---
title: Tunneling Spin Hall Effect
url: https://www.emergentmind.com/topics/tunneling-spin-hall-effect
type: topic
---

# Tunneling Spin Hall Effect

Searching arXiv for recent and foundational papers on tunneling spin Hall effect and closely related formulations.
The tunneling spin Hall effect denotes a class of transverse spin-transport phenomena in which quantum tunneling, evanescent propagation, or interface-selective coherent transmission generates a spin Hall response that is absent from, or parametrically distinct from, bulk diffusive spin Hall transport. Across the literature, the term encompasses several closely related settings: spin-polarized tunneling in biased magnetic tunnel junctions with Rashba spin-orbit coupling [1305.3467], spin transfer driven by spin Hall induced spin accumulation through an insulator or vacuum [1603.04240; 1507.06447], local sensing of spin Hall accumulation by scanning tunneling microscopy in tungsten films [1706.07882; 1707.06116], Zener-tunneling spin Hall currents in HgTe quantum wells [1107.3452], coherent-tunneling spin and valley Hall effects in graphene barriers [2409.17611], and transverse spin currents generated by spin-dependent Andreev reflection in normal-metal/\(p\)-wave-magnet/superconductor junctions [2507.09112]. The unifying feature is that the transverse response is controlled by tunneling amplitudes, interfacial phases, or evanescent-state structure rather than by conventional bulk scattering alone.

## 1. Conceptual scope and defining mechanisms

In the narrowest sense, the tunneling spin Hall effect refers to transverse spin currents produced during tunneling across a barrier. In a biased magnetic tunnel junction with Rashba spin-orbit coupling inside the barrier, the tunneling electrons experience a spin-orbit coupling inside the barrier due to the applied electrical field, and both charge and spin Hall currents are calculated as functions of position inside the barrier and the angle between the magnetizations of the electrodes [1305.3467]. In that formulation, the Hall response is carried by evanescent states and is localized inside the barrier near the interfaces.

A broader usage appears in spin Hall effect induced spin transfer through an insulator. There, charge current in a normal metal generates spin accumulation at the edge of the sample in the transverse direction, and this spin accumulation, or spin voltage, enables quantum tunneling of spin through an insulator or vacuum to reach a ferromagnet without transferring charge [1603.04240]. This is not a transverse Hall current inside the barrier in the same sense as the magnetic-tunnel-junction problem, but it is explicitly presented as a tunneling consequence of the spin Hall effect and provides the microscopic basis for spin-transfer torque and spin pumping across nominally insulating spacers.

A still wider family of tunneling Hall phenomena arises when the tunneling probability becomes asymmetric in the conserved transverse momentum. In HgTe quantum wells under Zener breakdown, the tunneling transition probability depends asymmetrically on the parallel momentum of the carriers to the barrier, and in HgTe the asymmetry is opposite for each spin, producing a spin current flowing in the perpendicular direction to the applied field [1107.3452]. In coherent graphene barriers with broken inversion symmetry and proximity-induced spin-orbit coupling, transmitted electrons acquire a finite spin- and valley-dependent backreflection geometric phase when the two interfaces of the barrier are asymmetric, causing spin- and valley-dependent skew coherent tunneling and hence transverse spin and valley Hall currents [2409.17611]. In normal-metal/\(p\)-wave-magnet/superconductor junctions, spin-dependent Andreev reflection exhibits strong asymmetry with respect to the transverse momentum, giving rise to a pure transverse spin Hall current with zero net charge [2507.09112].

These formulations suggest that “tunneling spin Hall effect” is best understood as a family of interfacial or barrier-mediated spin Hall responses whose microscopic origin lies in tunneling selection rules, spin-orbit-coupled phase accumulation, or skew transmission in \(k_y\).

## 2. Barrier-localized Hall currents in magnetic tunnel junctions

A canonical theoretical realization is the magnetic tunnel junction with Rashba spin-orbit coupling in the barrier [1305.3467]. The system consists of two semi-infinite ferromagnetic electrodes separated by an insulating barrier of thickness \(d\) along the \(y\)-direction, with a dc bias \(V\) applied so that an electric field \(E=V/d\) resides inside the barrier. In the free-electron approximation, the total Hamiltonian is
\[
H = -\frac{\hbar^2}{2m}\nabla^2 + V(y) + H_R,
\]
with
\[
H_R = \alpha_R \bigl[\boldsymbol{\sigma}\times\mathbf{p}\bigr]\cdot\hat{\mathbf{e}}_y,\qquad \alpha_R=\lambda E.
\]
The Rashba coupling therefore vanishes outside the barrier and is electrically induced within it [1305.3467].

