Gaididei, Y., Goussev, A., Kravchuk, V. P., …...Magnetization in narrow ribbons: curvature e ects Yuri Gaididei 1, Arseni Goussev2, Volodymyr P. Kravchuk , Oleksandr V. Pylypovskyi3,

Gaididei Y Goussev A Kravchuk V P Pylypovskyi O VRobbins J M Sheka D D Slastikov V amp Vasylkevych S (2017)Magnetization in narrow ribbons Curvature effects Journal of PhysicsA Mathematical and Theoretical 50(38) [385401]httpsdoiorg1010881751-8121aa8179

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Magnetization in narrow ribbons curvature effects

Yuri Gaididei1 Arseni Goussev2 Volodymyr P Kravchuk1Oleksandr V Pylypovskyi3 J M Robbins4Denis D Sheka3 Valeriy Slastikov4 andSergiy Vasylkevych4

1 Bogolyubov Institute for Theoretical Physics of National Academy of Sciencesof Ukraine 03680 Kyiv Ukraine2 Department of Mathematics Physics and Electrical Engineering NorthumbriaUniversity Newcastle upon Tyne NE1 8ST UK3 Taras Shevchenko National University of Kyiv 01601 Kyiv Ukraine4 School of Mathematics University of Bristol Bristol BS8 1TW UK

E-mail ybgbitpkievua arsenigoussevnorthumbriaacukvkravchukbitpkievua engraverknuua JRobbinsbristolacukshekaunivnetua svasylkevychbristolacuk

AbstractA ribbon is a surface swept out by a line segment turning as it moves along

a central curve For narrow magnetic ribbons for which the length of the linesegment is much less than the length of the curve the anisotropy induced by themagnetostatic interaction is biaxial with hard axis normal to the ribbon and easyaxis along the central curve The micromagnetic energy of a narrow ribbon reducesto that of a one-dimensional ferromagnetic wire but with curvature torsion andlocal anisotropy modified by the rate of turning These general results are appliedto two examples namely a helicoid ribbon for which the central curve is a straightline and a Mobius ribbon for which the central curve is a circle about which theline segment executes a 180 twist In both examples for large positive tangentialanisotropy the ground state magnetization lies tangent to the central curve Asthe tangential anisotropy is decreased the ground state magnetization undergoesa transition acquiring an in-surface component perpendicular to the central curveFor the helicoid ribbon the transition occurs at vanishing anisotropy below whichthe ground state is uniformly perpendicular to the central curve The transitionfor the Mobius ribbon is more subtle it occurs at a positive critical value of theanisotropy below which the ground state is nonuniform For the helicoid ribbonthe dispersion law for spin wave excitations about the tangential state is found toexhibit an asymmetry determined by the geometric and magnetic chiralities

PACS numbers 7570-i 7575-c 7510Hk 7530Et

Submitted to J Phys A Math Gen

Magnetization in narrow ribbons curvature effects 2

Introduction

The emerging area of magnetism in curved geometries encompasses a range offascinating geometry-induced effects in the magnetic properties of materials [1]Theoretical investigations in this area are providing new insights into the behaviourof curved magnetic nanostructures and the control of their magnetic excitations withapplications to shapeable magnetoelectronics [2] and prospective energy-efficient datastorage among others

In continuum models the magnetization is represented by a three-dimensionalunit-vector field m(r) The study of curvaturendashinduced effects in vector-field modelsin one- and two-dimensional geometries has a rather long history [3ndash6] In spite ofnumerous results [3ndash6] the problem is far from being fully solved In the majorityof these studies the vector field is taken to be tangent to the domain In particulara general expression for the surface energy of a tangential director field describinga nematic liquid crystal in a curvilinear shell was recently obtained [7ndash10] withpossible applications using different geometries and orientational ordering [11ndash13]The assumption of a strictly tangential field was also used in a study of the role ofcurvature in the interaction between defects in 2D XY -like models with applicationsto superfluids superconductors and liquid crystals deposited on curved surfaces [14]

Very recently a fully 3D approach was developed for thin magnetic shells andwires of arbitrary shape [15 16] This approach yields an energy for arbitrary curvesand surfaces and for arbitrary magnetization fields under the assumption that theanisotropy greatly exceeds the dipolar interaction so that

intdr (Eex + Ean) (1)

Here Eex is the exchange energy density and Ean is the density of effective anisotropyinteraction We consider the model of isotropic exchange Eex = (nablami) middot(nablami) wheremi with i = 1 2 3 describes the cartesian components of magnetization Thereforein cartesian coordinates the sample geometry appears only through the anisotropyterm via the spatial variation of the anisotropy axis for example in the case of auniaxial curved magnet Ean is given by K (m middot eA)

2 where the unit vector eA = eA(r)

determines the direction of the easy axisIn curvilinear coordinates adapted to the sample geometry the spatial variation of

the anisotropy axes is automatically accounted for and the anisotropy energy densityassumes its usual translation-invariant form Instead the exchange energy acquirestwo additional terms which describe contributions to (nablami) middot (nablami) due to thespatial variation of the coordinate frame [16] namely curvilinear-geometry-inducedeffective anisotropy and curvilinear-geometry-induced effective DzyaloshinskiindashMoriyainteraction For magnetic shells these contributions may be expressed in terms of localcurvatures [15] for magnetic wires in terms of curvature and torsion [16] Below wereview briefly some manifestations of these contributions which have been reportedelsewhere

(i) Curvilinear-geometry-induced effective anisotropy Geometry-inducedanisotropy can have a significant effect on the ground-state magnetization profilerendering it no longer strictly tangential even in the case of strong easy-tangentialanisotropy For example for a helical nanowire with strong anisotropy directed alongthe wire the ground-state magnetization is always tilted in the local rectifying surfacewith tilting angle dependent on the product of the curvature and the torsion [17 18]For two-dimensional geometries with nontrivial topology a striking manifestation

of geometry-induced anisotropy is shapendashinduced patterning In spherical shellsa strictly in-surface magnetization is forbidden due to the hairyndashball theorem [19]Instead the ground-state magnetization profile has two oppositely disposed vortices[20] Another nontrivial example is the Mobius ring Since a Mobius ring is anonorientable surface its topology forces a discontinuity in any nonvanishing normalvector field Recently we proposed that magnetic nanostructures shaped as Mobiusstrips possess non-volatility in their magneto-electric response due to the presenceof topologically protected magnetic domain walls in materials with an out-of-planeorientation of the easy axis of magnetization [21] In both of these examples the linkbetween surface topology and magnetization is a consequence of geometryndashdependentanisotropy

(ii) Curvilinear-geometry-induced effective DzyaloshinskiindashMoriya interactionRecently the role of curvature in domain wall pinning was elucidated [22] a localbend in a nanowire is the source of a pinning potential for transversal domainwalls Chiral symmetry-breaking due to a geometry-induced DzyaloshinskiindashMoriyainteraction strongly impacts the domain wall dynamics and allows domain wall motionunder the action of different spinndashtorques eg fieldndashlike torques [18] and antindashdampingtorques [23] In the particular case of a helical nanowire torsion can produce negativedomain wall mobility [18 23] while curvature can produce a shift in the Walkerbreakdown [23]

We have briefly described a theoretical framework for studying differentcurvilinear systems including 1D nanowires and 2D nanoshells In this approachwe suppose that the effects of nonlocal dipole-dipole interactions can be reduced to aneffective easy-surface anisotropy In the 1D case this reduction has been rigorouslyjustified in the limit where the diameter of the wire h is much smaller than its lengthL [24] Similar arguments have been provided in the 2D case for planar thin films [25]and thin shells [26] where the surface thickness h is much less than the lateral size L

In the current study we consider a ribbon which represents a curve with aninfinitesimal neighbourhood of a surface along it [27] For a narrow ribbon whosethickness h is much less than its width w which in turn is much less than its lengthL namely h w L another micromagnetic limit is realized We show that themicromagnetic energy can be reduced to the energy of a wire with modified curvaturetorsion and anisotropy We illustrate this approach with two examples namely anarrow helicoid ribbon and a Mobius ribbon The existence of a new nonhomogeneousground state is predicted for the Mobius ribbon over a range of anisotropy parameterK The prediction is confirmed by full scale spinndashlattice simulations We also analysethe magnon spectrum for a narrow helicoid ribbon unlike the magnon spectrum fora straight wire there appears an asymmetry in the dispersion law caused by thegeometric and magnetic chiralities

The paper is organized as follows In Section 1 we derive the micromagneticenergy for a narrow ribbon which may be interpreted as a modification of the 1Dmicromagnetic energy of its central curve We illustrate the model by two examples ahelicoid ribbon (Section 2) and a Mobius ribbon (Section 3) Concluding remarks aregiven in Section 4 The justification of the magnetostatic energy for ribbons and stripsis presented in Appendix A The spin-lattice simulations are detailed in Appendix B

1 Model of narrow ribbon vs thin wire

11 Thin ferromagnetic wire

Here we consider a ferromagnetic wire described by a curve γ(s) with fixed cross-section of area S parameterized by arc length s isin [0 L] where L is the length ofthe wire It has been shown [24] that the properties of sufficiently thin ferromagneticwires of circular (or square) cross section are described by a reduced one-dimensionalenergy given by a sum of exchange and local anisotropy terms

Ewire = 4πM2sS

ds(E wire

ex + E wirean

E wireex = `2|mprime|2 E wire

an = minusQ1

2(m middot et)

