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PHYSICAL REVIEW B **66**, 060404~R! ~2002!

**Spin battery operated by ferromagnetic resonance
**

Arne Brataas,1 Yaroslav Tserkovnyak,1 Gerrit E. W. Bauer,2 and Bertrand I. Halperin1
1*Harvard University, Lyman Laboratory of Physics, Cambridge, Massachusetts 02138
*

2*Department of Applied Physics and DIMES, Delft University of Technology, 2628 CJ Delft, The Netherlands
*~Received 1 May 2002; published 29 August 2002!

Precessing ferromagnets are predicted to inject a spin current into adjacent conductors *via *Ohmic contacts,
irrespective of a conductance mismatch with, for example, doped semiconductors. This opens the way to create
a pure spin source ~‘‘spin battery’’! by the ferromagnetic resonance. We estimate the spin current and spin bias
for different material combinations.

DOI: 10.1103/PhysRevB.66.060404 PACS number~s!: 76.50.1g, 72.25.Mk, 73.23.2b, 73.40.2c

The research field of magnetoelectronics or spinelectron-
ics strives to utilize the spin degree of freedom for electronic
applications.1 Devices made from metallic layered systems
displaying the giant2 and tunnel magnetoresistance3 have
been proven useful for read-head sensors and magnetic
random-access memories. Integration of such devices with
semiconductor electronics is desirable but difficult because a
large resistivity mismatch between magnetic and normal ma-
terials is detrimental to spin injection.4 Spin injection into
bulk semiconductors has been reported only in optical pump
and probe experiments,5 and with high-resistance ferromag-
netic injectors6 or Schottky/tunnel barriers.7 In these cases,
the injected spin-polarized carriers are hot and currents are
small. Desirable are semiconductor devices with an efficient
all-electrical cold-electron spin injection and detection *via
*Ohmic contacts at the Fermi energy, just as has been realized
by Jedema *et al. *for metallic devices.8

We introduce a concept for dc spin-current injection into arbitrary conductors through Ohmic contacts, which does not involve net charge currents. The spin source is a ferromag- netic reservoir at resonance with an rf field. Pure spin-current injection into low-density conductors should allow experi- mental studies of spintronic phenomena in mesoscopic, bal- listic, and nanoscale systems, which up to now has been largely a playground of theoreticians like Datta and Das,9

whose spin transistor concept has stimulated much of the present interest in spintronics.

The combination of a ferromagnet at the ferromagnetic resonance ~FMR! in Ohmic contact with a conductor can be interpreted as a ‘‘spin battery,’’ with analogies and differ- ences with charge batteries. For example, charge-current conservation dictates that a charge battery has two poles, plus and minus. A spin battery requires only one pole, since the spin current does not need to be conserved. Furthermore, the polarity is not a binary, but a three-dimensional vector. The important parameters of a charge battery are the maxi- mum voltage in the absence of a load, as well as the maxi- mum charge current, which can be drawn from it. In the following we present estimates for the analogous character- istics of the spin battery.

Central to our concept is a precessing ferromagnet, which acts as a source of spin angular momentum, when in contact with normal metals,10 see Fig. 1. This spin injection can be formulated in analogy with the adiabatic pumping of charge in mesoscopic systems.11,12 When the ferromagnet is thicker

0163-1829/2002/66~6!/060404~4!/$20.00 **66 **0604

than the ferromagnetic coherence length ~a few Å in transi-
tion metals such as Co, Ni, or Fe!, the spin current emitted
into the normal metal is determined by the *mixing
conductance*12 *g↑↓*5(*nm*@d*nm*2*rnm
*

*↑ *(*rnm
↓ *)*# in terms of the

spin-dependent reflection amplitudes *rnm
↑*(*↓*) between trans-

verse modes *m *and *n *in the normal metal at the interface to
the ferromagnet, where the latter is characterized by the mag-
netization direction **m**. The mixing conductance governs the
transport of spins that are noncollinear to the magnetization
direction in the ferromagnet12,13 and is also a material param-
eter proportional to the torque acting on the ferromagnet in
the presence of a noncollinear spin accumulation in the nor-
mal metal.10,12,14,15 For most systems ~with the exception of,
e.g., ferromagnetic insulators16! the imaginary part of the
mixing conductance can be disregarded due to the random-
ization of phases of spin-up and spin-down electrons in re-
ciprocal space14 and this is assumed in the following. The
spin current emitted into the normal metal is then, simply,10

**I***s
*source5

\

4p
*g↑↓***m**3

*d***m
**

*dt
*. ~1!

In our notation, the spin current is measured in units of me-
chanical torque. Equation ~1! is a time-dependent correction
to the Landauer-Büttiker formula for noncollinear
ferromagnetic–normal-metal (*F*-*N*) hybrid systems.12 A
simple physical picture can be inferred from the following
thought experiment.17 Suppose we have a *F*-*N *interface at

FIG. 1. Schematic view of the spin battery. Precession of the
magnetization **m**(*t*) of the ferromagnet *F *emits a spin current **I***s
*

source

into the adjacent normal-metal layer *N*. The spin accumulation in
the normal metal either relaxes by spin-flip scattering or flows back
into the ferromagnet, resulting in a net spin current **I***s*5**I***s
*

source

2**I***s
*back .

©2002 The American Physical Society04-1

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BRATAAS, TSERKOVNYAK, BAUER, AND HALPERIN PHYSICAL REVIEW B **66**, 060404~R! ~2002!

equilibrium and switch the magnetization instantaneously. The mismatch of the spin-up and spin-down chemical poten- tials leads to large nonequilibrium spin currents on the length scale of the spin-diffusion length. A slower magnetization reversal naturally induces smaller spin currents. Equation ~1! represents the adiabatic limit of the spin currents pumped by a slow magnetization dynamics. When the spin current ~1! is channeled off sufficiently rapidly, the corresponding loss of angular momentum increases the ~Gilbert! damping of the magnetization dynamics.10 Equation ~1! is the maximum spin current that can be drawn from the spin battery.

Next, we need the maximum spin bias obtained when the
load vanishes. When the spin-flip relaxation rate is smaller
than the spin-injection rate, a spin angular momentum **s **~in
units of \) builds up in the normal metal. We can neglect
spatical dependence within the ferromagnet when the film is
sufficiently thin.18 Under these conditions, one finds that the
component of the backflow spin current **I***s
*

back , from the nor-
mal metal to the ferromagnet, parallel to the instantenous
magnetization direction **m **is cancelled by an opposite flow
from the ferromagnet. The componenent of **I***s
*

back perpendicu-
lar to **m **is12

**I***s
*back5

*g↑↓
*2p*N
*

@**s**2**m**~**m**•**s**!# , ~2!

where *N *is the one-spin density of states. We note that the
mixing conductance in Eqs. ~1! and ~2! ought to be renor-
malized in highly transparent junctions.19

The relation between spin excess **s **and total spin current
**I***s*5**I***s
*

source2**I***s
*back in a normal diffusive metal is governed by

the spin-diffusion equation20

]**s
**

]*t
*5*D
*

]2**s
**

]*x*2
2

**s
**

ts , ~3!

where *D *is the diffusion coefficient, in three ~two! dimen-
sions *D*5*vF
*

2 t/6 (*D*5*vF
*2 t/4), and t ,ts are the elastic and

spin-flip relaxation times, respectively. We solve the diffu-
sion equation with boundary conditions at *x*50, where
(*DA*\)]*x***s**52**I***s *and at the end of the sample *x*5*L *, where
the spin current vanishes, ]*x***s**50. *A *is the cross section of
the system.

The precession of the magnetization vector of a ferromag-
net under a resonant rf electromagnetic field applied perpen-
dicularly to a dc magnetic field21 can be used to drive the
spin battery. The magnitude of the spin current **I***s
*

source and
spin bias Dm[2^**s**& *t */*N *as a function of the applied field **H**0
follows from the Landau-Lifshitz-Gilbert equation **ṁ5
**2g*m3H01am3ṁ, where g* is the gyromagnetic ratio,
a;0.0120.001 the Gilbert damping factor, and magnetic
anisotropies have been disregarded for simplicity. The spin
bias also has ac components. However, its frequency v har-
monics are strongly suppressed when *ls */(vt*s*)

1/2,*L*,*ls *,
which can be easily realized when vt*s*.1, e.g., t*s*.v

21

;10211*s*/*H*0 (T). *ls*5A*D*t*s *is the spin-diffusion length in
the normal metal. The dominant contribution to the spin bias
is then constant in time and directed along **H**0. The magni-

06040

tude of the time-averaged spin accumulation Dm
[2^**s**(*t*)& *t */*N *in the normal metal close to the *F*-*N *interface
then reads

Dm5\v0 sin2u

sin2u1h , ~4!

where the precession cone angle between **H**0 and **m **is u ,h
5(t *i */t*s*)tanh(*L*/*ls*)/(*L*/*ls*) is a reduction factor, and we have
introduced the spin-injection rate t *i
*

215*g↑↓ */(2p\*NAL*).
Large systems have a smaller injection rate since more states
have to be filled.

The ratio of the injection and spin-flip relaxation times
can be evaluated for a planar geometry. We consider a free-
electron gas in contact with a metallic ferromagnet. The mix-
ing conductance is *g↑↓*5k*AkF
*

2 /(4p)(*g↑↓*5k*AkF */p) for
spin injection into three-~two-!dimensional systems. First-
principles band-structure calculations show that for combina-
tions like Co/Cu or Fe/Cr, k remains close to unity.14 The
ratio between the injection and spin-flip relaxation times in
three ~two! dimensions can be calculated to be t *i */t*s
*5A8/3k21Ae(*L*/*ls*) @t *i */t*s*52k21Ae(*L*/*ls*)# . e5t/t*s *is the
ratio of the elastic scattering rate and the spin-flip relaxation
rate, which is usually much smaller than unity.

When the spin relaxation time is longer than the spin in-
jection time and the precession cone angle is sufficiently
large, sin2u.h, the spin bias saturates at its maximum value
Dm05\v0. In this regime the spin accumulation does not
depend on the material parameters. It should be feasible to
realize the full spin bias when *L*!*ls *since h
'A8/3k21Ae(*L*/*ls*), e.g., when *L*/*ls*50.1, A8/3k21Ae
50.1 the precession cone angle should be larger than 6°. For
small precession cone angles u'*H*1 /(a*H*0), so, for ex-
ample, *H*051.0 T,a510

23 this requires a *H*150.1 mT rf
field with a resulting spin bias of Dm50.1 meV. For a
smaller precession angle, e.g., u50.6° the spin bias is
smaller, Dm51 mV, but still clearly measurable. Epitaxially
grown clean samples with even longer spin-diffusion lengths
and smaller spin-flip to non-spin-flip relaxation ratios e will
function as spin batteries with smaller precession angles. The
precession cone angle u in FMR is typically small, but u
.15° can be achieved for a sufficiently intense rf field and a
soft ferromagnet, e.g., permalloy.22 The maximum dc spin
current source is u^**I***s
*

source& *t*u'\v0k*AkF
*2 sin2u/(4p), e.g., for a

precession cone angle of u56° the equivalent electrical spin
current (*e*/\)u^**I***s
*

source& *t*u is 0.1 nA per conducting channel
*AkF
*

2 /(4p). The total number of channels is a large number since the cross sections may be chosen very much larger than the Fermi wavelength thus ensuring that a large spin current may be drawn from the battery.

