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Preface

In recent years, with the advent of fine line lithographical methods, molecularbeam epitaxy, organometallic vapour phase epitaxy and other experimentaltechniques, low dimensional structures having quantum confinement in one,two and three dimensions (such as inversion layers, ultrathin films, nipi’s,quantum well superlattices, quantum wires, quantum wire superlattices, andquantum dots together with quantum confined structures aided by variousother fields) have attracted much attention, not only for their potential inuncovering new phenomena in nanoscience, but also for their interestingapplications in the realm of quantum effect devices. In ultrathin films, dueto the reduction of symmetry in the wave–vector space, the motion of thecarriers in the direction normal to the film becomes quantized leading to thequantum size effect. Such systems find extensive applications in quantumwell lasers, field effect transistors, high speed digital networks and also inother low dimensional systems. In quantum wires, the carriers are quantizedin two transverse directions and only one-dimensional motion of the carriersis allowed. The transport properties of charge carriers in quantum wires,which may be studied by utilizing the similarities with optical and microwavewaveguides, are currently being investigated. Knowledge regarding thesequantized structures may be gained from original research contributions inscientific journals, proceedings of international conferences and various re-view articles. It may be noted that the available books on semiconductorscience and technology cannot cover even an entire chapter, excluding a fewpages on the Einstein relation for the diffusivity to mobility ratio of thecarriers in semiconductors (DMR). The DMR is more accurate than any oneof the individual relations for the diffusivity (D) or the mobility (µ) of thecharge carriers, which are two widely used quantities of carrier transport insemiconductors and their nanostructures.

It is worth remarking that the performance of the electron devices at thedevice terminals and the speed of operation of modern switching transistorsare significantly influenced by the degree of carrier degeneracy present in thesedevices. The simplest way of analyzing such devices, taking into account the

VI Preface

degeneracy of the bands, is to use the appropriate Einstein relation to expressthe performances at the device terminals and the switching speed in terms ofcarrier concentration (S.N. Mohammad, J. Phys. C , 13, 2685 (1980)). It iswell known from the fundamental works of Landsberg (P.T. Landsberg, Proc.R. Soc. A, 213, 226, (1952); Eur. J. Phys, 2, 213, (1981)) that the Einsteinrelation for degenerate materials is essentially determined by their energyband structures. It has, therefore, different values in different materials havingvarious band structures and varies with electron concentration, the magnitudeof the reciprocal quantizing magnetic field, the quantizing electric field asin inversion layers, ultrathin films, quantum wires and with the superlatticeperiod as in quantum confined semiconductor superlattices having variouscarrier energy spectra.

This book is partially based on our on-going researches on the Einsteinrelation from 1980 and an attempt has been made to present a cross section ofthe Einstein relation for a wide range of materials with varying carrier energyspectra, under various physical conditions.

In Chap. 1, after a brief introduction, the basic formulation of the Ein-stein relation for multiband semiconductors and suggestion of an experimentalmethod for determining the Einstein relation in degenerate materials havingarbitrary dispersion laws are presented. From this suggestion, one can also ex-perimentally determine another two seemingly different but important quan-tities of quantum effect devices namely, the Debye screening length and thecarrier contribution to the elastic constants. In Chap. 2, the Einstein relationin bulk specimens of tetragonal materials (taking n-Cd3As2 and n-CdGeAs2as examples) is formulated on the basis of a generalized electron dispersionlaw introducing the anisotropies of the effective electron masses and the spinorbit splitting constants respectively together with the inclusion of the crys-tal field splitting within the framework of the k.p formalism. The theoreticalformulation is in good agreement with the suggested experimental methodof determining the Einstein relation in degenerate materials having arbitrarydispersion laws. The results of III–V (e.g. InAs, InSb, GaAs, etc.), ternary(e.g. Hg1−xCdxTe), quaternary (e.g. In1−xGaxAs1−yPy lattice matched toInP) compounds form a special case of our generalized analysis under certainlimiting conditions. The Einstein relation in II–VI, IV–VI, stressed Kane typesemiconductors together with bismuth are also investigated by using the ap-propriate energy band structures for these materials. The importance of thesematerials in the emergent fields of opto- and nanoelectronics is also describedin Chap. 2.

The effects of quantizing magnetic fields on the band structures of com-pound semiconductors are more striking than those of the parabolic one andare easily observed in experiments. A number of interesting physical featuresoriginate from the significant changes in the basic energy wave vector rela-tion of the carriers caused by the magnetic field. Valuable information couldalso be obtained from experiments under magnetic quantization regardingthe important physical properties such as Fermi energy and effective masses

Preface VII

of the carriers, which affect almost all the transport properties of the electrondevices. Besides, the influence of cross-field configuration is of fundamentalimportance to an understanding of the various physical properties of variousmaterials having different carrier dispersion relations. In Chap. 3, we study theEinstein relation in compound semiconductors under magnetic quantization.Chapter 4 covers the influence of crossed electric and quantizing magneticfields on the Einstein relation in compound semiconductors. Chapter 5 coversthe study of the Einstein relation in ultrathin films of the materials mentioned.

Since Iijima’s discovery (S. Iijima, Nature 354, 56 (1991)), carbon nan-otubes (CNTs) have been recognized as fascinating materials with nanometerdimensions, uncovering new phenomena in different areas of nanoscience andtechnology. The remarkable physical properties of these quantum materialsmake them ideal candidates to reveal new phenomena in nanoelectronics.Chapter 6 contains the study of the Einstein relation in quantum wires ofcompound semiconductors, together with carbon nanotubes.

In recent years, there has been considerable interest in the study of theinversion layers which are formed at the surfaces of semiconductors in metal–oxide–semiconductor field-effect transistors (MOSFET) under the influenceof a sufficiently strong electric field applied perpendicular to the surface bymeans of a large gate bias. In such layers, the carriers form a two dimensionalgas and are free to move parallel to the surface while their motion is quantizedin the perpendicular to it leading to the formation of electric subbands. InChap. 7, the Einstein relation in inversion layers on compound semiconductorshas been investigated.

The semiconductor superlattices find wide applications in many impor-tant device structures such as avalanche photodiode, photodetectors, electro-optic modulators, etc. Chapter 8 covers the study of the Einstein relation innipi structures. In Chap. 9, the Einstein relation has been investigated undermagnetic quantization in III-V, II-VI, IV-VI, HgTe/CdTe superlattices withgraded interfaces. In the same chapter, the Einstein relation under magneticquantization for effective mass superlattices has also been investigated. It alsocovers the study of quantum wire superlattices of the materials mentioned.Chapter 10 presents an initiation regarding the influence of light on the Ein-stein relation in optoelectronic materials and their quantized structures whichis itself in the stage of infancy.

In the whole field of semiconductor science and technology, the heavilydoped materials occupy a singular position. Very little is known regarding thedispersion relations of the carriers of heavily doped compound semiconductorsand their nanostructures. Chapter 11 attempts to touch this enormous field ofactive research with respect to Einstein relation for heavily doped materials ina nutshell, which is itself a sea. The book ends with Chap. 12, which containsthe conclusion and the scope for future research.

As there is no existing book devoted totally to the Einstein relation forcompound semiconductors and their nanostructures to the best of our knowl-edge, we hope that the present book will be a useful reference source for

VIII Preface

the present and the next generation of readers and researchers of solid stateelectronics in general. In spite of our joint efforts, the production of error freefirst edition of any book from every point of view enjoys the domain of im-possibility theorem. Various expressions and a few chapters of this book havebeen appearing for the first time in printed form. The positive suggestions ofthe readers for the development of the book will be highly appreciated.

In this book, from Chap. 2 to the end, we have presented 116 open and60 allied research problems in this beautiful topic, as we believe that a properidentification of an open research problem is one of the biggest problems inresearch. The problems presented here are an integral part of this book andwill be useful for readers to initiate their own contributions to the Einsteinrelation. This aspect is also important for PhD aspirants and researchers. Westrongly contemplate that the readers with a mathematical bent of mind wouldinvariably yearn for investigating all the systems from Chapters 2 to 12 andthe related research problems by removing all the mathematical approxima-tions and establishing the appropriate respective uniqueness conditions. Eachchapter except the last one ends with a table containing the main results.

It is well known that the studies in carrier transport of modern semicon-ductor devices are based on the Boltzmann transport equation which can, inturn, be solved if and only if the dispersion relations of the carriers of the dif-ferent materials are known. In this book, we have investigated various disper-sion relations of different quantized structures and the corresponding electronstatistics to study the Einstein relation. Thus, in this book, the alert readerswill find information regarding quantum-confined low-dimensional materialshaving different band structures. Although the name of the book is extremelyspecific, from the content one can infer that it will be useful in graduatecourses on semiconductor physics and devices in many Universities. Besides,as a collateral study, we have presented the detailed analysis of the effectiveelectron mass for the said systems, the importance of which is already wellknown, since the inception of semiconductor science. Last but not the least, wedo hope that our humble effort will kindle the desire of anyone engaged in ma-terials research and device development, either in academics or in industries,to delve deeper into this fascinating topic.

Acknowledgments

Acknowledgment by Kamakhya Prasad Ghatak

I am grateful to A.N. Chakravarti, my Ph.D thesis advisor, for introducingan engineering graduate to the classics of Landau Liftsitz 30 years ago, andwith whom I spent countless hours delving into the sea of semiconductorphysics. I am also indebted to D. Raychaudhuri for transforming a networktheorist into a quantum mechanic. I realize that three renowned books onsemiconductor science, in general, and more than 200 research papers of

Preface IX

B.R. Nag, still fire my imagination. I would like to thank P.T. Landsberg,D. Bimberg, W.L. Freeman, B. Podor, H.L. Hartnagel, V.S. Letokhov, H.L.Hwang, F.D. Boer, P.K. Bose, P.K. Basu, A. Saha, S. Roy, R. Maity,R. Bhowmik, S.K. Dasgupta, M. Mitra, D. Chattopadhyay, S.N. Biswas andS.K. Biswas for several important interactions.

I am particularly indebted to K. Mukherjee, A.K. Roy, S.S. Baral, S.K.Roy, R.K. Poddar, N. Guhochoudhury, S.K. Sen, S. Pahari and D.K. Basu,who acted as mentors in the difficult moments of my academic career. I thankmy department colleagues and the members of my research team for their help.P.K. Sarkar of the semiconductor device laboratory has always helped me. I amgrateful to S. Sanyal for her help and academic advice. I also acknowledge thepresent Head of the Department, S.N. Sarkar, for creating an environment forthe advancement of learning, which is the logo of the University of Calcutta,and helping me to win an award in research and development from the AllIndia Council for Technical Education, India, under which the writing of manychapters of this book became a reality. Besides, this book has been completedunder the grant (8023/BOR/RID/RPS-95/2007-08) as sanctioned by the saidCouncil in their research promotion scheme 2008 of the Council.

