Theory of the structural phase transition inA15compounds

The structural phase transition in the $A15$ compounds are investigated theoretically with the standpoint that the band Jahn-Teller effect of the twofold-degenerate ${\Gamma }_{12}$ subbands crossing the Fermi level is responsible for the instability. On the basis of $\overrightarrow{\mathrm{k}}·\overrightarrow{\mathrm{p}}$ perturbation theory, the ${\Gamma }_{12}$ subbands are revealed to be well described by two parabolic bands which couple not only to the bulk distortions, but also to the displacements of the ${\Gamma }_{12}$ optic modes. It is found that when the electron-lattice coupling exceeds the threshold of strength, the tetragonal phase with almost the same stabilities of $\frac{c}{a>\mathrm{}1\mathrm{}}$ and $\frac{c}{a<\mathrm{}1\mathrm{}}$ appears, accompanying one of the ${\Gamma }_{12}$ optic modes below a weak first-order phase transition temperature $T_M$. The temperature dependences of the elastic moduli are calculated; it is found that $c_{11}-c_{12}$ vanishes below $T_M$ while $c_{33}-c_{13}$ recovers from its softening partially or completely with decreasing temperature below $T_M$. The long-wavelength acoustic phonons are also investigated in order to clarify the relation between the phonon anomalies and the structural transition. The [110]${\mathrm{T}}_1$ mode ($\overrightarrow{\mathrm{q}}\parallel \mathrm{}\left[110\right]\mathrm{}$, $\overrightarrow{\mathrm{e}}\parallel \left[1\mathrm{}\overline{1}0\right]$) is considerably softened in the range $0<q\lesssim 2k_F$. This softening begins at high temperatures, remaining even at absolute zero. The theory explains successfully the various aspects of the phase transitions in ${\mathrm{V}}_3$Si and ${\mathrm{Nb}}_3$Sn. The comparison between them proves that the second-order Jahn-Teller effect occurs in both compounds.