Dynamic coupling of ferromagnets via spin Hall magnetoresistance

0.75 MB·

by  Tomohiro Taniguchi

Dynamic coupling of ferromagnets via spin Hall magnetoresistance
Tomohiro Taniguchi
National Institute of Advanced Industrial Science and Technology (AIST),
Spintronics Research Center, Tsukuba, Ibaraki 305-8568, Japan
(Dated: March 22, 2017)
Thesynchronizedmagnetizationdynamicsinferromagnetsonanonmagneticheavymetalcaused
by the spin Hall effect is investigated theoretically. The direct and inverse spin Hall effects near
7
theferromagnetic/nonmagneticinterfacegeneratelongitudinalandtransverseelectriccurrents. The
1
0 phenomenon is known as the spin Hall magnetoresistance effect, whose magnitude depends on the
2 magnetization direction in the ferromagnet due to the spin transfer effect. When another ferro-
magnet is placed onto the same nonmagnet, these currents are again converted to the spin current
r
by the spin Hall effect and excite the spin torque to this additional ferromagnet, resulting in the
a
M excitationofthecoupledmotionsofthemagnetizations. Thein-phaseorantiphasesynchronization
of the magnetization oscillations, depending on the value of the Gilbert damping constant and the
field-liketorquestrength,isfoundinthetransversegeometrybysolvingtheLandau-Lifshitz-Gilbert
0
2 equation numerically. On the other hand, in addition to these synchronizations, the synchroniza-
tion having a phase difference of a quarter of a period is also found in the longitudinal geometry.
] The analytical theory clarifying the relation among the current, frequency, and phase difference is
l also developed, where it is shown that the phase differences observed in the numerical simulations
l
a correspond to that giving the fixedpoints of the energy supplied bythe coupling torque.
h
- PACSnumbers: 85.75.-d,75.78.-n,05.45.Xt,72.25.-b
s
e
m
. I. INTRODUCTION respectively,where J0 is the electric currentdensity gen-
t erated by the external electric field. The definitions of
a
the dimensionless coefficients, χ, χ′, and χ′′, are given
m Dynamiccouplingofferromagnetshasbeenofinterest
below. It should be emphasized that the currents given
inthefieldofmagnetism. Thedipoleinteractionhasbeen
-
d the basic interaction to excite the coupled motion of the by Eqs. (1) and (2) depend on the magnetization direc-
n magnetizations, and is applied to magnetic recording1,2. tion m = (mx,my,mz). When another ferromagnet is
o placed onto the same nonmagnet, these currents will be
Anothermethodtorealizethecouplingistousethespin-
c transfereffect3,4,wheretheapplicationofanelectriccur- converted to spin current by the spin Hall effect again,
[
renttoferromagnetic/nonmagneticmultilayersresultsin and excite spin torque on this additional ferromagnet.

2 themagnetizationswitchingandself-oscillation5–14. The Then, the magnetization dynamics of two ferromagnets
v will be coupled through the angular dependencies of the
coupled dynamics through pure spin current generated
1 inspinpumping15,16 andnonlocal17 geometrieshavealso electric currentgivenby Eqs. (1) and(2). This coupling
0
is unavoidable whenever severalferromagnets are placed
2 been observed. It should be emphasized that these cou-
onto the same nonmagnet, and is not restricted by the
3 plingsarestronglyrestrictedbythe characteristiclength
distance between the ferromagnets because it is carried
0 scales. Forexample,thedipolecouplingdecaysaccording
. totheinversecubedetectionlaw,whereasthespintrans- by the electric current. Since the structure consisting
1
of severalferromagnets on the nonmagnetic heavy metal
0 fereffectbyaspin-polarizedelectricorpurespincurrent
willbeimportantfromtheviewpointsofbothfundamen-
7 occursinasystemsmallerthanthespindiffusionlength.
1 Recently, physical phenomena, such as the spin tal physics and practical applications based on the spin
v: torque18–25 and magnetoresistance effects26–34, due to Halleffect,suchasmagneticrandomaccessmemory,spin
i the spin Hall effect35–37 in bilayers consisting of an insu- torque oscillators, and bio-inspired computing38,39, it is
X of interest to clarify the role of this coupling.
lating or metallic ferromagnet and a nonmagnetic heavy
r metal have attracted much attention. The latter, known In this paper, we investigate the coupled dynamics of
a
as the spin Hall magnetoresistance, originates from the magnetizations in ferromagnets in the presence of the
charge-spin conversion of an external electric current by spin Hall effect by solving the Landau-Lifshitz-Gilbert
the direct and inverse spin Hall effects, and has been (LLG) equation numerically for both longitudinal and
observed by measuring the longitudinal and transverse transversegeometries. Inadditiontotheexternalelectric
electric currents, which are given by current, the current contributing to the spin Hall mag-
netoresistancealsoexcites the spin torque. The strength
J of this additional torque is estimated from the theory of
cx =1+χ′′+χm2, (1)
J y the spin Hall magnetoresistance extended to the system
0
consistingofseveralferromagnets. Theconventionalspin
torque is proportional to the spin Hall angle ϑ, whereas
J the new torque is on the order of ϑ3, and therefore, its
cy = χm m χ′m , (2)
J − x y− z value is two orders of magnitude smaller than the con-
0
2
(a) A. System description
longitudinal coupling F2
transverse coupling The systemwe consideris schematicallyshowninFig.
1(a),wherethreeferromagnetsF (ℓ=1,2,3)areplaced
F1 F3 ℓ
onto the same nonmagnet N. We assume that the ma-
terial parameters in the ferromagnets are identical, for
z simplicity. The external electric field, E , is applied
x x
N Ex Ex y to the x direction, inducing the electric current density
J = σ E , where σ is the conductivity of the non-
0 N x N
magnet. The directandinversespinHalleffects produce
(b)
electriccurrentsinthelongitudinal(x)andtransverse(y)
F1 F2 directions. These electric currents are converted to the
N SHE ISHE SHE ISHE σNEx+j1+j2 stopinthceusrpreinntHaanlldeiffnejecct,terdesiunlttointghienFt2heanedxcFit3atliaoynerosfdthuee
spintorque. Then,themagnetizationdynamicsintheF
ℓ
FIG. 1: (a) Schematic view of the system in this study. (ℓ=2,3)layerisaffectedbythatintheF layer,andvice
1
Three ferromagnets Fℓ (ℓ=1,2,3) are placed onto the same versa. WecallthecouplingbetweentheF1 andF2 layers
nonmagnet N. The external electric field is applied to the x
the longitudinal coupling, whereas that between the F
1
direction. (b) Schematic view of the generations of electric
and F layers the transverse coupling. We assume that
currents by the direct and inverse spin Hall effect (SHE and 3
both the ferromagnet and nonmagnet are metallic be-
ISHE)inthelongitudinalgeometry. Thetotalelectriccurrent
causemetallic bilayersare generallyusedto measure the
isthesumoftheconventionalelectriccurrentJ0 =σNEx and
thecurrent generated near theF1/N and F2/N interfaces, j1 magnetization switching and oscillation by the spin Hall
and j2. effect18–21,24. Although the spin Hall magnetoresistance
wasoriginallystudiedforinsulatingferromagnets26–29, a
large spin Hall magnetoresistancein metallic system has
ventional spin torque. Nevertheless, it is found that this
also been reported recently32–34.
