Imaging Polarity in Two Dimensional Materials by Breaking Friedel's Law

Pratiti Deb, Michael C. Cao, Yimo Han, Megan E. Holtz, Saien Xie, Jiwoong Park, Robert Hovden, David A. Muller

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


Friedel's law guarantees an inversion-symmetric diffraction pattern for thin, light materials where a kinematic approximation or a single-scattering model holds. Typically, breaking Friedel symmetry is ascribed to multiple scattering events within thick, non-centrosymmetric crystals. However, two-dimensional (2D) materials such as a single monolayer of MoS2 can also violate Friedel's law, with unexpected contrast between conjugate Bragg peaks. We show analytically that retaining higher order terms in the power series expansion of the scattered wavefunction can describe the anomalous contrast between hkl and hkl¯peaks that occurs in 2D crystals with broken in-plane inversion symmetry. These higher-order terms describe multiple scattering paths starting from the same atom in an atomically thin material. Furthermore, 2D materials containing heavy elements, such as WS2, always act as strong phase objects, violating Friedel's law no matter how high the energy of the incident electron beam. Experimentally, this understanding can enhance diffraction-based techniques to provide rapid imaging of polarity, twin domains, in-plane rotations, or other polar textures in 2D materials.

Original languageEnglish (US)
Article number113019
StatePublished - Aug 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Instrumentation
  • Atomic and Molecular Physics, and Optics


  • 2D Materials
  • Diffraction
  • Non-centrosymmetric
  • Pixelated detector
  • Polar


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