Part 3: Electron mobility of SrTiO\(_3\)
The conductivity strongly depends on the carrier density \(n\):
and thus the band-gap \(E_g\), since the intrinsic carrier density depends exponentially on \(E_g\) in zero order approximation:
Accordingly, semi-local xc-functionals can overestimate conductivities by orders of magnitude due the notorious band-gap problem. Conversely, mobilities \(\mu\) are an intrinsic property that is unaffected by the carrier density in the low-doping regime and are thus reasonably predicted already with semi-local xc-functionals. In experiments, the above definition also implies that the measured conductivity is sensitively depends on doping concentration. For these reasons, it is common both in theoretical and experimental studies of carrier transport to report the mobility in stead of conductivity.
In the last example, we learnt the ab inito KG workflow by a very simple example: 16-atom metallic Al. In this example, we will calculate a more meaningful and complicated case:
- the DC electron mobility \(\mu (\omega \to 0)\) of a 40-atom SrTiO\(_3\) supercell at 500K with ensemble average.
Prepare input files
Geometries
In a first step, it is necessary to obtain representative geometric configurations, so-called samples that cover the phase-space at the thermodynamic conditions of interest. These samples can be generated by stochastic methods at deseired temperature \(T\) such as harmonic sampling, ab initio molecular dynamics (aiMD), etc. In FHI-aims, the harmonic sampling can be performed via FHI-vibes; aiMD simulation can be performed via FHI-vibes or i-PI. Please refer to those tutorials if you are interested.
In this example, we picked 10 samples from a \(NVT\) aiMD trajectory at \(T=500\) K. These 10 geometries are under folder calculation\samples_500K\.
Control tags
xc pbe
k_grid 4 4 4
dos_kgrid_factors 5 5 5
# type[letter] temperature[eV]
occupation_type fermi 0.04309
# kubo_broadening[eV] , Fermi-Temperature[eV] , E_min[eV] , E_max[eV] , w_min[eV] , w_max[eV] , n_w_points[1] , spatial directions[letter]
compute_kubo_greenwood 0.020 0.04309 -8.8 -3.8 0.0 0.3 300 a a
kg_width 0.001 0.010 10
kg_charge -4.82e-4
compute_kubo_greenwood and kg_width are explained in the last example.
- Here we have a new tag
kg_chargeto control the carrier density in KG calculation.kg_charge -4.82e-4means adding negative charge 4.82e-4 in the simulation cell. In semiconductors the Fermi level lies in the band gap, this tag is equivalent to moving the Fermi level closer to the conduction bands.
When calculating mobility with ensemble average, constraining the charge-carrier density to a fixed value in KG calculation can be numerically beneficial, please refer to this paper for more details.
Run the calculation
The calculations in this example should be submitted to a HPC cluster, since they are hard to calculate on a personal laptop. If you do not have access to a HPC cluster or don't want to wait for the calculations, we also provide all output files in the repository, you can use these results for the following analysis.
Use provided script to submit all 10 jobs onto HPC:
Warning
Please modify the job submit script job_que.sh according to your HPC settings, such as --ntasks-per-node, --nodes, --time and path to your FHI-aims executable, etc.
Post-processing
Collect results
In the repository, we prepared a bash script called pp_all.sh to collect all output files.
Run
kg_result_e.txt and kg_result_h.txt which contains carrier densities and electron/hole conductivities for all the 10 samples and 10 broadening parameters.
