Floating Thermoelectric Probes in Quantum Transport

• Floating thermoelectric probes are idealized auxiliary reservoirs used in quantum transport theory to model inelastic scattering and dephasing in mesoscopic and nanoscale systems.
• They enforce a zero net charge and heat flow condition, which enables self-consistent determination of voltage and temperature, thereby modifying multi-terminal transport coefficients.
• These probes facilitate environment-assisted transport, symmetry breaking, and suppression of quantum interference, offering practical insights for designing quantum heat engines and nanoscale circuits.

Floating thermoelectric probes are idealized auxiliary reservoirs introduced in quantum transport theory to model inelastic and dephasing phenomena within mesoscopic and nanoscale conductors. By enforcing vanishing net charge and heat flow, these probes act as combined potentiometers and thermometers whose voltage and temperature are determined self-consistently. Their inclusion leads to modified multi-terminal Landauer–Büttiker and non-equilibrium Green function (NEGF) frameworks capable of capturing environment-assisted transport, symmetry breaking, thermoelectric efficiency bounds, and quantum interference suppression in phase-coherent systems.

1. Fundamental Theory and Formulation

Floating thermoelectric probes extend the multi-terminal Landauer–Büttiker formalism to incorporate inelastic scattering and dephasing in quantum conductors (Sanchez et al., 2011, Bergfield, 28 Aug 2025, Saha et al., 2022). The system is coupled to electron reservoirs labeled by $\alpha$. Usual transport involves biasing two leads (source/drain), while a third (or more) "probe" is introduced at selected positions in the device. The probe is enforced to carry neither net electrical nor net energy current:

$I_p = 0, \quad J_p = 0$

where $I_p$ and $J_p$ are the electrical and heat currents out of the probe.

Transport coefficients $(G, L, K)$ or Onsager blocks $L^{(\nu)}$ are constructed via energy integrals over probe-corrected transmissions:

$$G_{\alpha\beta} = -g_V \int dE [N_\alpha \delta_{\alpha\beta} - T_{\alpha\beta}(E)] f_0'(E)$$

$$L_{\alpha\beta} = -\frac{g_V}{eT} \int dE (E-E_F) [N_\alpha \delta_{\alpha\beta} - T_{\alpha\beta}(E)] f_0'(E)$$

$$K_{\alpha\beta} = \frac{g_V}{e^2T} \int dE (E-E_F)^2 [N_\alpha \delta_{\alpha\beta} - T_{\alpha\beta}(E)] f_0'(E)$$

where $T_{\alpha\beta}(E)$ is the transmission from $\beta$ to $\alpha$, $f_0'(E)$ is the Fermi function derivative, and $g_V=2e^2/h$. These integrals generalize to full NEGF implementations (Bergfield, 28 Aug 2025).

The floating condition determines probe voltage $V_p$ and temperature $\theta_p$ self-consistently, which are then substituted back to yield effective two-terminal linear response.

2. Determination of Probe Variables and Effective Currents

The probe's chemical potential $\mu_p$ and temperature $T_p$ adjust such that

$$I_p = (e/h) \sum_\alpha \int dE\, T_{p\alpha}(E)\,[f_p(E) - f_\alpha(E)] = 0$$

$$J_p = (1/h) \sum_\alpha \int dE\, (E-\mu_p) T_{p\alpha}(E)[f_p(E) - f_\alpha(E)] = 0$$

In linear response, expanding $f_\alpha(E)$ around equilibrium $(\mu,T)$ yields a system of equations for $(\mu_p,T_p)$ parameterized by Onsager-like coefficients $L_{11}^{p\alpha}$, $L_{12}^{p\alpha}$, $L_{22}^{p\alpha}$ (Saha et al., 2022). Solution of these provides probe conditions which filter into the final two-terminal conductance, thermopower, and thermal conductance coefficients.

The probe elimination via Schur complement yields effective lead-to-lead transport kernels:

$$L^\text{eff}_{LR} = L_{LR} - L_{Lp}[L_{pp}]^{-1}L_{pR}$$

All observables, such as conductance $G$, Seebeck $S$, and figure of merit $ZT$, derive from these renormalized coefficients (Bergfield, 28 Aug 2025).

3. Thermoelectric Coefficients and Bounds

Floating probes modify the definition of the two-terminal Seebeck coefficient:

$S = \frac{\Delta V}{\Delta T}\bigg|_{I=0}$

The algebraic expressions for $S$ are probe-dependent and contain cross-coefficients signifying inelastic and dephasing channels. For the adiabatic probe, the explicit formula (see Eq. 11 of (Sanchez et al., 2011)) involves products of $G$, $L$, $K$, and probe coefficients. In the isothermal case, terms proportional to $L_{13}$ or $L_{32}-L_{31}$ encode the effect of inelastic scattering.

