The GW approximation in 90 minutes or so. F. Bruneval Service de Recherches de Métallurgie Physique CEA, DEN
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1 The GW approxmaton n 90 mnutes or so Servce de Recherches de Métallurge Physque CEA, DEN DFT tutoral, Lyon december 2012
2 Outlne I. Standard DFT suffers from the band gap problem II. Introducton of the Green's functon III. The GW approxmaton IV. The GW code n ABINIT and the G0W0 method V. Some applcatons DFT tutoral, Lyon december 2012
3 Standard DFT has unfortunately some shortcomngs band gap Band gap problem! after van Schlfgaarde et al PRL (2008) DFT tutoral, Lyon december 2012
4 A pervasve problem Effectve masses for transport n semconductors Optcal absorpton onset Defect formaton energy, dopant solublty Photoemsson Exp. DFT tutoral, Lyon december 2012
5 How do go beyond wthn the DFT framework? Not easy to fnd mprovement wthn DFT framework There s no such thng as a perturbatve expanson Perdew's Jacob's ladder does not help for the band gap after J. Perdew JCP (2005). Need to change the overall framework! DFT tutoral, Lyon december 2012
6 Outlne I. Standard DFT suffers from the band gap problem II. Introducton of the Green's functon III. The GW approxmaton IV. The GW code n ABINIT and the G0W0 method V. Some applcatons DFT tutoral, Lyon december 2012
7 Many-body perturbaton theory Hstorcally older than the DFT (from the 40-50's)! Bg names: Feynman, Schwnger, Hubbard, Hedn, Lundqvst Green's functons = propagator G r t, r ' t ' = DFT tutoral, Lyon december 2012
8 The Green's functon N,0 Exact ground state wavefuncton: Creaton, annhlaton operator: 1 2 r t N,0 r t, r t s a (N+1) electron wavefuncton not necessarly n the ground state r ' t ' N,0 s another (N+1) electron wavefuncton Let's compare the two of them! DFT tutoral, Lyon december 2012
9 Green's functon defnton N,0 r t r ' t ' N, e = G r t, r ' t ' for t t ' Mesures how an extra electron propagates from (r't') to (rt). DFT tutoral, Lyon december 2012
10 Green's functon defnton N,0 r ' t ' r t N, h for = G r ' t ', r t t ' t Mesures how a mssng electron (= a hole) propagates from (rt) to (r't'). DFT tutoral, Lyon december 2012
11 Fnal expresson for the Green's functon G r t, r ' t ' = N,0 T [ r t r ' t ' ] N, 0 tme-orderng operator e G r t, r ' t ' =G r t, r ' t ' h G r ' t ', r t Compact expresson that descrbes both the propagaton of an extra electron and an extra hole DFT tutoral, Lyon december 2012
12 Lehman representaton G r, r ', t t ' = N,0 T [ r t r ' t ' ] N,0 Closure relaton M, M, M, Lehman representaton: G r, r ', = where { f r f r ' ± E N 1, E N,0 = E N,0 E N 1, Exact exctaton energes! DFT tutoral, Lyon december 2012
13 Related to photoemsson spectroscopy Ekn hν Energy conservaton: before after h E N,0 =E kn E N 1, Quaspartcle energy: =E N,0 E N 1, =E kn h DFT tutoral, Lyon december 2012
14 And nverse photoemsson spectroscopy hν Ekn Energy conservaton: before after E kn E N,0 =h E N 1, Quaspartcle energy: =E N 1, E N,0 =E kn h DFT tutoral, Lyon december 2012
15 Other propertes of the Green's functon Galtsk-Mgdal formula for the total energy: E total = 1 d Tr [ h0 Im G ] Expectaton value of any 1 partcle operator (local or non-local) O =lm Tr [ OG ] t t ' DFT tutoral, Lyon december 2012
