Magnetic domain wall coercivity and aftereffect in antiferromagnetic/ferromagnetic bilayers

Exchange bias and coercivity are investigated in antiferromagnetic (AFM)/ferromagnetic (FM) bilayers by steepest descent and Monte Carlo simulations. For a given range of the AFM anisotropy, a magnetic domain wall parallel to the interface is created in the AFM layer. This domain wall propagates along the AFM thickness and eventually gets pinned at the negative saturation due to the intrinsic Peierls potential of the AFM material. We demonstrate that this pinning and its associated dissipation is one possible source of coercivity in such exchange coupled bilayers. Monte Carlo simulations were also carried out to investigate the influence of thermal fluctuations, which induce a random walk of the domain wall throughout the AFM layer, leading to magnetic aftereffect. Depending on the experimental characteristic measurement time and temperature, the domain wall can eventually be expelled from the AFM layer leading to the reversal of the entire AFM spin lattice. This contribution to the exchange bias phenomenon particularly explains quite well the enhancement of coercivity in exchange coupled bilayers at low temperature, even in polycrystalline samples.

Figure 1

Calculated hysteresis loops using the steepest descent technique for $N=25$ and different anisotropy values.

Figure 2

Calculated hysteresis loops using the steepest descent technique for $N=25$ and increasing anisotropy values. In inset is shown a zoom close to the saturation where irreversible jumps of the magnetization of the order of 0.5% occur at very large field.

Figure 3

Variation of the exchange bias $\left(H_{EX}\right)$ and the coercivity $\left(H_C\right)$ with respect to the ratio between the AFM anisotropy $K$ and the exchange energy $J$, for various AFM thicknesses $(N$ in atomic plane numbers) deduced from the steepest descent calculations. In inset is shown the variation of the AFM thickness $N$ vs the DW width $\delta$ at $K=K_{DW}$.

Figure 4

DW progression inside the AFM thickness with respect to the applied field $H$. The continuous line in the figure are the spin angles calculated in the continuous approximation given by Eq. (7), $n\left(H\right)$ being adjusted for each field value. In (a) $K=0.129$ and the DW is pinned between the 12th and the 13th AFM atomic plane $\left(n_C\simeq 12.5\right)$. In (b) $K=0.128$ and the DW is expelled out from the AFM layer when the field is increased.

Figure 5

Variation of the DW center position $n_C$ versus the ratio between the anisotropy $K$ and the exchange energy $J$. In inset is shown a sketch of the atomic spin angles for $K=0.343$ at the negative saturation. When one of the central spin (here $i=3)$ is perpendicular to the easy axis, the total magnetic energy is maximum. The amount of energy needed to reach this metastable state from the equilibrium position at the saturation is the depinning energy.

Figure 6

(a) Averaged characteristic fields extracted from the MC simulated loops for $N=25$ and $K\left(0\right)=0.9$. Both exchange bias $\left(H_{EX}\right)$ and coercive field $\left(H_C\right)$ exhibit three different regimes with respect to $T/J$. (b) $H_{EX}$ extracted from all individual hysteresis loops with respect to $T/J$. The gray shaded region in both graphics delimits the transition between the first and the second regime.

Figure 7

Averaged hysteresis loops extracted from the MC simulations at different temperatures.

Figure 8

DW profile as a function of applied field at $T=0.2$ (a) and $T=0.48$ (b).

Figure 9

(a) Single hysteresis loop calculated at $T=0.2$. (b) Progression of the DW center with the applied field at $T=0.2$ for the particular hysteresis loop shown in (a). (c) Single hysteresis loop calculated at $T=0.48$. (d) Random walk of the DW center for the loop shown in (c). The shaded area corresponds to the field ranges where the DW has been expelled out from the AFM layer.

Figure 10

Variation of the characteristic fields $H_{EX}$ in (a) and $H_C$ in (b) with respect to $T/J$ and for different iteration number per spin.