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 earlier 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 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 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.

information from spectroscopic 3,4 and topographic 5 local probe tunneling experiments as well as from the periodic distortion of the atomic lattice measured by transmission electron microscopy 6,7 . However, extracting phase information from reciprocal-space data tends to suffer from the common problem of phase wrapping and fundamental numerical difficulties of finite sampling (see Supplementary Information Section 3). Unwrapping the phase in these cases leads to singularities that can easily be mistaken for physical phase variations that actually do not exist and it can also obscure real features, for instance domain walls in 1T-Cu x TiSe 2 . Combining reciprocal and real space scanning tunneling microscopy (STM) data to extract the full complex CDW order parameter enables to overcome these limitations.
The CDW state is characterized by a real-space periodic modulation of the charge density 8 : where is the constant background charge density in the metallic state, is the ordering vector and is the modulation amplitude which is proportional to the energy gap in the quasiparticle spectrum due to the formation of the CDW phase. The order parameter of the CDW state is a complex quantity: = ⋅ . In general, one expects its amplitude and phase to spatially vary ( = ( ) and = ( )), leading to collective excitations (amplitudons and phasons, respectively) and defects (domain walls, vortices, discommensurations, etc…) in the CDW condensate.
In this study, we focus on the complex CDW order parameter in three transition metal dichalcogenides (TMDCs) MSe 2 . They are layered materials where each slab is composed of a triangular metal layer M (Cu x Ti, Nb and V in the present study) sandwiched between two triangular selenium sheets. The layer stacking is maintained by weak van der Waals (vdW) forces allowing facile cleaving between the slabs exposing a triangular Se layer to the surface.
All three compounds studied here have a crystal structure with a three-fold symmetry in the ab-plane, regardless of their specific polytype (1T or 2H A rapid survey of the literature reveals that the CDW contrast in a given material varies significantly among published STM images, even in the absence of any structural defects.
While in several cases this can be explained with changing experimental conditions, either extrinsic, e.g. a change in the tip state, or intrinsic, e.g. bias dependence, we find compelling evidence of contrast variations which are real CDW features and not the consequence of particular tunneling conditions. In Fig. 1a, c-d we highlight two adjacent regions with a different contrast in the same STM micrograph. As we show in the following, these two contrasts can be reproduced straightforwardly by tuning the relative phases of the three CDWs discussed above. To quantify the changing configurations, we introduce a dephasing parameter defined as the sum of the three individual CDW phases modulo 2 : This dephasing parameter uniquely determines the general appearance of the CDW pattern.
Each value corresponds to a particular CDW imaging contrast, regardless of how individual phase shifts are distributed among the three . A more detailed description of the dephasing parameter is given in Supplementary Information Section 1.
In order to determine the phase of each required to compute the dephasing parameter, we developed a procedure based on the local -specific fitting of the real space CDW modulation measured by STM. We locally describe each CDW by a two-dimensional plane wave function ( ) = ( )cos ^ • + ( ) , where ^ is a unit vector in the direction of the n-th ordering vector. We consider a model where we allow the amplitude ( ( )) and the phase ( ( )) for each direction ( = 1,2,3) to vary spatially. The result is a complete local characterization of the CDW modulation.
To illustrate the fitting procedure described in details in Supplementary Information Section 2, we apply it to a large scale STM image of an in-situ cleaved 1T-VSe 2 single crystal measured at 40 K (Fig. 1). The atomic and the 4 × 4 CDW modulations are clearly resolved in the micrograph (Fig. 1a). The red and blue squares, magnified in Fig. 1c and d,   1T-VSe 2 for the same surface area as in Fig. 1a. a-c, The local amplitude of the CDWs in each direction (q 1 , q 2 , q 3 respectively) and, d-f, the corresponding local phase referenced to the lower left corner, whose phase is set to zero. Scale bar: 10 nm. Red arrows are indicating tightly bound vortex-antivortex pairs.
