Measuring interfacial Dzyaloshinskii-Moriya interaction in ultra thin films
Michaela Kuepferling, Arianna Casiraghi, Gabriel Soares, Gianfranco Durin, Felipe Garcia-Sanchez, Liu Chen, Christian H. Back, Christopher H. Marrows, Silvia Tacchi, Giovanni Carlotti
MMeasuring interfacial Dzyaloshinskii-Moriya interaction in ultra thin films
M. Kuepferling, A. Casiraghi, G. Soares, and G. Durin
Istituto Nazionale di Ricerca Metrologica,Strada delle Cacce 91, 10135, Turin,Italy
F. Garcia-Sanchez
Dep. of Applied Physics,University of Salamanca,Plaza de los Caidos s/n 37008, Salamanca,Spain
L. Chen and C. H. Back
Technical University Munich,James-Frank-Str. 1,85748 Garching,Germany
C. H. Marrows
School of Physics and Astronomy,University of Leeds, Leeds LS2 9JT,United Kingdom
S. Tacchi
CNR, Istituto Officina dei Materiali - Perugia,c/o Dipartimento di Fisica e Geologia- Univ. Perugia,Via A. Pascoli, 06123 Perugia,Italy
G. Carlotti
Dipartimento di Fisica e Geologia,Universit di Perugia,Via Pascoli, 06123 Perugia,Italy (Dated: September 25, 2020)
The Dzyaloshinskii-Moriya interaction (DMI), being one of the origins for chiral magnetism, is cur-rently attracting huge attention in the research community focusing on applied magnetism and spin-tronics. For future applications an accurate measurement of its strength is indispensable. In this work,we present a review of the state of the art of measuring the coe ffi cient D of the Dzyaloshinskii-Moriyainteraction, the DMI constant, focusing on systems where the interaction arises from the interfacebetween two materials. The measurement techniques are divided into three categories: a) domainwall based measurements, b) spin wave based measurements and c) spin orbit torque based measure-ments. We give an overview of the experimental techniques as well as their theoretical backgroundand models for the quantification of the DMI constant D . We analyze the advantages and disadvan-tages of each method and compare D values in di ff erent stacks. The review aims to obtain a betterunderstanding of the applicability of the di ff erent techniques to di ff erent stacks and of the origin ofapparent disagreement of literature values. Contents
I. Introduction 2II. Domain wall methods 3A. Method overview 3B. Theory and models 41. One-dimensional DW model 42. Current-driven domain wall motion 53. Field-driven domain wall motion 6 4. Equilibrium domain pattern 95. Magnetic stripe annihilation 106. Nucleation field 107. Domain wall stray fields 11C. Experimental results 121. Current-driven domain wall motion 122. Field-driven domain wall motion 123. Equilibrium domain pattern 154. Magnetic stripe annihilation 155. Nucleation field 156. Domain wall stray fields 16 a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p D. Advantages and limitations 161. DW motion 202. DW energy 20III. Spin wave methods 21A. Method overview 21B. Theory and models 221. Quantum spin wave theory 222. Classical spin wave theory 22C. Experimental results 251. Brillouin light scattering 252. Time resolved magneto optical imaging 273. All electric spin wave spectroscopy 274. Spin-polarized electron loss spectroscopy 31D. Advantages and Limitations 311. BLS 322. TR-MOKE 323. AESWS 334. SPEELS 33IV. Spin Orbit Torque method 33A. Method overview 33B. Theory and models 33C. Experimental Results 34D. Advantages and Limitations 36V. Comparison of the methods and problems in the determination of D 36A. Comparing the DMI constants obtained by the di ff erentmethods 36B. Comparing the DMI constants in di ff erent materials 37C. Influence of growth conditions on the DMI constant 40D. Outlook and conclusion 40 I. Introduction
Topological spin structures such as chiral domain walls(DWs) and skyrmions are emerging as promising informa-tion carriers for future spintronics technologies (Dieny et al. ,2020). Electric currents can drive such spin structures with anunprecedented level of e ffi ciency, which makes them partic-ularly attractive for innovative storage devices, including theracetrack memory (Parkin and Yang, 2015). The crucial in-gredient needed for stabilizing these chiral spin textures is theDzyaloshinskii-Moriya Interaction (DMI).The DMI is a chiral exchange interaction between localizedspins and has a net contribution only in systems where twoconditions are simultaneously met:1. presence of an indirect or long-range exchange inter-action: this can be superexchange as in original worksof Dzyaloshinskii (1957) and Moriya (1960), or Ruder-man Kittel Kosuya Yosida (RKKY) type as in Fert andLevy (1980)2. lack of structural inversion symmetry.Its origin is found in the spin-orbit coupling (SOC) which actsas a perturbation on localized orbital states. Given two neigh-boring spins (cid:126) S and (cid:126) S , the DMI energy can be described by − (cid:126) D · (cid:126) S × (cid:126) S (Moriya, 1960). This expression is part of ageneralized exchange interaction and in fact (cid:126) D is related tothe exchange constant J of the direct Heisenberg exchange − J (cid:126) S · (cid:126) S , but contrary to the latter which favours collinear FerromagnetHeavy Metal r ij D ij Figure 1 Diagram of a system having interfacial DMI. The couplingbetween the ferromagnetic spins is an indirect super-exchange me-diated by a heavy metal atom with large Spin Orbit Coupling. Inthis particular case, the DMI vector D ij is perpendicular to the planeformed by the ferromagnetic atoms with the heavy metal one. alignment, the DMI promotes an orthogonal arrangement be-tween (cid:126) S and (cid:126) S , with a chirality imposed by the direction of (cid:126) D , as sketched in Fig. 1.The DMI was first proposed in the 1950’s for antiferro-magnets such as α -Fe O to account for the existence of aweak ferromagnetism (Dzyaloshinskii, 1957; Moriya, 1960).In the following decades, several magnetic materials, suchas spin glasses, orthoferrites, manganites or superconductingcuprates (Bogdanov et al. , 2002; Co ff ey et al. , 1991; Fert andLevy, 1980; Luo et al. , 1999), were investigated for their non-collinear or helical magnetism and for the influence of DMIon the magnetic state. Theoretical models in non-collinearmagnetism are often based on ab-initio calculations and den-sity functional theory (Sandratskii, 1998). From an experi-mental point of view, the research was intensified at the be-ginning of the century, employing more sophisticated imag-ing techniques to visualize the magnetic state, highlighted inthe important works on DMI materials (Bode et al. , 2007;Heinze, 2000). In non-centrosymmetric single crystals withFeSi structure (or B20 according to classification of the jour-nal ”Strukturbericht”), as MnSi, FeGe, etc., it was shown thatDMI gives rise to skyrmion lattices and other exotic spin tex-tures at low temperature (R¨oßler et al. , 2006; Yu et al. , 2010).More recently, the interfacial DMI received broad atten-tion from the magnetic community (Hellman et al. , 2017),particularly in the context of systems composed of a heavymetal layer and a ultrathin ferromagnetic film with perpendic-ular magnetic anisotropy (PMA). Indeed, early experimentson current-induced domain wall motion were not able to fullyunderstand the driving mechanisms behind such motion. Onone hand, the conventional spin-transfer torque (STT) the-ory (Slonczewski, 1996) predicts an opposite direction of mo-tion. On the other hand, the newly proposed spin-orbit torques(SOTs) – the Rashba torque (Miron et al. , 2010, 2011) and thespin Hall e ff ect (SHE) torque (Haazen et al. , 2013; Liu et al. ,2012) – did not have the correct symmetry to drive Blochwalls (Khvalkovskiy et al. , 2013), which were believed to bepresent in these ultrathin PMA films from purely magneto-static considerations. These controversies were resolved whenit was suggested that the DMI can convert Bloch into N´eelwalls (Fig. 2) of a fixed chirality (Heide et al. , 2008; Thiaville et al. , 2012). A series of experimental works later confirmedthese findings (Emori et al. , 2013; Ryu et al. , 2013), sub-stantiating the view that DMI-stabilised N´eel walls are mostlymoved by the SHE torque. Ever since DMI-based phenomenahave created an extremely active research field, often referredto as chiral magnetism.Systems composed of a heavy metal and thin magnetic filmwith PMA are particularly interesting from the applicationpoint of view, due to the fast domain wall motion driven bycurrent in the presence of DMI. In these systems, the DMIarises due to the broken inversion symmetry at the interfacebetween the two materials and the large SOC of the heavymetal atoms, which mediate the interaction between neigh-bouring spins of the ferromagnet. It follows that the DMI hasan interfacial nature with a strength decreasing with the mag-netic film thickness. If the symmetry breaking is only dueto the interface, according to Moriya’s rules with symmetryplanes or axes are always normal to the film plane, the DMIvector has to be necessarily perpendicular to the film normal(which is also the direction of the magnetization). In such acase (not considering special crystal symmetries of the ferro-magnetic film itself) the DMI vector simplifies to (cid:126) D = D (ˆ r × ˆ z )(Moriya, 1960; Thiaville et al. , 2012) and can thus be de-scribed by a single interfacial DMI constant D , with ˆ r beingthe unit vector linking two neighbouring spins and ˆ z beingalong the film normal (see Fig. 1). The interfacial DMI canthan be thought of as equivalent to an e ff ective in-plane mag-netic field H DMI which, acting across the domain wall, causesa reorientation of spins from the a chiral Bloch configurationinto the N´eel configuration with a fixed chirality (Chen et al. ,2013; Thiaville et al. , 2012). The strength of the interfacialDMI is measured in terms of the DMI constant D , which isproportional to the DMI field (Thiaville et al. , 2012). Thesign of D dictates the chirality of the N´eel domain wall, andconsequently, the direction of the domain wall motion under acurrent-induced SHE torque.Because of the crucial role that the interfacial DMI playsin stabilizing chiral spin textures as well as in attaining ex-tremely e ffi cient domain wall motion, it is of utmost impor-tance for the magnetic community to be able to accuratelymeasure the magnitude and sign of D in commonly inves-tigated heavy metal (HM) / ferromagnet (FM) combinations.However, despite a rapidly increasing number of works quan-tifying the DMI through di ff erent experimental techniques, alot of disagreement is currently present in the literature andsystematic studies or reviews are still rare (Choe and You,2018). Not only di ff erent measuring techniques are found toprovide contradictory values for D , but controversies are alsopresent when utilizing the same method on nominally identi-cal stacks, or far di ff erent methods on the same sample (Shah-bazi et al. , 2019; Soucaille et al. , 2016).The most commonly used experimental techniques that inrecent years have been employed to measure D in HM / FMstructures can be broadly divided into three categories:
Néel wall
Bloch wall
Figure 2 Bloch and N´eel configurations for a perpendicularly mag-netized material. The N´eel wall has left-handed chirality ( ↑←↓ ). Domain wall methods (see Sec. II), where D is extractedby either measuring domain wall velocity or energy asa function of an in-plane magnetic field, or by measur-ing domain wall spacing in stripe domain phases, or bydirectly measuring the domain wall internal structure;2. Spin wave methods (see Sec. III), where D is extractedby measuring the non-reciprocity of propagating spinwaves in in-plane magnetized films;3. Spin-orbit torque methods (see Sec. IV), where D is ex-tracted by measuring the field shift of the out-of-planehysteresis loop under an in-plane field.With this literature review we aim at providing an accu-rate analysis of the current state-of-the-art regarding measure-ments of the interfacial DMI constant D based on the di ff er-ent techniques summarised above. We will discuss in detailsthe models that have been proposed to analyse the results ineach case, the advantages and limitations of each method, andfinally we will compare measurements of D obtained employ-ing the same or di ff erent techniques . II. Domain wall methodsA. Method overview
In ultra-thin films with PMA Bloch domain walls are mag-netostatically favoured over N´eel DWs . However, as said, theinterfacial DMI manifests itself as an e ff ective in-plane mag-netic field H DMI which, when large enough, converts BlochDWs into N´eel DWs with a chirality determined by the signof D (Chen et al. , 2013; Thiaville et al. , 2012). By tuning the Throughout this review we use the following conventions: positive D cor-responds to N´eel domain walls with the right-handed (clockwise) chirality( ↑→↓ or ↓←↑ ), while negative D corresponds to N´eel domain walls withthe left-handed (anticlockwise) chirality ( ↑←↓ or ↓→↑ ). Thicknesses infilm layers are in nm. However, N´eel DWs become energetically more favourable when the filmis patterned into very narrow wires (Koyama et al. , 2011). spin texture of the DWs towards a chiral N´eel configuration,the interfacial DMI changes all the DW properties, includingwidth, energy, and velocity.The first experimental technique able to extract D was themeasure of DW velocities driven by an electric current, as afunction of in-plane magnetic fields (Ryu et al. , 2013; Torre-jon et al. , 2014). This is not surprising, given the importanceof DMI in these systems in the context of current-induced do-main wall motion experiments (Emori et al. , 2013; Ryu et al. ,2013). Soon afterwards it was proven that D could also beinferred by measuring DW velocities driven by a perpendic-ular magnetic field (Je et al. , 2013). In any case, whetherthe driving mechanism is current or field (for advantages andlimitations - see Sec. II.D), H DMI is extracted by analysingthe dependence of DW velocity on the in-plane magneticfields. In the context of the one-dimensional DW model (seeSec. II.B.1), D is then derived according to Emori et al. , 2013: D = µ H DMI M S ∆ , (2.1)where M S is the saturation magnetization, and ∆ = (cid:112) A / K e f f is the DW width parameter, with A being the exchange sti ff -ness and K e f f = K U − µ M S the e ff ective perpendicularanisotropy constant, corresponding to the intrinsic perpendic-ular anisotropy constant K U decreased by the demagnetizingenergy µ M S .Following these early investigations (Je et al. , 2013; Ryu et al. , 2013; Torrejon et al. , 2014), several further works haveused DW motion, induced by either current or field, to esti-mate the strength and sign of the interfacial DMI in di ff erentmaterial systems. Moreover, the nucleation of DWs, which isa necessary step to study their motion, was shown to be de-pendent on D (Pizzini et al. , 2014), which established anotherpath to access the magnitude and sign of D . Finally, also thestatic properties of domains were shown to be altered by thepresence of DMI. This was observed from two points of view:firstly, the domain width is altered due the modification ofthe DW energy originated from DMI (Moreau-Luchaire et al. ,2016). Secondly, the DMI a ff ects the stability of reversed do-mains and their field of annihilation (Hiramatsu et al. , 2014).Both e ff ects have been used to estimate the magnitude of D .Here we provide a summary for the di ff erent DW-basedtechniques, explaining the models used to interpret the results(Sec. II.B), reviewing the current state-of-the-art regarding thedetermination of D (Sec. II.C), and highlighting the respectiveadvantages and limitations (Sec. II.D). B. Theory and models
1. One-dimensional DW model
DW dynamics are often analysed within the framework of aone-dimensional (1D) analytical model, which has been estab-lished as a useful tool to support computationally costly mi-cromagnetic simulations as well as to interpret experimentalresults. As the name suggests, the 1D model for DW dynam- χ q Δ zy xM Φ Figure 3 Sketch of the collective coordinates used to describe DWdynamics in di ff erent types of 1D models: q is the domain wall posi-tion, Φ the angle of the projection of the magnetization M in the x – y plane, ∆ the width, and χ the tilting of the DW. ics is based on the approximation that the magnetization M varies along one direction only – namely the axis of a narrowwire, usually identified with x , as depicted in Fig. 3.The simplest form of the 1D model was introducedby Walker and Slonczewski (Malozemo ff and Slonczewski,1979; Schryer and Walker, 1974) in the 1970s, to study DWdynamics in PMA materials under the influence of an appliedperpendicular field. Here DW dynamics were described interms of two time-dependent variables (see Fig. 3): the DWposition q ( t ) along the wire axis, and the DW angle Φ ( t ), de-fined as the in-plane ( x – y ) angle of the internal DW magneti-zation M with respect to the positive x axis . The q − Φ modelwas later extended by Sobolev et al. (1995, 1994) to describeDW dynamics in PMA materials under in-plane fields. Thiav-ille et al. (2002) next adapted the 1D model to systems with in-plane magnetic anisotropy, using the domain wall width ∆ ( t )as an additional time-dependent variable in a revised q − Φ − ∆ model.More recently, several other contributions have been madeto the 1D model to include newly discovered e ff ects, suchas the STTs (Thiaville et al. , 2005; Zhang and Li, 2004),the SOTs (Boulle et al. , 2012; Hayashi et al. , 2012; Mar-tinez et al. , 2013a,b; Seo et al. , 2012) (see Eq. 2.2), and theDMI (Emori et al. , 2013; Martinez et al. , 2013a; Thiaville et al. , 2012) (see Eq. 2.1). Additions of thermal fluctua-tions (Martinez et al. , 2007) and of spatially dependent pin-ning (Consolo and Martinez, 2012) helped to make the 1Dmodel more realistic. Furthermore, considering experimentsof fast current-driven DW motion (Ryu et al. , 2012), Boulle et al. (2013) proposed to include DW tilting as an additionaltime-dependent variable χ ( t ) (defined as the angle of the DWnormal plane with respect to the positive x -axis), which ledto the development of the q − Φ − χ model. Very recentlythe 1D model was extended to implement all four collectivecoordinates, namely q − Φ − ∆ − χ , with the aim of improv-ing the agreement with experimental observations and micro-magnetic simulations when large in-plane fields are applied According to these conventions,
Φ = π ) corresponds to a right (left)handed chiral N´eel DW, while Φ = ± π/ (Nasseri et al. , 2017). The same authors later showed that thesimple two coordinate q − Φ model can grant higher accuracywhen combined with an ansatz (which links collective coor-dinates to magnetization components) that takes into accountmagnetization canting within the domains under an in-planefield (Nasseri et al. , 2018).
2. Current-driven domain wall motion
The driving mechanism behind the current-induced motionof DWs in heavy metal / ferromagnet bilayers with PMA isnow widely believed to be due to a combination of SHE andDMI, while the Rashba torque is considered to be negligible(Emori et al. , 2013; Martinez et al. , 2014; Ryu et al. , 2013), asthe latter acts to stabilize Bloch walls, and does not have thecorrect symmetry to drive DWs directly (Emori et al. , 2013;Khvalkovskiy et al. , 2013). The SHE in the heavy metal con-verts the in-plane flow of charge current into spin accumu-lation at the interface between the two layers, with conse-quent spin-current di ff usion into the ferromagnet. This spin-current can interact with the local magnetization by exertinga damping-like torque on it, known as the SHE spin-transfertorque (SHE-STT). According to the 1D model, the ampli-tude of the e ff ective field associated with the SHE-STT is ex-pressed as (Khvalkovskiy et al. , 2013; Thiaville et al. , 2012): H S HE = (cid:126) θ S H | J | µ | e | M S t FM cos( Φ ) , (2.2)where θ S H is the spin Hall angle, J is the electron current den-sity, t FM is the thickness of the ferromagnetic film, and Φ is theinternal in-plane DW angle as defined in Fig. 3. As mentionedearlier, the SHE-STT can move DWs only if they possess aN´eel component (i.e. Φ (cid:44) π/
2) in their spin structure, due tothe interfacial DMI. The direction in which DWs move withcurrent depends both on the sign of the spin Hall angle θ S H ,determined by the spin-orbit coupling constant of the heavymetal, and on the sign of D (i.e. on the chirality of the DW).The 1D model predicts that the velocity of DWs driven by theSHE-STT increases with the magnitude of D (Thiaville et al. ,2012), as a consequence of the increase in Walker Breakdownfield (discussed in Sec. II.B.3).Current-driven DW dynamics are dramatically a ff ected bythe application of an in-plane magnetic field H x along thecurrent direction. It is indeed due to this that DW velocitymeasurements as a function of H x provide a means to quan-tify D . Given a fixed J , it is observed that N´eel DWs withthe same chirality ↑→↓ and ↓←↑ have the same velocity for H x =
0, while they move with di ff erent velocities under a non-vanishing H x . In particular, for a given sign of H x , one DWtype moves faster and the other slower with respect to their In principle, field-like and damping-like torques can occur as a result ofRashba or SHE e ff ects. H X =0 H X =- H DMI v DW1 = v DW2 v DW1 =0 v DW2 ≠ +H DMI H X V DW c)a) b) -H DMI
J HM
DW1 DW2 v DW1 v DW2 v DW1 v DW2 J DW1 DW2 v DW2 HM Figure 4 Sketch of the dynamics of DWs driven by current in theabsence (a) or presence (b) of an applied in-plane field H x . Thewhite arrows indicate DW velocity. The strength of H x applied in(b) matches that of the DMI field H DMI , thus stopping the motionof DW . A sketch of the velocity for the two DWs as a function ofapplied H x is illustrated in (c). velocity at H x =
0, and the situation is reversed by changingthe sign of the applied H x . This remarkable behaviour has animportant consequence: both DWs stop moving for a certain | H x | , equal in strength but opposite in sign for ↑→↓ and ↓←↑ DWs respectively, as schematically shown in Fig. 4.The applied H x under which these N´eel DWs become sta-tionary – typically referred to as “compensating” or “stop-ping” field – is the field that, opposing the DMI, restores aBloch DW configuration, for which indeed no motion is ex-pected via SHE-STT. As such, this “compensating” field H ∗ x can be considered equivalent in strength (but opposite in sign)to the H DMI acting locally across the DWs. In other words,N´eel DWs move faster or slower depending on whether thee ff ective in-plane field experienced by the DW is enhanced( H x + H DMI ) or decreased ( H x − H DMI ), respectively. Inthe latter case, DWs stop moving for H x = H ∗ x = H DMI ,when they become Bloch walls, and they reverse directionof motion for H x > H DMI , when the N´eel configuration isre-established but with an opposite chirality. However, identi-fying H ∗ x with H DMI is valid only if conventional STT, due tothe spin-polarized current in the ferromagnet, (Slonczewski,1996) can be neglected. When a significant STT is present,the relationship between the H ∗ x and H DMI in the 1D modelbecomes (Emori et al. , 2013; Ryu et al. , 2013): H ∗ x = H DMI + sgn( θ S H ) 2 π µ B P γ eM S ∆ | J | , (2.3)where µ B is the Bohr magneton, P is the spin current polariza-tion, and γ is the gyromagnetic ratio. Eq. 2.3 implies that H ∗ x can depend on the amplitude of the current used to drive DWs.When only a modest dependence of H ∗ x on J is observed, asfor instance in Karnad et al. (2018) and Ryu et al. (2014), it ispossible to conclude that the contribution from STT is small.In any case, once determined H DMI , the magnitude of D isderived through Eq. 2.1, while its sign is inferred from thedirection of DW motion.The dependence of DW velocity on an in-plane field for afixed current density can be analytically described in the con-text of the 1D model, taking into account STT, SHE-STT andDMI (Emori et al. , 2013; Ryu et al. , 2013; Thiaville et al. ,2012). As seen in Fig. 4, the DW velocity is expected to beapproximately linear with H x around the “compensating” field H ∗ x , and some experimental works derive H DMI by linearly fit-ting the data (Ryu et al. , 2013; Torrejon et al. , 2014). In somecases, it has been observed that DW velocity remain small ornull in a quite large range of H x around H ∗ x (Lo Conte et al. ,2017, 2015; Ryu et al. , 2014), consistent with thermally acti-vated creep regime, and strong pinning e ff ects. To account forit, the 1D model has been extended to include an e ff ective pin-ning potential both without (Lo Conte et al. , 2017; Ryu et al. ,2014), and with thermal fields (Lo Conte et al. , 2015) to de-scribe the influence of thermal fluctuations. In both cases, themodified 1D model has provided good agreement with the ex-perimental data and used to extract H DMI . It is also worthmentioning that the range of H x values for which the DWs re-main small has been observed to decrease upon increasing thecurrent density, due to a reduced influence from pinning (Ryu et al. , 2014).Finally, a few papers have shown that D can also be quan-tified through current-driven DW dynamics by measuring thedependence of the DW depinning e ffi ciency, rather than theDW velocity, on the in-plane field H x (Franken et al. , 2014;Kim et al. , 2018a). The e ffi ciency of DW depinning is definedas: (cid:15) = µ dH S HE dJ , (2.4)and is measured as the slope of the out-of-plane depinningfield as a function of J . The DW depinning e ffi ciency changesas a function of H x , due to the correspondent variation of theDW internal structure. In particular, (cid:15) is found to vanish at acertain H ∗ x , equal in strength but opposite in sign for ↑→↓ and ↓←↑ DWs. Similarly to what already discussed for the DWvelocity dependence on H x , this in-plane field H ∗ x for which (cid:15) = H DMI .
