Temperature dependence of the training effect in a $\mathrm{Co}∕\mathrm{Co}\mathrm{O}$ exchange-bias layer

The temperature dependence of the training effect is studied in a $\mathrm{Co}∕\mathrm{Co}\mathrm{O}$ exchange-bias bilayer and a phenomenological theory is presented. After field cooling the sample to below its blocking temperature, the absolute value of the exchange-bias field decreases when cycling the heterostructure through consecutive hysteresis loops. This decrease is known as the training effect and is studied in the temperature range $5⩽T⩽120\mathrm{K}$. An implicit sequence, which has been recently derived using the Landau-Khalatnikov approach of relaxation, fits the respective data set for each individual temperature. The underlying discretized dynamic equation involves an expansion of the free energy in powers of the interface magnetization of the antiferromagnetic pinning layer. The particular structure of the free energy with a leading fourth-order term is derived in a mean-field approach. The explicit temperature dependence of the leading expansion coefficient explains the temperature dependence of the training effect. The analytic approach is confirmed by the result of a best fit, which condenses the data from more than 50 measured hysteresis loops.

Figure 1

(Color online) (a) shows a Landau-type free energy (solid line) of the AF pinning layer at $T<T_B$ and the harmonic approximation (dashed line) in the vicinity of ${\eta }_e$. (b) shows a sketch of the temporal evolution of the AF interface magnetization with increasing number $n$ of cycles. (c) and (d) display sketches of the spin structures of the AF/FM bilayer after the second $(n=2)$ and third $(n=3)$ loop, respectively. Dashed vertical lines indicate AF domain walls. Parallel spin pairs at the interface (horizontal solid line) are highlighted by a gray background. The AF interface magnetization couples with the FM top layer via the exchange $J_{\mathrm{EB}}$.

Figure 2

$\theta -2\theta$ x-ray analysis of the ${\mathrm{Al}}_2{\mathrm{O}}_3$ substrate (a), the ${\mathrm{Al}}_2{\mathrm{O}}_3∕\mathrm{Co}∕\mathrm{Co}\mathrm{O}$ heterostructure as prepared (b), and after annealing for $4\mathrm{h}$ at $T=1000\mathrm{K}$ (c). All scans show the dominant (200) and (400) peaks of the single-crystalline ${\mathrm{Al}}_2{\mathrm{O}}_3$ substrate and its weaker (300) peak. There is no significant additional peak in the as-prepared structure (b). After annealing (c) two additional peaks are observed and assigned as (111) and (200) peaks of fcc Co.

Figure 3

Ratio $m_r∕m_S$ of the remanent and the saturation magnetic moment for various in-plane orientations $0⩽\varphi ⩽2\pi$ of the magnetic field. Data are determined from hysteresis loops of ${\mathrm{Al}}_2{\mathrm{O}}_3∕\mathrm{Co}∕\mathrm{Co}\mathrm{O}$ measured by alternating gradient force magnetometry at room temperature. The inset shows a typical loop. Dashed lines indicate the remanent and the saturation magnetic moment, respectively.

Figure 4

Training effect ${\mu }_0{H}_{\mathrm{EB}}$ versus $n$ for $T=25$ (open circles) and $75\mathrm{K}$ (diamonds) and the corresponding results of the best fits of Eq. (1) (solid squares and triangles, respectively). Note the different scales for $T=25$ and $75\mathrm{K}$, assigned by arrows. Inset shows the equilibrium EB field ${\mu }_0{H}_{\mathrm{EB}}^e$ versus $T$ which results from fitting of Eq. (1) to various data sets at $5<T<120\mathrm{K}$.

Figure 5

$\gamma$ vs $T$ obtained from fitting procedures of Eq. (1) to ${\mu }_0{H}_{\mathrm{EB}}$ vs $n$ data for temperatures $5⩽T⩽120\mathrm{K}$. The line is a one-parameter best fit of Eq. (7) to $\gamma$ vs $T$.