Scanning tunneling microscopy at the ${\text{NbSe}}_3$ surface: Evidence for interaction between ${\mathbf{q}}_{\mathbf{1}}$ and ${\mathbf{q}}_{\mathbf{2}}$ charge density waves in the pinned regime

We have investigated both ${\mathbf{q}}_{\mathbf{1}}$ and ${\mathbf{q}}_{\mathbf{2}}$ charge density wave (CDW) states taking place in ${\text{NbSe}}_3$ by means of low-temperature scanning tunneling microscopy (STM) under ultrahigh vacuum on the in situ cleaved $\left(\mathit{b},\mathit{c}\right)$ surface. High-resolution topographical images with atomic lattice resolution were obtained in the temperature range between 5 and 140 K. The careful and thorough analysis of the dependence of the STM images on bias polarity, energy, and temperature allowed us to identify unambiguously the three different types of chains composing the ${\text{NbSe}}_3$ unit cell at all temperatures, resolving contradictions from previous STM results. From two-dimensional Fourier transform of the STM images, we show that at the surface plane both CDW’s wave vectors are in very good agreement with bulk reported values projected on the $\left(\mathit{b},\mathit{c}\right)$ plane. The ${\mathbf{q}}_{\mathbf{1}}$ CDW has, for wave vector, ${\mathbf{q}}_{\mathbf{1}}=0.24{\mathit{b}}^{\ast }$. Spatially, the ${\mathbf{q}}_{\mathbf{1}}$ modulation is essentially developed on type III chains with a weak contribution on type II neighboring chains. The ${\mathbf{q}}_{\mathbf{2}}$ CDW has, for wave vector, the projected value ${\mathbf{q}}_{\mathbf{2}\mathit{p}}=0.26{\mathit{b}}^{\ast }+0.5{\mathit{c}}^{\ast }$. This modulation is mainly developed on type I chains but surprisingly has an important contribution on type III chains with an amplitude similar to the ${\mathbf{q}}_{\mathbf{1}}$ contribution on these chains. This simultaneous double modulation on chain III leads to a beating phenomenon between the ${\mathbf{q}}_{\mathbf{1}}$ and ${\mathbf{q}}_{\mathbf{2}\mathit{p}}$ periodicities and gives rise to a new domain superstructure developed along the chain axis which is characterized by the vector $\mathit{u}=2\times \left({\mathbf{q}}_{\mathbf{2}\mathit{p}}-{\mathbf{q}}_{\mathbf{1}}\right)-{\mathit{c}}^{\ast }=2\times \left(0.26-0.24\right){\mathit{b}}^{\ast }$. We believe that these new features give a clue of the coupling between the ${\mathbf{q}}_{\mathbf{1}}$ and ${\mathbf{q}}_{\mathbf{2}}$ CDWs in the pinned regime. Whereas most studies investigated the various characteristics of both CDWs by probing the Nb atoms properties, our results are consistent with the interpretation according to which the electronic local density of states probed by STM is mostly that of the surface Se atoms.

Figure 1

(Color online) Projection of the unit cell of ${\text{NbSe}}_3$ perpendicular to the chain axis $\mathit{b}$. It consists of three pairs of chains labeled I, II, and III. Small and large circles denote Nb and Se atoms, respectively. Hatched Se atoms are lying at $z=0$ and full Se atoms at $z=b/2$. For each type of chain, the Nb atoms are displaced by $\mathit{b}/2$ from the Se atoms. At the $\left(\mathit{b},\mathit{c}\right)$ plane, the chains I, II, and III are lying, exposing the Se atoms at the surface after cleavage.

