Lorenz Ratio Measurements

• Lorenz ratio measurements are defined as the ratio of thermal conductivity to the product of electrical conductivity and temperature, serving as a benchmark for the Wiedemann–Franz law.
• Experimental techniques such as magnetothermal methods and low-temperature extrapolation isolate electronic transport and diagnose scattering mechanisms.
• Advanced optical and contactless methods reveal deviations in the Lorenz ratio that highlight quantum criticality and optimize thermoelectric material performance.

The Lorenz ratio is a central quantity in the analysis of coupled thermal and electrical transport in metals, semiconductors, and correlated electron systems. Defined as the ratio of thermal conductivity to the product of electrical conductivity and temperature, the Lorenz ratio $L = \kappa/(T\sigma)$ provides a benchmark for the validity of the Wiedemann–Franz law and serves as a diagnostic of underlying scattering mechanisms and carrier dynamics. Its measurement and interpretation underpin the paper of Fermi liquids, non-Fermi liquids, quantum critical points, hydrodynamic regimes, and the design of thermoelectric materials.

1. Fundamental Concepts and Definitions

The Wiedemann–Franz law expresses a universal relation in conventional metals: $$\frac{\kappa}{\sigma T} = L_0 \qquad L_0 = \frac{\pi^2 k_B^2}{3e^2}$$ where $\kappa$ is the thermal conductivity, $\sigma$ is the electrical conductivity, $T$ is temperature, and $L_0$ is the Sommerfeld value, approximately $2.44 \times 10^{-8}\,\mathrm{W}\Omega/\mathrm{K}^2$. The law holds in the regime where heat and charge are transported by the same well-defined Landau quasiparticles and elastic scattering dominates. Deviations from $L_0$, especially in the low-temperature limit, signal anomalous transport: the presence of inelastic scattering, carrier fractionalization, or breakdown of quasiparticle integrity.

The Lorenz ratio can be generalized beyond the longitudinal channel, with transverse (Hall) components such as

$L_{xy} = \frac{\kappa_{xy}}{T\sigma_{xy}}$

and, in hydrodynamic systems or multi-band scenarios, multiple Lorenz ratios may be considered, each reflecting distinct transport channels and symmetry constraints.

2. Experimental Measurement Techniques

Magnetothermal Resistance Methods

Suppression of electronic thermal conduction via a large transverse magnetic field enables separation of phononic and electronic contributions. For single crystals (e.g., Al, Cu, Zn (Yao et al., 2017), Bi$_2$Te$_3$ (Yao et al., 2017)), $\kappa_\mathrm{tot}$ and $\sigma$ are measured as functions of field up to saturation. By plotting $\kappa_\mathrm{tot}$ vs.\ $\sigma$ for increasing field, extrapolation to $\sigma\to0$ yields the lattice thermal conductivity $\kappa_\mathrm{ph}$; fitting formulas such as

$$\kappa_\mathrm{tot}(H) = \kappa_e(H) + \kappa_\mathrm{ph}$$

and field-dependent conductivity models (e.g., $\sigma(H) = \sigma_0 / [1 + (\mu H)^2]$) are used to extract electronic transport coefficients and the Lorenz ratio.

Low-Temperature Transport and Extrapolation

In heavy-fermion compounds with quantum critical points, e.g., YbAgGe (Dong et al., 2013), electrical resistivity $\rho(T)$ and thermal conductivity $\kappa(T)$ are measured down to tens of millikelvin. Bosonic contributions (phonons, magnons) become negligible as $T\to0$; the Lorenz ratio $L(T)/L_0 = (\kappa(T)/T)\rho(T)/L_0$ often displays a linear-in-$T$ behavior below a threshold temperature (e.g., $T<120\,$mK), enabling robust extrapolation to zero temperature: $L(T)/L_0 = aT + L(0)/L_0$ where $L(0)/L_0$ indicates the zero-temperature limit, with deviations—such as $0.92 \pm 0.03$ at a quantum critical field—diagnosing violations of the WF law.

Optical and Contactless Methods

High-temperature measurements, especially in oxides or correlated materials (e.g., Sr$_3$Ru$_2$O$_7$, Sr$_2$RuO$_4$ (Sun et al., 2023)), employ optical heating and contactless temperature sensing. Thermal diffusivity $D$ is extracted from phase lag measurements, and thermal conductivity calculated via $K = cD$ (where $c$ is heat capacity). Combined with standard electrical resistivity, this enables construction of dimensionless Lorenz ratio curves across wide temperature ranges.

