Spin effects (spin polarized systems, spin-orbit ) G. Zérah CEA-DAM Ile de France Bruyères-le-Châtel

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1 Spin effects (spin polarized systems, spin-orbit ) G. Zérah CEA-DAM Ile de France Bruyères-le-Châtel 1

2 Macroscopic magnetization A crystal can be found in different magnetization states. The direction of macroscopic magnetization changes inside the crystal This can be observed by the magnetic Kerr effect (rotation of polarized light) Under the action of an external field, the magnetization changes its direction, not its amplitude. M () r = Ms M () r Under a very intense magnetic field B M = M s B When the magnetic field is turned off, the magnetization distribution is reestablished. The domain distribution has an energetic origin (and is not related to defects for instance) 2

3 Macroscopic Magnetism Magnetic energy Landau-Ginzburg like functional [ M (r)] = an [ M (r)] + ech [ M (r)] + E [ M ( r )] M ( r ). B ( r ) E E E m s The observed magnetization minimizes this functional. The first two terms are of microscopic origin The third term depends upon the shape of the sample. The fourth one describes the interaction with the external field. NB: If time is reversed, M () r M (), r B () r B () r but the energy Is constant ext 3

4 Macroscopic Magnetism Different terms in the magnetic energy Magnetic anisotropy energy depends upon orientation Can exist only if there is a coupling between the lattice and magnetization, through the spin-orbit coupling. [ ] E M ( r ) 1 µ e V / a to m an 4

5 Macroscopic Magnetism Anisotropy energy respects the symmetries of the crystal Cubic (K 1 >0, [100], K 1 <0 [111] etc..) E = K ( m m + m m + m m ) + K m m m + m = M / M an 1 x y x z z y 2 x y z Hexagonal K = = ( ) Ean K1( mx m y ) K2( m x m y ) ( mx imy ) +( mx imy ) m M / M 2 5

6 Objectives of a magnetic calculation 1. Microscopic determination of the saturation magnetization It might be non uniform inside the crystal cell (non-collinear magnetism) 2. Microscopic determination of the magneto crystalline anisotropy Its most important contribution from the spin-orbit coupling Other an isotropic interactions: spin-spin, spin-other orbits neglected 3. Exchange terms 6

7 Some Classes of Magnetic Systems Paramagnetism is favored by the presence of bands, which are doubly occupied. But some systems exhibit spontaneous magnetization. Collinear ferromagnets :3d Metals : Fe, Co, Ni. Mostly due to spin (orbital moment ~0) Stoner criterion: xc X D(Ef) large Antiferromagnet (anti parallel orientation): Cr, Mn (some phases), ordered alloys: Fe 3 Mn, insulating compounds MnO, NiS : different sublattices with opposite orientations. In case the moments do not compensate: ferrimagnetism yields a net magnetic moment. Magnetite: Fe 3 O 4,NiFe 2 O 4 Complicated structures: exchange mediated by oxygen Helimagnetism: some 4f Rare earth (RKKY interaction), MnAu 2 More complex non collinear structure: Mn, fcc Fe etc. 7

8 Example: Ferromagnetism of 3d metals Iron,Nickel and Cobalt: ferromagnets These are 3d elements, for which the atomic 3d WF strongly overlap and form a band of a given width W A qualitative understanding of the existence of magnetism is given by the Stoner criterion. Assume a constant DOS (D), and decompose the energy as band+exchange contributions Correlation effects has a structure: Un n If DU > 1, then one spin channel will be filled at the expense of the other. E = E + E = εd( ε) dε + εd( ε) dε + Un n B XC 0 0 ζ = + + = + + = + + 2D 2D D 2D D 4 D( ε ) = D= Cste Dε n 1 n 1 n (1 ) ( n n ) Un n ( U) n n ( U) n n + n = n ζ = = n n n ε ε 8

