# Extrinsic Spin-Orbit Coupling and Spin Relaxation in Phosphorene

## TL;DR

This paper derives an effective Hamiltonian for monolayer phosphorene to analyze how extrinsic spin-orbit coupling, influenced by external electric fields and substrate impurities, affects spin relaxation anisotropy.

## Contribution

It introduces a detailed derivation of extrinsic spin-orbit coupling in phosphorene considering symmetry and external fields, highlighting anisotropic effects on spin relaxation.

## Key findings

- Spin splitting is linear in electric field and atomic spin-orbit strength.
- Anisotropic Fermi contours influence spin relaxation times.
- Charged impurity scattering affects spin dynamics.

## Abstract

An effective Hamiltonian is derived to describe the conduction band of monolayer black phosphorus (phosphorene) in the presence of spin-orbit coupling and external electric field. Envelope function approximation along with symmetry arguments and Lowdin partitioning are utilized to derive extrinsic spin-orbit coupling. The resulting spin splitting appears in fourth order perturbation terms and is shown to be linear in both the magnitude of the external electric field and the strength of the atomic spin-orbit coupling, similar to the Bychkov-Rashba expression but with an in-plane anisotropy. The anisotropy depends on the coupling between conduction band and other bands both close and distant in energy. The spin relaxation of conduction electrons is then calculated within the Dyakonov-Perel mechanism where momentum scattering randomizes the polarization of a spin ensemble. We show how the anisotropic Fermi contour and the anisotropic extrinsic spin splitting contribute to the anisotropy of spin-relaxation time. Scattering centers in the substrate are considered to be charged impurities with screened Coulomb potential.