Understanding current-driven dynamics of magnetic N\'{e}el walls in heavy metal/ferromagnetic metal/oxide trilayers

We consider analytically current-driven dynamics of magnetic N\'{e}el walls in heavy metal/ferromagnetic metal/oxide trilayers where strong spin-orbit coupling and interfacial Dzyaloshinskii-Moriya interaction (i-DMI) coexist. We show that field-like spin-orbit torque (FL-SOT) with effective field along $\mathbf{n}\times\hat{\mathbf{J}}$ ($\mathbf{n}$ being the interface normal and $\hat{\mathbf{J}}$ being the charge current direction) and i-DMI induced torque can both lead to Walker breakdown suppression meanwhile leaving the wall mobility (velocity versus current density) unchanged. However, i-DMI itself can not induce the"universal absence of Walker breakdown"(UAWB) while FL-SOT exceeding a certain threshold can. Finitely-enlarged Walker limits before UAWB are theoretically calculated and well explain existing data. In addition, change in wall mobility and even its sign-inversion can be understood only if the anti-damping-like (ADL) SOT is appended. For N\'{e}el walls in ferromagnetic-metal layer with both perpendicular and in-plane anisotropies, we have calculated the respective modifications of wall mobility under the coexistence of spin-transfer torque, SOTs and i-DMI. Analytics shows that in trilayers with perpendicular anisotropy strong enough spin Hall angle and appropriate sign of i-DMI parameter can lead to sign-inversion in wall mobility even under small enough current density, while in those with in-plane anisotropy this only occurs for current density in a specific range.


Introduction
Pure current-induced domain wall propagation in magnetic nanostructures has attracted intensive attention for decades starting from academic interests in understanding the interplay between itinerant spinful electrons and localized magnetic moments [1,2,3,4]. In monolayer ferromagnetic nanostrips, in-plane currents drive domain walls to propagate along the direction of electron flow through the spin transfer process [5,6,7,8,9,10,11,12,13,14,15], which leads to promising applications in future magnetic racetrack memories [16,17], shift registers [18,19] and memristors [20,21], etc. However, in these monolayers the wall velocity is at most 10 2 m/s even when the current density is up to 10 8 A/cm 2 . This comes from the fact that spin transfer torques (STTs) therein can not be strong as the exchange energy avoids abrupt changes in magnetization texture. To improve the current efficiency, the current-perpendicular-to-plane (CPP) configuration in narrow and long spin valves is proposed [22,23,24]: to reach the same velocity level (10 2 m/s), the current density for "planar polarizer" case is reduced to 10 7 A/cm 2 while that for "perpendicular polarizer" can even be lowered to 10 6 A/cm 2 . However, the rapidly increasing CPP cross-section area largely offsets the decrease in current density. Even if the current is forced to focus on wall region, precise dynamical synchronization in real experiments remains challenging.
Meantime, analytics with Lagrangian functional [68,69,70] and simulations [71,72,73,74,75,76,77] based on Landau-Lifshitz-Gilbert (LLG) dynamical equation [78] have been performed to explain Néel wall dynamics in HM/FMM/Oxide trilayers in the framework of one-dimensional collective coordinate model (1D-CCM). All these works focus on two novel features in experiments: (i) Walker breakdown suppression thus high wall velocity and (ii) wall motion opposed to electron flow and the corresponding "polarity sensitivity". Historically, the Rashba-SOC-induced FL-SOT is first proposed to explain both novelties [71,72,73]. In addition, the novelty (ii) is also reproduced numerically by only ADL-SOTs from SHE in IPMA systems [74]. In 2017, Risinggåd and Linder proposed the "universal absence of Walker breakdown" (UAWB) of Néel walls in PMA systems for strong enough Rashba or SHE effect [79] with the coexistence of i-DMI. However, to our knowledge there are no explicit analytical expressions for finitely enlarged Walker limit before the occurrence of UAWB. In addition, theoretical criteria for sign-inversion in wall mobility (velocity versus current density) and the corresponding "polarity selection rule" for trilayers with both PMA and IPMA are absent. The role of i-DMI in all these processes is also unclear. Explorations to these issues constitute the main content of this paper.
The rest of this paper is organized as follows. In section 2 the system set up and its modelization are briefly introduced. Also, the static Néel wall configurations and the dynamical equations for FMM layers with both PMA and IPMA are presented. Then in section 3 within 1D-CCM we provide analytical expressions of finitely enlarged Walker limits by FL-SOT and/or i-DMI induced torque before UAWB. In section 4, theoretical criteria for sign-inversion in wall mobility and the corresponding "polarity selection rule" for FMM layers with both PMA and IPMA are provided under the coexistence of ADL-SOT and i-DMI. Finally, concluding remarks are provided in the last section.

