Behavior of the antiferromagnetic layer during training in exchange-biased bilayers within the domain state model

An analysis of the antiferromagnetic spins is used to study the training effect in exchange bias bilayers. We use an atomistic model for the magnetic interactions within a classical Heisenberg spin Hamiltonian for the investigation of the temperature dependence of the training effect. Various exchange bias bilayers with different kinds of anisotropy are presented. The spins in the antiferromaget are grouped as stable and unstable according to their behavior during the reversal. It is found that their population changes during the training effect. The stable spins result in a magnetization which is shown to be related to the exchange bias field behavior. The behavior of the interface spins is shown to be determined predominantly by the degree of geometric spin frustration at the interface.

Figure 1

(Color online) Magnetization of (a) the FM and (b) the AFM of the first and tenth hysteresis loop for $T/T_N=0.403$.

Figure 2

(Color online) Temperature dependence of the training effect for systems with (a) uniaxial AFM anisotropy and $t_{\text{AFM}}=2$, (b) uniaxial AFM anisotropy and $t_{\text{AFM}}=5$, (c) biaxial AFM anisotropy and $t_{\text{AFM}}=2$, and (d) biaxial AFM anisotropy and $t_{\text{AFM}}=5$. The temperature axis is scaled to the Néel temperature of every system. $H_{\text{eb}}^1$ and $H_{\text{eb}}^2$: EB field of the first and the second hysteresis loop, respectively. $M_{\text{irr}}^1$ and $M_{\text{irr}}^2$: Irreversible domain state magnetization of the AFM interface of the first and the second hysteresis loop, respectively. The colored arrows and markers show the temperatures that are going to be used in the Fig. 5, in which athermal (lower temperature) and thermal (higher temperature) training effect is exhibited in each system.

Figure 3

(Color online) AFM interfacial spin configurations for uniaxial (top) and a biaxial (bottom) AFM anisotropy. The snapshots were taken before the first reversal for the FM (a), (d), after the first reversal (b), (e) and after the second reversal (c), (f). The colors delineate the different AFM domains. For the uniaxial AFM (top) green and yellow indicates domains with antiparallel staggered magnetization in $x$ direction. For the biaxial AFM (bottom) green and yellow indicates domains with staggered magnetization pointing in the direction of first and third quadrant while blue and red stand for the second and fourth quadrant. The top (nonmagnified) circles show spins that become stable in the second loop while the bottom (nonmagnified) circles show spins that are unstable.

Figure 4

(Color online) Distributions of reversal angles of the first reversal for systems with (a) uniaxial AFM anisotropy and (b) biaxial AFM anisotropy for two temperatures. The AFM thickness is two monolayers. The lines indicate the cut-off angle separating stable and unstable spins.

Figure 5

(Color online) Training effect of the concentration of stable spins in the AFM interface for (a) uniaxial anisotropy and (b) biaxial anisotropy for two AFM thickness. One temperature for athermal and one for thermal training were chosen for every system from the data in Fig. 2 with the respective figure markers. The constant lines are guide to the eyes.

Figure 6

(Color online) Comparison of the magnetization of the stable interfacial spins $\left(M_{\text{sis}}\right)$ with the irreversible domain state magnetization $\left(M_{\text{irr}}\right)$ for systems with (a) uniaxial AFM anisotropy and $t_{\text{AFM}}=2$, (b) uniaxial AFM anisotropy and $t_{\text{AFM}}=5$, (c) biaxial AFM anisotropy and $t_{\text{AFM}}=2$, and (d) biaxial AFM anisotropy and $t_{\text{AFM}}=5$.

Figure 7

(Color online) Distribution of unstable interfacial spins in the AFM with different numbers of the nearest neighbors in the AFM. The FM neighbor is excluded. Data for two different temperatures and uniaxial as well as biaxial anisotropy are shown.

Figure 8

(Color online) Magnetization of the AFM interface of systems with (a) uniaxial and (b) biaxial AFM anisotropy for the first and a subsequent hysteresis loop. In (a) we show results at low temperature and (b) shows the loops for high and low temperature for comparison. The numbers in (b) are for reference in Fig. 8.

Figure 9

(Color online) First and tenth hysteresis loop of the in-plane component of the AFM interface magnetization for (a) $T/T_N=0.25$ and (b) $T/T_N=0.012$. The points show the start and the reversal of the ascending and the descending branch of the first and tenth hysteresis loop. The numbers indicate points of the hysteresis loop shown in Fig. 7.