Bubble domains in disc-shaped ferromagnetic particles

We study the fundamental magnetic states of disc-shaped ferromagnetic particles with a uniaxial anisotropy along the symmetry axis. Besides the monodomain, a bidomain state is also identified and studied both numerically and theoretically. This bidomain state in the symmetry broken system is the analog of the vortex state, consists of two coaxial oppositely magnetized cylindrically symmetric domains, and is stable at zero-bias field, unlike magnetic bubbles in ferromagnetic films. For a given disc thickness we find the critical radius above which the magnetization configuration falls into the bidomain bubble state. The critical radius depends strongly on the film thickness, especially for ultrathin films. In an external field the bidomain state remains stable over a range of field strengths. The switching from the bidomain to the monodomain takes place through an abrupt domain annihilation mechanism.

Figure 1

The bubble state illustrated at the middle $(z=0)$ plane of the disc and at the top $(z=d∕2)$. The magnetization is such that $m_z=-1$ at the center and $m_z\approx 1$ in the outer domain. The arrows show the projection of the magnetization on the $(x,y)$ plane, which has a significant value at the domain wall. This is a Bloch-like wall in the middle $(z=0)$ plane and it turns almost Néel-like at the top and bottom surfaces $(z=\pm d∕2)$. In all cases that the algorithm converges, the magnetization satisfies the parity relations $m_{\rho }(\rho ,z)=-m_{\rho }(\rho ,-z)$, $m_{\varphi }(\rho ,z)=m_{\varphi }(\rho ,-z)$, $m_z(\rho ,z)=m_z(\rho ,-z)$.

Figure 2

Energy per unit volume (in units of $2\pi M_0^2$) of the monodomain (dashed lines) and of the bubble state (solid lines) as a function of the disc radius $R$ for three values of the disc thickness $d$ ($R$ and $d$ in exchange length units). The bubble exists only for $R$ greater than a critical radius $R_1$ and it has a lower energy for $R>R_c$ where $R_c$ is yet another critical radius which corresponds to the intersection of the two lines for each value of $d$. Both $R_1$ and $R_c$ depend on $d$.

Figure 3

The critical radius $R_c$ as a function of the thickness of the disc $d$. For $R>R_c$ the bubble has a lower energy than the monodomain state.

Figure 4

The total magnetization $\mu$ per unit volume as a function of the applied field, for a bubble state in a disc of thickness $d=8$ and radius $R=24$.