Collective properties of indirect excitons in coupled quantum wells in a random field

The influence of a random field induced by impurities, boundary irregularities etc. on the superfluidity of a quasi-two-dimensional (2D) system of spatially indirect excitons in coupled quantum wells is studied. The interaction between excitons is taken into account in the ladder approximation. The random field is allowed to be large compared to the dipole-dipole repulsion between excitons. The coherent potential approximation (CPA) allows us to derive the exciton Green’s function for a wide range of the random field, and the CPA results are used in the weak-scattering limit, which results in the second-order Born approximation. The Green’s function of the collective excitations for the cases of (1) equal electron and hole masses and (2) the “heavy hole” limit are derived analytically. For quasi-two-dimensional excitonic systems, the density of the superfluid component and the Kosterlitz-Thouless temperature of the superfluid phase transition are obtained, and are found to decrease as the random field increases. This puts constraints on experimental efforts to observe excitonic superfluidity.

Figure 1

Diagrammatic representation of the equation for the vertex $\Gamma$ in the momenta frequency representation $(\mathbf{P},\Omega )$.

Figure 2

Dependence of temperature of Kosterlitz-Thouless transition $T_c=T_c\left(n\right)$ (in units of $K$; for $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$: $M=0.24m_0$, where $m_0$ is the free electron mass; $ϵ=13$) on the exciton density $n$ (in units of ${\mathrm{cm}}^{-2}$) at the interwell distance $D=15\mathrm{nm}$, for different random fields $Q$ (in units of meV): $Q=0$, solid curve (straight line); $Q=0.1\mathrm{meV}$, dotted curve; $Q=0.2\mathrm{meV}$, dashed curve; $Q=0.3\mathrm{meV}$, long-dashed curve; $Q=0.4\mathrm{meV}$, dashed-dotted curve; $Q=0.5\mathrm{meV}$, solid curve (bottom right hand corner).

Figure 3

Dependence of temperature of Kosterlitz-Thouless transition $T_c=T_c\left(Q\right)$ (in units of K; for $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$: $M=0.24m_0$; $ϵ=13$) on the random field $Q$ (in units of meV) at the interwell distance $D=15\mathrm{nm}$, for different exciton densities $n$: $n=5\times {10}^{10}{\mathrm{cm}}^{-2}$, solid curve; $n=1.0\times {10}^{11}{\mathrm{cm}}^{-2}$, dotted curve; $n=3.0\times {10}^{11}{\mathrm{cm}}^{-2}$, dashed curve.

Figure 4

Dependence of temperature of Kosterlitz-Thouless transition $T_c=T_c\left(D\right)$ (in units of K; for $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$: $M=0.24m_0$; $ϵ=13$) on the interwell distance $D$ (in units of nm) at the exciton density $n=1\times {10}^{11}{\mathrm{cm}}^{-2}$, for different random fields $Q$: $Q=0$, solid curve; $Q=0.3\mathrm{meV}$, dotted curve; $Q=0.5\mathrm{meV}$, dashed curve; $Q=0.6\mathrm{meV}$, dashed-dotted curve.

Figure 5

Temperature dependence of Kosterlitz-Thouless transition $T_c=T_c\left(n\right)$ based on CPA and the perturbation theory (PT) approximation used in Ref. 36 (in units of K; for $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$: $M=0.24m_0$; $ϵ=13$; $m_0$ is a mass of electron) on the exciton density $n$ (in units ${\mathrm{cm}}^{-2}$) at the interwell distance $D=15\mathrm{nm}$, for different random fields $Q$ (in units of meV): $Q=0.2\mathrm{meV}$, solid curve, full CPA vs dotted curve, PT; $Q=0.3\mathrm{meV}$, dashed curve, full CPA vs long-dashed curve, PT.

Figure 6

Temperature dependence of Kosterlitz-Thouless transition $T_c=T_c\left(Q\right)$ based on CPA and the perturbation theory (PT) approximation used in Ref. 36 (in units of K; for $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$: $M=0.24m_0$; $ϵ=13$) on the random field $Q$ (in units of meV) at the interwell distance $D=15\mathrm{nm}$, for different exciton densities $n$: $n=5\times {10}^{10}{\mathrm{cm}}^{-2}$, solid curve, full CPA vs dotted curve, PT; $n=1\times {10}^{11}{\mathrm{cm}}^{-2}$, dashed curve, full CPA vs long-dashed curve, PT.