Spin-orbit induced phenomena in Spin-orbit induced phenomena in

nanomnanomagnetiagnetismsm

László Szunyogh Department of Theoretical Physics

Budapest University of Technology and Economics, Hungary

Psik-Workshop on Magnetism, Vienna, 17th April, 2009

Coworkers

L. Udvardi, A. Antal, L. BaloghBudapest University of Technology and Economics, Hungary

B. Lazarovits, B. ÚjfalussyHungarian Academy of Sciences, Hungary

J.B. StauntonUniversity of Warwick, UK

B.L. GyörffyUniversity of Bristol, UK

U. Nowak, J. JacksonUniversity of Konstanz, Germany

University of York, UK

Outline of the talkOutline 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

Spin-orbit couplingSpin-orbit coupling

Paul A. M. Dirac (1928)

Expansion to first order in 1/c2

Central potential

(From classical electrodynamics: Uhlenbeck-Goudsmit 1926, Thomas 1927)

Spin-orbit interaction

Introduction Introduction

Ni films on W(110)Y.Li and K. Baberschke, PRL 68, 1208 (1992)

Magnetic anisotropy in thin films:Magnetic anisotropy in thin films: dimensional crossoverdimensional crossover

n: spin’s degree of freedom, n=1 Ising, n=2 XY, n=3 Heisenbergn=3 Heisenbergd: dimension of the lattice, d=1 chain, d=2 film, d=3 bulkd=3 bulk

Mermin-Wagner theorem (1966): for short-ranged interactions for n≥2 and d ≤2 there is no long-range order, i.e., spontaneous magnetization at finite temperatures.

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

Magneto-crystalline anisotropy

uniaxial (surface normal n):

Spin-model (classical) on a lattice

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

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

Constrained LSDA: first principles SD

First principles approaches to spin-dynamicsFirst principles approaches to spin-dynamics

Orientational state

P.H. Dederichs et al. PRL 53, 2512 (1984)

G.M. Stocks et al. Phil. Mag. B 78, 665 (1998)

gyromagnetic ratio, Gilbert damping factor

Landau-Lifshitz-Gilbert equation

Adiabatic decoupling of fast motion of electrons and slow motion of spins hopping (10-15 s) << spin-flip (10-13 s)

Rigid Spin Approximation

static LSDA can be used

Where to take from ?

Spin-model: multiscale approach

MultiscaleMultiscale approach approach

Classical spin Hamiltonian

exchange interaction

on-site anisotropy magnetic dipole-dipoleinteraction

First principles evaluation of Jij : the torque methodA.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

isotropic anisotropic symmetric antisymmetric

relativistic (spin-orbit) effects

Dzyaloshinskii-Moriya interaction

I. E. Dzyaloshinskii, Sov. Phys. JETP 5, 1259–1262 (1957)T. Moriya, Phys. Rev. 120, 91–98 (1960)

with

DMI prefers misalignment of spins!

12

Itinerant electron system → RKKY interaction in presence of spin-orbit coupling

Nonmagnetic host with spin-orbit coupling

Propagator without SOC

Magnetic impurities

Interaction between the impurities in first order of SOC:

1 2

SOC

Proportional to SOC strength Inversion symmetry →

Cn

C2

mirror plane

mirror plane

surface

Dzyaloshinskii-Moriya interaction

Simple tight-binding picture:Simple tight-binding picture:

Screened Korringa-Kohn-Rostoker Method for layered systems & Embedded Cluster Method for finite clusters

Grand canonical potential (frozen potential approximation)

compare with spin model

single-site t matrices: structure constants:

spherical potentials (ASA):

Example: uniaxial on-site anisotropy

Relativistic torque method

MultiscaleMultiscale approach approach

1. Mean field approach

2. Monte-Carlo simulations

3. Landau-Lifshitz-Gilbert equation

Determine ground-state spin structure

Finite temperature Curie/Néel temperature magnetic anisotropy

reorientation phase transitions

Spin Hamiltonian

MultiscaleMultiscale approach approach

1. Magnetic anisotropy of AFM bulk MnIr compounds1. 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

isotropic exchange two-site anisotropy on-site anisotropy

Bulk → sublattices, a=1,…,n

consider only the sublattices of Mn atoms

interactions between sublattices:

L10 MnIr

(100)

Ir

Mn

(001) (010)

1 22 1

n = 2 Global tetragonal symmetry

Collinear antiferromagnet (no frustration)

Magnetic anisotropy → rotating all spins around (100) axis

0 60 120 180 240 300 360-8

-6

-4

-2

0

(degree)

E (

meV

)

Keff = -6.81 meV

easy-plane anisotropy

Ab initio calculation

(easy excersize)

L12 Mn3Ir n = 3Each 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

Frustrated AFM → T1 spin-state within the (111) plane

1

2

3

Magnetic anisotropy → rotating around the (111) axis

with

1,3(111) plane

L12 Mn3Ir (contd.)

Keff = 10.42 meV (!)

0 60 120 180 240 300 3600

2

4

6

8

10

12

(degree)

E (

meV

)

Can the frustrated AFM state tilt with respect to the (111) plane?

