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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 2 / 33
Outline
1 Photovoltaic materials
2 Beyond Standard DFT
3 Beyond Standard GW
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 3 / 33
Photovoltaic materials
Present state of photovoltaic efficiency
from National Renewable Energy Laboratory (USA)
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 4 / 33
Photovoltaic materials
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.
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 5 / 33
Photovoltaic materials
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 6 / 33
Photovoltaic materials
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 7 / 33
Photovoltaic materials
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)
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 8 / 33
Photovoltaic materials
Modeling photovoltaic materials
ObjectivesPredict accurate values forfundamental opto-electronicalproperties of materials
Deal with complex materials (largeunit cells, defects)
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 9 / 33
Photovoltaic materials
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)
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 10 / 33
experimental gap (eV)
Photovoltaic materials
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 11 / 33
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 12 / 33
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 13 / 33
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).
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 14 / 33
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)
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 15 / 33
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)
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 16 / 33
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
van Schilfgaarde, Kotani, and Faleev, PRL 96 (2006)
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 17 / 33
experimental gap (eV)
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 18 / 33
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)
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 19 / 33
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)
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 19 / 33
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 20 / 33
Beyond Standard GW
Self-consistent GW a la Faleev
Make self-energy Hermitian and static
〈k i |Σ|k j〉 =14
(〈k i |Σ(εk j)|k j〉+ 〈k j |Σ(εk j)|k i〉∗
+〈k i |Σ(εk i)|k j〉+ 〈k j |Σ(εk i)|k i〉∗)
|k i〉 and εk i are self-consistent eigensolutions of the iterativeprocedureRepresentation of WFs on a restricted LDA basis setRequires sums over empty states
Faleev, van Schilfgaarde, and Kotani, PRL 93 2004
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 21 / 33
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!
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 22 / 33
Beyond Standard GW
Energy gap within sc GW
0 2 4 6 8
0
2
4
6
8
experimental gap (eV)
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
van Schilfgaarde, Kotani, and Faleev, PRL 96 (2006)
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 23 / 33
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 24 / 33
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 25 / 33
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)
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 26 / 33
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 27 / 33
Beyond Standard GW
K-point dependence of the correction
Corrections are independent of the k-point (within 0.1 eV)→ Effective masses remain unchanged
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 28 / 33
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 29 / 33
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 30 / 33
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 31 / 33
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
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 32 / 33
Beyond Standard GW
Thanks!!!
http://www.etsf.euhttp://etsf.polytechnique.fr
Silvana Botti (ETSF) Self-consistent GW for CIS TDDFT Workshop 33 / 33
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