From band structures to thermoelectric propertiesusing state-of-the-art ab initio methods

Silvana Botti

1LPMCN, CNRS-Université Lyon 1, France2European Theoretical Spectroscopy Facility

December 6, 2012 – Lyon

Silvana Botti Thermoelectricity 1 / 15

Outline

1 Calculations of thermoelectric properties

2 How to obtain reliable band structures?

3 Disilicides

4 Conclusions and perspectives

Silvana Botti Thermoelectricity 2 / 15

Thermoelectric properties

Thermoelectric power generation

Seebeck effect: converts temperature differences in electric voltages

Seebeck coefficient S = ∆V∆T

Silvana Botti Thermoelectricity 3 / 15

Thermoelectric properties

Thermoelectric power generation

Seebeck effect: converts temperature differences in electric voltages

Seebeck coefficient S = ∆V∆T

Silvana Botti Thermoelectricity 3 / 15

Thermoelectric properties

Thermoelectric power generation

Seebeck effect: converts temperature differences in electric voltages

Efficiency (< 10 – 15%)

ηmax =Thot − Tcold

Thot

√1 + ZT − 1

√1 + ZT + TcoldThot

Figure of merit:

ZT̄ =(Sp − Sn)2T̄

[(ρpκp)1/2 + (ρnκn)1/2]2

Seebeck coefficient S = ∆V∆T

Large ZT̄ are needed for high efficiency

Silvana Botti Thermoelectricity 3 / 15

Thermoelectric properties

Electronic terms of the figure of merit

ObjectivesPredict accurate values for thermoelectric propertiesDeal with complexity (large cells, doping, nanostructuring, . . .)

⇒ Find efficient, cheap and environment-friendly new thermoelectrics!

Silvana Botti Thermoelectricity 4 / 15

Thermoelectric properties

Boltzmann theory of transport

Within linear response:ji = σijEj

The conductivity tensor is

σij =1

4π3∑

n

∫τn,kvi(n,k)vj(n,k)

(−∂f (n,k,T )

∂E(k)

)dk

in terms of the group velocity and the relaxation time

The group velocity is given by the dispersion of the bands

Constant relaxation time approximation: τS is independent on τσ is proportional to τ

Silvana Botti Thermoelectricity 5 / 15

Thermoelectric properties

Kohn-Sham band structure

Kohn-Sham (KS) equations[−∇

2

2+ vKS (r)

]ϕKSi (r) = ε

KSi ϕ

KSi (r)

ρ (r) =occ.∑

i

|ϕKSi (r) |2

It is common to interpret the solutions of the Kohn-Shamequations as one-electron statesOften one obtains good band dispersions but band gaps aresystematically underestimatedSometimes one gets bad band dispersions and band gaps!This happens, e.g., when there are localized d or f states

W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).Silvana Botti Thermoelectricity 6 / 15

Thermoelectric properties

Kohn-Sham band structure

Kohn-Sham (KS) equations[−∇

2

2+ vKS (r)

]ϕKSi (r) = ε

KSi ϕ

KSi (r)

ρ (r) =occ.∑

i

|ϕKSi (r) |2

It is common to interpret the solutions of the Kohn-Shamequations as one-electron statesOften one obtains good band dispersions but band gaps aresystematically underestimatedSometimes one gets bad band dispersions and band gaps!This happens, e.g., when there are localized d or f states

W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).Silvana Botti Thermoelectricity 6 / 15

Reliable band structures

State-of-the-art of theorycom

puta

tional cost

# atoms

2000

200

20

Standard DFT (LDA,GGAs): severeunderestimation of gaps, often good structuralproperties, bad localized (d and f ) statesDFT + model U: corrects localization of d states,usually better gaps, reliability depends too muchon the systemHybrid functionals: good gaps (unfortunatelyprecision not systematic), excellent structuralproperties, better localized statesGW: excellent band structures, excellent localizedstates, self-consistent screening, hard to accessto total energies

accuracy

Silvana Botti Thermoelectricity 7 / 15

Reliable band structures

State-of-the-art of theorycom

puta

tional cost

# atoms

2000

200

20

Standard DFT (LDA,GGAs): severeunderestimation of gaps, often good structuralproperties, bad localized (d and f ) statesDFT + model U: corrects localization of d states,usually better gaps, reliability depends too muchon the systemHybrid functionals: good gaps (unfortunatelyprecision not systematic), excellent structuralproperties, better localized statesGW: excellent band structures, excellent localizedstates, self-consistent screening, hard to accessto total energies

accuracy

Silvana Botti Thermoelectricity 7 / 15

Reliable band structures

State-of-the-art of theorycom

puta

tional cost

# atoms

2000

200

20

Standard DFT (LDA,GGAs): severeunderestimation of gaps, often good structuralproperties, bad localized (d and f ) statesDFT + model U: corrects localization of d states,usually better gaps, reliability depends too muchon the systemHybrid functionals: good gaps (unfortunatelyprecision not systematic), excellent structuralproperties, better localized statesGW: excellent band structures, excellent localizedstates, self-consistent screening, hard to accessto total energies

