Table Of ContentSpin effects induced by thermal perturbation
in a normal metal/magnetic insulator system
I.I. Lyapilin, M.S. Okorokov, V.V. Ustinov∗
Institute of Metal Physics, UD RAS, Ekaterinburg, 620137, Russia
(Dated: January 14, 2015)
Usingoneofthemethodsofquantumnonequilibriumstatistical physicswehaveinvestigatedthe
spin transport transverse to the normal metal/ferromagnetic insulator interface in hybrid nanos-
5
tructures. An approximation of the effective parameters, when each of the interacting subsystems
1
(electron spin, magnon, and phonon) is characterized by its own effective temperature have been
0
considered. The generalized Bloch equations which describe the spin-wave current propagation in
2
thedielectrichavebeenderived. Finally,twosidesofthespintransport ”coin” havebeenrevealed:
n thediffusivenatureofthemagnonmotionandmagnonrelaxationprocesses,responsibleforthespin
a pumpingand thespin-torqueeffect.
J
3
1 I. INTRODUCTION Studying of the SSE in a non-conducting magnet in
the systemof a nonmagnetic conductor/ magnetic insu-
] lator(NM/FI)LaY Fe O 19 hasshownthatthiseffect
l 2 5 12
l Oneofthecentralissuesofspintronicsisthegeneration cannotbedescribedbystandardapproachesasregardsa
a
h andcontrolofspincurrentsinsolids. Therearedifferent description of thermoelectric effects18. As distinct from
- methods to implement the spin current: optical, mag- conductingcrystalswherethetransferofthespinangular
s
e netic, and using an electric current. The latter is espe- momentum is due to band charge carriers, the spin See-
m cially important for use in devices1 when spin-polarized beck effect can be realized in non-conducting magnetic
charge carriers are injected from a ferromagnetic ma- materials through the excitation of a localized spin sys-
.
t terial to a non-magnetic. At that, in the nonmagnetic tem. In this case, the excitations (magnons) underlying
a
m material at the spin diffusion length, spin accumulation the spin-wave current causes the transfer of the angular
arises. With an external perturbation usually affecting momentum. Thus, conducting crystals and a noncon-
-
d the kinetic degrees of freedom, the interaction between ductive magnetdiffer fromeachotherintype ofthe spin
n translational (kinetic) and spin degrees of freedom plays current, but namely - spin-wave one. It is a new type
o the main role in the formation of the spin response to of the spin current. The system of localized spin can be
c
the external perturbation. Combined electric dipole res- deviated from its equilibrium state in various ways. In
[
onancecanserveas anexample ofsuchaneffect. Inthis the experiment19, this has been achieved by passing an
1 case, the interaction of conduction electrons and an al- electric current through the nonmagnetic conductor Pt,
v ternatingelectricfieldresultsinresonanceattheZeeman which results in the spin accumulationand the spin cur-
3 frequency2. Another example of such a response is the rentinit. The interactionofthe spin-polarizedelectrons
8
spin Hall effect (SHE)3,4 that is exhibited as a spin cur- with the localized spins at the interface (NM/FI) is ac-
9
2 rent perpendicular to both the normal current and the companiedby the creation(or annihilation) ofmagnons,
0 spin accumulation. There also exist mechanisms of in- which in turn gives rise to a perturbation of the mag-
. teraction with external fields whose energy is simultane- netic subsystem. Due to small spin-wave damping, the
1
ously transferred to both electronic subsystems (kinetic spin-wave current propagates at much greater distances
0
5 and spin)5. It has been found that thermal perturba- thantheelectronspincurrent. Thisfactmakestheeffect
1 tions may also cause the spin effects to occur. The first possible to apply in practice20,21.
