Magnetic Heterostructures: Advances and Perspectives in Spinstructures and Spintransport (Springer Tracts in Modern Physics, 227)

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Springer Tracts in Modern Physics Volume 227 Managing Editor: G. H¨ohler, Karlsruhe Editors: A. Fujimori, Chiba J. K¨uhn, Karlsruhe Th. M¨uller, Karlsruhe F. Steiner, Ulm J. Tr¨umper, Garching C. Varma, California P. W¨olfle, Karlsruhe

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H. Zabel and S. D. Bader (Eds.)

Magnetic Heterostructures Advances and Perspectives in Spinstructures and Spintransport

123

Hartmut Zabel

Samuel D. Bader

Ruhr-Universit¨at Bochum Fakult¨at f¨ur Physik und Astronomie Institut f¨ur Experimentalphysik D 44780 Bochum, Germany [email protected]

Material Science Division Argonne National Laboratory 9700 South Cass Ave. Argonne, IL 60439, USA [email protected]

H. Zabel and S. D. Bader (Eds.), Magnetic Heterostructures, STMP 227 (Springer, Berlin Heidelberg 2008), DOI 10.1007/ 978-3-540-73462-8

Library of Congress Control Number: 2007936863 Physics and Astronomy Classification Scheme (PACS): Magnetic anis otropy 75.30.Gw Exchange bias 76.60.Es Magnetic tunnel junctions 85.75.Mm ISSN print edition: 0081-3869 ISSN electronic edition: 1615-0430 ISBN 978-3-540-73461-1 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2008 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and Integra using a Springer LATEX macro package Cover production: WMXDesign GmbH, Heidelberg Printed on acid-free paper

SPIN: 11687993

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Preface

Heterostructures consist of combinations of diﬀerent materials that are in contact through at least one interface. Heterostructuring can occur naturally, as in some phase-segregated systems, or artiﬁcially, as due to a layering process during growth. The excitement surrounding artiﬁcial heterostructures is that they can embrace new and unusual physical properties that do not otherwise exist in nature. They can introduce new periodicities and they can be used to create composite materials out of components whose properties tend to be mutually exclusive. This book is devoted to magnetic heterostructures. Magnetic heterostructures share the virtues of being both fascinating to basic researchers and potentially useful in many practical applications. Examples are ferromagnet/semiconductor, ferromagnet/superconductor, and ferromagnet/antiferromagnet. These combinations display unique physical properties that diﬀer from their individual building blocks. Interlayer exchange coupling, exchange bias (EB) eﬀect, proximity eﬀects, giant magneto-resistance (GMR), tunneling magneto-resistance (TMR), spin injection, and spin transport are examples of new physical phenomena that rely on combinations of various functional layers that include metals, semiconductors, and oxides. Such heterostructures are generated by stackwise deposition of layers of these materials and/or by laterally fabricating them via lithographic means. The history of magnetic ﬁlms research has been traced back over 150 years by Gr¨ unberg, in his article on GMR published in Physics Today [1]. Modern magnetic multilayer research started to emerge in the late 1970s and early 1980s at multiple research institutions. Notable early examples include multilayers for applications in magnetic recording [2] and as neutron spin ﬁlters [3], and to study elastic properties [4]. Exchange coupling in rare earth containing superlattice systems represented a turning point in tailoring fundamental properties and gaining new insights into magnetic coupling phenomenon. But the watershed event, in hindsight, was the exploration of exchange coupling in transition-metal superlattices by Gr¨ unberg, which led to the discovery of GMR and its practical applications.

VI

Preface

The chapters of this book are written 21 years after the discovery of exchange coupling in transition metal superlattices by Gr¨ unberg and coworkers [5], 50 years after the discovery of the exchange bias eﬀect by Meiklejohn and Bean [6], and 30 years after spin tunneling was ﬁrst observed by Julli`ere in a magnetic tunnel junction [7]. Therefore, it is fair to say that the ﬁeld of magnetic heterostructures has reached a certain stage of maturity. Nevertheless, magnetic heterostructures are still a timely topic. The reason for this is the much higher chemical purity, layer precision, and interface sharpness that can be reached today as compared to 20 or 30 years ago. Because of this increased interface deﬁnition, TMR values observed today are up to 400% at room temperature as compared to 14% at 4.2 K some 30 years ago [8], the oscillatory exchange coupling can be observed with monolayer precision, and the exchange bias can be “designed” to display a well-deﬁned exchange bias ﬁeld and coercivity. In addition, there are a number of new topics that have emerged in recent years, including (i) the spin injection from ferromagnetic metal electrodes into semiconductors, (ii) spin-transfer torque eﬀects leading to current-driven magnetization switching in nanostructures, and (iii) proximity eﬀects between superconducting and ferromagnetic ﬁlms. The literature on magnetic heterostructures is widely spread and highly specialized. This calls for a book that provides an overview on the basics and the state-of-the-art aspects of magnetic heterostructures. This book attempts to present a comprehensive overview of an exciting and fast-developing ﬁeld of research that has already resulted in numerous applications and that serves as a basis for future spintronic applications. Both young researchers entering this ﬁeld and those more experienced should beneﬁt from these overviews on the present status and future challenges of various aspects of magnetic heterostructures. Before starting to investigate the physical properties of magnetic heterostructure, there is always the issue of how to grow artiﬁcial heterostructures. The choice of the growth method depends on the material combination and other factors. In the early 1980s, the growth of metal superlattices with molecular beam epitaxy (MBE) was the most advanced method [9, 10]. Following the growth of semiconductor superlattices, Nb/Ta and rare earth superlattices were grown for the investigation of superconducting properties [11], hydrogen density modulations [12], and the exchange coupling of complex spin structures in rare-earth metals across interlayers [13, 14]. As the structural quality and chemical purity have steadily improved over the years, sputtering methods are now considered as competitive to MBE methods. This is in particular true for complex material combinations of metals and oxides or when alloy layers of complex stoichiometry are to be grown. Another important topic is the choice of a proper substrate before metal deposition, and the proper techniques to be chosen for in situ and ex situ characterization of the structural properties of the deposited layers. There is a tendency to go from the growth of “simple” elemental magnetic and non-magnetic transition metals to more complex oxides with rich functional properties. The growth

Preface

VII

of these oxide layers is particulary challenging. These and other topics are treated in the ﬁrst chapter by Hj¨orvarsson and Pentcheva. Magnetic systems always display an intrinsic magnetic anisotropy. How this anisotropy is aﬀected by ﬁlm thickness, surfaces, interfaces, and temperature is the topic of the second chapter by Lindner and Farle. This chapter focuses on the discussion of single-element ferromagnetic metallic epitaxial ﬁlms on single crystalline substrates. Even though the scope of the chapter is restricted, it shows how with a few epitaxial layers the magnetic anisotropy can artiﬁcially be controlled, adding to an improved understanding of the underlying physical mechanisms. After discussion of the main sources of the magnetic anisotropy energy, ferromagnetic resonance (FMR) is introduced as the main and most accurate source of information on the magnetic anisotropy of thin ﬁlms. This chapter concludes with a tutorial description of the magnetic anisotropy of three prototypical systems: Fe/MgO(001), Fe/GaAs(001), and Ni/Cu(001). Particularly important heterostructures are combinations of ferro- and antiferromagnetic materials. When cooled through the N´eel temperature in an external ﬁeld, Meijkeljohn and Bean [6] discovered in the early 1950s that the hysteresis of the ferromagnet is characteristically shifted, indicating an interaction with the antiferromagnet across the common interface. This exchange bias eﬀect has become very important in recent years for the fabrication of spin-valve devices. Therefore, much eﬀort has been spent to understand and control the EB eﬀect. In the third chapter, Radu and Zabel provide an introduction to the basic physical mechanism of the EB eﬀect. They focus on numerical calculations and analytical treatment of some basic models that are predicated on the Stoner–Wohlfarth model [15] and that describe the coherent magnetization reversal process. Subsequently, other models for the EB eﬀect are also discussed that consider various defects and domain walls in the antiferromagnet, and spin glass–type disorder at the ferromagnet/antiferromagnet interface. In the last part, these models are compared with recent experimental data. The phenomenon of exchange coupling is well known in the context of magnetic impurities in non-magnetic metal hosts and for explaining the rich spin order in rare-earth metals. But it was not until the discovery of interlayer exchange coupling that this eﬀect has been analyzed in such a systematic fashion. In the fourth chapter, by Heinrich, the phenomenological and the fundamental aspects of exchange coupling are described. Particular emphasis is given to the discussion of exchange coupling in Fe/Cr and Fe/Ag superlattices, which serve as model systems. In the last part of this chapter, aspects of the dynamic exchange coupling across spacer layers are discussed. These aspects, which were ﬁrst investigated by FMR techniques and which are now being investigated by time-resolved and element-speciﬁc x-ray magnetic circular dichroism techniques, have received much attention in recent years. The communication of spin across superconducting and ferromagnetic interfaces is an exquisite and most enlightening physical problem. When in

VIII

Preface

contact, Cooper pairs penetrate into the ferromagnetic layer while ferromagnetic spins diﬀuse into the superconducting ﬁlm. The mutual proximity eﬀect between feromagnetic and superconducting layers has evoked not only the idea of a re-entrant superconducting state when the thickness of the ferromagnetic layer is varied, known as the LOFF state [16], but also the generation of an odd-in-frequency spin triplet superconductivity at the interface that can penetrate deep into the ferromagnetic layer, and vice versa a ferromagnetic state induced in the superconducting layer when cooled below the superconducting transition temperature Tc . The latter eﬀect has been termed the “inverse proximity eﬀect.” These fundamental issues of the proximity eﬀect are described in the ﬁfth chapter by Efetov, Garifullin, Volkov, and Westerholt. They also review and discuss the most recent experimental investigations relevant to the theoretical predictions. For spin-polarized tunneling in magnetic heterostructures, we need a substrate and a ferromagnetic layer that is pinned via exchange bias, an oxide barrier, and a ferromagnetic counter electrode that can be readily rotated in an external magnetic ﬁeld. The preceding chapters provide the basic knowledge for these ingredients. The sixth chapter builds on this information and provides the latest up-to-date information on magnetic tunnel junctions, which are either very challenging to fabricate and/or provide the highest TMR eﬀects to date. At present the most important tunnel barriers are Al-O and MgO; as magnetic electrodes Fe, Co, and/or FeCoB are preferred, but recently also Heusler alloys are starting to move to center stage. Reiss and coauthors provide a brief discussion of the fundamental aspects of the TMR eﬀect. The main part is focused on applications of the TMR eﬀect in read heads of hard disk drives, in storage cells of magnetoresistive random access memory devices, in ﬁeld-programmable logic circuits, and in biochips. The success of TMR devices in modern spintronic products depends on the scalability of the junction. There is a fundamental problem, which could possibly be overcome by current-induced magnetization switching. Therefore, the last section of this chapter is devoted to a discussion of spin-transfer torque phenomena, which may lead to a ﬂip of the “free” ferromagnetic layer, thereby replacing the switching by external magnetic ﬁelds. Quantum size eﬀects will become crucial in semiconductor devices over the next 20 years with further decrease of structure size. In this limit the best solution for coping with the quantum limit may be the transport of spin rather than the transport of charge via electrons and holes. In the seventh chapter by Hofmann and Oestreich, the increasing attractiveness of utilizing electron and hole spin transfer for future semiconductor devices is described. Starting from the spin transistor as the prototype spintronic device, proposed 17 years ago by Datta and Das, the authors continue with a discussion of the principal problem of spin injection across interfaces. Resistive models versus tunneling across Schottky barriers are discussed as the basic mechanisms for spin injection.

Preface

IX

Clearly in the present book not all aspects of magnetic heterostructures can be covered because of size restrictions. Therefore, we refer to other reviews and topical monographs for further important topics that could not be covered in this book. We refer to some books that have recently been published and that cover topics related to those in the present book: Magnetic Nanostructures, edited Tagirov et al. [17], Ultrathin magnetic ﬁlms, Vol. I-IV, edited by Heinrich and Bland [18], and Advanced Magnetic Nanostructures, edited by Sellmyer and Skomski [19]. Finally, we thank our colleagues in the research community for their stimulating activities, which continue to heighten interest in this ever-fascinating topic. Simultaneously, we take responsibility for all mistakes and omissions that could have been corrected. Many omissions will undoubtedly blossom into topics for future books as the story of magnetic heterostructures has not entered anywhere near its ﬁnal chapter of development.

Bochum - Argonne, June 2007

Hartmut Zabel Samuel D. Bader

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Gr¨ unberg, P.: Physics Today, 31 May (2001) V Berghof, W.: IEEE Trans. Magn. 11, 1344 (1975) V Majkrzak C.F., Passell, L.: Acta Cryst. A41, 41 (1985) V Thaler, B.J., Ketterson, J.B., Hilliard, J.E.: Phys. Rev. Lett. 41, 336 (1978) V Gr¨ unberg, P., Schreiber, R., Pang, Y., Brodsky, M.B., Sowers, H.: Phys. Rev. Lett. 57, 2442 (1986) VI Meiklejohn W.H., Bean, C.P.: Phys. Rev. 102, 1413 (1956) VI, VII Julliere, M.: Phys. Lett. 54A, 225 (1975) VI Yuasa, S., Fukushima, A., Kubota, H., Suzuki, Y., Ando, K.: Appl. Phys. Lett. 89, 042505 (2006) VI Durbin, S.M., Cunningham, J.E., Mochel, J.E., Flynn, C.P.: J. Phys. F: Met. Phys. 11, L223 (1981) VI Kwo, J., Gyorgy, E.M., McWhan, D.B., Hong, M., DiSalvo, E.J., Vettier, C., Bower, J.E.: Phys. Rev. Lett. 55, 1402 (1985) VI Durbin, S.M., Cunningham, J.E., Flynn, C.P.: J. Phys. F: Met. Phys. 12 , L75 (1982) VI Miceli, P.F., Zabel, H., Cunningham, J.E.: Phys. Rev. Lett. 54, 917 (1985) VI Salamon, M.B., Sinha, S., Rhyne, J.J., Cunningham, J.E., Erwin, R.W., Borchers, J.A., Flynn, C.P.: Phys. Rev. Lett. 56, 259 (1986) VI Majkrzak, C.F., Cable, J.W., Kwo, J., Hong, M., McWhan, D.B., Yafet, Y., Waszczak, J.V., Grimm, H., Vettier, C.: Phys. Rev. Lett. 56, 2700 (1986) VI Stoner, E.C., Wohlfarth, E.P.: Nature 160, 650 (1947) VII Larkin A.I., Ovchinnikov, Yu.N.: Zh. Eksp. Teor. Fiz. 47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1965)]; Fulde, P., Ferrell, R.A.: Phys. Rev. 135, A550 (1964) VIII

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17. Tagirov, L., Aktas, B., Mikailov, F.: Magnetic Nanostructures. Springer, Berlin Heidelberg New York (2006) IX 18. Heinrich B., Bland, J.A.C.: Ultrathin Magnetic Films I–IV, Springer, Berlin Heidelberg New York (2001–2005) IX 19. Sellmyer D., Skomski R.: Advanced Magnetic Nanostructures. Springer, Berlin Heidelberg New York (2006) IX

Contents

1 Modern Growth Problems and Growth Techniques Bj¨ orgvin Hj¨ orvarsson and Rossitza Pentcheva . . . . . . . . . . . . . . . . . . . . . . . .

1

2 Magnetic Anisotropy of Heterostructures J¨ urgen Lindner and Michael Farle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3 Exchange Bias Eﬀect of Ferro-/Antiferromagnetic Heterostructures Florin Radu and Hartmut Zabel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4 Exchange Coupling in Magnetic Multilayers Bretislav Heinrich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5 Proximity Eﬀects in Ferromagnet/Superconductor Heterostructures Konstantin B. Efetov, Ilgiz A. Garifullin, Anatoly F. Volkov and Kurt Westerholt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 6 Magnetic Tunnel Junctions G¨ unter Reiss, Jan Schmalhorst, Andre Thomas, Andreas H¨ utten and Shinji Yuasa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 7 Ferromagnet/Semiconductor Heterostructures and Spininjection Martin R. Hofmann and Michael Oestreich . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

1 Modern Growth Problems and Growth Techniques Bj¨orgvin Hj¨ orvarsson1 and Rossitza Pentcheva2 1

2

Department of Physics, Uppsala University, Box 530, 75121 Uppsala, Sweden [email protected] Department of Earth and Environmental Sciences, Section Crystallography, University of Munich, Theresienstrasse 41, 80333 Munich, Germany [email protected]

Abstract. The growth and characterization of magnetic materials has progressed substantially during the last decades. In this chapter we give a brief overview of this vastly growing ﬁeld of research. We highlight some of the relevant growth techniques for diﬀerent materials classes but we do not intend to be complete with respect to technical details or materials systems. We also outline some of the concepts and theories of the growth of modern magnetic materials, emphasizing the role of ﬁrst principles calculations in providing microscopic understanding of the growth mechanisms. We discuss the growth of metallic and oxide single crystal ﬁlms, multilayers and superlattices and the inﬂuence of thickness, strain, crystallinity, structure and morphology on the resulting magnetic properties.

1.1 Growth and Characterization The growth and characterization of magnetic materials has progressed substantially during the last decades. In this chapter we give a brief overview of this vastly growing ﬁeld of research. We highlight some of the relevant growth techniques for diﬀerent materials classes but we do not intend to be complete with respect to technical details or materials systems. We also outline some of the concepts and theories of the growth of modern magnetic materials, emphasizing the role of ﬁrst principles calculations in providing microscopic understanding of the growth mechanisms. We discuss the growth of metallic and oxide single crystal ﬁlms, multilayers and superlattices and the inﬂuence of thickness, strain, crystallinity, structure and morphology on the resulting magnetic properties. 1.1.1 Concepts: The Thermodynamic Versus Kinetic Picture The ﬁrst concepts used to describe crystal growth were based on general thermodynamic considerations [1]. The resulting structures were assumed to

B. Hj¨ orvarsson and R. Pentcheva: Modern Growth Problems and Growth Techniques, STMP 227, 1–44 (2007) c Springer-Verlag Berlin Heidelberg 2007 DOI 10.1007/978-3-540-73462-8 1

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B. Hj¨ orvarsson and R. Pentcheva

be in thermal equilibrium and therefore determined by the minimum of the free energy. Depending on the balance between the surface energies of the substrate material, γS , the adsorbate layer, γA , and the interface energy, γI , three diﬀerent growth modes are distinguished (shown in Fig. 1.1): . Δγ = γS − γA − γI .

(1.1)

If Δγ > 0, wetting of the substrate by the deposited material is expected, resulting in layer-by-layer growth mode (Franck van der Merwe mode). In this case, a new layer starts to grow only after the ﬁrst one is completed. In the opposite case, Δγ < 0, the formation of three dimensional islands is likely to occur (Vollmer-Weber growth mode). An intermediate situation can appear when Δγ > 0 and the growing ﬁlm is strained. Initially the ﬁlm grows in a layer-by-layer mode, up to a critical thickness where the growth becomes island like. This mode is often referred to as Stransky-Krastanov growth mode. The transition from a layer-by-layer growth to island like growth can be viewed as governed by the strain state and the elastic properties of the growing material. Although useful, this classiﬁcation has its limitations, it does, for example, not include surface alloying. Furthermore in the homoeptiaxial case, i.e. when substrate and adlayer are of the same material, it predicts a layer-by-layer growth. This is not generally valid and diﬀerent growth modes are observed for homoepitaxial systems. For example, Ag forms three dimensional islands on strained Ag(111) [2, 3]. Furthermore, ferromagnetic materials (e.g. Co, Fe) have a higher surface energy than the noble metals (e.g. Cu, Ag). Accordning to this simpliﬁed view, Co and Fe should not grow in a layer-by-layer mode on Cu and Au substrates. However, Co deposited on Cu(001) grows up to twenty monolayers (ML) in a layer by layer mode [4]. These are just a few examples illustrating the possibilities and limitations in the thermodynamic description of growth of materials. The main reason for the failures is that the growth process is by deﬁnition a non − equilibrium process and the thermodynamic equilibrium condition is thereby not fulﬁlled. In this non-equilibrium kinetic process one or more steps can be rate limiting. By understanding the underlying processes, the growth procedures can be

γS

γA γI Frank-van der Merwe

Vollmer-Weber

Stranski-Krastanov

Fig. 1.1. The illustration on the left hand side describes the basis for the thermodynamic description of growth. The relative energy of the interface (I), the surface of the growing layer (A) and the substrate surface (S) is assumed to determine the resulting growth mode. Schematic illustration of the three basic types are also included in the ﬁgure

1 Modern Growth Problems and Growth Techniques

3

used to control the morphology and the crystallinity and thereby some of the emerging material properties. In an atomistic approach, the growth of a ﬁlm can be described as a result of a number of microscopic processes such as adsorption, diﬀusion and desorption of adatoms (cf. Fig. 1.2). Diﬀusion of atoms can take place on ﬂat regions of the substrate, along or across step edges or around island corners. Therefore, besides adatom diﬀusion also the adatom-adatom and adatomstep interaction determine island nucleation and growth. In the framework of transition state theory (TST) [5], surface diﬀusion is described by diﬀusion rates D which are determined by diﬀusion barriers, Ed , and prefactors, D0 . D = D0 e

Ed BT

−k

.

(1.2)

Time scales relevant for sample growth are of the order of seconds and minutes and the length scales of kinetically controlled structures and islands are of the order of 100 ˚ A and involve a large number of atoms (> 105 ). On the other hand the detailed quantum mechanical description of atomistic processes is currently restricted to relatively small system sizes, up to about 104 atoms. Typical time scales of e.g. ab initio molecular dynamics are of the order of picoseconds which limits its application to the determination of possible processes, probable paths, diﬀusion barriers and attempted jump rate (prefactors) of the adatoms. A phenomenological or statistical description of growth can for example be obtained by using nucleation theory [6] or kinetic Monte Carlo simulations. These methods are often based on empirical or semiempirical parameters and their predictive power is therefore limited. In nucleation theory, growth is described by rate equations, yielding the time evolution of the adatom and island density. When the desorption rate is negligible a simple relation between the saturation island density nx , deposition rate R, diﬀusion rate D and temperature is obtained [6]: nx ∝ (R/D)i/(i+2) .

(1.3)

deposition adsorption

diffusion via hopping

substitutional adsorption

Fig. 1.2. Atomistic picture of growth, including diﬀerent processes like deposition, adsorption, diﬀusion of adatoms on the terrace, incorporation into existing islands, as well as incorporation via exchange in the substrate layer

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Typically, the critical island size corresponds to i = 1, which implies that two adatoms form a stable conﬁguration. The linear dependence that follows between ln nx and 1/T is often used to extract the diﬀusion barrier from the island density at a constant deposition rate. It can also be used to determine the critical island size i from the deposition rate dependence of island density at a constant temperature. The graphical representation of ln nx (1/T ) is often referred to as Arrhenius plots. The rate equations express the time evolution of the average adatom and island density. Because it is a mean ﬁeld approach, immediate and constant adatom density in the vicinity the growing islands is assumed, while in reality the islands have “depletion zones” with lower adatom densities. For systems where medium-range interactions are important (e.g. on strained surfaces), island densities predicted from nucleation theory can diﬀer by as much as an order of magnitude when only short range interactions are considered. This was shown in a DFT-KMC-study [7] and emphasizes the need for including stress and strain in the theoretical considerations. For a review of the microscopic view on metal homoepitaxy, see [8]. Nucleation theory is restricted to adatoms forming islands on the surface and an exchange with the underlying material is not taken into account. However, exchange processes between adsorbate and substrate can signiﬁcantly inﬂuence the resulting heteroepitaxial growth modes. An attempt to include exchange mechanism, within nucleation theory, was obtained by introducing systems with a critical island size of zero, i = 0 (see e.g. [9]). Some further aspects of metal heteroepitaxy will be discussed in Sect. 1.2.1. When describing the growth of thin ﬁlms within the framework of statistical mechanics, the exact motion of an adatom is irrelevant. The motion is treated as a random process, while the probability as a functional of the energy of a particular conﬁguration is exact. The kinetic Monte Carlo approach provides a statistical description of the evolution with time, enabling realistic description of a non-equilibrium processes. Combining DFT results on diﬀusion barriers and chemical interactions on the atomic scale with a statistical description of the time evolution in a kinetic Monte Carlo simulation (ab initio kMC) is therefore a promising route to bridge the gap between the time and length scales of ﬁrst principle calculations and experimental observations [10, 11, 12] 1.1.2 Growth Techniques After this brief overview on the theoretical description of growth, we now consider some of the experimental aspects of the involved processes. The most commonly used deposition techniques for growing magnetic materials are: • molecular beam epitaxy (MBE); • magnetron sputtering • ion beam sputtering

1 Modern Growth Problems and Growth Techniques

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• pulsed laser deposition (PLD); • metal organic chemical vapor depostition (MOCVD); These deposition techniques have many similarities, but do also diﬀer substantially with respect to the underlying physical processes. In this section, the complementary aspects will be emphasized and general requirements for successful deposition of diﬀerent materials will be addressed. The crystal coherency and the chemical purity are the most important parameters describing the quality of as grown samples. Surface oxidation and intermixing at the substrate interface are issues of concern for single ﬁlms, while interface mixing and thickness variation become important when discussing multilayered structures. The chemical purity of the as grown structures depends strongly on the purity of the ambient. For this purpose, the vacuum conditions can be used as a qualiﬁer. Under ultra high vacuum (UHV) conditions, corresponding to 10−9 mbar or lower, the impinging rate of the residual gas is below 10−3 /s. Thus, with a growth rate of 1 monolayer (ML) per second, the impurity level of the samples originating from the vacuum environment can be below 10−3 (atomic ratio), which is comparable to a representative purity of commercially available deposition material. The residual gas in a tight UHV system is typically governed by H2 . The partial pressure of water (pH2 O ) is typically one or two orders of magnitude below that of H2 . pO2 is often below the detection limit of most residual gas analyzers (10−13 mbar) and can be ignored in this context. Although pH2 O is orders of magnitude below that of pH2 , the inﬂuence of H2 O can dominate the impurity levels. The sticking coeﬃcient of H2 O is close to unity while that of H2 is typically 1 at 300 K. Furthermore, the sticking coeﬃcient is strongly dependent on the chemical composition of the surface as well as the temperature and has therefore to be considered with extreme care. When using magnetron sputtering, the pressure during growth is typically in the 10−3 to 10−1 mbar range. Using UHV growth chambers and ultra pure gases, the partial pressure of impurities in the sputtering gas can be in the same range as in e.g. MBE systems. Impurity levels of bottled gases are at the ppm level at best which can result in adequate chemical purity, if the gas is not contaminated in the gas handling system. The composition of the low pressure ambient can inﬂuence the physical properties of the growing material. For example, the presence of water will inevitability cause H impurities, signiﬁcantly altering e.g. optical and elastic properties of oxide ﬁlms [13, 14]. Furthermore, when growing metallic layers, this will result in both H and O impurities in the ﬁlms. The partial pressure of active gases in the sample environment is therefore a good measure of the possibility to grow samples with high chemical purity. The noble gases are not to be considered in this context, due to their inertness with respect to chemical reactions. The use of all metal connections and thoroughly outgassing the gas lines as well as the deposition system (baking) can be a simple route to improve the

6

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sample quality. This approach reduces the contamination of the process gas, however, the purity of the commercially available gases can be insuﬃcient. The use of gas puriﬁers is therefore sometimes needed for obtaining the required purity levels. Getter materials and cold traps based on molecular sieves can be used for this purpose. The combination of all-metal bakeable gas lines and gas puriﬁers allows signiﬁcant reduction of impurity levels. Although the sputtering gases are chemically inert, the inclosure of the sputtering gases can cause signiﬁcant contributions to the chemical composition. Typically the sputtering gases (Ar, Kr, etc.) are found in voids created during growth [15]. This eﬀect can be profound in the context of high growth rates of polycrystalline materials but is almost always negligible in single crystal materials. The substrate temperature determines the sticking coeﬃcient, the surface mobility and the desorption rates, and is therefore one of the most important process parameters. To determine the actual growth temperature is experimentally challenging and the quoted numbers are typically rather inaccurate. This is of special importance with respect to the growth of heteroepitaxial materials. This is important especially in the case of heteroepitaxial growth. Interdiﬀusion and crystallization are competing processes which can result in a narrow temperature range available for the required growth processes. However, reproducible temperatures can easily be obtained in any growth system, which enables reproducibility of the growth while using the same setup. The concern is therefore the transfer of experimental procedures between growth systems, as temperature calibrations are often crude and are material and substrate dependent. Substrates Oxides and semicoductors are frequently used as substrates for growing magnetic ﬁlms, multilayers as well as superlattices. The beneﬁts from this choice are many. First of all, the availability and price. Single crystal MgO, Al2 O3 , SrTiO3 , Si, Ge, GaAs, etc., with diﬀerent orientations, are commercially available. The surface and bulk crystalline quality varies and the inﬂuence of exposure to ambient air diﬀers substantially. The variation is not limited to diﬀerent suppliers, large diﬀerences in substrate quality can be found from one and the same vendor. This is clearly seen in Fig. 1.3, which illustrates rocking curves around the MgO(002) peaks from two diﬀerent substrates. One of the substrates shows a well deﬁned peak (solid curve) consistent with a single crystal structure, while the second one exhibits number of peaks corresponding to crystallites which are only partially oriented. These substrates were obtained from the same vendor and represent two batches of the same material. The pre-treatment of substrates are both performed ex- and in-situ, depending on the purpose of the treatment. For example, ex-situ heat treatment of Al2 O3 (1500◦C for 1−5 hours, see Fig. 1.4) and SrTiO3 are commonly

1 Modern Growth Problems and Growth Techniques 20000

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Fig. 1.3. Rocking curve from a MgO(001) sample representing a good (solid) and a bad (dashed line) batch from the same supplier. The measurements were performed using Cu Kα radiation around the (002) peak of MgO [17]

used, increasing both the terrasse width and crystalline quality. Corresponding heat treatment of MgO destroys the surface completely, easily identiﬁed by an opaque appearance. In-situ treatment almost always involves extended annealing, removing water adsorbed at the surfaces. The required temperature and time depends on the adsorption energy of water on the surface, which can vary substantially between diﬀerent materials and crystallographic orientations [16]. High temperature annealing of semiconductor substrates can result in inward diffusion of the near surface oxides. Pre-sputtering is required to avoid this, but post annealing is required for obtaining smooth surface after sputter cleaning. Pre-sputtering is normally not utilized when working with oxide substrates.

Fig. 1.4. AFM pictures showing the development of a step pattern during the annealing in air of sapphire substrates for six hours at diﬀerent temperatures [18]

8

B. Hj¨ orvarsson and R. Pentcheva

Molecular Beam Epitaxy and Sputtering Molecular Beam Epitaxy (MBE) and sputtering techniques are commonly utilized for growing magnetic thin ﬁlms and superlattices. The main differences between these two techniques is the energy of the material ﬂux, reaching the substrate and the diﬀerent conditions of the evaporating material. In MBE growth, the material is heated to a temperature giving the desired vapor pressure, which can be either below or above the melting temperature of the material. For example, Mg has a high sublimation rate far below the melting temperature. Mg is therefore typically not melted during an evaporation process. This involves both limitations and possibilities, as in situ cleaning/outgassing of e.g. Mg becomes diﬃcult. Secondly, the evaporation temperature of the material deﬁnes the kinetic energy of the atoms reaching the substrate and later the growing ﬁlm. Thus, the energy of the impinging atoms are typically far below 0.2 eV in an evaporation process. In UHV based evaporation systems such as MBE, the mean free path is much larger than the system size. In magnetron based sputtering processes, the typical mean free path is of the order of centimeters. The ﬂux from the evaporation source can therefore be regarded as highly directional, while the ﬂux from magnetron sources is close to random at the substrate surface (covering 2π). This has profound implications on the possibilities of using masks to obtain patterned growth [19], for which e.g. MBE is much better suited as compared to magnetron sputtering. There are two main routes for evaporating materials, namely through direct heating as accomplished in eﬀusion cells (through radiation or direct heating from heating elements) and by direct bombardment using high energy electron beam (e-beam). Eﬀusion cells have much better stability with respect to the ﬂux of the evaporated material and is therefore the method of choice when well deﬁned layer thicknesses are required. The stability is critical when growing, for example multilayers and superlattices, where the layer thicknesses have to be extremely well deﬁned. The instability in the material ﬂux from an e-beam source originates from the dynamics of the melted region. Scanning the electron beam across the target material often increases the stability, but the resulting ﬂuctuations in the ﬂux are still much larger than obtained from eﬀusion cells. The limitations in the use of eﬀusion cells originate from the chemical and thermal stability of the crucibles which are in direct contact with the evaporating material. Typical crucible materials are Al2 O3 , BeO, C (pyrolytic graphite), Ta and W. For the growth of materials consisting of more than one element from a single source, the use of evaporation techniques can be inferior to that of sputtering. As the ﬂux of the evaporating material is determined by the vapor pressure of the elements, the growing ﬁlm is likely to have signiﬁcantly diﬀerent chemical composition, as compared to the composition of the source material. In a sputtering process, the surface composition changes initially but

1 Modern Growth Problems and Growth Techniques

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eventually compensates the diﬀerent cross sections of the elements, resulting in a close matching of the ﬁlm composition to that of the source (target material). Sputtering techniques can be viewed as complementary to MBE, not allowing any in situ treatment of the target material, but having wide ﬂexibility with respect to the kinetic energy of the material ﬂux. The kinetic energies relevant in the sputtering process must be divided into two classes, neutrals and charged particles. The energy of these can, in principle, be adjusted independently. The neutrals as well as the charged particles are directly aﬀected by collisions with the residual gas, while a bias of the sample only aﬀects the ratio of the kinetic energy of positive and negatively charged particles. Positive bias retards the positively charged sputtering gas and target material, while it accelerates the electrons and negatively charged atoms of the target material toward the growing ﬁlm. This technique has been used to alter the morphology as well as the crystallinity of thin ﬁlms and superlattices. The simplest and often used route to utilize the electric potential, is to keep the sample at a ﬂoating bias. An alternative mode of operation is reactive sputtering. By introducing reactive gases in the growth chamber, these will readily react with e.g. metals in an ionic or a neutral state. Reactive sputtering can be used to grow oxides, nitrides or any material where one of the components can be introduced in the gas phase feeding the sputtering process. The plasma chemistry can also be used for obtaining phases which can not be synthesized by regular chemistry under normal conditions. There are two main modes of sputtering, namely Radio Frequency sputtering (RF-sputtering) and Direct Current (DC-sputtering). Although the DC mode can be used for reactive sputtering, target poisoning poses a severe challenge, demanding high degree of control of the pressure and the electric potential at the target. RF sputtering is one possible route to remedy this, even allowing the use of an insulating target material. The corresponding MBE approach is denoted Oxygen Plasma Assisted Molecular Beam Epitaxy (OPAMBE), and is used for oxide growth as the name indicates. Although sputtering has a wide range of applications, there are some materials that cannot successfully be deposited using this technique. As an example, growth of high purity actinide and lanthanide ﬁlms is typically restricted to ultra high vacuum evaporation. The chemical purity of the purchased materials is speciﬁed for all elements but hydrogen, which is typically high. The processing required for obtaining good quality ﬁlms, involves therefore extended outgassing prior to deposition for reducing the hydrogen content. This processing is not compatible with standard sputtering techniques. The growth of the actinides and lanthanides is therefore typically restricted to UHV evaporation. Although great precaution is taken with respect to in-situ puriﬁcation, substantial amounts of hydrogen is inevitably present in rare earth ﬁlms, as will be discussed later.

10

B. Hj¨ orvarsson and R. Pentcheva

Other Techniques Both MOCVD and PLD are widely used for growing oxides and semiconductors. In the ﬁrst technique the chemistry in the reaction chamber can be tailored, while some restrictions apply with respect to the most reactive elements such as Sc, Y, Lu, Lr and the rare earth elements. The use of in-situ tools is restricted due to the high pressures in the reaction chamber. In the magnetic community MOCVD is mainly used for the growth of doped ferromagnetic semiconductors. The use of ion beam techniques for growth of magnetic samples for research purposes, has increased substantially. The main advantage of this technique is the versatility, allowing large number of target materials in the one and same setup. Although the use has increased, it is still not widely used. Pulsed-laser deposition (PLD) has been utilized for the growth of single ﬁlms and superlattices. This technique is often favored for the growth of complex oxides, allowing complete control of the ambient gas. Here, a laser beam is focussed on a target in an UHV chamber. Material ablated by the laser pulses is deposited on the substrate. This technique has two major advantages: First, the target has already the desired stoichiometry making the oxidation step superﬂuous and second, the amount of deposited material can be controlled/calibrated by the number of pulses, allowing high degree of precision of the thicknesses of the layers while growing thin ﬁlms and superlattices [20, 21]. A ﬁnite oxygen pressure is often required to obtain stoichiometric oxides. 1.1.3 Characterization Techniques In situ Characterization Both in-situ and ex-situ characterization tools are used for obtaining information on the composition and structure of the as-grown materials. These can be viewed, in many respects, as complementary. The big advantage of the in-situ techniques is the absence of the limitations invoked by the exposure to ambient, while the advantage of ex-situ characterization lies in the versatility. Some aspects of the structural quality of the growing materials can be determined in-situ, however, most of the relevant work is done ex-situ, when operating a production device for thin ﬁlms and heterostructures. The utilization of in-situ characterization is demanding and signiﬁcantly increases the complexity level of the operation. Therefore many production systems are only equipped with a limited amount of in-situ possibilities. Examples of common in-situ equipment are typical surface characterization tools such as Auger spectroscopy, Reﬂective High Energy Electron Diﬀraction (RHEED), Low Energy Electron Diﬀraction (LEED) as well as Scanning Probes such as Atomic Force Microscopy (AFM) and Scanning Tunneling Microscopy (STM). The most widely used probe for in-situ measurements is the electron. For example, probing the energy of the ejected Auger electrons gives information

1 Modern Growth Problems and Growth Techniques

11

about the near surface composition, diﬀraction from the surface yields the surface structure and electron energy loss spectra (EELS) even yields information on the oxidation state. When determining the surface structure, there are two basic conﬁgurations, the electrons arrive to the surface almost parallel to the surface normal (LEED) or close to parallel to the surface plane (RHEED). Consequently, these techniques probe diﬀerent directions and lengthscales, RHEED is highly sensitive with respect to steps and island growth while LEED yields information on the atomic distances and periodicity in the near surface region of the sample. An advantage of LEED with respect to other structural techniques such as e.g. XRD, is the higher sensitivity to the oxygen positions and the lower penetration depth making the method sensitive to the near surface structure. Thus, LEED is used to determine the crystal quality, in-plane lattice parameters, and the development of superstructures. By simulations of the LEED I − V curves, information on the atomic positions and interlayer distances of the outermost layers can be obtained. RHEED is mainly used to monitor the thickness of evolving ﬁlms and the growth quality. Oscillations in the RHEED intensity correspond to the formation of complete layers (closing of layers) and thus can be directly related to the thickness of the deposited ﬁlm. The use of STM and AFM allows the visualization of the real space surface morphology, thereby serving as a complementary tool to reciprocal techniques such as LEED and RHEED. Both techniques have been used to follow the diffusion of adatoms on surfaces as well as monitoring the surface quality of thin ﬁlms and heterostructures. Also diﬀusion parameters from island density measurements using Arrhenius plots are described in the literature. The combination of these techniques allows contact microscopy, enabling identiﬁcation of chemical elements with near to atomic resolution. Thus, these scanning probe techniques are exceedingly valuable tools on all growth systems. There are also techniques which have to be classiﬁed as in-situ techniques, although these are not used as a routine equipment for monitoring growth in conventional growth systems. One of the most important ones is the force ion microscopy (FIM) which allows the study of individual diﬀusion processes and determination of diﬀusion barriers. For a comprehensive review see Tsong [22]. Ex situ Characterization Techniques As mentioned above, some aspects of the structural quality of the growing materials is preferably determined in-situ, however, when operating a production device for thin ﬁlms and heterostructures most of the routine characterization is performed ex-situ. Two of the most important parameters describing the resulting ﬁlms are the chemical composition and the crystalline structure. The composition of thin ﬁlms is conveniently determined by Ion Beam Analysis techniques (IBA), such as Rutherford Back Scattering (RBS) and

12

B. Hj¨ orvarsson and R. Pentcheva

Nuclear Resonance Analysis (NRA). These techniques have been used to determine the composition of wide variety of materials such as metals [23, 24, 25], oxides [14], nitrides [26], carbides [27] and hydrides [28, 29, 30]. The signal in a RBS experiment scales as Z2 , thus, the sensitivity increases strongly with increasing atomic number. High energy scattering can be used for changing the scattering cross section, when a better sensitivity is required for low Z materials. The basic idea is to overcome the Coulomb barrier, entering a region where the cross section is highly varying with the energy of the impinging ions. The presence of resonances can even be used to obtain isotope selective depth proﬁling in materials. For example, the resonance between α and O16 can be used for depth proﬁling of oxygen [31] with a resolution and sensitivity which is far better than most other techniques. When the depth resolution is less important than the detection limit, Elastic Recoil Detection Analysis (ERDA) can be utilized [32]. These techniques can all be operated in an absolute mode, counting the number of ions hitting the sample. This in combination with the known scattering cross section allows absolute determination of the concentration with an accuracy only limited by the determined stopping power of the ions. Typical accuracies are below few atomic percent, while the precision can be much better. Most of the structural information is obtained by x-ray scattering using conventional laboratory sources. The combination of reﬂectivity and diﬀraction allows the probing of all relevant length scales, from the overall thickness of the ﬁlm to the the atomic distances. The x-ray scattering yields the crystal coherency of the material, the sharpness of the chemical modulations i n multilayers and superlattices as well as the thickness variation at all relevant lengthscales [33, 34, 35]. However, interdiﬀusion and thickness variation (see Fig. 1.5) in a superlattice can not be separated by solely performing specular scattering experiments. Only by combining oﬀ-specular and specular scattering the relative weight of the components can be separated. Simulations and ﬁtting routines for oﬀ-specular scattering are currently not generally available. The combination of x-ray scattering and Transmission Electron Microscopy (TEM) can be useful. For example, when investigating combinations of oxides and metals, the diﬀerence in the scattering cross section is often large enough to obtain near atomic resolution of the interface composition. This combined with large contrast in the x-ray scattering allows detailed comparison giving unique insight in the actual local and global variation in the chemical composition in samples. However, TEM is highly destructive technique. The sample preparation involves slicing and thinning of a cross sectional part of the sample. Consequently, the investigated samples can not be used for any other measurements. Examples from investigations using most of these techniques will be given in the following sections. Investigations of the structural quality is not restricted to the use of x-ray scattering. For example, the use of ion beam channeling can be highly rewarding. When the ions channel through a single crystal ﬁlm, the back scattering yield is small. If the ﬁlm and the substrate have a coherent interface, the yield

1 Modern Growth Problems and Growth Techniques

13

Fig. 1.5. Illustration of interdiﬀusion (left) and roughness (right) at interfaces. If it is possible to insert a surface which separates all the atoms of the two types, there is no interdiﬀusion

will remain small. On the other hand, if the interface is incoherent, there will be substantial back scattering from the region corresponding to the interface between the ﬁlm and the substrate. Ion beam scattering can thereby give highly relevant information about the interface quality as illustrated in the investigations of the initial growth of Cr on Fe [37]. Not only is the epitaxial relation between the substrate and the ﬁlm established, the thermal vibrations of the individual components can also be extracted. This has been demonstrated using high quality sputter deposited Fe/Cr(001) superlattices by R¨ uders et al. [23]. One important aspect of the use of nuclear scattering is the isotope selectivity. The cross section is unique for the isotope combination in the scattering process, allowing investigations of e.g. O in oxides by growing isotope layers of O16 and O18 . This region in the scattering cross section corresponds to scattering above the Coulomb barrier, at which the two particles can be viewed as a compound nucleus in the moment of scattering. For a comprehensive introduction to diﬀerent nuclear scattering techniques see for example [38] and [39]. The isotope selectivity is also prominent in neutron scattering experiments. Furthermore, the scattering cross section is strongly varying with the atomic number and the choice of isotopes, which makes neutron scattering extremely useful as a complement to x-ray scattering. Neutron reﬂectivity has been used to determine the composition variation in multilayers and superlattices as well as the magnetic proﬁles in thin ﬁlms, multilayers and superlattices. Neutron reﬂectivity is one of the few methods that allow the full determination of the magnetic structure in materials [40, 41]. Full determination of the magnetic order can be obtained using polarization analysis. This includes the determination of the magnetic ordering in layered magnets as well as stripes and islands. Magneto Optical Kerr Eﬀect (MOKE) is one of the most used ex- and insitu technique to determine changes in magnetization. For a review of the use of MOKE, see e.g. [42]. The MOKE is one of the most versatile approaches to

14

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probe the changes in magnetization with temperature, to measure anisotropies as well as the susceptibility. The absolute moment can not be determined by MOKE. Superconducting Quantum Interference Devices (SQUID) are therefore often used for calibrating MOKE results. The magnetic properties of the material can serve as a qualiﬁer with respect to the structural quality. For example, the ordering temperature of extremely thin layers is strongly depending on the thickness. Consequently, if the thickness is changing substantially at lengthscales equal or larger than the magnetic interaction, this would result in a ill deﬁned ordering temperature. In Fig. 1.6, the magnetization of 3.4 ML Fe on GaAs (001) is displayed. The results clearly support the presence of highly uniform layer thickness at all but extremely short length scales and thereby carries information about the uniformity of the grown ﬁlm.

Fig. 1.6. Remanent Kerr rotation (a) and magnetic susceptibility (b) versus temperature for 3.4 ML Fe on GaAs (001)–(2 × 6). Almost no tailing is observed in the magnetization and the susceptibility is narrow and well deﬁned. From [36]

1 Modern Growth Problems and Growth Techniques

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1.2 Growth of Metals The ﬁrst step towards fabricating a multilayer or superlattice stack is the growth of a thin ﬁlm. In this section we discuss the growth of transition metals on a noble metal substrate and of rare earths. The growth of transition metals on other transition metals as well as oxides is covered in Sect. 1.4. 1.2.1 Transition Metals on a Noble Metal Substrate Growth of thin magnetic ﬁlms on a nonmagnetic substrate provides the possibility to design materials that do not exist in the bulk phase: e.g. Co (hcp in bulk) grows fcc on a Cu substrate. Similarly, Fe (bcc in bulk) was inferred to adapt a fcc structure on Cu(001) up to a thickness of 10 ML, while the structure is transformed into a body centered tetragonal phase at thicknesses above 10 ML [43, 44, 45, 46]. Three (0–4 ML, 4–10 ML and > 10 ML) thickness regions of Fe on Cu(001) are identiﬁed exhibiting widely diﬀerent magnetic properties, closely connected with the structure of the ﬁlms. Recently the local atomic structure of the ﬁrst iron layers has been revisited and there is still a substantial controversy whether the structure is bct as determined using STM, LEED and DFT-calculations [47, 48, 49] or fcc [50]. The obtained structure is a result of an intricate balance between lattice mismatch, strain and bond strength between adsorbate and substrate, versus adsorbate-adsorbate. We will discuss some speciﬁc aspects of heteroepitaxial versus homoepitaxial growth using Co on Cu(001) as an example. Co and Cu are immiscible in the bulk and have only a small lattice mismatch (2%). Still, intermixing inﬂuences the obtained interface quality substantially. In the initial stages of metal homoepitaxy, nucleation theory [6] predicts the logarithm of the island density to decreases linearly with increasing temperature (cf. 1.3). Instead of a linear dependence, a complex N -shaped non-Arrhenius behavior of the island density (illustrated in Fig. 1.7) is obtained for Co/Cu(001) from an ab initio thermodynamics kinetic Monte Carlo study, using diﬀusion barriers from DFT [51, 52]. At low temperatures, the heteroepitaxial case resembles homoepitaxial growth with adatoms diﬀusing on the surface, forming nearly square islands (see STM image at 295 K in Fig. 1.7). At approximately 340 K the activation of atomic exchange leads to a minimum in ln nx and a subsequent increase of island density due to pinning at substitutionally adsorbed Co adatoms [53]. At higher temperatures, the exchange mechanism and diﬀusion of the substrate material on the surface results in a bimodal island size distribution with large Cu islands decorated with Co as well as a large number of small predominantly Co islands (see STM image at 415 K in Fig. 1.7). Island densities obtained from He-scattering experiments [51] conﬁrm the predicted N -shape of ln nx (1/T ) (cf. Fig. 1.7). This unusual behavior was also observed in STM-measurements [53, 54, 55]. Experimental results for

16

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T=415K

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1000/T [K ] Fig. 1.7. N -shaped non-Arrhenius behavior of island density in the initial growth of Co on Cu(001) obtained from DFT-kMC simulations (solid diamonds) and ion scattering experiments (solid circles) for F = 0.0045 ML/s [51]. Empty diamonds: theoretical results for F = 0.1 ML/s. Open circles: island densities derived from STM-images (shown above) for F = 0.0033 ML/s [53]. Inset: linear plot of island Cu density between 340 and 410 K for Co (nCo x ) (solid triangles) and Cu (nx ) (open triangles) islands. Experimental error bars comprise statistical and, for high and low temperatures, possible systematic errors

Fe/Cu(001) [9], Fe/Au(001) [56], Ni/Cu(001) [57], and Co on Ag(001) [58] imply, that the scenario described above could be relevant for a broader class of materials. Epitaxial growth of Co ﬁlms up to 20 ML has been reported in the literature. However, bilayer growth was observed for the ﬁrst two layers [4, 54], and the second layer starts to grow before the ﬁrst one is completed. At elevated temperatures a Cu capping layer is formed [4, 59]. To explain these experimental observations, DFT calculations were performed for diﬀerent conﬁgurations such as monolayers, bilayers and sandwich structures [60]. The corresponding formation energy for 1 ML of Co is shown in Fig. 1.8. The tendency towards

1 Modern Growth Problems and Growth Techniques

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Fig. 1.8. Formation energy of diﬀerent ferromagnetically ordered conﬁgurations for a total cobalt coverage of 1 ML as a function of the cobalt island thickness N . The structures consist of clean Cu(001) and a compact island with N Co layers (◦) or N Co-layers capped by copper. The area covered by the cobalt islands is N1 of the whole surface. Especially for the copper terminated systems the separation in higher than bilayer cobalt islands is unlikely because of a negligible energy gain

the growth of bilayer islands results from the aﬃnity of Co to maximize the number of Co-Co bonds versus Co-substrate bonds. This is also expressed in a strong relaxation (Δd/dCu = −13.4%) of the interlayer spacing between the two Co-layers. A Co double layer capped by 1 ML of Cu was found to be the thermodynamically most stable conﬁguration. Similar results were found for Co ﬁlms grown on Cu(111) [61]. A recent molecular dynamics simulation based on Tight Binding Second Moment Approximation (TBSMA) [62] identiﬁes an upward diﬀusion mechanism at island edges as the origin of bilayer growth [63]. The interaction between ultra thin transition metal ﬁlms and noble metal substrates is relatively weak. Consequently, the surface strain of such ﬁlms may be much larger that the one suggested from the bulk phases. For example in the extreme case of a free standing Co monolayer (missing interaction with the substrate) DFT-calculations predict an equilibrium lattice constant 12.2% smaller than the one for Cu, while the lattice constant of bulk fcc Co is only 2% smaller than Cu [60]. Related to this is the so called mesoscopic strain: based on ab initio calculations Stepanyuk et al. [64] found that the shape of the growing islands and the underlying substrate is strongly deformed by the inhomogeneous stress ﬁeld around Co islands.

1.2.2 Rare Earth The rare earth (RE) elements (actinides and lanthanides) are extremely reactive and therefore diﬃcult to purify and to maintain impurity free. These can be purchased in a reasonably pure form with respect to all elements but

18

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hydrogen. Typically, RE materials are reﬁned in situ, by extensive annealing and outgassing procedures to minimize the hydrogen content. This severely limits the possible deposition techniques and MBE appears to be the technique of choice for successful deposition. Here, we regard Y as representative for RE materials due to the similarities with respect to the outermost electron states, yielding similar chemical properties. The reactivity of the RE materials inﬂuences the sample design and most researchers use diﬀusion barriers to hinder e.g. oxygen transport from the substrate to the ﬁlm, forming RE oxide. For example, the growth of RE materials on Al2 O3 often involves a Nb layer serving simultaneously as a diﬀusion barrier and a seeding layer, as described by Kwo et al. [65]. This aspect will be discussed further when addressing the growth of RE superlattices. A comprehensive review of the growth of Nb (110) on Al2 O3 is found in [67]. The choice of capping layer is important for hindering deterioration of the material. The capping has to wet the RE ﬁlm and form a stable continuous layer hindering reactions with the ambient atmosphere. Even here, Nb has been used successfully, as Nb forms a self passivating oxide layer at ambient conditions [68]. A good counterexample is the use of gold as a capping layer. At ﬁrst glance, gold appears to be the ideal material choice for capping, it is inert and it is possible to form what appears to be continuous ﬁlms. However, Au is highly unsuitable as a capping layer as the RE ﬁlms deteriorate rapidly[66]. The root of the deterioration is the adsorption of H2 O on the Au surface. Water diﬀuses readily on grain boundaries, reaching the underlying ﬁlm where

(b) [H]/[Y] (Atomic Ratio)

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Energy (MeV)

Fig. 1.9. Illustration of the deterioration of a representative ﬁlm by H2 O. (a) The hydrogen content was determined by nuclear resonance analysis using the N-15 method, and the oxygen content by Rutherford back scattering spectrometry. The hydrogen diﬀuses deep into the ﬁlm, while the oxygen forms oxides in the near surface region. Notice the absence of hydrogen in the Nb layers. An illustration of the deduced structure is shown in (b) [66]

1 Modern Growth Problems and Growth Techniques Nb

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19

-0.005

-10 6.2

6.4

6.6

6.8

7

7.2

7.4

Energy (MeV)

Fig. 1.10. Illustration of the hydrogen content of a Y ﬁlm. The Nb capping is hindering the reaction of H2 O with the Y, however, substantial hydrogen concentration is still found in the ﬁlm

it dissociates. This results in the formation of oxides, hydroxides and hydrides. This is illustrated in Fig. 1.9 [66]. Although Nb appears to be an excellent capping material, substantial amount of hydrogen is still found in Nb capped MBE grown RE materials. Typical procedures for the MBE growth of RE materials involves extended (days) outgassing of the target materials, followed by evaporation onto a substrate at elevated temperatures. In-situ capping and subsequent dry oxidation of the Nb capping still results in signiﬁcant hydrogen content of the grown ﬁlms. Representative results on the determined hydrogen content are illustrated in Fig. 1.10 [69]. The hydrogen content of Y and RE ﬁlms are typically in the same range, as expected from the similar chemical properties, as conﬁrmed by measurements on Y, Gd and Ho. The hydrogen content of RE materials is frequently ignored, which seriously inﬂuences the reliability of the deduced ﬁlm properties.

1.3 Growth of Magnetic Oxides and Magnetic Semiconductors While detailed atomistic models for the homoepitaxial growth of metals have been put forward and attempts to incorporate some aspects of heteroepitaxial growth have been made (see discussion in Sect. 1.2.1), a kinetic description of oxide growth is largely lacking. One of the reasons is the complexity of the oxide structures, the diﬀerent nature of bonding and the variety of chemical species that are involved, leading to a multitude of potentially relevant diﬀusion processes. Besides the temperature and the deposition rate, the partial pressure of oxygen is an important parameter in the epitaxial growth of oxides.

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Post-growth treatment (e.g. annealing) in vacuum may lead to reduction of the oxide. Vice versa, post-growth annealing in oxygen atmosphere can help to reduce oxygen vacancies. Concerning the characterization of the grown ﬁlm, most of the surface science techniques require ultra high vacuum (UHV) conditions. The ﬁlm properties and structure may be altered by the ambient, which imposes a problem. Furthermore, the insulating nature of oxides hampers the application of imaging techniques, such as scanning tunneling microscopy which requires a reasonably conducting sample. This limitation can be circumvented by using thin oxide ﬁlms grown on a metal support, providing suﬃcient conductivity. The surface stoichiometry and structure has important consequences for the reactivity but also for the magnetic and electronic properties of the material. Oxides and their surfaces are typically classiﬁed according to electrostatic considerations. The most commonly used are Tasker’s scheme [70] and the autocompensation rule [71]. Originating from semiconductor physics and a covalent picture of bonding, the autocompensation rule states that on a stable surface all anion- (cation-) derived dangling bonds have to be ﬁlled (empty). Tasker’s classiﬁcation, which emphasizes the ionic nature of bonding, is shown in Fig. 1.11. Here, oxide surfaces are divided in three groups, according to the charge of the layers Q and the dipole moment μ perpendicular to the surface. Systems of type one have neutral layers and no dipole moment perpendicular to the surface (Q = 0, μ = 0). Systems of type two and three consist of charged layers. In type two systems the repeat unit has no dipole moment perpendicular to the surface, while in type three it has a nonvanishing dipole moment perpendicular to the surface. It should be noted that depending on the termination, surfaces of the same orientation can be polar or non-polar. For type three surfaces both the scheme of Tasker and the autocompensation rule postulate a diverging surface energy (“polar catasptrophy”) that can only be compensated by strong changes of the surface stoichiometry either by reconstructions or by faceting. Although usefull, these concepts have their limitations, because both models rely on the bonding and valence state in the bulk which may substantially be altered at the surface.

Type 1: Q=0, µ=0

Type 2: Q=0, µ=0

Type 3: Q=0, µ=0

Fig. 1.11. Classiﬁcation of polar oxide surfaces after the scheme of Tasker [70]. See text for details

1 Modern Growth Problems and Growth Techniques

21

DFT calculations have shown that lattice relaxations, where electronic charge redistribution often leads to metallization of the surface, can be an eﬀective mechanisms to reduce and even compensate surface polarity (see review of Noguera [72]). Other mechanisms emerging from the correlated nature of transition metal oxides will be discussed in Sect. 1.4.3. The development of ab initio thermodynamics [73, 74, 75] contributed substantially to identify cases where simple electrostatic arguments fail. In ab initio thermodynamics density functional theory is combined with concepts from thermodynamics to describe the surface stability at ambient pressures and temperatures. The main idea is that the most favorable surface conﬁguration minimizes the surfaces energy. The latter depends on the Gibbs free energy of the system Gslab Mx Oy (001) as well as on the chemical potentials of the constituents. γ(T, p) =

1 slab GMx Oy (001) − NM μM (T, p) − NO μO (T, p) . 2A

(1.4)

When the entropic contributions are small or cancel out, one can susbtitute Gslab Mx Oy (001) with the total energy from DFT-calculations. The chemical potentials μM (T, p) and μO (T, p) are not independent of each other. The condition that the surface is not only in equilibrium with the gas reservoir (e.g. oxygen pressure in the atmosphere) but also with the bulk oxide Mx Oy results in only one independent variable, the oxygen chemical potential which can be translated into partial pressures at a particular (growth) temperature. For further details, see e.g. [75]. A further aspect that has to be noted is that the treatment of transition metal oxides represents a challenge for DFT- methods due to the correlation eﬀects in the d states, localized oxygen orbitals and magnetism. For such systems all electron methods provide the highest accuracy and for the treatment of on-site Coulomb repulsion methods that go beyond the local density approximation (LDA) or the generalized gradient approximation (GGA) like e.g. the LDA+U method [77] are gaining importance. Such methods have mainly been applied to bulk systems and only recently to surfaces and interfaces. 1.3.1 Binary Oxides In this section we limit the discussion to few examples of the growth of oxides with ferro- or ferrimagnetic coupling. Prominent examples are the halfmetallic ferromagnets, Fe3 O4 and CrO2 . Fe3 O4 Magnetite is the oldest known magnetic material and its importance ranges from geology to magnetic recording. The predicted half metallic behavior [78] paired with a high Curie temperature makes it a prospective material for spintronics applications. This has generated substantial research activities on magnetite.

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Magnetite is a ferrimagnet, that crystallizes in the inverse spinel structure where oxygen ions form a slightly distorted fcc lattice. Trivalent iron ions occupy one fourth of the tetrahedral sites (FeA ), while 50% of the octahedral sites are occupied by mixed valence FeB -ions. While a p(1√× 1)-structure √ has been reported on the (111)-surface, Fe3 O4 (001) shows a ( 2 × 2)R45◦ reconstruction. The origin of the latter was subject of a controversial debate in the literature. Various models for a compensated (001)-surface were proposed, where the surface reconstruction was understood as an ordering of surface defects [79, 80, 81, 82, 83, 84]. The surface phase diagram, obtained within the framework of ab initio thermodynamics (shown in Fig. 1.12a) for all possible surface models, revealed that a modiﬁed bulk termination yields the lowest energy over the entire range of accessible oxygen pressures. In this so called modified B-layer (top and side view displayed in Fig. 1.12b) the surface reconstruction is a result of a wavelike Jahn-Teller-distortion and not an ordering of surfaces vacancies as in previous models. This termination does not fulﬁll the electrostatic models and was therefore ignored in the structural analysis so far. Experimental evidence for this structure is obtained from XRD [85], LEED and STM measurements [76]. The wave like pattern in the surface layer is clearly visible in the STM image of the Fe3 O4 (001)-surface and the STM simulation in Tersoﬀ-Hamann model using the charge density from the DFT-calculation shown in Fig. 1.12c). Both the DFT and spinpolarized photoemission measurements [76] show that the surface stabilization is accompanied by strong changes in the electronic properties: e.g. a half-metal to metal transition takes place from bulk to the surface. Besides natural samples synthetic single crystals as well as epitaxial ﬁlms are used in experiment. Koltun et al. [86] grew synthetic Fe3 O4 crystals using the ﬂoating zone technique from pre-sintered magnetite bars prepared from iron oxalate. After crystallization the samples were annealed at 1473 K for 20 h in a partial oxygen pressure of 3.2 × 10−6 mbar. a)

b)

c)

Fig. 1.12. (a)Surface phase diagram of the Fe3 O4 (001)-surface; (b) side and top view of the modified B-termination, oxygen, tetrahedral and octahedral iron are marked by white, grey and dark grey circles, respectively; (c) STM image of the Fe3 O4 (001)-surface [76] together with an STM simulation of the modified Btermination both showing the wave like structure in the 110-direction

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23

Epitaxial ﬁlms were grown on a variety of substrates. Due to the nearly perfect lattice match (0.31%), MgO is an excellent candidate for the growth of epitaxial magnetite ﬁlms. Fe3 O4 -ﬁlms in the (001)-orientation are typically grown on MgO(001). Other susbtrates used are SrTiO3 (100) [87](lattice mismatch -7.1%), MgAl2 O4 (100) [88] (lattice mismatch–3.8%), and GaAs(100) [89]. The latter substrate is particularly interesting for the incorporation of Fe3 O4 in spintronic devices. Two main techniques are used for the synthesis of Fe3 O4 : (i) oxidation of Fe ﬁlms with oxidizing agents as O2 , NO2 [84] or oxygen plasma; (ii) oxidation during deposition of Fe in an oxygen rich environment. On MgO(100), Fe3 O4 -ﬁlms were grown using O2 assisted MBE. With a substrate temperature of 525 K and a growth rate of 22.5 ˚ A /min good quality ﬁlms were obtained [90]. Oxygen plasma assisted MBE was used by Kim et al. to grow Fe3 O4 (001) on MgO(001) [92]. The best quality was obtained under oxygen poor conditions (pO2 = 4 × 10−6 mbar) and an iron deposition rate of 0.6 ˚ A/s with an electron-cyclotron resonance (ECR) plasma source runnung at 200 W. Voogt et al. [84] used NO2 as an oxidizing agent and a similar substrate temperature of 525 K. The growth was monitored by RHEED and the time to form a ML of Fe3 O4 (001) was estimated to be 46 s. The threshold of Mg interdiﬀusion at 625–675 K represents an upper limit for the growth temperature. For spintronics application it is desirable to combine the half-metallic oxides with semiconductor devices. Lu et al. [91] grew Fe3 O4 (001) on GaAs(100): initially a bcc epitaxial Fe-layer was grown on GaAs(100), suppressing the formation of secondry phases (e.g. FeAs) to avoid the development of a magnetically dead interface layer. Subsequently the Fe-ﬁlm was oxidized at pO2 = 5 × 10−5 mbar. An aspect that needs further investigation is whether the half-metallic behavior is preserved at the interface to MgO(001) or GaAs(001). For the growth of Fe3 O4 (111), diﬀerent substrates have been used, e.g. MgO (111) [94, 95] and Al2 O3 [96, 97], as well as metallic Pt(111). Weiss et al. [98] repeatedly deposited and oxidized layers of iron on Pt(111). The LEED analysis of well ordered ﬁlms suggested a termination with 1/4 monolayer of Fe over a distorted hexagonal oxygen layer. Dedkov et al. oxidized a Fe(110) ﬁlm grown on W(110) [99]. Depending on the oxygen pressure and the post growth annealing procedures, lattice parameters corresponding to the formation of FeO and Fe3 O4 (111) were obtained. A FeO(111)-surface was formed after 100 L oxygen exposure and post-annealing at 525 K. The FeO-ﬁlm was transformed into a Fe3 O4 (111) ﬁlm after subsequent exposure to 200 L oxygen and post-annealing at 525 K. Alternatively, Fe3 O4 (111) was obtained after an extended exposure to 900 L oxygen. Fonin et al. [93] grew Fe3 O4 (111) by oxidizing a Fe(110)-ﬁlm grown on Mo(110)/Al2O3 (11¯20) at 700◦ C and pO2 = 5 × 10−6 mbar. A TEM cross section of the ﬁlm demonstrating the four diﬀerent regions of the sample with sharp interfaces is shown in Fig. 1.13a). A lower spin-polarization of 60% was measured for this sample, as compared to the nearly fully spin-polarized (80%) one grown on W(110) [99]. This result

24

B. Hj¨ orvarsson and R. Pentcheva

is consistent with DFT calculations [100] of bulk magnetite showing that uniaxial strain reduces the degree of spin-polarization. A 100 ˚ A thick Fe3 O4 (111)-ﬁlm was grown on Al2 O3 (0001) [97] by codeposition of Fe from an eﬀusion cell and atomic O using a plasma source, at a substrate temperature of 450 K and postannealing at 900 K. Such interfaces are interesting as magnetic tunneling junctions (MTJ). For Fe3 O4 (111) ﬁlms grown on an (1 × 1)-OH terminated MgO(111)-surface [94] and Al2 O3 (0001) [96] a phase separation in Fe and FeO nanoinclusions were observed at the interface. Unlike MgO(100), the MgO(111)-surface is polar. Substrate polarity was identiﬁed as the driving mechanism towards phase separation at the interface. In regions between the Fe crystalites atomically abrupt interfaces were observed. A HRTEM image is shown in Fig. 1.13. DFT-GGA calculations [95] ﬁnd that these are stabilized through electronic screening and metallization at the interface in contrast to the stoichiometry change expected from classical electrostatic models. CrO2 Since the prediction of half metallic behavior of CrO2 [101], this oxide has attracted attention as a potential material for spintronic devices. Unfortunately, rutile CrO2 is unstable at room temperature and transforms irreversibly into

a)

b) MgO

Fig. 1.13. TEM cross section micrograph of a) the Fe3 O4 (111)/Fe(110/Mo(110) 20) system [93] with sharp interfaces and b) Fe3 O4 (111) grown on /Al2 O3 (11¯ MgO(111) showing the formation of Fe(110) nanocrystals at the interface [94]

1 Modern Growth Problems and Growth Techniques

25

Cr2 O3 , which is an insulating antiferromagnet. In order to stabilize the rutile structure, epitaxial ﬁlms were grown on TiO2 (001) and Al2 O3 (0001) using chemical vapor deposition CVD [102]. On Al2 O3 (0001) a 400 ˚ A Cr2 O3 -layer is formed prior to the growth of CrO2 (001). A 1000 ˚ A thick CrO2 (001) ﬁlm of good crystal quality was obtained on TiO2 with no Cr2 O3 formation. The magnetic ordering temperature of this ﬁlm was 385 K. Heterostructures based on the insulating TiO2 as a barrier are interesting for magnetic tunnel junctions. On the other hand, metallic RuO2 -spacers are interesting as GMR-elements. CrO2 (a = 4.421 ˚ A, c = 2.916 ˚ A) and RuO2 (a = 4.499 ˚ A, c = 3.107 ˚ A) have also a good lattice match. Upon deposition of RuO2 on CrO2 /TiO2 the Cr2 O3 termination of the surface is transformed to CrO2 but despite the restored conductivity a relatively low magnetoresistence of the sample indicates susbtantial chemical and magnetic disorder associated with this transformation [103]. Besides TiO2 and RuO2 , also SnO2 was used as a substrate, however the measured magnetoresistance was still relatively low. w-NiO Analogous to the growth of ferromagnetic materials on a non-magnetic substrate, an attractive possibility is opened by the growth of oxides on semiconductors where the oxide adopts the structure of the substrate, which does not exist in the bulk. Recently, Wu et al. [104] predicted, using LDA+U calculations, that NiO in the wurzite structure (w-NiO) should be halfmetallic and on substrates like ZnO or GaN the ferromagnetic coupling should be more stable than the antiferromagnetic. 1.3.2 Ferromagnetic Semiconductors The design of semiconductors with magnetic and/or spin-related properties and high Curie temperature for spintronic devices is a demanding and by far not completely resolved issue. In this subsection we will brieﬂy summarize the current knowledge on one of the most intensively studied III-V semiconductors, Mn doped GaAs (TC ∝ 170 K), and then discuss several systems where the prediction of room temperature ferromagnetism (e.g. Co:TiO2 , ptype Mn:ZnO, Mn:GaN) has envigorated a lot of research in the last years. A major issue in the fabrication of doped semiconductors is the incorporation of dopants in the lattice and whether a homogeneous distribution can be achieved. Mn:GaAs Unlike II-VI semiconductors where there is no solubility limit for 3d dopands, the solubility limit is very low for III-V systems (e.g. 0.1% for Mn). Therefore a secondary phase like MnAs is formed under typical growth conditions. To avoid this, MBE growth is performed at low temperatures (LT-MBE) of 473– 573 K. The Mn ions primarily occupy cation sites (MnGa ) where they act as

26

B. Hj¨ orvarsson and R. Pentcheva

acceptors introducing both magnetic moments and holes. The current understanding of the underlying mechanism is that of hole-mediated ferromagnetism in p-type materials. Since the growth proceeds at highly non-equilibrium conditions where kinetic eﬀects dominate, there are two main types of defects that form and inﬂuence the magnetic properties and conductivity: the As antisites (AsGa ) and the Mn-interstitials (MnI ). Both act as donors, i.e. reduce the number of holes. Additionally, MnI couple antiferromagnetically to MnGa and thus suppress ferromagnetism. Yu et al. [105] provided evidence that the reduction of TC is directly related to formation of MnI and the latter depends on the doping of the barrier layer on which Mn:GaAs is deposited. Therefore the reduction of both types of defects is the main path towards obtaining a higher TC . This is done through a control of the As ﬂux during growth and a post-deposition low temperature (473–573 K) annealing step. As shown by polarized neutron reﬂectometry on as grown and annealed samples, annealing improves substantially the homogeneity of Mn and thus the magnetic properties [106]. For further reading the reader is referred to several reviews e.g. [107, 108, 109]. Doped Thin Film Oxides Co:TiO2 : The interplay of growth conditions (temperature, deposition rate), but also the partial pressure of oxygen plays a decisive role on the quality of the samples. For example PLD growth of Co:TiO2 (anatase) from a mixed metal oxide target may result in Co -nanoinclusions within the TiO2 matrix if the O2 -pressure is too low or if the Co:Ti ratio is too high [110, 111, 112]. A continuous epitaxial ﬁlm with no signs for Co enrichment was obtained for depostion rates of 0.1 ˚ A/s and T = 650◦ C. On the other hand OPA-MBE material is found to be FM at RT for xCo ≈ 5 − 7% (1.1 μB /Co) [113]. Similar morphologies but a lower saturation magnetization of 0.6 μB /Co is obtained for Ar-sputtering of Ti and Co metal targets at T ≈ 650◦ C using water as an oxidant [114]. The incorporation of hydrogen could be of importance in this context, as discussed in Sect. 1.1.2 above. Another issue, similar to the ones arising at oxide interfaces, is the mechanism of charge compensation: e.g. in Co:TiO2 Co2+ substitutes for Ti4+ . DFT-GGA calculations predict Co2+ -segregation together with the formation of an oxygen vacancy [115]. Annealing in vacuum is not likely to lead to Co-oxidation. For example, XANES measurements revealed metallic Co at T > 750◦ C [116]. Post-growth annealing results in enrichment of Co at surfaces, grain boundaries and interfaces. Also the origin of room temperature ferromagnetism is not well understood, e.g. in Co:TiO2 crystallographically perfect samples were obtained for T ≈ 550◦ C and R = 0.014 ˚ A/s which turned out to be nearly nonmagnetic [117]. This leads to the assumption that defect formation at surfaces and interfaces plays a signiﬁcant role in triggering magnetism.

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Cr:TiO2 : PLD growth of Cr: TiO2 (rutile) on Al2 O3 (012) resulted in conducting ﬁlms (reﬂecting a ﬁnite density of O-vacancies) and a saturation moment of 2.9 μB /Co for x = 0.07. OPAMBE ﬁlms grown on TiO2 (110) were on the other hand insulating and nonmagnetic. XPS showed that Cr3+ substitutes for Ti4+ . Post growth annealing in vacuum reduced the ﬁlm and made it n-type, showing weak room temperature ferromagnetism with 1 μB /Co. Cr:TiO2 (anatase): LaAlO3 or SrTiO3 were chosen as substrates for OPAMBE growth. The in-plane lattice mismatch between anatase and the perovskite structure along (001)-direction diﬀers by an order of magnitude for LaAlO3 and SrTiO3 (–0.26 vs. –3.1%), respectively. For R ≈ 0.1 ˚ A/s and T ≈ 650◦ C the as grown ﬁlms exhibited room temperature ferromagnetism which was enhanced after annealing in vacuum. The ﬁlms exhibit a high Curie temperature (Tc =690 K). An improved crystalline quality is obtained for R ≈ 0.015 ˚ A/s and T ≈ 550◦ C [117] but again as for Co:TiO2 magnetic properties deteriorate. In summary, room temperature ferromagnetism in Co or Cr:TiO2 appears to be driven by defects and is not an intrinsic property of the material. Co:ZnO: The appearance of room temperature ferromagnetism in Co:ZnO depends critically on electron doping. Epitaxial ﬁlms of Co:ZnO grown on Al2 O3 (012) by MOCVD [118] were found paramagnetic and insulating. However, the interdiﬀusion of atomic Zn in these samples (Zn occupies interstitial sites) results in a weakly ferromagnetic semiconducting sample [119]. Ti:Fe2 O3 : A nontraditional candidate for room temperature ferromagnetism is Ti doped Fe2 O3 . The host is a (canted) antiferromagnetic insulator at room temperature, but the incorporation of Ti in the lattice is expected to lead to ferrimagnetism. Chambers and collaborators [116, 120] grew a 710 ˚ A α-Fe2 O3 -ﬁlm on Al2 O3 using a 130 ˚ A thick buﬀer layer of Cr2 O3 to reduce the in-plance lattice mismatch (5.8%). The OPAMBE growth was performed at Tsub = 550◦C with growth rate of 0.25 ˚ A/s at an oxygen pressure of 1.5 × 10−5 mbar. XRD measurements suggested a high degree of crystallinity. The magnetic signal depends sensitively on whether Ti is incorporated on one spin sublattice or randomly distributed on both spin sublattices. The measured saturation magnetization is however much lower (approximately 0.5 μB /Ti) than the expected 5 μB /Ti indicating that only about 12% of Ti contributes to a FM ordered phase and the the majority of Ti is randomly distributed in the lattice.

1.4 Multilayers and Superlattices 1.4.1 General Considerations A multilayer is a general term describing one dimensional variation in composition. A multilayer can be single crystalline, polycrystalline, amorphous or a combination of (poly-)crystalline and amorphous. When a multilayer is single crystalline and has many repetitions, it is denoted superlattice, see Fig. 1.14.

