🔬 materials science

Where heat flows and where to change it: gauge-free sensitivity maps of thermal conductivity in disordered solids

This paper develops and compares two local descriptions of thermal conductivity in disordered solids, revealing that while gauge-dependent atomic projections and gauge-free mass/stiffness sensitivity maps are initially anti-correlated due to local softness, the latter provides a robust, exact guide for optimizing material design and understanding heat transport limitations in amorphous and paracrystalline silicon.

Original author: D. A. Drabold, C. Ugwumadu

Published 2026-11-06✓ Author reviewed ⓘ

📖 6 min read🧠 Deep dive

CC BY 4.0

Original author: D. A. Drabold, C. Ugwumadu

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 by the author. For technical accuracy, refer to the original paper. Read full disclaimer

Heat is a constant traveler inside solid materials, moving from hot spots to cold ones, but in the disorderly world of amorphous solids—like glass or the silicon used in computer chips—this journey is chaotic. Unlike a crystal, where atoms sit in a perfect, repeating grid that allows heat to zip along in straight lines, an amorphous solid is a tangled mess of atoms with no long-range order. For decades, scientists have known how to calculate the total amount of heat a material can carry, a number called thermal conductivity. However, knowing the total is not enough for engineers trying to design better electronics or more efficient energy converters. They need to know exactly where the heat is flowing inside the material and, more importantly, where a small change to the material's structure would make the biggest difference. If you want to stop heat from escaping a device, or help it escape faster, you must know which specific atoms are doing the heavy lifting and which ones are merely bystanders.

This question has been difficult to answer because of a fundamental ambiguity in how scientists define the energy of a single atom within a group. When calculating heat flow, researchers must decide how to split the total energy of the system among the individual atoms. This choice is arbitrary, much like deciding how to split a bill among friends when the exact cost of each person's meal is unclear. Different ways of splitting the bill lead to different maps of where the heat is going, even though the total amount of heat remains the same. This has left scientists wondering if their maps of heat flow were describing the actual physics of the material or just the mathematical bookkeeping they used to get there.

To cut through this confusion, researchers at Ohio University and Los Alamos National Laboratory turned their attention to amorphous silicon, a material that is structurally simple yet complex enough to represent a wide range of disordered solids. They developed two distinct ways to look at the problem. The first approach was to map the heat flow directly, creating a picture of which regions of the material are carrying the most energy. They tested how much this map changed when they altered the arbitrary way of splitting the energy. They found that while the map looked slightly different at the level of a single atom, it became remarkably stable when viewed over small regions, about the size of a few atoms across. This confirmed that the map of heat flow is a real, physical feature of the material, not just a mathematical artifact.

The second, and more surprising, approach asked a different question: if you were to change the mass of a single atom or the stiffness of the bond connecting it to its neighbor, how would the total heat flow change? This question has no ambiguity because it does not require splitting the energy; it simply measures the sensitivity of the system to a local tweak. The researchers calculated this sensitivity map for thousands of atoms in their models. What they discovered was a striking reversal in the sign of the response. In the regions where the most heat was flowing, making an atom heavier tended to increase the total heat flow. Conversely, in the regions where very little heat was flowing, making an atom heavier actually reduced the total heat flow.

This counterintuitive result is driven by the local "softness" of the material. In amorphous silicon, some atoms are loosely held in place and vibrate with large amplitudes, while others are tightly constrained and vibrate with small amplitudes. The researchers found that the regions where heat flows the least are exactly the soft, loosely held regions where atoms vibrate wildly. In these soft spots, the atoms are already moving so much that adding mass to them disrupts the delicate mixing of vibrations that allows energy to pass through the material, effectively choking the flow. In the stiff regions where heat flows freely, the atoms are held tight, and adding mass tends to raise the conductivity. However, the researchers noted that while the sign of the response is reliably opposite to the heat flow, the magnitude of the sensitivity does not always perfectly match the heat flow; the two coincide only on average.

The study revealed that this relationship between softness and heat flow is the key to understanding the material. The researchers showed that if you account for how much an atom vibrates, the strange reversal in the sensitivity map largely disappears. The soft, large-amplitude regions act as bottlenecks, not because they block the heat directly, but because they mix the vibrational modes in a way that prevents energy from moving efficiently. When the researchers tested this on larger models and at different temperatures, they found that this effect is strongest at room temperature and becomes even more pronounced as the material size increases, allowing longer-wavelength vibrations to participate.

