Dynamics of coupled vortices in layered magnetic nanodots
K. Yu. Guslienko,* K. S. Buchanan, S. D. Bader, and V. Novosad Materials Science Division and Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439
Abstract The spin dynamics are calculated for a model system consisting of magnetically soft, layered nanomagnets, in which two ferromagnetic (F) cylindrical dots, each with a magnetic vortex ground state, are separated by a non-magnetic spacer (N). This permits a study of the effects of interlayer magnetostatic interactions on the vortex dynamics. The system was explored by applying the equations of motion for the vortex core positions. The restoring force was calculated taking into account the magnetostatic interactions assuming a realistic “surface charge free” spin distribution. For tri-layer F/N/F dots with opposite chiralities and the same core polarizations (lowest energy state), two eigenmodes are predicted analytically and confirmed via micromagnetic simulations. One mode is in the sub-GHz range for submicron dot diameters and corresponds to quasi-circular rotation of the cores about the dot center. A second mode is in the MHz range corresponding to a small amplitude rotation of the mean core position. The eigenfrequencies depend strongly on the geometrical parameters of the system, suggesting that magnetostatic effects play a dominant role in determining the vortex dynamics.
*
Corresponding author. E-mail:
[email protected]
1
Magnetically soft submicron structures have received considerable attention due to their tendency to support a stable vortex magnetization state. It is well established that a vortex (soliton-like) state appears for large enough lateral particle dimensions.1-3 The vortex consists of an in-plane, flux-closure magnetization distribution and a central core whose magnetization is perpendicular to the dot plane. The vortex magnetization distribution leads to considerable modification of the spin excitation spectrum in comparison with a uniformly magnetized dot. In particular, a low-frequency mode of displacement of the vortex as a whole appears.4, 5 Magnetostatic fields play an important role in determining the dynamic vortex excitations in magnetic dots. While the conditions necessary for a single-layer disk to support a vortex are well known, as is the reversal through vortex nucleation, displacement and annihilation,2,3,6 the properties of stacked ferromagnetic (F) disks separated by a thin, non-magnetic (N) spacer (F/N/F structures) have received less attention. Micromagnetic simulations indicate that the disks will each support oppositely directed vortices at remanence.7 This double vortex state can be created by choice of disk parameters or by including a small antiferromagnetic interlayer exchange contribution.7 The dynamic properties of F/N/F disks have not yet been considered. The disks are couple through both magnetostatic interactions and interlayer exchange (depending on the spacer material and thickness), which will result in unique vortex oscillations. It is only recently that the vortex excitation modes of single layer F structures have been observed experimentally, i.e. using Brillouin light scattering (BLS),8 time-resolved Kerr effect9,10 and X-ray magnetic circular dichroism techniques.11,12 The theory of vortex excitations in small cylindrical particles has been developed in Ref. 5, 8, 13 and micromagnetic simulations have been reported in Ref. 13-15. These works explore the lowest order vortex translation mode as well as other modes with radial and azimuthal symmetry. In previous studies on vortex dynamics, the only interaction effects that have been considered are weak lateral interactions between dots arranged on a two-dimensional (2D) plane.16 The approach16 is based on the simplified “rigid vortex” model, which assumes that the vortices are rigid 2
and do not deform as they move. In reality, however, when the vortex moves the spin configuration changes to reduce stray fields. This discrepancy leads to overestimation of the eigenfrequencies and the effects of inter-dot interactions on the vortex core trajectories have not been considered.16 In this paper, analytical and numerical investigations of the vortex oscillation modes are presented for magnetically soft tri-layer F/N/F disks consisting of two identical cylindrical magnetic dots with radii R and thicknesses L separated by spacer N of thickness d (Fig. 1). The effects of strong coupling between vertically stacked disks on the vortex dynamic behavior are explored assuming a realistic “surface charge free” spin distribution, and the vortex trajectories for the ground state dynamics are explicitly tracked. The magnetization distribution in each F layer is considered to be 2D. We limit our analysis to the case of magnetic vortices with identical core polarizations and opposite chiralities, assuming that the each layer supports the topologically simplest vortex state with a centered core at remanence. This state has the lowest energy among all possible states for sub-micron tri-layer