[Paper Review] Symmetry of Berry and spin Berry curvatures in ferromagnetic CoPt

Guanxiong Qu, Kohji Nakamura+1 more|arXiv (Cornell University)|Jan 17, 2019

Graphene research and applicationsMaterials Science1 references3 citations

TL;DR

This study investigates the intrinsic anomalous Hall and spin Hall conductivities in ferromagnetic L1₀-CoPt using first-principles calculations of Berry and spin Berry curvatures. It reveals that while the Berry curvature preserves the C₄ᵥ symmetry of the crystal, the spin Berry curvature exhibits reduced C₂ᵥ symmetry due to band crossing points with opposite spin characters, invalidating the simple scaling relation σ_AH = Pσ_SH with spin polarization P.

ABSTRACT

The intrinsic spin Hall conductivity and the anomalous Hall conductivity of ferromagnetic L1$_0$-CoPt are studied using first principle calculations of the spin Berry and Berry curvatures, respectively. We find that the Berry curvature and the spin Berry curvature exhibit different symmetry with respect to that of the band structure. The Berry curvature preserves the $C_{4v}$ crystal rotation symmetry along the c-axis whereas the symmetry of the spin Berry curvature reduces to $C_{2v}$. Contributions to the Berry curvature and the spin Berry curvature are classified by the spin character of bands crossing the Fermi level. We find that the reduced symmetry of the spin Berry curvature is due to band crossing points with opposite spin characters. From model Hamiltonian analyses, we show the universality of this distinct symmetry reduction of the spin Berry curvature with respect to the Berry curvature: it can be accounted for based on the form of spin current operator and velocity operator in the Kubo formula. Finally, we discuss the consequence of the reduced symmetry of the spin Berry curvature on the relationship between the anomalous Hall and spin Hall conductivity. When band crossing points with opposite spin characters are present in the reciprocal space, which is often the case, the anomalous Hall conductivity does not simply scale with the spin Hall conductivity with the scaling factor being the spin polarization at the Fermi level.

Motivation & Objective

• To investigate the intrinsic anomalous Hall conductivity (AHC) and spin Hall conductivity (SHC) in ferromagnetic L1₀-CoPt.
• To analyze the symmetry properties of Berry curvature and spin Berry curvature in momentum space with respect to crystal symmetry.
• To understand the origin of symmetry breaking in spin Berry curvature and its implications for the relationship between AHC and SHC.
• To test the validity of the two-current model and the scaling relation σ_AH = Pσ_SH in ferromagnetic systems with non-zero spin-orbit coupling.
• To establish the universality of spin Berry curvature symmetry reduction via model Hamiltonian analysis.

Proposed method

• First-principles density functional theory (DFT) calculations using the full-potential linearized augmented-plane-wave (FLAPW) method with generalized gradient approximation (GGA) for exchange-correlation.
• Inclusion of spin-orbit coupling via a second-variational method, with zero-temperature approximation to simplify the Fermi-Dirac distribution.
• Computation of Berry and spin Berry curvatures using the Kubo formula in the spectral representation, with velocity and spin current operators defined via generalized velocity operators vᵢ^α = ½{σ^α, vᵢ}.
• Projection of curvatures onto momentum space to analyze global symmetry properties, with k-space sampling up to 70×70×70 to ensure convergence.
• Classification of contributions to curvatures based on spin character of bands crossing the Fermi level (class I: pure spin, class II: mixed spin).
• Construction of a minimal model Hamiltonian to demonstrate the universality of the symmetry reduction in spin Berry curvature due to off-diagonal matrix elements of the velocity and spin current operators.

Experimental results

Research questions

• RQ1 How does the symmetry of the Berry curvature in L1₀-CoPt compare to the crystal symmetry of the system?
• RQ2 Why does the spin Berry curvature exhibit reduced symmetry (C₂ᵥ) compared to the Berry curvature (C₄ᵥ) in the same material?
• RQ3 What is the microscopic origin of the symmetry breaking in the spin Berry curvature, particularly in relation to band structure features?
• RQ4 How do band crossing points with opposite spin characters influence the relationship between anomalous Hall conductivity and spin Hall conductivity?
• RQ5 To what extent does the conventional two-current model, assuming σ_AH = Pσ_SH, fail in describing the intrinsic AHC and SHC in ferromagnetic CoPt?

Key findings

• The Berry curvature preserves the C₄ᵥ rotational symmetry of the L1₀-CoPt crystal along the c-axis, consistent with the band structure symmetry.
• The spin Berry curvature exhibits reduced C₂ᵥ symmetry, as evidenced by the inequality Ω_yx³⁰(Λk) ≠ Ω_yx³⁰(k) under C₄ᵥ operations, indicating symmetry breaking.
• The symmetry reduction in spin Berry curvature arises from band crossing points with opposite spin characters (class II states), where off-diagonal matrix elements of the generalized velocity and spin current operators differ.
• The intrinsic anomalous Hall conductivity (σ_AH) is -3 S/cm and the spin Hall conductivity (σ_SH) is 787 S/cm, with spin polarization P ≈ 0, invalidating the simple scaling σ_AH = Pσ_SH.
• The presence of ubiquitous class II states near the Fermi level in ferromagnets leads to non-trivial interplay between velocity and spin current operators, breaking the naive scaling between AHC and SHC.
• Model Hamiltonian analysis confirms the universality of the symmetry reduction in spin Berry curvature, rooted in the distinct transformation properties of the velocity and spin current operators under symmetry operations.