Here m(s) denotes the unit magnetization vector prime prime denotes derivative withrespect to s Ms is the saturation magnetization and ` =

radicA4πM2

s is the exchangelength with A being the exchange constant The local anisotropy is uniaxial with easyaxis along the tangent et = γprime The normalized anisotropy constant (or quality factor)Q1 incorporates the intrinsic crystalline anisotropy K1 as well as a geometry-inducedmagnetostatic contribution

Q1 =K1

2πM2s

Note that the shape-induced biaxial anisotropy is caused by the asymmetry of thecross-section In particular for a rectangular cross-section the anisotropy coefficientsare determined by Eq (A4) for elliptical cross-sections see [24]

It is convenient to express the magnetization in terms of the Frenet-Serret framecomprised of the tangent et the normal en = eprimet|eprimet| and the binormal eb = ettimesenThese satisfy the Frenet-Serret equations

eprimeα = Fαβeβ Fαβ =

0 κ 0minusκ 0 τ0 minusτ 0

where κ(s) and τ(s) are the curvature and torsion of γ(s) respectively Letting

m = sinΘ cosΦ et + sinΘ sinΦ en + cosΘ eb

where Θ and Φ are functions of s (and time t if dynamics is considered) one can show[16] that the exchange and anisotropy energy densities are given by

E wireex = `2 [Θprime minus τ sinΦ]

2+ `2 [sinΘ(Φprime + κ)minus τ cosΘ cosΦ]

E wirean = minusQ1

2sin2Θ cos2 Φ

12 Narrow ferromagnetic ribbon

As above let γ(s) denote a three-dimensional curve parametrized by arc lengthFollowing [27] we take a ribbon to be a two-dimensional surface swept out by a linesegment centred at and perpendicular to γ moving (and possibly turning) along γThe ribbon may parametrized as

ς(s v) = γ(s) + v cosα(s)en + v sinα(s)eb v isin[minusw

where w is the width of the segment (assumed to be small enough so that ς has noself-intersections) and α(s) determines the orientation of the segment with respect tothe normal and binormal We construct a three-frame e1 e2 e3 on the ribbon givenby

emicro =partmicroς

|partmicroς| micro = 1 2 e3 = e1 times e2 (6)

Here and in what follows we use Greek letters micro ν etc = 1 2 to denote indicesrestricted to the ribbon surface Using the FrenetndashSerret equations (4) one can showthat (6) constitute an orthonormal frame with e1 and e2 tangent to the ribbon ande3 normal to it It follows that the first fundamental form (or metric) gmicroν = partmicroς middotpartνςis diagonal The second fundamental form bmicroν is given by bmicroν = e3 middot part2

microνς TheGauszlig and mean curvatures are given respectively by the determinant and trace of||Hmicroν || = ||bmicroνradicgmicromicrogνν ||

We consider a thin ferromagnetic shell about the ribbon of thickness h where

h wL (7)

The shell is comprised of points ς(s v) + ue3 where u isin [minush2 h2] We express theunit magnetization inside the shell in terms of the frame eα as

m = sin θ cosφ e1 + sin θ sinφ e2 + cos θ e3

where θ and φ are functions of the surface coordinates s v (and time t for dynamicalproblems) but are independent of the transverse coordinate u The micromagneticenergy of a thin shell reads

Eshell = 4πM2s h

w2intminusw2

radicgdv

(E shell

ex + E shellan

)+ Eshell

ms (8a)

where g = det (gmicroν) The exchange energy density in (8a) is given by [15 16]

E shellex = `2 [nablaθ minus Γ (φ)]

[sin θ (nablaφminusΩ)minus cos θ

partΓ (φ)

partφ

where nabla equiv emicronablamicro denotes a surface del operator in its curvilinear form with

components nablamicro equiv (gmicromicro)minus12

partmicro the vector Ω is a spin connection with components

Ωmicro = e1 middot nablamicroe2 and the vector Γ (φ) is given by ||Hmicroν ||(

cosφsinφ

) The next term in

the energy functional E shellan is the anisotropy energy density of the shell

E shellan = minus K1

4πM2s

(m middot e1)2 minus K3

4πM2s

(m middot e3)2 (8c)

where K1 and K3 are the tangential and normal anisotropy coefficients of the intrinsiccrystalline anisotropy The magnetostatic energy Eshell

ms has in the general case anonlocal form The local form is restored in the limit of thin films [28ndash30] and thinshells [26 31]

We proceed to consider the narrow-ribbon limit

`le h w ` L (9)

Keeping leading-order terms in wL we obtain that the geometrical properties ofribbon are determined by∥∥gribbon

microν

∥∥ = diag(1 1)∥∥Hribbon

microν

∥∥ =

(minusκ sinα αprime + ταprime + τ 0

In the same way we obtain from (8) the following

Eribbon = 4πM2s hw

intds(E eff

ex + E effan

E effex = `2 (θprime minus Γ1)

[sin θ (φprime minusΩ1)minuscos θ

partΓ1

partφ

E effan = `2Γ 2

2 + `2 cos2 θ

(partΓ2

partφ

+ E ribbonan

where the effective spin connection Ω1 and vector Γ are given by

Ω1 = minusκ cosα Γ1 = minusκ sinα cosφ+ (αprime + τ) sinφ Γ2 = (αprime + τ) cosφ (10b)

The last term in the energy density E ribbonan is the effective anisotropy energy density

of the narrow ribbon Using arguments similar to those in [26 28 30] it can be shownthat

E ribbonan = minusQ1

2(m middot e1)

2 minus Q3

2(m middot e3)

2 (10c)

Here Q1 and Q3 incorporate the intrinsic crystalline anisotropies K1 and K3 as wellas geometry-induced magnetostatic contributions

Q1 =K1

2πM2s

+Qr Qr =h

πwlnw

h Q3 = minus1 +

2πM2s

+ 2Qr (10d)

see the justification in Appendix A In the particular case of soft magnetic materialswhere K1 = K3 = 0 the anisotropy E ribbon

an is due entirely to the magnetostaticinteraction From (10d) we get Q1 = Qr 1 and Q3 = minus1 + 2Qr

The induced anisotropy is biaxial with easy axis along the central curve as for athin wire (cf (3)) and hard axis normal to the surface as for a thin shell Indeed onecan recast the narrow-ribbon energy (10) in the form of the thin-wire energy (2) withbiaxial anisotropy as follows

E effex = `2

(θprime minus τ eff sin Ψ

)2+ `2

[sin θ

(Ψprime + κeff

)minus τ eff cos θ cos Ψ

E effan = minusQ

2sin2 θ cos2 φminus Qeff

2cos2 θ

In (11) the effective curvature and torsion are given by

κeff = κ cosαminus βprime τ eff =

radicκ2 sin2 β + (αprime + τ)2 (12)

the angle Ψ is defined by

Ψ = φ+ β tanβ = minusκ sinα

αprime + τ

and the effective anisotropies are given by

Qeff1 = Q1 minus 2`2 (αprime + τ)

2 Qeff

3 = Q3 minus 2`2 (αprime + τ)2 (13)

Magnetic moment

Ribbon

Local basis

0 1 2 30

Reduced wavenumber q

RibbonStraight wire

(a) Thin magnetic ribbon (b) Dispersion curve

Figure 1 Magnetic helicoid ribbon (a) A sketch of the ribbon(b) Dispersion curve according to Eq (20) (solid blue line) in comparison withthe dispersion of the straight wire Ωstr = 1 + q2

2 Helicoid ribbon

The helicoid ribbon has a straight line which has vanishing curvature and torsion asits central curve We take γ(s) = s z The rate of turning about γ is constant andwe take α(s) = Css0 where the chirality C is +1 for a right-handed helicoid and minus1for a left-handed helicoid From (5) the parametrized surface is given by

ς(s v) = x v cos

)+ y Cv sin

)+ z s v isin

[minusw

The boundary curves given by ς(splusmnw2) are helices see Fig 1 (a) It is well knownthat the curvature and torsion essentially influence the spin-wave dynamics in a helixwire acting as an effective magnetic field [17] One can expect similar behaviour in ahelicoid ribbon

From (12) and (13) the effective curvature torsion and anisotropies are given by

κeff = 0 τ eff =C

s0 Qeff

1 = Q1 minus 2

Qeff3 = Q3 minus 2

From (11) the energy density is given by

E effex =

[(s0θ

prime minus C sinφ)2 + (s0 sin θφprime minus C cos θ cosφ)2]

E effan = minusQ

2cos2 θ

Let us consider the particular case of soft magnetic materials (K1 = K3 = 0)Under the reasonable assumption ` s0 we see that Q3 asymp minus1 so that the easy-surface anisotropy dominates the energy density and acts as an in-surface constraint

Taking θ = π2 to accommodate this constraint we obtain the (further) reducedenergy density

E eff = `2φprime2 minus Q1

2cos2 φ

which depends only on the in-surface orientation φ The ground states have φ constantwith orientation depending on the sign of the tangential-axis anisotropy Q1 ForQ1 gt 0 the ground states are

θt =π

2 cosφt = C (16)

where the magnetochirality C = plusmn1 determines whether the magnetisation m isparallel (C = 1) or antiparallel (C = minus1) to the helicoid axis For Q1 lt 0 theground states are given by

θn =π

2 φn = C

where the magnetochirality C = plusmn1 determines whether the magnetisation m isparallel (C = 1) or antiparallel (C = minus1) to the normal en This behaviour is similarto that of a ferromagnetic helical wire which was recently studied in Ref [17]

21 Spin-wave spectrum in a helicoid ribbon

Let us consider spin waves in a helicoid ribbon on the tangential ground state (16)We write

