Gigantic enhancement of spin Seebeck effect by phonon drag

0.22 MB·English

by  Hiroto Adachi

Gigantic enhancement of spin Seebeck effect by phonon drag
Hiroto Adachi1,2,1,a) Ken-ichi Uchida3,1 Eiji Saitoh3,1,2,4,1 Jun-ichiro Ohe1,2,1 Saburo Takahashi3,2,1 and
Sadamichi Maekawa1,21
1 Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195,
Japan
2 CREST, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075,
Japan
3 Institute for Materials Research, Tohoku University, Sendai 980-8755, Japan
4 PRESTO, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075,
1 Japan
1
(Dated: 1 February 2011)
0
2 We investigate both theoretically and experimentally a gigantic enhancement of the spin Seebeck effect in a

n prototypicalmagnetLaY2Fe5O12 at low temperatures. Our theoreticalanalysissheds lightonthe important
a role of phonons; the spin Seebeck effect is enormously enhanced by nonequilibrium phonons that drag the
J
low-lying spin excitations. We further argue that this scenario gives a clue to understand the observation

1 of the spin Seebeck effect that is unaccompanied by a global spin current, and predict that the substrate
3
condition affects the observed signal.

]
l
l Whenatemperaturegradientisappliedtoaferromag- in the absence of global spin current flowing through
a
h net, a force is induced acting on electrons’ spin to drive GaMnAs. Obviously, the scenario of magnon-mediated
- spincurrents. ThisphenomenontermedthespinSeebeck SSE10,11 fails to explain the experiment, showing that
s
e effect (SSE)1–3 has recently drawntremendous attention the full understanding of SSE has not yet been reached.
m as a new source of spin currents needed for future spin- For a deep understanding of the physics behind SSE,
based electronics.4,5 SSE is now established as an uni- we here explore the low-temperature behavior of SSE in
.
at versal aspect of ferromagnetic materials as it has been an insulating magnet LaY2Fe5O12. Figure 1 shows a
m observed in a variety of materials ranging from a metal- schematic illustration of our device structure.12 An in-
lic ferromagnet1 Ni Fe and a semiconducting ferro- plane externalmagnetic field H and auniform tempera-
- 81 19
d magnet2GaMnAstoaninsulatingmagnet3LaY Fe O . ture gradient∇T were applied along the z direction [see
2 5 12
n Besides its impact on the technological application, SSE FIG. 1 (a)]. The ∇T generates a spin voltageacross the
o
offers a number of new topics on the interplay of heat LaY Fe O /Ptinterface,andinjects(ejects)aspincur-
c 2 5 12
and spin currents,6,7 and it triggered the emergence of rent I into (from) the Pt wire. In the Pt wire, a part of
[ s
the new field named “spin caloritronics”8 in the rapidly- the injected/ejected I is convertedinto a chargevoltage
s
2 growing spintronics community. through the so-called inverse spin-Hall effect (ISHE):13
v
A mystery concerning SSE was how conduction elec-
0 V =Θ (eI )(ρ/w), (1)
0 trons can sustain the spin voltage over so long range of ISHE H | | s
6 several millimeters in spite of the short conduction elec- where e,Θ ,ρandw arethe absolutevalueofelectron
H
4 trons’ spin-flip diffusion length λsf, which is typically of | |
. severaltensnanometers. Thisproblemhasrecentlybeen
0
resolvedbyaseriesofexperimentsonspincurrentsusing (a) LaY2Fe5O12 (b)
01 magnetic insulators. A recent experiment on the elec- Pt w V 0.5 H = 100 Oe
1 tric signal transmission through a magnetic insulator9 H Higher T
K)
