Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interaction
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Skyrmion confinement in ultrathin film nanostructures in the presence ofDzyaloshinskii-Moriya interaction
S. Rohart ∗ and A. Thiaville Laboratoire de Physique des Solides, Universit´e Paris-Sud,CNRS UMR 8502, F-91405 Orsay Cedex, France (Dated: October 3, 2013)We study the modification of micromagnetic configurations in nanostructures, due to theDzyaloshinskii-Moriya interaction (DMI) that appear at the interface of an ultrathin film. Weshow that this interaction leads to new micromagnetic boundary conditions that bend the magneti-zation at the edges. We explore several cases of ultrathin film nanostructures that allow analyticalcalculations (1D systems, domain walls, cycloids and skyrmions), compare with fully numerical cal-culations, and show that a good physical understanding of this new type of micromagnetics canbe reached. We particularly focus on skyrmions confined in circular nanodots and show that edgesallow for the isolation of single skyrmions for a large range of the DMI parameter.
PACS numbers: 75.70.Kw, 75.70.-i, 75.30.Et, 75.70.Tj
I. INTRODUCTION
Recent observations of chiral structures in mag-netic thin films have raised a great interest for theDzyaloshinskii-Moriya interaction (DMI) , as it favorsmagnetization rotations with a fixed chirality . Thiscoupling originates from the combination of low struc-tural symmetry and large spin-orbit coupling. It hasbeen first proposed in bulk materials lacking space in-version symmetry but it also exists at the interface be-tween a magnetic film and a high spin-orbit coupling ad-jacent layer. The most striking phenomenon inducedby DMI is the formation of skyrmion networks ,but its influence on domain walls is also at theorigin of interesting properties such as increased domainwall velocity versus magnetic field. Recently, interest hasalso been devoted to isolated skyrmions, which can be nu-cleated as a metastable state in thin films , openinga path to new concepts of magnetic memories based onskyrmion motion in nanotracks .While extensive work has already been performed onthe influence of DMI on micromagnetism for infinite sam-ples , no description is available for nanos-tructures, which is the aim of the present work. We showthat in nanostructures, DMI leads to a new form of mi-cromagnetic boundary conditions that should be imple-mented in micromagnetic numerical solvers. We describeseveral cases with analytical solutions that provide testsfor numerical codes, and help to get a physical feelingof the effects of this interaction. We particularly focuson the problem of skyrmions trapped in nanodots. Us-ing simple physical arguments based on the micromag-netic length scales, we discuss the different states thatare obtained. This should help future studies to designnew memories based on skyrmion motion . As mostof the recent advances in this field toward applicationin spintronics devices have been obtained for ultrathinfilms , we restrict our study to this case, using theinterfacial DMI coupling described by A. Fert and us-ing a 2D formulation, where any variation along the film normal is neglected. II. MICROMAGNETIC FRAMEWORK
The Dzyaloshinskii-Moriya interaction has been intro-duced in an atomic description as E DM = X h i,j i ~d ij . (cid:16) ~S i × ~S j (cid:17) (1)where ~d ij is the DM interaction vector for the atomicbond ij (in Joule), ~S i the atomic moment unit vector,and the summation is performed on the neighbor pairs h i, j i . The direction of ~d ij depends on the type of systemconsidered. We consider here magnetic ultrathin films,where DMI originates from the interaction with the highspin-orbit heavy metal of the adjacent layer . Inthis case, for isotropic films ~d ij is d~u ij × ˆ z ,where ~u ij is the unit vector between sites i and j andˆ z is the direction normal to the film oriented from thehigh spin-orbit layer to the magnetic ultrathin film. Inthe micromagnetic framework, the hypothesis that theatomic spin direction evolves slowly at the atomic scaleallows building a continuous form for the DMI. As weconsider films that are thinner than any micromagneticlength scale, variations along the surface normal are ne-glected so that, even if DMI originates from the inter-faces, we consider a uniform average value along the filmthickness. Given ~m ( ~r ) the magnetization direction at po-sition ~r , the DMI energy reads E DM = t Z Z D (cid:20)(cid:18) m x ∂m z ∂x − m z ∂m x ∂x (cid:19) + (cid:18) m y ∂m z ∂y − m z ∂m y ∂y (cid:19)(cid:21) d ~r (2)where D is the continuous effective DMI constant, inJ/m . The link between D and d depends on the typeof lattice, but scales as 1 /at ( a being the lattice con-stant and t the film thickness). The 1 /t scaling is dueto the assuption of interface induced DMI. We obtain D = d/at = d/N a for a simple cubic lattice orientedalong the (001) direction and D = d √ /at = 3 d/N a √ N is the number of atomic planes in thefilm). For example, given the value from the literaturefor 1 monolayer of Fe on Ir(111) d = − . a = 2 .
