Influence of magnetic quantization on the effective electron mass in Kane-type semiconductors

Summary We study the effective electron mass at the Fermi level in Kane-type semiconductors on the basis of fourth order in effective mass theory and taking into account the interactions of the conduction electrons, heavy holes, light holes and split-off holes, respectively. The results obtained are then compared to those derived on the basis of the well-known three-band Kane model. It is found, takingn-Hg_1−x Cd _x Te as an example, that the effective electron mass at the Fermi level in accordance with fourth-order model depends on the Fermi energy, magnetic quantum number and the electron spin respectively due to the influence of band nonparabolicity only. The dependence of effective mass on electron spin is due to spin-orbit splitting parameter of the valence band in three-band Kane model and the Fermi energy due to band nonparabolicity in two-band Kane model. The same mass exhibits an oscillatory magnetic-field dependence for all the band models as expected since the origin of oscillations in the effective mass in nonparabolic compounds is the same as that of the Shubnikov-de Hass oscillations. In addition, the corresponding results for parabolic energy bands have been obtained from the generalized expressions under certain limiting conditions.     

On the electron energy spectrum in heavily doped non-parabolic semiconductors

An attempt is made to present a simple theoretical analysis of the energy-wave vector dispersion relation of the conduction electrons in heavily doped non-parabolic semiconductors forming band tails. We observe that the complex energy spectrum in doped small-gap materials whose unperturbed conduction band is described by the three band model of Kane is due to the interaction of the impurity atoms in the tail with the spin-orbit splitting constant of the valence band (Δ), For band-gap (E_g)<Δ the imaginary part predominates which tails in to the conduction band. For the opposite inequality the real part comes in to play which tails in to the split-off band. In the absence of the band tailing effect, the imaginary part of the complex energy spectrum vanishes and the same is also true for doped two-band Kane-type and parabolic energy bands respectively. The present formulation helps us in investigating the Boltzmann transport equation dependent transport properties of degenerate semiconductors and are expected to agree better with experiments. The well-known results of unperturbed three and two band models of Kane together with wide-gap parabolic energy bands have been obtained as special cases of our generalized analysis under certain limiting conditions.     

Nonparabolicity and warping in the conduction band of GaAs

The dispersion of the conduction band in GaAs is calculated using k·p models which in different ways take into account the coupling to the p-bonding and p-antibonding states. Nonparabolicity, warping and spin-splitting are accurately described up to energies about 50 meV above the conduction band minimum by the 8×8 Kane model. For higher energies a 14×14 matrix is required.     

Simultaneous Effects of External Electric Field and Conduction Band Nonparabolicity on Optical Properties of a GaAs Quantum Dot Embedded at the Center of a GaAlAs Nano-Wire

An investigation of the optical properties of a GaAs spherical quantum dot which is located at the center of a Ga1-xAlx As cylindrical nano-wire has been performed in the presence of an external electric field. The band nonparabolieity effect is also considered using the energy dependent effective mass approximation. The energy eigenvalues and corresponding wave functions are calculated by finite difference approximation and the reliability of calculated wave functions is checked by computing orthogonality. Using computed energy eigenvalues and wave functions, the linear, third-order nonlinear and total optical absorption coefficients and refractive index changes are examined in detail. It is found that （i） Presence of electric field causes both blue and red shifts in absorption spectrum; （ii） The absorption coefficients shift toward lower energies by taking into account the conduction band nonparabolicity; （iii） For large values of electric field the effect of conduction band nonparabolieity is less dominant and parabolic band is estimated correctly; （iv） In the presence of electric field and conduction band nonparabolicity the nonlinear term of absorption coefficient rapidly increases by increasing incident optical intensity. In other words, the saturation in optical spectrum occurs at lower incident optical intensities.     

Magnetic-field-induced nonparabolicity of exciton dispersion in semiconductors with a nondegenerate valence band

The nonparabolicity of exciton dispersion due to the mixing of the ground and excited states of an exciton in an external magnetic field perpendicular to the direction of its motion is considered. A model describing this effect is proposed and the nonparabolicity for an exciton in a CdTe crystal is calculated. The magnetic-field induced exciton nonparabolicity is compared with the effect caused by the nonparabolicity of the electron energy dispersion in the conduction band.     

