Quantum Bipolar Thermoelectricity

TL;DR

The paper demonstrates a purely quantum mechanism for bipolar thermoelectricity in a thermal-equilibrium S-I-S' superconducting junction coupled to a cold electromagnetic environment. Using the $P(E)$ framework, it shows that an emission-absorption imbalance at energy $\hbar\omega$ induced by environmental coupling can generate a nonlinear thermoelectric current even when the leads share identical Fermi distributions. Two environmental realizations are analyzed: a dissipative Ohmic environment and a single-mode resonator, yielding Seebeck-like responses up to $\sim 100\ \mu$V/K and extractable power on the order of $10^{-13}$ W, with performance tunable via charging energy $E_C$ or resonator frequency $\omega_{LC}$. The findings suggest routes for environmentally engineered low-temperature thermoelectrics and for spectroscopic sensing of electromagnetic modes through quantum thermoelectric measurements.

Abstract

Thermoelectricity is generally understood as a classical effect emerging from energy-dependent transport asymmetries. Here we uncover a purely quantum mechanism, where a superconducting S-I-S' tunnel junction in thermal equilibrium develops a nonlinear bipolar thermoelectric response owing to the dynamical Coulomb blockade and the emission-absorption imbalance of a cold electromagnetic bath. Two representative environments are analysed, revealing Seebeck coefficients up to 100 $μ$V/K for realistic junction parameters. Because the response directly reflects the spectral properties of the surrounding environment, our results suggest that bipolar quantum thermoelectricity could provide a new route for spectroscopic sensing of electromagnetic modes and for designing low-temperature thermoelectric devices with environmentally engineered performance.

Figures (4)

• Figure 1: Quantum bipolar thermoelectricity from emission/absorption asymmetry a) Quasiparticle occupation at temperature $T_j$ for an S-I-S' junction biased at $\delta\mu$. Backward tunnelling processes $\reflectbox{\vec{\reflectbox{\Gamma}}}(\delta\mu)$ arise from photon-assisted transitions at $\hbar\omega$. Full (empty) red circles indicate particle (hole) excitations involved. The "STOP" sign denotes suppressed absorption due to the quantum nature of the environment at low $T_e$. Grey dashed lines represent the $\delta\mu=0$ condition. b) Same as in a), but for the forward tunnelling $\vec{\Gamma}(\delta\mu)$. c) Minimal circuit: an S-I-S' junction (capacitance $C$, tunnel resistance $R_T$) coupled to an environmental impedance $Z(\omega)$ and to an external bias. d) Forward tunnelling rate $\vec{\Gamma}(\delta\mu)$ for two different charging energies $E_C$ in the high-impedance (violet/blue) and low-impedance (red) regimes. Other parameters: $\Delta_0'=0.9\ \Delta_0$ and $T_j = 0.8\ T_C$, where $\Delta_0$ denotes the zero-temperature superconducting gap and $T_C$ the critical temperature of the S electrode. (e) Corresponding current-voltage ($I$-$V$) characteristics.
• Figure 2: Quantum thermoelectric performance (a) Thermoelectric power at the matching peak $P_{MAX}$ as a function of $T_j$ and $E_C$, for $r = 0.9$ (left) and $r = 0.7$ (right). (b) Top panel: $I$-$V$ characteristic under optimal conditions for $r=0.9$ and $r=0.7$. The parameters correspond to the stars shown in the respective colour plots. Bottom panel: $P_{MAX}$ computed at optimal $E_C$ and $T_j$ for different $r$.
• Figure 3: Quantum thermoelectricity in realistic conditions (a) $I$-$V$ characteristics for different values of $T_e$, with $g = 0.01$, $r = 0.9$, $E_C=0.15\Delta_0$ and $T_j = 0.8 T_C$. Inset: comparison between $g = 0.01$ and $g = 0.1$ at $T_e = 0.005 T_C$. (b) Density plot of the efficiency $\eta$ as a function of $E_C$ and $T_j$, computed for $T_e = 0.005T_C$, $g = 0.01$ and $r = 0.9$.
• Figure 4: Quantum thermoelectricity in a resonant cavity (a) Top Panel: Forward tunnelling rate $\vec{\Gamma}(\delta\mu)$ for three different environmental resonance frequencies, $\hbar\omega_{LC} = 0.1\Delta_0$ (dark green), $\hbar\omega_{LC} = 0.2\Delta_0$ (green), and $\hbar\omega_{LC} = 0.3\Delta_0$ (light green), with fixed parameters $T_e \to 0$, $T_j = 0.7T_C$, $r = 0.8$, and $\rho = 0.2$. Bottom panel: $\vec{\Gamma}$ as a function of energy for fixed $\hbar\omega_{LC} = 0.3\Delta_0$ and varying coupling strengths: $\rho = 0.2$ (light green), $\rho = 0.5$ (yellow), and $\rho = 1$ (brown), with all other parameters unchanged. (b) $I$-$V$ characteristic for three different environmental temperatures: $T_e \to 0$ (green), $T_e= 0.3 T_j$ (light green), and $T_e = 0.4 T_j$ (red), with $T_j = 0.7T_C$, $r = 0.8$, $\hbar\omega_{LC}=0.3\Delta_0$ and $\rho = 1$. Inset: maximum environmental temperature $T_e^{MAX}$ allowing thermoelectric operation as a function of $\omega_{LC}$ for $T_j=0.7 T_C$ and $r=0.8$. (c) Density plot of the extractable thermoelectric power at the peak ($P_{\mathrm{MAX}}$) as a function of $T_j$ and $\omega_{LC}$, for $r = 0.8$ and $\rho = 0.5$. (d) $P_{\mathrm{MAX}}$ as a function of $\rho$, for three different junction temperatures: $T_j = 0.6T_C$ (yellow), $T_j = 0.65T_C$ (orange), and $T_j = 0.7T_C$ (red), with fixed resonance frequency $\hbar\omega_{LC} = 0.25\Delta_0$ and $r = 0.8$.