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1. Spin-orbit induced phenomena in nanomagnetism László Szunyogh Department of Theoretical Physics Budapest University of Technology and Economics, Hungary Psik-Workshop on Magnetism, Vienna, 17thApril, 2009

2. Coworkers L. Udvardi, A. Antal, L. Balogh Budapest University of Technology and Economics, Hungary B. Lazarovits, B. Újfalussy Hungarian Academy of Sciences, Hungary J.B. Staunton University of Warwick, UK B.L. Györffy University of Bristol, UK U. Nowak, J. Jackson University of Konstanz, Germany University of York, UK

3. Outline of the talk Theoretical and computational concepts  Spin-orbit coupling, magnetic anisotropy  Classical spin Hamiltonian (Dzyaloshinskii-Moriya interaction)  The relativistic torque method Applications 1. Magnetic anisotropy of bulk antiferromagnets MnIr, Mn3Ir 2. Magnetic structure and magnon spectra of ultrathin films: Mn/W(110), Mn/W(001), Fe/W(110) 3.Magnetic nanoparticles: Cr trimer on Au(111) Ab initio Monte Carlo simulations: Cr and Co clusters Conclusions

4. Introduction Spin-orbit coupling Paul A. M. Dirac (1928) Expansion to first order in 1/c2 Central potential Spin-orbit interaction (From classical electrodynamics: Uhlenbeck-Goudsmit 1926, Thomas 1927)

5. Spin-model (classical) on a lattice Identify universality classes:critical exponent  M ~ (1-t ) (t=T/TC) d=2, n=1:  =1/8; d=3, n=1:  =0.325; d=3, n=2:  =0.345; d=3, n=3:  =0.365 Ni films on W(110) Y.Li and K. Baberschke, PRL 68, 1208 (1992) Magneto-crystalline anisotropy uniaxial (surface normal n): Magnetic anisotropy in thin films:dimensional crossover n: spin’s degree of freedom, n=1 Ising, n=2 XY, n=3 Heisenberg d: dimension of the lattice, d=1 chain, d=2 film, d=3 bulk Mermin-Wagner theorem (1966): for short-ranged interactionsfor n≥2 andd ≤2 there is no long-range order, i.e., spontaneous magnetization at finite temperatures.

6. n(E) EF E Simple phenomenological model for uniaxial anisotropy (P. Bruno, 1989 or so) Classical model: replace operators by its expectation values SOC as an effective field acting on the orbital magnetic moment Linear response → induced orbital moment Uniaxial system

7. Simple phenomenological model for uniaxial anisotropy Energy correction Direct proportionality between the anisotropy energy and the orbital moment: • Easy axis corresponds to the maximum of the orbital moment • MAE scales at best with 2 • Poorly applies to ab initio calculations

8. Adiabatic decoupling offast motion of electrons and slowmotion of spins hopping(10-15 s)<<spin-flip (10-13 s) static LSDAcan be used • Rigid Spin Approximation Orientationalstate gyromagnetic ratio, Gilbert damping factor First principles approaches to spin-dynamics • Landau-Lifshitz-Gilbert equation • Where to take from ? Constrained LSDA: first principles SD P.H. Dederichs et al. PRL 53, 2512 (1984) G.M. Stocks et al. Phil. Mag. B 78, 665 (1998) Spin-model: multiscale approach

9. Multiscale approach Classical spin Hamiltonian exchange interaction magnetic dipole-dipole interaction on-site anisotropy First principles evaluation of Jij : the torque method A.I. Liechtenstein et al. JMMM 67, 65 (1987) renormalized P. Bruno, PRL 90 , 087205 (2003) many-body M.I. Katsnelson et al. PRB 61, 8906 (2000) relativistic L. Udvardi et al. PRB 68 104436 (2003) Tensorial exchange interaction anisotropic symmetric antisymmetric isotropic relativistic (spin-orbit) effects

10. with DMI prefers misalignment of spins! 2 1 Dzyaloshinskii-Moriya interaction I. E. Dzyaloshinskii, Sov. Phys. JETP 5, 1259–1262 (1957) T. Moriya, Phys. Rev. 120, 91–98 (1960)

