Quasiparticle Self-Consistent GW Theory

Transcription

1 Quasiparticle Self-Consistent GW Theory Hands-on Course, Daresbury, March 2014 What is Quasiparticle self-consistency? How does it differ true self-consistency? PRL 96, ; PRB76, v Advantages - Optimal means to design one-body hamiltonians - An optimal starting point for many-body theory - Dual character makes a potentially powerful tool to study electronic structure without ambiguities inherent in LDA+U, or hybrid functionals, or LDA+GW, etc. v Limitations - Source of errors, and prospects for their amelioration - Very costly: N 4 scaling. How to surmount scaling problem? v Illustrations for representative systems.

2 Beyond the Local Density Approximation Ø Q: What is E xc [n]? All many-body effects lumped into it. Ø Kohn and Sham assumed E xc [n] to be a universal, local energy functional of n(r) : δ Exc n( r) εxc n( r) d r, Vxc( r) = n( r) n( r) δ n() r ( xc ) 3 [ ] ε [ ] Universal calculate almost exactly for model, e.g. jellium Ø LDA, variants have become standard workhorse --- now widely used in almost every branch of science, engineering Ø An extremely simple and universal theory all the complexity is folded into a simple and universal functional. Ø Key point: all electrons see same effective potential V eff (r). In contrast to Hartree-Fock (nonlocal V x (r,r ) each electron sees a different V eff (r)).

3 Failures in the Local Density Approximation Semi. bandgaps too small Poor Na bands Systematic overbinding LDA CoO is metalllic but real CoO is AFM insulator (~4 ev gap) ϕ B Schottky barriers at metal-semi contacts fall too close to Valence Band

4 GW Approximation Ø Hedin s GW approximation (1965): a major advance on Hartree-Fock theory. Conceptually, the Fock V x gets replaced by GW. G = Green s Function, W = screened coulomb interaction Ø Hartree Fock: e senses an effective potential V x owing to correlated motion. V x = functional derivative of E x and can be written in terms of Green s functions as: 1 3 Vx () r = G(, r r ) d r r r GW: Hartree-Fock like but the coulomb e e repulsion 1/ r r is dynamically screened: Vbare(, rr ) = W(, rr, ω) = r r ε(, r r, ω) r r Ø More rigorously: GW = the lowest order term in an exact expansion in W (many-body perturbation theory)

5 Two possible explanations for LDA error What is the dominant source of difficulty in the L(S)DA? Explanation I: Ansatz for E xc [n] is the primary cause. Explanation II: Kohn-Sham ψ i and eigenvalues ε i the Lagrange multipliers of the KS hamiltonian Ĥ σ KS = 2 2m 2 + [V σ KS (r) = V H (r) +V ext (r) +V σ xc (r)] are fictitious. Q: How do we assess the source of error? A: Density-functionalize nonlocal functionals and check. Not a strict division (there is an interplay between them). But roughly: For ground state properties, I is the primary problem For excited state properties II is the primary problem.

6 Connection between DFT and QP levels ψ i and ε i fictitious discontinuity Δ in V xc betw/ highest occ state and next higher one Grüning, Marini, Rubio, (J. Chem. Phys. 124, ) evaluated Δ xc by making OEP (density functionalized) GW for Si, LiF,Ar Results show: OEP gap (EXX+RPA) close to usual LDA gap. Thus Explanation II: the fictitious nature of ψ i and ε i are the primary problem

7 Many attempts to extend the LDA Ø Good ground-state properties in weakly correlated systems. Ø Excited state properties are much worse. Many attempts to extend, improve on the LDA Self-Interaction Correction (Perdew, Zunger, PRB 23, 5048 (1981)) LDA+U (Anisimov, Zaanen, Andersen, Phys. Rev. B 44, 943 (1991)) LDA+Screened exchange, (Seidl et al, PRB 53, 3764 (1996)) LDA+DMFT (Anisimov et al, J. Phys. C9, 7359 (1997) ) Mix Hartree-Fock with LDA (B3LYP, Becke; Scuseria) Optimized Effective Potential (Kotani, PRL 74, 2989 (1995) Exact exchange, EXX+RPA closest to LDA: local potential V eff (r). Ø All have significant successes to their credit, but improve one or another property in some special cases. Ø Removing locality is essential but without removing the ansatz LDA starts with, hard to systematically improve on the basic framework

