Anatomy of the modern theory of orbital magnetism from first principles: Term-by-term analysis in the gauge-covariant formalism

초록

We present an in-depth analysis of the orbital magnetism by means of the so-called modern theory based on the Berry phase across distinct classes of materials-d transition-metals, sp metals, and transition metal dichalcogenides-highlighting the importance of the microscopic nature of band structure characteristics on the orbital magnetization. We adopt a gauge-covariant formulation of the modern theory proposed in Lopez et al. [Phys. Rev. B 85, 014435 (2012)], which enables the calculation of orbital magnetism in a controlled manner in any chosen gauge of Wannier functions and gives the total contribution as a gauge-invariant measurable. This captures consistently the contributions due to the anomalous position, velocity, and orbital angular momentum of a Wannier basis, as well as the contributions due to Hamiltonians such that their sum is gauge-invariant. For d transition metals, we find that the atom-centered approximation captures the majority of the total contribution given by modern theory, which we attribute to the localized nature of d electrons. However, 5d metals tend to exhibit relatively larger deviation between the two methods than 3d metals do, as 5d electrons are more delocalized than 3d electrons. On the other hand, sp metals exhibit a strong deviation between the two methods, where the large kinetic energy of sp electrons is important. Finally, in 1H-MoS2, we find that the valley orbital moment at each valley far exceeds the atomic limit orbital moment of d electrons due to coherent hybridization between valence and conduction bands in direct band gaps. Our work elucidates the interplay of the chemical nature of electronic orbitals and the effect of band structures in a consistent manner and highlights the role of the Berry phase in orbital magnetism. The results suggest a promising direction of orbitronics beyond controlling atomic orbitals, in which the orbital magnetism can be greatly enhanced by exploiting the Berry phase.

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