PHYSICAL REVIEW B 81, 214418 共2010兲
Theory of magnon-driven spin Seebeck effect
Jiang Xiao (萧江兲,1,2 Gerrit E. W. Bauer,2 Ken-chi Uchida,3,4 Eiji Saitoh,3,4,5 and Sadamichi Maekawa3,6
1
Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China
2Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands
3Institute for Materials Research, Tohoku University, Sendai 980-8557, Japan
4Department of Applied Physics and Physico-Informatics, Keio University, Yokohama 223-8522, Japan
5
PRESTO, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan
6CREST, Japan Science and Technology Agency, Tokyo 100-0075, Japan
共Received 11 December 2009; published 14 June 2010兲
The spin Seebeck effect is a spin-motive force generated by a temperature gradient in a ferromagnet that can
be detected via normal metal contacts through the inverse spin Hall effect 关K. Uchida et al., Nature 共London兲
455, 778 共2008兲兴. We explain this effect by spin pumping at the contact that is proportional to the spin-mixing
conductance of the interface, the inverse of a temperature-dependent magnetic coherence volume, and the
difference between the magnon temperature in the ferromagnet and the electron temperature in the normal
metal 关D. J. Sanders and D. Walton, Phys. Rev. B 15, 1489 共1977兲兴.
DOI: 10.1103/PhysRevB.81.214418
PACS number共s兲: 85.75.⫺d, 73.50.Lw, 72.25.Pn, 71.36.⫹c
I. INTRODUCTION
The emerging field called spin caloritronics addresses
charge and heat flow in spin-polarized materials, structures,
and devices. Most thermoelectric phenomena can depend on
spin, as discussed by many authors in Ref. 1. A recent and
not yet fully explained experiment2 is the spin analogue of
the Seebeck effect—the spin Seebeck effect, in which a temperature gradient over a ferromagnet gives rise to an inverse
spin Hall voltage signal in an attached Pt electrode.
The Seebeck effect refers to the electrical current/voltage
that is induced when a temperature bias is applied across a
conductor. By connecting two conductors with different Seebeck coefficients electrically at one end at a certain temperature, a voltage can be measured between the other two ends
when kept at a different temperature. The spin counterpart of
such a thermocouple is the spin current/accumulation that is
induced by a temperature difference applied across a ferromagnet, interpreting the two spin channels as the two “conductors.” In Uchida et al.’s experiment,2 a temperature bias
is applied over a strip of a ferromagnetic film. A thermally
induced spin signal is measured by the voltage induced by
the inverse spin Hall effect3,4 in Pt contacts on top of the film
in transverse direction 共see Fig. 1兲. This Hall voltage is found
to be approximately a linear function 共possibly a hyperbolic
function with long decay length兲 of the position in longitudinal direction over a length of several millimeters. This result has been puzzling since spin-dependent length scales are
usually much smaller. The original explanation for this experiment has been based on the thermally induced spin accumulation in terms of the spin thermocouple analog mentioned above. However, the spin-flip scattering short circuits
the spin channels, and at which spin channels are short circuited, the signal should vanish on the scale of the spin-flip
diffusion length.5
In this paper, we propose an alternative mechanism in
terms of spin pumping caused by the difference between the
magnon temperature in the ferromagnetic film and the electron temperature 共assumed equal to the phonon temperature兲
in the Pt contact. Such a temperature difference can be gen1098-0121/2010/81共21兲/214418共8兲
erated by a temperature bias applied over the ferromagnetic
film.6
This paper is organized as follows: Sec. II describes how
a dc spin current is pumped through a ferromagnet
共F兲/normal 共N兲 metal interface by a difference between the
magnon temperature in F and electron temperature in N. In
Sec. III we calculate the magnon temperature profile in F
under a temperature bias. In Sec. IV we compute the spin
Seebeck voltage as a function of the position of the normalmetal contact.
II. THERMALLY DRIVEN SPIN PUMPING CURRENT
ACROSS F 円 N INTERFACE
In this section we derive expressions for the spin current
flowing through an F 兩 N interface with a temperature difference as shown in Fig. 2, starting with the macrospin approximation in Sec. II A and considering finite magnon dispersion
in Sec. II B.
Since the relaxation times in the spin, phonon, and electron subsystems are much shorter than the spin-lattice relaxation time,7,13 the reservoirs become thermalized internally
FIG. 1. 共Color online兲 A ferromagnet with thickness d, length L,
and magnetization M pointing in ẑ direction connects two N contacts at temperature TL/R at the left and right ends. A Pt strip of
dimension l ⫻ w ⫻ h on top of F converts an injected spin current
共Is = Isp + I fl兲 into an charge current Ic or Hall voltage VH by the
inverse spin Hall effect.