8 GW Approximation Ø Hedin s GW approximation (1965): a major advance on Hartree-Fock theory. Conceptually, the Fock V x gets replaced by GW. G = Green s Function, W = screened coulomb interaction Ø Hartree Fock: e senses an effective potential V x owing to correlated motion. V x = functional derivative of E x and can be written in terms of Green s functions as: V bare (r, r ) = V x (r) = i G(r, r ) 1 r r d 3 r 1 r r W (r, r,ω ) = 1 ε(r, r,ω ) = igv GW: Hartree-Fock like but the coulomb e e repulsion 1/ r r is dynamically screened: 1 r r ; Σ = igw Ø More rigorously: GW = the lowest order term in an exact expansion in W (many-body perturbation theory)

9 Advantages of the GW Approximation Ø The GW approximation can potentially redress the worst failings inherent in both Hartree-Fock and LDA: Ø HF : the nonlocality present, but not screened (disaster) Ø LDA: nonlocality pathologies in local potential. (Many problems, e.g. cannot break reversal symmetry). Ø But GW is a perturbation theory: first term in an expansion in W. Perturbation theory must be carried out around some starting point H 0. How choose H 0? Ø Major development (Hybertsen and Louie, 1987): use LDA as starting point H 0 =H LDA G=G LDA, W=W LDA ; Σ = ig LDA W LDA Ø Hugely successful in semiconductors

10 Failings in LDA-based GW If H 0 = H LDA, then GW G LDA W LDA NiO only slightly improved over LDA G LDA W LDA bandgaps are underestimated Fermi surfaces, magnetic structure in metals can be nonsensical, e.g. in FeTe. Many other problems; 10 see PRB B74, (2006)

11 Ambiguities in GW from ambiguities in H 0 The GW approximation significantly ameliorates errors in E G. Unlike many extensions to the LDA, (hybrid functionals) it is true ab initio. But GW is a perturbation theory usually calculated around a noninteracting H 0. But G LDA W LDA works well only in special cases. Change H 0 improve result, but not universal or predictive. Ambiguities cannot be avoided. From Patrick Rinke, CECAM Workshop Green's function methods, Toulouse, 5 June 2013

12 Quasiparticle self-consistent GW Approximation A new, first-principles approach to solving the Schrodinger equation within Hedin s GW theory. Principle : Can we find a good starting point H 0 in place of H LDA? How to find the best possible H 0? Requires a prescription for minimizing the difference between the full hamiltonian H and H 0. QSGW : a self-consistent perturbation theory where self-consistency determines the best H 0 (within the GW approximation) PRL 96, (2006)

13 QSGW: a self-consistent perturbation theory Partition H into H 0 + ΔV and (noninteracting + residual) in such a way as to minimize ΔV : G 0 1 GWA 1 = G = ω H ω H +ΔV ω 0 0 ( ) 0 If the GWA is meaningful, G 0 G Q: How to find G 0 that minimizes ΔV G 0? ( ( )) ω ( H +Δ V( ω)) G( ω) = δ( r r') We seek the G 0 (ω) that most closely satisfies Eqn. of motion ω ( H +ΔV( ω)) G ( ω) δ( r r') ( ) 0 0 ΔV( ω) G ( ω) 0 0

14 Optimal G 0 Start with some trial V xc (e.g. from LDA, or ). Defines G 0 : H 0 = 1 2m 2 +V ext (r) +V H (r) +V xc (r, r ) H 0 ψ i = E i ψ i G 0 (r, r,ω) = i ψ i (r)ψ * i ( r ) ω E i GWA determines ΔV and thus H : RPA GWA ε(ig 0 G 0 ) Σ(r, r,ω) = ig 0 W; ΔV = Σ V xc G 0 Find a new V xc that minimizes norm M, a measure of ΔV G 0. xc 1 ( ) V = ψi Re Σ ( Ei) +Σ ( Ej) ψ j 2 ij (approximate) result of min M Iterate to self-consistency. At self-consistency, E i of G matches E i of G 0 (real part). See Phys. Rev. B76,

15 Dual Nature of QSGW : framework for H 0 Except for specialized many-body effects, properties of interest are typically sufficiently described by H 0, e.g. semiconductor band offsets, magnetic moments, transport. QSGW (QSGW+BSE) generates a nearly optimal H 0 for many kinds of materials classes Graphene CuInSe 2 E G Electron pocket ErAs LDA QSGW T N Hole pocket

16 Dual Nature of QSGW : many body physics QSGW generates both an optimal H 0 or G 0 and an interacting G with dynamical, many-body effects. Impact ionization, ZnS Spin waves, NiO ARPES, BaFe 2 As 2 QSGW is relatively simple important many-body effects (superconductivity, Kondo physics, ) not captured. But it provides an ideal framework to handle such correlations. Key point: what correlations are depends on reference!