3. Theoretical Frameworks and First-Principles Calculations

Boltzmann Transport and Energy-Dependent Scattering

First-principles calculations based on DFT band structures and Boltzmann transport theory yield the Lorenz number through energy integrals over transport distributions (Wang et al., 2017): $$\sigma = \int_{-\infty}^{+\infty} \sigma'(E) dE, \qquad L = \frac{\kappa_e}{\sigma T}$$ where $\sigma'(E) = q^2 \Xi(E)(-\partial f_0/\partial E)$ and $\Xi(E)$ incorporates the number of conduction channels $M(E)_A$ and the energy-dependent mean-free-path $\lambda(E)$.

Crucially, energy-dependent scattering rates—often assumed to follow the density-of-states—can yield Lorenz numbers substantially below the canonical minimum $2(k_B/q)^2$ in non-degenerate limits, a result of the shape of the transport distribution. Accurate resolution near band edges, proper inclusion of bipolar conduction, and precise handling of energy-dependent scattering are all necessary for reliable Lorenz number computation.

Distinct Relaxation Times for Charge and Heat

At intermediate temperatures, models that distinguish momentum (electrical) and energy (thermal) relaxation rates provide insight into anomalous Lorenz ratios (Li et al., 2020). Usage of separate relaxation times,

$$\delta f_{nk} = \left(\frac{\partial f_{nk}^0}{\partial \epsilon_{nk}}\right)[e\,\tau_{nk}^{(p)}\,v_{nk}\cdot \mathbf{E} + (\epsilon_{nk}-\epsilon_F)\tau_{nk}^{(E)}/T (v_{nk}\cdot\nabla T)],$$

allows for substantial reduction of $L$ at low and intermediate $T$, reflecting enhanced energy relaxation via small-angle inelastic collisions.

Thermodynamic Reformulation and DFT

Thermodynamic descriptions relate the Lorenz number to ratios of heat capacity and electrochemical capacitance (Wang et al., 2020): $L T = C_\mathrm{el}/C_N$ This connection enables computation of $L$ directly from the electron density of states, bypassing kinetic details. In semiconductors, contributions from bipolar excitations can drive $L$ above the Sommerfeld value, important for separating electronic and lattice thermal conductivities in thermoelectrics.

4. Influence of Scattering Mechanisms

Electron–Phonon and Electron–Electron Scattering

Departures from the WF law are shaped by the interplay of elastic (impurities, defects) and inelastic (phonon, electron–electron) scattering (Lee et al., 2020, Wang et al., 2017, Li et al., 2020). In the compensated metal WP$_2$, strong inelastic interband “Baber” collisions—and weak phonon contributions at intermediate temperatures—lead to suppressed Lorenz ratios, sometimes below $1/4$ of $L_0$. With disorder, the Lorenz ratio is enhanced, approaching $L_0$ as $T\to0$.

In systems with dominant electron–electron scattering and suppressed momentum relaxation (the so-called Planckian regime (Sun et al., 2023)), Lorenz ratios at high temperature can greatly exceed $L_0$, reflecting efficient heat transport by phonons and ineffective charge transport due to rapid electron scattering.

Table: Manifestations of Scattering in Lorenz Ratio Measurements

System/type	Dominant scattering	Lorenz ratio behavior
Fermi liquid metal (low-$T$)	Elastic (impurities)	$L \to L_0$
Heavy-fermion QCP (YbAgGe)	Inelastic (critical)	$L/L_0 < 1$ at $H_c$
Conventional semiconductor	Electron–phonon, holes	$L > L_0$ w/ bipolar effects
Graphene hydrodynamic regime	Electron–electron	$L \gg L_0$ near neutrality
Non-Fermi liquids (MFL)	Inelastic, disorder	$L \to L_0$ as $T \to 0$, but $L(T)-L_0 \propto -T$ (Tulipman et al., 2022)
Contact-resistance limited	External bottleneck	False negatives for $L > L_0$ (Fong, 2017)

Effects of Screening and Dimensionality

In compensated metals with screened Coulomb interactions, the Lorenz ratio exhibits nonmonotonic dependence on the screening wave number $k_{TF}$, with a minimum at intermediate values. The lack of Galilean invariance in systems such as hydrodynamic graphene (Levchenko, 24 Mar 2025) introduces magnetic friction, driving nonmonotonic field dependence and an order-of-magnitude enhancement of $L$ near charge neutrality.

