Interaction of a Neel-type skyrmion and a superconducting vortex
IInteraction of a N´eel–type skyrmion and a superconducting vortex
E. S. Andriyakhina
1, 2 and I. S. Burmistrov
2, 3 Moscow Institute for Physics and Technology, 141700 Moscow, Russia L. D. Landau Institute for Theoretical Physics, acad. Semenova av. 1-a, 142432 Chernogolovka, Russia Laboratory for Condensed Matter Physics, HSE University, 101000 Moscow, Russia (Dated: February 11, 2021)Superconductor–ferromagnet heterostructures hosting vortices and skyrmions are new area ofan interplay between superconductivity and magnetism. We study an interaction of a N´eel–typeskyrmion and a Pearl vortex in thin heterostructures due to stray fields. Surprisingly, we findthat it can be energetically favorable for the Pearl vortex to be situated at some nonzero distancefrom the center of the N´eel–type skyrmion. The presence of a vortex–antivortex pair is found toresult in increase of the skyrmion radius. Our theory predicts that a spontaneous generation of avortex–anti-vortex pair is possible under some conditions in the presence of a N´eel–type skyrmion.
I. INTRODUCTION
Topological objects have been remaining at the focusof theoretical and experimental research for more thanhalf a century. The existence of topologically stable con-figurations in ferromagnets with Dzyaloshinskii–Moriyainteraction has been predicted by Bogdanov and Yablon-skii [1]. Now these topological excitations, termed asskyrmions, are intensively explored in an emergent fieldof skyrmionics [2].Research on an interplay between magnetism and su-perconductivity in heterostructures has long history [3–7]. Recently superconductor–ferromagnet bilayers host-ing skyrmions have attracted great theoretical inter-est. It was understood that skyrmions in proxim-ity with a superconductor can not only induce Yu-Shiba-Rusinov-type bound states [8, 9] but can alsohost Majorana modes [10–16]. It was found [17] thatthe presence of skyrmions affects strongly Josephsoncurrent via superconductor–ferromagnet–superconductorjunction. It has been also shown [18] that skyrmion con-figurations can be stabilized by a superconducting dotor antidot situated at the top of a ferromagnetic film.In ferromagnet–superconductor heterostructures super-conducting vortices and skyrmions can form bound pairseither due to interplay of proximity effect and spin-orbitcoupling [19, 20] or due to their interaction via stray fields[21–23].In this paper we study the interaction between a N´eel–type skyrmion and a superconducting vortex in a chiralferromagnet–superconductor heterostructure, see Fig. 1.We assume that the proximity effect is suppressed by thepresence of a thin insulating layer between ferromagnetand superconductor such that the interaction between askyrmion and a vortex is due to stray fields only. Atfirst, by solving Maxwell–London equation we determinethe Meissner current induced by a N´eel–type skyrmionin the superconductor. Contrary to the previous work[23], we consider the case of ferromagnet and supercon-ducting films of arbitrary widths. Analysis of the generalexpression, cf. Eq. (6), in the case of thin ferromagneticand superconducting films yields that the supercurrent has a maximum at distance of the order of the skyrmionsize from the center of the skyrmion. Secondly, for thinferromagnetic and superconducting films we compute theinteraction energy between a N´eel–type skyrmion and aPearl vortex due to stray fields. Contrary to previousresults, see Refs. [21–23], we find that in the case of aN´eel–type skyrmion with the positive and negative chi-ralities it can be energetically favorable for a vortex tosettle at some distance from the skyrmion’s center. Atthird, we study the effect of the presence of supercon-ducting vortex–anti-vortex pair on the skyrmion size inthin heterostructures. We find that a Pearl vortex leadsto increase of a skyrmion radius. Under some conditions,the spontaneous generation of a vortex–anti-vortex pairin a superconducting film is possible in the presence of askyrmion.
FIG. 1. Sketch of a ferromagnet (green) – superconductor(blue) bilayer. There is a thin insulating layer (black) whichsuppresses the proximity effect. The ferromagnetic layer hostsa N´eel–type skyrmion. The magnetic profile of the skyrmionwith the positive chirality is schematically shown. The su-perconducting layer hosts a vortex at some distance from theskyrmion’s center. The vortex is shown schematically by bluelines. d F and d S denote the width of the ferromagnet andsuperconductor film, respectively (see text). a r X i v : . [ c ond - m a t . s up r- c on ] F e b The outline of the paper is as follows. In Sec. II thesolution of the Maxwell–London equation is presented,and the results for the supercurrent are given. The inter-action energy between a N´eel–type skyrmion and a Pearlvortex is computed and analyzed in Sec. III. In Sec. IVthe effect of a Pearl vortex on the skyrmion radius isestimated. We end the paper with summary and conclu-sions in Sec. V. Some technical details of computationsare presented in Appendix.