Modeling and preparations
We consider an HM/FMM/Oxide trilayer with a domain wall formed in FMM layer. Generally, the FMM layer has strong PMA or IPMA. Typical example for the former (latter) case is Co (NiFe). Meanwhile, the HM layer is composed of Pt or Ta in most experiments. For both cases, the in-plane charge current flows along the long axis of the strip with density j a . As passing through the trilayer, the charge current splits into two parts. Suppose j F (j H ) to be the current density in FMM (HM) layer. A simple circuit model tells us that where t F (t H ) and σ F (σ H ) are the thickness and conductivity of the FMM (HM) layer, respectively. For the most common FMM (Co, Ni, Fe) and HM (Pt, Ta, Ir) materials, the conductivity varies from 10 to 20 (µΩm) −1 . For simplicity in this work, we set For trilayers with PMA, the coordinate system is depicted in figure 1a: e x is along the long axis of strip in which charge current flows, e z is the interface normal and e y = e z × e x . The easy (hard) axis of the FMM layer with PMA lies in e z (e y ) direction. While for trilayers with IPMA (see figure 1b), e z (e y ) is along the long axis (interface normal) of the strip which is the easy (hard) axis, and e x = e y × e z .

Dynamical equation
In thin enough strips, most of the nonlocal magnetostatic energy can be described by local quadratic terms of M x,y,z via three average demagnetization factors. Thus in the absence of any external magnetic field, the total magnetic energy density functional takes the following form in which E ex = J(∇m) 2 and J(> 0) is the exchange stiffness. The i-DMI contribution is where D is the magnitude of i-DMI vector and n is the interface normal. The total magnetic anisotropy energy density , in which n E (n H ) and k E (k H ) are the unit vector and total anisotropy coefficient in easy (hard) axis, respectively. The time evolution of magnetization texture M(r, t) ≡ M s m(r, t) with fixed saturation magnetization M s is governed by the generalized LLG equation where H eff = −(δE tot /δm)/(µ 0 M s ) is the effective field, γ and α are the gyromagnetic ratio and phenomenological damping coefficient, respectively. Note T STT only appears for inhomogeneous magnetization texture with [6,7] where B J = g e µ B P j F /(2eM s ) ≈ µ B P j a /(eM s ), with e, g e , µ B being the absolute value of electron charge, the electron g−factor and Bohr magneton, respectively. P is the spin polarization of j F . The two terms in the right hand side of equation (3) are the so-called adiabatic and non-adiabatic STTs, respectively. They are the continuous counterparts of the Slonczeswki [2] and FL STTs in spin valves. β is the dimensionless coefficient describing the relative strength of the nonadiabatic STT and usually of the same order as α. Previous works have verified that in traveling-wave mode STT-driven walls always move in the direction of electron flow, which is attributed to the existence of nonadiabatic ingredient (β−term). Generally SOTs have both FL and ADL components. Each component includes the contributions from both SHE and Rashba SOC. In this work we focus on domain wall dynamics rather than physical sources of SOTs, thus T SOT can be written as Both H FL and H ADL stem from various physical processes and have the unit of magnetic field. Their ratio varies in a wide range for different trilayer systems.