3. Field-driven domain wall motion
The simplest and common way to move DWs in PMA ma-terials is by a perpendicular field H z . To minimize the Zee-man energy associated with H z , domains with magnetizationalong the field direction expand at the expense of the others,leading to DW motion. Field-driven DW dynamics are typi-cally classified into three distinctive regimes – creep, depin-ning, and flow – which occur in succession upon increasing H z , as shown in Fig. 5a.For su ffi ciently low driving fields DWs move in the ther-mally activated creep regime where they interact stronglywith disorder, and their velocity grows exponentially as v ∼ exp( − H z ) − / (Chauve et al. , 2000; Lemerle et al. , 1998). creep depin. fl owT>0 T=0 H dep H z v DW steady precess H WB H z v DW a) b) creep depin. fl owT>0 T=0 H dep H z v DW steady precess H WB H z v DW a) b) Figure 5 Dependence of the DW velocity v DW as a function of in-creasing applied field. In (a) the three regimes - creep, depinning andflow - are visible. The depinning transition, which is abrupt at zerotemperature, shows some rounding due to thermal e ff ects. At higherapplied fields (b), two further regimes can be distinguished withinthe flow, namely a steady and a precessional flow. The drop in thevelocity is called Walker Breakdown. Upon increasing H z above a critical value, known as the de-pinning field H dep , disorder starts to become irrelevant and theDW velocity grows as v ∼ ( H z − H dep ) β (Chauve et al. , 2000),with β being the depinning exponent. Finally, for H z (cid:29) H dep the DW enters the flow regime where the velocity increaseslinearly with H z up to the so-called Walker Breakdown field H WB , which marks a significant decrease in DW velocity, dueto a change in its internal dynamics (Metaxas et al. , 2007).For H z (cid:29) H WB the DW recovers a linear flow with H z , albeitat a reduced mobility dv / dH .With the exceptions of a few works (Ajejas et al. , 2017;Ju´e et al. , 2016a; Pham et al. , 2016; de Souza Chaves et al. ,2019; Vaˇnatka et al. , 2015), the DMI has been mostly quan-tified through experiments of field-driven DW motion in thecreep regime, which is addressed below. Methods to extractthe DMI from DW dynamics in the flow regime will be dis-cussed later. a. Creep regime In the creep regime, DWs are driven by modest fields (typ-ically down to a few percent of H dep ) and move slowly bythermal activation, interacting strongly with disorder of vari-ous origin (pinning defects, film thickness variations, M S in-homogeneities, etc). The DW creep dynamics is understoodin terms of the motion of a one-dimensional elastic line in atwo-dimensional disordered potential. The dependence of the Figure 6 First observation of the asymmetric expansion of a bubbleunder in-plane magnetic field in a continuous film of Pt / Co / Pt. Ar-rows show the in-plane field directions. From Kabanov et al. , 2010.
DW velocity v on the applied field H z is described by the so-called creep law (Chauve et al. , 2000; Lemerle et al. , 1998): v = v exp[ − ζ ( µ H z ) − µ ] , (2.5)where v is the characteristic speed proportional to the attemptfrequency for DW propagation, ζ is a scaling constant, and µ = / H z issignificantly altered by the simultaneous presence of an in-plane field H x , similarly to what already discussed for thecurrent-driven case (see Sec. II.B.2). Indeed, it is observedexperimentally that when a circular magnetic bubble expandsunder the application of H z only, the radial symmetry is main-tained and the bubble grows isotropically – thus retaining itsoriginal shape. However, the symmetry is broken when thebubble is expanded under the application of both H z and H x ,as N´eel ↑→↓ and ↓←↑ DWs acquire di ff erent velocities alongthe direction of the applied H x . This circumstance was firstobserved in Kabanov et al. (2010) for continuous films ofPt / Co / Pt (see Fig. 6) and was attributed to interfacial DMI,even if only later it was fully understood and modelled in thecontext of DW creep.The first model proposed to explain the asymmetric motionof N´eel DWs with the in-plane field (Je et al. , 2013), suggestedto modify the creep law (Eq. 2.5) by changing the scaling pa-rameter ζ to take into account the dependence of DW energyon the in-plane field (Je et al. , 2013): ζ ( H x ) = ζ [ σ ( H x ) /σ (0)] / , (2.6)where ζ is a scaling constant and σ is the DW energy density.The dependence of DW energy density on the in-plane fieldhas been calculated as (Je et al. , 2013; Thiaville et al. , 2012): σ ( H x ) = σ − π ∆ µ M S K D ( H x + H DMI ) , (2.7) Figure 7 (a-b) Dynamics of bubble DWs driven by perpendicularfield H z = mT in the absence (a) or presence (b) of an in-planefield H x = mT in a Ta(5) / Pt(2.5) / Co(0.3) thin film with PMA.Each image is obtained by adding four sequential images with afixed time step (0.4 s), which are captured using a magneto-opticalKerr e ff ect microscope. The blue box in (b) designates the field ofmeasurement. From Je et al. , 2013. (c) A typical symmetric pro-file of left and right DWs as a function of the in-plane field H x inTa(5) / Co Fe B (0.8) / MgO(2) thin film. Here, the velocity min-ima occur at | H x | = H DMI . From Khan et al. , 2016. when the condition | H x + H DMI | < K D /πµ M S is satisfied. Inthis case the e ff ective field acting on the DW is not strongenough to fully convert it into a N´eel DW. Otherwise, forhigher fields and fully N´eel DWs the expression becomes: σ ( H x ) = σ + K D ∆ − π ∆ µ M S | H x + H DMI | . (2.8)In these equations, σ = (cid:112) AK e f f is the Bloch DW en-ergy density, ∆ = (cid:112) A / K e f f is the domain wall parameter,and K D = N x µ M S / ,with the demagnetising factor of the DW given by N x = ln(2) t / ( π ∆ ), with t the magnetic film thickness (Tarasenko et al. , 1998). In other words, considering H x to be appliedalong the left–right direction of a magnetic bubble, this modelpredicts that the e ff ective in-plane field acting locally on theN´eel DWs at the left and right side of the bubble can eitherbe increased ( H x + H DMI ) or decreased ( H x − H DMI ), result-ing in DWs moving faster on one side of the bubble than theother, as for the current-driven case (see Fig. 7). The field H ∗ x = H DMI at which N´eel DWs are converted into BlochDWs is then equal in magnitude but opposite in sign for ↑↓ and ↓↑ DWs, respectively. It is important to notice that, dif-ferently from the current-driven case, field-driven Bloch DWs K D represents the fact that Bloch DWs are magnetostatically more stablein the absence of DMI due to the high PMA of the films. do not stop moving under the in-plane field H ∗ x , since the per-pendicular field H z keeps expanding the magnetic bubble tominimize Zeeman energy. Rather, in field-driven experiments, | H ∗ x | = H DMI corresponds to a minimum in DW velocity, andthe velocity for both left and right DWs should be symmet-ric around this minimum, as in Fig. 7c, even if in most casesis not. It follows that in the absence of DMI, i.e. when thebubble DW is in the Bloch configuration, both left and rightside of the bubble have the same velocity dependence with H x , show a minimum at H x = H x = et al. , 2015a). This model can be used to fit thedependence of bubble DW velocities on in-plane field usingthree fitting parameters: v , ζ , and H DMI itself. Alternatively,the scaling parameters v and ζ can be extracted separatelyas the intercept and gradient of a linear fit to the curve ln( v )vs. H − / z for H x =
0, thus leaving H DMI as the only fitting pa-rameter of the v dependence on H x . Once determined H DMI ,the magnitude of D is derived through Eq. 2.1, while its signis inferred from the orientation of the bubble asymmetry withrespect to the H x direction.This modified creep model, in which the in-plane field af-fects DW dynamics only through a variation of domain wallenergy, has been successfully applied to fit several experimen-tal data and estimate the interfacial DMI constant D (Hrabec et al. , 2014; Je et al. , 2013; Khan et al. , 2016; Kim et al. ,2017a; Ku´swik et al. , 2018; Petit et al. , 2015; Shahbazi et al. ,2018; Wells et al. , 2017; Yu et al. , 2016b). However, fora growing number of experiments, as for instance in Fig. 8,the model fails to provide an adequate description of the data,which often show an asymmetric behaviour around the mini-mum velocity (Cao et al. , 2018a; Ju´e et al. , 2016b; Kim et al. ,2018a; Lau et al. , 2016; Lavrijsen et al. , 2015; Pellegren et al. ,2017; Shahbazi et al. , 2019; Shepley et al. , 2018; Soucaille et al. , 2016), a local peak in velocity (Balk et al. , 2017; Lavri-jsen et al. , 2015; Soucaille et al. , 2016), or even a maximumin velocity in the flow regime (Vaˇnatka et al. , 2015).To explain these non-symmetric results, several di ff erentmechanisms have been suggested. Ju´e et al. (2016b) pro-posed to neglect any DW energy contribution, and consider aDW chirality-dependent damping, which acts as a dissipativeSOT on the DW: this damping would modulate the attemptfrequency for DW motion, and thus modifying the character-istic speed v of the creep law (see Eq. 2.5). On the contrary,Lavrijsen et al. (2015) suggested that chiral damping couldnot be an explanation of the asymmetry of their own results,as v was found to be symmetric with respect to H x . In anycase, the importance to consider possible H x dependencies onboth creep parameters v and ζ (not necessarily related to chi-ral damping) has been highlighted in Balk et al. (2017) andShepley et al. (2018). While Shepley et al. showed that thedependence on H x is asymmetric for both v and ζ , Balk etal. modelled their results through a modified creep law thattakes into account a dependence on H x for both parameters.By combining Eqs. 2.5, and 2.6, with their expression for v ,they were able to fit velocity vs H x curves that showed a lo-cal peak around H x =
0, introducing a local anisotropy due to
Figure 8 Example of asymmetry of the domain wall ve-locity profile as a function of the in-plane field (reddots) for a set of multilayers with nominal structure
S i / S iO / X ( t ) / Co Fe B / MgO (2) / T a (1) (film thicknesses innm) with four underlayers X ( t ): W (2nm, (a)), W (3nm, (b)), TaN(1nm, (c)), and Hf (1nm, (d)). All cases are highly asymmetric in thevelocity profile, even showing a local maximum as for the latter case(d). Note that the velocity in the direction orthogonal to the in-planefield (black dots) remains symmetric. From Soucaille et al. , 2016. pinning K pin , and having ζ and H DMI as fitting parameters.Several analytical and numerical study were devoted to un-derstand these features. Kim et al. (2016a) attributed theasymmetry in the DW energy density σ , to the asymmetricvariation of DW width with H x , later confirmed by micro-magnetic simulations (Sarma et al. , 2018). Lau et al. (2016)the described the velocity asymmetry in terms of the Wul ff construction, which yields a methodology to determine theshape of a magnetic bubble, although it does not explicitlyprovide a model for the velocity as a function of in-plane field.They also speculated that nucleation and annihilation of Blochpoints may be responsible for a peculiar flattening of mag-netic bubbles, which could in turn cause the observed velocityasymmetry. This again was confirmed through micromagneticsimulations by Sarma et al. (2018). Pellegren et al. (2017) ar-gued that under an applied H x , where the DW energy density σ becomes anisotropic with respect to the DW orientation inthe film plane, the correct elastic energy that should be con-sidered to describe the creep regime does not simply identifywith σ , as typically assumed in the phenomenological modelof creep (Lemerle et al. , 1998). Rather, they propose that σ should be replaced by the DW sti ff ness (cid:101) σ = σ + σ (cid:48)(cid:48) , where σ (cid:48)(cid:48) is related to the local curvature of the DW energy. In thisway, by using (cid:101) σ instead of σ in Eq. 2.6, they are able to re-produce the asymmetry in their velocity curves. Interestingly,the H DMI value extracted using this sti ff ness model was foundto be higher than the field at which the minimum in veloc-ity is observed. In a later work (Lau et al. , 2018), the sameauthors proposed to upgrade the sti ff ness model, by takinginto account a possible variation of the characteristic speed v with H x , which they also speculate could be due to a chi-ral damping mechanism. Through this improved model, theycould fit velocity curves as a function of H x that are not onlyasymmetric about the minimum, but also show a crossoverbetween left and right DW velocities. More recently, anothermodel was advanced by Shahbazi et al. (2019) to explain thepresence of both asymmetry and left-right DW crossover inthe velocity curves. Here the DW depinning field H dep is al-lowed to vary with H x in a way determined from micromag-netic simulations. Notably, this work shows that the veloc-ity minimum underestimates H DMI , as was also found for thesti ff ness model (Pellegren et al. , 2017). Finally, Hartmann et al. (2019) reconsidered the change of the DW sti ff ness dueto deformation as an angular shape, and minimizing the en-ergy of the system with a semi-analytical approach they couldcalculate the velocity profile. Also in this case, contrary to Je et al. (2013), the minimum of the velocity does not occur at H DMI . b. Flow regime According to the 1D model (Thiaville et al. , 2012), thepresence of DMI significantly increases the Walker Break-down field H WB . Indeed, in samples with strong DMI ( D (cid:29) . µ M S t ), H WB is linearly proportional to H DMI : H WB ∼ α H DMI , (2.9)where α is the Gilbert damping (Thiaville et al. , 2012). TheDW velocity at the Walker Breakdown can thus be expressedas: v WB = γ ∆ α H WB ∼ π γ DM S , (2.10)where γ is the gyromagnetic ratio. In contrast to this pre-diction, it is found experimentally and confirmed by 2D mi-cromagnetic simulations that in samples with large DMI theDW velocity does not decrease at fields larger than the H WB ,but instead reaches a plateau (Ajejas et al. , 2017; Pham et al. ,2016). Its origin is the consequence of the complex meander-like structure that the DW takes on at velocities above H WB ,with continuously nucleation and annihilation of pairs of ver-tical Bloch lines (Pham et al. , 2016; Yoshimura et al. , 2016).Measurements of this roughly constant velocity, which cor-responds to v WB , provide a simple way to determine D forsamples with large DMI. Similarly to this technique, in other works (Ajejas et al. ,2017; Pham et al. , 2016; de Souza Chaves et al. , 2019;Vaˇnatka et al. , 2015) measurements of the minimum DW ve-locity as a function of H x in the flow regime have also beenused to quantify D , exactly as done in the creep regime. In-deed, it has been shown that under certain conditions the 1Dmodel provides an expression for the DW velocity that ex-hibits a parabolic dependence on ( H x + H DMI ) (Kim et al. ,2019b).
4. Equilibrium domain pattern
The demagnetized state of PMA materials consists of acomplex domain pattern, usually in the form of labyrinthicstructures of domains pointing either up or down, even forvery thin films. A couple of examples of the typical patternsare shown in Fig. 9 for a multilayer of Pt / Co / Al O , takenfrom Legrand et al. (2018). The exact demagnetized patterndepends on the direction and the history of the applied field:when an out of plane field is applied, a maze domain structureis created (Fig. 9a), while for an in plane field, domains re-main parallel and form a typical stripe structure (Fig. 9b). Inthe latter, the width and the density of domains can be usedto estimate the domain wall energy. As a matter of fact, assuggested by the theory developed for infinite parallel stripedomains (Kooy and Enz, 1960), the domain width is a func-tion of the domain wall energy, which includes magnetostatic,anisotropy, the Zeeman, and the exchange terms. N´eel walls,favored by DMI, have a reduced energy and, consequently,DMI interaction highly a ff ects the equilibrium domain width.For domain widths much larger than the domain wall width,as in general for PMA materials, the magnetostatic contribu-tion comes only from the surfaces charges at the top and bot-tom surfaces, so that DW energy is calculated as: σ DW = A / ∆ + K e f f ∆ − π cos ( Φ ) | D | (2.11)where Φ is the same angle of the internal DW magnetizationdefined in Fig. 3. The minimization of the energy with respectthe domain width ∆ and core angle Φ gives the value of theequilibrium domain width: a critical value D c = is obtained,with D ≤ D c the preferred configuration is a Bloch wall and D > D c it is a N´eel wall.Initially, the method was used for N´eel wall only, with D > D c , and cos ( Φ ) =
1, so that the DW energy density sim-plifies to σ DW = − π | D | . Later, the inclusion of the angle ofthe wall in the equation cos ( Φ ) = et al. , 2017; Meier et al. , 2017). A further improve-ment took into account the dipolar terms coming from the in-ternal structure of the DW or / and from the DW interaction(Lemesh et al. , 2017). This is important in thicker samples,as the dipolar energy makes the internal magnetization varyalong the thickness, even in the presence of DMI. To accountfor all these situations, the model was extended allowing adi ff erent angle Φ for each layer of the sample (Lemesh andBeach, 2018).0The analytical expression from the DW energy σ DW as afunction of the width d of the domain is σ DW µ M s t = d t π ∞ (cid:88) odd n = − (cid:16) + π ntd (cid:17) e − π ntd n (2.12)where t is the thickness. From the experimentally obtaineddomain width one can then calculate the domain wall energy,and estimate the DMI constant using one of the previouslyderived approximations.This general method has been applied in three di ff erentways: comparisons with micromagnetic simulations (Legrand et al. , 2018; Moreau-Luchaire et al. , 2016), analytical estima-tions (Wong et al. , 2018; Woo et al. , 2016, 2017; Yu et al. ,2017), and scaling of the energy of an experimental image(Ba´cani et al. , 2019). (a) (b) Figure 9 MFM domain patterns for an (a) out-of-planeand (b) in-plane demagnetized multilayer composed of[Pt(1) / Co(0.8) / Al O (1)] . From Supplementary Materials ofLegrand et al. , 2018.
5. Magnetic stripe annihilation
Two parallel domains in a thin film with DMI are homochi-ral N´eel walls. Since the type of domain wall is the same,either left-handed or right-handed, the cores of those paral-lel domain walls point in opposite directions. Such a pair ofdomains constitutes a topological structure. This situation isdi ff erent from the case of absence of DMI, where the cores ofthe Bloch walls point in the same direction but not in a fixedorientation. The annihilation of those two parallel N´eel wallsdepends on the strength of the DMI due to the topologicalconfiguration. This fact was confirmed by simulations (Hira-matsu et al. , 2014; Mart´ınez and Alejos, 2014). To annihilatethe walls an out of plane field is applied that reduces the sizeof the domain unfavored by the field. That domain achieves aminimum size until it collapses for a given out-of-plane field.From the determination of the field of annihilation and theminimum domain width the DMI can be extracted when com-pared with the corresponding simulations (Hiramatsu et al. ,2014). An example from (Benitez et al. , 2015) of the annihi-lation field dependence on DMI calculated by means of micro-magnetic simulations can be seen in Fig. 10. More recently,the formula for the minimum width of stripe domains was de-rived (Lemesh et al. , 2017), using the minimization procedure discussed in the previous section. This allowed the extractionof the DMI value without the performance of systematic mi-cromagnetic simulations.The analytical formula for the domain wall energy σ DW ob-tained from the minimum domain width d min is given by: σ DW µ M s t = π ( Ln [1 + ( d min / t ) ] + ( d min / t ) Ln [1 + ( d min / − ]) . (2.13)As in the previous method this allow extracting the value ofthe domain wall energy density which is also a function ofthe DMI constant. Using the same expressions for the energydensity like in the previous section one can estimate its value. Figure 10 Annihilation field of two homochiral walls as a functionof the DMI value calculated from micromagnetic simulations for twodi ff erent anisotropy values. Inset magnetic configuration of the twoN´eel walls squeezed together by the perpendicular field. From Ben-itez et al. , 2015.
