Tunneling Spin Hall Effect

• Tunneling spin Hall effect is an interfacial phenomenon where quantum tunneling yields transverse spin currents controlled by barrier parameters.
• It encompasses mechanisms such as Rashba-coupled magnetic tunnel junctions, spin accumulation through insulators, and Berry-phase-driven asymmetries.
• Experiments using STM, HgTe quantum wells, and p-wave superconducting junctions demonstrate tunable spin Hall responses with significant spin Hall angles.

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 (Vedyayev et al., 2013), spin transfer driven by spin Hall induced spin accumulation through an insulator or vacuum (Chen et al., 2016, Chen et al., 2015), local sensing of spin Hall accumulation by scanning tunneling microscopy in tungsten films (Xie et al., 2017, Xie et al., 2017), Zener-tunneling spin Hall currents in HgTe quantum wells (Lasia et al., 2011), coherent-tunneling spin and valley Hall effects in graphene barriers (Zeng, 2024), and transverse spin currents generated by spin-dependent Andreev reflection in normal-metal/$p$-wave-magnet/superconductor junctions (Zeng, 12 Jul 2025). 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 (Vedyayev et al., 2013). 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 (Chen et al., 2016). 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 (Lasia et al., 2011). 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 (Zeng, 2024). 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 (Zeng, 12 Jul 2025).

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 (Vedyayev et al., 2013). 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 (Vedyayev et al., 2013).

The barrier solutions are evanescent, and continuity of $\Psi$ and $p$0 at the interfaces fixes the spin-dependent amplitudes. Because of $p$1, a tunneling electron acquires a transverse velocity component. To first order in $p$2, only the Rashba term contributes to the transverse Hall current (Vedyayev et al., 2013). The charge-current density along $p$3 is written as

$p$4

with $p$5. The spin-current tensor yields nonzero components

$p$6

and $p$7 is identified as the spin-Hall current (Vedyayev et al., 2013).

Several specific consequences are emphasized. Both $p$8 and $p$9 peak sharply within a few Å of each interface, then fall off over $k_y$0–$k_y$1 Å. Near the right interface, both currents scale as $k_y$2, where $k_y$3 is the angle between the electrode magnetizations, while near the left interface the dependence on $k_y$4 is very weak because $k_y$5 is fixed (Vedyayev et al., 2013). Rotating $k_y$6 by $k_y$7 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 (Vedyayev et al., 2013).

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 (Chen et al., 2016, Chen et al., 2015, Ok et al., 2016). 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 (Chen et al., 2016).

The normal-metal Hamiltonian is written as

$k_y$8

where $k_y$9 encodes the spin accumulation, while the insulator has

$d$0

and the ferromagnetic insulator is described by

$d$1

(Chen et al., 2016). Matching wave functions across the trilayer yields the injected spin and hence the spin-transfer torque

$d$2

where $d$3 and $d$4 are the damping-like and field-like spin-mixing conductance components (Chen et al., 2016).

In the ferromagnetic-insulator case, the spin injection is proportional to

$d$5

so both damping-like and field-like torques decay exponentially with oxide thickness (Chen et al., 2016). 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 (Chen et al., 2016). 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 $d$6-$d$7 coupling, insulating gap, and layer thickness, while spin relaxation plays a minor role (Chen et al., 2015).

Within that minimal model, the normal metal carries a spin Hall generated “spin voltage” $d$8, and the spin current just inside the metal is written as

$d$9

so all subsequent spin currents and torques inherit a factor $y$0 (Chen et al., 2015). The conductance ratio is given by

$y$1

showing explicit dependence on barrier height, $y$2-$y$3 coupling, and Fermi wavevectors (Chen et al., 2015).

The same quantum-tunneling boundary condition also enters the theory of spin Hall magnetoresistance in normal-metal/ferromagnet bilayers (Ok et al., 2016). There, spin diffusion in the normal metal generates an interface spin current

$y$4

and the longitudinal and transverse resistivities obey

$y$5

(Ok et al., 2016). In this framework, the barrier thickness $y$6 and gap $y$7 enter only through $y$8, so quantum tunneling directly controls the observable magnetoresistance response (Ok et al., 2016).

