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The
Ferromagnetic Spin
Filter
A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH
for the
degree
of
Doctor of Natural Sciences
presented by
DANIEL OBERLI
Dipl. Phys. ETH
born February 15th, 1969
citizen of Willisau Stadt, LU (Switzerland)
accepted
on
the recommendation of
Prof. Dr. H. C.
Prof. Dr. D.
P.D. Dr. W.
Zurich,
October 1998
Siegmann,
Pescia,
Weber,
examiner
co-examiner
co-examiner
Contents
Abstract
3
Zusammenfassung
5
1
Introduction
7
2
Total Scattering Cross Section and Spin Motion
Energy Electrons Passing Through a Ferromagnet
3
15
2.1
Abstract
15
2.2
Introduction
15
2.3
Experimental Setup
17
2.4
Results and Discussion
19
2.5
Conclusion
25
The Electron
Analogue
to the
Faraday Rotation
27
3.1
Abstract
27
3.2
Introduction
27
3.3
Experimental Setup
31
3.4
Results and Discussion
32
3.5
Prospects
35
Spin
37
A The
B
of Low
Polarization of the Transmitted Electrons
The Contribution of
Spin Flip Scattering
41
References
43
Dank
47
Curriculum Vitae
49
Abstract
Transport properties of excited electrons in a metal are governed by electronscattering, which is spin-dependent in the case of a ferromagnetic
electron
transition metal. The
majority-
and
origin of this spin dependence is the imbalance between
minority-spin electrons in a ferromagnet. For a minority-spin
electron there
are more empty d-states available for scattering than for a
majority-spin electron, which results in a longer mean free path for majorityspin electrons as compared to minority-spin electrons.
In this work, experiments are presented in which the transmission of
spin-polarized free electron beam through a freestanding Au/Co/Au trilayer is investigated. The transmitted current depends on the relative orien¬
tation of the spin polarization with respect to the sample magnetization.
a
By measuring the transmitted current for both spin polarization parallel
and antiparallel to the sample magnetization, the transmission asymmetry is
determined. Transmission asymmetries of up to 80 percent
are
observed. The
important prerequisite for the observation of such large transmission
asymmetries is that the freestanding metal foil must have absolutely no holes.
most
Furthermore,
cross-section
can
we
show how the
spin-dependent part
of the total
scattering
be calculated from the transmission asymmetries. Remark¬
ably, the spin-dependent part of the total scattering cross-section, which is
governed by the scattering on the d-shell, is still large at energies as high as
16 eV above the Fermi level.
By additionally analyzing the spin polarization of the transmitted elec¬
beam, the contribution of spin productive scattering, such as Stoner
tron
excitations, is estimated.
The experiments show that the contribution of
Stoner excitations is below five percent, which proves that spin
productive
scattering is of minor importance in spin-dependent transmission.
In order to
in
completely
ferromagnetic solids,
We
are
able to show
spin polarization
describe the transport of
it is
important
experimentally
vector of the
sample magnetization, then
to
spin-polarized electrons
consider the spin motion
that if there is
a
as
well.
component of the
incoming electron beam perpendicular
to the
this component rotates into the direction of the
Abstract
4
sample magnetization and simultaneously also precesses around it. This is
completely analogous to the magneto-optic phenomena observed with light
passing through a ferromagnetic material.
The rotation is observed in the plane spanned by the spin polarization and
the sample magnetization and is caused by the spin-dependent absorption in
ferromagnetic film.
precession around the sample magnetization is the electron analog to
the Faraday rotation observed with linearly polarized light. It is caused by
the phase difference that develops between the majority- and minority-spin
wave function. This phase difference is a consequence of the spin-dependence
of the inner potential. The Faraday rotation observed with electrons is at
least two orders of magnitude larger as compared to the Faraday rotation
observed with light. It offers new prospects of studying magnetism in general.
the
The
Zusammenfassung
Die
Transporteigenschaften von angeregten Elektronen in einem Metall werElektron-Elektron-Streuung bestimmt, die in einem ferromagnetischen Ubergangsmetall aufgrund des Ungleichgewichts zwischen MajontatsWeil Minontatsspin- und Mmontatsspin-Elektronen spinabhangig wird
spm-Elektronen mehr leere d-Zustande haben, in die sie streuen konnen, als
Majontatsspm-Elektronen, ist die freie Weglange fur Majontatsspm-Elektronen grosser als fur Minoritatsspm-Elektronen
den durch
In dieser Arbeit werden
spinpolansierten
on eines
Au/Co/Au-Film
relativen
Expenmente vorgestellt,
in
denen die Transmissi¬
freien ElektronenstraMs durch
untersucht wird
einen
Der transmittierte Strom
Onentierung der Spmpolarisation
freistehenden
hangt
des ElektronenstraMs
zur
von
der
Magne-
tisierung der Probe ab
Durch Messung des transmittierten Stromes fur
Spmpolarisation parallel und antiparallel zur Probenmagnetisierung kann die
Transmissionsasymmetne bestimmt werden Dabei werden Transmissionsasymmetnen von bis zu 80 % gemessen Die wichtigste Voraussetzung fur
das Gehngen von Expenmenten dieser Art ist die Herstellung locherfreier
Metallfolien
aus den Transmissionsasymmetnen der spmabhangige
Wirkungsquerschmttes berechnet werden Es zeigt sich, dass
dieser immer noch gross ist fur Energien bis zu 16 eV oberhalb der Fermi
Energie Dies ist insofern bemerkenswert, als dass er hauptsacMich durch die
Streuung an der d-Schale bestimmt wird
Im weiteren kann
Teil des totalen
Spmpolarisation der transmittierten Elektronen ge¬
wird, kann der Beitrag an spin-produktiver Streuung, wie zum Bei-
Indem zusatzhch die
messen
Anregungen, abgeschatzt werden Die Expenmente zeigen, dass
Stoner Anregungen kleiner als funf Prozent ist Dies zeigt, dass
m Transrmssionsexpenmenten die spin-produktive Streuung von gennger Bedeutung ist
spiel
Stoner
der Anteil
an
Um den Transport von spinpolansierten Elektronen m ferromagnetischen
Festkorpern vollstandig zu beschreiben, ist es wichtig, zusatzhch die Bewegung des Spinpolansationsvektors zu berucksichtigen Wir konnen im Expe-
6
ZUSAMMENFASSUNG
zeigen, dass die Komponente des Spinpolarisationsvektors senkrecht
Magnetisierung der Probe in die Richtung der Magnetisierung hineinrotiert und gleichzeitig urn sie prazediert.
riment
zur
Die Rotation in die
Richtung der Magnetisierung ist eine unmittelbare
spinabhangigen Absorption im ferromagnetischen Film.
Die Prazession dagegen ist das Analogon zur Faradaydrehung von linearpolarisiertem Licht. Sie wird verursacht durch die Phasendifferenz zwischen
der Majoritatsspin- und der Minoritatsspin-Wellenfunktion, welche durch die
Spinabhangigkeit des inneren Potentials entsteht. Der Faradayeffekt mit
Elektronen ist mindestens zwei Grossenordnungen grosser als der Faraday¬
effekt mit Licht und scheint daher eine vielversprechende Methode zur Erforschung neuer magnetischer Eigenschaften zu sein.
Folge
der
Chapter
1
Introduction
During the last decade, magnetic nanostructures have become a hot topic in
physics. The quest for higher storage density in magnetic recording media
and for faster access to magnetically stored information triggered a lot of
basic research on the physical phenomena that are connected with magnetic
Another important prerequisite for the discovery of new
nanostructures.
magnetic properties of thin film structures was the improved technique to
high quality samples.
example for a discovery that involves both interesting basic physics
powerful applications is the giant magneto resistance (GMR). Here,
resistance in a metallic ferromagnet/non-magnet/ferromagnet trilayer
grow
One
and
the
depends upon the relative orientation of the magnetization of the two ferro¬
magnetic layers [1]. Because large effects in small fields are obtained yielding
a high signal to noise ratio, GMR is successfully used in applications such as
magnetic field sensors or read heads for hard disks [2]. Advanced
technology is possibly the most important technical impact of
Not even ten years after its discovery, hard disk equipped with GMR
sensitive
read head
GMR.
read heads
are
available
Besides these
on
the market.
applications,
a
lot of
interesting physics which
has been
extensively investigated in the past years, goes hand in hand with GMR. One
example is the oscillatory behavior of the GMR with increasing spacer layer
thickness, which is caused by the oscillatory interlayer exchange coupling.
The origin of these oscillations is the confinement of the electrons in the non¬
magnetic layer, in analogy to the particle in a box problem in fundamental
quantum mechanics
[3].
spin-dependent scattering which is the essential
According to the two-current model [4],
one spin component of the current is more strongly scattered than the other
component, if the two ferromagnetic layers are magnetized parallel to one
Another example is the
physical
process in GMR materials.