17 For weakly correlated sp systems, very well. How Well does it Work? Gap too large by ~0.3 ev Band dispersions ~0.1 ev Na GaAs Ga d level well described m * (QSGW) = m * (LDA) = m * (expt) = Na bandwidth reduced relative to LDA by 15% a key point! (see later)

18 Critical points, m* in sp bonded systems Γ Γ L L X X E 1 E 0 E 2 m* Oxides CP s always slightly overestimated ( ); m* mostly quite good 18

19 QSGW theory applied to d systems Fe: nearly ideal agreement with ARPES in FL regime Magnetic moments: well predicted in a GW context. But M overestimated in itinerant magnets. maj M (Calc) CaFe 2 As 2 (x) FeAl Ni min QSGW LDA Co NiO Fe Ni 0 3 Al MnAs M (Expt) * d band exchange splitting and bandwidths well described (exception: Ni )

20 Is Self-consistency important? G LDA W LDA vs LDA, Fe QSGW vs LDA, Fe Not always G LDA W LDA and QSGW scarcely distinguishable in Fe Sizeable modification of n(r); little effect on band structure

21 G LDA W LDA vs QSGW : FeTe superconductor G LDA W LDA vs LDA, FeTe QSGW vs LDA, FeTe G LDA W LDA : Fe d /As p alignment shifts by ~1/2 ev puts Fe d states too high. Fermi surface nonsensical. QSGW: marked narrowing of Fe d bandwidth. Velocities at E F 10-30% smaller. F.S. roughly similar to LDA.

22 Need for Self-Consistency even in sp Systems Cu(In,Ga)Se 2 solar cells * Modelling depends critically on band structure being realistically described. LDA is hopeless; even G LDA W LDA gap ~ 0 (as it is in InN) What about hybrid functionals, or LDA+U? Strong interplay between gap, dielectric response. Self-consistency essential to fully describe it. Vidal et al, PRL 104, (2010) E g [ev] DFT-LDA corrected DFT G 0 W 0 sc-cohsex sc-cohsex+g 0 W u sc-cohsex+g 0 W 0 HSE06 HSE06+ε 0 ¼ u

23 QSGW is not true self-consistent GW True self-consistent GW (scgw) Σ= igw G Π= igg W = v G = 1 1 ε ω T+ VH+ Vext+Σ ( ) True self-consistent GW looks good as formal theory: à Based on Luttinger-Ward functional. à Keeps symmetry for G à Conserving approximation But poor in practice, even for the electron gas Z-factor cancellation is not satisfied (next slides) G 0 W 0 GW Noninteracting B. Holm and U. von Barth, 23 PRB57, 2108 (1998)

24 Residual of this pole (loss of QP weight) is reduced by Z Write G as Also, Therefore, Z-factor cancellation in Σ Exact Σ=iGWΓ. Suppose W is exact. Then 1 G0 = ω H + iδ G = 0 1 xc ω H0 V +Σ ( ω0) + ( Σ/ ω) ( ω ω0) + iδ ω0 Z = 1 Σ/ ω GW ( ) 1 G 0 = ZG + (incoherent part) Γ = 1 Σ / ω = Z 1 for q, ω 0 0 Γ G W + (incoherent part) q ω (Ward identity) q ' ω ' G W q q' ω ω' 24 Γ

25 Z-factor cancellation in Π W=(1 Πv) 1 v is not exact, either. A similar analysis for proper polarization Π. Π= iggγ ig0g0 + (incoherent part) (See Appendix A in PRB76, (2007)). In the exact fully self-consistent theory, Z-factors cancel QP-like contribution in complicated ways. Self-consistent GW neglects Γ, so no Z-factor cancellation results rather poor. Higher order diagrams required to restore Z-factor cancellation. Complexity avoided by doing perturbation theory around a noninteracting H 0 : convergence more rapid for a given 25 level of approximation.

