All electron optimized effective potential method for solids

All electron optimized effective potential method for solids Institut für Theoretische Physik Freie Universität Berlin, Germany and Fritz Haber Institute of the Max Planck Society, Berlin, Germany. 22 May 2007

Synopsis Broad outline 1 Optimized effective potential method within DFT 2 Band gaps in semiconductors 3 Extension of method for non-collinear magnetic systems

Density Functional Theory Hohenberg-Kohn 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

Auxiliary non-interacting system in an effective potential Non-interacting kinetic energy T s [n] = min Φ T Φ Φ n Exchange-correlation energy functional E xc [n] F[n] T s [n] U[n] Kohn-Sham equations [ 1 ] 2 2 + v s (r) φ i (r) = ǫ i φ i (r) occ n(r) = φ i (r) 2. i

Auxiliary non-interacting system in an effective potential Non-interacting kinetic energy T s [n] = min Φ T Φ Φ n Exchange-correlation energy functional E xc [n] F[n] T s [n] U[n] Kohn-Sham equations [ 1 ] 2 2 + v s (r) φ i (r) = ǫ i φ i (r) occ n(r) = φ i (r) 2. i

Kohn-Sham equations DFT Kohn-Sham effective potential v s [n(r)] = v 0 (r) + dr n(r ) r r + v xc[n(r)] v xc [n(r)] = δe xc[n] δn(r). Many sins hidden in E xc [n]!

Kohn-Sham equations DFT Kohn-Sham effective potential v s [n(r)] = v 0 (r) + dr n(r ) r r + v xc[n(r)] v xc [n(r)] = δe xc[n] δn(r). Many sins hidden in E xc [n]!

Exchange-correlation functionals First generation : Local density approximation (LDA) Exc LDA [n] = dr n(r)e unif xc (n(r)) Second generation : Generalized gradient approximations Exc GGA [n] = dr f(n(r), n(r)) Exc Meta GGA [n] = dr g(n(r), n(r),τ(r))

Exchange-correlation functionals First generation : Local density approximation (LDA) Exc LDA [n] = dr n(r)e unif xc (n(r)) Second generation : Generalized gradient approximations Exc GGA [n] = dr f(n(r), n(r)) Exc Meta GGA [n] = dr g(n(r), n(r),τ(r))

Problems DFT Things that go wrong 1 Band-gap and band positions in semiconductors and insulators. 2 Band widths in simple magnetic metals. 3 Localization of f-electrons in rare earths.

Solutions DFT Beyond second-generation of functionals 1 Quasi particle: GW, LDA+DMFT... (band gaps improve) 2 Improve exchange-correlation: LDA+SIC, LDA+U... (localizes f-electrons) U[n] = 1 2 ij dr dr n i(r)n j (r ) r r

Expansion of total energy functional Goerling-Levy expansion E[n] = dr v 0 (r)n(r) + E j [n] j=0 [ E j [n] = T s [n] + dr dr n(r )n(r) r r j=0 ] + E x [n] + E j [n] j=2

Expansion of total energy functional Goerling-Levy expansion E[n] = dr v 0 (r)n(r) + E j [n] j=0 [ E j [n] = T s [n] + dr dr n(r )n(r) r r j=0 ] + E x [n] + E j [n] j=2

Orbital dependent exchange-correlation functionals Third generation: Exact exchange (EXX) Neglect correlation and use the Hartree-Fock exchange energy E x [n] = 1 2 occ This is automatically self-interaction free!! i,j Optimized Effective Potential dr dr φ i (r)φ j (r )φ j (r)φ i (r ) r r To solve the Kohn-Sham system we require v x [n](r) = δe x[n] δn(r)

Orbital dependent exchange-correlation functionals Third generation: Exact exchange (EXX) Neglect correlation and use the Hartree-Fock exchange energy E x [n] = 1 2 occ This is automatically self-interaction free!! i,j Optimized Effective Potential dr dr φ i (r)φ j (r )φ j (r)φ i (r ) r r To solve the Kohn-Sham system we require v x [n](r) = δe x[n] δn(r)

Optimized Effective Potential Using the functional derivative chain rule: Exact exchange local potential occ [ v x [n](r) = dr dr δex δφ i i (r ) occ unocc = dr i j δφ i (r ) δv s (r ) + δe x δφ i (r ) φ i ˆv NL x φ j φ j (r )φ i (r ) ǫ i ǫ j δφ i (r ) δv s (r ) ] δvs (r ) δn(r) + c.c. δv s(r ) δn(r), φ ik ˆv NL x φ jk = occ lk w k dr dr φ ik (r)φ lk (r)φ lk (r )φ jk (r ) r r.