6. Nucleation field
To perform domain wall experiments normally one needs tonucleate a reversed domain. While in the case of domain nu-cleation in wires the procedure can be done in di ff erent ways,in the case of thin films the nucleation usually involves theapplication of a perpendicular field opposed to the magnetiza-tion of the film. As stated above, the energy of a domain wallis reduced by the presence of DMI, provided it has the chi-rality favored by it. This modification also a ff ects the energybarrier relevant for nucleation of reversed domains. Pizzini et al. (2014) showed that the nucleation process is a ff ected bythe presence of DMI . They distinguished between nucleationat the edge of the patterned sample and in the center of themagnetic film. They showed that the nucleation of reverseddomains at the edge is dependent on the value of the in-planeapplied field and it is asymmetric with respect the combina-tion of DMI sign and in-plane field direction. The half-dropletmodel (Vogel et al. , 2006), applied in the paper, describes thenucleation of a magnetic domain at the side edge under the ap-plication of a magnetic field. The formula for the out-of-plane1 (a) (c)(d)(b) Figure 11 (a) Theoretical nucleation field as a function of the magnetic field applied perpendicular to the edge of the sample for edge andbubble (film) nucleation as a function of the field. (b) Experimental nucleation field as a function of the magnetic field applied perpendicularto the edge for edge and bubble (film) nucleation. (c) Schematics of the nucleation of bubble domain (red color) showing the orientation of thewall core parallel and anti-parallel to the applied field for in-plane fields below the DMI field. (d) Calculated domain wall energy as a functionof the in-plane field for di ff erent DMI fields. (a) and (b) are from Pizzini et al. , 2014, (c) and (d) from Kim et al. , 2017c. nucleation field for a reversed domain is: H n = πσ t µ M s pk B T (2.14)where σ is the domain wall energy associated with the bub-ble and p is the factor related to the waiting time accordingto τ = τ exp ( p ) with τ is the attempt frequency. The bubbleenergy is a function of the in-plane field and the DMI con-stant. From the best adjustment of their numerical model tothe experimental results, they were able to estimate the DMIconstant in Pt / Co / AlOx.Pizzini et al. (2014) concluded that the bubble energy gainin the middle of the sample in the presence of in-plane fieldswas compensated by the opposite magnetization directionalong the bubble (see the orientation of the magnetization inFig. 11(b)), while this is not true for an incomplete bubble nu-cleate at the edge. For the values used in that article the N´eelchirality is preferred. Therefore, this fact yields a nucleationfield that is independent of the in-plane field value for the edgenucleation. This fact agrees well with their model shown inFig. 11(a) and their measurements shown in Fig. 11(b).More recently, Kim et al. (2017c) realized that such argu-ment does not hold above a critical in-plane field value, whichcorresponds to the DMI field, where the energy of the bubbleis altered due to the fact that the magnetization at the bound-ary of the bubble aligns with the in-plane field. Hence, thepicture Fig. 11(c) is no longer valid. The implication is a re-duced value of the bubble magnetic energy and an associated reduction of the nucleation field as the in-plane field value isincreased beyond the DMI field. This was in agreement withtheir numerical calculations as shown in Fig. 11(d).Therefore, the critical field where there is a change froma constant nucleation field to a field dependent nucleation isa measure of the DMI constant, since it coincides with theDMI field H DMI . Experimentally, this can be determined fromthe dependence of the out-of-plane nucleation field on the in-plane field.
7. Domain wall stray fields
Another method taking into account the static DW structureis the direct measurement of the stray field of the DW. Thestray field has a typical profile along the axis perpendicular tothe DW including information on the angle φ and the strengthand sign of D (see Fig. 12). This method is limited up to lowvalues of D ≤ D c (e.g. in CoFeB wires D c ≈ . mJ / m ), i.e.when a significant angle is present. Since DW widths are ofthe order of 5-10 nm the stray field profile determination re-quires an excellent spatial resolution, which was experimen-tally tackled by nitrogen vacancy (NV) magnetometry (Gross et al. , 2016).2 Figure 12 Schematic of the stray field measurement using a NVmagnetometer. The internal DW magnetization angle ψ ( Φ , in ournotation) depends on the DMI constant, resulting in di ff erent profilesof the stray field component B ψ x ( x ). From Gross et al. , 2016. C. Experimental results
1. Current-driven domain wall motion
Measurements of current-driven DW motion are performedin wires with typical widths ranging between 1 µ m and 5 µ mand lengths of a few tens of µ m. The magnetization in the wireis initially saturated by applying an out-of-plane field H z . H z is then removed and a voltage pulse (with duration of a fewtens of ns) is applied either directly to the wire or to an Oerstedline patterned on top of the wire. In either case the voltagepulse nucleates a reversed magnetic domain and a DW is thusinjected into the wire.For measurements of DW velocities, current pulses are ap-plied directly to the wire to move a DW, while the DW posi-tion along the wire is imaged by magneto-optical Kerr e ff ect(MOKE) microscopy in polar configuration. The number andduration of the applied current pulses is chosen to obtain asignificant displacement of the DW, at least over a few µ m.Typical values of current densities fall in the range between10 A / m and 10 A / m . The velocity of the DW is calcu-lated as the ratio between the current-induced DW displace-ment and the total duration of the pulses. Finally, velocitiesare measured for both ↑↓ and ↓↑ DWs under di ff erent strengthof H x , keeping the current density fixed, in order to determinethe “compensating” field H ∗ x .For measurements of DW depinning e ffi ciencies, a constantcurrent J is applied continuously to the wire for a given in-plane field H x , while the out-of-plane field H z is ramped upuntil DW motion is detected by MOKE. The DW depinning e ffi ciency (cid:15) for a given H x is then determined from the slope ofthe depinning field H z , dep as a function of J . Modest currentvalues are used, up to about 10 A / m , in order to excludeJoule heating e ff ects. Finally, (cid:15) is measured for both ↑↓ and ↓↑ DWs under di ff erent strength of H x , and H DMI is determinedas the “compensating” field H ∗ x for which (cid:15) = H x have been used to extract H DMI inseveral material stacks, with Co / Ni / Co, Co Fe B or Coas ferromagnetic films, di ff erent heavy metals as underlayersand either a heavy metal or an oxide as overlayers. Tables I,II and III present a summary of the DMI values measured forCo / Ni / Co, CoFeB and Co, respectively.For Co / Ni / Co films the DMI seems to increase upon in-creasing the thickness of the Pt underlayer (Ryu et al. , 2013),while it is very low when using Pd or Ir as underlayers (Ryu et al. , 2014).For CoFeB films with Ta and MgO as underlayer and over-layer, respectively, results are in disagreement not only in themagnitude of the DMI but also in its sign (Karnad et al. ,2018; Lo Conte et al. , 2015; Torrejon et al. , 2014). Furher-more, Karnad et al. (2018) find di ff erent signs of DMI forTa / CoFeB / MgO depending on whether they measure DW mo-tion driven by current or field (see table V for the field-drivencase). Among the underlayers used with CoFeB, W providesthe highest (positive) DMI (Torrejon et al. , 2014).Finally, for Co films, Pt is the only underlayer investigated,while overlayers are either oxides or metals. In the symmetricstructure Pt / Co / Pt, either negative (Franken et al. , 2014) orvanishing (Kim et al. , 2018a) DMI values are measured. Forall the other stacks the DMI is always negative and largest inmagnitude when using Ti as overlayer (Kim et al. , 2018a).
2. Field-driven domain wall motion
Experiments of field-driven DW dynamics are mostly per-formed in continuous films where magnetization reversal pro-ceeds by nucleation and growth of magnetic bubble domains.Only few works report studies of flat DW dynamics in con-tinuous films (Kim et al. , 2017a; Pellegren et al. , 2017) or in µ m–wide wires (Ju´e et al. , 2016a; Kim et al. , 2018a). In anycase, the measurement technique is identical. The magnetiza-tion in the continuous film / wire is initially saturated by apply-ing a perpendicular field H z . A bubble DW (or a flat DW) isthen nucleated by applying a short H z pulse in the opposite di-rection, through either a coil or an electromagnet. The bubbleDW (or the flat DW) is expanded under simultaneous applica-tion of continuous H x (from an electromagnet) and pulsed orcontinuous H z (from a coil or an electromagnet). The initialand final positions of the DW are imaged through MOKE mi-croscopy in polar configuration. Typical bubble growths are inthe range of at least a few tens of µ m. The velocity of the DWis measured along the direction of the applied H x (typicallyalong the left–right direction in a magnetic bubble) and is cal-culated as the ratio between the DW displacement and the total3 FM Bottom NM Top NM | H DMI | D D s Ref (nm) (nm) (nm) (mT) (mJ / m ) (pJ / m)Co(0.3) / Ni(0.7) / Co(0.15) Pt(1) TaN(5) n.a. n.a. Ryu et al. , 2013Pt(1.5) n.a. n.a.Pt(3) n.a. n.a.Pt(5) n.a. n.a.Co(0.3) / Ni(0.7) / Co(0.15) Pd(5) TaN(5) 12 n.a. n.a.Ir(3) 18 n.a. n.a. Ryu et al. , 2014Pt(5) 140 n.a. n.a.Table I Overview of DMI measurements for Co / Ni / Co thin films via current-induced domain wall motion experiments. FM and NM standfor ferromagnetic and non-magnetic layer, respectively. H DMI is the DMI field, D is the interfacial DMI constant and D s = D · t FM , with t FM being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were eitherextracted from figures or calculated using the parameters provided. FM Bottom NM Top NM | H DMI | D D s Ref (nm) (nm) (nm) (mT) (mJ / m ) (pJ / m)Co Fe B (1) Hf(2.6–6) MgO(2) -0.38 – -0.05 -0.38 – -0.05 Torrejon et al. , 2014 a Ta(0.5–1.3) -0.08 – 0.07 -0.08 – 0.07
TaN(0.4–6.6)
W(2.1–3.6) Co Fe B (1) Ta(5) MgO(2) 7.8 0.06 Lo Conte et al. , 2015Co Fe B (0.8) Ta(5) MgO(2) -0.03 -0.02 Karnad et al. , 2018 a The highest and lowest DMI values reported do not necessarily correspond to the extremes of the bottom layer thickness range (for instance the highest DMIvalue could occur at mid thickness range).
Table II Overview of DMI measurements for
CoFeB thin films via current-induced domain wall motion experiments. FM and NM standfor ferromagnetic and non-magnetic layer, respectively. H DMI is the DMI field, D is the interfacial DMI constant and D s = D · t FM , with t FM being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were eitherextracted from figures or calculated using the parameters provided.Figure 13 Evolution of bubble domain shapes in an applied in-plane magnetic field. Co / Ni multilayers measured by perpendicular Kerr e ff ect,from Lau et al. , 2016. time during which H z is applied – whether in pulses or contin-uous. Finally, velocities are measured for both ↑↓ and ↓↑ DWs(i.e. left and right DWs in a bubble) under di ff erent strengthof H x , upon keeping H z constant. In this way velocity vs H x curves are constructed for both DWs and H DMI is determinedeither through fitting with one of the modified creep formulapreviously discussed (II.B.3), or simply by finding the field atwhich the velocity is minimum.Regarding the strength of the applied fields, values di ff ergreatly depending on the material and DW motion regime in-vestigated. For measurements in the creep regime only modest H z are needed – usually of a few mT depending on the depin-ning field – while H z of hundreds of mT are used to drive DWs in the flow regime. The maximum values of applied H x depend instead on the strength of the DMI. In sampleswith large DMI in-plane fields up to 350 mT have been used(Cao et al. , 2018a; Hrabec et al. , 2014; Lavrijsen et al. , 2015).However, since the use of in-plane fields can be accompaniedby artefacts due to misalignments and / or crosstalk betweenperpendicular and in-plane electromagnets, measuring largeDMI values can be problematic. It is worth mentioning thata simple scheme has been proposed to overcome the need forhigh H x values (Kim et al. , 2017a), whereby the DW velocityis measured at an angle θ > θ ), and becomes measurable also for samples4 FM Bottom NM Top NM | H DMI | D D s Ref (nm) (nm) (nm) (mT) (mJ / m ) (pJ / m)Co(0.36) Pt(4) Pt(1) 37 -0.24 -0.09 Franken et al. , 2014Co(0.36) Pt(4) Pt(2) 12.5 -0.10 -0.04
Co(0.5) Pt(4) Pt(2) 11 -0.09 -0.04
Co(0.5) Pt(2) Pt(4) 3 -0.03 -0.02
Co(0.8) Pt(4) AlO x (1.9) (cid:29)
40 n.a. n.a.Co(1) Pt(5) Gd(2) 280 -1.00 -1.00 Vaˇnatka et al. , 2015Co(0.93) Pt(4) AlO x (2) 99 -0.54 -0.50 Co(1.31)
54 -0.48 -0.63
Lo Conte et al. , 2017Co(1.37)
48 -0.47 -0.64
Co(0.9) Pt(2.5) Al(2.5) 107 -0.87 -0.78
Kim et al. , 2018aCo(0.9) Ti(2.5) -1.42 -1.27
Co(0.9) W(2.5) -1.35 -1.22
Co(0.5) Pt(1.5) 0 0 0Table III Overview of DMI measurements for Co thin films via current-induced domain wall motion experiments. FM and NM stand forferromagnetic and non-magnetic layer, respectively. H DMI is the DMI field, D is the interfacial DMI constant and D s = D · t FM , with t FM beingthe thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extractedfrom figures or calculated using the parameters provided. with large DMI by applying moderate in-plane fields.Field-driven DW dynamics in the creep regimes have beeninvestigated to extract H DMI in material systems with Co / Nimultilayers, CoFeB or Co as ferromagnetic layers, and di ff er-ent combinations of heavy metals and oxides as underlayersand overlayers. Regarding the flow regime, instead, H DMI hasbeen measured only for Co films. Tables II.C.2, V and VIpresent a summary of DMI measurements in the creep regimefor Co / Ni, CoFeB and Co, respectively, while Table VII con-cerns the flow regime. In the following we highlight some ofthe most interesting findings and observations of these exper-imental studies.For Co / Ni multilayers (see Tab. II.C.2), three works by thesame group (Lau et al. , 2018, 2016; Pellegren et al. , 2017)have reported on the asymmetry of DW velocity curves abouttheir minima and on the crossover between DW velocitiesfor left and right sides of magnetic bubbles, correspondingto a morphological change from flattened to teardrop bubbleshapes (Fig. 13). As previously mentioned, these velocitycurves have been fitted through a model that takes into ac-count both DW sti ff ness and a field-dependent prefactor v ,showing that measurements of the minimum in velocity canbe misleading to quantify the DMI (Lau et al. , 2018).Regarding CoFeB films (see Tab. V), both the respectiveCo and Fe compositions (Jaiswal et al. , 2017) and the post-growth annealing temperature (Khan et al. , 2016) have beenshown to play a crucial role in the magnitude of DMI. Fur-thermore, Soucaille et al. (2016) reported large discrepanciesbetween measurements of DMI performed through DW mo-tion and Brillouin Light Scattering (BLS), particularly for sys-tems with low DMI ( D (cid:46) / m ). Also worth mentioningis a recent study by Diez et al. (2019b) which showed thatthe DMI can be tuned in a Ta / CoFeB / MgO system throughlight He + irradiation, due to an increasing interface intermix-ing mostly between Ta and CoFeB layers.Co is the most widely studied material, specifically in com- bination with Pt as underlayer, which can provide high DMImagnitudes depending on the overlayer choice. DMI mea-surements performed in the creep regime (see Tab. VI) indi-cate that the nominally symmetric stack Pt / Co / Pt can havepositive values of D s as high as 0.58 pJ / m (Hrabec et al. ,2014), although the same work shows that upon ensuring epi-taxial growth, D s reduces to almost 0. An even larger valuevalue of 1.20 pJ / m was calculated by Hartmann et al. (2019).Two more studies report a vanishing DMI (Ajejas et al. , 2017;Pham et al. , 2016) for Pt / Co / Pt layers measured in the flowregime (see Tab. VII), while a small negative D s is found inShahbazi et al. (2018).Another system widely investigated in the literature is thePt / Co / Ir stack, which also gives rise to some controversial re-sults. First principle calculations predict opposite DMI signsfor the Pt / Co and Ir / Co interfaces, which would result in anadditive e ff ect for the Pt / Co / Ir stack (Yamamoto et al. , 2016;Yang et al. , 2015). However, Hrabec et al. (2014) show that D decreases when introducing a thin Ir top layer in a Pt / Co / Ptstack, and even changes sign upon increasing the Ir thick-ness. Similarly, Shahbazi et al. (2019) find that the magni-tude of D is smaller in Pt / Co / Ir / Ta than in Pt / Co / Ta alone. Onthe other hand, a huge increase of DMI has been observedin Pt / Co / MgO layers, upon insertion of a very thin Mg filmbetween Co and MgO (Cao et al. , 2018a), providing amongthe highest values of D s = .
32 reported in the literature.Another aspect worth mentioning is the influence of growthconditions on DMI, which has been extensively investigatedin Lavrijsen et al. (2015) and Wells et al. (2017). Both thesestudies report on a dramatic variation of the DW velocity de-pendence with in-plane field, observed upon changing sputter-deposition conditions, namely Ar gas pressure, substrate tem-perature or chamber base pressure. Indeed, it is speculatedthat such di ff erent growth conditions may lead to di ff erent de-grees of interfacial intermixing and / or quality, resulting in ahuge range of measured DMI values, which can even change5sign (Wells et al. , 2017). Ar + ion irradiation has also beendemonstrated to be an e ff ective way to tune the sign of DMIin Pt / Co / Pt films (Balk et al. , 2017).
3. Equilibrium domain pattern
As explained above, in the case of equilibrium domainpattern method, the DMI constant is estimated by measur-ing magnetic domain widths from nanometric magnetic mi-croscopy images. The values and systems analyzed using thismethod are presented in Table X for Co and FeCo and inTable VIII for CoFeB thin films. As the uncertainty of themeasurement depends heavily on the precision of the imagingtechniques, submicrometer range resolution is indispensable.The most commonly used experiments are: Scanning Trans-mission X-ray Microscopy (STXM) (Lemesh et al. , 2018;Moreau-Luchaire et al. , 2016), Magnetic Transmission X-ray Microscopy (MTXM) (Woo et al. , 2016, 2017), Magneticforce microscopy (MFM) (Agrawal et al. , 2019; Ba´cani et al. ,2019; B¨uttner et al. , 2017; Casiraghi et al. , 2019; Davydenko et al. , 2019; Dugato et al. , 2019; Legrand et al. , 2018; Schlot-ter et al. , 2018; Soumyanarayanan et al. , 2017), and or in a fewcases MOKE (Wong et al. , 2018; Yu et al. , 2017). Most of thematerials analyzed with this method have large DMI valuesbecause they are optimized to measure skyrmions (Moreau-Luchaire et al. , 2016; Soumyanarayanan et al. , 2017; Woo et al. , 2016). For example, Soumyanarayanan et al. (2017) re-ported that the replacement of the Co layer by a layer of FeCoincreases substantially the value of D, up to a maximum of D = . mJ / m which is compatible with hosting skyrmions.In these cases, the magnetic imaging for skyrmions can beused to measure the domain width. Many of the films anal-ysed are multilayers because the dipolar contribution reducesthe size of stable skyrmions (Moreau-Luchaire et al. , 2016).For instance, Legrand et al. (2018) studied the variations ofthe core of the magnetic structure along the thickness asso-ciated to the dipolar field in Co multilayers. With the sametechnique Pd / Co / Pd samples were also analyzed (Davydenko et al. , 2019; Dugato et al. , 2019). The origin of the DMI insuch symmetric trilayers is a di ff erent residual stress in thetop and bottom of the Co / Pd interfaces due to di ff erent latticematching (Davydenko et al. , 2019). Dugato et al. (2019) wereable to optimize the stack with the insertion of W between Coand Pd, resulting in an increase of D, with an optimal size fora W thickness of 0.2 nm and D = . mJ / m .Wedges of Ni on top a Fe layer with a Cu(001) substratewere analyzed with Threshold Photo-emission Magnetic Cir-cular Dichroism with PhotoEmission Electron Microscopy(TP-MCD-PEEM) by Meier et al. (2017), as shown in Ta-ble VIII. In these samples, the DMI is originated by the inter-faces of Ni and Fe, instead of the non magnetic material, andincreases when capped with a Pt layer. Using this method, thedependence of DMI on temperature was also measured forPt / Co / Cu multilayers up to the value of 500K (Schlotter et al. ,2018). It was found that DMI has a stronger dependence on temperature than other magnetic properties, like magnetocrys-talline anisotropy.
4. Magnetic stripe annihilation
With this method, the first step is to nucleate two domainwalls in a perpendicular magnetized sample. The two wallsare then manipulated with an out-of-plane field to minimizetheir distance until the two domains collapse. This annihila-tion field and the minimum width depend on D, those valuesare shown in Table IX. As for the equilibrium domain patterntechnique (Sec. II.C.3), this method is based on imaging andmake use of same set of measurements with the addition ofan out of plane field, requiring nanometric resolution. For thispurpose, MTXM (Jaiswal et al. , 2017; Litzius et al. , 2017),STXM (Woo et al. , 2016, 2017), where the two methods arecombined, and MOKE (Yu et al. , 2016a) have been used. Allthe samples are magnetic thin films or wide patterned tracks,to allow high number of domains and the manipulation of thewalls. In the original study (Benitez et al. , 2015), the domainwall were nucleated parallel and analysed using a combina-tion of Lorentz transmission electron microscopy (L-TEM)and polar Kerr. In later studies (Jaiswal et al. , 2017; Litz-ius et al. , 2017; Woo et al. , 2016, 2017; Yu et al. , 2016a), theannihilation field was studied independently of the shape andboundary of domains.