additional new torque affects the phase difference of the
The dimensionless coefficients, χ, χ′, and χ′′, in Eqs.
magnetizationintheself-oscillationstate. Thenumerical
(1)and (2)for single ferromagnetshavebeen derivedfor
simulationrevealsthatthein-phaseorantiphasesynchro-
an insulating30 or metallic34,40 ferromagnet, which are
nization is observed in the transverse geometry. On the
given by
otherhand,inadditionto them,the phasedifferencebe-
comesaquarterofaperiodinthelongitudinalgeometry. ϑ2λ g↑↓ g∗ d
It is found that these phase differences depend on the χ= N Re tanh2 N ,
d g +g↑↓coth(d /λ ) − g 2λ
values of the Gilbert damping constant and the dimen- N (cid:20) N N N N(cid:21) (cid:18) N(cid:19)
(3)
sionless field-like torque strength. An analytical theory
clarifyingthe relationamongthe current,frequency,and
phase difference is also developed. ϑ2λ g↑↓ d
χ′ = NIm tanh2 N ,
The paper is organized as follows. In Sec. II, we de- − d g +g↑↓coth(d /λ ) 2λ
N N N N (cid:18) N(cid:19)
scribe the system under consideration, and discuss the (4)
theoreticalformulaofthespintorqueexcitedbythespin
Halleffectinthepresenceoftheseveralferromagnetsand
2ϑ2λ d
the spin Hall magnetoresistance effect. In Sec. III, we χ′′ = N tanh N
study the phase differences in the synchronized state of dN (cid:18)2λN(cid:19)
the magnetizations for both the longitudinal and trans- ϑ2λ g↑↓ d
NRe tanh2 N ,
verse geometries by solving the LLG equation numeri- − d g +g↑↓coth(d /λ ) 2λ
cally. InSec. IV,thetheoryclarifyingtherelationamong N N N N (cid:18) N(cid:19)
(5)
the current,frequency, and phase difference is developed
based on the LLG equation averaged over constant en- where d and λ are thickness and spin diffusion
N N
ergy curves. The summary of the paper is given in Sec. length of the nonmagnet, respectively, whereas g /S =
N
V. hσ / 2e2λ with the cross section area of the ferro-
N N
magnetic/nonmagnetic interface S. The dimensionless
mixin(cid:0)g condu(cid:1)ctance g↑↓ = g↑↓+ig↑↓ consists of its real
r i
and imaginary parts41–43, and g∗ is defined as
II. SYSTEM DESCRIPTION AND LLG
EQUATION 1 2 1 1
= + + .
g∗ (1 p2)g g tanh(d /λ ) g tanh(d /λ )
− g F F F N N N
In this section,we describe the systemadoptedin this (6)
study, and show the spin torque formulas including the Here, g = g↑↑ + g↓↓ is the sum of the conductances
coupling torques between the ferromagnets. of the spin-up and spin-down electrons, whereas p =
g
3
(g↑↑ g↓↓)/g is its spin polarization. The ferromag- densityisdeterminedbytheconservationlawoftheelec-
−
netic/nonmagnetic interface resistance r is related to tric current, as follows. Considering in a similar manner
g via g/S = (h/e2)/r. We also introduce g /S = to the caseofthe singleferromagnet,the electriccurrent
F
h 1 p2 σ / 2e2λ , where σ is the conductivity of J +j is convertedto the spin currentby the spinHall
− σ F F F 0 1x
the ferromagnet and p is its spin polarization. The effectnearthe F /Ninterface,andthis spincurrentpro-
σ 2
th(cid:0)ickness(cid:1)and (cid:0)spin di(cid:1)ffusion length of the ferromagnet ducesanadditionalelectriccurrent(χ′′+χm2 )(J +j )
2y 0 1x
aredenotedasd andλ ,respectively. Thetermrelated by the inverse spin Hall effect. Therefore, the total lon-
F F
tog∗ isneglectedwhentheferromagnetisaninsulator30, gitudinalelectriccurrentdensityneartheF /Ninterface
2
i.e., r . The following quantities correspond to is (1+χ′′+χm2 )(J +j ). Similarly, the electric cur-
→ ∞ 2y 0 1x
the effective spin polarizations of the damping-like (or rent density near the F /N interface can be expressed
1
Slonczewski3) torque and the field-like torque, respec- as (1+χ′′+χm2 )(J +j ). These currents should be
1y 0 2x
tively, equalto the total electric currentdensity, J +j +j ,
0 1x 2x
accordingto the conservationlaw of the electric current.
d g↑↓
N Then, we find that j is given by
ϑ =ϑtanh Re(Im) . ℓx
R(I) 2λ g +g↑↓coth(d /λ )
(cid:18) N(cid:19) N N N
(7) χ′′+χm2 1+χ′′+χm2
ℓy ℓ′y
For the later discussion, we introduce j = J
ℓx 1(cid:16) χ′′+χ′(cid:17)m(cid:16)2 χ′′+χm2 (cid:17) 0 (9)
ϑI − ℓy ℓ′y
β = , (8)
−ϑ χ′′+(cid:16) χm2 J , (cid:17)(cid:16) (cid:17)
R ≃ ℓy 0
which corresponds to the ratio of the field-like torque to where we negl(cid:0)ect the high(cid:1)er order terms of ϑ. Then, the
the damping-like torque.
