Dzyaloshinskii-Moriya domain walls in magnetic nanotubes

We present an analytic study of domain-wall statics and dynamics in ferromagnetic nanotubes with spin-orbit-induced Dzyaloshinskii-Moriya interaction (DMI). Even at the level of statics, dramatic effects arise from the interplay of space curvature and DMI: the domains become chirally twisted leading to more compact domain walls. The dynamics of these chiral structures exhibits several interesting features. Under weak applied currents, they propagate without distortion. The dynamical response is further enriched by the application of an external magnetic field: the domain wall velocity becomes chirality-dependent and can be significantly increased by varying the DMI. These characteristics allow for enhanced control of domain wall motion in nanotubes with DMI, increasing their potential as information carriers in future logic and storage devices.

In recent years, ferromagnetic nanostructures featuring narrow and stable domain walls (DWs) have been in the spotlight of experimental and theoretical research with an overarching aim of making spintronic logic and memory devices more compact [1][2][3] . In particular, numerous efforts have been focused on DWs in ferromagnetic nanowires  , nanotubes [26][27][28][29][30][31][32] , and thin films with perpendicular magnetic anisotropy [33][34][35][36][37] featuring the Dzyaloshinskii-Moriya interaction (DMI) 38,39 . Here we report striking effects arising from the interplay between the curved geometry of a thin nanotube and the DMI, leading to narrow and stable DWs controllable, efficiently and reliably, by means of electric current and magnetic field.
Thin ferromagnetic nanotubes have attracted considerable attention from experimentalists owing to a number of technologically advantageous properties [30][31][32]40,41 . Certain advantages were pointed out, such as enhanced DW stability under strong external fields and suppression of Walker breakdown, allowing for higher DW velocities compared to flat geometries [26][27][28] ; increased DW velocities under electric current pulses 29 ; and the possibility of switching chirality in vortex domain walls through magnetic field pulses 42 .
We show that in thin ferromagnetic nanotubes, the DMI induces qualitatively different effects compared to those found in flat nanostructures, such as thin films and rectangular nanowires 43 . In nanotubes, DMI causes the domains to become twisted, exhibiting helical lines of magnetization as in Fig. 1, and DWs become narrower. This is in marked contrast to the behaviour in rectangular nanowires, where the magnetization far from the DW remains parallel to the wire axis while the DW becomes broader. This sharpening of DWs in nanotubes by DMI can enable substantial advances in downscaling future nanodevices.
We further demonstrate that in sufficiently thin nanotubes, DWs are topologically protected spin textures, which exhibit perfectly stable motion under an applied electric current, propagating without any distortion. Remarkably, the Walker-type breakdown (leading to lower velocities) is completely suppressed, allowing for robust and controllable DW transport effectively free from limitations on speed. Due to the topology of the problem the adiabatic spin-transfer torque 44,45 is fully absent in the spin dynamics equations, and the non-adiabatic term takes the form of the adiabatic one.
Complimentarily to the current, the application of an external magnetic field along the nanotube axis triggers a rich dynamical response in the magnetization texture. The DW velocity becomes strongly dependent on polarity and chirality 46 , and can be significantly enhanced by DMI, which is favorable for memory applications. Moreover, the onset of the magnonic 26 breakdown, impeding DW transport at high fields, can be efficiently suppressed by the presence of DMI.

A. Statics
We consider a ferromagnetic nanotube with inner radius R and thickness w (the outer radius of the nanotube is R + w). In the thin-nanotube regime w R, the micromagnetic energy 47 with DMI 38,39 takes the form: where the integral runs over the volume of the nanotube.