iv: hofiglholcigahlitzsedthsepirnosl,ei.oef.,tmheaglonwon-lsy,inbgymdeamgnoentsitcraetxicnigtatthioant T+T∆ T(K) ∇T T(V/∆µ 0
X magnons transmit the spin current over a long distance V/ Lower T
r of several millimeters. A subsequent experiment on SSE T -0.5
a foramagneticinsulator3LaY Fe O confirmedthatthe z (mm)
2 5 12 -4 0 4 0 100 200 300
magnon-based scenario can explain the SSE experiment T (K)
at room temperature, since the length scale associated
with magnons λ . However, a new issue on SSE was
sf
brought by a v≫ery recent experiment on a ferromagnetic FIG. 1. (Color online) Gigantic enhancement of SSE in
semiconductor2GaMnAs,whereitwasdemonstrated,by LaY2Fe5O12 at low temperatures. (a) Schematic illustra-
tionof theLaY Fe O /Ptsample andthetemperaturepro-
cutting the magnetic coupling in GaMnAs while keep- 2 5 12
file along the z direction. Here H denotes an external
ing the thermal contact, that SSE can be observed even
magnetic field (with magnitude H). The sample comprises
a LaY Fe O film with 8×4mm2 rectangular shape and
2 5 12
two separated Pt wires with the width w attached to the
LaY Fe O surface at the interval of 5.6 mm. (b) T depen-
2 5 12
a)[email protected] denceof V/∆T at H =100Oe.
2
(a) N N N ) 1.8
1 2 3 K 4
Jsd Jex Jex Jsd 50 1.6 K)3
P ( 1.4 50
1 HE 1.2 κ(2
F1∆TT1 < TF22 < T3 F3 V/IS 0 .18 κ/(T)100 100 T [K] 200 300
)
T 0.6
(b) N1 P3 N2 P’3 N3 (HE 0.4 phonon drag + magnon
S 0.2 magnon
VI
0
P P 0 50 100 150 200 250 300
2 4 T [K]
F1 F2 F3
(c) Ω Ω
Is 0 0 Is(N3)
z z z FIG.3. (Coloronline) Comparison ofexperimentalandthe-
1 2 3 z
oretical SSE signal. Solid circles: experimental spin See-
I(N )
Is(N1) s 2 beck data for LaY2Fe5O12 (Dotted line is a guide to the
eye). The solid curve: calculated T-dependence of V
ISHE
duetothesum ofthephonon-draggedSSEand themagnon-
FIG. 2. (Color online) Diagrammatic representation of the mediated SSE. The dashed curve: calculated T-dependence
thermal spin injection process. (a) Magnon-mediated SSE. ofVISHE dueonlytothemagnon-mediatedSSE.Wehaveas-
Here the system is composed of ferromagnet (F, in the ex- sumedT-independentΘH andα,andusedTD =565K23 and
periment LaY2Fe5O12) and nonmagnetic metals (N, in the TM = 560K. The data are normalized by its value at 50 K
experimentPt),whicharedividedintothreetemperaturedo- exceptthattheresultformagnon-mediatedSSEisplottedto
mains of F /N , F /N , and F /N with their local temper- reproduce the room-temperature signal. Inset: experimental
1 1 2 2 3 3
atures of T1, T2, and T3. The thin solid lines with arrows thermalconductivityκforY3Fe5O12takenfromRef.23(solid
(bold lines without arrows) are electron (magnon) propaga- circles) and theresult of thefit (solid curve)using Eq. (4).
tors. Here, J (J ) is the strength of the s-d coupling at
sd ex
theF/N interface(theexchangecouplinginF). (b)Phonon-
draggedSSEwherethedashedlinesarephononpropagators.
ment of magnon lifetime is not conclusively excluded,
The process P injects the spin current with the same mag-
4 judging from the ferromagnetic resonance linewidth in
nitude as (but opposite sign to) the process P due to the
relation T −T = −(T −T ), while no spin c2urrent is in- Y3Fe5O1215 asameasureofthe inversemagnonlifetime,
1 2 3 2
jected into N because of the cancellation between the two itdoesnotseemtobethecase. Therefore,weneedanew
2
relevant processes P3 and P3′. Here, Ω0 = pKph/Mion with mechanism to account for the observed low-T enhance-
theionmassM andtheelasticconstantK inF. (c)Cal- ment of SSE.