715 ˚A, we find D = − . for N = 1. Notealso that, although Eq. 2 has been derived from a sim-ple first neighbor description, it remains valid for a morecomplex formulation, as long as the system is isotropic.In such a case, only the link between D and d is modified.DMI needs to be included together with the other mi-cromagnetic energies so that the exchange energy den-sity A h(cid:0) ∂ ~m∂x (cid:1) + ( ∂ ~m∂y ) i and anisotropy energy density − K ( ~m. ˆ z ) are added to Eq.2 ( A being the micromag-netic exchange constant and K the anisotropy constant).In this paper, we consider the case with a perpendicu-lar easy axis ( K > K canbe seen as an effective anisotropy constant, which takesinto account the shape anisotropy ( K = K MC − µ M S ,with K MC the magnetocrystalline anisotropy and M S thespontaneous magnetization). This approximation is jus-tified by the fact that we are interested in ultrathin films,where dipolar coupling becomes local (shape anisotropy)in the zero thickness limit. See however Fig. 1 for acase where full dipolar coupling is included. As a firstapproximation, we also do not include any specific edgeenergies (enhanced edge anisotropy, modified exchangeor DMI constant, ...) as usual in continuous magnetism.For numerical applications, we consider in the fol-lowing the parameters of Pt/Co/AlOx samples [ A =16 pJ/m, K = 510 kJ/m (∆ = 5 . D c = 3 . - see paragraph III B)], thought to be good candidates toshow the importance of DMI . The value of D is var-ied in order to observe its influence on the micromagneticconfigurations. III. 1D CASE
We first consider the case where the magnetizationdirection only changes along the ˆ x direction. Such acase has already been considered for an infinite film andthe results presented in Sec. III B and III C are alreadyknown , but we recall them as they underline themicromagnetic meaning of the parameter D and its asso-ciated length ξ . Moreover, the results of this 1D modelare essential in order to understand results obtained onskyrmions.Given the fact that, in the case of ultrathin films, ~d ij is orthogonal to ~u ij , the DMI favors rotation in the (ˆ x, ˆ z )plane with a fixed chirality, so that a single angle θ is needed to describe the variation of ~m ( x ). Referring θ tothe ˆ z axis, the total micromagnetic energy density reads E [ θ ( x )] = Z x B x A " A (cid:18) ∂θ∂x (cid:19) − D ∂θ∂x − K cos θ dx, (3)where x A and x B are the boundaries of the sample inthe x direction. We note that, contrarily to the exchangeterm, the DMI term is chiral so that lowest energy statesare expected for ∂θ/∂x of the sign of D . Using standardvariation calculus , it can be shown that the function θ ( x ) which minimizes the energy is the solution of thefollowing equations d θdx = sin θ cos θ ∆ for x A < x < x B (4a) dθdx = 1 ξ for x = x A or x = x B (4b)where ∆ = p A/K and ξ = 2 A/D are the two charac-teristic lengths of the problem. The first one is the wellknown Bloch wall width parameter , while the secondone is . By integration of Eq. (4a) we obtain: (cid:18) dθdx (cid:19) = C + sin θ ∆ (5)where C is an integration constant. A. Magnetic edge structure and micromagneticboundary conditions