θ = θt + ϑ(χ t) φ = φt + ϕ(χ t) |ϑ| |ϕ| 1

where χ = ss0 and t = Ω0t with Ω0 = (2γ0Ms)(`s0)2 Expanding the energydensity (15) to quadratic order in the ϑ and ϕ we obtain

)2 [(partχϑ)

2+ (partχϕ)

(ϑpartχϕminus ϕpartχϑ)

[Q1 minusQ3 + 2

)2]ϑ2

The linearised LandaundashLifshits equations have the form of a generalized Schrodingerequation for the complex-valued function ψ = ϑ+ iϕ [17]

minus iparttψ = Hψ +Wψlowast H = (minusipartχ minusA)2

+ U (17)

where the ldquopotentialsrdquo have the following form

U = minus1

(2Q1 minusQ3) A = minusCC W =1

2minus 1

Q3 (18)

We look for plane wave solutions of (17) of the form

ψ(χ t) = ueiΦ + veminusiΦ Φ = qχminus Ωt+ η (19)

where q = ks0 is a dimensionless wave number Ω = ωΩ0 is a dimensionless frequencyη is an arbitrary phase and u v isin R are constant amplitudes By substituting (19)into the generalized Schrodinger equation (17) we obtain

Ω(q) = minus2CCq +

radic[q2 + 1 +

Q1 minusQ3

)2] [q2 +

)2] (20)

see Fig 1 (b) in which the parameters have the following values s0` = 5 Q1 = 02Q3 = minus06 and C = C = 1 The dispersion relation (20) for the helicoid ribbon issimilar to that of a helical wire [17] but different from that of a straight wire in thatit is not reflection-symmetric in q The sign of the asymmetry is determined by theproduct of the helicoid chirality C which depends on the topology of the ribbon andthe magnetochirality C which depends on the topology of the magnetic structureThis asymmetry stems from the curvature-induced effective DzyaloshinskiindashMoriyainteraction which is the source of the vector potential A = Aet where A = minusCC Inthis context it is instructive to mention a relation between the DzyaloshinskiindashMoriyainteraction and the Berry phase [32]

3 Mobius ribbon

In this section we consider a narrow Mobius ribbon The Mobius ring was studiedpreviously in Ref [21] The ground state is determined by the relationship betweengeometrical and magnetic parameters The vortex configuration is favorable in thesmall anisotropy case while a topologically protected domain wall is the ground statefor large easy-normal anisotropy Although the problem was studied for a wide rangeof parameters the limit of a narrow ribbon was not considered previously Below weshow that the narrow Mobius ribbon exhibits a new inhomogeneous ground state seeFig 2 (a) (b)

The Mobius ribbon has a circle as its central curve and turns at a constant ratemaking a half-twist once around the circle it can be formed by joining the ends of ahelicoid ribbon Letting R denote the radius we use the angle χ = sR instead of arclength s as parameter and set

γ(χ) = R cosχx+R sinχy α(χ) = π minus Cχ2 (21)

The chirality C = plusmn1 determines whether the Mobius ribbon is right- or left-handedFrom (5) the parametrized surface is given by

ς(χ v) =(R+ v cos

)cosχ x+

(R+ v cos

)sinχ y + Cv sin

2z (22)

Here χ isin [0 2π) is the azimuthal angle and v isin [minusw2 w2] is the positionalong the ring width From (10) the energy of the narrow Mobius ribbon reads

E = 4πM2s hwR

2πint0

E dχ where the energy density is given by

)2 [Cpartχθ +

2sinφ+ cosφ sin

)2 [sin θ

(partχφminus cos

)+C cos θ

2cosφminus sinφ sin

minus Qeff1

2cos2 θ

with effective anisotropies (cf (10d))

Qeff1 = Q1 minus

Qeff3 = Q3 minus

The effective curvature and torsion are given by (cf (12))

κeff = minus2 cos χ2R

1 + 2 sin2 χ2

1 + 4 sin2 χ2

τ eff = minus C

radic1 + 4 sin2 χ

05 1 15 2

minus06

minus04

minus02

Ribbon state

Vortex state

Reduced anisotropy coefficient k

0 π2 π 3π2 2π0

Azimuthal angle χ

k = kck = 1

4 magnetostatics

(a) (b)

(c) (d)

Figure 2 Magnetic Mobius ribbon (a) Magnetization distribution for theribbon state in the laboratory frame see Eq (25) (b) Magnetization distributionfor the ribbon state in the ribbon frame (c) The energy difference between thevortex and ribbon states when the reduced anisotropy coefficient k exceeds thecritical value kc see Eq (26) the vortex state is favourable while for k lt kcthe inhomogeneous ribbon state is realized (d) In-surface magnetization angleφ in the ribbon state Lines correspond to Eq (25) and markers correspondto SLaSi simulations see Appendix B for details Red triangles representthe simulations with dipolar interaction without magnetocrystalline anisotropy(k = 0) it corresponds very well to our theoretical result (solid red curve) foreffective anisotropy k = 14 induced by magnetostatics see Eq (3)

Let us consider the case of uniaxial magnetic materials for which K3 = 0 Underthe reasonable assumption ` R we have that Qeff

3 asymp minus1 so that the easy-surfaceanisotropy dominates the energy density and acts as an in-surface constraint (as forthe helicoid ribbon) Taking θ = π2 we obtain the simplified energy density

)2 (partχφminus cos

2sinφ+ cosφ sin

minus Qeff1

2cos2 φ

which depends only on the in-surface orientation φ The equilibrium magnetizationdistribution is described by the following EulerndashLagrange equation

partχχφ+ sinχ

2sin2φ+

(sin2 χ

2minus k)

sinφ cosφ = 0

φ(0) = minusφ(2π) mod 2π partχφ(0) = minuspartχφ(2π)(23)

where the antiperiodic boundary conditions compensate for the half-twist in theMobius ribbon and ensure that the magnetisation m is smooth at χ = 0 The reduced

anisotropy coefficient k in (23) reads

k =Qeff

4πM2s

2πw`2lnw

hminus 1

4 (24)

It is easily seen that if φ(χ) is a solution of (23) then φ(χ) + nπ is also a solutionwith the same energy Since solutions differing by multiples of 2π describe the samemagnetisation only φ(χ) + π corresponds to a configuration distinct from φ We alsonote that if φ(χ) is a solution of (23) then minusφ(minusχ) is also a solution with the sameenergy

By inspection φvor+ equiv 0 and φvor

minus equiv π are solutions of (23) the ground states are

θvor =π

2 cosφvor = C

where the magnetochirality C determines whether the magnetisation m is parallelor antiparallel to the circular axis We refer to these as vortex states Unlike thecase of the helicoid ribbon φ equiv plusmnπ2 is not a solution of (23) Numericallywe find two further solutions of the Euler-Lagrange equation denoted φrib

+ (χ) andφribminus (χ) equiv φrib

+ (χ) + π which we call ribbon states While we have not obtainedanalytical expressions for φrib(χ) good approximations can be found by assumingφrib

+ to be antiperiodic and odd so that it has a Fourier-sine expansion of the form

φrib+ (χ) =

infinsumn=1

cn sin((2nminus 1)χ2) (25)

The series is rapidly converging with the first four coefficients c1 = 2245 c2 = 00520c3 = minus00360 and c4 = minus00142 for k = 025 providing an approximation accurate towithin 003 (specifically the L2-norm difference between the numerically determinedφ as described in Appendix B and this expansion is 0003)

Numerical calculations indicate that the ground state of the Mobius ribbonlike the helicoid ribbon undergoes a bifurcation as the tangential-axis anisotropydecreases Unlike the helicoid ribbon the bifurcation occurs for positive anisotropykc given by

kc asymp 16934 (26)

For k gt kc the vortex state has the lowest energy whereas for k lt kc the ribbon statehas the lowest energy The energy difference between the vortex and ribbon states

∆E =Erib minus Evor

2M2s hwR

2πint0

Edχminus 5

is plotted in Fig 2(c) In some respects the ribbon state resembles an onion statein magnetic rings [16 33ndash35] in the laboratory reference frame the magnetizationdistribution is close to a spatially homogeneous state see Fig 2(a)

The in-surface magnetization angle φ(χ) for the ribbon state is plotted inFig 2(d) The plot shows good agreement between the analytic expression (25) andspinndashlattice SLaSi simulations (see Appendix B for details) The blue dashed line withsolid circles represents the case k = kc (the critical anisotropy value) The red solidline corresponds to the solution of Eq (23) for k = 14 an effective anisotropy inducedby magnetostatics It is in a good agreement with simulations shown by red triangleswhere the dipolendashdipole interaction is taken into account instead of easy-tangential

anisotropy The magnetization distribution for the ribbon state is shown in Fig 2(a)(3-dimensional view) and Fig 2(b) (an untwisted schematic of the Mobius ribbon)

Let us estimate the values of the parameters for which the ribbon state isenergetically preferable Taking into account (24) we find that the ribbon state isenergetically preferable provided

4πM2S

R2minus h

2πwlnw

This condition is a fortiori satisfied for the hard axial case ie when K1 lt 0 For softmagnetic materials (K1 = 0) the only source of anisotropy is the shape anisotropyThe ribbon state is the ground state when

2πw`2lnw

hlt kc +

which imposes constraints on the geometry and material parameters

4 Conclusion

We have studied ferromagnetic ribbons that is magnetic materials in the shape of thinshells whose median surface is swept out by a line segment turning as it moves alonga central curve Ferromagnetic ribbons combine properties of both 1D systems ienanowires and 2D systems ie curved films and nanoshells While the geometricalproperties of a narrow ribbon are described by its central curve and the rate ofturning of its transverse line segment its magnetic properties are determined bythe geometrical and magnetic properties of the ribbon surface The micromagneticenergy of the ribbon can be reduced to the energy of a 1D system (magnetic nanowire)with effective curvature torsion and biaxial anisotropy While the source of effectivecurvature and torsion is the exchange interaction only the biaxiality results from bothexchange and magnetostatics

