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

Transcription

1 Optimized Effective Potential method for non-collinear Spin-DFT: view to spin-dynamics Sangeeta Sharma 1,2, J. K. Dewhurst 3, C. Ambrosch-Draxl 4, S. Pittalis 2, S. Kurth 2, N. Helbig 2, S. Shallcross 5, L. Nordström 6 and E. K. U. Gross 2 1. Fritz-Haber nstitut of the Max Planck Society, Berlin, Germany 2. nstitut für Theoretische Physik, FU Berlin, Germany 3. School of Chemistry, The University of Edinburgh, UK 4. Department Materials Physics, University of Leoben, Austria 5. Lehrstuhl für Theoretische Festkörperphysik, Erlangen, Germany 6. Department of Physics, Uppsala University, Sweden 27 August 2007

2 Ground state of interacting system using DFT s Auxiliary non-interacting system in an effective potential. [ 1 ] v s (r) φ i (r) = ǫ i φ i (r) where v s (r) = v(r) + dr ρ(r ) r r + v xc(r) v xc (r) = δe xc[ρ] δρ(r). Many sins hidden in E xc [ρ]!

3 Ground state of interacting system using DFT s Auxiliary non-interacting system in an effective potential. [ 1 ] v s (r) φ i (r) = ǫ i φ i (r) where v s (r) = v(r) + dr ρ(r ) r r + v xc(r) v xc (r) = δe xc[ρ] δρ(r). Many sins hidden in E xc [ρ]!

4 Ground state of interacting system using DFT s Auxiliary non-interacting system in an effective potential. [ 1 ] v s (r) φ i (r) = ǫ i φ i (r) where v s (r) = v(r) + dr ρ(r ) r r + v xc(r) v xc (r) = δe xc[ρ] δρ(r). Many sins hidden in E xc [ρ]!

5 for DFT First generation : Local density approximation (LDA) Exc LDA [ρ] = dr ρ(r)e unif xc (ρ(r)) Second generation : Generalised gradient approximations Exc GGA [ρ] = dr f(ρ(r), ρ(r))

6 for DFT First generation : Local density approximation (LDA) Exc LDA [ρ] = dr ρ(r)e unif xc (ρ(r)) Second generation : Generalised gradient approximations Exc GGA [ρ] = dr f(ρ(r), ρ(r))

7 Functionals for Spin-DFT s for spin polarised case ( 1 ) v s (r) + µ B σ B s (r) Φ i (r) = ε i Φ i (r) B s (r) = B xc (r) + B ext (r) occ m(r) = µ B Φ i (r)σφ i(r) i E xc [ρ] E xc [ρ,m]

8 Functionals for Spin-DFT s for spin polarised case ( 1 ) v s (r) + µ B σ B s (r) Φ i (r) = ε i Φ i (r) B s (r) = B xc (r) + B ext (r) occ m(r) = µ B Φ i (r)σφ i(r) i E xc [ρ] E xc [ρ,m]

9 (Semi)-Local functionals for Spin-DFT Problems with extension of these functionals to spin-dft: These functionals are designed for spin unpolarised case. They are also used for spin-polarized collinear magnetic case by decoupling the two spins. Advantage: Two KS equations decoupled in spin space. This saves a lot of time. Disadvantage: Treatment of non-collinearity : local and semi-local functionals are designed for collinear magnetism. LSDA can still be extended for non-collinear magnetic case.

10 (Semi)-Local functionals for Spin-DFT Problems with extension of these functionals to spin-dft: These functionals are designed for spin unpolarised case. They are also used for spin-polarized collinear magnetic case by decoupling the two spins. Advantage: Two KS equations decoupled in spin space. This saves a lot of time. Disadvantage: Treatment of non-collinearity : local and semi-local functionals are designed for collinear magnetism. LSDA can still be extended for non-collinear magnetic case.

