Part 2: Mobility of Al
In the last example, we learnt the basic process of a KG calculation. In this example, we will briefly introduce the ab initio KG workflow by calculating the DC electron mobility \(\mu (\omega \to 0)\) of simple 16-atom metallic Al supercell with ensemble average.
By definition, the conductivity \(\sigma\) and mobility \(\mu\) are connected by the carrier density \(n\):
\[\sigma = qn \mu\]
The advantage of calculating mobility instead of conductivity alone will be discussed in detail in the next example Part 3.
Warning
This is a very simple showcase with no careful parameter convergence, designed to complete within tens of minutes. Therefore, the results should not be compared with any experimental data! For a more realistic example, please refer to Part 3.
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 simply picked 10 samples from harmonic sampling at \(T=1000\) K. For details of harmonic sampling with FHI-aims, please refer to: FHI-vibes tutorial. These 10 geometries are under the folder 2solutions-16atom-Al\calculation\samples_1000K\ in this tutorial repository and also displayed below, for convinience.
geometry.in files from 10 samples at \(T=1000\) K
#=======================================================
# FHI-aims file: geometry.in.supercell.1000K.000
# Created using the Atomic Simulation Environment (ASE)
# Thu Nov 14 15:36:15 2024
# Additional information:
# created from force constants
# T = 1000.0 K
# quantum: False
# deterministic: False
# plus_minus: False
# gauge_eigenvectors: False
# Random seed: 281513462
# Sample number: 1
# Harmonic E_pot: 4.418eV (1068.07K)
#=======================================================
lattice_vector 8.0778593799999996 0.0000000000000000 0.0000000000000000
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lattice_vector 0.0000000000000000 0.0000000000000000 8.0778593799999996
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atom 1.8027982395884421 5.9997291328293940 4.3170615462239770 Al
atom 5.9467648307632661 5.9592439659450065 4.1590108030533148 Al
#=======================================================
# FHI-aims file: geometry.in.supercell.1000K.001
# Created using the Atomic Simulation Environment (ASE)
# Thu Nov 14 15:36:15 2024
# Additional information:
# created from force constants
# T = 1000.0 K
# quantum: False
# deterministic: False
# plus_minus: False
# gauge_eigenvectors: False
# Random seed: 281513462
# Sample number: 2
# Harmonic E_pot: 3.830eV (926.04K)
#=======================================================
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#=======================================================
# FHI-aims file: geometry.in.supercell.1000K.002
# Created using the Atomic Simulation Environment (ASE)
# Thu Nov 14 15:36:15 2024
# Additional information:
# created from force constants
# T = 1000.0 K
# quantum: False
# deterministic: False
# plus_minus: False
# gauge_eigenvectors: False
# Random seed: 281513462
# Sample number: 3
# Harmonic E_pot: 4.514eV (1091.31K)
#=======================================================
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#=======================================================
# FHI-aims file: geometry.in.supercell.1000K.003
# Created using the Atomic Simulation Environment (ASE)
# Thu Nov 14 15:36:15 2024
# Additional information:
# created from force constants
# T = 1000.0 K
# quantum: False
# deterministic: False
# plus_minus: False
# gauge_eigenvectors: False
# Random seed: 281513462
# Sample number: 4
# Harmonic E_pot: 4.550eV (1100.00K)
#=======================================================