5. Nucleation field
A reversed domain is nucleated in a perpendicular magne-tized material and the out-of-plane nucleation field as a func-tion of the in-plane applied field is analyzed. This type ofmeasurements can be divided in two groups: edge nucleationin patterned wires (Pizzini et al. , 2014) or asymmetric mi-crostructures (e.g. triangles) (Han et al. , 2016) and bubblenucleation in extended films (Kim et al. , 2017c, 2018c). Inall the experiments, the microscopy images were obtained us-ing MOKE. The reversed domain can be enlarged by a out-of-plane field to increase the limit of the resolution of theimaging technique (Pizzini et al. , 2014), without altering thenucleation field. In Han et al. (2016), the hysteresis loopof asymmetric microstructures measured by wide-field polarKerr shows an asymmetry due to DMI similar to exchangebiasing. This asymmetry is attributed to the asymmetric nu-cleation when an in-plane field is present (applying a half-droplet model) and is independent of the structure size. Thevalues obtained for D are presented in Table XI, mostly sin-gle layers of Co (Han et al. , 2016; Kim et al. , 2018c; Pizzini et al. , 2014) or trilayers of Co / Ni / Co (Kim et al. , 2017c), Inparticular, this method was used to obtain the value of D as afunction of temperature in a Co layer (Kim et al. , 2018c).6
FM Bottom NM Top NM | H DMI | D D s Ref (nm) (nm) (nm) (mT) (mJ / m ) (pJ / m)Co(0.1) / [Ni(0.1) / Co(0.1)] Pt(4) MgO(2) 15.59 -0.05 -0.05 Yu et al. , 2016bCu(2) 19.23 -0.12 -0.11 Pt(2) 34.88 0.20
Ta(2) 103.86 -0.39 -0.35 [Co(0.2) / Ni(0.6)] / Co(0.2) Pt(2.5) Ta(0.5) / TaN(3) 60 -0.21 -0.37
Lau et al. , 2016[Co(0.2) / Ni(0.6)] / Co(0.2) Pt(2.5) Ta(0.5) / TaN(6) 106 -0.37 -0.66
Pellegren et al. , 2017[Co(0.2) / Ni(0.6)] / Co(0.2) Pt(1.2) Ta(0.8) / TaN(6) -0.52 -0.94
Lau et al. , 2018Ir(1.2) Ta(0.8) / TaN(6) -0.07 -0.12
Pt(2.5) Ir(2.5) -0.31 -0.56
Ir(2.5) Pt(2.5) 0.21
Table IV Overview of DMI measurements for Co / Ni multilayers via field-induced domain wall motion experiments in the creep regime.FM and NM stand for ferromagnetic and non-magnetic layer, respectively. H DMI is the DMI field, D is the interfacial DMI constant and D s = D · t FM , with t FM being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbersin italics were either extracted from figures or calculated using the parameters provided. FM Bottom NM Top NM | H DMI | D D s Ref (nm) (nm) (nm) (mT) (mJ / m ) (pJ / m)Co Fe B (0.8) Ta(5) MgO(2) a a a Khan et al. , 2016CoFeB(1) W(2) MgO(2) 35 0.23
Soucaille et al. , 2016W(3) 15 0.12
TaN(1) 5 0.05
Hf(1) 2 0.01 Co Fe B (0.6) W(5) MgO(2) 93 0.68 Jaiswal et al. , 2017Co Fe B (0.6) 0.03 Co Fe B (0.8) Ta(5) MgO(2) 8.8 0.03 Karnad et al. , 2018Co Fe B (1) Ta(5) MgO(2) 2.6 – 16 b b b Diez et al. , 2019b a Di ff erent DMI values correspond to di ff erent annealing temperatures. b Di ff erent DMI values correspond to di ff erent doses of He + ion irradiation. Table V Overview of DMI measurements for
CoFeB thin films via field-induced domain wall motion experiments in the creep regime.FM and NM stand for ferromagnetic and non-magnetic layer, respectively. H DMI is the DMI field, D is the interfacial DMI constant and D s = D · t FM , with t FM being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbersin italics were either extracted from figures or calculated using the parameters provided.
6. Domain wall stray fields
In Nitrogen Vacancy magnetometry, the D value is obtainedby measuring the magnetic stray field generated by a 180 o Bloch wall (in PMA materials) (Gross et al. , 2016; Tetienne et al. , 2015). As explained, the D values are here limited to D < D c , where D c = µ M s tln /π the critical D value abovewhich formation of Ne´el walls occurs ( t is the film thickness).Therefore only samples with relatively small D can be mea-sured since D c is typically of the range of 0.2 mJ / m (seeTab. XII. For samples with larger DMI, where not fully ori-ented Ne´el walls are present, only a lower limit of D can begiven together with its sign (Tetienne et al. , 2015). The NVmagnetometer measures actually the Zeeman shift in presenceof a small magnetic field in the electronic spin sublevels of aNV defect in a diamond crystal. The diamond nanocrystal isplaced on the tip of an AFM (atomic force microscope) andscanned across the DW at a distance of about d = B NV ). Bloch and Ne´el walls can be easily distinguished by the Zeeman shift profileperpendicular to the wall. Since the stray field depends onseveral parameters as d , M s t and the DW width ∆ = (cid:112) A / K e f f any error in these values reflects in an uncertainty of D . Themain source for uncertainty is considered A , the exchangesti ff ness, but also inhomogeneities in M s or thickness varia-tions contribute to the D uncertainty. D. Advantages and limitations
In general, methods that are based on domain walls requirean accurate estimation of several magnetic parameters in or-der to evaluate the interfacial DMI (see. Eq. 2.1). Amongthese, the exchange sti ff ness A is notoriously di ffi cult to mea-sure, as for instance in CoFeB, A ≈ − / m (Yamanouchi et al. , 2011)), adding an extra uncertainty in the quantifica-tion of D . Han et al. (2016) pointed out that in the nucleationmethod a variation of A from 5 to 15 pJ / m for Co results intoa 25% di ff erence for D obtained using the intermediate value10 pJ / m. This is not the case for methods based on spin waves7 FM Bottom NM Top NM | H DMI | D D s Ref (nm) (nm) (nm) (mT) (mJ / m ) (pJ / m)Co(0.3) Pt(2.5) Pt(1.5) 26.5 Je et al. , 2013Co(0.7) Pt(5) Pt(3)
104 0.83 0.58
Hrabec et al. , 2014Ir(0.23) / Pt(3)
10 0.08 0.05
Ir(0.69) / Pt(3)
155 -1.23 -0.84
Co(0.4) Pt(2) Pt(2) 83
Kim et al. , 2015aPt(2) Pd(2) >
200 n.a. n.a.Co(0.8) Pt(10) Pt(3) 50 n.a. n.a. Petit et al. , 2015Co(0.6) Pt(3) AlO x (1.6) 138 n.a. n.a. Kim et al. , 2017aMgO(2) 483 n.a. n.a.Co(0.54) Au(4) NiO(10) et al. , 2018Co(0.56) Pt(4) Ir(5)
53 -0.31 -0.17
Shepley et al. , 2018Co(1.05)
Co(1) Pt(3) MgO(0.65) / Pt(5)
Cao et al. , 2018aMg(0.2) / MgO(1.5-2) / Pt(5)
Co(1.8) Pt(5) W(1) / Pt(1) 25 0.19
Lin et al. , 2018Co(0.6) Pt(3) Pt(3) -0.07 -0.04
Shahbazi et al. , 2018Pt Au (3) -0.35 -0.21 Au(3) -1 -0.6 Co(0.8) Pt(2.2) Ta(4)
140 -1.12 -0.9
Shahbazi et al. , 2019 b Ir(0.2 – 2) / Ta(4) a -0.49 – -0.93 a -0.39 – -0.74 a Co(0.6) Ta(4) / Pt(4) Pt(4) 170
Hartmann et al. , 2019Co(0.8–1.2) Gd(3) / Pt(2) 217–77
Co(0.8–1) Ir(4) 156–92 a The highest and lowest DMI values reported do not necessarily correspond to the extremes of the Ir thickness range. b For the Pt / Co / Ta system, the value provided in the table is obtained considering H DMI to be the minimum in the velocity curves. However the work estimatesthe DMI also using a model that assumes H dep = H dep ( H x ). In this case D = -2 mJ / m and D s = -1.6 pJ / m. Table VI Overview of DMI measurements for Co thin films via field-induced domain wall motion experiments in the creep regime. FM andNM stand for ferromagnetic and non-magnetic layer, respectively. H DMI is the DMI field, D is the interfacial DMI constant and D s = D · t FM ,with t FM being the thickness of the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics wereeither extracted from figures or calculated using the parameters provided. FM Bottom NM Top NM | H DMI | D D D s1 D s2 Ref (nm) (nm) (nm) (mT) (mJ / m ) (mJ / m ) (pJ / m) (pJ / m)Co(1) Pt(5) Gd(5) 180 -1.48 n.a. -1.48 n.a. Vaˇnatka et al. , 2015Co(0.8) Pt(4) AlO x (3) 220 -1.63 / -1.91 a n.a. -1.3 / -1.53 a n.a. Pham et al. , 2016Co(1) GdOx(4) 200 -1.48 / -1.73 a n.a. -1.48 / -1.73 a n.a.Co(1) Gd(3) 300 -1.52 / -1.78 a n.a. -1.52 / -1.78 a n.a.Co(1) Pt(4) 0 0 n.a. 0 n.a.Co(0.6) Pt(2) Al(2) 200 / -1.52 / -2.02 b -1.38 / -2.2 b -0.91 / -1.21 b -0.83 / -1.32 b Ajejas et al. , 2017Pt(2) Ir(2) 96 / -0.5 / -0.92 b -0.37 / -0.67 b -0.3 / -0.55 b -0.22 / -0.4 b Pt(2) Cu(2) 200 -0.93 -1.03 -0.56 -0.62Pt(2) Pt(2) 0 0 0 0 0Ir(2) Pt(2) 106 x (1-3) n.a. n.a. -1 – -1.79 c n.a. -0.8 – -1.43 c de Souza Chaves et al. , 2019Co(1) Pt(4) GdOx(1-3) 150 – > c -0.6 – -1.42 c -0.62 – -1.34 c -0.6 – -1.42 c -0.62 – -1.34 ca The two values of DMI reported derive from two di ff erent values for the exchange constant A . b The two values of DMI reported are due to di ff erent growth temperatures for the Co layer, either room temperature or 100 ◦ C. c The highest and lowest DMI values reported do not necessarily correspond to the extremes of the top NM thickness range.
Table VII Overview of DMI measurements for Co thin films via field-induced domain wall motion experiments in the flow regime. FM andNM stand for ferromagnetic and non-magnetic layer, respectively. H DMI is the DMI field, D is the interfacial DMI constant and D s = D · t FM ,with t FM being the thickness of the ferromagnetic film. The subscripts 1 and 2 indicate values of D and D s extracted from H DMI and v sat ,respectively. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculatedusing the parameters provided. FM Bottom NM Top NM | D | | D s | Ref (nm) (nm) (nm) (mJ / m ) (pJ / m)[Pt(3) / Co Fe B (0.8) / MgO(1.5) x Ta(3) Ta(2) 1.66
Woo et al. , 2017[Pt(2.7) / Co Fe B (0.8) / MgO(1.5)] x Ta(2.3) / Pt(3.7) 1.5
B¨uttner et al. , 2017Co Fe B (1.2) Ta(5) TaO x (5) 0.17 Yu et al. , 2017Ni(6–12 ML) / Fe(1–3 ML) Cu(001) Pt(0.4) 0.28 ± et al. , 2017Ni(9 ML) / Fe(1-3 ML) / Ni(4-14 ML) 0Fe(1 ML) / Ni(6-12 ML) 0.38 ± / Ni(6-12 ML) 0.6 ± Fe B (1.2) MgO(1) Ta(5) 0.65 ± Wong et al. , 2018[Pt(2.7) / CoFeB(0.86)MgO(1.5)] x Ta(3.6) / Pt(1) Pt(2.7) 1.76
Lemesh et al. , 2018[MgO(2) / Co Fe B (1) / Ta(5)] x Ta(5) 0.08 ± ± et al. , 2019[MgO(2) / Co Fe B (0.6) / W(5)] x Ta(5) 0.61 ± ± / Co Fe B (0.8) / Pt(3.4)] x Ta(5.7) Ta(5) 1.0 ± ± / Co Fe B (0.8) / MgO(1.5)] x Ta(3) Ta(2) 1.6 ± Agrawal et al. , 2019Table VIII Overview of DMI measurements for
CoFeB and Ni / Fe thin films via equilibrium domain pattern . FM and NM stand forferromagnetic and non-magnetic layer, respectively. D is the interfacial DMI constant and D s = D · t FM , with t FM being the thickness of theferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures orcalculated using the parameters provided. FM Bottom NM Top NM | D | | D s | Ref (nm) (nm) (nm) (mJ / m ) (pJ / m)Co(0.8) Pt(3) AlO x (3) 0.33 ± a Benitez et al. , 2015[Pt(3) / Co(0.9) / Ta(4)] x Ta(3) 1.1 ± Woo et al. , 2016Co Fe B (1) Ta(5) Ta(0.74-0.9) 0.25 Yu et al. , 2016aCo Fe B (0.6) W(5) MgO(2) 0.73 ± Jaiswal et al. , 2017[Pt(3) / Co Fe B (0.8) / MgO(1.5)] x Ta(3) Ta(2) 1.35
Woo et al. , 2017[Pt(3.2) / CoFeB(0.7) / MgO(1.4)] x Ta(3) 1.35 ± Litzius et al. , 2017 a Lower limit
Table IX Overview of DMI measurements of
Co and CoFeB thin films via magnetic stripe annihilation . FM and NM stand for ferromagneticand non-magnetic layer, respectively. D is the interfacial DMI constant and D s = D · t FM , with t FM being the thickness of the ferromagneticfilm. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated usingthe parameters provided. FM Bottom NM Top NM D D s Ref (nm) (nm) (nm) (mJ / m ) (pJ / m)Co(0.6) / Pt(1) / [Ir(1) / Co(0.6) / Pt(1)] x Pt(10) Pt(3) 1.6 ± Moreau-Luchaire et al. , 2016Co(0.6) / Pt(1) / [Co(0.6) / Pt(1)] x Pt(10) Pt(3) 0.2 ± [Pt(3) / Co(0.9) / Ta(4)] x Ta(3) 1.5 ± Woo et al. , 2016[Ir(1) / Co(0.6) / Pt(1)] x Ta(3) / Pt(10) Pt(2)
Soumyanarayanan et al. , 2017[Ir(1) / Fe(0.2) / Co(0.6) / Pt(1)] x [Ir(1) / Fe(0.3) / Co(0.6) / Pt(1)] x [Ir(1) / Fe(0.2) / Co(0.5) / Pt(1)] x [Ir(1) / Fe(0.4 / Co(0.4) / Pt(1)] x [Ir(1) / Fe(0.4) / Co(0.6) / Pt(1)] x [Ir(1) / Fe(0.5) / Co(0.5) / Pt(1)] x [Ir(1) / Fe(0.6) / Co(0.6)) / Pt(1)] x [Ir(1) / Co(0.6)Pt(1)] x Pt(10) Pt(3) 2.30
Legrand et al. , 2018 a [Ir(1) / Co(0.8) / Pt(1)] x Pt(10) Pt(3) 2.00 [Co(0.8) / Ir(1) / Pt(1)] x Pt(11) Pt(3) -1.37 -1.1 [Co(0.8) / Ir(1) / Pt(1)] x Ta(5) / Pt(10) Pt(3) -1.63 -1.3 [Co(0.8) / Ir(1) / Pt(1)] x Pt(11) Pt(3) -1.52 -1.22 [Co(0.8) / Ir(1) / Pt(1)] x Ta(5) / Pt(10) Pt(3) -2.06 -1.65
Co(0.8) / [Pt(1) / Ir(1) / Co(0.8)] x Ta(15) Pt(3) 1.14 [Pt(1) / Co(0.6) / Al O (1)] x Ta(10) / Pt(7) Pt(3) -1.29 -0.77 [Pt(1) / Co(0.8) / Al O (1)] x Ta(10) / Pt(7) Pt(3) -1.01 -0.81 [Al O (1) / Co(0.6) / Pt(1)] x Ta(10) Pt(7) 1.94 [Al O (1) / Co(0.8) / Pt(1)] x Ta(10) Pt(7) 1.69 [Pt(2) / Co(1.1) / Cu(1)] x Ta(3) Pt(2)
Schlotter et al. , 2018
Co(0.6) / Pt(1) / [Ir(1) / Co(0.6) / Pt(1)] x Pt(10) Pt(3) 1.97 ± Ba´cani et al. , 2019[Pt(2.5-7.5) / Co(0.8) / Pt(1.5)] x Ta(3) Ta(2) 0 ± et al. , 2019[Co(0.8) / Pd(2)] x Cu(2) / Pd(3) Pd(3) -1.6 ± -1.28 Davydenko et al. , 2019 b [Co(0.8) / Pd(2)] x -1.85 ± -1.48 [Co(0.8) / Pd(2)] x -2.3 ± -1.84 [Pd(1) / Co(0.5) / Pd(1)] x ± Dugato et al. , 2019[Pd(1) / Co(0.5) / W(0.1) / Pd(1)] x ± [Pd(1) / Co(0.5) / W(0.2) / Pd(1)] x ± [Pd(1) / Co(0.5) / W(0.3) / Pd(1)] x ± [Pd(1) / Co(0.5) / W(1) / Pd(1)] x ± a The sign was measured using CD-XRMS (circular dichroism in x-ray resonant magnetic scattering). Sign convention used here is opposite to the used in thearticle. b Values assuming A =
20 pJ / m. Other values are also assumed in the article. The negatives sign of DMI is inferred in combination with other experiments. Table X Overview of DMI measurements for
Co and FeCo thin films via equilibrium domain pattern . FM and NM stand for ferromagneticand non-magnetic layer, respectively. D is the interfacial DMI constant and D s = D · t FM , with t FM being the thickness of the ferromagneticfilm. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated usingthe parameters provided. The method does not provide the sign. All the D values are absolute values unless noted. FM Bottom NM Top NM | H DMI | D D s Nucleation type Ref (nm) (nm) (nm) (mT) (mJ / m ) (pJ / m)Co(0.6) Pt(3) AlOx(2) -2.2 ∗ -1.32 ∗ Edge Pizzini et al. , 2014Co(1.15) AlO x (2.5) Pt(4) 1.43 ∗ ± ∗ ± et al. , 2016Co(1.2) Pt(4) Ir(4) -1.69 ∗ ± ∗ ± / Ni(0.6) / Co(0.3) Pt(2) MgO(1) 228 ±
60 0.45 ± Bubble Kim et al. , 2017cCo(0.5) Pt(2) MgO(2) 372 ± Bubble (100K) Kim et al. , 2018c324 ± Bubble (150K)245 ± Bubble (200K)166 ± Bubble (300K)Table XI Overview of DMI measurements of
Co and Co / Ni thin films via nucleation field dependence. FM and NM stand for ferromagneticand non-magnetic layer, respectively. H DMI is the DMI field, D is the interfacial DMI constant and D s = D · t FM , with t FM being the thicknessof the ferromagnetic film. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figuresor calculated using the parameters provided. Edge nucleation provides D sign, while bubble nucleation provides only the magnitude. Valueswith ∗ are according to the convention used in this paper and opposite to that in the original manuscript. A is not needed to extract the interfacialDMI. Furthermore, it is worth mentioning that methods thatrely on domain walls can be applied to measure D only in per-pendicularly magnetized materials, while methods based onspin waves (see Sec. III) are able to quantify D also for sys-tems with in-plane magnetization. Below we review the mainstrengths and weaknesses of all methods discussed in this Sec-tion.
1. DW motion
Both measurements of current-driven and field-driven DWdynamics allow to quantify not only the magnitude of D butalso its sign, which gives information on the DW chirality.Current-driven DW motion is overall a rather complex tech-nique to determine the interfacial DMI, both for experimentalrealization and for results interpretation. As a matter of fact,lithography is required to pattern films into wires of appropri-ate geometry and high in-plane fields need to be applied, es-pecially for samples with large DMI. A misalignment of thesein-plane fields with respect to the wire axis gives rise to un-wanted field components that could alter the motion of DWsand consequently the measurement of their velocity. Further-more, the torque generated by large applied currents can alsoresult in a tilt of the DW (Emori et al. , 2014). In any case,the biggest limitation lies in the interpretation of the results,which is particularly hard due to the complex interplay be-tween di ff erent current-induced spin torques (STT, SHE-SOT,Rashba torque) whose collective e ff ect on DW dynamics isstill not fully understood.Field-driven DW motion has the advantage of being a rel-atively simple method to implement experimentally, as it al-lows to measure interfacial DMI on continuous films, withoutthe need for any lithographic patterning. As a matter of fact,together with BLS field-driven DW motion is the most widelyemployed technique to quantify D . However, relying on theapplication of high in-plane fields, particularly for sampleswith significant DMI, this method has the same limitationsdiscussed for the current-driven case, related to possible mis-alignment of the in-plane field. Indeed, this aspect is likely tobe more relevant for the field-driven case, since measurementsof DW velocities are mostly performed in the creep regimeand an unwanted component of the field perpendicular to thesample plane would influence the DW velocity in an exponen-tial manner (see Eq. 2.5). Several precautions should be takeninto account to achieve an optimal alignment of the in-planefield (Cao et al. , 2018a; Je et al. , 2013; Lavrijsen et al. , 2015;Soucaille et al. , 2016). But even in this case, a perpendicularstray field component may still arise in MOKE set-ups, due toa cross-talk between in-plane and perpendicular electromag-nets or due to not completely non-magnetic objectives, whichwould alter the field distribution in a way di ffi cult to take intoaccount.Regarding measurements of DW velocities in the flowregime, here the main di ffi culties lie in the ability to mea- sure very high DW velocities and to determine accurately theWalker Breakdown field. In any case, aside from all these ex-perimental di ffi culties, the creep / flow field-driven techniquesu ff ers possibly more severe limitations due to inherent prob-lems in the interpretation of the results. While avoidingthe complications due to di ff erent torque e ff ects, which arisewhen DWs are driven by current, a unifying model able to un-derstand the wide variety of bubble morphology observed andrelated DW velocity trends is still under investigation.