Ferromagnetic resonance dissipates energy proportional to
the damping parameter a of the magnetization dynamics.
The power loss *dE*/*dt*5a\v0

2*Ns*sin
2u is proportional to the

volume of the ferromagnet through the number of spins in
the ferromagnet in units of \ ,*Ns *. The power loss can be
significant even for a thin film ferromagnet, e.g., for a 10
monolayer thick Fe film with a;1023,sin2u;1022, and
v0;10

11 s21, the power loss per unit area is (1/*A*)*dE*/*dt*

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SPIN BATTERY OPERATED BY FERROMAGNETIC RESONANCE PHYSICAL REVIEW B **66**, 060404~R! ~2002!

;0.1 W/cm2. The temperature can be kept low by, e.g., im- mersing the sample in superfluid helium. The heat transfer is then approximately 8 W/cm2 K for small temperature gradi- ents and increases for larger temperature gradients,23 which appears sufficient for the present purpose.

Schmidt *et al.*4 realized that efficient spin injection into
semiconductors by Ohmic contacts is close to impossible
with transition-metal ferromagnets since virtually all of the
applied potential drops over the nonmagnetic part and is
wasted for spin injection. The present mechanism does not
rely on an applied bias and does not suffer from the conduc-
tance mismatch, because the smallness of the mixing con-
ductance for a ferromagnet-semiconductor interface is com-
pensated by the small spin current that is necessary to
saturate the spin accumulation.

Possible undesirable spin precession and energy genera-
tion in the normal-metal parts of the system is of no concern
for material combinations with different *g *factors, as, e.g., Fe
(*g*52.1) and GaAs (*g*520.4), or when the magnetic an-
isotropy modifies the resonance frequency with respect to
electrons in the normal metal.

The optimal material combinations for a battery depend on the planned usage. From Eq. ~1! it follows that the largest spin current can be achieved when the conductor is a normal metal, whereas any material combination appears suitable when the load is small, as long as the contact is Ohmic and the system is smaller than the spin-diffusion length.

Standard metals, such as Al and Cu, are good candidate
materials, since the spin-diffusion length is very long, *ls
*;1 mm at low temperatures, and remains quite long at room
temperature.8,24 Indirect indications of spin accumulation in
Cu can be deduced from the absence of any enhancement of
the Gilbert damping in FMR when in contact with thin fer-
romagnetic films.10,25

Semiconductors have the advantage of a larger ratio of
spin bias to Fermi energy. Let us first consider the case of
GaAs. The spin-flip relaxation time in GaAs can be very
long, t*s*510

27 s at *n*5531016 cm3 carrier density.26 These
favorable numbers are offset by the difficulty to form
Ohmic contacts to GaAs, however. Large Schottky barrier
exponentially suppress the interfaces mixing conductance
parameter k .

InAs has the advantage of a natural accumulation layer at
the surface that avoids Schottky barriers when covered by
high-density metals. However, the spin-orbit interaction in a
narrow gap semiconductor like InAs is substantial, which
reduces t*s *. In asymmetric confinement structures, the spin-
flip relaxation rate is governed by the Rashba type spin-orbit
interaction, which vanishes in symmetrical quantum wells.27

The remaining D’yakonov-Perel scattering rate is reduced in
narrow quasi-one-dimensional channels of width *d *due to
waveguide diffusion modes by a factor of (*Ls */*d*)

2, where
*Ls*5*vF */u*h*(*kF*)u is the spin-precession length in terms of the
spin-orbit coupling *Hso*5**h**(**k**)•**s**,

28 which makes InAs based systems an interesting material for a spin battery as well.

In Si, the spin-flip relaxation time is long, since spin-orbit interaction is weak. Furthermore, the possibility of heavy doping allows control of Schottky barriers. So, Si appears to be a good candidate for spin injection into semiconductors.

06040

The spin bias can be detected noninvasively *via *tunnel
junctions with an analyzing ferromagnet having a switchable
magnetization direction. A voltage difference of *p*Dm is de-
tected for parallel and antiparallel configurations of the ana-
lyzing magnetization with respect to the spin accumulation
in the normal metal, where *p*5(*G↑*2*G↓*)/(*G↑*1*G↓*) is the
relative polarization of the tunnel conductances of the con-
tact. The test magnetic layer need not be flipped. It is suffi-
cient to reverse the direction of the dc static magnetic field.
The spin current, however, can be measured *via *the drop of
spin bias over a known resistive element.

