boltztrap2 - wien 2k

BoltzTraP2

Georg K. H. Madsen

Institute of Materials Chemistry, TU Wien, Austria

20-09-17 / WIEN2k-workshop

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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

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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.

��

��

��

��

kFλ� 1.

��

��

��

��

kFλ� 1.

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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

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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

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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

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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)

)

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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

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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.

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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)]

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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)

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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

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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

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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

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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

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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

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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

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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)

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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)

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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


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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


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p-type HHC: (V/Nb)FeSb

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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)

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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


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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


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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


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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)


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Installing Anaconda and BoltzTraP2

ssh -X [email protected] 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

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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

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Acknowledgements

Sandip Bhattacharya Jesús Carrete Matthieu Verstraete

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boltztrap2 - wien 2k

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