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Edge instabilities and skyrmion creation in magnetic layers Jan Mu¨ller,∗ Achim Rosch, and Markus Garst Institut fu¨r Theoretische Physik, Universita¨t zu Ko¨ln, D-50937 Cologne, Germany (Dated: January 27, 2016) We study both analytically and numerically the edge of two-dimensional ferromagnets with Dzyaloshinskii-Moriya (DM) interactions, considering both chiral magnets and magnets with interface-inducedDMinteractions. Weshowthatinthefield-polarizedferromagneticphasemagnon states exist which are bound to the edge, and we calculate their spectra within a continuum field theory. Upon lowering an external magnetic field, these bound magnons condense at a finite momentum and the edge becomes locally unstable. Micromagnetic simulations demonstrate that this edge instability triggers the creation of a helical phase which penetrates the field-polarized state 6 withinthebulk. Asubsequentincreaseofthemagneticfieldallowstocreateskyrmionsclosetothe 1 edge in a controlled manner. 0 2 PACSnumbers: 75.78.-n,75.75.-c,12.39.Dc n a I. INTRODUCTION J 6 2 ThepresenceoftheDzyaloshinskii-Moriya(DM)inter- actioninferromagnetsfavoursaspatialtwistofthemag- ] netization leading to modulated magnetic textures like l e helicesandskyrmionlattices,i.e.,closelypackedarrange- - ments of single skyrmions. Skyrmions are spatial con- r t figurations of the magnetization with a finite topological s . windingnumber. Theyarenotonlyfoundinthethermo- t a dynamic ground state but single skyrmions also arise as FIG. 1. Chain of skyrmions at the edge of a two- m topologically protected excitations of the field-polarized dimensionalchiralmagnetthatwerecreatedwiththehelpof - state. Skyrmion couple very efficiently to spin currents themagneticfieldprotocolshownintheinsetexploitingthe d sothatultralowcurrentdensitiesarealreadysufficientto local edge instability of the field-polarized state. These re- n drive skyrmion configurations1,2. Both properties, their sults were obtained with micromagnetic simulations of the o Landau-Lifshitz-Gilbert equation. The color code reflects topological protection as well as their mobility, identify c the z-component of the magnetization. [ skyrmions as promising candidates for elementary units of future spintronic devices, see the reviews of Refs. 3 1 and 4. v 2 A prerequisite for a skyrmion technology5 is, how- As skyrmion configurations are topologically pro- 2 ever, the stabilization of magnetic skyrmion configura- tected, theircreationorannihilationusuallyrequiresthe 9 tions at ambient conditions as well as their controlled climb of a topological energy barrier in some way or an- 6 writing and deleting. The existence of skyrmions far be- other. It has been theoretically discussed that magnetic 0 low room temperature has been experimentally demon- skyrmions can be nucleated by the local injection of a . 1 strated some years ago in the chiral magnets with space spin-polarized current17, local heating18,19, local appli- 0 group P2 3 like MnSi6,7, FeGe8 or Cu OSeO 9,10 as well cation of a magnetic19 or electric field20, and by ther- 1 2 3 6 as in magnetic mono- and bilayers11,12. Recently, var- mal fluctuations21. Experimentally, it has been already 1 ious groups have reported the observation of skyrmion demonstrated that skyrmions can be created in mag- : v configurations at ambient temperatures. Co-Zn-Mn al- netic multilayers with the help of local currents from an Xi loys with the chiral space group P4132 were shown to STM tip22. Magnetic skyrmions also materialize when possess the typical phase diagram of other chiral mag- driving stripe domains through a geometric constriction r a netsbutwithaskyrmionlatticephasestabilizedatroom with the help of a spin-current15,23. It was also shown temperature13. Inaddition,certainmagneticmultilayers that skyrmion crystals can be unwinded with the help of comprising magnetic Co and Fe atoms also show stable Bloch points that act as magnetic monopoles on itiner- skyrmion configurations at room temperatures14–16. In ant electrons24. Climbing the topological energy barrier allthesesystems,theDMinteractionarisesduetoalack and the concomitant singular Bloch point configuration ofinversionsymmetry. Theatomiccrystalofchiralmag- of the magnetization can be avoided however by feeding netsexplicitlybreaksinversionsymmetrysothattheDM in skyrmions from the edge of the sample. In fact, re- interaction is even present in bulk samples. In magnetic cent experiments on FeGe nanostripes have beautifully multilayers, on the other hand, the DM interaction is in- demonstrated already the edge-mediated nucleation of duced by the interface between a magnetic layer and a skyrmions25. In the present work we provide a theo- substrate containing heavy elements. retical