Envelope Function Approximation Research Articles

Solution of the Inverse Spectroscopic Ellipsometry Problem for an Absorbing Substrate with a Dielectric Layer

An analytical solution of the inverse spectroscopic ellipsometry problem for a transparent layer on an absorbing substrate is obtained based on the envelope function approximation. It is used to determine the spectral dependences of the refractive index n(λ) and absorption k(λ) for model and real silicon substrates with silicon dioxide layers. It was possible to determine analytically the effective optical characteristics of the silicon substrate in the approximation of a layer-substrate model with ideal interface boundaries. This result, together with the relative positions of the spectral curves for the ellipsometric angles tan ψ(λ) and cos Δ(λ) measured for a silicon substrate with a natural surface layer and for a silicon dioxide layer–silicon substrate structure, can be explained by the fact that the silicon dioxide layer is surrounded by transition and surface layers.

Excitonic transitions in Be-doped GaAs/AlAs multiple quantum well**Project supported by the National Natural Science Foundation of China (Grant No. 61178039) and the Natural Science Foundation of Shandong Province, China (Grant No. ZR2012FM028).

A series of GaAs/AlAs multiple-quantum wells doped with Be is grown by molecular beam epitaxy. The photoluminescence spectra are measured at 4, 20, 40, 80, 120, and 200 K, respectively. The recombination transition emission of heavy-hole and light-hole free excitons is clearly observed and the transition energies are measured with different quantum well widths. In addition, a theoretical model of excitonic states in the quantum wells is used, in which the symmetry of the component of the exciton wave function representing the relative motion is allowed to vary between the two- and three-dimensional limits. Then, within the effective mass and envelope function approximation, the recombination transition energies of the heavy- and light-hole excitons in GaAs/AlAs multiple-quantum wells are calculated each as a function of quantum well width by the shooting method and variational principle with two variational parameters. The results show that the excitons are neither 2D nor 3D like, but are in between in character and that the theoretical calculation is in good agreement with the experimental results.

Electronic states in spherical GaN nanocrystals embedded in various dielectric matrices: The k ⋅ p-calculations

Using the envelope-function approximation, the single-particle states of electrons and holes in spherical GaN nanocrystals embedded in different amorphous dielectric matrices (SiO2, Al2O3, HfO2 and Si3N4) have been calculated. Ground state energies of electrons and holes in GaN nanocrystals are determined using the isotropic approximation of the k ⋅ p -Hamiltonian. All the ground state energies are found to increase with lowering the nanocrystal size and are proportional to the R−n, where R is the nanocrystal radius, n =1.5-1.9 for electrons and 1.7-2.0 for holes. The optical gap of GaN nanocrystals changes from 3.8 to 5 eV for the nanocrystal radius ranging from 3 to 1 nm.

Insensitivity of spin dynamics to the orbital angular momentum transferred from twisted light to extended semiconductors

We study the spin dynamics of carriers due to the Rashba interaction in semiconductor quantum disks and wells after excitation with light with orbital angular momentum. We find that although twisted light transfers orbital angular momentum to the excited carriers and the Rashba interaction conserves their total angular momentum, the resulting electronic spin dynamics is essentially the same for excitation with light with orbital angular momentum $l=+|l|$ and $l=-|l|$. The differences between cases with different values of $|l|$ are due to the excitation of states with slightly different energies and not to the different angular momenta per se, and vanish for samples with large radii where a $k$-space quasi-continuum limit can be established. These findings apply not only to the Rashba interaction but also to all other envelope-function approximation spin-orbit Hamiltonians like the Dresselhaus coupling.

Exciton binding energy in an infinite potential semiconductor quantum well–wire heterostructure

An interacting electron–hole pair in a quantum well–wire is studied within the framework of the effective-mass approximation. An expansion to a 1 dimensional, quantum well model is presented to include another confinement dimension for a quasi-2 dimensional, quasi-1 dimensional quantum well–wire heterostructure. The technique is applied to an infinite well–wire confining potential. The envelope function approximation is employed in the approach, involving a three parameter variational calculation in which the symmetry of the component of the wave function representing the relative motion is allowed to vary from the one- to the two- and three-dimensional limits. Results to such a numerical calculation are presented. Quantitative comparisons with previous calculations for quantum wells is made (in the wire limit where Lz→∞) to find a good agreement between finite and infinite potential models up to a size of 100Å.

