Journals published by the American Physical Society can be found at http://journals.aps.org/Skyrmions are magnetic defects in ultrathin magnetic films, similar to the bubble domains in the thicker films. Even weak uniaxial anisotropy determines their radii unambiguously. We derive equations of slow dynamics for Skyrmions. We show that the discreteness of the lattice in an isotropic two-dimensional magnet leads to a slow rotation of the local magnetization in the Skyrmion and even a small dissipation leads to decay of the Skyrmion. The radius of such a Skyrmion as a function of time is calculated. We prove that uniaxial anisotropy stabilizes the Skyrmion and study the relaxation process. [S0163-1829(98)50438-9]

Abanov, Artem

Pokrovsky, Valery L.

arXiv

Texas A&amp;M Repository

RAPID COMMUNICATIONSPHYSICAL REVIEW B 1 OCTOBER 1998-IIVOLUME 58, NUMBER 14Skyrmion in a real magnetic filmAr. AbanovDepartment of Physics, Texas A&M University, College Station, Texas 77843-4242V. L. PokrovskyDepartment of Physics, Texas A&M University, College Station, Texas 77843-4242and Landau Institute of Theoretical Physics, Moscow, Russia~Received 10 July 1998!Skyrmions are magnetic defects in ultrathin magnetic films, similar to the bubble domains in the thickerfilms. Even weak uniaxial anisotropy determines their radii unambiguously. We derive equations of slowdynamics for Skyrmions. We show that the discreteness of the lattice in an isotropic two-dimensional magnetleads to a slow rotation of the local magnetization in the Skyrmion and even a small dissipation leads to decayof the Skyrmion. The radius of such a Skyrmion as a function of time is calculated. We prove that uniaxialanisotropy stabilizes the Skyrmion and study the relaxation process. @S0163-1829~98!50438-9#Skyrmions are topological excitations of two-dimensional~2D! magnet or ultrathin magnetic films similar to the well-known bubble domains in the industrial magnetic memorymaterials.1 The former play an important role as letters inmagnetic records. A natural question arises whether theSkyrmions can be utilized in a similar way. To answer thisquestion one must adjust the existing theory to a realisticcondition of real films with their anisotropy, defects, dissipa-tion, and discreteness. It is necessary also to consider howthe Skyrmion can be created and destroyed, i.e., the Skyr-mion dynamics, ignored in the previous studies. In this paperwe address these problems.The Skyrmion was first discovered by Skyrme2 who con-sidered it as a localized solution in a model of nuclear mat-ter. Belavin and Polyakov ~BP! ~Ref. 3! have shown that theSkyrmion is a topologically nontrivial minimum of energyfor the so-called n¢ field, the classical continuous limit of theHeisenberg model. It realizes the mapping of the plane inwhich the spins are placed onto sphere of the order parameterwith the degree of mapping 1.Though the n¢-field model was inspired by the studies ofmagnetic films, a more or less direct observation of the Skyr-mions was made on the 2D electron layers under the quan-tum Hall effect ~QHE! conditions4 following an earlier the-oretical prediction.5 An indirect observation of Skyrmioneffects in quasi-2D magnets was reported by F. Waldner6who found the Skyrmion energy from the heat capacity mea-surements in good agreement with the theoretical predictionby BP. This experimental observation is highly nontrivialsince the real magnets are at least weakly anisotropic. Due tothe existence of Goldstone modes in the Heisenberg magnet,even very weak anisotropy can change the excitations cru-cially.Let us first approach the problem with simple dimension-ality arguments. The Skyrmion is a static excitation of thehomogeneous ferromagnetic state localized in a circle of theradius R . From the dimensionality consideration and fromthe BP results the Skyrmion energy does not depend on itssize and is equal to 4pJumu where m is the degree of map-PRB 580163-1829/98/58~14!