Phonon-assisted tunneling in coupled semiconductor quantum dots

We present a theoretical study on the effects of electron-LO-phonon interaction in two coupled stacked $\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Al}\mathrm{As}$ quantum dots. The resonant and nonresonant electron-phonon coupling contributions to the polaron states are obtained in the framework of the Green function formalism and perturbation approaches at zero and finite temperatures. Ground state renormalization is achieved due to phonon virtual absorption at $T>0$. Tunneling effects between dots have been addressed and their influence on the electronic properties and optical absorption are analyzed. Topological modifications of electronic orbitals are found as a result of phonon-assisted tunneling.

Figure 1

Calculated polaron energy (solid curves) as a funcion of the relative size $W_2∕W_1$ of the two coulped $\mathrm{In}\mathrm{As}∕{\mathrm{In}}_{0.52}{\mathrm{Al}}_{0.48}\mathrm{As}$ quantum dots for (a) $T=0$ and (b) $T=300\mathrm{K}$ with $W_1=100\mathrm{\AA }$ and $B=0.3W_1$. The dotted curves are the bare electron energies and the dotted-dash curves indicate the bare electron energies + one LO-phonon. In (a) the bare electron levels are indicated by $(l,j)$ and in (b) the resonant virtual phonon absorption and emmision are highlighted where the resonant electron-phonon coupling occurs.

Figure 2

The tunneling factor $\Lambda$ at $T=300\mathrm{K}$ as a a funcion of the relative dot size $W_2∕W_1$. The solid (dotted) curve corresponds to the effective charge transfer due to virtual resonant phonon emission (absorption) from the excited state (0, 2) [the ground state (0, 1)] to the ground state (0, 1) [the excited state (0, 2)]. The inset shows the electron-phonon interaction strength as a function of the interdot barrier size $B∕W_1$.

Figure 3

Electronic orbitals for $W_2∕W_1=0.85$. The bare electronic orbitals are shown in (a) (0, 1), (b) (0, 2), and (c) (1, 1). The renormalized electronic orbitals due to polaron effects are given in (d) (0, 1) and (e) (0, 2). In (f) the solid (dashed) curve shows part of the orbital contour of the state (0, 2) in quantum dot $W_1$ with (without) polaron effects.

Figure 4

Absorption spectra with (a) ${\mathbf{e}}_L\Vert \mathbf{x}$ and (b) ${\mathbf{e}}_L\Vert \mathbf{z}$ for different relative dot sizes $W_2∕W_1$. The different curves are offset for clarity with a step of $\Delta (W_2∕W_1)=0.04$ in (a) and 0.046 in (b). The insets show the position of the corresponding absorption peaks. The dashed lines represent the position of the allowed optical transitions for bare electron states.