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PRL 111, 087205 (2013) PHYSICAL REVIEW LETTERS 23 AUGUST 2013

Evidence for a Magnetic Seebeck Effect

Sylvain D. Brechet,1,* Francesco A. Vetro,1 Elisa Papa,1 Stewart E. Barnes,2 and Jean-Philippe Ansermet1
1
Institute of Condensed Matter Physics, Station 3, Ecole Polytechnique Fédérale de Lausanne—EPFL,
CH-1015 Lausanne, Switzerland
2
James L. Knight Physics Building, 1320 Campo Sano Avenue, University of Miami, Coral Gables, Florida 33124, USA
(Received 5 June 2013; published 22 August 2013)
The irreversible thermodynamics of a continuous medium with magnetic dipoles predicts that a
temperature gradient in the presence of magnetization waves induces a magnetic induction field, which
is the magnetic analog of the Seebeck effect. This thermal gradient modulates the precession and
relaxation. The magnetic Seebeck effect implies that magnetization waves propagating in the direction
of the temperature gradient and the external magnetic induction field are less attenuated, while
magnetization waves propagating in the opposite direction are more attenuated.

DOI: 10.1103/PhysRevLett.111.087205 PACS numbers: 75.76.+j, 76.50.+g

The discovery of the spin Seebeck effects in ferromag- thermodynamic formalism does not allow for a direct
netic metals [1], in semiconductors [2], and in insulators estimation of
. The numerical value of this parameter
[3] has generated much research for spin transport in needs to be evaluated directly from the experimental data,
ferromagnetic samples of macroscopic dimensions sub- as shown below.
jected to temperature gradients. The interplay of spin, In the bulk of the sample, as shown in Ref. [5], the
charge, and heat transport defines the rich field known as magnetization force density has the structure of a
spin caloritronics [4]. Prompted by these recent develop- Lorentz force density [13] expressed in terms of the mag-
ments, we established a formalism describing the irrevers- netic bound current density jM ¼ r  M [14]
ible thermodynamics of a continuous medium with
magnetization [5]. M rBind ¼ jM  Bind : (2)
In this Letter, we test a particular experimental predic- Thus, using vectorial identities, the phenomenological
tion of this formalism on a yttrium iron garnet (YIG) slab. relations (1) and (2) imply that in the bulk of the system,
We argue that the thermodynamics of irreversible pro- the magnetic induction field Bind , induced by a uniform
cesses implies the existence of a magnetic counterpart to temperature gradient rT in the presence of a magnetic
the well-known Seebeck effect. We show how a thermally bound current density r  M, is given by, i.e.,
induced magnetic field modifies the Landau-Lifshitz equa-
tion and provide experimental evidence for the magnetic B ind ¼ "M  rT; (3)
Seebeck effect by the propagation of magnetization waves where the phenomenological vector "M is given by
in thin crystals of YIG. The effect of a temperature gradient
on the dynamics of the magnetization on a YIG slab with "M ¼

nkB ðr  MÞ
1 : (4)
and without Pt stripes was investigated recently by Obry
et al. [6], Cunha et al. [7], Silva et al. [8], Padrón- By analogy with the Seebeck effect, we shall refer to this
Hernández et al. [9,10], Jungfleisch et al. [11], and Lu phenomenon as the magnetic Seebeck effect.
et al. [12]. The time evolution of the magnetization M is given by
In general, irreversible thermodynamics predicts cou- the Landau-Lifschitz-Gilbert equation, i.e.,
plings between current and force densities. In Eq. (86) of
M_ ¼ M  Beff
 M  M; _ (5)
Ref. [5], we identified the magnetization force term mrB. MS
For an insulator like YIG, there is no charge current. As
explained in detail in Ref. [5], the transport equation (94) where  is the gyromagnetic ratio,  ’ 10
4 is the Gilbert
of Ref. [5] implies that the magnetization force density damping parameter of YIG [15], and MS ¼ 1:4 
MrBind induced by a thermal force density
nkB rT is 105 A m
1 is the magnitude of the effective saturation
proportional and opposite to this force density, i.e., magnetization of YIG at room temperature [16]. The ef-
fective magnetic induction field Beff includes the external
M rBind ¼
nkB rT; (1) field Bext , the demagnetizing field Bdem , the anisotropy
field Bani , which behaves as an effective saturation mag-
which corresponds to Eq. (155) of Ref. [5], where
> 0 is netization in the linear response [17], and finally a ther-
a phenomenological dimensionless parameter, kB is mally induced field Bind given by the relation (3). The
Boltzmann’s constant, and n ¼ 1:1  1028 m
3 is the exchange field Bint [18] is negligible in the following, as
Bohr magneton number density of YIG. The we consider magnetostatic modes [19]. The demagnetizing

