obtained by using the appropriate equations as formulated in this chapter. From Figure 2.1 it appears that the entropy in HD NWs of CdGeAs2 for all the models of the same material exhibit quantized variations with increasing film thickness. For a range of film thickness, the dependence exhibit trapezoidal variations and for higher values of film thickness, the length and width of the trapezoid increases. From Figure 2.2, it appears that the entropy decreases with increasing carrier concentration per unit length and the value of the entropy is least for the generalized band model and greatest for the parabolic band model of the same. From Figure 2.3, we observe that the entropy for HD NWs of InAs exhibits the lowest value in accordance with the three-band model of Kane model of the same whereas for parabolic energy bands it exhibits the highest value. It is apparent from plot (a) of Figure. 2.3 that the influence of the energy band gap is to reduce the value of the entropy as

Normalized entropy

100

10

(e) (d)

1

10

15 (c) (b)

20 30 35 25 Film thickness (in nm)

40

45

50

(a) 0.1

Figure 2.1: Plot of the normalized entropy for HD NWs of CdGeAs2 as a function of film thickness for all cases of Figure 1.1.

Normalized entropy

100

10 Carrier concentration (in ×108 m–1) 1

0

10

20

30

0.1

40

50 (d)

60 (e)

70

80

90 100

(a) 0.01

(b)

(c)

0.001 Figure 2.2: Plot of the normalized entropy for HD NWs of quantum wires of CdGeAs2,as a function of carrier concentration for all cases of Figure 1.1.

148

2 The entropy in nanowires of heavily doped materials

Normalized entropy

100

10 (c)

1

10

(b)

15

20

25

30

35

40

45

50

Film thickness (in nm) (a)

0.1 Figure 2.3: Plot of the normalized entropy for HD NWs of InAs as a function of film thickness for all cases of Figure 1.3.

compared with parabolic energy bands, and the influence of spin orbit splitting constant is to reduce the entropy further in the whole range of thickness as compared with two-band model of Kane. From Figure 2.4, we observe that entropy decreases with increasing concentration and the value of entropy in accordance with all the band models differs widely as concentration increases. Figures 2.5 and 2.6 exhibit the plot of the entropy for HD NWs of InSb as functions of thickness and concentration respectively. Nature of these two figures does not differ as compared with the plot of entropy in HD NWs of InAs as given in Figures 2.3 and 2.4, respectively. Important point to note here is that although the nature of the plots is same, the exact numerical values of the entropy are determined by the numerical values of the energy band constants of InSb and InAs, respectively. From Figure 2.7, we observe that the influence of the splitting of the two spin states by the spin orbit coupling and the crystalline field enhances the numerical values of the entropy in HD NWs of CdS as compared with
λ0 = 0. Besides, trapezoidal variations of entropy in HD NWs of CdS with respect to thickness as appearing from Figure 2.7 are found to be perfect. From Figure 2.8, we observe that entropy decreases with increasing carrier concentration per unit length and by comparing it with Figure 1.8 as the corresponding plot for HD NWs of CdS, we can state that although entropy decreases with increasing carrier degeneracy in the latter case the nature and rate of decrement with increasing concentration are totally different in the HD NWs of CdS. From Figure 2.9, we observe that the entropy for HD NWs of PbTe, PbSnTe and stressed InSb in accordance with the appropriate band models exhibit quantum steps and trapezoidal variations in the whole range of thickness as considered with widely different numerical values as apparent from thisfigure. From Figure 2.10, the entropy decreases with increasing carrier degeneracy for HD NWs of PbTe,PbSnTe and stressed InSb respectively. Figure 2.11 exhibits the plots of the normalized entropy in HD NWs of (a) GaP, (b) PtSb2, (c)Bi2Te3,and (d)cadmium antimonide, respectively, as a function of normalized film thickness.

2.3 Results and discussion

149

Normalized entropy

100 10 1

0

10

20

30 (c)

0.1

40

50

60

70

80

Carrier concentration (in

90

×108

100

m–1)

(b) 0.01

(a)

0.001 Figure 2.4: Plot of the normalized entropy for HD NWs of InAs as a function of carrier concentration for all cases of Figure 1.4.

Normalized entropy

100

10 (c) 1

10

0.1

15 (b)

20

25

30

35

40

45

50

Film thickness (in nm)

(a)

0.01 Figure 2.5: Plot of the normalized entropy for HD NWs of InSb as a function of film thickness for all cases of Figure 1.3.

Normalized entropy

100 10 1

0

10

20

(a)

(b)

0.1

30 40 50 60 70 80 90 100 (c) Carrier concentration (in x108 m–1)

0.01 0.001 0.0001

Figure 2.6: Plot of the normalized entropy for HD NWs of InSb as a function of carrier concentration for all cases of Figure 1.3.

150

2 The entropy in nanowires of heavily doped materials

Normalized entropy

336

100

(b) (a) 10 10

15

20

25

30

35

40

45

50

Film thickness (in nm) Figure 2.7: Plot of the normalized entropy for HD NWs of CdS as a function of film thickness for all cases of Figure 1.7.

Normalized entropy

10,000 1,000 100

(a)

10 1

(b) 0

Carrier concentration (in×108 m–1)

10

20 30 40 50 60 70 80 90 100

0.1

Normalized entropy

Figure 2.8: Plot of the normalized entropy for HD NWs of CdS as a function of carrier concentration for all cases of Figure 1.7.

500 450 400 350 300 250 200 150 100 50 0

12 10 8 6 (a)

(c)

4 2

(b) 0 10

15

20

25

30

35

Film thickness (in nm) Figure 2.9: Plot of the normalized entropy in HD NWs of (a) PbTe, (b) PbSnTe and (c) stressed InSb as a function of film thickness.

2.3 Results and discussion

151

10,000

Normalized entropy

1,000 100

(a)

10 1

0

10

20

30

40

50

60

70

80

0.1

90 (b)

100

(c)

0.01

Carrier concentration (in×108 m–1)

0.001 Figure 2.10: Plot of the normalized entropy in HD NWs of (a) PbTe, (b) PbSnTe and (c) stressed InSb as a function of carrier concentration.

Normalized entropy

1,000

(d)

(c)

100 (a)

(b)

25.5 15

17

19

21

23

25

27

29

31

33

35

Film thickness (in nm) Figure 2.11: Plot of the normalized entropy in HD NWs of (a) GaP, (b) PtSb2, (c) Bi2Te3, and (d) cadmium antimonide respectively, as a function of film thickness.

Normalized entropy

10

(d) (a)

(c)

1 0 10 20 30 40 50 60 70 Carrier concentration (×108 m–1) (b)

80 90 100

0.3 Figure 2.12: Plot of the normalized entropy for HD NWs of (a) GaP, (b) PtSb2, (c) Bi2Te3, and (d) cadmium antimonide respectively, as a function of carrier concentration.

152

2 The entropy in nanowires of heavily doped materials

The influence of 2D quantum confinement appears from Figures 2.1, 2.3, 2.5, 2.7, 2.9 and 2.11, respectively since the entropy depends strongly on the thickness of the quantum-confined materials which is in direct contrast with bulk specimens. The entropy increases with increasing film thickness in an oscillatory way with different numerical magnitudes. It appears from the aforementioned figures that the entropy in HD NWs exhibits spikes for particular values of film thickness which, in turn, is not only the signature of the asymmetry of the wave vector space but also the particular band structure of the specific material. Moreover, the entropy in HD NWs of different compounds can become several orders of magnitude larger than that of the 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 Figures. (2.2), (2.4), (2.6), (2.8), (2.10) and (2.12) that the entropy decreases with increasing carrier degeneracy for 1D quantum confinement as considered for the said Figures. For relatively high values of carrier degeneracy, the influence of band structure of a specific 1D material is large and the plots of entropy differ widely from one another whereas for low values of the carrier degeneracy, they exhibit the converging tendency. For bulk specimens of the same material, the entropy will be found to decrease continuously with increasing electron degeneracy in a non-oscillatory manner in an altogether different way. For HD NWs, the entropy increases with increasing film thickness in a step like manner for all the appropriate Figures. The appearance of the discrete jumps in the Figures for HD NWs 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 the others. With varying thickness, a change is reflected in the entropy through the redistribution of the electrons among the sizequantized levels. It should be noted that although, the entropy varies in various manners with all the variables in all the cases as evident from all the Figures, the rates of variations are totally band- structure dependent. The two different signatures of 2D and 1D quantization of the carriers of in HD NWs of all the materials as considered here are apparent from all the appropriate plots, values of entropy for NWs differ as compared with HD QWs and the nature of variations of the entropy also changes accordingly.

2.4 Open research problems (R2.1) Investigate the entropy for NWs of all of the HD materials in the presences of Gaussian, exponential, Kane, Halperian, Lax and Bonch-Burevich types of band tails for all systems whose unperturbed carrier energy spectra are defined in R1.1. (R2.2) Investigate the entropy in the presence of strain for NWs of all of the HD materials of the negative refractive index, organic, magnetic and other advanced optical materials.

2.4 Open research problems

153

(R2.3) Investigate the entropy for the NWs of HD negative refractive index, organic, magnetic and other advanced optical materials in the presence of an arbitrarily oriented alternating electric field. (R2.4) Investigate the entropy for the multiple NWs of HD materials whose unperturbed carrier energy spectra are defined in R1.1. (R2.5) Investigate the entropy for all the appropriate HD 1D of this chapter in the presence of finite potential wells. (R2.6) Investigate the entropy for all the appropriate HD 1D systems of this chapter in the presence of parabolic potential wells. (R2.7) Investigate the entropy for HD 1D systems of the negative refractive index and other advanced optical materials in the presence of an arbitrarily oriented alternating electric field and non-uniform light waves and in the presence of strain. (R2.8) Investigate the entropy for triangular HD 1D systems of the negative refractive index, organic, magnetic and other advanced optical materials in the presence of an arbitrarily oriented alternating electric field in the presence of strain. (R2.9) Investigate the entropy for all the problems of (R1.1) in the presence of arbitrarily oriented magnetic field. (R2.10) Investigate the entropy for all the problems of (R1.1) in the presence of alternating electric field. (R2.11) Investigate the entropy for all the problems of (R1.1) in the presence of alternating magnetic field. (R2.12) Investigate the entropy for all the problems of (R1.1) in the presence of crossed electric field and quantizing magnetic fields. (R2.13) Investigate the entropy for all the problems of (R1.1) in the presence of crossed alternating electric field and alternating quantizing magnetic fields. (R2.14) Investigate the entropy for HD NWs of the negative refractive index, organic and magnetic materials. (R2.15) Investigate the entropy for HD NWs of the negative refractive index, organic and magnetic materials in the presence of alternating time dependent magnetic field. (R2.16) Investigate the entropy for HD NWs of the negative refractive index, organic and magnetic materials in the presence of in the presence of crossed alternating electric field and alternating quantizing magnetic fields. (R2.17) (a) Investigate the entropy for HD NWs of the negative refractive index, organic, magnetic and other advanced optical materials in the presence of an arbitrarily oriented alternating electric field considering many body effects. (b) Investigate all the appropriate problems of this chapter for a Dirac electron.

154

2 The entropy in nanowires of heavily doped materials

(R2.18) Investigate all the appropriate problems of this chapter by including the many body, image force, broadening and hot carrier effects, respectively. (R2.19) Investigate all the appropriate problems of this chapter by removing all the mathematical approximations and establishing the respective appropriate uniqueness conditions.

References [1]

[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

Harrison P., Wells Q., Wires and Dots (John Wiley and Sons, Ltd, 2002); Ridley B.K., Electrons and Phonons in semiconductors multilayers, Cambridge University Press, Cambridge (1997); Bastard G., Wave mechanics applied to semiconductor heterostructures, Halsted; Les Ulis, Les Editions de Physique, New York (1988); Martin V.V., Kochelap A.A., Stroscio M.A., Quantum Heterostructures, Cambridge University Press, Cambridge (1999). Lent C.S., Kirkner D.J., J. Appl. Phys 67, 6353 (1990); Sols F., Macucci M., Ravaioli U., Hess K., Appl. Phys. Lett., 54, 350 (1980). Kim C.S., Satanin A.M., Joe Y.S., Cosby R.M., Phys. Rev. B 60, 10962 (1999). Midgley S., Wang J.B., Phys. Rev. B 64, 153304 (2001). Sugaya T., Bird J.P., Ogura M., Sugiyama Y., Ferry D.K., Jang K.Y., Appl. Phys. Lett. 80, 434 (2002). Kane B., Facer G., Dzurak A., Lumpkin N., Clark R., PfeiKer L., West K., Appl. Phys. Lett 72, 3506 (1998). Dekker C., Physics Today 52, 22 (1999). Yacoby A., Stormer H.L., Wingreen N.S., Pfeiffer L.N., Baldwin K.W., West K.W., Phys. Rev. Lett 77, 4612 (1996). Hayamizu Y., Yoshita M., Watanabe S., PfeiKer H.A.L., West K., Appl. Phys. Lett 81, 4937 (2002). Frank S., Poncharal P., Wang Z.L., Heer W.A., Science 280, 1744 (1998). Kamiya I., Tanaka I.I., Tanaka K., Yamada F., Shinozuka Y., Sakaki H., Physica E 13, 131 (2002). Geim A.K., Main P.C., LaScala N., Eaves L., Foster T.J., Beton P.H., Sakai J.W., Sheard F.W., Henini M., Hill G., et al., Phys. Rev. Lett 72, 2061 (1994). Melinkov A.S., Vinokur V.M., Nature 415, 60 (2002). Schwab K., Henriksen E.A., Worlock J.M., Roukes M.L., Nature 404, 974 (2000). Kouwenhoven L., Nature 403, 374 (2000). Komiyama S., Astafiev O., Antonov V., Hirai H., Nature 403, 405 (2000). Paspalakis E., Kis Z., Voutsinas E., Terziz A.F., Phys. Rev. B 69, 155316 (2004). Jefferson J.H., Fearn M., Tipton D.L.J., Spiller T.P., Phys. Rev. A 66, 042328 (2002). Appenzeller J., Schroer C., Schapers T., Hart A., Froster A., Lengeler B., Luth H., Phys. Rev. B 53, 9959 (1996). Appenzeller J., Schroer C., J. Appl. Phys 87, 31659 (2002). Debray P., Raichev O.E., Rahman M., Akis R., Mitchel W.C., Appl. Phys. Lett 74, 768 (1999). Solomon P.M., Proc. IEEE 70, 489 (1982); Schlesinger T.E., Kuech T., Appl. Phys. Lett. 49, 519 (1986). Kasemet D., Hong C.S., Patel N.B., Dapkus P.D., Appl. Phys. Letts 41, 912 (1982); Woodbridge T., Blood P., Pletcher E.D., Hulyer P.J., Appl. Phys. Lett. 45, 16 (1984); Tarucha S., Okamoto H.O., Appl. Phys. Letts. 45, 16 (1984); Heiblum H., Thomas D.C., Knoedler C.M., Nathan M.I., Appl. Phys. Letts. 47, 1105 (1985).

References

155

[24] Aina O., Mattingly M., Juan F.Y., Bhattacharyya P.K., Appl. Phys. Letts 50, 43 (1987). [25] Suemune I., Coldren L.A., IEEE, J. Quant. Electronic 24, 1178 (1988). [26] Miller D.A.B., Chemla D.S., Damen T.C., Wood J.H., Burrus A.C., Gossard A.C., Weigmann W., IEEE, J. Quant. Electron 21, 1462 (1985). [27] Blakemore J.S., Semiconductor Statistics (Dover, New York, 1987); Ghatak K.P., Bhattacharya S., Biswas S.K., Dey A., Dasgupta A.K., Phys. Scr., 75, 820, (2007).

3 Entropy in quantum box of heavily doped materials A person is measured by not what he says or does, but by what he becomes.

3.1 Introduction It is well known that as the dimension of the QWs increases from 1D to 3D, the degree of freedom of the free carriers decreases drastically and the density-of-states function changes from the Heaviside step function in OWs to the Dirac’s delta function in quantum box (QB) [1]. QBs can be used for visualizing and tracking molecular processes in cells using standard fluorescence microscopy [2–4]. They display minimal photobleaching [5], thus allowing molecular tracking over prolonged periods and consequently, single molecule can be tracked by using optical fluorescence microscopy [6]. The salient features of quantum dot lasers [7] include low threshold currents, higher power, and great stability as compared with the conventional one, and the QBs find extensive applications in nanorobotics [8], neural networks [9], and high-density memory or storage media [10]. QBs are also used in nanophotonics [11] because of their theoretically high quantum yield and have been suggested as implementations of Q-bits for quantum information processing [12]. QBs also find applications in diode lasers [13], amplifiers [14], and optical sensors [15]. High-quality QBs are well suited for optical encoding [16] because of their broad excitation profiles and narrow emission spectra. The new generations of QBs have far-reaching potential for the accurate investigations of intracellular processes at the single-molecule level, high-resolution cellular imaging, long-term in vivo observation of cell trafficking, tumor targeting, and diagnostics [17]. QB nanotechnology is one of the most promising candidates for use in solid-state quantum computation [18]. It may also be noted that QBs are being used in single electron transistors [19], photovoltaic devices [20], photoelectrics [21], ultrafast all-optical switches and logic gates [22], organic dyes [23], and other types of nano devices [24–29]. In this chapter in Sections 3.2.1–3.2.14, we have investigated the entropy in QBs of HD nonlinear optical, III–V, II–VI, stressed Kane-type, Te, GaP, PtSb2, Bi2Te3, Ge, GaSb, II–V, lead germanium telluride, zinc and cadmium diphosphides, respectively. Section 3.3 contains the result and discussions pertaining to this chapter. Section 3.4 presents 22 open research problems.

https://doi.org/10.1515/9783110661194-003

158

3 Entropy in quantum box of heavily doped materials

3.2 Theoretical background 3.2.1 Entropy in QB of HD nonlinear optical materials The dispersion relation in this case can be written following (2.1) as
y Þ2
z Þ2
x Þ2
y π=d
z π=d
x π=d
h2 ðn
2 ðn h
h2 ðn + +
1QBHD , η Þ 2m
1QBHD , η Þ 2m
1QBHD , η Þ = 1
21 ðE
22 ðE
21 ðE
*jj T
*? T
*jj T 2m g g g

(3:1)


1QBHD is the totally quantized energy in this case. where E The total density-of-states function in this case is given by
xmax n
ymax n n zmax X
X
X
= 2g v
−E
1QBHD Þ
0DT ðEÞ δ′ðE N
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:2)


−E
1QBHD Þ is the Dirac’s Delta function. where, δ′ðE Using (3.2) and Fermi–Dirac occupation probability factor, the total electron concentration can be written as
xmax n
zmax n ymax n X
X X 2g
v
− 1 ðη
0D = Re al Part of F n 31HD Þ


dx dy dz


nx = 1 ny = 1 nz = 1

(3:3)

where
F0DHD − E
1QBHD Þ
− 1 ðE η31HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.3) we can study the entropy in this case. For the purpose of comparison we shall also study the entropy in the absence of band tails in this case.
n ði =
x,
y and
zÞ be the quantized energy levels due to infinitely deep Let E i
i ð = 1, 2, 3, ...Þ as the size quantum numbers. potential well along ith-axis with n Therefore, from (1.2), one can write
x
n Þ πn
n Þ =
f1 ðE γðE x x
dx

n Þ =
f1 ðE
n Þ π ny γðE y y
dy
z
n Þ =
f2 ðE
n Þ πn γðE z z
dz

!2 (3:4) !2 (3:5) !2


QD1 Þ can be expressed as From (1.2), the totally quantized energy ðE

(3:6)

3.2 Theoretical background

2





x
QD1 Þ =
f1 ðE
QD1 Þ4 πn γðE
dx

2

!2 3 "
 #
y
z 2 πn πn

+
5 + f2 ðEQD1 Þ
z dy d

159

(3:7)

The total density-of-states function in this case is given by
xmax n
ymax n n zmax X
X
X
= 2g v
−E
QD1 Þ
0DT ðEÞ δ′ðE N
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:8)

The total electron concentration in this case can be written as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0D = n F 31
x d
y d
z d


nx = 1 ny = 1 nz = 1

(3:9)

where
F0D − E
QD1 Þ
− 1 ðE η31 ≡ ð
kB TÞ
F0D is the Fermi energy in this case. and E Using (1.44) and (3.9), we can find the entropy in this case. 3.2.2 Entropy in QB of HD III–V materials The dispersion relation of the conduction electrons of III–V materials are described by the models of Kane (both three and two bands together with parabolic energy band), Stillman et al. and Palik et al., respectively. For the purpose of complete and coherent presentation, the entropy in QBs of HD III–V compounds have also been investigated in accordance with the aforementioned different dispersion relations for relative comparison as follows: (a) The three-band model of Kane The dispersion relation in this case can be written as
y Þ2
h2 ðn
z Þ2 h
x Þ2
y π=d
z π=d
x π=d
2 ðn
2 ðn h
44 ðE
2QBHD , η Þ + + =T g
c
c
c 2m 2m 2m

(3:10)

where
2QBHD , η Þ = T
31 ðE
2QBHD , η Þ +
iT
31 ðE
2QBHD , η Þ
44 ðE T g g g
2QBHD is the totally quantized energy in this case. and E The total electron concentration can be written as
xmax n
zmax n ymax n X
X X 2g
v
− 1 ðη Þ
0D = Re al Part of F n 31


dx dy dz
x = 1 n
y = 1 n
z = 1 n

(3:11)

160

3 Entropy in quantum box of heavily doped materials

where
F0 DHD − E
2QBHD Þ
− 1 ðE η32 HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.11) we can study the entropy in this case.
n and E
n along x, y, and z directions, re
n , E The quantized energy levels (E x y z spectively) in the absence of band tails in QBs of III–V materials in accordance with the three-band model of Kane can be expressed as
2
n Þ = h
I11 ðE x
c 2m


x πn
dx


2
n Þ = h
I11 ðE y
c 2m


y πn
dy

!2 (3:12)

!2 (3:13)

and  2

z 2 πn
I11 ðE
n Þ =
h z
z
c d 2m
QD2 Þ in this case assumes the form The totally quantized energy ðE 2 !2 !2 !2 3 2 2


n h
π n n y z 5
QD2 Þ =
I11 ðE 4 x +
x
y + d
z
c 2m d d

(3:14)

(3:15)

The electron concentration in this case is given by
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0D = n F 32
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:16)

where
F0D − E
QD2 Þ
− 1 ðE η32 ≡ ð
kB TÞ
F0D is the Fermi energy in this case. and E Using (1.44) and (3.16) we can study the entropy in this case. (b) The two band model of Kane The dispersion relation in this case can be written following (2.24) as
y Þ2
h2 ðn
z Þ2 h
x Þ2
y π=d
z π=d
x π=d
2 ðn
2 ðn h
3 QBHD , η Þ + + = γ2 ðE g
c
c
c 2m 2m 2m and

(3:17)

3.2 Theoretical background

161


3QBHD is the totally quantized energy in this case. E The total electron concentration can be written as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη
0D = F n 33 HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:18)

where
F0DHD − E
3QBHD Þ
− 1 ðE η33 HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.18) we can find the entropy in this case.
n obeys the equation For two-band model of Kane in the absence of bandtails, E z
2
z
2 π2 n h

Enz ð1 + αEnz Þ =
z
c d 2m

(3:19)


QD3 Þ in this case is given by The totally quantized energy ðE 2 !2 !2 !2 3 2 2



QD3 Þ =
h π 4 nx + ny + nz 5
QD3 ð1 + αE E
x
y
z
c 2m d d d

(3:20)

The electron concentration in this case is given by
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
n0D =


F 33 dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:21)

where
F0D − E
QD3 Þ
− 1 ½ðE η33 ≡ ð
kB TÞ
F0D is the Fermi energy in this case. and E Using (1.44) and (3.21), we can find the entropy in this case. (c) The parabolic energy bands The dispersion relation in this case can be written as
y Þ
z Þ
x Þ
y π=d
z π=d
x π=d
2 ðn h
2 ðn h
h2 ðn
4QBHD , η Þ + + = γ3 ðE g
c
c
c 2m 2m 2m 2

2

2


4QBHD is the totally quantized energy in this case. and E The total electron concentration can be written as

(3:22)

162

3 Entropy in quantum box of heavily doped materials


xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη
0D = F n 34 HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:23)

where
F0DHD − E
4QBHD Þ
− 1 ðE η34HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.23) we can find the entropy in this case.
n and total electron
0D , E In the absence of bandtails, the expressions for Δn z
0D Þ, for QBs of wide-gap materials can, respectively, be written as concentration ðn 2g
v
 ′
0D = Δn

y d
z F − 1 η dx d
z π n
z d


2
n = h E z
c 2m

(3:24)

!2 (3:25)


xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη′Þ
0D = n F
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:26)

where, 2

!2 3
nz π 5
z d

2


− 1 4E
F0D −
h η′ ≡ ð
kB TÞ
c 2m


F0D is the Fermi energy in this case. and E Using (1.44) and (3.26) we can find the entropy in this case. (d) The Model of Stillman et al. The dispersion relation of the electrons in this case can be written as
y Þ
z Þ
x Þ
y π=d
z π=d
x π=d
2 ðn h
2 ðn h
h2 ðn
5QBHD , η Þ + + = θ4 ðE g
c
c
c 2m 2m 2m 2

2

2

(3:27)


5QBHD is the totally quantized energy in this case. and E The total electron concentration can be written as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη
0D = F n 35 HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

where

(3:28)

3.2 Theoretical background

163


F0DHD − E
5QBHD Þ
− 1 ðE η35 HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.28) we can find the entropy in this case.
n obeys the equation In the absence of band tails, E z
z Þ2 ≡
I12 ðE
n Þ
c Þðπn
z =d ð
h2 =2m z

(3:29)


QD5 in this case can be defined as E 2 !2
 3 2 2
2


z 2
QD5 Þ =
h π 4 nx + ny + n
I12 ðE 5
x
y
z
c 2m d d d

(3:30)

The electron concentration in this case is given by
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
n0D =


F 35 dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:31)

where
F0D − E
5 QDS Þ
− 1 ½ðE η35 ≡ ð
kB TÞ
F0D is the Fermi energy in this case. and E Using (1.44) and (3.31), we can find the entropy in this case. (e) The model of Palik et al. The dispersion relation of the electrons in this case can be written as
y Þ
z Þ
x Þ
y π=d
z π=d
x π=d
2 ðn h
2 ðn h
h2 ðn
6QBHD , η Þ + + = θ5 ðE g
c
c
c 2m 2m 2m 2

2

2

(3:32)


6QBHD is the totally quantized energy in this case. and E The total electron concentration can be written as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη
0D = F n 36 HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

where
F0DHD − E
6QBHD Þ
− 1 ½ðE η35 ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.33), we can study the entropy in this case.

(3:33)

164

3 Entropy in quantum box of heavily doped materials


n and E
QD7 are defined by the following equations: In the absence of band tails, E z
2
z
2 πn
n Þ = h
I13 ðE z

2 mc d z 2 !2
 3
2 2
y

z 2 π n
h π n πn x
QD7 Þ =
I13 ðE 4 5 +
+


c 2m dx dy dz

(3:34)

(3:35)

The electron concentration in this case is given by
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0D = n F 37
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

where η37 ≡

(3:36)


F0D − E
5QDS Þ ðE
kB T



F0D is the Fermi energy in this case. and E Using (1.44) and (3.36), we can study the entropy in this case.

3.2.3 Entropy in QB of HD II–VI materials The 0D electron dispersion law in QB of HD II–VI materials can be written following (2.56) as 2

2 !2 3 !2 31=2
2
2
z Þ2
y π

z π=d

π n n π π
2 ðn h n n y x x

4 5 4 5 ′ ± λ0 + γ3 ðE7QBHD , ηg Þ = a 0
x + d
x + d
y
y
*k 2m d d (3:37)
7QBHD is the totally quantized energy in this case. where E The total electron concentration can be written as
xmax n
zmax n ymax n X
X X g
v
− 1 ðη
0D = F n 37HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:38)

where
F0 DHD − E
7 QBHD Þ
− 1 ðE η37HD ≡ ð
kB TÞ Using (1.31f) and (3.38), we can study the entropy in this case.
QD10, ± in this case can In the absence of band tails the totally quantized energy E be expressed as

3.2 Theoretical background

2


QD10, E ±





x πn = a′0 4
dx

2

165

2 !2 3 !2 31=2


y
y
z 2

x 2 πn π n 1
h π n π n +
5+ ± λ0 4
+
5
z
*k d 2m dy dx dy (3:39)

The electron concentration can be written as
xmax n
zmax n ymax n X
X X g
v
− 1 ðη
0D = F n 40, ± Þ


dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:40)

where 1

QD10, ±  −E η40, ± ≡
½E
F0D kB T Using (1.44) and (3.40), we can study the entropy in this case.

3.2.4 Entropy in QBof HD IV–VI materials (a) Dimmock Model In this case, the dispersion relation of the electrons can be written following (2.65) as
z π

n


z = T36 ðE8QBHD , ηg , nx , ny Þ d

(3:41)


8QBHD is the totally quantized energy in this case. where E The total electron concentration can be written as
xmax n
ymax n n zmax X
X g
v X
− 1 ðη
0D = F n 38HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:42)

where
F0DHD − E
8QBHD Þ
− 1 ðE η38HD ≡ ð
kB TÞ Using (1.31f) and (3.42), we can study the entropy in this case. In the absence of band tailing, the electron dispersion relation in this case can be written following (2.71) as
z π

n


z = T40 ðEQD11 , nx , ny Þ d
QD11 is the totally quantized energy in this case. where E

(3:43)

166

3 Entropy in quantum box of heavily doped materials

The electron concentration per band can be written as
xmax n
zmax n ymax n X
X X g
v
− 1 ðη Þ
0D = ½F n 41


dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:44)

where 1

QD11  −E η41 ≡
½E
F0D kB T Using (1.44) and (3.44), we can study the entropy in this case. (b) Bangert and Kastner Model The electron dispersion relation in this case is given by following (2.77) as
z π n




z = T60 ðE9QBHD , ηg , nx , ny Þ d

(3:45)


9QBHD is the totally quantized energy in this case. where E The total electron concentration can be written as
xmax n
ymax n n zmax X
X g
v X
− 1 ðη
F n0D =


39HD Þ dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:46)

where
F0DHD − E
9QBHD Þ
− 1 ðE η39 HD ≡ ð
kB TÞ Using (1.31f) and (3.46), we can study the entropy in this case. In the absence of bandtails following (2.83) the dispersion relation is given by
z π

n


z = T61 ðE12QD , ηg , nx , ny Þ d

(3:47)


12QD is the totally quantized energy in this case. where E The total electron concentration can be written as
xmax n
zmax n ymax n X
X X g
v
− 1 ðη Þ
0D = F n 42


dx dy dz n
x = 1 n
y = 1 n
z = 1

where
F0D − E
12QD Þ
− 1 ðE η42 ≡ ð
kB TÞ
F0D is the Fermi energy in this case. and E Using (1.44) and (3.48), we can study the entropy in this case.

(3:48)

3.2 Theoretical background

167

3.2.5 Entropy in QB of HD stressed Kane-type materials The electron dispersion relation in this case is given by following (2.89) as
z π

n


z = T70 ðE10 QBHD , ηg , nx , ny Þ d

(3:49)


10QBHD is the totally quantized energy in this case. where E The total electron concentration can be written as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη
0D = F n 40 HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:50)

where
F0DHD − E
10QBHD Þ
− 1 ðE η40HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.50) we can study the entropy in this case.
QD23 in this case asIn the absence of band-tails, the totally quantized energy E sumes the form !2

2 − 2
y

x 2 

z πn πn
QD23 Þ− 2 + πn
QD23 Þ− 2 = 1
0 ðEQD23 Þ + ½ b ð E ½
c0 ðE a 0


dx dy dz

(3:51)

The electron concentration is given by
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0D = F n 53
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:52)

where
F0D − E
QD23 Þ
− 1 ðE η53 ≡ ð
kB TÞ
F0D is the Fermi energy in this case. and E Using (1.44) and (3.52), we can study the entropy in this case.

3.2.6 Entropy in QB of HD Te The 0D dispersion relation may be written in this case as
x π

n


x = T72 ðE11QBHD , nx , ny , ηg Þ d
11QBHD is the totally quantized energy in this case. where E

(3:53)

168

3 Entropy in quantum box of heavily doped materials

The total electron concentration can be written a
xmax n
zmax n ymax n X
X X g
v
− 1 ðη
0D = n F 42 HD Þ


dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:54)

where
F0DHD − E
12QBHD Þ
− 1 ðE η42HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.54) we can study the entropy in this case. In the absence of doping, the totally quantized energy can be written as 2 2 !2 3 2
 !2 331=2

2
2 2





π n π n π n π n π n π n y 5 4 2 y 55 z x z x
QD14, = ψ 4 ± ψ3
+ ψ24 4
+
E ± 1
z +ψ2
x + d
y d d dz dx dy (3:55) The electron concentration is given by
xmax n
zmax n ymax n X
X X g
v
− 1 ðη
n0D =


F 44, ± Þ dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:56)

where
F0D − E
QD14, ± Þ=ð
kB TÞ
η44, ± ≡ ðE
F0D is the Fermi energy in this case. and E Using (1.44) and (3.56) we can study the entropy in this case.

3.2.7 Entropy in QB of HD gallium phosphide The 0D dispersion relation may be written in this case following (2.109) as
x π n




x = u70 ðE14 QBHD , ny , nz , ηg Þ d

(3:57)


14QBHD is the totally quantized energy in this case. where E The total electron concentration can be written as
xmax n
ymax n n zmax X
X g
v X
− 1 ðη
0D = F n 44HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

where

(3:58)

3.2 Theoretical background

169


F0DHD − E
14QBHD Þ
− 1 ðE η44HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.58), we can study the entropy in this case.
QD16 ) In the absence of doping, the totally quantized energy (E be written as 2 2 !2 !2 3 !2 !2 2 2

y

x π n π n h
π n h
π n y x
4 5+ 4 EQD16 =
x + d
y
x + d
y +
*?
*jj 2m 2m d d 2

2 !2 4
2
k
h 0 4 π nx 4 −
x +
*2 m d jj


y πn
dy

!2 +


z πn
dz

in this case can


z πn
dz

!2 3 5

31=2 !2 3 5 + jVG j2 5 + jVG j

(3:59)

The electron concentration assumes the form
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðsη Þ
0D = n ½F 46
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:60)

where 1

QD16 Þ η46 ≡
ðE −E
F0D kB T
F0D is the Fermi energy in this case. and E Using (1.44) and (3.60), we can study the entropy in this case.

3.2.8 Entropy in QB of HD platinum antimonide The 0D dispersion relation may be written in this case following (2.119) as
x π n




x = V70 ðE15QBHD , ny , nz , ηg Þ d

(3:61)


15QBHD is the totally quantized energy in this case. where E The total electron concentration can be written as
xmax n
ymax n n zmax X
X g
v X
− 1 ðη
F n0D =


45HD Þ dx dy dz n
x = 1 n
y = 1 n
z = 1

where
F0DHD − E
15QBHD Þ
− 1 ðE η45HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.62), we can study the entropy in this case.

(3:62)

170

3 Entropy in quantum box of heavily doped materials

In the absence of band tailing the 0D dispersion relation in this case can be written using (2.124) as
x π

n


x = D71 ðEQD17 , ny , nz Þ d

(3:63)


1QD17 is the totally quantized energy in this case. where E The total electron concentration can be written as
xmax n
ymax n n zmax X
X g
v X
− 1 ðη Þ
0D = F n 50
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:64)

where
F0D − E
QD17 Þ
− 1 ðE η50 ≡ ð
kB TÞ
F0D is the Fermi energy in this case. and E Using (1.44) and (3.64) we can study the entropy in this case.

3.2.9 Entropy in QB of HD bismuth telluride The dispersion relation in this case can be written as
x π

n


x = J70 ðE18 QBHD , ny , nz , ηg Þ d

(3:65)


18QBHD is the totally quantized energy in this case. where E The total electron concentration can be written as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη
F n0D =


48 HD Þ dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:66)

where
F0DHD − E
18QBHD Þ
− 1 ðE η48HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.66), we can study the entropy in this case. The dispersion relation of the conduction electrons in Bi2Te3 in the absence of doping can be written as
2 2 2 2
+ αEÞ
= h ðα
k + α22
ky + α33
kz + 2α23
ky
kz Þ Eð1
0 11 x 2m where α11 , α22 , α33 and α23 are spectrum constants.

(3:67)

171

3.2 Theoretical background


QD33 can be written as The totally quantized energy E 2 !2 !3
2
2 2 2




π n π n n h
π n π n y y z 5 x z
QD33 Þ =
QD33 ð1 + αE 4α E
x + α22 d
y + α33 d
z + 2α23 d
y d
z
0 11 d 2m (3:68) The hole concentration is given by
xmax n
ymax n n zmax X
X 2g
v X
0D = p ½1 + expðη62 Þ − 1
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:69)

where 1

QD33  η62 =
½E −E
F0D kB T
F0D is the Fermi energy in this case. and E Using (1.44) and (3.69), we can study the entropy in this case.

3.2.10 Entropy in QB of HD germanium (a) Model of Cardona et al. The dispersion relation in accordance with this model in the present case can be written following (2.138) as
x π

n


x = L70 ðE20QBHD , ny , nz , ηg Þ d

(3:70)


20QBHD is the totally quantized energy in this case. where E The total electron concentration can be written as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη
0D = F n 50HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:71)

where
F0DHD − E
20QBHD Þ
− 1 ðE η50HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.71) we can study the entropy in this case.
QD30 in this case can be In the absence of doping the totally quantized energy E written as

172

3 Entropy in quantum box of heavily doped materials

2 !8
!2 931=2 2 2 < πn = 2
2
2
g E


E π n h
π n h
g y z x
g
QD30 = − 0 + 4 0 +E 5 + + E 0
z
x
y ;
*k d
*? : d 2 4 2m 2m d

(3:72)

The electron concentration assumes the form
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0D = ½F n 42
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:73)

where 1

QD30 Þ −E η42 ≡
ðE
F0D kB T
F0D is the Fermi energy in this case. and E Using (1.44) and (3.73) we can study the entropy in this case. (b) Model of Wang and Ressler The dispersion relation in accordance with this model in the present case can be written following (2.148) as
x π n



x = β70 ðE24QBHD , ny , nz , ηg Þ d

(3:74)


24QBHD is the totally quantized energy in this case. where E The total electron concentration can be written as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη
0D = ½F n 54HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:75)

where
F0DHD − E
24QDBHD Þ
− 1 ðE η54HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.75) we can study the entropy in this case. In the absence of doping, the totally quantized energy EQD40 in this case is given by 2
z
QD40 =
h πn E *

2mjj dz


11 −d

!2

8 8 !2 !2 9 !2 !2 9
y =
y =
x
x πn πn
h2 < πn
h2 < πn
+ Þ
x + d
y ; − c1 ð2m
x + d
y ;
*? : d
*? : d 2m

!8 !2
x
h2 < πn
x +
*? : d 2m

!2 9 ! !2 !2 !4
y =
h2
z
z πn πn
h2 πn
y ; 2m
z − e1 2 m
z
*jj
*jj d d d (3:76)

3.2 Theoretical background

173

The electron concentration assumes the form
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0D = ½F n 50
x d
y d
z d


nx = 1 ny = 1 nz = 1

(3:77)

where 1

QD40 Þ −E η50 ≡
ðE
F0D kB T
F0D is the Fermi energy in this case. and E Using (1.44) and (3.77) we can study the entropy in this case.

3.2.11 The entropy in quantum box (QB) of HD gallium antimonide The dispersion relation of the 0D electrons in this case can be written following (2.158) as
y Þ2
h2 ðn
d
x πÞ
z Þ2 h
y π=d
z π=d
2 ðn
2 ðn h x
30QBHD , η Þ + + =
I36 ðE g
c
c
c 2m 2m 2m 2

(3:78)


30QBHD is the totally quantized energy in this case. where E The total electron concentration can be written as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη
½F n0D =


60HD Þ dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:79)

where
F0DHD − E
30QBHD Þ
− 1 ðE η60HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.79) we can study the entropy in this case. In the absence of band tails, the dispersion relation in this case can be written as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 
= α9
k2 + Eg1 1 + α10
k − 1 (3:80) E 2 ! 
h2 ð2
h2 Þ 1 1 where α9 = and α10 =
−
0
c m
0 2m m ðEg1 Þ From (3.80), we get

k2 = E + α11 − ½α12 E
+ α13 1=2 α9 where

(3:81)

174

3 Entropy in quantum box of heavily doped materials

α11 =


2  E g1 8α29

! " # 
2
4 E E 10α29 8α9 α10 4α9 g1 g1 2
12 = α10 +
, α α +
2 −
, α13 = 64α49 10 E α39 Eg1 Eg1 g1


QD60 assumes the form The totally quantized energy E 2 !2
 3
2
y

z 4 π n π n πn x
QD60 = α9 4 5 + + E
x
y
z d d d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 2v 2 u !2
 3
 u 2 2
g1 6u
y
x
z πn πn πn E 5 − 17 + +
+
4t1 + α10 4
5 2 dx dy dz

(3:82)

The electron concentration is given by
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0D = n F 70
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:83)

where
F0D − E
QD60 Þ=ð
kB TÞ
η70 ≡ ðE
F0D is the Fermi energy in this case. and E Using (1.44) and (3.83) we can study the entropy in this case. 3.2.12 The entropy in quantum box (QB) of HD II–V materials The DR of the holes in QDs of HD II–V compounds can be expressed as !2

2

y π
x π 2

22 n n



12 nz π + A
13 nx π + A11
+A γ3 ðE100QBHD , ηg Þ = A10

z
x dx d d dy 20

1 31=2 !2 !
2
2
 2




π π n n π π π n n n y z x
14 x



18 y
2 5 A+A ± 4@ A
x + A15 d
z + A17 d
x
y + A16 d
y + A19 d d (3:84)
100QBHD , η is the totally quantized energy in this case. where E g The DOS function is given by
xmax n
zmax n ymax n X
X X

−E
100QBHD, ± Þ
D = gv δðE N


dx dy dz n
x = 1 n
y = 1 n
z = 1

The electron concentration can be expressed as

(3:85)

175

3.2 Theoretical background


xmax n
ymax n n zmax X
X g
v X
− 1 ðη
0D = F n 601HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:86)

where
F0DHD − E
10QBHD, ± Þ
− 1 ðE η601HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.86) we can study the entropy in this case. In the absence of band-tailing, the 0D hole energy spectrum in this case assumes the form
QD70,± E

!2
2
2

y π

z π
x π 22 n π n n n x



= A10
+ A11
+ A12
+ A13
dx dz dx dy

20

1 31=2 !2 !
2
2
 2
y π
y π


n n π π π n n n x z x 2
14



18
5 A+A ± 4@ A
x + A15 d
z + A17 d
x
y + A16 d
y + A19 d d (3:87)
QD70, is the totally quantized energy in this case. where E ± The DOS function is given by
xmax n
zmax n ymax n X
X X
= g
v
−E
QB70HD, ± Þ
0DT ðEÞ δðE N


dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:88)

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X g
v X
− 1 ðη
n0D =


F 602HD Þ dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:89)

where
F0D − E
QD70, ± Þ
− 1 ðE η602HD ≡ ð
kB TÞ
F0D is the Fermi energy in this case. and E Using (1.44) and (3.89) we can study the entropy in this case.

3.2.13 The entropy in quantum box (QB) of HD lead germanium telluride The 0D electron energy spectrum in n-type Pb1-xGexTe under the condition of formation of band tails can be written as

176

3 Entropy in quantum box of heavily doped materials

" 1 + Erf 2




#

2

E101QBHD, ± ηg


101QBHD,

θ0 ðE

±


101QBHD, ± , η Þ , ηg Þ + γ3 ðE g

!2 3
2
2
y π

n π n x
go  0.1954 4E 5  0.345 nz π +

z
dx d dy 2

2

2





x π n = 40.234
dx

2

3 !2 3
2
y π
z π n n 5 +
5 + 0.02
dz dy

2

2





x π
go + 0.0614 n ± 40.06E
x d 2

2


x π
go + 0.4114 n + 4E
x d 2

2


40.6064 nx π
x d

!2 +

!2 +


y π n
y d

2

32 !2 3 !2
2
y π
z π
x π n n n 54 +
5 + 0.0066

x + dz d dy

!2 31=2
y π n 5
y d

!2 3 !2 3
y π
z π n n 5 + 0.377 5
z
y d d

!2 #

!2 33
z π n 55 + 0.722
dz (3:90)


101QBHD, ± is the quantized energy in this case. where E The DOS function is given by
xmax n
zmax n ymax n X
X X
= g
v
−E
101QBHD, ±Þ
0DT ðEÞ δðE N


dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:91)

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X g
v X
− 1 ðη
0D = n F 603HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:92)

where
F0DHD − E
101QBHD, ± Þ
− 1 ðE η603HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.92) we can study the entropy in this case. The 0D dispersion law of n-type Pb1-xGexTe with x = 0.01 in the absence of band-tails can be expressed as

177

3.2 Theoretical background

2

2


QD70,± 4E 2

2


QD71, ± 4E 2

3 !2 3

2
y π
x π 2
z π n n n 5 − 0.6064
+
5 − 0.722
dx dz dy 3 !2 3
2
2
y π

n π π n n x z
go + 0.4114 5 + 0.377 5 +E
x + d
z
y d d

2

2 2 !2 3 !2 3

2
2
y π
y π
x π 2
z π
n n π n n n x
40.234 5 + 0.02 4 4 5
x + d
z ± 0.06Ego + 0.061
x + d
y
y d d d 2 !2 31=2
2 #

y π
z π
x π 2 n n n 4 5 + 0.0066
+
(3:93)
x dz d dy
QD71, ± is the totally quantized energy in this case. where E The DOS function is given by
xmax n
zmax n ymax n X
X X

= gv
−E
QD71, ±Þ
0DT ðEÞ δ′ðE N
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:94)

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X g
v X
− 1 ðη
0D = n F 604HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:95)

where
F0D − E
QD71, ± Þ
− 1 ðE η604HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.44) and (3.95) we can study the entropy in this case.

