mirror of https://github.com/abinit/abinit.git
921 lines
44 KiB
Markdown
921 lines
44 KiB
Markdown
---
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authors: GB, MG
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---
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# Phonon-limited mobility
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This tutorial discusses how to compute phonon-limited carrier mobilities in semiconductors within
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the self-energy relaxation time approximation (SERTA) and the momentum relaxation time approximation (MRTA),
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taking the specific case of AlAs as an example.
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It is assumed the user has already completed the two tutorials [RF1](/tutorial/rf1) and [RF2](/tutorial/rf2),
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and that he/she is familiar with the calculation of ground state and response properties,
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in particular phonons, Born effective charges and dielectric tensor.
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The user should have read the [introduction tutorial for the EPH code](/tutorial/eph_intro)
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before running these examples.
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This lesson should take about 1.5 hour.
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## Formalism
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Before starting, it is worth summarizing the most important equations implemented in the code.
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For a more detailed description of the ABINIT implementation, please consult [[cite:Brunin2020b]].
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Our goal is to find an approximated solution to the linearized
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Boltzmann transport equation (BTE) [[cite:Ashcroft1976]] within the SERTA/MRTA approximation [[cite:Giustino2017]].
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SERTA/MRTA are more accurate than the constant relaxation time approximation (CRTA) as the
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microscopic e-ph scattering mechanism is now included thus leading to carrier lifetimes $\tau$
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that depend on the band index $n$ and the wavevector $\kk$.
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Keep in mind, however, that both SERTA and MRTA are still a **approximated solutions to the BTE**
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and that a more rigorous approach would require to solve the BTE iteratively and/or
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the inclusion of many-body effects at different levels.
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For a review of the different possible approaches see the review paper [[cite:Ponce2020]].
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In the SERTA, the transport linewidth is given by
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the imaginary part of the electron-phonon (e-ph) self-energy evaluated at the KS energy [[cite:Giustino2017]].
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The linewidth of the electron state $n\kk$ due to the scattering with phonons is given by
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\begin{equation}
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\begin{split}
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\lim_{\eta \rightarrow 0^+} & \Im\{\Sigma^\FM_{n\kk}(\enk)\} =
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\pi \sum_{m,\nu} \int_\BZ \frac{d\qq}{\Omega_\BZ} |\gkkp|^2\\
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& \times \left[ (n_\qnu + f_{m\kk+\qq})
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\delta(\enk - \emkq + \wqnu) \right.\\
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& \left. + (n_\qnu + 1 - f_{m\kk+\qq})
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\delta(\enk - \emkq - \wqnu ) \right]
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\end{split}
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\label{eq:imagfanks_selfen}
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\end{equation}
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where $\nu$ is the phonon mode, $m$ the final electron state (after the scattering),
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$n_\qnu(T)$ is the Bose-Einstein occupation number, $f_{m\kk+\qq}(T, \ef)$ is the Fermi-Dirac occupation function,
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$\enk$ is the energy of the electron state, and $\wqnu$ is the phonon frequency for the phonon wavevector $\qq$.
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The integration is performed over the BZ for the phonon wavectors and $\gkkp$ is the e-ph matrix element.
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Only the Fan-Migdal (FM) part contributes to the linewidth as the Debye-Waller term is Hermitian.
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In the SERTA, the **transport lifetime** $\tau_{n\mathbf{k}}$ is inversely proportional to the e-ph self-energy linewidth:
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\begin{align}
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\frac{1}{\tau_{n\kk}} =
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2 \lim_{\eta \rightarrow 0^+} \Im\{\Sigma^\FM_{n\kk}(\enk)\}.
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\label{eq:fanlifetime}
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\end{align}
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In the MRTA, the back-scattering is included by expressing the transport lifetime as:
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\begin{equation}
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\begin{split}
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\frac{1}{\tau_{n\kk}} & =
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2 \pi \sum_{m,\nu} \int_\BZ \frac{d\qq}{\Omega_\BZ} |\gkkp|^2 \left( 1 - \frac{\vnk \cdot \vmkq}{|\vnk|^2} \right) \\
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& \times \left[ (n_\qnu + f_{m\kk+\qq})
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\delta(\enk - \emkq + \wqnu) \right.\\
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& \left. + (n_\qnu + 1 - f_{m\kk+\qq})
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\delta(\enk - \emkq - \wqnu ) \right].
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\end{split}
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\label{eq:mrta}
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\end{equation}
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where $\vnka$ is the $\alpha$-th Cartesian component of the velocity operator
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$$\vnk = \PDER{\enk}{\kk} = \langle \nk | \dfrac{\partial{H}}{\partial{\kk}} | \nk \rangle.$$
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that can be computed with DFPT.
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!!! important
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Note that the present formalism does not take into account contributions to the transport lifetime given by
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other scattering processes such as **defects, ionized impurities in doped semiconductors, e-e interaction,
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grain boundary scattering etc**.
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These effects may be relevant depending on the system and/or the temperature under investigation
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but they are not treated in this tutorial as here we are mainly focusing on **room temperature and non-degenerate
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semiconductors**, conditions in which e-ph scattering is one of the most important contributions.
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Last but not least, we are assuming that carriers can be described by Bloch states with a well-defined excitation
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energy (**band picture**). Polaronic effects such as those discussed in [this tutorial](/tutorial/eph4zpr)
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are not captured by the present approach.
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The generalized transport coefficients are defined by:
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\begin{equation}
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\mcL^{(m)}_{\alpha\beta} =
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- \sum_n \int \frac{d\kk}{\Omega_\BZ} \vnka \vnkb\, \tau_{n\kk}
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(\enk-\ef)^m
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\left.\frac{\partial f}{\partial\varepsilon}\right|_{\enk}
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\label{eq:transport_lc}
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\end{equation}
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These quantities can be used to obtain different transport tensors such as
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the electrical conductivity $\sigma$, Peltier ($\Pi$) and Seebeck coefficient (S),
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and charge carrier contribution to the thermal conductivity tensors [[cite:Madsen2018]].