2. DW energy
Among the methods that are based on DW energy to eval-uate the interfacial DMI, imaging the equilibrium domainpattern is particularly straightforward due to its experimen-tal simplicity. Indeed, this method is compatible with anymagnetic imaging technique like Kerr microscopy or MFM– which has to be chosen according to the expected domainwidth – and does not require the application of perpendicularnor in-plane fields. This is a huge advantage over the meth-ods that rely on DW dynamics. On the other hand, imagingthe equilibrium domain configuration does not provide infor-mation on the sign of D . Furthermore, for thin samples withsmall anisotropy values the domains can extend several mi-crometers, which restricts the possibility to achieve this typeof measurements in nanostructures. For thick samples withlarge DMI, high resolution imaging is needed in order to re-solve precisely the small domain widths. Experimentally, sev-eral problems can emerge that may render it unusable (seeSupplementary material of Lucassen et al. , 2017): for in-stance, the as-deposited samples may not be in the groundstate, thus requiring demagnetization to be performed. In pat-terned systems, the domain wall width and the domain sizeand configuration may depend on the geometry, adding dis-crepancies with respect to analytical formulations. Other dis-advantages include the fact that the simple analytical modelto extract D may not work for too thick samples which yieldneither N´eel nor Bloch wall types (hybrid wall with N´eel capsand Bloch core), although the DMI is expected to be low inthese systems.In principle, these problems may be solved by the use ofthe proper analytical model. As a matter of fact, the analyti-cal estimations are derived for perfect parallel stripe domains,while the real domain configuration consists of rounded mean-dering structures. Even more, refined expressions for the DWenergy may yield slightly di ff erent values of DMI (Lemesh et al. , 2017). From the experimental point of view, the domainwidths measured may be di ff erent for samples demagnetizedwith in-plane or out-of-plane fields, up to 20% (Davydenko et al. , 2019; Legrand et al. , 2018). For this reason, the in-plane demagnetized configuration is preferred, as it is closerto the parallel stripes described by the analytical theory.Similar considerations hold for for the method based onmagnetic stripe annihilation, although in this case applicationof a perpendicular magnetic field is necessary. Here too, it1 FM Bottom NM Top NM D D s Ref (nm) (nm) (nm) (mJ / m ) (pJ / m)Co Fe B (1) Ta(5) MgO(2) 0 ± ± Gross et al. , 2016Co Fe B (1) TaN . (4) 0.03 ± ± TaN . (1) 0.06 ± ± Table XII Overview of DMI measurements of
CoFeB thin films by stray field using NV magnetometry . FM and NM stand for ferromagneticand non-magnetic layer, respectively, D is the interfacial DMI constant and D s = D · t FM , with t FM being the thickness of the ferromagneticfilm. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated usingthe parameters provided. is only possible to extract information on the absolute valueof D , and not its sign. From an experimental point of view,this technique requires the presence of parallel DWs, whichmay not apply to the case of materials with coarse wall pro-files. Furthermore, this method is not useful for materials thatcan host skyrmion phases, due to the intrinsic faceting of theDWs. Moreover, as the domain walls are closer the energybarrier are reduced a ff ected by the presence of defects and bythermal fluctuations. Even in this case, analytical expressionsrelay on the assumption of perfect parallel DWs.Finally, the experimental disadvantages when using thebubble nucleation to extract the interfacial DMI are similar towhat already mentioned for the techniques involving DW mo-tion, as both perpendicular and in-plane fields are required.The latter can be particularly large, up to 1 T as in Kim et al. (2017c) . Furthermore, since it relies on the nucleation of areversed domain, the process of bubble nucleation is intrinsi-cally a statistical process and several repetitions of the sameexperiment need to be carried out in order to achieve a re-liable measurement of the nucleation field. In contrast, forinstance, imaging the equilibrium domain configuration in afilm can automatically provide averaged information of thedomain width, from which the DMI is quantified, if a largeenough area is imaged. Bubble nucleation experiments canbe performed both at the edge of patterned structures, such asnucleation pads or wires, as well as in continuous films. Inthe former case, the added complexity of lithographic pattern-ing is counterbalanced by the possibility of using this methodto determine both the absolute value of D as well as its sign.Specifically, the sign of D is related to the nucleated bubbleposition with respect to the direction of the applied in-planefield. On the other hand, patterning is clearly not requiredwhen considering bubble nucleation measurements in contin-uous films, and indeed the sample should be large enough toallow for bubble nucleation far away from the sample borders,which would change the type of nucleation. However, in thiscase only the absolute value of D can be measured. III. Spin wave methodsA. Method overview
As domain walls, also spin wave formation and propaga-tion depend on the magnetic energies present in a material or thin film, therefore spin waves (SW) are used as a tool tosense magnetic properties as susceptibility, anisotropy, Gilbertdamping, exchange sti ff ness, etc. These material propertiestogether with the geometry of the sample and the excitationdetermine the spin wave frequency, linewidth, amplitude andattenuation length. The DMI does not only modify the equi-librium magnetic ground state (Bak and Jensen, 1980) but alsothe spin wave dynamics (Zheludev et al. , 1999), essentiallybecause SW propagating in opposite directions have oppositechirality, so that the DMI contribution to the SW energy mayeither decrease or increase the SW frequency and results inan intrinsic non-reciprocity of the SW propagation. This isillustrated in Fig. 14. Note that, di ff erent from the case of thedomain walls analysed in the previous section, here the staticcomponent of the magnetization is fixed and aligned along thedirection of the applied field H (z direction in Fig. 14) whilethe dynamic component of the magnetization (small blue ar-rows in Fig. 14) exhibit a chirality that can be either the sameor opposite to the DMI-favoured chirality, depending on thepropagation direction of SWs. For magnetic films this DMI-induced non reciprocity was investigated the first time in Ud-vardi and Szunyogh (2009) and later a series of theoreticalinvestigations followed (Cort´es-Ortu˜no and Landeros, 2013;Costa et al. , 2010; Kostylev, 2014; Moon et al. , 2013). Exper-imentally, the relatively strong asymmetry in spin wave dis-persion is of interest, since the frequency of spin wave spectracan be measured with high accuracy. A pioneering investi-gation of DMI-induced non-reciprocity in spin wave propa-gation was performed analyzing SWs with large wavenum-ber (Zakeri et al. , 2010) by spin polarized electron loss spec-troscopy (SPEELS). Although a high wavenumber k is advan-tageous, leading to high frequency changes due to the DMI,the most versatile and exploited technique to determine DMI-related e ff ects on SW propagation is Brillouin light scattering(BLS), which relies on the inelastic scattering of photons byspin waves with micrometric or sub-micrometric wavelength.It is becoming more and more popular, thanks to the goodcompromise between sensitivity to SWs in a wide range of k and relatively simple experimental apparatus. Other two tech-niques that have been also applied to the study of SW prop-agation are the Time Resolved Magneto-Optical Kerr E ff ect(TR-MOKE) and the All Electrical Spin Wave Spectroscopy(AESWS). In the following we first recall some theoreticalbackground about the influence of DMI on SW characteris-tics and then we will review the main results achieved by2the four above mentioned experimental techniques, compar-ing their advantages and limitations. B. Theory and models
In the literature we find basically two theoretical ap-proaches to describe spin wave spectra in thin film sampleswith DMI. One is based on the classical theory of magneto-static spinwaves, and this is the approach mainly utilized foranalyzing experimental results obtained by BLS, AESWS orTR-MOKE. The other is based on the quantum spin wave the-ory, which is more suitable for high k measurements in ul-trathin films as done in SPEELS, where exchange interactiondominates the spin wave dispersion (Udvardi and Szunyogh,2009). In most experimental cases, the two formalisms areequivalent and slight di ff erences derive only from the choiceof energy terms. Some other authors use even a mixed ap-proach calculating the spin wave spectra by a quantization ofthe linearized Landau-Lifshitz equation (Udvardi et al. , 2003).The models used to analyze the experimental results (Moon et al. , 2013) derive typically the dispersion relation and donot consider the specific sensing technique, so that they areapplicable to all four techniques presented in the followingsection.
1. Quantum spin wave theory
The usual quantum formalism for spin waves uses the Hol-stein Primako ff method (Holstein and Primako ff , 1940) andstarts directly from the Schr¨odinger equation with a Hamilto-nian in a crystal lattice of spins (cid:126) S j that considers the Heisen-berg exchange energy at a spin site j with the nearest neigh-bors at position j + δ and the Zeeman energy in a magneticfield B ˆ z H = − J (cid:126) (cid:88) j ,δ (cid:126) S j · (cid:126) S j + δ − g µ B B (cid:126) (cid:88) j S jz . (3.1)where J is the exchange constant, g the gyromagnetic ratio, µ B the Bohr magneton. As described in textbooks (Stamps andHillebrands, 1991), using the spin raising and lowering opera-tors one can obtain the eigenstates and energies analytically incertain simplified cases (e.g. low temperature approximation)by employing the Holstein-Primako ff transformation and us-ing the Fourier transform of the harmonic oscillator raisingand lowering operators as creation and annihilation operatorsfor the quantized spin waves (magnons). For ka (cid:28) s with lattice constant a , and consider-ing only nearest neighbour interaction, one obtains the knownquadratic dispersion relation (cid:126) ω k = g µ B B + Jsa k . (3.2)In a quantistic approach the DMI is considered a general-ized exchange interaction leading to a non-collinear spin con-figuration described by (cid:126) D · ( (cid:126) S × (cid:126) S ), where the direction of (cid:126) D depends on the crystal symmetry. Usually it is written as anadditional term in the Hamiltonian (Moon et al. , 2013; Zhe-ludev et al. , 1999) H DMI = − D (cid:126) (cid:88) j ˆ z · ( (cid:126) S j × (cid:126) S j + ) (3.3)where ˆ z is perpendicular to the axis of symmetry breaking.This leads to an additional linear term in the spin wave disper-sion (Eq. 3.2) (cid:126) ω DMI = − Dsak . (3.4)A first study of DMI in bulk Ba CuGe O , an antiferro-magnet with a weak helimagnetic distortion (Zheludev et al. ,1999), was performed already 1999 using the quantum ap-proach, showing low energy spin wave spectra. This approachdoes not consider the sample geometry but only crystal sym-metry, so it is suitable for bulk materials. For thin films in-stead, its validity is restricted to monolayers. In the case ofultrathin films, it was shown theoretically about ten years laterthat the chiral degeneracy of the magnons can be lifted due tothe presence of the DMI (Udvardi and Szunyogh, 2009). Themethod used is based on relativistic first principle calculationsof the magnetic ground state similar to Bode et al. (2007) andFerriani et al. (2008), able to identify domain wall chirality.Therefore it was proposed to exploit this spin wave asymme-try for measuring DMI in ultrathin films. Although it wasknown already for a decade that the DMI stabilizes chiral spinstructures in bulk materials with a certain crystal symmetry(Bogdanov and R¨oßler, 2001; Cr´epieux and Lacroix, 1998),the work was stimulated by the discovery of homochiralityof domain walls in two monolayers of Fe on W(110) (Heide et al. , 2008; Kubetzka et al. , 2003).A quantistic approach was also used in Costa et al. (2010)where, going back to the microscopic origin of the DMI, thee ff ects of spin-orbit coupling on spin wave spectra are stud-ied. It is shown that the DMI leads to a linear term in (cid:126) k in thedispersion relation and that the linewidth of spin wave modesis increased by the spin-orbit coupling. The method used heregoes beyond the adiabatic approximation and directly operatesin the wavevector space, avoiding calculations in real space,taking into account large numbers of neighbor shells. Thestarting point is the multiband Hubbard model with Hamil-tonian (Costa et al. , 2003) where a spin-orbit interaction isadded. The spin wave dispersion is then obtained from thedynamic susceptibility.
2. Classical spin wave theory
Since in most experimental investigations (as in BLS) thewavelength of the detected SWs are in the range between afew microns and a few hundreds of nanometers, one can ig-nore the discrete nature of the spins. Therefore, a classicalformulation, assuming a continuum medium, is more suit-able. Traditionally, spin waves in magnetic films are treated3
Figure 14 Top: Sketch of a spin wave, in the Damon-Eshbach configuration, propagating towards the positive x direction, with a wavevector + k, in the FM film. All the individual magnetic moments (blue arrows) are precessing anticlockwise around the z axis, i.e. around the directionof the static magnetization and of the external field H . Due to the phase delay from one spin to the next, moving from left to right, the chiralityassociated with the SW is clockwise, i.e. opposite to the chirality associated to the DMI vector D (magenta vector). The latter couples twoneighbouring spins via a three-site exchange mechanism with the underlying atom of the HM (white atoms). As a consequence, the frequencyof spin waves with + k is down-shifted in frequency. Bottom: for a SW propagating along the negative x-direction. i.e. with wavevector -k, thespatial chirality of the spin wave is the same as that favoured by the DMI, resulting in an up-shift of its frequency. as magnetostatic spin waves, neglecting exchange interaction(Prabhakar and Stancil, 2009). Considering di ff erent geome-tries and their related boundary conditions one obtains ei-ther Forward Volume for perpendicular-to-plane magnetiza-tion or Backward Volume and Surface or Damon-Eshbach(DE) Waves for in-plane magnetization. The latter, wherethe wave vector (cid:126) k is perpendicular to the applied static field (cid:126) H , is the preferred configuration when exciting spin wavesvia antennas due to their more e ffi cient transduction. How-ever, as pointed out in several works, for the typical sam-ples with DMI the simple magnetostatic solution is not suf-ficient and often anisotropy energy or exchange energy haveto be considered, as in Gurevich (1996); Kalinikos (1981);and Kalinikos and Slavin (1986). Here the complete problemis solved for the magnetization (cid:126) M including the magnetostaticand dynamic regime using mixed boundary conditions, andgoing beyond the plane-wave approximation. Although theexchange term is often neglected when comparing to the ex-perimental conditions (low k -range, kd (cid:28)
1, with d is theFM film thickness) the spin wave modes obtained in nanos-tructured films require in general the complete solution, beingin the transition range from magnetostatic to exchange spinwaves (dipole-exchange spinwaves). One of the critical en-ergy terms is the dipolar field term, which can be obtained from this approach adding to the electrodynamics boundaryconditions the exchange boundary conditions. As mentionedin Moon et al. , 2013, the dipolar field term contains local andnon-local contributions, and can be divided in a stray fieldterm, related to the dipolar interactions between the spins inthe SW, and dipole or magnetostatic field term, related to thedemagnetizing field. In most cases, as in Moon et al. (2013),the dipolar term for unpinned exchange at the film surface (i.e. ∂∂ y (cid:126) m ( y ) = , y = ± d /
2) is applied (Kalinikos, ????). However,Kostylev (2014) points out that the DMI pins the circular com-ponents of the magnetization at the surface (interface) of themagnetic film and mixed boundary conditions have to be used,which require a numerical solution.In summary, considering a sample geometry as shown inFig. 14, (static component of the magnetization aligned alongthe z direction, i.e. parallel to the applied field H ) the disper-sion relation for small amplitude spin waves is usually derivedfrom the linearized Landau-Lifshitz (LL) equation ∂ (cid:126) M ∂ t = − γ ( (cid:126) Mx (cid:126) H e f f ) (3.5)The vector (cid:126) M has to be decomposed in one large static compo-nent, of modulus M s directed along the z -axis and two smalldynamic components that describe the precession around the4 Figure 15 Schematic diagram of the used geometry and notation. k is the wavevector and indicates the propagation direction of the spin-wave, H is the applied in-plane field and M the equilibrium magne-tization, pointing slightly out-of-plane. The DMI frequency dependson the angles φ k and φ M as given in Eq. 3.12. equilibrium direction: (cid:126) M = M s ˆ m ( (cid:126) x , t ) = M s ( m x ˆ x , m y ˆ y , ˆ z ) withvery small dynamic components, i.e. | m x | , | m y | (cid:28) x ,ˆ y ,ˆ z are the unitary vectors of the reference frame, chosen such thatthe y -axis is perpendicular to the sample plane, according toFig. 14. The e ff ective field to be considered in the LL equa-tion consists of di ff erent contributions, reflecting the di ff erentenergy terms as: (cid:126) H e f f = − µ ∂ E tot ∂ (cid:126) M = ± H ˆ z + J (cid:126) ∇ (cid:126) m + (cid:126) H dip + (cid:126) H ani + (cid:126) H DMI (3.6)where J = A /µ M s is the exchange constant. In case ofunpinned exchange boundary conditions the dipolar field foran thin film can be written as (Kalinikos, ????): (cid:126) H dip = − M s m x P ( kd ) ˆ x − M s m y (1 − P ( kd ))ˆ y , (3.7)with P ( kd ) = − − e −| kd | | kd | (3.8)that in the case of ultrathin films where kd <<
1, expandingthe exponential in series, reduces to P ( kd ) = | kd | /
2. While the anisotropy field, written for the case of a uniaxial perpendicu-lar anisotropy constant K u out-of-plane, is (cid:126) H ani = K u µ M s m y ˆ y The right expression for the DMI field H DMI in the con-tinuum theory is not that straightforward to obtain as it de-pends on the crystal symmetry considered. In a bulk, the firstapproach from a spin to a continuum model was performedby Dzyaloshinskii himself (Dzyaloshinskii, 1957), calculat-ing a thermodynamic potential for certain crystal symmetryclasses. This approach was used later by Bak and Jensen(1980) for MnSi and FeGe, and by Bogdanov and Hubert(1994) and Bogdanov and R¨oßler (2001) using the Lifshitzinvariants (cid:126) L ( (cid:126) M , (cid:126) ∇ × (cid:126) M ), where the k -th component of (cid:126) L is L ( k ) i j = M i ∂ M j ∂ x k − M j ∂ M i ∂ x k . The DMI energy can be described asa combination of Lifshitz invariants, depending on the crys-tal symmetry (Cort´es-Ortu˜no and Landeros, 2013; Cr´epieuxand Lacroix, 1998). For certain symmetry classes, as T (e.g.MnSi), the energy density due to DMI is a combination ofLifshitz invariants with a single coe ffi cient D . Only in thesecases the DMI strength is su ffi ciently described by a scalar.Another example is the class C nv . Its Lifshitz invariants are asuitable choice for planar systems (bilayers) with perpendic-ular anisotropy, where symmetry breaking occurs only alongthe perpendicular axis and only gradients in plane contributeto the DMI induced chirality (Bogdanov and R¨oßler, 2001;Bogdanov and Yablonskii, 1989). For a planar geometry inthe xz plane with perpendicular anisotropy along y , the DMIenergy can than be described by E DMI = D ( L zxy + L xyz + L yzx ) = − D (ˆ z · ˆ m × ∂∂ x ˆ m − ˆ x · ˆ m × ∂∂ z ˆ m )(3.9)The DMI can then be interpreted as a local field acting on themagnetization (cid:126) H DMI = − µ ∂ E DMI ∂ (cid:126) M = D µ M s (cid:32) ∂ m y ∂ x , − ∂ m x ∂ x , (cid:33) (3.10)The dispersion relation is then given by ω ( k ) = γµ (cid:113) ( H + Jk + M s P ( kd ) sin φ k )( H + Jk − H ani + M s (1 − P ( kd ))) + ω DMI = ω + ω DMI (3.11)that is valid also for values of φ k di ff erent from π/
2, as shownin Fig. 15. Moreover, it can be shown that the general expres-sion of ω DMI for a cubic crystal is given by: ω DMI = γ DM s k sin φ k cos φ M (3.12)where the angles are defined in Fig. 15. This means that, sim-ilar to the results of the quantum approach reported in the pre-vious paragraph, the DMI leads to a shift of dispersion rela-tion depending on the sign of the wavevector, i.e. a frequencynon-reciprocity due to DMI. This non reciprocity depends on the directions of both the sample magnetization and thespin wave propagation direction and is absent for k =
0, i.e.the condition for ferromagnetic resonance (FMR). To this re-spect, already in 2013 Cort´es-Ortu˜no and Landeros suggestedto use BLS measurements for determining the DMI constant(Cort´es-Ortu˜no and Landeros, 2013), pointing out that thelargest e ff ect can be observed applying a su ffi ciently large ex-ternal field H, so that the magnetization lies in-plane ( φ M = φ k = π/ et al. , 2015; Kostylev, 2014).Let us also notice that the above description is valid, strictlyspeaking, only for a film consisting of a single monolayer. Anattempt to take correctly into account the finite thickness ofthe film is shown in Kostylev (2014) by using mixed exchangeboundary conditions at the surface. The boundary conditionsresult then similar, but not equal, to the ones derived for auniaxial surface anisotropy (Soohoo, 1963). Di ff erent surfaceanisotropies at both interfaces of the magnetic film result in anintrinsic non reciprocity of DE spin waves (Hillebrands andG¨untherodt, 1987). In this case the two contributions are dif-ficult to be distinguished. In fact, already in Cr´epieux andLacroix (1998), the presence of DMI can be interpreted as acontribution to the surface anisotropy. In order to analyze filmthickness dependencies in a more rigorous way both have tobe taken into account, as discussed in Lucassen et al. (2020). C. Experimental results
1. Brillouin light scattering
