Numerical study of the influence of interfacial roughness on the exchange bias properties of ferromagnetic/antiferromagnetic bilayers

Exchange bias and coercivity are both studied numerically in antiferromagnetic/ferromagnetic (AFM/FM) bilayers in the presence of a rough interface. The roughness is modeled by an AFM atomic mesa of variable width, in a periodic bidimensional system. Unlike the flat interface case, roughness can favor the presence of magnetic interfacial frustration or the formation of sharp magnetic domain walls pinned within the first AFM planes, inside the AFM mesa, in a Peierls potential well. We demonstrate by using athermal steepest-descent calculations that irreversible processes can occur during the hysteresis loops, when the AFM mesa width is less than half of the system period. In this case, the depinning of the domain wall from the Peierls potential well during the descending branch is not followed by its rewinding in a certain range of the AFM anisotropy. This leads to a large increase of both exchange bias and coercivity at low temperature and to an athermal training effect. When the thermal activation is taken into account by using Monte Carlo simulations, we show that a random walk of the domain wall occurs within the AFM layer. These processes induce changes in the AFM spin configuration when the system is cycled several times and produce a thermally activated training effect. Our simulations, interpreted in the context of periodic Peierls potential, provide an explanation for two important features of the exchange bias phenomenon, i.e., the thermal variation of its characteristic fields and the different contributions giving rise to the training effect (AFM bulk vs interface). More generally, the presence of interfacial atomic roughness reduces both exchange bias and coercivity with respect to the perfect interface case.

Figure 1

Sketch of the FM/AFM studied bilayer. Two columns of spins of eight atomic planes each are considered, with five AFM and two FM planes being completely filled. In order to describe an atomic mesa of width $s$ at the rough interface, the spins within the left column are weighted by $s$ and by $p-s$ within the right column, with $p$ the system periodicity in the $x$ direction. The three configurations (a) $a$, (b) $b$, and (c) $c$ are the possible initial states of the system, whose respective energies $E_C$ are given in the figure.

Figure 2

Variation of the athermal normalized exchange bias $H_{EX}$ and coercivity $H_C$ with respect to $K/J$ for two initial configurations of the system, (a) $s=3$ and (b) $s=7$. For a large AFM mesa width in (b), the general trend of the $H_{EX}$ variation is very close to the perfect interface case plotted in the figure, with, however, another $K/J$ scale.

Figure 3

Characteristic hysteresis loops calculated with the steepest-descent technique for $s=3$ and three different $K/J$ ratios. In the inset of the figures, on the right, the initial spin state in the setting conditions is shown, and on the left, the spin configuration reached at the negative saturation. In (c), the bottom low inset is the spin configuration at the end of the hysteresis loop. The crosses in the different insets emphasize the location of domain-wall centers or of magnetic interfacial frustrations. The top left inset in (a) presents the angles in degree reached by the spins in both FM and AFM layers at the negative saturation. In (b) and (c), the spins angles are aligned with the easy axis and then not specified in the figure.

Figure 4

Characteristic hysteresis loops calculated with the steepest-descent technique for $s=7$ and two different $K/J$ ratios in the two regimes of Fig. 2. The insets in (a) present the spin angles at the positive saturation (right), at the remanent state (middle), and at the negative saturation (left).

Figure 5

(a) Thermal variation of $H_{EX}$ and $H_C$ for $s=3$ calculated with the Monte Carlo method. The ratio $K/J$ is kept constant during the simulations to 5.6. The three regimes depicted in the figure correspond to the regimes evidenced in Fig. 2. (b) Position of the DW center $n_C$ with respect to the iteration time at $T/J=3$ and $H{M}_s/2J_0=-15$. The simulations are performed on an AFM layer 25-atomic-planes thick in order to emphasize the random walk of the DW on a larger scale. The inset in the left side of the figure shows the spin configuration just after the magnetization reversal. The DW center is located in this case between $i=2$ and $i=3$ within the right column of spins ($n_C=2.5$). On the right side of the figure, the inset exhibits the presence of the same DW, but extending over the two columns. Its center has jumped several maxima of the Peierls potential, driven by thermal activation. This configuration implies the creation of another DW within the AFM mesa.

Figure 6

Hysteresis loops calculated using the Monte Carlo method for different $T/J$ ratio. In (a), the loops are calculated on the three regimes of Fig. 5.

Figure 7

Averaged normalized $H_{EX}$ and $H_C$ calculated using the Monte Carlo method. (a) The fields are averaged for $1<s<5$, describing the case where the AFM atoms are the minority species ($s<5$) at the FM/AFM interface on the plane $i=1$. (b) The fields are averaged for $5<s<9$. (c) Results for $s=1$ and $s=9$ compared to the perfect interface case.

Figure 8

(a) Thermal variation of the normalized $H_{EX}$ vs $T/J$ for the five successive hysteresis loops. In the inset, the characteristic loops obtained at low temperature for $T/J=0.9$ are shown. (b) Successive hysteresis loops calculated at $T/J=2.17$. In the inset, the variation of the normalized $H_{EX}$ vs the loop number $n$ is shown for different $T/J$ ratios.

Figure 9

(a) Sketch of two different spin configurations obtained after the first hysteresis loop, when the system is set in the $c$ configuration at the beginning of the calculations. If the DW is expelled out from the AFM layer at the negative saturation (case 2), a $b$-type configuration is found at the beginning of the second loop. (b) Successive hysteresis loops calculated using the Monte Carlo method at $T/J=2.4$. The loops presented in the figure are a single result of the 100 loops calculated per temperature. The first cycle is completely open and centered and corresponds to the path labeled 2 of Fig. 9. At the beginning of the following loops, the system is in a $b$-type configuration, which leads to a positive exchange bias.