🔬 mesoscale physics

Thermal Einstein-de Haas Effect Induced by Chiral Phonons in Carbon Nanotubes

This study predicts that chiral single-walled carbon nanotubes undergo thermally induced rigid-body rotation via the Einstein-de Haas effect, driven by finite phonon angular momentum arising from the splitting of degenerate transverse acoustic and optical modes in small-diameter, intermediate-chiral-angle nanotubes.

Original authors: Raimu Akimoto, Hiroyasu Matsuura, Takahiro Yamamoto

Published 2026-02-26

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CC BY 4.0

Original authors: Raimu Akimoto, Hiroyasu Matsuura, Takahiro Yamamoto

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a tiny, hollow tube made entirely of carbon atoms, so small that a million of them could fit across the width of a human hair. This is a Carbon Nanotube (CNT). Now, imagine that some of these tubes are perfectly straight and symmetrical, while others are twisted like a spiral staircase or a corkscrew. These twisted ones are called chiral nanotubes.

This paper is about a fascinating discovery: If you heat one end of a twisted carbon nanotube, it might actually start spinning on its own.

Here is the story of how that happens, broken down into simple concepts:

1. The "Dance" of Atoms (Phonons)

Inside any solid object, atoms aren't sitting still; they are constantly vibrating. In physics, we call these vibrations phonons. Think of phonons like a crowd of people doing the "wave" in a stadium. Usually, this wave just moves up and down or side to side.

However, in a chiral (twisted) nanotube, the geometry forces these atoms to do something special. Instead of just bobbing up and down, the atoms start moving in tiny circles, like dancers spinning on a stage. This circular motion gives the vibration a property called angular momentum (the same physics that makes a spinning top stay upright).

2. The "Traffic Jam" of Heat

In a perfect, symmetrical tube (straight up and down), these circular spins cancel each other out. For every atom spinning clockwise, another is spinning counter-clockwise, so the net result is zero.

But in a twisted tube, the symmetry is broken. When you apply a temperature gradient (making one end hot and the other cold), it's like creating a traffic jam for these vibrations. The "heat waves" rush from the hot side to the cold side. Because of the tube's twist, the atoms on the "right-handed" spin get a boost, while the "left-handed" ones get slowed down (or vice versa).

This creates an imbalance: More atoms are spinning one way than the other. Suddenly, the whole tube has a net "spin" built into its heat energy.

3. The Conservation Law (The Ice Skater Effect)

Here comes the magic trick of physics: Conservation of Angular Momentum.

Imagine an ice skater spinning. If she extends her arms, she slows down; if she pulls them in, she speeds up. But the total "spin" of the system must remain constant.

In our nanotube, the heat energy has created a "spin" inside the atoms (the phonons). To keep the total spin of the universe balanced, the tube itself (the physical object) must spin in the opposite direction to cancel out the internal atomic spin.

This is called the Thermal Einstein–de Haas Effect. It's like the tube is saying, "My atoms are spinning this way, so I, the tube, must spin that way to keep things fair."

4. Why Carbon Nanotubes are the Superstars

The researchers found that this spinning effect is strongest under two specific conditions:

Small Diameter: Think of a thin straw vs. a thick pipe. The thinner the tube, the easier it is to get it spinning. The atoms are closer to the center, making the "moment of inertia" (resistance to spinning) very low. It's like a figure skater pulling their arms in tight to spin faster.
The "Goldilocks" Twist: The tube shouldn't be too straight (zigzag) or too curved (armchair). It needs a "medium" twist (around 15 degrees) to maximize the circular motion of the atoms.

5. The Result: A Spinning Nanomachine

The paper calculates that if you heat a tiny, twisted carbon nanotube, it could spin at about 1 revolution per second.

While 1 revolution per second sounds slow to us, for a nanotube that is only a few nanometers wide, this is incredibly fast. It's like a tiny, invisible propeller driven purely by heat.

Why Does This Matter?

New Energy Tech: It suggests we could build microscopic motors that run on heat without any batteries or electricity, just by using temperature differences.
Understanding the Universe: It proves that heat isn't just "jiggling"; it can actually create mechanical motion and rotation in the right materials.
Future Sensors: These spinning tubes could be used as incredibly sensitive sensors to detect tiny changes in temperature or magnetic fields.

In a nutshell: The authors discovered that by twisting a carbon nanotube and heating it, they can turn heat energy into mechanical rotation. It's a tiny, heat-powered engine hidden inside a single molecule, waiting to be discovered.

1. Problem Statement

The paper addresses the theoretical prediction and quantification of the Thermal Einstein–de Haas (EdH) effect in chiral single-walled carbon nanotubes (SWCNTs).

Context: While the conventional EdH effect describes mechanical rotation induced by magnetization, the thermal EdH effect posits that a temperature gradient applied to a chiral crystal can induce macroscopic rigid-body rotation due to the conservation of angular momentum.
Gap: Although theoretically proposed, the thermal EdH effect has not yet been experimentally observed. Previous studies focused on bulk chiral crystals (e.g., $\alpha$ -quartz, $\alpha$ -HgS), but the magnitude of the effect in these materials is often too small to detect.
Objective: The authors investigate whether SWCNTs, specifically chiral ones, can exhibit a significantly enhanced thermal EdH effect due to their unique one-dimensional geometry and intrinsic chiral phonon modes.

2. Methodology

The study employs a combination of lattice dynamics and Boltzmann transport theory within a semiclassical framework.

