Bethe-Salpeter equation approach to excitations in transparent conducting oxides

Transcription

1 Bethe-Salpeter equation approach to excitations in transparent conducting oxides Friedhelm Bechstedt Friedrich-Schiller-Universität Jena Germany

2 Need: Prediction of spectra with photons / electrons with electrons with photons Resulting spectra dominated by spectral functionof electrons, holes or electronhole pairs F. Bechstedt, Many-body approach (Springer, 2015)

3 Need: Prediction of spectra Reflection anisotropy spectroscopy In/Si(111) quantum wire VUV ellipsometry wurtzite AlN S. Chandola et al., PRL 102, (2009) A. Riefer et al., PRB 84, (2011)

4 Absorption in SnO 2 Transparency A. Schleife et al., PRB 83, (2011) Electrons in conduction band n-conductivity A. Schleife, unpublished

5 Transparent conducting oxides (TCOs) (here: n-conductive) Definition high-energy absorption edge (ħω = 3 4 ev) unintentional n-doped (or easily intentional) Compounds doping: ITO, ATO, AZO, IGZO, ZITO, combination: layers, heterostructures, nanostructures, Applications transparent conductive electrodes (optoelectronics, photovoltaics) transparent circuits / electronics devices: displays, touch screens, solar cells, Ga 2 O 3 G. Hautier et al., Nature Commun. 4, 2292 (2013)

6 Challenges for ab initio theoretical description = complex quantum-mechanical many-electron problem (large unit cells) interfaces (e.g. to semiconductors) TCO polymorphism (In 2 O 3, Ga 2 O 3 ), alloys, heterostructures & atomic geometries doping and free carrier influence large number of atoms / electrons & semicore cationic d-electrons

7 Electronic excitations poles in G single quasiparticle (QP) poles in quasielectron-quasihole pair (irreducible part of G 2 = P) poles in -1 collective (plasmonic) F. Bechstedt, pss (b) 112, 9 (1982)

8 Possible realization: Many-body perturbation theory (MBPT) Hedin/Lundqvist (1969) System of fundamental equations dielectric function =1 vp (length gauge) polarization function P = GGΓ RPA (independent particle) vertex corrections (excitons) one-electron Green function Dyson equation ( t H ) G ΣG 1 / 1 H = XC self-energy Σ = GWΓ screened Coulomb potential W = v + vpw vertex function Bethe-Salpeter equation Γ = 1+ δσ δg GGΓ

9 Dream true predictive power calculation of atomic geometries, accompanying electronic excitations and resulting optical properties without parameter from experiment or adjustable parameter (ab initio) prize solution of many-body problem limits exchange & correlation in electron gas, system size (number of atoms), numerical accuracy, real structure (defects, structural & chemical disorder, free carriers)

10 Outline Theoretical-numerical three-step procedure (Pedestrian guide) (i) (ii) Determination of atomic geometry: Density functional theory (DFT) Band structures and eigenfunctions: Quasiparticle (QP) theory (iii) Frequency-dependent dielectric function: Exciton theory Applications

11 1. Ground-state properties: Geometry & energetics

12 Density functional theory in local (spin-) density approximation (or generalized gradient approximation) = working horse Ground-state energy: E [ n; { R }] = Ĥ = T + E + E + E + E ion S Problem: equilibrium geometry : E[ n; { R }] 0 S ion Electron density non-interacting particles H XC Hohenberg / Kohn (1964) Kohn / Sham (1965) ion Exchange & correlation Quantum Monte Carlo (LDA/AM05/GGA etc. parametrization) { eq} [ { eq} ] R R, E n R s s = s ; s n( x) occ = ϕ n, k 2 n k( x) 2 2m + V ion electronic structure (?) x Kohn-Sham equation ( x) + V H ( x) + ε n (k), ϕ nk (x) δe XC ϕ δn( x) nk ( x) = ε n ( k) ϕ nk ( x)