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The electrical conductivity tensor, for instance, is given by
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\begin{equation}
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\sigma_{\alpha\beta} =
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\frac{1}{\Omega} \mcL_{\alpha\beta}^{(0)} \label{eq:transport_sigma}
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\end{equation}
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and can be divided into hole and electron contributions
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\begin{equation}
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\sigma = n_e \mu_{e} + n_h \mu_{h} \label{eq:mobility}
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\end{equation}
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where $n_e$ and $n_h$ are the electron and hole concentrations in the conduction and valence bands respectively,
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and $\mu_e$ and $\mu_h$ are the electron and hole mobilities,
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which can be obtained by selecting the conduction or valences states $n$ in Eq. \eqref{eq:transport_lc}.
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For electrons, we have
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\begin{equation}
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\begin{split}
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n_e = \sum_{n\in \text{CB}} \int \dfrac{d\kk}{\Omega_\BZ} f_{n\kk}, \\
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\mu_e = \dfrac{1}{n_e \Omega}\, \mcL_{n\in \text{CB}}^{(0)}
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\end{split}
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\end{equation}
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where $n\in\text{CB}$ denotes states in the conduction bands.
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Similar expressions hold for holes.
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At zero total carrier concentration, the Fermi level $\ef$ is located inside the band gap so that $n_e = n_h$.
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A typical computation of mobilities requires different steps that are summarized
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in the [introduction page for the EPH code](/tutorial/eph_intro).
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Here we only describe the e-ph related part, i.e the blue-box in the workflow presented in the previous page.
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For this purpose, we use [[eph_task]] **-4** to compute **only the imaginary part of the SE at the KS energy**
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and explain other important aspects related to this kind of calculation.
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<!--
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* Ground state computation to obtain the DEN and the WFK file
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* DFPT calculation for a set of $\qq$-points in the IBZ associated to a homogeneous mesh.
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Each DFPT run reads the WFK file produced in the previous step and produces **partial** DDB and POT1 files.
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* Merging of the partial DDB and POT1 files with *mrgddb* and *mrgdv*, respectively
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* Computation of the GS wavefunctions on a $\kk$-mesh that is much denser than the one
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used for the DFPT part as e-ph properties converge much slower than phonons.
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* Interpolation of the e-ph scattering potentials in $\qq$-space and computation of the lifetimes
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for the relevant $\kk$-points contributing to the mobility
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* Computation of transport properties
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These steps can be summarized by the following graph:
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{: style="height:500px;width:400px"}
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-->
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All the results of the calculation are saved in netcdf format in the **SIGPEPH.nc** file,
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while the main output file is used to output selected quantities, mainly for testing purposes.
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Post-processing and visualisation tools are provided by AbiPy.
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See e.g. the README of [AbiPy](https://github.com/abinit/abipy)
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and the [AbiPy tutorials](https://github.com/abinit/abitutorials).
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[TUTORIAL_README]
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## Ground state and phonons of fcc AlAs
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*Before beginning, you might consider creating a different subdirectory to work in.
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Why not create Work_eph4mob ?*
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The file *teph4mob_1.abi* is the input file for the first step
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(GS + DFPT perturbations for all the $\qq$-points in the IBZ).
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Copy it to the working directory with:
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```sh
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cd $ABI_TUTORESPFN/Input
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mkdir Work_eph4mob
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cd Work_eph4mob
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cp ../teph4mob_1.abi .
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```
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{% dialog tests/tutorespfn/Input/teph4mob_1.abi %}
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This step might be quite time-consuming so you may want to immediately start the job in background with:
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```sh
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abinit teph4mob_1.abi > teph4mob_1.log 2> err &
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```
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The calculation is done for AlAs, the same crystalline material as in the first two DFPT tutorials.
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For further details about this first step, please refer to the first and second tutorials on DFPT.
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!!! important
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Since AlAs is a **polar semiconductor**, we need to compute with DFPT the Born effective charges $\bm{Z}^*$
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as well and the static dielectric tensor $\bm{\ee}^\infty$.
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These quantities are then used to treat the long-range (LR) part of the dynamical matrix in
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the Fourier interpolation of the phonon frequencies as well as in
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the Fourier interpolation of the DFPT potentials, as discussed in the EPH introduction.
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<!--
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Only the (partial) DDB and POT files produced at the end of the DFPT run
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are needed to perform e-ph calculation.
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The files containing the first order wavefunctions (*1WF*) due to an atomic perturbation are not needed.
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-->
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## Merging the derivative databases and potentials
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Once the DFPT calculation is completed, use the *mrgddb* tool to merge the eight **partial DDB files**
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corresponding to datasets 3-10 of *teph4mob_1*.
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These partial DDB files contain the dynamical matrices for the
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8 $\qq$-points in the IBZ, as well as the dielectric tensor and the Born effective charges.
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Name the new DDB file *teph4mob_2_DDB*.
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File *\$ABI_TUTORESPFN/Input/teph4mob_2.abi* is an example of input file for *mrgddb*.
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{% dialog tests/tutorespfn/Input/teph4mob_2.abi %}
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Copy the file in the *Work_eph4mob* directory, and run *mrgddb* using:
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```sh
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mrgddb < teph4mob_2.abi
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```
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!!! tip
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Alternatively, one can specify the name of the output DDB and the list of input DDB files
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to be merged directly via the command line.
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This approach is quite handy especially if used in conjuction with shell globbing and the "star" syntax:
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```sh
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mrgddb teph4mob_2_DDB teph4mob_1o_DS*_DDB
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```
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Use **mrgddb --help** to access the documentation.
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Now use the *mrgdv* tool to merge the 29 DFPT POT files corresponding to datasets 3-10 of *teph4mob_1*.
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Name the new file *teph4mob_3_DVDB*.
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File *\$ABI_TUTORESPFN/Input/teph4mob_3.abi* is an example of input file for *mrgdv*.
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{% dialog tests/tutorespfn/Input/teph4mob_3.abi %}
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You can copy it in the *Work_eph4mob* directory, and then merge the files with:
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```sh
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mrgdv < teph4mob_3.abi
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```
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!!! tip
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Alternatively, one can use the command line.
