Nonlinear Quantum Bipolar Thermoelectric Effect

• The phenomenon is defined by second-order thermoelectric responses arising from quantum coherence and the interplay of electron and hole transport channels.
• It leverages inversion-breaking mechanisms and Berry-curvature effects to enhance nonlinear Seebeck and Peltier coefficients in various material platforms.
• Experimental realizations in superconducting junctions and topological-insulator surfaces offer prospects for efficient, quantum-driven thermal management.

Nonlinear quantum bipolar thermoelectric effect denotes a class of nonequilibrium thermoelectric phenomena in which the thermoelectric response appears beyond linear response, has an explicitly quantum origin, and involves concurrent or competing electron-like and hole-like transport channels. In one operational definition, it is “the second-order thermoelectric response in a quantum (coherence-including) system where both electrons and holes (conduction and valence bands) contribute substantially (bipolarity), enabled by inversion breaking (warping/gap) and interband coherence” (Bhalla, 2020). In superconducting tunnel junctions, the same label is used for giant thermopower that emerges only in the nonlinear thermal-gradient regime through spontaneous particle–hole symmetry breaking, with two degenerate zero-current solutions $V=\pm V_{\mathrm{th}}$ for the same thermal bias (Germanese et al., 2022). A further extension identifies an equilibrium variant driven by dynamical Coulomb blockade, where a low-temperature electromagnetic environment supplies the essential emission–absorption asymmetry (Antola et al., 5 Aug 2025).

1. Constitutive structure and defining signatures

In non-centrosymmetric conductors, charge and heat currents are expanded to second order in electric field $E$ and thermal gradient $-\nabla T$ as

$$j_i=\sum_j \sigma^{(1)}_{ij}E_j+\sum_{jk}\sigma^{(2)}_{ijk}E_jE_k+\sum_j\alpha^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\alpha^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\alpha}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$

and

$$q_i=\sum_j \bar{\alpha}^{(1)}_{ij}E_j+\sum_{jk}\bar{\alpha}^{(2)}_{ijk}E_jE_k+\sum_j\kappa^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\kappa^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\kappa}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$

The explicitly evaluated second-order terms in the intrinsic non-centrosymmetric theory are proportional to $E^2$ and $(\nabla T)^2$, while mixed $E(\nabla T)$ terms are admitted by the same formalism but were not evaluated explicitly (Bhalla, 2020).

The characteristic observables are nonlinear Seebeck and Peltier coefficients. In the weakly nonlinear regime, the Seebeck coefficient along a collinear transport axis takes the form

$$S \approx \frac{\alpha_1}{\sigma_1}\left[1+\frac{\alpha_2}{\alpha_1}\nabla T-\frac{\sigma_2}{\sigma_1}E+\ldots\right],$$

with an analogous expression for $\Pi$. The leading nonlinear correction is therefore controlled by ratios of second- to first-order response coefficients, which provides the compact criterion emphasized in the intrinsic theory: large nonlinear effects require large ratios $E$0, $E$1, and $E$2 (Bhalla, 2020).

In superconducting junctions, the defining observable is the open-circuit thermovoltage determined by

$E$3

The linear-response Seebeck coefficient is $E$4, whereas the nonlinear definition used in the bipolar thermoelectric Josephson literature is $E$5, with $E$6 extracted by fitting the full $E$7–$E$8 curve to the tunneling model (Germanese et al., 2022). In mesoscopic diode language, a still more restrictive signature is “antireciprocity,” where the charge current flows in the same direction for both signs of the temperature bias, $E$9 and $-\nabla T$0 having the same sign (Balduque et al., 2024).

2. Symmetry principles and microscopic mechanisms

The symmetry requirements depend on the platform, but even-order response is the unifying theme. In non-centrosymmetric crystals, even-order electric and thermoelectric tensors appear only when inversion is broken; in the warped topological-insulator model, hexagonal warping explicitly breaks inversion and reduces the symmetry, enabling second-order electric and thermoelectric tensors (Bhalla, 2020). In time-reversal-symmetric but inversion-broken systems, the phenomenology is consistent with Berry-curvature-dipole control of nonlinear Hall-like currents.