spintorqueexcitedontheF layerbythespinHalleffect
ℓ
The values of the parameters used in the following
inthelongitudinalgeometryisobtainedbyreplacingthe
calculations are derived from recent experiments on the
externalelectric currentdensity J =σ E in the previ-
W/CoFeB heterostructure34, where ρ = 1/σ = 1.6 0 N x
kΩnm, pσ = 0.72, λF = 1.0 nm, ρNF= 1/σNF= 1.25 ous work30 with J0+jℓ′x ≃ 1+χ′′+χm2ℓ′y J0, and is
kΩnm, λ = 1.2 nm, and ϑ = 0.27, whereas the thick- given by (cid:16) (cid:17)
N
nessesareassumedtobed =2nmandd =3nm. The
isnutmerifnagceacotrnadnuscptaarnecnetsiwnteerFrefancoet.evIanlsutaetaedd,NiwneRuesf.e34tybpyiacas-l TLℓ =− γ2~eMϑRdJF0 1+χ′′+χm2ℓ′y mℓ×(ey×mℓ)
vfiarslut-epsrionfcitphleesinctaelrcfualcaetiocnons4d3u,crta=nc0e.s25obktΩaninme2d, fprgom=0t.h5e, − 2γe~MϑIdJF0 (cid:0)1+χ′′+χm2ℓ′y(cid:1)ey×mℓ,
and g↑↓/S =25 nm−2. We note that the imaginary part (cid:0) (cid:1) (10)
r
of the mixing conductance, g↑↓, is either positive or neg-
ative, depending on the matierial and thickness43. The where (ℓ,ℓ′) = (1,2) or (2,1). The unit vector pointing
signofg↑↓ determinesthoseofχ′ andϑ ,orequivalently, in the magnetizationdirectionof the Fℓ layeris mℓ, and
β. For iexample, when g↑↓/S = 1 nmI−2, χ 0.010, γ, M, and d are the gyromagnetic ratio, the saturation
χ′ 0.0002, χ′′ 0.035i, θ 0.167, and θ≃ 0.002 magnetization, and the thickness of the ferromagnet, re-
≃ − ≃ R ≃ I ≃ spectively.
(β 0.010); see Appendix A. In the following calcula-
≃− Note that the terms,
tions, we study the magnetization dynamics for several
values of β. γ~ϑ J γ~ϑ J
T(0) = R 0m (e m ) I 0e m ,
ℓ − 2eMd ℓ× y × ℓ − 2eMd y× ℓ
F F
(11)
B. Spin torques in longitudinal and transverse
geometries
in Eq. (10) are the conventional spin torques generated
by the external electric current J and the spin Hall ef-
0
Equation (1) was derived for a system having single fect,andareoftenreferredtoasthe damping-liketorque
ferromagnet. In this case, J0 is the external electric cur- andthefield-liketorque,respectively. Ontheotherhand,
rentdensity,whereas(χ′′+χmy)J0 isthecurrentdensity the terms
generatedasaresultofthecharge-spinconversionbythe
dgeiroemctetarnydinintvheerpseresspenintsHtuadllye,ffoenctths.eoItnhethrehalonndg,ittwuodifnear-l − γ2~eMϑRdJ0 χ′′+χm2ℓ′y mℓ×(ey×mℓ)
F (12)
rinotmerafganceest,icc/onnotnrmibaugtenettoictihnetegrefanceersa,tiio.en.,oFf1t/hNe alonndgiFtu2/dNi- − 2γe~MϑIdJ0 (cid:0)χ′′+χm2ℓ′y(cid:1)ey ×mℓ,
F
nal current through the direct and inverse spin Hall ef-
(cid:0) (cid:1)
fects, asschematicallyshowninFig. 1(b). Letusdenote inEq. (10)originatefromthecurrentjℓ′x,andarenewly
the electric current density generated by these effects introduced in this study. It should be emphasized that
near the F /N interface as j (ℓ = 1,2). This current Eq. (12) depends on the magnetization direction of the
ℓ ℓx
4
other ferromagnetFℓ′ (ℓ′ =1 or 2), mℓ′, resulting in the where the Gilbert damping constant is denoted as α.
coupling between the F and F layers. In the following calculations, we use the values of the
1 2
Letusnextshowthespintorqueformulasinthetrans- material parameters, γ = 1.764 107 rad/(Oe s) and
verse geometry. We denote the electric current den- α=0.005,derivedfromthe exper×iments44. For the later
sity flowing in the y direction generated near the F /N discussion, it is useful to show the explicit forms of the
ℓ
(ℓ = 1,3) interface by the inverse spin Hall effect as j . LLG equation in the longitudinal and transverse geome-
ℓy
Theconservationlawofthetotalelectriccurrentdensity, tries. Equation (15) for the longitudinal geometry is
j1y +j3y, gives (see also Appendix B) dm
1+α2 ℓ = γm H
(χmℓxmℓy+χ′mℓz) dt − ℓ× ℓ
jℓy =−(cid:20)1−(χ′′+χm2ℓx)(χ′′+χm2ℓ′x) (cid:0) γ(1+(cid:1)χ′′) ~ϑRJ0 mℓ (ey mℓ) αγmℓ (mℓ Hℓ)
+ χ′′+χm2ℓx (χmℓ′xmℓ′y +χ′mℓ′z) J (13) − ~ϑ2eJMdF × × − × ×
(cid:0) 1−(χ′′+(cid:1)χm2ℓx)(χ′′+χm2ℓ′x) # 0 −γχm2ℓ′y2eMRd0 mℓ×(ey×mℓ)
F
≃−(χmℓxmℓy+χ′mℓz)J0. γ(1+χ′′)(α+β) ~ϑRJ0 m e
ℓ y
Inadditiontotheconventionalspintorque,Eq. (11),this − 2eMdF ×
transverseelectriccurrentalsoexcitesthe spintorqueon +γ(1+χ′′)αβ ~ϑRJ0 m (e m )
the other ferromagnet. In total, the spin torque acting 2eMd ℓ× y× ℓ
F
oTnTℓth=e−Fℓγγ2l~~eaMyϑϑeRRrdJ(Fiχ0smmgiℓℓv′x×enm(bℓe′yyy×+[(ℓmχ,ℓ′mℓ′))ℓ−=′z)2(γJ1e~,0Mϑ3mI)dJℓoF0re(y(3e×,x1m)]ℓmℓ) −+γγα(αβχ+mβ2ℓ)′yχ2~meϑM2ℓR′yJd20~FeϑmMRℓJd×0Fm(eℓy××emy ℓ), (16)
− 2eMd × ×
F
γ~ϑI(χmℓ′xmℓ′y+χ′mℓ′z)J0e m , whereas that for the transverse geometry is
− 2eMdF x× ℓ (14) 1+α2 dmℓ = γmℓ Hℓ
dt − ×
Thelasttwotermsrepresentthecouplingtorquebetween (cid:0) γ ~ϑR(cid:1)J0 m (e m ) αγm (m H )
the F1 and F3 layers. Note that the direction of this − 2eMdF ℓ× y× ℓ − ℓ× ℓ× ℓ
couplingtorqueisdifferentfromthatoftheconventional ~ϑ J
torque because the currents J and j flow in different γ(χmℓ′xmℓ′y+χ′mℓ′z) R 0 mℓ (ex mℓ)
0 ℓy − 2eMd × ×
F
directions. ~ϑ J
In the following, the torques related to χ, χ′, and χ′′ γ(α+β) R 0 m e
ℓ y
− 2eMd ×
in Eqs. (10) and (14) are referred to as coupling torque. F
~ϑ J
The ratio of these new torques to the conventional spin +γαβ R 0 m (e m )
ℓ y ℓ
torque is on the order of χ ϑ2 10−2. Since the 2eMdF × ×
cHoanlvleanntgiolenaϑlRsp∝inϑt,ortqhueecdouuepli∝isngprtoopr∼oqruteioinsaplrtooptohretiospnianl −γ(α+β)(χmℓ′xmℓ′y +χ′mℓ′z)2~eϑMRJd0Fmℓ×ex
to ϑ3. Although the spin Hall angle is usually a small ~ϑ J
quantity, it will be shown that the coupling torques play +γαβ(χmℓ′xmℓ′y +χ′mℓ′z) R 0 mℓ (ex mℓ).
