Domain walls speed up in insulating ferrimagnetic garnets

Magnetic domain walls (DWs) are the finite boundaries that separate the regions of uniform magnetization in a magnetic material. They constitute a key research topic in condensed matter physics due to their intriguing physics and relevance in technological applications. A multitude of spintronic concepts for memory, logic


I. INTRODUCTION
Over the past several years, there have been tremendous scientific advancements in novel mechanisms and device designs for next-generation memory and data storage technology.Despite these advancements, digital data are predominately stored via hard disk drive and magnetic tape technologies, 1 concepts that originate from the 1970s or earlier.Traditional magnetic data recording is costeffective, can be intrinsically nonvolatile with long retention, and can produce the world's highest density memories. 1,2However, these technologies are inherently slow and power-hungry due to their mechanical nature and are not adaptive to beyond complementary metal-oxide-semiconductor (CMOS) fabrication demands or applications other than long-term storage.As a result, much effort has been devoted to creating solid-state forms of magnetic memory and logic devices.One promising pathway is using magnetic domains, which can be nucleated, driven, and annihilated using both magnetic fields and electrical currents.1][12][13] However, several challenges have stood in the way of implementation of such devices, such as high energy consumption and poor device performance.The most important challenges consist of, but not limited to, slow DW propagation velocities, low reliability of DW motion, and the lack of precise microscopic understanding of DW dynamics under external stimuli, such as spin-torque.To address these, innovation in the current-driven propagation of DWs has been at the forefront of spintronics.
The formulation of spin-transfer torques (STTs) in the late 1990s 14,15 formally initiated research into current-driven DW motion in conducting ferromagnets (FMs), 16,17 even though the first evidence of current-induced DW displacement had been reported as far back in 1978. 18In the STT scheme, spin-polarized conduction electrons, upon passing through a DW, conform to the local spin environment and transfer spin angular momentum to local magnetic moments, resulting in magnetic torque that displaces the DW along the electron flow [Fig.1(a)].The utilization of this concept has led to observations of DW velocities over 100 m/s in materials with in-plane magnetization. 3,19However, STTs are fundamentally limited to the transfer of angular momentum equivalent to 1 ̵ h per e − .As a result, detrimental electromigration effects of high current densities and highly dissipative precessional dynamics beyond the Walker threshold set the upper limit for the DW velocity using this method.
Recently, spin-orbit torques (SOTs) have emerged as a more flexible and powerful alternative to STTs for current-induced magnetic manipulation. 20Better understanding of charge-spin interrelations in electrical transport revealed that in the bulk and interfaces of certain material systems characterized by a large spin-orbit coupling, relativistic spin-dependent effects can generate pure spin currents from charge currents [Fig.1(b)].5][26] These mechanisms (and others) have been shown to generate significant current-induced torques in suitable ferromagnetic systems leading to magnetization switching, 6,27,28 DW motion, [29][30][31][32] and magnon generation/suppression. 33 The spin-torque arising from these charge-to-spin conversion mechanisms is collectively called SOTs, underlining their common physical origins in spin-orbit interactions.Similar to STTs, SOTs consist of damping-like and TABLE I. Summary of the current-induced DW velocity reported in the past two decades with corresponding current densities (j) and driving mechanism.(^) The maximum reported velocity here has an in-plane field applied.( * ) In this study, an in-plane field was applied to rotate the DW internal spin structure to the Néel-type for maximizing the SOT efficiency.( † ) In the original article, SOT was not considered as the DW driving mechanism, but the structure can generate strong SOTs.

