Kane Dispersion Law Research Articles

Spontaneous Exciton Collapse in a Strongly Flattened Ellipsoidal InSb Quantum Dot

Electronic and excitonic states in an InSb strongly flattened ellipsoidal quantum dot (QD) with complicated dispersion law are theoretically investigated within the framework of the geometric adiabatic approximation in the strong, intermediate, and weak quantum confinement regimes. For the lower levels of the spectrum, the square root dependence of energy on QD sizes is revealed in the case of Kane’s dispersion law. The obtained results are compared to the case of a parabolic (standard) dispersion law of charge carriers. The possibility of the accidental exciton instability is revealed for the intermediate quantum confinement regime. For the weak quantum confinement regime, the motion of the exciton's center-of-gravity is quantized, which leads to the appearance of additional Coulomb-like sub-levels. It is revealed that in the case of the Kane dispersion law, the Coulomb levels shift into the depth of the forbidden band gap, moving away from the quantum confined level, whereas in the case of the parabolic dispersion law, the opposite picture is observed. The corresponding selection rules of quantum transitions for the interband absorption of light are obtained. New selection rules of quantum transitions between levels conditioned by 2D exciton center of mass vertical motion quantization in a QD are revealed. The absorption threshold behavior characteristics depending on the QDs geometrical sizes are also revealed.

ВЫЧИСЛЕНИЕ ТЕМПЕРАТУРНОЙ ЗАВИСИМОСТИ ЭНЕРГЕТИЧЕСКОГО СПЕКТРА НОСИТЕЛЕЙ ЗАРЯДОВ В ПОЛУПРОВОДНИКАХ ПРИ ВОЗДЕЙСТВИИ СИЛЬНОГО МАГНИТНОГО ПОЛЯ

The energy spectrum of free holes in a st rong magnetic field is calculated for a non- quadratic dispersion law. The temperature dependen ce of the broadening of the Landau levels of holes in crystals with the Kane dispersion law has been studied. A new method for calculating the discrete energy spectrum in a quantizing magnetic field is proposed

Mathematical modeling determination coefficient of magneto-optical absorption in semiconductors in presence of external pressure and temperature

The calculation of the coefficients of magneto-optical absorption in semiconductors at different temperatures and pressures is carried out. A formula for the temperature dependence of the oscillations of the combined density of states by the Kane dispersion law is obtained. Mathematical modeling has been carried out that determines the magneto-optical absorption coefficient in semiconductors in the presence of external influences. A new method for determining the magneto-optical absorption coefficient in semiconductors in the presence of pressure and temperature is proposed. The correspondence of simulation results with experimental data is shown.

Resonant impurity level of Ni in the valence band of Pb1−xSnxTe alloys

The phase and elemental compositions and galvanomagnetic properties (4.2 K ≤ T ≤ 300 K, B ≤ 0.07 T) of samples from a single-crystal Pb1−x−ySnxNiyTe ingot (x = 0.08, y = 0.01) synthesized by the Bridgman–Stockbarger method were studied. Microscopic inclusions enriched in nickel were found. It is shown that in the main phase, the tin concentration increases exponentially along the ingot (x = 0.06–0.165), while the concentration of nickel impurity does not exceed 0.4 mol %. A significant increase in the concentration of holes along the ingot and an abnormal increase in the Hall coefficient with increasing temperature were found; both are due to the pinning of the Fermi level by the resonant nickel level located in the valence band. The dependences of the hole concentration and of the Fermi energy at T = 4.2 K on the tin concentration in alloys are calculated using the two-band Kane dispersion law. A qualitative model of electronic structure rearrangement is proposed. The model takes into account the movement of the nickel level into the depth of the valence band with an increase in tin concentration and the redistribution of electrons between the valence band and the level. The energy position of the nickel level and the speed of its movement relative to the top of the valence band with an increase in the tin content in Pb1−xSnxTe alloys are estimated.