/8889~4!/$15.00ping. Hence, the exchange energy 1/2*J(¹S)2d2x of a Skyr-mion does not depend on its radius R . The anisotropy energy21/2*lSz2d2x of the Skyrmion is proportional to R2 anddecreases together with R . Therefore the exchange energy ofthe fourth order in the derivatives 1/2*k(DS)2d2x is crucialfor the Skyrmion. It stabilizes the Skyrmion if k.0. Nowthe total energy of the Skyrmion depends on its radius andhas minimum at R;(k/l)1/4. The order of magnitude of k isJa2 where a is the lattice constant, but its sign is model-dependent. It is negative for the Heisenberg model with thenearest-neighbor interaction, but it is positive for a more re-alistic Ruderman-Kittel-Kasuya-Yosida interaction. Thus, ifthere exists a stable Skyrmion, its radius R is of the orderR;(Ja2/l)1/4;(lla)1/2!ll , where ll5AJ/l is the domainwall width. The energy of such a Skyrmion differs by aboutlR2;Akl from the classical Skyrmion energy 4pJ@Akl .This analysis shows that a kind of perturbation approachis applied for this problem. Below we develop nonlinear per-turbative theory working at rather general premises. Theclassical two-dimensional Heisenberg exchange ferromagnetin continuous approximation is described by the HamiltonianH0512 E J~¹S!2d2x ~1!with the constraint on the vector field S~r!: S2(r)51. Anobvious minimum of such a Hamiltonian is the homoge-neous ferromagnetic configuration in which all the spins areparallel S(r)5const. The simplest topologically nontrivialminimum of the Hamiltonian ~1! is given by the Skyrmesolution:S0x52RrR21r2cos~f1c!,S0y52RrR21r2sin~f1c!,R8889 © 1998 The American Physical SocietyRAPID COMMUNICATIONSR8890 PRB 58AR. ABANOV AND V. L. POKROVSKYS0z5R22r2R21r2. ~2!It describes a Skyrmion of radius R with the center placed inthe origin. The observation point is indicated by the polarcoordinates r and f; c is an arbitrary angle. We have alreadymentioned that the energy E of the Skyrmion ~2! does notdepend on its radius R . It also does not depend on the anglec.Any static distribution of magnetization S~r! satisfies theequilibrium equation:dHdS~r! 5S~r!S S~r! dHdS~r! D . ~3!The right-hand side of Eq. ~3! is added as a Lagrangian fac-tor which ensures that S2(r)51 at any point r. For H5H01H1 where H1 is a perturbation we look for a solution in aform S5S01S1 , where S0(r) is determined by Eq. ~2! andS1(r) is perpendicular to S0 and satisfies the linearized inho-mogeneous equation:Kˆ S1~r!5F dH1dS~r! 2S0~r!S S0 dH1dS D GS5S0, ~4!where the tensor kernel Ki j(r,r8) of the linear operator Kˆ isgiven byKi j~r,r8!52Jd~r2r8!3@d i jDr82d i j~S0DS0!2S0iS0 jDr8# .The derivatives ]S0 /]R and ]S0 /]c are the zero modesand, hence, they satisfy homogeneous equations Kˆ ]S0 /]R5Kˆ ]S0 /]c50. Therefore, the right-hand side of Eq. ~4!must be orthogonal to the vectorial functions ]S0 /]R and]S0 /]c . The two orthogonality conditions allow us to deter-mine both R and c fixed by small perturbation H1 . Consid-ering a special perturbation HamiltonianH1512 E @k~DS!21l~12Sz2!#d2x , ~5!we find from the orthogonality conditionR5R05S 8k3lL D1/4, ~6!where L5ln@(3l/8k)1/4ll# . The logarithm in Eq. ~6! comesfrom the divergent integral *(12S0z2 )]S0z /]Rd2x . It wascut off at a radius r5ll at which the perturbation theoryfails. Due to axial symmetry, c remains to be zero mode.Note that the Skyrmion does not exist for k,0. Now we arein a position to consider the dynamics of the Skyrmion.The dynamics of the unit vector field S(r,t) is given bythe Landau-Lifshitz equation:7S˙ ~r,t !52gS~r,t !3dH@S#dS~r,t !1nS~r,t !3FS~r,t !3 dH@S#dS~r,t !G . ~7!It can be checked that, at n50, the equation of motion ~7!conserves the magnetization and energy of the field S~r!, aswell as the local constraint S2(r)51. A small dissipationterm at nÞ0 allows for the relaxation processes.We consider only the slow Skyrmion dynamics. It