214418-1
©2010 The American Physical Society
PHYSICAL REVIEW B 81, 214418 共2010兲
XIAO et al.
current I fl from the normal-metal bath 共from N to F兲. The
latter is caused by the thermal noise in N and its effect on F
can be described by a random magnetic field h⬘ acting on the
magnetization10
I fl共t兲 = −
FIG. 2. 共Color online兲 F 兩 N interface with 共1兲 spin pumping
current Isp driven by the thermal activation of the magnetization in
F at temperature TF and 共2兲 fluctuating spin current I fl driven by the
thermal activation of the electron spins in N at temperature TN.
before they equilibrate with each other. Therefore, we may
assume that the phonon 共p兲, conduction electron 共e兲, and
magnon 共m兲 subsystems can be described by their local temp,e
in N.8 We furthermore assume
peratures: TFp,e,m in F, and TN
that the electron-phonon interaction is strong enough such
p
e
= TN
⬅ TN. However, the
that locally TFp = TFe ⬅ TF and TN
m
magnon temperature may deviate: TF ⫽ TF. This is illustrated
below by the extreme case of the macrospin model, in which
there is only one constant magnetic temperature, whereas the
electron and phonon temperatures linearly interpolate between the reservoir temperatures TL and TR. The difference
between magnon and electron/phonon temperature therefore
changes sign in the center of the sample. When considering
ferromagnetic insulators, the conduction electron subsystem
in F becomes irrelevant.
A. F 円 N contact
First, let us consider a structure such as shown in Fig. 2,
in which the magnetization is a single domain and can be
regarded as a macrospin M = M sVm, where m is the unit
vector parallel to the magnetization. We will derive in Sec.
II B the criteria for the macrospin regime. We assume
uniaxial anisotropy along ẑ, M s is the saturation magnetization, and V is the total F volume. The macrospin assumption
will be relaxed below but serves to illustrate the basic physics.
At finite temperature the magnetization order parameter in
F is thermally activated, i.e., ṁ ⫽ 0. When we assume N to
be an ideal reservoir, a spin-current noise Isp is emitted into
N due to spin pumping according to9
Isp共t兲 =
ប
关grm共t兲 ⫻ ṁ共t兲 + giṁ共t兲兴,
4␲
共1兲
where gr and gi are the real and imaginary parts of the spinmixing conductance gmix = gr + igi of the F 兩 N interface. The
thermally activated magnetization dynamics is determined
by the magnon temperature TFm while the lattice and electron
temperatures are TFp = TFe = TF. The term proportional to gr in
Isp has the same form as the magnetic damping phenomenology of the Landau-Lifshitz-Gilbert 共LLG兲 equation 关introduced below in Eq. 共6兲兴. The energy loss due to the spin
current represented by gr therefore increases the Gilbert
damping constant. According to the fluctuation-dissipation
theorem, the noise component of this spin-pumping-induced
current 共from F to N兲 is accompanied by a fluctuating spin
M sV
␥m共t兲 ⫻ h⬘共t兲.
␥
共2兲
In the classical limit 共at high temperatures kBT Ⰷ ប␻0, where
␻0 is the ferromagnetic resonance frequency兲 h⬘共t兲 satisfies
the time correlation
具␥hi⬘共t兲␥h⬘j 共0兲典 =
2 ␣ ⬘␥ k BT N
␦ij␦共t兲 ⬅ ␴⬘2␦ij␦共t兲.
M sV
共3兲
where 具 ¯ 典 denotes the ensemble average, i , j = x , y, and
␣⬘ = 共␥ប / 4␲ M sV兲gr is the magnetization damping contribution caused by the spin pumping. The correlator is proportional to the temperature TN.
The spin current flowing through the interface is given by
the sum Is = Isp + I fl 共see Fig. 2兲. Here we are interested in the
dc component
具Is典 =
M sV
关␣⬘具m ⫻ ṁ典 + ␥具m ⫻ h⬘典兴.
␥
共4兲
At thermal equilibrium, TN = TFm, and 具Is典 = 0. At nonequilibrium situation, the spin current component polarized along ṁ
with prefactor gi in Eq. 共1兲 averages to zero, thus does not
cause observable effects on the dc properties in the present
model. The x̂ and ŷ components of 具Is典 also vanish and
具Iz典 =
M sV
关␣⬘具mxṁy − myṁx典 − ␥具mxh⬘y − myhx⬘典兴.
␥
共5兲
We therefore have to evaluate the correlators: 具mi共0兲ṁ j共0兲典
and 具mi共0兲h⬘j 共0兲典.
The motion of m is governed by the LLG equation
ṁ = − ␥m ⫻ 共Heffẑ + h兲 + ␣m ⫻ ṁ,
共6兲
where Heff and ␣ are the effective magnetic field and total
magnetic damping, respectively. h accounts for the random
fields associated with all sources of magnetic damping, viz.,
thermal random field h0 from the lattice associated with the
bulk damping ␣0, random field h⬘ from the N contact associated with enhanced damping ␣⬘, and possibly other random
fields caused by, e.g., additional contacts. Random fields
from unrelated noise sources are statistically independent.
The correlators of h are therefore additive and determined by
the total magnetic damping ␣ = ␣0 + ␣⬘ + ¯
具␥hi共t兲␥h j共0兲典 =
2␣␥kBTFm
␦ij␦共t兲 ⬅ ␴2␦ij␦共t兲.
M sV
共7兲
The magnon temperature TFm is affected by the temperatures
of and couplings to all subsystems: ␣TFm = ␣0TF + ␣⬘TN + ¯.