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```sh
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mrgdv merge teph4mob_3_DVDB teph4mob_1o_DS*_POT*
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```
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Use **mrgdv --help** to access the documentation.
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We now have all the phonon-related files needed to compute the mobility.
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The DDB will be used to Fourier interpolate the phonon frequencies on an **arbitrarily** dense $\qq$-mesh while
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the DVDB will be used to Fourier interpolate the DFPT scattering potentials [[cite:Brunin2020b]].
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The only ingredient that is still missing is the WFK file with the GS wavefunctions on the dense $\kk$-mesh.
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!!! warning
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In real computations, you should always compute the electronic band structure along a $\kk$-path
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to have a qualitative understanding of the band dispersion, the position of the band edges,
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and the value of the band gap(s).
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Note also that there are several parts of the EPH code in which it is assumed that
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**no vibrational instability** is present so you should **always look
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at the phonon spectrum computed by the code**.
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Do not expect to obtain meaningful results if purely imaginary phonon frequencies
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(a.k.a **negative frequencies**) are present.
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## Calculation of the dense WFK file
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Converging transport properties requires careful convergence tests both for $\kk$-points and $\qq$-points.
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A dense $\qq$-mesh is needed to obtain **high-quality lifetimes**, whereas a dense $\kk$-sampling
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is needed to have a **good sampling of the electron (hole) pockets**.
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All these studies are explained later and left as an additional excercise.
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In this tutorial, indeed, we need to find some compromise between accuracy and computational cost
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hence a single $\kk$-mesh is used in all our examples.
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The computation of the dense WFK file is similar to a NSCF band structure computation.
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The main difference is that we need wavefunctions on a $\kk$-mesh instead of a $\kk$-path
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because these wavevectors are needed to evaluate integrals in the BZ.
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The file *\$ABI_TUTORESPFN/Input/teph4mob_4.abi* is an example of such computation.
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{% dialog tests/tutorespfn/Input/teph4mob_4.abi %}
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It consists of two parts:
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1. the first one (dataset 1) computes the GS wavefunctions,
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2. the second one (datasets 2-3) computes the dense WFK that will be used to compute mobilities.
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We also compute another (denser) WFK file that will be used with the **double-grid method** explained later.
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As we are mainly interested in electron mobilities (conduction bands)
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we need to include enough empty bands in the NSCF WFK computation ([[nband]] = 8).
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Copy the file in the *Work_eph4mob* directory, and run ABINIT:
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```sh
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abinit teph4mob_4.abi > teph4mob_4.log 2> err &
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```
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!!! important
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In the last part of the tutorial, we explain how to avoid the NSCF computation
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for all the $\kk$-points in the IBZ and produce a partial WFK file containing
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only the wavevectors relevant for transport properties. This trick is
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**crucial to reach dense meshes** but can also be used for coarser meshes if
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you want to accelerate the NSCF part and reduced the size of the WFK file.
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## Calculation of the mobility
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We begin by explaining how to specify the basic input variables required for a standard mobility calculation.
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The file *\$ABI_TUTORESPFN/Input/teph4mob_5.abi* is an example of such computation.
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{% dialog tests/tutorespfn/Input/teph4mob_5.abi %}
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First of all, we need to read the **WFK**, the **DDB** and the **DVDB** files produced previously.
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Since it is not possible to run mobility calculations with a single input file
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and datasets, we use **strings to specify the path of the input files**:
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```sh
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getwfk_filepath "teph4mob_4o_DS2_WFK"
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getddb_filepath "teph4mob_2_DDB"
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getdvdb_filepath "teph4mob_3_DVDB"
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```
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Now copy the input file in the *Work_eph4mob* directory, and run the code with:
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```sh
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abinit teph4mob_5.abi > teph4mob_5.log 2> err &
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```
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The job should take $\sim$15 seconds on a recent CPU.
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We now discuss the meaning of the e-ph variables in more detail:
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* [[optdriver]] 7 activates the EPH driver
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* [[eph_task]] -4 tells ABINIT that we only need the imaginary part
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of the e-ph self-energy at the KS energy.
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* The homogeneous $\kk$-mesh corresponding to the WFK file is specified by [[ngkpt]] 24 24 24.
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The code aborts with an error if [[ngkpt]] is not the same as the one found in the
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input WFK file. At present, multiple shifts ([[nshiftk]] > 1) are not supported.
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* [[ddb_ngqpt]] defines the initial $\qq$-grid used for the DFPT computation (4×4×4 in this example)
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* [[eph_ngqpt_fine]] defines the dense $\qq$-mesh where the scattering potentials are interpolated
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and the e-ph matrix elements are computed.
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!!! warning
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[[ngkpt]] and [[eph_ngqpt_fine]] should be commensurate.
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More specifically, [[ngkpt]] must be a **multiple** of the $\qq$-mesh ([[eph_ngqpt_fine]])
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because the WFK should contain all the $\kk$- and $\kq$-points.
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In most cases, [[ngkpt]] == [[eph_ngqpt_fine]].
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It is however possible to use fewer $\qq$-points.
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Note also that [[ngkpt]] does not necessarily correspond to the $\kk$-mesh
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used for the computation of transport quantities, see the following discussion.
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* We work within the rigid band approximation and introduce a small electron doping: [[eph_doping]] = -1e+15
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that corresponds to 1e+15 electrons per cm$^3$.
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To obtain results that are representative of the intrinsic mobility, we suggest to use a very small value,
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for instance $10^{15}$ to $10^{18}$ electrons per cm$^3$.
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Alternatively, one can specify the doping via [[eph_extrael]] or [[eph_fermie]].
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We also set [[occopt]] to 3 to correctly compute the location of the Fermi level
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using the Fermi-Dirac occupation function as we are dealing with the **physical temperature**
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and not a fictitious broadening for numerical integration purposes.
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* The list of physical temperatures is specified by [[tmesh]].
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!!! note
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The computational cost increases with the number of temperatures although not necessarily in a linear fashion.
