boltztrap2 - wien 2k
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BoltzTraP2
Georg K. H. Madsen
Institute of Materials Chemistry, TU Wien, Austria
20-09-17 / WIEN2k-workshop
1 / 29
Overview
The Boltzmann transport equationsBoltzTraP (Smoothed Fourier band interpolation)BoltzTraP2. A modern tool for modern workflows
AlgorithmIncluding the momentum matrix elementsCommand-line interface and Python3 library: CoSb3
Applications:Volumetric band alignmentp-doped half-Heusler Compounds
2 / 29
The Boltzmann equation
The steady state distribution f is constant in time(∂f∂t
)diff
+
(∂f∂t
)field
+
(∂f∂t
)scatt
= 0
Assumption:k should be a good quantum number. i.e. wavelength ofelectron small compared to mean free path. kFλ� 1.
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kFλ� 1.
3 / 29
Boltzmann Equation
(∂f∂t
)diff
+
(∂f∂t
)field
+
(∂f∂t
)scatt
= 0
Diffusion:(∂f∂t
)diff
=∂f∂T∇T v
Th Tc
x
+ -E
e-
Field:(∂f∂t
)field
= −∂f∂ε
vqE
v(−∂f∂ε
)(−ε− µ
T∇T + qE
)= −
(∂f∂t
)scatt
4 / 29
Relaxation time approximation
Phenomological assumption: Exponential decay of deviationfrom equilibrium with τ as the relaxation time.(
∂f∂t
)scatt
= − f − f (0)
τ
thereby
je =∑
n
∫qvnkvnkτnk
(−∂f (0)
∂ε
)(−ε− µ
T∇T + qE
)dk
8π3
jQ =∑
n
∫(ε− µ)vnkvnkτnkτk
(−∂f (0)
∂ε
)(−ε− µ
T∇T + qE
)dk
8π3
5 / 29
The transport distribution
Introduce the transport distribution
σ(ε) =∑
n
∫vnkvnkτnkδ(ε− εnk)
dk8π3
The generalized transport coefficients aremoments of the transport distribution
L(α)(T , µ) = q2∫σ(ε)(ε− µ)α
(−∂f∂ε
)dε
je = L(0)E +L(1)
qT(−∇T )
jQ =L(1)
qE +
L(2)
q2T(−∇T )
−(ε− µ)α∂f∂ε
0.3
0.2
0.1
0.0
0.1
0.2
0.3
(eV)
= 0= 1
6 / 29
Phenomenological transport coefficients
Identify two kinds of experimental situations∇T = 0:
je = L(0)E ⇒ σ = L(0)
jQ =L(1)
qE =
L(1)
qL(0)je ⇒ Π =
L(1)
qL(0)
je = 0:
L(0)E =L(1)
qT∇T ⇒ S =
1qTL(1)
L(0)
jq =1
q2T
((L(1))2
L(0)− L(2)
)∇T ⇒ κe =
1q2T
((L(1))2
L(0)− L(2)
)
7 / 29
Harvesting of waste heat
The thermoelectric effect is the direct conversion oftemperature differences to electric voltage and vice-versaApproximately 70% of energy is lost as waste heat whenburning fossil fuels for power generation
Figure of merit
zT =S2σTκe + κl
Electronic power factor
PF = S2σ
8 / 29
Shankland-Pickett algorithmConstrained optimization procedure: Minimize roughnessfunction with respect to the Fourier coefficients while exactlyreproducing calculated eigenvalues.
ε̃k =∑
Λ
cΛ
∑R∈Λ
exp(ik · R)
Minimize the Lagrangian
I =12
∑cΛρΛ +
∑k
λk
(εk − ε̃k
)Euwema, Shankland et al, Phys. Rev. 178 (1969) 1419–1423Shankland, Int. J. Quantum Chem. 5 (1971) 497–500.
Roughness function
ρ =(ε̃k − ε0 + C1∇2ε̃k
)2
Pickett et al. Phys. Rev. B 38 (1988) 2721–2726.
