Envelope Function Approximation Research Articles

Spin-Resolved Magneto-Tunneling and Giant Anisotropic g-Factor in Broken Gap InAs-GaSb Core-Shell Nanowires.

We experimentally and computationally investigate the magneto-conductance across the radial heterojunction of InAs-GaSb core-shell nanowires under a magnetic field, B, up to 30 T and at temperatures in the range 4.2-200 K. The observed double-peak negative differential conductance markedly blue-shifts with increasing B. The doublet accounts for spin-polarized currents through the Zeeman split channels of the InAs (GaSb) conduction (valence) band and exhibits strong anisotropy with respect to B orientation and marked temperature dependence. Envelope function approximation and a semiclassical (WKB) approach allow to compute the magnetic quantum states of InAs and GaSb sections of the nanowire and to estimate the B-dependent tunneling current across the broken-gap interface. Disentangling different magneto-transport channels and a thermally activated valence-to-valence band transport current, we extract the g-factor from the spin-up and spin-down dI/dV branch dispersion, revealing a giant, strongly anisotropic g-factor in excess of 60 (100) for the radial (tilted) field configurations.

Binding energies of shallow donors in polar ZnO/ZnBeO quantum well

This study examines a single wurtzite ZnO/ZnBeO quantum well structure oriented in the polar c-direction. The focus is on investigating the binding energies associated with an impurity donor atom within this system. To achieve this, a self-consistent solution to the Schrödinger and Poisson equations is obtained using the finite difference method. The framework employed involves the effective mass and envelope function approximations. The impurity is represented using a hydrogenic-type wave function, and donor binding energies are determined via a variational approach. The research analyzes the binding energies of the 1s and 2p± states, along with the transition energy between them. These quantities are explored as functions of the well width and Be concentration, considering donor positions at the right and left interfaces as well as at the well’s center. Furthermore, the impact of an external magnetic field oriented along the growth direction is assessed, spanning up to 10T, in order to quantify changes in the binding energies. The presence of a built-in electric field induces an asymmetric band profile and a triangular well configuration. This asymmetry results in a loss of symmetry within the binding energy curves. Ultimately, the investigation culminates in the computation of the oscillator strength governing transitions between the donor states. When the donor is situated at the right interface, the energy values remain relatively constant as the well width increases, and the oscillator strength values demonstrate a consistent linear rise.

A particular behavior of levels energy evolution and theoretical study of the quantum probability distribution in GaAs/AlGaAs asymmetric stepped semiconductor quantum wells for terahertz optoelectronics

The study of the evolution of the energy levels as a function of the step layer’s width Lw2 in an asymmetric stepped quantum well AlxGa1-xAs/GaAs/AlyGa1-yAs/AlxGa1-xAsx=24%,y=14%, revealed a particular behavior, marked by the presence of plateaus within certain deep well’s width, Lw2, ranges. We initially computed the eigenenergies and eigenfunctions by numerically solving the Schrödinger equation. This was done within the effective mass and envelope function approximation, taking into account the influence of non-parabolicity. To explain the origin of the plateaus and their impact on intersubband transition energies, we determined the distribution of the probability density of the presence of the electron in each region of the stepped quantum well. The results of these calculations demonstrated that the electron's probability density is higher in the step well than in the deep one, particularly for electrons with energy levels above 113.47 meV. This elevation in probability density exhibits maximum values that remain nearly constant within specific Lw2 domains, providing an explanation for the observed energy plateaus. Such plateaus allow more energy tunability and thus open up additional possibilities for applications in the THz domain either in the linear or nonlinear optics.

Interface Engineering and Electron-Hole Wave Function Overlap of InAs/AlSb Superlattice Infrared Detectors

InAs/AlSb is a material system that can be used as a low-noise avalanche detector and operates in the short-wave infrared band. The interface parameters determine the wave function overlap (WFO). Maximizing the WFO of InAs/AlSb superlattices improves the quantum efficiency (QE) of infrared avalanche photodetectors (APDs). However, this remains a huge challenge. Here, the 8-band k·p perturbation method based on Bloch wave envelope function approximation was used to calculate the energy level structure of InAs/AlSb superlattices. The results indicate that the WFO is enhanced with increasing InSb interface thickness or when the InSb (or AlAs) interface is far from the intersection of InAs and AlSb. As the AlAs interface thickness increases, the WFO enhances and then reduces. The maximum increase in WFO is 15.7%, 93%, and 156.8%, respectively, with three different models. Based on the stress equilibrium condition, we consider the interface engineering scheme proposed for enhancing WFO with an increase of 16%, 114%, and 159.5%, respectively. Moreover, the absorption wavelength shift is less than ±0.1 μm. Therefore, the interface layer thickness and position can be designed to enhance the WFO without sacrificing other properties, thereby improving the QE of the device. It provides a new idea for the material epitaxy of APDs.

