Spin filtering in ferromagnetic bilayers
Henri-Jean Drouhin and Nicolas Rougemaille
Citation: Journal of Applied Physics 91, 9948 (2002); doi: 10.1063/1.1476087
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JOURNAL OF APPLIED PHYSICS
VOLUME 91, NUMBER 12
15 JUNE 2002
Spin filtering in ferromagnetic bilayers
Henri-Jean Drouhin and Nicolas Rougemaille
Laboratoire de Physique de la Matière Condensée (UMR 7643-CNRS), Ecole Polytechnique,
91128 Palaiseau Cedex, France
共Received 6 February 2002; accepted for publication 12 March 2002兲
A general analysis of spin-polarized electron transmission through ultrathin ferromagnetic bilayers
is presented. The system is analyzed in terms of density operator and a compact description of
spin-filtering and spin-precession effects is given. The Sherman function, which characterizes the
spin selectivity, is precisely defined. Application to spin detectors is discussed and, in particular, the
two important cases where the layer magnetizations are collinear or orthogonal are studied in detail.
© 2002 American Institute of Physics. 关DOI: 10.1063/1.1476087兴
I. INTRODUCTION
A new field of the solid-state technology, spintronics, is
presently emerging and electron devices based on the electron spin manipulation in solids may expend our capacities in
key domains.1 Spin-dependent transport experiments have
rapidly raised a broad interest because of the new physics
involved and of their promises of application. Studies on
magnetoresistance effects and related phenomena are nowadays an active field of research stimulated by industrial applications to high-density recording or magnetic sensors,2
spin-valve transistor structures have been proposed,3 highperformance magnetoresistive random access memories
共MRAMs兲 based on spin-dependent tunneling are fabricated4
and very attracting new memories, based on magnetization
reversal triggered by a spin current are emerging.5 Progress
in the fabrication technologies enable us to associate the
properties of ferromagnetic thin films with those of metals,
semiconductors, or insulators, leading to original electron
devices6 and a new class of convenient spin polarimeters,
making use of the solid-state technology is under
development.7 This article deals with ultrathin ferromagnetic
layers, described in terms of spin filters, analogous to optical
polarizers. However, the electrons offer more possibilities
than the photons because the electron spin polarization possibly has both longitudinal and transverse components. The
inelastic electron mean free path 共IMFP兲 in metals, which
characterizes the escape depth, plays a crucial role in these
experiments, as in all electron spectroscopies. Its analysis in
the low-energy regime 共below a few tens of eV兲 remained a
puzzling question for tens of years. In Refs. 8 –10, it was
shown that an accurate analytical description of the IMFP is
obtained in a model based on density-of-state effects, in the
range 5–50 eV above the Fermi level. This does not depend
on the detail of the d bands, but only on the numbers of s and
d electrons, and the various electron relaxation channels are
disentangled. These results constitute the starting point of the
present analysis. Hereafter, we will carefully describe spin
filtering and spin precession in the density-operator formalism and we will focus on the ferromagnetic bilayer structure,
which associates a ‘‘polarizer’’ and an ‘‘analyzer.’’ This spinvalve structure plays a central role in the study of new phe-
nomena and presents some unusual properties, which are
conceptually related to Stern and Gerlach experiments. In
particular, the parameters which characterize the spin selectivity, known as the Sherman functions in spin polarimetry,11
will be carefully defined. The link between the overall Sherman function of the bilayer structure and the polarization of
the transmitted beam will be clarified.12
II. SPIN PRECESSION AND SPIN FILTERS
Spin-dependent transport of low-energy electrons in a
ferromagnetic metal arises because majority- and minorityspin electrons have different relaxation channels due to different final densities of states in the d spin subbands. Now,
consider the one-dimensional experiment where an electron
beam with a longitudinal polarization 共propagating along the
z axis兲 crosses a ferromagnetic layer magnetized along z 共the
vectors are indicated by boldface characters兲. Referring to
the initial polarization as P 0 , the polarization P of the transmitted beam is P⫽( P 0 ⫹S)/(1⫹S P 0 ). In this expression,
S⫽(t ⫹⫹ ⫺t ⫹⫺ )/(t ⫹⫹ ⫹t ⫹⫺ ) is the asymmetry of the transmission coefficients. The electron transmission only depends
on the relative orientation of the incident spin and the
