Spin Hall Magnetoresistance in Metals

• Spin Hall magnetoresistance (SMR) is a spin–orbit-driven magnetotransport effect where the resistivity of a normal metal is modulated by adjacent magnetic order.
• SMR enables quantitative extraction of key parameters such as the spin Hall angle, spin diffusion length, and spin-mixing conductance using diffusion and interface models.
• Recent research extends SMR analysis to metallic ferromagnets, antiferromagnets, and paramagnets, linking it to spin–orbit torque and device applications.

Spin Hall magnetoresistance (SMR) is a spin–orbit-driven magnetotransport effect in which the resistivity of a normal metal with strong spin–orbit coupling is modulated by the magnetic order at an adjacent interface. In the canonical geometry, a charge current in the metal generates, via the spin Hall effect (SHE), a transverse spin current and an interfacial spin accumulation; orientation-dependent spin absorption and reflection at the magnetic interface modify spin backflow, and the inverse spin Hall effect (ISHE) converts that change into longitudinal and transverse resistivity signals (Chen et al., 2013). In bilayers of Pt, Ta, W, or Pd with magnetic insulators such as yttrium iron garnet (YIG), SMR provides a linear-response electrical readout of interfacial magnetization and a route to extract the spin Hall angle, spin diffusion length, and spin-mixing conductance; later work extended the phenomenon to metallic ferromagnets, antiferromagnets, compensated ferrimagnets, and paramagnets (Chen et al., 2015).

1. Canonical phenomenology and symmetry

For the standard slab geometry, the charge current is taken along $x$, the film normal along $z$, and the SHE spin polarization along $y$. In that convention, the longitudinal and transverse resistivities of the normal metal assume the now-standard symmetry forms

$$\rho_{xx}(\mathbf{m})=\rho+\Delta\rho_0+\Delta\rho_1(1-m_y^2),$$

$$\rho_{xy}(\mathbf{m})=\Delta\rho_1 m_x m_y+\Delta\rho_2 m_z,$$

where $\mathbf{m}$ is the unit magnetization vector of the adjacent magnet, $\Delta\rho_1$ is the SMR amplitude, and $\Delta\rho_2$ is an anomalous-Hall-like spin Hall term associated with the imaginary part of the spin-mixing conductance (Chen et al., 2013).

Equivalent forms are widely used in experiments. In ferromagnet/metal bilayers one often writes

$\rho_{xx}=\rho_0+\Delta\rho(1-m_t^2),$

or, for in-plane rotations,

$R(\phi)=R_0+\Delta R\cos^2\phi,$

with $z$0 the magnetization component transverse to the current and parallel to the SHE spin polarization axis. The physical interpretation is that resistance is smallest when the magnetization is parallel to the interfacial spin accumulation, because spin reflection is then enhanced; when the magnetization is perpendicular to that spin accumulation, interfacial spin absorption is stronger and the longitudinal resistance increases (Oyanagi et al., 2020).

The angular selection rules are central to SMR metrology. Rotations that vary $z$1 generate the characteristic $z$2, $z$3, or $z$4 dependences of the longitudinal and transverse responses. Rotations that keep $z$5 ideally leave the longitudinal SMR constant. Much of the later literature is concerned with circumstances under which this ideal separation is modified by domain physics, longitudinal spin absorption, disorder, or additional magnetoresistive channels.

2. Drift–diffusion theory and microscopic interface physics

The minimal theory treats the normal metal by one-dimensional spin diffusion with weak spin–orbit coupling, augmented by SHE and ISHE terms. The spin accumulation $z$6 obeys

$z$7

where $z$8 is the spin diffusion length. For current along $z$9, the $y$0-directed spin current density is

$y$1

with $y$2 the spin Hall angle and $y$3 the conductivity of the normal metal (Chen et al., 2015).