Advancing carrier transport models for InAs/GaSb type-II superlattice mid-wavelength infrared photodetectors

To provide the best possible performance, modern infrared photodetector designs necessitate extremely precise modeling of the superlattice absorber region. We advance the Rode's method for the Boltzmann transport equation in conjunction with the $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ band structure and the envelope function approximation for a detailed computation of the carrier mobility and conductivity of layered type-II superlattice structures, using which we unravel two crucial insights. First, the significance of both elastic- and inelastic-scattering mechanisms, particularly the influence of the interface roughness and polar optical phonon scattering mechanisms in technologically relevant superlattice structures. Second, that the structure-specific Hall mobility and Hall scattering factor reveal that temperature and carrier concentrations significantly affect the Hall scattering factor, which deviates significantly from unity even for small magnetic fields. This reinforces the caution that should be exercised when employing the Hall scattering factor in experimental estimations of drift mobilities and carrier concentrations. Our research hence offers a comprehensive microscopic understanding of carrier dynamics in such technologically relevant superlattices. Our models also provide highly accurate and precise transport parameters beyond the relaxation-time approximation and thereby paving the way to develop physics-based device modules for mid-wavelength infrared photodetectors.

Photoreflectance studies of the band gap alignments in boron diluted BGaInAs/GaAs quantum wells

Band gap alignments of BGaInAs/GaAs quantum wells with mole fractions of indium around 40% and mole fractions of boron ranging from 0% up to 4.75% are studied experimentally by photoreflectance (PR) and photoluminescence (PL). Obtained results are explained within a k · p model within an envelope function approximation. The study shows an increase of the valence band offset with an addition of boron into the thin film at a rate of around 4.2% per 1% of boron incorporated. Non-zero bowing parameters of valence band offsets for ternary alloys with boron (BGaAs and BInAs) are estimated. Moreover, it was observed that unlike in other highly mismatched alloy systems the incorporation of boron does not significantly deteriorate the optical quality of the studied samples, i.e., the broadening of optical transitions observed in PR and PL is very comparable to that observed for the reference QW, and the PL properties of boron containing QWs are similar to the reference boron free QW. Some deterioration of optical quality due to the increased alloy inhomogeneity is observed only for the sample with the highest concentration of B (4.2%).

Donor binding energies in single ZnCdO/ZnO quantum well

The single wurtzite Zn1−xCdxO/ZnO quantum well structure in polar c-direction is studied and the binding energies of an impurity donor atom are obtained. The Schrödinger and Poisson equations are solved self consistently using finite difference method within the effective mass and envelope function approximations. Then, a hydrogenic type of wave function is assumed to represent the impurity, and donor binding energies are obtained using variational approach. The binding energies of the 1s and 2p± states and the transition energy between them are obtained as functions well width, Cd concentration and donor position in the well. Also, an external magnetic field along the growth direction up to 10T is applied to compute the changes in the binding energies. The built-in electric field causes an asymmetric band profile and a triangular well structure, and makes the binding energy curves lose symmetry. The binding energy increases with the donor position as the donor gets close to the wave function penetrated mostly inside the barrier. The built-in electric field is 2.5MV/cm in a 20− Å well with the Cd concentration of x=0.1, and the transition energy is 27.7meV(6.7THz) when the donor is at the well barrier interface close to the electron wave function, but it is 15.7meV(3.8THz) when the donor is at the well center. Also, the Zeeman splitting between the 2p− and 2p+ states is 4.64meV at 10T independent from the donor position.

Universality of the Förster's model for resonant exciton transfer in ensembles of nanocrystals.

