Thermal Hall conductivity of semimetallic graphite dominated by ambipolar phonon drag

This study reveals that the thermal Hall conductivity in highly oriented pyrolytic graphite is dominated by ambipolar phonon drag, resulting in a record-breaking Hall Lorenz number that is two orders of magnitude higher than expected from electronic carriers alone.

The Gist

Imagine a busy highway where two types of cars are driving in opposite directions: Electrons (negative charge) and Holes (positive charge, which are essentially empty spots where an electron could be). In most metals, one type of car dominates the traffic. But in Graphite, these two types of cars are perfectly balanced, driving at incredibly high speeds in both directions.

Now, imagine a strong wind blowing across this highway. In a normal situation, the wind would just push the cars sideways a little bit. But in this specific experiment, the researchers discovered something magical: the wind didn't just push the cars; it made the road itself (the atomic lattice) start moving sideways in a massive, coordinated way, carrying heat with it.

Here is the story of their discovery, broken down simply:

1. The Mystery: A Heat Signal Too Big to Be Just "Cars"

The scientists heated one end of a graphite strip and applied a magnetic field. They expected to see a sideways flow of heat (called the Thermal Hall Effect) caused by the moving electrons, just like the electric Hall effect.

However, the amount of heat moving sideways was 100 times larger than what the electrons alone could possibly explain. It was like seeing a tiny breeze create a hurricane. They calculated a "Hall Lorenz number" (a score that measures how well heat and electricity travel together) that was the highest ever recorded in any metal. This proved that something else was helping the heat move.

2. The Culprit: The "Phantom" Wind (Phonons)

In physics, heat in solids is often carried by vibrations in the atomic structure, called Phonons. Think of phonons as sound waves or ripples in a pond.

Usually, electrons and phonons (the ripples) don't interact much. But in graphite, because the electrons and holes are moving so fast and are so balanced, they act like a hydrodynamic fluid.

3. The Mechanism: The "Phantom Drag"

The paper proposes a mechanism called Ambipolar Phonon Drag. Here is a creative analogy:

• The Scene: Imagine a crowded dance floor (the graphite) where dancers (electrons) and empty spaces (holes) are spinning wildly.
• The Wind: A magnetic field acts like a strong wind blowing across the floor.
• The Interaction: The dancers get pushed by the wind. But because they are moving so fast and are so numerous, they start dragging the floorboards (the phonons) with them.
• The Result: The floorboards themselves start sliding sideways, carrying a massive amount of heat energy.

The term "Ambipolar" means that both the electrons and the holes are helping to drag the floorboards. They are working together, even though they are moving in opposite directions, to push the heat sideways.

4. The Twist: The Direction Flips

Here is the most fascinating part. As the scientists changed the temperature:

• At Cold Temperatures: The electrons were the main "draggers." They pulled the heat sideways in one direction (Negative).
• At Hot Temperatures: The holes took over as the main "draggers." They pulled the heat sideways in the opposite direction (Positive).
• The Flip: Around 100 Kelvin (about -173°C), the two forces canceled each other out, and the signal flipped from negative to positive.

This "flip" was the smoking gun. It proved that the heat wasn't just being carried by the electrons (which wouldn't flip) or just by the phonons (which wouldn't flip either). It was the interaction between the two types of charge carriers and the heat waves that caused the flip.

Why Does This Matter?

For over a century, scientists have relied on a rule called the Wiedemann-Franz Law, which says that in metals, heat and electricity are always linked in a predictable ratio.

This experiment broke that rule. Graphite showed that when you have a perfect balance of positive and negative charges moving fast, you can create a "super-highway" for heat that is completely different from the highway for electricity.

In Summary:
The researchers found that in graphite, the moving electric charges act like a powerful team of tugboats. They don't just move themselves; they drag the entire ocean (the atomic vibrations/phonons) along with them. This creates a massive, sideways flow of heat that is 100 times stronger than anyone expected, proving that in the quantum world, the whole can be much greater than the sum of its parts.

Technical Summary

1. Problem Statement

The thermal Hall effect (THE), the thermal analog of the electric Hall effect, has been observed in various quantum materials where the heat carriers are electrons, magnons, or phonons. However, in metals, the Wiedemann-Franz (WF) law typically dictates that the ratio of thermal to electrical conductivity is proportional to temperature, governed by the Sommerfeld Lorenz number ($L_0$).

• The Gap: While deviations from the WF law are known in specific contexts, the origin of large, non-electronic thermal Hall signals in compensated semimetals with high-mobility carriers and dominant phonon heat transport remains unclear.
• The Specific Challenge: Graphite is a compensated semimetal with equal densities of high-mobility electrons and holes, and it possesses an exceptionally high lattice thermal conductivity. Despite extensive study of its longitudinal transport properties, its transverse thermal Hall conductivity ($\kappa_{xy}$) had never been explored. The authors aim to determine the mechanism driving the thermal Hall effect in graphite and whether it adheres to or violates the WF law.

