Table Of ContentMagnetization in narrow ribbons: curvature effects
7 Yuri Gaididei1, Arseni Goussev2, Volodymyr P. Kravchuk1,
1
Oleksandr V. Pylypovskyi3, J. M. Robbins4,
0
Denis D. Sheka3, Valeriy Slastikov4, and
2
Sergiy Vasylkevych4
n
1 BogolyubovInstituteforTheoreticalPhysicsofNationalAcademyofSciences
a
J ofUkraine,03680Kyiv,Ukraine
2 DepartmentofMathematics,PhysicsandElectricalEngineering,Northumbria
6
University,NewcastleuponTyne,NE18ST,UK
3 TarasShevchenkoNationalUniversityofKyiv,01601Kyiv,Ukraine
] 4 SchoolofMathematics,UniversityofBristol,Bristol,BS81TW,UK
l
l
a E-mail: [email protected],[email protected],
h [email protected],[email protected],[email protected],
- [email protected],[email protected]
s
e
m Abstract.
A ribbon is a surface swept out by a line segment turning as it moves along
.
t a central curve. For narrow magnetic ribbons, for which the length of the line
a
segmentismuchlessthanthelengthofthecurve,theanisotropyinducedbythe
m
magnetostaticinteractionisbiaxial,withhardaxisnormaltotheribbonandeasy
- axisalongthecentralcurve. Themicromagneticenergyofanarrowribbonreduces
d tothatofaone-dimensionalferromagneticwire,butwithcurvature,torsionand
n localanisotropymodifiedbytherateofturning. Thesegeneralresultsareapplied
o totwoexamples,namelyahelicoidribbon,forwhichthecentralcurveisastraight
c line,andaMo¨biusribbon,forwhichthecentralcurveisacircleaboutwhichthe
[ linesegmentexecutesa180◦twist. Inbothexamples,forlargepositivetangential
anisotropy, the ground state magnetization lies tangent to the central curve. As
1 thetangentialanisotropyisdecreased,thegroundstatemagnetizationundergoes
v atransition,acquiringanin-surfacecomponentperpendiculartothecentralcurve.
1 Forthehelicoidribbon,thetransitionoccursatvanishinganisotropy,belowwhich
9 thegroundstateisuniformlyperpendiculartothecentralcurve. Thetransition
6 fortheM¨obiusribbonismoresubtle; itoccursatapositivecriticalvalueofthe
1 anisotropy,belowwhichthegroundstateisnonuniform. Forthehelicoidribbon,
0 thedispersionlawforspinwaveexcitationsaboutthetangentialstateisfoundto
. exhibitanasymmetrydeterminedbythegeometricandmagneticchiralities.
1
0
7
1 PACSnumbers: 75.70.-i,75.75.-c,75.10.Hk,75.30.Et
:
v
i
X
Submitted to: J. Phys. A: Math. Gen.
r
a
Magnetization in narrow ribbons: curvature effects 2
Introduction
The emerging area of magnetism in curved geometries encompasses a range of
fascinating geometry-induced effects in the magnetic properties of materials [1].
Theoretical investigations in this area are providing new insights into the behaviour
of curved magnetic nanostructures and the control of their magnetic excitations, with
applications to shapeable magnetoelectronics [2] and prospective energy-efficient data
storage, among others.
In continuum models, the magnetization is represented by a three-dimensional
unit-vector field m(r). The study of curvature–induced effects in vector-field models
in one- and two-dimensional geometries has a rather long history [3–6]. In spite of
numerous results [3–6], the problem is far from being fully solved. In the majority
of these studies, the vector field is taken to be tangent to the domain. In particular,
a general expression for the surface energy of a tangential director field describing
a nematic liquid crystal in a curvilinear shell was recently obtained [7–10], with
possible applications using different geometries and orientational ordering [11–13].
The assumption of a strictly tangential field was also used in a study of the role of
curvature in the interaction between defects in 2D XY-like models, with applications
to superfluids, superconductors, and liquid crystals deposited on curved surfaces [14].
