Effects of frequency mixing on Shapiro-step formations in sliding charge-density-waves

A one-dimensional charge density wave (CDW) is driven to slide by a dc electric field, carrying an electric current. In an additional ac field with frequency ${\omega }_{\mathrm{e}\mathrm{x}}$, it is known that the sliding CDW can be synchronized to ${\omega }_{\mathrm{e}\mathrm{x}}$, leading to the occurrence of Shapiro steps in the $I$-$V$ characteristics. Motivated by a recent experiment where ac fields with two frequencies ${\omega }_{\mathrm{e}\mathrm{x}}$ and ${\omega }_{\mathrm{e}\mathrm{x}}^{\mathrm{\prime }}$ are simultaneously applied, we theoretically investigate the effects of frequency mixing on the Shapiro-step formation. Based on the Fukuyama-Lee-Rice model, we show that in addition to the main steps induced by ${\omega }_{\mathrm{e}\mathrm{x}}$, satellite steps characterized by ${\omega }_{\mathrm{e}\mathrm{x}}^{\mathrm{\prime }}$ emerge. It is also found that with increasing the ac-field strength for ${\omega }_{\mathrm{e}\mathrm{x}}^{\mathrm{\prime }}$, each step width first exhibits a damped oscillation as in the one-frequency case, and then, exhibits a non-monotonic behavior. The origin of these behaviors and the relevance to the associated experiment are also discussed.