In superconducting tunnel junctions, the central constraint is particle–hole symmetry. In a simple PH-symmetric quasiparticle tunnel model, the linear thermoelectric coefficient vanishes. The nonlinear regime changes this because a sufficiently large thermal gradient drives the system far from equilibrium, and the $-\nabla T$1–$-\nabla T$2 curve develops absolute negative conductance together with multiple zero-current roots $-\nabla T$3. The experimentally emphasized consequence is the existence of two degenerate zero-current solutions with $-\nabla T$4 for the same $-\nabla T$5, selected by bias history or phase control, which is interpreted as spontaneous PH symmetry breaking (Germanese et al., 2022). In asymmetric SIS heat engines, the same physics is described in terms of a linear-in-bias thermoelectric component $-\nabla T$6 with $-\nabla T$7 and a strong nonlinear enhancement at the matching-peak singularity $-\nabla T$8 (Marchegiani et al., 2020).

The term “bipolar” has a band-transport meaning as well. In the intrinsic two-band theory, bipolarity refers to concurrent electron and hole participation when interband transitions transfer carriers from valence to conduction bands. This is enhanced when the chemical potential is near the gap or Dirac point, when a finite gap is present, and when thermal or optical excitation populates both bands (Bhalla, 2020). In Coulomb-blockaded quantum dots, bipolarity appears when large thermal bias causes electron-like and hole-like resonances to contribute simultaneously, leading to gate-dependent sign reversals and non-monotonic thermocurrent (Svilans et al., 2015). In nanowire quantum dots, pronounced nonlinear thermovoltage and thermocurrent were attributed chiefly to a $-\nabla T$9-dependent renormalization of the quantum-dot level, with a possible sharp contribution from the melting of Kondo correlations in the mixed-valence regime (Svensson et al., 2013).

A distinct quantum mechanism is furnished by dynamical Coulomb blockade. For a gap-asymmetric $$j_i=\sum_j \sigma^{(1)}_{ij}E_j+\sum_{jk}\sigma^{(2)}_{ijk}E_jE_k+\sum_j\alpha^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\alpha^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\alpha}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$0–$$j_i=\sum_j \sigma^{(1)}_{ij}E_j+\sum_{jk}\sigma^{(2)}_{ijk}E_jE_k+\sum_j\alpha^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\alpha^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\alpha}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$1–$$j_i=\sum_j \sigma^{(1)}_{ij}E_j+\sum_{jk}\sigma^{(2)}_{ijk}E_jE_k+\sum_j\alpha^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\alpha^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\alpha}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$2 junction in thermal equilibrium, coupled to a colder electromagnetic environment, the low-temperature bath obeys $$j_i=\sum_j \sigma^{(1)}_{ij}E_j+\sum_{jk}\sigma^{(2)}_{ijk}E_jE_k+\sum_j\alpha^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\alpha^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\alpha}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$3. This strongly suppresses absorption relative to emission and, combined with the energy-selective BCS densities of states, produces a nonlinear quantum bipolar thermoelectric effect even though the two superconductors are at the same temperature (Antola et al., 5 Aug 2025). This suggests that “nonlinear quantum bipolar thermoelectricity” is not a single mechanism but a family of symmetry-constrained nonequilibrium effects.

3. Theoretical frameworks

The intrinsic non-centrosymmetric theory employs a quantum kinetic density-matrix formalism. The density matrix is expanded as $$j_i=\sum_j \sigma^{(1)}_{ij}E_j+\sum_{jk}\sigma^{(2)}_{ijk}E_jE_k+\sum_j\alpha^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\alpha^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\alpha}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$4, with diagonal intraband and off-diagonal interband-coherence sectors. The driving term contains a covariant derivative involving the Berry connection

$$j_i=\sum_j \sigma^{(1)}_{ij}E_j+\sum_{jk}\sigma^{(2)}_{ijk}E_jE_k+\sum_j\alpha^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\alpha^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\alpha}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$5

