Interband Momentum Matrix Elements Research Articles

Extraction of eight-band k⋅p parameters from empirical pseudopotentials for GeSn

We extract the parameters for the eight-band k⋅p model for Ge1−xSnx (x≤0.15) from the calculation of the nonlocal empirical pseudopotential method with the modified virtual crystal approximation. The atomic pseudopotential form factors of Ge are improved such that the calculated Ge band structure has the commonly accepted bandgap and effective masses. The improved Ge parameters are used in proper interpolation to derive the parameters of GeSn for empirical pseudopotential calculation. The calculated band structures suggest that the Ge1−xSnx alloys exhibit a transition between an indirect bandgap semiconductor and a direct one at Sn composition xc=0.071. From the calculation, we extract the bandgap, the split-off energy, the interband momentum matrix element, and the effective masses of Ge1−xSnx (x≤0.15) as functions of x. From these results, we further derive the parameters used in the eight-band k⋅p model. These parameters are well expressed in quadratic form. The k⋅p model with the extracted parameters can give an interband tunneling current in a “pin” diode that is consistent with the current calculated by the empirical pseudopotential method.

Piezoelectric and deformation potential effects of strain-dependent luminescence in semiconductor quantum well structures

The mechanism of strain-dependent luminescence is important for the rational design of pressure-sensing devices. The interband momentum-matrix element is the key quantity for understanding luminescent phenomena. We analytically solved an infinite quantum well (IQW) model with strain, in the framework of the 6 × 6 k·p Hamiltonian for the valence states, to directly assess the interplay between the spin-orbit coupling and the strain-induced deformation potential for the interband momentum-matrix element. We numerically addressed problems of both the infinite and IQWs with piezoelectric fields to elucidate the effects of the piezoelectric potential and the deformation potential on the strain-dependent luminescence. The experimentally measured photoluminescence variation as a function of pressure can be qualitatively explained by the theoretical results.

Characterization of InN - In0.25Ga0.75N Quantum Well Laser Structure for 1330 nm Wavelength

A nitride based wurtzite-strained QW laser with 12Å InN well layer, 15Å In0.25Ga0.75N barrier layer and GaN SCH layer has been designed and characterized at 1330nm wavelength. To determine the electronic properties, a self-consistent method with 6-bands k.p formalism considering valence-band mixing effect, strain and polarization effect followed by Poisson's equation has been used. The interband momentum matrix elements, optical gain, spontaneous emission rate, and radiative current density have been calculated to analyze the optical properties of the laser. Due to the strain effect, the wave-function overlap integral was obtained as 43.27%. The structure is TE polarized with C1-HH1 and C1-LH1 dominating transitions, while the spontaneous emission rate per energy interval per unit volume was 7.21×1027 s-1cm-3eV-1 at 1329.55nm. Furthermore, the radiative recombination rate and the radiative current density were 7.77×1029 s-1cm-3 and 149.19 Acm-2, respectively. The optical gain of the structure is 5261.52cm-1 at 1336.7nm for TE-polarization.

Characterization of InN - In0.25Ga0.75N Quantum Well Laser Structure for 1330 nm Wavelength

In this work, a nitride based wurtzite-strained single quantum well laser has been designed and characterized at 1330 nm wavelength. Here, 12Å InN layer has been used as well material and 15Å In0.25Ga0.75N layer has been used as barrier material along with GaN as separate confinement heterostructure for better carrier and optical confinement. To determine the electronic properties of the designed structure, a developed self-consistent method has been solved. Conduction subband calculation has been performed with the solution of Schrodinger equation formed by single band Hamiltonian followed by Poisson's equation. 6-bands k . p Hamiltonian for wurtzite semiconductor has been solved for valence band Schrodinger's equation with Poisson's equation considering band mixing effect, strain due to lattice mismatch and spontaneous and piezoelectric polarization. The interband momentum matrix elements, optical gain, spontaneous emission rate, spontaneous radiative recombination rate and radiative current density have been calculated to analyze the optical properties of the designed laser and a good agreement with previously published works is obtained. All analysis has been performed using injection carrier density of 6×10-19 cm-3 at T = 300 K. Due to the strain effect, the wave-function overlap integral has been obtained as 43.27%. From the electronic properties analysis, it has been found that, the designed structure is TE polarized with C1-HH1 and C1-LH1 dominating transitions. From the analysis of optical properties, the obtained spontaneous emission rate is 5945.2 s-1 at 1329.55 nm and spontaneous emission rate per energy interval per unit volume is 7.21×1027 s-1cm-3eV-1 at same wavelength. The obtained radiative recombination rate is 7.77×1029 s-1cm-3 and radiative current density is 149.19 Acm-2. The optical gain for TE polarization of the designed QW structure is 5261.52cm-1 at 1336.7 nm. Figure 1

