Exploring anisotropic phases and spin transport in perovskite heterostructures: Insights into 3d/5d interfaces for antiferromagnetic spintronics

In recent years, the pursuit of innovative materials and architectures for next-generation spintronics devices has led to a burgeoning interest in antiferromagnetic (AFM) spintronics.^1–6 This emerging field harnesses the unique properties of AFM materials to manipulate spin currents, promising enhanced performance, reduced energy consumption, and increased functionality in spin-based applications.^2–4 One of the noteworthy features of AFM spintronics is its potential for achieving exceptionally high writing speeds, operating within the picosecond timescale. This accelerated performance is made feasible by the terahertz (THz) scale AFM resonance, a frequency that surpasses ferromagnetic resonance frequencies by three orders of magnitude (typically in the gigahertz range).^5,6 The study of anisotropy is crucial for selecting appropriate detection mechanisms in AFM spintronics. Anisotropic magnetoresistance (AMR), for instance, serves as a magneto-transport counterpart to magnetic anisotropy energy, offering a robust means of detecting AFM states.^7–9 

Transition metal oxides (TMOs) exhibit magnetic and transport anisotropy owing to their crystal structure and the presence of partially filled “d” orbitals in their valence shells.^7–9 These “d” orbitals participate in magnetic interactions and transport, showcasing various orientations within the crystal lattice. TMO structures encompass diverse configurations such as rocksalt, perovskite, spinel, layered, and rutile structures.^10,11 Anisotropy in TMOs can be either intrinsic, stemming from crystal structure and symmetry, or extrinsic, induced by epitaxial strain and structural asymmetry at TMO interfaces.^9,12–20

AFM TMOs exhibit remarkable properties, showcasing resilience against magnetic field perturbations, the absence of parasitic stray fields, ultrafast dynamics, and the generation of significant magnetotransport effects.^12–20 The detection and manipulation of AFM spin states can be achieved through spin–orbit torque (SOT) and optical methods.^21–25 Moreover, the AFM phase can be finely controlled or harnessed via canted phases.²⁵ As such, the systematic exploration of AFM perovskites and their anisotropic properties is imperative for the progressive development of AFM spintronics and their associated applications. Among the myriad materials under investigation, oxide heterostructures and interfaces, particularly those rooted in 3d/5d transition metal oxides, have emerged as promising platforms for advancing the frontier of AFM spintronics. These interfaces present unique opportunities for anisotropic phases due to reduced symmetry, increased correlations, and high surface-to-volume ratios in thin films.^16,25

Oxide perovskites based on 3d transition metals have played a crucial role in exploring different phenomena, particularly including colossal magnetoresistance (CMR),⁷ charge density waves (CDWs), high-temperature superconductivity, and metal–insulator transitions. For instance, La_0.7Ca_0.3MnO₃ single-crystal films exhibit CMR, showcasing resistance changes with temperature in the presence of a magnetic field.^26,27 Another intriguing perovskite material, Pr_0.5Sr_0.5MnO₃, demonstrates CDW behavior characterized by a periodic modulation of the charge distribution.^28,29 Moreover, the discovery of high-temperature superconductivity in compounds like YBa₂Cu₃O₇ and La_(2−x)Sr_xCuO₄ has revolutionized the field of superconductivity.^30,31 The pursuit of understanding and enhancing high-temperature superconductivity in perovskite materials remains an intense focus of research. In the domain of Mott insulators, compounds such as NiO and VO₂ undergo a Mott transition, representing a transition from a conducting to an insulating phase.^32–35 In 3d perovskites, higher Coulomb repulsion (U) among same-orbital electrons shapes electronic configurations and orbital behavior, despite weaker spin–orbit coupling (SOC). Conversely, in 5d perovskites, strong SOC due to relativistic effects, combined with crystal-field splitting from ligand interactions, alongside U, determines diverse electronic, orbital, and magnetic traits. The interplay of these factors defines nuanced properties in both 3d and 5d perovskite structures.^36–40 

Additionally, the introduction of TMOs and perovskite materials and their interfaces has led to the emergence of a novel class of compounds characterized by intriguing properties such as topological effects, semi-metallic behavior, Dirac materials, AMR, and the spin Hall effect (SHE), holding significant potential for advancing innovative applications in electronics, spintronics, and quantum computing.^25,41–48

Spintronics, a technology surpassing traditional CMOS computing, holds immense potential for advancements.^17,49 Various spin devices using interface technology have emerged, exploring applications in non-volatile Magnetic Random Access Memory (MRAM), following the discovery of Giant Magnetoresistance (GMR) and Tunnel Magnetoresistance (TMR).^20,50 Despite these advancements, Spin-Transfer Torque RAM (STT-RAM) surpasses MRAM in scalability.⁴⁷ Additionally, the Spin Field-Effect Transistor (Spin-FET), while not always requiring less energy than a Metal–Oxide–Semiconductor Field-Effect Transistor (MOSFET) to switch, is preferred due to faster switching speed.⁵¹ Consequently, attention has shifted toward interface physics based on spin scattering mechanisms.^5,17,26,27 The utilization of AFM materials and interfaces gained attention after the discovery of spin-valve-like magnetoresistance, exhibiting a magnetoresistance signal of over 100% for small (∼50 mT) applied fields in a NiFe/IrMn/MgO/Pt interfaces by Park et al.²⁰ Furthermore, AFM materials/interfaces are under investigation for their potential in Spin-Transfer Torque (STT) and Spin–Orbit Torque (SOT) based devices.^{17,19,21,48,51}

The interfaces of 3d/3d and 3d/5d perovskites explore anisotropic and topological phases, providing a unique opportunity to tune optical and electronic properties essential for applications in spintronics.^{17,19,20,25,44} The fabrication of high-quality films and interfaces holds paramount importance in accessing these phases and properties. However, the formation of various perovskite interfaces is a complex task due to chemical disparities and structural differences,^52,53 necessitating specialized techniques such as the RHEED-assisted pulsed laser deposition method.⁵⁴ One such notable example is the LaAlO₃/SrTiO₃ (3d/3d) perovskite interface, uniting lanthanum aluminate and strontium titanate, both non-magnetic and insulating materials. This interface has become a focal point in condensed matter physics, exhibiting a unique conductive layer in typically insulating materials.²⁶ It features emergent magnetism, high electron mobility, and potential for superconductivity. Additionally, in 3d/5d perovskite heterostructures, orbital reconstruction occurs at interfaces, leading to emerging anisotropy in transport phenomena like AMR and weak antilocalization (WAL).^44,55 This review primarily focuses on the development of AFM materials and interfaces possessing anisotropic magnetic and electronic properties, mainly governed by SOC and modifiable via charge transfer. The heterostructure of two-band insulators, LaAlO₃ and SrTiO₃, which exhibit anisotropic transport like AMR, serves as a foundational entry point into the field of AFM spintronics. Moreover, this comprehensive review encompasses several transport mechanisms, primarily focusing on describing anisotropy phenomena dependent on extrinsic and intrinsic scattering mechanisms in the realm of AMR and interface physics. The structure of this review comprises three main sections: Secs. II–IV, which progressively build upon these concepts.

In Sec. II, the transfer of charges across interfaces is crucial in causing exchange bias. This phenomenon occurs when the exchange coupling at the interface of different magnetic materials induces an asymmetric alignment of their magnetic moments.¹⁷ We will further explore these aspects in relation to AFM materials in the subsequent sections.

In Sec. III, the discussion revolves around WAL associated with two-dimensional transport anisotropy observed in LaAlO₃/SrTiO₃ and 3d/5d perovskite interfaces. Specifically, in interfaces lacking symmetry, Rashba SOC^56–58 induces WAL. WAL arises due to constructive interference among time-reversed electron paths in materials with robust SOC, enhancing electronic wave coherence and reducing resistance.⁵⁵ This review examines the impact of electric fields and disorder on WAL. Within LaAlO₃/SrTiO₃ interfaces, the presence of the D'yakonov–Perel spin relaxation mechanism indicates intrinsic scattering mechanisms.^59–62 Conversely, extrinsic scattering mechanisms,^63,64 notably observed in AMR while discussing in Sec. IV, are predominantly associated with LaAlO₃/SrTiO₃ interfaces. Expanding the concept of extrinsic spin scattering mechanisms to 3d/5d perovskite interfaces, such as LaMnO₃/SrIrO₃, the Elliott–Yafet mechanism is introduced.

Section IV investigates AMR, exploring transport anisotropy in conduction electrons concerning external fields due to extrinsic and intrinsic scattering mechanisms. It introduces a dilute semiconductor model using impurity spins as extrinsic scatterers.⁶³ The discussion delves into the origins of twofold and fourfold oscillations in AMR observed in LaAlO₃/SrTiO₃ and LaVO₃/KTaO₃ heterostructures, respectively, based on Rashba and Dresselhaus models. It also employs the Kondo model to characterize AMR in the LaMnO₃/SrIrO₃ superlattice, explaining the Kondo interaction between Mn moments and itinerant electrons. Additionally, it extends the concept of extrinsic spin scattering to 3d/5d interfaces, particularly in the CaMnO₃/CaIrO₃ interface, attributing AMR to spin–orbit scattering from spin-lattice coupling-induced lattice distortion. The review highlights intrinsic scattering mechanisms, notably anisotropic Hall resistance in CaMnO₃/CaIrO₃ interfaces, driven by underlying band topology associated with Rashba SOC.^59–62 

In a future perspective, it may be demonstrated that materials/interfaces within AFM exhibiting the topological Hall effect (THE) and spin Hall effect (SHE) provide avenues for harnessing AFM spins across diverse contexts.^{45,48,65–67} In conclusion, the exploration of AFM spintronics, particularly in oxide heterostructures and interfaces, promises an advanced spintronics application. The multifaceted interactions between electron correlations, SOC, and anisotropy provide a rich playground for the emergence of novel quantum phases and properties. This review aims to unravel the intricacies of these phenomena, paving the way for the effective utilization of AFM spins in next-generation spintronics technologies (Fig. 1).

