content_image/331247.jpg content_image/331248.jpg content_image/331249.jpg content_image/331250.jpg content_image/331251.jpg
components. 

A fourth order Landau-Ginzburg density energy is assumed for the spontaneous ferro- electric polarisation: 

<img src='content_image/331247.jpg'>

The coefficient ζ$_{1}$ is temperature dependent and an external static applied electric field E$_{y}$ along spontaneous polarisation is included. 

The magnetic contribution to the density energy is given by: 

where λ and K represent magnetic exchange and anisotropy energies, ⃗ m$_{i}$ is the magnetisation of sub-lattice i and H$_{o}$ is an external static magnetic field. The first and second terms in the right hand side Eq. (2) represent the exchange and anisotropy energy, while the third term is Zeeman energy contributed by external magnetic field. In this configuration, the easy axis is out of plane. We assume that there is zero nett magnetisation along the ˆ z direction and do not consider demagnetisation fields. 

<img src='content_image/331248.jpg'>

The magneto-electric coupling is assumed to be of the Dzyaloshinkii-Moriya (DM) form, appropriate to materials such as FeTiO$_{3}$[18] and BaMnF$_{4}$[9–11]. We suppose a linear cou- pling of the form 

<img src='content_image/331249.jpg'>

where 

<img src='content_image/331250.jpg'>

and α is the magneto-electric coupling constant. The component M$_{x}$ is the transverse component of magnetisation and represents the weak ferromagnetism. The component L$_{z}$ is the magnitude of the longitudinal component of the magnetisation. The first term in Eq. (3) represents the polarisation induced Dzyaloshinkii-Moriya interaction in our model, and we assume that this term is responsible for generating canting in the magnetic sublattices. 

The canting angle is calculated by minimizing the magnetic and magnetoelectric free energies. Writing ∂ ∂θ ( F$_{M}$ + F$_{ME}$ ) = 0, we obtain the condition: 

<img src='content_image/331251.jpg'>

