Enhanced spin-orbit coupling and orbital moment in ferromagnets by electron correlations

In atomic physics, the Hund's rule states that the largest spin and orbital state is realized due to the interplay of spin-orbit coupling (SOC) and Coulomb interactions. Here, we show that in ferromagnetic solids the effective SOC and the orbital magnetic moment can be dramatically enhanced by a factor of 1/[1 - (2U' - U - JH )rho 0], where U and U' are the on-site Coulomb interaction within the same orbitals and between different orbitals, respectively, JH is the Hund's coupling, and rho 0 is the average density of states. This factor is obtained by using the two-orbital as well as five-orbital Hubbard models with SOC. We also find that the spin polarization is more favorable than the orbital polarization, being consistent with experimental observations. The theory is also extended to study the spin fluctuations and long-range Coulomb interactions, and can be applied to understand the enhanced orbital magnetic moment and giant Faraday effect in ferromagnetic nanogranules in recent experiments. This present paper provides a fundamental basis for understanding the enhancements of SOC and orbital moment by Coulomb interactions in ferromagnets, which would have wide applications in spintronics.

Automated summary

In atomic physics, Hund's rule explains the realization of the largest spin and orbital state through the interplay of spin-orbit coupling and Coulomb interactions. This study demonstrates that in ferromagnetic solids, the effective spin-orbit coupling and orbital magnetic moment can be significantly enhanced by a specific factor derived from the two-orbital and five-orbital Hubbard models with spin-orbit coupling. The findings also suggest that spin polarization is preferred over orbital polarization, consistent with experimental observations. The research provides a fundamental understanding of the enhancement of spin-orbit coupling and orbital moment by Coulomb interactions in ferromagnets, with potential applications in spintronics.