Spin-Torque driven Vortex Core Motion
5.2 Influence of Current and Magnetic Field
The above analysis is only valid at zero external fields. Any permanent field will result in an offset of the VC displacement that has to be considered. Especially the Oersted field generated by the inhomogeneous current flow across the sample might influence the displacement. This can be modeled by a magnetic field H perpendicular to the electron flow thas is proportional to the current density and hence to u. This now results in the modified equation for the final displacement (see also section 5.3.1):
with a disk geometry dependent parameter ǫand the VC chirality c. By normal-izing the displacement, it can be simplified to:
xe
This displacement is schematically shown in Fig. 5.1(b) for the different com-binations of the VC polaritypand chiralityc. The displacement is not degenerate any more concerning the VC chirality.
By measuring the displacement for the two VC chiralities and averaging the VC displacement in the direction of the electron flow, the field-induced displacement cancels out and the pure non-adiabatic contribution remains. In addition, the difference between the two chiralities determines the Oersted field. Reversing the current direction or the VC polarity does not influence the slope and the angle of displacement only changes sign.
5.3 Micromagnetic Simulations 93
5.3 Micromagnetic Simulations
To study the effect of the spin-torque on the vortex structure, numerical simu-lations are carried out on a circular disk. To shorten the simulation time, the damping is chosen to be α = 0.2 except when stated differently. For each set of parameters, a constant current is applied and the final displacement of the VC is determined. To measure displacements smaller than the discretization size of 5 nm, the out-of-plane VC profile is fitted using a Gaussian bell shape. This allows for accuracies for the VC position below 1 nm.
5.3.1 Field-Induced VC Displacement
To be able to separate the effects of a magnetic field and of an injected current on the VC displacement, the field-induced VC displacement is studied.
The external field H will increase the region of the magnetization where the spins are already aligned to the field. This leads to a VC displacement perpendic-ular to the field and the direction of this displacement is determined by the VC chirality. Assuming a perfectly harmonic potential the displacement l is propor-tional to the applied fieldH and can be calculated as follows [GNO+01, GIN+02].
l
R =χ(0)H Ms
. (5.9)
χ(0) denotes the initial susceptibility at zero field and is given by [GNO+01, GIN+02]:
χ−1(0) = 2f(ln(8/f)−0.5), f ≪1. (5.10) with f being the ratio of the disk thickness t and the disk radius R (f = t/R).
Fig. 5.2 shows the results from a simulation on a 1µm wide and 30 nm thick disk.
A magnetic field along the x-direction is applied and the final displacement of the VC in the y-direction is plotted as a function of the applied field. In agreement with the theory no displacement in the x-direction is observed. For small fields the VC displacement depends linear on the field, for higher fields the VC displace-ment is reduced, due to the non-harmonic potential for large VC displacedisplace-ments.
The red curve shows a fit that was obtained by including third order term. Using Equation 5.9, the theoretical gradient is l/H = 1.2nm/G for the geometry inves-tigated, which is about 35% smaller than the numerical result of 1.7nm/G (blue dotted line in Fig. 5.2). Regardless of this deviation, the results show that for
94 Spin-Torque driven Vortex Core Motion
Figure 5.2: Tthe VC displacement is plotted as a function of the applied magnetic field (green squares). The disk is 1µm in diameter and 30 nm thick. The inset shows the shifted VC at a field of 200 G. The blue dotted line corresponds to a linear fit for fields below 50 G and the red curve to a cubic fit.
VC displacements smaller than about 40% of the disk radius the potential can be trated as harmonic in good approximation.
5.3.2 β Dependent Study
The validity of the theoretical spin-torque model is tested by studying the current-induced VC displacements as a function of β for various current densities. Al-though the angle of displacement should not depend on the current density, the current might have an indirect influence. At larger current densities and larger VC displacements, VC deformations can occur. These deformations alter the dis-sipation tensor D¯ and especially the off-diagonal elements will become non-zero.
Thus for large VC displacements the displacement angle cannot be expected to be constant. These changes are (first order) quadratic in the displacement, hence for small displacements, they should be negligible.
The simulations are carried out on a 1µm wide and 20 nm thick disk. β is varied between 0 and 1 and the final VC displacement is measured for different current densities. The results are shown in Fig. 5.3. Fig. 5.3(a) shows the final VC displacement in the disk. To allow for a better comparison between the different current densities, the VC displacements are divided by the corresponding current
5.3 Micromagnetic Simulations 95
Figure 5.3: (a) The two dimensional VC displacement in a 1µm wide and 20 nm thick disk is plotted as a function of β for different current densities [For the color code see (b)]. The VC displacement is normalized to the corresponding current density. The gray lines originating from the center correspond to the theoretical displacement angle for the differentβ values, whereas the vertical line indicates the constant perpendicular displacement due to the adiabatic torque. (b) Angle of displacement vs. β for different current densities.
density. For j = 0.5·1012A/m2 and β = 0 the real VC displacement is about 13 nm and for higher current densities it increases linearly.