Acknowledgment by Sitangshu Bhattacharya

I am indebted to H.S. Jamadagni and S. Mahapatra at the Centre for Electron-ics Design and Technology (CEDT), Indian Institute of Science, Bangalore,for their constructive guidance in spite of a tremendous research load and tomy colleagues at CEDT, for their constant academic help. I am also gratefulto my sister, Ms. S. Bhattacharya and my friend Ms. A. Chakraborty for theirconstant inspiration and encouragement for performing research work even inmy tough times, which, in turn, forms the foundation of this twelve-storiedbook project. I am grateful to my teacher K.P. Ghatak, with whom I workconstantly to understand the mysteries of quantum effect devices.

Acknowledgment by Debashis De

I am grateful to K.P. Ghatak, B.R. Nag, A.K. Sen, P.K. Roy, A.R. Thakur,S. Sengupta, A.K. Roy, D. Bhattacharya, J.D. Sharma, P. Chakraborty,D. Lockwood, N. Kolbun and A.N. Greene. I am highly indebted to my brotherS. De for his constant inspiration and support. I must not allow a special thankyou to my better half Mrs. S. De, since in accordance with Sanatan HinduDharma, the fusion of marriage has transformed us to form a single entity,where the individuality is being lost. I am grateful to the All India Council ofTechnical Education, for granting me the said project jointly in their researchpromotion scheme 2008 under which this book has been completed.

X Preface

Joint Acknowledgments

The accuracy of the presentation owes a lot to the cheerful profes-sionalism of Dr. C. Ascheron, Senior Editor, Physics Springer Verlag,Ms. A. Duhm, Associate Editor Physics, Springer and Mrs. E. Suer, assistantto Dr. Ascheron. Any shortcomings that remain are our own responsibility.

Kolkata, India K.P. GHATAKJune 2008 S. BHATTACHARYA

D. DE

2

The Einstein Relation in Bulk Specimensof Compound Semiconductors

2.1 Investigation on Tetragonal Materials

2.1.1 Introduction

It is well known that the A2IIIB

5II and the ternary chalcopyrite compounds

are called tetragonal materials due to their tetragonal crystal structures [1].These materials find extensive use in non-linear optical elements [2], photo-detectors [3] and light emitting diodes [4]. Rowe and Shay [5] showed thatthe quasi-cubic model [6] can be used to explain the observed splitting andsymmetry properties of the band structure at the zone center of k space of theaforementioned materials. The s-like conduction band is singly degenerate andthe p-like valence bands are triply degenerate. The latter splits into three sub-bands because of the spin–orbit and the crystal field interactions. The largestcontribution to the crystal field splitting is from the non-cubic potential [7].The experimental results on the absorption constants, the effective mass, andthe optical third order susceptibility indicate that the fact that the conduc-tion band in the same materials corresponds to a single ellipsoidal revolutionat the zone center in k-space [1, 8]. Introducing the crystal potential in theHamiltonian, Bodnar [9] derived the electron dispersion relation in the samematerial under the assumption of an isotropic spin–orbit splitting constant.It would, therefore, be of much interest to investigate the DMR in these com-pounds by including the anisotropies of the spin–orbit splitting constant and,the effective electron mass together with the inclusion of crystal field splitting,within the framework of k.p formalism since, these are the important physicalfeatures of such materials [1].

In what follows, in Sect. 2.1.2 on the theoretical background the expressionsfor the electron concentration and the DMR for tetragonal compounds havebeen derived on the basis of the generalized dispersion relation. In Sect. 2.1.3,it has been shown that the corresponding results for III–V, ternary and qua-ternary materials form special cases of our generalized analysis. The expres-sions for n0 and DMR for semiconductors whose energy band structures are

14 2 The Einstein Relation in Bulk Specimens of Compound Semiconductors

defined by the two-band model of Kane and that of parabolic energy bandshave further been formulated under certain constraints. For the purpose ofnumerical computations, n-Cd3As2 and n-CdGeAs2 have been used as exam-ples of A2

IIIB5II and the ternary chalcopyrite compounds, which are being

extensively used in Hall pick-ups, thermal detectors, and non-linear optics [3].In addition, the DMR has also been numerically investigated by taking n-InAs and n-InSb as examples of III–V semiconductors, n-Hg1−xCdxTe as anexample of ternary compounds and n-In1−xGaxAsyP1−y lattice matched toInP as example of quaternary materials in accordance with the three and thetwo band models of Kane together with parabolic energy bands, respectively,for the purpose of relative comparison. The importance of the aforementionedmaterials in electronics has been discussed in Sect. 2.1.3. Section 2.1.4 containsthe results and discussions.

2.1.2 Theoretical Background

The form of k.p matrix for tetragonal semiconductors can be expressed, ex-tending Bodnar’s [9] relation, as

H =[

H1 H2

H2+ H1

], (2.1)

where H1 ≡

⎡

⎢⎢⎣

Eg 0 P‖kz 00(−2∆||/3

) (√2∆⊥/3

)0

P‖kz

(√2∆⊥/3

)−(δ + 1

3∆‖)

00 0 0 0

⎤

⎥⎥⎦ and H2 ≡

⎡

⎢⎢⎣

0 −f,+ 0 f,−f,+ 0 0 00 0 0 0

f,+ 0 0 0

⎤

⎥⎥⎦ ,

in which Eg is the band gap, P|| and P⊥ are the momentum matrix elementsparallel and perpendicular to the direction of crystal axis respectively, δ isthe crystal field splitting constant, ∆|| and ∆⊥ are the spin–orbit split-ting constants parallel and perpendicular to the C-axis respectively, f,± ≡(P⊥/

√2)(kx ± iky) and i =

√−1. Thus, neglecting the contribution of the

higher bands and the free electron term, the diagonalization of the abovematrix leads to the dispersion relation of the conduction electrons in bulkspecimens of tetragonal compounds [1] as

ψ1 (E) = ψ2 (E) k2s + ψ3 (E) k2

z , (2.2)

where

ψ1(E) ≡ E(E + Eg)[

(E + Eg)(E + Eg + ∆||

)+ δ

(E + Eg +

23∆||

)

+29

(∆2

|| − ∆2⊥

) ], ks

2 = kx2 + ky

2,

2.1 Investigation on Tetragonal Materials 15

ψ2(E) ≡ �2Eg (Eg + ∆⊥)

[2m∗

⊥(Eg + 2

3∆⊥)][

δ

(E + Eg +

13∆||

)+ (E + Eg)

×(

E + Eg +23∆||

)+

19

(∆2

|| − ∆2||

) ],

ψ3 (E) ≡ �2Eg(Eg+∆||)

[2m∗||(Eg+ 2

3∆||)][(E + Eg)

(E + Eg + 2

3∆||)]

,m∗|| and m∗

⊥ are the

longitudinal and transverse effective electron masses at the edge of the con-duction band respectively.

The general expression of the density-of-states (DOS) function in bulksemiconductors is given by

D0(E) =2gv

(2π)3

(∂

∂E[V (E)]

), (2.3a)

where gv is the valley degeneracy and V (E) is the volume of k space. Using(2.2) and (2.3a), we get

D0 (E) = gv

(3π2)−1

ψ4(E), (2.3b)

ψ4 (E) ≡[

32

√ψ1 (E) [ψ1 (E)]′

ψ2 (E)√

ψ3 (E)− [ψ2 (E)]′ [ψ1 (E)]3/2

[ψ2 (E)]2√

ψ3 (E)

−12

[ψ3 (E)]′ [ψ1 (E)]3/2

ψ2 (E) [ψ3 (E)]3/2

],

[ψ1 (E)]′ ≡ [ (2E + Eg) ψ1 (E) [E (E + Eg)]−1 + E(E + Eg)

×(2E + 2Eg + δ + ∆||

)] ,

[ψ2 (E)]′ ≡[2m∗

⊥

(Eg +

23∆⊥

)]−1 [�

2Eg (Eg + ∆⊥)]

×[δ + 2E + 2Eg +

23∆||

],

and [ψ3 (E)]′ ≡[2m∗

||(Eg + 2

3∆||)]−1 [

�2Eg

(Eg + ∆||

)] [2E + 2Eg + 2

3∆||],

in which, the primes denote the differentiation of the differentiable functionswith respect to E.

Combining (2.3b) with the Fermi–Dirac occupation probability factor andusing the generalized Sommerfeld’s lemma [10], the electron concentration canbe written as

n0 = gv

(3π2)−1

[M (EF) + N (EF)] , (2.4)

where M(EF) ≡[

[ψ1(EF)]32

ψ2(EF)√

ψ3(EF)

], EF is the Fermi energy as measured from

the edge of the conduction band in the vertically upward direction in the


absence of any quantization, N(EF) ≡s∑

r=1L(r)M(EF), r is the set of real

positive integers whose upper limit is s, L(r) ≡[2 (kBT )2r (1 − 21−2r

)ξ (2r)

]

×[

∂2r

∂E2rF

]and ζ(2r) is the Zeta function of order 2r [11].

Thus the use of the (2.4) and (1.11) leads to the expression of DMR as

D

µ=

1|e|

[M (EF) + N (EF)][{M (EF)}′ + {N (EF)}′

] . (2.5)

2.1.3 Special Cases for III–V Semiconductors

(a) Under the substitutions δ = 0,∆|| = ∆⊥ = ∆(the isotropic spin–orbitsplitting constant) and m∗

|| = m∗⊥ = m∗(the isotropic effective electron

mass at the edge of the conduction band), (2.2) assumes the form [1]

�2k2

2m∗ = γ(E), γ(E) ≡E(E + Eg)(E + Eg + ∆)

(Eg + 2

3∆)

Eg(Eg + ∆)(E + Eg + 2

3∆) , (2.6)

which is the well-known three band model of Kane [1]. Equation (2.6) is thedispersion relation of the conduction electrons of III–V, ternary and quater-nary materials and should be used as such for studying the electron transportin n-InAs where the spin orbit splitting constant is of the order of band gap.The III–V compounds are used in integrated optoelectronics [12, 13], pas-sive filter devices [14], distributed feedback lasers and Bragg reflectors [15].Besides, we shall also use n-Hg1−xCdxTe and n-In1−xGaxAsyP1−y latticematched to InP as examples of ternary and quaternary materials respectively.The ternary alloy n-Hg1−xCdxTe is a classic narrow-gap compound and istechnologically an important optoelectronic semiconductor because its bandgap can be varied to cover a spectral range from 0.8 to over 30 µm by adjustingthe alloy composition [16]. The n-Hg1−xCdxTe finds applications in infrareddetector materials [17] and photovoltaic detector [18] arrays in the 8-12 µmwave bands. The above applications have spurred an Hg1−xCdxTe technologyfor the production of high mobility single crystals, with specially prepared sur-face layers and the same material is suitable for narrow subband physics be-cause the relevant material constants are within experimental reach [19]. Thequaternary compounds are being extensively used in optoelectronics, infraredlight emitting diodes, high electron mobility transistors, visible heterostruc-ture lasers for fiber optic systems, semiconductor lasers, [20], tandem solarcells [21], avalanche photodetectors [22], long wavelength light sources, detec-tors in optical fiber communications, [23] and new types of optical devices,which are being prepared from the quaternary systems [24].

