Nonlocal Thermoelectricity in a Superconductor–Topological-Insulator–Superconductor Junction in Contact with a Normal-Metal Probe: Evidence for Helical Edge States

We consider a Josephson junction hosting a Kramers pair of helical edge states of a quantum spin Hall bar in contact with a normal-metal probe. In this hybrid system, the orbital phase, induced by a small magnetic field threading the junction known as a Doppler shift, combines with the conventional Josephson phase difference and originates an effect akin to a Zeeman field in the spectrum. As a consequence, when a temperature bias is applied to the superconducting terminals, a thermoelectric current is established in the normal probe. We argue that this purely nonlocal thermoelectric effect is a unique signature of the helical nature of the edge states coupled to superconducting leads and it can constitute a useful tool for probing the helical nature of the edge states in systems where the Hall bar configuration is difficult to achieve. We fully characterize thermoelectric response and performance of this hybrid junction in a wide range of parameters, demonstrating that the external magnetic flux inducing the Doppler shift can be used as a knob to control the thermoelectric response and the heat flow in a novel device based on topological junctions.

Figure 1

(a) Sketch of the setup. A helical Kramers pair of edge states of the quantum spin Hall effect is contacted by two superconductors at different temperatures, $T_{S_L}=T+\delta T/2$ and $T_{S_R}=T-\delta T/2$, and with a normal-metal probe at temperature $T_N=T$ at which a bias voltage $V_N$ is possibly applied. The structure is threaded by a magnetic flux, which induces a Doppler shift in the edge states in addition to a Josephson phase difference applied between the two superconductors. (b) Dispersion curves for quasiparticles $e_{\pm }$ (solid lines) and quasiholes $h_{\pm }$ (dashed lines) in the proximized superconductor $S_L/S_R$ for $0<{\epsilon }_{\mathrm{DS}}(\mathrm{\Phi })<\mathrm{\Delta }$. Transport processes are depicted in (c) for $V_N=0,\delta T\ne 0$ and in (d) for $V_N\ne 0,\delta T=0$, when the spectrum for $e_{+},h_{-}$ is assumed fully gapped.

Figure 2

Onsager coefficients (a) $L_{11}$, (b) $L_{22}$, and (c) $L_{12}=-L_{21}$ as functions of ${\epsilon }_{\mathrm{DS}}(\mathrm{\Phi })/\mathrm{\Delta }$ and the coupling parameter $|t{|}^2$ for phase bias $\varphi ={\varphi }_{S_L}-{\varphi }_{S_R}=0$, $T/T_C=0.1$, and $L/\xi =1$. (d) $L_{12}$ as a function of ${\epsilon }_{\mathrm{DS}}(\mathrm{\Phi })/\mathrm{\Delta }$ and the phase difference $\varphi$ for $|t{|}^2=0.5$. Such quantities are normalized as follows: $L_{11}/(G_{0}T)$, $L_{22}/(G_T{T}^2)$, and $L_{12}/(\sqrt{G_0{G}_T{T}^3})$.

Figure 3

Seebeck coefficient as function of ${\epsilon }_{\mathrm{DS}}(\mathrm{\Phi })/{\mathrm{\Delta }}_0$ for different temperatures at (a) $\varphi =\pi /2$ and as function of ${\epsilon }_{\mathrm{DS}}(\mathrm{\Phi })/{\mathrm{\Delta }}_0$ and (b) $\varphi$ for $T/T_C=0.1$. The blue curve in (a) corresponds to the cut at $\varphi =\pi /2$ of the Seebeck coefficient depicted in (b) (dashed line) obtained for the same set of parameters: $L/\xi =1$ and $|t{|}^2={10}^{-2}$.