At the outer surface, the boundary condition is $y$4. At the magnetic interface, spin transfer is governed by the complex spin-mixing conductance $y$5. In conductance notation,

$y$6

with $y$7. Solving the diffusion equation with these boundary conditions yields the familiar $y$8 and $y$9 thickness dependences of $$\rho_{xx}(\mathbf{m})=\rho+\Delta\rho_0+\Delta\rho_1(1-m_y^2),$$0 and $$\rho_{xx}(\mathbf{m})=\rho+\Delta\rho_0+\Delta\rho_1(1-m_y^2),$$1, and reduces the experimental parameter set to $$\rho_{xx}(\mathbf{m})=\rho+\Delta\rho_0+\Delta\rho_1(1-m_y^2),$$2, $$\rho_{xx}(\mathbf{m})=\rho+\Delta\rho_0+\Delta\rho_1(1-m_y^2),$$3, $$\rho_{xx}(\mathbf{m})=\rho+\Delta\rho_0+\Delta\rho_1(1-m_y^2),$$4, $$\rho_{xx}(\mathbf{m})=\rho+\Delta\rho_0+\Delta\rho_1(1-m_y^2),$$5, and $$\rho_{xx}(\mathbf{m})=\rho+\Delta\rho_0+\Delta\rho_1(1-m_y^2),$$6 (Chen et al., 2015).

A microscopic reformulation expresses the interfacial spin conductance in terms of spin susceptibilities of the normal metal and magnetic insulator. In that treatment, SMR contains a nearly temperature-independent static part arising from interfacial spin flip and a dynamic part induced by magnon creation or annihilation; the dynamic contribution has the opposite sign from the static one and can produce a finite-temperature sign change of the SMR signal. The same framework derives an Onsager relation between spin conductance and thermal spin-current noise (Kato et al., 2020).

The canonical theory is therefore simultaneously a diffusion problem in the heavy metal and a quantum boundary-value problem at the interface. Its success explains why SMR became a standard route for extracting interfacial spin transparency and spin-transport parameters, while its limitations motivate later extensions involving magnetic disorder, longitudinal spin absorption, and fluctuating magnets.

3. Metallic ferromagnets, longitudinal spin absorption, and spin–orbit torque

Although SMR was first formulated for normal-metal|ferromagnetic-insulator bilayers, it also appears in metallic HM/FM stacks. In W/CoFeB/MgO, the longitudinal resistance follows the SMR symmetry

$$\rho_{xx}(\mathbf{m})=\rho+\Delta\rho_0+\Delta\rho_1(1-m_y^2),$$7

with $$\rho_{xx}(\mathbf{m})=\rho+\Delta\rho_0+\Delta\rho_1(1-m_y^2),$$8, showing that the response is determined by the magnetization orientation relative to the SHE spin polarization rather than relative to the current. For W(5 nm)/CoFeB(1.2 nm)/MgO(1.6 nm), the SMR is approximately $$\rho_{xx}(\mathbf{m})=\rho+\Delta\rho_0+\Delta\rho_1(1-m_y^2),$$9, nearly two orders of magnitude larger than typical Pt/YIG values. Thickness-dependent fitting gave $$\rho_{xy}(\mathbf{m})=\Delta\rho_1 m_x m_y+\Delta\rho_2 m_z,$$0, $$\rho_{xy}(\mathbf{m})=\Delta\rho_1 m_x m_y+\Delta\rho_2 m_z,$$1 nm, and $$\rho_{xy}(\mathbf{m})=\Delta\rho_1 m_x m_y+\Delta\rho_2 m_z,$$2; the SMR maximum at $$\rho_{xy}(\mathbf{m})=\Delta\rho_1 m_x m_y+\Delta\rho_2 m_z,$$3–$$\rho_{xy}(\mathbf{m})=\Delta\rho_1 m_x m_y+\Delta\rho_2 m_z,$$4 nm coincides with the maximum current-induced switching efficiency, linking SMR directly to spin–orbit torque in the same devices (Cho et al., 2015).

