Comparative study of the Dzyaloshinskii-Moriya interaction in ultrathin magnetic films: Cr, Mn, Fe on W(110)
B. Zimmermann, M. Heide ? , G. Bihlmayer and S. Bl¨ ugel
Institut f¨ur Festk¨orperforschung and Institute for Advanced Simulation, Forschungszentrum J¨ulich and JARA, 52425 J¨ulich, Germany
?Present address: Department of Precision Science & Technology, Osaka University, Japan
September 14, 2010 Ψ
k-conference 2010, Berlin, Germany
E-Mail: be.zimmermann@fz-juelich.de
MitgliedderHelmholtzgemeinschaft
Introduction Introduction
Magnetic thin films are a source of a great spectrum of fascinating mag- netic properties. Electrons propagating in the vicinity of inversion asym- metric environments such as ultra thin films can give rise to an important antisymmetric exchange interaction, known as Dzyaloshinskii-Moriya inter- action (DMI) [1, 2]. The DMI – favoring spiraling magnetic structures of unique rotational sense – competes with the Heisenberg interaction (sym- metric exchange interaction) and gives rise to a variety of possible magnetic ground states. Both, experiments and theory, give access to such struc- tures, e.g. by spin-polarized scanning tunneling microscopy (SP-STM) and density functional theory (DFT), respectively.
Here we determine the magnetic structure of the ground state for the magnetic systems Cr, Mn and Fe on W(110) by means of a micromagnetic model. The model parameters are obtained by first principles calculations of homogeneous spin spirals. We show that the DMI is strong enough to compete with other interactions, e.g. the anisotropy. An atom-resolved analysis shows that the main contribution to the DMI originates from the W interface, where spin-orbit coupling is strong due to the heavy nucleus and the surface layer induces high magnetic moments due to hybridization.
Dzyaloshinskii-Moriya interaction Dzyaloshinskii-Moriya interaction
The Dzyaloshinskii-Moriya interaction is the antisymmetric part of a gen- eral 2-spin interaction Si J
ij Sj,
H ≈ − X
i6=j
Ji,j Si · Sj
| {z }
Heisenberg
+ X
i,j
Di,j · Si × Sj
| {z }
DMI
+ X
i
STi KSi
| {z }
Anisotropy
.
The DMI
• favors canted spin structures (↑ → or ↑ ←),
• is sensitive to the rotational sense (Si × Sj = −Sj × Si),
• is a spin-orbit coupling effect,
• only occurs in systems with broken inversion symmetry, e.g. surfaces:
On structureless surfaces, DMI favors cycloidal spin spirals with D ⊥ q and D ⊥ n due to symmetry.
Micromagnetic model Micromagnetic model
ϕ x
D λ
In a micromagnetic model, the magnetization is treated as continuous vector field with constant modulus,
m(x) = sin ϑ cosϕ eˆ1 + sin ϑ sin ϕ eˆ2 + cosϑ eˆ3 .
The functions ϑ(x) and ϕ(x) which minimize the energy functional
E[m] = Z
dx (
A
dm dx
2
+ (D eˆ3) ·
m × dm dx
+ mTKm )
crucially depend on the direction of the easy axis (K = diag(K1, K2, K3)) and can give rise to different phases [3] with
1. collinear m(x) k D, i.e. ϑ = 0,
2. collinear m(x) ⊥ D, i.e. ϑ = π/2, ϕ = const., 3. non-collinear m(x) ⊥ D, i.e. ϑ = π/2, ϕ = ϕ(x),
4. truly 3-dimensional non-collinear m(x), i.e. ϑ = ϑ(x), ϕ = ϕ(x).
If we restrict our analysis to the cases 2 and 3, we find an analytical solution depending on one effective parameter κ, yielding a non-collinear ground state if and only if
0 ≤ κ =
16π2 A(KD2−K2 1)< 1
and the profile of the spiral is ( = eccentricity, λ = period length)
ϕ(x) = −signD am
√K(x − λ/4)
√A ,
!
, λ ∼ F(1)().
F(n)() is the complete elliptic integral of nth kind, κ =
F(2)()
2
, and
‘am()’ is the Jacobian elliptic amplitude function.
Homogeneous Spirals Homogeneous Spirals
Our computational method relies on electronic structure calculations of homogeneous spin spirals (dϕ/dx = const = 2π/λ). The corresponding energy dispersion (per atom) reads
Eat = 4π2A λ−2 + 2πD λ−1 + (K2 − K1)/2
The spirals propagation direction and period length λ = 2π/|q| is defined by a vector q in reciprocal space.
Cr/W(110) Cr/W(110)
Cr is an antiferromagnet. In SP-STM measurements [4], an additional modulation of contrast along [001] (arrows) is observed:
Is this long-scale modulation of contrast a spin spiral?
Electronic structure calculations:
0 0.1 0.2 0.3 0.4 0.5
λ-1 [nm-1] 0
5 10 15 20 25 30
E [meV]
without SOC
A from quadratic fits
-0.5 -0.4 -0.3 -0.2 -0.1 0
λ-1 [nm-1] -7
-6 -5 -4 -3 -2 -1 0
[001]
[110]
with SOC (odd part) D from linear fits
[001] [110]
4π2 A [meV nm2] 135 112 2π D [meV nm] 19.9 8.5 (K2 − K1) [meV] 0.9 1.2 κ 0.5 3.0 Note: K1 = easy axis = [110]ˆ
0 5 10 15
x [nm]
0 0.2 0.4 0.6 0.8 1
ϕ / 2π
Inhomogeneous spiral profile
=⇒ Left-handed spin spiral along [001] with
|λ/2| = 7 nm.