II. SUPERCURRENT GENERATED BY AN´EEL–TYPE SKYRMION
We start from calculation of the supercurrent inthe chiral ferromagnet – supercondutor heterostructurewhich is generated by a N´eel–type skyrmion (see Fig.1). The width of the chiral ferromagnet (superconduc-tor) film is d F ( d S ). We assume the presence of a thininsulating layer between the chiral ferromagnet and thesuperconductor that allows us to neglect the proximityeffect. The magnetization profile of a N´eel–type skyrmionin the chiral ferromagnet film in the cylindrical coordi-nate system with the origin at the center of the skyrmionis given as follows [24] M Sk = M s Θ( z )Θ( d F − z ) (cid:104) e r η sin θ ( r )+ e z cos θ ( r ) (cid:105) . (1)Here η = ± M s is the saturation magnetization of the chiral ferromagnetfilm, and e r and e z are unit vectors along the radialdirection and the z -axis (perpendicular to the interface),respectively.The spatial distribution of the vector potential A Sk isgoverned by the Maxwell–London equation: ∇× ( ∇ × A Sk )+Θ( − z )Θ( z + d S ) A Sk λ L = 4 π ∇× M Sk , (2)where λ L stands for the London penetration depth.The Maxwell–London equation should be supplementedby the boundary conditions of continuity of A Sk and ∂ A Sk /∂z at z = − d S , , d F [25].Since the right hand side of Eq. (2) is proportionalto the unit vector e ϕ , the vector potential A Sk has onlythe azimuthal component A Sk ,ϕ that depends on r and z .The solution for A Sk ,ϕ ( r, z ) can be cast in the followingform A Sk ,ϕ ( r, z ) = − ∞ (cid:90) dqJ ( qr ) G ( q ) q × κ V e − qz , z (cid:62) d F , κ F e qz + κ F e − qz , d F > z (cid:62) , κ S e Qz + κ S e − Qz , > z (cid:62) − d S , κ V e qz , − d s > z, (3) where J ( z ) stands for the Bessel function of the firstkind. Also we introduced Q = (cid:112) q + 1 /λ L and the func-tion G ( q ) = − πM s ∞ (cid:90) drrJ ( qr ) θ (cid:48) ( r ) sin θ ( r ) . (4)Using the continuity of the azimuthal component of thevector potential, A Sk ,ϕ , and its derivative, ∂A Sk ,ϕ /∂z , at z = − d S , , d F , we obtain κ V = 12 ( e qd F − − λ − L sinh( Qd S ) X , κ V = 2 Qe qd s X , κ F = − e − qd F / , κ F = − / − λ − L sinh( Qd S ) X , κ S = ( Q + q ) e Qd s X , κ S = ( Q − q ) e − Qd s X , X = q (1 − e − qd F )( Q + q ) e Qd S − ( Q − q ) e − Qd S . (5)The current density in the superconducting film, i.e.at − d S (cid:54) z (cid:54)
0, can be calculated by means of the Lon-don equation, j = − A Sk / (4 πλ L ). It is more convenientto trace the total supercurrent flowing in the supercon-ducting film, J ϕ ( r ) = (cid:82) − d S dzj ϕ ( r, z ). Then, we retrieve J ϕ = ∞ (cid:90) dq J ( qr )4 πλ L G ( q )(1 − e − qd F )(1 − e − Qd S ) Q [ q + Q − ( Q − q ) e − Qd S ] . (6)We mention that this expression is similar to the expres-sion for the current induced by a domain wall [26]. Inthe limit of a thick superconductor, d S (cid:29) λ L , R , Eq. (6)transforms into the result of Ref. [23].Below we shall focus on the case of a thin chiralferromagnet, d F (cid:28) R , and a thin superconductingfilm, d S (cid:28) λ L , R . As we shall demonstrate in thenext section, the asymptotic behavior of the supercur-rent can be found for an arbitrary smooth skyrmionprofile with θ (0) = π and θ ( r → ∞ ) →
0. Com-monly used variational examples with such kind behav-ior are the exponential ansatz θ ( r ) = ¯ θ ( r/R ) where¯ θ ( x ) = π exp( − x ) and the 360-degree domain wall ansatz¯ θ ( x ) = 2 arctan(sinh( R/δ ) / sinh( Rx/δ )). Also we shallconsider the linear ansatz with θ ( r ) = π (1 − r/R ) for r < R and zero overwise. A. The case of a smooth skyrmion profile
The behavior of the supercurrent with the distancefrom the center of the skyrmion is controlled by the func-tion G ( q ), see Eq. (4). It is convenient to introduce thedimensionless function, g , such that G ( q ) = 4 πM s R g ( qR ) ,g ( y ) = − ∞ (cid:90) dxxJ ( yx )¯ θ (cid:48) ( x ) sin ¯ θ ( x ) . (7) DW, R / δ = - y g ( y ) FIG. 2. The function g ( y ) in the cases of the exponential,domain wall and linear anzats. Then in the case of a thin superconducting film, d S (cid:28) λ L , R , and a thin chiral ferromagnet, d F (cid:28) R , Eq. (6)can be drastically simplified, J ϕ = M s d F R ∞ (cid:90) dy yg ( y ) J ( yr/R )1 + 2 yλ/R . (8)Here λ = λ L /d S denotes the Pearl penetration length[27]. The asymptotic behavior of the function g ( y ) isgiven as (see Appendix A), g ( y ) = (cid:40) c y , y (cid:28) , − θ (cid:48) (0)¯ θ (cid:48)(cid:48) (0) / (2 y ) , y (cid:29) , (9)where we introduced the numerical constants c k = − ∞ (cid:90) dx x k ¯ θ (cid:48) ( x ) sin ¯ θ ( x ) , k = − , , , . . . (10)For example, in the case of the exponential ansatz onefinds c ≈ .
51. The dependence of the function g ( y ) on y is shown in Fig. 2.Let us first consider the case of the skyrmion size muchsmaller than the size of the vortex, R (cid:28) λ . Evaluatingthe integral over q in Eq. (8), we obtain asymptotic be-havior of the supercurrent (see Appendix A), J ϕ = M s d F λ c − r/R, r (cid:28) R,c R /r , R (cid:28) r (cid:28) λ, c λ R /r , λ (cid:28) r. (11)For ¯ θ ( x ) = π exp( − x ) one finds c − ≈ .
