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Added documentation for solution of the Wigner transport equation.

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Michele Simoncelli 2022-05-25 01:00:44 +01:00
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doc/citation.md
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@ -48,3 +48,40 @@ following article:
url = {https://link.aps.org/doi/10.1103/PhysRevLett.110.265506}
}
```
## Citation of solution of the Wigner transport equation (`--wigner`)
If you have used the solution of the Wigner transport equation in phono3py, please cite the
following articles:
(citation_unified_theory)=
- "Unified theory of thermal transport in crystals and glasses"
Michele Simoncelli, Nicola Marzari, and Francesco Mauri, Nat. Phys., **15**, 809 (2019)
```
@article{NatPhys.15.809,
title={Unified theory of thermal transport in crystals and glasses},
author={Simoncelli, Michele and Marzari, Nicola and Mauri, Francesco},
journal={Nature Physics},
volume={15},
number={8},
pages={809--813},
year={2019},
publisher={Nature Publishing Group},
doi = {/10.1038/s41567-019-0520-x},
url = {https://www.nature.com/articles/s41567-019-0520-x}
}
```
(citation_wigner_formulation)=
- "Wigner formulation of thermal transport in solids"
Michele Simoncelli, Nicola Marzari, and Francesco Mauri, arXiv:2112.06897, (2021)
```
@article{arXiv:2112.06897,
title={Wigner formulation of thermal transport in solids},
author={Simoncelli, Michele and Marzari, Nicola and Mauri, Francesco},
journal={arXiv preprint arXiv:2112.06897},
url={https://arxiv.org/abs/2112.06897},
year={2021}
}
```

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doc/command-options.md
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@ -489,7 +489,7 @@ see {ref}`--ave-pp <ave_pp_option>`) is also given and stored.
(LTC_options)=
## Methods to solve phonon Boltzmann equation
## Methods to solve phonon Boltzmann equation, and Wigner formulation
### `--br` (`BTERTA = .TRUE.`)
@ -512,6 +512,19 @@ linearized phonon Boltzmann equation. The basis usage of this option is
equivalent to that of `--br`. More detail is documented at
{ref}`direct_solution`.
### `--wigner`
Run calculation of lattice thermal conductivity tensor computing the coherences (wave-like) contribution
to the thermal conductivity, obtained solving the Wigner transport equation equation.
This option can be combined with `--lbte` or `--br`; in the former case the populations
conductivity (particle-like, equivalent to the conductivity obtained solving the LBTE) is computed exactly,
in the latter case the populations conductivity is
computed in the relaxation-time approximation (RTA).
The coherences contribution to the conductivity is always computed exactly.
The coherences conductivity is usually non-negligible compared to the particle-like conductivity in materials
with ultralow or glass-like conductivity.
More details can be found at {ref}`wigner_solution`.
## Scattering
### `--isotope` (`ISOTOPE =.TRUE.`)

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doc/index.md
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@ -3,8 +3,9 @@
This software calculates phonon-phonon interaction and related properties using
the supercell approach. For example, the following physical values are obtained:
- Lattice thermal conductivity by relaxation time approximation and
direct-solution of phonon Boltzmann equation ({ref}`LTC_options`)
- Lattice thermal conductivity by relaxation time approximation,
direct-solution of phonon Boltzmann equation ({ref}`LTC_options`), and
solution of the Wigner transport equation ({ref}`LTC_options`)
- {ref}`Cummulative lattice thermal conductivity and related properties <auxiliary_tools_kaccum>`
- {ref}`self_energy_options` (Phonon lifetime/linewidth)
- {ref}`jdos_option`
@ -49,6 +50,7 @@ input-output-files
hdf5_howto
auxiliary-tools
direct-solution
wigner-solution
workload-distribution
cutoff-pair
external-tools

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@ -0,0 +1,87 @@
(wigner_solution)=
# Solution of the Wigner transport equation
This page explains how to compute the thermal conductivity from the solution of the Wigner transport equation (WTE)
[M. Simoncelli, N. Marzari, F. Mauri; Nat. Phys. 15, 809 (2019)](https://doi.org/10.1038/s41567-019-0520-x)
({ref}`citation <citation_unified_theory>`) and
[M. Simoncelli, N. Marzari, F. Mauri; arXiv:2112.06897 (2021)](https://arxiv.org/pdf/2112.06897)
({ref}`citation <citation_wigner_formulation>`).
The Wigner formulation of thermal transport in solids encompasses the emergence and coexistence of the particle-like propagation of phonon wavepackets discusses by Peierls for crystals [Peierls, Quantum theory of solids (Oxford Classics Series, 2001)], and the wave-like interband conduction mechanisms discussed by Allen and Feldman for harmonic glasses [
Allen and Feldman, Phys. Rev. Lett. 62, 645 (1989)]. As discussed in the references above, the Wigner formulation allows to describe the thermal conductivity of ordered crystals (where it yields practically the same result of the LBTE), of disordered glasses (where it generalizes Allen-Feldman theory accounting for anharmonicity), as well as of materials with intermediate characteristics (in this intermediate regime, both particle-like and wave-like conduction mechanisms are relevant, thus the Wigner formulation has to be used to obtain accurately predict the thermal conductivity).