Equation (4b) needs to be carefully considered. It cor-responds to a condition at the boundary of the sample.Note that no specific micromagnetic energy was consid-ered at the edges so that this is a ”natural” boundarycondition, that arises from the volume energies. It differsfrom usual micromagnetism (i.e. without DMI) where itwould be dθ/dx = 0 in the absence of surface term, orwhere the edge condition would be due to specific sur-face energies . A striking consequence is that, in afinite dimension structure with DMI, the uniform stateis never a solution of the micromagnetic problem as soonas D = 0.For more complex investigations, these boundary con-ditions have to be implemented in a micromagneticsimulation code, which we have done for two differ-ent codes (one homemade, ref. 37, and the public codeOOMMF, ref. 38). Similarly to previous works onmicromagnetism , a generalized calculation can beperformed for an arbitrary orientation ~n of the edge nor-mal, which leads to the boundary condition d ~mdn = 1 ξ (ˆ z × ~n ) × ~m. (6)This form ensures that the edge magnetization rotates ina plane containing the edge surface normal. Note thatthe condition does not depend on the definition of nor-mal vector ~n orientation. Similarly, the volume equation,Eq. (4a), can be replaced in a general description by aneffective field acting on the local magnetization . Thecontribution of DMI to this term is ~H DM = 2 Dµ M S h ( ~ ∇ . ~m )ˆ z − ~ ∇ m z i (7)In order to test the implementation, direct comparisonshave been performed between the numerical results andEq. (4).A particular case arises when the system under consid-eration has a magneto-crystalline anisotropy sufficientlylarge to avoid cycloid configurations in the structure(section III C). Then C = 0 in Eq. (5) and, combiningEqs. (4b) and (5), we find m x = sin θ = ± ∆ ξ (8)at the edge of the structure. The effect of the DMI spe-cific boundary condition is demonstrated in Fig. 1, witha perfect agreement between numerical and analyticalcalculations. We observe that, in the center of the struc-ture, the magnetization is uniform and perpendicular tothe film surface. At the edge, the magnetization tilts inthe (ˆ x, ˆ z ) plane. The influence of the edge is felt over alength scale ∆, which is the only characteristic length inthe volume equation, see Eq. (5).In reality, dipolar coupling may slightly modify thisresult. Indeed, as the magnetization turns out or inwardat the edges, magnetic charges are created which limitthe magnetization edge tilt. Using numerical simulation,we have calculated the profile with a full dipolar couplingcalculation (for that purpose, we use M S = 1 . × A/mand K MC = 1 . × J/m - which corresponds to K eff = K MC − µ M S = 510 kJ/m as in the previouscalculation). The results are plotted in Fig 1. A smallreduction of the edge tilt is indeed observed but the over-all shape of the magnetization profile is not dramaticallymodified as anticipated. B. Dzyaloshinskii domain walls (
D < D c ) We now consider an infinite system in the ˆ x direction,and have a closer look at Eq. (5). If D is small enoughnot to perturb too much the system (domain wall energyremains positive), the integration constant C must bezero so that no cycloid develops. It is striking to notethat DMI does not appear any longer in this equation.Eq. (5) now has two types of solution. The first one isuniform (far from the sample edges) with θ = 0 or π .The second one corresponds to a domain wall with θ ( x ) = 2 arctan (cid:20) exp (cid:18) ± x − x ∆ (cid:19)(cid:21) + nπ, (9)where x is the position of the domain wall and n an in-teger. The ± sign determines the chirality of the domain Analitical result (Eq. 8)Numerical calculation : Local dipolar approximation Explicit dipolar calculation m x D (mJ/m )(b) mz mx m x (nm) (a) D = 3 mJ/m†
FIG. 1: (color online) Magnetization rotation at the edgesof an ultrathin film with interface DMI. (a) Magnetizationprofile in a stripe infinite in ˆ y direction and with a 100 nmwidth in the ˆ x direction, with initial magnetization along ˆ z axis and for D = 3 mJ/m ( ξ = 10 .