17. The asymp-totic expression (11) suggests nonmonotonous spatial de-pendence of the supercurrent with a maximum at thedistance of order of the skyrmion size R (see Fig. 3).In the case of large skyrmion and small Pearl penetra-tion length, R (cid:29) λ , the supercurrent can be found to thelowest order in λ/R as (see Appendix A), J ϕ = − M s d F R ¯ θ (cid:48) ( r/R ) sin ¯ θ ( r/R ) . (12) DW, R / δ = r / R J φ λ / ( M s d F ) FIG. 3. The dependence of the supercurrent on the distancefrom the skyrmion center for d S /λ L = 0 . d F /λ L = 0 . R/λ L = 3 in the cases of the exponential, domain walland linear anzats. If the function ¯ θ ( x ) decays at x → ∞ faster than 1 /x ,the expression (12) determines the supercurrent at r (cid:28) r λ only. Then at distances r (cid:29) r λ (cid:29) R the asymptoticbehavior of the supercurrent is given as (cf. Eq. (11)), J ϕ = 12 c M s d F λR r , r λ (cid:28) r. (13)The length scale r λ can be estimated from the condition | ¯ θ ( r λ /R ) | ∼ λR /r λ . In the case of the exponentialansatz one finds r λ ∼ R ln( R/λ ) (cid:29) R . B. The case of the linear ansatz
In the case of the linear ansatz the expression (7) forthe function g ( y ) should be modified in order to havecontinuous solution for A ϕ at r = R , g ( y ) → g L ( y ), g L ( y ) = y (cid:90) dxxJ ( yx ) (cid:20) cos ( πx ) + 4 π (cid:21) = g ( y ) + δg ( y ) . (14)Here the function g ( y ) is given by Eq. (7) and δg ( y ) = − (1 − /π ) J ( y ). Therefore, the function g L ( y ) has thefollowing asymptotic behavior, g L ( y ) = π − π y , y (cid:28) ,π − π √ y + π/ √ πy , y (cid:29) . (15)We note that the abrupt change of θ ( r ) at r = R resultsin oscillating behavior of g ( y ) at y (cid:29)
1. The dependenceof the function g L ( y ) on y is shown in Fig. 2.With the help of Eqs. (8) and (15), we obtain thefollowing results for the asymptotic behavior of the su-percurrent in the case of R (cid:28) λ (see Appendix A), J ϕ = M s d F λ ( π Si( π ) − /π ) r/R, r (cid:28) R, − π − R / ( π r ) , R (cid:28) r (cid:28) λ, − π − R λ / ( π r ) , λ (cid:28) r. (16)Here Si( z ) stands for the sine integral. We note that inthe case of the linear ansatz the supercurrent decays at r (cid:29) R faster than in the case of smooth skyrmion profile.This occurs due to the fact that the contribution to thecurrent from δg ( y ) cancels the leading contributions from g ( y ). This occurs since c = ( π − / (2 π ). As in thecase of smooth skyrmion profile, Eq. (16) suggests non-monotonous behavior of J ϕ with r . There should be themaximum and the minimum in the supercurrent at thedistances of the order of the skyrmion size R . Contraryto the case of smooth skyrmion profile, Eq. (16) describesasymptotic behavior of the smooth part of the supercur-rent only. On the top of the monotononic dependencethere is also weak oscillating contribution to J ϕ with thetypical length scale of the order of R as shown in Fig.3. This oscillating contribution is the consequence of theabrupt boundary of the skyrmion configuration.The dependence J ϕ ( r ) in the case of large skyrmionsize, R (cid:29) λ , is more intricate. The supercurrent is givenas the sum of the contribution discussed above for thecase of the smooth skyrmion profile, cf. Eqs. (12) and(13), and the contribution due to δg ( y ). At short dis-tance, r (cid:28) R , we find (see Appendix A), J ϕ = π M s d F rR (cid:18) − π − π λR (cid:19) . (17)In the case of the long distance, r (cid:29) R the supercurrentis given as J ϕ = − π − π M s d F λR r . (18)We note that in the case of the linear ansatz the super-current is stronger suppressed at r (cid:29) R than in the caseof a smooth skyrmion profile. III. INTERACTION ENERGY BETWEENSKYRMION AND PEARL VORTEX
As above we focus on the case of a thin ( d S (cid:28) λ L )superconducting film with a superconducting vortex sit-uated at the distance a from the center of the N´eel-typeskyrmion (see Fig. 1). In order to compensate the mag-netic flux carried by the vortex we assume that thereexists anti-vortex located far away from the skyrmion–vortex pair. The free energy of this system, includingthe magnetic energy of the skyrmion can be written as F = F Sk + F V + F V + F Sk − V + F Sk − V + F V − V . (19)Here F Sk denotes the magnetic free energy of the isolatedchiral ferromagnet that leads to the appearance of the N´eel-type skyrmion (see its explicit form in the next sec-tion). F V and F V are the free energies of the isolated su-perconducting vortex and anti-vortex, respectively. Theelectromagnetic interaction between the skyrmion andthe vortex is described by the following free energy, F Sk − V = 14 π (cid:90) d r (cid:16) B Sk B V + λ L ( ∇ × B Sk )( ∇ × B V ) × Θ( − z )Θ( z + d s ) − π M Sk B V (cid:17) , (20)where B V = ∇ × A V and B Sk = ∇ × A Sk are the mag-netic fields generated by the vortex and the skyrmion,respectively. We note that the first two terms in theright hand side of the expression for F Sk − V compensateeach other in virtue of Eq. (2). Therefore, the distribu-tion of the supercurrent does not influence the interactionenergy between the skyrmion and the vortex. In what fol-lows, we shall neglect the free energies of the interactionof the anti-vortex with the skyrmion, F Sk − V , and withthe vortex, F V − V .The magnetic field of the superconducting vortex in athin film, d S (cid:28) λ L , can be written in a standard form[28], B V = φ sgn( z ) ∇ (cid:90) d q (2 π ) e − q | z | + i q ( r − a ) q (1 + 2 qλ ) . (21)Here φ = hc/ e is the flux quantum, a is the coordinatevector of the vortex center with respect to the skyrmioncenter. Since F Sk − V should depend on the distance a be-tween the skyrmion and the vortex only, we can averagethe magnetic field B V over directions of the vector a .This procedure implies that B V → − φ ∞ (cid:90) dq π q e − q | z | qλ J ( qa ) (cid:104) sgn( z ) J ( qr ) e r + J ( qr ) e z (cid:105) . (22)The free energy of the Pearl vortex (as well as anti-vortex) in a thin superconducting film is given by [27] F V = F V = φ π λ ln λξ , (23)where the superconducting coherence length is assumedto be much shorter than the Pearl length, ξ (cid:28) λ .Using Eqs. (1) and (22), we express the interactionpart of the free energy as F Sk − V = M s φ d F + M s φ ∞ (cid:90) dq − e − qd F q (1 + 2 qλ ) J ( qa ) × ∞ (cid:90) dr r (cid:104) ηq + θ (cid:48) ( r ) (cid:105) J ( qr ) sin θ ( r ) . (24)We note that the first term in the right hand side of Eq.