Position-dependent Effective Masses Research Articles

Pyramidal core-shell quantum dot under applied electric and magnetic fields

We have theoretically investigated the electronic states in a core/shell pyramidal quantum dot with GaAs core embedded in AlGaAs matrix. This system has a quite similar recent experimental realization through a cone/shell structure [Phys. Status Solidi-RRL 13, 1800245 (2018)]. The research has been performed within the effective mass approximation taking into account position-dependent effective masses and the presence of external electric and magnetic fields. For the numerical solution of the resulting three-dimensional partial differential equation we have used a finite element method. A detailed study of the conduction band states wave functions and their associated energy levels is presented, with the analysis of the effect of the geometry and the external probes. The calculation of the non-permanent electric polarization via the off-diagonal intraband dipole moment matrix elements allows to consider the related optical response by evaluating the coefficients of light absorption and relative refractive index changes, under different applied magnetic field configurations.

Intersubband Raman gain in strained zincblende III-nitride-based step asymmetric quantum wells: non-parabolicity effects

The conduction band states in step-like strained zincblende nitride-based quantum wells are theoretically investigated including effects of band non-parabolicity. In particular, the intersubband Raman gain is calculated in order to identify the possible use of this kind of structures as sources of THz Raman lasing. The theoretical procedure includes the use of the envelope function approximation, taking into account strain effects on the conduction band offset and position-dependent non-parabolic effective masses. Three-level Raman gain is reported for a group of possible intersubband transitions. The results are discussed in terms of the variation of the geometry of the heterostructure as well as the concentration of In in the well regions. It is shown that under a suitable geometric design and plausible detuning conditions it is possible to achieve–for certain intersubband transitions–values of the intersubband Raman gain above $$1000\,\hbox {cm}^{-1}$$ , quite larger than those previously reported in GaAs- and wurtzite GaN-based double asymmetric quantum wells. Besides, it is found that the band-nonparabolicity acts as a significant element that reduces the Raman gain response in this kind of systems, whereas strain affects it in the same way but the changes are mainly noticed for higher In contents. It turns out that it is necessary to take both effects into account for the better quantitative description on intersubband Raman response in zincblende nitride heterostructures.

Families of stable solitons and excitations in the PT-symmetric nonlinear Schr\xf6dinger equations with position-dependent effective masses

Since the parity-time-(pmb{mathscr{P}}pmb{mathscr{T}}-) symmetric quantum mechanics was put forward, fundamental properties of some linear and nonlinear models with pmb{mathscr{P}}pmb{mathscr{T}}-symmetric potentials have been investigated. However, previous studies of pmb{mathscr{P}}pmb{mathscr{T}}-symmetric waves were limited to constant diffraction coefficients in the ambient medium. Here we address effects of variable diffraction coefficient on the beam dynamics in nonlinear media with generalized pmb{mathscr{P}}pmb{mathscr{T}}-symmetric Scarf-II potentials. The broken linear pmb{mathscr{P}}pmb{mathscr{T}} symmetry phase may enjoy a restoration with the growing diffraction parameter. Continuous families of one- and two-dimensional solitons are found to be stable. Particularly, some stable solitons are analytically found. The existence range and propagation dynamics of the solitons are identified. Transformation of the solitons by means of adiabatically varying parameters, and collisions between solitons are studied too. We also explore the evolution of constant-intensity waves in a model combining the variable diffraction coefficient and complex potentials with globally balanced gain and loss, which are more general than pmb{mathscr{P}}pmb{mathscr{T}}-symmetric ones, but feature similar properties. Our results may suggest new experiments for pmb{mathscr{P}}pmb{mathscr{T}}-symmetric nonlinear waves in nonlinear nonuniform optical media.