explanation of this phenomenon. In addition, we 2 demonstratethatwiththehelpofacertainmagneticfield (cid:82) d2rF, reads protocol skyrmions can be created in a controlled manner at the edge by exploiting a local edge instability of F =A(∂αnî)2−Knˆ23−µ0HMnˆ3+FDM. (1) the field-polarized state, see Fig. 1. with α = x,y and i = 1,2,3, A is the stiffness, and K For this purpose, we investigate the edge of a single is the magnetic anisotropy with an easy-axis and easy- magnetic layer with DM interaction that is polarized by plane anisotropy for K > 0 and K < 0, respectively. aperpendicularmagneticfield. Theboundaryconditions Note that for a two-dimensional layer the dipolar in- arising from the DM interaction result in a twist of the teractions are effectively accounted for by a renormal- magnetization close to the edge which was discussed in- ized anisotropy K33. The third term is the Zeemann dependentlybyWilsonet al.26 andRohartandThiaville energy with an external field H applied perpendicular et al.27. This surface twist was experimentally investi- to the plane, H = Hzˆ with H > 0, and the magni- gated in Refs. 25 and 28, and it also plays an impor- tudeofthelocalmagnetizationM (integratedoverthez tant role in the stabilization of skyrmion configurations direction).There a two different types of Dzyaloshinskii- in thin films29–31. Here, we focus on the magnon exci- Moriya (DM) interactions F . For chiral magnets like tations of the field-polarized layer, and we demonstrate DM MnSi or FeGe where the inversion symmetry is broken thatthesurfacetwistresultinspin-waveexcitationsthat by the atomic crystal the DM interaction reads34 are bound to the edge of the magnetic layer. Within a continuum field-theory, we determineanalytically the ef- Fchiral =D(cid:15) nˆ ∂ nˆ , (2) DM iαj i α j fective Hamiltonian of these bound magnon modes and evaluatetheirspectrumforvariousvaluesoftheeffective whileforinterface-inducedDMinteractionsrelevant,e.g., magnetic field and the magnetic anisotropy. For vanish- for Co films grown on heavy-element layers (e.g. Pt or ingH =0,wereproducetheresultsofarecentnumerical Ir), we have35,36 study ofthese edge excitations32. Moreover, for finite H Finterface =D(nˆ ∂ nˆ −nˆ ∂ nˆ ) (3) and intermediate values of the magnetic anisotropy the DM α α 3 3 α α edge magnons become locally unstable and condense at with α = x,y and i,j = 1,2,3. It has been pointed out a finite momentum. It is this local instability that facili- in Ref. 37 that for two-dimensional systems both types tates the skyrmion creation. of DM interactions are actually equivalent because they Theoutlineofthepaperisasfollows. Insec.IIwede- can be transformed into each other by rotating nˆ by π/2 termine the magnetization at the edge of a single layer, around the z-axis, i.e., by nˆ → nˆ , nˆ → −nˆ . Mo- x y y x wederivethemagnonHamiltoniananddiscussitseigen- tivated by the experiment on FeGe films of Ref. 25, we modes that are bound to the edge of the magnetic layer. will use in the main part of this work the formulae ap- Wesummerizetheinstabilitiesofthefield-polarizedstate propriate for Fchiral and assume D > 0. Our results are DM in sec. IIE with a particular focus on the local instabil- straightforwardly generalized to interface-induced DM ity triggered by the bound states that become unstable interactions with the help of the above symmetry. at a finite transversal momentum. In sec. III we demon- The dynamics of the magnetization is governed by the strate with the help of micromagnetic simulations that Landau-Lifshitz equation this latter instability triggers the formation of a helical phase, and, in addition, how this instability can be used ∂tnˆ =−γnˆ×Beff (4) tocreateskyrmions. Weconcludeinsec.IVwithashort with the gyromagnetic ratio γ = gµ /(cid:126) > 0 and the ef- discussion. B fective magnetic field obtained by the functional derivative B = − 1 δF/δnˆ where F = (cid:82) d2rF. For our eff M micromagnetic simulations, we furthermore add Gilbert II. MAGNONS AT THE EDGE OF A CHIRAL damping to these equations, see Ref. 38. The typical MAGNET momentum, time and energy scales are given by D (cid:126)M (cid:126) A. The Free energy functional Q= 2A, tDM = gµ 2AQ2, εDM = t . (5) B DM For our study, we consider the one-dimensional edge In the main part of the paper, we will measure length, of a two-dimensional magnetic layer. For sufficiently low time and energy in dimensionless units by denoting the temperatures, one can neglect (thermal) amplitude fluc- variables with a tilde, e.g. tuations of the spins and the local direction of the mag- x˜=xQ, y˜=yQ, t˜=t/t , ε˜=ε/ε . (6) netization can be described by the unit vector nˆ. All DM DM modes discussed in the following including the boundary Moreover,itisconvenienttointroducethedimensionless