The current naturally decomposes into an intraband contribution $$j_i=\sum_j \sigma^{(1)}_{ij}E_j+\sum_{jk}\sigma^{(2)}_{ijk}E_jE_k+\sum_j\alpha^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\alpha^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\alpha}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$6 and two interband-coherence contributions $$j_i=\sum_j \sigma^{(1)}_{ij}E_j+\sum_{jk}\sigma^{(2)}_{ijk}E_jE_k+\sum_j\alpha^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\alpha^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\alpha}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$7 and $$j_i=\sum_j \sigma^{(1)}_{ij}E_j+\sum_{jk}\sigma^{(2)}_{ijk}E_jE_k+\sum_j\alpha^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\alpha^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\alpha}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$8. In this formulation, first order is organized so that intraband terms produce longitudinal conductivity while interband terms produce Hall conductivity; at second order the roles interchange, with interband coherence yielding longitudinal nonlinear conductivity and intraband terms yielding Hall-type nonlinear conductivity (Bhalla, 2020).

Superconducting realizations are usually formulated in terms of quasiparticle tunneling between BCS densities of states. The canonical current expression is

$$j_i=\sum_j \sigma^{(1)}_{ij}E_j+\sum_{jk}\sigma^{(2)}_{ijk}E_jE_k+\sum_j\alpha^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\alpha^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\alpha}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$9

with Dynes-broadened densities of states

$$q_i=\sum_j \bar{\alpha}^{(1)}_{ij}E_j+\sum_{jk}\bar{\alpha}^{(2)}_{ijk}E_jE_k+\sum_j\kappa^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\kappa^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\kappa}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$0

This tunneling model underlies the bipolar thermoelectric Josephson engine, the nonlinear superconducting heat engine, and the phase-control analysis of Josephson screening (Germanese et al., 2022). In the dynamical-Coulomb-blockade variant, the same quasiparticle rates are convolved with the $$q_i=\sum_j \bar{\alpha}^{(1)}_{ij}E_j+\sum_{jk}\bar{\alpha}^{(2)}_{ijk}E_jE_k+\sum_j\kappa^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\kappa^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\kappa}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$1 probability,

$$q_i=\sum_j \bar{\alpha}^{(1)}_{ij}E_j+\sum_{jk}\bar{\alpha}^{(2)}_{ijk}E_jE_k+\sum_j\kappa^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\kappa^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\kappa}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$2

so that environmental impedance and junction capacitance enter explicitly (Antola et al., 5 Aug 2025).

Quantum-dot and mesoscopic-conductor approaches are more diverse. Coherent conductors are described by weakly nonlinear scattering theory with self-consistent internal potentials $$q_i=\sum_j \bar{\alpha}^{(1)}_{ij}E_j+\sum_{jk}\bar{\alpha}^{(2)}_{ijk}E_jE_k+\sum_j\kappa^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\kappa^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\kappa}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$3, where gauge invariance and current conservation impose nonlinear sum rules on the transport coefficients (Meair et al., 2012). Thermal and bipolar thermoelectric diodes are modeled by Landauer currents together with either a thermalizing probe, fixed by $$q_i=\sum_j \bar{\alpha}^{(1)}_{ij}E_j+\sum_{jk}\bar{\alpha}^{(2)}_{ijk}E_jE_k+\sum_j\kappa^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\kappa^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\kappa}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$4 and $$q_i=\sum_j \bar{\alpha}^{(1)}_{ij}E_j+\sum_{jk}\bar{\alpha}^{(2)}_{ijk}E_jE_k+\sum_j\kappa^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\kappa^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\kappa}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$5, or screening equations for internal potentials in capacitively coupled regions (Balduque et al., 2024). Coulomb-blockaded nanowire dots are described by stationary master equations, whereas interferometric quantum dots and single-molecule junctions are treated by nonequilibrium Green functions and Fano lineshapes, enabling fully nonlinear thermopower calculations and explicit interference control (Svilans et al., 2015).

