Table Of ContentDomain wall propagation due to the synchronization with circularly polarized
microwaves
0 P. Yan and X. R. Wang
1 Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR, China
0
2 Findinganewcontrolparameterformagneticdomain wall(DW)motion inmagneticnanostruc-
turesis important in general and in particular for the spintronics applications. Here,we show that
n
a circularly polarized magnetic field (CPMF) at GHz frequency (microwave) can efficiently drive a
a
J DWtopropagatealongamagneticnanowire. Twomotionmodesareidentified: rigid-DWpropaga-
tion at low frequency and oscillatory propagation at high frequency. Moreover, DW motion under
2
aCPMFisequivalenttotheDWmotionunderauniformspincurrentinthecurrentperpendicular
1
to the plane magnetic configuration proposed recently by Khvalkovskiy et al. [Phys. Rev. Lett.
] 102, 067206 (2009)], and theCPMF frequency plays therole of thecurrent.
l
l
a
h Controlled manipulation of magnetic domain-wall of cm~ ~s (c-term). b and c are parameters roughly pro-
- ×
s (DW) propagation in nanowires has spurred intensive portionaltocurrent[5,6]. Theeffectsofb-andc-termson
e research in recent years[1–12] in nano-magnetism be- DW propagation are very different. b-term is incapable
m
cause of its fundamental interest and the potential im- of generating a sustained wall motion, except at very
. pact in spintronic device technology. Both static mag- largecurrent,while c-termcandrive a DW to propagate
t
a netic fields[1–4] and electric currents[5–12] can be the along the carrier direction[7]. Unfortunately, c-term is
m
control parameters. However, their control mechanisms normally much smaller than b-term[11]. A large current
- aredifferent,andtheyhavedifferentadvantagesanddis- density is needed to achieve a technologicallyuseful DW
d
n advantages. Thus,findinganynewcontrolparameterfor propagationvelocity[9], but the associatedJoule heating
o DW motion should be of great interest. couldaffectdeviceperformance. Thus,itshouldbeinter-
c esting if one can generate a large c-term (in comparison
Although it is known both theoretically[13] and
[
experimentally[14] that a microwave can be used to ma- with b-term) either in a new architecture or by using a
1 new control parameter. One solution along the first line
nipulate the motion of single domain magnetic particles
v ofthethinkingisprovidedbyKhvalkovskiyetal.[12]who
(macrospinsorStonerparticles),amicrowavewashardly
0
proposed a current perpendicular to the plane magnetic
0 thoughttobeaneffectivecontrolparameterforDWmo-
configuration in a sandwiched magnetic nanowire struc-
8 tion. In this letter, we show that a circularly polarized
1 magnetic field (CPMF) at GHz frequency (microwave) ture. Here we provide a solutionalongthe secondline of
1. can efficiently drive a magnetic DW to propagate along the thinking.
We consider a head-to-head (HH) DW in a magnetic
0 a nanowire at a high speed. Similar to a spin-polarized
0 current,aCPMFcangeneratebothSlonczewski-likeand nanowire whose easy axis is along the wire axis defined
1 as the z-axis shownin Fig. 1. The motionof the magne-
field-like spin-transfer torques (STTs) inside a DW. Un-
: tization M~ =M m~, is governed by the Landau-Lifshitz-
v like the STTs generated by a spin-polarized current, the s
i field-like STT is much bigger than the Slonczewski-like Gilbert (LLG) equation[16]
X
STT. Thus, a CPMF generates more useful STT[7] in
r ∂m~ ∂m~
a comparison with that of a spin-polarized current. More- = γm~ ~h +αm~ , (1)
eff
∂t − × × ∂t
over,itsdrivingmechanismisverydifferentfromthoseof
astaticaxialmagneticfieldandaspin-polarizedcurrent. here ~h = 1 δU/δM~ is the effective magnetic field
eff −µ0
In order to appreciate the CPMF-driven DW motion, thatisthevariationalderivativeofthefreeenergydensity
letusrecallthedrivingmechanismsofastaticaxialmag- U[M~]withrespecttomagnetizationM~,γ =g e /2m is
e
| |
netic field and a spin-polarized current. A DW propa- the gyromagnetic ratio, µ0 is the vacuum magnetic per-
gates along a wire under a static magnetic field because meabilityandαisthephenomenologicalGilbertdamping
the Zeeman energy must be released to compensate the constant[16] which measures dissipative effect. Eq. (1)
dissipatedenergydue to the non-existence[15] ofa static is a nonlinear partial differential equation that can be
DWinastaticmagneticfield. Anelectriccurrent,onthe solved exactly only in some special cases[1, 17].