0031-9007=13=111(8)=087205(5) 087205-1 Ó 2013 American Physical Society

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PRL 111, 087205 (2013) PHYSICAL REVIEW LETTERS 23 AUGUST 2013

field Bdem breaks the spatial symmetry and generates an 1

ðr  MÞ
1  rT ¼ ðr
1  mÞ  rT
elliptic precession cone. After performing the linear MS2
response of the magnetization in the presence of a ther-
1 1
mally induced field Bind , we shall describe how the demag- ¼ 2 ðrT  r
1 Þm
2 ðrTÞr
1 m;
netizing field Bdem affects the magnetic susceptibility. MS MS
We found evidence for the magnetic Seebeck effect by where r
1  r ¼ 1 and the last term on the right-hand side
exciting locally, at angular frequency ! ’ 2:74  1010 s
1 , vanishes since it averages out on a precession cycle.
the ferromagnetic resonance of a YIG slab of length The vectorial time evolution equation (6) is written
Lz ¼ 10
2 m, width Ly ¼ 2  10
3 m, and thickness explicitly in Cartesian coordinates as
Lx ¼ 2:5  10
5 m, subjected to a temperature gradient
as small as jrTj ’ 2  103 K m
1 generated by Peltier m_ x ¼ ð!0 þ !M kT  r
1 Þmy þ m_ y
!M 
1
0 by ;
(9)
elements. The excitation field is applied on the slab using m_ y ¼
ð!0 þ !M kT  r
1 Þmx
m_ x þ !M 
1
0 bx ;
a local antenna, as detailed in Ref. [20]. For signal trans-
mission experiments, two antennas are used, set approx- where the angular frequencies !0 and !M are defined,
imatively 8 mm apart, as shown in Fig. 1. Note that a respectively, as
similar setup for a gradient orthogonal to the YIG slab !0 ¼ B0 ; !M ¼ 0 MS : (10)
was investigated recently [7]. For reasons explained below,
these two setups can be expected to probe different In a stationary regime, the magnetic excitation field b and
mechanisms. the magnetization response m are oscillating at an angular
The external magnetic induction field Bext applied on the frequency !, which is expressed in Fourier series as
YIG film consists of a uniform and constant field B0 and a X X
small excitation field b ¼ bx x^ þ by y^ locally oscillating in bx ¼ bk ei½kx
!tþð=2Þ ; by ¼ bk eiðkx
!tÞ ;
a plane orthogonal to B0 ¼ B0 z^ . In the limit of a small k k
X X
excitation field, i.e., in the linear limit, the magnetization mx ¼ mk ei½kx
!tþð=2Þ ; my ¼ mk eiðkx
!tÞ ;
field M consists of a uniform and constant field MS ¼ MS z^ k k
and a response field m ¼ mx x^ þ my y^ locally oscillating in (11)
a plane orthogonal to MS such that m  MS . The linear
response of the magnetization to the excitation field, where the eigenstates bk and mk are complex valued and
according to the time evolution equation (5), is given by dephased.
 The Cartesian components of the eigenmodes kx;y;z sat-
m_ ¼ ðm  B0 þ MS  B1 Þ
M  m; _ (6) isfy the boundary conditions of null m at the surface of the
MS S
sample
where the first-order magnetic induction field B1 yields
nx;y;z 
B 1 ¼ b
0 ðkT  r
1 Þm; (7) kx;y;z ¼ ; (12)
Lx;y;z
and 0 is the magnetic permeability of vacuum and the
thermal wave vector where nx;y;z 2 N [20].
The eigenstates of the excitation field bk and the

nkB
kT ¼ rT: (8) response field mk are related through the magnetic suscep-
0 MS2 tibility k , i.e.,
To obtain the expressions (7) and (8), we used the linear mk ¼ 
1
0 k bk : (13)
vectorial identity
The time evolution equations (9), the definition (10),
Crystal and the Fourier series (11) in the stationary regime imply
RF pulse
Amplifier detector that the magnetic susceptibility k is given by
generator
Antennae Oscilloscope 1
k ¼
; (14)
Peltier


0 þ ið
þ kT  k
1 Þ
element
where the dimensionless parameters
and
0 are, respec-
B1 YIG tively, defined as
B0 ! !

