Elastic soft mode and charge ordering of ${\mathrm{Yb}}_4{\mathrm{As}}_3$

The elastic constant of $C_{44}$ in ${\mathrm{Yb}}_4{\mathrm{As}}_3$ relating to the transverse ultrasonic mode shows a pronounced softening around the structural phase transition point at $T_c=292\mathrm{}$ K associated with the charge ordering of ${\mathrm{Yb}}^{3+}$ and ${\mathrm{Yb}}^{2+}$ states. The group theoretical analysis leads to the charge fluctuation mode with ${\Gamma }_5$ symmetry coupled to the elastic strain ${\varepsilon }_{\mathrm{yz}},{\varepsilon }_{\mathrm{zx}},{\varepsilon }_{\mathrm{xy}}$ of the soft $C_{44}$ mode. In a framework of the Landau phenomenological theory for the phase transition, the order parameter $Q_{\mathrm{yz}}{=Q}_{\mathrm{zx}}{=Q}_{\mathrm{xy}}\ne 0$ below $T_c$ gives rise to the linear chain of ${\mathrm{Yb}}^{3+}$ ions among the body diagonal [111] direction as well as the trigonal distortion of ${\varepsilon }_{\mathrm{yz}}={\varepsilon }_{\mathrm{zx}}={\varepsilon }_{\mathrm{xy}}\ne 0.$ Furthermore, this theory explains experimental observations of the elastic softening $C_{44}{=C}_{44}^0\left(T-T_c^0\right)/\left(T-\Theta \right)$ above $T_c,$ the structural change from cubic phase of $T_d^6$ to trigonal phase of $C_{3v}^6,$ and the first-order nature of the transition.