Figure 2

(Color online) STM images of ${\text{NbSe}}_3$ $\left(\mathit{b},\mathit{c}\right)$ plane measured at 77 K on the same $7\times 7{\text{nm}}^2$ area, showing the dependency of the STM images on the polarity of the applied bias voltage. $I=100\text{pA}$. (a) $V_{bias}=+200\text{mV}$. Three types of chains can be identified, one of them carrying a strong ${\mathbf{q}}_{\mathbf{1}}$ CDW modulation and therefore identified as chain III. CDW modulation forms bright maxima along the chains, and the superlattice cell is indicated by a white rectangle. The two remaining chains are consistently identified as chains I and II. The transverse corrugation is of $0.2\text{Å}$ and the ${\mathbf{q}}_{\mathbf{1}}$ CDW corrugation is of $0.1\text{Å}$. (b) Change from $V_{bias}=+200\text{mV}$ in the bottom half of the image to $V_{bias}=-200\text{mV}$ in the upper half. There is a change of contrast on the chains II and only the chains I and III are clearly seen for $V_{bias}=-200\text{mV}$. Maxima of CDW hole and electron states present a $\pi$ phase shift (indicated by continuous lines), as expected for a quasi-one-dimensional system. (c) $V_{bias}=-200\text{mV}$. Only chains I and III are clearly visible. Transverse corrugation is of $0.4\text{Å}$ and ${\mathbf{q}}_{\mathbf{1}}$ CDW corrugation is of $0.1\text{Å}$.

Figure 3

(Color online) Dependence of the STM images on the probed energy of ${\text{NbSe}}_3$ $\left(\mathit{b},\mathit{c}\right)$ plane measured at 77 K on a $20\times 20{\text{nm}}^2$ area. Left images, top: $V_{bias}=+100\text{mV}$; bottom: $V_{bias}=-100\text{mV}$; right images, top: $V_{bias}=+250\text{mV}$; bottom: $V_{bias}=-250\text{mV}$. In the middle image, one can follow the change of polarity from $+100\text{mV}$ in the bottom, $-100\text{mV}$ in the median part, and $+100\text{mV}$ again in the top.

Figure 4

(Color online) STM image of the $\left(\mathit{b},\mathit{c}\right)$ plane of in situ cleaved ${\text{NbSe}}_3$ measured at 77 K. Scanned area: $20\times 20{\text{nm}}^2$. The atomic lattice corrugation is resolved and the corresponding unit cell is indicated together with the supercell of the ${\mathbf{q}}_{\mathbf{1}}$ CDW. $V_{bias}=+100\text{mV}$, $I=1\text{nA}$.

Figure 5

(Color online) 2D Fourier transform of the STM image is shown in Fig. 4. The lattice Bragg spots, corresponding to ${\mathit{b}}^{\ast }$ and ${\mathit{c}}^{\ast }$ reciprocal vectors are indicated by black arrows, and the ${\mathbf{q}}_{\mathbf{1}}$ CDW superlattice spots are indicated by purple (light) vectors. STM measurements lead to ${\mathbf{q}}_{\mathbf{1}}=0.24{\mathit{b}}^{\ast }$ at the surface, in very good agreement with bulk reported values. Weaker superlattice spots associated to the ${\mathbf{q}}_{\mathbf{2}}$ CDW are also visible at about 20 K above bulk transition temperature (see text).

Figure 6

(Color online) STM image of the $\left(\mathit{b},\mathit{c}\right)$ plane of in situ cleaved ${\text{NbSe}}_3$ measured at 5 K. $V_{bias}=+200\text{mV}$, $I=150\text{pA}$. Three types of chains are visible and the proposed identification of chains I, II, and III is consistent with our results obtained at 77 K shown in Figs. 2, 3, 4. The surface lattice unit cell and the ${\mathbf{q}}_{\mathbf{2}}$ CDW surface supercell are indicated by black and blue (gray) arrows (${\lambda }_2$ is the ${\mathbf{q}}_{\mathbf{2}}$ period along the chain axis). Surprisingly the ${\mathbf{q}}_{\mathbf{2}}$ CDW extends also on chains III making difficult the direct visualization of the ${\mathbf{q}}_{\mathbf{1}}$ CDW superlattice in the image (see text). However the 2D Fourier transform of this image shown in Fig. 7 clearly shows its existence.

Figure 7

(Color online) 2D Fourier transform of the STM image shown in Fig. 6. Lattice Bragg spots, corresponding to ${\mathit{b}}^{\ast }$ and ${\mathit{c}}^{\ast }$ reciprocal vectors are indicated by black arrows. ${\mathbf{q}}_{\mathbf{1}}$ and ${\mathbf{q}}_{\mathbf{2}\mathit{p}}$ CDW superlattice spots are indicated, respectively, by purple (light) and blue (gray) arrows. STM measurements yield ${\mathbf{q}}_{\mathbf{1}}=0.24{\mathit{b}}^{\ast }$ and ${\mathbf{q}}_{\mathbf{2}\mathit{p}}=0.26{\mathit{b}}^{\ast }+0.5{\mathit{c}}^{\ast }$ in very good agreement with bulk reported values projected on the $\left(\mathit{b},\mathit{c}\right)$ plane.

