Archetypical "push the band critical point" mechanism for peaking of the density of states in three-dimensional crystals: Theory and case study of cubic H3S

The point of zero gradient of the electronic band structure, the critical point, generally induces the singularity in the density of states (DOS), but no isolated critical point yields strict divergence of the DOS in three dimensions, differently from the lower-dimensional cases. In view of the band structure as a smooth hypersurface on the reciprocal space, we discuss the minimal deformation of the band structure that yields nondivergent but large sharp DOS peaks in three dimensions. By "pushing down" the energy level at the second-order saddle point (maximum), a continuous closed loop of saddle points (sphere of maxima) encircling the original position of the saddle point (maximum) emerges, with which the DOS peak is formed. Being high-dimensional features, the saddle loop and extremum shell thus formed are difficult to locate with standard band structure analysis on linear k-point paths. The Lifshitz transition occurring over a linear or planar manifold is discussed as an indicator of such features. We also find that the celebrated DOS peak in the recently discovered superconducting hydride H3S originates from the saddle loop. On this basis, we successfully extract the minimal model that explains how the DOS peak is formed. Our theory characterizes a large class of DOS peaks sometimes found in the three-dimensional electronic structures, building a basis for profound understanding of their origins.