First-principles study of lattice instabilities in the ferromagnetic martensite Ni₂MnGa

The phonon dispersion relations and elastic constants for ferromagnetic Ni$_2$MnGa in the cubic and tetragonally distorted Heusler structures are computed using density-functional and density-functional perturbation theory within the spin-polarized generalized-gradient approximation. For $0.9<c/a<1.06$, the TA$_2$ tranverse acoustic branch along $[110]$ and symmetry-related directions displays a dynamical instability at a wavevector that depends on $c/a$. Through examination of the Fermi-surface nesting and electron-phonon coupling, this is identified as a Kohn anomaly. In the parent cubic phase the computed tetragonal shear elastic constant, C$^\prime$=(C$_{11}-$C$_{12}$)/2, is close to zero, indicating a marginal elastic instability towards a uniform tetragonal distortion. We conclude that the cubic Heusler structure is unstable against a family of energy-lowering distortions produced by the coupling between a uniform tetragonal distortion and the corresponding $[110]$ modulation. The computed relation between the $c/a$ ratio and the modulation wavevector is in excellent agreement with structural data on the premartensitic ($c/a$ = 1) and martensitic ($c/a$ = 0.94) phases of Ni$_2$MnGa.