Wannier Functions: ab-initio tight-binding Jonathan Yates Cavendish Laboratory, Cambridge University Representing a Fermi Surface Accurate description of Fermi surface properties requires a detailed sampling of the Brillouin Zone Lead Fermi surface Al Fermi surface Representing a Fermi Surface • Accurate description of FermiFe fermi surface surface properties requires a detailedScalar sampling Relativistic of the Brillouin Zone with spin-orbit coupling • Full first-principles calculation expensive Lead Fermi surface Al Fermi surface Outline •Wannier Functions •One band •Isolated Set of Bands •Entangled Bands •Wannier Interpolation •Accurate and Efficient approach to Fermi surface and spectral properties •Examples •Anomalous Hall Effect •Electron-Phonon Coupling Wannier Functions V ik.R iφ(k) wnR(r) = ψnk(r)e− dk ψ (r) e ψ (r) (2π)3 nk − nk !BZ → Bloch state GaAs Wannier function Quantum # k -5 -5.5 Quantum # R -6 -6.5 -7 -7.5 -8 -8.5 -9 L G X K G Wannier Functions V ik.R iφ(k) wnR(r) = ψnk(r)e− dk ψ (r) e ψ (r) (2π)3 nk − nk !BZ → Bloch state GaAs Wannier function Quantum # k -5 -5.5 Quantum # R -6 -6.5 -7 -7.5 -8 -8.5 -9 L G X K G Si Multiband - Generalized WF 6 4 2 ψ (r) U (k)ψ (r) 0 nk → mn mk -2 n -4 ! -6 L G X K G Maximally Localised Wannier Functions Choose Uk to minimise V (k) k.R w (r) = U ψ (r) e− dk nR (2π)3 mn mk quadratic spread BZ " m $ ! # (Marzari, Vanderbilt) u u ! k| k+b" Ab-initio code Wannier code U (k) u WF defined in basis of | k! bloch states Si valence bands 6 4 2 0 -2 -4 Wannier localisation in R gives rot -6 (k) ψnk (r) = Umnψmk(r) Bloch smoothness in k L G X K G m ! Wannier Functions: A local picture 1- Intuitive picture of local bonding Si GaAs BaTiO3 2- Wannier centres give bulk polarisation in a ferroelectric Paraelectric Ferroelectric MLWF - Entangled Bands Disentanglement procedure (Souza, Marzari and Vanderbilt) Obtain “partially occupied” Wannier functions Essentially perfect agreement within given energy window Copper Wannier Interpolation Accurate and Efficient approach to Fermi surface and spectral properties •Exploit localisation of Wannier functions. •Require only matrix elements between close neighbours. Lead Fermi surface 1st principles accuracy at tight-binding cost Interpolation of any one-electron operator Wannier Interpolation 1- Obtain operator in Wannier basis ik R O (R) = Om Oˆ Rn = e− · UO(k)U † mn ! | | " !k Wannier Interpolation 1- Obtain operator in Wannier basis ik R O (R) = Om Oˆ Rn = e− · UO(k)U † mn ! | | " !k 2- Fourier transform to an arbitrary k-point, k’ ik. R Omn(k!) = e · Omn(R) !R Wannier Interpolation 1- Obtain operator in Wannier basis ik R O (R) = Om Oˆ Rn = e− · UO(k)U † mn ! | | " !k 2- Fourier transform to an arbitrary k-point, k’ ik. R Omn(k!) = e · Omn(R) !R 3- Un-rotate matrix at k’ ψm,k O ψn,k = [U † O(k!)Uk ] ! ! | | ! " k! ! What we need diagonalises H All operations involve small matrices (NwanxNwan) => FAST! Convergence Lead Wannier interpolated object converges to ab-initio value, exponentially with k-grid sampling. Band interpolation error 4x4x4 10x10x10 K-mesh size Wannier Interpolation - Summary Bloch States Wannier basis eg planewave basis • Each k-point expensive •Each k-point cheap • Calculation of MLWF •Introduce occupancies and converges exponentially compute properties (only polynomial convergence) Sample on coarse grid Sample on fine grid Wannier Interpolation - Fe •18 spinor Wannier functions •Keep up to 4th neighbour overlaps •Cost 1/2000 of full calculation Wannier Interpolation 1 0.5 0 -0.5 -1 -1.5 -2 ! H Wannier Interpolation pt2 1- k-derivatives can be taken analytically eg band gradient ik R Hk = e · H(R) !R ∂Hk ik R = i Rαe · H(R) ∂kα !R repeat for higher derivatives 2- Position operator matrix elements well defined Hall Effect ! !!!""