Significant softening of c₆₆ and hidden order parameter in the magnetic ordered state of La_2-xCa_xCoO₄ (0.4≤x≤0.7)

We have measured the longitudinal and transverse sound velocities of La_2-xCa_xCoO₄ (0.4≤x≤0.7) propagating along [100], [110], and [001] from 200 K down to 3 K with a frequency of 10 MHz. It was found that c₆₆ shows a remarkable softening below Co ²⁺ spin-ordering temperature T_N in the whole doping range, while c₁₁-c₁₂/2 exhibits a clear hardening below Co ³⁺ spin-ordering temperature T_N2 only when x=0.5. To explain the observed anomaly, the Landau free energy for the La _2-xCa_xCoO₄ system has been constructed, including the coupling energy between elastic strains and a two-dimensional order parameter (Q₁, Q₂), having a spatial modulation along [110] or [1̄ 10]. We argued that this represents orbital order, and it is this order that leads to the softening of c₆₆.