Nonlinear spin waves in cylindrical ferromagnetic nanowires

Nonlinear spin waves in cylindrical ferromagnetic nanowires

H. Leblond

Laboratoire POMA, Université d’Angers, UMR CNRS 6136, 2 Boulevard Lavoisier, 49045 Angers Cedex, France V. Veerakumar

Center for Magnetism and Magnetic Nanostructures, Department of Physics, University of Colorado at Colorado Springs, Colorado Springs, Colorado 80933, USA

M. Manna

Laboratoire de Physique Théorique et Astroparticules, Université Montpellier II, IN2P3/CNRS-UMR 5207, Place E. Bataillon, 34095 Montpellier Cedex 05, France

共Received 25 September 2006; revised manuscript received 9 March 2007; published 7 June 2007兲 We study the nonlinear evolution of bulk spin waves in a charge-free, isotropic ferromagnetic nanowire with negligible surface anisotropy, restricted to the modes with no azimuthal dependence. Using a multiple scale analysis, we find that the magnetization oscillations are always restricted to one particular plane for the Fourier component p= 1, while the Fourier component p= 2 comes out of the plane. Moreover, the magnetization excitations are governed by the cubic nonlinear Schrödinger equation. We also find that the ferromagnetic nanowire facilitates the propagation of dark solitons with the stable continuous wave.

DOI:10.1103/PhysRevB.75.214413 PACS number共s兲: 75.30.Ds, 75.75.⫹a, 05.45.Yv

I. INTRODUCTION

In recent years, magnetic nanostructures have attracted much attention as they have potential applications in ultrahigh-density memory devices and sensors.^1,2 The mag- netic excitations or spin waves of these nanostructures are best investigated using Brillouin light scattering 共BLS兲 techniques.³In particular, the linear and nonlinear evolutions of magnetostatic spin waves共SWs兲in charge-free and isotro- pic ferromagnetic nanotubes⁴have been recently studied. As a result, the ferromagnetic nanotubes facilitate the propaga- tion of elliptically polarized waves, which induces soliton excitations in the medium governed by the cubic nonlinear Schrödinger共NLS兲equation. The stability of soliton excita- tions is determined by the external field related to the mag- netized nanotube.

Apart from nanotubes, other forms of nanostructures in- clude nanowires. Two-dimensional arrays of nickel nano- wires were fabricated using a two-step electrochemical an- odization process and pulsed electrodeposition.^5,6 These nanowires are uniform in cross section with lengths of about 1␮m and diameters in the range of 30– 55 nm and, hence, can be realized as infinite in length to excellent approxima- tion. The diameter range considered for the isolated nano- wire prompts one to include exchange and dipolar contribu- tions to the excitation energy.

Recently, the linear theory of spin excitations in ferromag- netic nanowires with exchange and dipolar contributions has been studied.⁷ The bulk standing modes of the nanowires were obtained by using the general form of magnetic scalar potential calculated from the Bloch equation of motion with dipole and exchange fields within the nanowire. It showed the existence of discrete spin modes due to the quantization of bulk spin waves accordingly with the usual dipole- exchange theory, which has been successfully validated experimentally.⁸The boundary conditions generate the mix- ing of bulk SW with the surface SW. In Ref.7, this complex

mechanism was taken into account numerically, while Ref.8 gives approximate analytic expressions. The theory consid- ered a cylindrical cross section with the magnetization par- allel to the axis of the wire. A generalization to a magnetiza- tion not parallel to the axis has been proposed.⁹ The continuous description allows one to determine the number of guided modes.¹⁰ Other theoretical approaches have also been considered; using discrete spin lattices,^10–12the validity of which has been confirmed experimentally.¹³ From the nonlinear viewpoint, the existence of solitons in thin mag- netic films has been demonstrated long time ago,¹⁴and spin- wave bullets have been recently observed.¹⁵

In the present paper, we study the nonlinear theory of spin excitations in a long charge-free isotropic cylindrical ferro- magnetic nanowires by including contributions due to the dipolar and exchange fields. In Sec. II, we formulate the model, discuss the various length scales involved by the problem, and derive the linear propagation modes. Section III is devoted to the derivation of a nonlinear Schrödinger equation by means of a multiple scale expansion method, to the discussion of the existence of solitons, and to the descrip- tion of the corresponding wave profiles. The results are com- mented in Sec. IV, while the Appendix contains the details of the derivation.

II. SETTING THE PROBLEM A. Model and dynamical equations

We consider long charge-free isotropic ferromagnetic nanowire with circular cross section and the magnetization parallel to the symmetry axis of the nanowire, thezaxis. We consider that both the exchange and dipolar couplings be- tween the spins are comparable in magnitude, and we use the same macroscopic approach as in Ref. 7. Although a quan- tum theory approach could be thought to be more relevant at the nano scale, the linear theory of spin waves in nanowires

of Ref. 7 has been found to be in good agreement with experiment.⁸ Consequently, we consider that, in the nonlin- ear regime, the spin excitations in the nanowire are governed by the Landau-Lifshitz equation,¹⁶

⳵tM= −␦␮0M∧共H+␤⌬M兲, 共1兲 whereMis the magnetization of the medium,Hthe dipolar field,⌬the Laplacian operator,␦the gyromagnetic ratio,␮0

the magnetic permeability in vacuum,␤=D/MswithD be- ing the exchange stiffness, andMsthe saturation magnetiza- tion. The quantities M and H are rescaled to M/共␦␮0兲, H/共␦␮0兲, so that the coefficient␦␮0in Eq.共1兲is replaced by 1.