Structure
Magnetism v (m/s) j (×10  field-like components, of which the former is predominantly responsible for magnetization switching and DW motion. 34,357][38] Transition metal/FM bilayers constitute the basic structure for magnetic tunnel junctions, which are the fundamental building blocks of magnetic random access memories, 6,39,40 a faster and nonvolatile alternative to charge-based volatile random access memory technologies. 41Due to these attributes, conducting ferromagnetic structures have remained in the spotlight of SOT research for a long time.Moreover, in SOTs, spin and charge currents are decoupled and travel orthogonally; hence, they can also act on insulating ferro-/ferrimagnets and, more recently, antiferromagnets. 42,43This is fundamentally different from the STT, which requires the FM to be conducting, and hence, SOTs can be more versatile and comprehensive for spintronic research and device applications beyond metallic and semiconducting systems. The intersection of SOTs with DW studies has led to rapid progress.Initial work has shown that SOTs can drive DWs in archetypical ferromagnetic structures, such as Pt/Co, to velocities reaching 400 m/s. 22Later, it was understood that the efficiency of SOT-driven DW motion is owed to interfacial chiral Dzyaloshinskii-Moriya interaction (DMI) stabilizing Néel DWs in HM/FM systems, which maximizes the effective field of the damping-like SOT on the DW. 29,30,44Despite significant improvements in the current-induced DW velocities enabled by SOTs and the DMI, soon it became clear that DWs' precessional response to SOTs at higher current amplitudes sets a new maximum achievable DW velocity in conventional FMs (e.g., Co and CoFeB), fundamentally limited by the interplay of DMI and SOTs. 44In contrast, magnetically compensated systems have been found to elegantly circumvent this problem, 31 as we will explain later in detail.Therefore, the DW research has expanded to metallic ferrimagnets and synthetic antiferromagnets lately. 31,45,46In such systems, SOTs have propagated DWs to more than 1000 m/s when the net angular momentum of the material is reduced and precessional dynamics are suppressed. 31 , in particular ferrimagnetic garnets (FMGs), which can provide significant advantages over their metallic counterparts in the spintronics context. 47,48][51] FMGs' chemical structure takes the form A 3 Fe5O 12 , where the Asite typically consists of a nonmagnetic element, such as Bi or Y, or a magnetic rare-earth element, such as Tm, Gd, and Tb.The ferrimagnetic structure of garnets contains two magnetic sublattices (three in the case of magnetic A-site substitution), with two octahedrally coordinated Fe 3+ and three tetrahedrally coordinated Fe 3+ coupled antiferromagnetically per formula unit [Fig.2(a)].A rare-earth dodecahedral A-site, if magnetic, couples antiferromagnetically to the tetrahedral site.While most compositions of garnets retain their magnetic order parameter to high temperature (T ∼ 550 K), their magnetic properties can be significantly influenced by choice of A-site element.Not only can net magnetic moment (Ms) be tuned, but also the net angular momentum S(T), anisotropy, DMI, and α can be engineered dramatically, making garnets an ideal playground for optimizing DW properties leading to ultrafast motion.Although FMG films have been well-studied since 1960s, the dominant mechanism of thin film synthesis was liquid-phase epitaxy, which produced relatively thick (micrometer) films, where most magnetic properties of the film were similar to the bulk. 52Recently, atomically precise synthesis techniques, such as pulsed laser deposition (PLD), 53,54 molecular beam epitaxy, 55 and certain forms of magnetron sputtering, [56][57][58] have enabled the growth of ultrathin FMG films of a few unit cells in thickness, while retaining robust magnetization.Such ultrathin films enabled the engineering of material properties via interface design.Most notably was perpendicular magnetic anisotropy (PMA), 59 a requirement for efficient SOT-driven DW motion and long data retention in spintronic memory and logic devices.Unlike FMG films obtained by liquid-phase epitaxy, where PMA is achieved by reducing the magnetization (and shape anisotropy) combined with growth-induced anisotropy, 60 PMA in ultrathin FMGs is primarily achieved via exploitation of magnetostriction induced by the epitaxial growth on a lattice mismatched substrate [Figs.2(b)-2(d)].Such engineering of PMA and other magnetic properties in ultrathin FMGs enabled a new platform for studying DW dynamics in low-damping insulating systems.Recently, a range of PMA ultrathin films, such as SmIG, 61 TmIG, 59,62,63 TbIG, [64][65][66] and Bi-doped YIG, 67,68 and others have been reported. 64,65,69Both PLD and magnetron sputtering methods have been shown to conveniently produce FMG thin films. 56,59,64,66n this article, we will review the recent experimental progress in current-driven DW motion in magnetically compensated systems, with particular focus on FMGs.First, we will explain in detail the benefits of compensated systems for DW dynamics and the role of the DMI and magnetic damping in achieving high DW velocities.Then, we will showcase key results and their immediate broader impact on spintronics and fundamental physics.Finally, we will discuss open questions and promising future directions in this young, quickly accelerating research topic.