214418-2
PHYSICAL REVIEW B 81, 214418 共2010兲
THEORY OF MAGNON-DRIVEN SPIN SEEBECK EFFECT
We consider near-equilibrium situations, thus we may linearize the LLG equation. To first order in m⬜ Ⰶ 1
共with mz ⯝ 1兲
ṁx + ␣ṁy = − ␻0my + ␥hy ,
共8a兲
ṁy − ␣ṁx = + ␻0mx − ␥hx ,
共8b兲
where ␻0 = ␥Heff is the ferromagnetic resonance frequency.
With the Fourier transform into frequency space
g̃ = 兰gei␻tdt and the inverse transform g = 兰g̃e−i␻td␻ / 2␲, Eq.
共8兲 reads m̃i共␻兲 = 兺 j␹ij共␻兲␥h̃ j共␻兲, with i , j = x , y and the transverse dynamic magnetic susceptibility
冉
␻0 − i␣␻
− i␻
1
␹共␻兲 =
2
2
␻0 − i␣␻
共␻0 − i␣␻兲 − ␻
i␻
冊
␴2
2␣
冕
具mi共t兲h⬘j 共0兲典 =
␹ij共␻兲 − ␹ⴱji共␻兲 −i␻t d␻
,
e
2␲
i␻
␴⬘
␥
2
冕
␹ij共␻兲e−i␻t
共9兲
共10a兲
In extended ferromagnetic layers the macrospin model
breaks down and we have to consider magnon excitations at
all wave vectors. The space-time magnetization autocorrelation function for an extended quasi-two-dimensional film of
thickness d can be derived from the LLG equation 共see Appendix C兲
具mi共0,0兲mi共0,0兲典 =
d␻
.
2␲
␴2
2␣
冕
关␹ij共␻兲 − ␹ⴱji共␻兲兴e−i␻t
共10b兲
d␻
.
2␲
共11兲
By inserting Eqs. 共10b兲 and 共11兲 with t → 0 into Eq. 共5兲
具Iz典 =
冉
M sV ␣ ⬘ 2
␴ − ␴ ⬘2
␥
␣
冊冕
=
2 ␣ ⬘k B m
共T − TN兲
1 + ␣2 F
⯝
␥បgrkB m
共T − TN兲
2 ␲ M sV F
⬅ Ls⬘共TFm − TN兲,
关␹xy共␻兲 − ␹yx共␻兲兴
共13兲
where A is the contact area and Km
⬘ = ␻0␮BkB共gr / A兲 / ␲ M sV is
the interface magnetic heat conductance with Bohr magneton
␮ B.
In addition to the dc component of the spin pumping current, there is also an ac contribution to the frequency power
spectrum of spin current and spin Hall signal.11 A measurement of the noise power spectrum should be interesting for
insulating ferromagnets for which the imaginary part of the
mixing conductance can be much larger than the real part
共see Appendix B兲.
d␻
2␲
共12兲
where Ls⬘ is an interfacial spin Seebeck coefficient. In
Eq. 共12兲, we used 兰Im关␹ij共␻兲兴d␻ / 2␲ = 0 关since Eq. 共10b兲 is
real and Im ␹ changes sign when ␻ → −␻兴 and
兰关␹xy共␻兲 − ␹yx共␻兲兴共d␻ / 2␲兲 = 1 / 共1 + ␣2兲 共see Appendix A兲.
From Eq. 共12兲 we conclude that the dc spin pumping current
is proportional to the temperature difference between the
magnon and electron/lattice temperatures and polarized
along the average magnetization.
When the magnon temperature is higher 共lower兲 than the
lattice temperature, the dc spin pumping current flows from
F into N leading to a loss 共gain兲 of angular momentum that is
accompanied by the heat current
␥kBTFm
,
␻ 0 M sV a
共14兲
where we introduced the temperature-dependent magnetic
coherence volume
Equation 共10a兲 gives the mean-square deviation of m in the
equal time limit t → 0: 具mi共0兲m j共0兲典 = ␦ij␥kBTFm / ␻0M sV. The
time derivative of Eq. 共10a兲 is
具ṁi共t兲m j共0兲典 = −
2␮B
⬘ A共TFm − TN兲,
具Iz典Heff = Km
ប
B. Magnons
in terms of which, utilizing Eqs. 共3兲 and 共7兲
具mi共t兲m j共0兲典 =
Qm =
Va =
− 4␲Dd/ប␻0
兺n log共1 − e−␤ប␻ 兲
n
⬇
冉
冊
冉冊
1 kBTFm 4␲D
3 ប␻0 kBTFm
␨
2
3/2
共15兲
in which ប␻n = ប␻0 + Dk2n with kn = 2n␲ / d 共n = 0 , ⫾ 1 , . . .兲.
D = 2␥បAex / M s is the spin stiffness with the exchange constant Aex. The approximated form is the three-dimensional
limit that is valid when d Ⰷ 2␲冑D / kBTFm. This condition
turns out to be satisfied in the regimes of interest here. Physically, this coherence volume, or its cube root, the coherence
length, reflects the finite stiffness of the magnetic systems
that limits the range at which a given perturbation is felt.