Carriers confinement study of GaNAsBi/GaAs QWs emitting at 1.3 and 1.55 μm

Band structures of GaN0.58yAs1–1.58yBiy/GaAs quantum wells (QWs) were studied using the band anticrossing model and the envelope function approximation. The confined states energies and the oscillator strengths of interband transitions were determined for well widths LW and Bi composition y varying in the range of 4–10 nm and 0–0.07 respectively. The emissions 1.3 and 1.55 μm were reached for specific couples (LW, y). The band anticrossing effect on the in-plane carriers effective mass has been investigated at k = 0. The absorbance spectra were calculated for QWs operating at 1.3 and 1.55 μm.

Effectiveg-factor tensor for carriers in IV-VI semiconductor quantum wells

A theory for the electron (and hole) $g$ factor in multivalley lead-salt IV-VI semiconductor quantum wells (QWs) is presented. An effective Hamiltonian for the QW electronic states in the presence of an external magnetic field is introduced within the envelope-function approximation, based on the multiband $kp$ Dimmock model for the bulk. The mesoscopic spin-orbit (Rashba-type) and Zeeman interactions are taken into account on an equal footing and the effective $g$ factor in symmetric quantum wells (${g}_{\mathrm{QW}}^{*}$) is calculated analytically for each nonequivalent conduction-band (and valence-band) valley, and for QWs grown along different crystallographic directions.

CdSe/ZnCdSe Quantum Dot Heterostructures for Yellow Spectral Range Grown on GaAs Substrates by Molecular Beam Epitaxy

This paper reports on theoretical calculations and fabrication by molecular beam epitaxy of wide-gap II VI heterostructures emitting in the true yellow range (560 600 nm) at room temperature. The active region of the structures comprises CdSe quantum dot active layer embedded into a strained Zn1−xCdxSe (x = 0.2−0.5) quantum well surrounded by a Zn(S,Se)/ZnSe superlattice. Calculations of the CdSe/(Zn,Cd)Se/Zn(S,Se) quantum dot quantum well luminescence wavelength performed using the envelope-function approximation predict rather narrow range of the total Zn1−xCdxSe quantum well thicknesses (d ≈ 2−4 nm) reducing e ciently the emission wavelength, while the variation of x (0.2 0.5) has much stronger e ect. The calculations are in a reasonable agreement with the experimental data obtained on a series of test heterostructures. The maximum experimentally achieved emission wavelength at 300 K is as high as 600 nm, while the intense room temperature photoluminescence has been observed up to λ = 590 nm only. To keep the structure pseudomorphic to GaAs as a whole the tensile-strained surrounding ZnS0.17Se0.83/ZnSe superlattice were introduced to compensate the compressive stress induced by the Zn1−xCdxSe quantum well. The graded-index waveguide laser heterostructure with a CdSe/Zn0.65Cd0.35Se/Zn(S,Se) quantum dot quantum well active region emitting at λ = 576 nm (T = 300 K) with the 77 to 300 K intensity ratio of 2.5 has been demonstrated.

Understanding quantum confinement of charge carriers in layered 2D hybrid perovskites.

Layered hybrid organic perovskites (HOPs) structures are a class of low-cost two-dimensional materials that exhibit outstanding optical properties, related to dielectric and quantum confinement effects. Whereas modeling and understanding of quantum confinement are well developed for conventional semiconductors, such knowledge is still lacking for 2D HOPs. In this work, concepts of effective mass and quantum well are carefully investigated and their applicability to 2D HOPs is discussed. For ultrathin layers, the effective-mass model fails. Absence of superlattice coupling and importance of non-parabolicity effects prevents the use of simple empirical models based on effective masses and envelope function approximations. An alternative method is suggested in which 2D HOPs are treated as composite materials, and a first-principles approach to the calculation of band offsets is introduced. These findings might also be relevant for other classes of layered 2D functional materials.

Orbital magnetization in dilute ferromagnetic semiconductors

The relationship between the modern and classical Landau's approach to carrier orbital magnetization is studied theoretically within the envelope function approximation, taking ferromagnetic (Ga,Mn)As as an example. It is shown that while the evaluation of hole magnetization within the modern theory does not require information on the band structure in a magnetic field, the number of basis wave functions must be much larger than in the Landau approach to achieve the same quantitative accuracy. A numerically efficient method is proposed, which takes advantages of these two theoretical schemes. The computed magnitude of orbital magnetization is in accord with experimental values obtained by x-ray magnetic circular dichroism in (III,Mn)V compounds. The direct effect of the magnetic field on the hole spectrum is studied too, and employed to interpret a dependence of the Coulomb blockade maxima on the magnetic field in a single electron transistor with a (Ga,Mn)As gate.