MATHEMATICAL MODELING OF TEMPERATURE EFFECT ON THE FAN CHART OF MAGNETOABSORPTION SPECTRUM IN SEMICONDUCTORS

The article considers the oscillations of interband magneto-optical absorption in semiconductors with the Kane dispersion law. We have compared the changes in oscillations of the joint density of states with respect to the photon energy for different Landau levels in parabolic and non-parabolic zones. An analytical expression is obtained for the oscillation of the combined density of states in narrow-gap semiconductors. We have calculated the dependence of the maximum photon energy on the magnetic field at different temperatures. A theoretical study of the band structure showed that the magnetoabsorption oscillations decrease with an increase in temperature, and the photon energies nonlinearly depend on a strong magnetic field. The article proposes a simple method for calculating the oscillation of joint density of states in a quantizing magnetic field with the non-quadratic dispersion law. The temperature dependence of the oscillations joint density of states in semiconductors with non-parabolic dispersion law is obtained. Moreover, the article studies the temperature dependence of the band gap in a strong magnetic field with the non-quadratic dispersion law. The method is applied to the research of the magnetic absorption in narrow-gap semiconductors with nonparabolic dispersion law. It is shown that as the temperature increases, Landau levels are washed away due to thermal broadening and density of states turns into a density of states without a magnetic field. Using the mathematical model, the temperature dependence of the density distribution of energy states in strong magnetic fields is considered. It is shown that the continuous spectrum of the density of states, measured at the temperature of liquid nitrogen, at low temperatures turns into discrete Landau levels. Mathematical modeling of processes using experimental values of the continuous spectrum of the density of states makes it possible to calculate discrete Landau levels. We have created the three-dimensional fan chart of magneto optical oscillations of semiconductors with considering for the joint density of energy states. For a nonquadratic dispersion law, the maximum frequency of the absorbed light and the width of the forbidden band are shown to depend nonlinearly on the magnetic field. Modeling the temperature dependence allowed us to determine the Landau levels in semiconductors in a wide temperature spectrum. Using the proposed model, the experimental results obtained for narrow-gap semiconductors are analyzed. The theoretical results are compared with experimental results.

The electron structure of Pb1–x–ySnxFeyTe alloys

A study of phase and elemental compositions, galvanomagnetic properties (4.2 K ≤ T ≤ 300 K, B ≤ 0.07 T) and Shubnikov – de Haas oscillations (T = 4.2 K, B ≤ 6.5 T) in Pb1–x–ySnxFeyTe alloys with varying concentrations of tin and iron in single-crystal ingots synthesized using the Bridgman-Stockbarger method was performed. Scanning electron microscopy and X-ray fluorescence microanalysis revealed the presence of iron enriched microscopic inclusions, and the distribution of tin and iron along the ingots was determined. An increase in the concentration of holes with increasing concentrations of tin and iron, as well as anomalous temperature dependences of the Hall coefficient, which indicates a pinning of the Fermi level at the resonant level of iron, were detected. A model of electron structure rearrangement is proposed in order to explain the experimental dependences of the hole concentration and the Fermi energy relative to the valence band top as a function of tin concentration in the alloys. The model assumes that the iron level moves from the top to the depth of the valence band as the tin concentration increases. The compositional coefficient of iron level movement relative to the edges of the energy bands with increasing tin content in Pb1–x–ySnxFeyTe alloys is determined using the two-band Kane dispersion law.

Temperature and pressure coefficients of iron resonant impurity level in PbTe

We investigate temperature dependences of galvanomagnetic parameters in weak magnetic fields (4.2 ≤ T ≤ 300 K, B ≤ 0.07 T) in the p-Pb1−yFeyTe alloy from the middle part of the single-crystal ingot, where the Fermi level is pinned by the resonant impurity level lying under the top of the valence band. Experiments are performed under hydrostatic compression up to 10 kbar. Using scanning electron microscopy, we find microscopic inclusions of the secondary phase enriched with iron and show that the main phase is characterized by a good uniformity of the spatial distribution of impurities. A monotonous increase of the free hole concentration at liquid-helium temperature under pressure and anomalous temperature dependences of the Hall coefficient in the whole investigated pressure range are revealed. Experimental results are explained by a model assuming pinning of the Fermi level by the impurity level and a redistribution of electrons between the valence band and impurity states with increasing temperature and under pressure. In the framework of the two-band Kane dispersion law, theoretical temperature dependences of the Hall coefficient under pressure, which are in satisfactory agreement with the experimental ones at low temperatures, are calculated and temperature and pressure coefficients of the iron deep level are determined. Diagrams of the electronic structure rearrangement with increasing temperature for Pb1−yFeyTe at pressures up to 10 kbar are proposed.

Electronic and optical properties of a double quantum dot molecule with Kane’s dispersion law

In the framework of the adiabatic approximation the electronic states and direct interband absorption of light in the double quantum dot molecule (QDM) are discussed. Cases of both standard parabolic and Kane's dispersion law are considered. Analytical expressions for the wave functions and energy spectrum of the electron in the QDM are treated. The split of energy levels due to the possibility of the electron tunneling between quantum dots (QDs) in the molecule is revealed. The corresponding selection rules of quantum transitions for the direct interband absorption of light are obtained. The absorption edge behavior characteristics depending on the QDs geometrical sizes and the width of the QDs connecting region are revealed as well.