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For the initial convergence studies, we suggest to start from a relatively small number
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of temperatures **covering the range of interest**.
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The T-mesh can be densified aftwerwards while keeping the same T-range once converged parameters are found.
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Note that transport properties at low temperatures are more difficult to converge as the
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derivative of the Fermi-Dirac distribution is strongly peaked around the Fermi level hence
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a very dense sampling is needed to converge the BZ integrals.
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In a nutshell, **avoid low temperatures unless you are really interested in this regime**.
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<!--In this tutorial, we will use the same dense $\kk$- and $\qq$-meshes.
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As a rule of thumb, a $\qq$-mesh twice as dense in each direction as the $\kk$-mesh,
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is needed to achieve fast convergence of the integrals [[cite:Brunin2020b]].
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In this case, [[ngkpt]] = [[eph_ngqpt_fine]], but the use of [[sigma_ngkpt]]
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allows to downsample the $\kk$-mesh used for the integration and it should be set to half
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the values of [[ngkpt]].
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An example will be given in this tutorial.
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Possible exceptions are systems with very small effective masses (e.g. GaAs) in which
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a very dense $\kk$-sampling is needed to sample the electron (hole) pocket.
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In this case, using the same sampling for electrons and phonons may be enough to converge.-->
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* By default, the code use the tetrahedron method [[cite:Bloechl1994]]
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to perform the integration in $\qq$-mesh.
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This allows to efficiently filter out the $\qq$-points that do not contribute to the lifetimes
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since these transitions are not compatible with energy and crystalline-momentum conservation.
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The use of the tetrahedron method is **automatically activated** when [[eph_task]] is set to -4.
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It is possible to change this behaviour by using [[eph_intmeth]] albeit not recommended
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as the calculation will become significantly slower. If using the Gaussian method, one must
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converge the self-energy with respect to [[zcut]] (or [[eph_fsmear]] if [[eph_task]] is set to 1),
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By default, [[zcut]] is set to a very large value.
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* The [[sigma_erange]] variable defines the energy window, below the VBM and above the
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CBM, where the lifetimes will be computed.
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Since the mobility integrals involve the derivative of the Fermi-Dirac occupation function centered on the Fermi level,
|
||
it is possible to restrict the computation to those $\kk$-points that contribute to the mobility integral.
|
||
The value of the derivative, indeed, decreases rapidly
|
||
as we go further from the Fermi level hence only the states close to the band edges contribute.
|
||
**This variable should be subject to a convergence study** as explained in the next section.
|
||
<!--
|
||
A value of 0.2 eV represents a good starting point for further analysis.
|
||
Hopefully, in the next version this parameter will be automatically computed by the code.
|
||
-->
|
||
|
||
* [[boxcutmin]] and [[mixprec]] are used to accelerate the computation, see [the introductory tutorial](/tutorial/eph_intro).
|
||
|
||
We now examine the log file in detail.
|
||
After the standard output of the input variables, the code reports the different parameters
|
||
used for the treatment of the long-range part of the DFPT potentials:
|
||
the **Born effective charges, the high-frequency dielectric constant and the dynamical quadrupole tensor**.
|
||
Make sure to have all of them in order to obtain an
|
||
accurate interpolation of the scattering potentials, see discussion in [[cite:Brunin2020]].
|
||
|
||
!!! important
|
||
|
||
At present ( |today| ), the inclusion of the dynamical quadrupoles in the EPH code
|
||
is not available in the public version so you should have the following in the log file:
|
||
|
||
```sh
|
||
Have dielectric tensor: yes
|
||
Have Born effective charges: yes
|
||
Have quadrupoles: no
|
||
Have electric field: no
|
||
```
|
||
|
||
The code then outputs different quantities.
|
||
For instance, ABINIT finds the list of $\kk$-points belonging to the dense mesh that
|
||
are located within the energy window ([[sigma_erange]]):
|
||
|
||
```sh
|
||
Found 3 k-points within erange: 0.000 0.150 (eV)
|
||
```
|
||
Over the initial 413 $\kk$-points in the IBZ, only 3 will be computed!
|
||
|
||
The value of the Fermi level (a.k.a electronic chemical potential $\mu_e(T)$)
|
||
as a function of T is computed and printed afterwards.
|
||
Make sure that $\mu_e$ is far enough from the band
|
||
edges so that the computed mobility can be considered as intrinsic:
|
||
the values of *D_v* and *D_c* should be large compared to ~3 kT
|
||
else you enter the **degenerate** regime or the **highly-degenerate** case (when the Fermi level is inside the bands)
|
||
and additional physical phenomena start to play a role.
|
||
|
||
```sh
|
||
Position of CBM/VBM with respect to the Fermi level:
|
||
Notations: mu_e = Fermi level, D_v = (mu_e - VBM), D_c = (CBM - mu_e)
|
||
|
||
T(K) kT (eV) mu_e (eV) D_v (eV) D_c (eV)
|
||
5.0 0.000 3.521 1.165 0.004
|
||
64.0 0.006 3.475 1.118 0.051
|
||
123.0 0.011 3.428 1.071 0.098
|
||
182.0 0.016 3.379 1.023 0.146
|
||
241.0 0.021 3.328 0.972 0.197
|
||
300.0 0.026 3.274 0.918 0.251
|
||
```
|
||
|
||
ABINIT then reads the WFK file and interpolates the scattering potentials
|
||
to obtain the e-ph matrix elements.
|
||
The use of the tetrahedron method allows to significantly reduce the $\qq$-points:
|
||
|
||
```sh
|
||
qpoints_oracle: calculation of tau_nk will need: 15 q-points in the IBZ. (nqibz_eff / nqibz): 3.6 [%]
|
||
```
|
||
|
||
Once this is done, the code starts looping over the 3 $\kk$-points for which the lifetimes are needed.
