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BoltzWann: A code for the evaluation of thermoelectric and electronic transport properties with a maximally-localized Wannier functions basis
Giovanni Pizzi, Dmitri Volja, Boris Kozinsky, Marco Fornari, Nicola Marzari
TL;DR
Accurate electronic and thermoelectric transport calculations are hindered by costly dense sampling and unreliable finite-difference derivatives near band crossings. BoltzWann uses maximally-localized Wannier interpolation within semiclassical Boltzmann theory, and tests on CoSb3 and CoGe3/2S3/2 show very good agreement with BoltzTraP while enabling analytical band derivatives. Its transport treatment uses a constant relaxation-time approximation and covers electronic, not ionic, thermal transport.
Problem
Dense Brillouin-zone sampling is needed for converged transport properties, but costly first-principles calculations and finite-difference derivatives can be problematic near band crossings.
Method
BoltzWann solves semiclassical Boltzmann transport equations using maximally-localized Wannier functions to interpolate plane-wave bands and analytically evaluate band derivatives.
Results
The code shows very good agreement with BoltzTraP for CoSb3 and CoGe3/2S3/2 transport properties and is compared with experimental CoSb3 Seebeck data.
Takeaways & Limitations
Wannier interpolation provides accurate transport quantities from coarse k-point calculations at substantially lower computational cost, with stable derivatives near crossings.
Takeaways & Limitations
The method assumes a band- and wavevector-independent lifetime, treats only electronic thermal transport, and can disagree with BoltzTraP outside the frozen energy window.
Abstract
from arXiv · showhide
We present a new code to evaluate thermoelectric and electronic transport properties of extended systems with a maximally-localized Wannier function basis set. The semiclassical Boltzmann transport equations for the homogeneous infinite system are solved in the constant relaxation-time approximation and band energies and band derivatives are obtained via Wannier interpolations. Thanks to the exponential localization of the Wannier functions obtained, very high accuracy in the Brillouin zone integrals can be achieved with very moderate computational costs. Moreover, the analytical expression for the band derivatives in the Wannier basis resolves any issues that may occur when evaluating derivatives near band crossings. The code is tested on binary and ternary skutterudites CoSb_3 and CoGe_{3/2}S_{3/2}.
1. Introduction
Thermoelectric transport calculations require accurate band energies and velocities over dense Brillouin-zone grids, but direct first-principles evaluation is costly and finite-difference velocities can fail near band crossings. BoltzWann addresses these challenges by using maximally-localized Wannier functions to interpolate bands and analytically evaluate derivatives, with verification against other calculations and experiment.
- Thermal and electrical transport properties are studied to identify efficient thermoelectric materials for energy harvesting and waste heat recovery.
- ZT balances electrical conductivity, Seebeck coefficient, thermal conductivity, and temperature, requiring high conductivity and Seebeck coefficient alongside low thermal conductivity.Commercial bulk materials have ZT slightly above 1, while a laboratory nano- and microscale material reached 2.2.
- Transport calculations require band velocities in addition to band energies, because velocities enter the evaluation of electrical and thermal properties.
- Dense k-point grids and finite-difference derivatives are computationally demanding and can introduce errors from incorrect band ordering near crossings.First-principles eigenvalue calculations generally scale as O(N^3) with system size and increase with the number of k points.
- BoltzWann uses MLWFs to interpolate plane-wave band structures from coarse grids, reducing computational cost while retaining high accuracy through exponential localization.
- Analytical Wannier-basis band derivatives provide numerically stable results at band crossings and weak avoided crossings; the code is compared with BoltzTraP and experimental CoSb3 Seebeck data.
2. Semiclassical transport theory
The paper formulates electrical, thermoelectric, and electronic thermal transport within semiclassical Boltzmann theory using a transport distribution function. It adopts a constant relaxation-time approximation and distinguishes the electronic thermal contribution from the ionic lattice contribution.
- The transport equations relate electrical and electronic heat currents to electric fields and temperature gradients through conductivity, Seebeck, and thermal-conductivity tensors.These rank-two tensors reduce to scalars for isotropic media.
- The thermal current treated here contains only the electronic contribution; ionic thermal conductivity requires phonon–phonon scattering and often dominates in semiconductors and insulators.
- The transport tensors are expressed as energy integrals involving the transport distribution function, the Fermi–Dirac derivative, chemical potential, and temperature.
- The transport distribution function sums contributions over bands and the Brillouin zone using band energies and velocity components.
- The calculations assume a lifetime independent of band index and wavevector, with its value fitted to experimental electrical conductivity at a chosen temperature.This constant relaxation-time approximation is commonly used and can be effective for bulk materials.
3. Wannier functions basis
The paper constructs maximally localized Wannier functions to represent and interpolate electronic bands, including entangled bands through disentanglement. In this basis, band velocities can be evaluated analytically at arbitrary k points, while second derivatives require additional treatment.
- Wannier-function construction: Maximally localized Wannier functions are constructed from Bloch manifolds using a unitary transformation that minimizes the localization functional.For entangled bands, an outer window selects states and an optional frozen inner window preserves selected states during disentanglement.