8
Introduction
another
In the antiparallel alignment, both spin components are equally
strongly scattered This results in a lower resistance for the parallel than for
the antiparallel alignment
The origin of the spm-dependent scattering m
transition metals
be understood
by considering the fact that the charge
predominantly scattered into unoccu¬
sp
pied d-states [5] This final state effect leads to a spm-dependent scattering in
ferromagnetic transition metals, because there are more empty minority-spin
can
namely the
carriers,
electrons,
are
d-states available for scattering than empty majority-spin d-states
Another magnetoresistance effect is observed if the non-magnetic metal
layer in the tnlayer is replaced by a thin insulating layer Then, the tun¬
neling resistance for current flow perpendicular to the tnlayer depends on
the relative orientation of the magnetization of the two
ferromagnetic layers
tunneling junctions have been well known for more
than twenty years, large tunneling magnetoresistance (TMR) effects have
been observed only recently [6]
The mam reason for the absence of large
effects in the past years was the poor sample quality
A Gedanken experiment may help to understand the physics in TMR A
single ferromagnetic electrode is separated along a fictitious cleavage plane
into two halves Let us now assume that the magnetization of the right elec¬
Although
trode
of
a
is
these magnetic
rotated to become
antiparallel
given spin orientation impinging
of the electrons of the opposite spin
thus experience
tance
is
higher
Electrons
the interface will
potential
see
the
the other side of the interface and
this
This
completely analogous to the TMR, where
an insulating layer
is inserted between
electrodes So, the different phase space available for
minority-spin electrons leads again to a spm-dependent trans¬
occurs
tunneling barrier
the two ferromagnetic
—
majority- and
on
that of the left electrode
spm-dependent scattenng It is obvious that the resis¬
antiparallel configuration than in the parallel one, l e
a
in
magnetoresistance
a
to
on
for
is
example
—
port
Moreover,
tion
size
over
it has been shown that the TMR
more
devices are, when
for
is independent of the junc¬
magnitude [7] That is why TMR
magnetic memory cells, promising candidates
than five orders of
applied as
achieving very high storage
densities
A further
advantage of this so called
non-volatility, eliminating
the need for time- and power-consuming refresh-cycles
Both GMR and TMR are transport phenomena of conduction electrons
at the Fermi level To investigate these transport phenomena at higher ener¬
gies, Monsma et al [8] built a spin-valve transistor, l e a silicon metal-base
transistor with a base region made out of a GMR (Co/Cu) multilayer They
magnetic random
access
found evidence of
This observation
a
memory
(MRAM)
is
the
GMR effect for hot electrons 1 eV above the Fermi level
clearly
shows that
spm-dependent transport
is
also impor-
Introduction
9
tant for electrons excited
some
electron volts above the Fermi level.
In electron
spectroscopy, where electrons are excited above the vacuum
level, spin-dependent scattering has also been discussed for a while. Prob¬
ably
the first evidence of
spin-dependent electron scattering of excited elec¬
level in ferromagnetic materials was found with
spin-polarized photoemission [9], where an enhanced spin polarization of the
photoyield from Ni was observed as compared to the bulk value. Other
trons above the
vacuum
involving spin-polarized low-energy electron diffraction [10]
electron-energy-loss spectroscopy [11] pointed towards a
spin-dependent scattering as well. In addition, the polarization enhancement
at low energies in spin-polarized secondary electron spectroscopy [12-14] is
understood in terms of spin-dependent scattering. More recent experimen¬
tal evidence of spin-dependent scattering comes from overlayer experiments
with spin-polarized photoemission [15-17]. There, unpolarized electrons from
a non-magnetic substrate are found to become polarized after traversing a
thin ferromagnetic film. It was found that minority-spin electrons have a
shorter mean free path than majority-spin electrons, which results in a finite
spin polarization of the electron beam after transmission through the ferro¬
magnetic layer. Furthermore, it has been shown [18] by means of spin- and
time-resolved two-photon photoemission that the lifetime of excited electrons
between the Fermi level and the vacuum level is spin-dependent.
A different approach than the overlayer experiments to investigate the
relevant scattering processes are experiments where the transmission of a
measurements
and spin-resolved
spin-polarized free electron beam
recently Lassailly
electron beam
current
respect
across
depends
to
the
et al.
on
[19]
across
metal foils is
investigated. Only
measured the attenuation of
a
ultrathin ferromagnetic Co layers.
spin-polarized
very
free
The transmitted
the relative orientation of the spin polarization with
sample magnetization.
Several attempts have been made to
theoretically describe the spinFor example, a theory of renormalized one-electron states has been developed where it is assumed that
the excited electron does not interact appreciably with the other electrons
of the metal [20] and therefore only interacts with its own hole state left
behind. However, the fact that the intrinsic negative spin polarization of
the states near the Fermi energy in Co was never found in photoemission
experiments [21,22] cannot be explained by this theory.
To explain the polarization enhancement at low energies in spin-polarized
secondary electron spectroscopy, Glazer and Tosatti proposed a spin-flip
excitation across the ferromagnetic Stoner gap [23]. In this process
called
the incoming minority-spin electron scatters into an
a Stoner excitation
empty minority-spin state, and a majority-spin electron is reemitted with a
dependent scattering
of excited electrons.
—
—
10
Introduction
small energy loss
this process would enhance the spin
total,
In
of the transmitted electron beam
detected
polarization
Such Stoner excitations have indeed been
spm-polarized scattering experiments [24,25] However, they do
polarization enhancement is a very general
electron spectroscopy Finally, experiments have been performed
m
not account for the fact that the
feature
in
in
this work which show that Stoner
least
in
spin-dependent
excitations are of minor
importance,
at
transmission
A different origin for the
emission
spin
experiments
dependence
tion to the total
spin-dependent transmission observed m photoproposed by Gokhale and Mills [26] They suggest a
is
of the elastic scattering
Whereas
cannot be excluded
scattering
small elastic contribu¬
a
from the experiments, these
calculations fail to describe why the spin polarization is enhanced in any
investigated material, over a wide energy range, and independent of whether
the samples are amorphous, polycrystallme or smglecrystalline
All the above mentioned experiments
are well understood by applying
proposed by Mott [5] for the conductivity of transition metals to
excited electrons In fact, by compiling many attenuation experiments on a
number of materials, Siegmann and Schonhense [27] found an empirical rule
the model
for the scattering of hot electrons in transition metals (see Figure 1
justifies the application of this model to excited electrons
This rule says that the
the d-bands
occupied
—
is
absorption
divided into
a
coefficient
(5
—
n)
available to
a
=
a0 +
(7^(5
spin-dependent
because the number of holes
a±
with
=
scattering
term o& that
spin state
-
that
tib/2, and ns
this is analogous
=
m
is
into
un¬
proportional
to
the d-orbitals
n)
proportionality is that the absorp¬
ferromagnetic transition metals,
for
different for the two spin directions
a0 +
<Td(5
—
(n
±
An))
the Bohr magneton number
to
which
for electrons well above
(T* denoting the absorption coefficient for majority-(minority-)
trons, An
note
is
a
one
The most interesting consequence of this
tion coefficient becomes
—
term that accounts for
states other than d-states cto, and
the number of holes
a
1),
GMR and
It
is
spin elec¬
interesting
to
TMR, where the different density
of states for empty majority- and minority-spin states also leads to
a
spin-
dependent scattering
To further investigate the spm-dependent transport of hot electrons,
experiments have been performed m this work to measure the spm-dependent
part of the total scattering
a
function of energy
This
cross-section of thm
is
achieved
by
ferromagnetic
Co films
measuring the attenuation of
as
a
Introduction
5
11
1—
1
-
.,
..„!
1
i—
>
Gd
4
\
*^-
"
3
-
-
•*-"<»
Ta
Fe
to
2
Ag
Cr
Co
-
/Ni^*"''
1
i
\^
Cu
Au
0
i
1
12
0
3
# d-holes per
Figure
1.1: A
spin
4
5
state
compilation
of attenuation experiments shows that for
absorption coefficient c depends linearly on the
transition metals the
number of holes in the d-shell.
spin-modulated free electron beam impinging on a freestanding Au/Co/Au
trilayer. In addition, by analyzing the spin polarization of the transmit¬
ted electron beam, the contribution of spin-flip scattering is estimated. This
part of my thesis on the spin-dependent transport properties of ferromagnetic
films is described in chapter 2 [28].
spin-polarized electrons in
spin motion as
well. Our experiments show that the magneto-optic phenomena observed
with light passing through a ferromagnetic material are also observed with
In order to
solids,
fully describe
the transport of
it turned out that it is
spin-polarized
electrons passing
important
through
a
to consider the
ferromagnetic
foil.