3.2.14 The entropy in quantum box (QB) of HD zinc and cadmium diphosphides The DR in HD QDs of Zinc and Cadmium diphosphides can be written as 2 !2 !2 !2 3  
x , n
y , n
z Þ
y π

β β ð n n π π n n x z 2 31
102QBHD, ± , η Þ = β + 4 5 γ3 ðE g 1
x + d
z
y + d 8β4 d 82 2 !2
 33

2 <





z π 2 β β ð n , n , n Þ π n π n n y 55
x , n
y , n
z Þ β5 − 2 31 x y z 4 x ± 4β4 β31 ðn + +
x
z
y : 8β4 d d d

178

3 Entropy in quantum box of heavily doped materials

+ 8β4 2




2


4 nx π
x d

! !
x , n
y , n
z Þ
x , n
y , n
z Þ β31 2 ðn β31 2 ðn 1− − β2 1 − 4 4 2

) !2
 3 1=2 2

ny π nz π 5 +
+
dz dy

(3:96)

where 2

3 2 2
π 2 n ðn
d
x πÞ + ð d
y Þ − 2ðn
d
z πÞ 7 x y z
x , n
y , n
z Þ = 6 i5 β31 ðn 4h
y π 2 n
x π 2
z π 2 n n ð d
Þ + ð d
Þ + ð d
Þ x

y

z


102QBHD, ± is the totally quantized energy in this case. and E The DOS function is given by
xmax n
ymax n n zmax X
X g
v X

−E
102QBHD, ±Þ
δ′ðE N0DT ðEÞ =


dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:97)

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X g
v X
− 1 ðη
0D = n F 605HD Þ
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(3:98)

where
F0DHD − E
102QBHD, ± Þ
− 1 ðE η605HD ≡ ð
kB TÞ
F0DHD is the Fermi energy in this case. and E Using (1.31f) and (3.98) we can study the entropy in this case The 0D DR in QDs of Zinc and Cadmium diphosphides in the absence of bandtails can be written as 2 !2
 3  
2





z π 2 β β ð n , n , n Þ π n π n n y
72QB,± ,η = β + 2 31 x y z 4 x 5 + + E g 1
x
z
y 8β4 d d d 82 2 !2
 33

2 <





z π 2 β β ðn , n , n Þ ny π nπ n 55
x , n
y , n
z Þ β5 − 2 31 x y z 4 x ± 4β4 β31 ðn + +
x
z
y : 8β4 d d d +8β4

2

!2
 3)1=2 ! ! 2

x , n
y , n
z Þ
x , n
y , n
z Þ
y π
x π 2 n
z π 2 β31 2 ðn β31 2 ðn n n 4 5 1− + + −β2 1−
x
z
y 4 4 d d d (3:99)


QD72, ± is the quantized energy in this case. where E

3.3 Results and discussion

179

The DOS function is given by
xmax n
zmax n ymax n X
X X
= g
v
−E
QD72, ± Þ
0DT ðEÞ δ′ðE N


dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:100)

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X g
v X
− 1 ðη
n0D =


F 606HD Þ dx dy dz n
x = 1 n
y = 1 n
z = 1

(3:101)

where
F0D − E
QD72, ± Þ
− 1 ðE η606HD ≡ ð
kB TÞ
F0D is the Fermi energy in this case. and E Using (1.44) and (3.101) we can study the entropy in this case.

3.3 Results and discussion Using the appropriate equation and the band constants from appendix 15, the normalized entropy in HD QBs of CdGeAs2has been plotted as a function of film thickness for the generalized band model as shown by curve (a) where the curves (b), (c) and (d) are valid for three and two band models of Kane together with parabolic energy bands respectively. The case for δ = 0 has been plotted in the same figure and is represented by curve (e) for the purpose of assessing the influence of crystal field splitting on the entropy in HD QBs of CdGeAs2. The Figure 3.2 exhibits the variations of normalized entropy in HD QBs of CdGeAs2 as a function of electron concentration for all the cases mentioned as above for Figure 3.1 In Figures 3.3, 3.5 and 3.7, the normalized entropy in HD QBs of InAs, GaAs and InSb has been plotted as a function of film thickness in accordance with the three and two band models of Kane together with parabolic energy bands as shown by curves (a), (b) and (c) in the respective figures. The Figures 3.4, 3.6 and 3.8 demonstrate the concentration dependence of the normalized entropy in HD QBs of InAs, GaAs and InSb for all the cases of Figure 3.3. The Figures 3.9, 3.10 and 3.12 illustrate the film thickness dependence of the normalized entropy in HD QBs of InAs, GaAs and InSb in accordance with the models of Stillman et al., Palik et al. at T=5K and T=15K as represented by curves (a), (b) and (c) respectively. The Figures 3.11 and 3.13 demonstrate the influence of carrier concentration on the normalized entropy in HD QBs of GaAs and InSb for all the cases of Figure 3.11. The Figure 3.14 exhibits the film thickness dependence of the normalized entropy for HD QBs of InSb and InAs in accordance with the model of Palik et al. at T=1K as represented by the curves (a) and (b) respectively. The Figure 3.15 illustrates the concentration dependence of normalized entropy for all the cases of Figure 3.14.

180

3 Entropy in quantum box of heavily doped materials

Normalized entropy

103

(d) 102

(c) (b)

101 (a)

(e)

100 0

10 20 30 40 50 60 70 80 90 100 Film thickness (in nm)

Figure 3.1: Plot of the normalized entropy in HDQBs of CdGeAs2 as a function of film thickness has been shown in accordance with the (a) generalized band model (δ≠0), (b) three and (c) two band models of Kane together with (d) parabolic energy bands. The special case for δ = 0 (e) has also been shown to assess the influence of crystal field splitting.

7.15

(c) (d)

(e) (a)

4.3 0.001

0.01

0.1

Normalized entropy

(b)

1

10

Carrier concentration (1023 m–3) Figure 3.2: Plot of the normalized entropy in HDQBs of CdGeAs2 as a function of carrier concentration for all cases of Figure 3.1.

The Figure 3.16 demonstrates the film thickness dependence of the normalized entropy for HD QBs of InSb and InAs in accordance with the model of Stillman et al. at T=2K as represented by the curves (a) and (b) respectively. The Figure 3.17 shows the concentration dependence of normalized entropy for all the cases of Figure 3.16. The Figure 3.18 depicts the film thickness dependence of the normalized entropy for

3.3 Results and discussion

181

Normalized entropy

104

103

102 (a)

101

(b) (c)

100

0

10 20 30 40 50 60 70 80 90 100 Film thickness (in nm)

Figure 3.3: Plot of the normalized entropy in HDQBs of InAs as a function of film thickness in accordance with the (a) three and (b) two band models of Kane together with (c) parabolic energy bands.

10

0.01

0.1

1

(b) 10 1

(a)

Normalized entropy

(c)

0 Carrier concentration (1023 m–3) Figure 3.4: Plot of the normalized entropy in HDQBs of InAs as a function of carrier concentration for all the cases of Figure 3.3.

HDQBs of InSb and InAs in accordance with the model of Palik et al. as represented by the curves (a) and (b) in this context. The Figure 3.19 models the concentration dependence of normalized entropy for all the cases of Figure 3.11. The Figure 3.20 exhibits the variation of the normalized entropy with the film thickness in HDQBs of II–VI materials in accordance with Hopfield model, taking p-CdS as an example and considering both the cases
λ0 = 0 and
λ0 ≠0) and GaP in accordance with the model of

182

3 Entropy in quantum box of heavily doped materials

103

Normalized entropy

(b) 102 (c) 101 (a) 100 0

10 20 30 40 50 60 70 80 90 100 Film thickness (in nm)

Figure 3.5: Plot of the normalized entropy from HDQBs of GaAs as a function of film thickness in accordance with the (a) three and (b) two band models of Kane together with (c) the parabolic energy bands.

(c)

0.1

1

10 (b) 1

(a)

Normalized entropy

0.01

10

0.1 Carrier concentration (1023 m–3) Figure 3.6: Plot of the normalized entropy in HDQBs of GaAs as a function of carrier concentration for all the cases of Figure 3.5.

Rees) as shown by curves (a), (b) and (c) respectively. The Figure 3.21 demonstrates the concentration variation of the normalized entropy for all cases of Figure 3.20. In Figure 3.22, the normalized entropy has been plotted as a function of film thickness for HDQBs of Germanium for both the models of Wang et al. and Cardona et al. as shown by curves (a) and (b) respectively. The Figure 3.23 exhibits the normalized entropyas a function of film thickness in HDQBs of Tellurium (by using the models of Bouat et al. at T=10K and T=20K and stressed Kane type materials (taking n-InSb as

3.3 Results and discussion

183

Normalized entropy

104

103

102 (c) 101 (a) (b) 100 0 10 20 30 40 50 60 70 80 90 100 Film thickness (in nm)

Figure 3.7: Plot of the normalized entropy in HDQBs of InSb as a function of film thickness for all the cases of Figure 3.3.

10 (c)

0.01

0.1

1

10 1

Normalized entropy

(b)

(a) 0.1 Carrier concentration (1023

m

–3)

Figure 3.8: Plot of the normalized entropy in HDQBs of InSb as a function of carrier concentration for all the cases of Figure 3.4.

an example) in accordance with the models of Seiler et al. as shown by curves (a), (b) and (c) respectively. The Figure 3.24 shows the dependence of the normalized entropy on the carrier concentration for all the cases of Figure 3.23. In Figure 3.25, the normalized entropy has been plotted as a function of carrier concentration for HDQBs

184

3 Entropy in quantum box of heavily doped materials

Normalized entropy

104

(b)

103

102 (a)

101

(c)

0

10 20 30 40 50 60 70 80 90 100 Film thickness (in mm)

Figure 3.9: Plot of the normalized entropy in HDQBs of InAs as a function of film thickness in accordance with the models of (a) Stillman et al., (b) Newson et al. and (c) Rossler et al. respectively.

104

Normalized entropy

103

102

(b)

(a) 101

(c) 100 0

10 20 30 40 50 60 70 80 90 100 Film thickness (in nm)

Figure 3.10: Plot of the normalized entropy in HDQBs of GaAs as a function of film thickness for all the cases of Figure 3.9.

3.3 Results and discussion

185

10

0.1

0.01

(b)

10

(c) 0.1

Normalized entropy

(a) 1

0.01 Carrier concentration (1023 m–3) Figure 3.11: Plot of the normalized entropy in HDQBs of GaAs as a function of carrier concentration for all the cases of Figure 3.9.

104

Normalized entropy

103

(b)

102 (a)

101 (c) 100

0

10 20 30 40 50 60 70 80 90 100 Film thickness (in nm)

Figure 3.12: Plot of the normalized entropy in HDQBs of InSb as a function of film thickness for all the cases of Figure 3.9.

of Platinum Antimonide at T=5K, 10K and Lead Germanium Telluride at T=6K and 10K as shown by curves (a), (b), (c) and (d) respectively. The Figure 3.26 depicts the plot of normalized entropy as a function of film thickness of GaSb at T=5K for (a), T=10K for (b) and T=15K for (c) respectively. The Figure 3.27 exhibits the carrier concentration dependence of the normalized entropy

186

3 Entropy in quantum box of heavily doped materials

10

(c) (b)

1 0.1

0.001 0.01

1

10

Normalized entropy

(a)

0.1 Carrier concentration (1023 m–3) Figure 3.13: Plot of the normalized entropy in HDQBs of InSb as a function of carrier concentration for all the cases of Figure 3.9.

105

Normalized entropy

104

103

(b) (a)

102

101

100 0

10 20 30 40 50 60 70 80 90 100

10–1 Film thickness (in nm) Figure 3.14: Plot of the normalized entropy in HDQBs of (a) InSb and (b) InAs as a function of film thickness in accordance with model of Agafonov et al.

for all the cases of Figure 3.26. The Figure 3.28 illustrates the variation of the normalized entropy with film thickness in HDQBs of PbSe at T=5K, 10K, 14K and 16K as shown by curves (a), (b), (c) and (d) respectively. The Figure 3.29 shows the dependence of

3.3 Results and discussion

187

10

1 0.001 0.01

0.1

10

1 (b)

0.1

Normalized entropy

(a)

0.01 Carrier concentration (1023 m–3) Figure 3.15: Plot of the normalized entropy in HDQBs of (a) InSb and (b) InAs as a function of carrier concentration for the case of Figure 1.14.

105

Normalized entropy

104 103

(a)

102 101 100 (b) 10–1 0

10 20 30 40 50 60 70 80 90 100 Film thickness (in nm)

Figure 3.16: Plot of the normalized entropy in HDQBs of (a) InSb and (b) InAs as a function of film thickness in accordance with the model of Johnson et al.

normalized entropy on the carrier concentration in HDQBs of PbSe for all types of band models as stated in Figure 3.28. The Figure 3.30 shows the variation of entropy on film thickness for HDQBs of IV–VI materials (taking PbTe as an example) at T=5K, 10K and 15K together with Bismuth Telluride as shown by the curves (a), (b), (c) and (d)

188

3 Entropy in quantum box of heavily doped materials

(a)

1 0

0.01

1

0.1

10 (b)

Normalized entropy

10

0.1 1 Carrier concentration (1023 m–3) Figure 3.17: Plot of the normalized entropy in HD QBs of (a) InSb and (b) InAs as a function of carrier concentration for the case of Figure 3.16.

105

Normalized entropy

104 103 102

(a)

(b)

101 100 0 10 20 30 40 50 60 70 80 90 100 10–1 Film thickness (in nm)

Figure 3.18: Plot of the normalized entropy in HD QBs of (a) InSb and (b) InAs as a function of film thickness in accordance with the model of Palik et al.

respectively. The Figure 3.31 models the variation of the entropy with the carrier concentration for HDQBs of IV–VI materials and Bismuth Telluride in accordance with the band models of Figure 3.30. The Figure 3.32 shows the plot of the normalized entropy as a function of film thickness in HDQBs of (a) II–V compound (CdSb), (b) zinc diphosphide, (c) cadmium diphosphide at T=4K and (d) T=8K as a function of film thickness respectively. For the purpose of simplified numerical computation, broadening has been neglected for obtaining all the plots. The inclusion of broadening will change the numerical magnitudes without altering the physics inside.

3.3 Results and discussion

189

1 0.01

(a)

0.1

10 (b)

Normalized entropy

10

0.1 Carrier concentration (1023 m–3) Figure 3.19: Plot of the normalized entropy in HD QBs of (a) InSb and (b) InAs as a function of carrier concentration for the case of Figure 3.18.

105

104 Normalized entropy

(c) 103

102

(a)

101 (b) 100 50

60

70

80

90

100

Film thickness (in nm)
0 ≠0, (b) λ
0 = 0 and Figure 3.20: Plot of the normalized entropy in HD QBs of CdS with (a) λ (c) GaP as a function of film thickness in accordance with the models of Hopfield and Rees respectively.

The signature of 3D quantization is forthwith evident from the Figures 3.1, 3.3, 3.5, 3.7, 3.9, 3.10, 3.12, 3.14, 3.16, 3.18, 3.20, 3.22, 3.23, 3.26, 3.28, 3.30 and 3.32 for all materials as discussed having different band structures. It can be facilely discerned from the same that the normalized entropy oscillates with film thickness exhibiting spikes for various values of film thickness which are totally band structure dependent. The occurrence of peaks in the said figures originates from the totally quantized

190

3 Entropy in quantum box of heavily doped materials

10

(a) 0.001

0.01

0.1

10

1

Normalized entropy

(b)

(c)

0.1 Carrier concentration (1023 m–3)
0 ≠0, (b) λ
0 = 0 and (c) GaP as Figure 3.21: Plot of the normalized entropy in HD QBs of CdS with (a) λ a function of carrier concentration for all the cases of Figure 3.20.

104

Normalized entropy

103

102

(b)

101

100

(a) 0 10 20 30 40 50 60 70 80 90 100

10–1 Film thickness (in nm) Figure 3.22: Plot of the normalized entropy in HD QBs of Germanium as a function of film thickness in accordance with the models of (a) Wang et al. and (b) Cardona et al. respectively.

energy levels of the carriers of the concerned dots. The entropy spectra are found bearing composite oscillations as function of the nano-thickness. These are generally due to selection rules in the quantum numbers along the three confined directions. The dependence of the normalized entropy on the carrier concentration is manifested

3.3 Results and discussion

191

105

Normalized entropy

104 103 102

(c)

(b)

101 (a) 100 0

10 20 30 40 50 60 70 80 90 100

10–1 Film thickness (in nm) Figure 3.23: Plot of the normalized entropy in HD QBs of Tellurium and stressed Kane type material (n-InSb) as a function of film thickness in accordance with the models of (a) Bouat et al., (b) Ortenberg et al. and (c) Seiler et al. respectively.

2 1.8 (a) 1.6 1.4 1.2 (b)

1 0.8 0.6 0.4 0.2

(c)

0 0.1

1

10

Carrier concentration (1023 m–3) Figure 3.24: Plot of the normalized entropy in HD QBs of Tellurium and stressed Kane type material (n-InSb) as a function of carrier concentration for all the cases of Figure 3.23.

by the Figures 3.2, 3.4, 3.6, 3.8, 3.11, 3.13, 3.15, 3.17, 3.19, 3.21, 3.24, 3.25, 3.27, 3.29 and 3.31 for the different materials as considered here. It can be ascertained from the same figures that the entropy of all the corresponding materials decreases with increasing carrier concentration for relatively higher values of the carrier degeneracy. Although the entropy varies in various manners with all the variables in

192

3 Entropy in quantum box of heavily doped materials

(b)

0.1

(c)

10

1 (a)

Normalized entropy

10

(d)

0.1 Carrier concentration (1023 m–3) Figure 3.25: Plot of the normalized entropY in HD QBs of (a) Graphite, (b) Platinum antimonide, (c) zero gap (HgTe) and (d) Pb1-xGexTe as a function of carrier concentration in accordance with the models of Ushio et al., Emtage, Ivanov-Omskii et al. and Vassilev respectively.

103

Normzlied entropy

102 (c)

(b)

101 (a) 100

0 10 20 30 40 50 60 70 80 90 100

10–1 Film thickness (in nm) Figure 3.26: Plot of the normalized entropy in HD QBs of Gallium antimonide as a function of film thickness in accordance with the models of (a) Seiler et al., (b) Mathur et al. and (c) Zhang respectively.

all the limiting cases as evident from all the figures, the rate of variations in each case are totally band-structure dependent. The quantum oscillations of the entropy in HD QBs exhibit different numerical magnitudes as compared to the same in UFs and QWs. It may be comprehended that the HD QBs lead to the discrete energy levels, somewhat like atomic energy levels,

3.3 Results and discussion

193

(c) 0.001

0.01

0.1

10

1 (a)

Normzlied entropy

10

(b) 0.1 Carrier concentration (1023 m–3) Figure 3.27: Plot of the normalized entropy in HD QBs of Gallium antimonide as a of normalized carrier concentration for all cases of Figure 3.26.

104

Normzlied entropy

103

(a) (b)

102 (c) 101

100

(d)

0

10 20 30 40 50 60 70 80 90 100

10–1 Film thickness (in nm) Figure 3.28: Plot of the normalized entropy in HD QBs of Bismuth as a function of film thickness in accordance with the models of (a) McClure et al, (b) Takaoka et al. (Hybrid model), (c) Cohen and (d) Lax et al. respectively.

which produce very large changes. This is in accordance to the inherent nature of the quantum confinement of the carrier gas as dealt with here. In HDQBs, there remain no free carrier states in between any two allowed sets of size-quantized levels unlike that found for UFs and QWs where the quantum confinements are 1D and 2D respectively. Consequently, the crossing of the Fermi level by the size-quantized levels in HD QBs would have much greater impact on the redistribution of the carriers among the allowed levels, as compared to that found for UFs and QWs respectively.

194

3 Entropy in quantum box of heavily doped materials

(d) 10 0.001

0.01

10

0.1

(c) (b)

Normalized entropy

10

(a) 0.1 Carrier concentration (1023 m–3) Figure 3.29: Plot of the normalized entropy in HD QBs of Bismuth as a function of carrier concentration for all the cases of Figure 3.28.

107

Normalized entropy

106

(b)

105

(d) (c)

104 (a)

103 102 101 100 0

10 20 30 40 50 60 70 80 90 100 Film thickness (in nm)

Figure 3.30: Plot of the normalized entropy in HD QBs of PbTe as a function of film thickness in accordance with the models of (a) Dimmock, (b) Bangert et al. and (c) Foley et al. The plot (d) refers to Bi2Te3 in accordance with the model of Stordeur et al.

The effect of spin splitting of the carriers in HD QBs of p-CdS on entropy can be numerically investigated from Figures 3.20 and 3.23. It appears that the absence of the spin splitting constant decreases the numerical value of the entropy in CdS for a particular range of film thickness. Itappears from Figure 3. 22 that the entropy in Germanium in accordance with the model of Wang et al. is comparatively low with that of the Cardona et al. model. The entropies in HDQBs of Te and stressed Kane type materials are depicted in Figures 3.23 and 3.24 with respect to film thickness and carrier concentration respectively. It appears that at extremely low and high film thicknesses, the entropy in HDQBs of Te dominates over that of the corresponding

3.3 Results and discussion

195

25 Normalized entropy

20 (d) 15 (b) (c) 10 (a) 5 0 0.1

1

10

Carrier concentration (1023 m–3) Figure 3.31: Plot of the normalized entropy in HD QBs of PbTe and Bi2Te3 as a function of carrier concentration for all the cases of Figure 3.30.

104

Normalized entropy

103 (a) (c)

102

(b) 101 (d) 100 40

50

60

70

80

90

100

Film thickness (in nm) Figure 3.32: Plot of the normalized entropy in HD QBs of (a) II–V compound (CdSb), (b) zinc diphosphide, (c) cadmium diphosphide and (d) antimony as a function of film thickness in accordance with the models of Yamada, Chuiko and Ketterson respectively.

stressed InSb, although, at mid zone thickness, the entropy in stressed InSb exhibits a high peak together with the fact that the periods of oscillations of the entropy are comparatively higher in Te than that of stressed compounds. In Figure 3.25, the entropy in HD QBs of Platinum antimonide and Pb1-xGexTe materials have been plotted as a function of carrier concentration. From Figure 3.28,

196

3 Entropy in quantum box of heavily doped materials

it appears that the entropy in PbSe exhibits the high sharp peak in the mid thickness zone. From Figure 3.26, it is evident that the entropy in GaSb oscillates with film thickness exhibiting spikes. In Figures 3.30 and 3.31, the variation of the entropy in PbTe and Bi2Te3 against film thickness and carrier concentration has been demonstrated in accordance with the appropriate band models. In Figure 3.32, the variation of entropy in II–V compound, zinc and cadmium diphosphides as a function of film thickness has been shown in accordance with the appropriate band models. From Figure 3.30, it appears that the numerical values of entropy in HD QBs of PbTe and Bi2Te3 in accordance with all the band models are extremely higher than all other materials.

3.4 Open research problems (R3.1) Investigate the entropy for QBs of the HDS in the presences of Gaussian, exponential, Kane, Halperian, Lax and Bonch-Burevich types of band tails for all systems whose unperturbed carrier energy spectra are defined in R1.1 (R3.2) Investigate the entropy for QBs of all the HD materials as considered in R4.1.under non uniform strain (R3.3) Investigate the entropy in the presence of non uniform strain for QBs of HD negative refractive index, organic, magnetic and other advanced optical materials in the presence of an alternating electric field. (R3.4) Investigate the entropy for the QBs of HD negative refractive index, organic, magnetic and other advanced optical materials in the presence of an arbitrarily oriented alternating electric field. (R3.5) Investigate the entropy for the multiple QBs of HD materials whose unperturbed carrier energy spectra are defined in R1.1. (R3.6) Investigate the entropy for all the appropriate HD zero dimensional systems of this chapter in the presence of finite potential wells. (R3.7) Investigate the entropy for all the appropriate HD zero dimensional systems of this chapter in the presence of parabolic potential wells. (R3.8) Investigate the entropy for all the above appropriate problems in the presence of elliptical Hill and quantum square rings in the presence of strain. (R3.9) Investigate the entropy for parabolic cylindrical HD zero dimensional systems in the presence of an arbitrarily oriented alternating electric field for all the HD materials whose unperturbed carrier energy spectra are defined in R1.1 in the presence of strain. (R3.10) Investigate the entropy for HD zero dimensional systems of the negative refractive index and other advanced optical materials in the presence of an arbitrarily oriented alternating electric field and non-uniform light waves and in the presence of strain.

References

197

(R3.11) Investigate the entropy for triangular HD zero dimensional systems of the negative refractive index, organic, magnetic and other advanced optical materials in the presence of an arbitrarily oriented alternating electric field in the presence of strain. (R3.12) Investigate the entropy for all the problems of (R4.1) in the presence of arbitrarily oriented magnetic field. (R3.13) Investigate the entropy for all the problems of (R4.1)in the presence of alternating electric field. (R3.14) Investigate the entropy for all the problems of (R4.1)in the presence of alternating magnetic field. (R3.15) Investigate the entropy for all the problems of (R4.1)in the presence of crossed electric field and quantizing magnetic fields. (R3.16) Investigate the entropy for all the problems of (R4.1)in the presence of crossed alternating electric field and alternating quantizing magnetic fields. (R3.17) Investigate the entropy for HD QBs of the negative refractive index, organic and magnetic materials. (R3.18) Investigate the entropy for HD QBs of the negative refractive index, organic and magnetic materials in the presence of alternating time dependent magnetic field. (R3.19) Investigate the entropy for HD QBs of the negative refractive index, organic and magnetic materials in the presence of in the presence of crossed alternating electric field and alternating quantizing magnetic fields. (R3.20) a) Investigate the entropy for HD QBs of the negative refractive index, organic, magnetic and other advanced optical materials in the presence of an arbitrarily oriented alternating electric field considering many body effects. b) Investigate all the appropriate problems of this chapter for a Dirac electron. (R3.21) Investigate all the appropriate problems of this chapter by including the many body, image force, broadening and hot carrier effects respectively. (R3.22) Investigate all the appropriate problems of this chapter by removing all the mathematical approximations and establishing the respective appropriate uniqueness conditions.

References [1]

[2] [3]

Bimberg D., Grundmann M., Ledentsov N.N., Quantum Dot Heterostructures (John Wiley & Sons, 1999); Konstantatos G., Howard I., Fischer A., Howland S., Clifford J., Klem E., Levina L., Sargent E.H., (2006) Nature 442, 180. Jaiswal J.K., Mattoussi H., Mauro J.M., Simon S.M., Nat Biotechnol 21, 47 (2003). Watson A., Wu X., Bruchez M., Biotechniques 34, 296 (2003); Nakanishi J., Kikuchi Y., Takarada T., Nakayama H., Yamaguchi K., Maeda M., J. Am. Chem. Soc., 126, 16314 (2004).

198

[4] [5] [6] [7]

[8]

[9]

[10] [11] [12] [13] [14]

[15] [16]

[17] [18] [19]

[20]

[21] [22]

3 Entropy in quantum box of heavily doped materials

Michalet X., Pinaud F.F., Bentolila L.A., Tsay J.M., Doose S., Li J.J., Sundaresan G., Wu A.M., Gambhir S.S., Weiss S., Science 307, 538 (2005). van Sark W.G.J.H.M., Frederix P.L.T.M., Van Den Heuvel D.J., Bol H.C.G.A., van Lingen J.N.J., de Mello Donegá C., Meijerink A., J. Phys. Chem B 105, 8281 (2001). Sánchez E.J., Novotny L., Xie X.S., Phys Rev Lett 82, 4014 (1999), Bailey B., Farkas D.L., Taylor D.L., Lanni F., Nature 366, 44 (1993). Asryan L.V., Suris R.A., Selected Topics in Electronics and Systems Ed. by E. Borovitskaya, M. S. Shur, World Scientific Vol. 25 (Singapore, 2002); Asryan L.V., Suris R.A., Int. J. High Speed Electron. Syst., Special Issue on “Quantum Dot Heterostructures – Fabrication, Application, Theory,” 12 [1], 111 (2002); Asryan L.V., Luryi S., Future Trends in Microelectronics: The Nano Millennium, Ed. by Luryi S., Xu J.M., Zaslavsky A., Wiley Interscience, New York, 219 (2002). Freitas R.A., Jr, J. Comput. Theor. Nanosci 2, 1 (2005); A. Ferreira, C. Mavroidis, IEEE Robotics and Automation Magazine, 13, 78 (2006); Dubey A., Sharma G., Mavroidis C., Tomassone S. M., Nikitczuk K., Yarmush M.L., J. Comput. Theor. Nanosci.1, 18 (2004); Mavroidis C., Dubey A., Yarmush M.L., Annual Reviews of Biomedical Engineering, 6, 363 (2004). Liu Y., Starzyk J.A., Zhu Z., IEEE Trans. on Neural Networks (2008) [In press]; Starzyk J.A., He H., IEEE Trans. on Neural Networks, 18 (2), 344 (2007); Starzyk J.A., He H., IEEE Trans. Circuits Syst. II54 (2), 176 (2007). Hasaneen E.–.S., Heller E., Bansal R., Huang W., Jain F., Solid State Electronics 48, 2055 (2004). Kawazoe T., Tanaka S., Ohtsu M., J. Nanophoton 2, 029502 (2008). Krenner H.J., Stufler S., Sabathil M., Clark E.C., Ester P., Bichler M., Abstreiter G., Finley J.J., Zrenner A., New J. Phys 7, 184 (2005). Zhukov A.E., Kovsh A.R., Quantum Electron 38, 409 (2008). Sugawara M., Akiyama T., Hatori N., Nakata Y., Ebe H., Ishikawa H., Meas. Sci. Technol 13, 1683 (2002); van der Poel M., Birkedal D., Hvam J., Laemmlin M., Bimberg D., Conference on Lasers and Electro-Optics (CLEO) 1, 16 (2004). Costa-Fernandez J.M., Anal. Bioanal. Chem 384, 37 (2006), Djie H.S., Dimas C.E., Wang .D. –N., Ooi B. –S., Hwang J. C.M., Dang G.T., Chang W.H., IEEE Sensors Jour. 7, 251 (2007). Zhu X.–.X., Cao Y.–.C., Jin X., Yang J., Hua X.–.F., Wang H.–.Q., Liu B., Wang Z., Wang J.–.H., Yang L., Zhao Y.–.D., Nanotechnology 19, 025708; Gao X., Chan W. C.W., Nie S., (2002) J. Biomed. Opt. 7, 532 (2008). Jaiswal J.K., Goldman E.R., Mattoussi H., Simon S.M., Nature Methods 1, 73 (2004). Matsueda H., Internat. Jour. Circuit Theo. and Appl 31, 23 (2003); Hu X., Das Sarma. S., Phys. Stat. Sol. (b), 238, 360 (2003). Chen G.L., Kuo D.M.T., Lai W.T., Li P.W., Nanotechnology 18, 475402 (2007); Pogosov A.G., Budantsev M.V., Shevyrin A.A., Plotnikov A.E., Bakarov A.K., Toropov A.I., JETP Letts. 87, 150 (2008). Johnston K.W., Pattantyus-Abraham A.G., Clifford J.P., Myrskog S.H., MacNeil D.D., Levina L., Sargent E.H., Appl. Phys. Letts 92, 151115 (2008); Leschkies K.S., Divakar R., Basu J., Pommer E.E., Boercker J.E., Carter C.B., Kortshagen U.R., Norris, D.J.E.S. Aydil, Nano Lett. 7, 1793 (2007). Liu I.S., Lo H.H., Chien C.T., Lin Y.Y., Chen C.W., Chen Y.F., Su W.F., Liou S.C., J. Mater. Chem 18, 675 (2008). Hitoshi N., Sugimoto Y., Nanamoto K., Ikeda N., Tanaka Y.X., Nakamura Y., Ohkouchi S., Watanabe Y., Inoue K., Ishikawa H., Asakawa K., Optics Express 12, 6606 (2004); Yamamoto N., Matsuno T., Takai H., Ohtani N., Jpn. J. Appl. Phys. 44, 4749 (2005); Kinoshita T.Y., Kasai S., Hasegawa. Amemiya H., Y., Jpn. J. Appl. Phys. 40, 4485 (2001); Asakawa K., Sugimoto Y., Watanabe Y., Ozaki N., Mizutani A., Takata Y., Kitagawa Y., Ishikawa H., Ikeda N., Awazu K., Wang X., Watanabe A., Nakamura S., Ohkouchi S., Inoue K., Kristensen M., Sigmund O., Borel P. I, Baets R., New J. Phys. 8, 208 (2006).

References

199

[23] Clapp A.R., Medintz I.L., Fisher B.R., Mattoussi G.P., H., J. Am. Chem. Soc 127, 1242 (2005); Shi L., Hernandez B., Selke M., Am .J. Chem. Soc. 128, 6278 (2006); Wu. C., Zheng J., Huang. C., Lai J., Li S., Chen C., Zhao Y., Angewandte Chemie International Edition, 46, 5393 (2007). [24] Rossler U., Solid State Commun 49, 943 (1984). [25] Agafonov V.G., Valov P.M., Ryvkin B.S., Yaroshetskii I.D., Sov. Phys. Semi-cond 12, 1182 (1978). [26] Foley G.M.T., Langenberg P.N., Phys. Rev. B 15B, 4850 (1977). [27] Seiler D.G., Beeker W.M., Roth K.M., Phys. Rev 1, 764 (1970). [28] Mathur P.C., Jain S., Phys. Rev 19, 1359 (1979). [29] Zhang H.I., Phys. Rev. B 1, 3450 (1970).

4 Entropy in heavily doped materials under magnetic quantization If my only desire is to be desire-less then I will surely get the permanent visa to live in the wonderful world of SOLITUDE.

4.1 Introduction It is well known that the band structure of materials can be dramatically changed by applying external fields. The effects of the quantizing magnetic field on the band structure of compound materials are more striking and can be observed easily in experiments [1–3]. Under magnetic quantization, the motion of the electron parallel to the magnetic field remains unaltered while the area of the wave vector space perpendicular to the direction of the magnetic field gets quantized in accordance with the Landau’s rule of area quantization in the wave-vector space [3]. The energy levels of the carriers in a magnetic field (with the component of the wave-vector parallel to the direction of magnetic field be equated with zero) are termed as the Landau levels and the quantized energies are known as the Landau sub-bands. It is important to note that the same conclusion may be arrived either by solving the single-particle time independent Schrödinger differential equation in the presence of a quantizing magnetic field or by using the operator method. The quantizing magnetic field tends to remove the degeneracy and increases the band gap. A semiconductor, placed in a magnetic
m
c ÞÞ. This phe
0 = ðj
ejB= field B, can absorb radiative energy with the frequency ðω nomenon is known as cyclotron or diamagnetic resonance. The effect of energy quantization is experimentally noticeable when the separation between any two
A number of interesting transport pheconsecutive Landau levels is greater than
kB T. nomena originate from the change in the basic band structure of the semiconductor in the presence of quantizing magnetic field. These have been widely been investigated and also served as diagnostic tools for characterizing the different materials having various band structures [4–7]. 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 this chapter in Section 4.2.1, of the theoretical background, the entropy has been investigated in HD nonlinear optical materials in the presence of a quantizing magnetic field. Section 4.2.2 contains the results for HD III–V, ternary, and quaternary compounds in accordance with the three- and the two-band models of Kane. In the same section, the entropy in accordance with the models of Stillman

https://doi.org/10.1515/9783110661194-004

202

4 Entropy in heavily doped materials under magnetic quantization

et al. and Palik et al. has also been studied for the purpose of relative comparison. Section 4.2.3 contains the study of the entropy for HD II–VI materials under magnetic quantization. In Section 4.2.4, the entropy in HD IV–VI materials has been discussed in accordance with the models of Cohen, Lax, Dimmock, Bangert and Kastner, and Foley and Landenberg, respectively. In Section 4.2.5, the magneto-entropy for the stressed HDKane-type materials has been investigated. In Section 4.2.6, the entropy in HD Te has been studied under magnetic quantization. In Section 4.2.7, the magneto-entropy in n-GaP has been studied. In Section 4.2.8, the entropy in HD PtSb2 has been explored under magnetic quantization. In section 4.2.9, the magneto-entropy in HD Bi2Te3 has been studied. In Section 4.2.10, the entropy in HD Ge has been studied under magnetic quantization in accordance with the models of Cardona et al. and Wang and Rossler, respectively. In Sections 4.2.11 and 4.2.12, the magneto-entropy in HD n-GaSb and II–V compounds has respectively been studied. In Sections 4.2.13 the magneto-entropy in HD Pb1-xGexTe has been discussed. The Section 4.3 explores the result and discussions andSection 4.4 contains 11 open research problems.

4.2 Theoretical background 4.2.1 Entropy in HD nonlinear optical materials under magnetic quantization The dispersion relation in non-linear optical materials can be written as " #" # 2
2 2
2
g + Δ? Þ
hE
g ðE Δ2jj − Δ2?
k k h
h
e B
s z 0
f ðEÞ
f ðEÞ
=
+
±

γðEÞ
g + 2 Δ? Þ E + Eg0 + δ + 3Δjj
*? 3
*jj 4 6 2m 2m ðE 3 0 (4:1) where,

  


f3 ðEÞ
+E
g + 1 Δjj + 1 ðΔ2 − Δ2 Þ
= E g ðE g + Δ? Þ ðE
+E
g + 2 Δjj + δ E
+E
g Þ E ?
g + 2 Δ? Þ 3 3 9 jj ðE 3 and 



f4 ðEÞ = Eg ðEg + Δjj Þ ðE
+E
g + 2 Δjj
+E
g Þ E
g + 2 Δjj Þ 3 ðE 3 The (4.1) can be expressed as ! ! (" !

jj
c?
jj
jj ck − ab
jj Þ
jj 2 ða

cjj + b
b
b b a
h2
kz2
h2
ks2 1 a
+
+ 1− +
1− = E E
cjj
cjj
cjj
cjj
*jj
*?
cjj 2 2m 2m b?
cjj

203

4.2 Theoretical background

!
 
jj
jj 
jj ðΔ2k − Δ2? Þ 
b
b
b a 1 1 2 2 2a a 2 1− − 1− δE + − Δ Þ −  + ðΔ ?
+ 1Þ
cjj
cjj ð
cjj E
cjj
cjj 9 jj 9
cjj ð
cjj E + 1Þ !( !" ! !  2 2 2 2
jj c?
cjj
b
h2 ks2 δ Δjj − Δ? δ Δjj − Δ? a − + + −
+ 1Þ
+ 1Þ
E b?
cjj 6Δjj 6Δjj 2 2 2m*? ð
cjj E ða "

ρ ρ ±
e1
1
+

2 2 E + Eg E + Eg + 3 Δjj

# (4:2)

where, " # " #
1Þ
hðE
g + 2 Δjj Þ ðE
g + G
g + Δ? Þ Δ2jj − Δ2?
eB
ð−E 3 0 0

e1 = , G1 = Eg0 + δ +
g + 2 Δ ? Þ , ρ1 =
g + Δjj Þ ðE 3Δjj ð32 Δjj Þ 6ðE 3 0 and ρ2 =

  3 Δjj Δ2? −δ + 3Δjj 2Δjj 3

Therefore, the dispersion relation of the conduction electrons in HD nonlinear optical materials in the presence of a quantizing magnetic field B can be written following the methods as developed in Chapter 1 as
2
kz2
h
n
2, ± ðE,
n
, η g Þ + iU
, ηg Þ = U1, ± ðE,
*jj 2m where

! " # "(" 

Þ −1
jj

1 + Erf ðE=η bjj
c? ab −
eB 1 g
η Þ
θ ðE, + n+ g *
cjj 0
? 2 2
cjj b? m ! !" #
Þ
jj
jj
cjj − a
jj
b

cjj + b 1 + Erf ðE=η
b a 1 a g
γ0 ðE, ηg Þ + 1− 1−
cjj
cjj
cjj
c2 jj 2 ! "
jj
jj
b b a 1

η Þ

ð1 − cjj Þ 1 − cðβ1 , E, ηg Þ + δγ0 ðE, g
cjj
cjj
cjj " ) ##

1 + Erf ðE=η




Þ
jj
b 2 2 2a g 2 2 2

Δjj − Δ? c β1 , E, ηg − Δ − Δ? 2 9 jj 9
cjj !" ! !( 2 2
jj
c?
b
heB 1 δ Δjj − Δ?
η Þ


cðβ2 , E, n+ a + g
?
*? 6Δjj 2 2
cjj b m #) # ! 2 2



δ Δjj − Δ? ρ1
e1


ρ2
e1





±
c β2 , E, ηg ±
c β3 , E, ηg , cjj c β1 , E, ηg − 6Δjj 2 ðEg0 + 32 Δjj Þ Eg0


n
1, ± ðE,
, ηg Þ = U + − + − +

"

(4:3a)

204

4 Entropy in heavily doped materials under magnetic quantization

"

! !
jj
b 1 a
Þ
, E,η 1− 1− Dðβ 1 g
cjj
cjj
cjj !" ! !( 2 2
jj
c?
jj

b b 2a
heB 1 δ Δjj −Δ?
2 2


Þ
+
Dðβ2 , E,η ðΔ −Δ ÞDðβ1 , E,ηg Þ+ * n a + + g
?
? 9
cjj jj ? 2 2 6Δjj
cjj b m #) " ## ! 2 2 δ Δjj −Δ? ρ1 e1
ρ2 e1





cjj Dðβ1 , E,ηg Þ ∓ + −
g + 2 Δjj sÞ Dðβ3 , E,ηg Þ
g Dðβ2 , E,ηg Þ± ðE 2 6Δjj E 0 0 3


n
2,± ðE,
,ηg Þ= U


Þ 1+Erf ðE=η g 2

#−1 "

11 1
, β3 = β1 =
cjj , β2 =
= a
g + 2 Δjj Þ , ðE Eg0 3 0 "
, E,
η Þ= Cðβ i g

" # # ( )
2 ∞ X
expð −4p Þ 2 1 + βi E 2
u
i =
i Þ ×
i Þ , u pffiffiffi expð − u sinhðp
p βi ηg βi ηg π p=1

and " pffiffiffi # π 2
η Þ=
, E,
i Þ expð − u Dðβ i g βi ηg EEM at the Fermi level can be written from (4.3a) as
FBHD , n
FBHD , n
* ± ðE
, ηg Þ = m
*jj U ′1, ± ðE
, ηg Þ m

(4:3b)


FBHD is the Fermi energy in this case. where E Therefore, the double valued EEM in this case is a function of Fermi energy, magnetic field, quantum number and the scattering potential together with the fact that EEM exists in the band gap which is the general characteristics of HD materials. The complex density-of-states function under magnetic quantization is given by " # max
qffiffiffiffiffiffiffiffiffi n
X eB x′ i y′ *






pffiffiffi + pffiffiffi (4:4a) 2mjj NB ðEÞ = NBR ðEÞ + iNBI ðEÞ = x 2
y 2π2
h2
=0 2
n where
x =


y =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
1, ± ðE,
1, ± ðE,
n
2, ± ðE,
n
n
, ηg ÞÞ + ðU
, ηg ÞÞ + ðU
, ηg ÞÞ ðU 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
1, ± ðE,
1, ± ðE,
n
2, ± ðE,
n
n
, ηg ÞÞ + ðU
, ηg ÞÞ − ðU
, ηg ÞÞ ðU 2


and x′and y′ are the differentiations of
x and
y with respect to energy E. Therefore, from (4.4a), we can write

205

4.2 Theoretical background

max
qffiffiffiffiffiffiffiffiffi n
X x′
= eB
B Real ðEÞ
*jj pffiffiffi 2m N 2
x 4π2
h
=0 n

(4:4b)

max
qffiffiffiffiffiffiffiffiffi n
X y′ *
= eB
B Imaginary ðEÞ
pffiffiffi 2 m N jj 2 2
y 4π
h
=0 n

(4:4c)

and

The electron concentration is given by qffiffiffiffiffiffiffiffiffi
*jj n
X max gv eB 2m
0 = p ffiffi ffi n 2 n
= 0 2π2
h2 2"rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #

2

1=2 2





4


U 1, ± ðEFBHD , n, ηg Þ + U 2, ± ðEFBHD , n, ηg Þ + U1, ± ðEFBHD , n, ηg Þ
r =
s X
r = 1

"rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #

2



i1=2






, ηg Þ + U 2, ± ðEFBHD , n
, ηg Þ + U1, ± ðEFBHD , n
, ηg Þ U 1, ± ðEFBHD , n (4:5)

Using (1.31f) and (4.5), we can study the entropy in this case.

