Third-order Electrical Conductivity of the Charge-ordered Organic Salt $\alpha$-(BEDT-TTF)$_2$I$_3$

We performed third-order electrical conductivity measurements on the organic conductor $\alpha$-(BEDT-TTF)$_2$I$_3$ using an ac bridge technique sensitive to nonlinear signals. Third-order conductance $G_3$ is clearly observed even at low electric fields, and interestingly, $G_3$ is critically enhanced above the charge-order transition temperature $T_{\rm CO}=136$~K. The observed frequency dependence of $G_3$ is incompatible with a percolation model, in which a Joule heating in a random resistor network is relevant to the nonlinear conduction. We instead argue the nonlinearity of the relaxation time according to a phenomenological model on the mobility in materials with large dielectric constants, and find that the third-order conductance $G_3$ corresponds to the third-order electric susceptibility $\chi_3$. Since the nonlinear susceptibility is known as a probe for higher-order multipole ordering, the present observation of the divergent behavior of $G_3$ above $T_{\rm CO}$ reveals an underlying quadrupole instability at the charge-order transition of the organic system.