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.     

Generalized Kane model for the band spectrum of diamond-like semiconductors. Nonparabolicity and anisotropy of the dispersion relations

Available experimental data indicate that the Kane models for the band structure of diamond-like semiconductors do not sufficiently correctly describe the effects of the anisotropy and nonparabolicity of the valence bands. In [1], we carried out parallel examinations of the spectrum for all diamond-like semiconductors with a lower conduction band of the s-type within the four-band kp model, which is a generalization of both Kane models. We obtained a dispersion equation of fourth degree (DE-4s), allowing us to refine the range of applicability of the Kane models. This paper is a continuation of [1] in which we do a detailed analysis of the nonparabolicity and anisotropy of the dispersion relations for the holes. We show that neither of the Kane models correctly describe the state of the light-hole band over a broad energy range. At the same time, the heavy-hole band is correctly described by the Kane equation (1956); the conduction band and the split-off hole band are correctly described by the Kane theory (1957).Ukrainian Engineering Institute for Utilization and Control of Water. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 109–114, November, 1992.     

Band mixing in narrow gap semiconductors

The interaction of conduction and valence bands in narrow gap semiconductors such as InSb and HgCdTe influences the position and width of subband energy levels in space-charge layers. While a nonzero width can only occur if electrons from the conduction band can tunnel into approximately degenerate states of the valence band the level shifts due to band mixing are always present. We present a Green's function treatment which allows in a simple way to discuss the dependence of band mixing effects on the parameters of thek·p-Hamiltonian in particular the band gap. The essential qualitative feature of the level shifts is adecrease of subband energy separation withdecreasing effective mass. This agrees with recent experimental results for Hg_1-x Cd _x Te.     

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.     

The Kane oscillator

The energy spectrum and wave functions of the Kane oscillator are determined. The Kane oscillator equation describes the energy spectrum of electrons, light holes, and a spin-orbit split-off band of holes in a quantum dot with a parabolic confining potential.     

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.     

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.     

On the gate capacitance of MOS structures of kane-type semiconductors under magnetic quantization

An attempt is made to formulate the gate capacitance of MOS structures of Kane-type semiconductors under magnetic quantization, without any approximations of weak or strong electric field limits, on the basis of the fourth-order effective mass theory and taking into account the interactions of the conduction, light-hole, heavy-hole, and split-off bands. It is found, taking n-channel Hg_1–x Cd _x Te as an example, that the gate capacitance exhibits spiky oscillations with changing magnetic field, which is in qualitative agreement with experimental observations, reported elsewhere, in MOS structures of the same semiconductor. The corresponding results for n-channel inversion layers on parabolic semiconductors are also obtained from the expressions derived.     

Righi-Leduc effect in a quantizing magnetic field

The Righi-Leduc effect in semiconductors with a Kane dispersion law in the presence of strong, quantizing, magnetic fields is studied theoretically. The explicit form of the dependence on the magnetic field, temperature, and concentration in arbitrary quantizing magnetic fields is established for semiconductors with a nondegenerate electron gas in the approximation of small nonparabolicity. A simple formula that is applicable for all strong magnetic fields, including quantizing fields, is derived for the Righi-Leduc coefficient in the case of strongly degenerate semiconductors with an arbitrary nonparabolic band. It is shown that in order to determine the photon part of the thermal conductivity ,ph directly from experiment it is best to employ samples with a nondegenerate electron gas in strong, but nonquantizing, magnetic fields.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 102–107, July, 1988.     

Binding energies of helium-like impurities in parabolic quantum wells under an applied electric field

We present a variational method to compute the binding energies of helium-like impurities in finite parabolic GaAs- Ga_1 − xAl_xAs quantum wells. The effects of band nonparabolicity in the conduction band are taken into account within the effective mass approximation. The dependence of the impurity binding energy on the applied electric field and the impurity position is also discussed together with the polarization effect for all cases.     

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.     

Drag of carriers by photons in narrow-gap semiconductors

The current representing the drag of carriers by photons is calculated for cubic narrow-gap semiconductors allowing for the nonparabolicity and nonsphericity of the valence subbands. A quantitative Kane model is used to obtain a general tensor expression for the photocurrent related to the coefficients representing the nonparabolicity of the light-hole subband and the nonsphericity of the heavy-hole subband. The results are given of a numerical calculation of the principal parameters of the drag current in the form of components of the elementary tensors (as a function of temperature) and of the coefficient representing the nonparabolicity (as a function of the wavelength of the incident light) in the case of narrow-gap Hg_1–xCd_xTe solid solutions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 62–66, May, 1980.     