28

B. Hj¨ orvarsson and R. Pentcheva

Lc

Capping

c

! : LSL LA LB Seed layer Substrate

Fig. 1.14. Illustration of a typical superlattice structure. A seed layer is grown on a substrate, followed by the growth of the superlattice. The structure is thereafter typically covered by a capping layer, hindering the deterioration of the superlattice

A superlattice can therefore be thought of as a single crystal multilayer, with well deﬁned atomic distances as well as chemical repeat distance (Λ). Λ deﬁnes the unit cell in the growth direction, where Λ=La +Lb where La and Lb are the thicknesses of layer a and b, respectively. The variation in the chemical composition is a route to create a modulation in the electronic states, forming new material classes with unique properties. However, formation of a superlattice is only the ﬁrst step. Altering the chemical composition in 3 dimensions would allow much larger degree of freedom, forming unique electronic states deﬁned by the extension and the compositional variation in the material. This can be viewed as the ultimate task of materials processing of today. The growth of superlattices bears large similarities to the growth of single ﬁlms. However, there are also signiﬁcant diﬀerences both with respect to the growth procedures as well as the analysis of the samples. The inherent lattice parameters of the constituents have a special role, which is often used to judge the possibility for the growth of high quality superattices. The basic ideas resemble in many ways the criteria for the growth of single ﬁlms on a substrate, as will be apparent below. 1.4.2 Metallic Superlattices We will use the combination of Fe, Mo and V as examples for the possibilities and limitations of the growth of metallic superlattices. All these elements are bcc with a bulk lattice parameter of 2.86, 3.16 and 3.02 ˚ A, respectively and can therefore, in principle, form congruent single crystals. The diﬀerence in the lattice parameter of Fe and V is 5%, Mo and V is 4% and ﬁnally Fe and

1 Modern Growth Problems and Growth Techniques

29

Mo it is close to 10%. The initial growth of V on Fe has been investigated using RHEED [121] following the in-plane lattice parameter changes of the V, in a wide temperature range. A 200 ML Fe (001) layer was initially grown on a MgO(001) substrate, on which the V was deposited. Between 300 and 800 K, RHEED oscillations were observed up to 9 ML, consistent with a well deﬁned layer by layer growth of V. However, the in-plane lattice parameter at the V surface was observed to relax from that of Fe when the number of V layers exceeded 7 ML. At 800 K, no RHEED oscillations were observed, which indicates the presence of strong island formation and an upper boundary for a well deﬁned growth. Thus, it is possible to grow V on Fe up to 7 ML, although the diﬀerence of the lattice parameter is as large as 5%. When the number of repeats of the superlattice period is large, the average in-plane lattice parameter reﬂects the thickness ratio of the constituents. Thus, when growing e.g. Fe/V superlattices, the average in-plane lattice parameter can be estimated directly from the thickness ratio, as the elastic constants of Fe and V are rather similar. For example, when the layers have equal thicknesses (LFe=LV ), the average in-plane lattice parameter will be close to the average lattice parameter of the constituents. This reduces the in-plane strain from 5 to below 3%, which should be reﬂected in an increased critical thickness for the growth of coherent layers. This was observed in RHEED investigations of Fe(3)/V(x) (001) superlattices in which the critical layer thickness was determined to be around 16 ML [122]. The growth of Fe/V(001) superlattices has been thoroughly investigated by a number of authors (see for example [123, 124]). The temperature dependence of the growth on MgO (001) was established by Isberg et al. [124] where the quality of the SL were shown to depend strongly on the growth temperature. The best crystalline quality was obtained with a substrate temperature in the temperature range 570–600 K. The thickness variation of the layers was also at its best in the same range and was determined to be 1 ˚ A. This variation in layer thickness represents the lower limit for a non phase locked growth, resulting in incomplete formation of the individual Fe and V layers. Thickness variation corresponding to one monolayer of both the Fe and V layers is thus inevitable. Representative X-ray results are shown in Figs. 1.15 and 1.16. The optimal growth temperature of Mo/V superlattices is close to 1000 K, which is substantially higher than for Fe/V superlattices. This correlates with the substantially higher melting temperature of Mo compared to Fe. The growth of this material combination was pioneered by the group of Fisher [125, 126, 127], demonstrating both a (001) and (110) growth on MgO(001) and Al2 O3 (11¯ 20) respectively. The inﬂuence of the critical thickness of the layers in Mo/V superlattices on the superconducting properties was discovered by Karkut et al. [125]. A critical thickness of 16 ML was inferred for both the (001) and (110) grown Mo/V superlattices, when the ratio of the layer thicknesses was close to unity. The diﬀerence of the lattice parameters of Mo and Fe with respect to V is similar, but with diﬀerent sign. Thus, a comparable critical thickness for epitaxial growth is observed for compressive and tensile

30

B. Hj¨ orvarsson and R. Pentcheva

Fig. 1.15. Representative reﬂectivity data from a Fe/V(001) superlattice with Λ = 25.1 ˚ A. From [122]

biaxial strain in V, as expected from symmetry reasons. A compressive strain in one layer is balanced by a tensile strain in the second. Thus, the sign of the strain appears to be irrelevant, and a hint of the relation between the lattice mismatch and the critical thickness of the layers emerges. Above the critical thickness, the formation of dislocations and other defects results in buckling and increases therefore the variation in the layer thicknesses [126, 127, 128]. This leads to relaxation of the in plane lattice parameter, where the variation is accomplished by the presence of defects. The thickness variation is governed by the V layers, because V atoms have larger surface mobility at the actual growth temperature [128].

Fig. 1.16. Representative diﬀraction data from a Fe/V(001) superlattice with Λ = 25.1 ˚ A˙ The inset highlights the crystalline quality of the superlattice structure. The presence of Laue oscillations implies a interference between the scattering from the ﬁrst and the last monolayer of the Fe/V(001) stack. From [122]

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Let us now consider the growth of Fe on single crystal Mo(110), with a lattice mismatch of close to 10%. The growth results in a complicated defect generation already in the ﬁrst monolayer. The ﬁrst layer grows pseudomorphically [129] followed by a pronounced relaxation in the third layer. Thus, the diﬀerence in the lattice parameters of Fe and Mo appear to be beyond the limit for epitaxial growth. The growth and characterization of Fe/Mo multilayers is described in the literature [130, 131], but no reports are found on succesful growth of Fe/Mo superlattices. The choice of growth temperature is a compromise between two constraints, surface mobility and interdiﬀusion. Surface mobility of adatoms is increased with increased temperature, but so is the interdiﬀusion. This limitation is clearly seen in many of the transition metal superlattices and constitutes therefore a substantial challenge for optimizing the growth. One route to circumvent this limitation is the use of surfactants. The basic idea is to decrease the activation energy of the surface diﬀusion and thereby increasing the surface mobility. As seen in (1.2) the weight of the change in activation energy (Ed ) has the same inﬂuence on the diﬀusion rate (D) as the change in temperature. One possible surfactant is hydrogen. The inﬂuence of hydrogen on the diﬀusion of Pt adatoms on Pt(111) was investigated by STM [133]. A clear increase of the diﬀusion rate was demonstrated. However, the presence of hydrogen has also been found to inhibit surface mobility [134]. The main beneﬁt of the use of hydrogen in this context, is the compatibility with the vacuum processes. Hydrogen is simply removed by evacuation from the deposition chamber. The use of hydrogen as a surfactant was recently demonstrated by Remhof et al. [132], where a substantial increase of the quality of Fe/V(001) superlattices was obtained. Representative x-ray reﬂectivity results are illustrated in Fig. 1.17. Co/Cu Superlattices of Co/Cu have been widely studied during the last decades. Successful growth of an epitaxial Co/Cu on a single crystal Cu(001) under UHV conditions was demonstrated by Cebollada et al. [135]. The Cu substrates were cut from a single crystal bar and oriented within 0.3◦ of the [001] direction using Laue diﬀraction. The substrates were cleaned in-situ by cycles of Ar+ sputtering and annealing to 1000 K. The composition of the surface was investigated using Auger electron spectroscopy and the crystalline quality was investigated by thermal-energy atom scattering (TEAS). The resulting surface consisted of, on average, 300 ˚ A wide ﬂat terraces separated by monoatomic steps. The authors used extremely low deposition rate, 0.01 ˚ A/s as compared to few ˚ A/s, to reduce the amount of imperfections. The thickness of the evolving layers was monitored and determined by counting the number of oscillations in the TEAS signal. The thickness obtained this way was conﬁrmed by calibrated quartz balance. The samples were all covered by

32

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Fig. 1.17. Small angle x-ray reﬂectivity scans recorded at E = 8.048 keV. The upper curve displays the reﬂectivity of the sample sputter deposited at pH2 = 2×10−6 mbar at T = 320◦ C. The lower curve shows the reﬂectivity of a reference sample grown without the presence of hydrogen. The scans are shifted along the intensity axis for clarity. The inset shows the diﬀuse scattering recorded at the ﬁrst superlattice peak [132]

1000 ˚ A Cu, to hinder oxidation of the underlying superlattice. The quality of the resulting structure was established by neutron reﬂectivity and diﬀraction. Concerning the growth of Co/Cu superlattices, we tie up to discussion of the initial growth of Co on Cu(001) in Sect. 1.2.1. The lattice parameter of fcc Co at room temperature is 3.548 ˚ A, while the lattice parameter of Cu is 3.615 ˚ A under the same conditions. The lattice mismatch is therefore slightly below 2%. When growing thin Co layers on Cu(001) single crystal, the Co adapts the in-plane lattice parameter of Cu. This results in a tetragonal distortion of the fcc lattice, where the out-of-plane lattice parameter is contracted and the in plane lattice parameter is expanded, as compared to the bulk value [57]. The restoring force at the interface between the Co and the Cu substrate is ﬁxed but the strain energy associated to the tetragonal distortion increases linearly with the thickness of the Co ﬁlm. Thus, at some critical thickness, it will be energetically more favorable to form defects, relieving the strain and forming a non distorted fcc lattice. In the LEED study of Navas et al. [136] Co was found to grow in the strained state up to at least 10 monolayers. The inherent electronic structure of the Co layers is therefore signiﬁcantly diﬀerent from bulk Co. The relation between the structural and magnetic properties of single Co ﬁlms and Co/Cu superlattices is discussed by de Miquel et al. [137]. The subsequent growth of Cu on the thin Co ﬁlms retained the bulk lattice parameter of Cu. Thus, while growing Co/Cu(001) superlattices on a Cu(001)

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substrate, only the Co layer should exhibit a tetragonal distortion, as long as the superlattice is grown in a coherent mode. Fe/Cr The work on Fe/Cr superlattices was pioneered using evaporation techniques. For example, Etienne and collaborators used eﬀusion cells for the growth of both Fe and Cr on GaAs under UHV conditions [138]. The surface structure of the substrate was improved by growing a 3000–5000 ˚ A GaAs(001) buﬀer at ∼ 1000 K. The surface reconstructed from 2 × 4 to 2 × 6 after a brief Ga-exposure at 725 K. The metallic superlattices were grown on such buﬀer layers, starting with Fe. The growth temperatures of the superlattice were in the range 225−325 K, the drift reﬂecting the thermal balance with the cooling and heating devices in the MBE system. The presence of layering was conﬁrmed by a combination of Auger spectroscopy and sputtering and the surface crystallinity was investigated using RHEED. The chemical purity was investigated by Auger spectroscopy and the samples were found to be free of both oxygen and carbon. The authors did not present any x-ray data. The resulting giant magnetoresistance (GMR) of these structures was in the range of 100%, taking the high ﬁeld resistance as a reference. Epitaxial growth of Fe/Cr(001) by sputtering on MgO(001) was later demonstrated by Fullerton et al. [139]. In this case, the authors used a Cr buﬀer layer grown at 873 K which leads to an improved surface ﬂatness and wetting of the initial layers of the superlattice. The bcc Cr is rotated with respect to the MgO substrate in the same way as discussed for Fe and V on MgO. The Fe/Cr superlattice was grown at ∼450 K and the resulting structure was investigated by x-ray reﬂectivity and diﬀraction. Typical rocking curves of the (002) diﬀraction peak yielded a FWHM of 0.7◦ and a FWHM of 0.2◦ in 2θ, using Cu Kα radiation. The authors did not dwell on the choice of growth temperatures. This approach resulted in a 150% GMR, which is substantially higher than obtained by Etienne and collaborators. The structural quality is of large importance for the resulting physical properties, such as GMR. This is clearly seen when comparing the results from single crystal structures with polycrystalline samples. Parkin and York [140] grew polycrystalline Fe/Cr multilayers by sputtering using Si substrates. The structure was a combination of (110) and (001) textured crystallites, of which the (001) increased in weight with increasing temperature in the range 300 K to 475 K. These samples yielded a maximum 40% GMR eﬀect and the resistivity of the optimized samples was 4.4 μΩ cm at 4.2 K, as compared with 14 μΩ cm for single crystal superlattices [139]. The growth of Fe/Cr(001) on SrTiO3 using evaporation, was discussed by Ono and Shinjo [141]. The crystalline quality was substantially worse as compared with the results of Fullerton et al. [139], although great care was taken concerning the surface ﬂatness of the substrates. Chemical etching was used to obtain better ﬂatness and in-situ RHEED was used to investigate

34

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the substrate as well as the surface quality of the resulting ﬁlm. A strong inﬂuence of the substrate quality on the GMR was obtained, the chemically etched substrate yielded twice as large values for GMR. The x-ray contrast between Fe and Cr is poor due to the similarity in the electron density of these elements. This was a major obstacle for obtaining higher order Fourier components in, for example, x-ray reﬂectivity measurements. Bai et al. overcame this diﬃculty using resonant scattering techniques [142], allowing detailed simulations of the composition proﬁle as well as the roughness of the samples. The experiments were performed on sputtered superlattices, using MgO(001) as a substrate. Rare Earth Superlattices The main part of the magnetic moment in Rare Earth materials (RE, in short for actinides and lanthanides) stems from the localized f-electrons. In contrast to itinerant magnets, due to this localization one can often safely ignore the eﬀect of hybridization on the magnetic moment. Kwo et al. [65] demonstrated the growth of high quality Gd/Y superlattices and related the changes in moment and ordering temperature to the repeat distance in the samples. The growth of the superlattices was done in an UHV MBE system, with a base pressure in the 10−11 mbar range. A buﬀer layer of Nb (011) was grown on an Al2 O3 (11¯ 20) at ≈1170 K, hindering the transport of oxygen from the substrate to the RE ﬁlm. A seeding layer of Y(0001) was subsequently grown at ≈970 K, resulting in a ﬂat surface, enabling the growth of high quality Gd/Y(0001) superlattices at ≈470 K. Lower growth temperature is chosen for the superlattice to limit the interdiﬀusion of the constituents. An alloy interface region of 2 monolayers was established by combined x-ray and magnetic analysis. The interface region was inferred to be a GdY alloy, without magnetic ordering, even at low temperatures. The magnetic susceptibility is large, which conﬁrms the proposed model of the compositional modulation. The growth and characterization of other RE superlattices was established by a number of groups most of which followed the ideas of Kwo et al. with respect to diﬀusion barrier for oxygen and a seeding layer. McMorrow and collaborators investigated the chemical structure of Ho/Lu and Ho/Y superlattices [143] using high resolution x-ray scattering to address the nature of the interface imperfections. By detailed investigations of the shape of the superlattice Bragg peaks, the presence of conformal roughness of the interfaces was established. The basic idea behind these investigations was to establish not only the type of roughness, but to separate roughness and intermixing. This is of primary interest as the physical properties are highly dependent on the type of interface imperfections. Intermixing can for example result in an absence of ferromagnetic ordering as discussed above [65], while both conformal and uncorrelated roughness will give rise to local anisotropy ﬁelds. A thorough description of the growth and structural characterization of rare earth superlattices is given by Majkrzak et al. [144].

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1.4.3 Metal-Oxide Superlattices and Magnetic Tunnel Junctions Magnetic tunnel junctions containing an oxide barrier sandwiched between two ferromagnetic layers are a prototypic system to achieve high tunnel magnetoresistance (TMR) values. In this subsection we discuss several issues concerning the quality of the interface that have impact on the measured TMR value. For structurally perfect interfaces a TMR value of several 1000% was predicted theoretically for Fe/MgO/Fe(001) [145]. Experimentally, TMR values up to 188% at room temperature (RT) were achieved [146] for a junction where Fe was grown using MBE. The MgO barrier was epitaxially grown using electron-beam evaporation of a stoichiometric source material. To avoid the oxdation of the bottom Fe layer the ﬁrst MgO layer was deposited at pO2 = 2.5 × 10−5 mbar. The top Fe electrode was deposited at 473 K substrate temperature. A measured asymmetric current-voltage characteristic was attributed to an assymmetry of the interface structure. This is rationalized by thermodynamic arguments (cf. (1.1): the lower surface energy of MgO (1.1 J/m2 ) versus Fe (2.9 J/m2 )) suggests layer-by-layer growth of MgO on Fe but not vice versa. Tusche et al. [147] found a substantial eﬀect of the oxygen atmosphere on the quality of the interface. Starting with 2 MLs of MgO deposited on Fe(001) at a rate of R=0.125 ML/min by electron bombardment of a polycrystalline MgO rod under UHV conditions they deposited 8 ML of Fe using MBE at R=0.25 ML/min. In the sample where the Fe ﬁlm was deposited at UHV conditions, 30% of the interface layer was found to be FeO with a subsequent disordered Fe layer. In a second sample where the initial 0.5 ML Fe were deposited in oxygen pressure of 10−7 mbar, surface x-ray diﬀraction (SXRD) measurements and quantitative analysis revealed that a coherent Fe layer is formed attributed to the formation of a nearly complete FeO layer between the MgO spacer and the Fe ﬁlm. In a thermodynamic picture, the role of the FeO layer is understood as to reduce the interface energy (cf. (1.1)). DFT calculations found strong dependence of the transport properties on the structure of the interface [148]. Only a symmetric Fe/FeO/MgO/FeO/Fe junction was predicted to give rise to a giant TMR [147]. Parkin and coworkers measured 220% TMR at RT in a FeCo(001)/ MgO(001)/(Fe70Co30 )80 B20 sample where the bottom electrode was a polycrystalline bcc FeCo-layer with a (001)-texture [149]. Djayaprawira et al. [150] found a signiﬁcantly higher RT TMR using amorphous CoFeB electrodes as compared to polycrystalline CoFe (values of TMR were 230% vs. 62%). In this experiment the metal reference and the free layer were deposited by dc magnetron sputtering, while rf sputtering was used for the MgO ﬁlm. After annealing at 593 K a partial crystallization at the interface was observed in HRTEM. Microstructural analysis of the MgO(001) spacer shows a good crystal quality with a ﬁbre texture. The role of the recrystallization after annealing and the distribution of B is still not well understood. First principles calculations ﬁnd a preference for B to reside at the interface [151], which is expected to suppress the TMR.

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The authors conclude that inhibiting B segregation at the interface during processing is likely to enhance TMR. X-ray photoemission studies [152] on CoFeB/MgO bilayers ﬁnd evidence for CoFeB oxidation during MgO deposition, while annealing in vacuum leads to B interdiﬀusion into MgO and MgBx Oy formation. To avoid this a Mg-buﬀer layer is introduced between CoFeB and MgO. A Co/MgO/Co MTJ where Co is stabilized in the bcc structure was predicted from DFT to have an even higher MR than Fe/MgO/Fe [153]. Indeed, MBE grown Co-based MTJ showed a MR of 410% [154]. To retain the metastable bcc structure the thickness of the Co layers on both sides of the insulating MgO barrier were limited to 4 ML and grown at RT. The authors reported that Co does not wet the MgO(001) spacer but grows in a 3D manner. The subsequent annealing step at 525 K for 30 min was used to improve the crystalinity of the sample. Oxide Superlattices Transition metal oxide superlattices open new possibilities to make artiﬁcial materials with magnetic and electronic properties that diﬀer from the bulk components. Analogous to oxide surfaces (cf. Sect. 1.3.1) the question of polarity and disruption of charge neutrality arises also at oxide/oxide interfaces. For example perovskites possess a natural charge modulation in the [001]direction, e.g. in LaTiO3 positively charged (LaO)+ alternate with negatively charged (TiO2 )− , while in SrTiO3 both the SrO and TiO2 -layers are neutral. Thus the interface between these two insulators represents a simple realization of a polar discontinuity. Using PLD from a single crystal STO target and a polycrystalline La2 Ti2 O7 , Ohtomo et al. [155] fabricated superlattices of LaTiO3 and SrTiO3 with an atomically controlled number of layers of each material which therefore are refered to as ”digital”. The ﬁlms were grown at 970 K at an oxygen pressure of ∼ 10−5 mbar to stabilize both valence states of Ti and subsequently annealed at 670 K to ﬁll residual oxygen vacancies. An anular dark ﬁeld TEM image is shown in Fig. 1.18. RHEED oscillations were used to monitor growth. Ohtomo et al. found that although the parent compounds are a Mott and a band insulator respectively, the heterostructure is conducting with electron energy loss spectra suggesting mixed Ti-valence in the interface region. Based on Hubbard models, Okamoto and Millis [156] proposed an “electronic reconstructuion” of this interface. Another system showing unexpected behavior are heterostructures of the two simple band insulators LaAlO3 and SrTiO3 . Here, both the A and B sublattice cations in the perovskite structure change across the interface giving rise to two diﬀerent types of interfaces: an n-type between a LaO and a TiO2 layer that was found conducting with a high electron mobility and a p-type between a SrO and an AlO2 -layer that showed insulating behavior despite the charge mismatch [157]. Using PLD, the n-type LAO/STO IF was grown

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a)

37

b)

Ti Ti

4+

3+

Fig. 1.18. a) an anular dark ﬁeld STEM image of bright LTO layers in a STO host. The boosted up view above shows a 1 × 5 LTO/STO superlattice [155]; b) 45◦ checkerboard charge density distribution of the occupied 3d states in the charge and orbitally ordered TiO2 layer at the LTO/STO-IF. The positions of O-, Ti3+ and Ti4+ -ions are marked by white, black and grey circles, respectively [159]

on a TiO2 -terminated SrTiO3 -substrate. To grow a p-type LAO/STO interface a SrO-layer was deposited on the SrTiO3 -substrate prior to growth of LaAlO3 . The oxygen pressure is quite an important growth parameter in these systems that controls the oxygen stoichiometry and the underlying properties, e.g. STO alone can change from a wide band insulator to a metal with pO2 [20]. Nakagawa, Hwang and Muller [158] discussed recently that the p-type LAO/STO IF is ionically stabilized with an enhanced roughness attributed to oxygen vacancies while the n-type interface is electronically stabilized and hence sharp. In correlated materials with multivalent ions correlation driven charge order oﬀers an additional degree of freedom to accommodate the charge imbalance. To this end LDA+U calculations predict a compensation mechanism by a charge disproportionation: a charge and orbitally ordered IF-layer is found for the LTO/STO and the n-type LAO/STO interface with Ti3+ and Ti4+ ordered in a checkerboard manner [159, 160]. Such a correlation driven compensation mechanism is not present e.g. at polar semiconductor interfaces. Moreover, although both LaAlO3 and SrTiO3 are nonmagnetic and LaTiO3 is an antiferromagnet of G-type, the diluted layer of Ti3+ in the IF layer has a slight preference to couple antiferromagnetically with a magnetic moment of 0.71 μB . Recent experiments give ﬁrst indication for localized magnetic moments at the n-type LAO/STO interface [161]. Thus the violation of charge neutrality at interfaces of transition metal oxides can be used to generate novel charge and magnetically ordered phases that do not exist in the bulk.

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1.5 Conclusions and Outlook Growth of thin ﬁlms, multilayers and superlattices has provided extremely fruitful routes to realize systems with novel structural and physical properties distinct from the bulk phases. Within the ﬁeld of magnetism, thin ﬁlm processing is the single most important factor for the rapid development during the last decades, including the discovery of oscillatory exchange coupling, giant magneto resistance and tunneling magneto resistance. Thin ﬁlm technology will certainly continue to be of large importance, not least for the processing of materials for patterning devices, addressing both fundamental and applied research questions. In this chapter we have focussed on the modulation of the chemical composition in one dimension, the remaining two deﬁne the ﬁnal steps in the current paradigm of materials growth. Self assembly and organization which is certainly another important route to create materials with unusual properties goes beyond the scope of this chapter. We would like to summarize here some aspects we consider important in the growth of magnetic heterostructures: the window of optimal growth conditions in heteroepitaxial growth is much narrower and thus more challenging than the homoepitaxial case as higher tempertures lead to intermixing and a substantial interface roughness even for systems that are immiscible in the bulk. When the bonding to the substrate is weak (as on nobel metal substrates) ultrathin magnetic ﬁlms may experience a much stronger strain than what the lattice mismatch of the bulk phases would suggest. This may have a strong eﬀect on lattice relaxations and the strain ﬁeld in small islands (mesoscopic strain). The growth of oxide ﬁlms and heterostructures is intricately related to the question of surface and interface polarity. However, transition metal oxides provide a much richer variety of mechanisms to compensate excess charges than e.g. semiconductor systems: besides atomic reconstruction, electron redistribution via metallization, lattice distortion or even charge disporportionation can lead to novel properties.

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2 Magnetic Anisotropy of Heterostructures J¨ urgen Lindner and Michael Farle Fachbereich Physik and Center for Nanointegration, Universit¨ at Duisburg-Essen, Lotharstr. 1, 47048 Duisburg, Germany [email protected]

Abstract. The chapter provides a detailed introduction to magnetic anisotropy of ferromagnetic ultrathin ﬁlms and its analysis by ferromagnetic resonance on a tutorial level. While the microscopic origins of the magnetic anisotropy as well as recent developments in its theoretical description are shortly discussed, emphasis is put on a phenomenological description using the free energy of the system together with its symmetries. The formalism is used to describe ferromagnetic resonance experiments which present an extremely sensitive method to experimentally investigate magnetic anisotropy in thin ﬁlm heterostructures. Expressions for the free energy and the resonance equations are derived for the most widely used crystal symmetries such as cubic, tetragonal and hexagonal. The general equations are illustrated by giving selected examples of current research on thin metallic ﬁlms on diﬀerent kinds of substrates (MgO, GaAs and Cu).

2.1 Introduction Magnetic thin ﬁlms have provided a highly successful test ground for understanding the microscopic mechanisms which determine macroscopically observable quantitities like the magnetization vector, diﬀerent types of magnetic order (ferro-, ferri- and antiferromagnetism), magnetic anisotropy and ordering temperatures (Curie, N´eel temperature). The success has been based on the simultaneous development of the following techniques: a) the preparation of single-crystalline mono- and multilayers on diﬀerent types of substrates in ultrahigh vacuum systems, b) the development of vacuum compatible, monolayer-sensitive magnetic analysis techniques, c) the advance in computing power to provide ﬁrst-principles calculations of magnetic ground state properties [1]. Aside from these basic research orientated investigations the technological exploitation of thin ﬁlm magnetism has lead to huge increases in the hard disk’s magnetic data storage capacities [2] and new types of magneto-resistive angle and position sensitive sensors in the automotive industry, for example.

J. Lindner and M. Farle: Magnetic Anisotropy of Heterostructures, STMP 227, 45–96 (2007) c Springer-Verlag Berlin Heidelberg 2007 DOI 10.1007/978-3-540-73462-8 2

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The purpose of this article is twofold: a) an introductory level overview on magnetic anisotropy, b) characteristic examples of current research using magnetic resonance techniques to quantitatively determine magnetic anisotropy and explore its microscopic sources. Various aspects of ultrathin ﬁlm magnetism [3] have been discussed in extensive reviews and book chapters over the last few years (see for example [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]). There is no way to summarize all these issues in such limited space and the reader is referred to the reviews mentioned before. For an overview on the technological relevance or the many experimental techniques and methods that have been developed to investigate magnetic heterostructures the reader is referred to the series of books by Heinrich and Bland [6, 7, 8]. This review also excludes laterally structured samples (for excellent reviews see e.g. [9, 10, 11]) and epitaxially grown ﬁlms comprised of two or more elements (double- tri- and multilayers, see the article of B. Heinrich in this book or [6, 7, 8, 14]) in which coupling eﬀects lead to phenomena like the tunneling magneto-resistance (TMR), the giant magnetoresistance (GMR) or spin current related eﬀects (see e.g. [16, 17, 18] for a detailed discussion) such as current induced switching [19], current induced domain wall movement or spin torque induced magnetic damping [20, 21, 22]. The examples which will be discussed here are strictly restricted to the thickness and temperature dependent magnetic anisotropy of single element ferromagnetic metallic monolayers on diﬀerent kind of single crystalline substrates (metals, semiconductors and insulators). It will be shown that epitaxial ﬁlms consisting of few atomic layers provide an interesting playground for artiﬁcially controlling magnetic properties and hence improving the understanding of the underlying physical mechanisms. This chapter is divided into two basic sections. While the ﬁrst will shortly discuss the sources of magnetic anisotropy energy (MAE), explain Ferromagnetic Resonance (FMR) and introduce a phenomenological description of the MAE and its inﬂuence on the FMR resonance equations in terms of the magnetic part of the free energy of the system, the second section will give examples of FMR investigations of heterostructures. In the framework of a tutorial description, the prototype systems Fe/MgO(001), Fe/GaAs(001) and Ni/Cu(001) were chosen.

2.2 Origin of Magnetic Anisotropy Magnetic anisotropy describes the fact that the energy of the ground state of a magnetic system depends on the direction of the magnetization. The eﬀect occurs either by rotations of the magnetization vector with respect to the external shape of the specimen (shape anisotropy) or by rotations relative to the crystallographic axes (intrinsic or magneto-crystalline anisotropy). The direction(s) with minimum energy, i.e. into which the magnetization points in the absence of external ﬁelds are called easy directions. The direction(s)

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with maximum energy are called hard direction. The MAE between two crystallographic directions is given by the work WMAE needed to rotate the magnetization from an easy direction into the other direction. The MAE is a small contribution on the order of a few μeV/atom to the total energy (several eV/atom) of a bulk crystal. To estimate the magnitude of the MAE one can use as a rule of thumb, that the lower the symmetry of the crystal or of the local electrostatic potential (crystal or ligand ﬁeld) around a magnetic moment, the larger the MAE is. This becomes evident, if one remembers that in a crystal ﬁeld of cubic symmetry the orbital magnetic moment is completely quenched in ﬁrst approximation [23]. Only by calculating in higher order (2nd) or by allowing a slight distortion of the cubic crystal a small orbital magnetic moment, i.e. a nonvanishing expectation value of the orbital momentum’s z component is recovered. Without the presence of the orbital momentum which couples the spin degrees of freedom to the spatial degrees of freedom the MAE would be zero, since the exchange interaction is isotropic. One should also note, that the easy axis can deviate from crystallographic directions as for example in the case of Gd whose easy axis is temperature dependent and lies between the c-axis and the basal plane at T = 0 K [24]. Table 2.1 gives an overview about easy and hard axes and on the magnitude of the MAE for some elementary ferromagnetic materials with diﬀerent crystal symmetry. There are fundamentally two sources of magnetic anisotropy: (i) spin-orbit(LS) interaction and (ii) the magnetic dipole-dipole interaction. From the point of view of quantummechanics both interactions are relativistic corrections to the Hamilton-operator of the system that lift the rotational invariance of the quantization axis. Despite the fact that the dipole-dipole interaction as well as the LS coupling are much weaker than the exchange interaction (≈ 1 − 100 μeV/atom compared to ≈ 0.1 eV/atom), they link the magnetic Table 2.1. Anisotropic orbital moments, direction of easy axis of the magnetization, and magnetic anisotropy energy at T = 0 K for the four elemental ferromagnets as taken from standard references like Landolt-B¨ ornstein [32]. ΔμL is the diﬀerence of the orbital magnetic moment measured along the easy and hardest magnetization direction. μtot is the sum of orbital and eﬀective spin moment. The latter includes the so-called Tz contribution entering the sum rule analysis of x-ray magnetic circular dichroism measurements Material

Easy axis

|MAE| (μeV/atom)

ΔμL /μtot

bcc Fe hcp Co fcc Co fcc Ni hcp Gd

100 0001 111 111 tilted hcp

1.4 65 1.8 2.7 50

1.7 · 10−4 4.5 · 10−4 1.8 · 10−4 ≈ 10−3

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moments to position space. In the following we will use the term magnetic anisotropy energy or MAE for the magnetic anisotropy energy density, resulting from spin-orbit interaction which includes the so-called magnetocrystalline and also the magneto-elastic contributions [15]. Contributions originating from the magnetic dipole-dipole interaction will be called shape or magneto-static anisotropy. We note that the important aspect of magnetic domain formation is not discussed here, since we restrict our discussion to the intrinsic contributions to the magnetic anisotropy and the main technique which will be discussed in detail in this chapter is ferromagnetic resonance (FMR). This technique is – in most cases – performed in large enough ﬁelds that drive the magnetic ﬁlm into a single domain state. The reader should keep in mind, however, that the interplay of magnetostatic energy, exchange energy and magnetic anisotropy in general leads to an energetically favored multi-domain state. Especially, the analysis of hysteresis loops is complicated due to domain formation at small magnetic ﬁelds. Special imaging techniques have been developed [25] to obtain quantitative understanding of the many domain conﬁgurations. Due to the single-domain description used in this chapter, aspects of conﬁgurational magnetic anisotropy [26] which appear in submicron sized nanomagnets due to small deviations from the uniform state will not be discussed. Similarly, so-called exchange anisotropies which may arise from diﬀerent exchange coupling constants along diﬀerent crystallographic directions in a crystal will not be speciﬁcally addressed, since the experimental observations can be well described by the phenomenological approach presented in Sect. 2.3.2. Also a discussion of unidirectional anisotropy or exchange bias is beyond the scope of this chapter. Excellent overviews can be found in [27, 28, 29, 30]. Finally, we note that the aspect of a non- homogeneous magnetization across the thickness of a several nanometer thick ﬁlm does not enter the following discussion. An excellent overview on the magnetization proﬁle across a thin ﬁlm and its dependence on the ﬁlm’s morphology has been recently given by Jensen and Bennemann [31]. 2.2.1 Spin-orbit Interaction In a classical picture the orbital motion of the electrons in a perfect crystal is deﬁned by the potential that is predetermined by the crystal lattice. In case that there is an interaction between the orbital motion and the spin of the electrons (i.e. when spin-orbit interaction is present), the spins and thus the magnetization become coupled to the lattice. By using perturbation theory in which the LS coupling is described as perturbation of the exchange splitting Bruno [33] showed that the energy correction and the orbital moment of the ˆ · L↓ . Here ξ is the radial part of minority spins are related as ΔELS ∝ − 41 ξ S ˆ is the unit vector along the spin direction, deterthe spin-orbit interaction, S mining the magnetization direction and L↓ the orbital angular momentum of

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49

ˆ · L↓ is the projection of L↓ on the magnetithe minority spin band, so that S zation direction. The magnetic anisotropy energy for uniaxial symmetry, for which the energy diﬀers between the directions parallel () and perpendicular (⊥) to one anisotropy axis, is given by the anisotropy of the orbital angular momentum ΔL↓ = L↓ ⊥ − L↓ (or, respectively, the anisotropy of the orbital ↓ moment Δμ↓L = μ↓L⊥ − μL ): ξ ξ ⊥ − ΔELS ∝ − ΔL↓ = − Δμ↓L . MAE = ΔELS 4 4μB

(2.1)

According to Bruno the easy axis is the one, where the orbital moment is largest. This fact which is experimentally often overlooked yields information on the intrinsic origin of the macroscopically measured MAE by straightforward SQUID magnetometry measurements along diﬀerent crystallographic axes. The saturation magnetizations along the easy and hard axes are different! The eﬀect is very small, in bulk crystals–on the order of 10−4 , but measurable, and well documented (see Table 2.1 and [34]). Here, one should note that shape anisotropy is not involved. That is to say, that when taking the shape anisotropy (see Sect. 2.3.2) into account, the equilibrium (easy) direction of magnetization in zero ﬁeld may be a hard magnetocrystalline anisotropy direction with the smaller orbital moment. While Bruno assumed a fully occupied majority spin band in his model (exchange splitting much larger than the bandwidth), this restriction was dropped in the later work of van der Laan [35], who extended Bruno’s relation by including the majority spin band orbital moment μ↑L : ξ ↓ ΔμL − Δμ↑L . (2.2) MAE ∝ − 4μB Although approaches that employ perturbation theory have the advantage of being less complex, they often yield wrong results on a quantitative basis (in most cases too large values). Ab initio theories that consider the LS coupling within a fully relativistic ansatz lead to a clear improvement. They are, however, much more elaborate as the precision of the calculation of the total energy of the system has to be very high. The reason is that the overall total energy is of the order of 1 eV/atom, while the MAE is very much smaller and of the order of several μeV/atom. Nevertheless, a considerable progress on ab-initio calculations of the MAE was achieved within the last 10−15 years (see e.g. [31, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]). The correlation between the anisotropy of the orbital angular momentum and the MAE becomes strikingly evident in experimental results on ultrathin ﬁlms [50] and few atom nanostructures of Co [51], for which orbital anisotropies up to ΔμL /μL ≈ 20% have been experimentally conﬁrmed. In general, however, one should note that a direct proportionality between ΔμL and the MAE is only correct for ΔμL → 0 [41].

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2.2.2 Dipole-Dipole Interaction The magnetic ﬁeld produced by a dipole μi at the position r i is given by: Hi (r i ) =

3 (r i · μi ) · ri μi − 3 . ri5 ri

(2.3)

Due to this ﬁeld a second dipole μj in the distance rij with respect to the ﬁrst has the energy EDip = −μj · H i : EDip =

μi · μj 3 (rij · μi ) · (r ij · μj ) − . 3 5 rij rij

(2.4)

Since the dipoles are placed on periodic positions within a crystal lattice, the axis rij connecting the two dipoles is linked to the crystallographic directions and, in fact, the interaction energy is connected to the relative orientation of the crystallographic axes and the direction of the magnetic moments. This in turn leads to magnetic anisotropy.

2.3 Models of Magnetic Anisotropy Phenomenologically, the crystallographic easy axis of the magnetization is determined by the minimum of the free energy F 1 . Before the explicit expressions of F for various crystal symmetries are discussed, the microscopic origins that explain magnetic anisotropy will be shortly described. 2.3.1 Single Ion Anisotropy The single ion anisotropy is determined by the interaction between the orbital state of a magnetic ion and the surrounding crystalline ﬁeld, when the crystal ﬁeld is very strong. The anisotropy is the result of the quenching of the orbital moment by the crystalline ﬁeld. As this ﬁeld has the symmetry of the crystal lattice, the orbital moments can be strongly coupled to the lattice. This interaction is transferred to the spin moments via the spin-orbit coupling, giving a weaker electron coupling of the spins to the crystal lattice. When an external ﬁeld is applied the orbital moments may remain coupled to the lattice whilst the spins are more free to turn. The magnetic energy depends on the orientation of the magnetization relative to the crystal axes. 1

In the chapter by B. Heinrich in this book, the Gibbs free energy U is used instead of F . As long as no external energy contributions are incorporated into F , the free energy is the appropriate thermodynamic potential. Upon inclusion of the Zeeman energy of the external ﬁeld, however, U is the relevant potential. However, to avoid confusion, we will use the term F throughout the whole chapter.

2 Magnetic Anisotropy of Heterostructures

51

In a magnetic layer, the single-ion anisotropy is present throughout its volume, and contributes in general to the volume part of the MAE. In transition metals this contribution is usually much smaller than the shape anisotropy but can be comparable in magnitude in rare earth metals. However, in some cases also for 3d metals or alloys with small magnetization (and consequently shape anisotropy) the single ion contribution might overcome the shape anisotropy, leading to perpendicular magnetic anisotropy, for which the easy axis of magnetization is aligned perpendicular to the ﬁlm surface. The single-ion anisotropy can also contribute to the surface anisotropy via N´eel interface anisotropy [52], where the reduced symmetry at the interface strongly modiﬁes the anisotropy at the interface compared to the rest of the layer [53]. One may question the relevance of this purely phenomenological approach, which has also been extended in terms of two-ion anisotropy contributions. The importance lies in the fact that a simple model for the temperature dependence of the MAE and its correlation to the temperature dependence of the magnetization can be established. More recently, Mryasov et al. [54] have put this model on solid ground by showing that a model of magnetic interactions on the basis of ﬁrst-principles calculations of non-collinear magnetic conﬁgurations in FePt eﬀectively contains the observed single-ion and two-ion contributions and explains the observed unusual scaling exponent Γ between the magnetization and the MAE (K(T ) ∝ M (T )Γ , see also Sect. 2.4). Later on, in Sect. 2.5, we discuss that for ultrathin Fe ﬁlms a single-ion model (Γ = 3) yields an excellent explanation for the measured correlation of M (T ) and K(T ). 2.3.2 Free Energy Density As stated above the magnetic anisotropy energy is the work WMAE needed to rotate the magnetization between two diﬀerent directions. If this rotation is performed at constant temperature T , the MAE is given by the diﬀerence of the free energy F of the system with the magnetization pointing along the two directions. This is easy to see when one considers that for a closed system (no exchange of particles) dF = −dW − SdT , S being the entropy, at constant T reduces to dF = −dW . Setting dW ≡ dWMAE this in turn yields 2 F2 − F1 = 1 dWMAE = MAE, where 1 and 2 denote the initial (e.g. easy) and the ﬁnal direction of the magnetization. Provided that an expression for the magnetic part of F is given for the system under consideration, it can be used to interpret the FMR data on thin magnetic ﬁlms. The phenomenological expression for F is usually found by symmetry considerations. In the following we will summarize the expressions for F of the most often used symmetries. We discuss cubic, tetragonal as well as hexagonal symmetry as these are widely found in thin ﬁlm systems. As special case of hexagonal symmetry the uniaxial one is introduced, which in form of shape anisotropy always occurs in thin ﬁlms and which due to stray ﬁeld minimization favors an easy in-plane alignment of the magnetization. We will use anisotropy constants having suﬃxes according

52

J. Lindner and M. Farle

to the symmetry they describe, e.g. K2 for uniaxial (second order) and K4 for cubic symmetry. This seems for us to be more transparent than to just number the constants in a row. Note that the latter numbering is used by other authors, so that care has to be taken when comparing results of anisotropy constants. A ﬁrst order cubic anisotropy constant (denoted as K4 within our nomenclature) is often denoted as K1 by other authors. An expression for the anisotropy ﬁeld follows from considering the torque exerted on the magnetization by the eﬀective magnetic ﬁeld within the sample. We assume for simplicity that the free energy of the system only depends on the angle θ of the magnetization with respect to an anisotropy axis. If θB is the angle of the external ﬁeld relative to this axis, the free energy can be written as F = Fa − M · B cos(θ − θB ), where the ﬁrst term is the anisotropy energy and the second the Zeeman contribution of the external ﬁeld. The equilibrium angle of M can be found from ∂F = 0 = ∂Fa + M · ∂θ ∂θ B sin(θ − θB ) = ∂Fa + |M × B|. This equation means that in equilibrium ∂θ the torque M × B due to the external ﬁeld is balanced by the torque due to the magnetic anisotropy ﬁeld given by − ∂Fa (the opposite sign indicates that ∂θ the torques are antiparallel). When B causes a turn of M of δθ, the torque a due to the anisotropy ﬁeld is proportional to δθ and given by − ∂F = c · δθ. ∂θ 2 2 Thus, for δθ → 0 c = −∂ Fa /∂θ δθ=0 and the equilibrium condition becomes c · δθ + M · B sin δθ ≈ − ∂ 2 Fa /∂θ2 δθ=0 · δθ + M · Bδθ = 0. From this equation the anisotropy ﬁeld is found to be 1 ∂ 2 Fa Ba = − · . (2.5) M ∂θ2 δθ=0 Note that the derivative has to be taken at the equilibrium angle, for which δθ = 0. It is a quite common though not the only possibility to expand the free energy of a magnetically saturated single crystal (i.e. no domain walls in the crystal) as a series of the direction cosines αi of the magnetization vector relative to a rectangular Cartesian system of coordinate axes. The direction cosines are the projections of M onto the three unit vectors deﬁning the crystal lattice and given by αi = M /M · ei (i = 1, 2, 3) where the ei are the unit vectors. According to Birss one can write [55]: F = bi αi + bij αi αj + bijk αi αj αk + bijkl αi αj αk αl + . . . .

(2.6)

This series is a direct consequence of Neumann’s principle stating that any type of symmetry which is exhibited by the point group of the crystal, i.e. by the group of symmetry operations that describe the symmetry of the unit cell of the crystal, is possesed by every physical property tensor. Thus, the limitations of crystal symmetry must be reﬂected by the tensors bijk... . Note, that the higher order terms make F to oscillate rapidly with the angular orientation of the direction of magnetization. Since this is contray to

2 Magnetic Anisotropy of Heterostructures

53

experimental observations the higher order terms must be very small. The components of the tensors bijk... transform under a rotation of coordinate axes according to the relations bijk...n = lip ljq lkr . . . lnu bpqr...u (in many cases this transformation of a tensor is used as deﬁnition of the tensor as a physical object). Note that in the equation the Einstein notation is used, i.e. when a letter occurs as a suﬃx twice in the same term on one side of the equation, summation 3 3 with 3 to that suﬃx is to be understood, i.e. 3 respect bijk...n = . . . p=1 q=1 r=1 u=1 lip ljq lkr . . . lnu bpqr...u . The matrix [lip ] describes the symmetry operation. As special 3 case the equation contains the transformation of a vector given by bi = p=1 lip bp = lip bp . As example a right-handed rotation of 180◦ about the z-axis is described by the matrix ⎞ ⎛ ⎞ ⎛ cos 180◦ −1 0 0 0 0 0 cos 180◦ 0 ⎠ = ⎝ 0 −1 0 ⎠ . (2.7) [lip ] = ⎝ 0 0 1 0 0 cos 0◦ With this it follows that the requirement that bijk... is a property tensor and invariant under all permissible symmetry operation appropriate to the particular crystal class is equivalent to the requirement that the components bijk... n satisfy the set of equations: bijk...n = σip σjq σkr . . . σnu bpqr...u .

(2.8)

All the matrices [σ] correspond to permissible symmetry operations. It can be shown that there are only 9 so-called generating matrices that are needed to describe all crystal classes, i.e. all symmetry operations of the point groups can be described by these matrices and multiplications of them. The matrices are: ⎞ ⎞ ⎛ ⎛ 100 −1 0 0 σ(unit) = ⎝ 0 1 0 ⎠ σ(inv) = ⎝ 0 −1 0 ⎠ 001 0 0 −1 ⎞ ⎞ ⎛ ⎛ −1 0 0 −1 0 0 σ(2z) = ⎝ −1 0 0 ⎠ σ(2⊥z) = ⎝ 0 1 0 ⎠ 0 0 −1 001 ⎞ ⎞ ⎛ ⎛ 1 00 10 0 ¯ ¯ σ(2z) = ⎝ 0 1 0 ⎠ (2.9) σ(2⊥z) = ⎝ 0 −1 0 ⎠ 0 01 0 0 −1 √ ⎞ ⎞ ⎛ ⎛ 1 1 − 010 2 2 3 0 √ σ(3z) = ⎝ − 21 3 − 12 0 ⎠ σ(4z) = ⎝ −1 0 0 ⎠ 001 0 01 ⎞ ⎞ ⎛ ⎛ 0 −1 0 010 ¯ σ(4z) = ⎝ 1 0 0 ⎠ σ(3dia) = ⎝ 0 0 1 ⎠ . 0 0 −1 100 The ﬁrst matrix is the unity matrix, the second describes a point inversion through the unit cell of the crystal. The next two matrices describe a twofold

54

J. Lindner and M. Farle

rotation parallel to the z-axis and to an axis perpendicular to the z-axis, respectively. The other matrices describe in analogy fourfold and threefold rotational axes (a bar on top denoting a rotation followed by a point inversion). Finally, the last matrix describes a threefold rotation parallel to a cube-body diagonal. Other symmetry operations can be written as multiplications of the above E.g. rotation parallel to the z-axis can be described

matrices. a six-fold by σ(inv) σ(2z) σ(3z) . In the following expressions for the free energy in uniaxial, hexagonal, cubic and tetragonal crystals are derived. The cartesian coordinate system chosen to describe the crystal is shown in Fig. 2.1. The system is chosen so that the z-axis coincides with the ﬁlm normal. Consequently, the x- and y-axes are located within the ﬁlm plane. To obtain expressions that are a function of the external magnetic ﬁeld B 0 and the magnetization M , the polar angles θB and θ as well as the azimuthal angles ϕB and ϕ are introduced. In case of an additional distinguished direction (like the direction of step edges), the angle δ is deﬁned with respect to the x-axis. Cubic Symmetry

For crystals of cubic symmetry the generating matrices are: σ(inv) , σ(4[001])

(4[111]) and σ . While the ﬁrst matrix describes the fact that the cubic unit cell is centro-symmetric, the second and third matrix describe fourfold rotational axes

parallel to a cube-body edge and diagonal, respectively. Using σ(inv) within (2.8) yields bijk...n = −bijk...n for all tensors of odd rank (n is an odd number), thus making them vanish. We note that this

z

qB q

B0 M

j jB

y

d x Fig. 2.1. Cartesian coordinate system used to derive the expressions for the free energy for the diﬀerent crystal symmetries

2 Magnetic Anisotropy of Heterostructures

55

also follows directly from time inversion symmetry, which is valid as magne- tocrystalline anisotropy is a static property. Using the matrices σ(4[001])

and σ(4[111]) leads to relations showing that for the tensor bij all components, for which i = j vanish, while b11 = b22 = b33 which expresses the requirement that for cubic symmetry the energy must not change upon exchanging two αi (change of equivalent cubic axes). The ﬁrst non-vanishing term is thus given by bij αi αj = b11 α2x + b11 α2y + b11 α2z . Similarly one obtains the allowed terms for bijkl . Using again the symmetry matrices one gets b1111 = b2222 = b3333 and b1122 (6) = b2233 (6) = b1133 (6) ((6) means the 6 permutation that can be the term). Then, one has bijkl αi αj αk αl = made from b1111 α4x + α4y + α4z + 6b1122 α2x α2y + α2y α2z + α2z α2x . The factor 6 arises from the multiplicity implicit in the second relations of the bijkl . The ﬁrst nonvanishing contributions for cubic crystals therefore are: Fcub = b11 α2x + α2y + α2z + b1111 α4x + α4y + α4z + 6b1122 α2x α2y + α2y α2z + α2z α2x + . . . . (2.10) Using the relations α2x + α2y + α2z = 1 and 1 − 2 α2x α2y + α2x α2z + α2y α2z = α4x + α4y + α4z yields: Fcub = K0 + K4 α2x α2y + α2x α2z + α2y α2z + K6 α2x α2y α2z . . . .

(2.11)

Here the anisotropy constants K0 = b11 + b1111 and K4 = 6b1122 − 2b1111 were deﬁned. Note that also the next higher order term was introduced, for which K6 = 3b111111 − 45b111122 + 90b112233 (see [55] for details). Cubic Crystals with (001)-orientation Considering the coordinate system of Fig. 2.1 the direction cosines can be written as αx = sin θ cos ϕ, αy = sin θ sin ϕ, αz = cos θ. In the following we identify the z-axis with the [001]-direction, the x(y)-axis with the [100]([010])direction of the cubic crystal. Inserting the expressions for the αi into (2.11) one obtains: F001 = K4 sin2 θ cos2 ϕ sin2 θ sin2 ϕ + sin2 θ sin2 ϕ cos2 θ + sin2 θ cos2 ϕ cos2 θ = K4 sin2 θ cos2 θ + sin4 θ cos2 ϕ sin2 ϕ . (2.12) This expression can be transformed into another equivalent form: F001 = K4 sin2 θ cos2 θ + sin4 θ cos2 ϕ sin2 ϕ

= K4 sin2 θ 1 − sin2 θ + sin4 θ cos2 ϕ 1 − cos2 ϕ

= K4 sin2 θ − sin4 θ cos4 ϕ − cos2 ϕ + 1

(2.13)

56

J. Lindner and M. Farle [001]

[001]

[001]

[110]

[100] [110]

[100]

[110]

Fig. 2.2. Free energy for a cubic crystal with (001)-orientation with the 100(left panel), the 111- (middle panel) and the 110-axes (right panel) being the easy ones

= K4 sin θ − sin θ 2

4

1 3 1 1 1 cos 4ϕ + cos 2ϕ + − − cos 2ϕ 8 2 8 2 2

1 = K4 sin2 θ − K4 (cos 4ϕ + 7) sin4 θ. 8 The equation shows that for K4 > 0 the 100 -directions are the easy ones, whereas for K4 < 0 the 111 -directions are the easy ones. This is visualized in Fig. 2.2, where the free energy has been plotted as polar plot using the coordinates deﬁned by Fig. 2.1 (the z-axis coincides with the [001]-direction). For the case that K6 can not be neglegted, the 110 -directions can be the easy ones (see right panel in Fig. 2.2). Table 2.2 shows the combinations of K4 and K6 and the corresponding easy, intermediate and hard directions. A graphical representation in the form of stability or ﬂow diagramms of K4 versus other anistropy constants have been published by several authors [56, 57, 58, 59]. Cubic Crystals with (011)-orientation To derive an expression for F in the case of a cubic crystal with (011)orientation, one needs to rotate the (x, y, z)-coordinate system, for which the axes are parallel to the 100 -directions, yielding a new system (x , y , z ) Table 2.2. Conditions for easy, intermediate and hard axes in cubic symmetry K4

+

+

K6

−∞ to 4 − 9K 4

4 − 9K to 4 −9K4

Easy Interm. Hard

100 110 111

100 111 110

+

−

−

−

−9K4 to −∞

−∞ to

9|K4 | to 4 9 |K4 |

9 |K4 | to +∞

111 100 110

111 110 100

110 111 100

110 100 111

9K4 4

2 Magnetic Anisotropy of Heterostructures z [001] z’ [11 1]

[0

11 ]

z [001]

57

x’ [11 2]

45°

z’

y [010]

45°

y [010]

45°

0] [11 45° y’

[0 ] 11

x, x’ [100] (001)

y’

x [100]

(011)

(001)

(111)

Fig. 2.3. Rotation of the (001)-coordinate system

with its z -axis parallel to any of the {110}-planes. As example in Fig. 2.3 this rotation is shown for the case that the z -axis becomes parallel to the [011]-direction. Still the direction cosines in the new systems are given by αx = sin θ cos ϕ, αy = sin θ sin ϕ, αz = cos θ. θ is now deﬁned with respect to the z -axis ([011]-direction), ϕ is measured against the x -axis ([100]-direction). For (2.11), however, the direction cosines with respect to the (x, y, z)-coordinate system are needed, i.e., the direction cosines within the (x, y, z)-system have to be expressed by means of the direction cosines within the (x , y , z )-system. From Fig. 2.3 the direction cosines with respect to the axes of the two systems can be deduced. The relation are√summarized √ in Table 2.3. √ This yields αx = αx = sin θ cos √ϕ, αy = 1/√ 2 αy + 1/√2 αz = 1/ 2 (sin θ sin ϕ + cos θ) , αz = −1/ 2 αy + 1/ 2 αz = 1/ 2 (− sin θ sin ϕ + cos θ). Using these direction cosines within (2.11) one derives for the free energy of an (011) oriented cubic crystal:

K4 sin2 ϕ cos4 θ + sin4 θ sin4 ϕ + sin2 (2ϕ) + sin2 (2θ) cos2 ϕ − . F011 = 4 2 (2.14) We note that this equation describes the same polar plot that (2.12) does with the diﬀerence that it has been rotated such that the [011]-direction forms the Table 2.3. Direction cosines between (001) and (011)-coordinate system

x y z

x

y

z

cos 0◦ = 1 cos 90◦ = 0 cos 90◦ = 0

cos 90◦ = 0 √ cos 45◦ = 1/ 2 √ cos 45◦ = 1/ 2

cos 90◦ = 0 √ cos 135◦ = −1/ 2 √ cos 45◦ = 1/ 2

58

J. Lindner and M. Farle [001]

[100]

[110] [111]

[110]

[100]

[110]

[112] [011]

[011]

[100]

Fig. 2.4. Free energy for a cubic crystal with (001)-orientation (uppermost panel), (111)-orientation (middle panel) and (111)-orientation (lowest panel) for K4 > 0, i.e. for the 100-directions being the easy ones. The right column shows a cut of the polar plots on the left side, for which the [001] −, [111] − and [011] −direction point out of the paper plane. In this projection the in-plane symmetry can be better seen

ﬁlm normal (see lowest panel of Fig. 2.4). The azimuthal plane according to the coordinates introduced in Fig. 2.1 shows a dominating twofold symmetry as shown in the lowest panel (right plot) of Fig. 2.4. Cubic Crystals with (111)-orientation For (111) oriented cubic crystals one changes into a coordinate system ((x, y, z) → (x , y , z )), for which the z -direction is oriented parallel to the [111]-direction (see Fig. 2.3). This time θ denotes the polar angle with

2 Magnetic Anisotropy of Heterostructures

59

Table 2.4. Direction cosines between the (001) and (111)-coordinate system x

x y z

√ cos 114.1◦ = −1/ 6 √ cos 45◦ = 1/ 2 √ cos 54.7◦ = 1/ 3

y

√ cos 114.1◦ = −1/ 6 √ cos 135◦ = −1/ 2 √ cos 54.7◦ = 1/ 3

z

√ √ cos 35.3◦ = 2/ 3 cos 90◦ = 0 √ cos 54.7◦ = 1/ 3

respect to the z -axis ([111]-direction), ϕ the one with respect to the x -axis ([11¯ 2]-direction). The direction cosines between the two systems are given in Table 2.4. In analogy to the (011)-plane one gets expressions for the direction cosines within the (x, y, z)-system as function of the α’s within the rotated system: 1 1 1 sin θ cos ϕ sin θ sin ϕ cos θ √ √ αx = − √ αx + √ αy + √ αz = − + + √ 6 2 3 6 2 3 π √ 1 = √ cos θ − 2 sin θ sin ϕ + 6 3 1 1 1 sin θ cos ϕ sin θ sin ϕ cos θ √ √ αy = − √ αx − √ αy + √ αz = − − + √ 6 2 3 6 2 3 π √ 1 = √ cos θ − 2 sin θ cos ϕ + 3 3 √ √ 2 2 1 1 αz = √ αx + √ αz = √ sin θ cos ϕ + √ cos θ 3 3 3 3 (2.15) √ 1 = √ cos θ + 2 sin θ cos ϕ . 3 Here the trigonometric expressions for sin (x ± y) = sin x cos y ± cos x sin y, cos (x ± y) = cos x cos y√∓ sin x sin y and sin (π/6) = cos (π/3) = 1/2 and cos (π/6) = sin (π/3) = 3/2 have been used. Equation (2.11) yields for the free energy of a (111)-oriented cubic crystal: √ 2 1 1 F111 = K4 cos4 θ + sin4 θ − sin3 θ cos θ cos 3ϕ . (2.16) 3 4 3 F111 is visualized in Fig. 2.4 (middle panel). Now the [111]-direction coincides with the z-axis from Fig. 2.1. In the azimuthal plane the free energy is isotropic (see right plot of the middle panel of Fig. 2.4). An angular dependence within the azimuthal plane does only occur, when the next higher order term (K6 -term in (2.11)) is considered (see also Sect. 2.3.4 for the azimuthal dependence of F ).

60

J. Lindner and M. Farle

Cubic Films with In-plane Magnetization For thin layers of a cubic material and for the case that the magnetization is conﬁned to the ﬁlm plane, the direction cosines for the (001)-, (011)- and (111)-plane take the forms listed in Table 2.5. For the free energy also listed in the table in addition to the K4 -term the next higher order term (K6 ) according to (2.11) was considered. Table 2.5 shows that for the (001)-orientation a fourfold symmetry in the plane exists, while the (011)-orientation shows also twofold terms. For (111)-oriented ﬁlms the in-plane anisotropy in ﬁrst order vanishes and only to the next higher order shows a sixfold symmetry, which, however, in most cases is very small. Tetragonal Symmetry In case of tetragonal symmetry only the (001)-orientation will be discussed as this is the one most widely dicussed one in literature. The transformation to other orientations can be performed in the same way as discussed for cubic matrices for tetragonal systems are:

(inv) symmetry.

The symmetry σ , σ(2⊥z) and σ(4z) . The rotational axis perpendicular to the [001]direction now presents only a twofold symmetry and thus the terms in the expansion of the free energy reﬂect this lowering of symmetry. The ﬁrst allowed term is b11 α2x + α2y + b33 α2z and thus, the twofold symmetry does not vanish as for cubic symmetry. The ﬁrst terms are given by: Ftet = b11 α2x + α2y + b33 α2z + b1111 α4x + α4y + b3333 α4z + + 6b1122 α2x α2y + 6b1133 α2x α2z + α2y α2z + . . . . (2.17) Table 2.5. Direction cosines and free energy for the case that the magnetization is conﬁned in the ﬁlm plane Plane

Direction cosines

Free energy

(001)

αx = cos ϕ αy = sin ϕ αz = 0

(011)

αx = cos ϕ αy = √12 sin ϕ −1 sin ϕ αz = √ 2

F001 = K4 cos2 ϕ sin2 ϕ = K44 sin2 2ϕ = K84 (1 − cos 4ϕ) F011 = K44 sin4 ϕ + 4 sin2 ϕ cos2 ϕ K6 4 2 + 4 sin ϕ cos ϕ = K44 sin4 ϕ + sin2 2ϕ 6 +K sin2 ϕ · sin2 2ϕ 16 K4 = 32 (7 − 4 cos 2ϕ − 3 cos 4ϕ) 1 K6 (2 − cos 2ϕ − 2 cos 4ϕ + cos 6ϕ) + 128 6 9 cos2 ϕ − 24 cos4 ϕ + 16 cos6 ϕ F111 = K44 + K 54 K4 K6 = 4 + 108 (1 + cos 6ϕ)

(111)

αx = αy = αz =

− cos ϕ ϕ √ √ + sin 6 2 −√ cos ϕ sin ϕ − √2 √ 6 3 √ cos ϕ 2

2 Magnetic Anisotropy of Heterostructures

61

One can see that for b11 = b33 , b1111 = b3333 and b1122 = b1133 , i.e. for the case where x-,y- and z-axes are equivalent, the for cubic symmetry expression is retained. Using the relations α2z = 1 − α2x + α2y and α2x α2z + α2y α2z = 4 1 1 4 4 2 2 2 − 2 αx + αy + αz − αx αy the equation transforms to: 4 2 2 αx + α4y + K4 α2z + K4 αx αy + K4⊥ α4z + . . . , (2.18) Ftet = K0 − K2⊥ with K0 = b11 + 3b1133 , K2⊥ = b11 − b33 , K4 = b1111 − 3b1133 , K4 = 6 (b1122 − b1133 ) and K4⊥ = b3333 − 3b1133 . A further simpliﬁcation is made 2 2 through the relation α4x + α4y = α2x + α2y − 2α2x α2y or α2x α2y = 12 α2x + α2y − 1 4 4 2 αx + αy , leading to:

1 1 Ftet = K0 − K2⊥ α2z − K4 α4x + α4y − K4⊥ α4z + . . . , 2 2

(2.19)

= b11 + 3b1122 , K2⊥ = K2⊥ − K4 = b11 − b33 + with K0 = K0 + 12 K4 6 (b1122 − b1133 ), K4 = −2K4 + K4 = −2b1111 + 6b1133 + 6 (b1122 − b1133 ) and K4⊥ = −2K4⊥ − K4 = −2b3333 + 6b1133 − 6 (b1122 − b1133 ). Using the polar coordinates according to Fig. 2.1 ﬁnally yields:

1 1 Ftet = −K2⊥ cos2 θ − K4⊥ cos4 θ − K4 (3 + cos 4ϕ) sin4 θ . 2 8

(2.20)

Uniaxial and Hexagonal Symmetry

The symmetry matrices for hexagonal systems are σ(inv) , σ(2⊥z) , σ(2z)

(3z) and σ . The matrix σ(3z) describes threefold rotational symmetry about the z-axis (note that the sixfold symmetry of the hexagonal unit cell can be described by combinations of the matrices).

As for the other cases, we have a centrosymmetrical unit cell (due to σ(inv) ) and thus all odd rank tensors van ish. The use of σ(2⊥z) within (2.8) further shows that the σ‘s are separately non-zero only if i = p or j = q or . . . In addition, the product σip σjq σkr . . . σip is –1 when the subscript 2 appears an odd number of times (note that the number of σ’s within the product must be even). Since this means that dijkl... = −dijkl... all coeﬃcients, in which the subcript 2 appears an odd number of times, must vanish. Similarly, if σ(2z) is used, it can be shown that all coeﬃcients, in which the subscript 3 appears must vanish. These two restrictions then imply that the coeﬃcients, in which any subscript appears an odd number oﬀ times vanish. Thus, the ﬁrst non-vanishing term is the same as for tetragonal symmetry, i.e. bij αi αj = b11 α2x + α2y + b33 α2z . The last matrix yields several relations between the remaining dijkl... ’s, leading ﬁnally to 2 bijkl αi αj αk αl = b1122 α2x + α2y + 6b1133 α2x + α2y α2z + b3333 α4z (see [55] for details of the calculation). Taking even the next higher order term one obtains: 2 3 Fhex = K0 + K2⊥ α2x + α2y + K4⊥ α2x + α2y + K6⊥ α2x + α2y + + K6 α2x − α2y α4x − 14α2x α2y + α4y . (2.21)

62

J. Lindner and M. Farle

One sees that for K4 << Ki<4 there is no in-plane anisotropy and the hexagonal symmetry converts to an uniaxial one. Using polar coordinates yields: Fhex = K2⊥ sin2 θ + K4⊥ sin4 θ + K6⊥ sin6 θ + K6 sin6 θ sin 6ϕ .

(2.22)

Figure 2.5 shows a polar plot according to the coordinate system of Fig. 2.1 for (left panel) K2⊥ < 0, for which the [0001]-direction is the hard one and for (right panel) K2⊥ > 0, for which the [0001]-direction is the easy one. K6 leads to the sixfold anisotropy within the azimuthal plane. Shape Anisotropy At each lattice point within a cubic lattice the dipole ﬁelds of all neighbors cancel out. This is, however, only true for an inﬁnite system. As soon as surfaces are present, magnetic poles develop and thus, the dipole-dipole interaction leads to anisotropy. Since the shape of the sample determines this anisotropy, one usually calls it shape anisotropy. In systems with reduced dimension such as thin ﬁlms the shape anisotropy might be even the dominating contribution to the overall MAE. For a thin disc (realized by the thin ﬁlm), the shape anisotropy has, phenomenologically, the form of uniaxial anisotropy (being a special case of hexagonal anisotropy): shape Funi =

μ0 N⊥ − N M 2 cos2 θ , 2

(2.23)

where N⊥ and N are the demagnetizing factors parallel and perpendicular to the ﬁlm plane. In the limit of a homogeneous disc with inﬁnite diameter N⊥ = 1 and N = 0 holds and thus shape anisotropy always favors an easy in-plane orientation of the magnetization. The only exception occurs for rough [0001]

[0001]

Fig. 2.5. Polar plot of the free energy surface for a hexagonal system for (left panel) easy axis parallel to the hexagonal axis and (right panel) perpendicular to the hexagonal axis. For uniaxial systems (K6i = 0) the polar plot is the same with the only diﬀerence that no sixfold symmetry in the basal plane perpendicular to the uniaxial axis is present

2 Magnetic Anisotropy of Heterostructures

63

ﬁlms. This situation was theoretically studied by Bruno [60], who modeled the roughness by two parameters σ and ξ, the former being the mean vertical deviation from a reference plane, the latter describing the average lateral size of ﬂat areas. The calculation shows that the stray ﬁeld energy for perpendicular magnetization of a rough ﬁlm is given by: σ σ shape Frough . (2.24) = 0.5μ0 MS2 d 1 − f 2d ξ Here d is the average thickness, i.e. the thickness of the reference plane. MS is the saturation magnetization. The function f has the value 1 for a ﬂat ﬁlm, σ = 0, and it approaches 0 for increasing roughness, σ/ξ → 1. This model shows that ﬁlm roughness gives rise to a small dipolar surface anisotropy contribution of magnitude ∝ σ/ξ that favors an out-of-plane easy axis of magnetization. Such a contribution was indeed found in [61] for rough Ni ﬁlms on Cu(001). Another point raised by Heinrich and Cochran [4] and already earlier by Benson and Mills [62] is that for very thin ﬁlms of only a few atomic layers the continuum approximation fails to describe the dipolar shape anisotropy. The discreteness of the atomic moments results in a variation of the dipolar ﬁeld across the sample which depends on the number of atomic layers involved. The dipolar ﬁeld of a given layer decreases exponentially away from its surface with a decay length corresponding to the in-plane lattice spacing. Thus, the dipolar ﬁeld inside the ﬁlm decreases when approaching the sample surface from inside the ﬁlm and the value of the average dipolar ﬁeld decreases strongly when the thickness of the ﬁlm is reduced towards the monolayer regime. The larger the lattice spacing, the stronger this eﬀect will be. The reduced dipolar ﬁeld will appear as a reduced shape anisotropy and the reduction can be written as a reduced demagnetizing factor. For bcc(001) ﬁlms the demagnetizing factor is given as N⊥ = 1 − 0.425 N , while for the more densely packed fcc(001) surface N⊥ = 1 − 0.234 , N being the number of atomic planes of the ﬁlm. The N case of hcp structure is discussed in [24]. Except for very thin ﬁlms with a thickness of few monolayers this correction is less than 1% and will thus not be considered in the following. Uniaxial Symmetry – Surface Anisotropy In thin ﬁlms the presence of the symmetry breaking surface and interface to the substrate also introduces an uniaxial anisotropy term that can be written in the form: ⊥ Funi = K2⊥ sin2 θ = K0 − K2⊥ cos2 θ = K0 − K2⊥ α2z ,

(2.25)

2

with K0 = 1 . When an uniaxial distortion of the crystal lattice is present in the volume of the material, such a term may also contribute to volume 2

Note that as a potential energy contribution F is deﬁned as energy diﬀerence, so that adding constant (angular independent) terms has no inﬂuence on the value

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anisotropy. In general, magnetic anisotropies of thin ﬁlms can be decomposed in volume contributions, that are independent of thickness and surface state and can be explained as a superposition of shape, magnetocrystalline and residual strain anisotropies, and surface contributions, which scale with 1/d and depend sensitively on the state of the surface. In some cases N´eel’s phenomenological anisotropy model has provided a useful connection between diﬀerent components of surface anisotropies. For example, the role of epitaxial strain for the anisotropies of ultrathin ﬁlms was demonstrated for the case of Co(0001) ﬁlms on W(110) by measurements of anisotropies using torsion oscillation magnetometry combined with measurement of the strain by high-angular-resolution low-energy electron diﬀraction. Up to a thickness of d = 2 nm, the ﬁlms were found to grow in a state of constant strain, governed by pseudomorphism with a growth relation [1¯100] Co [1¯10] W, which results in a true volume-type strain anisotropy. Above 2 nm, a relaxation of strain is observed which scales roughly with 1/d and, therefore, results in an apparent surface-type contribution to strain anisotropy, superimposed on a reduced volume contribution [63]. Besides uniaxial anisotropy parallel to the ﬁlm normal uniaxial contributions frequently appear along a direction in the ﬁlm plane. This can be caused e.g. by preferential interactions due to oriented hybridization at the ﬁlm-substrate interface or an uniaxial in-plane distortion in the volume. Such a term can be described by: Funi = K2 sin2 θ cos2 (ϕ − δ) , (2.26) where ϕ is measured with respect to the [100]-direction (x-axis). To include any possible in-plane easy axis the angle δ was deﬁned as shown in Fig. 2.1. shape ⊥ One should note the following: The anisotropy due to Funi , Funi (and in many cases also Funi ) are direct results of the fact that the specimen has the shape of a thin ﬁlm with interfaces that break the translational symmetry of the system. Thus, all these contributions are inherently connected with the ﬁlm surface itself and do not depend on the crystallographic direction of the ﬁlm normal. Consequently, for other crystallographic orientations of the ﬁlm, shape ⊥ no transformation of Funi , Funi and Funi has to be made, in contrast to the case of crystalline anisotropy resulting from the volume symmetry of the ﬁlm material as discussed for cubic symmetry in Sect. 2.3.2. In order to separate the diﬀerent anisotropy contributions into volume and surface contributions one needs also to perform thickness dependent measurements. The thickness dependence of each anisotropy constant can be ﬁtted by a constant term representing a volume contribution (Kiv ) and an eﬀective surface/interface contribution (Kis,eﬀ ) being proportional to 1/d where d is the thickness of the ﬁlm.

of F . Therefore, the constant term K0 does not contribute to magnetic anisotropy and can be made to vanish upon normalizing F (i.e. subtracting the constant K0 ).

2 Magnetic Anisotropy of Heterostructures

Ki = Kiv +

Kis,eﬀ d

i = 2, 4.

65

(2.27)

Magnetoelastic Contributions pt Once one realizes that the MAE is a quantity describing the interaction between the electron spin and the lattice, it is intuitively clear that changes of the lattice constant will aﬀect the magnetic properties. In a magnetized body one has energy terms that depend both on the strain and the magnetization direction: the magneto-elastic energy. Although resulting from the same origin, namely the spin-orbit interaction, magneto-elastic anisotropies only exist when stress is exerted on the magnetic system. For instance in iron, the eﬀect of tension on a single crystal is to create a preferred direction of magnetization parallel to the direction of stress. The experimentally obtained magneto-elastic constants are signiﬁcantly larger than the crystalline anisotropy constants [64]. As a consequence, even small strains may give rise to an important anisotropy contribution. Moreover, this phenomenon may be of importance in epitaxial structures, where considerable strains may result from the epitaxial growth of the ﬁlm on a substrate or adjacent layers having a diﬀerent lattice parameter. With respect to the ﬁlm material, the strain in epitaxial ﬁlms is given by η = (asub − af ilm ) /af ilm , i.e. the misﬁt is determined by the lattice constants ai , which describe the atomic distances on the relevant surface orientation. If the lattice mismatch is not too large, below a critical thickness dc (coherent regime), the misﬁt is accommodated by introducing a tensile strain η in one layer and a compressive strain in the other such that both adopt the same in-plane lattice magnetic anisotropy parameter. A reasonable epitaxial match of the ﬁlm lattice with respect to the substrate one can also be achieved by the rotation of the two lattices against each other. This happens e.g. for bccFe(001) growing on fcc-Ag(001), for which the lattices are rotated by 45◦ with respect to each other. For relatively thin ﬁlms the strain and the magnetoelastic coupling are independent of thickness. Above the critical thickness dc , it becomes energetically more favorable to introduce misﬁt dislocations, which partially accommodate the lattice misﬁt, allowing the uniform strain to be reduced (incoherent regime). In the incoherent regime, the contribution to the magneto-elastic energy contains a reciprocal thickness dependence [65]. The ﬁlm strain can be isotropic in the plane of the ﬁlm (i.e. 11 = 22 = η, where the ii are the in-plane component of the strain tensor along two axes that are perpendicular to each other) or also anisotropic (11 = 22 ) The latter might occur when preferentially oriented misﬁt dislocations have formed or a strong interface hybridization between the ﬁlm and substrate along speciﬁc directions is present. According to continuum elasticity the in-plane strain leads to an out-of-plane variation of the lattice. From the requirement of a minimum of the elastic energy one can calculate the strain component 33 perpendicular to the ﬁlm plane. Table 2.6 lists the results for several symmetries of the ﬁlm lattice. The corresponding change in volume is given by ΔV /V = (11 + 22 + 33 ).

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J. Lindner and M. Farle

Table 2.6. Calculated out-of-plane strain 33 as function of in-plane strain components 11 , 22 (from [15]). The elastic stiﬀness constants cij are tabulated in [66] Cubic (001) 1:[100], 2:[010] −c12 (11 +22 ) c11

Cubic

(011) 1: 011 , 2:[100]

Cubic (111) 1,2:⊥ arbitrary axes

Hex. (0001) 1,2:⊥ basal plane

−(c11 +c12 −2c44 )(11 +2c12 22 ) −(c11 +2c12 −2c44 )(11 +22 ) −c13 (11 +22 ) (c11 +c12 +2c44 ) (c11 +2c12 +4c44 ) c33

The magneto-elastic part to the magnetic anisotropy for a cubic system can be written as [15]: cub FMEL (2.28) = B1 11 α2x + 22 α2y + 33 α2z + 2B2 (αx αy 12 + αy αz 23 + αx αz 31 ) + . . . . For hexagonal systems the magneto-elastic contribution is [15]: hex = B1 11 α2x + 22 α2y + 12 αx αy FMEL + B2 1 − α2z 33 + B3 1 − α2z (11 + 22 )

(2.29)

+ B4 (αy αz 23 + αx αz 13 ) + . . . , where the ij (i, j = 1, 2, 3) are the strain components, the Bi the magnetoelastic coupling constants and the αi the direction cosines (see coordinate system in Fig. 2.1) given by α1 = sin θ cos ϕ, α2 = sin θ sin ϕ and α3 = cos θ. One should note that while in general the Bi of ultrathin ﬁlms are diﬀerent than in the respective bulk material one ﬁnds in the case of Ni ﬁlms that the strain induced anisotropy contributions can be explained – even as a function of temperature – by the respective bulk values [5]. In Sect. 2.5 we will, however, show that the Bi -values for Fe ﬁlms on GaAs diﬀer from the ones of Fe bulk. In order to calculate the magneto-elastic part one needs to measure the Bi (volume values for 3d ferromagnets are found in e.g. in [15]). In (2.29) if i = j, the strain is along the cubic 100 axes, for i = j the strain is along the 110 axes. While the former type of strain leads to a change of the volume of the unit lattice cell, the latter is equivalent to a shearing of the lattice keeping the volume constant.