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 (Xie et al., 2017, Xie et al., 2017). The STM-based potentiometry technique exploits the built-in scanning tunneling spectroscopy capability of a conventional STM to map the local surface potential $y$9 with nanometer lateral resolution and sub-millivolt sensitivity (Xie et al., 2017). The core relation is

$V$0

and because the zero-current intercept $V$1 corresponds to $V$2, one finds $V$3 at zero current. In practice, the point of intersection of the measured $V$4-$V$5 curve with $V$6 yields the local surface potential directly (Xie et al., 2017).

For current-carrying $V$7-phase tungsten films, the local surface potential versus $V$8 is strictly linear, with a slope corresponding to $V$9 between tip position and reference contact. Rastering the tip along $E=V/d$0 at constant $E=V/d$1 mA yields a potential gradient $E=V/d$2 mV/$E=V/d$3m, in excellent agreement with the macroscopic resistivity. A single gold nanoparticle produces a local distortion of $E=V/d$4 resolved over a few nanometers (Xie et al., 2017). The reported precision is sub-mV, with calibration of the zero-current intercept accurate to better than $E=V/d$5 mV (Xie et al., 2017).

The spin Hall sensing protocol uses a thin $E=V/d$6-phase tungsten film driven by a charge-current density $E=V/d$7 along the $E=V/d$8-direction. The macroscopic spin Hall voltage across width $E=V/d$9 is given by

$H = -\frac{\hbar^2}{2m}\nabla^2 + V(y) + H_R,$0

with $H = -\frac{\hbar^2}{2m}\nabla^2 + V(y) + H_R,$1cm for tungsten (Xie et al., 2017). During pulsed-bias measurements, the STM tip freezes the tunneling gap at $H = -\frac{\hbar^2}{2m}\nabla^2 + V(y) + H_R,$2 V using the potentiometry compensation $H = -\frac{\hbar^2}{2m}\nabla^2 + V(y) + H_R,$3 (Xie et al., 2017). Two effects are then observed: a gradual increase of $H = -\frac{\hbar^2}{2m}\nabla^2 + V(y) + H_R,$4 during the current pulse attributed to local thermal expansion, and an asymmetry in $H = -\frac{\hbar^2}{2m}\nabla^2 + V(y) + H_R,$5 when $H = -\frac{\hbar^2}{2m}\nabla^2 + V(y) + H_R,$6 is switched between $H = -\frac{\hbar^2}{2m}\nabla^2 + V(y) + H_R,$7 V and $H = -\frac{\hbar^2}{2m}\nabla^2 + V(y) + H_R,$8 V under identical $H = -\frac{\hbar^2}{2m}\nabla^2 + V(y) + H_R,$9, which grows with pulse time and pulse amplitude (Xie et al., 2017).

That asymmetry is interpreted as a signature of the spin Hall effect: for $$H_R = \alpha_R \bigl[\boldsymbol{\sigma}\times\mathbf{p}\bigr]\cdot\hat{\mathbf{e}}_y,\qquad \alpha_R=\lambda E.$$0, electrons tunnel from the film toward the tip and carry the spin polarization induced by the spin Hall effect, whereas for $$H_R = \alpha_R \bigl[\boldsymbol{\sigma}\times\mathbf{p}\bigr]\cdot\hat{\mathbf{e}}_y,\qquad \alpha_R=\lambda E.$$1, electrons tunnel from the tip into the film and are unpolarized (Xie et al., 2017). The normalized difference

$$H_R = \alpha_R \bigl[\boldsymbol{\sigma}\times\mathbf{p}\bigr]\cdot\hat{\mathbf{e}}_y,\qquad \alpha_R=\lambda E.$$2