¼ ;
0 ¼ 0 : (15)
!M !M
Cu
The demagnetizing field Bdem ¼
0 mx x^ causes the
FIG. 1 (color online). Time-resolved transmission measure- damping and the magnetic susceptibility kx along
ment of magnetization waves. the x axis to differ respectively from the damping and

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PRL 111, 087205 (2013) PHYSICAL REVIEW LETTERS 23 AUGUST 2013

the magnetic susceptibility ky along the y axis. The than the opening angle for a magnetization wave propagat-
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
resonance frequency !0 ð!0 þ !M Þ is given by Kittel’s ing in the opposite direction, as shown in Fig. 2.
formula [21] to first order in  and kT . Thus, the magnetic This is confirmed experimentally by detecting induc-
susceptibilities kx;y yield tively at one end of the sample the signal that results
from an excitation pulse of 15 ns duration at the other
1 end. The signals obtained by sweeping the magnetic in-
kx;y ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; duction field B0 for the propagation of magnetization



0 ð
0 þ 1Þ þ irx;y ð
þ kT  k
1 Þ waves from the cold end to the hot end or from the hot
(16) end to the cold end are given in Fig. 3. Clearly, the waves
propagating from the cold to the hot side appear to decay
where rx;y > 0 are phenomenological damping scale fac- less rapidly than the waves propagating from the hot to the
tors accounting for symmetry breaking. cold side.
As shown by Cunha et al. in Fig. 1(a) of Ref. [7], the The time evolution of the signals for the waves propa-
propagating modes of the magnetization waves in the gating in the direction of the gradient or opposite to it is
bulk of YIG are magnetostatic backward volume modes obtained by averaging the signals over the range of the
propagating in the direction
k
1 . The expressions (8) magnetic induction field B0 displayed in Fig. 4. The signal
and (16) for the magnetic susceptibilities and the thermal is a convolution of kz modes that have different group
wave vector kT imply that the magnetization waves prop- velocities and decay exponentially due to the damping.
agating from the cold to the hot side, i.e., kT  k
1 < 0, are The peaks were identified in Ref. [22] as the result of the
less attenuated by the temperature gradient and the mag- propagation of odd modes. Since the peaks of the trans-
netization waves propagating from the hot to the cold sides, mitted signals are detected at the same time, the tempera-
i.e., kT  k
1 > 0, are further attenuated. ture gradient does not affect significantly the kz mode
Thus, the opening angle of the precession cone of the group velocities. Moreover, from the logarithmic scale
magnetization m for a magnetization wave propagating in for the signal in Fig. 4, a larger difference in attenuation
the direction of the temperature gradient decreases less between the signals for small kz modes is inferred. This is
in line with the theoretical prediction, made by Eq. (16), for
the magnetic Seebeck effect to be proportional to k
1 z .
Cold to Hot Moreover, since the relative difference between the signals
m(0)
is due to the temperature gradient, we can estimate the
relative difference between the damping terms 
and

k-1 k-1 B0 86
Cold to Hot
m(τ) 84

x 82
YIG
z
B0 (mT)

80

kT∝ T y 78

76
Hot to Cold
m(0) 74

Time (ns)
-20 0 20 40 60 80 100 120 140 160
k-1 86
k-1 B0
84
Hot to Cold
m(τ)
82

YIG
B0 (mT)

80

78
kT∝ T 76

74
FIG. 2 (color online). Propagation of magnetization waves
from the cold to the hot side (top) and vice versa (bottom).
The cones describe the precession of the magnetization at FIG. 3 (color online). Transmitted signals from the cold to the
excitation mð0Þ and at detection mðÞ. The amount of damping hot side and from the hot to the cold side as a function of the
depends on the relative orientation kT of the temperature magnetic field B0 and of the detection time after a 15 ns pulsed
gradient with respect to the magnetization wave propagation excitation at 4.36 GHz. The lighter areas correspond to a larger
direction
k
1 . signal.

