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Phono3py

Optional dependency

Phono3pyAnalyzer requires the phono3py extra.

uv add "materialsframework[phono3py]"

pip install "materialsframework[phono3py]"

Computes third-order (anharmonic) force constants and uses them to solve for lattice thermal conductivity.

Overview

Harmonic phonons (see Phonopy) don't scatter off each other, so on their own they predict infinite thermal conductivity. There's no mechanism to relax a phonon population back toward equilibrium. Phono3pyAnalyzer adds the third-order force constants needed to describe phonon-phonon scattering, then solves a phonon Boltzmann transport equation to get a finite lattice thermal conductivity tensor.

It builds two supercells: a larger one (supercell_matrix, default 2x2x2) for the third-order force constants, and a separate one (phonon_supercell_matrix, default 3x3x3) for the second-order (harmonic) force constants that phono3py also needs for the group velocities and mode heat capacities entering the transport equation. Atoms in both are displaced by distance (default 0.01 Å) and evaluated with the calculator, the same finite-displacement idea as PhonopyAnalyzer but doubled up for the extra derivative order.

Theory

Third-order force constants

The third-order force constants are the third derivatives of the potential energy with respect to three atomic displacements:

\[ \Phi_{\alpha\beta\gamma}(l\kappa, l'\kappa', l''\kappa'') = \frac{\partial^3 U}{\partial u_\alpha(l\kappa)\, \partial u_\beta(l'\kappa')\, \partial u_\gamma(l''\kappa'')} \]

Phono3py estimates \(\Phi\) numerically from the forces measured on systematically displaced pairs of atoms in the supercell_matrix supercell, the anharmonic analogue of the finite-displacement scheme used for the ordinary (second-order) force constants.

Phonon scattering and lifetimes

\(\Phi\) enters a three-phonon scattering rate for each mode \(\lambda = (\mathbf{q}, s)\), derived from Fermi's golden rule for absorption and emission of phonon pairs. The mode's linewidth \(\Gamma_\lambda\) sets its lifetime, \(\tau_\lambda = 1/(2\Gamma_\lambda)\): the shorter the lifetime, the more strongly that mode scatters and the less it contributes to heat transport. By default (is_lbte=False), lifetimes are obtained under the single-mode relaxation-time approximation (RTA), which treats every mode's population as relaxing independently back to equilibrium. Setting is_lbte=True instead solves the linearized Boltzmann transport equation (LBTE) directly, which additionally captures collective (normal-process) phonon drift and is more accurate, at significantly higher computational cost. is_isotope=True adds isotopic mass-disorder scattering, and boundary_mfp adds a simple grain/sample-boundary scattering contribution on top of the intrinsic phonon-phonon rates.

Thermal conductivity

Once mode lifetimes, group velocities \(\mathbf{v}_\lambda\), and mode heat capacities \(C_\lambda\) are known (the last two from the harmonic phonons on phonon_supercell_matrix), the lattice thermal conductivity tensor follows the standard BTE-RTA form:

\[ \kappa_{\alpha\beta}(T) = \frac{1}{N V_0}\sum_\lambda C_\lambda(T)\, v_{\lambda,\alpha}\, v_{\lambda,\beta}\, \tau_\lambda(T) \]

where \(N\) is the number of unit cells sampled on the mesh grid and \(V_0\) is the unit cell volume. transport_type selects an inter-band correction layered on top of this intra-band result: "SMM19" is the Wigner transport formulation of Simoncelli, Marzari & Mauri, which adds a coherent tunneling contribution between nearly degenerate bands (important in materials with low, glass-like conductivity); "IBDB19" is the quasi-harmonic Green-Kubo approach of Isaeva, Barbalinardo, Donadio & Baroni; "NJC23" is a further Green-Kubo-based formulation. kappa holds the resulting tensor with shape (sigmas, temperatures, 6), the last axis being the independent Voigt components (xx, yy, zz, yz, xz, xy).

References

  • Phono3py / phonon lifetimes: Togo, A., Chaput, L., & Tanaka, I. (2015). Distributions of phonon lifetimes in Brillouin zones. Physical Review B, 91, 094306. https://doi.org/10.1103/PhysRevB.91.094306
  • Wigner transport equation ("SMM19"): Simoncelli, M., Marzari, N., & Mauri, F. (2019). Unified theory of thermal transport in crystals and glasses. Nature Physics, 15, 809-813. https://doi.org/10.1038/s41567-019-0520-x
  • Quasi-harmonic Green-Kubo ("IBDB19"): Isaeva, L., Barbalinardo, G., Donadio, D., & Baroni, S. (2019). Modeling heat transport in crystals and glasses from a unified lattice-dynamical approach. Nature Communications, 10, 3853. https://doi.org/10.1038/s41467-019-11572-4