4.2.2 Entropy in QWs of HD III–V materials under magnetic quantization (a) The electron energy spectrum in III–V materials under magnetic quantization is given by 
+E
g + ΔÞðE
g + 2 ΔÞ

E
+E
g ÞðE
hΔ Eð 1
2
kz2 h eB
0 0 0 3


h ω + ± = n + 0 2




g + 2 ΔÞ


2 m 2 6mc ðE + E Eg0 ðEg0 + ΔÞðE + Eg0 + 3 ΔÞ c 3 0 (4:6) The (4.6) can be written as









b 2

a
+ 1 1− a 1− b − 1 1−
+ a
c + b
c − ab E E
c
c
c
c
c
c2
 2
2
hΔ 1
h kz eB

+
hω0 + + = n

g + 2 ΔÞ
c 6m
c ð1 +
cEÞðE 2m 2 3 0
= where, a

1
g E 0


= ,b

1
g + Δ E 0

and
c =

1
g + 2Δ E 0 3


a
c

 

 b 1 1−

c ð1 +
cEÞ

206

4 Entropy in heavily doped materials under magnetic quantization

Therefore







b
b + b
c − a
b
a
Ið4Þ + 1 1 − a
Ið5Þ + a 1−
c
c
c
c2

 

b 1
+
0 − n
hω
c 2

h i
2
k2 z

E,
C,
E,
Ið1Þ − g
± Gð
C,
η Þ =h
η Þ − iHð Ið1Þ g g
c 2m

(4:7)

where, "
1 1− g
± =
c


a
c

# 


hΔ b eB
± 1−
g + 2 ΔÞ ,
c
c ðE 6m 3 0

" # " ( ) #
2 ∞ X
expð −4p Þ 2 1 +
cE 2



u
Þ×
Þ , u
= , GðC, E, ηg Þ =
pffiffiffi expð − u sinhðp

p Cηg π Cη g p=1 " pffiffiffi # π 2
E,
C,
η Þ=
Hð g
expð − u Þ Cη g Therefore h i h1 + Erf ðE=η i−1

h i−1




gÞ
b a

η Þ 1 + Erf ðE=ηg Þ + a

c + b
c2
c − a
b γ0 ðE, g
c θ0 ðE, ηg Þ 2 2 i h 

 i h1 + Erf ðE=η i − 1h
 
gÞ
E,
C,
E,
C,
η Þ
η Þ − iHð
+ 21
hω
0 − g
± +
c1 1 − a

c 1 − b
c − n Gð g g 2 kz2
2
h
c 2m

=

(4:8) Therefore, the dispersion relation is given by
2
kz2
h
n
4, ± ðE,
η Þ
, η g Þ + iU = U3, ± ðE, g
c 2m

(4:9a)

where
n
3, ± ðE,
, ηg Þ = U



h1 + Erf ðE=η i−1

gÞ
b a

c θ0 ðE, ηg Þ × 2




1 a + 1− 1−
c
c

+

"




c − a


c + b

b a
c2


η Þ× γ0 ðE, g

h1 + Erf ðE=η i
Þ −1 g

2

" #  
Þ −1

1 + Erf ðE=η b 1 g




− n+
ω0 + g ± h GðC, E, ηg Þ
c 2 2

and
η Þ = g
±
4, ± ðE, U g




Þ 1 + Erf ðE=η g 2

#−1
E,
C,
η Þ Hð g

4.2 Theoretical background

207


nHD1 in this case can be obtained by substituding
kz = 0 The complex Landau energy E
=E
nHD in (4.9a) and E EEM at the Fermi level can be written from (4.9a) as
FBHD , n
FBHD , n
, ηg Þ = m
*jj U ′3, ± ðE
, ηg Þ
* ± ðE m

(4:9b)

Thus, EEM is a function of Fermi energy, Landau quantum number and scattering potential together with the fact it is double valued due to spin. The complex density of states function under magnetic quantization is given by " # max
qffiffiffiffiffiffiffiffi n
X eB x′ i y′ *






pffiffiffi + pffiffiffi (4:10a) 2 mc NB ðEÞ = NB Real ðEÞ + iNB Imaginary ðEÞ = x 2
y 2π2
h2
=0 2
n where
x =


y =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3, ± ðE,
3, ± ðE,
n
4, ± ðE,
n
n
, ηg ÞÞ2 + ðU
, ηg ÞÞ2 + ðU
, ηg ÞÞ ðU 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3, ± ðE,
3, ± ðE,
n
4, ± ðE,
n
n
, ηg ÞÞ2 + ðU
, ηg ÞÞ2 − ðU
, ηg ÞÞ ðU 2


. and x′1 and y′1 are the differentiations of
x and
y with respect to energy E From (4.10a), we can write

= eB
B Real ðEÞ N 2 4π
h2

qffiffiffiffiffiffiffiffi n
X max x′1
*c pffiffiffiffi 2m
x1
=0 n

(4:10b)

and

= eB
B Imaginary ðEÞ N 4π2
h2

qffiffiffiffiffiffiffiffi n
X max ′ y1
*c pffiffiffiffi 2m
y1
=0 n

(4:10c)

The electron concentration is given by max
qffiffiffiffiffiffi n
X g
v eB
*c m 2 2π2
h
=0 n "qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 2 2






, ηg Þ + ðU 4, ± ðEFBHD , n
, ηg ÞÞ + ðU 3, ± ðEFBHD , n
, ηg ÞÞ U 3, ± ðEFBHD , n


0 = n


r =
s X
r = 1



rÞ½ Lð

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2  2








ðU 3, ± ðEFBHD , n, ηg ÞÞ + ðU 4, ± ðEFBHD , n, ηg ÞÞ + ðU 3, ± ðEFBHD , n, ηg ÞÞ (4:11)

Using (1.31f) and (4.11) we can study the entropy in this case.

208

4 Entropy in heavily doped materials under magnetic quantization

(b) Two band model of Kane The magneto-dispersion law in this case is given by 



2
kz2 h 1 1
η − n

0 ∓ g
* μ0 B

ω h + = γ3 E, g
c 2m 2 2

(4:12a)

where g
* is the magnitude of the effective g
factor at the edge of the conduction band and μ0 is the Bohr magnetron. EEM at the Fermi level can be written from (4.12a) as
FBHD , η Þ = m
FBHD , η Þ
c γ′2 ðE
* ðE m g g

(4:12b)

Thus EEM is independent of quantum number. The electron concentration is given by  
r =
s max X
qffiffiffiffiffiffi n
X g
v eB 1=2 1=2 *









ðU5, ± ðEFBHD , n, ηg ÞÞ + LðrÞ½ðU5, ± ðEFBHD , n, ηg ÞÞ mc n0 = π2
h2
r = 1
=0 n (4:13) where


 1 1
FBHD , n
FBHD , η Þ − n

5, ± ðE
, ηg Þ = γ2 ðE

ω0 ∓ g
* μ0 B h + U g 2 2

Using (1.31f) and (4.13a), we can study the entropy in this case. (c) Parabolic Energy Bands The magneto-dispersion law in this case is given by

2
kz2 h 1 1
η Þ− n


hω0 ∓ g
* μ0 B = γ3 ðE, + g
c 2m 2 2

(4:14a)

EEM at the Fermi level can be written from (4.14a) as
FBHD , η Þ = m
FBHD , η Þ
*c γ′3 ðE
* ðE m g g

(4:14b)

Thus, EEM in heavily doped (HD) parabolic energy bands is a function of Fermi energy and scattering potential, whereas in the absence of bandtails the same mass is a constant quantity invariant of any variables. The electron concentration is given by  
r =
s max X
qffiffiffiffiffiffi n
X g
eB
6, ± ðE
FBHD , n
FBHD , n

rÞ½ðU
6, ± ðE
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
*c
0 = v 2 m ðU Lð n π2
h
r = 1
=0 n (4:15)

4.2 Theoretical background

209

where

FBHD , n
FBHD , η Þ − n + 1
hω0 ∓ 1 g
* μ B

6, ± ðE
, ηg Þ = γ3 ðE U g 0 2 2 Using (1.31f) and (4.15), we can find the entropy in this case. (d) The model of Stillman et al. The (1.107) under the condition of band tailing assumes the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 2
η Þ ½
t11 − ð
t11 Þ − 4
t12 γ3 ðE, g
k2 = 4 5 2
t12

(4:16)

Therefore, the magneto-dispersion law is given by
k2 = U
7 ðE,
n
, ηg Þ z

(4:17a)

where 2
n
7 ðE,
, ηg Þ = 4 U

½
t11 −

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 
η Þ ð
t11 Þ − 4
t12 γ3 ðE,

g 2eB 1 5
+ − n 2
t12
h 2

The EEM at the Fermi Level can be written from (4.17a) as
2 ′

FBHD , η Þ = h
, ηg Þ
* ðE U 7 ðEFBHD , n m g 2 The electron concentration is given by  
r =
s max X
qffiffiffiffiffiffi n
X g
eB
7 ðE
FBHD , n
FBHD , n

rÞ½ðU
7 ðE
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
0 = v 2 m
*c ðU n Lð π2
h
r = 1
=0 n

(4:17b)

(4:18)

Using (1.31f) and (4.18), we can study the entropy in this case. (e) The model of Palik et al To the fourth order in effective mass theory and taking into account the interactions of the conduction, light hole, heavy-hole and split-off hole bands, the electron energy spectrum in III–V materials in the presence of a quantizing magnetic field ~ B can be written as

210

4 Entropy in heavily doped materials under magnetic quantization


*


c 1
h2
kz2 1 m 1 *
k30 α
=
J 31 + n


0 Þ2
+

h ω
ω0 + h ð
hω ± ± E g n + 0 0
*c 4 m0 2m 2 2 ! " #2
h2
kz2 1
h2
kz2


+ Þ+

hω0

hω0 ðn ± k31 α + k32 α
*c
*c 2m 2m 2

(4:19)

where, h i
J31 = − 1 α
hω0 ð1 −
y11 Þ=ð2 +
x11 Þ2 ,
J32 , 2   
J32 = 1 ð1 −
x11 Þ2 − ð2 +
x2 ð2 +
x11 Þ.
y11 + 1 ð1 −
x2 Þð1 +
x11 Þð1 +
y11 Þ , 11 11 3 2     ð1 −
x11 Þ ð1 −
y11 Þ , g
0* = 2 1 −
y11 ð2 +
x11 Þ (" # )
k30 = ð1 −
y11 Þð1 −
x11 Þ ð2 + 3
x11 +
x2 Þ. ð1 −
y11 Þ − 2
y11 11 2 3 ð2 +
x11 Þ2 ( " # )  

ð1 − x Þ 3 ð1 − y Þ 2 11 11 2 . ð2 +
x11 +
x11 Þ. k
31 = ð1 −
y11 Þ − ð1 −
x11 Þ
y11 , ð2 +
x11 Þ 2 3 ð2 +
x11 Þ2 

k32 = − 1 + 1
x2 2 11

=

" !# − 1

* 1 Δ m 2 1 +
x11 ð1 −
y11 Þ
x11 = 1 +
and
y11 = c m0 2 E g0

Under the condition of heavy doping, the (A.20) assumes the form
J34
k4 +
J35, ± ðn
η Þ=0
Þkz2 +
J36, ± ðn
Þ − γ3 ðE, g z

(4:20)

where
J34 = α

k32


2 h
*c 2m

!2

"
Þ = ,
J35, ± ðn

#
2 h
h2
h2 1


+ Þ , ± αk31
hω0 . * + αk32
hω0 . * ðn
*c
c
c 2m 2m 2m 2

"
*
  2 #
1 1 1 m 2 c
J36, ± ðn
Þ =
J31 ±
+ Þ
+
hω0 g
0* ±
k30 αð
hω0 Þ n +
k32 α ð
hω0 Þðn 4 m0 2 2 The (4.20) can be written as
k2 = A
35, ± ðE,
n
, ηg Þ z where,

(4:21a)

211

4.2 Theoretical background

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  h i 2 −1



η Þ

, ηg Þ = ð2J34 Þ
Þ + ð
J 35, ± ðn
ÞÞ − 4
J 34
J 36, ± ðn
Þ − γ3 ðE, − J35, ± ðn A35, ± ðE, n g EEM at the Fermi level can be written from (A.22a) as
FBHD , n
, ηg Þ =
* ± ðE m


2
′ h
FBHD , n
, ηg Þ A 35, ± ðE 2

(4:21b)

Thus, EEM is a function of Fermi energy, Landau quantum number and the scattering potential. The electron concentration is given by
0 = n

i max h
g
v n
X eB
FBHD , n
34HD ðE
FBHD , n
34HD ðE

, η Þ + Z , η Þ Y g g 2π2
h2 n
= 0

(4:22)

where
FBHD , n
34HD ðE
, ηg Þ = Y

hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii
35HD, + ðE
FBHD , n
35HD, + ðE
FBHD , n
, ηg Þ + A
, ηg Þ A

and
34HD ðE
FBHD , n
, ηg Þ = Z


s X
r = 1


34HD ðE
FBHD , n
B ð
rÞ½Y
, ηg Þ L

Using (1.31f) and (4.22), we can study the entropy in this case.

4.2.3 Entropy in HD II–VI materials under magnetic quantization The magneto dispersion relation of the carriers in heavily doped II–VI materials are given by  

1=2

2eB 1 2eB 1 2


′ ′ ′


+ b 0 kz ± λ 0 γ3 ðE, ηg Þ = a 0 n+ n+
h 2
h 2

(4:23)

The (4.23) can be written as
k2 = U
8 ± ðE,
n
, ηg Þ z where "
 

1=2 #
a
′0 2eB 1 2eB 1 −1



′


∓ λ0 γ3 ðE, ηg Þ − n+ n+ U8 ± ðE, n, ηg Þ = ðb 0 Þ
h 2
h 2 EEM at the Fermi level can be written from (4.24a) as

(4:24a)

212

4 Entropy in heavily doped materials under magnetic quantization


2
′

FBHD , η Þ = h
, ηg Þ
*c ðE U 8 ± ðEFBHD , n m g 2

(4:24b)

The electron concentration is given by
0 = n

i max h
g
v n
X eB
FBHD , n
35HD ðE
FBHD , n
35HD ðE

, η Þ + Z , η Þ Y g g π2
h2 n
= 0

(4:25)

where
FBHD , n
35HD ðE
, ηg Þ = Y

hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii
8 + ðE
FBHD , n
8 − ðE
FBHD , n
, ηg Þ + U
, ηg Þ U

and
34HD ðE
FBHD , n
, ηg Þ = Z


s X
r = 1


35HD ðE
FBHD , n
B ð
rÞ½Y
, ηg Þ L

Using (1.31f) and (4.25), we can study the entropy in this case.

4.2.4 Entropy in HD IV–VI materials under magnetic quantization The electron energy spectrum in IV–VI materials are defined by the models of Cohen, Lax, Dimmock and Bangert and Kastner, respectively. The magneto entropy in HD IV–VI materials is discussed in accordance with the said model for the purpose of relative comparison. (a) Cohen Model In accordance with the Cohen model, the dispersion law of the carriers in IV–VI materials is given by
p

2y p
4y
2y ð1 + αEÞ αE αp
2x
2 p
+ αEÞ
= p + z − + + Eð1
1 2m
3 2m
2 2m 2m
′2
2m
′2 4m

(4:26)


1, m
2 and m
3 are the effective carrier masses at the
i =
h
ki , i =
x,
y,
z, m where, p ′
2 is the effective-mass band-edge along
x,
y and
z directions respectively and m tensor component at the top of the valence band (for electrons) or at the bottom of the conduction band (for holes). The magneto electron energy spectrum in IV–VI materials in the presence of
along
z-direction can be written as quantizing magnetic field B

 1 1 2 2

h2
kz2 2

± 1 g
* μ B
+ 3α n
+ αEÞ
= n

+
ωðEÞ h h
+ n + ω ð EÞ + Eð1 0
3 2m 2 2 8 2

(4:27a)

4.2 Theoretical background

213

where, " #1=2

2 m j e jB

ωðEÞ ≡ pffiffiffiffiffiffiffiffiffiffiffi 1 + αEð1 − Þ
1m
2 m m′2 Therefore, the magneto dispersion law in HD IV–VI materials can be expressed as
2
kz2
h
n
, ηg Þ = U16, ± ðE,
3 2m

(4:27b)

where, "




2 1
eB h 1 *
3α 2 1
heB

η Þ− n

, ηg Þ= γ2 ðE,



ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi p p + μ + n + ∓ B − g n U 16, ± ðE, n g 0
1m
1m
2 2
2 2 8 2 m m "



## 
 2 2


2
2 α 1 h
e B 3α 1 h
e B m m 2
η Þ
+
+n
+ pffiffiffiffiffiffiffiffiffiffiffi 1 − pffiffiffiffiffiffiffiffiffiffiffi + − γ3 ðE, 1− n n g
1m
1m
2
2 8 2 2 2
′2
′2 m m m m EEM at the Fermi level can be written from (4.27b) as
FBHD , n
FBHD , n
, ηg Þ = m
3 U ′16, ± ðE
, ηg Þ
*± ðE m

(4:27c)

Thus, EEM is a function of Fermi energy, Landau quantum number and the scattering potential. The carrier statistics in this case can be expressed as  
r =
s max X
pffiffiffiffiffiffi n
X g
eB
16 + ðE
FBHD , n
16 + ðE
FBHD , n
B ð
rÞðU
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
3
0 = v 2 m ðU L n π2
h
r = 1
=0 n (4:28) Using (1.31f) and (4.28), we can study the entropy in this case. (b) Lax Model In accordance with this model, the magneto dispersion relation assumes the form
 1
2
kz2 1
+h

+ αEÞ
= n
03 ðEÞ
+
ω h ± μ g
* B Eð1
3 2 0 2m 2

(4:29)

where
eB
≡ pffiffiffiffiffiffiffiffiffiffiffi ω03 ðEÞ
1m
2 m The magneto-dispersion relation in HD IV–VI materials, can be written following (A.30) as

214

4 Entropy in heavily doped materials under magnetic quantization


 1
2
kz2 1 *
h
η Þ= n
+ h
ω γ2 ðE, + ðEÞ + ± g
μ0 B 03 g
3 2 2m 2

(4:30)

(4.30) can be written as
2
kz2
h
n
, ηg Þ = U17, ± ðE,
3 2m

(4:31a)

where
 1


± 1 g
* μ B


, ηg Þ = γ2 ðE, ηg Þ − n
+
hω03 ðEÞ U17, ± ðE, n 0 2 2 EEM at the Fermi level can be written from (4.31a) as
FBHD , η Þ = m
FBHD , n
3 U ′17, ± ðE
, ηg Þ
* ðE m g

(4:31b)

The electron concentration can be written as  
r =
s max X
pffiffiffiffiffiffi n
X g
eB
17 + ðE
FBHD , n
17 + ðE
FBHD , n
B ð
rÞðU
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
0 = v 2 m
3 ðU n L π2
h
r = 1
=0 n (4:32) Using (1.31f) and (4.32), we can study the entropy in this case. (c) Dimmock Model The dispersion relation under magnetic quantization in HD IV–VI materials can be expressed in accordance with Dimmock model as    2
2


2eB 1 h
1 1 1 1
η Þ
x
h
+ + αγ − ð E, − n g 3
t+ m
t−
l+ m
l− 2 m
h 2 2 m " #
h2
ks2
h2
kz2
h2
ks2
h2
kz2
h4
ks4
h2
ks2
kz2
h2
kz2
ks2
h4
kz4 = + + + +α + + +
t− 2m
l−
t− m
l− m
t+ m
l− m
t+ 4m
l+ 4m
t− 4m
l+
*t
*l 2m 4m 2m 2m


η Þ + αγ ðE,
η Þ γ2 ðE, g g 3




2eB 1
h2 1
= n+
t+
h 2 2 m 


h4 2eB
+ ð +α n
t− m
t+
h 4m


 1
h2 1 1
− − +x − −
t
l
*l m 2 m m  4
  

1 2
h eB
h4 eB 1
h4 2
+
x Þ +
x + + n
l− m
l+ m
t+ m
t−
h 2m
l−
h
l+ 4m 2 2m 2 (4:33)

where
x =
kz2 Therefore, the magneto dispersion relation in HD IV–VI materials, whose unperturbed carriers obey the Dimmock Model can be expressed as

4.2 Theoretical background


k2 = U
170 ðE,
n
, ηg Þ z

215

(4:34)

where
n
n
n
n
170 ðE,
9 ðE,
, ηg Þ = ½2p
9 ðE,
, ηg Þ + ½q
29 ðE,
, ηg Þ + 4p
, ηg Þ1=2 ,
9  − 1 ½ − q
9 R (4:35) U "
  


α
h4
h2 1 1 α
h3 eB 1 1 1



9 =
+ , q ð E, n , η Þ = + + p n + g
l− m
l+ 9
l+ m
t+
l−
t− m
l− m
*l m 4m 2 m 2 2 m

η Þ 1 + 1
γ3 ðE, −α g
l+ m
l− m and "
 
2 # 2 1 1 1 α
h 1
n
η Þ + αeBγ
ðE,
η Þ n
n
9 ðE,
, ηg Þ = γ3 ðE,
+
+
h + − − − + eB R g g 3
l+ m
l
t m
t 2 m 2 m


(4:36a) EEM at the Fermi level can be written from (4.34) as
2
′
h
FBHD , n
, ηg Þ = U
, ηg Þ
* ðE m 17 ðEFBHD , n 2

(4:36b)

Thus, EEM is a function of Fermi energy, Landau quantum number and the scattering potential. The electron concentration can be written as  
r =
s max X
n
X g
eB
17 ðE
FBHD , n
FBHD , n

rÞðU
17 ðE
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
0 = v 2 (4:37) ðU Lð n π2
h n
= 0
r = 1 Using (1.31f) and (4.37), we can study the entropy in this case. (d) Model of Bangert and Kastner In accordance with this model [8], the carrier energy spectrum in HD IV–VI materials can be written following (3.68) as
k2 k
y2 s +
η Þ ρ2 ðE,
η Þ =1 ρ211 ðE, g g 12 where 1 1
η Þ = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
η Þ = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi, , ρ12 ðE, ρ11 ðE, g g
S1 ðE,
S2 ðE,
η Þ
η Þ g g

(4:38)

216

4 Entropy in heavily doped materials under magnetic quantization

" 2 i = 1 ðRÞ o ð
SÞ2 n
n
S1 ðE,
η Þ = 2γ ðE,
η Þ





E
g Þ


2 E,

c2 ðα ð α Þ − i D ð α Þ c E, E E, E 1 1 g 1 1 g g g 0
g E Δ′c h

o o
2n
2 ðα
3 ðα3 E,
E
g Þ + ðQÞ
c3 ðα
E
g Þ − iD
E
g Þ
2 E,
3 E, − iD Δ′′c

and " h i − 1 ð2AÞ
2
S2 ðE,
η Þ + 2γ ðE,
η Þ






g g 0
g c1 ðα1 E, Eg Þ − iD1 ðα1 E, Eg Þg E 
+ QÞ
2 ðS
3 ðα
E
g Þ − iD
E
g Þg
3 E,
3 E, + f
c3 ðα Δ′′c
η Þ and S
2 ðE,
η Þ are complex, the energy spectrum is also complex in Since
S1 ðE, g g the presence of Gaussian band tails. Therefore, the magneto dispersion law in the presence of a quantizing magnetic field B which makes an angle θ with kz axis can be written as
k2 = U
18 ðE,
n
, ηg Þ z

(4:39a)

where h i
n
η Þsin2 θ + ρ2 ðE,
η Þcos2 θ −
18 ðE,
, ηg Þ = ρ211 ðE, U g g 12 
i 2eB 1 h 2
−1 2 2
2
2
2 3=2
ðn + Þ ðρ11 ðE, ηg Þρ12 ðE, ηg ÞÞ fρ11 ðE, ηg Þsin θ + ρ12 ðE, ηg Þcos θg
h 2 EEM at the Fermi level can be written from (A.39a) as
2 h
FBHD , n
FBHD , n
18 ðE
, ηg Þ =
, ηg Þ
*c ðE Real part of ½U m 2

(4:39b)

Thus, EEM is a function of Fermi energy, Landau quantum number and the scattering potential and the orientation of the applied quantizing magnetic field. The electron concentration can be written as n
X  
r =
s max X
eg
B
18 ðE
FBHD , n
FBHD , n

rÞðU
18 ðE
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
0 = v 2 Real Part of ðU Lð n π2
h
r = 1
=0 n (4:40) Using (1.31f) and (4.40), we can study the entropy in this case. (e) Model of Foley and Langenberg The dispersion relation of the conduction electrons of IV–VI materials in accordance with Foley et al. can be written as [9]

217

4.2 Theoretical background

 1=2


+ ð
kÞ + Eg 2 + P
2
k2 + P
2
k2
0 ð
kÞ + ½E
+ Eg = E E ? s jj z 2 2
+ ð
kÞ = where E


h2
ks2
+ 2m ?

+

kz2 h2


+ 2m jj


− ð
kÞ = ,E

ks2 h2


− 2m ?

+

kz2 h2


− 2m jj

(4:41)

represents the contribution from the

interaction of the conduction and the valence band edge states with the more distant bands and the free electrons term, m
1± = 21 ½m
1 ± m
1 , m
1± = 21 ½m
1 ± m
1  tc tv lc lv ? jj Following the methods as given in chapter 1, the dispersion relation in HD IV–VI materials in the present case is given by "

" # #2 " # 2
2 2
2 2
2 2
2
2
g  E E k k k k
h h

h h
g0 0 s z s z
η Þ+ − + = + + γ3 ðE, g
?− 2m
jj−
?− 2m
jj− 2 2m 2m 4 " # 2
2 2
2 k k h
h
s z
g
2
k2
2
k2 + P +E + +P jj z ? s 0
?− 2m
jj− 2m

(4:42)

Therefore, the magneto-dispersion relation in HD IV–VI materials can be written as #2 2 h


h e B 1 h
x
η Þ+
η Þ
g γ ðE,
+ Þ+ ðn − 2 γ3 ðE, +E g g 0 3 − −
?
jj 4 m 2 2m " # " #
g 
heBð
n
n
+ 21Þ
+ 21Þ E
heBð
h2
x
h2
x + 0 + + = − − +
?
?
jj
jj+ 2 m m 2m 2m " #2  




heB 1
h2
x 1
2
x + P
2 2eB n

+ + + P n + + jj ?
?+
jj+ m 2 h
2 2m


η Þ+ γ23 ðE, g


2 E g0

"

(4:43)

where
kz2 =
x Therefore, the magneto dispersion relation in IV–VI HD materials, where unperturbed carriers follow the model of Foley et al. can be expressed as
k2 = U
19 ðE,
n
, ηg Þ z

(4:44a)

Where  n o1=2  −1 2






, ηg Þ = ½2p
91 ðE, n
, ηg Þ + q
91 ðE, n
, ηg Þ + 4p
, ηg Þ
91 
91 R91 ðE, n U19 ðE, n −q 2 3
4 4 1 h 1 5
91 = − p 4 ðm
jj− Þ2
2jj Þ2 ðm

218

4 Entropy in heavily doped materials under magnetic quantization

"

#    2
2
2

2

g 
+ 21 h
E E h
e B n
h e B 1
h g 2 0 0
+
n
η Þ+
, ηg Þ =
91 ðE,
+ +P − + − γ3 ðE, q n g jj
jj+
?− m
?− m
jj+
jj−
jj 2m 2 m m 2 m  
g 


E 2
heB 1 2
0






γ3 ðE, ηg Þ + R91 ðE, ηg , nÞ = γ3 ðE, ηg Þ + Eg0 γ3 ðE, ηg Þ − n+
?− 2 m 2



1 1
2 . 2eB n
g
heB n
+
+ −P −E ? 0 m?+ 2
h 2 EEM at the Fermi level can be written from (A.44a) as
2 h
FBHD , n
FBHD , n
, ηg Þ = U ′19 ðE
, ηg Þ
* ðE m 2

(4:44b)

Thus, as noted already in this case also EEM is a function of Fermi energy, Landau quantum number and the scattering potential. The electron concentration can be written as  
r =
s max X
n
X g
eB
19 ðE
FBHD , n
FBHD , n

rÞðU
19 ðE
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
0 = v 2 (4:45) ðU Lð n π2
h n
= 0
r = 1 Using (1.31f) and (4.45), we can study the entropy in this case.

4.2.5 Entropy in HD stressed Kane-type materials under magnetic quantization The dispersion relation of the conduction electrons in HDKane-type materials can be written following (1.206) of Chapter 1 as
k2
k2
k2 y x z + +
η Þ b
η Þ =1
2 ðE,
η Þ
c2 ðE,
2jj ðE, a g g g jj jj

(4:46)

where 1 1
jj ðE,
η Þ = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
η Þ = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jj ðE, ffi ,b a g g



η Þ Pjj ðE, ηg Þ Qjj ðE, g and 1
η Þ = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cjj ðE, ffi g
Sjj ðE,
η Þ g The electron energy spectrum in HDKane-type materials in the presence of an arbi
1 ,
β1 and
γ1 with trarily oriented quantizing magnetic field B which makes an angle α kx ,ky and kz axes respectively, can be written as

4.2 Theoretical background


41 ðE,
n
, ηg Þ ðk′z Þ2 = U

219

(4:47a)

Where
n
η Þ½1 −
I3 ðE,
n
41 ðE,
, ηg Þ =
I2 ðE,
, ηg Þ U g
11 ðE,
η Þ = ½½a
η Þ2 cos2 α1 + ½b
η Þ2 cos2 β + ½
c11 ðE,
η Þ2 cos2 γ 
I2 ðE,
11 ðE, g g g 1 g 1 and
2eB 1


I3 ðE,
n
η Þb
η Þ − 1 ½
I2 ðE,
η Þ1=2
, ηg Þ =
+ Þ½a
11 ðE, c11 ðE, ðn g 11 ðE, ηg Þ
g g h
2 EEM at the Fermi level can be written from (4.47a) as
FBHD , n
, ηg Þ =
* ðE m


2 ′
h
, ηg Þ U 41 ðEFBHD , n 2

(4:47b)

From (4.47b) we observe that EEM is a function of Fermi energy, Landau quantum number, the scattering potential and the orientation of the applied quantizing magnetic field. The electron concentration can be written as  
r =
s max X
n
X g
eB
41 ðE
FBHD , n
FBHD , n

rÞðU
41 ðE
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
0 = v 2 (4:48) ðU Lð n π2
h n
= 0
r = 1 Using (1.31f) and (4.48), we can study the entropy in this case.

4.2.6 Entropy in HD Te under magnetic quantization The magneto dispersion relation of the conduction electrons in HD Te can be expressed as
k2 = U
42 ± ðE,
n
, ηg Þ z

(4:49a)

where 

 n o
eB 1 2

Þ
2γ3 ðE,ηg Þψ1 +ψ3 −4ψ1 ψ2 ðn + Þ − ψ43 +4ψ1 ψ23 γ3 ðE,η g
h 2   −1=2
8eB 1
+ Þðψ21 ψ24 −ψ1 ψ2 ψ3 Þ + ðn
h 2


n
42,± ðE,
,ηg Þ=ð2ψ21 Þ−1 U

EEM at the Fermi level can be written from (4.49a) as

220

4 Entropy in heavily doped materials under magnetic quantization


2 h
FBHD , n
FBHD , n
, ηg Þ = U ′42 ± ðE
, ηg Þ
*± ðE m 2

(4:49b)

Thus from (4.49b), we note that EEM is a function of three variables namely Fermi energy, Landau quantum number and the scattering potential. The electron concentration can be written as  
r =
s max X
n
X g
eB
42, ± ðE
FBHD , n
FBHD , n

rÞðU
42, ± ðE
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
0 = v 2 ðU Lð n π2
h n
= 0
r = 1 (4:49c) Using (1.31f) and (4.49c), we can study the entropy in this case. 4.2.7 Entropy in HD Gallium Phosphide under magnetic quantization The magneto-dispersion relation in HD GaP can be written following (1.248) of chapter 1 as
k2 = U
43 ðE,
n
, ηg Þ z

(4:50a)

where  
1
2 Þ − 1 2γ ðE,
+
c − 2D
− 4a
eB ðn
n
η Þb
b
43 ðE,
, ηg Þ = ð2b

b U + Þ g 3
h 2 
1

cγ ðE,
− 8eB ðn
η Þ + 4D
2 b2 − 4
cD
b
+ Þ + ½
c2 + 4b g 3
h 2  − 1=2  2 2 2 2













ð2ab D + 4γ3 ðE, ηg Þb a + abc − 2b aγ3 ðE, ηg Þ − b cÞ 2
2 h
2
k0
b
=
h ,
c = h
= + b, a
*?
*jj
*jj 2m 2m m

!2
= jVG j , D

EEM at the Fermi level can be expressed from (4.50a) as
2
′
h
FBHD , n
, ηg Þ = U
, ηg Þ
* ðE m 43 ðEFBHD , n 2

(4:50b)

Thus, from (4.50b) it appears that EEM is the function of Fermi energy, Landau quantum number and the scattering potential. The electron concentration can be written as  
r =
s max X
n
X g
eB
43 ðE
FBHD , n
FBHD , n

rÞðU
43 ðE
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
0 = v 2 (4:50c) ðU Lð n π2
h n
= 0
r = 1 Using (1.31f) and (4.50c), we can study the entropy in this case.

4.2 Theoretical background

221

4.2.8 Entropy in HD platinum antimonide under magnetic quantization The magneto dispersion relation in HD PtSb2 can be written as
k2 = U
44 ðE,
n
, ηg Þ z

(4:51a)

where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1


n
2 ðE,
n
41 T
n
71 ðE,
44 ðE,
, ηg Þ =
, ηg Þ + T
, ηg Þ + 4T
, ηg Þ, ½ T ð E, n U 71 71
41 2T   1
n
51 ðE,
η Þ−T
31 2eB ðn
71 ðE,
, ηg Þ = T
T + Þ g
h 2 and   1


n
61 ðE,
η Þ + 2eB ðn
72 ðE,
, ηg Þ = T
ð E, η Þ + Þ T T 21 g g
h 2 EEM at the Fermi level can be written from (4.51a) as
2 h
FBHD , n
FBHD , n
, ηg Þ = U ′44 ðE
, ηg Þ
* ðE m 2

(4:51b)

Thus, from the above equation we infer that EEM is a function of Landau quantum number, the Fermi energy and the scattering potential. The electron concentration can be written as  
r =
s max X
n
X g
eB
45 ðE
FBHD , n
FBHD , n

rÞðU
45 ðE
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
0 = v 2 (4:52) ðU Lð n π2
h n
= 0
r = 1 Using (1.31f) and (4.52), we can study the entropy in this case.

4.2.9 Entropy in HD Bismuth Telluride under magnetic quantization The magneto-dispersion relation in HD Bi2Te3 can be written as
k2 = U
45 ðE,
η ,n g
Þ x

(4:53a)

where
η ,n
45 ðE, U g
Þ = and


e
hB
η Þ − ðn
+ 21Þ M γ2 ðE,
g 31

ω1

222

4 Entropy in heavily doped materials under magnetic quantization


31 = M


0 m
22 α
33 − α


23 Þ2 ðα 4

1=2

EEM at the Fermi evel can be written from (4.53a) as
FBHD , η Þ =
* ðE m g


2 ′
h
, ηg Þ U 45 ðEFBHD , n 2

(4:53b)

The electron concentration can be written as  
r =
s max X
n
X g
eB
44 ðE
FBHD , n
FBHD , n

rÞðU
44 ðE
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
0 = v 2 ðU n Lð π2
h n
= 0
r = 1

(4:54)

Using (1.31f) and (4.54), we can study the entropy in this case.

4.2.10 Entropy in HD Germanium under magnetic quantization (a) Model of Cardona et al. The magneto-dispersion relation in HD Ge can be written following (1.300) as
k2 = U
46 ðE,
η ,n g
Þ x

(4:55a)

where
n
46 ðE,
, ηg Þ = U


*jj 2m
2 h

"

" 2 ##1=2

g
g
h2 2eB
E E E 1 g0 0 0

γ3 ðE, ηg Þ + − + ðn + Þ
*?
h 2 4 2 m

EEM at the Fermi level can be written from (4.55a) as
2 h
FBHD , n
FBHD , n
, ηg Þ = U ′46 ðE
, ηg Þ
* ðE m 2

(4:55b)

From (4.55b) it appears that EEM is a function of Fermi energy and Landau quantum number due to band non-parabolicity. The electron concentration can be written as  
r =
s max X
n
X g
eB
46 ðE
FBHD , n
FBHD , n

rÞðU
46 ðE
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
0 = v 2 (4:56) ðU Lð n π2
h n
= 0
r = 1 Using (1.31f) and (4.56), we can study the entropy in this case.

223

4.2 Theoretical background

(b) Model of Wang and Ressler The magneto-dispersion relation in HD Gecan be written following (1.321) as
k2 = U
47 ðE,
η ,n g
Þ x

(4:57a)

where
n
47 ðE,
, ηg Þ = U


*jj m
6
2 α h

!" 1 − α5

# ! n o1=2
1 eB

Þ − 4α6 γ3 ðE, ηg Þ
+ n , ω? = *
hω? − θ7 ðn
? 2 m

and "

 2

 1 1 1
Þ = 1 + ðα5 Þ2 ðn
+ Þ
hω? − 2α5 n
+
+ θ7 ðn
hω? + 4α6 n
hω? 2 2 2  2  1
+ Þ
hω? − 4α6 α4 ðn 2 EEM at the Fermi level can be written from (4.57a) as
2 h
FBHD , n
FBHD , n
, ηg Þ = U ′47 ðE
, ηg Þ
* ðE m 2

(4:57b)

From (4.57b) we note that the mass is a function of Fermi energy and quantum number due to band non-parabolicity. The electron concentration under the condition of extreme degeneracy can be written as  
r =
s max X
n
X g
eB
47 ðE
FBHD , n
FBHD , n

rÞðU
47 ðE
, ηg ÞÞ1=2 +
, ηg ÞÞ1=2
0 = v 2 (4:58) ðU Lð n π2
h n
= 0
r = 1 Using (1.31f) and (4.58), we can study the entropy in this case.

4.2.11 Entropy in HD Gallium antimonide under magnetic quantization The magneto-dispersion relation in HD GaSbcan be written following (1.338) as
k2 = U
48 ðE,
η ,n g
Þ x

(4:59a)

where "

"( ) 
′g Þ2

α ð E 2e B 1 10 − 1 2 2 0
n
η Þ + α9 E
47 ðE,
′g +
, ηg Þ = −
+ + ð2α9 Þ 2α9 γ3 ðE, U n g 0 4
h 2

224

4 Entropy in heavily doped materials under magnetic quantization

!


 i
1
η Þg1=2 , ω? = eB
Þ − 4α6 γ3 ðE,
hω? − fθ7 ðn
1 − α n + 5 g 2
*? 2 m
6
h α ( )1=2 
′g Þ 4
′g Þ 3 α210 ðE α α ð E 9 10 2 2 2
′ 0 0
η ÞðE
′g Þ + − α9 ðE g0 Þ + + α9 α10 γ3 ðE, g 0 16 2
 2 2
h 2
h 1 1 α9 = and α10 = ′ −
0

2m m m c 0
g E 0
*jj m

EEM at the Fermi level can be written from (4.59a) as
2 ′

FBHD , η Þ = h
, ηg Þ
* ðE U 48 ðEFBHD , n m g 2

(4:59b)

The electron concentration can be written as  
r =
s max X
n
X g
v eB 1=2 1=2








ðU48 ðEFBHD , n, ηg ÞÞ + n0 = LðrÞðU48 ðEFBHD , n, ηg ÞÞ π2
h2 n
= 0
r = 1

(4:60)

Using (1.31f) and (4.60), we can study the entropy in this case.

4.2.12 The entropy in HD II–V materials under magnetic quantization The dispersion relation of the holes are given by [10] h i1=2 2
2 + θ2 k
2 + θ3 k
2 + δ4 k
x m
2 + θ6 k
2 + θ7 k
2 + δ5 k
x g2 + G2 k
2
= θ1 k
fθ5 k ± Δ3 E 3 y + Δ3 x y z x y z (4:61)
x , k
y and k
z are expressed in the units of where, k 1
1 Þ, θ2 = 1 ða
2 Þ, θ3 = 1 ða
3 Þ, δ4 = 1 ðxA
+ BÞ,

1 + b
2 + b
3 + b θ1 = ða 2 2 2 2 1
1 Þ, θ6 = 1 ða
2 Þ, θ7 = 1 ða
3 Þ, δ5 = 1 ðA
− BÞ,

1 − b
2 − x b
3 − b θ5 = ða 2 2 2 2
i , A,
3 and Δ3 are system constants
B,
G
i ði = 1, 2, 3, 4Þ, b a The hole energy spectrum in HD II–V materials can be expressed following the method of Chapter 1 as
ng Þ = θ1
k2 + θ2
k2 + θ3
k2 + δ4
kx ± y3 ðE, x y z

n 1 o2 2 2 2 2 2
2 2




θ5 kx + θ6 ky + θ7 kz + δ5 kx + G3 ky + Δ3 ± Δ3 (4:62)

the magneto dispersion law in HD II–V materials assumes the form

4.2 Theoretical background


k2 = U
49, ± ðE,
η ,n g
Þ y

225

(4:63a)

where,
n
η Þ +
I36, ± ðn
η Þ + γ ðE,
η Þ
I38, ± ðn
49, ± ðE,
, ηg Þ = − ½
I35 γ3 ðE,
Þ ± ½γ23 ðE,
Þ +
I39, ± ðn
Þ1=2  U g g g 3
I35 =

θ2 ðθ22

− θ25 Þ


Þ = ,
I36, ± ðn


I 33, ± ðnÞ
Þ = ð4θ25 Þ − 1 ½4θ2
I33, ± ðn
Þ + 8θ22
I31, ± ðn
Þ ,
I38, ± ðn 2ðθ22 − θ25 Þ


Þ, − θ25
I31, ± ðn
I39, ± = ð4θ2 Þ − 1 ½
I 2 33, ± ðn
Þ + 4θ32
I34, ± ðn
Þ − 4θ35
I34, ± ðn
Þ,
I33, ± ðn
Þ 5
2 + 2θ5
I32 ðn
Þ − 2θ2
I31, ± ðn
Þ, = ½G 3 2 2

I34, ± ðn
Þ = ½
I32,

± ðnÞ + Δ3 − I31, ± ðnÞ, " # " # δ2 δ2 1 1
I31, ± ðn
Þ = ðn
+ Þ
hω
Þ = ðn
+ Þ
hω
31 − 4 ± Δ3 ,
I32 ðn
32 − 5 , 4θ1 4θ5 2 2




eB
eB
2

2
h
h2
31 = h
32 = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , M
31 = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ω , M32 = , M33 = ω
31 M
33 M
32
34 2θ1 2θ3 2θ5 M M and
2
34 = h . M 2θ7 EEM at the Fermi level can be written from (4.63a) as
FBHD , n
, ηg Þ =
*± ðE m


2 ′ h
FBHD , n
, ηg Þ U 49, ± ðE 2

(4:63b)

From (4.63b) we note that EEM is a function of Fermi energy, Landau quantum number and the scattering potential. The electron concentration under extreme degeneracy can be written as  
r =
s max X
n
X g
v eB 1=2 1=2





, ηg ÞÞ +
, ηg ÞÞ
0 = ðU49, ± ðEFBHD , n Lð
rÞðU49, ± ðEFBHD , n n π2
h2 n
= 0
r = 1 (4:64) Using (1.31f) and (4.64), we can study the entropy in this case.

226

4 Entropy in heavily doped materials under magnetic quantization

4.2.13 Entropy in HD lead germanium telluride under magnetic quantization The dispersion relation of the carriers in n-type Pb1-xGaxTe with x=0.01 can be written following Vassilev [11] as
+E
g + 0.411
k2 + 0.0377
k2  = 0.23
k2 + 0.02
k2
− 0.606
k2 − 0.0722
k2 ½E ½E s z s z s z 0
g + 0.061
k2 + 0.0066
k2 ks ± ½0.06E 0

s

(4:65)

z


g = ð0.21eVÞis the energy gap for the transition point, the zero of the energy where E 0
E is at the edge of the conduction band of the Γpoint of the Brillouin zone and is
− 1. measured positively upwards,
kx ,
ky and
kz are in the units of 109 m The magneto-dispersion law in HD Pb1-xGexTe can be expressed following the methods as given in Chapter 1 as " #  
η Þ
2θ0 ðE, eB 1 g


Þ + γ3 ðE, ηg Þ Eg0 − 0.345x − 0.390
h ðn + 2Þ 1 + Erf ðE=η g

0.46eB 1 1
g + 0.122 eB ðn
+ Þ + 0.02
x ± ½0.06E
+ Þ ðn 0
h 2
h 2  

2eB 1 1=2 0.822e B 1
g +
+ ÞÞ + E

ðn ð n + Þ + 0.377 x + 0.0066
xð 0
h 2
h 2  
1.212eB 1
+ Þ + 0.722
x ðn (4:66)
h 2 =

The (4.66) assumes the form
k2 = U
50, ± ðE,
η ,n g
Þ z

(4:67a)

Where  n o1=2 
50, ± ðE,
n
n
n
n
10, ± ðE,
, ηg Þ = ð2p
10 Þ − 1 − q
10 ðE,
, ηg Þ + q
210 ðE,
, ηg Þ + 4p
10 R
, ηg Þ U " 1=2

η Þ ± 0.0066ð2B
ðn
10 ðE, n
, ηg Þ = 0.02 + 0.345γ3 ðE,
+ 21ÞÞ
10 ð0.377 × 0.722Þ, q p g h
" + 0.377 × and

"
10, ∓ ðE,
n
, ηg Þ = R


1.212
eB
1 h ðn + 2Þ + 0.722


##
g + 0.822 eB
ðn
+ 21Þ E h
0

 
η Þ

2θ0 ðE, 1 g
g − 0.390 eB n
ðE, η Þ E + γ + g 3 0
Þ
h 2 1 + Erf ðE=η g

 1=2

1 2eB 1
g + 0.122
eB n
+ Þ
+ ∓ ð0.06E ðn 0 2
h 2
h

227

4.3 Results and discussion


 


1 1.212eB 1 0.46eB 1
g + 0.822
eB ðn


− E Þ ð n + Þ − ð n + Þ + 0 2
h 2
h 2
h EEM at the Fermi level can be written from (4.67a) as
FBHD , η Þ =
*∓ ðE m g


2
′ h
FBHD , n
, ηg Þ U 50, ∓ ðE 2

(4:67b)

Thus, from (4.67b) we note that EEM is a function of the Fermi energy, Landau quantum number and the scattering potential. The electron concentration under extreme degeneracy can be written as  
r =
s max X
n
X g
v eB 1=2 1=2








ðU50, ± ðEFBHD , n, ηg ÞÞ + LðrÞðU50, ± ðEFBHD , n, ηg ÞÞ n0 = π2
h2 n
= 0
r = 1 (4:68) Using (1.31f) and (4.68), we can study the entropy in this case.

4.3 Results and discussion Using the appropriate equations and the energy band constants as given in Appendix 15, in Figure 4.1 the normalized entropy has been plotted as a function of inverse magnetic field for HD Cd3As2 as shown in plot (a) of Figure 4.1 where the plot (b) represents the case for δ = 0 and has been drawn to assess the influence of crystal field splitting on the entropy in HD Cd3As2. The plot (c) and (d) in the same Figure refer to the three and two band models of Kane; whereas the plot (e) exhibits the parabolic energy bands. Figure 4.2 exhibits the plot of the normalized entropy as a function of impurity concentration for HD Cd3As2 for all cases of Figure 4.1. The plot (a) of Figure 4.3 shows the variation of the normalized entropy in HD Cd3As2 as a function of orientation of the quantizing magnetic field for δ≠0 and the plot (b) refers for δ = 0. Figures 4.4 to 4.6 represent the variation of entropy as functions of inverse quantizing magnetic field, impurity concentration and angular orientation of the quantizing magnetic field for HD CdGeAs2 for the respective cases of Figures 4.1 to 4.3. It should be noted that under varying
− 3 , while, magnetic field, the concentration has been set to the value of 1024 m under varying electron concentration, the magnetic field is fixed to 2 tesla. It appears from Figures 4.1 and 4.4 that the entropy oscillates with1=B. It is well known that density-of-states in materials under magnetic quantization exhibits oscillatory dependence with inverse quantizing magnetic field, which is being reflected in this case. In fact, all electronic properties of electronic materials in the presence of quantizing magnetic field exhibit periodic variation with inverse quantizing magnetic field. The origin of oscillations of the entropy is the same as

228

4 Entropy in heavily doped materials under magnetic quantization

0.10 0.09 0.08

Normalized entropy

0.07

(c)

0.06

(a)

(d) 0.05

(b)

(e) 0.04 0.03 0.02 0.01 0

0

0.5

1

1.5

2

2.5

3

Inverse magnetic field (1/B) in tesla–1 Figure 4.1: Plot of the normalized entropy as function of inverse magnetic field for HD Cd3As2 in accordance with the (a) generalized band model (δ≠0), (b) δ = 0, (c) three- and (d) two-band models of Kane together with parabolic energy bands (e).

that of the Shubnikov de Haas oscillations. The influence of crystal field splitting, on the entropy can easily be conjectured by comparing the appropriate plots of Figures 4.1 and 4.4. Besides, the differences among three and two band models of Kane together with parabolic energy bands for entropy of HD Cd3As2, and HD CdGeAs2 can easily be assessed by comparing the appropriate plots of Figures 4.1 and 4.3. From Figures 4.2 and 4.5, it appears that entropy oscillates with impurity concentration in HD Cd3As2 and HD CdGeAs2 with different numerical values exhibiting the signature of the SdH effect. Although the rates of variations are different, the influence of spectra constants on all types of band models follows the same trend as observed in Figures 4.2 and 4.5, respectively. From Figures 4.3 and 4.6, it appears that the entropy shows sinusoidal dependence with increasing θ and the variation is periodically repeated which appears from these figures. For three- and two-band models of Kane together with parabolic energy bands, the entropy becomes θ invariant and for this reason these plots are not shown in Figures 4.3 and 4.6 respectively. The

4.3 Results and discussion

229

10,000

1,000

100 Normalized entropy

(d) 10

(b)

Impurity concentration (1024 m–3)

(c)

1 0

10

20

30

40

50 60

70

80

90 100

0.1 (a)

0.01 (e) 0.001 Figure 4.2: Plot of the normalized entropy as function of impurity concentration for HD Cd3As2 for all the cases of Figure 4.1.