13 Technical Supercell (superlattice, repeated slab) method = artificial translational symmetry Representation of eigenfunctions: plane-wave expansion generalization: projector augmented waves (PAWs) Electron-ion interaction: pseudopotentials generation within PAW method Brillouin-zone integration: special point summation problem: convergence bulk (3D) surface/interface (2D) superlattice (2D, 3D) chain (1D) nanocrystal, defect (0D) number of atoms maximum: 2744 (Ramos, PRB 66, (2002)) Vienna ab initio Simulation Package (VASP) k points, plane waves, conduction bands

14 Complex geometries monoclinic β-ga 2 O 3 (C 2 /m) C. Sturm et al., APL Materials 3, (2015) lattice constants (Å) and angles (degree) a b c γ theory experiment J. Furthmüller, F. B., PRB 93, (2016) + five internal cell parameters (Wyckhoff positions)

15 Formation energy (ev) Beyond covalent, ionic or metallic bonds: Hydrogen-bridge bond Example: Water dimer LDA GGA 0.00 O-O distance Experiment O-O distance (Å) O-O distance [Å] calc. deviation from Exp. bond energy [ev] calc. DFT-LDA % % DFT-GGA % % deviation from Exp. atomic distances and bond energy well described within DFT-GGA (better for ice)

16 Message For covalent / ionic / metallic bonds: Accuracy ± 1 % (no phonons & temperature effects) Improvements for dispersion forces: vdw Problem: Isolated H-bridge bonds

17 2. Excited-state properties: Single-quasiparticle electronic excitations

18 Dyson equation G(z) = G H (z) + G H (z)σ(z)g(z) Hartree: only classical interaction exchange-correlation (XC) self-energy = iħgwγ with W = W(G), Γ = Γ(G) screened Coulomb interaction vertex function Starting electronic structure G 0 (z) = G H (z) + G H (z)v XC G 0 (z) certain approximation for XC (KS, gks) G (z) = G 0 (z) + G 0 (z)[δσ(z)]g(z) δσ(z) = Σ(z) V XC importance of start point = ^ minimization of perturbation

19 More clever: Result: Problem: better starting point G 0 (z) G QP (z) = 1 ħz ε QP with G(z) = G QP (z) + G QP (z)[δσ(z) QP ]G (z) ε QP = ε + QP Green function with single, undamped QP pole shifted by QP against eigenvalue ε of starting electronic structure perturbation much smaller QP unknown Solution: first iteration Blomberg/Bergersen/Kus, Can. J. Phys. 50, 2286 (1972) + 51, 102 (1973) G (1) (z) = G QP ~ (z) + [δσ(z) ) QP ][G QP (z)] 2 with ~ Σ(z) = Σ(z) G = G QP

20 No unphysical double pole: QP ~ = Re δσ (ε QP ) shift clear procedure for self-consistency ~ A(ω) = 1+ Re δσ (ω) δ(ħω ε QP ) ω P 1 ~ + Im δσ(ω) ω ħω ε QP π undamped QP peak with reduced weight first satellite structures Γ 1 Γ = 1 (including vertex): further interations (Hedin GW approximation): stop for consistency

21 Standard GW schemes Γ = -1 GW approximation Start: G 0 (W 0 ) no iteration Self-consistency: G (W) but condition: one QP pole with full spectral weight, i.e., no satellites (self-consistency: energies or energies + wave functions)

22 Frequency-dependent self-energy (G 0 W 0, KS): H 2 O molecule crossings : QP energies strong variations: shakeup/shake-off satellites P.H. Hahn,, F.B., PRB 72, (2005)

23 Kohn-Sham particles quasiparticles here: spectral function ImG (schematic for occupied states) energy shift spectral broadening (finite lifetime) satellites: shake-up and shake-off structures (reduction of spectral weight) F. Bechstedt et al., pss (b) 246, 1877 (2009) (feature article) F. Bechstedt, Many-body approach to electronic excitations

