Electronic transport calculations with the BoltzWann module

By setting \(\verb#boltzwann#=\verb#TRUE#\), postw90 will call the BoltzWann routines to calculate some transport coefficients using the Boltzmann transport equation in the relaxation time approximation.

In particular, the transport coefficients that are calculated are: the electrical conductivity \(\bm{\mathrm{\sigma}}\), the Seebeck coefficient \(\bm{\mathrm{S}}\) and the coefficient \(\bm{\mathrm{K}}\) (defined below; it is the main ingredient of the thermal conductivity).

The list of parameters of the BoltzWann module are summarized in Table [parameter_keywords_bw]{reference-type="ref" reference="parameter_keywords_bw"}. An example of a Boltzmann transport calculation can be found in the wannier90 Tutorial.

Note: By default, the code assumes to be working with a 3D bulk material, with periodicity along all three spatial directions. If you are interested in studying 2D systems, set the correct value for the boltz_2d_dir variable (see Sec. [sec:boltz2ddir]{reference-type="ref" reference="sec:boltz2ddir"} for the documentation). This is important for the evaluation of the Seebeck coefficient.

Please cite the following paper ¹ when publishing results obtained using the BoltzWann module:

G. Pizzi, D. Volja, B. Kozinsky, M. Fornari, and N. Marzari,\ BoltzWann: A code for the evaluation of thermoelectric and electronic transport properties with a maximally-localized Wannier functions basis,\ Comp. Phys. Comm. 185, 422 (2014), DOI:10.1016/j.cpc.2013.09.015.

Theory

The theory of the electronic transport using the Boltzmann transport equations can be found for instance in Refs. ²³⁴. Here we briefly summarize only the main results.

The current density \(\bm{\mathrm{J}}\) and the heat current (or energy flux density) \(\bm{\mathrm{J}}_Q\) can be written, respectively, as \(\(\begin{aligned} \bm{\mathrm{J}} &= \bm{\mathrm{\sigma}}(\bm{\mathrm{E}} - \bm{\mathrm{S}} \bm{\mathrm{\nabla }}T) \\ \bm{\mathrm{J}}_Q &= T \bm{\mathrm{\sigma }}\bm{\mathrm{S}} \bm{\mathrm{E}} - \bm{\mathrm{K}} \bm{\mathrm{\nabla }}T, \end{aligned}\)\) where the electrical conductivity \(\bm{\mathrm{\sigma}}\), the Seebeck coefficient \(\bm{\mathrm{S}}\) and \(\bm{\mathrm{K}}\) are \(3\times 3\) tensors, in general.

Note: the thermal conductivity \(\bm{\mathrm{\kappa}}\) (actually, the electronic part of the thermal conductivity), which is defined as the heat current per unit of temperature gradient in open-circuit experiments (i.e., with \(\bm{\mathrm{J}}=0\)) is not precisely \(\bm{\mathrm{K}}\), but \(\bm{\mathrm{\kappa }}= \bm{\mathrm{K}}-\bm{\mathrm{S}} \bm{\mathrm{\sigma }}\bm{\mathrm{S}} T\) (see for instance Eq. (7.89) of Ref. ² or Eq. (XI-57b) of Ref. ³). The thermal conductivity \(\bm{\mathrm{\kappa}}\) can be then calculated from the \(\bm{\mathrm{\sigma}}\), \(\bm{\mathrm{S}}\) and \(\bm{\mathrm{K}}\) tensors output by the code.

