Nanoscale phase-slip domain walls in the charge density wave state of the Weyl semimetal candidate ${\mathrm{NbTe}}_4$

The transition-metal tetrachalcogenides are a model system to explore the conjunction of correlated electronic states such as charge density waves (CDWs) with topological phases of matter. Understanding the connection between these phases requires a thorough understanding of the individual states, which for the case of the CDW in this system, is still missing. In this paper we combine phonon-structure calculations and scanning tunneling microscopy measurements of ${\mathrm{NbTe}}_4$ in order to provide a full characterization of the CDW state. We find that, at short range, the superstructure formed by the CDW is fully commensurate with the lattice parameters. Moreover, our data reveals the presence of phase-slip domain walls separating regions of commensurate CDWs in the nanoscale, indicating that the CDW in this compound is discommensurate at long range. Our results solve a long-standing discussion about the nature of the CDW in these materials and provide a strong basis for the study of the interplay between this state and other novel quantum electronic states.

Figure 1

(a) Crystal structure of ${\mathrm{NbTe}}_4$, with space group P4/mcc (SG. 124) projected along the (a) [001] and (b) [010] directions. The Te atoms with red borders in (a) and (b) are atoms forming a cleaved {010} surface. (c) Powder x-ray diffraction measurements of ground single crystals (black line), with diffraction peaks consistent with the P4/mcc space group (refinement in red). (d) Temperature dependence of the resistivity, normalized by the room-temperature value. The residual resistivity ratio (RRR) is 6.3, similar to previously reported studies [9, 11, 12]. (e) Derivative of the normalized resistivity curve in (d). Vertical dashed lines indicate changes in the temperature dependence of the resistivity, previously associated to CDW transitions.

Figure 2

(a) Scanning tunneling topography image of a cleaved ${\mathrm{NbTe}}_4$ single crystal. The CDW superstructure formed at the Te-terminated surface is observed. The area enclosed by the red square is magnified in (b) in which the unit cell of the superstructure (pink square), with size $2a\times 3c$, can be clearly identified. (c) Fourier transform of the image shown in (a). Different peaks have been highlighted by different-color circles. The peaks highlighted by the pink circles correspond to the CDW superstructure unit cell. The peaks highlighted by the green circles correspond to the distance between adjacent “cashews” (Te trimers) with opposite curvature, along the $c$ direction, that is, with central $q=(0,\frac{2}{3}{c}^{*})$. Theses peaks are split, as better seen in the inset. (d) Closeup view of STM image, presented using a color scale that highlights the asymmetry in the charge distribution of the cashews or Te trimers. The maximum of the scale (yellow colors) is shifted toward the right side for all cashews, independent of their orientation (facing up or down).

Figure 3

(a) Calculated phonon-dispersion curves along various paths between high-symmetry points in the P4/mcc Brillouin zone of ${\mathrm{NbTe}}_4$. The unstable modes with imaginary frequencies are presented by negative values on the plot. The dispersion line color has been assigned to each point according to the contribution of each kind of atom to the associated eigenvector (red for Nb and blue for Te). (b) Schematic picture of the eigendisplacements of the $V_4$ unstable phonon at the $q=\frac{1}{2},\frac{1}{2},\frac{1}{3}$) vector. The atom displacements are shown as red arrows in the high-symmetry P4/mcc phase. (c) Fully relaxed modulated P4/ncc structure is presented.

Figure 4

All figures show the Te atoms (dark-yellow balls) of three chains in a $\left\{010\right\}$ cleaved surface, as well as the closest Nb atoms (green balls) underneath this surface. (a) Spatial configuration of the Nb trimers in the final modulated structure as calculated by our DFT model. Three nonequivalent distances between the Nb atoms are found in the model, with two short ($d_1$ and $d_2$) and one long ($d_3$), therefore forming Nb trimers as represented by the green solid lines. The unit cell of the Nb-trimer superstructure is indicated by the pink rectangle. A-B-A refers to the stacking of the chains. (b) Representation of the Te-trimer formation, which leads to the cashew shapes observed in the topographic images. The unit cell of the Te-trimer superstructure is indicated by the pink rectangle, matching the unit cell of the Nb-trimer superstructure. (c) Red arrows in this figure indicate a simplified representation of the Nb atom displacements, which lead to the CDW modulation in this compound. The charge density is maximum around the center of each Nb trimer, which coincides with the right side of the cashews, generating their asymmetric charge density (gradient color fill). The charge density amplitude variation is depicted by the black waves. The wavelength of the CDW is $\lambda =3c/2$, and the phase shift between adjacent chains is $\pm 2\pi /3$.

Figure 5

(a) Wide-area STM topographic image of a cleaved ${\mathrm{NbTe}}_4$ single crystal showing different domain walls (some are highlighted by the yellow arrows) which run over long distances along the $a$ direction. (b) and (g) show topographic images taken at a reduced area (therefore with an improved resolution), in which two different types of domain walls can be recognized (see Fig. 8 for five other types of domain walls). (c) and (h) show schematic representations of the cashew configuration across the domain walls in (b) and (g), respectively. The cashews forming the domain walls are represented in red. The blue dashed lines represent the sign of the phase shift of the CDWs of two adjacent chains: a change in inclination represents a change of phase sign. The schematic representations in (c) and (h) capture the change of sign of this phase difference at the left and right side of the domain walls. (d) and (i) show the atomic configuration of the Te atoms at the surface (and the Nb atoms underneath), which lead to the domain walls of [(b), (c)] and [(g), (h)], respectively. [(e), (f), (j), (k)] show the FFT of the images in [(b), (c), (g), (h)], respectively. Red ovals in all FFTs highlight the $(0,2c^{*}/3)$ peaks, in which splitting is reproduced in the real image FFTs and the schematic representation FFTs. Blue ovals in the FFTs of [(j), (k)] highlight the $(a^{*}/2,2c^{*}/3)$ peaks, which are split for the second type of domain wall [(g)–(k)] but not for the first type [(b)–(f)]. Green ovals in the FFTs of [(j), (k)] highlight peaks along the horizontal axis and close to the origin, which are present for the second type of domain wall but not for the first type.

Figure 6

(a) Topographic image of ${\mathrm{NbTe}}_4$ in which the internal structure of the charge distribution within the cashews can be resolved. (b) FFT of the image in (a). The $(0,2c^{*}/3)$ split peaks are highlighted in red circles. (c) Expanded view of one of the cashews in (a), in which three maxima of the charge density can be clearly resolved (green circles). This is consistent with the presence of three Te atoms per cashew.

Figure 7

(a) Large topographic image of ${\mathrm{NbTe}}_4$ in which two smaller areas are selected and expanded. (b) Highlights the region enclosed by the green square in (a), in which no domain walls can be identified. (c) The FFT of the image in (b). No peak splitting is recognized for the $(0,2c^{*}/3)$ peaks circled in red. (d) Highlights the region enclosed by the red square in (a), in which a single domain wall can be identified. (e) The FFT of the image in (d) with a clear splitting in the $(0,2c^{*}/3)$ peaks circled in red.

Figure 8

Topographic images (left-column figures) around different domain walls found in our ${\mathrm{NbTe}}_4$ samples and their corresponding FFTs (right-column figures). Al FFTs show the splitting of the $(0,2c^{*}/3)$ peaks, circled in red.

Figure 9

Tunneling conductance vs bias voltage curve taken at $T$ = 1.7 K. A gaplike spectral feature of $2\mathrm{\Delta }\approx$ 24 meV can be observed, similar to recent results in ${\mathrm{TaTe}}_4$ [6].