Topological textures and their bifurcation processes in 2D ferromagnetic thin films

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by  Jinlü Cao

Topological textures and their bifurcation processes in 2D ferromagnetic thin films
Jinlu¨ Cao,1,2 Guo-Hong Yang,1,3 and Ying Jiang1,2,3,∗
1Department of Physics, Shanghai University, Shanghai 200444, P.R. China
2Qian Weichang College, Shanghai University, Shanghai 200444, P.R. China
3Key Lab for Astrophysics, Shanghai 200234, P.R. China
In this paper, by the use of the topological current theory, the topological structures and the
dynamic processes in thin-film ferromagnetic systems are investigated directly from viewpoint of
6
topology. Itisfoundthatthetopologicalchargeofathin-filmferromagneticsystemcanbechanged
1
by annihilation or creation processes of opposite polarized vortex-antivortex pairs taking place at
0
space-time singularities of the normalized magnetization vector field of the system, the variation
2
of the topological charge is integer and can further be expressed in terms of the Hopf indices and

n Brouwerdegreesofthemagnetization vectorfieldaroundthesingularities. Moreover,thechangeof
a thetopologicalchargeofthesystemiscrucialtovortexcorereversalprocessesinferromagneticthin
J films. With thehelp of the topological current theory and implicit function theorem, theprocesses

4 of vortex merging, splitting as well as vortex core reversal are discussed in detail.
1
PACSnumbers: 75.70.Kw,75.60.Jk,03.75.Lm

]
l
l
a I. INTRODUCTION sal can be achieved by applying short bursts of alternat-
h
ingout-of-planemagneticfield[17,24]orin-planecurrent
-
s [1, 28, 29]. The reversal of the vortex core is achieved
e Many efforts have been devoted to the magnetic
by the creation of a vortex-antivortex pair of whose po-
m skyrmionsandvorticesinmagneticthinfilmsduetotheir
larity is opposite to the initial-existing vortex core [29]
potential application in spintronics [1–11]. In ferromag-
t. , the newly formed antivortex and the oppositely polar-
a netic systems,skyrmionsandvorticesaretwomostcom-
izedinitialvortexsubsequentlyannihilate,onlythenewly
m mon types of topological solitons[12, 13].
formed vortex survived, leaving the impression that the
- Actually, skyrmions or vortices can naturally arise in polarity of the vortex core is reversed.
d
2Dmagneticthinfilmsthatexceedcriticalsize[14]. Each
n Stimulated by the above mentioned progress, in this
ofthesetopologicaltexturesconsistsofacoreanditssur-
o paper,byutilizingthetopologicalcurrenttheory[30–33],
rounding magnetizationvector field [1, 12, 15]. The core
c wearegoingtoinvestigatethetopologicalstructuresand
[ is a small region in the 2D medium that has M~ vectors the dynamic processes in a ferromagnetic thin film theo-
1 pointing perpendicular to the plane of the film[16]. The retically from a more fundamental viewpoint, i.e. along
v magnetic skyrmions are magnetic configurations whose the avenue of topology.
7 magnetizationvectors at spatial infinity are out of plane
The rest of the paper is organized as follows. In sec-
8 andantiparalleltothemagnetizationvectorsatthecores,
tion II, by making use of the topological current theory,
4 while the magnetic vortices, on the other hand, have in-
the inner structure as well as the change of the topolog-
3 plane magnetic vectors at spatial infinity.
0 ical charge of the system is discussed, we find that an
. ThesignoftheMz componentsinthecoreregiongives annihilation or creation process of a vortex-antivortex
1
risetothepolarityλofatopologicaltexture. Apartfrom pairwithoppositepolarityleadstoanintegerleapofthe
0
the core,the M~ vectors around the core vary from point topological charge. In order to investigate the detail in-
6
1 to point. The total number of times the magnetization formation of the vorticity of magnetic vortices and the
: vectorswindaroundthe coreis the so-called2Dwinding dynamicprocesses,suchasvortexannihilation,merging,
v
number S or vorticity of the topological texture. as wellas splitting processes,the correspondingtopolog-
i
X Due to their potential applications for information icalcurrent is constructed in Section III, and the branch
r storage and information manipulation[17–19], the study processesaswell asthe bifurcationproperties ofthe sys-
a
of the static and dynamic properties of vortices has re- tem is discussed. Section IV is devoted to the discussion
centlybecomeoneofthemaintopicsinthisfield[20–26]. of the vortex core switching process by the use of the
In fact, magnetic vortex cores have indeed been consid- newly constructed topological current.
eredaspossiblecandidatesformagneticdatastorage[27].