: effect that has opened a new direction in spintronics -
v
the influence of thermal perturbations on spin effects, An important role in the study of thermal pertur-
i
X is the spin Seebeck effect (SSE)6,7. It has been turned bations is played by a lattice (phonons). Indeed, a
r out to be inherent in conducting crystals of Ni Fe . inhomogeneous temperature field may cause a devia-
81 19
a
Afterwards, the SSE could be observed in various mate- tion of both the localized spins subsystem (m) and the
rials such as (Ga,Mn)As7 semiconductors and metallic phonon subsystem (p) from their equilibrium state. If
Co MnSi ferromagnets8. Besides, later the spin Nerst the non-equilibrium state of each subsystem is charac-
2
effect (or the thermal spin Hall effect), the spin Peltier terized by an effective temperature T , i = m,p, the
i
effect, and others have been discovered9–14. As noted non-equilibrium state of the system as a whole can be
in the work15, there is much common between the spin clearly described by a set of parameters T . The dif-
i
effects implemented in an electric or an inhomogeneous ference in the effective temperatures of the subsystems
temperaturefields. So,inspintronics,studyingtheinter- can lead to the realization of the effect - the transfer of
action between charge and spin currents, the new direc- the angular momentum from the magnetic subsystem to
tion appears - spin caloritronics16,17. As far back as in thelatticeortotheelectronspinsubsystem22,23 andvice
the late 20th century, a few theoretical aspects of these versa. Thermalperturbationsarealsoresponsibleforthe
interactions were discussed18. drageffects, whoseroleis essentialinarangeofnotvery
2
high temperatures24,25. As the authors26 believe, it is In the given paper, we consider the spin-thermal ef-
the thermal perturbations which produce the observable fectsinanormalconductor/magneticinsulator,usingthe
anomaly in the temperature dependence of the SSE. NSO methodto describe thermalperturbations. The ar-
The spin effects in non-conductive magnetic materi- ticle is organized as follows. The first part formulates
als under thermal perturbations were studied in several a model at hand, represents a Hamiltonian of the sys-
papers22,27–29. In this case, the theoretical description temandintroducesbasicoperatorsandtheirmicroscopic
boiled largely down to the consideration of the evolu- equations of motion. The second part of the work cov-
tion of the localized moment subsystem. Given thermal ers constructing both the nonequilibrium entropy oper-
fluctuations derivedfrom the fluctuation-dissipativethe- ator including the thermal perturbations of the system
orem,theauthorsoftheworksmentionedabovemodeled and the NSO. The third section of the article focuses
the localizedmoments’ dynamics by the phenomenologi- on an analysis of macroscopic equations describing the
calLandau-Lifshitz-Gilbertequation. Astothespinden- spin-thermal effects.
sity dynamics in a nonmagnetic material, it is described
by the Bloch equation with phenomenological spin re-
II. THE HAMILTONIAN
laxation frequencies. A spin current that was injected
intothe non-magneticmaterial(NM),wascalculatedas
the average value of the rate of change of the spin den- Our model consists of a normal conductor (NM) and
sity in a nonmagnetic material. Spin-charge kinetics in aferromagneticinsulator(FI). Weholdthat,inthefirst
the systems (F/N), (F/N/F) under a temperature gra- conductor, the spin accumulation takes place. It can be
dient was thoroughly covered in the paper30 in terms of obtained and implemented in various ways, for example
a simple phenomenological model and the simplest ap- byusingthespinHalleffect. Theconductionelectronsof
proximations. thenormalmetalarecoupledwiththelocalizedspinsub-
Thestudyofthespin-thermaleffectsrequiresacorrect system of the ferromagnetic insulator through exchange
description of the thermal perturbation. There are sev- interaction. The inelastic scattering of the electrons by
eralpossiblescenariosaccountingforsuchperturbations. the localized spins, accompanied by magnon emission or
For example, the reaction of weakly nonequilibrium sys- absorption, alters the electron spin orientation and un-
tems on a disturbance of the thermal type can be repre- balancesthelocalizedspinsystem. Considerthemagnon
sentedas Fourier transformof time correlationfunctions scattering mechanism by the example of the magnon-
of appropriate flux operators over the statistical equi- phonon interaction. We assume the ferromagnetic in-
librium state of the system31. The structure of admit- sulatortobeinanon-uniformtemperaturefield. Letthe
tances of such a type is analogous to expressions for the system of conduction electrons in the normal conductor
transportcoefficients arisinginthe theoryofequilibrium consist of two subsystems: the kinetic and spin one. In
systems as the response to a mechanical perturbation. thiscase,thefirstischaracterizedbyanequilibriumtem-
These expressions can be represented as an additional perature T, and the second (the spin subsystem) - by a
summand in the system’s Hamiltonian. That is why the temperature Ts. The symbol Tm designates a tempera-
response to a thermal perturbation is often found using ture of the localized spin subsystem, Tp - a temperature
indirect methods. Such cases require introducing ficti- of the lattice (phonons).