The gray lines originating from the disk center correspond to the theoretical displacement angle for the different β values, whereas the vertical line indicates the constant perpendicular VC displacement caused by the adiabatic spin-torque that should be independent of β. Thus, the junctions between these curves mark the theoretical position of the VC for the different current densities and β values.
In Fig. 5.3(b) the angle of displacement is plotted as a function ofβ. The gray line corresponds to the analytical model using Eq. 5.6. Good agreement is found, but deviations become larger at higher current densities and highβvalues. At large VC displacements the VC deforms and the dissipation dyadicDchanges and especially becomes non-diagonal. Thus the agreement between theory and simulation might be improved by taken into account a position dependent dissipation tensor.
In conclusion, a linear dependence of the VC displacement angle on β is found as predicted by the theory. In the experimentally accessible region (j <
2·1012A/m2) and forβ <0.5the discrepancy from the linear behavior is negligi-ble.
96 Spin-Torque driven Vortex Core Motion
5.3.3 Disk Diameter Dependent Study
To further test the validity of Eq. 5.6 the current-induced VC displacement is studied in 30 nm thick disks with diameters ranging from 125 nm up to 4.5µm.
From the measured angle of the VC displacement the proportionality constant D˜ is calculated and plotted as a function of the disk diameter. The result is shown in the semilogarithmic plot in Fig. 5.4. The data is fitted by the function D˜ =aln(d/b) with fit parametersaand b. The fit resultD(d) = 0.76 ln(d/28nm)˜ corresponds to the blue curve in Fig. 5.4. Using directly Eq. 5.6 with the VC radius obtained from the simulations results in the green dotted line.
From the analytic model resulting in Eq. 5.6 only the VC radius is a free parameter. The deviation from the theoretical pre-factor 0.5 shows the limitations of the theoretical model. Nevertheless, the simulated data can very accurately be described by a logarithmic function. This points to a systematic error in the model, that could be due to VC deformations or off-diagonal elements occurring in D¯ that are not considered. The systematic difference between the simulation and the analytical expression using Eq. 5.6 with the VC diameter obtained from the simulations ranges between 5%-15% for a disk diameter between 3µm and 6µm.
Although the difference is not big, for the analysis of the experimental results,
Figure 5.4: Shown are the results of the simulated factor D˜ in 30 nm thick disks with different diameters. The blue line is a fit to the data points and the green dotted line corresponds to the analytical model.
5.3 Micromagnetic Simulations 97
the corresponding simulations are used to determine D˜ instead of the analytic expression.
5.3.4 Disk Thickness Dependent Study
In the analytical model, the thickness of the disk does not directly influence the VC displacement angle. However, it has an indirect influence via changing the diameter of the VC and hence the effective dissipation constant D.˜
In Fig. 5.5(a) the radius of the relaxed VC, rVC, is shown as a function of the disk thickness t that is varied between 5 and 200 nm for a 1µm wide disk.
The curve corresponds to a fit based on the theoretical model for the VC diame-ter [UP93]:
rVC(t) = 0.68 t
le 1/3
le, le: exchange length (5.7 nm for Py) (5.11) that should be valid for t > le. The full width at half maximum VC radius obtained from the simulation is not identical with the definition of the VC radius in Ref. [UP93], so the measured VC radius had to be increased by 20% to match the analytical model. After this correction, good agreement between theory and simulation is found for thicker disks. To estimate the influence of rVC on D˜ the current-driven VC displacement is measured. The diagram in Fig. 5.5(b) shows the parameterD˜ derived from the VC displacement angle as a function of the VC
Figure 5.5: (a) For a 1µm wide disk the thickness is varied and the width of the relaxed VC is determined (no field, no current). The dotted blue curve correspond to the ana-lytical model. (b) The angle of displacement is determined at a fixed current density of j = 4·1012A/m2andβ= 0.4and plotted vs. the VC radius for the different thicknesses.
The green dotted curve correspond to the analytical model and the blue line is fitted to the data points.
98 Spin-Torque driven Vortex Core Motion
radius. All data points correspond to a current density of j = 4·1012A/m2 and β = 0.4. The analytical model is represented by the green dotted line and the blue line is a fit to the data points using the functionD(r˜ VC) =aln(b/rVC)for VC radii larger than 8 nm.
