Attractive and repulsive dipolar interaction in bilayers of indirect excitons

We explore attractive dipolar interaction in indirect excitons (IXs). For one layer of IXs in a single pair of coupled quantum wells (CQWs), the out-of-plane IX electric dipoles lead to repulsive dipolar interaction between IXs. The attractive dipolar interaction between IXs is realized in a two-CQW heterostructure with two IX layers in two separated CQW pairs. We found both in experimental measurements and theoretical simulations that increasing density of IXs in one layer causes a monotonic energy reduction for IXs in the other layer. We also found an in-plane shift of a cloud of IXs in one layer towards a cloud of IXs in the other layer. This behavior is qualitatively consistent with attractive dipolar interaction. The measured IX energy reduction and IX cloud shift are higher than the values given by the correlated liquid theory.

Figure 1

Two-CQW heterostructure and $x$-energy image. (a) Diagram of the two-CQW structure with two CQW pairs. ${\mathrm{IX}}_2$ forms in ${\mathrm{CQW}}_2$, and ${\mathrm{IX}}_1$ forms in ${\mathrm{CQW}}_1$. Schematic below shows IX dipoles. $-$ electrons and $+$ holes. The intra-CQW interaction between ${\mathrm{IX}}_2$ (or ${\mathrm{IX}}_1$) side-to-side dipoles is repulsive. The inter-CQW interaction between ${\mathrm{IX}}_2$ and ${\mathrm{IX}}_1$ head-to-tail dipoles is attractive. (b) Differential $x$-energy luminescence image. The arrows indicate the excitation spot positions of the ${\mathrm{L}}_2$ and ${\mathrm{L}}_1$ lasers resonant to direct excitons in 15-nm ${\mathrm{CQW}}_2$ and 12-nm ${\mathrm{CQW}}_1$, respectively. ${\mathrm{L}}_2$ generates ${\mathrm{IX}}_2.{\mathrm{L}}_1$ generates ${\mathrm{IX}}_1$ and a smaller concentration of ${\mathrm{IX}}_2$. The laser powers $P_{\mathrm{L}1}=10\mu \mathrm{W},P_{\mathrm{L}2}=250\mu \mathrm{W}$. The differential $x$-energy image is obtained by subtracting the $x$-energy images created by only ${\mathrm{L}}_1$ on and by only ${\mathrm{L}}_2$ on from the $x$-energy image created by both lasers on. In the differential $x$-energy image, the yellow and blue colors indicate the enhancement and reduction of the IX luminescence intensity, respectively. For the energy axis, the blue region on the right side and the yellow region on the left (observed for ${\mathrm{IX}}_1$) correspond to an energy decrease. For the position axis, the blue region on the low side and the yellow region on the high side (observed for ${\mathrm{IX}}_1$) correspond to a cloud shift up. The differential $x$-energy image shows an increase in ${\mathrm{IX}}_2$ energy, a decrease in ${\mathrm{IX}}_1$ energy, and a spatial shift of the ${\mathrm{IX}}_1$ cloud towards the ${\mathrm{IX}}_2$ cloud.

Figure 2

Decrease and increase of IX energy due to attractive and repulsive dipolar IX interactions: Experiment. (a) The decrease in ${\mathrm{IX}}_1$ energy with increasing ${\mathrm{L}}_2$ power and, in turn, ${\mathrm{IX}}_2$ density. $P_{\mathrm{L}1}=10\mu \mathrm{W}$. An energy decrease corresponds to attractive ${\mathrm{IX}}_1\text{−}{\mathrm{IX}}_2$ interaction. (b) The redshift of the ${\mathrm{IX}}_1$ spectrum with turning on ${\mathrm{L}}_2$, which increases ${\mathrm{IX}}_2$ density. The black line shows the ${\mathrm{IX}}_1$ spectrum when only ${\mathrm{L}}_1$ is on, $P_{\mathrm{L}1}=10\mu \mathrm{W}$. The green line shows the ${\mathrm{IX}}_1$ spectrum when an additional ${\mathrm{L}}_2$ is on, $P_{\mathrm{L}2}=250\mu \mathrm{W}$. (c) ${\mathrm{IX}}_1$ energy as a function of both $P_{\mathrm{L}1}$ and $P_{\mathrm{L}2}$. (d) The increase in ${\mathrm{IX}}_2$ energy with increasing ${\mathrm{L}}_2$ power and, in turn, ${\mathrm{IX}}_2$ density. $P_{\mathrm{L}1}=0$. An energy increase corresponds to repulsive ${\mathrm{IX}}_1\text{−}{\mathrm{IX}}_2$ interaction. (e) The blueshift of the ${\mathrm{IX}}_2$ spectrum with increasing ${\mathrm{L}}_2$ power, which increases ${\mathrm{IX}}_2$ density. $P_{\mathrm{L}2}=25\mu \mathrm{W}$ (black line) and $250\mu \mathrm{W}$ (green line). (f) ${\mathrm{IX}}_2$ energy as a function of both $P_{\mathrm{L}1}$ and $P_{\mathrm{L}2}$. (g) The change in ${\mathrm{IX}}_1$ energy vs the change in ${\mathrm{IX}}_2$ energy. Each set of data corresponds to increasing ${\mathrm{L}}_2$ power. For the sets of data presented by orange, red, purple, blue, cyan, and green points, ${\mathrm{L}}_1=5,10,25,50,100,$ and $250\mu \mathrm{W}$, respectively.

