FourPhonon: An extension module to ShengBTE for computing four-phonon scattering rates and thermal conductivity

Authors and references for FourPhonon:

• Zherui Han [email protected]
• Xiaolong Yang [email protected]
• Wu Li [email protected]
• Tianli Feng [email protected]
• Xiulin Ruan [email protected]

References: please refer to our GitHub homepage

1. Feng, T. & Ruan, X. Quantum mechanical prediction of four-phonon scattering rates and reduced thermal conductivity of solids. Phys Rev B 93, 045202 (2016).
2. Feng, T., Lindsay, L. & Ruan, X. Four-phonon scattering significantly reduces intrinsic thermal conductivity of solids. Phys Rev B 96, 161201 (2017).
3. Han, Z., Yang, X., Li, W., Feng, T. & Ruan, X. FourPhonon: An extension module to ShengBTE for computing four-phonon scattering rates and thermal conductivity. Comput Phys Commun 270 (2022) 108179, https://doi.org/10.1016/j.cpc.2021.108179.

The original authors of ShengBTE and references:

• Wu Li [email protected]
• Jesús Carrete Montaña [email protected]
• Nebil A. Katcho [email protected]
• Natalio Mingo [email protected]

References: please refer to the ShengBTE link.

How to download and compile FourPhonon

FourPhonon is built within ShengBTE and updates ShengBTE to a new version. The codes are hosted at GitHub, and you can download the latest distribution from this repository: https://github.com/FourPhonon (you can also find this link from ShengBTE website). The compilation of this new version is the same as the previous ShengBTE: after setting proper paths in arch.make, one can then run make in the Src subdirectory. An executable ShengBTE will appear in the root directory of this distribution.

How to call FourPhonon function

Besides the routine inputs and CONTROL file of ShengBTE, FourPhonon requires a fourth-order force constants and some new flags in CONTROL file.

4th-IFCs files: FORCE_CONSTANTS_4TH

This file contains the fourth-order interatomic force constant matrix, and uses a sparse description to save space. To construct 4th-IFCs, one can refer to the Fourthorder python scripts. The format of this force constants is a direct extension of third-order force constants, and contains nb blocks of such matrix:

• A blank line
• A 1-based sequential index (from 1 to nb)
• A line with the Cartesian coordinates of the second unit cell in Å
• A line with the Cartesian coordinates of the third unit cell in Å
• A line with the Cartesian coordinates of the fourth unit cell in Å
• A line with the 1-based indices of the four atoms involved, each from 1 to natoms
• 81 lines of force constant matrix in $\frac{\textrm{eV}}{\textrm{Å}^4}$. The indexes at the beginning label the Cartesian axes.

An example block of this file:

1
0.0000000000e+00 0.0000000000e+00 0.0000000000e+00
0.0000000000e+00 0.0000000000e+00 0.0000000000e+00
0.0000000000e+00 0.0000000000e+00 0.0000000000e+00
     1      1      1      1
 1  1  1  1       -42.0002584218
 1  1  1  2         0.0000000000
 1  1  1  3         0.0000000000
 1  1  2  1         0.0000000000
 1  1  2  2        40.3875689003
 1  1  2  3         0.0000000000
 1  1  3  1         0.0000000000
 1  1  3  2         0.0000000000
 1  1  3  3        40.3875689003
 1  2  1  1         0.0000000000
 1  2  1  2        40.3875689003
 1  2  1  3         0.0000000000
 1  2  2  1        40.3875689003
 1  2  2  2         0.0000000000
 1  2  2  3         0.0000000000
 1  2  3  1         0.0000000000
 1  2  3  2         0.0000000000
 1  2  3  3         0.0000000000
 1  3  1  1         0.0000000000
 1  3  1  2         0.0000000000
 1  3  1  3        40.3875689003
 1  3  2  1         0.0000000000
 1  3  2  2         0.0000000000
 1  3  2  3         0.0000000000
 1  3  3  1        40.3875689003
 1  3  3  2         0.0000000000
 1  3  3  3         0.0000000000
 2  1  1  1         0.0000000000
 2  1  1  2        40.3875689003
 2  1  1  3         0.0000000000
 2  1  2  1        40.3875689003
 2  1  2  2         0.0000000000
 2  1  2  3         0.0000000000
 2  1  3  1         0.0000000000
 2  1  3  2         0.0000000000
 2  1  3  3         0.0000000000
 2  2  1  1        40.3875689003
 2  2  1  2         0.0000000000
 2  2  1  3         0.0000000000
 2  2  2  1         0.0000000000
 2  2  2  2       -42.0002584218
 2  2  2  3         0.0000000000
 2  2  3  1         0.0000000000
 2  2  3  2         0.0000000000
 2  2  3  3        40.3875689003
 2  3  1  1         0.0000000000
 2  3  1  2         0.0000000000
 2  3  1  3         0.0000000000
 2  3  2  1         0.0000000000
 2  3  2  2         0.0000000000
 2  3  2  3        40.3875689003
 2  3  3  1         0.0000000000
 2  3  3  2        40.3875689003
 2  3  3  3         0.0000000000
 3  1  1  1         0.0000000000
 3  1  1  2         0.0000000000
 3  1  1  3        40.3875689003
 3  1  2  1         0.0000000000
 3  1  2  2         0.0000000000
 3  1  2  3         0.0000000000
 3  1  3  1        40.3875689003
 3  1  3  2         0.0000000000
 3  1  3  3         0.0000000000
 3  2  1  1         0.0000000000
 3  2  1  2         0.0000000000
 3  2  1  3         0.0000000000
 3  2  2  1         0.0000000000
 3  2  2  2         0.0000000000
 3  2  2  3        40.3875689003
 3  2  3  1         0.0000000000
 3  2  3  2        40.3875689003
 3  2  3  3         0.0000000000
 3  3  1  1        40.3875689003
 3  3  1  2         0.0000000000
 3  3  1  3         0.0000000000
 3  3  2  1         0.0000000000
 3  3  2  2        40.3875689003
 3  3  2  3         0.0000000000
 3  3  3  1         0.0000000000
 3  3  3  2         0.0000000000
 3  3  3  3       -42.0002584218