ion ph
culated spatial dependence of the spin current injected into Here,thegiganticenhancementofSSEforLaY Fe O
2 5 12
Ni (i=1,2,3). below room temperature is analyzed in the light of
phonon-drag mechanism.16–18 Back in 1946 in the con-
text of thermoelectrics, Gurevich pointed out16 that the
charge, spin-Hall angle, resistivity and width of the Pt thermopower can be generated by a stream of phonons
wire, respectively. Therefore, the T-driven spin injec- driven by the temperature gradient, which then drag
∇
tion, or SSE, is electrically detectable. In FIG. 1 (b), electrons and cause their convection. This idea, known
we show the temperature (T) dependence of V /∆T nowadays as phonon-drag mechanism, has been estab-
ISHE
at H = 100Oe, measured when the Pt wires are at- lished17,18 as a principal mechanism causing the low-T
tached to the lower- and higher-temperature ends of the enhancement19 of the thermopower. Because SSE is a
LaY Fe O layer, respectively. The sign of V/∆T is spin counterpart of the Seebeck effect, it is natural to
2 5 12
reversed between these Pt wires, a situation consistent expect that a similar physics underlies SSE. It is this
with SSE-induced ISHE. Notable is that, at T = 50K, approachthat we adopt in the present work.
the magnitude of V/∆T is dramatically enhanced. Our theoretical analysis starts from considering the
A simple scenario of the magnon-mediated SSE10,11 model shown in FIG. 2 (a). The key point in our model
[FIG. 2 (a)] is unable to explain the observed low-T is that the temperature gradient T is applied over the
∇
enhancement. If such a scenario could explain the ex- ferromagnet, but there is locally no temperature differ-
periment, the low-T enhancement of V would come encebetweentheferromagnet(F)andtheattachednon-
ISHE
fromeithertheenhancementofthespin-HallangleΘ or magneticmetals(N). We assumethateachtemperature
H
thatofthemagnonlifetime;otherwiseI duetomagnon- domain is initially in local thermal equilibrium, then we
s
mediatedSSE is a monotonicincreasingfunction ofT as switchontheinteractionsamongthedomainsandcalcu-
discussedbelow. FromtheT-dependenceofthespin-Hall late the nonequilibrium dynamics of spin density in N.
conductivity,14weconcludethatthereisnoenhancement ThecentralquantitythatcharacterizesSSEisthespin
of Θ at low T. While the possibility of the enhance- current I injected into N (in experiment Pt), since it
H s
3
block spin current nonmagnetic where v is the phonon velocity, and C (T) =
ph ph
substrate 9N k (T/T )3 TD/T dw w3 is the phonon spe-
N1 N2 N3 D B D 0 4sinh2(w/2)
cific heat with thRe number of phonon modes N . After
D
getting the information on τ (T), we calculate the T-
ph
dependence of V resulting from the phonon-dragged
ISHE
F1 F2 F3 SSE. The result, plotted in FIG. 3 (the solid curve),
showsanexcellentdescriptionofthelow-T enhancement
of SSE.25 Our analysis demonstrates that the phonon-
T < T < T drag mechanism is of crucial importance to understand
1 2 3
SSE below the room temperature.
