Lattice thermal conductivity decomposition: Peierls vs. non-Peierls contributions

This study compares various methods for calculating lattice thermal conductivity across three crystalline systems, finding that quadratic and Peierls heat current approaches yield similar results, optical phonons can dominate acoustic modes in $\alpha$-quartz, and the relaxation time approximation consistently underestimates thermal conductivity.

The Gist

Imagine a solid block of material, like a piece of ice or a crystal, as a giant, crowded dance floor. The atoms are the dancers, and "thermal conductivity" is simply a measure of how efficiently they can pass a "heat message" (energy) across the room to the other side.

In this paper, the author, Andrey Pereverzev, is trying to figure out the best way to calculate exactly how fast that heat message travels. He compares three different "rulebooks" (mathematical formulas) used to describe how the dancers move and interact.

Here is a breakdown of his findings using simple analogies:

The Three Rulebooks

To measure the heat flow, scientists use a method called the "Green-Kubo" approach, which is like watching a movie of the dancers and averaging their movements over time. The author tested three different ways to write the script for this movie:

1. The Full Script (Full Heat Current): This includes every single detail of the dancers' movements, including their speed, their position, and how they push against each other. It's the most complete, messy, and realistic description.
2. The Quadratic Script (Quadratic Component): This is a simplified version. It ignores the very first, simple movements and focuses on the "middle" interactions—the way the dancers bump into each other in pairs. It's like looking at the dance floor through a slightly blurry lens that filters out the noise.
3. The Peierls Script (Peierls Heat Current): This is the most famous and commonly used rulebook in physics. It assumes the dancers move in perfect, independent lines (like waves). It's a very clean, idealized version of the dance.

The Experiment: Three Different Dance Floors

The author tested these three rulebooks on three different "dance floors" (crystals):

• Solid Argon: A simple floor where everyone is the same size and moves in a simple pattern.
• Solid Argon with Alternating Masses (SAAM): A floor where the dancers alternate between being very light and very heavy. This creates a more complex rhythm with different types of waves.
• Alpha-Quartz: A very complex floor with many different types of dancers (silicon and oxygen) and a complicated dance pattern.

The Big Findings

1. The "Blurry Lens" and the "Idealized Script" are almost the same.
For all three dance floors, the author found that the Quadratic Script and the Peierls Script gave almost identical results. Even though the Peierls script is a simplified, idealized version, it captures the heat flow just as well as the more complex quadratic version for these specific materials.

• Analogy: It's like trying to predict traffic flow. Whether you use a simple model that assumes cars move in straight lines (Peierls) or a slightly more detailed model that accounts for cars bumping into each other (Quadratic), you get the same estimate for how fast traffic moves.

2. The "Idealized Script" misses a hidden surprise in Quartz.
In the complex Alpha-Quartz crystal, the author discovered something surprising. Usually, we think heat is carried mostly by the "loud, low-pitched" sounds (acoustic modes). But in Quartz, the "quiet, high-pitched" sounds (optical modes) actually carried more heat than the loud ones.

• Analogy: Imagine a band where you expect the drums (acoustic) to carry the rhythm. But in this specific crystal, the violins (optical) were actually doing most of the heavy lifting. The Peierls script was able to catch this, showing that the high-pitched vibrations are doing the heavy work.

3. The "Relaxation Time" Guess is always too low.
The author also tested a very common shortcut method called the "Relaxation Time Approximation" (RTA). This is like guessing how fast traffic moves by assuming every car drives at a constant speed without ever slowing down or speeding up.

• Result: This shortcut consistently underestimated the heat flow for all three crystals. It told the author the heat would move slower than it actually did.
• Analogy: It's like a weather forecast that always predicts it will be 10 degrees colder than it actually is. It's a safe guess, but it's not accurate.

4. Why the "Full Script" is sometimes different.
For the simple crystals (Argon), the "Full Script" showed slightly higher heat flow than the simplified ones. However, for the complex Quartz, the difference was tiny. The author suggests that the extra heat seen in the "Full Script" comes from very complex, chaotic interactions (anharmonicity) that the simplified scripts ignore.

• Analogy: In a simple dance, the extra details don't matter much. But in a chaotic, complex dance (like a large unit cell with many atoms), ignoring the messy, chaotic bumps between dancers might make you miss a significant chunk of the energy transfer. The author notes that for very large and complex crystals (like explosives), this difference becomes huge, but for the small crystals tested here, the simplified scripts work fine.

The Bottom Line

If you want to know how well a crystal conducts heat, you don't always need the most complicated, messy math. For the materials tested in this paper, the simplified "Peierls" method works just as well as the more complex methods. However, you should avoid the "Relaxation Time" shortcut if you want an accurate number, because it will consistently tell you the heat moves slower than it really does.

The paper is essentially a quality check: it confirms that for many standard crystals, the simplified, elegant math we've been using for decades is actually quite accurate, but it warns us that in very complex systems, we might need to look closer at the messy details.