11. Nonmagnetic host with spin-orbit coupling Propagator without SOC Magnetic impurities Interaction between the impurities in first order of SOC: SOC • Proportional to SOC strength • Inversion symmetry → mirror plane Cn 2 1 C2 mirror plane surface Dzyaloshinskii-Moriya interaction Itinerant electron system → RKKY interaction in presence of spin-orbit coupling Simple tight-binding picture:

12. Multiscale approach Example: uniaxial on-site anisotropy Relativistic torque method Screened Korringa-Kohn-Rostoker Methodfor layered systems & Embedded Cluster Method for finite clusters Grand canonical potential(frozen potential approximation) single-site t matrices: structure constants: spherical potentials (ASA): compare with spin model

13. Multiscale approach Spin Hamiltonian • Mean field approach • Monte-Carlo simulations • Landau-Lifshitz-Gilbert equation Determine ground-state spin structure Finite temperature Curie/Néel temperature magnetic anisotropy reorientation phase transitions

14. 1. Magnetic anisotropy of AFM bulk MnIr compounds • Most widely used industrial antiferromagnet •  Knowledge of MAE is important to understand (increase) the stability of the AFM layer of • an exchange-bias device Theoretical model Bulk → sublattices, a=1,…,n consider only the sublattices of Mn atoms interactions between sublattices: isotropic exchangetwo-site anisotropyon-site anisotropy

15. Global tetragonal symmetry L10 MnIr n = 2 Ir 2 1 1 2 Mn (010) (001) (100)  Collinear antiferromagnet (no frustration)  Magnetic anisotropy→ rotating all spins around (100) axis Keff = -6.81 meV easy-plane anisotropy Ab initio calculation (easy excersize)

16. Each of the Mn atoms (sublattices) has local tetragonal symmetry symmetry axes: 1 → (001) 2 → (010) 3 → (100) Tab and Ka matrices have to be transformed accordingly 2 3 1 with L12 Mn3Ir n = 3 • Frustrated AFM → T1 spin-state within the (111) plane  Magnetic anisotropy→ rotating around the (111) axis

17. L12 Mn3Ir (contd.) ab initio calculation Keff = 10.42 meV (!) Can the frustrated AFM state tilt with respect to the (111) plane? → rotate around the (110) axis _ (111) plane 2 1,3 ‘Giant’ uniaxial MAE in the cubic bulk AFM Mn3Ir that stabilizes the frustrated T1 state within the (111) plane L. Szunyogh, B. Lazarovits, L. Udvardi, J. Jackson, U. Nowak, PRB (2009)

18. 2. Magnetic structure of ultrathin films Mn monolayer on W(110)M. Bode et al., Nature 447, 193 (2007) Constant current SP-STM image • row-by-row AF structure with a long-wavelength (12 nm) modulation • cycloidal spin-spiral  spins rotate around the (001) axis • theoretical explanation in terms of DM interactions

19. 1 Mn ML W(110) bcc(110) Biaxial magnetic anisotropy: Kx=-0.047 mRydKy=-0.037 mRyd 4 Nearest neighbors No DM interactions: 5 2 1 3 L. Udvardiet al.,Physica B 403, 402-404(2008) Mn monolayer on W(110) Calculated isotropic exchange interactions and length of DM vectors(all data in mRyd)

20. Mn monolayer on W(110) DM vectors MC simulations: • row-by-row AF arrangement modulated by a cycloidal spin-spiral • wavelength ~ 7.6 nm experiment~ 12 nm

21. Nearest neighbors Uniaxial magnetic anisotropy: K=-0.047 mRyd 2 1 3 3 Mn monolayer on W(100) Calculated isotropic exchange interactions and length of DM vectors(all data in mRyd) No DM interactions:

22. MC simulation Mn monolayer on W(100) DM vectors Spin-spiral wavelength ~ 2.2 nm Good agreement with experiment and the theoretical approach by P. Ferriani et al., PRL 101, 027201 (2008)

23. Fe monolayer on W(110)Domain wallsExperimental: M. Pratzer et al., PRL 87, 127201 (2001)

24. Isotropic exchange interactions Magnetic anisotropy: Ey - Ex = 2.86 meV, Ez - Ex = 0.41 meV easy axis x hard axis y (110) (001) Fe monolayer on W(110)(Fe layer→ 12.9 % inward relaxation) DM interactions  Dominating ferromagnetic interactions  Long-ranged → calculated up to a distance of 4 nm  Monte Carlo simulations indicate a Curie temperature of about 270-280 K. This is in nice agreement with experiment, TC ≈ 225 K.

