Computational
Materials Science
Read about my work under Dr. Anubhav Jain at the Hacking Materials Group of Lawrence Berkeley National Lab , studying negative thermal expansion as a SULI program intern.
Motivation
Thermal stress is detrimental to the performance of high-performance components, like semiconductors. Materials that exhibit negative thermal expansion (NTE) are of great interest, as understanding the physical mechanisms underpinning this behavior would enable the design of novel materials with near-zero thermal expansion. Components made from these materials would maintain consistent performance across a wider range of temperatures.
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In this project, I devise a workflow that employs first-principles techniques to study the atomic vibrations (lattice dynamics) that give rise to NTE in four materials. Ultimately, this method will be implemented as an automated high-throughput workflow to facilitate NTE material discovery.
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Training sets of perturbed supercells are used to determine the second and third-order interatomic force constants (IFCs) of each system. From the IFCs, we derive the thermal expansion coefficients that govern each material’s structural response to temperature changes.
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First Principles Study of Thermal Expansion
Studying lattice dynamics proves advantageous because they can be determined from \textit{ab initio} calculations and there exist many thermal properties that arise from lattice vibrations. This paper focuses on the phonon interactions underlying one such property: negative thermal expansion (NTE).
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Materials that exhibit NTE contract rather than expand while heated. A thorough description of the physical mechanisms that give rise negative thermal expansion serve as an essential first step towards the design of materials with near-zero thermal expansion. While there do exist non-vibrational mechanisms behind NTE, in the present study we focus on the vibrational mechanism in which a large, negative Grüneisen parameter of phonon modes leads to a strongly negative coefficient of thermal expansion (CTE).
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The CTE quantifies how the lattice volume or lattice parameter responds to changes in temperature. Previous theoretical investigations employ the quasi-harmonic approximation (QHA), which accounts for volume dependence of phonon frequencies but neglects temperature dependence. While sufficient for approximating temperatures far below the melting point or strongly harmonic structures, the QHA neglects higher order anharmonicity and therefore does not accurately reproduce experimental CTE values outside of these special cases. Inclusion of third-order anharmonicity is essential not only to explain experimental results, but also to ultimately enable the prediction and design of novel NTE materials based on lattice dynamics. This investigation carries out a study of NTE via a computationally efficient first-principles approach that incorporates anharmonic effects beyond the QHA. Reliance on the QHA is avoided by relating the CTE to the Grüneisen tensor, as determined by Phono3py.
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The components of the Grüneisen tensor (γ) are dimensionless quantities that depend on third-order phonon interactions and describe how phonon frequencies change with respect to volume. The tensor also relates the elastic stiffness tensor, heat capacity, volume of the primitive cell, and coefficient of thermal expansion.
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​The top equation gives the Grüneisen tensor, or the 3x3 Grüneisen tensor for an individual vibrational mode. The bottom equation expresses the total 3x3 Grüneisen tensor in terms of the elastic stiffness tensor (C), volume of the primitive cell (V), the total heat capacity at a fixed temperature (C_v), and the volumetric CTE tensor (α), whose components describe the thermal expansion along each direction. All quantities, with the exception of the elastic stiffness tensor, which is obtained from the Materials Project database, are determined from first principles at the level of density-functional theory (DFT).
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Studying vibrational effects to support materials design remains challenging, as balancing accuracy with the efficiency necessary for a high-throughput investigation is nontrivial. The finite displacement method, which Taylor expands the total energy with respect to atomic displacements (u), presents a computationally efficient way to describe lattice dynamics.
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Where F gives the interatomic forces of atom a in direction i and Φ gives the IFCs that will be used to describe phonon interactions.
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Each term in the expansion corresponds to various thermal properties. Second-order IFCs describe harmonic phonons that arise from pairwise atomic interactions and govern properties like heat capacity and entropy. These IFCs are directly determined from the finite displacement method using Phonopy. The higher order IFCs describe anharmonic phonons which are responsible for properties like thermal expansion. For the purposes of computing the CTE, truncating the expansion after third-order is sufficient. In contrast to harmonic IFCs, directly calculating anharmonic IFCs through finite displacement is not feasible as the number of tensor elements explodes as the order of IFCs increases. A promising alternative is to fit IFCs using a small representative training set of configurations with a sparse-recovery method. To this end, Compressive Sensing Lattice Dynamics (CSLD) is a employed through Pheasy, a calculator for high-order anharmonicity. This approach recovers physically meaningful solutions from incomplete data, by identifying and fitting only the most relevant anharmonic terms.
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Ultimately this workflow with be a part of a larger high throughput pipeline to facilitate materials discovery and design. In this study, we apply this workflow to four materials to demonstrate the efficacy of this method. ​​​​​​​​​​
