Journal of Superconductivity: Incorporating Novel Magnetism, Vol. 13, No. 1, 2000 Dependence of the Gap Ratio on the Fermi Level Shift in a Van Hove Superconductor S. Kaskamalas, 1 B. Krunavakarn, 1 P. Rungruang, 2 and S. Yoksan 3 Received 5 January 1999 The dependence of the zero temperature gap to the critical temperature for s- and d-wave superconductors on the Fermi level shift () from the Van Hove singularity is studied within the BCS theory. Exact numerical calculations for the s and d gap ratios are carried out and approximate analytic expressions for the ratio are given. We find that the maximum gap ratio occurs at  = 0 and it decreases slowly with increasing , and that this behavior is symmetric with respect to . KEY WORDS: Gap ratio; s-, d-wave superconductor; Van Hove singularity; Fermi level shift. In a previous paper [1] we gave numerical results for the reduced gap ratio (R) for an s-wave Van Hove superconductor as a function of the Fermi level shift (). The results were obtained by numerical computa- tions of R from an integral gap equation. We showed that the BCS formalism with the Van Hove singular- ity (VHS) density of states cannot be used to explain the large gap ratio of high T c cuprates [2,3]. However, recent experiments [4,5] indicate that the supercon- ducting order parameter has the d-pairing symmetry and this remarkable observation is now endorsed by most of the current microscopic theories [6]. Further- more, the high-resolution, angle-resolved photoemis- sion (ARPES) measurements [7] and band structure calculations [8] show that there is a logarithmic VHS in the normal-state density of states. As an extension of our previous work, in this paper we will study the variations of the gap ratio for a d-wave Van Hove superconductor. In particu- lar, we will investigate the gap ratio variation with the shift of the Fermi level from the VHS. Exact numerical results will be given and approximate 1 Department of Physics, Chulalongkorn University, Bangkok 10300, Thailand. 2 Department of Physics, Ramkamhaeng University, Bangkok 10110, Thailand. 3 Department of Physics, Srinakharinwirot University, Bangkok 10110, Thailand. 33 0896-1107/00/0200-0033$18.00/0  2000 Plenum Publishing Corporation analytic expressions for the gap ratio will be derived. For completeness, an approximate formula for the gap ratio of an s-wave superconductor will also be given. Therefore, we assume the density of states to be of the form N() = N(0) ln  E F - (E F - )  (1) where  is the displacement of the Fermi level with respect to the VHS, and  is the quasi-particle energy, here we are concerned with small values of . This form of the density of states was also used in [9]. The BCS expression for the gap equation is  k  =  k' V k  k  ,  k  , 2 ( k' - E F ) 2 + 2 k' tanh  ( k' - E F ) 2 + 2 k' 2T (2) where all the symbols have their usual meanings. For the sake of simplicity, we assume that V k  k  ' = Vg() g('), if E F -  D  k ,  k'  E F +  D  (3) where V is the strength of the attractive interacting interaction, and g() = 1 or cos2 depending on whether the superconductor is an s- or d-wave one,  is the angle between the momentum k  of the