5. Lorenz Ratio in Correlated and Mesoscopic Systems

Quantum Criticality and Breakdown of Quasiparticles

Measurements in YbAgGe (Dong et al., 2013) at the quantum critical field reveal $L(T\to0)/L_0 = 0.92\pm 0.03$, indicating moderate WF law violation due to strong inelastic scattering and the breakdown of heavy quasiparticles. By contrast, itinerant systems like Sr$_3$Ru$_2$O$_7$ and CeRu$_2$Si$_2$ obey WF near their QCPs, underscoring the role of magnetic frustration and local moment physics.

Exotic Universal Lorenz Ratios and Orthogonality Catastrophe

In multi-channel charge Kondo circuits (Kiselev, 2023), the generalized Wiedemann–Franz law takes the form

$$R_l = \frac{K}{TG} = L_0 \cdot \mathcal{R}_{n,m}, \qquad \mathcal{R}_{n,m} = \frac{3(M+2)(N+2)}{3NM+2N+2M}$$

where $M$, $N$ are numbers of quantum point contacts. These "magic" Lorenz ratios persist even in the strong correlation regime, reflecting the power-law tunneling density of states set by Anderson's orthogonality catastrophe. This phenomenology is directly accessible by charge and heat transport measurements in quantum simulators.

Hydrodynamic Transport and Magnetic Friction

In graphene electron liquids (Levchenko, 24 Mar 2025), diagonal and Hall Lorenz ratios deviate from each other due to hydrodynamic collective effects and lack of Galilean invariance. At charge neutrality, $L$ can exceed $L_0$ by an order of magnitude, with a strong nonmonotonic dependence on magnetic field governed by emergent magnetic friction and disorder-induced density fluctuations.

6. Practical and Methodological Considerations

Impact of Contact Resistance

Direct electronic thermal conductivity experiments are susceptible to contact resistance (Fong, 2017). Enhancement of $L$ may be artificially suppressed by finite contact resistance: the measured Lorenz number $\widetilde{\mathcal{L}}_m$ is bounded by the sample and contact resistances, with formulas such as

$$\widetilde{\mathcal{L}}_m^{(S)} = \left[1 - \left(1 - \frac{1}{\widetilde{\mathcal{L}}}\right)\left(\frac{1}{1+2\tilde{r}}\right)^3 \right]^{-1}$$

($\tilde{r} = R_c / R_s$). The asymmetric (external heater) method mitigates this bottleneck and is preferred for probing non-Fermi liquid deviations.

Construction of Confidence Intervals

Nonparametric inference for generalized Lorenz curves (income distribution) uses empirical likelihood (EL), adjusted EL, and transformed EL (Ratnasingam et al., 2023). Modified EL statistics are shown to converge to scaled $\chi^2_1$ limits, supporting robust interval estimation for Lorenz ordinates under complex, skewed distributions.

7. Implications for Thermoelectric Materials and Correlated Physics

Accurate extraction of the Lorenz number facilitates separation of electronic and lattice contributions to thermal conductivity, critical for evaluating and optimizing the thermoelectric figure of merit $zT$. In thermoelectrics, low Lorenz numbers are sought to suppress electron-mediated heat transport, thus boosting $zT$ (Wang et al., 2017, Wang et al., 2020). First-principles methods using DFT density of states provide high-throughput screening tools, while direct measurement approaches (magnetothermal, optical) are required for materials with complex scattering landscapes.

More broadly, Lorenz ratio anomalies serve as sensitive probes for strong correlation effects, quantum criticality, strange metallicity, and orthogonality catastrophe in mesoscopic systems. The temperature dependence and leading corrections to $L(T)$—not simply its absolute value—distinguish Fermi liquid and marginal Fermi liquid states (Tulipman et al., 2022); complementary longitudinal and transverse measurements further aid in classifying exotic phases.

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In summary, Lorenz ratio measurements integrate sophisticated experimental designs, theoretical analyses, and materials optimization strategies to unravel the intertwined dynamics of charge and heat transport. The depth and breadth of contemporary research—spanning first-principles thermodynamic approaches, quantum critical metals, hydrodynamic electron liquids, and income inequality curves—reflects the ongoing centrality of the Lorenz ratio in condensed matter, mesoscopic physics, and statistical inference.