When this length is small, a random field has a larger effect
on a smaller magnetic volume.
The results obtained in Sec. II A can be carried over simply by replacing V → Va in Eqs. 共3兲–共7兲. The corresponding
spin Seebeck coefficient is Ls⬘ = ␥បkB共gr / A兲 / 2␲ M sVa and the
heat conductance is Km
⬘ = ␻0␮BkB共gr / A兲 / ␲ M sVa. Here we assumed that the magnon temperature does not change appreciably in the volume Va Ⰶ V.
III. MAGNON-PHONON TEMPERATURE DIFFERENCE
PROFILE
Sanders and Walton 共SW兲 共Ref. 6兲 discussed a scenario in
which a magnon-phonon temperature difference arises when
a constant heat flow 共or a temperature gradient兲 is applied
over an F insulator with special attention to the ferrimagnet
yttrium iron garnet 共YIG兲. Its antiferromagnetic component
is small and will be disregarded in the following. This material is especially interesting because of its small Gilbert
214418-3
PHYSICAL REVIEW B 81, 214418 共2010兲
XIAO et al.
damping, which translates into a long length scale of persistence of a nonequilibrium state between the magnetic and
lattice systems. SW assume that, at the boundaries heat can
only penetrate through the phonon subsystem whereas magnons cannot communicate with the nonmagnetic heat baths
共i.e., Km
⬘ = 0 in our notation兲. Inside F bulk magnons interact
with phonons and become gradually thermalized with increasing distance from the interface. The different boundary
conditions for phonons and magnons lead to different
phonon and magnon temperature profiles within F. However,
according to Eq. 共13兲, the magnons in F are not completely
insulated as assumed by SW when the hear reservoirs
are normal metals. In this section, we follow SW and calculate the phonon-magnon temperature difference in an F insulator film induced by the temperature bias but consider also
the magnon thermal conductivity Km
⬘ of the interfaces. In
case of a metallic ferromagnet the conduction electron
system provides an additional parallel channel for the heat
current.
As argued above, the boundary conditions employed by
SW need to be modified when the heat baths connected to
the ferromagnet are metals. In that case, the magnons are
not fully confined to the ferromagnet since a spin current can
be pumped into or extracted out of the normal metal.
In Fig. 1, the two ends of F are at different temperatures: TL
and TR of the N contacts drive the heat flow. Let us ignore
the Pt contact on top of the F for now. When both phonons
and magnons are in contact with the reservoirs, by energy
conservation 共similar to Ref. 6兲 the integrated heat Qmp
共Q̄mp兲 flowing from the phonon to the magnon subsystem in
the range of −L / 2 ⱕ z⬘ ⱕ z 共z ⱕ z⬘ ⱕ L / 2兲 has the following
form:
Qmp共z兲 =
C pC m 1
CT ␶mp
Q̄mp共z兲 =
z
关Tp共z⬘兲 − Tm共z⬘兲兴dz⬘ , 共16a兲
−L/2
dTm
⬘ 关Tm共− L/2兲 − TL兴,
+ Km
dz
共16b兲
dTp
− Kp⬘关Tp共− L/2兲 − TL兴,
dz
共16c兲
=+ Km
=− Kp
冕
C pC m 1
CT ␶mp
L/2
关Tp共z⬘兲 − Tm共z⬘兲兴dz⬘ , 共16d兲
z
dTm
⬘ 关Tm共L/2兲 − TR兴,
+ Km
dz
共16e兲
dTp
− Kp⬘关Tp共L/2兲 − TR兴,
dz
共16f兲
=− Km
=+ Kp
冕
where Cp,m 共CT = Cp + Cm兲 are the specific heats, Kp,m
共KT = Kp + Km兲 are the bulk thermal conductivities for the
phonon and magnon subsystems, Kp,m
⬘ 共KT⬘ = Kp⬘ + Km
⬘ 兲 are the
respective boundary thermal conductivities. ␶mp is the
magnon-phonon thermalization 共or spin-lattice relaxation兲
time.7 The boundary conditions for Tm and Tp are set by
letting z = ⫾ L / 2 in Eq. 共16兲, i.e.,
冏
Km,p
dTm,p
dz
冏
⬘ 关Tm,p共⫾L/2兲 − TR/L兴. 共17兲
= ⫿ Km,p
z=⫾L/2
The solution to Eqs. 共16兲 and 共17兲 yields the magnon-phonon
temperature difference ⌬Tmp共z兲 = Tm共z兲 − Tp共z兲
z
z
⬘ Kp兲sinh ⌬T
sinh
KT共Kp⬘Km − Km
␭
␭
⬅␩
⌬Tmp共z兲 =
⌬T
L
1
L
L
2
2
sinh
⬘ Kp + 2Kp⬘Km + Kp⬘Km
⬘ LKT兲sinh
KmKp共KT⬘ L + 2KT兲cosh + 共2Km
2␭
␭
2␭
2␭
with ⌬T = TL − TR and
C p + C m K pK m
Km
␭2 =
␶mp ⬇
␶mp ,
C pC m K p + K m
Cm
Cm =
共19兲
where the approximation applies when Cp Ⰷ Cm and
Kp Ⰷ Km. Equation 共18兲 shows that the deviation of the magnon temperature from the lattice 共phonon兲 temperature is
proportional to the applied temperature bias and decays to
zero far from the boundaries with characteristic 共magnon diffusion兲 length ␭.