meansthat we present again the Hamiltonian H as a sum H01H1 , where H1 is a small perturbation to the exchangeHamiltonian H0 . We are looking for a solution in the formS5S01S1 , S0S150, where uS1u!uS0u and S0r;R(t),c(t)is the standard Skyrmion solution ~2! with the parameters Rand c slowly varying in time. We also consider the dissipa-tion and the term S˙ as a perturbation. Substituting onlyS0r;R(t),c(t) in the perturbation terms in Eq. ~7! and re-quiring the orthogonality of the perturbation terms to bothzero modes ]S0 /]R and ]S0 /]c , we obtain equations ofmotion for R and c :R˙R 52ng v , ~8!vR2lnR˜ 2R21gk1631R22glR2lnR˜ 2R250, ~9!where v5c˙ and R˜ is a scale at which the perturbation ex-pansion breaks down, i.e., uS0(r5R˜ )u'uS1(r5R˜ )u. Equa-tions ~8! and ~9! have a fixed point with v50 and R5R0 ,where R0 is determined by Eq. ~6!.If R deviates slightly from R0 so that DR5R2R0 is stillsmall enough, one can put R˜ 5ll into Eq. ~9! and obtain v54lgDR/R . Using Eq. ~8!, we findDR˙ 524lnDR . ~10!‘‘Small enough’’ DR means that we still can use R˜ 'll .However, if DR is large, the cutoff scale is determined byfinite frequency: R˜ 'lv5AgJ/uvu. Eq. ~10! is valid if ll,lv or DR/R,1/4. In the opposite case lv!ll and R@R0one can neglect the second term in Eq. ~9!. Then equations ofmotion ~8! and ~9! read as follows:v5lg; R˙ 52nlR . ~11!For lv!ll and R!R0 the last term in Eq. ~9! can be omitted.Thenv52163kgR41ln3R2J16k; R3R˙ 5163 kn1ln3R2J16k. ~12!Thus, the easy-axis anisotropy together with the fourth-order exchange term fix the radius ~6! of the Skyrmion if k.0. In the opposite case k,0 there is no stable configura-tion with nontrivial topology. We also have shown that theSkyrmion reaches its equilibrium radius within characteristictime tr5(nl)21.Another property of a real film which should be taken intoaccount is the lattice discreteness. We have mentioned earlierthat in the continuous model the Skyrmion is a topologicalexcitation and as such cannot dissipate. However, in the dis-crete lattice the continuity of the field S~r! is lost and theRAPID COMMUNICATIONSPRB 58 R8891SKYRMION IN A REAL MAGNETIC FILMvery notion of topological excitation becomes inconsistent.Therefore, the Skyrmion configuration in the discrete latticeis unstable. Moreover, it is sufficient to remove one plaquettein the center of the Skyrmion to make it unstable ~see, forexample, Ref. 8!. We will imitate the discreteness effect byconsidering a hole in the center of the Skyrmion. In the pic-ture due to BP the Skyrmion is described by a mesomorphicfunction which has a pole in the center of the Skyrmion. Thispole cannot be removed by any continuous variation of thefield S. However, the Skyrmion configuration with a holepunched in its center will dissipate.Let us first consider the Skyrmion without anisotropy.‘‘Punching a small hole in the center’’ means the substitutionJ!Ju(r2r0) in Eq. ~1! where r0!R is the radius of thehole and u(x) is the step function: u(x)50 for x,0 andu(x)51 for x.0. Hence, the perturbation H1 is given byH˜ 152JE u~r02r !~¹S!2. ~13!Employing the orthogonality condition of the right-hand sideof the linearized Eq. ~7! to the zero modes and neglecting allthe terms of the higher order in r0 , one finds that Eq. ~8! stillholds, but Eq. ~9! must be replaced byv lnR˜ 2R25gJr02R4, ~14!where R˜ 'lv5AgJ/uvu is the scale where the perturbationscheme breaks down. With the logarithmic accuracy one canwrite ln(R˜ 2/R2)5ln(R2/r02). Finally, substituting v from Eq.~14! into Eq. ~8!, we findR3R˙ 52nJr02lnR2r02. ~15!From this equation one can conclude that the Skyrmion’slifetime is roughly proportional to its radius in the fourthpower.Returning to the field with the easy-axis anisotropy andthe fourth-order term, let us introduce again a hole in thecenter. In this situation the perturbation is given by the sumof the two terms ~5! and ~13!. Using the same orthogonalitytrick one gets22vg R2lnR˜ 2R22323kR212R2l lnR˜ 2R2524Jr02R2. ~16!After substitution k!k˜ 5k2(3/8)Jr0 Eq. ~16! acquires thesame form as Eq. ~9!. It means that, as long as k.