Optical properties of Quantum Disks: Real density matrix approach

2008

We show how to compute the optical functions (the complex magnetosusceptibility, dielectric function, magneto-reflection and ellipsometric spectra) for semiconductor quantum disks exposed to a uniform magnetic field in the growth direction, including the excitonic effects. The optical response is calculated for an oblique incidence of the propagating electromagnetic wave and for input waves with different polarization. The method uses the microscopic calculation of nanostructure excitonic wave functions and energy levels, and the macroscopic real density matrix approach to compute the electromagnetic fields and susceptibilities. The electron-hole screened Coulomb potential is adapted and the valence band structure is taken into account in the cylindrical approximation, thus separating light-and heavy--hole motions. The novelty of our approach is that the solution is obtained in terms of known one-particle electron and hole eigenfunctions, since, in the considered nanostructure due to confinement effects accompanied by the e-h Coulomb interaction, the separation of the relative-and center-of-mass motion is not possible. We obtain both the eigenvalues and the eigenfunctions. The convergence of the proposed method is examined. We calculate the magnetooptical functions, including the optical Stokes parameters and ellipsometric functions for the case of oblique incidence. Numerical calculations were performed for InAs (disk)/ GaAs (barrier) disks. A good agreement with experiments was obtained.

“…We used the same band parameters (effective masses, band gap, effective Rydberg and Bohr radius) as in Ref. [2]. As we noticed in Ref.…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optical Anisotropy of Quantum Disks in the External Static Magnetic Field

2008

“…Here we consider semiconductor quantum dots of cylindrical symmetry, where the carriers move in the dot material in the step-like potentials in all directions. Such dots are also called the quantum disks (for example, [1] for references).…”

Section: Basic Equationsmentioning

confidence: 99%

Excitonic Magnetoabsorption of Cylindrical Quantum Disks

2011

We show how to compute the optical functions (the complex magnetosusceptibility, dielectric function, magnetoreflection) for semiconductor quantum disks exposed to a uniform magnetic field in the growth direction, including the excitonic effects. The method uses the microscopic calculation of nanostructure excitonic wave functions and energy levels, and the macroscopic real density matrix approach to compute the electromagnetic fields and susceptibilities. The electron-hole screened Coulomb potential is adapted and the valence band structure is taken into account in the cylindrical approximation, thus separating light-and heavy-hole motions. The confinement potentials are taken as step-like both in the z and in-plane directions. Numerical calculations have been performed for In0.55Al0.45As (disk)/Al0.35Ga0.65As (barrier) and InP/GaP disks and the results are in a good agreement with the available experimental data. PACS: 78.67.Hc, 73.22.Dj, 73.21.La, 71.35.Ji Basic equationsIn the effective mass approximation the exciton is treated as a hydrogen-like atom, where the electron and the hole interact via the Coulomb potential screened by the semiconductor dielectric constant. In typical II-VI and III-V semiconductors (GaAs, for example) the dielectric constant is large and the Wannier-Mott excitons occur, having large Bohr radius and small binding energy (a few meV). In bulk semiconductors the relative and the center-of-mass motion of the electron-hole pair separate and the excitonic energy levels follow the Rydberg formula −R * /n 2 . In semiconductor nanostructures the exciton is squeezed and its binding energy increases even by an order of magnitude. Due to confinement effects accompanied by the e-h Coulomb interaction, the separation of the relative-and center-of-mass motion is not possible which makes the calculation of the energy levels very difficult.On the other hand, potential applications of semiconducting nanostructures in novel optoelectronic devices make the determination of the excitonic energies and resulting optical properties important, since the excitonic resonances occur in the mostly used visible excitation region. In consequence, there is a motivation for developing methods of calculation of the excitonic states, wave functions and the resulting optical properties. Here we consider semiconductor quantum dots of cylindrical symmetry, where the carriers move in the dot material in the step-like potentials in all directions. Such dots are also called the quantum disks (for example, [1] for references).In addition, a constant magnetic field is applied along the symmetry axis. We use the effective-mass--approximation thus obtaining a two-particle Schrödinger equation where the nanostructure Hamiltonian contains the kinetic energy terms, the Coulomb potential and the * corresponding author; e-mail: psc@utp.edu.pl confinement potentials. The Schrödinger equation refers to a 6-dimensional configuration space and its analytical solution is not known. Also a direct numerical integration, due to the dimens...

“…In the present paper we consider a quantum dot of cylindrical shape, with the symmetry axis z (the growth direction), and with infinite hard wall potentials for electrons and holes located in the xy plane at the radius R. Such a quantum dot is also called a quantum disk (QDisk), see for example [2] and references therein. We compute the electrooptical functions of QDisks when an electric field F is applied in the z-direction, taking into account the electron-hole Coulomb interaction and the confinement effects.…”

Section: Introductionmentioning

confidence: 99%

Electrooptical Properties of Cylindrical Quantum Dots

2009

We show how to compute the optical functions (the complex electrosusceptibility tensor, dielectric tensor, electroreflection spectra) for semiconductor quantum dots exposed to a uniform static electric field in the growth direction, including the excitonic effects. The method uses the microscopic calculation of the quantum dot excitonic wave functions and energy levels, and the macroscopic real density matrix approach to compute the electromagnetic fields and susceptibilities. The electron-hole screened Coulomb potential is adapted and the valence band structure is taken into account in the cylindrical approximation, thus separating light-and heavy-hole motions. In the microscopic calculations, using the effective-mass approximation, we solve the 6-dimensional two-particle Schrödinger equation by transforming it into an infinite set of coupled second order 2-dimensional differential equations with the appropriate boundary conditions. These differential equations are solved numerically giving the eigenfunctions and the energy eigenvalues. Having them, we can compute the quantum dot electrooptical functions. Numerical calculations have been performed for an InGaAs quantum dot with a constant electric field applied in the growth direction. A good agreement with experiment is obtained.