Transient magnetic-domain-wall ac dynamics by means of magneto-optical Kerr effect microscopy

The domain wall response under constant external magnetic fields reveals a complex behavior where sample disorder plays a key role. Furthermore, the response to alternating magnetic fields has only been explored in limited cases and analyzed in terms of the constant field solution. Here we unveil phenomena in the evolution of magnetic domain walls under the application of alternating magnetic fields within the creep regime, well beyond a small fluctuation limit of the domain wall position. Magnetic field pulses were applied in ultrathin ferromagnetic films with perpendicular anisotropy, and the resulting domain wall evolution was characterized by polar magneto-optical Kerr effect microscopy. Whereas the dc characterization is well predicted by the elastic interface model, striking unexpected features are observed under the application of alternating square pulses: Magneto-optical images show that after a characteristic number of cycles, domain walls evolve toward strongly distorted shapes concomitantly with a modification of domain area. The morphology of domain walls is characterized with a roughness exponent when possible and contrasted with alternative observables which are more suitable for the characterization of this transient evolution. The final stationary convergence as well as the underlying physics is discussed.

Figure 1

Example of image processing. (a) Image of a magnetic domain obtained after background subtraction (saturated sample). (b) After applying a threshold averaging filter a binary image is constructed that allows us to obtain the DW contour. (c) Superposition of the resulting DW profile, indicated as a white line, and the original image (panel a). The scale bar is $40\mu \mathrm{m}$ width.

Figure 2

Protocols to probe and characterize the dc and ac dynamics. (a) Temporal evolution of the applied field $H$. The initial part corresponds to the dc protocol: After nucleation, the domains are grown by applying magnetic field pulses with the same polarity, $H>0$. After dc growth, the ac magnetic field pulses of alternating polarity are applied, with null average magnetic field after application of an integer number of ac cycles. In both cases, images are obtained at $H=0$, between (or after) pulses. (b) Example of the procedure followed to determine the DW mean velocity. The left panel shows the displacement $\mathrm{\Delta }x$ of the DW after $N$ dc pulses of duration $\tau$ each. The scale bar is $40\mu \mathrm{m}$. The velocity is obtained from the linear fit of $\mathrm{\Delta }x$ against the total time $\mathrm{\Delta }t=N\tau$ (right panel).

Figure 3

Procedure used to compute the roughness function $B\left(r\right)$: A linear coordinate $z=\rho \theta$ is defined from a circle centered in the centroid of the domain with the same total magnetized area [panels (a) and (b)]. $B\left(r\right)$ quantifies the correlation between $u\left(z\right)$ and $u(z+r)$ [panel (b)] as defined in Eq. (1). The roughness exponent $\zeta$ and roughness amplitude $B_0$ are defined in Eq. (2) and are obtained from the linear fit of $ln\left(B\right(r)/[\mu \mathrm{m}\left]\right)$ vs $ln(r/[\mu \mathrm{m}\left]\right)$. Error bars take into account the optical resolution and the dashed area indicates the region below resolution [panel (c)].

Figure 4

Dependence of velocity on applied magnetic field for samples S1 (blue circles) and S2 (red squares). The linear behavior of the creep plot, $lnv$ against $H^{-1/4}$, indicates that both systems are in the creep regime. The inset shows magnetic hysteresis loops for each sample.

Figure 5

(a) Examples of images taken during the ac evolution for sample S2, for different number of ac cycles, indicated in the top left corner. (b) Superimposition of the domain's contours as a function of the number of ac cycles, from the dc growth domain (blue) to 400 ac cycles (yellow) in sample S1 ($H=100$ Oe, $\tau =50$ ms) and S2 ($H=150$ Oe, $\tau =50$ ms).

Figure 6

Domains area as a function of the number of pulses for sample S1 (red) and sample S2 (blue). The average (AV) experimental curves (continuous color line) were fitted with a linear combination of an exponential decay function and a linearly decreasing function (black dashed line). AV and standard deviation (STD) were calculated over the 10 experimental realizations. The two curves that limit the shaded areas correspond to $\mathrm{AV}\pm \mathrm{STD}$.

Figure 7

Evolution of different observables as a function of the number of cycles for sample S1 (red) and sample S2 (blue). Average and standard deviation (AV and STD) for each observable were calculated over 10 realizations. The two curves that limit the shaded areas corresponds to $\mathrm{AV}\pm \mathrm{STD}$. Normalized correlation [see Eq. (3)] with the initial domain $\mathcal{C}(I_N,I_0)$ (a) and with a circular domain of the same area $\mathcal{C}(I_N,C_N)$ (b), and the ratio between actual area and the area of a circle of equal perimeter $\mathcal{R}\left(N\right)$ (c). In all cases the average curves were fitted with a linear combination of an exponential decay function and a linear decreasing function (black dashed line).

Figure 8

Example of domains evolution after applying a large number of cycles for sample S1 (a)–(d) and sample S2 (e)–(h). Scale bars are $40\mu \mathrm{m}$.