Introduction to spin and spin-dependent phenomenon

Transcription

1 Introduction to spin and spin-dependent phenomenon Institut für Theoretische Physik Freie Universität Berlin, Germany and Fritz Haber Institute of the Max Planck Society, Berlin, Germany. May 16th, 2007

2 Synopsis What are spins Broad outline 1 What are spins? 2 What do they do for us? 3 What to think of and how to treat spins? 1 Non-relativistic treatment 2 Relativistic treatment 3 Collinear, 4 Spin-dynamics 4 Future.

3 Dirac equation - relativistic Dirac Pauli Relativistic time-dependent equation (mc 2 ) 2 + (p c) 2 Ψ(x,t) = i Ψ(x,t) t pψ(x,t) i Ψ(x,t) Dirac equation HΨ(x,t) = i Ψ(x,t) t [ cα p + α0 mc 2] Ψ(x,t) = i Ψ(x,t) t α 0 = [ ] I 0 0 I α j = [ 0 ] σj σ j 0

4 Dirac equation - relativistic Dirac Pauli Relativistic time-dependent equation (mc 2 ) 2 + (p c) 2 Ψ(x,t) = i Ψ(x,t) t pψ(x,t) i Ψ(x,t) Dirac equation HΨ(x,t) = i Ψ(x,t) t [ cα p + α0 mc 2] Ψ(x,t) = i Ψ(x,t) t α 0 = [ ] I 0 0 I α j = [ 0 ] σj σ j 0

5 Dirac Pauli Pauli matrices and Dirac spinors Pauli matrices σ 1 = σ x = ( ) 0 1, σ = σ y = ( ) ( ) 0 i 1 0, σ i 0 3 = σ z = 0 1 Dirac spinor ψ (r,t) Ψ(x,t) ψ (r,t) χ (r,t) χ (r,t)

6 Dirac Pauli Pauli equation and spinors - non-relativistic Pauli equation: ( ) [ ] 1 ψ (r,t) 2 p 2 + µ B σ B(r,t) = i ψ (r,t) t [ ] ψ (r,t) ψ (r,t) [ ] ψ (r) σ ψ (r) [ ] ψ (r) = m(r) ψ (r) Magnetization density: the order parameter for magnetic phase transition in solids.

7 Dirac Pauli Pauli equation and spinors - non-relativistic Pauli equation: ( ) [ ] 1 ψ (r,t) 2 p 2 + µ B σ B(r,t) = i ψ (r,t) t [ ] ψ (r,t) ψ (r,t) [ ] ψ (r) σ ψ (r) [ ] ψ (r) = m(r) ψ (r) Magnetization density: the order parameter for magnetic phase transition in solids.

8 How to think of spins What are spins Dirac Pauli Magnetization σ = m(r)dr behaves like a classical vector under rotation m(r) for solid Fe Pauli matrices Generators of the rotation in spins.

9 What do spins do for us? Effects of spins Consequences of spins 1 Magnets 2 Invar effect 3 Kondo effect 4 spintronics 5 superconductivity 6 MOKE (Rödl) 7 GMR, CMR...

10 Non-relativistic ab-initio treatment of spins Spin Density Functional Theory (Barth and Hedin, JPC 5, 1629(72)) Total energy functional: E v/bext [n,m] = F[n,m] + dr v ext (r)n(r) + Kohn-Sham equation: [ 1 ] [ ] ψ i (r) v s (r) + µ B σ B s (r) ψ i (r) Kohn-Sham effective potential and field: dr B ext (r) m(r) [ ] ψ i (r) = ǫ i ψ i (r) v s (r) = v Cl (r) + v xc (r), B s (r) = B ext (r) + B xc (r)

11 Spin Density Functional Theory Densities and conjugate fields Density and scalar XC potential: v xc (r) = δe xc[n,m] δn(r) m Magnetization density and XC magnetic field B xc (r) = δe xc[n,m] δm(r) n

12 Collinear approximation What are spins Spin Density Functional Theory in Collinear approximation (upto 1990) Kohn-Sham equation: [ 1 ] [ ] ψ i (r) v Cl (r) + v xc (r) + µ B σ B s (r) ψ i (r) [ ] ψ i (r) = ǫ i ψ i (r) Collinear approximation: The spins are aligned parallel ( ) or anti-parallel ( ) to B(r). There is a universal direction of magnetization.