long that there is an analogy between the mathe¬
matical description of a polarized light beam and of a spin-polarized electron
beam [29]. In both cases a definite state of polarization can be fully described
It has been known since
by
a wave
function
l/>
in which
<f>i and fo
are a
=
Cl<£l
complete
+
C2<j>2
set of two
,
orthogonal
wave
functions. In
Introduction
12
the
electrons,
of
case
these two
correspond
functions
wave
to two
opposite
spin orientations, while in the case of photons, they may correspond to
right- and a left-circularly polarized wave
polarization
A different way to characterize the state of
electrons
to give the
is
One defines the
Two
polarization
orthogonal
states
(P
opposite direction
ization vector,
case
of
of
an
abstract
P
is
are
and
has to
one
electrons,
photons,
expectation values of the Pauh
P
a
is
a
matrices
photons
x, y,
=
or
z)
vector
then characterized
—P) However,
distinguish
direction
by polarization
between electrons and
in
space
physical
complete polarization analyzer
Pomcare
for
light
vectors with
for the interpretation of this
space,
suggested
onto
a
photons
whereas
vector of the 3-dimensional Pomcare
polarization
of
a,(i
a
in
the
case
representation
to map the
sphere
in
polar¬
In the
in
settings of
order to get
a
a
simple
intensity through the analyzer
analogy further, the mathematical description
of the well known Faraday rotation observed with linearly polarized light is
given first In a second step, this description is applied to a beam of totally
polarized electrons The general case of a partially polarized electron beam
with its spin polarization P0 in an arbitrary direction with respect to the
sample magnetization M is described in Appendix A
expression for the transmitted
In order to illustrate this
We
the
linearly polarized light beam which propagates along
plane of polarization lies in the xz-plane If it is a pure
light beam is described by the wave function
now
z-axis
state, this
consider
a
and whose
Vo
where fa
=
I
.
I
and
circularly polarized light
Let
us
further
assume
=
7^1
fa
=
Its
this
I
.
+
«
is
polarization
=
a
7i(i)
basis
vector
<">
describing right- and leftis
then
light beam passes through a ferromagnetic ma¬
along the propagation direction Because of the
terial with its magnetization
13
Introduction
spin orbit coupling, there will be
a different velocity for left- than for rightcircularly polarized light propagating along the magnetization direction. This
leads to phase difference e between the two components and thus to a new
wave function of the photons after leaving the ferromagnetic material:
Hence, its polarization
vector becomes:
(cose
\
(1.3)
sine
of the plane of
called Faraday rotation
corresponds to a rotation
polarization by an angle of e/2 around the magnetization direction.
If we now consider a totally polarized electron beam with its spin along
the x-axis, which passes through a ferromagnetic material magnetized along
the z-axis, then this electron beam is described mathematically in exactly
the same way as the linearly polarized light beam described above. The in¬
coming wave function is described as a coherent superposition V"o (eq. 1.1)
of a majority-spin wave function fa (spin parallel to the magnetization) and
a minority-spin wave function fa
(spin antiparallel to the magnetization).
This
Then,
—
if
a
phase
difference
develops
scribing the electron leaving
polarization
—
the
between the two, the
ferromagnet
is
r/i (eq.
vector of the transmitted electron is P
wave
1.2).
(eq. 1.3),
function de¬
Hence, the
which
corre¬
sponds to a precession of the polarization vector by an angle of e around the
sample magnetization.
In view of this analogy, it seems possible that the magneto-optic phenom¬
ena observed with polarized light should also be observed with spin-polarized
electrons.
In this work
we
optic" phenomena
choose
ments
present experiments which prove that these "magnetoare
indeed observed with spin-polarized electrons.
We
experimental geometry for the spin-dependent absorption experi¬
described above, where the spin polarization of the incoming electron
an
beam is
perpendicular
In this case,
we
to the
observe
a
sample magnetization.
spin polarization
rotation of the
into the direc¬
magnetization, which is caused by the spin-dependent absorption
in the ferromagnetic film.
Furthermore, we observe a precession of the spin polarization around the
sample magnetization, which is the electron analog to the Faraday rotation
observed with light. This precession is caused by the phase difference that
tion of the
14
Introduction
develops between the majority- and minority-spin
spin dependence of the inner potential.
This part of my thesis
spin-polarized
on
the
wave
function due to the
"magneto-optic" phenomena
chapter 3.
electrons is described in
observed with
Chapter
2
Scattering Cross Section
and Spin Motion of Low
Energy Electrons Passing
Through a Ferromagnet
Total
Abstract
2.1
spin asymmetry of the elastic transmission of electrons
can approach unity. The polycrystalline Co films
and saturated with the magnetization M in-plane. The
It is shown that the
through ferromagnetic films
few
are
nm
thick
contribution of spin-productive
spin
scattering events is below 5 %. If the electron
perpendicular to M, it rotates into the
at incidence is chosen to be
direction of M and also precesses around it.
2.2
Introduction
polarized electron beams to the study of magnetism took
spin-polarized electrons were obtained by photoemission from magnetic materials [30]. The most obvious way of looking
at photoemission of electrons theoretically is to assume that the fast photoelectron does not interact appreciably with the other electrons in the metal
so that the photoemission experiment often is thought of as measuring the
energy spectrum of its own hole state left behind. This theory of renormalized one-electron states has been discussed in the present context by
The
its
application
beginning
of
when the first
P.W. Anderson
[31],
However, it could
S. Doniach
never
explain
[32],
M. Gutzwiler
the fact that
no
[33],
and many others
[20].
negative spin polarization
is
Total Scattering Cross Section.
16
detected in
photoemission
from states
near
the Fermi energy Ef in Co
..
[21,22].
This and many other features observed in emission of low energy electrons
from transition metals
the excited electron
are now
on
the
understood
partially
by considering
the
scattering of
filled d-states of all the atoms
encoun¬
through the transition metal [34]. To study this important
phenomenon more thoroughly, we have measured the total scattering cross
tered in transport
section
as a
function of electron energy. In contrast to
numerous
earlier inves¬
large transmission asymmetries A of up
to 80% with an electron beam passing through a thin ferromagnet depending
Fur¬
on whether its spin is parallel or antiparallel to the magnetization M.
thermore, when the spin polarization vector Pa of the incident electron beam
is chosen to be perpendicular to M, then it rotates into the direction of M
and simultaneously also precesses around M. There is a complete analogy to
the magneto-optic phenomena observed when a light beam passes through
ferromagnetic material. But, even when measured on the length scale of
the penetration depth, the magneto-"optic" effects observed with electron
beams are at least one order of magnitude larger as compared to the ones
observed with light beams. This arises because the electron beam couples
directly to the magnetization, while the coupling of the light beam must be
tigations [35],
mediated
a
by
we
the
have observed very
spin-orbit interaction. The observations presented here have
number of immediate
cross
important implications. For instance, the scattering
nonequilibrium magnetization dynamics which is
section governs the
presently at the forefront of fundamental research in magnetism [18,36-38].
Furthermore, experiments of the type described here might help to improve
the performance of spin filters, spin transistors and spin tunneling, and may
also lead to magnetic imaging in transmission electron microscopy.
Experimental Setup
2.3
17
Experimental Setup
2.3
experiment is sketched in Figure 2.1. We have prepared a spin-modulated
a GaAs-type photocathode. By switching from right- to
left-circularly polarized light for excitation of the source, we can invert the
vector PQ of the spin polarization. By applying a combination of electric and
magnetic fields to the electron beam, we can also rotate P0 into any desired
The
electron beam with
direction in space. We
can produce an unpolarized electron beam as well by
applying linearly polarized light.
The spin-polarized electron beam impinges along the surface normal onto
a trilayer consisting of a supporting Au film 20 nm thick, a ferromagnetic
Co layer of varying thickness ranging from 1-6 nm, and a capping Au layer
of 2 nm thickness to prevent corrosion. In this geometry spin-orbit coupling
cannot produce any spin dependence of the transmission.
Au
P„
Spin
j Co
t >
•
Au
n
modulated
electron
11=1:1=1:1=1s ] ll
;!;!;!:;:
source
M
Figure 2.1: Tie principle of the experiment is shown. It is consisting
of a spin-modulated electron source of the GaAs-type, a Au/Co/Au
trilayer in which the ferromagnetic polycrystalline hep Co Sim is mag¬
netized remanently in-plane, and a detection system in which the in¬
tensity I and degree of spin polarization P perpendicular to the axis
of the electron beam is measured for the electrons transmitted by the
trilayer.
a separate chamber on a substrate consisting of
supported by a Si wafer with a number of 0.5 mm
wide apertures. The Au layer of 20 nm thickness is deposited on top of the
nitrocellulose by evaporation of Au from a heated Mo crucible. On top of
this layer, polycrystalline films of hep Co are deposited by electron bombard¬
trilayer
The
a
ment of
quartz
a
is made in
film of nitrocellulose
a
99.998 % pure Co rod. Their thickness
microbalance)
protecting Au layer
ranges from 1-6
of 2
nm
nm.