Displacement operator for quantum systems with position-dependent mass

A translation operator is introduced to describe the quantum dynamics of a position-dependent mass particle in a null or constant potential. From this operator, we obtain a generalized form of the momentum operator as well as a unique commutation relation for $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{\ensuremath{\gamma}}$. Such a formalism naturally leads to a Schr\"odinger-like equation that is reminiscent of wave equations typically used to model electrons with position-dependent (effective) masses propagating through abrupt interfaces in semiconductor heterostructures. The distinctive features of our approach are demonstrated through analytical solutions calculated for particles under null and constant potentials like infinite wells in one and two dimensions and potential barriers.

Effect of strain and variable mass on the formation of antibonding hole ground states in InAs quantum dot molecules

Using four-band $k\ensuremath{\cdot}p$ Hamiltonians, we study how biaxial strain and position-dependent effective masses influence hole tunneling in vertically coupled InAs/GaAs quantum dots. Strain reduces the tunneling and hence the critical interdot distance required for the ground state to change from bonding to antibonding. The reduced spin-orbit interaction in the GaAs matrix, which we account for using position-dependent Luttinger parameters, has the opposite effect. This compensation results in the critical distance being little affected. The possibility to induce the bonding-to-antibonding transition using longitudinal magnetic fields is also investigated. Luttinger-Kohn Hamiltonian predicts a magnetic enhancement of the heavy hole-light hole coupling which, in turn, leads to such transition. No such effect is, however, observed in magnetophotoluminescence experiments. An alternative implementation of the magnetic field in the envelope function Hamiltonian is given which retrieves the experimental behavior.

POSITION DEPENDENT EFFECTIVE MASSES FOR DONOR BINDING ENERGIES IN QUANTUM CONFINED SYSTEMS IN AN ELECTRIC FIELD

In our earlier works, the effective mass theory (EMT) was critically examined for a hydrogenic donor in a quantum well (QW) and in a quantum dot (QD) after introducing position dependent effective masses (PDEMs). Our results showed that the use of constant effective mass was justified for all well dimensions a*, the effective Bohr radius which is 100 Å in GaAs. Here the validity of EMT in the presence of an electric field is examined. The electric field dependent donor ionization energies in both the systems with finite and infinite barriers for different well dimensions have been calculated. It is shown that the electric field affects donor binding energies appreciably, the variation being larger in the case of finite barrier systems when PDEM is used. The donor polarizability has also been estimated in both the systems and the results are compared with other works in the literature. The use of EMT is justified if constant effective mass is used in general, even though large variation in the ionization energies occur in the finite barrier cases when position dependent mass is included correctly.

Effects of position-dependent effective mass of a hydrogenic donor impurity in a ridge quantum wire

In the present work, We calculate the donor binding energy for a V-groove GaAs / Al x Ga 1 − x As quantum wire using a variational procedure within the effective mass approximation. We present a relation for the position-dependent effective mass in V-groove quantum wire which is a function of wire parameters. Calculations are presented with both constant effective mass and position-dependent effective masses for different dimensions of ridge quantum wire. We find that the position-dependent effective mass plays an important effect in the binding energy of the donor impurity in a ridge quantum wire. Also, we observe that the position-dependent effective mass in x-direction depends on all wire parameters, but in y-direction it depends on only one wire parameter.

THE EFFECT OF POSITION DEPENDENT EFFECTIVE MASS OF HYDROGENIC IMPURITIES IN PARABOLIC GaAs/GaAlAs QUANTUM DOTS IN A STRONG MAGNETIC FIELD

The binding energy of shallow hydrogenic impurities in parabolic GaAs/GaAlAs quantum dots is calculated as a function of dot radius in the influence of magnetic field. The binding energy has been calculated following a variational procedure within the effective-mass approximation. Calculations are presented with constant effective-mass and position dependent effective masses. A finite confining potential well with depth is determined by the discontinuity of the band gap in the quantum dot and the cladding. The results show that the impurity binding energy (i) increases as the dot radius decreases for the infinite case, (ii) reaches a peak value around 1R* as the dot radius decreases and then diminishes to a limiting value corresponding to the radius for which there are no bound states in the well for the infinite case, and (iii) increases with the magnetic field. Also it is found that (i) the use of constant effective mass (0.067 m0) is justified for dot sizes ≥ a* where a* is the effective Bohr radius which is about 100 Å for GaAs , in the estimation of ionization energy and (ii) the binding energy shows complicated behavior when the position dependent mass is included for the dot size ≤ a*. These results are compared with the available existing literatures.