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#=======================================================
# FHI-aims file: geometry.in.supercell.1000K.004
# Created using the Atomic Simulation Environment (ASE)
# Thu Nov 14 15:36:15 2024
# Additional information:
# created from force constants
# T = 1000.0 K
# quantum: False
# deterministic: False
# plus_minus: False
# gauge_eigenvectors: False
# Random seed: 281513462
# Sample number: 5
# Harmonic E_pot: 4.697eV (1135.47K)
#=======================================================
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#=======================================================
# FHI-aims file: geometry.in.supercell.1000K.005
# Created using the Atomic Simulation Environment (ASE)
# Thu Nov 14 15:36:15 2024
# Additional information:
# created from force constants
# T = 1000.0 K
# quantum: False
# deterministic: False
# plus_minus: False
# gauge_eigenvectors: False
# Random seed: 281513462
# Sample number: 6
# Harmonic E_pot: 3.501eV (846.33K)
#=======================================================
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#=======================================================
# FHI-aims file: geometry.in.supercell.1000K.006
# Created using the Atomic Simulation Environment (ASE)
# Thu Nov 14 15:36:15 2024
# Additional information:
# created from force constants
# T = 1000.0 K
# quantum: False
# deterministic: False
# plus_minus: False
# gauge_eigenvectors: False
# Random seed: 281513462
# Sample number: 7
# Harmonic E_pot: 3.436eV (830.59K)
#=======================================================
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atom 4.1628832597203251 1.9349079193744221 6.1292440012622151 Al
atom 0.0741633672640634 6.3386029295739652 6.0782264935871009 Al
atom 4.0914668181837373 6.0244378233793796 6.1469078154604668 Al
atom 1.8375630619558443 -0.1811933070529585 2.1719398658922495 Al
atom 5.9554112837729951 -0.1047264502399202 2.1643023993935326 Al
atom 2.3063962537092522 3.9441646166519977 2.1668455298907783 Al
atom 6.2817453891105739 4.1405436152473403 1.7504321457642806 Al
atom 2.1592445274750700 -0.0475972561135777 5.9645959032967717 Al
atom 6.0280685810044732 -0.0311747568143748 6.0271337679932468 Al
atom 2.0051100192014264 3.8690159986493859 5.9322389744746884 Al
atom 5.8232037669686791 4.0565969760910265 6.0751954048743544 Al
atom 1.9223451440471739 1.9124821068278137 -0.0216036213125823 Al
atom 5.7915721810480436 1.8778064396170522 -0.1739056359166399 Al
atom 2.0614968474105106 6.1024309853155767 -0.2258683443963419 Al
atom 6.3901965541314363 5.9962746865489809 -0.0857749001895369 Al
atom 1.9899597389652315 1.9857991945992610 3.8846446932706256 Al
atom 6.1919982615833860 2.2344314609081213 3.9768287413778589 Al
atom 1.9952184497790340 6.1509562232816215 4.1572936374801968 Al
atom 5.9821155489248223 6.1549705850164766 3.7945314350144383 Al
#=======================================================
# FHI-aims file: geometry.in.supercell.1000K.007
# Created using the Atomic Simulation Environment (ASE)
# Thu Nov 14 15:36:15 2024
# Additional information:
# created from force constants
# T = 1000.0 K
# quantum: False
# deterministic: False
# plus_minus: False
# gauge_eigenvectors: False
# Random seed: 281513462
# Sample number: 8
# Harmonic E_pot: 3.971eV (960.11K)
#=======================================================
lattice_vector 8.0778593799999996 0.0000000000000000 0.0000000000000000
