Thermal rounding of the depinning transition in ultrathin Pt/Co/Pt films

We perform a scaling analysis of the mean velocity of extended magnetic domain walls driven in ultrathin Pt/Co/Pt ferromagnetic films with perpendicular anisotropy, as a function of the applied external field for different film thicknesses. We find that the scaling of the experimental data around the thermally rounded depinning transition is consistent with the universal depinning exponents theoretically expected for elastic interfaces described by the one-dimensional quenched Edwards-Wilkinson equation. In particular, values for the depinning exponent $\beta$ and thermal rounding exponent $\psi$ are tested, and the present analysis of the experimental data is compatible with $\beta =0.33$ and $\psi =0.2$, in agreement with numerical simulations.

Figure 1

Experimental velocity-field curves for driven domain wall motion in ultrathin Pt/Co/Pt films with perpendicular anisotropy. Each curve corresponds to a different thickness for the Co layer, as indicated. Continuous lines represent linear fits to the fast-flow regime at large fields. This experimental data was reported in Ref. 4.

Figure 2

Fitting the value of the depinning field $H_c$ for the sample with $s=0.8$, while searching for the best fit to the theoretically expected value for the depinning exponent $\beta \approx 1/3$. The main panel shows fitting curves for different test depinning fields in scaled variables, with $650<H_c^{\mathrm{test}}<810$ Oe from bottom to top in steps of $20\mathrm{Oe}$ (data were shifted upward for clarity). The inset presents the obtained values for $\beta \left(H_c^{\mathrm{test}}\right)$. The red square point in the inset corresponds to the estimated depinning field $H_c$ for $s=0.8$ nm.

Figure 3

Dependence of the depinning exponent $\beta$ on the test field $H_c^{\mathrm{test}}$ for different film thickness as indicated. This permits us to estimate the new values for the critical fields $H_c$ (indicated by the vertical arrows) as the values where $\beta$ is closer to the $\beta =1/3$ value (dashed line). The lowest field point of each curve corresponds to the lower bound for the depinning field $H_c^{\mathrm{test}}=H^{*}$.

Figure 4

Velocity-field curves normalized with respect to the fast-flow regime and the critical depinning field, where the values of $H_c$ obtained in this work have been used (see Table ).

Figure 5

Plot of the normalized velocity against the effective temperature used to fit the value of the thermal rounding exponent through $V\left(H_c\right)\sim {(T/T_c)}^{\psi }$, which gives $\psi =0.20\left(6\right)$.

Figure 6

Phenomenological determination of the critical depinning field as the field of maximum variation of the velocity. Each line corresponds to a five-order polynomial interpolation of the data in Fig. 1 around the depinning field. Symbols represent the obtained point in the maximum of each curve used to extract $H_c^{\mathrm{phen}}$, while the vertical dashed lines correspond to the values of $H_c$ reported in Table .

Figure 7

Universal scaling function of the velocity around the thermally rounded depinning regime, using scaling variables as in Eq. (5).

Figure 8

Universal scaling function of the velocity around the thermally rounded depinning regime. The main panel shows the same data as in Fig. 7 but in a log-linear representation which shows how the scaling fails far below the depinning threshold. For completeness, the inset shows the same data but in a log-log representation, emphasizing the good scaling close to the critical depinning region.