tious external forces identical to thermal perturbations The Hamiltonian of the system (NM/FI) canbe rep-
exerted on the system32. Response to the thermal per- resented as H =HNM +HFI +HL. Here
turbations can also be calculated based on the Onsager
hypothesis33 about of the nature of fluctuation damping HNM = dx(Hk(x)+Hs(x)),
Z
orbyusingthelocal-equilibriumdistributionasaninitial
p2
condition for the statistical operator of the system to be H (x)= { j ,δ(x−x )},
found34. k 2m j
Xj
The response of weakly nonequilibrium systems to
H (x)=−¯hω szδ(x−x ). (1)
thethermalperturbationscanbeuniversallyconstructed s s j j
throughthemethodofthe nonequilibriumstatisticalop- Xj
erator (NSO) and its modifications35. The latter rep-
The integration is over the volume occupied by (NM).
resent the NSO as a functional of the local-equilibrium sz and pγ are components of the spin and angular mo-
j j
distribution. Themethodallowscalculatingtheresponse
mentum of j-th electron operators, respectively. ω =
s
of not only equilibrium but also strongly nonequilibrium
g µ H/¯histheZeemanprecessionfrequencyoffreeelec-
s 0
systems described in terms of roughened macroscopic
tronsinanexternalmagneticfielddirectedalongtheaxis
variables. Within this method, the kinetic coefficients
z. (g , µ are the effective spectroscopic splitting fac-
s 0
are expressed via the Fourier transform of time correla-
tor for electrons and the Bohr magneton, respectively);
tion functions over the statistical distribution which de-
{A,B}=(AB+BA)/2.
scribes the unperturbed nonequilibrium process. Natu-
rally, in this case, electron scattering needs to be taken
H = dx(H (x)+H (x)) (2)
into account. FI Z m ms
3
is the Hamiltonian of of the localized spins subsystem the local-equilibrium value of the chemical potential of
. H (x) is the energy density operator of the magnetic electrons with spin α=↑,↓.
m
subsystem. It involves a sum of the exchange (over the
nearest neighbors) H and Zeeman energy H . nα(x)= δ(x−xα)
SS S j
Xj
H =−J S S , H =−¯hω Sz (3) is the operator of the particle number density with the
SS j j+δ S m j spin α. n(x)=n↑(x)+n↓(x).
Xjδ Xj
The operator δS(t) = dxδS(x,t) describes the sys-
J is the exchange integral. ω = g µ H/¯h. H (x) is tem deviation from its equRilibrium state.
m m 0 ms
the energy density operator of interaction between the s
and m subsystems at the interface. Then δS(t)=∆ dx{δβs(x,t)Hs(x)−βδµα(x,t)nα(x)+
Z
+δβ (x,t)(H (x)+H (x)+H (x))}, (6)
m m ms pm
H =−J dxs(x)S(R )δ(x−R ), (4)
ms 0 j j
Xj Z ∆A=A−<A>0, <...>0=Sp(...ρ0).
where J0 is the exchange integral, S(Rj) being the op- <...>t=Sp(...ρ(t)), ρ(t)=exp{−S(t)}.
erator of the localized spin with the coordinate R at
j
the interface. The integration in (2) is over the volume
occupied (FI). δµα(x,t)=µα(x,t)−µ, δβi(x,t)=βi(x,t)−β.
