Experimental Evidence for a Bragg Glass Density Wave Phase in a Transition-Metal Dichalcogenide

Analysis of the spatial dependence of current-voltage characteristics obtained from scanning tunneling microscopy experiments indicates that the charge density wave (CDW) occurring in ${\mathrm{NbSe}}_2$ is subject to locally strong pinning by a non-negligible density of defects, but that on the length scales accessible in this experiment the material is in a “Bragg glass” phase where dislocations and antidislocations occur in bound pairs and free dislocations are not observed. An analysis based on a Landau theory is presented showing how a strong local modulation may produce only a weak long range effect on the CDW phase.

Figure 1

(a) A topographic image of a $\sim 32\mathrm{nm}\times 32\mathrm{nm}$ region of ${\mathrm{NbSe}}_2$ at $22\mathrm{K}<T_c=34\mathrm{K}$ taken under conditions of constant sample-tip current and bias voltage. The heavy white spots are the strong pinning centers. (b) Delaunay analysis of the image. Diamonds (squares) represent CDW maxima with more (fewer) than six edges. Impurity locations are indicated by the circles. Thick broken lines that do not close indicate that some impurities create dislocations. Larger sized thick solid loops indicate that on larger length scales there are no free dislocations. (c) Main panel is the phase configuration of the phase ${\varphi }_1$ of one of the CDW components displayed over the same field of view as shown in (a) and (b). The right inset shows a typical smooth modulation near an impurity. The left inset shows a typical vortex-antivortex pair near an impurity. $X$’s are the locations of impurities. The lower panels of the insets are the phase profile along the dotted line.

Figure 2

Autocorrelations of the CDW component $\delta \rho$ parallel to and perpendicular to a CDW wave vector ${\overrightarrow{Q}}_1$, and of the amplitudes $\eta$.

Figure 3

Typical phase configurations obtained from Eq. (5) when $\epsilon =0$ in (a) two dimensions and (b) three dimensions. The lattice constant is $\xi \sim 1\mathrm{nm}$, and the linear system size is $31\xi$. The impurity concentration is $n_{\mathrm{imp}}\simeq 0.4\%$.