  A Hubbard model at half-filling on an anisotropic triangular lattice has been
proposed as the minimal model to describe conducting layers of
$\kappa-(BEDT-TTF)_2X$ organic materials. The model interpolates between the
square lattice and decoupled chains. The $\kappa-(BEDT-TTF)_2X$ materials
present many similarities with cuprates, such as the presence of unconventional
metallic properties and the close proximity of superconducting and
antiferromagnetic phases. As in the cuprates, spin fluctuations are expected to
play a crucial role in the onset of superconductivity. We perform a
weak-coupling renormalization-group analysis to show that a superconducting
instability occurs. Frustration in the antiferromagnetic couplings, which
arises from the underlying geometrical arrangement of the lattice, breaks the
perfect nesting of the square lattice at half-filling. The spin-wave
instability is suppressed and a superconducting instability predominates. For
the isotropic triangular lattice, there are again signs of long-range magnetic
order, in agreement with studies at strong-coupling.