2. Methodology

• Sample Preparation: The study utilized Highly Oriented Pyrolytic Graphite (HOPG) crystals. To overcome the challenge of graphite's high longitudinal thermal conductivity (which makes establishing a measurable temperature gradient difficult), the samples were cleaved into extremely thin sheets (e.g., ~7.3 µm thickness) with moderate width.
• Experimental Setup:
  • Transport Measurements: Experiments were conducted in high-vacuum chambers (Quantum Design PPMS and Oxford Teslatron PT) ranging from 28.2 K to 302.5 K.
  • Configuration: A "one-heater-four-thermocouples" method was employed. Four Type-E thermocouples were placed symmetrically on the sample to measure both longitudinal ($\nabla T_x$) and transverse ($\nabla T_y$) temperature gradients simultaneously. This setup allowed for cross-verification of data to ensure signal reliability.
  • Fields: Measurements were performed under magnetic fields up to 1 T (and higher for specific coefficients) to extract Hall and Nernst responses.
• Data Analysis:
  • Two-Band Model: The electric Hall conductivity ($\sigma_{xy}$) was fitted to a two-band model to extract electron ($n_e, \mu_e$) and hole ($p, \mu_h$) densities and mobilities.
  • Nernst Conductivity: The Nernst conductivity ($\alpha_{xy}$) was derived from measured resistivity ($\rho_{xx}, \rho_{xy}$) and thermoelectric coefficients ($S_{xx}, S_{xy}$).
  • Phonon Drag Quantification: The authors compared the measured $\kappa_{xy}$ against the electronic contribution predicted by the WF law ($L_0 \sigma_{xy} T$) and analyzed the relationship between the Nernst signal and the thermal Hall signal to isolate the phonon drag component.

3. Key Contributions and Results

A. Record-Breaking Thermal Hall Conductivity

• Magnitude: The measured thermal Hall conductivity ($\kappa_{xy}$) in graphite is two orders of magnitude larger than the electronic contribution predicted by the WF law at low temperatures (28.2 K).
• Hall Lorenz Number: This discrepancy yields a record Hall Lorenz number ($L_{xy} = \kappa_{xy} / \sigma_{xy} T$) of $164.9 \times 10^{-8} \, \text{V}^2\text{K}^{-2}$ (approximately $67 L_0$). This is the largest Hall Lorenz number ever observed in a metal, indicating a massive non-electronic contribution to heat transport.
• Sign Reversal: Unlike the electric Hall conductivity ($\sigma_{xy}$), which remains positive, the thermal Hall conductivity ($\kappa_{xy}$) exhibits a temperature-dependent sign reversal. It is negative at low temperatures (peaking at -818.9 mW/K·m at 28.2 K) and becomes positive at room temperature (reaching 3300 mW/K·m at 302.5 K).

B. Ruling Out Purely Phononic Origins

• In insulators, the thermal Hall signal often peaks at the same temperature as the longitudinal thermal conductivity ($\kappa_{xx}$). In graphite, $\kappa_{xy}$ and $\kappa_{xx}$ have distinct temperature dependencies and do not peak simultaneously.
• The sign reversal of $\kappa_{xy}$ further rules out a purely phononic origin, as phonon chirality alone would not typically cause such a dramatic sign change in a compensated system without carrier involvement.

C. Identification of Ambipolar Phonon Drag

• Mechanism: The authors propose that the dominant mechanism is ambipolar phonon drag. This is a hydrodynamic interplay where momentum is exchanged between the phonon bath and the reservoirs of both electrons and holes.
• Evidence:
  • The Nernst conductivity ($\alpha_{xy}$) shows a field and temperature dependence strikingly similar to $\kappa_{xy}$, including the sign reversal.
  • Using the relation $\kappa_{drag}^{xy} = S_{drag} T \alpha_{xy}$, the authors quantified the phonon-drag Seebeck coefficient ($S_{drag}$).
  • At low temperatures, the drag is dominated by electrons (negative signal), while at high temperatures, it is dominated by holes (positive signal). The sign reversal occurs around 100 K when these competing contributions cancel out.
  • The extracted $S_{drag}$ reaches -60 µV/K at 28.2 K, consistent with the phonon-drag peak observed in the longitudinal Seebeck coefficient.

4. Significance

• Violation of Wiedemann-Franz Law: The study demonstrates a severe violation of the WF law in a standard metal/semimetal, establishing that in systems with high-mobility carriers and strong phonon coupling, heat transport can be decoupled from charge transport to an extreme degree.
• New Paradigm for Thermal Transport: It identifies "ambipolar phonon drag" as a potent mechanism for generating large thermal Hall signals. This explains how a compensated semimetal can generate a transverse heat current significantly larger than what electrons alone could carry.
• Material Benchmark: Graphite is now established as the material with the highest observed Hall Lorenz number, providing a critical testbed for theories of hydrodynamic electron-phonon interactions.
• Theoretical Implications: The results support theoretical frameworks (such as those by Herring and recent Kubo formula applications) suggesting that electric currents can drive phonon energy flows, creating a "phonon drag Peltier effect" that amplifies thermal Hall responses.

In conclusion, the paper resolves the origin of the giant thermal Hall effect in graphite, attributing it to a complex, temperature-dependent momentum exchange between phonons and both types of charge carriers (electrons and holes), rather than a simple phonon or electron effect.