(24) corresponds to the homogeneous magnetization ofthe ferromagnetic film.Below we analyse general expression (24) in the caseof a thin ferromagnetic film, d F (cid:28) R . Then we find F Sk − V M s φ d F = 1 + ∞ (cid:90) dy J ( ya/R )(1 + 2 yλ/R ) ∞ (cid:90) dx x (cid:104) ηy + ¯ θ (cid:48) ( x ) (cid:105) × J ( yx ) sin ¯ θ ( x ) . (25)As in the case of the supercurrent, we start from thecase of a skyrmion of size R (cid:28) λ . Neglecting unity withrespect to 2 yλ/R in the denominator of the integrand inthe right hand side of Eq. (25), we obtain the followingasymptotic expression for the interaction free energy atshort distances, a (cid:28) λ , (see Appendix B) F Sk − V M s φ d F = 1 + R λ f η (cid:16) aR (cid:17) , (26)where the function f η ( z ) has the following asymptoticbehavior f η ( z ) = (cid:40) ηb − c + (cid:16) c − + η ¯ θ (cid:48) (0) (cid:17) z / , z (cid:28) , − c /z, z (cid:29) . (27)Here we introduced the numerical constants, b k = ∞ (cid:90) dx x k sin ¯ θ ( x ) , k = − , , , . . . (28)At very long distances, a (cid:29) λ , the free energy of interac-tion between the skyrmion and the vortex becomes (seeAppendix B), F Sk − V M s φ d F = 1 − c R λa . (29)We emphasize that at long distances a (cid:29) R F Sk − V becomes insensitive to chirality of the N´eel skyrmion.The coefficient c − is typically positive whereas ¯ θ (cid:48) (0) isnegative, therefore the interaction free energy may de-crease with increase of a for η = +1. Since the ratio F Sk − V / ( M s φ d F ) tends to unity at a → ∞ irrespectiveof chirality, one can expect the existence of the minimumof F Sk − V at some non-zero value of the distance a . Thissituation is realized for the exponential ansatz. In thecase of 360-degree domain wall ansatz with η = +1 thenontrivial minimum exists for δ/R (cid:38) .
64 only.Next we consider the opposite case of the skyrmionwith the size much larger than the size of the Pearl vor-tex, R (cid:29) λ . The interaction free energy can be writtenas a series in powers of λ/R (see Appendix B), F Sk − V M s φ d F = 1 + h η, (cid:16) aR (cid:17) + λR h η, (cid:16) aR (cid:17) + . . . (30) The function h η, that determines the magnitude of theinteraction free energy has the following asymptotic be-havior (see Appendix B), h η, ( z ) = ηb − − (cid:20) η ¯ θ (cid:48)(cid:48) (0) ln z + ¯ θ (cid:48) (0) + ηβ (cid:21) z , (31)at z (cid:28)
1, and h η, ( z ) = − ηb z , z (cid:29) . (32)Here the parameter β is given by the following lengthyexpression, β = 32 ¯ θ (cid:48) (0) + ¯ θ (cid:48)(cid:48) (0) (cid:104) − G )2 π − π (cid:90) dxx (cid:16) K ( x ) − π − πx (cid:17)(cid:105) + 32 ∞ (cid:90) dx sin ¯ θ ( x ) x + 32 (cid:90) dx (cid:104) sin ¯ θ ( x ) x + ¯ θ (cid:48) (0) x + ¯ θ (cid:48)(cid:48) (0)2 x (cid:105) , (33)where G ≈ .
916 denotes the Catalan’s constant and K ( x ) stands for the complete elliptic integral of thefirst kind. The function h η, ( z ) that determines the de-pendence on distance of the subleading contribution to F Sk − V has the following asymptotic behavior (see Ap-pendix B), h η, ( z ) = 4 (cid:0) c − + η ¯ θ (cid:48) (0) (cid:1) + 3 η ¯ θ (cid:48)(cid:48) (0) z − (cid:104)
94 ¯ θ (cid:48) (0)¯ θ (cid:48)(cid:48) (0) ln z + 43 η (cid:0) ¯ θ (cid:48) (0) − ¯ θ (cid:48)(cid:48)(cid:48) (0) (cid:1) − β (cid:105) z , z (cid:28) , (34)and h η, ( z ) = − c z , z (cid:29) . (35)Here the parameter β is given as β = 92 π (1 + 2 G )¯ θ (cid:48) (0)¯ θ (cid:48)(cid:48) (0) − ∞ (cid:90) dxx ∂ x (cid:0) x ¯ θ (cid:48) ( x ) sin ¯ θ ( x ) (cid:1) − (cid:90) dxx ∂ x (cid:0) x ¯ θ (cid:48) ( x ) sin ¯ θ ( x ) + ¯ θ (cid:48) (0) x + 32 ¯ θ (cid:48) (0)¯ θ (cid:48)(cid:48) (0) x (cid:1) + 18 π ¯ θ (cid:48) (0)¯ θ (cid:48)(cid:48) (0) (cid:90) dxx (cid:104) K ( x ) − π − πx (cid:105) +¯ θ (cid:48) (0) −
92 ¯ θ (cid:48) (0)¯ θ (cid:48)(cid:48) (0) . (36)We mention two discrepancies with the case of askyrmion of a small radius. At first, the short distancebehavior of the interaction free energy in the case of η = + η = - δ / R ( θ ′ ) ( ) + η β FIG. 4. The dependence of ¯ θ (cid:48) (0) + ηβ on δ/R (see text). R (cid:29) λ is not parabolic generically, see Eq. (31). Sec-ondly, the asymptotic behavior of F Sk − V at a (cid:29) R de-pends on the skyrmion’s chirality.Provided ¯ θ (cid:48)(cid:48) (0) >
0, the analytic results (31)–(32),suggest the existence of the global minimum of F Sk − V at a certain non-zero distance a in the case of positiveskyrmion’s chirality η = +1. For negative chirality, η = −
1, the minimum of the interaction free energy issituated at a = 0. Interestingly, the 360-degree domainwall ansatz is special since ¯ θ (cid:48)(cid:48) (0) = 0. Thus, for the 360-degree domain wall ansatz the existence of the minimumin F Sk − V is controlled by the sign of the term in the sec-ond line of Eq. (33). The dependence of this coefficienton the ratio δ/R for both chiralities is shown in Fig. 4.For δ (cid:38) . R ( δ (cid:46) . R ) the interaction free energy, F Sk − V , has the minimum at nonzero value of a for thecase of positive (negative) chirality, η = +1 ( η = − a/R for both chiralities, η = ± η = +1, theminimum of F Sk − V is reached at nonzero value of thedistance a . IV. THE EFFECT OF THE PEARL VORTEXON THE SKYRMION