We have studied two examples (i) a narrow helicoid ribbon and ii) a narrowMobius ribbon The helicoid ribbon has zero effective curvature but finite torsionwhich provides a paradigmatic model for studying purely torsion-induced effectsSimilar to a microhelix structure [17] a geometry-induced effective DzyaloshinskiindashMoriya interaction is a source of coupling between the helicoid chirality andthe magnetochirality which essentially influences both magnetization statics anddynamics The emergent magnetic field generated by the torsion breaks mirrorsymmetry so that the properties of magnetic excitations in different spatial directionsis not identical The narrow Mobius ribbon is characterized by spatially varyingeffective curvature and torsion We have predicted a new inhomogeneous ribbonstate for the Mobius ribbon which is characterized by an inhomogeneous in-surfacemagnetization distribution The existence of this state has been confirmed by spinndashlattice simulations

Acknowledgments

A G acknowledges the support of EPSRC Grant No EPK0241161V P Kacknowledges the Alexander von Humboldt Foundation for the support and IFWDresden for kind hospitality D D Sh thanks the University of Bristol where partof this work was performed for kind hospitality J M R and V S acknowledge thesupport of EPSRC Grant No EPK02390X1

Appendix A Magnetostatic energy of ribbons and strips

Here we justify formulae (10c) (10d) for the magnetostatic energy of a narrow ribbonTo this end we calculate the magnetostatic energy of the shell of reduced widthw = w` and reduced thickness h = h` in the regime

w2 le h le w 1

and then identify the leading contributions to the energy of a ribbon in the limit ofsmall aspect ratio

δ equiv h

For the sake of clarity before turning our attention to the general case we first considera flat strip Vs = [0 L]times [minusw2 w2 ]times [minush2 h2 ] The magnetostatic energy may be writtenin the form

Estripms = minusM

drprime (m(r) middotnabla) (m(rprime) middotnablaprime)1

|r minus rprime|

It is well known that the leading order contribution to the magnetostatic energyis coming from the interaction between the surface charges of the largest surfacesWe denote by T and B the pair of top and bottom surfaces of the strip (of surfacearea Lw) and by F R the front and rear surfaces of the strip (of surface area Lh)respectively It is straightforward to show (see eg [24 30]) that

Estripms =

intTcupB

dSprime(m(r) middot n) (m(rprime) middot nprime)

|r minus rprime|

intFcupR

|r minus rprime| + O(w2h

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρminus m3(s)m3(sprime)radic

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

intm(s u v) dudv is the average of magnetization m over the cross-

section of area wh n is the surface normal and ρ =

radic(sminus sprime)2

+ (uminus v)2We note that for an arbitrary smooth function f and a constant a

f(sprime)dsprimeradica2 + (sminus sprime)2

= f(L) ln(Lminus s+

radic(Lminus s)2 + a2

)+ f(0) ln

radics2 + a2

minus 2f(s) ln |a| minusLints

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Applying this formula and following the approach developed in [24] we can showthat the main contribution to the magnetostatic energy will be coming from the termminus2f(s) ln |a| in the last integral Therefore we obtain

Estripms

12intminus12

dv m23(s)

[lnradic

(uminus v)2 + δ2 minus ln |uminus v|]

12intminus12

dv m22(s)

[lnradic

(uminus v)2 + 1δ2 minus ln |uminus v|]

+ O(wh2

∣∣ln h∣∣) (A1)

By integrating over the cross-section variables the expression (A1) further simplifiesto

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Hence the magnetostatic energy of the flat strip is

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

Returning to the general case we recall from (5) that a ribbon may beparametrized as

ς(s v) = γ(s) + ve2(s) v isin[minusw

] s isin [0 L]

and consider a shell of thickness h around ς parametrized as

(s v u) = γ(s) + ve2(s) + ue3(s v)

where e2 e3 are defined in (6) and h is small enough so that does not intersectitself Then introducing m2 = m middot e2 and m3 = m middot e3 the energy of the shell up toterms of order O

∣∣ln h∣∣) is given by

Eribbonms

12intminus12

dvradicgm3(s wu)m3(s wv) ln

radic(uminus v)2 + δ2

|uminus v|

12intminus12

dv m22(s)

[lnradic

We remark that the formula (A3) yields the correct result both for a wire with arectangular cross-section (hw = const) and in the thin film limit (hw rarr 0) cf [24]and [26] respectively however in the latter case it resolves terms beyond the leadingorder

REFERENCES 15

Expanding the first integral in (A3) in w and integrating over cross-sectionvariables we obtain that the magnetostatic energy of the ribbon

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

is insensitive to curvature effects cf (A2) Finally using the constrain m2 = 1 weget the magnetostatic energy in the form (10c) (10d)

Appendix B Simulations

We use the in-house developed spin-lattice simulator SLaSi [36] A chain of classicalmagnetic moments mi |mi| = 1 i = 1 N is considered They are situated on a circle(21) which defines a central axis of the narrow Mobius ribbon (22) hence the periodiccondition mN+1 = m1 is used The following classical Hamiltonian is used

H = minusa`2Nsumi=1

(mi middotmi+1)minus a3

Nsumi=1

[Q1(mi middot e1i)

2 +Q3(mi middot e3i)2]

sumi6=j

[(mi middotmj)

minus 3(mi middot rij)(mj middot rij)

where a is the lattice constant e1i and e3i are unit basis vectors (6) in i-th site andthe coefficient d = 0 1 is used as a switch for dipolar interactions

To study the static magnetization distribution we minimize the energy by solvinga set of N vector LandaundashLifshitzndashGilbert ordinary differential equations for N = 100sites situated on a ring of radius R = aN(2π) and ` = R using the RungendashKuttandashFehlberg scheme (RKF45) see [37] for general description of the simulator Theequilibrium magnetization state is found starting the simulations from different initialdistributions (four different random ones uniformly magnetized states along plusmnx plusmnyplusmnz and along unit vectors ei

The simulations are performed using the high-performance computer clustersof the Taras Shevchenko National University of Kyiv [38] and the BayreuthUniversity [39]

References

[1] Streubel R Fischer P Kronast F Kravchuk V P Sheka D D Gaididei Y SchmidtO G and Makarov D 2016 Journal of Physics D Applied Physics 49 363001 URLhttpstacksioporg0022-372749i=36a=363001

[2] Makarov D Melzer M Karnaushenko D and Schmidt O G 2016 Applied PhysicsReviews 3 011101 URL httpdxdoiorg10106314938497

[3] Kamien R 2002 Reviews of Modern Physics 74 953ndash971 URL httpdxdoi

org101103RevModPhys74953

[4] Nelson D Weinberg S and Piran T 2004 Statistical Mechanics of Mem-branes and Surfaces (Second Edition) (Wspc) ISBN 9812387609 URL http

wwwamazoncomStatistical-Mechanics-Membranes-Surfaces-Edition

dp98123876093FSubscriptionId3D0JYN1NVW651KCA56C10226tag

3Dtechkie-2026linkCode3Dxm226camp3D202526creative3D165953

26creativeASIN3D9812387609

REFERENCES 16

[5] Bowick M J and Giomi L 2009 Advances in Physics 58 449ndash563 URL http

wwwtandfonlinecomdoiabs10108000018730903043166

[6] Turner A M Vitelli V and Nelson D R 2010 Rev Mod Phys 82 1301ndash1348 URLhttplinkapsorgdoi101103RevModPhys821301

[7] Napoli G and Vergori L 2012 Physical Review Letters 108 207803 URL http

dxdoiorg101103PhysRevLett108207803

[8] Napoli G and Vergori L 2012 Physical Review E 85 061701 URL httpdx

doiorg101103PhysRevE85061701

[9] Napoli G and Vergori L 2013 Soft Matter 9 8378 URL httpdxdoiorg10

1039c3sm50605c

[10] Napoli G and Vergori L 2013 International Journal of Non-Linear Mechanics 4966ndash71 URL httpdxdoiorg101016jijnonlinmec201209007

[11] Segatti A Snarski M and Veneroni M 2014 Physical Review E 90 URL http

dxdoiorg101103PhysRevE90012501

[12] Manyuhina O V 2014 Physical Review E 90 URL httpdxdoiorg101103

PhysRevE90022713

[13] de Oliveira E J L de Oliveira I N Lyra M L and Mirantsev L V 2016 PhysicalReview E 93 URL httpdxdoiorg101103PhysRevE93012703

[14] Vitelli V and Turner A M 2004 Phys Rev Lett 93(21) 215301 URL http

linkapsorgdoi101103PhysRevLett93215301

[15] Gaididei Y Kravchuk V P and Sheka D D 2014 Phys Rev Lett 112(25) 257203URL httplinkapsorgdoi101103PhysRevLett112257203

[16] Sheka D D Kravchuk V P and Gaididei Y 2015 Journal of PhysicsA Mathematical and Theoretical 48 125202 URL httpstacksioporg

1751-812148i=12a=125202

[17] Sheka D D Kravchuk V P Yershov K V and Gaididei Y 2015 Phys Rev B 92(5)054417 URL httplinkapsorgdoi101103PhysRevB92054417

[18] Pylypovskyi O V Sheka D D Kravchuk V P Yershov K V Makarov D andGaididei Y 2015 Rashba torque driven domain wall motion in magnetic helices(Preprint 151004725) URL httparxivorgabs151004725