statesattheedgearecharacterizedbylengthscalesmuch magnetic field h and the dimensionless anisotropy κ, larger than the lattice spacing, so that a description in terms of a continuum model is applicable. h= µ0HM, κ= K . (7) The free energy functional of the magnetic layer, F = 2AQ2 AQ2 3 Asweareinterestedinthepropertiesoftheedgeofthe The translational invariance along the edge ensures layered sample, we have to supplement the equations of that nˆ is only a function of y˜, i.e., the distance from motion(4)byaboundarycondition. Weassumetheedge the edge. We consider a chiral magnet described by Eqs. parallel to the x-axis. By varying the free energy F = (2) and (8) where the magnetization twists perpendicu- (cid:82)∞ (cid:82)∞ dx dyF,oneobtainsforchiralmagnetsdescribed lar to the propagation direction similar to a Bloch-type −∞ 0 by Eq. (2) the boundary condition28 domain wall. Hence we choose ∂ nˆ−Qyˆ×nˆ =0 at y =0 (8) nˆT =(sin(θ(y˜)),0,cos(θ(y˜))) (10) y 0 while for interface-DM interactions (3) one finds the as an ansatz for the magnetization. The equation of mo- boundary condition27 tion(4)inthestaticlimitandtheboundaryconditionof Eq. (8), respectively, then reduce to ∂ nˆ+Qxˆ×nˆ =0 at y =0. (9) y κ A subtle question is whether the continuum approach θ(cid:48)(cid:48)(y˜)=hsin(θ(y˜))+ sin(2θ(y˜)), (11) 2 advocated above is valid for real materials. In general, θ(cid:48)(y˜)| =1. (12) the chemistry and therefore the exchange couplings, the y˜=0 anisotropies, and DM interactions at the edge will dif- Moreover, θ(y˜),θ(cid:48)(y˜) → 0 for y˜ → ∞ as we assume the fer from the ones within the layer. These changes do, magnetization nˆ = zˆ to be aligned with the field within however, affect our results only weakly if (i) the modifi- the bulk of the layer. cation of the chemical properties is limited to distances of a few lattice constants a away from the edge and (ii) spin-orbit coupling effects are weak. The latter condi- 1. Kink soliton of the double sine-Gordon model tion implies that the typical length scale on which the magnetization varies is much larger than the lattice con- The equation of motion Eq. (11) is just the Euler- stant a and, furthermore, that anisotropy terms remain Lagrangeequationofthedoublesine-Gordonmodel39,40. smallcomparedtoexchangeinteractionsevenatthesur- It turns out that the magnetization at the edge can be face. Technically, one can check for the importance of obtained by placing a kink soliton of this model close to surface modification by adding surface, i.e., edge terms the edge. The analytical expression for this kink soli- to the continuum Lagrange density, e.g., of the form ton is known in closed form39,40 and we shortly discuss K δ(y)nˆ2, A δ(y)(∂ nˆ )2, or D δ(y)nˆ ∂ nˆ with cou- s 3 s y i s 1 x 2 its derivation in the following. As the derivative at the pling constants K , A , D . The δ-function takes into s s s edge is positive, θ(cid:48)(0) > 0, we will look for a kink that accountthatonlytheedgeisaffected. Theimportanceof interpolates θ from −2π to 0 for increasing y˜. The first such terms can be evaluated by doing power-counting in integral of the equation of motion is obtained in a stan- the strengthof spinorbitcoupling λ where weassume SO dard fashion by multiplying Eq. (11) with θ(cid:48), integrating that A∼A ∼λ0 , D ∼D ∼λ , and K ∼K ∼λ2 s SO s SO s SO andusingtheboundaryconditionswithinthebulkofthe suchthatthedimensionlessparametershandκareofor- layer, θ(∞)=θ(cid:48)(∞)=0, der 1. Under this condition all length scales are propor- tional to 1/λSO. As δ(y) = λSOδ(yλSO) all extra terms θ(cid:48)2 κ κ discussed above are suppressed by the small factor λ +hcosθ+ cos2θ =h+ . (13) SO 2 2 2 comparedtothebulkcontributions. Thisargumentdoes nominally not hold for the surface term δ(y)nˆ∂ nˆ which The kink is obtained by solving Eq. (13) for θ(cid:48)(y˜) > 0. y by powercounting is as important as the bulk contribu- Integrating Eq. (13) by separation of variables yields the tionsbutthistermisidenticaltozeroas∂ nˆ2 =0. Fora kink soliton y discussion of surface terms for a three-dimensional mag- (cid:16)(cid:114) h √ (cid:17) net see Meynell et al.28. θ(y˜)=−π+2arctan sinh( h+κ(y˜−y˜ )) h+κ 0 (14) B. The magnetization at the edge of the layer where y˜ is an integration constant that specifies the po- 0 sition of the kink. For later convenience, we will also We first consider the static magnetization close to the discuss the energy of the kink when it is placed deeply edge assuming that the magnetic layer is in a polarized withinthebulk,y˜ →∞. ThekinkenergyperlengthL state with nˆ =Hˆ =zˆwithin the bulk of the layer. Close 0 (cid:82) x isobtainedbyintegratingE /L = dy(F −F ) kink x kink FP to the edge of the two-dimensional chiral magnet, the where F is the free energy density evaluated for the kink magnetization has to twist due to the boundary condi- kink, and we have subtracted the energy density of the tions (8) or (9). An analytic solution for this twist has field-polarized state, F , FP already been derived by Meynell et al.28 for a system (cid:113) wizeiththoueitrmreasgunltetiincclaundisinogtraopfiynKite=K0..InBeolrodwe,rwtoe sgiemneprliafly- Ekink =−2π+4√h+κ+ 4harcta√nh h+κκ. (15) notations we use the rescaled coordinates of Eq. (6). 