Table Of ContentDynamic Feedback in Ferromagnet–Spin Hall Metal Heterostructures
Ran Cheng,1 Jian-Gang Zhu,2 and Di Xiao1
1Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
2Department of Electrical and Computer Engineering,
Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
In ferromagnet/normal metal heterostructures, spin pumping and spin-transfer torques are two
reciprocalprocessesthatoccurconcomitantly. Theirinterplayintroducesadynamicfeedbackeffect
interconnectingenergydissipationchannelsofbothmagnetizationandcurrent. Bysolvingthespin
diffusion process in the presence of the spin Hall effect in the normal metal, we show that the
6
dynamicfeedbackgivesriseto: (i)anonlinearmagneticdampingthatiscrucialtosustainuniform
1
0 steady-state oscillations of a spin Hall oscillator at large angles. (ii) a frequency dependent spin
2 Hall magnetoimpedance that reduces to the spin Hall magnetoresistance in the dc limit.
g
PACSnumbers: 75.78.-n,75.76.+j,75.47.-m,85.75.-d
u
A
Introduction.—A central concept in modern spintron- ever, recentexperimentsshowedthatthespinHalleffect
4
ics is the emergence of artificial electromagnetics due to (SHE) in the NM can drastically modify the dynamical
2
the interplay between magnetization dynamics and elec- behavior of the entire heterostructure [18, 19]. Taking
] trontransport. Forinstance,whenanelectronspinadia- into account the SHE, spin pumping and spin backflow
l
l baticallyfollowsaslowly-varyingmagnetization,itswave are also connected via the combined effect of the SHE
a
h function acquires a geometric phase changing with time. and its inverse process, which forms a feedback loop as
- This phase resembles a time-varying magnetic flux and illustrated in Fig. 1(a). This additional feedback mecha-
s
e produces a spin motive force (SMF) according to the nism, proportional to θs2 (θs is the spin Hall angle), was
m Faraday effect [1, 2]. As a feedback, electrons driven by completelyignoredinpreviousstudies[20,21]. Neverthe-
SMFs react on the magnetization via the spin-transfer less, the recently discovered spin Hall magnetoresistance
.
at torque (STT) [3–6], which enhances the magnetic damp- (SMR) [22–24] reveals that physics at the θs2 level is es-
m ing[7]tohinderthemagnetizationdynamicsthatcauses sential to the electron transport. As the reported spin
- the SMF. In a reciprocal sense, if a magnetic texture is Hall angle θs is getting larger [25, 26], it is tempting to
d driven into motion by a current, it in turn exerts SMFs askwhetherafeedbackeffectproportionaltoθ2 canalter
n s
ontheelectrons,modifyingtheelectricalresistivity[8,9]. the magnetization dynamics or the electron transport in
o
Thefeedbackmechanismpersistseveninthepresenceof a qualitative way.
c
[ thermal and mechanical forces [10], or when spin-orbit In this Letter, we show that our proposed feedback
3 interactions are strong [11]. These examples constituent mechanism manifests as a novel nonlinear damping ef-
v a general manifestation of Lenz’s law in artificial elec- fectintheFMdynamics. Itenablesuniform steady-state
8 tromagnetic, which states that a motive force induction auto-oscillationsofaspinHalloscillatorbypreventingit
1 alwaysopposesthechangeoffluxthatcausesthemotive from growing into magnetic switching. If our proposed
6
force, and vice versa [12]. In generic settings, Lenz’s law feedback effect is ignored, however, auto-oscillations are
1
imposesauniversalruleonhowaprocesscanbeaffected possible only for spin-valves without the participation of
0
. by its converse: feedback should be negative, otherwise the SHE [27], for materials with strong dipolar interac-
1
energy is not conserved. tions [28], or for spatially localized solitons in a FM/NM
0
6 Inallknownphenomenasofar,electronsandthemag- heterostructure [29, 30]. In a reciprocal sense, we show
1 netization couple locally in the bulk [13]. Therefore, one that the feedback loop also gives rise to a spin Hall mag-
v: isabletoeliminateeitherthemagnetizationdynamicsor netoimpedance in the electron transport which reduces
i the electron motion at arbitrary locations to derive the to the observed SMR in the dc limit.