2eMd × ×
F
a non-negligible role on the magnetization dynamics, as
(17)
shown below.
III. NUMERICAL ANALYSIS OF
C. LLG equation SYNCHRONIZATION
In the following sections, we study the magnetization In this section, we study the magnetization dynam-
dynamics excited by the spin torque given by Eq. (10) ics in the ferromagnets in the presence of the coupling
or (14) both numerically and analytically. We neglect torques by solving Eq. (15) numerically. The self-
the transversecouplingwhen the roleofthe longitudinal oscillation of the magnetization provides an interesting
coupling is studied, and vice versa, for simplicity, which example to understand the role of the coupling torque.
corresponds to considering a system consisting of the F1 Note that the coupling torques are proportional to the
and F2 layers, or F1 and F3 layers. The magnetization parametersχ,χ′,andχ′′,andtheirproductstootherpa-
dynamics in the Fℓ (ℓ=1,2,3) is described by the LLG rameters such as αβχ, the values of which are relatively
equation, small,ascanbeseeninEqs. (16)and(17). Nevertheless,
dm dm the coupling torques lead to the phase synchronization,
dtℓ =−γmℓ×Hℓ+αmℓ× dtℓ +TLℓ,T, (15) as shown below.
5
The self-oscillation of the magnetization in single fer-
romagnets by the spin Hall effect has been observed for (a)1.0 (b)1.0
in-plane magnetized ferromagnets19, and is induced by
the conventional spin torque given by Eq. (11). There- mz0
fore,in the following,we assumethat the magnetic field, mx 0
-1.0
Hℓ =HKmℓyey−4πMmℓzez, (18) -11..00 mx0 -1.01.0 0my -1.00 0.2 t0i.m4e (n0s.6) 0::.FF813 1.0
consists of an in-plane anisotropy field H along the y
K
direction and a demagnetization field 4πM in the z di- (c) (d)
1.0 1.0
rection,whichweassumeasH =200OeandM =1500
K
emu/c.c.34 in the following calculations. It is known for
the case of the single ferromagnet45 that the in-plane mx 0 my0.5
self-oscillation can be excited when the electric current
density J0 is in the range of Jc <J0 <J∗, where -1.0 0 ::FF13
999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000
2αeMd time (ns) time (ns)
F
J = (H +2πM), (19)
c ~ϑR K (e) (f)
1.0 1.0
4αeMd
J∗ = π~ϑ F 4πM(HK+4πM), (20) mx 0 my0.5
R
p :F1
which in this study are Jc 26 and J∗ 33 MA/cm2. -1.0 0 :F3
≃ ≃ 999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000
time (ns) time (ns)
A. Transverse geometry FIG. 2: (a) A typical in-plane self-oscillation trajectory of
the magnetization obtained by solving Eq. (15) numerically
Let us first investigate the magnetization dynamics for the transverse geometry. The dotted lines indicate the
in the transverse geometry because this geometry pro- oscillation direction. (b) The time evolutions of m1x(t) and
vides a simple example of the coupled motion. We start m1y(t) near the initial state. The time variations of the x
and y components of the magnetizations in the steady state
with solving the LLG equation given by Eq. (17) for are shown in (c) and (d) for β = −0.01 and (e) and (f) for
the F and F layers. Figure 2(a) shows an exam-
1 3 β =+0.01. The solid red lines correspond to time evolution
ple of the trajectory of the magnetization dynamics ob- inF1,anddottedbluelinestothoseinF3 for(b)through(f).
tained by solving Eq. (17) numerically, where J = 30
0
MA/cm2. As shown, the in-plane oscillation is observed
in the steady state. The initial conditions set for m results shown in Figs. 2(b) and 2(c) are shown for one
1
and m are different as m (0)= (cos80◦,sin80◦,0) and certain initial condition, the in-phase synchronizations
3 1
m (0)=(cos95◦,sin95◦,0). Therefore, the dynamics of are confirmed for the present set of the parameters even
3
m and m near the initial time are different, as shown when the initial conditions are changed.
1 3
in Fig. 2(b), where the time evolutions of m (t) and On the other hand, it was shown for the case of the
1x
m (t) in 0 t 1 ns are shown. Nevertheless, the dy- current-injectionlockingofthespintorqueoscillatorthat
3x
namics of m≤an≤dm synchronize gradually,andfinally, the phase difference between the magnetization oscilla-
1 3
synchronizationofm (t)=m (t)andm (t)=m (t) tionand the alternativecurrentdepends on the strength
1x 3x 1y 3y
is realized, as shown in Figs. 2(c) and 2(d). We empha- of the field-like torque53. The field-like torque in the
size here that the dynamics shown in these figures are present system can be either positive or negative, as
obtained for β = 0.01. mentioned above. These facts motivate us to study the
−
The mutual, as well as self, synchronization of the magnetization dynamics for different values of β. When
spin torque induced magnetization oscillation by using β = +0.01, synchronized dynamics is observed in a sim-
spin waves, electric current, microwave field, or dipole ilar manner, but in this case, the phase difference is an-
coupling has been an exciting topic from the viewpoints tiphase, i.e., m (t) = m (t), as shown in Figs. 2(e)
1x 3x
−
ofbothnonlinearscienceandpracticalapplications46–62. and 2(f).
