AC‐Stark effect in a semi‐spherical quantum dot

Abstract

We present a theoretical approach to study the effects of an ac-field applied to quantum dots with semi-spherical symmetry. Using the Floquet formalism for this periodically driven system, the timedependent Hamiltonian in the effective mass approximation is solved. We show that the Hilbert space of solutions is separated into orthogonal subspaces with different z-component of the angular momentum. We give an explicit analytical representation for electronic states as a function of the intensity and frequency of the electric field. Under the two level approximation, two particular cases are studied: the low-and high-frequency regimes, which result of comparing the ac-field frequency to the characteristic level splitting at zero field.The dynamics of charged particles in semiconductor structures under a time-dependent external electric field has been a topic of intense research [1,2,3,4,5]. The progress of techniques on the nanoscale has made possible the systematic study of effects only present in intense alternating fields in this domain.In this paper we address the problem of a quantum dot (QD) of semi-spherical shape with the ac-field along the axial symmetry of the dot (z-axis in this case) in a nonperturbative approach. To do so we make use of the Floquet formalism [1]. The Hamiltonian for an electron in the QD under an electric field of intensity F and frequency ω can be written aswhere H o = −(h 2 /2m * )∇ 2 + V o is the Hamiltonian for the nondriven system. Here V o = 0 inside the QD and we use hard-wall boundary conditions, m * is the effective mass of the electron in the QD. In the absence of the ac field the problem is exactly soluble and its eigenenergies are given by the square of the zeroes µ (l) n of the spherical Bessel functions, where n is the order of the zero and l the order of the Bessel function J l+1/2 (see details in [6]). The eigenfunctions, in spherical coordinates, are given bywhere P |m| l are the associated Legendre polinomials, m is the z-component of the angular momentum, and N J and N P are normalization constants.The dynamics of the driven system is governed by the time-dependent Schroedinger equationSince H is periodic in time with period τ = 2π/ω, it is possible to use the standard Floquet theory [1] and write the wavefunction as Ψ = exp(−i ε t/h)Φ, which allows us to rewrite Eq. (3) aswhere ε is a real-valued parameter termed the Floquet characteristic exponent, or the quasienergy. To solve Eq. (4), Φ can be expanded in a Fourier series for the temporal component and linear combinations of stationary states Φ(r, θ, φ, t) = ∞ p=−∞ exp(i p ω t) u p (r, θ, φ), p ∈ Z,

Some electronic and optical properties of self‐assembled quantum dots: asymmetries in a lens domain

“…The electronic properties of this so-called self-assembled quantum lens (SAQL) has been studied previously [5,6]. The study has included the effects of an external electric [7,8] and magnetic field [9]. Some optical properties of such structures have also been analyzed [10].…”

Effect of magnetic field and impurities in InAs/GaAs and GaN/AlN self-assembled quantum dots

“…When a negative/positive impurity is inserted, its potential attracts/repels holes, or in opposite form, to the electrons. This causes a redistribution of the carriers, i.e., the effect of the impurities is similar to apply a low electric field [31][32][33].…”

Stark Effect of Interactive Electron-hole pairs in Spherical Semiconductor Quantum Dots

“…It consists of an observable red-shift of an optical transition induced by the presence of a constant external electric field [37][38][39][40]. In recent years, some works have also dealt with ac-electric field [41,42]. Stark effect leads to an energy shift of the exciton photoluminescence as well as a corresponding enhancement of its recombination lifetime [43].…”