(as measured by a calibrated
The Co films
thickness. The first set of
are capped with
hysteresis loops is
Total Scattering Cross Section
18
right after deposition by m-situ Kerr-magnetometry The m-plane
hysteresis loops are square and exhibit full magnetic remanence After the
magnetic tests are completed, the whole sample is let to air The nitrocellu¬
lose on the apertures is removed in a solution of pentyl acetate The sample
is then introduced through a load-lock system into the chamber with the
GaAs electron source where the measurements are done There, the sample
is first exposed to mild sputtering designed to get rid of the contaminants
acquired m the process of letting it to air and dissolving the nitrocellulose
Further sputtering through the apertures thins the supporting Au layer until
measured
electrons of
a
primary energy of
10~5
attenuation of
—
10~6
~
6 eV above
Ep
are
transmitted at
The final thickness of the supporting Au
an
layer
The Kerr hysteresis loops taken later show no
18 nm
loops obtained just after deposition of the samples
In the actual measurements, the Co films are remanently magnetized mplane by applying a positive or negative magnetic field pulse The electrons
emerging from the Au/Co/Au multilayer are energy analyzed by a retarding
is
estimated at
~
difference to the
field,
and
subsequently accelerated
to
components of the spin polarization
electron beam
via
Mott scattering
an
energy of 100 keV to determine the
vector
perpendicular
to the
axis
of the
Results
2.4
Discussion
19
Results and Discussion
2.4
In
and
Figure
2.2
we
show data observed with
7 eV energy and Pg
perpendicular
to
an
incident electron beam of about
the electron beam.
5
"
Figure
the
2.2:
degree
This
figure shows the energy
polarization P/Po
of relative
traversed the supporting Au
polarization
The
graph
delivered
by
the
shows intensity and
layer
distribution
curve
1(E)
and
after the electron beam has
alone.
Pq is the degree of spin
source.
polarization
as a
the Co film in order to illustrate what kind of
function of energy without
an electron beam actually
ferromagnet. In the energy distribution curve 1(E) one distin¬
guishes still an elastic peak at 7 eV, but secondary electrons have of course
also been produced in Au at lower energies. However, the spin polarization
of the elastic electrons is not altered on passing through the Au film. Yet the
secondaries having suffered collisions with valence electrons in Au have a low¬
ered polarization that decreases with decreasing energy due to the increasing
admixture of unpolarized electrons excited from the conduction bands of Au.
enters
the
Total Scattering Cross Section
20
2 3 shows data when
Figure
Au-capping
a
/+
of the emerging electron beam
antiparallel
to M
(I+
=
4
—
nm
with its
The elastic part of the
-
I~)/(I+
+
I~)
curves
valid for P0
is
where the direction of M
,
of the majority spins
asymmetry A
Co film of thickness y
One observes two different energy distribution
added
is
for
a
parallel and /" for P0
is defined by the direction
beam displays a huge spin
On the other
pure spin state
hand, the inelastic part of the electron spectrum exhibits lower A This is
partly due to the lower polarization of the inelastic electrons generated in the
supporting Au layer In the following we would like to focus on the elastic
part of the spectrum which we can separate by applying a retarding field
a
l+(sTTM)
v
r(slTM)
3
AA"
4>AAAAAAA'
*AWAAA
UJ
^
-l-Vsbfo-J
6
5
E-EF[eV]
Figure
Tie intensity distribution
2 3
s&own for
a
Co film of 4
added
I+(E)
I~(E)
for spin
is
nm
valid for spin
antiparallel
I+(E)
curves
thickness and its 2
parallel
to the
absolutely
no
holes
relative intensity transmitted
I~(E)
and
thick
are
Au-cappmg
magnetization M, and
to M
The most important condition for observing the
must have
nm
This
through
E of the incident electron beam
is
large A is that the tnlayer
Figure 2 4 where the
evident from
the
Au/Co/Au
The attenuation
is
shown
increases
vs
by
the energy
3 orders of
2.4
Results
when E increases from 6 eV above Ef to 16 eV. If there is the
magnitude
tiniest
21
Discussion
and
signal
the main part of the elastic
hole,
observed at the backside of
passed through the hole. We
by
trilayer
suspect that this is the reason why much smaller j4-values were reported in
Ref. [39] at higher electron energies. The steep increase of the attenuation
with increasing E is in reasonable agreement with the energy dependence of
is caused
the
the electron
10-!
path
[40].
in Au
i
i
i
i
i
5
6
7
8
9
:
10"6
free
mean
electrons that have
i
i
i
i
i
i
i
:
t
c
o
CO
en
10"7
E
w
c
CO
H
"lO"8
10-'
12
11
10
14
13
15
16
E-EfM
Figure
of the
We
2.4:
The attenuation of the elastic electrons after
trilayer
now
vs
consider the attenuation of the elastic electron beam in the Co
separately.
film of thickness y for each spin direction
current
ct
I0 the transmitted
depends
on
the
<f>
A
{exp(Acry)
=
w
angle (j>
current is /
between
Pq
and the smallest <r+ with
with
—
penetration
the energy above the Fermi energy EF.
—
=
-
y
Ioe~"y.
and
4>
l}/{exp(A<7j/) + 1}.
Act
=
=
M,
The
the
In
largest value
0. With Act
One obtains
—
\i~
With the incident
absorption coefficient
=
c
<r~ occurs
we
have
Total Scattering Cross Section
22
2 5 shows
Figure
number of data obtained with
a
interpret this further,
scattering
the
on
and that the
d-shell,
tional to the number of holes
d-shell
is
not
a
is
that shell
in
prion known for atoms
romagnetic metals
which
To
various samples
spin-dependent scattering is
strength of the scattering is propor¬
that all the
assume
we
a
The number of holes
metal
the
in
However, with the fer¬
knows the spin part of the saturation magnetization
one
the difference
in
m
the occupancy of the d-shell between majority- and
mmority-spin electrons known
the number of Bohr magnetons, na, per
as
atom
0.6
'
1
0.5
i
1
1
i
1
I
**
.
*
-
*
*
•
1 5nm
o
2nm
*
*
04
1
0.3
*
•|fiDxJ
.
|
*
m
8°»0D
1
*
M
D
ft
0.2
01
*
n
0
<
-
2 5nm
°H
'.
B
D
3
*
4nm
X
6nm
nm
n
•
•
-
-
-
0.0
i
i
'
10
15
E-EF[eV]
Figure
Difference
2 5
in
the
and minority-spm electrons
with
a
absorption
vs
coefficient Act for majority-
electron energy for
With the present electron energies several eV above
available for scattering
coefficient for
well
one
supported by
With
ris
=
six
samples
each
different Co thickness
hep Co,
1 7 Bohr
This
yields
unoccupied
a
state
Acr
in
=
TiBCtd
Ep,
all the d-holes
where (Td
the 3d-shell
in
Co
number of quite different experiments
the
density
magnetons
of atoms
Hence
one
is
N
=
8 6
is
the
This
are
absorption
approach
is
[41]
1028Atoms/m3
obtains for the total scattering
and
cross
2.4
Results
and
Discussion
23
section
Q
=
—
ln|-
NnBy n\I-
Figure 2.6 shows Q calculated from the average of Act. The order of mag¬
Q reflects the fact that the d-shell is comparatively little extended
in space. For the interpretation one must be aware that Q is the sum of all
scattering on the d-shell, elastic and inelastic.
nitude of
Figure
2.6:
The average total scattering
cross
section
Q for
one
hole
in the 3d-shell of Co is shown.
Gokhale and Mills
[26]
on the example of a single crystalline
scattering plus crystal diffraction and channeling
can lead to sizeable contributions to the spin-dependent transmission. How¬
ever, these contributions favor both majority-spin and minority-spin trans¬
mission depending on the energy. Furthermore, they are generally not as
have shown
Fe film that effects of elastic
large
observed here and also tend to increase
on increasing the electron
Furthermore, crystal diffraction must cancel out for truly
polycrystalline samples. We believe therefore that the main contribution to
as
energy above 10 eV.
the total scattering
cross
age of Act
on
all
scattering
on
the d-shell.
section
Q
in
polycrystalline samples
Figure
2.6 obtained from the
reflects
predominantly
aver¬
the inelastic
Total Scattering Cross Section
24
spin-selective scattering m ferromagnets in more depth,
question of what happens after the minority-spin electron
has scattered into a hole of the d-shell forming one of the 3<f+1 multiplet
To
analyze
the
must ask the
one
tmg
a
that the
excess
energy
is
majority-spin electron which however has lost
the Stoner gap S
made out of
with
argued [23]
It has been
states
a
In
total, this
minority spin
small energy loss 8
a
process called
a
dissipated by
Stoner excitation, would have
the primary electron beam
in
reemit-
at least the energy of
majority spin
a
Such Stoner excitations have been detected
ex¬
perimentally [24,25] We can test how important these excitations are in the
spm-polarized transmission by making use of the theorem that a polarizing
spin filter must be equal to an analyzing spin filter in the absence of spinproductive scattering events such as Stoner excitations [42] The change in
the majority-spin current is dl+
—a+l+dy—adl~, and in the minority-spin
current dl~
—a~I~dy, where a is a constant The fraction of minorityspin electrons that has undergone a spin flip in a Stoner excitation but is still
detected in the elastic channel because S is small, typically a fraction of an
eV, is given by r
a/(l a + cr+/a~) The polarization P of an unpolanzed
electron beam passing through the ferromagnet will be
=
=
=
—
P
while it
is
P
=
A for
r
=
A +
=
P*(A,r,y)
Experimentally, the
0
shows that the contribution of Stoner excitations
minor
importance
We
case
a
now
spin-dependent
in
consider the situation
(s antiparallel
to
M)
a
which Po
perpendicular
function
(s parallel
function with
*°-7!
