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Journal ArticleDOI

Effect of magnetic quantization on the Einstein relation in kane-type semiconductors

01 Dec 1986-Czechoslovak Journal of Physics (Kluwer Academic Publishers)-Vol. 36, Iss: 12, pp 1396-1402
TL;DR: In this article, an attempt is made to formulate the Einstein relation for the diffusivity-mobility ratio of the carriers in degenerate semiconductors having Kane-type energy bands in the presence of a quantizing magnetic field, taking degenerate n-InAs as an example.
Abstract: An attempt is made to formulate the Einstein relation for the diffusivity-mobility ratio of the carriers in degenerate semiconductors having Kane-type energy bands in the presence of a quantizing magnetic field, taking degenerate n-InAs as an example. It is found on the basis of the three-band Kane model, which is the most valid model for n-InAs, that the same ratio oscillates with changing magnetic field only under degenerate conditions and remains unaffected otherwise. The corresponding results for parabolic semiconductors are also obtained from the expressions derived.
Citations
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Book ChapterDOI
01 Jan 2020
TL;DR: In this article, the authors derived the quantum capacitance in quantum wire field effect transistors (QWFETs) manufactured from completely different technologically vital nonstandard materials by using all types of anisotropies of band structures in addition to splitting of bands due to large fields of the crystals inside the framework of Kane's matrix methodology that successively generates new 1D dimensional electron energy versus wave vector relation.
Abstract: This chapter explores the quantum capacitance (\( C_{\text{g}} \)) in quantum wire field-effect transistors (QWFETs) manufactured from completely different technologically vital nonstandard materials by using all types of anisotropies of band structures in addition to splitting of bands due to large fields of the crystals inside the framework of Kane’s matrix methodology that successively generates new 1D dimensional electron energy versus wave vector relation. We derive the \( C_{\text{g}} \) under very low temperature so that the Fermi function tends to unity for QWFETs of \( {\text{Cd}}_{3} {\text{As}}_{2} ,{\text{CdGeAs}}_{2} ,{\text{InSb}},{\text{Hg}}_{1 - x} {\text{Cd}}_{x} {\text{Te}},{\text{InAs}},{\text{GaAs}},{\text{In}}_{1 - x} {\text{Ga}}_{x} {\text{As}}_{y} {\text{P}}_{1 - y} \) IV–VI, stressed materials,\( {\text{Te}},{\text{GaP,PtSb}}_{2} ,{\text{Bi}}_{2} {\text{Te}}_{3} ,{\text{Ge}},{\text{GaSb}} \) and II–V compounds using the appropriate band models. The \( C_{\text{g}} \) becomes the functions of the thickness of the quantum-confined transistors. The \( C_{\text{g}} \) varies with varying film thickness in various quantized steps and saw-tooth manners with different numerical values.

4 citations

Book ChapterDOI
01 Jan 2021
TL;DR: In this article, the carrier statistics in quantized extremely degenerate III-V, ternary, quaternary and tetragonal compounds were studied and the influence of photo-excitation and electric field on the Fermi energy was investigated.
Abstract: In this chapter, we study the carrier statistics (CS) in quantized extremely degenerate III–V, ternary, quaternary and tetragonal compounds respectively. We have also investigated the influence of photo-excitation and electric field on the Fermi energy. We note by taking various types of opto-electronic materials as examples that the Fermi energy oscillates with inverse magnetic field due to SdH effect, changes with changing electric field, light intensity, wave length and alloy composition in different ways which are totally energy band constants dependent.
References
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Journal ArticleDOI
Evan O. Kane1
TL;DR: The band structure of InSb is calculated using the k ·. p perturbation approach and assuming that the conduction and valence band extrema are at k = 0 as mentioned in this paper.

