Holographic imaging of the complex charge density wave order parameter

The charge density wave (CDW) in solids is a collective ground state combining lattice distortions and charge ordering. It is defined by a complex order parameter with an amplitude and a phase. The amplitude and wavelength of the charge modulation are readily accessible to experiment. However, accurate measurements of the corresponding phase are significantly more challenging. Here we combine reciprocal and real space information to map the full complex order parameter based on topographic scanning tunneling microscopy (STM) images. Our technique overcomes limitations of Fourier space based techniques to achieve distinct amplitude and phase images with high spatial resolution. Applying this analysis to transition metal dichalcogenides provides striking evidence that their CDWs consist of three individual unidirectional charge modulations whose ordering vectors are connected by the fundamental rotational symmetry of the crystalline lattice. Spatial variations in the relative phases of these three modulations account for the different CDW contrasts often observed in STM topographic images. Phase images further reveal topological defects and discommensurations, a singularity predicted by theory for a nearly commensurate CDW. Such precise real space mapping of the complex order parameter provides a powerful tool for a deeper understanding of the CDW ground state whose formation mechanisms remain largely unclear.

Figure 1

Real space CDW pattern and dephasing parameter. (a) High resolution STM micrograph $(V_{\mathrm{bias}}=-100\mathrm{mV},I_t=1$ nA) of an in situ cleaved 1T-$\mathrm{VS}{\mathrm{e}}_2$ surface at 40 K. The atomic lattice and the CDW modulation are clearly resolved. (b) Spatial variation of the dephasing parameter as determined by the fitting procedure. The areas marked by red and blue squares have distinct dephasing parameters corresponding to the different appearances of the CDW. (c) and (d) are magnified areas corresponding to the red and blue squares in (a) respectively. They are connected by single scan lines, the different contrasts can thus not be blamed on different tip state configurations. (e) and (f) are zooms into the large Fourier filtered image of the CDWs at the same location as shown in (c) and (d) respectively. (g) and (h) are reconstructed images from the mosaics of the fitting procedure, in excellent agreement with (e) and (f). The topographic images are all plotted with the exact same color scale. Scale bars: 10 nm in (a) and (b) and 1 nm in (c)–(h).

Figure 2

Coexisting charge density waves. Mapping of the full complex order parameter in $1\mathrm{T}\text{−}{\mathrm{VSe}}_2$ for the same surface area as in Fig. 1. (a)–(c) The local amplitude of the CDWs in each direction $({\mathbf{q}}_1,{\mathbf{q}}_2,{\mathbf{q}}_3$ respectively) and (d)–(f) the corresponding local phase. For easier comparison of the spatial evolution of the local phases of each ${\mathbf{q}}_n$, they are referenced to the lower left corner, whose phase is arbitrarily set to zero. Scale bar: 10 nm. Red arrows are indicating tightly bound vortex-antivortex pairs.

Figure 3

Phase domains of the CDW in $1\mathrm{T}\text{−}{\mathrm{Cu}}_{0.02}{\mathrm{TiSe}}_2$ at 1.2 K. (a) High resolution STM micrograph of an in situ cleaved surface $(V_{\mathrm{bias}}=-200\mathrm{mV},I_t=100$ pA). (b) and (c) are selected magnified areas from (a) at locations marked by red and blue squares, respectively. Blue $\left({\mathbf{q}}_1\right)$, orange $\left({\mathbf{q}}_2\right)$, and yellow $\left({\mathbf{q}}_3\right)$ parallel lines highlight the shifted CDW modulations for each direction. (d) and (e) show the Fourier filtered CDW seen in (b) and (c). (f) and (g) are reconstructed images from the mosaics of the fitting procedure. Note the excellent agreement with (d) and (e). (h) Vectors representing the CDW $q$ vectors. (i)–(k) are the fitted phase images for the three directions $({\mathbf{q}}_1,{\mathbf{q}}_2,{\mathbf{q}}_3)$. The red and blue squares mark the same areas as in (a). In the red region, ${\mathbf{q}}_2$ and ${\mathbf{q}}_3$ show a phase shift [see (b)]. This is very well reflected in (i)–(k): within the red region ${\mathbf{q}}_1$ is homogeneous (no shift) while the color of ${\mathbf{q}}_2$ and ${\mathbf{q}}_3$ changes abruptly at the DW. Similarly, in the blue region ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$ shift, while ${\mathbf{q}}_3$ does not [see (c)]. This is very well shown in (i)–(k): ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$ show an abrupt color change at the DW in the blue region, but ${\mathbf{q}}_3$ remains uniform. (l)–(o) are polar histograms of the CDW phase (in degrees) corresponding to (l) blue area in (i); (m) blue area in (j); (n) red area in (j); and (o) red area in (k). Scale bars=10 nm in (a) and (i)–(k); scale bars  = 1 nm in (b)–(g).

Figure 4

STM imaging of crystalline defects, CDW discommensurations and singularities in $2\mathrm{H}\text{−}{\mathrm{NbSe}}_2$. (a) STM micrograph with atomic defects. (b) Phase map of the corresponding ${\mathbf{q}}_2$ CDW. (c) Phase profile extracted from the diamond to the circle along the red dashed line in (b). (d) STM micrograph of a defect-free region on the same sample. (e) Phase and (f) amplitude maps of the corresponding ${\mathbf{q}}_2$ CDW. Both STM micrographs were measured at 1.2 K on an in situ cleaved surface. Tunneling parameters: $V_{\mathrm{bias}}=100$ mV, $I_t=100$ pA. Scale bars = 10 nm.

Figure 5

CDW phase mapping around topological defects. (a) Phase profile around a weakly bound vortex-antivortex pair in ${\mathrm{NbSe}}_2$ and (b)–(d) around strongly bound pairs in ${\mathrm{NbSe}}_2$ and ${\mathrm{VSe}}_2$. The red and blue profiles were taken along the circles with the corresponding color in each phase map to their left. Note the opposite winding polarities of the vortices (in blue) and antivortices (in red). The corresponding STM micrographs for ${\mathrm{NbSe}}_2$ and ${\mathrm{VSe}}_2$ are shown in Figs. 4 and 1, respectively. Scale bar = 10 nm.