The components of the dipolar field satisfy the magneto- static Maxwell equations,

ⵜ·共H+M兲= 0, 共2兲 ⵜ∧H= 0. 共3兲 Within the magnetostatic approximation, the dipolar field can be expressed as the gradient of the magnetic potential ⌽ according to

H=H₀−ⵜ⌽. 共4兲 Here,H₀is a uniform applied magnetic field, directed along thezaxis. The magnetization of the nanowire in the absence of wave is uniform and directed along the symmetry axis, it is, thus, M₀⬅m=共0 , 0 ,Ms兲. Hence, H₀=␣m, and ␣ mea- sures the strength of the applied field in units ofMs.

The boundary conditions are the magnetostatic ones共see detail in Appendix兲. However, since the exchange interaction is taken into account, we have to consider the pinning con- ditions describing the behavior of the magnetization at the surfaces. Using cylindrical coordinates 共r,␪,z兲, they read as^7,17

共⳵rM^␪兲_共r=R兲= 0, 共5兲

共⳵rM^r兲_共r=R兲=␥共M^r兲_共r=R兲, 共6兲 whereR is the radius of the nanowire, and␥ is the surface anisotropy parameter.

B. Length scales

The characteristic lengths of the nanowire are its lengthL, its radiusR, and the exchange length l=

冑

_␤ of the medium.

Typical values are L= 10– 20␮^{m, R}= 15– 250 nm,¹⁸ and l

= 5.8 nm for nickel. HencelⱗRandRⰆL. For spin waves, typical wavelengths ␭ can range from about 1 nm to 100␮m. Let us consider a typical value about ␭

= 0.5␮m; we have clearly lⰆ␭ⰆL. This is typical for a long-wave approximation, according to the reductive pertur- bation method.¹⁹ We introduce a perturbation parameter ␧ such that

␧⬃ l

␭⬃␭

LⰆ1, 共7兲

and take the exchange lengthlas a reference length. Hence, lis of order␧⁰, while the wavelength␭⬃l/␧ is very large,

corresponding to a long wave. The wave profile is accounted for by introducing the slow space variable,

␨⁼␧共z−Vt兲, 共8兲 which involves a characteristic length l/␧, and a velocityV to be determined, the velocity of the wave pulse.

Then we consider nonlinear and dispersive propagation of the wave profile on long distancesL⬃␭/␧⬃l/␧², or equiva- lently its evolution on large timest⬃t₀/␧², wheret₀is some zero-order reference time 共typically, 1 /t0 can be the ferro- magnetic resonance frequency兲. The evolution at such times is accounted for using a slow time variable,

␶=␧²t. 共9兲

Then the derivation operators with respect toxandyremain unchanged, while the derivation operators with respect to z andt become

⳵

⳵^z=␧

⳵

⳵␨^,

⳵

⳵^{t= −V␧}

⳵

⳵␨^+␧2

⳵

⳵␶^{. 共10兲}

With the above numerical values, we get ␭/l⯝86 and L/␭⯝40. According to relation共7兲, it allows us to take the value␧= 1 / 100 for the perturbation parameter. In many situ- ations, comparison between asymptotic models and numeri- cal resolution of the full initial set of equations have shown such a value of the perturbation parameter to be small enough to ensure the validity of the long-wave approxima- tion. The latter will obviously be improved if longer nano- wires are considered.

However, there is some important discrepancy between the present situation and the usual long-wave approximation.

Consider, indeed, the dispersion relation for the spin waves in nanowires, as can be found in Ref.7. As the wave number ktends to zero, the frequency␻tends to a finite limit⍀, with

⍀/M_s a few less than 1. For spin waves in nanowires, long waves does not imply slow oscillations in time. Hence, a standing carrier exp共i⍀t兲 must be introduced, and the long- wave approximation concerns the slow space-time evolution of the envelope of the standing carrier.

We expand, thus, the magnetization about the uniform magnetization M₀ and the scalar potential representing the dipolar field in a series of harmonics given by

M=M₀+␧M1共x,y,␨^,␶兲e^i⍀t+ c.c. +_j

兺

艌2,p

␧^jMj

p共x,y,␨^,␶兲e^ip⍀t, 共11兲

⌽=␧⌽1共x,y,␨^,␶兲e^i⍀t+ c.c. +_j艌2,p

兺

^{␧j⌽jp共x,y,␨,␶兲eip⍀t.}

共12兲 Here, c.c. stands for complex conjugate, ␧ is the small pa- rameter, and ⍀ is the frequency of the discrete standing modes which determine the transverse profile of the the guided spin waves.

