Theory of the Josephson current on a magnetically doped topological insulator

Proximity induced superconducting states in the surface of magnetically doped topological insulators can host chiral Majorana modes. We consider a Josephson junction in that system with changing the chemical potential, which drives a topological phase transition in the induced superconducting states as well as a metal-insulator transition in the surface states. The local density of states and the Josephson current are analytically calculated by McMillan's Green's function method in terms of the Andreev reflection coefficient. We show that although the magnitude of the Josephson current is greatly enhanced when the surface state changes from insulating to metallic, its temperature dependence drastically changes at the topological phase transition point, reflecting the appearance of the chiral Majorana modes.

Figure 1

Schematic illustration of the system. The surface state of a magnetically doped topological insulator (TI) is coupled to two conventional superconductors (SCs) which are placed with an interval of length $L$. The system preserves the translational symmetry parallel to the junction (along the $y$ direction).

Figure 2

Phase diagrams of the surface state (a), proximity induced superconducting state (b), and their superposition (c). In (a), the surface of the topological insulator in the normal state becomes insulating when the magnetic gap is larger than the chemical potential. The induced superconducting state becomes topologically nontrivial provided $|m_z|<\sqrt{{\mathrm{\Delta }}^2+{\mu }^2}$ as shown in (b). Then, there are three different phases; $\mu <\sqrt{m_z^2-{\mathrm{\Delta }}^2}$, $\sqrt{m_z^2-{\mathrm{\Delta }}^2}<\mu <\left|m_z\right|$, and $|m_z|<\mu$, named phases (I), (II), and (III), respectively.

Figure 3

$k_y$ resolved local density of states. Parameters are chosen so that the system is in phase (I) [(a),(d)], (II) [(b),(e)], and (III) [(c),(f)]. The top row is with a narrow interval $L/\xi =0.5$ while the bottom row is with a wider interval $L/\xi =\infty$, i.e., the result for a superconductor-normal metal junction. They show gapped surface states in (a) and (d), chiral Majorana mode(s) in (b) and (e), and chiral Majorana modes covered with continuous modes in the interval area of the junction in (c) and (f).

Figure 4

Current-phase relation in phase (I) (a), phase (II) (b), and phase (III) (c). They are plotted with varying temperature and the current is measured in units of ${\mathrm{\Delta }}_0/e{R}_{\mathrm{N}0}$ where $R_{\mathrm{N}0}$ is a resistivity in the normal state without magnetization and ${\mathrm{\Delta }}_0$ is the pair amplitude at absolute zero. (a) With the insulating junction in the absence of chiral Majorana modes, the relation is in sinusoidal form and has no temperature dependence below $T/T_{\mathrm{c}}<0.1$, being consistent with the Ambegaokar-Baratoff behavior. (b) With the insulating junction in the presence of chiral Majorana modes, the relation has similar form to Ambegaokar-Baratoff, though, the critical current keeps growing at low temperature, and it saturates around $T/T_{\mathrm{c}}\sim 0.01$. (c) With the metallic junction, the relation and its temperature dependence are the same as the Kulik-Omelyanchuk relation.

Figure 5

Maximum value of the Josephson current as a function of temperature. In the main panel, those with an insulating junction are plotted; the yellow dashed line for phase (I) and the green solid line for phase (II). The inset shows that with a metallic junction, i.e., in phase (III). We set $L/\xi =0.5$.

Figure 6

Ratio of the maximum current at zero temperature to that at $T=0.2T_{\mathrm{c}}$. It is nearly a constant close to unity in phase (I) and increases in phases (II) and (III).

Figure 7

Same quantity as in Fig. 6 but in one dimension.