We use the diffusion limited magnon thermal conductivity
and specific heat calculated by a simple kinetic theory 共assuming ប␻0 Ⰶ kBT兲 共Ref. 12兲
冉 冊冑
5
2
32
15␨
kB5 T3
3 3
␲D
and
Km =
冉 冊冑
7
2
16
35␨
共18兲
kB7 T5 ␶m
␲ 3D ប 2
with ␶m the magnon scattering time. Using these expressions
in the approximate form of Eq. 共19兲 we obtain
␭2 =
14␨共7/2兲 DkBT
␶m␶mp .
3␨共5/2兲 ប2
共20兲
In Appendix D, we estimate ␶mp ⯝ 1 / 共2␣␻0兲 for ferromagnetic insulators, assuming that magnetic damping ␣ is caused
by magnon-phonon scattering. It is difficult to estimate or
measure ␶m, and the values quoted in the literatures ranges
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PHYSICAL REVIEW B 81, 214418 共2010兲
THEORY OF MAGNON-DRIVEN SPIN SEEBECK EFFECT
from 10−9 to 10−7 s, depending on both material and
temperature.7,13 At present we cannot predict how ␭ varies
with temperature: Eq. 共20兲 seems to increase with temperature but the relaxation times likely decrease with T.
When Km
⬘ → 0 and Kp⬘ → ⬁, i.e., the boundary is thermally
insulated for magnons and has zero thermal resistivity for
phonons, the prefactor ␩ in Eq. 共18兲 reduces to SW’s result6
␩=
1
KT
⬇
,
L
L
L
L
coth
Kp coth + 2Km
␭
␭
2␭
2␭
共21兲
where the approximation is valid when Kp Ⰷ Km. ⌬Tmp is
obviously maximal in this limit.
The discussion in this section also applies to ferromagnetic metals when the electron-phonon relaxation is much
faster than the magnon-phonon relaxation. The electron and
phonon subsystem are then thermalized with each other and
can be treated as one subsystem. In this case, we may replace
Kp → Kpe = Kp + Ke and
C pC m 1
C pC m 1
C eC m 1
→
+
.
Cp + Cm ␶mp
Cp + Cm ␶mp Ce + Cm ␶me
␥
4␲ M s
D
␣
␻0
␶mp
␶m
gr / A
V1/3
a
␩
␭ 共th兲
␭ 共expt.兲
␰ 共th兲
␰ 共expt.兲
YIG
Py
Unit
1.76⫻ 1011
1.4⫻ 105 a
5.0⫻ 10−40 b
5 ⫻ 10−5 a
10a
10−6 c,d
10−9⬃7 c,e
1016 a
14.1
0.45
0.85–8.5
1.76⫻ 1011
8.0⫻ 105 f
5.5⫻ 10−40 f
0.01g
20g
−7
10 共Ni兲d
10−9 h
1018 i
11.7
0.27
0.3
4.0j
4.4
0.25j
1/T s
A/m
J m2
0.25
GHz
s
s
1 / m2
nm
mm
mm
␮V / K
␮V / K
aReference
共22兲
IV. SPIN SEEBECK VOLTAGE
In Uchida et al.’s experiment 共Ref. 2兲, a Pt contact is
attached on top of a Py film 共see Fig. 1兲 to detect the spin
current signals by the inverse spin Hall effect. A spin current
polarized in the ẑ direction that flows into the contact in the
ŷ direction is converted into an electric Hall current Ic and
thus a Hall voltage VH = IcR in the x̂ direction.
For the setup shown in Fig. 1 we assume that the Pt contact is small enough to not disturb the system, which is valid
when the heat flowing into Pt is much less than the heat
exchange between the magnons and phonons or
CC 1
⬘ ⌬Tmp共z兲共l ⫻ w兲 Ⰶ p m
⌬Tmp共z兲共l ⫻ w ⫻ d兲,
Km
CT ␶mp
which is well satisfied when d Ⰷ 1 nm for both YIG and Py
at low temperatures.
From Eq. 共18兲, we see that at the position below the contact 共z = zc兲 the magnon temperature deviates from the lattice
共and electron兲 temperatures by TFm − TFp = ⌬Tmp共zc兲. If we assume that the contact is at thermal equilibrium with the lattice 共and electrons兲 in the F film underneath, i.e.,
Tec = Tpc = TFp共zc兲, then a temperature difference between
the magnons in F and the electrons in Pt exists:
TFm共zc兲 − Tec = ⌬Tmp共zc兲. Therefore, by Eq. 共12兲, a dc spinpumping current 具Iz典 from F to Pt is driven by this temperature difference, which gives rise to a dc Hall current in the Pt
contact 共see Fig. 1兲
2e 具Iz典
ẑ ⫻ ŷ,
jcx̂ = ␪H
ប A
TABLE I. Parameters for YIG and Py.