The magnetic free energy of the chiral ferromagneticfilm is given by [1] F magn [ m ] = d F (cid:90) d r (cid:26) A ( ∇ m ) + K (1 − m z )++ D (cid:2) m z ∇ · m − ( m · ∇ ) m z (cid:3)(cid:27) . (37)Here m ( r ) denotes the unit vector of magnetization di-rection, A > D isthe Dzyaloshinskii–Moriya interaction, and K > F magn DW, R / δ = a / R F S k - V / M s ϕ d F FIG. 5. The dependence of the normalized interaction freeenergy, F Sk − V , on a/R for the chirality η = +1. The ratio ofthe skyrmion radius and the Pearl length is unity, λ/R = 1(see text). is zero for the ferromagnetic state, m z = 1. Substituting m = m Sk = M Sk /M s , see Eq. (1), into Eq. (37), wefind F Sk ≡ F magn [ m Sk ] = 2 πd F ∞ (cid:90) dr r (cid:40) A (cid:104) θ (cid:48) ( r ) + sin θ ( r ) r (cid:105) + Dη (cid:104) θ (cid:48) ( r ) + sin(2 θ ( r ))2 r (cid:105) + K sin θ ( r ) (cid:41) . (38)Assuming a scaling form of the skyrmion profile, θ ( r ) =¯ θ ( r/R ), we obtain F Sk = d F (cid:16) α A A − α D ηDR + α K KR / (cid:17) , (39)where α A = 2 π ∞ (cid:90) dx x (cid:104) ¯ θ (cid:48) ( x ) + sin ¯ θ ( x ) x (cid:105) ,α D = − π ∞ (cid:90) dx x (cid:104) ¯ θ (cid:48) ( x ) + sin(2¯ θ ( x ))2 x (cid:105) ,α K = 4 π ∞ (cid:90) dx x sin ¯ θ ( x ) . (40)We note that α A,D,K are positive constants in the case ofthe linear and exponential ansatz and are positive func-tions of the parameter
R/δ in the case of the 360-degreedomain wall ansatz. Minimizing F Sk with respect to R ,one can find the optimal radius of the skyrmion R = α D | D | / ( α K K ) (41)and the chirality η = sgn D . We note that the existenceof a skyrmion in a chiral ferromagnetic film is possibleunder the following condition, α A A < α K KR / . (42) DW, R / δ = - a / R F / M s ϕ d F FIG. 6. The dependence of the normalized interaction freeenergy, F Sk − V , on a/R for the chirality η = −
1. The ratio ofthe skyrmion radius and the Pearl length is unity, λ/R = 1(see text).
In order to simplify the presentation, we shall startour considerations below from the cases of the linear andexponential ansatz. In the presence of vortex anti-vortexpair the skyrmion radius is obtained by minimization of F Sk + F Sk − V with respect to R and a . Let us start fromthe case of a skyrmion of small radius, R ∗ (cid:28) λ . Forthe negative chirality the optimal distance between theskyrmion and the vortex is zero. Therefore, as it followsfrom Eqs. (26) and (39), for η = − R ∗ = R + (2 c + b / (cid:96) K /λ. (43)Here (cid:96) K = (cid:112) M s φ / ( α K K ) is the length scale associatedwith the anisotropy energy. In the case of the positivechirality the optimal distance between the vortex and theskyrmion is proportional to the skyrmion radius, a = ζ R , see Eq. (26). Interestingly, we find that in the caseof η = +1 the skyrmion radius is also enlarged due tointeraction with the vortex, R ∗ = R − f +1 ( ζ ) (cid:96) K / (2 λ ) . (44)We note that f +1 ( ζ ) <
0. The results (43) and (44) areapplicable for λ (cid:29) max { R , (cid:96) K } . In Table I we presentestimates of R ∗ for several ferromagnet structures. Asone can see from the Table I, the magnitude of the ratio R ∗ /R depends on a form of the ansatz for skyrmionprofile. For some materials the skyrmion radius can beenhanced by 4 times.In order a vortex–anti-vortex pair can be sponta-neously generated in the presence of a skyrmion the totalfree energy (19) should be negative. This implies the fol-lowing inequality, α A A − α K KR ∗ α K K(cid:96) K + φ π λd F ln λξ < . (45) Since R ∗ is larger than R this inequality can be fulfilledprovided the condition (42) holds. We note that then theradius of the skyrmion should satisfy λ (cid:29) R (cid:29) (cid:96) K . Inparticular, the vortex–anti-vortex pair cannot be gener-ated spontaneously in the absence of the Dzyaloshinskii–Moriya interaction, i.e. at D = 0. Indeed, in the lattercase R ∗ (cid:28) (cid:96) K and the left hand side of the inequality(45) is positive. In fact, there is a minimal value of theDzyaloshinskii–Moriya interaction at which the sponta-neous generation of a vortex–anti-vortex pair is possible, | D | > (cid:20) α K Kα D (cid:18) α A A + α K K(cid:96) K + φ ln( λ/ξ )8 π λd F (cid:19)(cid:21) / + f η ( ζ ) α K K(cid:96) K α D λ . (46)Now let us assume that the skyrmion radius is large, R (cid:29) λ . Then, Eqs. (30) and (39) result in the followingequation for the skyrmion radius modified by the inter-action with the vortex, R ∗ R − R ∗ R = h η, ( ζ ) λ(cid:96) K R . (47)For negative chirality, η = −
1, the optimal distancebetween the skyrmion and the vortex is zero, ζ = 0.We note that h − , (0) = 4[2 c − − ¯ θ (cid:48) (0)] >
0, see Eq.(34). For positive chirality, η = +1, the interactionbetween skyrmion and vortex has the minimum at fi-nite distance, ζ (cid:54) = 0. However, as one can check (seeEq. (B5)), h +1 , ( ζ ) >
0. Therefore, for both chiralitiesthe skyrmion–vortex interaction leads to increase of theskyrmion radius, R ∗ = R (cid:0) X − / + X / (cid:1) / , (48)where X = 1 + 27 u (cid:112) u + 81 u , u = h η, ( ζ ) λ(cid:96) K R . (49)We note that for R (cid:28) ( λ(cid:96) K ) / and (cid:96) K (cid:29) λ the skyrmion radius is parametrically enhanced, R ∗ ∼ ( λ(cid:96) K ) / (cid:29) R . For R (cid:29) ( λ(cid:96) K ) / , the radius of theskyrmion is only slightly increased, R ∗ ∼ R . In this caseEq. (48) holds under assumption R (cid:29) λ .A spontaneous generation of the vortex–anti-vortexpair requires the negative total free energy (19), α A A − α K KR ∗ α K K(cid:96) K (cid:20) h η, ( ζ ) + 2 h η, ( ζ ) λR ∗ (cid:21) + φ π λd F ln λξ < . (50)Since R ∗ > R the above inequality can be satisfiedprovided the condition (42) holds. However, it canoccur only for sufficiently large bare skyrmion radius, R (cid:29) λ (cid:29) (cid:96) K . In the case (cid:96) K (cid:29) R (cid:29) λ the skyrmionradius becomes R ∗ ∼ ( λ(cid:96) K ) / (cid:28) (cid:96) K . Therefore, the TABLE I. The parameters M s , A , K u , and D for a number of thin chiral ferromagnet films. The estimates for the a bare ( R )and effective ( R ∗ ) skyrmion radii and an anisotropic scale (cid:96) K for the linear (lin) and exponential (exp) ansatz are given. Inorder to obtain the estimate for R ∗ we choose λ = 100 nm.PtCoPt [29, 30] IrCoPt [31] PtCoNiCo [32] PdFeIr [33, 34]Saturation magnetization M s (10 A/m) 580 956 600 1100Exchange constant A (10 − J/m) 15 10 20 2.0Anisotropy constant K u (10 J/m ) 0.7 0.717 0.6 2.5DMI parameter D (10 − J/m ) +3 +1.6 +3 +3.9Bare radius (lin/exp) R (10 − m ) 13/4 7.0/2.1 20/5 5.0/1.5Skyrmion radius (lin/exp) R ∗ (10 − m ) 26/5.5 28/4.4 30/6 12/2.3Anisotropy scale (lin/exp) (cid:96) K (10 − m) 80/40 100/47 90/40 60/26 negative term − α K KR ∗ / α K K(cid:96) K and, consequently, spontaneous gen-eration of vortex–anti-vortex pair is not possible.In the case of the 360-degree domain wall ansatz Eqs.(43), (44), and (47) remain valid. However, the value of ζ depends on the ratio R ∗ /δ . The latter is determinedfrom the minimum of the total free energy with respectto δ . The corresponding analysis can be performed nu-merically. As one can check, the following inequalitieshold f η ( ζ ) < h η, ( ζ ) >
