College of Chemistry, Peking University, Beijing, China. Fritz-Haber-Institut der MPG, Berlin, Germany
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1 KITP Program Excitations in Condensed Matter Localized and Itinerant States in a Unified Picture beyond Density Functional Theory Hong Jiang 1, Patrick Rinke 2 and Matthias Scheffler 2 1 College of Chemistry, Peking University, Beijing, China 2 Fritz-Haber-Institut der MPG, Berlin, Germany
2 Outline Motivation: Why is it important to describe localized and itinerant states in an unified picture? Failure of LDA/GGA and its extensions What GW can do? and others Mott insulator from the GW perspective Concluding remarks
3 To localize or delocalize: that is the question localized VB CB itinerant E F
4 Oxygen vacancy in CeO 2 The release and storage of oxygen in CeO 2 relies on the localized f- states t a unified description of localized li and itinerant t states t is necessary Skorodumova et. al. PRL 89, (2002) Campbell and Peden, Science 309, 713 (2005) Esch et al. Science 309, 752 (2009)
5 Mott insulator NiO Insulating in both anti-ferromagnetic (T<TT c ) and paramagnetic (T>TT c ) Band theory without symmetry breaking is doomed to fail Nature of the gap opening: Slater vs Mott model Small gap from LDA in AFM phase 3d n-1 3d n L -1 3d n+1 VB CB E F
6 Failure of LDA/GGA and its extensions for correlated systems LDA VB CB VB CB E F E F
7 Fundamental deficiencies of LDA/GGA underestimates E tot for systems with factional charges delocalization li error (self-interaction) ti overestimates E tot for systems with factional spins static correlation error (strong correlation) A. J. Cohen, P. Mori-Sanchez and W. Yang, Science, 321, 798 (2008)
8 DFT band gap problem exact Concave (e.g. HF) Convex (e.g. LDA/GGA) More localized states have more severe delocalization error Open-shell d/f-electron systems: both fractional charge and fractional spin errors are present 8 Perdew et al. (1982); Mori Sanchez, Cohen and Yang, PRL 77, (2008)
9 Beyond LDA/GGA SIC-LDA (Perdew amd Zunger (1981) occ SIC LDA LDA tot = tot H + XC i ( ) i i E [ n( r)] E [ n( r)] E [ n( r)] E [ n( r)] Hybrid functionals(becke (1991); Perdew et al. (1996); Heyd et al. (2003)) ( ) ( ) E [ n ( r )] = a E + 1 a E + E PBE0 HF PBE PBE XC X X X X C E [ n( r)] = a E ( ω) + 1 a E ( ω) + E ( ω) + E HSE HF,SR ω PBE,SR ωpbe,lr PBE XC X X X X X C LDA/GGA+U (Anisimov et al. (1991)) LDA+DMFT (Kotliar et. al. 2006) LDA+Gutzwiller (Ho et al. 2008; Deng et al. 2008)
10 LDA+U: basic ideas M. Cococcioni & S. de Gironicoli, PRB 71, LDA U LDA+U U VB CB VB CB E F E F
11 LDA+U : a correction from the Hubbard Model Hubbard model for local interaction: Hartree-Fock approximation: For integer occupation Anisimov et al. (1993); Dudarev et 11al. (1998)
12 Assumptions within LDA+U 1. Identification of a local subsystem to which a correction is made 2. A static mean-field e approximation at o for the Hubbard model is adequate to describe many-body interaction in the local subsystem 3. For the double-counting correction, the LDA interaction ti energy for the local subsystem can be described by the same parameters U and J Assumptions similar to 1 and 3 are also made in LDA + DMFT or LDA + Gutzwiller Anisimov, Aryasetiawan and Lichtenstein, 12 JPCM (1997)
13 Example: Jahn-Teller distortion in in KCuF 3 Lichtenstein, Anisimov, Zaanen (1995)
14 Example: Mott insulator NiO U=5 5.2eVJ=0 0.9eV
15 for d/f-electron System localized itinerant
16 GW approximation for quasi-particle excitation 1 st term in a systematic MBPT expansion Screening in quasi-particle picture => accurate for itinerant electrons self-interaction treated at the exact exchange level Figure from P. Rinke et. al. Ψ k Newsletter No. 79, (2007)