These quantities depend on the value of the chemical potential \(\mu\) and on the temperature \(T\), and can be calculated as follows: \(\(\begin{aligned} _{ij}(\mu,T)&=e^2 \int_{-\infty}^{+\infty} d\varepsilon \left(-\frac {\partial f(\varepsilon,\mu,T)}{\partial \varepsilon}\right)\Sigma_{ij}(\varepsilon), \\ [\bm{\mathrm{\sigma }}\bm{\mathrm{S}}]_{ij}(\mu,T)&=\frac e T \int_{-\infty}^{+\infty} d\varepsilon \left(-\frac {\partial f(\varepsilon,\mu,T)}{\partial \varepsilon}\right)(\varepsilon-\mu)\Sigma_{ij}(\varepsilon), \label{eq:boltz-sigmas}\\ [\bm{\mathrm{K}}]_{ij}(\mu,T)&=\frac 1 T \int_{-\infty}^{+\infty} d\varepsilon \left(-\frac {\partial f(\varepsilon,\mu,T)}{\partial \varepsilon}\right)(\varepsilon-\mu)^2 \Sigma_{ij}(\varepsilon),\label{eq:boltz-thermcond} \end{aligned}\)\) where \([\bm{\mathrm{\sigma }}\bm{\mathrm{S}}]\) denotes the product of the two tensors \(\bm{\mathrm{\sigma}}\) and \(\bm{\mathrm{S}}\), \(f(\varepsilon,\mu,T)\) is the usual Fermi--Dirac distribution function \(\(f(\varepsilon,\mu,T) = \frac{1}{e^{(\varepsilon-\mu)/K_B T}+1}\)\) and \(\Sigma_{ij}(\varepsilon)\) is the Transport Distribution Function (TDF) tensor, defined as \(\(\Sigma_{ij}(\varepsilon) = \frac 1 V \sum_{n,\bm{\mathrm{k}}} v_i(n,\bm{\mathrm{k}}) v_j(n,\bm{\mathrm{k}}) \tau(n,\bm{\mathrm{k}}) \delta(\varepsilon - E_{n,k}).\)\)

In the above formula, the sum is over all bands \(n\) and all states \(\bm{\mathrm{k}}\) (including spin, even if the spin index is not explicitly written here). \(E_{n,\bm{\mathrm{k}}}\) is the energy of the \(n-\)th band at \(\bm{\mathrm{k}}\), \(v_i(n,\bm{\mathrm{k}})\) is the \(i-\)th component of the band velocity at \((n,\bm{\mathrm{k}})\), \(\delta\) is the Dirac's delta function, \(V = N_k \Omega_c\) is the total volume of the system (\(N_k\) and \(\Omega_c\) being the number of \(k\)-points used to sample the Brillouin zone and the unit cell volume, respectively), and finally \(\tau\) is the relaxation time. In the relaxation-time approximation adopted here, \(\tau\) is assumed as a constant, i.e., it is independent of \(n\) and \(\bm{\mathrm{k}}\) and its value (in fs) is read from the input variable boltz_relax_time.

Files

seedname_boltzdos.dat

OUTPUT. Written by postw90 if boltz_calc_also_dos is true. Note that even if there are other general routines in postw90 which specifically calculate the DOS, it may be convenient to use the routines in BoltzWann setting boltz_calc_also_dos = true if one must also calculate the transport coefficients. In this way, the (time-demanding) band interpolation on the \(k\) mesh is performed only once, resulting in a much shorter execution time.

The first lines are comments (starting with # characters) which describe the content of the file. Then, there is a line for each energy \(\varepsilon\) on the grid, containing a number of columns. The first column is the energy \(\varepsilon\). The following is the DOS at the given energy \(\varepsilon\). The DOS can either be calculated using the adaptive smearing scheme (see the following note) if boltz_dos_adpt_smr is true, or using a "standard" fixed smearing, whose type and value are defined by boltz_dos_smr_type and boltz_dos_smr_fixed_en_width, respectively. If spin decomposition is required (input flag spin_decomp), further columns are printed, with the spin-up projection of the DOS, followed by spin-down projection.

Note

Note that in BoltzWann the adaptive (energy) smearing scheme also implements a simple adaptive \(k-\)mesh scheme: if at any given \(k\) point one of the band gradients is zero, then that \(k\) point is replaced by 8 neighboring \(k\) points. Thus, the final results for the DOS may be slightly different with respect to that given by the dos module.

seedname_tdf.dat

OUTPUT. This file contains the Transport Distribution Function (TDF) tensor \(\bm{\mathrm{\Sigma}}\) on a grid of energies.

The first lines are comments (starting with # characters) which describe the content of the file. Then, there is a line for each energy \(\varepsilon\) on the grid, containing a number of columns. The first is the energy \(\varepsilon\), the followings are the components if \(\bm{\mathrm{\Sigma}}(\varepsilon)\) in the following order: \(\Sigma_{xx}\), \(\Sigma_{xy}\), \(\Sigma_{yy}\), \(\Sigma_{xz}\), \(\Sigma_{yz}\), \(\Sigma_{zz}\). If spin decomposition is required (input flag spin_decomp), 12 further columns are provided, with the 6 components of \(\bm{\mathrm{\Sigma}}\) for the spin up, followed by those for the spin down.