It is crucial to understand the charge-transfer mechanism when discussing the interface physics of heterostructures. Magnetic and electrical anisotropic properties, such as exchange bias, AMR, and WAL properties, are greatly influenced in the presence of charge transfer across the interface. These effects will be thoroughly discussed in this review.

A polarization catastrophe occurs at the interface between LaAlO₃ and SrTiO₃ layers when a certain number of LaAlO₃ unit cells are deposited on TiO₂-terminated SrTiO₃ substrate.^26,68 This is due to the accumulation of an electrostatic potential caused by the dipole moment associated charged LaO⁺ and AlO₂⁻ planes.⁶⁹ It can lead to the creation of an electric field at the interface. As a result, the system becomes unstable and reconstruction or other changes must occur to avoid a polarization catastrophe. Simple electrostatic considerations suggest that a LaAlO₃ film can grow on SrTiO₃ up to a critical thickness before reconstruction takes place, which is confirmed by the two-dimensional electron gas (2DEG) appearing only in samples with a LaAlO₃ thickness above 4 unit cells.^70,71 The electrons in the 2DEG are confined to the interface and have unique electronic properties, including high mobility and superconductivity at low temperatures.⁷² Density functional theory (DFT) calculations show that the transfer of electrons takes place when exactly 4 unit cells of LaAlO₃ are deposited on SrTiO₃, stabilizing the system and reducing the electrostatic potential in the LaAlO₃ film. The formation of the (2DEG) at the critical thickness is an important argument in favor of polar instability as the main driving force for its formation. Achieving charge balance at the LaAlO₃ surface and the LaAlO₃/SrTiO₃ interface allows for the effective reduction of the electrostatic potential at the critical thickness. The transfer of electrons from the LaAlO₃ surface to the SrTiO₃ interface is favored by the variable valence of transition metal titanium ions.^73,74 Also, the $d x y$ orbital in the $t 2 g$ band refers to a specific electron orbital (⁠ $d x y$⁠) within the $t 2 g$ band of the Ti atoms located at the interface. This $d x y$ orbital is the first available state in Ti atoms.⁷⁵ The $d x y$ orbital is believed to play a key role in the ferromagnetic behavior observed in the system, as it is strongly correlated with the magnetic ordering of the interface. The concept of the $d x y$ orbital in the $t 2 g$ band is primarily based on experimental and theoretical studies of the electronic structure of the interface, and how it relates to the magnetic behavior. One perspective suggests that a layer of localized $d x y$ electrons is magnetically coupled by mobile electrons in the $d x z$ and $d y z$ bands,^75,76 while alternative data indicate that itinerant electrons may not be required or effective in connecting the local moments. Within the realm of electronic bands, a fascinating duality emerges with regard to $d x y$ bands, which can be classified into two distinct types. The first type originates from the surface Ti atoms, while the second type extends across multiple layers within the bulk material, originating from Ti atoms. Interestingly, the former type of band displays localization, with its band minima residing far away from the Fermi level at gamma point. In stark contrast, the latter type positions itself near the Fermi level at the gamma point. Furthermore, there exists a band derived from the $d x z$ and $d y z$ orbitals, which also resides in close proximity to the Fermi level at the gamma point, as shown in Figs. 2(a) and 2(b).⁷⁷ This intriguing observation suggests the potential for strong localization in the $d x y$ band, originating from interfacial Ti atoms, thus giving rise to a moment. Conversely, the bands originating from bulk $d x y$ and $d x z$/ $d y z$ orbitals predominantly contribute to conduction. Hall effect measurements have shed light on a remarkable phenomenon wherein the magnetic field (H) exhibits nonlinearity with increasing gate voltage.⁷² This intriguing behavior has been attributed to the emergence of a second carrier with lower mobility in the framework of two-band theory.⁷⁸ We speculate that these carriers correspond to the $d x z$ and $d y z$ bands, as they experience confinement near the Fermi surface, as depicted in Figs. 2(a) and 2(b).⁷⁷ Notably, these bands exhibit high anisotropy, with a significant mass aligned along a specific direction, which could explain the observed lower mobility. It is worth mentioning that the first carriers originate from the bulk site of the SrTiO₃ layer. Also, the Stoner model predicts that the critical temperature for ferromagnetism is proportional to the density of states (DOS) at the Fermi level. For the LaAlO₃/SrTiO₃ interface, the existence of the interface results in a displacement of the Fermi level, causing a notable concentration of states at the Fermi level. This high density of states can enhance the exchange interaction between the electrons and promote ferromagnetic ordering without the coupling between local moments and conduction electrons.⁷⁹ However, the Stoner model does not account for many other important factors in real materials, such as the role of the crystal structure, magnetic anisotropy, and the effects of temperature and disorder. However, the origin of this ferromagnetism is still a matter of debate. However, oxygen vacancies created at the SrTiO₃ site provide additional electrons that can contribute to the formation of the 2DEG. One proposed explanation is the presence of oxygen vacancies or other defects at the interface, which can create local magnetic moments at Ti³⁺ and enhance the exchange interaction.^80,81 In an article, the relationship between electronic reconstruction, magnetism, and conductivity in LaAlO₃/SrTiO₃ heterointerfaces is discussed. The authors used scanning superconducting quantum interference device microscopy to study the behavior of the interface and find that magnetism only appears above a critical thickness of LaAlO₃, similar to the conductivity. It is also observed that ferromagnetism is not affected by gate voltage and can be detected even in a non-conducting p-type interface. The authors suggest that the carriers at the interface do not need to be itinerant to generate magnetism and disorder or local strain may play a role in inducing magnetism among a collective of interface carriers. The observation of inhomogeneity in the ferromagnetic patches suggests that magnetism is generated by local factors rather than a global mechanism.⁸² 

In conclusion, understanding the charge-transfer mechanism in LaAlO₃/SrTiO₃ is essential for identifying the source of magnetic scatterers and conduction electrons. This section has revealed that the origin of magnetism and magnetic scatterers primarily lies in the interfacial $d x y$ orbital, whereas the conduction channel is formed by the $d x z/ d y z$ orbitals and the Ti orbital at the bulk site of SrTiO_3.

This section delves into the intriguing realm of magnetic exchange coupling across an interface, with a particular focus on the paramount role played by interfacial charge-transfer phenomena and controlling dimension of materials in heterostructures. Additionally, it elucidates how the manipulation of the dimensions within the constituting layers of the heterostructure can exert a profound influence on charge transfer and the underlying mechanisms of exchange interactions. This intricate relationship is explored both empirically and theoretically, using CaMnO₃/CaRuO₃, LaMnO₃/LaNiO₃, CaMnO₃/CaIrO₃, and SrMnO₃/SrIrO₃ interfaces as illustrative examples. Furthermore, this section underscores the significance of employing AFM materials in inducing anisotropy within the system, further enriching our understanding of these intriguing phenomena.

This section focuses on the investigation of the charge-transfer mechanism occurring at the interface of a CaMnO₃/CaRuO₃ heterostructure, with a particular emphasis on magnetic and electronic properties. The journey of AFM spintronics in heterostructures using AFM materials began with the CaMnO₃/CaRuO₃ heterostructure. AFM magnetic ordering similar to bulk CaMnO₃ is observed in epitaxial films of CaMnO₃ grown on different substrates.⁸³ CaRuO₃, characterized as a paramagnetic (PM) metal, possesses nearly the same crystal structure, i.e., Pbnm.^84–86 When creating superlattices with alternating layers of CaRuO₃ and CaMnO₃, a distinct electronic structure is observed at the interface, leading to electron leakage and the accumulation of excess charge in the CaMnO₃ layers.^87–91 Experimental measurements, including temperature and magnetic field-dependent tests, reveal a logarithmic divergence in resistivity, indicating charge scattering caused by the presence of magnetic defects.^85,92 The essence of the charge-transfer mechanism lies in the exploration of the interplay between electronic and magnetic properties within the CaMnO₃/CaRuO₃ heterostructure. Theoretical investigation of the CaRuO₃/CaMnO₃ interface has unveiled a remarkable phenomenon: the exponential leakage of metallic electrons from the metal side to the insulator side. The pivotal role of these leaked electrons lies in their ability to govern the magnetism at the interface, engaging in a dynamic interplay between FM Anderson–Hasegawa double exchange and the competing AFM superexchange of the bulk CaMnO₃ site. Analyzing the layer projected density of states (DOS) for the interface of CaMnO₃/CaRuO₃, it is observed that the first interfacial MnO₂ layer is FM, while the remaining Mn moments exhibit a type G AFM configuration, as shown in Fig. 3(a). Figure 3(b) displays the total energy variation as a function of the canting angle for different electron concentrations (denoted as x) per lattice site. Notably, for the initial layer near the interface with x = 0.07, the AFM state becomes unstable, leading to the formation of a canted state. As demonstrated by the curve corresponding to x = 0.2 in Fig. 3(b), the FM configuration becomes highly favored with increasing electron concentration.⁹⁰ Surprisingly, the magnetization at the interface remains constant regardless of the CaRuO₃ layer thickness, suggesting that the FM behavior originates primarily from the interface itself. It is important to emphasize that the onset of exchange bias (H_EB) is observed when N (CaMnO₃ unit cell) is equal to 4, which means that the composite CaMnO₃ layers in superlattices with N greater than or equal to 4 consist of AFM core layers sandwiched between one unit cell of interfacial FM layer on each side. However, the case where N equals 3 is exceptional, as it contains only one unit cell of the non-interfacial CaMnO₃ layer, which is unable to produce a pinning effect essential for exchange bias.⁸⁸ Consequently, we can infer that the source of ferromagnetism originates from a single unit cell of CaMnO₃ near the interface, as supported by theoretical investigations also.^88,90

