An Effective Field Model to High Temperature Superconductors: Fitting of The Energy Gap to the Experimental Results of the Bi₂Sr₂CaCu₂O₈

We use an effective field model (transverse Ising model) to describe the dependence on the temperature of the energy gap of some two-dimensional $(2-D)$ superconducting systems. The order parameter of this model is put in a direct correspondence with that of the Ising model. Then, we use the exact relations for the spontaneous magnetization of the Ising model in some $2-D$ lattices as a means to fit the experimental results of $Bi_2Sr_2CaCu_2O_8 $ films obtained through reflectivity measurements by Brunel et al [Phys. Rev. Lett. 66, 1346 (1991)] and the argon-annealed $Bi_2Sr_2CaCu_2O_8$ samples investigated by Staufer et al [Phys. Rev. Lett. 68, 1069 (1992)] through Raman scattering. The zero temperature energy gap $E_g(0)$ relations were evaluated in various $2-D$ lattices and also in the $(3-D)$ simple cubic lattice, for comparison. In the square lattice case we obtained $E_g(0)=3.52kT_c$, coincidentally the same number as the BCS result, and in the triangular lattice case we get $E_g(0)=3.30kT_c$ in agreement with the experimental findings of Brunel et al.