Spin configurations in hard/soft coupled bilayer systems: Transitions from rigid magnet to exchange-spring

We investigate equilibrium properties of an exchange-spring magnetic system constituted of a soft layer (e.g., Fe) of a given thickness on top of a hard magnetic layer (e.g., FePt). The magnetization profile $M\left(z\right)$ as a function of the atomic position ranging from the bottom of the hard layer to the top of the soft layer is obtained in two cases with regard to the hard layer: (i) in the case of a rigid interface (the FePt layer is a single layer), the profile is obtained analytically as the exact solution of a sine-Gordon equation with Cauchy’s boundary conditions. Additional numerical simulations also confirm this result. Asymptotic expressions of $M\left(z\right)$ show a linear behavior near the bottom and the top of the soft layer. In addition, a critical value of the number of atomic planes in the soft layer, that is necessary for the onset of spin deviations, is obtained in terms of the anisotropy and exchange coupling between the adjacent plane in the soft layer. (ii) In the case of a relaxed interface (the FePt layer is a multilayer), the magnetization profile is obtained numerically for various Fe and FePt films thicknesses and applied field.

Figure 1

(Color online) (a) Representation of the system with N layers. (b) bcc iron structure, where the red spheres (atomic plane $n$ represents the spins in the same plane, and the gray spheres (atomic plan $n-1$ and $n+1$) represents spins in different atomic planes.

Figure 2

(Color online) Angle deviation $\xi \left(z\right)$ as a function of the layer position $z$ within the soft layer for different values of the soft layer thickness, $L$. The curve passing through the circles is the angle deviation at the free boundary for $d_{\text{S}}=0.01$. Inset: angle deviation at the free boundary against $L$, for $d_{\text{S}}=0.01;0.025;0.05$.

Figure 3

(Color online) Angular deviation of the individual spins as a function of the number of spins in the Fe (soft magnetic) layer with $D_{\text{Fe}}/J_{\text{Fe}}=0.05$. From left to right the two sets of arrows represent the rigid-magnet and ES regimes.

Figure 4

(Color online) Uppermost spin representation of the simulation of the RM-ES transition as a function of the Fe layer thickness (number of spins) and the $d_S=D_S/J$ ratio. The two figures show the same data with different scales.

Figure 5

(Color online) Full curves are the (analytical) angle deviation $\xi$ given in Eq. (9) as a function of the layer position $z$ for different values of the multilayer thickness $L$. Full circles are the numerical results. The dashed curve is the angle deviation at the free boundary against the multilayer.

Figure 6

(Color online) Angle deviation of the individual spins as a function of the number of spins in the Fe layer (relaxed interface) with a fixed number of spins in the FePt layer, for $J_{\text{FePt}}/J_{\text{Fe}}=1.44$. (Left) FePt layer with six spins and $D_{\text{Fe}}/J_{\text{Fe}}=0.1$, $D_{\text{FePt}}/J_{\text{Fe}}=0.2$. We can see the transition from the RM regime to the ES regime between four and five atomic layers of Fe. (Right) FePt layer with 25 spins and $D_{\text{Fe}}/J_{\text{Fe}}=0.01$, $D_{\text{FePt}}/J_{\text{Fe}}=0.02$. We can see that the transition from the RM to the ES regime is more gradual than the one on the left. The inset shows an expanded view of the region indicated in the main graph by a rectangle.

Figure 7

(Color online) False color plot of the angle variation as a function of the thickness of the Fe film and the thickness of the FePt film. The angle given corresponds to the upper free boundary spin orientation.

Figure 8

(Color online) OOMMF simulations in the case of a relaxed interface; the FePt layer has five spins while that for the Fe layer is varied.

Figure 9

(Color online) Variation in the spin configuration as a function of applied magnetic field. [(a)–(e)] show the effect of field direction. Note that (a) corresponds to the same data as in Fig. 6 (right). (f) Spin angle as a function of applied magnetic field for spins at the FePt outer surface (blue triangles), at the Fe-FePt interface (red circles) and at the outer Fe surface (black squares).