In order to store and manipulate information by utiliz-
ingvortexcores,mechanismsforacontrolledswitchingof
II. THE TOPOLOGICAL CHARGE IN A 2D
itsorientationarerequired. Experimentalandnumerical
THIN FILM AND ITS CHANGE
investigation shows that the magnetic vortex core rever-
Asiswellknown,inaferromagneticmediawhosetem-
perature is sufficiently below the Curie point [34], the
∗Correspondingauthor;Electronicaddress: [email protected] magnetization vectors M~ is governed by the Landau-
2
Lifshitz equation [1, 29, 34, 42] the singularities of m~ directly correspond to the zero
points of M~(t,x,y). The topological charge density in
∂m~ ∂m~
+m~ ×f~=κm~ × (1) Eq. (2) can now be rewritten in terms of M~ as
∂t ∂t
1
in terms of the normalized magnetization vector m~ = q = ǫ0µνǫ ma∂ mb∂ mc
abc µ ν
M~/||M||, f~ is the effective magnetic field, while the κ 8π
1 Ma
termservesasthe(Landau-Gilbert)dissipationterm[35]. = ǫ0µνǫ ∂ Mb∂ Mc. (8)
The topological charge of the magnetic configuration 8π abckM k3 µ ν
of the system is determined by the so-called skyrmion
Inthis expression,the Levi-Civitasymbolsareused,and
number [35]
forconvenience,thespace-timevariablesarerewrittenas
1 zµ = (t,x,y) where the superscript µ takes the value of
Q= m~ ·(∂ m~ ×∂ m~) d2r, (2)
x y 0,1 and 2. By making use of topological current method
4π
Z [31–33]andLaplacianGreen’sfunctionrelation,straight-
Itis a topologicalinvariantprovidingm~(t,x,y) isalways forward calculation shows that[30]
welldefinedinthewholesystem. Normally,m~(t,x,y)can
be factorizedbythe so-calledstereographicprojectionto ∂ q =D M δ(M~) (9)
spherical variables Θ=Θ(t,x,y) and Φ=Φ(t,x,y) as t z
(cid:18) (cid:19)
mx(t,x,y)=sinΘcosΦ, where D M is the Jacobian of M~ and zµ. Eq. (9)
z
 my(t,x,y)=sinΘsinΦ, (3) confirms that when M~ possess no zero points, i.e. when
m (t,x,y)=cosΘ, (cid:0) (cid:1)
 z m~ iswelldefinedinthewholespace-time,thetopological
chargeisconserved. However,whenm~ possessessingular
this leads to 
events, the topological charge Q of the system will be
1 changed.
Q = − dm dΦ (4)
x z
4π Suppose there are l singularities and the i-th one is
located at the space-time coordinates zµ = (t ,x ,y ), it
BynotingthatS = 1 dΦisjustthevorticity(winding i i i i
2π is not difficult to verify that
number) of the topological texture, it is not difficult to
R
get that l
β
δ(M~)= c δ(zµ−zµ), c = i (10)
1 i i i |D(M) |
Q=−2S·[mz|ρ=+∞−mz|ρ=0] (5) Xi=1 z z~i
whereβ istheHopfindex[37]ofM~-fieldaroundthei-th
we then have that the topological charge is expressed as i
zero point at z .
i
1λS, for vortices, Substituting Eq.(10) back to Eq.(9) yields
Q= 2 (6)
( λS, for skyrmions, l
∂ q = β η δ(zµ−zµ) (11)
t i i i
where λ = m | is the polarity of the topological tex-
z core i=1
ture. X
However,theaboveparametrizationtotallyneglectthe where η =sgnD(M) is the Brouwer degree [37] of M~-
singularities of the system, and cannot be used to in- i z zi
field,andβ η isthegeneralizedwindingnumberin2+1-
i i
vestigate the dynamic processes with topological charge
dimensional space-time wrapping around the i-th sin-
changed. Actually, the singularities may exist in 2d lat-
gular event. Therefore, the total change of topological
tice magnetic systems [3, 30, 36]. As is pointed out [36],
chargeisthesumoftheallthegeneralizedwindingnum-
for a magnetic system with lattice structure, when tak-
bers of the singular events emerging in the given time
ing continuum limit, singularities of the configurationm~
interval
away from the lattice sites may be allowed, and this
will violate the topological conservation in the system l
[30, 36]. Therefore, in order to investigate the the topo- ∆Q= dt d2x∂tq = βiηi, (12)
logicalstructureandthedynamicprocessofthemagnetic Z Z i=1
X
configuration generally, the condition for a well-defined
due to the second homotopy [38] of sphere π (S2) = Z,
m~(t,x,y) has to be relaxed. 2
this change is an integer.
As is known, the unit vector m~ is expressed in terms
Actually, these singular events correspond to annihi-
of the magnetization field as
lation (or creation) processes of vortex-antivortex pairs
Ma with opposite polarities in the system. In order to illus-
ma = , (7)
kM k trate this point clearly, let us consider an annihilation
3
process of a vortex-antivortex pair with opposite polari- topologicalchargeQwillkeepunchangedinthisprocess.
ties, the vorticity of the vortex (antivortex) is 1 (−1). (cid:3)
m
The annihilation process is characterized by the phe- z
nomena that cores of the vortex and antivortex collide
m (cid:32)(cid:14)1
and merge into one, the merging point is denoted as z
S(cid:32)(cid:16)1 S(cid:32)(cid:14)1
z =(t ,x ,y ). Apparently, when the annihilation pro-
0 0 0 0
cesshappens,duetotheoppositepolarity,atthemerging
point all three component of the magnetization M~ van-
ishes,leadingtoasingularityofm~ inthe2Dfilm,accord- m (cid:32)(cid:16)1
z
ingtoEq. (12),therewillbeanintegerchangeofQatz .