2.3.3 Landau-Lifshitz Equation of Motion and General Resonance Equation Magnetic excitations from the ground state that occur in the microwave regime and are detected within an FMR experiment (see Sect. 2.5 for details on the FMR technique) are usually described within the Landau-Lifshitz formalism. Due to the high number of spins that take part in the absorption process and the large quantum numbers associated with it classical and quantummechanical description lead to identical results [67] and thus a classical formulation of the process is usually considered.

2 Magnetic Anisotropy of Heterostructures

67

Upon considering the total angular momentum J n as a classical vector with continuous possibilities of adjustments, the time dependence of J n is – accordn ing to Newton’s law – given by the torque D n = dJ dt . In our case the torque is given by a magnetic ﬁeld acting on the magnetic moment μn that results from J n . Considering a magnetic part of the rf-ﬁeld due to the microwave int excitation brf μn , an external ﬁeld B 0 and also an internal ﬁeld B μn within the sample (e.g. magnetic anisotropy ﬁelds) as contributions to the overall magint rf netic ﬁeld, the torque is given by μn × B μn + B 0 + bμn . Taking the two expression for the torque together and using μn = −γJ n with γ = the gyromagnetic ratio (gJ : Land´e g-factor) one obtains: 1 dμn rf = −μn × bint . μn + b0 + bμn γ dt

gJ μB

being

(2.30)

Summing internal ﬁeld up all magnetic moments yields the macroscopic B int = n B int , that acts on the total magnetic moment μ = μn . Taking t μn into account that the magnetization M is deﬁned as magnetic moment per unit volume (μt /V ) and assuming a homogeneous microwave ﬁeld brf over the sample, one has: dM = −γ M × B eﬀ . (2.31) dt Here the abbreviation B eﬀ = B int + B 0 + brf was used. The latter equation is known as the Landau-Lifshitz(LL)-equation. We note that it can be also derived in the framework of quantummechanics [68]. The LL equation can be extended to include magnetic damping, leading to a ﬁnite linewidth of the FMR signal. However, throughout this paper, which focusses on magnetic anisotropy and thus on the ﬁeld needed for resonance (resonance ﬁeld) only, damping will be neglected. A straightforward but complex way to describe FMR is to solve the LL equation for given anisotropy ﬁelds. There is, however, an alternative route, which uses the LL equation to formulate a general equation on the basis of the magnetic part of the free energy of the system. As this approach is rather general, it is described in some detail in the following. The method to calculate the resonance frequency or, equivalently, the resonance ﬁeld for the uniform FMR mode (collective precession of all magnetic moments) was introduced by Smit and Beljers and independently by Suhl ([69, 70]). In this formalism the equation of motion is described by the free energy F . The same result was obtained by Gilbert by solving a Lagrange-Equation for the motion of M [71]. The magnetization is considered as classical gyroscope with moment of inertia I. Figure 2.6 shows the transformation from the laboratory (x, y, z) coordinate system to another cartesian one (x , y , z ), in which the z -axis rotates with the magnetization. The transformation is uniquely given by the three Euler Angles ϕ, θ, ψ. 2 The kinetic energy of the system is given by Ekin = I ψ˙ + ϕ˙ cos θ . 2

From this the Lagrangian function of the system follows to be L = Ekin − Epot (ϕ, θ), Epot being the potential energy. The Langrangian

68

J. Lindner and M. Farle z z’ M y’

x’

y

x

Fig. 2.6. Euler Angles to describe the rotation of the coordinate system

d ∂L ∂L equation of motion then is dt ∂ q˙i − ∂qi = 0, where the qi are the generalized coordinates, which in our case are given by ϕ, θ and ψ. This approach yields the following three equations: ∂E d ˙ pot I ψ + ϕ˙ cos θ cos θ + =0 dt ∂ϕ ∂Epot =0 (2.32) I ψ˙ + ϕ˙ cos θ ϕ˙ sin θ + ∂θ d I ψ˙ + ϕ˙ cos θ = 0. dt

As e.g. shown in [72] the term I(ψ˙ + ϕ˙ cos θ) describes the angular momentum of the magnetization within the (x , y , z )-system and thus, the last equation shows the time invariance of the angular momentum. Considering the LLG equation of motion the angular momentum of the magnetization is given by MS /γ. Therefore, the two remaining Lagrangian equations yield: MS ˙ θ sin θ = γ MS − ϕ˙ sin θ = γ

∂F ∂ϕ ∂F . ∂θ

(2.33)

Here it was taken into account that Epot is given by the free energy F of the system. We now assume that the precession angle of the magnetization is small, so that only small variations δθ and δϕ with respect to the equilibrium orientation θ0 , ϕ0 occur. This approach must be modiﬁed for high power microwave excitations which cause large precession angles and non-linear responses. In the small angle regime we have θ = θ0 + δθ, ϕ = ϕ0 + δϕ and one can expand the ﬁrst derivatives of F around the equilibrium position into a series of δϕ and δθ, in which only the linear terms have to be considered: Fθ = Fθθ δθ + Fθϕ δϕ , Fϕ = Fϕθ δθ + Fϕϕ δϕ .

(2.34)

2 Magnetic Anisotropy of Heterostructures

69

The derivatives have to be taken at the equilibrium positions, i.e. Fθθ θ0 and so on. Periodic solutions δθ, δϕ ∝ exp(iωt) have to be found, as the deviations around the equilibrium position are driven by the periodic excitation due to ˙ the microwave ﬁeld with a frequency ω. This yields θ = iωδθ and ϕ˙ = iωδϕ. Taking into account that for small deviations sin θ0 + δθ = sin θ0 cos δθ + cos θ0 sin δθ ≈ sin θ0 + cos θ0 δθ ≈ sin θ0 , one has from (2.33): iωMS sin θ0 − Fϕθ δθ − Fϕϕ δϕ = 0 (2.35) γ iωMS (2.36) − sin θ0 − Fϕθ δϕ − Fθθ δθ = 0 . γ In matrix formulation this can witten as: 0 Fϕϕ Fϕθ − iω δθ γ · MS sin θ =0. · 0 δϕ Fθθ Fϕθ + iω · M sin θ . S γ

(2.37)

The condition for a solution is Fθ2ϕ − Fθθ Fϕϕ + ω 2 γ−2 MS2 sin2 θ0 = 0, yielding the following equation for the resonance frequency: 2 Fθθ Fϕϕ − Fθ2ϕ ω 1 ω 2 = F F . − = 0 ⇒ − F ϕϕ θθ θϕ γ γ MS2 sin2 θ0 MS sin θ0 (2.38) If one has an expression for the free energy, from which also the equilibrium angles ϕ0 and θ0 can be determined, (2.38) yields the resonance condition ω(B) or an expression for the resonance ﬁeld Bres as function of the angles of the external ﬁeld for a ﬁxed frequency. Equation (2.38) shows that FMR is sensitive to the curvature of the free energy surface. As this surface strongly depends on the anisotropy ﬁelds, FMR is a very useful tool to quantitatively determine magnetic anisotropy. We ﬁnally note that (2.38) presents a singularity at the angle θ0 = 0. This problem is removed from the resonance equation by adding the ﬁrst derivatives as discussed in [73], the improved form being: 2 2 Fθϕ ω Fϕϕ 1 cos θ0 cos θ0 Fϕ . = 2 Fθθ + F − − γ MS sin2 θ0 sin θ0 θ sin θ0 sin θ0 sin θ0 (2.39) For θ0 = π/2, i.e. for the in-plane conﬁguration (see Fig. 2.1) the latter resonance equation has the same form as the original one, since the prefactor cos θ0 / sin θ0 vanishes. Also for other angles, except θ0 = 0, the original form is still numerically correct, since at equilibrium the ﬁrst derivatives Fθ and Fϕ are zero. The original equation is, however, not convenient anymore as the

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diﬀerent terms of the free energy are mixed, covering the symmetries. The form by Baselgia et al. [73] is therefore favorable compared to the older version of the resonance equation, in particular when out-of-plane dependencies are considered. 2.3.4 Resonance Equations for Angular and Frequency Dependent FMR In the following the resonance equations for tetragonal, cubic and hexagonal ﬁlms will be explicitly given. As for thin ﬁlms shape anisotropy and uniaxial inplane terms play an important role, they are included (see (2.26) and (2.23)). Moreover, the Zeeman energy FZee = −M · B due to the presence of the external ﬁeld is included. Only the equations for the so-called ‘saturated’ modes are given, being solutions, for which the precessional motion of the magnetization is mainly determined by the external ﬁeld. So called ‘unsaturated’ (or not aligned) modes are solutions, for which the motion of the magnetization vector is strongly inﬂuenced by internal ﬁelds (e.g. anisotropy ﬁelds). Such modes usually occur for small external magnetic ﬁelds and only a numerical description is possible. One should keep in mind that in the following resonance equations not aligned modes are excluded. This implies that we set ϕB = ϕ0 for the out-of-plane geometries (meaning that the magnetization is conﬁned to the same plane, in which the external magnetic ﬁeld is varied) and θB = θ0 for the in-plane geometry.

Cubic and Tetragonal Symmetry: Out-of-plane Geometry For the out-of-plane geometry the external magnetic ﬁeld is varied in a plane that comprises the ﬁlm normal and one principal in-plane crystallographic axis (see also Fig. 2.1 for the coordinate system in use). (001)-Orientation The free energy used to derive the resonance equations for tetragonal ﬁlms (of which cubic ones are a special case) is (see Sect. 2.3.2): F = − M B0 (sin θ sin θB cos (ϕ − ϕB ) + cos θ cos θB ) μ 0 N⊥ − N M 2 − K2⊥ sin2 θ + K2 sin2 θ cos2 (ϕ − δ) − 2 1 1 4 − K4⊥ cos θ − K4 (3 + cos 4ϕ) sin4 θ. (2.40) 2 8 The equation includes cubic systems, as can be seen by setting K4⊥ = K4 = K4 . Except for a constant term, which does not lead to anisotropy this yields the expression for cubic symmetry as given in Sect. 2.3.2. Then, (2.39) for

2 Magnetic Anisotropy of Heterostructures

71

the out-of-plane geometry, for which the external ﬁeld is varied from the ﬁlm normal [001] to the [100]-direction (ϕ0 = ϕB = 0) yields: 2 K4 ω K4⊥ res − cos 2 θ0 = B⊥ cos Δθ + Meﬀ + γ M M K4 K4⊥ 0 + cos 4 θ + u1 (2.41) + M M 4K4 res cos2 θ0 cos Δθ + Meﬀ − · B⊥ M 2K4 2K4 2K4⊥ 4 0 + cos θ + + u2 − u3 , + M M M with Δθ = θ0 − θB and Meﬀ = 2KM2⊥ − μ0 N⊥ − N M denoting the eﬀective out-of-plane anisotropy ﬁeld. For Meﬀ < 0 (> 0) the easy axis of the system lies in (normal to) the ﬁlm plane. Note that according to Fig. 2.1 the angles θ of the magnetization and θB of the external ﬁeld are measured with respect to the ﬁlm normal, while the in-plane angles ϕ and ϕB were deﬁned with respect to the [100]-direction. The terms ui resulting from an uniaxial in-plane anisotropy are listed in Table 2.7. The set of the ui being appropriate to a given out-of-plane geometry is determined by the in-plane angle of the external magnetic ﬁeld ϕB (being equal to the equilibrium angle of the magnetization ϕ0 for the reasons mentioned at the beginning of Sect. 2.3.4). For cubic systems one has to set K4⊥ = K4 = K4 within the equation. For the out-of-plane geometry, for which the external ﬁeld is varied from the ﬁlm normal [001] to the [1¯ 10]-direction (ϕ0 = ϕB = −π/4) the following equation results: 2 K4 ω K4⊥ res − cos 2 θ0 = B⊥ cos Δθ + Meﬀ + γ M 2M K4 K4⊥ 0 + cos 4 θ + u1 (2.42) + M 2M K4 res · B⊥ cos2 θ0 cos Δθ + Meﬀ + M 2K4 2K4⊥ K4 4 0 + cos θ − + u2 − u3 . + M M M Again the replacement K4⊥ = K4 = K4 leads to the special case of cubic symmetry. (011) and (111)-Orientation In this case the same free energy expression is used as for the (001)-orientation (2.40) with the only diﬀerence that the cubic anisotropy contribution being proportional to K4 is now given by (2.14) for (011)-oriented ﬁlms and by (2.16)

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J. Lindner and M. Farle

in case of (111)-orientation. This yields for the case that the external ﬁeld is varied from the ﬁlm normal ([011]- or [111]-direction) to the in-plane direction ([01¯ 1] or ϕB = 90◦ in case of (011)-orientation and [1¯10] or ϕB = 90◦ for the (111)-orientation): 2

res ω = B⊥ cos Δθ + Meﬀ cos 2θ0 + a + u1 γ

res · B⊥ cos Δθ + Meﬀ cos2 θ + b + u2 − u3 .

(2.43)

While the ui are the same as given by Table 2.7, the terms a and b are listed in Table 2.8. For the case that the external ﬁeld is varied from the ﬁlm normal ([011]-direction) to the [100]-direction (ϕB = 90◦ ) diﬀerent values for a and b result which are also given in Table 2.8. Cubic and Tetragonal Symmetry: In-plane Geometry Using the free energy expressions according to Table 2.5 within the general resonance equation ((2.39)), one obtains for the case that the magnetization is restricted to the ﬁlm plane: 2 ω Bres cos Δϕ + b − u2 − c2 , = Bres cos Δϕ − Meﬀ + a − u1 γ (2.44) with Δϕ = ϕ0 −ϕB . The relations for a, b and c are summarized in Table 2.9. If 2K an uniaxial in-plane anisotropy is present, the terms u1 = M2 cos2 ϕ0 − δ 2K and u2 = M2 cos 2 ϕ0 − δ have to be added. One should note that the angles ϕ and θ are measured with respect to diﬀerent crystallographic axes for the diﬀerent orientations, i.e. θ is measured either against the [111]-, the [011]- or the [001]-direction, ϕ against the [100]direction in case of the (011)- and (001)-orientation and with respect to the [11¯ 2]-direction in case of the (111)-orientation. Table 2.7. Uniaxial in-plane terms contributing to the resonance equations for a tetragonal (cubic) thin ﬁlm with (001)-orientation. The equilibrium angle θ0 from minimizing the free energy given by (2.40) ϕB = ϕ0 u1 0 π 4 π 2

2K2 M 2K2 M 2K2 M

u2 cos2 δ cos 2θ0 cos2 4π +δ cos 2θ0 cos2 δ cos 2θ0

2K2 M 2K2 M 2K2 M

u3 cos2δ cos2 θ0 − cos 2δ 2 π cos 4 +δ cos2 θ0 +sin 2δ 2 sin δ cos2 θ0 + cos 2δ

2 K2

M2 2 K2 M2 2 K2 M2

cos2 θ0 sin2 2δ cos2 θ0 cos2 2δ cos2 θ0 sin2 2δ

2 Magnetic Anisotropy of Heterostructures

73

Table 2.8. Cubic terms within the resonance equations for (011)- and (111)-oriented cubic systems. The equilibrium angle θ0 from minimizing the free energy given by (2.40) Plane

a

[011] to[100]

√ K4 0 2 0 cos θ − 3M √θ 160 2 sin θ cos 3 0 −10 2 sin θ + 28 cos θ 4 −27 cos θ0 − K M K4 12 cos4 θ0 − 13 cos2 θ0 + 2 M

[011] ¯ to[011]

4 8 cos4 θ − 8 cos2 θ + 1 − 2K M

[111] to[1¯ 10]

0

b √ K4 −√ cos θ0 4 2 sin θ0 cos2 θ0 3M −10 2 sin θ0 + 7 cos3 θ0 − 3 cos θ0 K4 M

3 cos4 θ0 − 7 cos2 θ0 + 2

4 3 cos4 θ − 3 cos2 θ − 1 −K M

Hexagonal Symmetry For hexagonal symmetry and (0001)-oriented ﬁlms the resonance equation for the in-plane variation of the external ﬁeld (in the plane perpendicular to the c-axis) is given by the same equation as for tetragonal symmetry ((2.44)) with a and b listed in Table 2.9. For the out-of-plane geometry one can in most cases neglect the very small sixfold anisotropy in the azimuthal plane given by K6 and only consider the out-of-plane constant of highest order (K2 ). Then, the resonance equation has the form of (2.42) and (2.43) when one sets K4i = 0.

2.4 Temperature Dependence of Magnetic Anisotropy The macroscopic anisotropy energy density is temperature dependent. This statement holds for the anisotropy contributions due to dipole-dipole and spinorbit interaction. The shape anisotropy which is proportional to the square of the magnetization (see Sect. 2.3.2) vanishes at the Curie temperature TC . Table 2.9. Resonance equations for a cubic thin ﬁlm with diﬀerent crystallographic orientations as well as for an (001)-oriented tetragonal and a (0001)-oriented hexagonal system. The equilibrium angle ϕ0 from minimizing the free energy given by (2.40) Plane

a

cos 4ϕ0 + 3 cos 4ϕ0 + 3

b

K4 2K4 (001) cos 4ϕ0 2M M 2K4 (001) K4 cos 4ϕ0 M tetra. 2M K4 4 3 cos4 ϕ0 + cos2 ϕ0 − 2 K 12 cos4 ϕ0 − 11 cos2 ϕ0 + 1 (011) M M 4 (111) − K 0 M 4K4⊥ +6K6⊥ +6K6 sin 6ϕ0 36K (0001) − M6 sin 6ϕ0 − M hex.

c 0 0 0√ −

0

2K4 M

sin3ϕ0

74

J. Lindner and M. Farle

Also, the intrinsic magneto-crystalline (spin-orbit) MAE is temperature dependent (see e.g. [5, 12, 34, 55]) and vanishes at TC . This has been often overlooked in the comparison of theoretical studies (usual performed at T = 0 K) and experimental investigations (usually conducted at room temperature). In regard to the microscopic origin of MAE, i.e. the anisotropy of the orbital moment, it is surprising to experimentally measure a temperature dependence of the MAE. When one considers that the spin-orbit interaction (approximately 70 meV in 3d ferromagnets) is temperature independent, and smearing out the exchange split states at the Fermi level (order of eV) [38] does not aﬀect the easy direction of the magnetization, one has to conclude that the diﬀerence of the orbital magnetic moment along the easy and the hard magnetic axis persists above TC . Unfortunately, there is no direct evidence for this, since the magnetic moment ﬂuctuates too vividly in space and time above TC . Most techniques will measure an averaged magnetic moment only. However, susceptibility measurements and paramagnetic resonance measurements (which actually measure the susceptibility at microwave frequencies) proof the existence of atomic magnetic moments above TC even in an intinerant ferromagnet like Ni. As the magnetic moment above TC is the same (except for polarization of the conduction electrons) as the one measured below TC (for T = 0 K), it is reasonable to conclude that the orbital magnetic moment is unchanged in the paramagnetic state. A direct proof of the existence of the orbital magnetic moment and its anisotropy in the paramagnetic state is obtained by angular dependent measurements in the paramagnetic phase of a ferromagnet in magnetic ﬁelds of several kOe. To our knowledge such measurements can be performed by electron spin resonance (ESR, EPR) only [74]. Here, the deviations of the spectroscopic splitting factor which is proportional to the ratio of orbital to spin magnetic moment was found to be diﬀerent for diﬀerent crystallographic directions and could be well described in the framework of crystal ﬁeld theory. How can one resolve the conceptual problem that the macroscopically measured MAE is temperature dependent while its microscopic origin is not? The classical theory of the temperature dependence of the intrinsic anisotropy (see for example [75] and references therein) was worked out based on the assumption that around each lattice site there exists a region of short-range magnetic order in which the local anisotropy constants are temperature independent. Due to thermal motion, the local instantaneous magnetizations of these regions will be distributed randomly, and they produce the average magnetization of the crystal as a whole which vanishes at TC . This does not mean that the magnetic moment vector vanishes, but it ﬂuctuates so quickly and uncorrelated to other moments that the spatially and timely averaged moment vanishes. Hence, also the macroscopically measurable MAE vanishes, it averages out above TC . This hand-waving argument has been quantiﬁed by expanding the MAE in a series of spherical harmonics Ylm (θ, ϕ), which reﬂects the role of crystal ﬁeld and spin-orbit interaction with temperature dependent coeﬃcients k2l,m (T ):

2 Magnetic Anisotropy of Heterostructures

F =

∞ 2l

k2l,m Y2l,m .

75

(2.45)

l=0 m=−2l

From a theoretical point of view the advantage of using spherical harmonics is the fact that they are orthogonal. As anisotropy is an even function of the magnetization, polynomials oﬀ odd degree vanish in the expansion. Accordingly, the expansion for cubic systems is given by [55] Fcub = K0 + k0,0 Y0,0 + k4,0 Y4,0 + k4,4 Y4,4 + k6,0 Y6,0 + k6,4 Y6,4 1 1 (21K4 + K6 ) Y0,0 − (11K4 + K6 ) Y4,0 = K0 + (2.46) 105 55 1 2 2 (11K4 + K6 ) Y4,4 + K6 Y6,0 − K6 Y6,4 , − 9240 231 41580 while for hexagonal symmetry the expansion is given by Fhex = K0 + k0,0 Y0,0 + k2,0 Y2,0 + k4,0 Y4,0 + k6,0 Y6,0 + k6,6 Y6,6 2 (35K2⊥ + 28K4⊥ + 24K6⊥) Y0,0 = K0 + 105 2 8 (11K4⊥ + 18K6⊥ ) Y4,0 − (7K2⊥ + 8K4⊥ + 8K6⊥ ) Y2,0 + 21 385 16 1 K6⊥ Y6,0 + K6 Y6,6 , − (2.47) 231 10395 with the Y2l,m listed in Table 2.10. The equation for hexagonal symmetry includes the special case of uniaxial symmetry. For uniaxial symmetry perpendicular to the ﬁlm plane, the equation is also valid when one sets k6,6 = 0. For uniaxial anisotropy in the ﬁlm plane (see (2.26)) and perpendicular to the ﬁlm plane (see (2.25)) the expression to ﬁrst order is: Funi = k0,0 Y0,0 + k2,0 Y2,0 + k2,2 Y2,2 (2.48) 1 1 1 70K2⊥ + 35K2 Y0,0 − 14K2⊥ + 7K2 Y2,0 + K2 Y2,2 . = 105 21 6 Table 2.10. Spherical harmonics used for the expansion of the free energy of cubic, uniaxial and hexagonal systems Y0,0 = 1 Y1,0 = αz Y1,1 = αx Y1,−1 = αy 4 2 Y4,0 = 18 35α z +3 4z − 30α 2 2 Y4,4 = 105 αx + α4y − x αy 6α 2 2 Y4,−4 = 105 (4αx αy ) αx − αy

Y2,0 = 12 3α2z − 1 2 2 Y2,2 = 3 αx − αy Y2,−2 = 3 (2αx αy ) 2 1 231α6z − 315α4z + 105α Y6,0 = 16 z −5 α4x + α4y − 11α2z − 1 Y6,4 = 945 x αy 2 4α 2 2 2 945 Y6,−4 = 2 (4α x αy ) αx − αy 11αz − 1 Y6,6 = 10395 α2x − α2y α4x − 14α2xα2y + α4y Y6,−6 = 10395 (2αx αy ) 3α2x − α2y α2x − 3α2y

76

J. Lindner and M. Farle

The relationship between the temperature variations of the anisotropy coeﬃcients ki and the magnetization M was theoretically [76] and experimentally [77] found to have the form: M (T )Γ k2l,m (T ) ∝ , k2l,m (0) M (0)

(2.49)

where Γ = l(l + 1)/2, l being the order of the sperical harmonics. This gives for example k2,m ∝ M (T )3 , k4,m ∝ M (T )10 . The Callen-Callen model does not identify the microscopic origin of the anisotropy coeﬃcients, but includes the contributions from magneto-elastic as well as magnetostrictive properties entering into the spin hamiltonian through the combination of spin-orbit coupling and crystal ﬁeld splitting. As the relation above holds for the anisotropy coeﬃcients k2l,m , one has to be careful when comparing to temperature dependencies of the experimentally measured anisotropy constants Ki . The relations between the k2l,m and the Ki can be found from (2.46), (2.47) and 2.48). Assuming a typical temperature dependence of the magnetization one can plot the anisotropy coeﬃcients as shown in Fig. 2.7. One sees that the k2,0 and k4,0 decrease monotonically with increasing temperature and vanish at TC . If one confuses these temperature dependent ki,0 with the usual magnetic anisotropy parameters Ki , one would draw the conclusion that a temperature change of the easy axis of magnetization is not possible [5]. However, one ﬁnds that if one rearranges the cos and sin terms in the Spherical harmonics in terms of increasing powers that the new parameters Ki (the ones used in the experimental analysis) can vary in sign so that their temperature dependent change of sign in Co

Ki /

Ki (arb. units)

K

arb.units

k

2

k

2

4 2

K

4

0 KK2 2(T) = 1.47 k2 - 3.3 k4(T)

K4(T) = 3.85 k4(T)

-2

0

100

200 300 Temperature (K)

T

C

400

Fig. 2.7. Temperature dependence of the coeﬃcients k2l,m used when expanding the free energy into spherical harmonics and the expected experimental Ki anisotropy parameters that are coeﬃcients of an expansion into direction cosines (reproduced from [5])

2 Magnetic Anisotropy of Heterostructures

77

arb. units

or Gd can be quantitatively understood. An illustrative example is plotted in Fig. 2.7 showing that K2⊥ changes its sign. This behavior becomes also clear from (2.47), where one can see that k2,0 is a linear combination of K2⊥ and K4⊥ (for K6⊥ = 0). There are only few experimental results on the power law dependence of the second order normalized MAE on the magnetization. Γ = 2.6(5) for 5.6 monolayer (ML) Fe on Cu(100) [78] and Γ = 6.5 for W(110)/Fe 6 nm/W(110) [79] was reported. While the value for Fe on Cu(100) shows reasonable agreement with the theory, the reason for the large value in the case of Fe on W(110) is unclear. One may speculate that higher order contributions of the anisotropy were not properly accounted for. An unusual exponent Γ = 2.1 was also reported for bulk like FePt ﬁlms [80, 81] and found to be the result of delocalized induced Pt moments leading to a two-ion anisotropy. This result has been explained by ab initio electronic structure theory for L10 ordered FePt [47, 54]. The importance of separating higher order terms from second order terms for this type of analysis was recently shown by Zakeri et al. [83]. Here, the thickness of Fe layers on GaAs(001) was tuned to a critical thickness so that higher order anisotropies present in thicker layers vanished. In this case perfect agreement was found within the error bar of the experiment with the theoretical prediction Γ = 3 (Fig. 2.8). A linear power law correlation with Γ = 2.9 is observed in the experiment. The inset shows the deviations from the linear behaviour, i.e. deviations from Γ = 2.9, for other ﬁlm thicknesses in which K4 contributions become important. Similarly, the

Fig. 2.8. Temperature dependence of the uniaxial out-of-plane anisotropy K2⊥ (ﬁlled circles) and the magnetization M (open squares) for 5 ML Fe/GaAs(001). The inset shows the dependence for Fe ﬁlms from 5 ML to 20 ML (‘Reprinted ﬁgure with permission from Kh. Zakeri et al., Phys. Rev. B, Vol. 73, 052405 (2006). Copyright (2006) by the American Physical Society’)

78

J. Lindner and M. Farle

temperature dependence of the surface anisotropy and its relation to the surface magnetization following a diﬀerent temperature dependence than the bulk one have been analyzed [5].

2.5 Selected Experimental Results Before discussing the experimental results, a typical FMR-signal ist shortly explained. Within an FMR experiment, the specimen is placed in a cavity, into which microwaves are coupled that excite the magnetic system. To generate resonant absorption from the microwave ﬁeld inside the cavity, the experiment is performed in an external magnetic dc-ﬁeld that is varied while the microwave frequency is kept constant. Detailed descriptions of various setups may be found elsewhere [4, 5, 7, 14]. FMR absorption spectroscopy measures the imaginary part of the high frequency susceptibility χ = mrf /hrf . mrf is the dynamic contribution of the magnetization that is created due to the high frequency magnetic ﬁeld hrf of the microwaves and, thus, χ determines the response of the magnetic system to the excitation (see [4, 5, 7, 14] for details). A typical FMR signal from a thin ﬁlm measured at a microwave frequency of 9 GHz is shown in Fig. 2.9. While the main plot shows the derivative of the signal obtained from the lock-in detection procedure, the inset shows the integral, i.e. the absorption signal itself. Three pieces of information can be directly extracted: (i) The resonance ﬁeld Bres that includes information on the internal ﬁelds, such as anisotropy ﬁelds. (ii) The linewidth ΔB that yields information on magnetic damping and the distribution of internal magnetic

c’’ (arb. units)

0.6

0.2

0

-0.2 -0.4 -0.6

250

300

350

Bres

DBpp

300

B0|| (mT) 400 500

600

Bres

A

0

Isotropic resonance field

dc’’/dB (arb. units)

0.4

DBpp 400

450

500

550

600

B0|| (mT) Fig. 2.9. Typical FMR spectrum of a thin ﬁlm. The spectrum is measured as derivative of the high frequency susceptibility with respect to the external magnetic ﬁeld The inset shows the integral of the spectrum (reproduced from [82])

2 Magnetic Anisotropy of Heterostructures

79

ﬁelds and (iii) the intensity of the signal that is proportional to the number of magnetic moments taking part in the resonance absorption. In the following we focus only on the analysis of the resonance ﬁeld, as this quantity is the one used to investigate the MAE in thin ﬁlm systems. From Fig. 2.9 it can be seen that the resonance ﬁeld is moved away from the so-called isotropic resonance ﬁeld (dotted vertical line in Fig. 2.9), which is the ﬁeld in case that no anisotropy ﬁeld is present. For a given microwave frequency ω = 2πν, the isotropic ﬁeld is given by ω/γ = B0 . This follows directly from (2.39) when one uses the expression for the free energy F without anisotropy terms ((2.40) with Ki = N⊥ = N = 0). 2.5.1 Films on Semiconducting and Insulating Substrates In the following sections some examples of diﬀerent systems are given, for which FMR was used to determine the MAE. Only metallic thin ﬁlms were chosen that were epitaxially grown on diﬀerent kind of single crystal substrates. Fe on MgO(001) Fe ﬁlms on MgO(001) are known to be a prototype system for epitaxial growth of a metal on an insulating substrate. The main reasons for this are the rather simple preparation of monoatomically ﬂat MgO substrates and the fact that the interface hybridization and intermixing between Fe and MgO is very small [84, 85]. The lattice mismatch of the Fe and the MgO lattice is reduced due to a rotation of the Fe lattice by 45◦ with respect to the MgO (meaning that the 110 -directions of Fe are parallel to the 100 -directions of the MgO) [86]. With the lattice constants of Fe (aF e = 0.287 nm) and MgO √ a

− 2a

Fe = 3.6%. (aMgO = 0.421 nm) this leads to a lattice misﬁt of M gO aM gO We do not want to give an overview over the magnetism of epitaxial Fe ﬁlms on MgO(001). For details we refer to some of the many publications on this system [85, 87, 88, 89]. In this chapter we will focus on how FMR is used to extract the MAE in form of the anisotropy constants that were introduced in the preceeding sections. In Fig. 2.10 (a) FMR spectra of 10 nm Fe on MgO(001) are shown that were taken at room temperature and at constant microwave frequencies of 9.2 and 24 GHz. The external magnetic ﬁeld B0 was applied along the direction given in the Figure (note that the directions refer to the Fe lattice, see also the inset of Fig. 2.10 (b), where a hard sphere model of the Fe/MgO(001) system is shown). While at 24 GHz only one peak can be measured for a given angle of the external ﬁeld, at 9.2 GHz two peaks are detectable. These two peaks only appear for external ﬁeld angles close to the Fe 110 -directions, while for other angles no signal is observed. This can be seen better in Fig. 2.10 (b) and (d), where the complete in-plane angular dependence of the resonance ﬁelds measured at the two frequencies is plotted (the angle ϕ is measured with respect

J. Lindner and M. Farle

0

100

Fe[110]

b) Fe[110] (001)

Fe

80

MgO

60

[110]

f=9.2GHz T=RT

40

24 GHz@RT

9.2 GHz@RT 0

30

100

200

saturated mode unsaturated mode

20 -60

300

-30

0

30

60

jB (°)

B0 (mT) 340

c)

25

320

20

300

Bres(mT)

(GHz)

B0 || [110]

]

B0 || [100]

B0 || [110]

Bres(mT)

d''/dB0 (arb. units)

a)

[1 10

80

15

d)

Fe[110]

Fe[110]

saturated mode

f=24GHz T=RT

280 260

10 B0|| [110] (hard in-plane axis) B0|| [100] (easy axis)

5

Fe[100]

240 220

0

0

100

200 300 B0 (mT)

2.5

400

500

3.0

Fe[001]

e)

-90

out of field range

2.0

-60

-30

0

jB (°)

30

60

90

Fe[001]

f)

out of field range

2.5

f=9.2GHz T=RT

1.5

1.0

saturated mode unsaturated mode

0.5 0

-90

Fe[110]

Bres(T)

Bres(T)

2.0 1.5

f=24GHz T=RT

sat. mode [110] sat. mode [100]

1.0 0.5

-60

-30

0

qB (°)

30

60

90

0 -90 Fe[110]

-60

-30

0

qB (°)

30

60

90

Fig. 2.10. In-plane angular dependence for 10 nm Fe/MgO(001) at room temperature at (b) 9.2 GHz and (d) 24 GHz. The solid lines are ﬁts to the data (see text). a) shows typical FMR spectra measured along the given directions of the external ﬁeld, c) shows the calculated dispersion relation. In (e) and (f) the out-of-plane dependence is shown for the two frequencies

to the Fe [100]-direction). The two signals at 9.2 GHz appear within an angle range of only 10◦ around the Fe 110 -directions. At 24 GHz one clearly observes a fourfold in-plane symmetry of Bres (note that only half of the whole dependence is shown) that is slightly disturbed by a twofold one. The latter leads to somewhat smaller resonance ﬁelds along the Fe [110]-direction com-

2 Magnetic Anisotropy of Heterostructures

81

¯ The same symmetry is observed at 9.2 GHz. The minimum pared to Fe [110]. of the dependence at 24 GHz indicates that the Fe 100 -directions are the easy ones of the ﬁlm. When the external ﬁeld is varied in the plane given by either the Fe [1¯10]direction and the ﬁlm normal ([001]-direction) or the Fe [100]-direction and the ﬁlm normal, the situation is diﬀerent. Figure 2.10 (e) shows the experimental result for 9.2 GHz, where (two) resonance signals are visible only within the [1¯10]/[001]-plane, while in Fig. 2.10 (f) shows the data for 24 GHz, where the angular dependence within the [1¯ 10]/[001]-plane (ﬁlled circles) as well as in the [100]/[001]-plane (open circles) can be measured. All out-of-plane angular dependencies show a twofold symmetry with a minimum resonance ﬁeld in the plane of the ﬁlm. This immediately implies that the ﬁlm normal is the hard axis of the system. The twofold symmetry results from the eﬀective magnetization Mef f that is the sum of the demagnetizing ﬁeld μ0 M and the intrinsic uniaxial out-of-plane anisotropy ﬁeld 2K2⊥ /M (see Sect. 2.3.2). The solid lines within the dependencies shown in Fig. 2.10 (b) (d)–(f) are ﬁts of the measured dependence according to (2.42) and (2.43) in case of the out-of-plane variation and according to (2.44) in case of the in-plane dependence. The equations were used for cubic(001) systems, including the eﬀective magnetization Mef f , a fourfold term K4 as well as an uniaxial in-plane term K2 as ﬁtting parameters. For the gyromagnetic ratio γ a g-factor of 2.09 (bulk Fe) was used. The equilibrium positions for a given angle of the external ﬁeld was found from numerically minimizing the free energy. All four angular dependencies could be ﬁtted with the same set of parameters that are Mef f = 2.1 T, 2μ0 K4 /M = 55 mT and 2μ0 K2 /M = 2 mT. The ﬁrst two contributions are very close to the Fe bulk value, showing that the ﬁlm has a dominating fourfold volume anisotropy with an in-plane easy direction due to shape anisotropy. The surprising contribution is the twofold in plane one. From the ﬁt one can conclude that its easy direction is parallel to Fe [110] or parallel to MgO [100]. As MgO substrates preferentially exhibit steps parallel to this direction, the small anisotropy with twofold symmetry can be attributed to steps on the substrate. The example shows that FMR is a very sensitive tool to disentangle and quantitatively determine even small anisotropy contributions. The nature of the modes detected at the two frequencies can be ﬁnally deduced from plotting the resonance dispersion ν(B0 ) using the anisotropy constants that have been determined from the angular dependencies. The resulting dispersion is shown in Fig. 2.10 (c). The expected resonance ﬁelds can be extracted from the intersections of the horizontal dashed lines with the dispersion branches. The upper frequency branch is the one for the case that the external ﬁeld is aligned parallel to the easy 100 -directions, while the lower branch gives the dispersion for the case that the external ﬁeld is aligned parallel to the hard in-plane axes ( 110 -directions). One sees that at 24 GHz one resonance signal along the easy as well as hard direction is expected, while for the lower frequency of 9.2 GHz no signal along the easy axes can be

82

J. Lindner and M. Farle

measured as the dispersion for this case starts at frequencies above the one used for the FMR experiment. The behavior of the dispersion along the hard axes leads to the occurence of the two resonances that have been observed in experiment. The change of a negative to a positive dispersion results from the fact that at small ﬁelds a not aligned mode occurs, for which the magnetization precesses around the internal anisotropy ﬁeld. As the external ﬁeld strength increases, this mode becomes an aligned mode that precesses around the external magnetic ﬁeld. Fe on GaAs(001) Fe/GaAs(001) can be considered as a prototype system for metallic ﬁlms on a semiconducting substrate and was investigated by many groups, mainly due to its possible application within spintronic devices. A good review about the diﬀerent approaches used to grow high quality thin ﬁlms and determine their magnetic properties may be found in [90]. In [91] Fe ﬁlms grown on the Ga rich 4 × 6-reconstructed GaAs(001) surface were investigated using in situ ultrahigh vacuum (UHV)-FMR. This technique can be used within the UHV environment and thus allows for the study of ﬁlms without protective layers that usually have to be deposited on top of the ﬁlm to avoid contamination. UHV-FMR is thus well suited for the analysis of surface anisotropy that might strongly diﬀer from the volume contribution. In the insets of Fig. 2.11 typical FMR spectra at 9.3 GHz with the external ﬁeld parallel to the [1¯ 10]-direction are shown for (a) a 5 ML and (b) a 20 ML thick Fe ﬁlm grown on 4 × 6-reconstructed GaAs(001). Note the very small linewidth ΔB=1.8 mT of the bulk like 20 ML ﬁlm shown in the insets of Fig. 2.11 (b) indicating excellent magnetic homogeneity of the ﬁlms. Figure 2.11 shows the polar angular dependence of the saturated resonance signal, Bres , for (a) 5 ML Fe and (b) 20 ML Fe measured at room temperature. The solid lines are ﬁts using (2.43) for cubic symmetry (i.e. for K4 = K4⊥ ) with the parameters given in Table 2.11. The maximum of the resonance ﬁeld along the ﬁlm normal indicates that the magnetization of the ﬁlms favors an in-plane alignment at both thicknesses. For the 20 ML ﬁlm the diﬀerence between the resonance ﬁeld in parallel and perpendicular conﬁguration is larger than for the 5 ML ﬁlm implying that the thin ﬁlm has a larger anisotropy. The ﬁts directly yield the anisotropy ﬁelds 2Ki /M . The g-value in (2.43) was chosen to be g = 2.09, which is the Fe-bulk value. From the anisotropy ﬁelds the anisotropy constants were extracted using the bulk saturation magnetization (M =1.71× 106 A/m). The results for the magnetocrystalline anisotropy constants are listed in Table 2.11 for diﬀerent ﬁlm thicknesses (d=5–20 ML). The value of K2⊥ in the ﬁrst column of Table 2.11 does not dominate over the shape anisotropy given by 12 μ0 M 2 (i.e. Mef f < 0). Consequently, the magnetization lies in the ﬁlm plane for all ﬁlms. As K4 is positive for ﬁlms above 11 ML, the magnetization is aligned along the [100]-direction within

2 Magnetic Anisotropy of Heterostructures

83

1.0

9.24 GHz 4.03 GHz

d''/dB [a.u.]

1.2

Bres (T)

0.8 0.6

1

5 ML 9.24 GHz

0 -1 225 B [mT]

250

a) 5 ML

0.4 0.2 0.0

-80° -60°-40°-20° 0° 20° 40° 60° 80° polar angle q

0.9

9.24 GHz 4.03 GHz

d''/dB [a.u.]

1.2 20 ML 9.24 GHz

1 0

x32

-1

Bres (T)

-2 0

0.6

20

40 100 B [mT]

120

b) 20 ML

0.3 0.0

-80° -60°-40°-20° 0° 20° 40° 60° 80° polar angle q

Fig. 2.11. Out-of-plane angular dependence of the resonance ﬁeld for (a) 5 ML Fe and (b) 20 ML Fe/GaAs(001) measured at room temperature at microwave frequencies of (open squares) 4 GHz and (open circles) 9.2 GHz. The inset shows typical FMR spectra measured in the ﬁlm plane (θB = 90◦ ) (from [91])

the thicker ﬁlms, which is the easy axis of bulk-Fe. For thinner ﬁlms d≤7 ML a strong thickness dependent in-plane uniaxial anisotropy K2 is observed (Table 2.11). The interplay between K4 and K2 leads to a change of the easy axis from the [100]- towards the [110]-direction around d≈7 ML. Qualitatively, the strong uniaxial in-plane anisotropy may be understood by considering the twofold surface symmetry of the Fe-GaAs interface due to the 4×6 reconstruction. The rectangular surface cell is supposed to be directly connected to the Fe-Ga and Fe-As bonds at the interface and thus to the atomic conﬁguration [92]. The uniaxial anisotropy could therefore be related to an uniaxial stress within the Fe ﬁlm or to a change of the Fe band structure at the interface due to hybridization. We come back to this point later. To investigate the in-plane anisotropy in more detail the in-plane angular dependence of the resonance ﬁeld close to the [1¯10]-direction was investigated. The result is presented in Fig. 2.12 (a), where the resonance ﬁeld as a function

84

J. Lindner and M. Farle

Table 2.11. Magnetic anisotropy constants of Fe on GaAs(001) for diﬀerent thickness at room temperature. The conversion to μeV/atom is given by 1×105 J/m3 =7.4 μeV/atom Thickness (ML)

K2⊥ (105 J/m3 )

K4 (10 J/m3 )

K2 (10 J/m3 )

Bulk 20 15 11 7 6 5

– 2.68 2.88 4.8 11.1 11.52 11.53

0.47 0.46 0.44 0.3 0 0 0

– –0.043 –0.08 –0.29 –0.59 –0.85 –1.02

5

5

K4 M

K2 M

(mT)

(mT)

(mT)

–1072 –910 –900 –785 –420 –390 –390

27.5 27 26 17.6 0 0 0

– –2.5 –3.5 –6.0 –32 –56 –60

Meff 2

of the in-plane angle is plotted for the 5 ML thick Fe ﬁlm. With rotation of the magnetic ﬁeld in the ﬁlm plane the saturated resonance mode (open squares) moves to lower ﬁelds, whereas the unsaturated resonance mode (solid circles) moves to higher ﬁeld values and within 2◦ of rotation the FMR signal disappears. Upon comparison to the case of Fe/MgO(001) (see Sect. 2.5.1) this directly shows that the [1¯ 10]-direction is the hard (in-plane) axis of the system and that the lower ﬁeld mode in an unsaturated one. Using the anisotropy constants, which have been determined by the out-of-plane angular dependent measurements the ﬁts for both saturated (closed circles) as well as unsaturated (open circles) modes reproduce the in-plane angular dependence around the hard direction very well.

10 3

.

120

-47°

-10 -20

-46°

[1 1 0]

-44°

in-plane angle jB

-43°

6

5 d (ML)

10 5 0 -5 -10 -15 -20

0

x20

-5 -15

110

8 7

5

5

Bres (mT)

130

20 15 11

b)

K4 K2II K2 0.03

0.06 0.09 0.12 -1

d (nm )

K i ( eV/atom)

KI (10 J/m )

140

15

a)

0.15

Fig. 2.12. (a) In-plane angular dependence of a 5 ML thick Fe ﬁlm on GaAs(001) measured at a microwave frequency of 9.3 GHz. (b) Surface and volume anisotropies for Fe/GaAs(001). The open triangles denote the uniaxial out-of-plane anisotropy, the open circles the uniaxial in-plane anisotropy and the ﬁlled squares the fourfold anisotropy (from [91])

2 Magnetic Anisotropy of Heterostructures

85

Table 2.12. Surface/interface and volume contributions to the magnetic constants of Fe on GaAs(001) at room temperature. The conversion to μeV/atom is given by 1×10−3 J/m2 =515 μeV/atom v K2⊥ 5 (10 J/m3 )

K4v (10 J/m3 )

v K2 (10 J/m3 )

s,eﬀ K2⊥ −3 (10 J/m2 )

K4s,eﬀ (10−5 J/m2 )

s,eﬀ K2 −5 (10 J/m2 )

–1.7

0.66

0.18

1.17

–6.1

–8.9

5

5

In order to separate volume and surface/interface anisotropy contributions we plot in Fig. 2.12 (b) the anisotropy constants as a function of ﬁlm thickness according to (2.27). The volume and surface/interface anisotropy constants resulting from this analysis are summarized in Table 2.12. We start the discussion with the uniaxial out-of-plane contribution K2⊥ . The volume v contribution K2⊥ = –1.7±0.8×105 J/m3 (–13±6 μeV/atom) is very small. It v is close to the value for Au capped Fe/GaAs reported by McPhail et al. (K2⊥ 5 3 v = 1.2 ±0.7×10 J/m = 9±6 μeV/atom) [93]. The negative sign of K2⊥ indicates a preferential alignment of the magnetization in the ﬁlm plane due to this volume term, which is enhanced by the larger contribution to Mef f , mainly resulting from the shape anisotropy. The much larger surface/interface term s,eﬀ K2⊥ =1.17±0.1×10−3 J/m2 (600±50 μeV/atom) is a superposition of the Fevacuum surface and Fe-GaAs interface anisotropy. As shown by the positive s,eﬀ sign of K2⊥ the interface anisotropy favors an easy axis out-of-plane, which is well known also to be the case for thin Fe ﬁlms on Cu(001) [94]. As the interface contribution gets more important for thinner ﬁlms, the reduced value of Mef f at lower ﬁlm thicknesses is a direct result of the interface anisotropy of second order. The analysis of Fig. 2.12 (b) shows that the fourfold anisotropy vanishes below 7 ML indicating a transition from cubic to predominantly uniaxial symmetry. This transition has also been observed by Brockmann et al. at a ﬁlm thicknesses of 8±1 ML [95]. The thickness dependence of the fourfold anisotropy constant (Fig. 2.12 (b)) yields a negative surface/interface contribution (K4s,eﬀ = –6.1±0.1×10−5J/m2 =–31±1 μeV/atom and a positive volume contribution (K4v = 0.66±0.1×105 J/m3 = 4.8±0.75 μeV/atom) close to bulk iron (K4,bulk = 0.47×105 J/m3 =3.5 μeV/atom). These values indicate that the interior part of the thinner ﬁlms exhibits a rather moderate strain. K4v is responsible for the alignment of the magnetization parallel to the [100]-direction (d>7 ML), which is the easy axis for bulk-Fe. The decrease of K4 at smaller ﬁlm thickness results from the negative interface contribution K4s,eﬀ . The negative sign indicates that the 110 directions are the favored easy axes, which is diﬀerent from the bulk easy axes 100 (positive K4 ).

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Similar to the uniaxial out-of-plane anisotropy, the thickness dependence of the uniaxial in-plane anisotropy K2 shows a very small value for the v volume contribution K2 = 0.18±0.25×105J/m3 (1.3±1.85 μeV/atom), proving that the uniaxial anisotropy is an interface eﬀect. It should be noted that v within the error bar this volume value of K2 is approximately zero. The surface/interface contribution is, however, large. From the ﬁt in Fig. 2.12 (b) s,eﬀ = –8.9 ±0.4×10−5 J/m2 (–46.3±2 μeV/atom). This value one gets K2 s,eﬀ was also found for Au capped samples studied by McPhail et al. (K2 −5 2 = 10 ±1×10 J/m =52±6 μeV/atom) [93] and also by Brockmann et al. s,eﬀ = 12 ±2×10−5J/m2 =62±11 μeV/atom) [95]. The negative sign of (K2 s,eﬀ shows that the easy axis given by the uniaxial in-plane anisotropy is K2 the [110]-direction, whereas the [1¯ 10] is the hard in-plane direction of our Fe ﬁlms. One should note that some confusion concerning the identiﬁcation of the crystallographic in-plane directions occurred in the literature (see e.g. [96]), where [1¯ 10]- and [110]-direction were erroneously exchanged. The correct description is given in [97]. The change of the easy axis for thicker ﬁlms towards the [100]-direction, which occurs at about 7 ML results therefore from the increasing inﬂuence of the volume part of the fourfold anisotropy K4v .

Using a magneto-elastic model the anisotropy constants can be investigated in more detail. As shown by quantitative studies of the stress evolution during Fe deposition [98] the Fe ﬁlms were found to present a compressive stress of –3.5 GPa at the initial stage of growth (ﬁrst 2–3 ML). This stress is even larger than the one resulting from an ideal coherent growth, for which the stress would be given by the 1.36% misﬁt between Fe (a =0.2866 nm) and GaAs (a/2 =0.2827 nm) yielding a compressive stress of –2.8 GPa [98]. This enhancement was explained in terms of surface stress changes when the substrate reconstruction changes to the new interface consisting of Fe, Ga and As atoms. Within the model used in [91] Fe is assumed to be uniaxially strained at the interface. The compression of the lattice parallel to the [110]-direction cub results in a contribution of the magneto-elastic energy FMEL per unit volume to the overall free energy density (see Sect. 2.3.2 for the discussion of magnetoelastic contributions to the free energy). For Fe/GaAs(001), the strain is given by 12 parallel to the (110)-direction. This leads to a contribution of cub FMEL given by 2B2 α1 α2 12 . Note that 12 is negative in our case due to the compressive stress [98]. In [99] the magneto-elastic constants of Fe on Ga-terminated GaAs(001) were measured. For 25 nm thick Fe ﬁlms values of B1 = 3.5 × 106 J/m3 and B2 = 7.2 × 106 J/m3 were obtained, while the Fe-bulk values are given by B1 = −3.44×106 J/m3 and B2 = 7.62×106 J/m3 , respectively. Using the direction cosines α1 = sin θ cos φ and α2 =√sin θ sin φ one gets FMEL = +B2 12 along the (110)-direction (α1 √= α2 = 2/2) and FMEL = −B2 12 along the (1¯ 10)-direction (α1 = −α2 = 2/2). With B2 > 0 and 12 < 0 a total energy reduction along the (110)-direction results, whereas

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the (1¯ 10)-direction becomes a hard one in excellent agreement to our results for the ﬁlms with d<7 ML. The contribution of the magneto-elastic anisotropy is directly given by the energy diﬀerence for the two directions, i.e. MEL K2 = FMEL,110 − FMEL,1¯10 = 2B2 12 . For 12 = −1.7% (corresponding to MEL a stress of –3.5 GPa)K2 = −2.6 × 105 J/m3 results. This has to be coms,eﬀ /d pared to the experimental value for the uniaxial interface anisotropy K2 5 3 = –6.2×10 J/m , where d is the thickness of the monolayer (d = 0.1433 nm). The value is of the correct order of magnitude and thus strain at the interface plays a crucial role for the uniaxial anisotropy in the ﬁlm plane that dominates at small ﬁlm thicknesses below ≈ 7 ML. The fact that the experimental value is larger than the estimated one due to strain indicates, however, that other sources possibly contribute to the uniaxial interface anisotropy as well. One reason could be hybridization of the Fe bands with the ones of GaAs, in particular when interface intermixing is present. However, to verify this model, theoretical support is needed. While the interface obviously exhibits a strain of uniaxial character, in the volume of the ﬁlms no uniaxial strain persists as can be concluded from the v vanishing value of K2 . The experiments in [98] showed that the Fe ﬁlms stay compressed even for ﬁlm thicknesses up to 20 nm. The magnitude of this stress is small due to the incorporation of misﬁt dislocations. An x-ray absorption ﬁne structure experiment in [100] showed that a 10 ML thick ﬁlm is strained by –1.1%. Since the experiments in [91] show that no uniaxial strain is present in the volume, the strain in the volume is governed by compressive strain of biaxial character. This is consistent with a nearly cubic bcc environment of the Fe in the inner part of the ﬁlms, and – unlike at the interface – the twofold symmetry of the reconstructed GaAs substrate has no eﬀect. Hence, the inplane biaxial strain in the volume is given by 11 = 22 ≡ . In [15] it has been shown that a biaxial strain does not lead to an in-plane anisotropy for cubic (001) systems. It leads, however, to an uniaxial anisotropy contribution perpendicular to the ﬁlm surface given by B1 ( − 33 ). The strain component perpendicular to the surface 33 can be calculated from elastic theory to be 12 −2 cc11 [15]. cij are the elastic constants of the material. For Fe c12 ≈ c11 , so that 33 ≈ −2 and one obtains a (perpendicular) uniaxial anisotropy term v given by K2⊥ ≈ 3B1 . Using the experimental value for B1 = 3.5 × 106 J/m3 v from [99] and ≈ −1% according to [100] K2⊥ ≈ –1.0×105 J/m3 results. v This is within the error bar the same value as the experimental one (K2⊥ 5 3 =–1.7±0.8×10 J/m ) with the same negative sign showing that strain is the v origin for K2⊥ . The fact that the ﬁlm strain becomes smaller for thicker ﬁlms is conﬁrmed by the observation of the fourfold volume anisotropy term K4v that is very close to the bulk Fe value.

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2.5.2 Films on Metallic Surfaces The growth of metallic ﬁlms on metallic substrates will be discussed using the example of the system Ni/Cu(001). This system became the focus of many studies, as it presents an out-of-plane easy axis of magnetization (see [5]). Usually the shape anisotropy overcomes the magneto-crystalline contribution, so that the magnetization will be aligned in the ﬁlm plane. Due to the small magnetic moment of Ni, shape anisotropy for Ni ﬁlms is small. Such spin reorientation transitions (SRTs) of the magnetization in ultrathin ferromagnetic ﬁlms have attracted much interest, experimentally [101, 102, 103, 104, 105, 106] as well as theoretically [57, 107, 108, 109, 110, 111, 112]. One reason for the interest is that temperature driven SRTs may become of technological importance, leading to new magnetic thin ﬁlm sensors or switches. From a basic research point of view, one should note that the existence of a perpendicular magnetization is non trivial since the demagnetizing energy and the entropy of disorder [9] favors in-plane magnetization in thin ﬁlms. In the case of Ni/Cu(001) it is not the surface contribution to the intrinsic anisotropy, but the volume part, that lead to the easy axis out of the ﬁlm plane. The complicated interplay between all contributions can be experimentally manipulated yielding direct access to the thickness of the reorientation transition from in to out-of-plane. Ni/Cu(001) Ni ﬁlms on Cu(001) with thicknesses below 10–11 ML exhibit a ﬁlm magnetization with easy axis in the ﬁlm plane. For larger thicknesses Ni ﬁlms show a spin reorientation transition (change in sign of Mef f ), which drives the magnetization out of the ﬁlm plane [5]. This transition is shifted down to about 7.5 ML when capping the ﬁlms with Cu [113]. In addition, the TC is reduced by a Cu overlayer. The complete out-of-plane angular dependent measurement for Ni at room temperature is shown in Fig. 2.13, where Bres as a function of θB for 8 ML of Ni/Cu(001) (a) and 9 ML of Ni/Cu(001) (b) Meff,8 = -4.80(5) kG

6

Meff,9 = -1.9(1) kG

3

4 2

1

Meff,8 = 1.336(8) kG

0

5

B res (102 mT)

B res (102 mT)

8

-45°

45°

0°

qB

a) 90°

0

Meff,9 = 2.56(4) kG -90° -45°

0°

45°

b) 90°

qB

Fig. 2.13. Out-of-plane angular dependence for (left panel) 8 ML and (right panel) 9 ML Ni/Cu(001) before (ﬁlled circles) and after (open circles) capping the ﬁlms with 4 ML of Cu (reproduced from [14])

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is plotted. In each panel we show the results before (ﬁlled circles) and after capping the ﬁlm with 4 ML of Cu (open circles). For the bare Ni ﬁlms the negative values of Mef f indicate an easy axis in-plane. The smaller value of the Ni9 ﬁlm3 shows that this ﬁlm is closer to the spin reorientation at about 10–11 ML, where Mef f is close to zero and thus, the higher order anisotropy constants become important [5]. After capping the two ﬁlms with Cu the minimum of Bres moves from θB = 90◦ to θB = 0◦ . This means that the easy axis has switched from in the ﬁlm plane towards the ﬁlm normal. This change of the easy axis is also reﬂected in the positive sign of Mef f . Besides the sign of Mef f its value is larger for the Cu4 Ni9 ﬁlm compared to the Cu4 Ni8 ﬁlm, which means an increase of the out-of-plane anisotropy within the Cu capped Ni ﬁlms. This trend is continued with increasing Ni thickness (Fig. 2.14). Figure 2.14 (c) shows the dispersion curves according to (2.42) with K4i = K2 = 0 as determined from experimentally measured Bres values for Ni ﬁlms with thicknesses ranging between 8 and 12 ML already capped with Cu. The external ﬁeld was aligned along the in-plane direction (θB = 90◦ ). The points of intersection of the 9 GHz line with the dispersion yield the resonance ﬁelds. Figure 2.14 (a) shows the original FMR spectra for the smallest (8 ML) and the largest (12 ML) Ni thickness. For 12 ML two resonances are observed (unsaturated and saturated mode, see Sect. 2.5.1 for details). This behavior can be explained with an increase in the perpendicular anisotropy given by K2⊥ and corresponds to the case of a ﬁlm having an out-of-plane easy axis of magnetization. The complete analysis presented in Fig. 2.14 (c) indicates that the positive values of Mef f increase with Ni thickness. Therefore, two resonances are observed for thicknesses larger than 10 ML. The transition between observing one and two signals is observed for the 10 ML ﬁlm, which was investigated at diﬀerent temperatures in the range 300–400 K. The transition between observing one and two signals can be seen directly in Fig. 2.14 (b), where the measured resonance ﬁelds of the ﬁlm are plotted as a function of the temperature. At room temperature one observes two signals, which are also shown in the inset for T = 313 K. The resonance ﬁeld for the signal at low ﬁeld values moves towards smaller ﬁelds until it vanishes at about T = 375 K. Above 375 K only one resonance is observed as shown in the inset for T = 380 K, which stays even up to the highest temperature of 400 K. As Fig. 2.14 (d) shows, this behavior can be explained quantitatively with the help of the dispersion calulations. Due to the decrease of Mef f at higher temperatures the left branch of the dispersion curve moves below the 9 GHz line, so that only one signal is observed at larger T values. We have seen that capping the Ni ﬁlms with Cu induces a shift of the thickness, at which (for constant temperature) the reorientation of the easy axis of the magnetization occurs. Cu tends to decrease the critical thickness. This 3

In the following the number of monolayers is given as subscript.

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dc¢¢/dB (arb. units)

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dc¢¢/dB (arb. units)

Cu4Ni10/Cu(001), f=9GHz, B0 II [110]

d) 0

0

200

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B0|| (mT)

600

800

Fig. 2.14. a) Spectra for 8 ML and 12 ML thick Ni ﬁlms capped by 4 ML of Cu. b) Resonance ﬁelds as a function of the temperature for the two signals observed in a 10 ML thick Ni ﬁlm capped by 4 ML of Cu. One of the signal vanishes at higher temperatures. The inset shows the spectra at two temperatures. c) Microwave resonance frequency as a function of the external ﬁeld for Ni ﬁlms with thicknesses from 8–12 ML. d) Microwave frequency as a function of the external ﬁeld for various temperatures calculated for the same ﬁlm shown in a). The calculation was performed such that the observed resonance ﬁelds are best reproduced by the intersections of the calculated curve with the 9 GHz line being the experimental frequency. From the calculation the anisotropy is derived despite the fact that not the whole angular dependence could be measured (reproduced from [14])

eﬀect was the motivation by several groups to investigate capping layers of diﬀerent kind and their inﬂuence on the reorientation thickness. Upon capping the Ni ﬁlms with Cu and also investigating the same ﬁlm without a protective layer within an UHV environment, one can disentangle the Ni/vacuum from the Ni/Cu interface anisotropy. In [113] the inﬂuence of CO on the interface anisotropy and in [114] the Ni ﬁlms were grown on O-reconstructed Cu(001). The latter procedure interestingly leads to a chemically bonded O layer ﬂoating on top of the Ni ﬁlm. The changes of the interface anisotropy are listed in Table 2.13. One sees that the growth on the O-reconstructed surface is most eﬀective. Using H2 has the advantage that the procedure is reversible as

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Table 2.13. Interface anisotropies for Ni ﬁlms on Cu(001) with various capping int layers. The error for K2⊥ is 10% and 15% for O/Ni. The values for CO and H2 are from [113]. Note that the measurements in [113] are performed at room temperature

Interface

int K2⊥ (μeV/Atom)

dC (ML)

Vakuum/Ni Cu/Ni CO/Ni H2 /Ni O/Ni

–107 –59 –81 –70 –17

10,8 7,6 7,3 6,8 4,9

by heating the hydrogen can be driven out of the ﬁlm. This example shows that adsorbates and (or) chemically bonded layers can be eﬀectively used to inﬂuence interface anisotropies and thus taylor magnetic properties of thin ﬁlms. The reason for the out-of-plane reorientation of Ni ﬁlms can be deduced from thickness dependent measurements. The result, plotted as function of the reciprocal Ni ﬁlm thickness, for bare Ni/Cu(001) (ﬁlled circles), Cu capped ﬁlms (open circles) and ﬁlms grown on a O-reconstructed Cu(001) surface (open squares, see [114] for details) are shown in Fig. 2.15. According to (2.27) one extracts a negative surface anisotropy (that is the sum of the two interfaces) and a positive volume anisotropy that even overcomes the 25 20 15

K2^(µeV/atom)

10 0 M2 2 5 0 Ni/Cu(001) -5

CuNi/Cu(001) Ni on O/Cu(001)

-10

t=0.75

-15 -20 0

0.05

0.10 0.15 1/d (1/ML)

0.20

Fig. 2.15. Surface and volume uniaxial out-of-plane anisotropy for (ﬁlled circles) Ni/Cu(001), (open circles) Cu/Ni/Cu(001) and (open squares) Ni on O-reconstructed Cu(001) (reproduced from [82])

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shape anisotropy for thicknesses above the critical thickness of the reorientation. Within the error the volume contribution is not aﬀected by capping the Ni ﬁlm. A negative surface anisotropy favors an in-plane easy alignment and, thus, it is the volume anisotropy that leads to the out-of-plane easy axis. This is contrary to many other systems, where the surface MAE favors an out-ofplane easy axis (as e.g. for Fe/GaAs(001), see Table 2.12). In such a case, for a thickness of only a few ML, the surface contribution does not dominate anymore and the easy axis turns into the ﬁlm plane. The fact that Ni ﬁlms on Cu(001) show the opposite eﬀect make them a very interesting system. In [115] it was shown that the positive volume anisotropy is a direct consequence of the tetragonal distortion of the Ni ﬁlms, showing again that the structure and symmetry of the system plays a major role for the MAE.

Acknowledgments We thank all our past and present coworkers who have contributed to the results presented here. Although we have tried to include all the relevant literature of the many groups working in this ﬁeld, the list of references must be incomplete due to the large number of publications and we apologize for any missing citation. This work was supported by the DFG, Sfb 290 and 491.

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3 Exchange Bias Eﬀect of Ferro-/Antiferromagnetic Heterostructures Florin Radu1 and Hartmut Zabel2 1

BESSY GmbH, Albert-Einstein-Str. 15, 12489 Berlin, Germany [email protected] 2 Institut f¨ ur Experimentalphysik/Festk¨ orperphysik, Ruhr-Universit¨ at Bochum, 44780 Bochum, Germany [email protected] Abstract. The exchange bias eﬀect, discovered more than ﬁfty years ago, is a fundamental interfacial property, which occurs between ferromagnetic and antiferromagnetic materials. After intensive experimental and theoretical research over the last ten years, a much clearer picture has emerged about this eﬀect, which is of immense technical importance for magneto-electronic device applications. In this review we start with the discussion of numerical and analytical results of those models which are based on the assumption of coherent rotation of the magnetization. The behavior of the ferromagnetic and antiferromagnetic spins during the magnetization reversal, as well as the dependence of the critical ﬁelds on characteristic parameters such as exchange stiﬀness, magnetic anisotropy, interface disorder etc. are analyzed in detail and the most important models for exchange bias are reviewed. Finally recent experiments in the light of the presented models are discussed.

3.1 Fundamental Aspects of Exchange Bias: Introduction The exchange bias (EB) eﬀect was discovered 50 years ago by Meiklejohn and Bean [1]. Meanwhile the EB eﬀect has become an integral part of modern magnetism with implications for basic research and for numerous device applications. The EB eﬀect was the ﬁrst of its kind which relates to an interface eﬀect between two diﬀerent classes of materials, here between a ferromagnet and an antiferromagnet. Later on the interlayer exchange coupling between ferromagnets interleaved by paramagnet layers was discovered [2], and the proximity eﬀect between ferromagnetic and superconducting layers was described [3, 4]. Recent reviews on these topics can be found in [5, 6, 7] and also in this book. The EB eﬀect manifests itself in a shift of the hysteresis loop to negative or positive direction with respect to the applied ﬁeld. Its origin is related to the magnetic coupling across the common interface shared by a ferromagnetic (F) and an antiferromagnetic (AF) layer. Extensive research is being carried out to unveil the details of this eﬀect, which has resulted in more

F. Radu and H. Zabel: Exchange Bias Eﬀect of Ferro-/Antiferromagnetic Heterostructures, STMP 227, 97–184 (2007) c Springer-Verlag Berlin Heidelberg 2007 DOI 10.1007/978-3-540-73462-8 3

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then 600 publications in the last ﬁve years and since the last comprehensive reviews by Nogues and Schuller [8], Kiwi [9], Berkowitz [10], and Stamps [11]. An EB bilayer consists of two key elements, with rather diﬀerent magnetic and structural properties: the ferromagnetic layer and the antiferromagnetic layer. While the ferromagnetic layer can be studied in detail by using laboratory equipment like SQUID, MOKE and MFM, this is not the case for the magnetic interface and for the antiferromagnetic layer. The interface embedded in between the F and AF layer has low volume, therefore it is diﬃcult to separate its contribution from the F layer. Still, for the exchange bias eﬀect the interfacial magnetic properties are essential for understanding the eﬀect. For this purpose polarized neutron scattering and soft-xray magnetic scattering techniques can reveal some of the key magnetic properties of the interface. The AF layer has in principle no macroscopic magnetization, even so the magnetic moment of individual atoms is rather high. The magnetic properties of the AF materials are traditionally studied by neutron diﬀraction. In thin ﬁlms, due to the reduced AF volume available for scattering, this method is rather diﬃcult to apply. Here, soft-xray magnetic scattering through the linear dichroism can reveal information about the magnetic properties of the AF layer, therefore providing useful insights into the origin of the EB eﬀect. From the application point of view the situation appears to be less complex. The eﬀect is being used in spin valves with one pinned and one free ferromagnetic layer which are embedded in devices such as storage media, readout sensors, and magnetic random access memory(MRAM). Nevertheless, robust, reliable and easy to control functional elements based on the exchange bias phenomenon require more understanding of the fundamentals of the eﬀect, which is further motivating research in this ﬁeld. Because of recent experimental veriﬁcations of the existence of interfacial layers by several groups [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], earlier models of EB need to be revisited and eventually modiﬁed to take into account the eﬀects, which are introduced by this layer. Here we ﬁrst review some of the basic models for exchange bias. We focus on numerical calculations and analytical treatment of those models which are based on the Stoner-Wohlfarth model [24, 25]. This has the advantage that analytical expressions can be derived and a numerical analysis can be much more eﬃciently performed as compared to micromagnetic simulations. It has, however, the disadvantage that only coherent magnetization reversal processes are described within this formalism. Nevertheless, for a large fraction of experimental situations the Stoner-Wolhfarth approach is adequate. The behavior of the F and AF spins during the magnetization reversal, as well as the dependence of the critical ﬁelds on the parameters of the F and AF layer are analyzed in detail. The Meiklejohn and Bean [1, 26, 27] model and the Mauri model [28] are revisited and numerical and analytical expressions are compared. We continue with describing the main features of the Random Field model (RF) of Malozemoﬀ [29, 30, 31] and the Domain State (DS) model [32, 33, 34, 35, 36, 84, 38, 39]. Then, we review the Kim and Stamp

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approach [40], which focuses on a spring-like behavior of the AF layers and coercivity enhancement. We continue with one of the most recent models for exchange bias, the Spin Glass (SG) model [41]. Assuming a realistic state of the interface between the F and AF layers, the SG model describes well most of the important features of EB heterostructures, including azimuthal dependence of exchange bias ﬁeld and coercivity, AF and F thickness dependence, the inverse linear dependence on the lateral extension, and training eﬀects. Finally we will discuss recent experiments in the light of the presented models. However, the main emphasis of this review is to describe the basic models in a systematic fashion and to compare them with recent experimental results.

3.2 Stoner-Wohlfarth Model The term anisotropy refers to the orientation of the magnetic moments with respect to given geometrical directions. In bulk materials the crystal axes are the reference directions, while in thin ﬁlms other reference systems become important. In order to account for the orientation of the magnetic moments in magnetic materials, the minimum energy state is provided by analysis of the diﬀerent contributing terms to the total magnetic energy: Zeeman term, anisotropy terms, and exchange coupling terms. This evaluation is performed by minimization of the total magnetic energy with respect to various parameters. In the following we use the simplest possible expression for the total magnetic energy for a ferromagnetic thin ﬁlm and calculate the magnetic hysteresis loops. We assume that all spins are conﬁned in the ﬁlm plane and that the ﬁlm has a uniaxial anisotropy. The response of the magnetization to an applied magnetic ﬁeld is then uniform. Therefore the spins will coherently rotate during the variation of the external ﬁeld. The direction of the magnetization can be described by only one parameter, namely the (θ − β) angle deﬁning the direction of the magnetization vector with respect to the applied ﬁeld (see Fig. 3.1). Many complexities of the magnetization reversal are neglected in this approach. Nevertheless, the Stoner-Wohlfarth (SW) model [24, 25, 42], named after the investigators who developed it for treating the magnetization reversal of a small single domain, is used with considerable success for various magnetic thin ﬁlms and heterostructures. The total magnetic energy per unit volume of a ferromagnetic ﬁlm with in-plane uniaxial anisotropy reads: EV (β) = −μ0 H MF cos(θ − β) + KF sin2 (β) ,

(3.1)

where the ﬁrst term is the Zeeman energy contribution and the second term is the magnetic crystalline anisotropy (MCA), here assumed to have an uniaxial symmetry. The other parameters are: H for the applied ﬁeld, MF for the saturation magnetization of the ferromagnet, KF for the volume anisotropy

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Ferromagnetic film

KF

MF

Easy axis

q

H

Hard axis Fig. 3.1. Deﬁnition of angles and vectors used in Stoner-Wohlfarth type model calculations. The reference direction of the ﬁlm is along the unidirectional anisotropy

constant of the ferromagnet, and θ for the orientation of the applied magnetic ﬁeld with respect to the uniaxial anisotropy direction, and β the orientation of the magnetization vector during the magnetization reversal. The minimization of the total magnetic energy with respect to the angle β and the stability equation: ∂EV (β) = 0, ∂β

∂ 2 EV (β) >0, ∂β2

(3.2)

leads to the following equations: − μ0 H MF sin(θ − β) + KF sin(2 β) = 0

(3.3)

μ0 H MF cos(β − θ) + 2KF cos(2 β) > 0.

(3.4)

By solving (3.3) with the condition imposed by the (3.4) one obtains the angle β, which determines the longitudinal component (m|| = cos(β − θ)) and the transverse component (m⊥ = sin(β − θ)) of the magnetization. Both components are plotted in Fig. 3.2 for diﬀerent in-plane orientations (θ). The evolution of the hysteresis loops for diﬀerent angles θ between the applied magnetic ﬁeld and the orientation of the uniaxial anisotropy is shown in the left column of Fig. 3.2 and reﬂects the typical behavior of thin ﬁlms with in-plane uniaxial anisotropy. Along the easy axis the hysteresis loop is square shaped and the transverse component is zero. When the applied ﬁeld is perpendicular to the anisotropy axis (hard axis), the hysteresis loop has a linear slope, whereas the transverse component is ovally shaped. The expression for the coercive ﬁeld can easily be inferred from (3.3): Hc (θ) =

2 KF | cos θ| . μ0 MF

(3.5)

For the hysteresis loops shown in Fig. 3.2, the coercive ﬁeld follows in detail the expression above. At the position of the easy axis (θ = 0, π) the coercive

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Fig. 3.2. The longitudinal (left column) and transverse (right column) components of the magnetization for a ﬁlm with in-plane uniaxial anisotropy. The curves are generated by numerical evaluation of (3.3)

ﬁeld is equal to the anisotropy ﬁeld Ha = 2 KF /μ0 MF , whereas along the hard axis (θ = ±π/2) the coercive ﬁeld is zero. Experimentally, this is an often encountered situation. For instance, polycrystalline magnetic ﬁlms grown on a-plane sapphire substrates show such uniaxial and growth induced anisotropy due to steps at the substrate surface. Aside from the coercive ﬁeld dependence as a function of the azimuthal angle θ, another critical ﬁeld can be recognized in Fig. 3.2. This is the ﬁeld where the magnetization changes irreversibly (i.e. where the hysteresis opens). This irreversible switching ﬁeld Hirr can be extracted by solving both (3.3) and (3.4). Expressing the applied magnetic ﬁeld H by its components along the easy and hard axis directions: H = ( Hx , Hy ) = ( H cos θ, H sin θ ), the solution of the system of (3.3) and (3.4) gives: Hx = − Ha cos3 β and Hy = + Ha cos3 β. Eliminating β from the previous two equations we obtain the asteroid equation [43, 44, 42]: 2/3

|Hx |

2/3

+ |Hy |

= Ha2/3 .

(3.6)

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Now, introducing back into the equation above the expression for the ﬁeld components we obtain for the irreversible switching ﬁeld the following expression [42, 43, 44]: Hirr 1 = . 2 2/3 Ha [(sin θ) + (cos2 θ)2/3 ]3/2

(3.7)

This ﬁeld is plotted in the right panel of Fig. 3.3. At the position of the easy axis (θ = 0, π) the irreversible ﬁeld is equal to the anisotropy ﬁeld Ha , whereas at θ = π/4, 3π/4, the irreversible ﬁeld is equal to half of the anisotropy ﬁeld (Hirr (π/4) = Ha /2). The irreversible switching ﬁeld can be experimentally extracted from the transverse components of the magnetization, whereas the coercive ﬁeld is extracted from the longitudinal component of the magnetization(see Fig. 3.2). Figure 3.6 shows the so called asteroid curve which deﬁnes stability criteria for the magnetization reversal (3.6). The asteroid method refers to an elegant geometrical solution of (3.1) introduced by Slonczewski [43]. An extended analysis can be found in [44]. The ﬁeld measured in units of Ha appears as a point in Fig. 3.4. Given a ﬁeld P1 outside the asteroid curve, two solutions can be found by drawing tangent lines to the critical curve. Only one is a stable solution and is given by the tangent closest to the easy axis, orienting the magnetization towards the ﬁeld. For ﬁelds inside the asteroid curve (P2) four tangents leading to four solutions can be drawn. Two solutions are stable and the other two are unstable. The magnetization is stable oriented along the corresponding tangent [43, 44].