Low-energy trions in graphene quantum dots

We investigate, within the envelope function approximation, the low-energy states of trions in graphene quantum dots (QDs). The presence of valley pseudospin in graphene as an electron degree of freedom apart from spin adds convolution to the interplay between exchange symmetry and the electron-electron interaction in the trion, leading to new states of trions as well as a low-energy trion spectrum different from those in semiconductors. Due to the involvement of valley pseudospin, it is found that the low-energy spectrum is nearly degenerate and consists of states all characterized by having an antisymmetric (pseudospin) $\ensuremath{\bigotimes}$ (spin) component in the wave function, with the spin (pseudospin) part being either singlet (triplet) or triplet (singlet), as opposed to the spectrum in a semiconductor whose ground state is known to be nondegenerate and always a spin singlet in the case of ${X}^{\ensuremath{-}}$ trions. We investigate trions in the various regimes determined by the competition between quantum confinement and electron-electron interaction, both analytically and numerically. The numerical work is performed within a variational method accounting for electron mass discontinuity across the QD edge. The result for electron-hole correlation in the trion is presented. Effects of varying quantum dot size and confinement potential strength on the trion binding energy are discussed. The ``relativistic effect'' on the trion due to the unique relativistic type electron energy dispersion in graphene is also examined.

Ground state energy and wave function of an off-centre donor in spherical core/shell nanostructures: Dielectric mismatch and impurity position effects

Ground state energy and wave function of a hydrogen-like off-centre donor impurity, confined anywhere in a ZnS/CdSe spherical core/shell nanostructure are determined in the framework of the envelope function approximation. Conduction band-edge alignment between core and shell of nanostructure is described by a finite height barrier. Dielectric constant mismatch at the surface where core and shell materials meet is taken into account. Electron effective mass mismatch at the inner surface between core and shell is considered. A trial wave function where coulomb attraction between electron and off-centre ionized donor is used to calculate ground state energy via the Ritz variational principle. The numerical approach developed enables access to the dependence of binding energy, coulomb correlation parameter, spatial extension and radial probability density with respect to core radius, shell radius and impurity position inside ZnS/CdSe core/shell nanostructure.

Molecular states of laterally coupled quantum dots under electric fields

The states of a single electron trapped in two laterally coupled quantum dots are studied theoretically in the framework of the effective mass and envelope function approximations. The electron tunneling between dots is studied by varying of inter-dot distance and we showed that the lateral quantum coupling between them allows the formation of molecular-like states, which exhibit similar characteristics to those of a molecule H2+. The effect of an in-plane electric field on the energy spectrum is analyzed and our results reveal that the wavelength of photons emitted from the system can be tuned by simply applying a low-intensity electric field. This latter feature is consistent with experimental observations.

Effect of spin-orbit coupling on the structure of the electron ground state in silicon nanocrystals

The effect of spin-orbit interaction on the structure of the ground state in the conduction band of spherical silicon nanocrystals is theoretically studied using the envelope-function approximation and the k · p method. It is shown that the arising weak spin-orbit coupling of the conduction- and valence bands leads to specific asymmetric hybridization of the s- and p-type envelope functions with opposite spin orientations caused by the anisotropy of spin mixing in the silicon conduction band. As a result, the wave functions of the ground-state transform which is accompanied by an insignificant decrease in its energy. In this case, the spin-mixing parameter in nanocrystals depends strongly on their size due to the quantum-confinement effect.

Magnetic field implementation in multiband k⋅p Hamiltonians of holes in semiconductor heterostructures

We propose an implementation of external homogeneous magnetic fields in k⋅p Hamiltonians for holes in heterostructures, in which we make use of the minimal coupling prior to introducing the envelope function approximation. Illustrative calculations for holes in InGaAs quantum dot molecules show that the proposed Hamiltonian outperforms the standard Luttinger model (Luttinger 1956 Phys. Rev. 102 1030) describing the experimentally observed magnetic response. The present implementation culminates our previous proposal (Planelles et al 2010 Phys. Rev. B 82 155307).