The most popular and widely employed technique for themeasurement of the DMI constant from the non-reciprocalspin wave propagation is Brillouin light scattering (BLS). Ashown in Tables XIII, XIV and XV, more than forty papershave been published during the last five years, reporting thevalue of DMI constant in samples where a FM material (usu-ally Co or CoFeB) is in contact with a HM material (such asPt, W, Ta, etc.).BLS is a classical tool to study spin wave dispersion in gen-eral and non-reciprocity in particular. As sketched in Fig. 16in a BLS experiment a monochromatic light beam is focusedon the surface of the specimen under investigation by a cameraobjective and the light that is back-scattered within a solid an-gle is collected by the same lens and analysed in frequency bya high-resolution spectrometer (Carlotti and Gubbiotti, 1999).The physical mechanism of BLS relies on the inelastic scat-tering of photons by spin waves, thanks to either the creation(Stokes process) or the annihilation (anti-Stokes process) ofa magnon. This implies that a red-shift or a blue-shift is ob-served in the scattered light with respect to the incident beam.In the wave vector space, magnons entering Stokes or anti-Stokes scattering processes correspond to either a positive ora negative wave vector. As a consequence, the non-reciprocitycaused by the presence of a DMI interaction leads to an asym-metry in frequency shift of the peaks corresponding to theStokes or the anti-Stokes process, as anticipated in the pre-vious paragraph (see Fig. 16).One should also notice that in typical experiments, laserwavelengths of 514nm or 532nm are typically exploited andthe angle of incidence can be varied between about 0 and 60 o . As a consequence, the corresponding interval of wavenumbersthat can be probed by BLS is in the range between zero and2.2 · rad / cm. As for the frequency resolution, this is limitedby the instrumental characteristics to about 0.1 GHz, puttinga lower limit to the minimum values of the DMI constant thatcan be measured by this technique. One should also considerthat usually several tens of mW of light are focused on themeasurement spot, so that, depending on the thermal proper-ties of the material under investigation (including the substratematerial), as well as on the numerical aperture of the exploitedlens, the real temperature of the probed sample region may belifted above room temperature and this can have an influenceon the values of the measured magnetic parameters, includ-ing D. Moreover, it is worth mentioning that the sign of the Dconstant obtained using BLS measurements is a ff ected by thechoice of the coordinate system and the method of calculat-ing the frequency asymmetry, therefore both of them shouldbe carefully specified to properly recognize the sign of D, i.e.which chirality is favoured by the DMI. Here we adopt the ref-erence frame presented in Fig. 14, where the z -axis is alignedwith the static magnetization, the x -axis coincides with thepropagation direction of SWs and the film normal ( y - axis)is oriented upward with respect to the free surface of the FMfilm. With such a choice, the frequency shift of the BLS peaksis f DMI = ω DMI / π = ( γ Dk ) / ( π M s ) (3.13)For the case of a FM film deposited over a HM one, assketched in Fig. 14 and Fig. 15, a negative (positive) value of f DMI , i.e. of D, indicates that a left (right)-handed chirality isfavoured by the DMI. Practically speaking, this means that inpresence of a positive (negative) value of D, in the measuredBLS spectra the absolute frequency of the anti-Stokes (Stokes)peak is larger (lower) than that of the Stokes (anti-Stokes) one,and as a consequence f DMI assumes positive (negative) values,as illustrated in the lower panel of Fig. 16. Please note that ifthe direction of the applied magnetic field is reversed, f DMI changes sign due to the reversal of the SW chirality.One of the first BLS experimental works was done on Co / Nimultilayers (Di et al. , 2015a,b; Zhang et al. , 2015). In thesame year a series of publications of DMI measurements byBLS on Py, Co and CoFeB followed (Belmeguenai et al. ,2015; Cho et al. , 2015; Kim et al. , 2015a; Nembach et al. ,2015; Stashkevich et al. , 2015; Vaˇnatka et al. , 2015) and theresearch continues to be very active.A comparison of the value and the sign of the DMI con-stant measured using BLS for Co, CoFeB, and a collection ofdi ff erent magnetic materials are reported in Tables XIII, XIVand XV, respectively. When the numbers were not explicitlygiven in the original manuscript, we have calculated or ex-tracted from the figures and reported in italic. Note that thesigns in the Tables are consistent with the convention of axes,field, D , k vector and frequency di ff erence as previously de-scribed in Fig. 14. If the authors in the original manuscripthave used a di ff erent coordinate system and / or method of cal-6 Figure 16 Schematic diagram of the BLS interaction geometry. k lighti and k lights represent the wavevectors of the incoming and of the backscat-tered light. Due to the conservation of the wavevector component parallel to the surface of the specimen, the length of the wavevector of theSW involved in the scattering process is 2 k lighti sin ( θ ), as shown in the right panel. SW propagating along the + k ( − k ) direction correspond tothose involved in the Stokes (anti-Stokes) process, i.e in the generation (annihilation) of a magnon. The bottom panel shows a typical BLSspectrum in the absence of DMI (dashed line) and with DMI (continuous line) for positive or negative value of D i.e. of f DMI , respectively. culating the frequency asymmetry we recalculated the sign ac-cording to our convention.Many papers discuss the dependence of the DMI constantas a function the ferromagnetic layer thickness (Belmegue-nai et al. , 2015; Chaurasiya et al. , 2016; Kim et al. , 2016a;Stashkevich et al. , 2015) and possible interactions with sur-face anisotropy (Stashkevich et al. , 2015) or Heisenberg ex-change (Nembach et al. , 2015). Theoretically, a 1 / d depen-dence of the DMI constant is predicted (Kostylev, 2014), as isthe case for all interface phenomena, and this makes it di ffi -cult to distinguish between the di ff erent contributions. Some-times a change in slope of DMI constant versus 1 / d is ob-served (Belmeguenai et al. , 2018, 2016; Cho et al. , 2015;Kim et al. , 2018b). Belmeguenai et al. (2018) discuss di ff er-ent origins. The first is a coherent-incoherent growth mecha-nism transition at a certain thickness accompanied by magne-toelastic anisotropy changes. Others are changes in surfaceroughness which are causing in-plane demagnetizing mag-netic fields. Another is interdi ff usion and mixing at the in-terfaces, which reduces interface anisotropy. According to theexperimental results the authors conclude that the first hypoth-esis is the most probable.Similarly, the dependence of the DMI on the heavy metal thickness has been studied for both Co (Kim et al. , 2017b) andCoFeB (Chen et al. , 2018b; Tacchi et al. , 2017) films in con-tact with a Pt layer. For both systems the DMI intensity wasfound to increase with the Pt thickness, d Pt , reaching a satu-ration value when d Pt approaches the Pt spin di ff usion length(about 2 nm). This behaviour has been explained by analyticalcalculations assuming that several Pt atoms, belonging to dif-ferent layers in the heavy metal, can contribute to the strengthof the interfacial DMI (Tacchi et al. , 2017). On the contrary, inMa et al. (2017), the DMI strength in the IrMn / CoFeB systemwas observed to increase even for IrMn layers thicker than theIrMn spin di ff usion length (about 0.7 nm). The authors claimthat this DMI enhancement can be ascribed to the reduction ofthe thermal fluctuations of the antiferromagnetic (AFM) spinarrangement when the IrMn thickness increases, suggesting adi ff erent microscopic origin with respect to HM / FM systems.The sign of the DMI constant has been investigated for sev-eral material combinations on changing both the stacking or-der and the multilayer composition (Cho et al. , 2017). In sys-tems with a Pt (W) underlayer, negative (positive) values ofthe DMI constant have been found with a quite overall agree-ment, indicating that a left-handed (right-handed) chirality isfavored. In agreement with the theoretical calculations, it has7been also observed that the sign of DMI is reversed when thestacking order is inverted, while its value becomes negligiblein symmetric structures where a FM film is sandwiched be-tween two identical HM layers. On the contrary, the DMI signinduced by other materials, such as Ir or IrMn, is still the sub-ject of controversial debate in the literature (see for instance(Belmeguenai et al. , 2018; Ma et al. , 2017, 2018)). Very re-cently, the influence of electric fields on the DMI strength hasbeen investigated in Ta / FeCoB / TaO x trilayer (Srivastava et al. ,2018). A strong variation of the D value has been observedand it has been explained taking into account that the maincontribution to the DMI come from the interface with the ox-ide layer. Among the Co articles, Khan et al. , 2018 inves-tigates the e ff ect of exchange bias with an antiferromagnetic(AFM) overlayer on Co and confirms that DMI is not influ-enced by exchange bias or the AFM spin order. Finally, thesinusoidal dependence of the frequency non-reciprocity as afunction of the angle between the sample magnetization andthe SW wave vector, theoretically predicted by Cort´es-Ortu˜noand Landeros (2013), has been experimentally verified for dif-ferent multilayer structures (Belmeguenai et al. , 2018; Cho et al. , 2017; Kim et al. , 2016a; Tacchi et al. , 2017; Zhang et al. , 2015).To conclude this paragraph devoted to BLS results, let usnotice that several studies were able to correlate the pres-ence of a sizeable DMI not only to the frequency posi-tion of the BLS peaks but also to their linewidth, i.e. tothe damping that a ff ects spin waves. Already Di et al. (2015b) studied the interfacial DMI in an in-plane anisotropicPt(4) / Co(1.6) / Ni(1.6) film by Brillouin spectroscopy, showingthat the measured linewidths of counterpropagating magnonsare di ff erent, with the di ff erence being more pronounced forlarger wave vectors. This could be ascribed to a DMI-inducedterm that is antisymmetric in the wave vector. Moreover an-alytical calculations showed that, due to the existence of theDMI, the magnon linewidth is no longer a monotonic func-tion of frequency. In Chaurasiya et al. (2016), asymme-try in the peak frequency, peak intensity and magnon life-time were observed in W / CoFeB / SiO . Also in this casethe linewidth for spin-wave propagating in + k direction issmaller than the same for spin-wave propagating in − k di-rection, indicating di ff erent lifetimes of magnon propagatingin opposite directions, induced by DMI. More recently, sev-eral studies where FMR measurements were compared withBLS results showed a clear correlation among the strength ofDMI and the spin wave damping: [Pt(1.5) / Co( d ) / W(1.5)] xN (Benguettat-El Mokhtari et al. , 2019); Py(5) / Cu − x Pt x bi-layers (Bouloussa et al. , 2019) combined with ferromagneticresonance; He + irradiated Ta / CoFeB / MgO lms (Diez et al. ,2019b); Ta / Pt / CoFeB / MgO / Ta (Zhang and Li, 2004).Very recently, several works reported that a sizeable in-terfacial DMI can also arise at the interface between anoxide layer and a ferromagnetic one (Arora et al. , 2020;Kim et al. , 2019c; Lin et al. , 2020; Nembach et al. ,2020). Nembach et al. (2020) studied the interfacial DMI inCu(3) / Co Fe (1.5) / Ta( d Ta ) / Oxide / Ta(3) system, where the Oxide layer was prepared by in-situ sample oxidation. Chang-ing the Ta thickness, d Ta , an Oxide layer of CoFeO x and / orTaO x was obtained following the sample exposure to oxygen.The authors found that both the magnitude and sign of theinterfacial DMI can be tuned by varying the thickness andthe composition of the Oxide layer, due to the changes ofthe electronic structure at the interface, induced by the oxi-dation process. Moreover, Lin et al. (2020) showed that in theBaTiO (BTO) / CoFeB system the DMI strength is stronglya ff ected by the termination of the oxide layer (TiO vs BaO).In particular, a higher value of the DMI constant was foundfor a TiO -BTO substrate, and this finding was attributed byfirst principle calculations to the di ff erent electronic statesaround the Fermi level at the oxide / FM interfaces. Finally,the inuence of an electric eld on the DMI present at theoxide / ferromagnetic metal interface was investigated in theMgO / Fe / Pt (Zhang et al. , 2018) and the Ta / FeCoB / TaO x sys-tems (Srivastava et al. , 2018). In both the cases the DMImagnitude was observed to linearly increase with electric eldstrength, and this behavior was ascribed to the enhancementof the Rashba spin-orbit coupling at oxide / ferromagnetic in-terface induced by the applied electric eld.
2. Time resolved magneto optical imaging
Time resolved magneto-optical Kerr microscopy (TR-MOKE) was employed by K¨orner et al. (2015) in order todetermine DMI in a Pt / Co / Py / MgO multilayer. For imag-ing spatially and time dependent spin wave signals a focusedlaser pulse (spot size ≤ λ = ≈ k -vectorsexcited are 2-10 µ m − . The out-of-plane magnetization is de-tected via MOKE along the direction normal to the CPW upto a distance of 5 µ m from the center of the CPW. The spinwave dispersion is determined by using the expression givenin Moon et al. (2015). The Co film is 0.4nm thick and hasa perpendicular magnetic anisotropy (PMA), however duringthe measurement it is magnetized in plane parallel to the CPW.The 5nm Py film on top of Co facilitates the SW propaga-tion. Damon-Eshbach modes (perpendicular H ) are detectedon both sides of the CPW. This allows to detect besides a fre-quency and amplitude non reciprocity, a non reciprocity inattenuation length of the spin waves, as predicted by theory.The DMI constant measured is D s = ± / m.
3. All electric spin wave spectroscopy
All electric spin wave spectroscopy (AESWS) (see Fig. 18)is a tool to characterize spin waves excited in magnetic thinfilms by measuring the dispersion relation and group veloc-ity all-electrically using a standard high frequency instru-8
FM Bottom NM Top NM D D s Sign Ref (nm) (nm) (nm) (mJ / m ) (pJ / m)Co(0.6-1.2) Ta(3) / Pt(3) AlO x / (2)Pt(3) 1.6-2.7 1.6-1.9 - Belmeguenai et al. , 2015Co(0-2) Pt(4) AlO x (2) n.a. Cho et al. , 2015Co(1) Pt(5) GdOx(2-5) / Al(2) 1.6 1.6 - Vaˇnatka et al. , 2015Co(1-1.7) Pt(4) AlO x (2) - ∗ Kim et al. , 2015bCo(1.3-1.8) Ta(4) / Pt(4) - ∗ Co(1.06) Ta(3) / Pt(3) MgO / Ta(2) 2.05 2.17 - Boulle et al. , 2016Co(1.3-2.9) Ta(4) / Ir(4) AlO x (2) n.a. Kim et al. , 2016bCo(2) Ta(4) Pt(4) 0.7 + Cho et al. , 2017Pt(4) Ta(4) 0.92 -Co(1.6) Ta(4) / Pt(0.8) AlO x (2) / Pt(1) 0.43 - Kim et al. , 2017bTa(4) / Pt(4.8) 1.57 -Co(1) Ta(5) MgO(2) / Ta(2) + Ma et al. , 2018W(5) + Ir(5) / Pt(2 /
6) Ru(3) 0.43 a a - Bouloussa et al. , 20180.33 b b -Co(1) Ta(5) / Pt(2) Ir Mn (1.1) 1.14 1.14 - Khan et al. , 2018Ir Mn (1.7) 1.14 1.14 -Ir Mn (2.4) 1.22 1.22 -Ir Mn (5) 1.11 1.11 -Co(0.6) Ta(5) / Pt(2) Fe Mn (1) 1.5 1.35 c - Khan et al. , 2018Fe Mn (2.6) 1.44 1.3 c -Co(2.5) Pt(5.4) Au (2.5) / Pt(2.6) 0.60 1.51 - Rowan-Robinson et al. , 2017Ir(2.5) / Pt(2.6) 0.60 1.51 -Co(1-2) Ta(3) / Pt(3) Ir(3) 0.8 – 0.3 0.8 - Belmeguenai et al. , 2019Cu(3) 0.9 – 0.4 1.05 -MgO(1) 0.9 – 0.4 0.95 -Pt(3) 0 0[Pt(1.5) / Co(1-2) / W(1.5)] x et al. , 2019[Pt(1.5) / Co(1-2) / W(1.5)] x / Co(1-2) / W(1.5)] x - Diez et al. , 2019aCo(0.9) Ta(5) / Pt(1.5) Ti(2.5) / Pt(2.5) 1.42 n.a. Kim et al. , 2019aCu(2.5) / Pt(2.5) 1.42 n.a.W(2.5) / Pt(2.5) 0.87 n.a.Ta(2.5) / Pt(2.5) 1.25 n.a.Al(2.5) / Pt(2.5) 0.99 n.a.Pt(2.5) / Pt(2.5) 0.02 n.a.Co(1.4) Ta(5) / Pt(5) MgO(2) / Ta(3) 1.20 n.a. Kim et al. , 2019cCu(2) / Ta(3) 1.05 n.a.Co(0.8) Ta(2) / Pt(2.2) Ir(0-2) / Ta(4) 1.8-0.88 1.64-0.7 - Shahbazi et al. , 2019[Pt(3) / Co(1.1) / Ta(4)] x - Saha et al. , 2019 a The sample is a stripe sample with stripe width 300nm and spacing 100nm b stripe width: 100nm, spacing 100nm c In the calculation of D s the authors have increased the e ff ective thickness of the FM layer, including a monolayer of Fe which forms at the Co / FeMninterface.
Table XIII Overview of DMI measurements for Co thin films via BLS . FM and NM stand for ferromagnetic and non-magnetic layer, respec-tively. D is the interfacial DMI constant and D s = D · t FM , with t FM being the thickness of the ferromagnetic film. Numbers in roman werequoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. Signswith ∗ are according to the convention used in this paper and opposite to that in the original manuscript. ment, a Vector Network Analyzer (VNA) (Bailleul et al. ,2001; Stamps and Hillebrands, 1991; W.Schilz, 1973). In thefield of magnonics it was widely applied, recently to studythe applicability of magnetic dot or antidot arrays constitut-ing magnonic crystals as band filters (Neusser et al. , 2010). Currently there are only a few experimental works publishedapplying AESWS to measure the DMI constant (Kasukawa et al. , 2018; Lee et al. , 2016; Lucassen et al. , 2020).In Lee et al. (2016), one of the first that applied AESWS,spin waves are excited by a meander shaped CPW (similar9 FM Bottom NM Top NM D D s Sign Ref (nm) (nm) (nm) (mJ / m ) (pJ / m)Co Fe B (1.6-3) Pt(4) AlO x (2) n.a. Cho et al. , 2015Co Fe B (0.8) MgO(2) / Pt(2) MgO(2) / SiO (3) 1 - Di et al. , 2015aCo Fe B (0.85, 1, 1.5, 2, 3) W(2) SiO (2) n.a. Chaurasiya et al. , 2016Co Fe B (1) W(2) MgO(1) / Ta(1) 0.25 + Soucaille et al. , 2016W(3) 0.27 + Ta N (1) 0.31 + Hf(1) 0.15 + Co Fe B (2) Si / SiO Pt(0.6) / Cu(3) 0.45 a + Tacchi et al. , 2017Co Fe B (2) Pt(4) Ta(4) 0.51 - Cho et al. , 2017Ta(4) Pt(4) 0.43 + Ta(4) Ta(4) 0.15 -Pt(4) Pt(4) 0.01 -Co Fe B (1.12) Ta(3) / Pt(3) Ru(0.8) / Ta(3) 0.84 - Belmeguenai et al. , 2017Pt(3) / Ru(0.8) MgO(1) / Ta(3) 0.3 -Co Fe B (1) Ta(5) MgO(2) / Ta(2) 0.04 + Ma et al. , 2018W(5) 0.07 + Ir(5) 0.21 + Pt(5) 0.97 -Au(5) 0.17 -[Pt(5) / Co Fe B (1) / Ti(1)] x Ti(5) Pt(5) 0.81 0.81 n.a. Karakas et al. , 2018[Pt(5) / Co Fe B (1) / Ti(1)] x / Co Fe B (1) / Ti(1)] x / Co Fe B (1) / Ti(1)] x / Pt(0.7-4) MgO(1) / Ta(1)
0- 0.87 0- 0.87 - Chen et al. , 2018bCo Fe B (0.9) W(1-13) MgO(1) / Ta(2) n.a. Kim et al. , 2018bCo Fe B (2) Ir Mn (1-7.5) MgO(2) / Ta(2) 0.02-0.13 - Ma et al. , 2017Co Fe B (0.8-2) Ir Mn (5) -Co Fe B (1.2) Ir(5) + Co Fe B (1) Ta(5) Pt(0.12-0.27) / MgO(2) + Ma et al. , 2016Co Fe B (0.9) Ta(3) TaO x (1) ∗ + Srivastava et al. , 2018 a for t Pt > Table XIV Overview of DMI measurements for
CoFeB thin films via
BLS . FM and NM stand for ferromagnetic and non-magnetic layer,respectively. D is the interfacial DMI constant and D s = D · t FM , with t FM being the thickness of the ferromagnetic film. Numbers in romanwere quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided.Numbers with ∗ indicate that a magnetic dead layer was taken into account. FM Bottom NM Top NM D D s Sign Ref (nm) (nm) (nm) (mJ / m ) (pJ / m)Co(1.6) / Ni(1.6) Pt(4) MgO(2) / SiO (3) 0.44 - Di et al. , 2015bCo(1.6) / Ni(1.6) MgO(2) / Pt(4) MgO(2) / SiO (3) 0.44 - Zhang et al. , 2015Ni Fe (4) Si substrate Pt(6) 1.2 [ ] a + Stashkevich et al. , 2015Ni Fe (1-13) Ta(3) / Pt(6) SiN - Nembach et al. , 2015Co FeAl(0.9-1.8) Ta(2) / Ir(4) Ti(2) ∗ - Belmeguenai et al. , 2018Py(5) Cu − x Pt x (6) b + Belmeguenai et al. , 2019Co Fe (1) / oxide c Ta(3) / Pt(6) Cu(3) / Ta(2) - Nembach et al. , 2020Ta(3) / Cu(6) -0.02-0.26 0.4 d -Fe(3) SiO Pt(4) 0.22 + Zhang et al. , 2018MgO(5) 0.35 + YIG(10) GGG substrate 0.01 n.a. Wang et al. , 2020 a Not realistic value due to the large FM thickness b x = c Oxidation: 0-1000s at 99% Ar and 1% O The value refers to the longest oxidation of 1000s
Table XV Overview of DMI measurements various FM materials via
BLS . FM and NM stand for ferromagnetic and non-magnetic layer,respectively. D is the interfacial DMI constant and D s = D · t FM , with t FM being the thickness of the ferromagnetic film. Numbers in romanwere quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided.The value indicated with ∗ was given for the thickest FM film. Figure 17 Scheme of a time resolved magneto-optic Kerr e ff ect ex-periment. The pulse train of a femtosecond laser system is syn-chronized to the sinusoidal excitation field which is generated bysending a microwave current through a coplanar wave guide. Thechange of polarisation of the reflected light is analysed in an opticalbridge and sent to a lock-in analyser synchronized to a microwavemixer / chopper.Figure 18 Scheme of an AESWS device. The horizontal stripe isthe FM / HM bilayer (purple), on top are two meander shaped CPWs.The measurements are performed by using a Vector Network Ana-lyzer for determination of the mutual inductances L and L . FromLucassen et al. , 2020. to Vlaminck and Bailleul (2010))), and detected in a sec-ond CPW at a certain distance, thus measuring the spin wavedispersion and group velocity by a VNA (Bailleul et al. ,2003). The magnetic multilayers measured are Pt / Co / MgO,MgO / Co / Pt and MgO / Co / MgO in order to compare sampleswith positive and negative DMI constant with one with zeroDMI. The films were patterned into 8 µ m wide stripes and a40nm thick AlO x layer for electric insulation was depositedon top. The CPW consists of Ti(5) / Au(150) and was realizedwith a primary spatial period of 800 nm ( k = µ m − ) and Figure 19 Mutual inductance spectra of Py / Pt bilayers (a-b). Fig-ure (c) shows the influence of a di ff erence in surface / interfaceanisotropies on the measured DMI value. From Gladii et al. , 2020. subsidiary spatial period of 2250 nm ( k = µ m − ). The Cothickness d was changed from 16 nm to 20 nm. The DMIconstant is considered here to be a pure interface e ff ect witha characteristic length λ = D i is calculated to be D = D i λ/ d . extrapolating to a Co thick-ness of 2 nm we obtain D s = / m.In Kasukawa et al. (2018), a similar filmTa(2) / Pt(3) / Co(20) / MgO(2) was investigated for electricfield e ff ects. Modulations of spin wave frequency wereobtained by the gate electric field and the DMI and interfacialperpendicular anisotropy were calculated to be 0.45 mJ / m and 0.71 mJ / m . In order to estimate the gate voltage depen-dence of the DMI and interfacial anisotropy the frequencydata were fitted with the dispersion relation obtained from theLandau-Lifshitz equation. In both works the values for D s arerather high, which might indicate that a simple normalization1with the film thickness is not reasonable.Lucassen et al. (2020) points out that especially in Co films,the gradual transition between di ff erent phases with the filmthickness and therefore a change in volume anisotropy hasto be considered. In their paper, Ta(4) / X(4) / Co(d) / Y(3) / Pt(2)stacks with d = = (Pt, Ir),(Ir, Pt) are inves-tigated by meander type CPWs with wave vectors k =
4, 5.5,7, 8.5, 10 µ m − . A detailed analysis of the frequency non-reciprocity due to DMI considering surface anisotropy andchanges in volume anisotropy is performed. The same prob-lem was pointed out by Gladii et al. , 2020, 2016 (see Fig.19) who performed an analysis of the influence of a di ff erencein surface (interface) anisotropy on both sides of a Permalloyfilm. In Gladii et al. (2016) the frequency non reciprocity dueto di ff erent surface anisotropies was calculated revisiting theapproach of Kalinikos and Slavin (1986). The important re-sult of the paper is that the frequency non reciprocity due to adi ff erence in surface anisotropies scales as d , where d is thethickness of the FM film, while the contribution due to DMIscales with d − . In this approach, the exchange spin wavesplay an important role and a purely magnetostatic approachwould lead to unrealistic modal profiles. In Gladii et al. (2020)additionally the spin wave relaxation rate was measured usingtwo CPWs as emitter and detector at di ff erent distance. Aninjected DC current in the Pt layer via the Spin Hall e ff ect in-duced STT showed to alter linearly the spin wave relaxationtime. From this it was possible to obtain additional informa-tion as the Spin Hall conductivity.On the brink of this review is the recent paper (Wang et al. , 2020) which investigates interfacial DMI not at the typ-ical HM / FM interface but at an oxide-oxide interface, hereYIG / GGG. This so called Rashba induced DMI was recentlyobserved in Avci et al. (2019) and Ding et al. (2019). HereYIG films with thicknesses 7,10, 20, 40, 80nm on a GGGsubstrate were measured by AESWS and from the asymmet-ric group velocities the DMI constant was calculated. Thevalues were comparable with the ones obtained from BLS.
