electronic excitations in materials for solar cells - tddft. · pdf fileks green’s...

Electronic excitations in materials for solar cellsbeyond standard density functional theory

Silvana Botti

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

3European Theoretical Spectroscopy Facility

October 21, 2010 – LASIM, Lyon

Silvana Botti Electronic excitations & photovoltaics 1 / 34

Outline

1 Electronic excitations: beyond standard DFT?

2 Band structuresChalcopyrite absorbersCu-based delafossite transparent conductive oxides

3 Optical spectra

4 Conclusions and perspectives


Problems of standard DFT

Modeling electronic excitations in complex systems

ObjectivesPredict accurate values forfundamental opto-electronicalproperties (gap, absorptionspectra, excitons, . . .)

Simulate real materials(nanostructured systems,large unit cells, defects,doping, interfaces, . . .)

Find a compromise between accuracy and computational effort



Density functional theory

In literature standard computational approach for band structures:

Kohn-Sham (KS) equations[−∇

2

2+ vext (r) + vHartree (r) + vxc (r)

]ϕi (r) = εiϕi (r)

it is necessary to approximate vxc (r),Structural parameters and formation energies are usually good inLDA or GGAKohn-Sham energies are not meant to reproduce quasiparticleband structures: one often obtains good band dispersions butband gaps are systematically underestimatedand how to calculate optical absorption?

Hohenberg&Kohn, PR 136, B864 (1964); Kohn&Sham, 140, A1133 (1965)



Excitation energies: photoemission

Photoemission process:

hν − (Ekin + φ) = EN−1,v − EN,0 = −εv



Excitation energies: photoemission

Inverse photoemission process:

hν − (Ekin + φ) = EN,0 − EN+1,c = −εc



Excitation energies: energy gap

Photoemission gap:Egap = I − A = mink ,l

(EN−1,k + EN+1,l − 2EN,0

)Silvana Botti Electronic excitations & photovoltaics 6 / 34


Excitation energies: energy gap

Optical gap:Egap = I − A− Eexc

binding



A intuitive path: propagation of electrons and holes

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In the many-body Green’s function framework:GW for electron addition and removal(one-particle G)Bethe-Salpeter equation for the inclusion ofelectron-hole interaction (two-particle G)

The price to pay is a more involved theoretical andcomputational framework

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



Green’s function and Hedin’s equations

Propagation of an extra particle (electron or hole):

G(r1, r2, t1 − t2) = −i〈N|T [ψ(r1, t1)ψ†(r2, t2)]|N〉

Electron density:

ρ (r) = G(r , r , t , t+)

Spectral function:

A(ω) = 1/πTr {Im G(r1, r2, ω)}

G has poles at addition and removal energies



Green’s function and Hedin’s equations

Propagation of an extra particle (electron or hole):

G(r1, r2, t1 − t2) = −i〈N|T [ψ(r1, t1)ψ†(r2, t2)]|N〉

Σ

G

ΓP

W

G=G 0+G 0 Σ G

Γ=1+

(δΣ/

δG)G

GΓ

P = GGΓ

W = v + vPW

Σ = GWΓ

P = GG

Hedin’s equationsshould be solvedself-consistently



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)


Band structures Chalcopyrite absorbers


J. Vidal, S. Botti, P. Olsson, J.-F. Guillemoles, and L. Reining, “Strong interplaybetween structure and electronic properties in CuIn(S,Se)2: a first-principlesstudy ”, Phys. Rev. Lett. 104, 056401 (2010).

I. Aguilera, S. Botti, J. Vidal, P. Wahnon, and L. Reining, “Band structure andoptical absorption of CuGaS2: a self-consistent GW study”, to be submitted(2010).

J. Vidal, “Ab initio Calculations of the Electronic Properties of CuIn(S,Se)2 andother Materials for Photovoltaic Applications”, Ph.D. Thesis, Ecole Polytechnique,France (2010).



Present state of photovoltaic efficiency



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

Wurth Elektronik GmbH & Co.Efficiency = 13 %



Energy converted by a silicon solar cell

Carriers below the gap are not absorbed



CIGS properties

Cu(In,Ga)(S,Se)2 absorbers:

high optical absorption⇒ thin-layer filmsoptimal photovoltaic gap(record efficiency 20.1%)self-doping with native defects⇒ p-n junctionsextraordinary stability underoperating conditions: tolerance tolarge off-stoichiometries, stress,defects (not well understood) 1 1.2 1.4 1.6 1.8 2

Eg [eV]

8

10

12

14

16

18

20

η [%

]

CuGaSe2

CuInS2

CuInSe2

Cu(In,Ga)Se2



LDA energies 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

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www.abinit.org


Perturbative GW: “best G, best W”

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



G0W0 energies 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 )Hybrid functionals (e.g. Fuchs et al. 2007)

EPBE0xc = EPBE

xc +14(EHF

x − EPBEx)



Beyond Standard GW


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



Beyond Standard GW


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

Our choice is to get a better starting point for G0W0 using scCOHSEX



COHSEX: approximation to GW self-energy

Statically screened exchange:

ΣSEX(r1, r2) = −∑

i

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

Induced classical potential due to an extra point charge:

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

L. Hedin and S. Lundqvist, Solid State Phys. 23, 1 (1969);

Bruneval et al. PRL 97, 267601 (2006); Gatti et al. PRL 99, 266402 (2007)



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

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www.abinit.org


Band structures of CuGaSe2

T Γ N-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

Ene

rgy

(eV

)

LDAscCOHSEX+G

0W

0

Se 4s

Ga-Se bond

Cu 3d - Se 4p

Comparison of LDA (red) andscGW (blue) bands of CuGaSe2

Analogous results for CuGaS2

No need to include the delectrons of Ga in the valenceAgreement with experimentalgap within less than 0.1 eV



Stability of the gap in In compounds

Anion displacement: u = 1/4 +(

R2Cu−(S,Se) − R2

In−(S,Se)

)/a2 6= 1/4.

1. Experiments measure the same value for the gap (within 10%)

0.22 0.23 0.24u

5.6

5.7

5.8

5.9

6

a [Å

]

GGA

LDA

PBE0HSE06

B3LYP

HF+cHSE03

0 0.2 0.4 0.6 0.8 1b

0.22

0.23

0.24

u

2. The experimental dispersion ofu is, however, large

Only hybrid functionals givestructural parameters whichoverlap with experiments

Jaffe, PRB 29, 1882 (1984); Merino, J. Appl. Phys. 80, 5610 (1996)Jaffe, PRB 27, 5176 (1983); Jiang, Sem. Sci. Technol. 23, 025001 (2008)






In−(S,Se)

)/a2 6= 1/4.

0.2 0.21 0.22 0.23 0.24 0.25u

0

1

2

3

Eg [

eV]

CuInS2

DFT-LDA

Strong variations in DFT-LDA(in agreement with literature)

www.abinit.org


www.abinit.org





In−(S,Se)

)/a2 6= 1/4.

0.2 0.21 0.22 0.23 0.24 0.25u

0

1

2

3

Eg [

eV]

CuInS2

DFT-LDAG

0W

0

G0W0 does not change theslope . . .

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www.abinit.org





In−(S,Se)

)/a2 6= 1/4.

0.2 0.21 0.22 0.23 0.24 0.25u

0

1

2

3

Eg [

eV]

CuInS2

DFT-LDAG

0W

0

. . . except if the gap is alreadyopen

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www.abinit.org





In−(S,Se)

)/a2 6= 1/4.

0.2 0.21 0.22 0.23 0.24 0.25u

0

1

2

3

Eg [

eV]

CuInS2

DFT-LDAG

0W

0scGW

sc-GW enhances the gapvariation

www.abinit.org


www.abinit.org





In−(S,Se)

)/a2 6= 1/4.

0.2 0.21 0.22 0.23 0.24 0.25u

0

1

2

3

Eg [

eV]

CuInS2

DFT-LDAG

0W

0scGWHSE06

HSE06 hybrid gives anintermediate slope

EHSE06xc = EGGA

xc +14

“EHF,sr

x − EGGA,srx

”

http://cms.mpi.univie.ac.at/vasp/







In−(S,Se)

)/a2 6= 1/4.

0.2 0.21 0.22 0.23 0.24 0.25u

0

1

2

3

Eg [

eV]

CuInS2

DFT-LDAG

0W

0scGWHSE06HSE06 ε

0

a modified-HSE06 (the mixingparameter of the screenedFock exchange is proportionalto the screening) gives thesc-GW slope








In−(S,Se)

)/a2 6= 1/4.

Is the gap stable under lattice distortion?

sc-GW and hybrid calculations predict even stronger variationsthan LDAThe gap is not stable under lattice distortion alone



Formation energy of Cu vacancies

The formation energy of VCu varies under lattice distortion:

∆Ef = ∆EDFTf −∆Esc−GW

VBM



Formation energy of Cu vacancies

The formation energy of VCu varies under lattice distortion:

∆Ef = ∆EDFTf −∆Esc−GW

VBM

conduction band minimum (CBM)

valence band maximum (VBM)

It is essential to go beyond DFT-LDALDA+U (blue lines) gives only constant shifts

Zhang et al. PRB 57, 9642 (1998); Lany et al. PRB 78, 235104 (2008).



Why is the experimental gap so stable?