0
Thismayalsobecross-checkedbyutilizingEq. (5). From
thatexpressionofthetopologicalcharge,itiseasytofind FIG.2: (Coloronline)Theredandbluearrowsrepresentthe
that the total topological charge before the annihilation mz componentsofthecoreofthevortexandtheanti-vortex,
isQ =λ(λ=±1isthepolarityofthevortexbeforethe respectively. Becausetheyhavethesamepolarity,thearrows
i
pointinthesamedirection. Astheymovecloserandcloserto
annihilation) while the total topologicalcharge after the
annihilation is Q = 0, thus ∆Q = Q −Q = −λ. The annihilationpoint,mz componentisidenticaltotheprevious
f f i core.
annihilation process is sketched in Fig. 1.
From above discussion, we see that since ∆Q = 0,
the annihilation/creation of vortex-antivortex pair par-
allel polarized cannot be properly reflected in this sense.
Evenfor∆Q6=0processes,i.e. theannihilation/creation
ofoppositepolarizedvortex-antivortexpair,the detailed
information also need to be decrypted. Hence, we shall
take a step further to construct a topological current
which is suitable for discussing the corresponding topo-
logical property systematically.
III. THE TOPOLOGICAL CURRENT THEORY
FOR VORTICES IN 2D THIN FILMS
Theabovediscussiontakesthetopologicalstructureof
thesystemasawhole,thedetailstructureofvorticesand
the corresponding processes can not be exposed by the
formula in previous fashion, especially for annihilation
(creation)processesofvortex-antivortexpairswithequal
polarities. Luckily, it is not necessary to take all three
componentsofthe magnetizationvectorsintoaccountto
constructasuitabletopologicalcurrentforvorticesinthe
system. First, it is sufficient to pinpoint the position of
thevortexcorebysolvingm =m =0forx(t)andy(t)
x y
FIG. 1: (Color online) The sketch of an annihilation process atagiventimet. Second,theprimarytopologicalfeature
ofavortex-antivortexpairwithoppositepolarities. Thecolor Q canbe decomposedinto twoparameters: the vorticity
gradient barontheright handsideindicates thevalueof the
S and the polarity λ. Among them, the polarity can be
mz component. Eachlayeroftheplanerepresentsatimeslice
treated independently, for it is just the value of m at
of the magnetic configuration at that moment. z
the vortex core which can be directly read out from the
configuration. Hence, only the property of the vorticity
Without any difficulty, it can also be seen that a cre- of the topological texture need to be concentrated on in
ation process of a vortex-antivortex pair with opposite the following.
polarities is also accompanied by ∆Q = ±1, this was
alsorecognizednumerically[1]. Moreover,processeswith
|∆Q| > 1 are also allowed, they correspond to multiple A. Constructing 2D topological current
annihilation(creation)eventshappening simultaneously.
Forannihilationprocessofvortex-antivortexpairwith By choosing (m (t,x,y),m (t,x,y)) as the two com-
x y
the same polarities, however, as shown in Fig. 2, the ponents of a vector field ψ~, i.e.
m component at the merging point will exactly be the
z
valueofthecorepolarityofeachonebeforethemerging, ψ1 =m (t,x,y)
x , (13)
and there is no singular event involved at all and the ψ2 =m (t,x,y)
y
(cid:26)
4
a topological current can be constructed as follows B. The branch process of topological current
1
jµ = ǫµλρǫ ∂ na∂ nb, (14) With the properly constructed topological current jµ
ab λ ρ
2π
in place, we can apply the branch process in φ-mapping
where na = ψa/ k ψ k. It can be easily verified that jµ theory[32, 33] to provide an understanding of the dy-
namical behaviors of the vortices at these topologi-
is identically conserved, i.e.
