Spin texture motion in antiferromagnetic and ferromagnetic nanowires

Davi R. Rodrigues,1 Karin Everschor-Sitte,2 Oleg A. Tretiakov,3, 4 Jairo Sinova,2, 5 and Ar. Abanov1

1Department of Physics & Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA2Institute of Physics, Johannes Gutenberg Universitat, 55128 Mainz, Germany3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

4School of Natural Sciences, Far Eastern Federal University, Vladivostok 690950, Russia5Institute of Physics ASCR, v.v.i, Cukrovarnicka 10, 162 00 Prag 6, Czech Republic

(Dated: July 21, 2018)

We propose a Hamiltonian dynamics formalism for the current and magnetic field driven dynamics of ferro-magnetic and antiferromagnetic domain walls in one dimensional systems. To demonstrate the power of thisformalism, we derive Hamilton equations of motion via Poisson brackets based on the Landau-Lifshitz-Gilbertphenomenology, and add dissipative dynamics via the evolution of the energy. We use this approach to studycurrent induced domain wall motion and compute the drift velocity. For the antiferromagnetic case, we showthat a nonzero magnetic moment is induced in the domain wall, which indicates that an additional applicationof a magnetic field would influence the antiferromagnetic domain-wall dynamics. We consider both cases of themagnetic field being parallel and transverse to the Neel field. Based on this formalism, we predict an orientationswitch mechanism for antiferromagnetic domain walls which can be tested with the recently discovered Neelspin orbit torques.

INTRODUCTION

The insensitivity of antiferromagnets (AFM) to externalmagnetic fields, together with their inherent faster dynamics,affords these materials a technological advantage over theirferromagnetic (FM) counterparts [1, 2]. However, the mag-netic invisibility is responsible also for the difficulty in detect-ing and manipulating spin-textures that can store informationin such materials. Making antiferromagnetic active compo-nents in spintronic devices is the focus of the emerging fieldof antiferromagnetic spintronics [3, 4]. This emerging fieldhas seen a lot of recent progress in experimental techniques[5–10], which has led to a raise in theoretical studies of thedynamics of AFM domain walls (DW) and other AFM spintextures [11–25]. Describing effectively the dynamics of thesetextures is a vital goal to connect experimental observables tothe AFM textures and their dynamics.

The analysis of FM DW dynamics in terms of a finite set ofparameters have been considered recently in Refs. [26, 27].The results obtained from such description based on a finiteset of collective coordinates representing soft modes agreewith experimental and numerical data. In this paper we showthat, under some conditions, it is possible to combine suchsoft modes in terms of conjugated parameters and a Hamil-tonian dynamics description. We apply this procedure to theelectrical current and magnetic field driven dynamics of bothFM and AFM DWs in nanowires. We derive the form of theeffective Hamiltonian for DW dynamics based on the symme-tries of the problem alone due to the transparent Hamiltonianstructure of the effective equations of motion.

A strength of the Hamiltonian dynamics formalism pro-vided in this paper is that it is insensitive to the details of themicroscopic magnetic Hamiltonian. This aspect makes thisformalism very powerful and rich as one can easily, with a fewassumptions, study many known aspects of AFM DW dynam-ics and also include various interactions. Moreover, within the

spirit of a phase space given by the conjugated soft modes, wemay consider interactions and scattering of DWs, as well asthermodynamic effects. As an application of the developedformalism, we will show that it is capable of describing: i)current-induced dynamics for both FM and AFM DWs, ii)magnetic field induced dynamics for both FM and AFM DWs,iii) orientation switching by current for AFM DWs. The orien-tation switching mechanism is a novel feature for AFM DWswhich is naturally derived within our approach. The switch ofconfigurations on antiferromagnets may have several applica-tions to magnetic memory devices [28]. We also show belowthat other effects such as different anisotropies, nanowire in-homogeneity, etc., can be incorporated within the same for-malism.

The paper is structured as follows. We will first describethe highly non linear dynamics of the magnetization field.We then define a simpler FM DW description through theuse of proper canonical Hamiltonian variables. After deriv-ing the Hamilton equations of motion for the spin polarizedcurrent driven FM DW in the dissipationless case, we showhow the dissipation must be included in these equations. Wealso demonstrate how the form of the effective DW Hamilto-nian can be understood from the symmetries of the problem.Finally, we apply this formalism to the case of AFM DWs,solve certain general cases for AFM DW dynamics, and makeproposals for future AFM DW experiments.

