Many-Body Perturbation Theory: (1) The GW Approximation Michael Rohlfing Fachbereich Physik Universit¨atOsnabr¨uck • Motivation MASP, June 29, 2012 • Excited states: electrons, holes • Equation of motion • Approximations and tricks • Examples Relevance of excited electronic states • Spectroscopy: – Electrons: Photoemission / Inverse PE / Tunneling spectroscopy – Light / Electron-hole pairs: Absorption / Emission / Reflectivity • Characterisation of systems: ••••••• – Geometric Structure • • • • • – Defects / interfaces / ... • ••• • ⇐⇒ • • • • • – Macroscopic signature of • • • • •• • microscopic details B B V B ¡ B • Short-time dynamics: B B¡ B B B hω¯ BBN – Electrons (femtoseconds) BN – Atomic structure (picoseconds) V V • Technological relevance / Materials science: – Optoelectronics / Molecular electronics / Photovoltaics – Manipulation of bonds / Photochemistry Density-Functional Theory: from 1960’s until today 1950’s/60’s: Energy of electronic systems (homogeneous electron gas, inhomogeneous systems), including many-body effects (exchange, correlation) Hohenberg, Kohn, Sham and others 1960’s: interrelation between wave function, Z 2 3 3 potential, energy, single-particle density ρ(r) = |Ψ(r, r2, .., rN )| d r2...d rN , ”single-particle orbitals” ψn(r) and ”single-particle energies” ²n =⇒ Intuitive understanding via effective single-particle quantum mechanics Technical issues (1970’s-today): Pseudopotentials, basis sets, iterations, geometry optimization, molecular dynamics, ... Approximation of Exchange+Correlation (1970’s-today): Local-density approximation (LDA), Generalized gradient approximation (GGA), ”Hybrid functionals” (including Hartree-Fock exchange) PBE0-BLYP-B3LYP-HSE-... , ... Today: 50-100 codes + 100-1000 developers + 1000-10000 users Total energy Etot[R], Geometric structure, vibrations, chemical reactions, ... Physics + Chemistry SIESTA, VASP, ABINIT, CRYSTAL, GAUSSIAN, TURBOMOLE, ... Interpretation of ²n as ”single-particle energies” without formal justification DFT = ”ground-state method”: not suited for electronic excitation processes Density-Functional Theory (DFT) [Hohenberg, Kohn, Sham (1964)] • Atomic postions {R}, Electron density ρ(r) • Etot[R,ρ] = Ekin[ρ] + ECoul[ρ] + Exc[ρ] + Eext,e[R,ρ] + Enucl,nucl Aluminium: Etot(alatt) Ekin kinetic energy of the electrons 0.15 ECoul[ρ] classical Coulomb energy of the electrons 0.10 Exc[ρ] Exchange-correlation functional (Local-density approx., gradient corrections) 0.05 Energy [eV] Eext,e Nucl.-electron interaction (+ ext. fields) 0.00 Enucl,nucl Nucl-Nucl interaction 3.8 3.9 4.0 4.1 4.2 ˚ X DFT 2 Lattice constant [A] • Optimize Etot[R,ρ] w.r.t. ρ(r) = |ψm (r)| 0 2 1 B p C B [ρ] [ρ] C DFT DFT DFT =⇒ Kohn-Sham equations: @ + V (r) + V (r) + V (r)A ψ (r) = ² ψ (r) 2m Coul xc ext,e m m m =⇒ Total energy Etot[R] and forces ∂Etot/∂Riα ←→ Geometry, bonds, phonons, ... DFT DFT ”Wave functions” |ψm i, ”Band-structure energies” ²m (often not really reliable...) DFT for bulk systems