0. These inequalitiesimply that the skyrmion radius increases always in thepresence of a vortex–anti-vortex pair.
V. SUMMARY AND CONCLUSIONS
To summarize, we have studied an interaction of aN´eel–type skyrmion and a vortex–anti-vortex pair due tostray fields in a chiral ferromagnet–superconductor het-erostructure. We computed the supercurrent in a super-conducting film induced by a skyrmion. For thin fer-romagnet and superconductor films we found that thesupercurrent has the maximum at the distance from thecenter of a skyrmion that is of the order of the skyrmionradius. It is worthwhile to mention that the supercurrentis sensitive to a profile of the skyrmion. For example,in the case of smooth profiles (exponential and domainwall ansatzes), the supercurrent decays monotonously atlarge distances from the skyrmion center. For the case ofa linear profile, there are decaying oscillations of the su-percurrent at large distances due to discontinuity in θ (cid:48) ( r )at r = R . Therefore, measurements of dependence of thesupercurrent on distance can allow one to extract infor-mation on the profile of a skyrmion. We mention thatthe behavior of the supercurrent with a distance fromthe center of the skyrmion is qualitatively similar to thebehavior of the supercurrent induced in a thin supercon-ducting film by a Bloch domain wall in a ferromagneticfilm [26]. The radius of the skyrmion plays the same roleas the width of a domain wall.We have also computed the energy of interaction be-tween a N´eel–type skyrmion and a Pearl vortex. Wefound that the interaction with a Pearl vortex is sensi- tive to the skyrmion chirality. In the case of a skyrmionwith negative chirality, typically, it is more energeticallyfavourable for a vortex to be attracted to the skyrmioncenter. This occurs in the cases of linear and exponen-tial skyrmion profiles and for a domain wall ansatz with δ (cid:38) . R . In the case of positive skyrmion chiralitya vortex is situated at a finite distance (of the order ofskyrmion radius) from the center of the skyrmion. Thishappens for linear and exponential profiles and in thecase of domain wall ansatz with δ (cid:38) . R .It is worthwhile to mention that in the case of a Bloch–type skyrmion it is always energetically favorable fora vortex to settle at the center of the skyrmion [21].Such a behavior is related with the absence of the radialcomponent of magnetization in a Bloch–type skyrmion.Therefore, the Bloch–type skyrmion interacts with the z -component of the magnetic field of a Pearl vortex only.This leads to the absence of terms proportional to thechirality η in Eqs. (27) and (31). As a result, the func-tion f η ( z ) and h η, ( z ) behave as increasing parabolas at z (cid:28)
1. Such a behavior implies the minimum of the in-teraction free energy at zero distance between the centerof the Bloch-type skyrmion and the Pearl vortex.The fact that it is energetically favourable for a Pearlvortex to take place at a finite distance from the centerof a N´eel-type skyrmion can have interesting implicationsfor skyrmion lattices [35] and dynamics of skyrmions [22]in superconductor – ferromagnet heterostructures [36].We have investigated how a Pearl vortex affects aN´eel-type skyrmion due to their mutual interaction. Wefound that a vortex–anti-vortex pair leads to an in-crease of the radius of the N´eel–type skyrmion. Wenote that this result can be contrasted with the case ofa Bloch–type skyrmion for which a vortex–anti-vortexpair can either increase or decrease the skyrmion ra-dius [21]. It is also possible that a vortex–anti-vortexpair will be spontaneously generated in the presence ofa N´eel-type skyrmion provided the skyrmion radius andPearl penetration length are large enough in comparisonwith the length associated with the anisotropy energyin a chiral ferromagnet, λ, R (cid:29) (cid:96) K . In the oppositecase of small bare skyrmion radius, R (cid:28) (cid:96) K , sponta-neous generation of a vortex–anti-vortex pair is not pos-sible. Unfortunately, the relation, R (cid:28) (cid:96) K , typicallyholds in a chiral ferromagnets (see Table I). However,for R (cid:28) ( λ(cid:96) K ) / (cid:28) (cid:96) K , we predict that a vortex–anti-vortex pair existing in a superconducting film cansubstantially increase the skyrmion radius: it becomesequal to R ∗ ∼ ( λ(cid:96) K ) / (cid:29) R . The typical values of R , (cid:96) K , and R ∗ are listed in Table I. Abrupt increase of theskyrmion radius can be used as indication of appearanceof vortex–anti-vortex pairs in superconducting films. It isan experimental challenge to detect enhancement of theskyrmion radius in a thin ferromagnet–superconductorheterostructure due to generation of vortex–anti-vortexpair in a superconducting film.Finally, we mention that it would be interesting to gen-eralize our results to the case of more exotic magneticexcitations, e.g. antiskyrmions, bimerons, biskyrmions,skyrmioniums, etc. [37] ACKNOWLEDGMENTS
The authors are grateful to M. Garst for useful com-ments. The work was funded in part by Russian ScienceFoundation under the grant No. 21-42-04410.
