Charge-density-wave pinning and finite-size effects in NbSe3.

We present a detailed experimental study of the effects of finite crystal size, impurities, and temperature on the properties of charge-density waves (CDW's) in ${\mathrm{NbSe}}_{3}$. Finite-size effects in ${\mathrm{NbSe}}_{3}$ are controlled by the crystal thickness t. They are large in crystals of ordinary thickness, and remain signficant in high-purity crystals having thicknesses approaching 0.1 mm. In sufficiently thick Ta- and Ti-doped crystals, the threshold electric field ${\mathit{E}}_{\mathit{T}}$ for CDW depinning is independent of thickness and varies with the residual resistance ratio ${\mathit{r}}_{\mathit{R}}$ as ${\mathit{E}}_{\mathit{T}}$\ensuremath{\propto}${\mathit{r}}_{\mathit{R}}^{\mathrm{\ensuremath{-}}1.9}$ and ${\mathit{r}}_{\mathit{R}}^{\mathrm{\ensuremath{-}}1.4}$, respectively. In thin crystals, ${\mathit{E}}_{\mathit{T}}$ increases with decreasing thickness as ${\mathit{E}}_{\mathit{T}}$=K/t, where K is roughly proportional to the impurity concentration. ${\mathit{E}}_{\mathit{T}}$ is strongly temperature dependent. The fractional increase in ${\mathit{E}}_{\mathit{T}}$ as T\ensuremath{\rightarrow}0 is independent of thickness but decreases rapidly with increasing impurity concentration. The divergence of ${\mathit{E}}_{\mathit{T}}$ near ${\mathit{T}}_{\mathit{P}2}$=59 K is described by ${\mathit{E}}_{\mathit{T}}$\ensuremath{\propto}(1-${\mathit{R}}_{\mathit{H}}$/${\mathit{R}}_{\mathit{L}}$${)}^{\mathrm{\ensuremath{-}}0.95}$, where ${\mathit{R}}_{\mathit{H}}$ and ${\mathit{R}}_{\mathit{L}}$ are the high-field and low-field resistances, respectively. In extremely thin crystals (t0.1 \ensuremath{\mu}m) near ${\mathit{T}}_{\mathit{P}2}$, the sharp threshold vanishes and nonlinear conduction occurs at arbitrarily small fields.Finite-size effects are observed in several other CDW properties. In thin crystals, the crossover frequency ${\mathrm{\ensuremath{\omega}}}_{\mathrm{co}}$ in the ac conductivity varies as ${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$, the low-frequency dielectric constant \ensuremath{\epsilon}(\ensuremath{\omega}=1 MHz) varies as t, and the width of the 1/1 Shapiro step varies as ${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$. Most ${\mathrm{NbSe}}_{3}$ crystals have irregular cross sections, in which the crystal thickness varies in a series of steps across the crystal width. Such crystals exhibit large ${\mathit{f}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ noise, complicated coherent oscillation spectra, and complicated mode-locking behavior, when compared with crystals with nearly rectangular cross sections. Analysis of these results indicates that (1) pinning of CDW's in Ta- and Ti-doped ${\mathrm{NbSe}}_{3}$ is weak; (2) the size dependence of CDW properties is due to a crossover in the dimensionality of the pinning from three to two dimensions which occurs when the crystal thickness becomes smaller than the CDW's bulk transverse phase-phase correlation length; (3) the vanishing of ${\mathit{E}}_{\mathit{T}}$ in thin crystals near ${\mathit{T}}_{\mathit{P}2}$ is due to thermally assisted depinning of the CDW; and (4) ${\mathit{f}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$-noise and complicated mode-locking behavior are a consequence of CDW velocity shear occurring along steps in crystal thickness. These results have significant implications for the study of nearly every aspect of CDW's in ${\mathrm{NbSe}}_{3}$.