Mapping Strain and Spatial-Modulation Phase from Local Wave Vectors in Scanning Tunneling Microscopy

Abstract

Spatially varying lattice distortions can produce local changes in electronic properties that are obscured by measurements averaging over extended regions. With its atomic-scale spatial resolution, scanning tunneling microscopy (STM) provides, in principle, a means of quantifying such distortions locally. A widely used approach for extracting local strain from STM images is the Lawler--Fujita (LF) algorithm [Lawler et al., Nature 466, 347 (2010)], which determines the displacement field from the phase of the lattice modulation and subsequently obtains the strain by differentiation. However, this approach requires two-dimensional phase unwrapping, which can fail in regions where the phase is ill-defined, particularly near topological defects such as edge dislocations. Here, we introduce an alternative approach that circumvents these limitations by directly measuring the local wave vector in real space. We derive the relationship between the local wave vector, strain, and the phase gradient, and show that the local wave vector mapping (LWM) algorithm remains robust even when the phase itself is ill-defined. This makes the method particularly well suited for imaging the phase-gradient textures associated with vortices and other topological defects. We benchmark the algorithm using simulated data containing known lattice distortions and artificially generated vortices. Finally, we apply the method to experimental STM data and resolve the charge density wave (CDW) incommensurabilities, lattice distortions, and CDW phase vortices in 4Ha-NbSe2 and 2H-NbSe2. The physical implications of these observations were recently reported by S. Sajan, X. Yang, et al. [arXiv:2607.20335 (2026)]. The program package and the example usage is publicly available at the GitHub: https://github.com/jsntxyang/LocalWavevectorMeasuremnt.