C. Linear modes

The Landau equation共1兲at order ␧ yields the following equations.

i⍀M₁^r= −m

冉

^␣−␤

冉

^⌬⬜−r12

冊冊

^{M1␪, 共13兲}

i⍀M₁^␪=m

冉

^{1 +␣−␤}

冉

^⌬⬜−r¹²

冊冊

^{M1r. 共14兲}

Hence, M₁^␪ and M₁^r are eigenfunctions of the operator

共

⌬_⬜

−_r¹2

兲

. Contrarily to Ref. 7which considered the modes with nonzero azimuthal dependence only, we restrict the study to the purely radial modes, i.e., we assume thatM₁^␪andM₁^r do not depend on␪. With this condition, the eigenfunctions of

共

⌬_⬜−_r¹2

兲

are well known, precisely

冉

^⌬⬜−r12

冊

^{J1共␬r兲= −␬2J1共␬r兲, 共15兲}

for any␬^,J1being the Bessel function. We assume␬^{real, so} that J₁共␬r兲 is regular at r= 0 关i.e.,J₁共0兲= 0兴. The transverse modes are, thus, given by

M₁^r= −m共␣⁺␤␬²兲f共␨^,␶兲J1共␬r兲, 共16兲 M₁^␪=i⍀f共␨^,␶兲J1共␬r兲. 共17兲

␬ can be interpreted as a transverse wave vector. Together with the frequency⍀, it satisfies the modal dispersion rela- tion,

⍀²=m²共␣⁺␤␬²兲共1 +␣⁺␤␬²兲. 共18兲 It is worth comparing the dispersion relation 共18兲 to the equivalent relation obtained in Ref.7 关Eq.共16兲 in the cited reference兴, which reads, with the notations of the present paper, as

␤^2m2共␬^2+k2兲³+␤^m2共2 +␣兲共␬^2+k2兲²+共␣共1 +␣兲m²−⍀²

−␤^m2k2兲共␬^2+k2兲=␣^m2k2. 共19兲 It is straightforwardly seen that relation共19兲reduces to Eq.

共18兲 as k= 0, which corresponds to the long-wave approxi- mation we use here. In the case of zero applied magnetic field, Eq.共17兲is similar to Eq.共4兲in Ref.8. It coincides with the dispersion relation of spin waves in a bulk medium共p. 19 of Ref. 20兲, which confirms the interpretation of ␬ ^{as the} transverse wave vector. Assuming azimuthal invariance, we show from Eq.共A4兲that M₁^z= 0. At this point, the first order

␧¹ of the perturbative scheme, involving the harmonics p

= ± 1, has been completed. The evolution law of the function f共␨^,␶兲will be determined at next order.

Using expressions 共16兲 and 共17兲 of M₁^r and M₁^␪ in the pinning conditions共5兲and共6兲, we obtain the two conditions J₁

⬘

共␬^R兲= 0, 共20兲 where the prime denotes the derivative, and

␥J₁共␬R兲= 0. 共21兲 Since a Bessel functionJ₁and its derivativeJ₁

⬘

have no zero in common, we get the condition␥= 0 as a solvability con- dition. In other words, the above ansatz cannot account for nonlinear spin-wave propagation if surface anisotropy is not

neglected. In real materials, surface anisotropy is not always negligible. Further, it has been shown that an effective non- zero pinning parameter may arise in thin stripes, even if the material itself does not present any exchange surface anisotropy.²¹ Moreover, if surface anisotropy is neglected, only the mode with the lowest azimuthal dependence can be excited by a uniform microwave field, as it appears from the linear theory.⁷ Nevertheless, we assume, thus, a completely unpinned surface in what follows. The inclusion of surface anisotropy requires a modification of the ansatz and is left for further investigation.

III. NONLINEAR ANALYSIS A. Nonlinear Schrödinger equation

At order ␧², the equations for the fundamental Fourier componentp= 1 give the solvability condition necessary to the determination of the velocity V. The justification is de- tailed in the AppendixA, together with some further relations of technical importance. In the frame of the reductive pertur- bation method for envelope evolution equations, V is the group velocity of the wave packet. The group velocity vg

=d⍀/dk is easily computed from Eq. 共19兲. It is seen that v_g= 0 at the limit k= 0, this is due to the fact that ⍀has a finite limit. Hence, v= 0 coincides with the value of the group velocity in the long-wave limit.