25. Néel wall normal to (110) LLG simulations Fe monolayer on W(110)Domain walls Bloch wall normal to (001) In both cases, Mx(L) = tanh(2L/w) could well be fitted, where w is the width of the domain wall. The BW’s are narrower than the corresponding NW’s. This can be understood in terms of a micromagnetic model → w=2√(A/K), where A and K are the stiffness and the anisotopy constants, respectively. For a bcc(110) surface A is anisotropic. Considering just nearest neighbor interactions, e.g., A(110) = 2A(001). For similar reasons, the energy of the Bloch wall is less than that of the Néel wall. The value, w=1.38 nm, for a Bloch wall and is in good agreement with the experiment of M. Bode et al. (unpublished).

26. Y [001] Brillouin zone _ P _ _ H X [110] _ N _ G Fe monolayer on W(110)Adiabatic spin-wave spectra Asymmetry Origin: DM interactions Considering just 2nd NN interactions: Possibility for a direct measurement of the DM interactions!

27. 2 S2 D12 S1 x S2 1 S1 D12’ S1 x S2 S2 2’ Simple explanation in terms of classical spin-waves q║ x (D12+ D12’)  (S1 x S2)= 0

28. 2 S2 D12 S1x S2 1 S1 D2’1 S2’x S1 S2’ 2’ Simple explanation in terms of classical spin-waves q║ y D12 (S1x S2)+ D2’1  (S2’x S1) = 2 D12 (S1x S2) < 0

29. General rules for the chiral asymmetry of spin-wave spectra in ferromagnetic monolayers with at least twofold rotational axis: No asymmetry • for normal-to-plane ground state magnetization, S0 • if S0 and q lie simultaneously in a mirror plane Otherwise, the asymmetry should be observed (?)

30. Asymmetry of the Fe/W(110) magnon spectrumExperiment (SPLEEM): J. Prokop, J. Kirschner (MPI Halle)

31. 2. Finite particles Equilateral Cr trimer on top of Au(111) AFM interactions → frustration G.M. Stocks et al. Prog. Mat. Sci. 52, 371-387 (2007) First principles spin dynamics simulation Magnetic moment of Cr atoms: 4.4 B 120o Néel state =120o small out-of-plane magnetization =90.6o

32. By using scf potentials: from ab initio SD ground state from out-of-plane ferromagneticstate Equilateral Cr trimer on top of Au(111) Deeper insight →scanning the band-energy along a given path in the configuration space: The magnetic ground state is sensitive on the reference state used to calculate the interactions!

33. Equilateral Cr trimer on top of Au(111) Chirality z = -1 z = 1 DM vectors Dz < 0 Dz > 0 Reference state for calculating the interactions ferromagnetic state Néel state True ground state confirmed by ab initio spin dynamics calculations

34. Monte-Carlo simulations by directly using ab initio grand canonical potential easy to calculate No spin Hamiltonian is needed (spin interactions up to any order included) Spin configuration is continuously updated to calculate  Efficient evaluation of thermal averages  correlation functions However, no self-consistency is included (use potentials from the ground state)

35. Cr3 Cr4 Cr36 no frustration as from spin-model nearly Néel type Co9 Co36 canted out of plane Cr clusters on Au(111) Co clusters on Au(111) Ground state spin-configuration depends on the size and the shape of the cluster

36. Co36 cluster on Au(111) Temperature driven spin-reorientation

37. Conclusions Multiscale approach using spin Hamiltonians derived from ab initio methods: useful to explain/predict spin structures on the atomic scale Relativistic (spin-orbit) effects play a pronounced role in nanomagnetism Dzyaloshinskii-Moriya interactions can overweight the magneticanisotropy:  spin spiral formation in thin films  asymmetry of the spin-wave spectra Care has to be taken when mapping the energy derived from first principles to a model Hamiltonian:  parameters should be obtained from the true ground state  higher order spin-interactions might be of comparable size  triaxial on-site anisotropies → use paramagnetic (DLM) state as reference (in progress) „Ab initio” Monte Carlo method→ towards first-principles (beyond spin Hamiltonian) theory of finite temperature magnetism

38. Thanks for attention!