ac-dc interference, complete mode locking, and origin of coherent oscillations in sliding charge-density-wave systems

We present the results of a detailed experimental study of ac-dc interference phenomena in charge-density-wave (CDW) transport. The dc I-V characteristic of high-quality ${\mathrm{NbSe}}_3$ crystals, measured in the presence of an applied ac voltage, exhibits a remarkable array of Shapiro steps, including at least 150 subharmonic steps between the dc threshold and the 1/1 harmonic step. The harmonic and low-order subharmonic steps mode lock completely for ac frequencies between 100 kHz and 300 MHz and for ac amplitudes from a fraction of the dc threshold voltage $V_T$ to 75$V_T$. The form and magnitude of the Bessel-like oscillations of the harmonic step widths with ac amplitude are independent of applied frequency at frequencies above 20 MHz, demonstrating conclusively that periodic CDW pinning by impurities does not vanish at high fields and frequencies. Surprisingly, even though CDW’s are randomly pinned many-degree-of-freedom systems, single-coordinate models reproduce nearly all qualitative features of the Shapiro steps. A comparison between our interference measurements and our earlier coherent oscillation (‘‘narrow-band noise’’) measurements yields strong evidence that both effects are of bulk origin. We also have observed mode locking in the frequency domain by measuring the coherent oscillation spectrum in the presence of an applied ac voltage. When mode locking occurs, the spectral width of the fundamental peak collapses dramatically, indicating that the CDW’s motion becomes almost perfectly periodic with no fluctuations. However, the amplitude of the fundamental does not change, suggesting that phase correlations are not dramatically enhanced. A detailed understanding of these results should yield important insights into the dynamics of charge-density waves and of other many-degree-of-freedom systems.