Optimized Effective Potential method for non-collinear Spin-DFT: view to spindynamics

Transcription

1 Optimized Effective Potential method for non-collinear Spin-DFT: view to spindynamics S. Sharma and E. K. U. Gross nstitut für Theoretische Physik, FU Berlin Fritz-Haber nstitut, Berlin 21 Sept 2006

2 Synopsis Plan of the talk 1 2 OEP for a general case non-collinear magnets 3 Results for magnetic materials 4 Summary and outlook

3 Density functional theory Theorems (1964) 1 External potential v(r) is uniquely determined by n(r) 2 The variational principle holds E 0 = E v0 [n 0 ] < E v0 [n] 3 E v0 [n] = F[n] + dr v 0 (r)n(r) F[n] = min Φ n Φ T + V ee Φ

4 s Auxiliary non-interacting system in an effective potential. T s [n] = min Φ T Φ Φ n E xc [n] = F[n] T s [n] U[n] [ 1 ] v s (r) φ i (r) = ɛ i φ i (r) and occ n(r) = φ i (r) 2. i

5 s Auxiliary non-interacting system in an effective potential. T s [n] = min Φ T Φ Φ n E xc [n] = F[n] T s [n] U[n] [ 1 ] v s (r) φ i (r) = ɛ i φ i (r) and occ n(r) = φ i (r) 2. i

6 s v s [n(r)] = v(r) + dr n(r ) r r + v xc[n(r)] where v xc [n(r)] = δe xc[n] δn(r). Many sins hidden in E xc [n]!

7 s v s [n(r)] = v(r) + dr n(r ) r r + v xc[n(r)] where v xc [n(r)] = δe xc[n] δn(r). Many sins hidden in E xc [n]!

8 for DFT First generation : Local density approximation (LDA) Exc LDA [n] = dr n(r)e unif xc (n(r)) Second generation : Generalised gradient approximations Exc GGA [n] = dr f(n(r), n(r)) Exc Meta GGA [n] = dr g(n(r), n(r), τ(r))

9 for DFT First generation : Local density approximation (LDA) Exc LDA [n] = dr n(r)e unif xc (n(r)) Second generation : Generalised gradient approximations Exc GGA [n] = dr f(n(r), n(r)) Exc Meta GGA [n] = dr g(n(r), n(r), τ(r))

10 (Semi)-Local functionals for Spin-DFT Problem: These functionals are designed for spin unpolarised case. They are also used for spin-polarized case by decoupling the two spins Advantage: Two decoupled KS equations are solved. This saves a lot of time. Disadvantage: Treatment non-collinearity : LDA is still justified but conventional semi-local functionals are designed for reduced density

11 (Semi)-Local functionals for Spin-DFT ρ (r) ( ρ (r) ρ (r) ρ (r) ρ (r) ) n absence of spincurrents (local moments) and external magnetic fields (spontaneous magnetisation) TD-SDFT equation of motion for the spin degrees of freedom dm(r, t) dt = 1 2c m(r, t) B xc(r, t) Within the LSDA B xc is locally parallel to m

12 (Semi)-Local functionals for Spin-DFT ρ (r) ( ρ (r) ρ (r) ρ (r) ρ (r) ) n absence of spincurrents (local moments) and external magnetic fields (spontaneous magnetisation) TD-SDFT equation of motion for the spin degrees of freedom dm(r, t) dt = 1 2c m(r, t) B xc(r, t) Within the LSDA B xc is locally parallel to m

13 Orbital exchange-correlation functionals Third generation: Exact exchange (EXX) Neglect correlation and use the Hartree-Fock exchange energy E x [n] = 1 2 occ i,j dr dr Φ i (r)φ i(r )Φ j (r )Φ j (r) r r Optimized Effective Potential To solve the s we require v x [n](r) = δe x[n] δn(r)