lattice_vector 0.0000000000000000 8.0778593799999996 0.0000000000000000
lattice_vector 0.0000000000000000 0.0000000000000000 8.0778593799999996
atom -0.0790151093049761 0.1485794378222844 -0.1194832556529174 Al
atom 4.2112050023637373 -0.0074948038793868 -0.0219340207225650 Al
atom -0.0939036062477678 4.0305949632256501 -0.0715820743029821 Al
atom 3.9254383584948465 4.0126347364185388 0.2088150529396906 Al
atom 0.2804488670350581 -0.0461430904289928 4.1742594747157309 Al
atom 3.9466067614905262 0.1904544473758693 3.8372573685423914 Al
atom 0.1809629912345928 3.9806665129558398 3.7425908416481390 Al
atom 4.1095109974594726 4.0607964252949271 4.1028167606838348 Al
atom 0.2183818807824085 1.9577715849781023 1.8191147950501187 Al
atom 4.1553219079111994 2.0871344578067927 2.3578864198457410 Al
atom -0.0453737571592037 6.0228428575001693 1.8081803721364007 Al
atom 3.7036175257018016 6.2195187486809678 2.0266667131554765 Al
atom -0.0410576727083087 1.9508527752533056 6.1236059755136729 Al
atom 4.1692483568329992 1.9255751455951673 6.1411999228505385 Al
atom -0.2719088098289064 5.9675949769235821 6.1103628498467453 Al
atom 4.0435315889104082 5.9426705117542253 5.8453008902228589 Al
atom 2.0581061201362765 -0.1033694200845422 1.9781926937243948 Al
atom 6.1164718236530549 0.3040327192526390 2.0760656847150281 Al
atom 2.0429160405118263 3.8574238263694953 1.9966869899196540 Al
atom 5.9021868884276909 4.0348393023527587 1.8630072821222372 Al
atom 2.1837084647503038 -0.1045344644170794 6.2416374603581399 Al
atom 5.9820791955043919 -0.2054720133557299 5.9987636693115585 Al
atom 1.5127696075159580 4.3139813076076399 6.2465859042088763 Al
atom 6.0788336158804164 4.0041183474625379 6.0049178590540917 Al
atom 2.1019776363216138 1.9424316619869417 0.0582803432881564 Al
atom 6.0872154299735737 2.2187023557638175 -0.0366172213224339 Al
atom 2.0235396034754065 6.0067162539890786 0.0257109541451550 Al
atom 5.9326044657045118 6.2942858867510543 -0.0462796689458253 Al
atom 1.9974569430923537 2.0106882955902314 4.4188449561840368 Al
atom 6.3621670175655760 2.0208379998087014 4.2812728061012564 Al
atom 2.0951188681334729 5.7573454054199065 3.9636491103580909 Al
atom 6.0441455563856801 6.1382354082255040 3.7785356503046992 Al
#=======================================================
# FHI-aims file: geometry.in.supercell.1000K.008
# Created using the Atomic Simulation Environment (ASE)
# Thu Nov 14 15:36:15 2024
# Additional information:
# created from force constants
# T = 1000.0 K
# quantum: False
# deterministic: False
# plus_minus: False
# gauge_eigenvectors: False
# Random seed: 281513462
# Sample number: 9
# Harmonic E_pot: 4.124eV (997.14K)
#=======================================================
lattice_vector 8.0778593799999996 0.0000000000000000 0.0000000000000000
lattice_vector 0.0000000000000000 8.0778593799999996 0.0000000000000000
lattice_vector 0.0000000000000000 0.0000000000000000 8.0778593799999996
atom 0.1257680518422207 0.0438648940679496 0.0914770179754121 Al
atom 4.0594809197237227 -0.0121701135873991 0.1226833213194142 Al
atom 0.0906769686829177 4.1316309138344653 0.1295065983648558 Al
atom 3.8357408336890586 3.9633522772924432 0.0186364616406736 Al
atom -0.2434770840437322 0.0295460109089710 4.0989907236936896 Al
atom 4.1919099853657826 -0.2357647749312366 4.3424517359047448 Al
atom 0.0341025975138814 3.9482266887343718 4.1461007631862863 Al
atom 3.9095787114182499 3.9316140800184280 3.8198952565753701 Al
atom -0.0247080638803107 2.0361396416425688 2.1143319276885144 Al
atom 3.9655105383425076 1.7684581324677078 2.1392729206315964 Al