Theory of confined states of positronium in spherical and circular quantum dots with Kane’s dispersion law

Confined states of a positronium (Ps) in the spherical and circular quantum dots (QDs) are theoretically investigated in two size quantization regimes: strong and weak. Two-band approximation of Kane’s dispersion law and parabolic dispersion law of charge carriers are considered. It is shown that electron-positron pair instability is a consequence of dimensionality reduction, not of the size quantization. The binding energies for the Ps in circular and spherical QDs are calculated. The Ps formation dependence on the QD radius is studied.

On the resonant level of chromium in the rhombohedral and cubic phases of Pb1 − x − y Ge x Cr y Te alloys

The temperature dependences of the Hall coefficient (4.2 K ≤ T ≤ 300 K, B ≤ 0.07 T) in Pb1 − x − y Ge x Cr y Te alloys (x = 0.03–0.08, y ≤ 0.01) are studied. An increase in the absolute value of the Hall coefficient with an increase in temperature is found. This fact is indicative of a decrease in the concentration of free electrons as a result of the motion of the resonant level of chromium stabilizing the Fermi level relative to the conduction-band bottom. The temperature dependences of the Hall coefficient, in satisfactory agreement with the experimental ones, are calculated in the context of the two-band Kane dispersion law allowing for the structural phase transition upon increasing temperature. The energy position and temperature coefficients of the motion of the resonant level of chromium relative to the middle of the band gap in the rhombohedral and cubic phases are determined.

Electronic structure of diluted magnetic semiconductors Pb1−x−ySnxCryTe

Abstract Galvanomagnetic effects in diluted magnetic semiconductors Pb 1− x − y Sn x Cr y Te at temperatures 4.2≤ T ≤300 K under variation of matrix composition ( x =0.06–0.235) and impurity content ( y =0.002–0.045) have been investigated. The p–n-conversion of the conductivity type, induced by the chromium doping, was observed. It is shown that in the heavily doped alloys temperature dependencies of resistivity and of the Hall coefficient have a “metallic” character, indicating pinning of the Fermi level by the chromium deep impurity level, situated in the conduction band. Free electron concentration and Fermi energy were calculated as functions of impurity concentration and of matrix composition by analyzing experimental data in the framework of the two-band Kane dispersion law. The impurity content coefficients of the free charge carrier concentration were determined and the diagram of electronic structure rearrangement under varying the alloy composition was obtained for the alloys under study.

EFFECT ON HEAT CAPACITY OF DEGENERATED ELECTRON GAS IN DIMENSIONALLY QUANTIZED THIN FILMS

In this work, the heat capacity in dimensionally quantized semiconductor thin films with Kane's dispersion law is investigated. Under certain conditions, quantum size effect occurs, depending on the thickness of the thin film, the concentration of conductive electrons, and the non-parabolicity parameters. In thin films having non-parabolic energy spectrum in degenerate electron gas, the film thickness depends on the subband number and the concentration. Therefore, heat capacity takes the form of the saw toothwise and changes as non-monotonous.

Metal-insulator transition in chromium-doped Pb1−x Ge x Te alloys

The electrical properties of chromium-doped n-Pb1−xGexTe alloys (x = 0.02–0.13) have been studied. A decrease in the free-electron concentration and a metal-insulator transition are observed as the germanium content of alloys increases. This is due to the Fermi level pinning by the chromium impurity level and to the flow of electrons from the conduction band to the impurity level. The experimental data obtained are used to calculate, in terms of the two-band Kane dispersion law, the dependences of the electron concentration and Fermi energy on the germanium content in the alloy. The motion rate of the chromium-related level with respect to the conduction band bottom is determined and a model of variation of the electronic structure with the matrix composition is suggested.

