Anisotropic magnetoresistance and planar Hall effect in correlated and topological materials

The anisotropic magnetoresistance (AMR) and the planar Hall effect (PHE) were first reported in ferromagnetic metals such as Fe and Ni by Thomson and Glascow in 1857.^1,2 The initially observed effect was very small, usually around a few mΩ, comparable to the resistance background.^2–5 However, due to easy accessibility and high sensitivity to the magnetic field, the AMR and PHE effects, along with their thermoelectric counterpart planar Nernst Hall effect,⁶ have been widely used to develop nanoscale material metrology,^2,7,8 magnetic sensors and transducers,^7,9–15 and spintronic devices.^16,17 Over the past decades, tremendous research progress has been made in the discovery of new quantum materials and emergent phenomena driven by strong correlation,^18–22 spin-orbit coupling (SOC),^23,24 and nontrivial band topology.^25–27 The complex magnetic energy landscape and intricate interplay between charge, spin, orbital, lattice, and topology in these materials make them versatile platforms for studying the AMR and PHE effects, whose temperature and magnetic field dependences can reveal critical information about these competing energy scales. Compared with conventional magnetometry methods using the vibrating-sample magnetometer and the superconducting quantum interference device, the AMR and PHE studies bear distinct advantages in probing samples with limited volume,²⁸ such as nanoscale and low dimensional systems, as well as surface and interface effects. Many of these materials also exhibit significantly enhanced AMR and PHE signals, which facilitate their technological applications.

In this article, we provide a comprehensive review of recent studies of the AMR and PHE phenomena, with an emphasis on two classes of quantum materials. The first class is the strongly correlated material based on transition metal oxides, including colossal magnetoresistance (CMR) manganites and high temperature superconductor cuprates, as well as correlated cobaltates, ruthenates, and iridates. Both single-phase bulk and thin film systems and interfaces in heterostructures and superlattices are discussed. The second class is the topological material, including topological insulators, Dirac and Weyl semimetals, topological superconductors, and topological interfaces.

The article is organized as follows. In Sec. II, we give a brief discussion of the phenomenological description of the AMR and PHE effects, including the galvanomagnetic tensor and transport theory approaches, and introduce the giant planar Hall effect (GPHE). Section III gives a comprehensive survey of the AMR and PHE behaviors in single-phase material systems, including ferromagnetic, antiferromagnetic, and topological materials. The effects of temperature, magnetic field, strain, chemical doping, and electric field effect tuning are discussed. Section IV focuses on the phenomena in artificial nanostructures, surfaces, heterointerfaces, and superlattices. Section V summarizes the review and provides an outlook to possible future directions in the AMR and PHE research.

In this section, we briefly review the AMR and PHE phenomena in magnetic conductors and their relation with magnetocrystalline anisotropy (MCA). In part A, we discuss the general form of the transport tensor in isotropic systems and how the MCA modifies the AMR and PHE. The former well describes the effects observed in polycrystalline systems and large magnetic field conditions. In single crystalline samples at low magnetic field, on the other hand, the angular dependences of AMR and PHE are affected by magnetization pinning, from which the magnetic anisotropy energy (MAE) can be deduced. Part B introduces the GPHE and its potential applications. We also discuss in part C the microscopic theories of AMR and PHE, including the SOC-induced intraband contribution to charge scattering and the interband contribution in systems with nontrivial Berry curvatures.

In a magnetic conductor, the resistivity measured for current along the magnetic field direction $(ρ∥)$ is different from that for current perpendicular to the magnetic field $(ρ⊥)$ [Fig. 1(a)]. For a magnetic field rotating within the plane of channel current and magnetization [Fig. 1(b)], the free energy F of the magnetic system is determined by the MCA and Zeeman energy,²⁸ 

Here, $θ$ is the angle between the magnetic field H and current; $φ$ is the angle between the magnetization M and current; K_u and K₁ are the uniaxial and biaxial magnetic anisotropy constants, respectively; and $φu$ and $φ1$ are the directions for the corresponding magnetic easy axes. In addition to the configuration of an in-plane magnetic field, AMR can also be characterized with the out-of-plane rotation of the magnetic field. Figure 1(b) illustrates two possible geometries, with the rotation plane either in parallel with or perpendicular to current. These types of measurement geometries have been widely adopted in studying epitaxial thin films, where the magnetic anisotropy is often dominated by strain, leading to distinct in-plane and out-of-plane MAE that deviates significantly from the bulk MCA. Combining the in-plane and out-of-plane AMR measurements can thus reveal the effect of epitaxial strain on the magnetic energy landscape.

Next, we consider the phenomenological description of the AMR and PHE. We first consider an isotropic medium, where magnetization is aligned with the external magnetic field $(θ=φ)$⁠. The system shows rotational symmetry around the direction of magnetization [SO(2)]. Along the magnetic field H direction (defined as $z^$⁠), the electric field E and the current density j are related by the galvanomagnetic tensor,

or

Here, $ρH$ is the Hall resistivity. Within the plane of magnetic field and current, the resistivity in Eq. (3) can be decomposed into the longitudinal magnetoresistance, or AMR, and transverse magnetoresistance, or PHE. The AMR depends on the angle $φ$ as⁴ 

also known as the Voigt–Thomson formula. The $φ$ dependence of the PHE is given by

The isotropic medium assumption can be well applied to polycrystalline films. For example, the sinusoidal $θ$-dependence of AMR and PHE has been observed in polycrystalline films of 3d ferromagnetic alloys.² For epitaxial thin films, the $θ=φ$ condition is only satisfied when the Zeeman energy is much larger than the MCA so that the system conforms to the SO(2) symmetry. As shown in Fig. 1(c), the high field in-plane AMR and PHE of a 5 nm La_0.5Sr_0.5MnO₃ film can be well described by Eqs. (4) and (5). At low magnetic fields, on the other hand, MCA becomes dominant and $θ≠φ$⁠. Figures 2(a) and 2(b) show the angular dependence of AMR in La_1−xCa_xMnO₃ (LCMO, x = 0.31) films. While the in-plane AMR (xy-plane) follows Eq. (4) well, the out-of-plane AMR (yz- and xz-plane) exhibits clear deviation,²⁹ showing sharp tips around the magnetic hard axis and broad curvature in the easy plane. This reflects the magnetization pinning close to easy axis.^30–53 The effect of magnetization pinning is more prominent in PHE, as it is not sensitive to magnetoresistance.^53–65 Figure 2(c) shows the angular-dependent PHE resistance in a 6 nm La_1−xSr_xMnO₃ (LSMO) (x = 0.33) film at different magnetic fields.⁵⁶ With decreasing magnetic field, PHE evolves from the sinusoidal θ-dependence to a bi-stable state, which reflects magnetization pinning around the [110] and $[11¯0]$ directions and sharp switching along the hard axes [100] and [010]. Similar MCA modulation of AMR and PHE includes the occurrence of switching hysteresis and jumps,^52,62–77 higher-order oscillations,^{32–34,47–54,68–73,78–94} and current direction-dependent amplitude and phase shift.^{31,55,68,79–81,95–99}

