Spin-orbit coupling effects on spin-phonon coupling in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="normal">Cd</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">Os</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>7</mml:mn></mml:msub></mml:mrow></mml:math>

Abstract

Spin-orbit coupling (SOC) is essential in understanding the properties of $5d$ transition metal compounds, whose SOC value is large and almost comparable to other key parameters. Over the past few years, there have been numerous studies on the SOC-driven effects of the electronic bands, magnetism, and spin-orbit entanglement for those materials with a large SOC. However, how the SOC affects the lattice dynamics is less studied and remains an unsolved problem. We therefore measured the phonon spectra of $5d$ pyrochlore ${\mathrm{Cd}}_{2}{\mathrm{Os}}_{2}{\mathrm{O}}_{7}$ over the full Brillouin zone to address the question by using inelastic x-ray scattering. Our main finding is a visible mode dependence in the phonon spectra, measured across the metal-insulator transition at 227 K. We examined the SOC strength dependence of the lattice dynamics and its spin-phonon (SP) coupling, with first-principles calculations. Our experimental data taken at 100 K are in good agreement with the theoretical results obtained with the optimized $U=2.0\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ with SOC. By scaling the SOC strength and the $U$ value in the density-functional theory calculations, we demonstrate that SOC is more relevant than $U$ in explaining the observed mode-dependent phonon energy shifts with temperature. Furthermore, the temperature dependence of the phonon energy can be effectively described by scaling SOC. Our work provides clear evidence of SOC producing a non-negligible and essential effect on the lattice dynamics of ${\mathrm{Cd}}_{2}{\mathrm{Os}}_{2}{\mathrm{O}}_{7}$ and its SP coupling.

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