Charge ordering and antiferromagnetic exchange in layered molecular crystals of theθtype

We consider the electronic properties of layered molecular crystals of the type $\theta -D_{2}A$ where A is an anion and D is a donor molecule such as bis-(ethylenedithia-tetrathiafulvalene) (BEDT-TTF), which is arranged in the $\theta$-type pattern within the layers. We argue that the simplest strongly correlated electron model that can describe the rich phase diagram of these materials is the extended Hubbard model on the square lattice at one-quarter filling. In the limit where the Coulomb repulsion on a single site is large, the nearest-neighbor Coulomb repulsion V plays a crucial role. When V is much larger than the intermolecular hopping integral t the ground state is an insulator with charge ordering. In this phase antiferromagnetism arises due to a novel fourth-order superexchange process around a plaquette on the square lattice. We argue that the charge ordered phase is destroyed below a critical nonzero value V, of the order of t. Slave-boson theory is used to explicitly demonstrate this for the $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ generalization of the model, in the large-$N$ limit. We also discuss the relevance of the model to the all-organic family ${\beta }^{″}-(\mathrm{BEDT}-\mathrm{TTF}{)}_2{\mathrm{SF}}_{5}Y{\mathrm{SO}}_3$ where $Y={\mathrm{CH}}_2{\mathrm{CF}}_2,$ ${\mathrm{CH}}_2,$ CHF.