Spin pumping into the normal metal can also have conse-
quences for the nuclei *via *the hyperfine interaction between
electrons and nuclear spins.26 An initially unpolarized collec-
tion of nuclear spins can be oriented by a spin-polarized
electron current, which transfers angular momentum by spin-
flop scattering. A ferromagnetically ordered nuclear-spin sys-
tem can lead to an Overhauser29 field on the electron spin.
This effect does not affect the spin bias Dm, but induces an
equilibrium spin density in the normal metal **s**0 *via *the
nuclear magnetic field, and can be exploited in experiments
where the the total spin-density **s**1**s**0 is an important param-
eter. The electron-nuclear interaction can be included by
adding21,29

**I***s
*nuc5

\**s***n
Tn
*

~5!

to the electron spin dynamics so that **I***→***I***s*source2**I***s*back1**I***s*nuc ,
where **s***n *is the nonequilibrium nuclear spin accumulation
and *Tn *is the electron-nuclear relaxation time. The nuclear
spin dynamics is described by

*d***s***n
dt
*

52
**s***n
Tn*8

1
**s
**

*Te
*, ~6!

where *Tn*8<*Tn *is the nuclear-spin relaxation time and *Te *is
the nuclear-electron relaxation. In steady state, **s***n*5(*Tn*8/*Tn*)
3(*Tn */*Te*)**s**. In the experimentally relevant regime *Te
*

21

!t *i
*21 the electron-nuclear interaction ~5! has a negligible

effect on the nonequilibrium spin accumulation **s **and thus
Eq. ~4! remains unchanged. *Tn */*Te*58*I*(*I*11)e*FnN */
(9*kBTne*) for small polarizations, where e*F *is the Fermi en-
ergy of the electron gas, *kBT *is the thermal energy, *nN *is the
nuclear density, and *ne *is the one-spin electron density.

29

Using *N*5(3/2)*ne */e*F *(*N*5*ne */e*F *in two dimensions! and
Eq. ~4! the relative enhancement of the dc nuclear spin po-
larization is

**s***n*5*nN
Tn*8

*Tn
*

2

3
*I*~*I*11 !

Dm

*kBT
*, ~7!

for Dm!*kBT *. The nuclear-spin polarization increases with
the spin bias and by lowering the temperature. The hyperpo-
larized nuclei, in turn, produce an effective nuclear field that
polarizes the *equilibrium *properties of the electron gas **s**0. In
bulk GaAs, the nuclear magnetic field is *Bn*55.3 T when the
nuclei are fully spin polarized, which should occur at suffi-
ciently low temperatures.30

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Berger31 proposed to generate a dc voltage by the FMR,
which bears similarities with our proposal. But Berger’s
mechanism of spin injection, originating from the spin-flip
scattering in the ferromagnet as induced by spin waves ap-
pears to be different from ours. We propose to achieve spin
injection *via *the modulation of the interface scattering matrix
by the coherent precession of the magnetization, which al-
lows, for example, quantitative calculations for various ma-
terials.

In conclusion, we present the concept of a spin battery, which is a source of spin, just as a conventional battery is a

06040

source of charge, and estimate its performance for different material combinations.

We are grateful to I. Appeli, B. Heinrich, A. D. Kent, D. Monsma, and H. C. Siegmann for stimulating discussions. This work was supported in part by the DARPA award No. MDA 972-01-1-0024, the NEDO International Joint Re- search Grant Program ‘‘ Nanomagnetoelectronics,’’ FOM, the Schlumberger Foundation, and NSF Grant No. DMR 99-81283. G. E. W. B. acknowledges the hospitality of Dr. Y. Hirayama and his group at the NTT Basic Research Laboratories.

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