The keyquantity ofthe synchronizationis the phasedif- Figure3summarizesthedependencesofthephasedif-
ference ∆ϕ between each magnetization to enhance the ference,∆ϕ,betweenthemagnetizationsonthefield-like
emission power from the spin torque oscillators and to torque strength β for the several values of the damping
investigate new practicalapplications such as neuromor- constant α. The vertical axis in this figure represents
phic architectures38,39. The synchronization found here, the phase difference in the unit of the oscillation period;
i.e., mℓx(t) = mℓ′x(t), is called the in-phase synchro- i.e.,∆ϕ=0correspondstothein-phasesynchronization.
nization. We should emphasize here that, although the Onthe otherhand,∆ϕ=0.50meansthatthe phasedif-
6
0.50 (a) (b)
1.0 1.0
α=0.005, 0.010, 0.015,
φ
Δ 0.020, 0.025, 0.030
e, mx 0 my0.5
c
n
ere 0.25 ::FF12
f -1.0 0
e dif 999.0 999.2 9t9im9.4e (9n9s9).6 999.8 1000 999.0 999.2 9t9i9m.4e (9n9s9).6 999.8 1000
s
a
h (c) (d)
p 1.0 1.0
0
-0.030 -0.020 -0.010 0 0.010 0.020 0.030 mx 0 my0.5
field-like torque, β
:F1
:F2
-1.0 0
FIG.3: Dependencesofthephasedifferences,∆ϕ,forseveral 999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000
valuesofαonthefield-liketorquestrengthβinthetransverse time (ns) time (ns)
geometry. ∆ϕ = 0 and 0.5 correspond to the in-phase and
(e) (f)
antiphase,respectively. Thevaluesofthecurrentdensity,J0, 1.0 1.0
is increased linearly, where J0 =30 MA/cm2 for α=0.005.
ference is half of a period, and thus, is antiphase. The mx 0 my0.5
algorithm evaluating ∆ϕ in the numerical simulation is
:F1
summarized in Appendix A. The results indicate that :F2
-1.0 0
the phase difference is antiphase for positive β, whereas 999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000
time (ns) time (ns)
it becomes in-phase when β becomes smaller than a cer-
tainvalue,exceptforthenarrowintermediateregionnear
FIG. 4: The variations of the x and y components of the
β α/2, where the phase difference is in between in-
∼ − magnetizationsintheF1 (solid,red)andF2 (dotted,blue)in
phase and antiphase.
the longitudinal geometry are shown in (a), (c) and (b), (d)
respectively, where β = −0.01. The initial conditions of m2
aredifferentbetween(a),(b)and(c),(d). Themagnetization
dynamicsfor β=+0.01 are shown in (e) and (f)
B. Longitudinal geometry
Next,westudythemagnetizationdynamicsinthelon- is a quarterof a period. The phase difference is found to
gitudinal geometry between F1 and F2 layers by solv- become a quarter ofthe period for positive β, whereas it
ing Eq. (16) numerically. Figure 4(a) and 4(b) show becomesin-phase orantiphasefor negativeβ, depending
mℓx(t) and mℓy(t) in a steady state, where β = 0.01. on the initial states of the magnetizations.
The initial conditions in these figures are m −(0) =
1
(cos80◦,sin80◦,0) and m (0) = (cos95◦,sin95◦,0).
2
An antiphase synchronization is observed in this case.
We notice, however, that the in-phase synchronization
can also be realized when the initial conditions are C. Summary of numerical simulations
changed. Figure 4(c) and 4(d) show such an exam-
ple, where m (0) = (cos80◦,sin80◦,0) and m (0) =
1 2
(cos85◦,sin85◦,0)are assumed. These numericalresults Let us summarize the results of the numerical simu-
lations here. In the transverse geometry, the coupling
indicate that both the in-phase and antiphase synchro-
torque induces the synchronized oscillation of the mag-
nizations are stable in this case, and whether the phase
netizations and finally stabilizes the configuration in the
difference finally becomes in-phase or antiphase depends
in-phase or antiphase state, depending on the values of
on the initial conditions, material parameters, and cur-
thefield-liketorquestrengthβ andthedampingconstant
rentmagnitude. We alsonoticethatthe phasedifference
α. The phase difference in the stable synchronized state
is changed when the value of β is changed. Figures 4(e)
in the longitudinal geometry also depends on the values
and 4(f) show m and m for β =+0.01. In this case,
ℓx ℓy
of β and α, as well as the initial conditions. In this case,
the phasedifference ofthe magnetizationsis aquarterof
however,inadditiontothein-phaseorantiphasestate,a
a precession period.
phase difference with a quarter of a period is generated.
The dependences of the phase difference on the field-
like torque strength β for several values of the Gilbert The phase difference can be measured from the spin
damping constant α are summarized in Fig. 5, where Hallmagnetoresistanceeffect. AccordingtoEq. (13),the
∆ϕ=0.25 in this figure means that the phase difference total electric current density in the transverse direction
7
0.50 as shown in Fig. 6(b). The Fourier transformation of
α=0.010 α=0.005 Eq.(22) has the peaks at f = 2nf for in-phase or an-
n 0
φ tiphase and f = 4nf when the phase difference is a
Δ α=0.025 n 0
e, quarter of a period.
c An interesting question regarding these numerical re-
n α=0.030
re sults is to clarify the reason why the phase difference
e 0.25
ff finally becomes a certain value for a given set of the
di parameters, i.e., which phase difference is an attrac-
e
s tor of the limit cycle. It is, however, difficult to an-
a
h
p α=0.015 swer this question directly due to the following rea-
son. We note that the present model includes sev-
α=0.020
0 eral small-valued parameters, α, β, χ, χ′, and χ′′,
-0.030 -0.020 -0.010 0 0.010 0.020 0.030 as shown in Eqs. (16) and (17), and is complicated.