2AQLx κ 4 There exists a critical magnetic field hcr for which the kink �� kink energy vanishes, )� ( � E | =0. (16) �� kink hcr � kink � � This critical field is defined within the range −1 < κ < �� (κ��) πze2r/o4. wFhoerreκit=m0onoonteonrieccaolvlyerdsehcrcrease=s frπo2m/1h6ckr(inRke=f. 136t)o. �� ((��)) Whereas for larger fields the kinkkiniks an excitation of the ��-�� (-(��)) field-polarized state, the commensurate-incommensurate (��/�) transitioncantakeplaceathcr ,andkinksbecomeener- -�� kink � � � � � geticallypreferredresultinginasolitonlattice. However, �� as the kink is a topological excitation it does not corre- spond to a local but rather a global instability of the FIG.2. Profileofthez-componentofthenormalizedmag- field-polarized state for h≤hcr . kink netization nˆ0,3 =cosθ(y˜), see Eq. (14), as a function of the dimensionless distance to the edge y˜= Qy for various val- uesofthedimensionlessfieldhandanisotropyκ. Notethat 2. Edge magnetization the profile for the values (κ,h) = (0,1/4) is unstable, see discussion in sec. IIE. The full magnetization profile is now easily obtained by placing the kink Eq. (14) close to the edge so that the boundary condition, θ(cid:48)(0) = 1, is fulfilled. This is with the angle θ = θ(y˜) given by Eq. (14). We use the achieved with the integration constant following parametrization of the field nˆ, 1 (cid:16)h+κ+(cid:112)(h+κ)2−κ(cid:17) nˆ = eˆ (cid:112)1−2|ψ|2+eˆ ψ+eˆ ψ∗ (23) y˜ =−√ arccosh √ . (17) 3 + − 0 κ+h h witheˆ± = √1 (eˆ1±ieˆ2). Achangeofphaseofthecomplex The center of the kink is positioned outside the sam- 2 magnon wavefunction ψ = ψ(x˜,y˜,t˜) naturally encodes ple y˜ < 0. Note that there is also a solution of the 0 the precession of the magnetization. boundary condition with a positive y˜ but it yields a 0 Expanding the Landau-Lifshitz equation (4) up to lin- state with larger energy. The result of Eq. (14) together ear order in the wavefunction ψ and ψ∗, we obtain the with Eq. (17) determines the magnetization close to the wave equation edge of the field-polarized magnetic layer. In the limit of vanishing anisotropy, κ=0, we recover i(cid:126)τz∂ ψ =Hψ (24) the results of Ref. 28, t˜ (cid:16) √ (cid:17) (cid:16) (cid:17) θ(y˜)|κ=0 =−4arctan e− h(y˜−y˜0) , (18) for the two-component spinor ψ = ψψ∗ , τi are Pauli- 1 (cid:18)1 1 (cid:19) matrices, and H is an effective Bogoliubov-deGennes y˜0|κ=0 =−√hlntan 2arcsin2√h . (19) Hamiltonian. It can be split into the bulk contribution H and the 0 We checked the result for various combinations of h and edgepotentialV =V(y˜)thatvanishesV →0fory˜→∞, κ also numerically by simulating the damped Landau- Lifshitz-Gilbert equation and find that numerics are in H =H +V. (25) 0 excellent agreement with the analytic expression. In Fig. 2 we show the analytic result for the z-component After performing a Fourier transformation of the spinor of the magnetization in the proximity of the edge. wavefunction, ψ(q˜x,y˜,t˜) = (cid:82) dx˜eiq˜xx˜ψ(x˜,y˜,t˜), for the x˜ direction along which the problem is translationally invariant, we obtain C. Effective magnon Hamiltonian H =−∂2+q˜2+∆ , (26) 0 y˜ x b Havingestablishedtheprofileofthemagnetization,we know consider the fluctuations around the saddle-point where q˜ is the dimensionless wavevector along x˜. The x solutiondefinedbyEqs.(10),(14)and(17)followingthe magnon gap within the bulk of the layer is given by approach of Ref. 41. We introduce a local basis ∆ =h+κ. (27) eˆT = yˆT =(0,1,0) (20) b 1 eˆT2 =− ∂θnˆT0=(−cosθ,0,sinθ) (21) For∆b <0,thefield-polarizedstatewithinthebulkislo- eˆT = nˆT =(sinθ,0,cosθ) (22) cally unstable with respect to a tilt of the magnetization 3 0 5 away from the field axis. The potential reads �� V(q˜x,y˜)=1(cid:16)−θ(cid:48)2(y˜)+ θ(cid:48)(y˜)−κsin2(θ(y˜))(cid:17) (28) )�� ==�� +τz2(cid:18)q˜xsin(θ(y˜)) (cid:19) ε/(ε�� ==�� 1 κ � +τx −2θ(cid:48)2(y˜)+θ(cid:48)(y˜)+ 2 sin2(θ(y˜)) �� with the angle θ(y˜) given in Eq. (14). We have used the � � firstintegral(13)tosimplifythepotentialand,inpartic- �� ular, to eliminate the explicit dependence on dimensionless magnetic field h. Moreover, an explicit calculation -�� showsthattheboundaryconditionsforthespinorissim- �� /(��/�) ply given by � FIG. 3. Dispersions ε (q )/ε = ε˜ (q˜ ) with q˜ = q /Q ∂ ψ| =0. (29) 0 x DM 0 x x x y˜ y˜=0 of the magnon edge modes with lowest energy, n = 0, as a function of momentum q along the edge for fields h = The bosonic Bogoliubov-de Gennes Hamiltonian H, x 1,0.65,0.45,0.33 and anisotropy κ = 0. The grey shaded Eq. (25), possesses scattering states that extend into the areaisthebulkcontinuumthatterminatesatthesolidblack bulk of the layer as well as magnon states that a bound line. to the edge by the potential V. In the following, we will concentrate on these magnon edge modes. �� κ=� D. Bound magnon edge modes εκ)�� κκ==�� /(� κ=�� ε Wewilllookforeigenstatesφ=φ (y˜)thatarelocal- � ε˜,q˜ �� ized at the edge of the chiral magnet, which requires the �� dimensionless energy ε˜to have values within the range, � � s0ta≤tioεña<ry∆wbav+eq˜ex2q.uaTtihone4l1ocalized eigenstates obey the �� Hφ=ε˜τzφ. (30) �� -�� /(�κ�/�) with the normalization condition � (cid:90) ∞ FIG. 4. Dispersions ε (q )/ε = ε˜ (q˜ ) with q˜ = q /Q φ†τzφdy =1. (31) 0 x DM 0 x x x of the magnon edge modes with lowest energy, n = 0, as 0 a function of momentum q along the edge for anisotropies x From the spinor-wavefunction φT = (φ ,φ ) then fol- κ = 1.05,1.30,1.77,4.00 and field h = 0. The grey shaded 1 2 areaisthebulkcontinuumthatterminatesatthesolidblack lows the corresponding time-dependent magnon wave- line. function in the parametrization of Eq. (23), that is ψ(t) = φ e−iεt/(cid:126) +φ∗eiεt/(cid:126) with ε = ε˜ε where ε 1 2 DM DM was defined in Eq. (5). and its bound states can be labelled by a discrete quan- We numerically search for bound state solutions using tum number n = 0,1,2,... that count the nodes of the the shooting method. Starting from the boundary con- wavefunction. Moreover, the solutions for given mo- dition at the edge φT(0)=c (cosα,sinα) and φ(cid:48)(0)=0 1 menta along the edge q define dispersing eigenenergies x wherec isanarbitraryconstantthatwillbefixedafter- 1 ε (q ) = ε ε˜ (q˜ ) with q˜ = q /Q for the magnon n x DM n x x x wards by the normalization condition (31), we vary the bound states that are discussed in the following. energy εãnd the parameter α until we find a solution of Eq. (30) that is bound to the edge so that it decays for y˜→∞. Theasymptoticdecayofthelocalizedwavefunc- 1. Lowest-energy magnon edge modes tion directly follows from H of Eq. (26), 0  √  The dispersion of magnon edge modes with lowest en- φ(y˜)∼c2e−√∆b+q˜x2−ε˜y˜ , for y˜→∞ (32) ergyε0(qx), i.e., quantumnumbern=0andmomentum c3e− ∆b+q˜x2+ε˜y˜ qx along the edge are shown for various values of the dimensionless magnetic field h and κ = 0 in Fig. 3 as with constants c and c . The problem (30) corre- well as for various values of the dimensionless magnetic 2 3 sponds to an effective one-dimensional wave equation, anisotropy κ and h = 0 in Fig. 4. We have rescaled the 6 energy by the dimensionless bulk gap ∆ = h+κ and �� √ b ⨯�� =� the momenta by ∆ such that the bottom of the bulk canolulnmtpieanrruiacumamles,tte∆urdsb.y+bOyq˜ux2rG,bariserscguiailvt-Sesnaanrbceyheianz esaitgnragell.eemibnelnaRtcekfw.lii3tn2he.tfhoer ε/(ε)�� 10 �� Generally,wealwaysfindarangeofmomentaqx where �� =� � magnon modes exist that are bound to the edge. In- �� 0 �� ⨯�� terestingly, the spectrum is chiral, i.e., it is not sym- � � 0 1 2 3 4 metric around q = 0 as the presence of the magnetic � � x �� field breaks all q → −q symmetries at the boundary. x x Especially for larger magnetic fields h the group veloc- � ity ∂ε (q )/∂q is mostly positive, which implies that -� � � � � � 0 x x magnonstravelalongtheedgeinapreferreddirection32, �� /(��/�) � for example, after they have been excited locally by a laser pulse. Note that the modes with the reversed dis- FIG. 5. Dispersions ε (q ) of the magnon edge mode with- n x persion ε (q )→ε (−q ) are found at opposite edges of out and a single node, n = 0 and n = 1, respectively, for n x n x the layer. Theoretically speaking, the dispersion could h=0.44andκ=0. Theinsetshowsthefirst(blue)andsec- be reversed at the same edge by reversing the sign of the ond(orange)componentoftheeigenfunctionφatqx =3Q, where the latter is multiplied by a factor 10 to make it vis- gyromagneticratioγ →−γ,e.g.,byconsideringholesin- ible. The grey shaded area is the bulk continuum that ter- steadofparticles. However,reversingthesignoftheDM minates at the solid black line. interaction, D → −D, modifies the edge magnetization but eventually leaves the dispersion ε (q ) invariant in n x contrast to the claim of Ref. 32. This also applies for in- E. Phase diagram and instabilities of the terfacialDMinteraction. Whileforκ=0(Fig.3)bound metastable field-polarized state states exist only for q > 0, one obtains edge magnons x at q =0 for h=0 and κ>1.005. In the latter case an x In the following we discuss the phase diagram of the homogeneously oscillating magnetic field can be used to