Based on the magnetization pinning effect, two phenomenological methods have been developed to quantify the magnetic anisotropy field. The first one is based on the Fourier transform of the AMR and PHE.^78,79 As MCA breaks the SO(2) symmetry, the magnetoresistance tensor in Eq. (2) needs to be modified to reflect the crystalline symmetry.^2–4 By expanding in the frame of magnetic field direction $α^$ and current direction $β^$⁠, the resistance tensor can be expressed by the following MacLaurin series:

where the tensor elements $cij$⁠, $ckij$⁠, $cklij$⁠,… should be invariant under symmetry operations of the crystal point group. Therefore, the AMR⁷⁸ and PHE can be expressed via Fourier expansion of the resistance tensor,

and

In Ref. 80, Bason et al. proposed a two-dimensional (2D) resistivity tensor with m3m symmetry to describe the in-plane AMR and PHE in pseudo-cubic LSMO films deposited on (001) SrTiO₃ (STO) substrates. The modeled AMR and PHE fit well the experimental results, clearly illustrating the higher order oscillations, amplitude tuning, and phase shift. The MCA energy can be deduced by using the relative ratios of the extracted coefficients for different Fourier terms. A similar method has been proposed in Ref. 69 by Li et al., where a three-dimensional resistivity tensor with 4/mmm (D_4h) tetragonal symmetry is adopted to describe LCMO thin films. This method is based on the MCA modified transport tensor and has been exploited to study a wide variety of material systems. Those include single-phase oxides such as manganites,^{29,33,55,68,82–84,97–99} cuprate- and iron-based superconductors,^39,50,91 Dirac antiperovskites Sr₃SnO,¹⁰⁰ and Fe₃O₄ and Ni_0.3Fe_2.7O₄^101,102; 3d ferromagnetic metals^2,4,103; trigonal crystals¹⁰⁴; kagome ferromagnet Fe₃Sn₂¹⁰⁵; two-dimensional (2D) materials such as graphene¹⁰⁶; and topological materials.^{58,94,107–111} Other nanoscale systems being studied include surface states of STO¹¹² and interface states of EuO/KTaO₃,⁹² LaAlO₃/STO,^113–116 and LaVO₃/STO.¹¹⁷ 

The second method is based on quantifying the relation between magnetization and magnetic field directions, i.e., the θ-φ relation, from the PHE resistance pinning effect. From Eq. (5), one can extract the magnetization direction using $φ=12arcsin(ρPHE/ρPHE,max)$⁠, with $ρPHE,max$ being the maximal value of $ρPHE$⁠. Figure 2(d)^,118 shows the φ versus θ relation calculated from the PHE resistance data similar to those shown in Fig. 2(c). When magnetization follows the direction of H, one expects a linear relation: θ = φ. Deviation from the linear relation reflects the magnetization pinning at the magnetic easy axes, which can be qualified by calculating the residual sum of squares (RSS) of its linear fit: $RSS=∑i[ϕ(θi)−θi]2$⁠.^56,118 As shown in Fig. 2(e), the magnetic anisotropy field marks where the RSS approaches zero. This method has been exploited to determine the magnetic anisotropy energy and magnetic easy axes in LSMO⁵⁷; Sr₂IrO₄ (SIO)⁵⁴; 2D ferromagnetic materials including CrCl₃, CrBr₃, and CrI₃¹¹⁹; Co/Pt heterostructures¹²⁰; and LaNiO₃ (LNO)/LSMO superlattices.⁴⁴ 

In simple magnetic metals such as Fe and Ni, the PHE is normally very small.^1–5 In Ref. 59, Tang et al. first discovered that the PHE resistivity in (Ga,Mn)As film is four orders of magnitude higher and named it as the GPHE. Similar size of effect has been observed in a wide range of materials, including strongly correlated oxide thin films^61,65 and interfaces,^92,121 ferromagnetic oxide Fe₃O₄,^60,122,123 and topological materials.^{46,58,76,94,108–110,124–139} Figure 3(a) shows the PHE resistance (R_PHE) of a 40 nm LSMO in a sweeping magnetic field along θ = 10°.⁶⁵ When the magnetic field exceeds the magnetic anisotropy field, magnetization switches abruptly between the two easy axes [110] and $[1¯10]$⁠, which leads to a sharp change of R_PHE with sign reversal. In Ref. 124, Burkov deduced this R_PHE jump from Eq. (5) in both normal and topological regions as $ΔρPHE=ρ(1−2La/Lx)$ and $ΔρPHE=ρ[(Lc/La)21+(Lc/La)2]$⁠, respectively, where $ρ$ is Drude resistivity and L_x, L_a, and L_c are the length scales of sample size, magnetic scattering, and chiral anomaly, respectively. In both cases, $ΔρPHE$ is on the order of $ρ$⁠, which indicates a large resistance jump during magnetization switching in more resistive materials. Due to its sensitivity to the magnetic field, the GPHE, combined with the magnetic flux concentrators, can be used to design magnetic sensors with low typical equivalent magnetic noise.^140–146 The GPHE-based magnetic sensor with high sensitivity has diverse applications, including magnetometer,^147–151 low-frequency detection,^151–153 magneto-memristor,¹⁵⁴ magnetofluidic,¹⁵⁵ magnetorelaxometry,¹⁵⁶ biosensing,¹⁵⁷ current sensing,¹⁵⁸ spin structure imaging,¹⁵⁹ and flux leakage inspection.¹⁶⁰ 

Due to the large signal magnitude and sign reversal feature, the GPHE has gained increasing research interests in designing arithmetic logic unit and storage devices.¹⁶¹ Figure 3(b) shows one strategy to design PHE magnetic memory using LSMO thin films.⁶¹ By applying magnetic field pulses alternating between the [110] and $[1¯10]$ directions [Fig. 3(b) lower panel], the PHE signal can be tuned between the positive and negative nonvolatile resistance states, which can be programmed as two logic states. The PHE-based logic and memory devices have also been proposed using conventional magnetic materials such as Fe¹⁶ and Fe₃O₄⁶⁰ as well as complex oxide superlattices with tunable noncollinear magnetization states.⁴⁴ 

An indepth understanding of how the MCA affects the AMR and PHE can be gained using the semiclassical transport theory. In Ref. 95, Fuhr et al. developed a microscopic theory to describe the intraband contribution of e_g orbitals to the magnetotransport in LSMO. Two terms are considered in the Hamiltonian: a tight-banding term considering the nearest-neighbor hopping of e_g orbital electrons, which naturally conforms to the crystalline symmetry, and a SOC term originated from the on-site coupling between e_g and t_2g term. The magnetoresistance is deduced by the intraband contribution of conductivity $σij$ from e_g orbitals,

where e is the elementary charge, τ is the isotropic relaxation time, $εn(k)$ is the dispersion relation in reciprocal space of the nth band, and f represents the Fermi–Dirac distribution function. Figure 4(a) shows the band structure of e_g orbitals with magnetization pointing along the (100) direction. The SOC leads to the nonequivalent band splitting between X_x and X_y points, which induces the angular dependence of integral intensity. Figure 4(b) shows that the integrand of σ_xx is larger than the corresponding value for σ_yy, which indicates a resistance minimum along the magnetization direction.

