Magnetism. Eric Bousquet. University of Liège. Abinit School, Lyon, France 16/05/2014

Transcription

1 Magnetism University of Liège Abinit School, Lyon, France 16/05/2014

2 Outline Origin of magnetism: Inside an atom Between 2 atoms Interaction with ligands Interaction through ligands Spin orientation Collinear magnetism in DFT Non-collinear magnetism in DFT Finite magnetic field Constrained magnetic moments

3 Origin of magnetism (Semi) Classical picture: Bohr atom model (L) + electron spin S Spin and Orbital contributions to the magnetization: M = (2 S + L ) µ B µ B = he 2mc In most cases S >> L magnetism comes from the spins: M 2 S µ B

4 Origin of magnetism: Inside an atom 2 electrons with the same l but different m l (says, Φ a and Φ b orbitals): H = H 1 + H 2 + H 12 H 12 = e2 4πε 0 r 12 and: φ a (r i ) H i φ a (r i ) = E 1, φ b (r i ) H i φ b (r i ) = E 2, φ i φ j = δ ij Each electron can have spin up and down states 4 spin-orbitals: Such as we have: K ab J ab H = E 1 + E K ab J ab 0 0 J ab K ab K ab J ab

5 Origin of magnetism: Inside an atom The diagonalization gives a triplet (FM) and a singlet (AFM) states: where we have the Coulomb and Exchange integrals: K ab = e2 4πε 0 d 3 r 1 J ab = e2 4πε 0 d 3 r 1 d 3 r 2 φ a (r 1 ) 2 φ b (r 2 ) 2 r 12 (= U) d 3 r 2 φ a(r 1 )φ b (r 1 )φ b (r 2)φ a (r 2 ) r 12 The Hamiltonian can be re-written in the Heisenberg form: H = constant 2J ab S 1 S 2

6 Origin of magnetism: Inside an atom To have S = 0 one needs partially field orbitals ex: d n orbitals Hund s rules Most of the crystals are magnetic because they contain Transition-Metal and Rare Earth atoms (d and f electrons)

7 Origin of magnetism: Between two atoms The FM state is lower in energy in an atom (Hund s rules). But in H 2 molecule: Similar Heisenberg Hamiltonian: H = constant 2J ab S 1 S 2 but with the 1s orbital overlap between the 2 atoms: J < 0 The singlet AFM state is lower in energy!

8 Origin of magnetism: Between two atoms The hopping process reinforce the AFM state:

9 Origin of magnetism: Interaction with ligands Atom in a solid: Crystal Field Splitting Atomic d (or f ) orbitals splitted due to the surrounding atoms. Hund s rules still apply.

10 Origin of magnetism: Interaction with ligands High-spin and low-spin configurations: Depending on the size of CF relative to U.

11 Origin of magnetism: Interaction with ligands Be aware of the Jahn-Teller effect: from Stöhr and Siegmann, Magnetism, Springer 2006 Very strong in Mn 3+ (d 4 ), Cr 2+ (d 4 ), Cu 2+ (d 9 )

12 Origin of magnetism: Interaction through ligands Atoms interacting through the ligands: Superexchange from Stöhr and Siegmann, Magnetism, Springer 2006 Heisenberg picture still holds (localised electrons, t <<): H = JS 1 S 2

13 Origin of magnetism: Interaction through ligands SE depends on the bonding: Goodenough-Kanamori rules

14 Origin of magnetism: Spin orientation 2 nd order Heisenberg model for localised magnetic moments: H = 2 [ JS S + D (S S ) + S Φ S ] J Superexchange interaction (favors S S ) D Dzyaloshinsky-Moriya interaction (favors S S ) Φ Single Ion Anisotropy (easy/hard spin orientation) One wants to estimate J, D and Φ from DFT! see for ex: PRB 84, p (2011), PRB 86, (2012)

15 : Collinear case DFT based on the charge density ρ(r) To enlarge DFT to (collinear) magnetism, decomposition of the density: ρ = ρ( ) + ρ( ) The Hohenberg and Kohn theorem generalizes with an energy functional: E = E [ρ( ),ρ( )] With 2 Kohn-Sham equations to be solved, one for each spin-channel σ: (T + V Ri (r) + V H (r) + V xc,σ )φ iσ (r) = ε iσ (r) with V xc,σ = δe xc [ρ( ),ρ( )] δρ σ (r)

16 : Collinear case Then minimizing the K-S equations we get the ground state with: ρ = ρ( ) + ρ( ) and magnetization m = ρ( ) ρ( ) Supposing the magnetic moments are localised around the atoms (this is often the case for d and f electrons), one can compute magnetization on each atom (prtdensph input flag in ABINIT): Integrated total density in atomic spheres: Atom Sphere radius Integrated_up_density Integrated_dn_density Total(up+dn) Diff(up-dn) Note: Diff(up-dn) can be considered as a rough approximation of a local magnetic moment. (FeF 2 example)

17 : Collinear case Superexchange constants J can be estimated. Ex: Rock-Salt oxides One needs to compute the energy for ferro and antiferro in order to extract J 1 and J 2 from: E = E 0 + S J i S i i see for ex: PRB 84, p (2011), PRB 86 86, (2012)

