Maximum intrinsic spin-Hall conductivity in two-dimensional systems with k-linear spin-orbit interaction

Area of Science:

• Condensed Matter Physics
• Spintronics
• Materials Science

Background:

• Spin-orbit interaction is crucial for spintronic devices.
• Understanding intrinsic spin-Hall conductivities (ISHCs) is key to controlling spin currents.
• Previous studies often assumed symmetry in ISHC.

Purpose of the Study:

• To analytically calculate ISHCs in a clean, 2D system with generic k-linear spin-orbit interaction.
• To investigate the relationship between spin-orbit matrix elements and ISHC properties.
• To explore the implications of spin-orbital decoupling on ISHC asymmetry and magnitude.

Main Methods:

• Analytical calculation of ISHCs (σ(z)(xy) and σ(z)(yx)).
• Analysis of the spin-orbit matrix (β̃) and its determinant (detβ̃).
• Investigation of the asymmetric response function (Δβ̃) and Berry curvature.

Main Results:

• The determinant of the spin-orbit matrix (detβ̃) governs spin-orbital coupling.
• Decoupling of spin and orbital motion leads to ISHC sign changes and band overlap.
• ISHC is generally asymmetric (σ(z)(xy) ≠ -σ(z)(yx)), determined by Δβ̃.
• The asymmetric response function implies ISHC can exceed e/8π, with an upper bound of e/4π.
• detβ̃ determines the SU(2) non-Abelian gauge field strength.

Conclusions:

• Spin-orbital decoupling is a key factor in ISHC asymmetry and magnitude.
• The Berry curvature provides insight into the unsymmetrical ISHC properties.
• The study offers a deeper understanding of spin-orbit interaction effects in 2D materials for spintronics.