For metallic ferromagnets, the standard HM|FI model is incomplete because the ferromagnet can absorb spin current whose polarization is longitudinal with respect to $$\rho_{xy}(\mathbf{m})=\Delta\rho_1 m_x m_y+\Delta\rho_2 m_z,$$5. A dedicated HM/FM theory therefore introduces a longitudinal spin-absorption term $$\rho_{xy}(\mathbf{m})=\Delta\rho_1 m_x m_y+\Delta\rho_2 m_z,$$6, in addition to the transverse interfacial term $$\rho_{xy}(\mathbf{m})=\Delta\rho_1 m_x m_y+\Delta\rho_2 m_z,$$7. In W/CoFeB, this model quantitatively describes the heavy-metal thickness dependence of SMR and accounts for the increase of SMR on cooling by allowing a temperature-dependent spin polarization $$\rho_{xy}(\mathbf{m})=\Delta\rho_1 m_x m_y+\Delta\rho_2 m_z,$$8 of CoFeB; in that description, increasing $$\rho_{xy}(\mathbf{m})=\Delta\rho_1 m_x m_y+\Delta\rho_2 m_z,$$9 reduces longitudinal spin absorption and raises the SMR amplitude (Kim et al., 2015).

The same correction becomes especially important when the nonmagnetic layer is highly resistive. In Co$\mathbf{m}$0Fe$\mathbf{m}$1B$\mathbf{m}$2/SrIrO$\mathbf{m}$3, where $\mathbf{m}$4–$\mathbf{m}$5, including longitudinal spin current absorption changes the effective spin Hall angle extracted from the SMR thickness dependence from $\mathbf{m}$6 to $\mathbf{m}$7, a relative correction of about $\mathbf{m}$8. The underlying reason is that $\mathbf{m}$9, so highly resistive nonmagnetic layers amplify the influence of FM-side spin absorption on the SMR response (Hori et al., 9 Sep 2025).

Not all metallic bilayers conform to the extended drift–diffusion picture. In Pt/Co, the SMR increases with increasing Co thickness, and the effective spin Hall angle inferred from conventional analysis exceeds reported values for Pt. An extended model including spin transport in Co, interface magnetoresistance, and textured induced anisotropic scattering does not reproduce that thickness trend; related measurements on W/Co and W/CoFeB led to the conclusion that the anomaly is associated with a particular property of Co rather than a generic HM/FM effect (Kawaguchi et al., 2018).

4. Antiferromagnets, paramagnets, and fluctuation-dominated regimes

In antiferromagnetic insulators, the relevant order parameter is the Néel vector rather than a net magnetization. For easy-plane antiferromagnets with two sublattices, the SMR acquires a characteristic phase reversal relative to ferrimagnets: in the single-domain limit,

$\Delta\rho_1$0

In real multi-domain films, the amplitude is controlled by a monodomainization field $\Delta\rho_1$1. Room-temperature measurements on $\Delta\rho_1$2-Fe$\Delta\rho_1$3O$\Delta\rho_1$4/Pt and NiO/Pt verified this framework: $\Delta\rho_1$5-Fe$\Delta\rho_1$6O$\Delta\rho_1$7/Pt reached an SMR amplitude of $\Delta\rho_1$8, about twice the value of YIG/Pt, with $\Delta\rho_1$9 T, whereas NiO/Pt showed a much smaller amplitude and $\Delta\rho_2$0 T (Geprägs et al., 2020).

Paramagnetic SMR established that spontaneous magnetic order is not necessary. In Pt/Gd$\Delta\rho_2$1Ga$\Delta\rho_2$2O$\Delta\rho_2$3, the interfacial boundary condition becomes