A. Theory
SOTs act differently on ferromagnetic and antiferromagnetic DWs.In the former, the DW precesses around the effective field produced by the damping-like SOTs when its value becomes comparable to the effective DMI field acting on the DW.At this point, the DW velocity reaches a plateau, and it is no longer possible to drive the DW faster by increasing the injected current.Extra energy from increasing the current density is dissipated into the lattice and does not aid in driving the DW's motion.This behavior is best understood by examining the one-dimensional model for the SOT-driven DW velocity in a ferromagnet, 44,70 where γ is the gyromagnetic ratio, D is the DMI energy, and j is the current density through the SOT generating material (usually a heavy metal, such as Pt or W).The critical current j 0 is defined as , where e is the electron charge, tFM is the FM thickness, θ eff SH is the effective spin Hall angle (here we only consider the spin Hall effect as the SOT source for simplicity), h is Planck's constant, and Δ is the domain wall width.From Eq. (1), at low current densities, the DW velocity scales linearly with the strength of the torque and, thus, the current density, v ∼ γΔ α H SHE . 44,70n the limit of high j, the velocity saturates, where D sets the DW speed limit, v ∼ γΔ π 2 H DMI . 44,70Here, H SHE and H DMI are the effective spin Hall and DMI fields, respectively, and are material-dependent properties.
In compensated systems, such as ferrimagnets, in the limit of large exchange coupling between the magnetic sublattices, the above-mentioned 1D model can be adapted to account for the reduced angular momentum S(T) of the opposing sublattice of the system, 31 where v FiM is the ferrimagnetic DW velocity and is independent of Ms. Rather, the denominator depends on the net spin density S(T).The net spin density refers to the net angular momentum, whose compensation can differ in temperature (or composition) from the net magnetic moment of a given system due to the differing gyromagnetic ratios of the elements constituting the magnetic sublattices.

B. Experiments
In the limit of vanishing net angular momentum (S → 0), the effective damping parameter goes to infinity and the system dissipates angular momentum instantaneously.This leads to extremely fast domain wall response to external stimuli such as magnetic fields.We note that although the effective damping diverges, the sitespecific Rayleigh dissipation remains finite. 71The first experimental signature of this phenomenon was observed in the field-driven dynamics of ferrimagnetic domain walls. 46Figure 3(a) shows the field driven DW velocity as a function of temperature in GdFeCo, a conducting ferrimagnet, for different applied field values.The maximum velocity reaches almost 2000 m/s for 100 mT applied field at the specific temperature of 310 K (blue vertical line), which is identified as the angular momentum compensation temperature, well different from the magnetic compensation temperature characterized at 220 K (red vertical line).Figure 3(b) shows micromagnetically computed DW speeds as a function of net spin density S, confirming maximum DW velocity at the angular momentum compensation temperature, in agreement with the experimental results in Fig. 3(a).
As a consequence, in the context of SOT-or STT-driven DW motion, S is the critical factor rather than Ms, and when S = 0, Eq. ( 2) takes the form v FiM = π 2 D S 0 j j 0 , where the DW velocity does not saturate and only scales linearly with current density, and the maximum velocity is no longer limited by the DMI.Even in the case of reduced (but non-zero) spin density, the saturation velocity still exists but increases significantly, 31,72 further illustrating the advantages of multi-sublattice systems for fast current-driven DW dynamics.Although more sophisticated multi-sublattice models are needed to fully account for all the dynamics present, 73 this generalized concept was experimentally verified in several rare earth-transition metal alloys, 31,72,74 as well as in synthetic antiferromagnets 45 following the seminal work of fast field-driven DW motion in Ref. 46 [Figs.3(a) and 3(b)].Figure 3(c) displays the temperature-dependent SOT-driven DW velocity measurements in ferrimagnetic GdCo alloys. 31,46Similar to the field-driven case, there is a sizable enhancement of DW velocity, reaching ∼1250 m/s upon crossing the angular momentum compensation temperature for a fixed current density.The non-saturating behavior of the velocity upon increasing the current is also observed and reported in Fig. 3(d), where the experimental data are overlaid with the ferrimagnetic model in Eq. ( 2) and both Joule heating and pinning are accounted for.The temperature dependence of the critical depinning current (j 0 ) in Fig. 3(d) explains the monotonic increase in vDW(T) at small j as shown in Fig. 3(c), where we would otherwise not expect an increase in velocity with temperature.Although the magnetic and angular momentum compensation temperatures are very similar, the DW velocity shows an onset of saturation behavior in the former above j = 1.5 × 1012 A/m 2 , while linearly increases in the latter within the experimentally tested limit.It is worth noting that the dynamics described in Eq. ( 2) does not exclude the possibility of STT as the DW driving force.Indeed, recent experiments in conducting ferrimagnetic Mn 4 N single layers and its Ni-doped variants, in the vicinity of angular momentum compensation, have reported an enhancement of STT-driven DW velocity up to 3000 m/s. 75,76The magnetic tunability of Mn 4 N offered by Ni doping and high Curie temperature (745 K) make this material attractive for both STT-and SOT-driven conducting DW devices.