Appendix A: Derivation of the asymptoticexpressions for the supercurrent
In this Appendix we present some details of derivationof expressions for the supercurrent.
1. The case of a smooth skyrmion profile
We start from the case of the smooth skyrmion profile.According to Eq. (8) the supercurrent is determined bythe function g ( y ), see Eq. (7).To find the asymptotic behavior of the function g ( y )in the case of a small argument, y (cid:28)
1, we approximatethe Bessel function J ( xy ) by xy/ g ( y ) = − y/ ∞ (cid:90) dxx ¯ θ (cid:48) ( x ) sin ¯ θ ( x ) (cid:39) c y, y (cid:28) . (A1)Asymptotic expression at large arguments, y (cid:29)
1, canbe found in the following way. Changing the variable x to xy under the integral sign in the definition of thefunction g ( y ), see Eq. (7), one can expand the function θ in powers of 1 /y . Then, we obtain g ( y ) = lim β → +0 ∞ (cid:90) dxJ ( x ) e − βx (cid:20) ¯ θ (cid:48) (0) + 3 x y ¯ θ (cid:48)(cid:48) (0) (cid:21) ¯ θ (cid:48) (0) x y (cid:39) − θ (cid:48) (0)¯ θ (cid:48)(cid:48) (0) / (2 y ) , y (cid:29) . (A2)Equations (A1) and (A2) are equivalent to Eq. (9). We will now present derivation of asymptotic expres-sions for the supercurrent in the case of a small skyrmion, R (cid:28) λ . At the shortest distances to the the center of askyrmion, one can neglect unity in the denominator ofthe expression (8) and expand the Bessel function in se-ries of r/R (cid:28)
1. Then we retrieve, J ϕ = M s d F r λR ∞ (cid:90) dy yg ( y ) (A3)The integral ∞ (cid:82) dy yg ( y ) can be simplified with the helpof the following identity yJ ( xy ) = − ∂ x ( J ( xy )). Then,we obtain ∞ (cid:90) dy yg ( y ) = − ∞ (cid:90) dxx ¯ θ (cid:48) ( x ) sin ¯ θ ( x ) ∞ (cid:90) dy yJ ( xy )= 4 c − . (A4)This results in J ϕ = ( M s d F /λ ) r/R for r (cid:28) R , cf. Eq.(11).For the case of long distances, r (cid:29) R , we rewrite theexpression for the supercurrent in a more convenient way,raising the denominator into exponent by means of anadditional integration, J ϕ = M s d F R ∞ (cid:90) dy ∞ (cid:90) dt e − t (1+2 yλ/R ) yg ( y ) J ( yr/R ) . (A5)Let us first consider the integration with respect to the y variable. Since for r/R (cid:29) y isdominated by small values of y , we obtain ∞ (cid:90) dy ye − ytλ/R J ( xy ) J ( yr/R ) (cid:39) x ∞ (cid:90) dy y e − y (2 tλ/R ) × J ( yr/R ) = x λt/r )[1 + (2 λt/r ) ] / R r . (A6)Hence for the supercurrent we write J ϕ = 2 c M s d F R r ∞ (cid:90) dt λt/r ) e − t [1 + (2 λt/r ) ] / . (A7)In this integral form for the supercurrent, one can clearlyfigure out the behavior of J ϕ ( r ) for r (cid:28) λ and r (cid:29) λ .For r (cid:28) λ we can substitute e − t by unity and, then, ob-tain J ϕ = ( M s d F /λ )( c R /r ), cf. Eq. (11). Otherwise,when r is much larger than λ , we neglect the term pro-portional to the small parameter λ/r in the denominatorof (A7). Then, one gets J ϕ = ( M s d F /λ )(12 c λ R /r ),cf. Eq. (11).Now we consider the case of large skyrmion R (cid:29) λ .We start from the limit of short distances r (cid:28) r λ . Inthis regime we neglect the unity in the denominator0in the right hand side of Eq. (8). Using the identity (cid:82) ∞ dy yJ α ( xy ) J α ( zy ) = δ ( x − z ) /x , we find J ϕ = − M s d F R ∞ (cid:90) dx ¯ θ (cid:48) ( x ) sin ¯ θ ( x ) δ ( x − r/R )= − M s d F R ¯ θ (cid:48) ( r/R ) sin ¯ θ ( r/R ) , r (cid:28) r λ . (A8)Equation (A8) is equivalent to Eq. (12). The asymptoticexpression (13) for r (cid:29) r λ can be easily derived from Eq.(A6).