Nonlinear terms are expected to appear at order␧², for the Fourier componentsp= 0 andp= 2. In fact, the only nonzero component is here

M₂^2,z=1

2m共␣+␤␬²兲f²共J1共␬r兲兲². 共22兲 At order␧³, for the fundamental Fourier componentp= 1, we get after some computation the following equation:

冋

^−⍀2+m2

冉

^{1 +␣−␤}

冉

^⌬⬜−r12

冊冊冉

^␣−␤

冉

^⌬⬜−r12

冊冊 册

^M31,␪

=F, 共23兲

in which we have set

F= 2⍀²⳵␶fJ₁共␬r兲−iC⳵_␨^2fJ1共␬r兲+G, 共24兲

C= −⍀m²

冉

^{2␤共␣+␤␬2兲−}␬^␣2

冊

^{, 共25兲}

G=−i

2 ⍀m²共␣+␤␬²兲f兩f兩²Q, 共26兲 and

Q= −

冉

^{1 +␣−␤}

冉

^⌬⬜−r12

冊冊

^{关␤J1*⌬⬜J12+␤␬2J1兩J1兩2兴}

+␤共␣+␤␬²兲J₁^*⌬_⬜J₁²+共1 +␤␬²兲共␣+␤␬²兲J1兩J1兩². 共27兲 The key problem of the derivation is the reduction of Eq.

共23兲. We consider the scalar product defined by

共␸兩␺兲=

冕

0 R

␸^*共r兲␺共r兲rdr 共28兲

关␬and thusJ₁共␬r兲are real; hence, the complex conjugate in 共28兲 has no influence below兴. Using the properties of the Bessel functions, we show that

共J1共␬r兲兩J1共qr兲兲= 0, 共29兲 ifq⫽␬, and, hence, the eigenmodesJ₁共qr兲, when theqRare the zeros ofJ₁

⬘

, yield an orthogonal family. It is, thus, rea- sonable to assume that both M₃^1,␪ and the right-hand side memberFof共23兲can be expanded on theJ₁共qr兲as

M₃^1,␪=

兺

_q ^{XqJ1共qr兲, F=}

兺

_q ^{FqJ1共qr兲. 共30兲}

Using this expansion, we can show that the solvability con- dition of Eq.共23兲is

共J1共␬r兲兩F兲= 0, 共31兲 which yields

2i⍀²⳵_␶^f+C⳵_␨^2f+Df兩f兩²= 0. 共32兲 The dispersion coefficientCis given by共25兲and

D=1

2⍀m²共␣+␤␬²兲 共J₁共␬^r兲兩Q兲

共J1共␬r兲兩J1共␬r兲兲. 共33兲 Equation共32兲is the NLS, which is well known to be com- pletely integrable by means of the inverse scattering trans- form method.²² Notice the unusual fact that the NLS equa- tion is obtained here not as describing envelope solitons but within a long-wave approximation.

B. Lighthill criterion

Recall that the existence of soliton solutions of the NLS equation 共32兲 is related to sign of the product CD of the coefficients, according to the so-called Lighthill criterion: the 共bright兲 solitons exist ifCD⬎0. In this case, an input con- tinuous wave suffers the modulational instability of Benjamin-Feir²³ type and breaks down into a train of soli- tons. On the other hand, ifCD⬍0, an input continuous wave is stable, while dark solitons can be formed.

It is straightforward that the dispersion parameterCvan- ishes for

␣^{= 2}␤²␬²

␬^{−2− 2}␤^{, 共34兲} and is negative for␣= 0. The sign of the nonlinear coefficient Dis the same as the one of the scalar product 共J1共␬r兲兩Q兲.

According to共27兲, the product J₁共␬r兲Q is an algebraic ex- pression involving the Bessel function J₁共x兲 and its deriva- tives, to be integrated on the interval关0 ,␬R兴, where ␬^{R is} some zero ofJ₁

⬘

. The integrals can be computed numerically.

For the first zero ofJ₁

⬘

^{, we get}

共J1共␬r兲兩Q兲=1

␬^{共0.079 65}␣^{+ 0.020 76}␤␬^{2− 0.2662}␤²␬⁴兲.

共35兲 Hence,D vanishes for a strength of the external field char- acterized by

␣^{= 38.41}␤

R²

冉

^{R␤2− 0.023}

冊

^{. 共36兲}

The sign of the coefficients is shown on Fig.1, which sum- marizes the results: it is seen that bright solitons exist for both the small and the large values of the ratio ␤^{/R2. A} domain of existence of dark solitons, and nonexistence of bright ones, appears for wires of small enough diameter, and a high enough applied field, with both signs of␣. Recall that a negative␣ corresponds to a magnetization and field with opposite directions, which is an unstable configuration in the bulk but can be eventually stable in nanowires of diameter small enough with respect to the exchange length, i.e.,␤/R² large enough.

The same discussion can be done for all linear propaga- tion modes. Recall that we only consider the modes with zero azimuthal dependence. Among them, each mode is char- acterized by a zero of J₁

⬘

, and we will refer to the mode corresponding to the first of these zeros as the fundamental one. Figures2and3present the results of the discussion for modes 1–4. It is seen that, varying the diameter of the wire and the applied field strength, it is possible to select one or several propagation modes for the propagation of bright soli- tons. There are domains in which only the fundamental mode can support solitons. For thick nanowires共Fig.2兲, it occurs at small applied field,␣close to 0, and forR/l= 15– 20, that is, e.g., in a nickel wire with diameter about 180– 240 nm. It can also occur for thinner nanowires, withR/l= 3 – 7, i.e., a diameter about 15– 40 nm共orR/l= 5–7 if negative␣^{are not} allowed兲, in a strong enough applied field 共Fig.3兲. Assume that only one mode, say, mode 1, can form bright solitons.