is
wave
majority-spin
wave
below 5 % and thus of
transmission
the spin part of the incident electron
coherent superposition of
spin
m
comparison of P and A
r is
to
to M
In this
can
be described
M)
and
a
as
minority-
equal amplitudes
;:
dependent absorption, the amplitude of the two wave functions
A phase difference e develops
on passing the ferromagnet
This yields for the wave function ip of the electrons leaving the
Due to spin
becomes different
as
well
ferromagnet
V2
The spin
VTTa{
polarization
[
e-'"2
+
Vn
vector P of the transmitted
the expectation values of the Pauh matrices
J
electrons
(Note
that the
is
determined
i-axis is
by
parallel
Conclusion
2.5
to
25
Po, the j/-axis parallel
This
to the electron
beam,
and the z-axis
parallel
to
M.)
yields:
/
P
=
P0v^^cos(e)
PoVT^^si^e)
\
,
and corresponds to two types of motion of the spin polarization vector,
namely, a rotation by an angle of <j> into the direction of M and a precession
by an angle of e around M.
The rotation takes
place in the plane spanned by P and M. This rotation
absorption in the ferromagnetic film, as discussed above, where the
minority-spin wave function is more strongly attenuated than the majorityspin wave function. The angle <j> of the rotation is given by
is due to
tan<^=
The direct measurement of
film with A
=
0.3, <f> for
a
.
(2.1)
_
<j> confirms equation (2.1). For example, for
pure
spin
state is
»
a
Co
17°.
The precession around M is the electron
analogue to the Faraday rotation
linearly polarized light. It is a quantity that does not depend
on A but is caused by the phase difference that develops between majorityand minority-spin wave functions due to the spin dependence of the inner
potential. We found that the precession angle e is 16 ± 2° per 1 ran of Co
observed with
film thickness for
an
electron energy of 7 eV. It will be discussed in
more
detail elsewhere.
2.5
Conclusion
In conclusion
note that the very strong spin dependence of the transmis¬
polycrystalline hep Co opens up the possibility to construct
highly efficient spin filters, and to determine the Bohr magneton number nj
of thin films. Furthermore, the precession e around the direction of M is
we
sion observed in
unique because
it
wise inaccessible.
measures
the
spin dependence of
potential other¬
spin observed here is
the inner
The overall motion of the electron
for the understanding of ultrafast magnetization dynamics. The
and
e are large considering that, depending on
tj>
angles
energy, the elec¬
0.3 10~15 sec per nanometer film thickness within the
trons spend only
important
~
ferromagnet.
Leer
-
Vide
-
Empty
Chapter
3
The Electron
Faraday
Analogue
Rotation
Abstract
3.1
Since the classic experiment of Davisson and Germer
tion
to the
[43],
it has been
suggested
on
electron diffrac¬
might be of interest to carry out
experiments with an electron beam analogous to optical experiments on
polarization. It was anticipated that the electron spin might appear in such
experiments as the analogue of the light polarization vector. However, many
experiments of this kind with spin-polarized electrons have not yet been real¬
ized. Among them are experiments on the magneto-optical phenomena, such
Here we show that the Faraday rotation, which is
as the Faraday rotation.
the rotation of the light polarization vector during the transmission of polar¬
ized light through a magnetic substance, has its analogue in experiments
with spin-polarized electrons. However the strength of the Faraday rotation
observed with electrons is two orders of magnitude larger as compared to the
one observed with light. We believe that exploiting this effect will offer new
prospects of studying magnetism in general.
3.2
So
that it
Introduction
in all experiments where the interaction of spin-polarized electrons
ferromagnetic materials has been investigated the spin polarization vec¬
tor P0 of the incoming electron beam has been chosen parallel or antiparallel
to the sample magnetization M. In this way, the spin-dependent scattering in
ferromagnets has been investigated [11,24,25]. In particular, the spin filtering
properties of ferromagnets have been established [19,28]. It has been shown
far,
with
The Electron Analogue
28
that
electron beam with Pq
an
is denned
than
a
by
beam with P0
of empty states for
tering
results in
electrons
antiparallel
the direction of the
a
parallel
to the
to M
majority spins
to M.
-
-
is
Faraday Rotation
where the direction of M
strongly
more
attenuated
It is believed that the different number
majority- and minority-spin electrons
mean free path for majority-
different
available for scat¬
and
minority-spin
[27,44].
fully
In order to
describe the transmission of spin-polarized electrons
through ferromagnets, it turned out that it is important to consider the
motion of the spin polarization vector as well. The experiments show that
if there is
a
component of the spin polarization
vector
perpendicular to M,
simultaneously also
then this component rotates into the direction of M and
precesses around it.
This is
completely analogous to
plane of polarization
polarized light.
ism and to the rotation of the
observed with
respectively,
the
absorptive dichroFaraday effect,
in the
long that there is an analogy between the mathematical
description of a polarized light beam and of a spin-polarized electron beam
[29]. For simplicity, we consider in the following discussion a pure spin state,
which is given by tpu
ei£i + c2$2- Here, £i]2 represent a complete set of
It is known since
=
two orthonormal
be
right-
£1,2
wave
functions. In the
of
case
may be chosen to
light £1,2
left-circularly polarized waves, while in the case of electrons
functions with two opposite spin orientations. In the following
and
are wave
spin orientations are chosen
respectively. The corresponding wave
these two
parallel M)
and
minority-spin
wave
to be
parallel
functions
function
are
antiparallel to M,
majority-spin (s
and
called
(J* antiparallel M).
As
we
are
interested in the motion of the spin polarization vector perpendicular to M,
we
consider the
wave
i>
=
function with
leaving
the
-±={VTTA
V2
with the energy
wave
wave
function after
•
e-**/*
+
VT^A & e-'Elt'h)
majority-
to
For
a
minority-spin
and
spent by the electrons within the
with /+ and I~ the transmitted current for
M, respectively.
The
(3.1)
•
and E2 for the
t the time
(£1 +^2)/V^-
=
and A the transmission asymmetry. A is denned
I~)/(I+ + I~)
antiparallel
fc
eigenvalues E\
function, respectively,
ferromagnet,
•
equal amplitudes ipo
ferromagnet is then
by A
=
(I+
spin J*parallel
detailed discussion of A
see
—
and
reference
[28].
Since the energy difference AEex
=
E2
—
E1 between majority and minor¬
ity spins, the so-called exchange splitting, is
phase shift, increasing with time, between the
In real space this
increasing phase
shift
in
non-zero
two
spin
corresponds
a
ferromagnet,
a
states is introduced.
to
a
precession of the
3.2
Introduction
29
spin polarization
vector around M with the frequency u
AEex/h. The
u>t. As t
angle of precession is then e
d/v, with d the thickness of the
ferromagnetic film and v the group velocity of the electrons, the precession
angle is given by e
AEexd/hv, i.e. it depends linearly on the ferromagnetic
=
=
=
=
film thickness.
Such precession must
difference between two
For
example, the
normal
occur
orthogonal spin states,
whenever there is
even
energy difference may be due to
an
energy
in
nonmagnetic materials.
spin-orbit coupling in off-
scattering.
The above discussion can be generalized
spin-polarized electron beam (|Po| < 1) [29].
Figure
3.1: Schematic
polarization
the two
drawing
vector for
wave
\P0\
=
to the
case
of
an
incompletely
of the two types of rotation of the spin
1.
Due
functions in eq. 3.1 the
to the
different amptitudes for
spin polarization
vector rotates
by an angle of <j> towards the sample magnetization M. The different
phase factors in eq. 3.1, on the other hand, cause the spin polarization
vector to precess around M
by
an
angle of t.
The Electron Analogue
30
If
we
choose,
for
spin polarization
example, Po along
to the
Faraday Rotation
along the z-axis, the
leaving the ferromagnet
the z-axis and M
vector of the electron beam after
is then
/
P=
and
corresponds
to two
\
.