The theory of charge transfer and exchange interactions is further extended to the LaMnO₃/LaNiO₃ heterostructure, where both constituent materials are AFM. Here, the phenomenon of interfacial charge redistribution in superlattices comprising LaNiO₃ (Ni³⁺) and LaMnO₃ (Mn³⁺) is investigated. LaMnO₃ is extensively studied as the parent compound for perovskite CMR oxides.⁷ In stoichiometric bulk LaMnO₃, the occupancy of the t_2g³e_g¹ orbitals leads to a cooperative Jahn–Teller distortion, which breaks the degeneracy of the half-filled “e_g” band and results in an A-type AFM insulating ground state with orbital ordering.⁷ However, thin films of LaMnO₃ often exhibit ferromagnetism with a Curie temperature of approximately 150 K and a saturation moment close to 4 μ_B/Mn. These characteristics are attributed to cation deficiency and strain effects.^52,53 On the other hand, LaNiO₃ is a paramagnetic metal where the Ni³⁺ ions adopt a low-spin, orbitally degenerate t_2g⁶e_g¹ electronic configuration. Utilizing ozone-assisted molecular beam epitaxy (MBE), researchers successfully fabricated epitaxial [(LaNiO₃)_n/ (LaMnO₃)₂]_m superlattices on (001) TiO₂-terminated SrTiO₃ single-crystal substrates, with n varying between 2 and 5 unit cells.^93, Figure 4 shows Mn L₃ and Ni L₃ edge x-ray absorption spectroscopy (XAS) and x-ray magnetic circular dichroism (XMCD) spectra for [(LaNiO₃)₂/(LaMnO₃)₂]₂₀ and [(LaNiO₃)₄/(LaMnO₃)₂]₁₃ superlattices. The XAS spectra demonstrate that the Mn valence in the superlattices is predominantly Mn⁴⁺, whereas Mn is in the 3+ oxidation state in bulk LaMnO₃ samples. For the Ni cations, the valence is close to Ni²⁺ in the superlattice with two layers, while a mixture of Ni²⁺ and Ni³⁺ is observed in the superlattice with four layers.⁹³ This observation provides conclusive evidence of charge transfer occurring at the interface between LaNiO₃ and LaMnO₃, driven by the difference in electronegativity. The resulting Ni²⁺−Mn⁴⁺ cations are anticipated to interact ferromagnetically following Goodenough–Kanamori rules.¹⁴ 

In another study, the unexpected observation of exchange bias in superlattices consisting of (111)-oriented layers of paramagnetic (PM) LaNiO₃ and FM LaMnO₃ is discussed. Epitaxial (LaNiO₃/LaMnO₃)_x, (n/m)_x superlattices, with n unit cells of LaNiO₃ and m unit cells of LaMnO₃, were grown on (111) SrTiO₃ substrate using off-axis radio frequency magnetron sputtering. The onset of magnetization was observed at approximately 200 K for superlattices with n ≥ 3 and m ≥ 3. The temperature dependence of the magnetization was similar to that of LaMnO₃ thin films grown on (111)-oriented SrTiO₃, suggesting the dominance of the inner ferromagnetic LaMnO₃ layers in determining the overall magnetization.⁹⁴ Similar dominance of the LaMnO₃ block in the magnetic properties was observed for superlattices grown on (001)-oriented SrTiO₃, consistent with previous reports on (001)-oriented LaNiO₃–LaMnO₃ superlattices.⁹³ However, the low-temperature field dependence of the magnetization in the (111)-oriented superlattice differed significantly from that of the (111)-oriented LaMnO₃ thin film when cooled in a −0.4 T field; the loop is shifted toward a positive direction, as shown in Fig. 5(a). The symmetry of the hysteresis loops of (111)-oriented LaMnO₃ thin films indicated that the LaNiO₃ layers were the driving force for the biasing effect observed in the heterostructures.⁹⁴ The exchange bias effect, shown in Fig. 5(b), decreased with increasing temperature and typically vanished within the range of 20–25 K, establishing a lower temperature limit for the onset of magnetism in the LaNiO₃ layers.⁹⁴ First-principles simulations revealed a more subtle magnetic order in (LaNiO₃/LaMnO₃) superlattices, showing that the interfacial Ni and Mn ions exhibited ferromagnetically coupled moments shown in Fig. 5(c).⁹⁵ However, the magnetic moment orientation of the core LaNiO₃ layer was opposite to that of the interfacial moments of LaNiO₃ near LaMnO₃. The distinctive behavior was particularly pronounced in the PM metal LaNiO₃, showcasing both insulating and magnetic characteristics in its ultra-thin state, spanning only a few monolayers. The overall coupling between neighboring LaMnO₃ layers, mediated by 7 monolayers of the LaNiO₃, was found to be AFM along the [111] direction, shown in Fig. 5(c).⁹⁵ 

In a distinct investigation focusing on [(SrMnO₃)_m/ (SrIrO₃)_n]_z, the magnetic field (H) dependence in symmetric samples with equal layer compositions (m = n) is studied.⁹⁶ In the thin film form, SrMnO₃ exhibits AFM behavior, while SrIrO₃ is characterized as a PM semimetal.^43,97 The exploration of iridates commenced with Sr₂IrO₄ (SIO), a canted AFM material that demonstrates remarkable anisotropic properties.⁸ Here, the experimental results on [(SrMnO₃)_m/(SrIrO₃)_n]_z, depicted in Fig. 6, uncover a remarkable observation.⁹⁶ The samples with the thinnest layers, comprising atomically thin superlattices, demonstrate the most pronounced magnetic response. Notably, while an FM response is observed at the SrMnO₃ (M)/SrIrO₃ (I) interface, (M_xI_y)_z where x and I represent the periodicity of SrMnO₃ and SrIrO₃ layers, respectively. It is intriguing to note that the overall magnetization of M₁I₁ is significantly higher than twice that of M₂I₂, while M₄I₄ displays no magnetization (M = 0).⁹⁶ Moreover, employing XAS, a shift in the Mn L₃ edge peak position toward a lower energy is identified, suggesting a lower Mn oxidation state in the heterostructures compared to stoichiometric SrMnO₃. Similarly, the Ir L₃ edge position shifts toward a higher energy, indicating an enhanced Ir oxidation state relative to Ir⁴⁺ in stoichiometric SrIrO₃, as shown in Fig. 6(a). These observations collectively suggest a charge transfer from the SrIrO₃ to the SrMnO₃ layers, resulting in electron (hole)-doped SrMnO₃ (SrIrO₃) layers. The inset of Fig. 6(a) presents the average oxidation states estimated from the peak shifts, with M₁I₁ exhibiting the most significant deviation from the nominal value, reflecting a charge transfer of approximately 0.5 electron/hole per perovskite unit cell.⁹⁶ Furthermore, Density Functional Theory (DFT) calculations are employed and experimental measurements are done to unravel intriguing findings concerning the strong electron transfer from SrIrO₃ to SrMnO_3.⁹⁸ This phenomenon can be elucidated by considering the electronic configurations of SrMnO₃ (⁠ $t 2 g 3$ electrons, S = 3/2) and SrIrO₃ (⁠ $t 2 g 5$ electrons). The strong SOC in SrIrO₃ splits the $t 2 g 5$ manifold into fourfold degenerate J_eff = 3/2 states (fully filled) and twofold degenerate J_eff = 1/2 states (half filled), leading to the presence of empty $e g$ states located well above the Fermi level. Figure 6(b) illustrates the alignment of the single-particle Density of States (DOS) for SrIrO₃ and SrMnO₃. Due to the substantial overlap of 3z² − r ² orbitals between neighboring Mn and Ir sites, molecular orbitals are anticipated to form. In this context, bonding orbitals, characterized by dominant Mn contribution, are situated below the lower Mn 3z² − r² level, while antibonding orbitals, exhibiting stronger Ir character, reside above the higher Ir 3z² − r² level. This molecular orbital description offers an explanation for the distinction observed between [SrIrO₃]_m[SrMnO₃]₁ and [SrIrO₃]_m[SrMnO₃]₃ superlattices. In the former case, where a SrMnO₃ layer is sandwiched between two SrIrO₃ layers, the bonding 3z² − r² orbital is more stable compared to the latter scenario. Consequently, in [SrIrO₃]_m[SrMnO₃]_n superlattices, where m remains fixed, the magnitude of electron transfer from SrIrO₃ to SrMnO₃ layers decreases as n increases.^96,98

Research on CaMnO₃/CaIrO₃ superlattices delves into the exchange bias (H_EB) as a distinctive feature characterizing the interface between FM and AFM phases.⁴⁴ The formed superlattices are denoted as (MIxy)z; CaMnO₃ and CaIrO₃ unit cells are represented as M and I, respectively, and x and y represent periodicity of M and I, respectively. CaIrO₃ thin films are reported to be PM semimetals,⁴³ while CaMnO₃ is known as an insulating AFM material. It was found that the charge transfer from CaIrO₃ to CaMnO₃ site is sensitive to the thickness of the constituent layers, with the (MI25)₅ superlattice displaying a maximum transfer of approximately 0.25 hole/electron per Ir/Mn ion, as summarized in Table I.⁴⁴ Additionally, Fig. 7(d) presents the H_EB observed in various superlattices, including (MI22)₁₀, (MI33)₅, (MI44)₅, and (MI84)₅, exhibiting corresponding H_EB values of 3, 15, 50, and 35 Oe. Achieving a higher H_EB necessitates a specific arrangement of the FM/AFM interface, which is absent in the initial, (MI22)₁₀ superlattice where only one CaMnO₃ layer is present at the interface as a pinning layer. The value of the magnetization is higher for the (MI22)₁₀ superlattice compared to other superlattices due to the canted AFM phase in CaIrO₃ at a lower dimension,⁹⁹ as shown in Fig. 7(b). However, in the latter three superlattices, as the CaMnO₃ layer thickness increases in conjunction with the FM interface, a more pronounced H_EB is observed, correlating with the thickness of the bulk CaMnO₃ layer apart from the interface. These findings provide insights into the potential formation of magnetic gradients, particularly where the AFM state remains unaltered away from the interface for both CaIrO₃ and CaMnO₃.