atom -0.0911464532102688 6.3406990986062723 1.9543695490675472 Al
atom 3.7818576406093900 5.9690031705186266 2.0802476907402654 Al
atom -0.0447289648798957 2.2700191838596830 6.0685637064369207 Al
atom 4.3673372326114466 2.0283421957126975 5.8861855806971377 Al
atom -0.0231213111061314 5.8760374655195546 6.1150305833459049 Al
atom 4.0084967339159618 5.9626281660131939 6.0851807795501633 Al
atom 2.2588487660112806 0.0416825057666120 2.2249567575298674 Al
atom 5.8248867800455537 0.0936877368051197 1.9105031045951366 Al
atom 1.8393979994327121 4.2005660871294310 1.9637738138577567 Al
atom 6.1673043727910368 3.8047871888546565 1.8300070664059409 Al
atom 1.9428464600239428 0.1028054610918658 6.0633666983779353 Al
atom 6.2300221026977152 0.2448861953698706 5.9805875416579246 Al
atom 2.3198982162646513 4.0515224994977377 6.1035285867258349 Al
atom 5.9263924412833848 4.0215944147548202 6.1407620499308511 Al
atom 1.9114205091675649 2.0271178831500540 0.0944137882163270 Al
atom 6.0396536801806935 2.0204744675544433 -0.0619171013791860 Al
atom 1.9826897855214602 6.2585594123356998 -0.1830575065611489 Al
atom 6.2019630012020306 6.0159703946437597 -0.2162267806603223 Al
atom 2.4462091276586784 2.1957603386710334 3.9352600529504649 Al
atom 6.2668009954523800 2.0635825316880543 4.0723759036173188 Al
atom 1.7985813954757806 5.8703093543704172 3.9279380524398295 Al
atom 5.8341185901963293 5.8693790576381231 3.9351159154769721 Al
#=======================================================
# FHI-aims file: geometry.in.supercell.1000K.009
# Created using the Atomic Simulation Environment (ASE)
# Thu Nov 14 15:36:15 2024
# Additional information:
# created from force constants
# T = 1000.0 K
# quantum: False
# deterministic: False
# plus_minus: False
# gauge_eigenvectors: False
# Random seed: 281513462
# Sample number: 10
# Harmonic E_pot: 4.628eV (1118.91K)
#=======================================================
lattice_vector 8.0778593799999996 0.0000000000000000 0.0000000000000000
lattice_vector 0.0000000000000000 8.0778593799999996 0.0000000000000000
lattice_vector 0.0000000000000000 0.0000000000000000 8.0778593799999996
atom -0.0027401104817948 0.0278305384937239 -0.1530342201569881 Al
atom 3.8643780944331039 -0.0264461239000665 0.4572359839668849 Al
atom 0.1055461636700433 3.7254524393103483 -0.1543952926328211 Al
atom 4.1211932826953461 4.1040813109282324 -0.0965770902149376 Al
atom -0.0312308521788747 0.0030555979102012 4.1188988280786951 Al
atom 4.0488519481536853 -0.0164613243526246 3.9227023169139592 Al
atom -0.0080286462179765 4.0955468070501215 3.8753140702861262 Al
atom 4.0160057929493913 4.1222822940853456 4.1085532892760908 Al
atom 0.0075081091540319 2.2678070785163440 2.1048334433294547 Al
atom 3.7831098531267653 1.9083049979077213 2.1511043472615867 Al
atom 0.2247555222786947 6.0629543388093214 1.9703354409711744 Al
atom 3.9651744393971233 5.8371256117741295 1.9741069623522651 Al
atom -0.1212556488389646 2.0117591411382860 5.8993067179097167 Al
atom 3.9509368282557187 1.8906695534250502 6.1675297239148819 Al
atom -0.0061628964700797 6.1702260241033420 6.3530752075030525 Al
atom 3.7559410733885676 6.3214893155983081 6.0535304188305785 Al
atom 1.9482264210218176 -0.1119905178144168 2.0044773230390538 Al
atom 6.0494860092798577 -0.1269755611679731 2.0107984297654302 Al
atom 2.1717357251571858 4.1390760447369974 1.8173618593154763 Al
atom 6.0067767449674170 4.0433359854630995 2.1286264108243533 Al
atom 1.9465534156973678 0.4946958303542788 6.0344114210470750 Al
atom 6.0245254375366466 0.0598309818983070 6.1862163177464966 Al