EFFECTIVE HAMILTONIAN DESCRIPTION

In the absence of spin-orbit torques (SOT) [29–38] the mag-netization dynamics of a FM due to magnetic fields and elec-tric current is described by the Landau-Lifshitz-Gilbert (LLG)equation [39, 40]:

~m = γ0 ~m×δHm

δ ~m− J∂ ~m+α~m× ~m+ β ~m× (J∂ ~m), (1)

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FIG. 1. A sketch of ferromagnetic tail-to-tail domain wall. X andφ denote the position and the tilt angle of the DW. J corresponds tothe injected spin polarized current. We assume the polarization of Jfollows ~m.

where ~m2 = 1 is the unitary vector along the magnetization,γ0 is the gyromagnetic constant, the spatial derivative ∂ isalong the nanowire, Hm is a magnetic Hamiltonian, α is theGilbert constant damping, and β is a dimensionless dampingparameter. J corresponds to the spin-polarized current alongthe nanowire, with the amplitude given by

J =jPgµB2eMs

, (2)

where j is the current density, P is the polarization of thecurrent, and Ms is the saturation magnetization. The current,α, and β terms generally depend on the microscopic details ofthe full system. In particular, the dissipative α and β terms de-pend on the strength and nature of the magnetization-electronand magnetization-phonon interactions. However, we assumelocality (both in space and time) of the dissipative effects. Thesymmetry demands a specific form of the the dissipative termsand the current interaction. As the current, α, and β termscouple with the variation of the magnetization, at the scalesof the magnetization configurations considered on this paper,only linear terms of these couplings are relevant. We also as-sume here the isotropic form of all current, α and β terms,although this requirement can be violated in some materials.Nevertheless, generally the LLG equation has been shown todescribe magnetization dynamics well in many different ma-terials, see e.g. Ref. 41.

The magnetic Hamiltonian Hm includes all interactions forthe local magnetization in the ferromagnet, such as the ex-change interaction and all the magnetic anisotropies. It mayalso include Dzyaloshinskii-Moriya, dipole-dipole, and otherinteraction terms. We assume that Hm does not explicitly de-pend on time.

Domain wall motion in a ferromagnet

A typical transverse DW in a ferromagnetic nanowire iswell approximated by a rigid object, which can move alongand rotate around the wire axis [42]. During this motion theDW shape changes a little, since in the presence of anisotropythe modes corresponding to the change of the DW shape aregapped. Here we derive the dynamics of the DW based on the

Hamiltonian equations of motion. To do so we calculate thePoisson brackets for two parameters of this object.

The Poisson brackets of the unit vector ~m, a representationof SO(3), are

mi(x),mj(x′) = εijkmk(x)δ(x− x′). (3)

The magnetization configuration ~m(x) of a single DW in ananowire may be described in terms of the soft modes X , theposition of the domain wall, and φ, the rotation of magneti-zation at the DW center around the nanowire axis, see Fig. 1.A variation of the configuration in terms of these parameterscorresponds to

dmi(x) = −dX∂mi(x) + dφεijkejmk(x), (4)

where ~e is the unit vector along the wire. Comparing the totalvolume of the phase space in terms of ~m and variables X andφ, we find

X,φ = ±1, (5)

where the + and − signs are for the tail-to-tail and head-to-head DWs, respectively [43]. Note that for a FM DW, theDW position X and the angle φ are canonically conjugatedHamiltonian variables independent of the microscopic formof the Hamiltonian.

Let us now consider the dependence of the total magneticenergy on time. As the Hamiltonian does not depend explic-itly on time, we can write

E =

∫δHm

δmi(x)midx (6)

and substitute Eq. (1) for ~m. Upon doing this we notice thatonly one of the terms is not dissipative,−J

∫δHm

δmi(x)∂midx =

J∂XE, where X is the DW center coordinate. The remainingterms are first order in dissipation. Using εijkmj

δHm

δmk= mi+

J∂mi we find

E − J∂XE = −∫

(mi + J∂mi)(αmi + βJ∂mi)dx. (7)

First we consider the dissipationless dynamics, where weset α = β = 0. We obtain then the equation of Hamiltoniandynamics E = J∂XE. Therefore, there exists an effectiveHamiltonian H(X,φ) such that E = E,H = J∂XE. Inthe absence of current, the Hamiltonian must be equal to themagnetic energy E(X,φ) of the DW, and thus we conclude

H(X,φ) = E(X,φ)± Jφ, (8)

as this Hamiltonian gives the correct E. This energy functionis all one needs to know to obtain the effective equations ofmotion for the FM DW.