further normalized by the mean tunneling current $$H_R = \alpha_R \bigl[\boldsymbol{\sigma}\times\mathbf{p}\bigr]\cdot\hat{\mathbf{e}}_y,\qquad \alpha_R=\lambda E.$$3, isolates the spin Hall induced component from thermal background (Xie et al., 2017). The reported signal size is $$H_R = \alpha_R \bigl[\boldsymbol{\sigma}\times\mathbf{p}\bigr]\cdot\hat{\mathbf{e}}_y,\qquad \alpha_R=\lambda E.$$4 of order $$H_R = \alpha_R \bigl[\boldsymbol{\sigma}\times\mathbf{p}\bigr]\cdot\hat{\mathbf{e}}_y,\qquad \alpha_R=\lambda E.$$5 to $$H_R = \alpha_R \bigl[\boldsymbol{\sigma}\times\mathbf{p}\bigr]\cdot\hat{\mathbf{e}}_y,\qquad \alpha_R=\lambda E.$$6 under $$H_R = \alpha_R \bigl[\boldsymbol{\sigma}\times\mathbf{p}\bigr]\cdot\hat{\mathbf{e}}_y,\qquad \alpha_R=\lambda E.$$7 of several mA (Xie et al., 2017).

A follow-up tungsten study extends the analysis to both bare W tips and Fe-coated W tips (Xie et al., 2017). For a bare W tip, reversing only the tunneling voltage yields

$$H_R = \alpha_R \bigl[\boldsymbol{\sigma}\times\mathbf{p}\bigr]\cdot\hat{\mathbf{e}}_y,\qquad \alpha_R=\lambda E.$$8

with $$H_R = \alpha_R \bigl[\boldsymbol{\sigma}\times\mathbf{p}\bigr]\cdot\hat{\mathbf{e}}_y,\qquad \alpha_R=\lambda E.$$9 the local spin polarization at the W surface (Xie et al., 2017). Quantitatively, $\Psi$0 mA gives $\Psi$1, $\Psi$2 mA gives $\Psi$3, and $\Psi$4 mA gives $\Psi$5 with saturation. At fixed $\Psi$6 mA, $\Psi$7 remains nearly constant for $\Psi$8, and thinner films show larger asymmetry, while at $\Psi$9 nm thickness $p$00 falls below $p$01 (Xie et al., 2017).

For an Fe-coated tip with spin-polarized density of states, reversing the film bias current introduces an additional asymmetry

$p$02

and experimentally $p$03 at $p$04 mA and $p$05 at $p$06 mA (Xie et al., 2017). Assuming $p$07, the measured $p$08 gives $p$09 at $p$10 A/cm$p$11, and using $p$12 with $p$13 s yields $p$14, in line with literature values for $p$15-W (Xie et al., 2017).

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 $p$16 along $p$17, the low-energy Hamiltonian for fixed spin block $p$18 is

$p$19

and the tunneling probability $p$20 for a valence-band hole to reach the conduction band is asymmetric in $p$21 (Lasia et al., 2011). To leading order,

$p$22

The peak occurs at

$p$23

so opposite spins see peaks shifted to opposite signs of $p$24 (Lasia et al., 2011). Since $p$25, more spin-up electrons tunnel into positive-$p$26 channels and more spin-down electrons into negative-$p$27 channels, producing a dc spin Hall current. The authors report that the resulting spin current can be as large as $p$28–$p$29 of the total charge current (Lasia et al., 2011). The origin is attributed to a spin- and $p$30-dependent Berry phase acquired upon adiabatic reflection in the gapped region (Lasia et al., 2011).

A conceptually similar but distinct coherent-tunneling mechanism is formulated for graphene with broken inversion symmetry and proximity-induced spin-orbit coupling (Zeng, 2024). The left and right electrodes are gapless graphene, while the central barrier carries a staggered sublattice potential $p$31 and intrinsic spin-orbit coupling $p$32. The net transmission amplitude has a Fabry–Pérot form

$p$33

where the total backreflection phase is split into a kinetic part $p$34 and a geometric phase

$p$35

The key symmetry is that $p$36 flips sign under $p$37, but $p$38 does not (Zeng, 2024). The result is skew tunneling: $p$39 which yields transverse conductances and a spin Hall angle

$p$40

Using $p$41 nm, $p$42 meV, $p$43 meV, $p$44 meV, $p$45 meV, and $p$46 meV, the spin Hall angle peaks at $p$47 meV with maximum $p$48, and the valley Hall angle peaks at $p$49 meV with maximum $p$50 (Zeng, 2024). Both effects vanish when $p$51, and as $p$52 both Hall angles vanish (Zeng, 2024).