Here A is the exchange constant, K is the easy-axis crystalline anisotropy, D is the DMI constant, M s is the saturation magnetization, and µ 0 is the magnetic permeability of vacuum. The cylindrical-coordinate unit vectors e z , e ρ , and e φ are shown in Fig. 1 (a). The last term on the right-hand side of Eq. (1) represents the shape anisotropy stemming from the thinness of the nanotube. In nanotubes with radius R much larger than the exchange length A/(µ 0 M 2 s ), this term forces the magnetization to lie tangent to the surface. In this case, the magnetization is described by its orientation Θ(z, ρ, φ; t) in the (z, φ)-tangent plane as follows: m = e z cos Θ + e φ sin Θ . ( It is convenient to substitute Eq. (2) into Eq. (1) and introduce the dimensionless coordinates s = r/R, ζ = z/R, ξ = ρ/R, anisotropy κ = KR 2 /A, and DMI constant η = DM 2 s R/(2A). Then for the dimensionless energy E = E/(2AR) we obtain where the two energy density terms are The "potential" V , which appears in ε 1 , is given by with a and δ defined as and δ is taken between −π/2 and π/2. Below we shall see that δ determines the orientation of the twisted domains, while 1/a is the DW width. Next we look for a magnetization profile Θ minimizing the energy E. The contribution from ε 2 can be made to be zero (and thus minimized) by taking ∂ ξ Θ = −η and ∂ φ Θ = 0, so that Θ = θ 0 (ζ) − η(ξ − 1). In the thin-nanotube limit, the contribution from ε 1 can be simultaneously minimized by taking θ 0 (ζ) to satisfy the Euler-Lagrange equation subject to the boundary conditions that θ 0 (±∞) correspond to maxima of V (not minima, as it is −V which appears in the energy ε 1 ). The maxima of V , given by θ = δ + nπ, describe the possible orientations of m in static domains. In the case of zero DMI, η = 0, the domains are parallel to the nanotube axis. However, for η = 0 they become helical, as shown in Fig. 1. Domain walls correspond to boundary conditions for Eq. (8) where θ 0 (−∞) and θ 0 (+∞) describe oppositely oriented domains. There are four distinct DW profiles, characterised by polarity σ = ±1 and chirality χ = ±1, see Fig. 1. The polarity determines whether the DW is head-to-head (σ = 1) or tail-to-tail (σ = −1), while chirality determines the sense of rotation of m with increasing ζ, so that sgn θ 0 = σχ. The exact solution of Eq. (8) in this case reads θ 0 = 2χ arctan e σaζ + δ.
The DW profiles may be understood qualitatively, in both the static and discussed below dynamic cases, in terms of a mechanical analogy, which is described in detail in Fig. 2. We regard θ(ζ) as the trajectory of a particle moving in a one-dimensional potential V (θ), with ζ playing the role of time. In the static case, the DW boundary conditions correspond to the particle approaching one of the maxima of V (located at δ mod π) as time ζ approaches −∞ and approaching a neighbouring maximum of V as time approaches +∞. During times in between, the particle traverses the intervening potential well (this is an example of a so-called instanton orbit). Such a trajectory exists because the V attains the same value at all of its maxima, or in other words, because oppositely oriented domains have the same energy.
FIG. 2. Mechanical analogy: The DW profile θ(ζ) may be regarded as the instanton orbit of a particle with position θ moving from one local maximum of V (θ) − he cos θ at time ζ = −∞ to a consecutive local maximum at ζ = +∞. κ = 1 throughout. Dashed curve: In the absence of DMI and an applied field, the local maxima are at θ = 0 and θ = π, corresponding to domains aligned along the nanotube axis. The instanton orbit, indicated by the arrows, describes a tail-totail DW with negative chirality. Dotted-dashed curve: With DMI parameter η = 1 but no applied field, the local maxima are shifted by δ = −π/8, corresponding to twisted domains. The instanton orbit shown corresponds to a head-to-head DW with positive chirality. Solid curve: With η = 1 and applied field he = 1, the values of V − he cos θ at consecutive maxima are no longer equal; a specific value j + v, the coefficient of linear damping in Eq. (14), is required to ensure that particle just reaches the second maximum without overshooting.
Hashed curve: For large enough he, a maximum and minimum of V − he cos θ coalesce, and the instanton orbit is destroyed.