9 / 29
Transport distribution. CoSb3
Transport distribution
σ(ε) =13
∑n
∫vnkvnkτnkδ(ε− εnk)
dk8π3
CoSb3
-2.0
-1.0
0.0
1.0
2.0
ε (e
V)
Γ H N Γ P
εF
−2
−1
0
1
2
0 1 2 3 4
ε [e
V]
n(ε) [#states/(f.u. eV spin)]
Sbt2geg
-2
-1
0
1
2
0 1 2 3
ε [e
V]
σ/τ [1018 S/(cm s)]
10 / 29
Automated search for new thermoelectrics
2kWI N
E ������@@@@@@������ÀÀÀÀÀÀ������@@@@@@������ÀÀÀÀÀÀ������@@@@@@������ÀÀÀÀÀÀ������@@@@@@������ÀÀÀÀÀÀ������@@@@@@������ÀÀÀÀÀÀ������yyyyyyBoltz TraP
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
ε (e
V)
W L Γ X W K
εF
0
100
200
300
400
500
600
|S|(µ
V/K
)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-10-4
-10-3
-10-2
-10-1
zT
N (e/Zn-Sb)
10-4
10-3
10-2
10-1
N (e/Zn-Sb)
StructureBaZn2Sb2CaZn2Sb2
LiZnSbZnSb
BaZn2Sb2 (b)
n-type p-type
BoltzTrap:All crystal structuresFull tensors quantitiesNumerically efficient and stable
GKHM JACS 128 p12140 (2006)GKHM, Singh, Comput. Phys. Commun. 175, p67 (2006)Bjerg, GKHM, Iversen, Chem. Mat. 23 p3907 (2011)
11 / 29
BoltzTraP2: A modern tool for modern workflows.BoltzTraP2
Design goals:All useful features fromBoltzTraPEasy installation, portabilitypip3 install BoltzTraP2
Command-line interface(no config files)
Speed:New algoritmesModularity, flexibilityStandard formats
Two use cases:1 I want to estimate the Onsager thermoelectric coefficients
from my DFT results⇒ BoltzTraP2 as a stand-alone tool
2 I need interpolated bands as inputs to my own algorithm⇒ BoltzTraP2 as a Python module
boltztrap.orgMadsen, Carrete, Verstraete Comp. Phys. Comm 231 p140 2018
12 / 29
BoltzTraP2 interpolation
Minimize roughness function with respect to the Fouriercoefficients while exactly reproducing calculated eigenvaluesand derivatives
I =12
∑cΛρΛ +
∑k
λk
(εk − ε̃k
)+∑
k
λ′k
(∇εk −∇ε̃k
)
Combine advantage of BoltzTraP (analytic bands) andScheidemantel-Sofo approach (exact derivatives atcalculated points)Potentially coarser k -mesh in ab-initio calculation
13 / 29
Example: Silicon band structure
X1.5
1.0
0.5
0.0
0.5
1.0
1.5
[eV]
CBM made up by degerate pocket along six-fold degenrateΓ− X lineInterpolated bands based on a coarse 9× 9× 9 k -pointmeshModified Lagrangian forces the fit to reproduce the exactderivatives at the calculated points.Position and derivatives at the pocket are well reproduced
14 / 29
Example: Silicon transport
20 35 56 84 120 165 220 286 364 455#kpts IBZ
65
60
55
50
45
S [1
014
V/K]
10
12
14
16
S2/
[1014
W
/(cm
K2 )
s]
Seebeck coefficient and thermoelectric power factorcalculated at a chemical potential close to the CBM usingthe CRTAThe results obtained by the modified Lagragian show botha faster and more systematic convergence towards theconverged valuesConvergence reached at about half the number of k -points
15 / 29
Example: Band and momentum dependent relaxationtimes
2 1 0 1 [eV]
0
250
500
750
1000
1250
[kSm
1 ]
CRTAe-phmodel
2 1 0 10.0
0.1
0.2
0.3
0.4
n [e
V1 ]
Including band and momentum dependent relaxation timeschanges the slope of the transport distribution (and therebythe Seebeck coefficient)
Madsen, Carrete, Verstraete Comp. Phys. Comm 231 p140 2018τbk from Xu, Verstraete, Phys. Rev. Lett. 112 p196603 2014
16 / 29
Some highlights of BoltzTraP2
FlexibilityUsable as a PythonmoduleExtensible scatteringmodelsAutomatic detection ofspace group
PortabilityStandard Python setuptoolchainDetection of compilers andlibrariesAdherence to C++11
SpeedHighly vectorized PythonSymmetry module in C++fftw