Figure 3

Attraction of the ${\mathrm{IX}}_1$ cloud to the ${\mathrm{IX}}_2$ cloud: Experiment. (a) The ${\mathrm{IX}}_1$ cloud profiles at different $P_{\mathrm{L}2}$'s and, in turn, ${\mathrm{IX}}_2$ densities. $P_{\mathrm{L}1}=10\mu \mathrm{W}$. The profiles of the ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ laser excitation spots are shown by black and purple lines, respectively. The dashed line indicates the center of the ${\mathrm{L}}_1$ excitation spot. (b) The center-of-mass position of the ${\mathrm{IX}}_1$ cloud as a function of $P_{\mathrm{L}2}$ for different $P_{\mathrm{L}1}$'s.

Figure 4

Decrease and increase in IX energy due to attractive and repulsive dipolar IX interactions: Theory. (a) The decrease in ${\mathrm{IX}}_1$ energy with increasing ${\mathrm{IX}}_2$ density. $n_1=1.5\times {10}^{10}{\mathrm{cm}}^{-2}$. An energy decrease corresponds to attractive ${\mathrm{IX}}_2\text{−}{\mathrm{IX}}_1$ interaction. (b) ${\mathrm{IX}}_1\text{−}{\mathrm{IX}}_2$ density correlation function. $n_1=n_2={10}^{10}{\mathrm{cm}}^{-2}$. (c) ${\mathrm{IX}}_1$ energy as a function of both ${\mathrm{IX}}_1$ and ${\mathrm{IX}}_2$ densities. (d) The increase in ${\mathrm{IX}}_2$ energy with increasing ${\mathrm{IX}}_2$ density. $n_1=0$. An energy increase corresponds to repulsive ${\mathrm{IX}}_2\text{−}{\mathrm{IX}}_2$ interaction. (e) ${\mathrm{IX}}_2\text{−}{\mathrm{IX}}_2$ (red solid line) and ${\mathrm{IX}}_1\text{−}{\mathrm{IX}}_1$ (black dashed line) density correlation functions. $n_1=n_2={10}^{10}{\mathrm{cm}}^{-2}$. (f) ${\mathrm{IX}}_2$ energy as a function of both ${\mathrm{IX}}_1$ and ${\mathrm{IX}}_2$ densities. (g) The change in ${\mathrm{IX}}_1$ energy vs the change in ${\mathrm{IX}}_2$ energy. Each line corresponds to increasing ${\mathrm{IX}}_2$ density. For the blue, orange, green, red, and purple lines, $n_1=0.5,1,1.5,2,$ and $2.5\times {10}^{10}{\mathrm{cm}}^{-2}$, respectively.

Figure 5

Attraction of the ${\mathrm{IX}}_1$ cloud to the ${\mathrm{IX}}_2$ cloud: Theory. (a) The ${\mathrm{IX}}_1$ cloud profiles simulated for different $P_{\mathrm{L}2}$'s and, in turn, ${\mathrm{IX}}_2$ densities. $P_{\mathrm{L}1}=250\mu \mathrm{W}$. The profiles of the ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ laser excitation spots are shown by black and purple lines, respectively. The dashed line indicates the center of the ${\mathrm{L}}_1$ excitation spot. (b) The center-of-mass position of the ${\mathrm{IX}}_1$ cloud as a function of $P_{\mathrm{L}2}$ for different $P_{\mathrm{L}1}$'s.

Figure 6

Model interaction potentials: $u_{22}$ (top) and $u_{11}$ (middle) are the intra-CQW potentials for ${\mathrm{IX}}_2$ and ${\mathrm{IX}}_1$, respectively; $u_{12}$ (bottom curve) is the inter-CQW interaction potential. Parameters: $d_1=20,d_2=25,D=43,a=5$ (all in nanometers); $\epsilon =13$.

Figure 7

Probability density distributions of inter-CQW biexcitons within the rigid-body approximation. The interaction potential $u_{12}$ (the same as in Fig. 6) is shown by the curve with a dip (black curve). The probability distributions (wave functions squared) of the bound states are depicted by the curves with the maxima. These curves are plotted in arbitrary units and are offset to show the binding energies $E_b$. They correspond (top to bottom) to the reduced masses of 0.5, 1, and 10 of the exciton mass.

