Magnetoelastic properties of antiferromagnetically coupled magnetic composite media

We study the magnetic response of a ferromagnetic bilayer with antiferromagnetic coupling, where the layers experience magnetostrictive strains and epitaxial misfit strains. These strains cause the layers to stretch and bend as the magnetic spins of the layers rotate, resulting in elastic energy that adds to the magnetic energy of the system. The magnetic and elastic energies are computed as a function of spin direction in each layer for a given set of material and geometric parameters. By finding the rotations that minimize the total energy, we compute magnetic hysteresis loops for different combinations of magnetic and elastic parameters. The elastic contribution is reflected in the transitions at the corners of the hysteresis curves as well as in the coercive field of the main loop. The details of the elastic contribution depend in a complicated way on the magnetostriction of the layers, the epitaxial strain, the magnetic anisotropies, and the system geometry.

Figure 1

Multilayer system schematic. Magnetization $M$ and applied field $H$ position with respect to the easy axis.

Figure 2

Magnetic hysteresis curves are shown for different values of the layer thickness ratio $\beta$ when ${\alpha }_1=0.15$ and ${\alpha }_2=0.3$.

Figure 3

The spin angles ${\theta }_1$ and ${\theta }_2$ are shown for the case corresponding to $\beta =0.2,{\alpha }_1=0.15$, and ${\alpha }_2=0.3$. The top figure shows the angles on increasing the external magnetic field from $-1$, while the lower curve shows the angles on decreasing the field from $+1$. The actual results from the minimization algorithm are shown, though ${\theta }_1$ and ${\theta }_2$ are $2\pi$ periodic.

Figure 4

Magnetic hysteresis curves are shown for different values of the layer thickness ratio $\beta$ when ${\alpha }_1=0.3$ and ${\alpha }_2=0.15$.

Figure 5

Magnetic hysteresis curves are shown for different values of magnetostrictive strains ${\lambda }_1$ and ${\lambda }_2$, with epitaxial misfit ${\epsilon }^T=0$ and thickness ratio $\beta =0.4$. The magnetic anisotropies ${\alpha }_1=0.15$ and ${\alpha }_2=0.3$.

Figure 6

Magnetic hysteresis curves are shown for different values of magnetostrictive strains ${\lambda }_1$ and ${\lambda }_2$, with epitaxial misfit ${\epsilon }^T=0$ and thickness ratio $\beta =0.4$. The magnetic anisotropies ${\alpha }_1=0.3$ and ${\alpha }_2=0.15$.

Figure 7

Magnetic hysteresis curves are shown for different values of magnetostrictive strains ${\lambda }_1$ and ${\lambda }_2$, with epitaxial misfit ${\epsilon }^T=0.002$ and thickness ratio $\beta =0.4$. The magnetic anisotropies ${\alpha }_1=0.3$ and ${\alpha }_2=0.15$.

Figure 8

Magnetic hysteresis curves are shown for different values of magnetostrictive strains ${\lambda }_1$ and ${\lambda }_2$, with epitaxial misfit ${\epsilon }^T=-0.002$ and thickness ratio $\beta =0.4$. The magnetic anisotropies ${\alpha }_1=0.3$ and ${\alpha }_2=0.15$.

Figure 9

Central loop coercive field $h_c$ vs epitaxial strain ${\epsilon }^T$ for different magnetic expansion coefficients. The magnetic anisotropies ${\alpha }_1=0.3$ and ${\alpha }_2=0.15$.