wien2wannier and woptic: From Wannier Functions to Optical Conductivity

Transcription

1 wien2wannier and woptic: From Wannier Functions to Optical Conductivity Elias Assmann Institute of Solid State Physics, Vienna University of Technology AToMS-2014, Bariloche, Aug 4

2 Outline brief introduction to maximally localized Wannier functions some applications of Wannier functions woptic: conductivity σ(ω), thermopower S from LDA+DMFT Big Thanks to: Original code authors As well as Karsten Held Peter Blaha Philipp Wissgott Waltzing Atoms Jan Kuneš Institute of Physics, Prague Starting Grant Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

3 From Bloch to Wannier w R = V (2π) 3 dk e ikr ψ k BZ Transform k R; r is spectator Properties: periodicity w R (r) w 0 (r R) orthonormality w nr w mr = δ nm δ RR from Marzari et al. Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

4 Gauge freedom But ψ k carries arbitrary phase φ(k) w R = V (2π) 3 dk e i(φ(k) kr) ψ k BZ strongly non-unique (shape, localization) choose φ(k) for maximum localization of w R from Marzari et al. Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

5 Maximally localized WF (Marzari-Vanderbilt) [Marzari et al., RMP (2012)] In the multiband case, e iφ(k) unitary matrix w nr = e BZdk ikr U (k) mn ψ mk m Total spread Ω := n r 2 n = Ω[U] + Ω I MLWF procedure: minimize Ω[U] (Wannier90) input: M (k,b) mn = ψ mk e ir(k+b) ψ nk optional input: A (k) mn = ψ mk g nk output: U (k) nm, ( H (R) nm,... ) gauge dependent gauge independent wien2wannier Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

6 Maximally localized WF (Marzari-Vanderbilt) [Marzari et al., RMP (2012)] In the multiband case, e iφ(k) unitary matrix w nr = e BZdk ikr U (k) mn ψ mk m Total spread Ω := n r 2 n = Ω[U] + Ω I MLWF procedure: minimize Ω[U] (Wannier90) input: M (k,b) mn = ψ mk e ir(k+b) ψ nk optional input: A (k) mn = ψ mk g nk output: U (k) nm, ( H (R) nm,... ) gauge dependent gauge independent wien2wannier Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

7 wien2wannier [Kuneš et al., Comp. Phys. Commun. (2010)] MLWF from full-potential LAPW code WIEN2k interface to Wannier90 A (k) mn, M (k,b) mn initial projections: atom-centered basis functions, rotations real-space plots (wplot) XCrySDen, VESTA available from wien2k.at unsupported to be included in WIEN2k 14.1 distribution Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

8 Applications of MLWF analysis of chemical bonding electric polarization and orbital magnetization BerryPI [Ahmed et al., Comp. Phys. Commun. (2013)] Wannier interpolation K K H(k) K F H(R) K 1 Wannier functions as basis functions tight-binding model H(k) = U + (k) ε(k) U(k) realistic dynamical mean-field theory (DMFT) F 1 H(k) K Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

9 Choice of Wannier subspace [Kuneš et al., Comp. Phys. Commun. (2010)] SrVO 3 : V-e g V-t 2g O-p 3-band projection V-centered t 2g orbital Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

10 Choice of Wannier subspace [Kuneš et al., Comp. Phys. Commun. (2010)] SrVO 3 : V-e g V-t 2g O-p 3-band projection 14-band projection V-centered t 2g orbital Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

11 U, U, J for t 2g Wannier orbitals [Ribic, EA, et al., arxiv: ] Local Coulomb interaction in cubic symmetry: U U J intra-orbital repulsion inter-orbital repulsion Hund s exchange [Held, Adv. Phys (2007)] U = U + 2J Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

12 U, U, J for t 2g Wannier orbitals [Ribic, EA, et al., arxiv: ] Local Coulomb interaction in cubic symmetry: U U J intra-orbital repulsion inter-orbital repulsion Hund s exchange U = U + 2J for atomic orbitals in crystal: hybridization 3 independent parameters U U 2J SrVO 3 BaOsO 3 (3d) (5d) Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

13 U, U 0, J for t2g Wannier orbitals [Ribic, EA, et al., arxiv: ] Local Coulomb interaction in cubic symmetry: U intra-orbital repulsion U 0 inter-orbital repulsion J Hund s exchange U = U 0 + 2J for atomic orbitals in crystal: hybridization U U0 2J 3 independent parameters SrVO3 (3d) BaOsO3 (5d) BaOsO3 t2g + p Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

14 woptic ω [ev] adaptive k-integration optical conductivity σ(k, ω) σk (ω) [10 6 Ω 1 ev 1 ] including local self-energy Σ(ω) (DMFT) thermopower full matrix elements ψ ψ from WIEN2k W L Γ X W K 2 ω [ev] 0 2 (Aluminum) [J. Sofo] Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

