SPRINGER SERIES IN MATERIALS SCIENCE 137 … · Thermoelectric Power in Nanostructured Materials ... quantum well and quantum wire transis- ... results in the realm of quantum effect

123

SPRINGER SERIES IN MATERIALS SCIENCE 137

Thermoelectric Power in Nanostructured Materials

Strong Magnetic Fields

Kamakhya Prasad GhatakSitangshu Bhattacharya

Springer Series in

materials science 137

Springer Series in

materials scienceEditors: R. Hull C. Jagadish R.M. Osgood, Jr. J. Parisi Z. Wang H. Warlimont

The Springer Series in Materials Science covers the complete spectrum of materials physics,including fundamental principles, physical properties, materials theory and design. Recognizingthe increasing importance of materials science in future device technologies, the book titles in thisseries ref lect the state-of-the-art in understanding and controlling the structure and propertiesof all important classes of materials.

Please view available titles in Springer Series in Materials Scienceon series homepage http://www.springer.com/series/856

123

Kamakhya Prasad GhatakSitangshu Bhattacharya

Thermoelectric Power

MaterialsStrong Magnetic Fields

With 174 Figures

in Nanostructured

Professor Dr. Kamakhya Prasad GhatakUniversity of CalcuttaDeptartment of Electronic ScienceAcharya Prafulla Chandra Rd. 92Kolkata, 700 009, India

Dr. Sitangshu BhattacharyaIndian Institute of ScienceCenter of Electronics Design and TechnologyNano Scale Device Research LaboratoryBangalore, 560 012, IndiaE-mail: [email protected]

Series Editors:

Professor Robert HullUniversity of VirginiaDept. of Materials Science and EngineeringThornton HallCharlottesville, VA 22903-2442, USA

Professor Chennupati JagadishAustralian National UniversityResearch School of Physics and EngineeringJ4-22, Carver BuildingCanberra ACT 0200, Australia

Professor R. M. Osgood, Jr.Microelectronics Science LaboratoryDepartment of Electrical EngineeringColumbia UniversitySeeley W. Mudd BuildingNew York, NY 10027, USA

Professor Jurgen ParisiUniversitat Oldenburg, Fachbereich PhysikAbt. Energie- und HalbleiterforschungCarl-von-Ossietzky-Straße 9–1126129 Oldenburg, Germany

Dr. Zhiming WangUniversity of ArkansasDepartment of Physics835 W. Dicknson St.Fayetteville, AR 72701, USA

Professor Hans WarlimontDSL Dresden Material-Innovation GmbHPirnaer Landstr. 17601257 Dresden, Germany

Springer Series in Materials Science ISSN 0933-033XISBN 978-3-642-10570-8 e-ISBN 978-3-642-10571-5DOI 10.1007/978-3-642-10571-5Springer Heidelberg Dordrecht London New York

© Springer-Verlag Berlin Heidelberg 2010

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Dedicated to the Sweet Memories of LateProfessor Sushil Chandra Dasgupta, D. Sc.,Formerly Head of the Department ofMathematics of the then Bengal EngineeringCollege (Presently Bengal Engineering andScience University), Shibpur, West Bengal,India, Late Professor Biswaranjan Nag,D. Sc., Formerly Head of the Departments ofRadiophysics and Electronics and ElectronicScience, University of Calcutta, Kolkata,West Bengal, India, and Late ProfessorSankar Sebak Baral, D. Sc., FormerlyFounding Head of the Department ofElectronics and TelecommunicationEngineering of the then Bengal EngineeringCollege (Presently Bengal Engineering andScience University), Shibpur, West Bengal,India, for their pioneering contributions inresearch and teaching of AppliedMathematics, Semiconductor Science, AndApplied Electronics, respectively, to whichthe first author remains ever grateful as astudent and research worker

.

Preface

The merging of the concept of introduction of asymmetry of the wave vector spaceof the charge carriers in semiconductors with the modern techniques of fabricat-ing nanostructured materials such as MBE, MOCVD, and FLL in one, two, andthree dimensions (such as ultrathin films, nipi structures, inversion and accumula-tion layers, quantum well superlattices, carbon nanotubes, quantum wires, quantumwire superlattices, quantum dots, magneto inversion and accumulation layers, quan-tum dot superlattices, etc.) spawns not only useful quantum effect devices but alsounearth new concepts in the realm of nanostructured materials science and relateddisciplines. It is worth remaking that these semiconductor nanostructures occupy aparamount position in the entire arena of low-dimensional science and technologyby their own right and find extensive applications in quantum registers, resonanttunneling diodes and transistors, quantum switches, quantum sensors, quantumlogic gates, heterojunction field-effect, quantum well and quantum wire transis-tors, high-speed digital networks, high-frequency microwave circuits, quantumcascade lasers, high-resolution terahertz spectroscopy, superlattice photo-oscillator,advanced integrated circuits, superlattice photocathodes, thermoelectric devices,superlattice coolers, thin film transistors, intermediate-band solar cells, microop-tical systems, high-performance infrared imaging systems, bandpass filters, thermalsensors, optical modulators, optical switching systems, single electron/moleculeelectronics, nanotube based diodes, and other nanoelectronic devices. Mathemati-cian Simmons rightfully tells us [1] that the mathematical knowledge is said to bedoubling in every 10 years, and in this context, we can also envision the extrap-olation of the Moore’s law by projecting it in the perspective of the advancementof new research and analyses, in turn, generating novel concepts particularly in thearea of nanoscience and technology [2].

With the advent of Seebeck effect in 1821 [3–6], it is evident that the investiga-tions regarding the thermoelectric materials, the subset of the generalized set mate-rials science have unfathomable proportions with respect to accumulated knowledgeand new research in multidimensional aspects of thermoelectrics in general [7–17].The timeline of thermoelectric and related research during the 200 years spanningfrom 1800 to 2000 is given in [18], and with great dismay, we admit that the citationof even pertinent references in this context is placed permanently in the gallery ofimpossibility theorems. It is rather amazing to observe from the detailed survey of

vii

viii Preface

almost the whole spectrum of the literature in this particular aspect that the avail-able monographs, hand books, and review articles on thermoelectrics and relatedtopics have not included any detailed investigations on the thermoelectric power innanostructured materials under strong magnetic field (TPSM).

It is well known that the TPSM is a very important quantity [19], since the changein entropy (a vital concept in thermodynamics) can be known from this relation bydetermining the experimental values of the change of electron concentration. Theanalysis of TPSM generates information regarding the effective mass of the carri-ers in materials, which occupies a central position in the whole field of materialsscience in general [20]. The classical TPSM (G0) equation is valid only under thenondegenerate carrier concentration, and the magnitude of the TPSM is given by(G0D .�2kB=3e/ (kB and e are Boltzmann’s constant and the magnitude of thecarrier charge, respectively; [21]). From this equation, it is readily inferred thatthis conventional form is a function of three fundamental constants only, beingindependent of the signature of the charge carriers in materials. The significantwork of Zawadzki [22–24] reflects the fact that the TPSM for materials havingdegenerate electron concentration is independent of scattering mechanisms and isexclusively 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 quantiza-tion, the nano thickness in ultrathin films, quantum wires and dots, the quantizingelectric field as in inversion layers, the carrier statistics in various types of quantum-confined superlattices having different carrier energy spectra, and other types oflow-dimensional field assisted systems.

This monograph, which is based on our 20 years of continuous and ongoingresearch, is divided into four parts. The first part deals with the thermoelectricpower under large magnetic field in quantum-confined materials and it contains fourchapters. In Chap. 1, we have investigated the TPSM for quantum dots of nonlin-ear optical, III–V, II–VI, n-GaP, n-Ge, Te, Graphite, PtSb2, zerogap, II–V, GalliumAntimonide, stressed materials, Bismuth, IV–VI, lead germanium telluride, Zincand Cadmium diphosphides, Bi2Te3, and Antimony on the basis of respective car-rier energy spectrum. In Chap. 2, the TPSM in ultrathin films and quantum wires ofnonlinear optical, Kane type III–V, II–VI, Bismuth, IV–VI, stressed materials, andcarbon nanotubes (a very important quantum material) have been investigated. InChap. 3, the TPSM in quantum dot III–V, II–VI, IV–VI, HgTe/CdTe superlatticeswith graded interfaces and quantum dot effective mass superlattices of the afore-mentioned materials have been investigated. In Chap. 4, the TPSM in quantum wiresuperlattices of the said materials have been studied.