III. INGREDIENTS FOR ULTRAFAST DOMAIN WALL MOTION A. Gilbert damping
A close examination of the DW mobility μ ∼ γΔ α reveals that only a few material properties can be tuned to enhance the currentdriven DW response in compensated systems.Of these, arguably, the most tunable is α, which scales inversely with DW mobility and can span several orders of magnitude depending on the choice of material.While metallic ferrimagnets have initially played a key role in understanding the physics governing the fast DW dynamics in compensated systems, they have inherent limitations, such as large α and pinning, undesired current shunting through the magnetic layer, reducing the SOT efficiency, and limited long-term structural and magnetic stability in some cases.Instead, FMGs, well-known for their low and tunable α, are an ideal material class for SOT-driven ultrafast DW dynamics.In addition, as we will discuss below, a handful of recent results have proven the vast potential of FMGs for ultrafast DW motion and related emerging physics.

B. Perpendicular magnetic anisotropy and Dzyaloshinskii-Moriya interactions
In PMA systems, conventional in-plane damping-like SOTs can only drive Néel-type domain walls due to the geometrical relationship between the damping-like SOT and the DW internal magnetization [Fig.1(b)].Néel DWs are stabilized by the interfacial DMI in systems with broken inversion symmetry. 29,30The DMI is substantial in metallic systems, such as Pt/Co 29 or W/CoFeB, 77 where it originates from the interfacial spin-orbit coupling occurring between 3d and 5d elements. 78,79Although garnet crystal structures are centrosymmetric and, therefore, do not host bulk DMI, the existence of the interfacial DMI and Néel-type DWs has recently been evidenced in TmIG and TbIG grown on gadolinium gallium garnet (GGG) through DW motion 32,80,81 and stray field imaging by nitrogen-vacancy center microscopy. 82Although still debated, later measurements using different metal capping on TmIG and TbIG layers indicated that the DMI in FMGs primarily originates at their interfaces with the GGG substrate but can also be modulated by both the capping layer material and epitaxial strain. 32,80,81,83his was the first direct demonstration of chiral DMI occurring at a magnetic/nonmagnetic oxide interface at room temperature.Later reports, through indirect evidence from topological Hall effect measurements, have claimed that the capping metal/FMG interface is the dominant source of DMI in this system. 84,85Nonetheless, the reported DMI effective fields were strong enough to turn the equilibrium DW configuration from the Bloch-type to the partially or fully Néel-type, enabling their SOT-driven motion.Further studies have shown that the DMI occurs exclusively in rare-earth containing FMGs, hence most likely linked to the rare-earth orbital magnetism. 80Despite experimental progress, the theoretical foundations of the interfacial DMI at magnetic/nonmagnetic garnet interfaces and methods to modulate it remain to be established.