H is the lattice hamiltonian
L (i = s, m). The quantities δµα(x,t),δβ (x,t) have the
m
meaningofnon-equilibriumadditionstothechemicalpo-
HL = dx(Hp(x)+Hpm(x)) tential µ and deviations of the magnetic subsystem tem-
Z
perature from the equilibrium temperature β−1 =T.
H (x) is the energy density operator of the phonon sub- In the linear approximation the deviation from the
p
system. H (x) is the energy density operator of inter- equilibrium, the NSO (or the density matrix) ρ(t) can
pm
action between the localized spins and phonons. In the be written as follows35
future,weareinterestedintheevolutionofthemagnetic
0 1
subsystem, so the scattering of conduction electrons by ρ(t)=ρ (t)+ dt eǫt1 dτρτS˙(t+t ,t )ρ−τ ρ . (7)
phonons for simplicity omitted from consideration. q Z 1 Z 0 1 1 0 0
−∞ 0
Here ρ (t)=exp{−S(t)} is the quasi-equilibrium statis-
q
THE ENTROPY OPERATOR tical operator, S˙(t) being the entropy production opera-
tor:
To analyze the kinetics of the spin-thermal effects, we
∂S(t) 1
employ a scheme developed in the NSO method applied S˙(t)=δS˙(t)= + [S(t),H].
to the case of a small deviation of the system from the ∂t i¯h
equilibrium Gibbs distribution ρ0 =exp{−S0}. The en- Afurtheralgorithmforconstructingtheoperatorρ(t)re-
tropyS0 oftheequilibriumsystemwiththe Hamiltonian duces to finding the entropy production operator. Com-
H can be written as mutingtheoperatorsnα(x), H (x)withtheHamiltonian
s
(1), we find the operator equations of motion
S =Φ +β(H +H −µn)+β(H +H +H +H ),
0 0 k s m p ms mp
where β−1 = T is the equilibrium temperature of the n˙α(x)=−∇Inα(x)+n˙α(ms)(x), (8)
system.
In terms of average densities, to the non-equilibrium H˙ (x)=−∇I (x)+H˙ (x). (9)
s Hs s(ms)
state of the systemthere correspondsthe entropy opera-
tor Here Inα(x) = m1 j{pj,δ(x−xαj)} is the flux density
of particles with spPin α. n˙α (x) determines the rate of
(ms)
S(t)=Φ(t)+ dx{β(Hk(x)−µα(x,t)nα(x,t))+ changeofthe numberofelectronswith spinαdue to the
Z
spin-flip electron scattering by the (m) -subsystem.
+βs(x,t)Hs(x,t)+βHp(x,t)+ Similarly, I (x) = −¯hω 1 sz{p ,δ(x − xα)}
Hs sm j j j j
+βm(x,t)(Hm(x,t)+Hms(x,t)+Hpm(x,t))= determines the flux density of Pthe Zeeman energy.
=S0+δS(t). (5) H˙s(sm)(x) is the rate of change of the electrons s-
subsystem local energy due to their interaction with
Φ(t) is the Massieu-Planck functional. βs(x,t), βm(x,t) magnons (m).
are the local-equilibrium values of the inverse tempera-
ture of the (s), (m) subsystems, respectively. µα(x,t) is A˙ (x)=(i¯h)−1[A (x),H ]. (10)
λ(γ) λ γ
4
Revealing the spin index explicitly, we write the expres-
sion for the spin current density J (x):
s
Js(x)=(1/2){n˙↑(x)−n˙↓(x)}=−∇Isz(x)+s˙z(ms)(x), Hms =−J∗ kX,k′,q{b+q a+k↑ak′↓+bqa+k′↓ak↑}δk′,k+q,
Isz(x)=(Is↑(x)−Is↓(x))/2, wherea+ (a )arethecreation(annihilation)opera(t1o4rs)
s˙z (x)=(n˙↑ (x)−n˙↓ (x))/2. (11) kα kα
(ms) (ms) (ms) for electrons with a certain spin value α =↑, ↓. The in-
elastic part of the exchange interaction (H ) leads to
Fromtheexpression(11)itfollowsthatJ (x)contains ms
s the angularmomentumtransferbetweenelectronsofthe
two parts: collisionless Iz(x), and collisional s˙z (x).