In the meantime, the MAE is also affected by the SOC. In Ref. 57, the MAE is calculated in LSMO films epitaxially strained on (001) STO substrates by adopting the second-order perturbation theory [Fig. 4(c)], which takes into account the SOC between the occupied (d_yz, d_xz) orbitals and partially occupied (d_xy, $dz2$⁠) orbitals. As the strained film takes the orthorhombic structure, the d_yz and d_xz orbitals are split [Fig. 4(c)], which induces a nonzero angular-dependent component in MAE. Additionally, the d-orbital population also modifies the magnitude of MAE, which will be discussed in detail in Sec. III A 4. As SOC affects both intraband electron scattering and MCA, the angular dependence of AMR and PHE can be used to deduce the strength and direction of MAE.

Recent studies have shown that giant AMR and PHE can also originate from nonvanishing Berry curvatures in materials such as Dirac and Weyl semimetals,^{46,58,76,94,109–111,126–131,162} topological insulators,^{41–43,75,77,107,108,133,137,163–165} and conducting interfaces,⁹² where the nontrivial topology gives rise to an interband contribution to conductivity.^{124,166–168} Starting from the Boltzmann transport equation and incorporating the Berry curvature $Ωk$ into the semiclassical equations of motion,¹⁶⁹ the current density can be expressed as¹⁶⁷ 

Here $v~k=vk−∇(m⋅B)$ is the canonical group velocity of Bloch electrons with m being the orbital magnetic moment, and $f~k$ is the distribution function solved from the Boltzmann equation. The AMR and PHE effects can then be expressed in terms of the longitudinal and transverse conductivities,

where $D=[1+(e/ℏ)(B⋅Ωk)]−1$ is the phase space factor, and f_eq represents the equilibrium distribution function in the absence of external fields. In Eqs (11) and (12), the angular dependence arises from the anomalous velocity factor term $(v~k⋅Ωk)$⁠, and a negative magnetoresistance effect can be predicted. This points to a topological chiral anomaly origin for the AMR and PHE in topological materials. The chiral anomaly-related AMR and PHE have been observed in a wide variety of topological materials. These include topological insulators, such as Bi_xSb_1−x,¹⁰⁷ Sb₂Te₃,⁴¹ Bi₂Te₃,^136,165 Bi₂Se₃,⁷⁷ Cr_x(Bi,Sb)_2−xTe₃,⁴² (Bi,Sb)₂Te₃,^75,108 Bi_1.1Sb_0.9Te₂S,¹³³ BiSbTeSe₂,⁴³ Bi₂Te_xSe_3−x,¹⁶⁴ Sr_0.06Bi₂Se₃,¹⁶³ and SmB₆¹³⁷; and Dirac and Weyl semimetals, such as Cd₃As₂,^46,126 VAl₃,¹²⁹ PdTe₂,^58,T_d-MoTe₂,¹⁰⁹ WTe₂,^110,127 ZrTe_5−d,⁹⁴ PtSn₄,¹¹¹ Na₃Bi,¹²⁵ GdPtBi,^125,132 Te,¹³⁹ and TaAs.¹³⁴ 

However, some recent studies of Dirac and Weyl semimetals, such as PdTe₂,¹³⁰ MoTe₂,¹²⁸ TaP,⁷⁶ PtSe₂,^135,162 NiTe₂,¹³⁸ and Co₃Sn₂S₂¹³¹ as well as the quasi-one-dimensional topological superconductor TaSe₃,¹⁷⁰ exhibit positive magnetoresistance associated with the AMR and PHE. This discrepancy between the theoretical prediction and experimental results suggests that the classical anisotropic orbital magnetoresistance can dominate AMR and PHE over the chiral anomaly. In addition, several anisotropic scattering mechanisms have been proposed to explain the AMR and PHE in topological materials, such as spin-momentum locking,^171,172 magnetic proximity,^53,77 magnetic clusters,⁴⁵ spin-orbit torque,¹⁷³ Landau quantization,¹⁷⁴ nonlinear lattice effect,¹⁷⁵ and coupling between surface states.^93,176 These transport mechanisms can coexist and compete with each other, making the AMR and PHE more challenging yet intriguing to understand in topological materials.

In this section, we discuss the AMR and PHE behaviors in bulk and thin film samples of strongly correlated oxides and topological materials. The correlated oxides include manganites, cobaltates, ruthenates, iridates, and cuprate superconductors. The topological materials of interest include the topological insulators and Dirac and Weyl semimetals. In part A, we review the AMR and PHE observed in ferromagnetic materials, discussing the effects of temperature, magnetic field, strain, and doping via chemical substitution and the electric field effect. Part B focuses on the antiferromagnetic phase and the effect of nanoscale phase separation. In part C, we discuss several mechanisms that render the AMR and PHE responses in topological materials, such as the competition between chiral and normal transport contributions, spin-flip back-scattering induced by time-reversal symmetry breaking, and spin-momentum locking.

