Topological Solitons versus Nonsolitonic Phase Defects in a Quasi-One-Dimensional Charge-Density Wave

We investigated phase defects in a quasi-one-dimensional commensurate charge-density wave (CDW) system, an In atomic wire array on Si(111), using low temperature scanning tunneling microscopy. The unique fourfold degeneracy of the CDW state leads to various phase defects, among which intrinsic solitons are clearly distinguished. The solitons exhibit a characteristic variation of the CDW amplitude with a coherence length of about 4 nm, as expected from the electronic structure, and a localized electronic state within the CDW gap. While most of the observed solitons are trapped by extrinsic defects, moving solitons are also identified and their novel interaction with extrinsic defects is disclosed.

Figure 1

(a) Schematic diagrams of three possible phase defects ($N=1$) on a $4\times 2$ CDW wire (solid ovals). Inset shows four degenerate CDW states in different colors for a single wire with two In atomic chains (black dots). The atomic structure is largely simplified [22]. (b) Schematics of possible phase defects: a phase slip defect (PSD, left), a phase flip defect (PFD) with the broken $8\times 2$ order (middle), and a line boundary of PFDs (right). Two red ovals indicate two out of four possible $8\times 2$ configurations at the given CDW orientation. STM images showing (c) two PSDs and (d) PFDs and composite defects with (e) a corresponding differential conductance map taken at $+0.15\mathrm{V}$ [22]. Two nearest normal CDW maxima are used to determine lengths of defects as denoted by slashes. The distinct common local features of the defects are marked by dots [(c)–(e)].

Figure 2

STM line profiles (dots) of (a) a long PSD and (c) a long PFD obtained along the dashed lines in Figs. 1c, 1d together with their best fits by the soliton shape with (red or gray) and without (green or light gray) considering the lattice modulation. The line profiles are high-pass filtered to get rid of slowly varying backgrounds. With the lattice modulation, the soliton shape is modified as $Z_S^{*}(x)=Z_S(x)+A_L\cos [2\pi (x-x_0)/a_0]$, where $A_L$ is the height modulation of underlying atoms. The inclusion of the lattice modulation results in a much better fit with almost the same coherence length. ${\xi }_{(\mathrm{a})}=3.75\pm 0.14(3.75\pm 0.17)\mathrm{nm}$ and ${\xi }_{(\mathrm{c})}=3.50\pm 0.23(3.40\pm 0.26)\mathrm{nm}$ with (without) the lattice modulation. (b), (d) Enlargements of (a) and (c) around the soliton center.

Figure 3

(a)–(c) A series of STM images taken consecutively with their start times indicated. Two defects indicated by white and black triangles are a reference and a hopping PSD, respectively. Sometimes, two abrupt CDW phase shifts ($0\to \pi \to 2\pi$) occur in a short time span as marked by a black arrow in the inset of (c). (d) Stacked STM line profiles (the newest line profile at topmost) taken along a dashed line in (c). Each black arrow indicates the two successive CDW phase shifts.

Figure 4

(a) Stacked STM line profiles showing the hopping of the PSD. Dotted lines and white arrows indicate the positions of the PSD and the hopping directions, respectively. (b) The STM profiles extracted from (a) are displayed in a time sequence, with the earliest on the top (0) and the latest on the bottom (11). Left (right) curves show the line profiles obtained by scanning from left (right) to right (left). The scan speed was set to 150 msec per line. Black triangles and vertical arrows indicate the positions of the PSD and abrupt CDW phase shifts, respectively. Gray boxes highlight emerging long PSDs or solitons. (c) Fitting of a soliton’s profile (dots) with a soliton solution (solid line, $\xi =3.56\pm 0.48\mathrm{nm}$). Note that raw data were used for fitting in contrast to the trapped solitons. (d) Schematics of the interaction between solitons ($S_{1,2}$) and a PSD. Black and white areas indicate different CDW states.