Having uncovered the physics that lead to intrinsic universal velocity oscillations, we turn now to extrinsic effects, such as defects and imperfections in the nanowire. These are
6.4. Effects of extrinsic pinning on domain wall velocity variations 119
a
1µm
0 1 2 3 4
0.8 0.9 1 Distance [µm]
Absorption [a.u.]
b
0 0.2 0.4 0.6 0.8 1 1.20.8
0.9 1
Distance [µm]
Absorption [a.u.]
c
Figure 6.15: Determination of the reproducibility. a) A time-resolved snapshot of domain walls in motion during a 5 MHz rotational field burst pulse is shown. The configuration of the vortex domain wall is shown by a snapshot of micromagnetic simulations (inset). b) The absorption profile along the saturated ring on top (red) and c) across the vortex domain wall in motion (blue) are plotted as a function of the distance and indicated by the horizontal lines.
expected to give rise to a spatially varying energy landscape for domain walls [128] and the interaction of the moving domain walls with the local pinning potential is expected to result in velocity variations. Two different pinning mechanisms exist, namely kinetic and static pinning of magnetic domain walls, introduced by Ahn et al. [133,134]. They show experimentally, by employing micromagnetic simulations and in the framework of the 1D model, that the static depinning field for domain walls is higher than the kinetic pinning field of domain walls in motion. Recently, Lewiset al.[135] demonstrated that the switching probability of T-shaped nanowires, which act as domain wall traps, depends on the strength of the external magnetic field above the Walker field.
However, open questions of particular interest include how the kinetic pinning field scales with the domain wall velocity and what is the role of domain wall inertia. For instance, in the analytical calculations of the kinetic pinning field within the 1D model, a basic assumption is sin (2ψ) ≈ 2ψ which only holds for small polar angles ψ [133]. However, in the previous section we found that the magnetostatic energy is given by a parabolic potential for the radial vortex core position, which is associated with the polar angle in the framework of the 1D model. Thus, the internal domain wall energy is maximal for large angles ψ.
We start by investigating numerically the interaction of domain wall with a parabolic pinning potential within the framework of the 1D model in circular nanowires and rotating magnetic fields, which is described in subsection 6.4.1. We can reproduce the pinning probability shown by Lewiset al. [135] and that the kinetic pinning regime is limited by a critical field rotation frequency fcrit, that depends on the polar angle of the domain wall
and therefore on the magnetostatic domain wall energy.
To test this theory and to quantify the interplay between the local pinning potential land-scape of the nanowire and moving domain walls at different velocities, we image the domain wall propagation for various field rotation frequencies between 2.5−17.5 MHz (subsection 6.4.2). Depending on the field rotation frequency, we identify three different propagation regimes: Pinning dominated domain wall propagation at low domain wall velocities and periodically oscillating domain motion only at high domain wall velocities below and above the Walker breakdown. This allows us to directly confirm the theory predictions of static and kinetic pinning. By direct imaging of the domain propagation, we directly probe the interplay between moving domain walls and the local pinning potential, revealingintrinsic and extrinsic domain wall velocity variations.
To reveal the underlying energetics, we performed micromagnetic simulations (subsection 6.5). We find that the magnetostatic domain wall energy oscillates and the maximum of the magnetostatic energy scales with the domain wall velocity, such that the magnetostatic energy dominates over the local pinning potential strength at high domain wall velocities.
We conclude that the magnetostatic energy can act as an energy reservoir, leading to domain wall inertia and that the stored energy allows the domain wall to overcome the local pinning potential.