In 3d ferromagnetic alloys, there is no prominent AMR above magnetic Curie temperature T_C. Below T_C, the resistance anisotropy ratio $Rρ=(ρ∥−ρ⊥)/ρ∥$ increases monotonically with increasing magnetic field and decreasing temperature,^2–4 which can be related to the change in magnetization. The former is known as Kohler's rule.¹⁷⁷ Similar monotonical temperature and magnetic field dependences have been observed in various magnetic materials, such as (Ga,Mn)As,⁵⁹ Fe₃O₄,^50,60,101 Fe_xNi_3−xO₄,⁵⁰ La_1−xSr_xCoO_3−δ,¹⁷⁸ SrRuO₃,¹⁷⁹ and Sr₄Ru₃O₁₀.⁶² In contrast, in CMR manganites, the AMR can occur well above T_C, and the AMR and PHE show a nonmonotonic dependence on temperature and magnetic field, which can be attributed to magnetic phase transition.⁶⁵ This is clearly illustrated in Figs. 5(a) and 5(b), where the AMR in single crystal LCMO peaks in the vicinity of T_C, coinciding with the metal-insulator transition.²⁹ This behavior has been widely observed in LCMO,^{29–33,69,180–186} LSMO,^{6,64–66,80,82,98,187} La_0.67(Ca_1−xSr_x)_0.33MnO₃,⁹⁶ La_0.7−xPr_xCa_0.3MnO₃ (LPCMO),^67,188–190 Nd_0.55−xSm_xSr_0.45MnO₃,¹⁹¹ Sm_0.53Sr_0.47MnO₃,¹⁹² and La_1.2Sr_1.8Mn₂O₇.¹⁹³ In Ref. 66, Yau et al. attributed this phenomena to the competing spin scattering mechanisms due to magnetization and magnetic inhomogeneity. This scenario is well supported by the chemical doping dependence of AMR in LSMO. Thus, the AMR occurs above T_C due to the existence of local ferromagnetic clusters, and peaks at the magnetic transition where the magnetic inhomogeneity reaches maximum. This scenario has been generalized to explain the anomalous granularity dependence of AMR in polycrystalline LCMO^185,186 and LSMO³⁷ films. The correlation between the magnetotransport anisotropy and magnetic transition provides a guideline in optimizing the AMR and PHE resistance for device applications.^146,194

Besides the variation in magnitude, the AMR and PHE resistance can also exhibit a sign reversal driven by temperature, magnetic field, and film thickness. Figures 5(c) and 5(d) show the AMR ratio versus the magnetic field in LCMO films.³¹ In contrast to the 6 nm film, the 4 nm thick LCMO exhibits a sign change in AMR. This difference has been attributed to the enhanced strain in the thinner film, which modifies the magnetic anisotropy from an easy axis to an easy plane.^31,184 The sign reversal phenomenon has been observed in various epitaxial manganite thin films, including LCMO,^{30,31,33,184,195} LSMO,^{80–82,85,196} La_0.67(Ca_1−xSr_x)_0.33MnO₃,⁹⁶ LPCMO,¹⁸⁹ Pr_0.67Sr_0.33MnO₃,¹⁹⁷ Pr_0.7Ca_0.3MnO₃,¹⁹⁸ Sm_0.5Ca_0.5MnO₃,¹⁹⁹ and Nd_1−xSr_xMnO₃^34,84 as well as magnetite thin films.^50,101

The strain state of epitaxial thin films depends on both the substrate type and the film thickness, and the resulting change in MCA is reflected in the AMR and PHE. For example, La_1−xSr_xMnO₃ film with $x>0.15$ deposited on (001) STO is subjected to tensile strain,^200,201 which leads to in-plane biaxial MCA with easy axes along in-plane [110] and $[1¯10]$ directions.^202,203 When deposited on (001) LaAlO₃ (LAO), LSMO is subjected to compressive strain, resulting in out-of-plane magnetic easy axis.^203,204 Figures 6(a) and 6(b) show the temperature dependence of the AMR ratio for La_0.7−xPr_xCa_0.3MnO₃ films deposited on STO and LAO substrates.¹⁸⁹ The AMR reverses sign at low temperature for the film on LAO and remains the same sign for the film on STO. This compressive strain-induced sign reversal has been observed in various magnetic oxides, including LCMO,^{31,32,184,195} LSMO,^81,85,196 Pr_0.67Sr_0.33MnO₃,^197,205 and SrRuO₃.¹⁷⁹ Alagoz et al. proposed that this sign reversal can be accounted for by the phase separation scenario, with the anisotropy evolving with temperature and magnetic field due to the competition between the percolative ferromagnetic phase and the charge-ordered insulating state.¹⁸⁹ 

It has been predicted that the AMR and PHE can be enhanced by charge localization,¹⁸⁸ an effect that can also be controlled by strain. As shown in Figs. 6(c) and 6(d), for LCMO deposited on STO and LAO, the AMR ratio increases with decreasing film thickness, which is attributed to the strain relaxation in thicker films.^{30–32,184,186} Similar thickness dependence of AMR and PHE ratios has been reported in LSMO,^81,82 La_0.7−xPr_xCa_0.3MnO₃,^188,189 and fractional doped Pr_xSr_1−xMnO₃.^70,197,205 This effect can be used to optimize the layer thickness ratio of heterostructure/superlattice configurations for designing devices based on AMR and PHE resistance.^44,146,206

The strain state can be also varied by depositing the epitaxial films along different crystalline directions. Figures 6(e) and 6(f) show the angular dependence of AMR at different magnetic fields for LSMO thin films deposited on yttria-stabilized zirconia-coated (001) Si substrates with different buffer layers.^83,87,207 The LSMO thin films are grown along (001), (011), and (111) directions on the STO/CeO₂, STO, and STO/TiN buffer layers, respectively, which correspond to different MCA. At high magnetic field [50 kOe, Fig. 6(e)], all three systems show similar sinusoidal angular dependence of AMR that is well described by Eq. (4). When the magnetic field is reduced to 1 kOe [Fig. 6(f)], the AMR shows distinct features for these three samples. For the (001) LSMO, the AMR oscillation exhibits a 90° phase shift while still follows the twofold oscillation, conforming to the uniaxial in-plane MCA. For the (011) film, the angular dependence exhibits a fourfold oscillation, as the [011] and $[11¯0]$ directions become the in-plane easy axes.²⁹ The AMR in the (111) film exhibits pinning close to 90° and 270° due to the emergence of a uniaxial anisotropy. The effect of film crystalline orientation on the AMR and PHE has been also studied in LCMO films grown on STO⁶³ and NdGaO₃³⁰ substrates as well as LSMO⁶⁸ and Sm_0.5Ca_0.5MnO₃¹⁹⁹  films.

In correlated oxides, chemical doping is an effective approach for tuning the AMR and PHE. Figures 7(a) and 7(b) show the temperature and magnetic field dependences of AMR ratio for La_1−xSr_xMnO₃ films (x = 0.16 and 0.35).⁶⁶ For the x = 0.16 sample, the AMR ratio peaks sharply close to the metal-insulator transition temperature T_MIT. At a higher doping of x = 0.35, the magnitude of the AMR ratio is almost one order of magnitude smaller and exhibits a much weaker dependence on temperature and magnetic field. As chemical substitution also modifies the lattice distortion due to the change in the A-site cation size,^200,208 the corresponding AMR can be affected by both charge and lattice effects. In Ref. 187, Hong et al. correlated the magnitude of AMR in manganite films with different compositions with the associated lattice distortion. As shown in Fig. 7(c), the maximum AMR ratio for LSMO and LCMO films (x = 0.16–0.33) increases monotonically with decreasing tolerance factor, $t=(rA+rO)/2(rMn+rO)$⁠, where r_A, r_O, and r_Mn represent the weighted average radius of A-site cations, Mn, and O ions, respectively.¹⁸⁷ The change is moderate in less distorted systems and rises sharply for t < 0.9. Similarly, the two distinct regimes in the variation of AMR have been observed in LPCMO^188,189 [Figs. 6(a) and 6(b)], La_0.67(Ca_1−xSr_x)_0.33MnO₃,⁹⁶ and antiferromagnetic materials such as Sr₂(Ir_1−xGa_x)O₄,²⁰⁹ which has been attributed to effects of distorted oxygen octahedral,¹⁸⁷ phase separation,^66,67,96 and charge localization.^188,189,209 Figures 7(d) and 7(e) show the maximum AMR ratio extracted from LPCMO films with various compositions, film thicknesses, and substrate types, which reveals a weak change in more itinerant systems and significant enhancement in highly localized systems.