atom 2.0126798529041952 3.8083806891972767 6.1565528554191484 Al
atom 6.1152043503684483 3.8692880554427296 6.1155194024073447 Al
atom 2.3040189354315670 2.0197456115608112 -0.1250302450633377 Al
atom 5.8363355190723176 1.8961664966754814 0.2623595501731578 Al
atom 2.0801730998440147 6.2171520746228905 -0.3603711039929702 Al
atom 6.3302562274300858 5.8566430481258385 0.0293666290040201 Al
atom 2.0605137702182184 2.2247033326849794 3.9154280085663271 Al
atom 6.0380008224223127 1.8620933534463613 4.1072094775938499 Al
atom 2.1603842994724194 6.3246648942258226 3.7525020650377510 Al
atom 6.1754589758613410 5.8518246397497302 4.1463640115270968 Al
In order to take into account electron-vibrational interaction in real sapce, we need to use large supercells to include the nuclei vibration. In this example, we use a \(2\times 2\times 2\) supercell with 16 atoms which is relatively small, in production calculation the supercell size should be significantly larger.
Control tags
For each geometry, we can use the same input file control.in from the first Part.
xc pbe
k_grid 6 6 6
dos_kgrid_factors 1 1 1
# type[letter] temperature[eV]
occupation_type fermi 1
# 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 1 -20.57 9.44 0.0 5.0 2000 a a
kg_width 0.01 0.10 10
[Add light species defaults for Al]
Warning
Here, we use \(2\times 2\times 2\) supercell, but the same control.in file as for the 4-atom Al primitive cell in Part 1. Thus, the \(\bf k\)-grid is denser then for the primitive cell in part 1.
In general, to maintain the same \(\bf k\)-grid density between a primitive cell and an \(N \times N \times N\) supercell, the \(\bf k\)-grid in the supercell need to be \(\frac{1}{N} \times \frac{1}{N} \times \frac{1}{N}\) sparser. Keep this relation in mind when making fair comparisons of \(\bf k\)-grid convergence between different supercell sizes.
For a more detailed discussion on the systematic convergence of the \(\bf k\)-grid density and supercell size, please refer to the next example in Part 3 and this paper.
Run the calculation
Create 10 separate directories for each of the samples, named from config_000 to config_009 and copy the relevant geometry.in file from above into them, as well as the control.in displayed above. Then run calculations from inside each of these folders. On your local PC, the each calculation with N processes can be launched by:
your_aims.x should be replaced by your FHI-aims binary file.
Alternatively, if you are on a HPC system, then you can use a job submission script to perform each of the calculations within their respective folders.
The estimated runtime for one single calculation with 4 CPU cores is approximately 15 minutes. Using more cores can further speed up the calculation.
(If you 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 repository, we prepared a bash script called pp_all.sh to collect all output files. This is also shown below: Note that this script assumed the 10 calculation directories are named config_000 to config_009
startp=000
endp=009
startw=100
endw=1000
# 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 "%03g" ${startp} 1 ${endp}` ; do
echo ${num}
cd "config_${num}"
if test ! -e KG_full_Mobility_out_Gaussian_0.0700_a_a.out; then
echo "skip ${num}"
cd ..
continue
fi
if [ ${num} = ${startp} ]; then
for iwidth in `seq -f "%04g" ${startw} 100 ${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} 100 ${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
Running
will generate the filekg_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.
