  We show how the area law for the entanglement entropy may be violated by free
fermions on a lattice and look for conditions leading to the emergence of a
volume law. We give an explicit construction of the states with maximal
entanglement entropy based on the fact that, once a bipartition of the lattice
in two complementary sets $A$ and $\bar{A}$ is given, the states with maximal
entanglement entropy (volume law) may be factored into Bell-pairs (BP) formed
by two states with support on $A$ and $\bar{A}$. We then exhibit, for
translational invariant fermionic systems on a lattice, an Hamiltonian whose
ground state is such to yield an exact volume law. As expected, the
corresponding Fermi surface has a fractal topology. We also provide some
examples of fermionic models for which the ground state may have an
entanglement entropy $S_A$ between the area and the volume law, building an
explicit example of a one-dimensional free fermion model where $S_A (L) \propto
L^\beta$ with $\beta$ being intermediate between $\beta = 0$ (area law) and
$\beta = 1$ (BP-state inducing volume law). For this model, the dispersion
relation has a "zig-zag" structure leading to a fractal Fermi surface whose
counting box dimension equals, for large lattices, $\beta$. Our analysis
clearly relates the violation of the area law for the entanglement entropy of
the ground state to the emergence of a non-trivial topology of the Fermi
surface.