The second part of this monograph deals with the thermoelectric power undermagnetic quantization in macro and micro electronic materials. In Chap. 5, the ther-moelectric power in nonlinear optical, Kane type III–V, II–VI, Bismuth, IV-VI, andstressed materials has been investigated in the presence of quantizing magnetic field.In Chap. 6, the thermoelectric power under magnetic quantization in III–V, II–VI,IV–VI, HgTe/CdTe superlattices with graded interfaces and effective mass super-lattices of the aforementioned materials together with the quantum wells of said

Preface ix

superlattices have been investigated. In Chap. 7, the thermoelectric power undermagnetic quantization in ultrathin films of nonlinear optical, Kane type III–V, II–VI,Bismuth, IV–VI, and stressed materials has been investigated.

The third part deals with the thermoelectric power under large magnetic fieldin quantum-confined optoelectronic materials in the presence of light waves. InChap. 8, the influence of light on the thermoelectric power under large magneticfield in ultrathin films and quantum wires of optoelectronic materials has been inves-tigated. In Chap. 9, the thermoelectric power under large magnetic field in quantumdots of optoelectronic materials has been studied in the presence of external lightwaves. In Chap. 10, the same has been studied for III–V quantum wire and quantumdot superlattices with graded interfaces and III–V quantum wire and quantum doteffective mass superlattices, respectively.

The last part of this monograph deals with thermoelectric power under mag-netic quantization in macro and micro optoelectronic materials in the presence oflight waves. In Chap. 11, the optothermoelectric power in macro optoelectronicmaterials under magnetic quantization has been investigated. In Chap. 12, the opto-thermoelectric power in ultrathin films of optoelectronic materials under magneticquantization has been studied. In Chap. 13, the magneto thermo power in III–Vquantum well superlattices with graded interfaces and III–V quantum well effectivemass superlattices have been studied. Chapter 14 discusses eight applications of ourresults in the realm of quantum effect devices and also discusses very briefly theexperimental results, and additionally, we have proposed a single multidimensionalopen research problem for experimentalists regarding the thermoelectric power innanostructured materials having various carrier energy spectra under different phys-ical conditions. Chapter 15 contains the conclusion and scope of future research.Appendix A contains the TPSM for bulk specimens of few technologically impor-tant materials. Each chapter except the last two contains a table highlighting thebasic results pertaining to it in a summarized form.

It is well known that the errorless first edition of any book is virtually impos-sible from the perspective of academic reality and the same stands very true forthis monograph in spite of the Herculean joint effort of not only the authors butalso the seasoned Editorial team of Springer. Naturally, we are open to accept con-structive criticisms for the purpose of their inclusion in the future edition. FromChap. 1 till end, this monograph presents to its esteemed readers 150 open researchproblems, which will be useful in the real sense of the term for the researchers inthe fields of solid state sciences, materials science, computational and theoreticalnanoscience and technology, nanostructured thermodynamics and condensed mat-ter physics in general in addition to the graduate courses on modern thermoelectricmaterials in various academic departments of many institutes and universities. Westrongly hope that the alert readers of this monograph will not only solve the saidproblems by removing all the mathematical approximations and establishing theappropriate uniqueness conditions, but will also generate new research problems,both theoretical and experimental and, thereby, transforming this monograph into amonumental superstructure.

x Preface

It is needless to say that this monograph exposes only the tip of the iceberg, therest of which will be worked upon by the researchers of the appropriate fields whomwe would like to believe are creatively superior to us. It is an amazing fact to observethat the experimental investigations of the thermoelectric power under strong mag-netic field in nanostructured materials have been relatively less investigated in theliterature, although such studies will throw light on the understanding of the bandstructures of nanostructured materials, which, in turn, control the transport phenom-ena in such low-dimensional quantized systems. Various mathematical analyses andfew chapters of this monograph are appearing for the first time in printed form. Wehope that our esteemed readers will enjoy the investigations of TPSM in a widerange of nanostructured materials having different energy-wave vector dispersionrelation of the carriers under various physical conditions as presented in this book.Since a monograph on the thermoelectric power in nanostructured materials understrong magnetic field is really nonexistent to the best of our knowledge even in thefield of nanostructured thermoelectric materials, we earnestly hope our continuouseffort of 20 years will be transformed into a standard reference source for creativelyenthusiastic readers and researchers engaged either in theoretical or applied researchin connection with low-dimensional thermal electronics in general to probe into thein-depth investigation of this extremely potential and promising research area ofmaterials science.

Acknowledgements

Acknowledgement by Kamakhya Prasad Ghatak

I am grateful to T. Roy of the Department of Physics of Jadavpur University forcreating the interest in the applications of quantum mechanics in diverse fieldswhen I was in my late teens pursuing the engineering degree course. I am indebtedto M. Mitra, S. Sarkar, and P. Chowdhury for creating the passion in me fornumber theory. I express my gratitude to D. Bimberg, W. L. Freeman, H. L. Hart-nagel, and W. Schommers for various academic interactions spanning the last twodecades. The renowned scientist P. N. Butcher has been a driving hidden force since1985 before his untimely demise with respect to our scripting the series in bandstructure-dependent properties of nanostructured materials. He insisted me repeat-edly regarding it and to tune with his high rigorous academic frequency, myselfwith my truly able students and later on colleagues wrote the Einstein Relation inCompound Semiconductors and their Nanostructures, Springer Series in MaterialsScience, Vol. 116 as the first one, Photoemission from Optoelectronic Materials andtheir Nanostructures, Springer Series in Nanostructure Science and Technology asthe second one, and the present monograph as the third one.

I am grateful to N. Guho Chowdhury of Jadavpur University, mentor of manyacademicians including me and a very pivotal person in my academic career, forinstigating me to carry out extensive research of the first order bypassing all the

Preface xi

difficulties. My family members and especially my beloved parents deserve a veryspecial mention for really forming the backbone of my long unperturbed researchcareer. I must express my gratitude to P. K. Sarkar of Semiconductor Device Lab-oratory, S. Bania of Digital Electronics Laboratory, and B. Nag of Applied PhysicsDepartment for motivating me at rather turbulent times. I must humbly concede theLate V. S. Letokhov with true reverence for inspiring me with the ebullient idea thatthe publication of research papers containing innovative concepts in eminent peer-reviewed international journals is the central cohesive element to excel in creativeresearch activity, and at the same time being a senior research scientist he substan-tiated me regarding the unforgettable learning for all the scientists in general fromthe eye opening and alarming articles by taking a cue from [25–27] immediatelyafter their publications. Besides, this monograph has been completed under the grant(8023/BOR/RID/RPS-95/2007–08) as jointly sanctioned with D. De of the Depart-ment of Computer Science and Engineering, West Bengal University of Technologyby the All India Council for Technical Education in their research promotion scheme2008.

Acknowledgement by Sitangshu Bhattacharya

It is virtually impossible to express my gratitude to all the admirable persons whohave influenced my academic and social life from every point of view; neverthe-less, a short memento to my teachers S. Mahapatra, at the Centre for ElectronicsDesign and Technology and R. C. Mallick at Department of Physics at Indian Insti-tute of Science, Bangalore, for their fruitful academic advices and guidance remainseverlasting. I am indebted to my friend Ms. A. Sood for standing by my side atdifficult times of my research life, which still stimulates and amplifies my effi-ciency for performing in-depth research. Besides, my sister Ms. S. Bhattacharya,my father I. P. Bhattacharya and my mother B. Bhattacharya rinsed themselves formy academic development starting from my childhood till date, and for their jointspeechless and priceless contribution, no tears of gratitude seems to be enough atleast when myself is considered. I am also grateful to the Department of Scienceand Technology, India, for sanctioning the project and the fellowship under ”SERCFast Track Proposal of Young Scientist” scheme-2008–2009 (SR/FTP/ETA–37/08)under which this monograph has been completed. As usual I am grateful to myfriend, philosopher and guide, the first author.

Joint Acknowledgements

We are grateful to Dr. C. Ascheron, Executive Editor Physics, Springer Verlag inthe real sense of the term for his inspiration and priceless technical assistance. Weowe a lot to Ms. A. Duhm, Associate Editor Physics, Springer and Mrs. E. Suer,assistant to Dr. Ascheron. We are grateful to Ms. S.M. Adhikari and N. Paitya for

xii Preface

their kind help. We are indebted to our beloved parents from every fathomable pointof view and also to our families for instilling in us the thought that the academicoutput D ((desire � determination � dedication) � (false enhanced self ego pre-tending like a true friend although a real unrecognizable foe)). We firmly believethat our Mother Nature has propelled this joint collaboration in her own unseen wayin spite of several insurmountable obstacles.

