[Analysis Example] Lattice thermal conductivity calculation using machine learning potential

In recent years, as semiconductor miniaturization and high integration have progressed, there is a growing need for thermal management of devices from the perspectives of safety and energy efficiency. This case introduces a method for analyzing thermal conductivity based on phonon analysis using machine learning potentials.
In solid crystals, the carriers of heat are phonons and electrons. In particular, phonons dominate the thermal properties of semiconductors and insulators, and the thermal conductivity of phonons is called lattice thermal conductivity. In solid crystals, the phonon properties inherent in the substance can be calculated using the atomic force constant that can be calculated from the interatomic potential. The thermal conductivity can be evaluated using the frequency \((ω_{s,k})\) and relaxation time \((τ_{s,k})\) of the phonon from the Boltzmann transport equation based on the relaxation time approximation.

where \(V\) is the volume of the unit cell, \(N_{k}\) is the number of wavenumber meshes, \(s\) is the phonon branch, \(k\) is the wavenumber vector, \(\hbar\) is the Dirac constant, \(Τ\) is the temperature, and \(f^{BE}\) is the Bose-Einstein distribution. In the calculation of phonon characteristics, a free software, ALAMODE, is used^[1]. ALAMODE consists of two programs: alm, which estimates the interatomic force constant, and anphon, which calculates phonon properties such as phonon dispersion and thermal conductivity. alm estimates the interatomic force constant (\(Φ\)) from the set of displacements (\(U\)) from the lattice point and the force (\(F\)) at that time:

In this article, we perform calculations for silicon systems with a diamond structure. In addition, the general-purpose machine learning potential (MLP)^[2] is used. To obtain fine-tuned potential, we used the dataset by Bartok et al.^[3].

The model used for the calculations is shown in Figure 1. Silicon crystals with 3×3×3 supercells (216 atoms) were used. Figure 2 shows the results of the phonon dispersion and the phonon density of states. The phonon dispersion was confirmed to be in good agreement with the experimental values. Figure 3 shows the temperature dependence of thermal conductivity, which also shows very good agreement. Additionally, the cumulative thermal conductivity with respect to the mean free path and frequency is shown. Cumulative thermal conductivity refers to the thermal conductivity considering only the contribution of phonons below a certain frequency, for example. These analyses can reveal which frequencies of phonons are carrying more heat. For instance, in the case of silicon, we can see that the contribution of low frequencies is very significant.

Other methods for analyzing the thermal conductivity of materials include those based on the RNEMD method. If you are interested in analyzing material properties using machine learning potentials, please feel free to contact us.

Fig. 1: Silicon 3×3×3 supercells used in calculations

Fig. 2: (Left) Comparison of phonon dispersion relationship with experimental values ^[4]
and (Right) Calculated phonon density of states.

Fig. 3: (Left) Comparison of thermal conductivity with experimental values ^[5], (Middle) Cumulative thermal conductivity with respect to mean free path, (Right) Cumulative thermal conductivity with respect to phonon frequency.