14 Orbital exchange-correlation functionals Third generation: Exact exchange (EXX) Neglect correlation and use the Hartree-Fock exchange energy E x [n] = 1 2 occ i,j dr dr Φ i (r)φ i(r )Φ j (r )Φ j (r) r r Optimized Effective Potential To solve the s we require v x [n](r) = δe x[n] δn(r)

15 Non-collinear magnetism within OEP terative solution of following equations R v (r) δe[ρ,m] δv s (r) R B (r) δe[ρ,m] δb s (r) occ un = Bs i j occ un = vs i j ( ) ρ ij (r) Λ ij + c.c. = 0 (1) ε i ε j ( ) m ij (r) Λ ij + c.c. = 0, (2) ε i ε j ( Λ ij = V NL ij ) ρ ij(r)v x (r)dr m ij(r) B x (r)dr,

16 Treatment of solids Synopsis Full-potential linearised augmented planewaves (FP-LAPW) potential is fully described without any shape approximation core is treated as Dirac spinors and valence as Pauli spinors space divided into interstitial and muffin-tin regions this is one of the most precise method available MT

17 Treatment of solids Synopsis Full-potential linearised augmented planewaves (FP-LAPW) potential is fully described without any shape approximation core is treated as Dirac spinors and valence as Pauli spinors space divided into interstitial and muffin-tin regions this is one of the most precise method available MT

18 Treatment of solids Synopsis Full-potential linearised augmented planewaves (FP-LAPW) potential is fully described without any shape approximation core is treated as Dirac spinors and valence as Pauli spinors space divided into interstitial and muffin-tin regions this is one of the most precise method available MT

19 Treatment of solids Synopsis Full-potential linearised augmented planewaves (FP-LAPW) potential is fully described without any shape approximation core is treated as Dirac spinors and valence as Pauli spinors space divided into interstitial and muffin-tin regions this is one of the most precise method available MT

20 Treatment of solids Synopsis Full-potential linearised augmented planewaves (FP-LAPW) potential is fully described without any shape approximation core is treated as Dirac spinors and valence as Pauli spinors space divided into interstitial and muffin-tin regions this is one of the most precise method available MT

21 Examples: Cr-monolayer Non-collinear magnetism in unsupported Cr-monolayer (S. Sharma et al. cond-mat/ )

22 Examples: Cr-monolayer Non-collinear magnetism in unsupported Cr-monolayer (S. Sharma et al. cond-mat/ )

23 Examples: Cr-monolayer m(r) B xc (r) for unsupported Cr-monolayer: EXX results (S. Sharma et al. cond-mat/ )

24 Examples: Cr-monolayer Magnetic metals: stringent tests for EXX Magnetic moment in Bohr magneton Metal LDA LAPW-OEP LMTO-OEP Expt Fe 2.12? Co 1.71? Ni 0.55? LMTO: T. Kotani JPCM (1998)

25 Examples: Cr-monolayer Magnetic metals: stringent tests for EXX Magnetic moment in Bohr magneton Metal LDA LAPW-OEP LMTO-OEP Expt Fe Co Ni LMTO: T. Kotani JPCM (1998)

26 Summary and outlook Examples: Cr-monolayer Summary 1 SDFT with OEP method using EXX functional implemented within FP-LAPW method. 2 This SDFT when applied to Cr-monolayer reveals a pronounced m(r) B xc (r), making this functional stuitable for the study of spin dynamics.

27 Summary and outlook Examples: Cr-monolayer Outlook 1 OEP is being generalised to handle spin-spirals. 2 Current Density Functional Theory with spin-orbit coupling for solids. 3 Correlation to go with exact exchange.

28 Examples: Cr-monolayer Code used: EXCTNG S. Sharma, K. Dewhurst, C. Ambrosch-Draxl and L. Nordström The code is released under GPL and is freely available at:

29 Acknowledgements Synopsis Examples: Cr-monolayer Collaborators Kay Dewhurst (Edinburgh, UK) Claudia Ambrosch-Draxl (Leoben, Austria) Sam Shallcross (Copenhagen, Denmark) Lars Nordström (Uppsala, Sweden) Stefan Kurth (Berlin, Germany)