Under the aforementioned limiting conditions, the density-of-states func-tion, the electron concentration, and the DMR in accordance with the threeband model of Kane assume the following forms


D0 (E) = 4πgv

(2m∗

h2

)3/2√γ (E) [γ1 (E)] , (2.7)

n0 =gv

3π2

(2m∗

�2

)3/2

[M1 (EF) + N1 (EF)] , (2.8)

and

D

µ=

1|e| [M1 (EF) + N1 (EF)]

[{M1 (EF)}′ + {N1 (EF)}′

]−1, (2.9)

where γ1 (E) ≡ γ (E)[

1E + 1

E+Eg+ 1

E+Eg+∆ − 1E+Eg+ 2

3∆

], M1 (EF) ≡

[γ (EF)]3/2 and N1 (EF) ≡s∑

r=1L (r) M1 (EF).

(b) Under the inequalities ∆ � Eg or ∆ Eg, (2.6) gets simplified as [1]

�2k2

2m∗ = E (1 + αE) , α ≡ 1/Eg, (2.10)

which is known as the two-band model of Kane [1]. Under the above con-straints, the forms of the DOS, the electron statistics and the DMR can,respectively, be written as,

D0 (E) = 4πgv

(2m∗

h2

)3/2√I (E) [I1 (E)] , (2.11)

n0 =gv

3π2

(2m∗

�2

)3/2

[M2 (EF) + N2 (EF)] , (2.12)

and

D

µ=

1|e| [M2 (EF) + N2 (EF)]

[{M2 (EF)}′ + {N2 (EF)}′

]−1, (2.13)

where I (E) ≡ E (1 + αE), I1 (E) ≡ (1 + 2αE), M2 (EF) ≡ [I (EF)]3/2 and

N2 (EF) ≡s∑

r=1L (r) M2 (EF).

(c) Under the constraints ∆ � Eg or ∆ Eg together with the inequalityαEF 1, we can write [1]

n0 = gvNc

[F1/2(η) +

(15αkBT

4

)F3/2(η)

], (2.14)

andD

µ=[kBT

|e|

] [ (F1/2(η) +

(15αkBT

4

)F3/2(η)

)

(F−1/2(η) +

(15αkBT

4

)F1/2(η)

)

]

, (2.15)


where NC ≡ 2(

2πm∗kBTh2

)3/2η ≡ EF

kBT and Fj(η) is the one parameter Fermi–Dirac integral of order j which can be written as [25],

Fj(η) =(

1Γ (j + 1)

) ∞∫

0

yj (1 + exp (y − η))−1 dy, j > −1, (2.16)

where Γ (j + 1) is the complete Gamma function or for all j, analyticallycontinued as a complex contour integral around the negative axis

Fj(η) = Aj

(0+)∫

−∞

yj (1 + exp (−y − η))−1 dy, (2.17)

in which Aj ≡ Γ(−j)

2π√−1

.

(d) For relatively wide gap materials Eg → ∞ and (2.14) and (2.15) assumethe forms

n0 = gvNcF1/2(η) (2.18)

andD

µ=(

kBT

|e|

)[F1/2(η)F−1/2(η)

]. (2.19)

Equation (2.19) was derived for the first time by Landsberg [1].

(e) Combining (2.18) and (2.19) and using the formula ddη [Fj(η)] = Fj−1(η)

[25] as easily derived from (2.16) and (2.17) together with the fact thatunder the condition of extreme carrier degeneracy

F1/2(η) =[

43√

π

](η)

3/2 , (2.20)

we can write

n0 =gv

3π2

[2m∗EF(1 + αEF)

�2

]3/2, (2.21)

andD

µ=

1|e|

(23

)EF

(1 + αEF)(1 + 2αEF)

, (2.22)

For α → 0, (2.21) and (2.22) assume the forms

n0 =gv

3π2

[2m∗EF

�2

]3/2, (2.23)

andD

µ=

2EF

3 |e| . (2.24)


(f) Under the condition of non-degenerate electron concentration η 0 and Fj(η) ∼= exp(η) for all j [25]. Therefore (2.18) and (2.19) assumethe well-known forms as [1]

n0 = gvNc exp(η), (2.25)

andD

µ=

kBT

|e| . (2.26)

2.1.4 Result and Discussions

Using n-Cd3As2 as an example of A2IIIB

5II compounds for the purpose of

numerical computations and using (2.4) and (2.5) together with the energyband constants at T = 4.2K, as given in Table 2.1, the variation of theDMR as a function of electron concentration has been shown in curve (a)of Fig. 2.1. The circular points exhibit the same dependence and have beenobtained by using (1.15) and taking the experimental values of the thermo-electric power in n−Cd3As2 in the presence of a classically large magneticfield [26]. The curve (b) corresponds to δ = 0. The curve (c) shows the depen-dence of the DMR on n0 in accordance with the three-band model of Kaneusing the energy band constants as Eg = 0.095 eV, m∗ =

(m∗

|| + m∗⊥

)/ 2

and ∆ =(∆|| + ∆⊥

)/ 2. The curves (d) and (e) correspond to the two-band

model of Kane and that of the parabolic energy bands. By comparing thecurves (a) and (b) of Fig. 2.1, one can easily assess the influence of crystalfield splitting on the DMR in tetragonal compounds. Figure 2.2 represents allcases of Fig. 2.1 for n-CdGeAs2 which has been used as an example of ternarychalcopyrite materials where the values of the energy band constants of thesaid compound are given in Table 2.1.

It appears from Fig. 2.1 that, the DMR in tetragonal compounds increaseswith increasing carrier degeneracy as expected for degenerate semiconductorsand agrees well with the suggested experimental method of determining thesame ratio for materials having arbitrary carrier energy spectra. It has beenobserved that the tetragonal crystal field affects the DMR of the electronsquite significantly in this case. The dependence of the DMR is directly deter-mined by the band structure because of its immediate connection with theFermi energy. The DMR increases non-linearly with the electron concentra-tion in other limiting cases and the rates of increase are different from that inthe generalized band model.

From Fig. 2.2, one can assess that the DMR in bulk specimens ofn-CdGeAs2 exhibits monotonic increasing dependence with increasing elec-tron concentration. The cases (b), (c) and (d) of Fig. 2.2 for n-CdGeAs2exhibit the similar trends with change in the respective numerical values ofthe DMR. The influence of spectrum constants on the DMR for n-Cd3As2and n-CdGeAs2 can also be assessed by comparing the respective variationsas drawn in Figs. 2.1 and 2.2 respectively.


Table 2.1. The numerical values of the energy band constants of few materials

Materials

n − Cd3As2

|Eg| = 0.095 eV, ∆|| = 0.27 eV, ∆⊥ = 0.25 eV,m∗

|| = 0.00697m0 (m0 is the free electron mass),m∗

⊥ = 0.013933m0, δ = 0.085eV, gv =1 [25,73] andεsc = 16ε0 (εsc and ε0 are the permittivity of thesemiconductor and free space respectively) [74]

n − CdGeAs2 Eg = 0.57 eV, ∆‖ = 0.30 eV, ∆⊥ = 0.36 eV,m∗

‖ = 0.034mo, m∗⊥ = 0.039mo, T = 4 K,

δ = −0.21 eV, gv = 1 [1,26] and εsc = 18.4ε0 [75]

n-InAs Eg = 0.36 eV, ∆ = 0.43 eV and m∗ = 0.026m0, gv = 1,εsc = 12.25ε0 [76]

n-InSb Eg = 0.2352 eV, ∆ = 0.81 eV and m∗ = 0.01359m0,gv = 1, εsc = 15.56ε0 [76]

n − Ga1−xAlxAs Eg (x) = (1.424 + 1.266x + 0.26x2)eV,∆ (x) = (0.34 − 0.5x)eV, m∗ (x) = [0.066 + 0.088x] m0

gv = 1, εsc (x) = [13.18 − 3.12x] ε0 [77]

Hg1−xCdxTe Eg (x) =(−0.302 + 1.93x + 5.35 × 10−4(1 − 2x)T

−0.810x2 + 0.832x3 ) eV,∆ (x) =

(0.63 + 0.24x − 0.27x2

)eV,

m∗ = 0.1m0Eg(eV)−1, gv = 1 andεsc =

[20.262 − 14.812x + 5.22795x2

]ε0 [78]

In1−xGaxAsyP1−y

lattice matched to InPEg =

(1.337 − 0.73y + 0.13y2

)eV,

∆ =(0.114 + 0.26y − 0.22y2

)eV,

m∗ = (0.08 − 0.039y) mo,y = (0.1896 − 0.4052x)(0.1896 − 0.0123x)−1, gv = 1 [79]and εsc = [10.65 + 0.1320y] ε0 [80]

CdS m∗‖ = 0.7mo, m∗

⊥ = 1.5mo and λ0 = 1.4 × 10−10 eVm,gv = 1 [76] and εsc = 15.5ε0 [81]

n-PbTe m−t = 0.070m0, m−

l = 0.54m0, m+t = 0.010m0,

m+l = 1.4m0, P|| = 141meV nm, P⊥ = 486 meV nm,

Eg = 190meV, gv = 4 [12] and εsc = 33ε0 [76, 82]n-PbSnTe m−

t = 0.063m0, m−l = 0.41m0, m+

t = 0.089m0,m+

l = 1.6m0, P|| = 137meV nm, P⊥ = 464 meV nm,Eg = 90meV, gv = 4 [12] and εsc = 60ε0 [76, 82]

n-Pb1−xSnxSe x = 0.31, gv = 4, m−t = 0.143m0, m−

l = 2.0m0,m+

t = 0.167m0, m+l = 0.286m0, P|| = 3.2 × 10−10 eVm,

P⊥ = 4.1 × 10−10 eVm, Eg = 0.137eV, gv = 4 [12] andεsc = 31ε0 [76, 83]

Stressed n-InSb m∗ = 0.048mo, Eg = 0.081 eV, B2 = 9 × 10−10 eVm,Cc

1 = 3 eV, Cc2 = 2 eV, a0 = −10 eV, b0 = −1.7 eV,

d0 = −4.4 eV, Sxx = 0.6 × 10−3 (kbar)−1,Syy = 0.42 × 10−3 (kbar)−1, Szz = 0.39 × 10−3 (kbar)−1

and Sxy = 0.5 × 10−3 (kbar)−1, εxx = σSxx, εyy = σSyy,εzz = σSzz, εxy = σSxy and σ is the stress in kilobar,gv = 1 [44]

(Continued)


Table 2.1. Continued

PtSb2 For conduction bands, along <111> direction,λ1 = 0.33 eV, l1 = 1.09 eV, ν1 = 0.17 eV, n1 = 0.22 eV, a =0.643 nm, I0 = 0.30 (eV)2, δ′0 = 0.33 eV, gv = 8 [56] andεsc = 30ε0 [56, 84]

n-GaSb Eg = 0.81 eV, ∆ = 0.80 eV, P = 9.48 × 10−10 eVm, ς0 =−2.1, v0 = −1.49, ω0 = 0.42, gv = 1 [53] andεsc = 15.85ε0 [53, 85]

HgTe m∗v = 0.028m0, gv = 1 and ε∞ = 15.2ε0 [52]

Bismuth Eg = 0.0153 eV, m1 = 0.00194m0, m2 = 0.313m0,m3 = 0.00246m0, m′

2 = 0.36m0, gv = 3 and gs = 2 [42]

Fig. 2.1. The plot of the DMR in the bulk specimens of n-Cd3As2 as a functionof electron concentration in accordance with (a) the generalized band model; (b)δ = 0; (c) the three band model of Kane; (d) the two band model of Kane and (e)the parabolic energy bands. The dotted circular points show the same dependencewhich have been obtained by using (1.15) and taking the experimental values ofthe thermoelectric power of the electrons in bulk n-Cd3As2 in the presence of aclassically large magnetic field [26]

Using the appropriate equations, one can numerically evaluate the DMR asa function of electron concentration for n-InAs, whose energy band constantsare presented in Table 2.1. This is shown in Fig. 2.3 by curves (a), (b) and(c) respectively, in accordance with the three and two band models of Kanetogether with the model of parabolic energy bands. Figure 2.4 exhibits all thecases of Fig. 2.3 for n-InSb whose energy band constants are given in Table 2.1.