The findings have immediate practical value for designing materials. The sensitivity map acts as a guide for engineers. If you want to reduce heat flow, for instance to keep a thermoelectric device efficient, you should not just add heavy atoms randomly. Instead, you should place them in the soft, high-amplitude regions where the map shows the greatest sensitivity. The researchers tested this by simulating the replacement of silicon atoms with heavier germanium atoms. They found that placing these heavy atoms in the specific soft spots predicted by their map reduced the heat flow significantly more than placing them at random or in the stiff, heat-carrying regions.

The study also looked at a boundary between a disordered amorphous region and an ordered crystal. They found that the heat flow across this boundary is controlled almost entirely by the disordered side. The crystal side is so stiff and uniform that changing its atoms has very little effect on the total heat transfer. The bottleneck is the amorphous region, specifically the soft spots just a few atomic layers deep into the disordered material. This means that to improve or degrade the thermal performance of such interfaces, one must focus on the disordered side, not the crystal.

Ultimately, this work provides a clear, unambiguous way to see inside a disordered solid. It separates the question of where heat flows from the question of where a change will matter. The two answers are often opposites: the places that carry the most heat are the places where you can change the least, and the places that carry the least heat are the places where a small change can have the biggest impact. By focusing on the local softness and the vibrational amplitude of the atoms, scientists can now predict exactly where to look to control the thermal properties of the materials that power our modern world.

Technical Summary: Gauge-Free Sensitivity Maps of Thermal Conductivity in Disordered Solids

Problem Statement
Designing materials for thermal management requires more than knowing the total thermal conductivity ( $\kappa$ ); it requires identifying where heat flows within a disordered network and where a local modification would alter that transport. While the Allen–Feldman (AF) harmonic theory successfully predicts the total $\kappa$ of glasses by treating heat carriers as diffusons, propagons, and locons, it yields only a single global number. Previous attempts to localize this quantity, such as the site-projected thermal conductivity (SPTC, $\zeta(\ell)$ ), suffer from a fundamental "gauge" ambiguity: the value assigned to an individual atom depends on how the potential energy is partitioned among atoms. This arbitrariness raises the question of whether local maps describe physical transport or merely bookkeeping conventions. Furthermore, it remains unclear how to predict the effect of local structural changes (e.g., mass substitution or bond stiffening) on the total conductivity without re-computing the entire system.

Methodology
The authors develop and test two complementary local descriptions of heat transport in amorphous silicon (a-Si) using the harmonic AF theory.

Site-Projected Thermal Conductivity ( $\zeta$ ): The authors analyze the robustness of the standard projection of $\kappa$ onto atoms. They quantify the dependence of $\zeta$ on the choice of local energy (the "gauge"), comparing the standard AF gauge against site-energy gauges derived from machine-learned potentials (ACE, GAP-18) and the Keating force field.
Gauge-Free Sensitivity Maps ( $\kappa_m$ and $s_b$ ): To eliminate gauge ambiguity, the authors derive the exact gradient of $\kappa$ $κ$ with respect to local parameters: the mass of each atom ( $m_\ell$ $m_{ℓ}$ ) and the stiffness of each bond. They define the mass sensitivity $\kappa_m(\ell) = 2m_\ell \partial \kappa / \partial m_\ell$ $κ_{m} (ℓ) = 2 m_{ℓ} \partial κ / \partial m_{ℓ}$ . This approach requires no partitioning of energy.
- Exact Decomposition: The mass sensitivity map is decomposed exactly into three terms: a current term (proportional to $\zeta$ ), an eigenvalue-shift term (sensitivity-weighted local density of states), and a rotation term (redistribution of sensitivity via mode mixing).
- Local Softness Proxy: The authors utilize the harmonic vibrational amplitude ( $\delta R$ ) of each atom, derived from the pseudo-inverse of the dynamical matrix, as a structural proxy for local softness.
Computational Models: The study employs continuous random networks of a-Si (4096, 512, and 32,768 atoms) and paracrystalline models (grains embedded in a matrix). Calculations utilize the ACE, GAP-18, and Keating potentials. The authors employ dense diagonalization for smaller systems and Chebyshev polynomial expansions for larger ones (up to 10,000 atoms).