In practice, the solution of the Wigner transport equation yields the following expression for the thermal conductivity tensor (we use $\alpha \beta$ to denote Cartesian directions) $\kappa^{\alpha \beta}_{\rm TOT}=\kappa_P^{\alpha \beta}+\kappa_{\rm C}^{\alpha \beta}$, where $\kappa_{\rm P}^{\alpha \beta}$ accounts for the particle-like propagation of phonon wavepackets and is exactly equivalent to the conductivity obtained solving the LBTE, while the other term $\kappa_{\rm C}^{\alpha \beta}$ is the "coherences" conductivity and accounts for the wave-like tunneling of phonons between bands with an energy differences smaller than their linewidths.
Specifically, the expression for $\kappa_{\rm C}^{\alpha \beta}$ reads:
$$
\kappa^{\alpha \beta}_{\rm C}=\frac{\hbar^2}{k_{B} {T}^2}\frac{1}{\mathcal{V}N_{\rm c}}\sum_{\mathbf{q}}\sum_{s\neq s'}\frac{\omega(\mathbf{q})_{s}+\omega(\mathbf{q})_{s'}}{2}{{V}^\alpha}(\mathbf{q})_{s,s'}{{V}}^\beta(\mathbf{q})_{s',s}
\frac{\omega(\mathbf{q})_{s}\bar{{N}}^{T}({\mathbf{q}})_{s}[\bar{{N}}^{T}({\mathbf{q}})_{s}+1]+\omega(\mathbf{q})_{s'}\bar{{N}}^{T}({\mathbf{q}})_{s'}[\bar{{N}}^{T}({\mathbf{q}})_{s'}+1]}{4[\omega(\mathbf{q})_{s'}-\omega(\mathbf{q})_{s}]^2+[\Gamma(\mathbf{q})_{s}+\Gamma(\mathbf{q})_{s'}]^2}[\Gamma(\mathbf{q})_{s}+\Gamma(\mathbf{q})_{s'}],
$$
where $k_{B}$ is the Boltzmann constant, $\mathcal{V}$ is the volume of the primitive cell, $N_{\rm c}$ is the number of phonon wavevectors $\mathbf{q}$ used to sample the Brillouin zone, $\hbar\omega(\mathbf{q})_s$ is the energy of the phonon with wavevector $\mathbf{q}$ and mode $s$, ${{V}^\alpha}(\mathbf{q})_{s,s'}$ is the velocity operator in direction $\alpha$, $\bar{{N}}^{T}({\mathbf{q}})_{s}$ is the Bose-Einstein distribution at temperature $T$, and $\Gamma(\mathbf{q})_{s}$ is the phonon linewidth (full width at half maximum, i.e. the inverse phonon lifetime $\Gamma(\mathbf{q})_{s}=[\tau(\mathbf{q})_{s}]^{-1}$).
As discussed in the references above, the term $\kappa_{\rm P}^{\alpha \beta}$ can be evaluated exactly or in the RTA approximation (the former corresponds to account for all the repumping/depumping scattering events, while the latter only for depumping scattering events). In contrast, the term $\kappa_{\rm C}^{\alpha \beta}$ depends only on the depumping scattering events, thus it remains unchanged if scattering is considered exactly or in the RTA approximation.
```{contents}
:depth: 2
:local:
```
## How to use
### Solution of the WTE, scattering in the RTA approximation
To compute the Wigner conductivity with scattering in the RTA approximation, specify `--br --wigner`. For `example/Si-PBEsol`, the command is:
```bash
% phono3py-load --mesh 11 11 11 --ts 1600 --br --wigner
```
and the output is
```bash
...
=================== End of collection of collisions ===================
----------- Thermal conductivity (W/m-k) with tetrahedron method -----------
# T(K) xx yy zz yz xz xy
K_P 1600.0 20.059 20.059 20.059 -0.000 -0.000 0.000
K_C 1600.0 0.277 0.277 0.277 -0.000 -0.000 0.000
K_T 1600.0 20.335 20.335 20.335 -0.000 -0.000 0.000
...
```
### Solution of the WTE, exact treatment of scattering
To compute the Wigner conductivity treating scattering exactly, specify `--lbte --wigner`. For `example/Si-PBEsol`, the command is:
```bash
% phono3py-load --mesh 11 11 11 --ts 1600 --lbte --wigner
```
and the output is
```bash
...
=================== End of collection of collisions ===================
- Averaging collision matrix elements by phonon degeneracy [0.035s]
- Making collision matrix symmetric (built-in) [0.000s]
----------- Thermal conductivity (W/m-k) with tetrahedron method -----------
Diagonalizing by lapacke dsyev... [0.148s]
Calculating pseudo-inv with cutoff=1.0e-08 (np.dot) [0.002s]
# T(K) xx yy zz yz xz xy
K_P_exact 1600.0 21.009 21.009 21.009 -0.000 -0.000 0.000
(K_P_RTA) 1600.0 20.059 20.059 20.059 -0.000 -0.000 0.000
K_C 1600.0 0.277 0.277 0.277 -0.000 -0.000 0.000
K_TOT=K_P_exact+K_C 1600.0 21.286 21.286 21.286 -0.000 -0.000 0.000
----------------------------------------------------------------------------
...
```
We also note that the examples above are performed at very high temperature for illustrative purposes.
The coherences conductivity is often a non-negligible fraction of the total conductivity in materials with glass-like or ultralow thermal conductivity ($\frac{1}{3}\sum_{\alpha=1}^3\kappa^{\alpha\alpha}_{\rm TOT}\lesssim 1 \frac{W}{m\cdot K}$).
## Computational cost
Using the code with the `--wigner` option has a negligible effect on the duration of the calculation.