The Hamiltonian equations for the coordinates X and φtake the form

X = X,H = ±∂H∂φ

, φ = φ,H = ∓∂H∂X

. (9)

3

Next, we include dissipative terms γX and γφ in these equa-tions:

X = X,H+ γX , φ = φ,H+ γφ. (10)

To calculate the dissipative terms, we expand Eq. (7) for theDW motion to the linear order in J

E − J∂XE = −αX2∆−1X + 2αφXΓ− αφ2∆φ

+(α+ β)JX∆−1X − (α+ β)φJΓ, (11)

where the constants are defined as

∆−1X =

∫(∂ ~m)2dx, ∆φ =

∫(1−m2

x)dx,

and Γ =

∫[~e× ~m]∂ ~mdx.

They are the parameters determining the DW dynamics anddepend on the DW shape. For a planar DW, when the mag-netic Hamiltonian has no Dzyaloshinskii-Moriya term [42],the constant Γ vanishes, Γ = 0.

We compare now Eq. (11) to the same expression we com-puted using (10), E − J∂XE = ∓Jγφ ± γφX ∓ γX φ, andfind for the dissipative terms

γφ = ∓αX∆−1X ± βJ∆−1

X , (12)

γX = ±αφ∆φ ∓ 2αXΓ± (α+ β)JΓ. (13)

We note that Eqs. (10), together with the constants γX and γφdefined above, completely determine the DW dynamics in aferromagnetic nanowire in terms of α, β, and the parametersof the magnetic Hamiltonian. In particular, many previousresults on ferromagnetic DW dynamics [26, 42, 44–46] canbe easily obtained within this formalism.

We emphasize that although in this paper we assume thedirection of the uniaxial magnetic anisotropy to be along thenanowire, this description is valid for any other anisotropy di-rection. The angle φ is then just the angle of the central spin inthe DW, which rotates around the anisotropy axis. In the caseof anisotropy axis perpendicular to the nanowire this angledescribes the oscillations between the Bloch and Neel DWs.

Domain wall motion in an antiferromagnet

Below we show how to extend the Hamiltonian formalismof DW dynamics to the important case of AFM DW. Both the-oretical and experimental studies [5–14, 18–24] of the AFMdynamics have been very active recently due to improvedexperimental capabilities to detect, produce, and manipulatethese systems. However, a more systematic approach to themotion of AFM DWs is still missing and here we make anattempt to fill in this gap.

We consider a collinear AFM on a bipartite lattice with sub-latticesA andB. A typical nanowire is quasi one-dimensionalfrom the point of view of magnetization and magnetizationdynamics, but still is three-dimensional from the point of view

FIG. 2. A sketch of antiferromagnetic domain wall. The AFMDW is a composite object consisting of two ferromagnetic DWs ontwo sublattices (A andB) with respective positionXA (XB) and thetilt angle of the ferromagnetic DW φA (φB) on respective sublattice.One ferromagnetic DW is tail-to-tail (A) and the other is head-to-head (B). The inset shows the AFM lattice with up- and down-spinsublattices and nearest (second nearest) neighbor hopping constantst (t′).

of the electronic degrees of freedom. An electron has hop-ping matrix elements both between the nearest neighbors andbetween next nearest neighbors on the lattice. In the pres-ence of the AFM order in the case of large on-site electronspin-magnetization interaction the hopping between the near-est neighbors – between a site of sublattice A and a nearestsite in sublattice B – is suppressed, while the hoping betweenthe next nearest neighbors – the sites of the same sublattice –is unaffected. Thus, we can assume that each electron lives onits own sublattice and interacts only with the magnetizationof the same sublattice. A hybrid systems of bilayer materi-als, i.e. what is known as an artificial antiferromagnet, whereboth layers have ferromagnetic in-layer order coupled anti-ferromagnetically to each other is a good example of such adescription.

In the presence of an AFM DW this separation of the elec-trons to different sublattices is not perfect, but under the as-sumption that the DW width is much larger than the electroniccoherence length, the corrections to this picture are expectedto be small. It is important to note that, if there is a percola-tion within each sublattice, the potential difference across thewire is the same on both sublattices, however the resistivityof each of them does not have to be the same. Thus the elec-trical currents on the two sublattices can be different. Thiscase is especially crucial in the situation where the sublatticesare made of different atoms, as it is on some hybrid systems.To simplify the equations, however, we assume that, given theassumptions of the scales considered, the currents on the twosublattices are the same [47].

