Vortices in ferromagnetic elements with perpendicular anisotropy

We study equilibrium properties of disc-shaped ferromagnetic elements with perpendicular anisotropy and quality factor less than unity. A vortex is a static state for sufficiently thin particles. For thicker particles we find a continuous transition to a bidomain state which is stable for a range of particle diameters. This bidomain state is akin to a magnetic “bubble” where the magnetization is directed along the symmetry axis at the center of the disc and it is oppositely magnetized at the periphery. The magnetization profile of the bidomain state presents the same winding around the symmetry axis as the vortex. Its signature is its magnetostatic field which consists of two concentric regions of opposite sign above the particle top surface. Higher-order states of multiple concentric domains with alternating magnetization direction are also found for particles of sufficiently large lateral dimensions. Finally, we explain how these ideas apply to particles with a very small perpendicular anisotropy.

Figure 1

The vortex profile at the top of a disc particle with radius $R=15.1$, thickness $t=3.1$, and quality factor $Q=0.4$. The arrows show the projection of the magnetization vector $\mathit{m}$ on the $(x,y)$ plane while the third component of the magnetization $\left(m_z\right)$ is coded in gray scale. The magnetization at the center is $m_z=-1$. There is little variation of the magnetization across the thickness. In this and following figures length is measured in exchange length units.

Figure 2

The magnetostatic field $\mathit{h}$ of the vortex in Fig. 1 just over the top particle surface. The field is axially symmetric and we present the two nonzero components $h_{\rho }$, $h_z$ as functions of the radial coordinate $\rho$.

Figure 3

The bidomain state (“monobubble”) illustrated in the middle plane (upper panel) and at the top (lower panel) of a disc particle with radius $R=15.1$, thickness $t=11$, and quality factor $Q=0.4$. The picture at the bottom surface is inferred by Eq. (8). The magnetization is $m_z=-1$ 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, while the third component $\left(m_z\right)$ is coded in gray scale. Note that the gray scale code adopted here is different from that in Fig. 1.

Figure 4

A cross section across the particle diameter for the monobubble shown in Fig. 3. The arrows show the projection of the magnetization on the cross section ($m_{\rho }$ and $m_z$ shown). The bar on the top color codes the sign of $h_z$ just over the top dot surface. Black indicates negative and gray indicates positive values.

Figure 5

The magnetostatic field $\mathit{h}$ of the monobubble in Figs. 3, 4 just above the top particle surface. The field is axially symmetric and we present the two nonzero components $h_{\rho }$, $h_z$ as functions of the radial coordinate $\rho$.

Figure 6

The total magnetization $\mu$ [in units of $M_s$, defined in Eq. (9)] of a particle with radius $R=15.1$ and varying thickness $t$ (in units ${\ell }_{\mathrm{ex}}$). We have a vortex for $t<5$, a monobubble for $t>7$, and an intermediate state for $5<t<7$.

Figure 7

Energy per unit volume (in units of $4\pi M_0^2$) of the single-domain (SD, dotted line), the monobubble [bidomain (BD), solid line], and the three-ring state [tridomain (TD), dashed line] as a function of disc radius $R$. The disc thickness is $t=11$, and the quality factor $Q=0.4$. The critical radii $R_{c1}$, $R_{c2}$ (see text) correspond to the intersections of the dotted and solid lines, and solid and dashed lines, respectively.

Figure 8

A tridomain (three-ring) state illustrated at the top surface of a dot with thickness $t=11$, radius $R=24.6$, and a quality factor $Q=0.4$. Arrows show the projection of the magnetization on the plane and $m_z$ is coded in gray scale.

Figure 9

The bidomain state profile in a cross section across the dot diameter for a particle with radius $R=59.25$, thickness $t=33.5$, and quality factor $Q=0.05$. The arrows show the projection of the magnetization on the cross section ($m_{\rho }$ and $m_z$ shown). The bar on the top color codes the sign of $h_z$ just over the top dot surface. Black indicates negative and gray indicates positive values.

Figure 10

The magnetostatic field $\mathit{h}$ over the top particle surface for the state shown in Fig. 9 as a function of the distance from the center $\rho$ of the dot.

Figure 11

Energy per unit volume (in units of $4\pi M_0^2$) of the single-domain (SD, dotted line), the bidomain (BD, solid line), and the three-ring state (TD, dashed line) as a function of disc radius $R$, for thickness $t=33.5$ and a quality factor $Q=0.05$.