Fig. 2.2. The plot of the DMR in the bulk specimens of n-CdGeAs2 as a functionof electron concentration in accordance with (a) the generalized band model; (b)δ = 0; (c) the three band model of Kane; (d) the two band model of Kane and (e)the parabolic energy bands

From Figs. 2.3 and 2.4 one can observe that the numerical values of the DMRfor n-InAs and n-InSb in accordance with the three band model of Kane areless than that of the corresponding parabolic energy bands for relatively largevalues of the electron concentration. Also, the numerical values of the DMR inaccordance with two-band model of Kane also changes as compared with thecorresponding three-band model of Kane for n-InAs. The influence of energyband constants on the DMR for n-InAs and n-InSb becomes apparent bycomparing Figs. 2.3 and 2.4 respectively.

For n-Hg1−xCdxTe together with the numerical values of the spectrumconstants as given in Table 2.1, the DMR has been plotted at T = 4.2Kas a function of electron concentration as shown in Fig. 2.5 for all cases ofFig. 2.3. Figure 2.6 represents the variation of the DMR with respect to alloycomposition x in this case.

It appears from Fig. 2.5 that the DMR of ternary materials increases withincreasing carrier degeneracy. From Fig. 2.6, it appears that the DMR internary materials decreases with increasing alloy composition. The plots ofFigs. 2.5 and 2.6 are valid for x > 0.17, due to the fact that for x < 0.17, theband gap becomes negative in n-Hg1−xCdxTe leading to a semimetallic state.

For n-In1−xGaxAsyP1−y lattice matched to InP together with the valuesof the energy band constants as given in Table 2.1, the DMR has been plottedas a function of electron concentration as shown in Fig. 2.7 in accordance withthe three and two band models of Kane together with the isotropic parabolicenergy band. It appears that the DMR monotonically increases with increas-ing n0. Figure 2.8 shows the dependence of the DMR in n-In1−xGaxAsyP1−y


Fig. 2.3. The plot of the DMR in bulk specimens of n-InAs as a function of electronconcentration in accordance with (a) the three band model of Kane, (b) the twoband model of Kane, and (c) the parabolic energy bands

Fig. 2.4. The plot of the DMR in bulk specimens of n-InSb as a function of electronconcentration in accordance with (a) the three-band model of Kane, (b) the two-band model of Kane, and (c) the parabolic energy bands


Fig. 2.5. The plot of the DMR in bulk specimens of n-Hg1−xCdxTe as a functionof electron concentration in accordance with (a) the three band model of Kane; (b)the two band model of Kane and (c) the parabolic energy bands (x = 0.3)

Fig. 2.6. The plot of the DMR in bulk specimens of n-Hg1−xCdxTe as a functionof alloy composition (x) in accordance with (a) the three band model of Kane; (b)the two band model of Kane and (c) the parabolic energy bands (n0 = 1022 percubic meter)


Fig. 2.7. The plot of the DMR in bulk specimens of n-In1−xGaxAsyP1−y latticematched to InP as a function of electron concentration in accordance with (a) thethree band model of Kane; (b) the two band model of Kane and (c) the parabolicenergy bands (y = 0.037)

Fig. 2.8. The plot of the DMR in bulk specimens of n-In1−xGaxAsyP1−y latticematched to InP as a function of alloy composition (x) in accordance with (a) thethree band model of Kane; (b) the two band model of Kane and (c) the parabolicenergy bands (y = 0.037)


lattice matched to InP on the alloy composition of x for all cases of Fig. 2.6.The DMR decreases with increasing x for all types of band models in thiscase. From Figs. 2.5 up to 2.8, one can infer the influence of energy bandconstants on the DMR for ternary and quaternary compounds respectively.

It may be noted that in recent years, the electron mobility in compoundsemiconductors has been extensively investigated, but the diffusion constant(a very important device quantity which cannot be easily determined exper-imentally) of such materials has been relatively less investigated. Therefore,the theoretical results presented in this chapter will be useful in determiningthe diffusion constants for even relatively wide gap materials whose energyband structures can be approximated by the parabolic energy bands.

We wish to point out that in formulating the basic dispersion relation wehave taken into account the combined influences of the crystal field-splittingconstant, the anisotropies in the effective electron masses, and the spin–orbitsplitting constants, respectively, since these are the significant physical fea-tures of the tetragonal compounds.

In the absence of crystal-field splitting together with the assumptions ofisotropic effective electron mass and isotropic spin–orbit splitting constantrespectively, our basic equation (2.2) converts to the well-known form of thethree-band model of Kane as given by (2.6). Many technologically importantcompounds obey the inequalities ∆ � Eg or ∆ Eg. Under these con-straints, (2.6) gets simplified into (2.10) and is known as the two-band modelof Kane. Finally, for Eg → ∞, as for relatively wide gap materials the aboveequation transforms into the well-known form E = �

2k2/2m∗. In addition,the DMR in ternary and quaternary materials has also been investigated inaccordance with the three and two band models of Kane together with theparabolic energy band for the purpose of relative assessment. Therefore, theinfluence of energy band constants on the DMR can also be studied from thepresent investigation and the basic equation (2.2) covers various materialshaving different energy band structures. Finally, one infers that, this simpli-fied analysis exhibits the basic features of the DMR in bulk specimens of manytechnologically important compounds and for n-Cd3As2, the theoretical resultis in good agreement with the suggested experimental method of determiningthe same ratio.

2.2 Investigation for II–VI Semiconductors

2.2.1 Introduction

The II–VI compounds are being extensively used in infrared detectors [27],ultra high speed bipolar transistors [28], optic fiber communications [29], andadvanced microwave devices [30]. These materials possess the appropriate di-rect band gap to produce light emitting diodes and lasers from blue to redwavelengths [31]. The Hopfield model describes the energy spectra of both

2.2 Investigation for II–VI Semiconductors 27

the carriers of II–VI semiconductors where the splitting of the two-spin statesby the spin orbit coupling and the crystalline field has been taken into ac-count [32]. The DMR in II–VI compounds on the basis of the Hopfield modelhas been studied by formulating the expression of carrier concentration inSect. 2.2.2. Section 2.2.3 contains the result and discussions for the numericalcomputation of the DMR taking p-CdS as an example.


The group theoretical analysis shows that, based on the symmetry propertiesof the conduction and valence band wave functions, both the energy bands ofII–VI semiconductors can be written as [32]

E = a′0k

2s + b′0k

2z ± λ0ks, (2.27)

where a′0 ≡ �

2

2m∗⊥

, b′0 ≡ �2

2m∗||

and λ0 represents the splitting of the two spin-states by the spin–orbit coupling and the crystalline field.

The volume in k-space enclosed by (2.27) can be expressed as

V (E)=π

2a′20

(E/b′0)1/2

∫

−(E/b′0)1/2

[λ2

0 + 2a′0E − 2a′

0b′0k

2z − λ0

(λ2

0 − 4a′0b

′0k

2z + 4a′

0E)1/2]dkz,

(2.28)From (2.28), one can write

V (E) =4π

3a′0

√b′0

⎡

⎢⎢⎣ E3/2 +

38

(λ0

)2 √E

a′0

−(

34

λ0√a′0

)(

E +

(λ0

)2

4a′0

)

× sin−1

⎡

⎢⎢⎣

√E

√

E + (λ0)2

4a′0

⎤

⎥⎥⎦

⎤

⎥⎥⎦ , (2.29)

Hence, the density of states function can be written using (2.3a) and (2.29) as

D0 (E) =gv

2π2a′0

√b′0

⎡

⎢⎢⎣√

E −(

λ0

2√

a′0

)

sin−1

⎡

⎢⎢⎣

√E

√

E + (λ0)2

4a′0

⎤

⎥⎥⎦

⎤

⎥⎥⎦ . (2.30)

Combining (2.30) with the Fermi–Dirac occupation probability factor, thecarrier concentration can be written as

n0 =4πgv

3a′0

√b′0

[τ1 (EF) + τ2 (EF)] , (2.31)


where τ1 (EF) ≡

⎡

⎣E3/2F + 3

8

(λ0)2

a′0

√EF −

⎧⎨

⎩34

λ0√a′0

(EF + (λ0)2

4a′0

)

sin−1

⎡

⎣√

EF√

EF+(λ0)

2

4a′0

⎤

⎦

⎫⎬

⎭

⎤

⎦ , and τ2 (EF) ≡s∑

r=1L (r) τ1 (EF).

Combining (2.31) and (1.11), the DMR in bulk specimens of II–VI semi-conductors assumes the form

D

µ=

1|e|

[τ1 (EF) + τ2 (EF)][[τ1 (EF)]′ + [τ2 (EF)]′

] , (2.32)

Under the condition λ0 → 0, (2.27) gets simplified as

E = a′0k

2s + b′0k

2z , (2.33)

Thus, under the condition λ0 → 0, (2.32) reduces to the well-known form asgiven by (2.19).


Using (2.31) and (2.32) together with the spectrum constants as given inTable 2.1, for p-CdS, the DMR has been plotted as a function of hole concen-tration p0 as shown by curve (a) of Fig. 2.9 in which the plot for λ0 = 0 (curve

Fig. 2.9. The plot of the dependence of the DMR on hole concentration p0 for bulkspecimens of p-CdS for (a) λ0 �= 0 and (b) λ0 = 0

2.3 McClure–Choi, the Cohen, the Lax, and the Parabolic Ellipsoidal Band 29

(b)) has also been drawn for the purpose of assessing the influence of the split-ting of the two spin states by the spin–orbit coupling and the crystalline fieldon DMR. From Fig. 2.8 it appears that the DMR increases with increasinghole concentration at a rate greater than that corresponding to the zero valueof λ0. For relatively low values of p0, the effect of λ0 increases whereas thesame constant affects the DMR less significantly for relatively higher valuesof carrier degeneracy. The presence of λ0 enhances the numerical values ofDMR in II–VI compounds for the whole range of concentration considered ascompared with that corresponding to λ0 = 0.

