Phonon Thermal Hall Effect in quartz and its absence in silica

This paper reports the observation of a phonon thermal Hall effect in crystalline quartz but not in amorphous silica, attributing the phenomenon to the misalignment of energy and entropy currents caused by magnetic-field-induced transverse forces on lattice drift, a mechanism confirmed by the effect's dependence on crystal quality rather than disorder.

The Gist

The Big Idea: Heat Has a "Side Step"

Imagine you are pushing a crowd of people (heat) through a hallway from a hot door to a cold door. Usually, the crowd moves straight ahead. But, if you turn on a giant magnet, something strange happens: the crowd doesn't just move forward; they start drifting sideways, like a group of dancers suddenly stepping to the left or right in unison.

In physics, this sideways drift of heat is called the Thermal Hall Effect.

For a long time, scientists thought this only happened in materials with special magnetic properties or electric charges. But recently, they found it happening in plain old insulators (materials that don't conduct electricity) like glass and quartz. The big mystery was: Why does it happen in some materials but not others?

This paper solves that mystery by comparing two "cousins" made of the exact same building blocks: Quartz (a crystal) and Silica (glass).

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The Experiment: The Crystal vs. The Glass

The researchers set up a race between two runners:

1. Quartz: Think of this as a perfectly organized marching band. Every atom is in its exact spot in a repeating pattern.
2. Silica (Glass): Think of this as a chaotic mosh pit. It's made of the same atoms (Silicon and Oxygen), but they are jumbled up with no long-range order.

They heated one end of both materials and applied a strong magnetic field to see if the heat would drift sideways.

The Results:

• Quartz (The Crystal): The heat drifted sideways! The researchers detected a clear signal. The cleaner the crystal (fewer defects), the stronger the sideways drift.
• Silica (The Glass): Nothing happened. The heat went straight from hot to cold, ignoring the magnetic field completely.

The Conclusion: To get this "side step" effect, you need order. You need a crystal lattice. If the material is a messy glass, the effect disappears.

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The "Why": The Two-Lane Highway Analogy

So, why does the crystal do it and the glass doesn't? The authors propose a brilliant analogy involving a two-lane highway.

Imagine heat traveling through a material isn't just one big blob of energy. Instead, it's like traffic moving in two different lanes:

• Lane A (The Fast Lane): Carries energy efficiently but doesn't create much "traffic noise" (entropy).
• Lane B (The Slow Lane): Carries energy but creates a lot of "traffic noise" (entropy) because the cars are bumping into each other.

In a Crystal (Quartz):
The magnetic field acts like a wind blowing across the highway. Because the two lanes behave differently (one is orderly, one is chaotic), the wind pushes them in different ways.

• The "Fast Lane" gets pushed one way.
• The "Slow Lane" gets pushed a different way.
• The Result: The total flow of heat (the sum of both lanes) gets twisted sideways. The energy is moving forward, but the "disorder" (entropy) is moving at a slight angle. This misalignment creates the Thermal Hall Effect.

In a Glass (Silica):
The highway is so messy and broken that the two lanes are indistinguishable. The wind (magnetic field) can't push them differently because they are already a chaotic mess. The result? No sideways drift.

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The "Berry Force": A Tiny Nudge

The paper also offers a simple picture of how the magnetic field pushes the heat.

Imagine the atoms in the crystal are like tiny balls on springs. When you heat them, they vibrate and drift slightly toward the cold side.

1. The Drift: The heat makes the atomic nuclei drift very slowly.
2. The Magnetic Kick: As these drifting nuclei move through the magnetic field, they feel a tiny sideways push called a Berry Force (named after a physicist).
3. The Counter-Force: Nature hates being pushed sideways without a reason. The temperature difference creates an "entropic force" (a desire to spread out) that pushes back.
4. The Balance: The sideways magnetic push and the entropic push balance out, creating a steady, tiny sideways temperature difference.

It's like a leaf floating down a river. The current (heat) pushes it down. A crosswind (magnetic field) tries to blow it sideways. The leaf drifts at an angle until the water's resistance balances the wind.

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Why This Matters

This discovery is a big deal for three reasons:

1. Order is Key: It proves that you don't need exotic magnetic materials to get this effect; you just need a clean, ordered crystal structure.
2. Disorder Kills It: If you mess up the crystal (make it "dirty" or turn it into glass), the effect vanishes. This rules out theories that said "disorder causes the effect."
3. A New Connection: It links the behavior of heat in solids to the behavior of gas molecules. Just like spinning molecules in a gas can be steered by a magnet, the "vibrations" (phonons) in a crystal can be steered too.

In a Nutshell:
Heat in a crystal is like a disciplined army marching in formation. If you blow a magnetic wind on them, they step sideways in unison. Heat in glass is like a chaotic crowd; the wind just blows through them, and they keep moving straight. This paper proves that structure creates the magic.

Technical Summary

1. Problem Statement

The Phonon Thermal Hall Effect (PTHE) refers to the observation of a transverse temperature gradient (misalignment between heat flux and temperature gradient) in insulating solids under a magnetic field. While observed in numerous crystalline insulators, its microscopic origin remains a subject of debate.

• Key Controversy: Is the effect driven by disorder (impurities, defects) or by intrinsic properties of the phonon gas (e.g., phonon-phonon interactions)?
• The Gap: Previous studies often compared different materials with distinct chemical compositions, making it difficult to isolate the role of structural order (crystalline vs. amorphous) versus chemical composition.
• Objective: To determine if crystalline order is a necessary condition for the PTHE by comparing two forms of Silicon Dioxide ($SiO_2$): Quartz (crystalline) and Silica (amorphous/glass), which share the same fundamental building blocks ($SiO_4$ tetrahedra) but differ in long-range order.