Effects of dielectric screening and position dependent effective mass on donor binding energies and on diamagnetic susceptibility in a quantum well

Abstract The binding energies of a shallow hydrogenic donor in the GaAs quantum well have been calculated using five different screening functions within the effective mass approximation. Both the infinite and finite confinements are used. Though all the screening functions yield results, which are nearly the same for all well dimensions studied, it is observed that the Resta dielectric function leads to smaller binding energies for well dimensions ≤100 A. The diamagnetic susceptibility of a hydrogenic donor has also been estimated and the results are in good agreement with the recently reported values. In addition, the position dependent effective masses are used for both confinements and the results for donor ionization energies and diamagnetic susceptibilities are shown to be appreciable for narrow well dimensions. Using the Resta dielectric function, pressure effects on donor ionization energies and diamagnetic susceptibilities are estimated for Q2D GaAs. It is found that while the ionization energies increase with pressure, | χ dia | decreases with pressure.

Effects of position-dependent effective mass and dielectric function of a hydrogenic donor in a quantum dot

Ionization energies of a shallow donor in a quantum dot of GaAs, using a variational procedure within the effective mass approximation, are obtained. Calculations are presented with a constant effective mass and position-dependent effective masses along with the spatially varying dielectric function. Donor binding energies are calculated using both the approximate method ( m 0 * ) and the spatially varying electron effective mass, m * ( r ) . Dielectric quantum dots are discussed and calculations are performed with the dielectric constants of the dot ε 1 and that of the barrier material, ε 2. It is found that (i) the use of a constant effective mass (0.067 m 0) is justified for dot size ⩾ a*, where a* is the effective Bohr radius, which is about 100 Å for GaAs, in the estimation of ionization energy, (ii) the ionization energy decreases as the dot increases in both the cases of constant and variable masses, (iii) an increase of binding energy is observed when the spatially varying mass and dielectric function are included, (iv) lower binding energies are observed when the average dielectric constant is included and (v) the binding energy shows complicated behaviour when the position-dependent mass is included for the dot size ⩽ a*.

Effective masses for donor binding energies in non-magnetic and magnetic quantum well systems: effect of magnetic field

Donor binding energies in quantum well systems using position dependent effective masses are obtained. Results are provided for GaAs-Ga1ixAlxAs and the semimagnetic CdTe-Cd1ix MnxTe systems. In the latter the barrier height reduces in a magnetic field. Using the available experimental data the variations of the barrier height with magnetic field has been obtained using a simple model. Our results are in good agreement with similar works in the literature in the case of constant effective mass for the donor electron. However, in the case of the semi magnetic system and in the case of the position dependent effective mass, the results obtained are shown to be appreciably different for narrow well dimensions. The validity of the effective mass approximation in the case of applied magnetic field is also critically examined.

Form-preserving Transformations for the Time-dependent Schrödinger Equation in (n + 1) Dimensions

We define a form-preserving transformation (also called point canonical transformation) for the time-dependent Schrodinger equation (TDSE) in (n+1) dimensions. The form-preserving transformation is shown to be invertible and to preserve L 2-normalizability. We give a class of time-dependent TDSEs that can be mapped onto stationary Schrodinger equations by our form-preserving transformation. As an example, we generate a solvable, time-dependent potential of Coulombic ring-shaped type together with the corresponding exact solution of the TDSE in (3+1) dimensions. We further consider TDSEs with position-dependent (effective) masses and show that there is no form-preserving transformation between them and the conventional TDSEs, if the spatial dimension of the system is higher than one.

