Voltage-Temperature Probe Theory

• VTP is a device that enforces dual zero-current conditions to define local voltage and temperature, ensuring local thermodynamic equilibrium in nonequilibrium conductors.
• It employs self-consistent nonlinear equations solved via fixed-point and Newton–Raphson iterations, guaranteeing rapid convergence and adherence to the second law of thermodynamics.
• By avoiding unphysical heat dissipation seen in traditional voltage probes, VTP is crucial for precise measurements in quantum transport, thermoelectric devices, and nanoscale thermometry.

A voltage-temperature probe (VTP) is a conceptual and computational device introduced in mesoscopic and quantum transport theory to establish local thermodynamic equilibrium in out-of-equilibrium conductors by enforcing simultaneously vanishing charge and heat currents into a fictitious probe. This dual equilibration guarantees that the local electrochemical potential (voltage) and temperature can be unambiguously defined far from equilibrium. The VTP constitutes the rigorous theoretical framework for local measurements of voltage and temperature, and for modeling dephasing, thermalization, and inelastic processes in quantum transport systems, building on and generalizing Büttiker’s voltage probe model.

1. Mathematical Formulation and Self-Consistency Conditions

The voltage-temperature probe is formulated within the Landauer–Büttiker scattering framework for quantum transport. The essential requirement is that both the net electrical (particle) current $I^\text{(0)}_P$ and the net heat (energy) current $I^\text{(1)}_P$ into the probe vanish: $$I^\text{(0)}_P(\mu_P, T_P) = 0,\quad I^\text{(1)}_P(\mu_P, T_P) = 0$$ with

$$I^\text{(ν)}_P = \frac{1}{h} \int_{-\infty}^{\infty} d\omega\, (\omega-\mu_P)^{\nu} \mathcal{T}_{PS}(\omega)\bigl[f_S(\omega) - f_P(\omega)\bigr]$$

where $\nu=0,1$, $\mathcal{T}_{PS}(\omega)$ is the energy-dependent probe-to-system transmission, $f_S(\omega)$ is the local nonequilibrium distribution of the system, and $f_P(\omega;\mu_P,T_P)$ is the equilibrium Fermi–Dirac distribution of the probe.

Solving these coupled nonlinear equations yields unique values for the local chemical potential and temperature that the probe would measure; this uniqueness is a mathematical manifestation of the second law of thermodynamics, as established rigorously via the Banach fixed-point theorem and the monotonicity of the currents with respect to probe parameters (Jacquet et al., 2011, Shastry et al., 2016).

For numerical solution, either one-dimensional (for voltage or temperature probe alone) or Newton–Raphson–type two-dimensional iterations (for full VTP) are employed, with the Jacobian given by the partial derivatives of $I_P$, $Q_P$ with respect to $\mu_P$, $T_P$. The contraction property ensures rapid convergence.

2. Physical Interpretation and Theoretical Significance

Physically, the VTP acts as an idealized “electron thermometer” that does not perturb the steady-state distribution of the system: it is coupled weakly (in the tunnel limit), and its dual zero-current conditions imply that neither charge nor energy is dissipated to or from the probe, making it noninvasive. This is crucial because voltage-only probes can serve as unphysical heat sources or sinks in general nonequilibrium conditions, violating thermodynamic consistency (Erdogan et al., 28 Aug 2025).

The uniqueness and existence conditions for VTP readouts are strongly constrained:

• If the average local system energy is below the centroid of the coupling spectrum, a positive temperature is registered; in cases of strong population inversion, a unique negative temperature is found.
• In the broad-band probe limit, uniqueness is guaranteed as long as the population inversion criterion is respected (Shastry et al., 2016).

The VTP thus provides foundational justification for defining intensive thermodynamic variables in arbitrary quantum steady states—far from equilibrium and with arbitrary interactions—enabling well-posed statements about entropy production, local equilibrium, and the extension of thermodynamics to nano and mesoscopic systems.

3. Comparison with Voltage Probe and Coherent Potential Approximation

A fundamental distinction exists between voltage-only probes (VPs) and VTPs. VPs impose zero net electrical current into the probe, adjusting only the chemical potential, but fix the temperature to that of an external reservoir. Under perfect spatial and thermal symmetry, the VP’s heat current may also vanish, making the models operationally similar. However, in general (whenever spatial/thermal asymmetry exists) the VP acts as an uncontrolled heat source or sink, violating conservation of energy and potentially resulting in spurious entropy production (Erdogan et al., 28 Aug 2025). The VTP, by contrast, consistently enforces both charge and energy conservation locally.

The analogy between voltage probes and the coherent potential approximation (CPA) in disordered systems has been formalized: the CPA’s local vertex function plays the role of an effective chemical potential, and the CPA formalism essentially simulates phase-breaking (inelastic) scattering in a manner equivalent to introducing a VP. Including both voltage and temperature equilibration can be anticipated to extend the CPA analogy to inelastic thermal processes (Zhuravlev et al., 2012).

