spin-orbit induced phenomena in nanomagnetism lászló szunyogh department of theoretical physics...
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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 (?)
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
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