A recent extension places nonlinear thermoelectricity inside the modern theory of quantum geometry. There the mixed $$q_i=\sum_j \bar{\alpha}^{(1)}_{ij}E_j+\sum_{jk}\bar{\alpha}^{(2)}_{ijk}E_jE_k+\sum_j\kappa^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\kappa^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\kappa}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$6–$$q_i=\sum_j \bar{\alpha}^{(1)}_{ij}E_j+\sum_{jk}\bar{\alpha}^{(2)}_{ijk}E_jE_k+\sum_j\kappa^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\kappa^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\kappa}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$7 responses are governed by the Berry-curvature dipole and the quantum-metric dipole. The low-temperature relations

$$q_i=\sum_j \bar{\alpha}^{(1)}_{ij}E_j+\sum_{jk}\bar{\alpha}^{(2)}_{ijk}E_jE_k+\sum_j\kappa^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\kappa^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\kappa}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$8

provide nonlinear analogs of Mott’s formula and the Wiedemann–Franz law, directly linking nonlinear thermoelectric coefficients to nonlinear Hall coefficients (Yang et al., 30 Apr 2025).

4. Canonical realizations and model systems

One canonical solid-state realization is the warped, gapped topological-insulator surface described by

$$q_i=\sum_j \bar{\alpha}^{(1)}_{ij}E_j+\sum_{jk}\bar{\alpha}^{(2)}_{ijk}E_jE_k+\sum_j\kappa^{(1)}_{ij}(-\nabla_jT)+\sum_{jk}\kappa^{(2)}_{ijk}E_j(-\nabla_kT)+\sum_{jk}\tilde{\kappa}^{(2)}_{ijk}(-\nabla_jT)(-\nabla_kT)+\ldots$$9

Its band energies are

$E^2$0

Within this model, the explicit coefficients $E^2$1, $E^2$2, $E^2$3, $E^2$4, $E^2$5, and $E^2$6 show that increasing warping $E^2$7 enhances both longitudinal and Hall nonlinear signals, whereas increasing the surface gap $E^2$8 enhances longitudinal nonlinear signals but decreases the nonlinear-to-linear Hall ratio. Representative values used were $E^2$9, $(\nabla T)^2$0, $(\nabla T)^2$1–$(\nabla T)^2$2, and $(\nabla T)^2$3 (Bhalla, 2020).

The experimentally best-developed platform is the asymmetric superconducting tunnel junction. In the bipolar thermoelectric Josephson engine, the core element is an $(\nabla T)^2$4 junction with an Al island of thickness $(\nabla T)^2$5, $(\nabla T)^2$6, $(\nabla T)^2$7, and an Al/Cu bilayer with $(\nabla T)^2$8, $(\nabla T)^2$9, $E(\nabla T)$0, $E(\nabla T)$1. Three such junctions form a double-loop SQUID with near-complete supercurrent suppression at specific flux. At $E(\nabla T)$2, the device shows $E(\nabla T)$3, $E(\nabla T)$4, and $E(\nabla T)$5 on $E(\nabla T)$6, corresponding to $E(\nabla T)$7 areal power density (Germanese et al., 2022). The phase-controlled version further shows that once the Josephson contribution is removed, the thermoelectric generation due to pure quasiparticle transport is phase-independent, while a residual Josephson channel introduces an additional metastable state at $E(\nabla T)$8 and modifies the hysteresis of the engine (Germanese et al., 2022).

