Low Dimensional Magnetism Workshop European School on Magnetism - PowerPoint PPT Presentation

Low Dimensional Magnetism Workshop European School on Magnetism Timisoara, Sept. 08, 2009 Pietro Gambardella ICREA and Centre d’Investigació en Nanociència i Nanotecnologia (ICN-CSIC), Barcelona, Spain Olivier Fruchart Institut Néel (CNRS), Grenoble, France Wulf Wulfhekel Physikalisches Institut, Universität Karlsruhe, Karlsruhe, Germany

a Magnetic moments: atoms vs. bulk

Magnetic moment of an atom: role of electron-electron and spin-orbit interactions 2 2 2 Z p Z Ze e Z ∑ ∑ ∑ ∑ l s = − + + ⋅ = + + H i ( )g(r ) H V V − i i i C e e s o. . 2 2 2m r − r r = = < = i 1 i 1 i j i 1 i i j … … 3d 7 (n=3, l=2) J=1/2 L=1, S=3/2 J=3/2 J=5/2 J=3/2 J=5/2 European School on Magnetism – P. Gambardella L=3, S=3/2 J=7/2 J=9/2 central field TERMS MULTIPLETS approximation electrostatic interaction spin-orbit coupling |LSM L M S > base |LSJM> base Hund's rules: 1) Total spin S = S i s i is maximized 2) Total orbital moment L = S i l i is maximized 3) L and S couple parallel ( J =| L+S |) if el. shell is more than half-filled, L and S couple antiparallel ( J =| L-S |) if el. shell is less than half-filled see lecture 2 by J.M.D. Coey

Spin and orbital magnetic moments in bulk solids Atom Magnetic metal Nonmagnetic metal European School on Magnetism – P. Gambardella tot d sp Material N holes m s m s m s m orb Fe 3.4 2.19 2.26 -0.07 0.09 Co 2.5 1.57 1.64 -0.07 0.14 Ni 1.5 0.62 0.64 -0.02 0.07 Data from O. Eriksson et al., Phys. Rev. B 42, 2707 (1990).

Free atoms vs. bulk: spin moment in 3 d metals 6 Atom 5 Bulk Magnetic moment ( μ B ) 4 European School on Magnetism – P. Gambardella 3 2 1 Spin 0 Sc Ti V Cr Mn Fe Co Ni Cu courtesy V. S. Stepanyuk, MPI Halle

Exchange-split electron band structure measured by UV photoemission European School on Magnetism – P. Gambardella Himpsel et al., J. Magn. Magn. Mat. 200 , 456 (1999).

Exchange splitting vs. spin magnetic moment Magnetic moment ~ exchange-splitting μ μ ≈ 1 B Δ eV exc Exchange-split density of states Δ Exchange splitting (eV) exc European School on Magnetism – P. Gambardella Magnetic moment (mu B /atom) Himpsel, Phys. Rev. Lett. 67, 2363 (1991)

Metal clusters in molecular beams Stern-Gerlach experiment European School on Magnetism – P. Gambardella I.M.L. Billas, A. Châtelain, W.A. de Heer, Science 265, 1682 (1994).

Band narrowing in low-dimensional systems bulk Fe DOS European School on Magnetism – P. Gambardella 1 ML Fe/Au(111) Fe 1 /Au(111) Moruzzi, Janak, and Williams, Sipr, Minar, and Ebert, Calculated electronic properties of metals Europhys. Lett., in press (2009) (Pergamon, 1978)

Enhanced magnetic moment at surfaces and 2D layers 1 ML Fe on W(110): Fe (100) and (110) surfaces 15 % increase in ground state (T=0) Symmetry-dependent increase in magnetic moment topmost layer magnetization European School on Magnetism – P. Gambardella S. Handschuh, PhD thesis, Uni Köln Elmers, Liu, and Gradmann, Phys. Rev. Lett. 1989