|
||
|
||
```sh
|
||
Computing self-energy matrix elements for k-point: [ 4.5833E-01, 4.5833E-01, 0.0000E+00] [ 1 / 3 ]
|
||
```
|
||
|
||
You can find various information for each $\kk$-point, such as:
|
||
|
||
* the total number of $\qq$-points in the irreducible zone defined by
|
||
the little group of $\kk$ (called IBZ(k) in the code),
|
||
|
||
* the number of $\qq$-point in the $\text{IBZ}_k$ contributing to the imaginary part of $\Sigma_\nk$
|
||
(in most cases, this number will be much smaller than the total number of $\qq$-points in the $\text{IBZ}_k$)
|
||
|
||
* the wall-time each step takes.
|
||
|
||
Finally, we have the results for the lifetimes (TAU) in the *teph4mob_5.abo* file:
|
||
|
||
```sh
|
||
K-point: [ 4.5833E-01, 4.5833E-01, 0.0000E+00], T: 5.0 [K], mu_e: 3.521
|
||
B eKS SE2(eKS) TAU(eKS) DeKS
|
||
5 3.573 0.000 36639.9 0.000
|
||
```
|
||
|
||
{% dialog tests/tutorespfn/Refs/teph4mob_5.abo %}
|
||
|
||
|
||
!!! tip
|
||
|
||
As already mentioned in the introduction, all the results are stored in the SIGEPH.nc file.
|
||
With |AbiPy|, one can easily access to all the data of the computation.
|
||
For instance, one can plot the electron linewidths as a function of the KS energy
|
||
using the |abiopen| script:
|
||
|
||
```sh
|
||
abiopen.py teph4mob_5o_SIGEPH.nc --expose
|
||
```
|
||
|
||
|
||
|
||
Well, the figure is not so impressive but this is normal as we are computing only 3 $\kk$-wavevectors
|
||
still there are some points that are worth discussing.
|
||
Note how the linewidths at the CBM are very small at low temperature.
|
||
For the CBM, indeed, only phonon absorption is allowed
|
||
and there are few vibrational states populated at low T.
|
||
The linewidth at the CBM increses with T since high energy phonon states
|
||
starts to be populated and more scattering channels become available.
|
||
|
||
|
||
At the end of the main output file, the diagonal elements of the SERTA/MRTA
|
||
mobility tensor $\sigma_{ij}$ are reported for the three **Cartesian directions** and all temperatures.
|
||
|
||
```sh
|
||
Cartesian component of SERTA mobility tensor: xx
|
||
Temperature [K] e/h density [cm^-3] e/h mobility [cm^2/Vs]
|
||
5.00 0.10E+16 0.00E+00 0.00 0.00
|
||
64.00 0.10E+16 0.00E+00 40.76 0.00
|
||
123.00 0.10E+16 0.00E+00 356.24 0.00
|
||
182.00 0.10E+16 0.00E+00 435.51 0.00
|
||
241.00 0.10E+16 0.00E+00 433.89 0.00
|
||
300.00 0.10E+16 0.25E+05 379.06 0.00
|
||
|
||
Cartesian component of MRTA mobility tensor: xx
|
||
Temperature [K] e/h density [cm^-3] e/h mobility [cm^2/Vs]
|
||
5.00 0.10E+16 0.00E+00 0.00 0.00
|
||
64.00 0.10E+16 0.00E+00 39.67 0.00
|
||
123.00 0.10E+16 0.00E+00 374.94 0.00
|
||
182.00 0.10E+16 0.00E+00 470.73 0.00
|
||
241.00 0.10E+16 0.00E+00 469.55 0.00
|
||
300.00 0.10E+16 0.25E+05 411.23 0.00
|
||
```
|
||
|
||
The temperature is first given then the electron (e) and hole (h) densities followed by
|
||
the corresponding (SERTA/MRTA) mobilities.
|
||
In our input file, we considered only electrons and this explains why the values for holes are zero.
|
||
In this particular case, the difference between SERTA and MRTA is not very large
|
||
but the two approximation may give significantly different results in other systems.
|
||
According to recent works the MRTA results are expected to be closer to ones obtained by iteratively solving the BTE.
|
||
|
||
!!! tip
|
||
|
||
You can also run the transport driver in standalone mode by setting [[eph_task]] 7,
|
||
provided you already have the lifetimes in an external SIGEPH.nc file specified via [[getsigeph_filepath]].
|
||
This task is relatively fast even in serial execution although some parts
|
||
(in particular the computation of DOS-like quantities) can benefit from MPI.
|
||
|
||
Now that we know how to obtain the mobility in a semiconductor for given $\kk$- and $\qq$-meshes,
|
||
we can give additional details about convergence studies and discuss extra tricks
|
||
to significantly decrease the computational cost.
|
||
|
||
## Convergence w.r.t. the energy range
|
||
|
||
The first convergence study consists in determining the energy range around the band edge
|
||
to be used for the computation of $\tau_{n\kk}$.
|
||
We can do that by performing mobility computations with fixed $\kk$- and $\qq$-meshes
|
||
and increasing values of [[sigma_erange]].
|
||
|
||
!!! tip
|
||
|
||
The code can compute both electron and hole mobilities in a single run
|
||
but this is not the recommended procedure as the $\qq$-point filtering is expected to be less efficient.
|
||
Moreover electrons and holes may require a different $\kk$-sampling to convergence depending on the dispersion
|
||
of the bands. As a consequence, **we suggest to compute electrons and holes with different input files**.
|
||
|
||
The file *$\$ABI_TUTORESPFN/Input/teph4mob_6.abi* is an example of such computation.
|
||
|
||
{% dialog tests/tutorespfn/Input/teph4mob_6.abi %}
|
||
|
||
Copy the input file in the *Work_eph4mob* directory, and run ABINIT:
|
||
|
||
```sh
|
||
abinit teph4mob_6.abi > teph4mob_6.log 2> err &
|
||
```
|
||
|
||
This run should take a few minutes.
|
||
|
||
We can now analyze the variation of the mobility with respect to [[sigma_erange]].
|
||
|
||
TODO: Add script to analyze convergence wrt sigma_erange
|
||
|
||
This study shows that an energy window of ~0.15 above the CBM is enough to converge
|
||
so we use this value in the subsequent calculations.