(3.2)
types of motion of the spin polarization vector,
an angle of e, discussed above, and a rotation
M, which takes place in the plane spanned by P and
M (see Fig. 3.1). The rotation into the direction of M is caused by the
spin filtering in the ferromagnet, which leads to the two different amplitudes
shown in eq. 3.1, and is analog to the ellipticity that is observed when light
namely,
the precession
P0Vr=~A^cos(e)
P0v^"=l?sin(e)
by
into the direction of
passes
through
a
medium with different
gonal polarization
<j>
arctan(^l/v/l
=
directions. The
-
A2).
absorption coefficients for two ortho¬
angle of rotation for a pure spin state is
3 3
Experimental Setup
3.3
31
Experimental Setup
In order to
experimentally verify
plete" spm-polarized
electron
the spin motion discussed
above,
scattering experiment has been
Au
a
"com¬
set up
*Au
Spin
modulated
electron
source
M
Energy
analyzer
Figure 3 2 Schematics of the experiment The experiment consists
of a spm-modulated electron source of the GaAs-type [45] with vari¬
able spin polarization direction, a free standing Au/Co/Au trilayer m
which the ferromagnetic polycrystalhne hep Co Sim is magnetized remanently m-plane, a retarding held energy analyser, and a detection
system
m
which the intensity and the
measured for the electrons transmitted
degree of spin polarization
by the trilayer
is
The expenment, which
detail elsewhere
[28]
3 2, is described in
is schematically shown in Fig
Briefly, a spin-modulated electron source produces a
transversely spin-polarized free electron beam having a spin polarization PQ
By applying a combination of electnc and magnetic fields to the electron
beam, P0 can also be rotated into any desired direction in space These
electrons impinge perpendicular onto a ferromagnetic polycrystalhne hep Co
layer of varying thickness sandwiched between Au layers, which serve both
The total thickness of the free standing
as support and protection layers
structure
is
around 25
nm
The transmitted electrons
spin
orbit
polarization
coupling
The Co film
are
energy
is remanently magnetized in-plane
analysed by a retarding field and their
detected
by Mott scattering It is emphasized that spmproduce any spin polarization in this normal geometry
only elastic electrons are leaving the trilayer, but also an
is
cannot
We note that not
energetically broad distribution of inelastically scattered electrons [28] In
the following, we focus on the elastic electrons, which can be separated by
applying a retarding field
The Electron Analogue
32
Faraday Rotation
Results and Discussion
3.4
Figure
3.3 shows the
experimentally determined precession angle e for differ¬
a primary energy E
Ep of 8 eV. Within the data
relationship between e and thickness d is found as expected.
ent Co thicknesses for
scatter
The
to the
a
linear
slope
is 16 ±
—
2°/nm.
O)
a
w
Figure
3.3:
The precession
angle
e as a
function of the Co thickness,
measured with elastic electrons of energy
at
zero
thickness
measured with
was
a
(E
—
Ep)
=
8 eV. The
point
pure Au film of about 20
nm
thickness. A linear fit to the data
confirms
eq.
We
a
yields a slope of 16 ± 2°/nm. This
the linear thickness dependence of the precession angle e (see
3.3).
can
estimate
free electron
energy range.
free electron
e
on
the basis of the above expression for
behaviour,
Then, the
which is
group
in the
by assuming
investigated
J2E/me,
with me the
justified for electrons
velocity
is
simply
v
=
t
and E the energy of the primary electron beam measured
with respect to the inner potential of Co (~16 eV [46]). One obtains:
mass
Results
3.4
and
Discussion
33
Putting in a value of about 0.2 eV for the exchange splitting AEex of the Co
7° per 1 nm Co film thickness.
sp-bands at low energies [47], one yields t
A value that is of the same order of magnitude as the experimental one.
~
We would like to note that the precession around M
Larmor precession of the electron spin around
of view is justified by the fact that the
in
spins
on
each
a
ferromagnet
spin,
acts in
a
way
as
the so-called Weiss field
a
exchange
if there
[48].
can
magnetic
as
a
the
point
interaction between the
magnetic
were a
fact,
In
be viewed
field. Such
a
field
acting
calculation of the
magnetic field, which would be necessary to explain the observed precession
angle, yields some 1000 Tesla, the same order of magnitude as found for the
Weiss field within the classical molecular field theory [49].
Though there is a complete analogy to the Faraday rotation observed with
the strength of this effect with electrons is two orders of magnitude
larger as compared to the effect with light. While a precession on the order
of 10°/nm is found for electrons, only 0.1°/nm is found for photons [50]. This
difference in the strength of the "magneto-optical" phenomena arises because
the electron spin couples directly to the sample magnetization whereas the
coupling of the photons to the magnetization must be mediated by the spinlight,
orbit interaction.
i
i
i
i
i
'
40
i
5
<
30
-30
$
:
3
£
20
10
I
-
4
<t> [deg]
i
g(D
CO
10
e
18
20
i
[deg]
c>
n
'
i
i
i
20
22
24
.i-
i
26
E[eV]
angles e and <j> vs the energy E (measured with respect
potential of Co) are shown. The Co thickness of the
investigated trilayer is 2.4 nm.
Figure
3.4: The
to the inner
The Electron Analogue to
34
The energy
dependence
of both
values for both types of rotation
values
are
obtained at
and
t
are
<j>
the
is shown in
high energies (see Fig. 3.5).
free
Fig.
found at low energies,
While
large
vanishingly
small
3.4.
We note that for energies
E between 30 and 130 eV the transmitted current
tected because of the
Faraday Rotation
was
too small to be de¬
path
minimum in this energy range [51]. The
energy is due to the decreasing matrix element for
mean
drop
in <j> with increasing
spin-dependent scattering into the Co 3d-shell [28]. In other words, the spin
filtering of the ferromagnetic Co film is strongly reduced at higher energies,
resulting in equal amplitudes in eq. 3.1 and hence a vanishing rotation angle
</>. It is however possible that some exchange potential may still recover at
special electron energies, for instance at 50 and 750 eV, where holes in the
3p
or
2p shells
can
resonantly
be excited.
""—
1
"
I
I
•
-j*\
0
20
10
e
[deg]
<t> [deg]
30
2.
40
i
.
s^l/2
1
-
-
30
-e-
20
I
CO,
10
-
Q
.
0
.
100
2
200
,r>
300
E[eV]
Figure
3.5:
The
angles
e
The sohd line shows the
The decrease in e,
that
eq.
develops
3.1).
If
we
From the data
on
and <f> are shown over a larger
E~1/2 dependence ofeq. 3.3.
energy range.
the other
hand, is solely caused by the phase difference
majority- and minority-spin wave function (see
eq. 3.3, we see that e is proportional to AE^/y/E.
between the
go back to
we readily recognize that the increase of
only partly explain the observed reduction of e. Thus,
the denominator
can
there must be also
a
strong reduction of AE^ with increasing energy. This is in accordance with
calculations showing a clear decrease of AEex with increasing energy [52].
Prospects
3.5
35
The higher the energy,
impinging hot electrons
the weaker the
exchange
interaction between the
and the electrons below the Fermi energy.
Prospects
3.5
The
Faraday rotation with electrons offers new prospects of studying mag¬
general. For instance, it opens up the possibility to measure the
exchange splitting AEcx with great sensitivity for energies above the vacuum
level, an energy range which is inaccessible for other experimental methods
such as spin-resolved inverse photoemission spectroscopy [53].
If, on the other hand, the Weiss field and hence the precession frequency
w is known, information on the time t spent by the electrons within the
ferromagnet can be gained. Therefore, the Faraday precession might serve
netism in
as
of
an
"internal clock".
potential wells
leads to
a
or
It
ferromagnet
resonance
well known that the existence
potentials
in
a
multilayer
resonant behaviour of electrons at certain
consequence, electrons with
the
is, for instance,
barriers due to the different
as
an
appropriate
energy
energies [54].
spend a longer time
As
a
within
compared to electrons with energies not matching the
Thus, information on the lifetime of such resonances
by tuning the primary energy of the electron beam.
condition.
should be obtained
Furthermore, the experiment described here
may be viewed
as
an
inter¬
ference experiments with electrons for the detection of quantum mechanical
phases. So far, interference experiments have only be done by using two dif¬
ferent geometrical pathways for the electrons. In our experiment, the phase
is probed by the spin part of the wave function using only one geometrical
pathway.
Experiments
of the kind
presented
The most obvious modification
here
can
be done
be modified in different ways.
by changing from transmission
to reflection geometry, which is the electron analogue to the magneto-optical
Kerr effect. In view of the wealth of information gained by the magnetooptical effects with light, their electron analogues promise to become powerful
tools to study magnetism.
can
Leer
-
Vide
-
Empty
Appendix A
Spin
The
Polarization of the
Transmitted Electrons
In this
appendix it is shown how the spin polarization of an electron beam
changed upon transmission through a ferromagnetic spin filter in the case
where the spin polarization Pq of the incoming electron beam has an arbit¬
is
rary direction with respect to the
consider
magnetization
M of the spin filter.
We
flying along the y-axis with its spin polarization P0
along the z-axis, which impinges perpendicular onto a ferromagnetic sample
with its magnetization M in the xz-plane. The angle between M and Pa is
6. Note that the only geometrical parameter determining the physics is 8.
an
electron beam
Every other geometry
the coordinate
{i, y, z}.
a
be calculated from the
one given here by rotating
spin polarization is measured with respect to
following calculations, it is furthermore useful to introduce
can
system.
