Intersublevel magnetoabsorption in the valence band of $p$-type $\mathrm{InAs}∕\mathrm{GaAs}$ and $\mathrm{Ge}∕\mathrm{Si}$ self-assembled quantum dots

The linear intraband (intersublevel) optical magnetoabsorption between the valence-band states in thin disk-shaped self-assembled quantum dots is studied. Strain is modeled with the continuum mechanical approach, while band mixing and the magnetic field are taken into account through the axially symmetric $\mathbf{k}·\mathbf{p}$ model. The absorption spectra in $\mathrm{InAs}∕\mathrm{GaAs}$ and $\mathrm{Ge}∕\mathrm{Si}$ quantum dots are computed for the case when a magnetic field perpendicular to the dot’s base is present for both $p$-polarization in the Voigt configuration and $s$-polarization in the Faraday configuration. Due to the selection rules, the transitions between the states which differ by $\pm 1$ in the total angular momentum $\left(F_z\right)$ dominate for $s$-polarized light, while the transitions between the states of different $F_z$ are strictly forbidden for $p$-polarization. In $\mathrm{InAs}∕\mathrm{GaAs}$ quantum dots, the magnetic field brings about a red shift of the absorption peak for $s$-polarized light, while the absorption peak for $p$-polarization is blue-shifted with respect to the zero field case, and also the absorption curves widen. In $\mathrm{Ge}∕\mathrm{Si}$, much smaller shifts of the absorption curves due to the magnetic field are found and almost no widening occurs, which is attributed to the larger number of energy levels in $\mathrm{Ge}∕\mathrm{Si}$ dots. The obtained results compare favorably with the spectroscopic measurements at zero magnetic field, especially with regard to the relative energies of the absorption peaks for $s$- and $p$-polarized light.

Figure 1

The energy spectrum in the valence band of $\mathrm{InAs}∕\mathrm{GaAs}$ single quantum dot as it varies with the magnetic field: (a) $S_{\pm 3∕2}^{\pm }$ and (b) $S_{\pm 1∕2}^{\pm }$ states. Even and odd hole states are shown by the solid and dashed lines, while the dotted-dashed lines denote the position of the Fermi level. The energy is measured from the valence-band top in the matrix upwards.

Figure 2

The same as in Fig. 1, but for $\mathrm{Ge}∕\mathrm{Si}$ quantum dots: (a) $S_{\pm 3∕2}^{\pm }$ and (b) $S_{\pm 1∕2}^{\pm }$ states.

Figure 3

(Color online) (a) The effective potentials for the $hh$ (blue line), the $\mathit{l}\mathit{h}$ (red line), and the $so$ band (green line) in $\mathrm{InAs}∕\mathrm{GaAs}$ SAQD, where the dot radius is $R=8\mathrm{nm}$, and the height is $h=3\mathrm{nm}$. The probability densities of (a) the $1S_{+3∕2}^{+}$ state, (b) the $1P_{+5∕2}^{+}$ state, and (c) the $4S_{+3∕2}^{-}$ state. The states displayed in (b) and (c) have the largest transition matrix elements for the initial $1S_{+3∕2}^{+}$ state at $B=0$ in the case of $s$- and $p$-polarized light. The dot boundary is indicated by the solid white line.

Figure 4

The dominant transition matrix elements (left panel) and the transition energies (right panel) in the valence band of $\mathrm{InAs}∕\mathrm{GaAs}$ single quantum dot as they vary with the magnetic field. Parity of the states is symbolically denoted by $+$ and $-$ sign, while the full state’s symmetry is shown in Tables . The three transitions for $s$-polarization (upper panel), (a) $\overline{{\mid M_{\perp }\mid }^2}$, (b) $E_{if}$. The three transitions for $p$-polarization (lower panel), (c) ${\mid M_{\Vert }\mid }^2$, (b) $E_{if}$.

Figure 5

The same as Fig. 4, but for the three dominant transitions in $\mathrm{Ge}∕\mathrm{Si}$ quantum dots. The symmetry of the states is shown in Tables and . $s$-polarization (upper panel), (a) $\overline{{\mid M_{\perp }\mid }^2}$, (b) the transition energies $E_{if}$. $p$-polarization (lower panel), (c) ${\mid M_{\Vert }\mid }^2$, (d) $E_{if}$.

Figure 6

The intersublevel absorption spectra in $\mathrm{InAs}∕\mathrm{GaAs}$ self-assembled quantum dots at $B=0$ (solid lines) and $B=40\mathrm{T}$ (dashed lines). $s$-polarization and homogeneous broadening for (a) $0\mathrm{meV}<\hslash \omega <120\mathrm{meV}$ and (b) $120\mathrm{meV}<\hslash \omega <200\mathrm{meV}$. (c) $p$-polarization and homogeneous broadening. (d) The case of inhomogeneous broadening. The highest absorption lines are explicitly labeled.

Figure 7

The same as in Fig. 6, but now for a $\mathrm{Ge}∕\mathrm{Si}$ single quantum dot. The case of inhomogeneous broadening for (a) $s$-polarization, (b) $p$-polarization in the energy range $280\mathrm{meV}<\hslash \omega <350\mathrm{meV}$, and (c) $p$-polarization in the energy range $380\mathrm{meV}<\hslash \omega <650\mathrm{meV}$. (d) The absorption for inhomogeneous broadening and for both $s$- and $p$-polarized light.