X
feedbackrenormalizationofvariousresponsecoefficients. Formalism.—Consider a FM/NM bilayer structure as
r
a In ferromagnet (FM)/normal metal (NM) heterostruc- shown in Fig. 1(a), where the layer thicknesses are dM
tures, however, nonlocal effects arise because conduction and d , respectively. The coordinate system is chosen
N
electrons and magnetization reside in different materials such that the magnetization direction at rest is along x,
and couple only at the interface. In this scenario, a pre- and the interface normal is along z. We assume that the
cessing FM can pump spin current into the NM [14, 15], FMisinsulating(e.g.,YIG),buttheessentialphysicsre-
whichsubsequentlyexperiencesabackflowandreactson mainsvalidforaconductingFM.Letµ /2betheelectro-
0
the FM via the STT [16]. The combined effect of spin chemicalpotentialandµthevectorofspinaccumulation
pumpingandthebackflow-inducedSTTrenormalizesthe intheNM,sobyOhm’slawthechargecurrentdensityis
interfacial transverse conductance [17], and captures a Jc = σ[∂ µ +θ ε ∂ µ ], and the spin current den-
i −2e i 0 s ijk j k
static feedback effect involving nonlocal processes. How- sity is Js = σ[∂ µ θ ε ∂ µ ] with i the transport
ij −2e i j − s ijk k 0
2
direction and j the direction of spin polarization. In our stationaryspindiffusionprocessateveryinstantoftime.
device geometry, only the spin current flowing along z- Retaining to the θ2 order, Eq. (1) is solved as
s
direction is relevant, thus we assume µ=µ(z,t). Corre-
spondingly, the spin current density reduces to a vector µ(z)= θ 2eλzˆ J sinh2z−2λdN
Js; we scale it in the same unit as the charge current s σ × c coshdN
2λ
density J . The electron and spin dynamics in the NM
are then dcescribed by three equations + 2eλ(cid:2)J +θ2zˆ (zˆ J )(cid:3)coshz−λdN , (6)
σ s0 s × × s0 sinhdN
λ
∂µ ∂2µ 1
=D µ , (1)
∂t ∂z2 − τsf where λ = √Dτsf is the spin diffusion length. Here, we
(cid:20) (cid:21) suppress the t variable in µ(z) since its time dependence
σ ∂µ
Jc =−2e ∇µ0+θszˆ× ∂z , (2) simply originates from Jc and Js0. Combining Eq. (1)—
Eq. (6), we can either eliminate the electron degrees of
(cid:20) (cid:21)
σ ∂µ
J = +θ zˆ µ , (3) freedomtoderiveaneffectivemagnetizationdynamics,or
s s 0
−2e ∂z ×∇ eliminate the time derivative of the magnetization (m˙ )
to get an effective magneto-transport of the electrons.
whereDisthediffusionconstant,τ isthespin-fliprelax-
sf These operations invoke our proposed dynamic feedback
ationtime,σ istheconductivity,eistheelectroncharge,
mechanism to the FM/NM heterostructure.
and θ is the spin Hall angle.
s Nonlinear damping.—Our goal is to express the spin
To solve the spin accumulation µ, we assume that the
current density flowing across the interface J in terms
chargecurrentdensityJ isanapplieddcchargecurrent s0
c ofthemagnetizationm(t),bywhichtheLLGEq.(5)will
densitywhichisfixedbyexternalcircuit. Itonlysupplies
no longer involve any electron degree of freedom except
a constant drive to the system but does not participate
J . Tothisend,wecombineEq.(4)andEq.(6)forz =0,
in the feedback process. To make it more specific, if we c
and obtain two convoluted relations of J and µ . By
instead consider a constant voltage drive µ =const., s0 s0
∇ 0 means of iterations truncating at θ2 order, we can solve
then J and µ will switch roles in Eq. (2) and (3). s
In othecr word∇s, e0ither J or µ must depend on z while Js0 as a function of Jc, m(t) and its time derivative.