|
||
|
||
!!! warning
|
||
|
||
One should perform this convergence study with a $\kk$-mesh that is already dense enough to
|
||
capture the band dispersion correctly.
|
||
In this case, we are using a 24×24×24 mesh, which is not very dense for such computations.
|
||
This means that, when increasing [[sigma_erange]],
|
||
no additional $\kk$-point is included as the sampling is **too coarse**.
|
||
This is the case for the first three datasets (3 $\kk$-points), and the last two datasets (6 $\kk$-points).
|
||
If a finer mesh was used, the number of $\kk$-points would have increased in a more monotonic way.
|
||
For instance, in Silicon, a 45×45×45 $\kk$-mesh could be used to determine [[sigma_erange]].
|
||
|
||
<!--
|
||
Other quantities (Seebeck etc may have a different convergence behaviour
|
||
In principle, one can run a single calculation with relatively large sigma_erange and then decrease
|
||
the window in the RTA part. This trick however is not yet implemented.
|
||
-->
|
||
|
||
## Convergence w.r.t. the k/q meshes
|
||
|
||
Once the energy window is set, we can start to converge the mobility with respect to the
|
||
dense $\kk$- and $\qq$-meshes.
|
||
The previous computations used 24×24×24 $\kk$- and $\qq$-meshes.
|
||
This is quite far from convergence.
|
||
Just to give you an idea, silicon requires a 45×45×45 $\kk$-mesh and 90×90×90 $\qq$-mesh
|
||
to reach convergence within 5%.
|
||
|
||
!!! tip
|
||
|
||
As a rule of thumb, a $\qq$-mesh twice as dense in each direction as the $\kk$-mesh,
|
||
is needed to obtain accurate values for the linewidth and achieve fast convergence
|
||
of the integrals in $\kk$-space [[cite:Brunin2020b]].
|
||
Possible exceptions are systems with very small effective masses (e.g. GaAs) in which
|
||
a very dense $\kk$-sampling is needed to sample the electron (hole) pockets.
|
||
In this case, using the same sampling for electrons and phonons may be enough to converge.
|
||
|
||
<!--
|
||
In this tutorial, we use [[ngkpt]] == [[eph_ngqpt_fine]], but the use of [[sigma_ngkpt]]
|
||
allows to downsample the $\kk$-mesh used for the integration and it should be set to half
|
||
the values of [[ngkpt]].
|
||
-->
|
||
|
||
To compute the mobility with a $\qq$-mesh twice as dense as the $\kk$-mesh,
|
||
there are two possible approaches:
|
||
|
||
1. Run a computation with:
|
||
|
||
* [[ngkpt]] 90 90 90 ([[nshiftk]] 1 and [[shiftk]] 0 0 0)
|
||
* [[eph_ngqpt_fine]] 90 90 90,
|
||
* [[sigma_ngkpt]] 45 45 45.
|
||
|
||
using [[sigma_ngkpt]] to select the $\kk$-points in $\Sigma_\nk$ belonging to the 45×45×45 mesh,
|
||
but now each lifetime is computed with a 90×90×90 $\qq$-mesh.
|
||
|
||
2. Run a computation with:
|
||
|
||
* [[ngkpt]] 90 90 90, ([[nshiftk]] 1 and [[shiftk]] 0 0 0)
|
||
* [[eph_ngqpt_fine]] 90 90 90,
|
||
* [[sigma_ngkpt]] 90 90 90.
|
||
|
||
Following this approach, we compute lifetimes on a 90×90×90 $\kk$- and the integration
|
||
is perfomed on the same $\qq$-mesh.
|
||
You can then run again the transport driver only,
|
||
by setting [[eph_task]] 7 and [[transport_ngkpt]] 45 45 45
|
||
to downsample the $\kk$-mesh used to integrate the mobility in $\kk$-space.
|
||
This second option has the advantage that it delivers **two mobilities** in one-shot
|
||
but it would be overkilling if a 45×45×45/90×90×90 k/q sampling is already enough.
|
||
|
||
You can run again the previous input files by densifying the different meshes.
|
||
For the densest grids, you might need to run with multiple MPI processes.
|
||
You should obtain something like this for $T$ = 300 K:
|
||
|
||
|
||
|
||
Sorry for repeating it again but,
|
||
the inputs of this tutorial have been tuned to make the computations quite fast,
|
||
but the final results are far from convergence.
|
||
In order to get sensible results, one should
|
||
use a denser DFPT $\qq$-mesh (around 9×9×9), and a larger cutoff energy [[ecut]].
|
||
Obviously, these parameters depend on the system under investigation.
|
||
|
||
## Double-grid technique
|
||
|
||
Another possibility to improve the results without increasing the computation time significantly
|
||
is the double-grid (DG) technique [[cite:Brunin2020b]].
|
||
In this method, a coarse sampling is used for the $\kk$- and the $\qq$-mesh for the e-ph matrix elements,
|
||
but a **finer mesh** is used to interpolate the weights for the delta functions associated to phonon absorption/emission.
|
||
This technique allows one to accelerate the convergence while keeping
|
||
the computational cost and the memory requirements at a reasonable level.
|
||
|
||
!!! important
|
||
|
||
The efficiency of the DG approach depends on the strength of the polar Frohlich divergence:
|
||
if the divergence is difficult to integrate numerically,
|
||
the coarse $\qq$-mesh for the e-ph matrix elements will have to be densified.
|
||
|
||
The DG technique requires a second WFK file, containing the KS eigenvalues on the fine mesh.
|
||
You can specify the path to the fine WFK file using [[getwfkfine_filepath]] as in:
|
||
|
||
```sh
|
||
getwfkfine_filepath "teph4mob_4o_DS3_WFK"
|
||
```
|
||
|
||
The file *$\$ABI_TUTORESPFN/Input/teph4mob_7.abi* (first dataset) is an example of such computation.