s (ms) normal metal and magnons of the ferromagnetic insula-
The former is due to the flux of particles with different
tor. Obviously,non-equilibriumofoneofthe subsystems
spinorientations,thelatterisdeterminedbythespin-flip
(electronic or magnon) makes the magnon/spin current
scattering.
via the interface. At the same time, the total angular
Let us turn to the consideration of the magnetic sub-
momentumconservationlawprovidesthe boundarycon-
system. Using the method of Holstein-Primakov36, the
dition for the current at the metal/insulator interface:
hamiltonianoflocalizedspinssubsystemcanberewritten
the continuity of the spin current.
with spin-wave (magnon) variables (using to the opera-
Thus,the equationofmotionfor the magnetic subsys-
torscreationb+ andannihilationb ). Expressedinterms
k k tem can be written as
of the magnon operators, the Zeeman and exchange in-
teractions are of the form H˙ (x)=−∇I (x)+H˙ (x)+H˙ (x). (15)
m Hm m(ms) m(mp)
HS =−¯hωdNS+¯hωdXk b+k bk. (12) HmeargenoIHnme(nxe)rg=y i−n¯htωhme I(mSz)(x-)suibssythsteemflu.xTdheensriteymaoifntinhge
summands on the right-hand side of the equation are
H ≃−JNzS2+ 2JzS(1−γ )b+b , responsible for the magnon-phonon scattering processes.
SS k k k
Xk Finally, the equation of motion for the lattice subsys-
tem has the form:
γ =(1/z) exp{ikδ}. (13)
k
Xδ H˙ (x)=−∇I (x)+H˙ (x). (16)
p Hp p(mp)
Here N is the number of localized moments with the
spin S,z is the number of the nearest neighbors. b+b Substituting the above equations of motion into the en-
k k tropy production operator S˙(t) and integrate by parts
are the creation and annihilation operators for magnons
the summands containing the divergence of fluxes. Hav-
with the wave vector k. For |kδ| << 1 , we have
z(1 − γ ) ≈ (1/2) (kδ)2 , i.e. ω = JS (kδ)2. ing discarded the surface integrals, we write the entropy
k δ k δ production operator in the form
For body-centered aPnd face-centered cubic lattPices with
alatticeconstantequaltoa,wehaveω =2JS(ka)2. In
otherwords,thecontributionoftheexckhangeinteraction S˙(t)=∆Z dx{−βInα(x)∇µα(x,t)−βδµα(x,t)n˙α(ms)(x)+
for the magnon frequency has the same form as the de
Broglie dispersion law for free particles of mass m∗ +IHm(x)∇βm(x,t)+(βs(x,t)−βm(x,t))H˙s(ms)(x)+
+(β (x,t)−β)H˙ (x)}. (17)
¯h 1 m m(pm)
ω = k2, m∗ = .
k 2m∗ 2JSa2 Revealing the spin index in the first two summands on
the right-hand side of the formula (17), we get
Thus, regarding the magnon gas as free, we arrive at
¯h2k2 Inα(x)∇µα(x,t)=Isz(x)∇µs(x,t). (18)
H = ε(k)b+b , ε(k)= .
m k k 2m∗
Xk δµα(x,t)n˙α (x)=
s(ms)
Thisexpressioncanbeinterpretedasthesumoftheener- =µ (x,t)s˙z (x)=−µ (x,t)S˙z (x). (19)
s s(ms) s m(ms)
giesofthe quasiparticles-ferromagnonshavingthe quasi-
momentum P with their own effective mass m∗ and the Here µ =µ↑−µ↓. It is obviousthat the expression(19)
s
magnetic moment37. provides the boundary conditions for the collisional part
The exchange interaction (4) includes both the elastic ofthespincurrentattheinterface. Theexpressions(18),
electronscatteringbythelocalizedspinsandtheinelastic (11) imply that the heterogeneity in the spin accumula-
scattering process, with the former preserving the spin tiondistributiongovernsthecollisionlesspartofthespin
orientation and the latter happening with the electron current.