26 Na as approximate realization of HEG The G 0 W 0 bandwidth of the HEG narrows by ~10%. The scgw bandwidth widens by ~20%. QSGW predicts the Na bandwidth to narrow relative to LDA by ~10%, G 0 W 0 GW Noninteracting in agreement with PE and standing wave measurements. Holm and von Barth PRB57, 2108 (1998) Shirley showed that the next order diagram, sc:gw+gwgwg approximately restores the G 0 W 0 bandwidth PRB 54, 7758 (1996) in true 26 scgw.

27 Where does QSGW Break Down? Unoccupied states universally too high ü ~0.2 ev for sp semicond; ü <~1eV for itinerant d SrTiO 3, TiO 2 ü >~1eV for less itinerant d NiO ü ~3 ev for f Gd,Er,Yb Peaks in Im ε(ω) universally too high ε universally 20% too small. Errors are all related!

28 Origin of Error: missing diagrams in W A blue shift in the plasmon peak has effect on Re[ε(0)] (Kramer s Kronig relation): Δ = δω ( ω ) = < 1 δω ( ωth ) exp Re χ1(0) dω 0 π ω ω π ωth ω exp ε too small because of blue shifts in plasmon peaks. GW uses RPA approximation for the polarizability Π =ig 0 G 0, and But e and h + are attracted via W, e.g. by ladder diagrams, G 0 G 0 W = ( 1 Πv) 1 v = ε 1 v G 0 G 0 + W + + (Ladders needed for good optical spectra) Conclusion: W calculated via RPA is too large, by 25% at q,ω=0.

29 NiO: a correlated TM oxide Bands of both sp and d character are present QSGW Expt Differentiate orbital character (sp vs d) : compare to peaks in BIS spectra. sp character states too high by ~0.3 ev (similar to tetrahedral semiconductors) d conduction band too high by ~1.1 ev (similar to SrTiO 3 )

30 Need for Nonlocality in Σ LDA+U, DMFT are site nonlocal (nonlocality in orbitals of different m), local betw/ sites: Σ is independent of k. Assess nonlocality by the substitution Σ(k) Σ = BZ dk Σ(k) Weak correlations: Si Σ(k) Σ V xc LDA Result: Σ V xc LDA Fe: Σ d bandwidth too large, similar to LDA. (Not selfconsistent: magnetic moment poor)

31 Schafer et al (PRB72, (2005) used k-dependence of ARPES peaks to extract v F for these peaks on the (100 line) in Fe : VI, V-VII, I, II (Measurement of limited accuracy ---reported 30%) Fermi Velocities in Fe Point v F (ARPES) v F (QSGW) (ev-a) VI V-VII I II LDA+DMFT has no nonlocality in Σ does not describe v F properly ARPES peaks overlap --- cannot resolve

32 Limitations to QSGW: cost QSGW is expensive: as now implemented scales as N 4 : 32 atoms feasible on 12 processor cluster 100 atoms on large facility (Cu 2 ZnSnS 4 ) 4 Example: 32 atom supercell of Kesterite Bulk E G = 1.5 ev (expt) Thermodynamic calc predict: Kesterite has numerous antisite defects, e.g. Cu Zn. Model as Cu 9 Zn 3 Sn 4 S 16. Cu Zn : shallow acceptor but cell too small to pinpoint level + Cu Zn

33 Large atom limit: Cut and paste Σ Near-sightedness principle: Σ is finite ranged and perturbed only in the vicinity of a defect. Prescription: 1. Calculate Σ in a small cell with a point defect. 2. Do the same calculation for the perfect crystal. 3. Write as a Bloch sum. Σ RL, R L (k) = Σ R+TL, R L e ik T Ansatz: assume Σ R+T,R given by the defect for all pairs (R+T,R ) that touch defect site or its NN, and Σ R+T,R is that of the bulk for all other pairs. Should be exact provided shell is large enough. Makes it possible to embed defects with good qualityσ in large supercells nearly as good as a full QSGW calculation.