A powerful technique to differentiate the charge and lattice effects is the electric field effect approach. Figure 8(a) shows the GPHE measured in a PbZr_0.2Ti_0.8O₃ (PZT)/4 nm LSMO heterostructure for the two polarization states of ferroelectric PZT.⁵⁷ For the polarization up (P_up) state, the second abrupt resistance jump occurs at a much higher field than the polarization down (P_down) state, indicating a larger anisotropy field in the accumulation region. The polarization-dependent anisotropy fields have been quantified using the method described in Sec. II B, which reveal a higher anisotropy energy E_MAE in accumulation. Figure 8 compares the E_MAE extracted from the P_up and P_down states of the heterostructure with those of single-layer LSMO samples with different chemical doping levels. While all samples show enhanced E_MAE with an increasing doping level, the change due to the ferroelectric field effect agrees well with the predicted doping effect via density functional theory (DFT) calculations combined with second-order perturbation to SOC. In contrast, the E_MAE for the chemically doped x = 0.33 sample is smaller than the theoretical value. This deviation can be understood by considering the smaller bulk lattice constant at higher chemical doping level,^57,200,210 which results in a larger tensile strain, suppressing the in-plane MAE.

The AMR and PHE study can reveal valuable information on the field effect modulation of the MCA. Figure 9(a) shows the temperature-dependence of in-plane AMR ratio obtained on 3–4 nm LSMO (x = 0.16 and 0.33) and LCMO (x = 0.3) films gated by PZT.¹⁸⁷ All three systems exhibit highly consistent in-plane AMR ratios for the accumulation and depletion states. A close inspection of the magnetic field dependence shows that modulation of the carrier density alone does not change the AMR ratio, i.e., the deviation only occurs when LSMO is tuned to be paramagnetic in the depletion state and ferromagnetic in the accumulation state [Fig. 9(b)]. In contrast, it has been shown that the out-of-plane AMR ratio in PZT-gated La_0.825Sr_0.175MnO₃ exhibits a much larger value [Fig. 9(c)] as well as a larger anisotropy field [Fig. 9(d)] in the accumulation state.²¹¹ The discrepancy between the in-plane and out-of-plane AMR can be explained by considering the effect of the orbital occupancy on MCA. In manganites, the ferroelectric polarization breaks the out-of-plane symmetry and modifies the occupancy of the $dz2$ orbital, which directly impacts the out-of-plane anisotropy. The polarization reversal thus leads to a pronounced field effect modulation of the out-of-plane AMR.^211–213 In contrast, the in-plane anisotropy is controlled by the occupancy difference between the $dxz$ and $dyz$ orbitals.⁵⁷ The corresponding modulation of E_MCA is proportional to the doping change, which scales with the channel resistance, resulting in a polarization-independent AMR ratio.

In addition to charge doping, the ferroelectric substrates can also be used to tune the MCA by electrically controlling the strain state.²¹⁴ In Ref. 36, Zhao et al. demonstrated nonvolatile magnetic switching and AMR phase shift in LSMO controlled by a piezoelectric 0.7Pb(Mg_1/3Nb_2/3)O₃/0.3PbTiO₃ (PMN-PT) (011) substrate. Applying an electric field between the two side electrodes of PMN-PT generates an in-plane strain in LSMO through the piezoelectric effect, switching its magnetic easy axis between (100) and $(011¯)$ directions. As shown in Fig. 10, the angular dependence of AMR exhibits similar characteristic for the negative polarization state $(PR−)$ with current along the $(011¯)$ direction [Fig. 10(a)] and the positive polarization state $(PR+)$ with current along the (100) direction [Fig. 10(d)]. Such similarity is also observed when interchanging the current directions [Figs. 10(b) and 10(c)]. This clearly illustrates the nonvolatile switching of the magnetic easy axis. The strain tuning of AMR via a piezoelectric substrate has also been demonstrated in LCMO/BaTiO₃,¹⁹⁵ (Ga,Mn)As/PZT,²¹⁵ and magnetic metals or alloys such as Ni²¹⁶ and Co/Cu/Ni heterostructures.²¹⁷ 

For correlated oxides, the antiferromagnetic phase often possesses a much higher MCA compared to the ferromagnetic phase, which can be associated with the spin canting or staggered spin texture in the charge- or orbital-ordered state. The staggered spin structures often give rise to more intriguing AMR and PHE features compared with the conventional ferromagnetic and paramagnetic materials. Figure 11(a) shows the angular dependence of PHE in SIO films.⁵⁴ At low magnetic fields, there are small dips at 135° and 315° superimposed on the twofold oscillation, which have been attributed to spin-canting-induced uniaxial MCA along the $[1¯10]$ direction.^{51,54,71–73,209} This feature disappears at high magnetic fields, where a weak ferromagnetic phase emerges [Fig. 11(b)]. Similar results have been observed in Pr_0.7Ca_0.3MnO₃¹⁹⁸ as well as (Sm,Ca)MnO₃,¹⁹⁹ (Pr,Sr)MnO₃,⁷⁰ and (Nd,Sr)MnO₃²¹⁸ close to half-doping. The AMR and PHE in antiferromagnetic materials can also change sign [Fig. 11(c)],¹⁹⁹ which has been explained by the competing magnetic anisotropy between the charged-ordered antiferromagnetic insulating (AFI) state and the percolative ferromagnetic clusters.

One interesting result is that the melting of the charge ordered antiferromagnetic state can cause irreversible change in the AMR and PHE. Figure 12(a) shows the measurement of AMR in Pr_0.7Ca_0.3MnO₃ with Néel temperature T_N = 110 K.¹⁹⁸ In the low magnetic field (≤4 T) and high magnetic field (≥7 T) regimes, the AMR exhibits twofold oscillation with an opposite sign. In the transition regime (5–6 T), however, the magnitude of AMR resistance decreases drastically with a rotating magnetic field, which has been attributed to the irreversible suppression of the charge-ordered AFI state. Similar irreversible change in AMR has been observed in antiferromagnetic LCMO on (001) NdGaO₃³⁰ and Nd_0.5Sr_0.5MnO₃ films²¹⁸ as well as ferromagnetic manganites with strong phase separation (e.g., La_0.3Pr_0.4Ca_0.3MnO₃⁶⁷) in the polycrystalline thin film form (e.g., LCMO¹⁸⁵ and LSMO³⁷ films). The latter can be attributed to the slow response of the inhomogeneous magnetic phase to the rotating magnetic field.