9 Spin DFT In principle, the DFT is an exact theory and so there is no need for a particular consideration for spin. But, the usual approximations generalize easily if we consider spin explicitly One complements the external potential V, with a magnetic field B along z (next for B ) The system will polarize, and one gets a up and a down electron density: The energy of the system will be a functional of the density and the magnetization (supposed along z). n ( r ), n ( r ) [ ], E = K n + K n + E n + E n n H xc + dn r () r V () r dm r () r B () r ext n() r = n () r + n () r m() r = n () r n (), r in units of µ ext B 9

10 The Kohn-Sham equations If we minimize the energy, we obtain two equations, one for each spin channel (we set B ext =0) σ i i i + V eff ( r) Ψ σ = ε σψ σ σ = 2 n i i () r = f( ε ) Ψ () r i σ σ V () r = V () r + V () r + V () r V σ eff ext h xc σ xc () r = δε xc σ ( n, n ) δ n σ n() r = n () r + n () r m() r = n () r n () r σ 2 10

11 The LSDA The total energy is also given by the sum over states: 1 σ E = ε ισ d n() Vh() d nσ() Vxc() Exc n, n, 2 r r r r r r + i, σ σ The exchange correlation energy of the electron gas can be written as: LDA Exc n, n n( ) ε xc( n, n ) d r r ε ( n, n ) = ε ( n, n ) + ε ( n, n ) ε xc x c 1/ /3 4/3 x( n, n ) = 3 ( n + n ) ε ( n, n ) c 4π by QMC n GGA Exc n, n n( ) ε xc( n, n, n, n ) d r r Perdew, Burke, Ernzerof PRL,77,3865(1996): PBE 11

12 Stoner revisited Each spin channel yields a different potential due to the xc term To study the stability of the paramagnetic phase, one can expand this potential for a small magnetization term V V m V 0 1 xc() r = xc() r ± () r xc() r 1 V r = V r ± IM 2 0 xc( ) xc( ) If second term ~constant: rigid shift ε f M = D + IM D IM d ( ε ) ( ε ) ε 2 2 ID ε > 0 Non zero solution at small M ( f ) 1 12

13 Stoner revisited More rigorously (Janak,1977): compute the spin susceptibility and the criterion is satisfied when it is negative. χ0 χ = 1 χ I 0 2 I = d r γ () r K() r 0 d D ( ε f ) γ( r) = ρ( r) dε f δ E ρ, m / δm( r) δm( r') = 2 K( r) δ( r r') { 2 [ ] } xc 13

14 Example of a Pseudopotential calculations (norm conserving) BCC Fe ~rigid shift of DOS Magnetization strongly dependent of cell parameter Magnetization as a function of cell (a.u.) Energy as a function of cell (a.u.) 14

15 Calculations using a Pseudo potential Iron has been much studied LSDA will predict a correct magnetic moment (2.17 vs 2.22 exp) BUT, the NM fcc structure has a lower energy than the FM bcc This is corrected but using the (PBE) GGA Pseudo potentials are generally constructed on non magnetic configurations and used for magnetic systems Also, the use of non local core corrections is essential (exercice: compare magnetization for bcc Fe, using 26fe.pspnc and 26fe.hgh) But, best results are obtained with PAW (cf talks by Jollet Torrent): this is an all electron theory with a psp look (and includes nlcc) 15

16 Results of calculations 3d ferromagnetic elements (Fe, Co, Ni): very good prediction of the magnetization with GGA 3d non ferromagnetic elements (Mn, Cr): complicated spin arrangement (e.g. spiral) non compatible with the hypothese of // spins Intermetallic compounds (that is compounds of Cu,Ni, Co, Fe, Mn, Cr), yields good values of magnetization Magnetic Oxydes: classical examples of failure of the LSDA (or GGA) This failure is presumably due to a neglect of the on site correlation (U term) of localized orbital. LDA includes this only on average (cf the example of Ce) This failure is the source of many improvements of LDA, GGA: LDA+U, SIC, Hybrids functionals (HF+LDA), etc 16

