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

Abstract

In atomic physics, the Hund rule says that the largest spin and orbital state is realized due to the interplay of the spin-orbit coupling (SOC) and the 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 )ρ0], where U and U ′ are the on-site Coulomb interaction within the same oribtals and between different orbitals, respectively, JH is the Hund coupling, and ρ0 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. This present work 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.

Enhanced Spin–Orbit Coupling in a Correlated Metal

“…In Ref. 50, an expression different from Eq. ( 8) was obtained for λ eff by using the eigenstates of l z , |l1 and |l − 1 , as the basis states instead of |x and |y [Eqs.…”

“…In Ref. 50, the combination of the interaction parameters (2 Ũ ′ − Ũ − J) appears instead of (U ′ − J ′ ) in Eq. ( 8).…”

“…The enhancement of the effective SOC in two-and three-orbital models has been studied using the dynamical mean-field theory, 20,21,23,24,47,48) Gutzwiller approximation, 49) and Hartree-Fock (HF) approximation. 50) However, systematic investigation of the effective SOC, such as dependence on the Coulomb interaction, Hund's coupling, and doping, has not been carried out.…”