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 (Zeng, 12 Jul 2025). In a normal-metal/$p$53-wave-magnet/superconductor junction, the central region has single-particle Hamiltonian

$p$54

where the term proportional to $p$55 shifts the spin-up and spin-down Fermi surfaces in momentum space by equal and opposite amounts along direction $p$56 (Zeng, 12 Jul 2025). 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 $p$57 is

$p$58

and time-reversal symmetry yields

$p$59

Thus the junction supports a pure spin current with zero net charge (Zeng, 12 Jul 2025). The spin Hall angle is

$p$60

The effect vanishes when the splitting direction is parallel to the junction normal, $p$61 or $p$62, and is largest when the splitting is perpendicular to the interface, $p$63 (Zeng, 12 Jul 2025). For reasonable parameters, the paper reports $p$64 (Zeng, 12 Jul 2025).

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 (Luo et al., 2010). In a prism–air–prism barrier, polarization-dependent transverse shifts arise in the transmitted beam, with opposite shifts for left- and right-circular polarization: $p$65 For $p$66, $p$67, and $p$68, varying $p$69 from $p$70 to $p$71 gives $p$72 of order $p$73–$p$74, compared with only a few $p$75 in single-interface refraction (Luo et al., 2010). The paper attributes the effect to total angular momentum conservation and emphasizes enhancement by the tunneling geometry (Luo et al., 2010). 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 (Vedyayev et al., 2013), HgTe Zener junctions (Lasia et al., 2011), coherent graphene barriers (Zeng, 2024), and $p$76-wave-magnet/superconductor structures (Zeng, 12 Jul 2025). 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 (Chen et al., 2016, Chen et al., 2015, Ok et al., 2016). Still others use tunneling chiefly as a detection channel for spin Hall accumulation, as in STM studies of tungsten films (Xie et al., 2017, Xie et al., 2017).

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 (Vedyayev et al., 2013). In coherent graphene and HgTe formulations, the defining ingredient is momentum-skew tunneling associated with geometric phase or Berry phase [(Lasia et al., 2011); (Zeng, 2024)]. In the superconducting case, the response is generated by asymmetric spin-dependent Andreev reflection (Zeng, 12 Jul 2025).

Second, the effect may or may not carry net charge. The superconducting $p$77-wave-magnet junction explicitly yields a pure transverse spin current with zero net charge (Zeng, 12 Jul 2025), whereas the magnetic tunnel junction with Rashba coupling hosts both charge and spin Hall currents inside the barrier (Vedyayev et al., 2013).

Third, the relevant control parameters differ by platform. These include barrier thickness and insulating gap in trilayers (Chen et al., 2016, Chen et al., 2015, Ok et al., 2016), magnetization angle in tunnel junctions (Vedyayev et al., 2013), electric field in Zener systems (Lasia et al., 2011), interface asymmetry and backreflection phase in graphene barriers (Zeng, 2024), and Fermi-surface-splitting direction in $p$78-wave magnets (Zeng, 12 Jul 2025).

The emerging direction is toward high-efficiency coherent or interfacial conversion. In graphene coherent tunneling, the spin Hall angle reaches $p$79 without disorder scattering (Zeng, 2024). In the superconducting $p$80-wave-magnet junction, $p$81 is likewise reported (Zeng, 12 Jul 2025). In tungsten STM studies, local spin polarization $p$82 is inferred at $p$83 A/cm$p$84, with estimated $p$85 for $p$86-W (Xie et al., 2017). 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.