References

References#

Methods used in phonopy#

Generation of supercell force constants#

In phonopy, force constants are generated using the supercell method with finite displacements. Several approaches can be employed to calculate supercell force constants. Technical details regarding supercell method can be found in the following paper:

  • A. Togo, L. Chaput, T. Tadano, I. Tanaka, J. Phys.: Condens. Matter 35 353001 (2023)

Systematic displacement method#

This is the traditional method that phonopy has employed for many years. Crystal symmetry is utilized to reduce both the computational cost and numerical noise in supercell force constant calculations. First, a symmetry-reduced set of atomic displacements is systematically generated. After calculating the atomic forces, the displacements are expanded using symmetry operations. The force constants between atoms in the primitive cell and the supercell are then fitted to the symmetry-expanded forces of atoms in the supercells using the Moore–Penrose pseudoinverse.

This procedure can be considered a variant of Parlinski-Li-Kawazoe method (see below). Unlike the Parlinski–Li–Kawazoe method, supercell force constants are initially computed without imposing the translational invariance constraint. The constraint is applied a posteriori. Additional implementation details in phonopy can be found in the appendix of the following paper:

  • L. Chaput, A. Togo, I. Tanaka, and G. Hug, Phys. Rev. B, 84, 094302 (2011)

Projector-based method#

This approach is implemented in the symfc code, allowing for the displacement of any number of atoms in the supercell. Typically, all atoms are displaced either in random directions with a fixed displacement magnitude or with both random directions and magnitudes. The former approach, using a small displacement (e.g., 0.01 to 0.03 Angstrom), is recommended. However, for estimating supercell force constants at finite temperatures, the latter approach may be used.

  • A. Seko and A. Togo, Phys. Rev. B 110, 214302 (2024)

Tadano-Tsuneyuki method#

This approach is implemented in the ALM code, allowing for the displacement of any number of atoms in the supercell.

  • T. Tadano and S. Tsuneyuki, J. Phys. Soc. Jpn. 87, 041015 (2018).

Parlinski-Li-Kawazoe method#

Supercell force constants are calculated using the Moore–Penrose pseudoinverse by fitting the symmetry-reduced elements of supercell force constants to the linear relationships between atomic forces and atomic displacements. When constructing the dynamical matrix, supercell boundary conditions are treated to preserve crystal symmetry by averaging the phase factors of atomic pairs that are equivalent under supercell lattice translations.

  • K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 (1997)

Thermal expansion using quasi-harmonic approximation#

In phonopy-qha, thermal properties at constant pressure is obtained from the thermodynamic definition. To achieve Legendre transformation, volume-energy function is generated from a set of Helmholtz free energies and pV terms at volumes by fitting to a smooth function for which equations of states are prepared in phonopy-qha.

The volume dependence of the Helmholtz free energy is included from quasi-harmonicity. When using DFT-GGA (-LDA), often we should have some amount of error in the absolute value since phonon frequencies are underestimated (overestimated). However the value of some ratio like thermal expansion coefficient is often very well estimated. An example is shown in the following paper:

  • A. Togo, L. Chaput, I. Tanaka, G. Hug, Phys. Rev. B, 81, 174301-1-6 (2010)

Non-analytical term correction#

Non-metallic crystals are polarized by atomic displacements and the generated macroscopic field changes force constants near \(\Gamma\) point. This contribution is included through non-analytical term correction.

  • R. M. Pick, M. H. Cohen, and R. M. Martin, Phys. Rev. B 1, 910, (1970)

Correction by dipole-dipole interaction#

  1. P. Giannozzi, S. Degironcoli, P. Pavone, and S. Baroni, Phys. Rev. B 43, 7231 (1991)

  2. X. Gonze, J.-C. Charlier, D.C. Allan, and M.P. Teter Phys. Rev. B 50, 13035(R) (1994)

  3. X. Gonze, and C. Lee, Phys. Rev. B 55, 10355 (1997)

Currently phonopy implements the method by Gonze et al. written in the above two papers (2 and 3) as the default method.

Interpolation scheme at general q-points with non-analytical term correction#

This is an interpolation scheme using phonons at \(\mathbf{q}\rightarrow \mathbf{0}\) with the correction by Pick et al. and other commensurate points.

  • Y. Wang , J. J. Wang , W. Y. Wang , Z. G. Mei , S. L. Shang , L. Q. Chen and Z K Liu, J. Phys.: Condens. Matter. 22, 202201 (2010)

The first derivative of this expression, which is for example used for group velocity calculation, is described in the following paper:

  • Atsushi Togo, Laurent Chaput, and Isao Tanaka, Phys. Rev. B, 91, 094306-1-31 (2015)

Other methods and software for calculating force constants#

Parlinsk-Li-Kawazoe method#

PHONON is the original implementation of the Parlinsk-Li-Kawazoe method.

Small displacement method#

  • Dario Alfè, Computer Physics Communications, 180, 2622 (2009)

PHON is based on the small displacement method.

DFPT#

  • Paolo Giannozzi, Stefano de Gironcoli, Pasquale Pavone, and Stefano Baroni, Phys. Rev. B, 43, 7231 (1991)

  • Xavier Gonze and Changyol Lee, Phys. Rev. B 55, 10355 (1997)

Currently there are several many implementations such as Abinit Quantum espresso Elk, etc. VASP can calculate force constants using DFPT however only at Gamma-point.

SSCHA#

Selected papers of SSCHA:

  • Ion Errea, Matteo Calandra, and Francesco Mauri, Phys. Rev. Lett. 111, 177002 (2013)

  • Lorenzo Monacelli, Raffaello Bianco, Marco Cherubini, Matteo Calandra, Ion Errea, and Francesco Mauri. J. Phys. Condens. Matter 33, 363001 (2021).

A kind of SSCHA calculation performed using phonopy and ALM is presented in the following paper:

  • Atsushi Togo, Hiroyuki Hayashi, Terumasa Tadano, Satoshi Tsutsui, Isao Tanaka, J. Phys.: Condens. Matter 34, 365401 (2022)

For the study of basics#

Phonons#

  • Introduction to Lattice Dynamics, Martin. T. Dove, Cambridge university press

  • Thermodynamics of Crystals, Duane C. Wallace, Dover Publications

  • Electrons and Phonons by J. M. Ziman, Oxford University Press

  • The Physics of Phonons by G. P. Srivastava, CRC Press

Symmetry#

  • International Tables for Crystallography - IUCr

  • Symmetry Relationships between Crystal Structures by Ulrich Müller, Oxford University Press

  • Bilbao crystallographic server, https://www.cryst.ehu.es/

  • Supplementary Material for the Lekeitio School, https://www.cryst.ehu.es/html/lekeitio.html, the presentation by B. Mihailova (phonons) is considered nice for beginners.