→ rotate around the (110) axis_

2

0 60 120 180 240 300 3600

2

4

6

8

10

12

(degree)

E (

meV

)

ab initio calculation

‘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)

Mn monolayer on W(110)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

2. Magnetic structure of ultrathin films2. Magnetic structure of ultrathin films

12

3

45

Nearest neighbors

Calculated isotropic exchange interactions and length of DM vectors (all data in mRyd)

Mn monolayer on W(110)Mn monolayer on W(110)

W(110)

1 Mn ML

bcc(110)

L. Udvardi et al., Physica B 403, 402-404 (2008)

Biaxial magnetic anisotropy:

Kx=-0.047 mRyd Ky=-0.037 mRyd

No DM interactions:

MC simulations:

row-by-row AF arrangement modulated by a cycloidal spin-spiral wavelength ~ 7.6 nm experiment ~ 12 nm

Mn monolayer on W(110)Mn monolayer on W(110) DM vectors

Mn monolayer on W(1Mn monolayer on W(1000)0)

1

2

3

3

Nearest neighbors Calculated isotropic exchange interactions and length of DM vectors (all data in mRyd)

Uniaxial magnetic anisotropy:

K =-0.047 mRyd

No DM interactions:

MC simulation

Spin-spiral wavelength ~ 2.2 nm

Good agreement with experiment and the theoretical approach by P. Ferriani et al., PRL 101, 027201 (2008)

Mn monolayer on W(1Mn monolayer on W(1000)0)DM vectors

FeFe monolayer on W(1 monolayer on W(1110)0) Domain walls

Experimental: M. Pratzer et al., PRL 87, 127201 (2001)

FeFe monolayer on W(1 monolayer on W(1110)0) ((Fe layer → → 12.9 % inward relaxation)

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.

0.0 0.5 1.0 1.5 2.0-25

-20

-15

-10

-5

0

5

10

J ij (m

eV

)

distance of pairs (nm)

Isotropic exchange interactions

0.0 0.5 1.0 1.5 2.00

2

4

6

Dij

(meV

)

distance of pairs (nm)

DM interactions

Magnetic anisotropy: Ey - Ex = 2.86 meV, Ez - Ex = 0.41 meV easy axis x hard axis y

(110)(001)

FeFe monolayer on W(1 monolayer on W(1110)0) Domain walls

Néel wall normal to (110) Bloch wall normal to (001)

-4 -3 -2 -1 0 1 2 3 4-1.0

-0.5

0.0

0.5

1.0

Néel wall Bloch wall

L (nm)

Mx

LLG simulations

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) = 2 A(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).

Y [001]

X [110]_

H_

_

P_

N_

Brillouin zone

FeFe monolayer on W(1 monolayer on W(1110)0) Adiabatic spin-wave spectra

Origin: DM interactions

Considering just 2nd NN interactions:

Possibility for a direct measurement of the DM interactions!

Asymmetry

-1.0 -0.5 0.0 0.5 1.00

50

100

150

200

250

E(q

) (

meV

)

q ( Å-1 )

-1.0 -0.5 0.0 0.5 1.00

20

40

60

80

100

120

E(q

) (

meV

)

q ( Å-1 )

-1.0 -0.5 0.0 0.5 1.0-30

-20

-10

0

10

20

30

q ( Å-1 )

E(q

) (

meV

)

-1.0 -0.5 0.0 0.5 1.0-30

-20

-10

0

10

20

30

q ( Å-1 )

E(q

) (

meV

)

S1 x S2

D12

S1 x S2D12’

S2

2

S22’

(D12+ D12’) (S1 x S2) = 0

Simple explanation in terms of classical spin-waves

q ║ x

S11

S1 x S2D12

S2

2

S2’

2’

D12 (S1 x S2) + D2’1 (S2’ x S1)

= 2 D12 (S1 x S2) < 0

Simple explanation in terms of classical spin-waves

q ║ y

S11

S2’ x S1D2’1

General rules for the chiral asymmetry of spin-wave spectra in

ferromagnetic monolayers with at least twofold rotational axis:

No asymmetry

(i) for normal-to-plane ground state magnetization, S0

(ii) if S0 and q lie simultaneously in a mirror plane

Otherwise, the asymmetry should be observed (?)

Asymmetry of the Fe/W(110) magnon spectrum

Experiment (SPLEEM): J. Prokop, J. Kirschner (MPI Halle)

Magnetic moment of Cr atoms: 4.4 B

120o Néel state =120o

small out-of-plane magnetization =90.6o

First principles spin dynamics simulation

Equilateral Cr trimer on top of Au(111)Equilateral Cr trimer on top of Au(111)

G.M. Stocks et al. Prog. Mat. Sci. 52, 371-387 (2007)

AFM interactions → frustration

2. Finite particles 2. Finite particles

Deeper insight →

scanning the band-energy along a given path in the configuration space:

from out-of-planeferromagnetic state

By using scf potentials:

from ab initio SDground state

Equilateral Cr trimer on top of Au(111)Equilateral Cr trimer on top of Au(111)

The magnetic ground state is sensitive on the reference state used to calculate the interactions!

DM vectors

Dz < 0Dz > 0

z = -1 z = 1

Chirality

ferromagnetic state

Néel state

Reference statefor calculating

the interactions

True ground stateconfirmed by ab initio spin

dynamics calculations

Equilateral Cr trimer on top of Au(111)Equilateral Cr trimer on top of Au(111)

Monte-Carlo simulations by directly using ab initio grand canonical potential

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)

easy to calculate

Cr clusters on Au(111)Cr clusters on Au(111)

Co clusters on Au(111)Co clusters on Au(111)

Co9

canted

Co36

out of plane

Ground state spin-configuration depends on the sizeand the shape of the cluster

Cr4

no frustration

Cr36

nearly Néel type

Cr3

as from spin-model

CoCo3636 cluster on Au(111) cluster on Au(111)

Temperature driven spin-reorientationTemperature driven spin-reorientation

ConclusionsConclusions

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 magnetic anisotropy: 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

Thanks for attention!