accuracy

Silvana Botti Thermoelectricity 7 / 15

Reliable band structures

State-of-the-art of theorycom

puta

tional cost

# atoms

2000

200

20

Standard DFT (LDA,GGAs): severeunderestimation of gaps, often good structuralproperties, bad localized (d and f ) statesDFT + model U: corrects localization of d states,usually better gaps, reliability depends too muchon the systemHybrid functionals: good gaps (unfortunatelyprecision not systematic), excellent structuralproperties, better localized statesGW: excellent band structures, excellent localizedstates, self-consistent screening, hard to accessto total energies

accuracy

Silvana Botti Thermoelectricity 7 / 15

Reliable band structures

Comparative test: kesterite Cu2ZnSnS4

T Γ N-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

Ene

rgy

(eV

)

T Γ N-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

self-consistent GW

standard DFT

hybrids

DFT+U

CBM

VBM

S3p-Cu3d

bonding

(Sn,Zn)-S

S 3s

S3p-(Zn,Sn)s

antibonding

SB, D. Kammerlander and M.A.L. Marques, APL 98, 241915 (2011)C. Sevik and T. Çağin, PRB 82, 045202 (2010)

Silvana Botti Thermoelectricity 8 / 15

Reliable band structures

Exploring new kesterites: Cu2ZnGe(S,Se)4

T Γ N-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

Ene

rgy

[eV

]

LDAscGW

Cu2ZnGeSe

2 kesterite

scGW gap (HSE geometry) = 1.44 eVscGW gap (GGA geometry) = 0.91 eV

T Γ N-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

Ene

rgy

[eV

]

LDAscGW

Cu2ZnGeS

2 kesterite

Silvana Botti Thermoelectricity 9 / 15

Reliable band structures

CuAlO2 delafossite

Γ F L Z Γ-2.0

0.0

2.0

4.0

6.0

8.0

Ene

rgy(

eV)

sc-GW

Γ F L Z Γ

HSE03

Γ F L Z Γ

LDA+U

Strong differences both in dispersion and energy gaps

J. Vidal, F. Trani, F. Bruneval, M.A.L. Marques, and S. Botti PRL 104, 136401 (2010)

Silvana Botti Thermoelectricity 10 / 15

Disilicides

Quality of the density of states

X-ray photoelectron spectroscopy (XPS) measured in BaSi2Calculated electronic density of states using GGA-PBE and thehybrid functional HSE06

J. Flores-Livas, Ph.D. thesis, University of Lyon 1 (2012).

Silvana Botti Thermoelectricity 11 / 15

Disilicides

Seebeck coefficient

GGA calculationsorthorhombicBaSi2 agrees withexp.S underestimatedfor SrSi2 (metallicin GGA)

Silvana Botti Thermoelectricity 12 / 15

Conclusions and perspectives

Conclusions and perspectives

In many cases the semi-classical theory of Boltzmann and therelaxation time approximation are reliable, provided that the banddispersions are good and the systems do not turn out metallic

Band-structure methods beyond ground-state DFT are well establishedand necessary for systems with d-states close to the Fermi energy

Hybrid functionals are often a reasonable compromise

Open problems:ab initio determination of the relaxation timesab initio determination of the thermal conductivity

Silvana Botti Thermoelectricity 13 / 15

Conclusions and perspectives

Conclusions and perspectives

In many cases the semi-classical theory of Boltzmann and therelaxation time approximation are reliable, provided that the banddispersions are good and the systems do not turn out metallic

Band-structure methods beyond ground-state DFT are well establishedand necessary for systems with d-states close to the Fermi energy

Hybrid functionals are often a reasonable compromise

Open problems:ab initio determination of the relaxation timesab initio determination of the thermal conductivity

Silvana Botti Thermoelectricity 13 / 15

Conclusions and perspectives

Conclusions and perspectives

In many cases the semi-classical theory of Boltzmann and therelaxation time approximation are reliable, provided that the banddispersions are good and the systems do not turn out metallic

Band-structure methods beyond ground-state DFT are well establishedand necessary for systems with d-states close to the Fermi energy

Hybrid functionals are often a reasonable compromise

Open problems:ab initio determination of the relaxation timesab initio determination of the thermal conductivity

Silvana Botti Thermoelectricity 13 / 15

Thanks!

Thanks to all collaborators at LPMCN

Miguel Marques, José Flores-Livas (theory)

Stéphane Pailhés, Régis Debord, Valentina Giordano(experiments)

More information on our group home page:http://www.tddft.org/bmg/

http://www.abinit.org http://www.yambo-code.org http://cms.mpi.univie.ac.at/vasp/

Silvana Botti Thermoelectricity 14 / 15

http://www.tddft.org/bmg/http://www.abinit.orghttp://www.yambo-code.orghttp://cms.mpi.univie.ac.at/vasp/

Calculations of thermoelectric propertiesHow to obtain reliable band structures?DisilicidesConclusions and perspectivesThanks!