Transcript Multiscale simulations of spin structures in nanomagnets

Spin-orbit induced phenomena in
nanomagnetism
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. 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
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
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)
Magnetic anisotropy in thin films: dimensional crossover
Spin-model (classical) on a lattice
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 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 
d=2, n=1:  =1/8;
d=3, n=1:  =0.325;
M ~ (1-t ) 
d=3, n=2:  =0.345;
(t=T/TC)
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):
Simple phenomenological model for uniaxial anisotropy (P. Bruno, 1989 or so)
Classical model: replace operators by its expectation values
n(E)
EF
E
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
First principles approaches to spin-dynamics
 Adiabatic decoupling of fast motion of electrons and slow motion of spins
hopping (10-15 s) << spin-flip (10-13 s)
static LSDA can be used
Orientational state
 Rigid Spin Approximation
 Landau-Lifshitz-Gilbert equation
gyromagnetic ratio,
 Where to take
Gilbert damping factor
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
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
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!
2
1
Dzyaloshinskii-Moriya interaction
Itinerant electron system → RKKY interaction in presence of spin-orbit coupling
Simple tight-binding picture:
Nonmagnetic host with spin-orbit coupling
Propagator without SOC
Magnetic impurities
Interaction between the impurities
in first order of SOC:
SOC
2
1
 Proportional to SOC strength
 Inversion symmetry →
Cn
mirror plane
C2
mirror plane
surface
Multiscale approach
Relativistic torque method
Screened Korringa-Kohn-Rostoker Method for layered systems
& Embedded Cluster Method for finite clusters
Grand canonical potential (frozen potential approximation)
single-site t matrices:
structure constants:
spherical potentials (ASA):
Example: uniaxial on-site anisotropy
compare with spin model
Multiscale approach
Spin Hamiltonian
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
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 exchange
two-site anisotropy
on-site anisotropy
L10 MnIr
Global tetragonal symmetry
n=2
Ir
2
1
1
2
Mn
(001) (010)
(100)
 Collinear antiferromagnet (no frustration)
 Magnetic anisotropy → rotating all spins around (100) axis
0
E (meV)
-2
-4
Keff = -6.81 meV
-6
easy-plane anisotropy
Ab initio calculation
-8
0
60
120
180
240
 (degree)
300
360
(easy excersize)
L12 Mn3Ir
n=3
Each of the Mn atoms (sublattices)
has local tetragonal symmetry
1 → (001)
2 → (010)
3 → (100)
symmetry axes:
Tab and Ka matrices have to be
transformed accordingly
2
 Frustrated AFM → T1 spin-state within the (111) plane
3
 Magnetic anisotropy → rotating around the (111) axis
with
1
L12 Mn3Ir
(contd.)
12
12
ab initio calculation
E
E (meV)
(meV)
10
10
8
8
Keff = 10.42 meV (!)
6
6
4
4
2
2
0
0
0
0
60
60
120
120
180
180
240
240
 (degree)
(degree)
300
300
360
360
Can the frustrated AFM state tilt with respect to the (111) plane?
_
→ rotate around the (110) axis
2
(111) plane
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)
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
Mn monolayer on W(110)
L. Udvardi et al., Physica B 403, 402-404 (2008)
Calculated isotropic exchange interactions
and length of DM vectors (all data in mRyd)
1 Mn ML
W(110)
bcc(110)
Biaxial magnetic anisotropy:
Kx=-0.047 mRyd
Nearest neighbors
4
5
2
1
3
Ky=-0.037 mRyd
No DM interactions:
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
Mn monolayer on W(100)
Nearest neighbors
Calculated isotropic exchange interactions
and length of DM vectors (all data in mRyd)
2
1
3
3
Uniaxial magnetic anisotropy:
K =-0.047 mRyd
No DM interactions:
Mn monolayer on W(100)
DM vectors
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)
Fe monolayer on W(110)
Domain walls
Experimental: M. Pratzer et al., PRL 87, 127201 (2001)
Fe monolayer on W(110)
(Fe layer → 12.9 % inward relaxation)
Isotropic exchange interactions
DM interactions
10
6
5
Dij (meV)
Jij (meV)
0
-5
-10
-15
4
2
-20
-25
0.0
0.5
1.0
1.5
2.0
0
0.0
distance of pairs (nm)
0.5
1.0
1.5
distance of pairs (nm)
 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.
Magnetic anisotropy: Ey - Ex = 2.86 meV, Ez - Ex = 0.41 meV
easy axis x (110)
hard axis y (001)
2.0
Fe monolayer on W(110)
Domain walls
Néel wall normal to (110)
Bloch wall normal to (001)
LLG simulations
1.0
Néel wall
Bloch wall
0.5
Mx
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.
0.0
-0.5
The value, w=1.38 nm, for a Bloch wall and is in good agreement with the
experiment of M. Bode et al. (unpublished).
-1.0
-4
-3
-2
-1
0
L (nm)
1
2
3
4
Fe monolayer on W(110)
Adiabatic spin-wave spectra
Brillouin zone
100
E(q) (meV)
250
Y [001]
_
H
80
_
60
P
40
_
20
0
G
-1.0
-0.5
0.0
0.5
_
_ X [110]
E(q) (meV)
120
N
200
150
100
50
0
1.0
-1
q(Å )
-1.0
-0.5
0.0
0.5
1.0
-1
q (Å )
Asymmetry
Origin:
DM interactions
E(q) (meV)
30
20
10
0
-10
Considering just 2nd NN interactions:
-20
-30
-1.0
-0.5
0.0
0.5
0.5
1.0
1.0
-1
-1
q (Å )
Possibility for a direct measurement
of the DM interactions!
Simple explanation in terms of classical spin-waves
q║x
2
S2
D12
S1 x S2
1
S1 x S2
(D12+ D12’)  (S1 x S2) = 0
S1
D12’
2’
S2
Simple explanation in terms of classical spin-waves
q║y
2
S2
D12
S1 x S2
D12  (S1 x S2) + D2’1  (S2’ x S1)
1
S1
D2’1
S2’ x S1
S2’
2’
= 2 D12  (S1 x S2) < 0
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)
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
Equilateral Cr trimer on top of Au(111)
Deeper insight →
scanning the band-energy along a given path in the configuration space:
By using scf potentials:
from ab initio SD
ground state
from out-of-plane
ferromagnetic state
The magnetic ground state is sensitive on the
reference state used to calculate the interactions!
Equilateral Cr trimer on top of Au(111)
Chirality
z = -1
z = 1
DM vectors
Dz < 0
Dz > 0
Néel state
True ground state
confirmed by ab initio spin
dynamics calculations
Reference state
for calculating
the interactions
ferromagnetic
state
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)
Cr clusters on Au(111)
Cr3
Cr4
as from spin-model
Cr36
no frustration
nearly Néel type
Co clusters on Au(111)
Co9
Co36
canted
out of plane
Ground state spin-configuration depends on the size
and the shape of the cluster
Co36 cluster on Au(111)
Temperature driven spin-reorientation
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 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!