Localization dynamics of excitons in disordered semiconductor quantum wells

Exciton transport in nanomaterials is sensitive to fluctuations in the confinement potential that are intrinsic to heterogeneous solid-state systems. Redistribution of exciton population manifests as spectral diffusion in which the exciton energy shifts. It is generally assumed that increase or decrease in the exciton energy are equally probable. We show that this assumption is not necessarily valid using two-dimensional coherent spectroscopy on a disordered GaAs quantum well. High-energy excitons relax into lower-energy localized states over a time scale of tens of picoseconds at low sample temperatures $\left(\sim 5\mathrm{K}\right)$. A transition to uniform spectral diffusion of excitons is observed as the temperature is increased to $\sim 20\mathrm{K}$. Numerical simulations reveal the contribution of exciton-phonon interactions to spectral diffusion of excitons. These results provide a perspective on the process of dynamic localization and the effect of the correlation length of disorder on spectral diffusion of excitons.

Figure 1

Calculated $1s$-exciton properties as a function of exciton energy. The exciton density of states (DOS) is plotted as a bar graph. The absorption spectrum (Abs), oscillator strength (OS), and exciton wave function size (Size) are also plotted. The approximate mobility-edge energy is indicated by the dashed line.

Figure 2

Energy level of $1s$-exciton states showing the ground $\left(g\right)$, one-exciton $(x_1$ and $x_2)$, and two-exciton $\left(f\right)$ states. One-exciton states $x_1$ and $x_2$ have energies $E_1$ and $E_2$, respectively. The two-exciton state $f$ has energy $E_1+E_2+{\mathrm{\Delta }}_{x_1,x_2}$, where ${\mathrm{\Delta }}_{x_1,x_2}$ is the interaction energy between excitons $x_1$ and $x_2$.

Figure 3

Calculated exciton decay rates as a function of the exciton energy for sample temperatures of (a) 5 K and (b) 20 K. The plot shows the total decay rate (black squares) and its radiative (blue triangles) and phonon-scattering (red circles) components. The decay rates for exciton states from 10 random realizations are shown.

Figure 4

(a) Schematic of the experimental setup. Pulses $A,B$, and $C$ are incident on the sample with wave vectors ${\mathbf{k}}_A,{\mathbf{k}}_B$, and ${\mathbf{k}}_C$, respectively. The FWM signal is emitted in the phase-matched direction ${\mathbf{k}}_S$. (b) Pulse timing for our 2DCS experiment. Excitation pulses $A,B$, and $C$ are separated by delays $\tau$ and $T$ in the rephasing time ordering. The signal is emitted during time $t$ as a photon echo.

Figure 5

The Liouville pathways that contribute to the photon-echo signal including the coherent and incoherent excited state emission (ESE), excited state absorption (ESA), and ground state bleach (GSB) pathways. ${\mathbf{k}}_A,{\mathbf{k}}_B$, and ${\mathbf{k}}_C$ are the wave vectors of the incident pulses; ${\mathbf{k}}_S$ is the wave vector of the emitted photon-echo signal.

Figure 6

(a)–(d) Experimental 2D spectra for $T=0.2$, 10, 20, and 30 ps, respectively, with the sample at a temperature of 5 K. All the spectra are normalized by the maximum of the peak of the spectrum for $T=0.2$ ps. The horizontal lines in (a) indicate excitation energies for the slices shown in Fig. 8. Note the relative increase in the signal within the circled region in (d).

Figure 7

(a)–(d) Simulated 2D spectra for $T=0.2$, 10, 20, and 30 ps, respectively, with the sample at a temperature of 5 K. All the spectra are normalized by the maximum of the peak of the spectrum for $T=0.2$ ps. The horizontal lines in (a) indicate excitation energies for the slices shown in Fig. 8.

Figure 8

(a)–(d) Horizontal slices from experimental and simulated 2D spectra for $T=0.2$, 10, 20, and 30 ps, respectively, with the sample at a temperature of 5 K. The slices are for the excitation energies indicated in Figs. 6 and 7. The lower excitation energy slices are on the top and the higher excitation energy slices are on the bottom. The experimental and simulated slices are shown as solid and dashed lines, respectively. The simulated slices are scaled by the number indicated in the top right corner of the panels.

Figure 9

(a)–(d) Experimental 2D spectra for $T=0.2$, 10, 20, and 30 ps, respectively, with the sample at a temperature of 20 K. All the spectra are normalized by the maximum of the peak of the spectrum for $T=0.2$ ps. The horizontal lines in (a) indicate excitation energies for the slices shown in Fig. 11.

Figure 10

(a)–(d) Simulated 2D spectra for $T=0.2$, 10, 20, and 30 ps, respectively, with the sample at a temperature of 20 K. All the spectra are normalized by the maximum of the peak of the spectrum for $T=0.2$ ps. The horizontal lines in (a) indicate excitation energies for the slices shown in Fig. 11.

Figure 11

(a)–(d) Horizontal slices from experimental and simulated 2D spectra for $T=0.2$, 10, 20, and 30 ps, respectively, with the sample at a temperature of 20 K. The slices are for the excitation energies indicated in Figs. 9 and 10. The lower excitation energy slices are on the top and the higher excitation energy slices are on the bottom. The experimental and simulated slices are shown as solid and dashed lines, respectively. The simulated slices are scaled by the number indicated in the top right corner of the figures.

Figure 12

(a)–(d) Simulated Green's function showing the probability distribution of change in the exciton energy for $T=0.2$, 10, 20, 30 ps, respectively, at a sample temperature of 5 K after averaging for different realizations. Note that the vertical energy axes in these plots are oriented opposite to those in 2D spectra.

Figure 13

(a)–(d) Simulated Green's function showing the probability distribution of change in the exciton energy for $T=0.2$, 10, 20, 30 ps, respectively, at a sample temperature of 20 K after averaging for different realizations. Note that the vertical energy axes in these plots are oriented opposite to those in 2D spectra.

Figure 14

Degree of dynamic localization to a particular final state as a function of the degree of localization of the final state for $T=0.2$, 10, 20, and 30 ps, with the sample at a temperature of (a) 5 K and (b) 20 K.

Figure 15

Calculated oscillator strength (OS), density of states (DOS), and absorption spectrum (Abs) of the $1s$-exciton resonance for disorder correlation lengths of (a) 10 nm and (b) 30 nm. The sample temperature was set at 5 K. The energy axis is shifted by the same amount as in Fig. 1.

Figure 16

(a)–(d) Simulated 2D spectra for $T=0.2$, 10, 20, and 30 ps, respectively, with the sample at a temperature of 5 K using disorder correlation length $\xi =10$ nm. All the spectra are normalized by the maximum of the peak of the spectrum for $T=0.2$ ps.

Figure 17

(a)–(d) Simulated 2D spectra for $T=0.2$, 10, 20, and 30 ps, respectively, with the sample at a temperature of 5 K using disorder correlation length $\xi =30$ nm. All the spectra are normalized by the maximum of the peak of the spectrum for $T=0.2$ ps.

Figure 18

(a)–(d) Experimental 2D spectra for $T=0.2$, 10, 20, and 30 ps, respectively, with the sample at a temperature of 10 K. All the spectra are normalized by the maximum of the peak of the spectrum for $T=0.2$ ps.

Figure 19

(a)–(d) Simulated 2D spectra for $T=0.2$, 10, 20, and 30 ps, respectively, with the sample at a temperature of 10 K. All the spectra are normalized by the maximum of the peak of the spectrum for $T=0.2$ ps.