0.037

Normalized entropy

0.036 0.035 0.034 0.033 (a) 0.032 0.031

(b)

0.030 0

50

100

150

200

250

300

350

θin degrees Figure 4.3: Plot of the normalized entropy as function of angular orientation of the quantizing magnetic field for HD Cd3As2 for (a) δ≠0 and (b) δ = 0.

230

4 Entropy in heavily doped materials under magnetic quantization

0.25

Normalized entropy

0.20

0.15 (a) (c)

0.10

(b)

0.05

0 0

0.5

1

1.5

2

2.5

3

Inverse magnetic field (1/B) in tesla–1 Figure 4.4: Plot of the normalized entropy as a function of inverse magnetic field for HD CdGeAs2 in accordance with the generalized band model (a) δ≠0, (b) δ = 0 (c) three- and (d) two-band models of Kane.

10,000 1,000

Normalized entropy

100

Impurity concentration(1024 m–3)

10

(c)

1 10

20

30 40 50

60

70

80

90 100

(a)

0.1 0 (b) 0.01

(d) 0.001 Figure 4.5: Plot of the normalized entropy as a function of impurity concentration for HD CdGeAs2 for all the cases of Figure 4.4.

4.3 Results and discussion

231

0.017 0.016 0.015

Normalized entropy

0.014 0.013 0.012 0.011 0.010 (a) 0.009 (b) 0.008 0

50

100

150 200 250 Θin degrees

300

350

Figure 4.6: Plot of the normalized entropy as a function of angular orientation of the quantizing magnetic field for HD CdGeAs2 for (a) δ≠0 and (b) δ = 0.

normalized entropy for InAs and InSb as a function of 1/B has been plotted in Figures 4.7 and 4.9 for three and two-band models of Kane together with parabolic energy bands, respectively. The normalized entropy as a function of impurity concentration for three and two band models of Kane for HD InAs and InSb has been plotted in Figures 4.8 and 4.10, respectively. It appears from the numerical values that the influence of the three band model of Kane in the energy spectrum of III-V, ternary and quaternary compounds are difficult to distinguish from that of the two band model of Kane. The normalized entropy has been plotted as a function of 1/B for HD p-CdS in Figure 4.11 where the plots (a) and (b) are valid for
λ0 = 0 and
λ0 ≠0 respectively. Figure 4.12 exhibits the plot of the same as a function of impurity concentration for all cases of Figure 4.11. The influence of the term
λ0 which represents the splitting of the two-spin states by the spin orbit coupling and the crystalline field is apparent from Figures 4.11 and 4.12. The normalized entropy in HD PbTe accordance the models of Cohen, Lax, Dimmock, Bangert and Kastner and Foley and Langenberg have been plotted in Figures 4.13 and 4.14, respectively, as functions of inverse quantizing magnetic

232

4 Entropy in heavily doped materials under magnetic quantization

0.050 0.045 0.040

Normalized entropy

0.035 (b)

0.030

(a) 0.025 0.020 0.015 0.010 0.005

(c)

0 0

0.5

1

1.5

2

2.5

3

Inverse magnetic field (1/B) in tesla–1 Figure 4.7: Plot of the normalized entropy as a function of inverse magnetic field for HD InAs in accordance with the (a) three and (b) two-band models of Kane together with (c) the parabolic energy bands.

field and impurity concentration, respectively. Figures 4.15 and 4.16 exhibit the plots of normalized entropy in this case as functions of inverse quantizing magnetic field and impurity concentration for HD PbSnTe. Figures 4.17 and 4.18 demonstrate the same for stressed HD InSb and it appears that the influence of stress leads to the enhancement of the entropy in this case. The influence of spin splitting has not been considered in obtaining the oscillatory plots since the peaks in all figures would increase in number with decrease in amplitude if spin splitting term is included in the respective numerical computations without introducing new physics. The effect of collision broadening has not been taken into account in this simplified analysis although the effects of collisions are usually small at low temperatures, the sharpness of the amplitude of the oscillatory plots would somewhat be reduced by collision broadening. Nevertheless, the present analysis would remain valid qualitatively since the effects of collision broadening can usually be taken into account by an effective increase in temperature. Although in a more rigorous statement

4.3 Results and discussion

233

1,000

Normalized entropy

100

10

Impurity concetration (1024 m–3)

1 10

0

20

30

40

50

60

70

80 90

100

0.1

(a)

0.01

(b) 0.001 Figure 4.8: Plot of the normalized entropy as a function of impurity concentration for HD InAs in accordance with the (a) three and (b) two-band energy models of Kane.

0.050 0.045

Normalized entropy

0.040 0.035 0.030 (c)

0.025

(a)

0.020 0.015

(b)

0.010 0.005 0 0

0.5

1

1.5

2

2.5

3

Inverse magnetic field(1/B) in tesla–1 Figure 4.9: Plot of the normalized entropy as a function of inverse magnetic field for HD InSb in accordance with the (a) three- and (b) two-band models of Kane together with (c) the parabolic energy bands.

234

4 Entropy in heavily doped materials under magnetic quantization

1,000

Normalized entropy

100

10 Impurity concentration (1024 m–3) 1 0

10

20

30

40

50

60

70

80

90 100

(a)

0.1

0.01 (b) 0.001 Figure 4.10: Plot of the normalized entropy as a function of impurity concentration for HD InSb in accordance with the (a) three and (b) two-band models of Kane.

4

3.5

Normalized entropy

3

2.5

2 (b) 1.5

(a)

1

0.5

0 0

0.5

1

1.5

2

2.5

3

Inverse magnetic field (1/B)in tesla–1 Figure 4.11: Plot of the normalized entropy as a function of inverse magnetic field in HD p-CdS for
0 ≠0.
0 = 0 and (b) λ (a) λ

4.3 Results and discussion

235

1,000,000 Normalized entropy

100,000 10,000 Impurity concentration (1024m–3)

1,000 100

(a) (b)

10 1 0

10

20

30

40

50

60

70

80

90 100

0.1 Figure 4.12: Plot of the normalized entropy as a function of impurity concentration field in HD
0 ≠0.
0 = 0 and (b) λ p-CdS for (a) λ

5 4.5

Normalized entropy

4

(b)

3.5 (a)

3 2.5

(e)

(c)

2 1.5 1 0.5 0 0

0.5

1

1.5

2

2.5

3

Inverse magnetic field (1/B) in telsa–1 Figure 4.13: Plot of the normalized entropy as a function of inverse magnetic field for HD PbTe in accordance with the models of (a) Cohen, (b) Lax, (c) Dimmock, (d) Bangert and Kastner and (e) Foley and Langenberg, respectively.

the effect of electron–electron interaction should be considered along with the self-consistent procedure, the simplified analysis as presented in this chapter exhibits the basic qualitative features of the entropy in the present case for degenerate materials having various band structures under the magnetic quantization with reasonable accuracy.

236

4 Entropy in heavily doped materials under magnetic quantization

1,000,000

Normalized entropy

100,000 10,000 1,000 100

(a)

(b)

10 1 0.1

(d)

0

(c)

10

20

(e) Impurity concentration (1024m–3) 30 40 50 60 70 80 90 100

Figure 4.14: Plot of the normalized entropy as a function of impurity concentration for HD PbTe for all the cases of Figure 4.13.

0.07

0.06

Normalized entropy

0.05

0.04

0.03

0.02

0.01

0 0

0.5

1

1.5

2

2.5

3

Inverse magnetic field (1/B) in telsa–1 Figure 4.15: Plot of the normalized entropy as a function of inverse magnetic field for HD PbSnTe.

4.4 Open research problems

237

100,000

Normalized entropy

10,000 1,000 100 10 Impurity concentration (1024 m–3)

1 0

10

20

30

40

50

60

70

80

90 100

0.1 0.01 Inverse magnetic field (1/B) in telsa–1 Figure 4.16: Plot of the normalized entropy as a function of impurity concentration field for HD PbSnTe.

0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 0

0.5

1

1.5

2

2.5

3

Inverse magnetic field (1/B) in telsa–1 Figure 4.17: Plot of the normalized entropy as a function of inverse magnetic field for stressed HD InSb.

4.4 Open research problems (R 4.1) Investigate the entropy both in the presence and the absence of an arbitrarily oriented quantizing magnetic field by considering all types of scattering mechanisms including broadening and the electron spin (applicable

238

4 Entropy in heavily doped materials under magnetic quantization

10,000

Normalized entropy

1,000 100 Impurity concentration (1024m–3)

10 1 0

20

40

60

80 100 120 140 160 180 200

0.1 0.01 0.001

Figure 4.18: Plot of the normalized entropy as a function of impurity concentration field for stressed HD InSb.

(R 4.2)

(R 4.3)

(R 4.4)

(R 4.5)

(R 4.6)

under magnetic quantization) for all the bulk materials whose unperturbed carrier energy spectra are defined in Chapter 1. Investigate the entropy considering all types of scattering mechanisms in the presence of quantizing magnetic field under an arbitrarily oriented (a) non-uniform electric field and (b) alternating electric field respectively for all the materials whose unperturbed carrier energy spectra are defined in Chapter 1 by including spin and broadening, respectively. Investigate the entropy by considering all types of scattering mechanisms under an arbitrarily oriented alternating quantizing magnetic field by including broadening and the electron spin for all the materials whose unperturbed carrier energy spectra as defined in Chapter 1. Investigate the entropy by considering all types of scattering mechanisms under an arbitrarily oriented alternating quantizing magnetic field and crossed alternating electric field by including broadening and the electron spin for all the materials whose unperturbed carrier energy spectra as defined in Chapter 1. Investigate the entropy by considering all types of scattering mechanisms under an arbitrarily oriented alternating quantizing magnetic field and crossed alternating nonuniform electric field by including broadening and the electron spin whose for all the materials unperturbed carrier energy spectra as defined in Chapter 1. Investigate the entropy in the presence and absence of an arbitrarily oriented quantizing magnetic field by considering all types of scattering mechanisms under exponential, Kane, Halperin, Lax, and Bonch–Bruevich band

References

(R 4.7)

(R 4.8)

(R 4.9)

(R 4.10)

(R 4.11)

239

tails [4] for all the materials whose unperturbed carrier energy spectra as defined in Chapter 1 by including spin and broadening (applicable under magnetic quantization). Investigate the entropy in the presence ofan arbitrarily oriented quantizing magnetic field by considering all types of scattering mechanisms for all the materials as defined in (R5.6) under an arbitrarily oriented (a) nonuniform electric field and (b) alternating electric field, respectively, whose unperturbed carrier energy spectra as defined in Chapter 1. Investigate the entropy by considering all types of scattering mechanisms for all the materials as described in (R5.6) under an arbitrarily oriented alternating quantizing magnetic field by including broadening and the electron spin whose unperturbed carrier energy spectra as defined in Chapter 1. Investigate the entropy by considering all types of scattering mechanisms as discussed in (R5.6) under an arbitrarily oriented alternating quantizing magnetic field and crossed alternating electric field by including broadening and the electron spin for all the materials whose unperturbed carrier energy spectra as defined in Chapter 1. Investigate all the appropriate problems of this chapter after proper modifications introducing new theoretical formalisms for functional, negative refractive index, macro molecular, organic and magnetic materials. Investigate all the appropriate problems of this chapter for HD p-InSb, p-CuCl and stressed materials having diamond structure valence bands whose dispersion relations of the carriers in bulk materials are given by Cunningham [6], Yekimov et al. [7], and Roman et al. [8], respectively. (replace the green part by (r1.14 to r1.17)

References [1]

[2]

Miura N., Physics of Materials in High Magnetic Fields, Series on Semiconductor Science and Technology (Oxford University Press, USA, 2007); Buschow K.H.J.F.R. de Boer, Physics of Magnetism and Magnetic Materials (Springer, New York, 2003); Sellmyer D. (Ed.), Skomski R. (Ed.), Advanced Magnetic Nanostructures (Springer, New York, 2005); Bland J.A.C. (Ed.), Heinrich B. (Ed.), Ultrathin Magnetic Structures III: Fundamentals of Nanomagnetism (Pt. 3) (Springer-Verlag, Germany, 2005); B.K. Ridley, Quantum Processes in Materials, Fourth Edition. Ghatak K.P., Mondal M., Zietschrift fur Naturforschung A41a, 881 (1986); Ghatak K.P., Mondal M., J. Appl. Phys., 62, 922 (1987); Ghatak K.P., Biswas S.N., Phys. Stat. Sol. (b), 140, K107 (1987); Ghatak K.P., Mondal M., (1988) Jour. of Mag. and Mag. Mat. 74, 203; Ghatak K.P.,Mondal. M., Phys. Stat. Sol. (b), 139, 195 (1987); Ghatak K.P., Mondal M., Phys. Stat. Sol. (b), 148, 645 (1988); Ghatak K.P., Mitra B., Ghoshal A., Phys. Stat. Sol. (b) 154, K121 (1989); Ghatak K.P., Biswas S.N., Jour. of Low Temp. Phys., 78, 219 (1990); Ghatak K.P., Mondal M., Phys. Stat. Sol. (b) 160, 673 (1990); Ghatak K.P., Mitra B., Phys. Letts. A156, 233 (1991); Ghatak K.P., Ghoshal A., Mitra B., Nouvo Cimento D 13D, 867 (1991); Ghatak K.P., Mondal M., (1989) Phys. Stat. Sol. (b) 148, 645;

240

4 Entropy in heavily doped materials under magnetic quantization

Ghatak K.P., Mitra B., Internat. Jour. of Elect. 70, 345 (1991); Ghatak K.P., Biswas S.N., J. Appl. Phys. 70, 299 (1991); Ghatak K.P., Ghoshal A., Phys. Stat. Sol. (b) 170, K27 (1992); Ghatak K.P., Nouvo Cimento D 13D, 1321 (1992); Ghatak K.P., Mitra B., Internat. Jour. of Elect. 72, 541 (1992); Ghatak K.P., Biswas S.N., Nonlinear Optics 4, 347 (1993); Ghatak K.P., Mondal M., Phys. Stat. Sol. (b) 175, 113 (1993); Ghatak K.P., Biswas S.N., Nonlinear Optics 4, 39 (1993); Ghatak K.P., Mitra B., Nouvo Cimento 15D, 97 (1993); Ghatak K.P., Biswas S.N., (1993) Nanostructured Materials 2, 91; Ghatak K.P., Mondal M., Phys. Stat. Sol. (b) 185, K5 (1994); Ghatak K.P., Goswami B., Mitra M., Nag B., Nonlinear Optics 16, 9 (1996); Ghatak K.P., Mitra M., Goswami B., Nag B., Nonlinear Optics 16, 167 (1996); Ghatak K.P., Nag B., Nanostructured Materials 10, 923 (1998). [3] Roy Choudhury D., Choudhury A.K., Ghatak K.P., Chakravarti A.N., Phys. Stat. Sol. (b) 98, K141 (1980); Chakravarti A.N., Ghatak K.P., Dhar A., Ghosh S., Phys. Stat. Sol. (b) 105, K55 (1981); Chakravarti A.N., Choudhury A.K., Ghatak K.P., (1981) Phys. Stat. Sol. (a) 63, K97; Chakravarti A.N., Choudhury A.K., Ghatak K.P., Ghosh S., Dhar A., Appl. Phys. 25, 105 (1981); Chakravarti A.N., Ghatak K.P., Rao G.B., Ghosh K.K., Phys. Stat. Sol. (b) 112, 75 (1982); Chakravarti A.N., Ghatak K.P., Ghosh K.K., Mukherjee H.M., Phys. Stat. Sol. (b) 116, 17 (1983); Mondal M., Ghatak K.P., (1984) Phys. Stat. Sol. (b) 133, K143; Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 126, K47 (1984); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 126, K41 (1984); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 129, K745 (1985); Mondal M., Ghatak K.P., Phys. Scr. 31, 615 (1985); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 135, 239 (1986); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 93, 377 (1986); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 135, K21 (1986); Mondal M., Bhattacharyya S., Ghatak K.P., Appl. Phys. A42A, 331 (1987); Biswas S.N., Chattopadhyay N., Ghatak K.P., Phys. Stat. Sol. (b) 141, K47 (1987); Mitra B., Ghatak K.P., Phys. Stat. Sol. (b) 149, K117 (1988); Mitra B., Ghoshal A., Ghatak K.P., Phys. Stat. Sol. (b) 150, K67 (1988); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 147, K179 (1988); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 146, K97 (1988); Mitra. B., Ghoshal A., Ghatak K.P., Phys. Stat. Sol. (b) 153, K209 (1989); Mitra B., Ghatak K.P.,) Phys. Letts. 142A, 401 (1989); Mitra B., Ghoshal A., Ghatak K.P., Phys. Stat. Sol. (b) 154, K147 (1989); Mitra B., Ghatak K.P., Sol. State Elect. 32, 515 (1989); Mitra B., Ghoshal A., Ghatak K.P., Phys. Stat. Sol. (b) 155, K23 (1989); Mitra B., Ghatak K.P., Phys. Letts. 135A, 397 (1989); Mitra B., Ghatak K.P., Phys. Letts. A146A, 357 (1990); Mitra B., Ghatak K.P., Phys. Stat. Sol. (b) 164, K13 (1991); Biswas S.N., Ghatak K.P., Internat. Jour. of Elect. 70, 125 (1991). [4] Wallace P.R., Phys. Stat. Sol. (b) 92, 49 (1979). [5] Nag B.R., Electron Transport in Compound Materials, Springer Series in Solid-State Sciences, Vol. 11 (Springer-Verlag, Germany, 1980). [6] Ghatak K.P., Bhattacharya S., De D., Einstein Relation in Compound Materials and Their Nanostructures, Springer Series in Materials Science, Vol. 116 (Springer-Verlag, Germany, 2009). [7] Wu C.C., Lin C.J., J. Low Temp. Phys 57, 469 (1984); Chen M.H., Wu C.C., Lin C.J., J. Low Temp. Phys. 55, 127 (1984). [8] Bangert E., Kastner P., Phys. Stat. Sol (b) 61, 503 (1974). [9] Foley G.M.T., Langenberg P.N., Phys. Rev. B 15B, 4850 (1974). [10] Singh M., Wallace P.R., Jog S.D., Arushanov E., J. Phys. Chem. Solids 45, 409 (1984); Yamada Y., Phys. Soc. Japan, 35, 1600, 37, 606 (1974). [11] Vassilev L.A., Phys. Stat. Sol. (b) 121, 203 (1984).

5 The entropy in heavily doped nanomaterials under magneto-size quantization I must pray as if everything depends on God but i must work as if everything depends on me.

5.1 Introduction In Section 5.2.1, of the theoretical background, the entropy has been investigated in ultra-thin films (UFs) of HD nonlinear optical materials in the presence of a quantizing magnetic [1–15] (Place the references at the indicated place). field.Section 5.2.2 contains the results for UFs of HD III–V, ternary, and quaternary compounds in accordance with the three- and the two-band models of Kane. In the same section, the entropy in accordance with the models of Stillman etal. and Palik etal. have also been studied for the purpose of relative comparison. Section 5.2.3 contains the study of the entropy for UFs of HD II–VI materials under magnetic quantization. In Section 5.2.4, the entropy in UFs of HD IV–VI materials has been discussed in accordance with the models of Cohen, Lax, Dimmock, Bangert and Kastner, and Foley and Landenberg, respectively. In Section 5.2.5, the magnetoentropy for the stressed UFs of HDKane-type materials has been investigated. In Section 5.2.6, the entropy in UFs of HD Te has been studied under magnetic quantization. In Section 5.2.7, the magnetoentropy in UFs of HD n-GaP has been studied. In Section 5.2.8, the entropy in ultra-thin film of HD PtSb2 has been explored under magnetic quantization. In Section 5.2.9, the magnetoentropy in UFs of HD Bi2Te3 has been studied. In Section 5.2.10, the entropy in UFs of HD Ge has been studied under magnetic quantization in accordance with the models of Cardona etal. and Wang and Rossler, respectively. In Sections 5.2.11 and 5.2.12, the magnetoentropy in UFs of HD n-GaSb and II–V compounds have respectively been studied. In Section 5.2.13, the magnetoentropy in UFs of HD Pb1 – xGexTe has been discussed.Section 5.3 explores the result and discussion and it contains 12 open research problems for this chapter.

5.2 Theoretical background 5.2.1 Electron energy spectrum in HD nonlinear optical materials under magneto-size quantization The entropy of the conduction electrons in UFs of HD nonlinear optical materials in the presence of a quantizing magnetic field B can be written following (4.3a) as

https://doi.org/10.1515/9783110661194-005

242

5 The entropy in heavily doped nanomaterials under magneto-size quantization


2 h




z π n
z d

2
1, ± ðe81 , n
2, ± ðe81 , n
, ηg Þ + U
, ηg Þ =U


*jj 2m

(5:1)

where e81 is the totally quantized energy in this case. The DOS function is given by
zmax max n X

n
X
− e81 Þ
Bn = gv
eB δ′ðE N z 2π
h n
= 0 n
z = 1

(5:2)

The electron concentration can be expressed as
0 = n


zmax
X n max n X
g
v eB
− 1 ðη Þ F Re al Part of 5, 1 2π
h
z = 1
=0 n n

(5:3)


F5 − e81 Þ, E
F5 is the Fermi energy in this case and F
j ðηÞ is the
− 1 ðE where η5, 1 = ð
kB TÞ one parameter Fermi-Dirac integral of order j as defined in (1.85). Thus, using (5.3) and (1.31f), we can study the entropy in this case.

5.2.2 Entropy in QWs of HD III–V materials under magneto-size quantization (a) Three-band model of Kane In accordance with three-band model of Kane, DR in the present case can be written as
2 h




z π n
z d
*jj 2m

2
3, ± ðe82 , n
4, ± ðe82 , n
, ηg Þ + U
, ηg Þ =U

(5:4)

where e82 is the totally quantized energy in this case. The DOS function is given by
zmax max n X

n
X
− e82 Þ
Bn = gv eB δ′ðE N z 2π
h n
= 0 n
z = 1

(5:5)

The electron concentration can be expressed as


0 = n


zmax
X n max n X
g
v eB
− 1 ðη Þ F Re al Part of 6, 1 2π
h
z = 1
=0 n n

where
F5 − e82 Þ,
− 1 ðE η6, 1 = ð
kB TÞ Thus, using (5.6) and (1.31f), we can study the entropy in this case.

(5:6)

5.2 Theoretical background

243

(b) Two band model of Kane The electron energy spectrum in this case is given by
2 h




z π n
z d

2


c 2m


 1 1

, ηg Þ − n
+
hω0 ∓ g
* μ0 B = γ2 ðe83 , n 2 2

(5:7)

where e83 is the totally quantized energy in this case. The DOS function is given by
zmax max n X
n
X g
v eB
− e83 Þ
δ′ðE NBnz = 2π
h n
= 0 n
z = 1

(5:8)

The electron concentration can be expressed as
0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 7, 1 2π
h n
= 0 n
z = 1

(5:9)

where
F5 − e83 Þ,
− 1 ðE η7, 1 = ð
kB TÞ Thus, using (5.9) and (1.31f), we can study the entropy in this case. (c) Parabolic Energy Bands The electron energy spectrum in this case is given by
2 h




z π n
z d


c 2m

2


 1 1

+
ω0 ∓ g
* μ0 B h = γ3 ðe84 , ηg Þ − n 2 2

(5:10)

where e84 is the totally quantized energy in this case. The DOS function is given by
zmax max n X

n
X
−
e84 Þ
Bn = gv eB δ′ðE N z 2π
h n
= 0 n
z = 1

(5:11)

The electron concentration can be expressed as
0 = n


zmax max n X
n
X g
v eB
− 1ð η Þ F 8, 1 2π
h n
= 0 n
z = 1

where
F5 − e84 Þ,
− 1 ðE η8, 1 = ð
kB TÞ Thus, using (5.12) and (1.31f), we can study the entropy in this case.

(5:12)

244

5 The entropy in heavily doped nanomaterials under magneto-size quantization

(d) The model of Stillman et al. The DR in the present case can be written as
2
z π n
7 ðe85 , n
, ηg Þ =U
z d

(5:13)

where e85 is the totally quantized energy in this case. The DOS function is given by
zmax max n X

n
X
− e85 Þ
Bn = gv eB δ′ðE N z π
h n
= 0 n
z = 1

(5:14)

The electron concentration can be expressed as
0 = n


zmax max n X
n
X g
v
eB
− 1 ðη Þ F 9, 1 π
h n
= 0 n
z = 1

(5:15)

where
F5 − e85 Þ,
− 1 ðE η9, 1 = ð
kB TÞ Thus, using (5.15) and (1.31f), we can study the entropy in this case. (e) The model of Palik et al. DR in the present case can be written as
2
z π n
, ηg Þ = A35, ± ðe86 , n
z d

(5:16)

where e86 is the totally quantized energy in this case. The DOS function is given by
zmax max n X
n
X g
v eB
− e86 Þ
δ′ðE NBnz = 2π
h n
= 0 n
z = 1

(5:17)

The electron concentration can be expressed as
0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 10, 1 2π
h n
= 0 n
z = 1

where
F5 − e86 Þ,
− 1 ðE η10, 1 = ð
kB TÞ Thus, using (5.18) and (1.31f), we can study the entropy in this case.

(5:18)

5.2 Theoretical background

245

5.2.3 Entropy in HD II–VI materials under magneto-size quantization DR in the present case can be written as
2
z π n
8, ± ðe87 , n
, ηg Þ =U
z d

(5:19)

where e87 is the totally quantized energy in this case. The DOS function is given by
zmax max n X

n
X
− e87 Þ
Bn = gv eB δ′ðE N z 2π
h n
= 0 n
z = 1

(5:20)

The electron concentration can be expressed as
0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 11, 1 2π
h n
= 0 n
z = 1

(5:21)

where
F5 − e87 Þ,
− 1 ðE η11, 1 = ð
kB TÞ Thus, using (5.21) and (1.31f), we can study the entropy in this case.

5.2.4 Entropy in HD IV–VI materials under magneto size-quantization The electron energy spectrum in IV–VI materials is defined by the models of Cohen, Lax, Dimmock and Bangert, and Kastner, respectively. The magnetoentropy in HD IV–VI materials is discussed in accordance with the said model for the purpose of relative comparison. (a) Cohen Model DR in the present case can be written as
2
z π n
16, ± ðe88 , n
, ηg Þ =U
z d

(5:22)

where e88 is the totally quantized energy in this case. The DOS function is given by
zmax max n X
n
X
− e88 Þ
Bn = g
v eB δ′ðE N z 2π
h n
= 0 n
z = 1

The electron concentration can be expressed as

(5:23)

246

5 The entropy in heavily doped nanomaterials under magneto-size quantization


0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 12, 1 2π
h n
= 0 n
z = 1

(5:24)

where
F5 − e88 Þ,
− 1 ðE η12, 1 = ð
kB TÞ Thus, using (5.24) and (1.31f), we can study the entropy in this case. (b) Lax Model DR in the present case can be written as
2
z π n
17, ± ðe89 , n
, ηg Þ =U
z d

(5:25)

where e89 is the totally quantized energy in this case. The DOS function is given by
zmax max n X
n
X
− e89 Þ
Bn = g
v eB δ′ðE N z 2π
h n
= 0 n
z = 1

(5:26)

The electron concentration can be expressed as
0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 13, 1 2π
h n
= 0 n
z = 1

(5:27)

where
F5 − e89 Þ,
− 1 ðE η13, 1 = ð
kB TÞ Thus, using (5.27) and (1.31f), we can study the entropy in this case. (c) Dimmock Model DR in the present case can be written as
2
z π n
170 ðe90 , n
, ηg Þ =U
z d

(5:28)

where e90 is the totally quantized energy in this case. The DOS function is given by
zmax max n X
n
X g
v eB
− e90 Þ
δ′ðE NBnz = π
h n
= 0 n
z = 1

The electron concentration can be expressed as

(5:29)

5.2 Theoretical background


0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 14, 1 π
h n
= 0 n
z = 1

247

(5:30)

where
F5 − e90 Þ,
− 1 ðE η14, 1 = ð
kB TÞ Thus, using (5.30) and (1.31f), we can study the entropy in this case. (d) Model of Bangert and Kastner DR in the present case can be written as
2 
z π n
18 ðe91 , n
, ηg Þ = U
z e=0 d

(5:31)

where e91 is the totally quantized energy in this case. The DOS function is given by
zmax max n X
n
X
− e91 Þ
Bn = g
v eB δ′ðE N z π
h n
= 0 n
z = 1

(5:32)

The electron concentration can be expressed as
0 = n


zmax
X n max n X
g
v eB
− 1 ðη Þ F Re al Part of 15, 1 π
h
z = 1
=0 n n

(5:33)

where
F5 − e91 Þ,
− 1 ðE η15, 1 = ð
kB TÞ Thus, using (5.33) and (1.31f), we can study the entropy in this case. (e) Model of Foley and Langenberg DR in the present case can be written as
2
z π n
19 ðe92 , n
, ηg Þ =U
z d

(5:34)

where e92 is the totally quantized energy in this case. The DOS function is given by
zmax max n X
n
X g
v eB
− e92 Þ
δ′ðE NBnz = π
h n
= 0 n
z = 1

The electron concentration can be expressed as

(5:35)

248

5 The entropy in heavily doped nanomaterials under magneto-size quantization


0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 16, 1 π
h n
= 0 n
z = 1

(5:36)

where
F5 − e92 Þ,
− 1 ðE η16, 1 = ð
kB TÞ Thus, using (5.36) and (1.31f), we can study the entropy in this case.

5.2.5 Entropy in HD stressed Kane type materials under magneto-size quantization DR in the present case can be written as
2 
z π n
41 ðe93 , n
, ηg Þ = U
z e=0 d

(5:37)

where e93 is the totally quantized energy in this case. The DOS function is given by
zmax max n X

n
X
− e93 Þ
Bn = gv eB δ′ðE N z π
h n
= 0 n
z = 1

(5:38)

The electron concentration can be expressed as
0 = n


zmax
X n max n X
g
v eB
− 1 ðη Þ F Re al Part of 17, 1 π
h
z = 1
=0 n n

(5:39)

where
F5 − e93 Þ,
− 1 ðE η17, 1 = ð
kB TÞ Thus, using (5.39) and (1.31f), we can study the entropy in this case.

5.2.6 Entropy in HD Te under magneto size-quantization DR in the present case can be written as
2
z π n
42 ± ðe94 , n
, ηg Þ =U
z d where e94 is the totally quantized energy in this case. The DOS function is given by

(5:40)

5.2 Theoretical background


zmax max n X
n
X
− e94 Þ
Bn = g
v eB δ′ðE N z π
h n
= 0 n
z = 1

249

(5:41)

The electron concentration can be expressed as
0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 18, 1 π
h n
= 0 n
z = 1

(5:42)

where
F5 − e94 Þ,
− 1 ðE η18, 1 = ð
kB TÞ Thus, using (542) and (1.31f), we can study the entropy in this case.

5.2.7 Entropy in HD gallium phosphide under magneto size quantization DR in the present case can be written as
2
z π n
43 ðe94 , n
, ηg Þ =U
z d

(5:43)

where e94 is the totally quantized energy in this case. The DOS function is given by
zmax max n X
n
X g
v eB
− e94 Þ
δ′ðE NBnz = π
h n
= 0 n
z = 1

(5:44)

The electron concentration can be expressed as
0 = n


zmax max n X
n
X g
v
eB
− 1 ðη Þ F 19, 1 π
h n
= 0 n
z = 1

where
F5 −
e94 Þ
− 1 ðE η19, 1 = ð
kB TÞ Thus, using (5.45) and (1.31f), we can study the entropy in this case.

(5:45)

250

5 The entropy in heavily doped nanomaterials under magneto-size quantization

5.2.8 Entropy in HD platinum antimonide under magneto size quantization The DR in the present case can be written as
2
z π n
44 ðe95 , n
, ηg Þ =U
z d

(5:46)

where e95 is the totally quantized energy in this case. The DOS function is given by
zmax max n X

n
X
− e95 Þ
Bn = gv eB δ′ðE N z π
h n
= 0 n
z = 1

(5:47)

The electron concentration can be expressed as
0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 20, 1 π
h n
= 0 n
z = 1

(5:48)

where
F5 − e95 Þ
− 1 ðE η20, 1 = ð
kB TÞ Thus, using (5.48) and (1.31f), we can study the entropy in this case.

5.2.9 Entropy in HD bismuth telluride under magneto size quantization DR in the present case can be written as
2
z π n
45 ðe96 , n
, ηg Þ =U
z d

(5:49)

where e96 is the totally quantized energy in this case. The DOS function is given by
zmax max n X
n
X
− e96 Þ
Bn = g
v eB δ′ðE N z π
h n
= 0 n
z = 1

(5:50)

The electron concentration can be expressed as
0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 21, 1 π
h n
= 0 n
z = 1

where
F5 − e96 Þ
− 1 ðE η21, 1 = ð
kB TÞ Thus, using (5.51) and (1.31f), we can study the entropy in this case.

(5:51)

5.2 Theoretical background

251

5.2.10 Entropy in HD germanium under magneto size quantization (a) Model of Cardona et al. DR in the present case can be written as
2
z π n
46 ðe97 , n
, ηg Þ =U
z d

(5:52)

where e97 is the totally quantized energy in this case. The DOS function is given by
zmax max n X
n
X
− e97 Þ
Bn = g
v eB δ′ðE N z π
h n
= 0 n
z = 1

(5:53)

The electron concentration can be expressed as
0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 22, 1 π
h n
= 0 n
z = 1

(5:54)

where
F5 − e97 Þ
− 1 ðE η22, 1 = ð
kB TÞ Thus, using (5.54) and (1.31f), we can study the entropy in this case. (b) Model of Wang and Ressler DR in the present case can be written as
2
z π n
47 ðe98 , n
, ηg Þ =U
z d

(5:55)

where e98 is the totally quantized energy in this case. The DOS function is given by
zmax max n X
n
X
− e98 Þ
Bn = g
v eB δ′ðE N z π
h n
= 0 n
z = 1

(5:56)

The electron concentration can be expressed as
0 = n

zmax max nX
nX g
v
eB
− 1 ðη Þ F 23, 1 π
h n = 0 n = 1 t

where
F5 −
e98 Þ
− 1 ðE η23, 1 = ð
kB TÞ Thus, using (5.57) and (1.31f), we can study the entropy in this case.

(5:57)

252

5 The entropy in heavily doped nanomaterials under magneto-size quantization

5.2.11 Entropy in HD gallium antimonide under magneto-size quantization DR in the present case can be written as
2
z π n
48 ðe99 , n
, ηg Þ =U
z d

(5:58)

where e99 is the totally quantized energy in this case. The DOS function is given by
zmax max n X

n
X
− e99 Þ
Bn = gv eB δ′ðE N z π
h n
= 0 n
z = 1

(5:59)

The electron concentration can be expressed as
0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 24, 1 π
h n
= 0 n
z = 1

(5:60)

where
F5 − e99 Þ
− 1 ðE η24, 1 = ð
kB TÞ Thus, using (5.60) and (1.31f), we can study the entropy in this case.

5.2.12 Entropy in HD II–V materials under magneto size quantization DR in the present case can be written as
2
z π n
49 ± ðe100 , n
, ηg Þ =U
z d

(5:61)

where e100 is the totally quantized energy in this case. The DOS function is given by
zmax max n X
n
X
− e100 Þ
Bn = g
v eB δ′ðE N z π
h n
= 0 n
z = 1

The electron concentration can be expressed as

(5:62)

5.3 Results and discussion


0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 25, 1 π
h n
= 0 n
z = 1

253

(5:63)

where
F5 − e100 Þ
− 1 ðE η25, 1 = ð
kB TÞ Thus, using (5.63) and (1.31f), we can study the entropy in this case.

5.2.13 Entropy in HD lead germanium telluride under magneto size quantization DR in the present case can be written as
2
z π n
50 ± ðe101 , n
, ηg Þ =U
z d

(5:64)

where e101 is the totally quantized energy in this case. The DOS function is given by
zmax max n X

n
X
− e101 Þ
Bn = gv eB δ′ðE N z π
h n
= 0 n
z = 1

(5:65)

The electron concentration can be expressed as
0 = n


zmax max n X
n
X g
v eB
− 1 ðη Þ F 26, 1 π
h n
= 0 n
z = 1

(5:66)

where
F5 − e101 Þ Thus, using (5.66) and (1.31f), we can study the entropy
− 1 ðE η26, 1 = ð
kB TÞ in this case.

5.3 Results and discussion Using the appropriate equations, in Figure 5.1, the normalized entropy in ultrathin films of tetragonal materials (taking HD Cd3As2 as an example) as a function of inverse magnetic field have been plotted in curve (a) where as the curve (b) of the same figure represents the same variation for HD CdGeAs2 (an example of non-linear optical material)in accordance with the generalized band model (δ ≠ 0). The curve (c) is valid for III-V materials (taking InSb as an example). The three-band energy model of Kane for InSb is valid for such highly non-parabolic material. The influence of energy band constants for the three

254

5 The entropy in heavily doped nanomaterials under magneto-size quantization

Normalized entropy

100 10–1 (c)

10–2

(a) (b)

10–3 10–4 10–5 10–6 0

1

2

3

4

5

Figure 5.1: Plot of the normalized entropy as a function of inverse magnetic field for HD UFs of (a) Cd3As2 and (b) CdGeAs2 in accordance with the generalized band model (δ ≠ 0). The plot (c) refers to n-InSb in accordance with the three band model of Kane (n0 = 1015 m − 2 and dz = 10nm).

aforementioned compounds can be estimated from the said curves. For all figure of
= 10K and consequently for the this chapter lattice temperature has been taken as T purpose of simplified numerical computation we have considered only the first subband occupancy in connection with the quantization due to the Born–Von Karman boundary condition for various Landau levels due to the quantizing magnetic field. It appears that the thermoelectric power exhibits a periodic oscillation with increase in the magnetic field, which has also been discussed in Chapter 5. In Figure 5.2, we have plotted the normalized entropy as a function of film thickness under constant magnetic field for all the cases of Figure 5.1. The entropy appears to exhibit composite oscillations because of the ad-mixture of size quantized levels with the Landau sub-bands. The nature of the variation of the entropy from a stair case to the highly zigzag can be explained as the combined influence of the magnetic quantization with the size quantization. As the thickness starts lowering, the influence of the field decreases due to which the stair case variation is retrieved. The entropy as function of carrier concentration for said materials for both mag
= 0) and size (n
z = 1) quantum limits has been plotted in Figure 5.3 from netic (n which we can conclude that the entropy decreases with carrier concentration for relatively large values where as for the relatively low values of the carrier degeneracy, the magneto thermo power shows the converging tendency. It appears from Figures 5.1 to 5.3 that HD InSb exhibits largest numerical entropy as compared to HD Cd3As2 and HD CdGeAs2 for UFs under magnetic quantization. In Figures 5.4 to 5.6, we have plotted the entropy for ultrathin films of HD II-VI and stressed III-V materials as functions of inverse magnetic field, thickness and carrier concentration
=2 respectively. The film thickness for Figures 5.4 and 5.6 are kept to 10nm, while B tesla for Figures 5.5 and 5.6 respectively.

5.3 Results and discussion

255

0.990

Normalized entropy

0.988 0.986 0.984

(c)

0.982 0.980

(b) (a)

0.978 10

20

30

40 50 60 70 80 Film thickness (in nm)

90

100

Figure 5.2: Plot of the normalized entropy as a function of film thickness for HD UFs of (a) Cd3As2 and (b) CdGeAs2 in accordance with the generalized band model (δ ≠ 0). The plot (c) refers to n-InSb in accordance with the three-band model of Kane.

1.0

Normalized entropy

0.8 (a) 0.6

(b)

0.4

(c)

0.2

0.0 10–2

10–1

10–0

101

102

Carrier concentration (x1014 m–2) Figure 5.3: Plot of the normalized entropy as a function of carrier concentration for HD UFs of (a) Cd3As2 and (b) CdGeAs2 in accordance with the generalized band model (δ ≠ 0). The plot (c) refers to n-InSb in accordance with the three-band model of Kane.

256

5 The entropy in heavily doped nanomaterials under magneto-size quantization

0.8

Normalized entropy

0.7 0.6

(b)

0.5 0.4 0.3 (a)

0.2 0.1 0.0 0

1

2

3

4

5

Inverse magnetic field (tesla–1) Figure 5.4: Plot of the normalized entropy as a function of inverse magnetic field for HD UFs of
0 ≠ 0) and (b) stressed HD InSb. (a) HD CdS (λ

1.000 (b)

Normalized entropy

0.995 0.990 0.985

(a) 0.980 0.975 10

20

30

40

50

60

70

80

90

100

Film thickness (in nm) Figure 5.5: Plot of the normalized entropy as a function film thickness for HD UFs of (a) HDCdS
0 ≠ 0) and (b) stressed HD InSb. (λ

It appears form the Figures 5.4 to 5.6 that the normalized entropy for UFs of stressed HD InSb exhibits higher numerical values as compared to the corresponding UFs of HD CdS. Figure 5.7 exhibits the plots of the normalized entropy as function of inverse magnetic field for UFs of HD PbSe in accordance with the models of (a) Lax and (b) Cohen, respectively. Besides the plot (c) in the same figure is valid for HD IV–VI materials (using HD PbTe as an example) whose carrier dispersion laws follow the Cohen model. The Figures 5.8 and 5.9 demonstrate the said variations as a function of film thickness and carrier concentration respectively. It

5.3 Results and discussion

257

1.1 1.0

Normalized entropy

0.9

(b)

0.8 0.7 (a)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 10–2

10–1

10–0

101

Carrier concentration (×1014 m–2) Figure 5.6: Plot of the normalized entropy as a function of carrier concentration for HD UFs of
0 ≠ 0) and (b) stressed HD InSb. (a) HD CdS (λ

0.96

100 (b)

0.94

Normalized entropy

10–1

0.92

(a)

10–2

0.90 10–3 0.88 (c)

10–4

0.86

10–5

0.84 0

0.5

1

1.5

2

2.5

3

3.5

4

10–6

4.5

5 0.82

Inverse magnetic field (tesla–1) Figure 5.7: Plot of the normalized entropy as a function of inverse magnetic field for UFs of HD PbSe in accordance with the (a) Lax and (b) Cohen models. The plot (c) refers to HD PbTe following Cohen model.

Normalized entropy

258

5 The entropy in heavily doped nanomaterials under magneto-size quantization

0.965 0.960 0.955 0.950 0.945 0.940 0.935 0.930 0.925 0.920 0.915 0.910 0.905 0.900 0.895

(a) (b)

(c)

10

20

30

40

50

60

70

80

90

100

Film thickness (in nm) Figure 5.8: Plot of the normalized entropy as a function of film thickness for UFs of HD PbSe in accordance with the (a) Lax and (b) Cohen models. The plot (c) refers to HD DHDPbTe following Cohen model.

1

Normalized entropy

0.95

(a)

0.90 0.85

(b)

0.8 (c)

0.75 0.70 0.01

0.1

1

Carrier concentration (x10

10 14 m–2)

Figure 5.9: Plot of the normalized entropy as a function of carrier concentration for UFs of HD PbSe in accordance with the (a) Lax and (b) Cohen models. The plot (c) refers to HD PbTe following Cohen model.

appears that the HD PbSe exhibits higher entropy than that of HD PbTe. For the purpose of simplicity the spin effects has been neglected in the computations. The inclusion of spin increases the number of oscillatory spikes by two with the decrement in amplitudes. The use of the data in the figures as presented in this chapter can also be used to compare the entropy for other types of materials.

5.4 Open research problems

259

5.4 Open research problems (R5.1) Investigate the entropy in the presence of arbitrarily oriented quantizing magnetic field in the presence of electron spin and broadening by considering all types of scattering mechanisms for UFs by considering the presence of finite, parabolic and circular potential wells applied separately for all the HD materials whose unperturbed carrier energy spectra are defined in Chapter 1. (R5.2) Investigate (R5.1) in the presence of an additional arbitrarily oriented (a) non-uniform electric field and (b) alternating electric field respectively for all the HD materials whose unperturbed carrier energy spectra are defined in this Chapter 1 by considering all types of scattering mechanisms. (R5.3) Investigate the entropy in the presence of arbitrarily oriented alternating quantizing magnetic field in the presence of electron spin and broadening by considering all types of scattering mechanisms for HD UFs by incorporating the presence of finite, parabolic and circular potential wells applied separately for all the HD materials whose unperturbed carrier energy spectra are defined in Chapter 1. (R5.4) Investigate the entropy under an arbitrarily-oriented alternating quantizing magnetic field and crossed alternating electric field by including broadening and the electron spin for HD UFs of all the materials whose unperturbed carrier energy spectra are defined in Chapter 1 by considering all types of scattering mechanisms. (R5.5) Investigate the entropy under an arbitrarily-oriented alternating quantizing magnetic field and crossed alternating nonuniform electric field by including broadening and the electron spin whose for HD UFs of all the materials unperturbed carrier energy spectra are defined in Chapter 1 by considering all types of scattering mechanisms. (R5.6) Investigate the entropy in the presence of a quantizing magnetic field under exponential, Kane, Halperin, Lax, and Bonch-Bruevich band tails [1] for HD UFs of all the materials whose unperturbed carrier energy spectra are defined in Chapter 1 by considering all types of scattering mechanisms. (R5.7) Investigate the entropy in the presence of quantizing magnetic field for HD UFs of all the materials as defined in (R5.6) under an arbitrarily oriented (a) non-uniform electric field and (b) alternating electric field, respectively, by considering all types of scattering mechanisms. (R5.8) Investigate the entropy for the HD UFs of all the materials as described in (R5.6) under an arbitrarily-oriented alternating quantizing magnetic field by including broadening and the electron spin by considering all types of scattering mechanisms. (R5.9) Investigate the entropy for HD UFs of all the materials as discussed in (R5.6) under an arbitrarily-oriented alternating quantizing magnetic field

260

5 The entropy in heavily doped nanomaterials under magneto-size quantization

and crossed alternating electric field by including broadening and the electron spin by considering all types of scattering mechanisms. (R5.10) Investigate all the appropriate problems after proper modifications introducing new theoretical formalisms for all types of HD UFs of all the materials as discussed in (R5.6) for functional, negative refractive index, macro molecular, organic, and magnetic materials by considering all types of scattering mechanisms in the presence of strain. (R5.11) Investigate all the appropriate problems of this chapter for all types of HD UFs for p-InSb, p-CuCl and materials having diamond structure valence bands whose dispersion relations of the carriers in bulk materials are given by Cunningham [2], Yekimov et al. [3], and Roman et al. [4], respectively by considering all types of scattering mechanisms in the presence of strain. (R5.12) Investigate the influence of deep traps and surface states separately for all the appropriate problems of all the chapters after proper modifications by considering all types of scattering mechanisms.