Effect of conduction band nonparabolicity on the optical properties in a single quantum well under hydrostatic pressure and electric field

The effect of conduction band nonparabolicity on the linear and nonlinear optical properties such as absorption coefficients, and changes in the refractive index are calculated in the Al_0.3Ga_0.7As/GaAs heterostructure-based symmetric rectangular quantum well under applied hydrostatic pressure and electric field. The electron envelope functions and energies are calculated in the effective mass equation including the conduction band nonparabolicity. The linear and nonlinear optical properties have been calculated in the density matrix formalism with two-level approximation. The conduction band nonparabolicity shifts the positions of the optical properties and decreases their strength compared to those without this correction. Both the optical properties are enhanced with the applied hydrostatic pressure. While the absorption coefficients are bleached under the combined effect of high pressure and electric field, the bleaching effect is reduced when nonparabolicity is included.     

Effective mass of heavy holes in diamond-like semiconductors

Nonparabolicity of the heavy hole band in diamond-like semiconductors, which occurs within the framework of the three band model with the perturbation from the other bands taken into account to the Löwdin prucedure, is studied. A direct dependence of nonparabolicity on the band anisotropy (caused by the different effect of _15c and _12c bands) and the inverse dependence on the magnitude of the spin-orbit splitting is established. A connection between the effective mass of heavy holes and their energy is obtained, which is valid for the majority of diamond-like semicondactors, except for materials with very strong nonparabolicity of the band of silicon type.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 66–70, February, 1987.     

Subband states inn-inversion layers on small-gap semiconductors

Subband states inn-inversion layers on small-gap semiconductors are subject to the coupling between valence and conduction band (nonparabolicity effects). In order to account for this coupling we study different models: two of them are based on Kane's 6×6 and 8×8 bulk-Hamiltonians, the third one takes into account higher order terms of the electron momentum in a 2×2 conduction band Hamiltonian. We perform selfconsistent calculations for these models with parameters characteristic for InSb and HgCdTe and electron concentrationsN _S , for which up to two subbands are occupied. The calculated subband separations and Fermi energies are independent of the models only if the same energy band dispersion is used and depend strongly on the applied boundary conditions.Work supported in part by the Deutsche Forschungsgemeinschaft     

Effect of Band Nonparabolicity on the Inter Band Tunneling in Semiconductors

A simple yet generalized theory is developed to study inter band tunneling property of narrow band gap III–V compound semiconductors. The band structures of these low band gap semiconductors with sufficiently separated split-off valance band are usually described by the three energy band model of Kane, so this has been adopted here for the analysis of interband tunneling property in the case of InAs, InSb, and In_1-xGa_xAs_yP_1-y lattice matched to InP as representative direct band gap semiconductors having varied split-off valence band compared to their bulk state band gap energy. It has been found that the magnitude of tunneling rate from heavy hole decreases with increasing band nonparabolicity and the impact is more significant at high electric field in the three-band model of Kane than those with simple parabolic energy band approximations reflecting the direct influence of energy band parameters on inter band tunneling transitions. With proper consideration of band nonparabolicity, the results of the analysis of tunneling rate of these narrow gap materials show significant deviations from the results when simple parabolic band approximation is considered. The exact physical basis of the sources of deviation in the nonparabolic case from the corresponding parabolic band approximations is discussed in association to band coupling effect, transverse energy dependence, and the interplay between them. Moreover, under certain limiting conditions, our results reduce to the well-known results of parabolic band approximation and thus providing an indirect test to the accuracy of our generalized formulations.     

On the photoemission from III–V, ternary and quaternary materials: Simplified theory and relative comparison

We study theoretically the electron energy spectrum and the photoemission from III–V, ternary and quaternary materials in the presence of light waves, whose unperturbed energy band structures are defined by the three-band model of Kane. The band gap of semiconductors increases as a consequence of incident light waves and we have suggested two new experimental methods of determining the band gap of semiconductors in the presence of photoexcitations. The solution of the Boltzmann transport equation on the basis of this newly formulated electron dispersion law will introduce new physical ideas and experimental findings in the presence of external photoexcitation. It has been found taking n-InAs, n-InSb, n-Hg_1−xCd_xTe and n-In_1−xGa_xAs_yP_1−y lattice matched to InP, as examples that the photoemission increases with the increase in electron concentration and decreases in increasing intensity, wavelength and alloy composition, respectively, in various manners. The numerical values of the photoemission in the presence of light waves is less than that of the same for unperturbed three- and two-band models of Kane together with parabolic energy bands for all types of external variables. The strong dependence of the photoemission on the light intensity reflects the direct signature of light waves on the dispersion relation of the conduction electrons, which is in contrast when compared with the corresponding bulk specimens for the unperturbed band models. The rate of change is totally band structure dependent and is significantly influenced by the presence of the different energy band constants. The well-known result of the photoemission from non-degenerate wide gap materials has been obtained as a special case of the present analysis under certain limiting conditions and this compatibility is the indirect test of our generalized formalism. Besides, we have suggested six important applications of our results in this context.     

Electronic states in a narrow band gap semiconductor microcrystal with parabolic confinement in magnetic field

The energy levels of electrons in a narrow band gap semiconductor microcrystal under the influence of magnetic field are investigated. The confinement potential of microcrystal is approximated as parabolic, and the electron dispersion law is considered within the framework of two-band Kane model. It has been shown that nonparabolicity of dispersion law results in the appearance of the “anharmonic” term in Hamiltonian. The values of magnetic field at which the “anharmonic” term can be considered as perturbation are found. Results of electron energy of nonperturbed Hamiltonian dependencies on values of magnetic field and frequency of microcrystal confinement potential are presented. A comparison of the obtained results with the other cases has been done.     

Application of band theory to experimental eigen-state energies of In0.53Ga0.47As quantum wells lattice-matched to InP

Nonparabolic band structure of InGaAs/InAlAs multi-quantum wells was studied theoretically and experimentally. The electron effective mass was derived even from eigen-states and this nonparabolicity was explicitly determined as a function of energy. Electron eigen-state energies applying Kane’s bulk band theory fitted very well with our experiments in the multi-quantum wells.     

Warping of the bulk dispersion of InSb

We calculate the warping of the bulk dispersion of InSb in thek-space using different models for the bulk band structure near the Γ point. It is shown that the dispersion of the conduction band Γ₆ is well described by the simplified six-band model, while the fourteen-band model is more accurate for the valence bands.     

Electronic-spin generation in multi-photon processes: Cubic anisotropy of pumping

We study the spin generation in multi-photon absorption processes in a bulk semiconductor, pumped it by a circularly polarized intense light by varying the angle of incidence. The generated spin polarization is calculated using the eight-band Kane model in the limit of large spin–orbit splitting and on the basis of the multi-photon photo-generation rate of the conduction electron spin density. It is found that the spin polarization strongly depends on the pumping photon energy outside the band edge. Cubic anisotropy in crystal pumping is also calculated. The results show that due to this anisotropy the spin generation differs by ∼8%, in consistence with earlier results obtained by others.     

Rashba spin-orbit splitting energy of a pressure induced hydrogenic donor impurity in a corrugated quantum well

Pressure induced hydrogenic donor impurity of ground and excited state in a GaAlAs/GaAs/GaAlAs corrugated quantum well is investigated. The calculations have been carried out using variational technique within the single band effective mass approximation taking into account the anisotropy and the corrections due to the conduction band nonparabolicity. The energy dependent effective mass and the position dependent quantity (Rashba spin-orbit splitting energy) are introduced to obtain the binding energy as a function of well width in the influence of pressure. The obtained results are compared with the other existing literature available.     

Tamm surface states of Hg1-xCdxTe

Using the Kane model, the wave functions and dispersion laws were obtained for Tamm surface states of electrons and holes arising on the ideal surface of Hg_1-xCd_xTe crystal in the parabolic and strong nonparabolic limits. The dependence of surface electrons and holes effective masses, in the parabolic limit, is determined from the parameters of the bulk states. The surface states of heavy holes is shown to be unsensitive to the degree of nonparabolicity of the electron spectrum.     

Quantum carpets of a slightly relativistic particle

We analyze the structures emerging in the spacetime representation of the probability density woven by a slightly relativistic particle caught in a one‐dimensional box. In particular, we evaluate the relativistic effects on the revival time and the specific changes produced in the intermode traces, which quantum carpets consist of. Moreover, we present a detailed mathematical analysis of such quantum carpets pursuing the approach of a kernel. Here we represent the probability distribution as a superposition of interfering Airy function‐type structures along straight world lines. We also show that this phenomenon can be enhanced by many orders of magnitude in semiconductors with narrow band‐gap (e.g. as in InSb) and small effective mass of the electron, whereby due to the strong nonparabolicity of the semiconductor conduction band, the electron energy vs momentum dispersion relation behaves in a pseudo‐relativistic way.