67 nm). (b) Variation of m x at the structure boundary versus ∆ /ξ . The calculationhas been stopped at D = D c = 3 . as beyond thisvalue, cycloids start to develop in the sample and C in Eq. 5is not zero. The continuous line is the solution (numericalintegration) of Eq. (4) for different strengths of the DMI. In(a) and (b) symbols correspond to numerical calculations: forthe open symbols, the local dipolar coupling approximationis used whereas, for the full symbols, the full dipolar energyis included. Note that in (a), both results are hardly distin-guishable. wall and n enables the two types of wall (from 0 to ± π orfrom ± π to ± π ). The shape of this domain wall is ex-actly the same as the Bloch wall obtained without DMI.Note however that in such calculation with schematicdipolar interaction term, the calculation without DMIwould not impose any condition on the orientation of therotation (N´eel and Bloch walls have the same energy),whereas DMI imposes here a rotation in the (ˆ x, ˆ z ) plane(N´eel walls). Note also that, if explicit dipolar interac-tion were included, small deviations to the Bloch wallprofile would occur, due to the magnetic charges createdin the wall. The energy of the domain wall can be calculated byinjecting Eq. (9) into Eq. (3). The integration of the DMIterm is straightforward as θ undergoes a ± π rotation,giving ∓ πD . The two other terms are the same as forthe wall without DMI , so that the domain wall energywith DMI is σ = 4 √ AK ∓ πD. (10)It is interesting to note that DMI does not change theshape of the 1D domain wall but introduces chirality, of asign fixed by that of D . For the most favorable chirality,it lowers the energy. This property is at the origin of quiteinteresting dynamic properties of Dzyaloshinskii domainwalls . The limit of this situation is when σ goes tozero. This defines the critical DMI energy constant D c =4 √ AK/π . Above it, the domain wall energy is negativeso that domain walls proliferate in the sample. In thiscase, the integration constant in Eq. (5) cannot be zeroanymore.
C. Cycloid state (
D > D c ) We now consider a large DMI ( D ≥ D c ). As domainwalls correspond to an energy gain, a cycloid develops inthe sample , with ~m rotating in the (ˆ x, ˆ z ) plane. Wefirst consider the simple case where K = 0 ( D c = 0).In this case, the constant in Eq. (5) is determined byminimizing the energy, integrated over one period L , tobe determined. This leads to θ ( x ) = xξ (11a) L = 2 πξ (11b)This equation corresponds to a pure cycloid with pe-riodicity L . Note that Eq. 11a is compatible withthe edge conditions so that the result is also valid innanostructures. This solution gives a physical meaningto the length scale ξ as it describes the period of cy-cloids, which develop due to DMI, in a zero anisotropysample. The larger the intensity of DMI, theshorter the period.If K = 0, a threshold D c is expected and, as stateswith θ = 0 or π are energetically favored, the pure cycloidshould be deformed . From Eq. (5) we obtain dθ p C + sin θ = dx ∆ (12)which, integrated over one period L , leads to L
4∆ = Z π/ dθ p C + sin θ . (13)Integrating the energy over one period and minimizingwith respect to L leads to DD c = π LL = Z π/ p C + sin θ dθ (14)This last equation determines C . Note that it has a so-lution only if D/D c ≥
1, which validates the previousintuition for the threshold, based on the domain wall en-ergy. For D = D c , C = 0 and the period L diverges.If D ≫ D c , C is large so that sin θ can be neglectedin Eqs. (13) and (14). This leads to L ≈ L . In thiscase, the solution is close to the anisotropy-free solutionin Eq. (11). Results for any value of D are plotted inFig. 2. IV. SKYRMIONS CONFINED IN NANODOTS
For
D > D c , the destabilization of the ferromagneticstate in 2D can lead to the formation of skyrmion net-works . While calculating such networks is beyond the -1.0-0.50.00.51.00.0 0.5 1.0 1.50.0 0.5 1.0 1.5-1.0-0.50.00.51.0 L / D (cid:18) (cid:6) (cid:5) D ¥ (cid:1) (cid:16)(cid:5)(cid:16) (cid:15) (cid:23)(cid:5)(cid:18) (cid:6) m z (cid:16)(cid:1) (cid:14)(cid:1) (cid:13)(cid:1) (cid:21)(cid:17)(cid:5)(cid:21)(cid:24)(cid:18) (cid:6) (cid:1) (cid:14)(cid:1) (cid:8)(cid:8)(cid:4)(cid:9)(cid:10)(cid:1) (cid:22)(cid:21)(cid:16)(cid:1) (cid:14)(cid:1) (cid:10)(cid:1) (cid:21)(cid:17)(cid:5)(cid:21)(cid:24)(cid:18) (cid:6) (cid:1) (cid:14)(cid:1) (cid:11)(cid:6)(cid:4)(cid:8)(cid:12)(cid:1) (cid:22)(cid:21) (cid:2)(cid:20)(cid:7)(cid:3)(cid:2)(cid:20)(cid:8)(cid:3) (cid:1) m z (cid:23)(cid:5)(cid:18) (cid:6) (cid:2)(cid:19)(cid:3) FIG. 2: (color online) (a) Variation of the cycloid period L asa function of the anisotropy-free period L (result of Eqs. (13)and (14)). (b) Shape of the cycloid (perpendicular magne-tization component m z ) in the presence of anisotropy (thedashed line is the reference cycloid with no anisotropy) for(b1) D/D c = 2 . D = 9 mJ/m , L = 22 .
