Thermopower and Mott formula for a Majorana edge state

We study the thermoelectric effect between a conducting lead and a Majorana edge state. In the tunneling limit, we first use the Landauer-Büttiker formalism to derive the Mott formula relating the thermopower and the differential conductance between a conducting lead and a superconductor. When the tunneling takes place between a conducting lead and a Majorana edge state, we show that a nonvanishing thermopower can exist. Combining measurements of the differential conductance and the voltage induced by the temperature difference between the conducting lead and the edge state, the Mott formula provides a unique way to infer the temperature of the Majorana edge state.

Figure 1

The upper panel schematically depicts a quantum dot weakly coupling to a Majorana edge state. The temperature and chemical potential of the quantum dot are $T_n$ and ${\mu }_n$, respectively, and those of the Majorana edge state are $T_s$ and ${\mu }_s$. (a) A quantum dot with a single energy level $\varepsilon$ is coupled to a Majorana edge state with tunneling strength $t$. (b) Multiple energy levels ${\varepsilon }_j$ of a quantum dot are coupled to a Majorana edge state with tunneling strengths $t_j$ for each state.

Figure 2

Schematic picture of a tunneling junction between a conducting lead and a superconductor. The conducting lead has $N_n^{\alpha }\left(E\right)$ channels at energy $E$ while the superconductor has $N_s^{\beta }\left(E\right)$, where $\alpha \left(\beta \right)=p,h$ represents the quasiparticle or quasihole channels. The temperature and chemical potential of the conducting lead are $T_n$ and ${\mu }_n$, respectively, and those of the superconductor are $T_s$ and ${\mu }_s$. $s\left(E\right)$ represents the scattering matrix across the tunneling junction.

Figure 3

(a) Schematic plot of a tip of a single-channel lead coupled to a Majorana edge state (red curve at the boundary of the superconductor). The tunneling strength is $t$. The temperature and chemical potential of the conducting lead are $T_n$ and ${\mu }_n$, respectively, and those of the Majorana edge state are $T_s$ and ${\mu }_s$. (b) A single-electron channel can be unfolded into a chiral electron channel that couples to a Majorana edge state at the point $x=0$. (c) The plot shows a toy model of a double-point-contact setup between an unfolded electronic channel and a Majorana edge state. Two tunneling points are at $x=\pm a$ along the chiral electron mode.