Brillouin zone summation

Brillouin zone sums appear at different two points for phonon lifetime calculation. First it is used for the Fourier transform of force constans, and then to obtain imaginary part of phonon-self-energy. In the numerical calculation, uniform sampling meshes are employed for these summations. To obtain more accurate result, it is always better to use denser meshes. But the denser mesh is more computationally demanding.

The second Brillouin zone sum contains delta functions. In phono3py calculation, a linear tetrahedron method (--thm, default option) and a smearing method (--sigma) can be used for this Brillouin zone integration. In most cases, the tetrahedron method is better, therefore it is the default choice in phono3py. Especially in high thermal conductivity materials, the smearing method results in underestimation of 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 demanding is not trivial. Extrapolation to 1/n \rightarrow 0 seems not a good idea, since it is converging. This plot shows that we have to decide how much value is acceptable as thermal conductivity value. What is important is that the obtained value has to be shown accompanied with the information of the computational settings. The BZ integration method and sampling mesh are definitely those of them.


In case the smearing method is necessary to use, the convergence of q-point mesh together with smearing width has to be checked carefully. Smearing parameter is used to approximate delta functions. Small sigma value is better to describe the detailed structure of three-phonon-space, but it requires a denser mesh to converge.

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. 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 parameter such as the smearing width, therefore it is easier to use than the smearing method and recommended to use. A drawback of using the tetrahedron method is that it is slower and consumes more memory space.

Numerical quality of force constants

Third-order force constants are much weaker to numerical noise of a force calculator than second-order force constants. Therefore supercell force calculations have to be done by enough high numerical accuracy.

The phono3py default displacement distance is 0.03 \text{\AA}. In some cases, accurate result may not be obtained due to the numerical noise of the force calculator. Usually increasing the displacement distance by --amplitude option reduces the numerical noise, but increases error from higher order anharmonicity.

It is not easy to check the numerical quality of force constants. It is suggested firstly to check deviation from the translational invariance condition by watching output where the output lines start with max drift of .... The drift value smaller than 1 may be acceptable but of course it is dependent on cases. The most practical way may be to compare thermal conductivities calculated with and without symmetrizing third-order force constants by --sym_fc3r, --sym_fc2, and --tsym options.

Mode-Gruneisen-parameters calculated from third-order force constants look very sensitive to numerical noise near the Gamma point. Therefore symmetrization is recommended.

Overall, numerical quality of forces given by force calculators is the most important factor for the numerical quality of the thermal conductivity. We may be able to apply symmetry constraints to the force constants during the calculation e.g. using statistical approach, but the quality of force constants will be bad if that of forces are bad. Just they suffice the symmetry and the intensity is not reliable. Therefore what we can do best is to use the best calculator as the first priority. If we use ab-initio code, the knowledge about the ab-initio calculation from practical points like usage to method and theory is mandatory for the good thermal conductivity calculation.

To reduce computational demands

Here it is assumed ab-initio code is used as the force calculator. Then the most heavy part of thermal conductivity calculation is a set of many supercell force calculations by ab-initio code.

The number of force calculation is reduced by employing crystal symmetry. This is only valid if the crystal we focus on has high symmetry. Therefore we need another strategy. Introducing cutoff distance to consider interaction among atoms is an idea. For this phono3py has a marginal option but it is not very recommended to use since there is a better code to do this task, which is the ALM code in alamode package. The ALM interface for phono3py is now preparing.