Non-collinear OEP for solids: SDFT vs CSDFT
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1 Non-collinear OEP for solids: SDFT vs Sangeeta Sharma 1,2, J. K. Dewhurst 3 S. Pittalis 2, S. Kurth 2, S. Shallcross 4 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. Lehrstuhl für Theoretische Festkörperphysik, Erlangen, Germany 21 September 2007 vs SDFT for solids: Sharma
2 Orbital exchange-correlation functionals Third generation: Exact exchange (EXX) 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 δn(r) = 0; m,jp δv s (r) = 0; Bs,A s E[ρ,m,j p ] δm(r) = 0; n,jp E[v s,b s,a s ] δb s (r) = 0; vs,a s δj p (r) = 0 n,m δa s (r) = 0 vs,b s vs SDFT for solids: Sharma
3 Orbital exchange-correlation functionals Third generation: Exact exchange (EXX) 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 δn(r) = 0; m,jp δv s (r) = 0; Bs,A s E[ρ,m,j p ] δm(r) = 0; n,jp E[v s,b s,a s ] δb s (r) = 0; vs,a s δj p (r) = 0 n,m δa s (r) = 0 vs,b s vs SDFT for solids: Sharma
4 Orbital exchange-correlation functionals Third generation: Exact exchange (EXX) 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 δn(r) = 0; m,jp δv s (r) = 0; Bs,A s E[ρ,m,j p ] δm(r) = 0; n,jp E[v s,b s,a s ] δb s (r) = 0; vs,a s δj p (r) = 0 n,m δa s (r) = 0 vs,b s vs SDFT for solids: Sharma
5 OEP equations SDFT R v (r) R B (r) R A (r) occ i occ i occ i [ dr δφ i (r ) [ dr δφ i (r ) δφ i (r ) δv s (r) + δφ i (r ) δb s (r) + [ dr δφ i (r ) δφ i (r ) δa s (r) + δφ i (r ) δφ i (r ) δφ i (r ) ] δφ i (r ) = 0 δv s (r) Bs,A s ] δφ i (r ) = 0 δb s (r) vs,a s ] δφ i (r ) = 0 δa s (r) vs,b s vs SDFT for solids: Sharma
6 OEP equations SDFT We need to solve: R v (r) occ un ( ) ρ ij (r) δv s (r) = Λ ij + c.c. = 0 (1) Bs,A s ε i ε j R B (r) R A (r) Λ ij δb s (r) = vs,a s i occ i i j un j j ( ) m ij (r) Λ ij + c.c. = 0 (2) ε i ε j occ un ( ) j ij (r) δa s (r) = Λ ij + c.c. = 0 (3) vs,b s ε i ε j = ( Vij NL ) ρ ij(r)v x (r)dr j ij(r) A x (r)dr m ij(r) B x (r)dr vs SDFT for solids: Sharma
7 terative solution of OEP equations At the solution the residue should be zero: R v (r) [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) vs SDFT for solids: Sharma
8 Treatment of solids SDFT 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 vs SDFT for solids: Sharma
9 Treatment of solids SDFT 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 vs SDFT for solids: Sharma
10 Treatment of solids SDFT 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 vs SDFT for solids: Sharma
11 Treatment of solids SDFT 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 vs SDFT for solids: Sharma
12 Treatment of solids SDFT 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 vs SDFT for solids: Sharma
13 Specific to solids (q 0 term) 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). vs SDFT for solids: Sharma
14 Non-collinear magnetism in unsupported Cr-monolayer (S. Sharma et al. Phys. Rev. Lett. 98, (2007)) vs SDFT for solids: Sharma
15 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) ( ) LSDA ṽ xc(r) 0 0 ṽxc(r) ) ( v U(r) = xc(r) vxc(r) vxc(r) vxc(r) ) ) m(r) LSDA B xc (r) vs SDFT for solids: Sharma
16 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!! vs SDFT for solids: Sharma
17 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!! vs SDFT for solids: Sharma
18 Non-collinear magnetism in unsupported Cr-monolayer (S. Sharma et al. Phys. Rev. Lett. 98, (2007)) vs SDFT for solids: Sharma
19 EXX results for unsupported Cr-monolayer (S. Sharma et al. Phys. Rev. Lett. 98, (2007)) m(r) B x (r) vs SDFT for solids: Sharma
20 Orbital moments for simple metals 0.2 S. Sharma et al. PRB 76, (R) (2007) Orbital moment (Bohr magneton) LSDA GGA Experiment LSDA GGA Experiment LSDA GGA Experiment 0 Fe Co Ni vs SDFT for solids: Sharma
21 Orbital moments for simple metals 0.2 S. Sharma et al. PRB 76, (R) (2007) Orbital moment (Bohr magneton) LSDA GGA EXX (EXX) Experiment LSDA GGA Experiment EXX (EXX) LSDA GGA EXX Experiment (EXX) 0 Fe Co Ni vs SDFT for solids: Sharma
22 Spin-orbit induced band splitting for semiconductors Spin-orbit induced band spliting (mev) PP-: Rohra, Engel, Goerling /xxx.lanl.gov /cond-mat/ LSDA GGA PP- Experiment LSDA GGA PP- Experiment LSDA GGA PP- Experiment 0 Ge (7v-8v) Ge (6c-8c) Si vs SDFT for solids: Sharma
23 Spin-orbit induced band splitting for semiconductors Spin-orbit induced band spliting (mev) LSDA GGA PP- FP- FP-SDFT Experiment FP-: S. Sharma et al. PRB 76, (R) (2007) LSDA GGA PP- FP- FP-SDFT Experiment LSDA GGA PP- FP- FP-SDFT Experiment 0 Ge (7v-8v) Ge (6c-8c) Si vs SDFT for solids: Sharma
24 Band gaps of semiconductors and insulators Staedele et al. PRB (1999) Magyar et al. PRB (2004) 2 0 PP-EXX FP-LDA E g KS - Eg (ev) Ge GaAs CdS Si ZnS C BN Ar Ne Kr Xe vs SDFT for solids: Sharma
25 Band gaps of semiconductors and insulators S. Sharma et al. Phys. Rev. Lett (2005) 2 0 FP-EXX PP-EXX LMTO-ASA-EXX FP-LDA E g KS - Eg (ev) Ge GaAs CdS Si ZnS C BN Ar Ne Kr Xe vs SDFT for solids: Sharma
26 Paramagnetic current density for solid Ge vs SDFT for solids: Sharma
27 Summary SDFT Summary OEP method extended to general case of non-collinear magnetism This when applied to Cr-monolayer reveals a pronounced m(r) B x (r), making this functional suitable for the study of spin dynamics within EXX functional, in absence of external magnetic fields, does not improve upon the SDFT results Correlation functional compatible with EXX is needed. vs SDFT for solids: Sharma
28 Code used: EXCTNG K. Dewhurst, S. Sharma, C. Ambrosch-Draxl and L. Nordström The code is released under GPL and is freely available at: vs SDFT for solids: Sharma
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