Fig. 3.3. a) The azimuthal dependence of the normalized coercive ﬁeld of a ferromagnetic ﬁlm with uniaxial anisotropy. The curve is calculated with (3.5). b) The azimuthal dependence of the normalized irreversible switching ﬁeld of a ferromagnetic ﬁlm with uniaxial anisotropy. The curve is calculated using (3.7)

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Fig. 3.4. The asteroid curve for a ﬁlm with uniaxial anisotropy. Two situations are depicted for ﬁnding a geometrical solution to the (3.6): a) The magnetic ﬁeld represented as a point P1 lying outside the asteroid region exhibits one stable solution (solid line with ﬁlled arrow) and one unstable one (solid line with open arrow). b) A magnetic ﬁeld P2 within the asteroid curve exhibits four solutions (see the tangent lines): two of them are stable (solid line with ﬁlled arrow) and the other two are unstable (solid line with open arrow). The dotted line show tangents for the unstable solutions [42, 43, 44]

3.3 Discovery of the Exchange Bias Eﬀect The exchange bias (EB) eﬀect, also known as unidirectional anisotropy, was discovered in 1956 by Meiklejohn and Bean [1, 26, 27] when studying Co particles embedded in their native antiferromagnetic oxide CoO. It was concluded from the beginning that the displacement of the hysteresis loop is brought about by the existence of an oxide layer surrounding the Co particles. This implies that the magnetic interaction across their common interface is essential in establishing the eﬀect. Being recognized as an interfacial eﬀect, the studies of the EB eﬀect have been performed mainly on thin ﬁlms consisting of a ferromagnetic layer in contact with an antiferromagnetic one. Recently, however, the lithographically prepared structures as well as F and AF particles are studied with renewed vigor.

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Fig. 3.5. a) Hysteresis loops of Co-CoO particles taken at 77◦ K. The dashed line shows the loop after cooling in zero ﬁeld. The solid line is the hysteresis loop measured after cooling the system in a ﬁeld of 10 kOe. b) Torque curve for Co particles at 300 K showing uniaxial anisotropy. b) Torque curve of Co-CoO particles taken at 77◦ K showing the unusual unidirectioal anisotropy. d) The torque magnetometer. The main component is a spring which measures the torque as a function of the θ angle on a sample placed in a magnetic ﬁeld. [1, 26]

In Fig. 3.5 the original ﬁgures from [1, 26] show the shift of a hysteresis loop of Co-CoO particles. The system was cooled from room temperature down to 77 K through the N´eel temperature of CoO (TN (CoO) = 291 K). The magnetization curve is shown in Fig. 3.5a) as a dashed line. It is symmetrically centered around zero value of the applied ﬁeld, which is the general behavior of ferromagnetic materials. When, however, the sample is cooled in a positive magnetic ﬁeld, the hysteresis loop is displaced to negative values (see continuous line of Fig. 3.5a)). Such displacement did not disappear even when extremely high applied ﬁelds of 70 000 Oe were used. In order to get more insight into this unusual eﬀect, the authors studied the anisotropy behavior by using a self-made torque magnetometer schematically shown in Fig. 3.5d). It consists of a spring connected to a sample placed in an external magnetic ﬁeld. Generally, torque magnetometry is an accurate method for measuring the magnetocrystalline anisotropy (MCA) of single crystal ferromagnets. The torque on a sample is measured as a function of the angle θ between certain crystallographic directions and the applied magnetic ﬁeld. In strong external ﬁelds, when the magnetization of the sample is almost parallel to the applied ﬁeld (saturation), the torque is equal to: T =−

∂E(θ) , ∂θ

where E(θ) is the MCA energy. In the case of Co, which has a hexagonal structure, the torque about an axis perpendicular to the c-axis follows a sin(2θ) function as seen in Fig. 3.5b). The torque and the energy density can then be written as: T = −K1 sin(2θ)

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K1 sin(2θ) dθ = K1 sin2 (θ) + K0 ,

EV =

where K1 is the MCA anisotropy and K0 is an integration constant. It is clearly seen from the energy expression that along the c-axis, at θ = 0 and θ = 180◦ , the particles are in a stable equilibrium. This typical case of a uniaxial anisotropy is seen for the Co particles at room temperature, where the CoO is in a paramagnetic state. At 77◦ K, after ﬁeld cooling, the CoO is in an antiferromagnetic state. Here, the torque curve of the Co-CoO system looks completely diﬀerent as seen in Fig. 3.5c). The torque curve is a function of sin(θ): T = −Ku sin(θ) , (3.8) hence,

EV =

Ku sin(θ) dθ = −Ku cos(θ) + K0 .

(3.9)

The energy function shows that the particles are in equilibrium for one position only, namely θ = 0. Rotating the sample to any angle, it tries to return to the original position. This direction is parallel to the ﬁeld cooling direction and such anisotropy was named unidirectional anisotropy. Now, one can analyze whether the same unidirectional anisotropy observed by torque magnetometry is also responsible for the loop shift. In Fig. 3.1 are shown schematically the vectors involved in writing the energy per unit volume for a ferromagnetic layer with uniaxial anisotropy having the magnetization oriented opposite to the ﬁeld. It reads: EV = −μ0 H MF cos(−β) + KF sin2 (β) ,

(3.10)

where H is the external ﬁeld, MF is the saturation magnetization of the ferromagnet per unit volume, and KF is the MCA of the F layer. The two terms entering in the formula above are the Zeeman interaction energy of the external ﬁeld with the magnetization of the F layer and the MCA energy of the F layer, respectively. Now, writing the stability conditions and assuming that the ﬁeld is parallel to the easy axis, we ﬁnd that the coercive ﬁeld is: Hc = 2 KF /(μ0 MF ) .

(3.11)

Next step is to cool the system down in an external magnetic ﬁeld and to introduce in (3.10) the unidirectional anisotropy term. The expression for the energy density then becomes: EV = −μ0 H MF cos(−β) + KF sin2 (β) − Ku cos(β) .

(3.12)

We notice that the solution is identical to the previous case (3.10) with the substitution of an eﬀective ﬁeld: H = H + Ku /MF . This causes the hysteresis loop to be shifted by −Ku /(μ0 MF ). Thus, Meiklejohn and Bean concluded that the loop displacement is equivalent to the explanation for the unidirectional anisotropy.

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Besides the shift of the magnetization curve and the unidirectional anisotropy, Meiklejohn and Bean have observed another eﬀect when measuring the torque curves. Their experiments revealed an appreciable hysteresis of the torque (see Fig. 3.9 and Fig.3.10 of [26] and Fig. 3.2 of [27] ), indicating that irreversible changes of the magnetic state of the sample take place when rotating the sample in an external magnetic ﬁeld. As the system did not display any rotational anisotropy when the AF was in the paramagnetic state, this provided evidence for the coupling between the AF CoO shell and the F Co core. Such irreversible changes were suggested to take place in the AF layer.

3.4 Ideal Model of the Exchange Bias: Phenomenology The macroscopic observation of the magnetization curve shift due to unidirectional anisotropy of a F/AF bilayer can qualitatively be understood by analyzing the microscopic magnetic state of their common interface. Phenomenologically, the onset of exchange bias is depicted in Fig. 3.6. A ferromagnetic layer is in close contact to an antiferromagnetic one. Their critical temperatures should satisfy the condition: TC > TN , where TC is the Curie temperature of the ferromagnetic layer and TN is the N´eel temperature of the antiferromagnetic layer. At a temperature which is higher than the N´eel temperature of the AF layer and lower than the Curie temperature of the ferromagnet (TN < T < TC ), the F spins align along the direction of the applied ﬁeld, whereas the AF spins remain randomly oriented in a paramagnetic state (see Fig 3.6(1)). The hysteresis curve of the ferromagnet is centered around zero, not being aﬀected by the proximity of the AF layer. Next, we saturate the ferromagnet by applying a high enough external ﬁeld HF C and then, without changing the magnitude or direction of the applied ﬁeld, the temperature is decreased to a ﬁnite value lower then TN (ﬁeld cooling procedure). After ﬁeld cooling the system, due to exchange interaction at the interface the ﬁrst monolayer of the AF layer will align parallel (or antiparallel) to the F spins. The next monolayer of the antiferromagnet will align antiparallel to the previous layer as to complete AF order, and so on (see Fig 3.6(2)). Note that the spins at the AF interface are uncompensated, leading to a ﬁnite net magnetization of this monolayer. It is assumed that both the ferromagnet and the antiferromagnet are in a single domain state and that they will remain in this single domain state during the magnetization reversal process. When reversing the ﬁeld, the F spins will try to rotate in-plane to the opposite direction. Being coupled to the AF spins, it takes a bigger force and therefore a stronger external ﬁeld to overcome this coupling and to rotate the ferromagnetic spins. As a result, the ﬁrst coercive ﬁeld is higher than it used to be at T > TN , where the F/AF interaction is not yet active (Fig 3.6(3)). On the way back from negative saturation to positive ﬁeld values (Fig 3.6(4)), the F spins require a smaller external force in order to rotate back (Fig 3.6(5)) to

3 Exchange Bias Eﬀect of Ferro-/Antiferromagnetic Heterostructures

M

1)

107

HFC TC

AF

H

F

FC

H

TN

3)

2)

M

4)

HEB

H

5)

0 Fig. 3.6. Phenomenological model of exchange bias for an AF-F bilayer. 1) The spin conﬁguration at a temperature which is higher than TN and smaller than TC . The AF layer is in a paramagnetic state while the F layer is ordered. Its magnetization curve (top-right) is centered around zero value of the applied ﬁeld. Panel 2): the spin conﬁguration of the AF and F layer after ﬁeld cooling the system through TN of the AF layer in a positive applied magnetic ﬁeld (HF C ). Due to uncompensated spins at the AF interface, the F layer is coupled to the AF layer. Panel 4): the saturated state at negative ﬁelds. Panel 3) and 5) show the conﬁguration of the spins during the remagnetization, assuming that this takes place through in-plane rotation of the F spins. The center of magnetization curve is displaced at negative values of the applied ﬁeld by Heb . (The description is in accordance with [1, 8, 26])

the original direction. A torque is acting on the F spins for all other angles, except the stable direction which is along the ﬁeld cooling direction (unidirectional anisotropy). As a result, the magnetization curve is shifted to negative values of the applied ﬁeld. This displacement of the center of the hysteresis loop is called exchange bias ﬁeld, and is negative in relation to the orientation of the F spins after ﬁeld cooling (negative exchange bias). It should be noted that in this simple description the AF spins are considered to be rigid and ﬁxed to the ﬁeld cooling direction during the entire reversal process.

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3.5 The Ideal Meiklejohn-Bean Model: Quantitative Analysis Based on their observation about the rotational anisotropy, Meiklejohn and Bean proposed a model to account for the magnitude of the hysteresis shift. The assumptions made are the following [1, 8, 42]: • • • •

The F layer rotates rigidly, as a whole; Both the F and AF are in a single domain state; The AF/F interface is atomically smooth; The AF layer is magnetically rigid, meaning that the AF spins remain unchanged during the rotation of the F spins; • The spins of the AF interface are fully uncompensated: the interface layer has a net magnetic moment; • The F and the AF layers are coupled by an exchange interaction across the F/AF interface. The parameter assigned to this interaction is the interfacial exchange coupling energy per unit area Jeb ; • The AF layer has an in-plane uniaxial anisotropy. In general, for describing the coherent rotation of the magnetization vector the Stoner-Wohlfarth [24, 25] model is used. Diﬀerent energy terms can be added as needed and to best account for the quantitative and qualitative behavior of the macroscopic magnetization reversal. In Fig. 3.7 is shown Ideal Meiklejohn-Bean Model KAF , KF MF H q

Fig. 3.7. Schematic view of the angles and vectors used in the ideal Meiklejohn and Bean model. The AF layer is assumed to be rigid and no deviation from its initially set orientation is allowed. K AF and K F are the anisotropy of the AF layer and F layer, respectively, which are assumed to be parallel oriented to the ﬁeld cooling direction. β is the angle between magnetization vector M F of the F layer and the anisotropy direction of the F layer. This angle is variable during the magnetization reversal. H is the external magnetic ﬁeld which can be applied at any direction θ with respect to the ﬁeld cooling direction at θ = 0 (see [1, 8, 26])

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schematically the geometry of the vectors involved in the ideal Meiklejohn and Bean model. H is the applied magnetic ﬁeld, which makes an angle θ with respect to the ﬁeld cooling direction denoted by θ = 0, KF and KAF are the uniaxial anisotropy directions of the F and the AF layer, respectively. They are assumed to be oriented parallel to the ﬁeld cooling direction. MF is the magnetization orientation of the F spins during the magnetization reversal. It is assumed that the AF spins are ﬁxed to their orientation deﬁned during the ﬁeld cooling procedure (rigid AF). In the analysis below the angle (θ = 0) for the applied ﬁeld is assumed to be parallel to the ﬁeld cooling direction. This condition refers to the direction along which the hysteresis loops is measured, whereas θ = 0 is used for torque measurements or for measuring the azimuthal dependence of the exchange bias ﬁeld. Within this model the energy per unit area assuming coherent rotation of the magnetization, can be written as [1, 8, 11]: EA = −μ0 H MF tF cos(−β) + KF tF sin2 (β) − Jeb cos(β) ,

(3.13)

where Jeb [J/m2 ] is the interfacial exchange energy per unit area, and MF is the saturation magnetization of the ferromagnetic layer. The interfacial exchange energy can be further expressed in terms of pair exchange interactions: F Eint = ij Ji j S AF i S j , where the summation includes all interactions within the range of the exchange coupling [29, 31, 45, 46]. The stability condition ∂EA / ∂θ = 0 has two types of solutions: one is β = cos−1 [(Jeb − μ0 H MF tF )/(2 KF )] for μ0 H MF tF − Jeb ≤ 2 KF ; the other one is β = 0, π for μ0 H MF tF − Jeb ≥ 2 KF , corresponding to positive and negative saturation, respectively. The coercive ﬁelds Hc1 and Hc2 are extracted from the stability equation above for β = 0, π: 2 KF tF + Jeb μ0 MF tF

(3.14)

2 KF tF − Jeb . μ 0 M F tF

(3.15)

Hc1 = − Hc2 =

Using the expressions above, the coercive ﬁeld Hc of the loop and the displacement Heb can be calculated according to: Hc =

−Hc1 + Hc2 Hc1 + Hc2 and Heb = 2 2

(3.16)

2 KF , μ0 M F

(3.17)

Jeb . μ 0 M F tF

(3.18)

which further gives: Hc = and Heb = −

Equation (3.18) is the master formula of the EB eﬀect. It gives the expected characteristics of the hysteresis loop for an ideal case, in particular the linear

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dependence on the interfacial energy Jeb and the inverse dependence on the ferromagnetic layer thickness tF . Therefore this equation serves as a guideline to which experimental values are compared. In the next section we will discuss some predictions of the model above. 3.5.1 The Sign of the Exchange Bias Equation (3.18) predicts that the sign of the exchange bias is negative. Almost all hysteresis loops shown in the literature are shifted oppositely to the ﬁeld cooling direction. The positive or negative exchange coupling across the interface produces the same (negative) sign of the exchange bias ﬁeld. There are, however, exceptions. Positive exchange bias was observed for CoO/Co, Fex Zn1−x F2 /Co and Cu1−x Mnx /Co bilayers when the measuring temperature was close to the blocking temperature [14, 47, 48, 49, 50]. At low temperatures positive exchange bias was observed in Fe/FeF2 [163] and Fe/MnF2 [51] bilayers. Speciﬁc of the last two systems is the low anisotropy of the antiferromagnet and the antiferromagnetic type of coupling between the F and AF layers. It was proposed that, at high cooling ﬁelds, the interface layer of the antiferromagnet aligns ferromagnetically with the external applied ﬁeld and therefore ferromagnetically with the F itself. As the preferred orientation between the interface spins of the F layer and AF layer is the antiparallel one (AF coupling), the EB becomes positive. Further theoretical and experimental details of the positive exchange bias mechanism are presented in [52, 53, 54]. In the original Meiklejohn and Bean model the interaction of the cooling ﬁeld with the AF spins is not taken into account. However this interaction can be easily introduced in their model. The positive exchange bias could also be accounted for in the M&B model by simply changing the sign of Jeb in (3.13) from negative to positive. 3.5.2 The Magnitude of the EB Often the exchange coupling parameter Jeb is identiﬁed with the exchange constant of the AF layer (JAF ). For various calculations a value ranging from JAF to JF was assumed. For CoO, JAF = 21.6 K = 1.86 meV [55]. Using this value, the expected exchange coupling constant Jeb of a CoO(111)/F layer can be estimated as [29, 56]: Jeb = N JAF /A = 4 mJ/m2 ,

(3.19)

2+ where N = 4 is the number of Co ions at the uncompensated CoO interface per unit area A= (3) a2 , and a = 4.27 ˚ A is the CoO lattice parameter. With this number we would expect for a 100 ˚ A thick Co layer, which shares an interface with a CoO AF layer, an exchange bias of:

Heb [Oe] =

Jeb [J/m2 ] 1011 MF [kA/m] tF [˚ A]

(3.20)

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0.004 1011 = 2740 Oe . 1460 × 100 This exchange bias ﬁeld is by far bigger than experimentally observed. So far an ideal magnitude of the EB ﬁeld as predicted by the equation (3.19) has not yet been observed, even so for some bilayers high EB ﬁelds were measured (see Table 3.1). We encounter here two problems: ﬁrst, we do not know how to evaluate the real coupling constant Jeb at the interfaces with variable degrees of complexity, and the second, in reality interfaces are never atomically smooth. The unknown interface was nicely labelled by Kiwi [9] as “a hard nut to crack”. Indeed, the features of the interfaces may be complex regarding the structure, the roughness, the magnetic properties, and domain state of the AF and F layers. In Table 3.1 are listed some EB data of systems with CoO as the AF layer. We focus on experimentally determined interfacial exchange coupling constants using Jeb = −Heb μ0 MF tF . The observed exchange coupling constant is usually smaller then the expected value of 4 mJ/m2 for CoO/Co bilayers by a factor ranging from 3 to several orders of magnitude. One anomaly is seen for the multilayer system Co/CoO which is actually ∼3 times higher then Heb =

Table 3.1. Experimental values related to Co/CoO exchange bias systems. The symbols used in the table are: ebe-electron-beam evaporation, rsp-reactive sputtering, msp-magnetron sputtering, mbe-molecular beam epitaxy, F-ferromagnet, AF antiferromagnet, tAF -the thickness of the AF, tF -the thickness of the F, Heb -measured exchange bias ﬁeld, Hc -measured coercive ﬁeld, TB -measured blocking temperature, Tmes -the measuring temperature, Jeb -the coupling energy extracted from the experimental value of exchange bias ﬁeld (Jeb = −Heb (μ0 MF tF )) AF

F

tAF tF Heb ˚] [˚ [A A] [Oe]

Hc TB Tmes Jeb Ref [Oe] [K] [K] [mJ/m2 ]

CoO (air) CoO (air) CoO (air) CoO (air) CoO (air) CoO (air) CoO (air) CoO (air) CoO (air) CoO (air) CoO (air) CoO (rsp) [CoO (rsp) CoO (in-situ) CoO(111)(mbe)

Co(rsp) Co(rsp) Co(rsp) Co(rsp) Co(rsp) Co(rsp) Co(rsp) Co(rsp) Co(rsp) Co(msp) Co(msp) Co(rsp) Co(rsp)]x25 Co(ebe) Fe(110)(mbe)

20 25 25 25 25 25 25 25 25 33 33 20 70 20 200

NA 3683 1751 1315 901 789 427 346 290 325 NA 295 5000 330 520

40 27 56 87 119 153 260 320 398 139 139 150 37 160 150

–3000 –2321 –1073 –675 –557 –443 –251 –202 –174 –145 –50 –25 –2500 –220 –150

180 180 180 180 180 180 180 180 180 291

4.2 10 10 10 10 10 10 10 10 5 30 20 5 10 10

1.75 0.91 0.88 0.86 0.97 0.99 0.95 0.94 1.00 0.29 0.1 0.055 13.5! 0.51 0.4

[57] [14] [14] [14] [14] [14] [14] [14] [14] [58] [59] [47] [60] [61] Sect. [62, 63]

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the expected value of 4 mJ/m2 , and to our knowledge is the highest value observed experimentally [60]. Such a variation of the experimental values for the interfacial exchange coupling constant is motivating further considerations of the mechanisms controlling the EB eﬀect. 3.5.3 The 1/tF Dependence of the EB Field Equation (3.18) predicts that the variation of the EB ﬁeld is proportional to the inverse thickness of the ferromagnet: Heb ≈

1 . tF

(3.21)

This dependence was subject of a large number of experimental investigations [8], because it is associated with the interfacial nature of the exchange bias eﬀect. For the CoO/Co bilayers no deviation was observed [61], even for very low thicknesses (2 nm) of the Co layer [14]. For other systems with thin F layers of the order of several nanometers it was observed that the 1/tF law is not closely obeyed [8]. It was suggested that the F layer is no longer laterally continuous [8]. Deviations from 1/tF dependence for the other extreme when the F layer is very thick were observed as well [8]. For this regime it is assumed that for F layers thicker than the domain wall thickness (500 nm for permalloy), the F spins may vary appreciably across the ﬁlm upon the magnetization reversal [64].

3.5.4 Coercivity and Exchange Bias According to (3.17) the coercivity of the magnetic layer is the same with and without exchange bias eﬀect. This contradicts experimental observations. Usually an increase of the coercive ﬁeld is observed.

3.6 Realistic Meiklejohn and Bean Model In [26] a new degree of freedom for the AF spins was introduced: the AF is still rigid, but it can slightly rotate during the magnetization reversal as a whole as indicated in Fig. 3.8. This parameter was introduced in order to account for the rotational hysteresis observed during the torque measurements. Allowing the AF layer to rotate is not in contradiction to the rigid state of the AF layer, because it is allowed only to rotate as a whole. Therefore, the fourth assumption of the ideal M&B model in Sect. 3.5 is removed. The new condition for the AF spins is: α = 0. With this new assumption, the equation (3.13) reads [8, 27]:

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Meiklejohn-Bean Model KAF , KF MAF

MF H q

Fig. 3.8. Schematic view of the angles and vectors used for the Meiklejohn and Bean model, allowing a rotation α of the AF layer as a whole with respect to the initially set orientation. M AF is the sublattice magnetization of the AF layer. K AF and K F are the anisotropy of the AF layer and F layer, respectively, which are assumed to be parallel oriented to the ﬁeld cooling direction. β is the angle between F magnetization vector M F and the anisotropy direction of the F layer. This angle is variable during the magnetization reversal. H is the external magnetic ﬁeld which can be applied at any direction θ with respect to the ﬁeld cooling direction at θ = 0 ([1, 8, 26, 27])

EA = −μ0 H MF tF cos(θ − β) +KF tF sin2 (β) + KAF tAF sin2 (α) −Jeb cos(β − α),

(3.22)

where tAF is the thickness of the antiferromagnet, and KAF is the MCA of the AF layer per unit area. The new energy term in the equation above as compared to (3.13) is the anisotropy energy of the AF layer. Equation (3.22) above can be analyzed numerically by minimization of the energy in respect to the α and the β angles. Below we will perform a numerical analysis of (3.23) and highlight a few of the conclusions discussed in [26, 27, 42]. The minimization with respect to α and β leads to a system of two equations: H ∞ sin(θ − β) + sin(β − α) = 0 Heb R sin(2 α) − sin(β − α) = 0 . where ∞ Heb ≡ −

Jeb μ 0 M F tF

(3.23) (3.24)

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is the value of the exchange bias ﬁeld when the anisotropy of the AF is inﬁnitely large, and KAF tAF R ≡ , (3.25) Jeb is the parameter deﬁning the ratio between the AF anisotropy energy and the interfacial exchange energy Jeb . As we will see further below, exchange bias is only observed, if the AF anisotropy energy is bigger than the exchange energy. The unknown variables α and β are numerically extracted as a function of the applied ﬁeld H. Note that for clarity reasons the anisotropy of the ferromagnet was neglected (KF = 0) in the system of equation above. As a result the coercivity, which will be discussed further below, is not related to the F layer anymore, but to the AF layer alone. Also, in order to simplify the discussion we consider ﬁrst the case θ = 0, which corresponds to measuring a hysteresis loop parallel to the ﬁeld cooling direction. Numerical evaluation of the (3.23) yields the angles: • α of the AF spins as a function of the applied ﬁeld during the hysteresis measurement • β of the F spins which rotate coherently during their reversal The β angle deﬁnes completely the hysteresis loop and at the same time the coercive ﬁelds Hc1 and Hc2 . These ﬁelds, in turn, deﬁne the coercive ﬁeld Hc and the exchange bias ﬁeld Heb (see equation (3.16)). The α angle inﬂuences the shape of the hysteresis loops when the R-ratio has low values, as we will see below. For high R values the rotation angle of the antiferromagnet is close ∞ to zero, giving a maximum exchange bias ﬁeld equal to Heb . The properties of the EB system originate from the properties of the AF layer, which are accounted for by one parameter, the R-ratio. We will consider the eﬀect of the R-ratio on the angels β and α which, as stated above, deﬁne the macroscopic behavior and the critical ﬁelds of the EB systems. Numerical simulations of (3.23) as a function of R-ratio are shown in Fig. 3.9 and in Fig. 3.10. We distinguish three physically distinct regions [26, 27, 42, 65]: • I. R ≥ 1 In this region the coercive ﬁeld is zero and the exchange bias ﬁeld is ﬁ∞ nite, decreasing from the asymptotic value Heb to the lowest ﬁnite value at R = 1. The AF spins rotate reversibly during the complete reversal of the F spins. The α angle has a maximum value as a function of the R-ratio, ranging from approximatively zero for R = ∞ to α = 45◦ at R = 1. Notice that as the maximum angle of the AF spins increases, a slight decrease of the exchange bias ﬁeld is observed. When the R-ratio approaches the critical value of unity, the exchange bias has a minimum. In the phase diagram, only range I can cause a shift of the magnetization curve. Simulated hysteresis loops for three diﬀerent values of the R-ratio are shown in Fig. 3.10. One notices that not only the size of the exchange

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Fig. 3.9. Left: The phase diagram of the exchange bias ﬁeld and the coercive ﬁelds as a function of the Meiklejohn-Bean parameter R. Right: Typical behavior of the antiferromagnetic angle α for the three diﬀerent regions of the phase diagram. Only region I can lead to a shift of the hysteresis loop. In the other two regions a coercivity is observed but no exchange bias ﬁeld

bias ﬁeld decreases when the R-ratio approaches unity, but also the shape of the hysteresis curve is changing. At high R-ratios the reversal is rather sharp, whereas for R-ratios close to unity it become more extended, almost resembling a spring-like behavior. • II. 0.5 ≤ R < 1 Characteristic for this region is that the AF spins are no more reversible. They follow the F spins and they change direction irreversibly, causing a coercive ﬁeld at the expense of the exchange bias ﬁeld, which becomes zero. Furthermore, depending on the ﬁeld sweeping direction, there is a hysteresis-like behavior of the AF spin rotation. At a critical angle β of the F spin rotation, the AF spins cannot withstand the torque by the coupling to the F spins and they jump in a discontinuous fashion to another angle (jump angle). The hysteresis loops corresponding to this region (see Fig. 3.10) are drastically diﬀerent from the previous case. The coercivity shows a strong dependence as function of the R-ratio and they are not shifted at all. Moreover, the AF jump angles are clearly visible as kinks in the hysteresis during the reversal. • III. R < 0.5 This region preserves the features of the previous one with one exception, namely that the AF spins follow reversibly the F spins, without any jumps. Therefore, no hysteresis-like behavior of the α angle is seen. The exchange bias ﬁeld is zero and the coercive ﬁeld is ﬁnite, depending on the R-ratio. The hysteresis loops shown in Fig. 3.10 are quite similar to a ferromagnet with uniaxial anisotropy. Within the Stoner-Wolhfarth model the resultant coercive ﬁeld can be roughly approximated as [42]: Hc ≈ 2 KAF tAF /(μ0 MF tF ).

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Fig. 3.10. Simulation of several hysteresis loops and antiferromagnetic spin orientations during the magnetization reversal. For the simulation we used the MeiklejohnBean formalism. Top row shows three hysteresis loops calculated for diﬀerent R ratios within region I. The graph in the last column to the right shows the α angle of the AF layer for the three R values. The middle raw shows corresponding hysteresis loops and α angles for R values in region II. The bottom row shows simulations for region III. Note that the scales for the α angle in the top panel is enlarged compared to those in the lower two panels

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It is easy to recognize that allowing the AF to rotate as a whole leads to an impressively rich phase diagram of the EB systems as a function of the parameters of the AF layer (and F layer). The R-ratio can be varied across the whole range from zero to inﬁnity by changing the thickness of the AF layer [8], by varying its anisotropy (dilution of the AF layers with non-magnetic impurities [32, 66, 67]), or by varying the interfacial exchange energy Jeb (low dose ion bombardment [68, 69, 70]). Recently [71], an almost ideal M&B behavior has been observed in Ni80 Fe20 /Fe50 Mn50 bilayers. At high thicknesses of the AF layer the hysteresis loop is shifted to negative values and the coercivity is almost zero, whereas for reduced AF thicknesses a strong increase of the coercive ﬁeld is observed together with a drastic decrease of exchange bias.

3.6.1 Analytical Expression of the Exchange Bias Field First we calculate analytically the expression of the exchange bias ﬁeld for θ = 0 and KF = 0. The exact analytical solution is obtained by solving the system of (3.23) for β = 0, which leads to: ⎧ ! 1 ∞ ⎪ ⎨Heb 1 − 4R R≥1 2 Heb = . (3.26) ⎪ ⎩ 0 R<1

This equation retains the 1/tF dependence of the exchange bias ﬁeld, but at the same time provides new features. The most important one is an additional term, which eﬀectively lowers the exchange bias ﬁeld when the R-ratio approaches the critical value of one. The R-ratio has three terms. One of them includes the thickness of the AF layer. The analytical expression for the exchange bias ﬁeld (3.26) predicts that there is a critical AF thickness tcr AF below which the exchange bias cannot exist. This is [72]: tcr AF =

Jeb . KAF

(3.27)

Below this critical thickness the interfacial energy is transformed into coercivity. Above the critical thickness the exchange bias increases as a function of ∞ the AF layer thickness, reaching the asymptotic (ideal) value Heb when tAF is inﬁnite. Most recent observation of an AF critical thickness can be found in [73]. A similar expression as (3.26) was derived by Binek et al. [127] using a ∞ series expansion of (3.22) with respect to α=0 : Heb ≈ Heb (1 − 8R1 2 ) for 1/R≥0. Note that close to the critical value of R=1, the α angle could reach high values up to 45◦ . Therefore, the series expansion with respect to α=0 is a good approximation for R>5.

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3.6.2 Azimuthal Dependence of the Exchange Bias Field In the following we consider the exchange bias ﬁeld in region I, where it acquires non vanishing values. The coercive ﬁelds and the exchange bias ﬁeld are extracted from the condition β = θ + π/2 for both Hc1 and Hc2 . This gives Hc1 = Hc2 = (−Jeb /μ0 MF tF ) cos(α(R, θ + π/2) − θ), where α(R, θ + π/2) is the value of the rotation angle of the AF spins at the coercive ﬁeld. With the notation: α0 ≡ α(R, θ + π/2), and using the expression 3.16, the angular dependence of the exchange bias ﬁeld becomes: Heb (θ) =

−Jeb cos( α0 − θ) . μ 0 M F tF

(3.28)

The equation above can be also written as: Heb (θ) = −

KAF tAF sin(2 α0 ) . μ0 MF tF

(3.29)

Interestingly, the exchange coupling parameter Jeb in (3.28) is missing in (3.29), leaving instead an explicit dependence of the exchange bias ﬁeld on the parameters of the antiferromagnet and the ferromagnet. The exchange coupling constant and the θ angle are accounted for by the AF angle α0 . Equations (3.28) and (3.29) are the most general expressions for an exchange bias ﬁeld. They include both, the inﬂuence of the rotation of the AF layer and the inﬂuence of the azimuthal orientation of the applied ﬁeld. Moreover, the anisotropy and the thickness of the AF layer are explicitly shown in (3.29). To illustrate their generality we consider below two special cases for the (3.28): • θ=0 In this case the hysteresis loop is measured along the ﬁeld cooling direction (θ = 0) and (3.28) becomes equivalent to the (3.26). • R→∞ When R is very large (R 1), α approximates zero, i.e. the rotation of the AF - layer becomes negligible. This is actually the original assumption of the Meiklejohn and Bean model. Such a condition (R → ∞) is approximately satisﬁed for large thicknesses of the AF layer. Then the exchange bias ﬁeld as function of θ can be written as [94, 127, 165]: α=0 Heb (θ) =

−Jeb cos(θ) . μ 0 M F tF

(3.30)

In order to get more insight into the azimuthal dependence of the exchange bias ﬁeld, we show in Fig. 3.11 the normalized exchange bias ﬁeld ∞ Heb (θ) /|Heb (0)| as a function of the θ angle, according to (3.28) and (3.29), and for three diﬀerent values of the R ratio (R = 1.1, R = 1.5, R = 20). The α0 angle (see Fig. 3.10) was obtained by numerically solving the system

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119

Fig. 3.11. Azimuthal dependence of exchange bias as a function of the θ angle. The curves are calculated by the (3.28) and (3.29)

of (3.23). For large values of R, the azimuthal dependence of the exchange bias ﬁeld follows closely a cos(θ) unidirectional dependence. When, however, the R-ratio takes small values but larger then unity, the azimuthal behavior of the EB ﬁeld deviates from the ideal unidirectional characteristic. There are two distinctive features: one is that at θ = 0 the exchange bias ﬁeld is reduced, and the other one is that the maximum of the exchange bias ﬁeld is shifted from zero towards negative azimuthal angle values. According to (3.28) this shift angle is equal to α0 . In other words, the exchange bias ﬁeld is not maximum along the ﬁeld cooling direction. Another striking feature is that the shifted maximum of the exchange bias ﬁeld with respect to the azimuthal angle θ does not depend on thickness and anisotropy of the AF layer: MAX Heb =−

Jeb . μ 0 M F tF

(3.31)

Summarizing we may state that, within the Meiklejohn and Bean model, a reduced exchange bias ﬁeld is observed along the ﬁeld cooling direction depending on the parameters of the AF layer (KAF and tAF ). However, for R ≥ 1 the maximum value for the exchange bias ﬁeld which is reached at θ = 0 does not depend on the anisotropy constant (KAF ) and thickness (tAF ) of the AF layer. The azimuthal characteristic of the exchange bias allows to extract all three essential parameters deﬁning the exchange bias ﬁeld: Jeb , KAF and, tAF . The condition for extracting the Hc1 and Hc2 from the same β angle hides an important property of the magnetization reversal of the ferromagnetic layer. This will be described next.

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3.6.3 Magnetization Reversal A distinct feature of exchange bias phenomena is the magnetization reversal mechanism. In Fig. 3.12 is shown the parallel component of the magnetization m|| = cos(β) versus the perpendicular component m⊥ = sin(β) for several R-ratios and for θ = 30◦ . The geometrical conventions are the ones shown in Fig. 3.8. We see that for R < 1 the reversal of the F spins is symmetric, similar to typical ferromagnets with uniaxial anisotropy. Although the regions 0.5 ≤ R < 1 and R < 0.5 exhibit diﬀerent reversal modes of the AF spins (see Fig. 3.10), there is little diﬀerence with respect to the F spin rotation. For both regions 0.5 ≤ R < 1 and R < 0.5 the F spins do make a full rotation, similar to the uniaxial ferromagnets. At the steep reversal branches one would expect magnetic domain formation. When R ≥ 1 another reversal mechanism is observed. The ferromagnetic spins ﬁrst rotate towards the unidirectional axis as lowering the ﬁeld from positive to negative values, and then the rotation proceeds continuously until the negative saturation is reached. On the return path, when the ﬁeld is swept from negative to positive values, the ferromagnetic spins follow the same path towards the positive saturation. The rotation is continuous without any additional steps or jumps. A similar behavior was observed theoretically within the domain state model [36, 39]. The magnetization reversal modes can be accessed experimentally by using the Vector-MOKE technique [74, 75, 76, 41] which allows to follow both the magnetization vector and its angle during the reversal process.

Fig. 3.12. Magnetization reversal for several values of the R ratio. The parallel component of the magnetization vector m|| = cos(β) is plotted as a function of the perpendicular component of the magnetization m⊥ = sin(β). The reversal for R < 1 resembles the typical reversal of ferromagnets with uniaxial anisotropy. For R ≥ 1 the reversal proceeds along the same path for the increasing and decreasing branch of the hysteresis loop. The angle θ = 30◦ is chosen arbitrarily

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3.6.4 Rotational Hysteresis We brieﬂy discuss again the rotational hysteresis deduced from torque measurements [1, 26, 27], now in the light of the analysis provided above. The torque measurements were carried out in a strong applied magnetic ﬁeld H. Therefore the applied ﬁeld H and the magnetization MF can be assumed to be parallel (β = θ). The torque is given by: T =−

∂E(θ) = Jeb sin(θ − α(θ)). ∂θ

This expression diﬀers from (3.8) for the ideal model by the rotation of the AF spins through the α angle. However, this does not explain the energy loss during the torque measurements, as observed in the experiment (3.6(b)). The torque curve would only be a bit distorted but completely reversible. The integration of the energy curve predicts a rotational hysteresis Wrot = 0. In order to account for a ﬁnite rotational hysteresis, one can assume that a fraction p of particles at the F-AF interface are uniaxially coupled behaving as in region II, whereas the remaining fraction (1 − p) of the F-AF particles are coupled unidirectionally, having the ideal behavior as described in regime I. As seen in Fig. 3.9(left), when the R-ratio of the uniaxial particle is in the range 0.5 ≤ R < 1, the AF spins will rotate irreversibly, showing hysteresislike behavior due to α jumps indicated in the Fig. 3.9 (right). A rotational hysteresis is not expected for unidirectional particles with R ≥ 1 because the AF structure changes reversibly with θ. With this assumption the uniaxial particles will contribute to the energy loss during the torque measurements, while the unidirectional particles are responsible for the unidirectional feature of the torque curve. This argument was used by Meiklejohn and Bean [27] when studying the exchange bias in core-Co/shell-CoO. A fraction p = 0.5 was inferred from the torque curves shown in Fig. 3.5.

3.7 N´ eel’s AF Domain Wall–Weak Coupling Both concepts, the rigid AF spin state and rigidly rotating AF spins impose a restriction on the behavior of the antiferromagnetic spins, namely that the AF order is preserved during the magnetization reversal. Such restriction implies that the interfacial exchange coupling is found almost entirely in the hysteresis loop either as a loop shift or as coercivity. Experimentally, however, the size of the exchange bias does not agree with the expected value, being several orders of magnitude lower then predicted. In order to cope with such loss of coupling energy, one can assume that a partial domain wall develops in the AF layer during the magnetization reversal. This concept was introduced by N´eel [77, 9] when considering the coupling between a ferromagnet and a low anisotropy antiferromagnet. The AF partial domain wall will store an important fraction of the exchange coupling energy, lowering the shift of the hysteresis loop.

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N´eel has calculated the magnetization orientation of each layer through a diﬀerential equation. The weak coupling is consistent with a partial AF domain wall which is parallel to the interface (N´eel domain wall). His model predicts that a minimum AF thickness is required to produce hysteresis shift. More importantly the partial domain wall concept forms the basis for further models which incorporate either N´eel wall or Bloch wall formation as a way to reduce the observed magnitude of exchange bias.

3.8 Malozemoﬀ Random Field Model Malozemoﬀ (1987) proposed a novel mechanism for exchange anisotropy postulating a random nature of exchange interactions at the F-AF interface [29, 30, 31]. He assumed that the chemical roughness or alloying at the interface, which is present for any realistic bilayer system, causes lateral variations of the exchange ﬁeld acting on the F and AF layers. The resultant random ﬁeld causes the AF to break up into magnetic domains due to the energy minimization. By contrast with other theories, where the unidirectional anisotropy is treated either microscopically [78, 79, 80] or macroscopically [1, 26, 28], the Malozemoﬀ approach belongs to models on the mesoscopic scale for surface magnetism. The general idea for estimating the exchange anisotropy is depicted in Fig. 3.13, where a domain wall in an uniaxial ferromagnet is driven by an applied in-plane magnetic ﬁeld H [29]. Assuming that the interfacial energy in one domain (σ1 ) is diﬀerent from the energy in the neighboring domain (σ2 ), then the exchange ﬁeld can be estimated by the equilibrium condition between the applied ﬁeld pressure 2 H MF tF and the eﬀective pressure from the interfacial energy Δσ: Heb =

Δσ , 2 M F tF

(3.32)

where MF and tF are the magnetization and thickness of the ferromagnet. When the interface is treated as ideally “compensated”, then the exchange bias ﬁeld is zero. On the other hand, if the AF/F interface is ideally uncompensated there is an interfacial energy diﬀerence Δσ = 2Ji /a2 , where Ji is Domain Wall

H

MF

tF

F AF

tAF

Fig. 3.13. Schematic side view of a F/AF bilayer with a ferromagnetic wall driven by an applied ﬁeld H [29]

3 Exchange Bias Eﬀect of Ferro-/Antiferromagnetic Heterostructures UNCOMPENSATED 2 2 = + J/a = - J/a a) b) x

x

x

123

COMPENSATED = 0 = 0 c) d)

x

x

x

x

x

Fig. 3.14. Schematic view of possible atomic conﬁgurations in a F-AF bilayer with ideal interfaces. Frustrated bonds are indicated by crosses. Compensated conﬁguration a) will result in conﬁguration b) by reversing the ferromagnetic spins through domain wall movement. It gives an exchange bias ﬁeld of Heb = Ji /a2 MF tF . The compensated conﬁguration c) will result in the compensated conﬁguration d). The exchange bias ﬁeld for this case is zero (Heb = 0). [29]

the exchange coupling constant across the interface, and a is the lattice parameter of a simple cubic structure assigned to the AF layer. The EB ﬁeld is Heb = Ji /a2 MF tF (see Fig. 3.14)1 . Estimating numerically the size of the EB ﬁeld using the equation above for an ideally uncompensated interface, results in a discrepancy of several orders of magnitude with respect to the experimental observation . Therefore, a novel mechanism based on random ﬁelds at the interface acting on the AF layer is proposed as to drastically reduce the resulting exchange bias ﬁeld. By simple and schematic arguments Malozemoﬀ describes how roughness on the atomic scale of a “compensated” AF interface layer can lead to uncompensated spins required for the loop to shift. An atomic rough interface depicted in Fig. 3.15a) containing a single mono-atomic bump in a cubic interface gives rise to six net antiferromagnetic deviations from a perfect compensation. A bump shifted by one lattice spacing as shown in Fig. 3.15b), which is equivalent to reversing the F spins, provides six net ferromagnetic deviations from perfect compensation. Thus a net energy diﬀerence of zi Ji with zi = 12 acts at the interface favoring one domain orientation over the other.

b)

a) x

x

x

x

x

c) x

x

x

x

Fig. 3.15. Schematic side view of possible atomic moment conﬁgurations for a nonplanar interface. The bump should be visualized on a two-dimensional interface. Conﬁguration c) represents the lower energy state of a). The conﬁguration b) is energetically equivalent to ﬂipping the ferromagnetic spins of a). The x signs represent frustrated bonds. [29] 1

For this example the energy is calculated as Ekl = −Ji S k S l per pair of nearest neighbor spins kl at the interface.

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Note that for an ideally uncompensated interface the energy diﬀerence is only 8Ji when reversing the F spins. This implies that an atomic step roughness at a compensated interface leads to a higher exchange bias ﬁeld as compared to the ideally compensated interface. The estimates of this local ﬁeld can be further reﬁned assuming a more detailed model. For example, by inverting the spin in the bump shown in Fig. 3.15c, the interfacial energy diﬀerence is reduced by 5 × 2Ji at the cost of generating one frustrated pair in the AF layer just under the bump. This frustrated pair increases the energy diﬀerence by 2JA , where JA is the AF exchange constant. Thus the energy diﬀerence between the two domains becomes 2Ji + 2JA or roughly 4J if Ji ≈ JA ≈ J. If one allows localized canting of the spins, one expects the energy diﬀerence to be reduced somewhat further. Each interface irregularity will give a local energy diﬀerence between domains whose sign depends on the particular location of the irregularity and whose magnitude is on the average 2zJ, where z is a number of order unity. Furthermore, for an interface which is random on the atomic scale, the local unidirectional interface energy σl = ±zJ/a2 will also be random and √ its average σ in a region L2 will go down statistically as σ ≈ σl / N , where N = L2 /a2 is the number of sites projected onto the interface plane. Therefore the eﬀective AF-F exchange energy per unit area is given by: 1 1 Jeb ≈ √ Ji ≈ Ji , L N where Ji is the exchange energy of a fully uncompensated AF-surface. Given a random ﬁeld provided by the interface roughness and assuming a region with a single domain of the ferromagnet, it is energetically favorable for the AF to break up into magnetic domains, as shown schematically in Fig. 3.16. A perpendicular domain wall is the most preferable situation. This perpendicular domain wall is permanently present in the AF layer. It should be distinguished from a domain wall parallel to AF/F interface, which according to the Mauri model [28] develops temporarily during the rotation of the F layer. By further analyzing the stability of the magnetic domains in the presence of random ﬁelds, a characteristic length L of thefrozen-in AF domains and their characteristic height are obtained: L ≈ π AAF /KAF and h = L/2, where AAF is the exchange stiﬀness and h is the characteristic height of the AF

x

x

x

Fig. 3.16. Schematic view of a vertical domain wall in the AF layer. It appears as an energetically favorable state of F/AF systems with rough interfaces [29, 81]

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125

domains. Once these domains are ﬁxed, ﬂipping the ferromagnetic orientation causes an energy change per unit area of Δσ = 4zJ/πaL, which further leads to the following expression for the EB ﬁeld [29]: √ 2 z AAF KAF Heb = . (3.33) π 2 M F tF Assuming a CoO/Co(100 ˚ A) ﬁlm, the calculated exchange bias using the (3.33) is: ! −19 [J] 2 × 1 0.0186×1.6×10 2.5 × 107 [J/m3 ] 4.27×10−10 [m] Heb = × 10 π2 × 1460 [kA/m] 100 × 10−10 = 580 Oe. (3.34) For the estimations above we used for the exchange stiﬀness the following value: AAF = JAF /a, where a is the lattice parameter of CoO (a = 4.27˚ A) and JAF = 1.86 meV is the exchange constant for CoO [55]. The characteristic length of the AF domains is for CoO: L = π AAF /KAF " 0.0186 × 1.6 × 10−19 [J] = π× 4.27 × 10−10 [m] × 2.5 × 107 [J/m3 ] = 16.6 ˚ A.

(3.35)

The height of the AF domains is h = L/2 = 8.3 ˚ A. Comparing this value to the experimental data on CoO(25˚ A)/Co studied in [82], we notice that the calculated EB ﬁeld agrees well with the value observed experimentally. For example, the exchange bias ﬁeld for CoO(25 ˚ A)/Co(119 ˚ A) is 557 Oe and the theoretical value calculated with (3.33) is 487 Oe. Also, the length and the height of the AF domains have enough space to develop. The diﬀerence between theory and experiment is, however, that experimentally AF domains can occur and vary size and orientation after the very ﬁrst magnetization reversal, whereas within the Malozemoﬀ model the AF domains are assumed to develop during the ﬁeld cooling procedure. Nevertheless, the agreement appears to be excellent.

3.9 Domain State Model The Domain State model (DS) introduced by Nowak and coworkers [32, 33, 35, 36, 83, 84, 38] is a microscopic model in which disorder is introduced via magnetic dilution not only at the interface but also in the bulk of the AF layer as in Fig. 3.17. The key element in the model is that the AF layer is a diluted Ising antiferromagnet in an external magnetic ﬁeld (DAFF) which exhibits a phase diagram like the one shown in Fig. 3.18 [35]. In zero ﬁeld the

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Fig. 3.17. Sketch of the domain state model with one ferromagnetic layer and three diluted antiferromagnetic layers. The dots mark defects [35]

system undergoes a phase transition from a disordered, paramagnetic state to a long-range-ordered antiferromagnetic phase at the dilution dependent N´eel temperature. In the low temperature region, for small ﬁelds, the long-range interaction phase is stable in three dimensions. When the ﬁeld is increased at low temperature the diluted antiferromagnet develops a domain state phase with a spin-glass-like behavior. The formation of the AF domains in the DS phase originates from the statistical imbalance of the number of impurities of the two AF sublattices within any ﬁnite region of the DAFF. This imbalance leads to a net magnetization which couples to the external ﬁeld. A spin reversal of the region, i. e., the creation of a domain, can lower the energy of the system. The formation of a domain wall can be minimized if the domain wall passes through nonmagnetic defects at a minimum cost of exchange energy. Nowak et al. [35] further argue that during the ﬁeld cooling below the irreversibility line Ti (B), in an external ﬁeld and in the presence of the interfacial exchange ﬁeld of the ferromagnet, the AF develops a frozen domain state with an irreversible surplus of magnetization. This irreversible surplus magnetization controls then the exchange bias. The F layer is described by a classical Heisenberg model with the nearestneighbor exchange constant JF . The AF is modelled as a magnetically diluted Ising system with an easy axis parallel to that of the F. The Hamiltonian of the system is given by [35]:

Fig. 3.18. Schematic phase diagram of a three-dimentional diluted antiferromagnet. AF is the antiferromagnetic phase, DS the domain state phase, and PM the paramagnetic phase [35]

3 Exchange Bias Eﬀect of Ferro-/Antiferromagnetic Heterostructures

H = −JF

S i .S j −

F

−JAF

i j σi σj −

127

2 2 (dz Siz + dx Six + μBS i )

iF

AF

−JIN T

μBz i σi

iAF

i σi Sjz ,

(3.36)

where the Si and σi are the classical spin vectors at the ith site of the F and AF, respectively. The ﬁrst line contains the energy contribution of the F, the second line describes the diluted AF layer, and the third line includes the exchange coupling across the interface between F and DAFF, where it is assumed that the Ising spins in the topmost layer of the DAFF interact with the z component of the Heisenberg spins of the F layer. In order to obtain the hysteresis loop of the system, the Hamiltonian in (3.36) is treated by Monte Carlo simulations. Typical hysteresis loops are shown in Fig. 3.19 [84], where both the magnetization curve of the F layer and of the interface monolayer of the DAFF are shown. The coercive ﬁeld extracted from the hysteresis curve depends on the anisotropy of the F layer, but it is also inﬂuenced by the DAFF. It, actually, depends on the thickness and anisotropy of the DAFF layer. The coercive ﬁeld decreases with the increasing thickness of the DAFF layer [84], which can be understood as follows: the interface magnetization tries to orient the F layer along its direction. The coercive ﬁeld has to overcome this barrier, and the higher the interface magnetization of the DAFF, the stronger is the ﬁeld required to reverse the F layer. The

Fig. 3.19. Simulated hysteresis loops within the domain state model. The top hysteresis belongs to the ferromagnetic layer and the bottom hysteresis belongs to an AF interface monolayer [84]

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interface magnetization decreases with increasing DAFF thickness due to a coarsening of the AF domains accompanied by smoother domain walls. The strength of the exchange bias ﬁeld can be estimated from the (3.36) using simple ground state arguments. Assuming that all spins in the F remain parallel during the ﬁeld reversal and some net magnetization of the interface layer of the DAFF remains constant during the reversal of the F, a simple calculation gives the usual estimate for the bias ﬁeld [35]: lμBeb = JIN T mIN T ,

(3.37)

where l is the number of the F layers and mIN T is the interface magnetization of the AF per spin. Beb is the notation for the exchange bias ﬁeld in [35] and is equivalent to Heb in this chapter. For an ideal uncompensated interface (mIN T = 1) the exchange bias is too high, whereas for an ideally compensated interface the exchange bias is zero. Within the DS model the interface magnetization mIN T < 1 is neither a constant nor is it a simple quantity [35]. Therefore, it is replaced by mIDS , which is a measure of the irreversible domain state magnetization of the DAFF interface layer and is responsible for the EB ﬁeld. With this, an estimate of exchange bias ﬁeld for l = 9, JIN T = −3.2 × 10−22 J, and μ = 1.7 μB gives a value of about 300 Oe. The exchange bias ﬁeld depends also on the bulk properties of the DAFF layer as shown by Milt´enyi et al. [32]. There the AF layer was diluted by substituting non-magnetic Mg in the bulk part and away from the interface. The representative results are shown in Fig. 3.20. It was shown experimentally that the EB ﬁeld depends strongly on the dilution of the AF layer. As a function of concentration of the non-magnetic Mg impurities, the EB evolves as following: at zero dilution the exchange bias has ﬁnite values, whereas by increasing the Mg concentration, the EB ﬁeld increases ﬁrst, showing a broad peak-like behavior, and then, when the dilution is further increased the EB ﬁeld decreases again. Simulations within the DS model showed an overall good qualitative agreement. The peak-like behavior of the EB as a function of the dilution is clearly seen in the simulations (see Fig. 3.20). However, it appears that at zero dilution, the DS gives vanishing exchange bias whereas experimentally ﬁnite values are observed. The exchange bias is missing at low dilutions because the domains in the AF cannot be formed as they would cost too much energy to break the AF bonds. This discrepancy [35, 85] is thought to be explained by other imperfections, such as grain boundaries in the AF layer which is similar to dilution and which can also reduce the domain-wall energy, thus leading to domain formation and EB even without dilution of the AF bulk. An important property of the kinetics of the DAFF is the slow relaxation of the remanent magnetization, i.e., the magnetization obtained after switching oﬀ the cooling ﬁeld [35]. It is known that the remanent magnetization of the DS relaxes nonexponentially on extremely long-time scales after the ﬁeld is switched oﬀ or even within the applied ﬁeld. In the DS model the EB is related to this remanent magnetization. This implies a decrease of EB due to slow

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129

Fig. 3.20. In the right side of the ﬁgure is shown the ﬁlm structure used to study the dilution inﬂuence on the exchange bias ﬁeld. a) EB ﬁeld as function of the Mg concentration x in the Co1−x Mgx O layer for several temperatures. b) EB ﬁeld as a function of diﬀerent dilutions of the AF volume. [32]

relaxation of the AF domain state. The reason for the training eﬀect can be understood within the DS model from Fig. 3.19 bottom panel, where it is shown that the hysteresis loop of the AF interface layer is not closed on the right hand side. This implies that the DS magnetization is lost partly during the hysteresis loop due to a rearrangement of the AF domain structure. This loss of magnetization clearly leads to a reduction of the EB. The blocking temperature 2 within the DS model can be understood by considering the phase diagram of the DAFF shown in Fig. 3.17. The frozen DS of the AF layer occurs after ﬁeld cooling the system below the irreversibility temperature Ti (b). Within this interpretation, the blocking temperature corresponds to Ti (b). Since Ti (b) < TN , the blocking temperature should be always below the Ne´el temperature and should be dependent on the strength of the interface exchange ﬁeld. The simulations within the DS model shows that EB depends linearly on the temperature, as observed experimentally in some Co/CoO systems, but no reason is given for this behavior [35]. In [86] the blocking temperature of a DAFF system (Fe1−x Znx F2 (110)/Fe/Ag with x=0.4) exhibits a signiﬁcant enhancement 2

The blocking temperature of an exchange bias system is the temperature where the hysteresis loop acquires a negative or positive shift with respect to the ﬁeld axis. It is always lower then the N´eel temperature of the AF layer.

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with respect to the global ordering temperature TN =46.9 K, of the bulk antiferromagnet Fe0.6 Zn0.4 F2 . Overall, it is believed that strong support for the DS model is given by experimental observations where nonmagnetic impurities are added to the AF layer in a systematic and controlled fashion [32, 87, 69, 88, 66, 85, 67]. Also, good agreement has been observed in [89], where the dependence of the EB as a function of AF thickness and temperature for IrMn/Co was analyzed. The asymmetry of the magnetization reversal mechanisms [36, 39] is shown to be dependent on the angle between the easy axis of the F and DAFF layers. It was found that either identical or diﬀerent F reversal mechanisms (domain wall movement or coherent rotation) can occur as the relative orientation between the anisotropy axis of the F and AF is varied. This is discussed in more detail in Sect. 3.12.4 and 3.14.1.

3.10 Mauri Model The model of Mauri et al. [28] renounces the assumption of a rigid AF layer and proposes that the AF spins develop a domain wall parallel to the interface. The motivation to introduce such an hypothesis was to explore a possible reduction of the exchange bias ﬁeld resulting from the Meiklejohn and Bean model. The assumptions of the Mauri model are: • • • • • •

both the F and AF are in a single domain state; the F layer rotates rigidly, as a whole; the AF layer develops a domain wall parallel to the interface; the AF interface layer is uncompensated (or fully compensated); the AF layer has a uniaxial anisotropy; the cooling ﬁeld is oriented parallel to the uniaxial anisotropy of the AF layer; • the AF and F spins rotate coherently, therefore the Stoner-Wohlfarth model is used to describe the system. Schematically the spin conﬁguration within the Mauri model is shown in Fig. 3.21. The F spins rotate coherently, when the applied magnetic ﬁeld is swept as to measure the hysteresis loop. The ﬁrst interfacial AF monolayer is oriented away from the F spins making an angle α with the ﬁeld cooling direction and with the anisotropy axis of the AF layer. The next AF monolayers are oriented away from the interfacial AF spins as to form a domain wall parallel to the interface. The spins of only one AF sublattice are depicted, the spins of the other sublattice being oppositely oriented for completing the AF order. At a distance ξ at the interface, a ferromagnetic layer of thickness tF follows. Using the Stoner-Wohlfarth model, the total magnetic energy can be written as [28]:

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Fig. 3.21. Mauri model for the interface of a thin ferromagnetic ﬁlm on a antiferromagnetic substrate [28]

E = −μ0 H MF tF cos(θ − β) +KF tF sin2 (β) −Jeb cos(β − α) −2 AAF KAF (1 − cos(α)),

(3.38)

where the ﬁrst term is the Zeeman energy of the ferromagnet in an applied magnetic ﬁeld, the second term is the anisotropy term of the F layer, the third term is the interfacial exchange energy and, the forth term is the energy of the partial domain wall. The new parameter in the equation above is the exchange stiﬀness AAF . As in the case of the Meiklejohn and Bean model, the interfacial exchange coupling parameter Jeb [J/m2 ] is again undeﬁned, assuming that it ranges between the exchange constant of the F layer to the exchange constant of the AF layer divided by an eﬀective area (see Sect. 3.5.2). √ The total magnetic energy can be written in units of 2 AAF KAF , which is the energy per unit surface of a 90◦ domain wall in the AF layer [28]: e = k (1 − cos(β)) + μ cos(β)2 +λ [1 − cos(α − β)] + (1 − cos(α)), (3.39) √ where λ = Jeb /( 2 AAF KAF ), is the interface exchange, with Jeb being redeﬁned as Jeb ≡ A12 /ξ, where A12 is the interfacial exchange stiﬀness and √ ξ is the thickness of the interface (see Fig. 3.21), μ = KF tF /√2 AAF KAF is the reduced ferromagnet anisotropy, and k = μ0 H MF tF / 2 AAF KAF is the reduced external magnetic ﬁeld. Mauri et al. [28] have calculated the magnetization curves by numerical minimization of the reduced total magnetic energy (3.39). Several values of the λ and μ parameters were considered providing quite realistic hysteresis loops. Their analysis highlights two limiting cases with the following expressions for the exchange bias ﬁeld: # f or λ 1 − (A12 / ξ)/ μ0 MF tF √ Heb = . (3.40) − 2 AAF KAF / μ0 MF tF f or λ 1

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In the strong coupling limit λ 1, the expression for the exchange bias ﬁeld is similar to the value given by the Meiklejohn and Bean model. For this situation, practically no important diﬀerences between the predictions of the two models exist. When the coupling is weak (λ 1), the Mauri model delivers a reduced exchange bias ﬁeld which is practically independent of the interfacial exchange energy. It depends on the domain wall energy and the parameters of the F layer. In either case the “1/tF ” dependence is preserved by the Mauri-model. 3.10.1 Analytical Expression of Exchange Bias Field In order to compare the predictions of the Mauri model and the Meiklejohn and Bean approach, we reconsider the analysis of the free energy. Starting from the expression of the free energy (3.38), the minimization with respect to the α and β angles leads to the following system of equations: ⎧ H ⎪ ⎨ − Jeb sin(θ − β) + sin(β − α) = 0 μ0 MF tF (3.41) √ ⎪ ⎩ 2 AAF KAF sin(α) − sin(β − α) = 0 . Jeb Similar to the Meiklejohn and Bean model we deﬁne the parameters P ≡ Jeb ∞ and Heb ≡ −μ M . Also we set θ = 0, meaning that the applied F tF 0 ﬁeld is swept along the easy axis of the AF layer. Also, we do not take into account the anisotropy of the ferromagnet (KF = 0), for two reasons. For one the coercive ﬁelds of the exchange bias systems are usually much higher then the coercive ﬁeld of the isolated F layer, and secondly it is easier to compare the results of the Mauri model and the M&B model when the anisotropy of the F layer is disregarded. From inspection of the (3.41) and (3.23) one can clearly see that the ﬁrst equations of the two systems are identical, while the second ones are diﬀerent in two respects. The ﬁrst diﬀerence is related to the term P , which includes the domain wall energy instead of the AF anisotropy term in the R ratio. The second diﬀerence is that instead of a sin(2 α) term in the second equation of (3.23), the Mauri model has a sin(α) term, which inﬂuences strongly the phase diagram shown in Fig. 3.22. The analytical expression for the exchange bias are obtained by solving the system of (3.41). First step in solving the (3.41) is to extract the α angle (α = ± arccos(± √ P −cos(β) )) from the second equation and to 2 √ 2 AAF KAF Jeb

1+P −2 P cos(β)

introduce it in the ﬁrst equation. Next we use the condition that at the coercive ﬁeld β = −π/2 to obtain both coercive ﬁelds Hc1 = Hc2 . Then, inserting them into the general expressions of (3.16), the coercive ﬁeld Hc is zero and the exchange bias ﬁeld becomes [11]:

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Fig. 3.22. Left: The phase diagram of the exchange bias ﬁeld and the coercive ﬁelds given by the Mauri formalism. Right: Typical behavior of the antiferromagnetic angle α for the two diﬀerent regions of the phase diagram. In both regions I and II a shift of the hysteresis loop can exist. The coercive ﬁeld is zero in both regimes

Heb

√ Jeb 2 AAF KAF =− 2 +4A μ0 MF tF Jeb AF KAF P Jeb √ =− . μ 0 M F tF 1 + P 2

(3.42)

This equation is plotted as a function of H/|HEB | and for diﬀerent P values ranging from P = 0 to P = 5. (compare Fig. 3.22 left panel). The behavior of the EB ﬁeld according to the (3.42) is monotonic with respect to the stiﬀness and anisotropy of the AF spins. At P 1 the exchange bias is equal Jeb P →∞ to Heb = −μ M , which is the well known expression given by M&B F tF 0 model. When, however, the P -ratio approaches low values, the exchange bias decreases, vanishing at P = 0, provided that the thickness of the AF layer is suﬃciently thick to allow a 180◦ wall. With some analytical analysis of the (3.42) one can easily reach the limiting cases of weak coupling (P 1) and strong coupling (P 1) discussed by Mauri et al. [28] (see (3.40)). In Fig. 3.22 right column is shown the representative behavior of the α angle of the ﬁrst interfacial AF monolayer as a function of the β orientation of the F spins during the magnetization reversal and for two representative P values (see the discussion below). In Fig. 3.23 the hysteresis loops (m|| = cos(β)) and the corresponding AF angle rotation during the magnetization reversal are plotted for several P -ratios. They were obtained by solving numerically the system of (3.41). For all the values of the P -ratio the magnetization curves are shifted to negative values of the applied magnetic ﬁeld. We distinguish two diﬀerent regions with respect to the behavior of the α angle of the ﬁrst AF monolayer. In the ﬁrst region, for P ≥ 1 (region I), the AF monolayer in the proximity of the F layer behaves similar to the Meiklejohn and Bean, namely the α angle deviates

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Fig. 3.23. Several hysteresis loops and antiferromagnetic spin orientations as plotted during the magnetization reversal. For the simulation we used the Mauri formalism. Top row shows three hysteresis loops calculated for diﬀerent P ratios of the region I shown in Fig. 3.22. The right hand panel in the top row shows the α angle of the antiferromagnetic layer for the three P parameters of the hysteresis loop. The bottom raw are the corresponding hysteresis loops and α angles for the P values within region II

reversibly from the anisotropy direction as function of β. The maximum value of the α angle acquired during the rotation of the F layer is two times higher for the Mauri model as compared to the M&B model, reaching a maximum value of 90◦ at P = 1. The coercive ﬁeld in this region is zero. The angle α in the region II where P < 1 has a completely diﬀerent behavior. It rotates with the ferromagnet following the general behavior depicted in Fig. 3.23. Notice that α follows monotonically the rotation of the F spins, with no jumps or hysteresis-like behavior in contrast to the M&B model(see Fig. 3.9). Very importantly, the exchange bias ﬁeld does not vanish in this region and therefore no additional coercive ﬁeld related to the AF is observed, provided that the

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AF layer is suﬃciently thick to allow for a domain wall as shown in Fig. 3.21. In this region (II) the EB ﬁeld is smaller as compared to the M&B model. This reduction is more clearly seen further below, when analyzing the azimuthal dependence of the EB ﬁeld within the Mauri model. Comparing the phase diagram of the Mauri model in Fig. 3.22 left to the corresponding one given by the M&B model in Fig. 3.9 left one can clearly see that region I of both models is very similar with respect to the qualitative behavior of the exchange bias ﬁeld as a function of the R-ratio and, respectively, P -ratio. However, we can compare those curves only when accounting for the variation of the EB ﬁeld as a function of the anisotropy of the AF layer. Both models predict that the EB ﬁeld depends on the anisotropy of the AF layer in a similar qualitative manner. Additionally, within the M&B model the EB ﬁeld includes also the dependence on the thickness of the AF layer, which is not visible in the Mauri model. The other regions of both phase diagrams are completely diﬀerent. Within the Mauri model, the exchange bias does not vanish at P < 1, but it continuously decreases, whereas the M&B model predicts that the exchange bias ﬁeld vanishes for R < 1 leading to enhanced coercivity. Also note that for the weak coupling region (II) of the Mauri model, the exchange bias would strongly depend on the temperature through the anisotropy constant of the AF layer [90]. 3.10.2 Azimuthal Dependence of the Exchange Bias Field Next we analyze the azimuthal dependence of the EB ﬁeld by deriving an analytical expression of the EB ﬁeld as a function of the rotation angle θ. By solving the second equation of the system of (3.41) with respect to β, one ﬁnds the angle α as function of β. Using the condition for the coercive ﬁeld as β = θ − π/2, and introducing it in the ﬁrst line of (3.41), one obtains the coercive ﬁelds Hc1 = Hc2 . It follows that the coercive ﬁeld Hc (θ) = 0 and the exchange bias ﬁeld as function of the azimuthal angle is [11]: √ 2 AAF KAF cos(θ) Jeb ! Heb (θ) = − . (3.43) μ0 MF tF J 2 + 4A K − 4J √A K sin(θ) AF AF eb AF AF eb In Fig. 3.24 is plotted the EB bias ﬁeld calculated by the expression above for diﬀerent values of the P -ratio, which was also conﬁrmed numerically. In region I the EB ﬁeld is maximum parallel to the ﬁeld cooling directions (θ = 0) only for very large P -ratios. When P approaches the unity, the maximum of the EB ﬁeld is shifted away from θ = 0, to higher azimuthal angles and has the value: Jeb MAX,P ≥1 Heb =− . (3.44) μ0 MF tF This expression is identical to (3.31) of the M&B model and shown in Fig. 3.11. The shape of the curves evolves from an ideal unidirectional shape at P → ∞ to a skewed shape at P ≥ 1.

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Fig. 3.24. Azimuthal dependence of exchange bias as a function of the θ angle. The dotted line for P = 20 can be considered an “ideal” case. The curves are plotted according to (3.43)

In region II (Fig. 3.24 left) a drastic change as compared to region I is seen for the maximum of the exchange bias ﬁeld as a function of azimuthal angle. Its value decreases monotonically towards zero according to the following expression: √ 2 AAF KAF MAX,P <1 Heb = −P = − . (3.45) Jeb In this region the shape of the curves is also skewed for P -ratios close to unity, but as P decreases towards zero, the curves acquire a more ideal unidirectional behavior. Similar to region I, the maximum EB value is shifted away from the ﬁeld cooling direction to higher azimuthal angles, whereas in M&B it is shifted to lower azimuthal angles. In the limit of strong R and P -ratios (R, P 1) the Mauri and M&B models give similar results. The diﬀerences appear for the R and P -ratios which are close to but higher than one. This region (0 < R, P ≤ 5) can be experimentally explored in order to decide in favor of one or the other model. In order to distinguish between the Mauri and the M&B model, the azimuthal dependence of the exchange bias oﬀers an excellent tool because it is visibly diﬀerent for the two models (shift to higher or lower azimuthal angles). The presence of the planar domain wall as described by Mauri et al. appears to have been conﬁrmed experimentally for ﬁrst time in [16]. Although the Co/NiO system studied by Scholl et al. did not exhibit a hysteresis loop shift, the rotation angle α of the AF planar wall was deduced as function of ﬁeld. Also, a recent publication by Gornakov et al. [91] show experimental results that are similar to the characteristic curves for the α angle shown in Fig. 3.22. An AF critical thickness which is often observed experimentally do not appear explicitly in the Mauri model. To account for the AF thickness

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dependence, Xi and White [92] proposed a model which assumes a helical structure for the AF spins during the magnetization reversal. The temperature dependence of EB is accounted for by the Mauri model through the √ anisotropy of the AF (HEB ≈ KAF ). Since the anisotropy constant is rather diﬃcult to measure for thin ﬁlms, the evidence for the predicted temperature dependence remains elusive [90]. One insuﬃciency of the Mauri model is the inability to predict any changes in coercivity. The domain wall produces only shifted reversible magnetization curves [40]. Therefore, further reﬁnements of the model were introduced which is described in the next section.

3.11 Kim-Stamps Approach – Partial Domain Wall The approach of Kim and Stamps [11, 93, 94, 95, 96, 40] follows from the work of N´eel and Mauri et al. extending the model of an extended planar domain wall to the concept of a partial domain wall in the AF layer. Biquadratic (spin-ﬂop) and bilinear coupling energies are used to describe exchange biased system, where the bias is created by the formation of partial walls in the AF layer. The model applies to compensated, partially compensated, and uncompensated interfaces. A typical hysteresis loop indicative for a partial domain wall in the AF layer is shown in Fig. 3.25 [40]. In saturation at h > 0, the F spins are

Fig. 3.25. (a) Magnetization curve for the ferromagnet/antiferromagnet system. (b) Calculated spin structure at three diﬀerent points of the magnetization curve. The creation of a partial antiferromagnet domain wall can be seen in (iii). Only the spins close to the F/AF interface are shown. From [40]

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aligned with the external ﬁeld while the AF spins are in a perfect N´eel state, collinear with the easy axis. The interface spins are antiparallel to the F layer due to a presumably antiparallel coupling. As the ﬁeld is reduced and reversed, the AF pins the F layer by interfacial exchange coupling until the critical value of the reversal ﬁeld is reached at hc , where the magnetization begins to rotate. When it is energetically more favorable to deform the AF, rather than breaking the interfacial coupling, a partial wall twists up as the F rotates. The winding and unwinding of the partial domain wall in the AF is reversible, therefore the magnetization is reversible (no coercivity). This mechanism is only possible if the AF is thick enough to support a partial wall. The magnitude of the exchange bias is similar to the one given by the Mauri model. Neiter the partial-wall theory nor the Mauri model account, however, for the coercivity enhancement that accompanies the hysteresis loop shift in single domain materials, which is usually observed in experiments. The enhanced coercivity observed experimentally, is proposed to be related to the domain wall pinning at magnetic defects. The presence of an attractive domain-wall potential in the AF layer, arising from magnetic impurities can provide an energy barrier for domain-wall processes that controls coercivity. Following the treatment of pinning in magnetic materials by Braun et al. [97], Kim and Stamps examined the inﬂuence of a pointlike impurity at an arbitrary position in the AF layer. As a result, the AF energy acquires, besides the domain wall energy, another term which depends on the concentration of the magnetic defects. These defects decrease the anisotropy locally and lead to an overall reduction of the AF energy. This reduction of the AF energy gives rise to a local energy minimum for certain defect positions relative to the interface. The domain walls can be pinned at such positions and contribute to the coercivity. Kim and Stamps argue that irreversible rotation of the ferromagnet due to a combination of wall pinning an depinning transitions, give rise to asymmetric hysteresis loops. Some examples are given in Fig. 3.26 [40]. The loops are calculated with an exchange defect at xL = 5, for three diﬀerent values of defect concentration ρJ , where xL denotes the defect positions in the antiferromagnet, with xL = 0 corresponding to the interface layer and xL = tAF − 1 being the free surface. At low defect concentrations, the pinning potential is insuﬃcient to modify partial-wall formation. The resulting magnetization curve, as shown in Fig. 3.26(a), is reversible and resembles the curve obtained with the absence of impurities. Pinning of the partial wall occurs during reversal for moderate concentrations, which appears as a sharp rotation of the magnetization at negative ﬁelds, as shown in Fig. 3.26(b). During remagnetization the wall is released from the pinning center at a diﬀerent ﬁeld, thus resulting in an asymmetry in the hysteresis loop. The release of the wall is indicated by a sharp transition in M. The energy barrier between wall pinning and release increases with defect concentration, resulting in a larger coercivity and reduced bias as in Fig. 3.26(c).

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Fig. 3.26. Defect-induced asymmetry in hysteresis loops. The hysteresis loops are shown for a reduced exchange defect at xL = 5 for three concentrations: (a) ρJ = 0.15,. (b) ρJ = 0.45, and (c) ρJ = 0.75. The components of magnetization parallel (M|| ) (dots) and perpendicular (M⊥ ) (open circles) to the ﬁeld direction are shown. The arrows indicate the directions for reversal and remagnetization. The spin conﬁguration near the interface is shown for selected ﬁeld values below the hysteresis curves [40]

Within this model, the asymmetry of the hysteresis loops is interpreted in terms of domain-wall pinning processes in the antiferromagnet. This explanation appears to be consistent with some recent work by Nikitenko et al. and Gornakov et al. on a NiFe/FeMn system [98, 91] who concluded that the presence of an antiferromagnetic wall at the interface is necessary to explain their hysteresis measurements.

3.12 The Spin Glass Model of Exchange Bias To overcame the theoretical diﬃculties in explaining interconnection between the exchange bias and coercivity, in [41, 63] is considered a magnetic state of the interface between F and AF layer which is magnetically disordered behaving similar to a spin glass system. The assumptions of the spin glass (SG) model are: • the F/AF interface is a frustrated spin system (spin-glass like); • frozen-in uncompensated AF spins are responsible for the EB shift; • low anisotropy interfacial AF spins contribute to the coercivity.

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Training Effect

KAF

KAF

INT

AF

F

Jeb

f Jeb

Frozen-in AF Spins Rotatable AF Spins

Fig. 3.27. Schematic view of the SG Model. At the interface between the AF and the F layer the AF anisotropy is assumed to be reduced leading to two types of AF states after ﬁeld cooling the system: frozen-in AF spins and rotatable AF spins. After reversing the magnetic ﬁeld, the rotatable spins follows the F layer rotation mediating coercivity. The frozen-in spins remain largely unchanged in moderate ﬁelds. But some of them will also deviate from the original cooling state. This could lead to training eﬀects and also to an open loop in the right side of the hysteresis loop. At larger applied ﬁelds in the negative direction, the frustrated frozen-in spins can further reverse leading to a slowly decreasing slope of the hysteresis loop. A more complex antiferromagnetic state consisting of frozen magnetic domains or/and AF grains can be also reduced to the basic concepts depicted here

Within this model, the AF layer is assumed to contain, in a ﬁrst approximation, two types of AF states (see Fig. 3.27). One part has a large anisotropy with the orientation ruled by the AF spins and another part with a weaker anisotropy which allows some spins to rotate together with the F spins. This interfacial part of the AF is a frustrated region (spin-glass-like) and gives rise to an increased coercivity. The presence of a low anisotropy AF region can be rationalized as follows: the interface between the F and AF layer is never perfect, therefore one may assume chemical intermixing, deviations from stoichiometry, structural inhomogeneities, low coordination, etc, at the interface to take place. This leads to the formation of a transition region from the pure AF state to a pure F state. On average, the anisotropy of such an interfacial region is reduced. In addition, structural and magnetic roughness can provide a weak AF interface region. Therefore, we assume that a fraction of the frustrated interfacial spins do rotate almost in phase with the F spins and that they mediate enhanced coercivity. We describe them by an eﬀective uniaxial ef f anisotropy KSG , because they are coupled to the presumably uniaxial AF layer.