spin-flip and the creation or annihilation for magnons. Beforeproceedingtoconstructingtheequationsforthe
Using the creation and annihilation operators for elec- averages,we show that the entropy production operator
trons and magnons, we rewrite the inelastic part of the S˙(t)canberepresentedboththroughthespinaccumula-
exchange interaction in the form tion µ (x,t) and through the spin temperature β (x,t).
s s
5
For this purpose, we find a linear relationship between average the operator equation (11) for the spin current
the deviations of thermodynamic coordinates and ther- <J (x)>=(d/dt)<sz(x)>. Then we arrive at
s
modynamic forces from their equilibrium values. In this
appapnrdotxhimeaqtuiaosni,-eitquisilsiburffiiucmienotpteoraptuotrρρ(t()t)∼=ρeqx(tp){a−nSd(et)x}- <Js(x)>=Z dx′{Dsγzsz(x,x′)β∇γµs(x′+
q
in powers of δS(t) . Then we come up with the result +(βm(x′)−βs(x′))¯hωsLzz,(ms)(x,x′)+
+βµ (x′,t+t )Lz (x,x′)},
s 1 z,(ms)
δ <H (x)>t=− dx′{δβ (x′,t)(H (x);H (x′)) −
s s s s 0 (25)
Z
−βδµα(x′,t)(Hs(x);nα(x′))0}, where
δ <n(x)>t=− dx′{δβ (x′,t)(n(x);H (x′)) − 0
Z s s 0 Lz (x,x′)= dt′eǫt′(s˙z (x); s˙z (x′,t′)),
−βδµα(x′,t)(n(x);nα(x′)) }, (20) z,(ms) Z (ms) (ms)
0
−∞
where 0
Dγ (x,x′)= dt′eǫt′∇λ(Iλ(x);Iγ (x′,t′)). (26)
δ <A>t=<A>t −<A> , szsz Z sz sz
0 −∞
HereAdenotestimeaveraging,β (x,t+t′)=β (x),(i=
i i
1 s,m); µ(x,t+t′)=µ(x).
(A,B)0 =Z dλSp{Aρλ0∆Bρ10−λ}. We have derived the generalized Bloch equation that
describes the motion of the spin magnetization density
0
- the spin diffusion and relaxation processes. The first
Going over to the Fourier components of the spatial co- summand on the right-hand side of the equation reflects
ordinates and taking into account that <n>t=<n>0, the former process; the second summand is responsible
we obtain for the latter that caused by the electron-magnetic im-
purity interaction at the interface. In doing so, we have
¯hω (n(q),sz(−q))
µ (q,t)=− s 0 δβ (q,t). (21) madeanallowancefortheeffectsoftemporalandspatial
s β (n(q),n(−q)) s
0 dispersionofthespindiffusiontensorandspinrelaxation
frequency.