Standard formatsJSON: Human readable &parsers for every languageFinal output as text
17 / 29
Volumetric Band Alignment
V1 Vopt V2
VBM
Zaitsev et al. PRB 74 p045207 (2006)
Optimize power factor by alignment of band edges
S2σ =1
q2T 2
(L(1) + L(1)
)2
L(0) + L(0)
18 / 29
Mg2SixSn1−x
1. Band structureCalculate volume depen-dence of S and σ/τ
2. ThermodynamicsCalculate mixing enthalpy(SQS)
Bhattacharya, GKHM Phys. Rev. B 92, p085205 (2015)
19 / 29
VBA screening
nitrogen
14.007
N7
helium
He4.0026
2
neon
Ne20.180
10f uorine
F18.998
9oxygen
O15.999
8carbon
C12.011
6boron
B10.811
5
argon
Ar39.948
18chlorine
Cl35.453
17sulfur
S32.065
16phosphorus
P30.974
15silicon
Si28.086
14aluminium
Al26.982
13
krypton
Kr83.798
36bromine
Br79.904
35selenium
Se78.96
34arsenic
As74.922
33germanium
Ge72.64
32gallium
Ga69.723
31zinc
Zn65.38
30copper
Cu63.546
29nickel
Ni58.693
28cobalt
Co58.933
27iron
Fe55.845
26manganese
Mn54.938
25chromium
Cr51.996
24vanadium
V50.942
23titanium
Ti47.867
22scandium
Sc44.956
21calcium
Ca40.078
20
potassium
K39.098
19
magnesium
Mg24.305
12
sodium
Na22.990
11
beryllium
Be9.0122
4
lithium
Li6.941
3
hydrogen
H1.0079
1
xenon
Xe131.29
54iodine
I126.90
53tellurium
Te127.60
52antimony
Sb121.76
51tin
Sn118.71
50indium
In114.82
49cadmium
Cd112.41
48silver
Ag107.87
47palladium
Pd106.42
46rhodium
Rh102.91
45ruthenium
Ru101.07
44technetium
Tc[98]
43molybdenum
Mo95.96
42niobium
Nb92.906
41zirconium
Zr91.224
40yttrium
Y88.906
39strontium
Sr87.62
38rubidium
Rb85.468
37
radon
Rn[222]
86astatine
At[210]
85polonium
Po[209]
84bismuth
Bi208.98
83lead
Pb207.2
82barium
Ba137.33
56caesium
Cs132.91
55
roentgenium
Rg[272]
111darmstadtium
Ds[271]
110meitnerium
Mt[268]
109hassium
Hs[277]
108bohrium
Bh[264]
107seaborgium
Sg[266]
106dubnium
Db[262]
105rutherfordium
Rf[261]
104radium
Ra[226]
88francium
Fr[223]
87
thallium
Tl204.38
81mercury
Hg200.59
80gold
Au196.97
79platinum
Pt195.08
78iridium
Ir192.22
77osmium
Os190.23
76rhenium
Re186.21
75tungsten
W183.84
74tantalum
Ta180.95
73hafnium
Hf178.49
72
M-XM-X
3150Initial M-X combinations
Thermodynamicallystable structures
Good thermoelectrics (zT > 0.4)
Candidates for VBA Checks for alloy stability
522
29
8 4
Bhattacharya, GKHM Phys. Rev. B 92, p085205 (2015)
20 / 29
Alloy Thermodynamics
Compound ∆Eh [∆ESnh ] Vopt xalloy ∆Emix(0.25) x
(meV/atom) (%) (kJ/mol) (800 K)Mg2Si1−xSnx 0[0] 2.0 0.09 1.997 0.171Ca2Si1−xSnx 0[0] 5.0 0.30 0.013 allCa9Ge5−xSnx 37.4[17.9] 6.1 0.28 4.495 0.019β−MoSi2−xSnx 27.3[180.8] 3.0 0.07 30.980 0.008o-Fe2Ge3−xSnx 0.1[22.5] 3.0 0.10 32.746 0.004
Bhattacharya, GKHM Phys. Rev. B 92, p085205 (2015)
21 / 29
p-type HHC: (V/Nb)FeSb
22 / 29
Screening. Band structure, abundance and stability.
75 18
zT0 > 1.3 & ΔEhull=0 & Horth<HHH
(ΔP 0-)AFLOWlib
Bandstructure& Abundance
EA>0.1 ppmDefects scanning
Stability
(Zr/Hf)CoSb(Nb/Ta)CoSn(V/Nb/Ta)FeSb
79,000
HfCoSbZrCoSb
TaFeSb
V/NbFeSb
NbCoSnTaCoSnTiNiSn
NaSrSb
Zr/HfNiSn
ZrNiPb
TaRhSnTaIrSn
NbCoSi
NbFeAs
V/TaCoGeTiNiGe
Zr/HfNiGe
Compound ∆Ehull orthmeV/atom phase
TaFeAs 33.03 yesTaFeSb 0.00 noNbFeAs 125.81 yesNbFeSb 0.00 noVFeSb 0.00 noZrCoAs 0.00 yesZrCoSb 0.00 noWFeGe 64.34 noNbCoGe 0.00 yesHfCoAs 0.00 yesTaCoSn 0.00 noVCoSn 90.77 noTiCoAs 0.00 yesNbCoSn 0.00 noTaCoGe 0.00 yesVCoGe 0.00 yesTaCoSi 91.47 yesHfCoSb 0.00 no
Bhattacharya, GKHM J. Mater. Chem. C, 4, p11261 (2016)
23 / 29
Screening. Band structure, abundance and stability.