c 0 Then we insert this J into Eq. (5), which yields the
the other is uniform in space. In the following, we focus s0
effective magnetization dynamics
on a constant current drive condition and allow µ =
0
µ0(z). Inaddition,wehavetwoboundaryconditions[21]: dm = γH m+ω m (cid:104)(zˆ jˆ) m(cid:105)
Js(dN)=0 and dt eff× s × × c ×
∂m
Js0 Js(0)= Gr [m (m µs0)+¯hm m˙ ] , (4) +(α0+αsp)m× ∂t
≡ e × × × (cid:18) (cid:19)
∂m ∂m
+α m2m + zm zˆ , (7)
where we used the macrospin model and m is the unit fb z × ∂t ∂t ×
vector of the magnetization. µ stands for µ(0) and G
s0 r
is the real part of the areal density of the spin-mixing where jˆ is the unit vector of J and
c c
conductance (the imaginary part G is neglected since
i
Greipr(cid:28)eseGnrt[S3T1]T).aTnhdespmin×pu(mmp×inµg,s0r)esapnedct¯hivmely×. Tm˙hetyeramres ωs =θsJceM¯hsγdM σ+λG2λrGtarnchotd2hNλdλN (8)
two fundamental ingredients bridging the electron (spin)
transportintheNMwiththeFM.Duetospinconserva- is the strength of the STT (driven by J ) scaled in the
c
tion, the spin current density J must be added to the frequency dimension. The two damping coefficients are
s0
Landau-Lifshitz-Gilbert (LLG) equation [16, 17]
¯h2γ σG
r
α = , (9)
ddmt =γHeff×m+α0m× ∂∂mt + 2eM¯hγd Js0 , (5) sp 2e2MsdM σ+2λGrcothdλN
s M ¯h2γ σλG2cothdN
α =θ2 r λ . (10)
where γ is the gyromagnetic ratio, ¯h is the reduced fb se2MsdM (σ+2λGrcothdλN)2
Planck constant, M is the saturation magnetization, α
s 0
istheGilbertdampingconstant, andH istheeffective Here, α describes the conventional enhanced damping
eff sp
magnetic field. from spin pumping with the spin backflow effects taken
Thetypicalfrequencyω ofmagnetizationoscillationis intoaccount[16,17,20,21];itisindependentoftheSHE.
much smaller than the spin relaxation rate 1/τ . As a By contrast, the α term is completely new. It reflects
sf fb
result,thespinaccumulationµ(z,t)adaptstotheinstan- the dynamic feedback realized by virtue of the combined
taneousmagnetizationandiskeptquasi-equilibrium[20], effectoftheSHEanditsinverseprocessasschematically
and the spin dynamics described by Eq. (1) reduces to a shown in Fig. 1(a). From Eq. (7), we see that this novel
3
p ✓ into Eq. (7) and setting Im[ω] = 0 yield the threshold
ISHE !s=105
m(t) STTstrength: ωth =(α +α +α /2)γH,whichcanbe
s 0 sp fb
Jspump ↵fb=4⇥10�4 Icnontvheertebdeytoonadtthhrreesshhoollddcruergriemnte,dJens>ityJJtcthh, bmy Eqs.ta(r8t)s.
✓ Jbackfeelodobpack rµ0 p2 !s=104 gtorogwrtohwweixllpuonlteimntaiatelllyyienvotlivmeei.ntIof αacfbm=agn0ce,thicowswe⊥vitecrh,itnhge.
s
x This is because the driving STT and the Gilbert damp-
!s=103 ingarebothlinearinm sothatiftheformerovercomes
y z FM NM SHE ⊥
the latter it wins at arbitrary angles θ = arcsinm . As
�dM 0 dN 0 4 8 12 t(ms) a result, whenever a spontaneous motion is triggere⊥d, its
(a) (b)
amplitude will grow indefinitely. The only way to enable
✓(t )
⇡p !1 stable oscillation at an intermediate configuration is to
make the overall damping grow faster than the driving
↵
fb STT with an increasing m , i.e., the damping has to be
0
⊥
⇡2p2 21⇥⇥1100��44 nteornmliinneaatreinatman⊥.anBgyledowinhgerseo,ththeetwamopcloitmupdeetginrgowmthecwhail-l
3⇥10�4 nisms compensate each other, and a steady-state oscil-
4⇥10�4 lation is realized there. The feedback-induced nonlinear
damping effect just fulfills this need. From the perspec-
0 5 11000 15 12001 25 13002 35 14003 45 15004 55!s6(2M050Hz) tiveofdynamicalstability,afterasteady-stateoscillation
(c)
is achieved, the damping (the STT) will dominate again
if the angle θ is getting larger (smaller) so that the mag-
FIG. 1. (Color online) (a) In a FM/NM bilayer, spin pump-
netization will be dragged back. We mention in passing
ing and spin backflow are connected by the SHE and its in-
that our proposed feedback mechanism is not exclusive
verse process. (b) and (c): Simulations of a spin Hall nano-
oscillator in the presence of the feedback-induced nonlinear to FMs, but applies to antiferromagnets as well when
dampingα ,withγH =10GHz,d =1nm,α +α =0.01, integrated with the SHE [32].