Equation (21) defines the relationship between the spin
Performingthe Fouriertransformoverthe coordinates
temperature T and the spin accumulation µ . In the
s s andtime andtakingintoaccountformulas(21),(23), we
long-wavelength limit q = 0 and the steady state, we
represent the right-hand side of expression as
obtain
− {qγqλDγλ(q,ω)+ν (q,ω)}P(q)δ <sz(q,ω)>,
µ =−¯hω (1−T/T ). (22) zz zz
s s S (n(q), sz(−q))
P(q)= 0. (27)
Substituting the expression (21) into the equation for (n(q),n(−q))0
δ < H (x)> , we get
s Here
δ <sz(q)>=δβs(q,t)h¯ωsCzz(q), 0
1
(n(q),sz(−q))2 Dγλ(q,ω)= dt e(ǫ−iω)t1(Iγ (q);Iλ(−q,t )),
C (q)=(sz(q),sz(−q)) − 0. (23) zz C (q) Z 1 sz sz 1
zz 0 (n(q),n(−q))0 zz −∞
(28)
Thedesiredrelationshipbetweenthecorrelationfunction describes the diffusion of the longitudinal spin respec-
Czz andthe differentialparamagneticsusceptibility χs50 tively alongthe Dzz andtransverseDxx of the magnetic
zz zz
of the electron gas is given by field, and
∂ 0
χs = ∂H gsµb <sz >=β(gsµb)2Czz. (24) ν (q,ω)= 1 dt e(ǫ−iω)t1(s˙z (q);s˙z (−q,t )),
zz C (q) Z 1 (ms) (ms) 1
zz
−∞
(29)
III. MACROSCOPIC EQUATIONS
describes the relaxation of the longitudinal spin
Let us construct the equations for the averages,which components50. In deriving the last formulas we have
have the meaning of local conservation laws of the aver- taken into consideration the spatial homogeneity of the
ageenergydensityforthe(s)and(m)subsystemsandof unperturbed state of the system. At that, the homo-
the particle number density. Inserting the entropy pro- geneity exhibits in the vanishing of the correlation func-
ductionoperatorintothe expressionfor the NSO(7), we tions of the type (A(q),B(q′)) only if (q′ = −q). It
6
is not hard to show that the combination of the corre- from (31), a non-zero contribution to the spin-wave cur-
lation functions for P(q) is proportional to the degree rent is the consequence of the difference in the temper-
of polarization of the electron gas in the normal con- atures of the above subsystems. The equation (31) im-
ductor: P(q) ∼ (n↑ −n↓)/n. It should be pointed out pliesthat,dependinguponitsdirection,themagnonflux
that the resulting expressions for the transport coeffi- produced by a temperature gradient can give rise to the
cients Dγλ(q,ω),ν (q,ω) are suitable for both classical angular momentum transfer from the magnon system to
zz zz
and quantizing magnetic fields and free from assump- the electron system. Besides, the heat fluxes contribute
tions about the nature of the spectrum, the the kind of to the emergence both of the thermally induced spin-
statistics,etc. Similarexpressionsforthe spinrelaxation torque effect52–54 andof the spin-torque effect generated
frequency are also inherent in the theory describing the by magnons55,56.
spin-lattice relaxation of a nonequilibrium spin system
with magnetic resonance saturation50.
Averaging the operator equation (15), we have CONCLUSIONS
<H˙ (x)>=−∇ <I (x)>+<H˙ (x)>+
m Hm m(ms) Using one of the methods of quantum nonequilib-
+<H˙ (x)>. (30) rium statistical physics (NSO), we investigated the
m(mp)
spin transport in hybrid nanostructures: normal
Or averagingtime
metal/ferromagneticinsulator. An approximationof the
effective parameters, when each of the interacting sub-
<H˙m(x)>=Z dx′ {DHγmHm(x,x′)∇γβm(x′)+ systems(electronspins,magnons,phonons)ischaracter-
ized by its effective temperatures was considered. We
+Lm (x,x′)(β (x′)−β (x′))+
m(ms) s m constructed macroscopic equations for the spin current
+Lm (x,x′)(β (x′)−β)}, (31) caused by both the unbalanced spin subsystem and an
m(mp) m
inhomogeneous temperature field in the ferromagnetic
where insulator. We derived the generalized Bloch equations
which describe the spin and spin-wave current propaga-
0
Lm (x,x′)= dt′eǫt′(H˙ (x); H˙ (x′,t′)), tion in the system. At that, these allow for both the
m(ml) Z m(ml) m(ml) diffusive nature of the magnon motion and the magnon
−∞ relaxationprocessesresponsibleforthespinpumpingand
0 the spin-torque effect.