Phase transition of N in GaAs: Forces on atoms silicon under pressure: Diamond → β-Sn Transition pressure P= • Ga • As • N dE/dV ∼ 10 GPa Ga (1.Nb.): 2.0 eV/A˚ As (2.Nb.): 0.1 eV/A˚ =⇒ Relaxations at defect Surfaces geometries from DFT Si(111)-(2×1) surface: π-bonded chains Adsorbed molecules: PTCDA on Ag(111) K.C. Pandey, Phys. Rev. Lett. 49, 223 (1982) 1 0.027 0.0046 ] 0.00342 −− B 0.0022a [ 0.001 0 −0.0002 −0.0014 dy ) r ( −0.0026ρ ∆ −0.0038Z 0 −0.005–0.027 0 1 Up Down Change of charge density during adsorption: ˚ 2.33 A˚ l ∆z = 0.51 A Red: charge accumulation ¡ ¡ 2.25 A˚ 2.28 A˚ Blue: charge depletion Many-Body Perturbation Theory: from 1960’s until today Key Idea of Many-Body Perturbation Theory: (0) (0) (0) −1 • H (including simplified el.-el. interac.) =⇒ G1 (E) = (E − H ) (exact) • plus rest of electron-electron interaction (= ”perturbation”): (0) (0) =⇒ G1(E) = G1 (E) + G1 (E) Σ(E) G1(E) Σ(E) = Self-energy operator (=single-particle signature of remaining interaction) 1950’s/60’s: Spectra of electronic systems (homogeneous electron gas) [ Hedin and others, 1960’s ] ”Forgotten” (success of DFT; computational effort), ”recovered” due to DFT band-gap problem. Band structure of bulk semiconductors (1980’s): Hybertsen, Louie, Godby, Schl¨uter,Sham, ... Surfaces, nanostructures (1980’s/90’s) Optical spectra (1990’s/2000’s): Onida, Reining, Benedict, Shirley, Rohlfing, Louie, ... Technical issues (1980’s-today): Basis sets, frequency integration, eigenvalue decomposition, self consistency, ... Today: 10-20 codes + 30-50 developers + 50-100 users Band structures + optical excitations Crystals, surfaces, molecules, polymers, ... States of a many-electron system N Electrons |N, 0i 1. Ground state |N, 0i : Density-functional theory =⇒ Geometry States of a many-electron system N − 1 Electrons N Electrons N + 1 Electrons V V |N − 1, 1i |N + 1, 1i • HYH ©©* • HH ©© HH ©© G HH ©© G o 1 H © 1 |N − 1, 0i H |N, 0i -© |N + 1, 0i G1 = Single-particle Green function 1. Ground state |N, 0i : Density-functional theory =⇒ Geometry Many-body perturbation theory: 0 0 + 0 0 2. |N, 0i → |N ± 1, mi : G1(xt, x t ) = −i hN, 0 |T (ψ(x, t)ψ (x , t ))| N, 0i (Propagation of electron or hole between (x, t) and (x0, t0) ) Hedin’s equations [ L. Hedin, Phys. Rev. 139, A796 (1965), L. Hedin and S. Lundqvist, Sol.St.Phys. 23, 1 (1969). ] ½ ∂ ¾ Z e2 • EOM of G : i + ∇2 − V (r) − V (r) G (xt, x0t0) + i G (x00t, x00t, xt, x0t0)dx00 = δ(x, x0)δ(t, t0) 1 ∂t ext Coul 1 |r − r00| 2 • Decoupling by self-energy operator Σ: Z 2 {∂t + ∇ − Vext+Coul(r)}G1(12) − Σ(13)G1(32)d(3) = δ(12) Σ = Self energy Z + Σ(12) = i W (1 3)G1(14)Γ(42; 3)d(34) W = Screened Coulomb interaction Z W (12) = ²−1(13)v(32)d(3) ² = Dielectric function Z χ0 = Polarizability ²(12) = δ(12) + v(13)χ0(32)d(3) Z G1 = Single-particle Green function χ0(12) = −i G1(23)G1(42)Γ(34; 1)d(34) Γ = Vertex function Z δΣ(12) (1) = (r ,σ ,t ) Position/Spin/Time Γ(12; 3) = δ(12)δ(13) + G1(46)G1(75)Γ(67; 3)d(4567) 1 1 1 δG1(45) 2 Z {∂t + ∇ − Vext+Coul(r)}G1(12) − Σ(13)G1(32)d(3) = δ(12) Σ(12) = iG (12)W (1+2) ”GW approximation”: 1 =⇒ Z −1 Γ(12; 3) = δ(12)δ(13) W (12)= ² (13)v(32)d(3) Z ²(12) = δ(12) + v(13)χ0(32)d(3) χ0(12) = −iG1(21)G1(12) Hedin’sPhysical equations significance of Σ: −1 [ L. Hedin,• in Phys.vacuum Rev.: 139,χ0=0, A796² (1965),=1, ² L.