In phase-separated manganites, the competition between the charge-ordered AFI state and the ferromagnetic clusters can lead to stabilized AMR and PHE magnitude over a broad temperature range.^219,220 As shown in Figs. 12(b) and 12(c), the AMR ratio in an LCMO film on (001) NdGaO₃ exhibits a plateau at the maximum value over about 100 K between the higher T_C and lower T_N.³⁰ Similar plateau at the AMR maximum has been observed in close to half-doped (Sm,Ca)MnO₃,¹⁹⁹ (Pr,Sr)MnO₃,⁷⁰ and (Nd,Sr) MnO₃²¹⁸ as well as Fe₂As²²¹ and SIO.⁵¹ This broadened AMR maximum can be desirable for designing the AMR and PHE resistance device that is robust against temperature and magnetic field variation.^44,146

The electric field effect has also been applied to modulate the AMR in SIO.⁷³ Figures 12(d) and 12(e) show the angular dependence of the out-of-plane AMR for thin films at different gate voltages (V_g).⁷³ At V_g = 0 V, the AMR reaches the minimum value along the in-plane direction, which can be explained by a smaller scattering rate in the weak in-plane ferromagnetic state induced by spin canting. Switching in-plane magnetization leads to a magnetic field hysteresis in AMR. At V_g = −2 V, in contrast, the AMR exhibits additional local maxima along the in-plane direction, which has been explained by the emergence of ferromagnetic clusters with an out-of-plane easy axis. As it suppresses the weak in-plane magnetization associated with spin canting, the magnetic field hysteresis is also diminished.^3,51

Distinct AMR and PHE behaviors have been observed in cuprate superconductors due to the much stronger spin-charge coupling and the inhomogeneous stripe phase within the CuO₂ conducting planes. Figures 13(a) and 13(b) show the angular dependences of out-of-plane and in-plane resistivities in a lightly doped La_2−xSr_xCuO₄ (x = 0.01) film, which reveal an extremely high magnetic anisotropy field (>15 T at 130 K).⁴⁷ At the zero magnetic field, the spins are aligned within the CuO₂ plane (ab-plane) along the easy axis (e.g., b-axis). The Dzyaloshinskii–Moriya interaction cants the in-plane spin toward c-axis, which induces a weak net moment within each layer. A moderate in-plane magnetic field can change the magnitude of net magnetization, while it is still confined within the bc-plane due to the large uniaxial MCA. Notably, when the magnetic field rotates in the ab-plane, the out-of-plane AMR exhibits a twofold sinusoidal oscillation [Fig. 13(a)]. This effect can be attributed to the relative alignment of interlayer magnetization, which prefers ferromagnetic (antiferromagnetic) coupling when the in-plane magnetic field is along a-axis (b-axis). The resulting change of interlayer conduction is thus similar to the spin valve effect.^40,222,223

The in-plane AMR, on the other hand, is dominated by the intrinsic stripe phase and antiphase antiferromagnetic domains within the CuO₂ plane^224–226 and exhibits a fourfold oscillation pattern [Fig. 13(b)]. It is strongly influenced by the spin-flop transition, around which the magnetic multidomain state emerges and the antiphase boundaries can lead to anisotropic electron scattering. The effects of antiphase boundary and spin-flop transition on AMR have been observed in various cuprate superconductors in the antiferromagnetic phase, including YBa₂Cu₃O_6+x,^226–228 YBa₂Cu₄O₈,^74,229 Nd_2−xCe_xCuO₄,^40,52,89 Pr_1.3−xLa_0.7Ce_xCuO₄,⁸⁸ Pr_2−xCe_xCuO₄,⁹⁰ La_2−xCe_xCuO₄,²³⁰ La_1.45Nd_0.4Sr_0.15CuO₄,⁴⁹ and Sr_1−xLa_xCuO₂^50,91 as well as some ferromagnetic materials such as CeSb,⁴⁸ Fe₃O₄,^50,101 and Ni_0.3Fe_2.7O₄.¹⁰² 

When the applied magnetic field exceeds the anisotropy field, the spin-flop transition can switch the magnetic easy axis, e.g., from b-axis to a-axis, which has been confirmed by angular-dependent magnetic susceptibility measurements.⁴⁹ In this case, the out-of-plane AMR also exhibits a fourfold oscillation pattern, with resistance minimized at the two equivalent easy axes. In Nd₂CuO₄, the spin structure shows two transitions between two types of configurations with varying temperature,^231–234 one type at the high (>75 K, type-I) and low (<30 K, type III) temperature regions and another one at intermediate temperature (type-II). Figure 13(c) shows the high field (12 T) angular dependence of out-of-plane AMR ratio at various temperatures in Nd_2−xCe_xCuO₄ (x = 0.025), whose anisotropy field is lower than 10 T.⁵² The AMR exhibits a fourfold feature within each type of state, which can be attributed to the switching of easy axes with the rotation of the magnetic field and emergency of antiphase boundaries during spin-flop transition. At the phase transition temperatures (30 and 70 K), on the other hand, the AMR shows a twofold oscillation due to the suppression of the long-range spin order. Also, the 45° shift of AMR maxima positions from the type-I to type-II states can be attributed to the switched magnetic easy/hard axes between these two phases. Similar effects of spin phase transitions on AMR have been observed by tuning the external magnetic field in Nd_2−xCe_xCuO_4+δ (x = 0.12).⁸⁹ 

In topological materials, the AMR and PHE are strongly influenced by the competition between the chiral anomaly and classical anisotropic orbital magnetoresistance. Figures 14(a) and 14(b) show the temperature and magnetic field dependences of PHE amplitude in the Weyl semimetal T_d-MoTe₂ crystal.^109,128 The PHE amplitude exhibits a moderate increase with decreasing temperature at high temperatures followed by a rapid growth below 50 K [Fig. 14(a)]. The turning point can be correlated with the occurrence of Lifshitz transition and band reconstruction around 50 K,^{170,235–239} suggesting a chiral contribution-induced PHE enhancement at low temperatures. The competition between the chiral and normal contributions has also been revealed in the magnetic field dependence. As shown in Fig. 14(b), below the transition temperature, the PHE amplitude shows a quadratic B-dependence below 4 T and a linear B-dependence at higher field. The T- and B-dependences of the PHE effect can be well fitted theoretically [e.g., Eq. (12)]^{94,124,166,167} by considering the transition between the classical and topological regions. Similar transition in AMR and PHE has been observed in topological insulators Bi_2−xSb_xTe₃,¹⁰⁸ Sr_0.06Bi₂Se₃,¹⁶³ SmB₆,¹³⁷ Bi₂Te_xSe_3−x,¹⁶⁴ Bi₂Te₃,¹⁶⁵ and Bi_xSb_1−x¹⁰⁷; topological superconductors TaSe₃¹⁷⁰; and the Dirac and Weyl semimetals PdTe₂,¹³⁰ PtSe₂,¹⁶² Co₃Sn₂S₂,¹³¹ Cd₃As₂,¹²⁶ WTe₂,¹²⁷ PtSn₄,¹¹² and ZrTe_5−δ.⁹⁴ 