fb m 0 sp
andotherparameterstakenfromRef.[33]. TheSTTstrength To justify the above prediction, we perform a series of
ω is scaled in Megahertz.
s numerical simulations. From Eq. (10), we know that a
smaller (larger) d (d ) leads to a larger α . Consider
M N fb
an YIG/Pt bilayer structure with d a few nanometers
M
dampingtermisnonlinear inm —thecomponentofm and d λ, and other material parameters taken from
transverse to the effective field H⊥eff, whereas the Gilbert a receNnt(cid:29)experiment [33], then αfb is estimated to be of
damping term is linear in m . order 10 4, comparable to the intrinsic Gilbert damping
⊥ −
Thefeedback-inducednonlineardampingeffectcanbe α in YIG. Assuming γH =10GHz, α +α =0.01 and
0 0 sp
understoodinanintuitiveway. Ifthemagnetizationpre- α =4 10 4, we plot in Fig. 1(b) the precession angle
fb × −
cession is getting larger, it will trigger a chain reaction: θ as a function of time for three different STT strengths
firstthepumpedspincurrentJs0 increases,thenthespin ωs (scaled in Megahertz). We also plot in Fig. 1(c) the
diffusion becomes stronger (i.e., |∂zµ| gets larger). This terminal angle θ(t→∞) as a function of ωs for four dif-
will necessarily lead to a larger emf ∇µ0 in the NM ac- ferentvaluesofαfb. InFig.1(c),twofeaturesareevident:
cording to Eq. (2), as we have fixed the current density (i) a larger ω (larger driving current density J ) results
s c
Jc. The change of the emf will eventually feed back into in a larger terminal angle, but at sufficiently large ωs,
Js0 according to Eq. (3), limiting its further growth. As theoscillatorinevitablyundergoesamagneticswitching.
a consequence, the growing magnetization precession is (ii) a lager α (stronger feedback) widens the window
fb
inhibited. Ifwedrawananalogybetweenthemagnetiza- of steady-state oscillations. These results have justified
tion oscillation and an electric motor, the feedback loop that the nonlinear damping effect described by Eq. (7)
realizes an effective back emf induction preventing the can indeed sustain stable oscillations.
electric motor from rotating faster. Next we comment on several side-effects that could
Example.—We demonstrate the physical significance potentially obscure the observation of our predictions.
of the nonlinear damping effect in a current-driven spin First, if the FM film is too thin, the dipolar interaction
Hall nano-oscillator. Consider that the magnetization is mightnotbenegligible,whichcancausemagnon-magnon
polarized by a magnetic field H =Hxˆ, and is driven by scatteringthatprovidesadifferentnonlinearitytobound
a dc current density J =J yˆ. To determine the thresh- aspontaneousexcitationfromblowingup[28]. Whenthe
c c
oldofauto-oscillationexcitation,weassumethatm(t)= dipolar effect dominates, the nonlinear damping effect is
xˆ+m eiωt where m =m +im and m 1, and undermined. However, if the magnon-magnon scattering
y z
⊥ ⊥ | ⊥|(cid:28)
regard ω as a complex frequency where the imaginary is negligible and the dipolar effect can be approximated
partrepresentsthedamping. InsertingtheaboveAnsatz by a hard-axis anisotropy, the nonlinear damping effect
4
shouldstillbeobservable,butthesteady-stateprecession 1.0
will become elliptical. Second, in existing realizations of
0.8
spin Hall oscillators such as Ref. [30], a point-contact is Re[�Z1]
oftenused. Aknownfactaboutsuchexperimentalsetup �⇢1
0.6
is that it can easily excite the spatially localized mode Im[�Z1]
(soliton) [29] rather than a uniform oscillation. Finally, 0.4 �⇢1
asteady-stateoscillationseemstobepossibleifweapply Im[�Z2]
0.2 �⇢1
the driving current density J parallel to m (so the spin
c
accumulation is perpendicular to m due to the device
0.0
geometry). However, in that case the oscillation cannot
be regarded as an auto-oscillation of the eigenmode with -0.2 !/!
H
a fixed frequency. Instead, the magnetization undergoes 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
consecutive precessional switching with a frequency pro-
FIG. 2. (Color online) Frequency dependence of the spin
portional to J [34]. While this still forms an oscillator,
c
Hall magnetoimpedance scaled by ∆ρ ≡ lim Re[∆Z ].
it is not able to directly verify the physical significance 1 ω→0 1
Re[∆Z ] is not shown since Re[∆Z ] = Im[∆Z ]. The plot
of our nonlinear damping effect. is based2 on an YIG/Pt structure [332] with α =12.3×10−4,
0
Spin Hall magnetoimpedance.—As a reciprocal effect, d =1nm, and d (cid:29)λ.
