Enhancement of superconductive critical temperatures in almost empty or full bands in two dimensions: Possible relevance to β-HfNCl, C₆₀ and MgB₂

We examine the possibility of enhancement of superconductive critical temperature in two-dimensions using the BCS theory. When the Fermi level is near an edge of a band the density of states (DOS) is not particularly high but k-dependent contribution to DOS (electron density on the Fermi surface) has a divergent peak at the bottom or top of the band (van Hove singularity). The attractive phonon-mediated interaction depends on q (momentum transfer) due to finite screening (non-local interaction). Enhancement of the critical temperatures is shown for the first time in this case. The results are qualitatively consistent with the superconductive behaviors of HfNCl (T_c ≤ 25 K) and ZrNCl (T_c ≤ 15 K), C₆₀ with a field-effect transistor configuration (T_c = 52 K), and MgB₂ (Tc ≈ 40 K) which have unexpectedly high critical temperatures.