Static wall configuration
In the absence of external charge current, the magnetization texture eventually evolves into some equilibrium state. The ground state is the one with a single domain which is of little interest. Alternatively, the metastable state with a wall separating two magnetic domains is of great importance for both academic and industrial interests. In this subsection, we provide static wall configurations for trilayers with both PMA and IPMA. First we focus on FMM layers with PMA (see figure 1a), thus n = n E = e z and n H = e y . For statics, the magnetization is no longer function of time but only varies with location along x−axis in 1D-CCM. By dropping a constant −k E µ 0 M 2 s /2, the total energy density turns to where θ (φ) is the polar (azimuthal) angle of the magnetization vector (see figure 1a) and a "prime" means d/dx. Physically, a static wall configuration should provide a minimum of the total magnetic energy. For this purpose, first we should have φ ′ ≡ 0 to suppress the exchange energy, which makes φ a collective coordinate. Then we introduce the Lagrangian functional L = Ld 3 r with Lagrangian density and the boundary condition with η = ±1 coming from the two-fold symmetry of magnetic anisotropy in easy axis and can be viewed as the topological charge of this wall. The corresponding Euler equation together with the static condition ∂φ/∂t = 0 lead to Its soliton solution is the well-known Walker profile [80] ln tan where x 0 denotes the wall center position. Putting it back into (5), one has The energy minimization strategy then naturally demands that sin φ = 0 and cos φ = −ηsgn(D), where "sgn" is the sign function. This means that in PMA systems, the i-DMI favors Néel wall and further selects wall polarity (sign of m y ).
Next we turn to IPMA systems (see figure 1b) in which n = n H = e y and n E = e z . After similar process, we obtain the same θ−profile as in equation (10) except for x(x 0 ) → z(z 0 ), under which the total energy density of IPMA systems becomes To minimize the first term in the right hand side of the above equation, one also need sin φ(z) ≡ 0 which eliminates the i-DMI term. This implies that in IPMA systems, the Néel wall is naturally the result of energy minimization strategy and the i-DMI does not select wall polarity.

General scalar LLG equations
By taking into account the conversion between Descartes and spherical coordinate systems, the vectorial LLG equation (2) is transformed into the following scalar pair For PMA systems, one has with where a "dot (prime)" means ∂/∂t (∂/∂x). While for IMPA system, alternatively one has in which a "prime" means ∂/∂z.

Walker breakdown suppression by FL-SOT and/or i-DMI
In this section we present analytical expressions of finitely enlarged Walker limit before UAWB in the absence of ADL-SOT. We will show the different roles of FL-SOT and i-DMI in modulating the STT-initiated traveling-wave mode of domain wall.

Brief review of wall dynamics under pure STT
For a Néel wall in an isolated FMM and driven by pure STTs from axial currents (H FL = H ADL ≡ 0, D ≡ 0 and B J = 0), the static wall profile can be generalized to [6,7] ln tan where r = x(z) for PMA (IPMA) case, ∆(φ) is the same as in equation (9) and v(t) is the wall velocity. Then for trilayers with both PMA and IPMA, we havẽ Putting back into the scalar LLG equations, one has v(t) with By settingφ = 0 in the above equation, the Walker limit is obtained as the result of constraint | sin 2φ| ≤ 1. Here we neglect the breathing effect of dynamical wall width ∆(φ). Equations (20) and (21) show that when |J e | ≤ J W the wall propagates along electron-flow direction in a traveling-wave mode with the velocity −βB J /α and the tilting angle where φ 0 = arccos[−ηsgn(D)] for PMA case and φ 0 = kπ, k ∈ Z for IPMA case.