C. Ferrimagnetic garnets in the spotlight
Current-driven DW dynamics has been studied in several FMGs; however, the most extensively studied system is TmIG.In TmIG, DMI-stabilized chiral DWs were found to move with velocities as fast as 800 m/s with current injection through Pt overlayer of the order of 1.2 × 1012 A/m 2 32 [Fig.4(a)].Despite the small H DMI found in this system (∼3 mT), v FiM was found to be non-saturating within the tested driving current range, validating the antiferromagnetic-like fast dynamics governing the motion, as described earlier and modeling in Fig. 4(b).A more recent ArXiv posting reports current-assisted DW motion in GdIG near angular momentum compensation, where the combined action of current and magnetic field has resulted in DW velocities reaching a record high value of 6000 m/s. 86These results are highly encouraging to engineer compensated FMGs at room temperature to achieve exceptionally large DW velocities with moderate current densities.
Even in the absence of DMI, fast current-driven DW motion can be achieved in FMGs if an in-plane field is applied for turning Bloch DWs into Néel DWs, as exemplified in Ref. 87.In this study, DWs in Bi:YIG, an FMG known for exceptionally low damping 67 due to the absence of magnetism in the rare-earth A-site, could be moved by currents up to an unprecedented velocity of over 4300 m/s without an out-of-plane assisting driving field [Figs.4(c) and 4(d)].Moreover, the current-driven DW motion in this material has revealed new and exciting physics governing the DW dynamics.The velocity was found to saturate above both a critical current density j and a critical in-plane DW stabilization field Hx, which cannot be explained by the precessional limitations discussed earlier [Eq.( 2)].Rather, calculations and simulations [Fig.4(d)] revealed that a universal DW speed limit had been reached, 67 regardless of driving force, caused by the Lorentz contraction of the DW.This was a direct consequence of spin wave group velocity that is fundamentally limited by relativistic effects.Although theorized, calculated, and predicted long ago, [88][89][90][91][92][93][94][95][96] the experimental verification that DWs obey Einsteinian special relativity had never been realized until this work.These results have revealed new limitations and challenges for the DW speeds that can be overcome with further material engineering.
Moreover, these findings opened new grounds for studying relativistic physics of quasiparticles with simple table-top experiments and in easily accessible materials.

IV. PERSPECTIVES
In the past decades, the fundamental understanding of domain wall physics and materials development has progressed together, which gave rise to numerous exciting device ideas and prototypes, such as racetrack memories and nonvolatile logic gates.We need to continue exploring the fundamental phenomena governing the DW dynamics, engineer materials to maximize the benefits DWs can offer, and develop new device concepts or improve the existing ones to bring DWs into the realm of microelectronics.Below are some possible directions we expect the DW research can take in the next years (summarized in Fig. 5).

A. Advances in materials
The family of FMGs is vast, and we have only studied a few systems.Expanding the existing studies to other engineered FMGs shall inevitably lead to more breakthrough knowledge, higher DW velocities, and useful device applications.A straightforward extension of the current-driven DW studies to temperatures other than room temperature to explore the role of magnetic and angular momentum compensation requires further exploration.Candidate materials for such studies are TbIG and GdIG with their magnetic compensation close to room temperature (248 and 286 K, respectively).While some studies point to the strong temperature dependence of spin-orbit coupling (high DMI) at low temperatures in REIG, 80 further direct measurements of DW dynamics could provide valuable fundamental insights.An alternative route to engineer the magnetic properties of FMGs to obtain superior DW properties is to create superlattices with alternating oxides stacked together.There are multiple degrees of freedom yet to be explored, such as combining magnetic and nonmagnetic garnet layers, or FMGs with different magnetic properties (e.g., chirally coupled in-plane and out-of-plane layers 97 ) to obtain a superior material performance not achievable by individual layers.
Extending the material spectrum to include other ferrites and cobaltites may be another promising route for DW studies and related phenomena.For instance, hexagonal ferrites, such as Ba(Pb)(Sr)Fe 12 O 19 , possess PMA arising from strong bulk magnetocrystalline anisotropy, rather than relatively weaker strain-induced effects observed in garnets, and are easily tunable with chemistry. 98nlike garnets, hexagonal ferrites are more easily integrated as magnetic materials with hexagonal topological insulators, taking advantage of potentially more efficient SOTs. 99Other oxide systems, such as perovskite oxides, host several novel functionalities, such as multiferroicity, superconductivity, Rashba effects, and tunable band structures, owing to the strong interplay between charge, spin, and lattice degrees of freedom.In the past few years, integration of these systems into modern spintronic devices has led to enhanced and tunable spin-charge interconversion, [100][101][102] nontraditional torques, 103 memory and logic devices, 104 and even nontrivial spin textures. 105

