^{1}

^{2}

^{3}

^{1}

^{1}

^{2}

^{3}

In this paper, we have performed a theoretical study on nonlinear optical rectification (OR) and second harmonic generation (SHG) for three-level dome-shaped InAs/GaAs quantum dots (QDs) in the presence of wetting layer (WL). We used the compact density matrix framework and effective mass approximation to investigate the second order nonlinear phenomena on InAs/GaAs QD. It is demonstrated that second harmonic generation (SHG), optical rectification (OR), and their mutual absorption and refractive index changes are quite sensitive to the size of QDs. The size variations have profound irregular behavior owing to distribution of envelope function on WL and QD simultaneously. Moreover it is found that

In the last few years the nonlinear optical properties of intersubband transitions in semiconducting materials have attracted remarkable attention due to the potential application in electronics and optoelectronics devices [

The early experimental research on mid-infrared and near-field SHG in semiconductor QDs was reported by Brunhes et al. [

The QDs in the practical growth method always are grown on wetting layer (WL) [

This work is concerned with the investigation of dome-shaped InAs QDs. First of all we calculated the transition frequencies between the levels and envelop function in single InAs QDs by making use of finite element method (FEM) numerical solution for Schrödinger equation within the effective mass approximation. Throughout this work the InAs QDs are grown on InAs WL which is embedded in a large GaAs matrix and the GaAs cap cover the grown QDs as Figure

(a) The three-dimensional InAs QDs/WL and GaAs matrixes, (b) the two-dimensional area with numbered boundaries for simplified Schrödinger equation.

By this definition we need to consider a dome-shaped InAs/GaAs QD with cylindrical symmetry whose Schrödinger wave equation in one-band envelop-function formalism is given by

Assume that our system is excited by linear

In this case OR is purely real, so it has no contribution in absorption coefficient but SHG collaborates in both refractive index changes and absorption coefficient. It is noteworthy to indicate that we avoid opening the summation because both OR and SHG consist of almost 76 and 144 successive expression, respectively. We admit removing any nonresonant expressions since the effect of all expression will leads to more practical results.

Figure

The variation of the first three transition frequency amplitudes in the sublevels of conduction band as a function of QDs size is shown in Figure

Transition frequencies for three levels in InAs/GaAs QDs versus field probe frequency. These curves show the variation of transition frequency for different radius of QDs ranging from 1 to 30 nm. The thickness of wetting layer is chosen to be 3 nm.

The absolute values of (a) diagonal and (b) off-diagonal elements of dipole momentum matrix versus dome radius.

Figure

As Figure

The

For

Optical rectification is purely real so according to (

The refractive index changes due to optical rectification versus the incident light frequency for different dimensions of QDs. The amplitude of transition frequencies and dipole moments matrix is applied according to the achievement in Figures

Figure

The absorption due to second harmonic generation (SHG) versus the incident light frequency for different dimensions of QDs. The physical parameters are set as in Figure

Figure

The refractive index changes due to second harmonic generation (SHG) versus the incident light frequency for different dimensions of QDs. The physical parameters are set as in Figure

Figure

The biggest peaks of (a) refractive index changes due to optical rectification, (b) absorption due to second harmonic generation, and (c) refractive index changes due to second harmonic generation.

In this work the effect of size on transition frequencies, all elements of dipole momentum matrix, ORs and SHG are investigated deeply. All resonant and none-resonant expressions of OR and SHG are taken into consideration, which makes the result more useful for practical applications. The WL caused weird behavior in all discussed parameters. The transition frequencies between subbands satisfy the general expectation, which quotes

At