An envelope function formalism for lattice-matched heterostructures

The envelope function method traditionally employs a single basis set which, in practice, relates to a single material because the k·p matrix elements are generally only known in a particular basis. In this work, we defined a basis function transformation to alleviate this restriction. The transformation is completely described by the known inter-band momentum matrix elements. The resulting envelope function equation can solve the electronic structure in lattice matched heterostructures without resorting to boundary conditions at the interface between materials, while all unit-cell averaged observables can be calculated as with the standard envelope function formalism. In the case of two coupled bands, this heterostructure formalism is equivalent to the standard formalism while taking position dependent matrix elements.

Effects of strain on the electron effective mass in GaN and AlN

Stress is known to strongly alter the effective mass in semiconductors, changing the mobility of carriers. Transport measurements on AlGaN/GaN heterostructures indicated a large increase in mobility under tensile strain [M. Azize and T. Palacios, J. Appl. Phys. 108, 023707 (2010)]. Using first-principles methods, we calculate the variation of electron effective mass in GaN and AlN under hydrostatic and biaxial stress. Unexpected trends are found, which are explained within k·p theory through a variation of the interband momentum matrix elements. The magnitude of the effective-mass reduction is too small to explain the experimentally reported increase in mobility.

Relation between the interband dipole and momentum matrix elements in semiconductors

It is shown that a frequently used relation between the interband momentum and dipole matrix elements (shortened to the ``p-r relation'') in semiconductors acquires an additional correction term if applied to finite-volume crystals treated with periodic boundary conditions. The correction term, which is a generalization of the one obtained by Yafet [Phys. Rev. 106, 679 (1957)] for infinite crystals, does not vanish in the limit of infinite volume. We illustrate this with numerical examples for bulk GaAs and GaAs superlattices. The persistence of the correction term is traced to the subtle nature of the dipole matrix element with spatially extended wave functions. In contrast, a straightforward application of the findings by Blount [Solid State Phys. 13, 305 (1962)] and Haug [Theoretical Solid State Physics (Pergamon, Oxford, 1972)] yields the usual p-r relation in the distribution sense, without any corrections, when Bloch wave functions normalized to delta functions in crystal momentum space are used. Our findings therefore show that, for the interband dipole matrix element, using Bloch wave functions under periodic boundary conditions is not the proper way to approach the infinite-volume limit. From our numerical evaluations, we find that the correction term is large in the case of interband transitions in bulk GaAs, and that it can be chosen to be small in the case of intersubband transitions in superlattices, which are important in the context of terahertz (THz) radiation. We also show that one can interpret the infinite-volume p-r relation in terms of a limiting procedure using progressively broadened wave packet states that approach delta-normalized Bloch wave functions. Finally, we discuss the p-r relation for nanostructures in the envelope function approximation and show that the cell-envelope factorization of the nanostructure dipole matrix element into a cell-matrix element and an envelope overlap integral involves the cell gradient-$k$ rather than the cell dipole matrix element.

Interband momentum matrix elements and a k·p interpolation method applied to PbTe

Interband momentum matrix elements and a k·p interpolation method applied to PbTe

Eigenstate fitting in the k · p method

We present a study of the decomposition of the conduction band (CB) eigenstates in terms of the Brillouin zone center solutions. The methods we use for the decomposition are, ranging from more empirical to more first-principle, a 40-band tight binding model, the empirical pseudopotential method and a quasiparticle self-consistent GW (QPscGW). We observe that on the order of 20 bands are needed to obtain a good description of the CB in the full zone. We extract univocally from those calculations the interband momentum matrix elements, useful for k·p models. Therefore, we can construct Hamiltonians that reproduce the correct components of the CB eigenstates, which is needed for the calculation of optical properties and spin dynamics, to name a few.