(3/8)Jr02, the Skyrmion is stable and its radius is definedby Eq. ~6! with k˜ instead of k. Equations ~10!, ~11!, and ~12!are valid as well after the same substitution. In the case k,(3/8)Jr02, however, the effective k˜ is negative and thestable Skyrmion exists no longer.The translation motion of a Skyrmion can be studied bythe same technique as well. In order to move the Skyrmion,a nonuniform magnetic field perpendicular to the film can beapplied. The perturbative part of the Hamiltonian readsH15E S 12 k~DS!21 12 l~12Sz2!2h~r!SzD d2x , ~17!where h(r)5mBSH(r), mB is the Bohr magneton, S is thespin per unit area, and H(r) is the magnetic field. The twonew zero modes associated with the translation motion are]S0 /]X and ]S0 /]Y , where X and Y are the coordinates ofthe center of the Skyrmion. Assuming that magnetic fieldvaries slowly, and employing the same orthogonality tech-nique, we find equations of translational motionvx52gR22 ]Yh0lnR˜ 2R21nR22 ]Xh0lnR˜ 2R2,vy5gR22 ]Xh0lnR˜ 2R21nR22 ]Yh0lnR˜ 2R2, ~18!where vx5X˙ , vy5Y˙ , h05h(X ,Y ), and R˜ 5min(lv ,ll ,AJ/h0). Equations ~18! are valid if u¹huR˜ /uhu!1.In conclusion, we proposed a general framework for de-scription of any slow motion in a system of Skyrmions. Wehave shown that in a real ultrathin ferromagnetic film witheasy-axis anisotropy the Skyrmion known for the isotropicmodel still exists, but it acquires a definite radius R0 givenby Eq. ~6!. This result was obtained in Ref. 9 by other meth-ods. By the order of magnitude R0;Aall!ll where ll is thedomain wall width. Once we made a domain with the re-versed magnetization in a ferromagnetic film, it shrinksdown to the size R0;1nm ~see also Ref. 9!. The magneticmoment of such a Skyrmion is still rather large, mS5mBp(ll /a)ln(ll /a);300mB , and can be observed experi-mentally by the methods employed to discover the Skyrmi-ons in QHE systems.4 The Skyrmion stability allows us tocreate it with a magnetic tip.The discreteness of the lattice in the isotropic model leadsto a finite Skyrmion lifetime which is roughly proportional tothe fourth power of its radius. However, anisotropy togetherwith the higher order exchange interaction stabilizes theSkyrmion. At finite temperature it can decay through an in-stanton configuration. Our results allow us to understandwhy the activation energy found by Waldner is so close to4pJ: the difference is expected to be of the relative ordera/ll;1022. The detailed thermodynamics of Skyrmionswill be published elsewhere.This work has been supported by the NSF Grant No.DMR-9705812 and by the DOE Grant No. DE-FGO3-96ER45598. We are indebted to B.I. Halperin for useful dis-cussions and to B. Ivanov who attracted our attention toRef. 9.RAPID COMMUNICATIONSR8892 PRB 58AR. ABANOV AND V. L. POKROVSKY1 See, for example, T. H. O’Dell, Ferromagnetodynamics: The Dy-namics of Magnetic Bubbles, Domains, and Domain Walls ~Hol-sted Press, 1981!, and references therein.2 T. H. R. Skyrme, Proc. R. Soc. London, Ser. A 247, 260 ~1958!.3 A. A. Belavin and A. M. Polyakov, Zh. Eksp. Teor. Fiz. Pis’maRed. 22, 503 ~1975! @JETP Lett. 22, 245 ~1975!#; see also A. M.Polyakov, Gauge Fields and Strings ~Harwood Academic Pub-lishers, Chur, Switzerland, 1987!.4 E. H. Aifer, B. B. Goldberg, and D. A. Broido, Phys. Rev. Lett.76, 680 ~1996!.5 S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi,Phys. Rev. B 47, 16 419 ~1993!.6 F. Waldner, J. Magn. Magn. Mater. 54–57, 873 ~1986!; Phys.Rev. Lett. 65, 1519 ~1990!.7 L. D. Landau and E. M. Lifshitz, Statistical Physics Part II ~Per-gamon, New York, 1984!, Vol. 9.8 F. D. M. Haldane, Phys. Rev. Lett. 61, 1029 ~1988!.9 B. Ivanov, V. A. Stephanovich, and A. A. Zhmudskii, J. Magn.Magn. Mater. 88, 116 ~1990!; Sov. Phys. JETP 64, 376~1986!.