6.4.1 One-dimensional model of static and kinetic domain wall pinning
The aim is to investigate numerically the interaction of a domain wall with the local pinning potential. We start with the equations of motion of the 1D domain wall model in straight nanowires, see section 1.6. We assume a periodic parabolic pinning potential of depth V0 and spatial extensionq0 [9]:
(q) = and the equations motion are
˙
First, we determine the pinning probability of a moving domain wall within a pinning potential to reproduce the findings from Lewis et al. [135]. The domain wall is initially placed before the pinning potential qinit = −2q0, the initial polar angle ψinit is chosen randomly between [0,2π[ and Pkin is the probability of a domain wall passing the pinning potential. In figure 6.16, the resulting kinetic transition probability Pkin is plotted as a
6.4. Effects of extrinsic pinning on domain wall velocity variations 121 function of the external magnetic field strengthB. We assume typical material parameters as listed in table 6.1. The spatial extent of the pinning potentials is q0 =w and q0 =w/2, and the pinning potential depth isV0 = 2 · 103 J/m3. At the kinetic pinning fieldBkin the transition probability makes a sharp transition from zero to one. The kinetic pinning field is always lower than the static depinning field Bstat = V0/2MS, and the transition field depends on the spatial extension q0 and the depth of the pinning potential, in agreement with [133]. Above the Walker field µ0HW = 0.6 mT, the transition probability can drop below 100 % because of Walker precession of the polar angle within the pinning potential.
0 0.5 1 1.5
Kinetic transition probability Pkin (q0=w/2) Kinetic transition probability Pkin (q0=w)
Figure 6.16: Kinetic pinning of magnetic domain walls. The kinetic transition probabilityPkin is plotted as a function of the magnetic field strength B. The spatial extensions of the pinning potential are q0 = w/2 and q0 = w. The kinetic pinning field Bkin is de-fined as the field, at which the transi-tion probability goes from zero to one and it is always smaller than the static depinning fieldBstat. The Walker field µ0HW is plotted as a red vertical line.
In the ring geometry, the domain wall velocity is controlled by the field rotation frequency f. Below the Walker breakdown, the steady-state motion is at constant velocity and without precession of the polar angle. The polar angle is directly connected to the domain wall velocity by equation (6.4)
˙
q =−γ0∆0
MS Kdsin(2ψ). (6.21)
Therefore, the polar angle ψ of moving domain walls in circular nanowires is determined by the field rotation frequency f. In figure 6.17 the kinetic transition probability Pkin is plotted as a function of the field amplitude B and field rotation frequency f. The initial domain wall position is qinit/r = −π/2 and the initial polar angle is ψinit = 0.
The field rotates CW and starts in the direction of the initial domain wall position. The transition probability is either one, if the domain wall passes the pinning potential of depth V0 = 1 · 103 J/m3 and spatial extension q0 = w/2, or zero, if the domain wall is trapped within the pinning potential.
If the field amplitude is smaller than the static depinning field Bstat = 1.25 mT, a critical field rotation frequency fcrit ≈ 12.5 MHz exists, where the polar angle is large enough, so that the domain wall passes the pinning potential. This transition from the static to the kinetic pinning regime is abrupt and directly related to the magnetostatic domain wall energy. This only occurs when the field strength exceeds the kinetic pinning field Bkin = 0.25 mT. The value of the static depinning field is proportional to the potential
µ0H
Field rotation frequency f [MHz]
Field amplitude B [mT]
Figure 6.17: Kinetic pinning of magnetic domain walls in ring struc-turesThe kinetic transition probability Pkinis plotted as a function of the mag-netic field strengthB and field rotation frequencyf. Dark (light) blue indicates the area, when Pkin = 1 (Pkin = 0).
The Walker field µ0HW is plotted as red horizontal line, the static and kinetic pinning fields are indicated by the hori-zontal dotted lines and the critical rota-tion frequencyfcritis plotted as vertical dotted line.
depth Bstat = V0/MS. The kinetic pinning field Bkin, as well as the critical field rotation frequencyfcrit are proportional to the square root of the static pinning field and the spatial extension of the pinning potential∝√
Bstatq0 (not shown), in agreement with [133].