References [1]

[2]

Miura N., Physics of Materials in High Magnetic Fields, Series on Semiconductor Science and Technology (Oxford University Press, USA, 2007); Buschow K.H.J., de Boer F.R., Physics of Magnetism and Magnetic Materials (Springer, New York, 2003); Sellmyer D., (Ed.), Skomski R., (Ed.), Advanced Magnetic Nanostructures (Springer, New York, 2005); Bland J.A.C., (Ed.), Heinrich B., (Ed.), Ultrathin Magnetic Structures III: Fundamentals of Nanomagnetism (Pt. 3) (Springer-Verlag, Germany, 2005); Ridley B.K., Quantum Processes in Materials, Fourth Edition (Oxford publications, Oxford, 1999); Davies J.H., Physics of low dimensional Materials (Cambridge University Press, UK, 1998); Blundell S., Magnetism in Condensed Matter, Oxford Master Series in Condensed Matter Physics (Oxford University Press, USA, 2001); Weisbuch C., Vinter B., Quantum Semiconductor Structures: Fundamentals and Applications (Academic Publishers, USA, 1991); Ferry D., Semiconductor Transport (CRC, USA, 2000); Reed M., (Ed.), Materials and Semimetals: Nanostructured Systems (Academic Press, USA, 1992); Dittrich T., Quantum Transport and Dissipation (Wiley-VCH Verlag GmbH, Germany, 1998); Shik A.Y., Quantum Wells: Physics & Electronics of Two dimensional Systems (World Scientific, USA, 1997). Ghatak K.P., Mondal M., Zietschrift fur Naturforschung A41a, 881 (1986); Ghatak K.P., Mondal M., J. Appl. Phys.62, 922 (1987); Ghatak K.P., Biswas S.N., Phys. Stat. Sol. (b) 140, K107 (1987); Ghatak K.P., Mondal M., Jour. of Mag. and Mag. Mat.74, 203 (1988); Ghatak K.P., Mondal M., Phys. Stat. Sol. (b) 139, 195 (1987); Ghatak K.P., Mondal M., Phys. Stat. Sol. (b) 148, 645 (1988); Ghatak K.P., Mitra B., Ghoshal A., Phys. Stat. Sol. (b) 154, K121 (1989); Ghatak K.P., Biswas S.N., Jour. of Low Temp. Phys. 78, 219 (1990); Ghatak K.P., Mondal M., (1990) Phys. Stat. Sol. (b) 160, 673; Ghatak K.P., Mitra B., Phys. Letts. A156, 233 (1991); Ghatak K.P., Ghoshal A., Mitra B., Nouvo Cimento D13D, 867 (1991); Ghatak K. P., Mondal M., Phys. Stat. Sol. (b) 148, 645 (1989); Ghatak K.P., Mitra B., Internat. Jour. of Elect.70, 345 (1991); Ghatak K.P., Biswas S.N., J. Appl. Phys.70, 299 (1991); Ghatak K.P., Ghoshal A., Phys. Stat. Sol. (b) 170, K27 (1992); Ghatak K.P., Nouvo Cimento D13D, 1321 (1992); Ghatak K.P., Mitra B.,

References

[3]

[4] [5] [6]

[7] [8] [9] [10] [11] [12] [13] [14] [15]

261

Internat. Jour. of Elect. 72, 541 (1992); Ghatak K.P., Biswas S.N., Nonlinear Optics 4, 347 (1993); Ghatak K.P., Mondal M., (1993) Phys. Stat. Sol. (b) 175, 113; Ghatak K.P., Biswas S.N., Nonlinear Optics 4, 39 (1993); Ghatak K.P., Mitra B., Nouvo Cimento 15D, 97 (1993); Ghatak K.P., Biswas S.N., Nanostructured Materials 2, 91 (1993); Ghatak K.P., Mondal M., Phys. Stat. Sol. (b) 185, K5 (1994); Ghatak K.P., Goswami B., Mitra M., Nag B., Nonlinear Optics 16, 9 (1996); Ghatak K.P., Mitra M., Goswami B., Nag B., Nonlinear Optics 16, 167 (1996); Ghatak K.P., Nag B., Nanostructured Materials 10, 923 (1998). Choudhury R.D., Choudhury A.K., Ghatak K.P., Chakravarti A.N., Phys. Stat. Sol. (b) 98, K141 (1980); Chakravarti A.N., Ghatak K.P., Dhar A., Ghosh S., Phys. Stat. Sol. (b) 105, K55 (1981); Chakravarti A.N., Choudhury A.K., Ghatak K.P., Phys. Stat. Sol. (a) 63, K97 (1981); Chakravarti A.N., Choudhury A.K., Ghatak K.P., Ghosh S., Dhar A., Appl. Phys. 25, 105 (1981); Chakravarti A.N., Ghatak K.P., Rao G.B., Ghosh K.K., Phys. Stat. Sol. (b) 112, 75 (1982); Chakravarti A.N., Ghatak K.P., Ghosh K.K., Mukherjee H.M., Phys. Stat. Sol. (b) 116, 17 (1983); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 133, K143 (1984); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 126, K47 (1984); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 126, K41 (1984); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 129, K745 (1985); Mondal M., Ghatak K.P., Phys. Scr. 31, 615 (1985); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 135, 239 (1986); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 93, 377 (1986); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 135, K21 (1986); Mondal M., Bhattacharyya S., Ghatak K.P., (1987) Appl. Phys. A42A, 331; Biswas S.N., Chattopadhyay N.,Ghatak K.P., Phys. Stat. Sol. (b) 141, K47 (1987); Mitra B., Ghatak K.P., (1988) Phys. Stat. Sol. (b) 149, K117; Mitra B., Ghoshal A., Ghatak K.P., (1988) Phys. Stat. Sol. (b) 150, K67; Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 147, K179 (1988); Mondal M., Ghatak K.P., Phys. Stat. Sol. (b) 146, K97 (1988); Mitra B., Ghoshal A., Ghatak K.P., Phys. Stat. Sol. (b) 153, K209 (1989); Mitra B., Ghatak K.P., Phys. Letts. 142A, 401 (1989); Mitra B., Ghoshal A., Ghatak K.P., Phys. Stat. Sol. (b) 154, K147 (1989); Mitra B., Ghatak K.P., Sol. State Elect. 32, 515 (1989); Mitra B., Ghoshal A., Ghatak K.P., Phys. Stat. Sol. (b) 155, K23 (1989); Mitra B., Ghatak K.P., Phys. Letts. 135A, 397 (1989); Mitra B., Ghatak K.P., Phys. Letts. A146A, 357 (1990); Mitra B., Ghatak K.P., Phys. Stat. Sol. (b) 164, K13 (1991); Biswas S.N., Ghatak K.P., Internat. Jour. of Elect. 70, 125 (1991). Wallace P.R., Phys. Stat. Sol. (b) 92, 49 (1979). Nag B.R., Electron Transport in Compound Materials, Springer Series in Solid-State Sciences, Vol. 11 (Springer-Verlag, Germany, 1980). Ghatak K.P., Bhattacharya S., De D., Einstein Relation in Compound Materials and Their Nanostructures, Springer Series in Materials Science, Vol. 116 (Springer-Verlag, Germany, 2009). Wu C.C., Lin C.J., J. Low Temp. Phys 57, 469 (1984); Chen M.H., Wu C.C., Lin C.J., J. Low Temp. Phys. 55, 127 (1984). Bangert E., Kastner P., Phys. Stat. Sol (b) 61, 503 (1974). Foley G.M.T., Langenberg P.N., Phys. Rev.B 15B, 4850 (1977). Singh M., Wallace P.R., Jog S.D., Arushanov E., J. Phys. Chem. Solids 45, 409 (1984). Yamada Y., Phys. Soc. Japan 35, 1600, 37, 606 (1974). Vassilev L.A., Phys. Stat. Sol. (b) 121, 203 (1984). Cunningham R.W., Phys. Rev 167, 761 (1968). Yekimov A.I., Onushchenko A.A., Plyukhin A.G., Efros Al L., J. Expt. Theor. Phys 88, 1490 (1985). Roman B.J., Ewald A.W., Phys. Rev B5, 3914 (1972).

Part II: Entropy in heavily doped quantum confined superlattices Every accomplishment starts with the decision to try.

6 Entropy in quantum wires of heavily doped superlattices The great aim of education is not knowledge but tremendous action in the positive direction.

6.1 Introduction In recent years, modern fabrication techniques have generated altogether a new dimension in the arena of quantum effect devices through the experimental realization of an important artificial structure known as semiconductor superlattice (SL) by growing two similar but different semiconducting compounds in alternate layers with finite thicknesses [1]. The materials forming the alternate layers have the same kind of band structure but different energy gaps. The concept of SL was developed for the first time by Keldysh [2] and was successfully fabricated by Esaki and Tsu [2]. SLs are being extensively used in thermal sensors [3], quantum cascade lasers [4], photodetectors [5], light emitting diodes [6], multiplication [7], frequency multiplication [8], photocathodes [9], thin film transistor [10], solar cells [11], infrared imaging [12], thermal imaging [13], infrared sensing [14], and also in other microelectronic devices. The most extensively studied III-V SL is the one consisting of alternate layers of GaAs and Ga1–xAlxAs owing to the relative easiness of fabrication. The GaAs and Ga1– xAlxAs layers form the quantum wells and the potential barriers, respectively. The III–V SLs are attractive for the realization of high-speed electronic and optoelectronic devices [15]. In addition to SLs with usual structure, other types of SLs such as II–VI [16], IV–VI [17], and HgTe/CdTe [18], SLs have also been investigated in the literature. The IV–VI SLs exhibit quite different properties as compared to the III–V SL due to the specific band structure of the constituent materials [19]. The epitaxial growth of II-VI SL is a relatively recent development and the primary motivation for studying the mentioned SLs made of materials with the large band gap is in their potential for optoelectronic operation in the blue [19]. HgTe/CdTe SLs have raised a great deal of attention since 1979, when as a promising new materials for long wavelength infrared detectors and other electro-optical applications [20]. Interest in Hg-based SLs has been further increased as new properties with potential device applications were revealed [20, 21]. These features arise from the unique zero band-gap material HgTe [22] and the direct band-gap semiconductor CdTe, which can be described by the three-band mode of Kane [23]. The combination of the aforementioned materials with specified dispersion relation makes HgTe/CdTe SL very attractive, especially because of the tailoring of the material properties for various applications by varying the energy band constants of the SLs. We note that all the aforementioned SLs have been proposed with the assumption that the interfaces between the layers are sharply defined, of zero thickness, that is, https://doi.org/10.1515/9783110661194-006

266

6 Entropy in quantum wires of heavily doped superlattices

devoid of any interface effects. The SL potential distribution may be then considered as a one-dimensional array of rectangular potential wells. The aforementioned advanced experimental techniques may produce SLs with physical interfaces between the two materials 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 exists for the electrons [24]. The influence of finite thickness of the interfaces on the electron dispersion law is very important, since the electron energy spectrum governs the electron transport in SLs. In addition to it, for effective mass SLs, the electronic subbands appear continually in real space [25]. In this chapter, the entropy in III–V, II–VI, IV–VI, HgTe/CdTe, and strained-layer quantum wire heavily doped SLs (QWHDSLs) with graded interfaces has been studied in Sections 6.2.1 to 6.2.5. From Sections 6.2.6 to 6.2.10, the entropy in III–V, II–VI, IV–VI, HgTe/CdTe, and strained-layer quantum wire HD effective mass SLs,respectively, has been presented. Section 6.3 contains the Results and Discussion pertinent to this chapter. Section 6.4 presents single open research problem.

6.2 Theoretical background 6.2.1 Entropy in III–V quantum wire HD SLs with graded interfaces The electron dispersion law in bulk specimens of the HD constituent materials of III–V SLs whose undoped energy band structures are defined by three-band model of Kane can be expressed as
2
k h
1j ðE,
Δj , E
gj , η Þ + iT
2j ðE,
Δj , E
gj , η Þ =T gj gj
*cj 2m 2

where
j =
cj Þ.θ0 ðE=η
j = 1, 2, T
Δj , E
gj , η Þ = ð2=ð1 − Erf ðE=η


1j ðE,
j b gj gj ÞÞ½ða gj Þ 2






+ ½ðaj cj + bj cj − aj bj Þ=c j j


j =
cj ÞÞ 1 ½1 + Erf ðE=η


j =
cj ÞÞð1 − ðb
j =
cj ÞÞ γ0 ðE=η cj Þð1 − ða cj Þð1 − ða gj Þ + ½1=
gj Þ − ð1=
2 " # ∞ X    pffiffiffi 2 2

=4Þ=p
Þ sinhðp
u
j Þ ,
Þ ð1 − bj =
cj Þ ð2=
cj η π expð − u ðexpð − p gj

j


=1 p

−1

1 +
cj E
j ≡ ðE
gj + Δj Þ − 1 ,
cj ≡ E
gj + 2 Δj
j ≡ b ,u
cj ηgj 3

(6:1)

267

6.2 Theoretical background

and 2
Δj , E
gj , η Þ ≡
2j ðE, T gj
1 + Erf ðE=η gj

!

1
cj


j α 1−
cj

!


j b 1−
cj

! pffiffiffi π exp
cj ηgj

!
2j −u

Therefore, the dispersion law of the electrons of HD quantum well III–V SLs with graded interfacescan be expressed as [25]
k2 = G
8 + iH
8 z

(6:2)

"

# " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#21
2 −D2 −1 C 7 7 2 −1 2 2

2
2 2
2



8= 2 G
2 − ks , C7 =cos ðω7 Þ,ω7 =ð2Þ ð1− G7 − H7 Þ− ð1− G7 − H7 Þ +4G7 L 0 "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#2 " !1 #
s2 ð0, η Þ
s2 ð0, η Þ 2 − 1 M M g2 g2
21
,a 4
a20 =
s1 ð0, η Þ + 1 M Ms1 ð0, ηg1 Þ g1 "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#2 " !1 #
s2 ð0, η Þ
s2 ð0, η Þ 2 − 1 M M g2 g2 = 4

s1 ð0, η Þ − 1 M Ms1 ð0, ηg1 Þ g1

h



i1=2 h

i − 1=2
2 E,
40 E,
S1 E,
1 E,

kx ,
ky , η = 1 − P
η
k2 − Q
η
k2
η C g1 g1 g1 g1 x y

h



i1=2 h

i − 1=2
2 E,
S1 E,
2 E,
40 E,

kx ,
ky , η = 1 − P
η
k2 − Q
η
k2
η D g1 g2 g2 g2 x y     
1 + ρ G
7 = G





G 5 2 =2 − ρ6 H2 =2 + ðΔ0 =2Þfρ6 H2 − ρ8 H3 + ρ9 H4 − ρ10 H4  
1

5 −ρ H

+ ρ11 H ,G 12 5 + ð1=12Þ ρ12 G6 − ρ14 H 6         
2
1 = cos h cos h cos g
1 cos g
2    
2 ðsinðg
ÞÞðsinðg
ÞÞ ,
1 ÞÞ sin h + ðsinðh 1 2 1 qffiffiffiffiffiffiffiffiffiffiffi !2 −1
1 = e1 ðb
0 − Δ0 Þ, e1 2 2
t2 +
t2 +
t1 , h 1

1


11 ðE,
E
g1 , Δ1 , η Þ −
k2 ,
t2 = ½ð2m
E
g1 , Δ1 , η Þ
t1 = ½ð2m
*c1 =
h2 Þ.T
*c1 =
h2 ÞT21 ðE, g1 g1 s ! ! qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 −1
1 ða
1 = 2 2
t12 +
t12 −
t1 , g
1 = d
0 − Δ0 Þ, d
x12 +
y21 +
x1 , 1


2 = e2 ðb
0 − Δ0 Þ, e2 = 2 −21 h

1


11 ðE
−V
0 , E
g2 , Δ2 , η Þ +
k2 ,
y1 = ½ð2m
22 ðE
−V
0 , E
g2 , Δ2 , η Þ
*c2 =
h2 Þ.T
*c2 =
h2 ÞT
x1 = ½ − ð2m g2 g2 s
2 ða
2 = 2 −21
0 − Δ0 Þ, d g
2 = d

qffiffiffiffiffiffiffiffiffiffiffiffiffi
x12 +
y21 −
x1

!1

2

, ρ5 = ðρ23 + ρ24 Þ − 1 ½ρ1 ρ3 − ρ2 ρ4 ,

268

6 Entropy in quantum wires of heavily doped superlattices


2 + e1 e2 , ρ = ½d
2 ,
2 + e2 − d
2 − e2 , ρ = ½d
1 e1 + d
2 e2 , ρ = 2½d
1 d
1 e2 − e1 d ρ1 = ½d 3 2 4 1 2 2 1
1 ÞÞðcos hðh
2 ÞÞðsinhðg
1 ÞÞðcosðg
2 ÞÞ
2 = ½ðsinðh G
2 ÞÞðcoshðg
1 ÞÞðsinðg
2 ÞÞ
1 ÞÞðsinhðh + ðcosðh ρ6 = ðρ23 + ρ24 Þ − 1 ½ρ1 ρ4 − ρ2 ρ3 ,
1 ÞÞðcoshðh
2 ÞÞðsinðg
2 Þðcoshðg
1 ÞÞ−ðcosðh
1 ÞÞðsinhðh
2 ÞÞðsinðg
1 ÞÞðcosðg
2 ÞÞ
2 =½ðsinðh H
2 e2  − 3e1 ,
2 − d
2 Þ − 2d
1 d ρ7 = ½ðe21 + e22 Þ − 1 ½e1 ðd 1 2
1 ÞÞðcoshðh
2 ÞÞðcoshðg
1 ÞÞðcosðg
2 ÞÞ+ðcosðh
1 ÞÞðsinhðh
2 ÞÞðsinðg
1 ÞÞðsinðg
2 Þ
3 =½ðsinðh G
2 − d
2 Þ + 2d1 d2 e1  + 3e2 , ρ8 = ½ðe21 + e22 Þ − 1 ½e2 ðd 1 2
1 ÞÞðcos hð
h
2 ÞÞðsinðg
2 ÞÞðsinhðg
1 ÞÞ
3 = ½ðsinðh H

2 ÞÞðcoshðg
1 ÞÞðcosðg
2 ÞÞ


hðh − ðcosðh1 ÞÞð
sin
2 e1  + 3 d
2 + d
2 Þ − 1 ½d
2 ðe2 − e2 Þ + 2e2 d
1 , ρ9 = ½ðd 1 2 2 1
1 ÞÞðcoshðh
2 ÞÞðcosðg
2 ÞÞðsinhðg
ÞÞ
4 = ½ðcosðh G 1
2 ÞÞðcoshðg
1 ÞÞðsinðg
2 ÞÞ
− ðsinðh1 ÞÞðsinhðh
2 e1  + 3 d
2 + d
2 Þ − 1 ½d
2 ð − e2 + e2 Þ + 2e2 d
2 , ρ10 = ½ − ðd 1 2 2 1
1 ÞÞðcoshðh
2 ÞÞðcoshðg
1 ÞÞðsinðg
2 ÞÞ
4 = ½ðcosðh H
2 ÞÞðsinhðg
1 ÞÞðcosðg
2 ÞÞ
+ ðsinðh1 ÞÞðsinhðh
2 + e2 − d
2 − e2 , ρ11 = 2½d 1 2 2 1
1 ÞÞðcoshðh
2 ÞÞðcosðg
2 ÞÞðcoshðg
1 ÞÞ
5 = ½ðcosðh G
2 ÞÞðsinhðg
1 ÞÞðsinðg
2 ÞÞ
− ðsinðh1 ÞÞðsinhðh
2 + e1 e2 ,
1 d ρ12 = 4½d
1 ÞÞðcoshðh
2 ÞÞðsinhðg
1 ÞÞðsinðg
2 ÞÞ
5 = ½ðcosðh H
2 ÞÞðcoshðg
1 ÞÞðcosðg
2 ÞÞ
1 ÞÞðsinhðh + ðsinðh 2

1 e3 − 3
e1
e2 d
3
2
2 − 1 ρ13 = ½f5ðd 1 2 1 Þ + 5d2 ðe1 − 3e1 e2 Þgðd1 + d2 Þ


3 − 3d

2 e2 d
3
2




+ ðe21 + e22 Þ − 1 f5ðe1 d 1 1 1 Þ + 5ðd2 e2 − 3d1 d2 e2 Þg − 34ðd1 e1 + d2 e2 Þ     
1 cosh h
2 ðsinhðg
ÞÞðcosðg
ÞÞ
6 = ½ sin h G 1 2   
1 ðsinhðh2 ÞÞðcoshðg
ÞÞðsinðg
ÞÞ, + cos h 1

2

3 2

1
e3 − 3
e2
e2 d

2
2 − 1 ρ14 = ½f5ðd 2 1 1 Þ + 5d2 ð − e1 + 3e2 e1 Þgðd1 + d2 Þ


3 + 3d

2 d
3
2


+ ðe21 + e22 Þ − 1 ð5ð − e1 d 2 1 2 e1 Þ + 5ð − d1 e2 + 3d2 d1 e2 ÞÞ + 34ðd1 e2 − d2 e1 Þ,

6.2 Theoretical background

269


1 ÞÞðcoshðh
2 ÞÞðcoshðg
1 ÞÞðsinðg
2 ÞÞ
6 = ½ðsinðh H
2 ÞÞðsinhðg
1 ÞÞðcosðg
2 ÞÞ,
1 ÞÞðsinhðh − ðcosðh

7 = ½H
1 + ðρ H
H 5 2 =2Þ + ðρ6 G2 =2Þ
3 + ρ H






+ ðΔ0 =2Þfρ8 G 7 3 + ρ10 G4 + ρ9 H4 + ρ12 G5 + ρ11 H5 + ð1=12Þðρ14 G6 + ρ14 H6 Þg,
1 ÞÞðsinhðh
2 ÞÞðcoshðg
1 ÞÞðcosðg
2 ÞÞ
1 = ½ðsinðh H
2 ÞÞðsinhðg
1 ÞÞðsinðg
2 ÞÞ,
+ ðcosðh1 ÞÞðcoshðh
7 D
8 = ð2C
2 Þ
7 =L
7 = sinh − 1 ðω
7 Þ, H D 0 The simplified DR of HD quantum wire III–V SLs with graded interfaces can be expressed as
k2 = ½G
8 + iH
8 j z


(6:3a)


y π n
kx = n
x π and k
y =
dx

dy

The DOS function can be written as    


′ ′
G + iH H E − E
ymax 13, 1 8 8 xmax nX   g
v nX
E
= pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N
8 + iH
8 π n
x = 1 n
y = 1 G

(6:3b)


13, 1 is the sub band energy and the sub-band equation in this case can be where E expressed as
8 + iH
8 j 0 = ½G


(6:3c)


y π n

−E
kx = n
x π and k
y =
and E 13, 1 dx

dy

The EEM in this case is given by
2 ′ h
η ,n


* ðE, , n Þ = G m x y g 2

(6:4)

The electron concentration can be written as
1D = Real part of n


zmax
X n max n X

½τ6 + τ7 

(6:5)


z = 1
=0 n n

where 2

31=2

6

τ 6 = 4½ G 8 + iH8 j
k

π
nz z=
,
ny

dz ky = π

y , E = EF61 d

7 5

,

270

6 Entropy in quantum wires of heavily doped superlattices

τ7 =


s X



rÞ½τ6  Lð


r = 1


F61 is the Fermi energy in this case and E Using (1.31f) and (6.5), we can study the entropy in this case.

6.2.2 Entropy in II–VI quantum wire HD SLs with graded interfaces The electron energy spectra of the HD constituent materials of II–VI SLs are given by
h2
ks2
h2
kz2

+ ± C0 k s *
?, 1 2m
*jj, 1 2m

(6:6)


h2
k
12 ðE,
Δ2 , E
g2 η Þ + iT
22 ðE,
Δ2 , E
g2 , η Þ =T g2 g2
*c2 2m

(6:7)


η Þ= γ3 ðE, g1 and 2


*?, 1 and m
*jj, 1 are the transverse and longitudinal effective electron masses, where m respectively, at the edge of the conduction band for the first material. The energywave vector dispersion relation of the conduction electrons in HD quantum well II–VI SLs with graded interfaces can be expressed as
k2 = G
19 + iH
19 z

(6:8)

where 
2
2  C18 − D18
2
− ks , G19 =
2 L 0
2 − H
18 = cos − 1 ðω18 Þ, ω18 = ð2Þ −21 ½ð1 − G
2 Þ − C 18 18

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 − H
2 21
2 Þ2 + 4G ð1 − G 18 18 18


12 +Δ0 ðG
13 + G
14 Þ+Δ0 ðG
15 + G
16 Þ, G
11 =2ðcosðg
1 ÞÞðcosðg
2 ÞÞðcosγ ðE,
18 = 1 ½G
11 + G

ks ÞÞ G 11 2 (" # * )1 2
2
jj, 1 2 2m k h
s
0
ks
k
s Þ = k
21 ðE,
k
s Þðb0 − Δ0 Þ, k
21 ðE,
k
s Þ =
η Þ− γ11 ðE, γ3 ðE, ±C g1 *
?, 1 2m
h2
2 − H
18 = cos − 1 ðω18 Þ, ω18 = ð2Þ −21 ½ð1 − G
2 Þ − C 18 18

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 − H
2 21
2 Þ2 + 4G ð1 − G 18 18 18


12 +Δ0 ðG
13 + G
14 Þ+Δ0 ðG
15 + G
16 Þ, G
11 =2ðcosðg
1 ÞÞðcosðg
2 ÞÞðcosγ ðE,
18 = 1 ½G
11 + G

ks ÞÞ G 11 2

271

6.2 Theoretical background

(


2m
2
ks2
0 − Δ0 Þ,
k21 ðE,
0
ks  jj, 1

ks Þ =
k21 ðE,

ks Þðb

ks Þ = ½γ ðE,
η Þ− h γ11 ðE, ±C g1 3 *
?, 1 2m
h2 *

)1

2


12 ð½Ω1 ðE,

ks Þðsinh g
1 Þðcos g
2 Þ − Ω2 ðE,

ks Þðsin g
2 Þðcosh g
1 Þðsin γ ðE,

ks ÞÞÞ G 11 " # " #
1
k21 ðE,
2
k21 ðE,
1
2

ks Þd

ks Þd d d



−
2
2 −
2
2 and Ω2 ðE, ks Þ =
Ω1 ðE, ks Þ =


ks Þ

ks Þ k21 ðE, d1 + d2 k21 ðE, d1 + d2
13 ð½Ω3 ðE,

ks Þðcosh g
1 Þðcos g
2 Þ − Ω4 ðE,
k
s Þðsinh g
1 Þðsin g
2 Þðsin γ ðE,

ks ÞÞÞ G 11 # " # "
2 − d
2
1 − d
2 d 2d 1 2






− 3k21 ðE, ks Þ , Ω4 ðE, ks Þ =
, Ω3 ðE, ks Þ =


ks Þ

ks Þ k21 ðE, k21 ðE, h i
14 ð Ω5 ðE,

ks Þðsinh g
1 Þðcos g
2 Þ − Ω6 ðE,

ks Þðsin g
1 Þðcosh g
2 Þ ðcos γ ðE,

ks ÞÞÞ G 11 " # " #
1
2 d d











Ω5 ðE, ks Þ = 3d1 −
2
2 k21 ðE, ks Þ , Ω6 ðE, ks Þ = 3d2 −
2
2 k21 ðE, ks Þ d1 + d2 d1 + d2 h i
15 ð Ω9 ðE,

ks Þðcosh g
1 Þðcos g
2 Þ − Ω10 ðE,

ks Þðsinh g
1 Þðsin g
2 Þ ðcos γ ðE,

ks ÞÞÞ G 11 i h i h
2
2 − 2d
2
k2 ðE,
1 d

ks Þ = 2d

ks Þ , Ω10 ðE,

ks Þ = 2d Ω9 ðE, 1 2 21 i h
16 Ω7 ðE,

ks Þðsinh g
1 Þðcos g
2 Þ − Ω8 ðE,

ks Þðsin g
1 Þðcosh g
2 Þ ðsin γ ðE,

ks Þ=12ÞÞ G 11 " #
1 Þ
3 − 3d
2 d 5d 5ð d 2 3 1 2





ks Þ =

ks Þd1 , − 34k21 ðE, Ω7 ðE,
2 + d
2 k21 ðE, ks Þ + k21 ðE,

ks Þ d 1 2 " #

3 − 3d
2 d 5d2
3

5ðd 2 2Þ 1





+ 34k21 ðE, ks Þd2 , Ω8 ðE, ks Þ =
2
2 k21 ðE, ks Þ +

ks Þ d1 + d2 k21 ðE,
12 + Δ0 ðH
13 + H
14 Þ + Δ0 ðH
15 + H
16 Þ,
11 + H
18 = 1 ½H H 2

ks Þðsinh g
1 Þðcos g
2 Þ + Ω1 ðE,

ks Þðsin g
2 Þðcosh g
1 Þðsin γ ðE,

ks ÞÞÞ,
12 = ð½Ω2 ðE, H 11

ks Þðcoshg
1 Þðcos g
2 Þ + Ω3 ðE,

ks Þðsinh g
1 Þðsin g
2 Þðsin γ ðE,

ks ÞÞÞ,
13 = ð½Ω4 ðE, H 11

ks Þðsinh g
1 Þðcos g
2 Þ + Ω5 ðE,

ks Þðsin g
1 Þðcosh g
2 Þðsin γ ðE,

ks ÞÞÞ,
14 = ð½Ω6 ðE, H 11

ks Þðcoshg
1 Þðcos g
2 Þ + Ω9 ðE,

ks Þðsinh g
1 Þðsin g
2 Þðcos γ ðE,

ks ÞÞÞ,
15 = ð½Ω10 ðE, H 11

ks Þðsinh g
1 Þðcos g
2 Þ + Ω7 ðE,

ks Þðsin g
1 Þðcosh g
2 Þðsin γ ðE,

ks Þ=12ÞÞ,
16 = ð½Ω8 ðE, H 11

272

6 Entropy in quantum wires of heavily doped superlattices




18 = sin − 1 ðω18 Þ
19 = 2C18 D18 and D H
2 L 0 The simplified DR of HD quantum wire III–V SLs with graded interfaces can be expressed as  
k2 = ½G
19 +
iH
19  (6:9a) z kx = ndx π and ky = ndy π x

y

The DOS function can be written as nX ymax

xmax nX
′19 +
iH
′19 
13, 2 Þ½G HðE − E
EÞ
= gv pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð
19 +
iH
19 π nx = 1 ny = 1 G

(6:9b)


13, 2 is the sub-band energy and the sub-band equation in this case can be where E expressed as   

(6:9c) 0 = ½G19 + iH19 
n
x π
π n kx =
and
ky =
y and E
− E
13, 2 dx

dy

The EEM in this case is given by
η ,n
y Þ =
* ðE, m g
x , n


2 ′ h G 19 2

(6:10)

The electron concentration can be written as
1D = Real part of n


zmax n n ymax X
X

½τ8 + τ9 


z = 1 n
y = 1 n

where 2

31=2    6
7

 π
nz τ 8 = 4½ G 5 , 19 + iH19 
kz =
, π
n y


dz ky = 
, E = EF62 dy

τ9 =


s X



rÞ½τ8  Lð


r = 1


F62 is the Fermi energy in this case and E Using (1.31f) and (6.11), we can study the entropy in this case.

(6:11)

273

6.2 Theoretical background

6.2.3 Entropy in IV–VI quantum wire HD SLs with graded interfaces The E-k dispersion relation of the conduction electrons of the HD constituent materials of the IV–VI SLs can be expressed as  
k2 = ½2p

ks , η Þ + ½q

ks , η Þ2 + 4p

ks , η Þ21
9, 1 ðE,
9, i  − 1 ½ − q
9, i ðE,
9, i ðE,
9, i R gi gi gi z

(6:12)

where,
l,−i m
l,+i Þ,
i = 1, 2,
9, i = ðαi
h4 Þ=ð4m
9, i = ðαi
h4 Þ=p p

ks , η Þ = ½ð
h2 =2Þðð1=m
*li Þ + ð1=m
li− ÞÞ +
9, 1 = ðE, q gi
η Þðð1=m
li+ m
li+ m
li+ Þ − ð1=m
li− ÞÞ
li− Þ + ð1=m
li− ÞÞ − αi γ3 ðE, αi ð
h4 =4Þ
ks2 ðð1=m gi and
9, i ðE,

ks , η Þ = ½γ ðE,
η Þ + γ ðE,
η Þ½ð
h2 =2Þαi k2 ðð1=m
*ti Þ − ð1=m
ti− ÞÞ R gi gi gi s 2 3
*ti Þ − ð1=m
ti− ÞÞ = αi ð
h6 =4Þ
ks4 ðð1=m
ti+ m
ti− ÞÞ − ½ð
h2 =2Þ
ks2 ðð1=m The electron dispersion law in HD quantum well IV–VI SLs with graded interfaces can be expressed as

ks Þ
0
kÞ 1 Φ2 ðE, (6:13) cosðL 2





 

ks Þ ≡ ½2 cos β ðE,

ks Þ cos γ ðE,

ks Þ + ε2 ðE,

ks Þ sinh β ðE,

ks Þ sin γ ðE,

ks Þ Φ2 ðE, 2 2 2 22 "
 

ks Þ 2
112 ðE,

  K
212 ðE,

ks Þ cosh β ðE,

ks Þ sin γ ðE,

ks Þ − 3 K + Δ0 2 22


K 212 ðE, ks Þ #



ks Þ 2 
212 ðE,



 K






sinh β2 ðE, ks Þ cos γ22 ðE, ks Þ + 3K112 ðE, ks Þ −

ks Þ
112 ðE, K


 




112 ðE,

ks Þ 2 − K
212 ðE,

ks Þ 2 Þ cosh β ðE,

ks Þ cos γ ðE,

ks Þ + Δ0 ½2ð K 2 22 # "
112 ðE,

ks Þg3 5fK
212 ðE,

ks Þg3 1 5fK



+ − 34K212 ðE, ks ÞK112 +

ks Þ

ks Þ
212 ðE,
112 ðE, 12 K K

ks Þg sinfγ ðE,

ks Þg, sinhfβ ðE, 2

22



ks Þ ≡ K
112 ðE,

ks Þ½a
0 − Δ0 , β2 ðE,  
k2 ðE,

ks Þ=½2p
V
0 ,
ks η Þ− ½q
V
0 ,
ks η Þ2+4p
V
0 ,
ks η Þ21 ,
9,2 ðE−
9,2 −1 ½−q
9,2 ðE−
9,2 ðE−
9,2 R g2 g2 g2 112

274

6 Entropy in quantum wires of heavily doped superlattices


0 − Δ0 ,


212 ðE,

ks Þ½b γ22 ðE, ks Þ ≡ K   1  2 −1 2
k2 ðE,




ks Þ = ½2p




9, 1 ðE, ks ηg1 Þ + ½q
9, 1 ðE, ks ηg1 Þ + 4p
9, 1 
9, 1 R9, 1 ðE, ks ηg1 Þ , −q 212 and # "

ks Þ K

ks Þ
112 ðE,
212 ðE, K

ε2 ðE, ks Þ ≡ − .
212 ðE,
112 ðE,

ks Þ K

ks Þ K The simplified DR in HD quantum wire IV–VI SLs with graded interfaces can be expressed as "  #   2  2 
k2 = 1 cos − 1 1 Φ2 ðE,


ks Þ − k (6:14a)
π 
n
x π n z s
2 kx = d
x and
ky = d
yy 2 L 0 The DOS function can be written as
ymax
xmax n n
k
s ÞgΦ2 ðE,
k
s ÞΦ′2 ðE,
k
s ÞHð
E
−E
13, 3 Þ X
X cos − 1 f21 Φ2 ðE,
EÞ
= gv qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð
2 2 2 πL0 n
x = 1 n
y = 1 ð cos − 1 f1 Φ ðE,

k Þg − L
2 fðn
x πÞ + ðn
x πÞ gÞ 1 − 1 Φ2 ðE,

k Þ 2

2

s

0


x d


x d

4

2

s

(6:14b)
13, 3 is the sub-band energy and the sub-band equation in this case can be where E expressed as "  #  2  1 −1 1 2 


− ks 
n
x π (6:14c) 0 =
2 cos Φ2 ðE13, 3 , ks Þ
π n kx = d
x and
ky = d
yy 2 L0 The EEM in this case is given by # − 1=2 " 2
h 1 1 − 1 2
η ,n

ks ÞΦ′2 ðE,

ks Þ½1 − Φ ðE,

ks Þ
y Þ = 2 cos m ðE, Φ2 ðE, g
x , n
2 4 2 2L 0
*

(6:15)

The electron concentration can be written as
1D = n


zmax n n ymax X
X

½τ10 + τ11 

(6:16)


z = 1 n
y = 1 n

where "" τ10 =

"

1 −1
2 cos L 0

(

)#2

1

ks Þ Φ2 ðE, 2

# #1=2  2  − ks k
= π
nz , , ny

 z d
z
ky = π

, E = EF63 dy

275

6.2 Theoretical background

τ11 =


s X



rÞ½τ10  Lð


r = 1


F63 is the Fermi energy in this case and E Using (1.31f) and (6.16), we can study the entropy in this case.

6.2.4 Entropy in HgTe/CdTe quantum wire HD SLs with graded interfaces The electron energy spectra of the constituent materials of HgTe/CdTe SLs are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # "
2
1E
1E
2 + 4A
−B
01 B
B01 + 4A 01
k2 = (6:17)
2 2A 1

and
h2
k
12 ðE,
Δ2 , E
g2 , η Þ + iT
22 ðE,
Δ2 , E
g2 , η Þ =T g2 g2
*c2 2m 2

(6:18)


01 = ð3jej2 =128εsc1 Þ, A
1 = ð
h2 =2m
*c1 Þ. εsc1 .εsc1 is the semiconductor permittivity where B of the first material. The energy-wave vector dispersion relation of the conduction electrons in HD quantum well HgTe/CdTe SLs with graded interfaces can be expressed as
k2 = G
192 + iH
192 z

(6:19)

where, "
192 = G

!
2 − D
2 Þ=L
2 ðC 182 182 0

# 2
− ks , "


182 = cos C

−1

−1 ðω182 Þ, ω182 = ð2Þ 2


2 ð1 − G 182


2 Þ − −H 182

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#21
2 − H
2
2 Þ2 + 4G ð1 − G 182

182

182


122 + Δ0 ðG
132 + G
142 Þ + Δ0 ðG
152 + G
162 Þ,
182 = 1 ½G
112 + G G 2
112 = 2ðcosðg
12 ÞÞðcosðg
22 ÞÞðcos γ ðE,

ks ÞÞ G 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # "
2
1E
1E
2 + 4A
−B
01 B
1=2 B01 + 4A 01
0 − Δ0 Þ,
k8 ðE,

ks Þ =
k8 ðE,

ks Þðb

ks Þ = γ8 ðE,
2 2A 1

276

6 Entropy in quantum wires of heavily doped superlattices


122 ð½Ω12 ðE,

ks Þðsinh g
12 Þðcos g
22 Þ − Ω22 ðE,
k
s Þðsin g
22 Þðcosh g
12 Þðsin γ ðE,

ks ÞÞÞ G 8 " " # #
12
k8 ðE,
22
k8 ðE,
12
22

ks Þd

ks Þd d d



Ω12 ðE, ks Þ =
−
2
2 −
2
2 , Ω22 ðE, ks Þ =


ks Þ

ks Þ k8 ðE, d12 + d22 k8 ðE, d12 + d22
132 = ð½Ω32 ðE,

ks Þðcosh g
12 Þðcos g
22 Þ − Ω42 ðE,

ks Þðsinh g
12 Þðsin g
22 Þðsin γ ðE,

ks ÞÞÞ, G 8 # " # "
2 − d
22
2
12 d d 2 d 12 22
k
s Þ =

ks Þ , Ω42 ðE,

ks Þ = − 3
k8 ðE, , Ω32 ðE,
k8 ðE,
k8 ðE,

ks Þ

ks Þ
142 = ð½Ω52 ðE,

ks Þðsinh g
12 Þðcos g
22 Þ − Ω62 ðE,

ks Þðsin g
22 Þðcosh g
22 Þðcos γ ðE,

ks ÞÞÞ, G 8 " # " #
12
22 d d 2

2









Ω52 ðE, ks Þ = 3d12 −
2
2 k8 ðE, ks Þ , Ω62 ðE, ks Þ = 3d22 +
2
2 k8 ðE, ks Þ , d12 + d22 d12 + d22
152 = ð½Ω92 ðE,

ks Þðcosh g
12 Þðcos g
22 Þ − Ω102 ðE,

ks Þðsinh g
12 Þðsin g
22 Þðcos γ ðE,

ks ÞÞÞ, G 8 h i h i
22 ,
2 − 2d
2 −
k2 ðE,
12 d

ks Þ = 2d

ks Þ , Ω102 ðE,

ks Þ = 2d Ω92 ðE, 12 22 8
162 ð½Ω72 ðE,

ks Þðsinh g
12 Þðcos g
22 Þ − Ω82 ðE,

ks Þðsin g
12 Þðcosh g
22 Þðsin γ ðE,

ks Þ=12ÞÞ G 8 " #
12 Þ
12
3 − 3d
2 d 5 d 5ð d 3 22 12



12 ,
k
s Þ =

ks Þd Ω72 ðE, − 34
k8 ðE,
2 + d
k8 ðE,
2 k8 ðE, ks Þ +

ks Þ d 12

22

"

#

12
3 − 3d
2 d 5d 5ðd 3

12 22 12 Þ






+ 34k8 ðE, ks Þd12 , Ω82 ðE, ks Þ =
2
2 k8 ðE, ks Þ +
k8 ðE,

ks Þ d12 + d22 i h
122 + Δ0 ðH
132 + H
142 Þ + Δ0 ðH
152 + H
162 Þ
182 = 1 H
112 + H H 2

ks ÞÞ,
112 = 2ðsinh g
12 sin g
22 cos γ ðE, H 8

ks Þðsinh g
12 Þðcos g
22 Þ + Ω12 ðE,

ks Þðsin g
22 Þðcosh g
12 Þðsin γ ðE,

ks ÞÞÞ,
122 = ð½Ω22 ðE, H 8

ks Þðcosh g
12 Þðcos g
22 Þ + Ω32 ðE,

ks Þðsinh g
12 Þðsin g
22 Þðsin γ ðE,

ks ÞÞÞ,
132 = ð½Ω42 ðE, H 8

ks Þðsinh g
12 Þðcos g
22 Þ + Ω52 ðE,

ks Þðsin g
12 Þðcosh g
22 Þðcos γ ðE,

ks ÞÞÞ,
142 = ð½Ω62 ðE, H 8

ks Þðcosh g
12 Þðcos g
22 Þ + Ω92 ðE,

ks Þðsinh g
12 Þðsin g
22 Þðcos γ ðE,

ks ÞÞÞ,
152 = ð½Ω102 ðE, H 8

ks Þðsinh g
12 Þðcos g
22 Þ + Ω72 ðE,

ks Þðsin g
12 Þðcosh g
22 Þðsin γ ðE,

ks Þ=12ÞÞ,
162 = ð½Ω82 ðE, H 8
182 D
2 Þ and D
182 = sinh − 1 ðω182 Þ
192 = ½ðð2C
182 Þ=L H 0 The simplified entropy in HD quantum wire HgTe=CdTe superllatices with graded interfaces can be expressed as

6.2 Theoretical background


k2 z

 
192 + iH
192  n
π = ½G
π n 
kx = d
xx and
ky = d
yy

277

(6:20a)

The DOS function can be written a
ymax
xmax n n X Hð
′192 
E
−E
13, 4 Þ½G′192 + iH
X
EÞ
= gv pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð
192 + iH
192 π n
x = 1 n
y = 1 G

(6:20b)


13, 4 is the sub band energy and the sub – band equation in this case can be where E expressed as   

(6:20c) 0 = ½G192 + iH192 
n
x π
n
y π kx = d
x , ky = d
y and E
= E
13, 4 The EEM in this case is given by
2 ′ h
η ,n


* ðE, , n Þ = G 192 m x y g 2

(6:21)

The electron concentration can be written as
1D = Real part of n


zmax n n ymax X
X

½τ12 + τ13 

(6:22)


z = 1 n
y = 1 n

where "" τ12 =

# #1=2 

G192 + iH192 
k = π
nz , ,  z d
z ky = π
n
y , E
= E
F64 dy

τ13 =


s X



rÞ½τ12  Lð


r = 1


F64 is the Fermi energy in this case. and E Using (1.31f) and (6.22), we can study the entropy in this case.

6.2.5 Entropy in strained layer quantum wire HD SLs with graded interfaces The dispersion relation of the conduction electrons of the constituent materials of the strained layer super lattices can be expressed as
−T
2i 
k2 + ½E
−T
3i 
k2 = q
iE
i E
3 − R
2 + V
+ζ
−T
1i 
k2 + ½E
i E ½E i x y z where

(6:23)

278

6 Entropy in quantum wires of heavily doped superlattices

"



i εxxi − bi εi +
c εi − ða
c Þεi + 3 b
gi − C
1i = θi , θi = E
i + C T 1i 1i 2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffi #
i εxyi 3d



i εxxi − bi εi −
c εi − ð a
c Þεi + 3 b
2i = ωi , ωi = E
gi − C
i + C T 1i 1i 2 2

2

,

qffiffiffiffiffiffiffiffiffiffiffiffiffi #
i εxyi 3d 2

,

# 
i εi b 3
c c




, T3i = δi , δi = Egi − C1i εi − ðai + C1i Þεi + bi εzzi − 2 2 3


c εi , q
c
i + C
i = q
i ½2A
i = 2 , A R i = Egi − C1i εi , 1i
2B 2i " " # 2
2 εxyi 2C ε 2 C xyi 2 c c 2 1i 1i
εi A
εi , ζ = q
−
iC

i = q
i A + 2A −C V i
i i 1i 1i i 3 3 Therefore, the electron energy spectrum in HD stressed materials can be written as
i ðE,
i ðE,
η Þ
k2 + Q
η Þ
k2 +
Si ðE,
η Þ
k2 = 1 P gi x gi y gi z

(6:24)

where
η Þ=
i ðE, P gi


η Þ −
I 0 T
1i  ½γ0 ðE, gi ,
η Þ
i ðE, Δ gi

"
Þ=
i ðE,η Δ gi


2
i η3gi −q −E pffiffiffi exp η2gi 2 π

!"