Skyrmion in a real magnetic film

Texas A&M University

RAPID COMMUNICATIONS
PHYSICAL REVIEW B 1 OCTOBER 1998-IIVOLUME 58, NUMBER 14Skyrmion in a real magnetic film
Ar. Abanov
Department of Physics, Texas A&M University, College Station, Texas 77843-4242
V. L. Pokrovsky
Department of Physics, Texas A&M University, College Station, Texas 77843-4242
and Landau Institute of Theoretical Physics, Moscow, Russia
~Received 10 July 1998!
Skyrmions are magnetic defects in ultrathin magnetic films, similar to the bubble domains in the thicker
films. Even weak uniaxial anisotropy determines their radii unambiguously. We derive equations of slow
dynamics for Skyrmions. We show that the discreteness of the lattice in an isotropic two-dimensional magnet
leads to a slow rotation of the local magnetization in the Skyrmion and even a small dissipation leads to decay
of the Skyrmion. The radius of such a Skyrmion as a function of time is calculated. We prove that uniaxial
anisotropy stabilizes the Skyrmion and study the relaxation process. @S0163-1829~98!50438-9#Skyrmions are topological excitations of two-dimensional
~2D! magnet or ultrathin magnetic films similar to the well-
known bubble domains in the industrial magnetic memory
materials.1 The former play an important role as letters in
magnetic records. A natural question arises whether the
Skyrmions can be utilized in a similar way. To answer this
question one must adjust the existing theory to a realistic
condition of real films with their anisotropy, defects, dissipa-
tion, and discreteness. It is necessary also to consider how
the Skyrmion can be created and destroyed, i.e., the Skyr-
mion dynamics, ignored in the previous studies. In this paper
we address these problems.
The Skyrmion was first discovered by Skyrme2 who con-
sidered it as a localized solution in a model of nuclear mat-
ter. Belavin and Polyakov ~BP! ~Ref. 3! have shown that the
Skyrmion is a topologically nontrivial minimum of energy
for the so-called n¢ field, the classical continuous limit of the
Heisenberg model. It realizes the mapping of the plane in
which the spins are placed onto sphere of the order parameter
with the degree of mapping 1.
Though the n¢-field model was inspired by the studies of
magnetic films, a more or less direct observation of the Skyr-
mions was made on the 2D electron layers under the quan-
tum Hall effect ~QHE! conditions4 following an earlier the-
oretical prediction.5 An indirect observation of Skyrmion
effects in quasi-2D magnets was reported by F. Waldner6
who found the Skyrmion energy from the heat capacity mea-
surements in good agreement with the theoretical prediction
by BP. This experimental observation is highly nontrivial
since the real magnets are at least weakly anisotropic. Due to
the existence of Goldstone modes in the Heisenberg magnet,
even very weak anisotropy can change the excitations cru-
cially.
Let us first approach the problem with simple dimension-
ality arguments. The Skyrmion is a static excitation of the
homogeneous ferromagnetic state localized in a circle of the
radius R . From the dimensionality consideration and from
the BP results the Skyrmion energy does not depend on its
size and is equal to 4pJumu where m is the degree of map-PRB 580163-1829/98/58~14!/8889~4!/$15.00ping. Hence, the exchange energy 1/2*J(¹S)2d2x of a Skyr-
mion does not depend on its radius R . The anisotropy energy
21/2*lSz
2d2x of the Skyrmion is proportional to R2 and
decreases together with R . Therefore the exchange energy of
the fourth order in the derivatives 1/2*k(DS)2d2x is crucial
for the Skyrmion. It stabilizes the Skyrmion if k.0. Now
the total energy of the Skyrmion depends on its radius and
has minimum at R;(k/l)1/4. The order of magnitude of k is
Ja2 where a is the lattice constant, but its sign is model-
dependent. It is negative for the Heisenberg model with the
