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FourPhonon: An extension module to ShengBTE for computing four-phonon scattering rates and thermal conductivity
Zherui Han, Xiaolong Yang, Wu Li, Tianli Feng, Xiulin Ruan
TL;DR
Three-phonon models can miss important anharmonic effects, motivating accessible four-phonon calculations. FourPhonon extends ShengBTE with fourth-order force-constant generation, adaptive four-phonon scattering calculations, and thermal-conductivity workflows, demonstrated on Si, BAs, and LiCoO2.
Problem
Four-phonon scattering can significantly affect thermal transport, but broader studies require accessible calculations beyond the conventional three-phonon treatment.
Method
FourPhonon extends ShengBTE with Fourthorder.py, adaptive broadening, four-phonon phase-space and scattering-rate calculations, and lattice thermal-conductivity evaluation.
Results
Examples on Si, BAs, and LiCoO2 demonstrate convergence behavior, normal and Umklapp contributions, and different four-phonon scattering channels.
Takeaways & Limitations
FourPhonon provides an open-source, ShengBTE-compatible tool for exploring four-phonon effects across common, strongly affected, and complex materials.
Takeaways & Limitations
The current package supports four-phonon calculations at the SMRTA level and cannot yet perform fully iterative four-phonon calculations because of memory limitations.
Abstract
from arXiv · showhide
FourPhonon is a computational package that can calculate four-phonon scattering rates in crystals. It is built within ShengBTE framework, which is a well-recognized lattice thermal conductivity solver based on Boltzmann transport equation. An adaptive energy broadening scheme is implemented for the calculation of four-phonon scattering rates. In analogy with $thirdorder.py$ in ShengBTE, we also provide a separate python script, $Fourthorder.py$, to calculate fourth-order interatomic force-constants. The extension module preserves all the nice features of the well-recognized lattice thermal conductivity solver ShengBTE, including good parallelism and straightforward workflow. In this paper, we discuss the general theory, program design, and example calculations on Si, BAs and $\mathrm{LiCoO_2}$.
PROGRAM SUMMARY
FourPhonon is an open computational package for lattice thermal conductivity and three- and four-phonon scattering calculations. It uses Fortran 90 and MPI, with productive runs requiring HPC resources and potentially lasting hours to days.
- FourPhonon calculates lattice thermal conductivity and both three-phonon and four-phonon scattering rates.Four-phonon scattering rates are computed at the RTA level using an adaptive broadening scheme.
- The package is implemented in Fortran 90 with MPI and distributed under the GNU General Public License, version 3.
- Productive calculations require HPC facilities and may take several hours to several days to complete.
1. Introduction
Four-phonon scattering extends the anharmonic description beyond the traditional three-phonon approximation, whose importance has been demonstrated across multiple materials and applications. FourPhonon addresses this need as a ShengBTE-compatible package for first-principles four-phonon calculations.
- Phonon-phonon interactions govern phonon properties and lattice thermal conductivity, which can be predicted from first principles using the phonon BTE.
- Four-phonon scattering can be significant beyond the conventional three-phonon treatment, including in BAs and diverse other systems.Experiments confirmed the significance of four-phonon scattering in BAs, and later studies extended its relevance across insulators, semiconductors, and semimetals.
- FourPhonon extends ShengBTE to compute four-phonon phase space, normal and Umklapp scattering rates, and lattice thermal conductivity.The package includes Fourthorder.py for calculating fourth-order interatomic force constants through first-principles calculations.
- The paper presents theoretical background, computational details, and case studies on Si, BAs, and LiCoO2.
2. Methodology and computational details
FourPhonon extends ShengBTE with fourth-order force constants and four-phonon scattering calculations while retaining the ShengBTE workflow. Its methodology combines iterative-BTE formalism, symmetry-reduced finite differences, adaptive broadening, and convergence checks.
- Workflow integration: FourPhonon adds fourth-order force constants and a CONTROL-file flag to ShengBTE, enabling four-phonon utilities within the existing workflow.An additional input file supplies fourth-order force constants, and the four_phonon flag activates the added utilities.
- BTE solution: FourPhonon treats four-phonon scattering rates at the RTA level rather than including four-phonon phase space in the iterative BTE scheme.The full formalism contains iterative terms, but the current package excludes those terms because of exceptionally high memory demands.
- Scattering formalism: The implemented formalism computes three- and four-phonon scattering rates using normal and Umklapp processes, with energy conservation enforced through a delta function.Four-phonon processes use scattering probability matrices and distinguish reciprocal-lattice-vector conditions for normal and Umklapp events.
- Fourth-order force constants: Fourthorder.py calculates fourth-order IFCs with a real-space finite-difference method based on displaced-atom forces.The displacement h is set by default to 0.04 Bohr radius and can be changed in the program header.
