Spin Hall Conductivities in Spintronics

• Spin Hall conductivity is the constant linking an applied electric field to a generated transverse spin current in solid-state systems.
• It arises from intrinsic band-structure effects and extrinsic impurity scattering, paving the way for efficient spin current generation.
• Advanced evaluation methods using Berry curvature and linear response theory enable tunable analyses across diverse material systems.

Spin Hall conductivity quantifies the linear response of a transverse spin current to an applied electric field in a solid-state system. This tensorial quantity captures both the intrinsic and extrinsic mechanisms by which charge flow is converted into flow of spin angular momentum, and plays a central role in spintronics for the efficient generation and manipulation of spin currents. The underlying physics of spin Hall conductivity (SHC), as well as its evaluation and tunability, encompasses band-structure effects, topological invariants, impurity scattering, symmetry constraints, and many-body interactions.

1. Definitions, Fundamental Mechanisms, and Model Systems

Spin Hall conductivity σSH refers to the proportionality constant connecting an electric field Eβ to the generated spin current J^γ_α (flow of the γ component of spin along the α direction): J^γ_α = σ^γ_{αβ} E_β. It is commonly computed in the linear response/Kubo formalism as a Brillouin-zone or real-space integral over the spin Berry curvature, or from appropriate current-current correlation functions, depending on the nature of the system (periodic, disordered, or topologically nontrivial).

Two broad mechanisms contribute to the SHC:

• Intrinsic: Originates from the electronic structure—in particular, band crossings with strong spin–orbit interaction and associated Berry curvature—without relying on disorder.
• Extrinsic: Arises from impurity scattering, through side-jump and skew-scattering processes, and is particularly important in systems with significant disorder or half-metallic ferromagnets.

Representative model systems where SHC has been analyzed include:

• 2DEGs with Rashba spin–orbit interaction and magnetic disorder (Berg et al., 2011).
• Strong spin–orbit semimetals such as Bi₁₋ₓSbₓ (Şahin et al., 2014).
• Noncentrosymmetric quantum spin Hall insulators, exemplified by PbBiI (Nobahari et al., 6 May 2024).
• Topological models such as the Kane–Mele–Hubbard and Bernevig–Hughes–Zhang models (Lessnich et al., 2023, Monaco et al., 2020).

2. Theoretical Formalism and Microscopic Computation

The SHC is typically evaluated using a combination of the Kubo formula and Berry curvature analysis. The general linear response expression is

$$\sigma^{\text{SH}}_{\alpha\beta} = \frac{e}{\hbar} \sum_{n}\int_{\text{BZ}} \frac{d^dk}{(2\pi)^d} f_n(k)\, \Omega^{z}_{n,\alpha\beta}(k)$$

with the spin Berry curvature

$$\Omega^{z}_{n,\alpha\beta}(k) = 2 \operatorname{Im} \sum_{m \neq n} \frac{\langle u_{n k} |\hat{J}^z_\alpha| u_{m k}\rangle \langle u_{m k} |\hat{v}_\beta| u_{n k}\rangle}{[\epsilon_{n k} - \epsilon_{m k}]^2}$$

where $u_{n k}$ are Bloch eigenstates, $\hat{J}^z_\alpha$ is the spin current operator, $\hat{v}_\beta$ the velocity operator, and $f_n(k)$ the Fermi–Dirac distribution (Nobahari et al., 6 May 2024, Ryoo et al., 2019). In alloys and disordered systems, the Kubo–Bastin formulation and coherent potential approximation (CPA) are often adopted to treat configuration averaging and separate intrinsic and extrinsic contributions (Turek et al., 2019).

Numerical evaluation leverages:

• Chebyshev expansion kernel polynomial methods for large tight-binding systems (Berg et al., 2011).
• Maximally-localized Wannier function interpolation to efficiently reconstruct Kubo integrals on dense k meshes (Ryoo et al., 2019).
• Self-consistent field and many-body approaches (e.g., TPSC, GW) to include electronic correlations and vertex corrections, necessary for capturing effects like quantization breakdown or band-gap renormalization in the presence of interactions (Lessnich et al., 2023, Mir et al., 2022, Monaco et al., 2020).

3. Influence of Symmetry, Berry Curvature, and Topological Invariants

The topology of the band structure and the overall symmetry constraints fundamentally shape the tensor structure and quantization of SHC:

• Quantum Spin Hall phases exhibit a quantized SHC linked to topological invariants (e.g., the spin Chern number $C_s$):

$\sigma_{\text{SH}} = \frac{e}{2\pi} C_s$

which can be robust even in the absence of spin conservation, provided that spin Berry curvature can be diagonalized in a suitable basis (Dayi et al., 2014).

• Symmetry constraints from the space group dictate which tensor components are allowed. When the local point group at an atom is lower than the global symmetry, "staggered" or layer-resolved SHCs may emerge that are significant for interfacial spin-orbit torques and not captured in a bulk-averaged response (Xue et al., 2020).
• Noncommutative geometry introduces a tunable parameter (θ) into the effective Lorentz force, modifying Hall conductivities and interpolating to standard models in the limit θ → 0 (Asli et al., 2013).

