Laser radiation effects on optical absorptions and refractive index in a quantum dot

Abstract

An investigation of the laser radiation effects of a hydrogenic impurity in a quantum dot has been performed by using the matrix diagonalization method. We find that the laser field amplitude has an important influence on the linear, third-order nonlinear, and total absorption coefficients as well as the refractive index changes.

Introduction

In the last years, the study of the electronic and optical properties of low-dimensional semiconductor systems, such as quantum wells, quantum wires, quantum dots (QDs) and superlattices has been of great interest due to their importance for potential applications in electronic and optoelectronic devices. Of these systems, QDs have attained considerable theoretical and experimental attention due to their potential application in microelectronics and future laser technology [1], [2], [3]. QDs confine carriers in all three spatial dimensions and the many-body effects of particle–particle interactions show a broad range of electric structures similar to those of real atoms. The main features to consider in relation to QDs are geometrical shape, size, and the confining potential.

Especially the last two decades have witnessed an increasing interest in the theoretical and experimental investigations about the behavior of shallow hydrogenic donor, and acceptor impurities in semiconductor quantum heterostructures [4], [5], [6], [7], [8], [9], [10]. The study of impurities is one of the main problems in semiconductor low-dimensional systems because the presence of impurities in nanostructures influences greatly the electronic mobility and their optical properties [11]. Hence, the study of the impurity states in QDs has been extensively reported in the last few years [12], [13], [14], [15], [16], [17].

The linear and nonlinear optical properties are of particular interest, since they provide detailed information on the microscopic interactions of the quasiparticles. In semiconductor QDs, the nonlinear optical properties associated with optical absorptions are known to be greatly enhanced as compared to nonlinearity in bulk semiconductors. Hence, the nonlinear optical properties of QDs have been investigated both experimentally and theoretically by many authors [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. In the nonlinear optical absorption of QDs, the analysis of impurity states is inevitable because the confinement of quasiparticles in such structure leads to the enhancement of the oscillator strength of electron-impurity excitations. Since the operation of optoelectronic devices relies on the interaction of carriers with electromagnetic radiation, it is intriguing to investigate the effects of an intense laser field on the impurity states. Some investigations have been published about the effect of laser fields on nano-structure [29], [30]. However, to the best of our knowledge, there are only a few studies on the effect of an intense laser field on the optical transitions in a QD. In present work, we will focus on studying the effect of an intense, high-frequency laser field on the linear and nonlinear optical absorption coefficients (ACs) as the refractive index (RI) changes of a hydrogenic impurity in a disc-like QD by using the matrix diagonalization method and the compact density-matrix approach.

We consider one electron with a hydrogenic impurity confined by a disc-like QD, subjected to a laser field of frequency Ω, whose vector potential is given by $\overrightarrow{A}$(t) = A₀ê_xcosΩt. Here ê is the unit vector along the X axis. Following Refs. [29], [31], in the high-frequency limit, the Hamiltonian of a hydrogenic impurity confined by a disc-like parabolic QD can be written as

$$H=\frac{p^2}{2m_{\text{e}}}+V^d\left(\overrightarrow{r},{\overrightarrow{\alpha }}_0\right)+V_C^d\left(\overrightarrow{r},{\overrightarrow{\alpha }}_0\right)\text{,}$$

with ${\overrightarrow{\alpha }}_0={\alpha }_0{\widehat{e}}_x$, where ${\alpha }_0={(8\pi /\Omega c)}^2\sqrt{I}e/m_{\text{e}}$ is the amplitude of the electron oscillation in the laser field, m_e being the effective mass of an electron and I the laser intensity. $\overrightarrow{r}\left(\overrightarrow{P}\right)$ is the position vector (the momentum vector) of the electron originating from the center of the dot. The dressed parabolic and Coulomb potentials are defined as follows

$$V^d\left(\overrightarrow{r},{\alpha }_0\right)=\frac{1}{4}{m}_{\text{e}}{\omega }_0^2\left[\left(\overrightarrow{r}-{\overrightarrow{\alpha }}_0\right)+\left(\overrightarrow{r}+{\overrightarrow{\alpha }}_0\right)\right]\text{,}$$

