Harmonic oscillator model for current- and field-driven magnetic vortices

In experiments, the distinction between spin-torque and Oersted-field-driven magnetization dynamics is still an open problem. Here, the gyroscopic motion of current- and field-driven magnetic vortices in small thin-film elements is investigated by analytical calculations and by numerical simulations. It is found that for small harmonic excitations, the vortex core performs an elliptical rotation around its equilibrium position. The global phase of the rotation and the ratio between the semiaxes are determined by the frequency and the amplitude of the Oersted field and the spin torque.

Figure 1

(a) Scheme of the magnetization in a square magnetic thin-film element with a vortex that is deflected to the right. (b) Magnetization of a vortex in its static ground state. The height denotes the $z$ component, while the gray scale corresponds to the direction of the in-plane magnetization.

Figure 2

(Color online) Dependence of the free frequency $\omega$ and the damping constant $\Gamma$ on the length $l$ for various thicknesses $t$ of the system. The symbols denote numerical results, while the lines are fits with the analytical results.

Figure 3

(Color online) Amplitude of the (a) current-driven and (b) field-driven vortex oscillations in the $x$ direction (solid red line, pluses) and the $y$ direction (dashed blue line, crosses) for a spin-polarized current density of $jP=2.5\times {10}^{10}\mathrm{A}∕{\mathrm{m}}^2$ and a field of $H=250\mathrm{A}∕\mathrm{m}$. The insets show the phases between the maximum of the applied current or field and the core displacement in the $x$ direction (solid red line, pluses) and the $y$ direction (dashed blue line, crosses). The symbols denote numerical results, while the lines are derived from the analytical expression in Eq. (16).

Figure 4

(Color online) Analytically calculated phase between the maximum current or magnetic field and the $x$ deflection of the vortex core for a $200\times 200\times 20{\mathrm{nm}}^3$ permalloy square excited with a frequency of $\Omega =4.8\mathrm{GHz}$ (above the resonance frequency of $\omega =4.4\mathrm{GHz}$). The inset shows a section of the sample with the simulated trajectories of the vortex core excited with (i) (solid red line) a spin-polarized current density with an amplitude of $jP=1.2\times {10}^{11}\mathrm{A}∕{\mathrm{m}}^2$ and (ii) (dashed blue line) a magnetic field with an amplitude of $H=1000\mathrm{A}∕\mathrm{m}$. Points denote the position of the vortex at the maximum current (i) and the magnetic field (ii), respectively.