Bulk Spin Hall Effect–Induced SIT

• Bulk Spin Hall Effect–Induced SIT is a phenomenon where spin currents generated by intrinsic and extrinsic SHE produce angular momentum transfer, modulating magnetic states.
• It integrates first-principles calculations and quantum kinetic methodologies to quantify contributions from Berry curvature, side-jump, and skew scattering.
• Optimizing the balance between bulk and interfacial effects in materials like topological insulators is key for enhancing spin–orbit torque in spintronic devices.

Bulk spin Hall effect–induced spin-injection torque (SIT) refers to the generation of spin currents through the intrinsic or extrinsic spin Hall effect (SHE) within the bulk of a material, and the resultant transfer of angular momentum as a torque onto an adjacent or intrinsic magnetic system. In spintronics, this phenomenon plays a central role in current-driven magnetization switching, damping, and related effects in topological insulators (TIs), heavy metals, ferromagnetic alloys, and their heterostructures. The mechanism integrates first-principles physics (e.g., Berry-curvature SHE), disorder-driven extrinsic effects (side jump, skew scattering), and complex interface phenomena, with experimental signatures observable in both pure metals/ferromagnets and multilayer heterostructures.

1. Fundamental Theory of the Bulk Spin Hall Effect

The spin Hall effect arises when an applied electric field $\mathbf{E}$ drives a transverse spin current $\mathbf{j}_s$ due to spin–orbit coupling. The general relation is

$\mathbf{j}_s = \sigma_{\mathrm{SH}} \mathbf{E}$

where $\sigma_{\mathrm{SH}}$ is the spin Hall conductivity tensor. The intrinsic contribution to $\sigma_{\mathrm{SH}}$ is captured by the Kubo formula in terms of the spin Berry curvature $\Omega^γ_{αβ,n}(\mathbf{k})$ over occupied states:

$$\sigma^γ_{αβ} = -(e^2/\hbar) \cdot (\hbar/2e) \int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3} \sum_n f(\varepsilon_{n\mathbf{k}})\,\Omega^γ_{αβ,n}(\mathbf{k})$$

with

$$\Omega^γ_{αβ,n}(\mathbf{k}) = \hbar^2 \sum_{m \neq n} \frac{-2\,\mathrm{Im} \langle n\mathbf{k}|J^γ_α|m\mathbf{k}\rangle \langle m\mathbf{k}|v_β|n\mathbf{k}\rangle}{(\varepsilon_{n\mathbf{k}}-\varepsilon_{m\mathbf{k}})^2}$$

where $J^γ_α = \frac{1}{2}\{v_α,σ_γ\}$, $v_β$ is the velocity operator, and $f(\varepsilon)$ the Fermi occupation.

In extrinsic SHE, subtle disorder effects introduce side-jump and skew-scattering terms. In topological insulators, the full quantum kinetic equation is required to correctly capture these, with the proper (conserved) spin-current operator defined as:

$$\mathcal{J}_j^i = \frac{1}{2}\{s_i,v_j\} + \frac{i}{\hbar}[H_0,s_i]\,r_j$$

where $H_0$ includes spin-orbit coupled bands and disorder potentials (Cullen et al., 2023).

2. Mechanisms for Bulk Spin Hall Effect–Induced SIT

The generated spin current from the bulk SHE can flow into an adjacent ferromagnet (in heterostructures) or interact directly with local magnetization (in ferromagnets with strong spin–orbit coupling). The critical process is the conversion of the spin current into an angular momentum transfer—i.e., spin–orbit torque—on the magnetic order parameter. The primary expression for the damping-like SIT is:

$$\tau_{\mathrm{SIT}} = (\hbar/2e)\,\mathbf{j}_s \times \hat{\sigma}$$

where $\hat{\sigma}$ is the spin polarization direction.

In vectorial form, a charge current density $j_c$ along $y$ (for a bulk TI) creates a transverse spin current along $z$ with spins polarized along $x$, and the injected spin current density is:

$j_s = (\hbar/2e)\,\theta_{\mathrm{SH}}\,j_c$

$$\theta_{\mathrm{SH}} = (2e/\hbar)\,\sigma^γ_{αβ}/\sigma_{ββ}$$ is the spin Hall angle.

When this spin current is absorbed by an adjacent ferromagnetic layer, the spin transfer torque can switch or damp the magnetization, with the critical switching current density in the macrospin approximation given by:

$$J_c \simeq (2e/\hbar) \alpha M_s t_F \mu_0 H_K / \theta_{\mathrm{SH}}$$

where $M_s$ is the saturation magnetization, $t_F$ the ferromagnet thickness, $\alpha$ the Gilbert damping, and $\mu_0 H_K$ the effective anisotropy field (Farzaneh et al., 2020, Cullen et al., 2023, Gladii et al., 2019).

3. First-Principles and Quantum Kinetic Methodologies

First-principles calculations of the intrinsic SHE utilize density functional theory (DFT) with generalized gradient approximation (GGA) and projector augmented-wave (PAW) pseudopotentials to obtain Bloch eigenstates. These are mapped to maximally localized Wannier functions for high-resolution Brillouin zone integration (k-mesh), allowing for computation of Berry curvature and velocity-operator matrix elements. Iterative interpolation is deployed for optimal k-point sampling ($\sim 10^7$ points) (Farzaneh et al., 2020).

For extrinsic SHE, quantum kinetic (Liouville) equations incorporating disorder via Born approximation allow for separation of the disorder-averaged density matrix into band-diagonal and off-diagonal terms, facilitating calculation of side-jump and skew-scattering corrections to the spin Hall conductivity (Cullen et al., 2023).

In ferromagnetic metals (e.g., permalloy), SPR-KKR Kubo-formalism and first-principles numerical fitting yield the temperature and composition dependence of skew and side-jump contributions to $\sigma_{\mathrm{SH}}$ (Gladii et al., 2019).

4. Quantitative Results, Material Dependence, and Experimental Signatures

First-principles calculations for bulk topological insulators report principal spin Hall conductivity components for Sb$_2$Se$_3$, Sb$_2$Te$_3$, Bi$_2$Se$_3$, and Bi$_2$Te$_3$ at stoichiometric Fermi levels to be in the range of $94$–$218$ $(\hbar/2e)\,\mathrm{S/cm}$ (Farzaneh et al., 2020). The corresponding spin Hall angles, for typical measured longitudinal conductivities $σ_{yy} = 10^2$–$10^3$ S/cm, reach $0.1$–$0.8$, on par with or exceeding those of Pt or Ta, despite smaller absolute $\sigma_{\mathrm{SH}}$.

In Bi$_2$Se$_3$-class TIs, quantum kinetic theory gives intrinsic $\sigma_{\mathrm{SH}} \approx 1 \times 10^3$ and extrinsic (side-jump dominated) $\sigma_{\mathrm{SH}} \approx 2 \times 10^3$ $(\hbar/2e)\,\Omega^{-1}\,\mathrm{m}^{-1}$—again, comparable magnitudes but both are at least one order below “effective spin conductivity” values extracted from experimental SOT measurements in TI/FM heterostructures ($1$–$2 \times 10^5$ $(\hbar/2e)\,\Omega^{-1}\,\mathrm{m}^{-1}$) (Cullen et al., 2023).

In ferromagnet-only systems, such as permalloy, the self-induced inverse SHE (self-ISHE) creates a transverse voltage scaling linearly with film thickness, directly confirming a bulk, rather than interface, origin. The spin Hall conductivity $\sigma_{\mathrm{SH}}(T)$ shows a non-monotonic temperature dependence due to sign-changing skew and side-jump/intrinsic contributions, resulting in vanishing SIT near $T\approx100\ \rm K$ (Gladii et al., 2019).

5. Limitations, Interfacial Phenomena, and Model Deficiencies

Experimental analyses in heavy-metal/magnetic-insulator systems reveal a key deficiency in the pure bulk-SHE “spin-current” model: the ratio of field-like torque (FLT) to damping-like torque (DLT) predicted from SHE-based magnetoresistance signals (spin Hall-induced anomalous Hall effect [SH-AHE] and spin Hall magnetoresistance [SMR]) does not match measurements. In Tm$_3$Fe$_5$O$_{12}$/Pt bilayers, $H_{\rm FL}/H_{\rm DL}$ is at least $2\times$ larger than $\Delta\rho_{\rm SH-AHE}/\Delta\rho_{\rm SMR}$, and the bulk SHE underestimates the FLT efficiency by a factor of $2$ or more (Li et al., 2017).

Comprehensive agreement requires inclusion of interfacial contributions:

• Rashba spin–orbit coupling at the interface ($\mathrm{MI/HM}$), generating a large interfacial torque (Rashba field).
• Magnetic proximity effect in the heavy metal, inducing moments and enlarging the interface region's torque efficiency.

These mechanisms coexist with bulk SHE and dominate or enhance the overall spin–orbit torque, reconciling the observed large FLT and the violation of the bulk-SHE torque relation.

6. Comparative Impact of Intrinsic and Extrinsic Contributions

In TIs and their heterostructures, both intrinsic (Berry curvature) and extrinsic (side-jump, skew) mechanisms must be considered to accurately assess the bulk SHE. Quantum mechanical calculations show that, over a wide Fermi energy range, extrinsic and intrinsic contributions are of comparable magnitude, with side-jump typically dominant within the extrinsic channel (Cullen et al., 2023, Gladii et al., 2019).

However, even combined, the total bulk SHE is consistently too small to account for the large spin torques measured in experiments. This suggests that in most TI/FM device geometries, surface-state-induced phenomena such as the Rashba-Edelstein effect and proximity-enhanced transfer torque are the principal contributors to the observable SIT.

7. Practical Considerations and Device Implications

Managing the balance between bulk and surface-state effects is crucial for device optimization:

• Maximizing $\sigma_{\mathrm{SH}}$ via choice of TI compound (e.g., Bi$_2$Te$_3$ for larger values), tuning Fermi level placement (into the gap), and adjusting thicknesses enhances bulk-mediated torques (Farzaneh et al., 2020).
• Exploiting the tunable, non-monotonic temperature response of $\sigma_{\mathrm{SH}}$ in ferromagnetic devices (e.g., permalloy) may allow temperature-specific control of SIT for applications without heavy-metal layers (Gladii et al., 2019).
• Interface engineering, such as enhancing spin-mixing conductance or the Rashba parameter, is imperative to fully harness and control the contributions to total device torque.

A plausible implication is that, while the bulk Spin Hall effect—both intrinsic and extrinsic—sets the lower bound for spin-torque phenomena in relevant materials, device-level efficiencies and practical switching currents are overwhelmingly determined by interface-specific effects and the properties of surface states. Future research directions include advanced first-principles torque calculations, alloy engineering (e.g., (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$), and hybrid modeling combining bulk and interface contributions (Farzaneh et al., 2020, Li et al., 2017, Cullen et al., 2023, Gladii et al., 2019).