In this section, magnetic-order modulation in LaMnO₃/SrIrO₃ superlattices grown on a SrTiO₃ (100) substrate is discussed by atomically changing the stacking of the LaMnO₃ and SrIrO₃ period in the LaMnO₃/SrIrO₃ superlattice.¹⁰⁰ These superlattices are denoted as SLnmz, where n represents the periodicity of LaMnO₃, m represents the periodicity of SrIrO₃, and z represents the repetition of the heterostructure. This modulation resulted in changing magnetization, rotation of the FM easy axis, and transition of AMR symmetry discussed later. When a polar material, like LaMnO₃, is grown epitaxially on a nonpolar substrate such as SrTiO₃, an electric potential builds up, leading to electron accumulation at the bottom of the LaMnO₃ layer. When the LaMnO₃ thickness exceeds 3 unit cells, this accumulation triggers a phase transition from AFM to FM.¹⁰¹ This explains why the reference sample, [(SrTiO₃)1-(LaMnO₃)1]8, known as [(STO)1-(LMO)1]8, does not exhibit ferromagnetism since the individual LaMnO₃ thickness remains below 3 unit cells, as shown in Fig. 8. However, when the Ti in the structure is replaced with Ir in SL118 superlattice, ferromagnetism is observed. This transition confirms the presence of interfacial coupling between SrIrO₃ and LaMnO₃, establishing it as the source of ferromagnetism in SL118 superlattice. Thus, the presence of ferromagnetism and magnetic exchange bias in CaMnO₃/CaRuO₃ heterostructures depends on the charge transfer mechanism. Controlling the dimensions of the materials leads to magnetic exchange effects in LaMnO₃/LaNiO₃ and CaMnO₃/CaIrO₃ superlattices. Additionally, the importance of interfacial exchange interactions is demonstrated in the LaMnO₃/SrIrO₃ superlattice. All of the constituent materials are AFM, except for CaRuO₃, which exhibits paramagnetism in nature. Hence, studying the magnetic anisotropy in AFM materials and interfaces helps to understand the governing mechanisms in AFM spintronics.

The Rashba SOC intertwines electron spin and momentum, giving rise to intriguing phenomena like WAL, a quantum interference phenomenon, in two-dimensional materials and interfaces, which are highly anisotropic in nature. In our investigation of the LaAlO₃/SrTiO₃ heterostructure, we have delved into WAL phenomena that arise from Rashba SOC. Additionally, we have found that charge transfer effects have a significant impact on WAL in the context of this specific heterostructure. Furthermore, we have expanded this research paradigm to encompass AFM materials and their interfaces, as exemplified in superlattices such as LaMnO₃/SrIrO₃, SrIrO₃/SrTiO₃, and CaMnO₃/CaIrO₃. This expansion has pushed the boundaries of AFM spintronics research, opening up new avenues for exploration. Notably, Vagadia and colleagues worked on AFM heterostructures to probe anisotropies within AFM systems.

At sufficiently low temperatures where the phase coherence length greatly exceeds the diffusion length, the phenomenon of quantum interference between different electron paths becomes significant. The diffusion length of an electron is a distance that can be traversed before experiencing a scattering event. In systems that give rise to spin momentum locking due to Rashba SOC, surface electrons traverse a space through diffusive scattering, resulting in their spin orientation being rotated by 2π. Additionally, the wave function associated with these electrons accumulates a π Berry phase due to the presence of a two-component spinor wave function. Consequently, destructive interference occurs between the wave functions corresponding to the forward and reverse paths in a closed loop, leading to an enhancement in conductivity. This remarkable behavior, wherein the electron's motion is topologically protected, is known as WAL.^103,105 The theory of WAL has been proposed to describe the magnetoconductance curve for Dirac materials such as topological insulators where spin momentum locking is expected due to the specific band structure near the Fermi level.^105–107 The interplay between SOC and U in correlated materials has garnered significant interest. Theoretical studies have suggested the potential existence of a Dirac node through the appropriate tuning of U and SOC in the orthorhombic Pbnm symmetry of the SrIrO₃ crystal structure,^108,109 although no relevant experimental evidence has been reported to date. However, other topological signatures, such as WAL, confirm the existence of SOC-mediated transport phenomena in this material. Notably, a thickness-dependent experimental study has demonstrated the transition from WL to WAL, offering insights into the spin transport mechanism near the insulator-to-metal transition point.⁴³ Additionally, a theoretical study has shown the attainment of a metallic-to-insulating phase transition via a semimetallic phase by tuning U and SOC. The band structure calculations for the semimetallic phase revealed the presence of a Dirac node near the Fermi surface, although such a topological phase was not observed in the conducted experiment. Nevertheless, the observed WAL effect is expected to be governed by strong SOC, underscoring its significance in SrIrO_3.⁴³ Further experimental investigations are warranted to explore the interplay between SOC and U in SrIrO₃ and to validate the presence of predicted Dirac-like topological phases.

The structural configuration employed in the study of LaAlO₃/SrTiO₃ breaks the inversion symmetry, leading to the generation of a strong electric field perpendicular to the conduction plane in the electron gas confined near a polar interface. As we discussed previously, the Rashba Hamiltonian (H_R) captures a new class of physical phenomena arising due to the effective electric field. This Hamiltonian characterizes the coupling between the spin of electrons and an internal magnetic field (B_eff = E × P/mc²), known as Rashba field (B_Ry) experienced in their rest frame. B_Ry is perpendicular to their wave vector and lies in the interface plane. An important consequence of this interaction is the splitting of the electron dispersion relation into two branches separated by a spin splitting.^21,55 A notable feature of this interaction is its coupling constant $α R= μ B E zℏ/ m c 2$⁠, which is connected to the electric field experienced by the electrons and can be tuned by applying an external gate voltage. The external gate voltage helps to confine the electrons at the interface as well as modification of the electric field (E_z) is expected.¹⁰⁶ The conductance's dependence on the magnetic field highlights the intriguing correlation between spin dynamics and transport. Figure 10(a) depicts the magnetoconductance as a function of the applied magnetic field, denoted as H measured at a temperature of T = 1.5 K for gate voltages (V) ranging from −300 to +200 V, with measurements performed in a perpendicular field configuration.¹¹⁰ The magnetoconductance measurements are carried out using a standard four-point direct current (dc) technique. As shown in Fig. 10(a), a large positive magnetoconductance exceeding +25% at 8 T is observed for significantly negative and highest gate voltages (−300 V). With increasing voltage (V ∼ 200 V), a low-field regime characterized by a negative magnetoconductance emerges. This behavior has been observed in multiple samples, and similar modulations of magnetoconductance have been previously documented in semiconductor thin films.⁴³ The value of $τ so$ and $τ p$ can be obtained by fitting the Maekawa–Fukuyama equation.^43,111 Also, Fig. 10(c) shows that $τ so$ is inversely proportional with $τ p$⁠, i.e., $τ so α 1 τ p$⁠, indicating D'yakonov–Perel spin relaxation mechanism. The presence of D'yakonov–Perel mechanism indicates that the scattering mechanism is mainly dominated by the D'yakonov–Perel mechanism compared to the Elliott–Yafet mechanism.

The present study investigates the impact of the disorder on the LaAlO₃/SrTiO₃ interface and its correlation with the sheet carrier concentration. In the case of a sharp interface without oxygen vacancies synthesized at high oxygen pressure, it is observed that the carrier density remained constant beyond a critical thickness, i.e., 6 unit cells reported in several experiments.^70,71 The relationship between the degree of disorder and the metallic behavior as well as WAL of LaAlO₃/SrTiO₃ interfaces,¹¹² which arises from the polar interface instability, has been explored here. The findings indicate that the presence of static scattering centers near the interface leads to an increase in the number of carriers. To vary the sheet carrier concentration, the thickness of the LaAlO₃ film is manipulated. Table II illustrates the dependence of sheet carrier concentration on the LaAlO₃ film thickness (t) at 275 K. Thinner LaAlO₃ films in this heterostructure generally exhibit lower sheet carrier electron-like concentrations, which saturate above a thickness of 5 nm. Electrical transport data at 3 K demonstrate a general trend of decreasing electron mobility with increasing interfacial sheet carrier concentration.¹¹² At low temperatures, where inelastic scattering is minimized, the mobility serves as a measure of the degree of disorder associated with static elastic scattering sites for carriers. Also, the mobility values obtained at low temperatures align with those observed by other research groups.^71,78 These values, however, are significantly lower than those of bulk electron-doped SrTiO₃ crystals.¹¹³ This suggests that the effects of disorder must be taken into account in the LaAlO₃/SrTiO₃ heterointerfacial metallic channels. As shown in Table II, the increase in sheet carrier concentration from a lower thick LaAlO₃-based heterostructure, i.e., D, to a higher thick LaAlO₃-based heterostructure, i.e., A, accompanied by a decrease in mobility and an increase in disorder, implies that the generation of carriers is associated with a greater number of scattering centers. The formation of hole polarons, which has been postulated in related oxides, is likely to occur. In this context, the LaAlO₃ layer is considered as the source of carriers, resulting in the presence of immobile holes on the LaAlO₃ side.¹¹² Furthermore, the magnetoresistance (MR) anisotropy highlights the quasi-2D nature of the metallicity at the LaAlO₃/SrTiO₃ interface. Also, a crossover from positively to negatively sloped MR from A to D sample is observed, similar to previous observations in 2D disordered metals, as shown in Fig. 11(a). This phenomenon has been attributed to antilocalization effects arising from increasing spin–orbit interaction while increasing the LaAlO₃ thickness from D to A heterostructure. Thus, the strength of the SOC is higher in samples with higher sheet carrier density. The reason for the augmentation of the interfacial electric field arises from the aggregation of mobile electrons on the SrTiO₃ facet and potentially localized holes on the LaAlO₃ facet, resulting in a pronounced SOC phenomenon.¹¹² This phenomenon is enhanced in thicker sample A due to a higher number of hole polarons compared to the other samples.