An important characteristic of the DW is its width ∆ given by 1/a, or in physical units, As is clear from this expression, DWs in thin nanotubes become narrower in the presence of DMI, in marked contrast to the case of rectangular nanowires 43 . DWs also become sharper in nanotubes with higher curvature 1/R. These properties may be significant for building smaller spintronic nanodevices.

B. Dynamics
Under an applied current, the magnetization dynamics in a ferromagnet far below the Curie temperature is described by the Landau-Lifshitz-Gilbert (LLG) equation 45,[48][49][50] : where H = −(µ 0 M s ) −1 δE/δm is the effective magnetic field, γ is the gyromagnetic ratio, α is the Gilbert damping constant, J is the current along the nanotube, and β is the non-adiabatic spin transfer torque parameter. As in the static case, in the thin-nanotube limit m lies tangent to the nanotube surface, and it can be shown that Eq. (11) reduces to Here τ = 2γA αMsR 2 t is the dimensionless time, h t is the tangential component of the dimensionless effective field h = µ0MsR 2 2A H, and j = MsR 2γA βJ is the dimensionless current. Note that j is proportional to non-adiabatic spin transfer torque parameter β, whereas the current term itself looks like adiabatic spin transfer torque. This has an important consequence, as described below.
Proceeding as in Eq. (2), we write m = e z cos Θ + e φ sin Θ with Θ(ζ, ξ, τ ) = θ(ζ, τ ) + η(ξ − 1) to obtain To find the solutions of this equation, we look for traveling waves of the form θ(ζ, τ ) = ϑ(ζ − vτ ), which describe azimuthally symmetric DW profiles propagating with velocity v. From Eq. (13), these satisfy subject to the same boundary conditions, parameterized by polarity and chirality, as in the static case. It is easy to see that the moving DW profile ϑ coincides with the static profile θ 0 in Eq. (9), with velocity v = j or, in physical units, Thus, under an applied current, the DW propagates without distortion and with velocity independent of its polarity, chirality and DMI. In thin nanowires, the same response is found for small currents only. In thin nanotubes, this behaviour persists for any (including large) current, making current-driven DWs in nanotubes much more robust and allowing them to be driven at higher velocities.
Next we study the effect of an external magnetic field on DW dynamics. We take the field to be uniform along the nanotube axis, H e = H e e z . In the thin-nanotube limit, the applied field generates an additional term in Eq. (14): where h e = µ0MsR 2

2A
H e . The boundary conditions are modified so that ϑ(±∞) correspond now to consecutive maxima of the modified potential V (ϑ) − h e cos ϑ. In terms of the mechanical analogy of Fig. 2, the solution ϑ(ξ) describes the trajectory of a particle moving from one potential maximum to another, as in the static case. However, the term (j −v)ϑ plays here the role of an additional friction (or antifriction) force, which compensates for the fact that, in the presence of the applied field, the potential energies at consecutive maxima are different.
Numerical solutions of Eq. (16), shown in Fig. 3, indicate that in the presence of an applied field, the DW velocity depends strongly on DMI parameter η as well as on chirality and polarity. For a given field strength, the velocity achieves a maximum for a nonzero value of η, and varies with η by a factor exceeding two. Equation (16) cannot be solved analytically, but it is straightforward to develop an expansion in powers of h e . It turns out that an expansion through quadratic order is sufficient to capture the leading-order dependence on polarity and chirality. Letting p(ϑ(ζ)) = ϑ (ζ), we may write Eq. (16) as the equivalent first-order equation In terms of the mechanical analogy of Fig. 2, equation (17) corresponds to energy balance. Letting = h e , we expand p = p 0 + p 1 + 2 p 2 + · · · and v = v 0 + v 1 + 2 v 2 + · · · . At zeroth order we deduce that p 0 (ϑ(ζ)) = θ 0 (ζ), where θ 0 is the static profile (9), i.e.