Figure 8

Repulsion between the ${\mathrm{IX}}_2$ clouds: Experiment. (a) The ${\mathrm{IX}}_2$ cloud profiles when only the left ${\mathrm{L}}_2$ is on (L ${\mathrm{L}}_2$ on, green line), when only the right ${\mathrm{L}}_2$ is on (R ${\mathrm{L}}_2$ on, red line), and when both left and right ${\mathrm{L}}_2$'s are on [(L ${\mathrm{L}}_2 +$ R ${\mathrm{L}}_2$) on, blue line]. The sum of L ${\mathrm{L}}_2$ on profile and R ${\mathrm{L}}_2$ on profile (L ${\mathrm{L}}_2$ on $+$ R ${\mathrm{L}}_2$ on) is shown by the orange line. Profile (L ${\mathrm{L}}_2 +$ R, ${\mathrm{L}}_2$) on is wider than the sum of L, ${\mathrm{L}}_2$ on profile and R, ${\mathrm{L}}_2$ on profile, indicating the repulsion between the ${\mathrm{IX}}_2$ clouds. The profiles of the left and right ${\mathrm{L}}_2$ laser excitation spots are shown by purple and brown lines, respectively. The dashed lines indicate the centers of the excitation spots. (b) The width of the ${\mathrm{IX}}_2$ cloud when both left and right ${\mathrm{L}}_2$'s are on [(L ${\mathrm{L}}_2 +$ R ${\mathrm{L}}_2$) on, blue points] in comparison to the width of the ${\mathrm{IX}}_2$ cloud obtained as the sum of L ${\mathrm{L}}_2$ on the cloud and R ${\mathrm{L}}_2$ on the cloud [L ${\mathrm{L}}_2$ on $+$ R ${\mathrm{L}}_2$ on, orange points] as a function of $P_{\mathrm{L}2}$.

Figure 9

Repulsion between the ${\mathrm{IX}}_2$ clouds: Theory. (a) The ${\mathrm{IX}}_2$ cloud profiles simulated when only the left ${\mathrm{L}}_2$ is on (L ${\mathrm{L}}_2$ on, green line), when only the right ${\mathrm{L}}_2$ is on (R ${\mathrm{L}}_2$ on, red line), and when both left and right ${\mathrm{L}}_2$'s are on [(L ${\mathrm{L}}_2 +$ R ${\mathrm{L}}_2$) on, blue line]. The sum of simulated L ${\mathrm{L}}_2$ on profile and R ${\mathrm{L}}_2$ on profile (L ${\mathrm{L}}_2$ on $+$ R ${\mathrm{L}}_2$ on) is shown by the orange line. The profile (L ${\mathrm{L}}_2 +$ R ${\mathrm{L}}_2$) on is wider than the sum of L ${\mathrm{L}}_2$ on the profile and R ${\mathrm{L}}_2$ on the profile, indicating the repulsion between the ${\mathrm{IX}}_2$ clouds. The profiles of the left and right ${\mathrm{L}}_2$ laser excitation spots are shown by purple and brown lines, respectively. The dashed lines indicate the centers of the excitation spots. (b) The width of the simulated ${\mathrm{IX}}_2$ cloud when both left and right ${\mathrm{L}}_2$'s are on [(L ${\mathrm{L}}_2 +$ R ${\mathrm{L}}_2$) on, blue points] in comparison to the width of the ${\mathrm{IX}}_2$ cloud obtained as the sum of simulated L ${\mathrm{L}}_2$ on the cloud and R ${\mathrm{L}}_2$ on the cloud [L ${\mathrm{L}}_2$ on $+$ R ${\mathrm{L}}_2$ on, orange points] as a function of $P_{\mathrm{L}2}$.

Figure 10

Position-energy images of IX luminescence. (a) Both ${\mathrm{L}}_2$ and ${\mathrm{L}}_1$ lasers are on. (b) Only ${\mathrm{L}}_2$ is on. (c) Only ${\mathrm{L}}_1$ is on. The arrows indicate the excitation spot positions of the ${\mathrm{L}}_2$ and ${\mathrm{L}}_1$ lasers resonant to direct excitons in 15-nm CQW and 12-nm CQW, respectively. ${\mathrm{L}}_2$ generates ${\mathrm{IX}}_2$. ${\mathrm{L}}_1$ generates ${\mathrm{IX}}_1$ and a smaller concentration of ${\mathrm{IX}}_2$. The laser powers $P_{\mathrm{L}1}=10\mu \mathrm{W},P_{\mathrm{L}2}=250\mu \mathrm{W}$. Gate voltage $V_{\mathrm{g}}=-2.0\mathrm{V}$ and temperature $T=1.7\mathrm{K}$.

Figure 11

Cross section of the two-CQW heterostructure. The thicknesses and doping concentrations of the layers are indicated.