The most general input wave packet will expand on several modes and then propagate. If we neglect the nonlinear inter- action between modes, all modes except mode 1 suffer a FIG. 1. 共Color online兲The sign of the coefficients of the NLS equations and the domains were either bright or dark solitons exist in the ␣vs ␤/R² plane. We consider here the fundamental mode only共hence,␬Ris the first zero ofJ₁⬘兲.

nonlinear dispersive effect, and are spread out, while mode 1 forms a bright soliton. Then the final wave packet entirely belongs to mode 1: the nonlinear effects transforms the mul- timode waveguide into an effectively monomode one.

In contrast to this situation, the nanowires of large diam- eter allow, in general, propagation of solitons for any modes.

They are multimode even from the nonlinear point of view.

C. Wave profiles The expression of the NLS soliton is

f共␨^,␶兲=p

冑

^2CD sechp

冉

^␨−k⍀^C2␶

冊

^{ei共k␨+共p2−k2兲C/⍀2␶+␸兲,}

共37兲 p,kand␸are arbitrary real parameters, which represent the inverse of the wave duration, proportional to its amplitude, a wave number, and a phase. For k= 0, the modulus 兩f兩 ac- counts for a sech-shaped deformation of the magnetization

and magnetic field which does not propagate, but oscillates uniformly in time. The exponential factor in 共37兲 corre- sponds to a frequency shift, such that ␻⁼⍀+C/共d²⍀²兲, whered= 1 /共␧p兲is the pulse duration in the unit ofz.

For nonzero ␭, slow spatial oscillation adds to the sech profile, and the pulse propagates with velocity

V=kC

⍀². 共38兲

However, the pulse is no merely a sech-shaped envelope modulating a fast carrier, since the carrier does not evolve spatially. The wave magnetization can be defined as Mw

=␧M₁e^i⍀t+ c.c. . Itsrcomponent, e.g., is

M_w^r = −␧m共␣⁺␤␬²兲J1共␬r兲f共␨^,␶兲e^i⍀t+ c.c., 共39兲 according to共37兲, it can be written as

M_w^r =AJ₁共␬r兲sechp␨

⬘

^cos共k␨

⬘

⁺␺共t,␶兲兲 共40兲 whereAis the amplitude,␨

⬘

the slow spatial coordinate in a frame at the wave velocity, and␺共t,␶兲a fast varying phase.

Figure4 shows the␨dependency of the wave profile during one period of␺共t,␶兲, for a few values ofk.

IV. CONCLUSIONS

Propagation of spin waves in ferromagnetic nanowires has been studied, neglecting damping, conductivity, and sur- face anisotropy, by means of a long-wave approximation. For the purely radial modes, an asymptotic model of NLS type has been derived by means of the reductive perturbation method. From the point of view of asymptotic methods, this result is unusual since the NLS equation is typically related to the slowly varying envelope approximation, while the asymptotic models relevant for long waves are typically of the same type as the Korteweg–de Vries equation. The exis- tence of soliton solutions to the NLS is determined by means of the Lighthill criterion. It depends on two parameters, which are the radius of the nanowire and the strength of the external field. We specify the values of these quantities with respect to the exchange length of the ferromagnetic material and its saturation magnetization, respectively. Domains in the space of these parameters where solitons can be formed have been characterized, for the few first linear guided modes. Therefore, the mode共s兲 for which solitons can be formed can be controlled by means of the two parameters.

Some domains exist where only the fundamental mode al- lows soliton propagation. In this case, the nanowire would act nonlinearly as a monomode waveguide, although it is strongly multimode from the linear point of view. The wave profile corresponding to the solitons has also been described;

it can be a single hump, or a few-cycle pulse relatively to the space variable, which oscillates quickly in time.

ACKNOWLEDGMENT

This work has been done as part of the programme GDR PhoNoMi2.

FIG. 2.共Color online兲The domains where bright solitons exist in the␣vsR/l_exchplane, for small applied fields. In each domain, the numbers indicate the modes which can support bright solitons.

The curveC= 0 andD= 0 for thejth modes are labeledCjandDj, respectively.

FIG. 3. 共Color online兲 The same as Fig.2, but for larger field strength␣.