Modeling and process control of MOCVD growth of InAlGaAs MQW structures on InP

We have developed a model which integrates calculation of InAlGaAs multiple quantum well (MQW) transition energies using the envelope function approximation with a statistical analysis of the PL emission wavelength, net strain and MQW period measured for a variety of MQW designs grown by MOCVD. The model relates the measured MQW parameters directly to MOCVD process parameters, allowing an accurate prediction of the process parameters required to grow a specified MQW design. This greatly reduces the need to grow and characterize individual calibration layers. The difference of the measured and predicted MQW parameters is recorded run-to-run over time, which allows process variability to be analyzed across a number of process parameters with intentional variations to grow different MQW designs.

Electronic States and Optical Gap of Phosphorus-Doped Silicon Nanocrystals Embedded in a Silica Host Matrix

Using the envelope-function approximation the electronic states and the optical gap of silicon nanocrystals heavily doped with phosphorus have been calculated. Assuming the uniform impurity distribution over the crystallite volume we have found the fine structure of the electron ground state (induced by the valley-orbit interaction) and the optical gap as a function of the crystallite size and donor concentration. It is shown that the energy of the ground singlet state decreases almost linearly as the concentration increases, while the valley-orbit splitting increases nonlinearly. Phosphorus doping also results in the decrease of the nanocrystal gap with increasing the impurity concentration.

Achievement of tailored laser frequencies by fine-tuning the structural parameters of Fibonacci’s in AlxGa1−xAs/GaAs superlattices

We investigate the tunneling conduction in AlxGa1−xAs/GaAs Fibonacci superlattices. Using the transfer matrix formalism, the effective mass and envelope function approximations, we calculate the transmission coefficient of quasiperiodic multibarrier system. We examine numerically the effect of the Fibonacci’s structural parameters on the electronic energy spectra of Fibonacci Superlattices (FSL). Ours results show that increasing the width of Fibonacci’s wells af allows to the confinement of subminibands with a widening of minigaps, this causes a consistent and coherent fragmentation. The barrier thickness of Fibonacci bf acts on the width of subminibands by controlling the interaction force between neighboring eigenstates. Its increase gives rise to singularly extended states. The barrier height Fibonacci Vf permit to control the degree of structural disorder in these structures. The variation of these parameters permits the design of laser with modulated wavelength.

Optical Modulation by Conducting Interfaces

We analyze the interaction of a propagating guided electromagnetic wave with a quantum well embedded in a dielectric slab waveguide. First, we design a quantum well based on InAlGaAs compounds with the transition energy of 0.8eV corresponding to a wavelength of 1.55um. By exploiting the envelope function approximation, we derive the eigenstates of electrons and holes and the transition dipole moments, through solution of the Luttinger Hamiltonian. Next, we calculate the electrical susceptibility of a three-level quantum system (as a model for the two-dimensional electron gas trapped in the waveguide), by using phenomenological optical Bloch equations. We show that the two-dimensional electron gas behaves as a conducting interface, whose conductivity can be modified by controlling the populations of electrons and holes the energy levels. Finally, we design a slab waveguide in which a guided wave with the wavelength of 1.55um experiences a strong coupling to the conducting interface. We calculate the propagation constant of the wave in the waveguide subject to the conducting interface, by exploiting the modified transfer matrix method, and establish it linear dependence on the interface conductivity. By presenting a method for controlling the populations of electrons and holes, we design a compact optical modulator with an overall length of around 60um.

Interband tansition and electronic structures in strained In x Ga1−x N/GaN multiple quantum well

In x Ga1−x N/GaN multiple quantum wells (MQWs), composed of 28-A In0.29Ga0.71N wells and 24-A In0.23Ga0.77N wells, were grown by using metal-organic chemical vapor deposition. Temperaturedependent photoluminescence (PL) spectra showed that the energies of the two dominant peaks in the In x Ga1−x N/GaN MQWs decreased with increasing temperature. The electronic subband energies and the wavefunctions in the In x Ga1−x N/GaN MQWs were calculated by solving the Schrodinger equation in the 8-band envelope function approximation. The effects of the spontaneous polarization, strain and piezoelectric polarizations on the electronic structures of the In x Ga1−x N/GaN MQWs were examined to explain the PL data. The calculated interband transition energies from the ground electronic subbands to the ground hole subbands in In x Ga1−x N/GaN MQWs were in reasonable agreement with those obtained from the PL spectra.