# " !# #
2
ζi E E



1+ 2 − Ri θ0 ðE,ηgi Þ+ Vi γ0 ðE,ηgi Þ+ 1+Erf , 2 ηgi ηgi


η Þ −
I 0 T
η Þ −
I 0 T
2i 
3i  ½γ0 ðE, ½γ ðE, gi gi


η Þ= 0
I0 = 1 ½1 + Erf ðE=η
and
i ðE, gi Þ, Qi ðE, ηgi Þ = gi



2 Δi ðE, ηgi Þ Δi ðE, ηgi Þ The energy-wave vector dispersion relation of the conduction electrons in heavily doped strained layer quantum well SLs with graded interfaces can be expressed as
ðE,

ks Þ
0
kÞ = 1 ϕ cosðL 2 6

(6:25)


ðE,

ks Þ=½2cosh½T
4 ðE,η









ϕ 6 g2 Þcos½T5 ðE,ηg1 Þ+½T6 ðE, ks Þsinh½T4 ðE,ηg2 Þsin½T5 ðE,ηg1 Þ  

2 
        k0 E, ηg2 Þ
4 E,
η
η
5 E,
η   − 3
k′ E, cosh T + Δ0 sin T g1 g2 g1
k′ E,
η g1 ′2   

η      k
E,     g1
4 E,
η − 
η
5 E,
η  sinh T + 3k
0 E, cos T g2 g2 g1
k0 E,
η g2

6.2 Theoretical background

279


η Þ −
k′2 ðE,
η ÞÞ cosh½T
4 ðE,
η Þ sin½T
5 ðE,
η Þ + Δ0 ½2ð
k02 ðE, g2 g2 g2 g1 !
′ 3

1 5
k03 ðE,η g2 Þ 5k ðE,ηg1 Þ




4 ðE,η


′ + + −34k0 ðE,ηg2 Þk ðE,ηg1 Þ sinh½T g2 Þsin½T5 ðE,ηg1 Þ

12
k′ðE,η k0 ðE,η g2 Þ g1 Þ
4 ðE,
η Þ =
k0 ðE,
η Þ½a ½T g2 g2
0 − Δ0 ,
k0 ðE,
2 ðE
η Þ = ð
S2 ðE
−V
0 , η ÞÞ − 1=2 ½P
2 ðE
−V
0 , η Þ
k2 + Q
−V
0 , η Þ
k2 − 11=2 , g2 g2 g2 x g2 y
0 − Δ0 ,
η Þ =
k′ðE,
η Þ½b
5 ðE, T g1 g1
k′ðE,
2 ðE,
η Þ = ½
S1 ðE,
η Þ − 1=2 ½1 − P
1 ðE,
η Þ
k2 − Q
η Þ
k2 1=2 g1 g1 g1 x g1 y and " #
k0 ðE,
η Þ
η Þ
k′ðE, g1 g1

ks Þ =
6 ðE, − T
k′ðE,
η Þ
η Þ
k0 ðE, g1 g1 Therefore the entropy of the conduction electrons in HD strained layer quantum well SL with graded interfaces can be expressed as "  #   2  2 
k2 = 1 cos − 1 1 Φ6 ðE,


ks Þ − k (6:26a)
π n z s 

2 kx = n
d
xxπ and
ky = d
yy 2 L 0 The DOS function can be written as
xmax n n ymax

ks ÞgΦ
6 ðE,

ks Þ½Φ
6 ðE,

ks Þ′Hð
E
−E
13, 5 Þ
6 ðE, X
X cos − 1 f21 Φ
EÞ
= gv
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð
0 πL
x = 1 n
y = 1

ks Þg2 − L
2 fðnx πÞ2 + ðnx πÞ2 g
6 ðE,

ks Þ2 n
6 ðE, cos − 1 f21 Φ 1 − 41 ½Φ 0 dx dx

(6:26b)
13, 5 is the sub – band energy and the sub – band equation in this case can where E be expressed as "  #  2 
y π
x π n 1 n −1 1 2 



− ks kx =
and
ky =
(6:26c) 0 =
2 cos Φ6 ðE13, 3 , ks Þ  2 L0 dx dy EEM in this case is given   
h2 1


2 − 1=2 *
−1 1





′


m ðE, ηg , nx , ny Þ =
2 cos Φ6 ðE, ks Þ½Φ6 ðE, ks Þ 1 − ½Φ6 ðE, ks Þ 2 4 2L0 The electron concentration can be written as

(6:27)

280

6 Entropy in quantum wires of heavily doped superlattices


1D = n


zmax n n ymax X
X

½τ17 + τ18 

(6:28)


z = 1 n
y = 1 n

where 2

"



6 1 τ17 = 4
2 cos − 1 L0 τ18 =


s X



31=2 #  2  1


7 −
ks2 
k = πn
z , Φ6 ðE, ks Þ 5 , ny

2  z d
z
ky = π
, E = E F66
dy



rÞ½τ17  Lð


r = 1


F66 is the Fermi energy in this case. and E Using (1.31f) and (6.28), we can study the entropy in this case.

6.2.6 Entropy in III–V quantum wire HD effective mass SLs Following Sasaki [24], the electron dispersion law in III– V heavily doped effective mass superlattices (EMSLs) can be written as  


k2 = 1 cos − 1 ð
f ðE,

ky ,
kz ÞÞ 2 −
k2 (6:29) 21 x ?
2 L 0 in which
0 D
21 ðE,



k? , η Þ + b

k? , η Þ
21 ðE,
1 cos½a
0 C ð
f21 ðE, ky ,
kz Þ = a g1 g2
0 D

k? , η Þ − b

k? , η Þ,
k2 =
k2 +
k2 ,
21 ðE,
0 C21 ðE,
1 cos½a −a g1 g2 ? y z #−1 "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#2 "
2 ð0, η Þ
2 ð0, η Þ M M g2 g2 1=2
1 = 4ð
a Þ
1 ð0, η Þ + 1 M M1 ð0, η Þ g1

g1

#−1 "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#2 "
2 ð0, η Þ
2 ð0, η Þ M M g2 g2 1=2
2 = 4ð
Þ a
1 ð0, η Þ − 1 M M1 ð0, η Þ g1


iz ð0,η Þ= m
*ci M gi



g1







i −αi b
i  bi −2
αi bi ηgi 1 αi
ci +
ci b 1 αi pffiffiffi Tð0,η ffiffiffiffiffiffi ffi p ffiffiffi p + 1− Þ+2 + 1− gi
ci π 2
ci
ci
c2i π π
ci

6.2 Theoretical background

281

! !! 
2 


 α
X

bi 1 αi 2 −2 1 −p 1 p pffiffiffi 1− − 1− exp 2 2 exp sinh
ci
ci
ci ηgi π
ci ηgi
ci

ci ηgi
ci ηgi 4 p p=1 ! ! ! 
2  α
X

−1 −p 1 p , + exp 2 2 exp cosh
ci ηgi
ci ηgi 4 ηgi p=1 "


i η2gi
αi
ci + b
i  ηgi
i 
i
ci − αi b b αi b 1 αi
p ffiffiffi 1 − Tð0, ηgi Þ = 2 1 − + +
ci 4
ci
ci
c2i 2 π 2
ci ! !

α
i  X
2 =4Þ
b 1 αi 2 1 expð − p p pffiffiffi exp 2 2 , 1− 1− − sinh
ci

ci
ci
ci ηgi π
ci ηgi
ci ηgi p = 1 p
21 ðE,

k? , η Þ = e1 +
i
e2 , D
21 ðE,

k? , η Þ = e3 + ie4 , C g1 g2 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 2



t12 +
t22 +
t1 =2 e= , e2 = t1 + t2 + t1 =2  * 



*

2



t2 = 2mc1 T
t1 = 2mc1 T , ð E, Δ , η , E Þ − k 11 1 g1 21 ðE, Δ1 , ηg1 , Eg1 Þ g1 ?
h2
h2 2qffiffiffiffiffiffiffiffiffiffiffiffi 31=2 2qffiffiffiffiffiffiffiffiffiffiffiffi 31=2
t32 +
t42 +
t3
t32 +
t42 −
t3 5 , e4 = 4 5 e=4 2 2  * 



*

2



t4 = 2mc2 T
t3 = 2mc2 T , ð E, Δ , η , E Þ − k 12 1 g2 22 ðE, Δ1 , ηg1 , Eg2 Þ, g1 ?
h2
h2 Therefore, (6.29) can be expressed as kz2 = δ7 + iδ8

(6:30)

where   1 2 , δ5 = cos − 1 p5 , δ7 =
2 ðδ25 − δ26 Þ −
k? L0 2
5 = p

1 − δ23 4

− δ24

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31=2′ 2 ð1 − δ23 − δ24 Þ + 4δ24 5 2


1 cos Δ1 cosh Δ2 − a
2 cos Δ3 cosh Δ4 Þ, δ3 = ða
1 sin Δ1 sinhΔ2 − a
2 sin Δ3 sinhΔ4 Þ, δ4 = ða
0 e3 Þ, Δ2 = ða
0 e4 Þ, Δ3 = ða
0 e3 Þ, Δ4 = ða
0 e4 Þ,
0 e1 + b
0 e2 + b
0 e1 − b
0 e1 − b Δ1 = ða

282

6 Entropy in quantum wires of heavily doped superlattices


2 
5 and δ8 = ½2δ5 δ6 =L δ6 = sinh − 1 p 0 The entropy in III–V HD effective mass quantum wire SLs can be written as  
y π
x π n
k2 = ½δ7 + iδ8 
kx = n and
ky =
j (6:31a) z
 dx dy The DOS function can be written as
xmax n n ymax

X
X
13, 6 Þ½δ′7 + iδ′8  HðE − E
EÞ
= g
v pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð π n
x = 1 n
y = 1 δ7 + iδ8

(6:31b)

where E13, 6 is the sub-band energy and the sub-band equation in this case can be expressed as  
y π
x π n n 
−E
13, 6 (6:31c) 0 = ½δ7 + iδ8 
kx =
,
ky =
and E  dx dy The EEM in this case is given by
η ,n
y Þ =
* ðE, m g
x , n


2 ′ h δ7 2

(6:32)

The electron concentration can be written as
1D = Real part of n


zmax n n ymax X
X

½τ15 + τ16 

(6:33)


z = 1 n
y = 1 n

where 2

31=2    6 7 τ15 = 4½δ7 + iδ8 
k = π
nz , 5 , z
π
ny


d  z ky =
, E = EF67 dy

τ16 =


s X



rÞ½τ15  Lð


r = 1


F67 is the Fermi energy in this case. and E Using (1.31f) and (6.33), we can study the entropy in this case.

6.2.7 Entropy in II–VI quantum wire HD effective mass SLs Following Sasaki [24], the electron dispersion law in HD II–VI EMSLs can be written as

6.2 Theoretical background


k2 = Δ13 + iΔ14 , z   1 2 2 2
Δ13 =
2 ðΔ11 − Δ12 Þ − ks , L0

283

(6:34)


6 , Δ5 = cos − 1 p 2
5 = 4 p

1 − Δ29 − Δ210

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31=2 ′ 2 ð1 − Δ29 − Δ210 Þ + 4Δ210 5 2


1 cos Δ6 cosh Δ7 − a
2 cos Δ8 cosh Δ7 Þ, Δ9 = ða
1 sin Δ6 sinhΔ7 + a
2 sin Δ8 sinhΔ7 Þ, Δ10 = ða
0
e3 , Δ7 = b
0
e4 , Δ8 = ½a
0
e3 
22 ðE,
22 ðE,



ks , η Þ − b
0 C
0 C ks , ηg1 Þ + b Δ6 = ½a g1 "
22 = ðE,

ks , η Þ = C g1

2m*jj, 1
2 h




2
ks2

η Þ− h
C0 ks γ3 ðE, + g1 2m*?, 1

"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#2 2
2 ð0, η Þ M g2 44
1 = a
1 ð0, η Þ + 1 M g1


2 ð0, η Þ M g2
1 ð0, η Þ M g1

#1=2 ,

!1=2 3 − 1
 2 *
1 ð0, η Þ = m 5 ,M
c1 1 − , g1 π

! 3−1 "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#2 2
2 ð0, η Þ
2 ð0, η Þ 1=2 M M g2 g2 44 5
2 = a
1 ð0, η Þ − 1
1 ð0, η Þ M M g1 g1
6 , Δ14 = Δ12 = cos − 1 p

2Δ11 Δ12
2 L 0

Entropy in III–V HD effective mass quantum wire SLs can be written as    2 k
z = ½δ13 + iδ14 
n
x π
π n kx = d
x and
ky = d
yy

(6:35a)

The DOS function can be written as nxmax nX ymax


13, 6 Þ½δ′13 + iδ′14  g
v X HðE − E

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NðEÞ = π nx = 1 ny = 1 δ13 + iδ14

(6:35b)


13, 7 is the sub band energy and the sub – band equation in this case cane be where E expressed as    (6:35c) 0 = ½δ13 + iδ14 
n
x π
n
y π kx = d
x , ky = d
y and E
− E
13, 71

284

6 Entropy in quantum wires of heavily doped superlattices

The EEM in this case is given by
2 h
η ,n
y Þ = δ′
* ðE, m g
x , n 2

(6:36)

The electron concentration can be written as
1D = Real part of n


zmax n n ymax X
X

½τ19 + τ20 

(6:37)


z = 1 n
y = 1 n

where 2

31=2    6 7 τ19 = 4½Δ13 + iΔ14 
k = π
nz , 5 ,
y z

π n

dz ky = 
, E = EF610 dy

τ20 =


s X



rÞ½τ19  Lð


r = 1


F610 is the Fermi energy in this case and E Using (1.31f) and (6.37), we can study the entropy in this case.

6.2.8 Entropy in IV–VI quantum wire HD effective mass SLs Following Sasaki [24], the electron dispersion law in IV–VI, EMSLs can be written as  
k2 = 1 fcos − 1 ð
f23 ðE,

kx ,
ky ÞÞg2 −
k2 (6:38) z s
2 L 0 where
f23 ðE,
0 D
23 ðE,



kx ,
ky η Þ + b

kx ,
ky η Þ
23 ðE,
3 cos½a
0 C kx ,
ky Þ = a g1 g1
0 D
23 ðE,

kx ,
ky η Þ − b

kx ,
ky η Þ,
23 ðE,
0 C
4 cos½a −a g2 g2 "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#2 "
3 ð0, η Þ M g2
3 = 4 a
3 ð0, η Þ + 1 M g1 "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#2 "
3 ð0, η Þ M g2
3 = a 4
3 ð0, η Þ − 1 M g1


3 ð0, η Þ M g2
3 ð0, η Þ M g1

!1=2 # − 1

! #
3 ð0, η Þ 1=2 − 1 M g2
3 ð0, η Þ M g1

,

285

6.2 Theoretical background

"( αi

2 1− π

αi

2 1− π


3 ð0, η Þ = ð4p
9, 1 Þ − 1 M gi #1=2 "
9, i ð0, η Þ
9, 1 ÞR + ð4p gi

!

!

1 1 − − +

ml, i ml, i

#2

!) ""
9, i ð0, ηgi Þ q

+

!## ! 1 1 2 αi ηgi
ð0, ηgi Þ + 2p
9, i 1 − + pffiffiffi − − q ,
l,+i m
l, i 9, i m π π

" ! αi
h4
h2 1 1
ð0, ηgi Þ =
9, i = ,q − − p
l,+i m
l,−i 9, i
l,+i m
l, i 4m 2 m # " # αi ηgi 1 ηgi αi η2gi 1
9, 1 ð0, η Þ = pffiffiffi + − pffiffiffi ð + − − Þ , R , gi
l, i m
l, i 2 π m π
23 ðE,

kx ,
ky , η Þ = C g1

hh


9, 1 2p

i − 1h

h

kx ,
ky , η Þ + ðq

kx ,
ky , η ÞÞ2
9, 1 ðE,
9, 1 ðE, −q g1 g1


9, 1 ðE,

kx ,
ky , η Þ
9, 1 ÞR + ð4p g1 ""

kx ,
ky , η Þ =
23 ðE, D g2

# − 1"
9, 2 2p

i1=2 ii1=2

, "



kx ,
ky , η Þ +
9, 2 ðE, −q g2

n



kx ,
ky , η Þ
9, 2 ðE, q g2

o2

#1=2 ##1=2
9, 2 ðE,

kx ,
ky , η Þ
9, 2 ÞR + ð4p g2 "

2

kx ,
ky , η Þ =
h
9, i ðE, q gi 2

!


4 2 h k + αi
4 s !# 1 1
η Þ − αi γ3 ðE, − − , gi
l,+i m
l, i m 1 1 + −
l,+i m
l, i m

, 1 1 + + −
l,−i m
l, i
l,+i m
l, i m m

!

" !
h2 2 1 1





− − R9, i ðE, kx , ky , ηgi Þ = γ2 ðE, ηgi Þ + γ3 ðE, ηgi Þα ks
l,+i m
l, i 2 m ! #
h2 2 1 1 α
h6 ks4 − ks − − − ,
l,+i m
l, i 2 4 mt,−i mt,+i m "sffiffiffiffiffiffiffiffiffiffiffiffiffi#2 "
1=2 # − 1
*2
*2 m m
5 = + 1 4 a
*1
*1 m m Therefore, the entropy in HD IV–VI, quantum wire EMSLs can be written as     2 2 
y π
x π 1 2 − 1
n
k
x , k
y g − k
k
x = n k
z =
2 ½cos
f23 ðE, (6:39a) s
x and ky = d
y  L d 0

286

6 Entropy in quantum wires of heavily doped superlattices

The DOS function can be written as
ymax
xmax n n
EÞ
¼ g
v P P Nð πL 0

y ¼1 nx ¼1 n

n

o

h

i0
n

x π ; n

y π
f 23 E;
n

x π ; n

y π
f 23 E;
n

x π ; n

y π Hð
E
E
13;8 Þ cos1 21
f 23 E; dx dx dx dy dy dy vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ( ! u 



2 2  rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h

i2 u 2

tcos1 1
f E;
2 n

x π þ n

x π
n

x π ; n
y π
n

x π ; n
y π L 1  1
f 23 E; 23 2

dx

dy

0

dx

4

dx

dx

dy

(6:39b)
13, 8 is the sub-band energy and the sub-band equation in this case can be where E expressed as  
y π
x π n 1 n 2
2 
−1



(6:39c) 0 =
2 fcos ðf23 ðE13, 8 , kx , ky ÞÞg − ks kx =
and
ky =
 L0 dx dy EEM in this case is given by " !# 2

π n
h 1 π n y x
η ,n

y Þ = 2 cos − 1
f23 E,
* ðE, m g
x , n

x , d
y 2 2L d 0 2 2 2 !#′ !#2 3 − 1=2



π π n n π 1 π n n y 6

x 6
x , y 5 4f23 E,
,
41 − 4
f23 E,
x d
y 4 dx dy d

(6:40)

The electron concentration can be written as
1D = n


xmax n n ymax X
X

½τ21 + τ22 

(6:41)


x = 1 n
y = 1 n

where "" τ21 =

#2 " # 

 1 −1
2 



f 23 ðE, kx , ky Þ − ks  cos 2
L0
kx = π
nx ,
x d

τ22 =


s X

#1=2 , π
ny


ky =
, E = EF611 dy



rÞ½τ21  Lð


r = 1


F611 is the Fermi energy in this case. and E Using (1.31f) and (6.41), we can study the entropy in this case.

6.2 Theoretical background

287

6.2.9 Entropy in HgTe/CdTe quantum wire HD effective mass SLs Following Sasaki [24], the electron dispersion law in HDHgTe/CdTeEMSLs can be written
k2 = Δ13H + iΔ14H , z

(6:42)

where, 

1 Δ13H =
2 ðΔ211H − Δ212H Þ −
ks2 L



0

2
6H = 4
6H , p Δ11H = cos − 1 p

1 − Δ29H − Δ210H −

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31=2 2 ð1 − Δ29H − Δ210H Þ + 4Δ210H 5 , 2


1H cos Δ5H cosh Δ6H − a
2H cos Δ7H cosh Δ6H Þ, Δ9H = ða
1H sin Δ5H sinh Δ6H + a
2H sin Δ7H sinh Δ6H Þ, Δ10H = ða
0
e3 , Δ6H = b
0
e4 , Δ7H = ½a
0
e3 
22H ðE,
22H ðE,

ks, , η Þ + b

ks, , η Þ − b
0 C
0 C Δ5H = ½a g1 g1 
2



2

1=2
22H ðE,

ks, , η Þ = B01 + 2A1 E − B01 ðB01 + 4A1 EÞ , C g1
2 2A 1 "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #2 " ! #
2 ð0, η Þ
2 ð0, η Þ 1=2 − 1 M M g2 g2
1H = , +1 , 4 a m*c1 m*c1 "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #2 " ! #
2 ð0, η Þ
2 ð0, η Þ 1=2 − 1 M M g2 g2
2H = +1 , 4 a m*c1 m*c1
6H , Δ14H = Δ12H = cos − 1 p

2Δ11H Δ12H L20

The entropy in HDHgTe/CdTe QWEMSLs can be written as
k2 = ½Δ13H + Δ14H j z






y π n

kx = n
x π and
ky =
dx dy

(6:43a)

The DOS function can be written as nxmax nX ymax


13, 9 Þ½Δ′13H + iΔ′14H 
X HðE − E
EÞ
= gv pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð π nx = 1 ny = 1 Δ13H + iΔ14H

(6:43b)


13, 9 is the sub-band energy and the sub-band equation in this case can be where E expressed as

288

6 Entropy in quantum wires of heavily doped superlattices

 i  0 = Δ13H + iΔ14H 
n
x π
n
y π kx = d
x , ky = d
y and E
= E
13, 9 h

(6:43c)

The EEM in this case is given by
2 ′ h Δ 13H 2


η ,n
y Þ =
* ðE, m g
x , n

(6:44)

The electron concentration can be written as
1D = Real part of n


xmax n n ymax X
X

½τ23 + τ24 

(6:45)


x = 1 n
y = 1 n

where 2

31=2    6 7 τ23 = 4½Δ13H + iΔ14H 
k = π
nx , 5 , x
π
n y


dx ky =  , E = E F614
dy

τ24 =


s X



rÞ½τ23  Lð


r = 1


F614 is the Fermi energy in this case and E Using (1.31f) and (6.45), we can study the entropy in this case.

6.2.10 Entropy in strained layer quantum wire HD effective mass SLs The dispersion relation of the constituent materials of HD III–V SLs can be written as
i ðE, η Þ
k2 +
Si ðE, η Þk2 = 1
i ðE, η Þk2 + Q P gi x gi y gi z

(6:46)

where
η Þ = ðγ ðE,
η Þ −
I0 T
η ÞÞ − 1 ,
I0 = ð1=2Þ½1 + Erf ðE=η

i ðE,
1i ÞðΔ
i ðE, P gi gi gi gi Þ, 0 qffiffiffiffiffiffiffiffiffiffiffiffiffi
i εxxi − ðbε
i =2Þ + ð 3d
i εxyi =2Þ,
c εi − ða
c Þεi ð3=2Þb
gi − C
li = ½E
i + C T li li pffiffiffi
η Þ = ½ð − q
2 =η2 ÞÞ½1 + ðE2 =η2 Þ − R
i θi ðE,
η Þ+V
i γ ðE,
η Þ
i η3gi =2 πÞ expð − ðE Δi ðE, gi gi gi gi gi 0
c εi , A
c εi ,

2 Þ, R
i = q
i + C
i = E
gi − C
i = ð3=2B
i ½2A + ðζ i =2Þ½1 + Erf ðE=η gi Þ, q 2i li li
2 εxyi =3Þ + 2A
2
c
2
c εi , ζ = q
2 − ð2C
iC
i = q
i ½A V i
i ½ð2C2i εxyi =3Þ − Cli εi Ai , i 2i li

289

6.2 Theoretical background


i ðE,
η Þ = ðγ ðE,
η Þ −
I0 T
η ÞÞ − 1 ,
2i ÞðΔ
i ðE, Q gi gi gi 0 
qffiffiffiffiffiffiffiffiffiffiffiffiffi  c c


i εxyi =2 ,




i + Cli Þεi + ð3=2Þbi εxyi − ðbi εi =2Þ − 3d T2i = Egi − Cli εi − ða
Si ðE,
η Þ = ðγ ðE,
η Þ −
I0 T
η ÞÞ − 1
3i ÞðΔ
i ðE, gi gi gi 0
i εzzi − ðb
i εi =2Þ,
c εi + ð a
c Þεi + ð3=2Þb
gi − C
3i = ½E
i + C T li li Therefore entropy in HD IV–VI, quantum wire EMSLs can be written as   1 2 2 −1

kz =
2 



2 fcos ðf40 ðE, kx , ky ÞÞg − ks 
kx = n

x π and
ky = n

y π L dx dy 0

(6:47a)

The DOS function can be written as
ymax n n ymax X
X
EÞ
¼ g
v Nð πL0 n
x ¼1 n
x ¼1
n

o

h

i0
n

x π ; n

y π
f 40 E;
n

x π ; n

y π
f 40 E;
n

x π ; n

y π Hð
E
E
13;10 Þ cos1 21
f 40 E; dy dy dy dx dx dx ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

n

o2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h

i2ffi 2
y π
y π
x π n
x π
x π
x π n n n n n 1
1
2 1


cos f 40 E;
;
L þ
1  f 40 E;
;

2

dx

dy

0

dx

4

dx

dx

dy

(6:47b)
13, 10 is the sub-band energy and the sub-band equation in this case can be where E expressed as   2 2  1 −1



(6:47c) 0 =
2 cos ðf 40 ðE13, 10 , kx , ky ÞÞ − ks 
n
x π
π n kx = d
x and
ky = d
yy L0 The EEM in this case is given by 

h2 −1 1

η ,n



* ðE, f , n Þ = cos m 40 E, g x y
2 2 2L 0


x π n
x , d


y π n
y d



′  


y π
x π n 1


f40 E,
n 1 − f40 E, ,
x d
y 4 d


y π
x π n n ,

dx dy

2  − 1=2

(6:48a)

The electron concentration can be written as
1D = n


xmax n n ymax X
X
x = 1 n
y = 1 n

where

½τ40 + τ41 

(6:48b)

290

6 Entropy in quantum wires of heavily doped superlattices

2



6 1

kx ,
ky Þgt2 − k2 τ40 = 4
2 ½cos − 1 f
f40 ðE, s L 0

τ41 =


s X

 31=2   
π
nx 5 , kx =
,
π
ny
=E
dx ky =  and E F615 d y



rÞ½τ40  Lð


r = 1


F615 is the Fermi energy in this case. and E Using (1.31f) and (6.48b), we can study the entropy in this case.

6.3 Results and discussion Using the appropriate equations and the band constants from Appendix 15, the entropy in HD QW III–V SLs (taking GaAs/Ga1–xAlxAs and InxGa1−xAs∕InP QW SLs) with graded interfaces has been plotted as functions of the film thickness and impurity concentration at 10 K, respectively, as shown in Figures 6.1 and 6.2, respectively. 0.89690 0.89685

(b)

Normalized entropy

0.89680 0.89675 0.89670 0.89665

(a)

0.89660 0.89655 0.89650 0.89645 10

15

20

25

30

35

40

45

50

Film thickness (in nm) Figure 6.1: Plot of the normalized entropy in (a) GaAs/Ga1–xAlxAs and (b) InxGa1–xAs∕InP HD quantum wire SLs with graded interfaces as a function of film thickness.

The normalized entropy has been plotted for (a) CdS/ZnSe with
λo = 0, (b) CdS/ZnSe with
λo ≠0 (c) HgTe/CdTe and (d) PbSe/PbTe HD quantum wire SLs with graded interfaces as functions of film thickness and impurity concentration in Figures 6.3 and 6.4, respectively. The entropy in GaAs/Ga1–xAlxAs, HgTe/CdTe, CdS/ZnSe, HgTe/Hg1-xCdxTe and PbSe/PbTe quantum wire effective mass SLs have been plotted as functions of film thickness and impurity concentration in Figures 6.5 and 6.6, respectively.

291

6.3 Results and discussion

0.1

1 1

Normalized entropy

0.01

10

(b)

(a) 0.1

0.01

0.001 Impurity concentration (109 m–1) Figure 6.2: Plot of the normalized entropy in (a) GaAs/Ga1–xAlxAs and (b) InxGa1−xAs∕InP HD quantum wire SLs with graded interfaces as a function of impurity concentration.

0.8144 0.6903 0.7942 0.6902 0.7940 0.6901 0.7938 0.69(c)

Normalized entropy

0.8143 0.8142 0.8141

(b) 0.6899

0.8140

(d)

0.6898

0.8139

0.7934

0.6896 0.7930

(a)

0.8137

0.75170 0.75168

0.7936

0.6897 0.7932

0.8138

0.75172

0.75166 0.75164 0.75162

0.6895 0.75160

0.8136 10

15

20

25

30

35

40

45

50 0.6894 0.7928

Film thickness (in nm)
o = 0, (b) CdS/ZnSe with λ
o ≠ 0 (c) HgTe/CdTe Figure 6.3: Plot of the entropy in (a) CdS/ZnSe with λ and (d) PbSe/PbTe HD quantum wire SLswith graded interfaces as a function of film thickness.

The effect of size quantization is clearly exhibited by Figures 6.1 and 6.3, in which the composite fluctuations are due to the combined influence of the Landau quantization effect (due to magnetic field) with the size quantization effect. It also appears from the same figures that the entropy bears step functional dependency function of film thickness due to the Van Hove Singularity. Since the Fermi level decreases with the increase in the film thickness, the entropy increases. This physical fact

292

6 Entropy in quantum wires of heavily doped superlattices

0.1

1 1

10

Normalized entropy

(d) (a) (c) (b) 0.01

0.001

0.0001 Impurity concentration (109 m–1)
o = 0, (b) CdS/ZnSe with λ
o ≠ 0 (c) HgTe/CdTe Figure 6.4: Plot of the entropy in (a) CdS/ZnSe with λ and (d) PbSe/PbTe HD quantum wire SLs with graded interfaces as a function of impurity concentration.

0.98

0.9804 0.965

0.97

0.960 (c) 0.955 (d) 0.950

Normalized entropy

0.96 (a)

0.95

(b)

0.9802 0.98

0.945

0.94

0.940

0.93

0.9798

0.935 0.930

0.92

0.9796

0.925 0.91

0.920

0.90

0.9794

0.915 0.910

0.89 10

11

12

13

14

15

16

17

18

19

0.9792

20

Film thickness (in nm) Figure 6.5: Plot of the entropy in (a) GaAs/Ga1–xAlxAs, (b) CdS/ZnSe, (c) HgTe/CdTe and (d) PbSe/ PbTe HD quantum wire effective mass SLs as a function of film thickness.

also governs the nature of oscillatory variation of all the curves where the change in film thickness with respect to entropy for all type of SLs appears. The entropy changes with film thickness in oscillatory manner, where the nature of oscillations is totally different. It should also be noted that the entropy decreases with the increasing carrier degeneracy exhibiting different types of oscillations as is observed

6.4 Open research problem

0.1

100

10–1

10–2

1

10

(c) (b) (b)

(d)

Normalized entropy

0.001

293

10–3 (a) 10–4

Figure 6.6: Plot of the entropy in (a) GaAs/Ga1-xAlxAs, (b) CdS/ZnSe, (c) HgTe/CdTe and (d) PbSe/ PbTe HD quantum wire effective mass SLsas a function of impurity concentration.

from Figure 6.2. It may be also noted that due to the confinement of carriers along two orthogonal directions, the entropy exhibits the composite oscillations in Figures 6.1 and 6.3, while in Figures 6.2 and 6.4, the absence of composite oscillation are due to the suppression of the size quantization number along one direction by another. It appears from Figure 6.5 that the entropy in GaAs/Ga1–xAlxAs, CdS/ ZnSe, HgTe/CdTe and PbSe/PbTe HD quantum wire effective mass SLs also exhibits such composite oscillations with increasing film thickness. The nature of oscillation in effective mass SLs are radically different than that of the corresponding graded interfaces which is the direct signature of the difference in band structure in the respective cases as found from all the respective corresponding figures. From Figure 6.6, we observe that the entropy in the aforementioned case decreases with increasing impurity concentration and differ widely for large values of impurity concentration, whereas for relatively small values of the carrier degeneracy, the entropy converges to a single value in the whole range of the impurity concentration considered.

6.4 Open research problem (R6.1) Investigate all the appropriate problems of Chapter 3 for all types of quantum wire SLs in the presence of strain.

294

6 Entropy in quantum wires of heavily doped superlattices

References [1]

Mukherjee S., Mitra S.N., Bose P.K., Ghatak A.R., Neoigi A., Banerjee J.P., Sinha A., Pal M., Bhattacharya S., Ghatak K.P., Compu. Journal of Theor. Nanosci 4, 550 (2007); Anderson N.G., Laidig W.D., Kolbas R.M., Lo Y.C., Journal of Appl. Phys. 60, 2361 (1986); Paitya N., Ghatak K.P., Jour. Adv. Phys. 1, 161 (2012); Paitya N., Bhattacharya S., De D., Ghatak K.P., Adv. Sci. Engg. Medi. 4, 96 (2012); Bhattacharya S., De D., Adhikari S.M., Ghatak K.P., Superlatt. Microst. 51, 203 (2012); De D., Bhattacharya S., Adhikari S.M., Kumar A., Bose P.K., Ghatak K.P., Beilstein Jour. Nanotech. 2, 339 (2012); De D., Kumar A., Adhikari S.M., Pahari S., Islam N., Banerjee P., Biswas S.K., Bhattacharya S., Ghatak K.P., Superlatt. and Microstruct. 47, 377 (2010); Pahari S., Bhattacharya S., Roy S., Saha A., De D., Ghatak K.P., Superlatt. and Microstruct. 46, 760 (2009); Pahari S., Bhattacharya S., Ghatak K.P., Jour. of Comput. and Theo. Nanosci. 6, 2088 (2009); Biswas S.K., Ghatak A.R., Neogi A., Sharma A., Bhattacharya S., Ghatak K.P., Phys. E: Low-dimen. Sys. and Nanostruct. 36, 163 (2007); Singh L.J., Choudhury S., Baruah D., Biswas S.K., Pahari S., Ghatak K.P., Phys. B: Conden. Matter, 368, 188 (2005); Chowdhary S., Singh L.J., Ghatak K.P., Phys. B: Conden. Matter, 365, 5 (2005); Singh L. J., Choudhary S., Mallik A., Ghatak K.P., Jour. of Comput. and Theo. Nanosci. 2, 287 (2005). [2] Keldysh L.V., Sov. Phys. Solid State 4, 1658 (1962); Esaki L., Tsu R., IBM J. Research and Develop. 14, 61 (1970); Bastard G., Wave mechanics applied to heterostructures, (Editions de Physique, Les Ulis, France, 1990). [3] Fürjes P., Dücs C., Ádám M., Zettner J., Bársony I., Superlattices and Microstructures 35, 455 (2004); Borca-Tasciuc T., Achimov D., Liu W.L., Chen G., Ren H.W., Lin C.H., Pei S.S., Microscale Thermophysical Engg. 5, 225 (2001). [4] Williams B.S., Nat. Photonics 1, 517 (2007), Kosterev A., Wysocki G., Bakhirkin Y., So S., Lewicki R., Tittel F., Curl R.F., Appl. Phys. B90, 165 (2008); Belkin M.A., Capasso F., Xie F., Belyanin A., Fischer M., Wittmann A., Faist J., Appl. Phys. Lett. 92, 201101 (2008). [5] Brown G.J., Szmulowicz F., Linville R., Saxler A., Mahalingam K., Lin C.–H., Kuo C.H., Hwang W.Y., IEEE Photonics Technology Letts. 12, 684 (2000); Haugan H.J., Brown G.J., Grazulis L., Mahalingam K., Tomich D.H., Physics E: Low-dimensional Systems and Nanostructures 20, 527 (2004). [6] Nikishin S.A., Kuryatkov V.V., Chandolu A., Borisov B.A., Kipshidze G.D., Ahmad I., Holtz M., Temkin H., Jpn. J. Appl. Phys 42, L1362 (2003); Su Y.K., Wang H.C., Lin C.L., Chen W.B., Chen S.M., Jpn. J. Appl. Phys. 42, L751 (2003). [7] Endres C.P., Lewen F.T., Giesen F., Schleemer S., Paveliev D.G., Koschurinov Y.I., Ustinov V.M., Zhucov A.E., Rev. Sci. Instrum 78, 043106 (2007). [8] Klappenberger F., Renk K.F., Renk P., Rieder B., Koshurinov Y.I., Pavelev D.G., Ustinov V., Zhukov A., Maleev N., Vasilyev A., Appl. Phys. Letts 84, 3924 (2004). [9] Jin X., Maeda Y., Saka T., Tanioku M., Fuchi S., Ujihara T., Takeda Y., Yamamoto N., Nakagawa Y., Mano A., Okumi S., Yamamoto M., Nakanishi T., Horinaka H., Kato T., Yasue T., Koshikawa T., J. of Crystal Growth 310, 5039 (2008); Jin X., Yamamoto N., Nakagawa Y., Mano A., Kato T., Tanioku M., Ujihara T., Takeda Y., Okumi S., Yamamoto M., Nakanishi T., Saka T., Horinaka H., Kato T., Yasue T., Koshikawa T., Appl. Phys. Express 1, 045002 (2008). [10] Lee B.H., Lee K.H., Im S., Sung M.M., Organic Electronics 9, 1146 (2008). [11] Wu P.H., Su Y.K., Chen I.L., Chiou C.H., Hsu J.T., Chen W.R., Jpn. J. Appl. Phys 45, L647 (2006); Varonides A.C., Renewable Energy 33, 273 (2008). [12] Walther M., Weimann G., Phys. Stat. Sol. (b) 203, 3545 (2006).

References

295

[13] Rehm R., Walther M., Schmitz J., Fleiβner J., Fuchs F., Ziegler J., Cabanski W., Opto-Electronics Rev 14, 19 (2006); Rehm R., Walther M., Scmitz J., Fleissner J., Ziegler J., Cabanski W., Breiter R., Electronics Letts. 42, 577 (2006). [14] Brown G.J., Szmulowicz F., Haugan H., Mahalingam K., Houston S., Microelectronics Jour 36, 256 (2005). [15] Vaidyanathan K.V., Jullens R.A., Anderson C.L., Dunlap H.L., Solid State Electron 26, 717 (1983). [16] Wilson B.A., IEEE, J. Quantum Electron 24, 1763 (1988); 1988. [17] Krichbaum M., Kocevar P., Pascher H., Bauer G., IEEE, J. Quantum Electron 24, 717 (1988). [18] Schulman J.N., McGill T.C., Appl. Phys. Letts 34, 663 (1979). [19] Kinoshita H., Sakashita T., Fajiyasu H., J. Appl. Phys 52, 2869 (1981). [20] Ghenin L., Mani R.G., Anderson J.R., Cheung J.T., Phys. Rev B 39, 1419 (1989). [21] Hoffman C.A., Mayer J.R., Bartoli F.J., Han J.W., Cook J.W., Schetzina J.F., Schubman J.M., Phys. Rev. B 39, 5208 (1989). [22] Yakovlev V.A., Sov. Phys. Semicon 13, 692 (1979). [23] Kane E.O., J. Phys. Chem. Solids 1, 249 (1957). [24] Jiang H.X., Lin J.Y., J. Appl. Phys 61, 624 (1987). [25] Sasaki H., Phys. Rev B 30, 7016 (1984).

7 Entropy in quantum dot HDSLs It is much better to know something about your own pin pointed topic of research than to know everything about one thing.

7.1 Introduction In this chapter, the entropy from III–V, II–VI, IV–VI, HgTe/CdTe and strained layer quantum dot heavily doped superlattices (QDHDSLs) with graded interfaces [1–10] has been studied in Sections 7.2.1 to 7.2.5. From Sections 7.2.6 to 7.2.10, the entropy from III–V, II–VI, IV–VI, HgTe/CdTe and strained layer quantum dot heavily doped effective mass SLs [6–10], respectively, has been presented. Section 7.3 contains the summary and conclusion pertinent to this chapter. Section 7.4 presents 14 open research problems.

7.2 Theoretical background 7.2.1 Entropy in III–V quantum dot HD SLs with graded interfaces The simplified DR of heavily doped quantum dot III–V SLs with graded interfaces can be expressed as
  
z π n
8 + iH
8  = G (7:1)
y π n



−E
kx = n
x π and
ky =
E dz 14, 1 dx dy
14, 1 is the totally quantized energy in this case. where E The DOS function is given by
xmax n
ymax n n zmax X
X 2g
v X

−E
14, 1 Þ
0QDSL ðEÞ = δ′ðE n
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(7:2)

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0 = n F 7, 1
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n





E −E
F7, 1 is the Fermi energy in this case. where η7, 1 = F7,
k1 T
14, 1 and E B Using (1.31f) and (7.3), we can study the entropy in this case.

https://doi.org/10.1515/9783110661194-007

(7:3)

298

7 Entropy in quantum dot HDSLs

7.2.2 Entropy in II–VI quantum dot HD SLs with graded interfaces The simplified DR of heavily doped quantum dot III–V SLs with graded interfaces can be expressed as
2  
z π n
19 + iH
19  = G
y π n


z
−E
kx = n
x π and
ky =
E d 14, 2 dx

(7:4)

dy


14, 2 is the totally quantized energy in this case. where E The DOS function is given by
= 2g
v
0QDSL ðEÞ n
x d
y d
z d


xmax n
zmax n ymax n X
X X


−E
14, 2 Þ δ′ðE

(7:5)


x = 1 n
y = 1 n
z = 1 n

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0 = n F 7, 2
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n


(7:6)




E −E
F7, 2 is the Fermi energy in this case. where η7, 2 = F7, 2
k T
14, 2 and E B Using (1.31f) and (7.6), we can study the entropy in this case.

7.2.3 Entropy in IV–VI quantum dot HD SLs with graded interfaces The simplified DR in heavily doped quantum dot IV–VI SLs with graded interfaces can be expressed as #
2 "   2 
z π 1 n −1 1 2
k 
14, 3 ,
ks Þ = cos ð E − (7:7) Φ 2
y π n s 2
z

kx = n
x π and
ky =
2 L d 0

dx

dy


14, 3 is the totally quantized energy in this case. where E The DOS function is given by
xmax n
ymax n n zmax X
X 2g
v X

−E
14, 3 Þ
0QDSL ðEÞ = δ′ðE n
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(7:8)

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0 = n F 7, 3
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(7:9)

7.2 Theoretical background




299




−E E
F7, 3 is the Fermi energy in this case. where η7, 3 = F7, 3
k T
14, 3 and E B Using (1.31f) and (7.9), we can study the entropy in this case.

7.2.4 Entropy in HgTe/CdTe quantum dot HD SLs with graded interfaces The simplified DR of heavily doped quantum dot III–V SLs with graded interfaces can be expressed as
2 
z π n
192 +
iH
192  = ½ G
y π n


z
−E
ky =
and E kx = n
x π ,
d 14, 4 dx

(7:10)

dy


14, 4 is the totally quantized energy in this case. where E The DOS function is given by
xmax n
ymax n n zmax X
X 2g
v X

−E
14, 4 Þ
δ′ðE n0QDSL ðEÞ =


dx dy dz n
x = 1 n
y = 1 n
z = 1

(7:11)

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0 = n F 7, 4
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n


(7:12)




−E E
F7, 4 is the Fermi energy in this case. where η7, 4 = F7, 4
k T
14, 4 and E B Using (1.31f) and (7.12), we can study the entropy in this case.

7.2.5 Entropy in strained layer quantum dot HD SLs with graded interfaces DR of the conduction electrons in heavily doped strained layer quantum dot SL with graded interfaces can be expressed as #
 "   2 
z π 1 n −1 1 2
k 
14, 5 , k
s Þ = cos ð E − (7:13) Φ 6
y π n s
2
z

kx = n
x π and
ky =
2 L d 0 d d x

y


14, 5 is the totally quantized energy in this case. where E The DOS function is given by nxmax nymax nzmax X X X
= 2g
v
−E
14, 5 Þ
0QDSL ðEÞ δ′ðE n
x d
y d
z d nx = 1 ny = 1 nz = 1

The electron concentration can be expressed as

(7:14)

300

7 Entropy in quantum dot HDSLs


xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0 = F n 7, 6
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n


(7:15)




−E E
F7, 6 is the Fermi energy in this case. where η7, 5 = F7, 6
k T
14, 6 and E B Using (1.31f) and (7.15), we can study the entropy in this case.

7.2.6 Entropy in III–V quantum dot HD effective mass SLs DR in III–V heavily doped effective mass quantum dot SLs can be written as
 
z π n  = ½δ7 + iδ8 
n
x π
n
y π

−E
kx =
, ky =
and E dz 14, 6 dx dy

(7:16)


14, 6 is the totally quantized energy in this case. where E The DOS function is given by
xmax n
ymax n n zmax X
X 2g
v X

−E
14, 6 Þ
0QDSL ðEÞ = δ′ðE n
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(7:17)

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0 = n F 7, 6
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n


(7:18)




E −E
F7, 6 is the Fermi energy in this case. where η7, 6 = F7, 6
k T
14, 6 and E B Using (1.31f) and (7.18), we can study the Entropy in this case.

7.2.7 Entropy in II–VI quantum dot HD effective mass SLs DR in III–V heavily doped effective mass quantum dot SLs can be written as
 
z π n  = ½δ + iδ  
n
x π
n
y π 13 14
z
−E
kx =
, ky =
and E d 14, 6 dx dy

(7:19)


14, 7 is the totally quantized energy in this case. where E The DOS function is given by
xmax n
zmax n ymax n X
X X
= 2g
v
−E
14, 7 Þ
0QDSL ðEÞ δ′ðE n


dx dy dz n
x = 1 n
y = 1 n
z = 1

(7:20)

7.2 Theoretical background

301

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0 = n F 7, 7
x d
y d
z d


nx = 1 ny = 1 nz = 1

(7:21)







−E E
F7, 7 is the Fermi energy in this case. where η7, 7 = F7, 7
k T
14, 7 and E B Using (1.31f) and (7.21), we can study the entropy in this case.