Kolkata K.P. GhatakBangalore S. BhattacharyaApril 2010

References

1. G.E. Simmons, Differential Equations with Application and Historical Notes, InternationalSeries in Pure and Applied Mathematics (McGraw-Hill, USA, 1991)

2. H. Huff (ed.), Into the Nano Era – Moore’s Law beyond Planar Silicon CMOS, Springer Seriesin Materials Science, vol 106 (Springer-Verlag, Germany, 2009)

3. T.J. Seebeck, Abh. K. Akad. Wiss. Berlin, 289 (1821)4. T.J. Seebeck, Abh. K. Akad. Wiss. Berlin, 265 (1823)5. T.J. Seebeck, Ann. Phys. (Leipzig) 6, 1 (1826)6. T.J. Seebeck, Schweigger’s J. Phys. 46, 101 (1826)7. A.F. Ioffe, Semiconductor Thermoelements and Thermoelectric Cooling (Inforsearch, London,

1957)8. R.W. Ure, R.R. Heikes (eds.), Thermoelectricity: Science and Engineering (Interscience,

London, 1961)9. T.C. Harman, J.M. Honig, Thermoelectric and Thermomagnetic Effects and Applications

(McGraw-Hill, New York, 1967)10. R. Kim, S. Datta, M.S. Lundstrom, J. Appl. Phys. 105, 034506 (2009)11. C. Herring, Phys. Rev. 96, 1163 (1954)12. T.M. Tritt (ed.), Semiconductors and Semimetals, Vols. 69, 70 and 71: Recent Trends in

Thermoelectric Materials Research I, II and III (Academic Press, USA, 2000)13. D.M. Rowe (ed.), CRC Handbook of Thermoelectrics (CRC Press, USA, 1995)14. D.M. Rowe, C.M. Bhandari, Modern Thermoelectrics (Reston Publishing Company, Virginia,

1983)15. D.M. Rowe (ed.), Thermoelectrics Handbook: Macro to Nano (CRC Press, USA, 2006)16. I.M. Tsidil’kovski, Thermomagnetic Effects in Semiconductors (Academic Press, New York,

1962), p. 29017. J. Tauc, Photo and Thermoelectric Effects in Semiconductors (Pergamon Press, New York,

1962)18. G.S. Nolas, J. Sharp, J. Goldsmid, Thermoelectrics: Basic Principles and New Materials

Developments, Springer Series in Materials Science, vol 45 (Springer-Verlag, Germany, 2001)19. J. Hajdu, G. Landwehr, in Strong and Ultrastrong Magnetic Fields and Their Applications,

Topics in Applied Physics, vol 57, ed. by F. Herlach (Springer-Verlag, Germany, 1985), p. 9720. I.M. Tsidilkovskii, Band Structures of Semiconductors (Pergamon Press, London, 1982)21. K.P. Ghatak, S. Bhattacharya, S. Bhowmik, R. Benedictus, S. Choudhury, J. Appl. Phys. 103,

034303 (2008)22. W. Zawadzki, in Two-Dimensional Systems, Heterostructures and Superlattices, Springer

Series in Solid-State Science, Vol. 53, ed. by G. Bauer, F. Kuchar, H. Heinrich (Springer-Verlag,Germany, 1984)

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23. W. Zawadzki, 11th International Conference on the Physics of Semiconductors, vol 1 (ElsevierPublishing Company, Netherlands, 1972)

24. S.P. Zelenin, A.S. Kondrat’ev, A.E. Kuchma, Sov. Phys. Semiconduct. 16, 355 (1982)25. M. Gad-el-Hak, Phys. Today 57(3), 61 (2004)26. P. Gwynne, Phys. World 15(5), 5, (2002)27. T. Choy, M. Stoneham, Mater. Today 7(4) 64 (2004)

.

Contents

Part I Thermoelectric Power Under Large Magnetic Field in QuantumConfined Materials

1 Thermoelectric Power in Quantum Dots Under LargeMagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Theoretical Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Magnetothermopower in Quantum Dotsof Nonlinear Optical Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 Magnetothermopower in Quantum Dotsof III–V Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.3 Magnetothermopower in Quantum Dotsof II–VI Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.4 Magnetothermopower in Quantum Dotsof n-Gallium Phosphide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2.5 Magnetothermopower in Quantum Dotsof n-Germanium .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.2.6 Magnetothermopower in Quantum Dots of Tellurium.. . . . 231.2.7 Magnetothermopower in Quantum Dots of Graphite . . . . . . 251.2.8 Magnetothermopower in Quantum Dots

of Platinum Antimonide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.2.9 Magnetothermopower in Quantum Dots

of Zerogap Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.2.10 Magnetothermopower in Quantum Dots

of II–V Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2.11 Magnetothermopower in Quantum Dots

of Gallium Antimonide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.2.12 Magnetothermopower in Quantum Dots

of Stressed Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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xvi Contents

1.2.13 Magnetothermopower in Quantum Dots of Bismuth . . . . . . 351.2.14 Magnetothermopower in Quantum Dots

of IV–VI Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.2.15 Magnetothermopower in Quantum Dots

of Lead Germanium Telluride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.2.16 Magnetothermopower in Quantum Dots

of Zinc and Cadmium Diphosphides . . . . . . . . . . . . . . . . . . . . . . . 431.2.17 Magnetothermopower in Quantum Dots

of Bismuth Telluride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.2.18 Magnetothermopower in Quantum Dots of Antimony . . . . 45

1.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.4 Open Research Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

2 Thermoelectric Power in Ultrathin Films and QuantumWires Under Large Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.2 Theoretical Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

2.2.1 Magnetothermopower in Quantum-ConfinedNonlinear Optical Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

2.2.2 Magnetothermopower in Quantum-ConfinedKane Type III–V Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

2.2.3 Magnetothermopower in Quantum-ConfinedII–VI Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103

2.2.4 Magnetothermopower in Quantum-Confined Bismuth . . . .1052.2.5 Magnetothermopower in Quantum-Confined

IV–VI Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1122.2.6 Magnetothermopower in Quantum-Confined

Stressed Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1162.2.7 Magnetothermopower in Carbon Nanotubes .. . . . . . . . . . . . . .117

2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1192.4 Open Research Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142

3 Thermoelectric Power in Quantum Dot SuperlatticesUnder Large Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1453.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1453.2 Theoretical Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146

3.2.1 Magnetothermopower in III–V Quantum DotSuperlattices with Graded Interfaces . . . . . . . . . . . . . . . . . . . . . . .146

3.2.2 Magnetothermopower in II–VI Quantum DotSuperlattices with Graded Interfaces . . . . . . . . . . . . . . . . . . . . . . .149

3.2.3 Magnetothermopower in IV–VI QuantumDot Superlattices with Graded Interfaces . . . . . . . . . . . . . . . . . .151

Contents xvii

3.2.4 Magnetothermopower in HgTe/CdTeQuantum Dot Superlattices with Graded Interfaces . . . . . . .155

3.2.5 Magnetothermopower in III–V Quantum DotEffective Mass Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .158

3.2.6 Magnetothermopower in II–VI Quantum DotEffective Mass Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .159

3.2.7 Magnetothermopower in IV–VI QuantumDot Effective Mass Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . .160

3.2.8 Magnetothermopower in HgTe/CdTeQuantum Dot Effective Mass Superlattices . . . . . . . . . . . . . . . .162


4 Thermoelectric Power in Quantum Wire SuperlatticesUnder Large Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1734.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1734.2 Theoretical Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173

4.2.1 Magnetothermopower in III–V QuantumWire Superlattices with Graded Interfaces . . . . . . . . . . . . . . . . .173

4.2.2 Magnetothermopower in II–VI QuantumWire Superlattices with Graded Interfaces . . . . . . . . . . . . . . . . .174

4.2.3 Magnetothermopower in IV–VI QuantumWire Superlattices with Graded Interfaces . . . . . . . . . . . . . . . . .175

4.2.4 Magnetothermopower in HgTe/CdTeQuantum Wire Superlattices with Graded Interfaces . . . . . .176

4.2.5 Magnetothermopower in III–V QuantumWire Effective Mass Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . .177

4.2.6 Magnetothermopower in II–VI QuantumWire Effective Mass Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . .178

4.2.7 Magnetothermopower in IV–VI QuantumWire Effective Mass Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . .179

4.2.8 Magnetothermopower in HgTe/CdTeQuantum Wire Effective Mass Superlattices . . . . . . . . . . . . . . .180

4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1814.4 Open Research Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .187References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .187

Part II Thermoelectric Power Under Magnetic Quantization in Macroand Microelectronic Materials

5 Thermoelectric Power in Macroelectronic MaterialsUnder Magnetic Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1915.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1915.2 Theoretical Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191

xviii Contents

5.2.1 Magnetothermopower in Nonlinear Optical Materials . . . .1915.2.2 Magnetothermopower in Kane Type III–V Materials . . . . .1935.2.3 Magnetothermopower in II–VI Materials . . . . . . . . . . . . . . . . . .1955.2.4 Magnetothermopower in Bismuth . . . . . . . . . . . . . . . . . . . . . . . . . .1965.2.5 Magnetothermopower in IV–VI Materials . . . . . . . . . . . . . . . . .1985.2.6 Magnetothermopower in Stressed Materials . . . . . . . . . . . . . . .198


6 Thermoelectric Power in Superlattices Under Magnetic Quantization 2156.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2156.2 Theoretical Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .215

