Dynamic converse magnetoelectric effect in ferromagnetic nanostructures with electric-field-dependent interfacial anisotropy

Magnetization oscillations excited by microwave voltages in ferromagnetic nanostructures having electric-field-dependent interfacial anisotropy are described theoretically. By calculating frequency dependencies of complex susceptibilities we show that this dynamic magnetoelectric effect acquires anomalous characteristics near thickness-induced spin reorientation transitions (SRTs) in nanolayers of cubic ferromagnets. Most importantly, the peak magnetoelectric susceptibilities may reach giant values exceeding ${10}^{-6}$ s/m, and the tunability of the resonance frequency of the magnetization precession by a dc voltage increases drastically near the critical thickness or voltage inducing an SRT.

Figure 1

Schematic representation of a magnetic nanostructure having the form of a ferromagnet-dielectric-metal trilayer connected to a voltage source.

Figure 2

Magnitudes of the angular deviations δϕ (curve 1) and δθ (curve 2) of the magnetization from the static orientation calculated as a function of frequency ν of the applied ac voltage. The voltage amplitude $\delta V_{\max }$ is taken to be 0.01 V, and the out-of-plane magnetic field $H_3$ equals 100 Oe. The CoFeB layer has the thickness of 1.629 nm and almost in-plane magnetization orientation (${\varphi }_0$ = 0, ${\theta }_0$ = 77°).

Figure 3

Frequency dependencies of the real (curve 1) and imaginary (curve 2) parts of the dynamic magnetoelectric coefficients ${\alpha }_{13}$ (a), ${\alpha }_{23}$ (b), and ${\alpha }_{33}$ (c) calculated for the CoFeB-MgO nanostructure subjected to the out-of-plane magnetic field $H_3$ = 100 Oe. The CoFeB layer has the thickness $t_f$ = 1.629 nm, negligible lattice strains, demagnetizing factors $N_{11}$ = $N_{22}$ = 0.008, $N_{33}$ = 0.984, $N_{12}$ = $N_{13}$ = $N_{23}$ = 0, and almost in-plane magnetization orientation (${\varphi }_0$ = 0, ${\theta }_0$ = 77°). The MgO thickness is assumed to be 0.8 nm.

Figure 4

Evolution of the ME susceptibility ${\alpha }_{13}$ with the CoFeB layer thickness near the size-induced SRT. The real (a) and imaginary (b) parts of ${\alpha }_{13}$ are plotted as functions of the frequency ν of the driving ac voltage. The thickness of the CoFeB layer subjected to the magnetic field $H_3$ = 100 Oe increases from $t_f^{*}$ = 1.624 nm (curve 1) to $t_f$  = 1.674 nm (curve 3) with the step $\delta t_f$ = 0.025 nm.

Figure 5

Evolution of the ME coefficient ${\alpha }_{23}$ with the CoFeB layer thickness near the size-induced SRT. The real (a) and imaginary (b) parts of ${\alpha }_{23}$ are plotted as functions of the frequency ν of the driving ac voltage. The thickness of the CoFeB layer subjected to the magnetic field $H_3$ = 100 Oe increases from $t_f^{*}$ = 1.624 nm (curve 1) to $t_f$ = 1.674 nm (curve 3) with the step $\delta t_f$ = 0.025 nm.

Figure 6

Evolution of the ME coefficient ${\alpha }_{33}$ with the CoFeB layer thickness near the size-induced SRT. The real (a) and imaginary (b) parts of ${\alpha }_{33}$ are plotted as functions of the frequency ν of the driving ac voltage. The thickness of the CoFeB layer subjected to the magnetic field $H_3$ = 100 Oe increases from $t_f^{*}$ = 1.624 nm (curve 1) to $t_f$ = 1.674 nm (curve 3) with the step $\delta t_f$ = 0.025 nm.

Figure 7

Peak values of the ME susceptibilities Re[${\alpha }_i$${}_3$(ν)] plotted as a function of the dc voltage applied to the CoFeB-MgO heterostructure. Panel (a) corresponds to the CoFeB layer with the thickness $t_f$ = 1.545 nm < $t_f^{*}$ subjected to the magnetic field $H_1$ = 100 Oe, whereas panel (b) shows the results obtained for the CoFeB thickness $t_f$ = 1.73 nm > $t_f^{*}$ at the field $H_3$ = 700 Oe. Curves 1, 2, and 3 show Re[${\alpha }_{13}$(${\nu }_{\mathrm{res}}$)], Re[${\alpha }_{23}$(${\nu }_{\mathrm{res}}$)], and Re[${\alpha }_{33}$(${\nu }_{\mathrm{res}}$)], respectively. The dashed line indicates the critical dc voltage $V^{*}\left(t_f^{}\right)$ inducing SRT.

Figure 8

Resonance frequency of the magnetization precession as a function of dc voltage applied to the CoFeB-MgO heterostructure. The thickness of the CoFeB nanolayer equals 1.579 nm ($t_f<t_f^{*}$) (a) and 1.629 nm ($t_f$ > $t_f^{*}$) (b). The applied magnetic field is $H_1$ = 100 Oe (a) and $H_3$ = 700 Oe (b). The dashed line here and below indicates the critical dc voltage $V^{*}\left(t_f\right)$ inducing SRT.

Figure 9

FWHM of the peak of Im[${\alpha }_{33}$(ν)] plotted as a function of dc voltage applied to the CoFeB-MgO heterostructure. The thickness of the CoFeB nanolayer equals 1.579 nm ($t_f<t_f^{*}$) (a) and 1.629 nm ($t_f$ > $t_f^{*}$) (b). The applied magnetic field is $H_1$ = 100 Oe (a) and $H_3$ = 700 Oe (b).

Figure 10

Tunability $|\partial {\nu }_{\mathrm{res}}/\partial E_3|$ of the resonance frequency as a function of dc voltage applied to the CoFeB-MgO heterostructure. The thickness $t_f$ of the CoFeB nanolayer equals 1.545 nm $(t_f<t_f^{*})$ (a) and 1.73 nm $(t_f>t_f^{*})$ (b). The applied magnetic field is $H_1$ = 100 Oe (a) and $H_3$ = 700 Oe (b).

Figure 11

Dependence of the energy density on the magnetization orientation in a CoFeB layer with the critical thickness $t_f^{*}(\mathbf{H}=0)$ = 1.604 nm.

Figure 12

Effect of the applied magnetic field $H_3$ on the orientation angle θ characterizing inclined magnetization orientation in a CoFeB layer with the critical thickness $t_f^{*}\left(H_3\right)$ (a) and the variation of $t_f^{*}$ with the field intensity (b). The vertical dashed line indicates the critical field intensity $H_3^{*}=770\mathrm{Oe}$ above which the SRT disappears.

Figure 13

Typical trajectories of the end of the unit vector m = $\mathbf{M}/M_s$ projected on the plane perpendicular to an equilibrium direction of M. The left panel corresponds to the unsmooth motion around the in-plane [100] direction induced by a sequence of unipolar voltage pulses with $V_{\max }>V^{*}\left(t_f^{}\right)$, whereas the right panel characterizes the magnetization precession around an inclined equilibrium direction driven by a weak ac voltage $V_{\mathrm{ac}}\ll V^{*}\left(t_f\right)$.