34 Test: As Antisite in GaAs Experiment: Localized level at VBM ev. LDA puts As Ga at ~0.35eV w/ E G = 0.3eV Different variants LDA+XX may predict the same bandgap, yet place the deep level at different positions within the gap. Defect level QSGW-derived : band CG comes out near exptl position

35 Other ways to Bridge Length Scales Algorithmic improvements ~ atoms with efficient use of parallel architectures. Sufficient for many key properties at the nanoscale level, e.g. band offsets, energy levels of defects. Alternate strategies: Use QSGW H 0 to: 1) Map onto reduced classical H 0, or quantum H 0, (e.g. Wannier functions, many-body context) 2) Use QSGW as a parameter generator for empirical hamiltonians, classical or quantum type. An electronic device simulator requires energy bands, scattering matrix elements.

36 QSGW as an engine for Classical Simulations Classical transport: Sophisticated techniques developed to solve Boltzmann transport equation, which can model real electron devices. Example: band Cellular Monte Carlo (CMC) (Saraniti, ASU). CMC simulators can feed (in principle) into higher-level simulators, e.g. circuit simulators, or solar cell simulators. CMC requires as input: energy bands, scattering matrix elements (electron-phonon, impurity scattering) These can be supplied by QSGW. Makes feasible an ab initio device simulator. (Future area of research) Device I(V) CMC ε(k), σ QSGW

37 Bridging Length Scales: feasibility demo First step: feed QSGW bands into Cellular MC simulator. Test case: GaAs (bands very well known from experiment) Velocity-field characteristics of bulk GaAs, computed by CMC QSGW and empirical PP energy bands. Electron energy distribution function in a 3D CMC simulation of GaAs MESFET Corresponding current-voltage characteristics. Slight differences with a calculation using an EPM band structure.

38 Clean Fe/MgO/Fe tunnel junction Tunneling Magnetoresistance: Trilayer has two possible magnetic configurations Apply small voltage V to trilayer tunneling current. Conductance G depends on relative alignment of top and bottome spins. Define tunneling magnetoresistance Fe MgO Fe or T MR = (G G ) / G An efficient way to electrically detect the magnetic state of a spot on a surface (bit on disk drive) Basis for modern read heads.

39 Clean Fe/MgO/Fe tunnel junction The LSDA predicts a zero bias anomaly as a consequence of a surface resonance near E F. Apply small voltage V: E F swept off resonance minority channel DOS reduced coupling to the leads T MR. Experiment: zero bias anomaly never observed. Instead, a peak in di /dv is seen near V=0.15V (Zermatten et al, PRB 78 (2008)) QSGW finds 2 surface resonances, one at E F +0.15eV. Is the experiment explained by the QSGW band structure? Need to transport calculation to find out.

40 Clean Fe/MgO/Fe tunnel junction Challenge: QSGW is already expensive. Also, standard transport by Landauer-Buttiker theory requires semi-infinite leads, not well suited to QSGW. Solution: Derive analytic formula for T MR in terms quantities accessible to a method with periodic BC. Landau T σ σ D (E) = 1 N k=k ψ σ ke (L) 2 e 2γ ke d LR ψ σ (R) 2 ke Replace normalized ψ 2 with D(E); compensate with constant scaling λ. λ N D(L), D(R), are any pair of planes in the MgO, separated by d LR. k=k D σ ke (L) e 2γ ke d LR D σ (R) ke γ = wave number of most slowing decaying 40 evanescent state.

41 Clean Fe/MgO/Fe tunnel junction Compare various calculations of T D. In the LDA cases FP and ASA) confirm zero bias anomaly of other LDA calculations. QSGW T D tracks measured I-V characteristics very well. So, the observed I-V is an intrisinsic property of the ideal interface, not an artifact of nonidealities T _AP D 10 4 LDA QSGW d 2 I/dV E-E v Correlation effects critically affect H 0 in this case. 0

42 Conclusions Ø The QSGW approximation - has some formal justification. Ø Unique features: - Reliably treats variety of properties in a wide range of materials in a true ab initio manner. A kind of gold standard at the 1-particle level - The errors are systematically improvable. " - Truly predictive when correlations are not strong. Limitations: - Does not handle strong correlations properly ü Include extra diagrams, or combine with DMFT. - Cost: N 4 scaling ü Build reduced or model hamiltonians for defect studies, ab initio device simulation ü Redesign algorithms.