|
||
|
||
{% dialog tests/tutorespfn/Input/teph4mob_7.abi %}
|
||
|
||
Copy the input file in the *Work_eph4mob* directory, and run ABINIT:
|
||
|
||
```sh
|
||
abinit teph4mob_7.abi > teph4mob_7.log 2> err &
|
||
```
|
||
|
||
In the log file, you will now find information about the double-grid method:
|
||
|
||
```sh
|
||
coarse: 24 24 24
|
||
fine: 48 48 48
|
||
```
|
||
|
||
The SERTA mobility obtained at 300 K is 163.84 cm$^2$/V/s.
|
||
Using a 48×48×48 $\qq$-mesh for the matrix elements as well would give 96.97 (using [[sigma_ngkpt]] 24 24 24).
|
||
The result is indeed improved, since using a 24×24×24 mesh both for electrons and phonons gives 379.06.
|
||
You can also use a finer mesh, but always multiple of the initial coarse mesh
|
||
(in this case, 72×72×72, 96×96×96, etc).
|
||
It is worth noticing that, according to our tests,
|
||
there is very little use to go beyond a mesh three times as dense as the coarse one.
|
||
Using a 72×72×72 fine mesh for the energies gives a mobility of 152.30 cm$^2$/V/s,
|
||
and a 96×96×96 mesh leads to 149.38 cm$^2$/V/s: the improvement is indeed rather limited.
|
||
|
||
!!! important
|
||
|
||
As a rule of thumb, consider to use the DG method for systems in which the tetrahedron filter
|
||
is not able to reduce the number of $\qq$-points in the integrals below 5% for a significant fraction
|
||
of the $\kk$-points in the [[sigma_erange]] energy window.
|
||
This may happen if there are multiple equivalent pockets and thus many intra-valley scattering channels.
|
||
In this case, the computation of $\tau_\nk$ may require several minutes (2-10) per $\kk$-point and calculations
|
||
performed with the same $\kk$- and $\qq$-mesh start to be expensive when the BZ sampling gets denser.
|
||
|
||
## In-place restart
|
||
|
||
All the results of the calculation are stored in a single SIGEPH.nc file
|
||
for all the $\kk$-points (and spins) considered.
|
||
The list of $\kk$-points is initialized at the beginning of the calculation and an internal table
|
||
in the netcdf file stores the status of each $\kk$-point (whether it has been computed or not).
|
||
This means that calculations that are killed by the resource manager due to time limit can reuse
|
||
the SIGEPH file to perform an **automatic in-place restart**.
|
||
Just set [[eph_restart]] to 1 in the input file and rerun the job
|
||
|
||
!!! important
|
||
|
||
There is no harm in setting [[eph_restart]] to 1 from the begining but keep in mind that
|
||
the code will restart the calculation from scratch if all the $\kk$-points in the SIGEPH.nc have
|
||
been computed (a backup copy is kept).
|
||
So we do not recommended the use of this option in MultiDataset mode.
|
||
Again, MultiDataset are evil when it comes to high-performance!
|
||
|
||
## Transport calculation from SIGEPH.nc
|
||
|
||
The routine that computes carrier mobilites is automatically invoked when [[eph_task]] -4 is used
|
||
and a **RTA.nc** file with the final results is produced.
|
||
There are however cases in which one would like to compute mobilities starting from
|
||
a **pre-existent SIGEPH.nc** without performing a full calculation from scratch.
|
||
In this case, use [[eph_task]] 7 and specify the name of the SIGEPH.nc file with [[getsigeph_filepath]].
|
||
The advanced input variable [[transport_ngkpt]] can be used to downsample
|
||
the $\kk$-mesh used in the mobility integrals.
|
||
|
||
## MPI parallelism and memory requirements
|
||
|
||
There are **five different MPI levels** that can be used to distribute the workload
|
||
and the most memory-demanding data structures.
|
||
By default, the code tries to reach some compromise between memory requirements and time to solution
|
||
by activating the parallelism over perturbations and then the $\qq$-point parallelism
|
||
if no input is provided by the user.
|
||
You can however specify manually the MPI distribution across the five different levels
|
||
by using [[eph_np_pqbks]] (a list of 5 integers).
|
||
The product of these five numbers **must be equal to the total number of MPI processes**.
|
||
|
||
The first entry defines the number of processes for the parallelization over perturbations.
|
||
The allowed value range between 1 and 3 × [[natom]], and should be a divisor
|
||
of 3 × [[natom]] to distribute the work equally.
|
||
The higher this number, the lower the memory requirements at the price of increased MPI communication.
|
||
The second entry determines the parallelization over the $\qq$-points in the IBZ.
|
||
This parallelization level allows one to decrease both the computational time as well as memory although
|
||
it's not always possible to distribute the load equally among the processes.
|
||
The parallelization over bands is usually not relevant for mobility computations
|
||
as only a few states close to the VBM or CBM are usually included.
|
||
The MPI parallelism over $\kk$-points and spins is very efficient
|
||
but it requires HDF5 with MPI-IO support and, besides, memory won't scale.
|
||
Use these additional two levels if the memory requirements are under control
|
||
and you want to boost the calculation.
|
||
|
||
## How to reduce the memory requirements
|
||
|
||
As mentioned above, the memory will scale with the number of MPI processors used for the $\qq$-point and
|
||
the perturbation communicators.
|
||
However, there might be tricky systems in which you start to experience memory shortage that
|
||
prevents you from running with many MPI processes.
|
||
This problem should show up for very dense $\kk$/$\qq$ meshes.
|
||
As a rule of thumb, mobility calculations with meshes denser than e.g 200x200x200 start to be very memory demanding
|
||
and the execution will slow down because several algorithms and internal tables for the BZ sampling
|
||
and the tetrahedron method start to dominate.
|
||
The double grid technique helps mitigate this bottleneck.
|
||
In some cases, you may try to reduce slightly the value of [[sigma_erange]] to reduce the memory requirements.
|
||
|
||
Note also that the EPH code allocates a relatively small buffer to store the Bloch states involved
|
||
in transport calculations but unfortunately the $\kk$-points are not easy to distribute with MPI.
|
||
Moreover the size of this array depends on the electronic dispersion:
|
||
systems with several relatively flat bands around the band edges require more memory.
|
||
To reduce the memory for the wavefunctions, the code uses internal buffers in single precision.