7.2.8 Entropy in IV–VI quantum dot HD effective mass SLs DR_in heavily doped IV–VI, quantum dot EMSLs can be written as
2  

2 2 
z π 1  n −1




f 23 ðE14, 8 , kx , ky Þ =
2 cos − ks 
n
x π
y π n
z kx =
and
ky =
L d 0

dx

(7:22)

dy


14, 8 is the totally quantized energy in this case. where E The DOS function is given by
xmax n
ymax n n zmax X
X 2g
v X

−E
14, 8 Þ
δ′ðE N0QDSL ðEÞ =


dx dy dz n
x = 1 n
y = 1 n
z = 1

(7:23)

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0 = n F 7, 8
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(7:24)

where η7, 8 =


14, 8
F7, 8 − E E
kB T



F7, 8 is the Fermi energy in this case. and E Using (1.31f) and (7.24), we can study the entropy in this case.

7.2.9 Entropy in HgTe/CdTe quantum dot HD effective mass SLs DR in heavily doped HgTe/CdTe QWEMSLs can be written as
2 
z π n  = ½Δ + iΔ  
n
x π
n
y π 13H 14H
z
=E
kx =
, ky =
and E d 14, 9 dx dy
14, 9 is the totally quantized energy in this case. where E

(7:25)

302

7 Entropy in quantum dot HDSLs

The DOS function is given by
xmax n
zmax n ymax n X
X X
= 2g
v
−E
14, 9 Þ
QDSL ðEÞ δ′ðE N


dx dy dz n
x = 1 n
y = 1 n
z = 1

(7:26)

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
n0 =


F 7, 9 dx dy dz n
x = 1 n
y = 1 n
z = 1


(7:27)




−E E
F7, 9 is the Fermi energy in this case. where η7, 9 = F7, 9
k T
14, 9 and E B Using (1.31f) and (7.27), we can study the entropy in this case.

7.2.10 Entropy in strained layer quantum dot HD effective mass SLs DR in heavily doped IV–VI, quantum dot EMSLs can be written as
2  

2 2 
z π 1  n −1




f = cos ð E , k , k Þ − k
y π 40 14, 10 x y n s 
n
x π
z
2 kx =
and
ky =
L d 0 dx dy

(7:28)


14, 10 is the totally quantized energy in this case. where E The DOS function is given by
xmax n
ymax n n zmax X
X
X
= 2g v
−E
14, 10 Þ
QDSL ðEÞ δ′ðE N
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n

(7:29)

The electron concentration can be expressed as
xmax n
ymax n n zmax X
X 2g
v X
− 1 ðη Þ
0 = n F 7, 10
x d
y d
z d
x = 1 n
y = 1 n
z = 1 n


(7:30)




−E E
F7, 10 is the Fermi energy in this case. where η7, 10 = F7, 10
k T
14, 10 and E B Using (1.31f) and (7.30), we can study the entropy in this case.

7.3 Results and discussion Using the band constants from appendix 15, the normalized entropy in this case in HgTe=Hg1 − x Cdx Te, CdS=ZnSe, PbSe=PbTe and HgTe=CdTeHD quantum dot SLs with graded interfaces have been plotted as a function of film thickness as shown by curves (a), (b), (c), and (d), respectively, in Figure 7.1. Figure 7.2 demonstrates the

7.3 Results and discussion

303

0.940 0.935

Normalized entropy

0.930

(c)

0.925 0.920

(b)

0.915 0.910

(a)

0.905 (d)

0.900 0.895 0.890 20

30

40

50

60

70

80

90

100

Figure 7.1: Plot of the entropy in (a) HgTe/Hg1–xCdxTe, (b) CdS/ZnSe, (c) PbSe/PbTe, and (d) HgTe/CdTe HD quantum dot SLs with graded interfaces as a function offilm thickness.

0.01

1 1

10

0.1

0.1

Normalized entropy

(c)

(b) (a) (d)

0.03

Figure 7.2: Plot of the entropy in (a) HgTe/Hg1–xCdxTe, (b) CdS/ZnSe, (c) PbSe/PbTe, and (d) HgTe/CdTe HD quantum dot SLs with graded interfaces as a function of carrier concentration.

normalized entropy for the said quantized structures as a function of impurity concentration. Figure 7.3 exhibits the normalized entropy as a function of film thickness in HgTe=Hg1 − x Cdx Te, CdS=ZnSe, PbSe=PbTe and HgTe=CdTe HD quantum dot effective mass SLs as shown by curves (a), (b), (c), and (d), respectively. The normalized entropy in HgTe=Hg1 − x Cdx Te, CdS=ZnSe, PbSe=PbTe and HgTe=CdTeHD quantum dot effective mass SLs has been plotted as a function of electron concentration in Figure 7.4.

304

7 Entropy in quantum dot HDSLs

0.8 0.7

(c)

0.6

(b)

0.5

(a)

0.4 (d)

0.3 0.2 0.1 0 20

30

40

50

60

70

80

90

100

Figure 7.3: Plot of the entropy in (a) HgTe/Hg1–xCdxTe, (b) CdS/ZnSe, (c) PbSe/PbTe, and (d) HgTe/CdTeHD quantum dot effective mass SLs as a function of film thickness.

Normalized entropy

0.01

0.1

10

100

10–1

(c) (b) (a) (d)

10–2

10–3

Figure 7.4: Plot of the entropy in (a) HgTe/Hg1–xCdxTe, (b) CdS/ZnSe, (c) PbSe/PbTe, and (d) HgTe/ CdTeHD quantum dot effective mass SLs as a function of carrier concentration.

It appears from Figure 7.1 that the entropy in HgTe=Hg1 − x Cdx Te, CdS=ZnSe, PbSe=PbTe and HgTe=CdTe HD quantum dot SLs with graded interfaces increases with increasing film thickness exhibiting quantum jumps for fixed values of film thickness depending on the values of the energy band constants of the particular quantized structures. It is observed from Figure 7.2 that the entropy in quantum dots of aforementioned SLs decreases with increasing carrier degeneracy and differ widely for large values of same whereas for relatively small values of electron concentration, the entropy exhibits a converging behavior. From Figure 7.3, it is observed that the TPSM in HgTe=Hg1 − x Cdx Te, CdS=ZnSe, PbSe=PbTe and HgTe=CdTe quantum dot effective

7.4 Open research problems

305

mass SLs oscillates with increasing film thickness. From Figure 7.4, it appears that the entropy of the aforementioned SLs decrease with increasing concentration. It should be noted that all types of variations of entropy with respect to thickness and concentration are basically band structure dependent. It may further be noted that the entropy of a two-dimensional electron gas in the presence of a periodic potential has already been formulated in the literature. SL is a three-dimensional system under periodic potential. There is a radical difference in the dispersion relations of the 3D quantized structures and the corresponding carrier energy spectra of the 2D systems. From the dispersion relations of various SLs as discussed in this chapter, the energy spectra of the various other types of low-dimensional systems can be formulated and the corresponding entropy can also be investigated. The results will be fundamentally different in all cases due to system asymmetry together with the change in the respective wave functions exhibiting new physical features in the respective cases. Therefore, it appears that the dispersion law and the corresponding wave function play a cardinal role in formulating any electronic property of any electronic material, since they change in a fundamental way in the presence of dimension reduction. Consequently, the derivations and the respective physical interpretations of the different transport quantities change radically. It is imperative to state that our investigations excludes the many-body, hot electron, spin, broadening and the allied quantum dot and SL effects in this simplified theoretical formalism due to the absence of proper analytical techniques for including them for the generalized systems as considered here. Our simplified approach will be appropriate for the purpose of comparison when the methods of tackling the formidable problems after inclusion of the said effects for the generalized systems emerge. Finally, it may be noted that the inclusion of the said effects would certainly increase the accuracy of the results although the qualitative features of the entropy would not change in the presence of the aforementioned effects.

7.4 Open research problems (R 7.1)

(R 7.2)

(R 7.3)

Investigate the entropy in the absence of magnetic field by considering all types of scattering mechanisms for III–V, II–VI, IV–VI and HgTe/CdTe SLs with graded interfaces and also the effective mass SLs of the aforementioned materials with the appropriate dispersion relations as formulated in this chapter. Investigate the entropy in the absence of magnetic field by considering all types of scattering mechanisms for strained layer, random, short period, Fibonacci, polytype and saw-toothed SLs, respectively. Investigate the entropy in the absence of magnetic field by considering all types of scattering mechanisms for (R3.1) and (R3.2) under an arbitrarily

306

7 Entropy in quantum dot HDSLs

oriented (a) nonuniform electric field and (b) alternating electric field, respectively. (R 7.4) Investigate the entropy by considering all types of scattering mechanisms for (R3.1) and (R3.2) under an arbitrarily oriented alternating magnetic field by including broadening and the electron spin, respectively. (R 7.5) Investigate the entropy by considering all types of scattering mechanisms for (R3.1) and (R3.2) under an arbitrarily oriented alternating magnetic field and crossed alternating electric field by including broadening and the electron spin, respectively. (R 7.6) Investigate the entropy by considering all types of scattering mechanisms for (R3.1) and (R3.2) under an arbitrarily oriented alternating magnetic field and crossed alternating non-uniform electric field by including broadening and the electron spin, respectively. (R 7.7) Investigate the entropy in the absence of magnetic field for all types of SLs as considered in this chapter under exponential, Kane, Halperin, Lax and Bonch-Bruevich band tails [30], respectively. (R 7.8) Investigate the entropy in the absence of magnetic field for the problem as defined in (R3.7) under an arbitrarily oriented (a) nonuniform electric field and (b) alternating electric field, respectively. (R 7.9) Investigate the entropy for the problem as defined in (R3.7) under an arbitrarily oriented alternating magnetic field by including broadening and the electron spin, respectively. (R 7.10) Investigate the entropy for the problem as defined in (R3.7) under an arbitrarily oriented alternating magnetic field and crossed alternating electric field by including broadening and the electron spin, respectively. (R 7.11) Investigate the problems as defined in (R3.1) to (R3.10) for all types of quantum dot SLs as discussed in this chapter. (R 7.12) Investigate the problems as defined in (R3.1) to (R3.10) for all types of quantum dot SLs as discussed in this chapter in the presence of strain. (R 7.13) Introducing new theoretical formalisms, investigate all the problems of this chapter in the presence of hot electron effects. (R 7.14) Investigate the influence of deep traps and surface states separately for all the appropriate problems of this chapter after proper modifications.

References [1]

[2]

Ghatak K.P., Mukhopadhyay J., Banerjee J.P., SPIE Proceedings Series 4746, 1292 (2002); Ghatak K.P., Dutta S., Basu D.K., Nag B., Il Nuovo Cimento. D, 20, 227 (1998); Ghatak K.P., Basu D.K., Nag B., Jour. of Phys. and Chem. of Solids, 58, 133 (1997). Ghatak K.P., De B., Mat. Resc. Soc. Proc 300, 513 (1993); Ghatak K.P., Mitra B., Il Nuovo Cimento. D 15, 97 (1993); Ghatak K.P., Inter. Soci. Opt. and Photon. Proc. Soc. Photo Opt. Instru. Engg., 1626, 115 (1992).

References

[3]

307

Ghatak K.P., Ghoshal A., Phys. Stat. Sol. (b) 170, K27 (1992); Ghatak K.P., Bhattacharya S., Biswas S.N., Proc. Soc. Photo opt. instru. Engg., 836, 72 (1988); Ghatak K.P., Ghoshal A., Biswas S.N., Mondal M., Proc. Soc. Photo Opt. Instru. Engg. 1308, 356 (1990). [4] Ghatak K.P., De B., Proc. Wide bandgap semi. Symp., Matt. Res. Soc 377 (1992); Ghatak K.P., De B., Defect Engg. Semi. Growth, Processing and Device Tech. Symp., Mat. Res. Soc. 262, 911 (1992); Biswas S.N., Ghatak K.P., Internat. Jour. Electronics Theo. Exp. 70, 125 (1991). [5] Mitra B., Ghatak K.P., Phys. Lett. A 146, 357 (1990); Mitra B., Ghatak K.P., Phys. Lett. A. 142, 401 (1989); Ghatak K.P., Mitra B., Ghoshal A., Phy. Stat. Sol. (b), 154, K121 (1989). [6] Mitra B., Ghatak K.P., Phys. Stat. Sol. (b) 149, K117 (1988), Ghatak K.P., Biswas S.N., (1987) Proc. Soc. Photo Optical Instru. Engg. 792, (239); Bhattacharyya S., Ghatak K.P., Biswas S., (1987) OE/Fibers’ 87, Inter. Soc. Opt. Photon. (73). [7] Mondal M., Ghatak K.P., Czech. Jour. Phys. B 36, 1389 (1986); Ghatak K.P., Chakravarti A.N., Phys. Stat. Sol. (b), 117, 707 (1983). [8] Ivchenko E.L., Pikus G., Superlattices and other heterostructures (Springer-Berlin, 1995); Tsu R., Superlattices to nanoelectronics, (Elsevier, The Netherlands, 2005). [9] Liu C.H., Su Y.K., Wu L.W., Chang S.J., Chuang R.W., Semicond. Sci. Technol 18, 545 (2003). [10] Che S.B., Nomura I., Kikuchi A., Shimomura K., Kishino K., Phys. Stat. Sol. (b) 229, 1001 (2002).

8 Entropy in HDSLs under magnetic quantization Real gentleness in a person is the power that sees, understands and yet interferes in a positive way.

8.1 Introduction In this chapter, the magneto entropy in III–V, II–VI, IV–VI, HgTe/CdTe, and strained layerheavily doped superlattices (HDSLs) with graded interfaces [1–10] has been studied in Sections 8.2.1 to 8.2.5. From Sections 8.2.6 to 8.2.10, the magnetoentropy in III–V, II–VI, IV–VI, HgTe/CdTe and strained layer heavily doped (HD) effective mass superlattices (SL), respectively, has been presented. Section 8.3 contains the result and discussions pertinent to this chapter. Section 8.4 presents 14 open research problems.

8.2 Theoretical background 8.2.1 Entropy in III–VHD SLs with graded interfaces under magnetic quantization The simplified DR of HD quantum well III–V SLs with graded interfaces under magnetic quantization can be expressed as
k2 = G
8, E, n + iH
8E, n z where "
8E, n = G



 #
2 − D
2 C 2eB 1 7 E, n 7 E, n
7 E, n = cos − 1 ðω7E, n Þ,
− n+ ,C
2
h 2 L 0

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi21 1
2 − H
2 − H
2
2 Þ − ð1 − G
2 Þ2 + 4 G ω7E, n = ð2Þ − 2 ð1 − G 7E, n 7E, n 7E, n 7E, n 7E, n 
7E, n = G

1E, n + ðρ G
G 5E, n 2E, n =2Þ − ðρ6E, n H2E, n =2Þ
2E, n − ρ H


+ ðΔ0 =2Þfρ6E, n H 8E, n 3E, n + ρ9E, n H4E, n − ρ10E, n H4E, n 
6E, n − ρ
5E, n − ρ
5E, n + ð1=12Þðρ
6E, n Þg , ρ11E, n H H H G 12E, n 12E, n 14E, n

https://doi.org/10.1515/9783110661194-008

(8:1a)

310

8 Entropy in HDSLs under magnetic quantization


1E, n ÞÞðcoshðh
2E, n ÞÞðcoshðg
1E, n ÞÞðcosðg
2E, n ÞÞ
1E, n = ½ðcosðh G
2E, n ÞÞðsinhðg
1E, n ÞÞðsinðg
2E, n ÞÞ,
1E, n ÞÞðsinhðh + ðsinðh
1E, n = e1E, n ðb
0 − Δ0 Þ, e1E, n = 2 − 21 h

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1
t2 +
t22 +
t1E, n 2 1E, n

 

 1
11 ðE,
E
g1 , Δ1 η Þ − 2eB n
t1E, n = ð2m
*c1 =
h2 ÞT
+ , g1 h
2
21 ðE,
E
g1 , Δ1 η Þ
t2 = ð2m
*c1 =
h2 ÞT g1
2E, n = e2E, n ðb
0 − Δ0 Þ, e2E, n = 2 − 21 h

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1
t2 +
t22 −
t1E, n 2 , 1E, n


1E, n ðb
0 − Δ0 Þ, d
1E, n = 2 − 21 g
1E, n = d

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

21
x21E, n +
y21 −
x1E, n

 
 1
11 ðE
−V
0 , E
g0 , Δ0 η Þ − 2eB ðn
*c2 =
h2 ÞT

x1E, n = − ð2m + Þ , g2
h 2
−V
0 , E
g2 , Δ2 η Þ
*c2 =
h2 ÞT22 ðE
y2 = ð2m g2
2E, n ðα0 − Δ0 Þ, d
2E, n = 2 − 21 g
2E, n = d

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

21 2 2 −


x1E, + y , x 1E, n 1 n

  ρ5E, n = ðρ23E, n + ρ24E, n Þ − 1 ρ1E, n ρ3E, n − ρ2E, n ρ4E, n ,
2 + e2 − d
2 − e2 , ρ

ρ1E, n = ½d 3E, n = ½d1E, n e1E, n + d2E, n e2E, n , 1E, n 2E, n 2E, n 1E, n
2E, n + e1E, n e2E, n , ρ

1E, n d
ρ2E, n = 2½d 4E, n = ½d1E, n e2E, n − e1E, n d2E, n ,
1E, n ÞÞðcoshðh
2E, n ÞÞðsinhðg
1E, n ÞÞðcosðg
2E, n ÞÞ
2E, n = ½ðsinðh G
2E, n ÞÞðcoshðg
1E, n ÞÞðsinðg
2E, n ÞÞ,
1E, n ÞÞðsinhðh + ðcosðh   ρ6E, n = ðρ23E, n + ρ24E, n Þ− 1 ρ1E, n ρ4E, n + ρ2E, n ρ3E, n ,
1E, n ÞÞðcoshðh
2E, n ÞÞðsinðg
2E, n ÞÞðcoshðg
1E, n ÞÞ
2E, n = ½ðsinðh H
2E, n ÞÞðsinhðg
1E, n ÞÞðcosðg
2E, n ÞÞ,
1E, n ÞÞðsinhðh − ðcosðh
2E, n e2E, n  − 3e1E, n ,
2 − d
2 Þ − 2d
1E, n d ρ7E, n = ½ðe21E, n + e22E, n Þ − 1 ½e1E, n ðd 1E, n 2E, n
1E, n ÞÞðcoshðh
2E, n ÞÞðcoshðg
1E, n ÞÞðcosðg
2E, n ÞÞ
3E, n = ½ðsinðh G
2E, n ÞÞðsinhðg
1E, n ÞÞðsinðg
2E, n ÞÞ,
1E, n ÞÞðsinhðh + ðcosðh
2E, n e1E, n  + 3e2E, n ,
2 − d
2 Þ − 2d
1E, n d ρ8E, n = ½ðe21E, n + e22E, n Þ − 1 ½e2E, n ðd 1E, n 2E, n

8.2 Theoretical background

311


1E, n ÞÞðcoshðh
2E, n ÞÞðsinðg
2E, n ÞÞðsinhðg
1E, n ÞÞ
3E, n = ½ðsinðh H
2E, n ÞÞðcoshðg
1E, n ÞÞðcosðg
2E, n ÞÞ,
1E, n ÞÞðsinhðh − ðcosðh
2E, n e1E, n  + 3d
2 + d
2 Þ − 1 ½d
2E, n ðe2 − e2 Þ + 2e1E, n d
1E, n , ρ9E, n = ½ðd 1E, n 2E, n 1E, n 2E, n
1E, n ÞÞðcoshðh
2E, n ÞÞðcosðg
2E, n ÞÞðsinhðg
1E, n ÞÞ
4E, n = ½ðcosðh G
2E, n ÞÞðcoshðg
1E, n ÞÞðsinðg
2E, n ÞÞ,
1E, n ÞÞðsinhðh − ðsinðh
2E, n ð − e2 + e2 Þ + 2e1E, n d
2E, n e1E, n  + 3d
2 + d
2 Þ − 1 d
2E, n , ρ10E, n = ½ − ðd 1E, n 2E, n 1E, n 2E, n
1E, n ÞÞðcoshðh
2E, n ÞÞðcoshðg
1E, n ÞÞðsinðg
2E, n ÞÞ
4E, n = ½ðcosðh H
2E, n ÞÞðsinhðg
1E, n ÞÞðcosðg
2E, n ÞÞ,
1E, n ÞÞðsinhðh + ðsinðh
2 + e2 − e2 − d
2 , ρ11E, n = 2½d 1E, n 2E, n 2E, n 1E, n
1E, n ÞÞðcoshðh
2E, n ÞÞðcosðg
2E, n ÞÞðcoshðg
1E, n ÞÞ
5E, n = ½ðcosðh G
2E, n ÞÞðsinhðg
1E, n ÞÞðsinðg
2E, n ÞÞ,
1E, n ÞÞðsinhðh − ðsinðh
1E, n + d
2E, n − e1E, n + e2E, n , ρ12E, n = 4½d
1E, n ÞÞðcoshðh
2E, n ÞÞðsinhðg
1E, n ÞÞðsinðg
2E, n ÞÞ
5E, n = ½ðcosðh H
2E, n ÞÞðcoshðg
1 ÞÞðcoshðg
2 ÞÞ,
1E, n ÞÞðsinhðh + ðsinðh
1E, n + e3 − 3e1E, n e2 + d
2E, n Þ + 5d
2E, n , ðe3 − 3e2 e2E, n Þg ρ13E, n = ½f5ðd 1E, n 2E, n 2E, n 1E, n
2 Þ − 1 + ðe2 + e2 Þ − 1 f5ð − e1E, n + d
3 − 3e2E, n e2 + d
1E, n Þ
2 + d ðd 1E, n 2E, n 1E, n 2E, n 2E, n 1E, n
2 + d
2 Þ − 1
2E, n , ð − e3 − 3e2 e1E, n Þgðd + 5d 1E, n 2E, n 1E, n 2E, n
1E, n e2E, n − d
2E, n e1E, n Þ + 34ðd
1E, n ÞÞðcoshðh
2E, n ÞÞðcoshðg
1E, n ÞÞðsinðg
2E, n ÞÞ
6E, n = ½ðsinðh H
1E, n ÞÞðsinhðh
2E, n ÞÞðsinhðg
1E, n ÞÞðcosðg
2E, n ÞÞ, − ðcosðh
7E, n = ½H
1E, n + ðρ H
H 5E, n 2E, n =2Þ
2E, n =2Þ + ðΔ0 =2Þfρ G



+ ðρ6E, n G 8E, n 3E, n + ρ7E, n H 3E, n + ρ10E, n G4E, n + ρ9E, n H 4E, n
5E, n + ρ


+ ρ12E, n G 11E, n H 5E, n + ð1=12Þðρ14E, n G6E, n + ρ13E, n H 6E, n Þg
1E, n ÞÞðsinhðh
2E, n ÞÞðcoshðg
1E, n ÞÞðcosðg
2E, n ÞÞ
1E, n = ½ðsinðh H
2E, n ÞÞðsinhðg
1E, n ÞÞðsinðg
2E, n ÞÞ,
1E, n ÞÞðcoshðh + ðcosðh
7E, n D
8E, n = ð2C
2 Þ
7E, n = sinh − 1 ðω7E, n Þ, H
7E, n =L D 0 The DOS function can be written a

312

8 Entropy in HDSLs under magnetic quantization

max
n
X
E
−E
15, 1 Þ ðG′8E, n + iH ′8E, n ÞHð
EÞ
= eB pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð 2
8E, n + iH
8E, n 2π
h n
= 0 G

(8:1b)


15, 1 is the sub-band energy in this case and is given by where E 
8E, n +
iH
8E, n  0 = ½G

(8:1c)

E = E15, 1

EEM can be written as
η ,n
* ðE, m g
Þ =


2 ′ h G 8E, n 2

(8:2)

The electron statistics can be expressed as
0 = n

nX   s max X g
v eB
8, E, n +
iH
8, E, n +
iH
8, E, n Þ1=2 
8, E, n Þ1=2  ½ð G LðrÞ½ð G Real part of EF1321 EF1321 π2
h r=1 n=0

where EF1321 is the fermi energy in this case. Using (1.31f) and (8.4), we can study the entropy in this case.

8.2.2 Entropy in II–VI HD SLs with graded interfaces under magnetic quantization The simplified DR in HD II–VI SLs with graded interfaces under magnetic quantization can be expressed as
k2 = G
19E, n +
iH
19E, n z

(8:3a)

where "

#
2
2 C 2eB 1 18E, n − D18E, n

+ − G19E, n = n
2
h 2 L 0
18E, n = cos − 1 ðω18E, n Þ, C −1


2
2 ω18E, n = ð2Þ 2 ½ð1 − G 18E, n − H18E, n Þ −

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2 21 ,
2 Þ2 + 4G ð1 − G −H 18E, n

18E, n

180D


12E, n + Δ0 ðG
13E, n + G
14E, n Þ + Δ0 ðG
15E, n + G
16E, n Þ,
18E, n = 1 ½G
11E, n + G G 2
0 − Δ0 Þ
11E, n = 2ðcosðg
1E, n ÞÞðcosðg
2E, n ÞÞðcos γ ðE,
n
n
n
ÞÞ, γ11 ðE,
Þ =
k21 ðE,
Þðb G 11 ("
k21 ðE,
n
Þ =


 
1 # * )1=2 2  2 2mjj, 1
h 2eB 1 2eB 1
0
η Þ−
+
+ ±C γ3 ðE, , n n g1 *
?, 1
h 2
h 2 2m
h2

8.2 Theoretical background

313


12E, n = ð½Ω1 ðE,
n
Þðsinh g
1E, n Þðcos g
2E, n Þ G

n
Þðsin g
2E, n Þðcosh g
1E, n ÞÞðsin γ11 ðE,
ÞÞÞ − Ω2 ðE, n " #
1E, n
k21 ðE,
1E, n
k21 ðE,
1E, n
n
n
Þd
Þd1E, n d d


Þ =
−
2 −
2 Ω1 ðE, n
k21 ðE,
2 , Ω2 ðE, nÞ = ½
k21 ðE,
2
n
n
Þ
Þ d1E, n + d d1E, n + d 2E, n 2E, n
13E, n = ð½Ω3 ðE,
n
Þðcosh g
1E, n Þðcos g
2E, n Þ G

n
Þðsin g
2E, n Þðcosh g
1E, n ÞÞðsin γ11 ðE,
ÞÞÞ − Ω4 ðE, n " # 


2 − d
2 d 2d1E, n d2E, n 2E, n
21 ðE,
n
n
n
Þ = 1E, n
Þ , Ω4 ðE,
Þ = − 3 k Ω3 ðE,
k21 ðE,
k21 ðE,
n
n
Þ
Þ
14E, n = ð½Ω5 ðE,
n
Þðsinh g
1E, n Þðcos g
2E, n Þ G
n
n
Þðsin g
2E, n Þðcosh g
1E, n ÞÞðcos γ11 ðE,
ÞÞÞ − Ω6 ðE, " #
1E, n
2E, n d d 2 2
1E, n







n


Þ = 3d Ω5 ðE,
2 − d
2 − d
2 − k21 ðE, nÞ, Ω6 ðE, nÞ = ½3d2E, n d
2 − k21 ðE, nÞ d 1E, n 2E, n 1E, n 2E, n
15E, n = ð½Ω9 ðE,
n
Þðcosh g
1E, n Þðcos g
2E, n Þ G
n
n
Þðsinh g
1E, n Þðsinh g
2E, n ÞÞðcos γ11 ðE,
ÞÞÞ − Ω10 ðE,
2E, n 
2 − 2d
2 −
k21 ðE,
1E, n d
n
n
n
Þ = ½2d
Þ, Ω10 ðE,
Þ = ½2d Ω9 ðE, 1E, n 2E, n
16E, n = ð½Ω7 ðE,
n
Þðsinh g
1E, n Þðcos g
2E, n Þ G
n
n
Þðsin g
1E, n Þðcosh g
2E, n ÞÞðsin γ11 ðE,
Þ=12ÞÞ − Ω8 ðE, " #

3 − 3d
2 d
1E, n 5ðd 5d 2E, n 1E, n 1E, n 3






Þ =

Þd1E, n , Ω7 ðE, n − 34k21 ðE, n
2 + d
k21 ðE,
2 k21 ðE, nÞ +
n
Þ d 1E, n 2E, n "

#
1E, n
3 − 3d
2 d
2E, n 5ðd 5 d 2E, n 2E, n 3



n


n
Þ =
Þd1E, n , − 34k21 ðE, Ω8 ðE,
2 + d
k21 ðE,
2 k21 ðE, nÞ +
n
Þ d 1E, n 2E, n  
12E, n + Δ0 ðH
13, E, n + H
14E, n Þ + Δ0 ðH
15E, n + H
16E, n Þ
18, E, n = 1 H
11, E, n + H H 2
11E, n = 2ðsinh g
1E, n Þðsin g
2E, n Þðcos γ ðE,
n
ÞÞÞ H 11
n
12E, n = ð½Ω2 ðE,
Þðsinh g
1E, n Þðcos g
2E, n Þ H
n
n
Þðsin g
2E, n Þðcosh g
1E, n Þðsin γ11 ðE,
ÞÞÞ, + Ω1 ðE,
n
13E, n = ð½Ω4 ðE,
Þðcosh g
1E, n Þðcos g
2E, n Þ H

n
Þðsinh g
1E, n Þðsin g
2E, n Þðsin γ11 ðE,
ÞÞÞ, + Ω3 ðE, n

314

8 Entropy in HDSLs under magnetic quantization


n
14E, n = ð½Ω6 ðE,
Þðsinh g
1E, n Þðcos g
2E, n Þ H

n
Þðsin g
1E, n Þðcosh g
2E, n Þðcos γ11 ðE,
ÞÞÞ + Ω5 ðE, n
n
15E, n = ð½Ω10 ðE,
Þðcosh g
1E, n Þðcos g
2E, n Þ H

n
Þðsin g
1E, n Þðsin g
2E, n Þðcos γ11 ðE,
ÞÞÞ, + Ω9 ðE, n
n
19E, n = ð½Ω8 ðE,
Þðsinh g
1E, n Þðcos g
2E, n Þ H
n
n
Þðsin g
1E, n Þðcosh g
2E, n Þðsin γ11 ðE,
Þ=2ÞÞ + Ω7 ðE, 


18E, n = sinh − 1 ðω18E, n Þ
19E, n = 2C18E, n D18E, n and D H
2 L 0 The DOS function can be written as nX max
′
′19E, n ÞHðE
− E15, 2 Þ ðG 19E, n +
iH
EÞ
= eB pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð 2 2π
h n = 0 G19E, n + iH19E, n

(8:3b)

where E15, 2 is the sub-band energy in this case and is given by 
19E, n + H
19E, n  0 = ½G

(8:3c)

E = E15, 2

EEM can be written as
η ,n
* ðE, m g
Þ =


2 ′ h G 19E, n 2

(8:4)

The electron statistics can be expressed as
0 = n

g
v eB Real part of π2
h " nX

1=2  max 
19, E, n + +
iH
19, E, n G  n=0

s X EF1322

h LðrÞ


19, E, n + +
iH
19, E, n G

r=1

1=2

jEF1322

# i

(8:5)
F1322 is the fermi energy in this case. where E Using (1.31f) and (8.7), we can study the entropy in this case.

8.2.3 Entropy in IV–VI HD SLs with graded interfaces under magnetic quantization The simplified DR in HD IV–VI SLs with graded interfaces under magnetic quantization can be expressed as

315

8.2 Theoretical background

    2

2eB 1
k2 = 1 cos − 1 1 Φ2 ðE,
n

Þ − n + z
2 2
h 2 L 0

(8:6)

where
n
n
n
n
n
n
Þ ≡ ½2 coshfβ2 ðE,
Þg cosfγ2 ðE,
Þg + ε2 ðE,
Þsinhfβ2 ðE,
Þg sinfγ22 ðE,
Þg Φ2 ðE,
112 ðE,
n
212 ðE,
n
212 ðE,
n
n
n
Þg2 =K
ÞÞ − 3K
ÞÞcoshfβ2 ðE,
Þg sinfγ22 ðE,
Þg + Δ0 ½ððfK
ðE,
n
Þg2 fK
112 ðE,
n
n
n
Þ − 112
Þg cosfγ22 ðE,
Þg + ð3K sinhfβ2 ðE,
n
112 ðE,
Þ K
112 ðE,
n
212 ðE,
n
n
n
Þg2 − fK
Þg2 coshfβ2 ðE,
Þg cosfγ22 ðE,
Þg + Δ0 ½2fK 


212 ðE,
n
Þg3 5fK
Þg3 1 5fK 112 ðE, n + +
n
n
212 ðE,
112 ðE,
Þ
Þ 12 K K
n
112 ðE,
n
n
n
212 ðE,
ÞK
Þ sinhfβ2 ðE,
Þ sinfγ22 ðE,
Þg − 34K
n
112 ðE,
n
Þ ≡ K
Þ½a
0 − Δ0 , β2 ðE,
k2 ðE,
n
−V
0 η Þ − ½½q
−V
0 η Þ2
Þ = ½2p
9, 2n  − 1 ½ − q
9, 2n ðE
9, 2n ðE g2 g2 112 1


−V
0 η Þ2 
9, 2n ðE
9, 2n R + 4p g2
 2eB 1 2 −
−V
0 η Þ½ð
h2 =2Þðð1=m
l2+ m


t2− Þ
9, 2n ðE
ðð1=m ÞÞ + a ð
h =4Þ q n + 2 g2 l2
h 2
−V
0 η Þðð1=m
l2+ Þ − ð1=m
t2− Þ
t2+ m
l2− ÞÞ − α2 γ3 ðE + ð1=m g2 

2 h 2eB 1
*t2 Þ
+ α2 ðð1=m n 2
h 2 
2  
h 2
*t2 Þ + ð1=m
t2− ÞÞÞ
t2− ÞÞ − ks0 ðð1=m − ð1=m 2
6 
2
h 2eB 1
t2+ m
t2− ÞÞ,
+ ðð1=m − α2 n 4
h 2


η Þ + ½γ ðE
−V
0 η Þ + γ ðE
−V
0η Þ
9, 2n ðE, R g2 g2 g2 2 3





0 − Δ0 , K
n
212 ðE,
n
2 ðE,
n
Þ = K
Þ½b
Þ γ2 ðE, 212 1


η Þ + ½½q
η Þ2 + 4p
η Þ2 
9, 1n ðE,
9, 1n ðE,
9, 1n ðE,
9, 1n R
9, 1n  − 1 ½ − q = ½2p g1 g1 g1
η Þ=
9, 1n ðE, q g1


6


2 h
h 2eB 1 * −

l1+ m

t1− Þ
ðð1=ml1 Þ + ð1=ml1 ÞÞ + α1 ðð1=m n+ 2 4
h 2  + −
η Þðð1=m


l1+ m
t1− Þ − α1 γ3 ðE, Þ − ð1= m Þ , + ð1=m g1 l1 t1




316

8 Entropy in HDSLs under magnetic quantization


η Þ = ½γ ðE,
η Þ + γ ðE,
η Þ
9, 1n ðE, R g1 g1 g1 2 2




 

2 h 1
hÞ n
*t1 Þ
+ α1 ð2eB=
ðð1=m 2 2


2
2 h
*t2 Þ + ð1=m
t2− ÞÞ ks0 ðð1=m 2
6 



2 
h 2eB 1
t2+ m
t2− ÞÞ
+ ðð1=m − α1 n 4
h 2


t1− ÞÞ − − ð1=m




and 




nÞ ≡ K 112 ðE, nÞ − K 212 ðE, nÞ ε2 ðE,
nÞ K
nÞ
212 ðE,
112 ðE, K The DOS function can be written as
X n max cos − 1 ½1 ’ ðE,
n
n
n
E
−E
15, 3 Þ

Þð1 − 41 ’2 2 ðE,
Þ − 1=2 ’′2 ðE,
ÞHð 2 2
EÞ
= eBg
v Nð 1=2
0 2π2
hL
n
0 2 2eB ð1 + 1Þ
=0 n
Þ2 − L ½½cos − 1 ½21 ’2 ðE, h
2

(8:7)


15, 3 is the sub-band energy in this case and is given by where E    2

1 2eB 1 −1 1


− 0 =
2 cos Φ2 ðE15, 3 , nÞ n+ 2
h 2 L0

(8:8)

EEM can be written as
η ,n
* ðE, m g
Þ =

  − 1=2
2 h 1 2
−1 1

n


Þ cos ð E, n Þð1 − ð E, n Þ ’′2 ðE, ’ ’ 2 2
0 2 2 4 2L

(8:9)

The electron concentration can be written as
0 = n
s X
r = 1


X n max
g
v eB Real Part of π2
h
=0 n



rÞ Lð

  n
X max
=0 n



1=2  i2 2eB

1 h 1  −1 1
n

ð E, Þ − cos ϕ n + 

2 EF1323 2 2
h 2 L 0



 2 1 2eB 1 1=2  −1 1

Þ −
+ ϕ ðE, n n 

2 cos EF1323 2 2
h 2 L 0

(8:10)


F1323 is the fermi energy in this case. where E Using (1.31f) and (8.10) we can study the entropy in this case.

8.2.4 Entropy in HgTe/CdTe HD SLs with graded interfaces under magnetic quantization The simplified DR in HDHgTe=CdTeSLs with graded interfaces under magnetic quantization can be expressed as

317

8.2 Theoretical background


192E, n + iH
192E, n ð
kz Þ2 = G

(8:11a)

where  
2



2 1
192E, n = C182E, n − D182E, n − 2eB
, n + G
2
h 2 L 0
1820D = cos − 1 ðω182E, n Þ, ω182E, n = ð2Þ − 21 ½ð1 − G
2
2 C 182E, n − H182E, n Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2
2
2
2 2 − ð1 − G 182E, n − H182E, n Þ + 4G182E, n 
0 − Δ0 Þ,
112E, n = 2ðcosðg
12 ÞÞðcosðg
22 ÞÞðcos γ ðE,
n
n
n
ÞÞ, γ8 ðE,
Þ = k8 ðE,
Þðb G 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 B
1E
1E
2 + 4A
2 + 4A
−B
01 B



01 2eB 1 01
k8 ðE,
n
Þ =
+ − , n
2
h 2 2A 1


120D = ð½Ω12 ðE,
n
Þðsinh g
12E, n Þðcos g
12E, n Þ G
n
n
Þðsinh g
12E, n Þðcos g
12E, n Þðsin γ8 ðE,
ÞÞÞ, − Ω22 ðE, 


k8 ðE,
k8 ðE,
12E, n 
22E, n 
n
n
Þd
Þd d12E, n d22E, n
n
n
Þ =
Þ = , Ω Ω12 ðE, − − ð E, 22
k8 ðE,
12E, n + d
k8 ðE,
12E, n + d
22E, n
22E, n ,
n
n
Þ d
Þ d
1320D = ð½Ω32 ðE,
n
Þðcosh g
12E, n Þðcos g
22E, n Þ G
n
n
Þðsinh g
12E, n Þðcos g
12E, n Þðsin γ8 ðE,
ÞÞÞ, − Ω42 ðE,
n
Þ = Ω32 ðE,



   d212E, n − d22E, n 2d12E, n d22E, n
k8 ðE,
n
n
Þ , Ω42 ðE,
Þ = − 3
n
k8 ðE,
n
Þ
Þ k8 ðE,


1420D = ð½Ω52 ðE,
n
Þðsinh g
12E, n Þðcos g
22E, n Þ G
n
n
Þðsinh g
12E, n Þðcosh g
22E, n Þðcos γ8 ðE,
ÞÞÞ, − Ω62 ðE,   d
k2 ðE,
n
n
n
Þ = 3d12E, n − 2 12E, n2
Þ , Ω62 ðE,
Þ Ω52 ðE, d12E, n + d22E, n 8   d22E, n
k2 ðE,
22E, n +
n
= 3d Þ , d212E, n + d222E, n 8
1520D = ð½Ω72 ðE,
n
Þðcosh g
12E, n Þðcos g
22E, n Þ G
n
n
Þðsinh g
12E, n Þðcosh g
22E, n Þðsin γ80D ðE,
Þ=12ÞÞ, − Ω102 ðE,
n
n
n
Þ = ½2d212E, n − 2d222E, n −
k82 ðE,
Þ, Ω102 ðE,
Þ = ½2d12E, n d22E, n  Ω92 ðE,
162E, n = ð½Ω72 ðE,
n
Þðsinh g
12E, n Þðcos g
22E, n Þ G
n
n
Þðsinh g
12E, n Þðcosh g
22E, n Þðsin γ80D ðE,
Þ=12ÞÞ, − Ω82 ðE,

318

8 Entropy in HDSLs under magnetic quantization


n
Þ = Ω72 ðE,
n
Þ = Ω82 ðE,

 

5ðd312E, n − 3d222E, n d12E, n Þ 5d12E, n
k3 ðE,
n
n

Þd12E, n − 34
k8 ðE, Þ + 8
k8 ðE,
2
n
d212E, n + d Þ 22E, n 5ðd312E, n − 3d222E, n d12E, n Þ 5d22E, n
k3 ðE,
n
n

Þd12E, n − 34
k8 ðE, Þ +
k8 ðE,
n
Þ d212E, n + d222E, n 8

 

 
122E, n + Δ0 ðH
132E, n + H
142E, n Þ + Δ0 ðH
152E, n + H
162E, n Þ ,
182E, n = 1 H
112E, n + H H 2
n
112E, n = 2ðsinh g
12E, n Þðsinh g
22E, n Þðcos γ ðE,
ÞÞÞ, H 8
n
1220D = ð½Ω22 ðE,
Þðsinh g
12E, n Þðcos g
22E, n Þ H
n
n
Þðsinh g
22E, n Þðcosh g
12E, n Þðsin γ8 ðE,
ÞÞÞ, + Ω12 ðE,
132E, n = ð½Ω42 ðE,
n
Þðcosh g
12E, n Þðcos g
22E, n Þ H
n
n
Þðsinh g
12E, n Þðsinh g
22E, n Þðsin γ8 ðE,
ÞÞÞ, + Ω32 ðE, 
132E;n = ½Ω42 ðE;
n
Þðcosh g
12E;n Þðcos g
22E;n Þ H 
n
n
Þðsinh g
12E;n Þðsinh g
22E;n Þðsin γ8 ðE;
ÞÞ , + Ω32 ðE; 
142E;n = ½Ω62 ðE;
n
Þðsinh g
12E;n Þðcos g
22E;n Þ H  
n
n
Þðsinh g
12E;n Þðcosh g
22E;n Þ cos γ8 ðE;
Þ , + Ω52 ðE;
n
142E, n = ð½Ω62 ðE,
Þðsinh g
12E, n Þðcos g
22E, n Þ H
n
n
Þðsinh g
12E, n Þðcosh g
22E, n Þðcos γ8 ðE,
ÞÞÞ, + Ω52 ðE,
1520D = ð½Ω102 ðE,
n
Þðcosh g
12E, n Þðcos g
22E, n Þ H
n
n
Þðsinh g
22E, n Þðc sinh g
12E, n Þðsin γ8 ðE,
ÞÞÞ, + Ω92 ðE,
182E, n d182E, n Þ=L
2 Þ and D
182E, n = sinh − 1 ðω182E, n Þ
192E, n = ½ðð2C H 0 The DOS function can be written as max
nX
E
−E
15, 4 Þ ðG′192E, n + iH ′192E, n ÞHð
EÞ
= eB pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð 2
192E, n + iH
192E, n 2π
h n = 0 G

(8:11b)


15, 4 is the sub-band energy in this case and is given by where E
192E, n + H
192E, n j

0 = ½G E = E15, 4

(8:11c)

EEM can be written as
2 ′ h
η ,n
* ðE, G 192E, n m g
Þ = 2

(8:12)

319

8.2 Theoretical background

The electron statistics can be expressed as g
v eB Real part of π2
h 
X n  max
192, E, n + +
iH
192, E, n Þ1=2 
ðG


0 = n

EF1324


=0 n


s X
r = 1

 h i
192, E, n + +
iH
192, E, n Þ1=2 


rÞ ðG Lð EF1324 (8:13)


F1324 is the fermi energy in this case. where E Using (1.31f) and (8.13), we can study the entropy in this case.