Dγ (x,x′)= dt′eǫt′∇λ(Iλ (x);Iγ (x′,t′)).(32)
HmHm Z Hm Hm
−∞
appendix
The first term on the right-hand side of the equation
(30) is the energy density change in the magnon sub-
Let us give the results coming from the expressions
system due to the temperature gradient, which in turn
for the spin diffusion tensor (28), including the temporal
leadsto the magnonflux. Asinthe caseofthe spinelec-
andspatialdispersionandfindtheirrelationshipwiththe
tronsubsystem,thistermdescribesthemagnondiffusion.
conductivity tensor components. The correlation func-
Thesecondandthirdtermsoftheexpression(30)arere-
tionsofthe fluxesinthe formula(28)involvetheFourier
sponsible for the impact of the electronic (spin) and lat-
components of the isothermal Green’s functions
tice (phonon) subsystems on the magnon energy change
through their interaction. The role of the spin electron +∞
subsystem reduces to the generation and annihilation of G (t)=θ(−t)eǫt(A, B(t)) = dω eiωtG (ω)
AB 0 BA
the magnonsby the inelastic spin-flipelectronscattering Z 2π
at the normal metal/ferrodielectric interface. Accord- −∞
(33)
ing to (22), this contribution is proportional to the spin
Differentiating (33) over t, we obtain the chain of equa-
accumulation µ . The phonon subsystem proves to af-
s tions
fect the magnonenergy changein a twofoldmanner. On
the one hand, the magnon-phonon scattering processes ∂
−ǫ−iω G=−δ(t)(B,A)+G ,
makethemselvesfeltintheenergyrelaxationbehaviorof 0 1
(cid:18)∂t (cid:19)
the magnon subsystem, on the other hand, the phonon
∂
subsystem often acts as a ”heating” of the heat/charge −ǫ−iω G =−δ(t)(B,A˙)−G ,
0 1 2
transfer processes by means of drag effects24. It should (cid:18)∂t (cid:19)
be emphasized that the booster role can also be studied ......... , (34)
by the NSO method. The energy transfer efficiency be-
tween the phonon and magnon subsystems depends on G(t)=GAB(t), G1(t)=θ(−t)eǫt(A, B˙(t))0.
temperatures of the appropriate subsystems and the de-
gree of their mutual interaction. As can easily be seen G (t)=θ(−t)eǫt(A˙, B˙(t)) .
2 0
7
Here A˙ =(i¯h)−1[A,H ], H is the scatterer Hamilto- where F (x) are the Fermi integrals. The expression
V V m
nian. The formal solution of the chain is νzz coincides exactly with the formula for the relaxation
zz
frequency ν of the electron momentum51. Thus, the
p
longitudinal spin diffusion coefficient is given by
G (ω)=[M (ω)+ǫ−i(ω−ω )]−1(A,B), (35)
AB AB 0
Dzz =D νp , D = 2T F−1/2(µ/T). (37)
where M(ω) = G G−1 is the mass operator for the zz 0 νp−iω 0 νp F1/2(µ/T)
1
Green’s function.
The components of the spin diffusion tensor can be ex-
Let us consider the frequency dispersion by the ex- pressed in terms of the components of the conductivity
ample of the longitudinal spin diffusion coefficient. Re- tensor σ (q,ω), which in our notation is
ik
stricting ourselves to the Born approximation of the in-
teraction with the lattice in the expression for the mass 0
e2
operator, we have σ = dte(ǫ−iω)t(Iγ(q),Ik(−q,t)). (38)
γk T Z N N
−∞
1 (Iz ,Iz )
Dzz = sz sz 0 . (36) Simple calculations show that
zz C νzz(ω)−iω
zz zz
T
Dzz(q,ω)= σ (q,ω).
ik 4e2C (q) ik
Calculating the correlation functions (A,B) , we come zz
0
up with
The given work has been done as the part of the state
task on the theme ”Spin” 01201463330 (project 12-T-2-
1011) with the support of the Ministry of Education of
(Iz ,Iz ) =n/2m, C = nF−1/2(µ/T), the Russian Federation (Grant 14.Z50.31.0025)
sz sz 0 zz 8 F (µ/T)
1/2
∗ [email protected] Ser. 200, 062030 (2010).
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