共23兲
where ␪H is the Hall angle. In Eq. 共23兲 we disregarded the
spin diffusion backflow and finite thickness corrections to the
Hall effect, which is reasonable when thickness h is comparable to the spin diffusion length lsd.
16.
Reference 14.
cReference 13.
d
Reference 15.
eReference 7.
f
Reference 17. D is derived from Aex = 13 pJ/ m
gReference 18.
h
Reference 19.
iRefrence 20.
jReference 2.
b
According to Eq. 共12兲, the thermally driven spin pumping
current can flow from F to Pt
具Iz典 = Ls⬘关Tm共zc兲 − Tc兴 = Ls⬘关Tm共zc兲 − Tp共zc兲兴.
共24兲
Making use of Eq. 共18兲, the electric voltage over the two
transverse ends of the Pt contact separated by distance l is
VH共zc兲 = ␳ljc共zc兲
zc
zc
sinh
␭
␭
␩␪H␳lgr␥ekB
VH =
⌬T ⬅ ␰
⌬T.
L
L
␲ M sV aA
sinh
sinh
2␭
2␭
sinh
共25兲
Using the numbers in Tables I and II and a value of
␭ ⯝ 7 mm that reflects the low magnetization damping in
YIG, we have ␩ ⯝ 0.47 from Eq. 共21兲, and ␰ ⯝ 0.25 ␮V / K
from Eq. 共25兲 for YIG. For Py, we estimate ␰ ⯝ 4.4 ␮V / K,
which is one order of magnitude larger than the experimental
value of 0.25 ␮V / K given in Ref. 2.
TABLE II. Parameters for the Pt contact.
Quantity
Values
Reference
␪H
␳
l⫻w⫻h
0.0037
0.91 ␮⍀ m
4 mm⫻ 0.1 mm⫻ 15 nm
4
2
2
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PHYSICAL REVIEW B 81, 214418 共2010兲
XIAO et al.
The crucial length scale ␭ is determined by two thermalization times: the magnon-magnon thermalization time ␶m
and the magnon-phonon thermalization time ␶mp as shown in
Eq. 共20兲. Knowledge of these two times is essential in estimating ␭. The quoted values of ␶m,mp are rough order of
magnitude estimates. Yet, in order to completely pin down
the value of ␭ for this material, a more accurate determination of ␶m,mp and Km , Cm as a function of temperature by
both theory and experiments is required. As seen in Eq. 共25兲,
the spatial variation in the Hall voltage over the F strip is
determined by the magnon-phonon temperature difference
profile as calculated in the previous section. From Eq. 共20兲
and Table I 共where the values of ␶m and ␶mp are very uncertain兲, we estimate ␭ ⯝ 0.85– 8.5 mm for YIG. For Py, we
estimate ␭ ⯝ 0.3 mm 共using the ␶mp value for Ni instead of
Py兲, which is about one order of magnitude smaller than its
measured value.
V. DISCUSSION AND CONCLUSION
A magnon-phonon temperature difference drives a dc spin
current, which can be detected by the inverse spin Hall effect
or other techniques. This effect can, vice versa, be used to
measure the magnon-phonon temperature difference. Because the spatial dependence of this effect relies on various
thermalization times between quasiparticles, which are usually difficult to measure or calculate, this effect also accesses
these thermalization times. On a basic level, we predict that
the spin Seebeck effect is caused by the nonequilibrium between magnon and phonon systems that is excited by a temperature basis over a ferromagnet. Since the inverse spin Hall
effect only provides indirect evidence, it would be interesting
to measure the presumed magnon-phonon temperature difference profile by other means. Such measurements would also
give insight into relaxation times that are difficult to obtain
otherwise.
In principle, the theory holds for both ferromagnetic insulators and metals. However, as shown above, the theory fails
for ferromagnetic metal Py, which underestimates the length
scale ␭ and overestimates the magnitude ␰. This might have
two reasons: 共i兲 the lack of reliable information about relaxation times ␶mp,m for Py and 共ii兲 the complication due to the
existence of conduction electrons in ferromagnetic metals.
In conclusion, we propose a mechanism for the spin Seebeck effect based on the combination of: 共i兲 the inverse spin
Hall effect, which converts the spin current into an electrical
voltage, 共ii兲 thermally activated spin pumping at the F 兩 N
interface driven by the phonon-magnon temperature difference, and 共iii兲 the phonon-magnon temperature difference
profile induced by the temperature bias applied over a ferromagnetic film. Effect 共ii兲 also introduces an additional magnon contributed thermal conductivity of F 兩 N interfaces. The
theory holds for both ferromagnetic metals and insulators.
The magnitude and the spatial length scale for YIG is predicted to be in microvolt and millimeter range. The lack of
agreement for the length scale of the spin Seebeck effect for
permalloy remains to be explained.
ACKNOWLEDGMENTS
This work has been supported by EC under Contract No.
IST-033749 “DynaMax,” a Grant for Industrial Technology
Research from NEDO, Japan, and the National Natural Science Foundation of China 共Grant No. 10944004兲. J.X. and
G.B. acknowledge the hospitality of the Maekawa Group at
the IMR, Sendai.