cally nontrivial sites with the evolution of time. For a
∗ ∗ ∗
∂µjµ =0. (15) vo∗rtex-a∗ntivortex pa∗ir loca∗ted at∗ p~i and p~j, i.e. p~i =
(x (t),y (t)) and p~ = (x (t),y (t)) are corresponding
i i j j j
By defining the Jacobian determinants Jµ as solutions of
ψ1(t,x,y)=0
ψ (21)
ǫabJµ =ǫµλρ∂ ψa∂ ψb, (16) ψ2(t,x,y)=0
λ ρ
x (cid:26)
(cid:18) (cid:19)
For the branch process, say, an annihilation event, to
with the help of the relationbetween na and ψa and the
occur, the cores of the two vortices have to move closer
2D Laplacian Green’s function relation, a further calcu- ∗ ∗
andcloser,untilp~ andp~ overlap. Asaresult,theanni-
lationshowsthat the topologicalcurrentinEq. (14) can i j
hilation event can be characterized by p~ = (x (t),y (t))
then be rewritten in terms of ψa as i i i
being a repeated root for Eq.(21), thus
ψ
jµ =Jµ δ(ψ~). (17) ψ ∂(ψ1,ψ2)
x J0 = =0. (22)
(cid:18) (cid:19) x ∂(x,y)
(cid:18) (cid:19)(cid:12)p~∗i (cid:12)p~i∗
(cid:12) (cid:12)
It is obvious that jµ is nonzero only when ψ~ = 0. Ac- (cid:12) (cid:12)
Therefore, the sp(cid:12)atial-temporal (cid:12)coordinates of the
tually, this expression indicates that all the vortices are
located at the zero points of ψ~, i.e. at the zero points of branch process is the solution of
ma =(m ,m ), as we expected. Moreover,the vorticity
x y ψ1(t,x,y)=0
of each vortex in the system can also be decrypted out
of the topological current. ψ2(t,x,y)=0 (23)
Suppose ψ~ possessl zeroes,an∗dthe∗i-tho∗ne is located ψ3(t,x,y)≡J0 ψ =0
at the spatial-temporal point ~p = (x (t),y (t)), by de-
i i i x
choemlppoofstihnegiδm(ψp~l)iciintftuenrmctsioonftδh(e~xo−remp~∗i,)w, htoegnep~th∗earrewritehgutlhaer Infact, there aretwopossible br(cid:18)anch(cid:19)points, namely the
i
points, the topological current jµ can further be rewrit- limit points and bifurcation points [32, 33]. satisfying
ten as [30–33]
ψ
J1( ) 6=0, (24)
jµ = l β η δ(~x−p~∗(t))Jµ ψx , (18) x (cid:12)(cid:12)(t∗,p∗i)
Xi=1 i i i J0(cid:16)(cid:16)ψx(cid:17)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)~x=p~∗i and (cid:12)(cid:12)
(cid:12) ψ
β is the Hopf index of the vector field ψ~(cid:12)at zero point J1( ) =0, (25)
i x
p~∗i and ηi = ±1 the corresponding Brouwer degree [37]. (cid:12)(cid:12)(t∗,p∗i)
Actually, the winding number W = β η just represents (cid:12)
i i i ∗ respectively. Inthefollowi(cid:12)ng,wearegoingtodiscussthe
the vorticity S of the i-th vortex located at ~p , η = 1
i i branchprocessestaking place atthese twodifferent type
for vortex and η =−1 for antivortex. The density of the
of branch points separately.
vorticity of the topological texture of the system reads
• Branch Process at Limit Point
l
ρ=j0 = β η δ(~x−~p∗(t)), (19)
i i i
Fromimplicit function theorem, we can see that when
i=1
X
J0(ψ) =0, the solution of the i-th singular event
while the velocity of i-th vortex(antivortex) moving in x (t∗,p∗i)
the ferromagnetic thin film is xi =x(cid:12)(cid:12)i(t),yi =yi(t) is not unique. However, in the case
of limi(cid:12)t point, Eq. (24) is satisfied, it implies that x and
Jµ ψ t can change their roles, and the system has a unique
x =vµ,µ=1,2 (20) solution of the form
D (cid:16)ψ (cid:17)(cid:12)(cid:12)
(cid:16)x(cid:17)(cid:12)(cid:12)~x=p~∗i ti =ti(x), yi =yi(x) (26)
(cid:12)
(cid:12)
5
∗ ∗
in the vicinity of the limit point (t ,p~ ). By the use of thatthereexistsoneandonlyonefunctionrelationsolved
i
the third equationinEq. (23)andEq. (24), wesee from from ψ1(t,x,y)=0, that is
dt ∂(ψ1,ψ2) ∂(ψ1,ψ2) y =f(x,t) (29)
= (27)
dx ∂(x,y) ∂(t,y)
(cid:30) We denote the partial derivatives as f = ∂f,f =
1 ∂x t
that the first term in the Taylor expansion for ti =ti(x) ∂∂ft,f11 = ∂∂2xf2,f1t = ∂∂x2∂ft,ftt = ∂∂2t2f. From Eqs. (23)
intheneighborhoodoflimitpoint(t∗,p~∗)vanishes,thus, and (29), we have
i
theTaylorexpansionuptothesecondorderinthe vicin-
ity reads ψ1 =ψ1(t,x,f(x,t))=0 (30)
1 d2t which give
t−t∗ = (x−x∗)2 (28)
i 2 (dx1)2(cid:12)(cid:12)(cid:12)p∗i i ∂∂ψx1 =ψ11+ ∂∂ψf1∂∂fx =0 (31)
Thisisaparabolainthex-tplan(cid:12)e,witheachofitsbranch
∂ψ1 ∂ψ1∂f
corresponds to a trajectory of a vortex core hurtling to- =ψ1+ =0 (32)
∗ ∗ ∂t t ∂f ∂t
wards the merging point p at time t , and an annihila-
i
tion event occurs.
Generally, (ddx2t)2|p∗i can be either positive or negative, Banyddti,ffearpepnltyiaintigngthEeqGs.a(u3s1si)ananedlim(3i2n)atwioitnhmreestpheocdt,tothxe
corresponding to the generation and annihilation of the second order derivatives f ,f and f can be found.
