Part 3: Electron mobility of SrTiO\(_3\)
The conductivity strongly depends on the carrier density \(n\):
\[\sigma = qn \mu\]
and thus the band-gap \(E_g\), since the intrinsic carrier density depends exponentially on \(E_g\) in zero order approximation:
\[{n} \propto \exp\left(-\frac{ E_g}{2{\rm k_B} T}\right) \ .\]
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 3solutions-40atom-SrTiO3\calculation\samples_500K\ and are also shown below, with the numbering referring to the sample number from the aiMD trajectory:
geometry.in files from 10 samples at \(T=500\) K
#=======================================================
# FHI-aims file: geometry.in.00218
# Created using the Atomic Simulation Environment (ASE)
# Tue Aug 22 10:07:44 2023
# Additional information:
# Sample no.: 218
#=======================================================
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#=======================================================
# FHI-aims file: geometry.in.00236
# Created using the Atomic Simulation Environment (ASE)
# Tue Aug 22 10:07:44 2023
# Additional information:
# Sample no.: 236
#=======================================================
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#=======================================================
# FHI-aims file: geometry.in.00254
# Created using the Atomic Simulation Environment (ASE)
# Tue Aug 22 10:07:44 2023
# Additional information:
# Sample no.: 254
#=======================================================
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#=======================================================
# FHI-aims file: geometry.in.00272
# Created using the Atomic Simulation Environment (ASE)
# Tue Aug 22 10:07:44 2023
# Additional information:
# Sample no.: 272
#=======================================================
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#=======================================================
# FHI-aims file: geometry.in.00290
# Created using the Atomic Simulation Environment (ASE)
# Tue Aug 22 10:07:44 2023
# Additional information:
# Sample no.: 290
#=======================================================
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#=======================================================
# FHI-aims file: geometry.in.00308
# Created using the Atomic Simulation Environment (ASE)
# Tue Aug 22 10:07:44 2023
# Additional information:
# Sample no.: 308
#=======================================================
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atom_frac 0.7501573043101065 0.7575306734810984 0.2654494135234196 Ti
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#=======================================================
# FHI-aims file: geometry.in.00326
# Created using the Atomic Simulation Environment (ASE)
# Tue Aug 22 10:07:44 2023
# Additional information:
# Sample no.: 326
#=======================================================
lattice_vector 7.8402046709000004 0.0000000000000000 0.0000000000000000
lattice_vector 0.0000000000000000 7.8402046709000004 0.0000000000000000
lattice_vector 0.0000000000000000 0.0000000000000000 7.8402046709000004
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#=======================================================
# FHI-aims file: geometry.in.00344
# Created using the Atomic Simulation Environment (ASE)
# Tue Aug 22 10:07:44 2023
# Additional information:
# Sample no.: 344
#=======================================================
lattice_vector 7.8402046709000004 0.0000000000000000 0.0000000000000000
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lattice_vector 0.0000000000000000 0.0000000000000000 7.8402046709000004
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#=======================================================
# FHI-aims file: geometry.in.00362
# Created using the Atomic Simulation Environment (ASE)
# Tue Aug 22 10:07:44 2023
# Additional information:
# Sample no.: 362
#=======================================================
lattice_vector 7.8402046709000004 0.0000000000000000 0.0000000000000000
lattice_vector 0.0000000000000000 7.8402046709000004 0.0000000000000000
lattice_vector 0.0000000000000000 0.0000000000000000 7.8402046709000004
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#=======================================================
# FHI-aims file: geometry.in.00380
# Created using the Atomic Simulation Environment (ASE)
# Tue Aug 22 10:07:44 2023
# Additional information:
# Sample no.: 380
#=======================================================
lattice_vector 7.8402046709000004 0.0000000000000000 0.0000000000000000
lattice_vector 0.0000000000000000 7.8402046709000004 0.0000000000000000
lattice_vector 0.0000000000000000 0.0000000000000000 7.8402046709000004
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atom_frac 0.0058132320192535 0.5047514479723989 1.0205281577319696 Sr
atom_frac 0.5039820447961247 0.5079087150529786 1.0073888692201565 Sr
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atom_frac 0.4928466838575897 -0.0299979258835394 0.4933160584811704 Sr
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atom_frac 0.7496165491451227 0.2593756534885488 0.2668742897619972 Ti
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atom_frac 0.2488304751735738 0.2475325624503015 0.7338181380382528 Ti
atom_frac 0.7360959039241196 0.2607472209900902 0.7676955053695638 Ti
atom_frac 0.2445919927047908 0.7409512487836768 0.7486434396863546 Ti
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atom_frac 0.4990865543017160 0.7643625840599713 0.2205249327702382 O
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atom_frac 0.4918667795303716 0.2483932922676311 0.7590724300984077 O
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atom_frac 0.4910300507257246 0.7552676051385810 0.7403738671052555 O
Please create directories named config_00218, config_00236, ... to config_00380 to match the trajectory numbers, and copy the respective geometry.in files from above into each of the directories.