A feedback loop can explain the stability of the band gap:

∆u ∆Eg

∆Hf(V

Cu) ∆[V

Cu]

∆Eg





∆u ∆Eg

∆Hf(V

Cu) ∆[V

Cu]

∆Eg

-3 -2.5 -2 -1.5 -1ln([V

Cu]/N

Cu)

1

1.1

1.2

1.3

1.4

Eg [

eV]

Eg=1.6663+0.231*ln([V

Cu]/N

Cu)

0.2 0.21 0.22 0.23 0.24 0.25u

0

0.5

1

1.5

2

2.5

3

Eg [

eV]

Eg=-5.975+32.1*u

0.2 0.21 0.22 0.23 0.24 0.25u

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

∆VB

M [

eV] ∆E

v

scGW= 2.1335-11.06*u





∆u ∆Eg

∆Hf(V

Cu) ∆[V

Cu]

∆Eg

Experimental variation of u is 0.02⇒ ∆Eg ≈ 0.65 eVConsidering variations of u and [VCu]⇒ ∆Eg ≈ -0.04 eV


Band structures Cu delafossites TCOs


J. Vidal, F. Trani, F. Bruneval, M. A. L. Marques and S. Botti, “Effects ofpolarization on the band-structure of delafossite transparent conductive oxides”,Phys. Rev. Lett. 104, 136401 (2010).

F. Trani, J. Vidal, S. Botti and M. A. L. Marques, “Band structures of delafossitetransparent conductive oxides from a self-consistent GW approach ”, Phys. Rev.B 82, 085115 (2010).



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



The long dispute about delafossite gaps

LDA LDA+U B3LYP HSE03 HSE06 G0W

0scGW scGW+P0

1

2

3

4

5

6

Eg [

eV]

Eg

indirect

Eg

direct

∆=Eg

direct-E

g

indirect

exp. direct gap

exp. indirectgap

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)]



Comparison with hybrid functional calculations

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 gapsAre hybrids a good compromise?


Optical spectra

Bethe-Salpeter equation: electron-hole interaction

The BSE for the reducible 4-point polarizability L:

L = L0 + L0

(4v −4 W

)L

4v(1, 2, 3, 4) = δ(1, 2)δ(3, 4)v(1, 3) and 4W = δ(1, 3)δ(2, 4)W (1, 2)

The measurable χ is obtained via a two-point contraction of L

χred(1,2) = −L(1,1,2,2)

In transition space and using the only-resonant approximation:

H2p,exc(vc)(v ′c′)A

v ′c′

λ = Eexcλ Av ′c′

λ

The ingredients are:Kohn-Sham wavefunctions and energiesGW corrected energiesscreening matrix ε−1

GG′(q)

Salpeter and Bethe, Phys. Rev. 84, 1232 (1951)Silvana Botti Electronic excitations & photovoltaics 29 / 34

Optical spectra

Optical absorption of CuGaS2

Excitonic binding energy of about 0.05 eVExperimental optical gap at 2.5 eV

2 3 4 5Energy (eV)

0

5

10

15

20O

ptic

al a

bsor

ptio

n

scGW gap

⊥||

Exp. 1

Exp. 2

BSE calculations

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Optical spectra

Optical absorption of CuInO2

Strong excitonic effects also for the In compoundExp. absorption edge at 3.9 eVPolaronic effects should also be strong

0

5

10

15

20

0

5

10

15

20

ε 2

0 1 2 3 4 5 6 7 8Energy (eV)

0

5

10

15

20

RPA (NLF)

scGW(NLF)

scGW+BSE

solid lines: xy component, dashed: z component

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www.yambo-code.org

Conclusions and perspectives


Interpretation of experiments is often not straightforward – manycoupled effects!

Methods that go beyond ground-state DFT are by now well established

A better starting point than LDA is absolutely necessary for d-electronsSelf-consistent COHSEX+G0W0 gives a very good description ofquasi-particle statesHybrid functionals can be a good compromiseLDA+U does not work when there is hybridization of p − d states

In progress now:Absorption spectra from the Bethe-Salpeter equation for allcompoundsDefects using VASP and “improved” hybrid functionals



Building-Integrated Photovoltaics

Projects realised so far do not yet represent the full range ofproducts available on the market: these include integratablecrystalline modules, thin-layer modules, transparent and shadingmodules, solar roof tiles, photovoltaic roof foils or complete solarroofsPlaying with light and glass: building-integrated photovoltaicsopens new creative opportunities for architects


Thanks!

Thanks to all collaborators! Thank you!

LSI – Ecole Polytechnique

Lucia Reining

Julien Vidal

Irene Aguilera

IRDEP – Paris

J.-F. Guillemoles

Par Olsson

LPMCN – Universite Lyon 1

Miguel Marques

Fabio Trani

Guilherme Vilhena

David Kammerlander

CEA – Saclay

Fabien Bruneval

http://www.etsf.euhttp://www.abinit.orgwww.yambo-code.org


http://www.etsf.eu

http://www.abinit.org

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