Finally,weshowinFIG.4ourinterpretationontheob-
FIG. 4. (Color online) Schematic illustration of SSE unac- servationof SSE that is unaccompanied by a globalspin
companied by a global spin current. The phonon-drag pro- current,2 where the heat is carried by phonons through
cess which explains the experiment2 is shown. The meaning the nonmagnetic substrate while the spin is injected lo-
of each line (propagator) is thesame as in FIG. 2. cally at the F/N interface. This interpretation is rein-
forcedwhenwerecallthatthemagnitudeofthespinSee-
beck signal is enhanced with decreasing T even well be-
is proportional to the experimentally-detectable electric low the Curie temperature, whose tendency is consistent
voltage via ISHE [Eq. (1)]. Following Ref. 11, the spin with the phonon-drag mechanism as is seen in FIG. 3.
current Imag(N ) injected into N due to the magnon- Furthermore, the fact that the experiment was done be-
s 1 1
mediated SSE10,11 is calculated as lowtheroomtemperaturesupportsthethephonon-drag-
based scenario, since the phonon-drag process becomes
Imag(N )/∆T = P TM/T ds(T/TM)s2, (2) more effective at low T as emphasized in the previous
s 1 (cid:18)α(cid:19)Z 4sinh2(s) paragraph. All these considerations strongly support
0 2
thattheSSEexperimentforGaMnAscanbeinterpreted
where α is the Gilbert damping constant, T is the in terms of the phonon-drag mechanism, and results in
M
characteristictemperature correspondingto the magnon a prediction that the substrate condition affects the ob-
high-energy cutoff, and P is a nearly-temperature- served signal. Our demonstration opens a new route to
independent coefficient.20 Equation (2) means that the controlspincurrentsbymeansofphononsandstimulates
magnon-mediated SSE cannot explain the low-T en- further progresses in spin caloritronics.
hancement of the signal (the dashed curve in FIG. 3). This work was supported by a Grant-in-Aid for Scien-
Now we proceed to a detailed analysis of the T- tific Research in Priority Area ’Creation and control of
dependence of SSE in terms of the phonon-drag mech- spin current’ (19048009, 19048028), a Grant-in-Aid for
anism. The Feynman diagram for the phonon-drag pro- Scientific Research A (21244058), the Global COE for
cessinthepresentsituationisshowninFIG.2(b),where the ’Materials Integration International Centre of Edu-
the phonons feel the temperature difference between F cationandResearch’,aGrant-in-AidforYoungScientists
1
and F , and drag magnons through the magnon-phonon (No. 22740210) all from MEXT, Japan, a Grant for In-
2
interaction. Since the nonequilibrium phonons affect the dustrial Technology Research from NEDO, Japan, Fun-
magnon dynamics, this process injects spin current into damentalResearchGrantsfromCREST-JST,PRESTO-
N . TheimportantpointisthatthespincurrentIdragin- JST, TRF, and TUIAREO, Japan.
1 s
jectedinthisprocessbecomesproportionaltothephonon
lifetime τph as12 1K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K.
Ando,S.Maekawa,andE.Saitoh, Nature455,778(2008).
Idrag(N )/∆T =P′τ , (3) 2C.M.Jaworski,J.Yang,S.Mack,D.D.Awschalom,J.P.Here-
s 1 phB1B2 mans,andR.C.Myers,NatureMater.9,898(2010).
3K.Uchida,J.Xiao,H.Adachi,J.Ohe,S.Takahashi,J.Ieda,T.
w(Th/eT4rπMe2)9/B21(kB=T~Mτs(fT)/43TπD30T)5MR/0TTDd/vTtadnuvh7(s/vi2n/h22u)(6uw/2i)thanthde BD2eby=e 4IOM.taZ˘aeu,ktaYiw´c.,aKJ,.aajnFidwabaEira.an,S,aHai.tnodUhm,SN.ezDaat.wuSrae,arMHm.aatK,earR.we9av,i.,8MG94o.d(E.2.0PW1h0y)..sB.a7u6e,r3,2S3.
temperature T , aRnd P′ is a nearly-temperature- (2004).
