Time-dependent rotatable magnetic anisotropy in polycrystalline exchange-bias systems: Dependence on grain-size distribution

Angular-resolved measurements of the exchange-bias field and the coercive field are a powerful tool to distinguish between different competing magnetic anisotropies in polycrystalline exchange-bias layer systems. No simple analytical model is as yet available which considers time-dependent effects such as enhanced coercivity in magnetic easy and hard axis configurations arising from the grain-size distribution of the antiferromagnet. In this work, we expand an existing model class describing polycrystalline exchange-bias systems by a rotatable magnetic anisotropy taking into account the relaxation time of thermally unstable grains. Our calculations show that coercivity mediated by the rotatable magnetic anisotropy can be distinguished from coercivity arising from ferromagnetic anisotropy by the shape of the angular dependence. Additionally, we performed angular-resolved magnetization curve measurements using vectorial magneto-optic Kerr magnetometry. Fitting the proposed model to the experimental data shows excellent agreement and reveals the ferromagnetic anisotropy and properties connected to the grain-size distribution of the antiferromagnet. Therefore, a distinction between the different influences on coercivity and magnetic anisotropy becomes available.

Figure 1

Schematic probability distribution of energy barriers in AF grains. F1 and F2 are log-normal functions related to systems with a different average energy barrier. AF grains may be divided into four categories (see text) depending on their energy barriers.

Figure 2

Schematic view of a magnetization curve of an EB layer system. The characteristic quantities EB field $H_{\mathrm{EB}}$ (the curve shift) and coercive field $H_{\mathrm{C}}$ (the broadness of the hysteretic curve) are calculated with the two coercive fields $H_{\text{C1}}$ and $H_{\text{C2}}$, at which the magnetization along the measurement direction equals zero.

Figure 3

Graphical illustration of the angles and vectors used in the proposed model in a polar coordinate system. ${\vec{M}}_{\mathrm{F}}$ is the magnetization vector of the F layer and ${\beta }_{\mathrm{F}}$ its azimuth. $K_{\mathrm{F}}$ is the energy density per unit volume of the F uniaxial magnetic anisotropy (FUMA) of the F and ${\gamma }_{\mathrm{F}}$ the azimuth of its magnetic easy direction. ${\vec{H}}_{\mathrm{ext}}$ is the external magnetic field vector with the azimuth ${\phi }_{\mathrm{ext}}$. ${\vec{m}}_{\mathrm{AF},\mathrm{C}2}$ and ${\vec{m}}_{\mathrm{AF},\mathrm{C}3}$ are the surface magnetization moments associated to AF grains of class II and class III, respectively. ${\gamma }_{\mathrm{RMA}}$ and ${\gamma }_{\mathrm{EB}}$ are the corresponding azimuths of the vector sums, which are connected to the RMA and UDA with the respective surface energy densities $J_{\mathrm{C}}^{\mathrm{eff}}$ and $J_{\mathrm{EB}}^{\mathrm{eff}}$.

Figure 4

(a) $H_{\mathrm{EB}}$ and (b) $H_{\mathrm{C}}$ as a function of angle calculated using Eq. (7) for different $K_{\mathrm{F}}$. Material parameters used for the FUMA were $K_{\mathrm{F}}=500\text{J}/{\text{m}}^3$ (red line), $K_{\mathrm{F}}=1000\text{J}/{\text{m}}^3$ (black dotted line), $K_{\mathrm{F}}=2000\text{J}/{\text{m}}^3$ (blue dash-dotted line), and $K_{\mathrm{F}}=4000\text{J}/{\text{m}}^3$ (green dashed line). Other material parameters can be found in Table 1.

Figure 5

(a) $H_{\mathrm{EB}}$ and (b) $H_{\mathrm{C}}$ as a function of angle calculated using Eq. (7) for different $J_{\mathrm{C}}^{\mathrm{eff}}$. Material parameters used for the RMA were $J_{\mathrm{C}}^{\mathrm{eff}}=0.1\text{mJ}/{\text{m}}^2$ (red line), $J_{\mathrm{C}}^{\mathrm{eff}}=0.05\text{mJ}/{\text{m}}^2$ (black dotted line), $J_{\mathrm{C}}^{\mathrm{eff}}=0.1\text{mJ}/{\text{m}}^2$ (blue dash-dotted line), and $J_{\mathrm{C}}^{\mathrm{eff}}=0.2\text{mJ}/{\text{m}}^2$ (green dashed line). For the FUMA $K_{\mathrm{F}}=0\text{J}/{\text{m}}^3$ (red line) and $K_{\mathrm{F}}=2000\text{J}/{\text{m}}^3$ (others) was used. Other material parameters can be found in Table 1.

Figure 6

(a) $H_{\mathrm{EB}}$ and (b) $H_{\mathrm{C}}$ as a function of angle calculated using Eq. (7) for different ${\tau }_{\mathrm{avg}}$. Material parameters used for the RMA were ${\tau }_{\mathrm{avg}}/t_{\mathrm{H}}={10}^{-3}$ (red line), ${\tau }_{\mathrm{avg}}/t_{\mathrm{H}}=3\times {10}^{-2}$ (black dotted line), and ${\tau }_{\mathrm{avg}}/t_{\mathrm{H}}=1\times {10}^{-2}$ (blue dash-dotted line). Other material parameters can be found in Table 1.

Figure 7

Schematic view of the RMA during a hysteresis loop along the magnetic hard axis where the magnetization reversal takes place via a stepwise rotation of $\vec{M}$. Due to the relaxation times of class II grains in the AF there is a misalignment between RMA and $\vec{M}$.

Figure 8

(a) $H_{\mathrm{EB}}$ and (b) $H_{\mathrm{C}}$ as a function of angle (black symbols) of a system of ${\text{Cu}}^{50\mathrm{nm}}/{\text{Ir}}_{17}{\text{Mn}}_{83}^{30\mathrm{nm}}/{\text{Co}}_{70}{\text{Fe}}_{30}^{15\mathrm{nm}}/{\text{Si}}^{20\mathrm{nm}}$. The dashed line (green) corresponds to numerical calculations based on Eq. (7) using a resolution of $0.5^{\circ }$ for ${\beta }_{\mathrm{F}}$. Material parameters used can be found in Table 2. The red line corresponds to a numerical calculation based on Eq. (8) using the same parameters. For $H_{\mathrm{C}}$ there is almost no difference for both calculations.

Figure 9

(a) $H_{\mathrm{EB}}$ and (b) $H_{\mathrm{C}}$ as a function of angle of a system of ${\text{Cu}}^{50\mathrm{nm}}/{\text{Ir}}_{17}{\text{Mn}}_{83}^{30\mathrm{nm}}/{\text{Co}}_{70}{\text{Fe}}_{30}^{15\mathrm{nm}}/{\text{Si}}^{20\mathrm{nm}}$ for different measurement times of the magnetization curves $t_{\mathrm{H}}$.

Figure 10

Minimum of the relation $H_{\mathrm{C}}\left({\phi }_{\mathrm{ext}}\right)$ for different $t_{\mathrm{H}}$. The error corresponds to the uncertainty in the determination of $H_{\mathrm{C}}$. The line is serving as a guide to the eye.