majority-spin direction in the ferromagnetic layer and
t ⫹⫹ (t ⫹⫺ ) is the transmission coefficient of an up-共down-兲
spin electron when the majority spins in the ferromagnetic
layer are parallel to the direction of the z quantization axis
共indicated by the first⫹index兲. Following the terminology
widely used in spin polarimetry, we refer to the S parameter
as the Sherman function.11 In the case where the incident
beam polarization 共along the z axis, the propagation direction兲 and the layer magnetization axis 关in the (x,y) plane, say
along the x axis兴 are orthogonal, the electron spin both undergoes spin filtering in the x direction, which tends to align
it in the x direction, and spin precession around the x axis, in
the exchange field of the ferromagnet, which results in the
emergence of a y component. This effect was indeed observed by Oberli and co-workers.13 In their experiment, a
longitudinally spin-polarized electron beam was injected
trough a free-standing ultrathin ferromagnetic film with an
in-plane magnetization. Because the primary beam is injected into the sp bands 共at a few eV above the metal Fermi
0021-8979/2002/91(12)/9948/4/$19.00
9948
© 2002 American Institute of Physics
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J. Appl. Phys., Vol. 91, No. 12, 15 June 2002
H.-J. Drouhin and N. Rougemaille
level兲, spin precession originates from the sp-band splitting,
whereas spin filtering is a consequence of the spin-dependent
inelastic scattering, i.e., of the d-band splitting. An analysis
of these effects was given by N. Rougemaille et al.14
A. Density-operator formalism
The interest to use a density-matrix description of the
spin filters was pointed out by H. C. Siegmann.15 More generally, the electron beam entering a ferromagnetic layer is
described by the density operator
共1兲
ˆ the Pauli
where 具S典 is the mean value of the electron spin, ␴
operator, and I the identity.16 The time evolution of the density operator in the spin filter is given by
D共 t 兲 ⫽U共 t 兲 D U共 t 兲 ⫹ ,
共2兲
where the evolution operator U(t) is related to the Hamiltonian H through the relation U(t)⫽exp⫺i(Ht/ប). 16 Consider a ferromagnetic layer where an orbital eigenstate with a
given wave vector corresponds to two different energies E ⫹
and E ⫺ depending whether its spin state is a majority- 共⫹兲 or
a minority- 共⫺兲 spin state. An electron injected at this wave
vector but with a spin state which is not a pure state will
undergo spin precession. As long as only precession is concerned, H is diagonal with E ⫹ and E ⫺ real eigenvalues; the
energy spin splitting is ⌬E⫽E ⫺ ⫺E ⫹ ⫽ប⍀ and, as the majority spin states should correspond to a lower energy, ⍀
should be positive. Electron absorption, i.e., spin filtering,
can be empirically taken into account by adding imaginary
energy contributions.17 Thus, we write
冉
E ⫹ ⫺i␧ ⫹
0
0
E ⫺ ⫺i␧ ⫺
冊
共3兲
.
Let us define T ⫹ (t)⫽exp⫺i(E⫹⫺i␧⫹)t/ប and T ⫺ (t)⫽exp
⫺i(E⫺⫺i␧⫺)t/ប. From Eq. 共2兲, we obtain
D共 t 兲 ⫽
冉
R⍀ ␶ 共 u兲 P⫹S
,
1⫹S•P
d ⫹⫹ 兩 T ⫹ 共 t 兲 兩 2
d ⫹⫺ T ⫹ 共 t 兲 T ⫺ 共 t 兲 *
* T ⫹共 t 兲 *T ⫺共 t 兲
d ⫹⫺
d ⫺⫺ 兩 T ⫺ 共 t 兲 兩 2
冊
,
共4兲
where d i j (i, j⫽⫾) are the matrix elements of D. Let us denote as D⬘ the density matrix of the beam leaving the ferromagnetic layer, then the number of transmitted electrons with
a majority or a minority spin is Tr(D⬘ 兩 ⫾ 典具 ⫾ 兩 ), the transmitted intensity is TrD⬘ , and the spin polarization component of
the emerging beam in the ␣ direction ( ␣ ⫽x,y,z), P ␣
⫽Tr(D⬘ ␴ˆ ␣ )/Tr D⬘ . Note that this statistical picture is conceptually different to the coherent quantum transmission for
pure spin states proposed in Ref. 13. The electrons cross the
layer in the time ␶ ⫽d/ v , where d is the layer thickness and
v , their group velocity. The relations 兩 T ⫹ ( ␶ ) 兩 2 ⫽exp
⫺(2␧⫹d/vប)⫽exp⫺(␴⫹ d) and 兩 T ⫺ ( ␶ ) 兩 2 ⫽exp⫺(2␧⫺ d/vប)
⫽exp⫺(␴⫺ d) connect the imaginary energy components ␧ ⫾
to the spin-dependent scattering cross sections ␴ ⫾ calculated
in Ref. 9. From the D⬘ expression, it is straightforward to
show that the polarization P⬘ of the emerging beam is14
共5兲
where S⫽S u, u being the unit vector parallel to the direction of the majority spins in the ferromagnetic layer and
R⍀ ␶ (u) is the matrix corresponding to the composition of a
clockwise spin rotation of the angle ⍀␶ around u 共spin precession兲 with the multiplication of the spin component in the
plane normal to u by the (1⫺S2 ) 1/2 homothety ratio 共spinfilter effect兲. It is also readily checked that the transmitted
intensity is
I 共 u兲 ⫽I 共 1⫹S•P兲 ,
1
1
ˆ,
D⫽ I⫹ 具 S典 • ␴
2
ប
H⫽
P⬘ ⫽
9949
共6兲
where I is the intensity which would be transmitted if the
layer were nonmagnetic (S⫽0) or equivalently if the primary beam were unpolarized (P⫽0).