For nanocrystals in a strong quantum confinement regime, it has been confirmed analytically that resonant exciton transfer proceeds in full accordance with the Förster mechanism. This means that the virtual exciton transitions between the nanocrystals of close sizes are governed only by the dipole-dipole interaction of nanocrystals even in very dense ensembles, while the contributions of all other higher-order multipoles are negligibly small. Based on a simple isotropic model of the envelope function approximation and neglecting the electron-hole interaction inside each nanocrystal, we have computed the rate of the resonant exciton transfer between two nanocrystals. Using the obtained result, we have estimated, for some arbitrarily chosen nanocrystal, the total rate of the exciton non-radiative annihilation caused by the possibility of its resonant virtual transitions into all other nanocrystals of the ensemble. The total rate dependence on the nanocrystal size is determined only by the size distribution function of nanocrystals in the ensemble.

Simulation of the Band Structure of InAs/GaSb Type II Superlattices Utilizing Multiple Energy Band Theories

Antimonide type II superlattices is expected to overtake HgCdTe as the preferred materials for infrared detection due to their excellent photoelectric properties and flexible and adjustable band structures. Among these compounds, InAs/GaSb type II superlattices represent the most commonly studied materials. However, the sophisticated physics associated with the antimonide-based bandgap engineering concept started at the beginning of the 1990s gave a new impact and interest in the development of infrared detector structures within academic and national laboratories. InAs/GaSb superlattices are a type II disconnected band structure with electrons and holes confined in the InAs and GaSb layers, respectively. The electron miniband and hole miniband can be regulated separately by adjusting the thickness of InAs and GaSb layers, which facilitates the design of superlattice structures and optimizes the value of band offset. In recent years, both domestic and foreign researchers have made many attempts to quickly and accurately predict the bandgaps of superlattice materials before superlattice materials grow. These works constituted a theoretical basis for the effective utilization of the InAs/GaSb system in material optimization and designing new SL structures; they also provided an opportunity for the preparation and rapid development of InAs/GaSb T2SLs. In this paper, we systematically review several widely used methods for simulating superlattice band structures, including the k·p perturbation method, envelope function approximation, empirical pseudopotential method, empirical tight-binding method, and first-principles calculations. With the limitations of different theoretical methods proposed, the simulation methods have been modified and developed to obtain reliable InAs/GaSb SL energy band calculation results. The objective of this work is to provide a reference for designing InAs/GaSb type II superlattice band structures.

Phonon-assisted pressure-dependent opto-electronic properties in a CdS/CdSe/CdS asymmetric quantum well

ABSTRACT Pressure-associated phonon-induced opto-electronic properties are brought out in a CdS/CdSe/CdS double quantum well. The effects of pressure on the interband associated with the changes of refractive index and optical absorption coefficients are investigated in the asymmetric well by altering its right well size and the barrier size. Aldrich and Bajaj polaronic effect is employed in the Hamiltonian. The effects of pressure on the optical properties due to the interband transition in the double quantum well are carried out. The single band effective mass, envelope function approximation and density matrix are used as theoretical tools in order to obtain these properties. The pressure produces a reduction in a radiative life time. The enhancement in the magnitude of absorption coefficients occurs due to the applications of hydrostatic pressure. The results bring out the fact that the optical susceptibilities suffer a blue shift and its magnitudes are determined by the matrix elements.

Electron–electron scattering rate in CdTe/CdMnTe single quantum well

CdMnTe is demonstrated to be a good candidate in the X-ray and [Formula: see text]-ray detector application, however, there are few reports on theoretical analysis of electron scattering rate in CdMnTe quantum well. Within the framework of effective mass approximation and envelope function approximation, the influence of the Mn alloy composition ([Formula: see text], the well width ([Formula: see text], the electron temperature ([Formula: see text] and the electron density ([Formula: see text] on the electron–electron scattering rate (1/[Formula: see text] in the CdTe/Cd[Formula: see text]Mn[Formula: see text]Te single quantum well (SQW), are simulated by shooting method and Fermi’s Golden Rule. The results show that 1/[Formula: see text] is significant inverse proportional to [Formula: see text], but positively proportional to [Formula: see text] and [Formula: see text]. Except for a small peak at 20 K, 1/[Formula: see text] is not sensitive to [Formula: see text]. The above differential dependency of 1/[Formula: see text] on [Formula: see text] and [Formula: see text] can be interpreted by sub-band separation ([Formula: see text], which is proportional to [Formula: see text] but inversely proportional to [Formula: see text]. When [Formula: see text] decreases gradually, the electron transition becomes easier, which leads to 1/[Formula: see text] increases. The dependency of 1/[Formula: see text] on [Formula: see text] can be interpreted by kinetic energy of electrons. The larger the electron kinetic energy is, the more difficult the electron transition from first excited state to ground state is, which leads to 1/[Formula: see text] decreasing. The dependency of 1/[Formula: see text] on [Formula: see text] can be interpreted by the Coulomb interaction between electrons, i.e., the increase of electron collision probability caused by the increase of [Formula: see text].

