Transition from Population to Coherence-dominated Non-diffusive Thermal Transport

This paper presents a Wigner Transport Equation-based framework to model non-diffusive thermal transport driven by both phonon populations and coherences, predicting significant size-dependent thermal conductivity deviations in low-conductivity materials like CsPbBr$_3$ and La$_2$Zr$_2$O$_7$ at experimentally accessible length scales.

The Gist

Imagine heat moving through a solid material like a crowd of people trying to walk through a hallway.

The Old Way: The "Bottleneck" Theory

For a long time, scientists understood heat using a model called the Boltzmann Transport Equation. Think of this like a crowd of people walking down a hallway, bumping into each other and the walls.

• The Analogy: If the hallway is wide and the people are far apart, they walk in a straight line, bumping occasionally. This is diffusive transport. The heat spreads out slowly and predictably, like a drop of ink spreading in water.
• The Limit: This model works great for materials like silicon (used in computer chips) where the "people" (phonons, or heat vibrations) are distinct and don't interact too weirdly.

The New Discovery: The "Ghost Dance"

However, the authors of this paper looked at "messy" materials (like certain crystals used in solar cells or thermal barriers) where the hallway is crowded, the walls are wobbly, and the people are moving very fast. In these materials, the old model breaks down.

They found that heat doesn't just move like a crowd; it moves like a wave or a ghost dance.

• The Analogy: Imagine two people in the crowd are holding hands and dancing in perfect sync. Even if they bump into a wall, they don't stop; they "tunnel" through the obstacle because they are connected. In physics, this is called coherence.
• The Problem: The old math assumed everyone was an individual. It couldn't account for these "dancing pairs" (coherences) that move together. The new math, called the Wigner Transport Equation, treats the heat as a mix of individual walkers and these synchronized dancing pairs.

What They Did: The "Flashlight" Experiment

To test this, the researchers didn't just look at how heat moves in a big block of material. They imagined shining a super-fast, tiny flashlight (a heat source) on the material and watching how the heat ripples out.

1. The Setup: They used a "thermal grating" (like a comb made of light) to create a pattern of hot and cold spots.
2. The Twist: They changed the size of the comb teeth (the distance between hot spots) and how fast they flashed the light.
3. The Result:
   • In Silicon: The heat acted like a normal crowd. The "dancing pairs" (coherences) didn't matter much.
   • In "Messy" Crystals (CsPbBr3 and La2Zr2O7): When they made the hot spots very close together (nanometers apart) or flashed the light very fast, the heat behaved completely differently. The "dancing pairs" took over! The heat didn't just diffuse; it flowed in a coordinated, wave-like manner.

Why This Matters: The "Traffic Jam" vs. The "Highway"

The paper predicts that in these special materials, if you try to move heat over very short distances (a few hundred nanometers, which is smaller than a human hair), the material becomes much more efficient at moving heat than we thought.

• The Metaphor: Imagine a traffic jam.
  • Old View: Cars (heat) are stuck, inching forward one by one.
  • New View: In these specific materials, the cars can link up into a "train" (coherence) and zip through the traffic jam without stopping.
• The Catch: This "train" only works if the road (the material) is short enough. If the road is too long, the train breaks apart, and the cars go back to inching forward.

The Big Picture

This research is a game-changer for two reasons:

1. Better Electronics: It helps us design materials that can handle heat better in tiny, high-speed computer chips, preventing them from overheating.
2. New Physics: It proves that heat isn't just a random walk; sometimes, it's a synchronized dance. The authors have built a new mathematical "map" (the Wigner Transport Equation) that allows scientists to predict exactly how this dance happens in real-world experiments.

In short: They found that in certain materials, heat doesn't just wander; it can "tunnel" and "dance" in sync, allowing it to move faster and more efficiently over tiny distances than previously thought possible. This opens the door to designing better materials for energy and computing.

Technical Summary

1. Problem Statement

Traditional heat transport in crystalline insulators is typically modeled using the Phonon Boltzmann Transport Equation (BTE). The BTE assumes that heat is carried by incoherent phonon populations and is valid when the system size and timescales are much larger than phonon mean free paths (MFPs) and lifetimes.

However, the BTE fails to accurately describe thermal transport in materials with:

• Low thermal conductivity (e.g., complex unit cells, strong anharmonicity).
• Overlapping phonon bands where phonon linewidths ($\Gamma$) become comparable to the inter-branch energy spacing ($\Delta\omega$).

In these regimes, phonon wave-like tunneling between branches becomes significant. The BTE, which treats the phonon density matrix as diagonal (populations only), cannot capture phonon coherences (off-diagonal elements of the density matrix) and interference effects. The authors aim to develop a framework to model non-diffusive transport where these coherence effects dominate, specifically addressing arbitrary space- and time-dependent heat sources.

2. Methodology

The authors generalize the Wigner Transport Equation (WTE) to include arbitrary heat sources and solve it in the frequency domain.

