Optical-phonon scattering and theory of magneto-polarons in a quantum well in a strong magnetic field

We report a theoretical study of the carrier relaxation in a quantum-well structure subjected to a strong magnetic field. Both the alloy (GaInAs) disorder effects and the Fröhlich interaction are taken into account when the electron energy differences are tuned to the longitudinal optical (LO) phonon energy. In the weak electron-phonon coupling regime, a Fermi’s golden rule computation of LO phonon-scattering rates shows a very fast nonradiative relaxation channel for the alloy-broadened Landau levels (LL’s). In the strong electron-phonon coupling regime, we use a magneto-polaron formalism and compute the electron survival probabilities in the upper LL’s including increasing numbers of LO phonon modes for a large number of alloy disorder configurations. Our results predict a nonexponential decay of the upper-level population once electrons are injected in this state.

Figure 1

(Color online) Typical in-plane electron-density distributions (in ${\text{nm}}^{-2}$) for four states in the set of independently broadened LL’s at 24.58 T: (a) DOS tail of $|E_2,0⟩$; (b) DOS max of $|E_2,0⟩$; (c) DOS tail of $|E_1,2⟩$; and (d) DOS max of $|E_1,2⟩$.

Figure 2

(Color online) Position uncertainties of the independently alloy-broadened LL’s at 24.58 T: (a) ${\delta }_x$ of $|E_2,0⟩$; (b) ${\delta }_y$ of $|E_2,0⟩$; (c) ${\delta }_x$ of $|E_1,2⟩$ ; and (d) ${\delta }_y$ of $|E_1,2⟩$.

Figure 3

LO phonon-emission rate from alloy-broadened LL’s $|E_2,0⟩$ to alloy-broadened LL’s $|E_1,2⟩$ at $B_2=24.58\text{T}$, as a function of the energy of the initial state, $T=0\text{K}$.

Figure 4

Illustrations of magneto-polaron states in ideal systems. (a) A schematic picture of the formation of magneto-polaron states. Inset: energy anticrossing around $B_2=24.58\text{T}$. (b) Polaron-splitting energies as a function of $B_p$ for $p=1–10$ (the line is only a guide for the eyes).

Figure 5

(Color online) Alloy-disorder broadening of magneto-polarons: (a) SCBA calculation of the polaron DOS for $p=1–4$ at the resonant fields $B_p$; (b) SCBA calculation of the polaron DOS (in unit of $m^{\ast }/\left(\pi {\hslash }^2\right)$) around $B_2=24.58\text{T}$; and (c) and (d) are numerical results at 23.50 and 24.58 T, respectively, to compare with the SCBA calculations.

Figure 6

Survival probability $P_s$ for $B_2=24.58\text{T}$ with the initial state as the central level of the broadened $|E_2,0⟩$ LL’s while neglecting all the uncoupled states, $T=0\text{K}$.

Figure 7

(Color online) Electron-LO-phonon coupling element related to Fröhlich factor versus phonon wave vector.

Figure 8

(Color online) Taking increasing numbers of the uncoupled one-phonon states into account, computations of the electron survival probabilities with initial state in center of the alloy-broadened LL’s $|E_2,0⟩$: (a) remaining in $|E_2,0⟩$ at $B_2=24.58\text{T}$; (b) in polaron states $|\pm ⟩$ at $B_2$; (c) in $|E_2,0⟩$ at 20.00 T; and (d) in $|E_2,0⟩$ at 23.50 T (see text).

Figure 9

(Color online) Computations with 36 phonon modes of the long-time limit of the survival probability $P_s\left(\infty \right)$ in the $|E_2,0⟩$ LL’s: (a) Initial states in center of LL’s $|E_2,0⟩\otimes |0_{\text{LO}}⟩$ for each sample; (b) Average over all initial LL’s $|E_2,0⟩\otimes |0_{\text{LO}}⟩$ for each sample. Also the ensemble averages are taken over the $N=100$ samples for each case (jointed by lines, which are guides for the eyes).