- Fourth-order force constants: Symmetry analysis reduces the otherwise impractical fourth-order IFC workload by applying index-transposition, space-group, and translational constraints.A fourth-order IFC calculation may require tens of thousands of DFT simulations for supercells exceeding 100 atoms, motivating symmetry reduction.
- Convergence and cost: Fourth-order IFC convergence with respect to cutoff distance must be checked manually, although second-nearest-neighbor cutoffs often suffice for many three-dimensional crystals.Increasing the cutoff substantially raises computational cost; convergence can be assessed from force constants or thermal conductivity.
3. Examples
Examples on Si, BAs, and LiCoO2 demonstrate how FourPhonon evaluates four-phonon scattering and its effects on thermal transport. The calculations show material-dependent scattering strengths, convergence behavior, and agreement with reference data.
- Si: 17 × 17 × 17 q-points suffice for converged Si thermal conductivity with four-phonon scattering, compared with 25 × 25 × 25 for three-phonon scattering.This reduced mesh requirement is beneficial because four-phonon calculations are computationally expensive.
- Si: In Si, Umklapp processes dominate four-phonon scattering, while normal three-phonon processes do not overwhelm Umklapp processes.The authors therefore consider an RTA treatment acceptable for Si thermal-conductivity calculations.
- BAs: Four-phonon scattering rates in BAs can be comparable to or larger than three-phonon rates even at room temperature.A single-temperature scattering-rate calculation requires about 1680 CPU hours.
- BAs: BAs thermal conductivity calculations give 1221.9 W/mK with scalebroad=1 and 1266.9 W/mK with scalebroad=0.1, a relative difference within 4%.The temperature-dependent results show larger reductions when four-phonon scattering is included at higher temperatures and agree well with experimental data.
- LiCoO2: LiCoO2 has four-phonon scattering rates generally comparable to three-phonon rates across the frequency range at room temperature.Unlike BAs, four-phonon rates are less significant in optical branches; redistribution contributes most, with Umklapp dominating that channel.
- LiCoO2: At 300 K, LiCoO2 thermal conductivity is 9.35 W/mK in-plane and 1.39 W/mK through-plane, compared with reference values of 9.7 and 1.4 W/mK.The calculations use a 3 × 3 × 3 supercell, a 10 × 10 × 10 q-point grid, and scalebroad=0.01.
4. Conclusion
FourPhonon extends ShengBTE with open-source four-phonon calculations, adaptive broadening, and fourth-order force-constant support. Examples across Si, BAs, and LiCoO2 demonstrate technical capabilities, while full four-phonon iterative calculations remain future work.
- FourPhonon provides an open-source tool for four-phonon scattering calculations while preserving ShengBTE features.
- The package extends adaptive broadening to four-phonon scattering and supplies Fourthorder.py for fourth-order force constants using point-group and translational symmetries.
- Examples on Si, BAs, and LiCoO2 cover common materials, systems with significant four-phonon effects, and newly reported complex systems.
- The examples demonstrate convergence with broadening factors and q-point grids, plus contributions from process types and scattering channels.
- Current calculations use the Single Mode Relaxation Time Approximation and can iterate with three-phonon scattering rates, but not fully four-phonon iterations.
Appendix A. Input and Output Files of FourPhonon
FourPhonon adds fourth-order force constants and a control-file flag to the ShengBTE workflow. Its outputs include process counts, phase-space measures, and RTA four-phonon scattering rates separated by process type.
- FourPhonon requires the FORCE_CONSTANTS_4TH file, generated by Fourthorder.py, in addition to ShengBTE input files.
- Users enable four-phonon calculations by setting four_phonon=.TRUE. in the CONTROL file.
- BTE.Numprocess_4ph reports the number of allowed four-phonon scattering processes for each mode.
- BTE.P4 and BTE.WP4 report available and weighted phase space for four-phonon processes.
- BTE.w_4ph, BTE.w_4ph_Umklapp, and BTE.w_4ph_normal report total, Umklapp, and normal four-phonon RTA scattering rates.
Appendix B. Derivation of Adaptive Broadening Formulas
The adaptive broadening derivation describes three participating phonons as independent or dependent in reciprocal space, formulates energy mismatches for three four-phonon processes, and obtains a common absolute-value expression.
- For a mode (q, ω), three other phonons occupy q′, q′′, and q′′′ with corresponding frequencies, where q′′′ depends on q′ and q′′.
- The second and third phonons are independent, while the fourth phonon is determined by their choice.
- The recombination process uses ∆ω = −ω′ −ω′′ + ω′′′ and q + q′ + q′′ = q′′′.
- The redistribution process uses ∆ω = −ω′ + ω′′ + ω′′′ and q + q′ −q′′ = q′′′.
- The splitting process uses ∆ω = ω′ + ω′′ + ω′′′ and q −q′ −q′′ = q′′′, after which absolute values give a common formula.