System Modeling:
- The authors model semiconducting chiral SWCNTs defined by chiral indices $(n, m)$ .
- Atomic interactions are modeled using the Tersoff potential, known for accurately describing carbon-carbon interactions.
- The system is treated as a rigid body for the final rotation calculation, neglecting electron spin contributions (justified by the large bandgap of semiconducting SWCNTs and weak spin-orbit coupling in carbon).
Theoretical Framework:
- Phonon Angular Momentum ( $L_{ph}$ ): Calculated using the formula $L_{ph} = \sum_{q\lambda} l_{q\lambda} (n(\omega_{q\lambda}) + 1/2)$ , where $l_{q\lambda}$ is the angular momentum of a specific phonon mode derived from the cross product of displacement eigenvectors.
- Non-Equilibrium Transport: The deviation from equilibrium ( $\delta n$ ) caused by a temperature gradient ( $\nabla T$ ) is calculated using the Boltzmann transport equation in the relaxation-time approximation.
- Thermal Angular Momentum Coefficient ( $\kappa_{\alpha\beta}$ ): Defined as the proportionality constant between the induced phonon angular momentum density and the temperature gradient ( $J_{ph} = \kappa (-\nabla T)$ ).
- Conservation of Angular Momentum: The total angular momentum is conserved ( $L_{lat} + L_{ph} = \text{const}$ ). Therefore, the generation of phonon angular momentum ( $\Delta L_{ph}$ ) induces an equal and opposite rigid-body angular momentum ( $\Delta L_{lat}$ ), resulting in rotation.
Parameters:
- Calculations were performed at 300 K.
- A constant phonon relaxation time ( $\tau_{ph}$ ) was assumed to isolate the geometric dependencies on tube diameter ( $d_t$ ) and chiral angle ( $\theta$ ).

3. Key Contributions

Mechanism Elucidation: The paper establishes that the splitting of degenerate transverse acoustic (TA) and optical phonon modes in chiral SWCNTs (which are degenerate in zigzag and armchair tubes) generates finite phonon angular momentum associated with circular atomic motion.
Geometric Scaling Laws: The study derives specific scaling laws for the thermal angular momentum coefficient ( $\kappa_{AM}$ ) regarding the tube diameter and chiral angle.
Quantitative Prediction: It provides a concrete estimate of the angular velocity for a specific SWCNT configuration, predicting values significantly larger than those in bulk chiral crystals.

4. Key Results

A. Phonon Dispersion and Angular Momentum

Mode Splitting: In chiral SWCNTs, the TA and certain optical phonon modes split into two branches with opposite signs of angular momentum. This splitting is absent in highly symmetric zigzag ( $\theta=0^\circ$ ) and armchair ( $\theta=30^\circ$ ) tubes.
Enantiomeric Sign: Left-handed (L) and Right-handed (R) SWCNTs exhibit phonon angular momenta of opposite signs, reflecting their mirror-image nature.
Diameter Dependence: As the tube diameter ( $d_t$ ) increases, the splitting between chiral phonon branches decreases, causing the angular momentum to diminish.

B. Dependence on Tube Diameter ( $d_t$ )

The thermal angular momentum coefficient scales approximately as $|\kappa_{AM}| \propto d_t^{-2}$ .
Implication: Smaller diameter nanotubes are far more efficient at generating phonon angular momentum. This is attributed to the stronger confinement and the specific behavior of transverse acoustic modes in 1D systems.

C. Dependence on Chiral Angle ( $\theta$ )

At a fixed diameter, the magnitude of the coefficient follows a sinusoidal dependence: $|\kappa_{AM}| \propto \sin(6\theta)$ .
Maximum Efficiency: The effect is maximized at $\theta = 15^\circ$ , which represents an intermediate chiral angle between the zigzag ( $0^\circ$ ) and armchair ( $30^\circ$ ) limits. This suggests that "strong helicity" is required for the effect, but the extreme limits (pure zigzag/armchair) yield zero effect due to symmetry.

D. Predicted Angular Velocity

Using realistic parameters for a (6,5) SWCNT ( $d_t = 0.76$ $d_{t} = 0.76$ nm, $\theta = 27^\circ$ $θ = 2 7^{\circ}$ ) with a temperature gradient of $\nabla T = 10$ $\nabla T = 10$ K and a length of $1.24$ $\mu$ $μ$ m:
- The estimated angular velocity is $\omega \sim 1$ rad/s.
Comparison: This value is orders of magnitude larger than previously reported values for bulk chiral crystals.
Reason for Enhancement: The large velocity is attributed to:
1. High group velocities of phonon modes (including TA modes) folded at the Brillouin zone boundaries.
2. The extremely small moment of inertia of the 1D cylindrical structure compared to bulk crystals.

5. Significance

Experimental Feasibility: The predicted angular velocity ( $\sim 1$ rad/s) is within the range of experimentally observable values, suggesting that the thermal EdH effect in SWCNTs could be the first material system where this phenomenon is successfully detected.
Spintronics and Chiral Physics: The work reinforces the concept of chiral phonons as a carrier of angular momentum that can be manipulated via thermal gradients, offering new avenues for phononics and spintronics (via phonon-spin coupling).
Material Design: The findings provide clear design rules for maximizing the effect: use small-diameter SWCNTs with intermediate chiral angles (near $15^\circ$ ).

In conclusion, this paper theoretically demonstrates that chiral SWCNTs are ideal candidates for observing the thermal Einstein–de Haas effect, overcoming the sensitivity limitations of previous bulk material studies through their unique 1D geometry and chiral phonon properties.