Theory of spin-selective Andreev reflection in the vortex core of a topological superconductor

Majorana zero modes (MZMs) have been predicted to exist in a topological insulator (TI)/superconductor (SC) heterostructure. A recent spin-polarized scanning tunneling microscope (STM) experiment [Sun et al., Phys. Rev. Lett. 116, 257003 (2016)] has observed a spin-polarization dependence of the zero bias differential tunneling conductance at the center of a vortex core. Here, we consider a helical electron system described by a Rashba spin-orbit coupling Hamiltonian on a spherical surface with an $s$-wave superconducting pairing due to proximity effect. We examine the in-gap excitations of a pair of vortices with one at the north pole and the other at the south pole. While the MZM is not a spin eigenstate, the spin wave function of the MZM at the center of the vortex core, $r=0$, is parallel to the magnetic field, and the local Andreev reflection of the MZM is spin selective, namely, occurs only when the STM tip has the spin polarization parallel to the magnetic field, similar to the case in a one-dimensional nanowire [He et al., Phys. Rev. Lett. 112, 037001 (2014)]. The total local differential tunneling conductance consists of the normal term proportional to the local density of states and an additional term arising from the Andreev reflection. We also discuss the finite size effect, for which the MZM at the north pole is hybridized with the MZM at the south pole. We apply our theory to examine the recently reported spin-polarized STM experiments and show good agreement with the experiments.

Figure 1

Sphere geometry for the vortex problem. There exists a pair of vortex and antivortex. The vortex is located at the north pole and the antivortex is located at the south pole.

Figure 2

Energy spectra for the vortex problem. $E_m$ is plotted as a function of angular momentum $m$ [quantum number of $K_z$ in Eq. (15)]. Red hollow $\text{circular}(\circ )$ represents localized in-gap bound states, and black solid $\text{circular}(•)$ represents bulk states. The energy discretization for bulk states is due to finite size effect. Here we choose parameters as ${\mathrm{\Delta }}_0=1$ meV, ${\xi }_0=35$ nm, $R=50{\xi }_0,\alpha =30$ meV, and $\mu =90$ meV.

Figure 3

Wave function for Majorana zero mode $E_0^{-}\approx -{10}^{-4}$ meV, where $u_1\left(\theta \right)=-u_1(\pi -\theta ),u_2\left(\theta \right)=u_2(\pi -\theta ),v_1\left(\theta \right)=-v_1(\pi -\theta )$, and $v_2\left(\theta \right)=v_2(\pi -\theta )$.

Figure 4

Wave function for the first excited state $E_1^{-}\approx -0.05$ meV, where $u_1\left(\theta \right)=-u_1(\pi -\theta ),u_2\left(\theta \right)=u_2(\pi -\theta ),v_1\left(\theta \right)=-v_1(\pi -\theta )$, and $v_2\left(\theta \right)=v_2(\pi -\theta )$.

Figure 5

Wave function for the second (a), third (b), fourth (c), and fifth (d) excited states.

Figure 6

Two-vortex hybridization effect. (a) The energy of the $m=0$ state (MZM) for different sphere radii $R$. (b) The wave function of MZM for a small size $R=13{\xi }_0$.

Figure 7

(a) The normal metal (lead)/helical metal/helical metal (lead) junction used to estimate the normal conductance in STM/STS experiments. (b) Calculated normal conductance, ${\sigma }_n(\overline{E}\gg {\mathrm{\Delta }}_0)\approx 0.88e^2/h$, where we choose the hopping integral in the normal lead $t^{\prime }=24$ and the coupling between the helical metal and left and right leads as $t_L=t^{\prime }$ and $t_R=0.85t^{\prime }$.

Figure 8

Local density of states. (a) The smearing factor defined in Eq. (26) $\eta =0.005{\mathrm{\Delta }}_0$. (b) $\eta =0.04{\mathrm{\Delta }}_0\sim E_1$, comparable to the first excited state energy. (c) $\eta =0.3\mathrm{\Delta }\approx 6E_1$. The LDOS for spin-up ${\mathcal{N}}_{\uparrow }\left(E\right)$ and spin-down ${\mathcal{N}}_{\downarrow }\left(E\right)$ coincide to each other.

Figure 9

Illustration of spin-selective Andreev reflection. An incoming spin-down electron will be reflected as a spin-down electron because of the mismatch of the spins of the incoming electron and the MZM.

Figure 10

Calculated AR conductance $dI/dV$ at vortex core center $r=0$. (a) $R=50{\xi }_0$, the incoming electron spin polarization is parallel to local MZM spin. (b) $R=13{\xi }_0$ the incoming electron spin polarization is parallel to local non-MZM spin. (c) $R=50{\xi }_0$, the incoming electron spin polarization is antiparallel to local MZM spin. (d) $R=13{\xi }_0$, the incoming electron spin polarization is antiparallel to the local non-MZM spin.

Figure 11

(a) Calculated total conductance with $\eta =0.04{\mathrm{\Delta }}_0$. (b) Calculated total conductance with $\eta =0.3{\mathrm{\Delta }}_0$. (c) Experiment data from Ref. [30].

Figure 12

The angle between the local MZM spin and external magnetic field ${\theta }_M$ and the amplitude of MZM wave function $\sqrt{|u_1{\left(r\right)|}^2+{\left|u_2\left(r\right)\right|}^2}$. ${\theta }_M=0$ is for parallel and ${\theta }_M=\pi$ is for antiparallel.

Figure 13

Energy spectra for the vortex-free case. Only $E_m$ for non-negative angular momentum are calculated numerically and plotted here, with used parameters ${\mathrm{\Delta }}_0=1$ meV, $\eta =20$ meV, ${\xi }_0=35$ nm, $R=50{\xi }_0,\alpha =0.1$ meV, and $\mu =32$ meV.

Figure 14

Ordinary Andreev reflection calculated using $T_A=\text{Tr}\left[{\tilde{r}}_{he}^{†}{\tilde{r}}_{he}\right]$ to reproduce the BTK theory (two channels/particles scattering process). Increase the coupling $t_c$ from $0.4{\mathrm{\Delta }}_0$ (high barrier case, large $Z)$ in (a) to $t_c=1.38{\mathrm{\Delta }}_0$ (transparent limit, tiny $Z)$ in (h).