11 1t tt
vortex-antivortex pair, respectively. But in either cre- The aboveresultsaremade outquite independent ofthe
ationorannihilationcase the winding number S (vortic- last component ψ2(t,x,y). In order to find the different
ity), due to Eq. (15), should be conserved. trajectoriesin the bifurcationprocess,the Taylorexpan-
sionofthe remainingcomponentψ2(t,x,y)inthe neigh-
x1 borhood of (t∗,~p∗i) should be studied. By substituting
Eq. (29) into ψ2(t,x,y), we define
(cid:3) F(x,t)=ψ2(t,x,f(x,t)) (33)
x1(cid:32)x1*
i
t By using the condition for branch process, i.e. Eq. (23),
we have
t(cid:32)t*
∗ ∗
F(t ,p~ )=0. (34)
i
Besides, from Eq. (33), the 1st order derivatives are
FIG. 3: (Color online) A parabola in the x1−t plane, with
each of its branch corresponds to a trajectory of a vortex ∂F ∂ψ2 ∂F ∂ψ2
coremovingtowardsthemerging pointp∗i at timet∗,andan ∂x1 =ψ12+ ∂f f1 =0, ∂t =ψt2+ ∂f ft =0 (35)
annihilation eventemerges.
By utilizing Eqs. (31) and (32), the Jacobian determi-
nant J0(ψ/x) in the third equation of Eq. (23) can be
• Branch Process at Bifurcation Point rewritten as
∂F
Apart from the limit point, what constitutes the larger ∂xJ10|t∗,p~∗i =0. (36)
variety of the vortices’ behavior comes from the bifur-
cation points. A bifurcation process occurs when the As J0 6=0, the above equation gives
1
Jacobian J1(ψ/x) = 0. From Eq. (27), one can
easily see that the v(te∗l,opc∗ii)ty of the vortex core is undeter- ∂F
(cid:12) =0 (37)
mined. Therefore,(cid:12)it is easy to picture the situation that ∂x(cid:12)t∗,p~∗i
the trajectory of the core at the bifurcation point will (cid:12)
(cid:12)
diverge into multiple curves, because the function rela- Similarly, (cid:12)
tionship between t and x is not unique in the vicinity of
∗ ∗ ∂F
the bifurcation point (t ,p~i). =0 (38)
Because the Jacobian J0(ψ/x) = 0, the rank of the ∂t (cid:12)t∗,p~∗i
Jacobian will be smaller than 2. As in our case, for (cid:12)
(cid:12)
practical reasons, we consider the rank to be 1. As- The second order partial(cid:12)derivatives of F, can be cal-
sume J0 = ψ1 ≡ ∂ψ1 is nonzero in the Jacobian matrix culated from Eq.(35), and we further denote them by
J0(ψx) =1 0. T2hen t∂hye implicit function theorem ensures A= (∂∂2xF)2 t∗,p~∗i,B = ∂∂x2∂Ft t∗,p~∗i,C = ∂∂2tF2 t∗,p~∗i.
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
6
By virtues of Eqs. (34), (37), and (38) the Taylor processes have been numerically simulated successfully
expansionofF(x,t)inthevicinityofbifurcationpointis [29,39]. Immediatelyafterafieldpulse[39]orapolarized
currentpulse [29] is applied to a 2D nanodisk possessing
A(x−x∗i)2+2B(x−x∗i)(t−t∗)+C(t−t∗)2 =0 (39) a vortex with positive polarity, i.e. λ = mz|core = +1,
the magnetization of the nanodisk is heavily distorted
∗
Divide Eq. (39) by (t−t ), and take the limit of t →
and a magnetic bubble [34, 35, 40] with negative polar-
∗ ∗
t ,x→x , we get
i ityisformed,leadingtoacreationofavortex-antivortex
pair. The newly formed antivortex and the oppositely
dx dx
A( )2+2B +C =0 (40) polarized initial vortex subsequently annihilate, leaving
dt dt
the new vortex oppositely polarized with respect to the
or initial one.
Due to its potentialapplicationin informationstorage
dt dt
C( )2+2B +A=0 (41) and manipulation, it is worthto investigate the topolog-
dx dx
ical property of the system in the vortex core switching
while the remaining component dy is determined from processes.
dt Although multiple switching process can be discussed
y =f(x,t), which yields
in the same fashion, for simplicity and clarity, we only
dy dx set out to investigate the single switching process. By
=f +f (42)
dt 1dt t the use of the topological current theory, we will show
that the first step in the core switching process, i.e. the
Theaboveequationshowsthatthesecondcomponentsof creation of a vortex-antivortex pair, can be classified as
thetrajectoryatthebifurcationpointcandivergeaswell, a∆Q=0bifurcationprocesswhilethelattersubprocess
for ddxt may choose different values. This can be used to is a ∆Q = 1 vortex-antivortex pair annihilation process
explain phenomena which is categorizedas a bifurcation at a limit point.
process. For instance, a case when two vortices coming In order to study the detail topological information
into a merging point and then diverge, a single vortex of the dynamic process, an initial state need to be con-
due to certain reasons spilt into several, a number of structed to describe the magnetic configuration of the
verticesmergeintooneatthebifurcationpoint,etc. Fig. diskrightafterthemagneticbubbleinducedbythepulse.