17 Non collinear Magnetism Up to now, we considered moments all parallel along a given axis But there are cases where it is not justified: In some systems, the ground state is non collinear: in Mn, in solid oxygen, in Mn 4 N, in fcc Fe etc Spin excitations ( spin waves ) must take into account non parallel configurations The exchange coupling parameter of the Heisenberg Hamiltonian requires a non collinear arrangement This requires a generalization of spin DFT to account for a vector m(r) 17

18 Spin ½ reminder Spin state a s = a + + b = b Magnetic moment along x S = µ s σ s x B x i 1 0 σ =, σ, σ x 1 0 = y = i 0 z

19 The WF as a two component spinor In each point in space, the wave function posses two spin components, and the KS principle is generalized towards a duality between (density, magnetization) and external fields (potential, magnetic field) ψ ( r ) ψ ( r ) = ψ ( r ) The electronic density can be written: ψ () r ρ() r = ψ () r ψ () r I = ψ () r + ψ () r * 2 2 ψ () r 19

20 Direction of magnetization in space The magnetization along x, for instance: * ψ ( r ) M x( r) = ψ ( r) ψ ( r) S x ψ ( r ) * ψ ( r ) = µ B ψ ( r) ψ ( r) σ x ψ ( r ) * 0 1 ψ ( r ) = µ B ψ ( r) ψ ( r) 1 0 ψ ( r ) = µ ψ r ψ r + ψ r ψ r * * B ( ( ) ( ) ( ) ( ) ) 20

21 Another view: the spin density matrix Magnetization * ψ ( r ) M ( r) = ψ ( r) ψ ( r) S ψ ( r ) Spin density matrix ρ = ψ ψ ρ = ψ ψ * * kj kj ( r) α ( r) β ( r) ( r) f kj α ( r) β ( r) kj 21

22 Magnetization and spin density matrix We can equivalently use each object: αα m( r) ρ ( r) σ ρ (r)= ρ ( r) = 2 ρ ( r) = ρ ( r) δ +m(r). σ * Note the * 22

23 Energy functional (GLSDA) Energy is now a functional of energy and magnetization vector m [ ] E ρ () r = T ρ () r + EH ρ() r + Exc ρ () r + Vext() r ρ() r dr T ρ () r = f ( ψ () r ψ () r + ψ () r ψ ()) r dr 2 * α * α * β * β kj kj kj kj kj 2m jk [ ρ m ] Exc ρ () r = Exc (), r () r One can equivalently use the spin density matrix. The variational principle applied to it leads to the new Hamiltonian As usual, one needs to construct the exchange correlation 23

24 The «local spin density approximation»(lsda) One extends the usual (LDA) exchange correlation term to include the presence of vector magnetization Exc = ρ() r εxc( ρ(), r m() ) r dr = ρ() r εxc( ρ ()) r dr If one uses the electron gas as a reference, the magnetization direction is immaterial (note, this is easy to express with the help of the magnetization vector) We can use the same formula for the LDA as the usual collinear magnetic case, since it only depends on m In principle, the GGA should include the rotationally invariant parts of m, but is not yet done 24

25 Kohn-Sham Hamiltonian H 1 δ E = δ + V eff = 2 δρ () r V = V ( ρ( r)) δ + V ( ρ ( r)) + V ( r) eff H xc ext 25

26 Kohn-Sham Hamiltonian (contd) δexc( ρ ( r)) Vxc ( ρ ( r)) = δρ () r V ( ρ ( r)) = V ( ρ ( r)) δ µ b( ρ ( r)). σ xc xc B δexc( ρ ( r)) εxc( ρ ( r)) b( ρ ( r)) = = mˆ ( r) ρ( r) δ m() r m() r 26

27 Energy functional Expression with the help of eigenvalues E ρ () r = f( εk j) εk j Veff() r ρ() r dr+ Beff() r m() r dr kj [ ] [ ] + V () r ρ() r dr+ E ρ() r + E ρ(), r m() r ext H xc 27