Rashba splitting in MIS structures HgCdTe

The measured parameters of spin-orbit spectral splitting in HgCdTe-based MIS structures with positive and negative Kane gap E _g are compared with the parameters calculated using the three-and four-band Kane model. The disregard of the finite spin-orbit splitting Δ of the valence band in calculations leads to exaggerated values of Rashba splitting (especially for E _g < 0) even for small ratios |E _g|/Δ, although the subband parameters averaged over two spin branches of the spectrum in the two-, three-, and four-band Kane approximations for the same concentrations are practically identical. In the zero-gap HgCdTe, the measured as well as calculated values are noticeably higher, but the four-band approximation leads to values of splitting for both materials which are 20–40% lower than the experimental value. The inclusion of the interband interaction reduces these discrepancies, but does not eliminate them completely. It is shown that the approximations of the 2D spectrum with spin-orbit splitting linear in quasimomentum, which are conventionally used in the analysis, may lower the effective Rashba parameter by a factor of 2–4.     

Nonparabolicity corrections to the valence band energy levels of Si/GexSi1-x quantum wells

In earlier work, subband energies of Si/Ge_xSi_1-x quantum wells were obtained neglecting valence band nonparabolicity. Here we present an algorithm for solving for the energies and wavefunctions of the mixed light-hole and split-off subbands under stress, which incorporates the energy dependence of the effective mass exactly. We show that this gives corrections to the energies that are of the order of tens of meV, and results in small nonorthogonality corrections to the envelope wavefunctions. These corrections are discussed for several cases of physical interest and are compared to the results obtained assuming a constant bandedge effective mass.     

A Novel Approach for the Evaluation of Band Gap Energy in Semiconductors

On the basis of absorption nature of semiconductors, we present a novel and simple method to determine the band gap energies of semiconductors directly from their absorption spectra at any temperatures, without any fitting processes and restrictions of sample thickness. The key point of the approach is the different dependence of the absorption coefficient derivative on the photon energy at different absorption regions in semiconductors. We first demonstrate and verify the approach by detailed temperature-dependent absorption measurements, combined with photoluminescence measurements and empirical band gap equations for the direct band gap of uniform InAs films, and then extend successfully to the indirect band gap of elemental Ge and to the ternary HgCdTe alloys with composition gradient. Furthermore, we have also shown that our approach can not only evaluate the average band gap energy for ternary semiconductor alloys, but also estimate their composition uniformity to monitor the material quality.     

A model of enhanced STM current due to semiconductor optical absorption

We present a simple model for the change in tunneling current between a semiconductor surface and a metal tip under spectroscopic illumination in a scanning tunneling microscope. This model predicts a sharp increase in the tunneling current due to the increase in the conduction band carrier density when the photon energy exceeds the optical band gap. The tunneling current for a large diffusion length has a more pronounced onset than for a small length. Our model should provide, when combined with experiments, a method of determining localized effective stoichiometry, and therefore provides a localized alternative to the use of optical absorption measurements. Our theoretical tunneling current versus photon energy curves are in good qualitative agreement with the existing experimentally measured curves for Si, GaAs, and InP obtained by Qian and Wessels. In addition, we have examined the effects of temperature, surface recombination velocity, and degeneracy on our theoretical results for the Hg_1−xCd_xTe, Hg_1−xZn_xTe, and Hg_1−xZn_xSe ternary narrow gap semiconductor systems.     

Electronic,optical and bonding properties of MgCO3

The electronic, optical and bonding properties of MgCO₃ (magnesite, rhombohedral calcite-type structure) are calculated using a first-principles density-functional theory (DFT) method considering the exchange-correlation function within the local density approximation (LDA) and the generalized gradient approximation (GGA). The indirect band gap of magnesite is estimated to be 5.0 eV, which is underestimated by ～1.0 eV. The fundamental absorption edge, which indicates the exact optical transitions from occupied valence bands to the unoccupied conduction band, is estimated by calculating the photon energy dependent imaginary part of the dielectric function using scissors approximations (rigid shift of unoccupied bands). The optical properties show consistent results with the experimental calcite-type structure and also show a considerable optical anisotropy of the magnesite structure. The density of states and Mulliken population analyses reveal the bonding nature between the atoms.     

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.