75 18
zT0 > 1.3 & ΔEhull=0 & Horth<HHH
(ΔP 0-)AFLOWlib
Bandstructure& Abundance
EA>0.1 ppmDefects scanning
Stability
(Zr/Hf)CoSb(Nb/Ta)CoSn(V/Nb/Ta)FeSb
79,000
HfCoSbZrCoSb
TaFeSb
V/NbFeSb
NbCoSnTaCoSnTiNiSn
NaSrSb
Zr/HfNiSn
ZrNiPb
TaRhSnTaIrSn
NbCoSi
NbFeAs
V/TaCoGeTiNiGe
Zr/HfNiGe
Compound ∆Ehull orthmeV/atom phase
TaFeAs 33.03 yesTaFeSb 0.00 noNbFeAs 125.81 yesNbFeSb 0.00 noVFeSb 0.00 noZrCoAs 0.00 yesZrCoSb 0.00 noWFeGe 64.34 noNbCoGe 0.00 yesHfCoAs 0.00 yesTaCoSn 0.00 noVCoSn 90.77 noTiCoAs 0.00 yesNbCoSn 0.00 noTaCoGe 0.00 yesVCoGe 0.00 yesTaCoSi 91.47 yesHfCoSb 0.00 no
Bhattacharya, GKHM J. Mater. Chem. C, 4, p11261 (2016)
23 / 29
Screening. Band structure, abundance and stability.
75 18
zT0 > 1.3 & ΔEhull=0 & Horth<HHH
(ΔP 0-)AFLOWlib
Bandstructure& Abundance
EA>0.1 ppmDefects scanning
Stability
(Zr/Hf)CoSb(Nb/Ta)CoSn(V/Nb/Ta)FeSb
79,000
HfCoSbZrCoSb
TaFeSb
V/NbFeSb
NbCoSnTaCoSnTiNiSn
NaSrSb
Zr/HfNiSn
ZrNiPb
TaRhSnTaIrSn
NbCoSi
NbFeAs
V/TaCoGeTiNiGe
Zr/HfNiGe
Compound ∆Ehull orthmeV/atom phase
TaFeAs 33.03 yesTaFeSb 0.00 noNbFeAs 125.81 yesNbFeSb 0.00 noVFeSb 0.00 noZrCoAs 0.00 yesZrCoSb 0.00 noWFeGe 64.34 noNbCoGe 0.00 yesHfCoAs 0.00 yesTaCoSn 0.00 noVCoSn 90.77 noTiCoAs 0.00 yesNbCoSn 0.00 noTaCoGe 0.00 yesVCoGe 0.00 yesTaCoSi 91.47 yesHfCoSb 0.00 no
Bhattacharya, GKHM J. Mater. Chem. C, 4, p11261 (2016)
23 / 29
Intrinsic defects
VB CB
Ef
-ΔμD μeVBM0
εg
D'
(1)
∆µD′ D′(q) ∆µD′ D′(q) ∆µD′ D′(q)
(eV) (eV) (eV)NbCoSn -0.27 Co(3)
Int VFeSb -0.18 Fe(2)Int ZrCoSb -0.51 Sb(1)
ZrTaCoSn -0.26 Co(2)
Int NbFeSb -0.60 Fe(2)Int HfCoSb -0.37 Co(2)
Int
No intrinsic doping limits
Bhattacharya, GKHM J. Mater. Chem. C, 4, p11261 (2016)
24 / 29
Extrinsic doping
VB CB
Ef
-ΔμD μeVBM0
εg
D'
(1)
EDf (µVB) D(q) ∆ED′−D D′(q)
(eV) (eV)
NbFeSb -0.01 Hf(−1)Nb 1.20 Fe(2)
Int0.09 Ti(−1)
Nb 1.10 Fe(2)Int
0.21 Mn(−1)Fe 0.72 Vac(2)
Fe0.41 Zr(−1)
Nb 0.78 Fe(2)Int
0.55 Sn(−1)Sb 0.70 Fe(2)
Int
ZrCoSb 0.17 Sc(−1)Zr 0.60 Sb(1)
Zr0.61 Sn(−1)
Sb 0.35 Vac(1)Co
0.61 Fe(−1)Co -0.48 Vac(1)
Co
NbCoSn 0.32 Hf(−1)Nb 0.50 Co(3)
Int0.62 Fe(−1)
Co -0.15 Fe(2)Int
0.68 Ti(−1)Nb 0.14 Co(3)
Int0.72 Zr(−1)
Nb 0.10 Co(3)Int
TaCoSn 0.44 Hf(−1)Ta 0.33 Co(2)
Int0.87 Fe(−1)
Co -0.10 Fe(2)Int
Known carrier inducing defects reproduced in NbFeSb andZrCoSbA new system with favorable extrinsic dopants identified
