3.3 Dynamics of vortices with static fields
3.3.1 In-plane field
IP vortices
If we look for static and planar solutions ( ˙φ = 0 and ˙m = m = 0) of eqs. (3.13), then eq. (3.13a) requires b⊥ = 0 ( i.e. there are no static planar solutions when a perpendicular field is applied), and eq. (3.13b) becomes a static sine-Gordon equation
∆φ = bk
a2 sin(φ−β0) (3.75)
which is probably the most studied equation of nonlinear physics. It has been shown to admit vortex solutions by Hud´ak [34], but with the vorticity Q=±4, a case which is not very relevant to dynamics, since vortices with Q=±4 are seen in the simulations to decay rapidly to several vortices withQ=±1, which are then at least in simulations
observed to be stable. Eq. (3.75) was also studied in connection with IP vortices of the XY-model (D= 0, λ= 0) by Gouvˆea et al. [23, 24].
It was found in [23] by numerical simulations that the main effect of the IP applied field, while forcing the spins to align towards its direction, is to deform the vortex shape, so that a region where the spin distribution is homogeneous (and pointing in the direction of the field) grows, pushing the vortex into the other region towards the border of the system. In other words, the system with an IP static field wants to be homogeneous and, for a field strong enough, it manages to expel out the inhomogeneities.
The trajectories observed in the simulations are in general a quite irregular “drift”
movement, determined by several factors. First, there are two driving forces: the force exerted by the image vortex (or the effect of the boundary) and the (effective) force exerted by the applied field.
Secondly, the vortex suffers accelerations and decelerations in the background of the potential produced by the discrete lattice. It is known that IP vortices are much more sensitive to the discreteness than OP vortices. Discreteness is the first factor which breaks the rotational symmetry in the XY plane (in the 2D case, it defines two
“cristallographic” special directions). Thus, the trajectory towards the border will be a straight line only in a case of extreme symmetry14. Eventually, the effect of discreteness may be so strong that the vortex can get pinned in a fixed position during the course of its trajectory.
The direction of the movement is, theoretically, affected by the relative phases of the vortex and the applied field. We note that the constant β0 is easily removed from the equation (3.75) by rotating the XY plane such thatφ →φ0 =φ−β0, but in any case the final solution will contain this constant, which corresponds to the fact that the applied IP field additionally breaks the rotational symmetry in the XY plane, in the discrete as well as in the continuum system. Since the applied IP field (4.5) couples directly to theφ field, with the interaction density
−S(x, y)~ ·~h =−h Sc cos(φ−β0) , (m= 0) , (3.76) the initial phase of the vortex ϕ0 (providing that the form (3.36) is used as initial condition), will contribute to determine the initial direction of the vortex trajectory.
In the simulations, however, strong effects of discreteness turn the trajectory to be practically unpredictable after few time steps (see below).
14 Providing, for instance, that the vortex is initially at the center of the lattice, a “valley” of the Peierls-Nabarro potential, and the field is applied either in ˆx or in ˆy direction, the vortex moves perpendicularly to the field, always between two lines of spins. If the initial position is shifted from the center of the system, the trajectory is more complicated.
20 25 30 35 40 20
25 30 35 40
Fig. 3.8: Trajectories of an IP vor-tex under the action of a static field in ˆx direction, hx= 0.007, starting from 3 different points, in a system of radius L= 20 with anisotropy λ= 0. The vortex launched from X~ = (20.5,20.5) (the center of the system) approaches the border vertically, the vortex start-ing at X~ = (20.5,24.5) gets pinned at X~ ≈(27.2,22.5), and the vortex starting at X~ = (25.5,27.5), gets to the border on a very irregular path.
The grid of points represents the spin sites, only a section of the circular system is shown.
The numerical work in [23] was carried out on square systems. We have confirmed this scenario for circular systems in our simulations, and next we illustrate it with some examples.
The Fig. 3.8 shows some trajectories in a small system of radius L = 20, obtained with an applied field~h =hxˆx, with intensity hx = 0.007. For this field intensity, the discreteness can cause the pinning of the vortex in some point of the lattice, an effect which depends on the initial conditions (IC): for the same value of the magnetic field, some IC result in trajectories which end up in a pinned state, and some other IC result in trajectories which get up to the border.