[19] Milnor J 1978 The American Mathematical Monthly 85 521ndash524 URL http

wwwjstororgstable2320860

[20] Kravchuk V P Sheka D D Streubel R Makarov D Schmidt O G and GaidideiY 2012 Phys Rev B 85(14) 144433 URL httplinkapsorgdoi101103

PhysRevB85144433

[21] Pylypovskyi O V Kravchuk V P Sheka D D Makarov D Schmidt O G andGaididei Y 2015 Phys Rev Lett 114(19) 197204 URL httplinkapsorg

doi101103PhysRevLett114197204

[22] Yershov K V Kravchuk V P Sheka D D and Gaididei Y 2015 Phys Rev B92(10) 104412 URL httplinkapsorgdoi101103PhysRevB92104412

[23] Yershov K V Kravchuk V P Sheka D D and Gaididei Y 2015 Curvatureand torsion effects in the spin-current driven domain wall motion (Preprint151102193) URL httparxivorgabs151102193

[24] Slastikov V V and Sonnenberg C 2012 IMA Journal of Applied Mathematics 77220ndash235 ISSN 1464-3634 URL httpdxdoiorg101093imamathxr019

REFERENCES 17

[25] Kohn R and Slastikov V 2005 Proc R Soc A 461 143ndash154 URL httpdx

doiorg101098rspa20041342

[26] Slastikov V 2005 Mathematical Models and Methods in Applied Sciences15 1469ndash1487 (Preprint httpwwwworldscientificcomdoipdf101142S021820250500087X) URL httpwwwworldscientificcomdoiabs10

1142S021820250500087X

[27] Sternberg S 2012 Curvature in mathematics and physics (Mineola New YorkDover Pubications Inc) ISBN 9780486478555

[28] Gioia G and James R D 1997 Proc R Soc Lond A 453 213ndash223 URLhttpwwwjournalsroyalsocacukopenurlaspgenre=articleampid=

doi101098rspa19970013

[29] Carbou G 2001 Mathematical Models and Methods in Applied Sciences (M3AS)11 1529ndash1546 URL httpdxdoiorg101142S0218202501001458

[30] Kohn R V and Slastikov V V 2005 Archive for Rational Mechanics and Analysis178 227ndash245 URL httpdxdoiorg101007s00205-005-0372-7

[31] Fratta G D ArXiv e-prints (Preprint 160908040v1) URL httparxivorg

abs160908040

[32] Freimuth F Blugel S and Mokrousov Y 2014 J Phys Condens Matter 26104202 ISSN 1361-648X URL httpdxdoiorg1010880953-89842610

104202

[33] Klaui M Vaz C A F Lopez-Diaz L and Bland J A C 2003 Journal of PhysicsCondensed Matter 15 R985ndashR1024 URL httpstacksioporg0953-8984

15R985

[34] Kravchuk V P Sheka D D and Gaididei Y B 2007 J Magn Magn Mater 310116ndash125 URL httpdxdoiorg101016jjmmm200607035

[35] Guimaraes A P 2009 Principles of Nanomagnetism NanoScience and Technology(Berlin Springer-Verlag Berlin Heidelberg) ISBN 978-3-642-01481-9 URL http

linkspringercombook101007978-3-642-01482-6page1

[36] SLaSi spinndashlattice simulations package httpslasiknuua accessed 2016-04-17 URL httpslasiknuua

[37] Pylypovskyi O V Sheka D D Kravchuk V P and Gaididei Y 2014 Journal ofMagnetism and Magnetic Materials 361 201ndash205 ISSN 0304-8853 URL http

wwwsciencedirectcomsciencearticlepiiS0304885314002157

[38] Highndashperformance computing cluster of Taras Shevchenko National University ofKyiv URL httpclusterunivkievuaeng

[39] Bayreuth University computing cluster URL httpwwwrzuni-bayreuthde



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



Magnetization in narrow ribbons curvature effects

Yuri Gaididei1 Arseni Goussev2 Volodymyr P Kravchuk1Oleksandr V Pylypovskyi3 J M Robbins4Denis D Sheka3 Valeriy Slastikov4 andSergiy Vasylkevych4

1 Bogolyubov Institute for Theoretical Physics of National Academy of Sciencesof Ukraine 03680 Kyiv Ukraine2 Department of Mathematics Physics and Electrical Engineering NorthumbriaUniversity Newcastle upon Tyne NE1 8ST UK3 Taras Shevchenko National University of Kyiv 01601 Kyiv Ukraine4 School of Mathematics University of Bristol Bristol BS8 1TW UK

E-mail ybgbitpkievua arsenigoussevnorthumbriaacukvkravchukbitpkievua engraverknuua JRobbinsbristolacukshekaunivnetua svasylkevychbristolacuk

AbstractA ribbon is a surface swept out by a line segment turning as it moves along

a central curve For narrow magnetic ribbons for which the length of the linesegment is much less than the length of the curve the anisotropy induced by themagnetostatic interaction is biaxial with hard axis normal to the ribbon and easyaxis along the central curve The micromagnetic energy of a narrow ribbon reducesto that of a one-dimensional ferromagnetic wire but with curvature torsion andlocal anisotropy modified by the rate of turning These general results are appliedto two examples namely a helicoid ribbon for which the central curve is a straightline and a Mobius ribbon for which the central curve is a circle about which theline segment executes a 180 twist In both examples for large positive tangentialanisotropy the ground state magnetization lies tangent to the central curve Asthe tangential anisotropy is decreased the ground state magnetization undergoesa transition acquiring an in-surface component perpendicular to the central curveFor the helicoid ribbon the transition occurs at vanishing anisotropy below whichthe ground state is uniformly perpendicular to the central curve The transitionfor the Mobius ribbon is more subtle it occurs at a positive critical value of theanisotropy below which the ground state is nonuniform For the helicoid ribbonthe dispersion law for spin wave excitations about the tangential state is found toexhibit an asymmetry determined by the geometric and magnetic chiralities

PACS numbers 7570-i 7575-c 7510Hk 7530Et

Submitted to J Phys A Math Gen

Introduction

Ewire = 4πM2sS

ds(E wire

ex + E wirean

an = minusQ1

2(m middot et)

radicA4πM2

Q1 =K1

2πM2s

E wirean = minusQ1

2sin2Θ cos2 Φ

emicro =partmicroς

h wL (7)

Eshell = 4πM2s h

w2intminusw2

radicgdv

(E shell

ex + E shellan

)+ Eshell

ms (8a)

partΓ (φ)

partφ

cosφsinφ

) The next term in

4πM2s

(m middot e3)2 (8c)

`le h w ` L (9)

microν

∥∥ =

Eribbon = 4πM2s hw

intds(E eff

ex + E effan

partΓ1

partφ

E effan = `2Γ 2

2 + `2 cos2 θ

(partΓ2

partφ

+ E ribbonan

2(m middot e1)

2 minus Q3

2(m middot e3)

2 (10c)

Q1 =K1

2πM2s

+Qr Qr =h

πwlnw

h Q3 = minus1 +

2πM2s

+ 2Qr (10d)

E effex = `2

)2+ `2

[sin θ

(Ψprime + κeff

E effan = minusQ

2cos2 θ

αprime + τ

2 Qeff

3 = Q3 minus 2`2 (αprime + τ)2 (13)

Magnetic moment

Ribbon

Local basis

0 1 2 30

RibbonStraight wire

2 Helicoid ribbon

ς(s v) = x v cos

)+ y Cv sin

)+ z s v isin

[minusw

κeff = 0 τ eff =C

s0 Qeff

1 = Q1 minus 2

Qeff3 = Q3 minus 2

E effex =

[(s0θ

E effan = minusQ

2cos2 θ

2cos2 φ

θt =π

2 cosφt = C (16)

θn =π

2 φn = C

θ = θt + ϑ(χ t) φ = φt + ϕ(χ t) |ϑ| |ϕ| 1

)2 [(partχϑ)

2+ (partχϕ)

[Q1 minusQ3 + 2

)2]ϑ2

+ U (17)

U = minus1

2minus 1

Q3 (18)

Ω(q) = minus2CCq +

radic[q2 + 1 +

Q1 minusQ3

)2] [q2 +

)2] (20)

3 Mobius ribbon

ς(χ v) =(R+ v cos

)cosχ x+

(R+ v cos

)sinχ y + Cv sin

2z (22)

E = 4πM2s hwR

2πint0

)2 [Cpartχθ +

2sinφ+ cosφ sin

)2 [sin θ

(partχφminus cos

)+C cos θ

minus Qeff1

2cos2 θ

Qeff1 = Q1 minus

Qeff3 = Q3 minus

1 + 2 sin2 χ2

1 + 4 sin2 χ2

τ eff = minus C

radic1 + 4 sin2 χ

05 1 15 2

minus06

minus04

minus02

Ribbon state

Vortex state

0 π2 π 3π2 2π0

Azimuthal angle χ

k = kck = 1

4 magnetostatics

(a) (b)

(c) (d)

2sinφ+ cosφ sin

minus Qeff1

2cos2 φ

partχχφ+ sinχ

2sin2φ+

(sin2 χ

2minus k)

sinφ cosφ = 0

k =Qeff

4πM2s

2πw`2lnw

hminus 1

4 (24)

θvor =π

2 cosφvor = C

φrib+ (χ) =

infinsumn=1

kc asymp 16934 (26)