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Generally, one can visualize a spin glass system [37] as a collection of spins which remains in a frozen disordered state even at low-temperatures. In order to achieve such a state, two ingredients are necessary: a) there must be a competition among the diﬀerent interactions between the moments, in the sense that no single conﬁguration of the spins is uniquely favored by all interactions (this is commonly called ‘frustration’); b) these interactions must be at least partially random. This partial random state will be introduced in the M&B model as an eﬀective uniaxial anisotropy. Adding this eﬀective anisotropy to the M&B model, the free energy reads [41]: E = −μ0 H MF tF cos(θ − β) + KF tF sin2 (β) + KAF (tAF ) tAF sin2 (α) ef f ef f + KSG sin2 (β − γ) − Jeb cos(β − α) ,

(3.46)

ef f is an eﬀective uniaxial SG anisotropy related to the frustrated where, KSG ef f AF spins with reduced anisotropy at the interface, Jeb is the reduced interfacial exchange energy, and γ is the average angle of the eﬀective SG anisotropy. KAF (tAF ) is the anisotropy constant of AF layer. To avoid further complications for the numerical simulations, we neglect the thickness dependence of the KAF anisotropy (KAF (tAF ) ≡ KAF ). We mention though, that this dependence could become important for low AF thicknesses due to ﬁnite size eﬀects and due to structurally non-ideal very thin layers. From now on, the MCA anisotropy of the ferromagnetic layer (KF = 0) will also be neglected as to highlight more clearly the inﬂuences of AF layer and the SG interface onto the general properties of the EB systems. Note that the Zeeman energies of the ferromagnetic-like AF interfacial spins are neglected in the model since they are usually much smaller as compared to Zeeman energy of the F layer. Nevertheless, they can be seen as a vertical shift of the hysteresis loop (frozen-in AF spins in Fig. 3.27) and as an additional contribution to the total magnetization (rotatable AF spins in Fig. 3.27). The model is depicted schematically in Fig. 3.27. At the interface two rather distinct AF phases are assumed to occur in an EB system: the rotatable AF spins, depicted as open circles and frozen-in AF spins shown as ﬁlled circles. After ﬁeld cooling, a presumably collinear arrangement is depicted in the right hand panel. After reversing the magnetic ﬁeld, the rotatable AF spins follow the F layer rotation mediating coercivity. The frozen-in spins remain largely unchanged in moderate ﬁelds. But some of them could also deviate from the original cooling state. Irreversible changes of the frustrated AF spins lead to training eﬀects and also to an open loop in the right side of the hysteresis loop. At larger negative applied ﬁelds, the frustrated frozen-in spins could further reverse leading to a slowly decreasing slope of the hysteresis loop. A more complex antiferromagnetic state consisting of frozen magnetic domain state or/and AF grains can be also reduced to the basic two spin

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components depicted in Fig. 3.27. Basically, additional frozen-in spins can occur in the AF layer extending to the interface and, therefore, leading to even more interfacial disorder. Next, we evaluate numerically the resulting hysteresis loops and azimuthal ef f dependence of the exchange bias within the SG model. When the KSG parameter is zero, the system behaves ideally as described by the M&B model discussed in Sect. 3.6: the coercive ﬁeld is zero and the exchange bias is ﬁnite. In the other case, when the interface is disordered we relate the SG eﬀective anisotropy to the available interfacial coupling energy as follows: Kef f = (1 − f ) Jeb ef f Jeb = f Jeb ,

(3.47)

where Jeb is the total available exchange energy and f is a conversion factor describing the fractional order at the interface, with f = 1 for a perfect interface and f = 0 for perfect disorder. Some basic models to calculate the available exchange energy were discussed in the previous sections. For even more complicated situations when the AF consists of AF grains and/or AF domains the exchange energy can be further estimated as described in [46]. With these notations we write the system of equations resulting from the minimization of the (3.46) with respect to the angles α and β: (1 − f ) sin(2 (β − γ)) + sin(β − α) = 0 f R sin(2 α) − sin(β − α) = 0 , h sin(θ − β) +

(3.48)

where, h=

H −μ

0

ef f Jeb

=

MF tF

H − μ fMJFebtF 0

,

is the reduced applied ﬁeld and R≡

KAF tAF ef f Jeb

=

KAF tAF , f Jeb

is the R-ratio deﬁning the strength of the AF layer. The system of equations above can easily be solved numerically, but it does not provide simple analytical expressions for the exchange bias. Numerical evaluation provides the α and β angles as a function of the applied magnetic ﬁeld H. The reduced longitudinal component of magnetization along the ﬁeld axis follows from m|| = cos(β − θ) and the transverse component from m⊥ = sin(β − θ). With the assumptions made above the absolute value of the exchange bias ﬁeld is directly proportional to f . The parameter f can be called conversion factor, as it describes the conversion of interfacial energy into coercivity. For example, in the M&B phase diagram in the region II and III corresponding

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to reduced R-ratios, the exchange bias ﬁeld is zero and the coercive ﬁeld is enhanced as a result of such a conversion of the interfacial energy into coercive ﬁeld. The idea of reduced interfacial anisotropy at the interface can be traced back to the the N´eel weak ferromagnetism at the surface of AF particles. N´eel [77] discussed the training eﬀect as a tilting of the superﬁcial magnetization of the AF domains. Later, Schlenker et al. [99] suggested that successive reversals of the F magnetization could lead to changes of the interface uncompensated AF magnetization and therefore provide means of going from one ground state to another. Such multiple interface conﬁgurations are similar to a spin glass system. The spin-glass interface is further discussed by other authors [100, 101, 102, 103, 104, 105, 106, 107, 50]. Exchange bias has been observed recently for spin-glass/F system [108]. The interfacial magnetic disorder was observed through hysteresis loop widening below a critical temperature point [109]. Non-collinearity have been observed at the AF/F in remanence [110] and even in saturation [111, 60, 112]. The frozen spins at the interface were also observed by MFM [113]. Using element speciﬁc techniques such as soft x-ray resonant magnetic dichroism (XMCD) and of x-ray resonant magnetic scattering (XRMS), both frozen and rotatable AF spins can be studied [13, 114, 115, 116, 15, 117, 17, 19, 20]. The frozen-in spins appear as a shift of the hysteresis loop along magnetization axis, whereas the AF rotatable spins exhibit a hysteresis loop. Moreover, an evidence for SG behavior is recently reported in thin ﬁlms [118] and AF nanoparticles [119]. Therefore, we believe that there is enough experimental evidence to consider the interface between the AF/F layer as a disordered state behaving similar to a spin-glass system. 3.12.1 Hysteresis Loops as a Function of the Conversion Factor f If Fig. 3.28 we show simulations of several hysteresis loops as a function of the conversion factor f . We assume a strong antiferromagnet in contact with a ferromagnet, where the interface has diﬀerent degrees of disorder depicted in the right column of Fig. 3.28. For the R ratio we assume the following value: tAF R = KAF = 62.5/f which corresponds to a 100 ˚ A thick CoO antiferromagf Jeb netic layer. The ﬁeld cooling direction and the measuring ﬁeld direction are parallel to the anisotropy axis of the AF. The anisotropy of the ferromagnet is neglected in the simulations below. For the interface we have chosen a SG anisotropy oriented 10 degrees away from the unidirectional anisotropy orien∞ tation (γ = 10◦ ). On the abscissa the reduced exchange bias ﬁeld h = H/|Heb | J ef f

∞ (Heb ≡ − μ MebF tF ) is plotted, which then can easily be compared to the M&B 0 model. With this assumption the system of (3.48) was solved numerically. The left column shows the longitudinal component of the magnetization (parallel to the measuring ﬁeld direction) (m|| = cos(β)) whereas the middle column shows the transverse component of the magnetization (m⊥ = sin(β)).

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Fig. 3.28. Longitudinal (m|| = cos(β)) and transverse (m⊥ = sin(β)) components of the magnetization for a F(KF = 0)/AF(R = 62.5/f, γ = 10◦ ,θ = 0) bilayer for diﬀerent values of the conversion factor f (f = 80%, 60%, 20%). We observe, that when the AF layer is strong (R 1), the hysteresis loops are symmetric when measured along the ﬁeld cooling direction. The hysteresis loops are simulated by solving numerically the (3.48). In the right column is schematically depicted the layer structure, here the emphasis is given to the disorder state at the interface. The AF layer is depicted as consisting of magnetic domains which also contribute to interface disorder

We observe that with decreasing conversion factor f the exchange bias vanishes linearly. The reduction of the EB ﬁeld is accompanied by an increased coercivity. The shape of the hysteresis loop is close to the results found in literature. For instance the hysteresis loop with f = 60% is similar to the data shown in [120, 61]. The hysteresis loop with f = 20% is similar to the data shown in [14, 109]. The longitudinal and transverse components of the magnetization show that the reversal mechanism is symmetric. The symmetry is directly related to the strength of the AF layer, when no training eﬀect is involved. For the examples depicted in Fig. 3.28, the R-ratio is much larger than 1 (R 1), and therefore the hysteresis loops are symmetric when measured along the ﬁeld cooling direction and along the anisotropy axis of the AF layer. The asymmetry of the hysteresis loops is discussed further below.

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3.12.2 Phase Diagram of Exchange Bias and Coercive Field Within the Spin Glass Model In this section we discuss the phase diagram for the exchange ﬁeld and coercive ﬁeld as a function of the R-value within the SG-model. The additional parameter is the conversion factor f . In Fig. 3.29 phase diagrams are shown for reduced exchange bias and reduced coercive ﬁelds as a function of the R-ratio for four diﬀerent values of the conversion factor f . This allows us to compare directly the behavior of exchange bias ﬁelds as predicted in the SG model and the M&B model. The reduced exchange bias ﬁeld plotted in Fig. 3.29 (left panel) is deﬁned: heb =

Heb ef f Jeb μ0 MF tF

=

Heb ∞, f Heb

where the Heb is the absolute value of the exchange bias within the SG model Jeb and the denominator term μ M is the exchange bias ﬁeld within the ideal F tf 0 M&B model. The reduced coercive ﬁeld shown in Fig. 3.29 is deﬁned: hc =

HcSG ef f Jeb μ0 MF tF

,

where HcSG is the absolute value of the coercive ﬁeld within the SG model. It has no relation to the coercive ﬁeld of the M&B model because the coercive

Fig. 3.29. The dependence of the reduced exchange bias ﬁeld heb and the reduced coercive ﬁeld hc as a function of the R-ratio for four diﬀerent values of the conversion factor f

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ﬁeld within the M&B model is considered to be a constant when the exchange bias is ﬁnite. 3.12.3 Azimuthal Dependence of Exchange Bias and Coercive Field Within the Spin Glass Model In this section, the azimuthal dependence of exchange bias and coercive ﬁelds within the SG model are discussed and compared with experimental results of polycrystalline Ir17 M n83 (15 nm)/Co70 F e30 (30 nm) exchange bias system [41]. In Fig. 3.30 calculated magnetization components are plotted together with the experimental data points, and in Fig. 3.31b the azimuthal dependence of the coercive ﬁeld and exchange bias ﬁeld are plotted and compared to the experimental data in Fig. 3.31a. The hysteresis loops were calculated by numerical minimizing the expressions in (3.48). The parameters used in the simulation f = 80%, R = 5.9/f, γ = 20◦ do best reproduce the experimental data. Furthermore, it is assumed that the AF layer has a uniaxial anisotropy. The MCA anisotropy of the F layer is neglected (KF = 0). Therefore, the coercivity which appears in the simulations is not related to the F properties, but to the interfacial properties of the F/AF bilayer. First we discuss the hysteresis loops shown in Fig. 3.30. The system is cooled down in a ﬁeld oriented parallel to the AF anisotropy direction. The hysteresis loops (solid lines) are simulated for diﬀerent azimuthal angles θ of the applied ﬁeld in respect to the ﬁeld cooling orientation. In Fig. 3.30 representative hysteresis curves are shown for the longitudinal (m|| ) and transverse (m⊥ ) magnetization. At θ = 0◦ , the magnetization curves are symmetric and shifted to negative ﬁelds. At θ = 180◦ , the magnetization curves are also symmetric but shifted to positive ﬁelds. At θ = 3◦ , however, the longitudinal hysteresis loop becomes asymmetric. The ﬁrst reversal at Hc1 is sharp and the reversal at Hc2 is more rounded. This asymmetry is also seen in the transverse component of the magnetization. The F spins rotate asymmetrically: the values of the β angle depend on the external ﬁeld scan direction, being diﬀerent for swaps from negative to positive saturation as compared with swaps from positive to negative saturation. As the azimuthal angle increases, the coercive ﬁeld becomes zero. For instance, at θ = 20◦ , 90◦ and 160◦ there is almost no coercivity. Also, the transverse component of the magnetization shows that the F spins do not follow a 360◦ path, but they rotate within the 180◦ angular space. In Fig. 3.31 the coercive ﬁeld and the exchange bias ﬁeld are extracted from the experimental and simulated hysteresis loops using (3.16). We distinguish the following characteristics of the Hc and Heb : the unidirectional behavior (≈ cos(θ)) of the Heb as a function of the azimuthal angle is (see Fig. 3.11) clearly visible; additionally, the behavior of the Heb as a function of the azimuthal angle shows sharp modulations close to the orientation of the AF uniaxial anisotropy; the coercive ﬁeld Hc has a peak-like behavior close

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Fig. 3.30. Experimental (open circles)and simulated hysteresis loops (black lines) for diﬀerent azimuthal angles. The simulated curves are calculated by the (3.48) with the following parameters: f = 80%, R = 5.9/f, γ = 20◦ [41]

to the orientation of the AF uniaxial anisotropy, at θ = 0◦ and θ = 180◦. In all cases we ﬁnd an astounding agreement between calculated curves and experimental data. It is remarkable, that the EB ﬁeld and the coercive ﬁeld are completely reproduced by the SG model Experimentally the azimuthal dependence of the exchange bias ﬁeld was ﬁrst explored for NiFe/CoO bilayers [121]. It was suggested that the experimental results can be better described with a cosine series expansions, with odd and even terms for Heb and Hc , respectively, rather than being a simple sinusoidal function as initially suggested by Meiklejohn and Bean [1, 26]. The simulations shown in this section are diﬀerent with respect to the previous reports on the angular dependence of exchange bias ﬁeld [121, 122, 123,

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Fig. 3.31. a) The experimental coercive ﬁeld (open symbols ) and exchange bias (ﬁlled symbols) ﬁeld as a function of the azimuthal angle θ.d) Simulated coercive ﬁeld (dotted line) and exchange bias ﬁeld (continuous line) as a function of the azimuthal angle [41]

124, 125]. One diﬀerence is that the MCA anisotropy of the F is supposed to be negligible when compared with the coercive ﬁelds obtained experimentally, and the sharp features of the Heb are reproduced numerically rather then being described by cosine series expansions. Recently, Camarero et al. [76] reports on very similar azimuthal dependent hysteresis loops as shown here. There, an elegant way based on asteroid curve is used to describe the intrinsic asymmetry of the hysteresis loops close to the 0◦ and 180◦ azimuthal angle. A unidirectional anisotropy displaces the asteroid critical curve from the origin. Therefore, if the applied ﬁeld is not parallel to the unidirectional anisotropy, the ﬁeld sweep line does not pass through the symmetry center of the asteroid critical curve leading to inequivalent switching ﬁelds and consequently asymmetric reversals [76]. 3.12.4 Dependence of Exchange Bias Field on the Thickness of the Antiferromagnetic Layer After a short inspection of the phase diagram of EB and coercive ﬁeld (Fig. 3.29) we notice that there is a critical value for the R-ratio at which

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the exchange bias vanishes and the coercive ﬁeld is enhanced. This critical value R=1 depends on four parameters: the anisotropy of the antiferromagnet, the interfacial exchange coupling parameter, the thickness of the ferromagnet, and the conversion factor. The conversion factor further depends on the AF domain and/or AF grain size, if any. In Fig. 3.32 the normalized exchange bias ﬁeld is plotted as a function of the AF thickness and for several conversion factors. The anisotropy of the AF layer (KAF ) and the Jeb parameter are assumed to be constant.We notice two main characteristics of the EB ﬁeld dependence on the AF thickness: when f has high values close to unity, the EB ﬁeld decreases with decreasing AF thickness. However, when f is reduced, a completely diﬀerent behavior of the EB is observed. The EB ﬁeld increases as the thickness of the AF decreases, developing a peak-like feature. This peak-like behavior for the EB ﬁeld at critical AF thickness is a result of enhanced coercivity which is accounted for by the f-factor. Also, an essential parameter is the α angle, which describes the rotation of the AF spins during the magnetization reversal. The critical thickness is preserved by the SG model, but it diﬀers in magnitude as compared to the M&B model. Since some interfacial coupling energy is dissipated as coercivity, the critical thickness within SG model is lower as compared to the corresponding one given by the M&B model. The critical AF thickness within the SG model follows from the condition R = 1:

∞ Fig. 3.32. The normalized exchange bias heb /f = Heb /Heb as a function of the tAF KAF /Jeb for diﬀerent values of the conversion factor f . The anisotropy constant KAF and Jeb are assumed to be constant as the AF thickness is being varied for each f . An asymmetric peak like behavior of the exchange bias ﬁeld develops for high values of the conversion factor

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tSG AF,cr =

f Jeb = f tMB AF,cr , KAF

(3.49)

MB where tSG AF,cr and tAF,cr are the AF critical thickness predicted by the SG and M&B models, respectively. Experimentally the R-ratio can be tuned by changing the thickness of the AF and keeping the other three parameters constant. As a result one observes a critical thickness of the AF layer for which the EB disappears [126, 72, 127, 73, 128, 81, 89]. This AF critical thickness can be qualitatively understood within the M & B model. When the hardness of the AF layer is reduced, the AF spins will rotate under the torque created by the F layer trough the interfacial coupling constant. The shape of the EB as function of AF thickness, however, can be diﬀerent from one system to another depending on the other three parameters. The most prominent experimental feature of the EB dependence on the AF thickness is the development of a peak close to the critical thickness. Several proposals were made to describe the peculiar shapes of EB ﬁeld dependence on AF thickness. According to the Malozemoﬀ model, a change in the AF domain size as function of AF thickness results in a change of exchange bias magnitude. Other inﬂuences on the AF dependence of the EB and coercive ﬁeld are of structural origin [126, 73, 18]. It has been shown by Kuch et al. [18] that at the microscopic level, the coupling between the AF an F layers depends on the atomic layer ﬁlling and on the morphology of the interface. The AF-F coupling was observed to vary by a factor of two between ﬁlled and half-ﬁlled interface. Moreover, islands and vacancy islands at the interface lead to a quite distinct coupling behavior. Therefore, structural conﬁgurations are indeed contributing to the EB-dependence as function of AF thickness. The peak-like behavior of the EB ﬁeld as function of the AF thickness is strongly dependent on temperature. An almost complete set of curves, showing a monotonous development of the AF peak from high to low temperatures was measured by Ali et al. [89]. The data is reproduced in Fig. 3.33 together with the simulations based on DS model. Although the DS model does describe well some experimental observed features, some discrepancies still exists. For instance the development of the AF peak as well as the critical AF thickness as function of temperature are more pronounced in the experimental data as compared to the DS simulations. The SG model, through the conversion factor f, appears to be able to describe the evolution of the critical AF thickness(see Fig. 3.32 and Fig. 3.33 (left)). Also the shape of the EB dependence on the AF thickness, from an almost ideal M&B type at T=290 K to a pronounced peaked curve at T=2 K is qualitatively reproduced. Although not considered so far, the conversion factor seems to be temperature dependent. This can be understood if we consider the basic assumption of the SG model, namely the frustration at the interface. Temperature ﬂuctuations acting on metastable spin states cause a variation of the SG anisotropy as function of tempera-

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Fig. 3.33. Left: IrMn thickness dependence of the exchange bias ﬁeld Heb for a number of temperatures. Lines between the points are a guide to the eye. Right: Prediction of the DS model for the AF thickness dependence of the exchange bias ﬁeld from the stability analysis of the interface AF domains at diﬀerent temperatures. (from [89])

ture. We will not further speculate on the exact temperature dependence of the f-factor, but we mention that a numerical analysis for the EB dependencies shown in Fig. 3.33 would allow to untangle all the parameters in the R-ratio. The conversion factor is given by the shape of the EB curves, f Jeb can be extracted from the temperature dependence of the EB ﬁeld at high AF thicknesses, and ﬁnally, the anisotropy constant will be deduced from the critical AF thickness. Note also, that the SG model has the potential to even describe the diﬀerent temperature dependent shapes of the EB ﬁeld, namely linear dependence versus more rounded shape: for an AF thickness close to the critical region, the temperature dependence of the EB bias will be clearly steeper (linear-like) as compared to the corresponding one at higher AF thicknesses(more rounded). 3.12.5 The Blocking Temperature for Exchange Bias Experimentally it is found that the temperature where the exchange bias eﬀect ﬁrst occurs is usually lower than the N´eel temperature of the AF layer (TN ) [8]. This lower temperature is called blocking temperature (TB ). For thick AF layers TB ≤ TN , whereas for thin AF layers TB TN [8]. Furthermore, the coercive ﬁeld increases starting just below TN (with some exceptions) in contrast to the EB ﬁeld, which appears only below TB . These three experimentally observed characteristic features can qualitatively be explained within the M&B and SG models. In order to have a nonvanishing EB ﬁeld in the region with R ≥ 1, the following condition has to be be fulﬁlled: f Jeb KAF > = KAF,crit , (3.50) tAF

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where KAF,crit is the critical AF anisotropy for the onset of the EB ﬁeld. For a ﬁxed AF layer thickness, the condition above sets a critical value for the AF anisotropy for which the EB can exist. Considering that the AF anisotropy increases steadily below TN , for large AF layer thicknesses the condition of (3.50) is fulﬁlled just below TN , whereas for thinner AF layers this condition is fulﬁlled at a correspondingly lower temperature TB . It is clear from the phase diagrams in Fig. 3.29 and in Fig. 3.32 that there is a region of anisotropy crit 0 < KAF < KAF where the EB ﬁeld is zero and the coercive ﬁeld is enhanced. It follows that the enhancement of the coercive ﬁeld should be observed above the blocking temperature and below the N´eel temperature of the AF. This situation is indeed observed experimentally. For the case of CoO(25 ˚ A)/Co layers the coercive ﬁeld increases starting from the TNCoO = 291 K, whereas the exchange bias ﬁeld ﬁrst appears below TB = 180 K [14]. Further possible causes for a reduced blocking temperature and for the behavior of the EB and coercive ﬁelds as a function of temperature are discussed elsewhere: ﬁnite size eﬀects [129], stoichiometry [130] or multiple phases [131], AF grains [132] and diluted AF [38].

3.13 Training Eﬀect The training eﬀect refers to the dramatic change of the hysteresis loop when sweeping consecutively the applied magnetic ﬁeld of a system which is in a biased state. The coercive ﬁelds and the resulting exchange bias ﬁeld versus n, where n is the nth measured hysteresis loop, displays a monotonic dependence [133, 134, 134, 99]. The absolute value of Hc1 and of the EB ﬁeld decreases from an initial value at n = 1 to an equilibrium value at n = ∞. The absolute value of the coercive ﬁeld Hc2 , however, displays an opposite behavior, i.e. it increases with n. These features of the training eﬀect is referred to as Type I by Zhang et al. [135]. The other case when both |Hc1 | and |Hc2 | decrease is called Type. II. In this section we deal only with the so-called Type I training eﬀect. Several mechanisms were suggested as a possible cause of the eﬀect. While it is widely accepted that the training eﬀect is related to the unstable state of the AF layer and/or F/AF interface prepared by ﬁeld cooling procedure, it is not yet well established what mechanisms are dominantly contributing to the training eﬀect. N´eel [77] discussed the training eﬀect as a tilting of the superﬁcial magnetization of the AF domains. This would lead to a Type I training eﬀect. N´eel also discussed that a creeping eﬀect could lead to a Type II training eﬀect. Micromagnetic simulations within the DS model [32, 35] show that the hysteresis curve is not closed after a complete loop. The lost magnetization is directly related to a partial loss of the superﬁcial magnetization of the AF domains, which further leads to a decreased exchange bias.

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Zhang et al. [136] suggested that the training eﬀect can be explained by incorporating into the Fulcomer and Charap’s model [137] positive and negative exchange coupling between the grains constituting the AF layers. In [138], the authors found direct evidence for the proportionality between the exchange bias and the total saturation moment of the heterostructure. The ﬁndings were related to the prediction of the phenomenological M&B approach, where a linear dependence of the exchange bias on the AF interface magnetization is expected. Binek [139] suggested that the phenomenological origin of the training eﬀect is a deviation of the AF interface magnetization from its equilibrium value. Analytical calculations in the framework of non-equilibrium thermodynamics leads to a recursive relation accounting for the dependence of the Heb ﬁeld on n. Hoﬀmann [140] argues that only biaxial AF symmetry can lead to training eﬀects, reproducing important features of the experimental data, while simulation with uniaxial AF symmetry show no diﬀerence between the ﬁrst and second hysteresis loops. Experiments performed by PNR, AMR and Kerr Microscopy [111, 47, 14, 141, 59, 112, 142, 143] also support the irreversible changes taking place at the F/AF interface and in the AF layer. It has been observed that during the very ﬁrst reversal at Hc1 , interfacial magnetic domains are formed and they do not disappear even in positive or negative “saturation”. The interfacial domains serve as seeds for the subsequent magnetization reversals. These ferromagnetic domains at the interface have to be intimately related to the AF domain state [144]. Therefore, the irreversible changes of the AF domain state are responsible for the training eﬀect. Furthermore measurements have detected out-of-plane magnetic moments [82, 59] hinting at the existence of perpendicular domain walls in the AF layer, as originally suggested by Malozemoﬀ. Therefore, irreversible changes of the AF magnetic domains and of the interfacial domains during the hysteresis loops play an important role for the training eﬀect. 3.13.1 Interface Disorder and the Training Eﬀect In the following we analyze the AF domains and interface contributions to the training eﬀect. We assume that a gradual increase of the interfacial disorder of the F/AF system leads to a training eﬀect. Within the SG model, the magnetic state of the F/AF interface can be mimiced through a unidirectional induced anisotropy Kef f , which is allowed to have an average direction γ, where γ is related to the spin disorder of the interface. Also, we will consider the inﬂuence of a progressive rotation of an AF domain anisotropy during the reversal. Both situations will be treated below. In Fig. 3.34a) and b) we show ﬁrst and second hysteresis loops (longitudinal and transverse components of magnetization) calculated with the help of (3.48). In these calculations we set the conversion factor to f = 60% and

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Fig. 3.34. Simulations of training eﬀect within the SG model. Longitudinal a) and transverse b) components of magnetization for a scenario (see text) involving irreversible changes of the AF domain angle θAF during the magnetization reversal. Longitudinal c) and transverse d) components of magnetization for the case when only the interface disorder parameter γ increases and the AF state remains unchanged

R = 62.5/f . We consider a drastic change of an AF domain which progressively rotates its anisotropy axis during the magnetization reversal. This situation can be accounted for in the SG model by replacing the α angle in (3.48) with α − θAF , where the θAF is the orientation of the AF domain anisotropy. We also set the γ angle to be almost zero. Following closely the experimental observations [111, 14], before the ﬁrst reversal θAF is zero, and just after the ﬁrst reversal θAF increases towards an equilibrium value. The ﬁrst branch of the hysteresis loop appears rather sharp, therefore we assume that the AF spins and F spins are collinear immediately after cooling in a ﬁeld (θAF = 0◦ ). For the second branch of the 1st hysteresis loop we consider that θAF = 20◦ , therefore the second leg appears more rounded. The transition from θAF = 0◦ to θAF = 20◦ is assumed to happen right after or during the ﬁrst reversal at Hc1 . This is in accordance with the observation that for thin CoO layers [111, 14] where the disordered interface appears after the ﬁrst reversal at Hc1 . Now, the ﬁrst branch of the second hysteresis loop is simulated with θAF = 20◦ . At the third reversal, we again

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assume that the AF domain angle further increases. Therefore, the second branch of the hysteresis loop is simulated assuming a new value of θAF = 30◦ . The hysteresis loops bear all the features observed experimentally [99, 136, 135, 58, 61, 111, 60, 138]. More strikingly, the transverse component of magnetization shows a small step increase at Hc1 and larger increase at Hc2 , on the reverse path. These are typical features observed experimentally by PNR [111, 14], AMR [47, 143, 112] and MOKE [141]. Moreover, the transverse component at saturation behaves very close to recent observations of Brems et al. [112] on Co/CoO bilayers and lithographically nanostructured wires. After ﬁeld cooling and before passing through the ﬁrst magnetization reversal in the descending ﬁeld branch, the resistance in saturation (which is proportional to the square of the orientation of transverse component of magnetization) reaches its maximum because all spins are oriented along the cooling ﬁeld. After going through a complete hysteresis loop, the resistance at saturation is reduced, indicating that spins in the F are rotated away from the cooling ﬁeld. After reversing the ﬁeld back to positive saturation the resistance does not recover its initial value. Moreover, the untrained state can be partially reinduced by changing the orientation of the applied magnetic ﬁeld [112] which can be interpreted as a further indication of AF domain rotation during the reversal. Next, we consider only the interface disorder through a progressive change of γ angles. In Fig. 3.34b) and c) we show ﬁrst and second hysteresis loops calculated with the help of (3.48). In these calculations we consider that the AF is strong, R = 62.5/f . For the conversion factor we take a value of f = 60%. Also, we assume the average AF orientation to be parallel to the ﬁeld cooling orientation (θAF = 0). Following closely the experimental observations, before the ﬁrst reversal γ is zero, while just after the ﬁrst reversal, γ increases towards an equilibrium value. The ﬁrst branch of the hysteresis loop appears rather sharp, therefore we assume that the AF spins and F spins are collinear immediately after cooling in a ﬁeld (γ = 0). For the second branch of the 1st hysteresis loop we consider that γ = 10◦ , therefore the second leg appears rather rounded. The transition from γ = 0 to γ = 10◦ is assumed to happen right after or during the ﬁrst reversal at Hc1 . The ﬁrst branch of the second hysteresis loop is simulated with γ = 10◦ . At the third reversal, we again assume that the disorder of the interface increases. Therefore, the second branch of the hysteresis loop (and right after during during the reversal at Hc1 of the second loop) is simulated assuming a new value of γ = 20◦ . The simulations above implies a viscosity-like behavior to the disordered interface [145]. For example, when the F magnetization acquires an angle with respect to the unidirectional anisotropy, the torque exerted on the interfacial spins will drag them away from the initial direction set by the ﬁeld cooling. Reversing the magnetization back to positive directions the Kef f spins will not follow (completely), they remain close to this position (viscosity). This is because the maximum torque exerted by the F spins was already acting at negative ﬁelds, while for positive ﬁelds it is much reduced. When measuring

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again the hysteresis loop, at negative coercive ﬁeld, the Kef f spins will rotate even further and so on. Therefore, the angle of the Kef f anisotropy and/or of the AF domains increases after each hysteresis loop, similar to a rachet, causing a decreased exchange bias ﬁeld. Comparing the hysteresis curves shown in Fig. 3.34a) with the ones in Fig. 3.34c) one notices the same qualitative characteristics. This is not the case for the transverse magnetization curve. The F spins rotate only on the positive side for the ﬁrst case, whereas for the second case the F magnetization rotation covers the entire 360◦ angular range. The anisotropic magnetoresistance (AMR) and PNR hides the chirality of the ferromagnetic spin rotation as they provide sin2 (β) information, whereas MOKE is sensitive to the chirality as it provides sin(β) information. Therefore, measuring both hysteresis components by MOKE, can help to distinguish between the dominant inﬂuence on the training eﬀect: AF domain (or/and grain) rotation versus SG interface instability. The training eﬀect is discussed furthermore in Sect. 3.13.2. 3.13.2 Empirical Expression for the Training Eﬀect The very ﬁrst empirical expression for training [133, 134] eﬀects suggested a power law dependence of the coercive ﬁelds and the exchange bias ﬁeld as a function of cycle index n: k n ∞ Heb = Heb +√ , n

(3.51)

where k is an experimental constant. This expression follows well the experimental dependence of the EB ﬁeld for n ≥ 2, but when the very ﬁrst point is included to the ﬁt, then the agreement is poor. Binek [139] has shown by using non-equilibrium thermodynamics that using a recursive relation, the evolution of the EB ﬁeld as a function of n, can be well reproduced for all cycle indexes (n ≥ 1). The recursive expression reads: n+1 n n ∞ 3 Heb − Heb = −γ (Heb − Heb ) ,

(3.52)

where γ is a physical parameter which, for n 1, was directly related to the k parameter of (3.51). It was shown that a satisfactory agreement between the (3.51) and (3.52) is achieved for n ≥ 3. Therefore the approach of Binek appears to provide the phenomenological origin of the hitherto unexplained power-law decay of the EB ﬁeld with increasing loop index n > 1. The analytic expression (3.52) was further tested for temperature dependent training eﬀect [146]. More recently the equation (3.52) has been further reﬁned by extending the free energy expansion with a correction of the leading term. The new equation for training eﬀect reads [147]: n+1 n n ∞ 3 n ∞ 5 Heb − Heb = −γb (Heb − Heb ) − γc (Heb − Heb ) ,

(3.53)

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where the new γc parameter results from the higher order expansion of the free energy and hence γc γb . The γb parameter is similar to the γ in (3.52). Both parameters γb and γc exhibits a exponential dependence on the sweep rate for measuring the hysteresis loop. A number of three ﬁt parameters is required for both (3.52) and (3.53), but the last equation provides better ﬁtting results for moderate sweep rates. In the following we analyze another type of expression which reproduces the dependence of the coercive ﬁeld and exchange bias ﬁeld as a function of the loop index n and for diﬀerent temperatures. It is based on the simulations shown in the previous section. There it was argued that the training eﬀect is related to the interfacial spin disorder. With each cycle the spin disorder increases slightly, thereby decreasing the exchange bias ﬁeld. Additional eﬀects are related to the AF domain size that also aﬀects the magnitude of the EB and Hc ﬁelds. Both contributions cause a gradual decrease of exchange bias as a function of cycle n. They can be treated probabilistically. We suggest the following expression to simulate the decrease of the EB as a function of n: n ∞ Heb = Heb + Af exp(−n/Pf ) + Ai exp(−n/Pi ) ,

(3.54)

n where, Heb is the exchange bias of the nth hysteresis loop, Af and Pf are parameters related to the change of the frozen spins, Ai and Pi are parameters related to the evolution of the interfacial disorder. The A parameters have dimension of Oersted while the P parameters have no dimension but they are similar to a relaxation time, where the continuous variable “time” is replaced by a discrete variable n. We expect that the interfacial contribution sharply decreases with n as the anisotropy of the interfacial spins is reduced (low AF anisotropy spins), while the contribution from the “frozen” AF spins belonging to the AF domains (“frozen-in” uncompensated spins) appear as a long decreasing tail as they are intimately embedded into a much stiﬀer environment. In the following we show ﬁts to the “trained” exchange bias ﬁeld. In Fig. 3.35a) the EB ﬁeld of thirteen consecutive hysteresis loops were measured at T = 10 K and are plotted as a function of loop index for a CoO(40 ˚ A)/Fe(150 ˚ A)/Al2 O3 bilayer. Three ﬁts are shown: one using the empirical relation (3.51), the second one is a ﬁt performed by Binek [148] using the equation (3.53), and the third one using (3.54). We observe that ∞ the ﬁt using the (3.51) (Heb = 146.6 Oe, k = 44 Oe) follows well the experimental curve for n ≥ 2. However, the best ﬁts are obtained using (3.53) ∞ and (3.54). The best ﬁt parameters using (3.53) are [148]: Heb = 148.632 Oe, b c −8 γeb = 0.00029472, and γeb = −2.772 10 . The parameters obtained from ∞ ﬁts to the data using the (3.54) are: Heb = 158 Oe, Af = 25.87 Oe, Pi = 4.33, Ai = 739.14 Oe, Pi = 0.39. Within the SG approach, we distinguish, indeed, a sharp contribution due to low anisotropy AF spins at the interface and a much weaker decrease from the “frozen-in” uncompensated spins. The temperature dependence of the training eﬀect for a epitaxial Fe(150 ˚ A)(110)/CoO(300 ˚ A)(111)/Al2 O3 bilayer [149, 62, 63] is shown in

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Fig. 3.35. a) Exchange bias as function of the loop index n. The gray line is the best ﬁt to the data using (3.51). The open circles are the best ﬁt [148] to the data using (3.53). The black line is the best ﬁt to the data using (3.54). b)Temperature dependence of the training eﬀect. The lines are the best ﬁt to the data using (3.54)

Fig. 3.35b). The sample was ﬁeld cooled in saturation to the measuring temperature where 31 consecutive hysteresis loops were measured. The ﬁts to the data using (3.54) are shown as continues lines in Fig. 3.35b). We distinguish three main characteristics related to the temperature dependence of the training eﬀect: • each curve shows two regimes, a fast changing one and a slowly decreasing tail; • the “relaxation times” (Pi and Pf ) do not visibly depend on the temperature; • the interface transition towards the equilibrium state is approximatively ten times faster then the transition of the “frozen” spins towards their stable conﬁguration.

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3.14 Further Characteristics of EB-systems 3.14.1 Asymmetries of the Hysteresis Loop A curios characteristic of EB systems is the often observed asymmetry between the two branches of the hysteresis loop for descending and ascending magnetic ﬁelds. The hysteresis loop shape of an isolated ferromagnet is with no exceptions symmetric with respect to the ﬁeld and magnetization axis. This is not the case for an exchange bias system, where the unidirectional anisotropy and the stability of the AF can result in asymmetries of the hysteresis loops and of the magnetization reversal modes. One can distinguish two diﬀerent classes of the hysteresis loop asymmetries. One of them can be assigned to intrinsic properties of the EB systems which lacks training eﬀects, and the another one is intimately related to irreversible changes of the AF domain structure during the magnetization reversal. i) In the ﬁrst category we encounter four diﬀerent situations of asymmetryic magnetization reversal all related to a stable interface without training eﬀect: a) the ﬁrst branch of the hysteresis loop is much extended compared to the ascending branch (see Fig. 3.36). This asymmetric hysteresis loop was observed in FeNi/FeMn bilayers [98]. The underlying mechanism is related to a Mauri type mechanism for exchange bias where a parallel domain wall (exchange spring) is formed in the AF layer. The reversal is understood in terms of domain wall pinning in the antiferromagnet [98, 40, 91]. b) coherent rotation during the ﬁrst reversal at Hc1 , domain wall nucleation and propagation at Hc2 (see Fig. 3.37). Such asymmetric magnetization reversal has been observed by PNR in Fe/FeF2 and Fe/MnF2 systems [150]. This asymmetry depends on the relative orientation of the ﬁeld cooling direction with respect to the twin structure of the AF layer. The reversal asymmetry mentioned above takes place when the FC is parallel oriented with a direction bisecting the anisotropy axes of the two AF structural domains. When the ﬁeld cooling orientation is parallel to the anisotropy axes of one AF domain,

Fig. 3.36. a) Asymmetric hysteresis loop (a) of a NiFe/FeMn bilayer and schematics of domain structure at diﬀerent stages of magnetization reversal [98]

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Fig. 3.37. a) Asymmetric hysteresis loop (b) of a Fe/MnF2 bilayer and and the corresponding neutron reﬂectivity curves: a) the PNR curves recorded at the ﬁrst coercive ﬁeld, Hc1 and c) the PNR curve recorded at Hc2 . The lack of SF reﬂectivity at Hc2 suggests that the reversal proceeds by domain wall nucleation and propagation, whereas at Hc1 the magnetization reverses by rotation [150, 90]

the reversal mechanism is symmetric, i.e. for both branches of the hysteresis loop magnetization rotation prevails. c) sharp reversal on the descending branch and rounded reversal on the ascending one (see Fig. 3.38). This asymmetry has recently been clariﬁed by studies of the azimuthal dependence of exchange bias in IrMn/F bilayers [76, 41, 151]. It is an intrinsic property of the EB bilayer systems and it takes place whenever the measuring external ﬁeld is oﬀset with respect to the ﬁeld cooling orientation. By simply analyzing the geometrical asteroid solutions, it becomes obvious that the sweep line does not symmetrically cross the shifted asteroid when the ﬁeld is not parallel to the unidirectional anisotropy. Actually, this is a peculiar case of asymmetry which can be understood even within the phenomenological model for exchange bias [76] which assumes a rigid AF spin structure. Within the SG and M&B model [41] such asymmetric reversals can be simulated over a wide range of AF thicknesses and anisotropies. Also,

Fig. 3.38. Asymmetric hysteresis loop of a IrMn/CoFe bilayer along an oﬀset θ = 3◦ angle [41]

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within the DS model [36, 39], the eﬀect of an oﬀset measuring ﬁeld axis with respect to the anisotropy axes of the AF and F layer can result in asymmetric reversal modes. d) the descending part is steeper, while the ascending branch is more rounded (see Fig. 3.39). This asymmetry of the hysteresis loop needs to be distinguished from the previous ones, since it occurs when the external ﬁeld is oriented parallel with respect to the anisotropy axis. It is observed in EB bilayers with thin antiferromagnetic layers or for systems containing low anisotropy AF layers. We call these antiferromagnets weak antiferromagnets and characterize them by the R-ratio. When the R-ratio is slightly higher than one (weak AF layers), then the asymmetry of the hysteresis loop can be reproduced within the SG model. When the R-ratio is much higher than one (strong AF layers), then the hysteresis loops are symmetric as shown in Fig. 3.28. To account for the asymmetry we consider the following example where it is essential that the AF layer is weak but the R ratio is higher than 1: R = 1.1, f = 60%, γ = 5◦ , and θ = 0◦ . For these values the minimization of the free energy is evaluated numerically. The results are plotted in Fig. 3.39. The longitudinal and transverse components of the magnetization vector is shown ∞ as a function of the reduced ﬁeld h = H/|Heb |. We clearly recognize that the hysteresis loop is asymmetric: steeper on the descending leg and more rounded on the ascending leg. The asymmetry is due to the large rotation angle of the AF spins during the F magnetization reversal. This asymmetry has not received experimental recognition so far, therefore it remains a prediction of the SG model. ii) The second class of asymmetric hysteresis loop is directly related to the training eﬀect, and therefore to the stability of the AF layer and AF/F magnetic interface during the magnetization reversal.

Fig. 3.39. Longitudinal (left) and transverse components (right) of the magnetization vector for an week antiferromagnet: R = 1.1. The hysteresis loop is asymmetric: the descending part is steeper than the ascending part. The asymmetry is clearly seen also in the transverse component of the magnetization

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a) sharp reversal at Hc1 and more rounded reversal on the ascending branch, at Hc2 (see Fig. 3.40). Measuring subsequent hysteresis loops, the rounded reversal character does not change but appears also at Hc1 . This type of asymmetry, being related to the training eﬀect, is frequently reported in diﬀerent exchange bias systems [133, 99, 61, 58, 136, 152, 135, 111, 60, 138, 141, 59, 153, 154, 142]. Its underlying microscopic origin, has been recently demonstrated to stem in irreversible changes that occurs in the AF layer. Polarised Neutron Reﬂectivity measurements have revealed that at Hc1 the reversal proceeds by domain wall nucleation and propagation [111, 155, 60, 156, 59, 153, 154, 142]. At the second coercive ﬁeld, magnetization rotation is the reversal mechanism. Moreover, by analyzing the AF/F interface [111, 60], it has been observed experimentally that a transition from a collinear state to an non-collinear disorder state occurs. It suggests that the AF layer in (CoO thin layer)/F evolves from a single AF to a multiple AF domain state. In a AF layer that exhibits a domain state the anisotropy orientation in diﬀerent domains is laterally distributed causing a reduced coercive and exchange bias ﬁeld. Note that a thick CoO ﬁlm is suggested to be

Fig. 3.40. (a) MOKE hysteresis loop of a CoO/Co bilayer after ﬁeld cooling to 50 K in an external ﬁeld of 2000 Oe. The black dots denote the ﬁrst hysteresis loop, the dotted line the second loop. Any further loops are not signiﬁcantly diﬀerent from the second. (b) and (c) Hysteresis loops recorded by polarized neutrons from the same sample but at 10 K. I+ +, I− −, I+ − and I− + refer to non-spin ﬂip and spin-ﬂip intensities as a function of external magnetic ﬁeld. [111]

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already in a domain state [35] after ﬁeld cooling, therefore one would expect a variation of the loop asymmetry (and training eﬀect) with respect to the thickness of the AF layer. 3.14.2 Temperature Dependence of the Rotatable AF Spins (Coercivity) We discuss here the temperature dependence of the interfacial properties. Experimentally this can best be studied by an element selective method [114, 115, 117, 17, 19] to distinguish between the hysteresis of the F and AF layer. Element speciﬁc hysteresis loops have been studied for Fe/CoO [19], which highlights the behavior of the rotatable interfacial AF spins. The exchange bias hysteresis loops measured at the L3 absorption edges of Co (E=780 eV, closed symbols) and Fe (E=708.2 eV, open symbols) and for diﬀerent temperatures are shown in Fig. 3.41. After FC to 30 K, several

Fig. 3.41. The temperature dependence of the exchange biased hysteresis loops measured at L3 absorption edges of Co (E=780 eV, closed symbols) and Fe (E=708.2 eV, open symbols). Scattering angle is 2θ = 32◦ [19]

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hysteresis loops were measured in order to eliminate training eﬀects. Subsequentially the temperature was raised stepwise, from low to high T. For each temperature an element-speciﬁc hysteresis loop at the energies corresponding to Fe and CoO, respectively, was measured. The hysteresis loops of Fe as a function of temperature show a typical behavior. At low temperatures an increased coercive ﬁeld and a shift of the hysteresis loop is observed. As the temperature is increased, the coercive ﬁeld and the exchange bias decrease until the blocking temperature is reached. Here, the exchange bias vanishes and the coercive ﬁeld shows little changes as the temperature is further increased. A ferromagnetic hysteresis loop corresponding to the CoO layer is observed for all temperatures, following closely the hysteresis loop of Fe, with some notable diﬀerences. It appears that the ferromagnetic components of CoO develop higher coercive ﬁelds than Fe below the blocking temperature. This is an essential indication that the AF rotatable spins mediate coercivity between the AF layer and the F one, justifying the conversion factor introduced in the SG model. After careful analysis of the element speciﬁc reﬂectivity data [157, 19], one can conclude that a positive exchange coupling across the Fe/CoO interface. The ferromagnetic moment of CoO is present also above the N´eel temperature. Here, the AF layer is in a paramagnetic state, therefore the coercive ﬁelds for the Fe and CoO rotatable spins are equal. 3.14.3 Vertical Shift of Magnetization Curves (Frozen AF Spins) A vertical shift of magnetization has been observed frequently [53, 158, 138, 66, 35, 159, 15, 85, 20, 160, 161, 160, 119, 162] and is considered to have several origins related to the diﬀerent mechanisms for exchange bias. Within the M&B model a AF monolayer in contact to the F layer is assumed to be uncompensated, but still being part of the AF lattice. At most one could expect a contribution to the macroscopic or microscopic magnetization equal to that of the net magnetization of an AF monolayer and this only by probing an AF layer consisting of an odd number of monolayers. The Mauri mechanism for exchange bias is not likely to result in a vertical shift of the hysteresis loop, since the AF interface is compensated. Within the SG, Malozemoﬀ, and DS models for EB, a small vertical shift is intrinsic. At the interface between the AF and F layer, a number of frozen AF spins will be uncompensated due to the proximity of the F layer. Their orientation is either parallel or antiparallel oriented with respect to the F spins, depending on the type of coupling (direct or indirect exchange). They contribute to the magnetization of the system. In case of an indirect exchange coupling mechanism, the hysteresis loop should be shifted downwards [163, 158], whereas in case of direct exchange coupling the magnetization curve should be shifted upwards. The other part of the interface magnetization, namely the rotatable AF spins, do not cause any shift since they rotate in phase with the F layer. Within the DS model AF domains cause an additional

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shift to the macroscopic magnetization curve along the magnetization axis [35, 66, 85]. To demonstrate the shift of the magnetization we discuss a recent experiment by Ohldag et al. [20] using XMCD [164]. The evolution of the dichroic signal as function of the magnetic ﬁeld, providing the element speciﬁc hysteresis loops, is shown Fig. 3.42. The sample structure is Pd(2 nm)/Co(2.8 nm)/FeF2 (68 nm)(110)/ MgF2 (110)/substrate and has been grown by via molecular beam epitaxy. The F layer is a polycrystalline Co whereas the AF layer is a FeF2 (110) untwined single crystalline layer. The system exhibits a positive exchange bias at large cooling ﬁelds [163]. For weak cooling ﬁelds the exchange bias curve is shifted to negative ﬁelds, as usual for all EB systems. The microscopic origin of the positive exchange bias is an antiferromagnetic coupling at the F/AF interface [163]. Although, the mechanism was clearly demonstrated by magnetometery measurements, the microscopic investigation of the AF interface provides more detailed information. The hysteresis loop of Co is typical and appears symmetric with respect to the magnetization axis (see Fig. 3.42). This is not the case for the AF interface magnetization which shows a twofold behavior: a) some interfacial AF spins are parallel oriented with respect to the F spins and both are rotating almost in phase; the AF hysteresis loop is shifted downwards with respect to the magnetization axis. This seems to be a direct proof of the preferred antiparallel coupling between the F Co and the AF Fe magnetization at the AF-F interface. The interfacial AF Fe moments are aligned during FC by the exchange interaction which acts as an eﬀective ﬁeld on the uncompensated

Fig. 3.42. Element-speciﬁc Co (black) and Fe (gray) hysteresis loops acquired at T= 15 K after ﬁeld cooling in +200 Oe along the FeF2 [001] axis, parallel to the AF spin axis. The direction of the cooling ﬁeld and the vertical shift of the Fe loop at T = 15 K is indicated by arrows. Reproduced from [20]

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spins, most likely conﬁned to the interface [20]. An AF uncompensated magnetization in FeF2 can also occur due to piezomagnetism, which is allowed by symmetry in rutile-type AF compounds and may be induced by stresses occurring below the N´eel temperature [165, 148]. These experimental observations can also be described by the SG model. During the ﬁeld cooling procedure the frozen-in spins depicted in Fig. 3.27 will be pointing opposite to the F spins as to fulﬁll the indirect exchange condition at the F/AF interface. The rotatable ones remain unchanged in the ﬁgure, being parallel aligned with the F spins. The orientation of the frozen spins is governed by the exchange interaction at the F/AF interface, whereas the rotatable spins are aligned by the F layer and the external ﬁeld. Upon reversal the rotatable spins will follow the ferromagnet whereas the frozen spins remain pinned, leading to a shift downwards of the interfacial AF hysteresis loop. As the system described above does not exhibit training eﬀects, no irreversible changes occur during the magnetization reversal. In high enough cooling ﬁelds, the frozen-in spins will align parallel to the cooling ﬁeld becoming also parallel with the rotatable AF and F spins. This will cause a positive shift of the hysteresis loop (positive exchange bias), since the orientation of the AF frozen spins is negative with respect to the coupling sign [163]. The experimental results described above could most likely be described also by the DS model and the Malozemoﬀ model if magnetic domains in the AF layer will be conﬁrmed. A relative vertical shift of the magnetization related to training eﬀects is described by Hochstrat et al. [138] for a NiO(0001)/Fe(110) exchange bias system. The antiferromagnet is a single crystal NiO whereas the ferromagnetic material is an Fe layer deposited under ultrahigh vacuum condition. Upon successive reversals of the F layer a decrease of the total magnetization was observed by SQUID magnetometery. The variation of the vertical shift as a function of the hysteresis loop index n was correlated to a decrease of the AF magnetization. A linear correlation between the AF magnetization, deduced from the vertical shift, and the exchange bias ﬁeld during training was found, suggesting that the training eﬀect may be related to a reduced AF magnetization along the measuring ﬁeld axis and as a function of the loop index. In Fig. 3.43 we show another situation where a vertical shift is visible in macroscopic magnetization curves measured by SQUID magnetometery [166]. The system is a polycrystalline CoO(2.5 nm)/Co(15 nm) bilayer grown by magnetron sputtering [112]. The hysteresis loop at room temperature and along the easy magnetization axis of the F layer is symmetric with respect to the magnetization axis and shows a low coercive ﬁeld. Upon ﬁeld cooling the system through the N´eel temperature t of the CoO layer (TN = 291 K) to the measuring temperature T=4.2 K the system is set in an exchange bias state. Then, a hysteresis loop is measured by sweeping the ﬁeld from 2000 Oe to –2000 Oe and back. By comparing this hysteresis loop with the

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Fig. 3.43. Hysteresis loops of CoO(25 ˚ A)/Co(180 ˚ A). The dashed line is the hysteresis loop recorded at 300 K which is above the N´eel temperature of the CoO layer. After ﬁeld cooling the system to 4.2 K, a hysteresis loop is recorded sweeping the applied ﬁeld from 9000 to –9000 Oe and back to 9000 Oe (black line, long loop). A second loop is measured under the same conditions as the previous one, but the ﬁeld range is shorter, namely between 2000 Oe to –2000 Oe and back to 2000 Oe. The cooling ﬁeld was positive. A vertical shift of the hysteresis loop is clearly visible for the short loop, whereas the long loop appears to be centered with respect to magnetization axis. [166]

one measured at room temperature one clearly observes a vertical shift up along the magnetization axis. Moreover, this up-shift is due to the ferromagnetic Co spins which do not fully saturate at –2000 Oe. Previous studies have shown that after the reversal at Hc1 , the AF CoO layer breaks into AF domains exhibiting an anisotropy distribution. Due to the strong coupling between the F and the AF domains, the F layer cannot easily be saturated. Next, after repeating the ﬁeld cooling procedure, another hysteresis loop was measured by sweeping the external ﬁeld to a much larger negative value, namely to –9000 Oe. Now one observes that the hysteresis loop becomes more symmetric with respect to the magnetization axis. The saturation ﬁeld, where the hysteresis loops closes, is about –8000 Oe. The example above shows that a vertical shift of magnetization can be related to a non homogeneous state of the F layer due to non-collinearities at the AF/F interface. The relation to the training eﬀect is clearly seen as diﬀerent coercive ﬁelds Hc2 depending on the strength of the applied ﬁelds. When a stronger ﬁeld is applied in the negative direction, the exchange bias decreases due to a larger degree of irreversible changes into the AF layer.

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A distinct class of vertical shifts is found in exchange bias systems where a diluted antiferromagnet [167, 66, 35, 85] acts as a pinning layer. As function of dilution of a CoO layer by Mg impurities, as well as, partial oxygen pressure during the deposition, the AF layer acquires an excess magnetization seen as a vertical shift of the hysteresis loop. This strongly supports the fact that the domain state in the AF layer as well as the EB eﬀect is caused and controlled by the defects [66, 85]. Vertical shifts in nanostructures and nanoparticles are often observed due to uncompensated AF spins. For a detailed discussion we refer to recent reviews by Nogues et al. [161] and Iglesias et al. [160] and also recent papers on AF nanoparticles [119, 162].

3.15 Further Evidence for Spin-Glass Like Behavior Observed in Finite Size Systems Although nanoparticles are not covered in this review, we, nevertheless, discuss two recent instances which in AF nanoparticles conﬁrm the SG behavior. One is Co3 O4 nanowires [119] and the other is CoO granular structure [118]. 3.15.1 AF Nanoparticles Nanoparticles of antiferromagnetic materials have been predicted by N´eel [168] to have a small net magnetic moment due to an unequal number of spins on the two sublattices as a result of the ﬁnite size [169]. Hysteresis loops of AF nanoparticles have been observed and several suggestions were made to account for their weak ferromagnetism [169]. One important ﬁnite size eﬀect of AF magnetic nanoparticle is the breaking of a large number of exchange bonds for surface atoms. This can have a particularly strong eﬀect on ionic materials, since the exchange interactions are superexchange interactions. The deﬁcit of exchange bonds could lead to a spin disordered shell exhibiting spin-glass like behavior. The last eﬀect is demonstrated most recently by Salabas et al. [119] and discussed further below. In Fig. 3.44 two hysteresis loops of Co3 O4 nanoparticles (8 nm diameter) prepared by a nanocasting route are shown [119]. Both curves were measured at T = 2 K after cooling the system in zero ﬁeld (ZFC hysteresis loop) and in an external applied ﬁeld of +4 T (FC hysteresis loop). Whereas the ZFC loop shows typical weak ferromagnetic properties speciﬁc to AF particles, after FC a completely diﬀerent behavior is observed. The hysteresis loop is exchange biased, it is vertically shifted and shows training eﬀect. Moreover, the temperature dependence (not shown) of the coercive ﬁeld and exchange bias ﬁeld increases with decreasing temperature, which is also a usual behavior of EB systems. More strikingly the FC hysteresis loop does not close on the right side at positive ﬁelds. This open hysteresis loop is a direct indication of a spin-glass like behavior, similar to the loops observed in pure spin glass

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Fig. 3.44. Field cooled and zero ﬁeld cooled hysteresis loops of Co3 O4 nanowires. From [119]

systems [37]. This behavior appears to support the SG model (Fig. 3.27), where a reduced anisotropy is assumed to occur at the F/AF interface. Both frozen-in spins and rotatable spins are directly seen in the FC hysteresis loop. Moreover, the irreversible changes of the surface AF spins are causing the open loop. A similar open loop is also predicted by the DS model. Within the DS model the hysteresis loop of the AF interface layer is not closed on the right hand side because the DS magnetization is lost partly during F reversal due to a rearrangement of the AF domain structure. The AF particles, however, are supposed to be single AF domains, therefore irreversible changes are due to surface eﬀects rather than caused by AF domain kinetics. Certainly, at very large cooling ﬁelds, the bulk structure of the AF particle should be also aﬀected. The spin-glass like behavior was also recently observed for cobalt ferrite nanoparticles [162], for (Mn,Fe)2 O3−t nanograins [170], and for a εFe3 NCrN nanocomposite system [171]. 3.15.2 Extended Granular AF Film Another direct experimental evidence of a spin-glass like behavior was observed by Gruyters [118] on CoO/Au multilayers, CoO/Cu/Fe trilayers, and CoO/Fe bilayers with granular structure. In Fig. 3.45a) the ZFC hysteresis loop of a CoO/Au multilayer and for diﬀerent temperature is shown. One notices that the coercive ﬁeld is enhanced at low temperatures, but no hysteresis shift develops. The FC hysteresis loop (see Fig. 3.45b)), however, is almost completely shifted to one side of the ﬁeld axis. These observations are explained by Gruyters as an eﬀect of the uncompensated AF spins of the granular ﬁlm structure. The saturation magnetization of this granular CoO ﬁlm is about 60–62 emu/gCoO which would result in an enormous amount of uncompensated spins equivalent to 22% for the observed particles volume. No stoichiometry inﬂuences are considered to contribute to this value. Although the uncompensation level is unclear, the evidence of a spin-glass behavior

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Fig. 3.45. a) ZFC hysteresis loops of [CoO [x ˚ A]/Au (60 ˚ A)] multilayers. b) FC hysteresis loop of a [CoO [20 ˚ A]/Au (60 ˚ A)] multilayer. c) ZFC and FC magnetization curves as a function of increasing temperature T in diﬀerent ﬁelds for a [CoO [15 ˚ A]/Au (60 ˚ A)] multilayer. d) Field dependence of Tirr raised to the 2/3 power for two diﬀerent multilayers. ([118])

is well demonstrated. In Fig. 3.45c) ZFC and FC magnetization curves are measured as function of temperature and for diﬀerent external ﬁelds. Two main characteristic features are observed for these curves: an irreversibility temperature Tirr , where the ZFC and FC branches of MCoO (T) coalesce, and a pronounced peak due to superparamagnetic blocking in the ZFC magnetization. The direct evidence of a spin-glass behavior is the ﬁeld dependence of Tirr shown in Fig. 3.45d). The existence of critical lines spanned by the variables temperature and magnetic ﬁeld can be explained by mean-ﬁeld theory. One of these lines has been predicted by de Almeida and Thouless for Ising spin systems [118]. The Tirr raised to 2/3 power as function of ﬁeld exhibits a linear dependence in agreement with the predictions of Almeida and Thouless.

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3.15.3 Dependence of the Exchange Bias Field on Lateral Size of the AF Domains The relation between the exchange bias and the reduced size eﬀects due nano-structuring of the AF-F systems is important from both fundamental and technological points of view. From a fundamental point of view, the reduced lateral size of both F and AF objects induces signiﬁcant changes of the exchange bias ﬁeld, coercive ﬁelds and also the asymmetry of the hysteresis loops [172, 173, 174, 175, 176, 180, 179, 177, 182, 181, 178, 154, 183]. In some systems an increased exchange bias ﬁeld occurs for reduced lateral sizes, whereas in other cases an opposite behavior is reported, the exchange bias ﬁeld decreases with decreasing the lateral length scales [161]. We refer in the following to the last situation. In Fig. 3.46 are depicted schematically several lateral systems which are commonly used to study the inﬂuences of the nano-structuring onto the exchange bias properties. A reduced lateral size of the ferromagnet (Fig. 3.46b) and Fig. 3.46c)) gives rise to additional shape anisotropies for the ferromagnet leading to a change of both coercive and exchange bias ﬁelds as well as a change of the hysteresis shape. In Fig. 3.46a) these additional anisotropies are minimized and therefore the dependence of exchange bias as function of the AF lateral size is more transparent. For all three situations we can assume that at the borders deﬁned by the geometrical nanostructures, there is an additional disorder extending to the interface. Even for the case b) where the F is nano-structured we may expect that the lithographic process will not always be stopped exactly at the interface but also aﬀecting the AF layer around the dot. Due to nano-structuring it is natural to expect that at the edges of the dot there are AF spins with reduced anisotropy. These spins will contribute to the coercivity at the expense of the interfacial exchange energy. The eﬀective ef f interfacial exchange energy can be written as: Jeb = F f Jeb , where F is a conversion factor related to size eﬀects, similar to the f deﬁned for the interface. It is easy to estimate the F -parameter, as the fraction of the outer shell area divided by the total dot area: F ≈ A1/A2 ≈ (π(D − d)2 )/(πD2 ) = 1 − 2d/D + d2 /D2 , where d is the lateral thickness of the outer shell and D is the diameter of the dot itself. Assuming that d << D and that d is constant,

a) AF

F

b)

c)

F AF

F AF

Fig. 3.46. A schematic view of diﬀerent nanostructered systems used to study the ﬁnite size eﬀects on the exchange bias properties

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we obtain the general expression for the variation of the exchange bias ﬁeld as D: D→∞ −Heb + Heb 1 , (3.55) ≈1−F ≈ D→∞ D Heb D→∞ is the exchange bias of an extended ﬁlm. Within this simple where Heb model the exchange bias ﬁeld decreases as the lateral size of the AF structure decreases. This 1/D dependence is often observed experimentally [161]. The general behavior of the hysteresis loops as a function of the f parameter is depicted in Fig. 3.28. We see that for high f corresponding to a large diameter AF dot, the coercive ﬁeld is small and the exchange bias is high. When, however, the lateral size of a AF object becomes smaller, the f ratio decreases leading to an increased coercive ﬁeld and reduced exchange bias ﬁeld. Transition from the top hysteresis to the bottom hysteresis of Fig. 3.28 are usually observed due to nanostructuring of the AF layers [172, 177, 178, 161].

3.16 Conclusions The results presented in this chapter provides an overview of the physics of a F/(Interface)/AF exchange bias systems. Fundamental properties of the unidirectional anisotropy are considered and discussed. The Meiklejohn and Bean model as well as the Mauri model are considered in detail and compared, both analytically and numerically. The N´eel approach, Malozemoﬀ model, Domain State model and a model of Kim and Stamps are discussed as they provide novel and fundamental ideas on the EB phenomenon. The Spin Glass model is discussed in even more details. Further experimental results touching the fundamentals of exchange bias are described. We distinguish several outcomes of our overview: • The exchange bias is an interface eﬀect, as clearly proven by the 1/tF dependence. Deviations from this law were observed in the literature, but fundamentally this expression is clearly well established. • The AF anisotropies in the bulk of the AF layer and at the interface to a soft ferromagnetic layer give rise to an impressively rich behavior of the magnetic properties: the hysteresis loops can be shifted along both ﬁeld and magnetization axis and in both positive and negative directions, the azimuthal dependence of exchange bias exhibits non-intuitive behavior such as a shift of its maximum with respect to the ﬁeld cooling orientation, the hysteresis loops are asymmetrically shaped, etc. Some of the exchange bias characteristics mentioned above are not fully and consistently revealed experimentally which leaves an open window for further quests. • The analytical formulae for the exchange bias within the M&B model depends mainly on the properties of the AF layer, namely on its anisotropy and thickness.

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• The peak-like increase of the EB close to the critical thickness of the AF layer can be reproduced by the SG model. This eﬀect is also accounted for by the DS model. • The azimuthal dependence of the exchange bias eﬀect could help to distinguish between two ideal mechanisms for exchange bias: Mauri versus M&B models. • The SG model describes the conversion of coupling into coercivity.

List of acronyms AMR FC ZFC EB DS SG SW M&B MCA MOKE PNR SQUID AF F MFM MRAM DAFF XMCD XRMS

Anisotropic Magneto-Resistance Field Cooling Zero Field Cooling Exchange Bias Domain State Model Spin Glass Model Stoner-Wohlfarth Model Meiklejohn and Bean Model Magneto-Crystalline Anisotropy Magneto-Optical Kerr Eﬀect Polarized Neutron Reﬂectivity Superconducting Quantum Interference Device Antiferromagnet Ferromagnet Magnetic Force Microscopy Magnetic Random Access Memory Diluted Antiferromagnets in an External Magnetic Field Soft X-ray Resonant Magnetic Dichroism Soft X-ray Resonant Magnetic Scattering

Acknowledgments We gratefully acknowledge support by the DFG Sonderforschungsbereich 491: Magnetic Heterostructures: spin structure and spin transport, and by BESSY. We have beneﬁtted from discussions with: Werner Keune, Wolfgang Kleemann, Kurt Westerholt, Ulrich Nowak, Klaus D. Usadel, Christian Binek, Kristiaan Temst, Ivan K. Schuller, Robert L. Stamps.

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4 Exchange Coupling in Magnetic Multilayers Bretislav Heinrich Physics Department Surion Fraset University Burnaby, BC, V5AIS6, Canada

Abstract. Spintronics devices employ a wide range of magnetic multilayer structures. In this chapter the coupling between the magnetic layers through non-magnetic spacers will be reviewed. It starts with phenomenology of interlayer magnetic coupling. This is followed by experimental techniques and a detailed interpretation of measurements. The main emphases will be given to the static and rf measurement techniques. Theory of magnetic coupling covers the basic ideas of interlayer exchange coupling and dipolar coupling. The additional contributions caused by imperfections in realistic samples are discussed in sessions on orange peel coupling, pinhole coupling, and biquadratic exchange coupling. Experimental results and their interpretation were carried out to emphasize the richness of magnetic interactions in a wide range of systems. A particular attention was given to the Fe/Ag,Au/Fe, Co/Cu/Co and Fe/Cr,Mn,Pd/Fe structures allowing one to demonstrate the importance of interfaces on magnetic coupling in multilayer systems. Finally, it is shown that the time retarded response of interlayer exchange interaction leads to an entirely new coupling which is based on spin pumping and spin sink concepts. This contribution arises only during spin dynamics and compared to the static interlayer exchange coupling is long ranged allowing one in principle move information by means of a spin current via a mechanism that does not directly involve a net transport of electron charge.

4.1 Introduction Spintronics and high density magnetic recording employ a number of magnetic layers separated by non magnetic spacers. The magnetic layers are coupled through a non-magnetic spacer. The coupling between the magnetic layers can be caused by intrinsic and extrinsic mechanisms and be of static and dynamic origin. The purpose of this Chapter is to review the basic concepts of magnetic coupling in 3d transition elements. A number of excellent review articles have been published by Hathaway [1], Slonczewski [2], Stiles [3], and Edwards and Umerski [4]. An extensive review of the behavior of spin density waves in Fe/Cr/Fe trilayers and multilayers is provided by Fishman [5].

B. Heinrich: Exchange Coupling in Magnetic Multilayers, STMP 227, 185–250 (2007) c Springer-Verlag Berlin Heidelberg 2007 DOI 10.1007/978-3-540-73462-8 4

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The Chapter is organized in the following way: Section 4.2 describes the phenomenology of magnetic coupling. Section 4.3 deals with the basic techniques employed in the study of magnetic coupling. A brief summary of theoretical concepts of magnetic coupling is carried out in Sect. 4.4. A summary of experimental results on Fe/Ag,Au/Fe(001), Co/Cu/Co(001), and Fe/Cr,Mn,Pd/Fe(001) systems is presented in Sect. 4.5. The dynamic coupling will be described in Sect. 4.6.

4.2 Phenomenology of Magnetic Coupling The description of magnetic coupling will be mostly restricted to ultrathin magnetic ﬁlms. In ultrathin ﬁlms magnetic variations across the thickness of the ﬁlm are mainly suppressed. This means that the magnetic moments on lattice sites across the ﬁlm thickness are nearly parallel to each other. This is not exactly correct, but greatly simpliﬁes the treatment of magnetic properties. The limits of this concept are described in [6] and [7]. The ﬁlm can be fairly well considered as ultrathin when its thickness does not much exceed the exchange length δ, see [6], 0.5 A δ= , (4.1) 2πMs2 where A is the exchange stiﬀness coeﬃcient (for Fe A =2×10−6 ergcm−1 ) and Ms is the saturation magnetization. The exchange length in Fe is 3.2 nm. The simplest form of magnetic coupling per unit area of an ultrathin ﬁlm trilayer structure of two ferromagnetic ﬁlms (FM) separated by a normal metal (NM) spacer, FM1/NM/FM2, can be described by a bilinear form E1 = −J1 n1 · n2 ,

(4.2)

where J1 is the exchange coupling coeﬃcient in erg cm−2 and n1 and n2 are unit vectors along the magnetic moments in layers 1 and 2, respectively. Another common coupling equation has a biquadratic form 2

E2 = +J2 (n1 · n2 ) ,

(4.3)

where J2 describes the strength of biquadratic coupling. Cases of the bilinear coupling J1 are found having either a + or a–sign. For a + sign the minimum of the bilinear energy term is reached for a parallel orientation of the magnetic moments; for a–sign the minimum energy corresponds to antiparallel magnetic moments. J2 is almost exclusively found to be positive and the minimum of the biquadratic energy term is reached when the magnetic moments are oriented perpendicularly to each other. A detailed description of bilinear and biquadratic coupling terms will be carried out in Sect. 4.4. The equilibrium of the magnetic moments and their dynamic response can be found by using the Landau-Lifshitz-Gilbert (L.L.G) equation of motion [8]

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G1,2 ∂n1,2 1 ∂M1,2 = − [M1,2 × Heﬀ ,1,2 ] + 2 , M1,2 × γ1,2 ∂t γ1,2 Ms,1,2 ∂t

(4.4)

where γ1,2 , M1,2 , and G1,2 are the the absolute values of the electron gyromagnetic ratios, magnetic moments per unit area , and Gilbert damping parameters of layers 1 and 2, respectively. The damping parameter is expressed often in a dimensionless parameter α = G/γMs . The ﬁrst term on the righthand side of (4.4) represents the precessional torque per unit area and the second term represents the well known Gilbert damping torque per unit area. The eﬀective ﬁelds Heﬀ ,1,2 are given by the derivatives of the magnetic Gibbs energy density, F, with respect to the components (Mx,1,2 , My,1,2 , Mz,1,2 ) of the magnetization vector densities M1,2 , see [9],[10],[6], and [11], Heﬀ = −

∂F . ∂M

(4.5)

F includes the Zeeman energy of the dc applied magnetic ﬁeld, demagnetizing ﬁelds Hdem , rf magnetic ﬁeld h, magnetic anisotropies, and the inter-layer magnetic coupling energy. In order to carry out appropriate partial derivatives in (4.5) the magnetic bilinear and biquadratic coupling terms (4.2) and (4.3) are rewritten by replacing n1,2 by M1,2 /Ms,1,2 . In thick ﬁlms (∼100 nm) the internal rf magnetic ﬁeld has to be evaluated by using Maxwell’s equations in the presence of the externally applied rf magnetic ﬁeld. This means that the rf ﬁeld and rf magnetization vary across the ﬁlm thickness, see e.g. [7, 12]. In the ultrathin ﬁlm approximation the variations across the ﬁlm thickness are neglected. For small precession angles (| m | Ms corresponding to an angle of precession less than a few Degrees) the magnetization vector can be linearized by setting M = m+Ms , where Ms and m are the longitudinal and transverse components of M, see Fig. 4.1. This allows one to linearize the equation of motion (4.4).

4.3 Experimental Techniques for Magnetic Coupling Measurements The magnetic coupling can be determined by measuring the dependence of the net magnetic moment on the applied dc ﬁeld. That can be done using low and high-frequency techniques. SQUID, sample vibrating magnetometer, and Magneto Optical Kerr Eﬀect (MOKE)[13] are the most common techniques allowing one to measure the total magnetic moment as a function of the applied ﬁeld. X-ray Magnetic Circular Diochroism (XMCD)[14] is an element sensitive technique allowing one to follow the ﬁeld dependence of a particular layer in a multilayer structure. The neutron scattering is a very powerful tool for investigating the structural and magnetic properties of multilayer ﬁlms [15, 16]. Polarized Neutron Reﬂection (PNR) measurements allow one to investigate the spatial distribution of the magnetic moment inside the sample

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Fig. 4.1. The coordinate system for the instantaneous magnetization M and external ﬁeld H. X,Y, and Z axes are oriented along the principle crystallographic axes. θ and ϕ are the polar and azimuthal angles with respect to the crystallographic axes. The x,y,z coordinate system is oriented along the magnetization M with the x axis directed along the saturation magnetization moment. For a small precessional angle the components My , Mz Mx

[17, 18, 19]. Photo Electron Emission Microscopy (PEEM) [20], Low Energy Electron Microscopy (LEEM) [21, 22] and Secondary Electron Microscopy with Polarization Analysis (SEMPA) [23] are other techniques suitable for the study of magnetic coupling. Ferromagnetic Resonance [10] and Brillouin Light Scattering (BLS) [24] are rf techniques allowing one to determine quantitatively all magnetic macroscopic parameters including interlayer and intralayer exchange couplings. In this Section the discussion will be limited to the most common dc and rf techniques: SQUID, vibrating sample magnetometer, MOKE, FMR, and BLS. The experimental details can be found in the above reference. Here the discussion will be limited to a quantitative interpretation of the experimental measurements. SQUID and vibrating sample magnetometer measure the total sample response. At FMR the microwave penetration (skin) depth is ∼ 100 nm. In MOKE and BLS the total depth of studies is given by the depth of penetration of the laser beam into metallic samples. For visible light this is approx. 15 nm. The depth resolution of MOKE signal was investigated by Hamrle et al. [25] using magnetic multilayers. They determined both the rotational and elliptical contributions to MOKE as a function of depth.