Bhattacharya, GKHM J. Mater. Chem. C, 4, p11261 (2016)
25 / 29
Extrinsic doping
VB CB
Ef
-ΔμD
(1)
μeVBM0
εg
(-1)
D'
ΔED'-D
D
(1)
(-1)
D'
D
-ΔED'-D
EDf (µVB) D(q) ∆ED′−D D′(q)
(eV) (eV)
NbFeSb -0.01 Hf(−1)Nb 1.20 Fe(2)
Int0.09 Ti(−1)
Nb 1.10 Fe(2)Int
0.21 Mn(−1)Fe 0.72 Vac(2)
Fe0.41 Zr(−1)
Nb 0.78 Fe(2)Int
0.55 Sn(−1)Sb 0.70 Fe(2)
Int
ZrCoSb 0.17 Sc(−1)Zr 0.60 Sb(1)
Zr0.61 Sn(−1)
Sb 0.35 Vac(1)Co
0.61 Fe(−1)Co -0.48 Vac(1)
Co
NbCoSn 0.32 Hf(−1)Nb 0.50 Co(3)
Int0.62 Fe(−1)
Co -0.15 Fe(2)Int
0.68 Ti(−1)Nb 0.14 Co(3)
Int0.72 Zr(−1)
Nb 0.10 Co(3)Int
TaCoSn 0.44 Hf(−1)Ta 0.33 Co(2)
Int0.87 Fe(−1)
Co -0.10 Fe(2)Int
Known carrier inducing defects reproduced in NbFeSb andZrCoSbA new system with favorable extrinsic dopants identified
Bhattacharya, GKHM J. Mater. Chem. C, 4, p11261 (2016)
25 / 29
TE Heusler. Band structure.
ε-ε F
W L Γ XWK
b) Co t2g
ε-ε F
W L Γ XWK
a) Co eg
ε-ε F
W L Γ XWK
c) Ta eg
ε-ε F
W L Γ XWK
d) Ta t2g
i)
CB
VB
t2g
eg
Ta
Ta t2g(Γ)
t2g
eg
Co eg
(L,W)
Co eg
(X)
Ta eg (Γ-X)
εg
t2
a1
*
*
5s
5p
Sn
4s
4pCo
t2
a1ii)
Alignment of pockets at L and W (and Γ)
Bhattacharya, GKHM J. Mater. Chem. C, 4, p11261 (2016)
26 / 29
Installing Anaconda and BoltzTraP2
ssh -X wienXXX@psiXX.theochem.tuwien.ac.atwget https://repo.anaconda.com/archive/Anaconda3-2019.07-Linux-x86_64.shchmod u+x Anaconda3-2019.07-Linux-x86_64.shbash./Anaconda3-2019.07-Linux-x86_64.sh# Complete the installation procedure and init# Log out of and log into the machine.conda config --set auto_activate_base falseconda install cmake git vtk pytestpip install pyfftwpip install boltztrap2# Now, to download the example data:git clone https://gitlab.com/sousaw/BoltzTraP2.gittar -xvf BoltzTraP2/data.tar.xz
27 / 29
Command Line Interface
Make working directory and copy data. Activate Conda if you have switchedthe autoconfig off.Fit the calculated eigenvalues. The interpolated k -mesh should be five timesas dense as the original:btp2 interpolate . -m 5 -o CoSb3.bt2Integrate to get the transport propertiesbtp2 integrate CoSb3.bt2 50:500:50Plot the resultsbtp2 plot -c ’["xx"]’ CoSb3.btj S
Read the wiki
28 / 29
Acknowledgements
Sandip Bhattacharya Jesús Carrete Matthieu Verstraete
29 / 29
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