6.2.1 Magnetothermopower in III–V Superlatticeswith Graded Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .215

6.2.2 Magnetothermopower in II–VI Superlatticeswith Graded Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .217

6.2.3 Magnetothermopower in IV–VI Superlatticeswith Graded Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .218

6.2.4 Magnetothermopower in HgTe/CdTeSuperlattices with Graded Interfaces . . . . . . . . . . . . . . . . . . . . . . .220

6.2.5 Magnetothermopower in III–V EffectiveMass Superlattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .221

6.2.6 Magnetothermopower in II–VI EffectiveMass Superlattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .222

6.2.7 Magnetothermopower in IV–VI EffectiveMass Superlattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .223

6.2.8 Magnetothermopower in HgTe/CdTeEffective Mass Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .224

6.2.9 Magnetothermopower in III–V QuantumWell Superlattices with Graded Interfaces . . . . . . . . . . . . . . . . .225

6.2.10 Magnetothermopower in II–VI QuantumWell Superlattices with Graded Interfaces . . . . . . . . . . . . . . . . .226

6.2.11 Magnetothermopower in IV–VI QuantumWell Superlattices with Graded Interfaces . . . . . . . . . . . . . . . . .226

6.2.12 Magnetothermopower in HgTe/CdTeQuantum Well Superlattices with Graded Interfaces . . . . . .227

6.2.13 Magnetothermopower in III–V QuantumWell-Effective Mass Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . .228

6.2.14 Magnetothermopower in II–VI QuantumWell-Effective Mass Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . .228

6.2.15 Magnetothermopower in IV–VI QuantumWell-Effective Mass Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . .229

6.2.16 Magnetothermopower in HgTe/CdTeQuantum Well-Effective Mass Superlattices . . . . . . . . . . . . . . .229

Contents xix


7 Thermoelectric Power in Ultrathin FilmsUnder Magnetic Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2417.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2417.2 Theoretical Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .241

7.2.1 Magnetothermopower in Ultrathin Filmsof Nonlinear Optical Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .241

7.2.2 Magnetothermopower in Ultrathin Filmsof Kane Type III–V Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .242

7.2.3 Magnetothermopower in Ultrathin Filmsof II–VI Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .244

7.2.4 Magnetothermopower in Ultrathin Films of Bismuth . . . . .2457.2.5 Magnetothermopower in Ultrathin Films

of IV–VI Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2477.2.6 Magnetothermopower in Ultrathin Films

of Stressed Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2477.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2477.4 Open Research Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .256

Part III Thermoelectric Power Under Large Magnetic Fieldin Quantum Confined Optoelectronic Materials in the Presenceof Light Waves

8 Optothermoelectric Power in Ultrathin Filmsand Quantum Wires of Optoelectronic MaterialsUnder Large Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2598.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2598.2 Theoretical Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .260

8.2.1 Optothermoelectric Power in Ultrathin Filmsof Optoelectronic Materials Under LargeMagnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .260

8.2.2 Optothermoelectric Power in Quantum Wiresof Optoelectronic Materials Under LargeMagnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .272

8.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2748.4 Open Research Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .291References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .294

xx Contents

9 Optothermoelectric Power in Quantum Dotsof Optoelectronic Materials Under Large Magnetic Field . . . . . . . . . . . . . . .2959.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2959.2 Theoretical Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .295

9.2.1 Magnetothermopower in Quantum Dotsof Optoelectronic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .295

9.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2969.4 Open Research Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .299Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .299

10 Optothermoelectric Power in Quantum-ConfinedSemiconductor Superlattices of Optoelectronic MaterialsUnder Large Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30110.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30110.2 Theoretical Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .301

10.2.1 Magnetothermopower in III–V QuantumWire Effective Mass Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . .301

10.2.2 Magnetothermopower in III–V Quantum DotEffective Mass Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .303

10.2.3 Magnetothermopower in III–V QuantumWire Superlattices with Graded Interfaces . . . . . . . . . . . . . . . . .303

10.2.4 Magnetothermopower in III–V Quantum DotSuperlattices with Graded Interfaces . . . . . . . . . . . . . . . . . . . . . . .305

10.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30610.4 Open Research Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .312Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .315

Part IV Thermoelectric Power Under Magnetic Quantization in Macroand Micro-optoelectronic Materials in the Presence of Light Waves

11 Optothermoelectric Power in Macro-OptoelectronicMaterials Under Magnetic Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31911.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31911.2 Theoretical Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .319

11.2.1 Magnetothermopower in Optoelectronic Materials . . . . . . . .31911.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32111.4 Open Research Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .332Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .332

12 Optothermoelectric Power in Ultrathin Filmsof Optoelectronic Materials Under Magnetic Quantization . . . . . . . . . . . . .33312.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33312.2 Theoretical Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .333

12.2.1 Magnetothermopower in Ultrathin Filmsof Optoelectronic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .333

Contents xxi

12.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33412.4 Open Research Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .338Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .338

13 Optothermoelectric Power in Superlatticesof Optoelectronic Materials Under MagneticQuantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33913.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33913.2 Theoretical Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .339

13.2.1 Magnetothermopower in III–V QuantumWell-Effective Mass Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . .339

13.2.2 Magnetothermopower in III–V QuantumWell Superlattices with Graded Interfaces . . . . . . . . . . . . . . . . .341


14 Applications and Brief Review of Experimental Results . . . . . . . . . . . . . . . . .34914.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34914.2 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .349

14.2.1 Effective Electron Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34914.2.2 Debye Screening Length .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35014.2.3 Carrier Contribution to the Elastic Constants . . . . . . . . . . . . . .35114.2.4 Diffusivity–Mobility Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35114.2.5 Diffusion Coefficient of the Minority Carriers . . . . . . . . . . . . .35314.2.6 Nonlinear Optical Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35314.2.7 Third-Order Nonlinear Optical Susceptibility . . . . . . . . . . . . .35314.2.8 Generalized Raman Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .354

14.3 Brief Review of Experimental Works. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35414.3.1 Bulk Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35414.3.2 Nanostructured Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .358

14.4 Open Research Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .362References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .362

15 Conclusion and Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .367References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .370A.1 Nonlinear Optical Materials and Cd3As2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .371A.2 III–V Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .372

A.2.1 Three Band Model of Kane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .372A.2.2 Two Band Model of Kane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .372A.2.3 Parabolic Energy Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .373A.2.4 The Model of Stillman Et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .373A.2.5 The Model of Palik Et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .374A.2.6 Model of Johnson and Dicley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .374

A.3 n-Type Gallium Phosphide .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .375A.4 II–VI Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .376

xxii Contents

A.5 Bismuth Telluride .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .376A.6 Stressed Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .376A.7 IV–VI Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .377

A.7.1 Bangert and Kastner Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .377A.7.2 Cohen Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .377A.7.3 Dimmock Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .378A.7.4 Foley and Langenberg Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .380

A.8 n-Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .382A.8.1 Model of Cardona Et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .382A.8.2 Model of Wang and Ressler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .382

A.9 Platinum Antimonide .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .384A.10 n-GaSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .385A.11 n-Te . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .385A.12 Bismuth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .386

A.12.1 McClure and Choi Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .386A.12.2 Hybrid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .387A.12.3 Lax Ellipsoidal Nonparabolic Model . . . . . . . . . . . . . . . . . . . . . . .388A.12.4 Ellipsoidal Parabolic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .388

A.13 Open Research Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .388Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .388

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .389

Material Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .393

List of Symbols

˛ Band nonparabolicity parameter˛11; ˛12 Energy-band constants˛11; ˛22; ˛33; ˛23 Spectrum constants˛11; ˛22; ˛33; ˛23

ˇ1; ˇ2; ˇ4; ˇ5 System constants�11; �12 Energy-band constantsˇ11; ˇ12; �1; �5

ı Crystal field splitting constantı0 Dirac’s delta functionı0 Band constant�jj Spin–orbit splitting constant parallel to the C-axis�? Spin–orbit splitting constant perpendicular to the C-axis� Isotropic spin–orbit splitting constant�0 Interface width in superlattices�1 Energy-band constant�0c ; �00c Spectrum constants� Wavelength�0 Band constantO" Strain tensor" Trace of the strain tensor" Energy as measured from the center of the band gap Eg

0

"sc Semiconductor permittivity"0 Permittivity of vacuum�.2r/ Zeta function of order 2r�0 Constant of the spectrum�.j C 1/ Complete Gamma function!0 Cyclotron resonance frequency� Frequency1.k/ Warping of the Fermi surface2.k/ Inversion asymmetry splitting of the conduction band�i Broadening parameter Thermodynamic potentiala Constant of the spectrum

xxiii

xxiv List of Symbols

a Lattice constantac Nearest neighbor C–C bonding distancea13 Nonparabolicity constanta15 Warping parametera0 The width of the barrier for superlattice structuresA Spectrum constantAi Energy band constantsb0 The width of the well for superlattice structuresB Quantizing magnetic fieldB2 Momentum matrix elementc Velocity of lightNc Constant of the spectrumC0 Splitting of the two-spin states by the spin orbit coupling and

the crystalline fieldC1 Conduction band deformation potentialC2 Strain interaction between the conduction and valance bandsdx; dy ; dz Nano thickness along the x, y, and z-directionse Magnitude of electron chargeE Total energy of the carrierE 0 Sub-band energy

