Electron-hole pairing in a topological insulator thin film

We consider the pairing of massless Dirac electrons and holes located on opposite surfaces of thin films of “strong” three-dimensional topological insulators. Such pairing is predicted to give rise to a topological exciton condensate with unusual properties. We estimate the quantitatively achievable critical temperature of the pairing while taking into account the self-consistent screening of the Coulomb interaction, disorder, and hybridization of electron and hole states caused by tunneling through the film. The increase of the gap above the hybridization value when the temperature is lowered can be an observable signature of the pairing. System parameters required to observe the electron-hole pairing are discussed.

Figure 1

TI film of thickness $d$ and dielectric permittivity ${\varepsilon }_2$ surrounded by trivial insulators with permittivities ${\varepsilon }_1$ and ${\varepsilon }_3$. Electric potentials $\pm V$ on opposite surfaces of the film create electron and hole gases of equal concentrations.

Figure 2

Band picture of the pairing. (a) The applied potential difference shifts energies of Dirac cones on opposite surfaces in opposite directions in such a way that electron and hole gases with chemical potentials $\mu$ and $-\mu$ are formed. (b) The pairing opens a gap $2\Delta$ on the Fermi level.

Figure 3

The coupling constant of Eq. (8) for the pairing in a ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ film with ${\varepsilon }_2=80$ as a function of $p_{F}d$ at ${\varepsilon }_1={\varepsilon }_3=1$ (solid line), at ${\varepsilon }_1={\varepsilon }_3=4$ (dashed line), and for the TI-vacuum-TI system with ${\varepsilon }_1={\varepsilon }_3=80$ and ${\varepsilon }_2=1$ (dotted line).

Figure 4

The BCS critical temperature of the pairing of Eq. (7) in a suspended ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ film as a function of the chemical potential $\mu$ at different film thicknesses $d$, indicated near the corresponding curves.

Figure 5

Conceptual example of the behavior of the gap $\Delta$ (solid line) as a function of temperature $T$ in a system with hybridization at $\mu =0.1$ eV, $\lambda =0.15$, and ${\Delta }_T=0.3$ K. The initial hybridization gap ${\Delta }_T$ (dashed line) and the BCS gap calculated without hybridization (dotted line) are shown for comparison.

Figure 6

Characteristics of the pairing in suspended ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ films with hybridization: the increase ${\Delta }_0-{\Delta }_T$ of the total gap (solid line) above the hybridization gap ${\Delta }_T$ (long-dashed line), the characteristic temperature range $T_{\mathrm{char}}$ (dotted line) at which the gap increases, and the BCS gap ${\Delta }_0^{\mathrm{BCS}}$ (short-dashed line) calculated without hybridization. The data on hybridization are taken from Ref. 25 for (a) $d=2$ QLs and (b) $d=5$ QLs and from Ref. 29 for (c) $d=5$ QLs and (d) $d=8$ QLs.

Figure 7

The increase ${\Delta }_0-{\Delta }_T$ of the zero-temperature total gap above the hybridization gap with no disorder (solid line), at ${\mu }_c={10}^4$ cm${}^2$/V s (short-dashed line), and at ${\mu }_c=500$ cm${}^2$/V s (dotted line). The hybridization gap (long-dashed line) in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ films is taken from Ref. 25 for (a) $d=2$ QLs and (b) $d=5$ QLs and from Ref. 29 for (c) $d=5$ QLs and (d) $d=8$ QLs.

Figure 8

Phase diagram of the pairing (without hybridization) in a ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ film at different dielectric constants of the surrounding medium ${\varepsilon }_1={\varepsilon }_3$ and different dimensionless thicknesses $p_{F}d$. The regions when the gap equation at $T=0$ gives one small gap, three gaps, and one large gap are shown. Inset: Typical behavior of solutions $\Delta$ of the gap equation at ${\varepsilon }_1,{\varepsilon }_3<4.5$.

Figure 9

The increase ${\Delta }_0-{\Delta }_T$ of the zero-temperature total gap in suspended ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ films above the hybridization gap with (solid line) and without (short-dashed line) taking into account self-consistent suppression of the screening. The hybridization gap (long-dashed line) in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ films is taken from Ref. 25 for (a) $d=2$ QLs and (b) $d=5$ QLs and from Ref. 29 for (c) $d=5$ QLs and (d) $d=8$ QLs.