The barrier solutions are evanescent, and continuity of \(\Psi\) and \(\partial_y\Psi\) at the interfaces fixes the spin-dependent amplitudes. Because of \(H_R\), a tunneling electron acquires a transverse velocity component. To first order in \(\alpha_R\), only the Rashba term contributes to the transverse Hall current [1305.3467]. The charge-current density along \(x\) is written as
\[
J_c^x(y,\theta)=J_0^L e^{-2\kappa_L y}+J_0^R e^{-2\kappa_R(d-y)},
\]
with \(J_0^p\propto [T_{p,\uparrow}-T_{p,\downarrow}]\cos\theta_p\). The spin-current tensor yields nonzero components
\[
J_s^{zx}(y)=-J_s^{xz}(y), \qquad J_s^{yz}(y),
\]
and \(J_s^{zx}\) is identified as the spin-Hall current [1305.3467].

Several specific consequences are emphasized. Both \(J_c^x(y)\) and \(J_s^{zx}(y)\) peak sharply within a few Å of each interface, then fall off over \(5\)–\(10\) Å. Near the right interface, both currents scale as \(\cos\theta\), where \(\theta\) is the angle between the electrode magnetizations, while near the left interface the dependence on \(\theta\) is very weak because \(\mathbf M_L\) is fixed [1305.3467]. Rotating \(\mathbf M_R\) by \(90^\circ\) fully suppresses the Hall currents from the right side, making the effect magnetization-controllable.

This formulation is important because it makes explicit that a tunneling spin Hall response need not be a bulk spin Hall current entering a barrier from outside. Instead, the barrier itself can host a transverse spin current carried by evanescent states, with the relevant control parameters being bias-induced Rashba coupling, interfacial exchange polarization, and magnetization geometry [1305.3467].

## 3. Spin Hall induced tunneling through insulators and the torque formalism

A second major formulation concerns spin transport generated by a spin Hall spin accumulation in a normal metal and transmitted by quantum tunneling through an insulating barrier [1603.04240; 1507.06447; 1607.03409]. In a normal metal/insulator/ferromagnetic insulator trilayer, when charge current passes through a normal metal that exhibits spin Hall effect, spin accumulates at the edge of the sample in the transverse direction. This spin accumulation, or spin voltage, enables quantum tunneling of spin through an insulator or vacuum to reach a ferromagnet without transferring charge [1603.04240].

The normal-metal Hamiltonian is written as
\[
H_N=\frac{p^2}{2m}-\mu_x^\sigma,
\]
where \(\mu_x^\sigma=\pm \tfrac12 |\boldsymbol\mu(x)|\) encodes the spin accumulation, while the insulator has
\[
H_O=\frac{p^2}{2m}+V_1,
\]
and the ferromagnetic insulator is described by
\[
H_{FI}=\frac{p^2}{2m}+V_0+\Gamma\,\mathbf S\cdot\boldsymbol\sigma
\]
[1603.04240]. Matching wave functions across the trilayer yields the injected spin and hence the spin-transfer torque
\[
\boldsymbol\tau
=\frac{\Gamma S a^2 N_F}{\hbar}
\Bigl[
G_r\,\mathbf S\times(\mathbf S\times\boldsymbol\mu_0)
+G_i\,\mathbf S\times\boldsymbol\mu_0
\Bigr],
\]
where \(G_r\) and \(G_i\) are the damping-like and field-like spin-mixing conductance components [1603.04240].

In the ferromagnetic-insulator case, the spin injection is proportional to
\[
e^{-(q_++q_-)d},
\]
so both damping-like and field-like torques decay exponentially with oxide thickness [1603.04240]. In the ferromagnetic-metal case, by contrast, propagating solutions in the ferromagnet introduce oscillatory integrals in the ferromagnet thickness, leading to non-monotonic dependence on both ferromagnet and barrier thickness via interference [1603.04240]. The minimal tunneling model of spin-transfer torque and spin pumping caused by spin Hall effect reaches the same conclusion that the ratio of damping-like to field-like component depends on the tunneling wave function and is strongly influenced by interface \(s\)-\(d\) coupling, insulating gap, and layer thickness, while spin relaxation plays a minor role [1507.06447].