General framework under the coexistence of STT, FL-SOT and i-DMI
First we focus on trilayers with PMA. As illustrated, a traveling wave described by equation (18) can always be adopted to perform analytics [79]. Under this wall profile, A andB in equation (14) becomes Putting back to the scalar LLG equations, and then integrating over the whole strip ( with the functional Obviously, equation (25) shares the same structure with equation (20), except for the substitution of "H K sin 2φ" by the functional H(H K , H FL , D, φ), thus leads to the rediscovery of equation (21). This means neither the FL-SOT nor the i-DMI can change the wall mobility. However they do suppress the Walker breakdown thus increase the upper limit of wall velocity in traveling-wave mode.
To see this, we first define Then by settingφ = 0, the second line in equation (25) provides Next we set (H FL ) W as the absolute effective field strength when |j a | = J W . Then by defining equation (28) is rewritten as By maximizing the absolute value of its right hand side, we get the modified Walker limit J FL+DM W . If a W > 1, j a /J W → ∞ when cos φ = −ηsgn(α − β)/a W thus leading to infinite J FL+DM W (i.e. UAWB), which is essentially the same as that from equation (8) of Ref. [79]. When a W = 1 and b = 2, without losing generality we set "ηsgn(α − β) ≡ −1". As φ → 0 + one has |j a /J W | ≈ |2φ −1 (2 − φ 2 − b)| → +∞. Note that a special parameter combination "a W = 1 and b = 2" leads to |j a /J W | = 2| sin φ|, which gives a doubled Walker limit. Except for this special case, one would see that finite J FL+DM W can only exist for 0 < a W < 1.
On the other hand for trilayers with IPMA, the only difference is that equation meanwhile leaving all other definitions and discussions unchanged. However, for both PMA and IPMA cases, it is hopelessly complicated in mathematics if finite a and b coexist. In the following subsections, we provide explicit analytics on finite enlargement of Walker limit in the presence of either finite a (FL-SOT) or finite b (i-DMI).
To make the above mathematics more physical, here we take a limit case in which the Rashba effective field [46,47] solely contributes to H FL . Here α R is the Rashba parameter describing the Rashba SOC strength. The resulting λ is the conversion factor from current density to the Rashba field and is about 10 −8 ∼ 10 −9 T cm 2 /A [25,26]. Then the definition of a W in equation (29) can be rewritten as For magnetic parameters of Co-Ni FMM which is a typical PMA material (see the first column of Table 1, adopted from Ref. [79]), one has α 0 R = 9.54 meV · nm. Hence the above result means that for strong enough Rashba SOC (|α R | ≥ α 0 R ), the UAWB occurs, while for |α R | < α 0 R the Walker limit is finitely enlarged as described in equation (34). For this reason, the horizontal axis of figure 2 can also be set as "α R /α 0 R ". In addtion, the critical condition "a W = 1" leads to "α th R ∝ |β/α − 1|" relationship in the threshold above which no Walker breakdown occurs, thus explicates existing numerical simulations, for example figure 3b in Ref. [73].
At last, for IPMA systems all the definitions and discussion are the same except for that the enlarged Walker limit achieves at φ = arcsin u 2 . When a W → 1 − the asymptotic behavior of φ turns to On the other hand, for magnetic parameters of permalloy FMM which is a typical IPMA material (see the second column of Table 1, adopted from Ref. [74]), one has a smaller critical Rashaba parameter α 0 R = 0.64 meV · nm due to the relatively large wall width. This means that in IPMA systems, the UAWB is more likely to occur.

Coexistence of STT and i-DMI
In this subsection, we study whether i-DMI itself can lead to UAWB, thus we have a = 0 and b = 0. As usual, we first focus on PMA systems. By setting cos φ ≡ s ∈ [−1, +1], another function G(s) can be defined from (28) as G(s) is also nonnegative and only equals to zero when s = ±1 or b/2. By requiring G ′ (s) = 0, three extremal sites are obtained Standard calculus tells us that function G(s) always approaches maximum at s = s 3 when b > 0 (s = s 2 for b < 0). Then the modified Walker limit [maximum absolute value of the right hand side of equation (40)] and the corresponding φ reads Meanwhile, simple calculation yields that 7b 2 /8 + 1 < G(s 3 ) < b 2 + 5, which confirms the Walker breakdown suppression effect at finite b (i.e. i-DMI). However since no singularities appear, UAWB does not occur under the pure action of finite i-DMI. For IPMA systems, again the definitions and discussions are similar. The enlarged Walker limit can be described by equation (42) except for that the corresponding extremum point locates at φ = arcsin s 3 .
Also, one should note that although the enlarged Walker limits in PMA and IPMA cases share the same analytical form, they will take different value under the same i-DMI strength D. To see this, we rewrite the definition of b in equation (27)  The qualitative role of FL-SOTs and i-DMI has been extensively discussed and now is clear. The driving current pulls the magnetization out of the easy plane, while the demagnetization field (H K ) tends to prevent this from happening, leading to the classical Walker limit H W . The extra effective field from SOC (H FL ) and i-DMI in e y axis (hold for both PMA and IPMA cases) also helps to prevent the magnetization from leaving the easy plane, thus extending the traveling-wave region of walls. This is the physical origin of Walker breakdown suppression. Our analytics here provides detailed and solid foundation for the above physical picture.

