Indicators for phonon hydrodynamics from first principles predictions of thermal conductivity

This paper proposes a computationally efficient indicator, defined as the ratio of thermal conductivity calculated via the full linearized Peierls-Boltzmann equation to that from the relaxation time approximation, to identify and accelerate the discovery of materials exhibiting phonon hydrodynamics, while also highlighting the necessity of rigorous Brillouin zone sampling convergence for accurate predictions.

The Gist

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

The Usual Way: The Diffusive Crowd
In most materials (like the silicon in your computer chip), heat moves like a chaotic crowd. People bump into each other constantly, changing direction randomly. They don't move as a group; they just jostle their way forward. This is called "diffusive" heat flow. It's slow, messy, and follows the standard rules of physics we learned in school (Fourier's Law).

The Special Way: The Hydrodynamic River
In some special materials (like graphite or diamond), something magical happens. The "people" (which are actually tiny vibrations called phonons) stop bumping into each other randomly. Instead, they start moving together in a synchronized, fluid-like stream, like a river flowing smoothly. This is called hydrodynamic heat flow. It's incredibly fast and efficient. Scientists have seen this happen in graphite at room temperature, but finding other materials that do this is like looking for a needle in a haystack.

The Problem: The Expensive Search
To find these special materials, scientists use powerful computers to simulate how phonons behave.

• The "Easy" Method (RTA): This is like guessing how the crowd moves by only looking at how fast individuals get tired. It's fast to calculate but often wrong for these special materials because it misses the fact that the crowd is moving together.
• The "Hard" Method (Full Solution): This simulates every single interaction between every person in the crowd. It's incredibly accurate but takes a massive amount of computer power and time. It's like trying to simulate every single step of a million people in a stadium just to see if they are marching in sync.

The Discovery: A Simple "Litmus Test"
The authors of this paper found a clever shortcut. They discovered a simple ratio you can calculate that tells you if a material has this special "river-like" heat flow, without needing to do the super-expensive, full simulation.

They call this ratio $\kappa_{LPBE} / \kappa_{RTA}$.

Here is the analogy:

• Imagine you have two ways to predict how fast a river flows.
  • Method A (RTA): Predicts the speed based only on how fast a single swimmer can paddle.
  • Method B (Full Solution): Predicts the speed by simulating the entire river's current, including how the water pushes the swimmers together.
• The Indicator: If Method B gives you a result that is much higher than Method A (a high ratio), it means the water is pushing the swimmers together. The crowd is moving as a team! This high ratio is the "smoking gun" that the material has hydrodynamic heat flow.
• If the two methods give similar results (a ratio close to 1), the crowd is just jostling randomly (diffusive flow).

Why This Matters
Before this, scientists had to run the super-expensive "Method B" simulations to know if a material was special. Now, they can run the cheap "Method A" simulation, multiply it by a factor, and check the ratio. If the ratio is high, they know they've found a winner. This acts like a low-cost filter to quickly scan thousands of materials to find the ones that might have this super-efficient heat flow.

A Crucial Warning
The paper also warns that this test is very sensitive to how you set up your computer simulation. If you don't look at enough details (like zooming in too little on the material's structure), you might get a fake "high ratio" that disappears when you look closer. It's like taking a blurry photo of a crowd and thinking they are marching in sync, only to realize that when you zoom in, they are actually just walking randomly. You have to be careful to get the "resolution" just right to trust the result.

In Summary
The paper provides a simple, cheap, and fast way to spot materials where heat flows like a fluid rather than a gas. By comparing a simple calculation to a slightly more complex one, scientists can now quickly identify new materials that might revolutionize how we manage heat in electronics, without needing to run expensive, time-consuming simulations for every single candidate.

Technical Summary

Technical Summary: Indicators for Phonon Hydrodynamics from First Principles Predictions

Problem Statement
Heat transport in semiconductors is typically described by Fourier's law, where phonons diffuse due to scattering. However, in materials where momentum-conserving Normal (N) scattering dominates over momentum-dissipating Umklapp (U) scattering, phonons can exhibit collective drift, leading to a hydrodynamic heat flow regime. This regime, characterized by phenomena such as second sound and thermal rectification, has been experimentally observed in materials like graphite, sodium fluoride, and boron arsenide. Despite these observations, the discovery of new materials and experimental conditions capable of sustaining hydrodynamic heat flow at technologically relevant temperatures (e.g., near room temperature) is hindered by a lack of computationally efficient, microscopic guidelines. Existing methods to predict thermal conductivity ($\kappa$) often fail to distinguish between diffusive and hydrodynamic regimes or are too computationally expensive for high-throughput screening.

