Theory of surface-induced multiferroicity in magnetic materials, thin films, and multilayers

We present a theoretical study of the onset of electric polarization close to a surface in magnetic materials and in thin films and multilayers. We consider two different paths that lead to the onset of multiferroic behavior at the boundary in materials that are bulk collinear ferromagnets (or even antiferromagnets). These two paths are distinguished by the presence or absence of a surface-induced Dzyaloshinskii-Moriya interaction, which can be taken into account through Lifshitz invariants in the free energy of the system. Experimental consequences are discussed in light of the developed theory.

Figure 1

Polarization as a function of the distance from the surface.

Figure 2

Magnetization ($\varphi$) and polarization as a function of the distance from the surface for in-plane anisotropy with $n=0$ (a) and $n=4$ (b). The used parameters are $A=1, D=1, k=1, \gamma =1$, and ${\chi }_E=1.5$. The peak of the polarization is at a distance from the surface in agreement with Fig. 1.

Figure 3

Numerical results for $K=1$. (a),(b) $\theta$ as a function of $z$ and $\rho$ with boundary conditions as explained in the text. The surface are at $z=L$ and $z=-L$, while the bulk is at $z=0$. (c) The component of polarization $p_y$ as a function of $z$ and $\rho$, (d) $p_y$ at fixed $\rho =0.05\mathrm{R}$, and (e) at fixed $z=0.96L$ (and $L=4\lambda$), where it shows oscillatory behavior as a function of $\rho$ and $\varphi$ for the second layer from the surface. The $x$-component of the polarization is out of phase by $3\pi /2$ in $\varphi$. The rest of the parameters are the same as in Fig. 2.

Figure 4

Computational grid.