Good agreement with experiment.
Layer-resolved analysis
0 2 4 6
8 1 Cr + 7 W
1 Cr + 8 W 1 Cr + 9 W
1 (Cr)
2 (W)
3 (W)
4 (W)
5 (W)
6 (W)
7 (W)
8 (W)
9 (W)
10 (W) 0
2 4
2πDµ [meV nm]
spirals along [001]
spirals along [110]
Magnetic moments:
µS µL
(µB) (10−2µB) (Cr) 2.41 100 % −2.42 (W) −0.21 8.7 % 1.70 (W) 0.04 1.7 % −0.36 (W) −0.01 0.4 % 0.28 (W) 0.00 0.4 % −0.38
Main contribution from W-interface layer (induced magnetic moment + high Z)
Comparison to Mn, Fe on W(110) Comparison to Mn, Fe on W(110)
Model parameter
Cr Mn [5] Fe [6] 2 Fe [7]
A001 [meV] 48 ? 19 59
D001 [meV/nm] 44 ? † −8
K001 [meV/nm2] 12 20 4 1.1
A110 [meV] 39 33 49 51
D110 [meV/nm] 19 53 † 7
K110 [meV/nm2] 17 8 −37 2.3
?: no appropriate fit possible due to non-parabolic behavior
†: no results available
Resulting magnetic structures
rotational dir. propagation dir. λ [nm]
Cr: left-handed [001] 14
Mn: left-handed [110] 8
2 Fe: right-handed [110]
SP-STM measurements
Mn/W(110) [5]
In good agreement with experimental spin spiral period
length λ = 12 nm
2 Fe/W(110) [8]
Instead of inducing a spin spiral, the DMI determines the
direction and rotational sense of domain walls [7].
Computational Method Computational Method
FLAPW method
We perform ab-initio DFT calculations as implemented in the Fleur code.
z
x vacuum
vacuum
muffin tin interstitial
unit cell
LAPW basis extended to film calculations:
interstitial: eik·r muffin-tin sphere: X
`,m
Ak,`,m u`(r) + Bk,`,m u˙`(r)
Y`,m(ˆr) vacuum: Akk vkk(z) + Bkk v˙kk(z)
eikk·rk
Spin spirals without SOC
A generalization of Bloch’s theorem can be used to circumvent large mag- netic supercells [9]: A generalized translation (= translation + spin rota- tion) maps all atoms of a homogeneous spin spiral to one chemical unit cell (in absence of SOC). The wavefunctions read
ψkq(r) = ψ↑(r) ψ↓(r)
!
= eikr e−iq2ru↑k(r) eiq2ru↓k(r)
!
where the Bloch functions uσk(r) = uσk(r + R) exhibit the periodicity of the chemical lattice.
Many k-points (> 4000) are required to resolve the correct behavior for small energies (< 1meV).
0 1×10-2 2×10-2 3×10-2 λ-2 [nm-2]
0 2 4 6
E SR [meV]
ground state
Nkpts=2048 Nkpts=4608 Nkpts=10368 Nkpts=18432
Spin spirals with SOC
SOC is only considered in the muffin tin spheres. We use first order per- turbation theory to estimate the effect of SOC on band energies:
δεµ =
ψkq
HSOCµ
ψkq
We do not need supercells, but calculate in the chemical unit cell. About 2000 k-points are sufficient to obtain converged results.
SOC in collinear states (Second variation)
First solve problem neglecting SOC:
H0[n0] ψν,0 = εν,0 ψν,0 with ψν,0 = X
µ
cν,µ ϕLAPWµ
Now take eigenstates as new basis for perturbed wavefunctions:
(H0 + HSOC) ψν = εν ψν with ψν = X
µ
dν,µ ψµ,0
This is a suitable basis to describe the perturbed system and the number of basis functions can be reduced.
We need about 1600 k-points & 8 layers to determine the magneto- crystalline anisotropy up to a resolution of 0.1 meV.
Local Force Theorem
Small perturbation added to Hamiltonian (e.g. small deviation of q-vector):
H = H0 + ∆H. Self-consistent solution of unperturbed Hamiltonian:
H0[n0] Ψν,0 = ν,0Ψν,0 Non-self-consistent solution of perturbed system:
(H0 + ∆H)[n0] Ψν,ft = ν,ftΨν,ft
Total energy difference can be approximated by the force theorem [10]
∆E ≈
occ.
X
ν
ν,ft −
occ.
X
ν
ν,0
References References
[1] I. E. Dzialoshinskii, Sov. Phys. JETP 5 (1957), 1259 [2] T. Moriya, Phys. Rev. 120 (1960), 91
[3] M. Heide, et al., J. of Nanoscience and Nanotechnology (2010, accepted) [4] B. Santos, et al., New Journal of Physics 10 (2008), 013005
[5] M. Bode, et al., Nature 447 (2007), 190 [6] M. Heide, Ph.D. Thesis (2006)
[7] M. Heide, et al., Phys. Rev. B 78 (2008), 140403
[8] A. Kubetzka, et al., Phys. Rev. Lett. 88 (2002), 057201 [9] L. M. Sandratskii, J. Phys.: Condens. Matter 3 (1991), 8565
[10] A. R. Mackintosh and O. K. Andersen, Cambridge University Press (1980)