E Energy of the hole as measured from the top of the valanceband in the vertically downward direction

EF Fermi energyEF1

Fermi energy as measured from the mid of the band gap in thevertically upward direction in connection with nanotubes

EFB Fermi energy in the presence of magnetic fieldNEi Energy-band constantEnz Energy of the nth sub-bandEFQD Fermi energy in the presence of 3D quantization as measured

from the edge of the conduction band in the vertically upwarddirection in the absence of any quantization

EF1D Fermi energy in the presence of two-dimensional quantiza-tion as measured from the edge of the conduction band in thevertically upward direction in the absence of any quantization

EF2D Fermi energy in the presence of size quantization as measuredfrom the edge of the conduction band in the vertically upwarddirection in the absence of any quantization

EFQDSLGI Fermi energy in quantum dot superlattices with graded inter-faces as measured from the edge of the conduction band in thevertically upward direction in the absence of any quantization

EFQDSLEM Fermi energy in quantum dot effective mass superlattices asmeasured from the edge of the conduction band in the verti-cally upward direction in the absence of any quantization

List of Symbols xxv

EFQWSLGI Fermi energy in quantum wire superlattices with graded inter-faces as measured from the edge of the conduction band in thevertically upward direction in the absence of any quantization

EFQWSLEM Fermi energy in quantum wire effective mass superlatticesas measured from the edge of the conduction band in thevertically upward direction in the absence of any quantization

EFQWSLEML Fermi energy in quantum wire effective mass superlattices inthe presence of light waves as measured from the edge ofthe conduction band in the vertically upward direction in theabsence of any quantization

EFQDSLEML Fermi energy in quantum dot effective mass superlattices in thepresence of light waves as measured from the edge of the con-duction band in the vertically upward direction in the absenceof any quantization

Eg0Band gap in the absence of any field

EB Bohr electron energyEQD Totally quantized energyEFQD Fermi energy in quantum dots as measured from the edge of

the conduction band in the vertically upward direction in theabsence of any quantization

EF2DL Fermi energy in ultrathin films in the presence of light wavesas measured from the edge of the conduction band in thevertically upward direction in the absence of any quantization

EF0DL Fermi energy in quantum dots in the presence of light wavesas measured from the edge of the conduction band in thevertically upward direction in the absence of any quantization

EFBL Fermi energy under quantizing magnetic field in the presenceof light waves as measured from the edge of the conductionband in the vertically upward direction in the absence of anyquantization

EF2DBL Fermi energy in ultrathin films under quantizing magnetic fieldin the presence of light waves as measured from the edge ofthe conduction band in the vertically upward direction in theabsence of any quantization

EFBQWSLEML Fermi energy in quantum well effective mass superlatticesunder magnetic quantization in the presence of light wavesas measured from the edge of the conduction band in thevertically upward direction in the absence of any quantization

EFBQWSLGIL Fermi energy in quantum well superlattices with graded inter-faces under magnetic quantization in the presence of lightwaves as measured from the edge of the conduction band in thevertically upward direction in the absence of any quantization

Enx; Eny

; Enz The quantized energy levels due to infinity deep potential wellalong the x, y and z-directions

f .E/ Fermi–Dirac occupation probability factor

xxvi List of Symbols

Fj .�/ One parameter Fermi–Dirac integral of order jF0.�nz/ Special case of the one parameter Fermi–Dirac integral of

order jF1.�nz/ Special case of the one parameter Fermi–Dirac integral of

order jgv Valley degeneracyG0 Thermoelectric power under strong magnetic fieldh Planck’s constant„ Dirac’s constant .� h=.2�//

H Heaviside step functioni IntegerI0 Light intensityk0 Constant of the energy spectrumNk0 Inverse Bohr radiuskB Boltzmann’s constantk Electron wave vectorlx Sample length along x directionNl ; Nm; Nn Matrix elements of the strain perturbation operatorl Band constantL0 Period of the superlatticesm0 Free electron massm� Isotropic effective electron mass at the edge of the conduction

bandm�jj Longitudinal effective electron mass at the edge of the conduc-

tion bandm�? Transverse effective electron mass at the edge of the conduc-

tion bandm1 Effective carrier mass at the band-edge along x directionm2 Effective carrier mass at the band-edge along y directionm3 The effective carrier mass at the band-edge along z directionm02 Effective-mass tensor component at the top of the valence band

(for electrons) or at the bottom of the conduction band (forholes)

m�t The transverse effective mass at k D 0

m�l

The longitudinal effective mass at k D 0

m�?;1; m�k;1 Transverse and longitudinal effective electron mass at the edge

of the conduction band for the first material in superlatticemr Reduced massmv Effective mass of the heavy hole at the top of the valance band

in the absence of any fieldm�v Effective mass of the holes at the top of the valence bandmt Contributions to the transverse effective mass of the external

LC6 and L�6 bands arising from the�!k � �!p perturbations with

the other bands taken to the second order

List of Symbols xxvii

m˙l

Contributions to the longitudinal effective mass of the external

LC6 and L�6 bands arising from the�!k � �!p perturbations with

the other bands taken to the second ordermtc Transverse effective electron mass of the conduction electrons

at the edge of the conduction bandmlc Longitudinal effective electron mass of the conduction elec-

trons at the edge of the conduction bandmtv Transverse effective hole mass of the holes at the edge of the

valence bandmlv Longitudinal effective hole mass of the holes at the edge of the

valence bandnx ; ny ; nz Size quantum numbers along the x, y, and z-directionsn0 Carrier degeneracyn Landau quantum number/chiral indicesn Band constantN.E/ Density of states in bulk specimensNc Effective number of states in the conduction bandN2D.E/ 2D density-of-states function per sub-bandN2DT .E/ Total 2D density-of-states functionP0 Momentum matrix element. NP / Energy-band constantPk; P? Momentum matrix elements parallel and perpendicular to the

direction of C-axisNQ; NR Spectrum constantsr0 Radius of the nanotubeNs Spectrum constants0 Upper limit of the summationS0 Entropy per unit volumetc Tight binding parameterti Energy band constantsT Temperaturev Band constantNv0; Nw0 Constants of the spectrumV0 Potential barrierjVG j Constants of the energy spectrumx, y Alloy compositions

Part IThermoelectric Power Under

Large Magnetic Field in QuantumConfined Materials

Chapter 1Thermoelectric Power in Quantum DotsUnder Large Magnetic Field

1.1 Introduction

In recent years, with the advent of Quantum Hall Effect (QHE) [1,2], there has beenconsiderable interest in studying the thermoelectric power under strong magneticfield (TPSM) in various types of nanostructured materials having quantum confine-ment of their charge carriers in one, two, and three dimensions of the respectivewave-vector space leading to different carrier energy spectra [3–38]. The classicalTPSM equation as mentioned in the preface is valid only under the condition ofcarrier nondegeneracy, is being independent of carrier concentration, and reflectsthe fact that the signature of the band structure of any material is totally absent inthe same.

Zawadzki [8] demonstrated that the TPSM for electronic materials having degen-erate electron concentration is essentially determined by their respective energyband structures. It has, therefore, different values in different materials and changeswith the doping; with the magnitude of the reciprocal quantizing magnetic fieldunder magnetic quantization, quantizing electric field as in inversion layers, andnanothickness as in quantum wells, wires, and dots; and with the superlattice periodas in quantum-confined semiconductor superlattices with graded interfaces havingvarious carrier energy spectra and also in other types of field-assisted nanostructuredmaterials. Some of the significant features that have emerged from these studies are:

(a) The TPSM decreases with the increase in carrier concentration.(b) The TPSM decreases with increasing doping in heavily doped semiconductors

forming band tails.(c) The nature of variations is significantly influenced by the spectrum constants of

various materials having different band structures.(d) The TPSM exhibits oscillatory dependence with inverse quantizing magnetic

field because of the Shubnikov–de Haas effect.(e) The TPSM decreases with the magnitude of the quantizing electric field in

inversion layers.(f) The TPSM exhibits composite oscillations with significantly different values in

superlattices and various other quantized field aided structures.