Very recently a fully 3D approach was developed for thin magnetic shells and
wires of arbitrary shape [15, 16]. This approach yields an energy for arbitrary curves
and surfaces and for arbitrary magnetization fields under the assumption that the
anisotropy greatly exceeds the dipolar interaction, so that
(cid:90)
E = dr(E +E ). (1)
ex an
Here E is the exchange energy density and E is the density of effective anisotropy
ex an
interaction. Weconsiderthemodelofisotropicexchange,E =(∇m ) (∇m ),where
ex i i
·
m with i = 1,2,3 describes the cartesian components of magnetization. Therefore
i
in cartesian coordinates, the sample geometry appears only through the anisotropy
term via the spatial variation of the anisotropy axis; for example, in the case of a
uniaxialcurvedmagnet,E isgivenbyK(m e )2,wheretheunitvectore =e (r)
an A A A
·
determines the direction of the easy axis.
Incurvilinearcoordinatesadaptedtothesamplegeometry,thespatialvariationof
the anisotropy axes is automatically accounted for, and the anisotropy energy density
assumes its usual translation-invariant form. Instead, the exchange energy acquires
two additional terms, which describe contributions to (∇m ) (∇m ) due to the
i i
·
spatial variation of the coordinate frame [16], namely curvilinear-geometry-induced
effectiveanisotropyandcurvilinear-geometry-inducedeffectiveDzyaloshinskii–Moriya
interaction. Formagneticshells,thesecontributionsmaybeexpressedintermsoflocal
curvatures [15]; for magnetic wires, in terms of curvature and torsion [16]. Below we
review briefly some manifestations of these contributions, which have been reported
elsewhere.
(i) Curvilinear-geometry-induced effective anisotropy. Geometry-induced
anisotropy can have a significant effect on the ground-state magnetization profile,
rendering it no longer strictly tangential, even in the case of strong easy-tangential
anisotropy. For example, for a helical nanowire with strong anisotropy directed along
thewire,theground-statemagnetizationisalwaystiltedinthelocalrectifyingsurface,
with tilting angle dependent on the product of the curvature and the torsion [17, 18].
For two-dimensional geometries with nontrivial topology, a striking manifestation
Magnetization in narrow ribbons: curvature effects 3
of geometry-induced anisotropy is shape–induced patterning. In spherical shells,
a strictly in-surface magnetization is forbidden due to the hairy–ball theorem [19].
Instead, the ground-state magnetization profile has two oppositely disposed vortices
[20]. Another nontrivial example is the M¨obius ring. Since a M¨obius ring is a
nonorientable surface, its topology forces a discontinuity in any nonvanishing normal
vector field. Recently we proposed that magnetic nanostructures shaped as M¨obius
strips possess non-volatility in their magneto-electric response due to the presence
of topologically protected magnetic domain walls in materials with an out-of-plane
orientation of the easy axis of magnetization [21]. In both of these examples, the link
between surface topology and magnetization is a consequence of geometry–dependent
anisotropy.
(ii) Curvilinear-geometry-induced effective Dzyaloshinskii–Moriya interaction.
Recently, the role of curvature in domain wall pinning was elucidated [22]; a local
bend in a nanowire is the source of a pinning potential for transversal domain
walls. Chiral symmetry-breaking due to a geometry-induced Dzyaloshinskii–Moriya
interactionstronglyimpactsthedomainwalldynamicsandallowsdomainwallmotion
undertheactionofdifferentspin–torques,e.g. field–liketorques[18]andanti–damping
torques[23]. Intheparticularcaseofahelicalnanowire,torsioncanproducenegative
domain wall mobility [18, 23], while curvature can produce a shift in the Walker
breakdown [23].
We have briefly described a theoretical framework for studying different
curvilinear systems, including 1D nanowires and 2D nanoshells. In this approach
wesupposethattheeffectsofnonlocaldipole-dipoleinteractionscanbereducedtoan
effective easy-surface anisotropy. In the 1D case, this reduction has been rigorously
justified in the limit where the diameter of the wire h is much smaller than its length
L[24]. Similarargumentshavebeenprovidedinthe2Dcaseforplanarthinfilms[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 an
infinitesimal neighbourhood of a surface along it [27]. For a narrow ribbon whose
thickness h is much less than its width w, which in turn is much less than its length
L, namely h w L, another micromagnetic limit is realized. We show that the
(cid:28) (cid:28)
micromagneticenergycanbereducedtotheenergyofawirewithmodifiedcurvature,
torsion and anisotropy. We illustrate this approach with two examples, namely a
narrowhelicoidribbonandaM¨obiusribbon. Theexistenceofanewnonhomogeneous
ground state is predicted for the M¨obius ribbon over a range of anisotropy parameter
K. The prediction is confirmed by full scale spin–lattice simulations. We also analyse
the magnon spectrum for a narrow helicoid ribbon: unlike the magnon spectrum for
a straight wire, there appears an asymmetry in the dispersion law caused by the
geometric and magnetic chiralities.
