Temperature-dependent Barkhausen volume in two-dimensional Ising-like ferromagnetic films

We report our systematic investigation of temperature-dependent Barkhausen volume behavior for CoFeB/Pd with perpendicular magnetic anisotropy by means of magneto-optical Kerr microscopy. In the temperature range where two-dimensional (2D) Ising-like and single-domain features are sustained, hysteresis parameters such as coercivity, hysteresis area, and saturation magnetization are quantitatively analyzed with respect to the temperature. Interestingly it is demonstrated that the Barkhausen length is directly proportional to the domain-wall width via the magnetic anisotropy in this single-domain and 2D Ising-like model system.

Figure 1

Hysteresis loop measured by (a) MOKE measurement and by (b) VSM at different temperatures from 230 to 300 K with color codes. Coercivity ($H_c$) with respect to the temperature by (c) MOKE measurement with $\delta =0.57\pm 0.04$ and by (d) VSM with $\delta =0.57\pm 0.03$ with variation of temperature. Other $\delta$ values of 1/2 (black), 2/3 (green), and 3/4 (blue) are shown as dashed line. Hysteresis loop area measured by (e) MOKE with $\gamma =1.26\pm 0.11$ and by (f) VSM resulting $\gamma =1.25\pm 0.08$ (red dashed line). Inset $log\left(A-A_0\right)$ vs $logT$ with MOKE $\gamma =1.26$ and VSM $\gamma =1.25$. Hysteresis loop area fitted with various $\gamma$ exponents from 2D Ising (black dashed), 3D Ising (blue dashed), and 3D Heisenberg (green dashed) models.

Figure 2

Typical domain evolution pattern observed by MOKE microscopy at zero external field state at various temperatures. (a) Circular-shaped single domain initiated by external field, then observed with variation of temperatures. (b) Circular-shaped single domain prepared by proper external field at each temperature. (c) Fractal dimension ($D_f$) with respect to temperatures. Inset: superposition image of domain-wall boundary with three times measurement repetition and perimeter-area relation on log-log scale at 230 and 300 K.

Figure 3

(a) Domain displacement $\left(\mathrm{\Delta }r\right)$ at every 30 ° of circle domain at 270 K where the inset figure is circle domain configuration. (b) Domain-wall velocity, $v\left(H,T\right)$ vs applied magnetic field ($H$) at different temperatures, with fitting by the Arrhenius law (dashed line). (c) $v\left(H,T\right)$ vs $H^{-1/4}$ with linear fitting on a log-normal scale.

Figure 4

(a) Barkhausen volume $\left(V_B\right)$ at various temperatures. (b) Temperature-dependent saturation magnetization ($M_s$). The best fitting is with $\beta =0.18\pm 0.02$. Values from 2D Ising (0.125), 2D XY (0.26), 3D Ising (0.325), 3D XY (0.345), and 3D Heisenberg (0.365) models are shown as well. (c) Magnetic anisotropy ($K_u$) at different temperatures. Inset: Comparison of hysteresis area between hard axis (in-plane) and easy axis (out-plane) hysteresis loop at 230 K. (d) Barkhausen length (${\xi }_B$) and domain-wall width $\left({\delta }_w\right)$ with respect to the temperature.