Exciton Supersolidity in Hybrid Bose-Fermi Systems

We investigate the ground states of a Bose-Einstein condensate of indirect excitons coupled to an electron gas. We show that in a properly designed system the crossing of a roton minimum into the negative energy domain can result in the appearance of the supersolid phase, characterized by periodicity in both real and reciprocal space. Accounting for the spin-dependent exchange interaction of excitons we obtain ferromagnetic supersolid domains. The Fourier spectra of excitations of weakly perturbed supersolids show pronounced diffraction maxima which may be detected experimentally.

Figure 1

The excitation spectrum (top) and effective exciton-exciton interaction potential (bottom) for three values of the exciton-electron layer separation $L$. We use the exciton mass $m_{\mathrm{ex}}=0.21m_{\mathrm{e}}$, and the average exciton density ${\overline{n}}_{\mathrm{tot}}={10}^{11}{\mathrm{cm}}^{-2}$. The exciton dipole separation, $l=12\mathrm{nm}$, and the correction factor, $\xi =0.05$.

Figure 2

Examples of density profiles of the exciton condensate in real space (left column) and reciprocal space (right column, density in logarithmic scale). The figures (a),(b) show the homogeneous ground state for separation $L=30\mathrm{nm}$, and the figures (c),(d) depict the supersolid ground state for $L=18\mathrm{nm}$, characterized by periodicity in both real and momentum spaces. The orientation of the supersolid lattice is chosen spontaneously. (e),(f) A metastable state with lattice imperfections, corresponding to a local energy minimum. In momentum space, such a state is characterized by concentric rings.

Figure 3

Numerically determined spectra of a weakly perturbed (a) noninteracting exciton condensate, (b) condensate interacting through a local (momentum-independent) potential, and (c) supersolid condensate in the presence of coupling with electron gas, interacting through an effective potential $V_{\mathrm{ex}-\mathrm{ex}}^{\mathrm{eff}}$ [parameters as in Figs. 2c, 2d]. While in the cases (a) and (b) we obtain the parabolic and linear Bogoliubov spectra as predicted by the theory, in the case (c) the spectrum is distorted as a result of interference of branches starting from different $k$ components of the ground state in the momentum space, corresponding to the vertical lines. The density is shown in a logarithmic intensity scale.

Figure 4

Numerically calculated density plot of the exciton condensate in the supersolid phase, showing the ferromagnetic domain structure. Here ${\overline{n}}_{+}={\overline{n}}_{-}={10}^{11}{\mathrm{cm}}^{-2}$, $L=18\mathrm{nm}$, and $\alpha$ is chosen in such a way that for $q=0$ the interaction of excitons with opposite spin is 10% stronger than the interaction of excitons with parallel spins. The green and orange coloring corresponds to the $+1$ and $-1$ spin components.