Charge Density Wave And The Analysis On Grüner's Equation

Charge density-waves is a type of electron collective behaviors in a condensed mode. In the CDW materials there are a lot of anomalous characters with it. Nonlinear conductivity, narrow-band noise, threshold field and so on are found by experiments. Grüner and his collaborators first proposed the single-particle model to explain them. In the model they gave a nonlinear differential equation to describing the behaviors of electronic transportation. Because this differential equation which is named as Grüner's equation doesn't have analytical solution, they derived the conductivity formula by over-damping approximation. Although this formula satisfied with the experiences very well in some degree, the derivative of conductivity is divergent at the threshold. This is contradictory with the result of experiment, but we think that the contradictory is due to neglecting the second order differential term of the equation.In this paper, we analyze the equation by the means of the rotating vector field theory. Using the mathematical analysis method we obtain the relation between the applied field E and the periodical solution of the equation. It is concluded that in the over-damping situation the E0 in Grüner's equation is just the critical value of the applied field E. we also obtain a conclusion that the equation has a unique and stable periodical solution when the applied field E is large than the critical value of the applied field E. According to the conclusion, it could be naturally derived that the sliding current of a segment of charge density wave obeys the Ohm's law. Sequentially we apply these conclusions with Portis's multiple segment model. We put forward a idea of elastic connection accounting for the connection of different segments. The theoretical result we derive from these for describing nonlinear conductivity of charge density wave is consistent with the experiential formula given by Fleming.