In another study, the engineering of magnetism and SOC at the LaMnO₃/SrIrO₃ (3d-5d) oxide interface is explored by tuning the growth conditions of LaMnO₃, which influences spin-correlated interfacial coupling through charge transfer.¹¹⁴ In recent research, there has been a surge of interest in studying combinations of 3d-5d oxide interfaces, particularly in the context of SOC manipulation for potential applications in spintronics memory devices. LaMnO₃, which is the parent compound for manganite and contains the 3d element Mn, exhibits interesting magnetic properties. While it is an A-type AFM insulator in bulk, it can behave like a ferromagnet in epitaxial thin films due to vacancies or epitaxial strain.^52,53 However, the oxygen gas atmosphere pO₂ (37.5 mTorr) during the deposition process enhances the formation of Mn⁴⁺ ions, promoting double exchange-mediated FM ordering in epitaxial high-pressure LaMnO₃ film in LaMnO₃/SrIrO₃ heterostructures, known as HP (High Pressure)-LaMnO₃/SrIrO₃, grown on the (LaAlO₃)_0.3(Sr₂TaAlO₆)_0.7 (LSAT) substrate. Conversely, when the oxygen partial pressure pO₂ (37.5 × 10⁻³) during the deposition decreases, increased formation of Mn³⁺ leads to an AFM ground state in LaMnO₃ film in LaMnO₃/SrIrO₃ heterostructures, as known as LP (low pressure)-LaMnO₃/SrIrO_3.¹¹⁴ The magnetoresistance measurements reveal negative magnetoresistance for both low-pressure and high-pressure LaMnO₃/SrIrO₃ samples, attributed to dominating magnetic contributions. A crossover from negative to positive magnetoconductance in ultrathin SrIrO₃ films grown on SrTiO₃ (001) substrates is attributed to the competition between WL and strong SOC-based WAL.^43,105 However, a crossover from negative to positive magnetoconductance is observed at low magnetic fields and higher temperatures in both low-pressure and high-pressure LaMnO₃/SrIrO₃ samples.¹¹⁴ 

The effective fields of elastic-, inelastic-, and SOC-induced scattering terms, denoted as B_e, B_i, and B_so, respectively, are determined and plotted in Fig. 12. Also, a crossover from the negative to positive magnetoconductance curve in LP, HP-LaMnO₃/SrIrO₃ samples is indicating the presence of WAL, a signature of Rashba SOC at the interface. Recent studies on LaMnO₃/SrIrO₃ superlattices have shown that the strain-induced Ir–O–Ir bond angle leads to an internal electric field at the LaMnO₃–SrIrO₃ interface.¹⁰⁰ Interestingly, the behavior of the B_so parameter, related to the SOC-induced scattering, at the LP-LaMnO₃/SrIrO₃ sample, varies with temperature and saturates above 25 K. The values of B_so throughout the temperature range for the HP-LaMnO₃/SrIrO₃ sample are higher than that of the LP- LaMnO₃/SrIrO₃ sample, shown in Fig. 12(e), indicating the influence of interfacial magnetic moment arising from the LaMnO₃ layer to the spin scattering mechanism in HP-LaMnO₃/ SrIrO₃ sample. The magnitude of B_so is significantly higher compared to SrIrO₃ directly grown on LSAT substrates, as shown in Fig. 12(e).¹⁰³ These results suggest that the Mn spins at the interface and their magnetic moment play a vital role in tuning the scattering mechanism at the interface. To gain further insights into the influence of interface-induced Rashba SOC on spin relaxation mechanisms in these bilayers, two commonly observed mechanisms, namely the D'yakonov–Perel and Elliot–Yafet dominant spin relaxation mechanisms are discussed. In the HP-LaMnO₃/SrIrO₃ samples, an Elliot–Yafet-type spin relaxation mechanism, attributed to interfacial defects and screening from Mn magnetic moments, is observed. The Elliott–Yafet mechanism specifically focuses on the interactions between electron spins and spin–orbit-coupled lattice ions in spin–orbit scattering processes, emphasizing the mechanism of spin relaxation.¹⁰² While impurities, such as magnetic impurities or defects, can also contribute to spin-flip scattering, the Elliott–Yafet mechanism itself does not primarily address interactions with impurity spins. For a comprehensive understanding of spin relaxation in the presence of impurities, more generalized frameworks like Spin-Relaxation Time Approximation (Spin-RTA) may be necessary to account for a broader range of scattering mechanisms, including those associated with impurity spins.^63,64 In contrast, the LP- LaMnO₃/SrIrO₃ sample exhibits a D'yakonov–Perel-type spin relaxation mechanism, as the $τ so$ is inversely proportional to the $τ p$⁠, shown in Fig. 12(g). The D'yakonov–Perel spin relaxation mechanism is considered an intrinsic spin relaxation mechanism rather than an extrinsic one. Intrinsic spin relaxation via intrinsic scattering mechanisms arises from the material's inherent properties, such as crystal symmetry and SOC, rather than from external factors like impurities or defects.^55,116 The presence of structural inversion asymmetry causes spatial variations in the effective electric field and, consequently, the effective magnetic field across the material. These variations can occur due to the specific arrangement of atoms or the confinement potential of the structure. Extrinsic mechanisms like the Elliott–Yafet mechanism, on the other hand, involve interactions with impurities, defects, and other external factors that are not inherent to the crystal lattice's symmetry and properties. Thus, this mechanism is extrinsic.

Here, the study on CaMnO₃/CaIrO₃ superlattices, which was discussed in Sec. IV B 4, explores the precise control and manipulation of WAL through charge transfer-induced Rashba SOC at interfaces.¹¹⁷ In Figs. 13(a) and 13(b), the presence of WAL manifests as a distinctive cusp-like feature and positive magnetoresistance at low magnetic fields, providing clear evidence for the existence of Rashba SOC in the CaMnO₃/CaIrO₃ heterostructure. Examining the magnetoresistance data in Fig. 13(b), it is observed that (MI84)₅ exhibits superior magnetoresistance loop properties compared to (MI58)₄ and (MI22)₈. When analyzing the magnetoresistance and normalized magneto-conductance Δ(H)/G(0) [ΔG = G(H) – G(0)] data for CaMnO₃/CaIrO₃ heterostructures at 10 K, as shown in Figs. 13(a) and 13(b), respectively. We notice an increased cusp and critical magnetic field (Bmin) with higher charge transfer from (MI22)₈ to (MI58)₄ and finally to (MI84)₅, where the parameter “B_min” represents the magnetic field associated with minima of the magnetoconductance curve. This progression suggests a corresponding increase in the strength of Rashba SOC for these three superlattices. Moreover, the quantum-corrected magnetoconductance of the 2D system, considering SOC and WAL at low perpendicular magnetic fields, complies with the Hikami–Larkin–Nagaoka (HLN) equation, as denoted by Eq. (2).^111,114,115 By fitting the experimental data to the magnetoconductance curve, Rashba SOC values (B_so) exhibit an incremental rise in the order of (MI22)₁₀ < (MI58)₄ < (MI84)₅, as illustrated in Fig. 13(c). The Rashba coefficient (α) is directly proportional to B_so, following the relation α = (eħ³B_so)^1/2/m*. This demonstrates that the strength of Rashba SOC (α) increases with the rise in B_so values, as shown in Fig. 13(d).

In conclusion, our exploration of the physical phenomena arising from SOC in systems lacking inversion symmetry for the 2D interface has shed light on intriguing behaviors, particularly with regard to the Rashba SOC. The emergence of WAL, where electron motion is topologically protected, underscores the remarkable nature of this phenomenon. As we delve into the interplay between SOC and U in SrIrO₃ and strive to validate the predicted Dirac topological state, further experimental investigations are imperative. The manipulation of the interfacial environment through external gate voltage and electric field modifications holds promise in confining electrons at the interface. The intricate relationship between spin dynamics and transport, highlighted by the dependence of conductance on the magnetic field, offers a unique insight into the underlying mechanisms. Within the context of the LaAlO₃/SrTiO₃ heterointerface, the intensified interfacial electric field resulting from the accumulation of mobile electrons and potentially trapped holes on respective sides has unveiled a significant SOC effect. Also, the growing interest in studying the 3d-5d oxide interface combinations, driven by recent AFM spintronics research, particularly in the realm of SOC manipulation, holds great promise for potential applications in spintronics memory devices. The convergence of various phenomena and the pursuit of deeper understanding at these interfaces mark a captivating frontier in condensed matter physics, with the potential to shape future technological advancements.