The paper is organized as follows. In Section 1 we derive the micromagnetic
energy for a narrow ribbon, which may be interpreted as a modification of the 1D
micromagneticenergyofitscentralcurve. Weillustratethemodelbytwoexamples,a
helicoidribbon(Section 2)andaM¨obiusribbon(Section3). Concludingremarksare
giveninSection4. Thejustificationofthemagnetostaticenergyforribbonsandstrips
is presented in Appendix A. The spin-lattice simulations are detailed in Appendix B.
Magnetization in narrow ribbons: curvature effects 4
1. Model of narrow ribbon vs thin wire
1.1. 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 [0,L], where L is the length of
∈
the wire. It has been shown [24] that the properties of sufficiently thin ferromagnetic
wires of circular (or square) cross section are described by a reduced one-dimensional
energy given by a sum of exchange and local anisotropy terms,
L
(cid:90)
Ewire =4πM2S ds(cid:0)Ewire+Ewire(cid:1),
s ex an
(2)
0
Q
Eewxire =(cid:96)2|m(cid:48)|2 Eawnire =− 21 (m·et)2.
Here, m(s) denotes the unit magnetization vector, prime (cid:48) denotes derivative with
(cid:112)
respect to s, M is the saturation magnetization, and (cid:96)= A/4πM2 is the exchange
s s
lengthwithAbeingtheexchangeconstant. Thelocalanisotropyisuniaxial,witheasy
axisalongthetangentet =γ(cid:48). Thenormalizedanisotropyconstant(orqualityfactor)
Q incorporates the intrinsic crystalline anisotropy K as well as a geometry-induced
1 1
magnetostatic contribution,
K 1
1
Q = + . (3)
1 2πM2 2
s
Note that the shape-induced biaxial anisotropy is caused by the asymmetry of the
cross-section. In particular, for a rectangular cross-section, the anisotropy coefficients
are determined by Eq. (A.4); for elliptical cross-sections, see [24].
It is convenient to express the magnetization in terms of the Frenet-Serret frame
comprisedofthetangentet,thenormalen =e(cid:48)t/e(cid:48)t ,andthebinormaleb =et en.
| | ×
These satisfy the Frenet-Serret equations,
0 κ 0
e(cid:48)α =Fαβeβ, (cid:107)Fαβ(cid:107)= −κ 0 τ , (4)
0 τ 0
−
where κ(s) and τ(s) are the curvature and torsion of γ(s), respectively. Letting
m=sinΘcosΦet+sinΘsinΦen+cosΘeb,
whereΘ andΦarefunctionsofs(andtimet,ifdynamicsisconsidered),onecanshow
[16] that the exchange and anisotropy energy densities are given by
Ewire =(cid:96)2[Θ(cid:48) τsinΦ]2+(cid:96)2[sinΘ(Φ(cid:48)+κ) τcosΘcosΦ]2,
ex − −
Q
Ewire = 1 sin2Θcos2Φ.
an − 2
1.2. Narrow ferromagnetic ribbon
As above, let γ(s) denote a three-dimensional curve parametrized by arc length.
Following [27], we take a ribbon to be a two-dimensional surface swept out by a line
segment centred at and perpendicular to γ, moving (and possibly turning) along γ
The ribbon may parametrized as
(cid:104) w w(cid:105)
ς(s,v)=γ(s)+vcosα(s)en+vsinα(s)eb, v , , (5)
∈ −2 2
Magnetization in narrow ribbons: curvature effects 5
where w is the width of the segment (assumed to be small enough so that ς has no
self-intersections) and α(s) determines the orientation of the segment with respect to
thenormalandbinormal. Weconstructathree-frame e ,e ,e ontheribbongiven
1 2 3
{ }
by
∂ ς
µ
e = , µ=1,2, e =e e . (6)
µ 3 1 2
∂ ς ×
µ
| |
Here and in what follows, we use Greek letters µ,ν,etc = 1,2 to denote indices
restricted to the ribbon surface. Using the Frenet–Serret equations (4), one can show
that (6) constitute an orthonormal frame, with e and e tangent to the ribbon and
1 2
e normaltoit. Itfollowsthatthefirstfundamentalform(ormetric),g =∂ ς ∂ ς,
3 µν µ ν
·
is diagonal. The second fundamental form, b , is given by b = e ∂2 ς. The
µν µν 3 · µ,ν
Gauß and mean curvatures are given respectively by the determinant and trace of
Hµν = bµν/√gµµgνν .