4. Spin-polarized electron loss spectroscopy
Following the theoretical work of Udvardi and Szunyogh(2009), the spin wave dispersion asymmetry due to DMI wasfirstly measured by spin-polarized electron loss spectroscopy(SPEELS) on a double layer of Fe(110) grown on W(110) sin-gle crystal (Zakeri et al. , 2010). A monochromatized spin-polarized electron beam is scattered from the sample and en-ergy loss or gain as a function of in-plane momentum transferis measured. The minority electrons create surface SW bya virtual spin flip scattering process and loose energy, whilemajority electrons may annihilate thermally excited SW andgain energy. One measures a peak in the minority spin chan-nel in the energy loss region and a peak in the majority spinchannel in the energy gain region (analog to the Stokes andAnti-Stokes peak in BLS). At an electron energy of 4.163 eV(resolution 19 meV) a wave vector of ± − is employed by changing the angle of the incident beam. A small SW dis-persion asymmetry is expected due to Damon-Eshbach sur-face modes of about 0.1 meV. The energy di ff erence for ± k is evaluated by inverting the magnetization direction. A DMIconstant of 0.9 meV in [1¯10] direction and 0.5 meV in [¯110]direction was measured (for (cid:126) k along [001]). With an inter-atomic distance of a = D s = D · a of0.9 pJ / m and 0.5 pJ / m. D. Advantages and Limitations
The main advantage of spinwaves methods for determina-tion of the DMI constant is the rather simple expression for themeasurable frequency di ff erence that occurs when the sign of k is changed, as seen in in Eq. 3.11. This also means thatnot only the size, but also the sign of the constant can beeasily determined, provided that the geometry of interactionand the polarity of the applied field are specified. However,attention has to be payed to the following aspects. First ofall, the exact gyromagnetic factor and the saturation magne-tization should be determined for the given sample material.This can be done by FMR and vibrating sample magnetome-try (VSM). Morever, in the expression of ω DMI the thicknessof the magnetic active layer has to be evaluated, which notalways coincides with the nominal layer thickness due to in-terlayer mixing (dead layer) (Belmeguenai et al. , 2018). Fi-nally, it is considered that ω in Eq. 3.11 is independent ofthe sign of k . This is in general the case, but when com-paring di ff erent samples (e.g. with di ff erent thicknesses),this might be not perfectly true, for example due to di ff er-ent surface / interface anisotropies at the both sides of the mag-netic film which influence di ff erently the ± k spinwaves and ∆ ω = ω ( k ) − ω ( − k ) (cid:44)
0. Therefore, such a di ff erenceis necessarily reflected in the measured frequency asymme-try. This problem comes from the fact that DE waves travelon opposite surfaces at opposite k directions (Camley, 1987)and therefore any di ff erence in magnetic properties at the twosurfaces, as e.g. surface anisotropy, will reflect in a non-reciprocity of the spin wave spectra. By comparison with the-oretical models and the correct measurement procedure theircontribution has to estimated as far as possible. For exam-ple, the asymmetry due to di ff erences in surface anisotropy orsaturation magnetisation is evaluated in Di et al. (2015b) bymeasuring additionally to the Pt / Co / Ni sample a Co / Ni anda Co / MgO / Ni sample. While in the latter ∆ ω is negligible,the former shows an opposite sign for ∆ ω , meaning that theanisotropy plays a major role, compare to asymmetry in M s counts, but might be distinguished from DMI by the oppositesign. However typically kd < et al. , 2015) and magneticproperties can be averaged over the film thickness (Soohoo,1963). Instead any surface e ff ect (as surface anisotropy or alsoDMI in bilayers) scales as 1 / d (Hillebrands, 1990; Stampsand Hillebrands, 1991). The frequency non reciprocity ob-2 FM Bottom NM Top NM D D s Sign Ref (nm) (nm) (nm) (mJ / m ) (pJ / m)Co(0.4) / Py(5) Ta(2) / Pt(2) MgO(5) 0.16 0.89 - K¨orner et al. , 2015Co(14-20) Ta(3) / Pt(3) MgO(1.8) / Ta(3) [ ] a + Lee et al. , 2016Ta(3) / MgO(1.8) Ta(3) / Pt(3) [ ] a -Ta(3) / Pt(3) Pt(3) / Ta(3) [ ] a -Co(20) Ta(2) / Pt(3) MgO(2) 0.45 [ ] a + Kasukawa et al. , 2018Co(4-26) Ta(4) / Pt(4) Ir(3) / Pt(2) n.a. 1.0 ± et al. , 2020Ta(4) / Pt(4) Pt(3) / Pt(2) n.a. 0.1 ± / Ir(4) Pt(3) / Pt(2) n.a. 1.0 ± + Py(6) Al O (21) Al O (5) + Gladii et al. , 2016Py(4-20) Ti(5) Pt(5,10) n.a. 0.25 + Gladii et al. , 2020YIG(10) GGG substrate 0.0099 ± n.a. Wang et al. , 2020Fe(2ML) b W(110) single crystal c , d c , d + Zakeri et al. , 2010 a Not realistic value due to the large FM thickness b ML...atomic monolayer c magnetization along [1¯10] d magnetization along [¯110] Table XVI Overview of DMI measurements of various FM materials via spin wave methods di ff erent from BLS . K¨orner et al. (2015)use TR-MOKE, Zakeri et al. (2010) use SPEELS, and all the others AESWS. FM and NM stand for ferromagnetic and non-magnetic layer,respectively. D is the interfacial DMI constant and D s = D · t FM , with t FM being the thickness of the ferromagnetic film. Numbers in romanwere quoted in the reviewed papers, while numbers in italics were either extracted from figures or calculated using the parameters provided. served for DMI and surface anisotropy are intrinsically di ff er-ent. The first depends directly on k and therefore on its signwhile the latter does not depend on k . This means that thenon reciprocity due to surface anisotropy becomes importantfor thicker films, independently on k , while DMI non reci-procity might be strong also for thin films, depending on k .Furthermore at k = et al. (2015) that the surface anisotropy non reciprocity in the spinwave spectrum vanishes for thin films ( d Py < D values of about1 mJ / m the DMI non reciprocity is about one order of mag-nitude stronger than the one due to surface anisotropy. On thebasis of the above arguments, in the literature it is usually con-sidered that at kd << ω on the sign ofk can be neglected and that, in absence of DMI, the ± k spin-waves are subjected to an average magnetic energy equal atboth sides of the magnetic film.Care has to be taken not only with respect to frequency non-reciprocity, but also to amplitude non-reciprocity. Also herethere might be e ff ects due to di ff erent boundary conditions atthe interfaces of the ferromagnetic film. For example a metal-lic boundary condition at one side would attenuate stronglythe spin wave travelling close to it. Furthermore, methods asAESWS show an amplitude non-reciprocity due to the factthat the CPW couples more to one wavevector than to theother due to the symmetry of the CPW (Serga et al. , 2010).The strongest limitation of the methods based on spinwaves might be that the sample has to be magnetized in-planein order to be able to detect DE modes. Therefore for sam- ples with strong out-of-plane anisotropy (perpendicular mag-netic anisotropy, PMA) it is necessary to apply a su ffi cientlyhigh in-plane field. Furthermore, ultrathin PMA films, have alow signal to noise ratio and fitting procedures may have quitelarge uncertainties.
1. BLS
BLS has rapidly a ffi rmed as the most used technique forstudying DMI using spin waves. A reasonable complexity ofthe experimental setup and a high accuracy in determinationof the spin wave dispersion, together with the fact that no ex-ternal excitation is needed (relying on thermal spin waves, nat-urally present within the medium), are in favour of this tech-nique. A rather wide range of k vectors is available, up toabout 2.2 · rad / cm. The specific value of k can be changedeasily by changing the angle of the incident beam of light,while reversing the applied magnetic field permits to cross-check for the sign of the DMI constant. Attention should bepaid to the overheating of the portion of specimen under in-vestigation, caused by the focused laser light.
2. TR-MOKE
The complexity of the experimental technique is compa-rable to BLS, what the optical signal acquisition concerns.However, magnetostatic SW are excited here by a CPW whichhas to be prepared by lithography directly on top of the sam-ple. The k -vectors that can be excited are limited by the CPWgeometry. Broadband excitation can be obtained by a singleline (as e.g. in Ciubotaru et al. (2016)) with the drawback that3the amplitude is reduced with respect to a monochromatic ex-citation. Since high k values are preferable the CPW lineshave to be rather thin ( < µ m) which requires electron beamlithography (EBL). Boundary conditions for the generation ofplane waves have to be taken into account and it is prefer-able to have stripes of magnetic material (see supplementalmaterial of Chauleau et al. (2014)). The Gilbert damping ofthe magnetic material (Bauer et al. , 2014) limits strongly thetechnique, since the attenuation length becomes too small forthe SW to be detectable.
3. AESWS
The all-electric measurement is rather straightforward anddoes not need sophisticated equipment. Among the SW tech-niques it has a very good frequency resolution of a few MHz.The excitation of SWs is performed by CPW as in the TR-MOKE experiment, which has to be prepared by EBL in or-der to obtain high wave numbers, and therefore larger DMIsignals. Another CPW is used for detecting the spin wave sig-nal and the signal to noise ratio is in generally low, so thatonly thick FM films ( > et al. (2016)). Usually only few wavevectors are available, since each wave vector requires its CPWgeometry and values are limited to about 10 µ m − with con-ventional techniques. Only sophisticated techniques allow toreach higher values (Liu et al. , 2018).
4. SPEELS
The main advantage of SPEELS is that the wave vector k = µ m − is hundreds of times larger than for theabove techniques, allowing to explore exchange-based spinwaves. Since the energy di ff erence of spin waves with op-posite k vector grows linear with k the asymmetry detected isabout 100 times larger, so that even very small D constantscan be measured. Among the limitations, one can recall thenecessity of ultra-high-vacuum conditions and the impossi-bility of applying magnetic fields. Furthermore, spin waveenergies are usually calculated from a Heisenberg model nottaking into account the itinerant character of the spins in fer-romagnetic metals. This may lead to errors in the estimationof D . IV. Spin Orbit Torque methodA. Method overview
When a charge current is applied to a HM / FM bilayer, spintorque e ff ects occur due to the strong spin-orbit interaction atthe interface. A charge current J e flowing in HM along the x axis will generate a transverse spin accumulation σ alongthe y direction through the spin Hall e ff ect and / or inverse spinGalvanic e ff ect (Manchon et al. , 2019). The spin accumula-tion acts on the magnetic moment M and results in an e ff ec-tive damping-like magnetic field H e f f ≈ σ × M . Of particu-lar interest is the out-of-plane e ff ective magnetic field H e f f , z induced at the N´eel-type DW with typical Walker profile (Pai et al. , 2016). As shown in Fig. 20, H e f f , z depends on the angle φ between the DW moment and the x axis, and can be quan-titatively written as H e f f , z = χ DL J e cos φ , where the charge-to-spin conversion e ffi ciency (e ff ective damping-like field perunit current density) χ DL = ( π/ (cid:126) ξ DL / e µ M S t FM ). Here, ξ DL , M S , t FM are the e ff ective damping-like torque e ffi ciency,the saturation magnetization of the FM and thickness of theFM layer. B. Theory and models
Pai et al. (2016) first proposed a method to simultaneouslydetermine the spin-torque e ffi ciency and DMI field H DMI inHM / FM with PMA, based on current assisted DW propaga-tion model. As shown in Fig. 20(a), in the case of homochiralN´eel-type DW, the net H e f f , z which acts on the DW magne-tization is expected to be zero due to the opposite signs of H e f f , z for up-to-down ( cos φ =
1) and down-to-up ( cos φ = − H x is applied that is large enough to overcome H DMI , the DWin the N´eel-type walls will align parallel to H x as shown inFig. 20(b) and the resulting H e f f , z will point along the samedirection for both up-to-down and down-to-up DWs. Thus,it is expected that the out-of-plane hysteresis loop of the bi-layer can be shifted by H e f f , z , and the shift depends on themagnitude and the polarity of H x and J e . Therefore, by mea-suring the shift of the hysteresis loop as a function of H x and J e , one can determine the magnitude of charge-to-spin con-version e ffi ciency χ as well as H DMI . As shown in Fig. 21(a),an Anomalous Hall (AH) loop, i.e., an anomalous Hall volt-age versus out-of-plane magnetic field, is measured by chang-ing both the magnitude and polarity of I dc and H x . Underan in-plane bias-field of 2500 Oe the AH-loop shifts to op-posite directions at ± H e f f , z due to the spin-transfer torque e ff ect. I dc dependence of the up-to-down switching field H S W , up − to − down and down-to-up switching field H S W , down − to − up is summarizedin Fig. 21(c). Two current-related e ff ects should be consid-ered to explain the switching fields, i.e., the e ff ect of Jouleheating and H e f f , z . The Joule heating reduces the coerciv-ity H C which is proportional to I dc , i.e., H C ( I dc ); while H e f f , z H X = 0 D H C = 0 H z V AHE H X < 0 D H C > 0 V AHE H z a) b) c) d) Figure 20 (a) Schematics of the current induced e ff ective magnetic field H ef f , z at the N´eel-type chiral domain wall in a HM / FM bilayerwith perpendicular magnetic anisotropy. In the absence of an external magnetic field, H ef f , z at the DW points along the opposite direction( H ef f , z =
0) thus the motion of a domain wall v DW is along the x direction and there is no domain expansion. (b) In the presence of an externalmagnetic field H x large enough to align the N´eel-type DW along H x , the induced H ef f , z points along the same direction, which leads to domainexpansion due to the domain-wall motion toward to the opposite directions. (a) and (b) are taken from Pai et al. , 2016. (c) In the absence of H x ,there is no shift of anomalous Hall loop. (d) However, a shift of anomalous Hall loop is expected with the application of H x , which provides ameasure of DMI and spin-torque e ffi ciency. is expected to show a linear behaviour with respect to I dc ,i.e., H e f f , z ( I dc ). Thus, for a fixed I dc , the switching fieldcan be written as H S W , up − to − down = H e f f , z + H C for up-to-down switching, and H S W , down − to − up = H e f f , z − H C for down-to-up switching. By eliminating the e ff ect of Joule heating,the magnitude of H e f f , z can be easily obtained as H e f f , z = ( H S W , up − to − down + H S W , down − to − up ) /
2. As shown in Fig. 21(c), H e f f , z linearly scales with I dc , indicating that H e f f , z is indeedinduced by current-induced spin accumulation due to interfa-cial spin-orbit interaction. One can quantify the conversione ffi ciency by the slope χ ( = H e f f , z / j e ), and Fig. 21(d) showsthe magnitude and polarization of χ as a function of in-planebias magnetic-field along x and y directions. One can see that χ remains zero with the application of H y , but χ increases lin-early with H x and saturates at H x = < cos φ > = H x = < cos φ > = H x fully aligns the DWmoment, which provides a measure of H DMI . Moreover, themagnitude of χ saturates for H x > H DMI , from which one candetermine the charges-to-spin conversion e ffi ciency ξ DL . C. Experimental Results
Data of DMI constant using the SOT method are reported inTab. XVII. Initially, Pai et al. (2016) quantified the magnitudeof H DMI in Pt / Co / MgO, Pt / CoFeB / MgO and Ta / CoFeB / MgOmultilayers, and demonstrated opposite signs of the spin Hallangle for Pt and Ta. They also show sizeable DMI also existsin wedged Pt / Co(wedge) / MgO samples, where an additionalspin-orbit torque appears due to the structure lateral asym-metry and can be used for in-plane bias magnetic field freeswitching. By inserting a Pt spacer between Ta and CoFeBinterface, Chen et al. (2018b) found that both the spin-torquee ffi ciency and the DMI constant D gradually decrease as thePt thickness increases to 1 nm, which is due to the oppositesign of the spin Hall angle of Ta and Pt. To generate largerinterfacial DMI to support a skyrmion phase, an antisymmet-ric superlattice structure [HM1 / FM / HM2] N with PMA is usu-ally adopted because of the enhanced interlayer exchange cou-pling. Typical examples are Pt / Co / Ir, which has been shownto have enhanced DMI due to the same chirality at the Pt / Coand Co / Ir interface. Iin other words, the chirality is oppo-5 a) a) b)c) d) Figure 21 (a) Schematics of the measurement of the DMI field utilizing the anomalous Hall (AH) e ff ect. (b) AH-loops for Pt(4 nm) / Co(1nm) / MgO (2 nm) with I dc = ± H x = ff sets are introduced for bothAH loops. (c) Switching fields (coercive field) HSW for down-to-up (red triangles) and up-to-down (blue circles) magnetization reversal asa function of I dc . H = H ef f , z represent the centre of the AH-loop. (d) Charge to spin conversion e ffi ciency as a function of in-plane biasfield for Pt(4 nm) / Co(1 nm) / MgO (2 nm). Blue squares and red circles represent the data obtained with in-plane magnetic field along x and ydirections, respectively. All the figures are taken from Pai et al. , 2016. site for Pt / Co and Ir / Co interfaces (Moreau-Luchaire et al. ,2016). However, Ishikuro et al. (2019) demonstrated that theDMI at Ir / Co and Pt / Co interfaces shows comparable magni-tude with the same sign, leading to a reduced DMI in Pt / Co / Irtrilayer. Khadka et al. (2018) show that the DMI in trilayersof Pt / Co / Ru is comparable to Pt / Co / Ir, which could be anotheralternative material system to host skyrmions.The DMI field not only depends on the HM, but also de-pends on the details of FM, as noted before. In particular,since DMI is an interfacial e ff ect, the magnitude of H DMI is expected to be inversely proportional to the thickness ofFM. However, no clear experimental results with the SOTmethod show this trend indicating DMI has a complicated re-lation to the FM thickness. Interestingly, it has been shownin W / FM / MgO structures, with FM = CoFeB, FeB, that H DMI changes sign upon increasing the thickness of FM (Dohi et al. ,2019), indicating that there could be competing mechanismscontributing to DMI, and to the interfacial spin-orbit interac-tion (Cao et al. , 2018b; Chen et al. , 2018a; Yun et al. , 2018).As shown above, the magnitude of DMI has been widely measured in heavy metal / ultrathin ferromagnetic metal sys-tem. Recently, measurements have also demonstrated thepresence of DMI in heavy metal / magnetic insulator het-erostrucures, and the DMI is strong enough to stabilizeskyrmions (Ding et al. , 2019; Shao et al. , 2019) and for fastdomain motion (Avci et al. , 2019). Magnetic insulators areattractive due to their lower Gilbert damping in comparisonwith ultrathin ferromagnetic metals. Usign SOT, Ding et al. (2019) quantified the DMI in thulium iron garnet (TmIG) / Ptbi-layers, and have shown that the magnitude of the DMI con-stant is about 1-2 orders smaller that in metallic heterostruc-tures because of the much lower saturation of TmIG (50 –110 emu · cm − ). They also showed that the magnitude of theDMI constant is inversely proportional to the TmIG thickness,indicating that the DMI in TmIG / Pt bi-layer is indeed an in-terfacial e ff ect.6 FM Bottom NM Top NM | H DMI | | D | | D s | Ref (nm) (nm) (nm) (mT) (mJ / m ) (pJ / m)Co(1) Pt(4) MgO(2) 500 3.0 Pai et al. , 2016CoFeB(1) Pt(4) 250 1.8
CoFeB(1) Ta(6) 25 0.6
Co(0.65) Pt(4) 110 1.45
Co(0.80) Pt(4) 200 1.99
Co(0.92) Pt(4) 290 2.12
Co(1.00) Pt(4) 400 2.64
Co(1.10) Pt(4) 450 2.91
Co(1.23) Pt(4) 310 2.49
Co(1.43) Pt(4) 160 1.62
Co(1.52) Pt(4) 100 1.37 [Pt(0.6) / Co(0.9) / Ir(0.6)] x Pt(2) MgO(2) 80 0.4
Ishikuro et al. , 2019[Pt(0.6) / Co(0.9) / Cu(0.6)] x Pt(2) MgO(2) 180 1.8 [Pt(0.6) / Co(0.9) / Ir(0.6)] x Pt(2) MgO(2) 110 0.7
Co(0.9) Ir(7) MgO(2) 220 1.6
Ir(1) / Co(0.9) Pt(2) Ru(1) 150 1.1
Co(1.2) Pt(4) Ir(1) 65 ± ± ± Khadka et al. , 2018Co(1.2) Pt(4) Ir(2) 81 ± ± ± Co(1) Pt(4) Ir(1) 130 ±
13 2.06 ± ± Co(1) Pt(4) Ru(2) 140.4 ±
14 2.66 ± ± Co(0.8) Pt(4) Ru(2) 212 ±
21 2.40 ± ± Co(0.8) Pt(4) Ru(3) 218 ±
22 2.30 ± ± Co(0.8) Pt(4) Ru(4) 238.6 ±
24 2.56 ± ± [Pt(1) / Co(0.8) / Ru(1.3)] x Pt(3) 220 ±
22 2.07 ± ± FeB(0.96) α -W(4) MgO(1.6) 25 n.a. n.a. Dohi et al. , 2019FeB(0.86) 30 n.a. n.a.FeB(0.76) 18 n.a. n.a.FeB(0.56) 50 n.a. n.a.Co(0.5) Pt(4) Ta(2) 110 ±
10 1.01 ± ± Yun et al. , 2018Ta(4) 110 ±
10 0.75 ± ± Ta(6) 80 ±
10 0.70 ± ± Ta(8) 190 ±
10 1.4 ± ± CoFeB(1.4) Mo(4) MgO(2) 20 0.35
Chen et al. , 2018bCoFeB(1.3) Ta(3) MgO(1) 30 0.22