2. The case of the linear ansatz
Let us start from the case of R (cid:28) λ and derive theasymptotic expression (16) for the supercurrent. Atshortest distances, r (cid:28) R , one writes similar to the caseof the smooth profile, J ϕ = M s d F λ ∞ (cid:90) dy g L ( y ) J ( yr/R ) . (A9)Since g L ( y ) = g ( y ) + δg ( y ), for the first contribution to J ϕ we can use the expression (A4). While for the secondterm, δg ( y ) = − c J ( y ), we apply the identity ∞ (cid:90) dyJ ( y ) J ( yr/R ) = 2 Rπr (cid:2) K ( R /r ) − E ( R /r ) (cid:3) (cid:39) r/ (2 R ) + O ( r /R ) , r (cid:28) R. (A10)Here K ( z ) and E ( z ) denotes the complete elliptic inte-grals of the first and second kinds. Together, these twocontributions give the final result, cf. Eq. (16), J ϕ = (cid:18) π Si( π ) − π (cid:19) M s d F r λR . (A11)To find the behaviour of the superconducting currentat large distances we use the method as described aroundEq. (A6) above. The only difference is that instead of theexpression for the smooth profile function g ( y ) we needto use the expression (14) for g L ( y ). Then, we retrieve, ∞ (cid:90) dy y e − ytλ/R J ( xy ) J ( yr/R ) (cid:39) − x ∞ (cid:90) dy y e − ytλ/R × J ( yr/R ) = x R r λt/r ) (cid:0) λt/r ) − (cid:1) [1 + (2 λt/r ) ] / . (A12)This leads to the following approximate expression forthe supercurrent, J ϕ = M s d F R r − π π ∞ (cid:90) dt e − t λt (cid:0) λt/r ) − (cid:1) r [1 + (2 λt/r ) ] / . (A13) In the case of R (cid:28) r (cid:28) λ , the exponent e − t in the righthand side of Eq. (A13) can be approximated by the unity.Then, we obtain, cf. Eq. (16), J ϕ = − π − π M s d F R λr . (A14)In the limit of longest distances, r (cid:29) λ , we neglect theterms (2 λt/r ) in the enumerator and denominator underthe integral in Eq. (A13). Then, we find, cf. Eq. (16), J ϕ = − π − π M s d F λR r . (A15)Finally, we derive asymptotic expressions for the super-current for the case of a large skyrmion, R (cid:29) λ . As it wasexplained in the main text, we are to compare the con-tributions from the term g ( y ), given in (7), and from the δg ( y ) = − c J ( y ). Let us start from the limit r (cid:28) R .We can use Eq.(A8) for the asymptotic expression, cor-responding to the contribution due to g ( y ). In the caseof the linear ansatz it reads ( M s d F /R ) π ( r/R ). In orderto find the contribution due to the second term, δg ( y ),we replace the Bessel function J ( yr/R ) by yr/ (2 R ) andexpand denominator in powers of yλ/R . Then, we find − c rR lim β → +0 ∞ (cid:90) dy e − βy yJ ( y ) (cid:20) − yλR + O (cid:18) λ R (cid:19)(cid:21) (cid:39) − c λrR . (A16)Bringing these two contributions together, we retrieve,cf. Eq. (17), J ϕ = π M s d F rR (cid:18) − π − π λR (cid:19) , r (cid:28) R. (A17)For r (cid:29) R (cid:29) λ one can repeat derivation followingEqs. (A12) and (A13). Then one arrives eventually atthe expression (A15). Appendix B: Derivation of the asymptoticexpressions for the interaction energy
In this appendix we present some details of derivationof the asymptotic expressions for F Sk − V .We start from the case of a small skyrmion and a largevortex, R (cid:28) λ . In the regime of short distances, a (cid:28) R , we can neglect the unity in comparison to the largeparameter λ/R in the denominator under the integralsign in the right hand side of Eq. (25). Expanding theBessel function J ( ya/R ) in series of ya/R , we obtain F Sk − V M s φ d F (cid:39) ∞ (cid:90) dy − ( a/R ) y / yλ/R ∞ (cid:90) dx x (cid:104) ηy + ¯ θ (cid:48) ( x ) (cid:105) × J ( yx ) sin ¯ θ ( x ) . (B1)1This expression can be easily simplified to the form ofEq. (26).For the intermediate distances, R (cid:28) a (cid:28) λ , one cansimplify Eq. (25) by using the following identities, ∞ (cid:90) dy J ( yz ) J ( xy ) = Θ( x − z ) /x, (B2) ∞ (cid:90) dyy J ( y ) J ( xy ) = 2 πx (cid:104) E (cid:0) x (cid:1) − (1 − x ) K (cid:0) x (cid:1)(cid:105) × Θ(1 − x ) + 2 π E (cid:0) x − (cid:1) Θ( x − . (B3)Here Θ( x ) denotes the Heaviside step function. Aftersome simplifications, the expression for the interactionenergy can be brought to the form of Eq. (26).The case of the longest distances, a (cid:29) λ can be stud-ied in the following way. One can transform the ex-pression in the denominator under the integral sign inthe right hand side of Eq. (25) into the exponent withthe help of an additional integration, 1 / (1 + 2 yλ/R ) = (cid:82) ∞ dt e − t (1+2 yλ/R ) . Then expanding the Bessel function J ( xy ) in its argument to the lowest order, we derive Eq.(29).Now let us consider the opposite case of large skyrmionradius, R (cid:29) λ . Making in Eq. (25) expansion in powersof λ/R , we obtain Eq. (30) with the functions h η, ( z )and h η, ( z ) that are given as h η, = ∞ (cid:90) dxdy xJ ( yz ) J ( yx ) (cid:104) ηy + ¯ θ (cid:48) ( x ) (cid:105) sin ¯ θ ( x ) (B4)and h η, = − ∞ (cid:90) dxdy xyJ ( yz ) J ( yx ) (cid:104) ηy + ¯ θ (cid:48) ( x ) (cid:105) sin ¯ θ ( x ) . (B5)We shall start with the asymptotic behavior of thefunctions h η, ( z ) and h η, ( z ) at z (cid:28)
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