Effective Mass Hamiltonian Research Articles

Application of pseudopotential theory to magnetic semiconductor heterostructures

The results of empirical pseudopotential calculations for the semiconductor compound Cd1−xMnxTe are presented. The effective electron and hole masses obtained from the pseudopotential calculations are then employed in an envelope function approximation, using two different effective mass Hamiltonians to evaluate the transition energies of the excitonic ground state in CdTe– Cd1−xMnxTe quantum wells of variable width. It is shown that in non-magnetic systems it is not possible to utilize exciton energies alone to either distinguish between different model Hamiltonians or to quantify the interface roughness. However, it is shown that the latter can be quantified in magnetic systems via the resulting Zeeman effect.

Magnetic field effects on quantum ring excitons

We study the effect of magnetic field and geometric confinement on excitons confined to a quantum ring. We use analytical matrix elements of the Coulomb interaction and diagonalize numerically the effective-mass Hamiltonian of the problem. To explore the role of different boundary conditions, we investigate the quantum ring structure with a parabolic confinement potential, which allows the wavefunctions to be expressed in terms of center of mass and relative degrees of freedom of the exciton. On the other hand, wavefunctions expressed in terms of Bessel functions for electron and hole are used for a hard-wall confinement potential. The binding energy and electron-hole separation of the exciton are calculated as function of the width of the ring and the strength of a external magnetic field. The linear optical susceptibility as a function of magnetic fields is also discussed. We explore the Coulomb electron-hole correlation and magnetic confinement for several ring width and size combinations. The Aharanov-Bohm oscillations of exciton characteristics predicted for one-dimensional rings are found to not be present in these finite-width systems.

Treating some solid state problems with the Dirac equation

The ambiguity involved in the definition of effective-mass Hamiltonians for nonrelativistic models is resolved using the Dirac equation. The multistep approximation is extended for relativistic cases allowing the treatment of arbitrary potential and effective-mass profiles without ordering problems. On the other hand, if the Schrodinger equation is supposed to be used, our relativistic approach demonstrate that both results are coincidents if the BenDaniel and Duke prescription for the kinetic-energy operator is implemented. Applications for semiconductor heterostructures are discussed.

Size effects onexcitons in nano-rings

The size effects on an exciton in a nano-ring are investigatedtheoretically by using an effective-mass Hamiltonian which can beseparated into terms in centre-of-mass and relative coordinates.The binding energy and oscillator strength of the ground state arecalculated for two different ring radii as functions of the ringwidth. The resulting linear optical susceptibility of the low-lyingexciton states is also discussed.

Amount of hole conversion across AlxGa1−xN/GaN heterojunctions

Starting with the 6×6 valence band Burt Hamiltonian for semiconductor heterojunctions with wurtzite symmetry that was recently derived [F. Mireles and S. E. Ulloa, Phys. Rev. B 60, 13659 (1999)], we study the tunneling properties of holes across an AlxGa1−xN/GaN heterojunction while including the effects of strain on the valence band edges. We analyze the amount of hole conversion between the heavy-, light-, and crystal-field split-off-valence bands during tunneling across the heterointerface. The results are compared with an approach based on a widely used symmetrized form of the Luttinger–Kohn effective-mass Hamiltonian and associated boundary conditions for wurtzite materials

Spin-Orbit Interaction in Carbon Nanotubes

An effective-mass Hamiltonian is derived for carbon nanotubes in the presence of a weak spin-orbit interaction. The spin-orbit interaction gives rise to terms corresponding to a kind of a spin-Zeeman energy due to magnetic fields in the circumference and axis direction. As a result spin scattering is induced even by spin-independent scatterers.

Envelope Function Approximation (EFA) Bandstructure Calculations for III-V Non-square Stepped Alloy Quantum Wells Incorporating Ultra-narrow (~5Å) Epitaxial Layers

We describe Envelope Function Approximation (EFA) bandstructure calculations based on a 4-band electron (EL), heavy-hole (HH), light-hole (LH) and split-off hole (SO) effective mass Hamiltonian, with Burt-Foreman hermitianisation, which can handle III-V quantum well structures that incorporate ultra-narrow epi-layers. The model takes into account the coupling of EL, HH, LH and SO bands and is suitable for describing quantum wells tuned to the 1.0 - 1.55 µm window exploited by optical fibre communication devices. We have used the multi-band solver to calculate the bandstructure of an illustrative InGaAsSb-AlGaSb non-square quantum well that incorporates 6Å potential “spikes” in its well region. Calculations based on the Burt-Foreman hermitianised Hamiltonian and those based on a Hamiltonian with standard “symmetrised” hermitianisation are presented and compared. When coupling to the conduction band is excluded from the calculation, the latter formulation leads to anomalous electron-like curvature of the dispersion curves for our spiked non-square quantum well structure.

