[Paper Review] Theory of Spin Hall Effects in Semiconductors

Hans‐Andreas Engel, É. I. Rashba+1 more|ArXiv.org|Mar 10, 2006

Quantum and electron transport phenomena60 references5 citations

TL;DR

This paper provides a comprehensive theoretical framework for spin Hall effects in semiconductors, distinguishing between intrinsic and extrinsic spin-orbit coupling mechanisms. It identifies that spin accumulation and spin currents arise from spin-orbit coupling in bulk and at boundaries, with key results showing robust quantized spin Hall conductivity in topological insulators like HgTe/CdTe quantum wells and graphene under specific conditions, despite disorder and finite spin-orbit coupling strength.

ABSTRACT

Spin Hall effects are a collection of phenomena, resulting from spin-orbit coupling, in which an electrical current flowing through a sample can lead to spin transport in a perpendicular direction and spin accumulation at lateral boundaries. These effects, which do not require an applied magnetic field, can originate in a variety of intrinsic and extrinsic spin-orbit coupling mechanisms and depend on geometry, dimension, impurity scattering, and carrier density of the system--making the analysis of these effects a diverse field of research. In this article, we give an overview of the theoretical background of the spin Hall effects and summarize some of the most important results. First, we explain effective spin-orbit Hamiltonians, how they arise from band structure, and how they can be understood from symmetry considerations; including intrinsic coupling due to bulk inversion or structure asymmetry or due to strain, and extrinsic coupling due to impurities. This leads to different mechanisms of spin transport: spin precession, skew scattering, and side jump. Then we present the kinetic (Boltzmann) equations, which describe the spin-dependent distribution function of charge carriers, and the diffusion equation for spin polarization density. Next, we define the notion of spin currents and discuss their relation to spin polarization. Finally, we explain the electrically induced spin effects; namely, spin polarization and currents in bulk and near boundaries (the focus of most current theoretical research efforts), and spin injection, as well as effects in mesoscopic systems and in edge states.

Motivation & Objective

• To systematically classify and explain the origins of spin Hall effects in semiconductors through intrinsic and extrinsic spin-orbit coupling mechanisms.
• To clarify the role of symmetry, band structure, and impurity scattering in generating spin accumulation and spin currents without external magnetic fields.
• To establish a theoretical basis for understanding electrically induced spin transport and spin injection in low-dimensional systems.
• To investigate the stability of quantized spin Hall conductivity in the presence of disorder and finite spin-orbit coupling in materials like graphene and HgTe/CdTe quantum wells.
• To identify parameter regimes where topological spin Hall effects remain robust, particularly in the quantum spin Hall phase of two-dimensional systems.

Proposed method

• Derives effective spin-orbit Hamiltonians from band structure and symmetry considerations, distinguishing between intrinsic (bulk inversion, strain) and extrinsic (impurity scattering) mechanisms.
• Applies kinetic (Boltzmann) equations and spin diffusion equations to model the spin-dependent distribution function and spin polarization density in non-equilibrium conditions.
• Introduces and analyzes the concept of spin current, relating it to spin polarization and transport in the absence of net charge flow.
• Uses the Landauer-Büttiker formalism to compute spin Hall conductivity in mesoscopic systems, modeling disorder via uncorrelated on-site potentials with uniform disorder strength.
• Constructs a phase diagram in the parameter space of spin-orbit coupling and sublattice potential to identify the quantum spin Hall phase in two-dimensional systems.
• Evaluates the robustness of edge state transport using numerical simulations with varying disorder strength and spin-orbit coupling, comparing results to theoretical predictions.

Experimental results

Research questions

• RQ1 What are the distinct microscopic mechanisms—such as skew scattering, side jump, and intrinsic spin-orbit coupling—that give rise to spin Hall effects in semiconductors?
• RQ2 How does the interplay between intrinsic spin-orbit coupling and extrinsic impurity scattering affect the magnitude and robustness of spin Hall conductivity?
• RQ3 Under what conditions does a two-dimensional system exhibit a quantized, dissipationless spin Hall effect, and how stable is this effect against disorder and finite-size effects?
• RQ4 What is the role of edge states in the quantum spin Hall effect, and how do they differ topologically from trivial insulating phases?
• RQ5 Can the spin Hall effect be observed in graphene and HgTe/CdTe quantum wells, and what parameter regimes support robust, nearly quantized spin transport?

Key findings

• The spin Hall effect in semiconductors arises from both intrinsic mechanisms (due to bulk inversion asymmetry or strain) and extrinsic mechanisms (via impurity scattering), with distinct contributions from skew scattering and side-jump processes.
• Numerical simulations show that spin Hall conductivity remains within a few percent of its quantized value when disorder strength W is less than the hopping integral t, and remains stable for spin-orbit coupling ratios λR/λSO ≤ 0.2t.
• The quantum spin Hall phase in HgTe/CdTe quantum wells and graphene-like systems is stabilized by strong spin-orbit coupling λSO and protected by time-reversal symmetry, forming an ovaloid region in the λv/λSO–λR/λSO parameter plane.
• In graphene, the spin gap ΔSO is estimated to be ~2.4 K, but more recent calculations suggest it is significantly smaller, while momentum relaxation times τ imply a broadening of the gap due to disorder (ℏ/τ ≳ 25 K), suppressing coherent edge transport.
• Edge states in the quantum spin Hall phase are Kramers-degenerate and topologically protected, with an even number of pairs ensuring backscattering is forbidden in the absence of time-reversal symmetry breaking.
• Theoretical models predict that exchange-enhanced gaps in the quantum Hall regime can reach up to 100 K, suggesting potential for high-temperature spin transport in engineered heterostructures.