The energy \(\varepsilon\) is in eV, while \(\bm{\mathrm{\Sigma}}\) is in \(\displaystyle\frac{1}{\hbar^2}\cdot\frac{\mathrm{eV}\cdot\mathrm{fs}}{\text{\AA}}\).

seedname_elcond.dat

OUTPUT. This file contains the electrical conductivity tensor \(\bm{\mathrm{\sigma}}\) on the grid of \(T\) and \(\mu\) points.

The first lines are comments (starting with # characters) which describe the content of the file. Then, there is a line for each \((\mu,T)\) pair, containing 8 columns, which are respectively: \(\mu\), \(T\), \(\sigma_{xx}\), \(\sigma_{xy}\), \(\sigma_{yy}\), \(\sigma_{xz}\), \(\sigma_{yz}\), \(\sigma_{zz}\). (The tensor is symmetric).

The chemical potential is in eV, the temperature is in K, and the components of the electrical conductivity tensor ar in SI units, i.e. in 1/\(\Omega\)/m.

seedname_sigmas.dat

OUTPUT. This file contains the tensor \(\bm{\mathrm{\sigma}}\bm{\mathrm{S}}\), i.e. the product of the electrical conductivity tensor and of the Seebeck coefficient as defined by Eq. [eq:boltz-sigmas]{reference-type="eqref" reference="eq:boltz-sigmas"}, on the grid of \(T\) and \(\mu\) points.

The first lines are comments (starting with # characters) which describe the content of the file. Then, there is a line for each \((\mu,T)\) pair, containing 8 columns, which are respectively: \(\mu\), \(T\), \((\sigma S)_{xx}\), \((\sigma S)_{xy}\), \((\sigma S)_{yy}\), \((\sigma S)_{xz}\), \((\sigma S)_{yz}\), \((\sigma S)_{zz}\). (The tensor is symmetric).

The chemical potential is in eV, the temperature is in K, and the components of the tensor ar in SI units, i.e. in A/m/K.

seedname_seebeck.dat

OUTPUT. This file contains the Seebeck tensor \(\bm{\mathrm{S}}\) on the grid of \(T\) and \(\mu\) points.

Note that in the code the Seebeck coefficient is defined as zero when the determinant of the electrical conductivity \(\bm{\mathrm{\sigma}}\) is zero. If there is at least one \((\mu, T)\) pair for which \(\det \bm{\mathrm{\sigma}}=0\), a warning is issued on the output file.

The first lines are comments (starting with # characters) which describe the content of the file. Then, there is a line for each \((\mu,T)\) pair, containing 11 columns, which are respectively: \(\mu\), \(T\), \(S_{xx}\), \(S_{xy}\), \(S_{xz}\), \(S_{yx}\), \(S_{yy}\), \(S_{yz}\), \(S_{zx}\), \(S_{zy}\), \(S_{zz}\).

NOTE: therefore, the format of the columns of this file is different from the other three files (elcond, sigmas and kappa)!

The chemical potential is in eV, the temperature is in K, and the components of the Seebeck tensor ar in SI units, i.e. in V/K.

seedname_kappa.dat

OUTPUT. This file contains the tensor \(\bm{\mathrm{K}}\) defined in Sec. 3.1{reference-type="ref" reference="sec:boltzwann-theory"} on the grid of \(T\) and \(\mu\) points.

The first lines are comments (starting with # characters) which describe the content of the file. Then, there is a line for each \((\mu,T)\) pair, containing 8 columns, which are respectively: \(\mu\), \(T\), \(K_{xx}\), \(K_{xy}\), \(K_{yy}\), \(K_{xz}\), \(K_{yz}\), \(K_{zz}\). (The tensor is symmetric).

The chemical potential is in eV, the temperature is in K, and the components of the \(\bm{\mathrm{K}}\) tensor are the SI units for the thermal conductivity, i.e. in W/m/K.

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1. G. Pizzi, D. Volja, B. Kozinsky, M. Fornari, and N. Marzari. Boltzwann: a code for the evaluation of thermoelectric and electronic transport properties with a maximally-localized wannier functions basis. Comput. Phys. Commun., 185:422, 2014. doi:doi:10.1016/j.cpc.2013.09.015. ↩

2. J. Ziman. Principles of the Theory of Solids. Cambridge University Press, 2^nd ed. edition, 1972. ↩↩

3. G. Grosso and G. P. Parravicini. Solid State Physics. Academic Press, 2000. ↩↩

4. G. D. Mahan. Transport properties. In Intern. Tables for Crystallography, volume D, chapter 1.8, pages 7828. 2006. ↩