: band-gaps PP-EXX : Städele et al. PRB 59 10031 PP-EXX: Magyar et al. PRB 69 045111 2 0 PP-EXX FP-LDA E g KS - Eg (ev) 1 0-1 -2-4 -6-2 Ge GaAs CdS Si ZnS C BN Ar Ne Kr Xe

The fundamental band-gap for materials: E g = A I = (E N+1 E N ) (E N E N 1 ) E g = δe δn + δe δn E[n] = T[n] + U s [n] + E xc [n] E g = ( δt δn + δt δn ) + (δv xc δn + δv xc δn ) E g = E KS g + xc

Numerical method DFT Full-potential linearized 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 I I MT I I

Numerical method DFT Full-potential linearized 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 I I MT I I

Numerical method DFT Full-potential linearized 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 I I MT I I

Numerical method DFT Full-potential linearized 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 I I MT I I

Numerical method DFT Full-potential linearized 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 I I MT I I

: band-gaps Sharma et al. Phys. Rev. Lett. 95 136402 (2005) 2 0 FP-EXX PP-EXX LMTO-ASA-EXX FP-LDA E g - E g KS (ev) 1 0-1 -2-4 -6-2 Ge GaAs CdS Si ZnS C BN Ar Ne Kr Xe

Spin Density Functional Theory collinear magnetism spin-dynamics occ m(r) = µ B Φ i (r)σφ i(r) Total energy functional (Barth and Hedin, JPC 5, 1629 (72)) i E v/b0 [n,m] = F[n,m]+ dr v 0 (r)n(r)+ dr B 0 (r) m(r) Kohn-Sham equations [ 1 ] 2 2 + v s (r) + σ B s (r) Φ i (r) = ǫ i Φ i (r)

Spin Density Functional Theory collinear magnetism spin-dynamics occ m(r) = µ B Φ i (r)σφ i(r) Total energy functional (Barth and Hedin, JPC 5, 1629 (72)) i E v/b0 [n,m] = F[n,m]+ dr v 0 (r)n(r)+ dr B 0 (r) m(r) Kohn-Sham equations [ 1 ] 2 2 + v s (r) + σ B s (r) Φ i (r) = ǫ i Φ i (r)

Decoupled Kohn-Sham equations collinear magnetism spin-dynamics [ 1 ] 2 2 + v s (r) ± µ B B sz (r) φ i (r) = ǫ i φ i (r) Pros and cons Advantage: Two decoupled KS equations are solved. This saves a lot of time. Functionals designed for spin-unpolarized case can be used. Disadvantage: Treatment non-collinearity : LDA is still justified but conventional semi-local functionals are designed for reduced density

collinear magnetism spin-dynamics Functionals for non-collinear magnetism LSDA and Kübler trick: ( ρ U (r) ρ (r) ρ (r) ρ (r) ρ (r) ( ρ (r) ρ (r) ρ (r) ρ (r) ( ρ (r) 0 ) 0 ρ (r) ( U ṽxc(r) 0 0 ṽxc(r) ) ) ( U ρ = (r) 0 0 ρ (r) m(r) LSDA ) ( ) LSDA ṽ xc(r) 0 0 ṽxc(r) ) ( ) v U = xc(r) vxc(r) vxc(r) vxc(r) B xc (r)

collinear magnetism spin-dynamics Functionals for non-collinear magnetism LSDA and Kübler trick: ( ρ U (r) ρ (r) ρ (r) ρ (r) ρ (r) ( ρ (r) ρ (r) ρ (r) ρ (r) ( ρ (r) 0 ) 0 ρ (r) ( U ṽxc(r) 0 0 ṽxc(r) ) ) ( U ρ = (r) 0 0 ρ (r) m(r) LSDA ) ( ) LSDA ṽ xc(r) 0 0 ṽxc(r) ) ( ) v U = xc(r) vxc(r) vxc(r) vxc(r) B xc (r)