In this example, all data in file kg_result_h.txt are zero since we shift the Fermi level closer to the conduction bands to have high electron density and nearly zero hole density. The resulting kg_result_e.txt looks like:
#
# 0.495456674219E+18 0.527508974424E+18 0.510711992269E+18 0.491744112468E+18 0.469357030874E+18 0.468678429345E+18 0.561842279160E+18 0.551072862638E+18 0.545915028059E+18 0.500805097355E+18
# 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00
1.000000E-003 5.134103E-005 9.042815E-007 7.038302E-007 2.828424E-007 1.635544E-002 1.265616E-010 8.195798E-010 6.955893E-012 1.816593E-006 5.234981E-006
2.000000E-003 1.956573E-004 3.394807E-006 2.599010E-005 1.573936E-006 3.329689E-002 3.185261E-009 2.359895E-008 2.390767E-011 2.633976E-005 1.109301E-005
3.000000E-003 3.657397E-004 6.270061E-006 4.855487E-004 1.302612E-005 3.435108E-002 4.328498E-008 1.288286E-006 8.905788E-011 2.298024E-004 2.109896E-005
4.000000E-003 2.829500E-004 5.135368E-006 3.778452E-003 1.127759E-004 2.142022E-002 2.660254E-007 5.418163E-005 1.039213E-009 9.408184E-004 4.153642E-005
5.000000E-003 8.600912E-005 7.409761E-006 1.155391E-002 7.425872E-004 2.404659E-002 7.636100E-007 1.091952E-003 2.652858E-008 1.661272E-003 1.597793E-004
6.000000E-003 1.342031E-005 1.586880E-004 1.356736E-002 4.042183E-003 3.524718E-002 1.157915E-006 9.568887E-003 8.319912E-007 1.652968E-003 9.834619E-004
7.000000E-003 4.295673E-005 2.595016E-003 6.609196E-003 1.582086E-002 7.219506E-002 1.612432E-006 3.577235E-002 2.198875E-005 2.653422E-003 3.958493E-003
8.000000E-003 2.020558E-004 1.770369E-002 6.481737E-003 3.436404E-002 1.448759E-001 3.661515E-006 6.004392E-002 2.555108E-004 4.149066E-003 6.964293E-003
9.000000E-003 3.616280E-004 5.199614E-002 1.887933E-002 3.935279E-002 1.797088E-001 5.770565E-006 6.111007E-002 1.130990E-003 3.134408E-003 5.604679E-003
1.000000E-002 2.443383E-004 7.484042E-002 2.456573E-002 2.648016E-002 1.366210E-001 4.788819E-006 5.147252E-002 1.876220E-003 2.555616E-003 7.176369E-003
1.100000E-002 1.147042E-004 6.919876E-002 1.294101E-002 1.667489E-002 7.947553E-002 4.491101E-006 2.518837E-002 1.173136E-003 9.588940E-003 1.674195E-002
1.200000E-002 5.749219E-004 9.261560E-002 4.289181E-003 1.976343E-002 7.307298E-002 7.616082E-006 8.207821E-003 3.415292E-004 2.251195E-002 2.158761E-002
1.300000E-002 2.605288E-003 1.558592E-001 7.182781E-003 2.597682E-002 8.814640E-002 8.308044E-006 1.882435E-002 1.102072E-003 2.203307E-002 2.038265E-002
1.400000E-002 5.847713E-003 1.586776E-001 2.428621E-002 3.039256E-002 7.269113E-002 1.414011E-005 5.665850E-002 7.615359E-003 9.912925E-003 2.447664E-002
...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......
#
# 0.495456674219E+18 0.527508974424E+18 0.510711992269E+18 0.491744112468E+18 0.469357030874E+18 0.468678429345E+18 0.561842279160E+18 0.551072862638E+18 0.545915028059E+18 0.500805097355E+18
# 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00
1.000000E-003 2.839925E-004 3.120748E-005 2.205344E-003 2.577133E-004 3.668919E-002 1.658840E-007 4.815068E-004 3.025233E-007 4.669193E-004 7.532573E-005
2.000000E-003 2.369076E-004 1.027967E-004 3.281230E-003 6.088961E-004 3.053674E-002 2.603785E-007 1.216554E-003 1.287281E-006 6.324241E-004 1.541055E-004
3.000000E-003 2.078294E-004 3.933829E-004 5.115693E-003 1.585938E-003 3.125592E-002 4.551710E-007 3.237172E-003 5.715104E-006 9.430681E-004 3.755591E-004
4.000000E-003 1.678513E-004 1.371023E-003 7.156328E-003 3.806134E-003 3.703914E-002 7.756457E-007 7.684508E-003 2.229359E-005 1.352353E-003 8.643589E-004
5.000000E-003 1.333877E-004 4.061700E-003 8.823364E-003 7.927515E-003 5.043223E-002 1.266861E-006 1.551985E-002 7.238276E-005 1.837881E-003 1.740579E-003
6.000000E-003 1.277269E-004 1.000765E-002 1.003967E-002 1.395375E-002 7.216442E-002 1.969066E-006 2.625166E-002 1.910587E-004 2.390151E-003 3.015825E-003
7.000000E-003 1.583980E-004 2.046521E-002 1.136125E-002 2.054240E-002 9.763556E-002 2.853456E-006 3.709417E-002 4.058837E-004 3.049367E-003 4.600237E-003
8.000000E-003 2.199785E-004 3.519821E-002 1.308736E-002 2.538125E-002 1.173823E-001 3.793968E-006 4.398522E-002 6.962332E-004 4.038155E-003 6.510796E-003
