Electron spectrum topology and giant singularities of the electron density of states in cubic lattices

The topology of energy surfaces in reciprocal space is studied in detail for simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc) lattices in the tight-binding approximation, taking into account hopping integrals $t$ and $t'$ between the nearest and next-nearest neighbor sites, respectively. It is shown that lines and surfaces formed by van Hove $\mathbf{k}$ points can arise at values $\tau = t'/t = \tau_\ast$ corresponding to a change in the surface topology. At a small deviation of $\tau$ from these special values, the spectrum near the van Hove line (surface) only slightly depends on $\mathbf{k}$. This corresponds to a giant effective mass proportional to $|\tau - \tau_\ast|^{-1}$ near several van Hove points. Singular contributions to the density of states near these special t values are analyzed and explicit expressions are obtained for the density of states in terms of elliptic integrals. It is shown that, in some cases, the maximum density of states is achieved at energies corresponding to $\mathbf{k}$ points in high-symmetry directions inside the Brillouin zone rather than at its edges. The corresponding contributions to electronic and magnetic characteristics are discussed, in particular, in application to itinerant weak magnets.