other hand, moves DWs by the STT due to the transfer AmacrospincansynchronizeitsmotionwithaCPMF.
of spin angular momentum from conduction electrons to A synchronized motion generates a damping field that
local spins[5]. A spin-polarized current can transfer two forces the spin to move perpendicularly to the synchro-
types of torques to a local magnetization M~ = M m~. nized motion, leading to a dramatic effect of a CPMF
s
One is the Slonczewski-like STT [5] of bm~ (m~ ~s) (b- on a Stoner particle[13, 14]. Thus, it is natural to study
× ×
term), where ~s is the polarization direction of the spin- the CPMF effect on DW motion since a DW may also
polarized current. The other one is a field-like torque[6] synchronize its motion with a CPMF due to the DW
2
texture nature if the motion of its constitute local spins where the effective field is ~h′eff = h0xˆ′ + f(cosθ)zˆ′ +
are synchronized. To demonstrate that this can indeed 2A ∂2m~′ that does not depend on time (In general,the
happen,letusstudy the motionofaHHDWinauniax- µ0Ms ∂z′2
field due to the magnetic anisotropy depends on time
ialwireunder aCPMF~h(t)=h0(cosωtxˆ+sinωtyˆ)with in a rotation frame, but it is time-independent for the
frequency ω and amplitude h0. The free energy density uniaxial wire).
U(M ) is a functional function of M , and the effective
z z Eq. (3) does not have any explicit time-dependent
field takes a form of~h =~h(t)+f(M )zˆ+ 2A ∂2M~ ,
eff z µ0Ms2 ∂z2 term,andthe originalproblembecomesaDWsubjected
wheref istheanisotropyfieldandthelasttermistheex- to a transverse field h0xˆ′, a Slonczewski-like torque (b-
changefieldwiththeexchangecoefficientA. Thephysics term) αm~′ (m~′ ~ω), and a field-like torque (c-term)
is thatDWsynchronizeits motionwith aCPMFsothat m~′ ~ω. Th×us a D×W under a CPMF behaves like the
DW plane rotates around wire axis at angular velocity DW×under a current-induced STT if one views ~ω as a
ω. According to the LLG equation (Eq. (1)), the pre- spin polarizedcurrent. Unlike the STT froma realspin-
cession motion around z axis gives rise to a damping polarized current, c-term is much larger than b-term.
torque Td = α m~ × ∂∂m~t −= −αωsinθ~eθ (~eθ is unit vec- Thus a CPMF is very efficient in driving DW propaga-
tor of θ directi(cid:0)on) show(cid:1)n in Fig. 1. Also, because of tionalongthewire. Itshouldbenoticedthattheequiva-
the lag of the DW motion with the field (δ is the angle lentspin-polarizedcurrentisuniformlyappliedtoaDW,
between the DW plane and CPMF plane defined by ~h0 instead of spatially dependent spin-polarized current re-
andz axis),the fieldexertsatorquewithθ-component lated to ∂m~ inside a DW[6, 7]. Interestingly, Eq. (3) is
− ∂z
Tp = γ m~ ~heff ~eθ = γh0sinδ shown in Fig. 1. exactlythe same asthatina recentlystudiedsystem[12]
− (cid:16) × (cid:17)· of current-driven DW motion in a sandwiched long and
Later analysis (Eq. (5) derived later) shows that T al-
p
narrow spin valve, in which the magnetization of refer-
ways overwhelms T so that the HH DW propagates to
d
ence magnetic layer plays the role of field polarity ω~.
the leftalongthe wire(acollectivemotionofspins along
ThusaDWofuniaxialwireunder aCPMFisequivalent
θ-directioncorrespondstotheDWpropagationalongthe
to the DW in composite spin valves under a current[12].
wire).