|| || || ||
We consider a thin ferromagnetic shell about the ribbon of thickness h, where
h w,L. (7)
(cid:28)
The shell is comprised of points ς(s,v)+ue , where u [ h/2,h/2]. We express the
3
∈ −
unit magnetization inside the shell in terms of the frame e as
α
m=sinθcosφe +sinθsinφe +cosθe ,
1 2 3
where θ and φ are functions of the surface coordinates s,v (and time t, for dynamical
problems), but are independent of the transverse coordinate u. The micromagnetic
energy of a thin shell reads
L w/2
(cid:90) (cid:90)
Eshell =4πM2h ds √gdv(cid:0)Eshell+Eshell(cid:1)+Eshell, (8a)
s ex an ms
0 −w/2
where g =det(g ). The exchange energy density in (8a) is given by [15, 16]
µν
(cid:20) (cid:21)2
∂Γ(φ)
Eshell =(cid:96)2[∇θ Γ(φ)]2+(cid:96)2 sinθ(∇φ Ω) cosθ , (8b)
ex − − − ∂φ
where ∇ e denotes a surface del operator in its curvilinear form with
µ µ
≡ ∇
components (g )−1/2∂ , the vector Ω is a spin connection with components
µ µµ µ
∇ ≡ (cid:18) (cid:19)
cosφ
Ω =e e , and the vector Γ(φ) is given by H . The next term in
µ 1·∇µ 2 || µν|| sinφ
the energy functional, Eshell, is the anisotropy energy density of the shell:
an
K K
Eshell = 1 (m e )2 3 (m e )2, (8c)
an −4πM2 · 1 − 4πM2 · 3
s s
whereK andK arethetangentialandnormalanisotropycoefficientsoftheintrinsic
1 3
crystalline anisotropy. The magnetostatic energy, Eshell, has, in the general case, a
ms
nonlocal form. The local form is restored in the limit of thin films [28–30] and thin
shells [26, 31].
We proceed to consider the narrow-ribbon limit,
w2
h w (cid:96)(cid:46)L. (9)
(cid:96) ≤ (cid:28) (cid:28)
Magnetization in narrow ribbons: curvature effects 6
Keeping leading-order terms in w/L we obtain that the geometrical properties of
ribbon are determined by
(cid:13) (cid:13) (cid:13) (cid:13) (cid:18) κsinα α(cid:48)+τ (cid:19)
(cid:13)gribbon(cid:13)=diag(1,1), (cid:13)Hribbon(cid:13)= − ,
µν µν α(cid:48)+τ 0
In the same way, we obtain from (8) the following:
(cid:90)
Eribbon =4πM2hw ds (cid:0)Eeff +Eeff(cid:1),
s ex an
(cid:20) (cid:21)2
∂Γ
Eeexff =(cid:96)2(θ(cid:48)−Γ1)2+(cid:96)2 sinθ(φ(cid:48)−Ω1)−cosθ ∂φ1 , (10a)
(cid:18) (cid:19)2
∂Γ
Eeff =(cid:96)2Γ2+(cid:96)2cos2θ 2 +Eribbon,
an 2 ∂φ an
where the effective spin connection Ω and vector Γ are given by
1
Ω = κcosα, Γ = κsinαcosφ+(α(cid:48)+τ)sinφ, Γ =(α(cid:48)+τ)cosφ. (10b)
1 1 2
− −
The last term in the energy density, Eribbon, is the effective anisotropy energy density
an
ofthenarrowribbon. Usingargumentssimilartothosein[26,28,30],itcanbeshown
that
Q Q
Eribbon = 1 (m e )2 3 (m e )2. (10c)
an − 2 · 1 − 2 · 3
Here Q and Q incorporate the intrinsic crystalline anisotropies K and K as well
1 3 1 3
as geometry-induced magnetostatic contributions:
K h w K
Q = 1 +Q , Q = ln , Q = 1+ 3 +2Q , (10d)
1 2πM2 r r πw h 3 − 2πM2 r
s s
see the justification in Appendix A. In the particular case of soft magnetic materials,
where K = K = 0, the anisotropy Eribbon is due entirely to the magnetostatic
1 3 an
interaction. From (10d), we get Q =Q 1 and Q = 1+2Q .