3

4 1 Thermoelectric Power in Quantum Dots Under Large Magnetic Field

In this chapter, an attempt is made to investigate the TPSM in quantum dots ofnonlinear optical, III–V, II–VI, GaP, Ge, Te, Graphite, PtSb2, zerogap, II–V, GaSb,stressed materials, Bismuth, IV–VI, Lead Germanium Telluride, Zinc and Cadmiumdiphosphides, Bi2Te3, and Antimony from Sects. 1.2.1 to 1.2.18, respectively. Inthis context, it may be noted that with the advent of fine line lithography [39],molecular beam epitaxy [40, 41], organometallic vapor-phase epitaxy [42], andother experimental techniques, low-dimensional structures [43–55] having quan-tum confinement of the charge carriers in one, two, and three dimensions [such asultrathin films (UFs), nipi structures, inversion and accumulation layers, quantumwell superlattices, carbon nanotubes, quantum wires (QWs), quantum wire super-lattices, quantum dots (QDs), magnetoinversion and accumulation layers, quantumdot superlattices, etc.] have, in the last few years, created tremendous passion amongthe interdisciplinary researchers not only for the potential of these quantized struc-tures in uncovering new phenomena in nanostructured science but also for theirnew diverse technological applications. As the dimension of the UFs increasesfrom one dimension to three dimension, the degree of freedom of the free carriersdecreases drastically and the density-of-states function changes from the Heavisidestep function in UFs to the Dirac’s delta function in QDs [56, 57].

The QDs can be used for visualizing and tracking molecular processes in cellsusing standard fluorescence microscopy [58–61]. They display minimal photo-bleaching [62], thus allowing molecular tracking over prolonged periods, and con-sequently, single molecule can be tracked by using optical fluorescence microscopy[63,64]. The salient features of quantum dot lasers [65–67] include lower thresholdcurrents, higher power, and greater stability compared with that of the conventionalone, and the QDs find extensive applications in nanorobotics [68–71], neural net-works [72–74], and high density memory or storage media [75]. The QDs are alsoused in nanophotonics [76] because of their theoretically high quantum yield andhave been suggested as implementations of qubits for quantum information process-ing [77]. The QDs also find applications in diode lasers [78], amplifiers [79, 80],and optical sensors [81, 82]. High-quality QDs are well suited for optical encod-ing [83, 84] because of their broad excitation profiles and narrow emission spectra.The new generations of QDs have far-reaching potential for the accurate investiga-tions of intracellular processes at the single-molecule level, high-resolution cellularimaging, long-term in vivo observation of cell trafficking, tumor targeting, and diag-nostics [85,86]. The QD nanotechnology is one of the most promising candidates foruse in solid-state quantum computation [87, 88]. It may also be noted that the QDsare being used in single electron transistors [89, 90], photovoltaic devices [91, 92],photoelectrics [93], ultrafast all-optical switches and logic gates [94–97], organicdyes [98–100], and in other types of nanodevices.

Section 1.2.1 investigates the TPSM in QDs of nonlinear optical materials (tak-ing n-CdGeAs2 as an example), which find applications in nonlinear optics andlight-emitting diodes [101]. The quasicubic model can be used to investigate thesymmetric properties of both the bands at the zone center of wave-vector space ofthe same compound [102]. Including the anisotropic crystal potential in the Hamil-tonian and the special features of the nonlinear optical compounds, Kildal [103]

1.1 Introduction 5

formulated the electron dispersion law under the assumptions of isotropic momen-tum matrix and the isotropic spin–orbit splitting constant, respectively, although theanisotropies in the two aforementioned band constants are the significant physicalfeatures of the said materials [104–106].

In this context, it may be noted that the III–V compounds find potential applica-tions in infrared detectors [107], quantum dot light-emitting diodes [108], quantumcascade lasers [109], quantum well wires [110], optoelectronic sensors [111], highelectron mobility transistors [112], etc. The III–V, ternary and quaternary materialsare called the Kane-type compounds, since their electron energy spectra are beingdefined by the three-band model of Kane [113]. In Sect. 1.2.2, the TPSM from QDsof III–V materials has been studied, and the simplified results for two-band modelof Kane and that of wide gap materials have further been demonstrated as specialcases. Besides Kane, the conduction electrons of III–V materials also obey anothersix different types of electron dispersion laws as given in the literature. The TPSMhas also been investigated for all the cases for the purpose of complete presentationand relative assessment among the energy band models of III–V compounds.

The II–VI compounds are being extensively used in nanoribbons, blue greendiode lasers, photosensitive thin films, infrared detectors, ultrahigh-speed bipo-lar transistors, fiber-optic communications, microwave devices, photovoltaic andsolar cells, semiconductor gamma-ray detector arrays, and semiconductor detectorgamma camera and allow for a greater density of data storage on optically addressedcompact discs [114–121]. The carrier energy spectra in II–VI materials are definedby the Hopfield model [122], where the splitting of the two-spin states by the spin–orbit coupling and the crystalline field has been taken into account. Section 1.2.3contains the investigation of the TPSM in QDs of II–VI compounds, taking p-CdSas an example.

The n-Gallium Phosphide (n-GaP) is being used in quantum dot light-emittingdiode [123], high-efficiency yellow solid state lamps, light sources, and high peakcurrent pulse for high gain tubes. The green and yellow light-emitting diodesmade of nitrogen-doped n-GaP possess a longer device life at high drive currents[124–126]. In Sect. 1.2.4, the TPSM in QDs of n-GaP is studied. The importance ofGermanium is already well known since the inception of transistor technology, andin recent years, memory circuits, single photon detectors, single photon avalanchediode, ultrafast all-optical switch, THz lasers, and THz spectrometers [127–130] aremade of Ge. In Sect. 1.2.5, the TPSM has been studied in QDs of Ge.

Tellurium (Te) is also an elemental semiconductor which has been used asthe semiconductor layer in thin-film transistors (TFT) [131]. Te also finds exten-sive applications in CO2 laser detectors [132], electronic imaging, strain sensitivedevices [133, 134], and multichannel Bragg cell [135]. Section 1.2.6 contains theinvestigation of TPSM in QDs of Tellurium. The importance of graphite is alreadywell known in the whole spectrum of materials science, and the low-dimensionalgraphite is used instead of carbon wire in many practical applications. Graphiteintercalation compounds are often used as suitable model for investigation oflow-dimensional systems and, in particular, for investigation of phase transitionin such systems [136–139]. In Sect. 1.2.7, the TPSM in QDs of graphite has


been explored. Platinum Antimonide (PtSb2/ finds applications in device minia-turization, colloidal nanoparticle synthesis, sensors and detector materials, andthermo-photovoltaic devices [140–142]. The TPSM in QDs of p-PtSb2 has beeninvestigated in Sect. 1.2.8.

Zerogap compounds are used in optical waveguide switch or modulators that canbe fabricated by using the electro-optic and thermo-optic effects for facilitating opti-cal communications and signal processing. The gapless materials also find extensiveapplications in infrared detectors and night vision cameras [143–147]. Section 1.2.9contains the study of TPSM in QDs of the same taking p-HgTe as an example.

The II–V materials are used in photovoltaic cells constructed of single crystalmaterials in contact with electrolyte solutions. Cadmium selenide shows an open-circuit voltage of 0.8 V and power conservation coefficients of nearly 6% for 720-nm light [148]. They are also used in ultrasonic amplification [149]. The thin filmtransistor using cadmium selenide as the semiconductor has been developed [150,151]. In Sect. 1.2.10, the TPSM in QDs of II–V materials has been studied takingCdSb as an example. Gallium antimonide (GaSb) finds applications in the fiber-optic transmission window, heterojunctions, and quantum wells. A complementaryheterojunction field effect transistor (CHFET) in which the channels for the p-FETdevice and the n-FET device forming the complementary FET are formed fromGaSb. The band gap energy of GaSb makes it suitable for low power operation[152–157]. In Sect. 1.2.11, the TPSM in QDs of GaSb has been studied.

It may be noted that the stressed materials are being widely investigated forstrained silicon transistors, quantum cascade lasers, semiconductor strain gages,thermal detectors, and strained-layer structures [158–161]. The TPSM in QDs ofstressed materials (taking stressed n-InSb as an example) has been investigated inSect. 1.2.12. In recent years, Bismuth (Bi) nanolines are fabricated, and Bi alsofinds use in array of antennas which leads to the interaction of electromagneticwaves with such Bi nanowires [162,163]. Several dispersion relations of the carriershave been proposed for Bi. Shoenberg [164,165] experimentally verified that the deHaas–Van Alphen and cyclotron resonance experiments supported the ellipsoidalparabolic model of Bi, although, the magnetic field dependence of many physicalproperties of Bi supports the two-band model [166]. The experimental investiga-tions on the magneto-optical [167] and the ultrasonic quantum oscillations [168]support the Lax ellipsoidal nonparabolic model [166]. Kao [169], Dinger and Law-son [170], and Koch and Jensen [171] demonstrated that the Cohen model [172]is in conformity with the experimental results in a better way. Besides, the Hybridmodel of bismuth, as developed by Takoka et al., also finds use in the literature[173]. McClure and Choi [174] devised a new model of Bi and they showed that itcan explain the data for a large number of magneto-oscillatory and resonance exper-iments. In Sect. 1.2.13, we have formulated the TPSM in QDs of Bi in accordancewith the aforementioned energy band models for the purpose of relative assessment.