2M2s hwR

2πint0

Edχminus 5

4πM2S

R2minus h

2πwlnw

2πw`2lnw

hlt kc +

4 Conclusion

Acknowledgments

w2 le h le w 1

δ equiv h

Estripms = minusM

|r minus rprime|

Estripms =

intTcupB

|r minus rprime|

intFcupR

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

= f(L) ln(Lminus s+

)+ f(0) ln

radics2 + a2

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



Introduction

Ewire = 4πM2sS

ds(E wire

ex + E wirean

an = minusQ1

2(m middot et)

radicA4πM2

Q1 =K1

2πM2s

E wirean = minusQ1

2sin2Θ cos2 Φ

emicro =partmicroς

h wL (7)

Eshell = 4πM2s h

w2intminusw2

radicgdv

(E shell

ex + E shellan

)+ Eshell

ms (8a)

partΓ (φ)

partφ

cosφsinφ

) The next term in

4πM2s

(m middot e3)2 (8c)

`le h w ` L (9)

microν

∥∥ =

Eribbon = 4πM2s hw

intds(E eff

ex + E effan

partΓ1

partφ

E effan = `2Γ 2

2 + `2 cos2 θ

(partΓ2

partφ

+ E ribbonan

2(m middot e1)

2 minus Q3

2(m middot e3)

2 (10c)

Q1 =K1

2πM2s

+Qr Qr =h

πwlnw

h Q3 = minus1 +

2πM2s

+ 2Qr (10d)

E effex = `2

)2+ `2

[sin θ

(Ψprime + κeff

E effan = minusQ

2cos2 θ

αprime + τ

2 Qeff

3 = Q3 minus 2`2 (αprime + τ)2 (13)

Magnetic moment

Ribbon

Local basis

0 1 2 30

RibbonStraight wire

2 Helicoid ribbon

ς(s v) = x v cos

)+ y Cv sin

)+ z s v isin

[minusw

κeff = 0 τ eff =C

s0 Qeff

1 = Q1 minus 2

Qeff3 = Q3 minus 2

E effex =

[(s0θ

E effan = minusQ

2cos2 θ

2cos2 φ

θt =π

2 cosφt = C (16)

θn =π

2 φn = C

θ = θt + ϑ(χ t) φ = φt + ϕ(χ t) |ϑ| |ϕ| 1

)2 [(partχϑ)

2+ (partχϕ)

[Q1 minusQ3 + 2

)2]ϑ2

+ U (17)

U = minus1

2minus 1

Q3 (18)

Ω(q) = minus2CCq +

radic[q2 + 1 +

Q1 minusQ3

)2] [q2 +

)2] (20)

3 Mobius ribbon

ς(χ v) =(R+ v cos

)cosχ x+

(R+ v cos

)sinχ y + Cv sin

2z (22)

E = 4πM2s hwR

2πint0

)2 [Cpartχθ +

2sinφ+ cosφ sin

)2 [sin θ

(partχφminus cos

)+C cos θ

minus Qeff1

2cos2 θ

Qeff1 = Q1 minus

Qeff3 = Q3 minus

1 + 2 sin2 χ2

1 + 4 sin2 χ2

τ eff = minus C

radic1 + 4 sin2 χ

05 1 15 2

minus06

minus04

minus02

Ribbon state

Vortex state

0 π2 π 3π2 2π0

Azimuthal angle χ

k = kck = 1

4 magnetostatics

(a) (b)

(c) (d)

2sinφ+ cosφ sin

minus Qeff1

2cos2 φ

partχχφ+ sinχ

2sin2φ+

(sin2 χ

2minus k)

sinφ cosφ = 0

k =Qeff

4πM2s

2πw`2lnw

hminus 1

4 (24)

θvor =π

2 cosφvor = C

φrib+ (χ) =

infinsumn=1

kc asymp 16934 (26)

2M2s hwR

2πint0

Edχminus 5

4πM2S

R2minus h

2πwlnw

2πw`2lnw

hlt kc +

4 Conclusion

Acknowledgments

w2 le h le w 1

δ equiv h

Estripms = minusM

|r minus rprime|

Estripms =

intTcupB

|r minus rprime|

intFcupR

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

= f(L) ln(Lminus s+

)+ f(0) ln

radics2 + a2

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



Ewire = 4πM2sS

ds(E wire

ex + E wirean

an = minusQ1

2(m middot et)

radicA4πM2

Q1 =K1

2πM2s

E wirean = minusQ1

2sin2Θ cos2 Φ

emicro =partmicroς

h wL (7)

Eshell = 4πM2s h

w2intminusw2

radicgdv

(E shell

ex + E shellan

)+ Eshell

ms (8a)

partΓ (φ)

partφ

cosφsinφ

) The next term in

4πM2s

(m middot e3)2 (8c)

`le h w ` L (9)

microν

∥∥ =

Eribbon = 4πM2s hw

intds(E eff

ex + E effan

partΓ1

partφ

E effan = `2Γ 2

2 + `2 cos2 θ

(partΓ2

partφ

+ E ribbonan

2(m middot e1)

2 minus Q3

2(m middot e3)

2 (10c)

Q1 =K1

2πM2s

+Qr Qr =h

πwlnw

h Q3 = minus1 +

2πM2s

+ 2Qr (10d)

E effex = `2

)2+ `2

[sin θ

(Ψprime + κeff

E effan = minusQ

2cos2 θ

αprime + τ

2 Qeff

3 = Q3 minus 2`2 (αprime + τ)2 (13)

Magnetic moment

Ribbon

Local basis

0 1 2 30

RibbonStraight wire

2 Helicoid ribbon

ς(s v) = x v cos

)+ y Cv sin

)+ z s v isin

[minusw

κeff = 0 τ eff =C

s0 Qeff

1 = Q1 minus 2

Qeff3 = Q3 minus 2

E effex =

[(s0θ

E effan = minusQ

2cos2 θ

2cos2 φ

θt =π

2 cosφt = C (16)

θn =π

2 φn = C

θ = θt + ϑ(χ t) φ = φt + ϕ(χ t) |ϑ| |ϕ| 1

)2 [(partχϑ)

2+ (partχϕ)

[Q1 minusQ3 + 2

)2]ϑ2

+ U (17)

U = minus1

2minus 1

Q3 (18)

Ω(q) = minus2CCq +

radic[q2 + 1 +

Q1 minusQ3

)2] [q2 +

)2] (20)

3 Mobius ribbon

ς(χ v) =(R+ v cos

)cosχ x+

(R+ v cos

)sinχ y + Cv sin

2z (22)

E = 4πM2s hwR

2πint0

)2 [Cpartχθ +

2sinφ+ cosφ sin

)2 [sin θ

(partχφminus cos

)+C cos θ

minus Qeff1

2cos2 θ

Qeff1 = Q1 minus

Qeff3 = Q3 minus

1 + 2 sin2 χ2

1 + 4 sin2 χ2

τ eff = minus C

radic1 + 4 sin2 χ

05 1 15 2

minus06

minus04

minus02

Ribbon state

Vortex state

0 π2 π 3π2 2π0

Azimuthal angle χ

k = kck = 1

4 magnetostatics

(a) (b)

(c) (d)

2sinφ+ cosφ sin

minus Qeff1

2cos2 φ

partχχφ+ sinχ

2sin2φ+

(sin2 χ

2minus k)

sinφ cosφ = 0

k =Qeff

4πM2s

2πw`2lnw

hminus 1

4 (24)

θvor =π

2 cosφvor = C

φrib+ (χ) =

infinsumn=1

kc asymp 16934 (26)

2M2s hwR

2πint0

Edχminus 5

4πM2S

R2minus h

2πwlnw

2πw`2lnw

hlt kc +

4 Conclusion

Acknowledgments

w2 le h le w 1

δ equiv h

Estripms = minusM

|r minus rprime|

Estripms =

intTcupB

|r minus rprime|

intFcupR

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

= f(L) ln(Lminus s+

)+ f(0) ln

radics2 + a2

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



Ewire = 4πM2sS

ds(E wire

ex + E wirean

an = minusQ1

2(m middot et)

radicA4πM2

Q1 =K1

2πM2s

E wirean = minusQ1

2sin2Θ cos2 Φ

emicro =partmicroς

h wL (7)

Eshell = 4πM2s h

w2intminusw2

radicgdv

(E shell

ex + E shellan

)+ Eshell

ms (8a)

partΓ (φ)

partφ

cosφsinφ

) The next term in

4πM2s

(m middot e3)2 (8c)

`le h w ` L (9)

microν

∥∥ =

Eribbon = 4πM2s hw

intds(E eff

ex + E effan

partΓ1

partφ

E effan = `2Γ 2

2 + `2 cos2 θ

(partΓ2

partφ

+ E ribbonan

2(m middot e1)

2 minus Q3

2(m middot e3)

2 (10c)

Q1 =K1

2πM2s

+Qr Qr =h

πwlnw

h Q3 = minus1 +

2πM2s

+ 2Qr (10d)

E effex = `2

)2+ `2

[sin θ

(Ψprime + κeff

E effan = minusQ

2cos2 θ

αprime + τ

2 Qeff

3 = Q3 minus 2`2 (αprime + τ)2 (13)

Magnetic moment

Ribbon

Local basis

0 1 2 30

RibbonStraight wire

2 Helicoid ribbon

ς(s v) = x v cos

)+ y Cv sin

)+ z s v isin

[minusw

κeff = 0 τ eff =C

s0 Qeff

1 = Q1 minus 2

Qeff3 = Q3 minus 2

E effex =

[(s0θ

E effan = minusQ

2cos2 θ

2cos2 φ

θt =π

2 cosφt = C (16)

