Controlling mesoscopic phase separation near electronic topological transitions via quenched disorder in ternary diborides

A phase separation driven by the negative compressibility of the electron gas, near electronic topological transitions (ETT), could drive the system at the verge of a catastrophe. We show here that the metastable phases very close to the ETT transition are observed in a mesoscopic phase separation (MePhS) driven by the quenched lattice disorder. By using high resolution synchrotron radiation x-ray powder diffraction we have identified the MePhS for the intermetallic ternary Mg_1-xAl_xB₂ in the proximity of two ETTs: the first at x₁ = 0.1 and the second at x ₂ = 0.3. We have identified the competition between a first 'relaxed' (R) hole poor and a second 'tense' (T) hole rich phase, and by micro-Raman we observe the splitting of the in-plane phonon E_2g mode in the proximity of the first ETT at x = 0.1. The anisotropic quenched disorder due to a random distribution of Al³⁺ and Mg²⁺ ions both in the axial (c axis) direction and planar (ab plane) direction, probed by x-ray diffraction and Raman data, is proposed to be the physical variable that allows the formation of metastable phases near the critical points of electronic topological transitions, where Feshbach shape resonances in interband pairing amplifies the superconducting critical temperature.