28 Spin-orbit: The Dirac Hamiltonian Dirac s equation (φ and ψ two components spinors) E+ mc V + c p = E V ψ + cσ. pφ = 0 [ ] 2 2 φ σ. ψ 0 Low E: scalar relativistic approximation + spin-orbit coupling φ ( v / c) ψ ordre (1/c) = (1/137) 2 p F + V ( r) φ 2 4 p 1 F V ( r). φ 2 2 8c 4c 1 + σ.( V ( r) pφ ) = εφ 2 4 c

29 Spin-orbit coupling Spherical potential 1 dv V ( r) = r r dr 1 1 dv 1 dv σ.( V ( r) p ) = σ.( r p ) = σ.l 2 4 c r dr r dr Order of magnitude for a valence orbital a 1/ Z, V Z < ψ r dv / dr ψ > ( Z / c) >Much better for the valence pseudo orbital ( Z / c) eff 2 29

30 Spin orbit projector The L.S operator The angular momentum operator has invariant spaces indexed by l and of dimension (2l+1). A projector on this space is denoted l >< l The same is true for spin (with l=1/2) s >< s A projector on spin and angular momentum space is ls >< ls 30

31 The L.S operator On this space the L+S=J operator has two invariant subspaces Angular momentum l+1/2 and l-1/2 of dimensions 2l+2 and 2l Now, J = ( L+ S) = L + S + 2 LS. So we derive the meaning of L.Sls >< ls Project on the ls space, and apply the spin-orbit coupling 31

32 How to build Pseudo potentials For a potential with a spherical symmetry, the solutions of Dirac s equation have the form: φ 1/2 1/2 l+ m+ 1 m l m+ 1 m 1 Yl Yl 2l 1 2l 1 () r + φ () r + m Y l 2l+ 1 2l+ 1 l+ 1/2 1/2 l 1/2 1/2 l m m+ 1 l+ m Yl One can built a pseudo potential for which the wave functions have the same shape beyond a certain radius, by only adding a spin orbit term to it SR SO VNL = Vl (, r r ') ls ls + Vl (, r r ') L.S ls ls l m m V (, r r') = α P() r P(') r SR( O) l ij i j ij 32

33 How to use a plane wave basis Each component is developped on a basis For instance, in plane waves, the basis is G 2 iπ G. r e et G 2 iπ G. r = = 0 e 0 The components of the wave function are: ig. r () ik. r ψ r a e ik. r G G ψ () r = e e ig. r ψ () r = a e G G That is two times more coefficients, and matrices 4 times bigger 33

34 Expression in the plane wave basis The important term is GVNL ( ) ( ) GQ() rq(') r ls lsg' = 4(2 π l+ 1) f G f G' PGG ( ˆˆ. ') G' i j i i l ( ) ( ) Gs Q() rq(') r L.S ls lsg' s = i4(2 π l+ 1) f G f G' P'( GG ˆˆ. ') < s S s >. G G' 1 i j 2 i i l 1 2 ( ) 2 fi G = 4 π + Qi( r)exp( igr) r dr 0 See Gonze at al. Comp. Mat. Sci., 25 (2002) 34

35 Effect of spin-orbit coupling upon total energy Heavy metals «norm conserving» pseudo potentials Experimental equilibrium phases N. Richard et S. Bernard(PRB,66,2002) GGA GGA+SO EXP GGA GGA+SO EXP U 145GPa 132Gpa 135Gpa Np 180Gpa 165Gpa 120Gpa 18,

36 Conclusions Magnetic moments generally well reproduced by GLSDA But magnetic oxides exhibit localized states which are not well described by LDA (Mott insulators e.g. due to correlations are described as metals) Anisotropy is well described for hard magnets, less well for soft magnets Temperature effects on K are nearly untouched Spin-orbit coupling has a smaller effect on total energy in a psp context than in all electrons calculations But changes the DOS substantially for heavy metals 36

37 Bibliography General: R. Martin s Book Magnetism Introductory (with emphasis on materials and properties) Magnetic Material, Nicola Spaldin, Cambridge (2003) Advanced Magnetism and the electronic structure of crystals, Gubanov et al., Springer