Electric-field-induced magnetic easy-axis reorientation in ferromagnetic/ferroelectric layered heterostructures

A phenomenological model is presented for the electric-field-induced reorientation of magnetic easy axis in ferromagnetic (FM)/ferroelectric (FE) layered heterostructures, where the FM layers include either polycrystalline or (001) preferentially oriented films of widely used Fe, ${\text{CoFe}}_2{\text{O}}_4$ (CFO), Ni, and ${\text{Fe}}_3{\text{O}}_4$. The magnetic easy axes could rotate within the film plane at relatively low applied electric fields in CFO, Ni, and ${\text{Fe}}_3{\text{O}}_4$ films. As the applied electric field increases, the easy axes tend to switch from an in-plane direction to an out-of-plane one in CFO and Ni films. Such electric-field-induced easy-axis reorientations (EARs) can be abrupt and gradual, defined as first-order and second-order transitions, respectively. The influences of residual strain, different electric-field orientations, and nonlinear strain vs electric-field behavior of the FE substrates, on the EARs are also addressed.

Figure 1

(Color online) FM/FE heterostructure used in the model. The arrows denote the orientation of magnetization (polarization) in the FM (FE) layer. An out-of-plane polarization $P_3$ or in-plane polarization $P_1$ can be generated by an external electric field applied along the out-of-plane (i.e., longitudinal electric field $E_3$) or in-plane (i.e., transverse electric-field $E_1$) direction. Vector diagram of an out-of-plane EAR from $\overrightarrow{O{M}_0}$ to $\overrightarrow{OM}$ is shown on the right-hand side of the panel.

Figure 2

(Color online) Schematic of an electric-field-induced in-plane EAR by a transverse electric field $E_1$. The crossed arrows (yellow) in the upper layer and rightward arrows (red) in the bottom layer denote the orientations of magnetization and polarization, respectively.

Figure 3

(Color online) (a) Free energy of (001)-oriented Fe films as a function of the direction cosine $m_3$ at $E_3=0$ and $E_3=3\text{MV}/\text{cm}$. Direction cosine $m_3$ of the magnetic easy axis in (b) (001)-oriented and (c) polycrystalline Fe films as a function of the electric field in the $E_3$ and $E_1$ modes at ${\epsilon }_0=0$. (d) Dependence of $E^{cr}$ on ${\epsilon }_0$ in the $E_3$ and $E_1$ modes in (001)-oriented Fe films. The FE layer is PZN-PT. The arrows within ellipses denote the magnetic easy axis.

Figure 4

(Color online) $m_3$ of the magnetic easy axis in (a) (001)-oriented and (c) polycrystalline CFO films as a function of electric field in the $E_3$ and $E_1$ modes at ${\epsilon }_0=0$. Dependence of $E^{cr}$ on ${\epsilon }_0$ in the $E_3$ and $E_1$ modes in (b) (001)-oriented and (d) polycrystalline CFO films. The FE layer is PZN-PT. The arrows within ellipses denote the magnetic easy axis.

Figure 5

(Color online) $m_3$ of the magnetic easy axis in (a) (001)-oriented and (c) polycrystalline Ni films as a function of electric field in the $E_3$ and $E_1$ modes at ${\epsilon }_0=0$. Dependence of $E^{cr}$ on ${\epsilon }_0$ in the $E_3$ and $E_1$ modes in (b) (001)-oriented and (d) polycrystalline Ni films. The FE layer is PZN-PT. The arrows within ellipses denote the magnetic easy axis.

Figure 6

(Color online) $m_3$ of the magnetic easy axis in (a) (001)-oriented and (c) polycrystalline ${\text{Fe}}_3{\text{O}}_4$ films as a function of electric field in the $E_3$ and $E_1$ modes at ${\epsilon }_0=0$. Dependence of $E^{cr}$ on ${\epsilon }_0$ in the $E_3$ and $E_1$ modes in (b) (001)-oriented and (d) polycrystalline ${\text{Fe}}_3{\text{O}}_4$ films. The FE layer is PZN-PT.

Figure 7

(Color online) (a) In-plane piezostrains ${\epsilon }_{11}^p$ vs applied longitudinal electric field $E_3$ for a polarized PZT ceramic; the dashed lines indicates the smallest piezostrains required for the out-of-plane EAR in the (001) CFO films. (b) $m_3$ of the magnetic easy axis in the (001)-oriented CFO films at ${\epsilon }_0=0.05\%$ as a function of $E_3$.

Figure 8

(Color online) Electric-field-induced in-plane EAR: $m_1$ of the magnetic easy axis in (001)-oriented (a) Fe, (b) CFO, (c) Ni, and (d) ${\text{Fe}}_3{\text{O}}_4$ films as a function of a transverse electric field $E_1$ at ${\epsilon }_0=0$. The arrows within ellipses denote the magnetic easy axis.