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Received 17 May 2000; published in the issue dated 15 December 2000 A systematic method is described for the optimized design of quantum-well structures, in respect to maximizing the stimulated gain in optically pumped intersubband lasers. It relies on applying supersymmetric quantum mechanics to an initial Hamiltonian, in order to (i) add one bound state below the spectral range of the initial Hamiltonian, and (ii) to generate a parameter-controlled family of isospectral Hamiltonians with the desired energy spectrum. By changing the value of the control parameter one changes the potential shape and thus the values of dipole matrix elements and electron-phonon scattering matrix elements. The use of this procedure is demonstrated by designing an appropriate smooth or stepwise-constant graded AlxGa1-xAs ternary alloy quantum well.

A method is described for the optimized design of quantum-well structures, in respect to maximizing the second-order susceptibilities relevant for electro-optic applications. It relies on applying the isospectral (energy structure preserving) transformations to an initial Hamiltonian, in order to generate parameter(s) controlled family of Hamiltonians that (i) are isospectral to the initial one and, (ii) have the potential variation proportional to the effective mass variation, being thus realizable by graded ternary alloys like ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}.$ By changing the values of control parameters one changes the potential shape and thus the values of matrix elements relevant for the susceptibility to be maximized. The use of the method is demonstrated by designing optimal quantum wells for the Stark effect or quantum interference derived electro-optical effects.

A systematic procedure is proposed for the design of ternary alloys based quantum-well structures optimized for double-resonance second-harmonic generation. The method relies on the supersymmetric quantum mechanics as derived here for the case of position-dependent effective mass. Starting from a symmetric, truncated quasiparabolic potential, itself lacking any second-order nonlinearity, we generate a family of asymmetric potentials, fully isospectral with the starting potential, and choose the one which maximizes the product of transition matrix elements relevant for the second-order nonlinearity. Realization of the optimized potential (in an approximate manner) by grading the ternary alloy AlxGa1-xAs is then described. The best value of nonlinear susceptibility obtained exceeds those reported in the literature.

A self-consistent procedure for calculating the energy structure, wave functions, and charge distribution in spherically symmetric semiconductor quantum dots is presented that takes account of both bound and free-electron states. The Schrödinger and Poisson equations are solved iteratively while using the Morse-type parametrized potential to keep the charge neutrality in each iterative step. Numerical calculations performed for a GaAs-Al0.3Ga0.7As based quantum dot indicate that under realistic doping conditions bound states account for most of the charge accumulated in the dot. However, the self-consistent potential very significantly modifies the free-state wave functions and hence the bound-free transition matrix elements.

The existence of bound electron states above the barrier in Bragg-confined structures is proposed as a means of extending the range of photon energies suitable for resonant intersubband second-harmonic generation beyond what is available in conventional quantum-well structures. Within the envelope-function approximation the expressions are first derived for energies and wave functions of bound states in Bragg structures with an asymmetric perturbation layer. A systematic procedure, based on supersymmetric quantum mechanics, is then described for the design of optimized structures to get the maximum nonlinear susceptibility. An example of a calculation for the case of pump photon energy \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\approx}240 meV is also presented and the design aspects of such structures are considered.

We report a systematic and quantitative study of near-infrared intersubband absorption in strained AlGaN/GaN and lattice-matched AlInN/GaN superlattices grown by plasma-assisted molecular-beam epitaxy as a function of Si-doping profile with and without \ensuremath{\delta} doping. For AlGaN/GaN, we obtained good theoretical agreement with experimental measurements of transition energy, integrated absorbance and linewidth by considering many-body effects, interface roughness, and calculations of the transition lifetime that include dephasing. For the AlInN/GaN system, experimental measurements of the integrated absorbance due to the superlattice transitions produced values more than one order of magnitude lower than AlGaN/GaN heterostructures at similar doping levels. Furthermore, observed transition energies were roughly 150 meV higher than expected. The weak absorption and high transition energies measured in these structures is attributed to columnar alloy inhomogeneity in the AlInN barriers observed in high-angle annular dark-field scanning transmission electron microscopy. We simulated the effect of these inhomogeneities using three-dimensional band-structure calculations. The inhomogeneities were modeled as AlInN nanorods with radially varying In composition embedded in the barrier material of the superlattice. We show that inclusion of the nanorods leads to the depletion of the quantum wells (QWs) due to localization of charge carriers in high-In-containing regions. The higher energy of the intersubband transitions was attributed to the relatively uniform regions of the QWs surrounded by high Al (95$%$) composition barriers. The calculated transition energy assuming Al${}_{0.95}$In${}_{0.05}$N barriers was in good agreement with experimental results.