34 nm) and (b2)
D/D c = 1 . D = 4 mJ/m , L = 50 .
26 nm). possibilities of the present formalism, we consider thesimple case of an isolated skyrmion in a circular nanodotof radius R , similarly to the model of the vortex studiedby Feldtkeller and Thomas . The skyrmion being cen-tered in the dot, the circular geometry allows consideringradial variations only. Furthermore, the thin film expres-sion for DMI imposes again a magnetization rotation inthe (ˆ r, ˆ z ) plane (ˆ r is the radial unit vector), which pro-duces a hedgehog skyrmion. The rotation is described bya unique angle θ ( r ) referenced from the ˆ z axis. The dotenergy is E [ θ ( r )] = 2 πt Z R ( A "(cid:18) dθdr (cid:19) + sin θr − D (cid:20) dθdr + cos θ sin θr (cid:21) + K sin θ (cid:27) rdr (15)where t is the dot thickness. A variational calculationleads to the equations for θ ( r ): d θdr = − r dθdr + sin 2 θ (cid:18) r + 1∆ (cid:19) + 2 sin θξr (16a) dθdr = 1 ξ for r = R (16b)We note that the edge condition Eq. (16b) is equiva-lent to that found for the 1D case. Equation (16a)describes the variation of θ in the dot. Its solutionshave been extensively studied in the case of infinite thinfilms. It has no trivial solution respectingthe edge condition. In particular, the uniform state isno more a solution of the problem as soon as D = 0, inanalogy with the 1D case. It has to be integrated numer-ically with initial value θ ( r = 0) = 0. The initial valuefor dθ/dr ( r = 0) is adjusted so as to fulfill the boundarycondition (shooting method).For D = 0, only one solution is found, the uniformstate. Indeed, when no magnetic field is applied and inthe absence of dipolar coupling, no energy can stabilizea reversed domain (magnetic bubble) in the dot. When D increases, this uniform solution is slightly modifiedto fulfill the boundary condition. We further note thatchirality also appears: for this solution, dθ/dr is of thesign of D .Other solutions also exist. An example is given inFig. 3 for D = 4 . . Four solutions have been -1.0-0.50.00.51.0 0 10 20 30 40 500 10 20 30 40 5001 p p p m z Quasi-uniform 2 p state Skyrmion 3 p state q (r ad ) r (nm)(a)(b) FIG. 3: (color online) Results of numerical integrationof Eq. (16) for a 100 nm diameter nanodot with D =4 . ( D/D c = 1 . ), given for comparison. In (b), the variationof θ shows the chirality imposed by DMI in the micromagneticconfiguration. For this set of parameters, three solutions arefound: quasi-uniform (black), skyrmion (red) and 2 π (green)and 3 π (blue) rotation states. found: the uniform one, a skyrmion ( π rotation) and two other solutions with larger magnetization rotation (2 π and 3 π rotation). In order to test the 2D micromagneticsolvers, simulations have been performed and comparedwith these results, as shown in Fig. 3(a). Each state is re-produced with a perfect agreement (for the configurationas well as for the energy), when the energy minimizationis started from an initial configuration close enough tothe targeted one.The skyrmion solution is similar to a bubble centeredin the dot so that the center and