Protected gap closing in Josephson trijunctions constructed on ${\mathrm{Bi}}_2{\mathrm{Te}}_3$

We carried out a phase-sensitive experiment on Josephson trijunctions constructed on the surface of three-dimensional topological insulator ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. Through local flux contol and contact resistance measurement, we observed that the minigap at the ends of the trijunctions approaches zero linearly with varying phase difference and closes completely at $\pi$ phase difference. The minigap at the center of the trijunction is protected to close over extended regions in phase space. These results agree with the theory by Fu and Kane [Phys. Rev. Lett. 100, 096407 (2008)].

Figure 1

(a) False-color SEM image of the first device. A Pb Josephson trijunction connected with two superconducting loops (in blue) was fabricated on the surface of a ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ flake. Two half-turn coils (in silver) were used to apply local magnetic flux in the loops. The Au electrodes (in gold) were used to detect the ABSs at the center and the ends of the trijunctions. (b) The central part of the device. (c) Schematic of the trijunction device and the three-terminal configuration for contact resistance measurement. (d) Fu and Kane's Majorana phase diagram [5] for the center of the trijunction in the loop-flux space. MZMs are expected in the shadowed regions where minigap inversion occurs in odd numbers of single junctions (shown in the brackets are the signs of the minigap in the left, the right, and the bottom single junctions).

Figure 2

The contact resistance $dV/d{I}_{\mathrm{b}}$ measured at the ends of the first trijunction at $T=0.5\mathrm{K}$. (a) The $dV/d{I}_{\mathrm{b}}$ measured at the left end, as functions of global magnetic field $B$ and bias current $I_{\mathrm{b}}$. (b) The vertical line cuts of (a) at magnetic fields indicated by the arrows in corresponding color in (a). The data are fitted by using the BTK theory (black line). (c) The horizontal line cut of (a) at $I_{\mathrm{b}}=0$. (d)–(f) and (g)–(i) Similar data measured at the right end and the bottom end of the trijunction, respectively. (j) Effective magnetic flux ${\varphi }_{\mathrm{e}}$ in the superconducting loop as a function of applied magnetic flux $\varphi$, when the screening parameter of the loop $\beta =0$ (blue line) and $\beta =0.5$ (black line). (k) and (l) The theoretical flux dependences of the minigap in the left/right junctions (k) and in the bottom junction (l), when $\beta =0$ (blue lines) and $\beta =0.5$ (black lines). By using the functional forms of these black lines, the magnetic field dependences of $dV/d{I}_{\mathrm{b}}$ in (c), (f), and (i) can be simulated (black lines).

Figure 3

The 2D data of $dV/d{I}_{\mathrm{b}}$ measured at (a) the left end, (b) the right end, and (c) the bottom end of the first trijunction at $I_{\mathrm{b}}=0, T=0.5\mathrm{K}$, and by feeding currents $I_{\mathrm{L}}, I_{\mathrm{R}}$ to the half-turn coils to control the magnetic flux in the superconducting loops. (d)–(f) The simulated $dV/d{I}_{\mathrm{b}}$ at the left end, the right end, and the bottom end of the trijunction, respectively.

Figure 4

The $dV/d{I}_{\mathrm{b}}$ measured at the center of the trijunctions. (a) The $dV/d{I}_{\mathrm{b}}$ measured on the second trijunction at $T=0.15$ K, as functions of $I_{\mathrm{b}}$ and $B$ varying along the dashed line in (h). (b) The vertical line cuts of (a) at magnetic fields indicated by the arrows in (a) with corresponding colors. The data at $B=0$ are fitted by using the BTK theory (black lines). (c) The horizontal line cut of (a) at $I_{\mathrm{b}}=0$. (d)/(h) The measured/simulated $dV/d{I}_{\mathrm{b}}$ at the center of the second trijunction in 2D flux space at $T=0.25$ K. The region enclosed by the gray square in (h) corresponds to Fu and Kane's Majorana phase diagram shown in Fig. 1. (e) The minigap expected from the effective Hamiltonian via lattice-model numerical simulation, in the long junction limit (blue line) and with a finite junction width of $1.5\mu \mathrm{m}$ (black line). Using the functional form of this black line, the data in (c) can be simulated (black line). (f), (g) The $dV/d{I}_{\mathrm{b}}\text{vs}{I}_{\mathrm{b}}$ and $dV/d{I}_{\mathrm{b}}\text{vs}B$ curves measured at the center of the third trijunction whose contacts were in the tunneling regime. $T=30\mathrm{mK}$.