11 Functionals for non-collinear magnetism occ ( ρ (r) Φ i (r)φ ρ i (r) = (r) ρ (r) ρ (r) ρ (r) LSDA and Kübler trick: ( ρ U(r) (r) ρ (r) ρ (r) ρ (r) ( ρ ) (r) 0 0 ρ (r) ( U ṽ (r) xc(r) 0 0 ṽxc(r) i ) ) ( U ρ (r) = (r) 0 0 ρ (r) m(r) LSDA ( ) LSDA ṽ xc(r) 0 0 ṽxc(r) ) ( v U(r) = xc(r) vxc(r) vxc(r) vxc(r) B xc (r) ) )

12 Functionals for non-collinear magnetism occ ( ρ (r) Φ i (r)φ ρ i (r) = (r) ρ (r) ρ (r) ρ (r) LSDA and Kübler trick: ( ρ U(r) (r) ρ (r) ρ (r) ρ (r) ( ρ ) (r) 0 0 ρ (r) ( U ṽ (r) xc(r) 0 0 ṽxc(r) i ) ) ( U ρ (r) = (r) 0 0 ρ (r) m(r) LSDA ( ) LSDA ṽ xc(r) 0 0 ṽxc(r) ) ( v U(r) = xc(r) vxc(r) vxc(r) vxc(r) B xc (r) ) )

13 Ab-initio spin-dynamics Landau-Lifschitz equation for spin dynamics: m(r, t) t = γm(r, t) (B xc (r, t) + B ext (r, t)) J(r, t) n absence of external field and spin-currents: m(r, t) t = γm(r, t) B xc (r, t) Within the LSDA B xc is locally parallel to m, so no spin dynamics!!

14 Ab-initio spin-dynamics Landau-Lifschitz equation for spin dynamics: m(r, t) t = γm(r, t) (B xc (r, t) + B ext (r, t)) J(r, t) n absence of external field and spin-currents: m(r, t) t = γm(r, t) B xc (r, t) Within the LSDA B xc is locally parallel to m, so no spin dynamics!!

15 Orbital exchange-correlation functionals Third generation: Exact exchange () Neglect correlation and use the Fock exchange energy E x [Φ] = 1 2 occ i,j dr dr Φ i (r)φ j(r)φ j (r )Φ i (r ) r r E[ρ, m] δe δe δn(r) = 0; m δm(r) = 0 n E[v s,b s ] δe δe δv s (r) = 0; Bs δb s (r) = 0 vs

16 Orbital exchange-correlation functionals Third generation: Exact exchange () Neglect correlation and use the Fock exchange energy E x [Φ] = 1 2 occ i,j dr dr Φ i (r)φ j(r)φ j (r )Φ i (r ) r r E[ρ, m] δe δe δn(r) = 0; m δm(r) = 0 n E[v s,b s ] δe δe δv s (r) = 0; Bs δb s (r) = 0 vs

17 Orbital exchange-correlation functionals Third generation: Exact exchange () Neglect correlation and use the Fock exchange energy E x [Φ] = 1 2 occ i,j dr dr Φ i (r)φ j(r)φ j (r )Φ i (r ) r r E[ρ, m] δe δe δn(r) = 0; m δm(r) = 0 n E[v s,b s ] δe δe δv s (r) = 0; Bs δb s (r) = 0 vs

18 Optimised effective potential R v (r) R B (r) occ i occ i [ dr δe δφ i (r ) [ dr δe δφ i (r ) δφ i (r ) δv s (r) + δφ i (r ) δb s (r) + δe δφ i (r ) δe δφ i (r ) ] δφ i (r ) = 0 δv s (r) Bs ] δφ i (r ) = 0 δb s (r) vs

19 Non-collinear magnetism within OEP (S. Sharma et al. Phys. Rev. Lett. 98, (2007)) We need to solve following integral equations: R v (r) δe[v s,b s ] δv s (r) R B (r) δe[v s,b s ] 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 = ( Vij NL ) ρ ij(r)v x (r)dr m ij(r) B x (r)dr

20 terative solution of OEP equations At the solution the residue should be zero: R v (r) δe[v s] δv s (r) = 0 Therefore by repeatedly adding the intermediate residues to the exchange potential v (i) we can converge to the solution. x (r) = v x (i 1) (r) τr v (i) (r) S. Kümmel and J. Perdew Phys. Rev. Lett. 90, (2003) S. Sharma et al. Phys. Rev. Lett. 98, (2007)

21 Treatment of solids Theory and motivation 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 methods available for solids MT

22 Treatment of solids Theory and motivation 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 methods available for solids MT

