gw approximation: one- and two-particle excitations in...
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GW approximation: One- and two-particle excitations
in solids and molecules
Friedhelm BechstedtFriedrich-Schiller-Universität JenaGermany
Goaltheoretical / computational approach to calculate optical and other excitation spectra including (i) quasiparticle effects due to excitation
(ii) electron-hole atraction & exchange
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c
v
GW
DFT-LDA
c
v
Eg
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applicable tosystems of arbitrary dimension
0D3D
O
N 6
C 5
N 7
N 4
N 10
C 9
C 3
C 2
C 1
N 8
Guanine
Jena: e.g. organic functionalization
Si bulk
2DIn/Si(111)4x1
Spectra
Absorption Reflectance Anisotropy (RA)
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)(RR
ω∆)(Im
)(cn)( ω∈
ωω
=ωαd)(e~ ωα−
System of fundamental equationsHedin/Lundqvist (69): Many-body perturbation theory (MBPT)
dielectric function (length gauge)
• polarization function
• one-electron Green function ⇒ Dyson equation
• XC self-energy
• screened Coulomb potential
• vertex function ⇒ Bethe-Salpeter equation
vP1−∈=
Γ= GGP
RPA (independent particle) vertex corrections (excitons)
( ) 1GGHt/ H1 =Σ−−∂∂
Γ=Σ GW
vPWvW +=
ΓδΣδ
+=Γ GGG
1
Outline: Unified treatment of solids and molecules
• GW: reformulation
• single-particle excitations
• two-particle excitations
• summary
Single-particle problem
Quasiparticle QP approachone-electron Green function ⇒ Dyson equation XC self-energy
screened Coulomb potential
GW approximation
Standard treatment First-order perturbation theory(start wave functions = KS wave functions)diagonal self-energy only energy shifts
Update of wave Wrong energetical ordering, band crossingsfunctions → off-diagonal elements
( ) 1GGHt/ H1 =Σ−−∂∂ Γ=Σ GW
( ) ν−εΣν=∆ νν XCQP V
Γ = 1, W = WRPA
Computational method
• start: DFT-LDA, Kohn-Sham equationnc pseudopotentials, supercell approach (Bloch picture) ⇒ also for confined systems
two exceptions: all-electron PAW wave functions ⇒ futurenc pp + plane waves
• real-space multigrid methodE.L. Briggs, D.J. Sullivan, J. Bernholc, PRB 14362 (1996)
- Basis functions: no plane waves instead: 3D mesh in real space GaP: 0.238 Å (= 24 Ry)
• advantage (i): massive parallelization(ii): matrix elements
- ⟨ ck | grad | vk ⟩ → 6-point method- real-space cutoff for surface optical properties
• screened potential: model or Ehrenreich-Cohen formula for matrix
Special treatments
• Blomberg-Bergersen method for G in GWRPA
⇒ correct positions of first satellites (no plasmaron!)
• Avoid sum over intermediate states
∑ν ν
νν
ε−ϕϕ
= QP
*
z)'()()z;',(G
h
xxxx
( )[ ] ( )
( ) ⎥⎦
⎤⎢⎣
⎡ω
−ωε−ε
++ω++∈πω
+
⎥⎦
⎤⎢⎣
⎡++−−δ−++∈+
δ++⎩⎨⎧−
++π
=εΣ
νµν
ννν
−∞
µνν
νννµ−
µνν
νννµ
∑∫
∑
∑∑
hmhm
h 11)'(B)(B;',d
)'(B)(B'B210;',
)'(B)(B'
e4V1)(
QP'
*'
''
1
0
*'
occ
'''GG
1
'*
'
occ
''
',,
2
GqGqGqGq
GqGqGGGqGq
GqGqGqGq GG
GGq
a
EX
COHSEX-EX
dyn
main problem
Wrong ordering of bands:Example: 2H-InN
• too shallow In4d electrons• overestimation of pd repulsion• “negative gap“ (Γ6v, Γ1v, Γ1c)
• In4d frozen in the core
→ automatically gap opening
Ene
rgy
(eV
)
Γ K H A Γ M L A-8
-4
0
4
8
Ene
rgy
(eV
)
23
23
33
1,2
13
1,3
3
5
165,6
1
1,5
3
65,6 3
31
1,3
1,3
41,3
3
22,4
1
1,33
1
Kohn-Sham
KS + Quasiparticle
Wrong ordering of bands:Example: C(111)2×1 π-bonded chain model without
buckling and dimerizationKohn-Sham Quasiparticle
⇐ diagonal GW fails (more or less equal shifts)
⇐ non-perturbation GW fails(matrix elements small)
M. Marsili et al., PRL (submitted)
⇒ self-consistent GW with respect to occupation(updating the quasiparticle energies including their occupation till self-consistency)⇒ gap of 0.8 eV (≈1.0 eV: onset of EELS)
Diagonal quasiparticle approach:Example: Silan SiH4 (Td)
Computation:dSi-H = 1.477 ÅLsupercell = 11.9 Å48 grid pointsdiel. Matrix: 4v, 256c
|G| = 90eV^
Comparison: Rohlfing/Louie, PRB 62, 4927 (2000)
exp.: Itoh, J. Chem. Phys. 85, 4867 (1986)Quasiparticle energies (eV):
LDA HF GW *) GW **) Exp.
