Interlevel electromagnetic response of systems of spherical quantum dots

Interlevel electromagnetic response of individual spherical quantum dots (SQDs) as well as of two-dimensional (2D) rectangular lattice of spherical quantum dots is considered employing the self-consistent field approach in the quasistatic limit. It is established that the response can be considerably affected by the dynamic direct intradot and interdot electron-electron interaction. It is shown that the effects of these intradot and interdot interactions on the response can be analyzed separately. We show that correct description of the Coulomb coupling must take into account relevant umklapp processes. Fundamental importance of the problem of the electron self-interaction in quantum dot systems is established. The values are found which can be used for instant approximate estimation of the resonant dynamic screening (the depolarization shift) in any single SQD and 2D square lattice of SQDs. The effect of the size parameters of the systems, the SQD radius $R$ and the lattice period $d$ (and, in part, the three-dimensionality of the systems) on the depolarization shift is thoroughly investigated. Effect of polarization of incident radiation is investigated too. It is shown that the difference between the resonant photon energies for the normal and in-plane light polarization can be treated as independent of the static interdot interaction. Two-state and four-state electron systems are considered. Numerical calculations of the absorption spectra are performed for short period $(d∕2R⩽5)$ lattices of GaAs spherical quantum dots in the ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$ medium. It is established that the approximation of the point dipole-dipole interaction can be used for adequate representation of the dynamic interdot electron-electron interaction in the lattice. Also it is shown that the approach of the modified oscillator strength reproduces the absorption spectra of the considered systems with interacting modes of the collective excitation very well.

Figure 1

Dependence of the DE shift associated with the interdot dynamic $e\text{−}e$ interaction in a square lattice of SQDs upon the lattice period calculated within different approximations. One electron is per QD. $R=95\mathrm{\AA }$. Square symbols represent SQDs with finite potential barrier. Triangular symbols represent infinite deep SQDs (the Bessel wave functions). The lines stand for the DDA, with the dipole matrix element being of the finite (solid lines) or infinite deep (dash lines) SQD. Positive (negative) values represent the normal (in-plane) polarization of the incident radiation. The zero level is also labelled by symbols in order to emphasize that the “zero” is also calculated (it is the self-interaction value). For comparison the change of the interlevel spacing because of the static interdot $e\text{−}e$ interaction is presented by the dot line with circle symbols (see Fig. 8).

Figure 2

Dependence of the DE shift caused by the interdot dynamic $e\text{−}e$ interaction in a square lattice of SQDs upon the SQD radius calculated within different approximations. One electron is per QD. $d=600\mathrm{\AA }$. The legend is the same as in Fig. 1.

Figure 3

Dependence of $\left({\overline{\beta }}_{\mathbf{a},{\mathbf{a}}^{\prime };\mathbf{a},{\mathbf{a}}^{\prime }}^{\left(j\right)}d\right)$ (a), $\left({\overline{\beta }}_{\mathbf{b},{\mathbf{b}}^{\prime };\mathbf{b},{\mathbf{b}}^{\prime }}^{\left(j\right)}d\right)$ (b), $-\left({\overline{\beta }}_{\mathbf{a},{\mathbf{a}}^{\prime };\mathbf{b},{\mathbf{b}}^{\prime }}^{\left(j\right)}d\right)$ (c), and $\left({\overline{\beta }}_{\mathbf{g},{\mathbf{g}}^{\prime };\mathbf{g},{\mathbf{g}}^{\prime }}^{\left(j\right)}d\right)$ (d) upon $d∕2R$ for normal (circle symbols) and in-plane (square symbols) incident light polarizations for a square lattice of SQDs at $R=95\mathrm{\AA }$ with the Bessel wave functions and one electron per QD. The solid lines represent the DDA. The symbols labelling the zero level are to emphasize that the “zero” is also calculated (it is the self-interaction value).