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4.3.1 Dc Magnetometry In SQUID and vibrating sample magnetometers the total magnetic moment is measured, usually along the direction of the applied ﬁeld. In MOKE the magnetic moment is commonly measured in the polar and longitudinal conﬁgurations. In the longitudinal conﬁguration the dc ﬁeld is applied parallel to the ﬁlm surface and it is usually applied in the plane of incidence of the laser beam. In the longitudinal conﬁguration one is sensitive to three contributions. The ﬁrst contribution arises from the magnetic moment that is parallel to the dc ﬁeld (longitudinal Kerr eﬀect), the second contribution comes from the magnetic component perpendicular to the ﬁlm surface (polar Kerr eﬀect), and in a non collinear conﬁguration of magnetic moments the third contribution arises from the product of the parallel and perpendicular components of the magnetic moments with respect to the applied ﬁeld. This contribution is called the quadratic magneto-optic eﬀect. It arises from the second order magneto-optical eﬀect [26]. This eﬀect is the reﬂection analogue of the Voigt eﬀect [27]. The magnetization components in the quadratic magneto-optic eﬀect are conﬁned to the ﬁlm surface. The quadratic magneto-optic eﬀect contribution often becomes important in magnetization measurements using either antiferromagneticaly coupled ﬁlms or with the ﬁeld applied along the hard magnetic axis. The quadratic magneto-optic contribution is a nuisance in MOKE measurements. The quadratic magneto-optic eﬀect is an even function of the applied magnetic ﬁeld. It does not change its sign with reversal of the magnetic moment and creates a profound asymmetry in magnetization measurements as a function of applied ﬁeld. It can be removed by adding the measured MOKE signal to its inverted counterpart. The signal is inverted around a point that is the intersection of the MOKE signal axis with a line parallel to the ﬁeld axis and located midway between the saturated MOKE signals for positive and negative magnetic ﬁelds [6]. The even part, proportional to the quadratic eﬀect, can be obtained by subtracting the measured and inverted signals. The discussion in this Section will be limited to the micromagnetics of trilayer structures consisting of a crystalline ultrathin ﬁlm FM1/NM/FM2 structure. For simplicity these calculations will be limited to cubic materials with the ﬁlm surface oriented in the (001) plane, see Fig. 4.1. The ﬁlms of 3d transition group elements are often but not exclusively grown on (001) templates. This is a particular case but it involves all the ingredients needed to formulate calculations for any other conﬁgurations involving an arbitrary direction of the applied ﬁeld and crystalline orientation. The discussion below should be viewed as an example taken from a user’s manual. The total Gibbs energy per unit area for ultrathin ﬁlms in a distorted cubic material with the saturation magnetic moments, Ms1 and Ms2 , and the applied ﬁeld H can be written in the following form

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⊥ K1,ef f,i 4 α4X,i + α4Y,i − αZ,i − 2 2 i=1,2 2 (nu,i · Mi ) ⊥ 2 − K α − M · H di − J1 m1 · m2 + − Ku,ef f,i i u,ef f,i Z,i 2 Ms,i

F=

−

K1,ef f,i

2

+ J2 (m1 · m2 ) ,

(4.6)

where αX,i , αY,i and αZ,i are the directional cosines between the magnetization vector Mi and the crystallographic axes [100], [010], and [001], re ⊥ ⊥ spectively. K1,ef f,i ,K1,ef f,i , Ku,ef f,i , and Ku,ef f,i are the parameters of the in-plane eﬀective four-fold (cubic), perpendicular eﬀective four-fold, in-plane uniaxial, and perpendicular uniaxial anisotropies, respectively. nu,i are the directions of the in-plane uniaxial axes and di are the ﬁlm thicknesses. m1 and m2 are unit vectors directed along the magnetizations of the coupled ﬁlms: J1 and J2 are the bilinear and biquadratic coupling coeﬃcients. The indices i =1 and 2 describe the properties of the layers FM1 and FM2, respectively. In ultrathin ﬁlms the magnetic moments across the ﬁlm are locked together by intralayer exchange coupling and they can be considered to be giant magnetic molecules [6]. For ultrathin ﬁlms the role of the interface anisotropies of a uniformly magnetized sample can be included in the eﬀective anisotropy parameter,

K1,s d Ku,s Ku,ef f = Ku,bulk + d ⊥ K u,s 2 ⊥ , (4.7) = −2πM + Ku,ef s f d where the subscript s represents the interface contributions. The units of the interface anisotropies are erg/cm2, see [6]. The energy expression in (4.6) is valid for a wide range of magnetic ultrathin ﬁlms such as Fe on Ag(001) [6] and GaAs(001) [28, 29, 30] templates. One can easily generalize it by using the appropriate ﬁlm symmetry An example for an arbitrary orientation of mag⊥ netic moments can be found in [31]. The expression for K1,ef f is not included in (4.7). It originates in variations of the perpendicular uniaxial anisotropy, ⊥ Ku,s , across the ﬁlm surface. It leads to a higher order dependence on 1/d, see [32] and the end of subsection Biquadratic exchange coupling. The ﬁeld dependence of the magnetic moment can be found by minimizing the total Gibbs energy. The static equilibrium is found by minimizing the total energy with respect to the angles ϕ1 , ϕ2 , θ1 , and θ2 for the given angles ϕH and θH , see Fig. 4.1. In this Chapter the calculations will be restricted to the in-plane geometry, this means θ1 = θ2 = θH = π/2. There are a number of minimization procedures available and they are usually implemented by individual groups according to their liking. One should

K1,ef f = K1,bulk +

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realize that minimum energy solutions can exhibit metastable states. Usually one looks for the lowest energy state. This means no hysteresis is present in magnetization measurements. This is not often the case in experiments especially those using ﬁlms grown on GaAs(001) substrates [29]. Therefore it is imperative to carry out MOKE measurements in the lowest energy state. The lowest energy state for a given magnetic ﬁeld can be achieved by cycling the magnetic state at the given applied ﬁeld with a transverse ac magnetic ﬁeld which increases to some preselected maximum and then the amplitude is gradually decreased to zero. This has to be repeated for each applied dc ﬁeld. An example of such procedure can be found in [29] using exchange coupled GaAs/Fe/Au/Fe/Au(001) structures. Several examples of hysteresis loops using exchange coupled magnetic bilayers FM1/NM/FM2 are shown in Figs. 4.2 and 4.3. In order to avoid complexities associated with a general direction of the applied ﬁeld one should employ two magnetic ﬁlms having diﬀerent thicknesses and with the external ﬁeld applied along the magnetic easy axis of the thicker ﬁlm. In that case the thicker ﬁlm remains oriented close to the easy axis. It is the thinner ﬁlm that undergoes a full angular dependence, see the insets in Fig. 4.2. Since the two magnetic moments are diﬀerent it is easy to identify the contributions from the individual layers. The family of curves describing the role of biquadratic magnetic coupling is shown in Fig. 4.2. When the biquadratic magnetic coupling becomes comparable to the bilinear coupling one is not able to achieve an antiparallel conﬁguration of magnetic moments in low magnetic ﬁelds. The biquadratic coupling can lead in zero ﬁeld to an angle between the two magnetic moments ranging from–180 to 90 degrees. The biquadratic magnetic coupling can also aﬀect the approach to saturation. For even small J2 the saturation point is reached without a jump in the magnetic moment; this is also true even in the presence of magnetic anisotropies, see Fig. 4.2. The interpretation of the data when there is no discontinuity in the magnetization becomes quite simple. The deviation from saturation can be treated using a small angle expansion. It is easy to show that the saturation ﬁeld along the easy direction can be described by the simple expression 2Kef f J1 − 2J2 1 1 Hsat + , (4.8) =− + Ms Ms d1 d2 where it has been assumed that the saturation magnetization Ms is the same for both ﬁlms. Kef f is an eﬀective anisotropy ﬁeld obtained from minimization of (4.6), and d1 and d2 are the thicknesses of the layers FM1 and FM2, respectively. In samples where one is not able to get both easy axes along the same direction the magnetization curves can be kept relatively simple if one orients the ﬁeld along the easy axis of the ﬁlm having a large in-plane anisotropy. An example is shown in Fig. 4.3. Notice that the thinner ﬁlm 8Fe with a large

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Fig. 4.2. Simulations of magnetization curves using a set of exchange coupling parameters. The lines 1, 2, 3, 4 and 5 correspond to J2 =0.0, 0.01, 0.04, 0.06, and 0.08 erg/cm2 , respectively. The bilinear coupling was kept constant, J1 = –0.1. The calculations were carried out for the magnetic parameters obtained in GaAs/8Fe/Au/16Fe/Au(001) structures [30], where the integers represent the number of atomic layers. 16Fe: K1,ef f =3.1×105 erg/cm3 ; 8Fe:K1,ef f =1.33×105 3 erg/cm . 4πMs =21.5 kOe. In-plane uniaxial anisotropies were omitted in order to keep the easy magnetic axis in both ﬁlms along the [100] crystallographic direction. The applied ﬁeld was oriented along the magnetic easy axis [100]. The inset shows the ﬁeld dependence of the magnetization angle for J1 =–0.1 erg/cm2 and J2 =0. The dashed line corresponds to the 16Fe ﬁlm and solid line to the 8Fe ﬁlm. Note that the ﬁrst jump brings the magnetic moment of the thinner 8Fe ﬁlm from the parallel orientation over the ﬁrst hard axis, [110], close to the second easy axis, [010]. The second jump corresponds to pulling the magnetic moment over the second hard axis, [110] resulting in an antiparallel conﬁguration of the magnetic moments

in-plane uniaxial anisotropy rotates from the easy axis by a small angle, while the thick 16Fe ﬁlm undergoes a large angular rotation. In FM1/NM/FM2 structures one can measure by MOKE only negative bilinear J1 (antiferromagnetic coupling). One is able to measure a positive (ferromagnetic) bilinear coupling by using spin “engineered structures”. An additional FM0 layer with a normal metal spacer creating a large antiferromagnetic coupling between FM0 and FM1 is needed [33, 34], e.g. FM0/NM0/FM1/NM/FM2. In these structures the magnetic moments in

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Fig. 4.3. Simulations of magnetization curves for the magnetic anisotropies corresponding to GaAs/8Fe/Au/16Fe/Au(001) structures [30], the integers represent the number of atomic layers. The following magnetic parameters were used. 16Fe: K1,ef f =3.1×105 erg/cm3 , Ku,ef f =3.3×104 erg/cm3 ; 8Fe: K1,ef f =1.33×105

erg/cm3 , Ku,ef f =–1.14×106 erg/cm3 . The hard axis of the in-plane uniaxial anisotropy in the 8Fe ﬁlm is oriented along the [110] direction. 4πMs =21.5 kOe. The solid, dashed, and dotted lines in (a) correspond to J1 =0.0, –0.1, and –0.3 erg/cm2 , respectively. The applied ﬁeld was oriented along the magnetic easy axis, [110], of the 8Fe ﬁlm. Note that this ﬁlm has a large in-plane uniaxial anisotropy. This means that for the 16Fe ﬁlm the applied ﬁeld was oriented along the hard axis. (b) shows the ﬁeld dependence of the magnetization angle with respect to the [100] axis for J1 =–0.3 erg/cm2 . The dotted line corresponds to the 16Fe ﬁlm and the solid line to the 8Fe ﬁlm. Note that below 1.5 kOe the magnetic moments are oriented antiparallel to each other with the magnetic moment of the 8Fe ﬁlm oriented along its easy magnetic axis [110](45 Degree), and the magnetic moment of the 16Fe ﬁlm oriented along its hard magnetic axis [110] (–135 Degree)

FM1 and FM2 in zero applied ﬁeld are oriented antiparallel to that in FM0. One needs to apply a dc ﬁeld along the direction of the magnetic moment of the FM0 layer to overcome the ferromagnetic coupling and thus to orient the moments in FM1 and FM2 in an antiparallel conﬁguration. Strong antiferromagnetic coupling can be achieved by using ultrathin Ru spacers [33], see Sect. 4.5. 4.3.2 FMR and BLS Techniques The magnetic coupling can be measured by rf techniques, see [6, 10, 24, 35, 36, 37]. In FMR one usually sets the microwave frequency and sweeps the ﬁeld.

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However in this age of network analyzers (NA) this is not limitation; one can set the ﬁeld and sweep the frequency. In BLS one sets the applied ﬁeld and sweeps the frequency. When the ﬁeld is held constant the angle between the magnetic moments is ﬁxed. This is a simpler situation compared to a regular FMR measurement (holding constant frequency and sweeping the ﬁeld) where the angle between the magnetic moments changes in non-collinear conﬁgurations. For a saturated sample the diﬀerence between constant ﬁeld and constant frequency sweeps is minimal. The main diﬀerence is that in BLS one measures spinwaves having an in-plane wave-vector q-component coresponding to that of the in-plane wavelength of the incoming laser light whereas in FMR the measured spinwave mode corresponds to q0 (homogeneous mode). This means that the ferromagnetic ﬁlms in BLS are always coupled by a dipolar interaction, see Subsection 4.2. Interpretation of FMR and BLS results can be carried out by using rf solutions of (4.4). Let us brieﬂy outline a method for setting up the equations of motion for the rf magnetization components. The coordinate system for an arbitrary orientation of the magnetic moment with respect to the crystallographic axes is shown in Fig. 4.1. Further discussion will be limited to the in-plane conﬁguration where the saturation moment and external ﬁeld are oriented in the X-Y plane, θi = π/2. For a noncollinear conﬁguration with static magnetic moments in the ﬁlm plane the directional cosines αX,i , αY,i , and αZ,i with respect to the crystallographic axes are given by Mx,i cos(ϕi ) − Ms Mx,i = sin(ϕi ) + Ms,i

αX,i = αY,i

My,i sin(ϕi ) Mi,s My,i cos(ϕi ) Ms Mz,i αZ,i = , Ms

(4.9)

where Mx,i , My,i and Mz,i are the instantaneous magnetization components in the coordinate systems with the x-axis parallel to the saturation magnetization. The eﬀective ﬁelds in the frame of the moments FM1 and FM2 can be obtained by inserting (4.9) into (4.6) and using (4.5). However (4.6) is not the density of energy. This requires conversion of the interlayer coupling energy to the energy density. This conversion is well described on pp. 569–570 in [6]. Assuming an even sharing of the interlayer coupling by all atomic layers inside the ﬁlm the density of energy, Ui , for the individual layers FM1 and FM2 are given by

Ui +

K1,ef f,i 4 (ni · Mi )2 αX,i + α4Y,i − Ku,ef f,i =− + 2 2 Ms,i 2 ⊥ 2 /di Ku,ef f,i αZ,i − Mi H − J1 m1 · m2 − J2 (m1 · m2 )

Hef f,i = −

∂Ui . ∂Mi

(4.10)

4 Exchange Coupling in Magnetic Multilayers

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The partial derivatives are taken with respect to Mx,i , My,i and Mz,i . In FMR and BLS the rf transversal components are negligible compared to the magnetization component parallel to the x-axis (static eﬀective ﬁelds). The replacement of the x-component of the magnetization by the saturation magnetization Ms can be done only after the partial derivatives have been carried out. The equations of motion (torque equations) for FM1 and FM2 are obtained by using the above eﬀective ﬁelds in the LLG (4.4). One can ignore all terms which are quadratic in the small rf components. Notice that the perpen⊥ dicular 4-fold anisotropy, K1,ef f,i , is not included in the parallel conﬁguration; it leads to 3rd order terms in the small rf magnetic component Mz,i . This leads to a coupled system of equations for the transverse magnetization components for FM1 and FM2. The z-components of the torque include also expressions which contain only large static magnetization Ms,i components. These expressions correspond to static equilibrium and are equal zero. They provide solutions for ϕi (H) for the applied ﬁeld H. The static equilibrium can be also obtained from minimizing the total energy given by (4.6). The coupled equations for the rf magnetization components leads to two solutions. The precessional motions are coupled and result in an acoustic mode in which the magnetic moments in the two layers precess in phase and in an optical mode in which the magnetic moments precess in antiphase. The simplest interpretation of the coupling can be obtained in the saturated case when the dc magnetic moments are parallel to the applied ﬁeld. The isolated ﬁlms FM1 and FM2 must have diﬀerent resonant frequencies (ﬁelds) in order to be able to observe acoustic and optical modes. If the resonance frequencies (ﬁelds) of the two ﬁlms are exactly same than the strength of the optical mode is zero. Diﬀerent resonant ﬁelds are easy to establish by choosing diﬀerent ﬁlm thicknesses for the ﬁlms FM1 and FM2. The interface anisotropies scale with 1/d and in consequence the isolated ﬁlms exhibit two separate resonance frequencies (ﬁelds) and the optical mode becomes observable. The sign of coupling can be determined from the relative positions of the acoustic and optical modes. In FMR the microwave frequency is usually ﬁxed. For antiferromagnetic coupling (J1 < 0) and ferromagnetic coupling (J1 > 0) the optical modes are located at higher and lower ﬁelds than the acoustic modes, respectively. The positions and intensities of these two modes are nontrivial functions of the magnetic anisotropies and the strength of the interlayer coupling. However the resonance spectrum can be easily evaluated by using the coupled L.L.G. equations of motion, see Fig. 4.4. In the saturated state (collinear magnetic moments) the overall strength of the interlayer coupling, Jef f is given by the superposition of bilinear and biquadratic interlayer couplings, Jef f = J1 − 2J2 .

(4.11)

The optical mode has its magnetization components out of phase and consequently a homogeneous rf driving ﬁeld inside the ultrathin ﬁlms makes

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Fig. 4.4. Simulations of acoustic and optical resonance peaks at f=36 GHz as a function of bilayer exchange coupling in an FM1/NM/FM2 structure. In panel (a) J1 =0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 ergs/cm2 . In panel (b) J1 =0.0, –0.1, –0.2, –0.3, and –0.4 ergs/cm2 . Note that the antiferromagnetic interlayer coupling moves the resonant peaks to larger ﬁelds. For the antiferromagnetic coupling the acoustic and optical peaks move to higher magnetic ﬁelds at a ﬁxed FMR frequency. The acoustic peaks keep increasing their intensity with increasing coupling while the optic peaks get weaker with increasing coupling. The acoustic peaks gradually approach a ﬁxed point which is located between the resonance peaks of the uncoupled ﬁlms. Calculations were carried out for the magnetic parameters obtained in GaAs/8Fe/Au/16Fe/Au(001) structures [30], where the integers represent the number of atomic layers. The following magnetic pa 2 rameters were used: 16Fe: K1,ef f =3.1×105 erg/cm3 , K⊥ u,s =0.88 erg/cm , and

Ku,ef f =3.3×104 erg/cm3 ; 8Fe:K1,ef f =1.33×105 erg/cm3 , K⊥,s =0.82 erg/cm2 , and

Ku,ef f =–1.14×106 erg/cm3 . 4πMs =21.5 kG, g=2.09, and α=0.009. The in-plane uniaxial easy axes for the 16Fe and 8Fe ﬁlms were along the [110] and [110] directions, respectively. The applied ﬁeld was oriented along the [110] crystallographic axis. The damping parameter was increased approx. 3 fold, compared to the measured values, to make the FMR lines wide for easy viewing

excitation of optical modes ineﬀective. The optical mode signal rapidly decreases with the strength of the interlayer coupling, see Fig. 4.4. It is relatively easy to measure the strength of the interlayer coupling up to 0.5 ergs/cm2 [38]. In the saturated state one is not able to measure the interlayer coupling strength if the two ﬁlms have the same magnetic properties. The diﬀerence in thickness does not help. However in a non-collinear conﬁguration of the magnetic moments one can measure the exchange coupling even in ﬁlms having the same magnetic properties. In that case in FMR one gets only one resonant mode which depends strongly on the exchange coupling, see Fig. 4.5. This is strictly only true for the rf ﬁeld oriented perpendicular to the dc applied ﬁeld.

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197

Fig. 4.5. The dependence of the FMR absorption peak on the bilinear magnetic coupling. Simulations were carried out at 10 GHz for a FM/NM/FM structure. The magnetic ﬁlms were of the same thickness. The magnetic anisotropies were assumed to be zero,4πMs =21.5 kG, g=2.09, and α=0.009. The numbers above the absorption peaks represent the strength of bilinear magnetic coupling. One needs to use a low enough microwave frequency to bring the FMR resonance to low ﬁelds where the magnetic moments are not parallel (the unsaturated state). In the saturated state the FMR signal does not depend on the interlayer coupling

The eﬀectiveness of the coupling between a homogeneous rf ﬁeld and the optical mode can be increased if the magnetic moments in the two ﬁlms are noncollinear, see Fig. 4.6. It was shown by Z. Zhang et al. [31] that for the rf ﬁeld oriented parallel to the dc ﬁeld one gets the projected rf ﬁeld components in phase with the optical rf magnetization components resulting in an enhancement of the optical resonance. Note in Fig. 4.6 that the acoustic peak is completely absent for the rf ﬁeld parallel to the dc ﬁeld while the optical peak reaches its maximum. The eﬀective rf ﬁeld components (perpendicular to the dc magnetic moments) in the magnetic layers are antiparallel. This way one is not coupled to the acoustic mode but the optical peak is fully excited. The strength of biquadratic coupling can not be measured independently in the saturated state, see (4.11). However in a non-collinear state the contributions of bilinear and biquadratic interlayer couplings in FMR and BLS measurements can be separated, see Fig. 4.7 and [39]. There is an alternative approach to evaluate the resonance modes using the Smit and Beljers method which is based on the partial derivatives of the Gibbs energy with respect to the magnetization angles. The details of this approach can be found in [35]. An excellent theoretical treatment of rf excitations in a wide range of multiayers with complex spin conﬁgurations can be found in the review article by Camley and Stamps [40].

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Fig. 4.6. The dependence of the FMR signal on magnetic coupling in a non-collinear conﬁguration. Simulations were carried out at 10 GHz for a FM/NM/FM structure. The magnetic ﬁlms were of the same thickness. The magnetic anisotropies were assumed to be zero, 4πMs =21.5 kG, g=2.09, and α=0.009. (a) J1 =0.0 (b) J1 =–0.4 erg/cm2 . The rf magnetic ﬁeld is perpendicular to the applied dc ﬁeld. Only the acoustic mode is excited. (c) J1 =–0.4 erg/cm2 . The rf ﬁeld is oriented 45 Degrees with respect to the dc applied ﬁeld. Note that with this rf driving one can see both the acoustic and optical modes. (d) J1 =–0.4 erg/cm2 . The rf ﬁeld is oriented parallel to the dc applied dc ﬁeld. Only the optical mode is excited. For (b), (c) and (d) the magnetic moments are non-collinear. Their magnetic moments are canted symmetrically away from the dc magnetic ﬁeld. The FMR signal in (a) is 4.5x larger than those in (b), (c) and (d)

In order to obtain a reliable interpretation of the magnetic coupling in MOKE and FMR measurements one needs to know reasonably well the magnetic anisotropies. Independent measurements of the magnetic anisotropies in individual ﬁlms are extremely useful. The interlayer coupling parameters are then the only parameters left to ﬁt the measured data obtained for a pair of coupled thin ﬁlms.

4.4 Theory 4.4.1 Interlayer Exchange Coupling The ﬁrst successful model of interlayer exchange coupling was introduced by Mathon, Villeret and Edwards [41]. They correctly pointed out that exchange coupling is primarily a property of the normal metal (NM) spacer and is

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199

Fig. 4.7. BLS spectra for an FM1/NM/FM2 structure. The in-plane magnetic anisotropies are assumed to be zero. The upper curves correspond to acoustic peaks, and the lower curves correspond to optical peaks. The calculations were carried out for J1 =–0.2 ergs/cm2 (solid line), and J1 =–0.1 ergs/cm2 and J2 =0.05 ergs/cm2 (dashed line) providing an identical coupling in the saturated state to J1 =–0.2 ergs/cm2 , see (4.11). Note that the resonant frequencies are indeed identical for ﬁelds greater than that required to align the magnetizations in the two ﬁlms (H=1.38 kOe). However signiﬁcant diﬀerences in resonant frequencies occur in the non-collinear state allowing one to separate the contributions from the bilinear and biquadratic exchange couplings. Similar behavior would be obtained for FMR measurements carried out as a function of microwave frequency at ﬁxed magnetic ﬁeld

related to the conﬁnement of Fermi surface electrons in the NM. This model was quickly extended to include the spin dependent electron reﬂectivity at the FM/NM interfaces [42, 43, 44]. One has to include the itinerant nature of the 3d, 4sp electrons in the FM layers. The interlayer bilinear coupling, J, is given by the diﬀerence in energy between the antiparallel and parallel alignment of the magnetic moments in FM/NM/FM structure, J=

1 (E↑↓ − E↑↑ ) , 2A

(4.12)

where A is the area of the ﬁlm. Calculations of energy diﬀerences are simpliﬁed by using the force theorem. The main problem is how to treat electron correlations self consistently. The force theorem says that the energy diﬀerence between the two conﬁgurations is well accounted for by taking the difference in single particle energies. It is adequate to take an approximate spin dependent potential and to calculate the single particle energies in the parallel

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and antiparallel conﬁgurations. This diﬀerence in energy is very close to that obtained from self-consistent calculations, see the further discussion in [3]. In fact this section follows closely Stiles’s Sect. 4.4 in [3]. This procedure signiﬁcantly simpliﬁes the calculation of exchange coupling and interface magnetic anisotropies. In calculations of the interlayer exchange coupling one does not create a big error by neglecting spin orbit interactions, while in calculations of the interface anisotropies spin orbit coupling is the crucial ingredient. Single particle energy calculations require one to evaluate the electron energy in four quantum well states (QWS), see Fig. 4.8. For thick FM layers one ﬁnds large energy contributions. However these large contributions cancel out in the diﬀerence (4.12). In order to avoid mistakes in this procedure it is better to calculate the cohesive energy of the QWS by subtracting the bulk contributions, ΔEQW S = Etot − EF M VF M − EN M VN M ,

(4.13)

where VF M,N M and EF M,N M are the total volumes and bulk energies for FM and NM layers, respectively. Quantum interference Let us consider a simple one dimensional model in which an electron with a wave vector k⊥ travels inside the NM spacer and is partially reﬂected at the FM/NM (interface A) and NM/FM (interface B) interfaces with reﬂection coeﬃcients RA,B = rA,B exp(iφA,B ). After multiple interference the electron density of states (EDS) changes. The phase of the wavefunction after a round trip changes by Δφ = 2k⊥ d + φA + φB . (4.14)

Antiparallel Alignment

Parallel Alignment EF

Spin ↑ M↑

M↓

E ↓

M↑

M↑

Unoccupied states EF

↑ k k

EF Spin ↓

M↑

M↓

M↑

M↑

Ferromagnet M↑

Spacer

Fig. 4.8. Quantum wells employed in the calculation of the interlayer exchange coupling. These spin dependent potentials correspond reasonably well to a Co/Cu/Co(001) system. On the left side the four panels show quantum wells for spin up and spin down electrons for parallel and antiparallel alignment of the magnetic moments. The grey regions show the occupied states

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201

The amplitude after multiple reﬂections is given by a sum of round trips is ∞

[rA rB eiΔφ ]n =

1

rA rB eiΔφ . 1 − rA rB eiΔφ

(4.15)

The denominator becomes small when one obtains a constructive interference Δφ=2nπ. For energies less than the potential barrier at the interface rA =rB =1 and one gets perfect QWS. For energies above the barrier hight the QWS become broader resonances by partly transmitting its amplitude to the surrounding FM layers. By changing the NM spacer thickness these states pass through the Fermi energy, see Fig. 4.9, which leads to an oscillatory behavior of the cohesive energy and consequently to an oscillatory interlayer exchange coupling. The ﬁrst clear experimental observation of QWS was presented by Himpsel and Ortega [45, 46] using photoemission and inverse photoemission using a nonmagnetic layer on top of a magnetic layer. In ﬁrst approximation the change in the density of states due to interference, Δ n(ε), should be proportional to rA rB cos (Δφ) and to the spacer width d and the density of states per unit length, (2/π)(dk⊥ /dε) [44]. Therefore Δ n(ε) per spin can be written as 2d dk⊥ dk⊥ 1 Δn(ε) rA rB cos(Δφ) = Im i2d rA rB eiΔφ . (4.16) π dε π dε For multiple scattering one has to use the expression in (4.15). It is relatively easy to show that 4.16 can be generalized to [47] 1 d Δn(ε) = − Im ln(1 − rA rB eiΔφ ) , π dε note that (4.17) equals to (4.16) for small reﬂection coeﬃcients. The cohesive energy is than given by EF Ecoh = − dε(ε − EF )Δn(ε) .

(4.17)

(4.18)

−∞

EF

D

D + π / 2k F

D + 2π / 2k F

Fig. 4.9. Evolution of quantum well (QW) states as a function of the ﬁlm thickness. The solid lines represent bound states (localized in the QW) and resonance states are shown in “fuzzy ellipses”. EF is the Fermi energy

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Using integration by parts one gets Ecoh =

1 Im π

EF

−∞

dεln(1 − rA rB eiΔφ ) .

(4.19)

For ﬁxed thickness d the integral oscillates rapidly as a function of k⊥ . Only contributions close to the Fermi level will leave non-zero contributions, see Fig. 4.10. It can be shown that in these regions for large d [3] Ecoh =

vF 1 Re (rA rB )n einΔφ(kF ) . 2πd n n

(4.20)

For small reﬂection coeﬃcients Ecoh

vF rA rB cos(2kF d + φA + φB ) . 2πd

(4.21)

The interlayer exchange energy, J, is then given by adding all cohesive energies in Fig. 4.9, assuming the same reﬂection coeﬃcients at the A and B interfaces J

vF vF Re(R↑ R↓ + R↓ R↑ − R↑2 − R↓2 )ei2kF d = − Re(R↑ − R↓ )2 ei2kF d . 4πd 4πd (4.22)

The exchange coupling in this simple one dimensional limit is inversely proportional to the ﬁlm thickness, d, and its oscillatory period is given by the Fermi spanning vector 2kF . In 3D space one has to take into account the k-vectors parallel to the surface. These k-vectors due to the lattice periodicity are conserved in going from FM to NM regions. In this 3D case the one dimensional QWS have additional k-wave-vectors parallel to the interface. The total cohesive energy per unit area involves integration of the QWS over the interface Brillouin zone. In the limit of large d [3] (asymptotic limit) d2 k vF i2kF z (k)d Ecoh Re e R (k)R (k) , (4.23) R L 2πd IBZ (2π)2 where RR (k), RL (k) are the reﬂectivity coeﬃcients at the right and left interfaces, and kF2 kF z is the perpendicular k-vector at the Fermi surface. The integrand in (4.23) oscillates rapidly with the argument 2kFz (k)d except on areas of the Fermi surface where opposite sheets of the Fermi surface are nearly parallel, see Fig. 4.10. The vector connecting these parts of the Fermi surface are called critical spanning vectors. The spanning k-vectors for (001) interfaces for simple metals such as Cu and complex spin density Cr are shown in Figs. 4.14 and 4.21. The exchange coupling involves the diﬀerence in cohesive energies for parallel and antiparallel conﬁguration of magnetic moments. In its asymptotic form this coupling can be written as

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203

2k F Integrand

2π/2k F

Thickness Fig. 4.10. The right hand side shows a slice through a spherical Fermi surface. The solid double headed arrows represent spanning k vectors. The left hand side shows their oscillatory contributions to the cohesive energy. The sum of these contributions is shown by a heavy line. The constructive interference (heavy line) comes mostly from the belly area of the Fermi surface. Note that this constructive interference decreases with increasing ﬁlm thickness

J

v i κi i i i 2 iq⊥ d iχi ⊥ , Re (R − R ) e e ↑ ↓ 16π2 d2 i

(4.24)

i i are Fermi velocities at the spanning vectors, q⊥ is the length where the v⊥ i of a critical spanning wave-vector, κ is the phase associated with the type of the critical point, and R↑i , R↓i are corresponding reﬂectivities. The periods of the observed exchange coupling oscillations as the ﬁlm thickness is varied are in good agreement with those obtained in de Haas-van-Alphen measurements, see Table 4.1 in [3]. A detailed discussion of calculations of exchange coupling for Co/Cu/Co(001), Fe/Au/Fe(001) and Fe/Ag/Fe(001) systems can be found in [3]. The quantitative agreement for the exchange coupling between theory and experiment is far from being good. The main reason is that the interfaces in real samples are far from being ideal and measurements are often not carried out in the asymptotic limit.

4.4.2 Dipolar Coupling In measurements involving an inhomogeneous distribution of magnetization one has to include dipolar coupling. BLS measurements in the backscattered conﬁguration [24, 37, 48] represents perhaps one of the best deﬁned cases of dipolar couplings. In this case the in-plane k-vector of the rf magnetization is given by k = 2qcos(θ), where q is is the length of the laser wave-vector, and θ is the angle of the incoming laser beam with respect to the ﬁlm surface. For a ﬁlm with its normal oriented along the z-axis, the in-plane dc magnetic moment oriented along the x-axis, and the rf magnetization components in form of my exp(i(k x+k⊥ y)), mz exp(i(k x+k⊥ y)), the spatially averaged dipolar ﬁeld components inside the ﬁlm in the limit as kd→ 0 are given by

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Table 4.1. Comparison of oscillation periods measured in magnetic multilayers with those expected from the critical spanning extracted from Fermi surfaces measured in de Haas-van Alphen measurements (dHvA). This Table is a copy of Table 4.1 in reference [3], see further references contained therein interface

Period (ML)

Period (ML)

Technique

Ag/Fe(100)

2.38 2.37±0.07

5.58 5.73±0.05

dHvA SEMPA

Au/Fe(100)

2.51 2.48±0.05

8.6 8.6±0.3

dHvA SEMPA

Cu/Co(100)

2.56 2.6±0.05 2.58 to 2.77

5.88 8.0±0.5 6.0 to 6.17

dHvA MOKE SEMPA

Cr/Fe(100)

11.1 12±1 12.5

dHvA SEMPA MOKE

Cr/Fe(112)

14.4 15.4

dHvA MOKE

k k⊥ kdei(k x+k⊥ y) hx = −2πmy k2 2 k⊥ hy = −2πmy kdei(k x+k⊥ y) k hz = (−4πmz + 2πmz kd) ei(k x+k⊥ y) ,

(4.25)

2 0.5 and k = (k2 + k⊥ )

Outside the ﬁlm for z ≥ d: k k⊥ k kdei(k x+k⊥ y) e−k(z−d) + 2πim hx = − 2πmy z 2 k k 2 k⊥ k⊥ kdei(k x+k⊥ y) e−k(z−d) hy = − 2πmy + 2πimz k k k⊥ − 2πmz kdei(k x+k⊥ y) e−k(z−d) . hz = − 2πimy (4.26) k

Outside the ﬁlm for z ≤ 0.:

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205

k k⊥ k kdei(k x+k⊥ y) ekz − 2πimz hx = − 2πmy k2 k 2 k⊥ k⊥ kdei(k x+k⊥ y) ekz hy = − 2πmy − 2πimz k k k⊥ + 2πmz kdei(k x+k⊥ y) ekz . hz = 2πimy k

(4.27)

k and k⊥ being in-plane wave-vector components parallel and perpendicular to the saturation magnetization, see Fig. 4.11. Notice that the dipolar ﬁeld outside the ﬁlm decays exponentially with the decay length of 1/k. A general treatment of dipolar ﬁeld can be found in [49, 50]. Dipolar ﬁelds play an important role in rf measurements using coplanar and slotted transmission lines. The distribution of k-vectors depends on the geometry of the transmission line. These inhomogeneous dipolar ﬁelds lead to both a shift of the resonant frequency and a broadening of the FMR line [51, 52]. Orange peel coupling Rough interfaces lead to a surface magnetic charge density and consequently to dipolar coupling. This coupling was introduced by Neel [53]. It is often called “orange peel” coupling [54, 55]. It leads to an additional dipolar magnetic coupling. Figure 4.12 indicates that the interface roughness creates a lower energy for the parallel orientation of the ﬁlm magnetic moments than that for the antiparallel conﬁguration. Usually the surface roughness is slowly varying and its amplitude is much smaller than the ﬁlm thickness. Calculations then become simple. The surface charge can be distributed over a ﬂat surface [3]. Assuming that the surfaces vary along the x-direction as zs = Δzcos(2πx/L)

Fig. 4.11. The coordinate system and the ﬁlm geometry corresponding to dipolar ﬁelds generated by a spinwave with the wave vector k. The magnetic layer has its normal in the z-direction. d is the layer thickness. The saturation magnetization and spin wave vector k are oriented in the ﬁlm surface. The k−vector propagates with the angle ϕ with respect to the saturation magnetization Ms

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-++ ++ ++

-- ++ -++ -+

-- ++ -++ -+

--

++ ++ ++

++ ++ - ++

--

--

-- ++ -++ -+

+ ++ - ++

--

--

-- ++ -++ -+

++ -++ ++ - - ++ ++ ++

++ -++ -+ -

-- ++ -++ -+

++ ++ +

-- ++ -++ -+

(b) Fig. 4.12. A schematic picture demonstrating the presence of interface eﬀective magnetic charges for an in-phase corrugated interface roughness. The solid short arrows represent the local induced magnetic dipoles. For the parallel orientation of the ﬁlm magnetic moments the magnetic dipoles form a closed magnetic pattern. In the antiparallel conﬁguration this pattern changes to a head to head and tail to tail conﬁguration

and zs = d+Δzcos(2πx/L), see Fig. 4.12. For the magnetization perpendicular to corrugation the ferromagnetic coupling strength is given by [3] J1,dip ∼ 4πMs2

Δz 2 −2πd/L e . L

(4.28)

When the interface roughness is completely uncorrelated the bilinear dipolar exchange coupling goes to zero. Pinhole coupling Magnetic coupling can arise from pinholes. Basically parts of the FM ﬁlms are in a direct contact that results in an overall ferromagnetic coupling [56]. However this coupling is not homogeneously distributed over the surface. Fluctuations of pinhole coupling over the ﬁlm surface can result in a contribution to biquadratic exchange coupling. 4.4.3 Biquadratic Exchange Coupling The presence of biquadratic exchange coupling was observed at the same time by Heinrich et al. [57] on Co/Cu/Co(001) and by Ruehrig et al. [58] on Fe/Cr/Fe(001). The evidence for biquadratic exchange coupling in [57] was found in the magnetization loops. In order to properly explain the observed critical ﬁelds one needed to use an angular dependent bilinear exchange coupling in the form of J(θ) = J1 − J2 cos(θ) , (4.29) where θ is the angle between the magnetic moments. Consequently the corresponding exchange energy was given by E(θ) = −J(θ)cos(θ) = −J1 cos(θ) + J2 cos2 (θ) .

(4.30)

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Ruehrig et al. observed a perpendicular orientation of the magnetic moments in an Fe/wedge Cr/Fe(001) sample in which the Cr interlayer was grown with a linearly variable thickness. They explained the observed perpendicular conﬁguration by using E(θ) = −J(θ)cos(θ) = −J1 cos(θ) − J2 sin2 (θ) .

(4.31)

Clearly these two concepts are identical. Slonczewski soon after that proposed a theoretical interpretation [59]. He realized that ﬂuctuations in the interlayer thickness could result in an additional coupling term. The ferromagnetic layers at diﬀerent parts of the sample have diﬀerent thicknesses and consequently diﬀerent strengths of coupling, see Fig. 4.13. Short-wavelength oscillations can even result in changing the coupling from ferromagnetic to antiferromagnetic. His model is applicable when lateral variations in the bilinear coupling strength are on a shorter length scale than the lateral exchange length. This means that local angular variations from the average direction of the magnetic moments are small so that in this case the problem can be treated by perturbation theory. The magnetic moments are frustrated across the ﬁlm surface by a variable interlayer coupling. Consequently there is an additional energy term which prefers to orient the magnetic moments in the FM layers perpendicularly to each other. This additional coupling has then an angular dependence given by cos2 (θ), for which the

Fig. 4.13. A schematic picture demonstrating variations of the local magnetic moment across a ﬁlm surface. The local magnetic moments (solid black arrows) are partly rotated away from the average direction of the magnetic moment (light grey arrows) in an attemp to decrease the overall interlayer exchange coupling energy. As a result, moments in FM coupled regions rotate a little towards each other whereas in AFM coupled regions the magnetizations rotate away from each other

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name “biquadratic exchange coupling” was coined. Its strength is given by the competition between variations in the interlayer exchange coupling ﬁeld, ΔJ1 /Ms d , and the in-plane intralayer exchange ﬁeld, 2Ak 2 /Ms . The length of the k-vector, k, is given by the average lateral variations of the interlayer exchange coupling, and ΔJ1 is the average variation of the interlayer exchange coupling. Slonczewski has shown that the exchange coupling ﬂuctuations are decreased by a factor due to exchange averaging. In the simplest form the strength of the biquadratic exchange coupling can be expressed by ΔJ

J2 =

1 4 Ms d ΔJ1 2Ak 2 . π M

(4.32)

s

Notice that the large fraction describes the exchange averaging eﬀect. A more general description can be found in [59]. The above expression shows that biquadratic coupling has only a positive sign that encourages a perpendicular orientation of the magnetic moments. The angle between the magnetic moments is given by a competition between the bilinear, biquadratic magnetic couplings, and the magnetic anisotropies, see Sect. 4.3.1 In zero ﬁeld this angle can range continuously from 0 to π. It is often believed that biquadratic exchange coupling occurs only from short wavelength exchange coupling oscillations where the exchange coupling changes its sign between two subsequent atomic terraces. This is not correct. Any lateral variations in magnetic coupling strength (including ferromagnetic coupling) will result in biquadratic exchange coupling. Once the magnetic moments are in a non-collinear state the magnetic frustrations due to an inhomogeneous magnetic coupling strength result in biquadratic magnetic coupling. Further details about biquadratic exchange coupling can be found in Demokritov’s review article on “Biquadratic exchange coupling in layered magnetic systems” [60]. The Slonczewski idea of additional energy terms due to imperfect interfaces is more general. It was shown [32] that “in any system that exhibits a lateral inhomogeneity, one can expect additional energy terms. It originates from intrinsic magnetic energy terms that ﬂuctuate in strength across the sample interface. These additional terms have a next higher angular power compared with that of the intrinsic term, and they should have only one sign. The power of a higher order angular term has to satisfy the requirements of sample symmetry including time inversion symmetry. Variations of the interlayer exchange coupling results in a cos2 (θ) angular term; variations in a uniaxial interface anisotropy results in an angular dependent term having the form cos4 (ϑ), where ϑ is the angle between the magnetic moment and the ﬁlm normal.

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4.5 Experimental Studies Early studies The early stages of interlayer coupling are well described in review articles [3, 6]. The ﬁrst measurements of interlayer coupling were carried out by Gruenberg’s group [61]. Using BLS measurements they showed that Cr can couple Fe layers antiferromagnetically. This result was expected considering that Cr contains a spin-density wave having a period of approximately 2 ML. It was not clear that one could expect antiferromagnetic coupling through simple metal spacers such as Cu. The ﬁrst indication that the exchange coupling through Cu could be antiferromagnetic was found by Cellobata et al. [62] in superlattices of fcc [6Co/8Cu](001) and [9Co/5Cu](001) using spin polarized neutron scattering and magnetometric techniques. Soon after that several FMR and BLS experiments established a cross-over from ferromagnetic to antiferromagnetic coupling through bcc Cu(001). The ﬁrst cross-over was observed at 8 ML of Cu and the ﬁrst antiferromagnetic maximum at 11 ML [63, 64]. These measurements were quickly followed by a number of measurements that identiﬁed exchange coupling terms having long range oscillatory periods of 10 ML and 5.5–8 ML for bcc and fcc Cu(001) [38, 65, 66, 67, 68, 69], respectively. Systematic studies of multilayers grown by means of sputtering revealed oscillatory couplings having oscillation periods in the range of 0.9 nm to 1.2 nm for V,Nb,Mo,Rh,Ru,Ta,W,Re and Ir spacer layers [70, 71, 72, 73, 74]. The Co/Ru/Co and Co/Rh/Co systems became very useful in forming antiparallel pinned spin valves that are employed in GMR read out heads [75], and Magnetic Random Access Memories(MRAM) using the spin tunelling eﬀect. In Co/Ru/Co and Co/Rh/Co structures the ﬁrst antiferromagnetic coupling was found at 0.3 and 0.8 nm with the strength of 5 and 1.6 ergs/cm2 for Ru and Rh, respectively [70]. These results were obtained by monitoring the Giant Magneto Resistance (GMR) eﬀect. The resistance of an FM/NM/FM structure increases for the antiparallel orientation of the magnetic moments (antiferromagnetic coupling) due to the GMR eﬀect. By following the maxima of the resistance one can determine the regions of antiferromagnetic coupling as a function of the spacer layer thickness [76]. The multilayer structures for GMR studies are mostly prepared by means of sputtering. In the work carried out by the Strasbourg group [72] crystalline Co/Ru/Co(0001) hcp ﬁlms were prepared using MBE. The interlayer exchange coupling strength was investigated using FMR. The strongest coupling was found to be 6 ergs/cm2 observed at 0.5 nm of Ru at RT. The period of oscillations in the coupling strength was found to be 1.2 nm. Preparation of the ﬁlms using sputtering resulted in a weaker exchange, see above. This indicates that smoother interfaces result in a stronger coupling. Ru is used in antiferromagnetic coupled multilayers which are attractive for use as high density recording media. [Co(0.4)/Pt(0.7)]X−1 multilayer structures, where X represents the number of repetitions, and the numbers are

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in nm, possess a strong perpendicular uniaxial anisotropy that allows the Co magnetic moment to be oriented perpendicular to the ﬁlm surface. In [[Co(0.4)/Pt(0.7)]X /Co(0.4/Ru(0.9)]]N structures one can create vertical and laterally coherent antiferromagnetic ﬁlms by changing X [77]. The strong antiferromagnetic coupling through Ru requires large applied ﬁelds to saturate FM/Ru/FM trilayers. For a Py(5 nm)/Pd(.5 nm)/Py(5 nm) structure the magnetic ﬁeld required to achieve saturation of the magnetic moments is in excess of 10 kOe at RT, [70, 76]. A FM/Ru/FM ﬁlm having zero net magnetic moment can be eﬀectively pinned by an exchange bias ﬁeld from an antiferromagnet (AFM) [75, 76]. Such a hard magnetic layer composed of AFM/Fe/Ru/Fe is extensively used in spin valve structures. The presence of short wavelength oscillations in the exchange coupling were observed for the ﬁrst time using perfect single crystals of Fe whiskers as substrates. Whiskers were prepared by means of chemical vapor deposition using FeCl2 and H2 as a transport gas. Under correct conditions which required a proper temperature and a proper ﬂow of hydrogen gas one could sometimes get single crystals of Fe in the form of whiskers having {001} crystalline facets. Whiskers were usually between several mm to 1–2 cm long and 10–100 μm across. The facets were smooth with atomic terrace sizes in excess of several μm. Fe whiskers proved to be ideal templates for the observation of short wavelength oscillations. Approximately at the same time the NIST group of Unguris et al. [78], and Purcell et al. [79] (Philips group), observed short wavelength oscillations having a period of 2 ML. The exchange coupling through spin-density wave Cr will be highlighted in detail in Sect. 4.5.3. After realizing that short wavelength oscillations can be observed in carefully prepared samples a large number of papers were devoted to Co/Cu/Co ﬁlms oriented along all the main (001), (110) and (111) crystallographic orientations. A detailed account of this work can be found in [6]. In the following Section the emphasis will be put on several prototypes of exchange coupling covering the simple metal spacers Cu, Ag, Au, Cr and Mn. 4.5.1 Simple Interlayers: Cu, Ag and Au The observation of short wavelength oscillations required a very smooth interface. Convincing evidence of short-wavelength oscillations was presented by the Philips group [80]. Fcc Co/Cu/Co(001) grown on Cu(001) bulk substrates and bcc Fe/Cu/Fe(001) grown on Fe whiskers were investigated by means of MOKE. The thickness dependence of the exchange coupling was achieved by using a Cu wedge grown between the ferromagnetic layers. The spacer thickness varied continuously from 4 to 19 atomic layers. In the fcc Co/Cu/Co(001) system the variation of the magnetic coupling with Cu thickness was described by a superposition of two oscillatory terms having periods of 2 and 8 atomic layers. In bcc Fe/Cu/Fe(001) grown on a Ag(001) crystal the observed interlayer coupling oscillated with a period of 2 atomic layers. One does not have to use

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Fe whiskers to be able to observe two monolayer exchange coupling oscillations using bcc Cu(001) spacers. The interface smoothness of Fe/Cu/Fe(001) systems was signiﬁcantly improved by growing an Fe ﬁlm on a Ag(001) single crystal substrate at 415 K [81]. The terrace size was increased from 3 to 15 nm and resulted in the presence of short wavelength oscillations. The FMR and MOKE data were interpreted by the simultaneous use of bilinear and biquadratic exchange coupling terms [6, 81]. The exchange coupling was found to have maxima of ferromagnetic coupling at 9,11 and 13 atomic layers. The maxima for antiferromagnetic coupling occurred for 10 and 12 atomic layers of Cu. No ferromagnetic coupling was observed in the Philips data. The maxima for antiﬀeromgnetic coupling were observed at 12,14 and 16 atomic layers in the Philips work. There was an obvious diﬀerence reported in the phase of the coupling between the Phillips and B. Heinrich et al. (SFU groups). Comprehensive studies of exchange coupling and its relationship to quantum well states, QW, were carried out by the Qiu group at the University of California at Berkeley [82] (see references within) using wedged Cu Co/Cu/ Co(001)structures grown on Cu(001) single crystal substrates. This system was particularly convenient for such studies because Cu has a simple Fermi surface whose sp bands can be easily separated from the other energy bands, see the Fermi surface of Cu in Fig. 4.14. Cu and Co can be grown in the (001) orientation with atomically ﬂat interfaces. Angular resolved Photoemission Spectroscopy (ARPES) of QW states was carried out at the Advanced Light Source (ALS) of the Lawrence Berkeley National Lab: the orientation of the

Fig. 4.14. The (110) cross-section of the fcc Cu Fermi surface. The hexagon of straight lines outlines the ﬁrst Brillouin zone. The solid dots represent reciprocal kvectors. All three important orientations are present. The critical spanning vectors in the extended Brillouin zone are outlined by the solid arrows. Along the [001] direction the two critical spanning vectors are located at the belly and neck of the Fermi surface

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magnetic moments was determined using Magnetic X-ray Linear Diochroism (MXLD) from the Co 3p level, and the the coupling strength was determined by means of the MOKE technique. The density of states (DOS) is signiﬁcantly increased at energies corresponding to the QW states, see also [46]. This allows one to follow the QW states as a function of energy for diﬀerent Cu thicknesses. In Fig. 4.15 ARPES measurements show the formation of QW states corresponding to the belly direction of the fcc Cu Fermi surface, see Fig. 4.14. The study was carried out for 20 ML thick Co grown on a Cu(001) substrate and with a Cu wedge grown on top of the Co layer. The ARPES oscillations have clearly shown the QW states corresponding to the sp electrons in the Cu layer. The periodicity of the oscillations was found to be 5.88 atomic layers at the Fermi level and this is exactly the periodicity of the long period of the interlayer exchange coupling in Co/Cu/Co(001) systems. Photoemission intensity along the belly and the neck directions (with k vectors oriented 30 Degrees with respect to the ﬁlm surface) of the Cu Fermi surface are shown in Fig. 4.16. The belly, 5.88, and neck, 2.67, atomic layer periodicities can be explained by employing the extended Brillouin zone picture, see the arrowed solid lines in Fig. 4.14. In this case one subtracts from the regular spanning vector inside the ﬁrst Brillouin zone a k vector with the atomic layer periodicity (4π/a). The oscillatory period in Cu is given by 2k e dCu − φA − φB = 2πn ,

(4.33)

Fig. 4.15. Photoemission spectra obtained along the surface normal corresponding to the belly direction of the fcc Cu Fermi surface [82]. Oscillations in intensity as a function of the Cu layer thickness and electron energy demonstrate the formation of quantum well states (QW)

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Fig. 4.16. Photoemission intensity along the belly direction (a) and neck direction (b) of the fcc Cu Fermi surface, see Fig. 4.14, as a function of the ﬁlm thickness and electron energy below the Fermi level. Two distinct oscillatory periods are present. The dotted curves are calculated using the phase accumulation method [82]

where k e = kBZ − k, kBZ = 2π/a is a Brillouin-zone vector, n is an integer, and φA,B are the phase shifts of electron wavefunctions upon reﬂection at the two boundaries of the potential well formed by the Cu layer surrounded by Co and vacuum, and a is the lattice spacing of Cu. Equation (4.33) explains the long and short wavelength oscillation periods by the belly and neck spanning k vectors, respectively. It is caused by evaluating the strength of the exchange coupling at the discrete atomic layer separations. This is often called aliasing eﬀect. Simple calculations using an image potential and the work function at the Cu/vacuum interface allow one to determine the phase shift at the Cu/vacuum interface. The phase shift at the Co/Cu interface is determined by the conﬁnement of Cu electrons by the minority spin energy band of Co. The Cu sp conduction band can be approximated by a nearly-free-electron model. The solutions of (4.33) are shown by dotted lines in Fig. 4.16. Clearly this simple model can account well for the QW states in Cu. The QW states are the underlying basis for the presence of the interlayer exchange coupling. To insure the direct comparison of the exchange coupling periodicity with the QW states as a function of the Cu spacer thickness a half of the wedge sample was covered by 3 ML thick Co. MXLD measurements are only surface sensitive [83] and consequently the FM and AFM coupling can be determined by monitoring the MXLD signal coming from the 3 ML thick Co. Images of the DOS (by photo-emission measurements) at the belly and neck of the Fermi surface were obtained by scanning the photon beam across the Cu wedge on the Co/Cu side of the wedge. Figure 4.17c shows the observed MXLD signal with maxima and minima intensities corresponding to AFM and FM couplings respectively. Clearly long and short wavelength oscillations are easily visible.

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Fig. 4.17. (a) QW states at the belly of the Cu Fermi surface. (b) QW states at the neck of the Cu Fermi surface. (c) XMLD from the top 3 atomic layers of Co evaporated over the Cu wedge spacer. See further details in [82]. The dark and light regions correspond to ferromagnetic and antiferromagnetic coupling. (d) Calculated interlayer coupling. Notice remarkable agreement between theoretical predictions and experiment for the sign of the exchange coupling

The coupling between the layers is determined by the energy diﬀerence between the parallel (P) and antiparallel (AP) alignment of the magnetic moments 2J ∼ EP − EAP =

EF

EΔDdE ,

(4.34)

−∞

where ΔD = DP − DAP is the diﬀerence of the DOS between P and AP alignment of the magnetic moments. For the P conﬁguration of the magnetic moments the minority spins are conﬁned and form well deﬁned QW states. At the neck of the Fermi surface the minority spins are completely conﬁned by the spin potential of Co. At the belly of the Fermi surface they are only partially conﬁned. Whenever the energy of a QW state crosses the Fermi level it adds energy to EP making the P conﬁguration of the magnetic moments unfavorable. Fitting of the MXLD data with A1 A2 2π 2π J = − 2 sin (4.35) + Φ1 − 2 sin + Φ2 , d Λ1 d Λ2 resulted in Λ1 =5.88 ML, Λ2 =2.67 ML, A1 /A2 =1.2, Φ1 =–86π, and Φ2 =64π. MXLD is not able to determine the strength of the coupling. The coupling strength was investigated using MOKE [82]. Only saturation ﬁelds were given allowing one to estimate of the strength of the AFM coupling in (4.35). At d=6

4 Exchange Coupling in Magnetic Multilayers

215

ML JAF M 0.1 erg/cm2. Surprisingly this is a weak coupling considering that the QW states were so well deﬁned. In addition the MOKE results mostly have shown oscillations with a periodicity of Λ1 =5.88 ML. The short wavelength oscillations are very weakly present with saturation ﬁelds less than 100 Oe implying that J<0.06 erg/cm2 . Clearly the interface roughness is a big factor in the exchange coupling measurements but is much less pronounced in the QW state studies. The recent studies of interlayer exchange coupling in Cu/Ni/Cu/Ni/Cu(001) and Ni/Cu/Co/Cu(001) structures by FMR were carried out by J. Lindner and K. Baberschke [36] Ag and Au: Ag, Au, and Cu have similar Fermi surfaces. Excellent work using Ag and Au spacers was carried out by the NIST group using Fe whiskers as substrates [84, 85]. They used wedged samples with the Au and Ag spacers ranging in thickness from 0 to 15 nm (equivalent of 75 atomic layers). The NM spacers were covered by 1–2 nm of Fe. Again these structures were oriented along (001) and displayed an excellent crystalline quality and large atomic terraces. The magnetic state of the top Fe ﬁlm was monitored using scanning electron microscopy with polarization analysis (SEMPA). SEMPA is a surface sensitive technique that allows one to measure all three magnetization components [23]. The exchange coupling was found to oscillate between FM and AFM coupling over a range of 50–65 atomic layers. This long range of oscillations allowed one to determine with a high degree of accuracy the short and long wavelength periods. SEMPA measurements revealed the short and long wavelength periodicities 2.38, 5.73 ML and 2.48, 8.6 ML for the Fe/Ag/Fe(001) and Fe/Au/Fe(001) ﬁlms, respectively. See further details in Table 4.1 in Stiles review chapter [3]. These oscillatory periods are in very good agreement with those measured using de Haas-van Alphen (dHvA) and cyclotron resonance measurements of the Fermi surface extremal areas [3, 86]. The strength of the bilinear coupling in the Fe whisker/Au/Fe(001) system as a function of the Au spacer thickness is shown in Fig. 4.18 Theoretical estimates of the asymptotic value of the coupling at 5 ML of Au is about a factor 3 higher than that measured. Stiles discusses this difference in detail in his review article [3]. One should note that the measured exchange coupling is always substantially smaller than theoretical predictions. In my view the interlayer coupling is very sensitive to interface structure and this is the origin of the discrepancy between theory and experiment. Averaging exchange coupling by taking a statistical distribution of terraces only partly explains this diﬀerence. Interface mixing can even aﬀect in a profound way the sign of the coupling, see Sect. 4.5.3. The heights of the atomic steps on Au(001) are poorly matched to the Fe(001)interlayer spacing, thus atomic steps lead to a strong vertical mismatch. The aliasing mechanism is no longer eﬀective and can even wipe out long wavelength oscillations [88]. In Fe/Ag/Fe(001) trilayers

216

B. Heinrich 10.0

best fit measurement

2

- Javg (mJ/m )

1.00

0.10

0.01

0.001

0

5

10 15 20 25 Au spacer thickness (layers)

30

Fig. 4.18. (a) The solid circules represent the measured values of the antiferromagnetic interlayer exchange coupling in Fe whisker/Au/Fe(001)using MOKE and the grey areas show the best ﬁt function over a wide range of coupling strength and spacer layer thickness using short, λ1 =2.48 ML, and long wavelength, λ1 =8.6 ML, oscillatory periods [87]

entirely grown at room temperature (RT) the ferromagnetic exchange coupling decreased rapidly with increasing thickness. It reached zero at 7 ML [38]. Growing the Fe layers at elevated temperatures, Tsub ∼ 180◦, signiﬁcantly decreased the density of atomic steps at the Fe/Ag interface and recovered the oscillatory behavior of the exchange coupling [89]. Good quantitative agreement with theory is very hard to reach. The NIST group results are unique; they were able to establish a well deﬁned oscillatory behavior of the interlayer exchange coupling over a wide range of thicknesses. This allowed them to determine the true exchange coupling periods and showed that well deﬁned QW states exist in simple metals even for large spacer thicknesses. Quantitative agreement with theory proves to be more elusive. The spin-dependent potential in multilayer ﬁlms creates electron conﬁnement and resonant states which are responsible for the oscillatory behavior of the exchange coupling. According to theoretical calculations [42, 90] such states are not restricted to the nonmagnetic spacers, but are also present inside the ferromagnetic layers, and the coupling cannot be entirely described as an interaction that is localized at the interfaces. The energy terms coming from the electron conﬁnement in the ferromagnetic layers due to multiple reﬂections and the interference of such states with the states inside the spacer layer results in a variation of the interlayer exchange coupling with the ferromagnetic layer thickness. Calculations and experimental studies on the Co/Cu/Co(001) system [74] have shown that the exchange coupling contains a component that oscillates as a function of the ferromagnetic layer thickness. However, the oscillatory part is smaller than the total strength of the exchange coupling so that the sign of the coupling is determined by the thickness of the nonferromagnetic spacer layer.

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One can expect that the quantum well states aﬀect also other magnetic properties. Bayreuther et al. [91] using Ni(0.73 nm) covered by a Au overlayer with a variable thickness (from 0.4 to 8 nm), have shown that the Curie point Tc in the Ni layer oscillates with the Au thickness overlayer in agreement with QW states in Au. The observed variations in Tc were as large as ±20 K. Oscillations in Tc of 40 K were also observed by Ney et al. [92] in 2.8Co/2-4Cu/4.8Ni trilayers. The integers represent the number of atomic layers. These results are supported by theoretical calculations by Pajda et al. [93] of the ordering temperature Tc for a single atomic layer of Fe or Co covered by a variable thickness of Cu(001). The Green’s function method using the Random Phase Approximation have shown that the QW states in Cu signiﬁcantly aﬀect Tc . The amplitude of oscillations in Tc could be as large as 30 K which is 10 percents of the average value of Tc . The thickness dependence of Tc is particularly spectacular in fcc Fe(001) grown on fcc Cu(001). In these studies by Vollmer et al. [94] the Fe and Cu layer thicknesses were continuously changed between 0 to 12 and 0 to 9 atomic layers, respectively, using a double wedge Fe/Cu(001) structure. It is interesting to note that the oscillations in Tc were completely suppressed after the lattice transformation at 10 atomic layers from the fcc structure to the stable bcc Fe structure. 4.5.2 Temperature Dependence of the Exchange Coupling The temperature dependence of the interlayer exchange coupling provides a great theoretical challenge. It is always diﬃcult to account correctly for the role of spin ﬂuctuations in any system and nanosystems provide a great palate ranging from 1D to 3D behavior including topological complexities related to the interface roughness. The temperature dependence of the bilinear and biquadratic exchange coupling has usually been investigated for magnetic trilayer FM/NM/FM systems. These studies were usually carried out in systems having a signiﬁcant degree of interface roughness such that the bilinear coupling was substantially smaller than that in samples having atomically ﬂat interfaces, see above. This roughness can be expected to aﬀect the role of spin ﬂuctuations at the FM/NM and NM/FM interfaces. Small values of the interlayer exchange coupling implies a strong cancellation of short and long wavelength oscillatory exchange contributions. It can be expected under these circumstances that the the bilinear exchange coupling would be particularly aﬀected by thermal ﬂuctuations having a wavelength comparable to the average separation between atomic terraces. The temperature dependence of the exchange coupling was measured over the interval from 77 to 400 K by Celinski et al. [95] using the system Ag-substrate/9Fe/10,12bccCu/16Fe(001) and by Lindner and Baberschke [35] over the temperature range from 55 to 350 K using the system fcc Cu-substrate/7Ni/5,9Cu/2Co(001). The integers represent the number of atomic layers. It was found that the strength of the bilinear exchange coupling was increased by a factor of 3 to 4 times to a maximum of 0.18 erg/cm2 in the Fe/Cu/Fe system as the temperature was reduced from

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400 K to 77 K, [95]. In the Ni/Cu/Co system the bilinear exchange coupling strength increased by a factor of 4 to a maximum of 0.06 erg/cm2 as the temperature was reduced from 350 K to 55 K; moreover, this increase was found to scale with an inverse 3/2 power law in temperature [35]. It is interesting to note that in the Ag-substrate/9Fe/10,12bccCu/16Fe(001)samples no evidence of a 3/2 power law was found. In the Cu-substrate/7Ni/5,9Cu/2Co(001) samples the Co layer was only 2 atomic layers thick and therefore a strong dependence on thermal ﬂuctuations would be expected. In fact, in a recent paper by Schwieger et al. [96] it has been shown that an extended Heisenberg model accounts for 75% of the reduction of the bilinear exchange coupling strength from 0 K to RT. However a similar increase in the bilinear exchange coupling strength with decreasing temperature using relatively thick Fe layers [95] suggests that there are some other contributions which are signiﬁcant and not caused by thermal ﬂuctuations. In fact for such thick Fe ﬁlms one can’t expect to have a strong temperature dependence since the saturation magnetization changes by less than 10%. This view is even more supported by measurements by Milton et al. [97] on the Fe-whisker/Cr/Fe(001) samples having the best available interfaces. Here the temperature dependence of the bilinear coupling strength is a linear function of the temperature and it increased by a factor of 2 over the temperature interval from 300 K to 110 K. In the Ag-substrate/9Fe/10,12bccCu/16Fe(001) samples the temperature dependence was measured independently for the bilinear and biquadratic coupling strengths. Surprisingly the biquadratic coupling strength was found to be linearly dependent on the bilinear exchange coupling strength over the above temperature interval. This result does not explicitly follow the prediction by Slonczewski’s formula 4.32. In this formula the biquadratic exchange coupling strength is proportional to the square of the mean deviation of variations of the bilinear coupling strength and therefore one would expect that the biquadratic exchange coupling strength should follow a quadratic power law in the bilinear exchange coupling strength. The biquadratic exchange coupling is deﬁnitely caused by lateral variations in the bilinear coupling strength. In many samples it has been found that the biquadratic exchange coupling strength decreases signiﬁcantly more rapidly with decreasing terrace size than the bilinear coupling strength, see [6, 81], in agreement with formula 4.32. Therefore the observed linear dependence of the biquadratic exchange coupling strength on temperature implies that the exchange averaging factor in (4.32) is nearly temperature independent. Co(3.2 nm)/Ru(0.9 nm)/Co(3.2 nm)(0001) trilayer structures prepared by means of MBE by Zhang et al. [31] have provided good samples for testing the temperature dependence of the exchange strength in samples having a large interlayer exchange coupling strength. In these samples the exchange coupling strength increased from RT to He temperatures by 30%. This temperature dependence was well described by the theoretical predictions of Edwards [98] and Bruno [86]. Agreement with theory indicated that the interlayer exchange

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coupling through Ru was closely related to the spatial conﬁnement of d holes in the Ru layer due to spin splitting of the d band in the Co layers. The magnetic order in a [2Fe/13(VHx )]x200 superlattice was investigated by Leiner et al. [99] as a function of the hydrogen content. It was shown that the interlayer exchange coupling strength could be continuously and reversibly changed via absorption of hydrogen into the nonmagnetic V. The magnetic phase diagram of a [2Fe/13(VHx )]x200 superlattice exhibited a critical concentration xc 0.022 at which the magnetic order changed from antiferromagnetic to ferromagnetic order. In the vicinity of xc 0.022 the transition temperature strongly depended on x and decreased continuously as the concentration approached xc 0.022. The hydrogenation of V was used by Paernaste et al. [100] to investigate the magnetic order in a 3Fe/14.4VHx/3Fe trilayer as a function of the oscillatory interlayer exchange coupling strength through the VHx spacer. A pure V interlayer showed a 2D FM (X-Y) behavior which is consistent with a weak interlayer coupling strength. As a result of the introduction of H into the V the critical point Tc exhibited an oscillatory dependence on an increasing H content. The minima and maxima of Tc were expected to be directly associated with an oscillatory interlayer exchange coupling strength. An FM exchange coupling could be expected to be associated with an increase in Tc , and AF exchange coupling associated with a decrease of Tc . 4.5.3 Spin Density Wave (SDW) in Cr Fe-whisker/Cr/Fe(001) wedge structures, see Fig. 4.19, have played a crucial role in the study of interlayer exchange coupling. Unguris, Celotta, and Pierce studies [78, 101] using scanning electron microscopy with polarization analysis (SEMPA) on Fe-whisker/Cr/Fe(001) samples showed that the exchange coupling oscillates with a short-wavelength period of 2 monolayers ( ML). The SEMPA images revealed in a very explicit way the presence of both short-wavelength and long-wavelength oscillations in the thickness range from 5 to 80 ML, see Fig. 4.20. The period of the short-wavelength oscillations, λ=2.11 ML, was found to be slightly incommensurate with the Cr lattice spacing; the period of the long wavelength oscillations was found to be 12 ML. These two basic periods are expected for the complex Cr Fermi surface, see Fig. 4.21. The incommensurate nature of the short wavelength oscillations in Cr, λ=2.11 ML, results in the phase slips of exchange coupling at 24 and 44 atomic layers of Cr, see Fig. 4.20 and further details in [23]. Heinrich and co-workers (SFU group) carried out quantitative exchange coupling measurements on Fe-whisker/Cr/Fe(001) samples [6, 102, 103]. The objective of the SFU group was to grow samples having the best available interfaces, to measure quantitatively the strength of the exchange coupling, and to compare these coupling strengths with ab initio calculations that explicitly include the presence of spin-density waves in the Cr. The requirement of smooth interfaces limited our study to samples which were grown on

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Fig. 4.19. A schematic view of Fe whisker/Cr/Fe(001) sample containing a wedged Cr spacer employed by the NIST group in SEMPA measurements. The arrows show the directions of the magnetization. The Fe whisker is demagnetized by a single 180 degree domain wall. The direction of the magnetization in the Fe ﬁlm follows the sign of the local interlayer exchange coupling

Fe-whisker templates with the Cr spacers terminated at an integral number of Cr atomic layers. It was found that the strength of the exchange coupling through the Cr(001) spacer even for these smooth Cr layers was extremely sensitive to small variations in the growth conditions. Cr ultrathin ﬁlms do not possess a robust spin-density wave; the spatial variation of the spin moments is extremely sensitive to the interfaces. In our studies we concentrated on samples for which the Cr thickness ranged from 4 to 13 atomic layers where the role of interfaces was most pronounced. The measured exchange coupling was found to be reproducible only in those samples that exhibited true layer by layer growth. The existence of unattenuated RHEED intensity oscillations of the RHEED specular spot during the growth of the Cr spacer did not guarantee reproducible values of the exchange coupling between a thin Fe(001) ﬁlm and the whisker(001) substrate. It was necessary to establish conditions such that new layers were initiated at a repeatable pattern of nucleation sites and subsequent attachment of adatoms to the atomic steps of the newly formed atomic islands. This condition was monitored by examining RHEED intensity oscillation amplitudes together with the width of the RHEED specular spot proﬁle. The best results were obtained when the RHEED intensity oscillations exhibited sharp cusps at the RHEED intensity maxima (ﬁlled atomic layers)and the RHEED spot proﬁle oscillated repeatedly between narrow peaks (ﬁlled atomic layers) and wider, split intensity peaks (half ﬁlled layers), see [103]. These requirements were achieved by maintaining the substrate temperature in a narrow range of temperatures, 280 C
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Fig. 4.20. The direction of the magnetic moment in the Fe ﬁlm grown over a Cr wedge is shown in the second SEMPA image. The top SEMPA image shows the presence of a magnetic moment in the Cr wedge using an Fe whisker/Cr(001) sample. White and black contrast indicate parallel and antiparallel orientations of the magnetic moment with respect to the Fe whisker magnetization. Note in the bottom SEMPA image that at the boundaries between the parallel and antiparallel orientation of the magnetic moments a perpendicular component of the magnetic moment is present. This is caused by magnetic frustrations due to a partial completion of the top Cr/Fe interface which results in a strong biquadratic exchange coupling

parallel with the RHEED streaks for half ﬁlled atomic layers indicated that new atomic layers were formed from nucleation centers which were separated by a well deﬁned mean distance of ∼ 80–90 nm. In most depositions of Fe layers the ﬁrst 5 ML were grown at 100 C. The substrate temperature was then increased to 120–240 C. All samples prior to their removal from the vacuum system were covered by a 20-ML-thick epitaxial Au (001) layer. The growth of Au exhibited well deﬁned RHEED intensity oscillations and the surface was terminated by a 5×1 reconstruction typical for the Au(001) surface. Magnetic studies A great review of magnetic studies on Fe/Cr/Fe systems that includes experiments and underlying theories can be found in the review paper by Fishman [5]. Here I will summarize mostly studies on the best available samples which were prepared on Fe whiskers. Quantitative BLS studies of the exchange coupling in Fe-whisker/Cr/ Fe(001) have been discussed in [6, 103, 104]: the main results are shown in Fig. 4.22. These studies showed that the exchange coupling through Cr(001) contains both oscillatory bilinear J1 and positive biquadratic J2 exchange

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k010 k010

k100

k100

Fig. 4.21. The left panel shows a representation of the Cr Fermi surface in the paramagnetic state. The gray shaded areas are ellipsoids centered at the N points in the bcc reciprocal lattice. The right panel shows a slice through the Fermi surface as indicated in the left panel. The gray shaded arrows are the critical spanning vectors at the N-centered ellipsoids and the white arrows indicate the nested parts of the Fermi surface that give rise to spin density wave antiﬀeromagnetism [3]

Fig. 4.22. The thickness dependence of the bilinear J1 (•) and biquadratic J2 (+) exchange coupling. The biquadratic coupling can be measured only for AFM coupled samples. The values of J2 for FM coupled samples (10 and 12 ML) were assumed to be the same as those for the AFM coupled samples having 9,11, and 13 ML of Cr. Note that the coupling becomes AFM for thicknesses greater than 4 ML and the thickness dependence of J1 has a broad maximum around 7 ML of Cr. This behavior is caused by long wavelength oscillations. Large short wavelength oscillations appear for Cr thicknesses greater than 9 ML

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coupling terms. The exchange coupling ﬁrst becomes antiferromagnetic at 4 ML of Cr. For Cr spacer thicknesses dCr <8 ML the strength of the shortwavelength oscillations is quite weak, ∼ 0.1 ergs/cm2. The exchange coupling in this range is antiferromagnetic. This is due to the presence of an antiferromagnetic (AFM) long-wavelength bias. This AFM bias is peaked in the range between 6 to 7 ML. It is interesting to note that the strength of the long-wavelength AFM bias is very nearly the same as that observed in Fe/Cr/Fe(001) epitaxial multilayers prepared by sputtering where relatively large interface roughness annihilates the presence of the shortwavelength oscillations [105]. Exchange coupling in the sputtered ﬁlms showed long-wavelength oscillations with a rapidly decreasing strength of the coupling for thicknesses greater than 10 ML of Cr. For Cr spacers thicker than 8 ML, dCr >8 ML, the exchange coupling in ﬁlms grown on a whisker substrate is dominated by the short-wavelength oscillations. In this thickness range the samples are antiferromagnetically coupled for an odd number of Cr atomic layers, Jtot = |J1 -2J2 | ∼ 1.5 erg/cm2, and ferromagnetically (FM) coupled for an even number of Cr atomic layers. It is interesting to note that the strength of the exchange coupling, 1 erg/cm2 in our samples, is very close to that measured by the NIST group, see Fig. 4.23, indicating that the Fe-whisker treatment and growth procedure carried out by the SFU and NIST groups were similar. MOKE measurements on Fe/wedgeCr/Fe(001) samples were carried out by the NIST group [106]. They covered a wide range of Cr thicknesses, see Fig. 4.23, showing that antiferromagnetic coupling (AFM) was observed at an odd number of atomic layers of Cr. This changes after the ﬁrst phase slip between 24–25 ML of Cr. Interface atom exchange (interface alloying) The coupling between the Fe and Cr atoms at the Fe/Cr interface is expected to be strongly antiferromagnetic [107] and consequently the spin-density wave in Cr is locked to the orientation of the Fe magnetic moments. Since the period of the short-wavelength oscillations is close to 2 ML one would expect AF coupling for an even number of Cr atomic layers and FM coupling for an odd number of Cr atomic layers. For the period 2.11 ML the ﬁrst phase slip in the shortwavelength coupling is predicted to occur at 20 ML. Surprisingly the BLS and SEMPA measurements showed clearly that the phase of the shortwavelength oscillations is exactly opposite to that expected. It is also important to note that the strength of the exchange coupling was found to be much less than that obtained from the ﬁrst-principles calculations, J1 ∼30 ergs/cm2 [108]. Our studies showed that the strength of the bilinear exchange coupling J1 is very sensitive to the initial growth conditions: a lower initial substrate temperature resulted in a larger exchange coupling strength. The bilinear exchange coupling could be changed by as much as a factor of 5 by varying the substrate temperature during the growth of the ﬁrst Cr atomic layer [109]. This behavior led us to believe that the atomic formation

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Fig. 4.23. The saturation ﬁeld as a function of Cr thickness using an Fe-whisker/Cr wedge/15Fe(001) sample. The integer represents the number of atomic layers. These measurements were carried out using MOKE measurements [106]. These measurements were possible only for AFM coupling. Note that antiferromagnetic coupling corresponds to an odd number of Cr atomic layers in agreement with the SFU measurements. No quantitative interpretation of the bilinear and biquadratic coupling was provided

of the Cr layer was more complex than had been previously acknowledged. Angular resolved Auger spectroscopy (ARAES) [110, 111], STM [112], and proton induced Auger electron spectroscopy (ARAES) [113] have shown that the formation of the Fe/Cr(001) interface is strongly aﬀected by an interface atom exchange mechanism (interface alloying). The above studies revealed that the Cr adatoms undergo interface mixing when the substrate temperature is adjusted for optimum growth. The ARAES data showed that the interface mixing was conﬁned mainly to two Fe atomic layers; see Fig. 4.24, and interface alloying during the growth already starts at low substrate temperatures, Tsub ∼100 C. Using temperatures less than Tsub ∼100 C one was not able to establish a proper layer by layer growth even by subsequently raising the substrate temperature. The interface alloying increases with increasing substrate temperature; see Fig. 4.24. It should be noted that interface alloying due to the atom exchange mechanism is not, in general symmetric; it occurs chieﬂy at one interface [114]. Interface

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Fig. 4.24. The substrate temperature dependence of the fraction of Cr atoms in the ﬁrst layer deposited on an Fe whisker substrate (), the fraction of Cr atoms contained in the whisker surface layer (•), and the fraction of Cr atoms in the ﬁrst whisker subsurface layer ( ). The fractional coverages were obtained from ﬁtting the angular dependence of the Auger Cr (529 eV) peak intensity using angular resolved Auger electron spectroscopy [110]

alloying is driven by the diﬀerence in binding energies between the substrate and adatoms. The binding energies are proportional to the melting points of the solids. Interface alloying has been observed in systems for which the substrates have lower melting points than do the adatom solids [114, 115]. Freyss, Stoeﬄer, and Dreysse [116] investigated the phase of the exchange coupling for intermixed Fe/Cr interfaces. The calculations were carried out using a tight-binding d-band Hamiltonian and a real-space recursive method for two mixed layers: Fe(001)/CrxFe1−x /Cr1−x Fex /Crn , where n represents the number of pure Cr atomic layers. This simulates our experimental studies which were carried out on specimens for which the ﬁrst few atomic Cr layers were grown at lower substrate temperatures where the surface alloying was mainly conﬁned to the two interface atomic layers. The calculations were able to account for two important experimental observations. First, the crossover to antiferromagnetic coupling and onset of short-wavelength oscillations was predicted to occur at 4 to 5 ML of Cr, in good agreement with our observations, see Fig. 4.22, and in agreement with the NIST studies using the SEMPA imaging technique. Second, the phase of the coupling was reversed for a concentration x>0.2. AFM and FM coupling was obtained at an odd and an even number of Cr atomic layers, respectively, in perfect agreement with experiment. We tested the role of interface mixing by terminating the Cr growth with the co-deposition of Cr plus Fe to produce a Cr-Fe alloy. Two alloy concentrations were used: Cr 85%-Fe 15% and Cr 65%-Fe 35%. BLS and MOKE measurements revealed that the sign of the exchange coupling between the iron ﬁlm and the whisker substrate was not changed by this alloyed atomic

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layer. The sign was given by the number of Cr atomic layers deposited without any Fe. The system behaved as if the Cr alloy layer formed part of the iron ﬁlm. This result strongly supports the idea that interface alloying at the Fe-whisker/Cr interface leads to the observed phase reversal of the shortwavelength oscillations in the strength of the exchange coupling relative to a system having perfect interfaces. A large number of measurements showed that a part of J2 was proportional to the measured value of J1 . We found that J2 ∼0.1+0.16|J1 |. This is an interesting result that requires a brief comment. Stoeﬄer and Gautier predicted the presence of a biquadratic exchange coupling term for Fe layers coupled through Cr(001)[108]. This could indicate that the main part of the biquadratic coupling is of an intrinsic origin. However in subsequent calculations by the Strasburg group [117] for a spin system that was completely relaxed the exchange coupling through Cr(001) spacers with ideal interfaces was formally similar to Slonczewski s proximity (torsion) model [118] in which the exchange energy increases quadratically with the angle between the magnetic moments of the Fe layers: E = J(Δϕ − π)2 ,

(4.36)

where in zero ﬁeld Δϕ=0 and 180◦ for the odd and even number of Cr atomic layers, respectively. For AF coupling ϕ varies between 180◦ and 0◦ , consequently the total magnetic moment approaches saturation gradually; there is no torque free solution in high ﬁelds. The BLS and MOKE measurements using Fe-whisker/Cr/Fe(001) samples having a low density of atomic steps exhibited a much weaker coupling than that predicted by theory and at the same time it was possible to fully saturate the magnetic moments in these samples in suﬃciently large external ﬁelds. All our MOKE and BLS measurements carried out on Fe-whisker/Cr/Fe(001) samples that were prepared with the best possible interfaces (low density of atomic steps) are consistent with the use of bilinear and biquadratic exchange coupling terms. In particular, we found that in MOKE and BLS measurements for ﬁelds somewhat above the critical ﬁeld H2 , see Fig. 4.26, the iron ﬁlm and whisker moments were parallel, whereas for ﬁelds slightly less than the critical ﬁeld H1 the thin ﬁlm and whisker moments were antiparallel. The discrepancy between experiment and theory was very likely caused by the presence of interface alloying at the Fe/Cr interface. Interface alloying not only signiﬁcantly decreases the exchange coupling but even changes the angular dependence of the coupling. Neutron-diﬀraction and MOKE studies by the Bochum group [119] using Fe/Cr/Fe(001) samples having a high density of atomic steps, showed that the ground state in Fe/Cr/ Fe(001) multilayers exhibited a noncollinear orientation of the magnetic moments for which the approach to saturation could be described by the Slonczewski magnetic proximity model in which the exchange energy depends quadratically on the angle between the magnetic moments of the ferromagnetic layers.

