Spin Hall effect and Its Renormalization Due to Surface Roughness on Surface of the Topological Insulator

Chia-Hsin Chen^2*, Po-Hao Chou¹, Chung-Yu Mou²

¹Physics Division, National Center for Theoretical Sciences, Taiwan
²Physics, NTHU, Hsinchu, Taiwan

* Presenter:Chia-Hsin Chen, email:log800920@gmail.com

The surface states of a topological insulator are governed by massless Dirac Hamiltonian. In realistic materials, however, no surface is perfectly smooth—surface roughness is inevitable. To explore how such geometric irregularities affect the surface Dirac states, we adopt the tetrad formalism to formulate the massless Dirac equation on curved surfaces. A rough surface can be modeled as an ensemble of small bumps and dips superimposed on an ideal plane, giving rise to nonzero Gaussian and mean curvatures. These curvature fields act as effective magnetic fields with directions opposite to the surface electron spins. Consequently, electrons with opposite spins experience opposite Lorentz forces when traversing curved regions, leading to spin separation and a curvature-induced spin Hall effect.
We investigate whether these bumps and dips can modify the spin Hall effect through the renormalization of surface Hamiltonian. By treating bumps and dips as quenched curvature-impurities and using the replica technique, we perform the momentum-shell renormalization group analysis to one loop so that the renormalization group flow is derived with possible fixed points being identified. Our analysis indicates that these curvature-induced impurities generally renormalize the surface Dirac Hamiltonian, resulting in enhanced spin Hall effect. Implication for other transport coefficients is also discussed.

Keywords: surface roughness, spin Hall effect, curvature-induced effects, Topological insulator, surface Dirac states