Modeling heat transport in crystals and glasses from a unified lattice-dynamical approach (1904.02255v3)

Abstract: We introduce a novel approach to model heat transport in solids, based on the Green-Kubo theory of linear response. It naturally bridges the Boltzmann kinetic approach in crystals and the Allen-Feldman model in glasses, leveraging interatomic force constants and normal-mode linewidths computed at mechanical equilibrium. At variance with molecular dynamics, our approach naturally and easily accounts for quantum mechanical effects in energy transport. Our methodology is carefully validated against results for crystalline and amorphous silicon from equilibrium molecular dynamics and, in the former case, from the Boltzmann transport equation.

Unified Lattice-Dynamical Approach to Heat Transport in Solids

The paper presents a unified approach to modeling heat transport in solid insulators, whether crystalline or disordered, that elegantly integrates the strengths of the Boltzmann Transport Equation (BTE) and the Allen-Feldman (AF) model. This methodology bridges classical treatments of heat transport, applicable in crystalline solids through BTE, and the complex dynamics in amorphous materials captured by the AF model. The authors employ the Green-Kubo (GK) theory of linear response, allowing the method to naturally incorporate quantum mechanical effects, which are crucial for accurate heat transport modeling below the Debye temperature.

The cornerstone of this approach is the quasi-harmonic Green-Kubo (QHGK) approximation, which is used to describe the thermal conductivity tensor. It synthesizes the relaxation time approximation (RTA) for BTE in crystals with a generalized AF model applicable in glasses, utilizing interatomic force constants (IFCs) and normal-mode lifetimes computed at mechanical equilibrium. This framework is validated through comparisons with equilibrium molecular dynamics (EMD) and BTE calculations for crystalline silicon, as well as EMD for amorphous silicon.

Key Methodological Details

1. Quantum Effects and Statistical Mechanics: The QHGK approach incorporates quantum mechanical effects in the calculation of heat transport, which is crucial for systems below the Debye temperature. This quantum treatment contrasts with standard molecular dynamics, which struggles with quantum effects, especially in disordered systems.
2. Green-Kubo Linear Response: The authors utilize the GK theory to relate heat conductivity to the ensemble average of heat flux autocorrelation functions. They express the macroscopic heat flux in terms of atomic displacements and energies, leveraging Gaussian integrals for evaluation under harmonic approximations.
3. Vibrational Mode Lifetimes: Important to the QHGK model are the normal-mode lifetimes calculated from perturbative anharmonic corrections to harmonic IFCs, using the Fermi’s golden rule. These lifetimes are vital in extending classical harmonic models to account for anharmonic and quantum effects.
4. Validation and Comparisons: The computational results show that the QHGK methodology yields thermal conductivities consistent with BTE-RTA predictions for crystalline silicon. Moreover, for amorphous silicon, QHGK provides results in line with empirical data above 100K, illustrating its robustness in capturing the complex heat transport behaviors across different material structures.

Implications and Future Directions

The unified approach outlined in the paper has significant implications for the theoretical and practical understanding of thermal transport phenomena. It provides a framework applicable to a wide range of systems, from perfectly ordered crystals to highly disordered amorphous solids. This applicability makes it a valuable tool for examining materials with more complex disorder, including systems with point defects or nanostructuring, which stand beyond the scope of traditional BTE and AF models.

The method’s full quantum mechanical treatment enables accurate predictions of thermal conductivities at low temperatures, which are crucial for exploring materials used in quantum technologies and low-temperature applications. Furthermore, the direct correlation between normal-mode properties and macroscopic heat transport characteristics offers deeper insight into the vibrational dynamics underlying heat transport in complex systems.

This unified model lays the groundwork for future studies aiming to explore the thermal properties of novel materials without reliance on empirical parameters or symmetry assumptions. It opens pathways for further refinement and application to next-generation materials, specifically designed for improved thermal management in electronics and thermoelectric devices.

Conclusion

Through the development and validation of a robust unified lattice-dynamical approach, this paper contributes significantly to the nuanced understanding of heat transport across different solid states. The QHGK method, with its elegant amalgamation of BTE and AF models underpinned by quantum GK theory, positions itself as a pivotal framework for future research in heat transport, especially within disordered and quantum-material domains.