Control tags
For the control.in file, we can use the following keywords:
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.
Attach light species defaults for the Sr, Ti and O atoms in this control.in file, and place a copy in each of the config_xxxxx directories.
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.
Post-processing
Collect results
In the solutions folder of the repository, we prepared a bash script called pp_all.sh to collect all output files. This is also displayed below. Note that this script will only work if you named your directories config_00218, config_00236, ..., config_00380. If you called them something different, you will have to modify the below script accordingly.
########## JKQ
#
# Post-process KG output files
# Here I trim Col1: frequency, Col5: hole conductivity, Col8: electron conductivity
# from each output file of each sample into one kg_result_h/e.txt files.
#
##############################################################################################
startp=00218
endp=00380
startw=10
endw=100
# Filename prefix of your structure:
kgresult='KG_full_L11_out_Gaussian_0.'
touch kg_result_e.txt kg_result_h.txt
for num in `seq -f "%05g" ${startp} 18 ${endp}` ; do
echo ${num}
cd "config_${num}"
if test ! -e KG_full_L11_out_Gaussian_0.0070_a_a.out; then
echo "skip ${num}"
cd ..
continue
fi
if [ ${num} = ${startp} ]; then
for iwidth in `seq -f "%04g" ${startw} 10 ${endw}` ; do
echo ${iwidth}
awk '{print $1,$5}' ./${kgresult}${iwidth}'_a_a.out' >> ../kg_result_tmp_h.txt
awk '{print $1,$8}' ./${kgresult}${iwidth}'_a_a.out' >> ../kg_result_tmp_e.txt
done
else
for iwidth in `seq -f "%04g" ${startw} 10 ${endw}` ; do
awk '{print $5}' ./${kgresult}${iwidth}'_a_a.out' >> ../kg_result_tmp_h.txt
awk '{print $8}' ./${kgresult}${iwidth}'_a_a.out' >> ../kg_result_tmp_e.txt
done
fi
cd ..
paste kg_result_h.txt kg_result_tmp_h.txt >> kg_result_new_h.txt
paste kg_result_e.txt kg_result_tmp_e.txt >> kg_result_new_e.txt
#
mv kg_result_new_h.txt kg_result_h.txt
mv kg_result_new_e.txt kg_result_e.txt
rm kg_result_tmp_h.txt kg_result_tmp_e.txt
done
echo "Start add charge carrier density in kg_result_h.txt"
# remove the first 3 lines in kg_result_h.txt
sed -i '1,3d' kg_result_h.txt
# copy the first 3 lines from kg_result_e.txt
head -n 3 kg_result_e.txt > temp_charge.txt
# combine them to kg_result_h.txt
cat temp_charge.txt kg_result_h.txt > temp_kg_result_h.txt
mv temp_kg_result_h.txt kg_result_h.txt
rm temp_charge.txt
# trim unnecessary lines
#sed -i '4,5d' kg_result_h.txt
#sed -i '4,5d' kg_result_e.txt
sed -i -e '/# Full/d' -e '/# cm/d' kg_result_e.txt
sed -i -e '/# Intra/d' -e '/# (Ohm/d' kg_result_h.txt
In the folder which has the config_xxxxx folders as sub-directories, 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:
\[\langle \mu(\omega) \rangle_T = \langle \frac{\sigma(\omega)}{ n_e q} \rangle_T\]
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 solutions folder of the repository, named part3-post-process.ipynb, reading kg_result_e.txt as input.
A standalone Python script, capable of plotting the averaged electron mobility, plot_SrTiO3_mobility.py is shown below
import numpy as np
import matplotlib.pyplot as plt
import math
import matplotlib
def get_carrier(nsample, filename):
ftxt = open(filename,"r")
carrier = []
for iline in range(0,3):
line = ftxt.readline()
if iline == 0:
header1 = line.strip('#')
elif iline == 1:
for isample in range(1,nsample+1):
carrier.append(float(line.split()[isample]))
elif iline == 2:
for isample in range(1,nsample+1):
carrier.append(float(line.split()[isample]))
carrier = np.asarray(carrier)
carrier = carrier.reshape((2,nsample))
return carrier
def pp_mobility(filename, n_width, n_freq, carrier):
'''
Return sample averaged data.