Carrier localization and miniband modeling of InAs/GaSb based type-II superlattice infrared detectors

Microscopic features of carrier localization, minibands, and spectral currents of InAs/GaSb based type-II superlattice (T2SL) mid-infrared detector structures are studied and investigated in detail. In the presence of momentum and phase-relaxed elastic scattering processes, we show that a self-consistent non-equilibrium Green’s function method within the effective mass approximation can be an effective tool to fairly predict the miniband and spectral transport properties and their dependence on the design parameters such as layer thickness, superlattice periods, temperature, and built-in potential. To benchmark this model, we first evaluate the band properties of an infinite T2SL with periodic boundary conditions, employing the envelope function approximation with a finite-difference discretization within the perturbative eight-band framework. The strong dependence of the constituent material layer thicknesses on the band-edge positions and effective masses, offers a primary guideline to design performance-specific detectors for a wide range of operation. Moving forward, we demonstrate that using a finite T2SL structure in the Green’s function framework, one can estimate the bandgap, band-offsets, density of states and spatial overlap which comply well with the results and the experimental data. Finally, the superiority of this method is illustrated via a reasonable estimation of the band alignments in barrier-based multi-color non-periodic complex T2SL structures. This study, therefore, provides deep physical insights into the carrier confinements in broken-gap heterostructures and sets a perfect stage to perform transport calculations in a full-quantum picture.

The impurity states in InGaAsP/InP coaxial double quantum well wires with the effects of electric and magnetic fields

The hydrogen donor impurity states are calculated in [Formula: see text] coaxial double quantum well wires by the plane wave method under the theoretical framework of effective mass envelope function approximation. The binding energies of impurity in [Formula: see text] state and [Formula: see text] state are obtained as the functions of impurity position, distance between the inner and outer quantum wires, magnetic and electric field strengths. Transition energies are calculated as the functions of impurity position, distance between the inner and outer quantum wires. The effects of quantum wire thickness and distance of quantum wires on impurity states are analyzed in detail. It is found that the effects of electric field and magnetic field on binding energy of [Formula: see text] state are different for impurity located at different positions.

Self-Consistent Schrödinger-Poisson Study of Electronic Properties of GaAs Quantum Well Wires with Various Cross-Sectional Shapes.

Quantum wires continue to be a subject of novel applications in the fields of electronics and optoelectronics. In this work, we revisit the problem of determining the electron states in semiconductor quantum wires in a self-consistent way. For that purpose, we numerically solve the 2D system of coupled Schrödinger and Poisson equations within the envelope function and effective mass approximations. The calculation method uses the finite-element approach. Circle, square, triangle and pentagon geometries are considered for the wire cross-sectional shape. The features of self-consistent band profiles and confined electron state spectra are discussed, in the latter case, as functions of the transverse wire size and temperature. Particular attention is paid to elucidate the origin of Friedel-like oscillations in the density of carriers at low temperatures.

Долинно-орбитальное взаимодействие в германии, легированном донорами V группы: количественный анализ

In the framework of the envelope function approximation, the wave functions of electrons localized at shallow donors P, As, Sb in Ge are calculated taking into account the valley-orbit coupling caused by the donor short-range potential. It is proposed an approach that makes it possible to include inter-valley mixing in the equation for a multi-component envelope function. The calculation of the effects of the valley-orbit interaction was carried out according to the perturbation theory, while the "bare" single-valley functions were found using the Ritz method. The parameters of the short-range part of the potential and the coefficient of inter-valley mixing were found individually for each donor, making it possible to obtain the best agreement with the results of experimental measurements of the energies of the singlet and triplet states. The envelope functions of the 1s(A1) and 1s(T2) states are calculated. The parameters of the valley-orbit interaction are found for each donor. It is also shown how the functions of the excited 2s, 2p0, 2p±, 3p0 states should be modified in order to remain orthogonal to the singlet and triplet functions within the framework of a more rigorous multivalley model.