EFFECTIVE MASSES FOR DONOR BINDING ENERGIES IN QUANTUM WELL SYSTEMS

The donor ionization energies in a quantum well and quantum dot with finite and infinite barriers are estimated for different well dimensions. Using the effective mass (EM) approximation, calculations are presented with constant effective mass and position dependent effective masses that are different for finite and infinite cases. Our results reduce to an approximate form used by X. H. Qi et al., Phys. Rev. B58 (1998) 10578 in the finite barrier model and that of L. E. Oliveira and L. M. Falicov, Phys. Rev. B34 (1986) 8676 in the infinite barrier case. Results are presented by taking the GaAs quantum well as an example. The use of constant effective mass of 0.067m0 is justified for well dimensions ≥a* where a* is an effective Bohr radius which is about 100 Å. While Qi et al. found a maximum of 22% variation in the binding energies due to mass variation, we obtained nearly 100% variation when mass variations are included correctly.

Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem

We show that there exist some intimate connections between three unconventional Schrodinger equations based on the use of deformed canonical commutation relations, of a position-dependent effective mass or of a curved space. This occurs whenever a specific relation between the deforming function, the position-dependent mass and the (diagonal) metric tensor holds true. We illustrate these three equivalent approaches by considering a new Coulomb problem and solving it by means of supersymmetric quantum mechanical and shape invariance techniques. We show that in contrast with the conventional Coulomb problem, the new one gives rise to only a finite number of bound states.

Position-dependent effective masses in semiconductor theory. II.

We consider a compound semiconductor possessing a slowly varying position-dependent chemical composition. We derive an effective-mass equation governing the dynamics of electron (or hole) motion using the Kohn-Luttinger representation and canonical transformations. We show that, as long as the variation in chemical composition may be treated as a perturbation, the effective masses become constant, position-independent quantities. The effective-mass equation derived here is identical to the effective-mass equation derived previously by von Roos, using a Wannier representation.

Comment on "Position-dependent effective masses in semiconductor theory"

In a recent paper, von Roos criticized the concept of a position-dependent mass in semiconductor theory. His remarks are analyzed. It is shown that the Wannier Hamiltonian in the non-effective-mass approximation as devised by the authors is unique. The concept of position-dependent effective mass is a classical concept which in its classical form is unique and very useful for semiconductor device analysis.

Position-dependent effective masses in semiconductor theory

Position-dependent effective masses in semiconductor theory

Effects of quantum confinement and compositional grading on the band structure of heterojunctions

The band structure of a heterojunction may have discrete electronic states arising from quantum confinement; in addition the band edge is perturbed by grading in the composition. A new method is introduced to analyse these effects by an effective field “ F eff ” in the transition region, including the position-dependence of space charge and of composition. The F eff is position-dependent and is zero-valued at the positions which correspond to extrema of the band edges. For most practical cases it is shown that F eff can be approximated as a linear function of position near the zero-valued positions, thus yielding harmonic oscillator states. The effective force constant can be expressed in terms of position-dependent material parameters. The effects of position-dependent effective masses are also considered. The analysis is applied to the p- n GaAsAl 0.4Ga 0.6As heterojunction; changes of the order of 1 10 eV in the effective band gap are predicted.

Theory of Electronic States and Transport in Graded Mixed Semiconductors

Semiconductors which are slowly graded in composition can be shown to have position-dependent band gaps and position-dependent effective masses, describable in terms of an effective Hamiltonian in an effective-mass equation. The effective Hamiltonian previously obtained is, in the present work, rendered Hermitian. Electronic minority-carrier transport for graded systems is described in terms of an effective field which includes the electrostatic field plus a term in the gradient of the band edge and another in the gradient of the effective mass. The local radiative-recombination lifetime and local density of states for inhomogeneous semiconductors are discussed. The equation for the excess minority-carrier concentration in an inhomogeneous semiconductor is deduced and is found to differ from that in an homogeneous system, by the effective field replacing the electric field, by the position dependences of lifetime and mobility, and by terms in the mobility gradient. Some phenomena specific to graded mixed semiconductors are considered on the basis of the theoretical analysis.