HOMOLUMOQP gap
-8.42 (-8.42)-0.50 (0.57)7.93 (7.85)
12.77 (12.98)1.13 (1.49)13.90 (14.47)
-11.680.7212.40
-12.56 (-12.69)0.50 (1.10)13.06 (13.79)
-12.6
8.8
*) unclever treatment of intermediate states:1028 states shifted by 0.7eV**) use of completeness in static part
increase supercell: further shift by 0.1eV
HFHOMOε
Problem ⇔ Off-diagonal elements of δΣ- in contrast to LDA, LUMO above Evac in QP (diag)
- KS wave function too localized- indeed mixing of LUMO with LUMO + 4 of same symmetry + drastic lowering of its energy by 0.63eV
however: depends on description of “continuum“ above Evac(supercell size, number of states)
preliminarysolution: only 8 bands
Energy0 = E vac
LUMO
HOMO
KS QP (diag) QP (non-diag)
⇒ (with Coulomb effects) reliable reproduction of pair excitation energies
H2O molecule: QP approachdH-O = 0.966 Å , < = 104.49°)
Lowest ionization energies (eV): (512 bands, 6000 G vectors, Lcell = 10 Å, 64 grid points)
MO LDA HF GW Exp
1b1
3a1
1b2(HOMO)
EgQP
13.119.277.21
6.2
19.1415.4713.16
14.1
18.79 (18.51)14.42 (14.33)11.94 (12.04)
12.5
18.72, 18.5514.83, 14.7312.78, 12.61
comparison: Shigeta, Int. J. Quant. Chem. 85, 411 (2001)
exp: Handbook of He I Photoelectron Spectra of Fundamental MoleculesBrundle/Turner, Proc. R. Soc. London Ser. A 307, 27 (1968)
Pair excitations(Optical spectra)
Two-particle problem (quasielectron-quasihole pair excitations, excitons)
_P
1 1 12 2 23 4
1' 2' 1' 2' 1' 2'3' 4'=
=
+
+3 43' 4'
3 4
3' 4'
3
3'
4
4'
_P Ξ
Ξ
• Macroscopic optical polarizability → polarization function
• BSE
0q→=α Pv
PLLP 00 Ξ+=
vG/ +δΣδ=Ξ
GGL0 ⋅=v - short range (electron-
hole exchange, local fields)• Standard approximations
(i) GW: δΣ/δG = W + GδW/δG = ladder approximationcommon believe, for extended systems (Strinati, Nuov. Cim. 11, 1 (1988))
(ii) neglect of dynamical screening (K. Shindo, J. Phys. Soc. J. 29, 287 (1970))
(iii) neglect of non-particle conserving terms and coupling between resonant and non-resonant terms ⇒ no effect on optical absorption of
Si (S. Albrecht, Ph.D. thesis)
^
Bloch-Kohn-Sham representation of two-particle problem
approximate BSE
( ) ( ){ } ( )
''vv'cc
''k,''v,''c''''vv''cc ;''v'c,''''v''cPi''''v''c,cvH
kk
kk kkkk
δδδ−=
ωδδδγ+ω−∑ h
Pair Hamiltonian
(i) independent-particle approximation (RPA)
(ii) independent-quasiparticle approximation
(iii) Coulomb-correlated electron-hole pairs
( ) [ ] ''vv'ccvc )()(''v'c,cvH kkkkkk δδδε−ε=
)()()( nnQPn kkk ∆+ε=ε kkkk nV))((n)( xc
QPn n
−εΣ=∆
[ ])''v'c,cv(v2)''v'c,cv(W
)()()''v'c,cv(H ''vv'ccQPv
QPc
kkkkkkkk kk
+−δδδε−ε=
Representation for polarizability• eigenvalue problem (homogeneous BSE)
(Albrecht et al, PRL 80, 4510 (1998); Rohlfing, Louie, PRB 62, 4927 (2000))
∑λ λ
λλλλ γ+ω−
φµ=ωα→φ=φ
)i(E)(EH
2
h
small surface slab ≈ 28 atoms→ rank of H < NcNvNk ≈ 400,000 !!