Figure 4

Dependence of ${\overline{\beta }}_{\mathbf{a},{\mathbf{a}}^{\prime };\mathbf{a},{\mathbf{a}}^{\prime }}^{\left(j\right)}$ (a), ${\overline{\beta }}_{\mathbf{b},{\mathbf{b}}^{\prime };\mathbf{b},{\mathbf{b}}^{\prime }}^{\left(j\right)}$ (b), $-{\overline{\beta }}_{\mathbf{a},{\mathbf{a}}^{\prime };\mathbf{b},{\mathbf{b}}^{\prime }}^{\left(j\right)}$ (c), and ${\overline{\beta }}_{\mathbf{g},{\mathbf{g}}^{\prime };\mathbf{g},{\mathbf{g}}^{\prime }}^{\left(j\right)}$ (d) for normal (circle symbols) and in-plane (square symbols) incident light polarizations upon the interdot distance in a square lattice of finite deep SQDs at $R=95\mathrm{\AA }$. $n_{\mathbf{a},{\mathbf{a}}^{\prime }}=2$, $n_{\mathbf{a},{\mathbf{b}}^{\prime }}=4$, and $n_{\mathbf{g},{\mathbf{g}}^{\prime }}=2$. The intradot dynamic $e\text{−}e$ interaction is represented by the dot lines with triangular symbols. The solid lines represent the DDA.

Figure 5

Dependence of ${\overline{\beta }}_{\mathbf{a},{\mathbf{a}}^{\prime };\mathbf{a},{\mathbf{a}}^{\prime }}^{\left(j\right)}$ (a), ${\overline{\beta }}_{\mathbf{b},{\mathbf{b}}^{\prime };\mathbf{b},{\mathbf{b}}^{\prime }}^{\left(j\right)}$ (b), $-{\overline{\beta }}_{\mathbf{a},{\mathbf{a}}^{\prime };\mathbf{b},{\mathbf{b}}^{\prime }}^{\left(j\right)}$ (c), and ${\overline{\beta }}_{\mathbf{g},{\mathbf{g}}^{\prime };\mathbf{g},{\mathbf{g}}^{\prime }}^{\left(j\right)}$ (d) for normal (circle symbols) and in-plane (square symbols) incident light polarizations upon the radius of finite deep SQDs at $d=600\mathrm{\AA }$. $n_{\mathbf{a},{\mathbf{a}}^{\prime }}=2$, $n_{\mathbf{b},{\mathbf{b}}^{\prime }}=4$, and $n_{\mathbf{g},{\mathbf{g}}^{\prime }}=2$. Intradot dynamic $e\text{−}e$ interaction is represented by the dot lines with triangular symbols. The solid lines represent the DDA.

Figure 6

Dependence of ${\alpha }_{\perp }^{″}\left(\omega \right)$ (a) and ${\alpha }_{\Vert }^{″}\left(\omega \right)$ (b) upon photon energy of incident radiation for square lattice of finite deep SQDs described in the text. $R=95\mathrm{\AA }$ and $d=400\mathrm{\AA }$. $n_{\mathbf{a},{\mathbf{a}}^{\prime }}=2$ and $n_{\mathbf{b},{\mathbf{b}}^{\prime }}=4$. The solid line represents results obtained including intermode coupling into the dynamic $e\text{−}e$ interaction. Dash lines correspond to the diagonal approximation (i.e., no intermode coupling). Dash-dotted lines are obtained neglecting dynamic Coulomb interaction. In the two last approximations the separate contributions associated with $a$-mode and $b$-mode to ${\alpha }_{\perp }^{″}\left(\omega \right)$ and ${\alpha }_{\Vert }^{″}\left(\omega \right)$ are also presented by thin lines.

Figure 7

Near-peak absorption spectra of the four-state system including the intermode coupling predicted by the MOS approach (solid lines with diamonds) are compared with the corresponding exact spectra (solid lines) represented by solid lines in Fig. 6. The position of the peaks is marked by the vertical dash line.

Figure 8

Shift of the $E_{1,0}$ (square symbols) and $E_{1,1}$ (triangle symbols) energy, and modification of the gap between them $E_{1,1}-E_{1,0}$ (circle symbols) due to the static interdot $e\text{−}e$ interaction as function of the interdot distance. $R=95\mathrm{\AA }$. One electron is per QD.