The sign reversal of AMR and PHE has also been reported in topological materials, including topological insulators Bi_xSb_1−x¹⁰⁸ and SmB₆¹³⁷ and semimetals HfTe₅¹⁷⁶) and ZrTe_5−δ.⁹⁴ As shown in Figs. 14(c) and 14(d), in topological insulator Bi₈₅Sb₁₅ films, the angular dependences of both AMR and PHE change from a negative amplitude $(ρ∥−ρ⊥<0)$ to a positive one $(ρ∥−ρ⊥>0)$ as the temperature is decreased to about 150 K.¹⁰⁷ From Eqs. (11) and (12), at high temperatures, the chiral charge pumping is suppressed when the magnetic field is tilted away from the current, so that $ρ⊥$ is larger than $ρ∥$⁠. In the low temperature region, on the other hand, the transport contribution from Eqs. (11) and (12) diminishes, and resistivity is dominated by back-scattering. By applying a magnetic field, the spin-flip back-scattering is allowed by breaking the time reversal symmetry along the direction of the magnetic field but remains prohibited at the direction perpendicular to the magnetic field. The sign reversal at low temperatures can thus be attributed to anisotropic back-scattering, which lifts topological protection.^{53,93,108,133,171}

In topological insulators, due to the spin-momentum locking of the surface states, the AMR and PHE can be tuned by the electric field effect. Figures 15(a) and 15(b) show the angular dependence of the in-plane AMR ratio in BiSbTeSe₂ flakes under various magnetic fields at V_g = +10 V [Fig. 15(a)] and −50 V [Fig. 15(b)].⁴³ The AMR oscillation reverses sign between these two gate voltages, which can be attributed to the reversal of spin polarization.^2,240–242 In topological insulators, the magnetization is dominated by the net spin polarization from the conductive surface states. The Dirac electrons from the top and bottom surface states [the two Dirac cones in Figs. 15(c) and 15(d)] can be coupled via either the side surface states or the bulk states.^{164,243–252} Due to spin-momentum locking, the magnetic field can split these two surface states similarly as the Rashba effect, which induces net spin polarization. As the electron and hole branches have opposite spin helicities, the net spin polarization reverses direction when the Fermi level goes across the Dirac point [Figs. 15(c) and 15(d)], leading to the sign reversal of AMR. The electric field effect-modulated AMR and PHE have also been observed in Cr_x(Bi,Sb)_2−xTe₃ films⁴² and Bi_2−xSb_xTe₃ films¹⁰⁸ and heterostructures.⁷⁵ 

In this section, we discuss the recent progress of AMR and PHE studies in artificial nanostructures, 2D electron gas (2DEG) formed at oxide interfaces, heterostructures, and superlattices. Part A discusses a giant enhancement of uniaxial MCA in nanostructured LSMO revealed by the PHE pinning effect. In part B, we review the AMR and PHE in oxide-based interfaces, which exhibit features of strong SOC and interface crystalline symmetry. Part C outlines the spin-coupling and exchange-spring effects detected by AMR and PHE in the heterostructure and superlattice samples.

In Ref. 118, Rajapitamahuni et al. reported a giant enhancement of MCA in nanostructured LSMO films on (001) STO, where the top layer of the 6 nm LSMO thin films are periodically etched into 200–400 nm wide, 2 nm deep nanostrips [Figs. 16(a) and 16(b)]. Compared with the unpatterned films [Figs. 2(c)–2(e)], the nanostructured LSMO shows strong pinning of R_PHE along the $[110]$ direction [Fig. 16(c)], revealing the emergence of strong uniaxial MCA. Systematic studies of the magnetic field dependence of the $θ$ versus $φ$ relation [Figs. 16(d) and 16(e)] show that the corresponding magnetic anisotropy energy is about 50-fold of the biaxial MCA in unpatterned LSMO. This giant enhancement has been attributed to a strong strain gradient sustained in the nanostripes, where the transmission electron microscopy imaging reveals a drastic suppression of the c-axis lattice constant approaching the nanostripe surface. This scenario is well corroborated by the first-principles DFT calculations. Figure 16(f) shows the angular dependence of R_PHE in a nanostructured LSMO thin film, where only half of the current channel between the Hall terminals are patterned into nanostripes.⁵⁶ Besides the strong pinning at the R_PHE maximum values around the emergent uniaxial easy axes, the PHE exhibits a sharp switching to an additional pinning state of R_PHE ∼ 0 Ω. This zero PHE resistance state can be attributed to domain formation due to magnetization pinning in the patterned side competing with rotating magnetization in the unpatterned side, leading to compromised net magnetization. The nanostructure approach with enhanced MCA can be exploited to improve the thermal stability of AMR- and PHE-based applications as well as designing multilevel memory devices.^44,146

The 2DEG formed at the interface of two insulating complex oxides often possesses strong Rashba-type SOC and MCA, which can lead to more prominent and intriguing AMR and PHE features compared with the single-phase materials.²⁵³ For example, oxides-based interface 2DEG, such as Co/STO,²⁵⁴ LaAlO₃/STO,^113–116 LaVO₃/STO,¹¹⁷ LaTiO₃/STO,²⁵⁵ LaTiO₃/KTaO₃ (KTO),²⁵⁶ EuO/KTO,^92,257 and LaVO₃/KTO,¹²¹ exhibit large AMR ratios (e.g., >60% in LaVO₃/STO) and PHE at low temperatures. This effect is absent in the symmetric STO/Nb:STO/STO heterostructure,¹¹⁶ which highlights the critical role of SOC. Figure 17(a) shows the angular dependence of the AMR ratio in (001) LaAlO₃/STO. The sample exhibits twofold oscillation at low field (3 T), with a fourfold oscillation signal superimposed at a high field (9 T). Unlike the single-phase materials, where MCA is dominated by the Zeeman term at high field, the interface MCA is also increasing with the magnetic field due to the enhanced SOC. Recent studies of the AMR and PHE in EuO/KTO^92,257 and LaVO₃/KTO¹²¹ also reveal the presence of the Berry phase effect, which can be attributed to the emergence of nontrivial interband contribution from the Dirac points generated by the Rashba effect.