dots with a thin spacer7 and should also be the simplest state to observe experimentally. The theoretical description of the vortex oscillations utilizes the effective equation for vortex collective coordinates.17 We focus on the vortex translation modes, where the vortex center (a topological soliton) oscillates about its equilibrium position in each layer. These modes have the lowest frequencies in the vortex excitation spectra. The 2D (independent of the z-coordinate along the dot thickness) magnetization distributions m j (ρ, t ) = M j (ρ, t ) / M sj , m j 2 = 1 are assumed for each of the F dots (j= 1, 2 is the F layer number, Msj is the saturation magnetization) and the angular parameterization for the dot magnetization components m z j = cos Θ j , mx j + im y j = sin Θ j exp(iΦ j ) , Φ j = q jϕ + Φ 0 j is used. The phase is Φ 0 j = C jπ / 2 , where C j = ±1 describes the vortex chirality. The signs C j = ±1
correspond to counter-clockwise (clockwise) rotation of the vectors mj in the dot plane. The total energy of a vortex does not depend on chirality for an isolated disk; however, the energy for coupled disks depends on the chirality product C1C2. The integer parameter qj is the vortex topological charge 3
(vorticity),17,18 and pj =±1 is the vortex core polarization (direction of the mz component in the vortex center ρ =0). In some sense the solitons can be represented as quasi-particles and can be characterized by their coordinates (X1, X2), masses and other collective variables. In order to describe the translation modes of the vortex motion we apply Thiele’s equations4 for each layer, neglecting the small vortex masses:
G
1
×
where
dX dt
Xj
1
−
∂W
=(Xj,
(X 1 , X 2 ) = ∂X
Yj)
0
,
G
2
×
1
is
the
vortex
dX dt
2
−
center
∂ W (X 1 , X ∂X 2
position
2
)=
in
0
,
j-th
(1)
layer,
W (X1 , X 2 ) = W1 (X1 ) + W (X 2 ) + Wint (X1 , X 2 ) is the potential energy of the vortices shifted from their equilibrium positions Xj=0. Here the interaction energy term Wint (X1 , X 2 ) includes only magnetostatic contributions. Inclusion of interlayer exchange effects could be achieved by rescaling some of the magnetostatic coupling energy terms but would not affect the form of the result. The first term in Eq. (1) is the gyroforce determined by the vortex magnetization distribution in equilibrium.18 The gyroforce is proportional to the gyrovector G j = −G j zˆ , Gj = 2πqj pj LMs ⁄γ, where γ is the gyromagnetic ratio. The topological charges q1 = q2 =1 are considered below, which describes the single vortices observed experimentally in soft magnetic cylinders at remanence.1-3 The gyrovector, an intrinsic property of the vortex,17 is of principal importance in describing the vortex dynamics and can be calculated by integration over the vortex core.5 The second terms in Eq. (1) describe the restoring forces acting on vortices that are shifted from their equilibrium positions at the dot centers. The restoring force is a consequence of the finite lateral size of the dot and is directed toward the dot center. For submicron dot radii, the dot magnetostatic energy (shifting the vortex induces magnetic charges) provides the main contribution to the shifted-vortex energy W (X1 , X 2 ) . 4
The static energy dependence of a vortex on its core position was calculated for a cylindrical dot
3,5
on the basis of a model with no magnetic side surface charges,19 where the complex function
(
)
(
) has the form6 w(ζ , ζ ) = f (ζ ) if f (ζ ) < 1 (within the vortex core) and w(ζ , ζ ) = f (ζ ) / f (ζ )
w ζ , ζ = tan (Θ(x, y ) / 2 )exp(iΦ (x, y )) , with the complex variable ζ = (x + iy ) / R , was used. The function w ζ ,ζ
where
f (ζ )
is
an
appropriate
analytical
function.
In
this
case
if f (ζ ) ≥ 1 ,
the
function
f (ζ ) = (1 / c )(iζC + (a − a ζ 2 ) / 2) was used, where c=b/R is the relative core radius, which corresponds to the “side surface charge free” model [ (m ⋅ n)S = 0 ].19 The function f (ζ ) for the complex parameter a ( a < 1 ) describes magnetization field of two vortices. The first vortex has its center at ζ 1 = iaC / 2 , and a second image vortex is centered at ζ 2 = 2iaC / a . Their coordinates are connected by the inversion 2
transformation ζ 1ζ 2 = 1 . It is convenient to consider the complex parameter a = a ′ + ia ′′ as a 2D vector a = (a x , a y ) with components a x = a ′ and a y = a ′′ . The vector a corresponds to the volume-averaged magnetization of the shifted vortex m (r ) V = a / 3 . It is perpendicular to the vortex core shift vector s=X/R and can be expressed as a= -2C z × s. The value a=0 corresponds to a centered vortex.3 For a small displacement of the vortex center from its equilibrium position (X=0), one can write 2 W j (X j ) = W j (0 ) + 1 / 2κX j for each F layer, where the stiffness coefficient κ can be determined from the
decomposition of the vortex energy in a power series of a.5 For the present model, the interlayer magnetostatic coupling can be represented as the sum of contributions from the volume and face surface magnetic charges. Due to the small size of the vortex core c