nearest-neighbor interaction, but it is positive for a more re-
alistic Ruderman-Kittel-Kasuya-Yosida interaction. Thus, if
there exists a stable Skyrmion, its radius R is of the order
R;(Ja2/l)1/4;(lla)1/2!ll , where ll5AJ/l is the domain
wall width. The energy of such a Skyrmion differs by about
lR2;Akl from the classical Skyrmion energy 4pJ
@Akl .
This analysis shows that a kind of perturbation approach
is applied for this problem. Below we develop nonlinear per-
turbative theory working at rather general premises. The
classical two-dimensional Heisenberg exchange ferromagnet
in continuous approximation is described by the Hamiltonian
H05
1
2 E J~¹S!2d2x ~1!
with the constraint on the vector field S~r!: S2(r)51. An
obvious minimum of such a Hamiltonian is the homoge-
neous ferromagnetic configuration in which all the spins are
parallel S(r)5const. The simplest topologically nontrivial
minimum of the Hamiltonian ~1! is given by the Skyrme
solution:
S0x5
2Rr
R21r2
cos~f1c!,
S0y5
2Rr
R21r2
sin~f1c!,R8889 © 1998 The American Physical Society
RAPID COMMUNICATIONS
R8890 PRB 58AR. ABANOV AND V. L. POKROVSKYS0z5
R22r2
R21r2
. ~2!
It describes a Skyrmion of radius R with the center placed in
the origin. The observation point is indicated by the polar
coordinates r and f; c is an arbitrary angle. We have already
mentioned that the energy E of the Skyrmion ~2! does not
depend on its radius R . It also does not depend on the angle
c.
Any static distribution of magnetization S~r! satisfies the
equilibrium equation:
dH
dS~r! 5S~r!S S~r! dHdS~r! D . ~3!
The right-hand side of Eq. ~3! is added as a Lagrangian fac-
tor which ensures that S2(r)51 at any point r. For H5H0
1H1 where H1 is a perturbation we look for a solution in a
form S5S01S1 , where S0(r) is determined by Eq. ~2! and
S1(r) is perpendicular to S0 and satisfies the linearized inho-
mogeneous equation:
Kˆ S1~r!5F dH1dS~r! 2S0~r!S S0 dH1dS D GS5S0, ~4!
where the tensor kernel Ki j(r,r8) of the linear operator Kˆ is
given by
Ki j~r,r8!52Jd~r2r8!
3@d i jDr82d i j~S0DS0!2S0iS0 jDr8# .
The derivatives ]S0 /]R and ]S0 /]c are the zero modes
and, hence, they satisfy homogeneous equations Kˆ ]S0 /]R
5Kˆ ]S0 /]c50. Therefore, the right-hand side of Eq. ~4!
must be orthogonal to the vectorial functions ]S0 /]R and
]S0 /]c . The two orthogonality conditions allow us to deter-
mine both R and c fixed by small perturbation H1 . Consid-
ering a special perturbation Hamiltonian
H15
1
2 E @k~DS!21l~12Sz2!#d2x , ~5!
we find from the orthogonality condition
R5R05S 8k3lL D
1/4
, ~6!
where L5ln@(3l/8k)1/4ll# . The logarithm in Eq. ~6! comes
from the divergent integral *(12S0z2 )]S0z /]Rd2x . It was
cut off at a radius r5ll at which the perturbation theory
fails. Due to axial symmetry, c remains to be zero mode.
Note that the Skyrmion does not exist for k,0. Now we are
in a position to consider the dynamics of the Skyrmion.
The dynamics of the unit vector field S(r,t) is given by
the Landau-Lifshitz equation:7
S˙ ~r,t !52gS~r,t !3
dH@S#
dS~r,t !
1nS~r,t !3FS~r,t !3 dH@S#dS~r,t !G . ~7!It can be checked that, at n50, the equation of motion ~7!
conserves the magnetization and energy of the field S~r!, as
well as the local constraint S2(r)51. A small dissipation
term at nÞ0 allows for the relaxation processes.
We consider only the slow Skyrmion dynamics. It means
that we present again the Hamiltonian H as a sum H0
1H1 , where H1 is a small perturbation to the exchange
Hamiltonian H0 . We are looking for a solution in the form
S5S01S1 , S0S150, where uS1u!uS0u and S0r;R(t),c(t)
is the standard Skyrmion solution ~2! with the parameters R
and c slowly varying in time. We also consider the dissipa-
tion and the term S˙ as a perturbation. Substituting only
S0r;R(t),c(t) in the perturbation terms in Eq. ~7! and re-
quiring the orthogonality of the perturbation terms to both
zero modes ]S0 /]R and ]S0 /]c , we obtain equations of