In summary, we find that the kinetic pinning field Bkin is always lower than the static depinning field Bstat in agreement with recent reports [133,135]. Furthermore, in the ring structure, the polar angle ψ is directly coupled to the field rotation frequency. If the field rotation frequency exceeds a critical frequency f > fcrit, the polar angle is large enough, such that the domain wall passes the pinning potential. Since the polar angle is associated with the magnetostatic energy, it also shows that the domain wall can pass the pinning potential for sufficiently large magnetostatic energy.
6.4.2 Direct imaging of pinning dominated domain wall propagation
We have confirmed that, in the framework of the 1D model, the pinning field of moving domain walls is smaller than the depinning field of a trapped domain wall (kinetic and static pinning). To test this theory and to quantify the interplay between the local pinning potential landscape of the nanowire and moving domain walls at different velocities, we image the domain wall propagation for various field rotation frequencies between 2.5− 17.5 MHz, corresponding to average domain wall velocities of 31 −220 m/s. The field amplitudes are B = 6.8 mT and 5.4 mT, respectively, and the ring geometry corresponds to sample type B. In figure6.18, the domain wall velocities are plotted as a function of the domain wall azimuthal position. Depending on the field rotation frequency, and therefore on the domain wall velocity, we identify three different domain wall propagation regimes.
For f = 2.5 MHz (black) we observe pinning dominated stop-and-go motion, where the domain walls are sometimes statically pinned. Between f = 3.75−5 MHz (green) the domain walls propagate freely with oscillating domain wall velocity due to the intrinsic domain wall spin structure oscillations with a constant period. Above 7.5 MHz (blue) we observe domain wall motion above the Walker breakdown, where the domain wall spin
6.4. Effects of extrinsic pinning on domain wall velocity variations 123
Figure 6.18: Domain wall propagation regimes. The domain wall velocity is plotted as a function of the azimuthal domain wall position for field rotation frequencies f = 2.5−17.5 MHz and field amplitudes a)6.8 mTand b)5.4 mT. The transition between extrinsically pinning dominated domain wall motion and oscillating domain wall propagation, due to intrinsic domain wall spin structure oscillations, occurs at approximately 40 m/s, indicated by the saturation of the background. The Walker breakdown velocity is about100m/s [61]. The errors are listed in table 6.3. (From [146])
0 2 4 6 8 10 12 14 16 18 20
Field rotation frequency f [MHz]
Field amplitude B [mT] Maximum phase lag θmax [rad]
0 Sustained rotation with θ
max No sustained rotation
Figure 6.19: Phase diagram of the maximum phase lag. The maximum phase lagθmaxas a function of the field amplitude B and rotation frequencyf, shown by the colorscale, if known. Oth-erwise the observation of sustained rota-tion is indicated by the shape of the data points. The light blue area is a guide for the eye, indicating where sustained rotation of the domain walls can be ob-served experimentally. (From [146])
structure undergoes periodic transformations into a transverse domain wall. We estimate the Walker breakdown velocity to be at approximately 100 m/s for this geometry [61].
The shape of the velocity curves above the Walker breakdown for field rotation frequencies above f = 7.5 MHz is significantly different compared to domain wall propagation below the Walker breakdown. In particular, for higher field rotation frequency the domain wall travels a larger distance until it reaches the maximum domain wall velocityvmax, indicated by the red circles in figure6.18a. This phenomenon can be attributed to finite acceleration of the domain wall, and hence it is direct evidence of domain wall inertia.