APPENDIX: MULTIPLE SCALE ANALYSIS 1. Coordinates and boundary conditions

The cylindrical coordinates 共r,␪,z兲 are defined by x

=rcos␪^,y=rsin␪^{, andz=}z. We denote byu^r,u^␪ the radial and orthogonal components of a vectoru. Then the magne- tostatic boundary conditions in atr=R, whereRis the radius of the nanowire, can be given as

共H^r+M^r兲_{共r=R−0兲}=共H^r兲_共r=R+0兲, 共A1兲 共H^␪兲_共r=R−0兲=共H^␪兲_共r=R+0兲, 共A2兲

共H^z兲_{共r=R−0兲}=共H^z兲_共r=R+0兲. 共A3兲 We now substitute the expansions 共11兲 and 共12兲 in the basic equations共1兲,共2兲, and共4兲, and collect the terms pro- portional to different powers of ␧ and solve the resultant equations. At order␧⁰, we find that both the uniform fields are collinear to each other.

2. Solutions at order␧¹

At the next order␧¹, we obtain the following equations from Eqs.共1兲and共2兲, respectively:

FIG. 4. 共Color online兲 The wave profile corresponding to the fundamental soliton, for fixedr, as a function of␨, for one period of the fast oscillating phase ␺, and for some value of the wave num- berk.

i⍀M₁¹= −m∧共−ⵜ_⬜⌽1+␤⌬_⬜M₁兲−M₁∧共␣m兲, 共A4兲 and

−⌬_⬜⌽1+ⵜ_⬜·M₁= 0. 共A5兲 In order to solve the equations, we assume that the fields are uniform about zaxis and so the operators in Eqs. 共A4兲 and共A5兲take the following form for a given scalar f and a vectoru as

ⵜ_⬜f=

冢

^{1r⳵⳵0r␪ff}

冣

^{, 共A6兲}

⌬_⬜u=

冢 ^{冉 冉}

^{⌬⌬⬜⬜−−rr1122⌬}

^{冊 冊}

^{⬜uuur␪z−+r2r222⳵⳵␪␪uu␪r}

冣

^{, 共A7兲}

ⵜ_⬜·u=⳵ru^r+1 ru^r+1

r⳵␪u^␪, 共A8兲 and⌬_⬜=⳵r

2+¹_r⳵r+_r¹2⳵_␪^2.

Assuming azimuthal invariance Eq. 共A4兲 reduces to M₁^z

= 0 and Eqs.共13兲and共14兲. The latter are solved to yield the linear modes, Eqs.共16兲and共17兲, and the dispersion relation 共18兲arises as a solvability condition.

Again with the azimuthal invariance, we find that Eq.

共A5兲is satisfied if

⳵r⌽1=M₁^r. 共A9兲 In order to complete our analysis at the first order, we consider the usual magnetostatic boundary conditions for the present geometry given in Eqs.共A1兲–共A3兲,共5兲, and 共6兲. We find that inside the wireH₁^␪= 0 because of the azimuthal in- variance, H₁^z= 0, and from Eq. 共A3兲 it can be shown that H₁^r+M₁^r= 0, which shows that the magnetostatic boundary conditions are satisfied at this order of analysis, with a mag- netic fieldH⬅␣^moutside the nanowire.

3. Solutions at order␧²

At order␧², for any Fourier componentp, we obtain the following equations from Eqs.共1兲and共2兲:

i⍀M₂^p−V⳵_␨^M1 p+⳵_␶^M0

p= −m∧共−⳵_␨⌽1

pe_z−ⵜ_⬜⌽2 p

+␤⌬_⬜M₂^p兲

−

兺

q+s=p

M₁^q∧共−ⵜ_⬜⌽₁^s+␤⌬_⬜M₁^s兲

−M₂^p∧共␣^m兲, 共A10兲 in which we have set M₁¹=M₁^−1*=M₁, ⌽₁¹=⌽₁^−1*=⌽1, and M₁^p=⌽₁^p= 0 for p⫽± 1, and

−⌬_⬜⌽₂^p+⳵␨M₁^p,z+ⵜ_⬜·M₂^p= 0. 共A11兲 a. Fourier component p= 1:Determination of V. Now we collect the coefficients of the fundamental Fourier compo- nentp= 1 at the same order␧². From Eq.共A10兲, we find that M₂^1,z= 0 and

i⍀M21,r

−V⳵␨M₁^r= −m

冉

^␣−␤

冉

^⌬⬜−r12

冊冊

^M21,␪,

共A12兲 i⍀M₂^1,␪−V⳵␨M₁^␪=m

冉

^{1 +␣−␤}

冉

^⌬⬜−r¹²

冊冊

^M21,r.

共A13兲 We decomposeM₂^1,r andM₂^1,␪as

M₂^1,r=B^r共␨^,␶兲J₁共␬^r兲+W^r共␨^,␶^,r兲, 共A14兲 M₂^1,␪=B^␪共␨^,␶兲J1共␬r兲+W^␪共␨^,␶,r兲, 共A15兲 in whichW^r and W^␪ may have nonzero component on any characteristic or eigenspace of the operator

共

⌬_⬜−_r¹2

兲

^{, except}

the one which corresponds to the eigenvalue −␬² of the se- lected mode. Substituting Eqs. 共A14兲 and 共A15兲 in Eqs.