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However one should realize that dc magnetometry can be aﬀected by sample inhomogeneities. The approach to saturation at the critical ﬁeld H2 as observed using MOKE, and the onset of the antiferromagnetic conﬁguration at the critical ﬁeld H1 , see Fig. 4.25, is more gradual than is expected using the sum of bilinear and biquadratic exchange coupling terms; see Fig. 4.26, curve A. The calculated approach to saturation clearly exhibits a well deﬁned kink at the ﬁeld H2 , while the experimental measurements usually show a concave gradual approach to saturation; see Fig. 4.25. According to the calculations the antiferromagnetic conﬁguration of the Fe magnetic moments is reached via a ﬁrst-order phase jump; the experimental measurements show a more gradual s-shaped change. However these experimental MOKE features can be explained by an inhomogeneous distribution of the exchange coupling strength. A 10% variation in the exchange coupling across the measured area would result in hysteresis loops that are very similar to those observed using MOKE; compare Fig. 4.25 with Fig. 4.26. In fact an inhomogeneous distribution of the exchange coupling also explains the observed diﬀerences between values of the critical ﬁelds H2 that have been obtained using the MOKE and the BLS techniques. The BLS measurements on Fe/Cr/Fe(001) always yield lower values for the critical ﬁeld H2 , with corresponding lower values of the exchange coupling strength, compared with those obtained using MOKE. The BLS thin-ﬁlm resonant modes were also visibly broadened for external ﬁelds greater than the saturation ﬁeld H2 , where the Fe ﬁlm magnetic moment is parallel to the Fe-whisker moment. The Fe ﬁlm resonance mode was observed to be much narrower at a ﬁeld midway between H1 and H2 where the Fe ﬁlm and whisker magnetizations are nearly orthogonal [104]. The diﬀer-

Fig. 4.25. The longitudinal MOKE signal for the sample Fe-whisker/11Cr/ 20Fe(001) as a function of applied ﬁeld.The integers represent the number of atomic layers. The saturation ﬁeld measured using MOKE was 300 Oe higher than that measured using BLS. The diﬀerence between the MOKE and BLS measurements was caused by lateral inhomogeneities in the exchange coupling [103]

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Fig. 4.26. Calculated MOKE signal for a 20 ML Fe(001) ﬁlm exchange coupled to a bulk Fe(001) substrate and assuming an inhomogeneous distribution of the bilinear coupling strength J1 . The biquadratic coupling strength J2 was set to 0.3 ergs/cm2 for all curves. The spread of bilinear coupling was assumed to satisfy Gauss’s distribution with J1 =–1 ergs/cm2 , and the distribution width Δ J. Curve A Δ J=0, B Δ J=0.2 ergs/cm2 , and C Δ J=0.5 ergs/cm2

ence between critical ﬁelds measured using BLS and MOKE, as well as the broadening of the BLS signal for ﬁelds greater than H2 , can be explained by inhomogeneous distributions of J1 and J2 . The BLS technique measures the frequencies of the rf resonance modes. The mode corresponding to the thin Fe ﬁlm covering the Cr spacer exhibits a resonance at a ﬁeld that is the algebraic average of local inhomogeneous resonance ﬁelds, corresponding to the distribution of local exchange coupling strengths. The broadening of this BLS resonance peak is related to the distribution of the local resonance ﬁelds. On the other hand, in the MOKE studies a complete saturation is observed only after the external ﬁeld reaches the value corresponding to the maximum value of the H2 critical ﬁeld distribution. The observed diﬀerences between the MOKE and BLS measurements clearly indicate that the coupling through Cr is not even homogeneous across the small area only a few micrometers in diameter corresponding to the laser spot size in the BLS measurements.

Role of multiple scattering The role of electron interface scattering was tested using heterogeneous Cr spacers. Two specimens were grown with a Cu interface layer between the Cr spacer and the Fe thin ﬁlm: Fe-whisker/ 11Cr/1Cu/20Fe(001)/20Au and Fewhisker/11Cr/2Cu/20Fe(001)/20Au, where the integers represent the number

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of atomic layers. Two specimens were grown with a silver interface layer: Fewhisker/11Cr/1Ag/20Fe(001)/20Au and Fe-whisker/11Cr/2Ag/20Fe(001)/ 20Au. Measurements carried out using BLS and MOKE revealed that the exchange coupling in the Fe-whisker/ 11Cr/1−2Cu/Fe(001) was found to increase twofold compared to that observed in samples having a simple 11 ML Cr spacer layer. No signiﬁcant diﬀerence was found for the samples having a Ag interlayer where the exchange coupling decreased by 20%. Mirbt and Johansson [120] presented calculations that are in accord with these results. Their calculations showed that the enhanced coupling strength in Fe/Cr/Cu/Fe(001) samples is due to a change in the spin dependent reﬂectivity of the Cr spacer electrons at the Cr/Cu/Fe interface. The presence of the Cu atoms changes the spin dependent interface potential due to hybridization of the Cu electron states with the Fe electron states. Since the Fe majority spin band lies closest to the Fermi level the eﬀect of the hybridization will be most pronounced for the majority spin Fe band. The hybridization with Cu results in a downward energy shift that moves the Fe majority spin band below the Fermi level. An energy gap is created at the Cu/Fe interface, and consequently the majority spin electrons in Cr undergo a nearly perfect reﬂection. The states for minority spin electrons are very little aﬀected by the Cu, and therefore their reﬂectivity is left unchanged. It follows that the spin reﬂection asymmetry is increased leading to an increased coupling. The eﬀect of a Ag spacer on the coupling in Fe/Cr/Ag/Fe ﬁlms is less dramatic. Calculations show that the spin asymmetry in reﬂectivity is somewhat decreased leading to an overall decrease in the exchange coupling. The theoretical calculations of Mirbt and Johansson suggested that a proper model for exchange coupling through spin-density waves in Cr has to include two contributions: (a) a spin dependent potential due to the magnetic moments on the antiferromagnetic Cr atoms; (b) a spin dependent potential at the Fe/Cr and Cr/Fe interfaces. The ﬁrst contribution for Cr layers thinner than 24 ML can be described by a Heisenberg-like Hamiltonian [108] with AFM coupling between the Cr magnetic moments on adjacent (001) planes and a strong AFM coupling between the Cr and Fe atomic moments at the interfaces. The experimental result implies that the spin-density and multiple scattering contributions act in phase to increase the total exchange coupling. Dependence of exchange coupling on the thickness of the Fe layer Okuno and Inomata [121, 122] reported a strong oscillatory dependence on iron thickness of the exchange coupling in Fe/Cr/ Fe(001) multilayer specimens. The period of the oscillation was 6 ML. However, the specimens used in their work were multilayers characterized by rough interfaces, and the exchange coupling exhibited no short period ∼2 ML variations with Cr thickness. We have measured the dependence of the exchange coupling strength on iron ﬁlm thickness using an Fe(001) whisker substrate, an 11 ML layer of Cr, and a wedge-shaped Fe ﬁlm deposited on the Cr. This structure,

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Fe-whisker/11Cr/wedge Fe(001), was then capped by 20 ML layers of gold. The Cr was deposited using optimum conditions for a smooth growth. No evidence was found for either a short or a long wavelength oscillatory dependence of the coupling strength on the iron ﬁlm thickness. We concluded that Fe/Cr/Fe/Au(001) samples having a low density of interfacial steps, and that exhibit 2 ML oscillations in the exchange coupling as a function of the Cr layer thickness, displayed no measurable variations of the exchange coupling strength as a function of the Fe ﬁlm thickness. It appears that the exchange coupling between Fe layers separated by a Cr spacer can be ascribed to interactions that are localized to the interfaces. However, our subsequent experiments did indicate that the exchange coupling is sensitive to the Fe ﬁlm thickness when the iron ﬁlm is capped with Cr. For a whisker/11Cr/20Fe/20Au structure J1 =–0.82 erg/cm2 , J2 =0.3 erg/cm2 ; and for a whisker/11Cr/20Fe/11Cr/20Au(001) J1 =–1.6 erg/cm2 , J2 =0.33 erg/cm2. Clearly the insertion of Cr between the Fe and Au ﬁlms caused a substantial increase in the coupling strength. Note that the biquadratic coupling term is essentially the same for both specimens. Since the exchange coupling for Cr/Fe/Cr ﬁlms depends upon the Fe ﬁlm thickness, it follows that the Fe/Cr interfaces support the formation of electron resonance states in the Fe ﬁlms. On the other hand, the Fe/Au interfaces tend to suppress electron resonance states. Neutron studies Extensive work on Cr has been carried out by a large number of groups. I would like particularly to point out the work done using neutron scattering. Neutron studies have been carried out on thick Cr layers ranging from 2.0 nm to 300 nm and over a wide range of temperatures down to cryogenic temperatures where one can study the Cr SDW phase diagram. The conclusion of these studies is that the SDW magnetism in Cr ﬁlms is complex and exquisite. The SDW in Cr ﬁlms is not a rigid property but depends on the ﬁlm thickness, the temperature, and is crucially aﬀected by interface roughness, and interface exchange coupling. Theoretical and experimental investigations have been mainly limited to the (001) orientation. The ﬁrst reliable SDW data in Cr ﬁlms were reported in papers by Schreyer et al. [123, 124] on [Fe/Cr/Fe](001) superlattices prepared using MBE. Soon after neutron studies on superlattices grown in the [001], [011], and [111] crystallographic orientations and prepared by sputtering were performed by Fullerton et al. [125] and Adenwalla et al. [126]. The results on the MBE and sputtered samples agree on a few points. The MBE grown superlattices exhibited both incommensurate collinear (I) and commensurate spiral SDW (C) magnetic states for a Cr spacer thicker than 3.5 nm [127]. The non-collinear commensurate C state results from magnetic frustration at the Fe/Cr interfaces. This mode is equivalent to the H SDW in Fishman’s review article, see Fig. 4.15 in [5]. Schreyer et al. suggested that the exchange interlayer coupling energy

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in C follows the proximity (torque) model suggested by Slonczewski [118], see (4.36) and (4.37). This torque model resulted in a spiral conﬁguration of the Fe magnetic moments. For dCr <4.5 nm only the C state exists. The transition temperature from I to C starts from 0 K at 4.5 nm and then gradually increases to 300 K with increasing dCr . The Neel ordering temperature TN 500 K was found to be independent of the Cr thickness, dCr . See the full phase diagram in [127]. The studies by Fullerton et al. on sputtered specimens [125] yielded diﬀerent results. They found a collinear I SDW state with the SDW nodes near the Fe/Cr and Cr/Fe interfaces. No C and H SDW states were observed. The presence of SDW nodes at the interfaces can result in the loss of magnetic contact with the neighboring Fe layers. The Cr layer is either in an incommensurate state for thick enough layers or becomes paramagnetic with decreasing ﬁlm thickness or with increasing temperature. One needs at least one SDW period to create a collinear I SDW. The SDW period was found to be nearly the same as the bulk value. The Neel ordering temperature TN vanished for ﬁlms having a thickness less than a critical thickness of about 30 ML [128]. Clearly the diﬀerent results obtained using MBE and sputtered samples is due to the diﬀerent interface roughness. The sputtered samples have most likely a greater interface roughness. The SDW behavior in Cr is very adaptable and easily aﬀected by the interfaces. The spin ﬂip transition from a transverse to a longitudinal SDW observed in bulk Cr at T=123 K is not observed in superlattices in the thickness range so far studied. Although in thick Cr layers (above 30 nm) covered by an Fe(001) layer the transverse SDW with the q-vector parallel to the ﬁlm surface and with the magnetic moment of Cr perpendicular the ﬁlm surface was observed [129, 130]. In this case the interface frustrations are minimized by ﬂipping the magnetic moment of Cr perpendicular to the Fe magnetic moment. This is analogous to the biquadratic exchange coupling mechanism in interlayer exchange coupling. Layer speciﬁc measurements of the magnetization reversal in AFM coupled Fe/Cr/7Fe/Cr/Fe(001) layers were carried out by L’abbe et al. [131] using the isotope (7 Fe) and nuclear resonant scattering of circularly polarized synchrotron radiation. This work provided interesting results. The total magnetic hysteresis loop with the ﬁeld applied along the magnetic easy axes indicated a typical antiferromagnetic coupling in the presence of 4-fold in-plane magnetic anisotropy, compare Fig. 4.14b in [131] with line 3 in Fig. 4.2. The absence of a jump in order to reach full saturation indicated that biquadratic exchange coupling was present, but the existence of a well deﬁned jump at low ﬁelds indicated that the bilinear coupling was strong enough to orient the magnetic moments antiparallel to each other and parallel to the magnetic easy axes. However the resonant scattering results in [131] suggested that the magnetic moment in the central Fe ﬁlm was oriented away from the magnetic easy axis and this observation indicated a non-trivial conﬁguration of magnetic moments.

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Magnetic domains in the remanent state in [Fe/Cr] superlattices were investigated by means of Polarized Neutron Reﬂectivity (PNR) [132] and Synchrotron Moessbauer spectroscopy techniques [133]. It is not possible to discuss these studies in greater detail within the space available for this chapter. There are excellent review articles by Zabel [134], Fullerton et al. [135], Boedeker et al. [129], and Fishman [5] that cover both theory and experiment including SWD phase diagrams in Cr ﬁlms; see also the references contained therein. 4.5.4 Antiferromagnetic Mn The growth of Mn is challenging by virtue of its ability to adopt various forms of crystalline structures ranging from the bulk αMn (with many atoms per unit cell) to bcc and fcc structures stabilized by alloying or by epitaxial growth on suitable substrates [136]. Kim et al. [137] found that Mn grows on Fe(001) in a strained body-centered tetragonal (bct) structure with lattice spacing parameters a=0.2866 nm and c=0.3228 nm for thicknesses greater than 4 atomic layers of Mn. The measured RT values of the Mn magnetic moment for a thin ﬁlm on Fe range from 1.7 to 4.5 μB [138, 139]. The magnetic state of Mn ﬁlms thinner than 5 ML is rather complicated. Magnetic Circular X-ray Dichroism (MCXD indicates that the fully ﬁlled atomic layer of Mn is magnetically compensated [139, 140]. For submonolayer coverage the Mn moments were oriented antiparallel to the Fe magnetic moments [140]. This observation is in agreement with calculations by Wu and Freeman [141]. The ground state of a Mn atomic layer is a c(2 × 2) magnetically compensated state. The situation is simpler for Mn ﬁlms thicker than 4 atomic layers. SEMPA measurements by the NIST group showed that Mn(001) planes are ferromagnetically ordered with the adjacent atomic layers oriented antiferromagneticaly, thus the magnetization exhibits 2 ML oscillations [142]. This conclusion is also supported by spin polarized STM (spSTM) studies [143, 144, 145]. Schlickum et al. [145] found that in regions where Mn overgrows Fe substrate steps a frustration of the antiferromagnetic order results in a 180◦ domain wall in the Mn ﬁlm. The width of the domain wall increases linearly with the Mn layer thickness and reaches 7 nm in a 20 ML thick ﬁlm of Mn. The ability of spSTM to observe the ferromagnetic order in the top Mn layer is not obvious: it is a weak eﬀect (a few %). Calculations indicate that in ballistic transport the spin current asymmetry can be attributed mainly to the symmetry breaking at the surface [146]. The exchange coupling in Fe whisker/Mn/Fe(001) was investigated by the NIST group using SEMPA. The SEMPA images indicated that for the ﬁrst three Mn atomic layers the magnetization of the top Fe layer pointed in the same direction as the Fe whisker magnetic moment, ie. FM coupling. For Mn thicknesses greater than 4 ML the magnetic moment of the Fe ﬁlm was not collinear with the magnetic moment of the Fe whisker. Between 4 to 8 layers of Mn the direction of the Fe ﬁlm magnetic moment lay at an angle of 60–80◦

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relative to the magnetization of the Fe whisker. Beginning at the 9th Mn layer, the direction of the coupling oscillated with a two ML period between 90◦ − φ and 90◦ +φ, where φ was sample dependent. Values of φ were found between 10 to 30◦ . Similar behavior was found by means of MOKE measurements using Fe/Mn/Fe wedged samples deposited on a GaAs/Fe/Ag(001) template [147]. For Mn thicknesses greater than 1.2 nm (7 ML) a non-collinear conﬁguration (φ=0) was observed. Hysteresis loops in these measurements were well described by Slonczewski s proximity magnetism model which accounts for interface roughness in strong antiferromagnets. The total coupling energy E is given by E = C+ θ2 + C− (θ − π)2 , (4.37) where θ is the angle between the magnetic moments of the two Fe layers. The strength of such proximity coupling showed weak oscillations having a 2 ML period. It should be pointed out that Mn can not be grown with interfaces as smooth as Cr, and therefore it is not surprising that Mn spacers provide an excellent system where the consequences of magnetic frustration are fully demonstrated. It is known that even small concentrations of Mn in Cr results in a strong and commensurate antiferromagnetism [148]. It was therefore thought to be of interest to investigate the phase and the strength of the exchange coupling between Fe layers separated by layers of a Cr-Mn alloy. To this end the SFU group attempted to grow Fe/Cr-Mn/Fe structures. Unfortunately we found that the Mn atoms have a very strong tendency to segregate on the surface of Cr during the growth. It was necessary to maintain a substrate temperature greater than 200 C in order to obtain a good layer by layer growth. At this temperature the top surface layer contained a strongly enhanced concentration of Mn (∼50%). In view of this surface segregation, and since the interfaces play a very crucial role in exchange coupling Heinrich et al. [103] decided to grow pure layers of Mn between the Cr and the Fe layers. Eleven and twelve ML of Cr were grown on an Fe(001) whisker using growth conditions optimizing layer-by-layer growth. Mn layers were deposited on the Cr at a substrate temperature of 120◦C. The substrate was allowed to cool to room temperatures and 20 ML of Fe(001) were deposited on the Mn layer, and a protective layer of 20 ML of Au(001) were deposited on the iron. At 120 C the deposition of the ﬁrst two atomic layers of Mn proceeded in a good layer by layer growth with large RHEED intensity oscillations at the second anti-Bragg scattering condition. At substrate temperatures well below 100 C the Mn does not segregate on Fe. In this way one is able to grow well deﬁned Fe/Cr/Mn/Fe(001) structures having smooth and abrupt interfaces. It should be pointed out that the top Cr surface atomic layer is smooth, with large atomic terraces corresponding to those of the Fe whisker, and is unaﬀected by interface alloying during the deposition of the Mn. Therefore variations in the exchange coupling due to the addition of the Mn layers are primarily due to the presence of the Mn atomic layers and their magnetic state. There is no intermediate Cr-Mn mixed region similar to the Cr-Fe mixed

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region that occurs at the Fe-whisker/Cr interface. The following samples were studied: Fe(001)/11,12Cr/ 1,2,3Mn/20Fe(001)/20Au. This selection of samples allowed one to determine the eﬀect of Mn on the exchange coupling. The coupling was found to be AFM for the 11 ML Cr spacer and FM for the 12 ML Cr spacer. Therefore the phase of the coupling was only determined by the thickness of the Cr layer. Assuming that the Mn is an AFM state with an uncompensated magnetic moment in the (001) plane the coupling should have changed its sign as each Mn atomic layer was added. Such a phase reversal was not observed. The exchange coupling oscillated with the same phase and the same periodicity of 2 ML as was observed using pure Cr spacer layers. The assumption that the magnetic state of Mn in Fe/Cr/ Mn/Fe(001) structures can be described by commensurate antiferromagnetism with uncompensated (001) planes is not necessarily correct. In recent calculations by Krueger et al. [149] the magnetic structure of bct Mn in bulk was studied as function of c/a, a measure of the tetragonal distortion. The calculations showed that bct Mn grown on Fe(001) is in a magnetic state that is at the border line between the AFM1 conﬁguration, in which the magnetic moments are parallel in the (001) planes, and the AFM3(110) conﬁguration, having ferromagnetic planes oriented along (110), and fully compensated (001) planes having zero net magnetic moment. The AFM3(110) state would not lead to an alternating sign of exchange coupling with increasing Mn thickness in agreement with our experimental observations. 4.5.5 Loose Spins Fe atoms can be dispersed in NM interlayers [150, 151] and result in additional coupling between two ferromagnetic ﬁlms. Slonczewski developed a model incorporating the exchange interaction between the FM layers and “loose spins” inside the NM spacer [152]. He assumed that the exchange ﬁeld on the “loose spins” is given by H1,2 = H1,2 (z, d − z)m1,2, where m1,2 are unit vectors along the magnetic moments of layers FM1 and FM2 and z, and d − z determine the distance of a loose spin from the interfaces. The strength of the exchange ﬁeld, H1,2 (z) can be estimated from the thickness dependence of the interlayer exchange coupling. The energy levels of a loose spin are given by εm = −U m/S with m=–S, –S+1,...S, where 1/2 U (θ) = U12 + U22 + 2U1 U2 cos(θ) .

(4.38)

U1,2 are energies associated with the exchange ﬁelds H1,2 . U is expected to be weak and the total energy of a loose spin is given by thermal excitations. The total energy is then multiplied by the concentration of loose spins. This results in bilinear and biquadratic contributions. Heinrich et al. [153] and Schaefer et al. [154] investigated the role of loose spins in Fe/Cu/Fe(001) and Fe/Ag/Fe(001) structures, respectively, by inserting less than 1 ML of Fe in well prescribed positions inside the NM spacer. The strengths of U1 and U2

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were estimated from the thickness dependence of the interlayer coupling using pure Cu spacers. We found that the presence of an alloyed Cu-Fe ML inside the Cu spacer signiﬁcantly decreased the strength of the bilinear coupling while leaving the biquadratic coupling almost unchanged. This decrease in the bilinear coupling can then result in the coupled Fe magnetic moments becoming oriented in mutually perpendicular directions. Both groups have shown that the consideration of loose spins to describe the data requires the inclusion of clusters of Fe atoms inside the NM. The application of Slonczewski’s theory in the work by Heinrich et al. [153] required the inclusion of a molecular exchange ﬁeld in the alloyed atomic layer in order to account for the measured temperature dependence of the bilinear exchange coupling. Recently the magnetic state of Cr spacers in [Fe/Cr/Fe](001) superlattices was investigated by means of perturbed angular correlation (PAC) spectrometry [155]. These measurements revealed that above the blocking temperature of Cr the Cr magnetic moment exhibits superparamagnetic spin ﬂuctuations. In this temperature regime the interlayer coupling in these samples is mostly given by the biquadratic contribution. It was suggested [155] that the superparamagnetic clusters acted as “loose spins” and were responsible for the presence of biquadratic exchange coupling. In my view this is a rather strong claim. “Loose spins” as treated using the Slonczewski model create a strong bilinear coupling term that surpasses the strength of the biquadratic contribution. In order to obtain a dominant biquadratic exchange coupling contribution one needs to compensate the bilinear coupling contribution by some other mechanism. The absence of interlayer coupling below the Neel ordering temperature in [Fe/Cr/Fe] superlattices is caused by the nodes in the Cr magnetic moment around the Fe-Cr interfaces, see the above section on Neutron studies. In the Cr paramagnetic regime one can get a direct coupling as is commonly observed in other simple metal spacers. In that case the interface roughness will signiﬁcantly decrease the bilinear exchange coupling strength in paramagnetic Cr and will introduce biquadratic coupling due to interface magnetic frustrations. Most likely the biquadratic coupling strength observed in [Fe/Cr/Fe] superlattices is a combination of all of these contributions. We found a typical loose spin-like behavior in Co/Cu/Co(001)/Fe and Fe/Pd/Fe(001) systems. The sample 4Co/6Cu/4Co(001) grown on Cu(001) was found to be coupled antiferromagnetically. With the addition of a 3 ML ﬁlm of Fe deposited on the top of the second Co layer the top Fe surface developed a strong reconstruction and the exchange coupling became ferromagnetic [38, 57]. Moreover, the temperature dependence of the exchange coupling followed a Curie-Weiss type of dependence, proportional to 1/T . In our view the strong lattice reconstruction resulted in lattice defects and subsequent penetration of Co atoms into the Cu spacer. These Co atoms then acted as “loose spins” inside the Cu spacer. These loose spins were subjected to the exchange ﬁeld of the surrounding FM Co layers and their contribution to the overall exchange coupling scaled with their magnetic moments. The magnetic moments of true loose spins are expected to follow a Curie-Weiss law.

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4.5.6 Lattice Strained Pd in Fe/Pd/Fe(001) Structures In the Fe/Pd(001) system, a lattice strain results from the 4.2% mismatch between the bcc Fe(001) and the fcc Pd(001) surface nets. Ultrathin Pd(001) ﬁlms grown on Fe(001) are expanded laterally to match the Fe mesh. The structure and magnetism of fcc Pd in Fe/Pd/Fe (001) trilayers grown on Ag(001) substrates were studied in [156, 157]. Using RHEED and x-ray diﬀraction, Fullerton et al. [157], have shown that the Pd ultrathin ﬁlms grew on Fe(001) with a 4.2% latterly expanded lattice accompanied by an out-of-plane contraction of 7.2% (c/a=0.89). The ultrathin ﬁlms of Pd grown on Fe(001) had a pronounced face centered tetragonal (fct) structure. Theoretical ab initio calculations of ultrathin ﬁlm Pd layers on an Fe(001) template have shown that the ground state of the lattice strained Pd(001) has a fct structure such that the bulk Pd atomic volume is maintained [157]. This theory is in good agreement with the results of RHEED and x-ray diﬀraction measurements. Polarized neutron reﬂectivity measurements on an Fe(5.6 ML)/Pd(7 ML)/Au(20 ML) sample determined the average moment per Fe atom to be 2.66 μB [157]. Ab initio spin density calculations for the same structure showed that this value is consistent with the observed induced Pd polarization. It is interesting to point out that a 4.2% lattice expansion of bulk metallic fcc Pd would result in long range ferromagnetic order. The main conclusion of the magnetic measurements and calculations was that the Pd was ferromagnetic only for two adjacent Pd atomic layers at the Fe/Pd and Pd/Fe interfaces. By increasing the thickness of the Pd by one additional atomic layer (a total thickness of 5 ML) the long range ferromagnetic order in Pd was lost. Clearly the lattice vertical relaxation has to be taken into account in order to explain the real magnetic properties of strained epitaxial structures. The exchange coupling through Pd exhibited an oscillatory behavior as a function of the Pd thickness [156]. The period of oscillations was 4 ML. This period is close to the 3 ML period corresponding to the large Pd 4d belly Fermi surface sheets [158]. The temperature dependence of the coupling for 5 ML thick Pd showed a perfect Curie Weiss dependence. A strong temperature dependence is also observed for 6 ML Pd. For thicker Pd layers the temperature dependence is weak. This again indicates the presence of ﬂuctuating magnetic moments in the Pd layers in the range of thicknesses from 5 to 6 ML. In my view the above two systems, Co/Cu/Co/Fe(001) and Fe/Pd/Fe(001), represent the best examples of self assembled loose spins.

4.6 Time Dependent Exchange Coupling 4.6.1 Multilayers The interlayer exchange coupling had so far no explicit time dependent behavior. The spin dynamics studies in magnetic FM/NM/FM trilayers revealed that the coupling between the magnetic layers can also have a purely dynamic

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character. The role of interface damping has been investigated in high quality crystalline Au/Fe/Au/Fe(001) structures grown on GaAs(001) substrates, see the details in [30]. The in-plane FMR experiments were carried out using 10, 24, and 36 GHz systems [159]. The in-plane resonance ﬁelds and resonance linewidths were measured as a function of the azimuthal angle ϕ between the external dc magnetic ﬁeld and the Fe in-plane cubic axes. Single Fe ultrathin ﬁlms with thicknesses of 8, 11, 16, 21, and 31 monolayers (ML) were grown directly on GaAs(001). They were covered by a 20 ML thick Au(001) cap layer for protection under ambient conditions. FMR measurements were used to determine the in-plane four-fold and uniaxial mag netic anisotropies, K1,ef f and Ku,ef f , and the eﬀective demagnetizing ﬁeld perpendicular to the ﬁlm surface, 4π Mef f (=4πMs -2K⊥ u,s /Ms ), as a function of the ﬁlm thickness d [30, 160]. The magnetic anisotropies were well described by a linear dependance on 1/d. The constant and linear terms represent the bulk and interface magnetic properties, respectively. The ultrathin Fe ﬁlms grown on GaAs(001) have magnetic properties nearly equal to those of bulk Fe, modiﬁed only by sharply deﬁned interface anisotropies; this indicates that the Fe layers are of a high crystalline quality with well deﬁned interfaces. The lineshapes of the FMR peaks are Lorentzian and the FMR linewidths are small (less than 100 Oe for our microwave frequencies) and only weakly dependent on the ﬁlm thickness. The reproducible magnetic anisotropies and small FMR linewidths provided an excellent opportunity for the investigation of non-local relaxation processes in magnetic multilayer ﬁlms. The ultrathin Fe ﬁlms which were studied in the single layer structures were regrown as a part of magnetic double layer structures. The thin Fe ﬁlm (F1) was separated from the second thicker layer (F2), 40 ML thick, by a 40 ML thick Au spacer. The magnetic double layers were covered by a 20 ML Au(001) layer for protection under ambient conditions. The thickness of the Au spacer layer was much smaller than the electron mean free path in gold (38 nm) [161], and hence ballistic spin transfer between the magnetic layers is allowed. The interface magnetic anisotropies separated the FMR ﬁelds of F1 and F2 by a large margin ( 1 kOe), see [30]. That separation allowed us to carry out FMR measurements on F1 without exciting a large response in F2: the angle of precession in F2 was negligible compared to that in F1. The FMR linewidths in single and double layer structures were only weakly dependent on the azimuthal angle ϕ of the saturation magnetization with respect to the in-plane crystallographic axes. The 16Fe ﬁlm (F1) in the single and double layer structures had the same FMR ﬁeld showing that the interlayer exchange coupling [6] through the 40 ML thick Au spacer was negligible, and the magnetic properties of the Fe ﬁlms grown by MBE on well prepared GaAs(001) substrates were fully reproducible. The FMR linewidth in the thin ﬁlms always increased in the presence of the thick layer F2. The additional FMR linewidth, ΔHadd , followed an inverse

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Fig. 4.27. The FMR linewidth measurements (half width half maximum (HWHM)) in the parallel conﬁguration. (a) The dependence of the additional FMR linewidth ΔHadd = ΔHd − ΔHs on 1/d at f = 36 GHz. ΔHd and ΔHs represent the FMR linewidths for the Fe ﬁlms in the double and single layer magnetic structures, respectively. d is the thickness of the F1 layer in a GaAs/nFe/40 Au/40Fe/20Au(001) structure. The integers in the structure notation represent the number of atomic layers. (b) The frequency dependence of the FMR linewidth in the 16 ML Fe layer: ΔHd (•) in a GaAs/16Fe/40Au/40Fe/20Au(001) structure; ΔHs (◦) in a single magnetic layer GaAs/16 Fe/20Au(001) structure; and the additional FMR linewidth ΔHadd is shown in ( )

dependence on the thin ﬁlm thickness d, see Fig. 4.27a. The non-local damping originates at the ﬁlm interface (F1/Au). The linear dependence of ΔHadd on the microwave frequency for both the parallel and perpendicular conﬁguration with negligible zero-frequency oﬀset, see Fig. 4.27b is equally important. This means that the additional contribution to the FMR linewidth can be described by an interface Gilbert damping. The additional Gilbert damping for the 16Fe ﬁlm was found to be weakly dependent on the crystallographic direction, Gadd = 1.2 × 108 s−1 along a cubic axis. Discussion of the interface torque Tserkovnyak, Brataas and Bauer [162, 163] showed that interface damping can be generated by a spin current ﬂowing from a ferromagnet into adjacent normal metallic layers. The spin current is generated by a precessing magnetic moment in F1. The spin current was calculated using Brouwer’s scattering matrix [164] which evolves under a time dependent parameter (phase angle of precession). Normal metal (NM) layers surrounding a magnetic layer were viewed as reservoirs in common thermal equilibrium as result of contact with an inﬁnite thermal bath. The calculations were carried out assuming that the reservoirs acted as ideal spin sinks. The resulting spin current per unit area is given by ∂n ∂n jspin = ReA↑↓ n × + ImA↑↓ , (4.39) 4π ∂t ∂t

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where n is a unit vector along the magnetic moment M, and A↑↓ is the complex spin-pumping conductance per unit area given by the diﬀerence between the r t reﬂection (g↑↓ ) and transmission (g↑↓ ) mixing conductances per unit area, r t − g↑↓ . A↑↓ = g↑↓

(4.40)

The spin dependent reﬂection and transmission matrices for the ferromagnetic ﬁlm are given by r ↑ ↓∗ g↑↓ δm,n − rm,n = rm,n m,n t = g↑↓

t↑m,n t↓∗ m,n ,

(4.41)

m,n ↑↓ where rm,n are the reﬂection parameters for spin up and spin down electrons in the NM reservoirs, and t↑↓ m,n are the transmission parameters into the reservoirs. The indices m and n in (4.40) label the modes (channels) corresponding to k,⊥ wave-vectors (parallel and perpendicular to the interface) at the Fermi energy. The Gilbert damping is given by the conservation of the total spin momentum per unit area

jspin −

1 ∂Mtot =0, γ ∂t

(4.42)

where Mtot is the total magnetic moment of F1. After simple algebraic steps one obtains an expression for the dimensionless damping parameter G 1 gμB 1 α= = αbulk + Re(A↑↓ ) γ Ms 4πMs d1 1 gμB 1 1 1− = Im(A↑↓ ) , (4.43) γef f γ 4πMs d1 ferromagnetic where d1 is the thickness of the layer F1. The imaginary part ↑ ↓∗ of A↑↓ arises from the m,n rm,n rm,n and m,n t↑m,n t↓∗ m,n sums and is very close to zero due to cancellation of phases. Therefore spin-pumping mostly eﬀects the damping, the renormalization of the gyromagnetic ratio γ due to spin pumping is very small. The inverse dependence of the Gilbert damping on the ﬁlm thickness clearly testiﬁes to its interfacial origin. The layer F1 acts as a spin pump. Now another important point has to be answered: how is the generated spin current dissipated? This answer can be found in the recent article by Stiles and Zangwill in Anatomy of spin transfer torque [165], see Sect. 4.4.1. They showed that the transverse component of the spin current in a normal layer (NM) is entirely absorbed at the NM/FM interface. For small precessional angles the spin current jspin is almost entirely transverse. This means that the N/F2 interface acts as an ideal spin sink, and provides an eﬀective spin brake for F1, see Fig. 4.28(a). For ferromagnetic layers that

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Fig. 4.28. A cartoon representing the dynamic coupling between two magnetic layers which are separated by a non-magnetic spacer. (a) represents two magnetic layers with diﬀerent FMR ﬁelds. F1 is at resonance, and F2 is nearly stationary. A bow like arrow in the normal spacer describes the direction of the spin current. The dashed line represents the instantaneous direction of the spin momentum. F1 acts as a spin pump, F2 acts as a spin sink, and consequently F1 acquires an additional Gilbert damping. (b) represents a situation when F1 and F2 resonate at the same ﬁeld. Both layers act as spin pumps and spin sinks. In this case the net spin momentum transfer across each interface is zero. No additional damping is present as the precession is in phase

˚), and the electron are thicker than the spin lateral coherence length( 2–4 A scattering at the interfaces is partly diﬀuse, the coeﬃcient A↑↓ is nearly equal to the number of transverse channels in the normal metal NM, m,n δm,n , see [166], [167], [168]. In simple metals this sum per unit area is given by A↑↓ =

kF2 = 0.85n2/3 , 4π

(4.44)

where kF is the Fermi wavevector and n is the density of electrons per spin in the normal metal N. The spin current has the form of Gilbert damping. It can be shown that the coeﬃcient A↑↓ is proportional to the interface mixing cond conductance, A↑↓ eh2 g↑↓ , see [166], [167]. The layers F1 and F2 act as mutual spin pumps and spin sinks. The equation of motion for F1 can be written as 1 ∂M 1 G1 ∂M1 = − [M1 × Hef f,1 ] + 2 2 M1 × γ ∂t γ Ms,1 ∂t ∂n1 ∂n2 − , (4.45) + g↑↓,1 n1 × g↑↓,2 n2 × 4πd1 ∂t 4πd1 ∂t

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where g↑↓,1,2 are the real parts of A↑↓,1,2 for the layers F1 and F2. M1 is the magnetization vector of F1, n1,2 are the unit vectors along M1,2 , and d1 ,d2 are the thicknesses of layers F1 and F2. The exchange of spin currents is a symmetric concept and the equation of motion for the layer F2 is obtained by interchanging the indices 1 ⇔ 2. Equation (4.45) can be tested by investigating the FMR linewidth around an accidental crossover of the resonance ﬁelds for F1 and F2 [169]. In this case the resonant ﬁeld of F1 approaches the resonant ﬁeld of F2. When they reach the same resonant ﬁeld the rf magnetization components of F1 and F2 are parallel with each other. Each precessing magnetization creates its own spin current which is pumped across the NM spacer. The electron mean free path in Au thick ﬁlms is 38 nm [161], and consequently the spin transport is purely ballistic even in an 80 ML thick Au spacer. At the same time both interfaces F1/NM and NM/F2 act as spin sinks. It follows that the net ﬂow of the spin current through each interface can be zero, and the contribution of spin pumping to the FMR linewidth can disappear, see Fig. 4.28(b). The bulk Gilbert damping in F1 and F2 were very nearly equal which resulted in the marked disappearance of the additional FMR linewidth at the crossover of the resonance ﬁelds, see Fig. 4.29. The good agreement between theory and experiment clearly shows the validity of the spin pumping theory based on (4.45). Even in the absence of static interlayer exchange coupling the magnetic layers are coupled by dynamic interlayer exchange. The increase in the additional FMR linewidth for 16Fe in Fig. 4.29 which appears for angles near the accidental crossover in the resonance ﬁelds shows that the spin pumping eﬀect can be enhanced when the rf magnetic moments are partly out of phase. This eﬀect is present in ferromagnetic ﬁlms that are exchange coupled. In the exchange coupled case the optical mode exhibits a larger linewidth than that expected from simple spin pumping in which the spin sink has a negligible precessional amplitude, see [170, 171]. The quantitative comparison between experiment and spin pumping theory is very good [170]. First principles electron band calculations [168] resulted in g↑↓ ≈1.1×1015cm−2 for a clean Cu/Co(111) interface. By scaling this value to Au using (4.44) one obtains a value for the Gilbert damping parameter Gsp,cal =1.4×108s−1 for a 16 ML thick Fe ﬁlm; this value is very close to the experimentally observed value Gsp,exp =1.2×108s−1 measured at RT using FMR. This agreement is amazing considering the fact that calculations of the intrinsic damping in bulk metals have been carried out over the last three decades and have not been able to produce a comparable agreement with experiment. The spin pumping can be also found in single Fe ﬁlms surrounded by normal metal layers provided that the spin current diﬀuses away from the FM/NM interface. Interface damping was observed in NM/Py/NM sandwiches by Mizukami et al. [172] where NM=Pt,Pd and Ta non-magnetic layers were surrounding a permalloy (Py) layer.

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Fig. 4.29. The FMR ﬁeld and linewidth at 24 GHz as a function of the angle ϕ of the applied dc ﬁeld. The measurements were done on GaAs/16Fe/40Au/40Fe/20Au(001), where the integers represent the number of atomic layers. The upper Figure shows the FMR ﬁelds, the symbols ( ) correspond to 40Fe and (◦) symbols to 16Fe. Notice that the 16Fe ﬁlm grown on GaAs(001) exhibits a strong in-plane uniaxial anisotropy. The presence of the in-plane uniaxial anisotropy allows one to get an accidental crossover of the resonance ﬁelds at the angle ϕ=112◦ . The lower Figure shows the FMR linewidth. The solid lines were obtained from calculations using (4.45). The symbols (•) correspond to F1 (16Fe) and ( ) correspond to F2(40 ML). Note that the FMR linewidth for the thinner sample, 16Fe, ﬁrst increases before it reaches its minimum value corresponding to its single GaAs/16Fe/20Au(001) layer structure. Note also that the additional line broadening due spin pumping scales inversely with the ﬁlm thickness

Spin pumping allows one to take a new look at the new ﬁeld of spintronics. One can in principle move information by means of a spin current at frequencies in the GHz range via a mechanism that does not directly involve a net transport of electron charge. This potentially represents an approach to electronics that is truly diﬀerent from that employed using semiconductors. The spin pump model is a rather exotic theory for those who are used to magnetic studies. One would expect that there should be a more direct connection with common concepts used to understand the behavior

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of magnetic multilayers. The obvious choice would be a generalization of interlayer exchange coupling. One would expect that a dynamic part of exchange coupling could create magnetic damping. A ferromagnetic sheet surrounded by a normal metal can be investigated using a contact exchange interaction between the ferromagnetic spins and the electrons in the normal metal [173, 174]. A similar model was used by Yafet [175] in order to calculate the static interlayer coupling. One can extend Kubo’s linear response theory [176] in order to treat a slow precessional motion using a linear approximation for the retarded magnetic moment, ∂S(t) S(t − τ ) ∼ , = S(t) − τ ∂t

(4.46)

where S(t) is the spin moment of the magnetic sheet at the instantaneous time t and τ is the time delay of the retarded response. The induced moment in the N metal at the F/N interface results in an eﬀective damping ﬁeld which is proportional to the imaginary part of the rf transverse susceptibility of N and the time derivative of the magnetic moment ∞ dq dM(t) ∂ sd Hdamp ∼ Imχ(q, ω) . (4.47) ∂ω −∞ 2π dt ω→0 This damping term also satisﬁes the phenomenology of Gilbert damping. By using the same interaction potential it was shown [173, 174] that the Gilbert damping in dynamic interlayer exchange coupling, Gs−d , is identical to that calculated using the spin-pumping theory [162] combined with a perfect spin sink. This leads to an important conclusion: The spin pumping theory is directly related to the dynamic response of the interlayer exchange coupling.

Acknowledgements The author would like to thank his colleagues Professor J.F. Cochran, Mr. B. Kardasz, and Mr. O. Mosendz for stimulating discussions and invaluable help in the preparation of this Chapter. I also owe a debt of gratitude to all those with whom I have held extensive discussions and who provided valuable help during the preparation of this manuscript: Dr. M. Stiles, Dr. J. Unguris, Professor Z. Qiu, Dr. E. Fullerton, Professor H. Zabel, and Referees of this Chapter. Without this help and these discussions the Chapter would have been less complete. I would like to thank Professor Dr. M. Stiles (Figs. 4.8, 4.9, 4.10 and 4.21), Dr. J. Unguris (Figs. 4.18, 4.19, 4.20 and 4.23), Professor Z. Qiu (Figs. 4.15, 4.16 and 4.17), Dr. G. Woltersdorf (Fig. 4.1), and Prof. P. Bruno (Fig. 4.14) for allowing me to include their Figures in this chapter. I thank Dr. Stiles for allowing me to include some of his presentation which beautifully demonstrates theory of interlayer exchange coupling (section Quantum interference). The ﬁeld of magnetic coupling is a very diversiﬁed ﬁeld covering

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at least 2–3 decades. It is not possible for a single Chapter to describe all the complexities of this ﬁeld. I apologize to those who contributed to this ﬁeld and whose results are not explicitly included in this chapter. The omission is not intentional but merely reﬂects my inability to include all studies in a Chapter that has to be constrained to a reasonable length. I would like to thank especially the Natural Sciences and Engineering Research Council of Canada (NSERC), and the Canadian Institute for Advanced Research (CIAR) for continued research funding which makes my work possible. I would also like to express my thanks to the Alexander von Humboldt Foundation and Professor J. Kirschner, Max Planck Institute in Halle, for providing me with generous support during my recent summer research semesters in Germany where this manuscript was partly prepared.

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5 Proximity Eﬀects in Ferromagnet/Superconductor Heterostructures Konstantin B. Efetov1,2 , Ilgiz A. Garifullin3 , Anatoly F. Volkov1,4 and Kurt Westerholt5 1

2 3

4

5

Institut f¨ ur Theoretische Physik III, Ruhr-Universit¨ at Bochum D-44780 Bochum, Germany [email protected] L. D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia Zavoisky Physical-Technical Institute RAS, 420029 Kazan, Russia ilgiz [email protected] Institute for Radioengineering and Electronics RAS, 125009 Moscow, Russia [email protected] Institut f¨ ur Experimentalphysik/Festk¨ orperphysik, Ruhr-Universit¨ at Bochum D-44780 Bochum, Germany [email protected]

Abstract. We review the present status of the experimental and theoretical research on the proximity eﬀect in heterostructures composed of superconducting (S) and ferromagnetic (F) thin ﬁlms. First, we discuss traditional eﬀects originating from the oscillatory behavior of the superconducting pair wave function in the F-layer. Then, we concentrate on recent theoretical predictions for S/F layer systems. These are a) generation of odd triplet superconductivity in the F-layer and b) ferromagnetism induced in the S-layer below the superconducting transition temperature Tc (inverse proximity eﬀect). The second part of the review is devoted to discussion of experiments relevant to the theoretical predictions. In particular, we present results of measurements of the critical temperature Tc as a function of the thickness of F-layers and we review experiments indicating the existence of the odd triplet superconductivity, cryptoferromagnetism and inverse proximity eﬀect.

5.1 Introduction If a superconducting layer S is brought into contact with a non superconducting metallic layer N, the superconducting critical temperature Tc of S decreases with increasing the thickness of the N-layer and the superconducting condensate penetrates into the N-layer over a long distance. This phenomenon, the conventional proximity eﬀect, has been studied since the 1960’s (see reviews [1, 2]).

K. B. Efetov et al.: Proximity Eﬀects in Ferromagnet/Superconductor Heterostructures, STMP 227, 251–289 (2007) c Springer-Verlag Berlin Heidelberg 2007 DOI 10.1007/978-3-540-73462-8 5

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Although attractive electron-electron interactions may be absent in the N-layer, the condensate wave-function (or the Cooper pair wave function) f (t − t ) penetrates into N over a distance ξN , much exceeding the interatomic spacing. In a metal with a high impurity concentration (τ T / << 1, where τ is the momentum relaxation time), ξN , the correlation length, is given by ξN = D/2πT , where D = vF l/3 is the diﬀusion coeﬃcient, vF is the Fermi velocity and l = vF τ is the mean free path of the conduction electrons. Magnetic impurities or a magnetic ﬁeld signiﬁcantly reduce the values of ξN . An impressive manifestation of the induced superconductivity in a normal metal is the Josephson eﬀect in S/N/S junctions. If the thickness of the N-layer L is of the order of ξN , the critical current jc decays exponentially with L as jc ∼ exp(−L/ξN ), which means that the characteristic length of the decay is ξN and not interatomic distances. Due to this eﬀect the Josephson critical current can still be observed even if the thickness of the N-layer exceeds 1 μm. Replacing the normal metallic layer N in an S/N proximity eﬀect structure by a metallic ferromagnetic layer F, one basically has the same eﬀect: The pair wave function from S penetrates into F and makes the F-layer superconducting. However, there are important diﬀerences, rendering the S/F proximity an interesting subject on its own. The ﬁrst important diﬀerence is that the penetration depth of the pair wave function into the F-layer is drastically reduced as compared to the Nlayer. As will be explained in the theoretical sections below, in the diﬀusive limit the penetration depth into the ferromagnet is given by the correlation length ξF = DF /2h with the diﬀusion coeﬃcient of the ferromagnet DF and the exchange ﬁeld in the ferromagnet h. For strong ferromagnets like Fe, Co and Ni, the length ξF has a typical value of 0.7 nm, i.e. the superconducting pairing function decreases in F exponentially on a nearly atomic length scale. The basic physical reason for this is that the exchange ﬁeld in the F-layer h tends to align the spins of a Cooper pair and this leads to a strong pair breaking eﬀect. However, a faster decay of the superconducting condensate in the ferromagnet is not the only diﬀerence in comparison with the normal metals. Actually, there are other novel features of the S/F proximity eﬀect that are less obvious and they are the main subject of the present review. Since in the F-layer the spin-up and spin-down bands are split by the exchange ﬁeld h, the electrons of a Cooper pair at the Fermi energy have necessarily diﬀerent k-vectors for the up and down spins and thus the Cooper pairs acquire a ﬁnite momentum Δk. As a consequence, the condensate function in the ferromagnet oscillates in space. As described in the theoretical chapter below, this leads to oscillations of the superconducting transition temperature Tc as a function of the Fe-layer thickness. For the same reason, in Josephson junctions with an S/F/S structure, where the insulating barrier of a conventional tunnel junction is replaced by a ferromagnetic layer, the condensate function may change sign when crossing the F-layer, which leads to π-type coupling of

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the two S-layers. This eﬀect has been predicted long ago theoretically [3] but only recently conﬁrmed experimentally [4, 5]. Following the same line it has been demonstrated that, for a superconducting ﬁlm covered from both sides by ferromagnetic layers (F/S/F- trilayer structure), the critical temperature Tc depends on the mutual orientation of the ferromagnetic layers [6, 7]. This is the superconducting spin valve eﬀect, the latter term originating from the possibility to switch the resistivity between zero and a ﬁnite value by changing the mutual magnetization direction of the two ferromagnetic ﬁlms. First experiments demonstrating this eﬀect will be reviewed in the subsequent experimental section. A fascinating new aspect of S/F structures discovered recently is a possibility of generation of a new unconventional superconducting pairing state. The original Bardeen-Cooper-Schrieﬀer (BCS) theory [8] leads to a conventional s-wave pairing and for several decades this type of superconductivity had been the only one observed experimentally. On the other hand, the superconductivity in high-Tc cuprates discovered later shows a d-wave symmetry or a mixture of s- and d-wave components of the order parameter [9]. Both the s-wave and d-wave types of the symmetries of the order parameter usually imply singlet pairing, which means that the total spin of the Cooper pair is zero. In this case, the order parameter Δαβ has the form Δαβ = Δ (k) · (iσ2 )αβ , where σ2 is the second Pauli matrix in the spin space and Δ (k) is a function of the momentum k. As the spin part is antisymmetric with respect to transposition of the spin indices, the antisymmetricity of the order parameter following from the Pauli principle is fulﬁlled provided the function Δ (k) is even (Δ (k) = Δ (−k)). Another type of pairing, spin-triplet superconductivity, has been discovered in materials with strong electronic correlations, namely, in heavy fermion intermetallic compounds as well as in organic materials (for a review see [10]). Recently, much work has been devoted to studying superconducting properties of Sr2 RuO4 and convincing experimental data in favor of the triplet p-wave superconductivity have been obtained. We refer the reader to the review articles [11] and [12]. In contrast to singlet superconductivity, the spin part of the order parameter for triplet superconductivity is symmetric with respect to exchanging the spin indices. Assuming that the order parameter (or the condensate function) does not depend on frequency (which is a standard assumption) one comes to the conclusion that the order parameter must be antisymmetric with respect to the inversion of the momentum or, equivalently, to the transposition of the coordinates. That is the type of superconductivity which has been observed e.g. in heavy fermions. Still, there is one more, non-trivial possibility for triplet pairing ﬁrst predicted in [42] that may be realized in S/F systems. It turns out that triplet pairing is also possible when the condensate is an even function of momentum and an odd function of the Matsubara frequency. As will be described in the theoretical sections below, a corresponding component of the condensate

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function can be generated at the S/F interface by a spatially inhomogeneous magnetization. First experimental evidence in favor of the existence of such an odd triplet superconductivity has been reported recently [13, 14, 15]. A hallmark of triplet superconductivity in S/F systems is its large penetration depth in the F-layer, which follows from the fact that the exchange ﬁeld in the ferromagnet does not break the triplet pairs. Moreover, in contrast to the “conventional” triplet pairing, the odd triplet superconductivity is not sensitive to non-magnetic disorder and therefore is very robust. Last but not least, in S/F layer systems the conventional proximity effect is not the only interesting phenomenon caused by the mutual inﬂuence of ferromagnetic and superconducting order. As will be explained below, the ferromagnetic state of the F-layer can in turn be strongly modiﬁed by the presence of the superconductor, an eﬀect that is usually referred to as cryptoferromagnetism [16, 17, 18]. Remarkably, not only supercoductivity can penetrate ferromagnets but also the S-layers can become ferromagnetic [19]. For the latter phenomenon the term inverse proximity eﬀect was coined. Thus, one can see that there is not only one proximity eﬀect in S/F structures but there are many of them. This makes these systems extremely interesting for both theorists and experimentalists. In the following sections we analyze the proximity eﬀects in S/F systems from both theoretical and experimental points of view. Although several reviews on the proximity eﬀects have been recently published [20, 21, 22, 23, 24, 25], we emphasize novel developments and open problems. At the same time, we focus mainly on our own recent work.

5.2 Proximity Eﬀect: Theory 5.2.1 S/F Structures: Uniform Magnetization of the Ferromagnet In this section we review the main theoretical results on what happens if one replaces the normal metal N in a N/S proximity structure by a ferromagnetic metal F. The eﬀective ferromagnetic exchange ﬁeld acts on spins of the conˆ ex for duction electrons in the ferromagnet resulting in an additional term H ˆ tot describing the proximity eﬀect: this interaction in the total Hamiltonian H ˆ ˆ ˆ ˆ Hex = − d3 rψα+ (r) h (r) σαβ ψβ (r) dr , (5.1) Htot = H + Hex , where ψ + (ψ) are creation and destruction operators, h is the exchange ﬁeld, ˆ stands σαβ are Pauli matrices, and α, β are spin indices. The operator H for a non-magnetic part of the Hamiltonian. It includes the kinetic energy, impurities, external potentials and is suﬃcient to describe all properties of the system in the absence of the exchange ﬁeld h. The energy of spin-up electrons diﬀers from the energy of spin-down electrons by the Zeeman energy 2h. All functions, including the condensate

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Green’s function f , become matrices in the spin space with non-zero, diagonal and oﬀ-diagonal elements. In this subsection we consider the case of a homogeneous magnetization M . In this situation the matrix fˆ is diagonal and can be represented in the form fˆ = f3 σ ˆ 3 + f0 σ ˆ0 , (5.2) where f3 is the amplitude of the singlet component and f0 is the amplitude of the triplet component with zero projection of the magnetic moment of the Cooper pairs on the z axis (S = 0). Note that for S/N structures the condensate function has a singlet structure only, i.e. it is proportional to σ ˆ3 . The presence of the exchange ﬁeld h leads to the appearance of the triplet term proportional to σ ˆ0 . In the general (non-homogeneous magnetization) case, the ˆ1,2 . matrix fˆ contains not only the matrices σ ˆ0,3 but also the matrices σ The amplitudes of$ the singlet and triplet components are related to the % correlation functions ψα ψβ as follows [26, 27]: f0 (t) ∼ ψ↑ (t)ψ↓ (0) + ψ↓ (t)ψ↑ (0) , f3 (t) ∼ ψ↑ (t)ψ↓ (0) − ψ↓ (t)ψ↑ (0) . (5.3) One can see that a permutation of spins does not change the function f3 (0), whereas such a permutation leads to a change of the sign of f0 (0). This means that the amplitude of the triplet component taken at equal times is zero in agreement with the Pauli exclusion principle. Later we will see that in the case of a non-homogeneous magnetization all triplet components including ψ↑ (t)ψ↑ (0) and ψ↓ (t)ψ↓ (0) diﬀer from zero. Let us begin with a discussion of properties of F/S systems with homogeneous magnetization. The exchange interaction tends to align the spins of the free electrons in one direction, whereas the superconducting correlations result in formation of Cooper pairs consisting of electrons with opposite spins. Therefore, the superconducting transition temperature Tc of an S/F bilayer system should be considerably reduced in S/F structures provided the interface transparency is high. In this case the electron can travel freely from the S to the F side and vice versa. However, it turns out that the dependence of Tc on the exchange ﬁeld h and on the thickness of S- or F-layers is nontrivial: Tc may vary with increasing dF in a non-monotonic way. The critical temperature for S/F bilayer and multilayer structures was calculated in many works [22, 28, 29, 30, 31, 32]. In most theoretical papers it is assumed that the transition to the superconducting state is of a second order, i.e. the order parameter Δ varies continuously from zero to a ﬁnite value with decreasing temperature T . However, generally this is not the case. If the phase transition is of the second order, one can linearize the corresponding equations (the Eilenberger or Usadel equations) for the matrix Green’s function fˆ assuming that T is close to Tc . This is this case that was considered in most papers on this topic. The critical temperature of an S/F structure can be calculated using an equation obtained from the selfconsistency condition. Close to Tc , the self-consistency condition can be linearized in Δ, and in the Matsubara representation it acquires the form (see, e.g. [24] and references therein)

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ln

1 Tc − ifω Δ) , = (πTc∗ ) ( ∗ Tc |ωn | ω

(5.4)

where Tc is the critical temperature in the absence of the proximity eﬀect and Tc∗ is the critical temperature when the proximity eﬀect is taken into account. The condensate function in (5.4) is the (11) element of the matrix Green’s function fˆω . In the diﬀusive case the matrix condensate function obeys the linearized Usadel equation DF ∂ 2 fˆF /∂x2 − 2(|ω|ˆ σ0 + ihω σ ˆ3 )fˆF = 0 ,

(5.5)

in the F-layer and the equation DS ∂ 2 fˆS /∂x2 − 2(|ω|ˆ σ0 fˆS + iˆ σ3 Δ) = 0 ,

(5.6)

in the S-layer; where DF,S is the diﬀusion coeﬃcient in the F- or S-layer, hω = hsgnω, h is the value of the exchange ﬁeld. These equations should be supplemented by boundary conditions that near Tc have the form (see, for example, [24, 25] and references therein) γF,S ∂ fˆF,S /∂x = −(fˆS − fˆF ) ,

(5.7)

where γF,S = 2Rb σF,S , Rb is the S/F interface resistance per unit area, σF,S are the conductivities of the F- and S-ﬁlms in the normal state. The Usadel equation is applicable to systems with a short mean free path l, which, in other words, means that the inverse momentum relaxation time τ −1 should be larger than max{h, 2πT } in the ferromagnet and τ −1 should be larger than Tc in the superconductor. If these conditions are not met, one has to solve the more complicated Eilenberger equation. At the ﬁrst glance, (5.5) and (5.6) look simple and seem to allow a straightforward solution. However this is not so, because the order parameter Δ depends on x : Δ = Δ(x). In order to solve these equations, a single-mode approximation has been introduced in several papers [6, 22, 29, 30]. Using this approximation one comes to an interesting non-monotonic dependence of Tc on dF . A more reﬁned multi-mode method leads to a change of this dependence that can be signiﬁcant for some values of parameters [33]. If the interface transparency is low, (5.5) and (5.6) can be solved. The low transparency limit means that the condition |κh γF | << 1 is fulﬁlled, where κ2h = 2(|ω| + ihω )/DF . In this case the condensate function in S is not aﬀected in the main approximation by the proximity eﬀect and is equal to fˆS = −iˆ σ3 Δ/|ω|, where Δ is approximately constant in space. A solution for (5.5) can be found f± (x) = ±

fS exp(−κ± x) , γF,S κ±

(5.8)

where κ± = 2(|ω| ± ihω )/DF , fS = –iΔ/|ω| and f± (x) are the (11) and (22) elements of the matrix fˆF .

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Usually the exchange energy h is much larger than the temperature T and the expression for κ± shows that the condensate function f± (x) decays in a strong ferromagnet (h >> T ) on a rather short length ξF = DF /2h and experiences oscillations with the same period (to be more precise, the period of oscillations is 2πξF ). This oscillatory behavior of the condensate function f (x) in F leads to a non-monotonic dependence of Tc on dF and to oscillations of the critical current in S/F/S Josephson junctions (see below). We would like to emphasize here an important point. The characteristic length of the oscillations and decay of the condensate function is equal to ξF = DF /2h only in the diﬀusive limit (hτ << 1). In the opposite limit (hτ >> 1) the situation is diﬀerent: the period of the condensate oscillations in the ferromagnet is 2πvF /h, whereas the decay length is of the order of the mean free path l [34, 35]. One can see that the singlet component f3 = (f+ − f− )/2 is an even function of ω and the triplet component f0 = (f+ + f− )/2 with zero total spin (S = 0) is an odd function of ω. However, both these components, singlet and triplet with S = 0, coexists in the ferromagnet over a short length of the order of ξF . In the next Section we will see that in case of a non-monotonic magnetization a triplet component with S = ±1 arises and penetrates the ferromagnet over much larger distance of order ξN . Of course, in the more complicated F/S/F structure the critical temperature is also suppressed. However, this suppression depends also on the mutual orientation of the magnetization in the left and right side ferromagnets. This property has led to the idea to switch the system between the superconducting and normal states by varying the magnetization orientation, [6, 7]. Qualitatively, this eﬀect can be understood as follows. Consider an F/S/F structure with thin F- and S-layers. Assuming that the S/F interfaces are highly transparent, one can average the Usadel equation over the thickness. In this case the condensate function is continuous across the S/F interfaces, and after averaging one can obtain an equation with an eﬀective order parameter Δef f = Δds /d and an eﬀective exchange ﬁeld [36] hef f = (hl + hr )dF /d, where d = 2dF + dS is the total thickness of the trilayer and hl and hr are the exchange ﬁelds from the left and from the right, respectively. Thus, the F/S/F structure is similar to a magnetic superconductor with an eﬀective exchange ﬁeld hef f . It is known that the critical temperature of this superconductor decreases with increasing hef f and may be even a multivalued function of hef f [37]. If the magnetizations of the left and right side ferromagnet are opposite to each other, we obtain hef f = 0 and therefore Tc is larger than in the case of parallel orientations of the magnetization when hef f = 0. In order to ﬁnd Tc in the general case, one has to solve (5.5) and (5.6) with the boundary conditions (5.7). These calculations have been performed in [6, 7]. The oscillations of the condensate function in the ferromagnet, (5.8), lead to interesting peculiarities not only in the dependence Tc (dF ) but also for the Josephson eﬀect in the S/F/S junctions. It turns out that under certain conditions the Josephson critical current Ic changes sign and becomes negative.

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In this case the energy of the Josephson coupling EJ = (Ic /e)[1 − cos ϕ] has a minimum in the ground state when the phase diﬀerence ϕ is equal not to 0, as in conventional Josephson junctions, but to π (the so called π−junction). This eﬀect was predicted for the ﬁrst time by Bulaevskii et al. [3]. The authors considered a Josephson junction consisting of two superconductors separated by a region containing magnetic impurities. Later on, the Josephson current was calculated for a S/F/S junction [38]. Recently, this interesting theoretical prediction has been conﬁrmed experimentally [4, 5, 39, 40]. 5.2.2 Exotic Superconductivity in S/F Structures As mentioned in the Introduction, there is strong experimental evidence for the realization of non-BCS type of superconducting states in several highly correlated electron systems like the high-Tc cuprates, heavy fermions and SrRuthenates. In this section, we will demonstrate that under certain conditions triplet pairing is expected also in S/F systems for an arbitrary superconductor S. In other words, the triplet component can be artiﬁcially generated by the exchange ﬁeld. Before presenting explicit calculations, let us summarize certain general features of superconducting condensates using symmetry arguments. Due to anticommutation of the fermionic creation ψ + and annihilation ψ operators, the condensate function < ψα (r, t)ψβ (r , t ) > must be at equal times, t = t , an odd function with respect to permutations α β, r r . The triplet pairing means that the spins of the Cooper pairs are parallel to each other and the transposition of the spin indices does not change the condensate function. Provided this function remains ﬁnite at t = t it must change the sign under transposition of the coordinates r and r . So, the triplet Cooper pair has to be an odd function of the orbital momentum or, in other words, the orbital angular momentum L is an odd number. The dependence of the condensate function on the direction in the space makes such a superconductivity very sensitive to disorder. The p-wave condensate (as well as d-wave pairing, etc.) is strongly suppressed by non-magnetic impurities. Of course, then order parameter Δαβ = k Δαβ (kF ) ∼ k < ψα (r, t)ψβ (r , t) >k is also suppressed. The s-wave (L = 0) singlet condensate is an exception, because it is a scalar and therefore is not destroyed by non-magnetic impurities (Anderson theorem). At ﬁrst glance, any non-singlet pairing should be suppressed by impurities, which makes an experimental observation very diﬃcult. However, one more non-trivial possibility for triplet pairing exists. The previous conclusion about the antisymmetricity of the orbital part of the condensate function remains ﬁnite at equal times, which excluds functions antisymmetric in t − t . At the same time, nothing forbids the function < ψα (r, t)ψβ (r , t) > to change sign under the transposition t t . In the frequency representation, this property is realized if the correlator < ψα (r, τ )ψβ (r , τ ) >k,ω is

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an odd function of the Matsubara frequency ω. However, if the condensate function is odd in frequency, it may be even in the momentum and we have the triplet pairing again. In this case, the correlation function < ψα (r, τ )ψβ (r , τ ) >k,ω equals zero at coinciding times (the sum over all frequencies is zero) and therefore the Pauli principle for the equal-time correlators is not violated. This type of pairing was suggested by Berezinskii [41] as a possible mechanism of superﬂuidity of 3 He. He assumed that the order parameter Δ(ω) is an odd function of ω : Δ(ω) = −Δ(−ω). However experiments on superﬂuid 3 He have shown that the Berezinskii’s state was not realized in this system. Now it is well known that the condensate in 3 He is antisymmetric in the momentum space and symmetric (triplet) in spin space. Thus, the Berezinskii hypothetical pairing mechanism remained unrealized for a few decades. Recent theoretical studies have shown that a superconducting state similar to the one suggested by Berezinskii might be induced in S/F systems due to the proximity eﬀect [42, 43]. In the next sections we will analyze this new type of superconductivity with triplet pairing that is odd in frequency and, in the diﬀusive limit, even in momentum. This can be s-wave pairing and therefore this type of superconductivity is not sensitive to impurities. We note, however, that there is a qualitative diﬀerence between this new superconducting state in S/F structures and the one proposed by Berezinskii. In S/F structures both singlet and triplet types of the condensate coexist and the order parameter Δ existing only in the S region (we assume that the superconducting coupling in the F region is zero) is determined solely by the singlet part of the condensate. Note that, while theories of unconventional superconductivity often imply strongly correlated systems, the triplet state induced in S/F structures can be derived within the framework of the BCS theory valid in the weak-coupling limit. This fact not only drastically simpliﬁes theoretical considerations but also helps in designing experiments, since well known elemental superconductors prepared under controlled growth procedures may be used in order to detect the triplet superconductivity. To ﬁnish this subsection, let us summarize the properties of this new type of superconductivity that we call odd triplet superconductivity: • It contains a triplet component. In particular the components with projection S = ±1 are insensitive to the presence of an exchange ﬁeld and therefore long-range proximity eﬀects arise in S/F structures. • In the dirty limit it has an s-wave symmetry. The condensate function is even in p and therefore, contrary to unconventional superconductors with triplet pairing, is not destroyed by the presence of non-magnetic impurities. • The triplet condensate function is odd in frequency. Before we turn to a more detailed theoretical analysis of triplet superconductivity, we remark that in the F-regions of the S/F structures no attractive

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electron-electron interaction exists, and therefore Δ = 0 in F. This means that only the superconducting condensate function f in the ferromagnet exists and, as it will become clear later, it arises only due to the proximity eﬀect. 5.2.3 Triplet Odd Superconductivity Induced by an Inhomogeneous Magnetization in S/F Structures As discussed in Sect. 5.2.1, the presence of an exchange ﬁeld results in the formation of the triplet component of the condensate function. In a homogeneous exchange ﬁeld, only the component with the projection S = 0 is induced. Then the natural question arises: Can the other components with S = ±1 also be induced? If they could, this would lead to a long range penetration of the superconducting correlations into the ferromagnet because these components correspond to the correlations of the type < ψ↑ ψ↑ > with parallel spins and they are not as sensitive to the exchange ﬁeld as the other ones. In what follows, we analyze a few examples of S/F structures in which all the projections of the triplet component are induced. The common feature of these structures is that the magnetization should be non-homogeneous. F/S/F Trilayer Structure We start with considering the F/S/F system shown schematically in Fig. 5.1. The structure consists of one S-layer and two F-layers with the magnetization inclined at the angle ±α with respect to the z-axis (in the yz plane). As we have seen in the previous section, each of the layers generates the triplet component with the zero total projection of the spin, S = 0, in the direction of the exchange ﬁeld. If the magnetic moments of the layers are collinear (parallel or antiparallel), the total projection remains zero. However, if the moments of the ferromagnetic layers are not collinear, the superposition of the triplet components coming from the diﬀerent layers should have all the possible projections of the total spin.

−α

α

F

−( dS + dF )

F

S

−dS

dS

dS + dF x

Fig. 5.1. Trilayer geometry. The magnetization of the left (right) side F-layer makes an angle α (−α) with the z-axis

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From this qualitative argument we can really expect the non-trivial eﬀect of the generation of the triplet components with all the projections of the total spin, provided the thickness of the S-layer is not too large. The point is that the triplet component decays in S on a length of the order of the coherence length ξS ≈ DS /πTc , (5.6). We assume that the thickness of the S-layer does not much exceed this length. In order to ﬁnd all types of the condensate (singlet and triplet), one has to solve the linearized Usadel equation in the F-region (we assume a weak proximity eﬀect) [43] for the condensate function fˇ that is a 4 × 4 matrix in the particle-hole and spin spaces (∂ 2 fˇ/∂x2 ) − κ2ω fˇ + iκ2h {τ0ˆ⊗[ˆ σ3 , fˇ]+ cos α ± τ3ˆ⊗[ˆ σ2 , fˇ] sin α} = 0 ,

(5.9)

ˇ+ = σ where [ˆ σ3 , f] ˆ3 ⊗ fˇ + fˇ ⊗ σ ˆ3 . The wave vectors κω and κh entering (5.9) have the form κ2ω = 2|ω|/DF ,

κ2h = 2hsgn(ω)/DF .

(5.10)

The magnetization vector M lies in the (y, z)-plane and has the components: M = M {0, ± sin α, cos α}. The sign “+” (“-”) corresponds to the right (left) side F-ﬁlm. We consider here the simplest case of a highly transparent S/F interface and temperatures close to Tc . In this case the function fˇ, being small, obeys a linear equation in S similar to (5.6). The boundary conditions at the S/F interfaces are obtained by a generalization of (5.7) (see [43]). A solution for (5.9) can be found. The matrix fˇ can be represented as fˇ = iˆ τ2 ⊗ fˆ2 + iˆ τ1 ⊗ fˆ1 ,

(5.11)

σ1 , fˆ2 = b3 (x)ˆ σ3 + b0 (x)ˆ σ0 . where fˆ1 = b1 (x)ˆ For the left side F-layer the functions bk (x) are to be replaced by ¯bk (x). For simplicity we assume that the thickness of the F-ﬁlms dF exceeds ξF (the case of an arbitrary dF was analyzed in [43]). Using the representation, (5.11), we ﬁnd the functions bi (x) and ¯bi (x). They are decaying exponential functions and can be written as bk (x) = bk exp(−κ(x − dS )),

¯bk (x) = ¯bk exp(κ(x + dS )) .

(5.12)

Substituting (5.12) into (5.9), we obtain a set of linear equations for the coefﬁcients bk that should be complemented by expressions for the eigenvalues κ. In the limit of large exchange energy h ({T, Δ} << h, but h << τ −1 ), the eigenvalues κ are equal to κ = κω ,

κ± ≈ (1 ± i)κh .