Correlated electron materials display a complex interplay involving spin, charge, orbital, and lattice properties, which imparts them with distinctive functionality and tunability. In the realm of FM materials, the phenomenon of AMR has been acknowledged for an extended duration, with the earliest report by Lord Kelvin nearly two centuries ago.¹⁵ Notably, experiments conducted on Ni-based alloys have elucidated that the discrepancy in resistivity between currents perpendicular and parallel to the magnetization direction arises due to the spin–orbit-induced transfer of resistivity from spin-down to spin-up electron currents.¹⁵ AMR can be categorized into two distinct types: crystalline anisotropy and non-crystalline anisotropy.^27,118,119 Crystalline anisotropy: Crystal anisotropy is associated with materials possessing a well-defined crystal structure, where the spin or band couples with the crystal's symmetry direction, depending on the arrangement of atoms in the lattice.²⁷ In a crystal, the arrangement of atoms is characterized by its symmetry, which includes rotations, reflections, and translations. Anisotropic magnetic and transport properties are obtained through various mechanisms, such as SOC, Jahn Teller distortion, and others, where the magnetic moments and electronic transport align along specific symmetric directions in the crystal. For instance, in a material with tetragonal symmetry, the magnetic properties may vary along different crystallographic axes. Additionally, in certain materials, the magnetic moments of electrons can spontaneously align in a particular pattern known as magnetic ordering, leading to macroscopic magnetization. This alignment breaks the rotational symmetry of the magnetization. Non-crystalline magnetic anisotropy: AMR observed in polycrystalline materials is referred to as non-crystalline anisotropy.^118,119

In this section, a few AMR mechanisms relating magnetism with transport properties are discussed. The discussion aims to provide a better understanding of the AMR mechanisms in thin films and heterostructures, focusing specifically on LaAlO₃/SrTiO₃ and 3d/5d perovskite interfaces based on AFM materials.

A simplified phenomenological model aids in comprehending AMR phenomenon in manganites. In manganites, Jahn–Teller distortions and double-exchange interactions, which are closely interconnected, play crucial roles in both the transport and magnetic properties.^7,9 The metal–insulator transition in doped manganites, like La_0.699Ca_0.31MnO₃, is driven by two main factors. Above the transition temperature of 220 K, Jahn–Teller distortions favor a PM insulating state with an orthorhombic structure, while below this temperature, double-exchange interactions promote a metallic FM phase with a cubic structure. In La_0.69Ca_0.31MnO₃, the magnetic transition temperature (Tc) aligns with the metal–insulator transition temperature. In a cubic lattice, the response to an external magnetic field is uniform in all directions, leading to no AMR due to magneto-elastic coupling. However, beyond the metal–insulator transition temperature, Jahn–Teller distortion introduces an anisotropic magneto-elastic response when subjected to a magnetic field, though the AMR diminishes due to the material's insulating nature. The most significant AMR and colossal CMR response occurs near the metal-to-insulator transition temperature, driven by both Jahn–Teller distortion and the presence of a conducting magnetic phase in the material, as shown in Fig. 14.⁹ 

In order for an electron to participate in conduction, it needs to occupy an available state near the Fermi surface that lies within the energy range where conduction occurs. In the scattering phenomena, if all states near the Fermi energy are already filled, the scattered electrons will not contribute to conduction because they cannot transition to unoccupied states and gain the necessary energy to participate in the conduction process. The anisotropy based on SOC scattering phenomena lies in the simultaneous presence of s- and d-bands in proximity to the Fermi energy, combined with a substantial SOC. The manipulation of magnetization by an external magnetic field induces alterations in the population of unoccupied d-states relative to the current direction, consequently causing modifications in the scattering rate between s- and d-bands.¹⁵ This model offers a comprehensive explanation of the AMR behavior observed in atomic spin–orbit-coupled materials. In these materials, the response of the perovskite structure to a rotating magnetic field induces octahedral distortion, resulting in modifications to the scattering rate between s- and d-bands. These modifications, in turn, lead to the emergence of transport anisotropy.⁴⁴ 

Figure 15(b) shows the probable transition states after the backscattering of electrons in the presence of magnetic impurities. Considering the current direction along (100), the possible transition states will be majority to majority and minority to minority bands when magnetization is along the current direction, here along left to right (100). Given a current flowing along the x axis direction, we can deduce the backscattering amplitudes of states that possess a group velocity with a k-vector aligned parallel to the current. The scattering matrix elements could be written as Eq. (5). Also, in the case of a perpendicular magnetization direction, the possible transition states are majority to minority and minority to majority bands, represented by Eq. (5). For the case of K₋  $≫$ K₊, the analysis suggests that backscattering is significantly suppressed when the magnetization is perpendicular to the current, resulting in lower resistivity compared to the case with parallel magnetization. Therefore, the Rashba model predicts a positive AMR according to Eq. (3).

Spin valves are a specific type of magnetoresistive device that exploits the spin-dependent transport of electrons to achieve a significant change in electrical resistance in response to an external magnetic field. They are essential components in various magnetic sensors and read heads used in modern data storage devices, such as hard disk drives. The spin valve structure typically consists of two FM layers separated by a non-magnetic spacer (typically a thin metallic layer) in a sandwich-like configuration.^20,50,122 One of the FM layers has its magnetic moment fixed, while the other has a free magnetic moment that can rotate in response to an external magnetic field. When a current flows through the spin valve, electrons with different spin orientations experience different scattering rates as they pass through the magnetic layers. The majority of spin electrons (with spins parallel to the fixed magnetic layer) experience less scattering, while the minority spin electrons (with spins antiparallel to the fixed magnetic layer) experience more scattering. When there is no external magnetic field or when the external magnetic field is aligned in the same direction as the fixed magnetic layer, the majority spins have an easier path to traverse through both magnetic layers, leading to a low resistance state. Conversely, when the external magnetic field is aligned in the opposite direction to the fixed magnetic layer, the majority spins experience more scattering, and the resistance increases, resulting in a high resistance state. Spin valves are used as magnetic field sensors because the relative alignment of the magnetic moments in the two layers changes the resistance of the device. By applying an external magnetic field, the free magnetic layer aligns itself with the field, altering the resistance and allowing the sensor to detect changes in the external magnetic field.

In various distinct studies exploring the NiFe/IrMn/MgO/Pt stack, which incorporates an antiferromagnet on one side and a non-magnetic metal on the opposite side of the tunnel barrier, researchers have made fascinating observations.²⁰ They have witnessed a spin valve-like signal surpassing an impressive 100% AMR signal. Remarkably, this remarkable phenomenon can be attributed to the exchange spring effect, whereby the FM NiFe layer undergoes rotation under the influence of an externally applied magnetic field (as low as 50 mT). As a consequence of this rotation, the AFM IrMn layer responds, leading to the observation of AMR. The significance of these findings lies in the revelation of spin orientation-dependent AMR, which goes beyond the conventional reliance solely on magnetization. This discovery opens up exciting possibilities, suggesting that AFM materials can play a vital role in the development of spintronics devices.¹⁷ Building upon this groundbreaking work, Fina et al. have taken the investigation further, extending the understanding of this mechanism to perovskite Sr₂IrO₄/La_2/3Sr_1/3MnO₃ heterostructures.²⁵ The electrical behavior of this system in the in-plane (CIP) configuration primarily depends on the conduction occurring throughout the La_2/3Sr_1/3MnO₃ (LSMO) metal layer, holding true across a wide temperature range. Conversely, the out-of-plane current (CPP) measurement reveals semiconducting transport characteristics within the Sr₂IrO₄ film, consistent with previous findings in Sr₂IrO₄ (SIO) epilayers⁴⁰ [depicted in Fig. 17(a)]. Figure 17(b) illustrates that the AMR response in the CPP setup is exclusively observed in the SIO/LSMO heterostructure, contrasting with its absence in the Sr₂IrO₄/LaNiO₃/La_2/3Sr_1/3MnO₃ (SIO/LNO/LSMO) superlattice. This underlines the crucial role of the interfacial coupling between the Sr₂IrO₄ and La_2/3Sr_1/3MnO₃ layers in influencing the electronic behavior. This work promises to advance the frontier of AFM spintronics research.

Over the past few decades, several research studies have been conducted on LaAlO₃/SrTiO₃ in order to comprehend the underlying origins of its magnetic and transport properties. Additionally, the AMR mechanism has been previously explored in dilute semiconductors.⁶³ In this section, we will investigate transport anisotropy in heterostructures composed of two-band insulators, drawing insights from the AMR studies conducted on dilute semiconductors.

Symmetry analysis of in-plane AMR reveals deviations from twofold oscillation for positive gate voltages, and residual data indicate a clear sixfold structure with equally spaced peaks separated by 60°. AMR coefficients, i.e., C₂, C₄, and C₆ as a function of gate voltage and temperature are presented in Fig. 20(c). C₂ associated with noncrystalline AMR is governed by the Rashba Model in AMR, while C₄ and C₆ terms are associated with crystalline symmetry. Previously, it was discussed that the origin of the C₄ term is governed by the Dresselhaus Model in AMR.⁶³ The magnitude of the sixfold symmetry in AMR (C₆) is greater than the magnitude of the fourfold symmetry in AMR (C₄), which aligns with the expectations for a hexagonal crystal structure. C₆ becomes significant at high gate voltages, shown in Fig. 20(c). The onset of the observed sixfold crystalline AMR coincides with the saturation in 1/|eR_H (Vg = 20 V), suggesting a common mechanism. Also, LaAlO₃/SrTiO₃ (111) does not possess any Hall effect. These results indicate that the coupling of the higher electronic band with hexagonal crystal sites [Fig. 20(d)] contributes to the magnetic effect, introducing a potentially stronger sixfold AMR response.