the boundary have op-posite magnetization. However, the stabilization of thisstate is given by DMI only, whereas bubbles are stabilizedby external field and/or dipolar coupling . Moreover,we note that this state is different from usual bubbles asthe magnetization rotation is chiral, with a dθ/dr signimposed by D . The magnetization rotation is not pro-gressive along the radius but occurs in a narrow rangeof radius like for a domain wall, and the minimization ofanisotropy energy imposes that this transition occurs ona length scale of ∆. The skyrmion core radius R s (theline with m z = 0) is mainly controlled by the DMI andincreases with D (see Fig. 4(a)): as D lowers the domainwall energy, the skyrmion expands to larger diameterswhen D is large.To discuss the results, we first consider a singleskyrmion, represented as a bubble of radius R s , in aninfinite film. Two ranges have to be considered, accord-ing to the value of D compared to D c . For D < D c ,the domain wall energy σ ( D ), as described in Eq. (10),is positive so that the skyrmion radius should be zero.However, the domain wall is circular so that a curvatureenergy cost needs to be included. This term arises fromthe terms A sin θdr/r and D cos θ sin θdr in Eq. (15),which do not appear in the 1D case (Eq. (3)). As fora domain wall, sin θ = 0 only for r ≈ R S , if R S ≫ ∆the variation of r can be neglected in the integral. Usingthe 1D solution for θ ( r ) (Eq. (9)), the skyrmion energyis then E s ≈ πR s tσ ( D ) + 4 πtA ∆ R s (17)The first term is the domain wall energy cost, the secondone the curvature energy cost. The minimisation of thisequation gives the skyrmion equilibrium size R s ≈ ∆ p − D/D c ) . (18)This solution is plotted as a dotted line in Fig 4(d). When D tends toward D c the skyrmion radius diverges. Forsmall D , the radius is small compared to ∆, so thatEq. (18) cannot be used; numerical calculations showthat R s goes to zero, as demonstrated previously . Thistype of skyrmions are soliton solutions and have beencalled isolated skyrmions . Note that for the smallest D , the skyrmion radius is so small that the magneti-zation profile is close to an arrow shape rather thanto that of a magnetic bubble. However, the transitionfrom one shape to the other is continuous in D so thatno strict semantic difference can be made between thetwo shapes, which both are skyrmions. In the secondrange, for D > D c , the domain wall energy being neg-ative, the previous description does not hold; in infinitefilms skyrmions or cycloids should proliferate, as de-scribed previously. m z r (nm) R = 25 nm R = 50 nm R = 75 nm R = 100 nm Infinite film limite Eq. (18)0 1 2 3 4 5 6 7
D (mJ/m²) r (nm)
D/DC = 0.83D = 3 mJ/m² R = 25 nm R = 50 nm R = 75 nm R = 100 nm r (nm)
D/DC = 1.25D = 4.5 mJ/m² R = 25 nm R = 50 nm R = 75 nm R = 100 nm (a) (b) (c)(d) R s ( n m ) D/D C FIG. 4: (color online) (a) Variation of the skyrmion profilefor different values of D , for a 100 nm diameter nanodot.(d) and (c) Variation of the skyrmion profile versus the dotradius for D = 3 and 4.5 mJ/m ( D/D c = 0 .