Within that minimal model, the normal metal carries a spin Hall generated “spin voltage” \(\mu_0\), and the spin current just inside the metal is written as
\[
J_s^{in}\simeq (\hbar/2e)\,\theta_{SH}\,j_c\,\hat\sigma\times\hat z,
\]
so all subsequent spin currents and torques inherit a factor \(\mu_0\propto \theta_{SH}E\) [1507.06447]. The conductance ratio is given by
\[
\frac{G_r}{G_i}
=
\frac{k_{0\downarrow}(q_+-q_-)}{k_{0\downarrow}^2+q_+q_-},
\]
showing explicit dependence on barrier height, \(s\)-\(d\) coupling, and Fermi wavevectors [1507.06447].

The same quantum-tunneling boundary condition also enters the theory of spin Hall magnetoresistance in normal-metal/ferromagnet bilayers [1607.03409]. There, spin diffusion in the normal metal generates an interface spin current
\[
\boldsymbol j_s^{int}
=
\frac{\hbar}{2e^2}
\bigl[
G_r\,\hat{\mathbf m}\times(\hat{\mathbf m}\times\boldsymbol\mu_0)
+
G_i\,\hat{\mathbf m}\times\boldsymbol\mu_0
\bigr],
\]
and the longitudinal and transverse resistivities obey
\[
\rho_{long}=\rho_0+\Delta\rho_0+\Delta\rho_1\sin^2\theta,\qquad
\rho_{trans}=\Delta\rho_1\sin\theta\cos\theta\sin\varphi-\Delta\rho_2\sin\theta\cos\varphi
\]
[1607.03409]. In this framework, the barrier thickness \(t\) and gap \(V_0-\epsilon_F\) enter only through \(G_{\uparrow\downarrow}(t,V_0,\Gamma)\), so quantum tunneling directly controls the observable magnetoresistance response [1607.03409].

A plausible implication is that this line of work defines a “tunneling spin Hall effect” less by a literal Hall current inside the barrier than by spin Hall generated spin bias whose transmission is tunnel-limited and whose observables are torque, pumping, and magnetoresistance.

## 4. Local sensing by STM and tungsten-film experiments

A distinct experimental strand uses scanning tunneling microscopy to probe spin Hall accumulation locally in current-carrying tungsten films [1706.07882; 1707.06116]. The STM-based potentiometry technique exploits the built-in scanning tunneling spectroscopy capability of a conventional STM to map the local surface potential \(V(x,y)\) with nanometer lateral resolution and sub-millivolt sensitivity [1706.07882]. The core relation is
\[
V_s - V(x,y) = V_{ts} + I_t R_{tip},
\]
and because the zero-current intercept \(I_t=0\) corresponds to \(V_{ts}=0\), one finds \(V_s=V(x,y)\) at zero current. In practice, the point of intersection of the measured \(I\)-\(V\) curve with \(I=0\) yields the local surface potential directly [1706.07882].

For current-carrying \(\beta\)-phase tungsten films, the local surface potential versus \(I_{bias}\) is strictly linear, with a slope corresponding to \(\sim 260.7\,\Omega\) between tip position and reference contact. Rastering the tip along \(x\) at constant \(I_{bias}=10\) mA yields a potential gradient \(\partial V/\partial x \approx 1.20\) mV/\(\mu\)m, in excellent agreement with the macroscopic resistivity. A single gold nanoparticle produces a local distortion of \(V(x,y)\) resolved over a few nanometers [1706.07882]. The reported precision is sub-mV, with calibration of the zero-current intercept accurate to better than \(\pm 0.2\) mV [1706.07882].

The spin Hall sensing protocol uses a thin \(\beta\)-phase tungsten film driven by a charge-current density \(J_c\) along the \(x\)-direction. The macroscopic spin Hall voltage across width \(w\) is given by
\[
V_{SH}=\theta_{SH}\rho J_c w,
\]
with \(\rho\approx 240\,\mu\Omega\cdot\)cm for tungsten [1706.07882]. During pulsed-bias measurements, the STM tip freezes the tunneling gap at \(V_{ts}=\pm 0.5\) V using the potentiometry compensation \(V_s=V_{ts}+V(x,y)\) [1706.07882]. Two effects are then observed: a gradual increase of \(I_t\) during the current pulse attributed to local thermal expansion, and an asymmetry in \(I_t\) when \(V_{ts}\) is switched between \(+0.5\) V and \(-0.5\) V under identical \(J_c\), which grows with pulse time and pulse amplitude [1706.07882].