24 Electronic problem in ground state (KS) Many-body perturbation after excitation Perturbation theory = quasiparticle energy shift E g QP equation (from Dyson equation) -only for main peakc v KS QP c v Hedin 1965 (reaction by screening renormalization) Σ = GW Gap opening

25 QP shifts in G 0 W 0 approximation: Dependence on KS/gKS starting point filled gap empty LDA small-gap system (Si): KS close to QP (occupied) hybrid functional starting pointreduces shifts HF: too large shifts, opposite sign F. Fuchs et al., PRB 76, (2007) F.B., F. Fuchs, G. Kresse, pss(b) 246, 1877 (2009) feature article

26 Quasiparticle band gaps novel (self-consistent) approach: better starting point including nonlocal exchange opening of fundamental gaps = agreement with experiment accuracy: ev prediction: E g (InN) 0.7 ev F. Fuchs et al., PRB 76, (2007)

27 Quasiparticle band structure & ARPES: wz-zno Excellent agreement for valence bands (absolute with respect to ε F + dispersion) especially for k c-axis 6 B i n d in ge n erg y (ev ) 8 H L A GΓ K M GΓ QP theory: A. Schleife et al., PRB 80, (2009) ARPES: M. Kobayashi et al., Proc. 29th, ICPS2008 (Rio de Janeiro)

28 wz-zno: Occupied states (DOS) PES (XPS) excellent description of O 2p-derived bands (van Hove singularities) underestimation of Zn 3d binding energy by 0.7 ev (vertex corrections?, consequence: gap shrinkage) P.D.C. King et al., PRB 79, (2009)

29 Messages Energies: G 0 W 0 is fine if resonable starting point Bad starting point Self-consistency (but condition: one QP pole with full spectral weight) Do GW in a clever way! As long you do not touch satellites, the physics (perhaps not the values) is correctly described

30 3. Excited-state properties: Electron-hole pair excitations & optical properties

31 Longstanding problem for parameterfree calculations of pair excitations (excitonic, local-field effects) Theory: L.J. Sham, T.M. Rice, PR 144, 708 (1966) W. Hanke, L.J. Sham, PRB 21, 4656 (1980) G. Strinati, Riv. Nuovo Cimento 11, 1 (1988) numerical S. Albrecht et al., PRL 80, 4510 (1998) calculations: L.X. Benedict et al., PRL 80, 4514 (1998) (bulk) M. Rohlfing, S.G. Louie PRL 81, 2312 (1998) new methods, excotic materials (surfaces, polymers, ice ) W.G. Schmidt et al., PRB 67, (2003) (time development) P.H. Hahn et al., PRL 88, (2002) surface F. Fuchs et al. PRB 78, (2008) P.H. Hahn et al., PRL 94, (2005) water (efficient method for bound states) P.H. Hahn et al., PRB 72, (2005) molecule C. Rödl et al., PRB 77, (2008) magnetic systems

32 Bethe-Salpeter equation for quasielectron-quasihole pair excitations (excitons) Macroscopic optical polarizability polarization function q 0 (irreducible part of e-h G) Bethe-Salpeter equation α = vp 1 _ _ 2 P = + Ξ P 1' 2' 1' 2' 1' 3' 4' 2' Ξ = 3' 4' 3' 4' 3 + 3' screened e-h attraction unscreened e-h exchange 4 ^ 4' L L P = 0 + 0Ξ Ξ = δσ / δg + L 0 = G G P 2v (singlets) v - short range (e-h exchange, local fields)

33 Kernel Ξ of e-h interaction (standard treatment?) (i) GW: Γ = 1, Σ = GW no additional vertex corrections in interaction (ii) ladder approximation: δσ/δg = W + G δw/δg Strinati, Riv. Nuov. Cim. 11, 1 (1988) W W W Im[ M (ω)] beyond ladder approximation: Silicon

