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

The density-of-states at the Fermi energy, $N(E_F)$, is low in doped superconducting semiconductors and high-$T_C$ cuprates. This contrasts with the common view that superconductivity requires a large electron-boson coupling $lambda$ and therefore also a large $N(E_F)$. However, the generic Fermi surfaces (FS) 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 $N(E_F)$. The typical behavior of the 2-dimensional FS for cuprates, and small 3-dimensional FS pockets in doped semiconductors and diamond, leads to $T_C$ variations as a function of doping in line with what has been observed. Diamond is predicted to attain higher $T_C$ from electron doping than from hole doping, while conditions for superconductivity in Si and Ge are less favorable. A high-$T_C$ material should ideally have few flat and parallel FS sheets with a reasonably large $N(E_F)$.