16 Outlne I. Standard DFT suffers from the band gap problem II. Introducton of the Green's functon III. The GW approxmaton IV. The GW code n ABINIT and the G0W0 method V. Some applcatons DFT tutoral, Lyon december 2012
17 How to calculate the Green's functon? Feynman dagrams Hedn's functonal approach PRA (1965). DFT tutoral, Lyon december 2012
18 Hedn's coupled equatons 6 coupled equatons: 1= r 1 t 1 1 2= r 2 t 2 2 G 1,2 =G 0 1,2 d34 G0 1,3 3,4 G 4,2 1,2 = d34 G 1,3 W 1,4 4,2,3 1,2,3 = 1,2 1,3 d 4567 Dyson equaton self-energy 1,2 G 4,6 G 5,7 6,7,3 G 4,5 vertex 0 1,2 = d34 G 1,3 G 4,1 3,4,2 polarzablty 1,2 = 1,2 d3 v 1,3 0 3,2 delectrc matrx 1 W 1,2 = d3 1,3 v 3,2 screened Coulomb nteracton DFT tutoral, Lyon december 2012
19 Smplest approxmaton 1,2 = G 1,2 v 1,2 t Fock exchange Dyson equaton: v 1,2 G=G0 G 0 G G=G0 G 0 G0... G 1,2 Not enough: Hartree-Fock s known to perform poorly for solds DFT tutoral, Lyon december 2012
20 Hartree-Fock approxmaton for band gaps DFT tutoral, Lyon december 2012
21 Hedn's coupled equatons 6 coupled equatons: G 1,2 =G 0 1,2 d34 G0 1,3 3,4 G 4,2 1,2 = d34 G 1,3 W 1,4 4,2,3 1,2,3 = 1,2 1,3 d 4567 Dyson equaton self-energy 1,2 G 4,6 G 5,7 6,7,3 G 4,5 0 1,2 = d34 G 1,3 G 4,1 3,4,2 1,2 = 1,2 d3 v 1,3 0 3,2 1 W 1,2 = d3 1,3 v 3,2 screened Coulomb nteracton DFT tutoral, Lyon december 2012
22 Hedn's coupled equatons 6 coupled equatons: G 1,2 =G 0 1,2 d34 G0 1,3 3,4 G 4,2 1,2 = d34 G 1,3 W 1,4 4,2,3 1,2,3 = 1,2 1,3 d 4567 Dyson equaton self-energy 1,2 G 4,6 G 5,7 6,7,3 G 4,5 0 1,2 = d34 G 1,3 G 4,1 3,4,2 1,2 = 1,2 d3 v 1,3 0 3,2 1 W 1,2 = d3 1,3 v 3,2 screened Coulomb nteracton DFT tutoral, Lyon december 2012
23 Hedn's coupled equatons 6 coupled equatons: G 1,2 =G 0 1,2 d34 G0 1,3 3,4 G 4,2 1,2 = d34 G 1,3 W 2 1,4 2 4,2,3 1,2,3 = 1,2 1,3 d 4567 Dyson equaton self-energy 1,2 G 4,6 G 5,7 6,7,3 G 4,5 0 1,2 = d34 G 1,3 G 3,4,2 2 4,1 2 1,2 = 1,2 d3 v 1,3 0 3,2 1 W 1,2 = d3 1,3 v 3,2 screened Coulomb nteracton DFT tutoral, Lyon december 2012
24 Here comes the GW approxmaton 1,2 = G 1,2 W 1,2 GW approxmaton 0 1,2 = G 1,2 G 2,1 RPA approxmaton 1,2 = 1,2 d3 v 1,3 0 3,2 W 1,2 = d3 1 1,3 v 3,2 DFT tutoral, Lyon december 2012
25 What s W? Interacton between electrons n vacuum: 2 1 e v ( r, r ' )= 4 πε0 r r ' Interacton between electrons n a homogeneous polarzable medum: 2 1 e W (r, r ' )= 4 π ε0 ε r r r ' Delectrc constant of the medum Dynamcally screened nteracton between electrons n a general medum: 2 1 e ε ( r, r ' ', ω) W (r, r ', ω)= d r ' ' 4 π ε0 r ' ' r ' DFT tutoral, Lyon december 2012
26 W s frequency dependent W can measured drectly by Inelastc X-ray Scatterng W q=0.80 a.u, Slcon Plasmon frequency ω [ev] Zero below the band gap H-C Wessker et al. PRB (2010) DFT tutoral, Lyon december 2012
27 GW has a super Hartree-Fock GW Approxmaton Hartree-Fock Approxmaton x r 1, r 2 = xc r 1, r 2, = d ' G r 1, r 2, ' v r 1, r 2 2 d ' G r 1, r 2, ' W r 1, r 2, ' 2 = bare exchange x r 1, r 2 Bare exchange c r 1, r 2, + correlaton GW s nothng else but a screened verson of Hartree-Fock. Non Hermtan dynamc DFT tutoral, Lyon december 2012