18 : Non-collinear case Wave functions are described by spinors: φ i (r) = ( φi Such as the density is a 2 2 matrix: ( ρ ρ ρ = ) ρ ρ = 1 ( ) n + mz m x im y 2 m x + im y n m z with n the electron density and m i the magnetization density along the direction i φ i ) n(r) = 1 2 Trρ(r) = ρ αα (r) α m(r) = ρ αβ (r) σ αβ αβ with the Pauli matrices σ αβ = (σ x,σ y,σ z ): ( ) ( i σ x = σ 1 0 y = i 0 ) ( 1 0 σ z = 0 1 )

19 : Non-collinear case Kohn-Sham equations with spinors: H αβ φ β i = ε i φi α β where the Hamiltonian is a 2 2 matrix: and: n(r H αβ ) = T δ αβ + V (r)δ αβ + r r dr δ αβ + Vxc αβ (r) V αβ xc (r) = δe xc [ρ(r)] δρ αβ (r) ρ αβ is diagonal when m = m z collinear case. However one needs Spin-Orbit coupling in order to couple directions (space rotations) to the spins.

20 : Non-collinear case Spin-Orbit coupling: H = H KS + λ(r)l S = H KS + h2 1 2m 2 c 2 r dv dr L S from Stöhr and Siegmann, Magnetism, Springer (2006)

21 : Non-collinear case Spin-Orbit coupling important for heavy elements: Phonons of Pb, Phys. Rev. B 78, (2008): Phonons of Bi, Phys. Rev. B 76, (2007):

22 Magnetism in practice with Abinit nsppol, nspinor and nspden input flags: nsppol nspinor nspden Non-magnetic Collinear FM Collinear AFM Non-Collinear When the non-collinear flags are "on the SOC coupling is switch on (controlled by so_psp or pawspnorb flags). With SOC better not to use time-reversal symmetry (kptopt = 3 or 4). Initialize spinat = (m x, m y, m z ) for each atom.

23 Magnetism in practice with Abinit Ex: NiF 2 spinat = (0,0,m) no canting spinat = (m,0,0) canting along z prtdensph x_mag y_mag z_mag prtdensph x_mag y_mag z_mag

24 Applied magnetic field Applying a (Zeeman) magnetic field on the spins gives (zeemanfield flag): ( ) Hz H V H = µ B µ x + ih y 0 H x ih y H z Allows to access to: Linear magnetic susceptibility tensor M = χh M Linear magnetoelectric tensor P = αh P H H As well as non-linear responses!

25 Linear magnetoelectric response Free energy of a crystal under E and H fields: F (E,H) = ε 0ε ij E i E j µ 0µ ij H i H j + α ij E i H j +... Polarization: P j = F(E,H) E j =... + ε 0 ε ij E i + α ij H i +... Magnetization: M j = F(E,H) H j =... + µ 0 µ ij H i + α ij E j +... α = magnetoelectric tensor

26 Ex: Linear ME in Cr 2 O 3 Collinear AFM oxide First experimental evidence of linear ME effect: D. N. Astrov (1961) P (10-6 C/m 2 ) 30 P tot (Berry Phase) P ion ( Z δ ion Ω ) B (T) P elec or clamped ion (Berry Phase) Precision on energy/potential is crucial for P elec (toldfe Ha ) Precision on forces is crucial for P ion (toldmxf 10 7 Ha/Bohr )

27 Linear ME in Cr 2 O 3 Collinear AFM oxide First experimental evidence of linear ME effect: D. N. Astrov (1961) P (10-6 C/m 2 ) 30 ME coef: = 1.45 ps.m 1 (α exp = 1 4 ps.m 1 ) α spin tot α spin ion = 1.1 ps.m B (T) α spin elec = 0.34 ps.m 1 PRL 106, p (2011)

28 Constrained magnetic moment calculations (β version) Constrain the direction of the magnetic moments (magconon flag =1) : [ ] 2 Lagrange multiplier: E = E KS + λ m i m 0 i (m0 i m i ) i Constrain the direction and the amplitude of the magnetic moments (magconon flag =2) : [ ] 2 E = E KS + λ m i m 0 i i with λ the strength of the Lagrange multiplier (magcon_lambda flag) and mi 0 the desired magnetic moment on each atom i (given by spinat).

29 Constrained magnetic moment calculations (β version) Useful to explore the spin energy landscape "by hand" (in case of multiple local minima of complex and flat energy landscapes): And also to compute the magnetocrystaline anisotropy energy.

30 Ex: Effect of distortions on SIA and MCA in LaFeO 3 Octahedra distortions (a 0 a / ): LaFeO 3 10 a 0 a 0 c + Energy (µev) 5 a 0 a 0 c (z) angle ( ) (y) SIA in the xy plane with a 0 a 0 10 and along z axis for a 0 a

31 Effect of distortions on SIA and MCA in LaFeO 3 Octahedra distortions (a 0 a 0 10 ): SIA local easy axis follows octahedra LaFeO 3 site B site A 0 Energy (µev) -2 MCA sum angle ( ) Global easy-axis = [110] PRB 86, p (2012)

32 Conclusions DFT + spins: Collinear magnetism: easy to handle Non-Collinear magnetism: often less easy Often DFT+U or Hybrid functionals are better for magnetic systems Allows to compute: (Super) Exchange interaction between spins (J). Spin canting / Dzyaloshinsky-Moriya interaction (D) Magnetic anisotropy Response under Zeeman field (magnetic and magnetoelectric susceptibilities) Future/on-going works: Orbital magnetism: J. Zwanziger and X. Gonze Phys. Rev. B 84, (2011) Constrained moments, spin dispersion DFPT with Zeeman field