$$V_C^d\left(\overrightarrow{r},{\alpha }_0\right)=-\frac{e^2}{2ϵ}\left[\frac{1}{\left|\overrightarrow{r}-{\overrightarrow{\alpha }}_0\right|}+\frac{1}{\left|\overrightarrow{r}+{\overrightarrow{\alpha }}_0\right|}\right]\text{,}$$

where ω₀ measures the strength of the confinement, and ∊ is the dielectric constant. The Hamilton can be rewritten as

$$H=\frac{p^2}{2m_{\text{e}}}+\frac{1}{2}{m}_{\text{e}}{\omega }_0^2{r}^2+\frac{1}{2}{m}_{\text{e}}{\omega }_0^2{\alpha }_0^2-\frac{e^2}{2ϵ}\left[\frac{1}{\left|\overrightarrow{r}-{\overrightarrow{\alpha }}_0\right|}+\frac{1}{\left|\overrightarrow{r}+{\overrightarrow{\alpha }}_0\right|}\right]\text{.}$$

From Eq. (4) a hydrogenic impurity in a two-center Coulomb field becomes apparent. The separation between the centers is 2α₀, being proportional to $\sqrt{I}$.

The Hamiltonian has cylindrical symmetry which implies the total orbital momentum L (i.e., L_z) is a conserved quantity, i.e., a good quantum number. Hence, the eigenstates of a hydrogenic impurity in a disc-like QD can be classified according to the total orbital angular momentum L. To obtain the eigenfunction and eigenenergy associated with the hydrogenic impurity in a disc-like QD, the Hamiltonian is diagonalized in the model space spanned by two-dimensional harmonic states

$${\Psi }_{\text{L}}=\sum _i{c}_i{\varphi }_i^{\omega }\left(\overrightarrow{r}\right)\text{,}$$

where is ${\varphi }_i^w\left(\overrightarrow{r}\right)=R_{n_i{\ell }_i}\left(r\right)\text{exp}(-i{l}_i\theta )$ ith two-dimensional harmonic oscillator eigenstate with a frequency ω and an energy (2n_i + |$\mathcal{\ell }$_i| + 1) ℏω. R_nℓ(r) is the radial wave function, given by

$$R_{\mathit{n\ell }}(r)=Nexp\left(-r^2/\left(2a^2\right)\right)r^{\left|\ell \right|}{L}_n^{\left|\ell \right|}\left(r^2/a^2\right)\text{,}$$

in which N is the normalization constant, $a=\sqrt{\hslash /(m_{\text{e}}\omega )}$, and L_n^k(x) is the associated Laguerre polynomial. The radial and orbital angular momentum quantum members can have the following values

$$n=1,2,\cdots ,\ell =0,\pm 1,\pm 2,\cdots \text{.}$$

This single-particle basis is used by the matrix diagonalization method in order to expand the Hamiltonian H. Here ω is the adjustable parameter, and in general not equal to ω₀.

Let N = 2n + $\mathcal{l}$. Let {Ψ_K} denote the set of basis functions including all the Ψ_K having their N smaller or equal to an upper limit N_max. It is obvious that the total number of basis functions of the set is determined by N_max. After the diagonalization we obtain the eigenvalues and eigenvectors. Evidently, the eigenvalues depend on the adjustable parameter ω. In our calculation, ω serves as a variational parameter to minimize the low-lying state energy. The matrix diagonalization method consists in spanning the Hamiltonian for a given basis and extracting the lowest eigenvalues of the matrix generated. The better the basis describes the Hamiltonian, the faster the convergence will be. The most common basis chosen is the one that describes the Hamiltonian at zero order.