28 Summary: DFT vs GW Electronc densty r Local and statc exchange-correlaton potental v xc r Approxmatons:, GGA, hybrds Green's functon G r t, r ' t ' Non-local, dynamc Depends onto empty states exchange-correlaton operator = self-energy xc r, r ', GW approxmaton GW r t, r ' t ' =G r t, r ', t ' W r t, r ' t ' DFT tutoral, Lyon december 2012
29 GW approxmaton gets good band gap No more a band gap problem! after van Schlfgaarde et al PRL (2008) DFT tutoral, Lyon december 2012
30 Outlne I. Standard DFT suffers from the band gap problem II. Introducton of the Green's functon III. The GW approxmaton IV. The GW code n ABINIT and the G0W0 method V. Some applcatons DFT tutoral, Lyon december 2012
31 Avalable GW codes DFT tutoral, Lyon december 2012
32 Avalable GW codes has a GW code nsde DFT tutoral, Lyon december 2012
33 How to get G? Remember the Lehman representaton: f r f r ' ± G r, r ', = where the f r and the are complcated quanttes But for ndependent electrons lke Kohn-Sham electrons: G r, r ', = KS KS KS KS r r' ± Ths can be consdered as the best guess for G One can get W and GW DFT tutoral, Lyon december 2012
34 GW as a perturbaton wth respect to GW quaspartcle equaton: [ h0 xc ] = GW GW GW GW KS equaton: [h 0 v Approxmaton : xc ] = GW DFT tutoral, Lyon december 2012
35 GW as a perturbaton wth respect to GW quaspartcle equaton: [ h 0 xc GW ] = GW KS equaton: [ h v ] = GW = xc 0 [ xc GW v xc ] DFT tutoral, Lyon december 2012
36 Lnearzaton of the energy dependance GW = [ xc GW v xc ] Not yet known Taylor expanson: GW xc = xc GW xc... Fnal result: GW = where Z [ xc xc Z =1/ 1 v xc ] DFT tutoral, Lyon december 2012
37 Quaspartcle equaton A typcal ABINIT ouptput for Slcon at Gamma pont k = Band E0 <Vxc> SgX SgC(E0) E^0_gap E^GW_gap GW = Z dsgc/de Sg(E) E-E E Z [ xc v xc ] DFT tutoral, Lyon december 2012
38 Flow chart of a typcal GW calculaton DFT, occuped AND empty states calculate W If self-consstent GW GW, Egenvalues calculate G * W GW k DFT tutoral, Lyon december 2012
39 Outlne I. Standard DFT suffers from the band gap problem II. Introducton of the Green's functon III. The GW approxmaton IV. The GW code n ABINIT and the G0W0 method V. Some applcatons DFT tutoral, Lyon december 2012
40 GW approxmaton gets good band gap No more a band gap problem! after van Schlfgaarde et al PRL (2008) DFT tutoral, Lyon december 2012
41 Clusters de sodum Na 4 e Na 4 4 E 0 Na 4 E 0 Na = { HOMO, Na4 LUMO, Na4 + Na4 /Na4 Bruneval PRL (2009) DFT tutoral, Lyon december 2012
42 Defect calculaton wthn GW approxmaton Up to 215 atoms Cubc slcon carbde DFT tutoral, Lyon december 2012
43 Photolumnescence of VS Bruneval and Roma PRB (2011) DFT tutoral, Lyon december 2012
44 Band Offset at the nterface between two semconductors Very mportant for electroncs! Example: S/SO2 nterface for transstors GW correcton wth respect to R. Shaltaf PRL (2008). DFT tutoral, Lyon december 2012
45 Summary The GW approxmaton solves the band gap problem! The calculatons are extremely heavy, so that we resort to many addtonal techncal approxmatons: method named G0W0 The complexty comes from Dependance upon empty states Non-local operators Dynamc operators that requres freq. convolutons There are stll some other approxmatons lke the Plasmon-Pole model... that I'll dscuss durng the practcal sesson... DFT tutoral, Lyon december 2012
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