Key Contributions

Resolution of Gauge Ambiguity: The paper demonstrates that while $\zeta(\ell)$ at a single site is moderately gauge-dependent (uncertainty ~23% of spatial spread), it becomes robust over regions of radius 4 Å (uncertainty ~9.5%). The ranking of sites is nearly invariant beyond a few Ångströms.
Derivation of Exact Sensitivity: The authors provide the first exact gradient of AF conductivity with respect to local masses and bonds, establishing an Euler sum rule that the sensitivity maps must satisfy.
Discovery of Anti-Correlation: A central finding is that the heat-flow map ( $\zeta$ ) and the mass-sensitivity map ( $\kappa_m$ ) are strongly anti-correlated. Regions where little heat flows (soft, large-amplitude regions) exhibit a large positive $\kappa_m$ (adding mass lowers $\kappa$ ), while regions where heat flows most exhibit negative or small $\kappa_m$ (adding mass raises $\kappa$ ).
Mechanism Identification: The anti-correlation is mediated by local softness ( $\delta R$ ). The rotation term in the sensitivity decomposition, which involves the mixing of vibrational modes separated by 1–2 meV, is the primary driver of this effect.
Application to Interfaces and Design: The maps are applied to an amorphous/crystalline interface and paracrystalline silicon, identifying that the disordered matrix, not the crystalline grains, controls the cross-plane transport response.

Results

Robustness of $\zeta$ : Across different potentials (ACE, GAP-18, Keating) and network sizes, the heat-flow map $\zeta$ is a robust structural descriptor at the 4 Å scale. The Spearman correlation between different gauges or potentials is high (0.86–0.98).
Anti-Correlation Mechanism: In a-Si, the 12% of the volume with the highest heat flow shares only 0.27% of the volume with the 12% of the volume having the highest mass sensitivity.
- Mediation by Softness: At room temperature (300 K), the anti-correlation largely vanishes when conditioning on the local vibrational amplitude $\delta R$ . This confirms that "soft" regions (large $\delta R$ ) carry little heat, and changing the mass there significantly alters $\kappa$ by mixing nearby vibrational modes.
- Temperature and Size Dependence: The anti-correlation strengthens with system size (more low-frequency modes) and is strongest at low temperatures. Below ~100 K, a residual anti-correlation persists even after conditioning on $\delta R$ , suggesting that long-wavelength propagons contribute a component not captured by local softness alone.
Paracrystalline Silicon: In models with nanocrystalline grains, the grains conduct more heat per atom than the amorphous matrix. However, the sensitivity map shows that the amorphous matrix controls the response to mass changes. The sign of $\kappa_m$ flips from positive (matrix) to negative (grain interiors), but the magnitude of the response is governed by the disordered fraction.
Design Implications:
- Substituent Placement: For dispersed heavy substituents (e.g., Ge in Si), the sensitivity map serves as an effective guide. Placing substituents in regions of high positive $\kappa_m$ (soft regions) reduces $\kappa$ significantly more than random placement or placement in hard regions.
- Compact Clusters: For compact clusters of substituents, the map loses predictive power; the effect depends primarily on the total number of substituted atoms, acting as a single heavy defect.
- Interfaces: In an a-Si/c-Si interface, the sensitivity map identifies the amorphous side (4–6 Å from the interface) as the region where local perturbations most affect cross-plane conductivity, whereas the site projection $\zeta$ remains uniform across the series-resistance layers.

Significance and Claims
The paper claims to resolve a long-standing ambiguity in local descriptions of heat transport by shifting the focus from "how heat is shared" (which is gauge-dependent) to "how heat transport responds to change" (which is gauge-free). The authors argue that the sensitivity map provides a constructive answer to the design question: where would a change matter?

The work establishes that in disordered solids, the regions that carry the least heat are precisely the regions where the conductivity is most sensitive to local perturbations. This counter-intuitive result is explained by the mixing of vibrational modes in soft regions. The authors position the sensitivity map as a practical tool for material design, particularly for optimizing thermoelectrics (low $\kappa$ ) or thermal management in nanoscale devices, noting that the local vibrational amplitude ( $\delta R$ ) can serve as a cheap, computationally efficient proxy for the full sensitivity map. The study is limited to harmonic networks and the $\Gamma$ -point, and the authors note that physical lifetimes (anharmonicity) may require larger cells to resolve the sensitivity maps without artificial broadening, but the qualitative anti-correlation and design principles are presented as robust findings within the harmonic framework.

Technical Summary: Gauge-Free Sensitivity Maps of Thermal Conductivity in Disordered Solids

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