from which it follows that v 0 = j. The equations for the next two corrections p 1 and p 2 can be readily solved, and v 1 and v 2 are then obtained by integrating Eq. (17) over the interval δ < ϑ < π + δ and noting that p 0 vanishes at the endpoints. Up to terms of order h 3 e /a 5 , we obtain v = j + σ For the limiting case j = η = 0, Eq. (19) yields v = σh e /( √ 2a), which coincides with a previously obtained exact result 51 . As can be seen from Fig. 4, the quadratic approximation is in a good agreement with numerical results. The dependence of the DW velocity on applied field and chirality is shown in Fig. 5. As is evident from Eqs. (19) and (7), for small applied fields the DW velocity is suppressed by the DMI. However, for larger fields and DWs with the appropriate chirality, namely χ = σ sgn(ηh e ), the velocity may be enhanced by DMI (but DWs with the "wrong" chirality are always slower).
At a certain critical magnetic field h c , a bifurcation occurs, beyond which the DW velocity becomes suppressed. In terms of the mechanical analogy of Fig. 2, as h e approaches h c , a maximum and minimum of the potential function V − h e cos ϑ coalesce, the instanton orbit is destroyed, and the character of the traveling DW changes. This phenomenon has been discussed in terms of the spin-Cherenkov effect 26,52 , and more recently in terms of pulled wavefronts of the KPP equation 53 . One can derive 54 an analytic expression for h c in terms of η, see Fig. 6. For η 1 + κ the leading-order behaviour is given by h c = 1 + κ − 3(cη) 2/3 , with c = √ 1 + κ/(2 √ 2), while for η 1+κ the leading-order behaviour is h c = η. The important conclusion is that the critical field can be enhanced by increasing DMI (beyond certain threshold value), thus allowing for faster DW propagation.

II. DISCUSSION
In recent years there have been ongoing efforts to use ferromagnetic materials with perpendicular anisotropy 34,35,37 to produce sharp and stable domain walls, with a view to make potential spintronic logic and memory devices more compact. Here we have described an alternative approach to this objective via domain walls in thin nanotubes with Dzyaloshinskii-Moriya interaction. These DWs are found to have novel properties: The domains themselves become twisted about the nanotube axis; moreover, DWs become sharper with increasing DMI, the opposite of what is seen in thin nanowires 43 , where they become broader. Under an applied current, these DWs propagate without distortion with a velocity proportional to the current -there is no Walker-type breakdown. Introducing an external magnetic field, we uncover a rich dependence of DW velocity on polarity, chirality, DMI and field strength, which may provide enhanced control for future spintronic devices. The DW velocity can be significantly increased by DMI, and the onset of the magnonic regime can be suppressed.
This work provides the favorable material trends for engineering nanotubes with DMI for faster and more robust DW operation. Using the DMI parameter from Ref. 34 (DM 2 s = 0.5 · 10 −3 J/m 2 ), A = 10 −11 J/m, and taking the nanotube radius R ≈ 100 nm, we estimate the dimensionless DMI parameter η ≈ 2. For the same material parameters κ = 1 is reached for K = 10 3 J/m 3 . These estimates show that the regime where DMI has visible effects is experimentally feasible.
In the thin nanotube limit, we are able to treat leading contributions of dipolar interactions exactly. We have derived explicit analytic expressions for the DW profiles and their velocities under applied currents and fields in this regime, which are in a good agreement with numerical solutions of the LLG equation. These results are robust and potentially applicable even beyond the thinnanotube limit, which is hinted by recent micromagnetic studies 26 .
Our finding of novel Dzyaloshinskii-Moriya domain walls on a nanotube surface points to new routes of exploring topologically protected and compact spin textures. These DWs can be driven robustly by electric currents without Walker breakdown and manipulated by magnetic fields depending on the DW chirality. This offers the perspective of new types of experiments and nanodevices where one can move bits of information associated with the DWs of different chirality and polarity in a precise and controllable fashion.