83 and 1.25)respectively. In (c) the skyrmion radius is independent of thedot radius, except for very small radius which compresses theskyrmion. Note that all these profiles, although θ ( x ) is notrepresented, it corresponds to a monotonic increasing functionas in Fig 3(b), thus to chiral solutions. (d) Variation of theskyrmion core radius R s versus D for different dot radius.The radius is defined at the m z = 0 line. The line is thesolution for an infinite thin film and the dotted line is theapproximate solution described in the text (Eq. 18). In nanostructures, the situation is rather different asedges play a major role. For the smallest D , we foundthat the skyrmion diameter is independent on the dotdiameter and coincides with the infinite film solution(see Fig 4(b)). These skyrmions are so small that theirshape is not impacted by the edge. For D ∼ D c , wedo not observe the divergence of the skyrmion diame-ter and the transition across D c looks rather continuous.These skyrmions are in fact confined in the dot whichlimits the diameter increase. Moreover, for D > D c and if the dot diameter is not too large compared to the cy-cloid period L (see section III C), a single skyrmion canbe isolated in the dot. This sheds light on the impor-tant role of the edges, which limit the expansion of theskyrmions. We have identified two main aspects of thisconfinement. First, for D > D c , the negative domainwall energy means that nothing is expected to limit thegrowth of an isolated skyrmion. However, in a nanostruc-ture, unlimited increase of the skyrmion radius would letthe domain wall move out of the structure, which wouldturn the dot in the uniform state, with a higher energy(Fig. 5). This contradiction proves that necessarily, theedge must limit the growth of the skyrmion and pro-vides a confinement. The skyrmion radius is then fixedby the dot radius (see Fig 4(c)). Beyond this, anothermechanism also needs to be taken into account, as for D < D c , the domain wall energy being positive, the pre-vious reasoning does not hold. Indeed, if the skyrmionradius increases, as soon as the predicted radius (Eq. 18)is larger than the dot radius, the dot would turn intothe uniform state, with a lower energy (Fig. 5). The factthat these metastable skyrmions exist even for D closeto D c is the signature of an other confinement energy.It is due to the edge tilting previously described: havingthe same chirality as the skyrmion, it provides a topolog-ical barrier and limits the skyrmion diameter increase.Note the importance of this barrier has been observedin a previous study where metastable skyrmions weremoved in a track using spin-transfer torque and where itwas observed that the edge repels the skyrmions.In this study, we have considered only a local dipolarcoupling due to the ultrathin film character needed toobserve interface induced DMI effects. However, in an-other study using purely numerical calculations , similarresults have been obtained with a true dipolar energy cal-culation, which proves that most of the physics can becaptured without the need for sophisticated argumentson this rather complicated energy term.Other solutions, with more magnetization rotationalong the radius also appear. Note that such solutionshave been recently observed in skyrmion networks in in-finite films. This is similar to the problem of the cy-cloid, so that the length scale is again L . Depending onthe value of D they can be more or less stable than theskyrmion. In the example of Fig. 3, the third solutionwith 2 π rotation has an energy slightly higher than theskyrmion state. Indeed the dot radius being R ≈ L ,it seems reasonable to obtain more magnetization rota-tion. Finally, the last solution with 3 π rotation is quiteunfavorable. When D is changed the energy of each statechanges. In Fig. 5, we plot the energy of each state versus D . The four states described previously are not neces-sarily found for each D . It is interesting to note thatthe quasi uniform state no longer exists as a metastablestate above D ≈ , and that the skyrmion ex-ists as metastable state between ≈ . ≈ .However, in the absence of thermal excitation, it be-comes more stable than the quasi uniform state as soonas D & D c . As expected, considering the absolute min-imum, larger D favors larger spin rotation so that nπ solutions (with n >
