Electronic properties of materials for solar cells: - Which ab initio

Electronic properties of materials for solar cells:Which ab initio approaches can we trust?

Silvana Botti

1LSI, CNRS-CEA-Ecole Polytechnique, Palaiseau, France2LPMCN, CNRS-Universite Lyon 1, France

3European Theoretical Spectroscopy Facility

June 19, 2009 – Donostia, “DIPC Seminars”

Silvana Botti Electronic excitations in solar cells 1 / 35

Collaborators

Ecole Polytechnique Universite Lyon 1Julien Vidal, Lucia Reining Fabio Trani, Miguel Marques

EDF Paris CEA SaclayPar Olsson, J.-F. Guillemoles Fabien Bruneval


Outline

1 Thin-film photovoltaic materials

2 Why do we need to go beyond standard DFT?

3 How to compare with experiments?


Thin-film photovoltaic materials

Outline






Present state of photovoltaic efficiency

from National Renewable Energy Laboratory (USA)



CIGS solar cell

Devices have to fulfill 2 functions:Photogeneration of electron-hole pairsSeparation of charge carriers to generate a current

Structure:

Molybdenum back contactCIGS layer (p-type layer)CdS layer (n-type layer)ZnO:Al TCO contact

Efficiency = 13 %Wurth Elektronik GmbH & Co.



CIGS properties

Cu(In,Ga)(S,Se)2 are among the best absorbers:

high optical absorption⇒ thin-layer filmsoptimal photovoltaic gap (record efficiency 19.9 %)self-doping with native defects⇒ p-n junctionselectrical tolerance to large off-stoichiometries:not yet understoodbenign character of defects:not yet understood



Delafossite TCO properties

Cu(Al,In,Ga)O2 thin-films are transparent and conducting:

p-type or even bipolar conductivitycombination of n- and p-type TCO materials allows

→ stacked cells with increased efficiency→ functional windows→ transparent transistors



Modeling photovoltaic materials

ObjectivesPredict accurate values forfundamental opto-electronicalproperties of materials

Deal with complex materials(large unit cells, defects)


Why do we need to go beyond standard DFT?

Outline






Density functional theory

DFT in its standard form is a ground state theory

Structural parameters: lattice parameters, internal distortions areusually ok in LDA or GGAFormation energies for defects calculated from total energies arereliableKohn-Sham energies are not meant to reproduce quasiparticleband structures: often one obtains good band dispersions butband gaps are systematically underestimatedKohn-Sham DOS is not meant to reproduce photoemission



LDA Kohn-Sham energy gaps

0

2

4

6

8

calc

ula

ted g

ap (

eV

)

:LDA

HgT

e

InS

b,P

,InA

sIn

N,G

e,G

aS

b,C

dO

Si

InP

,GaA

s,C

dT

e,A

lSb

Se,C

u2O

AlA

s,G

aP

,SiC

,AlP

,CdS

ZnS

e,C

uB

r

ZnO

,GaN

,ZnS

dia

mond

SrO A

lN

MgO

CaO

van Schilfgaarde, Kotani, and Faleev, PRL 96 (2006)


experimental gap (eV)


LDA Kohn-Sham energy gaps for CIS

CuInS2DFT-LDA exp.

Eg -0.11 1.54In-S 6.5 6.9

S s band 12.4 12.0In 4 d band 14.6 18.2

CuInSe2DFT-LDA exp.

Eg -0.29 1.05In-Se 5.8 6.5

Se s band 12.6 13.0In 4 d band 14.7 18.0

www.abinit.org


www.abinit.org


Beyond standard DFT

For photovoltaic applications we areinterested in evaluating

quasiparticle band gapoptical band gapdefect energy levelsoptical absorption spectra

All these quantities require going beyond standard DFT



Solution

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Π0

!