Mobility change by ADL-SOT
To understand the mobility change and even its sign-inversion of Néel walls in FMM layer of trilayers in quite a lot experiments and simulations, the ADL-SOT must be appended. With the coexistence of STT, FL-SOT, ADL-SOT and i-DMI, the travelingwave ansatz in equation (18) is again selected as the start-point of investigation. We will show that unlike the similarity in "Walker breakdown suppression" part, the mobility changes by ADL-SOT take quite different forms for PMA and IPMA cases.

General framework under the coexistence of STT, FL-SOT, ADL-SOT and i-DMI
As usual, we first concentrate on PMA cases. Under the traveling-wave ansatz, one has After putting back into the scalar LLG equations and integrating over the whole strip, it turns out v(t) with the same functional H(H K , H FL , D, φ) defined in equation (26). Note that the structure of equation (45) is different from that of equation (20) due to the presence of H ADL −terms. By requiringφ = 0 in equation (45), one gets Since H FL and H ADL are both proportional to j a , we set (H FL ) W and (H ADL ) W as the absolute effective-field strengths of FL-and ADL-SOTs when |j a | = J W , respectively. Then after defining equation (46) is rewritten as The rest discussion on Walker breakdown suppression is the same as those in section 3.2 to section 3.4. We define the enlarged Walker limit as J all W . For |j a | ≤ J all W , from equation (45), the wall velocity reads As for IPMA systems, after similar discussion, the existence condition of travelingwave mode ,φ = 0, provides and the corresponding wall velocity reads

Mobility change in PMA systems
In principle, to obtain the wall velocity in traveling-wave mode, one should solve φ from its existence condition [for PMA sytems, equation (46)] and then put it into the velocity formula [see equation (49)]. However the calculation process is hopelessly complicated. Inspired by the asymptotic approach [81,82,83,84,85], we consider the case where |j a | ≪ J all W thus the wall must be in traveling-wave mode (φ = 0) and φ is not far from its static position (φ 0 = arccos[−ηsgn(D)]). Next we introduce the azimuthal deviation (see figure 1a) Since |ψ| ≪ 1, thus sin φ ≈ [−ηsgn(D)]ψ, cos φ ≈ −ηsgn(D) and sin 2φ ≈ 2ψ. Putting them into equation (46), the azimuthal deviation ψ can be solved as Meanwhile the wall velocity in equation (49) becomes Since H ADL is proportional to j a , thus the wall mobility can be changed. One extreme case is that ADL-SOT is induced solely by SHE. Thus H ADL = H SHE = θ SH /(2µ 0 eM s t F ). Then equation (54) in which θ 0 SH ≡ 2βP µ 0 et F /(γπm e ∆) and m e is the electron mass. As long as [−sgn(D)]θ SH > θ 0 SH , the motion direction of the wall will be reversed (v PMA /v STT < 0). Note that to achieve this, not only the strength but also the sign of spin Hall angle should be specified. This former is controlled by the ADL-SOT strength while the latter is selected by the sign of i-DMI parameter. For Co-Ni FMM, one has sgn(D) = −1 and θ 0 SH ≈ 0.048 < 0.1 = [−sgn(D)]θ SH . Therefore Néel walls in this trilayer will move along charge current direction. In addition, by carefully arranging magnetic parameters [β, ∆, t F and sgn(D)] of trilayer systems, |v PMA /v STT | can be much higher than 1. This will help to explain the relatively high velocity of domain walls in Pt(3 nm)/Co(0.6 nm)/AlO x (2 nm) trilayer (∼ 400 m/s when j a ∼ 10 8 A/cm 2 )[37].

