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

Abstract

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 microscopic nature of band structure characteristics. 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 Wannier basis, as well as the contributions due to Hamiltonian 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 localized nature of d electrons. However, 5d metals tend to exhibit 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 large kinetic energy of sp electrons is important. Finally, in 1H-MoS2, we find that the valley orbital moment far exceeds the atomic limit 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 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 Berry phase.

Figures (12)

• Figure 1: Schematic illustration of the atom-centered approximation (ACA) and the modern theory of orbital magnetism. (a) ACA accounts only for the orbital magnetism arising from the local orbital angular motion of electrons inside each MT sphere. (b) The modern theory evaluates not only the local contribution within the MT sphere but also the contribution from the interstitial region and the itinerant motion of electrons.
• Figure 2: Schematic illustration of the space selection. The band structure of bcc Fe is presented with the inner window (black dotted line). The energy eigenvalues of the Bloch states and the Bloch-like states are plotted as the blue solid lines and red dotted lines, respectively. The green box shows the spectrum of the bands at $\mathbf{k} = \text{N}$ point. The subspaces corresponding to the projection operators $\hat{P}_{\text{N}}$, $\hat{Q}_{\text{N}}$, and $\hat{\mathbb{P}}_{\text{N}}$, $\hat{\mathbb{Q}}_{\text{N}}$ are depicted.
• Figure 3: Schematic illustration of the intuitive meaning of the Wannier gauge objects.$\mathbb{A}_{\alpha}$, $\mathbb{B}_{\alpha}$, and $\mathbb{C}_{\alpha \beta}$ describe the position, velocity, and orbital angular motion of the Wannier basis, respectively, while the gauge correction $J_{\alpha}$ captures the inter-state mixing. Here, $\ket{\phi_{n}}$ and $\ket{\psi}$ denote Wannier-gauge states and Hamiltonian-gauge states, respectively.
• Figure 4: Modern theory and ACA of orbital magnetization in ferromagnetic $d$-transition metals with DFT+U. Comparison between the modern theory and ACA for (a) bcc Fe with spin-quantization axis [001], (b) fcc Ni [111], (c) hcp Co [0001], and (d) fcc Co [111]. The energy dependence of the modern theory orbital magnetization $M_{z}$ (blue), $M^{(0)}_{z}$ (yellow), $M^{(1)}_{z}$ (purple), $M^{(2)}_{z}$ (green), self-rotation contribution $M^{\text{SR}}_{z}$ (brown line), and ACA $M^{\text{ACA}}_{z}$ (red dashed line) are shown. Experimental values of the orbital magnetization at $\mathcal{E}_{\text{F}}$Ceresoli10PRB are presented by red stars.
• Figure 5: Modern theory and ACA for orbital magnetization in various $d$-transition metals. Comparison between the modern theory and ACA for (a) magnetic bcc W with spin-quantization axis [001], (b) fcc Pt [111], (c) fcc Ti [111], and (d) bcc V [001]. The energy dependence of $M_{z}$ (blue), $M^{(0)}_{z}$ (yellow), $M^{(1)}_{z}$ (purple), $M^{(2)}_{z}$ (green), $M^{\text{SR}}_{z}$ (brown), and $M^{\text{ACA}}_{z}$ (red dashed line) are shown.