Electric Field Control of the Skyrmion Hall Effect in Piezoelectric-Magnetic Devices

Relying on both electromechanical and micromagnetic simulations, we propose a method to control the trajectory of current-driven skyrmions using an electric field in hybrid piezoelectric-magnetic systems. By applying a voltage between two lateral electrodes, a transverse strain gradient is created, as a result of the nonuniform electric field profile in the piezoelectric material. Due to magnetoelastic coupling, this transverse gradient leads to a lateral force on the skyrmions that can be used to suppress the skyrmion Hall angle for any given current density, if a proper voltage is applied. We show that this method works under realistic conditions, such as the presence of disorder in the ferromagnet, and that skyrmion trajectories can be controlled with moderate voltages. Moreover, our method allows the maximum current density that can be injected before the skyrmion is annihilated at the nanostrip edge to be increased, which leads to an increase in the maximum achievable velocities.

Figure 1

(a) Schematic representation of the simulated device. (b) Device cross section showing the electric field lines in the PZ substrate. (c) Profile of the vertical component, $E_z$, along the PZ/HM interface. Inset is a magnification of the profile in the central region of the nanostrip.

Figure 2

(a) Elastic strain $({\epsilon }_{yy})$ profile transferred to the FM layer across the central region for different PZ layer widths. Blue line represents a linear profile obtained by extrapolating the slope at the origin $(y=0)$ for $w_{\mathrm{PZ}}=1\mu \mathrm{m}$. (b) Strain gradient in the center of the FM $(y=0)$ for different PZ-layer widths, $w_{\mathrm{PZ}}$. Voltage, $V=10\mathrm{V}$, between the electrodes is applied in all cases.

Figure 3

(a) Schematic representation of the force contributions to skyrmion dynamics. (b) Skyrmion trajectories obtained from micromagnetic simulations (dots) and analytical calculations (lines) for a current density of $J_{\mathrm{HM}}=10\mathrm{GA/}{\mathrm{m}}^2$ and different strain gradients. In the legend, “s” stands for “strain gradient” $s\equiv d{\epsilon }_{yy}/dy$. (c) Evolution of the skyrmion Hall angle with strain gradient for different current-density values.

Figure 4

(a) Strain-gradient slope required to compensate for the skyrmion Hall angle for each current-density, $J_{\mathrm{HM}}$, value. (b) Skyrmion speed versus applied current in the absence of strain and in the presence of a strain compensating for the skyrmion Hall angle.

Figure 5

(a) Skyrmion speed versus strain gradient for a disordered FM film (red dots) and for an ideal FM film (blue dots). (b) Skyrmion Hall angle evolution versus strain gradient for a disordered FM film (red dotes) and for an ideal FM film (blue dots). Graphs show a comparison between disordered and ideal films in the presence of a current density of $J_{\mathrm{HM}}=30\mathrm{GA/}{\mathrm{m}}^2$.

Figure 6

Skyrmion trajectory for different applied current densities (a) without and (b) in the presence of a strain gradient. Both the results of micromagnetic simulations (symbols) and Thiele’s model (lines) are shown. Time evolution of the skyrmion radius is shown in the insets. (c) Threshold current for skyrmion annihilation as a function of the strain gradient.