Far infrared response of InAs–GaSb type-II quantum dots

We have investigated theoretically the ground state and electromagnetic response of InAs–GaSb type-II quantum dots in the presence of a vertical magnetic field. The ground state is calculated within the Hartree approximation using a 2-band k· p effective mass Hamiltonian to represent the coupling between the two spin-split sets of dot levels derived from the lowest InAs electron subband and the highest GaSb heavy hole subband. The FIR response is calculated within the RPA. We show that the interlayer Coulomb interaction between the electrons and the holes combined with the k· p mixing breaks the generalized Kohn's theorem leading to a complex FIR spectrum.

Electronic structure of quantum spheres with wurtzite structure

The hole effective-mass Hamiltonian for the semiconductors with wurtzite structure is given. The effective-mass parameters are determined by fitting the valence-band structure near the top with that calculated by the empirical pseudopotential method. The energies and corresponding wave functions are calculated with the obtained effective-mass Hamiltonian for the CdSe quantum spheres, and the energies as functions of sphere radius R are given for the zero spin-orbital coupling (SOC) and finite SOC cases. The energies do not vary as ${1/R}^{2}$ as the general cases, which is caused by the crystal-field splitting energy and the linear terms in the Hamiltonian. It is found that the ground state is not the optically active S state for the R smaller than $30 \mathrm{\AA{}}\mathrm{},$ in agreement with the experimental results and the ``dark exciton'' theory.

Exciton Absorption in GaN

B, C associated with split valence bands of Γ9, Γ7 and Γ, symmetries using a spherically symmet- ric approximation for the standard effective-mass Hamiltonian. The exci- tonic equations are solved within the subspace of optically active states of s-symmetry. The effects of crystal field and spin-orbit coupling are discussed. The results show that the spherical approximation is in perfect agreement with experimental findings.

Band-mixing and coupling in single and double quantum wire structures

The electronic states in the conduction and valence bands of quantum wires are studied by means of the effective mass hamiltonian and the Luttinger hamiltonian. The band mixing effects due to the wire geometry are fully taken into account by an accurate finite element solution of the resulting Schrödinger equations. The quantum wire profiles measured by transmission electron microscopy are used. The comparison of theoretical results with photoluminescence excitation spectra of various nanostructures yields a good agreement and, thus, demonstrates the usefulness of the model in terms of general spectral absorption shapes as a function of wire dimensions and light polarization. Finally, the coupled states of a double quantum wire structure demonstrating simultaneous electron and hole coupling are investigated. Efficient hole tunneling is predicted due to the ubiquitous role of valence band mixing.

Valence subbands and optical gain in wurtzite and zinc-blende strained GaN/AlGaN quantum wells

In this communication, the valence band structure, the momentum matrix elements and the optical gain of the wurtzite strained GaN/AlGaN quantum wells are compared with those of the zinc-blende ones. The 3\sX3 effective mass Hamiltonian in the basis set of |3/2, m) and |1/2, m) for both wurtzite and zinc-blende semiconductors is employed in this study. We also present the momentum matrix elements which can be used for both wurtzite and zinc-blende semiconductors. The optical gain of a wurtzite GaN/AlGaN quantum well is smaller than that of a zinc-blende GaN/AlGaN quantum well because of its larger electron and hole effective masses and its strong TE-polarization on account of the small spin-orbit splitting energy.

Population Inversion of Landau Levels in the Valence Band of Silicon in Crossed Electric and Magnetic Fields

The Landau level structure of the degenerate valence band of silicon in crossed electric and magnetic fields (B = 5 T, E = 0 to 5 kVcm—1) has been calculated using the complete effective mass Hamiltonian for the three valence subbands. The calculations reveal a population inversion between light hole Landau levels at cryogenic temperatures due to a strongly level-dependent scattering on optical phonons, which is caused by quantum mechanical mixing of light and heavy hole states in crossed E and B fields. The possibility of amplification of far-infrared radiation on light hole cyclotron resonance transitions is discussed.

Non-Markovian gain of strained-layer wurtzite GaN quantum-well lasers with many-body effects

A theoretical model for the optical gain of strained-layer wurtzite GaN quantum-well (QW) lasers is developed taking into account valence-band mixing, many-body effects and non-Markovian relaxation. The valence-band structure is calculated from a 6/spl times/6 multiband effective mass Hamiltonian for the wurtzite structure taking into account built-in strain due to lattice mismatch. The theoretical foundation for the optical processes is based on the time-convolutionless reduced-density operator formalism given in previous papers for an arbitrary driven system coupled to a stochastic reservoir. Many-body effects are taken into account within the time-dependent Hartree-Fock approximation and the optical gain with Coulomb (or excitonic) enhancement is derived by integrating the equation of motion for the interband polarization. It is predicted that the Coulomb enhancement of gain is pronounced with increasing magnitude of compressive strain in the QW.

