Interaction driven transverse thermal resistivity in a phonon gas

This paper challenges the non-interacting paradigm of the phonon thermal Hall effect by proposing that magnetic-field-modulated phonon-phonon interactions generate a transverse thermal resistivity, a mechanism that successfully explains experimental data across seven crystalline insulators.

The Gist

The Big Idea: Heat Has a "Hall Effect" Too?

You probably know about the Hall Effect in electricity. If you send an electric current through a wire and apply a magnetic field, the electrons get pushed to the side, creating a voltage across the wire. It's like a crowd of people walking down a hallway who suddenly get pushed sideways by a strong wind.

For a long time, scientists thought this "sideways push" only happened to charged particles (like electrons). But in the last 20 years, they discovered that heat (carried by vibrations called phonons) also gets pushed sideways in a magnetic field, even in materials that don't conduct electricity at all. This is called the Phonon Thermal Hall Effect.

The big question this paper answers is: Why does this happen?

The Old Theory vs. The New Idea

The Old Theory (The Solo Dancer):
Most scientists previously thought this happened because the atoms in the material were arranged in a special, "chiral" (handed) way, like a spiral staircase. They imagined the heat vibrations were dancing alone, and the magnetic field just twisted their path because of their shape.

The New Theory (The Crowd Mosh Pit):
The authors of this paper say, "No, that's not the whole story." They argue that the effect is driven by interactions—specifically, how these heat vibrations bump into each other.

To explain this, they use a brilliant analogy involving molecular gases (like air in a balloon).

The Analogy: The Spinning Top and the Billiard Table

Imagine a room full of spinning tops (molecules) bouncing off each other.

1. The Spin: If you apply a magnetic field, it makes these tops wobble and precess (spin like a gyroscope).
2. The Collision: When two spinning tops collide, the way they bounce off each other depends on how they were spinning.
3. The Result: Even if the tops aren't "handed" (chiral), the combination of their spin and the magnetic field makes them bounce slightly to the left or right. This creates a sideways flow of heat.

The authors say phonons (heat vibrations in a solid) act just like these spinning tops. They aren't just dancing alone; they are constantly crashing into one another. The magnetic field changes how they crash, creating a sideways drift.

The "Drifting Nucleus" Mechanism

So, how does this actually work in a solid crystal like WS2 (Tungsten Disulfide)?

1. The Heat Flow: When you heat one side of the crystal, the atoms vibrate and move heat to the cold side. This creates a tiny, collective "drift" of the atoms themselves, moving very slowly in the direction of the heat.
2. The Magnetic Twist: As these atoms drift, the magnetic field acts on them. Because the electrons surrounding the nucleus shield the nucleus, the magnetic field creates a weird, quantum force called a Berry Force.
3. The Push: This Berry Force acts like a Lorentz force (the force that pushes electrons), but it pushes the nuclei sideways.
4. The Balance: The sideways push creates a temperature difference across the material. The system reaches a balance where the sideways "thermal force" cancels out the magnetic "Berry force."

The Metaphor:
Imagine a river flowing downstream (heat flow). The magnetic field is like a strong crosswind.

• In the old view, the water molecules were shaped like propellers, so the wind pushed them sideways.
• In this new view, the water molecules are just round balls, but the wind changes how they bump into each other. These collisions create a "traffic jam" that forces the whole river to drift sideways.

What Did They Find?

The team tested this theory on WS2 and compared it to six other insulators (like Silicon, Germanium, and Black Phosphorus).

1. The Peak: They found that the sideways heat flow peaks at almost the same temperature as the straight-ahead heat flow. This suggests they are linked by the same "traffic rules" (collisions).
2. The Universal Rule: They discovered a simple formula that predicts the size of this sideways effect for all these different materials.
   • It depends on how fast sound travels in the material.
   • It depends on the distance between atoms.
   • It depends on the magnetic field.
3. The Surprise: This simple formula worked surprisingly well, even for materials that aren't "chiral." It proved that you don't need special spiral shapes to get this effect; you just need atoms that interact and bump into each other.