The resulting text file kg_result_e.txt has the following content:
#
# 0.179505314282E+23 0.180362079085E+23 0.181043813377E+23 0.180027736242E+23 0.181852508509E+23 0.181445140065E+23 0.180577348314E+23 0.182572276313E+23 0.182107982559E+23 0.180805474730E+23
# 0.159479405674E+23 0.158403845963E+23 0.157680253335E+23 0.158947832445E+23 0.157083621546E+23 0.157378916563E+23 0.158092117597E+23 0.156749394161E+23 0.156812098001E+23 0.158320244012E+23
2.500000E-003 8.495217E+003 8.148546E+003 5.477885E+003 6.523649E+003 8.907962E+003 1.002064E+004 1.079687E+004 1.194417E+004 7.120126E+003 9.894707E+003
5.000000E-003 6.438347E+003 6.216643E+003 4.246308E+003 5.056675E+003 6.687284E+003 7.604281E+003 7.964200E+003 9.010525E+003 5.473586E+003 7.124721E+003
7.500000E-003 6.269923E+003 6.124878E+003 4.251543E+003 5.060463E+003 6.503440E+003 7.437250E+003 7.631502E+003 8.780709E+003 5.463632E+003 6.655162E+003
1.000000E-002 6.642949E+003 6.582644E+003 4.648832E+003 5.520270E+003 6.926764E+003 7.917551E+003 8.036046E+003 9.344086E+003 5.975679E+003 6.824271E+003
1.250000E-002 7.285391E+003 7.328216E+003 5.274650E+003 6.230062E+003 7.672778E+003 8.709798E+003 8.836728E+003 1.031304E+004 6.790578E+003 7.299702E+003
1.500000E-002 8.102780E+003 8.268527E+003 6.076987E+003 7.113780E+003 8.641595E+003 9.684485E+003 9.925713E+003 1.154581E+004 7.832061E+003 7.965532E+003
1.750000E-002 9.049047E+003 9.358129E+003 7.033583E+003 8.129578E+003 9.779089E+003 1.077039E+004 1.125458E+004 1.296343E+004 9.059736E+003 8.761529E+003
2.000000E-002 1.009304E+004 1.057062E+004 8.131056E+003 9.247546E+003 1.104167E+004 1.192230E+004 1.278730E+004 1.450439E+004 1.044424E+004 9.643257E+003
2.250000E-002 1.120693E+004 1.188978E+004 9.357430E+003 1.044367E+004 1.238555E+004 1.311395E+004 1.448353E+004 1.611133E+004 1.196129E+004 1.057158E+004
2.500000E-002 1.236266E+004 1.330530E+004 1.069870E+004 1.169796E+004 1.376527E+004 1.433665E+004 1.629443E+004 1.772874E+004 1.359091E+004 1.151316E+004
2.750000E-002 1.353291E+004 1.480857E+004 1.213720E+004 1.299298E+004 1.513734E+004 1.559706E+004 1.816669E+004 1.930572E+004 1.531696E+004 1.244566E+004
3.000000E-002 1.469468E+004 1.638775E+004 1.365142E+004 1.431202E+004 1.646694E+004 1.691184E+004 2.005173E+004 2.080140E+004 1.712510E+004 1.336347E+004
...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......