The above described situation leads to the following pic-ture. A single AFM DW is composed of two coupled FMDWs on each of the sublattices with its own current. One ofthe DWs is tail-to-tail (DW on sublattice A in Fig. 2) and theother one is head-to-head. We thus introduce two sets of vari-

4

ables, XA and φA for the head-to-head DW on sublattice Aand XB and φB for the tail-to-tail DW on sublattice B. Tosimplify the expressions below we assume that the DW pa-rameters ∆φ, ∆X , α, and β are the same for the two DWs,and that the DWs are planar, i.e. Γ = 0.

The magnetic Hamiltonian depends now on the two sets ofcoordinates H(XA, φA, XB , φB). The center of the DW isat X = 1

2 (XA + XB). The value of XA − XB gives themagnetization of the AFM DW along the nanowire. φB − φAis the angle between the directions of the spins at the centerof the DWs. It corresponds to the magnetic moment of theAFM DW perpendicular to the AFM DW plane. The coupledequations of motion are:

XA,B = XA,B , H ∓ αφA,B∆φ, (14)

φA,B = φA,B , H ±α

∆XXA,B ∓

β

∆XJA,B , (15)

where upper and lower signs are for sublattices A and B, re-spectively, and the Poisson brackets are XA, φA = −1,XB , φB = 1. The Hamiltonian H is

H(XA, φA, XB , φB)=E(XA, φA, XB , φB)− hXA + hXB

−JAφA + JBφB , (16)

where we have explicitly written the coupling of the DWson the sublattices to the magnetic field h, and E is thenthe energy of the two DWs in the absence of the magneticfield. It includes the magnetic interaction of the two DWsand thus depends on the two sets of coordinates. In a trans-lationally invariant nanowire the energy E is independent ofXA+XB . However, in the problems for the DW dynamics ina nanowire with a non-uniform shape [46], one should add aterm Ω(X2

A + X2B)/2 to the energy E, where Ω is a constant

inversely proportional to the nanowire curvature.One can write the function E(XA, φA, XB , φB) in a very

general form. We notice that the minimum of the total AFMDW energy is reached at XA −XB = 0 and φA − φB = π.Expanding for small |XA −XB | and the first harmonic of theangular dependence, we find

E(XA, φA, XB , φB) =∆1

2(XA−XB)2+∆2 cos(φA−φB).

(17)The constants ∆1 and ∆2 are of the order of JAF∆−2

X andJAF , respectively, where JAF is the antiferromagnetic ex-change constant. One also can add a transverse anisotropyby adding K(sin2 φA + sin2 φB) to the energy.

Using the Hamiltonian (16) with energy (17) to calculatethe Poisson brackets, we obtain

XA,B = ∆2 sin(φA − φB)∓ αφA,B∆φ + J, (18)

φA,B = ∆1(XA −XB)− h± α

∆XXA,B ∓

β

∆XJ, (19)

where we set JA = JB = J . We point out that for AFM mate-rials with different compositions of their sublattices JA 6= JB .In the limiting case that one sublattice is insulating, passing

the current through this material would lead to the rotation ofthe AFM DW as in the case of a ferromagnetic DW above theWalker breakdown. In this paper we assume that the easy-axismagnetic anisotropy is along the wire, and the magnetic fieldh in the above equations is also along the wire – along theanisotropy axis. However, the general formalism is valid forany direction of the anisotropy after a trivial change of nota-tions.

The equations of motion (18) and (19) are one of our mainresults. They provide the dynamics of AFM DW interactingwith both magnetic field h and electrical current J . These twointeractions may be studied independently.

Current induced AFM DW dynamics.- Many aspects ofthe current driven dynamics of magnetization configurationsin antiferromagnetic is well known, see for example Refs.[15, 16, 23, 25, 28]. We show that some results may be eas-ily derived within the Hamiltonian formalism. From Eqs. (18)– (19) we consider next the case with no magnetic field, i.e.h = 0, and

XA − XB = −α(φA + φB)∆φ, (20)

φA + φA = 2∆1(XA −XB) +α

∆X(XA − XB). (21)

As XA − XB is already first order in the dissipation, the sec-ond term on the right-hand side (RHS) of Eq. (21) should bedropped. Then, by substituting Eq. (21) into (20) we obtain,

XA − XB = −2α∆φ∆1(XA −XB). (22)

It has an exponentially decaying solution for XA − XB ∝exp(−2α∆φ∆1t) and thus in the steady state φA = −φB .We also obtain the other two equations from the system (18) -(19):

φA − φB =2α

∆XX − 2β

∆XJ,

X = ∆2 sin(φA − φB)− α∆φφA − φB

2+ J.

