Brillouin zone summation

Brillouin zone (BZ) summations appear at different two points in phonon lifetime calculation. First it is used for the Fourier transform of force constants, and second to obtain imaginary part of phonon-self-energy. For the summation, usually uniform sampling meshes are employed. To obtain more accurate result, it is always better to use denser meshes. But the denser mesh requires more computational demand.

The second BZ summation contains delta functions. In phono3py calculation, a linear tetrahedron method (thm option, default option) and a smearing method (sigma option) can be used for this BZ integration. In most cases, the tetrahedron method is better. Especially in high thermal conductivity materials, the smearing method results in underestimation of lattice thermal conductivity.

The figure below shows Si thermal conductivity convergence with respect to number of mesh points along an axis from n=19 to 65. This is calculated with RTA and the linear tetrahedron method. Within the methods and phono3py implementation, it is converging at around n=55, however this computational demand is not trivial. As far as observing this result, an extrapolation to \(1/n \rightarrow 0\) seems not a good idea, since it gives overestimation in the case of this Si example. This plot tells that we have to decide how much value is acceptable as lattice thermal conductivity value. Therefore it important to describe the number of sampling mesh and method of BZ integration to let other people reproduce the computational results.


In case the smearing method is necessary to use, the convergence of q-point mesh together with smearing width has to be checked carefully. Since smearing parameter is used to approximate delta functions, small sigma value is better to describe the detailed structure of three-phonon-space, however it requires a denser sampling mesh to converge the result. To check the convergence with respect to the sigma value, multiple sigma values can be set. This can be computationally efficient, since it is avoided to re-calculate phonon-phonon interaction strength for different sigma values in this case. Convergence with respect to the sampling mesh and smearing parameter strongly depends on materials. For Si example, a \(20\times 20\times 20\) sampling mesh (or 8000 reducible sampling points) and 0.1 THz smearing value for reciprocal of the volume of an atom may be a good starting choice. The tetrahedron method requires no such parameter as the smearing width.

Importance of numerical quality of force constants

Third-order force constants (fc3) are much weaker to numerical noise of a force calculator than second-order force constants (fc2). Therefore supercell force calculations have to be done very carefully.

Numerical quality of forces given by force calculators is the most important factor for the numerical quality of lattice thermal conductivity calculation. We may be able to apply symmetry constraints to force constants, however even if force constants fulfill those symmetries, the numerical quality of force constants is not guaranteed since elements of force constants just suffice the symmetries but most of those intensities are not constrained.

It is important to use the best possible force calculator in the possibly best way. The knowledge of the force calculator from the theory and method to the practical usage is required to obtain good results of lattice thermal conductivity calculation.

In the following, a few things that may be good to know are presented.

A practical way to check lattice thermal conductivity result

Some feeling whether our calculation result is OK or not may be obtained by comparing lattice thermal conductivities calculated with and without symmetrizations of force constants. If they are enough different, e.g., more than twice different, it is better to re-consider about the force calculation. In the case of DFT calculations, the choice of input settings such as k-point sampling mesh, plane-wave energy cutoff, and exchange-correlational potential, etc, should be reconsidered.

Displacement distance of atoms

The phono3py default displacement distance is 0.03 \(\text{Angstrom}\). In some cases, accurate result may not be obtained due to the numerical noise of the force calculator. Usually increasing the displacement distance by the amplitude option reduces the numerical noise, but as its drawback higher order anharmonicity is involved (renormalized) into fc3 and fc2.

File format compatibility with phonopy

  • disp_fc3.yaml and disp_fc2.yaml are not compatible with phonopy’s phonopy_disp.yaml.

  • FORCES_FC3 and FORCES_FC2 are not compatible with phonopy’s FORCE_SETS.

  • FORCE_SETS can be created using –cfs from FORCES_FC3 and disp_fc3.yaml or FORCES_FC2 and disp_fc2.yaml (needs to specify --dim-fc2).

  • FORCES_FC2 and disp_fc2.yaml can be created using –fs2f2 from FORCE_SETS.

  • fc2.hdf5 can be used in phonopy in the hdf5 mode when it is renamed to force_constants.hdf5. In the previous combinations of phonopy and phono3py, depending on the physical unit of force constants of calculators, the direct compatibility is not guranteed.