4 depicts one of the possible scenarios described above. Then, by plugging it back into Landau-Liftshitz-Gilbert
equation as an initial state, the property of the vortex
system will be revealed.
For simplicity but without loss of generality and the
topology of the vortex state, the trial state may be
constructed by the use of the general static solutions
of the Landau-Lifshitz equation, that is the well-known
Belavin-Polyakov instanton [35, 41, 42]. This is a
staticsolutionofthesimplifiedLandau-Lifshitzequation,
which only includes the exchange interactions [35]:
∂m~
=m~ ×∆m~, |m~ |=1,
∂t
By re-parameterizing the magnetization vector m~ with
the complex quantity
m +im
x y
Ω=
1+m
z
and using the complex variables z = x + iy, the LLE
reduces to
FIG. 4: (Color online) A sketch of a single vortex splitting
i ∂Ω 2ΩΩ Ω
z z
into multiple vortices at thebifurcation point. +Ω = . (43)
zz
4 ∂t 1+ΩΩ
Allsolutionsofthisequationcanbeconstructedfromthe
followingfourtypesofbasicsolutionswhich,byproperly
IV. THE VORTEX CORE SWITCHING IN A 2D choosing the coordinate system, take forms as
FERROMAGNETIC THIN-FILM
z a
2
Ω = , Ω = ,
1 2
a z
With the above established topological current in (cid:18) 1(cid:19) (cid:16) (cid:17)
z a
hand, we can now investigate the vortex core switching Ω = , Ω = 4 ,
3 4
process in detail theoretically. The vortex core reversal (cid:18)a3(cid:19) (cid:16) z (cid:17)
7
inthisexpression,thecorrespondingcoreforeachtypeof
basicconfigurationisexactlylocatedattheoriginalpoint
of the coordinate system, and a represents the relative
i
sizeofeachtype ofvortex. It isnotdifficult tocheckthe
polarityλ=limρ→0mz =limρ→0 1−ΩΩ ofeachtype and
1+ΩΩ
find
λ =1, λ =−1,
Ω1 Ω2
λ =1, λ =−1.
Ω3 Ω4
The vorticities for Ω and Ω are 1, while both Ω and
1 2 3
Ω have vorticity of −1.
4
The pulse creates a magnetic bubble with opposite
FIG.5: (Coloronline)ThemagneticconfigurationinEq. (45)
polarity with respect to the initial-existing vortex, i.e.
λ = −1, only Ω and Ω are suitable candidates to con-
2 4
struct the corresponding magnetic configuration of the
whereαistheGilbertdampingconstant,β isthedimen-
induced magnetic bubble. Moreover, due to the conser-
vation property ofthe topologicalcurrent, i.e. ∂ jµ ≡0, sionless parameter describing the nonadiabatic process,
µ
and
the 2D winding number of the system should be con-
served, hence the configuration for the magnetic bubble H~ =∆m~ +γ′(∂ (∂ m ),∂ (∂ m ),0) (46)
can only be a superposition of Ω and Ω which reads e x µ µ y µ µ
2 4
a 2 is the effective field containing the exchangeinteractions
Ω=Ω2Ω4 = (44) and the magnetostatic energy. By substituting the de-
ρ
(cid:18) (cid:19) tailed expression of the magnetic configuration in Eq.
where ρ = zz = x2 + y2, a2 = a2a4 representing the (45) into above equations, direct calculation shows that
relative size of the magnetic bubble [35].
′
By taking the initially existing vortex with λ=1 and ∂m~ 4bγ(γ +2)
= (1,−1,0), (47)
S = 1 into account, the total configuration of the entire ∂t a2d(1+α2)
(cid:12)ρ→0,t=0
disk is Ω = Ω1Ω2Ω4, for simplicity but without loss of (cid:12)
(cid:12)
topologicalinformation, the magnetic configurationpos- Further calcul(cid:12)ation leads to
sessing the same topological feature as in the simulation
[29] reads J1 ψ = ∂(mx,my) =0 (48)
m =− 2by x2a+2y2 n (cid:18)x(cid:19)(cid:12)(cid:12)(cid:12)ρ→0,t=0 ∂(t,y) (cid:12)(cid:12)(cid:12)ρ→0,t=0
mxy = (((d(−d−2xb)x(2x)+2−+yd2y))(cid:16)2(cid:16))xx(cid:16)22aax++222yya+(cid:17)222y(cid:17)2n2(cid:17)n2+n+b2b2 (45) Nbittnhiroogeanwnsmp,cehrcawotgpicinoteohnesitn,si.btcw,(cid:12)oDebatiuhcrsbceeocDbetrldtea(cid:16)hipntψxaphgt(cid:17)leiticmos=aytmtsihto0eeendmaitoanhftwdetehi(cid:12)dlJaelifs1butce(cid:16)inufrduψxsetrs(cid:17)hcirogaenot=icoiarnne0abpptiarrfituoeocncrtecehdaoses--f
in topologicalcurrenttheory yields the differentialequa-
wheredmiszth=ebd(2(isd−t−a(cid:0)n(xcde)2−b+extwy)22e)+e(cid:16)(cid:16)nyxt22ha(cid:1)+e2(cid:16)yi2nx(cid:17)(cid:17)2iat2+i2nay2l+(cid:17)vob2n2rtexandthe ttihoensnetihgahtbogrohvoeroAnds(cid:18)otfhddtexth(cid:19)ter2ab+jiefuc2trBocardidtexitson+ofpCtohi=netv,0owrtheixchcoisre(s49in)
newly formed magnetic bubble, b reflects the size of the
core of the initial vortex. This configuration is shown in dy = ∂f dx + ∂f
dt ∂x dt ∂t
Fig. 5.