1 r 3 r
(cid:28) −
The induced anisotropy is biaxial, with easy axis along the central curve as for a
thin wire (cf (3)) and hard axis normal to the surface as for a thin shell. Indeed, one
can recast the narrow-ribbon energy (10) in the form of the thin-wire energy (2) with
biaxial anisotropy, as follows:
Eeff =(cid:96)2(cid:0)θ(cid:48) τeffsinΨ(cid:1)2+(cid:96)2(cid:104)sinθ(cid:0)Ψ(cid:48)+κeff(cid:1) τeffcosθcosΨ(cid:105)2,
ex − −
(11)
Qeff Qeff
Eeff = 1 sin2θcos2φ 3 cos2θ.
an − 2 − 2
In (11), the effective curvature and torsion are given by
(cid:113)
κeff =κcosα β(cid:48), τeff = κ2sin2β+(α(cid:48)+τ)2, (12)
−
the angle Ψ is defined by
κsinα
Ψ=φ+β, tanβ = ,
−α(cid:48)+τ
and the effective anisotropies are given by
Qeff =Q 2(cid:96)2(α(cid:48)+τ)2, Qeff =Q 2(cid:96)2(α(cid:48)+τ)2. (13)
1 1− 3 3−
Magnetization in narrow ribbons: curvature effects 7
(a)Thinmagneticribbon (b)Dispersioncurve
Ribbon
Straightwire
Localbasis 10
e
e 3
2
e1 Ω
y
c
n
e e
3 u
q
re 5
f
d
e1 ce
u
Magneticmoment e2 ed
R
1
0
Ribbon 0 1 2 3
Reducedwavenumberq
Figure 1. Magnetic helicoid ribbon: (a) A sketch of the ribbon.
(b) Dispersion curve according to Eq. (20) (solid blue line) in comparison with
thedispersionofthestraightwireΩstr=1+q2.
2. Helicoid ribbon
The helicoid ribbon has a straight line, which has vanishing curvature and torsion, as
its central curve. We take γ(s) = szˆ. The rate of turning about γ is constant, and
we take α(s)=Cs/s , where the chirality C is +1 for a right-handed helicoid and 1
0
−
for a left-handed helicoid. From (5), the parametrized surface is given by
(cid:18) s (cid:19) (cid:18) s (cid:19) (cid:104) w w(cid:105)
ς(s,v)=xˆvcos +yˆCvsin +zˆs, v , .
s s ∈ −2 2
0 0
Theboundarycurves,givenbyς(s, w/2),arehelices,seeFig.1(a). Itiswellknown
±
that the curvature and torsion essentially influence the spin-wave dynamics in a helix
wire, acting as an effective magnetic field [17]. One can expect similar behaviour in a
helicoid ribbon.
From (12) and (13), the effective curvature, torsion and anisotropies are given by
C (cid:18) (cid:96) (cid:19)2 (cid:18) (cid:96) (cid:19)2
κeff =0, τeff = , Qeff =Q 2 , Qeff =Q 2 . (14)
s 1 1− s 3 3− s
0 0 0
From (11), the energy density is given by
Eeff = (cid:96)2 (cid:2)(s θ(cid:48) Csinφ)2+(s sinθφ(cid:48) Ccosθcosφ)2(cid:3),
ex s2 0 − 0 −
0 (15)
Qeff Qeff
Eeff = 1 sin2θcos2φ 3 cos2θ.
an − 2 − 2
Let us consider the particular case of soft magnetic materials (K = K = 0).
1 3
Under the reasonable assumption (cid:96) s , we see that Q 1, so that the easy-
0 3
(cid:28) ≈ −
surface anisotropy dominates the energy density and acts as an in-surface constraint.