4. Experimental Determination and Material Dependence

Experimental determination of SHC employs:

• Spin pumping and ferromagnetic resonance (FMR) techniques, in which a spin current is injected (via FMR or via spin torque) and detected via the inverse spin Hall effect or harmonic Hall voltage measurements (Liu et al., 2011, Skowroński et al., 2021, Kumar et al., 2018).
• Lateral spin injection and non-local detection geometries, sometimes complicated by current shunting and spin diffusion, which must be carefully modeled to avoid underestimation of the signal (Liu et al., 2011).
• Thickness and interface dependence studies, optimizing the layer thickness near the spin diffusion length to maximize spin injection efficiency and torque, particularly in heavy metal/ferromagnet bilayers (Kumar et al., 2018, Skowroński et al., 2021).

Spin Hall conductivities in benchmark materials:

Material/System	σ_SH [(ħ/e)Ω⁻¹cm⁻¹]	Method	Notes
Platinum (Pt)	1.4–3.4×10⁵	ST-FMR, spin pumping	λ_sf = 1.4 ± 0.3 nm (Liu et al., 2011)
Bismuth (Bi)	474	Kubo + Berry curvature	More than twice of Pt (Şahin et al., 2014)
Bi₀.₈₃Sb₀.₁₇ (3D TI)	~190	Kubo + Berry curvature	TI regime, tunable by gate (Şahin et al., 2014)
Ta (mixed α+β phase)	–2439	FMR + ISHE	Dominant extrinsic, low resistivity (Kumar et al., 2018)
WTe₂ (intrinsic, conv/unconv)	100/20	Layer-resolved theory	Conventional/staggered (Xue et al., 2020)
Orthorhombic PbTe	> 1000	Layer-resolved theory	Conventional/staggered (Xue et al., 2020)
CoSi (B20 chiral semimetal)	52	SMR/harmonic, DFT+Kubo	Unique Fermi-level dependence (Tang et al., 2021)

This spectrum highlights the diversity not only in mechanism (intrinsic vs extrinsic, topology-driven vs orbital hybridization) but also in the scaling with chemical potential, band engineering, and microstructural factors.

5. Tunability and Control

The SHC can be effectively tuned by:

• Fermi level adjustment: either by chemical doping or electrostatic gating, affecting the occupancy near band crossings with large Berry curvature (Şahin et al., 2014, Tang et al., 2021).
• Electric-field or ferroelectric control: In polar Rashba semiconductors such as GeTe, the magnitude and sign of the SHC can be tuned through the ferroelectric polarization, which alters the spin-orbital texture and the Rashba parameter α_R (Zhang et al., 2019).
• External perturbations: Staggered exchange fields and applied electric fields modify the band structure and Berry curvature distribution, directly affecting the SHC—particularly in noncentrosymmetric or quantum spin Hall systems (Nobahari et al., 6 May 2024).
• Temperature and magnetic excitations: In half-metals, the SHC is strongly enhanced by thermal activation of minority-spin states by magnons, with a characteristic T^3/2 scaling (Ohnuma et al., 2016).
• Many-body interactions: Electron–electron correlations can enhance or even reverse the sign of SHC, as demonstrated in magnetic 2DEGs with Rashba SOI and in correlated topological insulators, especially under strong exchange or close to magnetic instabilities (Mir et al., 2022, Lessnich et al., 2023).

6. Sample-to-Sample Fluctuations, Local and Layer-Resolved Responses

Even in large, mesoscopic samples, the static SHC can display wide Gaussian distributions centered near the intrinsic value, particularly in systems with disorder or finite size. Sample-to-sample fluctuations are substantial, with standard deviations on the order of 10% of the mean (Berg et al., 2011). These fluctuations are relevant for device repeatability and performance.

Advanced theoretical approaches allow the decomposition of SHC into local, spatially resolved contributions. This formalism, applicable to heterogeneous, finite, or nanoribbon geometries, provides insight into how surfaces, interfaces, or inhomogeneous regions locally contribute—or suppress—the overall spin Hall effect (Rauch et al., 2021, Xue et al., 2020). For example, in low-symmetry systems such as WTe₂ and orthorhombic PbTe, unconventional (staggered) SHC components can be comparable to or even exceed the uniform bulk response, with direct implications for spin-orbit torque generation.

7. Technological Implications and Material Engineering

Robust, high-efficiency spin Hall conductivity is necessary for the realization of spin-orbit torque memory, spin-logic circuits, and coherent manipulation of magnetic order by electrical means. The capacity to modulate SHC by material architecture, external fields, and microstructural control enables optimization for low-power, high-efficiency spintronics. Candidates such as Pt- and Ta-based alloys (with controlled composition and microstructure), topological semimetals (e.g., Bi₁₋ₓSbₓ, CoSi), and ferroelectric semiconductors (GeTe, PbBiI) offer a spectrum of tunable performance. However, challenges remain in achieving stable, reproducible, and scalable SHC, particularly under device-relevant conditions, and in systems where the dynamical (ac) spin Hall response is intrinsically weak without auxiliary magnetic elements (Nobahari et al., 6 May 2024).

The precise understanding of how SHC arises, how it can be engineered, and how it fluctuates in real structures is central to advancing both foundational physics and spintronic applications.