θn =π

2 φn = C

θ = θt + ϑ(χ t) φ = φt + ϕ(χ t) |ϑ| |ϕ| 1

)2 [(partχϑ)

2+ (partχϕ)

[Q1 minusQ3 + 2

)2]ϑ2

+ U (17)

U = minus1

2minus 1

Q3 (18)

Ω(q) = minus2CCq +

radic[q2 + 1 +

Q1 minusQ3

)2] [q2 +

)2] (20)

3 Mobius ribbon

ς(χ v) =(R+ v cos

)cosχ x+

(R+ v cos

)sinχ y + Cv sin

2z (22)

E = 4πM2s hwR

2πint0

)2 [Cpartχθ +

2sinφ+ cosφ sin

)2 [sin θ

(partχφminus cos

)+C cos θ

minus Qeff1

2cos2 θ

Qeff1 = Q1 minus

Qeff3 = Q3 minus

1 + 2 sin2 χ2

1 + 4 sin2 χ2

τ eff = minus C

radic1 + 4 sin2 χ

05 1 15 2

minus06

minus04

minus02

Ribbon state

Vortex state

0 π2 π 3π2 2π0

Azimuthal angle χ

k = kck = 1

4 magnetostatics

(a) (b)

(c) (d)

2sinφ+ cosφ sin

minus Qeff1

2cos2 φ

partχχφ+ sinχ

2sin2φ+

(sin2 χ

2minus k)

sinφ cosφ = 0

k =Qeff

4πM2s

2πw`2lnw

hminus 1

4 (24)

θvor =π

2 cosφvor = C

φrib+ (χ) =

infinsumn=1

kc asymp 16934 (26)

2M2s hwR

2πint0

Edχminus 5

4πM2S

R2minus h

2πwlnw

2πw`2lnw

hlt kc +

4 Conclusion

Acknowledgments

w2 le h le w 1

δ equiv h

Estripms = minusM

|r minus rprime|

Estripms =

intTcupB

|r minus rprime|

intFcupR

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

= f(L) ln(Lminus s+

)+ f(0) ln

radics2 + a2

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



emicro =partmicroς

h wL (7)

Eshell = 4πM2s h

w2intminusw2

radicgdv

(E shell

ex + E shellan

)+ Eshell

ms (8a)

partΓ (φ)

partφ

cosφsinφ

) The next term in

4πM2s

(m middot e3)2 (8c)

`le h w ` L (9)

microν

∥∥ =

Eribbon = 4πM2s hw

intds(E eff

ex + E effan

partΓ1

partφ

E effan = `2Γ 2

2 + `2 cos2 θ

(partΓ2

partφ

+ E ribbonan

2(m middot e1)

2 minus Q3

2(m middot e3)

2 (10c)

Q1 =K1

2πM2s

+Qr Qr =h

πwlnw

h Q3 = minus1 +

2πM2s

+ 2Qr (10d)

E effex = `2

)2+ `2

[sin θ

(Ψprime + κeff

E effan = minusQ

2cos2 θ

αprime + τ

2 Qeff

3 = Q3 minus 2`2 (αprime + τ)2 (13)

Magnetic moment

Ribbon

Local basis

0 1 2 30

RibbonStraight wire

2 Helicoid ribbon

ς(s v) = x v cos

)+ y Cv sin

)+ z s v isin

[minusw

κeff = 0 τ eff =C

s0 Qeff

1 = Q1 minus 2

Qeff3 = Q3 minus 2

E effex =

[(s0θ

E effan = minusQ

2cos2 θ

2cos2 φ

θt =π

2 cosφt = C (16)

θn =π

2 φn = C

θ = θt + ϑ(χ t) φ = φt + ϕ(χ t) |ϑ| |ϕ| 1

)2 [(partχϑ)

2+ (partχϕ)

[Q1 minusQ3 + 2

)2]ϑ2

+ U (17)

U = minus1

2minus 1

Q3 (18)

Ω(q) = minus2CCq +

radic[q2 + 1 +

Q1 minusQ3

)2] [q2 +

)2] (20)

3 Mobius ribbon

ς(χ v) =(R+ v cos

)cosχ x+

(R+ v cos

)sinχ y + Cv sin

2z (22)

E = 4πM2s hwR

2πint0

)2 [Cpartχθ +

2sinφ+ cosφ sin

)2 [sin θ

(partχφminus cos

)+C cos θ

minus Qeff1

2cos2 θ

Qeff1 = Q1 minus

Qeff3 = Q3 minus

1 + 2 sin2 χ2

1 + 4 sin2 χ2

τ eff = minus C

radic1 + 4 sin2 χ

05 1 15 2

minus06

minus04

minus02

Ribbon state

Vortex state

0 π2 π 3π2 2π0

Azimuthal angle χ

k = kck = 1

4 magnetostatics

(a) (b)

(c) (d)

2sinφ+ cosφ sin

minus Qeff1

2cos2 φ

partχχφ+ sinχ

2sin2φ+

(sin2 χ

2minus k)

sinφ cosφ = 0

k =Qeff

4πM2s

2πw`2lnw

hminus 1

4 (24)

θvor =π

2 cosφvor = C

φrib+ (χ) =

infinsumn=1

kc asymp 16934 (26)

2M2s hwR

2πint0

Edχminus 5

4πM2S

R2minus h

2πwlnw

2πw`2lnw

hlt kc +

4 Conclusion

Acknowledgments

w2 le h le w 1

δ equiv h

Estripms = minusM

|r minus rprime|

Estripms =

intTcupB

|r minus rprime|

intFcupR

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

= f(L) ln(Lminus s+

)+ f(0) ln

radics2 + a2

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



microν

∥∥ =

Eribbon = 4πM2s hw

intds(E eff

ex + E effan

partΓ1

partφ

E effan = `2Γ 2

2 + `2 cos2 θ

(partΓ2

partφ

+ E ribbonan

2(m middot e1)

2 minus Q3

2(m middot e3)

2 (10c)

Q1 =K1

2πM2s

+Qr Qr =h

πwlnw

h Q3 = minus1 +

2πM2s

+ 2Qr (10d)

E effex = `2

)2+ `2

[sin θ

(Ψprime + κeff

E effan = minusQ

2cos2 θ

αprime + τ

2 Qeff

3 = Q3 minus 2`2 (αprime + τ)2 (13)

Magnetic moment

Ribbon

Local basis

0 1 2 30

RibbonStraight wire

2 Helicoid ribbon

ς(s v) = x v cos

)+ y Cv sin

)+ z s v isin

[minusw

κeff = 0 τ eff =C

s0 Qeff

1 = Q1 minus 2

Qeff3 = Q3 minus 2

E effex =

[(s0θ

E effan = minusQ

2cos2 θ

2cos2 φ

θt =π

2 cosφt = C (16)

θn =π

2 φn = C

θ = θt + ϑ(χ t) φ = φt + ϕ(χ t) |ϑ| |ϕ| 1

)2 [(partχϑ)

2+ (partχϕ)

[Q1 minusQ3 + 2

)2]ϑ2

+ U (17)

U = minus1

2minus 1

Q3 (18)

Ω(q) = minus2CCq +

radic[q2 + 1 +

Q1 minusQ3

)2] [q2 +

)2] (20)

3 Mobius ribbon

ς(χ v) =(R+ v cos

)cosχ x+

(R+ v cos

)sinχ y + Cv sin

2z (22)

E = 4πM2s hwR

2πint0

)2 [Cpartχθ +

2sinφ+ cosφ sin

)2 [sin θ

(partχφminus cos

)+C cos θ

minus Qeff1

2cos2 θ

Qeff1 = Q1 minus

Qeff3 = Q3 minus

1 + 2 sin2 χ2

1 + 4 sin2 χ2

τ eff = minus C

radic1 + 4 sin2 χ

05 1 15 2

minus06

minus04

minus02

Ribbon state

Vortex state

0 π2 π 3π2 2π0

Azimuthal angle χ

k = kck = 1

4 magnetostatics

(a) (b)

(c) (d)

2sinφ+ cosφ sin

minus Qeff1

2cos2 φ

partχχφ+ sinχ

2sin2φ+

(sin2 χ

2minus k)

sinφ cosφ = 0

k =Qeff

4πM2s

2πw`2lnw

hminus 1

4 (24)

θvor =π

2 cosφvor = C

φrib+ (χ) =

infinsumn=1

kc asymp 16934 (26)

2M2s hwR

2πint0

Edχminus 5

4πM2S

R2minus h

2πwlnw

2πw`2lnw

hlt kc +

4 Conclusion

Acknowledgments

w2 le h le w 1

δ equiv h

Estripms = minusM

|r minus rprime|

Estripms =

intTcupB

|r minus rprime|

intFcupR

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

= f(L) ln(Lminus s+

)+ f(0) ln

radics2 + a2

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



Magnetic moment

Ribbon

Local basis

0 1 2 30

RibbonStraight wire

2 Helicoid ribbon

ς(s v) = x v cos

)+ y Cv sin

)+ z s v isin

[minusw

κeff = 0 τ eff =C

s0 Qeff

1 = Q1 minus 2

Qeff3 = Q3 minus 2

E effex =

[(s0θ

E effan = minusQ

2cos2 θ

2cos2 φ

θt =π

2 cosφt = C (16)

θn =π

2 φn = C

θ = θt + ϑ(χ t) φ = φt + ϕ(χ t) |ϑ| |ϕ| 1

)2 [(partχϑ)

2+ (partχϕ)

[Q1 minusQ3 + 2

)2]ϑ2

+ U (17)

U = minus1

2minus 1

Q3 (18)