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PRL 111, 087205 (2013) PHYSICAL REVIEW LETTERS 23 AUGUST 2013
6.0
kT· k-1 < 0
against the temperature gradient and that the effect of the
4.0 temperature is proportional to k
1 z .
kT· k-1 > 0
For a temperature gradient orthogonal to the YIG plane,
Cunha et al. [7] showed that the temperature gradient
2.0 affects the propagation of magnetization waves only
when Pt is deposited on the YIG slab. The effect is
accounted for by a model of spin injection and spin pump-
1.0
0.8
ing, detailed by Ando et al. [24], at the interface between Pt
and YIG. The quantitative analysis of the data is presented
0.6
in Ref. [8]. In Ref. [7], it is stated clearly that the effect
0.4 does not occur in the absence of Pt on the surface. When Pt
-20 0 20 40 60 80 100 120 140 160 is removed in such a setup where kT  k
1 ¼ 0, the mecha-
Time (ns)
nism invoked by Cunha et al. is not operative and our
FIG. 4 (color online). Transmitted signal as a function of time mechanism is not effective either.
after a 15 ns pulsed excitation at 4.36 GHz. In summary, we point out that thermodynamics of irre-
versible processes implies a coupling between heat current
and magnetization precession in a temperature gradient.
kT  k
1 appearing in the expression (16) for the magnetic This effect can be expressed by an induced magnetic field
susceptibilities. Comparing the signals at t ¼ 40 ns, we Bind proportional to the applied temperature gradient.
find that the dimensionless parameter
’ 6  10
7 , which Thus, we suggest to refer to it as a magnetic Seebeck
corresponds to a thermal damping ratio jkT  k
1 j=
’ effect, since it is the magnetic analog of the regular
0:3 less that an order of magnitude below the self- Seebeck effect. It is distinct from the magneto-Seebeck
oscillation threshold. effect, which refers to a change in the Seebeck coefficient
The difference in attenuation between the signals is also due to the magnetic response of nanostructures [25]. We
shown on the ferromagnetic resonance (FMR) spectrum analyze how the Landau-Lifshitz equation is modified and
detected 70 ns after the pulse and displayed in Fig. 5. The find a contribution to the dissipation that is linear in rT.
spectral linewidth 0:2 mT corresponds to inhomogene- Hence, this effect can increase or decrease the damping,
ous broadening, since it is much larger than the homoge- depending on the orientation of the wave vector of the
neous linewidth Beff [23]. excited magnetostatic mode with respect to the tempera-
As is rightly pointed out in Ref. [6], the temperature ture gradient. If the temperature gradient could be made
dependence of the saturation magnetization affects the strong enough, i.e., kT  k
1 > 
, then the damping
amplitude of the magnetization waves. However, since would be negative and the magnetization would undergo
our experimental setup is sufficiently close to the self- self-oscillation. This would be analogous to the magneti-
oscillation threshold for a temperature gradient that is zation self-oscillation described in Chap. 7 of Ref. [26] and
small enough, we expect the dynamic contribution kT  the heat equivalent of Berger’s spin amplification by simu-
k
1 to be larger than the static contribution due to the lated emission of radiation (SWASER) predicted for
temperature dependence of the saturation magnetization. charge-driven spin polarized currents [27].
Moreover, in contrast to the claim made in Ref. [6], Fig. 4 We thank François A. Reuse, Klaus Maschke, and
shows that magnetization waves can propagate with and Joseph Heremans for insightful comments and acknowl-
edge the following funding agencies: Polish-Swiss
Research Program NANOSPIN PSRP-045/2010 and
kT· k-1 < 0 Deutsche Forschungsgemeinschaft SS1538 SPINCAT,
2.0 kT· k-1 > 0 Grant No. AN762/1.

1.5

*sylvain.brechet@epfl.ch
1.0
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