Magnetization in narrow ribbons: curvature effects 8
Taking θ = π/2 to accommodate this constraint, we obtain the (further) reduced
energy density
Q
Eeff =(cid:96)2φ(cid:48)2 1 cos2φ,
− 2
whichdependsonlyonthein-surfaceorientationφ. Thegroundstateshaveφconstant,
with orientation depending on the sign of the tangential-axis anisotropy Q . For
1
Q >0, the ground states are
1
π
t t
θ = , cosφ =C, (16)
2
where the magnetochirality C = 1 determines whether the magnetisation m is
±
parallel (C = 1) or antiparallel (C = 1) to the helicoid axis. For Q < 0, the
1
−
ground states are given by
π π
n n
θ = , φ =C ,
2 2
where the magnetochirality C = 1 determines whether the magnetisation m is
±
parallel (C = 1) or antiparallel (C = 1) to the normal en. This behaviour is similar
−
to that of a ferromagnetic helical wire, which was recently studied in Ref. [17].
2.1. 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,
| | | |(cid:28)
where χ = s/s and t˜ = Ω t with Ω = (2γ /M )((cid:96)/s )2. Expanding the energy
0 0 0 0 s 0
density (15) to quadratic order in the ϑ and ϕ, we obtain
(cid:18) (cid:96) (cid:19)2(cid:104) (cid:105) (cid:18) (cid:96) (cid:19)2
E = (∂ ϑ)2+(∂ ϕ)2 +2CC (ϑ∂ ϕ ϕ∂ ϑ)
χ χ χ χ
s s −
0 0
(cid:34) (cid:18) (cid:96) (cid:19)2(cid:35)ϑ2 ϕ2
+ Q Q +2 +Q .
1 3 1
− s 2 2
0
The linearised Landau–Lifshits equations have the form of a generalized Schr¨odinger
equation for the complex-valued function ψ =ϑ+iϕ [17],
i∂ ψ =Hψ+Wψ∗, H =( i∂ A)2+U, (17)
− t˜ − χ−
where the “potentials” have the following form:
1 1(cid:16)s (cid:17)2 1 1(cid:16)s (cid:17)2
U = + 0 (2Q Q ), A= CC, W = 0 Q . (18)
1 3 3
−2 4 (cid:96) − − 2 − 4 (cid:96)
We look for plane wave solutions of (17) of the form
ψ(χ,t˜)=ueiΦ+ve−iΦ, Φ=qχ Ωt˜+η, (19)
−
whereq =ks isadimensionlesswavenumber,Ω=ω/Ω isadimensionlessfrequency,
0 0
η is an arbitrary phase, and u,v R are constant amplitudes. By substituting (19)
∈
into the generalized Schr¨odinger equation (17), we obtain
(cid:115)
(cid:20) Q Q (cid:16)s (cid:17)2(cid:21)(cid:20) Q (cid:16)s (cid:17)2(cid:21)
Ω(q)= 2CCq+ q2+1+ 1− 3 0 q2+ 1 0 , (20)
− 2 (cid:96) 2 (cid:96)
Magnetization in narrow ribbons: curvature effects 9
see Fig. 1 (b), in which the parameters have the following values: s /(cid:96)=5, Q =0.2,
0 1
Q = 0.6, and C = C = 1. The dispersion relation (20) for the helicoid ribbon is
3
−
similar to that of a helical wire [17], but different from that of a straight wire, in that
it is not reflection-symmetric in q. The sign of the asymmetry is determined by the
product of the helicoid chirality C, which depends on the topology of the ribbon, and
the magnetochirality C, which depends on the topology of the magnetic structure.
This asymmetry stems from the curvature-induced effective Dzyaloshinskii–Moriya
interaction, which is the source of the vector potential A=Aet, where A= CC. In
−
thiscontext,itisinstructivetomentionarelationbetweentheDzyaloshinskii–Moriya
interaction and the Berry phase [32].
3. M¨obius ribbon
In this section we consider a narrow M¨obius ribbon. The M¨obius ring was studied
previously in Ref. [21]. The ground state is determined by the relationship between
geometrical and magnetic parameters. The vortex configuration is favorable in the
small anisotropy case, while a topologically protected domain wall is the ground state
for large easy-normal anisotropy. Although the problem was studied for a wide range
of parameters, the limit of a narrow ribbon was not considered previously. Below we
show that the narrow M¨obius ribbon exhibits a new inhomogeneous ground state, see
Fig. 2 (a), (b).
The M¨obius ribbon has a circle as its central curve and turns at a constant rate,
making a half-twist once around the circle; it can be formed by joining the ends of a
helicoid ribbon. Letting R denote the radius, we use the angle χ=s/R instead of arc
length s as parameter, and set
γ(χ)=Rcosχxˆ+Rsinχyˆ, α(χ)=π Cχ/2. (21)
−
The chirality C= 1 determines whether the M¨obius ribbon is right- or left-handed.