Together with Eqs. (13) and (16), for the above mag- where y = f(x,t), and the two solutions of the differ-
netic configurationit is found that J0(ψ/x)|ρ→0 = 0, in- ential equations will corresponds to the equation of mo-
dicatingthatthesystemwillundergoabranchprocessat tion of the two vortex cores split from the zero-winding-
the instant the pulsed current induces the zero-winding numbermagneticbubbleinthevicinityofthebifurcation
number vortex. To determine the type of the branch point. Moreover, according to the conserved 2D topo-
process, the dynamical equationwhich governs the vor- logicalcurrent,the newly-generatedvorticesshouldhave
tex evolution, should be brought into the fold, that is equalbutoppositevorticity,i.e. thesetwovorticesshould
the Landau-Lifshitz-Gilbert equation be a vortex-antivortex pair. The formation of vortex-
∂m~ ∂m~ antivortex pairs after application of short current pulses
=−γm~ ×H~e+αm~ × −(~u·▽)m~ +βm~ ×[(~u·▽)m~], is consistent with recent experimental observations [43].
∂t ∂t
8
After the bifurcation process,the system possesses to- the splitting vortices drift apart. In order to investigate
tally three vortices, one is the initially existing vortex the following dynamic process, the interactions between
with λ = +1, the other two are the vortex-antivortex the newly generatedvortex (antivortex)and initially ex-
pair with λ = −1, at this stage the topological current isting vortex then need to be taken into account.
describing the system now reads Asisknown[44],theeffectiveinteractionbetweenvor-
tices in the magnetic environment is
l−1 Jµ ψ
jµ =Xi=0βiηiδ(~x−p~∗i(t)) D(cid:16)(cid:16)ψxx(cid:17)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)~x=p~∗i F~ij = 2πJXSijiSjXˆij (51)
=δ(~x−p~∗(t))Jµ(ψ/x) (cid:12)(cid:12) where J is the magnetic coupling constant in the mag-
0 D(ψ/x) netic system (for our ferromagnetic thin-film system,
∗ Jµ(ψ/x)(cid:12)(cid:12)(cid:12)~x=p~∗0 J >0), Xˆij istheunitvectorpointingfromthe centerof
+δ(~x−p~ (t)) (cid:12) i-th vortex to the center of j-th vortex, and X the dis-
1 D(ψ/x) ij
(cid:12)~x=p~∗1 tancebetweenthesetwovortices,SiandSj arethecorre-
(cid:12)
∗ Jµ(ψ/x)(cid:12) sponding vorticities of the two vortices. It is then easily
−δ(~x−p~ (t)) (cid:12) (50)
2 D(ψ/x) verifiedthattheinteractionbetweenthenewlygenerated
(cid:12)~x=p~∗2 antivortex and the initially existing vortex is attractive
(cid:12)
(cid:12) ∗ whiletheinteractionbetweenthenewlygeneratedvortex
The core positions of the vortices(cid:12)are located at ~p (t),
i
and the initially existing vortex is repulsive.
respectively,andthetrajectoriesofthevorticesinthe2D
nanodiskcanbeobtained. Ifwedenotethetimewhenthe
y
first branch process event happens, i.e. the bifurcation
discussed above, to be t , then one can arrive at the
1
sketch in Fig. 6 depicting the motion of x component of
(cid:3)
the vortex core as a function of time. S(cid:32)(cid:14)1 S(cid:32)(cid:16)1
(cid:3) (cid:79)(cid:32)(cid:14)1 (cid:79)(cid:32)(cid:16)1
(cid:3)
(cid:3) x
(cid:3) annihilation point P
p i p
x1 x2
(cid:3)
(cid:3)
FIG. 7: (Color online) A sketch of a head on collision of a
vor(cid:3)tex-antivortexpair with ∆Q=−1
(cid:3)
The attractive interaction between the antivortex and
the(cid:3) vortex may lead to a collision and merge. Dur-
ing the process, due to the conservation of the vortex
momentum [35], in the center-of-mass reference frame,
the total momentum p~ = p~ +p~ should be
vortex antivortex
zero all the time during the process. Meanwhile, for the
vortex-antivortexpair,thetotalvorticityiszero,thecor-
respondingvirialrelationreads[35]~v·p~=−E,whereEis
the total energy of this vortex antivortex pair. By prop-
erlychoosingthereferencecoordinateasshowninFig. 7,
alsotakingintoaccountthatthetotalmomentumiszero
while the total energy is finite, we then have 1/v = 0.