#
# 0.179505314282E+23 0.180362079085E+23 0.181043813377E+23 0.180027736242E+23 0.181852508509E+23 0.181445140065E+23 0.180577348314E+23 0.182572276313E+23 0.182107982559E+23 0.180805474730E+23
# 0.159479405674E+23 0.158403845963E+23 0.157680253335E+23 0.158947832445E+23 0.157083621546E+23 0.157378916563E+23 0.158092117597E+23 0.156749394161E+23 0.156812098001E+23 0.158320244012E+23
2.500000E-003 3.967877E+004 4.326409E+004 3.516766E+004 3.722001E+004 4.360884E+004 4.681503E+004 5.288660E+004 5.534676E+004 4.408378E+004 3.802817E+004
5.000000E-003 2.421576E+004 2.656323E+004 2.180372E+004 2.286227E+004 2.665970E+004 2.857890E+004 3.245301E+004 3.365266E+004 2.723434E+004 2.311534E+004
7.500000E-003 1.953317E+004 2.155115E+004 1.786369E+004 1.855277E+004 2.153858E+004 2.306288E+004 2.632272E+004 2.703186E+004 2.222660E+004 1.858925E+004
1.000000E-002 1.757627E+004 1.949874E+004 1.632197E+004 1.678692E+004 1.940852E+004 2.076484E+004 2.381622E+004 2.421005E+004 2.022248E+004 1.669358E+004
1.250000E-002 1.673281E+004 1.865796E+004 1.577270E+004 1.606199E+004 1.850035E+004 1.978296E+004 2.279533E+004 2.292873E+004 1.945142E+004 1.587761E+004
1.500000E-002 1.646387E+004 1.844356E+004 1.574588E+004 1.587504E+004 1.822234E+004 1.948159E+004 2.254438E+004 2.243146E+004 1.931964E+004 1.562471E+004
1.750000E-002 1.653688E+004 1.860195E+004 1.603829E+004 1.600841E+004 1.831879E+004 1.958631E+004 2.275336E+004 2.239064E+004 1.956857E+004 1.571356E+004
2.000000E-002 1.683388E+004 1.900350E+004 1.654614E+004 1.635118E+004 1.865964E+004 1.995780E+004 2.326359E+004 2.263922E+004 2.006456E+004 1.603338E+004
2.250000E-002 1.728763E+004 1.957337E+004 1.720931E+004 1.683967E+004 1.917046E+004 2.051637E+004 2.398306E+004 2.308140E+004 2.072881E+004 1.652230E+004
2.500000E-002 1.785602E+004 2.026398E+004 1.798915E+004 1.743380E+004 1.980438E+004 2.121191E+004 2.485258E+004 2.365696E+004 2.150949E+004 1.714263E+004
2.750000E-002 1.851048E+004 2.104257E+004 1.885831E+004 1.810633E+004 2.052939E+004 2.201016E+004 2.583042E+004 2.432510E+004 2.236923E+004 1.786957E+004
3.000000E-002 1.923016E+004 2.188507E+004 1.979577E+004 1.883760E+004 2.132200E+004 2.288598E+004 2.688480E+004 2.505640E+004 2.327899E+004 1.868544E+004
...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......
This file is divided into blocks by the lines started with #, these commented lines are the electron and 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 \((1999, 10)\) stored all electron conductivity data of 10 samples at 1999 frequency points for each broadening width. The file kg_result_h.txt have exactlly the same structure.
(P.S. we requested 2000 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 total mobility \(\mu(\omega)\) is calculated by:
\[\langle \mu(\omega) \rangle_T = \langle \frac{\sigma_e(\omega)}{ n_e q} \rangle_T + \langle \frac{\sigma_h(\omega)}{ n_h q} \rangle_T \]
where \(\langle \cdots \rangle = \frac{1}{N}\sum_{I}\) is averaging over all samples.
A simple post-processing and visualization script pp_mobility_Al.ipynb is available as a Jupyter Notebook in the solutions folder of the repository using kg_result_e.txt and kg_result_h.txt as input files. A standalone python plotting script, plot_Al_mobility.py is also 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]))
np.shape(eachk)
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 # 10 for 222, 7 for 555
nn_freq = 1999
nn_samples = 10 # 100samples for 222, 10samples for 555
charge = 1.60217663e-19
filename1 = f"kg_result_e.txt"
filename2 = f"kg_result_h.txt"
carrier_den = get_carrier(nn_samples, filename1)
avrg_mobility_e, std_mobility_e, freq = pp_mobility(filename1, nn_width, nn_freq, carrier_den[0,:])
avrg_mobility_h, std_mobility_h, freq = pp_mobility(filename2, nn_width, nn_freq, carrier_den[1,:])
avrg_mobility = avrg_mobility_e + avrg_mobility_h
j = 2
end_point = 200
start_point = 1
fig, ax = plt.subplots()
# plot figure
ax.plot(freq[start_point:end_point],avrg_mobility[j,start_point:end_point]
, label=r"16 atom ")
plt.xlabel(r'$\hbar\omega$ (eV)',fontsize=12)
plt.ylabel(r'Mobility (${\rm cm^2/ (Vs)}$)',fontsize=12)
plt.legend()
plt.xlim(0,0.55)
plt.savefig('Al_16atom_mobility.png',dpi=250)
The averaged mobility \(\mu (\omega)\) with broadening \(\eta=0.03\) eV should look like the following figure:
Divergence can be observed in the DC limit \(\omega \to 0\), which arises from the \(1/\omega\) term in the Kubo-Greenwood formula. This divergence is eliminated only in the thermodynamics limit (infinitely large supercell), but will always exist in our finite-size simulation.