Coordination-dependent spin magnetic moment in metal clusters European School on Magnetism – P. Gambardella SP-KKR calculations, Mavropoulos, Lounis, Zeller, and Blügel, Appl. Phys. A 2006

Trends with band structure > ( ) 1 U n E Stoner criterion F W Rectangular-shaped DOS 1 ∫ ⇒ ε ε = ( ) n E ∼ ( ) . W n d const F W E F E Tight-binding: In transition metals: 1 European School on Magnetism – P. Gambardella ≈ ≈ 2 ( ) n E ( ) n ( E ) ∼ W N h r F d F d nn d nn W d nearest neighbors lattice spacing hopping S. Blügel, FZ Jülich

b 4d, 5d magnetism

Nonzero magnetic moments in low-dimensional 4 d and 5 d metal structures European School on Magnetism – P. Gambardella S. Blügel and P. Dederichs, FZ Jülich – Phys. Rev. Lett. 1992, Solid State Comm. 1994.

Nonzero magnetic moments in 4 d metal clusters Free clusters Adatoms, adclusters European School on Magnetism – P. Gambardella nonmagnetic (superpara)magnetic if < 100 atoms Honolka et al., Cox et al., PRB 1994 Phys. Rev. B 76 , 144412 (2007).

Induced interface magnetization in nonmagnetic metals Increase in total magnetization due to Pd deposition on a Fe film European School on Magnetism – P. Gambardella Janak, PRB 16, 255 (1977) Gradmann, Dürkop, and Elmers, JMMM 165, 56 (1997).

c Orbital moment

Quenched orbital moment in metals Electron hybridization reduces m L European School on Magnetism – P. Gambardella Eriksson et al., Phys. Rev. B 42, 2707 (1990). m L atom ( μ B ) m L bulk ( μ B ) Fe (-bcc) 2.0 0.09 Co (-hcp) 3.0 0.15 Ni (-fcc) 3.0 0.05 E.P. Wohlfart, in Ferromagnetic Materials , Vol. 1, E.P. Wohlfart ed., North Holland, Amsterdam (1980).

Quenched orbital moment in cubic compounds Cubic symmetry d -wavefunctions xz ( ) 15 1 ψ ≡ = = − + d Y Y − 1 xz 21 2 1 2 r π 4 2 − xy i ( ) 15 ψ ≡ = = − d Y Y − 2 xy 22 2 2 2 r π 4 2 yz i 15 ( ) ψ ≡ = = + d Y Y − 3 21 2 1 yz 2 r π 4 2 European School on Magnetism – P. Gambardella − 2 2 3 z r 5 ψ ≡ = = d Y 4 2 2 20 − 2 3 z r r π 16 − 2 2 1 x y ( ) 5 ψ ≡ = = + d Y Y − 5 2 2 22 2 2 − 2 x y r π 16 2 = − μ m L L B = − ⋅ = μ H L B m B Zeeman L B z Electrons described by real 3 d wavefunctions have zero orbital moment: 1 1 ( ) ( ) ( ) ψ ψ = + + − + + − = + − = 2 2 2 2 2 2 0 L L 5 z 5 z 2 2

Orbital moment in a cubic crystal field z d z 2 ,d x 2 -y 2 y x d xy ,d xz , d yz European School on Magnetism – P. Gambardella In the presence of a cubic crystal field larger than the s.o. or Zeeman field: eigenvectors t 2g subspace: ( ) − − 1 Partial quench ⎛ 0 0 i ⎞ ⎛ 1 0 0 ⎞ = ψ − ψ = t i Y − 1 1 3 2 1 2 ⎜ ⎟ ⎜ ⎟ = ⎯⎯⎯⎯ → diagonalize L 0 0 0 0 1 0 1 ( ) = − ψ + ψ = t i Y ⎜ ⎟ ⎜ ⎟ z 2 1 3 21 ij 2 ⎜ ⎟ ⎜ ⎟ i 0 0 0 0 0 − i ⎝ ⎠ ⎝ ⎠ ( ) = ψ = − t Y Y − 3 2 22 2 2 2 e g subspace: Total quench ⎛ 0 0 ⎞ = ψ = e Y = ⎜ 1 4 20 L ⎟ 1 z ( ) ij 0 0 = ψ = + e Y Y ⎝ ⎠ 2 5 22 2 2 − 2