±
From (5), the parametrized surface is given by
(cid:16) χ(cid:17) (cid:16) χ(cid:17) χ
ς(χ,v)= R+vcos cosχxˆ+ R+vcos sinχyˆ+Cvsin zˆ. (22)
2 2 2
Here χ [0,2π) is the azimuthal angle and v [ w/2,w/2] is the position
∈ ∈ −
along the ring width. From (10) the energy of the narrow M¨obius ribbon reads
2π
E =4πM2hwR(cid:82) Edχ, where the energy density is given by
s
0
(cid:18)(cid:96)(cid:19)2(cid:20) 1 χ(cid:21)2 (cid:18)(cid:96)(cid:19)2(cid:104) (cid:16) χ(cid:17)
E = C∂ θ+ sinφ+cosφsin + sinθ ∂ φ cos
χ χ
R 2 2 R − 2
(cid:18)1 χ(cid:19)(cid:21)2 Qeff Qeff
+Ccosθ cosφ sinφsin 1 sin2θcos2φ 3 cos2θ.
2 − 2 − 2 − 2
with effective anisotropies (cf (10d))
(cid:18) (cid:19)2 (cid:18) (cid:19)2
1 (cid:96) 1 (cid:96)
Qeff =Q , Qeff =Q .
1 1− 2 R 3 3− 2 R
The effective curvature and torsion are given by (cf (12))
2cosχ 1+2sin2 χ C (cid:114) χ
κeff = 2 2, τeff = 1+4sin2 .
− R 1+4sin2 χ −2R 2
2
Magnetization in narrow ribbons: curvature effects 10
e e
1 2
(a) (b)
e
3 e
3 e
2
e
1
e
2
e
3
e k =k
1 c
k = 1, magnetostatics
(c) (d) 4
0.2
φ
Vortex state e
E∆ 0 ngl 2
n Ribbon state a
ai 0.2 on
g− ti
y a
erg−0.4 etiz 1
n n
E g
0.6 a
− M
0
0.5 1 1.5 kc 2 0 π/2 π 3π/2 2π
Reduced anisotropy coefficient k Azimuthal angle χ
Figure 2. Magnetic M¨obius ribbon: (a) Magnetization distribution for the
ribbonstateinthelaboratoryframe,seeEq.(25). (b)Magnetizationdistribution
for the ribbon state in the ribbon frame. (c) The energy difference between the
vortex and ribbon states; when the reduced anisotropy coefficient k exceeds the
critical value kc, see Eq. (26), the vortex state is favourable, while for k < kc,
the inhomogeneous ribbon state is realized. (d) In-surface magnetization angle
φ in the ribbon state. Lines correspond to Eq. (25) and markers correspond
to SLaSi simulations, see Appendix B for details. Red triangles represent
the simulations with dipolar interaction without magnetocrystalline anisotropy
(k = 0); it corresponds very well to our theoretical result (solid red curve) for
effectiveanisotropyk=1/4inducedbymagnetostatics,seeEq.(3).
Letusconsiderthecaseofuniaxialmagneticmaterials, forwhichK =0. Under
3
the reasonable assumption (cid:96) R, we have that Qeff 1, so that the easy-surface
(cid:28) 3 ≈ −
anisotropy dominates the energy density and acts as an in-surface constraint (as for
the helicoid ribbon). Taking θ =π/2, we obtain the simplified energy density
(cid:18)(cid:96)(cid:19)2(cid:16) χ(cid:17)2 (cid:18)(cid:96)(cid:19)2(cid:18)1 χ(cid:19)2 Qeff
E = ∂ φ cos + sinφ+cosφsin 1 cos2φ,
χ
R − 2 R 2 2 − 2
which depends only on the in-surface orientation φ. The equilibrium magnetization
distribution is described by the following Euler–Lagrange equation
χ (cid:16) χ (cid:17)
∂ φ+sin sin2φ+ sin2 k sinφcosφ=0,
χχ
2 2 − (23)
φ(0)= φ(2π) mod2π, ∂ φ(0)= ∂ φ(2π),
χ χ
− −
where the antiperiodic boundary conditions compensate for the half-twist in the
M¨obiusribbonandensurethatthemagnetisationmissmoothatχ=0. Thereduced