Chen et al. , 2018aTmIG a (2.9) Pt(7) n.a. 20 ± ± ± et al. , 2019 b thulium iron garnet Table XVII Overview of DMI measurements of various FM materials by spin-orbit torque induced e ff ective field under an in-plane biasmagnetic field. FM and NM stand for ferromagnetic and non-magnetic layer, respectively. D is the interfacial DMI constant and D S = D · t FM ,where t FM is the thickness of the FM. Numbers in roman were quoted in the reviewed papers, while numbers in italics were either extractedfrom figures or calculated using the parameters provided. D. Advantages and Limitations
Being di ff erent from the methods based on DW motionand asymmetric spin-wave propagation, the method basedon current-induced shifts of the anomalous Hall loops is astraightforward way for determining the DMI field withoutinvolving complicated mathematical models. However, thismethod requires perpendicular magnetic anisotropy of the fer-romagnetic material, which typically limits this method toultrathin ferromagnetic materials. Non-square shapes of theAH-loops also lead to inaccuracies in the determination of H DMI . To determine the DMI constant, one requires a numeri-cal value for the exchange sti ff ness; usually an assumed valueis used. Moreover, this method is not capable to determine thesign of DMI. V. Comparison of the methods and problems in thedetermination of DA. Comparing the DMI constants obtained by the differentmethods
Many papers discuss the advantages and disadvantages ofdi ff erent methods to measure the interfacial DMI. One impor-tant point is to distinguish static or quasi static methods (as themethods discussed in sections II.C.3 to section II.C.6) fromdynamic methods as current- (section II.C.1) or field-driven(section II.C.2) DW motion, spin wave-based methods (sec-tion III) or SOT-based methods (section IV).Within the dynamic methods one still has to di ff erentiatedi ff erent dynamic regimes, such as the creep and flow regimefor DW motion, and the spin wave regime that is concerned7with much shorter time scales. Quasi static methods basedon imaging usually rely on observing the DW or a number ofdomains in a certain position. Therefore they may depend onpinning sites and the result of the measurement may vary dueto inhomogeneities and defects. Analytical models and simu-lations usually do not take these imperfections into account,which may lead to intrinsic errors, although considering alarger sample area may reduce such an e ff ect. Also nucleationof domains occurs at pinning sites and single measurementsmay not be representative. In addition, strain in the thin filmsmight play a role and in fact, several recent works investi-gate defect- or strain-induced DMI (Deger, 2020; Fern´andez-Pacheco et al. , 2019; Michels et al. , 2019). More research hasto be dedicated to this complex, and a bit exotic topic, in orderto understand defect related di ff erences in the measurements.Dynamic methods often average out these e ff ects. Anintermediate regime can be considered the creep DW mo-tion, where DWs move due to thermal activation from onepinning site to the next. These measurements show gener-ally good agreement with static methods, such as the strayfield measurements by NV center magnetometry (Gross et al. ,2016; Soucaille et al. , 2016), nevertheless measurements inthe creep regime require models with strong assumptions. Infact, concerning measurements of DW motion in the creepregime by asymmetric bubble expansion, there is an ongoingdiscussion on how to evaluate the DMI from the domain ve-locity profile, and how to fit the data to extract the DMI field.In addition, the evaluation of the DMI constant depends onthe DW width or the DW energy, parameters known only upto a certain accuracy. They are often calculated from the ex-change sti ff ness and the anisotropy constant. Both parame-ters should be therefore determined experimentally, which re-quires a substantial e ff ort especially for the exchange sti ff nessin thin films, and the direct measurement of the DW width byimaging may be an alternative.Spin wave methods have the advantage of a more straight-forward evaluation of the DMI constant by measuring a fre-quency shift, which is possible with high accuracy. The most”handy” method currently is BLS, as it does not require anyexternal spin wave transduction or any kind of patterning, con-trary to TR-MOKE and AESWS. The BLS approach is com-pletely di ff erent from domain and DW methods, as the in-fluence of defects, grain boundaries, inclusions and surfaceroughness inhomogeneities play a marginal role. In fact, thesample has to be magnetically saturated in the film plane, evenif this can be challenging for very thin films with strong PMA.Moreover, for materials with relatively high damping, the spinwave signal can be di ffi cult to measure or can be a ff ected byrather large errors (due to broad peaks in the BLS spectra, orsmall signal in TR-MOKE and AESWS), setting a lower limitto the values of D that may be measured by spin wave meth-ods.These various experimental di ffi culties explain why a di-rect comparison of di ff erent methods on the same sample arerarely reported in the literature. Only in four recent papers(Kim et al. , 2019a; Shahbazi et al. , 2019; Soucaille et al. , 2016; Vaˇnatka et al. , 2015), a quantitative comparison of thevalues of D is reported, measured by both BLS and DW-basedmethods. In general, a reasonable agreement is attained forthe di ff erent systems if BLS results are compared to those ex-tracted from analysis of DW motion in the flow-regime. In-stead, when DW motion is analysed in the creep regime, thevalues obtained by DW-based methods are generally signif-icantly lower than those obtained by BLS. In the latter case,the level of discrepancy may be also strongly influenced by thematerial properties and layers combination, as well as by thespecific thickness and chosen substrate (Shahbazi et al. , 2019;Soucaille et al. , 2016). In fact, since in the thin film systemsDMI is an interface e ff ect, details concerning the structuraland chemical quality of the materials, in particular of the in-terfaces, play a crucial role.Concerning methods that determine the DMI field underthe influence of a current (such as current induced DW mo-tion and SOT field measurements), the main di ffi culty is theinterpretation of the result, taking into account the di ff erentcontributions of spin torques and spin orbit torques that occur.Damping and field like torques may have di ff erent origins asthe spin Hall e ff ect and the Rashba e ff ect, and both contributeto the observed dynamic. Furthermore, in metallic stacks, thecurrent flow in the plane is not well defined as it is partly in theFM and partly in the HM material, depending on the relativeresistances and the interface scattering and transparency.An attempt to classify the applicability of three methods(BLS, SOT e ffi ciency and DW velocity) according to DMIstrength and FM layer thickness was performed in Kim et al. (2019a) (see Fig. 22). In a certain range all three methodsare applicable and a direct comparison of di ff erent methodson the same sample is possible. In fact on the investigatedPt(2.5) / Co(0.9) / X(2.5) (X = Ti, Ta, Al, Pt) they find an excel-lent agreement within the 5% di ff erence. Only in the caseof W 10% of di ff erence between DW based and SW basedmethod occurs, while Cu presents a much larger 40% of dif-ference.The above considerations are corroborated by a synopticview of the Tables of results and Figs. 23 and 24 discussedlater: one may find that in several examples of nominally sim-ilar systems, di ff erent groups obtained relatively di ff erent re-sults. In this context, more systematic multi-technique investi-gation of the DMI on the same samples, for di ff erent materialscombination and layer thicknesses (including a cross-checkconcerning the mutual consistency of results relative to di ff er-ent regions on the same sample) would be highly desirable. B. Comparing the DMI constants in different materials
The two most popular stacks available in the literature arebased on CoFeB / MgO and Pt / Co interfaces. Fig. 23 show thedata for the X / CoFeB / MgO thin films with six di ff erent, andmost used, bottom layers X (with X = IrMn, Hf, Pt, Ta, TaN,W). The methods not able to determine the sign are shownin gray, and in general have values compatible in magnitude8
Figure 22 Scheme of the applicability range of BLS, the SOT e ffi ciency (cid:15) ST and the field induced DW velocity measurement method. Themeasurements were performed on trilayers consisting of Pt(2.5nm) / Co(0.9nm) / X (2.5nm) with X = (Al, Au, Cu, Pt, Ta, Ti, and W). From Kim et al. , 2019a.Figure 23 Most popular literature data of the DMI constant D s for the X / CoFeB / MgO thin films, where the bottom layers X (IrMn, Hf, Pt, Ta,TaN, W) are the most popular in the literature. The shapes of the symbols refer to di ff erent experimental methods, while their size and colorreflect the amplitude and the sign of D s , respectively, as specified in the right legends. Data for methods not able to determine the sign of D s (i.e, domain pattern, stripe annihilation and spin torque) are in grey. Most of the data ( > with the other methods. The CoFeB / MgO bilayer alone hasa small positive D s = .
13 pJ / m (Chen et al. , 2018b), which is compensated in the case of Pt and IrMn showing a negative D s for the largest thickness, up the maximum value of about9 Figure 24 Most popular literature data of the DMI constant D s for the Pt / Co / X thin films, where the top layers X (Pt, Ir, Au, Ta, Al, Gd, Mg,AlO x , GdO, MgO). The shapes of the symbols refer to di ff erent experimental methods, while their size and color reflect the amplitude and thesign of D s , respectively, as specified in the right legends. Data for methods not able to determine the sign of D s (i.e, domain pattern, stripeannihilation, nucleation, and spin torque) are in grey. The thickness of the bottom layer of Pt varies between 0.8 nm and 30 nm, with morethan 75% of the data for a thickness larger that 3 nm. -1.5 for the Pt layer. On the contrary, W and TaN increasethe positive value of D s , up to about 0.7 pJ / m for the W layer.Less clear is the contribution of Hf and Ta. Hf has negativeD for current induced domain wall motion, while positive forcreep regime and BLS. Ta has a variation of D from negativeto positive when increasing the thickness, measured by currentinduced domain wall motion (Torrejon et al. , 2014). On theother hand, Karnad et al. (2018) reported di ff erent signs whenmeasuring by creep regime and by current induced domainwall motion. This contradiction may originate from the smallvalue of D in those Ta samples.Even more intriguing is the situation of Pt / Co / X film, asshown in Fig. 24. Also in this case, the gray data are for meth-ods non defining the sign. The sign of D is mostly negative be-cause it is dominated by the Pt / Co interface. The oxide groupsof AlO x , GdO and MgO shows the largest (negative) DMI val-ues, with the record of | D s | = pJ / m in Pt(4) / Co(1) / MgO(2)thin films measured using the spin orbit torque method. Notethat Pt / Co / MgO films with a thick Co (up to 20 nm) have apositive sign for D (not shown, as D s is not available) whenmeasured by spin waves methods di ff erent from BLS (Ka-sukawa et al. , 2018; Lee et al. , 2016). The other elementsshow more or less contradicting results, as for Ta, Pt, and Ir.The popular stacks Pt / Co / Pt and Pt / Co / Ir / X (where X is Pt or Ta) are analysed in more detail in Figs. 25. These plots clearlyshow all the problems due to film preparation, di ff erences ininterface properties of top and bottom layers, etc. which ex-plain the high variability in magnitude and sign of the D s . Forinstance, films with the same thickness of the two Pt layersshould, in principle, have a zero DMI, as for epitaxial films.As a matter of fact, a perfect compensation of the top / bottomcontributions is rare, and small values of DMI are found, bothpositive and negative. Even more, one expects to see o ff thediagonal, i.e. when one of the layers is thicker than the other,a prevalence of the sign, which is not occurring in the data.As a matter of fact, current induced motion experiments shownegative values independent of the thickness of top or bottomlayer (Franken et al. , 2014), while creep regime experimentsgive mostly positive values.A similar puzzle in the interpretation of the results concernsthe Pt / Co / Ir / X stack, which show a large variety in magnitudeand sign. For example, the introduction of an Ir thin layer inPt / Co / Ir / Pt reduces the magnitude of D and then changes thesign for thicker Ir (Hrabec et al. , 2014).Despite all these di ffi culties, a few conclusions can bereached concerning the sign of D. If the order of the stackis reversed, then one finds a reversal of the sign by symme-try. The strength of D may be di ff erent due to the di ff erent0 Figure 25 Literature data of the DMI constant D s for Pt / Co / Pt, and Pt / Co / Ir / X thin films, where the top layer X is Pt or Ta (not specified). Theshapes of the symbols refer to di ff erent experimental methods, while their size and color reflect the amplitude and the sign of D s , respectively,as specified in the right legends. Data for D s = growth conditions of a reversed stack, but the sign reversal ise ff ectively verified as in Pt / Co / Ta and Ta / Co / Pt trilayers mea-sured by BLS (Cho et al. , 2017). All the BLS experiments areconsistent with a negative (positive) value of D for a Pt under-layer (top layer) regardless the FM composition, as veried forCoFeB,Co, Co / Ni, Ni Fe , independently whether the toplayer is an oxide or another heavy metal. This result is alsoconsistent with the sign detected for current induced domainwall motion and field driven motion in the flow regime.As noted before, the contribution of Ir is less understood.Ir has a positive D measured by BLS in Ir / CoFeB / MgO anda negative value for Ir / Co / MgO. This fact is in contrast withthe theoretical prediction that the contribution of the interfaceIr / Co was opposite to that of Pt / Co (Yamamoto et al. , 2016;Yang et al. , 2015). This would make it possible to increase theDMI by fabricating Pt / Co / Ir trilayers. As discussed in Sec-tions II.C.2 and IV, the combination Pt / Co / Ir seems to givesmaller values that the simple addition of the D values.
C. Influence of growth conditions on the DMI constant
Due to the large spread of used materials, the investiga-tion of the influence of growth condition on DMI is notconclusive so far. Wells et al. (2017) studied the e ff ect ofsputter-deposition condition on DMI in Pt / Co / Pt structures.They found that the growth temperature modifies the inter-facial roughness. The di ff erent quality of the top Pt / Co in-terface and the lower Co / Pt one introduces a structural in-version asymmetry, which results in a net DMI field in thissymmetric structure. Regarding the e ff ect of post annealing,Khan et al. (2016) studied the influence of annealing on DMIin Ta / CoFeB / MgO. Here the DMI field H DMI is determinedby magnetic field driven domain pattern in the creep regime.They found that both H DMI and the DMI constant D varies with temperature, reaching a peak at 230 o C and then de-creasing as the temperature is further increased. They alsofound that the dependence of interfacial perpendicular mag-netic anisotropy field H K on annealing temperature follows asimilar trend as DMI, suggesting a connection between theseparameters. They suggested that the increase of H DMI and H K is due to an improved ordering of atoms at the Ta / CoFeB in-terface. Higher annealing temperature leads to di ff usion ofB atoms out of CoFeB as well as intermixing at the inter-face, which significantly reduces H DMI and H K . Cao et al. (2020) also reported a similar trend in annealing temperaturedependence of H DMI in Pt / Co / x / MgO structures (x = Mg orTa) investigated by magnetic field driven domain motion. Amaximum H DMI is obtained at an annealing temperature of300 o C, which is independent of the MgO thickness. Simi-lar to Khan et al. (2016), they also propose that the enhanced H DMI is due to the improved crystalline quality upon anneal-ing. However, Furuta et al. (2017) used current-driven do-main wall motion to study the e ff ect of annealing on H DMI inPt / [Co / Ni] structures. They can show that annealing causesa significant reduction of H DMI , domain wall velocity, per-pendicular magnetic anisotropy, as well as spin-orbit torques,which is attributed to the di ff usion of Co atoms across thePt / Co interface.
D. Outlook and conclusion
In this review more than 100 articles measuring interfacialDMI were analyzed and the number of publications is stillgrowing, indicating the considerable interest in this topic. Re-cently the interest shifts away from the traditional HM / FM bi-layers towards synthetic antiferromagnets (SAF) and oxides.SAFs exploit the RKKY exchange interaction between twoultrathin FM layers separated by a non magnetic spacer layer,1which can be tuned by the spacer layer thickness (Duine et al. ,2018). In these systems, with a HM as spacer layer, it wasshown that the DMI is enhanced by the dipolar field betweenthe FM layers (Fern´andez-Pacheco et al. , 2019; Meijer et al. ,2020). Besides having the advantages of antiferromagnets,such as negligible strayfields and stability against magneticfields (Baltz et al. , 2018), these SAFs exhibit asymmetricDWs and spin wave dynamics. The possibility of tailoringDMI in SAFs makes them extremely interesting for appli-cations of chiral magnetism and topological spin structures(Legrand et al. , 2020; Vedmedenko et al. , 2020). Since thistopic goes slightly beyond the review, it is not included inthe tables, however it is worth mentioning the upcoming in-terest in SAFs (Bollero et al. , 2020; Fern´andez-Pacheco et al. ,2019; Han et al. , 2019; Meijer et al. , 2020; Tanaka et al. , 2020;Tsurkan and Zakeri, 2020).A second topic that is coming up recently, and which bringsus to the microscopic origin of DMI, is DMI in FM oxides(Wang et al. , 2020) and oxidized metallic FM (Nembach et al. ,2020). In the first it is suggested that the di ff erent band gapsin YIG and GGG lead to a band o ff set in conductance and va-lence bands and therefore cause a Rashba-induced DMI (Yang et al. , 2018b). In the latter the enhanced DMI is suggestedto be caused by an electric dipole moment induced by hy-bridization and charge transfer at the oxygen / FM metal inter-face (Belabbes et al. , 2016) This Rashba-induced DMI (Kim et al. , 2013) was also observed for a Co / graphene interface(Yang et al. , 2018b). In both cases density functional the-ory (DFT) calculations were fundamental to interpret the re-sults and the calculations showed that the DMI originates fromthe FM layer, instead of the HM layer (Yang et al. , 2018a,b).However, the fact that hybridization at the interface plays anessential role makes the analysis complex and categorizingmaterials or stacks according to their DMI strength becomesalmost impossible. Interface intermixing, interface roughness,dead layers and proximity e ff ects all were known to a ff ectthe DMI. In fact, a detailed interface characterization shouldbe performed in order to obtain a complete picture. E ff ec-tively, the interfacial DMI may depend on the interface prop-erties much more than on material properties themselves. TheRashba-induced DMI is closely related to materials for spin-to-charge conversion which recently attracted attention (Ding et al. , 2020; Rojas-S´anchez and Fert, 2019). A key feature isagain the mechanism of SOC, but not necessarily a materialwith strong SOC, as in HM, is required, but SOC can also beinduced at the interface by the Rashba e ff ect. It is thereforeobvious that SOC related phenomena cannot be disentangledand have to be treated in a common theoretical approach. TheDMI was firstly described by a phenomenological thermody-namic theory, but it appears now fundamental, for develop-ing materials designed for future applications, that the micro-scopic origin of DMI has to be understood in more detail. Acknowledgment
This work was supported by the EMPIR project 17FUN08TOPS. This project has received funding from the EMPIRprogramme co-financed by the Participating States and fromthe European Unions Horizon 2020 research and innovationprogramme. F.G.S. acknowledges the support from ProjectNo. SA299P18 from the Consejer´ıa de Educaci´on of Junta deCastilla y Le´on. L.C. and C.B. acknowledge the support fromDFG through SFB 1277. M.K. thanks H. T. Nembach, J. M.Shaw and A. Magni for fruitful discussions.
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