Band parameters for GaAs and Si in the 24-k ⋅ p model

A 24-k ⋅ p model is used to compute the principal effective-mass parameters at Γ, X and L valleys in a direct-band-gap semiconductor (GaAs) as well as an indirect-band-gap semiconductor (Si). The values of the effective masses for electrons, heavy holes and light holes in the Γ, X and L valleys are in very good agreement with those reported in other publications. Satisfactory agreement with available experimental data is also obtained by this model. The Luttinger parameters and interband momentum matrix elements proposed in this work are consistent with the previous publications.

Band parameters for III–V compound semiconductors and their alloys

We present a comprehensive, up-to-date compilation of band parameters for the technologically important III–V zinc blende and wurtzite compound semiconductors: GaAs, GaSb, GaP, GaN, AlAs, AlSb, AlP, AlN, InAs, InSb, InP, and InN, along with their ternary and quaternary alloys. Based on a review of the existing literature, complete and consistent parameter sets are given for all materials. Emphasizing the quantities required for band structure calculations, we tabulate the direct and indirect energy gaps, spin-orbit, and crystal-field splittings, alloy bowing parameters, effective masses for electrons, heavy, light, and split-off holes, Luttinger parameters, interband momentum matrix elements, and deformation potentials, including temperature and alloy-composition dependences where available. Heterostructure band offsets are also given, on an absolute scale that allows any material to be aligned relative to any other.

Theory of the electronic structure of Ga1-yInyNxAs1−x and related alloys

We present a quantitative k.P Hamiltonian which describes analytically the composition dependence of the energy gap, interband momentum matrix element, band edge effective masses and conduction band dispersion of GaNXAs1−x alloys for low N concentrations (x < ∼ 0.05). The model has been confirmed using an sp3s* tight-binding Hamiltonian whose results agree well both with experiment and with previous pseudopotential calculations. The model should be of wide use to guide the future development of this material system and its applications.

Theory of enhanced bandgap non-parabolicity in GaN xAs 1− x and related alloys

We present a quantitative two-band theory which describes analytically the composition dependence of the energy gap, interband momentum matrix element and band edge effective masses in GaN x As 1− x alloys for x<∼0.1. The model is confirmed using an sp 3s ∗ tight-binding Hamiltonian, whose results agree well, both with experiment and previous pseudopotential calculations.

Magnetoreflectivity ofPb1−xEuxTe epilayers and PbTe/Pb1−xEuxTe multiple quantum wells

Molecular-beam epitaxy grown n-type ${\mathrm{Pb}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Eu}}_{\mathrm{x}}$Te epilayers (x\ensuremath{\leqslant}0.034) and PbTe/${\mathrm{Pb}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Eu}}_{\mathrm{x}}$Te (x\ensuremath{\leqslant}0.039) multiple-quantum-well (MQW) samples were studied by magnetoreflectivity in the Faraday configuration (B\ensuremath{\Vert}[111]) for magnetic fields up to 6 T at 4.2 K. Since the IV-VI lead salt compounds are quite polar semiconductors, resonant electron-longitudinal-optic- (LO-) phonon coupling (Fr\"ohlich coupling) modifies the cyclotron resonance (CR) energies in the ${\mathrm{Pb}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Eu}}_{\mathrm{x}}$Te single epilayers for the three-dimensional (3D) case. Due to the many-valley band structure two different Fr\"ohlich coupling constants are relevant. However, the CR energies of quasi-two-dimensional (2D) carriers in PbTe wells [${\mathrm{n}}^{2\mathrm{D}}$=(1.5-3)\ifmmode\times\else\texttimes\fi{}${10}^{11}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}2}$] of PbTe/${\mathrm{Pb}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Eu}}_{\mathrm{x}}$Te MQW samples do not exhibit a significant resonant electron-LO-phonon interaction. This observation is attributed to finite-electron concentration effects, in particular, to a partial filling of the lowest 2D Landau spin level. The static and dynamic screening of the polar interaction are considered as well, but are ruled out as an explanation for the absence of any remarkable polaron correction to the CR energies of electrons in the PbTe quantum wells for the range of carrier concentrations investigated. The magnetoreflectivity spectra of ${\mathrm{Pb}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Eu}}_{\mathrm{x}}$Te single layers and PbTe/${\mathrm{Pb}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Eu}}_{\mathrm{x}}$Te quantum well samples are simulated numerically, using a model for the dielectric response of IV-VI compounds in a magnetic field, which also includes the electron-LO-phonon interaction. The transverse and longitudinal masses, and thus also the interband momentum matrix elements are determined for ${\mathrm{Pb}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Eu}}_{\mathrm{x}}$Te as a function of the composition up to x0.034. It is found that the transverse mass increases with Eu content, whereas the longitudinal one nearly stays constant. The 2D CR masses of electrons in the PbTe wells increase with decreasing well width, i.e., with increasing quantum-well interband energies, a behavior which results from the strong band nonparabolicity.