B. New physics
Understanding the dynamic properties of DWs is one of the most challenging problems in solid state physics due to the spatial and temporal resolution required for their experimental examination.Thus far, efforts have focused on magnetic field and spin-torque driven DW motion in FMGs.High crystallinity and low damping in FMGs are essential ingredients for efficient DW motion, which could be driven with other external stimuli, such as light or heat.The use of laser can provide a detection mechanism through both magneto-optic Kerr and Faraday effects but also a driving mechanism through a (laser-induced) heat-assisted change in DW propagation properties. 106,107This concept can be merged with laser-induced ultrafast demagnetization and/or transfer of angular momentum in suitable devices.Another unexplored territory is the use of coherent and incoherent magnons to manipulate domain walls in FMGs.These magnons can be generated optically, 108,109 electrically (using antennas for coherent magnons 110 and spin currents for incoherent magnons 111 ), or thermally 112 to act on DWs in a nonlocal manner.Remote control of DWs could provide essential ingredients for post-CMOS technologies, such as in-memory computing.Yet another interesting research direction is the local/dynamic manipulation of the strain state to control DW velocities through changes in the magnetic anisotropies.As explained previously, the PMA sensitively depends on the strain state with the substrate in FMGs.A slightest change in the strain might induce a strong modulation of PMA 59 and, hence, DW properties in FMGs.By coupling FMGs to, for example, piezoelectric materials, in principle, it is possible to locally and reversibly strain the film using an electric field.An alternative route is to develop techniques to grow FMGs on flexible substrates.Through controlled mechanical bending, one can continuously tune the magnetic anisotropy and potentially the interfacial DMI and, hence, control and optimize the DW properties.The local control of static and dynamic DW properties can be exploited in standard racetrack devices to combine memory and logic functionalities in a single device.The research into these aspects is still primitive 113 but highly promising for the future.

C. Innovative device concepts
The knowledge gathered through the DW studies summarized here can be readily transferred to other magnetic solitons, namely, skyrmions or merons.FMGs possess all the essential ingredients to host skyrmions (DMI, PMA, and tunable magnetostatics), which can be easily moved by SOTs.Indirect evidence of skyrmion formation through the topological Hall effect measurements has been reported earlier, 85 but in a recent study on YIG/TmIG bilayers, magnetic skyrmion bubbles were obtained and directly imaged in TmIG by lowering its magnetic anisotropy via exchange coupling with the proximate in-plane magnetized YIG layer. 114Through SOTs from a Pt overlayer, the skyrmions were subsequently driven into motion, albeit at relatively low speeds.It was noted that, surprisingly, pinning and thermal effects played a large role in the low skyrmion velocities despite the typically soft magnetic properties.
In addition to two-dimensional DW and skyrmion racetrack, the prospects of three-dimensional spintronics have garnered significant attention for potential advanced functionalities and improved scaling.Recently, current-driven DW dynamics in freestanding racetrack devices has been demonstrated using metallic ferromagnetic systems but with slower dynamic performance than their two-dimensional counterparts. 115In a parallel effort, in engineered three-dimensional structures, geometrically driven domain wall automotion has been demonstrated; a concept that can be used as spintronic interconnectors or signal transmission devices. 116Both these concepts can also be expanded to FMGs, materials with stronger structure-property relationships to exploit in three-dimensions and where freestanding continuous films have recently been fabricated. 117

FIG. 1 .
FIG. 1. Schematics of (a) spin transfer torque and (b) spin-orbit torque motion of domain walls.The grayscale arrows indicate magnetic moments comprising the magnetic material domain wall.The blue and red arrows indicate the spin orientation of flowing electrons.Note the Néel domain wall in panel (b), implying that only Néel walls are driven with conventional spin-orbit torques.(c) Seminal progress in current-driven domain wall motion over the past two decades.STT, spin transfer torque; SOT, spin-orbit torque; FM, ferromagnet; FiM, ferrimagnet; FMG, ferrimagnetic garnet.