Earlystudy[12]onEq. (3)showedthatDWpropagates
h T like a rigid body under small torques, corresponding to
0 M(z) p Td y a perfect synchronized motion with the CPMF in the
δ x current case. It is noted that ̥~ = αm~′ ~ω is a non-
z γ ×
ω conservative field since ∇m~′ ×̥~ = −2γα~ω 6= 0. In order
tofindouthowtheDWpropagationvelocitydependson
theamplitudeandthefrequencyoftheCPMF,weadopt
FIG.1: Dashedarrow denotesthedampingtorqueTd on the
the generalized analysis of Schryer and Walker[1]. It is
local spins (long arrows) when the DW plane moves around
(indicatedbythecurvedarrow)thewireaxisduringsynchro- requiredtofirstfindthestaticDWsolutionsofEq. (3)at
nization. Short arrow denotes torque Tp arising from the lag ~ω =0(thecaseofaconstanttransversefield~h0). Forthe
of DW motion and thefield. conventionaluniaxial anisotropy U(cosθ)= 1Kcos2θ,
′ −2
astaticDWcenteredatz =QexistswithfollowingDW
For the motion of a HH DW under the CPMF, it is ′
profile θ =θ(z Q),η =0, where[18]
convenient to use the rotation frame of −
′
xyzˆˆˆ′′ = −cosis0nωωtt csoins0ωωtt 100xyzˆˆˆ, (2) sinθ(z′)=sinθ0+ cosh z∆′ ccooss2θθ00+sinθ0, (4)
(cid:2) (cid:3)
where xˆ′, yˆ′, zˆ′ are the unit vectors of Cartesian′co- here θ0 = sin−1(µ0Msh0/K) is the tilted polar angle of
ordinates in the rotation frame. m~ becomes m~ = two domains due to transverse magnetic field ~h0, and
sinθcosη, sinθsinη, cosθ in the rotation frame,
∆ = 2A/K is the static DW width without external
w(cid:0)here η = δ is the azimut(cid:1)hal angle of m~ in the ro-
− magneptic field. Assume the profile of the moving DW
tationframewhichis relatedto azimuthalangleφ inthe
is the same as that of the static one, then the moving
laboratory frame by η = φ ωt. Polar angle θ is the
− DW is given by Eq. (4) with collective coordinates[18–
same in the two frames. The time derivatives of m~ in
20] (Q, η) being functions of time. Substituting Eq. (4)
the two reference frames are connected to each other by
into Eq. (3), the equations for Q and η are
∂m~ = ∂m~′ m~′ ~ω,hereangularvelocityvector~ω =ωzˆis
∂t ∂t − ×
assumed(Fig. 1). Eq. (1)inthisrotationframebecomes
αγh0
∂m~′ = γm~′ ~h′ + αm~′ ~ω ~ω +αm~′ ∂m~′, η˙ = −ω− 1+α2 sinη, (5)
∂t − ×(cid:18) eff γ × − γ(cid:19) × ∂t ω+η˙
Q˙ = ∆. (6)
(3) −α(1 sinθ0)
−
3
Solution of Eq. (5) with initial condition η =0 is
cotη =−ωωc −q ωωc 2−1coth√ωc22−ω2t, ω <ωc 2.5 0.9
2 ωc (cid:0)1 (cid:1)ωc 2cot√ω2−ωc2t, ω >ω
−ω −q −(cid:0)ω (cid:1) 2 (c7) 1/2 ) 2.0 /h0c0.6
where ωc = 1α+γαh02 is the critical frequency that separates /)0 0.3
two modes: η approaches exponentially a fixed value A 1.5
sin−1(−ω/ωc)inatime scaleof1/ ωc2−ω2 forω <ωc. ( 0.0 0.5 1.0 1.5 2.0
DT(cid:0)αhWi+spα1irs(cid:1)opωtahgeisatfaeulswllytaoysstyhlnaecrhlgerefortntaihzseadmneTmnpdtoito=inoenαd.ωesaiSrnliynθ.c,etηhTekpeHep=Hs elocity 1.0
V
rotatingaroundz-axiswithaperiodof2π/ ω2 ω2and 0.5
− c
a variable velocity for ω > ωc, corresponpding to an in-
complete synchronization. For ω < ωc, the steady DW 0.0
0.0 0.5 1.0 1.5 2.0
propagation velocity is given by Eq. (6) when η˙ = 0 is
reached, Frequency ( Ms)
υ =Q˙ (t )= ω∆ . (8) FIG. 2: DW velocity versus CPMF frequency with α = 0.1
→∞ −α(1 sinθ0) and K = 8. Symbols are numerical data for field amplitude
− h0 = 3 (squares), 1 (circles), and 0.5 (stars). Solid curves
Similar to the low current velocity, DW propagation ve- are the theoretical results of Eqs. (8) and (9). Inset: Ratio
locity is linear in ω as shown by the solid lines in Fig. ωc/γh0 versus α. Solid curve is the theoretical formulation
α as we get.