Drude fitting
From the averaged electron mobility spectra, we can also observe that in the low frequency region (before the numerical divergence), when the frequency decreases, 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 look like a Drude function, but a converged ensemble will show a Drude-like behaviour.
A simple Drude fitting code is provided in the same Jupyter Notebook, by fitting with the electron mobility data between the first peak and 0.5 eV. This is also shown in an extended version of the standalone python plotting script 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]))
np.shape(eachk)
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 # 10 for 222, 7 for 555
nn_freq = 1999
nn_samples = 10 # 100samples for 222, 10samples for 555
charge = 1.60217663e-19
filename1 = f"kg_result_e.txt"
filename2 = f"kg_result_h.txt"
carrier_den = get_carrier(nn_samples, filename1)
avrg_mobility_e, std_mobility_e, freq = pp_mobility(filename1, nn_width, nn_freq, carrier_den[0,:])
avrg_mobility_h, std_mobility_h, freq = pp_mobility(filename2, nn_width, nn_freq, carrier_den[1,:])
avrg_mobility = avrg_mobility_e + avrg_mobility_h
j = 2
end_point = 200
start_point = 10
fig, ax = plt.subplots()
# drude fit
freq_start = 10
freq_end = 200
print('max condcutivity index:', np.argmax(avrg_mobility[j, freq_start:]))
freq_start = np.argmax(avrg_mobility[j, freq_start:]) + freq_start
#freq_start = 15
freq_in = freq[freq_start: freq_end]
cond_in = avrg_mobility[j, freq_start:freq_end]
popt = drude_fit(freq_in, cond_in)
# plot figure
ax.plot(freq[start_point:end_point],avrg_mobility[j,start_point:end_point], label=r"16 atom ")
ax.plot(freq[0:freq_end], drude_func(freq[0:freq_end], *popt), color='tomato',label=f'Drude fit')
plt.xlabel(r'$\hbar\omega$ (eV)',fontsize=12)
plt.ylabel(r'Mobility (${\rm cm^2/ (Vs)}$)',fontsize=12)
plt.legend()
plt.xlim(0,0.55)
plt.savefig('Al_16atom_drude_fit.png',dpi=250)
The resulting fit is displayed in the figure below:
In this case, the extrapolated DC mobility is \(\mu_0 \sim 28\) \(cm^2/(Vs)\).
Warning
The above result is not converged. In production calculation, to get reliable vibrational-limited mobility one should carefully converge all parameters including:
-
supercell size
-
number of samples
-
k-grid density
-
broadening width.
Convergence test
In this example, we only use a small 16-atom supercell, sparse \(6\times 6\times 6\) k-grid and 10 samples for simplicity. These settings are insufficient for production calculations. A detailed discussion on convergence tests for all these parameters can be found in this paper.
The next example, which uses a 40-atom SrTiO\(_3\) supercell, also provides insights on the convergence behavior of these parameters.
Solutions
You find all the solution to all the above exercises by clicking on the button below.