magneticlayerasafunctionofmagneticfieldhandmag- excitealeft-movingedgemode. Inthepresenceofdisor- netic anisotropy κ, see Fig. 6. First, we review the ther- der at the edge, i.e., when the momentum q parallel to x modynamically stable phases, and, afterwards, we focus the edge is not conserved, also right-moving modes can onthemetastablefield-polarizedphaseanditsglobaland be excited by such an ac field. local bulk and local edge instabilities. With lower magnetic fields the minimum of the spectrum, 1. Thermodynamic phase diagram ∆ (h,κ)=minε˜ (q˜ ) (33) e 0 x Thebulkthermodynamicphasediagramofamagnetic is shifted down to lower energies and to larger wave vec- layer in the presence of a perpendicular magnetic field tors. At a critical dimensionless magnetic field h = c has been studied in Refs. 37, 42, and 43. For a two- 0.4067 at κ = 0 the minimum of the spectrum goes to dimensional chiral magnet, there exist three stable ther- zero energy, ∆ =0, at a finite momentum q . The sys- e x modynamic phases for values of κ that are of interest tem therefore experiences a local instability at the edge here: the field-polarized state (FP), the skyrmion crys- which we discuss in further detail in Sec. IIE. Similarly, tal (SkX), and the chiral soliton lattice (CSL), i.e., a we find such a local edge instability for κ = 1.005 for c helix that in general possesses higher harmonics44. The h=0. black dashed lines in Fig. 6 show the phase boundaries that were given by Wilson et al.42. (We have slightly ex- trapolated the phase transition lines of Ref. 42 to more 2. Higher-order magnon edge modes negative values of κ.) There exists a triple point (black dot) at (κ,h)≈(1.9,0.1) where the three phases meet42. For larger values of q , the gap between the lowest x bound magnon mode ε (q ) and the bulk continuum in- 0 x creases approximately linear. This dependence can also 2. Global, local bulk and local edge instabilities of the be estimated from the potential, Eq. (28), which grows metastable field-polarized state linearinq forlargeq . Asthepotentialbecomesdeeper, x x higher-orderboundstates,ε (q )withn>0,ariseatthe Importantly,thetransitionbetweenthefield-polarized n x edge which are characterized by one (or more) nodes in state (FP) and the skyrmion crystal (SkX) corresponds their wave function. The spectrum of the bound states toaglobalinstabilityasthetwostatesbelongtodifferent with zero and one node is shown in Fig. 5 for the mag- topological sectors. If the magnetic field is reduced adi- netic field h=0.44. abatically at low enough temperatures to a value within 7 �� III. EDGE INSTABILITY AND CREATION OF �� SKYRMIONS �� Δ�=� � Δ�=� �� In the following we demonstrate that the local edge �� Δ�=Δ� �� instability of the metastable field-polarized state at the � �� red line in Fig. 6 triggers the formation of a helical state �� (CLS) although in some region of the phase diagram �� theskyrmioncrystal(SkX)isinfactthermodynamically more stable. Moreover, we show that this edge instabil- �� ity can be exploited to create skyrmions in a controlled �� manner. Inordertoinvestigatetheevolutionofthemag- -�� -�� neticstateintheregimewherethefield-polarizedstateis ��κ locally unstable, we have performed micromagnetic simulations using the Landau-Lifshitz-Gilbert (LLG) equa- FIG. 6. Overview of the local and global stability of the field-polarizedstate(FP).Theblackdashedlinesarerepro- tion at T = 0, for details see Ref. 38. Typically we use ducedfromRef.42anddenotethermodynamicphasetransi- α=0.1 or 0.4 as a damping constant in our simulations. tionsbetweentheFPstate,theskyrmioncrystal(SkX)and Wesimulatedatwo-dimensionalstripewithopenbound- the chiral soliton lattice (CSL). Within the white regime ary conditions in one and periodic boundary conditions the FP state remains metastable, but it is locally unstable in the perpendicular direction. within the grey shaded regime. The green dotted line indi- We focus on the range of magnetic anisotropies where catesaglobalinstabilityhcr wheretheenergyofthekink kink the edge magnons locally destabilize the FP state, i.e., solitonvanishes,seeEq.(16). Onthegreysolidlineh=−κ −0.61 < κ < 1.005 for the two-dimensional system, see themagnongap∆ vanisheswithinthebulkwhiletheedge b Fig. 6. For a film of finite thickness for which the phase gap ∆ is zero on the red solid line. On the blue solid line e diagram also contains a conical phase42,43, the values of the magnon gap within the bulk and at the edge are equal, ∆ =∆ . κwouldberestrictedtoasmallerinterval. Thefollowing e b protocolforthetime-dependentmagneticfieldallowsfor acontrolledcreationofaskyrmionchainclosetotheedge  the SkX regime, there is the possibility that the mag- hi for t<0 netic layer remains in a metastable field-polarized state. h(t)= h for 0<t<t (34) 0 f As the field is reduced further, another global instability h for t <t. ofthemetastableFPstateisencounteredathcr within f f kink therange−1<κ<1.9thatisshownasthedashedgreen Theinitialfieldvalueh islocatedabovetheupperblack i line in Fig. 6. Here the energy of a kink soliton vanishes, dashedlineinFig.6sothatwestartwithamagnetization see Eq. (16), triggering the formation of a chiral soliton which is field-polarized. At t = 0 the field is reduced lattice. The phase boundary between FP and CSL state h < h below the red line in Fig. 6 so that ∆ < 0. 