motion for R and c :
R˙
R 52
n
g v , ~8!
vR2ln
R˜ 2
R2
1gk
16
3
1
R2
2glR2ln
R˜ 2
R2
50, ~9!
where v5c˙ and R˜ is a scale at which the perturbation ex-
pansion breaks down, i.e., uS0(r5R˜ )u'uS1(r5R˜ )u. Equa-
tions ~8! and ~9! have a fixed point with v50 and R5R0 ,
where R0 is determined by Eq. ~6!.
If R deviates slightly from R0 so that DR5R2R0 is still
small enough, one can put R˜ 5ll into Eq. ~9! and obtain v
54lgDR/R . Using Eq. ~8!, we find
DR˙ 524lnDR . ~10!
‘‘Small enough’’ DR means that we still can use R˜ 'll .
However, if DR is large, the cutoff scale is determined by
finite frequency: R˜ 'lv5AgJ/uvu. Eq. ~10! is valid if ll
,lv or DR/R,1/4. In the opposite case lv!ll and R@R0
one can neglect the second term in Eq. ~9!. Then equations of
motion ~8! and ~9! read as follows:
v5lg; R˙ 52nlR . ~11!
For lv!ll and R!R0 the last term in Eq. ~9! can be omitted.
Then
v52
16
3
kg
R4
1
ln
3R2J
16k
; R3R˙ 5
16
3 kn
1
ln
3R2J
16k
. ~12!
Thus, the easy-axis anisotropy together with the fourth-
order exchange term fix the radius ~6! of the Skyrmion if k
.0. In the opposite case k,0 there is no stable configura-
tion with nontrivial topology. We also have shown that the
Skyrmion reaches its equilibrium radius within characteristic
time tr5(nl)21.
Another property of a real film which should be taken into
account is the lattice discreteness. We have mentioned earlier
that in the continuous model the Skyrmion is a topological
excitation and as such cannot dissipate. However, in the dis-
crete lattice the continuity of the field S~r! is lost and the
RAPID COMMUNICATIONS
PRB 58 R8891SKYRMION IN A REAL MAGNETIC FILMvery notion of topological excitation becomes inconsistent.
Therefore, the Skyrmion configuration in the discrete lattice
is unstable. Moreover, it is sufficient to remove one plaquette
in the center of the Skyrmion to make it unstable ~see, for
example, Ref. 8!. We will imitate the discreteness effect by
considering a hole in the center of the Skyrmion. In the pic-
ture due to BP the Skyrmion is described by a mesomorphic
function which has a pole in the center of the Skyrmion. This
pole cannot be removed by any continuous variation of the
field S. However, the Skyrmion configuration with a hole
punched in its center will dissipate.
Let us first consider the Skyrmion without anisotropy.
‘‘Punching a small hole in the center’’ means the substitution
J!Ju(r2r0) in Eq. ~1! where r0!R is the radius of the
hole and u(x) is the step function: u(x)50 for x,0 and
u(x)51 for x.0. Hence, the perturbation H1 is given by
H˜ 152JE u~r02r !~¹S!2. ~13!
Employing the orthogonality condition of the right-hand side
of the linearized Eq. ~7! to the zero modes and neglecting all
the terms of the higher order in r0 , one finds that Eq. ~8! still
holds, but Eq. ~9! must be replaced by
v ln
R˜ 2
R2
5g
Jr0
2
R4
, ~14!
where R˜ 'lv5AgJ/uvu is the scale where the perturbation
scheme breaks down. With the logarithmic accuracy one can
write ln(R˜ 2/R2)5ln(R2/r02). Finally, substituting v from Eq.
~14! into Eq. ~8!, we find
R3R˙ 52n
Jr0
2
ln
R2
r0
2
. ~15!
From this equation one can conclude that the Skyrmion’s
lifetime is roughly proportional to its radius in the fourth
power.
Returning to the field with the easy-axis anisotropy and
the fourth-order term, let us introduce again a hole in the
center. In this situation the perturbation is given by the sum
of the two terms ~5! and ~13!. Using the same orthogonality
trick one gets
22
v
g R
2ln
R˜ 2
R2
2
32
3
k
R2
12R2l ln
R˜ 2
R2
524J
r0
2
R2
. ~16!
After substitution k!k˜ 5k2(3/8)Jr0 Eq. ~16! acquires the
same form as Eq. ~9!. It means that, as long as k
.(3/8)Jr02, the Skyrmion is stable and its radius is defined
by Eq. ~6! with k˜ instead of k. Equations ~10!, ~11!, and ~12!
are valid as well after the same substitution. In the case k
,(3/8)Jr02, however, the effective k˜ is negative and the
