  The Casimir-Lifshitz interaction, a long-range force that arises between
solids and molecules due to quantum fluctuations in electromagnetic fields, has
been widely studied in solid-state physics. The degree of polarization in this
interaction is influenced by the dielectric properties of the materials
involved, which in turn are determined by factors such as band-to-band
transitions, free carrier contributions, phonon contributions, and exciton
contributions. Gapped metals, a new class of materials with unique electronic
structures, offer the potential to manipulate dielectric properties and,
consequently, the Casimir-Lifshitz interaction. In this study, we theoretically
investigate the finite temperature Casimir-Lifshitz interaction in
La$_3$Te$_4$-based gapped metal systems with varying off-stoichiometry levels.
We demonstrate that off-stoichiometric effects in gapped metals can be used to
control the magnitude and, in some cases, even the sign of Casimir-Lifshitz
interactions. We predict measurable corrections due to stoichiometry on the
predicted Casimir force between a La$_3$Te$_4$ surface and a gold sphere,
attached to an atomic force microscopy tip.