Thus, the most straightforward medium for a purely noncrystalline AMR can be found in the tangential spin-1/2 texture of the Rashba model in the LaVO₃/KTaO₃ heterostructure.

Demonstrating the connection between a radial spin-1/2 texture and AMR, the Dresselhaus spin–orbit interaction serves as a minimal model that also showcases the emergence of crystalline AMR in the LaAlO₃/SrTiO₃ heterostructure. In the realm of LaAlO₃/SrTiO₃ and LaVO₃/SrTiO₃ heterostructures, the remarkable influence of two-dimensional Rashba and Dresselhaus spin band splitting leaves a profound impact on resistance facilitated by spin backscattering. Notably, within the confined 2DEG scenario, a distinct fourfold oscillation emerges, underscoring the predominant magnetic ordering near the interface in LaAlO₃/SrTiO₃. On the other hand, in the context of LaVO₃/KTaO₃, the presence of strong Rashba SOC is suggested by the twofold oscillation observed in AMR. These fascinating phenomena shed light on the intricate interplay between spin textures and magnetic scattering within these systems.

In the two-dimensional Ruddlesden–Popper (RP) structure type compound Sr_n+1Ir_nO_3n+1 (specifically Sr₂IrO₄ with n = 1), a SOC-driven J_eff = 1/2 band structure model was proposed.⁸ This model forms the basis for discussing many of the novel physical phenomena observed in perovskite iridates. It results from the octahedral crystal-field splitting induced by strong SOC, leading to the division of the t_2g band into a J_eff = 1/2 doublet and a J_eff = 3/2 quartet. The moderate value of U further separates the J_eff = 1/2 doublet band into an upper Hubbard band (UHB) and a lower Hubbard band (LHB), resulting in a half-filled J_eff = 1/2 band model.⁸ Another notable phenomenon is the significantly large magnetoresistance observed in Sr₂IrO₄ with robust AFM order, highlighting the remarkable functionalities of the J_eff = 1/2 state in the context of AFM spintronics.^127–129 These phenomena emphasize the unique promise and functional potential of iridates with the J_eff = 1/2 state known as pseudospin. A comparison was made between the magnetic ground states of Sr₂IrO₄, SrIrO₃/SrTiO₃, and CaMnO₃/CaIrO₃ superlattices, as shown in Figs. 21(a)–21(c). All of them exhibit a canted AFM in-plane spin structure, which has the potential to be utilized in AFM spintronics. The pseudospins in Sr₂IrO₄ exhibit a preference for AFM ordering within the ab plane, with a noticeable contribution from their orbital properties as discussed previously. Additionally, they collectively deviate from the a-axis, defining a pseudospin canting angle of approximately 13°. This canting angle, denoted as φ, closely follows the lattice distortion parameter α, as proposed by Jackeli et al.^128,130,131 The pseudospin canting results in a net magnetic moment within each IrO₂ layer, while the overall structure's net moment is canceled out due to the compensated AFM order along the C axis. By applying a small magnetic field within the ab plane, it is possible to induce a spin-flip transition, leading to a weak FM phase in Sr₂IrO₄. This offers a convenient means to manipulate the planar AFM orders using external magnetic fields in this material.

Matsuno et al. proposed that the ground state of the SrIrO₃/SrTiO₃ superlattice is FM rather than AFM.¹³² In this context, the interlayer coupling between the Ir local moments is found to be FM, rather than exhibiting canted moments, as shown in Fig. 21(b). To achieve this interlayer FM coupling while also considering canted in-plane moments, it is necessary for the IrO₆ octahedra in adjacent IrO₂ layers to rotate in the same direction. This rotation pattern results in a distinct magnetic and lattice ordering arrangement, as illustrated in Fig. 21(b). This canted structure concept can be extended to the CaMnO₃/CaIrO₃ superlattice.⁴⁴ Given that both CaMnO₃ and CaIrO₃ have the same octahedral distortion pattern (pbnm symmetry), it is expected that the spin canting follows a consistent direction between consecutive layers of CaMnO₃ and CaIrO₃, thereby promoting FM coupling between these layers. Also, a net FM moment arises from spin canting in the ab plane, represented by the red and green arrows in Fig. 21(c). The significant contrast in ground states between Sr₂IrO₄ and two other superlattices, SrIrO₃/SrTiO₃ and CaMnO₃/CaIrO₃, provides valuable insights into the role of canted AFM moments in spintronics applications.

Furthermore, the current direction aligns along the (100) axis of the sample. The magnetic and electrical characteristics of (MIxy)z (x = y = 2–4), i.e., (MI22)₁₀, (MI33)₅, and (MI44)₅ superlattices are presented in this work, indicating an enhancement of “U” due to the influence of dimensionality and charge transfers.⁴⁴ In the canted AFM phase of these superlattices, the mechanism of domain scattering, which is responsible for the AMR, is controlled by interlayer coupling. The term “domain scattering” refers to the phenomenon where electrons are scattered when they traverse regions with different magnetic orientations within a material. In the case of these superlattices, the canted AFM phase creates the domains, and it is believed that the scattering of electrons within these domains gives rise to the observed AMR. The strength of the interlayer coupling depends on the magnitude of the interface coupling and the dimensions of the individual layers. Also, the AMR in this system is related to s-d scattering mechanism mentioned in Sec. IV B 2.

This equation represents two possible states of spin canting, i.e., for α = 0 and α = 45°. Sinusoidal ϕ -AMR emerges as a result of lattice deformation under an external magnetic field known as S–L coupling. The regime (25–100 K) of the sinusoidal ϕ-AMR in (MI22)₁₀ superlattice is shown in Fig. 22(b). S–L coupling involves pseudospin–octahedral coupling and the response of pseudospins to temperature-dependent lattice vibrations has been shown previously.¹⁴⁰ Notably, the structural transition due to octahedral deformations with varying temperature is yet to be reported in layered systems like SrIrO₃/SrTiO₃ or CaIrO₃/SrTiO₃ superlattices. The strength of S–L coupling is divided into two regimes based on the magnetic moment of the superlattices: (i) Lower magnetic moment-based superlattices, while S–L coupling governs ϕ-AMR near the transition temperatures. (ii) Higher magnetic moment-based SLs, such as (MI22)₁₀, show a step-like AMR pattern at 10 K [Fig. 22(c)] without sinusoidal variation, indicating the suppression of S–L coupling and the dominance of field–pseudospin coupling. Conversely, in this work, it is shown that (MIx2)₅ (x = 4, 5, 8) SLs lack the kink pattern, suggesting the absence of field–pseudospin coupling, with orthorhombic distortion through S–L coupling primarily restoring sinusoidal resistance. The sinusoidal ϕ-AMR pattern with a kink in the (MI22)₁₀ SL from 15 to 22 K in Fig. 22(c) indicates the coexistence of S–L coupling and field–pseudospin coupling. Both (MI22)₁₀ and (MI33)₅ superlattices show increasing ϕ-AMR amplitudes with decreasing temperature. The competition between these couplings, varying with temperature, determines the ϕ-AMR behavior in (MI22)₁₀ superlattice. At high temperatures, S–L coupling reorients moments, while at lower temperatures, the direct coupling of field pseudospin dominates due to a rigid lattice and larger moments. In Sec. IV D 4, we will show how the field–pseudospin coupling induces spin-flop transition.⁴⁴ 

Also, a spin-flop transition in a nearly Mott region introduced an additional two orders of magnitude in AMR amplitude in (MI22)₁₀ superlattice, achieving an exceptional 70% AMR at 10 K at 9 T, shown in Fig. 22(c), surpassing the previous record.^25,44 Obtaining a spin-flop transition in thin films is challenging due to reduced thickness, surface, interface effects, and defects.^141–143 The ϕ-AMR in (MI22)₁₀ superlattice, a biaxial anisotropic system, is characterized by trough and crest peaks that correspond to different scattering by soft (100) and hard (110) crystal axes for 14 and 25 K, as shown in Fig. 22(c). The sinusoidal symmetry of the ϕ-AMR deviates due to the emergence of a fourfold pattern of kinks at close temperature intervals from (100) to (110) crystal axes at 22, 18, and 15 K at 9 T field, as shown in Fig. 22(c). At 9 T, a 70% step-like AMR, while at 5 T, a modest 10% increase in AMR is observed. The kink-like and step-like AMR patterns are the symbol of spin-flop-induced transition. The mechanism of spin-flop transition is depicted in Fig. 22(e). Thus, the diverse characteristics of ϕ -AMR arise from biaxial magneto-crystalline anisotropy and the spin-flop transition. Also, the competition between pseudospin–lattice coupling and field–pseudospin coupling explains the anomalous AMR, with different symmetries observed in SrIrO₃/SrTiO₃ and CaIrO₃/SrTiO₃ superlattices.⁹⁹ 