Lead chalcogenides (PbTe, PbSe, and PbS) are IV–VI compounds whose stud-ies over several decades have been motivated by their importance in infrared IRdetectors, lasers, light-emitting devices, photovoltaics, and high-temperature ther-moelectric [175–179]. PbTe, in particular, is the end compound of several ternary

1.2 Theoretical Background 7

and quaternary high-performance high-temperature thermoelectric materials [180–184]. It has been used not only as bulk but also as films [185–188], quantumwells [189], superlattices [190, 191], nanowires [192], and colloidal and embeddednanocrystals [193–196]. PbTe films doped with various impurities have also beeninvestigated [197–200]. These studies revealed some of the interesting features thathave been observed in bulk PbTe, such as Fermi level pinning, and in the case ofsuperconductivity [201]. In Sects. 1.2.14 and 1.2.15, the TPSM in QDs of IV–VImaterials Pb1�xGexTe has been studied.

The diphosphides find prominent role in biochemistry where the foldingand structural stabilization of many important extracellular peptide and proteinmolecules, including hormones, enzymes, growth factors, toxins, and immunoglob-ulin, are concerned [202–204]. Besides, artificial introduction of extra diphosphidesinto peptides or proteins can improve biological activity [205, 206] or conferthermostability [207]. The asymmetric diphosphide bond formation in peptides con-taining a free thiol group takes place over a wide pH range in aqueous buffers andcan be crucially monitored by spectrophotometric titration of the released 3-nitro-2-pyridinethiol [208, 209]. In Sect. 1.2.16, the TPSM in QDs of zinc and cadmiumdiphosphides has been investigated.

Bismuth telluride (Bi2Te3/ was first identified as a material for thermoelectricrefrigeration in 1954 [210] and its physical properties were later improved by theaddition of bismuth selenide and antimony telluride to form solid solutions [211–215]. The alloys of Bi2Te3 are very important compounds for the thermoelectricindustry and have extensively been investigated in the literature [211–215]. In Sect.1.2.17, the TPSM in QDs of Bi2Te3 has been considered. In recent years, antimonyhas emerged to be very promising, since glasses made from antimony are beingextensively used in near infrared spectral range for third- or second-order nonlinearprocesses. The chalcogenide glasses are in general associated with high nonlinearproperties for their Infrared transmission from 0.5–1 �m to 12–18 �m [216–221].Alloys of Sb are used as ultrahigh-frequency indicators and in thin-film thermocou-ple [216–221]. In Sect. 1.2.18, the TPSM in QDs of Sb has been studied. Section1.3 contains results and discussion for this chapter. Section 1.4 contains the openresearch problems pertinent to this chapter.

1.2 Theoretical Background

1.2.1 Magnetothermopower in Quantum Dots of NonlinearOptical Materials

The form of k.p matrix for nonlinear optical compounds can be expressed extendingBodnar [222] as

H D�H1 H2

HC2 H1

�; (1.1)


where

H1 �

266664Eg0

0 Pkkz 0

0��2�jjı3� �p

2�?.3�0

Pkkz

�p2�?

.3�� ı C 1

3�k�0

0 0 0 0

377775

and

H2 �

26640 �f;C 0 f;�f;C 0 0 0

0 0 0 0

f;C 0 0 0

3775

in which Eg0is the band gap in the absence of any field; Pk and P? the momen-

tum matrix elements parallel and perpendicular to the direction of crystal axis,respectively; ı the crystal field splitting constant; and �jj and �? are the spin–orbit splitting constants parallel and perpendicular to the C -axis, respectively,f;˙ � .P?=

p2/�kx ˙ iky

�and i D p�1. Thus, neglecting the contribution of

the higher bands and the free electron term, the diagonalization of the above matrixleads to the dispersion relation of the conduction electrons in bulk specimens ofnonlinear optical compounds [223] as

� .E/ D f1 .E/ k2s C f2 .E/ k

2z ; (1.2)

where

�.E/ � E.E C Eg0/

� �E C Eg0

� �E C Eg0

C�jj�

C ı�E C Eg0

C 2

3�jjC 2

9

��2jj ��2?

��; k2

s D k2x C k2

y ;

f1.E/ � „2Eg0

�Eg0

C�?�

2m�?

�Eg0

C 23�?

�� ı�E C Eg0C 1

3�jj

C �E C Eg0

� �E C Eg0

C 2

3�jjC 1

9

��2jj ��2

jj��;

f2 .E/ � „2Eg0

�Eg0

C�jj�

h2m�jj

�Eg0

C 23�jj�i��E C Eg0

� �E C Eg0

C 2

3�jj�

and m�jj and m�? are the longitudinal and transverse effective electron masses at theedge of the conduction band, respectively.

Let Eni.i D x; y and z/ be the quantized energy levels due to infinitely deep

potential well along i th axis with ni D 1; 2; 3 : : : as the size quantum numbers.Therefore, from (1.2), one can write


� .Enx/ D f1 .Enx

/

��nx

dx

2

(1.3)

��Eny

� D f1

�Eny

� ��ny

dy

2

(1.4)

��Enz

� D f2

�Enz

� ��nz

dz

2

(1.5)

From (1.2), the totally quantized energy�EQD1

�in this case can be expressed as

��EQD1

� D f1

�EQD1

� "��nx

dx

2

C��ny

dy

2#C f2

�EQD1

� "��nz

dz

2#

(1.6)

The total electron concentration per unit volume in this case assumes the form

n0 D 2gv

dxdydz

nxmaxXnxD1

nymaxXnyD1

nzmaxXnzD1

L11

M11

; (1.7)

where gv is the valley degeneracy,

L11 D Œ1C A1 cos H1 (1.8)

M11 D 1C A21 C 2A1 cos H1 (1.9)

in which

A1 D exp

�EQD1 � EFQD

kBT

�;

EFQD is the Fermi energy in the presence of three-dimensional quantization as mea-sured from the edge of the conduction band in the vertically upward direction inthe absence of any quantization; T the temperature; H1 D �1=kBT ; and �1 is thebroadening parameter in this case.

The TPSM (G0/ can, in general, be expressed as [3]

G0 D 1

e

@S0

@n0

!EF ;T

; (1.10)

where EF is the Fermi energy corresponding to the electron concentration n0 andS0 is the entropy per unit volume which can be written as

S0 D � @

@T

ˇˇEDEF

(1.11)

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


D �kBTX

ln

ˇˇ1C exp

�EF � Eı0

kBT

�ˇˇ; (1.12)

where the summation is carried out over all the possible ı0 states.Thus, combining (1.10)–(1.12), the magnitude of the TPSM can be written in a

simplified form as [10]

G0 D��2k2

BTı3en0

� � @n0

@EF

(1.13)

It should be noted that being a thermodynamic relation and temperature-inducedphenomena, the TPSM as expressed by (1.13), in general, is valid for electronicmaterials having arbitrary dispersion relations and their nanostructures. In additionto bulk materials in the presence of strong magnetic field, (1.13) is valid under one-,two-, and three-dimensional quantum confinement of the charge carriers (such asquantum wells in ultrathin films, nipi structures, inversion and accumulation lay-ers, quantum well superlattices, carbon nanotubes, quantum wires, quantum wiresuperlattices, quantum dots, magnetoinversion and accumulation layers, quantumdot superlattices, magneto nipis, quantum well superlattices under magnetic quan-tization, ultrathin films under magnetic quantization, etc.). The formulation of G0

requires the relation between the electron statistics and the corresponding Fermienergy which is basically the band-structure-dependent quantity and changes underdifferent physical conditions. It is worth remarking to note that the number

��2ı3�

has occurred as a consequence of mathematical analysis and is not connected withthe well-known Lorenz number. For quantum wells in ultrathin films, nipi structures,inversion and accumulation layers, quantum well superlattices, magnetoinversionand accumulation layers, magneto nipis, quantum well superlattices under magneticquantization and magnetosize quantization, the carrier concentration is measuredper unit area, whereas, for quantum wires, quantum wires under magnetic field,quantum wire superlattices, and such allied systems, the same can be measured perunit length. Besides, for bulk materials under strong magnetic field, quantum dots,quantum dots under magnetic field, quantum dot superlattices, and quantum dotsuperlattices under magnetic field, the carrier concentration is expressed per unitvolume.