##$$%%&&''##((!!))'**!+,,+-.!/)'****!!0011223344 Ordinary Hall Effect (Hall 1879) ! !!!BCrossedB##669999++$$!!!"" !E'!'& &$and$!!!!! ,!,%B+%+ *$*fields$99 ! !!!B!! !??breaks##++''@@99!!..%%7 7time-reversal++CC##++AA++##99''** ! !!!;;66!!77''>>&&++..%%--!6!6##$$++##!&!&++++$$++$$ !! Anomalous!!!!55&&6677''**66 88Hall99!)' *Effect*!+,,+-.! /)(Hall)''****!!00 11880)111::44!! !!!No!;;66! !!B!!, ,%field%++**$$!&!& ++needed!++$$++$$<< !!!Ferromagnetism!==++####6677''>>&&++.%.%9977!?!?##++' '@@99!! ..%%breaks77++!!##++AA++# #9time-reversal9''** !! Mechanisms for the AHE Intrinsic 1954 Karplus and Luttinger property of electron motion in perfect lattice “dissipationless” current (spin-orbit effect) Extrinsic 1955 Smit “skew scattering” 1970 Berger “side-jump” Intrinsic (again) 1996- Rexamination of KL in terms of Berry’s phases Niu (Texas), Nagaosa (Tokyo) Electron Dynamics Textbook wavepacket dynamics 1 ∂εnk r˙ = k˙ Ωn(k) ! ∂k − × !k˙ = eE er˙ B − − × Electron Dynamics Textbook wavepacket dynamics missing a term: “anomalous velocity” 1 ∂εnk r˙ = k˙ Ωn(k) ! ∂k − × “k-space magnetic field” caused by lattice potential “k-space Lorentz force” even when B=0 !k˙ = eE er˙ B − − × Anomalous Hall Conductivity e2 σ = − dk f (k) Ω (k) xy (2π)2h n n,z n BZ ! " Fermi function Berry Curvature Berry Curvature Berry Ω (k) = Im u u Curvature n − "∇k n,k| × |∇k n,k% cell periodic Berry Bloch state An(k) = i unk k unk Ω (k) = A (k) Potential ! |∇ | " n ∇ × n Ab-initio Calculation Finite-differences difficult: phases, band-crossings bcc iron (LAPW) Yao et al (PRL 2004) vnm,x(k) vmn,y(k) Kubo formula Ωn,z(k) = 2Im 2 − (ωm(k) ωn(k)) m=n !! − Extremely computationally expensive! final number within 20% of experiment Jonathan Yates Tyndall Institute 27th March 2007 Band-structure bcc Fe 4 2 0 -2 -4 -6 -8 ! H P N ! H N ! P N DFT (PBE) + spin-orbit Spin up Spin down Berry curvature bcc Fe 1e5 4 2 1e3 0 1e1 -2 -1e1 -4 -6 -1e3 Berry Curvature (atomic units) -8 -1e5 ! H P N ! H N ! P N Berry curvature bcc Fe 1e5 4 2 1e3 0 1e1 -2 -1e1 -4 -6 -1e3 Berry Curvature (atomic units) -8 -1e5 ! H P N ! H N ! P N Berry Curvature Total Berry Curvature Ωtot,z = fn(k) Ωn,z(k) n ! 1e5 0.4 1e3 0.2 1e1 0 -1e1 -0.2 -0.4 -1e3 -1e5 Berry Curvature (atomic units) ! H P Our Calculation Ab-initio k-grid: 8x8x8 Interpolation k-grid: 320x320x320 + 7x7x7 “adaptive refinement” (when Ω> 100 a.u.) Wannier Interpolation Existing Calculation* planewaves + Kubo formula + pseudopotentials LAPW 758 (Ω cm)-1 751 (Ω cm)-1 Experiment ~1000 (Ω cm)-1 Differences: Scattering, Approximate DFT, Experimental uncertainty *Yao et al PRL (2004) Berry curvature in ky=0 plane Total Berry curvature (atomic units) Berry curvature in ky=0 plane Contributions opposite 26% spin 21% like spin smooth 53% background Total Berry curvature (atomic units) Wannier Interpolation •Map low energy electronic structure onto “exact tight binding model” •Compute quantities with ab-initio accuracy and tight-binding cost Basis set independent (planewaves, LAPW, guassians etc) Any single particle level of theory (LDA, LDA+U, GW) •Many applications: Accurate DOS, joint-DOS, Fermi Surfaces Spin-relaxation Magneto-crystalline anisotropy NMR in metals (Knight shift) Electron-phonon interaction Acknowledgments Ivo Souza : UC Berkeley Wannier Interpolation: Phys Rev B 75 195121 (2007) Anomalous Hall Effect Xinjie Wang, David Vanderbilt : Rutgers University Phys Rev B 74 195118 (2006) Wannier90 code Arash Mostofi, Nicola Marzari MIT Released under the GPL at www.wannier.org Electron-Phonon Feliciano Guistino, Marvin Cohen, Steven Louie : UCB Phys Rev Lett 94 047005 (2007) Funding - Lawrence Berkeley National Laboratory Corpus Christi College, Cambridge.