FIG. 2 .
FIG. 2. (a) Magnetic structure of rare-earth iron garnets, indicating up to three magnetic sublattices (two Fe and one rare earth).(b) 2θ−ω x-ray diffraction scan of epitaxial 19.7 nm Tm 3 Fe 5 O 12 film around the (444)-Gd 3 Ga 5 O 12 peak, at various sputter target-substrate distances.(c) and (d) HAADF STEM images of the TmIG/GGG structure at the interface in the (c) [112] direction and in the (f) [101] direction.EELS line scans at the interface show the distribution of Ga, Gd, Fe, and Tm atoms as black-white contrast.RE, rare earth; Dod, dodecahedral; Tet, tetrahedral; Oct, octahedral; and a, lattice parameter.Panels (b) and (c) adapted with permission from Quindeau et al., Adv.Electron.Mater.3, 1600376 (2017).Copyright 2017 Author(s), licensed under a Creative Commons Attribution 4.0 License.

Figure 1 (
c) and Table I summarize the seminal progress in current-driven DW motion over the past several decades and highlight significant breakthroughs enabled by new spin torque mechanisms and materials engineered for fast DW dynamics.The immense progress in metallic systems [Fig.1(c), open symbols] has sparked interest into magnetic insulators [Fig.1(c), closed symbols]

FIG. 3 .
FIG. 3. (a) Experimentally measured domain wall velocity v DW as a function of temperature T under various applied magnetic fields (μ 0 H) and (b) micromagnetically calculated v DW as a function spin density S for various μ 0 H in a GdFeCo thin film sample.T A and T M indicate the angular momentum and magnetic compensation temperatures, respectively.(c) Experimentally measured v DW as a function of T for various current densities j in a Pt/GdCo/Ta heterostructure.(d) Open symbols show experimental v DW as a function of j for various T in the same Pt/GdCo/Ta heterostructure as panel (c).The overlaid solid lines are fits to a one-dimensional model adapted from Eq. (2) to account for Joule heating and temperature-dependent pinning.Panels (a) and (b) are adapted with permission from Kim et al., Nat.Mater.16, 1187-1192 (2017).Copyright 2017 Springer Nature.Panels (c) and (d) are adapted with permission from Caretta et al., Nat.Nanotechnol.13, 1154-1160 (2018).Copyright 2018 Springer Nature.

FIG. 4 .
FIG. 4. (a) DMI-stabilized chiral domain wall velocity v DW as a function of current density j in two separate studies of Tm 3 Fe 5 O 12 /Pt.(b) A comparison of a ferromagnetic and a two-sublattice ferrimagnetic model describing current-driven domain wall dynamics in (a).The ferrimagnetic model more accurately describes the dynamics in (a).(c) v DW as a function of j for various applied in-plane magnetic fields (Hx) in a Bi-doped Y 3 Fe 5 O 12 /Pt heterostructure.(d) Numerical and analytical models of domain wall dynamics of the material in (c), which include ferrimagnetic dynamics and Lorentz contraction.Data in panel (a) are adapted with permission from Vélez et al., Nat.Commun.10, 4750 (2019).Copyright 2019 Author(s), licensed under a Creative Commons Attribution 4.0 License, and from Avci et al., Nat.Nanotechnol.14, 561-566 (2019).Copyright 2019 Springer Nature.Data in panel (b) are adapted with permission from Avci et al., Nat.Nanotechnol.14, 561-566 (2019).Copyright 2019 Springer Nature.Data in panels (c) and (d) are adapted with permission from Caretta et al., Science 370, 1438-1442 (2019).Copyright 2019 AAAS.

FIG. 5 .
FIG. 5. New driving mechanisms, innovative device concepts, and materials development research directions of domain walls in ferrimagnetic garnets. 12