Electrically controllable spin filtering based on superconducting helical states

The magnetoelectric effects in the surface states of the three-dimensional (3D) topological insulator (TI) are extremely strong due to the full spin-momentum locking. Here the microscopic theory of S/3D TI bilayer structures in terms of quasiclassical Green's functions is developed. On the basis of the developed formalism it is shown that the DOS in the S/TI bilayer manifests giant magnetoelectric behavior and, as a result, S/3D TI heterostructures can work as nonmagnetic fully electrically controllable spin filters. It is shown that due to the full spin-momentum locking the amplitudes of the odd-frequency singlet and triplet components of the condensate wave function are equal. The same is valid for the even frequency singlet and triplet components. We unveil the connection between the odd-frequency pairing in S/3D TI heterostructures and magnetoelectric effects in the DOS.

Figure 1

(a) Fermi surface of the TI surface states. The quasiparticle spin $\mathit{S}$ is locked at the right angle to its momentum $\mathit{p}$. (b) Sketch of the system under consideration. The thick top superconductor S induces the superconductivity in the surface layer of the 3D topological insulator TI.

Figure 2

DOS for the ballistic system. (a) Momentum-resolved DOS. The circle means the Fermi surface and the energy axis is directed out of this Fermi surface and normally to it at each momentum direction. (b) Momentum-averaged DOS as a function of the quasiparticle energy. $H_{\text{eff}}/\mathrm{\Delta }=0$ (black line), 0.3 (red), 0.6 (blue), 0.9 (green), and 1.2 (pink). (c) Spin-resolved DOS as a function of the quasiparticle energy. $H_{\text{eff}}/\mathrm{\Delta }=0.6$ (blue), 0.9 (green), and 1.2 (pink). The quantization axis is along the $y$ axis, while ${\mathit{j}}_s=j_s\widehat{x}$. The spin-up DOS $N_{\uparrow }$ is plotted by solid lines, and the spin-down DOS $N_{\downarrow }$ by dashed lines.

Figure 3

DOS for the diffusive system. $\mathrm{\Delta }\tau =0.15$. (a) DOS as a function of the quasiparticle energy. $H_{\text{eff}}/\mathrm{\Delta }=0$ (black line), 0.6 (red), 1.2 (green), 1.8 (blue), 2.4 (tan), and 3.0 (pink). (b) Spin-resolved DOS as a function of the quasiparticle energy. $H_{\text{eff}}/\mathrm{\Delta }=0.6$ (red), 1.8 (blue), and 3.0 (pink). The spin-up DOS $N_{\uparrow }$ is plotted by solid lines, and the spin-down DOS $N_{\downarrow }$ by dashed lines.

Figure 4

Sketch of the possible setup for the realization of the electrically controllable spin filter. The spin polarized current flows though the TI/N interface in response to the voltage bias $V$ applied between the normal electrode N and the S/TI bilayer.

Figure 5

Ballistic system. (a)–(c) $I_{\uparrow }$ (solid lines) and $I_{\downarrow }$ (dashed lines) as functions of $eV/\mathrm{\Delta }$ for $H_{\text{eff}}/\mathrm{\Delta }=0.5$ (a), 1.0 (b), and 1.5 (c). (d) Degree of the current polarization for all $H_{\text{eff}}$, considered in (a)–(c). Different colors correspond to the same $H_{\text{eff}}$, as in (a)–(c).

Figure 6

Diffusive system. $\mathrm{\Delta }\tau =0.15$. (a)–(c) $I_{\uparrow }$ (solid lines) and $I_{\downarrow }$ (dashed lines) as functions of $eV/\mathrm{\Delta }$ for $H_{\text{eff}}/\mathrm{\Delta }=0.6$ (a), 1.8 (b), and 3.0 (c). (d) Degree of the current polarization for all $H_{\text{eff}}$, considered in (a)–(c).

Figure 7

Degree of the current polarization as a function of $eV/\mathrm{\Delta }$. Different curves correspond to different values of impurity strength $\mathrm{\Delta }\tau =0.03$ (tan line), 0.06 (pink), 0.09 (green), 0.12 (red), 0.15 (blue), and 0.18 (black). $H_{\text{eff}}=1.0\mathrm{\Delta }$.

Figure 8

(a) DOS calculated in the framework of the microscopic model. Different curves correspond to different values of the supercurrent with $H_{\text{eff}}=0$ (black), 0.1 (red), 0.2 (blue), 0.3 (pink), and 0.4 (green) in units of $\mathrm{\Delta }$, the true exchange field $h=0$. (b) DOS at $j_S\ne 0$ (corresponding to $H_{\text{eff}}=0.1\mathrm{\Delta }$) and $h=0$ (blue line) in comparison to DOS at $j_S=0$ and $h=0.1\mathrm{\Delta }$ (red line). $t^2/\left(2\right|v_{F,z}\left|\right)=0.2\mathrm{\Delta }$ for both panels.

Figure 9

Spin-filtering properties, calculated in the framework of the microscopic model. $t^2/\left(2\right|v_{F,z}\left|\right)=0.2\mathrm{\Delta }$. Supercurrent-driven regime. (a) $I_{\uparrow }$ (solid line) and $I_{\downarrow }$ (dashed line) as functions of $V/\mathrm{\Delta }$ for $H_{\text{eff}}=0.1\mathrm{\Delta }$. (b) Degree of the current polarization for the same $H_{\text{eff}}$.

Figure 10

Results, calculated in the framework of the microscopic model, high-transparency case $t^2/\left(2\right|v_{F,z}\left|\right)=2.0\mathrm{\Delta }$. (a) DOS for $H_{\text{eff}}=0$ (black line), 0.15 (red), 0.3 (blue), 0.45 (pink), and 0.6 (green). (b) Spin-resolved DOS for the same $H_{\text{eff}}$. As before, the spin-up DOS is plotted by solid lines and the spin-down DOS by dashed lines. (c) $I_{\uparrow }$ (solid lines) and $I_{\downarrow }$ (dashed lines) as functions of $V/\mathrm{\Delta }$ for the same $H_{\text{eff}}$. (d) Degree of the current polarization for the same $H_{\text{eff}}$.

Figure 11

(a) Averaged over momentum directions $\langle \mathrm{Re}{f}_s^R\rangle$ as a function of $\varepsilon$. (b) $\langle \mathrm{Im}{f}_s^R\rangle$. For the both panels $\varepsilon$ is normalized to the gap of the top superconductor $\mathrm{\Delta }$. Different curves correspond to different $H_{\text{eff}}=0$ (black), $0.1\mathrm{\Delta }$ (red), 0.2 (green), 0.3 (pink), 0.5 (tan), and 0.75 (green). Microscopic model with $t^2/\left(2\right|v_{F,z}\left|\right)=0.2\mathrm{\Delta }$.

Figure 12

(a) Averaged over momentum directions $-\langle n_{F,x}\mathrm{Re}{f}_a^R\rangle$ as a function of $\varepsilon$. (b) $-\langle n_{F,x}\mathrm{Im}{f}_a^R\rangle$. The colors marking $H_{\text{eff}}$ and the other parameters are the same as in Fig. 11.