17 GW for d/f-electron systems QSGW: T. Kotani, et al., PRB 76, (2007). HSE03-G 0 W 0 : C. Roedl et al. PRB 79, (2009).
18 Assumptions LDA+U as an approximation to GW static ti approximation: frequency dependence in W can be neglected LDA for itinerant states: QP correction are important only for localized d/f-states fully localization limit : localized d/f-states are orthogonal from itinerant states Anisimov et al. JPCM(1997)
19 method LDA+U as the starting point of G 0 W 0 Simplest extension of GW@LDA LDA+U overcome the major failure of LDA for localized states LDA+U U single-particle Hamiltonian as the reference, all states are treated on the same footing no double counting issue Dependence on U implicitly
20 method
21 U-dependence: CeO 2 and Ce 2 O 3 LDA+U: f-d gap strongly depends on U G 0 W both f-d and p-d gaps depend on U weakly Around physical value of U (~5-7 ev), the uncertainty ~ 0.2 ev
22 U-dependence: MnO and NiO CB 3d n+1 E F VB 3d n AFM-II
23 Ce 2 O 3 : Compare G 0 W 0 and Expt. U=5.4 ev? f occ state binding energy is underestimated!!
24 NiO: G 0 W vs expt. U=5.4 ev U=5.44 ev Satellite is missing!! i Jiang, Gomez-Abal, Rinke and Scheffler, in preparation.
25 vs LDA+DMFT Both consider dynamic correction to LDA+U DMFT: strong correlation among localized states only GW considers all states on the same footing Ce 2 O 3 DMFT: D. Jacob et al. EPL 84, 57009(2008); Expt.: J. W. Allen, J. Magn. Magn. Mater. 47/48, 168 (1985)
26 U-dependence Limitations of Allow operational investigation of interaction ti effects Often quite weak for Ln 2 O 3, but stronger for late TMO For quantitative prediction, one needs first-principles approaches to determine U Change of wave functions LDA+U wave functions may not be good enough off-diagonal contributions of self-energy energy are important More Fundamental problems (fail for strong correlation?) Binding energy of occupied localized states significantly underestimated Satellite structure missing
27 Mott insulator in GW? localized VB CB E F itinerant
28 What is a Mott insulator? A material which exhibits a radical breakdown of conventional band theory A material whose insulating gap is due primarily to the one-site Coulomb repulsion U A material with properties similar to NiO An ordinary magnetic insulator B. H. Brandow, Advance in Physics, 26, 651 (1977)
29 Slater vs Mott: Are they really incompatible? Slater model: gap opening due to the anti-ferromagnetic ordering (breaking the spin symmetry) Can not explain the gap in the paramagnetic ordering What is the cause of the anti-ferromagnetic ordering? Mott model: gap opening due to the strong on-site interaction The gap opening is not related to magnetic ordering, but there is always a local magnetic moment even in the paramagnetic phase symmetry is broken How to incorporate this basic physics in a band theoretical framework?
30 Ce 2 O 3 in FM and AFM Ce 2 O 3 Electronic density of states is essentially impendent of spinordering!
31 NiO in different spin configuration
32 Role of symmetry breaking: a unified view? static correlation can be (partly?) described by breaking symmetry (spin, orbital, structure) e.g. spin unrestricted LSDA or HF for H 2 dissociation The essence of the (generalized) Slater model is symmetry breaking, not anti-ferromagnetism Symmetry breaking is real: Peiers transition, Jahn-Teller distortion... Strong correlation can be regarded as one possible mechanism to drive such symmetry breaking Strong correlation is compatible with the band theory
33 A proposal: strong interaction vs strong correlation Weak correlation LDA/GGA for total energy LDA-G 0 W 0 for quasiparticle energy works well Strong interaction Delocalization (self-interaction) error is severe, but can be essentially overcome by orbital-dependent corrections GW with proper starting point or approximate self-consistency for QP energies Broken symmetry if necessary Strong correlation Truly many-body treatment beyond single-particle picture needed 33
34 Concluding remarks Accurate first-principles modelling of d/f-electron systems requires a unified treatment of localized and itinerant states The state-of-the-art GW, either using improved reference as the input for G 0 W 0, or with approximate self-consistency consistency, can generally describe electronic band structure of d/f-electron systems quite well, but the satellite structure is still absent, and occupied localized states binding energy tend to be overestimated high order correlation effects are important Symmetry breaking makes the band-theory description of d/f-electron systems compatible with strong local Coulomb interaction a unified treatment of localized and itinerant states is possible
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