Upper critical field for anisotropic superconductivity. A tight-binding approach

We study the problem of the upper critical field ($H_{c2}$) for tight-binding electrons in a two-dimensional lattice. The external magnetic field is introduced into the model Hamiltonian both via the Peierls substitution and the Zeeman term. Carrying out calculations for finite systems we analyze the influence of the external field in the commensurable and incommensurable case on an equal footing. The upper critical field has been discussed for intrasite as well as anisotropic intersite pairing that, in the absence of magnetic field, has a $d_{x^2-y^2}$ symmetry. A comparison of $H_{c2}$ determined for different symmetries shows that the on-site pairing is more affected by the external field i.e., the critical temperature for the on-site pairing decreases with the increase of the magnetic field faster than in the anisotropic case. Moreover, we have shown that the tight-binding form of the Bloch energy can lead to the upward curvature of $H_{c2}$, provided that the Fermi level is close enough to the van Hove singularity.