D
independentcoefficient.22Sinceτ inahigh-purityspec- 5Maekawa, S. (ed) Concepts in Spin Electronics (Oxford Univ.
ph
Press,Oxford,U.K.,2006).
imenisknowntoincreasesteeplyatlowT becauseofthe
6J.C.Slonczewski,Phys.Rev.B82,054403 (2010).
rapidsuppressionofumklappscattering,21itleadstothe 7A.Slachter,F.L.Bakker,J-P.Adam,andB.J.vanWees,Nature
drastic enhancement of the phonon-draggedSSE. In our Phys.6,879(2010).
analysis, the T-dependence of τ is extracted from the 8SpinCaloritronics,editedbyG.E.W.Bauer,A.H.MacDonald,
ph
thermal conductivity data for Y Fe O 23 (see the inset andS.Maekawa,specialissueofSolidStateCommun.,150,459
3 5 12 (2010).
of FIG. 3) using21,24
9Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M.
Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S.
κ(T)=(1/3)vp2hCph(T)τph(T), (4) Maekawa,andE.Saitoh, Nature464,262(2010).
4
10J.Xiao,G.E.W.Bauer,K.Uchida,E.Saitoh,andS.Maekawa, 2. Derivation of Eq. (3)
Phys.Rev.B81,214418(2010).
11H.Adachi,J.Ohe,S.Takahashi,andS.Maekawa,Phys.Rev.B
Following Ref. [S1], the spin current I (N ,t) injected
(inpress);arXiv:1010.2325. s i
12See supplemental material for experimental and theoretical de- into the nonmagnetic metal Ni (i = 1,2,3) is calculated
tails. as
13E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys.
Lett.88,182509(2006). 4 k−q√S
14L. Vila, T. Kimura, and Y. Otani, Phys. Rev. Lett. 99, 226604 Is(Ni,t)=− √J2sNd N ~0ReCk<,q(t,t), (S1)
(2007). Xq,k F N
15C. Vittoria, P. Lubitz, P. Hansen, and W. Tolksdorf, J. Appl.
Phys.57,3699(1985). where N (N ) is the number of lattice sites in F (N),
F N
1167LF..GJ.uBrelvaitcth,,PZ.hA..ESkcshpr.oTeedoerr.,FCiz..L1.6F,o1i9le3s,(1a9n4d6)D. . Greig, Ther- S0 is the size of the localized spins in F, and Jskd+q is
moelectric Power of Metals(PlenumPress,NewYork,1976). the Fourier transform of the s-d interaction at the F/N
18E.M.LifshitzandL.P.Pitaevskii,PhysicalKinetics(Pergamon, interface. Here, C< (t,t′) = i a+(t′)s−(t) measures
NewYork,1982). thecorrelationbetwke,qenthemag−nohnqoperaktoria+ andthe
19H.P.R.Frederikse,Phys.Rev.92,248(1953); C.Herring,ibid. q
96, 1163 (1954); T. H. Geballe and G. W. Hull, ibid. 94, 1134 spin-density operator s−k = (sxk −isyk)/2. Note that the
(1954). time dependence of I (N ,t) vanishes in the steady state
s i
20P = (kBτsf/8π5~3) × L with L = 0.1 × and it is hereafter discarded. Introducing the frequency
Nint(Js2dS0)χN(a/λsf)3(aS/Λ), where Nint and Jsd are the representation C< (t t′) = ∞ dωC< (ω)e−iω(t−t′)
number of localized spins and the strength of the s-d exchange k,q − −∞ 2π q,k
coupling at the N1-F1 interface; χN, τsf, and λsf are the and adopting the representationR [S2] Cˇ = CR,CK as
paramagnetic susceptibility, the spin relaxation time, and the (cid:16)0 ,CA(cid:17)
spin diffusion length in N1; Λ and aS are the dimension and well as using the relation C< = 21[CK −CR +CA], we
latticeconstantoftheeffective blockspinofF1. obtain
21N.W.AshcroftandN.D.Mermin,SolidStatePhysics(Saunders
College,Philadelphia,1976). 2 k−q√S ∞ dω
22Pm′ag=non0-.p5h×ongeo2nkcBoTuDplLin/g(πc2oMnsitoannvtp2hge~a3n)dwtihtehiothnemdaismseMnsiioonn.less Is(N1)=Xq,k −√2JNsdFNN~0 Z−∞ 2πReCkK,q(ω) (S2)
23G.A.SlackandD.W.Oliver,Phys.Rev.B4,592(1971).