Thermoelectric efficiency, commonly bounded by Carnot $\eta_C = \Delta T/T$, is subject to tighter linear-response bounds in probe-inclusive systems. For example (Saha et al., 2022):

$$\eta \leq \frac{\Delta\mu}{eT} \frac{1}{S} \leq \eta_C$$

and a stronger bound incorporating $ZT$:

$$\eta \leq \frac{\Delta\mu}{eT} \frac{1}{S} \sqrt{\frac{ZT}{ZT+1}} \leq \eta_C$$

with $ZT = L_{12}^2/(L_{11}L_{22}-L_{12}^2)$.

4. Magnetic-Field Asymmetry and Sommerfeld Expansion

Floating probes induce novel symmetry breaking in thermoelectric response. In contrast to two-terminal systems where the conductance $g(B) = g(-B)$ and $S_0 = -L_{11}/G_{11}$ is even in $B$, floating-probe Seebeck coefficients acquire odd components in $B$:

$\Phi = S(B) - S(-B)$

A crucial feature is that $\Phi=0$ in the lowest Sommerfeld order, because $$K_{\alpha\beta} \simeq -\frac{\pi^2k_B^2 T}{3e^2} G_{\alpha\beta}$$ causes numerator cancellation. Only higher-order Sommerfeld terms survive, scaling as $\Phi \propto T^3$ for $T\to 0$ (Sanchez et al., 2011). This generates magnetic-field asymmetry in Seebeck even for structurally symmetric systems, if floating probes are present.

5. Scaling Laws, Diffusive Crossover, and Environment-Assisted Transport

Floating probes convert ballistic and coherent transport to diffusive regimes as probe–system coupling is increased. In lattice models such as Aubry-André chains, the conductance $G$ displays universal scaling behaviors in the probe coupling strength $\gamma$ (Saha et al., 2022):

• At small $\gamma \ll 1$ in "no-transport" regimes (energy gaps), $G \propto \gamma^4$—termed environment-assisted transport.
• At large $\gamma \gg$ bandwidth, both $L_{11} \sim 1/\gamma^4$ and $G \sim 1/(N\gamma^4)$ (with $N$ the system length), typical of diffusive scaling.
• These power laws persist at finite $T$, with thermal broadening smoothing small-$\gamma$ enhancements.

This framework holds in molecular junctions and quantum dots, where decoherence destroys quantum thermopower fluctuations and suppresses mean $S, ZT$ (Sanchez et al., 2011, Bergfield, 28 Aug 2025). For multiple probes, the variance of thermopower decays rapidly ($\sim N_{probe}^{-4}$).

6. Quantum Interference, "Supernodes," and Dephasing via Probes

Probes introduce incoherent bypass channels which profoundly influence quantum interference features ("supernodes") (Bergfield, 28 Aug 2025). Near a destructive node at energy $E_0$, the transmission scales as $T_{\alpha\beta}(E)\sim |E-E_0|^{2n_{\alpha\beta}}$. In the presence of floating voltage–temperature probes:

• Effective nodal order becomes $n_\text{eff} = \min(a, \max(b,c))$ where $a$ is the coherent L–R node order, $b,c$ are probe-assisted channel orders.
• Single-site probe coupling typically collapses supernode order to lowest available (floor-building).
• Distributed probes generate an energy-independent floor suppressing $S_\text{max} \propto \Gamma_p^{-1/2}$ and $(ZT)_\text{max} \propto \Gamma_p^{-1}$.
• Once the incoherent floor dominates ($B\sim A(k_BT)^{2a}$), the fractional suppression is order-agnostic and universal for $S, ZT$.

Examples include benzene (meta-junction) and biphenyl (torsion angle $\theta$) systems, where probe-placement controls whether node reduction or incoherent floor dominates.

7. Applications and Perspectives

Floating thermoelectric probes serve as model environments for decoherence and inelastic processes, enabling rigorous benchmarking of theoretical efficiency limits and direct simulation of experimental scanning-probe devices. In quantum point contacts, scanning tips acting as floating probes can induce local and nonlocal Seebeck oscillations, mapping phase-coherent thermoelectric response and leading to perfect rectification (thermoelectric diode behavior) for specific probe placements (Fleury et al., 2021).

Real-space mapping via floating probes provides access to local density of states, direct measurement of decoherence rates, and manipulation of heat and charge flow in nanoscale circuits. This formalism facilitates analysis of thermal logic, energy harvesting devices, and quantum heat engines under realistic, environment-coupled conditions.