2.3 Investigation for Bi in Accordancewith the McClure–Choi, the Cohen, the Lax,and the Parabolic Ellipsoidal Band Models

2.3.1 Introduction

It is well-known that the carrier energy spectra in Bi differ considerably fromthe simple spherical energy wave vector dispersion relation of the degenerateelectron gas and several models have been developed to describe the energyspectra of Bi. Earlier works [33, 34] demonstrated that the physical proper-ties of Bi could be described by the ellipsoidal parabolic energy band model.Shoenberg [33] showed that the de Haas-Van Alphen and cyclotron resonanceexperiments supported the ellipsoidal parabolic model, though the latter workshowed that Bi could be described by the two-band model due to the fact thatthe magnetic field dependence of many physical properties of Bi supports theabove model [35]. The experimental results of the magneto-optical [35] andthe ultrasonic quantum oscillations [36] favor the Lax ellipsoidal non-parabolicmodel [35]. Kao [37], Dinger and Lawson [38] and Koch and Jensen [39] ob-served that the Cohen model [40], is in better agreement with the experimentalresults. McClure and Choi [41] presented a new model of Bi, which was moreaccurate and general than those that were currently available. They showedthat it can explain the data for a large number of magneto-oscillatory andresonance experiments. We shall study the influence of different energy bandmodels on the DMR in bulk specimens of Bi which have been investigated byformulating the carrier concentration in Sect. 2.3.2. Section 2.3.3 contains theresult and discussions in this context.


(a)The McClure and Choi modelThe carrier energy spectra in Bi can be written, following McClure and

Choi, [42] as


E (1 + αE) =p2

x

2m1+

p2y

2m2+

p2z

2m3+

p2y

2m2αE

{1 −(

m2

m′2

)}+

p4yα

4m2m′2

−αp2

xp2y

4m1m2−

αp2yp2

z

4m2m3, (2.34)

where m1, m2 and m3 are the effective carrier masses at the band-edge alongx, y and z directions respectively and m′

2 is the effective- mass tensor com-ponent at the top of the valence band (for electrons) or at the bottom of theconduction band (for holes).

The area of the ellipse in the kx−kz plane can be expressed as

A (E, ky) = T0

[E (1 + αE) − θ2 (E) k2

y − θ3k4y

1 − θ4k2y

]

, (2.35)

where T0 ≡(

2π√

m1m3

�2

), θ2 (E) ≡

[(αE�

2

2m2

)(1 − m2

m′2

)+ �

2

2m2

], θ3 ≡

(α�

4

4m2m′2

)and θ4 ≡

(α�

2

2m2

).

The volume of k-space enclosed by (2.34) can be written as

V (E) =(

4π√

m1m3

�2θ4

) p0(E)∫

0

[h9 (E)

θ52 − ky

2 + θ2 (E) + θ3

(θ5

2 + ky2)]dky,

(2.36)

where p0 (E) ≡(√

2m2m′2

�2√α

)[−θ2 (E) +

√θ22 (E) + θ3E (1 + αE)

]1/2

, θ5 ≡

(θ4)−1/2 and h9 (E) ≡

[E (1 + αE) − θ2 (E) θ5

2 − θ3θ54].

From (2.36) one obtains

V (E) =4π

√m1m3

�2θ4

[h9 (E)

2θ5ln∣∣∣∣θ5 + p0 (E)θ5 − p0 (E)

∣∣∣∣+[θ2 (E) + θ3θ5

2]p0 (E)

+θ3

3[p0 (E)]3

]. (2.37)

The density-of-states function in this case can be expressed using (2.3a) as

D0 (E) =

(gν√

m1m3

)

(π2�2θ4)

[{h9 (E)}′

2θ5ln∣∣∣∣θ5 + p0 (E)θ5 − p0 (E)

∣∣∣∣+

h9 (E) {p0 (E)}′

[θ25 − p2

0 (E)]

+ {θ2 (E)}′ p0 (E)+[θ3 {p0 (E)}′ p2

0 (E)+[θ2 (E)+θ3θ

25

]{p0 (E)}′

],

(2.38)

where {h9 (E)}′ ≡[1 + 2αE −

(θ25α�

2

2m2

)(1 − m2

m′2

)], {θ2 (E)}′ ≡

(α�

2

2m2

)

×(1 − m2

m′2

)and


{p0 (E)}′ ≡[

12p0 (E)

[

−{θ2 (E)}′ +2θ2 (E) {θ2 (E)}′ + θ3 (1 + 2αE)

(2)√

θ22 (E) + θ3E (1 + αE)

]

×[−θ2 (E) +

√θ22 (E) + θ3E (1 + αE)

]−1]

.

Therefore the electron concentration is given by

n0 = θ6

[M2 (EF) + N2 (EF)

]. (2.39)

where θ6 ≡(

gν√

m1m3

π2�2θ4

),

M2 (EF) ≡[

h9 (EF)2θ5

ln∣∣∣∣θ5 + p0 (EF)θ5 − p0 (EF)

∣∣∣∣+ [ θ2 (EF)

+θ3θ52 ] p0 (EF) +

θ3

3[p0 (EF)]3

],

and N2 (EF) ≡s∑

r=1L (r)

[M2 (EF)

].

Thus, combining (2.39) and (1.11), we can write the expression of DMRin Bi in accordance with the McClure and Choi model as

D

µ=

1|e|

[M2 (EF) + N2 (EF)

{M2 (EF)

}′+{N2 (EF)

}′

]

. (2.40)

(b) The Cohen modelIn accordance with Cohen [40], the dispersion law of the carriers in Bi is

given by

E(1 + αE) =p2

x

2m1+

p2z

2m3−

αEp2y

2m′2

+

(αp4

y

4m2m′2

)

+p2

y

2m2(1 + αE), (2.41)

In this case the area of the ellipse in the kx−kz plane can be written as

A(E, ky) =2π

√m1m3

�2

[

E(1 + αE) −αp4

y

4m2m′2

+αEp2

y

2m′2

−p2

y

2m2(1 + αE)

]

.

Therefore the volume enclosed by (2.41) is given by

V (E) =4π

√m1m3

�2

p0(E)∫

0

[E(1 + αE) −

α�4k4

y

4m2m′2

+�

2k2y

2

[αE

m′2

− 1m2

(1 + αE)] ]

dky, (2.42)


where p0(E) ≡ 1�

[α

2m2m′2

]−1/2 [−[

1+αE2m2

− αE2m′

2

]+[ [

1+αE2m2

− αE2m′

2

]2

+ αEm2m′

2(1 + αE)

]1/2 ]1/2

.From (2.42) we can write

V (E) =4π

√m1m3

�2

[

E (1 + αE) p0 (E) − α�4 [p0 (E)]5

20m2m′2

+�

2 [p0 (E)]3

6

[αE

m′2

− 1m2

(1 + αE)]]

. (2.43a)

The density-of-states function can be expressed using (2.3a) and (2.43a) as

D0 (E) =gv√

m1m3

π2�2[ (1 + 2αE) p0 (E) + E (1 + αE) [p0 (E)]′

−α�4 [p0 (E)]4 [p0 (E)]′

5m2m′2

+�

2

2[p0 (E)]2 [p0 (E)]′

×[αE

m′2

− 1m2

(1 + αE)]

+�

2 [p0 (E)]3

6α

(1

m′2

− 1m2

)]

, (2.43b)

where [p0 (E)]′ ≡[

m2m′2

α�2p0(E)

] [α2

(1

m′2− 1

m2

)+ 1

2

[[1+αE2m2

− αE2m′

2

]2+ αE

m2m′2

× (1 + αE)]−1/2

.

[α

m2m′2

(1 + 2αE) + α2

(1

m2− 1

m′2

)(1+αE

m2− αE

m′2

)]]. Thus,

using (2.43b) with the Fermi–Dirac occupation probability factor and usingthe generalized Sommerfelds lemma [10], the electron concentration in thiscase can be expressed as

n0 =(

gv√

m1m3

π2�2

)[M3 (EF) + N3 (EF)] , (2.44)

where M3 (EF) ≡[

EF (1 + αEF) p0 (EF)− α�4[p0(EF)]5

20m2m′2

+ �2[p0(EF)]3

6

[αEFm′

2−

1m2

(1 + αEF)] ]

, and N3 (EF) ≡s∑

r=1L (r) [M3 (EF)] .

Thus, combining (2.44) and (1.11), we can write the expression of theDMR in bismuth in accordance with the Cohen model as

D

µ=

1|e|

[M3 (EF) + N3 (EF)][{M3 (EF)}′ + {N3 (EF)}′

] . (2.45)

(c) The Lax modelThe carrier spectrum of Bi in accordance with the Lax model is given

by [35]


E (1 + αE) =px

2

2m1+

py2

2m2+

pz2

2m3. (2.46)

For this model, under the condition αEF 1, the expressions for n0 andDMR in Bi, in accordance with the Lax non-parabolic and ellipsoidal model,get simplified to (2.14) and (2.15) where

Nc ≡ 2[2π (m1m2m3)

13

kBT

h2

] 32

.

(d) Ellipsoidal parabolic modelThe carrier energy spectra for this type of band model of Bi can be written

as [33]

E =px

2

2m1+

py2

2m2+

pz2

2m3. (2.47)

For this model, the expressions of carrier concentration and DMR arerespectively given by (2.18) and (2.19) where Nc has been defined as above.


Using (2.39) and (2.40) and taking the spectrum constants from Table 2.1,in curve (a) of Fig. 2.10, the DMR in Bi has been plotted as a function ofelectron concentration by using the McClure and Choi model. The curves (b),

Fig. 2.10. The plot of the DMR in bulk specimens of bismuth as a function ofelectron concentration in accordance with the model of (a) the McClure Choi; (b)the Cohen; (c) the Lax and (d) the ellipsoidal parabolic energy bands


(c) and (d) exhibit the same dependence in accordance with the Cohen, theLax, and the parabolic ellipsoidal models respectively.

From Fig. 2.10, it appears that the DMR increases with increasing n0 forall the models of Bi. In accordance with the model of McClure and Choi, theDMR exhibits the least numerical values as compared to the other models ofBi. For various energy band models, the values of the DMR with respect tothe electron concentration are different. The rates of variations of the DMRwith respect to n0 are also different for different types of energy band models.It should be noted that under the condition α → 0, the models of McClureand Choi, the Cohen and the Lax reduce to (2.47). Thus, under certain con-straints, all the three energy models are reduced to the ellipsoidal parabolicenergy bands and the expression for the DMR under the same condition getssimplified to the well-known equation (2.19) as given for the first time byLandsberg [1].

The Cohen model is often used to describe the dispersion relation of thecarriers of IV–VI semiconductors. The model of Bi, by Lax, under the con-dition of the isotropic effective mass of the carriers of the band edge (i.e.m1 = m2 = m3 = m∗.) reduces to the two-band model of Kane, which is usedto investigate the physical features of III–V compounds, in general, excludingn-InAs. Thus, the analysis is valid not only for bismuth, but also for all leadchalcogenides, III–V compounds excluding n-InAs, and wide-gap materialsrespectively. The influence of the energy band models on the DMR of Bi canalso be assessed from the Fig. 2.10. It can be noted that the present analysisis valid for the holes of Bi with the appropriate values of the energy bandconstants.