The TPSM in this case using (1.7) and (1.13) can be written as

G0 D �2kB

3e

24nxmaxX

nxD1

nymaxXnyD1

nzmaxXnzD1

L11

M11

35�124nxmaxX

nxD1

nymaxXnyD1

nzmaxXnzD1

Q11

.M11/2

35 ; (1.14)

whereQ11 D A1

�1C A2

1

�cos H1 C 2A1

�: (1.15)


1.2.2 Magnetothermopower in Quantum Dots of III–V Materials

The dispersion relation of the conduction electrons of III–V compounds aredescribed by the models of Kane (both three and two bands) [224, 225], Stillmanet al. [226], Newson and Kurobe [227], Rossler [228], Palik et al. [229], Johnson andDickey [230], and Agafonov et al. [231], respectively. For the purpose of completeand coherent presentation, the TPSM in QDs of III–V compounds has also beeninvestigated in accordance with the aforementioned different dispersion relationsfor the purpose of relative comparison as follows.

1.2.2.1 The Three Band Model of Kane

Under the conditions, ı D 0; �k D �? D � (isotropic spin–orbit splitting con-stant), and m�k D m�? D m� (isotropic effective electron mass at the edge of theconduction band), (1.2) gets simplified into the form

„2k2

2m�D I.E/; I.E/ � E

�E C Eg0

� �E C Eg0

C�� Eg0C 2

3��

Eg0

�Eg0

C�� E C Eg0C 2

3�� (1.16)

which is known as the three band model of Kane [224, 225] and is often used tostudy the electronic properties of III–V materials.

The totally quantized energy�EQD2

�in this case assumes the form

I�EQD2

� D „2�2

2m�

"�nx

dx

2

C�ny

dy

2

C�nz

dz

2#: (1.17)

The electron concentration is given by

n0 D 2gv

dxdydz

nxmaxXnxD1

nymaxXnyD1

nzmaxXnzD1

L12

M12

; (1.18)

where L12 D Œ1C A2 cos H2 , A2 D expŒEQD2 � EFQD=kBT , H2 D �2=kBT ,�2 is the broadening parameter in this case, and M12 D 1C A2

2 C 2A2 cos H2.The TPSM in this case, using (1.13) and (1.18), can be expressed as

G0 D �2kB

3e

24nxmaxX

nxD1

nymaxXnyD1

nzmaxXnzD1

L12

M12

35�124nxmaxX

nxD1

nymaxXnyD1

nzmaxXnzD1

Q12

.M12/2

35; (1.19)

where Q12 D A2

�1C A2

2

�cos H2 C 2A2

�.


1.2.2.2 The Two Band Model of Kane

Under the inequalities�� Eg0or �� Eg0

, (1.16) assumes the form

E.1C ˛E/ D �„2k2ı2m�

�; ˛ � 1

ıEg0

: (1.20)

Equation (1.20) is known as the two-band model of Kane and should be as such forstudying the electronic properties of the materials whose band structures obey theabove inequalities [224, 225].

The totally quantized energy EQD3 in this case is given by

EQD3

�1C ˛EQD3

� D „2�2

2m�

"�nx

dx

2

C�ny

dy

2

C�nz

dz

2#: (1.21)

The electron concentration can be written as

n0 D 2gv

dxdydz

nxmaxXnxD1

nymaxXnyD1

nzmaxXnzD1

L13

M13

; (1.22)



G0 D �2kB

3e

24nxmaxX

nxD1

nymaxXnyD1

nzmaxXnzD1

L13

M13

35�124nxmaxX

nxD1

nymaxXnyD1

nzmaxXnzD1

Q13

.M13/2

35 ; (1.23)

where Q13 D A3

�1C A2

3

�cos H3 C 2A3

�.

1.2.2.3 The Model of Stillman et al.

In accordance with the model of Stillman et al. [226], the electron dispersion law ofIII–V materials assumes the form

E D t11k2 � t12k

4 (1.24)

where t11 � „2=2m� and

t12 ��1 � m

�

m0

2 � „2

2m�

2

��3Eg0

C 4�C 2�2

Eg0

:˚�Eg0

C�� 2�C 3Eg0

��1�


Equation (1.24) can be expressed as

„2k2

2m�D I11 .E/ ; (1.25)

where

I11.E/ � a11

h1 � .1 � a12E/

1=2i;

a11 �� „2t11

4m�t12

; a12 � 4t12

t211

:

The EQD4 in this case can be defined as

I11

�EQD4

� D „2�2

2m�

"�nx

dx

2

C�ny

dy

2

C�nz

dz

2#: (1.26)


n0 D 2gv

dxdydz

nxmaxXnxD1

nymaxXnyD1

nzmaxXnzD1

L14

M14

; (1.27)



G0 D �2kB

3e

24nxmaxX

nxD1

nymaxXnyD1

nzmaxXnzD1

L14

M14

35�124nxmaxX

nxD1

nymaxXnyD1

nzmaxXnzD1

Q14

.M14/2

35 ; (1.28)

where Q14 D A4

�1C A2

4

�cos H4 C 2A4

�.

1.2.2.4 The Model of Newson and Kurobe

In accordance with the model of Newson and Kurobe, the electron dispersion law inthis case assumes the form as [227]

E D a13k4z C

� „2

2m�C a14k

2s

�k2

z C„2

2m�k2

s C a14k2xk

2y C a13

�k4

x C k4y

�;

(1.29)

where a13 is the nonparabolicity constant, a14 .� 2a13C a15/ and a15 is known asthe warping parameter.


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

EQD5 D a13

��nz

dz

4

C"„2

2m�C a14

��nx

dx

2

C��ny

dy

2!#

:

��nz

dz

2

C „2

2m�

"��nx

dx

2

C��ny

dy

2#C a14�

4

�nxny

dxdy

2

C a13�4

"�nx

dx

4

C�ny

dy

4#: (1.30)


n0 D 2gv

dxdydz

nxmaxXnxD1

nymaxXnyD1

nzmaxXnzD1

L15

M15

; (1.31)

where L15D Œ1C A5 cos H5 , A5D expŒEQD5 �EFQD=kBT , H5D�5=kBT , �5

is the broadening parameter in this case, and M15D 1C A25 C 2A5 cos H5.

The TPSM in this case, using (1.13) and (1.31), can be expressed as

G0 D �2kB

3e

24nxmaxX

nxD1

nymaxXnyD1

nzmaxXnzD1

L15

M15

35�124nxmaxX

nxD1

nymaxXnyD1

nzmaxXnzD1

Q15

.M15/2

35 ; (1.32)

where Q15 D A5

�1C A2

5

�cos H5 C 2A5

�.

1.2.2.5 The Model of Rossler

The dispersion relation of the conduction electrons in accordance with the model ofRossler can be written as [228]

E D „2k2

2m�C Œ˛11 C ˛12k k

4 C .ˇ11 C ˇ12k/k2

xk2y C k2

yk2z C k2

z k2x

�˙ Œ�11 C �12k

k2�k2

xk2y C k2

yk2z Ck2

z k2x

� � 9k2xk

2yk

2z

�1=2; (1.33)

where ˛11, ˛12, ˇ11, ˇ12, �11, and �12 are energy-band constants.


EQD6;˙ in this case assumes the form

EQD6;˙ � „2�2

2m�

"�nx

dx

2

C�ny

dy

2

C�nz

dz

2#

C24˛11 C ˛12

"��nx

dx

2

C��ny

dy

2

C��nz

dz

2#1=2

35

�"�

�nx

dx

2

C��ny

dy

2

C��nz

dz

2#2

C24ˇ11 C ˇ12

"��nx

dx

2

C��ny

dy

2

C��nz

dz

2#1=2

35

�"�4

�nxny

dxdy

2

C �4

�nynz

dydz

2

C �4

�nznx

dzdx

2#

˙24�11 C �12

"��nx

dx

2

C��ny

dy

2

C��nz

dz

2#1=2

35

�""�

�nx

dx

2

C��ny

dy

2

C��nz

dz

2#

�"�4

�nxny

dxdy

2

C �4

�nynz

dydz

2

C �4

�nznx

dzdx

2#

� 9�6�nxnynz=dxdydz

�6 #(1.34)


n0 D gv

dxdydz

nxmaxXnxD1

nymaxXnyD1

nzmaxXnzD1

L16;˙M16;˙

; (1.35)

where L16;˙ D1C A6;˙ cos H6

�, A6;˙ D expŒEQD6;˙ � EFQD=kBT , H6D

�6=kBT , �6 is the broadening parameter in this case, and M16;˙ D 1 C A26;˙ C

2A6;˙ cos H6.The TPSM in this case, using (1.13) and (1.35), can be expressed as

G0 D �2kB

3e

24nxmaxX

nxD1

nymaxXnyD1

nzmaxXnzD1

L16;˙M16;˙

35�124nxmaxX

nxD1

nymaxXnyD1

nzmaxXnzD1

Q16;˙�M16;˙

�235 ; (1.36)

where Q16;˙ D A6;˙

h�1C A2

6;˙

�cos H6 C 2A6;˙

i.

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