Ω(q) = minus2CCq +

radic[q2 + 1 +

Q1 minusQ3

)2] [q2 +

)2] (20)

3 Mobius ribbon

ς(χ v) =(R+ v cos

)cosχ x+

(R+ v cos

)sinχ y + Cv sin

2z (22)

E = 4πM2s hwR

2πint0

)2 [Cpartχθ +

2sinφ+ cosφ sin

)2 [sin θ

(partχφminus cos

)+C cos θ

minus Qeff1

2cos2 θ

Qeff1 = Q1 minus

Qeff3 = Q3 minus

1 + 2 sin2 χ2

1 + 4 sin2 χ2

τ eff = minus C

radic1 + 4 sin2 χ

05 1 15 2

minus06

minus04

minus02

Ribbon state

Vortex state

0 π2 π 3π2 2π0

Azimuthal angle χ

k = kck = 1

4 magnetostatics

(a) (b)

(c) (d)

2sinφ+ cosφ sin

minus Qeff1

2cos2 φ

partχχφ+ sinχ

2sin2φ+

(sin2 χ

2minus k)

sinφ cosφ = 0

k =Qeff

4πM2s

2πw`2lnw

hminus 1

4 (24)

θvor =π

2 cosφvor = C

φrib+ (χ) =

infinsumn=1

kc asymp 16934 (26)

2M2s hwR

2πint0

Edχminus 5

4πM2S

R2minus h

2πwlnw

2πw`2lnw

hlt kc +

4 Conclusion

Acknowledgments

w2 le h le w 1

δ equiv h

Estripms = minusM

|r minus rprime|

Estripms =

intTcupB

|r minus rprime|

intFcupR

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

= f(L) ln(Lminus s+

)+ f(0) ln

radics2 + a2

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



2cos2 φ

θt =π

2 cosφt = C (16)

θn =π

2 φn = C

θ = θt + ϑ(χ t) φ = φt + ϕ(χ t) |ϑ| |ϕ| 1

)2 [(partχϑ)

2+ (partχϕ)

[Q1 minusQ3 + 2

)2]ϑ2

+ U (17)

U = minus1

2minus 1

Q3 (18)

Ω(q) = minus2CCq +

radic[q2 + 1 +

Q1 minusQ3

)2] [q2 +

)2] (20)

3 Mobius ribbon

ς(χ v) =(R+ v cos

)cosχ x+

(R+ v cos

)sinχ y + Cv sin

2z (22)

E = 4πM2s hwR

2πint0

)2 [Cpartχθ +

2sinφ+ cosφ sin

)2 [sin θ

(partχφminus cos

)+C cos θ

minus Qeff1

2cos2 θ

Qeff1 = Q1 minus

Qeff3 = Q3 minus

1 + 2 sin2 χ2

1 + 4 sin2 χ2

τ eff = minus C

radic1 + 4 sin2 χ

05 1 15 2

minus06

minus04

minus02

Ribbon state

Vortex state

0 π2 π 3π2 2π0

Azimuthal angle χ

k = kck = 1

4 magnetostatics

(a) (b)

(c) (d)

2sinφ+ cosφ sin

minus Qeff1

2cos2 φ

partχχφ+ sinχ

2sin2φ+

(sin2 χ

2minus k)

sinφ cosφ = 0

k =Qeff

4πM2s

2πw`2lnw

hminus 1

4 (24)

θvor =π

2 cosφvor = C

φrib+ (χ) =

infinsumn=1

kc asymp 16934 (26)

2M2s hwR

2πint0

Edχminus 5

4πM2S

R2minus h

2πwlnw

2πw`2lnw

hlt kc +

4 Conclusion

Acknowledgments

w2 le h le w 1

δ equiv h

Estripms = minusM

|r minus rprime|

Estripms =

intTcupB

|r minus rprime|

intFcupR

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

= f(L) ln(Lminus s+

)+ f(0) ln

radics2 + a2

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



3 Mobius ribbon

ς(χ v) =(R+ v cos

)cosχ x+

(R+ v cos

)sinχ y + Cv sin

2z (22)

E = 4πM2s hwR

2πint0

)2 [Cpartχθ +

2sinφ+ cosφ sin

)2 [sin θ

(partχφminus cos

)+C cos θ

minus Qeff1

2cos2 θ

Qeff1 = Q1 minus

Qeff3 = Q3 minus

1 + 2 sin2 χ2

1 + 4 sin2 χ2

τ eff = minus C

radic1 + 4 sin2 χ

05 1 15 2

minus06

minus04

minus02

Ribbon state

Vortex state

0 π2 π 3π2 2π0

Azimuthal angle χ

k = kck = 1

4 magnetostatics

(a) (b)

(c) (d)

2sinφ+ cosφ sin

minus Qeff1

2cos2 φ

partχχφ+ sinχ

2sin2φ+

(sin2 χ

2minus k)

sinφ cosφ = 0

k =Qeff

4πM2s

2πw`2lnw

hminus 1

4 (24)

θvor =π

2 cosφvor = C

φrib+ (χ) =

infinsumn=1

kc asymp 16934 (26)

2M2s hwR

2πint0

Edχminus 5

4πM2S

R2minus h

2πwlnw

2πw`2lnw

hlt kc +

4 Conclusion

Acknowledgments

w2 le h le w 1

δ equiv h

Estripms = minusM

|r minus rprime|

Estripms =

intTcupB

|r minus rprime|

intFcupR

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

= f(L) ln(Lminus s+

)+ f(0) ln

radics2 + a2

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



05 1 15 2

minus06

minus04

minus02

Ribbon state

Vortex state

0 π2 π 3π2 2π0

Azimuthal angle χ

k = kck = 1

4 magnetostatics

(a) (b)

(c) (d)

2sinφ+ cosφ sin

minus Qeff1

2cos2 φ

partχχφ+ sinχ

2sin2φ+

(sin2 χ

2minus k)

sinφ cosφ = 0

k =Qeff

4πM2s

2πw`2lnw

hminus 1

4 (24)

θvor =π

2 cosφvor = C

φrib+ (χ) =

infinsumn=1

kc asymp 16934 (26)

2M2s hwR

2πint0

Edχminus 5

4πM2S

R2minus h

2πwlnw

2πw`2lnw

hlt kc +

4 Conclusion

Acknowledgments

w2 le h le w 1

δ equiv h

Estripms = minusM

|r minus rprime|

Estripms =

intTcupB

|r minus rprime|

intFcupR

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

= f(L) ln(Lminus s+

)+ f(0) ln

radics2 + a2

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



k =Qeff

4πM2s

2πw`2lnw

hminus 1

4 (24)

θvor =π

2 cosφvor = C

φrib+ (χ) =

infinsumn=1

kc asymp 16934 (26)

2M2s hwR

2πint0

Edχminus 5

4πM2S

R2minus h

2πwlnw

2πw`2lnw

hlt kc +

4 Conclusion

Acknowledgments

w2 le h le w 1

δ equiv h

Estripms = minusM

|r minus rprime|

Estripms =

intTcupB

|r minus rprime|

intFcupR

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

= f(L) ln(Lminus s+

)+ f(0) ln

radics2 + a2

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



4πM2S

R2minus h

2πwlnw

2πw`2lnw

hlt kc +

4 Conclusion

Acknowledgments

w2 le h le w 1

δ equiv h

Estripms = minusM

|r minus rprime|

Estripms =

intTcupB

|r minus rprime|

intFcupR

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

= f(L) ln(Lminus s+

)+ f(0) ln

radics2 + a2

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



w2 le h le w 1

δ equiv h

Estripms = minusM

|r minus rprime|

Estripms =

intTcupB

|r minus rprime|

intFcupR

dsprimew2intminusw2

w2intminusw2

[m3(s)m3(sprime)

ρ2 + h2

dsprimeh2intminush2

h2intminush2

[m2(s)m2(sprime)

ρ2 + h2

where m(s) = 1wh

= f(L) ln(Lminus s+

)+ f(0) ln

radics2 + a2

f prime(sprime) ln

(|sminus sprime|+

)dsprime

f prime(sprime) ln

(|sminus sprime|+

)dsprime

Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



Estripms

12intminus12

dv m23(s)

[lnradic

12intminus12

dv m22(s)

[lnradic

+ O(wh2

∣∣ln h∣∣) (A1)

Estripms

(2 arctan

δ+ δ ln δ +

2δminus δ

)ln(1 + δ2)

) Lint0

m23(s) ds

(minusδ ln δ +

δarctan δ +

2minus 1

)ln(1 + δ2)

) Lint0

m22(s) ds

Estripms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

] s isin [0 L]

(s v u) = γ(s) + ve2(s) + ue3(s v)

Eribbonms

12intminus12

|uminus v|

12intminus12

dv m22(s)

[lnradic

REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



REFERENCES 15

Eribbonms = 2πM2

πln δ

3(s)minus δ

πln δ m2

]ds+ O(δ)

H = minusa`2Nsumi=1

Nsumi=1

[Q1(mi middot e1i)

sumi6=j

[(mi middotmj)

References

REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



REFERENCES 16

1039c3sm50605c

PhysRevE90022713

1751-812148i=12a=125202

PhysRevB85144433

REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



REFERENCES 17

1142S021820250500087X

doi101098rspa19970013

abs160908040

104202

15R985



2 Helicoid ribbon


3 Moumlbius ribbon

4 Conclusion



Documents

Gaididei, Y., Goussev, A., Kravchuk, V. P., …...Magnetization in narrow ribbons: curvature e ects Yuri Gaididei 1, Arseni Goussev2, Volodymyr P. Kravchuk , Oleksandr V. Pylypovskyi3,