$\Delta\rho_2$4

where $\Delta\rho_2$5, $\Delta\rho_2$6, and $\Delta\rho_2$7 are field- and temperature-dependent spin-transfer, field-like, and spin-sink conductances. At $\Delta\rho_2$8, $\Delta\rho_2$9 while $\rho_{xx}=\rho_0+\Delta\rho(1-m_t^2),$0 is maximal; with increasing field, $\rho_{xx}=\rho_0+\Delta\rho(1-m_t^2),$1 and $\rho_{xx}=\rho_0+\Delta\rho(1-m_t^2),$2 grow and $\rho_{xx}=\rho_0+\Delta\rho(1-m_t^2),$3 is suppressed by the Zeeman gap. Experimentally, the longitudinal SMR is of order $\rho_{xx}=\rho_0+\Delta\rho(1-m_t^2),$4 by $\rho_{xx}=\rho_0+\Delta\rho(1-m_t^2),$5 T at $\rho_{xx}=\rho_0+\Delta\rho(1-m_t^2),$6 K, and simultaneous fitting of SMR and spin Hall anomalous Hall effect yields at $\rho_{xx}=\rho_0+\Delta\rho(1-m_t^2),$7 K $\rho_{xx}=\rho_0+\Delta\rho(1-m_t^2),$8, $\rho_{xx}=\rho_0+\Delta\rho(1-m_t^2),$9, and $R(\phi)=R_0+\Delta R\cos^2\phi,$0 (Oyanagi et al., 2020).

A related paramagnetic extension was demonstrated in Pt/NdGaO$R(\phi)=R_0+\Delta R\cos^2\phi,$1, where in-plane ADMR follows the standard $R(\phi)=R_0+\Delta R\cos^2\phi,$2 and $R(\phi)=R_0+\Delta R\cos^2\phi,$3 forms, the amplitude scales linearly with current bias, and the temperature dependence tracks the magnetization of NdGaO$R(\phi)=R_0+\Delta R\cos^2\phi,$4. The interpretation emphasizes torque on localized Nd$R(\phi)=R_0+\Delta R\cos^2\phi,$5 moments together with crystal-field-induced intermultiplet transitions, allowing SMR signatures up to about $R(\phi)=R_0+\Delta R\cos^2\phi,$6 K despite the absence of spontaneous magnetization (Phanindra et al., 2022). In noncrystalline paramagnetic YIG/Pt, a clear SMR-like angular dependence was also observed; comparison between models favored a net-moment picture over an ensemble of independent moments because the measured resistivity at $R(\phi)=R_0+\Delta R\cos^2\phi,$7 matches the prediction $R(\phi)=R_0+\Delta R\cos^2\phi,$8 rather than the finite offset expected from $R(\phi)=R_0+\Delta R\cos^2\phi,$9 (Lammel et al., 2019).

A further generalization appears near ferromagnetic criticality. A 2025 theory introduced “longitudinal” SMR (LSMR), in which the magnetization remains collinear with the spin Hall accumulation and the resistance change is instead driven by the magnetic-field suppression of spin fluctuations in the ferromagnet. Unlike conventional SMR, which vanishes near $z$00, LSMR is suppressed at low temperature but becomes critically enhanced near $z$01, reaching a magnitude comparable to conventional SMR amplitudes (Tang, 1 Jan 2025).

5. Surface sensitivity, disorder, and competing interpretations

Because the magnetic layer in many benchmark systems is insulating, SMR is intrinsically interfacial. This surface sensitivity is explicit in Pt/CoFe$z$02O$z$03, where SMR-derived loops do not follow bulk magnetometry. The surface magnetization lacks the zero-field steps seen in vibrating-sample magnetometry, remains non-saturated up to $z$04 T, and is nearly independent of CoFe$z$05O$z$06 thickness from $z$07 to $z$08 nm. In the same system, a significant $z$09-rotation signal was traced not to SMR but to ordinary magnetoresistance of Pt, isolated through Kohler scaling with exponent $z$10; proximity-induced AMR in Pt was ruled out by the symmetry of the field sweeps (Isasa et al., 2015).

The same interfacial selectivity enables spatially resolved probes of complex oxides. In strained SrMnO$z$11|Pt, small $z$12 devices and a large $z$13 device on the same film yielded different ADMR phases, revealing local antiferromagnetic domains with easy axes near $z$14 and $z$15 in the small devices and a more ferromagnetic-like average response in the large device. From this contrast, the predominant antiferromagnetic domain area was inferred to be approximately $z$16 (Rijn et al., 2023).

