Thermopower Induced by a Supercurrent in Superconductor–Normal-Metal Structures

We examine the thermopower $Q$ of a mesoscopic normal-metal ($N$) wire in contact with superconducting ($S$) segments and show that even with electron-hole symmetry $Q$ may become finite due to the presence of supercurrents. Moreover, we show how the dominant part of $Q$ can be directly related to the equilibrium supercurrents in the structure. In general, a finite thermopower appears both between the $N$ reservoirs and the superconductors and between the $N$ reservoirs themselves. The latter, however, strongly depends on the geometrical symmetry of the structure.

Figure 1

$N\mathrm{\text{−}}S$ thermopower at different temperatures $T_1$ and $T_2$ of the normal electrodes, for a setup with $L_k$ and $A_k$ assumed equal for all wires, and in the limit $\Delta \gg E_T$, $eV$, $k_{B}T$. The analytical approximation (8) with $\mathcal{T}$-correction (9) deviates at worst 15% from these results, but the difference is over 5% only where $Q_{NS}$ is small, i.e., $k_B{T}_1,k_B{T}_2\gtrsim 10E_T$. In all of the figures, $E_T\equiv \hslash D/(L_3+L_4+L_5{)}^2$, and we plot the results for $\varphi =\pi /2$ yielding a near-maximal supercurrent.

Figure 2

$N\mathrm{\text{−}}S$ thermopower along the cross sections of Fig. 1. Solid line: $Q_{NS}$ at $T_1\approx T_2$. Dotted line: Approximation (8). Correction (9) accounts for most of the difference. Dashed line: $Q_{NS}$ at $k_B{T}_1=3.6E_T$ with varying $T_2$. Dash-dotted line: the corresponding approximation. Inset: The setup under consideration: two superconducting ($S$) reservoirs with phase difference $\varphi$ connected to two normal-metal ($N$) reservoirs through diffusive normal-metal wires. We assume that the lengths $L_k$, $k=1,\dots ,5$ of the wires satisfy ${\xi }_0\lesssim L_k\ll l_{\varphi },l_E$, where ${\xi }_0=\sqrt{\hslash D/(2\Delta )}$ is the superconducting coherence length, $l_{\varphi }$ the phase-coherence length, and $l_E$ the energy-relaxation length. We also consider the wires as quasi-1D structures [8].

Figure 3

(a) $N\mathrm{\text{−}}S$ thermopower at $k_B{T}_1=3.6E_T$ for various values of $\Delta$: $\infty$ (solid line), $54E_T$ (dotted line), $27E_T$ (dashed line). Superconductor temperature $T_0=3.6E_T$ is fixed. (b) Potential $e{V}_2$ induced due to the energies above $\Delta$, with $\Delta =27E_T$ . Here, $k_B{T}_0=3.6E_T$ but $T_1=T_2$ vary. (c) Phase oscillation of $Q_{NS}$ (solid line) and $Q_{NN}$ (dashed line), at $k_B{T}_1=4.5E_T$, $k_B{T}_2=1.8E_T$. Because of a numerical convergence problem, there are no data for $|\varphi |>2.1$. (d) Spectral variables: $L_3{j}_S$ (solid line), $⟨{\mathcal{T}}_1⟩$ (dashed line), $⟨{\mathcal{T}}_3⟩$ (dotted line). In each figure, all the lengths $L_k$ and areas $A_k$ are assumed equal.

Figure 4

Left: $N\mathrm{\text{−}}N$ thermopower induced by setup asymmetry, at fixed $k_B{T}_1=3.6E_T$. One wire (1 or 3) is assumed to have a different resistance than the others. Right: $N\mathrm{\text{−}}N$ thermopower with the same parameters, but with $T_1\approx T_2$.