x
Together with Eq. (20), we finally draw the conclusion
that at the merging point P∗ we have J0(ψ/x)|P∗ = 0
FIG. 6: (Color online) A sketch of process of the magnetic while J1(ψ/x)|P∗ 6=0, indicating that the merging point
bubblewithnegativepolaritycreatedbyapulsedcurrentand in this stage is a limit point. By the use of the topolog-
thensplitsintoavortex-antivortexpair. Duringthisprocess, ical current theory at limit point which is discussed in
thetotal topological charge, i.e. the skyrmion numberof the preceding section, the relation between x-components of
system, keepsunchanged, ∆Q=0
the vortex-corepositions and t is
Intheabovebifurcationprocess,thetrajectoriesofthe t−t∗ = 1 d2t (x−x∗)2 (52)
newly generated vortex and the antivortex diverge and 2 dx2
(cid:12)P∗
(cid:12)
(cid:12)
(cid:12)
9
which is a parabola in the x-t plane.
From Fig. 3, it can be recognized that the upper
and lower branch of the parabola correspond to the x-
components of trajectories of the vortex and antivortex
cores respectively. More over, if d2t < 0, then the
dx2 P∗
opening of the curve is to the left, t(cid:12)herefore with the
(cid:12)
passing oftime the positionof the vor(cid:12)tex andantivortex
corewillbecomecloserandclosertothe limitpoint, and
∗
both vortices cease to exist the moment after t , indi-
∗
cating that an annihilation event takes place at P . On
the other hand, if d2t > 0, then this limit process
dx2 P∗
will correspondto the c(cid:12)reationof vortex-antivortexpair.
(cid:12)
Actually, it is not diffic(cid:12)ult to verify that
d2t 1
sgn(dx2 )=sgn(v (x−x∗)|P∗). (53)
(cid:12)P∗ x
(cid:12)
In our case, due (cid:12)to the attractive force between the
(cid:12)
two vortices of the vortex-antivortex pair, the sign of
∗
v is always opposite to the sign of (x−x ), therefore,
x
d2t < 0, indicating that the branch process at this
dx2 P∗
limi(cid:12)t point is a vortex-antivortexannihilation process.
(cid:12)
W(cid:12)ith the final piece in place, a complete sketch of
the single switching process of magnetic vortex core in FIG. 8: (Color online) A sketch of the system’s development
a ferromagnetic thin film is drawn in Fig. 8. Long be- throught theswitching process.
fore a pulse is applied to the magnetic thin film, there
is only one vortex with vorticity S = 1 and polarity
0
λ = 1 in the system. At time t , a pulse is applied
1
whichinducesanegativepolarizedmagneticbubblewith
zero winding number. Immediately after the creation the corresponding singular events. We found that such
of such a magnetic bubble, it quickly bifurcates into a integer leap of the topological charge of the system is
vortex-antivortex pair oppositely polarized with respect alwaysassociatedwithannihilationorcreationprocesses
to the initially existing vortex. Since so far m~ is well de- of vortex-antivortex pair with opposite polarities, these
fined in the whole system, the topological charge of the processescanbelookeduponasspace-timeskyrmionsor
system keeps unchanged during the first sub-process of monopole events in the system.
the vortex-core switching. Due to the attractive inter-
action between the newly formed antivortex and the ini- In order to investigate the topological properties of
tialvortex,they annihilateeachotherattime t , leaving magneticvorticesin2Dferromagneticthinfilms,wehave
2
the newly formed negatively polarized vortex survived constructedatopologicalcurrentoutofthein-planecom-
in the system, and the vortex-core switching is accom- ponents of the normalized magnetization vector field of
plished. Clearly, the topological charge of the system is the system. We found that the vorticity of a vortex can
Q = 1/2 before the annihilation event while Q = −1/2 further be expressedin terms of the Brouwerdegree and
afterthat,henceduringthesecondsub-processofvortex- Hopf index of the normalized magnetization vector field
core switching, ∆Q = −1, indicating that there is a around the vortex core. We found that the creation
space-time singular event involved in the process. (annihilation) of vortex-antivortex pairs correspond to
the topological current branch processes at limit points,
while the merging and splitting of vortices can be ex-
V. CONCLUSION plained by the topological current branch theory at bi-
furcation points. By the use of this topological current
In summary,by the use ofthe topologicalcurrentthe- theory, the vortex-core switching process in a 2D fer-
ory, we have discussed the topological structure and dy- romagnetic nanodisk has been discussed in detail. Our
namic processes in a thin-film ferromagnetic system. discussionisconsistentwithrecentnumericalsimulations
We found that the topological charge of the magnetic [17, 29].
system can be changed when space-time singularities of
normalizedmagnetizationfieldofthesystemareallowed, We believe that the topologicalcurrenttheory used in
and the variation of the topological charge can only be this paper may shed a light on theoretical study of the
integer which is exactly the total wrapping number of magnetic system directly from topological viewpoint.
10
Acknowledgments portedbyNationalNaturalScienceFoundationofChina
underGrantNo. 11275119andbyPh.D.ProgramsFoun-
TheauthorsgreatlyacknowledgeYaowenLiuforstim- dation of Ministry of Education of China under Grant
ulating and fruitful discussions. This Work was sup- No. 20123108110004.
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