2,905 citations

Book
01 Jan 1962

1,408 citations

Book
06 May 1980
TL;DR: In this paper, the Boltzmann Transport Equation is used to calculate the collision probability of the Sphalerite and the Chalcopyrite structures, and the Brillouin Zone is used for the Wurtzite structure.
Abstract: 1. Introduction.- 1.1 Historical Note.- 1.2 Applications.- 1.3 Transport Coefficients of Interest.- 1.4 Scope of the Book.- 2. Crystal Structure.- 2.1 Zinc-Blende Structure.- 2.2 Wurtzite Structure.- 2.3 Rock-Salt Structure.- 2.4 Chalcopyrite Structure.- 3. Energy Band Structure.- 3.1 Electron Wave Vector and Brillouin Zone.- 3.2 Brillouin Zone for the Sphalerite and Rock-Salt Crystal Structure.- 3.3 Brillouin Zone for the Wurtzite Structure.- 3.4 Brillouin Zone for the Chalcopyrite Structure.- 3.5 E-k Diagrams.- 3.5.1 Energy Bands for the Sphalerite Structure.- 3.5.2 Energy Bands for the Wurtzite Structure.- 3.5.3 Energy Bands for the Rock-Salt Structure.- 3.5.4 Band Structure of Mixed Compounds.- 3.6 Conclusion.- 4. Theory of Efiergy Band Structure.- 4.1 Models of Band Structure.- 4.2 Free-Electron Approximation Model.- 4.3 Tight-Binding Approximation Model.- 4.4 Energy Bands in Semiconductor Super!attices.- 4.5 The k-p Perturbation Method for Derivating E-k Relation.- 4.5.1 Single-Band Perturbation Theory.- 4.5.2 Two-Band Approximation.- 4.5.3 Effect of Spin-Orbit Interaction.- 4.5.4 Nonparabolic Relation for Extrema at Points Other than the r Point.- 4.6 External Effects on Energy Bands.- 4.6.1 Effects of Doping.- 4.6.2 Effects of Large Magnetic Fields.- 5. Electron Statistics.- 5.1 Fermi Energy for Parabolic Bands.- 5.2 Fermi Energy for Nonparabolic Bands.- 5.3 Fermi Energy in the Presence of a Quantising Magnetic Field.- 5.3.1 Density of States.- 5.3.2 Fermi Level.- 5.4 Fermi Energy and Impurity Density.- 5.4.1 General Considerations.- 5.4.2 General Formula.- 5.4.3 Discussion of Parabolic Band.- 5.4.4 Effect of Magnetic Field.- 5.5 Conclusions.- 6. Scattering Theory.- 6.1 Collision Processes.- 6.2 Transition Probability.- 6.3 Matrix Elements.- 6.4 Free-Carrier Screening.- 6.5 Overlap Integrals.- 6.6 Scattering Probability S(k).- 6.6.1 S(k) for Ionised Impurity Scattering.- 6.6.2 S(k) for Piezoelectric Scattering.- 6.6.3 S(k) for Deformation-Potential Acoustic Phonon Scattering.- 6.6.4 S(k) for Polar Optic Phonon Scattering.- 6.6.5 S(k) for Intervalley and Nonpolar Optic Phonon Scattering.- 6.7 Scattering Probabilities for Anisotropic Bands.- 6.7.1 Herring-Vogt Transformation.- 6.7.2 Scattering Integrals.- 6.8 S(k) for Neutral Impurity, Alloy, and Crystal-Defect Scattering.- 6.8.1 Neutral-Impurity Scattering.- 6.8.2 Alloy Scattering.- 6.8.3 Defect Scattering.- 6.9 Conclusions.- 7. The Boltzmann Transport Equation and Its Solution.- 7.1 The Liouville Equation and the Boltzmann Equation.- 7.2 The Boltzmann Transport Equation.- 7.3 The Collision Integral.- 7.4 Linearised Boltzmann Equation.- 7.5 Simplified Form of the Collision Terms.- 7.5.1 Collision Terms for Elastic Scattering.- 7.5.2 Collision Terms for Inelastic Scattering.- 7.6 Solution of the Boltzmann Equation.- 7.6.1 Relaxation-Time Approximation.- 7.6.2 Variational Method.- 7.6.3 Matrix Method.- 7.6.4 Iteration Method.- 7.6.5 Monte Carlo Method.- 7.7 Method of Solution for Anisotropic Coupling Constants and Anisotropic Electron Effective Mass.- 7.7.1 Solution for Elastic Collisions.- 7.7.2 Solution for Randomising Collisions.- 7.7.3 Solution for Nonrandomising Inelastic Collisions.- 7.8 Conclusions.- 8. Low-Field DC Transport Coefficients.- 8.1 Evaluation of Drift Mobility.- 8.1.1 Formulae for Relaxation-Time Approximation.- 8.1.2 Evaluation by the Variational Method.- 8.1.3 Evaluation by Matrix and Iteration Methods.- 8.1.4 Evaluation by the Monte Carlo Method.- 8.2 Drift Mobility for Anisotropic Bands.- 8.2.1 Ellipsoidal Band.- 8.2.2 Warped Band.- 8.3 Galvanomagnetic Transport Coefficients.- 8.3:1 Formulae for Hall Coefficient, Hall Mobility, and Magnetoresistance.- 8.3.2 Reduced Boltzmann Equation for the Galvanomagnetic Coefficients.- 8.3.3 Solution Using the Relaxation-Time Approximation Method.- 8.3.4 A Simple Formula for the Low-Field Hall Mobility.- 8.3.5 Numerical Methods for the Galvanomagnetic Coefficients for Arbitrary Magnetic Fields.- 8.3.6 Evaluation of the Galvanomagnetic Transport Coefficients for Anisotropic Effective Mass.