First-principles study of lattice instabilities in ferromagnetic ${\mathrm{Ni}}_2\mathrm{MnGa}$

The phonon-dispersion relations and elastic constants for ferromagnetic ${\mathrm{Ni}}_2\mathrm{MnGa}$ 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 $0.9<c/a<\mathrm{}1.06,$ the ${\mathrm{TA}}_2$ transverse- acoustic branch along $\left[110\right]$ and the symmetry-related directions exhibit a dynamical instability at a wave vector that depends on $c/a.\mathrm{}$ 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 $\left[110\right]$ modulation. The computed relation between the $c/a$ ratio and the modulation wave vector is in excellent agreement with structural data on the premartensitic $\mathrm{}(c/a=1\mathrm{})\mathrm{}$ and martensitic $\mathrm{}(c/a=0.94\mathrm{})\mathrm{}$ phases of ${\mathrm{Ni}}_2\mathrm{MnGa}.$