Theoretical aspects of simple and nested Fermi surfaces for superconductivity in doped semiconductors and high-TC cuprates

•The doping dependence of TC as a function of doping in doped semiconductors can be understood.•TC in electron doped diamond should be better than with holes.•The higher TC in the cuprate BCO compared to LCO might be due to less warping of their Fermi surfaces.

The density-of-states at the Fermi energy, N(EF)N(EF), is low in doped superconducting semiconductors and high-TC cuprates. This contrasts with the common view that superconductivity requires a large electron–boson coupling λ   and therefore also a large N(EF)N(EF). However, the generic Fermi surfaces (FSs) of these systems are relatively simple. Here is presented arguments showing that going from a 3-dimensional multi-band FS to a 2-dimensional and simple FS is energetically favorable to superconductivity. Nesting and few excitations of bosons compensate for a low N(EF)N(EF). The typical behavior of the 2-dimensional FS for cuprates, and small 3-dimensional FS pockets in doped semiconductors and diamond, leads to TC variations as a function of doping in line with what has been observed. Diamond is predicted to attain higher TC from electron doping than from hole doping, while conditions for superconductivity in Si and Ge are less favorable. A high-TC material should ideally have few flat and parallel FS sheets with a reasonably large N(EF)N(EF).