共A12兲 and 共A3兲 and making use of Eqs. 共16兲 and 共17兲 we obtain, along this eigenmodeJ₁共␬^R兲,

i⍀B^r+m共␣⁺␤␬²兲V⳵_␨^{f= −}m共␣⁺␤␬²兲B^␪, 共A16兲 i⍀B^␪−i⍀V⳵␨f=m共1 +␣+␤␬²兲B^␪. 共A17兲 EliminatingB^␪, and making use of Eq.共18兲, Eqs.共A16兲and 共A17兲reduce to

− 2i⍀m共␣+␤␬²兲V⳵␨f= 0, 共A18兲 and, hence, the velocityV is zero.

In the hyperplane supplementary to the eigenspace corre- sponding to −␬², the above substitution gives the equations

i⍀W^r= −m

冉

^␣−␤

冉

^⌬⬜−r¹²

冊冊

^{W␪, 共A19兲}

i⍀W^␪= −m

冉

^{1 +␣−␤}

冉

^⌬⬜−r12

冊冊

^{Wr. 共A20兲}

It follows from Eqs. 共A19兲 and 共A20兲 that W^r and W^␪ are eigenfunctions of

共

⌬_⬜−_r¹2

兲

. Let us denote by共−␬

⬘

^{2兲 the ei-} genvalue

i⍀W^r= −m共␣⁺␤␬

⬘

²兲W^␪, 共A21兲 i⍀W^␪= −m共1 +␣⁺␤␬

⬘

²兲W^r, 共A22兲 and, hence,␬

⬘

must satisfy the same dispersion relation, Eq.

共18兲, as ␬, with the same frequency ⍀. It follows that ␬

⬘

=␬, which was excluded initially. We conclude that W^r

=W^␪= 0, and M₂^1,r and M₂^1,␪ belong to the same transverse mode as the first-order componentM₁.

Equations共A16兲and共A17兲are easily solved to yield B^r共␨^,␶兲= −g共␨^,␶兲m共␣⁺␤^k2兲, 共A23兲

B^␪共␨^,␶兲=i⍀g共␨^,␶兲, 共A24兲 whereg共␨^,␶兲is an arbitrary function. Then from Eqs.共A12兲 and共A13兲, we obtain

M₂^1,r= −m共␣⁺␤␬²兲g共␨^,␶兲J1共␬r兲, 共A25兲 M₂^1,␪=i⍀g共␨^,␶兲J1共␬r兲. 共A26兲 Since M₁^z= 0, from Eq. 共A11兲 for the Fourier component p

= 1, we find that

⳵r⌽₂¹=M₂^1,r. 共A27兲

Also using the same arguments as in the order␧, one can show that the boundary and the pinning conditions are satis- fied at order␧²for p= 1, withH⬅␣^moutside the nanowire.

b. Nonlinear term for Fourier components p= 0 and p

= 2. For the Fourier componentp= 0, we find that Eq.共A10兲 reduces to

−m

冉

^␣−␤

冉

^⌬⬜−r12

冊冊

^{M20,␪=R0r, 共A28兲}

m

冉

^{1 +␣−␤}

冉

^⌬⬜−r¹²

冊冊

^{M20,r0 =R0␪, 共A29兲}

R₀^z= 0. 共A30兲 The nonlinear term is

R₀=M₁^*∧共−ⵜ_⬜⌽1+␤⌬_⬜M₁兲+ c.c. . 共A31兲 Using Eqs.共13兲 and共14兲, it is seen that R₀= 0 and, hence, M₂^0,r=M₂^0,␪= 0, while M₂^0,z remains free. We set M₂^0,z= 0 for simplicity.

Similarly, forp= 2, we find

2i⍀M₂^2,r= −m

冉

^␣−␤

冉

^⌬⬜−r¹²

冊冊

^{M22,␪−R2r, 共A32兲}

2i⍀M₂^2,␪=m

冉

^{1 +␣−␤}

冉

^⌬⬜−r12

冊冊

^{M22,r−R2␪, 共A33兲}

2i⍀M₂^2,z= −R₂^z. 共A34兲

The nonlinear term is

R₂=M₁∧共−ⵜ_⬜⌽1+␤⌬_⬜M₁兲, 共A35兲 which reduces to

R₂= −m共␣⁺␤␬²兲i⍀f²共J1共␬r兲兲²ez. 共A36兲 Hence, Eq. 共A32兲 and 共A34兲 are homogeneous, and M₂^2,r

=M₂^2,␪= 0. From Eqs.共A34兲and共A36兲, we get the expression 共22兲ofM₂^2,z.

The divergence equation共A11兲yields

⳵r⌽2

p=M₂^p,r, 共A37兲

for p= 2 and 0, and hence⌽₂⁰=⌽₂²= 0. As the only nonzero component ofM₂²andM₂⁰ is parallel to the wire axisz, the boundary conditions do not produce a nonzero correction to

the magnetic field outside the wire at this order.