34 Quasielectron-quasihole interaction: Pair Hamiltonian H Hedin s GW approximation + neglect of (i) δw/δg (ii) dynamical screening Sham, Rice, PR 144, 708 (1966); Hanke, Sham, PRL 33, 582 (1974) Ĥ (eh,e' ε M = ε h' ) = 1 [ ] QP QP ε ε e W(eh,e' h' ) + 2v(eh,e' h' ) 1 ( q + G, q + G' ) h δ ee' δ hh' G= G' = 0 (iii) resonance-antiresonance interaction (Tamm-Dancoff) (iv) non-particle-conserving terms quasielectron-quasihole pair screened Coulomb attraction bare electron-hole exchange equivalent to local fields (Adler, Wiser, 1962/63) homogeneous Bethe-Salpeter equation for electron-hole pairs (excitons) pair amplitude e',h' Ĥ(eh,e' h' )A (e' h' ) Λ = E Λ A Λ (eh) ^ pair energy

35 Band-edge excitons: Binding energy of Wannier-Mott exciton Test: Hydrogenic model (Energies = mev/n 2 ) New method: Lowest eigenvalues with conjugate-gradient scheme (Kalkreuter, Simma) main problem: BZ sampling (toward k points) InN (full band structure, electronic screening) Prediction of exciton binding energy for InN < 5 mev (dynamical lattice screening?) F. Fuchs et al., PRB 78, (2008)

36 Computational methods for optical polarizability eigenvalue problem (homogeneous BSE) (Albrecht et al, PRL 80, 4510 (1998); Rohlfing, Louie, PRB 62, 4927 (2000)) H A Λ = E Λ A Λ α( ω) = Λ E Λ µ A Λ 2 ( ω + iγ) rank of H = N c N v N k !! small surface slab 28 atoms bulk (400 days, single processor) initial value problem (inhomogeneous BSE) (Hahn et al., PRL 88, (02); Schmidt et al., PRB 67, (2003)) α( ω) i = 0 dte with : iωt i e γt d dt µ Ψ(t) Ψ(t) = H Ψ(t) ; Ψ(0) central-difference ( leap-frog ) method dipole operator for one band pair = µ = ε c ck v vk ( k) ε v ( k) boundary-value problem (Haydock method): (Benedict et al., PRL 80, 4514 (98))

37 Result: Macroscopic dielectric function * πe ck pj vk AΛ cvk Λ cv,, c( ) v( ) k k k Λ Λ εjj( ω ) = 1 + ( ) + m Ω ε ε E ω iγ E + ω+ iγ Exciton eigenvector & eigenvalue (bound & scattering states) Challenges: number of non-interacting pair states cvk (especially k-points in BZ) - Bulk spectra: ~ Surface spectra < Excitonic bound states (Wannier-Mott) ~ k-points d-electrons spin polarization

38 Solution of the gap problem: rt-sno 2 Selection rules according to D 4h (band gap forbidden) Strong polarization dependence of absorption Band gap E g = 3.65 ev in agreement with 2- photon spectroscopy (D. Fröhlich et al., PRL 41, 1750 (1987)) Absorption edge ħω = 4.36 ev Good agreement with measured valence-band distances (still controversial band ordering) A. Schleife et al., PRB 83, (2011)

39 Quasiparticle and exciton influence Example: MgO blueshift Experiment [Bortz et al., Phys. Scr. 41, 537 (1990)] redshift redistribution of spectral strenght bound exciton states at absorption edge small oscillator strength Im ε ~ 2 computable with high accuracy A. Schleife et al., Phys. Rev. B 80, (2009)

40 Frenkel versus Wannier-Mott exciton (ice) Absorption Ψ(r e, r h ) 2 with r h = R 0 Exp.: Kobayashi, J. Phys. Chem. 87, 4317 (83) Calc.: P.H. Hahn et al., PRL 94, (2005) Virtual J. Biological Physics Research, February 1, 2005 Compensation of QP and excitonic shifts but Coulomb enhancement bound state (Frenkel exciton)

