Specular Andreev reflection in thin films of topological insulators

We theoretically reveal the possibility of specular Andreev reflection in a thin film topological insulator normal-superconductor (N/S) junction in the presence of a gate electric field. The probability of specular Andreev reflection increases with the electric field, and electron-hole conversion with unit efficiency happens in a wide experimentally accessible range of the electric field. We show that perfect specular Andreev reflection can occur for all angles of incidence with a particular excitation energy value. In addition, we find that the thermal conductance of the structure displays exponential dependence on the temperature. Our results reveal the potential of the proposed topological insulator thin-film-based N/S structure for the realization of intraband specular Andreev reflection.

Figure 1

(a) Schematic illustration of the TI thin film N/S junction in the presence of a perpendicular electric field $\mathit{E}$, which can be realized by pairs of gate electrodes ($U_{\mathrm{top}}=U$, $U_{\mathrm{bottom}}=-U$). (b) The dispersion relation along the $x$ axis in momentum space [$\varepsilon =\varepsilon \left(k_x\right)$] of N (left panel) and S (right panel) regions, when $\omega =0.05$ eV, $U=0.2$ eV, ${\mu }_N={\mu }_c$ (${\mu }_c=\sqrt{U^2+{\omega }^2}$), ${\mu }_S=1$ eV, and ${\mathrm{\Delta }}_S=0.5$ eV. An incident electron from the $A$ ($A^{\prime }$) point [“$-$” (“$+$”) branch of the spectrum] with excitation energy $\varepsilon >0$ can be normally reflected as an electron from $B$ (“$-$” branch) and $C$ (“$+$” branch) points and Andreev reflected as a hole from $D$ (“$-$” branch) and $F$ (“$-$” branch) points, with excitation energy $-\varepsilon$.

Figure 2

The normal and Andreev reflection probabilities for an incident electron from (a) $A$ and (b) $A^{\prime }$ points versus the angle of incidence, when there is no coupling between the top and bottom surfaces of the TI thin film ($\omega =0$), $U=0.05$ eV, and $\varepsilon /{\mathrm{\Delta }}_S=0.5$. The superscript $e,B\left(C\right)$ denotes the normal reflection of the electron from the $B$ ($C$) point, and $h,D\left(F\right)$ denotes the retro (specular) AR from the $D$ ($F$) point. The inset of (a) shows the dispersion relation (along the $x$ axis in momentum space) of the uncoupled top (red lines) and bottom (blue lines) surfaces of the N region. An incident electron from $A$ ($A^{\prime }$) point [bottom (top) surface] can be normally reflected as an electron from the $B$ ($C$) point and Andreev reflected as a hole from the $D$ ($F$) point.

Figure 3

Probabilities of the normal and Andreev reflection processes for four possible scattering modes of an incident electron from (a) $A$ and (b) $A^{\prime }$ points to the TI thin-film-based N/S interface versus the angle of incidence, when $\omega =0.05$ eV, $U=\omega$, and $\varepsilon /{\mathrm{\Delta }}_S=0.5$. The inset of (a) shows the normal and Andreev reflection probabilities for the incident electron from the $A$ point, when $\omega =0.2$ eV.

Figure 4

Dependence of the specular AR probability for a near-normally incident electron (with ${\alpha }_{A^{\prime }}=\pi /90$) from $A^{\prime }$ point on (a) the top gate potential $U$, when the coupling parameter of the two surface states is $\omega =0.05$ eV, and (b) the coupling parameter $\omega$, when $U=0.2$ eV.

Figure 5

Probability of specular AR versus the angle of incidence for an incident electron from the $A^{\prime }$ point with different values of the excitation energy, when $U=0.4$ eV and $\omega =0.05$ eV. Of particular interest is the perfect specular AR for all angles of incidence to the N/S interface with $\varepsilon ={\mathrm{\Delta }}_S$, when $U$ is large.

Figure 6

Retro and specular Andreev conductances, respectively, for an incident electron from $A$ and $A^{\prime }$ points to the N/S junction versus the bias voltage $eV/{\mathrm{\Delta }}_S$ (in units of the superconducting gap ${\mathrm{\Delta }}_S$), when $U=0.2$ eV and $\omega =0.05$ eV.

Figure 7

Andreev conductance of the N/S junction as a function of the bias voltage $eV/{\mathrm{\Delta }}_S$ for different values of $U$ and $\omega$.

Figure 8

The energy dependence of the normal (left panel) and Andreev (right panel) reflection probabilities for different angles of incidence of the electron to the N/S interface from $A$ [(a) and (b)] and $A^{\prime }$ [(c) and (d)] points, when $U=0.1$ eV and $\omega =0.05$ eV.

Figure 9

Thermal conductance of the N/S junction as a function of the temperature $T/T_C$ (in units of the critical temperature of the superconducting order parameter ${\mathrm{\Delta }}_S$) for different values of $U$ and $\omega$.