Several methods have been proposed to tune the interface SOC, which can be detected by AMR and PHE. Figure 17(b) shows the evolution of the AMR ratio in (001) LaAlO₃/STO at different gate voltages, which exhibits an enhancement of the fourfold component with increasing electron accumulation.^114,115 This can be explained by considering the second-order perturbation to the SOC between the e_g and t_2g orbitals, which increases with the rising Fermi level, leading to a larger MAE. The electric field effect tuning of SOC has also been studied in the Co₂FeSi/BaTiO₃ heterostructure.²⁵⁸ By applying an electric field, the interfacial SOC and magnitude of AMR ratio have been tuned by the ferroelectric domain wall motion in BaTiO₃. The Rashba SOC field can also be tuned by current, which in turn modifies the MCA. Figure 17(c) shows that in LaAlO₃/STO interface 2DEG, the AMR resistance at θ = 0° and 180° becomes progressively more asymmetric with increasing current, indicating the effect of enhanced SOC on MCA.¹¹³ 

The effect of MCA on AMR and PHE has also been investigated in oxide interface 2DEG at different crystal orientations. Figure 17(d) shows the angular dependence of the AMR ratio measured on (111) LaAlO₃/STO.¹¹⁵ A sixfold oscillation has been clearly identified by subtracting the two- and fourfold components from the original data, which can be attributed to the triangular crystalline symmetry for the (111) orientation. Such higher order oscillation conforms to the in-plane symmetry of the materials systems¹⁰⁴ and has been reported in the (111) LaVO₃/STO interface,¹¹⁷ surface 2DEG at (111) STO,¹¹² Dirac antiperovskite Sr₃SnO,¹⁰⁰ pristine and doped graphene,^35,106 PdSe₂,²⁵⁹ Nd-Sn artificial honeycomb spin ice,²⁶⁰ and kagome ferromagnet Fe₃Sn₂.¹⁰⁵ 

In magnetic heterostructures and superlattices, spin in different layers can be correlated via the spin-coupling and exchange-spring effects,²⁶¹ which can be detected by the AMR and PHE. Figure 18(a) shows the angular dependence of the AMR ratio in SIO/LSMO and SIO/LaNiO₃/LSMO heterostructures.²⁰⁶ The magnetic field (100 mT) is lower than the magnetic anisotropy field of the antiferromagnetic SIO, but much higher than the coercive field of LSMO (∼5 mT). Due to the exchange-spring effect, the spin structure in the LSMO layer is highly affected by the MCA of SIO, leading to a giant fourfold AMR signal in the SIO/LSMO heterostructure. This fourfold AMR signal is absent in the SIO/LNO/LSMO heterostructure, as the nonmagnetic LNO buffer layer blocks the spin exchange between SIO and LSMO.

Similar spin coupling effect on AMR has been observed in LCMO/YBa₂Cu₂O₈ (YBCO) heterostructures. Figure 18(b) shows the angular dependence of AMR resistance in LCMO/YBCO/LCMO heterostructures and a (7.5 nm LCMO/13 nm YBCO)₁₀ superlattice.^74,229 While the sandwiched structures show twofold AMR oscillations, the AMR in the multilayer exhibits a period of 2π. The latter effect has been attributed to the enhanced spin coupling between LCMO and YBCO, where the magnetization in LCMO is entirely pinned by the extremely high MCA of YBCO and can no longer rotate with the magnetic field.

In Ref. 44, Hoffman et al. used the AMR behavior to probe the interfacial spin coupling and noncollinear spin alignment in LNO/LSMO superlattices. Figures 18(c) and 18(d) show the angular dependence of AMR in this sample, which exhibits the pinning effect and magnetic hysteresis at low field (5 and 15 mT) and low temperature (110 K). Increasing the magnetic field or temperature recovers the sinusoidal oscillation and further reverses the sign of the AMR signal, which has been attributed to the evolving spin textures. In this system, the LNO serves as a spacer layer that tunes the RKKY interaction, which settles the coupling between LSMO layers to be antiferromagnetic. At low field/temperature, MCA competes with the interlayer antiferromagnetic coupling, so that the spins in adjacent layers are close to antialigned and perpendicular to the magnetic field, giving rise to a negative AMR signal $(ρ⊥>ρ∥)$⁠. With the increasing magnetic field and temperature, the Zeeman energy becomes dominant. A net magnetization is developed due to the spin canting toward the magnetic field direction [Fig. 18(d) inset], leading to a positive AMR $(ρ∥>ρ⊥)$⁠.

In this review, we aim at providing a comprehensive picture of the progress in understanding the AMR and PHE effects in various quantum materials, with a focus on their connection with magnetic anisotropy, charge correlation, spin-orbit coupling, nontrivial band topology, and interface coupling. A wide range of tuning parameters has been outlined, including temperature, magnetic field, strain, and chemical and electrical doping, which can be exploited to elucidate the underlying spin scattering mechanisms and competing energy scales. Understanding these effects can also facilitate the material design for their implementation in magnetic sensing and spintronic applications.

Given the high sensitivity to various control parameters and the simplicity in the measurement setup, it is conceivable that the AMR and PHE effects can be utilized in the future to probe a wide range of emergent spin phenomena in nanoscale or surface/interface systems,^262,263 such as spin-polarized edge states,^264–272 spin wave and magnon states,^273,274 spin torque,^{114,275–282} spin injection and pumping,^76,283–291 spin tunneling,^292–296 and complementary spin Hall^297–303 and inverse spin Hall effects.^{290,304–309}

In the meantime, an in-depth understanding of the AMR and PHE effects in quantum materials is yet to be gained. To date, the theoretical descriptions of these effects mostly remain at phenomenological and semiclassical levels. A more detailed quantum transport theory incorporating the microscopic picture should be developed. In strongly correlated oxides, there are only qualitative discussion of the relevant mechanisms controlling AMR and PHE, while the hypotheses are to be further examined. It thus calls for further theoretical and computational efforts as well as experimental studies in structural imaging, material characterization, and metrology. The studies of AMR and PHE in topological materials have mainly focused on the topological insulators and semimetals. The anisotropic transport phenomena in other topological materials, such as topological superconductors, are also of high research interests. We hope our review can motivate such efforts in understanding the AMR and PHE effects and optimizing them for material metrology and device applications.

The authors would like to thank Yuanyuan Zhang and Wuzhang Fang for valuable discussions. This work was supported by the National Science Foundation (NSF) through Grant No. DMR-1710461 and NSF-EPSCoR RII Track-1: Emergent Quantum Materials and Technologies (EQUATE), Award No. OIA-2044049.

The data that support the findings of this study are available from the corresponding author upon reasonable request.