The intraband light absorption between conduction-band states in symmetric semiconductor quantum wells in the presence of a perpendicular magnetic field is discussed. By theoretical analysis three types of transitions are found: bound bound, bound free, and free free. Numerical calculations are given for a rectangular GaAs quantum well in bulk ${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As. The absorption in this structure strongly depends on structure parameters (well thickness, mole fraction, and effective masses), and also on temperature and external magnetic field. Analysis and numerical results such as those presented here may be important for the design of infrared detectors.

The local empirical pseudopotential theory in the S-matrix implementation was used to study the structure of (Zn,Cd)(S,Se) polytypes considered as natural heterocrystalline superlattices. Hexagonal polytype superlattices of (Zn,Cd)(S,Se) are predicted to be characterized by direct band gaps and have minibands derived from evanescent bulk states, which collapse into the heterocrystalline interface bound states or resonances above the valence-band edge. There is a linear relationship between various band gaps and hexagonality h. The superlattice wave function is strongly localized in the cubic portions of superlattice unit cell.

The relationship between the Fr\"olich electron-LO phonon interaction and the electron-light interaction in the conduction band of quantum dots (QDs) based on polar semiconductors is investigated and used to parametrize the intersublevel polaron lifetime. Based on this, the ratio of the optical gain cross section and nonradiative lifetime is described in terms of the QD geometrical and compositional parameters, which is important for possible intraband lasing transitions in QDs.

The electronic properties of twinning boundaries, stacking faults, and a recently proposed structure, the twinning superlattice, in group IV and III-V diamond-type and zinc-blende-type semiconductors are calculated and discussed

A procedure for the self-consistent calculation of the electronic structure of effective-mass quantum wells is described. The procedure uses a parametrized electrostatic potential, and also takes into account the exchange-correlation effects within the local-density approximation. Numerical calculations, performed for InP/${\mathrm{In}}_{0.69}$${\mathrm{Ga}}_{0.31}$As wells at differing temperatures and donor doping levels, indicate that the self-consistent procedure is indeed necessary if reasonably accurate calculations of the charge density, potential, etc., are required.

Block diagonalization of 4\ifmmode\times\else\texttimes\fi{}4 Luttinger Hamiltonians, to be applied for hole-state quantization in [100]-, [110]-, and [111]-grown semiconductor quantum wells, is discussed and the accuracy of the axial approximation is analyzed.

Scattering rate calculations in two-dimensional $\mathrm{Si}∕{\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}$ systems have typically been restricted to rectangular Ge profiles at interfaces between layers. Real interfaces, however, may exhibit diffuse Ge profiles either by design or as a limitation of the growth process. It is shown here that alloy disorder scattering dramatically increases with Ge interdiffusion in (100) and (111) $n$-type quantum wells, but remains almost constant in (100) $p$-type heterostructures. It is also shown that smoothing of the confining potential leads to large changes in subband energies and scattering rates, and a method is presented for calculating growth process tolerances.

We report the direct determination of nonradiative lifetimes in Si/SiGe asymmetric quantum well structures designed to access spatially indirect (diagonal) interwell transitions between heavy-hole ground states, at photon energies below the optical phonon energy. We show both experimentally and theoretically, using a six-band k center dot p model and a time-domain rate equation scheme, that, for the interface quality currently achievable experimentally (with an average step height >= 1 A), interface roughness will dominate all other scattering processes up to about 200 K. By comparing our results obtained for two different structures we deduce that in this regime both barrier and well widths play an important role in the determination of the carrier lifetime. Comparison with recently published experimental and theoretical data obtained for mid-infrared GaAs/AlxGa1-xAs multiple quantum well systems leads us to the conclusion that the dominant role of interface roughness scattering at low temperature is a general feature of a wide range of semiconductor heterostructures not limited to IV-IV materials.

A theoretical model of electron transport in quantum cascade lasers subjected to a magnetic field is developed. The Landau level electronic structure was calculated from the envelope-function Schr\"odinger equation within the effective-mass approximation. The electron transport in a magnetic field was modeled using the self-consistent rate-equation description for the full period of the cascade and its interaction with adjacent periods. The scattering processes included in the model are electron\char21{}longitudinal-optical-phonon, electron\char21{}longitudinal-acoustic-phonon, and electron-electron scattering. All these processes show oscillatory behavior with magnetic field, and their interplay determines the electron transport and the output characteristics of quantum cascade lasers in magnetic field. The model was applied to investigate the influence of magnetic field on the performance of a $\mathrm{GaAs}∕\mathrm{AlGaAs}$ quantum cascade laser emitting at $\ensuremath{\lambda}\ensuremath{\approx}11.4\phantom{\rule{0.3em}{0ex}}\mathrm{\ensuremath{\mu}}\mathrm{m}$ [P. Kruck et al., Appl. Phys. Lett. 76, 3340 (2000)]. The calculated results show good overall agreement with the available experimental data.