'''
charge = 1.60217663e-19
eachk_raw = np.loadtxt(filename)
eachk = eachk_raw[:,1:]
freq = eachk_raw[:,0]
eachk = eachk.reshape((n_width,n_freq,np.shape(eachk)[1]))
for isample in range(np.shape(eachk)[2]):
eachk[:,:,isample] = eachk[:,:,isample] / charge / carrier[isample]
avrg_mobility = np.mean(eachk,axis=2)
std_mobility = np.std(eachk,axis=2)
return avrg_mobility, std_mobility,freq
nn_width = 10
nn_samples = 10 # JKQ: Aware that we skip the first point (omega = 0) !!!
nn_freq = 299
charge = 1.60217663e-19
data_file = 'kg_result_e.txt'
carrier_density = get_carrier(nn_samples, data_file) # carrier_density[0,:] for electron, carrier_density[1,:] for hole.
avrg_mobility, std_mobility, freq = pp_mobility(data_file, nn_width, nn_freq, carrier_density[0,:])
j = 6 # plot the spectrum with the 5th kubo_broadening
fig, ax = plt.subplots()
ax.plot(freq[5:299],avrg_mobility[j,5:299],'-',color='m',linewidth=2.5,label='averaged over 10 samples')
plt.xlabel(r'$\hbar\omega$ (eV)',fontsize=12)
plt.ylabel(r'Mobility (${cm^2/Vs}$)',fontsize=12)
plt.legend()
plt.savefig('STO_40atom_10samples.png',dpi=250)
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, 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:
\[ \mu(\omega \to 0) \approx \frac{\mu_0}{(\omega\tau)^2 + 1} \ ,\]
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 provided in the Jupyter Notebook, part3-post-process.ipynb by fitting with the electron mobility data between 0.052 eV and 0.13 eV, as well as a small modification to plot_SrTiO3_mobility.py shown below in plot_SrTiO3_mobility_drude.py
import numpy as np
import matplotlib.pyplot as plt
import math
import matplotlib
def get_carrier(nsample, filename):
ftxt = open(filename,"r")
carrier = []
for iline in range(0,3):
line = ftxt.readline()
if iline == 0:
header1 = line.strip('#')
elif iline == 1:
for isample in range(1,nsample+1):
carrier.append(float(line.split()[isample]))
elif iline == 2:
for isample in range(1,nsample+1):
carrier.append(float(line.split()[isample]))
carrier = np.asarray(carrier)
carrier = carrier.reshape((2,nsample))
return carrier
def pp_mobility(filename, n_width, n_freq, carrier):
'''
Return sample averaged data.
'''
charge = 1.60217663e-19
eachk_raw = np.loadtxt(filename)
eachk = eachk_raw[:,1:]
freq = eachk_raw[:,0]
eachk = eachk.reshape((n_width,n_freq,np.shape(eachk)[1]))
for isample in range(np.shape(eachk)[2]):
eachk[:,:,isample] = eachk[:,:,isample] / charge / carrier[isample]
avrg_mobility = np.mean(eachk,axis=2)
std_mobility = np.std(eachk,axis=2)
return avrg_mobility, std_mobility,freq
def drude_func(x, a, b):
y = a / (1 + b * (x**2))
return y
def drude_fit(x, y):
from scipy.optimize import curve_fit
popt, pcov = curve_fit(drude_func, x, y)
print('drude fit coefficient: ', popt)
return popt
nn_width = 10
nn_samples = 10 # JKQ: Aware that we skip the first point (omega = 0) !!!
nn_freq = 299
charge = 1.60217663e-19
data_file = 'kg_result_e.txt'
carrier_density = get_carrier(nn_samples, data_file) # carrier_density[0,:] for electron, carrier_density[1,:] for hole.
avrg_mobility, std_mobility, freq = pp_mobility(data_file, nn_width, nn_freq, carrier_density[0,:])
j = 6 # plot the spectrum with the 5th kubo_broadening
fig, ax = plt.subplots()
### Find the peak position as the first point to do drude fitting
### I.e. Drude fitting between the first peak position and freq_end.
freq_start = 5 # ! start point you want
freq_end = 129 # ! end point you want
freq_start = np.argmax(avrg_mobility[j,freq_start:]) + freq_start
freq_in = freq[freq_start: freq_end]
data_in = avrg_mobility[j,freq_start:freq_end]
popt = drude_fit(freq_in, data_in)
# plot figure
fig, ax = plt.subplots()
ax.plot(freq[5:299],avrg_mobility[j,5:299],'-',color='m',linewidth=2.5,label='averaged over 10 samples')
ax.plot(freq[0:299], drude_func(freq[0:299], *popt), color='tomato',label=f'Drude fit')
plt.xlabel(r'$\hbar\omega$ (eV)',fontsize=12)
plt.ylabel(r'Mobility (${cm^2/Vs}$)',fontsize=12)
plt.legend()
plt.savefig('STO_40atom_10samples_drude.png',dpi=250)
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.