M N
the dynamic feedback also affects the electron transport.
If we apply an ac current density J (t) = J˜eiωt to
c c
an FM/NM heterostructure longitudinally, the SHE will components
drivethemagnetizationprecessionviatheSTT,whichin
turncanpumpspincurrentbackintotheNMandrenor- ∆Z1(ω) = θ2λ2ρGr (1+U +Pω)tanh2 d2Nλ , (13)
malize the resistivity by means of the inverse SHE. This ρ s d (1+U + )2+ 2
N Pω Qω
is analogous to an ac electric motor accommodating the ∆Z (ω) λ2ρG tanh2 dN
counteractive motive force induced by the simultaneous 2 = θ2 r Qω 2λ , (14)
ρ − s d (1+U + )2+ 2
dynamotor effect. Although the feedback received by an N Pω Qω
accurrentdrivehasbeenstudiedfromtheangleofSTT- where U =2ρG λcothdN and
induced ferromagnetic resonance [18, 35–37], we explore r λ
its phenomenology from the feedback perspective, which ¯h2γG iω(ω +iα ω)
r H 0
is conceptually advanced and reveals new insights. = , (15a)
Consider that the magnetization m(t) is oscillating Pω 2e2MsdF (ωH +iα0ω)2−ω2
uniformly around an applied magnetic field H = Hhˆ. = ¯h2γGr ω2 . (15b)
By performing a Fourier transformation, we can rewrite Qω 2e2MsdF (ωH +iα0ω)2 ω2
−
Eq. (5) in the frequency domain (where quantities are
In the dc limit ω 0, the above results reduce to the
capped with tildes) and obtain
→
recently discovered SMR [22–24]. Eq. (13) and (14) give
usarelationRe[∆Z ]=Im[∆Z ],whichwillbreakdown
¯hγ iωJ˜ +(ω +iα ω)hˆ J˜ 2 1
m˜ = s0 H 0 × s0 , (11) if the imaginary part of the spin-mixing conductance Gi
⊥ 2eMsdF (ωH +iα0ω)2 ω2 is included in our calculation [31]. In Fig. 2, we plot
−
∆Z (ω)and∆Z (ω)asfunctionsofthefrequencyωwith
1 2
where ωH =γH. Combining Eq. (11) with the spin cur- allquantitiesscaledbythelongitudinalSMR∆ρ1. Fig.2
rent density flowing through the interface [Eq. (4)], the shows that a pronounced deviation of the SMI from the
spin accumulation [Eq. (6)], and Ohm’s law [Eq. (2)], SMRtakesplaceonlyinthevicinityoftheSTT-induced
we are able to solve the (spatially) averaged electric field ferromagnetic resonance. This deviation, according to
E˜ 1 (cid:82)dN µ˜ dz. Choosing the in-plane coordi- Eqs. (13), (14), and (15), scales roughly as d /d when
≡ −2edN 0 ∇ 0 N M
nates such that J˜c =J˜cxˆ, we obtain dN is small. In a recent measurement [35], the observed
deviation of the SMI from the SMR is negligibly small,
E˜ =[ρ+∆ρ +∆Z (ω)(1 h2)]J˜ , (12a) probablybecausetheirFMistoothick(dM=55nm)while
x 0 1 − y c the NM is too thin (d =4nm).
N
E˜y =[∆Z1(ω)hxhy+∆Z2(ω)hz]J˜c , (12b) The Oersted field generated by Jc is also responsible
for the SMI [36]. But one can distinguish the feedback
where ρ=1/σ is the intrinsic bulk resistivity of the NM contribution and the Oersted field contribution from the
withoutincludinganyfeedbackeffect. Here,thespinHall symmetry pattern of SMI with respect to (ω ω ). For
H
−
magnetoimpedance (SMI) consists of three distinct con- instance, Re[∆Z ] due to the dynamic feedback is sym-
1
tributions: one frequency independent (dc) component metric around ω , whereas it becomes antisymmetric
H
∆ρ /ρ = θ22λ tanhdN, and two frequency dependent when the Oersted field is dominating. The relative ratio
0 − sdN 2λ
5
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[18] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,
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Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman,
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N Science 336, 555 (2012).
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heterostructures,see,forexample,J.Sinova,S.O.Valen-
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