Resonant impurity states in PbTe‐based alloys doped with gallium and chromium under pressure

AbstractThe galvanomagnetic effects in the n‐Pb1–x Snx Te:Ga (x = 0.21) and n‐Pb1–y Gey Te:Cr (y = 0.10) alloys at temperatures 4.2 ≤ T ≤ 300 K and under hydrostatic compression up to 17 kbar have been investigated. It is found that in almost the whole investigated pressure range in the samples investigated temperature dependencies of resistivity and Hall coefficient have a “metallic” character, indicating stabilization of the Fermi level by the impurity resonant levels. Using the experimental data in the framework of a two‐band Kane dispersion law the dependencies of the free electron concentration and the Fermi‐level position upon the pressure were calculated. The pressure coefficients of gallium and chromium deep‐level movement were obtained and the models of the electronic‐structure reconstruction under pressure were developed. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Magnetoresistance of Carbon Nanotube with the Kane Dispersion Law

Considering the nanotube as effective conducting media the magnetoresistance is calculated. For an electron motion along nanotube axis the spin–orbital interaction is considered as perturbation and the Dirac equation is solved for longitudinal wave functions. It is shown that in this case a magnetic field can be considered as perturbation for energy spectrum and density matrix diagonal components for each spin orientation are needed to calculate the nanotube conductivity. For narrow gap nanotube the magnetoresistance depends linearly on a weak magnetic field. Our results show that in nanotubes the negative magnetoresistance (NMR) can be described without theory of weak localization.

Binding energy of hydrogenic impurities in quantum well wires of InSb/GaAs

The binding energy of a hydrogen-like impurity in a thin size-quantized wire of the InSb/GaAs semiconductors with Kane's dispersion law has been calculated as a function of the radius of the wire and the location of the impurity with respect to the axis of the wire, using a variational approach. It is shown that when the wire radius is less than the Bohr radius of the impurity, the nonparabolisity of dispersion law of charge carriers leads to a considerable increase of the binding energy.

Binding Energy of Hydrogen-Like Impurities in a Thin Semiconductor Wire with Complicated Dispersion Law in a Magnetic Field

The binding energy of a hydrogen-like impurity in a quantum well wire (QWW) of A 3 B 5 -type semiconductors with Kane's dispersion law in a magnetic field B parallel to the wire axis has been calculated as a function of the radius of the wire and magnitude of B, using a variational approach. It is shown that the nonparabolicity of the dispersion law leads to a considerable increase of the binding energy in the magnetic field, as well as to a more rapid nonlinear growth of binding energy with B.

Binding Energy of Hydrogen-Like Impurities in a Thin Semiconductor Wire with Complicated Dispersion Law

The binding energy of a hydrogen-like impurity in a thin size-quantized wire of A 3 B 5 -type semiconductors with Kane's dispersion law has been calculated as a function of the radius of the wire and the location of the impurity with respect to the axis of the wire, using a variational approach. It is shown that when the wire radius is less than the Bohr radius of the impurity, the nonparabolicity of the dispersion law of charge carriers leads to a considerable increase of the binding energy.

Extended moment method for electrons in semiconductors

The semiclassical Boltzmann equation for electrons in semiconductors with the Kane dispersion law or the parabolic band approximation is considered and systems of moment equations with an arbitrary number of moments are derived. First, the paper deals with spherical harmonics in the formalism of symmetric trace-free tensors. The collision frequencies are carefully studied for the physical properties of silicon. Then, for the parabolic band approximation, the hierarchy of equations for full moments of the phase density and the corresponding closure problem is discussed. In particular, a set of 2 R scalar and vectorial moments is considered. To answer the question which number R one has to chose in order to retain the physical contents of the Boltzmann equation, the moment equations are examined in the drift-diffusion limit and in an infinite crystal in a homogeneous electric field (transient and stationary cases) for increasing number of moments R. The number R must be considered to be sufficient, if its further increase does not change the result considerably and the appropriate numbers for the processes are given.

Tunneling conductivity oscillations in a magnetic field in metal-insulator-narrow-gap-HgCdTe structures: The energy spectrum and spin-orbit splitting of 2D states

We study tunneling conductivity oscillations in a magnetic field in narrow-gap p-HgCdTe-oxide-metal (Yb, Al) structures. In tunnel structures with Yb we detect two types of tunneling conductivity oscillations. The first is related to the crossing of the Landau levels of two-dimensional (2D) states localized in the surface quantum well of the semiconductor, and has an energy EF+eV, where EF is the Fermi energy of the semiconductor and V is the bias voltage; the second has an energy EF. We find that in such structures with an asymmetric quantum well there is strong spin-orbit splitting in the spectrum of the 2D states. In p-HgCdTe-oxide-Al tunnel structures the surface potential is much weaker and only oscillations of the first type are observed. We find that in such structures there is only one spin state of the 2D carriers, while the second is pushed into the continuous spectrum because of strong spin-orbit coupling. To analyze the experimental results we calculate the spectrum of 2D states localized in the surface quantum well in a semiconductor with a Kane dispersion law. We find that all the experimental results are in good agreement with the results of calculations. Finally, we discuss the features of “kinematically coupled” states in an asymmetric quantum well.