Low-frequency response of pinned charge-density-wave condensates

The frequency-dependent conductivity σ(ω) was investigated in the 10 Hz to -500 MHz range in materials with an incommensurate charge-density wave. ${\mathrm{NbSe}}_3$, orthorhombic ${\mathrm{TaS}}_3$, and (${\mathrm{TaSe}}_4$ ${)}_2$I. Over a wide range of frequencies both Reσ(ω) and Imσ(ω) are described by the expression σ(ω)=C(iω/ω¯${)}^{\alpha }$ with αω-dependent response is in clear disagreement with descriptions which neglect the internal degrees of freedom of the condensate. The excess low-frequency conductivity is due to the disorder caused by random distribution of pinning centers. The results are compared with calculations based on a microscopic phase Hamiltonian, which takes impurity pinning into account. The results are in semiquantitative agreement with a modified form of the Mott-Berezhinskii law for one-dimensional hopping conductivity, and they are in qualitative agreement with the classical, relaxational dynamics approach in the ω→0 limit. We also discuss the relation of our experimental findings to other studies of the frequency-dependent response in these materials.