=1, HedinW and=v S. Lundqvist, Sol.St.Phys. 23, 1 (1969). ] ½ ∂ ¾ Z e2 • EOM of G : i=⇒+Σ∇ =2 −iGV 1v(r) −ExchangeV (r) G (Fock)(xt, x0t0) operator + i / Hartree-FockG (x00t, x00t, xt, theoryx0t0)dx00 = δ(x, x0)δ(t, t0) 1 ∂t ext Coul 1 |r − r00| 2 • Decoupling• byin self-energymatter: Dielectric operator Σ: polarizability χ0 =⇒ ”W <v” (weaker) Z 2 {∂t + ∇ − Vext+Coul(r)}G1(12) − Σ(13)G1(32)d(3) = δ(12) Σ=== Self= energy =⇒Z Σ = iG1W ”Screened Exchange” = = + - Σ(12) = i W (1 3)G1(14)Γ(42; 3)d(34) = W = Screened Coulomb interaction Z W (12)• =Nomenclature:²−1(13)v(32)d(3) ² = Dielectric function Z χ0 = Polarizability ²(12) =Σδ(12) = iG + 1vv(13)+ iGχ0(32)1(Wd(3)− v) = ”Exchange” + ”Correlation” Z G1 = Single-particle Green function χ0(12) = −i G1(23)G1(42)Γ(34; 1)d(34) Γ = Vertex function Z δΣ(12) (1) = (r ,σ ,t ) Position/Spin/Time Γ(12; 3) = δ(12)δ(13) + G1(46)G1(75)Γ(67; 3)d(4567) 1 1 1 δG1(45) 2 Z {∂t + ∇ − Vext+Coul(r)}G1(12) − Σ(13)G1(32)d(3) = δ(12) Σ(12) = iG (12)W (1+2) ”GW approximation”: 1 =⇒ Z −1 Γ(12; 3) = δ(12)δ(13) W (12)= ² (13)v(32)d(3) Z ²(12) = δ(12) + v(13)χ0(32)d(3) χ0(12) = −iG1(21)G1(12) Hedin’s equations [ L. Hedin, Phys. Rev. 139, A796 (1965), L. Hedin and S. Lundqvist, Sol.St.Phys. 23, 1 (1969). ] ½ ∂ ¾ Z e2 • EOM of G : i + ∇2 − V (r) − V (r) G (xt, x0t0) + i G (x00t, x00t, xt, x0t0)dx00 = δ(x, x0)δ(t, t0) 1 ∂t ext Coul 1 |r − r00| 2 • Decoupling by self-energy operator Σ: Z 2 {∂t + ∇ − Vext+Coul(r)}G1(12) − Σ(13)G1(32)d(3) = δ(12) Σ = Self energy Z + Σ(12) = i W (1 3)G1(14)Γ(42; 3)d(34) W = Screened Coulomb interaction Z W (12) = ²−1(13)v(32)d(3) ² = Dielectric function Z χ0 = Polarizability ²(12) = δ(12) + v(13)χ0(32)d(3) Z G1 = Single-particle Green function χ0(12) = −i G1(23)G1(42)Γ(34; 1)d(34) Γ = Vertex function Z δΣ(12) (1) = (r ,σ ,t ) Position/Spin/Time Γ(12; 3) = δ(12)δ(13) + G1(46)G1(75)Γ(67; 3)d(4567) 1 1 1 δG1(45) 2 Z {∂t + ∇ − Vext+Coul(r)}G1(12) − Σ(13)G1(32)d(3) = δ(12) Σ(12) = iG (12)W (1+2) ”GW approximation”: 1 =⇒ Z −1 Γ(12; 3) = δ(12)δ(13) W (12)= ² (13)v(32)d(3) Z ²(12) = δ(12) + v(13)χ0(32)d(3) χ0(12) = −iG1(21)G1(12) Some pragmatic issues 0 0 • Time homogeneity: G1(t, t ) = G1(t − t ) ; Fourier transform to energy: G1(E), Σ(E) etc. • ”Quasiparticle (QP) approximation”: QP peak ¡ Ignore background, satellites, etc. in G1(E) Background ¡ - QP QP QP 0 ∗ Satellite 2Im(Em ) 0 X φm (x)(φm (x )) =⇒ G1(x, x ,E) = m QP + E − Em ± i0 QP QP Re(Em ) = QP band-structure energy, Re(Em ) QP QP Im(Em ) = spectral width, 1/Im(Em ) = lifetime 2 QP Z 0 QP QP 0 QP QP • EOM: { − ∇ + Vext(r) + VCoul(r)}φm (x) + Σ(x, x ,Em )φm (x )dx = Em φm (x) • Self-consistency problem similar to DFT, Hartree-Fock, etc.