[Paper Review] The Edelstein effect in the presence of impurity spin-orbit scattering

Amin Maleki, Roberto Raimondi+1 more|arXiv (Cornell University)|Oct 26, 2016

Quantum and electron transport phenomena3 citations

TL;DR

This paper presents a diagrammatic Kubo formula analysis of the Edelstein effect in a two-dimensional electron gas, incorporating both intrinsic Rashba and Dresselhaus spin-orbit interactions and extrinsic impurity spin-orbit scattering via side-jump and skew-scattering mechanisms. The key result is a modified Edelstein conductivity that exhibits anisotropic current-induced spin polarization, with cancellation between Rashba and Dresselhaus contributions when their strengths are equal, and tunability via the ratio of D’yakonov-Perel’ and Elliott-Yafet spin relaxation rates.

ABSTRACT

In this paper we study the current-induced spin polarization in a two-dimensional electron gas, known also as the Edelstein effect. Compared to previous treatments, we consider both the Rashba and Dresselhaus spin-orbit interaction as well as the spin-orbit interaction from impurity scattering. In evaluating the Kubo formula for the spin polarization response to an applied electric field, we explicitly take into account the side-jump and skew-scattering effects. We show that the inclusion of side-jump and skew-scattering modifies the expression of the current-induced spin polarization.

Motivation & Objective

• To extend prior work on current-induced spin polarization by including both Rashba and Dresselhaus spin-orbit interactions alongside impurity scattering effects.
• To systematically account for extrinsic contributions—side-jump and skew-scattering—using diagrammatic techniques within the Kubo formalism.
• To determine how the interplay of intrinsic and extrinsic spin-orbit mechanisms modifies the Edelstein conductivity and leads to anisotropic spin polarization.
• To provide a closed-form analytical expression for the Edelstein conductivity valid in the Born approximation and first-order beyond it.

Proposed method

• The study employs the Kubo formula for spin polarization response to an electric field, using the diagrammatic approach to evaluate the Edelstein conductivity.
• The Hamiltonian includes Rashba and Dresselhaus spin-orbit coupling terms, as well as impurity potential with short-range disorder characterized by white-noise statistics.
• The retarded and advanced Green's functions are expanded in the Pauli matrix basis, with momentum-dependent spin-orbit coupling strength γ = √(α² + β² + 2αβ(ˆpxˆpy + ˆpyˆpx)).
• Side-jump and skew-scattering contributions are derived from third-order disorder vertex corrections in the self-energy, with explicit evaluation of vertex corrections involving momentum and spin matrices.
• The spin Hall conductivity from extrinsic mechanisms (σ̂SHE,ext = σ̂SHE,sj + σ̂SHE,ss) is incorporated into the final expression for the Edelstein conductivity.
• The final expressions for spin polarization (Sext^y and Sext^x) are derived by combining intrinsic and extrinsic contributions, with a denominator involving the sum and difference of spin relaxation rates.

Experimental results

Research questions

• RQ1 How does the inclusion of both Rashba and Dresselhaus spin-orbit interactions affect the current-induced spin polarization in a two-dimensional electron gas with impurity scattering?
• RQ2 What is the role of side-jump and skew-scattering contributions in modifying the Edelstein conductivity beyond the intrinsic spin-orbit effect?
• RQ3 Can the interplay between intrinsic and extrinsic spin-orbit mechanisms lead to cancellation or tuning of the Edelstein effect?
• RQ4 How does the ratio of D’yakonov-Perel’ and Elliott-Yafet spin relaxation rates influence the magnitude and anisotropy of the spin polarization?
• RQ5 Is the Edelstein effect anisotropic when both Rashba and Dresselhaus terms are present, and can this be experimentally probed via reciprocal charge current generation?

Key findings

• The Edelstein conductivity is modified by extrinsic contributions: side-jump and skew-scattering, which are incorporated via the total extrinsic spin Hall conductivity σ̂SHE,ext.
• The total spin polarization along the y-direction is given by a formula involving the inverse of a combination of relaxation rates, with a prefactor dependent on the difference (1/τα - 1/τβ + 1/τEY).
• When Rashba and Dresselhaus spin-orbit coupling strengths are equal (α = β), the intrinsic contribution to the Edelstein effect cancels due to the (1/τα - 1/τβ) factor.
• The skew-scattering contribution to the Edelstein effect is proportional to the spin Hall conductivity σ̂SHE,ss, with a final expression S^y = -2mατsσ̂SHE,ssE_x.
• The anisotropy of the Edelstein effect arises from the interplay of Rashba and Dresselhaus terms, leading to different polarization responses along x and y directions.
• The Onsager reciprocity suggests that a spin polarization along y can induce a transverse charge current, providing a testable signature of the effect.