13 Decoupled equations What are spins Decoupled Kohn-Sham equations: [ 1 ] v Cl (r) + v xc (r) ± µ B B sz (r) Φ i± (r) = ǫ i± Φ i± (r) Φ + (r) [ ] [ ] ψ (r) 0, Φ 0 (r) ψ (r) Advantages of decoupled equations: Two decoupled Kohn-Sham equations are easy to solve. This saves a lot of time. Functionals designed for (spin unpolarized) DFT can be used.

14 Classic collinear magnets Magnetic moment for Transition metals Magnetic moment in Bohr magneton Metal Expt SDFT-LSDA Fe Co Ni

15 DOS for a collinear magnet (Alloy of Fe-Al) using LSDA Kohn-Sham is spectrum is no good!! Quasi-particle spectrum needed (Fausti)! Expt LSDA DOS (arb units) Energy [ev]

16 Non- Non-collinearity: there is no universal direction of magnetization. Spin-orbit coupling: [ v s (r) + µ B σ B s (r) + 1 ] 4c 2L S Φ i (r) = ǫ i Φ i (r) Problem In both cases decoupling of spins not possible!! What about functionals??

17 Functionals for 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)

18 Functionals for 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)

19 Functionals for 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)

20 Non-collinearity in Cr-monolayer Sharma et. al Phys. Rev. Lett (2007)

21 Spin-dynamics What are spins Landau-Lifschitz equation for spin dynamics m(r, t) t = γm(r,t) (B xc (r,t) + B ext (r,t)) J In abscence 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!!

22 Spin-dynamics What are spins Landau-Lifschitz equation for spin dynamics m(r, t) t = γm(r,t) (B xc (r,t) + B ext (r,t)) J In abscence 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!!

23 Orbital exchange-correlation functionals (Stefano) Third generation: Exact exchange (EXX) Neglect correlation and use the Hartree-Fock exchange energy E x [Φ(r)] = 1 2 occ i,j dr dr Φ i (r)φ i(r )Φ j (r )Φ j (r) r r Treating magnetization and XC field as unconstraint vector fields

24 Non- in unsupported Cr-monolayer Sharma et. al Phys. Rev. Lett (2007)

25 Magnons What are spins Magnon A magnon is a collective excitation of spins in a crystal lattice. In contrast, a phonon is a collective excitation of atoms or ions in a crystal lattice. A magnon can be viewed as a quantized spin wave.

26 Magnons or spin-waves What are spins Pauli spinor for a spin spiral of vector q (Frozen magnon) [ ] ψ Φ(r) (r)e iq 2 r m x (r)cos(q r) ψ (r)e iq 2 r, m(r) = m y (r)sin(q r) m z (r)

27 Magnons or spin-waves What are spins Pauli spinor for a spin spiral of vector q (Frozen magnon) [ ] ψ Φ(r) (r)e iq 2 r m x (r)cos(q r) ψ (r)e iq 2 r, m(r) = m y (r)sin(q r) m z (r)

28 Magnon spectra for Fe: SDFT results Halilov et. al Phys. Rev. B (1998)

29 Magnon spectra for Tb, TD-SDFT needed (Oliviera) Perlov et. al Phys. Rev. B (2000)

30 Outlook What are spins Future developments New functionals spin transport: spintronic devices spin fluctuations and quantum criticality spin-dependent GW applications...

31 Code used: EXCITING K. Dewhurst, S. Sharma, C. Ambrosch-Draxl and L. Nordström The code is released under GPL and is freely available at: