Chromo-Spin Hall Effect in Chiral Materials

• Chromo-Spin Hall Effect is an umbrella term for spin Hall phenomena modulated by chiral band topology, magnetic order, or detection symmetry.
• It highlights mechanisms such as Fermi-level tuning in spin-1 semimetals and SOC-free SHE in noncollinear magnets to achieve efficient spin-charge conversion.
• Innovative methods like interface-free XMCD-PEEM and mixed spin-voltage correlators provide practical ways to isolate intrinsic spin transport properties in complex materials.

In current arXiv literature, “Chromo-Spin Hall Effect” is not an established term. A plausible usage is to denote spin Hall phenomena in which the transverse spin response is shaped by chiral band topology, chiral magnetic order, or chiral-like signal symmetries, together with measurement strategies designed to isolate such responses from interface artifacts. In the conventional spin Hall effect (SHE), a longitudinal charge current is converted into a transverse spin current through spin-orbit coupling, producing spin accumulation of opposite sign at opposite edges or surfaces over approximately the spin diffusion length; the inverse SHE is the reciprocal conversion from spin current to transverse charge current. A standard efficiency measure is the spin Hall angle, $\alpha = \frac{\sigma_{xy}^{s}}{\sigma_{xx}}$, where $\sigma_{xy}^{s}$ is the spin Hall conductivity and $\sigma_{xx}$ is the charge conductivity (Ruiz-Gómez et al., 2021).

1. Terminology and transport framework

The baseline SHE framework is spin–charge conversion in a nonmagnetic conductor. Opposite spin species are deflected in opposite transverse directions, so the lateral charge flow cancels while a transverse spin current and boundary spin accumulation remain. This definition underlies the interface-free CuBi spectroscopy study, the CoSi spin-1 chiral semimetal study, the noise-correlation proposal, the asymmetric Pt detector, the nanomechanical torque proposal, and the graphene/Pt inverse-SHE measurements (Ruiz-Gómez et al., 2021).

What distinguishes the “chromo-” qualifier, in the broad sense suggested by these papers, is not a separate universal transport equation but the origin and symmetry of the spin Hall response. In one line of work, the response is tied to a chiral topological band structure, as in B20-CoSi. In another, it emerges from a chiral magnetic lattice that breaks spin-rotation symmetry even without spin-orbit coupling. In a third, the relevant novelty lies in the detection symmetry itself, for example even-in-current nonequilibrium spin-Hall signals or spin–charge fluctuation correlators. This suggests that the term is best understood as an umbrella description rather than a sharply delimited named effect.

A recurrent theme across the literature is that the apparent size of the SHE depends strongly on how it is probed. Interface-based electrical schemes can mix in interface spin-orbit effects, Rashba effects, spin mixing conductance, and sample-specific interface quality, while alternative probes attempt to access the material response more directly (Ruiz-Gómez et al., 2021).

2. Chiral band topology in spin-1 semimetals

A concrete topological realization is provided by B20-CoSi, described as a prototypal spin-1 chiral semimetal hosting a threefold degeneracy formed by a dispersive band and a flat band with topological charge $\pm 2$. In CoSi/CoFeB/MgO heterostructures, the SHE was investigated by spin Hall magnetoresistance and harmonic Hall measurements, while first-principles calculations were used to compute the intrinsic SHC through the Kubo formalism and spin Berry curvature. The calculated SHC at the Fermi level is about $52\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$, consistent with experiment. Experimentally, the thickness-dependent analysis gives a damping-like spin Hall efficiency of about $3.4 \pm 0.2\%$ and a spin diffusion length of about $4.4 \pm 0.6$ nm. The same work reports thickness-extrapolated room-temperature conductivities on the order of $45\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$ for damping-like and $95\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$ for field-like contributions (Tang et al., 2021).

The most distinctive result is the Fermi-energy dependence of the SHC. Unlike the peak-like structure often associated with Dirac and Weyl fermion-mediated Hall responses, the CoSi SHC is odd-like: it crosses zero at the spin-1 node and develops two antisymmetric extrema of opposite sign on the two sides of the crossing. In a rigid-band picture, maxima of about $-119$ and $\sigma_{xy}^{s}$0 appear when the Fermi level is shifted downward by $\sigma_{xy}^{s}$1 eV or upward by $\sigma_{xy}^{s}$2 eV, respectively (Tang et al., 2021).

Microscopically, the SHC is attributed to hybridization between Co $\sigma_{xy}^{s}$3 and Si $\sigma_{xy}^{s}$4 orbitals together with spin-orbit coupling. The Si $\sigma_{xy}^{s}$5 weight near the Fermi level is only about $\sigma_{xy}^{s}$6, yet it is essential because it enables the interband transitions responsible for the spin Berry curvature. The temperature dependence is also unusual: both damping-like and field-like efficiencies decrease strongly on cooling and nearly vanish near 10 K, a behavior the authors state is inconsistent with simple intrinsic or extrinsic skew-scattering scaling. The central design implication is that Fermi-level tuning, rather than pinning the chemical potential exactly at the topological node, is necessary to optimize spin-current generation in this class of chiral semimetal (Tang et al., 2021).

3. Noncollinear magnetic order as a spin-Hall source without spin–orbit coupling

A separate line of work shows that intrinsic SHE can arise in a magnetic crystal even when spin-orbit coupling is absent, provided the magnetic order is noncollinear and breaks spin-rotation symmetry. This mechanism is explicitly distinguished from the topological Hall effect (THE): it does not require noncoplanarity, nonzero scalar spin chirality, or the real-space Berry-phase mechanism usually associated with THE. The starting point is the double-exchange or $\sigma_{xy}^{s}$7–$\sigma_{xy}^{s}$8 Hamiltonian

$\sigma_{xy}^{s}$9

together with the spin-current operator $\sigma_{xx}$0 and the linear response $\sigma_{xx}$1 (Zhang et al., 2017).

The symmetry argument is central. In coplanar noncollinear magnets, combined operations such as $\sigma_{xx}$2 can suppress the anomalous Hall effect while still allowing selected SHC tensor components. For the kagome-lattice $\sigma_{xx}$3 order, which is coplanar and has zero scalar spin chirality, the symmetry-allowed component is

$\sigma_{xx}$4

By contrast, in collinear magnets the remaining symmetries are sufficient to force all SHC components to vanish in the absence of spin-orbit coupling. This makes the effect less restrictive than the anomalous Hall effect and conceptually distinct from chirality-driven THE (Zhang et al., 2017).

The ab initio realization is given by Mn$\sigma_{xx}$5 compounds. Without SOC, the reported Fermi-level SHC values are $\sigma_{xx}$6 for Mn$\sigma_{xx}$7Ga, $\sigma_{xx}$8 for Mn$\sigma_{xx}$9Ge, and $\pm 2$0 for Mn$\pm 2$1Sn, while Mn$\pm 2$2Ir is symmetry-forbidden in the SOC-free limit (Zhang et al., 2017). The significance is twofold. First, it shows that heavy elements are not a prerequisite for large intrinsic SHC. Second, it grounds a broader “chromo-spin” interpretation in magnetic symmetry rather than in relativistic spin–orbit entanglement alone.

4. Interface-free and element-specific observation in CuBi

The most direct spectroscopic observation among the cited works is the interface-free detection of the direct SHE in highly Bi-doped Cu, specifically Cu$\pm 2$3Bi$\pm 2$4. The samples were CuBi thin films of thickness 20–50 nm grown on SiO$\pm 2$5/Si, capped with 3 nm Al$\pm 2$6O$\pm 2$7, patterned into electrode loops, and driven at current densities up to about $\pm 2$8. Soft X-ray PEEM with circularly polarized light at the Cu $\pm 2$9 and $52\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$0 edges was used to measure XMCD at the top surface. The information depth is about 5 nm, effectively preventing cancellation between opposite surfaces and avoiding magnetic-interface contributions. The XMCD asymmetry was defined as

$52\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$1

with $52\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$2 and $52\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$3 the two circular polarizations (Ruiz-Gómez et al., 2021).

The observed signal has the spectral hallmarks expected for SHE-induced surface spin accumulation. It is maximal at $52\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$4, inverted at $52\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$5, changes sign with current direction, has a narrow peak with $52\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$6 eV, and peaks about 0.2 eV before the $52\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$7 absorption maximum, consistent with spin accumulation near the Fermi edge. Linear fits of XMCD versus current density gave $52\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$8 at Cu $52\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$9 and $3.4 \pm 0.2\%$0 at Cu $3.4 \pm 0.2\%$1. From these values the induced magnetic moment per Cu atom, averaged over the PEEM probing depth, was extracted as

$3.4 \pm 0.2\%$2

corresponding at typical current densities to an average detected moment of about $3.4 \pm 0.2\%$3 (Ruiz-Gómez et al., 2021).

This measured moment is of the same order as Pt values obtained previously by magneto-optics, quoted in the paper as about $3.4 \pm 0.2\%$4 for the topmost atomic layer and $3.4 \pm 0.2\%$5 for the upper half of the sample. Using a Zhang/Stamm spin-diffusion model with depth weighting $3.4 \pm 0.2\%$6 and $3.4 \pm 0.2\%$7 nm, the data imply that for reasonable spin diffusion lengths the spin Hall angle in CuBi is at least around $3.4 \pm 0.2\%$8 for thicker films and could be larger if the spin diffusion length is shorter; for the thinnest 20 nm sample, matching the data would require $3.4 \pm 0.2\%$9 nm and $4.4 \pm 0.6$0. The sign is consistent with a negative spin Hall angle in CuBi (Ruiz-Gómez et al., 2021). The broader significance is methodological: interface-free, element-specific spectroscopy directly constrains the material parameter rather than an interface-dressed effective response.

5. Detection schemes, correlated observables, and device geometries

Several works expand SHE metrology beyond standard transport. One proposal introduces hybrid spin noise spectroscopy in a semiconductor channel with a longitudinal field $4.4 \pm 0.6$1, optical Faraday detection of spin fluctuations, and electrical detection of a transverse voltage. In this framework, the standard spin–spin correlator is not sensitive to the spin Hall coefficient $4.4 \pm 0.6$2 at leading order, but the mixed spin–voltage correlator $4.4 \pm 0.6$3 is linear in $4.4 \pm 0.6$4, while the voltage autocorrelator $4.4 \pm 0.6$5 is quadratic in $4.4 \pm 0.6$6. The physical mechanism is that a spin fluctuation is converted by the SHE into a charge dipole, which produces a transverse electric field and hence a measurable voltage. This establishes correlated fluctuations, rather than mean transport alone, as a probe of spin–charge coupling (Slipko et al., 2013).

A distinct electrical scheme uses an asymmetric Pt sample driven by an unpolarized alternating current, $4.4 \pm 0.6$7. In rolled Pt foils of size $4.4 \pm 0.6$8 with edge thicknesses differing by about $4.4 \pm 0.6$9 mm, the asymmetry makes the opposite edges inequivalent, so edge spin accumulation does not cancel electrically. The nonequilibrium spin-Hall contribution is argued to be even in current and symmetric under magnetic-field reversal in the measurement geometry, whereas the ordinary Hall contribution is odd in both. Under AC drive, this produces a characteristic $45\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$0 transverse voltage. The experiments used $45\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$1 mA at 22 Hz and reported a crossover in the extracted double-frequency amplitude together with an estimated Pt spin Hall angle $45\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$2 (Chiang et al., 2021).

Another route relies on angular-momentum conservation. Because a spin current carries angular momentum, its divergence generates a local torque density,

$45\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$3

which can actuate a nanomechanical resonator. For a rectangular film, the torque localizes near the surfaces, and for large dimensions relative to the spin diffusion length the paper gives $45\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$4 and $45\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$5. In a doubly clamped beam with a Pt top layer and piezoelectric readout, COMSOL-based simulations found a DC deflection of about $45\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$6 fm/mA and strong AC excitation at 19.18, 56.91, and 139.24 MHz. For the 19.18 MHz mode, the estimated output was approximately $45\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$7 nV at 300 K and $45\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$8 nV at 4 K, assuming $45\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$9 (Boales et al., 2015).

Hybrid graphene/Pt nanostructures illustrate how device architecture can amplify inverse-SHE detection. In monolayer graphene/Pt devices, the signal exceeds $95\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$0 and is stated to be two orders of magnitude larger than the largest reported values in fully metallic systems. The analysis attributes this to efficient spin injection, suppressed current shunting, complete spin absorption in Pt, and the large resistivity of graphene. The extracted Pt spin Hall angle is reported as $95\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$1, with values across devices and gate voltages in the range $95\,(\hbar/e)\,\Omega^{-1}\,\mathrm{cm}^{-1}$2. The same platform was used to characterize spin precession and extract graphene spin-relaxation times in the hundreds-of-picoseconds range by fitting inverse-SHE signals with a 1D diffusive model (Torres et al., 2017).

6. Conceptual limits, common misconceptions, and design implications

The first conceptual boundary is terminological. None of the cited papers presents a separate universally accepted phenomenon explicitly named “Chromo-Spin Hall Effect.” Instead, the literature supports several related classes of SHE behavior: conventional spin–orbit-driven SHE, chiral-band-topology-modified SHC in a spin-1 semimetal, noncollinear-magnetism-induced SHE without SOC, and unconventional detection symmetries in fluctuation, AC, mechanical, and hybrid spin-transport experiments (Tang et al., 2021).

A second boundary concerns frequent conflations with neighboring Hall phenomena. The SOC-free SHE in a chiral magnetic lattice is not the topological Hall effect: it does not require noncoplanarity or nonzero scalar spin chirality (Zhang et al., 2017). Likewise, in B20-CoSi the topological node does not maximize the SHC; the SHC vanishes at the spin-1 band crossing and changes sign above and below it, so Fermi-level tuning away from the node is the indicated route to enhancement (Tang et al., 2021). In semiconductors, conventional spin noise spectroscopy is not, by itself, a sensitive SHE probe at leading order; the sensitivity emerges in mixed spin–voltage or voltage–voltage correlators (Slipko et al., 2013).

A third issue is metrological uncertainty. Interface-based electrical methods can obscure intrinsic material parameters, which motivates interface-free XMCD-PEEM in CuBi (Ruiz-Gómez et al., 2021). More generally, reported values of the spin Hall angle in materials such as Pt and Pd vary widely, in some cases by orders of magnitude, owing to growth conditions, thickness, impurity concentration, frequency, spin diffusion length, and interfacial assumptions (Boales et al., 2015). This explains why the literature places considerable weight on orthogonal probes—spectroscopic, fluctuation-based, mechanical, and hybrid-channel transport.

Taken together, these results suggest a coherent research program. Chiral semimetals motivate Fermi-level engineering of odd-in-energy SHC profiles. Noncollinear antiferromagnets provide a route to large SHE without heavy elements. Interface-free spectroscopy constrains intrinsic spin accumulation directly. Correlation measurements and nanomechanical devices supply probes that are complementary to standard Hall or spin-torque measurements. In that sense, “Chromo-Spin Hall Effect” is most usefully understood as a compact label for the subset of spin Hall physics in which chirality—whether encoded in band topology, magnetic order, or response symmetry—plays the organizing role.