* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download The Ferromagnetic Spin Filter - ETH E
Survey
Document related concepts
Electrical resistivity and conductivity wikipedia , lookup
Hydrogen atom wikipedia , lookup
Introduction to gauge theory wikipedia , lookup
Thomas Young (scientist) wikipedia , lookup
Bell's theorem wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Density of states wikipedia , lookup
Nuclear physics wikipedia , lookup
Electron mobility wikipedia , lookup
Condensed matter physics wikipedia , lookup
Spin (physics) wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Transcript
Diss. ETH Nr. 12933 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