2.4 Investigation for IV–VI Semiconductors

2.4.1 Introduction

The IV–VI compounds are being extensively used in thermoelectric devices,superlattices, and other quantum effect devices [43]. The dispersion relationof the carriers of the IV–VI compounds could be described by the Cohenmodel [40], which includes the band non-parabolicity and the anisotropies ofthe effective masses of the carriers. The DMR in bulk specimens of IV–VImaterials has been studied, taking n-PbTe, n-PbSnTe, and n-Pb1−xSnxSe asexamples. Sections 2.4.2 and 2.4.3 contain the theoretical background and theresult and discussions in this context.


The expressions of n0 and the DMR in this case are given by (2.44) and (2.45)in which the energy band constants correspond to the IV–VI compounds.

2.5 Investigation for Stressed Kane Type Semiconductors 35


Using (2.44) and (2.45) together with the spectrum constants as given inTable 2.1 for n-PbTe, n-PbSnTe, and n-Pb1−xSnxSe, the DMR has been plot-ted as a function of electron concentration n0 as shown in Fig. 2.11 where theplots (a), (b) and (c) correspond to the said materials respectively in accor-dance with the Cohen model. From Fig. 2.11, it appears that the DMR for afixed n0 is maximum for n-PbTe and minimum for n-Pb1−xSnxSe. For rela-tively low values of n0, the values of the DMR for the three materials exhibitconvergence behavior whereas for relatively large values of n0, the numericalvalues differ from each other. Besides, our present analysis is also valid forp-type IV–VI materials with the proper change in the energy band constants.

2.5 Investigation for Stressed Kane TypeSemiconductors

2.5.1 Introduction

In recent years there has been a considerable interest in studying the variouselectronic properties of stressed materials because of their important physi-cal characteristics [44]. In this section we shall study the DMR in stressed

Fig. 2.11. The plot of the DMR in bulk specimens of (a) n-PbTe (b) n-PbSnTeand (c) n-Pb1−xSnxSe as a function of electron concentration in accordance withCohen model


semiconductors, taking stressed n-InSb as an example for numerical compu-tations. Sections 2.5.2 and 2.5.3 contains the theoretical background and theresult and discussions in this context.


The electron energy spectrum in stressed Kane type semiconductors can bewritten [44] as

(kx

a0 (E)

)2

+(

ky

b0 (E)

)2

+(

kz

c0 (E)

)2

= 1, (2.48)

where

[a0 (E)]2 ≡ K0 (E)A0 (E) + 1

2D0 (E), K0 (E) ≡

[

E − C1ε −2C2

2ε2xy

3E′g

](3E′

g

2B22

),

C1 is the conduction band deformation potential, ε is the trace of the strain

tensor ε which can be written as ε =

⎡

⎣εxx εxy 0εxy εyy 00 0 εzz

⎤

⎦, C2 is a constant which

describes the strain interaction between the conduction and valance bands,E′

g ≡ Eg + E − C1ε,B2 is the momentum matrix element,

A0 (E) ≡[1 − (a0 + C1)

E′g

+3b0εxx

2E′g

− b0ε

2E′g

], a0 ≡ −1

3(b0 + 2m

),

b0 ≡ 13(l − m

), d0 ≡ 2n√

3,

l, m, n are the matrix elements of the strain perturbation operator, D0 (E) ≡(d0

√3) εxy

E′g,

[b0 (E)

]2 ≡ K0 (E)A0 (E) − 1

2D0 (E), [c0 (E)]2 ≡ K0 (E)

L0 (E)

and L0 (E) ≡[

1 − (a0 + C1)E′

g

+3b0εzz

E′g

− b0ε

2E′g

],

The density-of-states function in this case can be written using (2.3a) and(2.48) as

D0 (E) = gv

(3π2)−1

[ a0 (E) b0 (E) [c0 (E)]′ + a0 (E)[b0 (E)

]′c0 (E)

+ [a0 (E)]′ b0 (E) c0 (E) ] , (2.49)

2.5 Investigation for Stressed Kane Type Semiconductors 37

Combining (2.49) with the Fermi–Dirac occupation probability factor andusing the generalized Sommerfelds lemma [10], the electron concentration inthis case can be expressed as

n0 = gv

(3π2)−1

[M4 (EF) + N4 (EF)] , (2.50)

where M4 (EF) ≡[a0 (EF) b0 (EF) c0 (EF)

]and N4 (EF) ≡

s∑

r=1L (r) M4 (EF).

Thus using (2.50) and (1.11) we get

D

µ=

1|e|

[M4 (EF) + N4 (EF)][{M4 (EF)}′ + {N4 (EF)}′

] . (2.51)

In the absence of stress together with the substitution B22 ≡ 3�

2Eg4m∗ , (2.51) gets

simplified to (2.13).


Using (2.50) and (2.51) together with the energy band constants as given inTable 2.1 at T = 4.2K, the DMR has been plotted as a function of electronconcentration n0 in stressed n-InSb as shown in Fig. 2.12 where the curves(a), and (b) correspond to the presence and absence of stress respectively.

Fig. 2.12. The plot of the DMR in bulk specimens of stressed n-InSb as a functionof the electron concentration both in the presence and absence of stress as shownby the curves (a) and (b) respectively


It appears that stress enhances the numerical values of the DMR to a largeextent as compared to that of the stress free condition. The rates of increaseof DMR for both the cases are different as concentration increases.

2.6 Summary

In Chap. 2, an attempt is made to present the DMR in tetragonal com-pounds on the basis of a generalized electron dispersion law by considering theanisotropies of the effective electron masses and the spin–orbit splitting con-stants together with the inclusion of crystal field splitting constant within theframe work of k.p formalism. The theoretical result is in agreement with thesuggested experimental method of determining the DMR for materials havingarbitrary dispersion laws. Under certain limiting conditions, the results forIII–V materials as defined by the three and two band models of Kane havebeen obtained as special cases of the generalized analysis. The concentrationdependence of the DMR has also been numerically computed for n-Cd3As2,n-CdGeAs2, n-InAs, n-InSb, n-Hg1−xCdxTe, and n-In1−xGaxAsyP1−y latticematched to InP respectively. The II–VI compounds obey the Hopfield modeland p-type CdS has been used for numerical computation. The DMR has alsobeen investigated for Bi in accordance with the models of the McClure andChoi, the Cohen, the Lax, and the parabolic ellipsoidal energy bands respec-tively. The IV–VI materials obey the Cohen model and n-PbTe, n-PbSnTe,and n-Pb1−xSnxSe have been used for investigations. The chapter ends withthe study of the DMR in stressed Kane type semiconductors taking stressedn-InSb as an example, which obey the dispersion relation as suggested bySeiler et al. [44].

Thus, a wide class of technologically important materials has been coveredin this chapter whose energy band structures are defined by the appropriatecarrier energy spectra. Under certain limiting conditions, all the results of theDMRs for different materials having various band structures lead to the well-known expression of the DMR for degenerate semiconductors having parabolicenergy bands as obtained for the first time by Landsberg. For the purpose ofcondensed presentation, the specific electron statistics related to a particularenergy dispersion law for a specific material and the Einstein relation havebeen presented in Table 2.2.

2.7 Open Research Problems

The problems under these sections of this book are by far the most impor-tant part for the readers. Few open and allied research problems are presentedfrom this chapter onward to the end. The numerical values of the energybandconstants for various materials are given in Table 2.1 for the related com-puter simulations.

Table

2.2

.T

he

carr

ier

stati

stic

sand

the

Ein

stei

nre

lati

on

inbulk

spec

imen

softe

tragonal,

III–

V,te

rnary

,quate

rnary

,II

–V

I,all

the

model

sofB

ism

uth

,IV

–V

Iand

stre

ssed

mate

rials

Type

of

mate

rials

The

carr

ier

stati

stic

sT

he

Ein

stein

rela

tion

for

the

diff

usi

vity

mobility

rati

o

1.Tetragonal

com

pounds

Inaccord

ance

wit

hth

egenera

lized

dis

pers

ion

rela

tion

as

form

ula

ted

in

this

chapte

r

n0

=gv

( 3π2) −

1[M

(EF)+

N(E

F)]

,(2

.4)

D µ=

1 |e|

[M(E

F)+

N(E

F)]

[{M

(EF)}

′ +{N

(EF)}

′ ](2

.5)

2.II

I–V

,te

rnary

and

quate

rnary

com

pounds

Inaccord

ance

wit

hth

eth

ree

band

modelofK

ane

whic

his

asp

ecia

l

case

ofour

genera

lized

analy

sis

n0

=gv

3π2

(2

m∗

�2

) 3/2

[M1

(EF)+

N1

(EF)]

(2.8

)D µ

=1 |e|[M

1(E

F)+

N1

(EF)]

[{M

1(E

F)}

′

+{N

1(E

F)}

′]−

1(2

.9)

Equati

on

(2.8

)is

asp

ecia

lcase

of(2

.4)

Equati

on

(2.9

)is

asp

ecia

lcase

of(2

.5)

Under

the

condit

ions

∆�

Eg

or

∆�

Eg,

Under

the

condit

ions

∆�

Eg

or

∆�

Eg,

n0

=gv

3π2

(2

m∗

�2

) 3/2

[M2

(EF)+

N2

(EF)]

(2.1

2)

D µ=

1 |e|[M

2(E

F)+

N2

(EF)]

[{M

2(E

F)}

′

+{N

2(E

F)}

′]−

1(2

.13)

Equati

on

(2.1

2)

isa

specia

lcase

of(2

.8)

and

isvalid

for

the

two

band

modelofK

ane

Equati

on

(2.1

3)

isa

specia

lcase

of(2

.9)

and

isvalid

for

the

two

band

modelofK

ane

Under

the

const

rain

ts∆

�E

gor

∆�

Eg

togeth

er

wit

hth

econdit

ion

αE

F�

1

Under

the

const

rain

ts∆

�E

gor

∆�

Eg

togeth

er

wit

hth

econdit

ion

αE

F�

1

n0

=gv

Nc

[ F1/2(η

)+(

15

αkB

T4

)F3/2(η

)](2

.14)

D µ=[

kB

T|e

|

]

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎛ ⎜ ⎝F1/2(η

)+(

15

αkB

T4

)F3/2(η

)

⎞ ⎟ ⎠

(F−

1/2(η

)+(

15

αkB

T4

)F1/2(η

))

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(2.1

5)

For

Eg

→∞

,For

Eg

→∞

,

n0

=gv

NcF1/2(η

)(2

.18)

D µ=(

kB

T|e

|

)[

F1/2(η

)

F−

1/2(η

)

](2

.19)

Equati

on

(2.1

8)

isa

specia

lcase

of(2

.14)

and

isvalid

for

para

bolic

energ

ybands

Equati

on

(2.1

9)

isa

specia

lcase

of(2

.15)

Under

the

condit

ion

ofextr

em

ecarr

ier

degenera

cy

Under

the

condit

ion

ofextr

em

ecarr

ier

degenera

cy

n0

=gv

3π2

[2

m∗

EF

(1+

αE

F)

�2

] 3/2

(2.2

1)

D µ=

1 |e|

(2 3

)E

F(1

+α

EF)(1

+2α

EF)−

1(2

.22)

(Continued

)