The ability to determine the local amplitude and phase of each CDW enables a much more thorough characterization of the CDW ground state in real space. Beyond explaining the variety of STM imaging contrasts and demonstrating the individual nature of the three CDWs developing in TMDCs, our fitting procedure reveals several intrinsic CDW properties. Of particular interest are singularities in the amplitude and phase, such as domain walls and topological defects (vortices). They are important to understand the interplay of ordered electronic phases 3,17 . For example, the model analysis discussed here provides a unique contrast mechanism to identify and locate domain walls, which have been proposed to promote superconductivity 18,19 .
We now focus on three selected CDW features to illustrate the augmented experimental phase space accessible by fitting the complex order parameter in real space. The first example is a detailed analysis of -phase shift CDW domain walls ( DWs) developing in TiSe 2 when intercalating either Cu 9 or large amounts of Ti 20 . These DWs are usually identified by eye, seeking for one atomic row shifts in the characteristic sequence of bright and dark atoms of the 2 × 2 CDW. Such domains can be easily found in Fig 3a. Their topographic structure is magnified in Fig. 3b and 3c corresponding to the red and blue outlined regions in Fig. 3a.
The real space CDW modulation in these two regions highlighted in the corresponding filtered images (Fig. 3d and 3e) is perfectly reproduced by our fitting procedure in Fig. 3f and 3g. Most instructive are the local phases of the three independent CDWs involved in the above fitting procedure (Fig. 3i-k). Each phase is essentially constant within any given domain and changes abruptly at the domain wall. An original insight of our analysis is that the domain walls do not necessarily correspond to a -shift as shown by the polar histograms in  h, vectors representing the CDW q-vectors. i-k, are the fitted phase images for the three directions (q 1 , q 2 , q 3 ). The red and blue squares mark the same areas as in a. In the red region, q 2 and q 3 undergo a phase shift (see panel b). This is very well reflected in panels i-k: within the red region q 1 is homogeneous (no shift) while q 2 and q 3 undergo an abrupt colour change.
Similarly, in the blue region q 1 and q 2 shift, while q 3 does not (see panel c). This is very well shown in i-k: q 1 and q 2 show an abrupt colour change in the blue region, but q 3 remains uniform. l-o, 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, i-k; Scale bars=1 nm in b-g.
As a second example, we address discommensurations (DCs), a particular type of defects associated with nearly commensurate CDW (NC-CDW) found for example in 2H-NbSe 2 or 2H-TaSe 2 . Discommensurations were first proposed in the seminal work of McMillan 14 and searched for, e.g in the NC-CDW phase of 1T-TaS 2 21 . DCs are domain walls where the order parameter phase is changing rapidly between small phase locked regions, which allow the CDW system to lower its energy. Our holographic analysis reveals precisely such a phase texture in the CDW images acquired on in-situ cleaved surfaces of 2H-NbSe 2 (Fig. 4a). Fig.   4b is a phase map of one of the three CDWs in the same field of view as Fig. 4a. It reveals areas of rather constant phase separated by nanometer-sized regions (domain walls) where the phase is changing rapidly. This staircase structure is best seen in a phase profile (Fig. 4c) taken along the red dashed line in Fig. 4b which perfectly mimics the prediction by  Fig 6a-c), whereas it is vanishing along the DWs (Suppl. Fig. 5a-c), suggesting the different nature and origin of these two CDW defects.
While DWs can be readily seen in topographic STM images, DCs can only be detected using our holographic fitting scheme. Note that in order to detect DCs, one has to control the phase wrapping in the fitting procedure. Otherwise, cumulative errors introduce an artificial phase gradient in the phase-locked domains delimited by the DCs, making them essentially undetectable (see Suppl.  analysis, is that a single atom defect can trigger a topological defect in the phase of one CDW, while only producing a smooth phase variation for another one (Fig. 2, Fig. 4a and 4b and Suppl. Fig. 6). While CDW defects are expected in the presence of structural defects, the holographic analysis discussed here reveals that the CDW order parameter landscape ( Fig. 4e and 4f) can be remarkably inhomogeneous on a 2H-NbSe 2 surface with no obvious structural defects (Fig. 4d). This is likely the result of discommensurations not readily detectable in the topographic STM images. It may also reflect 3D CDW correlations with defects below the surface not apparent in the STM micrograph. More generally, Fig. 4d-f suggests the ability of the CDW to develop a spatial structure essentially independent of the supporting crystalline lattice.