General rules for constructing valence band effective mass Hamiltonians with correct operator order for heterostructures with arbitrary orientations

Working from the recently developed exact envelope-function theory, the question of how the envelope functions connect across a boundary is addressed for arbitrary growth directions. Using angular momentum basis functions that are expressed in terms of the desired growth orientation, a general form for a six-band Hamiltonian is presented. A set of general rules has been obtained which describe how unordered bulk elements can be replaced to conform to the ordering determined from the exact theory, thereby avoiding the use of ad hoc symmetrization procedures. One important result of this study is that the reduced interface coupling between the heavy-hole state and light/split-off-hole state found previously for [100] and [110] orientation is a general property valid for all growth directions.

Effect of the Stark confinement on the excitonic optical spectrumof an array of coupled quantum dots

We investigate theoretically the influence of a uniform electric field on the optical properties of a linear array of coupled quantum dots. We study the excitonic spectrum of this system by solving the electron-hole effective-mass Hamiltonian. The potential along the array is modeled by a periodic square-well potential and we assume parabolic lateral confining potentials for electrons and holes. We calculate the excitonic absorption coefficient as a function of the quantum dot size and the applied electric field. We discuss the different confinement regimes. The results illustrate the competing effects of the lateral confinement, the electric-field confinement, and the Coulomb interaction. We show that in the weak-lateral confinement regime the natural coordinates are the relative and the center-of-mass coordinates which are weakly coupled by the lateral potential. In the strong-lateral confinement regime the electrons and holes are weakly correlated and the spectrum can be explained in a single-particle picture.

On the Theory of Optical Gain of Strained-Layer Hexagonal and Cubic GaN Quantum-Well Lasers

The optical gains of strained-layer hexagonal and cubic GaN quantum wells are calculated within the multiband effective mass approximation. The 6×6 multiband effective-mass Hamiltonians are used to calculate the band structures of hexagonal and cubic quantum wells. Non-Markovian relaxation is taken into account in the optical gain calculation. Calculated results show that the optical gains of the cubic quantum well are larger in magnitudes than those of the hexagonal GaN quantum well over the wide range of carrier densities.

Optical gain of strained hexagonal and cubic GaN quantum-well lasers

In order to estimate and compare the would-be performances of both hexagonal and cubic GaN quantum-well lasers, the optical gains of strained-layer hexagonal and cubic GaN quantum wells are studied theoretically taking into account non-Markovian relaxation. The 6×6 multiband effective-mass Hamiltonians are used to calculate the band structures of both hexagonal—and cubic—quantum wells. As a numerical example, the optical gains of strained cubic and hexagonal quantum wells with various well widths are calculated. It is expected that the optical gains of the cubic-phase quantum well are larger in magnitudes than those of the hexagonal GaN quantum well over the wide range of carrier densities due to the heavier effective mass of the HH1 band of the latter at the zone center.

Optical gain of strained wurtzite GaN quantum-well lasers

A theory for the electronic band structure and the free-carrier optical gain of wurtzite-strained quantum-well lasers is presented. We take into account the strain-induced band-edge shifts and the realistic band structures of the GaN wurtzite crystals. The effective-mass Hamiltonian, the basis functions, the valence band structures, the interband momentum matrix elements, and the optical gain are presented with analytical expressions and numerical results for GaN-AlGaN strained quantum-well lasers. This theoretical model provides a foundation for investigating the electronic and optical properties of wurtzite-strained quantum-well lasers.

K

We derive the effective-mass Hamiltonian for wurtzite semiconductors, including the strain effects. This Hamiltonian provides a theoretical groundwork for calculating the electronic band structures and optical constants of bulk and quantum-well wurtzite semiconductors. We apply Kane's model to derive the band-edge energies and the optical momentum-matrix elements for strained wurtzite semiconductors. We then use the k\ensuremath{\cdot}p perturbation method to derive the effective-mass Hamiltonian, which is then checked with that derived using an invariant method based on the Pikus-Bir model. We obtain the band structure ${\mathit{A}}_{\mathit{i}}$ parameters in the group theoretical model explicitly in terms of the momentum-matrix elements. We also find the proper definitions of the important physical quantities used in both models and present analytical expressions for the valence-band dispersions, the effective masses, and the interband optical-transition momentum-matrix elements near the band edges, taking into account the strain effects. \textcopyright{} 1996 The American Physical Society.