Table

2.2

.C

ontinued

Type

of

mate

rials

The

carr

ier

stati

stic

sT

he

Ein

stein

rela

tion

for

the

diff

usi

vity

mobility

rati

o

For

α→

0,

For

α→

0,

n0

=gv

3π2

[2

m∗

EF

�2

] 3/2

(2.2

3)

D µ=

2E

F3|e

|(2

.24)

Under

the

condit

ion

ofnon-d

egenera

teele

ctr

on

concentr

ati

on

Under

the

condit

ion

ofnon-d

egenera

teele

ctr

on

concentr

ati

on

n0

=gv

Nc

exp(η

)(2

.25)

D µ=

kB

T|e

|(2

.26)

3.II

–V

I

com

pounds

n0

=4

πgv

3a′ 0

√b′ 0

[τ1

(EF)+

τ2

(EF)]

(2.3

1)

D µ=

1 |e|

[τ1(E

F)+

τ2(E

F)]

[[τ1(E

F)]

′ +[τ

2(E

F)]

′ ](2

.32)

4.Bis

muth

(a)

The

McC

lure

and

Choim

odel:

n0

=θ6[ M

2(E

F)+

N2(E

F)]

(2.3

9)

D µ=

1 |e|

[M2(E

F)+

N2(E

F)]

[ {M2(E

F)}

′ +{N

2(E

F)}

′](2

.40)

(b)

The

Cohen

model:

n0

=gv√

m1

m3

π2

�2

[M3

(EF)+

N3

(EF)]

(2.4

4)

D µ=

1 |e|

[M3(E

F)+

N3(E

F)]

[{M

3(E

F)}

′ +{N

3(E

F)}

′ ](2

.45)

(c)

The

Lax

model:

n0

=gv

Nc

[ F1/2(η

)+(

15

αkB

T4

)F3/2(η

)],

(2.1

4)

where

Nc≡

2

[ 2π

(m1m

2m

3)1 3

kB

T

h2

]3 2

The

DM

Rin

this

case

are

giv

en

by

(2.1

5)

The

DM

Rin

this

case

are

giv

en

by

(2.1

9)

(d)

The

para

bolic

ellip

soid

alm

odel:

The

ele

ctr

on

stati

stic

sin

this

case

are

giv

en

by

(2.1

8),

wit

hN

cas

defined

above

5.IV

–V

I

com

pounds

The

expre

ssio

ns

of

n0

inth

iscase

are

giv

en

by

(2.4

4)

inw

hic

hth

e

const

ants

ofth

eenerg

yband

spectr

um

corr

esp

ond

toth

ecarr

iers

of

the

IV–V

Ise

mic

onducto

rs

The

expre

ssio

ns

ofD

MR

inth

iscase

are

giv

en

by

(2.4

5)

inw

hic

hth

e

const

ants

ofth

eenerg

yband

spectr

um

corr

esp

ond

toth

ecarr

iers

ofth

e

IV–V

Ise

mic

onducto

rs

6.Stress

ed

com

pounds

n0

=gv

( 3π2) −

1[M

4(E

F)+

N4

(EF)]

(2.5

0)

D µ=

1 |e|

[M4(E

F)+

N4(E

F)]

[{M

4(E

F)}

′ +{N

4(E

F)}

′ ](2

.51)

2.7 Open Research Problems 41

(R2.1) Investigate the Einstein relation for the materials having the re-spective dispersion relations as given below:

(A) The conduction electrons of n-GaP obey two different dispersion lawsas given in the literature [45, 46]. In accordance with Rees [45], the electronenergy spectrum is given by

E =�

2k2s

2m∗⊥

+�

2

2m∗||

[℘k2

s + k2z

]−[

�4k2

0

m∗2||

(k2

s + k2z

)+ |VG|2

]1/2

−|VG| , (R2.1a)

where k0 and |VG| are constants of the energy spectrum with m∗|| =

0.92m0,m∗⊥ = 0.25m0, k0 = 1.7 × 1019 m−1, |VG| = 0.21 eV, gv = 6, gs =

2 and ℘ = 1.(1) In accordance with Ivchenko and Pikus [46], the electron dispersion lawcan be written as

E =�

2k2z

2m∗||

+�

2k2s

2m∗⊥

∓ ∆2

±[(

∆2

)2

+ P1k2z + D1k

2xk2

y

]1/2

, (R2.1b)

where ∆ = 335meV, P1 = 2×10−10 eVm,D1 = P1a1 and a1 = 5.4×10−10 m.

(B) In addition to the Cohen model, the dispersion relation for the conduc-tion electrons for IV–VI compounds can also be described by the models ofDimmock [47], Bangert et al. [48], and Foley et al. [49] respectively.(1) In accordance with the Dimmock model [47], the carrier energy spectrumof IV–VI materials assumes the form[∈ −Eg

2− �

2k2s

2m−t

− �2k2

z

2m−l

] [∈ +

Eg

2+

�2k2

s

2m+t

+�

2k2z

2m+l

]= P 2

⊥k2s +P 2

‖ k2z , (R2.2)

where ∈ is the energy as measured from the center of the band gapEg,m

±t and m±

l represent the contribution of the transverse and longitu-dinal effective masses of the external L+

6 and L−6 bands arising from the k.p

perturbations with the other bands taken to the second order and gv = 4.(2) In accordance with Bangert et al. [48] the dispersion relation is given by

Γ (E) = F1 (E) k2s + F2 (E) k2

z , (R2.3)

where Γ (E) ≡ 2E,F1 (E) ≡ R21

E+Eg+ S2

1E+∆′

c+ Q2

1E+Eg

, F2 (E) ≡ 2C25

E+Eg+ (S1+Q1)

2

E+∆′′c

,

R21 =2.3 × 10−19 (eVm)2 , C2

5 =0.83 × 10−19 (eVm)2 , Q21 =1.3R2

1,

S21 = 4.6R2

1,∆′c = 3.07 eV,∆′′

c = 3.28 eV and gv = 4. It may be noted that un-der the substitutions S1 = 0, Q1 = 0, R2

1 ≡ �2Egm∗

⊥, C2

5 ≡ �2Eg

2m∗||

, (R2.3) assumes

the form E (1 + αE) = �2k2

s

2m∗⊥

+ �2k2

z

2m∗||

which is the simplified Lax model.


(3) The carrier energy spectrum of IV–VI semiconductors in accordance withFoley et al. [49] can be written as

E +Eg

2= E− (k) +

[[E+ (k) +

Eg

2

]2+ P 2

⊥k2s + P 2

||k2z

]1/2

, (R2.4)

where E+ (k) = �2k2

s

2m+⊥

+ �2k2

z

2m+||

, E− (k) = �2k2

s

2m−⊥

+ �2k2

z

2m−||

represents the contribution

from the interaction of the conduction and the valance band edge states withthe more distant bands and the free electron term,

1m±

⊥=

12

[1

mtc± 1

mtv

],

1m±

||=

12

[1

mlc± 1

mlv

],

For n-PbTe, P⊥ = 4.61 × 10−10 eVm, P|| = 1.48 × 10−10 eVm, m0mtv

= 10.36,m0mlv

= 0.75, m0mtc

= 11.36, m0mlc

= 1.20 and gv = 4.

(C) The importance of Germanium is well known since the inception of semi-conductor physics. The conduction electrons of n-Ge obey two different disper-sion laws since band non-parabolicity has been included in two different waysas given literature [50, 51]. In accordance with Cardona et al. [50] and Wanget al [51] the electron dispersion laws in Ge can respectively, be expressed as

E = −Eg

2+

�2k2

z

2m∗‖

+

[E2

g

4+ Egk

2s

(�

2

2m∗⊥

)]1/ 2

, (R2.5)

andE = a9k

2z + l9k

2s − c9k

4s − d9k

2sk2

z − e9k4z , (R2.6)

where a9 =(�

2 / 2m∗||

),m∗

|| = 1.588m0, l9 =(�

2 / 2m∗⊥),m∗

⊥ = 0.0815m0,c9 = 1.4A9,

A9 ≡ 14(�

4 / Egm∗2⊥)[1 − m∗

⊥m0

]2, Eg = 2.2 eV,

d9 = 0.8A9, e9 = 0.005A9, gv = 4 and gs = 2.

(D) The dispersion relation of the conduction electrons of zero-gap materials(e.g. HgTe) is given by [52]

E =�

2k2

2m∗ +3e2

128ε∞k −(

2EB

π

)ln∣∣∣∣k

k0

∣∣∣∣ , (R2.7)

where ε∞ is the semiconductor permittivity in the high frequency limit, EB ≡m0e2

2�2ε2∞

and k0 ≡ m0e2

�2ε∞.


(E) The conduction electrons of n-GaSb obey the following three dispersionrelations:

(1) In accordance with the model of Seiler et al. [53]

E =[−Eg

2+

Eg

2[1 + α4k

2]1/ 2

+ς0�

2k2

2m0+

v0f1(k)�2

2m0± ω0f2(k)�2

2m0

],

(R2.8)where α4 ≡ 4P 2

(Eg + 2

3∆) [

E2g (Eg + ∆)

]−1, P is the isotropic momentum

matrix element, f1(k) ≡ k−2[k2

xk2y + k2

yk2z + k2

zk2x

]represents the warping of

the Fermi surface, f2(k) ≡[{

k2(k2

xk2y + k2

yk2z+ k2

zk2x ) − 9k2

xk2yk2

z }1/ 2k−1 ]

represents the inversion asymmetry splitting of the conduction band and ς0, v0

and ω0 represent the constants of the electron spectrum in this case.It should be noted that under the substitutions,ς0 = 0, v0 = 0, ω0 = 0

and P 2 ≡ �2

2m∗Eg(Eg+∆)

(Eg+ 23∆) , (R2.8) assumes the form of (2.10), which represents

the well known two band model of Kane.(2) In accordance with the model of Mathur et al. [54],

E =�

2k2

2m0+

Eg1

2

[(1 +

2�2k2

Eg1

{1

m∗ − 1m0

})1/2

− 1

]

, (R2.9)

where Eg1 ={

Eg +[(

5 × 10−5T 2) (

2 (112 + T )−1)]}

eV.

(3) In accordance with the model of Zhang et al. [55]

E =[E

(1)2 + E

(2)2 K4,1

]k2 +

[E

(1)4 + E

(2)4 K4,1

]k4

+k6[E

(1)6 + E

(2)6 K4,1 + E

(3)6 K6,1

]. (R2.10)

where K4,1 ≡ 54

√21[

k4x+k4

y+k4z

k4 − 35

], K6,1 ≡

√639639

32

[k2

xk2yk2

z

k6 + 122(

k4x+k4

y+k4z

k4 − 35

)− 1

105

], the coefficients are in eV, the values of k are

10(

a2π

)times those of k in atomic units (a is the lattice constant),

E(1)2 = 1.0239620, E

(2)2 = 0, E

(1)4 = −1.1320772,

E(2)4 = 0.05658, E

(1)6 = 1.1072073, E

(2)6 = −0.1134024

and E(3)6 = −0.0072275.