24For the T-dependence of τph, we introduce a model for the spin current Is(N1) in the steady state.
τtwhpiheth(fiTrfi)s/ttτtppeharr(maTmr=eetper0re)sse=(nat0[s1,tb+h0e,ca1l00o)we−=-Tb0((e2TnDh×/aTn1)c0]e−−m41,e[1n0.t+55(r,ce100f..(0T211)/,)TDwτph)h]e−r∼1e iδsXˇgWqe(nhωee)rn,atwllhyeeieinxntptreroredfsauscecedecaaosrrerneloartmioanliCzˇedapmpaeganrionngpinroEpqa.g(aSt2o)r
eb0TD/T originating from the steep suppression of the umklapp
25TsNcoabtteethetrahivnaitgotraht(erToerfe.∼t2ibc10a)TllτDyph,thw∼ehs1iil/geTntah.ledsueecotnodthteerpmhocnaopntu-drerasgtgheedhSigShE- Cˇk,q(ω)= √JsNkd−qN√S~0χˇk(ω)δXˇq(ω), (S3)
N F
hasaclose-to-linearspatialdependence [seeFIG.2(c)]asisob-
s(ber)v.eIdninadedxipteiorinm,etnhtes,ltehnegtrheasscoanleofawsshoiccihatisedobwviiothustfhreomphFoInGo.n2- where χˇk(ω)= χRk0(,ω),χχKkA((ωω)) is the spin-density propa-
(cid:16) k (cid:17)
dragged SSE is set by that related to the energy conservation, gator satisfying the local equilibrium condition:
suchthatitcanbeaslongasseveralmillimeters.
χA(ω)=[χR(ω)]∗; χK(ω)=2iImχR(ω)coth( ~ω ).
k k k k 2kBT
(S4)
Here the retarded component of χˇ (ω) is given by
k
χR(ω)=χ /(1+λ2k2 iωτ ) [S3] where χ , λ , and
k N sf − sf N sf
τ arethe paramagneticsusceptibility, thespindiffusion
sf
SUPPLEMENTAL MATERIAL length, and spin relaxation time.
We now consider the phonon-dragged SSE [the pro-
cess P shown in FIG. 2 (b)]. The renormalizedmagnon
2
1. Experimental details propagator δXˇ (ω) in the present case is given by
q
δXˇ (ω)=Xˇ (ω)Σˇ (ω)Xˇ (ω), (S5)
The single-crystal LaY Fe O (111) film with the q q q q
2 5 12
sthuibcsktnraestse obfy3li.q9uµimd pwhaassegreopwitnaxoyn. aThGed135G-na5mO-t1h2ic(k11P1t) whereXˇq(ω)= XqR0(,ω),XXqKA((ωω)) isthebaremagnonprop-
(cid:16) q (cid:17)
wires were then sputtered in an Ar atmosphere. The agator satisfying the equilibrium condition:
length and width of each Pt wire are 4 mm and 0.1 mm,
respectively. The temperatures of the lower- andhigher- XqA(ω)=[XqR(ω)]∗; XqK(ω)=2iImXqR(ω)coth(2k~BωT).
temperature ends of the sample were respectively stabi- (S6)
lized to T and T +∆T, where T was controlled in the Here, the retarded component is given by XqR(ω) =
range of 300-50 K by means of a closed-cycle helium re- (ω ω +iαω)−1, where ω = γH +ω is the magnon
q q 0 q
−
frigerator. frequency for uniform mode γH and exchange mode
0
e e
5
ω = D q2/~. In Eq. (S5), the selfenergy Σˇ due to the bare phonon propagator satisfying the equilibrium
q ex
phonons is given by condition:
i Γ 2 DA(ν)=[DR(ν)]∗; DK(ν)=2iImDR(ν)coth( ~ν ),
Σˇq(ω)= 2NF XK (cid:18) K~,q(cid:19) ZνnδDR(ν)Xˇq−(ω−)τˇ1 wiKth its retaKrded componKent and the pKhonon life(tS2i1km1Be)T
+δDA(ν)τˇ1Xˇq−(ω−)+δDK(ν)Xˇq−(ω−) , (S7) givenbyDKR(ν)=(ν−νK+i/τph)−1−(ν+νK+i/τph)−1
o and τ .