41 Optical properties of wz-zno (- - -): Experiment (ordinary / extraordinary) P. Gori et al., PRB 81, (2010) (- - -): Theory A. Schleife et al., Phys. Rev. B 80, (2009) Peaks: E: bound exciton A (7.7 ev): two-uppermost VBs CBs B (10 12 ev): two-lower VBs CBs C (13 15 ev): Zn 3d contributions DKs: Exp.: 4.08 (4.01) Theory: 3.75 (3.70)

42 Influence of degenerate electron gas (single-particle level) BGR (band gap renormalization) shrinkage BM shift (Burstein-Moss shift) blue shift of absorption Compensation (up to cm -3 )

43 Modified: Pair Hamiltonian: generalized eigenvalue problem BGR Hˆ ( eh, e ' h ') = ε ε δ δ QP QP e h ee' hh ' { (, ' ') 2 v(, ' ')} + f f W eh e h + eh e h f f e h e' h' quasielectron-quasihole pair screened Coulomb attraction + bare electron-hole exchange additional screening by free carriers Pauli blocking homogeneous Bethe-Salpeter equation for electron-hole pairs (excitons) e', h' Hˆ ( eh, e ' h ') A ( e ' h ') = E A ( eh ) Λ Λ Λ pair amplitude pair energy

44 Result: Macroscopic (complex, frequencydependent) dielectric function 2 = δ + + ( kk * kk ) ( ) Ω j M * jj '(ω) jj ' M eh M e' h ' (, ' ' ';ω) ( ' ' ', ; ω) j' P ehk h e k P e h k hek eh,, k e', h', k' with polarization function * AΛ( ehk) AΛ( e' h' k') e h e' h' Λ EΛ ω iη P( ehk, e' h ' k';ω) = f f f f Exciton eigenvector & eigenvalue factors f e f h guarantee Pauli blocking of optical transitions

45 Optical absorption in n-doped ZnO no free carriers Mott transition? Burstein- Moss Result n e = n e = cm n e = cm -3 A. Schleife et al., Phys. Rev. Lett. 107, (2011)

46 Comparison & Mahan exciton? A. Schleife et al., Phys. Rev. Lett. 107, (2011) Wannier-Mott peak disappears but Sommerfeld enhancement less influenced Binding energy and oscillator strength are reduced. Indication for Mahan exciton? Exp.: T. Makino et al., Phys. Rev. B 65, (R)(2002) reasonable agreement (position + absolute values)

47 Summary: Influence of many-body effects rs-mgo Conclusion: Optical spectra not treatable by independent KS particles but QP effects are also insufficient independent-particle independent - QP approximation: blueshift, no lineshape change excitonic effects: -LFE weak for bulk -redshift -redistribution form higher to lower photon energies -excitonic bound states below E g (QP) A. Schleife et al., PRB 80, (2009)

48 Prediction of spectra single-qp energies (accuracy ev but localized states) predictive power (oscillator strengths 5%, pair energies 0.2 ev) excitonic effects (extremely important, also far away from edge) refined picture (excitons van Hove singularities / defects, free carriers) To do / Open problems dynamical screening (including lattice polarization) vertex corrections in Σ and W (semicore states) non-collinear spins, spin-orbit interaction convergence (k-point sampling, number of bands)

49 Collaboration/Acknowledgements Theory: Jena: W.G. Schmidt, P.H. Hahn, C. Rödl, F. Fuchs, J. Furthmüller, A. Schleife, A. Riefer, L.C. de Carvalho Rome: M. Marsili, O. Pulci, R. Del Sole Vienna: G. Kresse, M. Shiskin, J. Paier Experiment: Berlin: M. Rakel, Ch. Cobet, A. Navarro-Quezada, N. Esser Magdeburg: P. Schley, G. Rossbach, R. Goldhahn Warwick: P.D.C. King, T. Veal, C.F. McConville Grants: EU: RTN NANOPHASE, NoE NANOQUANTA, e-i3 ETSF, ITN RAINBOW Federal Government: Projects on solar cells of third generation DFG: Projects on nitrides, oxides, spin polarization, GRK Quantum fields Austrian FFW: SFB IR-ON Carl-Zeiss-Stiftung: Scholarship