In-plane magnetic anisotropies in ${\mathrm{Fe}}_3{\mathrm{O}}_4$ films on vicinal MgO(100)

Ferromagnetic resonance was used to study the influence of vicinal (miscut) angle and film thickness on in-plane fourfold and uniaxial magnetic anisotropies in epitaxial ${\mathrm{Fe}}_3{\mathrm{O}}_4$ films grown on vicinal MgO(100) surfaces. The in-plane fourfold anisotropy constant $K_{4\parallel }$ is approximately the same for all films but the dominant in-plane uniaxial constant $K_{2\parallel }$ varies linearly with the inverse ${\mathrm{Fe}}_3{\mathrm{O}}_4$ layer thickness and approximately quadratically with the vicinal angle. A second, weaker, in-plane uniaxial term is evident for the film on a larger miscut (10°) substrate. The easy axis of the dominant in-plane uniaxial term is perpendicular to the step edges. The dominant in-plane uniaxial anisotropy has one term inversely proportional to the film thickness that is associated with anisotropy localized at the interface and a second term that is independent of film thickness; the latter may arise from the preferential alignment of antiphase boundaries with the step edges.

Figure 1

Schematic diagram showing one possibility of formation of a step induced APB. The large and small circles represent ${\mathrm{O}}^{2-}$ and $B$-site ${\mathrm{Fe}}^{3+}∕{\mathrm{Fe}}^{2+}$ ions, respectively. The boxes indicate the magnetite unit cell and show the change in structure as the APB is crossed.

Figure 2

Schematic diagram of (a) the vicinal MgO (100) substrate characterized by (011) terraces with steps parallel to $\left[0\overline{1}1\right]$ and (b) the film. $\widehat{n}$ is the normal to the film and $x^{\prime }$ and $y^{\prime }$ are the projections of [010] and [001], respectively, within the film plane. $\mathit{H}$ lies in the film plane.

Figure 3

The in-plane ferromagnetic resonance field, $H_R$, as a function of angle ${\varphi }_H^{\prime }$ for ${\mathrm{Fe}}_3{\mathrm{O}}_4$ on 2° miscut vicinal MgO(100) with film thicknesses (a) 70, (b) 45, and (c) $30\mathrm{nm}$. The solid lines are fits using Eq. (4) with parameters given in Table . Note that the curvatures at points $A$ and $B$ are very similar.

Figure 4

The in-plane ferromagnetic resonance field, $H_R$, as a function of angle ${\varphi }_H^{\prime }$ for $45\mathrm{nm}$ ${\mathrm{Fe}}_3{\mathrm{O}}_4$ on vicinal MgO(100) with miscut angles of (a) 2°, (b) 5°, and (c) 10°. The solid lines are fits using Eq. (4) with parameters given in Table .

Figure 5

The in-plane ferromagnetic resonance linewidth, $\Delta H_{\mathrm{p.p.}}$, as a function of angle ${\varphi }_H^{\prime }$ for films with $\alpha =2°$ and thicknesses, $d$, of (a) 70, (b) 45, and (c) $30\mathrm{nm}$; the lines are guides to the eyes. (d) $H_R$ versus angle ${\varphi }_H^{\prime }$ for the film with $\alpha =2°$, $d=45\mathrm{nm}$ with theoretical fit.

Figure 6

The in-plane ferromagnetic resonance linewidth, $\Delta H_{\mathrm{p.p.}}$, as a function of angle ${\varphi }_H^{\prime }$ for ${\mathrm{Fe}}_3{\mathrm{O}}_4$ films with $d=45\mathrm{nm}$ and vicinal angles, $\alpha$, of (a) 2°, (b) 5°, and (c) 10°; the lines are guides to the eyes. (d) $H_R$ versus ${\varphi }_H^{\prime }$ for the film with $d=45\mathrm{nm}$, $\alpha =5°$ with theoretical fit.

Figure 7

Dependence of the uniaxial anisotropy field $H_{2\parallel }$ on 1/film thickness; the line is a best fit with parameters given in the text. $\alpha =2°$.

Figure 8

Dependence of $H_{2\parallel }$ on vicinal angle $\alpha$ shown with (a) linear scales and (b) log scales. Shown are fits in (a) of $-H_{2\parallel }=4.8{\alpha }^2$ and in (b) of $\log \left(\mid H_{2\parallel }\mid \right)=1.89\log \alpha +0.80$. Each film has a thickness of $45\mathrm{nm}$.

Figure 9

The in-plane ferromagnetic resonance linewidth, $\Delta H_{\mathrm{p.p.}}$, as a function of angle ${\varphi }_H^{\prime }$ for ${\mathrm{Fe}}_3{\mathrm{O}}_4$ films with $d$, $\alpha$ values of (a) $45\mathrm{nm}$, 2°; (b) $45\mathrm{nm}$, 5°; (c) $45\mathrm{nm}$, 10°; and (d) $30\mathrm{nm}$, 2°. The lines show fits to $\Delta H_{\mathrm{p.p.}}=\Delta H_0+(d{H}_R∕d{\varphi }_U)\Delta {\varphi }_U$ with parameters given in the text.