Similarly φA − φB is already of the first order in dissipation,so we can neglect the second term on the RHS of the secondequation to find:

X = ∆2 sin(φA − φB) + J. (23)

This set of equations has the simple solutions φA− φB = 0and X = V given by

V =β

αJ, sin(φA − φB) = − J

∆2(1− β/α).

There is a critical current Jc = α∆2

|α−β| up to which this solu-tion exists. This current is generally large, since ∆2 is of theorder of the exchange constant. Physically this critical currentcorresponds to the situation when the magnetizations on thetwo sublattices rotate with respect to each other and eventu-ally point in the same direction. For small currents J Jcthis solution shows that a moving AFM DW is not rotating and

5

has a magnetic moment of the order of ∆XJJc

perpendicularto the plane of the AFM DW.

For J > Jc the AFM DW will not rotate, but the magneticmoment perpendicular to the AFM DM oscillates in time withT = ∆X

2α∆2

2π√(J/Jc)2−1

. The AFM DW velocity is also not

constant in time. The average AFM DW velocity is given by〈V 〉 = J − J∆2

Jc

(1−

√(J/Jc)2 − 1

).

Magnetic field parallel to the nanowire.- Next, we con-sider the AFM DW dynamics under the action of the mag-netic field along the wire in the absence of current. In thissituation we find from Eqs. (18) – (19) that in the steady stateφA = φB = 0, and φB = π+φA (a small deviation from thisdecays exponentially with time), while XA = XB = 0, andXA − XB = h/∆1, which show that the induced magneticmoment is∼ h∆X/∆1. Also, notice that unlike in a FM DW,there is no motion induced by an external magnetic field inthis configuration.

Magnetic field perpendicular to the nanowire with current.-Let us consider the AFM DW dynamics under the magneticfield perpendicular to the current, i.e. the nanowire axis. Thismagnetic field couples to the angles φA and φB , the corre-sponding term in the Hamiltonian is −h(sinφA + sinφB),where h is the magnetic field multiplied by the perpendicularmagnetization of a single DW. Such term in the Hamiltoniandoes not change the equations for φA and φB , see Eqs. (18)– (19), but adds a term h cosφA to the RHS of equation forXA and −h cosφB to the RHS of equation for XB . Then, itfollows,

XA,B = ∆2 sin(φA − φB)∓ αφA,B∆φ + J ± h cosφA,B

(24)

φA,B = ∆1(XA −XB)± α

∆XXA,B ∓

β

∆XJ . (25)

A simple solution for the steady state at small J and h is givenby XA = XB and XA = β

αJ for the coordinates. Note thatthis is the same result as for the current driven AFM DW mo-tion. For the angles, we obtain φA = π + φB and

cosφA = −Jh

(1− β/α). (26)

We notice from Eqs. (24) – (25) that as we switch the cur-rent with constant magnetic field, the angles could also beswitched.

AFM DW orientation switching mechanism.- Given a staticAFM DW without current or magnetic fields applied to it,from Eqs. (18) – (19) the configuration is given by XA = XB

and φA = φB or φA = φB + π. The first corresponds to aferromagnetic state. The second configuration, as we apply amagnetic field perpendicular to the nanowire, may representtwo different drift velocities, see Eqs. (18) – (19). These dif-ferent behaviors suggest that it would be interesting to study areorientation mechanism for an AFM DW.

First, we notice that a weak magnetic field applied parallelto the nanowire would induce a precession of the AFM DW.This precession, however, would be extremely slow. As we

are interested in practical use for the reorientation, we need toconsider faster processes. This may be obtained by consider-ing a time dependent current and a magnetic field perpendic-ular to the nanowire.