• initial value problem (inhomogeneous BSE)(Hahn et al., PRL 88, 016402 (02); Schmidt et al., PRB 67, 084307 (2003))
)()(vc
)0(;)t(H)t(dtdi :with
)t(edtei)(
vc
0
tti
kkkvkε−ε
=µ=ΨΨ=Ψ
Ψµ=ωα ∫∞
γ−ω
h
h
central-difference („leap-frog“) method
• boundary-value problem: (Benedict et al., PRL 80, 4514 (1998))
Solve initial-value problem...
… with central difference method:
• matrix diagonalization → matrix-vector multiplications
tnt);t(Ht2
)t()t(i n1nn2n ∆=Ψ=
∆Ψ−Ψ
++h
( ) ( )4at
6at NONO →•
• parallelization
Example: CPU time for bulk Si (Pentium PC)
HP9000 with 8 CPUs / 32 GB sufficient for surface calculation
Standard approachExample: ionic compound InP
(200 random k-points, 8-atom sc supercell)
Exp.: P. Lautenschläger et al., PRB36, 4813 (87)Calc.: P.H. Hahn, PhD Thesis, Jena (04)
Influence of wave function representationReal-space grid versus PAW
Example: Si bulk
good agreementpreliminary: advantage: first-row elements
disadvantage: too slow
Kernel beyond ladder approximation
Example: Si crystal(sc supercell, 25 random points)
[ ])31(W)24(W)41(W)23(W)43(Gi)34(G)12(W
+−=δδ
h
standard approximations (dynamics, particle conservation)
- reduction of e-h attraction- zero for Wannier-Mott exciton
here:- reduction of E2 → E1 redistribution- shift by 0.1 eV toward higher energies
Influence of dynamical screeningProblem: no closed BSE
{ }∫ ωωω−ωω−ωω+ω≈ωω )',''(P~)''(W''d1)(G)'(G),'(P~
SilanShindo approximation
(Shindo, J. Phys. Soc. Japan 29, 287 (1970))
[ ] )(P~),'(P~)''(G)''(G''d
)(G)'(G ω∫
≈ωωω−ω+ωω
ω−ω+ω
only influence on higher transitions
partial compensation of dynamical effects and resonant-nonresonant coupling
Monohydride Si(001)2×1-H:Model system for a passivated surface
no π/π* states in the gapSaturation of one dangling bond per atom
only electron-hole pairs in the energy range of bulk optical transitions
RA of Si(100)2×1-Hcalculation: 12-layer slab, 200 k points,
spectrum < 6eV (50c, 50v) ≈ 5×105 pair states
• GW: not only rigid shift
• local fields: surprisingly small
• screened electron-hole attraction:
- enhancement of optical anisotropynear E1
- redistribution of oscillator strengthE2 → E1
⇒ surface modification of bulk excitons(analog Si(110)1×1:H)
Hexagonal ice (Ih)
phase diagram of water structural model
Frenkel versus Wannier-Mott exciton
Absorption |Ψ(re, rh)|2 with rh = R0
Exp.: Kobayashi, J. Phys. Chem. 87, 4317 (1983)Calc.: P.H. Hahn et al., PRL 94, 037404 (2005)
• Compensation of QP and excitonic shifts• but Coulomb enhancement ⇒ bound state
(Frenkel exciton)
Virtual J. Biological Physics Research, February 1, 2005
Many-body versus condensation effects (H2O)
lowest pair excitation (eV)
phase DFT-GGA QP QP + BSE Exp.molecule
solid6.28.0
12.512.5
7.29.2
7.48.7
peak (not onset)
• strange result: E (molecule) < E (solid)
• reduction of many-body effects by 30 (QP) or40 (Ex) % in solid
• reason: increased screening delocalization of excitonrex = 2.27 → 4.02 Å
Exp: Chan, Chem. Phys. 178, 387 (93)Kerr, PRA 5, 2523 (72)
Conclusions• in principle: MBPT in GW applicable independent of dimensionality
even using supercellsneed: screening with dynamics and local fields
• shown for 3D (bulk semiconductors), 2D (C and Si surfaces), and 0D (SiH4 and H2O molecules)
• critical: • wrong energetical ordering of states• supercell size• description of states above ionization edge?• completeness of KS basis set (self-energy,
screening function) • in general: convergence
We are on the right way!?
more or less correct oscillator& spectral strengths
but accuracy on energy scale 0.1 ... 0.2 eV
Acknowledgements
Collaboration: W.G. Schmidt, P.H. Hahn, M. Marsili, O. Pulci, R. Del Sole, J. Furthmüller, L.E. Ramos, H.-Ch. Weissker
Grants: EU: RTN NANOPHASE, NoE NANOQUANTA
Federal Government: Junior scientist groupComputational Materials Science
DFG: Project Si & Ge nanocrystals