In Sec. IV D 3, we discussed atomic SOC-driven fourfold oscillation in AMR within the CaMnO₃/CaIrO₃ superlattice.⁴⁴ Additionally, we delved into the topic of magnetic impurity scattering based on RTA, analyzing its connection to the Rashba spin–orbit-coupled Fermi surface and the resulting twofold oscillation in AMR, as observed in the KTaO₃/LaVO₃-based heterostructure.^57,63,64,123 In this section, we will expand our discussion to the 3d/5d AFM interface, specifically within the LaMnO₃/SrIrO₃ (LaMnO₃: AFM below 3 unit cell/SrIrO₃: Canted AFM) heterostructure,¹⁰⁰ where we will explore the concept of Rashba SOC due to inversion asymmetry and its impact on AMR, elucidating the underlying twofold oscillation pattern using insights from the Kondo lattice model and spin scattering. In this study, superlattices denoted as SLnmz, where “n” represents the periodicity of LaMnO₃, “m” signifies the periodicity of SrIrO₃, and “z” indicates that the repetition of the heterostructure is investigated. The influence of interfacial coupling on magnetism mentioned earlier in this article is discussed. Notably, this research unveils a remarkable discovery regarding the AMR symmetry, as depicted in Fig. 23. By adjusting the repeating period from (SL442) to (SL118), a transition from fourfold to twofold AMR symmetry is observed. Also, it is established that atomic SOC generates the fourfold oscillation in AMR, while the enhancement of the Rashba effect, due to the asymmetric interfacial structure, was implicated in the twofold AMR transition shown in this work.¹⁰⁰ 

In this expression, we have operators for creating $c †iσ$ and annihilating $( c i σ)$ electrons, which describe electron behavior at a specific site (r_i) with a particular spin (σ). The first term accounts for the kinetic energy associated with electrons hopping (t) between nearest-neighbor sites, while the second term introduces the Rashba SOC effect. The chemical potential (μ) regulates the electron density. Moving on, the Kondo interaction term elucidates the interaction between the electron spin (⁠ $s i= ∑ α β c i σ † σ → α β c i β$⁠) at site r_i and the localized magnetic moments (S_i) associated with Mn ions. Last, an external magnetic field within the plane is incorporated, represented as $B=B n ^(Bcosθ,Bsinθ,0)$⁠.

The sum encompasses all energy levels, with “N” representing the total site count. “ $( f m , n)$” corresponds to the Fermi–Dirac distribution for single-particle eigenstates |m⟩ and |n⟩, characterized by their energy levels $ε m , n$⁠. “η” denotes the scattering rate governing the interaction between conduction electrons and local moments. The transverse conductivity $σ x y= ρ x y ρ x y 2 + ρ x x 2≈0$⁠, since $ρ x y≪ ρ x x$ the longitudinal resistivity is given by $ρ x x= σ x x σ x y 2 + σ x x 2≈1/ σ x x$⁠. Upon increasing the strength of the SOC term λ_R from 0.4t to 0.5t, the primary symmetry of the AMR shifts from a fourfold pattern to a twofold one, as shown in Fig. 24(c). This alignment with experimental findings lends support to the qualitative agreement between theory and observation. In summary, the interfacial coupling between the spin–orbit-coupled Ir⁴⁺ and Mn³⁺ was found to significantly influence the magnetism and magnetotransport properties of the heterostructure of AFM materials.

Figure 26 presents a comparison of the AMR at different temperatures under a magnetic field of 7 T across the three studied superlattices: (MI84)₅, (MI58)₄, and (MI22)₈. An increase in the strength of Rashba SOC from (MI22)₈ to (MI84)₅ results in a shift in the phase transition temperature of AMR. Specifically, this transition occurs at higher temperatures for (MI84)₅ and (MI58)₄. In contrast, the superlattice (MI22)₈, possessing the weakest Rashba SOC, completely loses this transition under both temperature and magnetic field conditions.¹⁴⁶ These findings emphasize the crucial role played by Rashba SOC in tailoring the magnetic anisotropy within oxide heterostructures. Also, transport anisotropy induced by the Berry phase mechanism is established in this section. However, it is worth noting that the observed twofold oscillation in AMR has been reported in previous studies, attributed to factors like exchange bias and ferromagnetism,^44,153 which are more subtle in (MI84)₅ [Fig. 7(d)]. Importantly, the absence of exchange bias in (MI58)₄ and (MI22)₈ superlattices is a notable distinction.¹¹⁷ It is important to highlight that all AMR measurements were carried out using a zero-field cooling protocol, erasing any magnetic history to mitigate the effects of exchange bias. Moreover, previous research supports the idea that Rashba SOC is indeed the origin of the observed twofold oscillation governed by extrinsic scattering mechanism.^60,123 Thus, we can conclude that the twofold oscillation depends on Rashba SOC and is influenced by the charge transfer mechanism. In the CaMnO₃/CaIrO₃ superlattice, the coexistence of a twofold AMR oscillation and anisotropic AHE points to the presence of both intrinsic and extrinsic scattering mechanisms. Interestingly, the observed quadratic dependence of AHE with longitudinal resistivity implies that the intrinsic scattering mechanism plays a more dominant role compared to the extrinsic scattering mechanism within this material system.

In summary, our review explores the profound impact of charge transfer and orbital reconstruction at the heterostructure and interfaces, leading to the emergence of anisotropic effects, notably magnetic exchange anisotropy. Also, charge-transfer phenomena offer valuable insights into the realm of quantum phenomena in spintronics and various device applications. A key focus of our discussion centers on the interplay of interfacial asymmetry, particularly within 3d/5d AFM interfaces, leading to the generation of Rashba SOC. This Rashba SOC governs the locking of spin momentum, ensuring the preservation of spin phase coherence. It further engenders quantum interference, thereby influencing conductivity and effecting corrections in magnetoresistance. Also, our review provides a comprehensive exploration of AMR mechanisms, driven by various scattering mechanisms. We extensively discuss impurity-based extrinsic scattering models in the context of Fermi surfaces, particularly within the Rashba and Dresselhauss models. We apply these concepts to interfaces of two-band insulators and 3d/5d heterostructures, exemplified by the LaMnO₃/SrIrO₃ and CaMnO₃/CaIrO₃ heterostructures.

Furthermore, we investigate intrinsic scattering mechanisms grounded in Berry phase considerations, with special emphasis on the CaMnO₃/CaIrO₃ superlattice and its Hall resistance measurements. The pivotal role of atomic SOC, especially in 3d/5d heterostructures owing to the presence of heavy atomic nuclei in 5d atoms, is underscored as we aim to provide a comprehensive understanding of AMR. We explore the intriguing response of perovskite octahedra to a rotating magnetic field, a phenomenon contributing to the generation of a fourfold oscillation in AMR, as exemplified in the CaMnO₃/CaIrO₃ heterostructure. Additionally, spin-flop transitions are highlighted for their role in achieving an impressive 70% AMR in this specific superlattice. In conclusion, our review endeavors to offer a comprehension of AFM spintronics, with a specific focus on anisotropy within AFM interfaces. We rely on strong spin transport models, supported by extensive experimental studies, to help us understand the complex realm of spin-related phenomena in heterostructures. Ongoing investigations into AFM materials and interfaces are driven by the concurrent progress of interconnected factors, including the evolution of THE and SHE effects, with direct implications for memory devices, as elaborated below.

THE, stemming from the intriguing chiral interaction of lattice spins, continues to captivate researchers in the realm of spintronics as we look ahead. This mysterious phenomenon, driven by the Dzyaloshinskii–Moriya interaction (DMI), arises from the asymmetry caused by the loss of inversion symmetry in both non-centrosymmetric bulk crystal structures and interfaces. Interfaces play a crucial role in controlling chiral spin configurations in real space. The chiral spin configuration helps us to generate Lorentz force acting on conducting electrons, introducing a Berry phase in their trajectories while a finite Berry phase collection in momentum space was associated with the Berry curvature. Experimentally, the quest lies in detecting these topological spin textures, particularly skyrmions. A breakthrough has come in the form of the THE, which manifests as distinctive features in the transverse Hall resistivity (ρ_xy) due to this hypothetical Lorentz force. This effect has become an invaluable tool, offering a means to identify the presence of skyrmionic contributions in various materials, spanning single crystals, thin films, and heterojunctions of AFM materials.^154–158 Our objective is to pioneer the development of an AFM heterostructure that serves as a promising platform for showcasing the THE as a future research frontier.

The trajectory of anisotropy and AFM spintronics devices presents a promising avenue in the realm of future electronic systems. While traditional electronics have focused on charge currents for their functionalities, the continuous miniaturization of devices confronts us with challenges such as escalating heat generation and power dissipation. In this context, SHE emerges as a transformative alternative, harnessing both charge and electron spin, introducing a new era of efficiency and versatility. SHE is a fundamental quantum phenomenon in condensed matter physics where an electric current can induce a net spin polarization perpendicular to the direction of the current flow. The concept of charge-to-spin conversion stands as a pivotal enabler, offering the seamless transformation between charge and spin currents, with aligned electron spins giving rise to coherent spin currents. Notably, the introduction of spin-transfer torques (STT) represents a significant milestone in this journey.^{47,49,51,159–161} STT effectively controls magnetic bits in magnetic random access memories (MRAMs).^51,161 These bits, which utilize nanomagnetic orientations for data storage, have been revolutionized by STT, enabling rapid and energy-efficient data manipulation without the need for external magnetic fields. Currently, this is being observed experimentally in AFM heterostructures and devices.

D.S.R. acknowledges the Department of Science and Technology (DST) Nanomission for financial support under Research Project No. SM/NM/NS-84/2016 and the Science and Engineering Research Board (SERB) Technology, New Delhi under Project No. CRG/2020/002338. J.S. acknowledges the DST-INSPIRE for Fellowship (No. DST/INSPIRE/03/2018/000699).

The authors have no conflicts to disclose.

Suman Sardar: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Megha Vagadia: Conceptualization (equal); Data curation (equal); Writing – original draft (supporting); Writing – review & editing (equal). Tejas M. Tank: Conceptualization (equal); Data curation (equal); Writing – original draft (supporting); Writing – review & editing (equal). Jayaprakash Sahoo: Conceptualization (equal); Data curation (equal); Writing – original draft (supporting); Writing – review & editing (equal). D. S. Rana: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from cited articles. Restrictions apply to the availability of these data, which were used under license for this study.