field-like torque, β The torque related to the lowest order of β in these
equations is the conventional field-like torque given by
FIG.5: Dependencesofthephasedifferences,∆ϕ,forseveral [γβ~ϑ J /(2eMd )]m e . It should be noted that
R 0 F ℓ y
×
values of α on thefield-like torque strength β in the longitu- thephasedifferenceisnotdeterminedsolelybythisterm
dinal geometry, where ∆ϕ = 0 and 0.5 correspond to the because this lowest order term of the field-like torque
in-phase and antiphase, respectively. ∆ϕ = 0.25 means that does not include the coupling between the magnetiza-
the phase difference is a quarter of a period. The values of
tions. Similarly, the attractor is not determined solely
the current density, J0, is increased linearly, where J0 = 30 by the lowest order term of the coupling torque, which
MA/cm2 for α=0.005.
is [γ~ϑ J /(2eMd )]χm2 m (e m ) in Eq. (16)
R 0 F ℓ′y ℓ × y× ℓ
2cm)(a)00..32 2cm)(b)33 a(1n7d),[γb~eϑcRauJs0e/(t2heiMs tdoFr)q]uχemdℓo′xems nℓ′oytminℓ×clu(edxe×thme fiℓ)elidn-lEikqe.
610 A/0.01 610 A/ tinocrlquudei.ngTbhoetihr βcomanbdintahteiocnosuporlintghetohriqguheesrroerladteerdtteormχs,
ensity (--00..21 ensity ( qχu′,esatniodnχ, ′w′ hshicohulids bdieffitcaukletndiunetotoactchoeunntontolinaenasrwiteyratnhde
d d
ent -0.9399.0 999.2 999.4 999.6 999.8 1000 ent 32999.0 999.2 999.4 999.6 999.8 1000 complexity of the LLG equation.
urr time (ns) urr time (ns) Nevertheless, it is possible to reveal the relation be-
c c
tween the current and frequency in the synchronized os-
FIG. 6: (a) The transverse electric current densities given cillation state by assuming a certain value of the phase
by Eq. (21) for in-phase synchronization (solid line) and an- difference between the magnetizations. The current-
tiphase synchronization (dotted line). (b) The longitudinal frequency relation has been often measured in the ex-
electric current densities given by Eq. (22) for in-phase or periments, and therefore, it will be useful to develop a
antiphasesynchronization (solid line) and phasedifferenceof theory clarifying the role of the coupling on the current-
a quarterof a period (dotted line). frequency relation. In the next section, we will investi-
gate this subject.
is
JT =j +j
c 1y 3y
= (χm m +χ′m )J (21) IV. THEORETICAL ANALYSIS OF
1x 1y 1z 0
−(χm m +χ′m )J . CURRENT-FREQUENCY RELATION
3x 3y 3z 0
−
Then,anoscillatingcurrentappearsinthetransversedi- The purpose of this section is to develop an analyti-
rection for in-phase synchronization, whereas the trans- cal theory of the synchronization revealing the relation
versecurrentbecomeszeroforantiphasesynchronization, amongthecurrent,frequency,andthephasedifferenceof
as shown in Fig. 6(a). The Fourier transformation of the magnetizations in the synchronized oscillation state.
Eq. (21) for in-phase synchronization has peaks at the
frequencies off =(2n 1)f (n=1,2,3, ), where f
n 0 0
− ···
is the lowest frequency; see Appendix C. Similarly, the A. Basis of analysis
totalelectriccurrentdensityinthelongitudinaldirection
is
Here, let us mention the basis of our theoretical anal-
JL =J +j +j ysis. It is difficult to solve the LLG equation exactly
c 0 1x 2x
(22)
=J + χ′′+χm2 J + χ′′+χm2 J . because of its nonlinearity. Instead, we employ the aver-
0 1y 0 2y 0 agingtechnique of the LLG equation onconstantenergy
The oscillation(cid:0)frequency(cid:1)of this(cid:0)current is d(cid:1)ifferent for curves63. This approachhas been used to study the spin
the synchronizations having the phase difference of in- torque switching in thermally activated regions64–68 and
phase or antiphase and that of a quarter of a period, spin torque oscillators69–77, as well as the microwave as-
8
sisted magnetization reversal78,79, but has not been ap- where K(k) is the first kind of complete elliptic integral
pliedtothecoupledsystem. Thisapproachisvalidwhen with the modulus
the magnetic energy is changed slowly compared with
the oscillation period. We note that only the lowest or- 4πM(HK+2E/M)
k = . (30)
der terms in the LLG equationis necessaryto derive the sHK(4πM 2E/M)
−
current-frequency relation, as far as several parameters
such as β are small. Thus, we use the simplified LLG Note that Eq. (29) reproduces the ferromagnetic res-
equationmaintainingonlythedominantterms. TheLLG onance (FMR) frequency, γ H (H +4πM)/(2π), in
K K
equation used in this section for the longitudinal geome- thelimitofE E . IdentifyingE andϕ corresponds
min 0
→ p
try is to the determination of the initial condition.
dm Theaveragedtechniqueinvestigatestheenergychange
ℓ γmℓ Hℓ αγmℓ (mℓ Hℓ) during a precessionon a constant energy curve, which is
dt ≃− × − × ×
obtained from the LLG equation as
γ~ϑ J
− 2eMRd0 1+χ′′+χm2ℓ′y mℓ×(ey×mℓ), dE
F dt =W +WL,T+W , (31)
(cid:0) (cid:1) (23) dt s s α
I
whereas that for the transverse geometry is where W is the energy change by the conventional spin
s
dmℓ γmℓ Hℓ αγmℓ (mℓ Hℓ) tsoipraqtuieondudeuetotothtehsepdinamHpailnlgeffteocrtq,uweh.eTrehaesiWntαegisratlhreadnigse-
dt ≃− × − × ×
γ~ϑ J is over the precession period. The explicit forms of Ws
− 2eMRd0mℓ×(ey×mℓ) (24) and Wα are given by67
F
γ~ϑ J γ~ϑ J
− 2eMRd0χmℓ′xmℓ′ymℓ×(ex×mℓ). Ws = dt 2edR 0 [ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)]
In the self-oscillatFion state, the damping torque dur- Iπ~θRJ0(HFK+2E/M) (32)
= ,
ing the precession is balanced with the spin torque, and ed H (H +4πM)
F K K
thetorqueduetothemagneticfield,correspondingtothe
p
firsttermontherighthandsideofEq. (15),becomesthe
dominanttermdeterminingthemagnetizationdynamics. W = dtαγM H2 (m H )2
The dynamic trajectory given by this field torque corre- α − ℓ − ℓ· ℓ
I h i
sponds to constant energy curves of the energy density,
4πM 2E/M 2E
E =−M dmℓ·Hℓ, where its explicit form is =−4αMs H−K (cid:20)M K(k)+HKE(k)(cid:21),
R E = MHKm2 +2πM2m2 . (25) (33)
− 2 ℓy ℓz
whereE(k)isthesecondkindofcompleteellipticintegral.