- 8.4 Transport Coefficients for Nonuniform conditions.- 8.4.1 Diffusion.- 8.4.2 Thermal Transport Coefficients.- 8.4.3 Formula for Thermoelectric Power.- 8.4.4 Electronic Thermal Conductivity.- 8.5 Conclusions.- 9. Low-Field AC Transport Coefficients.- 9.1 Classical Theory of AC Transport Coefficients.- 9.1.1 Solution for the Relaxation-Time Approximation.- 9.1.2 Solution for Polar Optic Phonon and Mixed Scattering.- 9.1.3 Solution for Nonparabolic and Anisotropic Bands.- 9.2 AC Galvanomagnetic Coefficients.- 9.3 Cyclotron Resonance and Faraday Rotation.- 9.3.1 Propagation of Electromagnetic Waves in a Semiconductor in the Presence of a Magnetic Field.- 9.3.2 Cyclotron Resonance Effect.- 9.3.3 Faraday Rotation.- 9.4 Free-Carrier Absorption (FCA).- 9.4.1 Classical Theory of FCA.- 9.4.2 Quantum-Mechanical Theory of FCA.- 9.5 Concluding Remarks.- 10. Electron Transport in a Strong Magnetic Field.- 10.1 Scattering Probabilities.- 10.2 Mobility in Strong Magnetic Fields.- 10.3 Electron Mobility in the Extreme Quantum Limit (EQL).- 10.3.1 Electron Mobility for Polar Optic Phonon Scattering in the EQL.- 10.4 Oscillatory Effects in the Magnetoresistance.- 10.4.1 Shubnikov-de Haas Effect.- 10.4.2 Magnetophonon Oscillations.- 10.5 Experimental Results on Magnetophonon Resonance.- 10.6 Conclusions.- 11. Hot-Electron Transport.- 11.1 Phenomenon of Hot Electrons.- 11.2 Experimental Characteristics.- 11.3 Negative Differential Mobility and Electron Transfer Effect.- 11.4 Analytic Theories.- 11.4.1 Differential Equation Method.- 11.4.2 Maxwellian Distribution Function Method.- 11.4.3 Displaced Maxwellian Distribution Function Method.- 11.5 Numerical Methods.- 11.5.1 Iteration Method.- 11.5.2 Monte Carlo Method.- 11.6 Hot-Electron AC Conductivity.- 11.6.1 Phenomenological Theory for Single-Valley Materials.- 11.6.2 Phenomenological Theory for Two-Valley Materials.- 11.6.3 Large-Signal AC Conductivity.- 11.7 Hot-Electron Diffusion.- 11.7.1 Einstein Relation for Hot-Electron Diffusivity.- 11.7.2 Electron Diffusivity in Gallium Arsenide.- 11.7.3 Monte Carlo Calculation of the Diffusion Constant.- 11.8 Conclusion.- 12. Review of Experimental Results.- 12.1 Transport Coefficients of III-V Compounds.- 12.1.1 Indium Antimonide.- 12.1.2 Gallium Arsenide.- 12.1.3 Indium Phosphide.- 12.1.4 Indium Arsenide.- 12.1.5 Indirect-Band-Gap III-V Compounds.- 12.2 II-VI Compounds.- 12.2.1 Cubic Compounds of Zinc and Cadmium.- 12.2.2 Wurtzite Compounds of Zinc and Cadmium.- 12.2.3 Mercury Compounds.- 12.3 IV-VI Compounds.- 12.4 Mixed Compounds.- 12.5 Chalcopyrites.- 12.6 Conclusion.- 13. Conclusions.- 13.1 Problems of Current Interest.- 13.1.1 Heavily Doped Materials.- 13.1.2 Alloy Semiconductors.- 13.1.3 Transport Under Magnetically Quantised Conditions.- 13.1.4 Inversion Layers.- 13.1.5 Superlattices and Heterostructures.- 13.2 Scope of Further Studies.- Appendix A: Table of Fermi Integrals.- Appendix B: Computer Program for the Evaluation of Transport Coefficients by the Iteration Method.- Appendix C: Values of a. and b. for Gaussian Quadrature Integration. 417 Appendix D: Computer Program for the Monte Carlo Calculation of Hot-Electron Conductivity and Diffusivity.- List of Symbols.- References.

716 citations

Journal ArticleDOI
TL;DR: In this paper, the ratio of the diffusion coefficient of electrons to their mobility is kT/q, in close analogy with a result first obtained by Einstein for Brownian motion.
Abstract: For pt.II see ibid., vol.2, p.213 (1981). The ratio of the diffusion coefficient of electrons to their mobility is kT/q, in close analogy with a result first obtained by Einstein for Brownian motion. A similar relation holds for electrons and holes in semiconductors and it has been generalised in various ways. It also enters into theories for the screening length and for the velocity of sound in solids.

98 citations

Book
01 Jan 1971
TL;DR: In this article, a composite table of aluminum antimonide, aluminum arsenide, and aluminum nitride is presented, along with aluminum phosphide and gallium arsenide, respectively.
Abstract: Composite Table.- Aluminum Antimonide.- Aluminum Arsenide.- Aluminum Nitride.- Aluminum Phosphide.- Boron Arsenide.- Boron Nitride.- Boron Phosphide.- Gallium Antimonide.- Gallium Arsenide.- Gallium Nitride.- Gallium Phosphide.- Indium Antimonide.- Indium Arsenide.- Indium Bismuth.- Indium Nitride.- Indium Phosphide.

62 citations