ph
where ΓK,q = gωqq2M~iνonKvp2h is the magnon-phonon in- anNdouwsesuEbqs.ti(tSu2ti)n,gththeesspeinexcpurrersesniotnisnijnetcoteedquinattoioNn1(Sb5y)
teractionvertexewithνK,vph andMionbeingthephonon the phonon-drag process is calculated as
frequency, phonon velocity and the ion mass, τˇ is the
Pauli matrix in the Keldysh space, and we have intro- Idrag(N )= R dν ν4 Γ 2A (ν)
danudcedt=he s∞hortdhνa.nIdnnEoqta.t(iSo7n)s,ωth−e=fuωll−phνo,nqo−n=prqop−agKa-, s 1 NNNF kX,q,KZ K K(cid:0) K,q(cid:1) k,q
tor δRDνK [Rt−he∞w2hπole of the phonon lines for P2 in FIG. 2 × coth(2~kνBKT2)−coth(2~kνBKT1) , (S12)
(b)] isbwritten as [S4] where R = √2(cid:2)(Js2dS0)Ω20Nint(aS/Λ)τph/(4(cid:3)π3~4νD6) with
ν =v /a , and
D ph S
δD (ν)=δDl-eq(ν)+δDn-eq(ν). (S8)
K K K
Here, δDl-eqb(ω) = bδDl-eq,R(ν),bδDl-eq,K(ν) is the Ak,q(ν)=ZωImχRk(ω)ImXqR−(ω−)|XqR(ω)|2
local-equilibrium propa(cid:16)gator0, satiδsDfyli-enqg,A(νt)h(cid:17)e local- [coth( ~ω− ) coth( ~ω )]. (S13)
b × 2kBT1 − 2kBT1
equilibrium conditions δDl-eq,A(ν) = [δDl-eq,R(ν)]∗ and
K K By setting T = T , ∆T = T T , Ω = ν and after
δDl-eq,K(ν) = [δDl-eq,R(ν) δDl-eq,A(ν)]coth( ~ν ) 2 1 − 2 0 D
K K − K 2kBT some algebra, we obtain Eq. (3) in the main text.
with its retarded component given by
δDKl-eq,R(ν)= N (ΩΛ20/a ) DKR(ν) 2DKR′(ν), (S9) References
F S XK′ (cid:2) (cid:3)
S1: H. Adachi, J. Ohe, S. Takahashi, and S. Maekawa,
while δDn-eq = (0,δDn-eq,K) is the nonequilibrium prop- Phys. Rev. B (in press); arXiv:1010.2325.
0, 0
agator with its Keldysh component given by
b S2: A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp.
Ω2 Teor. Fiz. 68, 1915 (1975) [Sov. Phys. JETP 41,
δDKn-eq,K(ν)= N (Λ0/a ) [DKR′(ν)−DKA′(ν)]|DKR(ν)|2 960 (1975)].
F S XK′
coth( ~ν ) coth( ~ν ) , (S10) S3: P.FuldeandA.Luther,Phys. Rev. 175,337(1968).
× 2kBTF2 − 2kBTF1
(cid:2) (cid:3) S4: K. Michaeli and A. M. Finkel’stein, Phys. Rev. B
In the above equations, Ω0 = Kph/Mion with the elas- 80, 115111(2009).
tic constant Kph in F, and DpK(ν) = DKR0(,ν),DDKKA((νν)) is
(cid:16) K (cid:17)
b