Figure 8

(Color online) Inverse Fourier transform of Fourier-filtered STM image. This image is obtained from the original STM image shown in Fig. 6 by multiplying the band-pass filter indicated in black and white below the IFT image by the Fourier transform shown in Fig. 7 (white corresponds to values equal to one and black to zero). This procedure allows to disentangle the various spatial frequencies present in the original image. This filter keeps only the atomic lattice and the ${\mathbf{q}}_{\mathbf{1}}$ CDW superlattice.

Figure 9

(Color online) Same as in Fig. 8 but with a filter keeping only the atomic lattice and the ${\mathbf{q}}_{\mathbf{2}}$ CDW superlattice.

Figure 10

(Color online) STM image measured at 5 K showing the same area as the one seen in Fig. 6 but showing here the occupied states. $V_{bias}=+200\text{mV}$, $I=150\text{pA}$. The inset in the bottom left part allows a detailed comparison with Fig. 6. It shows a smaller part of the same area, the bottom half being scanned with $V_{bias}=+200\text{mV}$, the upper half with $V_{bias}=-200\text{mV}$, highlighting the polarity dependence of chains II.

Figure 11

(Color online) 2D Fourier transform of the STM image shown in Fig. 10. Lattice Bragg spots, corresponding to ${\mathit{b}}^{\ast }$ and ${\mathit{c}}^{\ast }$ reciprocal vectors, together with ${\mathbf{q}}_{\mathbf{1}}$ and ${\mathbf{q}}_{\mathbf{2}\mathit{p}}$ CDW superlattice spots are indicated. The same CDW wave vectors are found than in Fig. 7, in very good agreement with bulk reported values projected on the $\left(\mathit{b},\mathit{c}\right)$ plane. New numerous satellite spots are observed including first, second, and third harmonics of ${\mathbf{q}}_{\mathbf{1}}$ and ${\mathbf{q}}_{\mathbf{2}\mathit{p}}$, as well as multiple combinations of ${\mathbf{q}}_{\mathbf{1}}$ and ${\mathbf{q}}_{\mathbf{2}\mathit{p}}$.

Figure 12

(Color online) Zoom of the central part of the Fourier transform shown in Fig. 11. ${\mathit{c}}^{\ast }$ reciprocal lattice vector together with the ${\mathbf{q}}_{\mathbf{1}}$ and ${\mathbf{q}}_{\mathbf{2}\mathit{p}}$ CDW wave vectors is indicated. Numerous superlattice peaks are visible, forming a superstructure involving both CDWs through the vector $\mathit{u}=2\times \left({\mathbf{q}}_{\mathbf{2}\mathit{p}}-{\mathbf{q}}_{\mathbf{1}}\right)-{\mathit{c}}^{\ast }$, indicated by orange (light) short lines.

Figure 13

(Color online) Inverse Fourier transform of Fourier-filtered STM images such as shown in Fig. 11 (sizes $20\times 20{\text{nm}}^2$ and $60\times 60{\text{nm}}^2$). The band-pass filter being used is indicated on the upper right side (white corresponds to values equal to one and black to zero). This filter selects the atomic lattice and the doublet of satellite peaks formed by ${\mathbf{q}}_{\mathbf{1}}$ and $2\times \left({\mathbf{q}}_{\mathbf{2}\mathit{p}}-0.5{\mathit{c}}^{\ast }\right)-{\mathbf{q}}_{\mathbf{1}}$. This procedure makes visible a domain superstructure developed on chains III, indicated by red arrows, which is hardly seen in original Fig. 10. The period is approximately defined by twice the difference in commensurability between the ${\mathbf{q}}_{\mathbf{1}}$ and ${\mathbf{q}}_{\mathbf{2}}$ CDW components along the chains, i.e., $\mathit{u}=2\times \left(0.26-0.24\right){\mathit{b}}^{\ast }=0.04{\mathit{b}}^{\ast }$.