Mobility change in IPMA systems
As indicated in section 2.2, in IPMA cases the i-DMI does not select wall polarity thus φ 0 = kπ, k ∈ Z, i.e. cos φ 0 = (−1) k . Again we introduce the azimuthal deviation ψ ≡ φ − φ 0 under small current density. Since |ψ| ≪ 1, thus sin φ ≈ (−1) k ψ, cos φ ≈ (−1) k and sin 2φ ≈ 2ψ. Putting them into equation (50), one has Meanwhile the wall velocity is reduced to which is complicated due to the coexistence of H FL , H ADL and D.
Next we consider an extreme case where the i-DMI is neglected. After simple algebra, the wall velocity in equation (57) becomes with ζ ≡ H FL /H ADL and can be assumed positive without losing generality. Obviously, the presence of H ADL as well as the condition α = β provides us the possibility of changing wall mobility. Interestingly, the direction of wall motion can even be reversed (v IPMA /v STT < 0) under the following condition In addition, the wall halts at (−1) k+1 πH ADL = 2H K /(ζ + β −1 ) and a velocity divergence occurs at (−1) k+1 πH ADL = 2H K /(ζ + α −1 ). Equation (59) shows that only walls with polarity satisfying (−1) k+1 H ADL > 0 can be reversed from electron flow to charge current direction, which well explains the "polarity sensitivity" phenomena in IPMA systems. On the other hand, in real IPMA materials α, β ≪ 1. Therefore the current density under which equation (59) holds can be small enough to ensure the approximation for obtaining equation (56), thus makes the whole deduction coherent.
To numerically check our analytics, in the simplest case we set H FL = 0 (thus ζ = 0) and suppose that H ADL solely comes from SHE, which is exactly the case in Ref. [74]. Under the magnetic parameters in the second column of Table 1, the "v IPMA (v STT ) ∼ j a " curves are plotted in figure 4. The black line indicates the linear dependence of v STT on j a , while the red curve represent the wall velocity v IPMA when only SHE-induced ADL-SOT is considered. The reversal region is −αJ 0 < j a < −βJ 0 with α = 0.02, β = 0.01 and J 0 ≡ 4µ 0 ek H M 2 s t F /(π θ SH ) = 5.32 × 10 13 A/m 2 . The wall-halt current is −βJ 0 = −5.32×10 11 A/m 2 and the velocity-divergence current is −αJ 0 = −1.064×10 12 A/m 2 . All these results reproduce very well the "θ SH = +0.1" case in figure 3a of Ref. [74]. In addition, the positively divergent part for "j a < −αJ 0 " in figure 4 indicates the

Discussions
All analytics in this work are performed based on the traveling-wave ansatz in equation (18). One must bear in mind that it is rigorous only in the absence of any SOTs and i-DMI, otherwise in principle it fails to provide the rigorous wall profile. However, it may serve as a "not-bad" approximation of the actual magnetization texture in trilayers. This has been numerically tested in Ref. [79]. Second, as this ansatz can not hold everywhere along the long axis of strip, to obtain the collective behaviors we then integrate it over r ∈ (−∞, +∞) which is transferred to the integration of θ ∈ (0, π). However, when j a increases, effective transverse fields from SOTs and i-DMI will pull the magnetization in two faraway domains away from strip axis. Then the integration of r ∈ (−∞, +∞) should be converted to that of θ over (θ 0 , π − θ 0 ), where θ 0 is positively correlated with j a with some complicated mathematical dependence. For simplicity, we have not considered this θ 0 in the above sections. Further investigation on this issue is out of the scope of this work. At last, by "small quantity analysis" we succeed in explaining the mobility change in both PMA and IPMA systems. In particular, the sign-inversion of mobility as well as the "polarity sensitivity" therein is recovered analytically. In real experiments, the motion-direction-reversal behavior is observed in a relatively wide range of j a (thus H ADL ). This should not be regarded as a contradiction with equation (59) since it is obtained under the assumption |j a | ≪ J all W . In fact, the necessity of ADL-SOTs for mobility sign-inversion, as well as the polarity selection rule therein, should be the main focus in this subsection. Also, this is one of the main reasons why this part of SOT is named as "anti-damping-like" since they can input energy into the system, not just dissipate it.

Summary
In this work, we analytically investigate the current-induced domain wall dynamics in HM/FMM/Oxide trilayers with strong SOCs and i-DMI. We show that in both PMA and IPMA systems, FL-SOT can induce UAWB but i-DMI can not. For moderate FL-SOT and arbitrary i-DMI strength, we provide analytical expressions of the finitely enlarged Walker limits. On the other hand, the wall mobility change can be explained only when ADL-SOT is included under the coexistence of STT, SOT and i-DMI. In particular, for PMA systems strong enough spin Hall angle and appropriate sign of i-DMI parameter will lead to sign-inversion in wall mobility even under small enough current density, while for IPMA systems this will only occur when current density falls into a finite range. These analytical results provide insights not only for explaining existing experimental and numerical data (in fact a numbers of them have been explained in the main text), but also for the development of future domain-wall-based magnetic nanodevices.