For the
The
second coordinate system
around the
j/-axis
so
An ensemble of electrons
be written
as
an
incoherent
ing a totally spin-polarized
unpolarized beam [29]
Ptotal
Let
us
start with
be written
{x',y,z'},
that the z'-axis is
a
can
which is
parallel
a
foPpol
(1
=
+
totally polarized
y,
z}
a density matrix, which can
density matrix ppoi describ¬
density matrix punpoi describing a
be described
superposition
beam and
given by rotating {x,
to M.
of
—
by
a
-PfOPunpol
electron beam. Its
wave
function
can
as
*-(0
To describe this
wave
function in the coordinate system
{x1, y, z'},
we
have
Appendix
38
to transform it
by
the
Wigner
matrix
/
V
Hence the
The
wave
[55]
sin(0/2)
cos(6>/2)
sin(6>/2) cos(0/2)
-
function reads
ferromagnetic spin
filter is described
by the following
(
VTTAe-"'2
0_
[
0
v/T3^e+,£/2
matrix
This spin filter matrix describes how the wave function is modified by the
ferromagnetic spin filter. There is a stronger damping, which is described
by the transmission asymmetry A, for the minority-spin wave function than
for the majority-spin wave function (see chapter 2). Furthermore a phase
difference e develops as well between the two wave functions (see chapter 3).
The
wave
function after the spin filter is then
.„
*
Because the spin
the transmitted
tl>'"
=
=^
/
,
=
polarization
wave
VTTAe-"'2 cos(0/2)
(-vT^eW2sm(0/2)
is measured in the coordinate system
function
0"
must be
Try
VTTAe-"'2cos2(fl/2)
iv/TTAe-'/2 sin(0)
The
density
matrix
The
density
matrix is then
describing
Let
+
-
V"'
y/l^Ae+K'2sin2(0/2)
|vT^4e+"/2 sin(fl)
can
De
( N2
\CIC2
calculated
=
\
)
by writing
ci^ \
\C2\2)-
proceed with an unpolarized electron beam.
{x1, y, z'} it is described by JV wave functions
us now
6
{x, y, z},
by i?_1
given by
Ppol
system
the state
transformed back
qt(j)+4(j)
In the coordinate
*=!,...,JV
39
Appendix
or
by the density
matrix
Pmn
N2~,ckCk-
—
With respect to the basis described above
1/10
p=2 V
The fc-th
wave
&
The
ing
density
function is modified
=
F&
matrix
by
VTTAe-cl
=
describing
this
0
1
the
ferromagnetic spin
(Jj
+
ensemble,
the transmitted electron beam of
an
VT^Ae-cl
i.e. the
filter
density
matrix describ¬
initially unpolarized beam,
P'n
P22
(l + A)/2
(1+
=
^E(Vl + ie-»cl)'(VPie«4)
1
=
=
k=l
•'*
P'21
(l + A)Pll
=
k=\
=
^E{VT^Ae-clY (VT+Ae^cl)
=
^K^^
JV
=
k=l
(l-A)if;j^ja
=
,_ini+A)
P~2l
0
(l-A)pB
=
0
(l-A)T
follows
(J)
given by
A)± j:\ttf
as
(l-A)/2,
is then
Appendix
40
Because the spin
the
density
measured
m
the coordinate system
transformed
as
follows
polarization
matrix has to be
is
Punpol
Now, the density
can
be written
=
R
{x, y, z},
p R
matrix of the electron beam with spin
polarization Po
as
Aotal
PoPpol
=
The three components of the spin
(1
+
—
-Po)Punpol
polarization
vector
are
then
easily
calcu¬
lated
_
X
Asm(6)
Ti(axP)
Trip)
*~
2
"
Tr(p)
In the
special
transmitted spin
A
cos(6>)
+
P0
1 +
of 8
=
0,
polarization
is
case
P
spin
l e
ferromagnetic
Another special
to M
+
P0 v/1
-
A2
cos(e) sai*(0)
AP0 cos(e)
polarization Pa parallel
to
M, the
\
0
0
=
A + P0
V
the
AP0 cos(6»)
cos2(6)
/
l e
VT^A2 cosje)) P0sm(2fl)
-
-P0y/1-A2sm(e)sm(9)
1 + AP0 cos(6)
Tr(CTzp)
Tr(p)
p
(l
\
1 +
Tr(avp)
p
+
_
1 +
AP0
J
spin filter enhances the spin
case is
6
The transmitted spin
P
=
7r/2,
spin
i e
polarization
is
polarization as expected
polarization Po perpendicular
then
-PaVl-A2sm(e)
=
Pov/r^42cos(e)
A result used
in
chapter
3
However,
note that
Po and M
thus Formula 3 2 reads
/
P0v/T^45cos(e)
Pov/l
-
A2
A
sin(e)
\
are
interchanged,
Appendix
B
The Contribution of
Spin Flip
Scattering
In
first experiment the transmission asymmetry
a
A
jp
/a
_
—
Ir + Ia
by measuring the transmitted current Ip^
incoming electron beam parallel(antiparallel)
is determined
for
Po
to the
of the
spin polarization
sample
mag¬
netization.
In a second experiment with an unpolarized primary electron beam, the
spin polarization P of the transmitted electrons is measured. If there were
no spin flip scattering at all the two experiments would yield the same results
due to the well known theorem that
polarizing spin filter [24].
r
—
rate
defined
—
one
as
an
analyzing spin
ratio of the
spin flip scattering rate and
polarization P
has to know how the spin
electrons depends
number of minority-spin electrons. After
changed according
where a± is the
and
a
is
a
equal to a
spin flip scattering
the total
scattering
of the transmitted
on r.
To calculate P let I+ be the number of
has
filter must be
To determine the contribution of
majority-spin
a
electrons and I~ the
slab of thickness dx their number
to
dl+
=
-a+l+dx-adl~
(B.l)
dr
=
-aTdx,
(B.2)
absorption
constant, which
coefficient for
majority-(minority-) spin
accounts for the net amount of
the minority-spin channel into the majority-spin channel.
electrons
scattering from
Appendix
42
The solution of equation B 2
is
/-(*)
where
for
Iq
is
I~(x)
Equation
the initial number of mmonty-spm electrons
B 3
is
1^
easily integrated
ct+/<t~
defined
yieldsr
I+(x)
I+(x)
electrons
ln((l
=
in
as
=
aa-I^eT"
Its solution
1
\
the
r
a/(l
=
—
-
+
I-(x)
I-(x)
1/
—
A)to/2)/ln((l
+
film
\adI~\/{\dI+\
+
a+a+ /a~)
Thus,
we
are
the spin
is
a/(a+/<T--l)
aA/{a+/<r--l)'
—
A)w/2)
and
w
the spin
The contribution of spin
\dl~\)
Hence,
Because
Iq
=
l +
l +
Using equation
the spin
polarization
integrated
fhp scattering
B 1 and B 2, this
of the transmitted
is
r(a+/a- + l)/((r + l)(g+/gAr(<,+/<r- + l)/((r + l)(«r+/«r-
1 +
spin
is
beam, Iq
primary electron
ferromagnetic
1 +
Now
(B 3)
xdx
a+/a
transmitted electron beam
W
attenuation
—
unpolarized
m an
polarization of the
is
With this solution
the initial number of majority-spin electrons
is
interested
-a+I+dx +
=
a+1 a
where
J0-e- *,
equation B 1 reads
dl+
where
=
by comparing the experimental values
flip scattering r can be calculated
-
-
1))
1))
of A and P the contribution of
Bibliography
[1]
M.N. Baibich et
[2]
S.S.P.
2710
[3]
[4]
[5]
Rev. Lett.
61,
Z.G. Li and D.J. Smith,
2472
(1988).
Appl. Phys.
Lett. 58,
(1991).
S.S.P.
2304
Parkin,
al, Phys.
Parkin,
N. More and K.P.
Roche, Phys.
Rev. Lett.
64,
(1990).
T. Valet and A.
See for
Fert, Phys. Rev.
48, 7099 (1993).
B
example: N.F. Mott and H. Jones, The Theory of the Properties
Alloys, Clarendon Press, Oxford (1936).
of Metals and
[6]
J.S. Moodera et
[7]
S.S.P.
[8]
D.J. Monsma et
[9]
A.
al., Phys.
Parkin, talk given
Bringer
[10]
R.J. Celotta et
[11]
D. Venus and J.
[12]
J.
Unguris
[13]
E.
Kisker,
Rev. Lett.
74,
5260
Rev. Lett. 42, 1705
al., Phys.
Rev. Lett.
Kirschner, Phys.
43,
728
W. Gudat and K.
(1995).
(1979).
(1979).
37, 2199 (1988).
Rev. B
al, Phys. Rev. Lett. 49, 72
et
(1995).
(1982).
Schroder, Solid
State Comm.
44,
(1982).
591
[14]
H.
[15]
D.P.
[16]
3273
at the Heraeus Seminar 1998.
al., Phys.
al, Phys.
et
74,
Rev. Lett.
Hopster
et
Pappas
M.
Getzlaff,
467
(1993).
al., Phys. Rev.
et
al, Phys.
Lett.
50, 70 (1983).
Rev. Lett.
J. Bansmann and G.
66,
504
(1991).