collinear magnetism spin-dynamics Functionals for non-collinear magnetism LSDA and Kübler trick: ( ρ U (r) ρ (r) ρ (r) ρ (r) ρ (r) ( ρ (r) ρ (r) ρ (r) ρ (r) ( ρ (r) 0 ) 0 ρ (r) ( U ṽxc(r) 0 0 ṽxc(r) ) ) ( U ρ = (r) 0 0 ρ (r) m(r) LSDA ) ( ) LSDA ṽ xc(r) 0 0 ṽxc(r) ) ( ) v U = xc(r) vxc(r) vxc(r) vxc(r) B xc (r)

Non-collinearity in Cr-monolayer collinear magnetism spin-dynamics Sharma et. al Phys. Rev. Lett. 98 196405 (2007)

Spin-dynamics DFT collinear magnetism spin-dynamics Landau-Lifschitz equation for spin dynamics m(r, t) t = γm(r,t) (B xc (r,t) + B ext (r,t)) J In 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!!

Spin-dynamics DFT collinear magnetism spin-dynamics Landau-Lifschitz equation for spin dynamics m(r, t) t = γm(r,t) (B xc (r,t) + B ext (r,t)) J In 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!!

collinear magnetism spin-dynamics Non-collinear magnetism within Iterative solution of following equations R v (r) δe[ρ,m] δv s (r) R B (r) δe[ρ,m] δb s (r) = Bs occ i un j occ un = vs i j ( ) ρ ij (r) Λ ij + c.c. = 0 ε i ε j ( ) m ij (r) Λ ij + c.c. = 0, ε i ε j Λ ij = ( Vij NL ) ρ ij (r)v x(r)dr m ij (r) B x(r)dr, Sharma et. al Phys. Rev. Lett. 98 196405 (2007)

collinear magnetism spin-dynamics Non-collinear magnetism in unsupported Cr-monolayer Sharma et. al Phys. Rev. Lett. 98 196405 (2007)

collinear magnetism spin-dynamics m(r) X B XC (r) for unsupported Cr-monolayer

collinear magnetism spin-dynamics Code used: EXCITING K. Dewhurst, S. Sharma, C. Ambrosch-Draxl and L. Nordström The code is released under GPL and is freely available at: http://exciting.sourceforge.net/

EXCITING DFT collinear magnetism spin-dynamics Features 1 General FP-LAPW basis with local-orbitals APW radial derivative matching to arbitrary orders (super-lapw, etc.) Arbitrary number of local-orbitals (all core states can be made valence) LSDA and GGA functionals available 2 Structure and symmetry Determination of lattice and crystal symmetry groups from input lattice and atomic coordinates Determination of atomic coordinates from space group data Automatic reduction from conventional to primitive unit cell Full symmetrisation of density and magnetisation and their conjugate fields

EXCITING DFT collinear magnetism spin-dynamics 1 Magnetism Spin-orbit coupling included in second-variational scheme Non-collinear magnetism with arbitrary on-site magnetic fields Spin-spirals for any q-vector 2 Plotting Band structure plotting with angular momentum character Total and partial density of states Charge density plotting (1/2/3D) Plotting of exchange-correlation and Coulomb potentials (1/2/3D) Electron localisation function (ELF) plotting (1/2/3D) Fermi surface plotting (3D) Magnetisation and exchange-correlation magnetic field plots (2/3D) Wavefunction plotting (1/2/3D) Simple scanning tunnelling microscopy (STM) imaging

EXCITING DFT collinear magnetism spin-dynamics 1 Advanced Exact exchange (EXX) optimised effective potential () method (with SOC and NCM) EXX energies (with SOC and NCM) Hartree-Fock for solids (with SOC and NCM) Moessbauer hyperfine parameters: isomer shift, EFG and hyperfine contact fields First-order optical response Kerr angle and Magneto-Optic Kerr Effect (MOKE) output L, S, and J expectation values Effective mass tensor Equation of state fitting