0 i e within the range 1.9<κ<π2/4 is also defined by hcr . Finally, at time t we increase the field again to a value, kink f h > hcr , that is located above the green dashed line The field-polarized state is stable or metastable in the f kink inFig.6. Theresultofthemicromagneticsimulationfor whole white regime of Fig. 6. It becomes locally unsta- such a protocol at κ=0 is shown in Fig. 7. ble however when one of its magnon excitations reaches Initially, the magnetization is fully polarized except zero energy upon decreasing the magnetic field h. This close to the lower edge of Fig. 7(a) that shows the sur- instability can occur either in the bulk or at the edge. facetwistofEq.(14). Asthefieldisloweredfort>0,the For κ < −0.61, first the bulk becomes locally unstable edgemagnonbecomessoftatafinitetransversalmomen- at h=−κ (grey solid line) where the bulk gap vanishes, tum q and destabilizes the magnetization whose time ∆ = 0, see Eq. (27). For −0.61 < κ < 1.005, on the x b evolution is shown in Fig. 7(b)-(d). To trigger the edge other hand, the gap of the magnon edge modes vanishes instability in the numerics, we explicitly broke transla- first, ∆ = 0, see Eq. (33), at the red solid line. This e tion symmetry by introducing a tiny perturbation by edge instability occurs at a finite transversal momentum canting one spin at the right-hand side of the bound- q , see Fig. 3 and Fig. 4. x ary by 1%. First, a periodic modulation of the edge Finally,thebluesolidlineinFig.6indicateswherethe spins grows in amplitude – the edge magnon with neg- bulk and edge gap have equal size. Below that line the ative energy becomes macroscopically occupied at finite gap ∆ of the edge magnons is smaller than the gap ∆ momentum q . This state evolves smoothly into a heli- e b x of the bulk magnons, and the spin wave excitation with cal state which penetrates into the field-polarized state lowest energies are located at the edge of the sample. of the bulk. The interface between the helical and po- Within this regime, for frequencies ∆ ≤ (cid:126)ω˜ < ∆ only larized phase is thereby described by a chain of merons, e b edge magnons of the stable or metastable FP state are i.e.,half-skyrmionswithwindingnumber1/2. Asafunc- thus excited. tionof time, theinterface moves furtherandfurther into 8 h =h . Remarkably, the initial state is however not re- f i covered. An interesting local dynamics governs the fate ofthehelicalfingersastheyarepushedtowardstheedge. When the merons, i.e., the half-skrymions approach the boundary, each of them pulls a second meron out of the edge. Both combine to a skyrmion which gets repelled from the edge by the surface twist of the magnetization. As a result, one obtains a chain of equally spaced skyrmions. Note that h =h is here located within the f i regime where the FP phase is thermodynamically stable. The interaction of the receding meron with the spin configuration at the edge thus results in a final state, Fig. 7(g), that possesses a larger energy than the initial field-polarized state of Fig. 7(a). The spin configuration at the boundary with its surface twist apparently acts as a repulsive potential for the merons hindering them to leave the sample so that the field-polarized thermodynamic groundstate becomes dynamically inaccessible. The systems is thus trapped in a metastable state containing a chain of skyrmions. The advantage of the protocol (34) is that it is ex- tremely robust. The final state is, for example, com- pletelyindependentofthetimet forwhichthemagnetic f field is lowered. The precise field values and the size of thedampingtermisalsonotimportant. Anotherimpor- tantaspectofthisprotocolisthatduringtheprocess,the magneticconfigurationisalwayssmoothandsingularin- termediate spin configurations, i.e., Bloch points are not needed. Thisisincontrasttoprocesseswhereskyrmions are created within the bulk20,24. Thesecondpartofourprotocol,thecreationofachain of skyrmions by pushing the helical state towards the FIG. 7. Creation of a chain of skyrmions using the local edge with the help of a magnetic field has been realized edge instability of the field-polarized state. The panels show recently for FeGe nanostrips in a beautiful experiment snapshots at various times obtained by LLG simulations of by Du et al. in