2 for low frequency. For ω > ωc, the precession velocity 1+α2
of DW plane change with time. According to Eq. (6),
DW velocity will also change with time. Since the aver-
age angular velocity of η is Ω= ω2 ω2, the averaged find that the DW position line is a straight line at low
− c frequency, and it becomes an oscillation curve at high
DW velocity is p
frequency.
ω ω2 ω2 The symbols in Fig. 2 are numerical data for K = 8,
υ¯= − − c∆, (9)
− α(1p sinθ0) α = 0.1 and various h0 = 3, 1 and 0.5. It is clear that
− h0 affects DW velocity. The larger h0 is, the higher the
monotonically decreasing with ω shown by the solid velocitywillbe, asgivenbyEqs. (8)and(9)becausethe
curves in Fig. 2 for large ω. The results can also be un- DW width is increased by a factor 1/(1 sinθ0). The
−
derstood from the energy consideration[4]. For ω < ωc, agreementbetweenEq. (8)(straightlinesinFig. 2with-
the non-conservative field does not do any work to the outanyfittingparameters)andnumericalresultsforlow-
systemforarigidDWpropagation,andtheenergydissi- frequency are good. Eq. (9) (curves in Fig. 2) captures
pationmustbecompensatedbytheenergyreleasedfrom qualitatively the decay feature for large frequency, but
the DW propagation[4]. According to Ref. [4], the DW systematically smaller than the numerical values. This
velocity is proportional to DW width and the axial field should not be a surprise since Eqs. (5) and (6) are de-
which is the frequency in the present case. For ω > ωc, rived from rigid DW assumption that, strictly speaking,
DW plane precesses around the wire while it propagates does not hold for ω > ω . To further test our assump-
c
alongthe wire. Thenon-conservativefieldpumps energy tionsforEqs. (5)and(6),wecomparenumericalvaluesof
tothe systemsothatthesystemneedstoreleaselessen- ωc/(γh0)withtheoreticalpredictionofα/(1+α2)(curve
ergy, corresponding to a low DW velocity shown in Eq. intheinsetofFig. 2). Thesymbols(squares,circles,and
(9). stars for h0 =3, 1, and 0.5, respectively) in the inset of
To test validity of our analytical results, we solve Fig. 2 are numerical data for various α that compares
Eq. (1) numerically for a uniaxial 1D wire of magnetic well with the theoretical prediction.
anisotropy U = 1Kcos2θ. We first scale time, length, Itispredictedandconfirmed[21,22]thataDWmotion
−2 −1
energy density and field amplitude in units of (γM ) , with its plane precessing around wire axis can generate
s
A/µ0Ms2, µ0Ms2 and Ms, respectively, so that velocity an electromotive potential of ~eddφt between the two sides
ips in the unit of γ A/µ0. We then adopt a standard of the DW, where ddφt is the precession velocity of the
arithmetic Methodp-of-Lines to discretize space with an DW. However, a rigid body propagation along the wire
adaptivetime-step control. The length ofthe wire is 100 willnotgenerateanelectromotiveforce. Anaiveapplica-
and the mesh size is 0.2. The density plot of m in z t tion of this theory to our DW motion seems lead to zero
z
−
plane determines the DW position. The DW velocity is electromotive force in the rotation frame and non-zero
extracted from the slope of the DW position line. We electromotive force in the laboratory frame. This obvi-
4
ous contradiction is due to the neglect of a Coriolis field can be mapped to the DW under STT due to uniform
of B~ = ω~ along wire axis in the rotation frame. Thus, spin-polarizedcurrent. Inthemap,theCPMFfrequency
−γ
aconductionelectronvirtuallymovesacrosstheDWwill plays the role of the spin polarized current.
experience an static potential gain of E = 2µ~·ω~ = ~ω,
s γ e
exactly what is predicted by Niu’s theory[22] in the lab-
This work is supported by Hong Kong UGC/CERG
oratory frame.
grants (#603007, 603508 and SBI07/08.SC09). We
A realistic magnetic wire will not be completely sym-
thank J. Lu for helping numerical simulations.
metric around wire axis. In order to show the robust-
ness of the physics discussed here, we consider a biaxial
anisotropyfield~han = µ01Ms (K1mzzˆ−K2mxxˆ),K1 and
K2 describe the anisotropies along the easy- and hard-
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FIG. 3: DW propagation velocity υ as a function of CPMF may be a better quantity to understand the origin of
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