B. Electron transmission through ferromagnetic
bilayers
Consider a ferromagnetic bilayer with an arbitrary magnetization of each layer. Since in the absence of quantum
interferences the transmitted current I through the multilayer
is the product of the transmitted current through each layer,
I 共 u,v兲 ⫽I 共 1⫹S1 •P0 兲共 1⫹S2 •P1 兲 .
共7兲
In this expression S1 ⫽S 1 u (S2 ⫽S 2 v) is the Sherman vector
of the first 共second兲 layer and Pi⫺1 (Pi ) is the polarization of
the beam entering 共emerging of兲 the ith layer. I is the current
which would be transmitted through the structure if the layers were not magnetized (S 1 ⫽S 2 ⫽0), which is no longer
equivalent to the transmitted current for an unpolarized primary beam because the first layer acts as a polarizer. Using
Eq. 共5兲, we obtain
I 共 u,v兲 ⫽I 关 1⫹S 1 S 2 u•v⫹S 1 P0 •u⫹S 2 R⍀ ␶ 共 u兲 P0 •v兴 . 共8兲
Equation 共8兲 can be rewritten in the form
I 共 u,v兲 ⫽I 0 共 u,v兲关 1⫹S共 u,v兲 •P0 兴 ,
共9兲
where I 0 (u,v)⫽I 关 1⫹S 1 S 2 u•v兴 . Analogously to Eq. 共6兲, Eq.
共9兲 defines the overall Sherman vector S(u,v) of the structure and I 0 (u,v) is the transmitted intensity when the primary beam is unpolarized, which depends on the 兵u,v其 configuration. This dependence is closely related to a
magnetoresistance effect. It can be easily shown that the last
term in Eq. 共8兲 verifies the relation
R⍀ ␶ 共 u兲 P0 •v⫽R⍀ ␶ 共 u兲 v•P0⬘ ,
共10兲
where P0⬘ is the mirror symmetric of P0 with respect to the
共u,v兲 plane, i.e., changing the sign of the polarization component along (u⫻v) when u and v are not collinear. When u
and v are collinear, the P0⬘ component in the plane perpendicular to u is arbitrary and we can take P0⬘ ⫽P0 ; this case is
obvious. In the first case, the relation is derived after developing P0 on the 共u,v,u⫻v兲 basis 共which, in general, is nonorthogonal兲. The following properties have been used:
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9950
J. Appl. Phys., Vol. 91, No. 12, 15 June 2002
H.-J. Drouhin and N. Rougemaille
R⍀ ␶ (u) u⫽u; R⍀ ␶ (u) (u⫻v)⫽ 兵 关 R⍀ ␶ (u)u兴 ⫻ 关 R⍀ ␶ (u)v兴 其 ;
R⍀ ␶ (u)v•u⫽u•v. We define p(u,v) the polarization of the
transmitted beam when the primary beam is unpolarized and
first impinges onto the 2nd layer 共v axis兲, then crosses the
first layer 共u axis兲. From Eq. 共5兲, we have
p共 u,v兲共 1⫹S 1 S 2 u•v兲 ⫽S 1 u⫹S 2 R⍀ ␶ 共 u兲 v.