8.2.5 Entropy in strained layer HD SLs with graded interfaces under magnetic quantization DR of the conduction electrons in HD strained layer SL with graded interfaces can be expressed as    2

2jejB 1
k2 = 1 cos − 1 1 ’
n


ð E, Þ − n + z
2 2 6
h 2 L 0

(8:14a)

Where
ðE,
n
4 ðE,
n
5 ðE,
n
Þ = ½2 cosh½T
, ηg2 Þ cos½T
, ηg1 Þ ϕ 6
n
4 ðE,
n
5 ðE,
n
6 ðE,
Þ sinh½T
, ηg2 Þ sin½T
, ηg1 Þ + ½T 

2
 
, ηg2 Þ k0 ðE, n
k0 ′ðE,
4 ðE,
n
n
n


+ Δ0 Þ cos½ T , η Þ sin½T ð E, , η Þ , η − 3 1 g1 g2 g1 ′
k0 2 ðE,
n
, ηg1 Þ


2


k′ ðE,
n
, ηg1 Þ 0


+ 3k0 ðE, n, ηg2 Þ −

n
, ηg2 Þ k0 ðE,




4 ðE,
n
5 ðE,
n
, ηg2 Þ cos½T
, ηg1 Þ sin½T

′2
n
n
4 ðE,
n
5 ðE,
n
, ηg1 Þ −
k0D
, ηg1 ÞÞ cos½T
, ηg2 Þ cos½T
, ηg1 Þ + Δ0 ½2ð
k02 ðE, ðE,



3
′

, ηg2 Þ 5
k03 ðE,
, ηg1 Þ 5k0 ðE, n n +

k0 ′ðE,

n
, ηg2 Þ
, ηg1 Þ k0 ðE, n 
n
4 ðE,
n
n
n
, ηg2 Þk0 ′ðE,
, ηg2 Þ sinðE,
, ηg1 Þ
, ηg1 Þ sinh½T − 34
k0 ðE,

+

1 12


n
n
4 ðE,
, ηg2 Þ =
k0 ðE,
, ηg2 Þ½a
0 − Δ0 , ½T

320

8 Entropy in HDSLs under magnetic quantization

 
 1=2 1 pffiffiffiffiffi

k0 ðE,
n
n
−1
, ηg2 Þ = ½
S2 ðE,
, ηg2 Þ − 1=2 .
+
heb=ð ρ1 ðEÞρ ð EÞÞ n 2 2
2 ðE
=
h2 =ð2p
−V
0 , η ÞÞ, ρ ðEÞ
=
h2 =ð2Q
−V
0 , η ÞÞ
2 ðE ρ1 ðEÞ 2 g2 g2
0 − Δ0 ,
n
n
5 ðE,
, ηg1 Þ =
k0′ðE,
, ηg1 Þ½b T  
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2
k0′ðE,
n
ðEÞ

n
+ 1=2Þ
heB=
, ηg1 Þ − 1=2 1 − ðn
, ηg1 Þ = ½
S1 ðE, ρ3 ðEÞρ 4
1 ðE,
=
h2 =ð2p
η ÞÞ, ρ ðEÞ
=
h2 =ð2Q
η ÞÞ
1 ðE, ρ3 ðEÞ 4 g2 g1 " #
k0 ðE,
n
n
, ηg2 Þ
k0′ðE,
, ηg1 Þ
n
6 ðE,
Þ = − T

k0′ðE,
n
n
, ηg2 Þ
, ηg1 Þ k0 ðE, The DOS function can be written as nX max cos

EÞ
= eBg
v Nð
0 2π2
hL
=0 n


h

  i − 1=2  
n
n
E
−E
15, 5 Þ
Þ 2
Þ ′Hð ’6 ðE, ’6 ðE, h i 1=2  
n
0 2 2eB
ð1 + 1Þ
Þ 2 − L cos − 1 21 ’6 ðE,
h 2

−1 1 1

2 ’6 ðE, nÞð1 − 4

(8:14b)
15, 5 is the sub-band energy in this case and is given by where E     2

1 1 2jejB 1
15, 5 , n
Þ
+
6 ðE − 0 =
2 cos − 1 ’ n 2
h 2 L0

(8:14c)

EEM can be written as   − 1=2

2 h 1 1 2 − 1



n
Þ =
Þ 1 − ½’
Þ
Þ′
6 ðE,
6 ðE, n
6 ðE, n cos ½’ m ðE, ηg , n ’
0 2 2 4 2L
*

(8:15)

The electron concentration can be written as
0 = n
s X
r = 1

 
 2
X n max


1 2eB 1 1=2  g
v eB −1 1
n

cos ð E, Þ − Real Part of ϕ n + 

2 EF1325 π2
h 2 2
h 2 L 0
=0 n

rÞ Lð

  n
X max
=0 n


 1=2  2

1 2eB 1  −1 1
n

cos ð E, Þ − ϕ n + 
2
E 2 2
h 2 L F1325 0


F1325 is the Fermi energy in this case. where E Using (1.31f) and (8.16), we can study the entropy in this case.

(8:16)

8.2 Theoretical background

321

8.2.6 Entropy in III–V HD effective mass SLs under magnetic quantization DR in this case assumes the form ð
kx Þ2 = δ7E, n + iδ8E, n

(8:17a)

Where 



  1 2 2eB 1 2
, δ5E, n = cos − 1 p5E, n , δ5E, n =
2 ðδ5E, n − δ6E, n Þ − n+
h 2 L0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31=2 2 1 − δ23E, n − δ24E, n − ð1 − δ23E, n − δ24E, n Þ + 4δ24E, n 5 ,
5E, n = 4 p 2
1 cos Δ1E, n cos Δ2E, n − a
2 cos Δ3E, n cos Δ4E, n Þ δ3E, n = ða
2 sin Δ1E, n sin Δ2E, n − a
2 sin Δ3E, n sin Δ4E, n Þ δ4E, n = ða
0 e3E, n Þ, Δ2E, n = ða
0 e4E, n Þ, Δ3E, n
0 e1E, n + b
0 e2E, n + b Δ1E, n = ða
0 e3E, n Þ, Δ4E, n = ða
0 e4E, n Þ,
0 e1E, n − b
0 e2E, n − b = ða
2 ,
5E, n and δ8E, n = ½2δ5E, n δ6E, n =L δ6E, n = sinh − 1 p 0 e1E, n =

h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

i1 h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

i1
t2 +
t22 +
t1E, n =2 2 , e2E, n =
t2 +
t22 −
t1E, n =2 2 1E, n 1E, n

2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=2 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=2
t2 +
t42 +
t3E, n
t2 +
t42 −
t3E, n 3E, n 3E, n 5 , e4E, n = 4 5 , e3E, n = 4 2 2  * 




2eB 1

t1E, n 2mc1 T
, ð E, Δ , η , E Þ − n + 11 1 g1 g1
h 2
h2  * 




2eB 1

t3E, n 2mc2 T
ð E, Δ , η , E Þ − n + 12 2 g2 g2
h 2
h2 The DOS function can be written as
X n max
E
−E
15, 6 Þ ðδ′7E, n + iδ′8E, n ÞHð
EÞ
= eB pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð 2π2
h n
= 0 δ7E, n + iδ8E, n

(8:17b)


15, 6 is the sub-band energy in this case and is given by where E 0 = δ7E15, 6 , n + iδ8E15, 6 , n EEM can be written as

(8:17c)

322

8 Entropy in HDSLs under magnetic quantization


2 ′ h
η ,n

* ðE, Þ = δ 7E, n m g 2

(8:18)

The electron statistics can be expressed as 
X n max
g
v eB
0 = 2 Real part of ðδ8, E, n + + δ8, E, n Þ1=2 jE
F1326 n π
h
=0 n 
s X

rÞ½ðδ8, E, n + δ8, E, n Þ1=2 j
 Lð EF1326

(8:19)


r = 1


F1326 is the Fermi energy in this case. where E Using (1.31f) and (8.19) we can study the entropy in this case.

8.2.7 Entropy in II–VI HD effective mass SLs under magnetic quantization DR in HD II–VI EMSL can be written as ð
kz Þ2 = Δ13E, n + iΔ14E, n ,

(8:20a)

where, 



 1 2 2eB 1 2
+ Δ13E, n =
2 ðΔ11E, n − Δ12E, n Þ − n
h 2 L0 2
6E, n = 4
6E, n , p Δ11E, n = cos − 1 p

1 − Δ29E, n − Δ210E, n −

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31=2 2 ð1 − Δ29E, n − Δ210E, n Þ + 4Δ210E, n 5 2


1 cos Δ6E, n cosh Δ7E, n − a
2 cos Δ8E, n cosh Δ7E, n Þ, Δ9E, n = ða
1 sin Δ6E, n sinh Δ7E, n + a
2 sin Δ8E, n sinh Δ7E, n Þ, Δ10E, n = ða
0
e3E, n , Δ7E, n = b
0
e4E, n , Δ8E, n
22E, n ðE
E, n , η Þ + b
0 C Δ6E, n = ½a g1
0
e3E, n ,
22E, n ðE
E, n , η Þ − b
0 C = ½a g1

"
22E, n ðE
E, n, η Þ = C g1

(




2 h 2eB 1
n+
*?, 1
h 2 2m
h2 

1=2 1=2 1
0 2eB n
+ , ∓C
h 2 2m*jj, 1


E, n, η Þ − γ3 ðE g1


6E, n , Δ14E, n = Δ12E, n = cos − 1 p

2Δ11E, n Δ12E, n , L20

The DOS function can be written as

8.2 Theoretical background

max
nX
E
−E
15, 7 Þ ðΔ′13E, n + iΔ′14E, n ÞHð
EÞ
= eB pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð 2 2π
h n = 0 Δ13E, n + iΔ14E, n

323

(8:20b)


15, 7 is the sub-band energy in this case and is given by where E 0 = Δ13E15, 7, n + iΔ14E15, 7, n

(8:20c)

EEM can be written as
2 ′ h
η ,n

* ðE, Þ = Δ 13E, n m g 2

(8:20d)

The electron statistics can be expressed a
0 = n


g
v eB Real part of π2
h  
X
s n max X 1=2 1=2
ðΔ13, E, n + + Δ14, E, n Þ jE
F1327 Lð
rÞ½ðΔ13, E, n + + Δ14, E, n Þ jE
F1327 
r = 1


=0 n

(8:21)
F1327 is the Fermi energy in this case. where E Using (1.31f) and (8.21), we can study the entropy in this case.

8.2.8 Entropy in IV–VI HD effective mass SLs under magnetic quantization DR in HD IV–VI, EMSL sunder magnetic quantization can be written as 




2 2eB 1 2 2 −1





n+ ðkz Þ = ½1=L0  cos ðf 23 ðE, nÞÞ −
h 2 where,
f ðE,
0 D
23E, n ðE,
n
n
n
23E, n ðE,
Þ = a
3 cos½a
0 C
, ηg1 Þ + b
, ηg1 Þ 23




4 cos½a
0 C23E, n ðE, n
, η Þ − b0 D23E, n ðE, n, η Þ, −a g2

g2


23 ðE;
n
n
n
; ηg1 Þ = ½½2p
9;1  ½ − q
9;1 ðE;
; ηg1 Þ + ½fq
9;1 ðE;
; ηg1 Þg2 C −1


9;1 ðE;
n
9;1 ÞR
; ηg1 Þ1=2 1=2 ; + ð4p
n
23 ðE;
; ηg2 Þ = ½½2p
9;2 ðE; n; ηg2 Þ
9;2  − 1 ½ − q D
9;2 ðE; n; η Þ1=2 1=2 ;
9;2 ðE; n; ηg2 Þg2 + ð4p
9;2 ÞR + ½fq g2

(8:22a)

324

8 Entropy in HDSLs under magnetic quantization




  
 2

h 1 1
h4 2eB 1 1 1
n
, ηgi Þ =
9, i ðE,
+ α − + q n + i
l,+i m
l,−i
l,+i m
t,+i m
t,−i m
l,−i 2 m 4
h 2 m 

η Þ 1 − 1 , − αi γ3 ðE, gi
l,+i m
l,−i m


 

h2 2eB 1 1 1




, ηgi Þ = γ2 ðE, ηgi Þ + γ3 ðE, ηgi Þ + αi
+ R9, i ðE, n − + n
t,+i m
t, i 2
h 2 m
h2 − 2




2  




+ 21ÞÞ 2eB 1 1 1 α
h6 ð2e
hB ðn
+ − + − n
t,−i m
t,+i
t,+i m
t, i 4 m
h 2 m

The DOS function can be written as − 1=2

1
2

max cos − 1 ½1

f ′ ðE,


n
E
−E
15, 8 Þ
g
v n
X
ÞHð eB 23 2 f 23 ðE, nÞð1 − 4 f 23 ðE, nÞ

NðEÞ = 2
1=2 2 2π
hL0 n
= 0
n
0 2 2eB
ð1 + 1Þ
Þ − L ½½cos − 1 ½21
f 23 ðE, h
2

(8:22b)
15, 8 is the sub-band energy in this case and is given by where E 




2 2eB 1
15, 8 , n
0 2  cos − 1 ð
f 23 ðE
ÞÞ −
+ 0 = ½½1=L n
h 2

(8:22c)

EEM can be written as 
 − 1=2
2 h 1
2
−1 1



η ,n
n


Þ
* ðE, f f Þ = cos ð E, n Þ 1 − ð E, n Þ f ′23 ðE, m 23 23 g
0 2 2 4 2L

(8:23)

The electron concentration can be written as
0 = n

nX max
eB ½Θ8, 15 + Θ8, 16  Real Part of 2 π
h
=0 n


where 



 1 2eB 1 1=2
F8, 8 , n
Þ 1=2 −
+ Þ , Θ8, 15 =
2 ½cos − 1
f 23 ðE ðn
h 2 L0 Θ8, 16 =


s X



rÞ½Θ8, 15  Lð


r = 1


F8, 8 is the Fermi energy in this case. and E Using (1.31f) and (8.24), we can study the entropy in this case.

(8:24)

8.2 Theoretical background

325

8.2.9 Entropy in HgTe/CdTe HD effective mass SLs under magnetic quantization DR in HDHgTe/CdTe EMSLs under magnetic quantization can be written as ð
kz Þ2 = Δ13HE, n + iΔ14HE, n

(8:25a)

where   

1 2eB 1
+ Δ13HE, n =
2 ðΔ211HE, n − Δ212HE, n Þ − n
h 2 L0 Δ11HE, n = cos − 1 p6HE, n , p6HE, n 2 =4

1 − Δ29HE, n − Δ210HE, n −

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31=2 2 ð1 − Δ29HE, n − Δ210HE, n Þ + 4Δ210HE, n 5 , 2


1H cos Δ5HE, n cosh Δ6HE, n − a
2H cos Δ7HE, n cosh Δ6HE, n Þ, Δ9HE, n = ða
1H sin Δ5HE, n sinh Δ6HE, n + a
2H sin Δ7HE, n sinh Δ6HE, n Þ, Δ10HE, n = ða
0
e3 , Δ6HE, n = b
0
e4 , Δ7HE, n
22HE, n ðE
E, n , η Þ + b
0 C Δ5HE, n = ½a g1
0
e3 ,
22HE, n ðE
E, n , η Þ − b
0 C = ½a g1 
2 

1=2


2
1
22HE, n ðE
E, n, η Þ = B01 + 2A1 EE, n − B01 ðB01 + 4A1 EE, n Þ − 2eB n
, C + g1
2
h 2 2A 1
6HE, n , Δ14HE, n = Δ12HE, n = cos − 1 p

2Δ11HE, n Δ12HE, n ,
2 L 0

The DOS function can be written as max
n
X
E
−E
15, 9 Þ ðΔ′13HE, n + iΔ′14HE, n ÞHð
EÞ
= eB ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p Nð 2π2
h n
= 0 Δ13HE, n + iΔ14HE, n

(8:25b)


15, 9 is the sub-band energy in this case and is given by where E 0 = Δ13HE15, 9 , n + iΔ14HE15, 9 , n

(8:26a)

The EEM can be written as
2 ′ h
η ,n
* ðE, Δ 13HE, n m g
Þ = 2

(8:26b)

326

8 Entropy in HDSLs under magnetic quantization

The electron concentration can be written as
0 = n


X n max
eB ½Θ8, 17 + Θ8, 18  Real part of π2
h
=0 n

(8:27)

where Θ8, 17 = ½Δ13HEF8, 9 , n + iΔ14HEF8, 9 , n 1=2 , Θ8, 18 =


s X



rÞ½Θ8, 17  Lð


r = 1


F8, 9 is the Fermi energy in this case. and E Using (1.31f) and (8.27) we can study the entropy in this case.

8.2.10 Entropy in strained layer HD effective mass SLs under magnetic quantization DR in HD strained layer effective mass SLs under magnetic quantizationcan be expressed as 


 2 1 2eB 1
n
ÞÞ −
+ (8:28a) n ð
kz Þ2 =
2 cos − 1 ð
f 40 ðE,
h 2 L0 where
f ðE,
0 D
40 ðE,
n
n
n
40 ðE,
Þ = a
20 cos½a
20 C
, ηg1 Þ + b
, ηg1 Þ 40
0 D
40 ðE, n, η Þ − b
n
40 ðE,
0 C
, ηg2 Þ, − a21 cos½a g2 "


, ηg1 Þ = 1 − C40 ðE, n
η Þ= ϕ50 ðE, g1


#1=2

eB h 1
η Þ = 1=2 ,
½
S1 ðE, g1
η Þ n+ 2 ϕ50 ðE, g1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
η Þψ η , ψ50 ðE, g1 51 g1


h2
h2
η Þ=
η Þ= ð E, , ψ ψ50 ðE, g1 51 g1
1 ðE,
1 ðE,
η Þ
η Þ 2P 2Q g1 g1 "


#1=2

h e B 1
n
η Þ − 1=2 ,
40 ðE,
, ηg2 Þ = 1 −
½
S2 ðE, D g2
η Þ n+ 2 ϕ50 ðE, g2
η Þ= ϕ501 ðE, g2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
η Þψ η ψ501 ðE, g2 511 g2

327

8.3 Results and discussion


h2
h2
η Þ=
η Þ= ψ501 ðE, ð E, , ψ g2 g2
2 ðE,
1 ðE,
η Þ 511
η Þ 2P 2Q g2 g2 The DOS function can be written as

EÞ
= eBg
v Real Part of Nð
0 2π2
hL
X n max

2
n
Þð1 − 41
f 40 ðE, nÞ cos − 1 ½21
f 40 ðE,

− 1=2


n
0
Þ − L ½½cos − 1 ½21
f 40 ðE, 2


=0 n


f ′ ðE,
n
E
−E
15, 10 Þ
ÞHð 40

(8:28b)

1=2 2 2eB
1 h ð1 + 2Þ



15, 10 is the sub-band energy in this case and is given by where E 


 2 2eB 1 2 −1



ÞÞ −
+ 0 = ½1=L0  cos ðf 40 ðE15, 10 , n n
h 2

(8:28c)

EEM can be written as 
 − 1=2
2 h 1

1
2
− 1
f ′ ðE,

n
Þ =
Þ 1 − f40 ðE, n
Þ
Þ f40 ðE, n cos m ðE, ηg , n 40 2
2 4 2L0
*

(8:29)

The electron concentration can be written as
0 = n


X n max
eB ½Θ8, 19 + Θ8, 20  Real Part of π2
h
=0 n

(8:30)

where 2 Θ8, 19 =

41

2


L 0


F8, 10 , nÞg ½cos f
f 40 ðE −1

1=2

3
 1=2
s X 2eB 1 5

rÞ½Θ8, 19 
+ − , Θ8, 20 = Lð n
h 2
r = 1


F8, 10 is the Fermi energy in this case. and E Using (1.31f) and (8.30) we can study the entropy in this case.

8.3 Results and discussion Using Appendix 15, we have plotted in Figures 8.1 and 8.2 the entropy as functions of inverse quantizing magnetic field and impurity concentration, respectively, for HgTe/ CdTe, PbTe/PbSnTe, CdS/CdTe, and GaAs/Ga1–xAlxAs HD SLs with graded interfaces. With decreasing magnetic field intensity, the thermoelectric power increases periodically as a result of SdH periodicity. However, with increasing impurity concentration, the thermoelectric power increases to some extent exhibiting spikes for higher values,

328

8 Entropy in HDSLs under magnetic quantization

Normalized entropy

1,000

(a) 100

(e)

(d)

(b)

(c)

10 0.2

0.7

1.2

1.7

2.2

2.7

3.2

Inverse magnetic field (tesla–1) Figure 8.1: The plot of the entropy as a function of inverse quantizing magnetic field for (a) HgTe/CdTe, (b) PbTe/PbSnTe, (c) CdS/CdTe, and (d) GaAs/Ga1–xAlxAs HDSLs with graded interfaces.

Normalized entropy

10,000 1,000 (c) 100 (b) (d)

(a) 10 1

0.001

0.01

0.1

1

Impurity concentration (1022 m–3) Figure 8.2: Plot of the entropy as a function of impurity concentration for all the cases of Figure 8.1.

a result which already been discussed in previous chapter. It appears that the entropy is lower in magnitude for HD GaAs/Ga1–xAlxAs and higher in magnitude for HD HgTe/ CdTe for all the cases. In Figures 8.3 and 8.4, the entropy as functions of inverse quantizing magnetic field and impurity concentration for HgTe/CdTe, PbTe/PbSnTe, CdS/CdTe, and GaAs/Ga1– xAlxAs effective mass HDSLs structures. The concentration has been fixed at a value 1022m–3 for varying magnetic field intensity, while 10 tesla

8.3 Results and discussion

329

300

Normalized entropy

250

(a)

200 150

(c) (d)

100

(b)

50 0 0

0.5

1

1.5

2

2.5

3

Inverse magnetic field (1/B) tesla–1 Figure 8.3: Plot of the entropy as a function of inverse quantizing magnetic field for (a) HgTe/CdTe, (b) PbTe/PbSnTe, (c) CdS/CdTe, and (d) GaAs/Ga1–xAlxAs effective mass HD SLs.

Normalized entropy

10,000 1,000 (a)

100 (b)

(d)

(c) 10 1

0.01

0.1

1

Impurity concentration (1022 m–3) Figure 8.4: Plot of the entropy as a function of impurity concentration for all the cases of Figure 8.3.

was fixed for varying impurity concentration. With decreasing magnetic field intensity, the thermoelectric power increases periodically as a result of SdH periodicity. However, with increasing impurity concentration, the entropy decreases. In Figures 8.5 and 8.6, the entropy as functions of film thickness and 2D carrier concentration for HgTe/CdTe, PbTe/PbSnTe, CdS/CdTe, and GaAs/Ga1–xAlxAs for HD QWSLs with graded interfaces. It appears that the entropy in this case signatures an increasing step like variation with increasing film thickness and decreases with increasing 2D carrier concentration. In Figures 8.7 and 8.8, the magneto thermoelectric power as function of film thickness and 2D carrier concentration for HgTe/CdTe, PbTe/PbSnTe, CdS/CdTe, and GaAs/Ga1–xAlxAs for HD QW effective mass SLs.

330

8 Entropy in HDSLs under magnetic quantization

0.010

Normalized entropy

0.009 0.008 (b)

0.007

(d)

(a)

0.006

(c)

0.005 0.004 0.003 0.002 20

30

40

50

60

70

80

90

100

Film thickness (in nm) Figure 8.5: Plot of the normalized entropy as a function of film thickness for (a) HgTe/CdTe, (b) PbTe/PbSnTe, (c) CdS/CdTe, and (d) GaAs/Ga1–xAlxAsquantum well HD SLs with graded interfaces.

(a)

0.1

(b) (c) (d) 0.01

0.1

0.001

Normalized entropy

1

1

10

Impurity concentration (1015 m–2) Figure 8.6: Plot of the normalized entropy as a function of impurity concentration for all the cases of Figure 8.6.

Finally, it may be remarked from the Figures 8.7 and 8.8 and Figures 8.5 and 8.6 that the nature of variations of the entropy for all types of HD QW effective mass SLs does not differ widely as compared with the corresponding HD QWSLs with graded interfaces.

8.4 Open research problems

331

0.035

Normalized entropy

0.030 0.025

(a) (b)

0.020

(c)

0.015 (d)

0.010 0.005 0 20

30

40

50

60

70

80

90

100

Film thickness (in nm) Figure 8.7: Plot of the normalized entropy as a function of film thickness for (a) HgTe/CdTe, (b) PbTe/PbSnTe, (c) CdS/CdTe, and (d) GaAs/Ga1–xAlxAsquantum well effective mass HD SLs.

0.12 0.10 (a) Normalized entropy

0.08 0.06 0.04 0.02

(b) (c) (d)

0 0.1

1

10

100

Impurity concentration (1015 m–2) Figure 8.8: Plot of the normalized entropy as a function of impurity concentration for all the cases of Figure 8.7.

8.4 Open research problems (R8.1) Investigate the entropy in the absence of magnetic field by considering all types of scattering mechanisms for HD III–V, II–VI, IV–VI, and HgTe/CdTe quantum well and quantum wire SLs with graded interfaces and also the effective mass SLs of the aforementioned materials.

332

8 Entropy in HDSLs under magnetic quantization

(R8.2) Investigate the entropy in the absence of magnetic field by considering all types of scattering mechanisms for HD strained layer, random, short period and Fibonacci, polytype and saw-tooth quantum well and quantum wire SLs. (R8.3) Investigate the entropy in the presence of an arbitrarily oriented quantizing magnetic field in the presence of spin and broadening by considering all types of scattering mechanisms for (R8.1) and (R8.2) under an arbitrarily oriented (a) nonuniform electric field and (b) alternating electric field respectively. (R8.4) Investigate the entropy by considering all types of scattering mechanisms for (R8.1) and (R8.2) under an arbitrarily oriented alternating magnetic field by including broadening and the electron spin, respectively. (R8.5) Investigate the entropy by considering all types of scattering mechanisms for (R8.1) and (R8.2) under an arbitrarily oriented quantizing alternating magnetic field and crossed alternating electric field by including broadening and the electron spin, respectively. (R8.6) Investigate the entropy by considering all types of scattering mechanisms for (R8.1) and (R8.2) under an arbitrarily oriented alternating quantizing magnetic field and crossed alternating non-uniform electric field by including broadening and the electron spin respectively. (R8.7) Investigate the entropy in the absence of magnetic field for all types of quantum well and quantum wire SLs as considered in this chapter under exponential, Kane, Halperin, Lax, and Bonch-Bruevich band tails [2], respectively. (R8.8) Investigate the entropy in the presence of quantizing magnetic field including spin and broadening for the problem as defined in (R8.7) under an arbitrarily oriented (a) nonuniform electric field and (b) alternating electric field, respectively. (R8.9) Investigate the entropy for the problem as defined in (R8.7) under an arbitrarily oriented alternating quantizing magnetic field by including broadening and the electron spin, respectively. (R8.10) Investigate the entropy for the problem as defined in (R8.7) under an arbitrarily oriented alternating quantizing magnetic field and crossed alternating electric field by including broadening and the electron spin, respectively. (R8.11) Investigate all the appropriate problems as defined in (R8.1) to (R8.10) for all types of quantum dot SLs. (R8.12) Investigate all the appropriate problems as defined in (R8.1) to (R8.10) for all types of quantum dot SLs in the presence of strain. (R8.13) Introducing new theoretical formalisms, investigate all the problems of this chapter in the presence of hot electron effects. (R8.14) Investigate the influence of deep traps and surface states separately for all the appropriate problems of this chapter after proper modifications.

References

333

References [1]

Paitya N., Bhattacharya S., De D., Ghatak K.P., Adv. Sci. Engg. Medi 4, 96 (2012); Bhattacharya S., De D., Adhikari S.M., Ghatak K.P. Superlatt. Microst. 51, 203 (2012). [2] De D., Bhattacharya S., Adhikari S.M., Kumar A., Bose P.K., Ghatak K.P., Beilstein Jour. Nanotech 2, 339 (2012); De D., Kumar A., Adhikari S.M., Pahari S., Islam N., Banerjee P., Biswas S.K., Bhattacharya S., Ghatak K.P., Superlatt. and Microstruct., 47, 377 (2010). [3] Pahari S., Bhattacharya S., Roy S., Saha A., De D., Ghatak K.P., Superlatt. and Microstruct 46, 760 (2009); Pahari S., Bhattacharya S., Ghatak K.P., Jour. of Comput. and Theo. Nanosci., 6, 2088 (2009). [4] Biswas S.K., Ghatak A.R., Neogi A., Sharma A., Bhattacharya S., Ghatak K.P., Phys. E: Low-dimen. Sys. and Nanostruct 36, 163 (2007); Singh L.J., Choudhury S., Baruah D., Biswas S.K., Pahari S., Ghatak K.P., Phys. B: Conden. Matter, 368, 188 (2005). [5] Chowdhary S., Singh L.J., Ghatak K.P., Phys. B: Conden. Matter 365, 5 (2005); Singh L.J., Choudhary S., Mallik A., Ghatak K.P., Jour. of Comput. and Theo. Nanosci. 2, 287 (2005). [6] Ghatak K.P., Mukhopadhyay J., Banerjee J.P., SPIE Proceedings Series 4746, 1292 (2002); Ghatak K.P., Dutta S., Basu D.K., Nag B., Il Nuovo Cimento. D, 20, 227 (1998). [7] Ghatak K.P., Basu D.K., Nag B., Jour. of Phys. and Chem. of Solids 58, 133 (1997); Ghatak K.P., De B., Mat. Resc. Soc. Proc. 300, 513 (1993); Ghatak K.P., Mitra B., Il Nuovo Cimento. D 15, 97 (1993). [8] Ghatak K.P., Inter. Soci. Opt. and Photon. Proc. Soc. Photo Opt. Instru. Engg 1626, 115 (1992); Ghatak K.P., Ghoshal A., Phys. Stat. Sol. (b), 170, K27 (1992); Ghatak K.P., Bhattacharya S., Biswas S.N., Proc. Soc. Photo Opt. Instru. Engg. 836, 72 (1988). [9] Ghatak K.P., Ghoshal A., Biswas S.N., Mondal M., Proc. Soc. Photo Opt. Instru. Engg 1308, 356 (1990); Ghatak K.P., Mazumder G., Proc., Matt. Res. Soc., 484 (1998); Ghatak K.P., De B., Defect Engg. Semi. Growth, Processing and Device Tech. Symp., Mat. Res. Soc., 262, 911 (1992). [10] Biswas S.N., Ghatak K.P., Internat. Jour. Electronics Theo. Exp 70, 125 (1991); Mitra B, Ghatak K.P., Phys. Lett. A., 146, 357 (1990); Mitra B., Ghatak K.P., Phys. Lett. A., 142, 401 (1989); Ghatak K.P., Mitra B., Ghoshal A., Phy. Stat. Sol. (b), 154, K121 (1989); Mitra B., Ghatak K.P., Phys. Stat. Sol. (b)., 149, K117 (1988); Ghatak K. P. Biswas S.N., Proc. Soc. Photo Optical Instru. Engg. 792, 239 (1987); Bhattacharyya S., Ghatak K.P., Biswas S., OE/Fibers’ 87,Inter. Soc. Opt. Photon. 836, 73 (1988); Mondal M., Ghatak, Czech. Jour. Phys. B., 36, 1389 (1986); Ghatak K.P., Chakravarti A.N., Phys. Stat. Sol. (b)., 117, 707 (1983).

9 Conclusion and scope for future research The greatest pleasure in life is doing what people say we cannot do.

This monograph deals with the entropy in various types of HD materials and their quantized counter parts. The quantization and strong electric field alter profoundly the basic band structures, which, in turn, generate pinpointed knowledge regarding entropy in various HDS and their nanostructures. The in-depth experimental investigations covering the whole spectrum of nano materials and allied science in general, are extremely important to uncover the underlying physics and the related mathematics in this particular aspect. We have formulated the simplified expressions of entropy for few HD quantized structures together with the fact that our investigations are based on the simplified ~ k.~ p formalism of solid-state science without incorporating the advanced-field theoretic techniques. In spite of such constraints, the role of band structure, which generates, in turn, new concepts are truly amazing and discussed throughout the text. We present the last bouquet of open research problem in this pin-pointed topic of research of modern physics. (R9.1) Investigate the entropy in the presence of a quantizing magnetic field under exponential, Kane, Halperin, Lax and Bonch-Bruevich band tails [1] for all the problems of this monograph of all the HD materials whose unperturbed carrier energy spectra are defined in Chapter 1 by including spin and broadening effects. (R9.2) Investigate all the appropriate problems after proper modifications introducing new theoretical formalisms for the problems as defined in (R9.1) for HD negative refractive index, macro molecular, nitride and organic materials. (R9.3) Investigate all the appropriate problems of this monograph for all types of HD quantum confined p-InSb, p-CuCl and materials having diamond structure valence bands whose dispersion relations of the carriers in bulk materials are given by Cunningham [2], Yekimov et. al. [3] and Roman et. al. [4], respectively. (R9.4) Investigate the influence of defect traps and surface states separately on the entropy of the HD materials for all the appropriate problems of all the chapters after proper modifications. (R9.5) Investigate the entropy of the HD materials under the condition of nonequilibrium of the carrier states for all the appropriate problems of this monograph. (R9.6) Investigate the entropy for all the appropriate problems of this monograph for the corresponding HD p-type materials and their nanostructures. (R9.7) Investigate the entropy for all the appropriate problems of this monograph for all types of HD materials and their nanostructures under mixed conduction in the presence of strain. https://doi.org/10.1515/9783110661194-009

336

9 Conclusion and scope for future research

(R9.8) Investigate the entropy for all the appropriate problems of this monograph for all types of HD materials and their nanostructures in the presence of hot electron effects. (R9.9) Investigate the entropy for all the appropriate problems of this monograph for all types of HD materials and their nanostructures for nonlinear charge transport. (R9.10) Investigate the entropy for all the appropriate problems of this monograph for all types of HD materials and their nanostructures in the presence of strain in an arbitrary direction. (R9.11) Investigate all the appropriate problems of this monograph for strongly correlated electronic HD systems in the presence of strain. (R9.12) Investigate all the appropriate problems of this chapter in the presence of arbitrarily oriented photon field and strain. (R9.13) Investigate all the appropriate problems of this monograph for all types of HD nanotubes in the presence of strain. (R9.14) Investigate all the appropriate problems of this monograph for HD Bi2Te3Sb2Te3 superlattices in the presence of strain. (R9.15) Investigate the influence of the localization of carriers on the entropy in HDS for all the appropriate problems of this monograph in the presence of crossed fields. (R9.16) Investigate entropy for HD p-type SiGe under different appropriate physical conditions as discussed in this monograph in the presence of strain and crossed fields. (R9.17) Investigate entropy for HD GaN under different appropriate physical conditions as discussed in this monograph in the presence of strain and crossed fields. (R9.18) Investigate entropy for different disordered HD conductors under different appropriate physical conditions as discussed in this monograph in the presence of strain and crossed fields. (R9.19) Investigate all the appropriate problems of this monograph for HD Bi2Te3-xSex and Bi2-xSbxTe3, Respectively, in the presence of strain and crossed fields. (R9.20) Investigate all the appropriate problems of this monograph in the presence of crossed electric and alternating quantizing magnetic fields. (R9.21) Investigate all the appropriate problems of this monograph in the presence of crossed alternating electric and quantizing magnetic fields. (R9.22) Investigate all the appropriate problems of this monograph in the presence of crossed alternating non uniform electric and alternating quantizing magnetic fields. (R9.23) Investigate all the appropriate problems of this monograph in the presence of alternating crossed electric and alternating quantizing magnetic fields.

9 Conclusion and scope for future research

337

(R9.24) Investigate all the appropriate problems of this monograph in the presence of arbitrarily oriented pulsed electric and quantizing magnetic fields. (R9.25) Investigate all the appropriate problems of this monograph in the presence of arbitrarily oriented alternating electric and quantizing magnetic fields. (R9.26) Investigate all the appropriate problems of this monograph in the presence of crossed in homogeneous electric and alternating quantizing magnetic fields. (R9.27) Investigate all the appropriate problems of this monograph in the presence of arbitrarily oriented electric and alternating quantizing magnetic fields under strain. (R9.28) Investigate all the appropriate problems of this monograph in the presence of arbitrarily oriented electric and alternating quantizing magnetic fields under light waves. (R9.29) (a) Investigate the entropy for all types of HD materials of this monograph in the presence of many body effects, strain and arbitrarily oriented alternating light waves, respectively. (b) Investigate all the appropriate problems of this chapter for the Dirac electron. (c) Investigate all the problems of this monograph by removing all the physical and mathematical approximations and establishing the respective appropriate uniqueness conditions. The formulation of entropy for all types of HD materials and their quantumconfined counterparts considering the influence of all the bands created due to all types of quantizations after removing all the assumptions and establishing the respective appropriate uniqueness conditions is, in general, an extremely difficult problem. Around 200 open research problems have been presented in this monograph and we hope that the readers will not only solve them but also generate new concepts, both theoretical and experimental. Incidentally, we can easily infer how little is presented and how much more is yet to be investigated in this exciting topic which is the signature of coexistence of new physics, advanced mathematics combined with the inner fire for performing creative researches in this context from the young scientists since like Kikoin [5] we firmly believe that “A young scientist is no good if his teacher learns nothing from him and gives his teacher nothing to be proud of.” In the mean time, our research interest has been shifted and we are leaving this particular beautiful topic with the hope that (R9.29) alone is sufficient to draw the attention of the researchers from diverse fields and our readers are surely in tune with the fact that “Exposition, criticism, appreciation is the work for second-rate minds” [6].

338

9 Conclusion and scope for future research

References [1] [2] [3] [4] [5] [6]

Nag B.R., Electron Transport in Compound Materials, Springer Series in Solid State Sciences, Vol. 11 (Springer-Verlag, Germany, 1980). Cunningham R.W., Phys. Rev 167, 761 (1968). Yekimov A.I., Onushchenko A.A., Plyukhin A.G., Efros A.L., J. Expt. Theor. Phys 88, 1490 (1985). Roman B.J., Ewald A.W., Phys. Rev. B 5, 3914 (1972). Kikoin I.K., Science for Everyone: Encounters with Physicists and Physics, 154 (Mir Publishers, Russia, 1989). Hardy G.H., A mathematician’s Apology, 61 (Cambridge University Press, 1990).

10 Appendix A: The entropy under intense electric field in HD Kane type materials My life equation is Mission = Vision = Passion = Creation

10.1 Introduction With the advent of modern nano devices, there has been considerable interest in studying the electric field-induced processes in materials having different band structures. It appears from the literature that the studies have been made on the assumption that the carrier dispersion laws are invariant quantities in the presence of intense electric field, which is not fundamentally true. In this chapter, we shall study the entropy in quantum-confined optoelectronic materials under strong electric field. In Section 10.2.1, an attempt is made to investigate the entropy in the presence of intense electric field in HD III–V, ternary and quaternary materials Section 10.2.2, contains the investigation of the entropy under magnetic quantization in HD Kane-type materials in the presence of intense electric field. In Section 10.2.3, entropy in QWs of HD Kane-type materials in the presence of intense electric field has been studied. In Section 10.2.4, we investigatethe entropy in NWs of HD Kane-type materials in the presence of intense electric field. In Section 10.2.5, the magneto entropy in QWs in HD Kane-type materials in the presence of intense electric field has been studied. In Section 10.2.6, the entropy in accumulation and inversion layers of Kane-type materials in the presence of intense electric field has been studied. In Section 10.2.7,the entropy in doping superlattices of HD Kane-type materials in the presence of intense electric field has been studied. In Section 10.2.8, the entropy in QWHD effective mass superlattices of Kane-type materials in the presence of intense electric field has been investigated. In Section 10.2.9, the entropy in NWHD effective mass superlattices of Kane type materials in the presence of intense electric field has been studied. In Section 10.2.10, the magneto entropy in QWHD effective mass superlattices of Kanetype materials in the presence of intense electric field has been studied. In Section 10.2.11, the entropy in QWHD superlattices of Kane-type materials with graded interfaces in the presence of intense electric field has been studied. In Section 10.2.12, the entropy in NWHD superlattices of Kane-type materials with graded interfaces in the presence of intense electric field has been investigated. In Section 10.2.13, the entropy in Quantum dot HD superlattices of Kane-Type materials with graded interfaces in the presence of intense electric field is studied. In Section 10.2.14, the magneto entropy in HD superlattices of Kane-type materials with graded interfaces in the presence of intense electric field has been investigated. In Section 10.2.15, the magneto

https://doi.org/10.1515/9783110661194-010

340

10 Appendix A: The entropy under intense electric field in HD Kane type materials

entropy in QWHD superlattices of Kane-type materials with graded interfaces in the presence of intense electric field has been investigated. Section 10.3 presents Six open research problems that challenge the first-order creativity of the readers from diverse fields.

10.2 Theoretical background 10.2.1 Entropy in the presence of intense electric field in HD III–V, ternary, and quaternary materials
12 Þ in this case can be The expression of the inter-band transition matrix element ðX written as ð ∂
3
r
12 = i u

*k1 ð
rÞ.

k2 ð
rÞd u (10:1) X ∂
kx
1 ð
k,
rÞ and u

k2 ð
rÞ ≡ u
2 ð
k,
rÞ in which u
1 ð
k,
rÞ and u
2 ð
k,
rÞare given by

k1 ð
rÞ ≡ u where u
s along x-axis, the interIn the case of the presence of an external electric field, F
12 , has finite interaction band same band, e.g., band transition matrix-element, X    


Y
n


S
Sn ~=a X ~=a Y ~=a Z ~=1 a Xn Zn  
Y
n


X
n X ~=a Y ~=a Zj ~=0 a Zn  
n

Sn 
SX ~=a X ~ = 0; a    
n

Sn
n


SY ~=a Y ~ = 0 and a 
SZ ~=a Z ~ = 0: a Sn Using the appropriate equations, we can write ) ! # " ( ð
0
0 − iY ∂ X 3 0 0 0 0





k + ½ðiSÞ #  + bk + pffiffiffi " +
ck + ½Z #  X 12 = i d
r a ∂
kx 2 ) ! # ( "
0
0 −
iY X 0 0 0 0



k − ½ðiSÞ "  − bk − pffiffiffi # +
ck − ½Z "  a 2 ! ! "( ð ih i h ∂ 3 0 0



k +
k − : ðiSÞ # ðiSÞ " = i d
r a a ∂
kx !

 h i
k ∂ b +
0 − iY
0 "0 : ði
SÞ "0
k − X + pffiffiffi
a 2 ∂k x )# ! i h i h ∂ 0 0 0


k − Z # : ðiSÞ " +
ck +
a ∂kx

10.2 Theoretical background

−

− ð


k b +

∂ ∂k
x


k ! b −

2
ck +

∂ ∂
kx

pffiffiffi 2 "(


3
r =i d +
ck +


k ! b −

341


0 Þ "0  :½ðX
0 − iY
0 Þ #0 
0 − iY ½ðX


0 − iY
0 Þ #0 g
0 #0 Þ :½ðX ½ðZ

! ∂ 
0 0
k +
ck − :½ði
SÞ #0  :½Z " − −a
∂k x )# ! ∂ 
0 0 0 0

ck − ½Z #  :½Z "  ∂
kx

!
k ∂ b +
0 Þ "0  :½Z
0 "0 
0 − iY
ck − ½ðX pffiffiffi
2 ∂k x

(10:2) Therefore, ! " ! # "("
k ∂
12       b ∂ X +





′ ′ ′ ′ ′ ′
k +
k − . iSjiS # j "  +
k − ðX − iY ÞjiS " j # pffiffiffi = a a a i ∂
kx 2 ∂
kx " ! #)#    ∂

′ ′ ′
k − Z jiS # j " +
ck +
a ∂kx "(" ! #   
k + ∂
a


′ ′ ′ ′ pffiffiffi
bk − . iSjðX − iY Þ # j # − 2 ∂kx " ! #
k ∂    b +




′ ′ ′ ′ ′ ′ bk + ðX − iY ÞjðX + iY Þ " j # 2 ∂
kx + " ! #)#   
ck + ∂


′ ′ ′ ′ ′
k − Z jðX − iY Þ # j " pffiffiffi + a 2 ∂
kx (" ! # " ! #
k ∂       b ∂ +




′ ′ ′ ′ ′ ′ ′ ′
k +
ck − . iSjZ # j "
ck − . ðX − iY ÞjZ " j " pffiffiffi + a + ∂
kx 2 ∂
kx " ! #)    ∂

′ ′ ′ ′ +
ck +

ck − . Z jZ # j " ∂kx (10:3) Therefore, we can write, 

  ∂ ∂
12 = i − a
k +
k − h# ′j " ′i +
ck +
ck − h# ′j " ′i X a ∂
kx ∂
kx

(10:4)

342

10 Appendix A: The entropy under intense electric field in HD Kane type materials

We can prove that 1 h#′j "′i = ð^r1 + i^r2 Þ 2

(10:5)

Therefore, by (10.4) and (10.5), we get 

   ∂ ∂
12 = i − a
k +
k − +
ck +
ck − #′j "′ X a ∂k
x ∂k
x 

 ∂ ∂ 1
k +
k − +
ck +
ck − = −i − a . ð^r1 + i^r2 Þ a

2 ∂kx ∂k x =

(10:6)

− iAð
kÞ ð^r1 + i^r2 Þ 2

where

 ∂ ∂


k +
k − −
ck +
ck − AðkÞ = a a ∂
kx ∂
kx

(10:7)

From (10.6), we find,
12 j2 = jX

1
2
1
2
ðkÞ½since, j^r1 j = j^r2 j = 1 A ðkÞð1 + 1Þ = A 4 2

(10:8)

considering spin-up and spin-down, we have to multiply by 2
2 ð
kÞ
12 j2 = 2 × 1 A
2 ð
kÞ = A jX 2

(10:9)


22 in the following way:
11 and X We can evaluate X ð ∂
3
r
11 = i u
*
k1 ð
rÞ.

k1 ð
rÞ.d u X ∂k
x 


 ð ∂ ∂
3
r a
k
k ∂ b

k +
k − + b
+ c a c =i d k+
k− +
− ∂
kx ∂kx ∂kx     ð ð 1
3 ∂
3
r ∂ ð1Þ = 0, since a
2 +
c2 Þ = 1
i d
2 +
c2 = 1.
2k + b
2k + + b
r = i d ða k k k+ k+
2 2 ∂kx ∂
kx
11 = 0, and similarly we can prove X
22 = 0. Thus, we conclude that intraTherefore, X
cc ) is zero. From the band momentum matrix element due to external electric field (X
k ± we can write expression of a "
2

a

k+

=
r0

2


g − δ′Þ
g − γ2 ðE E k+ 0
g + δ′ E 0

#2

" and


2k − a

=
r0

2

#
g − δ′Þ 2
g − γ2 ðE E k− 0 0 .
g + δ′ E 0

343

10.2 Theoretical background


 2

− δ′
g − δ′ ∂γ
k − E
r0 2 E 1 0 k − and ∂a
k − =
r0 2 −
g0 ′ = − a


k −
g + δ′ a 2 ∂k x ∂kx Eg0 + δ E 0

− δ′
2 E ∂γ2
k + ∂
a
k − = −
r02
g0 ′ aa
k +
k − . Combining the above, we can write a ∂kx Eg0 + δ k − ∂k x Similarly,
ck + =
tγk + and
ck − =
tγk − . Therefore,
k − Therefore, 2a

∂ ∂
kx

∂γ2 k− . ∂
kx


t
ck + ∂γ2k − ∂
ck − =
ck + 2 ck − ∂
kx ∂
kx # " ) ( 2 E
g − δ′ a
t2
ck + ∂γ2k −

r 0 k + 0
= −
kÞ − Að
g + δ′ a
k − 2 E 2
ck − ∂
kx 0 2

Now, a


k+

2


k − a





g − δ′Þ
g − η − Eg0 ðE
g − δ′Þ E
g − γ2 ðE E 0 0 2ðη + δ′Þ k+ 0 0 = =
2 η + E

g

g − Eg0 − γk − ðEg0 − δ′Þ E ′ ′ ðEg − δ Þ 0 2ðη + δ Þ

=


g − ðη + δ′Þ − ðη − E
g ðE
g − δ′Þ ηðE
g + δ ′Þ + E
g ðE
g + δ′Þ 2E 0 0 0 0 0 =





′ ′ ′ 2Eg ðη + δ Þ − ðη + Eg0 ÞðEg0 − δ Þ ηðEg0 + δ Þ − Eg0 ðEg0 − 3δ′Þ

Therefore, a


k+

22


k − a

=


g η+E
0

Eg0 − 3δ′ ′
Eg0 + δ


g η−E 0

=


g η+E 0 ′


g η−E 0

where

′
g = Eg0 ðEg0 − 3δ Þ E 0
′ Eg0 + δ

Thus,


k + a
k − a

rffiffiffiffiffiffiffiffiffiffiffi =


g η+E 0
g η−E 0

Similarly,

8