That asymmetry is interpreted as a signature of the spin Hall effect: for \(V_{ts}>0\), electrons tunnel from the film toward the tip and carry the spin polarization induced by the spin Hall effect, whereas for \(V_{ts}<0\), electrons tunnel from the tip into the film and are unpolarized [1706.07882]. The normalized difference
\[
\Delta I_t = I_t(+V_{ts})-I_t(-V_{ts}),
\]
further normalized by the mean tunneling current \(\langle I_t\rangle\), isolates the spin Hall induced component from thermal background [1706.07882]. The reported signal size is \(\Delta I_t/\langle I_t\rangle\) of order \(10^{-3}\) to \(10^{-2}\) under \(I_{bias}\) of several mA [1706.07882].

A follow-up tungsten study extends the analysis to both bare W tips and Fe-coated W tips [1707.06116]. For a bare W tip, reversing only the tunneling voltage yields
\[
\frac{\Delta I}{I}
\equiv
\frac{I(+V_g)-I(-V_g)}{[I(+V_g)+I(-V_g)]/2}
\approx \beta P_s,
\]
with \(P_s\) the local spin polarization at the W surface [1707.06116]. Quantitatively, \(I_{bias}=0\) mA gives \(\Delta I/I\approx 0\), \(I_{bias}=4\) mA gives \(\Delta I/I\approx 1.5\%\), and \(I_{bias}=7\) mA gives \(\Delta I/I\approx 3.5\%\) with saturation. At fixed \(I_{bias}=7\) mA, \(\Delta I/I\approx 3.5\%\) remains nearly constant for \(0.15\,\text{V}\le |V_g|\le 0.5\,\text{V}\), and thinner films show larger asymmetry, while at \(40\) nm thickness \(\Delta I/I\) falls below \(1\%\) [1707.06116].

For an Fe-coated tip with spin-polarized density of states, reversing the film bias current introduces an additional asymmetry
\[
\frac{\Delta I'}{I}
\equiv
\frac{I(I_{bias})-I(-I_{bias})}{\langle I\rangle}
\approx \alpha P_s P_t,
\]
and experimentally \(\Delta I'/I\approx 1\%\) at \(I_{bias}=4\) mA and \(\approx 2.5\%\) at \(I_{bias}=7\) mA [1707.06116]. Assuming \(\beta\approx 1\), the measured \(\Delta I/I\approx P_s\) gives \(P_s\approx 0.035\) at \(J_c\approx 1\times 10^7\) A/cm\(^2\), and using \(P_s\approx \theta_{SH}(\hbar/2e)J_c\tau_s\) with \(\tau_s\approx 10^{-13}\) s yields \(\theta_{SH}\approx 0.3\), in line with literature values for \(\beta\)-W [1707.06116].

These experiments do not directly image a transverse spin current in the barrier. Rather, they establish a nanometer-scale tunneling probe of spin Hall induced spin accumulation at a conducting surface. This suggests that, experimentally, “tunneling spin Hall effect” may denote either a transport mechanism or a tunneling-based detection modality.

## 5. Momentum-asymmetric tunneling, Berry phase, and coherent skew transmission

Another major route to a tunneling spin Hall effect is asymmetry in transmission with respect to conserved transverse momentum. In HgTe quantum wells under a uniform electric field \(F\) along \(x\), the low-energy Hamiltonian for fixed spin block \(s_z=\pm1\) is
\[
H_{s_z}(\mathbf k)=\epsilon(k)\mathbf 1 + M(k)\sigma_z + A(k_y\sigma_y+s_z k_x\sigma_x),
\]
and the tunneling probability \(P(k_y,s_z)\) for a valence-band hole to reach the conduction band is asymmetric in \(k_y\) [1107.3452]. To leading order,
\[
P(k_y,s_z)
\simeq
1-\frac{\pi M_0^2}{FA}
-\frac{\pi}{FA}\Bigl[(Ak_y)^2+s_z F B\,k_y+\mathcal O(k_y^2,B^2)\Bigr].
\]
The peak occurs at
\[
k_y^{max}
=
-\,s_z\,\frac{F B}{A^2}
\Bigl[1-\tfrac52 \tfrac{B M_0}{A^2}\Bigr],
\]
so opposite spins see peaks shifted to opposite signs of \(k_y\) [1107.3452]. Since \(P(k_y,s_z)\neq P(-k_y,s_z)\), more spin-up electrons tunnel into positive-\(k_y\) channels and more spin-down electrons into negative-\(k_y\) channels, producing a dc spin Hall current. The authors report that the resulting spin current can be as large as \(20\)–\(30\%\) of the total charge current [1107.3452]. The origin is attributed to a spin- and \(k_y\)-dependent Berry phase acquired upon adiabatic reflection in the gapped region [1107.3452].