APPENDIX A: INTEGRAL
Here we evaluate the frequency integral in Eq. 共12兲. To
this end, we need to reintroduce the time dependence and
then take the limit t → 0. When t ⬎ 0, the integral in Eq. 共10b兲
can be calculated using contour integration 共real axis
+ semicircle in the lower plane so that e−i␻t → 0, where the
integral over the semi-circle vanishes by Jordan’s Lemma兲
冕
␹e−i␻t
冋
册
d␻
e−i␻−t
共1̂ − ␴ˆ y兲 ⌰共t兲.
= − i Im
2␲
1 − i␣
共A1兲
The minus sign comes from the counterclockwise contour,
and ⌰共t兲 is the Heaviside step function, which vanishes
when t ⬍ 0 and is unity otherwise. This integral is discontinuous at t = 0, therefore the value at t = 0 is given by the average
of the values at t = 0⫾
冕
␹e−i␻0
冉
冊
␣ 1
d␻ 1 1
=
.
2␲ 2 1 + ␣2 − 1 ␣
共A2兲
APPENDIX B: SPIN-MIXING CONDUCTANCE FOR F 円 N
INTERFACE
In this appendix, we use a simple parabolic band model to
estimate the spin pumping at an F 兩 N interface, where F can
be a conductor or an insulator. The Fermi energy in N is EF,
and the bottom of the conduction band for spin up and spin
down electrons in F are at EF + U0 and EF + U0 + ⌬, respectively, where U0 is the bottom of the majority band and ⌬ is
the exchange splitting. The reflection coefficient for an electron spin ␴ 共↑ or ↓兲 from N at the Fermi energy reads
r␴共k兲 =
k − k␴
,
k + k␴
共B1兲
where k = 冑2m0EF / ប2 − q2 is the longitudinal wave vector
in
N
共q
is
the
transverse
wave
vector兲.
2
2
冑
冑
k↑ = 2m↑共EF − U0兲 / ប − q and k↓ = 2m↓共EF − U0 − ⌬兲 / ប2 − q2
are the longitudinal wave vectors 共or imaginary decay constants兲 in F for both spins. m0 is the effective mass in N and
m␴ is the effective mass for spin ␴ in F. The mixing conductance reads
gmix =
A
4␲2
冕
共1 − r↑rⴱ↓兲d2q = gr + igi .
共B2兲
We evaluate Eq. 共B2兲 with m0 = m␴ having the free electron
mass, EF = 2.5 eV, U0 = 0 – 4 eV.21 A plot of the mixing conductance is shown in Fig. 3 for ⌬ = 0.3, 0.6, 0.9 eV.
U0 = 0 corresponds to a ferromagnetic metal 共with
majority band matching the N electronic structure兲, and
U0 ⱖ EF = 2.5 eV corresponds to a ferromagnetic insulator
共with 兩r␴兩 = 1 for both spin types兲.
214418-6
PHYSICAL REVIEW B 81, 214418 共2010兲
THEORY OF MAGNON-DRIVEN SPIN SEEBECK EFFECT
spin-mixing conductance: 4πgr,i / kF2A
1
具mi共␨, ␶兲m j共0,0兲典 =
0.1
⫻e−i␻␶
gr: ∆ = 0.3 eV
冕
d␻
2␲
␹ij − ␹ⴱji ប兩␻兩/kBT
.
i␻ eប兩␻兩/kBT − 1
gr: ∆ = 0.6 eV
-gi: ∆ = 0.6 eV
gr: ∆ = 0.9 eV
具mi共0,0兲m j共0,0兲典 =
-gi: ∆ = 0.9 eV
冉 冊
Li3/2共e−ប␻0/kBT兲 ␥ប kBT
8␲3/2
Ms D
3/2
␦ij ,
EF
0.001
0
1
2
barrier height: U0 (eV)
3
4
FIG. 3. 共Color online兲 gr,i vs U0 in log scale for different ⌬
values. Curves starting with values close to unity give the real part
gr and curves starting from very low values display the imaginary
part −gi. The vertical axis is in units of the Sharvin conductance.
共C6兲
where Li3/2共x兲 is the polylogarithm function, which is
bounded by the Zeta function ␨共3 / 2兲 ⯝ 2.6 when x → 1 or
ប␻0 / kBT → 0.
For a quasi-two-dimensional film with thickness d,
V−1兺k → d−1兺kz兰d2k / 共2␲兲2
具mi共0,0兲m j共0,0兲典 =
APPENDIX C: MAGNETIZATION CORRELATION FOR
MACROSCOPIC SAMPLES
⯝
The dynamics of m is governed by the LLG equation
ṁ共r,t兲 = − ␥m共r,t兲 ⫻ Heff + ␣m共r,t兲 ⫻ ṁ共r,t兲, 共C1兲
where Heff = H0 + 共D / ␥ប兲ⵜ2m共r , t兲 + h共r , t兲 is the total effective magnetic field with: 共i兲 the external field plus the
uniaxial anisotropy field H0 = H0ẑ = 共␻0 / ␥兲ẑ, 共ii兲 the exchange field 共D / ␥ប兲ⵜ2m due to spatial variation in magnetization, and 共iii兲 the thermal random fields h共r , t兲.