The minimum and saddle points of Eq. (25) are mmin = On the other hand, WL,T represents the work done
e andm = e ,wherethecorrespondingenergy s
± y saddle ± x by the coupling torque in the longitudinal or transverse
densities are E = MH /2 and E = 0. The
solution of mℓ mprinecess−ing onKa constantsaedndelergy curve is goero(m24e)t.ry,Tchoerreexspploicnidtinfogrmtos tohfeWlaLstantedrmWTinaEreq.sh(o2w3n)
described by the Jacobi elliptic function80 as (see also s s
in the following sections. For both cases, the relation
Appendix C summarizing the derivations)
betweenthe currentandfrequency is clarifiedas follows.
2E 4K(k) Intheself-oscillationstate,sincethespintorquebalances
mx = 1+ sn t+ϕ0,k , (26) the damping torque, the following condition should be
MH τ(E)
r K (cid:20) (cid:21) satisfied,
4πM 2E/M 4K(k) dtdE =0. (34)
my = − dn t+ϕ0,k , (27) dt
s HK+4πM (cid:20) τ(E) (cid:21) I
The currentdensity J satisfying Eq. (34) is the current
0
H +2E/M 4K(k) necessarytoexcitetheself-oscillationontheconstanten-
K
m = cn t+ϕ ,k , (28) ergy curve of E, and is denoted as J (E). The relation
z 0 0
s HK+4πM (cid:20) τ(E) (cid:21) between the current and frequency in the self-oscillation
whereϕ0 istheinitialphase. Theperiodofmx andmz is state is given by this J0(E) and f(E) givenby Eq. (29).
τ(E),whereasthatofmy isτ(E)/2,wheretheprecession It should be emphasized that the current density J0(E)
frequency f(E)=1/τ(E) is given by depends on the phase difference between the magnetiza-
tionsthroughWL,T. Wewillthereforestudytherelation
s
γ HK(4πM 2E/M) between the phase difference and the current-frequency
f(E)= − , (29)
4K(k) relation in line with this deduction.
p
9
B. Transverse geometry (a) (b)
rtWhesleITantc=iootuhnIpisilndinstgetγch~tte2oϑioerRtndqrχF,uaJnew0sievmsedirℓn′sexvefimengseℓte′dioyg[meaaxtseet·rHtyh.ℓeT−chu(emrrweℓno·tre-kfxr)de(qomunee(ℓn3·bc5Hyy)ℓ)]. 62current density, J(E) (10 A/cm)0332222086.0freq3u.n0ecy (4G.0Hz)5.0 6.00 p1h.0ase di2f.f0erenc3e.,0 Δφ/K4.(0k) 62current density, J(E) (10 A/cm)022228888....012300000 pha1s.0e differ2en.0ce, Δφ/3K.0(k) 4.0
Before advancing the calculation, let us briefly men- (c) (d)
taprItifnenirephoraspaenEpjtnrsehoqeotccenshated.doecpiisrh(ffsrdy2.eeted6osrifis)eeeI,nfinnf∆di(tnctet2ϕeeishco7dcaei)n=rs,siuebooeas,πsaefincnd,sotdiighlnlwbly(eatyh2tthdpe8hiteoerh)ehfien.aeeanolsIslteeetnirhdpl∆alde,ttijphiirϕeicfft.icesiehtic=.rnao,cetfnratnue0ydhscgne,eerici,sasttaitlihntnhoaheKtnetei,(hcpopkieahisrhn)csca,ap-ilslpsserwlehea,ehhostcadteiewosohinrrefnneet--. 62current density, J(E) (10 A/cm)0322219752.0freq3u.n0ecy (4G.0Hz)5.0 6.00 p1h.0ase di2f.f0erenc3e.,0 Δφ/K4.(0k) 62current density, J(E) (10 A/cm)0222777...0001230 pha1s.0e differ2en.0ce, Δφ/3K.0(k) 4.0
∆ϕ = 0 for the in-phase synchronization, and the an-
tiphase synchronization corresponds to ∆ϕ = 2K(k). It FIG. 7: (a) The current density, J0(E), necessary to excite
theself-oscillation inthetransversegeometryasafunctionof
isusefultonotethat∆ϕ=2K(k)becomesπ inthe limit
the oscillation frequency f(E) and the phase difference ∆ϕ
of k 0, corresponding to the case that the oscillation of the magnetizations. The phase difference is in the unit
traje→ctorybecomesacircle. Similarly,∆ϕ=K(k)means of K(k). (b) Dependence of J0(E) for the transverse geome-
that the phase difference is a quarter of a period. try, on ∆ϕ at f(E) = 4.6 GHz. The dotted line represents
Equation(35)foranarbitraryphasedifference iseval- J0(E)intheabsenceofthecoupling. (c)Therelationamong
uated by numerically calculating the integral with the J0(E), f(E), and ∆ϕ in the longitudinal geometry. (d) The
solutions of mℓ and mℓ′ shown in Appendix D. It is, currentdensityJ0(E)forf(E)=4.6GHzinthelongitudinal
geometry.
however, useful to derive the analytical solutions of Eq.
