Enhanced intraband carrier relaxation in quantum dots due to the effect of plasmon–LO-phonon density of states in doped heterostructures

The effect of surface plasmon–LO-phonon excitations of the doped elements of heterostructures in the electronic dynamics of quantum dots has been theoretically studied in a double heterostructure. It has been found that, in contrast to a single heterostructure, critical points can arise in the surface plasmon–LO-phonon density of states for layered structures. This results in enhanced quantum-dot intraband carrier relaxation as compared with the single heterostructure. It has been shown that the relaxation rates and spectral positions of relaxation windows strongly depend on the thickness of the layer containing the quantum dots. These effects of the critical points of density of states have been demonstrated using a model n-$\mathrm{Ga}\mathrm{As}∕\mathrm{Ga}\mathrm{As}∕\text{air}$ heterostructure with an InAs quantum dot embedded in the GaAs layer.

Figure 1

The double heterostructure considered in the model. $b$ is the thickness of the undoped layer, and $a$ is the distance between the doped semiconductor and the quantum dot.

Figure 2

Left: The dispersion branches $\hslash {\omega }_{s\pm }\left(q\right)$ for the SPLP modes in the double $\mathrm{n}\text{−}\mathrm{Ga}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ heterostructure (Fig. 1). The concentration of free electrons equals $n_0={10}^{18}{\mathrm{cm}}^{-3}$. The thickness of the undoped GaAs layer is $50\mathrm{nm}$. Right: The density of states (DOS) corresponding to the SPLP branches. The symbol ∞ marks the critical points in which the DOS diverges as ${(\hslash \omega -\hslash {\omega }_{d\pm })}^{-1∕2}$.

Figure 3

Top: The depth of the minima in the high (+) and low (−) energy dispersion branches $\hslash {\omega }_{s\pm }\left(q\right)$ of the SPLP modes in the double $n-\mathrm{Ga}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ heterostructure (Fig. 1) as a function of the thickness $\left(b\right)$ of the undoped GaAs layer. Bottom: The position of the minima as a function of the thickness $\left(b\right)$ of the undoped GaAs layer. In both cases the concentration of free electrons equals $n_0={10}^{18}{\mathrm{cm}}^{-3}$.

Figure 4

The scheme of a cylindrical quantum dot. Several low energy hole states for the strong confinement regime are shown.

Figure 5

The hole relaxation rates as a function of the level spacing $\Omega$ for three intraband transitions: the solid lines, $\mid 2⟩\to \mid 1⟩$; the dotted lines, $\mid 3⟩\to \mid 2⟩$; and the dashed lines, $\mid 4⟩\to \mid 2⟩$ (see the text). In calculations the following parameters were used: $a=40\mathrm{nm}$, $b=50\mathrm{nm}$, $n_0={10}^{18}{\mathrm{cm}}^{-3}$, and $\gamma =0.05\mathrm{meV}$.

Figure 6

The hole relaxation rates as a function of the level spacing $\Omega$ of the intraband transition $\mid 2⟩\to \mid 1⟩$ for different undoped layer thicknesses $\left(b\right)$: the dotted lines, $b=50\mathrm{nm}$; the dashed lines, $b=80\mathrm{nm}$; and the dash-dotted lines, $b=120\mathrm{nm}$. The solid lines show the corresponding relaxation rates for a single heterostructure. In calculations the following parameters were used: $a=40\mathrm{nm}$, $n_0={10}^{18}{\mathrm{cm}}^{-3}$, and $\gamma =0.05\mathrm{meV}$.

Figure 7

Hole relaxation rates as a function of the level spacing $\Omega$ of the intraband transition $\mid 2⟩\to \mid 1⟩$ for different dephasing rates $\left(\gamma \right)$: the solid lines, $\gamma =0.05\mathrm{meV}$; the dashed lines, $\gamma =0.5\mathrm{meV}$. The symbols SH and DH indicate single and double heterostructures, respectively. In calculations the following parameters were used: $a=40\mathrm{nm}$, $b=50\mathrm{nm}$, and $n_0={10}^{18}{\mathrm{cm}}^{-3}$.

Figure 8

Top: The maximum hole relaxation rates for the transition $\mid 2⟩\to \mid 1⟩$ as a function of the distance $\left(a\right)$ between the QD and the doped substrate. Signs (+) and (−) correspond to the high and low energy relaxation windows, respectively; $b=50\mathrm{nm}$, $n_0={10}^{18}{\mathrm{cm}}^{-3}$, and $\gamma =0.05\mathrm{meV}$. Bottom: The hole relaxation rates as a function of the level spacing $\Omega$ of the intraband transition $\mid 2⟩\to \mid 1⟩$ for different free carrier concentrations $\left(n_0\right)$: the dotted lines, $n_0=0.75\times {10}^{18}{\mathrm{cm}}^{-3}$; the solid lines, $n_0={10}^{18}{\mathrm{cm}}^{-3}$; and the dashed lines, $n_0=1.25\times {10}^{18}{\mathrm{cm}}^{-3}$. In calculations the following parameters were used: $a=40\mathrm{nm}$, $b=50\mathrm{nm}$, and $\gamma =0.05\mathrm{meV}$.