The calculation of the electronic structure of the GaN stacking faults in the framework of the local empirical pseudopotential theory is presented. The stacking faults in both zinc-blende and wurtzite GaN are predicted to introduce electronic levels within the band gap, with the energy (0.13\ifmmode\pm\else\textpm\fi{}0.01) eV above the valence-band top. These levels are found to originate from interface states of heterocrystalline wurtzite (0001)-zinc-blende (111) interfaces. The possible implications of stacking fault electronic states on the luminescence properties of GaN are discussed.

A method for the calculation of the electronic structure of pyramidal self-assembled InAs/GaAs quantum dots is presented. The method is based on exploiting the C-4 symmetry of the 8-band k·p Hamiltonian with the strain taken into account via the continuum mechanical model. The operators representing symmetry group elements were represented in the plane wave basis and the group projectors were used to find the symmetry adapted basis in which the corresponding Hamiltonian matrix is block diagonal with four blocks of approximately equal size. The quantum number of total quasiangular momentum is introduced and the states are classified according to its value. Selection rules for interaction with electromagnetic field in the dipole approximation are derived. The method was applied to calculate electron and hole quasibound states in a periodic array of vertically stacked pyramidal self-assembled InAs/GaAs quantum dots for different values of the distance between the dots and external axial magnetic field. As the distance between the dots in an array is varied, an interesting effect of simultaneous change of ground hole state symmetry, type, and the sign of miniband effective mass is predicted. This effect is explained in terms of the change of biaxial strain. It is also found that the magnetic field splitting of Kramer's double degenerate states is most prominent for the first and second excited state in the conduction band and that the magnetic field can both separate otherwise overlapping minibands and concatenate otherwise nonoverlapping minibands.

Time-resolved studies of the dynamics of intersubband transitions are reported in three different strain symmetrized p-Si∕SiGe multiple-quantum-well and quantum cascade structures in the far-infrared wavelength range (where the photon energy is less than the optical phonon energy), utilizing the FELIX free-electron laser. The calculated rates for optical and acoustic phonon scattering, alloy disorder scattering, and carrier-carrier scattering have been included in a self-consistent energy balance model of the transient far-infrared intersubband absorption, and show good agreement with our degenerate pump-probe spectroscopy measurements in which, after an initial rise time determined by the resolution of our measurement, we determine decay times ranging from ∼2 to ∼25 ps depending on the design of the structure. In all three samples the lifetimes for the transition from the first light hole subband to the first heavy hole subband are found to be approximately constant in the temperature range 4–100 K.

A detailed analysis of hole transport in cascaded $p\text{\ensuremath{-}}\text{Si}∕\text{SiGe}$ quantum well structures is performed using ensemble Monte Carlo simulations. The hole subband structure is calculated using the $6\ifmmode\times\else\texttimes\fi{}6\phantom{\rule{0.3em}{0ex}}\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ model, and then used to find carrier relaxation rates due to the alloy disorder, acoustic and optical phonon scattering. The simulation accounts for the in-plane $\mathbf{k}$-space anisotropy of both the hole subband structure and the scattering rates. Results are presented for prototype terahertz $\text{Si}∕\text{SiGe}$ quantum cascade structures.

Using the $6\ifmmode\times\else\texttimes\fi{}6 \mathbf{k}\ensuremath{\cdot}\mathbf{p}$ method we calculate hole-phonon and alloy disorder scattering rates in SiGe quantum wells, and how these depend on the various parameters of the system. The relative importance of different branches of nonpolar optical and acoustic phonons is discussed, and a comparison is made with alloy scattering. The latter is found to be an important mechanism for intersubband hole relaxation, particularly for low-energy transitions at low temperatures, where it dominates over phonon scattering, while losing significance in the opposite case. The results are relevant for the design and operation of SiGe-based quantum cascade lasers relying on intersubband transitions in the valence band.

A method for finding bound states in quantum systems described by coupled Schro\ifmmode\ddot\else\textasciidieresis\fi{}dinger equations is discussed, and applied to the case of hole-state quantization in semiconductor quantum wells, within the axial approximation.

This paper treats the influence of indirect minima on electron concentration in GaAs-${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$As superlattices. It is shown that the population of the minizones arising from indirect minima significantly depends on superlattice parameters. For higher values of the mole fraction x and for thinner wells, the relative population of indirect-minima minizones is remarkable, and may even be equal to unity for x=1 and for very thin layers (\ensuremath{\sim}3 nm). This effect is of importance in the determination of superlattice parameters.

The energy spectrum and bound-bound intraband transitions in semiconductor quantum dots are analyzed. Numerical results for the GaAs-${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$As system indicate that a considerable absorption at a number of distinct wavelengths in the infrared range (\ensuremath{\sim}7--25 \ensuremath{\mu}m) may be obtained, which greatly exceeds free-carrier absorption.