(35) for specific values of the phase difference. Equa-
The current J in the transverse geometry is defined
tion (35) for both the in-phase (∆ϕ = 0) and antiphase 0
[∆ϕ=2K(k)] becomes as
WsT =∓2edF Hπ~Kϑ(RHχKJ0+4πM)(cid:18)−2ME(cid:19)(cid:18)1+ M2HEK(cid:19), J0(E)= 2α~eϑMRdFDNT, (37)
(36) where and are defined as
where the doupble sign means the upper for the in-phase N DT
(∆ϕ=0)synchronizationandthelowerfortheantiphase =γ dt H2 (m H )2 , (38)
[∆ϕ = 2K(k)] synchronization. Equation (36) is zero at N ℓ − ℓ· ℓ
E = E and E , and is negative (positive) for the I h i
min saddle
energy density E in the rage of E < E < E
min saddle
when ∆ϕ = 0 [2K(k)]. This means that the coupling =γ dt[e H (m e )(m H )]
T y ℓ ℓ y ℓ ℓ
torque acts as a damping (an antidamping)torque when D · − · ·
I
thephasedifferenceisin-phase(antiphase). Wealsonote
that WT =0 when ∆ϕ=K(k); i.e., the phase difference +γχ dtmℓ′xmℓ′y[ex·Hℓ−(mℓ·ex)(mℓ·Hℓ)].
s
I
is a quarter of a period. The calculations necessary to (39)
obtain these specific values of WT are also summarized
s
inAppendixD.WenotethatthesignchangeofWT with The explicit form of is obtained from Eq. (33) as
s N
respect to the phase difference is related to the fact that
thecouplingtorqueinthetransversegeometry,Eq. (14), 4πM 2E/M 2E
Bhaescatuhseeaonfgtuhlaisradnegpuelnadrednecpeeonfdmenℓc′xe,mtℓh′yemcoℓu×pl(ienxg×tomrqℓu)e. N =4s H−K (cid:20)M K(k)+HKE(k)(cid:21). (40)
acts as an anti-damping (a damping) torque when m
ℓx On the other hand, for the in-phase or antiphase is
T
andmℓ′x havethe opposite(same)signs,resultinginthe obtained from Eqs. (D32) and (36) as
increase (decrease) of the energy supplied to the ferro-
magnets by the coupling torque. 2π(H +2E/M)
K
In summary, the work done by the coupling torque, T =
D H (H +4πM)
WsT,isnegativeandminimizedatthein-phase(∆ϕ=0), K K (41)
zero for ∆ϕ = K(k), and positive and maximized at the pπχ(−2E/M)[1+2E/(MHK)],
antiphase [∆ϕ=2K(k)]. ∓ H (H +4πM)
K K
p
10
where the double sign means the upper for the in-phase Ontheotherhand,whenthephasedifferenceisaquarter
synchronizationandthe lowerforthe antiphasesynchro- of a period [∆ϕ=K(k)], Eq. (45) becomes
nization.
tciuoFrnriegfnruter-qefure7eqn(uac)eynsfchy(oErwe)slaatJni0od(nEtih)netahpsehftaursnaecntdsivioffenerssreeongfceetohm∆eeϕtor.sycTicllahane- WsL = π~ϑedRFχJ0HKH(KH+K2+E4/πMM)s−2ME (cid:18)4πM − 2ME(cid:19).
(47)
be obtained from this figure. To reveal the role of the
ItshouldbeemphasizedthatWLisalwayspositiveforan
phase difference more clearly, we show J (E) for a cer- s
0 arbitrary phase difference. This is because the coupling
tain value of f(E)(= 4.6GHz) in Fig. 7(b). Note that
torqueinEq. (10)alwaysactsasananti-dampingtorque.
J (E)issmallerthanthatintheabsenceofthecoupling,
0
WecandeterminethecurrentdensityJ (E)satisfying
which is shown by the dotted line, and minimized when 0
∆ϕ=2K(k),i.e.,theantiphase. Thisisbecausethework Eq. (34) in the longitudinal geometry, as in the case of
the transverse geometry, by replacing in Eq. (37)
donebythecouplingtorqueispositiveandmaximizedat DT
with
the antiphase. On the other hand, J (E) is maximized
0
at the in-phase, and is larger than that in the absence
of the coupling because the work done by the coupling L =γ(1+χ′′) dt[ey Hℓ (mℓ ey)(mℓ Hℓ)]
D · − · ·
torqueisnegativeandminimizedatthein-phase. Weno- I
tice that the phase differences observedin the numerical +γχ dtm2ℓ′y[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)].
simulation in Sec. III, i.e., the in-phase and antiphase, I
(48)
correspond to ∆ϕ satisfying
The explicit form of for both the in-phase and an-
∂J L
0 =0, (42) tiphase is obtained froDm Eqs. (32) and (46) as
∂∆ϕ
2π(1+χ′′)(H +2E/M)
or equivalently, = K
L
D H (H +4πM)
K K
∂WsT =0. (43) + πχ(pHK+2E/M)[4πM(HK−2E/M)−2HK(2E/M)],
∂∆ϕ [H (H +4πM)]3/2
K K
(49)
In other words,the phase differences observedin the nu-
merical simulations correspond to those giving the ex-
whereas that when the phase difference is a quarter of a
trema of J (WT).
0 s period is obtained from Eqs. (32) and (47) as
2π(1+χ′′)(H +2E/M)
K
C. Longitudinal geometry L =
D H (H +4πM)
K K
Let us investigate the theoretical relation between the 2pπχ(H +2E/M) 2E 2E
K
+ 4πM .
current and frequency in the longitudinal geometry. In HK(HK+4πM) s−M (cid:18) − M (cid:19)
this case, the averagedLLG equation is given by
(50)
dE
dt =(1+χ′′)W +WL+W , (44) Figure7(c)showsJ (E) asfunctions off(E) and∆ϕ.
dt s s α 0
I The current-frequencyrelationinthe longitudinalgeom-
etry can be obtained from this figure. It is noted that
where W and W are given by Eqs. (32) and (33). On
s α J (E) is always smaller than that in the absence of the
the other hand, WL representing the energy change due 0
s coupling because the coupling torque in the longitudinal
to the longitudinal coupling is defined as
geometry always points to the anti-damping direction,
γ~ϑ χJ and therefore, the work done by the coupling torque is
WsL = dt 2eRd 0m2ℓ′y[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)]. always positive. Figure 7(b) shows J0(E) as a function
I F (45) of ∆ϕ at a certain value of f(E). As shown, J0(E) has
minimaatboththein-phase(∆ϕ=0)andtheantiphase
For both the in-phase and antiphase, Eq. (45) becomes
[∆ϕ = 2K(k)], whereas it is maximized when the phase
(see Appendix D)
differenceisaquarterofaperiod[∆ϕ=K(k)]. Weagain
π~ϑ χJ notice that these phase differences found in the numer-
WsL = R 0 ical simulations in Sec. III correspond to ∆ϕ satisfying
2ed H (H +4πM)
F K K ∂J (E)/∂∆ϕ=0, or equivalently,
0
(H +2E/M)[4πM(H 2E/M) 2H (2E/M)]
pK K− − K .
× H (H +4πM) ∂WL
K K s =0. (51)
(46) ∂∆ϕ