Impurity states in a GaN/AlxGa1−xN spherical quantum dot under an applied electric field

In the framework of effective mass envelope function approximation, the impurity states in GaN / AlxGa1 − xN spherical quantum dot (QD) are calculated by plane-wave expansion method. The binding energies of the impurity in 1s and 2p± states and the transition energy between 1s state and 2p± state are calculated and analyzed in detail. Several critical influencing factors of impurity states, including geometry, impurity position, material composition, and applied electric field, are considered in the calculation. The effects of QD radius, the impurity position, and the electric field strength on binding energies of 1s state and 2p± state are different. However, the effect of Al component on binding energies of 1s state and 2p± state is consistent. The transition energy is strongly dependent on the QD radius, impurity position, Al composition, and electric field strength. The change trend of transition energy is similar to that of binding energy in 1s state.

Band structure and end states in InAs/GaSb core-shell-shell nanowires

Quantum wells in InAs/GaSb heterostructures can be tuned to a topological regime associated with the quantum spin Hall effect, which arises due to an inverted band gap and hybridized electron and hole states. Here, we investigate electron-hole hybridization and the fate of the quantum spin Hall effect in a quasi one-dimensional geometry, realized in a core-shell-shell nanowire with an insulator core and InAs and GaSb shells. We calculate the band structure for an infinitely long nanowire using $\mathbf{k \cdot p}$ theory within the Kane model and the envelope function approximation, then map the result onto a BHZ model which is used to investigate finite-length wires. Clearly, quantum spin Hall edge states cannot appear in the core-shell-shell nanowires which lack one-dimensional edges, but in the inverted band-gap regime we find that the finite-length wires instead host localized states at the wire ends. These end states are not topologically protected, they are four-fold degenerate and split into two Kramers pairs in the presence of potential disorder along the axial direction. However, there is some remnant of the topological protection of the quantum spin Hall edge states in the sense that the end states are fully robust to (time-reversal preserving) angular disorder, as long as the bulk band gap is not closed.

The impurity states in different shaped quantum wells under applied electric field

In this paper, the impurity states in square, parabolic and V-shaped quantum wells (QWs) are calculated by the plane wave method under the theoretical framework of effective mass envelope function approximation. In different shaped QWs, the impurity binding energies in 1s-like state and 2p-like state have the same trend with the increase of QW width. However, the transition energies of impurity states between 1s-like state and 2p-like state in three shaped QWs are different. When the width of QW is fixed, the transition energy in square QW is the minimum and that in V-shaped QW is the maximum. The effect of electric field is larger for the on-center donor impurity. The effect of electric field on the impurity states is the largest in square QW and the smallest in V-shaped QW.

General-purpose open-source 1D self-consistent Schrödinger-Poisson Solver: Aestimo 1D

We present a general-purpose numerical quantum mechanical solver using Schrödinger-Poisson equations called Aestimo 1D. The solver provides self-consistent solutions to the Schrödinger and Poisson equations for a given semiconductor heterostructure built with materials including elementary, binary, ternary, and quaternary semiconductors and their doped structures. The software can be used to calculate electronic band structures of heterostructures either using a single-band or multi-band k.p envelope function approximation. The software is fully open-source and it is released under the GNU general public license version 3 for full freedom of usage for applications in the fields of nano-electronics, optoelectronics, and solid-state device simulations.

Electron States of Group-V Donors in Germanium: Variational Calculation Taking into Account the Short-Range Potential

The wave functions of low-lying 1s (A1), 2s, 2p0, 2p±, and 3p0 states of P, As, and Sb shallow donor centers in germanium are calculated in the scope of the envelope-function approximation taking into account the short-range impurity potential. The latter is constructed individually for each impurity allowing for the spatial permittivity dispersion and the difference between the ion cores of germanium and the impurity center. The envelope-function equation is solved using the Ritz variational method; herewith, the selected test functions of orbitally nondegenerate s states are characterized by two spatial scales. The first scale, on the order of the donor effective Bohr radius, corresponds to the long-range part of the potential, and the second scale, which is smaller by an order of magnitude, simulates the electron response to the short-range part of the donor potential. The electron density in the donor ground state is shifted to the nucleus, which is due to allowance made for the attracting “central cell” potential. The envelope functions of p states are in turn constructed so that they are orthogonal to envelopes in the ground states for each impurity centers and are different for various donors in contrast with previous works.