Technical Summary

Technical Summary: Lattice Thermal Conductivity Decomposition: Peierls vs. Non-Peierls Contributions

Problem Statement
The calculation of lattice thermal conductivity (TC) in crystalline solids using the Green-Kubo (GK) formalism relies on the heat current correlation function (HCCF). While the full classical heat current provides a rigorous definition, theoretical analyses often utilize approximations, specifically the quadratic component of the heat current or the Peierls heat current (derived from normal mode coordinates). A critical ambiguity exists regarding the extent to which these approximations capture the total TC, particularly in complex polyatomic crystals where the HCCF exhibits decaying oscillatory terms. Previous studies have suggested that linear terms in the Taylor expansion of the heat current do not contribute to TC, while quadratic terms are separated into "Peierls" (diagonal in phonon branch index) and "non-Peierls" (off-diagonal) contributions. The latter are responsible for oscillatory behavior in the HCCF. The specific dependence of calculated TC on the choice of heat current definition (full vs. quadratic vs. Peierls) and how this dependence varies with system complexity (monatomic vs. polyatomic) remains a subject of investigation.

Methodology
The study employs classical molecular dynamics (MD) simulations combined with the Green-Kubo formalism to compute the thermal conductivity tensor ($\kappa_{\alpha\beta}$). The analysis focuses on three distinct crystalline systems:

1. Solid Argon (SA): A monatomic face-centered cubic (fcc) lattice modeled with a Lennard-Jones potential.
2. Solid Argon with Alternating Masses (SAAM): A model with the same Lennard-Jones potential but alternating atomic masses (ratio of 10) to create a tetragonal lattice with both acoustic and optical phonon branches.
3. $\alpha$-Quartz: A trigonal crystal with nine atoms per unit cell, modeled using the Beest-Kramer-van Santen (BKS) potential.

For each system, the TC is calculated using three different definitions of the heat current:

• Full Heat Current ($J_{full}$): The complete classical expression including all terms.
• Quadratic Heat Current ($J_{quad}$): The second-order Taylor expansion of the heat current with respect to atomic displacements and velocities.
• Peierls Heat Current ($J_{Peierls}$): A subset of the quadratic current obtained by transforming to normal mode coordinates and retaining only terms diagonal in the phonon branch index.

Additionally, the Relaxation Time Approximation (RTA) is evaluated by applying the GK approach to fluctuations of individual normal-mode energies. Simulations were conducted at specific temperatures (20 K for SA/SAAM, 300 K for $\alpha$-quartz) and zero pressure using the LAMMPS package. The anharmonicity of the systems was quantified using a relative measure ($\eta$) comparing the adjusted average potential energy to the harmonic limit.

Key Results

• Comparison of Heat Current Definitions: For all three systems, the thermal conductivities calculated using the quadratic and Peierls heat currents differ only slightly from each other. However, the TC calculated using the full heat current is consistently higher than that of the quadratic/Peierls approximations.
  • In SA and SAAM, the quadratic/Peierls values are approximately 10–13% lower than the full current values.
  • In $\alpha$-quartz, the difference between the full and quadratic/Peierls values is significantly smaller, falling within numerical uncertainty.
• Acoustic vs. Optical Contributions:
  • In the SAAM model, acoustic modes dominate the TC, with optical contributions being negligible (especially along the $c$-axis).
  • In $\alpha$-quartz, the optical phonon contribution to the Peierls TC exceeds that of the acoustic modes along the $a$-axis (optical $\approx$ 6.7 W m$^{-1}$K$^{-1}$ vs. acoustic $\approx$ 2.1 W m$^{-1}$K$^{-1}$). Along the $c$-axis, optical and acoustic contributions are comparable.
• Relaxation Time Approximation (RTA): The RTA systematically underestimates the thermal conductivity in all three systems compared to the full GK calculations. For SA, RTA yields values ~15% lower than the Peierls result.
• Anharmonicity: The estimated anharmonicity ($\eta$) is small for all systems (~1% for SA/SAAM and ~1.6% for $\alpha$-quartz), yet sufficient to produce finite and sizable thermal conductivity.

Significance and Claims
The paper claims that for systems with small unit cells (few atoms), the Peierls and quadratic approximations provide TC values very close to the full classical heat current, though the full current consistently yields slightly higher values. The discrepancy between the full current and the Peierls approximation is attributed to anharmonic (cubic and higher-order) terms in the heat current that are not captured by the quadratic expansion.

A significant finding is the reversal of the dominant heat-carrying mechanism in $\alpha$-quartz compared to the SAAM model: optical modes contribute more than acoustic modes in $\alpha$-quartz, whereas the opposite is true for the SAAM model. The authors attribute this to the larger number of optical branches in $\alpha$-quartz (24 branches) compared to SAAM (3 branches).

Furthermore, the study highlights a limitation in previous literature (specifically Ref. [5] regarding $\alpha$-quartz) that used heuristic fitting of the HCCF to partition acoustic and optical contributions. The authors argue that such fitting approaches may yield incorrect estimates because they attempt to fit functions with finite time integrals to data containing oscillatory components whose time integrals vanish (and thus do not contribute to TC). By directly evaluating mode contributions without ad hoc fitting, this work suggests that optical contributions in $\alpha$-quartz are significantly larger than previously reported.

Finally, the authors note that while the approximations hold for the small-unit-cell systems studied here, their recent results on systems with large unit cells (e.g., $\beta$-HMX and $\alpha$-RDX) show that the Peierls TC can be substantially smaller than the full current TC, suggesting the validity of the Peierls approximation is system-dependent.