Model	Enforces $I^\text{(0)}{=}0$	Enforces $I^\text{(1)}{=}0$	Thermodynamic Consistency
Voltage Probe	Yes	No	Only under symmetric/asymptotic cases
VTP	Yes	Yes	Always

4. Computational Frameworks: Iterative Schemes and Fixed-Point Algorithms

In both the linear and nonlinear regime, the determination of probe parameters proceeds by mapping the nonlinear self-consistent condition onto a contractive fixed-point problem. For the voltage (or temperature) probe: $X_j(\mu_j) = \int f(E,\beta,\mu_j) M_j(E)\,dE$ is a strictly monotonic mapping that can be inverted. Self-consistency is enforced via a fixed-point iteration: $\mathbf{X}^{(n+1)} = F(\mathbf{X}^{(n)})$ with $F$ defined via the transmission and Fermi functions, as described in (Jacquet et al., 2011). The Banach fixed-point theorem guarantees uniqueness and rapid convergence. In the full VTP case, two-dimensional Newton–Raphson updates are employed: $$\left(\begin{array}{c} \mu_P^{k+1} \ T_P^{k+1} \end{array}\right) = \left(\begin{array}{c} \mu_P^{k} \ T_P^{k} \end{array}\right) - \mathbf{D}^{-1} \left(\begin{array}{c} I_P(\mu_P^k, T_P^k) \ Q_P(\mu_P^k, T_P^k) \end{array}\right)$$ where $\mathbf{D}$ is the Jacobian matrix.

These algorithms are robust, with rapid convergence observed even for strong nonlinearities. For example, in the three-terminal system, the convergence of probe potential is independent of probe–system coupling parameter $\epsilon$ (Jacquet et al., 2011).

5. Experimental and Theoretical Applications

The VTP formalism has been instrumental in several contexts:

• Mesoscopic Conductors: In three-terminal and Aharonov–Bohm ring systems, the VTP enables local readout of chemical potential and temperature, yielding physical predictions for measured phases (e.g., the AB phase shift tracks the intrinsic quantum dot phase even in the nonlinear regime) (Jacquet et al., 2011, Bedkihal et al., 2013).
• Heat Engines and Broken Time-Reversal Symmetry: In quantum heat engines such as the triple-dot Aharonov–Bohm device, imposing VTP conditions leads to a reduced effective Onsager description and enables analytical and numerical exploration of efficiency at maximum power (EMP). In this nonlinear context, the VTP efficiency can surpass the Curzon–Ahlborn limit if time-reversal symmetry is broken and suitable asymmetry and nonlinear dissipation are present:

$$\eta^*(P_{\rm max}) = \eta_C \frac{x^2}{2(2x^2-3x+2)-\alpha T X_L^T x^2}$$

where $x$ is an asymmetry parameter linked to Seebeck coefficients, and $\alpha$ quantifies nonlinear dissipation. In contrast to voltage probes, VTPs require additional system anisotropy for TRS breaking to impact efficiency (Behera et al., 23 Oct 2024).

• Decoherence, Dephasing, and Thermodynamic Consistency: VTPs provide a physically consistent way to enforce decoherence in both charge and heat channels. In molecular junctions, e.g., benzene-based systems, only the VTP yields realistic lifting of interference nodes in both charge and heat currents under practical conditions; the VP provides incorrect predictions for heat currents under asymmetry (Erdogan et al., 28 Aug 2025).
• Thermoelectric Corrections and Precision Voltage Measurements: Voltage measurements at finite temperature are subject to thermoelectric corrections (Peltier and Seebeck effects) only accurately accounted for if the probe's temperature is measured and adjusted concurrently with its potential. For example, the voltage correction $\Delta V_p$ can reach up to 24% of the peak voltage in ballistic quantum wires; accurate correction is given by:

$\Delta V_p = S_{ps}(T_p-T_0)$

where $S_{ps}$ is the junction thermopower (Bergfield et al., 2014).

• Implementation in Integrated Circuits and Thermometry: VTPs are realized (for temperature measurement) as noise thermometers in CMOS ICs, leveraging Johnson–Nyquist noise and cross-correlation techniques to perform dissipationless, spatially distributed temperature mapping in quantum and cryogenic electronics (Mishonov et al., 2022, Ridgard et al., 23 Feb 2025). For combined high-resolution electrical and thermal measurements at high temperatures, contactless ultra-high-temperature probe stations use the 3ω/2ω method integrated with VTP measurement strategies (Jalabert et al., 27 Jun 2025).

6. Design Principles, Constraints, and Practical Realization

Key design requirements for a VTP are:

• Noninvasiveness: Coupling to the system must be weak (tunnel) and broadband, avoiding perturbation of system's nonequilibrium distribution.
• Dual Equilibration: Both chemical potential and temperature must be adjusted self-consistently so that no net charge or energy enters or leaves the probe.
• Calibration and Thermal Coupling: In experimental realizations, minimizing extrinsic thermal coupling is essential to maximize measurement fidelity. Residual backgrounds can be subtracted, but uncertainties in offset noise or coupling can limit absolute accuracy (Mishonov et al., 2022, Ridgard et al., 23 Feb 2025).
• Electronic Implementation: Practical VTPs span a range of architectures, from analog noise thermometers, which map measured voltage to absolute temperature via $\langle U_f^2 \rangle = 4 R_N k_B T \mathcal{B}$, to digitally integrated on-chip temperature sensors with automatic voltage regulation and frequency-coded outputs robust to DVFS conditions (Zambrano et al., 2022).

7. Impact, Limitations, and Future Directions

The VTP formalism is essential for all theoretical and simulation studies seeking thermodynamic consistency in quantum and nanoscale transport. The accurate modeling of decoherence effects, symmetry breaking, and the full interplay of charge and energy conservation requires the application of voltage-temperature probe conditions. For thermoelectric and spintronic device optimization, only VTP-based models can predict correct efficiency, rectification, and entropy production in realistic, structurally asymmetric systems.

Limitations include the complexity of extracting system-specific transmission functions for strongly correlated or interacting systems, and the practical difficulties of simultaneous non-invasive voltage and temperature measurement at nanoscales or under extreme conditions. Nevertheless, the VTP framework is now foundational in the modeling of energy and information flow at the quantum level, in both conceptual and applied studies.