9.000000E-003 3.342032E-004 5.249142E-002 1.448804E-002 2.692307E-002 1.232059E-001 4.721395E-006 4.431801E-002 9.986474E-004 5.812538E-003 8.994372E-003
1.000000E-002 6.025739E-004 7.102330E-002 1.466678E-002 2.585784E-002 1.150446E-001 5.967967E-006 3.935834E-002 1.347051E-003 8.639929E-003 1.222910E-002
1.100000E-002 1.201244E-003 9.094244E-002 1.427075E-002 2.448432E-002 1.006011E-001 8.724216E-006 3.429164E-002 2.097550E-003 1.189225E-002 1.597981E-002
1.200000E-002 2.257289E-003 1.118690E-001 1.562726E-002 2.462804E-002 8.786331E-002 1.550728E-005 3.532520E-002 3.945885E-003 1.412833E-002 1.986464E-002
1.300000E-002 3.659604E-003 1.310927E-001 2.080138E-002 2.654189E-002 7.955245E-002 2.972294E-005 4.535707E-002 7.417693E-003 1.439551E-002 2.387239E-002
1.400000E-002 4.993497E-003 1.458657E-001 2.985528E-002 2.974053E-002 7.461250E-002 5.265570E-005 6.154363E-002 1.201687E-002 1.325547E-002 2.815969E-002
...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......
This file is divided into blocks by the lines started with #, these commented lines are the electron, hole carrier density of the 10 samples. There are number of broadening width blocks in total, in this example 10. And each block is in shape (n_freqency, n_samples), in this example \((299, 10)\) stored all electron conductivity data of 10 samples at 300 frequency points for each broadening width.
(P.S. we requested 300 frequency points in the input tag but remember that we skipped the first frequency point \(\omega = 0\) to avoid potential numerical issues in the denominator.)
Ensemble average
The electron mobility \(\mu(\omega)\) is calculated by:
where \(\langle \cdots \rangle = \frac{1}{N}\sum_{I}\) is averaging over all samples.
A simple post-processing and visualization script is available as a Jupyter Notebook in the repository, reading kg_result_e.txt as input. The averaged electron mobility \(\mu (\omega)\) should looks like:
Drude fitting
From the averaged electron mobility spectra, we can observe that in the low frequency region, when frequency decrease, the mobility shows a peak and then quickly drops down when \(\omega \to 0\), which is due to the finite-size effect of the simulation cell.
In practical calculations, the 'infinite supercell' limit cannot be reached, thus it is important to employ strategies to extrapolate to the DC limit. Here we use the Drude fitting approach. In the low-frequency limit, the mobility can be fitted by a Drude function:
whereby \(\mu_0\) and \(\tau\) are two parameters to be fitted. In practice, the Drude function should be fitted using frequency data from the first peak and usually extending through a small frequency window beyond it. Please refer to this paper for more details about the finite size effect and Drude fitting.
Warning
The Drude fitting must be performed after ensemble average! The individual spectra of one sample may not looks like a Drude function, but as long as the ensemble average is converged it will show a Drude-like behaviour.
A simple Drude fitting code is also provided in the same Jupyter Notebook, by fitting with the electron mobility data between 0.052 eV and 0.13 eV, the result looks like:
In this case the extrapolated DC mobility is \(\mu_0 \sim 2.97\).
Warning
The above result is not converged. In production calculation, to get reliable mobility of SrTiO\(_3\) at \(500K\) one should carefully converge all parameters including: supercell size, number of samples, k-grid density and broadening width.
Convergence test
In this example, we only use a small 40-atom supercell and 10 samples for simplicity. In production calculation, both the cell size and number of samples are not enough. As for 40-atom SrTiO\(_3\), the converged mobility spectra by averaging over 100 samples looks like:
which is smoother than our example.
Similarly, the supercell convergence behaviour is:
the DC mobility increase with the supercell size, since more indirect transitions are consider by increasing supercell size. It indicates that the 625-atom supercell might be considered as converged in this case. Please refer to this paper for more details about the convergence tests.
Solutions
You find all the solution to all the above exercises by clicking on the button below.