Quadriexcitons and excitonic condensate in a symmetric electron-hole bilayer with valley degeneracy

Using quantum Monte Carlo simulations we have mapped out the zero-temperature phase diagram of a symmetric electron-hole bilayer with twofold valley degeneracy, as a function of the interlayer distance $d$ and in-layer density $n$. We find that the effect of the valley degeneracy is to shrink the region of stability of the excitonic condensate, in favor of quadriexcitons at small $d$ and of the four-component plasma at large $d$, with minor effects on the value of the excitonic condensate fraction. The enclosure of the condensate in a density window possibly explains why anomalous tunneling conductivity, interpreted as a signature of condensation, is observed only between two finite values of carrier density in graphene bilayers. Our phase diagram may provide directions to select device parameters for future experiments.

Figure 1

(a) Phase diagram of an electron-hole bilayer with twofold valley degeneracy for small to moderate values of the intralayer coupling $r_s$ and interlayer distance $d$. The red dots indicate states that are studied in Figs. 2 and 3. The regions of stability of the competing (fluid) phases are displayed in different colors. (b) Comparison of the phase diagrams of electron-hole bilayers with one ($1v$) and two ($2v$) valleys. The region of stability of the excitonic phase in the one-valley system [25, 38] (union of the gray and blue areas) is substantially reduced in going to the two-valley system (blue area).

Figure 2

PCFs (solid lines) between electrons (holes) with different flavors (${\sigma }^{\prime }\ne \sigma$) in a symmetric electron-hole bilayer with twofold valley degeneracy, at $r_s=4$; interlayer distances $d$ correspond to various phases (see Fig. 1). The inset reports the RCNs defined in Eq. (6). Electron-hole PCFs and RCNs are also shown for the two smaller distances (dashed lines), where they turn out to be independent of flavors, which have therefore been dropped.

Figure 3

Electron-hole PCFs in a symmetric electron-hole bilayer with twofold valley degeneracy, at $r_s=4$ and interlayer distances $d$ corresponding to various phases (see Fig. 1). The PCFs of quadriexcitons and plasma are independent of flavors $\sigma ,{\sigma }^{\prime }$.

Figure 4

Condensate fraction vs interlayer distance $d$ in a symmetric electron-hole bilayer, at $r_s=5$: (a) Present results for $g_v=2$ (red dots), with vertical dashed lines providing boundaries between phases; (b) $g_v=1$ [25] (blue circles), with the dashed blue line a guide to the eyes; (c) BCS mean field [10] predictions (solid curves).