4. Solutions at order␧³

Collecting the coefficients of␧³, we obtain the following equations from the Landau-Lifshitz equation 共1兲 and the magnetostatic equation共2兲, respectively,

i⍀M₃¹−V⳵␨M₂¹+⳵␶M₁= −m∧关−⳵␨⌽₂¹ez−ⵜ_⬜⌽₃¹ +␤共⌬_⬜M₃¹+⳵_␨^2M1兲兴

−

兺

q+s=p

M₁^q∧共−⳵_␨⌽1

se_z−ⵜ_⬜⌽2 s

+␤⌬_⬜M₂^s兲−

兺

q+s=p

M₂^q∧共−ⵜ_⬜⌽₁^s +␤⌬_⬜M₁^s兲−␣^M3

p∧m, 共A38兲

and

−⳵_␨²⌽1−⌬_⬜⌽₃¹+⳵␨M₂^1,z+ⵜ_⬜·M₃¹= 0. 共A39兲 a. Nonlinear term for Fourier component p= ± 1. Using the expressions共16兲,共17兲,共A25兲, and共A26兲of the compo- nents ofM₁ andM₂¹ 共andM₁^z=M₂^1,z= 0兲, we can reduce Eq.

共A38兲 for the fundamental Fourier component p= 1 to the following set:

i⍀M₃^1,r+⳵␶M₁^r=m␤

冉

^⌬⬜−r12

冊

^{M31,␪−m␣M31,␪}

+m␤⳵_␨^2M1␪−N₃^1,r, 共A40兲 i⍀M₃^1,␪+⳵␶M₁^␪=m⳵⌽₃¹−m␤

冉

^⌬⬜−r12

冊

^{M31,r+m␣M31,r}

−m␤⳵_␨^2M1

r−N₃^1,␪, 共A41兲

i⍀M₃^1,z+⳵␶M₁^z= −N₃^1,z. 共A42兲 The nonlinear termN₃¹is defined by

N₃¹=M₁∧共−ⵜ_⬜⌽₂⁰+␤⌬_⬜M₂⁰兲+M₁^*∧共−⳵␨⌽₁²ez−ⵜ_⬜⌽₂² +␤⌬_⬜M₂²兲+M₂⁰∧共−ⵜ_⬜⌽1+␤⌬_⬜M₁兲

+M₂²∧共−ⵜ_⬜⌽₁^*+␤⌬_⬜M₁^*兲. 共A43兲 Using the results of previous orders, it reduces to

N₃¹=m

2共␣⁺␤␬²兲f兩f兩²关J₁^*⌬_⬜J₁²P+J₁兩J1兩²T兴, 共A44兲 with

P=␤

冢

^{m共␣−+0i⍀␤␬2兲}

冣

^{, 共A45兲}

T=

冢

^{m共1 +−␤␬i⍀20兲共␤␬␣2+␤␬2兲}

冣

^{. 共A46兲}

Since N₃^1,z= 0 and M₁^z= 0, we obtain from Eq. 共A42兲 that M₃^1,z= 0. The divergence equation 共A39兲reduces to

−⳵_␨²⌽1+

冉

^⳵r+1r

冊

^⳵r⌽31+

冉

^⳵r+1r

冊

^{M31,r= 0. 共A47兲}

We observe that

⌽1=− 1

␬²

冉

^⳵r+1r

冊

^{M1r, 共A48兲}

which solves共A47兲as

M₃^1,r=⳵r⌽₃¹− 1

␬²⳵_␨^2M1

r. 共A49兲

The remaining harmonics p= 0 , 2 , 3 of order ␧³ can be useful to derive higher-order evolution equations, which can be expected in the present case to be higher-order NLS ones.

However, such a derivation would necessitate the introduc- tion of higher-order slow time variables, which are not present in Eqs.共8兲and共9兲. Anyways, the computation of the higher harmonics do not yield additional conditions at the order considered.

5. Nonlinear Schrödinger equation

Substituting Eq.共A49兲 into Eq. 共A41兲, we get Eq. 共23兲, with the right-hand-side member:

F= −i⍀⳵␶M₁^␪+m

冉

^{1 +␣−␤}

冉

^⌬⬜−r12

冊冊

⫻关−⳵_␶^M1

r+m␤⳵_␨^2M1␪−N₁^1,r兴

+i⍀m

冉

␬¹²⁻␤

冊

^{⳵␨2M1r−i⍀N31,␪. 共A50兲}

Using expressions共16兲and共17兲ofM₁, the expression共A50兲 ofFis reduced to formulas共24兲–共27兲.

Then Eq.共23兲is solved using expansion共30兲. We get

Xq= Fq

−⍀²+m²共1 +␣⁺␤^q2兲共␣⁺␤^q2兲, 共A51兲 forq⫽␬ and

0 .X_␬=F_␬. 共A52兲 Since F_␬=共J1共␬r兲兩F兲, condition 共A52兲 is nothing else than Eq.共31兲.

To reduce the coefficients, the following formula is use- ful:

共J₁共␬^r兲兩J₁共␬^r兲兲=R²

2

冉

^{1 −}␬^21R2

冊

^{共J1共␬R兲兲2. 共A53兲}

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