Microscopic theory of the martensitic transition inA−15compounds based on a three-dimensional band structure

A two-band tight-binding model of the electronic bands near the Fermi surface is formulated for the $A-15$ compounds, based on Mattheiss's result that the density of states at the Fermi surface is composed mainly of the ${\delta }_1(x^2-y^2)$ orbitals on the transition-metal sites. The model involves two tight-binding parameters (one intrachain and one interchain), one electron-phonon coupling constant and a bare elastic constant, and exhibits both an $X$-point Peierls gap and Jahn-Teller effects at a couple of saddle points in the bands, which are found to be jointly responsible for the instability of the electronic spectrum. For parameters determined by Mattheiss' calculation, and fits to the martensitic-transition temperature, specific heat at low temperatures, and $c_{11}-c_{12}$ above the transition, the model is able to predict the temperature variation of the susceptibility, distortion in the tetragonal phase, and the specific-heat jump at the transition. The results are in very good agreement with experiment for ${\mathrm{V}}_3$Si, but not so good for ${\mathrm{Nb}}_3$Sn, where because of the extremely short zero-temperature coherence length, lattice entropy, not included in the present model, could play a major part. A qualitative discussion of the softening of the shear modulus $c_{44}$, the marked softening of the [110] transverse [110] polarized phonon away from $\overrightarrow{\mathrm{k}}=0\mathrm{}$, and effects of non-transition-metal site alloying is included. Modifications of the highly successful Landau theory of the martensitic transition in $A-15$ compounds by McMillan and the author, based on the Gor'kov model, are discussed in light of the present model and found to lead to no essential changes in its results.