|
||
This option is enabled at configure time by using `enable_gw_dpc="no"` (this is the default behaviour).
|
||
|
||
If these tricks do not solve your problem, consider using OpenMP threads.
|
||
The code is not highly-optimized for OpenMP but a couple of threads may be useful to avoid replicating memory at the MPI level.
|
||
As a rule of thumb, 2-4 OpenMP threads should be OK provided you link with threaded FFT and BLAS libraries.
|
||
To compile ABINIT with OpenMP support and link with a threaded library see the
|
||
corresponding section in the [ABINIT_build tutorial](/tutorial/abinit_build).
|
||
|
||
!!! warning
|
||
|
||
Last but not least, **do not use datasets**: split the calculation into different input files
|
||
and optimize the number of MPI processes according to the dimension of the problem.
|
||
You have been warned!
|
||
|
||
<!-- part of the discussion can be moved to the eph_intro as SKW will be used also in phgamma -->
|
||
|
||
## How to compute the WFK only for k-points close to the band edges
|
||
|
||
As we have already seen in the previous sections, a relatively small number of $\kk$-points
|
||
close to the band edges is usually sufficient to converge mobilities.
|
||
Yet, in the NSCF run, we computed a WFK file for all the $\kk$-points in the IBZ
|
||
hence we spent a lot of resources to compute and store states that
|
||
are **not needed for phonon-limited mobilities**.
|
||
|
||
In principle, it is possible to restrict the NSCF calculation to the relevant $\kk$-points
|
||
provided we have a cheap and good-enough method to predict whether the wavevector
|
||
is inside the energy window **without solving the KS eigevalue problem exactly**.
|
||
For example, one can use the star-function interpolation by Shankland-Koelling-Wood (SKW)
|
||
[[cite:Shankland1971]], [[cite:Koelling1986]], [[cite:Madsen2006]], [[cite:Madsen2018]]
|
||
which requires as input a set of eigenvalues in the IBZ and a single parameter defining the basis set for the interpolation.
|
||
|
||
There are several technical problems that should be addressed at the level of the internal implementation but
|
||
the idea is relatively simple and goes as follows:
|
||
|
||
1. Compute the KS eigenvalues on a relatively coarse $\kk$-mesh in the IBZ
|
||
2. Use this *coarse* WFK file to interpolate the eigenvalues on a **much denser** $\kk$-mesh specified by the user.
|
||
3. Find the wavevectors of the dense mesh inside an energy window specified by the user and
|
||
store the list of $\kk$-points in a external file (**KERANGE.nc**).
|
||
4. Use this external file to run a NSCF calculation only for these $\kk$-points.
|
||
At the end of the NSCF job, ABINIT will produce a **customized** WFK file on the dense mesh that
|
||
can be used to run calculations for phonon-limited mobilities
|
||
|
||
An example will help clarify.
|
||
|
||
Suppose we have computed a WFK file with a NSCF run using a 16x16x16 $\kk$-mesh (let's call it *161616_WFK*)
|
||
and we want to compute mobilites with the much denser 64x64x64 $\kk$-mesh.
|
||
In this case, use
|
||
|
||
- [[optdriver]] = 8
|
||
- [[wfk_task]] = "wfk_kpts_erange"
|
||
|
||
to read the WFK file specified by [[getwfk_filepath]], find the wavevectors belonging to the [[sigma_ngkpt]] $\kk$-mesh
|
||
inside the energy window defined by [[sigma_erange]] and produce a *KERANGE.nc* file.
|
||
The parameters defining the SKW interpolation are specified by [[einterp]].
|
||
|
||
A typical input file for this step looks like:
|
||
|
||
```sh
|
||
optdriver 8
|
||
wfk_task "wfk_kpts_erange"
|
||
getwfk_filepath "161616_WFK"
|
||
|
||
# Define fine k-mesh for the interpolation
|
||
sigma_ngkpt 64 64 64
|
||
sigma_nshiftk 1
|
||
sigma_shiftk 0 0 0
|
||
|
||
sigma_erange 0.0 0.2 eV # Select kpts in fine mesh within this energy window.
|
||
einterp 1 5 0 0 # Parameters for star-function interpolation
|
||
```
|
||
|
||
This input produces a *KERANGE.nc* file (let's call it *out_KERANGE*) that can be used
|
||
via the [[getkerange_filepath]] variable as a starting point to perfom a NSCF run with the following variables:
|
||
|
||
```sh
|
||
getkerange_filepath "out_KERANGE.nc"
|
||
getden_filepath "161616_DEN" # Read DEN file to initialize NSCF run
|
||
getwfk_filepath "161616_WFK" # Init GS wavefunctions from this file (optional)
|
||
iscf -2
|
||
tolwfr 1e-18
|
||
kptopt 0 # Important!
|
||
|
||
# These variables must be consistent with the values of
|
||
# sigma_ngkpt, sigma_shiftk used in the previous step
|
||
ngkpt 64 64 64
|
||
nshiftk 1
|
||
shiftk 0.0 0.0 0.0
|
||
```
|
||
|
||
This part will produce a **customized Fortran WFK file** with [[ngkpt]] = 64 64 64 in which
|
||
only the states listed in the **KERANGE.nc** netcdf file have been computed.
|
||
This WKF file can then be used in the EPH code to compute mobilites.
|
||
For further examples see [[test:v9_57]], and [[test:v9_61]].
|
||
|
||
Note that the two tests **cannot be executed in multidataset mode with a single input file**.
|
||
Also, keep in mind that the quality of the interpolation depends on the initial coarse $\kk$-mesh
|
||
so we recommended to look at the interpolant, see discussion at the end of [the introductory tutorial](/tutorial/eph_intro).
|
||
|
||
!!! important
|
||
|
||
It is also a good idea to use an energy window that is **larger** than the one that will be employed
|
||
to compute the mobility in the EPH code.
|
||
As a rule of thumb, increase [[sigma_erange]] by 0.15 eV when computing the KERANGE.nc file.
|
||
This should be however tested in each case.
|