Strain-induced darkening of trapped excitons in coupled quantum wells at low temperature

In GaAs/AlGaAs coupled quantum wells, strain-induced traps may be used to confine excitons in in-plane, harmonic traps. Using these traps, we have pursued Bose-Einstein condensation (BEC) of long-lived, spatially indirect excitons. Here, we report a remarkable transition of the indirect exciton luminescence pattern with increasing strain, increasing exciton density, and decreasing temperature, to a spatial pattern exhibiting a large dark spot at the trap center, where we expect the exciton density to be maximum. The mechanism of particle loss is ruled out as an explanation for this dark spot. While the onset criteria are approximately consistent with the conditions for BEC of a weakly interacting gas, the conspicuous proximity in energy of the indirect light-hole states suggests that an explanation employing the single-particle physics of light-hole–heavy-hole mixing may explain the phenomenon. The effect of the strain is modeled, and the resulting landscape of indirect exciton spin states is discussed. The relative oscillator strengths of these states are predicted by an exact numerical solution of the two-particle Schrödinger equation for electrons and holes in coupled quantum wells and an electric field. The contrast in oscillator strengths is sufficient to produce this luminescence pattern, but this analysis suggests a strongly diminished lifetime as stress is increased. The opposite lifetime dependence is observed experimentally. Additionally, the temperature dependence eludes explanation by this mechanism.

Figure 1

(a) The stressor mechanism. Strain is produced by applying a localized force to the substrate side of the wafer. The optical side remains free to flex under the force of the pin, allowing expansion of the unit cell and, hence, lowering of the carrier energies. (b) A typical stressed spectrum of recombination PL after laser excitation. In order of decreasing energy, we see the heavy-hole DX, heavy-hole biexciton, and heavy-hole IX.

Figure 2

Conduction and valence band in the evaporated structure. Solid lines show the energy of states with no voltage across the structure. Dotted lines show the effect of an applied electric field. The regions of the 125-$\mu$m-thick n-GaAs substrate and p-GaAs cap are not pictured.

Figure 3

Conduction and valence band dispersion relations for a GaAs quantum well. The lh-hh splitting is exaggerated by showing the effect for 5-nm wells.

Figure 4

PL images showing darkening with increasing applied stress, for symmetric coupled quantum wells with a width of 12 nm and temperature $T=2.2$ K. The laser excitation photon energy was 4 meV above the heavy-hole DX absorption at trap center; $P_{\text{avg}}=120$ $\mu$W. The trap depth changes from 4 to 5 meV from $F=2.5$ to 2.8 N. An external electric potential difference of 0.50 V was applied to augment the built-in field. The horizontal axis (with weaker confinement) corresponds to the [110] crystal axis, and the vertical axis corresponds to [1$\stackrel{̲}{1}$0].

Figure 5

The appearance of the dark spot is shown with increasing density in a series of spatially resolved spectra with increasing laser excitation power, as stress, temperature, applied bias, and excitation laser duty-cycle remain fixed. The bright vertical line at the center of the trap is impurity luminescence from the substrate, demonstrating the position and profile of the pumping laser. The well width is 12 nm, and $T=1.5$ K. The external bias was 0.6 V.

Figure 6

The appearance of the dark spot with decreasing temperature, in IX PL images with decreasing temperature at high stress. Excitation $P_{\mathrm{avg}}=50\hspace{0.5em}\mu \text{W}$; $\text{bias}=0.5$ V.

Figure 7

Measured and predicted linear polarization. Experimental linear polarizations at very high stress are displayed in the upper half, with an underlying image of the IX PL intensity. The line orientation and length indicate polarization angle and degree of linear polarization, respectively. The maximum degree of polarization, about 75%, is reached in the dimmed intensity region on the horizontal axis. The intensity image displays the reappearance of PL from the trap center at very high stress. The lower half displays the theoretical prediction for linear polarization due to lh IX–hh IX mixing, as discussed in Sec. IV.

Figure 8

IX spectrum at a stress surpassing that of the PL dark spot, showing trap-center luminescence reemerging.

Figure 9

Profiles of IX PL linewidth and spectrally integrated counts across the weak-confinement axis of the trap. The linewidth is used here as a qualitative measure of density. $T=2.0$ K, $P_{\text{avg}}=2$ mW, $\text{and bias}=1$ V.

Figure 10

Low-stress, spatially resolved spectrum in the trap, showing, in order of decreasing energy, lh DX, hh DX, lh IX, and hh IX. The excitation incidence angle was about 55${}^{ˆ}$ from normal. Excitation $\lambda =$ 797 nm.

Figure 11

Results of the numerical solution to the two-particle Schrödinger equation in coupled quantum wells. Points marked with a circle or triangle are the results of numerical simulation. (a) The magnitude of the confined Stark shift of the hh and lh IX energies as a function of electric field, with a linear fit in our region of interest to extract the electric dipole moment. (b) The oscillator strength of the hh and lh excitons as a function of applied electric field. A rough fit in the region around our built-in field suggests near exponential dependence ($I_0{e}^{-\alpha E}$). The lh value is roughly ten times greater than that of the heavy hole, a ratio which is relatively independent of the field in the region of interest.

Figure 12

A fit (solid line) to the measured trap profile of the direct hh exciton at low stress (higher energy points) using the discussed model. Also shown is the measured trap profile for the corresponding biexciton line (lower energy points).

Figure 13

The hh and lh indirect exciton energy at the trap center for 14-nm (solid lines) and 12-nm (dashed lines) coupled quantum wells as a function of applied force. The higher energy state of each pair at low stress corresponds to the lh IX, while the lower energy state is the hh IX.

Figure 14

The spatial dependence of the ${\epsilon }_{\mathit{xy}}$ component of the shear strain as predicted by calculations (a) compared to the energy difference between the indirect and direct lines (minus an overall constant) (b). (c) The calculated value of the strain (solid line) fit to the measured difference between the indirect and direct lines, using a multiplicative constant as a fitting parameter.

Figure 15

A numerical model of the expected spatial dependence of the lh fraction in the lower energy IX state as a function of applied stress.

Figure 16

Predicted IX recombination PL intensity profiles with increasing force applied to the sample, using the theory of a spatially varying oscillator strength.

Figure 17

Predicted IX recombination PL intensity profiles with increasing exciton density.

Figure 18

(a) IX luminescence as a function of excitation laser $P_{\mathrm{avg}}$. No significant deviations from the linear fit emerge as the new luminescence pattern appears. (b) IX lifetime vs trap depth. Trap depth is indicative of total applied force and the magnitude of strain. Lifetimes plotted are decay constants of single-exponential decays fit to the time-resolved IX PL. The inset shows this fitting for one point in the lifetime plot.

Figure 19

Relative particle number required for dark spot onset vs temperature for two different trap depths. The $N^{1/2}$ line suggested by the noninteracting BEC theory is shown for comparison. Data are shown for 12-nm wells.