In figure 6.19 the maximum phase difference θmax is plotted as a function of the field rotation frequency f and the field amplitude B. The phase lag increases for faster field rotation and lower field amplitude, as expected from the 1D model below the Walker breakdown (see figure 6.2). Further, we can check if sustained rotation of the domain wall
Static
Figure 6.20: Intrinsic and extrinsic domain wall velocity variations. The domain wall velocity is plotted as a function of the domain wall azimuthal coordinate, recorded for a full field rotation at a)7 MHz and b)f = 15 MHzwith different start and stop angles0◦ (blue), 45◦ (red) and 90◦ (green). a) We observe two static pinning events coinciding in all three velocity curves at around 140◦and310◦. Furthermore between180◦−270◦the domain wall velocities are strongly influenced by the local pinning potential landscape at non zero velocity (dynamic pinning dominated domain wall motion). b) The domain wall velocity oscillates and no indication of extrinsic pinning can be observed. The errors are listed in table6.3. (Data from [146])
occurs by recording the time resolved transmission intensity at one point, without taking a movie of the magnetization dynamics, indicated by the diamond shaped or crossed data point markers, respectively. The light blue region is a guide for the eye, indicating where sustained rotation of the domain walls can be observed experimentally. In analogy to the 1D model, we observe a linear dependence of the sustained domain wall rotation as a function ofBandf. However, in the one-dimensional model this proportionality originates from the maximum domain wall velocity below the Walker breakdown, whereas in the experiment, this dependence is observed above the Walker breakdown. The maximum field amplitude and the minimum rotation frequency are limited by the maximum power of the electronic equipment and the heat dissipation of the crossed striplines.
To directly identify the extrinsic velocity variations at low domain wall velocities due to the local pinning potential landscape, we record the domain wall velocity for a complete field rotation with different start angles of 0◦ (blue), 45◦ (red) and 90◦ (green), as shown in figure 6.20. The field rotation frequencies are 7 MHz and 15 MHz, and the ring geometry corresponds to sample type D. Forf = 7 MHz we observe pinning dominated domain wall propagation at an average domain wall velocity of 110 m/s. The domain wall velocities are plotted as a function of the domain wall azimuthal coordinate. We observe two static pinning events at 140◦ and 310◦, where the domain wall velocity drops close to zero. For f = 15 MHz the domain wall velocity oscillates due to the intrinsic periodic domain wall spin structure changes and there is no indication of extrinsic pinning. This allows us to confirm the theory predictions of static and kinetic pinning. And by direct imaging of the domain propagation, we also probe directly the interplay between moving domain walls and the local pinning potential, revealing intrinsic and extrinsic domain wall velocity variations.
6.4. Effects of extrinsic pinning on domain wall velocity variations 125
Figure 6.21: Depth of the pinning potential. a) The Zeeman energy is calculated for the relaxed vortex domain wall in equilibrium and an external field B = 4.8 mTat an angle θ to the domain wall center position. b) The measured phase lagθ and the corresponding Zeeman energy are plotted as a function of time. The maximum phase difference is θpin = 21◦±1◦, corresponding to an average pinning potential depthVpin= 122±19 J/m3.
To complete the picture, we estimate the depth of the pinning potentialVpin by the Zeeman energy density associated with a vortex domain wall at an angleθpin with the external field where the phase difference θ is maximal for pinning dominated domain wall propagation.
We assume a domain wall spin structure of a vortex domain wall in equilibrium, which is a reasonable assumption at small fields and low domain wall velocities. For a field rotation frequency of f = 2.5 MHz the maximum phase difference isθpin = 21◦±1◦, corresponding to a Zeeman energy density of Vpin = 120±20 J/m3.
Error considerations
The error of the domain wall velocity δvdw is determined from the error of the vortex core positions δr (6.13), see section 6.3.1. We neglect systematic uncertainties of the sample dimensions w, t and r and we assume that the field amplitude B and rotation frequency f are known. The resulting errors of the domain wall velocities δvdw for figures 6.18 and 6.20 are listed in table 6.3.
6.4.3 Scaling of the magnetostatic domain wall energy
To reveal the underlying energetics, we calculate the magnetostatic energy Edem from the domain wall velocity vdw and the phase difference θ, both of which can be measured experimentally. We assume a parabolic potential with stiffness κr = 8.9 · 10−4 kg/s2 and κt= 1.5 · 10−4 kg/s2 as determined in subsection 6.3.3. In figure6.22, the resulting radial potential energy Vr = κrδr2/2 from the radial vortex core displacement is plotted as a function of the domain wall position for field rotation frequencies f = 2.5−5 MHz, below the Walker breakdown. The radial potential energy oscillates with the relative vortex
0 Field rotation frequency f [MHz]
Energy density [J/m3]
Vr,max(f) Linear fit
Figure 6.22: Potential energy of moving domain walls. The radial po-tential energy Vr = κrδr2/2 is plot-ted as a function of the domain wall position, for field rotation frequencies f = 2.5−5 MHz and domain wall propagation below the Walker break-down. The maximum radial potential energy Vr,max(f) (red circles) is plot-ted as a function of the field rotation frequency (inset). The errors are listed in table6.4.
