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6.5 Lattice thermal conductivity: external third-order tools

These are unexecuted teaching inputs and starting settings to test. Original figures are schematics, not computed results. Use licensed VASP and PAW data, replace every placeholder, record the executable version and validate convergence.

6.5.1 Model, units and provenance

Use eV for energy, Å for length and eV/Å for force; 1 kbar = 0.1 GPa. State normalization per atom, molecule, primitive cell or simulation cell. Record PAW identifiers, release, ZVAL, ENMAX and permitted hashes; never redistribute POTCAR. SCF convergence addresses the chosen electronic problem; convergence of the target property requires separate tests.

Lattice thermal conductivity: external third-order tools — original conceptual schematic

Original schematic. Curves explain concepts; blank data areas await verified learner results. No calculation is claimed.

\[ \kappa_{\alpha\beta}^{\mathrm{RTA}}=\frac1V\sum_{\mathbf q\nu}w_{\mathbf q}C_{\mathbf q\nu}v_{\mathbf q\nu,\alpha}v_{\mathbf q\nu,\beta}\tau_{\mathbf q\nu} \]

6.5.2 Worked case: procedure, interpretation and checks

Scope and physical idea

Heat capacity measures how much vibrational energy a crystal stores. Thermal conductivity measures how effectively that energy travels down a temperature gradient. A perfectly harmonic crystal has no intrinsic phonon–phonon scattering in that model. To obtain finite intrinsic lattice conductivity through three-phonon processes, one needs third-order force constants and a transport solver.

VASP is the first-principles force engine here. Phono3py is an external package that can generate displacement sets, determine third-order interactions, and solve phonon transport. The separate thirdorder_vasp.py utility from the ShengBTE ecosystem also generates/reconstructs third-order data, primarily in the ShengBTE format. “thirdorder plus phono3py” is not one built-in VASP mode, and a thirdorder output file is not automatically interchangeable with phono3py's native fc3.hdf5. Prefer one internally consistent pipeline unless a verified format conversion is genuinely required.

Mathematical map

The third-order tensor is Ψiα,jβ,kγ = ∂³E/(∂uiα∂ujβ∂ukγ), with units eV/ų. In a simple phonon relaxation-time approximation (RTA),

καβ = (1/Nq Vcell) Σλ Cλ vλα vλβ τλ.

Here Cλ is mode heat capacity, v group velocity, τ lifetime, and Vcell the primitive-cell volume. With C in J/K, v in m/s, τ in s, and V in m³, κ is W/(m K). Check whether mesh weights are already included before dividing by Nq again. RTA treats each mode's relaxation independently; a direct linearized Boltzmann solution accounts for coupling between nonequilibrium populations and can differ when momentum-conserving normal processes matter.

Route A: native phono3py displacement workflow

Begin only after harmonic phonons are reliable. Record the Phono3py version because v4 separates setup from transport execution.

  1. Choose second- and third-order supercells and third-order interaction range. fc2 may need a larger supercell than fc3, but this must be tested.
  2. Generate the native displacement manifest and supercells. An illustrative current setup command is phono3py-init -d --dim 2 2 2 -c POSCAR-unitcell. It selects a size, not a convergence guarantee. Choose primitive mapping explicitly when needed.
  3. Run VASP static force jobs for every displacement using the 6.1 Finite-displacement phonons with Phonopy force template, stringent force accuracy, and consistent atom ordering. Do not relax them.
  4. Collect XML force outputs in manifest order using the installed phono3py-init --cf3 interface. Collect any separate fc2 set using its own matching metadata.
  5. Build and validate fc2/fc3. Inspect translational/permutation symmetry residuals and harmonic stability before computing expensive scattering grids.
  6. Perform an RTA calculation, for example phono3py --mesh 13 13 13 --br with the corresponding native data present, then compare larger q meshes and, when scientifically motivated, the direct transport solution.

Route B: thirdorder workflow and interoperability

The ShengBTE thirdorder workflow uses a “sow → VASP forces → reap” pattern: generate inequivalent displaced supercells, compute forces, reconstruct third-order constants. Its specific supercell/cutoff arguments and force-file ordering must follow the installed official README. It commonly produces FORCE_CONSTANTS_3RD, whose real-space triplet and lattice-vector conventions differ from dense or compact phono3py arrays.

If importing this information into phono3py, use only a documented, version-compatible converter or API route. Validate atom order, primitive/supercell mapping, lattice translations, force-constant units, tensor-index order, and compact versus full shape. Compare a sample of reconstructed forces for independent small displacements in the source and target representations. Renaming FORCE_CONSTANTS_3RD to fc3.hdf5 does not convert it. For a beginner, staying within phono3py's own generation/collection route is safer than adding an unverified converter; thirdorder naturally pairs with ShengBTE when that solver is intended.

Checks and interpretation

Converge force precision, displacement amplitude, supercell interaction range, and scattering q mesh independently. Tightening only the last one can produce a stable answer to an inaccurate force model. Inspect κxx, κyy, κzz and off-diagonal components in the correct Cartesian frame; cubic symmetry offers a useful equality test. Lifetimes and linewidths have convention-dependent 2π/factor-of-two relations, so convert using the tool's definitions, not a memorized rule.

State included scattering sources: three-phonon anharmonicity, isotope disorder if enabled, and any boundary model. Electronic thermal conductivity, four-phonon effects, defects, grain boundaries, and finite-size transport are absent unless explicitly treated. Do not compare lattice κ to total measured κ in a metal without separating the electronic contribution. Low-dimensional κ depends on thickness/volume conventions; report them. Very short lifetimes or strongly overdamped modes can undermine the phonon quasiparticle/Boltzmann assumptions.

Exercise: make a convergence matrix of two fc3 ranges and three q meshes. Compare changes in κ and in the dominant mode lifetimes. Identify whether an apparent anisotropy survives tighter numerical settings. No numerical “expected κ” is supplied because that would depend on the material and complete physical model.

6.5.3 Unexecuted inputs and analysis scaffolds

These are unexecuted teaching inputs and starting settings to test. Original figures are schematics, not computed results. Use licensed VASP and PAW data, replace every placeholder, record the executable version and validate convergence.

This case uses a validated parent calculation and the stated post-processing workflow. The worked lesson defines its data requirements rather than supplying an unvalidated universal executable.

6.5.5 Technical sources