15 Tetrahedral mesh management tetrahedron T has integration error estimate ɛ T refine T if ɛ T Θ max T ɛ T harshness Θ [0, 1] enforce regularity at most one hanging node for stable convergence enforce shape stability no heavily-distorted octahedra avoid large integration errors X X L Γ W K L Γ W K Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

16 Tetrahedral mesh management tetrahedron T has integration error estimate ɛ T refine T if ɛ T Θ max T ɛ T harshness Θ [0, 1] enforce regularity at most one hanging node for stable convergence enforce shape stability no heavily-distorted octahedra avoid large integration errors Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

17 Tetrahedral mesh management tetrahedron T has integration error estimate ɛ T refine T if ɛ T Θ max T ɛ T harshness Θ [0, 1] enforce regularity at most one hanging node for stable convergence enforce shape stability no heavily-distorted octahedra avoid large integration errors [Ong, SIAM J. Sci. Comput. (1994); Endres & Krysl, Int. J. Numer. Meth. Engng. (2004)] Kuhn triangulation Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

18 Some results for Na x CoO 2 [Wissgott et al., PRB (2010)] Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

19 Some results for Na x CoO 2 [Wissgott et al., PRB (2010)] Thank you for your attention Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 12

20 Bloch waves Bloch s theorem Ĥ = 2 + V(r) with V(r + R) V(r) has solutions Ĥ ψ nk = ε n (k) ψ nk with ψ nk (r) = e ikr u nk (r); u n,k (r + R) u n,k (r) and ε n (k + K) ε n (k) (Simultaneous eigenbasis of Ĥ and translation operators T R ) Usual basis for solid-state calculations But for many applications, a localized basis is preferable Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 8

21 r-localization and k-smoothness sin x + 1 sin 3x sin 9x 3 9 w R = V (2π) 3 dk e i(φ(k) kr) ψ k BZ sin x sin x sin 3x + + sin 49x 49 1 sin 3x + + sin 249x 249 Idea: choose φ(k) for maximum localization of w R Fourier series converges faster for smoother functions: f C p f n const n p [Wikipedia, Gibbs phenomenon ] Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 8

22 From bands to WF isolated band isolated group of bands entangled bands E(k) k k k w k = e iφ(k) ψ k w nk = U (k) mn ψ mk w nk = U (k) in V(k) mi ψ mk Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 8

23 Disentanglement Cu (fcc): What to do when #bands > #WF? Ansatz: Select optimally smooth subspace rectangular matrix V k (#bands(k) #WF)! Ω I = Ω I [V] minimize ΩI [V] from Marzari et al. 5 d-like WF, 2 interstitial s-like WF Ω = Ω[U] + Ω I [V] woptic not implemented with disentanglement Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 8

24 Anatomy of a calculation WIEN2k optic Kohn-Sham states momentum matrix ψ nk (r), ε n (k) ψ nk ψ mk woptic BZ integration σ(ω), σ(0), S wien2wannier k-space overlaps M (k,b) mn = ψ m,k ψ n,k+b Wannier90 max. loc. WF w nr (r) DMFT self-energy Σ(ω) Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 8

25 Transport properties with WIEN2k BoltzTrap semi-classical (Boltzmann) band velocities ε(k)/ k instead of momentum matrix elements ψ ψ BoltzWann similar, with Wannier functions woptic quantum-mechanical linear response (Kubo) adaptive BZ integration inclusion of local self-energy Σ(ω) more information: WIEN2k.at reg. users unsupported wien2wannier woptic preprint Wissgott et al., PRB (2012) Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 8

26 Calculating optical conductivity and thermopower Very schematically: linear response (Kubo formula): Ĥ = Ĥ0 + λ(t) B Â (t) = Â 0 i h t dt λ(t ) [A(t), B(t ) ] 0 + σ, S: current operators Ψ + Ψ ( Ψ + ) Ψ [ĵ(r, χ ret el el (r r, t t ) θ(t t ) t), ĵ(r, t ) ] 0 [ĵ(r, χ ret el heat (r r, t t ) θ(t t ) t), Q(r, t ) ] 0 el. current operator heat-current operator Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 8

27 Interpolation and random-phase problem DMFT in Wannier basis self-energy Σ i (ω) spectral function A(k, ω) in evaluation of χs: { } tr v(k) A(k) v(k) A(k) Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 8

28 Interpolation and random-phase problem DMFT in Wannier basis self-energy Σ i (ω) spectral function A(k, ω) in evaluation of χs: { } tr v(k) A(k) v(k) A(k) from DMFT momentum matrix w nk w mk ; from WIEN2k ψ nk carries random phase φ n (k), which propagates to v but not A problem with w w (v wx A xy v yz A zw ) and w ψ (v wi A ii v ix A xw ) terms interpolate v(k) v( k) Elias Assmann (TU Vienna) wien2wannier and woptic Bariloche / 8