Asymmetric SIS junctions have also been analyzed as active caloritronic elements beyond heat engines. A zero-bias superconducting voltage amplifier based on the bipolar thermoelectric effect uses an Al/AlO$E(\nabla T)$9/Al–Cu SIS junction with $$S \approx \frac{\alpha_1}{\sigma_1}\left[1+\frac{\alpha_2}{\alpha_1}\nabla T-\frac{\sigma_2}{\sigma_1}E+\ldots\right],$$0, $$S \approx \frac{\alpha_1}{\sigma_1}\left[1+\frac{\alpha_2}{\alpha_1}\nabla T-\frac{\sigma_2}{\sigma_1}E+\ldots\right],$$1, $$S \approx \frac{\alpha_1}{\sigma_1}\left[1+\frac{\alpha_2}{\alpha_1}\nabla T-\frac{\sigma_2}{\sigma_1}E+\ldots\right],$$2, and $$S \approx \frac{\alpha_1}{\sigma_1}\left[1+\frac{\alpha_2}{\alpha_1}\nabla T-\frac{\sigma_2}{\sigma_1}E+\ldots\right],$$3. Numerical simulations predict $$S \approx \frac{\alpha_1}{\sigma_1}\left[1+\frac{\alpha_2}{\alpha_1}\nabla T-\frac{\sigma_2}{\sigma_1}E+\ldots\right],$$4 voltage gain, a $$S \approx \frac{\alpha_1}{\sigma_1}\left[1+\frac{\alpha_2}{\alpha_1}\nabla T-\frac{\sigma_2}{\sigma_1}E+\ldots\right],$$5 compression point at an input amplitude of $$S \approx \frac{\alpha_1}{\sigma_1}\left[1+\frac{\alpha_2}{\alpha_1}\nabla T-\frac{\sigma_2}{\sigma_1}E+\ldots\right],$$6, total harmonic distortion below $$S \approx \frac{\alpha_1}{\sigma_1}\left[1+\frac{\alpha_2}{\alpha_1}\nabla T-\frac{\sigma_2}{\sigma_1}E+\ldots\right],$$7, input-referred noise of approximately $$S \approx \frac{\alpha_1}{\sigma_1}\left[1+\frac{\alpha_2}{\alpha_1}\nabla T-\frac{\sigma_2}{\sigma_1}E+\ldots\right],$$8, and a $$S \approx \frac{\alpha_1}{\sigma_1}\left[1+\frac{\alpha_2}{\alpha_1}\nabla T-\frac{\sigma_2}{\sigma_1}E+\ldots\right],$$9 cutoff around $\Pi$0 (Trupiano et al., 24 Mar 2026).

A higher-temperature hybrid realization is the bilayer-graphene–superconductor tunnel junction. There the BLG gap $\Pi$1 and chemical potential are controlled independently by gates, while the superconductor provides the BCS singularity. The matching condition is $\Pi$2, and the predicted performance reaches up to $\Pi$3 regarding the Seebeck coefficient and a power density of $\Pi$4 for temperature gradients of tens of kelvins (Bernazzani et al., 2022).

Mesoscopic dots and interferometers supply a different realization. In Coulomb-blockaded InAs/InP nanowire quantum dots, thermocurrent sign reversals occur at Coulomb-peak centers and the thermocurrent becomes strongly nonlinear and non-monotonic at large $\Pi$5, consistent with concurrent electron-like and hole-like channels (Svilans et al., 2015). In quantum-dot interferometers, Fano interference reshapes the transmission into an asymmetric line shape with nodes, enabling controlled bipolar thermopower sign reversals with gate voltage and Aharonov–Bohm phase (Taniguchi, 2019). Quantum Hall antidot systems and helical edge states extend the same logic to nonlinear Peltier response, Wiedemann–Franz-law breakdown, and spin thermoelectricity beyond linear response (Lopez et al., 2015).

5. Observables, scaling laws, and bipolar signatures

The most compact scaling statement in the intrinsic theory is that nonlinear Seebeck and Peltier effects are controlled by ratios of nonlinear to linear conductivities. For the warped topological-insulator surface, the explicit coefficients show $\Pi$6, $\Pi$7, and $\Pi$8 all scale linearly with the warping parameter $\Pi$9, while linear Hall conductivities scale as $E$00. This produces the experimentally relevant trend already stressed in the original abstract: enhancement in the longitudinal and Hall effects on increasing the warping strength, and opposite behavior with the surface gap for Hall ratios (Bhalla, 2020).

In asymmetric superconducting junctions, the dominant spectral marker is the matching-peak singularity

$E$01

Near this bias, the BCS coherence peaks align and both the thermoelectric current and the delivered power are maximized. For realistic Al-based parameters, the nonlinear superconducting heat-engine theory reports $E$02, $E$03, a Seebeck voltage approaching $E$04, and a nonlinear Seebeck coefficient up to $E$05 (Marchegiani et al., 2020). In the experimentally realized Josephson-engine geometry, the output is smaller because of the specific load-line constraints and the deliberate suppression of supercurrent, but the same matching-peak logic controls ignition, hysteresis, and polarity selection (Germanese et al., 2022).