The method for determining the position of the vortex center (more details in Appendix D) relies on fitting the form (3.36) for the four innermost spins, after having found in which plaquette of 4 spins in the lattice the circulation (the integral (3.20) in discrete version) is non-zero. This is clearly just an approximation, for the innermost plaquette in the vortex, in the case that this is deformed by the applied field. In any case, it is enough to show us the main tendency of the movement. In order to complete the picture, in the Fig. 3.9 we depict the spin field for two points of the trajectory of Fig. 3.8 which starts at X~ = (25.5,27.5) and goes to the boundary, at the instants t = 0 (left) and t = 40 (right), where we can see that an increasing portion of the lattice is turning into an almost homogeneous state pointing in the direction of the magnetic field. The final state looks like the bottom part of the figure at the right panel, when the vortex has left the system.
If we want to get some information about this dynamics from the Thiele eq. (3.43), the
0 10 20 30 40 0
10 20 30 40
0 10 20 30 40
0 10 20 30 40
Fig. 3.9: Two snapshots of the IP field, corresponding to the trajectory of Fig. 3.8 which starts atX~ = (25.5,27.5), taken at timest= 0 (left) andt= 40.(right).
first thing we should do is to try to estimate from eq. (3.46) the effective force made by the applied field, which for the magnetic interaction density (3.76), can be written as:
F~~h =Sc2 Z
d2r
·
∇θ~ ∂
∂θ +∇φ~ ∂
∂φ
¸ µ
− h sinθcos(φ−β0)
¶
, (3.77)
where, for IP vortices, sinθ = 1, and the first term vanishes. Already Huber in 1982 saw the trouble to do so in the fact that, unfortunately, we do not know the mathematical expression for the spin distributionφ of the Fig. 3.9. In the words of Huber [32],
“The response to in-plane applied fields is less certain. If the vortex moves without distortion (which is probably a reasonable approxima-tion only at very low fields, if at all) then the corresponding static force takes the form [... of eq. (3.77) ...]”.
Here I have emphasized “if at all”, because the integral in eq. (3.77), and the Thiele equation itself, assume already a stationary movement. Therefore, we restrict us here to show just the basic facts of the phenomenology as seen in the simulations.
OP vortices
Naturally, in the case of OP vortices in the circular system, in addition to this applied field, there is always the action of the gyrotropic force and the force exerted by the image vortex. But, in contrast, the OP vortex, in a range of anisotropies of 0.9−0.99, is
much less sensitive to the discreteness, and therefore the trajectories are much smoother than those of the IP vortex. The resultant of the magnetic force caused by the applied field, the gyrotropic force, and the force of the image vortex, gives a complex drift movement for strong enough fields, but a “damping dominated” spiral-like movement for weak fields. The damping can cause a large difference in the movement of the vortex.
To illustrate this effect, we show some simulation results for an OP vortex in presence of an IP field applied in ˆx direction. The rest of parameters are chosen to be L= 20, λ= 0.95 and ε= 0.005.
10 20 30 40
10 20 30 40
a
b c
10 20 30 40
10 20 30 40
Fig. 3.10: Trajectories of an OP vortex, starting from the center of the system, (Left Panel) corresponding to three values of an applied field (a)hx= 0.0055 , (b)hx = 0.0054 and (c) hx = 0.0053. (Right Panel) The trajectory (c) up to time t = 3490. Other parameters are: L= 20 ,λ= 0.95 ,ε= 0.005.
We see inFig. 3.10that for slightly different intensities of the field, we have the vortex approaching the border in different ways. For hx = 0.0053 the trajectory develops into the spiral typical for the damping-dominated regime, but for hx = 0.0054 and larger intensities, the drift towards the border is more direct. This picture is changed in absence of the damping. Damping provides a way to decrease the energy, which otherwise is constant, also in presence of the static field. At zero damping, the extra energy left in the system when the vortex abandons it, is transferred continuously into spin waves, which are thus more pronounced than in the case with damping.