A conceptually similar but distinct coherent-tunneling mechanism is formulated for graphene with broken inversion symmetry and proximity-induced spin-orbit coupling [2409.17611]. The left and right electrodes are gapless graphene, while the central barrier carries a staggered sublattice potential \(\Delta\) and intrinsic spin-orbit coupling \(\lambda\). The net transmission amplitude has a Fabry–Pérot form
\[
t_{s,\tau}(\theta)=\frac{t_L t_R}{1-r_L r_R e^{i\phi(\theta)}},
\]
where the total backreflection phase is split into a kinetic part \(\phi_0(\theta)=2p_{b,x}d\) and a geometric phase
\[
\phi_G(\theta)
=
\arctan\!\Bigl[\tau\,\Delta\,(\mu_R-\mu_L)\,k_y\,f_{s,\tau}(\theta)\Bigr].
\]
The key symmetry is that \(\phi_G\) flips sign under \(k_y\to -k_y\), but \(\phi_0\) does not [2409.17611]. The result is skew tunneling:
\[
T_{s,\tau}(+k_y)\neq T_{s,\tau}(-k_y),
\]
which yields transverse conductances and a spin Hall angle
\[
\theta_H^s
\approx
\frac{\sum_{s,\tau} s\,\sigma_{s,\tau}^\perp}{\sum_{s,\tau}\sigma_{s,\tau}^\parallel}.
\]
Using \(d=25\) nm, \(\mu_L=+1\) meV, \(\mu_b=+6.5\) meV, \(\mu_R=-5.6\) meV, \(\Delta=3.25\) meV, and \(\lambda=2.5\) meV, the spin Hall angle peaks at \(\epsilon\approx 1.5\) meV with maximum \(\simeq 0.20\), and the valley Hall angle peaks at \(\epsilon\approx 0.6\) meV with maximum \(\simeq 0.20\) [2409.17611]. Both effects vanish when \(\mu_R=\mu_L\), and as \(\Delta\to 0\) both Hall angles vanish [2409.17611].

These examples establish a coherent-tunneling paradigm in which skew transmission originates not from impurity scattering but from geometric phase, pseudospin rotation, or Berry curvature effects encoded in the barrier.

## 6. Superconducting and analogous tunneling Hall phenomena

A recent extension places the tunneling spin Hall effect in superconducting hybrid structures [2507.09112]. In a normal-metal/\(p\)-wave-magnet/superconductor junction, the central region has single-particle Hamiltonian
\[
h(\mathbf k)
=
\xi_{\mathbf k}\sigma_0
+
\mathcal J (k_x\cos\phi+k_y\sin\phi)\Theta(x)\Theta(L-x)\sigma_z,
\]
where the term proportional to \(\mathcal J\) shifts the spin-up and spin-down Fermi surfaces in momentum space by equal and opposite amounts along direction \(\phi\) [2507.09112]. The transverse spin conductance is derived using the nonequilibrium Green’s function approach, and the spin-conserving Andreev reflection probability is identified from the corresponding Green-function blocks.

At zero temperature and small bias, the transverse conductance for spin \(s\) is
\[
\sigma_s^{yx}
=
\frac{e^2}{h}\,
\mathrm{Im}\!\sum_{k_y}
\mathcal M_s
\Bigl[
(G_{he}^r\Gamma_{L,e}G_{eh}^a)_{ss}
+
(G_{eh}^r\Gamma_{L,h}G_{he}^a)_{ss}
+
(G^r\Gamma_R G^a)_{ee,ss}
\Bigr],
\]
and time-reversal symmetry yields
\[
\sigma_\uparrow^{yx}=-\sigma_\downarrow^{yx},\qquad
\sigma_{charge}^{yx}=0.
\]
Thus the junction supports a pure spin current with zero net charge [2507.09112]. The spin Hall angle is
\[
\theta_{SH}=2\,\frac{\sigma_\uparrow^{yx}}{\sigma^{xx}}.
\]
The effect vanishes when the splitting direction is parallel to the junction normal, \(\phi=0\) or \(\pi\), and is largest when the splitting is perpendicular to the interface, \(\phi=\pm \pi/2\) [2507.09112]. For reasonable parameters, the paper reports \(\theta_{SH}\sim 0.2\) [2507.09112].