The optical absorption calculation is based on the compact density-matrix approach, for which the total AC is given by [32]

$$\alpha (\upsilon ,I)={\alpha }^{(1)}(\upsilon )+{\alpha }^{(3)}(\upsilon ,I)\text{,}$$

where

$${\alpha }^{(1)}(\upsilon )=\omega \sqrt{\frac{\mu }{{ϵ}_R}}\frac{{\sigma }_{\text{s}}{\left|M_{f\text{i}}\right|}^2\hslash {\Gamma }_{\mathit{fi}}}{{\left(h\upsilon -\Delta E_{\mathit{fi}}\right)}^2+{\left(\hslash {\Gamma }_{\mathit{fi}}\right)}^2}\text{,}$$

and

$${\alpha }^{(3)}(\upsilon ,I)=-\upsilon \sqrt{\frac{\mu }{{ϵ}_{\text{R}}}}\left(\frac{I}{2{ϵ}_0{n}_{\text{r}}c}\right)\frac{{\sigma }_{\text{s}}{\left|M_{\mathit{fi}}\right|}^2\hslash {\Gamma }_{\mathit{fi}}}{{\left[{\left(h\upsilon -\Delta E_{\mathit{fi}}\right)}^2+{\left(\hslash {\Gamma }_{\mathit{fi}}\right)}^2\right]}^2}\left\{4{\left|M_{\mathit{fi}}\right|}^2-\frac{{\left(M_{\mathit{ff}}-M_{\mathit{ii}}\right)}^2\left[3\Delta E_{\mathit{fi}}^2-4\Delta E_{\mathit{fi}}h\upsilon +\left(h^2{\upsilon }^2-{\hslash }^2{\Gamma }_{\mathit{fi}}^2\right)\right]}{\Delta E_{\mathit{fi}}^2+{(\hslash {\Gamma }_{\mathit{fi}})}^2}\right\}\text{,}$$

are the linear and third-order nonlinear optical ACs, respectively. In the above equations, ΔE_fi = E_f − E_i denotes difference of the energy between lower and upper levels, c is the speed of light, I is the intensity of electromagnetic field, σ_s is the electron density, μ is the permeability of the system, n_r is the RI and ϵ_R is the real part of the permittivity. M_fi = 〈Ψ_i|x|Ψ_f〉 is the electric dipole moment of the transition from the Ψ_i state to the Ψ_f state. Here Γ is the phenomenological operator. Diagonal matrix element Γ_ff of operator Γ, which is called as relaxation rate of fth state, is the inverse of the relaxation time T_f for the state |f〉, namely Γ_ff = 1 / T_ff. Whereas nondiagonal matrix element Γ_fi (f ≠ i) is called as the relaxation rate of fth state and kth state. In a disc-like QD the dipole transitions are allowed only between states satisfying the selection rules Δ$\mathcal{l}$ = ∓ 1, where $\mathcal{l}$ is the angular momentum quantum number. Hence, in the present work, we restrict our study to the transition of the L = 0 state to the L = 0 state.

The linear, and third-order nonlinear RI changes are given by

$$\frac{\Delta n^{(1)}(\upsilon )}{n_{\text{r}}}=\frac{1}{2n_r^2{ϵ}_0}\frac{{\sigma }_{\text{s}}{\left|M_{\mathit{fi}}\right|}^2\left(E_{\mathit{fi}}-h\upsilon \right)}{{\left(E_{\mathit{fi}}-h\upsilon \right)}^2+{\left(\hslash {\Gamma }_{\mathit{fi}}\right)}^2}\text{,}$$

and

$$\frac{\Delta n^{(3)}(\upsilon )}{n_{\text{r}}}=-\frac{\mu cI}{4{ϵ}_0{n}_{\text{r}}^3}\frac{{\sigma }_s{\left|M_{\mathit{fi}}\right|}^2}{{\left[{\left(E_{\mathit{fi}}-h\upsilon \right)}^2+{\left(\hslash {\Gamma }_{\mathit{fi}}\right)}^2\right]}^2}\left\{4\left(E_{\mathit{fi}}-h\upsilon \right){\left|M_{\mathit{fi}}\right|}^2-\frac{{\left(M_{\mathit{ff}}-M_{\mathit{ii}}\right)}^2}{E_{\mathit{fi}}^2+{\left(\hslash {\Gamma }_{\mathit{fi}}\right)}^2}\left(E_{\mathit{fi}}-h\upsilon \right)\left[E_{\mathit{fi}}\left(E_{\mathit{fi}}-h\upsilon \right)-{\left(\hslash {\Gamma }_{\mathit{fi}}\right)}^2\right]-{\left(\hslash {\Gamma }_{\mathit{fi}}\right)}^2\left(2E_{\mathit{fi}}-h\upsilon \right)\right\}\text{.}$$