Table 6.3: List of errors of the experimental domain wall velocitiesδvdw. Frequencyf B= 6.8 mT B = 5.4 mT
-Sample typeB, figure 6.18
Field angle f = 7 MHz f = 15 MHz 0◦ 26 m/s 63 m/s 45◦ 19 m/s 66 m/s 90◦ 13 m/s 59 m/s Sample typeD, figure6.20
core displacement. It is the gyroforce G× v and the force Ft ∝ sinθ which push the vortex core radially and increase the potential energy. Both the gyroforce and Ft are on average proportional to the field rotation frequency, because both the domain wall velocity and θ depend linearly on f (below the Walker breakdown). Therefore the maximum of the radial potential energy oscillations is higher for higher field rotation frequencies, as shown in figure 6.22 (inset). However the contribution from the radial potential energy is small compared to the local pinning potential depth Vpin = 122 J/m3. The tangential potential energy from the radial field component dominates (not shown) and is in the order of Vt = 280±10 J/m3.
To corroborate the experimental results, we performed micromagnetic simulations mod-eling the the rotation of the domain walls for various field rotation frequencies from f = 5−10 MHz. The dimensions of the ring structure correspond to sample type C. The field amplitude is B = 5 mT, the cellsize is 5×5×30 nm3 and the material parameters are listed in table 6.1.
Figure6.23a shows the time evolution of the normalized magnetostatic energy density de-termined from the simulations for field rotation frequencies of 8 and 10 MHz. The timescale
6.4. Effects of extrinsic pinning on domain wall velocity variations 127
Field rotation frequency f [MHz]
Energy density [J/m3]
b
Epin=122±19 J m−3
Energy peak maximum Emax Linear interpolation
Linear interpolation
Energy oscillation amplitude Eamp
Figure 6.23: Scaling of the magnetostatic energy. a) The time evolution of the normalized magnetostatic energy density is plotted as a function of the normalized time t·f. Emax is the peak maximum magnetostatic energy density and Eamp is the maximum amplitude of the energy oscillations, shown by the blue arrow forf = 10 MHz. b) The maximum peak heightEmaxand the oscillation amplitudeEamp of the magnetostatic energy density are plotted as a function of the field rotation frequency f. The dotted lines are a guide for the eye. Additionally, the Zeeman pinning energy density is indicated by the horizontal red bar.
is normalized to the period of one rotation T = 1/f, in order to visualize the correspond-ing maximum peak height Emax and the oscillation amplitude Eamp of the magnetostatic energy density. As expected, the oscillation amplitude and the energy maximum of the magnetostatic energy increase linearly between 5−10 MHz, as plotted in figure6.23b. The magnetostatic energy is the main contribution to the internal domain wall energy of the vortex domain wall, because the anisotropy energy is zero and the exchange energy is con-centrated in the vortex core and is small and constant. Therefore the internal domain wall
is normalized to the period of one rotation T = 1/f, in order to visualize the correspond-ing maximum peak height Emax and the oscillation amplitude Eamp of the magnetostatic energy density. As expected, the oscillation amplitude and the energy maximum of the magnetostatic energy increase linearly between 5−10 MHz, as plotted in figure6.23b. The magnetostatic energy is the main contribution to the internal domain wall energy of the vortex domain wall, because the anisotropy energy is zero and the exchange energy is con-centrated in the vortex core and is small and constant. Therefore the internal domain wall