Radiative carrier lifetime, momentum matrix element, and hole effective mass in GaN

By using picosecond time-resolved photoluminescence we have measured the lifetime of excess charge carriers in GaN epitaxial layers grown on sapphire at temperatures up to 300 K. The decay time turns out to be dominated by trapping processes at low excitation levels. The radiative lifetime derived from our data is dominated by free excitons at temperatures below 150 K, but also clearly shows the gradual thermal dissociation of excitons at higher temperatures. From our data, we are able to determine the free exciton binding energy and the free carrier radiative recombination coefficient. By combining these data with optical absorption data, we find the interband momentum matrix element and an estimate for the hole effective mass, which is much larger than previously thought.

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.

Near-Bandgap Photoluminescence Decay Time in GaN Epitaxial Layers Grown on Sapphire

ABSTRACTUsing picosecond time-resolved photoluminescence we have studied the decay time of excess carriers in GaN epitaxial layers over a wide range of temperatures from 4 K up to 400 K. At low temperature, a thermal dissociation of donor-bound excitons is observed. At higher temperatures up to room temperature, the luminescence decay at moderate excitation is governed by trapping of photogenerated electrons in ionized shallow donor levels. Using measured luminescence intensities to determine the quantum efficiency, we obtain the radiative lifetime of free excitons from low temperature up to room temperature. We use these data to determine the radiative recombination coefficient and the interband momentum matrix element.

On the estimation of matrix elements for optical transitions in semiconductors

A semi-empirical method is used to calculate the numerical values of the interband momentum matrix elements of the allowed optical transitions in semiconductors. This method is based on the evaluation of the ratio of the two-photon and one-photon absorption coefficients and to compare the result with the corresponding experimental values in a number of semiconductors both for direct and indirect transition processes. The numerical values of the momentum matrix elements are compared with the convenient theoretical calculations available. The result is found to agree fairly well with the corresponding values computed using the k· p perturbation theory.

Shubnikov-De Haas oscillations in n-CuInSe 2

The results of the Shubnikov-De Haas (SDH) oscillation measurements on CuInSe 2-doped (CIS) crystals are presented for the first time. The measurements were performed in the temperature range from 1.9 to 40 K in pulsed magnetic fields up to 40 T. It was found that the Fermi surface of electrons consisted of one sphere. The values of the cyclotron effective mass at the bottom of the conduction band, the inter-band momentum matrix element and the g∗ factor were calculated.

Shubnikov–de Haas oscillations in n-CuInSe2

The results of the Shubnikov–de Haas oscillations measurements on CuInSe2 crystals are presented for the first time. The measurements were performed in the temperature range from 1.9 to 40 K in pulsed magnetic field up to 40 T. It was found that the Fermi surface of electrons consisted of one sphere. The values of cyclotron effective masses, the effective mass at the bottom of the conduction band, the interband momentum matrix element and g*-factor were calculated.