Why Does This Matter?

This paper changes how we think about heat.

• It's not just about shape: You don't need a "twisted" crystal to get a thermal Hall effect.
• It's about interaction: The way particles bump into each other is the key driver.
• It's a new tool: By understanding this "Berry force" on drifting atoms, scientists can better design materials for cooling electronics or harvesting waste heat.

In a nutshell: The authors showed that heat in insulators behaves like a crowd of people in a magnetic field. The magnetic field doesn't just push the individuals; it changes how they bump into each other, causing the whole crowd to drift sideways. This simple "interaction-driven" picture explains the mystery of the Thermal Hall Effect across many different materials.

Technical Summary

1. Problem Statement

The phonon thermal Hall effect (the generation of a transverse temperature gradient in an insulator under a longitudinal heat current and a magnetic field) has been observed in numerous materials over the last two decades. However, the theoretical understanding remains fragmented:

• Current Theories: Most existing theories rely on either intrinsic mechanisms (topological properties of phonon bands, Berry curvature) or extrinsic mechanisms (scattering off defects).
• The Gap: These approaches largely ignore phonon-phonon interactions (anharmonicity), which are the fundamental drivers of longitudinal thermal conductivity ($\kappa_{xx}$).
• The Analogy: In neutral molecular gases, the Senftleben-Beenakker (SB) effect demonstrates that a magnetic field can induce transverse thermal transport through interactions (collisions) between non-spherical molecules, without requiring chirality.
• The Question: Can a similar interaction-driven mechanism explain the thermal Hall effect in solid-state phonon gases, where particle number is not conserved?

2. Methodology

The authors employed a combination of experimental measurements, comparative analysis, and theoretical modeling:

• Experimental Subject: The study focused on 2H-WS$_2$ (Tungsten Disulfide), a layered transition metal dichalcogenide (TMD) insulator with a bandgap of ~1.2 eV.
• Measurements:
  • Measured longitudinal ($\kappa_{xx}$) and transverse ($\kappa_{xy}$) thermal conductivities using a three-thermocouple setup to monitor temperature gradients.
  • Applied magnetic fields up to 9 T.
  • Verified that electronic contributions to heat transport are negligible ($\kappa_{xx} \gg L_0 T \sigma_{xx}$) via Wiedemann-Franz law analysis.
• Comparative Analysis:
  • Compared the thermal Hall response of the phonon gas in WS$_2$ with the SB effect in molecular gases.
  • Analyzed data from seven different crystalline insulators (WS$_2$, Black P, Si, Ge, SrTiO$_3$, Nd$_2$CuO$_4$, and SiO$_2$) to test for universal scaling laws.
• Theoretical Framework:
  • Developed a model linking heat flux to nuclear drift velocity.
  • Applied the concept of Berry forces arising from the breakdown of the Born-Oppenheimer approximation in the presence of a magnetic field.
  • Derived an expression for transverse thermal resistivity based on the competition between the Berry force and a thermal restoring force.

3. Key Contributions

A. Experimental Characterization of WS$_2$

• Peak Alignment: Found that $\kappa_{xx}$ and $\kappa_{xy}$ peak at nearly the same temperature (~34 K and ~27 K, respectively), suggesting a common underlying mechanism.
• Scaling Law: Confirmed the scaling relation $\kappa_{xy} \propto \kappa_{xx}^2$ observed in other insulators. The authors argue this arises because the transverse thermal resistivity ($W_\perp$) is nearly temperature-independent over a wide range, rather than $\kappa_{xy}$ itself being universal.
• Upper Bound: The ratio $\kappa_{xy}/\kappa_{xx}$ (thermal Hall angle) respects a known upper bound related to the product of phonon wavelength and atomic displacement asymmetry.