2. Methodology

The authors employed a rigorous comparative experimental approach using identical setups and samples to eliminate systematic errors.

• Samples:
  • Quartz #1 & #2: Two single-crystal samples with different levels of purity (disorder). Quartz #2 is "dirtier" (lower longitudinal thermal conductivity) than Quartz #1.
  • Silica: An amorphous glass sample.
  • Note: All samples share the same chemical composition ($SiO_2$) and basic structural unit ($SiO_4$ tetrahedra).
• Experimental Setup:
  • Geometry: A "one-heater, three-thermometer" configuration. A longitudinal heat flux ($J_x$) is applied, and both longitudinal ($\Delta T_x$) and transverse ($\Delta T_y$) temperature differences are measured simultaneously.
  • Thermometry: Three Cernox 1070 thermometers were used. The transverse signal ($\Delta T_y$) is derived from the resistance difference between two thermometers ($R_3 - R_1$).
  • Signal Extraction: The magnetic field was swept from -10 T to +10 T. The genuine thermal Hall signal is the antisymmetric (odd) component of the response with respect to the magnetic field ($B$). The symmetric component (due to magnetoresistance of thermometers and misalignment) was subtracted.
• Measurement Range: Temperatures from 5 K to 40 K, covering the peak thermal conductivity regime for quartz.

3. Key Contributions

1. Definitive Test of Disorder vs. Order: By comparing crystalline and amorphous $SiO_2$ with identical chemistry, the study isolates the role of structural periodicity.
2. Disproving Disorder as the Driver: The study demonstrates that the "cleaner" crystal (higher $\kappa_{xx}$) exhibits a larger thermal Hall signal, directly contradicting theories that attribute the effect to impurity scattering or side-jump mechanisms.
3. Universal Transverse Resistivity: The authors identify that the transverse thermal resistivity ($W_\perp$) is nearly identical in both quartz samples and independent of disorder, suggesting a universal mechanism.
4. Theoretical Framework (Two-Channel Model): The paper proposes a general mechanism for PTHE based on the Senftleben-Beenakker effect in molecular gases, adapted for phonons. It argues that a thermal Hall effect arises whenever heat is conducted via two channels that differ in:
   • Entropy production rates.
   • Coupling to the magnetic field.
5. Berry Force Mechanism: A simple physical picture is proposed where the heat flux induces a tiny drift velocity of lattice nuclei. The magnetic field exerts a transverse Berry force on these drifting nuclei, which is balanced by an entropic restoring force.

4. Key Results

A. Longitudinal Transport ($\kappa_{xx}$)

• Quartz: Shows a peak in thermal conductivity at ~16 K, characteristic of phonon-phonon scattering (Umklapp processes) in crystals. Quartz #1 (cleaner) has a higher peak than Quartz #2.
• Silica: Shows a monotonic decrease in $\kappa_{xx}$ with decreasing temperature (typical of glasses), with no peak dominated by phonon-phonon scattering.

B. Transverse Transport (Thermal Hall Effect)

• Quartz (Crystalline): A finite, linear, antisymmetric transverse temperature gradient ($\Delta T_y$) was detected.
  • Disorder Correlation: The cleaner sample (Quartz #1) exhibited a larger thermal Hall conductivity ($\kappa_{xy}$) than the dirtier sample (Quartz #2). This rules out disorder as the primary driver.
• Silica (Amorphous): No detectable thermal Hall signal was observed within the experimental resolution.
  • Conclusion: Crystalline order is a necessary ingredient for the emergence of the phonon thermal Hall effect in $SiO_2$.

C. Transverse Thermal Resistivity ($W_\perp$)

• The transverse thermal resistivity, defined as $W_\perp = \nabla_y T / J_x$, was found to be independent of temperature and sample purity in the crystalline samples.
• Value: $W_\perp / B \approx 10^{-6} \, \text{m}\cdot\text{K}\cdot\text{W}^{-1}\cdot\text{T}^{-1}$.
• Comparison: This value is consistent across different insulators, suggesting a universal bound.

5. Significance and Theoretical Implications

• Mechanism of PTHE: The paper argues that the PTHE is not an intrinsic property of a single phonon type but arises from the misalignment between energy flux and entropy flux.
  • In a phonon gas, heat is carried by two reservoirs (low-$q$ "Normal" phonons and high-$q$ "Umklapp" phonons).
  • These reservoirs produce entropy differently and couple differently to the magnetic field (potentially via a geometric Berry phase).
  • This mismatch causes the entropy current to deviate from the energy current, generating a transverse temperature gradient.
• Analogy to Molecular Gases: The authors draw a parallel to the Senftleben-Beenakker effect in molecular gases, where magnetic fields alter collision rates of rotating molecules, creating a transverse thermal conductivity.
• Quantitative Prediction: The authors derive a simple formula for transverse thermal resistivity based on a Berry force acting on drifting nuclei:
  $$W_\perp \approx \frac{Ze|B|}{k_B \rho v_s^2}$$
  Using material parameters for $SiO_2$, this theoretical estimate ($1.7 \times 10^{-6}$) matches the experimental order of magnitude ($10^{-6}$) remarkably well, despite the simplicity of the model.

Conclusion

This study establishes that crystallinity is essential for the phonon thermal Hall effect in $SiO_2$, as the effect vanishes in the amorphous phase. It refutes the hypothesis that disorder drives the effect and instead supports a mechanism where the interplay between distinct heat-conducting channels (differing in entropy production and magnetic coupling) generates a transverse response. The findings suggest a universal origin for PTHE in insulators, potentially rooted in the geometric (Berry) phases of lattice dynamics.