Photoelectronic properties of chalcopyrites for photovoltaic · PDF filePhotoelectronic...

Photoelectronic properties of chalcopyrites forphotovoltaic conversion:self-consistent GW calculations

Silvana Botti

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

3European Theoretical Spectroscopy Facility

September 13, 2008 – 3rd International Workshop “Time dependentDensity-Functional Theory: Prospects and Applications”

Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 1 / 33

Collaborators

Julien Vidal Lun Mei HuangMatteo Gatti Par OlssonLucia Reining Christophe Domain

Jean-Francois Guillemoles

LSICNRS-Ecole Polytechnique

IRDEPCNRS-EDF


Outline

1 Photovoltaic materials

2 Beyond Standard DFT

3 Beyond Standard GW


Photovoltaic materials

Present state of photovoltaic efficiency

from National Renewable Energy Laboratory (USA)



CIS solar cell

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

Structure:

Molybdenum back contactCIS layer (p-type layer)CdS layer (n-type layer)ZnO:Al transparent contact

Efficiency = 13 %Wurth Elektronik GmbH & Co.



CIS properties

CuIn(S,Se)2 are among the best photovoltaic absorbermaterials:

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



Density functional theory

DFT in its standard form is a ground state theory

Structural parameters: lattice parameters, internal distortions areOK, even 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



State of the art for CuIn(S,Se)2

First ab initio calculation for chalcopyriteJaffe et al., PRB 28,10 (1983)

Formation energies of intrinsic defects and defect levels in the gapZhang et al., PRB 57, 9642 (1998)

Correction of the bandgap DFT+U ∆Ev=- 0.37 eVLany et al., PRB 72 035215 (2005)

Metastability caused by the vacancy complex VSe-VCuLany et al., JAP, 100,113725 (2006)



Modeling photovoltaic materials

ObjectivesPredict accurate values forfundamental opto-electronicalproperties of materials

Deal with complex materials (largeunit cells, defects)



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


Beyond Standard DFT

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


Beyond Standard DFT

Solution

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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


Beyond Standard DFT

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).


Beyond Standard DFT

Self-energy and screened interaction

Self-energy: nonlocal, non-Hermitian, frequency dependent operatorIt allows to obtain the Green’s function G once that G0 is known

Hartree-Fock Σx(r1, r2) = iG(r1, r2, t , t+)v(r1, r2)

GW Σ(r1, r2, t1 − t2) = iG(r1, r2, t1 − t2)W (r1, r2, t2 − t1)

W = ε−1v : screened potential (much weaker than v !)

Ingredients:KS Green’s function G0, and RPA dielectric matrix ε−1

G,G′(q, ω)

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


Beyond Standard DFT

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

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


Beyond Standard DFT

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




Beyond Standard DFT

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


Beyond Standard GW

Beyond Standard GW

Looking for another starting point:DFT with another approximation for vxc : GGA, EXX,...(e.g. Rinke et al. 2005)Semi-empirical hybrid functionals (e.g. Fuchs et al. 2007)

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


Beyond Standard GW

Self-consistent COHSEX

Coulomb hole:

ΣCOH(r1, r2) =12δ(r1 − r2)[W (r1, r2, ω = 0)− v(r1, r2)]

Screened Exchange:

ΣSEX(r1, r2) = −∑

i

θ(µ− Ei)φi(r1)φ∗i (r2)W (r1, r2, ω = 0)

The COHSEX self-energy is static and HermitianSelf-consistency can be done either on energies alone or on bothenergies and wavefunctionsRepresentation of WFs on a restricted LDA basis set


Beyond Standard GW

Self-consistent COHSEX

Advantages of COHSEX:Old approximation physically motivated: accounts forCoulomb-hole and screened-exchangeComputationally “inexpensive”: static; only sums over occupiedstatessc-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!


Beyond Standard GW

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



Beyond Standard GW

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


Beyond Standard GW

Results for CIS

Self-consistency in the energies suffices to correct the gap

Self-consistency in the wave-functions is necessary to correctdeeper states

Spin-orbit coupling is important for Se compound→ Can be added perturbatively within DFT→ Experiment: 0.2 eV Calculation: 0.16 eV


Beyond Standard GW

Example: Insulating phase of Vanadium Oxide

LDA valence WF

metal

sc-GW: variation of WF

Eg=0.65 eV

The variation of the wave-functions is essential to open up the gap(metal both in LDA and G0W0!)

Gatti et al., PRL 99 (2007)


Beyond Standard GW

Quasi-particle corrections for CuInS2

-4

-2

0

EG

W -

ED

FT (

eV)

-15 -10 -5 0 5E

DFT (eV)

T

Γ

N

In-S bondS 3s

In 4d

Bandgap regionConduction bands

Cu 3d S 3p

Cu 3d S 3p

Corrections depend on the character of the band→ Models for quasi-particle corrections for defects


Beyond Standard GW

K-point dependence of the correction

Corrections are independent of the k-point (within 0.1 eV)→ Effective masses remain unchanged


Beyond Standard GW

Quasi-particle corrections for CuInSe2

-6

-4

-2

0

2

EG

W-E

DFT

[eV

]

-16 -12 -8 -4 0 4 8KS energies [eV]

DO

S [a

.u.]

blue: G0W0

black: sc-COHSEXred: sc-COHSEX + G0W0


Beyond Standard GW

Quasi-particle corrections for CuInS2

-6

-4

-2

0

2

EG

W-E

LD

A [

eV]

-16 -12 -8 -4 0 4 8KS Energies [eV]

DO

S [a

.u]

blue: G0W0

black: sc-COHSEXred: sc-COHSEX + G0W0


Beyond Standard GW

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)

Gap in the valence disappears in the presence of defects


Beyond Standard GW

Conclusions and perspectives

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

Self-consistency is absolutely necessary for d-electronsSelf-consistent GW gives a very good description of quasi-particlestatesSelf-consistent COHSEX good starting point for one-shot GW→ 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


Beyond Standard GW

Thanks!!!

http://www.etsf.euhttp://etsf.polytechnique.fr


http://www.etsf.eu

http://etsf.polytechnique.fr

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