B. Theoretical Paradigm Shift: Interaction-Driven Mechanism

The paper proposes a novel mechanism analogous to the SB effect in gases:

1. Non-Conservation of Phonon Number: Unlike molecular gases, phonon number is not conserved. A heat flux ($\vec{J}_q$) creates a net phonon flux density ($\vec{J}_n$) and a tiny drift velocity ($\langle \dot{R} \rangle$) of the crystal lattice nuclei.
2. Berry Force: In the presence of a magnetic field, this nuclear drift induces a Berry force (analogous to a Lorentz force) on the nuclei due to the screening of nuclear charge by electrons.
3. Transverse Resistivity: This Berry force causes a rigid rotation of the phonon flow. A transverse temperature gradient ($\nabla_y T$) builds up to counteract this force via an entropic (thermal) force.
4. Role of Anharmonicity: The mechanism relies on phonon-phonon interactions (anharmonicity) to allow the heat flux to drive the lattice drift. It does not require chiral phonons.

C. Quantitative Prediction

The authors derived a formula for the field-normalized transverse thermal resistivity:
$$\frac{W_\perp}{|B|} \approx \frac{\tilde{b}\tilde{a}}{\tilde{c}} \frac{e}{2k_B m_p} \frac{a^3}{v_s^2}$$
Where:

• $a$ is the interatomic distance.
• $v_s$ is the sound velocity.
• $\tilde{a}, \tilde{b}, \tilde{c}$ are dimensionless parameters related to momentum conservation, charge screening, and thermal force efficiency.

4. Results

• Universal Scaling: The derived formula successfully accounts for the experimentally measured $W_\perp$ across seven different insulators (Si, Ge, Black P, WS$_2$, SiO$_2$, SrTiO$_3$, Nd$_2$CuO$_4$).
• Parameter Extraction:
  • For simple, harmonic crystals (Si, Ge, Black P, WS$_2$, SiO$_2$), the extracted dimensionless parameter combination ($\frac{\tilde{b}\tilde{a}}{\tilde{c}}$) is between 0.1 and 0.8, validating the model's assumptions.
  • For highly anharmonic perovskite oxides (SrTiO$_3$, Nd$_2$CuO$_4$), the value is >1, suggesting that additional time scales or extrinsic skew scattering play a role in these specific materials.
• Thermal Hall Angle vs. Resistivity:
  • The Thermal Hall Angle ($\Theta_H = \kappa_{xy}/\kappa_{xx}B$) peaks at a universal value (~$10^{-4}$ T$^{-1}$) across all materials, suggesting a fundamental bound on the rotation of phonon flux independent of dissipation details.
  • The Thermal Resistivity ($W_\perp$) varies significantly between materials, depending on the specific dissipation mechanisms (anharmonicity).

5. Significance

1. Unification of Phenomena: The paper bridges the gap between the thermal Hall effect in solids and the Senftleben-Beenakker effect in gases, establishing that interactions (collisions/anharmonicity) are central to the phenomenon, not just topology or defects.
2. Chirality Not Required: It challenges the prevailing view that chiral phonons are a necessary condition for the thermal Hall effect, showing that non-chiral, non-spherical interactions can generate transverse transport.
3. New Physical Picture: It introduces a clear physical picture where a heat flux drives a minute nuclear drift, which couples to the magnetic field via a Berry force, generating the transverse signal.
4. Predictive Power: The simple analytical expression derived (Eq. 13) provides a quantitative framework to predict thermal Hall resistivity based on basic material parameters (sound velocity, interatomic distance), successfully explaining data across diverse material classes.
5. Distinction of Mechanisms: It clarifies the difference between the rotation of the phonon flux (universal bound) and the misalignment of heat/entropy flux (material-dependent dissipation), resolving discrepancies in previous literature regarding the universality of the effect.

In conclusion, the authors argue that the phonon thermal Hall effect is fundamentally an interaction-driven phenomenon mediated by Berry forces on drifting nuclei, offering a simpler and more universal explanation than previous topological or defect-scattering models.