Although not an electron-spin Hall effect in the usual condensed-matter sense, the spin Hall effect of light in photon tunneling provides an optical analogue [1004.4345]. In a prism–air–prism barrier, polarization-dependent transverse shifts arise in the transmitted beam, with opposite shifts for left- and right-circular polarization:
\[
\Delta y_\pm
=
\pm \frac{1}{k_0}
\frac{\cot\theta_i}{|t_p|^2+|t_s|^2}
\Bigl[(|t_p|^2+|t_s|^2)-2\eta |t_p||t_s|\cos\Delta\phi\Bigr].
\]
For \(n_1=1.5\), \(d=0.2\lambda\), and \(\theta_i=45^\circ\), varying \(n_2\) from \(1.3\) to \(1.7\) gives \(\Delta y_\pm\) of order \(0.1\)–\(0.3\lambda\), compared with only a few \(0.01\lambda\) in single-interface refraction [1004.4345]. The paper attributes the effect to total angular momentum conservation and emphasizes enhancement by the tunneling geometry [1004.4345]. This suggests a broader structural analogy: tunneling can amplify spin-orbit-coupled transverse responses across fermionic and photonic systems alike.

## 7. Interpretive issues, related effects, and emerging directions

The literature grouped under tunneling spin Hall effect is not terminologically uniform. Some papers reserve the phrase for a genuine transverse spin current generated during tunneling, as in Rashba-coupled magnetic tunnel junctions [1305.3467], HgTe Zener junctions [1107.3452], coherent graphene barriers [2409.17611], and \(p\)-wave-magnet/superconductor structures [2507.09112]. Others emphasize spin Hall induced tunneling of spin through an insulating barrier, where the primary observables are spin-transfer torque, spin pumping, or spin Hall magnetoresistance rather than a barrier-internal Hall current [1603.04240; 1507.06447; 1607.03409]. Still others use tunneling chiefly as a detection channel for spin Hall accumulation, as in STM studies of tungsten films [1706.07882; 1707.06116].

Several distinctions are therefore necessary.

First, a tunneling spin Hall effect need not involve a bulk spin Hall current inside the barrier. In the magnetic-tunnel-junction theory, the transverse response is localized near interfaces and carried by evanescent states [1305.3467]. In coherent graphene and HgTe formulations, the defining ingredient is momentum-skew tunneling associated with geometric phase or Berry phase [1107.3452; 2409.17611]. In the superconducting case, the response is generated by asymmetric spin-dependent Andreev reflection [2507.09112].

Second, the effect may or may not carry net charge. The superconducting \(p\)-wave-magnet junction explicitly yields a pure transverse spin current with zero net charge [2507.09112], whereas the magnetic tunnel junction with Rashba coupling hosts both charge and spin Hall currents inside the barrier [1305.3467].

Third, the relevant control parameters differ by platform. These include barrier thickness and insulating gap in trilayers [1603.04240; 1507.06447; 1607.03409], magnetization angle in tunnel junctions [1305.3467], electric field in Zener systems [1107.3452], interface asymmetry and backreflection phase in graphene barriers [2409.17611], and Fermi-surface-splitting direction in \(p\)-wave magnets [2507.09112].

The emerging direction is toward high-efficiency coherent or interfacial conversion. In graphene coherent tunneling, the spin Hall angle reaches \(\simeq 0.20\) without disorder scattering [2409.17611]. In the superconducting \(p\)-wave-magnet junction, \(\theta_{SH}\sim 0.2\) is likewise reported [2507.09112]. In tungsten STM studies, local spin polarization \(P_s\approx 0.035\) is inferred at \(J_c\approx 1\times10^7\) A/cm\(^2\), with estimated \(\theta_{SH}\approx 0.3\) for \(\beta\)-W [1707.06116]. These values arise in distinct observables and should not be conflated, but they collectively indicate that tunnel-mediated spin Hall responses can be quantitatively significant.

A plausible implication is that the field is converging on a common viewpoint: quantum tunneling is not merely a passive bottleneck for spin Hall transport, but an active generator, modulator, and probe of transverse spin phenomena.

Source: https://www.emergentmind.com/topics/tunneling-spin-hall-effect