- Hamiltonian representation: The Hamiltonian is projected onto the Wannier subspace, where its matrix elements form a tight-binding representation that is diagonalized through a gauge transformation.Within the inner energy window, the resulting eigenvalues and Hamiltonian-gauge states coincide with the original first-principles quantities.
- Velocity interpolation: Band velocities are obtained analytically from the k-derivative of the Wannier-gauge Hamiltonian and the unitary transformation to the Hamiltonian gauge.The diagonal velocity elements reduce to those of the transformed Hamiltonian derivative because terms involving derivatives of the gauge transformation cancel.
- Velocity interpolation: The analytic velocity expression is simple for first derivatives, whereas second derivatives such as effective masses require non-diagonal terms and additional treatment, including for degeneracies.The paper refers to a separate discussion for the proper treatment of second derivatives and band degeneracies.
4. Details of the code
BoltzWann operates as a Wannier90 post-processing module: it uses MLWFs from a standard calculation to interpolate bands on dense k meshes and evaluate transport-related quantities. Its inputs control mesh, energy-grid, DOS, band-shift, and parallel-processing options.
- Module workflow: BoltzWann is developed as a Wannier90 module, with its control flags placed in the shared seedname.win input file and prefixed by boltz.A standard wannier90.x calculation produces the MLWFs, after which postw90.x activates the BoltzWann module.
- Transport-distribution calculation: The code evaluates the transport distribution function on a Γ-containing n1 × n2 × n3 Monkhorst-Pack mesh or from a specified minimum k-point spacing.The energy grid is controlled by the boltz tdf energy step parameter, and dense-grid band interpolation is the time-consuming operation.
- Additional outputs: BoltzWann can calculate the density of states on the same interpolated k grid, avoiding a second band-interpolation step when DOS data are also needed.DOS smearing can be selected through the boltz dos smr type flag, including Gaussian and Marzari–Vanderbilt options.
- Input parameters: Table 1 lists the input keywords controlling the BoltzWann module within the Wannier90 seedname.win file.The table serves as the module’s input-parameter reference.
- Additional controls: The code supports shifting bands above a selected first-band index to account for effects such as minority intrinsic carriers when comparing simulations with experiments.The shift is applied to all bands with indices larger than boltz bandshift firstband.
- Parallelization: The calculation is fully parallelized by distributing k points for TDF evaluation and (µ, T) pairs for the subsequent calculation.Processors locally accumulate TDF contributions before the results are summed and distributed.
5. Applications and verification
BoltzWann was applied to CoSb3 and CoGe3/2S3/2 using MLWF interpolations, reproducing first-principles bands and benchmarking transport properties against BoltzTraP. Across both skutterudites, the codes showed very good agreement, with comparisons also made to experimental CoSb3 Seebeck data.
- Wannier functions basis in CoSb3: CoSb3 calculations used 56 MLWFs after excluding the lowest 12-band manifold, with a frozen window about 1–1.5 eV above the conduction-band minimum.The initial orbitals were Co d and Sb p states.
- Wannier functions basis in CoGe3/2S3/2: CoGe3/2S3/2 calculations excluded two 12-band valence manifolds and used 112 MLWFs, with the frozen-window top set 1 eV above the conduction-band minimum.The final spreads were 2.1–4.4 Å^2.
- Code results and verification: Transport calculations interpolated bands from a 4 × 4 × 4 coarse grid onto a 40 × 40 × 40 mesh, using 1 meV energy bins and τ = 10 fs.BoltzTraP comparisons started from a 20 × 20 × 20 mesh.
- Code results and verification: Electrical conductivity, Seebeck coefficient, and K showed very good agreement between BoltzWann and BoltzTraP for both skutterudites at 300, 500, and 700 K.The comparisons are reported in Figures 4–6 as functions of chemical potential.
- Code results and verification: BoltzWann Seebeck coefficients were also compared with experimental results for n- and p-doped CoSb3 samples at reported doping levels.The theoretical calculations assumed the experimental doping values and excluded intrinsic-carrier effects.
- Code results and verification: Discrepancies between the two codes are expected when the chemical potential lies outside the frozen window because the Wannier interpolation is accurate only within that window.
6. Conclusions
The authors implemented and tested a Wannier-based Fortran module for semiclassical electronic and thermoelectric transport, finding strong agreement with BoltzTraP. Analytical band derivatives improve accuracy near band crossings while reducing computational cost, especially for large systems.
- The Fortran module computes electrical conductivity, Seebeck coefficient, and electronic thermal conductivity using Wannier-interpolated bands and velocities.
- The code reproduced BoltzTraP results with very good agreement for CoSb3 and CoGe3/2S3/2.
- Analytical band derivatives in the Wannier basis properly treat band crossings, improving accuracy and reducing computational cost.
- These benefits are particularly relevant for large systems containing many intersecting folded bands in the Brillouin zone.
- The code can be extended beyond constant relaxation time when the relaxation time depends on band index and quasimomentum.