We define Fourier transforms
g̃共k, ␻兲 =
g共r,t兲 =
冕
dre
ik·r
1
兺 e−ik·r
V m,n,l
冕
冕
i␻t
dte g共r,t兲,
共C2兲
d␻ −i␻t
e g̃共k, ␻兲,
2␲
共C3兲
where k = 2␲共m / L , n / W , l / H兲 and L , W , H are the length,
width, and height of the ferromagnetic film. Near thermal
equilibrium, m⬜ Ⰶ 1 共and so mz ⯝ 1兲, we may linearize the
LLG equation. After Fourier transformation, the linearized
LLG equation becomes m̃i共k , ␻兲 = 兺 j␹ij共k , ␻兲␥h̃ j共k , ␻兲 with
i , j = x , y and
␹=−
1/共1 + ␣2兲
共␻ −
␻k+兲共␻
−
共C5兲
The limit ␨ = 0 and ␶ = 0 can be obtained for threedimensional systems by replacing the sum over k by an integral V−1兺k → 兰d3k / 共2␲兲3 in Eq. 共C5兲
-gi: ∆ = 0.3 eV
0.01
2 ␥ k BT
兺 e−ik·␨
M sV k
␻k−兲
冉
␻k − i␣␻
− i␻
␻k − i␣␻
i␻
冊
共C4兲
with ប␻k = ប␻0 + D兩k兩2 and ␻k⫾ = ⫾ ␻k / 共1 ⫾ i␣兲.
The spectrum of random motion in the linear response
regime xi = 兺 j␹ij f j with “current” x and random force f
共x and f are chosen such that xf is in the units of energy兲 is
comprehensively studied in Ref. 22. In our problem, f → ␥h,
x → 共M s / ␥兲m, and ␹ → 共M s / ␥兲␹, the autocorrelation of the
magnetization then becomes 共␨ = r1 − r2 , ␶ = t1 − t2兲
冉
冊
1
k BT
log
␦ij
兺
4␲AM sd n
1 − e −␤ប␻n
冉 冊
ប␻0
k BT
log
␦ij ,
4␲AM sd
k BT
共C7兲
2
where ប␻n = ប␻0 + Dkz,n
with kz,n = 2n␲ / d. The approximation in the last line consists of retain only the n = 0 term,
allowed when d ⱕ 2␲冑D / kBT and ប␻0 Ⰶ kBT. The first line
of Eq. 共C7兲 approaches Eq. 共C6兲 when the film becomes
thick 共d Ⰷ 2␲冑D / kBT兲.
APPENDIX D: RELATION BETWEEN ␣ AND ␶mp
Here we derive a relationship between the magnetic
damping constant ␣ and the magnon-phonon thermalization
time ␶mp. When the magnon and phonon temperatures are Tm
and Tp, the magnon-phonon relaxation time is phenomenologically defined by6
Tp − Tm
d
共Tp − Tm兲 = −
.
dt
␶mp
共D1兲
If the phonon system is attached to a huge reservoir 共substrate兲 such that its temperature Tp is fixed 共dTp / dt = 0兲, Eq.
共D1兲 becomes
Tm − Tp
dTm
⯝−
.
dt
␶mp
共D2兲
In metals we should consider three subsystems 共magnon,
phonon, and electron兲 leading to
dTs
= − 兺 kst共Ts − Tt兲
dt
t
共D3兲
with s , t = m, p, and e for magnon, phonon, and electron. The
coupling strength kst = ␶st−1Ct / 共Cs + Ct兲 with the s-t relaxation
time ␶st and the specific heat Cs. When the magnon specific
heat is much less than that of phonons and electrons
共Cm Ⰶ Cp , Ce兲
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PHYSICAL REVIEW B 81, 214418 共2010兲
XIAO et al.
Tm − Tp Tm − Te
dTm
⯝−
−
.
dt
␶mp
␶me
共D4兲
−
kB dTm
= 具ṁz典
HeffM sV dt
= 具my␥hx − mx␥hy典 + ␣具mxṁy − myṁx典
We may parameterize the magnon temperature by the
thermal suppression of the average magnetization
=
HeffM sV共1 − 具mz典兲 = kBTm ,
共D5兲
1 2 ␥ k B共 ␣ T m − ␣ pT p − ␣ eT e兲
1 + ␣2
M sV
共D8兲
or
where Heff points in the ẑ direction. The LLG equation provides us with the information about mz
2␻0
dTm
=−
共 ␣ T m − ␣ pT p − ␣ eT e兲
dt
1 + ␣2
ṁ = − ␥m ⫻ 共Heff + h兲 + ␣m ⫻ ṁ,
with ␻0 = ␥Heff. Comparing Eq. 共D9兲 with Eq. 共D4兲, we find
共D6兲
2␣␻0 ⯝
where the random thermal field h = hp + he from the lattice
and electrons are determined by the phonon/electron temperature:
i
j
共t兲␥hp/e
共0兲典
具␥hp/e
2␥␣p/ekBTp/e
=
M sV
共D7兲
with ␣p/e the magnetic damping caused by scattering with
phonons/electrons and ␣ = ␣p + ␣e. Therefore
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