Schonhense,
Solid State Comm.
87,
44
Bibliography
[17]
E. Vescovo et al, Phys. Rev. B 52, 13497
[18]
M. Aeschlimann et al, Phys. Rev. Lett. 79, 5158
[19]
Y.
[20]
For
29
Lassailly
a
et
review
al, Phys.
L.
see:
Rev. B
50,
(1995).
13054
Kleinman, Comments
(1997).
(1994).
Solid State
Phys. 10,
(1981).
[21]
G. Busch et al.,
[22]
J.C. Grobli et al,
[23]
J. Glazer and E. Tosatti, Solid State Commun. 52, 905
[24]
J. Kirschner, D. Rebenstorfl and H. Ibach,
698
[25]
[26]
Phys.
Rev. Lett. 28, 611
(1972).
Physica (Amsterdam), 204B,
Phys.
359
(1995).
(1984).
Rev. Lett.
53,
(1984).
H. Hopster, R. Raue and R.
M.P. Gokhaie and D.L.
Clauberg, Phys.
Mills, Phys.
Rev. Lett.
Rev. Lett.
66,
53,
2251
695
(1984).
(1991),
and
references therein.
[27]
G. Schonhense and H.C.
[28]
Publication note:
Siegmann, Ann. Physik 2,
465
(1993).
This chapter has been published in Physical Re¬
Oberli, R. Burgermeister, S. Riesen, W. Weber,
and H.C. Siegmann, Phys. Rev. Lett. 81, 4228 (1998).
view Letters.
D.
[29]
See for example: H.A. Tolhoek, Review of Modern Physics 28,
277 (1956).
[30]
U.
[31]
P.W.
[32]
S.
Banninger
al, Phys.
et
Anderson,
Doniach,
Philos.
in
Rev. Lett.
Mag. 24,
Magnetism
D.C. Graham and J.J.
Rhyne,
203
and
25,
585
(1970).
(1971).
Magnetic Materials,
AIP Conf. Proc. No. 5,
(AIP,
edited
New
by
York,
1971) p.549.
[33]
M.C.
Gutzwiler,
in
Magnetism and Magnetic Materials, edited by
Rhyne, AIP Conf. Proc. No. 10, (AIP, New
D.C. Graham and J.J.
York, 1972)
[34]
H.C.
p. 1197.
Siegmann,
J. Electron
Spectrosc.
Relat. Phenom.
68,
505
(1994).
45
Bibliography
[35]
A.
Filipe
al.,
et
Phys.
Lett.
Rev.
80,
2425
(1998),
and
references therein.
J. Hohlfeld et
[37]
A. Scholl et
ai., Phys.
[36]
[38] Ganping
al., Phys.
Ju et
H.J. Drouhin et
[40]
O.
[41]
H.C.
[42]
Paul,
al,
57,
Rev. B
5146
(1997).
R700
(1998).
Appl. Phys. 79,
J.
(1997).
4734
Dissertation ETH Zurich No. 9210
(1996).
(1990).
M.
Siegmann, Selected Topics on Electron Physics, edited by
Campbell and H. Kleinpoppen (Plenum, New York, 1996).
J.
Kirschner,
Modern
Polarized Electrons
Physics
Vol. 106
C.J. Davisson and L.H. Germer,
[44]
H.-J.
[45]
D.T. Pierce and F.
[46]
B.W. Lee et
[47]
V.L.
Drouhin, Phys.
Phys.
Rev. B
Rev. B
17,
1510
30,
705
of
(1927).
13,
5484
(1976).
(1978).
J.F. Janak, A.R. Williams, Calculated
Metals, Pergamon (New York, 1978).
Weiss, J. Phys. Radium 6, 661
Tracts in
p. 60.
(1997).
14886
56,
Meier, Phys.
Rev.
Moruzzi,
Properties
P.
Rev. B
al, Phys.
Surfaces, Springer
at
(Springer-Verlag, Berlin, 1985),
[43]
[48]
79,
Rev. Lett.
al, Phys.
[39]
78, 4861
Rev. Lett.
(1907).
Electronic
The Weiss field is also called
molecular field.
[49]
a contribution to the precession angle by the dipole field of
ferromagnetic layer can be neglected. Angles of the order 0.01°/nm
even smaller are expected.
We note that
the
or
[50]
V.
J.
A. Maziewski,
5939
73,
(1993).
Appl. Phys.
Grolier,
J.
Ferre,
[51]
M.P. Seah and W.A.
[52]
R.
Feder,
[53]
B.
Gubanka,
[54]
J.E.
J.
Ortega
Phys.
C
Stefanowicz,
and D.
Dench, Surface and Interface Analysis 1,
14,
2049
M. Donath and F.
et
E.
al., Phys.
Renard,
2
(1979).
(1981).
Passek, Phys. Rev. B 54, 11153 (1996).
Rev. B
47,
1540
(1993).
46
[55]
Bibliography
example: R.P. Feynman, R.B. Leighton and M.L. Sands,
Feynman Lectures on Physics, Vol. Ill, chapter 6, (AddisonWesley, 1989).
See for
The
Dank
Grosser Dank
gebuhrt
Hans-Christoph Siegmann,
Prof Dr
der mich
in sei¬
Forschungsgruppe aufgenommen und mir die Durchfuhrung emer mterMit seiner Grundessanten und vielseitigen Doktorarbeit ermoghcht hat
emstellung, dass der Kernpunkt jedes physikalischen Problems auch in emfachen Worten beschrieben werden kann, war er mir em grosses Vorbild
ne
Darulo Pescia mochte ich danken fur die Freundlichkeit
Professor Dr
das Koreferat
zu
Seme
ubernehmen
Vorlesung
fur mich
war
Einstieg
ins
faszimerende Gebiet des Magnetismus
geht an PD Dr Wolfgang Weber, mit dem
spannende Messstunden verbrmgen durfte In unzahligen Diskus-
Em grosses Dankeschon
ich viele
sionen
konnte ich viel
von
ihm lernen
Auch mochte ich mich bedanken fur
die Ubernahme des Koreferats und fur das Korrekturlesen
on
Ich wunsche ihm viel
Erfolg
und Freude bei
seiner
memer
Dissertati¬
weiteren Karriere als
Physiker
Em
spezieller
Dank
Burgermeister, der mit mir Freud
Seine ruhige Art war oft die ideale
etwas ungeduldigen Wesen Ich wunsche ihm viel
geht
an
Dr
Rolf
und Leid wahrend der Doktorarbeit teilte
Erganzung zu memem eher
Gluck bei
Bankkarnere
seiner
Nachfolger Stefan Riesen mochte ich danken fur seme Hilfe
seine Begabung zur Losung mechamscher Probleme hat uns
Meinem
besondere
geholfen
Ich wunsche ihm weiterhin viel
Spass
gewesen
sungen, sondern
sohden Losungen
es uns
Obwohl
nur
er
der
Memung ist,
Problemtransfer
verblufft,
gebe,
die das Problem
viel
beim Forschen
Ohne die techrnsche Unterstutzung durch Kurt Brunner
unmoglich
Ins-
hat
dass
er
in eine
es
uns
Ecke
ware
diese Arbeit
kerne Problemlo-
immer
wieder mit
transfenerten,
wo
mcht mehr storte
Im weiteren mochte ich Prof
Moglichkeit,
bei ihm
sungen machen
zu
zusammen
Dr
Martm Aeschlimann danken fur die
mit Rolf
interessante, zeitaufgeloste Mes-
durfen
Auch den anderen
"Spinnern"
Prof
Dr
Martin
Landolt,
Peter Walser
und Michael Hunziker mochte ich danken fur die gute Zusammenarbeit und
48
das
Dank
angenehme Arbeitsklima.
An die
"BoUologen"
schon fur die nette
der
Zeit,
Gruppe Siegmann geht auch ein herzliches Dankeverbringen durfte.
die ich mit ihnen
Zu guter Letzt mochte ich mich bei Urs Maier und Andreas Vaterlaus
bedanken,
die mir immer wieder mit Rat und Tat
zur
Seite standen.
Curriculum Vitae
(Switzerland)
15 Feb 1969
born in Cham
1976
1982
Primary School
1989
High
1982
-
-
27 Jun 1989
1989
-
1994
in
Zug
School at the Kantonsschule
Zug
High School Diploma (Matura Typus B)
Student at the
faculty of mathematics and physics of the
Technology (ETH) in Zurich
Swiss Federal Institute of
1993
-
1994
H.C.
Dr.
Diploma student in the group of Prof.
Siegmann under guidance of Prof. Dr. F. Meier (f)
Diploma thesis: Polarization Resonances of Optically
Spin-Oriented Photoelectrons Emitted from Strained
Semiconductor Photocathodes
1 Nov 1994
since 1994
Physics diploma
of ETH Zurich
teaching assistant in the group of
Siegmann under guidance of Prof. Dr.
Ph.D. student and
Prof. Dr. H.C.
F. Meier
(f)
and PD Dr. W. Weber