stable Skyrmion exists no longer.
The translation motion of a Skyrmion can be studied by
the same technique as well. In order to move the Skyrmion,a nonuniform magnetic field perpendicular to the film can be
applied. The perturbative part of the Hamiltonian reads
H15E S 12 k~DS!21 12 l~12Sz2!2h~r!SzD d2x , ~17!
where h(r)5mBSH(r), mB is the Bohr magneton, S is the
spin per unit area, and H(r) is the magnetic field. The two
new zero modes associated with the translation motion are
]S0 /]X and ]S0 /]Y , where X and Y are the coordinates of
the center of the Skyrmion. Assuming that magnetic field
varies slowly, and employing the same orthogonality tech-
nique, we find equations of translational motion
vx52
gR2
2 ]Yh0ln
R˜ 2
R2
1
nR2
2 ]Xh0ln
R˜ 2
R2
,
vy5
gR2
2 ]Xh0ln
R˜ 2
R2
1
nR2
2 ]Yh0ln
R˜ 2
R2
, ~18!
where vx5X˙ , vy5Y˙ , h05h(X ,Y ), and R˜ 5min(lv ,ll ,
AJ/h0). Equations ~18! are valid if u¹huR˜ /uhu!1.
In conclusion, we proposed a general framework for de-
scription of any slow motion in a system of Skyrmions. We
have shown that in a real ultrathin ferromagnetic film with
easy-axis anisotropy the Skyrmion known for the isotropic
model still exists, but it acquires a definite radius R0 given
by Eq. ~6!. This result was obtained in Ref. 9 by other meth-
ods. By the order of magnitude R0;Aall!ll where ll is the
domain wall width. Once we made a domain with the re-
versed magnetization in a ferromagnetic film, it shrinks
down to the size R0;1nm ~see also Ref. 9!. The magnetic
moment of such a Skyrmion is still rather large, mS
5mBp(ll /a)ln(ll /a);300mB , and can be observed experi-
mentally by the methods employed to discover the Skyrmi-
ons in QHE systems.4 The Skyrmion stability allows us to
create it with a magnetic tip.
The discreteness of the lattice in the isotropic model leads
to a finite Skyrmion lifetime which is roughly proportional to
the fourth power of its radius. However, anisotropy together
with the higher order exchange interaction stabilizes the
Skyrmion. At finite temperature it can decay through an in-
stanton configuration. Our results allow us to understand
why the activation energy found by Waldner is so close to
4pJ: the difference is expected to be of the relative order
a/ll;1022. The detailed thermodynamics of Skyrmions
will be published elsewhere.
This work has been supported by the NSF Grant No.
DMR-9705812 and by the DOE Grant No. DE-FGO3-
96ER45598. We are indebted to B.I. Halperin for useful dis-
cussions and to B. Ivanov who attracted our attention to
Ref. 9.
RAPID COMMUNICATIONS
R8892 PRB 58AR. ABANOV AND V. L. POKROVSKY1 See, for example, T. H. O’Dell, Ferromagnetodynamics: The Dy-
namics of Magnetic Bubbles, Domains, and Domain Walls ~Hol-
sted Press, 1981!, and references therein.
2 T. H. R. Skyrme, Proc. R. Soc. London, Ser. A 247, 260 ~1958!.
3 A. A. Belavin and A. M. Polyakov, Zh. Eksp. Teor. Fiz. Pis’ma
Red. 22, 503 ~1975! @JETP Lett. 22, 245 ~1975!#; see also A. M.
Polyakov, Gauge Fields and Strings ~Harwood Academic Pub-
lishers, Chur, Switzerland, 1987!.
4 E. H. Aifer, B. B. Goldberg, and D. A. Broido, Phys. Rev. Lett.
76, 680 ~1996!.5 S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi,
Phys. Rev. B 47, 16 419 ~1993!.
6 F. Waldner, J. Magn. Magn. Mater. 54–57, 873 ~1986!; Phys.
Rev. Lett. 65, 1519 ~1990!.
7 L. D. Landau and E. M. Lifshitz, Statistical Physics Part II ~Per-
gamon, New York, 1984!, Vol. 9.
8 F. D. M. Haldane, Phys. Rev. Lett. 61, 1029 ~1988!.
9 B. Ivanov, V. A. Stephanovich, and A. A. Zhmudskii, J. Magn.
Magn. Mater. 88, 116 ~1990!; Sov. Phys. JETP 64, 376
~1986!.


Skyrmion in a real magnetic film

Abstract

Similar works

Full text

Available Versions

Texas A&M Repository

Texas A&M University

Skyrmion in a real magnetic film

Abstract

Similar works

Full text

Available Versions

Texas A&amp;M Repository

Texas A&M University

Texas A&M Repository