共11兲
Consequently, we find that Eq. 共8兲 writes
共12兲
where p⬘ (u,v) is obtained from p(u,v) by mirror symmetry,
in the same way P⬘0 is obtained from P0 . When u and v are
collinear, p⬘ (u,v)⫽p(u,v). This demonstrates that the overall Sherman vector S(u,v) is nothing but p⬘ (u,v). This vector is easily expressed in the set 兵 u, 关 v⫺(u•v)u兴 ,(u⫻v) 其 ,
which constitutes an orthogonal basis if u and v are not
collinear.
关 1⫹S 1 S 2 共 u•v兲兴 S共 u,v兲
共13兲
When using a bilayer as a spin polarimeter, different combinations of the transmitted intensities, which will be easily
measured, allow the determination of P0 . The following
equations are of particular interest because they are directly
related to the three polarization components:
I 共 u,v兲 ⫹I 共 u,v̄兲 ⫽2I 关 1⫹S 1 共 P0 •u兲兴 ,
共14兲
I 共 u,v兲 ⫹I 共 ū,v兲 ⫽2I 兵 1⫹S 2 共 u•v兲共 P0 •u兲
⫹S 2 冑1⫺S 21 cos ⍀ ␶ P0 • 关 v⫺u共 u•v兲兴 其
⫹S 2 冑
sin ⍀ ␶ P0 • 共 u⫻v兲兴 ,
I 共 u,v兲 ⫹I 共 ū,v兲 ⫹I 共 u,v̄兲 ⫹I 共 ū,v̄兲 ⫽4I,
共20兲
This relation allows us to determine (P0 •u) once S(u,v) is
known. It is easy to verify that
I 0 共 u,v兲 2 ⫺
冉
⌬I 共 u,v兲
共 P0 •u兲
冊
2
⫽I 0 共 u,v̄兲 2 ⫺
冉
⌬I 共 u,v̄兲
共 P0 •u兲
冊
2
⫽I 2 共 1⫺S 21 兲共 1⫺S 22 兲
共22兲
冉
共23兲
so that one obtains
冊
⌬I 共 u,v̄兲 2
I 0 共 u,v̄兲 2
2
.
2 ⫽S共 u,v 兲 1⫺
I 0 共 u,v兲
⌬I 共 u,v兲 2
This relation allows us to determine the two Sherman functions S(u,u) and S(u,ū) for parallel and antiparallel layer
magnetizations, only from intensity measurements, with no
need of independent characterization of the bilayer. In that
sense, the structure appears as ‘‘self-calibrated.’’
D. The bilayer with perpendicular magnetizations
This case, where u•v⫽0, has been discussed in Ref. 14.
There, Eqs. 共14兲–共17兲 yield
I 共 u,v兲 ⫹I 共 u,v̄兲 ⫽2I 关 1⫹S 1 共 P0 •u兲兴 ,
共24兲
共15兲
I 共 u,v兲 ⫹I 共 ū,v兲 ⫽2I 关 1⫹S 2 冑1⫺S 21 cos ⍀ ␶ 共 P0 •v兲兴 ,
共16兲
I 共 u,v兲 ⫹I 共 ū,v̄兲 ⫽2I 关 1⫹S 2 冑1⫺S 21 sin ⍀ ␶ P0 • 共 u⫻v兲兴 ,
共26兲
共17兲
I 共 u,v兲 ⫹I 共 ū,v兲 ⫹I 共 u,v̄兲 ⫹I 共 ū,v̄兲 ⫽4I.
I 共 u,v兲 ⫹I 共 ū,v̄兲 ⫽2I 关 1⫹S 1 S 2 共 u•v兲
1⫺S 21
I 共 u,v兲 ⫺I 共 ū,v̄兲
⫽I 共 P0 •u兲关 S 1 ⫹S 2 共 u•v兲兴
2
and consequently
1⫺
⫽ 共 S 1 ⫹S 2 u•v兲 u⫹S 2 冑1⫺S I2 cos ⍀ ␶ 关 v⫺ 共 v•u兲 u兴
⫹S 2 冑1⫺S 21 sin ⍀ ␶ 共 u⫻v兲 .