Methodology
The authors employ first-principles predictions of thermal conductivity by solving the linearized Peierls–Boltzmann equation (LPBE) for the non-equilibrium deviational phonon distribution function ($f_\lambda$). The study compares three distinct approaches to solving the LPBE:

1. Full LPBE Solution ($\kappa_{LPBE}$): Obtained via iterative solvers that account for the complete, dense collision matrix ($\Omega$), including off-diagonal terms that couple different phonon modes. This captures the collective drift of phonons.
2. Relaxation Time Approximation (RTA) ($\kappa_{RTA}$): A simplified solution that considers only the diagonal terms of the collision matrix, effectively ignoring mode coupling and collective drift.
3. Callaway Approximation ($\kappa_{Callaway}$): An intermediate approach that separates N- and U-scattering processes but retains the computational simplicity of diagonal-only scattering rates, failing to fully capture the off-diagonal coupling.

The authors analyze the ratio $\kappa_{LPBE}/\kappa_{RTA}$ across various materials (including 10BP, isotopically enriched diamond, silicon, and monolayer graphene) and temperatures. They correlate this ratio with the spectral contributions of the eigenmodes (relaxons) of the collision matrix to the total thermal conductivity. A key aspect of the methodology involves rigorous convergence studies regarding Brillouin zone (BZ) sampling density, as the authors note that coarse grids can artificially inflate hydrodynamic signatures.

Key Contributions and Results

• Identification of a Low-Cost Indicator: The paper establishes that the ratio $\kappa_{LPBE}/\kappa_{RTA}$ serves as a computationally inexpensive and robust indicator for the strength of phonon hydrodynamics. A high ratio correlates with a strong hydrodynamic signature (collective drift), while a ratio near unity indicates predominantly diffusive transport.
• Failure of Diagonal-Only Approximations: The study demonstrates that methods relying solely on diagonal scattering rates (RTA and Callaway approximations) are inadequate for predicting hydrodynamic signatures. These methods underpredict $\kappa$ in ultrahigh-$\kappa$ materials and fail to capture the spectral redistribution of heat flow into a few drifting eigenmodes, which is the hallmark of hydrodynamics.
• Spectral Analysis of Hydrodynamics: By decomposing $\kappa_{LPBE}$ into contributions from the eigenmodes of the collision matrix, the authors show that strong hydrodynamic regimes are characterized by a few eigenmodes with nearly-degenerate, vanishingly small eigenvalues contributing the majority of the thermal conductivity. In contrast, diffusive regimes (e.g., Silicon at 60 K) show broad contributions from many eigenmodes with larger eigenvalues.
• BZ Sampling Convergence: The authors reveal a critical dependency on BZ sampling density. For ultrahigh-$\kappa$ materials at low temperatures, the ratio $\kappa_{LPBE}/\kappa_{RTA}$ and the strength of hydrodynamic signatures decrease as BZ sampling density increases. Coarse grids can overpredict hydrodynamic effects, necessitating careful convergence studies to ensure accurate predictions.
• Material Specific Findings:
  • 10BP and Enriched Diamond: These materials exhibit a ratio of $\sim4$ at specific low temperatures, with spectral contributions dominated by drifting eigenmodes, confirming strong hydrodynamic behavior.
  • Silicon: Even at low temperatures (60 K), Silicon shows a ratio $\sim1$ and a broad spectral distribution, confirming its diffusive nature.
  • Graphene: In suspended monolayer graphene, the ratio increases significantly as temperature decreases (from $<4$ at 300 K to $\sim13$ at 150 K), correlating with the strengthening of hydrodynamic signatures driven by anharmonic renormalization and four-phonon scattering.

Significance
The paper claims that the identification of $\kappa_{LPBE}/\kappa_{RTA}$ as a computationally efficient indicator will accelerate the data-driven search for new materials and experimental conditions that exhibit unconventional, non-diffusive heat flow regimes. By providing a surrogate metric that avoids the prohibitive computational cost of full collision matrix diagonalization, this approach enables the screening of material databases for candidates suitable for applications such as thermal rectification, thermal cloaking, and thermal shielding. The work underscores the necessity of moving beyond standard scattering rate approximations to fully capture the physics of phonon hydrodynamics.