Tuning Incommensurate Charge Order in Ba${}_{1-x}$Sr${}_x$Al${}_4$ and Ba${}_{1-y}$Eu${}_y$Al${}_4$

Abstract

The BaAl${}_4$-type structure family is home to a vast landscape of interesting and exotic properties, with descendant crystal structures hosting a variety of electronic ground states including magnetic, superconducting and strongly correlated electron phenomena. BaAl${}_4$ itself hosts a non-trivial topological band structure, but is otherwise a paramagnetic metal. However, the other members of the $A$Al${}_4$ family ($A$= alkali earth), including SrAl${}_4$ and EuAl${}_4$, exhibit symmetry-breaking ground states including charge density wave (CDW) and magnetic orders. Here we investigate the properties of the solid solution series Ba${}_{1-x}$Sr${}_x$Al${}_4$ and Ba${}_{1-y}$Eu${}_y$Al${}_4$ using transport, thermodynamic and scattering experiments to study the evolution of the charge-ordered state as it is suppressed with Ba substitution to zero near 50% substitution in both systems. Neutron and x-ray diffraction measurements reveal an incommensurate CDW state in SrAl${}_4$ with $c$-axis-oriented ordering vector (0, 0, 0.097) that evolves with Ba substitution toward a shorter wavelength. A similar progression is observed in the Ba${}_{1-y}$Eu${}_y$Al${}_4$ series that also scales with the ordering temperature, revealing a universal correlation between charge-order transition temperature and ordering vector that points to a critical wavevector that stabilizes CDW order in both systems. We study the evolution of the phonon band structure in the Ba${}_{1-x}$Sr${}_x$Al${}_4$ system, revealing the suppression of the CDW phase matches the suppression of a phonon instability at precisely the same momentum as observed in experiments, confirming the electron-phonon origin of charge order in this system.