Exploring the Charge-Ordering Transition in (TMTTF)$_2$X via Thermal Expansion Measurements

We report results of high-resolution measurements of the \emph{c$^*$}-axis expansivity ($\alpha_{c^{*}}$) at the charge-ordering (CO) transition for the quasi-1D (TMTTF)$_{2}$X compounds with X = SbF$_6$ and Br and make a comparison with previous results for the X = PF$_6$ and AsF$_6$ salts. For X = SbF$_6$, due to the screening of the long-range Coulomb forces, a sharp $\lambda$-type anomaly is observed at $T_{CO}$, which contrasts with the step-like mean-field anomaly at $T_{CO}$ for PF$_6$ and AsF$_6$, where CO occurs in the Mott-Hubbard charge-localized regime. For the latter two salts, a negative contribution to $\alpha_{c^{*}}$ is observed above $T_{CO}$. This feature is assigned to the anions' rigid-unit modes, which become inactive for $T$ $<$ $T_{CO}$. Our $\alpha_{c^{*}}$ results for the X = Br salt, where such rigid-unit modes are absent, reveal no traces of such negative contribution, confirming the model based on the anions' rigid-unit modes for the X = PF$_6$ and AsF$_6$ salts.