To illustrate the reorientation mechanism, we consider thatinitially XA = XB = X and φA = π − φB = φ. FromEqs. (24) – (25), we note that these relations are valid for theentire switching process as long as we apply a time-dependentcurrent of the form

J(t) =1

α− β

[∆X φ+ α cosφ (2∆2 sinφ− h)

], (27)

where we neglected the terms proportional to α2. Other typesof currents may be considered for the reorientation. However,the dynamics involved will be a lot more complex and may notbe possible to obtain the exact time dependence analytically.For initial angle φ0 we consider that we have a static AFMDW profile with no current. This implies

sinφ0 =h

2∆2. (28)

In order to obtain the minimum Ohmic losses for the switch-ing process with finite time of switching T , we find that thetime dependence of φ is given by

t =∆X

γ

∫ φ(t)

φ0

dφ√ET

γ2 + cos2 φ (sinφ− sinφ0)2, (29)

where γ = 2∆2α and ET = ∆2X φ

20 is the constant related to

the time of switching T by

T =∆X

γ

∫ π−φ0

φ0

dφ√ET

γ2 + cos2 φ (sinφ− sinφ0)2. (30)

Therefore, with the current given by Eq. (27) and Eq. (29) sat-isfying Eq. (28), we are able to efficiently switch the orienta-tion of the AFM DW within a finite time T given by Eq. (30).It is important to notice, however, that with the time dependentcurrent, Eq. (27), it is also possible to obtain other φ(t) evolu-tions that also corresponds to the switching process, but withhigher energy losses. The integral can be solved in terms of el-liptic functions, see Fig. 3(a). Once we are able to switch theorientation in a controllable fashion, one must then measurethe switching of this 180 AFM DW. Whereas most measure-ments cannot directly observe such DW, a system that containsthe newly discovered Neel spin-orbit torque, [17, 28], can inprinciple be sensitive to the orientation of a 180 DW as shownin Ref. [17]. The measurement of the DW orientation was alsoconsidered in Refs. [45, 48]. One must also notice that dur-ing the switching process a finite magnetic moment arises, seeFig. 3(b), which can be measured experimentally allowing anindirect measure of the process.

In the present formulation the direction of the magneticfield in the plane perpendicular to the nanowire is arbitrary.However, in the presence of the transverse anisotropy this ef-fect will be the largest if the field direction is perpendicular

6

FIG. 3. (a) A plot of the numerical solution of integral (30) givingthe time of switching T ′ = Tγ/∆X as a function of the parameterH = ET /γ

2 for different initial angles φ0. As expected, the time ofswitching decreases as we increase the initial φ. (b) A sketch of thereorientation mechanism for AFM DWs. The different sublattices’orientation will cross, producing a temporary magnetic moment per-pendicular to the wire during the process.

to the transverse anisotropy axis. The current required for theswitching in this case will be determined by both the magni-tude of the magnetic field and by the anisotropy.

CONCLUSION

We have developed a Hamiltonian approach to the currentand magnetic field driven dynamics of both ferromagnetic andantiferromagnetic DWs, which describes the domain walls asrigid topological objects. We have shown how dissipation isincluded in this description by means of LLG equation for-malism. The dynamics can be described by a set of univer-sal equations which depend only on a few parameters. Theseparameters can be measured in real nanowires either throughmagnetoresistance or through electrical means as shown inRef. [45].

We have shown that the developed formalism allows tosolve various problems of both FM and AFM DW dynamicson the same footing and extend it to different geometries. Asthe Hamiltonian formalism does not depend on microscopicaspects, it allows to easily introduce new interactions. In par-ticular, it can be used to describe both FM and AFM DW dy-namics induced by parallel magnetic field and current. As aconsequence of this analysis, we were able to obtain a novelorientation switch mechanism for AFM DWs. With the devel-opments of measuring techniques, the mechanism describedhere may be useful for memory devices.

ACKNOWLEDGMENTS

Ar. A. is very grateful to the INSPIRE group in JohannesGutenberg-Universitat, Mainz, Germany. We are thankful toH. Gomonay for valuable discussions, to O. Tchernyshyovfor explaining to us the Poisson brackets (5) as the Pois-son brackets of the component of the total angular momen-tum and the corresponding angle and to B. McKeever for re-viewing the calculations on this paper. O.A.T. acknowledgessupport by the Grants-in-Aid for Scientific Research (Nos.25247056, and 15H01009) from the Ministry of Education,Culture, Sports, Science and Technology (MEXT) of Japan.K. E.-S. acknowledges funding from the German ResearchFoundation (DFG) under the Project No. EV 196/2-1. J.S. acknowledges funding from the Alexander von HumboldtFoundation, the ERC Synergy Grant SC2 (No. 610115), theTransregional Collaborative Research Center (SFB/TRR) 173SPIN+X, and Grant Agency of the Czech Republic grant no.14-37427G.

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