Is there any evidence that spin-orbit coupling (SOC) leads to a crystal lattice distortion? Using Quantum Espresso and DFT, I am getting different relaxed lattice constants when I take SOC into account and when I neglect it. Is this a bug or a feature? Are there any known materials where SOC leads to structural distortion?
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It's hard to tell without any information about the material, but it could be a feature. It is known that transition metal compounds can have an involved interplay between spin-orbit coupling and the Jahn-Teller (JT) effect. The paper S. V. Streltsov and D. I. Khomskii, "Jahn-Teller Effect and Spin-Orbit Coupling: Friends or Foes?", Phys. Rev. X 10, 031043 (2020) gives some general guidelines that depend on the electron configuration.
for some cases, such as $d^4$ and $d^5$, but also $d^2$ configurations, strong SOC suppresses JT distortion, whereas in other cases, notably $d^3$, SOC promotes or generates the JT effect in the configuration which is usually considered as "orbitally dead.”
The $d^4$ and $d^5$ states they consider are low-spin $t_{2g}$ states. They also have a recent follow-up paper: S. V. Streltsov, F. V. Temnikov, K. I. Kugel, and Daniel I. Khomskii, "Interplay of the Jahn-Teller effect and spin-orbit coupling: The case of trigonal vibrations", Phys. Rev. B 105, 205142 (2022).