⌬I 共 u,v兲 ⫽
共19兲
S 1 ⫹S 2 共 u•v兲
⌬I 共 u,v兲
⫽
共 P •u兲 ⫽S共 u,v兲共 P0 •u兲 . 共21兲
I 0 共 u,v兲 1⫹S 1 S 2 共 u•v兲 0
I 共 u,v兲 ⫽I 0 共 u,v兲关 1⫹p共 u,v兲 •P0⬘ 兴
⫽I 0 共 u,v兲关 1⫹p⬘ 共 u,v兲 •P0 兴 ,
I 共 u,v兲 ⫹I 共 ū,v兲
⫽I 关 1⫹S 1 S 2 共 u•v兲兴 ⫽I 0 共 u,v兲 ,
2
where ū⫽⫺u(v̄⫽⫺v). Note that only two layers may allow
to determine the three components of the incident polarization. Other combinations, obtained from intensity differences
are also useful, such as the following:
I 共 u,v兲 ⫺I 共 ū,v̄兲 ⫽2I 兵 关 S 1 ⫹S 2 共 u•v兲兴共 P0 •u兲
⫹S 2 冑1⫺S 21 cos ⍀ ␶ P0 • 关 v⫺u共 u•v兲兴 其 .
共18兲
C. The bilayer with collinear magnetizations
In this case, we have u⫽⫾v. The bilayer in the collinear
geometry allows measuring the polarization component
along the magnetization axis, P0 •u. This case has been analyzed in detail in Ref. 9. The simplest way to perform a
polarization measurement makes use of Eqs. 共16兲 and 共18兲,
which write
共25兲
共27兲
The structure allows to determine the three components of
the polarization, as if there were a third ‘‘virtual’’ layer magnetized along the (u⫻v) axis, provided the Sherman function is known. This additional calibration can be performed
either by starting with a primary beam of given polarization
or by measuring the polarization of the transmitted beam
when an unpolarized primary beam impinges on the flipped
structure 关see Eq. 共13兲兴. It has been shown in Ref. 14 that,
with a proper choice of materials and thickness, the spin
sensitivity of such a structure can be as high as 0.6 in the
three directions, thus competing with the best existing polarimeters.
III. CONCLUSION
The efficiency of ferromagnetic thin films as spin filters
was experimentally demonstrated in Ref. 7. There, a low-
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J. Appl. Phys., Vol. 91, No. 12, 15 June 2002
energy 共a few eV兲 spin-polarized electron beam emitted from
a GaAs photocathode under optical pumping18 was impinging onto a free-standing Au/Co/Au multilayer under ultrahigh vacuum. The transmitted current was energy analyzed
and Sherman functions as high as 0.6 were measured. The
case of ferromagnetic bilayers with collinear magnetizations
is reported in Refs. 19 and 20, in complete agreement with
the above analysis. Going a step further towards application
requires deposing the films on convenient substrates. The
possibility to grow spin filters on semiconductors was demonstrated in Ref. 21, in the case of Fe on n-type GaAs. There,
a spin-polarized electron beam was injected into a Pd/Fe/
GaAs structure and the current collected in the semiconductor was detected. The system appears analogous to a transistor where the emitter is spatially separated from the base/
collector part. A large spin asymmetry in the collected
current, corresponding to a Sherman function of about 0.6,
was measured when the ‘‘base’’ and the ‘‘collector’’ were
maintained at the same potential. The absolute current asymmetry ⌬I, which corresponds to the difference of the transmitted current for two opposite polarizations of the primary
beam obviously do not depend on the ‘‘base’’ access resistance. On the contrary, it was observed that the relative
asymmetry ⌬I/I 0 , where I 0 is the averaged collector current,
decreases when the access resistance increases. This is not
surprising, because the electrons which are not transmitted
through the ferromagnetic layer accumulate and polarize the
structure, resulting in a current flow with no memory of the
primary beam polarization, and emphasizing the importance
of the boundaries conditions, which lead from ballistic filtering to the spin diffusion regime where the transport is governed by a chemical-potential diffusion equation.22,23 The
electrons captured by the ferromagnetic layer transfer their
charge, but also their spin momentum, which may lead to
magnetization switching,5 although this may also result from
the torque exerted by the ballistically transmitted spin current on the magnetization.24,25 The complete microscopic description of the bilayer structure with noncollinear magnetizations in the diffusion regime may bring valuable
information, and these effects are presently under investigation. Finally, let us emphasize that we have seen evidence of
H.-J. Drouhin and N. Rougemaille
9951
the potential of ferromagnetic bilayers as three-dimensional
spin polarimeters and that a convenient and physical description has been given.
ACKNOWLEDGMENT
H.J.D. thanks the Délégation Générale pour l’Armement
for support.
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