Thermoelectric processes of quantum normal-superconductor interfaces

TL;DR

This review examines how thermoelectric effects arise at normal-superconductor interfaces across zero-, one-, and two-dimensional systems, highlighting how Andreev reflections, quasiparticle tunneling, and nonlocal Cooper-pair processes generate sizable heat-to-charge conversions despite intrinsic electron-hole symmetry. By organizing the discussion around dimensionality—from quantum dots and nanoisolated islands to helical edge states and Dirac materials—it shows how symmetry breaking (via Coulomb blockade, multiterminal geometries, or proximity-induced pairing) enables finite thermoelectric coefficients and even nonlocal engines. Key mechanisms include Cooper pair splitting, nonlocal thermoelectric currents in three-terminal devices, and phase- and angle-dependent effects in topological and Dirac materials, with practical signatures such as Seebeck signals, cooling powers, and Joule spectroscopy dips. The work highlights potential applications in heat management and quantum technologies, including thermoelectric heat engines, refrigerators, and thermal diodes that operate at millikelvin temperatures using nanoscale NS hybrids.

Abstract

Superconducting interfaces have recently been demonstrated to contain a rich variety of effects that give rise to sizable thermoelectric responses and unexpected thermal properties, despite traditionally being considered poor thermoelectrics due to their intrinsic electron-hole symmetry. We review different mechanisms driving this response in hybrid normal-superconducting junctions, depending on the dimensionality of the mesoscopic interface. In addition to discussing heat to power conversion, cooling and heat transport, special emphasis is put on physical properties of hybrid devices that can be revealed by the thermoelectric effect.

Figures (6)

• Figure 1: Scope of the review. (a) Scheme of a hybrid configuration composed of normal (gray) and superconducting (cyan) terminals separated by a mesoscopic region (yellow). Each terminal $l$ is in a local equilibrium situation defined by its chemical potential, $\mu_l$, and temperature, $T_l$, resulting in particle, $I_l$, and heat, $J_l$, currents. (b) Transport occurs due to quasiparticle (electron or hole-like) tunneling or to Andreev reflection, here represented by the creation of a Cooper pair in the superconductor. (c) The mesoscopic region can be of different dimensionalities: 0D (e.g., quantum dots), 1D (e.g., quantum wires, topological edge states), 2D (e.g., graphene, semiconductor 2DEGs, surfaces of 3D topological insulators), depending on the properties of the propagating particles.
• Figure 2: (a) Scheme of a two-terminal thermoelectric engine with a proximized quantum dot. (b) Superconducting density of states for $\Delta=2k_\text{B}T_S$, and (c) generated current for different gaps in the quasiparticle regime, for $U=2k_\text{B}T_S$ and $T_N=2T_S$. The gray line in (c), corresponding to the all-normal case $\nu_S=\nu_N$, is shown for reference. The tunneling rate through the normal contact is $\Gamma_N=2\pi|\tau_N|^2\nu_N/\hbar$. (d) Nonlocal three-terminal heat engine generating current between terminals $N$ and $S$ from the heat injected from $H$ via the capacitive coupling of two quantum dots. (e) Cooper pair splitter serving as a nonlocal thermoelectric and refrigerator.
• Figure 3: Sketch of the setup considered in Refs. Blasi2020JunBlasi2021Jun. A Kramers pair of helical edge states of the quantum spin Hall effect is contacted by two superconductors at different temperatures, $T_{S_L, S_R}= T \pm \delta T/2$ and with a normal-metal probe at temperature $T_N=T$ at which a bias voltage $V_N$ is eventually applied/generated. 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) within the proximized spin-Hall region. Transport processes are depicted in (c) for $V_N = 0$; $\delta T \neq 0$ and in (d) for $V_N \neq 0$; $\delta T = 0$, when the spectrum for $e_+; h_-$ is assumed fully gapped. With courtesy of Ref. Blasi2020Jun.
• Figure 4: (a) Sketch of the setup considered in Ref. Mateos2024Aug. A semiconducting wire with SOC and magnetic field is contacted to a superconductor. A temperature bias $T \pm \delta T/2$ is imposed and the electrical current is generated at a normal-metal probe at temperature $T_N=T$ at which a bias voltage $V_N$ is applied. (b) Spectrum for different angles between the SOC and the magnetic field. The central panel corresponds to a perpendicular orientation of the magnetic field with respect to the direction of the SOC, while the other panels correspond to departures from this configuration, leading to the formation of Bogoliubov Fermi points. This situation is similar to the one illustrated in Fig. \ref{['fig:s-ti-s']}(b) and favors a nonlocal thermoelectric response. With courtesy of Ref. Mateos2024Aug
• Figure 5: (a) Schematic of a highly transmitting SNS junction based on a hybrid superconductor-semiconductor nanowire. The temperatures of the S leads at the junction, $T_{0,i}$, can increase substantially due to the dissipated Joule power, $P_i$. (b) $V_{dip,i}$ are the voltages at which the S leads turn to the normal state due to Joule heating. They appear as dips in $dI/dV$, reflecting the suppression of excess current. (c) Gate dependence of $V_{dip,i}$. The dashed lines are fits to Eq. \ref{['eq:joule-spectroscopy']}, confirming the $\sqrt{R_J}$-dependence of the dips. (d) Joule spectroscopy of a device with a floating superconducting island and two grounded superconducting leads. In this case, three $dI/dV$ dips are observed, one for each superconductor-to-normal metal transition. The magnetic field-dependence of the dips of the floating and grounded superconductors are different, owing to their distinct dominant heat dissipation mechanisms. With courtesy of Ref. Ibabe2023May.