The ${\mathrm{BaAl}}_4$-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. ${\mathrm{BaAl}}_4$ itself hosts a nontrivial topological band structure, but is otherwise a paramagnetic metal. However, the other members of the $A{\mathrm{Al}}_4$ family ($A$ = alkali earth), including ${\mathrm{SrAl}}_4$ and ${\mathrm{EuAl}}_4$, exhibit symmetry-breaking ground states including charge density wave (CDW) and magnetic orders. Here we investigate the properties of the solid solution series ${\mathrm{Ba}}_{1-x}{\mathrm{Sr}}_x{\mathrm{Al}}_4$ and ${\mathrm{Ba}}_{1-y}{\mathrm{Eu}}_y{\mathrm{Al}}_4$ 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 ${\mathrm{SrAl}}_4$ 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 ${\mathrm{Ba}}_{1-y}{\mathrm{Eu}}_y{\mathrm{Al}}_4$ 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 wave vector that stabilizes CDW order in both systems. We study the evolution of the phonon band structure in the ${\mathrm{Ba}}_{1-x}{\mathrm{Sr}}_x{\mathrm{Al}}_4$ system, revealing that 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.

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• Charge density waves
• Electrical properties
• Phonons
• Topological materials