Structure of exciton condensates in imbalanced electron-hole bilayers

We investigate the possibility of excitonic superfluidity in electron-hole bilayers. We calculate the phase diagram of the system for the whole range of electron-hole density imbalance and for different degrees of electrostatic screening, using mean-field theory and a Ginzburg–Landau expansion. We are able to resolve differences on previous work in the literature which concentrated on restricted regions of the parameter space. We also give detailed descriptions of the pairing wave function in the Fulde–Ferrell–Larkin–Ovchinnikov paired state. The Ginzburg–Landau treatment allows us to investigate the energy scales involved in the pairing state and discuss the possible spontaneous breaking of two-dimensional translation symmetry in the ground state.

Figure 1

Schematic diagram of potentials of the electron-hole bilayer. The upper layer of holes has chemical potential ${\mu }_h=\mu +h$ while the lower layer of electrons has ${\mu }_e=\mu -h$.

Figure 2

Ground states of the electron-hole bilayer as a function of chemical potential $\mu$ and bias $h$ at layer separation $d=a_0$. ZP: no particles, FP: fully polarized normal state with only electrons or only holes, PP: partially polarized normal state, SF: zero-momentum $\mathbf{Q}=0$ superfluid (SF), FFLO: exciton condensation at nonzero momenta ($\mathbf{Q}\ne 0$). Dotted line represents the SF-FFLO phase boundary (first-order transition). Dashed line represents FFLO-normal phase boundary (continuous transition). Left panel is for $\alpha =m_h/m_e=1$ with screening wave vectors $k_{\text{TF}}{a}_0=0.1$ and 0.5. There is a direct transition between FFLO phase and the fully polarized phase. Right panel is for $\alpha =4$ with $k_{\text{TF}}{a}_0=0.1$ and 1.0. The SF phase is centered around the line for which the normal state would have equal electron and hole populations. At small $k_{\text{TF}}{a}_0$, the FFLO phase extends all the way to the fully polarized (FP) limit. At higher $k_{\text{TF}}{a}_0$, a small region of PP normal state intervenes between the FFLO and FP states.

Figure 3

Continuous transition between normal and FFLO state at $h=0.5$ (left) and $h=-0.5$ (right). Top panels show maximum value of the gap function ${\mathrm{\Delta }}_{\max }$ as a function of bias $h$ at $\mu =0.5$ and $k_{\text{TF}}{a}_0=0.1$ ($m_h/m_e=4$). Bottom panels show energy difference $\delta F$ between the SF or FF phase and the normal state.

Figure 4

Free-energy difference $\delta F$ between FF state with wave vector $Q$ and normal state (of energy $F_N$) at $\mu =0.5$ at different values of the bias $h$. $m_h/m_e=1,k_{\text{TF}}{a}_0=0.1$. The SF-FF transition occurs at $h=0.085$.

Figure 5

The ordering wave vectors (points) of the FF state at the two FFLO-normal phase boundaries ($\alpha =m_h/m_e=4$) as a function of chemical potential. $k_{\text{TF}}{a}_0=0.1$ (left), 1 (right). Solid line shows naive estimate $|k_{\text{F}h}-k_{\text{F}e}|a_0$.

Figure 6

Gap function ${\mathrm{\Delta }}_{\mathbf{k}\mathbf{Q}}$ at chemical potential $\mu =0.5$ for $m_h/m_e=4,k_{\text{TF}}{a}_0=0.1$. Top left panel shows the uniform ($Q=0$) SF phase. Top right and bottom left and right panels show FF states. Dashed lines represent the normal-state Fermi surfaces shifted by $\pm \mathbf{Q}/2$ to touch at energetically degenerate points. Hole Fermi surface (red, smaller): ${\xi }_{-\mathbf{k}+\frac{\mathbf{Q}}{2},h}=0$. Electron Fermi surface (black, larger): ${\xi }_{\mathbf{k}+\frac{\mathbf{Q}}{2},e}=0$.

Figure 7

Ground states of the electron-hole bilayer as a function of electron separation $r_s$ and fractional density imbalance $(n_h-n_e)/(n_h+n_e)$ at layer separation $d=a_0$. SF shows where zero-momentum $\mathbf{Q}=0$ superfluid is found when there are equal numbers of electrons and holes. FFLO labels region of exciton condensation at nonzero momenta ($\mathbf{Q}\ne 0$). PS labels region of phase separation. PP labels region of partially polarized normal state. Left panels are for $\alpha =m_h/m_e=1$ with screening wave vectors $k_{\text{TF}}{a}_0=0.1$ and 0.5. Right panels are for $\alpha =4$ with $k_{\text{TF}}{a}_0=0.1$ and 1.0. Parts of the FFLO-PP boundary are extrapolated (dashed line) because this corresponds to very small regions in $\mu -h$ space and we do not have the numerical resolution to determine the boundary precisely.

Figure 8

The free-energy difference per unit area between the FFLO and normal states near the FFLO-normal phase boundaries (upper panel is for majority holes, lower panel is for majority electrons) over a range of chemical potentials $\mu$. $\alpha =m_h/m_e=4,k_{\text{TF}}{a}_0=0.1$.