Holographic imaging of the complex charge density wave order parameter

TL;DR

This paper introduces a novel method combining reciprocal and real space data to map the full complex charge density wave order parameter in solids, revealing detailed phase and amplitude information with high spatial resolution.

Contribution

It presents a new technique that overcomes previous limitations to accurately measure both amplitude and phase of CDWs using STM images, enabling detailed real space analysis.

Findings

Identified three charge modulations in transition metal dichalcogenides with symmetry-connected ordering vectors.

Revealed spatial phase variations explain contrast differences in STM images.

Detected topological defects and discommensurations consistent with theoretical predictions.

Abstract

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.