• Theoretical Framework:

  • They utilize the Wigner Transport Equation, which describes the evolution of the phonon density matrix $N(R, q, t)$ (a rank-2 tensor) rather than just a scalar distribution function.
  • The equation includes a commutator term $[\Omega, N]$ representing coherent evolution (tunneling) and an anti-commutator term $\{V, \nabla N\}$ representing drift.
  • A source term $Q(R, q, t)$ is introduced, capable of generating both phonon populations (diagonal) and coherences (off-diagonal).

• Mathematical Formulation:

  • The equation is Fourier transformed in space ($k$) and time ($\omega$) to yield a linear system: $\tilde{L}\tilde{N} = \tilde{Q}$.
  • Green's Function Approach: The system response is characterized by a Green's function $\tilde{G} = \tilde{L}^{-1}$, which encodes the response to an impulsive excitation.
  • Collision Term: The authors employ the Relaxation Time Approximation (RTA) for the collision operator. Crucially, in this approximation, the collision term annihilates coherences but does not couple populations to coherences; coherences are driven solely by the velocity operator's off-diagonal elements.

• Numerical Implementation:

  • Two strategies are implemented to solve the linear system:
    1. Explicit Green's Function Inversion: Efficient for simple unit cells or when responses to many different sources are needed.
    2. Iterative Solvers (GMRES): More efficient for complex unit cells with many phonon branches or when only a few specific sources are required.
  • Dynamical Thermal Conductivity: Defined via the linear response to a harmonic temperature gradient. For arbitrary sources (e.g., pump-probe), the local temperature is solved self-consistently to enforce energy conservation.

3. Key Contributions

• Generalized WTE Framework: Developed a frequency-domain formulation of the WTE that accommodates arbitrary space-time heat sources, bridging the gap between theoretical transport models and time-resolved experimental techniques (like transient thermal grating).
• Decomposition of Transport: Introduced a dimensionless parameter, transport character ($\chi$), defined as $\chi = (\kappa_P - \kappa_C) / (\kappa_P + \kappa_C)$, where $\kappa_P$ is the population contribution and $\kappa_C$ is the coherence contribution. This allows for a quantitative distinction between population-dominated ($\chi \to 1$) and coherence-dominated ($\chi \to -1$) transport.
• Prediction of Size Effects: Demonstrated that coherence contributions persist at length scales where population contributions are suppressed, leading to a transition in the dominant transport mechanism.

4. Key Results

The framework was applied to Silicon (Si), Cesium Lead Bromide (CsPbBr$_3$), and Lanthanum Zirconate (La$_2$Zr$_2$O$_7$).

• Material Comparison:

  • Silicon: Coherence contributions are negligible; transport is purely population-dominated, consistent with standard BTE results.
  • CsPbBr$_3$ and La$_2$Zr$_2$O$_7$: These materials exhibit dense clusters of flat optical branches with small energy splittings ($\Delta\omega \approx \Gamma$). Here, coherence contributions are significant, becoming the dominant mechanism for thermal conductivity above 240 K (CsPbBr$_3$) and 800 K (La$_2$Zr$_2$O$_7$).

• Size Effects (Spatial Filtering):

  • As the grating period ($\Lambda$) decreases (increasing $k$), long-mean-free-path population modes are suppressed.
  • Coherence Robustness: Coherence contributions are controlled by mode-pair coherence propagation lengths rather than single-mode MFPs. Consequently, they remain largely unchanged until grating periods reach the 50 nm scale.
  • This leads to a regime where the total thermal conductivity is dominated by coherences at length scales of hundreds of nanometers to microns, even at room temperature.

• Dynamical Effects (Temporal Filtering):

  • As the excitation frequency ($\omega$) increases, long-lifetime population modes are suppressed.
  • Similar to spatial filtering, coherence channels survive longer. At high frequencies (e.g., 31 GHz for La$_2$Zr$_2$O$_7$), the transport character shifts from population-dominated to coherence-dominated.
  • The shift is not due to individual modes changing their nature, but rather the suppression of population-dominated channels.

• Experimental Accessibility:

  • The predicted deviations from bulk diffusivity occur at length scales (hundreds of nm to microns) and time scales (GHz frequencies) accessible via current Transient Thermal Grating (TTG) techniques, including Extreme Ultraviolet (EUV) and hard X-ray implementations.

5. Significance

• Theoretical Advancement: This work moves beyond the limitations of the BTE for low-thermal-conductivity materials, providing a rigorous quantum mechanical description of heat transport that includes wave interference and tunneling.
• Experimental Guidance: It predicts specific regimes (nanoscale lengths and high frequencies) where coherence dominates, offering clear targets for experimental validation using ultrafast pump-probe and transient grating techniques.
• Material Design: Understanding the role of coherence is crucial for engineering thermal properties in complex materials (e.g., thermoelectrics, thermal barrier coatings) where traditional scattering models fail.
• Computational Tool: The open-source Python package greenWTE provides a robust tool for calculating dynamical thermal conductivities and size effects in complex crystalline systems.

In conclusion, the paper establishes that in materials with strong anharmonicity and complex unit cells, thermal transport transitions from a particle-like (population) regime to a wave-like (coherence) regime at experimentally accessible scales, fundamentally altering our understanding of heat conduction in these systems.