Disorder can also produce deceptively canonical SMR-like signals. In Pt/MnPSe$z$17, the ADMR exhibits the expected $z$18 symmetry in the $z$19 and $z$20 planes and amplitudes of order $z$21 to $z$22, but the signal grows with temperature and persists well above the MnPSe$z$23 Néel temperature. Cross-sectional TEM revealed a $z$24 nm amorphous Pt–Se interlayer, and the transport behavior was attributed to a disordered magnetic system formed at the interface rather than to the intrinsic antiferromagnetic lattice of MnPSe$z$25. This result explicitly challenges the assumption that interfacial disorder contributes only electrical noise and not a substantial SMR-like response (Catalano et al., 2022).

Alternative mechanisms have therefore remained part of the SMR discourse since the early review literature. The 2015 theoretical review identifies ferromagnetic proximity effects and Rashba spin-orbit torques as possible competing explanations in some metallic systems (Chen et al., 2015). A more radical alternative, the multi-conduction-channel model, proposes that four SHE-induced spin channels at the top, bottom, left, and right surfaces of a thin film generate an “intrinsic SMR” even without a magnetic insulator, predicting $z$26 and a non-constant $z$27-rotation response (Zhang et al., 2015). Standard diffusion-plus-spin-mixing theory remains the dominant framework, but the later literature shows that interface morphology, parasitic magnetoresistances, and material-specific transport channels must be addressed explicitly in any quantitative interpretation.

6. Metrology, devices, and current directions

One of the principal uses of SMR is parameter extraction. In Pt/YIG bilayers grown on both Gd$z$28Ga$z$29O$z$30 and thermally oxidized Si, the optimized structures reach $z$31 at room temperature, with the Pt-thickness dependence peaking near $z$32 nm. Fits yield $z$33, $z$34 on GGG and $z$35 on Si, and $z$36–$z$37 nm. The same study finds that the SMR correlates with YIG magnetization, interface roughness, and carrier density, showing that high SMR can be achieved on Si even without the crystallinity of epitaxial YIG on GGG, provided the interface is smooth and the magnetic properties are optimized (Fukushima et al., 2023).

SMR has also moved into device engineering. In ultrathin NiFe/Pt bilayers, the coexistence of SMR and spin–orbit torque enables a Wheatstone-bridge sensor with built-in AC excitation and rectification. For a bridge of NiFe(1.8 nm)/Pt(2 nm), the measured detectivity is around $z$38 at $z$39 Hz under AC bias, with essentially zero DC offset, negligible hysteresis, a sensitivity of about $z$40, and demonstrated operation in angle sensing, vibration detection, and finger-motion monitoring (Xu et al., 2018). This device architecture exploits the same interfacial physics that underlies SMR metrology, but uses it for built-in linearization and noise suppression.

Across the literature, the principal open directions are consistent. Thickness series remain necessary when only a single normal-metal thickness is available, especially in paramagnetic systems where $z$41 and geometry factors are otherwise underconstrained (Oyanagi et al., 2020). Interface engineering is essential for van der Waals magnets and other chemically fragile materials, where sputter-induced amorphous interlayers can dominate the signal (Catalano et al., 2022). In antiferromagnets and fluctuation-dominated systems, combining SMR with direct magnetic imaging or complementary spin-transport probes is increasingly important for separating domain physics, disorder, and genuine interfacial exchange.

SMR has therefore evolved from a bilayer magnetoresistance effect into a broader interfacial spectroscopy of spin transport. Its canonical formulation remains the SHE–diffusion–spin-mixing framework of normal-metal|magnet heterostructures, but its present scope includes metallic ferromagnets with longitudinal spin absorption, antiferromagnets with Néel-vector readout, paramagnets with field-tunable torque efficiencies, fluctuating magnets near criticality, and device platforms in which the same effect serves simultaneously as readout, calibration tool, and functional transducer.