Bipolarity itself has several experimentally distinct forms. In superconducting SIS junctions it denotes two stable nonzero solutions $E$06 at identical thermal bias, selected by ignition current or external phase; in BLG–superconductor junctions it denotes sign-reversible thermovoltage and thermocurrent under the same thermal gradient, controlled by bias and gating; in mesoscopic thermoelectric diodes it denotes the stronger condition that the charge current flows in the same direction irrespective of the sign of the temperature difference (Germanese et al., 2022). A plausible implication is that “bipolarity” is best regarded as a response-property descriptor, not a unique microscopic taxonomy.

The quantum-geometry formulation supplies a particularly sharp observable. In chiral Weyl systems, the cyclic sum of the mixed electric-field–temperature-gradient coefficient becomes quantized in the high-frequency clean limit,

$E$07

thereby directly measuring the total chirality $E$08 of Weyl points below the Fermi level (Yang et al., 30 Apr 2025). This is not a power-generation metric but a transport probe of band topology through nonlinear thermoelectricity.

6. Limitations, interpretive issues, and directions

Several restrictions recur across the literature. The intrinsic non-centrosymmetric theory focuses on coherence contributions, treats collisions in a constant relaxation-time approximation with a single $E$09 for diagonal and off-diagonal sectors, and neglects higher-order terms beyond second order (Bhalla, 2020). Weakly nonlinear scattering theory likewise assumes phase-coherent elastic transport and keeps only quadratic corrections in voltage and temperature biases (Meair et al., 2012). In the quantum-geometry formulation, extrinsic skew and side-jump mechanisms, phonon drag, and lattice contributions are not included in the transverse Hall-like sector (Yang et al., 30 Apr 2025).

Superconducting devices add further interpretive complications. In the BTJE literature, the standard figure of merit $E$10 is said to be “not straightforwardly applicable” because the observed thermoelectricity is nonlinear, history-dependent, and phase-tunable, while superconductors support dissipationless condensates and gapped quasiparticle spectra (Germanese et al., 2022). The phase-control analysis shows that Josephson coupling can short-circuit the quasiparticle thermoelectric response, so a clean identification of the bipolar effect requires either deep supercurrent suppression or explicit subtraction of the equilibrium Josephson contribution (Germanese et al., 2022). In the dynamical-Coulomb-blockade proposal, the effect relies on maintaining a colder environment than the junction itself; increasing $E$11 restores emission–absorption symmetry and suppresses the response (Antola et al., 5 Aug 2025).

A conceptual point of continuing interest is the distinction between explicit and spontaneous particle–hole asymmetry. Conventional superconducting thermoelectricity was long associated with explicit PH-symmetry breaking by spin splitting or ferromagnetism. The nonlinear bipolar thermoelectric Josephson engine instead emphasizes spontaneous PH-symmetry breaking under large thermal bias, whereas the BLG–superconductor proposal combines a nonequilibrium “spontaneous” mechanism with the option of externally controlled doping, and the DCB-based equilibrium proposal replaces thermal-bias asymmetry across the junction by bath-induced emission–absorption asymmetry (Germanese et al., 2022). This suggests that future work will continue to differentiate spontaneous, structural, interaction-driven, and environment-induced routes to the same observable class.

The direction of development is toward function rather than mere detection. Arrays of bipolar thermoelectric Josephson engines can be connected in series or parallel so that, with $E$12 elements, $E$13 or $E$14, yielding $E$15 (Germanese et al., 2022). Zero-bias voltage amplification, thermoelectric memory, thermal and bipolar diodes, and nonlinear probes of quantum geometry all extend the topic beyond conventional heat-to-electricity conversion (Trupiano et al., 24 Mar 2026). A plausible implication is that the nonlinear quantum bipolar thermoelectric effect has become a general framework for studying how symmetry, coherence, interaction, and spectral selectivity reorganize thermoelectric transport once the linear-response regime is abandoned.