Size constraints on a Majorana beam-splitter interferometer: Majorana coupling and surface-bulk scattering

Topological insulator surfaces in proximity to superconductors have been proposed as a way to produce Majorana fermions in condensed matter physics. One of the simplest proposed experiments with such a system is Majorana interferometry. Here we consider two possibly conflicting constraints on the size of such an interferometer. Coupling of a Majorana mode from the edge (the arms) of the interferometer to vortices in the center of the device sets a lower bound on the size of the device. On the other hand, scattering to the usually imperfectly insulating bulk sets an upper bound. From estimates of experimental parameters, we find that typical samples may have no size window in which the Majorana interferometer can operate, implying that a new generation of more highly insulating samples must be explored.

Figure 1

The Majorana beam-splitter interferometer proposed by Fu and Kane, and Akhmerov et al. [17, 18]. It consists of a 3D strong TI in proximity to a superconductor and magnets of opposite polarization.

Figure 2

Top view of the interferometry device with a superconductor (SC) and magnetic domains (with magnetization $M_{\uparrow /\downarrow }$) that probes charge transport from the source (S) to the drain (D) through a pair of chiral Majorana modes. (a) Single-point coupling of strength $\lambda$ at coordinate $x=0$. Here ${\xi }_{1R}\equiv {\xi }_1(x=0^{+})$ and ${\xi }_{1L}\equiv {\xi }_1(x=0^{-})$. (b) The interferometer coupled to a single-mode conducting lead representing leakage to the bulk of the TI.

Figure 3

The differential conductance from Eq. (13) as a function of the voltage. Here $\nu =1$ $\mu \mathrm{eV}$ and $\hslash v_m/\delta L=2$ $\mu \mathrm{eV}$. The dotted lines show the conductance with $\nu =0$. At low voltage the even-odd effect is reversed.

Figure 4

(a) The three lowest-lying eigenstates in a TI film of thickness $L=40$ nm. The surface state localized on the TI top is displayed in purple; the bottom surface state is not shown. Inset: A sketch of the typical transverse surface Majorana wave function. Its effect on the scattering rate is discussed in Appendix pp4-s1. The model parameters are adjusted to those of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ except for the gap, which is here set to ${\mathrm{\Delta }}_{\mathrm{TI}}=0.1$ eV to model puddles from the conduction band [44, 45]. (b) The dispersion of the states displayed in (a) but with two additional bulk states. In black: The typical Majorana surface dispersion. (c) The differential scattering rate $d{\sigma }_{\mathrm{long}}^{(n^{\prime }=1)}\left({\epsilon }_F\right)/d{k}_z^{\prime }$ defined in Eq. (D5) shown for three values of the Fermi energy (measured in eV). (d) The electronic scattering rate as given in Eq. (23) with $n_{3\mathrm{D}}={10}^{19}{\mathrm{cm}}^{-3}$. Van Hove singularities are seen whenever the Fermi energy hits the bulk bands, but they are softened by the $k_{\parallel }^{\prime }$ factor that goes to zero at these points. Here we used ${\epsilon }_r=100$ [46].

Figure 5

The scattering states at the two contact points.

Figure 6

The reduced drain current $R$ as function of $\beta \nu$ for symmetric arms $\delta L=0$. The dotted lines represent the weak coupling result from Eq. (B13) and the full lines are numerical results for $\beta eV$ being 5 (purple), 10 (orange), and 25 (green).