3) are expected for D larger thanthe explored range. (cid:9)(cid:1) (cid:2)(cid:18)(cid:10)(cid:5)(cid:18)(cid:26)(cid:3) E ne r g y ( x - J ) (cid:1) (cid:11)(cid:24)(cid:13)(cid:22)(cid:16)(cid:4)(cid:24)(cid:19)(cid:16)(cid:15)(cid:20)(cid:21)(cid:18)(cid:1) (cid:22)(cid:23)(cid:13)(cid:23)(cid:14)(cid:1) (cid:12)(cid:17)(cid:25)(cid:21)(cid:18)(cid:16)(cid:20)(cid:19)(cid:1) (cid:22)(cid:23)(cid:13)(cid:23)(cid:14)(cid:1) (cid:6) p (cid:21)(cid:20)(cid:23)(cid:13)(cid:23)(cid:16)(cid:20)(cid:19)(cid:1) (cid:22)(cid:23)(cid:13)(cid:23)(cid:14)(cid:1) (cid:7) p (cid:21)(cid:20)(cid:23)(cid:13)(cid:23)(cid:16)(cid:20)(cid:19)(cid:1) (cid:22)(cid:23)(cid:13)(cid:23)(cid:14) (cid:9)(cid:5)(cid:9) (cid:8) FIG. 5: (color online) Variation of the energy of the differentstates versus D , in a 100 nm diameter dot. Note that eachline does not cover the full explored D range, as we only plotthe solution where a (meta)stable solution has been found. V. CONCLUSION
In conclusion, we have considered the effect of theDzyaloshinskii-Moriya interaction on the micromagneticconfiguration in nanostructures, made of ultrathin mag-netic films. One of the most striking effects is the modi-fication of boundary conditions at the edge of nanostruc-tures, which tilts the edge moments.The formalism has been applied to describe confinedskyrmions in nanodots. The results show that edges areessential to understand such a situation as they providea confinement and limit the skyrmion expansion. Thisconfinement is rather important for future developmentof skyrmions-based memories and should deserve fur-ther studies in order to be quantitatively understood. AppendixA. Generalization to other forms for DMI
We have limited ourselves to the DMI form for ul-trathin films. Much experimental work has also beenperformed on bulk materials lacking inversion symmetry,belonging to the D n symmetry group. In these, DMIis homogeneous in the volume and ~d ij = d~u ij . For athin film where magnetization direction variation along the film normal can be neglected, the continuous DMIenergy becomes E DM = Z Z Z D (cid:20)(cid:18) m y ∂m z ∂x − m z ∂m y ∂x (cid:19) − (cid:18) m x ∂m z ∂y − m z ∂m x ∂y (cid:19)(cid:21) d ~r (19)For a 1D system, this interaction favors spin rotation inthe (ˆ y, ˆ z ) plane (which means Bloch walls, spirals andvortex-type skyrmions for the different cases consideredabove), so that θ has to be defined in this plane. In thiscase, all other equations remain the same, in particularthe boundary condition in Eq. (4b). Only the generalform of the boundary condition for this form of DMI ismodified, though the derivation follows the same proce-dure: d ~m/dn = ( ~m × ~n ) /ξ. (20)Compared to the boundary condition in Eq. 6, this oneensures that the edge magnetization rotates in a planeparallel to the edge surface. B. Extension to thicker samples
In this paper, we considered the case of DMI ultra-thin films with interface DMI. As the sample consideredis thinner than ∆ and ξ , we assumed a uniform effec-tive DMI constant across the thickness. This assumptiondoes not hold for thicker samples. In these, the DMIis expressed as a surface term, with D S,i the interfaceDMI constant (in J/m) where i accounts for the bottomand top interfaces. The micromagnetic energy, limitedto DMI and exchange, reads: E = X i D S,i
Z Z (cid:20)(cid:18) m x ∂m z ∂x − m z ∂m x ∂x (cid:19) + (cid:18) m y ∂m z ∂y − m z ∂m y ∂y (cid:19)(cid:21) d ~r + A Z Z Z (cid:16) ~ ∇ ~m (cid:17) d ~r (21)where the surface integral is performed at the inter-faces only (assumed normal to ˆ z ). Using variationalcalculation we extract interface conditions ∂m x ∂z = ε i D S,i
A ∂m z ∂x (22a) ∂m y ∂z = ε i D S,i
A ∂m z ∂y (22b) ∂m z ∂z = − ε i D S,i A (cid:18) ∂m x ∂x + ∂m y ∂y (cid:19) (22c)with ε bottom = 1 and ε top = − D S, top = − D S, bottom . Indeed, in the atomic formulation,DMI is proportional to ( ~u × ˆ z ), ˆ z being oriented from thehigh-spin orbit layer to the magnetic layer, thus oppositefor both interfaces. As a consequence, magnetization isbend the same way (i.e. with the same chirality) at bothinterfaces.While in such situation, DMI should not be sufficient todestabilize the ferromagnetic state, such boundary condi-tion should modify the structure of domain walls. Indeed,in the volume, Bloch rotation is expected and, at the fer-romagnetic film surfaces, N´eel rotation is expected, withopposite chirality for bottom and top interfaces. This ef-fect, which is purely related to DMI, should add to similar effects due to dipolar coupling. Acknowledgments
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