Individual versus collective pinning in charge-density-wave systems with impurties

A phenomenological model is proposed here to show that the pinning of the charge-density-wave (CDW) phase by individual impurities yields an energy gain even for very small values of the Fukuyama-Lee-Rice (FLR) parameter. In three-dimensional systems the individual pinning is energetically more favored than the collective (weak in the FLR sense) pinning in all situations that can be described within a phase-only theory. The present results agree well with those of numerical simulation. Unlike the collective pinning, the individual pinning can account for the +2${\mathbf{k}}_{\mathit{F}}$/-2${\mathbf{k}}_{\mathit{F}}$ asymmetry of the x-ray satellite lines recently reported in many CDW compounds. In three dimensions, two regimes are found for the depinning by an applied electric field, characterized by a different dependence of the threshold field on the CDW gap and CDW-impurity coupling strength. The extension to the spin-density wave is straightforward.