The total RI change can be written as

$$\frac{\Delta n(\upsilon )}{n_{\text{r}}}=\frac{\Delta n^{(1)}(\upsilon )}{n_{\text{r}}}+\frac{\Delta n^{(3)}(\upsilon )}{n_{\text{r}}}\text{.}$$

In what follows the energy unit is meV and the length unit is nm. All calculations were performed using the following parameters: m_e = 0.067m₀, where m₀ is the free electron mass, ϵ = 12.4, σ_s = 5.0 × 10²⁴ m^− 3, I = 1.5 × 10¹⁹W/m², Γ_ff = 1 ps^− 1, Γ_fi = 1/0.14 ps^− 1, and n_r = 3.2. These parameters are suitable for GaAs–Ga_1 − xAl_xAs QDs. Fig. 1, Fig. 2 show the linear, third-order nonlinear and total optical ACs and the total optical RI changes as a function of the incident photon energy hυ for four different laser field values, i.e., α₀ = 0.0, 1.0, 3.0 and 5.0 nm, respectively. The confining energy ℏω₀ is set to be 15.0 meV. When α₀ = 0, there is no laser field. The laser-field-induced effect is clear. It is readily seen that as the laser field value increases, the AC and the RI peaks will move to the left side, which show a laser-field-induced red shift of the resonance in QDs. This result is obviously different from that of quantum wells [29]. This redshift is because the energy difference between the ground state and the excited state in QDs decreases with increasing α₀. Obviously this is the result of the laser-dressed Coulomb interaction. However, the peak values of the ACs and the RI changes are not monotonic functions of α₀. As seen in Fig. 1, Fig. 2, we find that the magnitude of the ACs and the RI changes first increases with α₀ up to a critical laser field value, α₀ ≈ 3.0 nm, and for further large α₀ value they begin to decrease. On the one hand, we find that in the magnitude of ACs and RI considerably increases as the laser field inducing. Hence, we can say that in order to obtain the larger ACs and the RI changes in QDs, we should induce the laser field.

In Fig. 3, Fig. 4, in order to see clearly the quantum confinement effect on the optical properties, we set ℏω₀ = 50.0 meV and plot the linear, third-order nonlinear and total optical ACs and the total optical RI changes as a function of the incident photon energy hυ for four different laser field values, i.e., α₀ = 0.0, 1.0, 3.0 and 5.0 nm, respectively. Fig. 3, Fig. 4 show that the qualitative properties of ACs and RI changes are in good agreement with those of Fig. 1, Fig. 2. However, the quantitative differences are also obvious. By comparing the AC and the RI peaks of Fig. 3, Fig. 4 with those of Fig. 1, Fig. 2 we can see that the AC peak increases and the RI peak decreases in magnitude for large confinement strength. On the other hand, we note that as the confinement strength ℏω₀ increases, the ACs and the RI peak positions will move to the right side, which show a confinement-induced blue shift of the resonance in QDs.

In conclusion, we have calculated the ACs and the RI changes of a hydrogenic impurity in a disc-like QD in an intense laser field by using the matrix diagonalization method and the compact density-matrix approach. The results obtained have been presented as a function of the incident photon energy for the different values of laser field and confinement strength. We find that the linear, nonlinear third-order, and total optical ACs as well as the RI changes are strongly affected by the laser radiation and confinement strength of QDs. In order to obtain the larger ACs and the RI changes in QDs, we should induce the laser field.