Excess velocity of magnetic domain walls close to the depinning field

Magnetic field driven domain wall velocities in [Co/Ni] based multilayers thin films have been measured using polar magneto-optic Kerr effect microscopy. The low field results are shown to be consistent with the universal creep regime of domain wall motion, characterized by a stretched exponential growth of the velocity with the inverse of the applied field. Approaching the depinning field from below results in an unexpected excess velocity with respect to the creep law. We analyze these results using scaling theory to show that this speeding up of domain wall motion can be interpreted as due to the increase of the size of the deterministic relaxation close to the depinning transition. We propose a phenomenological model to accurately fit the observed excess velocity and to obtain characteristic values for the depinning field $H_d$, the depinning temperature $T_d$, and the characteristic velocity scale $v_0$ for each sample.

Figure 1

PMOKE images for the CoNiTa sample. (a) Bubble domain after nucleation. Symbols $\odot$ and $\otimes$ indicate the direction of the magnetization in the two observed domains. (b) The same bubble domain after the first square field pulse of $30\mathrm{Oe}$ and $100\mathrm{ms}$. (c) Differential image corresponding to the two consecutive images (a) and (b).

Figure 2

Domain wall velocity as a function of magnetic field for [Co/Ni] multilayers: (a) Semilogarithmic scale and (b) creep plot, $lnv$ against $H^{-1/4}$ [for simplicity, error bars were only included in (a)]. Dashed lines in (b) are fits to the creep law $lnv\sim H^{-1/4}$, which gives a good fit at low magnetic fields. The corrected creep law described in Sec. 3 (continuous lines) gives a good description in an extended magnetic field range. Vertical dotted lines indicate the obtained depinning field $H_d$ for each curve. The corrected creep law was fit using fields $H<H_d$ (vertical dotted lines).

Figure 3

Characteristic length scales and corresponding sizes. $L_{\mathrm{eve}}$ is the longitudinal length of an individual creep event. $L_{\mathrm{opt}}$ corresponds to the optimal length associated to the effective energy barrier the domain wall should overcome in order to trigger a forward movement. $S_{\mathrm{opt}}$ and $S_{\mathrm{rel}}$ are the sizes of the corresponding magnetization reversed regions. On one hand, when $H\to 0$ one expects that $S_{\mathrm{opt}}\sim L_{\mathrm{opt}}^{{\zeta }_{\mathrm{eq}}+1}\sim H^{-{\nu }_{\mathrm{eq}}({\zeta }_{\mathrm{eq}}+1)}$, diverging at zero field. On the other hand, $S_{\mathrm{rel}}\sim L_{\mathrm{rel}}^{\zeta +1}\sim {(H_d-H)}^{-\nu (\zeta +1)}$ is diverging when approaching the depinning field from below. The size of the full event can thus be written as $S_{\mathrm{eve}}=S_{\mathrm{opt}}+S_{\mathrm{rel}}$.

Figure 4

Proposed characteristic dependence of the relaxation size $S_{\mathrm{eve}}$ with magnetic field. $S_{\mathrm{eve}}$ has two contributions: $S_{\mathrm{opt}}$ and $S_{\mathrm{rel}}$ (see Fig. 3). $S_{\mathrm{eve}}$ goes to a constant value $S_0$ at low fields. This is a good approximation since at low fields ($H<H_r$) the domain wall velocity is dominated by the exponential time scale and at higher fields ($H>H_r$) it is dominated by $S_{\mathrm{rel}}$. A saturation of $S_{\mathrm{rel}}$ is considered in order to arrest the divergence of the domain wall velocity near the depinning field. For $H<H_{\varepsilon },S_{\mathrm{rel}}$ is barely changed, but for $H>H_{\varepsilon }$ a saturation to $S_{\varepsilon }$ is reached.

Figure 5

Three different velocity-field characteristics identified below $H_d$: (i) the classical creep regime for fields $0<H<H_r$, where the velocity goes as $v_0{e}^{-T_d/T{(H/H_d)}^{-\mu }}$, (ii) the field-dependent correction to the velocity prefactor coming from the relaxation contribution gives the excess velocity for fields $H_r<H<H_{\varepsilon }$, and (iii) the saturation of the relaxation length at $L_{\varepsilon }$, for fields $H_{\varepsilon }<H<H_d$, arrests the divergence of the velocity with an $\varepsilon$-dependent velocity prefactor.

Figure 6

(a) Velocity-field characteristics in a creep plot, $lnv$ against $H^{-1/4}$, for Pt/Co/Pt at room temperature [28]. The usual creep law (black dashed lines) fits the low magnetic field data. The continuous black line corresponds to a fit using the corrected creep law, which permits us to extract the parameters shown in Table 1. The corrected creep law was fit using fields $H<H_d$ (vertical dotted line). The thin continuous line shows the divergence of the velocity due to the divergence of $S_{\mathrm{rel}}$, plotted using the same parameters of Table 1 but with $\varepsilon =0$. Finally, the upper straight dashed line shows the effect of the saturation which gives an extra contribution to the prefactor $v\left(H_d\right)$ very close to $H_d$, Eq. (27). (b) Same data as in (a) but with a flattened low field creep regime [see Eq. (26)], which emphasizes the excess velocity.