(F) The dispersion relation of the carriers in p-type Platinum antimonide(PtSb2) following Emtage [56] can be written as(

E + λ1a2

4k2 − l1k

2s

a2

4

)(E + δ′0 − υ1

a2

4k2 − n1

a2

4k2

s

)= I0

a4

16k4, (R2.11)

where λ1,l1, δ′0, ν1, n1 and I0 are the energy band constants and a is the lattice

constant.


(G) In addition to the well known three band model of Kane, the conductionelectrons of n-GaAs obey the following three dispersion relations:(1) In accordance with the model of Stillman et al. [57]

E =�

2k2

2m∗ −(

1 − m∗

m0

)2(�

2k2

2m∗

)2⎡

⎣3Eg + 4∆ +

(2∆2

Eg

)

(Eg + ∆) (2∆ + 3Eg)

⎤

⎦ , (R2.12)

(2) In accordance with the model of Newson et al. [58]

E = α0k4z +[

�2

2m∗ +(2α0 + β0

)k2

s

]k2

z +�

2

2m∗ k2s +(2α0 + β0

)k2

xk2y

+α0

(k4

x + k4y

), (R2.13)

where α0

(= −1.97 × 10−37 eVm4

)is the non-parabolicity constant and

β0

(= −2.3 × 10−37 eVm4

)is the wrapping constant.

(3) In accordance with the model of Rossler [59]

E =�

2k2

2m∗ + α10k4 + β10

[k2

xk2y + k2

yk2z + k2

zk2x

]

±γ10

[k2(k2

xk2y + k2

yk2z + k2

zk2x

)− 9k2

xk2yk2

z

]1/2, (R2.14)

where α10 = α11 + α12k, β10 = β11 + β12k and γ10 = γ11 + γ12k, in which,

α11 = −2132 × 10−40 eVm4, α12 = 9030 × 10−50 eVm5,

β11 = −2493 × 10−40 eVm4, β12 = 12594 × 10−50 eVm5,

γ11 = 30 × 10−30 eVm3 and γ12 = −154 × 10−42 eVm4.

(H) In addition to the well known three band model of Kane, the conductionelectrons of n-InSb obey the following three dispersion relations:

(1) To the fourth order effective mass theory, and taking into account theinteractions of the conduction, the heavy hole, the light hole, and the split-offbands, the electron energy spectrum in n-InSb is given by [60]

E =�

2k2

2m∗ + b1k4, (R2.15)

where b1 ≡ K2�4

4Eg(m∗)2, K2 ≡ −

[(1+ 1

2 x21)

1+ 12 x1

](1 − y1)

2, x1 ≡

[1 +(

∆Eg

)]−1

and

y1 ≡ m∗

m0.

(2) In accordance with Johnson and Dickey [61], the electron energy spectrumassumes the form

E = −Eg

2+

�2k2

2

[1

m0+

1mγb

]+

Eg

2

[1 + 4

�2k2

2m′c

f1 (E)Eg

]1/2

, (R2.16)


where m0m′

c≡ P 2

0

[(Eg+ 2∆

3 )Eg(Eg+∆)

], P 2

0 is the energy band constant, f1 (E) ≡(Eg+∆)(E+Eg+ 2∆

3 )(Eg+ 2∆

3 )(E+Eg+∆),m′

c = 0.139m0 and mγb =[

1m′

c− 2

m0

]−1

.

(3) In accordance with Agafonov et al. [62], the electron energy spectrum canbe written as

E =η − Eg

2

[

1 − �2k2

2ηm∗

{D√

3 − 3B2(

�2

2m∗

)

}[k4

x + k4y + k4

z

k4

]]

, (R2.17)

where η ≡(E2

g + 83P 2k2

)1/2, B ≡ −21 �2

2m0and D ≡ −40

(�2

2m0

).

(I) The dispersion relation of the carriers in n-type Pb1−xGaxTe with x = 0.01following Vassilev [63] can be written as

[E − 0.606k2

s − 0.0722k2z

] [E + Eg + 0.411k2

s + 0.0377k2z

]

= 0.23k2s + 0.02k2

z ±[0.06Eg + 0.061k2

s + 0.0066k2z

]ks, (R2.18)

where Eg (= 0.21 eV) is the energy gap for the transition point, the zero of theenergy E is at the edge of the conduction band of the Γ point of the Brillouinzone and is measured positively upwards, kx, ky and kz are in the units of109 m−1.(J) The charge carriers of Tellurium obey two different dispersion laws asgiven in the literature [64,65].(1) The dispersion relation of the conduction electrons in Tellurium, followingBouat [64] can be written as

E = A6k2z + B6k

2⊥ ±

[ϑk2

z + �k2⊥]1/2

, (R2.19a)

where A6

(= 6.7 × 10−16 meVm2

), B6

(= 4.2 × 10−16 meVm2

), ϑ(= (6 ×

10−8 meVm)2) and �(=(3.8 × 10−8 meVm

)2) are the band constants.(2) The energy spectrum of the carriers in the two higher valance bands andthe single lower valance band of Te can respectively be expressed as [65]

E = A10k2z + B10k

2s ±[∆2

10 + (β10kz)2]1/2

and E = ∆|| + A10k2z + B10k

2s ± β10kz (R2.19b)

where E is the energy measured within the valance bands, A10 =3.77 × 10−19 eVm2, B10 = 3.57 × 10−19 eVm2,∆10 = 0.628 eV, (β10)

2 =6 × 10−20 (eVm)2 and ∆|| = 1004 × 10−5 eV are the spectrum constants.

(K) The dispersion relation for the electrons in graphite can be written fol-lowing Brandt [66] as

E =12

[E2 + E3] ±[14

(E2 − E3)2 + η2

2k2

]1/2

, (R2.20)


where E2 ≡ ∆ − 2γ1 cos φ0 + 2γ5 cos2 φ0, φ0 ≡ c6kz

2 , E3 ≡ 2γ2 cos2 φ0,

η2 ≡(√

32

)a6 (γ0 + 2γ4 cos φ0) , in which the band constants are ∆ =

−0.0002 eV, γ0 = 3 eV, γ1 = 0.392 eV, γ2 = −0.019 eV, γ4 = 0.193 eV, γ5 =0.194 eV, a6 = 2.46 A and c6 = 6.74 A.

(L) The dispersion relation of the conduction electrons in Antimony (Sb) inaccordance with Ketterson [67] can be written as

2m0E = α11p2x + α22p

2y + α33p

2z + 2α23pypz, (R2.21)

and

2m0E = a1p2x + a2p

2y + a3p

2z + a4pypz ± a5pxpz ± a6pxpy, (R2.22)

where a1 = 14 (α11 + 3α22) , a2 = 1

4 (α22 + 3α11) , a3 = α33, a4 = α33, a5 =√3 and a6 =

√3 (α22 − α11) in which α11 = 16.7, α22 = 5.98,

α33 = 11.61 and α23 = 7.54 are the system constants.(M) The dispersion relation of the holes in p − Bi2Te3 can be written [68] as

E

(1 +

E

Eg

)=

�2

2m0

[α11k

2x + α22k

2y + α33k

2z + 2α23kykz

], (R2.23)

where x, y and z are parallel to binary, bisectrix and trigonal axes respectively,Eg = 0.145 eV, α11 = 32.5, α22 = 4.81, α33 = 9.02, α23 = 4.15, gs = 2 andgv = 6.(N) The dispersion relation of the holes in p-InSb in accordance withCunningham [69] can be written as

E = c4 (1 + γ4f4) k2 ± 13

[2√

2√

c4

√16 + 5γ4

√E4g4k

], (R2.24)

where E is the energy of the hole as measured from the top of the valanceand within it, c4 ≡ �

2

2m0+ θ4, θ4 ≡ 4.7 �

2

2m0, γ4 ≡ b4

c4, b4 ≡ 3

2b5 + 2θ4, b5 ≡2.4 �

2

2m0, f4 ≡ 1

4

[sin2 2θ + sin4 θ sin2 2φ

], θ is measured from the positive

z-axis, φ is measured from positive x-axis, g4 ≡ sin θ[cos2 θ + 1

4 sin4 θ sin2 2φ]

and E4 = 5 × 10−4 eV.(O) The dispersion relation of the valance bands of II–V compounds in accor-dance with Yamada [70] can be written as

E =12

(t1 + t2) k2x +

12

(t3 + t4) k2y +

12

(t5 + t6) k2z +

12

(t7 + t8) kx

± [{

12

(t1 − t2) k2x +

12

(t3 − t4) k2y +

12

(t5 − t6) k2z +

12

(t7 − t8) kx

}2

+t29k2y + t210 ]1/2

, (R2.25)

where ti (i = 1to8) , t9 and t10 are the constants of the energy spectra.


For p − CdSb, t1 = −32.3 × 10−20 eVm2, t2 = −60.7 × 10−20 eVm2, t3 =−1.63 × 10−19 eVm2, t4 = −2.44 × 10−19 eVm2, t5 = −9.19 × 10−19 eVm2,t6 = −10.5 × 10−19 eVm2, t7 = 2.97 × 10−10 eVm, t8 = −3.47 × 10−10 eVm,,t9 = 1.3 × 10−10 eVm and t10 = 0.070 eV.

(P) The energy spectrum of the valance bands of CuCl in accordance withYekimov et al. [71] can be written as

Eh = (γ6 − 2γ7)�

2k2

2m0, (R2.26)

and

El,s = (γ6 + γ7)�

2k2

2m0− ∆1

2±[

∆21

4+ γ7∆1

�2k2

2m0+ 9(

γ7�2k2

2m0

)2]1/2

,

(R2.27)where γ6 = 0.53, γ7 = 0.07,∆1 = 70meV.(Q) In the presence of stress, χ6 along <001> and <111> directions, the en-ergy spectra of the holes in semiconductors having diamond structure valancebands can be respectively expressed following Roman [72] et al. as

E = A6k2 ±[B2

7k4 + δ26 + B7δ6

(2k2

z − k2s

)]1/2, (R2.28)

and

E = A6k2 ±[B2

7k4 + δ27 +

D6√3δ7

(2k2

z − k2s

)]1/2

, (R2.29)

where A6, B7,D6 and C6 are inverse mass band parameters in whichδ6 ≡ l7

(S11 − S12

)χ6, Sij are the usual elastic compliance constants,

B27 ≡

(B2

7 + c265

)and δ7 ≡

(d8S44

2√

3

)χ6. For gray tin,d8 = −4.1 eV,

l7 = −2.3 eV,

A6 = 19.2�

2

2m0, B7 = 26.3

�2

2m0,D6 = 31

�2

2m0and c2

6 = −1112�

2

2m0.

R2.2 Investigate the Einstein relation for all materials of problem (R2.1), inthe presence of an arbitrarily oriented non-quantizing and (a) non-uniformelectric field (b) alternating electric field respectively.

Allied Research Problems

R2.3 Investigate the Debye screening length, the carrier contribution to theelastic constants, the heat capacity, the activity coefficient, and the plasmafrequency for all the materials of problem (R2.1).

R2.4 Investigate in detail, the mobility for elastic and inelastic scatteringmechanisms for all the materials of problem (R2.1).

R2.5 Investigate the various transport coefficients in detail for all the ma-terials of problem (R2.1).


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