Especially for small VC radii in thin disks the discrepancy between the an-alytical model and simulation is relatively large. Since the simulation is done always at the same current density, the VC displacement in these disks is rather large compared to the thicker ones (the confining potential is weaker). Thus the deviations from the analytical model are stronger. For larger VC radii good agree-ment is found within 10% accurancy. However a better fit can be accieved by D˜ = 0.9 ln(100nm/rVC) for VC radii larger than 9 nm (blue line). This deviation is consitent with the results of the previous section, showing that the theoretical factor of0.5in Eq. 5.6 has to be increased to adequately fit the simulation results.
5.3.5 VC Dynamics for Long Current Rise Times
It has been predicted that VC deformations can induce a switching of the VC polarity and that they are directly related to the VC velocity [GLK08]. Recently, these VC deformations were experimentally verified by direct imaging [VCW+09].
The critical velocity at witch VC switching can occur is estimated to be around 300 m/s [GLK08].
To achieve these high VC velocities, steep current rise times are required.
Otherwise, if the current changes on timescales much longer than the magnetiza-tion damping timescale, the magnetic configuramagnetiza-tion can adiabatically adapt to the changing current. In this regime the VC velocity directly depends on the current slew rate. This is verified by a micromagnetic simulation on a 1µm large and 20 nm thick disk. The damping is chosen to be α= 0.02 and the non-adiabaticity parameter is β = 0.1. A 100 ns long current pulse is injected into the disk with a rise time of 50 ns followed by a steep fall time. The peak current density is j = 0.7·1012A/m2. The resulting VC trajectory is plotted in Fig. 5.6 (a). The initially centered VC is linearly displaced as the current density increases and finally reaches its new equilibrium position (red curve). When the current is in-stantaneously switched off, the VC relaxes back to the center position conducting a normal precessional motion (blue curve). As a result the velocities in the quasi-static regime are much lower than in the latter. This is shown in more detail in Fig. 5.6(b) where the velocity of the VC is plotted (top curve). The corresponding pulse shape is shown in the lower part of the diagram. During the slow current
5.3 Micromagnetic Simulations 99
Figure 5.6: (a) The VC trajectory is plotted for a current pulse with a 50 ns long rise time. The long rise time results in a straight displacement along the red line, whereas the very short fall time induces a precessional motion of the VC (blue curve). (b) The VC velocity (green center curve), the pulse shape (red bottom curve) as well as the total magnetization components Mx andMy (top curves) is plotted as a function of the time
rise time the VC velocity never exceeds 0.5 m/s, for a total displacement of 20 nm in about 50 ns. When the current is switched off, the VC velocity is determined only by the magnetic properties of the sample and is initially as high as 20 m/s.
Assuming a maximum slew rate of dj/dt = 5·108A/m2/ns for the pulses used in the experiments (see also Fig. 3.12) this corresponds to VC velocities far below 1 m/s and thus dynamic VC switching can be ruled out in the experiment.
5.3.6 Cell Size Dependent Study
Since the VC radius is similar in size with the exchange length, the finite cell size used in the simulations of 5x5 nm might influence the results. To verify that this is not the case, simulations on a 500 nm wide and 30 nm thick disk are carried out with cell sizes varying between 1.25 and 10 nm. The current density is j = 4·1012A/m2 and the non-adiabaticity parameter is chosen to beβ = 0.4.
Fig. 5.7(a) shows the displacement and the angle of the VC displacement θ versus the cell size is plotted in Fig. 5.7(b). For cell sizes smaller than approximatly 7 nm, the displacement angle is about θ= 40◦ in agreement with the theory. For larger cell sizes the displacement starts to deviate. Since the cell size becomes larger than the exchange length (le = 5.7nm for permalloy [HHK03]) the vortex structure is destabilized and chaotic VC polarity switching occurs. As soon as the
100 Spin-Torque driven Vortex Core Motion
Figure 5.7: (a) Plotted is the VC displacement for different cell sizes. The electrons are flowing in the bottom-top direction. Only the inner 100 nm of the disk is shown. The disk is 500 nm in diameter and 30 nm thick. The current density isj = 4·1012A/m2,α= 0.2 andβ = 0.4. (b) The VC displacement angleθis plotted as a function of the cell size.
VC starts to move in one direction, the VC polarity switches and the adiabatic torque pushes the VC now in the opposite direction. Hence the VC cannot conduct a stable gyroscopic motion, but is effectively pushed in the direction of the electron flow, thusθ≈90◦. The error bars in Fig. 5.7(a) indicate the unstable VC position in this regime. These results verify that the 5 nm cell size used in the simulations before is a good trade-off between accuracy and simulation speed.