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In the many-body framework, we know how to solvethese problems:

GW for quasi-particle propertiesBethe-Salpeter equation for the inclusion ofelectron-hole interaction

The first step can be substantially more complicatedthan the second, so in the following we will focus onGW



Hedin’s equations

Σ

G

ΓP

W

G=G 0+G 0 Σ G

Γ=1+

(δΣ/

δG)G

GΓ

P = GGΓ

W = v + vPW

Σ = GWΓ

L. Hedin, Phys. Rev. 139 (1965).



Standard one-shot GW

Kohn-Sham equation:

H0(r)ϕKS (r) + vxc (r)ϕKS (r) = εKSϕKS (r)

Quasiparticle equation:

H0(r)φQP (r) +

∫dr ′Σ

(r , r ′, ω = EQP

)φQP

(r ′) = EQPφQP (r)

Quasiparticle energies 1st order perturbative correction with Σ = iGW :

EQP − εKS = 〈ϕKS|Σ− vxc|ϕKS〉

Basic assumption: φQP ' ϕKS

L. Hedin, Phys. Rev. 139 (1965)

Hybersten and Louie, PRB 34 (1986); Godby, Schluter and Sham, PRB 37 (1988)



Energy gap within standard one-shot GW

0

2

4

6

8

calc

ula

ted g

ap (

eV

)

:LDA

:GW(LDA)

HgT

e

InS

b,P

,InA

sIn

N,G

e,G

aS

b,C

dO

Si

InP

,GaA

s,C

dT

e,A

lSb

Se,C

u2O

AlA

s,G

aP

,SiC

,AlP

,CdS

ZnS

e,C

uB

r

ZnO

,GaN

,ZnS

dia

mond

SrO A

lN

MgO

CaO





Quasiparticle energies within G0W0 for CIS

CuInS2DFT-LDA G0W0 exp.

Eg -0.11 0.28 1.54In-S 6.5 6.9 6.9

S s band 12.4 13.0 12.0In 4 d band 14.6 16.4 18.2

CuInSe2DFT-LDA G0W0 exp.

Eg -0.29 0.25 1.05In-Se 5.8 6.15 6.5

Se s band 12.6 12.9 13.0In 4 d band 14.7 16.2 18.0

www.abinit.org


www.abinit.org


Beyond Standard GW

Looking for another starting point:DFT with another approximation for vxc : GGA, EXX,...(e.g. Rinke et al. 2005)LDA/GGA + U (e.g. Kioupakis et al. 2008, Jiang et al. 2009 )Semi-empirical hybrid functionals (e.g. Fuchs et al. 2007)

Self-consistent approaches:GWscQP scheme (Faleev et al. 2004)scCOHSEX scheme (Hedin 1965, Bruneval et al. 2005)



Beyond Standard GW

Looking for another starting point:DFT with another approximation for vxc : GGA, EXX,...(e.g. Rinke et al. 2005)LDA/GGA + U (e.g. Kioupakis et al. 2008, Jiang et al. 2009 )Semi-empirical hybrid functionals (e.g. Fuchs et al. 2007)

Self-consistent approaches:GWscQP scheme (Faleev et al. 2004)scCOHSEX scheme (Hedin 1965, Bruneval et al. 2005)



Self-consistent COHSEX

Advantages of COHSEX:Old approximation physically motivated: accounts forCoulomb-hole and screened-exchangeComputationally “inexpensive”: hermitian, static (only sums overoccupied states)sc-COHSEX wave-functions very similar to sc-GW

Disadvantages of COHSEX:Dynamical correlations are missingQuasiparticle gaps are better (10-20% higher than experiment),but still not OK

One-shot GW on top of sc-COHSEX corrects the energy gap!



Energy gap within sc GW

0 2 4 6 8

0

2

4

6

8


QP

scG

W g

ap

(e

V)

MgO

AlN

CaO

HgT

e InS

b,I

nA

sIn

N,G

aS

b

InP

,Ga

As,C

dT

eC

u2

O Zn

Te

,Cd

SZ

nS

e,C

uB

rZ

nO

,Ga

NZ

nS

P,Te

SiGe,CdO

AlSb,SeAlAs,GaP,SiC,AlP

SrOdiamond




Quasiparticle energies within sc-GW for CIS

CuInS2DFT-LDA G0W0 sc-GW exp.

Eg -0.11 0.28 1.48 1.54In-S 6.5 6.9 7.0 6.9

S s band 12.4 13.0 13.6 12.0In 4 d band 14.6 16.4 18.2 18.2

CuInSe2DFT-LDA G0W0 sc-GW exp.

Eg -0.29 0.25 1.14 1.05 (+0.2)In-Se 5.8 6.15 6.64 6.5

Se s band 12.6 12.9 13.6 13.0In 4 d band 14.7 16.2 17.8 18.0

sc-GW is here sc-COHSEX+G0W0

www.abinit.org


www.abinit.org

How to compare with experiments?

Outline






Is the gap stable under lattice distortion?