Flags in CONTROL file

This file contains all the user-specified settings and parameters, including crystal structural information, broadening factor, q-mesh, temperature, etc. To call FourPhonon capabilities, one should add a new &flags namelist:

• four_phonon (logical, default=.false.): compute four-phonon phase space and four-phonon scattering rates.

Here, we show some practical usage of this flag in combination with other existing flags.

• onlyharmonic=.true. and four_phonon=.true.: only compute four-phonon phase space
• convergence=.false. and four_phonon=.true.: compute thermal conductivity at RTA level for both three- and four-phonon scatterings
• four_phonon=.true.(convergence is default to be .true.): compute thermal conductivity with three-phonon iterative scheme but treat four-phonon scatterings at RTA level

*Note that: all other parameters in CONTROL file, like temperature or q-mesh, apply to both three- and four-phonon processes. nanowires function is not supported in FourPhonon package.

Output files

Besides the routine output files from previous ShengBTE program, FourPhonon generates these output files:

• BTE.Numprocess_4ph: number of allowed four-phonon scattering processes, for each irreducible q point and phonon band
• BTE.P4: phase space available for four-phonon processes, for each irreducible q point and phonon band
• BTE.P4_total: total volume in phase space available for four-phonon processes
• BTE.P4_plusplus*, BTE.P4_plusminus*, BTE.P4_minusminus*: similar to BTE.P4 but only includes contributions from ++/+-/- - processes

*Note for four-phonon scatterings, there are three different channels: recombination (++), redistribution (+-) and splitting (- -) processes.

Under temperature-dependent directories: (the unit of scattering rates is ps$^{-1}$ if not specified)

• BTE.WP4: weighted phase space available for four-phonon processes ([rad/ps]$^{-5}$, 2nd column) vs angular frequency (rad/ps, 1st column) for each irreducible q point and phonon band
• BTE.WP4_plusplus*, BTE.WP4_plusminus*, BTE.WP4_minusminus*: similar to BTE.WP4 but only includes contributions from ++/+-/- - processes
• BTE.w_3ph: three-phonon scattering rates for each irreducible q point and phonon band, this file replaces the original output BTE.w_anharmonic. Absorption and emission processes are written out into BTE.w_3ph_plus and BTE.w_3ph_minus
• BTE.w_4ph: four-phonon scattering rates for each irreducible q point and phonon band. Similarly, we provide the contributions from different channels: BTE.w_4ph_plusplus, BTE.w_4ph_plusminus and BTE.w_4ph_minusminus
• BTE.w_4ph_normal: four-phonon scattering rates of normal processes, for each irreducible q point and phonon band. This file has 5 columns and each one represents: angular frequency in rad/ps, recombination channel, redistribution channel, splitting channel, overall scattering rates from normal processes
• BTE.w_4ph_Umklapp: four-phonon scattering rates of Umklapp processes, for each irreducible q point and phonon band. The format is the same as BTE.w_4ph_normal
• BTE.kappa*: thermal conductivity results are written out as usual