Theory of vortex states in magnetic nanodisks with induced Dzyaloshinskii-Moriya interactions

Broken inversion symmetry in magnetic nanostructures induces Dzyaloshinskii-Moriya couplings. In the presence of surfaces/interfaces the magnetism of magnetic nanodisks can be essentially affected by these chiral couplings. Within a micromagnetic approach we calculate the equilibrium sizes and shape of the vortices as functions of magnetic field, the material, and the geometrical parameters of nanodisks. It was found that the Dzyaloshinskii-Moriya coupling can considerably increase sizes of vortices with “right” chirality and suppress vortices with opposite chirality. This allows to form a bistable system of vortices with the same chirality as a basic element for storage applications.

Figure 1

(Color online) (a) Vortex states in a circular magnetic nanodisk of radius $R_d$ with axisymmetric magnetization structure. (b) Geometry and definition of variables of the problem. (c) The four possible vortex states are characterized by the indices (polarity $p=\pm 1$; chirality $c=\pm 1$) Dzyaloshinskii-Moriya couplings with chirality $\tilde{c}=1$ $\left(D>0\right)$ widen vortices with the same chirality, $c=1$, and squeeze vortices with opposite chirality, $c=-1$.

Figure 2

(Color online) (a) Typical phase space trajectories for the solutions of the auxiliary Cauchy problem [Eqs. (7, 9)]. (b) Corresponding profiles $\theta \left(\rho \right)$. The shown trajectories pertains to the case $H=0$, ${\theta }_h=\pi /2$. Due to the localized character of the vortex states the equilibrium solutions $\theta \left(\rho \right)$ of the boundary problem [Eqs. (7, 8)] are close to the localized profiles given by the (dashed) separatrix lines in the phase portrait. [In (b) the values of the initial slope, Eq. (9), are $\alpha =1.3\left(1\right)$, 1.25 (2), 1.193 (3) 1.923 (4, separatrix), 1.19 (5), 1.1 (6), and 1.05 (7).]

Figure 3

(Color online) Typical solutions $\theta \left(\rho \right)$ for different values of $D$ in zero applied field. The vortices widen when their chirality $c=1$ coincides with the chirality of the DM interaction, $D>0$. In the opposite case, $D<0$, the vortex cores shrink, and the local chirality $\chi \left(\rho \right)$ changes its sign outside the vortex core (inset). The core with unfavorable chirality is encircled by a ring with negative local polarization and favorable chirality. For strong negative DM coupling $\left(D/D_0=-1.5\right)$ the circulation of the projected magnetization in the plane changes the sense of rotation from positive in the core to negative at the edge, $\theta \to -\pi /2$ at $R_d$.

Figure 4

(Color online) (a) Schematic representation of vortex states with unfavorable chirality $D<0$ with corresponding magnetization profile (b).

Figure 5

(Color online) Typical solutions $\theta \left(\rho \right)$ for different values of the applied field and the reduced values of the Dzyaloshinskii constant: $D/D_0=0$ (solid lines), 0.5 (dashed lines), and $-0.5$ (dash-dotted lines).

Figure 6

(Color online) Magnetization profiles $m_z\left(\rho /l_c\right)$ for different values of $D$. For $D>0$ the perpendicular magnetized central spot broadens with increasing $D$. For negative $D$ an increasing $\left|D\right|$ widens the ring with negative $m_z$ and compresses a central spot with positive $m_z$ (inset).

Figure 7

(Color online) The equilibrium sizes of vortex core $R_0$ [Eq. (10)] as functions of reduced magnetic field $H/H_a$ for different values of $D/D_0$. Inset shows $R_0$ as functions of $D/D_0$ for different values of the applied magnetic fields.

Figure 8

(Color online) The first three excitation modes for $D/D_0=0.5$ and $H/H_a=0.25$ and corresponding eigenvalues: ${\lambda }_1=2.966$, ${\lambda }_2=8.428$, and ${\lambda }_3=15.685$. The structure is radially stable because the smallest eigenvalue ${\lambda }_1$ is positive. Inset shows the first eigenvalue as a function of the applied field and for different values of $D/D_0$.

Figure 9

(Color online) The first excitation mode ${\xi }_1\left(\rho \right)$ at zero field and for different values of the Dzyaloshinskii constant. Inset shows ${\xi }_1\left(\rho \right)$ for $D=0$ and different values of the applied field. Corresponding eigenvalues ${\lambda }_1\left(H/H_a\right)$: ${\lambda }_1\left(-0.5\right)=0.9556$, ${\lambda }_1\left(0\right)=1.7124$, and ${\lambda }_1\left(0.5\right)=2.3644$.