0.2 0.22 0.24u

0

1

2

Eg [

eV]

CuInS2

0.2 0.22 0.24u

0

1

2CuInSe

2

Expt

Expt

The gap is not stable!Self-consistency enhances thegap variationsPrevious corrected-LDA (dots)results have LDA slopesc-COHSEX only in energiesis enough for the gap

DFT-LDA, G0W0, sc-COHSEX, sc-COHSEX+G0W0

www.abinit.org


www.abinit.org


Shifts of the band edges under lattice distortion

Note: Zhang et al. showed that an upward (downward) shift of theVBM favors (inhibits) the formation of VCu

conduction band minimum (CBM)

valence band maximum (VBM)

LDA+U (blue lines) gives only constant shiftsself-consistency in energies (dashed) is not enough

Zhang et al. PRB 57, 9642 (1998)



Cu-Se bond under lattice distortion

Contribution of the VBM to |ρCOHSEX − ρLDA|

u = 0.2 u = 0.215 u = 0.235 u = 0.25

Small u: the Cu-Se bond is weakenedLarge u: the Cu-Se bond is strenghtened

Once again VCu formation is favored at small u



Defects

a) Perfect crystal b) VCu c) VSe d) 2VCuInCu

0

500

1000

0

500

1000

DO

S

0

500

1000

-7 -6 -5 -4 -3 -2 -1 0 1 2Energy (eV)

0

500

1000

(a)

(b)

(c)

(d)

The presence of VCu opens up the gap!



Why the experimental gap is stable

0.2 0.22 0.24u

0

1

2

Eg [

eV]

CuInS2

0.2 0.22 0.24u

0

1

2CuInSe

2

Expt

Expt

In conclusion:In experiments where u is smaller than in the bulk and the gap isstable we can expect the concentration of VCu to be higher andcompensate for the decrease of the gap



The long dispute about delafossite gaps

The most studied compound is CuAlO2:

LDA LDA+U B3LYP HSE03 HSE06 G0W

0scGW

0

1

2

3

4

5

6

Eg [

eV]

Eg

indirect

Eg

direct

∆=Eg

direct-E

g

indirect Indirect gapMinimum directgap at L: dipoleallowedExperimental datafar from sc-GWcalculations!

Is sc-GW wrong in this case?



The long dispute about delafossite gaps

LDA LDA+U B3LYP HSE03 HSE06 G0W

0scGW

0

1

2

3

4

5

6

Eg [

eV]

Eg

indirect

Eg

direct

∆=Eg

direct-E

g

indirect

Experimental dataare for optical gap:exciton bindingenergy ≈ 0.5 eV[Laskowski et al. PRB 79,

165209 (2009)]

Strong latticepolaron effects areexpected ≈ 1 eV[Bechstedt et al. PRB 72,

245114 (2005)]



Bands of CuAlO2 from LDA+U

Γ F L Z Γ

-2

0

2

4

6

8

Ene

rgy(

eV)

2.91

2.892.60

1.97

LDA+U direct gap close toexperimentCB are rigidly shifted



Bands of CuAlO2 from sc-GW calculations

Γ F L Z Γ

-2

0

2

4

6

8

Ene

rgy(

eV)

2.91

2.892.60

1.97

5.01

5.05

7.16

5.41

GW corrections stronglyk-dependentCBM moves from Γ to Lgap becomes quasi-directdirect gap 1.5 eV larger thanexperiment



Conclusions and perspectives

Interpretation of experiments is often not straightforward

Methods that go beyond ground-state DFT are by now well establishedGW and BSE

A better starting point is absolutely necessary for d-electronsSelf-consistent COHSEX+G0W0 gives a very good description ofquasi-particle states→ In all cases we studied this proved to be at the level of scGW→ Much more friendly from the computational point of view

In progressDefectsAbsorption spectra from the Bethe-Salpeter equation


Thanks!

Thanks!

J. Vidal, S. Botti, P. Olsson, J.-F. Guillemoles and L.Reining, “Strong interplay between structure and electronicproperties in CuIn(S,Se)2: a first-principle study”,submitted.

J. Vidal, F. Trani, F. Bruneval, M. A. L. Marques and S.Botti, “Accurate band structure calculations of delafossitetransparent conductive oxides”, in preparation.

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http://www.abinit.org


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http://etsf.polytechnique.fr

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Electronic properties of materials for solar cells: - Which ab initio