Ref. [25]. In this experiment, the system the field-protocol in Eq. (34) as sketched in the inset with was first prepared in a helical state. After an increase of Gilbert damping α = 0.4, κ = 0, h = 1, h = 0.39, h = 1, i 0 f the magnetic field, helices ultimately reconstruct yield- and t˜ = 40.5. (a)-(d) Triggered by the edge instability, a f ing skyrmions. For nanostrips with a small width, the chainofmeronsformsthatpenetratesintothefield-polarized skyrmionsrecombinedinthecenterofthestripwhilefor stateofthebulk. (e)-(g)Themeronsarepushedbacktowards theedgeandtheirlocaldynamicsclosetotheedgeresultsin largerwidthpreciselythesameprocesswasobservedthat theformationofachainofskyrmions. Steps(e)-(g)havebeen we discuss above. observed experimentally in Ref. 25. The color code denotes The same protocol can also be used to create a single the z-component of the magnetization. skyrmion by reducing the magnetic field however only within a confined region close to the edge as indicated by the dashed box in Fig. 8(a). Experimentally, one can the field-polarized state of the bulk. For this simulation use, e.g., a magnetic tip to change locally the magnetic the intermediate field value h0 of Eq. (34) was chosen to field. PreviousmicromagneticsimulationbyKoshibaeet be located within the regime where the skyrmion crys- al.19 have already demonstrated that pulses of magnetic tal(SkX)isthermodynamicallystable. Nevertheless,the fieldinasmallareacantriggerthecreationofaskyrmion local edge instability prompts the formation of a helical close to the edge. Here, we identify the edge magnon in- state which is metastable in this case. stability as the underlying principle of this phenomenon. In a second step, the magnetic field is again increased Asthefieldislocallyreducedbelowtheedgemagnonin- to a value h at t = t , and, as a consequence, the he- stability,ameronisprotrudingintothesystemasshown f f lical phase is pushed back towards the edge. For this to in Fig. 8(b)-(d). As the magnetic field is increased again happen, it is required that h > hcr so that the field- a single skyrmion forms, see Fig. 8(e) and (f). In panel f kink polarized state is energetically favoured compared to the (g) we show the time evolution of the total skyrmion CSL state and can exert a positive pressure on the inter- winding number. As the field is decreased at time t=0, face. For the simulation in Fig. 7(e)-(g), we have chosen the winding number increases and saturates to a value 9 IV. DISCUSSION The Dzyaloshinskii-Moriya (DM) interaction imposes boundaryconditionsonthemagnetization26,27 whichre- sultsinareconstructionofthemagnetizationprofileclose to surfaces. For the field-polarized state this just trans- lates to a twist of the magnetization along or perpendicular to the surface normal depending on the type of DMinteraction. Wehavedemonstratedthatthissurface twist can act as an attractive potential on the spin wave excitations leading to magnon modes that are bound to the edge of the sample. The energy of these edge modes as a function of momentum transverse to the edge has been calculated and is shown in Figs. 3 and 4. These bound magnons become soft at a finite momentum thus locallydestabilizingthemetastablefield-polarizedphase, and this edge instability triggers the formation of a helical state as shown in Fig. 7. We have shown by micromagnetic simulations that the process of helix forma- �� tion via the edge instability is dynamically irreversible (�) allowing for the creation of skyrmions close to the edge. �� Ourresultsexplaintheexperimentalobservationofedge- �� mediated nucleation of skyrmions by Du et al.25. � �� Whereas we focused here on the properties of the � �� field-polarized state, the surface reconstruction is also �� expected29–31 for the other thermodynamically stable �� phases, i.e., the skyrmion crystal and the helix. It is ��=� ��=�� =� an interesting open question how it influences the spin �� wave spectrum and whether or not bound magnon edge ��/�� modes also exist in these other phases. At least for the skyrmion crystal phase, magnon edge modes are indeed FIG. 8. Creation of a single skyrmion using the edge insta- expectedbutforaverydifferentreason. Magnonsexperi- bility. We use a similar protocol as in Fig. 7 with α = 0.4, enceanemergentorbitalmagneticfieldwhentheyscatter κ = 0, h = 1, h = 0.39, h = 1 and t˜ = 90 but this time i 0 f f offatopologicalskyrmionconfiguration3,41,45. Inamag- themagneticfieldisonlyreducedinthefiniteregionmarked neticskyrmioncrystalthisshouldgiverisetoatopologi- bythedashedrectangleinpanel(a). Wehavechosentheex- calmagnonbandstructure characterizedbyfinite Chern tentofthisregionalongtheedgetobehalfofthewavelength, λ /2 = π/q , at which the edge magnon softens. 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