Crystallization of an exciton superfluid

Indirect excitons—pairs of electrons and holes spatially separated in semiconductor bilayers or quantum wells—are known to undergo Bose-Einstein condensation and to form a quantum fluid. Here we show that this superfluid may crystallize upon compression. However, further compression results in quantum melting back to a superfluid. This unusual behavior is explained by the effective interaction potential between indirect excitons, which strongly deviates from a dipole potential at small distances due to many-particle and quantum effects. Based on first-principles path-integral Monte Carlo simulations, we compute the complete phase diagram of this system and predict the relevant parameters necessary to experimentally observe exciton crystallization in semiconductor quantum wells.

Figure 1

Exciton interaction potential $V_{\mathrm{XX}}$ for a dipole moment $d=13.3a_{\mathrm{B}}^{*}$. (a) The interaction potential $V_{\mathrm{XX}}$ (point-dashed line) is compared to the exciton interaction energy $E_{\mathrm{XX}}$ in several approximations: average interaction of two excitons $\langle E_{\mathrm{XX}}\rangle$ evaluated by PIMC simulations using the Hamiltonian, Eq. (4); the BO model with the infinite hole mass, symmetric ($E_{\mathrm{XX}}^S)$, and antisymmetric ($E_{\mathrm{XX}}^A$) electronic states; the improved BO model, $E_{\mathrm{XX}}=E_{\mathrm{e}}+V_{\mathrm{hh}}$, with a realistic mass ratio $m_{\mathrm{h}}/m_{\mathrm{e}}=2.46$ (ZnSe-based QW). Also shown are the electronic contribution $E_{\mathrm{e}}$ and the dipole potential $d^2/r^3$. Two vertical lines indicate the boundaries of the exciton crystal. (b) Radial electron density $n_{\mathrm{e}}\left(r\right)$ for several hole separations $r_{\mathrm{hh}}$, relative to the midpoint of two holes located at ${\mathbf{R}}_{1,2}=\pm \frac{1}{2}{\mathbf{r}}_{\mathrm{hh}}$. (c) Electron pair distribution function $g_{\mathrm{ee}}\left(r_{\mathrm{ee}}\right)$ for the values $r_{\mathrm{hh}}$ in (b).

Figure 2

Constant density freezing. Top row: 2D PDF $g\left({\mathbf{r}}_{ij}\right)$ (relative to a fixed particle in the center) for $n{a}_{\mathrm{B}}^{*2}=0.0035$ and temperatures $k_{\mathrm{B}}T/{\mathrm{Ha}}^{*}$ of $1.74\times {10}^{-3}$ (a), $1.38\times {10}^{-3}$ (b) and $1.08\times {10}^{-3}$ (c). Bottom: Radial distribution function $\left|g\right(r)-1|$ (left) and bond angular order distribution function $g_6\left(r\right)$ (right) at $n{a}_{\mathrm{B}}^{*2}=0.0022$ for $N=500$. Lines are a guide to the eye to visualize an algebraic decay in this log-log plot.

Figure 3

Temperature dependence of the defect fraction at density $n{a}_{\mathrm{B}}^{*2}=2.2\times {10}^{-3}$ and particle numbers $N=501-505$ (vertically aligned dots) characterizing the ordered and disordered phases. The defect fraction ($1-P_6$) shows a sharp jump at $T\sim {10}^{-3}$Ha${}^{*}$. This is in disagreement with the KTNHY theory, which predicts a continuous unbinding of dislocations in the hexatic phase indicated by a continuous variation of the critical exponent ${\eta }_6\left(T\right)⩽1/4$ and the bond angular correlation function $g_6\left(r\right)\sim r^{-{\eta }_6\left(T\right)}$. In contrast, we observe an abrupt transition from the LR angular order ($T<T_c$) to a quasi-long-range angular order with ${\eta }_6\left(T\right)\sim 2$ ($T⩾T_c$); see Fig. 2.

Figure 4

Isothermal freezing and melting of indirect excitons. (a)–(f) 2D PDF $g\left(\mathbf{r}\right)$ for $k_{\mathrm{B}}{T}_1=0.001{\mathrm{Ha}}^{*}$ at densities $n{a}_{\mathrm{B}}^{*2}$ of $0.84\times {10}^{-3}$ (a), $1.3\times {10}^{-3}$ (b), $1.7\times {10}^{-3}$ (c), $3.2\times {10}^{-3}$ (d), $3.6\times {10}^{-3}$ (e), and $4.0\times {10}^{-3}$ (f). Bottom panel: Superfluid fraction ${\gamma }_{\mathrm{s}}$, Eq. (25), vs density for two temperatures. Symbols are PIMC results, lines are a guide to the eye. The increase of ${\gamma }_s$ at high density extends over a small finite range of solid-liquid coexistence, which is due to the finite particle number in the simulations.

Figure 5

(a) Temperature dependence of the winding number $\langle W^2\rangle \left(T\right)$ for the exciton numbers $N=56$ and 170. Density $n{a}_{\mathrm{B}}^{*2}=5\times {10}^{-3}$. The Berezinskii-Kosterlitz-Thouless transition temperature $T_{\mathrm{KT}}\left(N\right)$ is determined by the condition (Refs.18 and35) $\langle W^2\rangle \left(T_{\mathrm{KT}}\right)=4/\pi$, shown by the horizontal dashed line. (b) System size dependence of $T_{\mathrm{KT}}\left(L\right)$ for three densities: $n{a}_{\mathrm{B}}^{*2}=5\times {10}^{-3},{10}^{-2}$, and $2\times {10}^{-2}$. Values of $T_{\mathrm{KT}}$ are rescaled to fit into a single plot.

Figure 6

Phase diagram of 2D indirect excitons with $d=13.3a_{\mathrm{B}}^{*}$. Circles and squares mark our PIMC results, data for triangles are from Ref. 18. Vertical dashed lines ($D=17\pm 1$ and $r_{\mathrm{s}}=9.4\pm 0.3$) indicate the two density induced quantum freezing (melting) transitions. Filled symbols mark the two triple points. The normal-fluid–superfluid phase boundary is marked by the full (red) line with triangles and squares, respectively, and is below the ideal estimate $T_{\mathrm{KT}}$ according to Eq. (24); cf. thick solid line labeled $\chi =4$. The line $T_{\mathrm{dip}}$ marks the freezing transition of a classical 2D dipole system. The e-h plasma phase is beyond the present analysis.

Figure 7

Boundaries of the exciton crystal for different dipole moments $d$. Lower abscissa: density range given by $D^{\mathrm{c}}=17$ and $r_{\mathrm{s}}^{\mathrm{c}}=9.4$. Upper abscissa: maximum temperature estimated from $T_{\mathrm{dip}}^{\max }$. No solid phase exists for $d⩽d_{\mathrm{c}}\approx 9.1a_{\mathrm{B}}^{*}$.