23 Treatment of solids Theory and motivation 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 methods available for solids MT

24 Treatment of solids Theory and motivation 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 methods available for solids MT

25 Treatment of solids Theory and motivation 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 methods available for solids MT

26 Examples: Cr-monolayer simple magnets Non-collinear magnetism in unsupported Cr-monolayer (S. Sharma et al. Phys. Rev. Lett. 98, (2007))

27 Examples: Cr-monolayer simple magnets Non-collinear magnetism in unsupported Cr-monolayer (S. Sharma et al. Phys. Rev. Lett. 98, (2007))

28 Examples: Cr-monolayer simple magnets results for unsupported Cr-monolayer (S. Sharma et al. Phys. Rev. Lett. 98, (2007)) m(r) B x (r)

29 Examples: Cr-monolayer simple magnets Magnetic moments for simple metals LMTO: T. Kotani JPCM (1998) Hartree-Fock:. Schnell et al. PRB (2003) Spin magnetic moment (Bohr magneton) LSDA Hartree-Fock LMTO-OEP Experiment LSDA Hartree-Fock LMTO-OEP Experiment LSDA Hartree-Fock LMTO-OEP Experiment 0 Fe Co Ni

30 Examples: Cr-monolayer simple magnets Magnetic moments for simple metals LAPW: S. Sharma et al. PRL 98, (2007) Spin magnetic moment (Bohr magneton) LSDA LAPW-OEP Hartree-Fock LMTO-OEP Experiment LSDA LAPW-OEP Hartree-Fock LMTO-OEP Experiment LSDA LAPW-OEP Hartree-Fock LMTO-OEP Experiment Fe Co Ni

31 Examples: Cr-monolayer simple magnets Optimised effective potential for collinear magnets occ = i [ dr dr δe x δφ i (r ) v x (r) = δe x[φ] δρ (r) δφ i (r ) δv s (r ) + δe x δφ i (r ) ] δφ i (r ) δv s (r ) δv s (r ) δρ (r) T. Kotani J. Phys. Cond. Mat. 10, 9241 (1998)

32 nverting response matrix Examples: Cr-monolayer simple magnets χ (r,r ) = δρ occ (r) δv s (r ) = δv s (r ) δρ (r) = χ 1 (r,r ) i unocc j φ i (r)φ j (r)φ j (r )φ i (r ) + c.c. ǫ i ǫ j Response is inverted in an auxiliary basis set without constant function. (M. Städele et al. Phys. Rev. B (1999))

33 What we learned Theory and motivation Examples: Cr-monolayer simple magnets Positives: Leads to natural extension for the case of non-collinear magnetism Opens door for ab-initio treatment of spin-dynamics. Negatives: Lack of correlations lead to overestimation of quantities Being explicit functionals of orbitals, quite time consuming.

34 Examples: Cr-monolayer simple magnets Band gaps of semiconductors and insulators Staedele et al. PRB (1999) Magyar et al. PRB (2004) 2 0 PP- FP-LDA E g KS - Eg (ev) Ge GaAs CdS Si ZnS C BN Ar Ne Kr Xe

35 Examples: Cr-monolayer simple magnets Band gaps of semiconductors and insulators S. Sharma et al. Phys. Rev. Lett (2005) 2 0 FP- PP- LMTO-ASA- FP-LDA E g KS - Eg (ev) Ge GaAs CdS Si ZnS C BN Ar Ne Kr Xe

36 Summary Theory and motivation Examples: Cr-monolayer simple magnets Summary OEP method extended to general case of non-collinear magnetism This method using functional implemented within FP-LAPW scheme. This when applied to Cr-monolayer reveals a pronounced m(r) B x (r), making this functional stuitable for the study of spin dynamics.

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

38 Examples: Cr-monolayer simple magnets Spin current N J(r) = µ 0 Φ ˆσ i ĵ i (r) Φ i ĵ(r) = 1 N k δ(r r k ) + δ(r r k ) k 2i k

39 Examples: Cr-monolayer simple magnets Require integral over k-point differences (q = k k ): E x = BZ f(q) q 2 d 3 q ntegrable but very slow convergence w.r.t. k-points. Precalculate weights using many q-points (millions) 1 W q = B q q 2 d3 q, then E x q W q f(q).