Anharmonicity-assisted multiphonon transitions between distant levels in semiconductor quantum dots

We calculate the multiphonon transition rate from the excited state of a single-occupied two-level quantum dot. The electron interacts with certain optical phonon modes which in turn transfer the transition energy to the bath of other phonon modes decoupled from the electron states. Our theory covers the previously unexplored range of transition energies several times larger than the optical phonon energy and systematically studies the role of quantum interference of the processes involving different virtual polaron states.

Figure 1

(a)–(c) Graphical representation of six processes entering the multiphonon matrix element (4.1). Wavy arrows denote the intraladder couplings originating from the anharmonic interaction (3.3), while straight arrows denote the interladder couplings due to the off-diagonal electron-phonon interaction (2.5). Six processes $(2,\nu )\to (1,\mu )$ are grouped into three pairs (a)–(c) each with different energy denominator; see Eqs. (4.5)–(4.8). Due to destructive interference, the contributions (4.5)–(4.8) are of the same order in Fröhlich couplings $u_{ik}$ despite individual terms in Eq. (4.1) being of different order. (d) Illustration of allowed multiphonon transitions entering the sum (4.24) for $E=2.75$ (straight arrows). In this case, $|{\Delta }_M|\equiv |E-M|<2$ for $M=1,2,3,4$; rare transitions to the intermediate states $(1,M)$ trigger fast intraladder relaxation (4.23) to the ground state $(1,0)$ (wavy arrows). Direct transition to the ground state is impossible since the energy $E=2.75$ cannot be transferred to the phonon bath in a single 3-phonon collision.

Figure 2

Graphical representation of the matrix element $\langle {\Psi }_{1,2}\left|\mathbf{x}\right|{\Psi }_{2,0}\rangle$, Eq. (4.28), illustrating 12 quantum paths connecting states (2,0) and (1,2) via different intermediate virtual states for $E=5/2$. The vertical distance between the states (n,k) and (m,l) represents their energy separation in units $\hslash \Omega$.

Figure 3

Resonant coupling of the polaron states (2,0) and (1,K) [panel (a)] leads to their mixing and splitting [panel (b)]. Note that (1,K) includes several degenerate states, in particular, $|r_{\parallel }\rangle =|{\varphi }_{1,K,0}\rangle$ and $|r_{\perp }\rangle =|{\varphi }_{1,K-1,1}\rangle$. Panels (c) and (d) illustrate two off-resonant terms in Eq. (4.24) for $M=K-1$ which can be ignored in the vicinity of the resonance.

Figure 4

Diagrammatic illustration of the relation $\gamma =2{\gamma }_{nd}$ between the longitudinal [$\gamma$, panel (a)] and the transverse [${\gamma }_{nd}$, panel (b)] relaxation rate in the situation when the state 1 is everlasting, while the state 2 decays with probability $\gamma$ to the third state 3. Panel (c): An equivalent description in terms of incoherent tunneling is applicable when the level width $\gamma$ exceeds the Rabi frequency $t$.

Figure 5

The anharmonic decay rate $w\left(\Delta \right)$, Eq. (6.1) [in units of $w_0\left(1\right)$], vs the energy $\Delta$ transferred to the bath (in units of the LO phonon energy $\hslash \Omega$) for $T=0$ (lower line) and $T=300$ K (upper line).

Figure 6

The inelastic relaxation rate [in units of $w_0\left(1\right)$] vs the separation $E$ of the lowest polaron levels in two ladders (in units of LO phonon energy) for $T=0$. The dashed lines illustrate the off-resonant results (4.26), (4.27), and (4.31) where $E$ is given by Eq. (4.2). Solid line: the anticrossing of the resonant levels near $E=1$ and $E=2$, Sec. 5a, and the anharmonicity-induced broadening of levels near $E=3$, Sec. 5b, are taken into account.

Figure 7

Contributions ${\Gamma }_n^{\left(M\right)}$ of different channels to the inelastic relaxation rate ${\Gamma }_T$, Eq. (4.25), at $T=300$ K as a function of the level separation $E$. Different curves correspond to $n=0,1,2$ (from top to bottom) and to $M=0$ (curves peaked at $E\sim 1$) and $M=1$ (peaked at $E\sim 2$ and $E\sim 0$). The corrections to Eq. (4.25) near $E=2$ due to anharmonicity-induced broadening of levels, Sec. 5b, are taken into account.

Figure 8

The inelastic relaxation rate ${\Gamma }_T$ [in units of $w_0\left(1\right)$] vs the level separation $E$ for $T=0\text{K},300\text{K}$, and 1000 K (from bottom to top).