* Orbital moment in 2nd order perturbation theory ψ 0 = 0 ψ 0 unperturbed Schroedinger equation: H E 0 gnd gnd gnd + λ ψ = ψ small perturbation: (H V) E 0 = (0) λ (1) + λ 2 (2) + + ..., E E E E ψ = ψ λ ψ + λ ψ + (0) (1) 2 (2) + ... ψ ψ (1) (0) (0) European School on Magnetism – P. Gambardella = E V gnd gnd ψ (0) ψ (0) V ∑ exc gnd ψ (1) = ψ (0) exc (0) − (0) E E ≠ exc gnd λ = λ ⋅ exc gnd V L S 2 ψ ψ = (0) (0) = 0 L L ψ (0) ψ (0) V gnd gnd gnd ∑ exc gnd = (2) ψ λ ⋅ ψ E (0) (0) L S ∑ exc gnd ψ ψ ≈ ψ (0) ψ (0) ≠ = 0 (0) − (0) E E L L L gnd exc (0) − (0) E E ≠ exc gnd exc gnd exc gnd ≠ exc gnd

Orbital moment in 2nd order perturbation theory Tetragonal distorted d 9 state (Cu 2+ , 10 Dq=2 eV, D s =0.1 eV, D t = 0) d xz ,d yz E g 0.3 eV B 2g d xy 0.8 eV 2 D λ L S ∓ ± European School on Magnetism – P. Gambardella λ L S z z 1.2 eV A 1g d z 2 0.4 eV B 1g d x 2 -y 2 Unperturbed ground state has d x2-y2 symmetry, i.e., <L> = 0, however ( ) λ ⋅ = λ + + L S L S L S L S + − − + z z mixes excited states into d x2-y2 inducing nonzero <L> ~ λ / Δ E

Orbital moment in low-dimensional metal films Orbital d-shell moment z z in an atom L z = +1, +2 m L e e e - European School on Magnetism – P. Gambardella m L L z = -1, -2 Directional quenching of orbital moment in a free-standing metal layer: + + − d ∼ 2 2 + − − d ∼ 2 2 2 2 − xy x y

Orbital moment in low-dimensional metal films bulk metal in-plane out-of-plane orbitals orbitals Band Narrowing W � Different in-plane and out-of-plane bandwidth d W ⊥ d European School on Magnetism – P. Gambardella Δ = − E d ( , d , d ) E d ( , d ) CF xz yz 2 2 xy 2 2 − − 3 z r x y P. Bruno, PhD thesis Δ CF 1 1 ⊥ ∝ ∝ � m m L W ⊥ L � W d d Free-standing metal layer

Orbital moment in metal films: interface effects European School on Magnetism – P. Gambardella > ⊥ ⇒ ⊥ < � � W W m m Co film free-standing d d L L ⊥ ⊥ < ⇒ > � � W W m m Co film on Au d d L L

enhanced orbital magnetic moment from 3D to 2D and 1D European School on Magnetism – P. Gambardella m L 0.68 μ B 0.37 μ B 0.33 μ B 0.31 μ B P. Gambardella et al., Phys. Rev. Lett. 93 , 077203 (2004).

Orbital moment and magnetocrystalline anisotropy in 3 d metals crystal field electron orbits fixes L relative to the crystal lattice Different L values along different crystal directions Direction with the easy direction Lowest spin-orbit energy largest component of L of magnetization European School on Magnetism – P. Gambardella L L see, e.g., P. Bruno, PRB 39 , 865 (1989); H. A. Dürr et al., Science 277 , 213 (1997).

Magnetocrystalline anisotropy