Probing the Topological Exciton Condensate via Coulomb Drag

The onset of exciton condensation in a topological insulator thin film was recently predicted. We calculate the critical temperature for this transition, taking into account screening effects. Furthermore, we show that the proximity to this transition can be probed by measuring the Coulomb drag resistivity between the surfaces of the thin film as a function of temperature. This resistivity shows an upturn upon approaching the exciton-condensed state.

Figure 1

The drag resistivity ${\rho }_{\mathrm{D}}$ [in units of ${\overline{\rho }}_{\mathrm{D}}=({\nu }_{0}U{)}^2\hslash /4e^2$, with ${\nu }_0$ the density of states at the Fermi level and $U$ the interlayer interaction strength] calculated from Eq. (3) as a function of the reduced temperature $T/T_c$. With this choice of units ${\rho }_{\mathrm{D}}/{\overline{\rho }}_{\mathrm{D}}$ depends on two parameters—the chemical potential $\mu$ and the critical temperature $T_c$—which implicitly determine the interaction strength $U$ in Eq. (1). The curves correspond to the case of equal chemical potentials in the two layers ($\mu =0.04\mathrm{eV}$). From top to bottom the critical temperatures are $T_c/T_{\mathrm{F}}=(1/150,1/200,1/250)$. The vertical dashed lines are located at $T=T_c$ (left) and at the “upturn” temperature scale (see text) $T=T_u$ (right).

Figure 2

The Bethe-Salpeter equation for the effective interaction $V_{\mathrm{eff}}$ in Eq. (1). The thin wavy line is $V_0$ and the arrows are the electron propagators in the top layer (top arrows) and the bottom layer (bottom arrows). Furthermore, $\mathit{K}={\mathit{k}}_1-{\mathit{k}}_3={\mathit{k}}_4-{\mathit{k}}_2$ is the center-of-mass momentum of the electron-hole pair and $\mathit{q}={\mathit{k}}_1-{\mathit{k}}_4={\mathit{k}}_3-{\mathit{k}}_2$ is the interlayer momentum transfer, which is responsible for Coulomb drag. Finally, $\mathit{p}$ is an integration variable. To make contact with the drag resistivity in Eq. (3) replace $\mathit{k}={\mathit{k}}_1$ and ${\mathit{k}}^{\prime }={\mathit{k}}_2$.

Figure 3

Critical temperature $T_c$ (in $\mathrm{K}$) versus interlayer distance $d$ (in nm) in the absence of density imbalance ($h=0$). From right to left, the lines correspond to carrier densities $n=(0.25,0.5,1)\times {10}^{10}{\mathrm{cm}}^{-2}$. Inset: The critical lines $T_c/\mu$ versus $h/\mu$. From top to bottom, the lines correspond to $k_{\mathrm{F}}d=(0.05,0.075,0.1)$. The critical lines terminate on the dashed line, i.e., the locus of the points where the fourth order term of the Ginzburg-Landau expansion of the free energy in powers of the order parameter vanishes.