# Direct band-gap energy of semiconductors

TL;DR: In this article, the direct band-gap energy of semiconductors and the mean atomic number of the constituent atoms were derived for all III-V compounds except those containing indium antimonide.

Abstract: Empirical formulae are obtained relating the direct band-gap energy of semiconductors and the mean atomic number of the constituent atoms. The formulae are used to indicate doubtful values and to obtain the values of bowing parameters of energy band-gaps of ternaries and quaternaries. Derived values of the bowing parameter give band-gap energies of ternaries and quaternaries agreeing with experiments to within 0.3-7.3% for all III-V compounds except those containing indium antimonide.

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TL;DR: In this paper, basic knowledge of thermoelectric materials and an overview of parameters that affect the figure of merit ZT are provided, as well as the prospects for the optimization and their applications are also discussed.

Abstract: Developing thermoelectric materials with superior performance means tailoring interrelated thermoelectric physical parameters – electrical conductivities, Seebeck coefficients, and thermal conductivities – for a crystalline system. High electrical conductivity, low thermal conductivity, and a high Seebeck coefficient are desirable for thermoelectric materials. Therefore, knowledge of the relation between electrical conductivity and thermal conductivity is essential to improve thermoelectric properties. In general, research in recent years has focused on developing thermoelectric structures and materials of high efficiency. The importance of this parameter is universally recognized; it is an established, ubiquitous, routinely used tool for material, device, equipment and process characterization both in the thermoelectric industry and in research. In this paper, basic knowledge of thermoelectric materials and an overview of parameters that affect the figure of merit ZT are provided. The prospects for the optimization of thermoelectric materials and their applications are also discussed.

663 citations

### Cites background from "Direct band-gap energy of semicondu..."

...This spreading should cause a lowering of the band-gap energy according to two theories that describe these energy bands: the free-electron theory and the tight-binding approximation theory [113]....

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TL;DR: An overview of the understanding of correlations between energy gap and refractive index of semiconductors is presented in this article, where the Ravindra relation is discussed in the context of alternate approaches that have been presented in the literature.

Abstract: An overview of the understanding of correlations between energy gap and refractive index of semiconductors is presented here. The atomic approach of Moss, the nearly free electron model of Penn, the oscillator concept of Wemple and the optical polarizability approach of Finkenrath are considered in this study. The Ravindra relation is discussed in the context of alternate approaches that have been presented in the literature. Case studies of applications of these relations to infrared materials and wide band gap semiconductors are presented.

374 citations

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TL;DR: This paper presents a meta-analyses of the determinants of infectious disease in eight operation theatres of the immune system and three of them were connected to each other by a simple probabilistic test.

Abstract: John Androulakis , Sebastian C. Peter , Hao Li , Christos D. Malliakas , John A. Peters , Zhifu Liu , Bruce W W. essels , Jung-Hwan Song , Hosub Jin , Arthur J. reeman , F and Mercouri G. Kanatzidis *

163 citations

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TL;DR: In this paper, a zinc tin sulfide aerogel was constructed by metathesis reactions between Zn(acac)2·H2O and tetrahedral thiostannate cluster salts containing discrete [SnS4]4-, [Sn2S6] 4-, and [Sn4S10]4- units.

Abstract: Porous zinc tin sulfide aerogel materials were constructed by metathesis reactions between Zn(acac)2·H2O and tetrahedral thiostannate cluster salts containing discrete [SnS4]4-, [Sn2S6]4-, and [Sn4S10]4- units. Self-assembly reactions of the Zn2+ linker and anionic thiostannate clusters yielded polymeric random Zn/Sn/S networks with gelation properties. Supercritical drying of the gels and solvent/counterion removal resulted in a metal sulfur framework. Zn2SnxS2x+2 (x = 1, 2, 4) aerogels showed high surface areas (363−520 m2/g) and pore volumes (1.1−1.5 cm3/g), and wide bandgap energies (2.8−3.2 eV). Scanning and transmission electron microscopy studies show the pores are micro- (d 50 nm) regions. The zinc chalcogenide aerogels also possess high affinities toward soft heavy metals and reversible absorption of strong electron-accepting molecules.

81 citations

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TL;DR: In this article, the electronic band structure of GaxIn1−xAs alloy is calculated by using the local empirical pseudo-potential method including the effective disorder potential in the virtual crystal approximation.

Abstract: The electronic band structure of GaxIn1−xAs alloy is calculated by using the local empirical pseudo-potential method including the effective disorder potential in the virtual crystal approximation. The compositional effect of the electronic energy band structure of this alloy is studied with composition x ranging from 0 to 1. Various physical quantities such as band gaps, bowing parameters, refractive indices, and high frequency dielectric constants of the considered alloys with different Ga concentrations are calculated. The effects of both temperature and hydrostatic pressure on the calculated quantities are studied. The obtained results are found to be in good agreement with the available experimental and published data.

29 citations

##### References

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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

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TL;DR: In this paper, the problem of the band structure of semiconductor alloy systems is treated by both the dielectric two-band method and by the use of an empirical (local) pseudopotential.

Abstract: The problem of the band structure of semiconductor alloy systems is treated by both the dielectric two-band method and by the use of an empirical (local) pseudopotential. With both methods, calculations are made in the virtual-crystal approximation assuming linear dependence on alloy concentration of the lattice constant and the parameters of the two methods. Contrary to some previous assertions, both methods predict, in general, a nonlinear dependence of the interband gaps on concentration. An estimate is also made of the effects of second-order perturbations to the virtual-crystal approximation, i.e., the effect of disorder. Of particular interest are the lowest direct and indirect energy gaps and the deviations of these from linearity. The treatment is confined to alloys of compounds having the formula ${A}^{N}{B}^{8\ensuremath{-}N}$, but quaternary and more complicated alloys may be treated as easily as the ternary alloys to which most previous experimental work has been confined. Results are compared to experiment and to the empirical formula of Thompson and Woolley. We find that, with one free parameter, the dielectric method gives good agreement with experiment, but that the local-pseudopotential method apparently does not yield satisfactory results for this problem.

548 citations

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TL;DR: In this article, the lattice constant, the lowest direct and indirect gap energies, and the refractive index of a quaternary lattice matched to GaSb and InAs were calculated using an interpolation scheme and the effects of compositional variations were properly taken into account in calculations.

Abstract: The methods for calculation of material parameters in compound alloys are discussed, and the results for AlxGa1−xAsySb1−y, GaxIn1−xAsySb1−y, and InPxAsySb1−x−y quaternaries lattice matched to GaSb and InAs are presented. These quaternary systems may provide the basis for optoelectronic devices operating over the 2–4‐μm wavelength range. The material parameters considered are: the lattice constant, the lowest direct‐ and indirect‐gap energies, and the refractive index. The model used is based on an interpolation scheme, and the effects of compositional variations are properly taken into account in the calculations. Key properties of the material parameters for a variety of optoelectronic device applications are also discussed in detail.

385 citations