(5.13)

We see from (5.13) that the solutions κω and κ± are completely diﬀerent. The roots κ± proportional to κh are very large and therefore the corresponding

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solutions bk (x) decay very fast (similar to the singlet component). This is the solution that exists for a homogeneous magnetization (collinear magnetization vectors). In contrast to the roots κ± , the value for κω given by (5.13) does not depend on the exchange energy h and is much smaller. It is this eigenvalue that leads to a slow decay of the superconducting correlations. The solutions corresponding to the root κω describe the long-range penetration of the triplet component into the ferromagnetic region. The function b1 (x) is the amplitude of the triplet component penetrating the F-region over a long distance of the order of κ−1 ω ∼ ξN . Its value, as well as the values of the other functions bk (x), is to be found from the boundary conditions at the S/F interfaces. Matching the solutions in S and F at the S/F interfaces, we obtain the coeﬃcients bk and ¯bk . Note that b3± = ¯b3± and bω = −¯bω . Although the solution can be found for arbitrary parameters entering the equations, we present here the expressions for b3± and bω in some limiting cases only. For example, if the parameter γκh /κS is small, the amplitudes of the long-range triplet component bω and singlet components b3± can be written in a rather simple form 2Δ γκh sin α cos2 α Δ , b3+ ≈ b3− ≈ ( ) , (5.14) Eω κS sinh(2ΘS ) 2iEω √ where Θ = κS dS and Eω = ω 2 + Δ2 , γ = σF /σS and σF (σS ) is the conductivity in the ferromagnet (superconductor). At the S/F interface the amplitude of the triplet component bω is small compared to the magnitude of the singlet one b3+ . However, the triplet component decays over a long distance ξN , while the singlet one vanishes at distances exceeding the short length ξF . The amplitudes bω and b3± become comparable if the parameter γκh /κS is of the order of unity. It follows also from (5.14) that the amplitude of the triplet component bω is zero in the case of collinear vectors of magnetization, what means at α = 0√or α = π/2. It reaches the maximum at the angle αm for which sin αm = 1/ 3. Therefore the maximum angle-dependent factor in (5.14) is sin αm cos2 αm = √ 2/3 3 ≈ 0.385. One can see from (5.14) that bω becomes exponentially small if the thickness dS of the S-ﬁlms signiﬁcantly exceeds the coherence length ξS ≈ DS /πTc . This means that in order to have a considerable penetration of the superconducting condensate into the ferromagnet, one should not make the superconducting layer too thick. On the other hand, if the thickness dS is too small, Tc is suppressed. In order to avoid this suppression, one has to use, for instance, a F/S/F structure with a small thickness of the F-ﬁlms. In Fig. 5.2 we plot the spatial dependence of the singlet and triplet components in F/S/F structure. bω ≈ −

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1

0

-(dS+dF)

-dS

dS+dF

dS

Fig. 5.2. The spatial dependence of Im(b3 (x)) (dashed line) and the long-range part of Re(b(x)) (solid line). We have chosen σF /σS = 0.2, h/Tc = 50, σF Rb /ξF = 0.05, dF Tc /DS = 2, dS Tc /DS = 0.4 and α = π/4. The discontinuity of the triplet component at the S/F interface is because the short-range part is not shown in this ﬁgure. Taken from [43]

Domain Wall at the S/F Interface and Helical Ferromagnets Now we consider another example of an S/F structure in which the long-range triplet component (LRTC) also arises. This structure is shown schematically in Fig. 5.3. It consists of an S/F bilayer with a non-homogeneous magnetization in the F-layer. We assume for simplicity that the magnetization vector M = M (0, sin α, cos α) rotates in the F-ﬁlm starting from the S/F interface (x = 0), and the rotation angle has a simple, piece-wise x-dependence: α(x) = Qx, in the domain wall for 0 < x < w and α(x) = Qw for w < x. This means that the M vector is directed parallel to the z-axis at the S/F interface and rotates by the angle α(w) over the length w (w may be the width of a domain wall). At x > w the orientation of the vector M is ﬁxed. This structure was considered α αw

F S

0

w

L

F

S F

F S

Fig. 5.3. S/F structure with a domain wall in the region 0 < x < w. In this region α = Qx, where Q is the wave vector which describes the spiral structure of the domain wall. For x > w it is assumed that the magnetization is homogeneous, i.e., α = Qw

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ﬁrst in [42] and later in [44]. The Usadel equation for this case has been solved in [42]. The solution is found in the region of the domain wall 0 < x < w and in the region of a constant magnetization: w < x < ∞. For this conﬁguration of the magnetic moment the LRTC arises in the domain wall and spreads into the ferromagnet over a long distance. The characteristic decay length of the LRTC inside the domain wall is ξQ = (Q2 + κ2ω )−1/2 ,

(5.15)

whereas its value equals κ−1 ω outside the domain wall. The singlet component penetrates the ferromagnet over a short length of the order ξF . Although the amplitude of the LRTC at the S/F interface may be comparable with the amplitude of the singlet component, the decay length of the LRTC is much larger (see Fig. 5.4). One more system where the LRTC arises is a helical ferromagnet [45] (see Fig. 5.15). Such a structure is realized, for example, in several heavy rare earth metals. In this ferromagnet the magnetization vector rotates around the z-axis and has a non-zero projection Hz on this axis. It was shown that in this case the LRTC penetrates the ferromagnet over a length of the order of ξQ , (5.15). What is interesting, the monotonic decay of the LRTC in this case occurs only if the the cone angle θ is less than sin−1 (1/3) ≈ 19◦ . At larger θ the decay of the LRTC is accompanied by oscillations. In the quasi-ballistic case (hτ > 1), the characteristic length of the LRTC penetration into the ferromagnet changes. In the case of Neel-type domain walls the LRTC vanishes provided the magnetization vector M rotates continuously [46]. However, in an S/F structure with several Neel domain walls (the vector M rotates only inside the domain walls) the LRTC arises at the domain walls and decays in the domains over a large distance [47].

0.4

α w = π/5

0.3

0

αw = π

x=w

x=L .

Fig. 5.4. Spatial dependence of amplitudes of the singlet (dashed line) and triplet (solid line) components of the condensate function in the F wire for diﬀerent values of αw . Here w = L/5, = ET , and h/ET = 400. ET = DF /L2 is the Thouless energy, = iω is the energy (From [42])

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5.2.4 Other Proximity Eﬀects in S/F Structures Up to now we implicitly assumed that the proximity eﬀect in S/F structures changes the superconducting properties but leaves the magnetization of the F-layer unchanged. However, this is not always true and experiments performed by [48] and [49] indicate that the ferromagnetic magnetization of S/F bilayers may decrease when lowering the temperature below Tc . At that time it was not quite clear what physical mechanism causes this decrease of the magnetization. Here we review two diﬀerent and independent mechanisms that may explain the eﬀect. 5.2.5 Cryproferromagnetism In a classic paper Anderson and Suhl [16] proposed an idea that at some circumstances superconductivity might coexist with a non-homogeneous magnetic ordering. They called this magnetic non-homogeneously ordered state cryptoferromagnetic. The basic reasoning leading to this suggestion was that superconductivity could survive in a ferromagnetic background, if the magnetization direction varied on a scale smaller than the superconducting coherence length. The cryptoferromagnetic state in S/F structures was considered ﬁrst in [17] in the case of a weak ferromagnet. In a more recent theoretical paper on cryptoferromagnetism in S/F bilayers [18] a more realistic case of a strong ferromagnet was considered. It was shown that even if the exchange ﬁeld is large; the cryptoferromagnetic state is still possible provided the ferromagnetic ﬁlm is suﬃciently thin. A phase diagram containing the cryptoferromagnetic state has been drawn depending on the stiﬀness of the ferromagnet J , the thickness of the F-ﬁlm dF and the exchange ﬁeld h of the system. This phase diagram (a, λ) for the S/F system is represented in Fig. 5.5, where a = 2h2 d2F /(DF Tc η 2 ), η = vF /vS , λ = (J dF /NF 2Tc DF3 )(7ς(3)/2π2 ), and NF is the density-of-states (DOS) in the ferromagnet. Estimates of the parameters (J , h and dF ) for the samples used in the experiments [49] in which a reduction of the eﬀective magnetization was observed show that the results of [18] agree with the experimental data. The calculations show that the proximity eﬀect may lead to a magnetic spiral structure in the F-ﬁlm even if the exchange energy h is much larger than the characteristic energy of Tc . This cryptoferromagnetic ordering is related to the existence of low lying states in the ferromagnet. The spiral structure increases the magnetic energy only by a small amount, whereas the energy of interaction between the exchange ﬁeld and the superconductivity can essentially be reduced. At the same time, there exists another mechanism that can reduce the total magnetization in S/F structures and it is also due to the proximity eﬀect. This is the inverse proximity eﬀect describing the situation when the orientation of the magnetization remains unchanged, while its magnitude changes both in the S- and F-layers.

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λ

0.01

|τ|=0.6

|τ|=0.4 0.005

|τ|=0.2

0 0

0.5

1

1.5

2

2.5

3

a Fig. 5.5. Phase diagrams (λ, α) for diﬀerent values of |τ | = (Tc − T )/Tc . The area above (below) the curves corresponds to the F (CF) state

5.2.6 Inverse Proximity Eﬀect The inverse proximity eﬀect is due to a contribution of free electrons both in the ferromagnet (δMF ) and in the superconductor (MS ) to the total magnetization. On one hand, the DOS in the F-ﬁlm is reduced due to the proximity eﬀect, thus decreasing of the magnetization in F by δMF . On the other hand, the Cooper pairs in S are polarized in the direction opposite to MF , giving rise to a magnetization (MS ) with a direction opposite to MF . So, the S-layer becomes ferromagnetic and this is the reason for calling this eﬀect the inverse proximity eﬀect. For a more detailed qualitative explanation of this mechanism we consider the S/F structure with a thin F-layer in Fig. 5.6. We assume that the exchange ﬁeld of F is homogeneous and directed along the z-axis.

S

−dS

F

0

dF

x

Fig. 5.6. S/F structure and schematic representation of the inverse proximity eﬀect. The dashed curves show the local magnetization

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If temperature exceeds Tc , the total magnetization of the system Mtot equals M0 dF , where dF is the thickness of the F-layer. When the temperature is lowered below Tc , the S-layer becomes superconducting and Cooper pairs with a size of the order of ξS appear in the superconductor. Due to the proximity eﬀect the Cooper pairs cross the interface and penetrate into the ferromagnet. In the case of a homogeneous magnetization in F the Cooper pairs are composed of electrons with opposite spins, such that the total magnetic moment of the pair equals zero. The exchange ﬁeld is assumed to be not too strong, otherwise the Cooper pairs would be destroyed. It is clear from this simple picture that pairs located entirely in the superconductor cannot contribute to the magnetic moment of the superconductor. However, some pairs are distributed in space in a more complicated manner: one of the electrons of the Cooper pair stays in the superconductor, whereas the other one enters the ferromagnet. These are these pairs that create the magnetic moment in the superconductor. Energetically it is favorable for the electron of the Cooper pair with the spin parallel to the magnetization of the ferromagnet to have a higher probability density in F. This means that the electron with opposite spin has a higher probability density in S. This is the reason why these pairs form a magnetic moment in the S-layer. As a result, the ferromagnetic order is created in the S-layer with a direction of the magnetic moment opposite to the direction of M in F-layer. The induced magnetic moment penetrates the superconductor over the size of the Cooper pairs, which may be much larger than dF . Using similar arguments we can predict a related eﬀect: the magnetic moment in the ferromagnet should be reduced in the presence of superconductivity because some of the electrons located entirely in the ferromagnet condense into Cooper pairs and do not contribute to the magnetization. From this qualitative, simpliﬁed picture one can expect that the total magnetization of an S/F system will be reduced for temperatures below Tc . A quantitative analysis based on the Usadel equation (diﬀusive case) [19] or on the Eilenberger equation (quasiballistic case) [50] supports the qualitative picture. It turned out that at low temperatures the magnetic moment MF in F is screened completely by the spin-polarized Cooper pairs in S if MF is due to free electrons (ideal itinerant ferromagnet) i.e. MS = −MF . This conclusion is valid in the limit h < DF /d2F . With increasing the exchange energy h the induced magnetic moment decreases monotonically in the diﬀusive limit [19] or non-monotonically in the clean limit [50]. It should be stressed that both the mechanism discussed here and that of the last section lead to a decrease of the total magnetization. The spin polarization of Cooper pairs in the superconductor in F/S/F structure with a non-collinear magnetization in F was studied in [51].

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5.3 S/F Proximity Eﬀect: Experiments 5.3.1 Superconducting Transition Temperature in F/S Systems Following the theoretical predictions (see Sect. 5.2.2), a ferromagnetic ﬁlm deposited on a superconducting ﬁlm should drastically suppress the superconducting Tc . In experimental systems, however, this is often not the case, the Tc - suppression appears rather moderate. This is due to two diﬀerent reasons. First, in real thin ﬁlm systems there is often an intermediate alloy layer caused by interdiﬀusion that is weakly magnetic or even non-magnetic. This is the case, e.g., in Fe/Nb [52, 53] and, probably, in Gd/Nb [54]. This interlayer prevents the direct contact between the F- and S-layer and weakens the suppression of Tc . Second, the quantum mechanical transparency of a real S/F interface is often quite small, i.e. the coeﬃcient γF,S in (5.8) is small and the Tc -suppression is much weaker than that with an ideally transparent interface. An interesting feature of the S/F proximity eﬀect that has recently been under intensive discussion in the literature, is oscillation of the superconducting transition temperature as a function of the F-layer thickness dF . There are quite diﬀerent physical mechanisms that may cause Tc (dF ) oscillations or a non monotonic Tc (dF ) behavior. An indirect mechanism, not directly related to the proximity eﬀect, has been observed in Fe/Nb bilayers [52, 53] (Fig. 5.7c). Here an alloying at the interface leads to a nonferromagnetic NbFe interlayer of about 0.7 nm thickness and therefore the minimum in Tc (dF ) just correlates with the onset of ferromagnetism. The explanation of this phenomenon is that strong longitudinal spin ﬂuctuations exist in the NbFe interlayer with a concentration close to the onset of ferromagnetic long range order. They are responsible for the strong initial Tc suppression when increasing the Fe-thickness from 0 to 0.7 nm in Fig. 5.7c. When the ﬁrst ferromagnetic Fe layer appears above dF e =0.7 nm, the spin ﬂuctuations in the NbFe interlayer are suppressed by the exchange ﬁeld of the Fe-layer and result in an increase of Tc . This is a rather indirect inﬂuence of the ferromagnetic Fe-layer on the superconductivity of the Nb layer. Now, coming to oscillations in Tc (dF ) induced by the S/F proximity eﬀect, we recall that, as discussed in Sect. 5.2.2, oscillations of the condensate function fˆs in space may directly lead to a non-monotonic Tc -dependence on h or dF . Actually the reason for this non monotonic behavior may be diﬀerent for S/F bilayers and S/F/S trilayers, since the boundary conditions in these two cases are diﬀerent. For bilayers, only one side of the F-layer is in contact with the superconductor, whereas in the S/F/S trilayers the F-layer is in contact with the superconducting layers on the both sides. For the case of trilayers (but not for bilayers) oscillations of Tc may be due to the appearance (or disappearance) of Josephson π-coupling. As mentioned in Sect. 5.2.3, due to the oscillation of the superconducting pairing function in the F-layer, the phase diﬀerence in the superconducting pairing function on

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(a) dNb= 60 nm

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4

dGd (nm)

7

(c) dNb= 40 nm

Tc (K)

6 5 4 3 0

0.5

1

1.5

2

2.5

dFe (nm)

Fig. 5.7. Dependence of superconducting transition temperature on the ferromagnetic layer thickness in (a and b) two series of Nb/Gd multilayers (Jiang et al. [54]), and (c) Fe/Nb/Fe trilayers (M¨ uhge et al. [52, 53])

both sides of the F-layer may have opposite phases at certain F-layer thicknesses. This means that the phase diﬀerence between the neighboring S-layers may be equal to π. Radovi´c et al. [28] concluded from their calculations that Tc for π-coupling most probably is higher than for the vanishing phase diﬀerence. Jiang et al. [54] claimed that the observed oscillations of Tc (dGd ) (Fig. 5.7a and 5.7b) are due to this type of the Josephson π-coupling. Several other works on S/F multilayers have reported a single peak in Tc (dF ), and have attributed this feature to “π-switching” (see, e.g., [55, 56, 57]). Whereas the role of π-coupling for the non monotonous Tc (dF ) in S/F multilayers has not been ﬁnally clariﬁed and an alternative explanation exists (see below), a clear experimental evidence for π-coupling across an F-layer

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comes from the study of Josephson junctions using F-layers as barriers [4, 39]. Tunnelling spectroscopy revealed damped oscillations of the superconducting order parameter induced in the F-ﬁlm by the proximity eﬀect [39]. Ryazanov et al. [4] performed measurements of the critical current in Josephson junctions consisting of superconducting Nb and weakly ferromagnetic interlayers and found that the character of the junction changed from 0-phase at high temperatures to π-phase at low temperatures. This result was later conﬁrmed by Blum et al. [5]. A diﬀerent phase sensitive experiment [58] also gave evidence for the oscillatory behavior of the critical supercurrent of S/F layered system when varying the F-layer thickness. Without invoking π-coupling, oscillations of Tc (dF ) can simply originate from oscillations of the condensate amplitude in space within the F-layer. As shown theoretically by [28, 30, 59], due to these oscillations and taking the boundary conditions for the pairing wave function at the S/F interfaces into account, the Tc (dF )-curve may have an oscillatory character with the oscillation period of the order ξh = vF /h (see Sect. 5.2.2). The physical origin of the oscillatory character of Tc (dF ) can qualitatively be traced back to the propagating character of the superconducting pairing wave function in the ferromagnet. If the thickness of the F-layer is smaller than the penetration depth of the pairing wave function, this function, when transmitted through the S/F interface into the F-layer, will interfere with the wave reﬂected from the opposite surface of the ferromagnet. As a result, the ﬂux of the pairing wave function crossing the S/F interface varies with the thickness of the F-layer dF . Then, the coupling between the electrons of the ferromagnet and the superconductor will be modulated and Tc will oscillate with dF . If the interference at the S/F interface is essentially constructive (this corresponds to a minimal jump of the pairing function amplitude at the S/F interface), the coupling is weak, and one expects Tc to be maximal. When the interference is destructive, the coupling is maximized and Tc (dF ) is minimal. It should be noted that this model explaining Tc (dF ) oscillations applies to the case of the S/F bilayers as well as to F/S/F trilayers or S/F multilayers, whereas the π-coupling concept does not apply for bilayers. Aarts et al. [60] studied V/V1−x Fex multilayers without interdiﬀusion at the interface. They showed that Tc strongly depends on the interface transparency and presented experimental evidence for an intrinsically reduced interface transparency. From the dependence of Tc on the magnetic layer thickness they calculated the penetration depth of Cooper pairs into the F-layer and found it to be inversely proportional to the eﬀective magnetic moment per Fe atom. For the interpretation of the observed peculiarities they introduced a ﬁnite transparency of the S/F interface and argued, based on their experimental data, that with an increasing the exchange splitting of the conduction band in the F-layer the transparency of the S/F interface for Cooper pairs decreases.

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Lazar et al. [61] experimentally studied the role of the interface transparency in the Fe/Pb/Fe system and, for comparison, in the Fe/V/Fe system, too [62]. In contrast to the case of Fe/Nb/Fe discussed above, in Fe/Pb/Fe and Fe/V/Fe the intermixing at the interfaces is much weaker. Figure 5.8 shows the dependence of Tc on the thicknesses of the Fe layers for Fe/Pb/Fe trilayers. A theoretical analysis of the curves using model calculations revealed that the experimental results can only be described assuming Pb/Fe interfaces that are not perfectly transparent. The Tc for the case of S/F interfaces with a non-perfect transparency has been calculated by Golubov [60] and Tagirov [30]. A ﬁt to the experimental points using the model calculations [30] is plotted on Fig. 5.8 as a solid line. The quality of the ﬁt is satisfactory and reproduces the details of the Tc (dF e )-curve. The most important parameter obtained from this ﬁt is the value of Tm , characterizing the transparency of the interface. The ﬁt gives Tm = 0.4. This value corresponds to a quantum mechanical transmission coeﬃcient T = Tm /(1 + Tm ) = 0.3 [61] that is considerably reduced as compared to the ideally transparent interface with T = 1. Lazar et al. [61] concluded that the exchange splitting of the conduction band of the F-layer is the main physical reason for the strongly reduced interface transparency. In principle, the calculation of the interface transparency is a standard quantum mechanical problem of reﬂection and transmission of electrons at the interface of two metals with diﬀerent Fermi energies. It is obvious that two electrons with opposite spins forming a Cooper pair can never match the Fermi momenta of the exchange-split subbands of a ferromagnet simultaneously and there will always be a Fermi vector mismatch reducing 7

dPb=73 nm

6

T c (K)

5 4 3 2 1

0

1

2

3

4

dFe (nm) Fig. 5.8. Tc dependence on the Fe thickness at ﬁxed value dP b = 73 nm for Fe/Pb/Fe trilayers. The dashed line is obtained by the Radovi´c et al. [28] theory which supposes an ideally transparent interface. The solid line takes a ﬁnite transparency of the interface into account [30]

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the transmission. Additionally, a decrease of T is expected due to a chemical mismatch of Pb and Fe giving rise to a contact potential barrier at the interface. The barrier height should be larger for immiscible metals like Pb and Fe in comparison to metals that form solid solutions in the whole concentration range as, e.g., V and Fe. Measurements of dependence of Tc on the thickness were performed also for Fe/V/Fe trilayers [62]. For two series of samples at small iron thicknesses the Tc drops sharply when increasing dF e up to 0.5 nm. Then, at dF e ∼0.7 nm for the series with dV = 31 and 29 nm, a clear minimum of Tc is observed. The deepness of this minimum increases with decreasing dV . For these two series the residual resistivity ratio RRR 4, meaning that the mean free path of the conduction electrons in the S-layer is lS ∼ 4 nm [63]. The parameters resulting from a theoretical ﬁt of these curves are the superconducting coherence length ξS = ξ0 lS /3.4 = 4 nm (here ξ0 = 44 nm is the BCS coherence length), as estimated from the resistivity data. For the transparency parameter one obtains Tm = 1.6 and for the exchange length in the Fe ﬁlm ξh = 0.7 nm. Using the theoretical model calculations [30] as a guideline, one can extract the important physical parameters necessary for an observation of the theoretically predicted rather spectacular re-entrant behavior of the superconductivity, i.e. superconductivity vanishing for a certain range of dF and coming back for larger dF . The system should possess a large electron mean free path in the F- as well as in the S-layer, a high quantum-mechanical transparency of the S/F interface and a geometrically ﬂat interface without introducing too much diﬀuse scattering of the electrons. The last two conditions are well fulﬁlled in Fe/V/Fe trilayers, so one could try to further increase the electron mean free path lF or lS . Whereas this is hardly possible for the F-layers since lF is limited by the very small layer thickness dF , improved growth conditions of the V-layer is a promising perspective to increase lS This was accomplished by samples prepared on single crystalline MgO (100) with nearly epitaxial quality and an RRR-value of the order of 10. For this set of samples re-entrant superconductivity was observed (Fig. 5.9) for the ﬁrst time. From the RRR-value for this series we estimate the mean free path of the conduction electrons lS ∼ 12 nm and the corresponding coherence length ξS ∼ 13 nm. The latter was used for the theoretical ﬁt of the Tc (dF e )-curve in Fig. 5.9. From the transparency parameter Tm = 1.6 the average quantum mechanical transmission coeﬃcient T [61] can be estimated to be T 0.6. This value of T is about twice as large as the T -value for the Pb/Fe interface discussed above [61]. This relatively high transparency of the Fe/V interface is an essential ingredient for observing re-entrance behavior. As mentioned above, a highly transparent S/F interface is diﬃcult to achieve with strong ferromagnets, since problems with the matching of the Fermi momentum necessarily occur for at least one spin direction. An even higher transparency of the S/F interface can, in principle, be achieved combining a superconductor with a ferromagnet weakened by

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5 dV=33.9 nm

TC (K)

4 3 2 1 0

2

1

3

dFe (nm)

Fig. 5.9. Superconducting transition temperature vs Fe thickness at ﬁxed V thickness for the Fe/V/Fe series with dV = 33.9 nm. The drawn line is a theoretical curve with the parameters given in the text

dilution. Recently [64], the re-entrant superconductivity has been observed for the Nb/Cu1−x Nix bilayers. It is important to note that the results described above for the Fe/V system were ﬁtted using the values of the mean free path of the conduction electrons in the F- and S-layers calculated from the resistivity data. In contrast, for the Nb/Cu1−x Nix system Zdravkov et al. [64] had to use surprisingly large values for the mean free path for the conduction electrons in the F-layer in their ﬁtting procedure. Finishing this section, we would like to mention also study of the interplay between magnetism and superconductivity in epitaxial structures of half metal-colossal magnetoresistive La2/3 Ca1/3 MnO3 (LCMO) and high-Tc superconducting YBa2 Cu3 O7−δ (YBCO) [65, 66, 67, 68, 69]. Jacob et al. [65] demonstrated the possibility of preparation of hybrid perovskite highTc superconductor/ferromagnet superlattices. The superlattices consisting of YBCO and LBMO (La2/3 Ba1/3 MnO3 ) layers with the thickness of a few unit cells, showed both strong colossal magnetoresistance at room temperature and superconductivity at low temperatures. Yeh et al. [66] reported phenomena manifesting nonequilibrium superconductivity induced by spin-polarized quasiparticles in F/I/S (I is insulator) structures. Sefrioui et al. [67] based on their measurements of Tc vs S- and F-layer thickness speculate that the injection of spin-polarized carriers from LCMO into YBCO may add a new source of superconductivity suppression: pair breaking by spin-polarized carriers. This pair breaking eﬀect extends over the spin diﬀusion length into the S, which can be very long (it can be as long as 8 nm for YBCO). As a result in the YBCO layer superconductivity is suppressed by the presence of manganite layers with a characteristic length scale much longer than the one predicted by existing theories of the S/F proximity eﬀect.

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The same result has been obtained by Holden et al. [68] using ellipsometry measurements of the far-infrared dielectric properties of superlattices composed of thin layers of YBCO and LCMO. Finally, Soltan et al. [69] studied the role of spin-polarized self injection from LCMO into the YBCO layer. They concluded that the nearly full spin polarization at the Fermi level of LCMO leads to quenching of the proximity eﬀect since it prevents the Cooper pairs to tunnel into the magnetic layer. Thus, one can see that the results for superlattices consisting of YBCO and LCMO presented above are somewhat contradicting each other. Nevertheless, they provide an avenue for future theoretical studies of the F/S proximity eﬀect in presence of the spin-polarized ferromagnets. 5.3.2 Superconducting Spin Valve In recent years much attention has been devoted to experimental realization of the superconducting spin valve. As described in Sect. 5.2.1, a consequence of the S/F proximity eﬀect is that the superconducting transition temperature of a F/S/F sandwich depends on the mutual orientation of the magnetization of the two F-layers, the antiparallel orientation having a higher Tc than the parallel one [6]. In an ideal superconducting spin valve the superconductivity of the S-layer can be switched on and oﬀ by rotating the magnetization of one of the F-layers relative to the other, giving an inﬁnite magnetoresistance for the switching ﬁeld. The device is similar to the well known conventional spin valve F/N/F system with a normal metallic layer N interleaved between two ferromagnetic layers F. In this device the antiparallel magnetization state usually has a larger resistance than the parallel one. It turned out that the realization of a superconducting spin valve is diﬃcult experimentally and the eﬀects obtained until now are quite small. There are two recent reports in the literature on the successful realization a F/S/F superconducting spin valve. In the CuNi/Nb/CuNi trilayers system [70, 71] the maximum shift is only 6 mK of the superconducting transition temperature Tc by changing the mutual orientation of the two ferromagnetic layers from parallel to antiparallel. Actually, such a small shift may also be due to changes of the domain structure of the ferromagnetic layers under the inﬂuence of the external magnetic ﬁeld [72]. Pena et al. [14] measured the magnetoresistance of F/S/F trilayers combining the ferromagnetic manganite La0.7 Ca0.3 MnO3 with the high-Tc superconductor YBa2 Cu3 O7 . They observed a magnetoresistance in excess 1000% for the superconducting state of YBa2 Cu3 O7 that vanished in the normal state. There is another possible design for the realization of the superconducting spin valve eﬀect proposed by Sungjun Oh et al. [73] that found less attention until now. It has the layer structure S/F1/N/F2, i.e. two ferromagnetic layers F1 and F2 separated by a nonmagnetic (N) layer are deposited on the one

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side of the superconductor with F1 and N thin enough to allow the superconducting pair wave function to penetrate into F2. The authors have shown that changing the mutual magnetization direction of F1 and F2 from parallel to antiparallel results in a substantial diﬀerence ΔTc when the microscopic parameters for S- and F-ﬁlms are optimized. For the realization of the F/S/F spin valve design [6] it would be optimal to use a system where the re-entrant Tc (dF )-behavior is observed. As discussed above, Fe/V/Fe fulﬁlls this criterion (see Fig. 5.9). However, an acceptable performance of the spin valve with a sizable shift of Tc can only be expected if the S-layer thickness dS is close to the superconducting coherence length ξS . The studies of the Fe/V/Fe system however revealed that the superconductivity vanishes typically already at dS < 3ξS . A possibility to overcome this problem and maintain superconductivity at dS ∼ ξS is to introduce very thin non-ferromagnetic layers between the S- and Flayers that should screen to some extent the very strong exchange ﬁeld of the F-layers. A proper Fe/Cr/V/Cr/Fe system, where the Cr layers play the role of such screening layers, has been studied in detail [74]. In Fig. 5.10 the Tc values measured for the samples from series with a ﬁxed dF e = 5 nm and dCr varied are plotted. In the other three series dCr has been kept constant at dCr = 1.5, 2.8 and 4.7 nm and the thickness of the Fe-layer was varied. The results for the transition temperatures of these series are reproduced in Figs. 5.11b–5.11d and compared to previous results on Fe/V/Fe trilayers [62] (Fig. 5.11a). The salient features of the results shown in Fig. 5.11 are as follows.

Fig. 5.10. The superconducting transition temperature as a function of the Cr-layer thickness for all samples from the series (5.1). The solid line is a theoretical curve (see main text)

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Fig. 5.11. Superconducting transition temperature as a function of the Fe-layer thickness for samples from series with dCr =1,5 nm (b); with dCr =2.8 nm (c); with dCr = 4.7 (d). The corresponding curve for Fe/V/Fe trilayers is taken from [62] and shown in (a) for comparison. The solid lines are calculations according to a model explained in the main text

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(1) The overall shape of the Tc (dF e )-curve is similar to that obtained for Fe/V/Fe. (2) The amplitude of the initial drop in Tc decreases with increasing the thickness of the interleaved Cr-layer. (3) At dCr = 4.7 nm in Fig. 5.11d the Fe-layers have virtually no inﬂuence on Tc anymore, indicating that the amplitude of the pair wave function in the Fe-layer is negligible. This allows the estimation of the penetration depth of the pair wave function in Cr of about 4 nm, consistent with the results on Cr/V/Cr trilayers [75]. These features are due to the expected screening eﬀect of the Cr-layer, since with increasing dCr the Cooper pair density reaching the Fe-layer is continuously reduced and the eﬀect of the strong exchange ﬁeld in Fe on the superconductivity is weakened. The results of the model calculations are shown by the solid lines in Figs. 5.10 and 5.11a. The complications caused by the spin density wave (SDW) state of antiferromagnetic Cr are neglected in these calculations. The standard procedure described in the literature (see, e.g., [75] and references therein) was applied and the proximity eﬀect of the V/Cr interface was treated by the conventional theory for S/N metal ﬁlms originally developed by de Gennes [1]. In addition, pair breaking scattering of Abrikosov-Gor’kov type [76] at magnetic defects in the Cr-layer is characterized by a spin-ﬂip scattering time τs i.e. Cr is treated as a paramagnetic (P) layer. Theory of the proximity eﬀect for S/P/F layer systems has been developed by Vodopyanov et al. [77]. With certain assumptions [74] and the microscopic parameters known from studies of Fe/V/Fe trilayers [62] all data points in Figs. 5.10 and 5.11 have been ﬁtted simultaneously, with τs being the only ﬁtting parameter. All curves can be best described with τs = 5 · 10−13 s. The overall shape of the curves is well reproduced, including the penetration depth of about 4 nm for the superconducting pairing wave function in Cr. This remarkably small penetration depth in Cr is thus clearly proven to result from strong inelastic pair breaking scattering leading to an exponential damping of the pair wave function amplitude within the Cr-layer. There is, however, an additional interesting experimental detail in Fig. 5.10, which the applied model fails to describe even in qualitative terms. This is the drop of Tc (dCr ) for dCr ≥ 4 nm, clearly seen in Fig. 5.10. This feature was attributed to a transition of the entire Cr-layer from a non-magnetic state to an incommensurate SDW state at dCr ∼ 4 nm. The assumption of a strong suppression of the Cooper pair density by the transition of the Cr layer from a non-magnetic to a spin density wave state is plausible by the following reason. BCS-ordering and SDW-ordering in the same region of the Fermi surface can be considered as competing electronic ordering phenomena. In a theoretical paper on studying this problem (see, e.g., [78]), it was shown that those parts of the Fermi surface where the nesting feature leads to a SDW state the formation of the BCS-gap is suppressed and the superconducting transition temperature is reduced.

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The study of the superconducting proximity eﬀect in Fe/Cr/V/Cr/Fe gave new results concerning the magnetic phase transition in the Cr-layer, demonstrated a strong screening of the ferromagnetic exchange ﬁeld of Fe by the interleaved Cr-layers and allowed an estimate of the upper limit of the thickness of the screening Cr-layers for a spin valve to operate. A novel approach for a realization of the superconducting spin valve design originally proposed by Sungjun et al. [73] was also undertaken recently. The idea [79] for the realization of such a device was to choose as the non-magnetic interlayer N in the S/F1/N/F2/ layer scheme an interlayer with a thickness corresponding exactly to an antiferromagnetic interlayer exchange coupling between F1 and F2 [80]. Then, one can rotate the relative magnetization direction of F1 and F2 from antiparallel to parallel in an external ﬁeld and observe the accompanying shift of Tc . The experimental system of choice was the epitaxial superlattice system MgO(100)/[Fe2V11 ]20 /V (dV ). (The index denotes the number of monolayers.) There are several reasons that make the choice of the epitaxial (V/Fe)-system favorable for demonstrating the superconducting spin valve eﬀect: First, it is the superior quality of the Fe/V interface in the superlattice [81, 82, 83, 84] that guarantees a high interface transparency and weak diﬀusive pair breaking scattering at the interface. Second, the Fe2 layers have a thickness dF of about 0.3 nm only, whereas for the decay length of the superconducting pair density ξF ∼ 0.7 nm holds (see, e.g., [61]). Thus the pair wave function within the Fe2 -layer will only be weakly damped and the condition dF /ξF < 0.5 optimal for observing the superconducting spin valve eﬀect will be fulﬁlled [73]. In Fig. 5.12 we reproduce the magnetization curve of a [Fe2 V11 ]20 - superlattice measured at 10 K. The shape of the hysteresis shows that the interlayer exchange coupling is antiferromagnetic with a ferromagnetic saturation ﬁeld of Hsat = 6 kOe. The upper critical magnetic ﬁeld for the ﬁeld direction parallel and perpendicular to the ﬁlm plane is plotted in Fig. 5.13 for several samples.

Fig. 5.12. Magnetization hysteresis loop of the sample [Fe2 V11 ]20 /V(18 nm) measured at 10 K

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Hc2 (kOe)

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Fig. 5.13. Upper critical magnetic ﬁeld versus temperature with the ﬁeld applied parallel and perpendicular to the ﬁlm plane for three samples [Fe2 V11 ]20 /V(dV ). The thickness dV is given in the ﬁgure, the open symbols refer to the magnetic ﬁeld direction parallel to the ﬁlm plane, the solid symbols refer to the direction perpendicular to the plane

For a two dimensional (2D) thin ﬁlm with the magnetic ﬁeld perpendicular or parallel to the ﬁlm plane the classic result for the upper critical ﬁeld is [85]: T Φ0 perp 1 − , (5.16) = Hc2 2πξ 2 (0) Tc par Hc2

√ " T Φ0 12 1− , = 2πξ(0) ds Tc

(5.17)

with the ﬂux quantum Φ0 , the thickness of the ﬁlm dS and the GinzburgLandau correlation length ξ(0) related to Pippard’s correlation length ξs as ξ(0) = 1.6ξs . The measurements of the upper critical ﬁeld for Fe/V/Fe trilayers for parallel orientation of the magnetic ﬁeld relative to the ﬁlm plane is perfectly described by (5.16), as it was observed earlier [86, 87]. In Figs. 5.14a and 5.14b the square of parallel upper critical ﬁeld are plotted together with the straight line that describes the temperature dependence for ﬁelds above 6 kOe perfectly. Below H = 6 kOe there is an increasing deviation from the straight line. From the extrapolation of the straight line one gets a Tc that is more than 0.1 K below the true transition temperature measured at zero ﬁeld. A comparison with the magnetization curve in Fig. 5.12 shows that the ferromagnetic saturation ﬁeld of 6 kOe is correlated with the ﬁrst deviation 2 of Hc2 (T ) from the straight line in Fig. 5.14a and 5.14b. From this one can infer that the deviation of the upper critical ﬁeld from the 2D-behavior in Fig. 5.14 is caused by the gradual change of the sublattice magnetization direction of the [Fe2 /V11 ]20 -superlattice from parallel above 6 kOe to antiparallel in zero ﬁeld. For the sample with dV = 16 nm in Fig. 5.14a Tc = 1.78 K

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0 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 T (K) Fig. 5.14. Square of the parallel upper critical magnetic ﬁeld versus temperature for the sample [Fe2 /V11 ]20 /V(16 nm) (a) and [Fe2 /V11 ]20 /V(30 nm) (b). The full straight line is the extrapolation of the linear temperature dependence for higher ﬁelds, the dashed line is the theoretical curve expected if the magnetization of the superlattice would not change. ΔTc is the shift of the superconducting transition temperature between the superlattice in the antiferromagnetic state and in ferromagnetic saturation. The inset in panel (a) depicts the shift of the superconducting transition temperature with the magnetization of the [Fe2 /V11 ]20 superlattice

in antiferromagnetic state, while in ferromagnetic saturation it extrapolates to Tc = 1.67 K. The temperature diﬀerence ΔTc = 0.11 K is the anticipated superconducting spin valve eﬀect. These experiments demonstrate that the superconducting transition temperature of the V-ﬁlm reacts sensitively to the mutual magnetization orientation of the Fe2 layers of an antiferromagnetically coupled [Fe2 V11 ]20 superlattice. Actually the ferromagnetic layers in this system cannot be switched from the parallel to the antiparallel state, since the parallel state needs the application of a strong external magnetic ﬁeld. At the same time, it should in principle be possible to construct a switching device by replacing

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the antiferromagnetically coupled [Fe/V] superlattice by a conventional spin valve trilayer system. 5.3.3 Odd Triplet Superconductivity in S/F/S Structures In Sect. 5.2.3 a theoretical model predicting a possible robust triplet proximity eﬀect in S/F structures has been described. The mechanism is operational in the presence of a rotating magnetization at the S/F interface. Recently Sosnin et al. [13] presented the ﬁrst experimental indication of this type of proximity eﬀect using an Andreev type of interferometer and an S/F/S mesoscopic thin ﬁlm structure. The design of the interferometer is depicted intergerometer in Fig. 5.15a. It consists of a superconducting Al-loop with an area of 20 μm2 with a narrow gap bridged by a ferromagnetic Ho-stripe. The distance between the two Al/Ho contact points was more than one order of magnitude larger than the singlet magnetic coherence length ξF 0 . A rotating magnetization at S/F interface is established here by the intrinsic conical ferromagnetism of Ho (see Fig. 5.15b). The essential experimental ﬁnding is that below the Tc of Al the resistance of the Ho wire exhibits oscillations as a function of the superconducting phase diﬀerence between the two interfaces of the Ho-stripe with the superconducting Al ring, as shown in Fig. 5.16a. The phase diﬀerence was generated by varying the magnetic ﬂux penetrating the Al-loop. The period of the oscillations corresponds to the ﬂux quantum Φ0 = 2 ·10−7 Gcm2 and gives rise to the sharp peaks in the Fourier spectrum of oscillations (Fig. 5.16b). Estimates show that for the relative amplitude of the conductance oscillations ΔR/RF 10−4 ( RF is the resistivity of the ferromagnetic wires) is expected. These oscillations were observed for the samples with a distance between the (a)

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Fig. 5.15. a) Experimental set-up and SEM micrograph of S/F/S junction area prepared by shadow evaporation. b) Magnetic structure of Ho: magnetization M rotates by 30◦ each atomic layer along c-axis at an angle of 80◦ to this axis

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(a) Δ R (mΩ )

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Fig. 5.16. a) Magnetoresistance oscillations of the sample shown in Fig. 5.15a measured at T=0.27 K as a function of normalized external ﬂux through the loop. The sample resistance is 94.3 Ω. b) Fourier spectrum of the oscillations conﬁrming the hc/2e periodicity

Al/Ho contact points interfaces of up to LF = 160 nm. Such a long-range phase coherence cannot be explained by the proximity eﬀect involving the penetration of the ordinary singlet pairs, since the upper limit for the singlet penetration depth ξF 0 is equal to the electron mean free path l which for was l ≈ 6 nm. Thus, the observed oscillations of the magnetoresistance seem to originate from the long-range penetration of a helical triplet component of superconductivity generated in a ferromagnetic conductor and induced by the presence of a rotating magnetization. Recently Keizer et al. [15] studied lateral S/F/S Josephson junctions combining the strong ferromagnet CrO2 that belongs to the group of half-metals with full spin polarization of the electrons at the Fermi level and the conventional s-wave superconductor NbTi. They observed a Josephson supercurrent prevailing over very long length scales up to ∼ 1 μm. This is orders of magnitude larger than expected for singlet correlations, which is of the order of 1 nm. In addition to the long-range penetration of the superconducting pair density into CrO2 , they found that the supercurrent strongly depended on the magnetization direction in the ferromagnet. On the basis of these ﬁndings

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Keizer et al. attributed the long-range supercurrent to the triplet correlations. In this case of a half-metallic ferromagnet it is reasonable to assume that the LRTC is created at the S/F interface where spin-ﬂip processes may happen [88]. Hints of the realization of the triplet proximity eﬀect also came from recent magnetization data on hybrid structures consisting of multilayers of manganites [La0.33 Ca0.67 MnO3 /La0.60 Ca0.40i MnO3 ]15 in contact with a low-Tc Nb superconductor [89]. 5.3.4 Other Proximity Eﬀects It seems natural and actually it is theoretically well established that the penetration of superconductivity from the S- into the F-layers is not the only possible proximity eﬀect in S/F systems (see Sect. 5.2.5 and 5.2.6). The proximity eﬀect can also work in the reverse direction, i.e. the ferromagnetism from the F-layer can leak into the S-layer (inverse proximity eﬀect) or the S-layer can modify the ferromagnetic state of the F-layer (cryptoferromagnetism). However, these eﬀects are more subtle from the experimental point of view and are still less well established. Cryptoferromagnetism in S/F Layers As shown in Sect. 5.2.5, under certain conditions the ferromagnetic order in F-layers may be reconstructed by the action of the S-layer into a new magnetic domain state [17]) or a cryptoferromagnetic state [18]. The basic physical reason for this behavior is that the destructive inﬂuence of the ferromagnetic exchange ﬁeld on the superconductivity can be considerably reduced if the ferromagnetic state is modiﬁed in such a manner that the exchange ﬁeld cancels when averaged over the superconducting coherence length. The ﬁrst hint in favor of a reconstruction of the ferromagnetic state below the superconducting transition temperature was obtained from the anomalous temperature dependence of the eﬀective magnetization extracted from the ferromagnetic resonance (FMR) line position in epitaxial Fe/Nb bilayers below Tc [48]. However, a quantitative estimate using the theory of Buzdin and Bulaevskii [17] rises doubts about this interpretation, since the eﬀect in Fe/Nb should only occur at an Fe-layer thickness an order of magnitude smaller than observed experimentally. Later Bergeret et al. [18] studied theoretically the possibility of a non-homogeneous magnetic order of a ferromagnetic ﬁlm placed on top of a bulk superconductor. They also concluded that due to the large magnetic stiﬀness constant in Fe, the cryptoferromagnetic state can hardly be realized using pure Fe ﬁlms. These considerations suggested that the tendency to a reconstruction of the ferromagnetic state observed experimentally in Fe/Nb might be caused by a granular structure of the very thin Fe layers.

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Quantitative estimates by Bergeret et al. [18] showed that the transition from the ferromagnetic to the cryptoferromagnetic state should be observable in a ferromagnet with a magnetic stiﬀness constant an order of magnitude smaller than that of pure Fe. This can be achieved by dilution of Fe in suitable alloy systems, a favorable choice being Pd1−x Fex at small x due to its low and tunable Curie temperature. In an FMR study for a series of samples V/Pd1−x Fex the temperature dependence of the eﬀective magnetization 4πMef f = 4πM − (2Ku /M ) (M is the saturation moment of the ferromagnet and Ku is the perpendicular anisotropy constant) was measured [49]. The low-temperature part of 4πMef f (T ) is depicted in Fig. 5.17. One observes a decrease of the eﬀective magnetization 4πMef f below Tc for the sample 2 (Fig. 5.17) but not for sample 1. A decrease of 4πMef f can be caused by a decrease of the saturation magnetization M or by an increase of the perpendicular uniaxial anisotropy constant Ku . A comparison of FMR results for ﬁlms with diﬀerent thickness of the ferromagnetic layers leads to the conclusion that Ku is very small and the decrease of 4πMef f must be caused by a decrease of the saturation magnetization M . This suggests that the decrease of the saturation magnetization below Tc is caused by a reconstruction of the ferromagnetic state. An estimate following the phase diagram by Bergeret et al. [18] (Fig. 5.5) gave the parameters a ∼ 1.2 and λ ∼ 1.3 · 10−3 for sample 2 with dM ∼1.2 nm and TCurie ∼ 100 K. In accordance with the phase diagram of Bergeret et al. (Fig. 5.5) this implies that starting from τ ∼ 0.2 (T ∼ 3.2 K) a transition from the ferromagnetic to the cryptoferromagnetic state should take place, as it is actually observed experimentally. For the sample 1 with dM ∼ 4.4 nm and TCurie ∼ 250 K we have a ∼ 20 and λ ∼ 1.4 · 10−2 . With these parameter values the ferromagnetic state should be stable at any temperature, in agreement with the experimental result. 5

4πMeff (kG)

4 TC

3 2

TC sample 1 sample 2

1 0

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2

3

4

5

T (K)

Fig. 5.17. Low-temperature parts of 4πMef f (T ) for the sample 1 with dV = 37.2 nm, dP d−F e = 4.4 nm, Tc = 4.0 K and sample 2 with dV = 40 nm, dP d−F e = 1.2 nm, Tc = 4.2 K. The arrows show the Tc -values at the resonance ﬁeld H0

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Thus, these estimates support the conclusion that a phase transition from the ferromagnetic state to the cryptoferromagnetic state occurs in sample 2. However, one cannot completely exclude that the anomalous temperature dependence of Mef f might be due to the screening of the magnetic moments of the ferromagnetic layer by the polarized Cooper pairs, as discussed in Sect. 5.2.6 and in the next section. Inverse Proximity Eﬀect Up to now any unequivocal experimental evidence for the penetration of the magnetization from the ferromagnetic side into the superconducting side of an S/F bilayer, as discussed in Sect. 5.2.6, does not exist. First interpretations in this direction have been published recently [89, 90]. Stahn et al. [90] studied the magnetization proﬁle of [YBa2 Cu3 07 /La2/3 Ca1/3 MnO3 ] multilayers using neutron reﬂectometry. From a change of the Stahn et al. [90] argue in favor of the ﬁrst possibility, but the situation is not yet settled. Stamopoulos et al. [89] presented magnetization measurements on multilayers of manganites [La0.33 Ca0.67 MnO3 /La0.60 Ca0.40 MnO3 ]15 in contact with a low-Tc superconductor. They came to the conclusion that the superconductor below Tc becomes ferromagnetically coupled to the multilayer. Since it is expected that for singlet pairing the magnetization of F penetrates into S antiferromagnetically, the authors conclude that a spin-triplet superconducting component forms and penetrates into the F-layer thus inducing the ferromagnetic coupling observed experimentally.

5.4 Summary and Conclusions The main purpose of the present paper was to review the status of research on proximity eﬀects in S/F layer systems from the experimental as well as from the theoretical point of view. Peculiarities of the S/F proximity eﬀect originating from the penetration of the condensate function into the ferromagnet that have been discussed controversially in the beginning seem to be well established now. The S/F/S Josephson junctions with π-coupling are, e.g., even suggested as basic units for the realization of Q-bits for quantum computing [91]. It has become traditional in the ﬁeld of the S/F proximity eﬀect that theory is somewhat ahead of experiment. The situation persists and intriguing theoretical predictions are still waiting for the ﬁrst experimental veriﬁcations or further experimental support. One of these predictions concerns the unconventional superconductivity in S/F systems. The experimental realization is diﬃcult, since the unconventional superconductivity expected here, namely odd triplet superconductivity, can only be generated by a rotating magnetization at the interface. Nevertheless, the ﬁrst experimental indications of its existence have already been reported. What is important, the odd triplet

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superconductivity is insensitive to scattering on non-magnetic impurities and this is certainly helpful for an experimental observation. We should also mention further recent ideas on how to identify the odd triplet superconductivity [92, 93, 94]. The inverse proximity eﬀect, i.e. the penetration of the magnetic order parameter into a superconductor, has not been clearly observed experimentally yet. However, indications on the closely related eﬀect, namely the decreasing of the total ferromagnetic moment below Tc , already exist. Yet, it is not easy to clarify to what extent the non-homogeneous distribution of MF produced in the ferromagnet below Tc contributes to the eﬀect. The best way to observe the spin screening of MF is either probing directly the spatial distribution of the magnetic ﬁeld using neutron scattering or by measuring muon spin resonance. Since the magnetic moment M varies on the macroscopic length ξS , it should be possible to detect it. A considerable work still remains to be done on the experimental side. Only careful material selection, optimization of ﬁlm preparation and device design will enable a clariﬁcation of all the complex phenomena that may occur in the S/F proximity systems. The study of F/S structures with comparable ferromagnetic Curie and superconducting transition temperatures seems very promising. Combining elemental superconductors with elemental ferromagnets, as was done in the majority of papers on the S/F proximity eﬀect published until now, is not the best way for the observation of the proximity effects because the ferromagnetic exchange energy is orders of magnitude larger than the superconducting condensation energy. In this case the ferromagnetic state can hardly be modiﬁed by the superconductor. Rare earth based ferromagnetic compounds with low Curie temperatures would, in principle, be a better choice. Combining high-Tc superconductors with ferromagnetic oxides is another promising option.

Acknowledgement The authors are grateful for the support by the Deutsche Forschungsgemeinschaft (DFG) within SFB 491. One of us (AFV) would like to acknowledge ﬁnancial support from DFG within the project

References 1. P. G. de Gennes: Superconductivity of Metals and Alloys (Benjamin, New York, 1966) 251, 277 2. G. Deutscher: Rev. Mod. Phys. 77, 109 (2005) 251 3. L. N. Bulaevskii, V. V. Kuzii, and A.A. Sobyanin, Pis’ma Zh. Eksp. Teor. Fiz. 25, 314 (1977) (JETP 25, 290 (1977) 253, 258 4. V. V. Ryazanov, V. A Oboznov, A. Yu. Rusanov, A. V. Veretennikov, A. A. Golubov, and J. Aarts: Phys. Rev. Lett. 86, 2427 (2001) 253, 258, 270

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6 Magnetic Tunnel Junctions G¨ unter Reiss1 , Jan Schmalhorst1 , Andre Thomas1 , Andreas H¨ utten1 , 2 and Shinji Yuasa 1

2

Thin Films and Physics of Nanostructures, Department of Physics, Bielefeld University, Bielefeld, Germany [email protected] National Institute of Advanced Industrial Science and Technology (AIST), Nanoelectronics Research Institute, Tsukuba, Japan [email protected]

Abstract. In magnetoelectronic devices large opportunities are opened by the spin dependent tunneling resistance, where a strong dependence of the tunneling current on the relative orientation of the magnetization of the electrodes is found. Within a short time, the amplitude of the resistance change of the junctions increased dramatically. We will cover Al-O and MgO based junctions and present highly spinpolarized electrode materials such as Heusler alloys. Furthermore, we will give a short overview on applications such as read heads in hard disk drives, storage cells in MRAMs, ﬁeld programmable logic circuits and biochips. Finally, we will discuss the currently growing ﬁeld of current induced magnetization switching.

6.1 Introduction Tunneling is a quantum mechanical phenomenon based on the wave character of particles [1]. An, e.g., electron impinging on a wall of potential energy of energy height Φ and width d will be reﬂected with a certain probability R and transmitted with T = 1 − R. In classical mechanics, R will be 1 (i.e. T = 0) if the kinetic energy of the electron is smaller than Φ and 0 otherwise. In quantum mechanics, however, the wave character of the electron produces a non-vanishing probability for transmission already for a kinetic energy smaller than Φ. This eﬀect – called tunneling – forms the basis for the so-called magnetic tunnel junctions (MTJs). Whereas tunneling is usually demonstrated in a trilayer system consisiting of two simple metallic (non-magnetic) electrodes separated by a thin insulator, in MTJs the two electrodes are ferromagnets such as Fe or Co. In this case, the tunneling current obtained through such structures not only depends on the averaged density of electronic states in the two electrodes but additionally on the relative direction of the magnetizations in the two ferromagnets. In a simple theoretical description, one obtains for the transmission probability for the majority (↑) and minority carriers

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(↓), respectively: T↑,↓ ∝ exp(−k↑,↓ Φ1/2 d), i.e. the transmission becomes spin dependent [2] due to the exchange splitting of the band structure in the ferromagnetic metalls (k↑,↓ : Complex wave vector of the electrons). However, it was not before 1975 that a spin dependence of the tunneling current was observed experimentally: Julli`ere [3] found in the tunneling transport properties of Fe/Ge/Co MTJs at low temperature an unambigous signature of the magnetic states of the two ferromagnetic electrodes. Again, it took a long time, until this eﬀect was also measured at room temperature. Moodera and Miyazaki published simultaneously [4, 5] results for Co/Al2 O3 /NiFe MTJs, where they observed a signiﬁcant change of around 15% of the tunneling resistance between the parallel (R↑↑ )and anti-parallel (R↑↓ ) alignment of the magnetizations of the electrodes, if one deﬁnes the TMR ratio as TMR = (Rmax − Rmin )/Rmin . Some years before, a similar eﬀect was found in all metallic structures such as Fe/Cr/Fe or Co/Cu/Co [6, 7] (GMR, Giant Magnetoresistance), which triggered enormous research activities on spin dependent transport properties. First attempts to form reliable MTJ’s used one relatively hard and one relatively soft ferromagnetic electrode. This leads typically to minor loops for a tunnel junction with about 10 μm × 10 μm size as shown in Fig. 6.1 [8]. Here, only the soft magnetic electrode switches its magnetization, whereas that of the hard electrode is supposed to stay constant. As can be already seen from Fig. 6.1, the dependence of the resistance on the external ﬁeld shows a curved crossover to the saturation values and therefore does not correspond to 100% remanence of the magnetization. Moreover, it exhibits small kinks which additionally are not reproducible. This simple hard / soft architecture turned out to be not stable with respect to magnetically cycling the soft electrode, because the domain splitting of the soft electrode causes large stray ﬁelds which induce a deterioration of the hard magnetic material [9]. The

Fig. 6.1. Resistance as a function of an external magnetic ﬁeld (minor loop) for a tunneling junction Co / Al2 O3 / Ni80 Fe20 from 1999 showing about 30% TMR amplitude

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same remains true, if the hard electrode is additionally stabilized by an antiferromagnetically RKKY-coupled [10, 11, 12] trilayer as, e.g., Co / Cu / Co [13]. Moreover, the Cu turned out to diﬀuse to the barrier between Co and Al2 O3 giving rise to a rapid decrease of the TMR while maintaing ‘good’ tunneling properties. For a further stabilization of the hard magnetic electrode, a direct contact with a natural antiferromagnet can be used. Due to the exchange interaction between antiferro- and ferromagnet, an annealing of this structure in an external ﬁeld leads to a unidirectional anisotropy in the ferromagnet [14] with a shift of its hysteresis of several 100 Oe. Thus, the nowadays standard stack design is a sequence of conductor- and seed layers followed by a natural anitferromagnet. The subsequent artiﬁcial anti-ferromagnet usually is a CoFe / Ru / CoFe trilayer in the second maximum of the anti-ferromagnetic coupling. This combination of exchange bias and anti-ferromagnetic coupling further both stabilizes the hard electrode and gives the possibilty to tailor its net magnetic moment. The tunneling barrier is ususally made by depositing an Al ﬁlm with a thickness between 0.6 nm and 1.6 nm which then is oxidized by a mild treatment in an oxygen- or an Ar-O plasma. From intensive investigations, it is known, that the energy of the ions impinging on the ﬁlm during this process should be kept well below 50 eV in order to avoid O implantation in the underlying ferromagnet. A considerable increase of this amplitude was reached by replacing the crystalline electrode deposited prior to the barrier layer by an amorphous CoFe-B ferromagnet [15]. For these amorphous Co-Fe-B electrodes, TMR values of more than 70% have been shown (see Fig. 6.2). The reason for this large TMR being not yet known causes some uncertainty concerning the maximum TMR reachable with conventional 3d ferromagnets. The next step in the preparation routine is then the initialization of the exchange bias which is usually done by annealing the ﬁlm stack in a magnetic ﬁelds at a typical temperature of about 300◦ C for some minutes.

Fig. 6.2. TMR major loop of a layer sequence Ta5 / Cu30 / Ta5 / Cu5 / Mn-Ir12 / Co-Fe-B3.5 / Al1.2 + oxidation Co-Fe-B3.5 / Ni-Fe3 / Ta5 / Cu20 / Au50 (subscripts: layer thickness in nm) with the amorphous ferromagnet Co-Fe-B (12% B)

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Considerable further improvement of the theoretical understanding of the spin dependence of the tunneling was obtained by taking into account the complex band structure of the insulator and the lattice match between ferromagnets and barrier material. In this contribution, we will highlight some of the major aspects of the development of MTJs. The discussion starts with some of the present and possible future applications of MTJs which put forward some challenges for these new devices such as amplitude and magnetic switching. In the next sections, possible answers to these challenges will be sketched.

6.2 Applications Magnetoresistance has been used for many years in conventional devices for sensing magnetic ﬁelds. These sensors use the so called Anisotropic Magnetoresistance (AMR) which produces a change of the resistance of the order of a few percent during magnetization switching. One mass market for these products was the application as read heads in hard disk drives. For storing information, magnetic devices also played a major role from the mid-1950s to around 1970 as magnetic core memory. Afterwards, however, the semiconductor technology produced rapid advances in down-scaling, performance and price so that – although devices such as bubble memories have been developed simultaneously – storing data by magnetics was essentially restricted to hard disks and tapes [16]. The discovery of the giant magnetoresistance in magnetic multilayers and sandwiches in 1986 [6, 7] and the development of the related spin valve device [17], then showed great promise for read-head sensors for hard disk drives and thus boosted the development of magnetoresistive devices. Spin valve based principles have been also proposed for storing information, but it was the discovery of a large tunneling magnetoresistance at room temperature [4, 5], which opened the ﬁeld of data storage in the so called Magnetic Random Access Memory (MRAM). For this purpose, ﬁrst a ‘cross point architecture’ was proposed. This consist just of two arrays of conducting wires running perpendicularly to each other on a chip. The wire arrays are separated by an insulating ﬁlm. At the crossing points, magnetic tunnel junctions connect the wires (see Fig. 6.3). If the high resistance state of the junctions is related with, e.g., logic 1 and

Fig. 6.3. A sketch of the crossed wire architecture of a random access memory with magnetic tunnel junctions

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the low resistive with logic 0, one bit of information can be stored in this device. It, however, turned out that the feasibility of this simple crossed wire architecture is questionable, because the selected path between two wires is shunted by the other crossing point MTJs. Thus, an additional select transistor has to be integrated for reliably selecting one MTJ. The downscaling of the MTJs to sizes used in nowadays microelectronics turned out to be a challenge for the development of the technology. The possible beneﬁts of non-volatility of the information, unrestricted read- and write access and potential down-scaling capabilities down to the 32 nm node, however, initiated a participation of several major companies of the electronic industry in the development of the MRAM. For the related downscaling, magnetostatic coupling by stray ﬁelds [18, 19], a homogeneous magnetization switching and the reproducible preparation of the insulating barrier turned out to be critical issues for the applicability of the MRAM. Whereas stray ﬁeld eﬀects can be minimized by tailoring the net magnetization of the antiferromagnetically coupled hard electrode, the homogeneity of the switching and the barrier resistance are challenging for both the ﬁlm deposition and the lithography. By using elliptically shaped MTJ cells with smooth edges, the distribution of the switching ﬁelds was successfully optimized. Also, the soft magnetic layer is often replaced by an additional antiferromagnetically coupled trilayer in order to drastically reduce the total stray ﬁeld of the device. The tunneling barriers can be deposited nowadays with a homogeniety of around 2% on 12 inch wafers [20], leading to a variance of the MTJ resistance of around 5%. In Fig. 6.4, we show the distribution of the resistance values of MTJs found both for the high resistive (antiparallel) and the low resistive state. A further application related with MRAM is the use of arrays of MTJs to perform logic operations [21, 22, 23]. The realization of a logic based on the same principles and technologies as the – potentially universal – memory MRAM is of large interest, because it opens the way to a common technology

Fig. 6.4. The distribution of the area resistance products of MTJs in the high- and low resistive states, respectively. The gap between these two distributions is large enough for a reliable operation in an MRAM matrix

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Fig. 6.5. Bridge conﬁguration of MTJs. The input/programming lines produce a magnetic ﬁeld that rotates the soft magnetic electrode’s magnetization, changing the output voltage Vout, which represents the logic function of the inputs

platform for both storing as well as computing data. Moreover, magnetic logic gate arrays can be ﬁeld programmable, leading to ﬁeld programmable logic gate arrays (FPGAs). Such FPGAs are programmable ‘on the ﬂy’ and, thus, open also a path to fast reconﬁgurable computing [24]. In Fig. 6.5, we show the principle of the operation of an FPGA based on four MTJs. Here, the input is represented by currents on two input lines, which can change the magnetization state of the MTJs soft electrodes. Two neighbouring lines are used to set the resistance states of the other two MTJs which ‘program’, i.e. deﬁne the value Vout obtained as logic function of the two inputs. For such logic operations, however, the requirements concerning the quality of the MTJs are more stringent than for the MRAM. An estimation of the yield of producing logic gate arrays with for MTJs shows [25], that at least a TMR eﬀect amplitude of 100% is necessary in order to enable a production with realistic tolerances concerning the variances of resistance and TMR. Thus, although these challenges have been successfully solved, there are prototype MRAM chips available only up to a density of 16 Mb in the 180 nm node technology [26]. One of the main reasons for this lack in density is the current need to perform the magnetization switching on the chip: At the location of the MTJ two perpendicular magnetic ﬁelds are applied by running current through perpendicular metallization wires in two metallization levels below the MTJ level. The two related ﬁeld pulses, one in hard- and one in easy axis direction of the soft electrode then switch the magnetization, because the sum of the ﬁeld vectors is large enough to overcome the Stoner-Wohlfarth astroid [18]. The currents needed for this are, however, in the range of some mA. Thus both the wires in the metallization levels below the MTJ as well as the transistors for switching these large currents have to be much larger than the mimimum feature size in these cells.

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Fig. 6.6. Principle of the magnetic biochip (left): Biotin marked DNA-molecules hybridize with complementary strands attached to the surface, streptavidin coated beads then bind to the biotin. By applying a magnetic ﬁeld perpendicular to the surface, only the in plane components of the dipole stray ﬁeld of the beads are detected. A TMR sensor surface covered by magnetic beads (middle) and the TMR signal measured during applying a magnetic ﬁeld perpendicular to the sensor surface and an in plane ﬁeld which is close to the switching ﬁeld of the soft electrode (right) [27]

As last application related example, we now turn to a completely diﬀerent ﬁeld: In biotechnology and medical applications, molecules like DNA or proteins are frequently marked by magnetic spheres called ‘beads’ [28]. This opens the possibility to measure the presence or absence of these biomolecules by detecting the magnetic beads with an MTJ. Baselt et al. [29] already described this technique using Giant Magnetoresistance sensors. Figure 6.6 shows the principle of this method and the result of the measurement of diﬀerent bead concentrations with a 100 μm wide TMR cell [27]. As can be seen in Fig. 6.6, reasonable signals as in dependence of the perpendicular ﬁeld are obtained at a surface coverage of only a few percent, if an in plane ﬁeld is additionally applied which brings the soft electrode close to switching. Comparisons with the established optical method of marking with ﬂuorescent molecules showed, that the magnetic biosensor can be more sensitive at low concentrations of the analyte molecule, which is the most interesting area of application. The detection of such magnetic markers, however, requires both a high sensitivity and a non-hysteretic answer of the sensor. Thus again very high TMR amplitudes are needed, but now in devices responding in an unambiguous manner to an external magnetic ﬁeld. At the moment, the use of TMR sensors with very soft magnetic free layers and a strong shape anisotropy seems to be the most promising approach. Work towards a single molecule detection using such systems is currently in progress. In summary, these and other applications puts forward challenges for the further development of magnetic tunnel junctions. We have used the MRAM, magnetic logic and a magnetic biosensor to point out the major ﬁelds, where the properties of the MTJs need improvements: • The TMR ratio of around 75% obtained for standard Al2 O3 based MTJs is enough to prepare some working FPGAs in the laboratory, but it is not

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satisfying the needs of a high yield production. Here, the lower limit is at about 100% TMR. • The currents in the clock- and wordlines needed to switch the tunneling cells by crossed ﬁeld pulses is too large and does not scale down with the size of the MTJs. This is regarded to be the most severe obstacle for the MRAM and applications in logic, because a successful introduction of a product in this ﬁeld must have perspectives well below around 50 nm minimum feature size. • If the sensors which can be realized with magnetic tunneling cells are not only used as counters then they also need a higher sensitivity over a relatively broad range of external ﬁelds. Moreover, for most sensor applications, the hysteretic answer of a TMR cell to an external ﬁeld is a disadvantage, because it limits the accuracy of the ﬁeld detection.

6.3 Current Induced Magnetization Switching As addressed in the foregoing section, the scalability of the MTJs is a crucial point for their successful introduction in microelectronic products. Especially the traditional switching mechanism for the soft layer’s magnetization by two perpendicular ﬁeld pulses is not scalable, because the current creating the ﬁelds does not go down as the TMR cell size shrinks. The solution for this obstacle could be current induced magnetization switching [30, 31] which was recently demonstrated by several groups [31]. In this technique, a spin polarized current is driven into a ferromagnetic thin ﬁlm. If the spin of the incoming electrons and the magnetization of the ferromagnet is not alligned, these two systems will exchange torque, which can above a critical current density lead to a ﬂip of the magnetization of the ferromagnet. This method has a high potential in magnetic tunneling cells, because the current needed to obtain the switching scales down in the same way as the TMR cell shrinks. Mandatory, however, are fast switching and critical current densities lower than both the breakdown properties of the tunnel junctions as well as the electromigration threshold which lies between 106 A/cm2 and 107 A/cm2 . The torque exterted by the spin polarized current on the soft electrode’s magnetization MsF is included in its equation of motion (Landau-LifshitzGilbert equation) by an additional term introduced by Slonzewski [30]. 1 dMSF α = MSF × [H − ( M F × (H + HS )] γ dt | MSF | S ε HS = JA 2eMSF α JCr ∝

αMSF tF (H + HK + 2πMS ) . ε

(6.1) (6.2) (6.3)

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The classical equation of motion (6.1) describes both the precession of the magnetization MSF upon applying a magnetic ﬁeld H and the momentum transfer from a spin polarized electrical current with a density J, which occurs in the LLG equation (6.1) as an additional ﬁeld term HS (6.2). If a critical current density Jcr is applied to the device at zero temperature (6.3), the soft magnetization layer should switch, i.e., MSF changes its direction. Further important parameters are the gyromagnetic ratio γ, the damping parameter α and the spin torque eﬃciency ε which contains the microscopic description of the transfer of momentum between the incoming spin polarized electrons and the magnetization of the soft layer. If directly related with the TMR eﬀect, ε should strongly decrease with decreasing ΔR; for large TMR values, this decrease should be nearly linearly [31]. For common materials and tunnel junctions,this results in some 106 A/cm2 –107 A/cm2 critical current density of the injected electrons. In order to prepare samples capable of current induced magnetization switching (CIMS), however, both a large TMR as well as a very low resistive tunneling barrier are required. For these purposes, we prepared MgO based [32, 33] tunnel junctions with a stack sequence of Pt-Mn20 nm / Co-Fe2 nm / Ru0.75 nm / Co-Fe-B2 nm / MgO1.3 nm / Co-Fe-B3 nm both with low and a high area resistance product (6.5 Ωμm2 and 51 Ωμm2 , respectively) and a tunneling magnetoresistance ratio between 120% and 140% by sputter deposition in a Singulus TIMARIS PVD tool. In Fig. 6.7, we show a CIMS curves of the low and high resistive tunnel junctions. This is recorded by applying ﬁrst a voltage (i.e. current-) pulse of 100 msec and then measuring the sample’s resistance at a low bias voltage (20 mV). For the low resistive junctions, switching occurs at a bias voltage between 0.5 V and 0.7 V at a current density of around 107 A/cm2 , which is well within the expected range but too large as compared with the electromigration threshold. For the samples with the larger area resistance product of

Fig. 6.7. Magnetoresistance of a 100 nm × 200 nm large tunneling junction as a function of the amplitude of the current pulses applied prior to the individual resistance measurements for the low and high resistive (area resistance product: 6.5 Ωμm2 and 51 Ωμm2 ) MgO based magnetic tunnel junctions

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51 Ωμm2 , where switching occurs between 0.8 V and 1.2 V bias pulse voltage, the shape of the switching curves are very similar. A siginiﬁcant diﬀerence, however, occurs for the current density needed to switch the magnetization. Whereas for the low resistive samples around 107 A/cm2 are necessary, the high resistive tunnel junctions already switch between 1 × 106 A/cm2 and 2 × 106 A/cm2 , i.e., at a value about ﬁve times lower. During switching, the role of thermal heating also needs to be considered, because both the coercive ﬁeld as well as the saturation magnetization are temperature dependent. In this case, however, the power dissipated by the current is considerably lower for the high resistive samples. Thus, although the temperature during current pulsing is not reliably known, it should be larger for the low resistive samples and, therefore, these results cannot be explained by the diﬀerence in the power dissipation in the two types of samples. In terms of the theoretical description of current induced magnetization switching this means, that the spin torque eﬃciency ε in (6.3) is at least 5 times larger for the spin current injected at 1 V bias compared to that at 0.6 V. Whereas the physical origin for this behavior is not yet known, this eﬀect, nevertheless, opens a way to further lower the current density needed for switching such MTJs. If it is possible to enhance the scattering rate by tailoring the band structures of the involved materials, it would enable nanoscale MTJ electronics.

6.4 Heusler Alloys 6.4.1 Introduction to Highly Spin Polarized Materials Two complementary avenues are currently followed so as to realize the vision of spin electronics: to employ MTJs with TMR-eﬀect amplitudes of up to several thousand percent. One of which is determined by evaluating diﬀerent tunnel barrier materials such as Al2 O3 and MgO and the other is aiming at 100% spin polarized magnetic electrodes in MTJs. Hence, one of the crucial issues today in the development of spin electronic devices is the search for new materials that exhibit large carrier spin polarization [34]. Potential candidates include half-metallic ferromagnets oxides [35] or ferromagnetic half-metals such as Heusler alloys [36]. Characteristic for these magnetic materials named ‘half-metallic’ [37] is their unusual band structure with only one spin direction being metallic. Electrons of the opposite spin have a gap in their density of states (DOS) at the Fermi level (EF ) and hence are insulating. Consequently with only one spin band present at EF half-metallic ferromagnets are 100% spin-polarized and allow the transport of only one spin carrier across an interface or tunnel barrier into an adjacent material. Therefore, the spectacular theoretical prediction of 100% spin polarization in an entire class of materials, the half-Heusler XYZ [37, 38, 39, 40] as well as full-Heusler X2 Y Z [41, 42, 43, 44] alloys is currently the driving force for

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evaluating the potential of MTJs with at least one magnetic electrode made of a Heusler alloy. However, the earlier experimental eﬀorts to realize MTJs did not show any evidence for a true enhancement of the TMR-eﬀect amplitude when using Heusler alloys as magnetic electrodes. The spin polarization of the half-Heusler compound NiMnSb which has been integrated in a MTJ [45] was measured to be 25% at 4.2 K and is related to a TMR-eﬀect amplitude of 19.5%. The corresponding TMR value at RT was found to be 9% only. Using a full-Heusler alloy of type Co2 Cr0.6 Fe0.4 Al as one of the magnetic electrodes in a MTJ slightly larger TMR values of 16% at RT and 26% at 4.2 K could be realized [46]. Nevertheless, the list of promising half- and full-Heusler alloys is long [40, 44]. One interesting candidate is Co2 MnSi which is also characterized by a 100% spin polarization as was predicted [41] from band structure calculations. In addition, Co2 MnSi possesses with TC = 985 K [47] a large Curie temperature identifying it to be an excellent candidate for technological applications; materials with large Curie temperatures should have a high remnant magnetization at RT. Recently, the spin polarization of Co2 MnSi was determined to be 54% at 4.2 K using point contact Andreev reﬂection spectroscopy [48] which is fairly comparable with the spin polarization of the 3d-based magnetic elements or their alloys, i.e., at RT a maximum spin polarization of 44% was found for Co50 Fe50 , whereas that of Ni80 Fe20 reached 53% at 10 K [49]. By that time the potential of Co2 MnSi integrated as one ferromagnetic electrode in technological relevant magnetic tunnel junctions had yet to be proven. To cope with this challenging task magnetic tunnel junctions had been fabricated [50] consisting of one Co70 Fe30 and one Co2 MnSi electrode separated by a very thin insulating AlOx barrier so as to determine spin polarization of Co2 MnSi and the resulting TMR-eﬀect amplitude. The TMR-eﬀect amplitude achieved in these MTJs is 94.6% at 20 K [51] applying a bias voltage of 1 mV. This corresponds to a 65.5% spin polarization of Co2 MnSi which clearly exceeds that of the 3d-based magnetic elements or their alloys but is also well below the predicted 100%. These results have triggered enormous research activities on the integration of Co2 MnSi as magnetic electrodes in MTJs. Today we are looking on an impressive development [52, 53] which is crowned by the TMR ratio of 570% at 2 K [54], which is the largest value so far reported in combination with an amorphous AlOx tunneling barrier. 6.4.2 Structure and Ordering The key to achieve large TMR-eﬀect amplitudes in MTJs with integrated magnetic Co2 MnSi electrodes is to induce a high degree of atomic order within these Co2 MnSi layers. From a crystallographic point of view, the evolution of the long-range order parameter as a function of preparation conditions such as annealing time and temperature can be determined by monitoring the intensity change of superlattice reﬂections in x-ray or neutron or transmission

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electron diﬀraction patterns of thin polycrystalline or epitaxial Co2 MnSi layers. Thus, the question of interest to be answered in the following is about the experimental possibility to employ these diﬀraction techniques so as to determine the disorder-order transition in Co2 MnSi. The unit cell of Co2 MnSi [55] contains 16 atoms which are located on four interpenetrating fcc sublattices A, B, C and D where each of the sublattices is occupied by atoms of one element: A by Co, B by Mn, C by Co and D by Si when fully ordered. This arrangement corresponds to the L21 structure type giving rise to non-zero Bragg reﬂections only when the Miller indices of the corresponding scattering planes are either all even or all odd. The planes with all even Miller indices can further be divided in those for which (h + k + l)/2 is odd and those for which (h + k + l)/2 is even. The intensities of these allowed reﬂections can be calculated by the squares of the relevant structure factors as given by [56]: h, k, l all odd : for example F 2 (111) = 16[(fA − fC )2 + (fB − fD )2 ] h+k+l = 2n + 1 for example : F 2 (200) = 16(fA − fB + fC − fD )2 (6.4) 2 h+k+l = 2n for example : F 2 (220) = 16(fA + fB + fC + fD )2 , 2 where the fi are the averaged atomic scattering factors for the four sublattices sites. The squared structure factor F 2 (220) is characterized by the sum of all averaged atomic scattering factors for the four sublattices sites. Hence, even in the presence of complete disorder the resulting reﬂection intensity will remain unchanged and identiﬁes this class of scattering planes as the principal reﬂections. In contrast, the intensities of the two other classes of scattering planes F 2 (111) and F 2 (200) are very sensitive to any ordering / disordering process whereby one atom of one sublattice is interchanged by one atom of another sublattice and identify the superlattice reﬂections. Since Co2 MnSi is a ternary alloy it is not possible to describe the state of order by one single order parameter as is usually done or binary alloyed phases. On the contrary, the possible ways of disorder have to be associated with a certain disorder parameter α which deﬁnes the fraction of either x or y atoms not on the correct sublattice. The four sublattices present in Co2 MnSi allow six most probable types of disorder as is summarized in Fig. 6.8. The same relations are also valid for the iso-structural Co2 FeSi when Mn is replaced by Fe. Inserting these fi into (6.4) enables to calculate the intensity change of the superlattice reﬂections (111) and (200) for all six probable ways of disorder as a function of the disorder parameter α. The results are summarized in Fig. 6.9. As it can be seen the unique (111) and (200) intensity changes with increasing atomic disorder can experimentally be used as a ﬁngerprint so as to determine which way the atomic disorder is following upon annealing, despite the fact that way 2 and 4 are identical. Nevertheless, to quantify the state of atomic ordering using X-ray or electron diﬀraction is quite demanding since

6 Magnetic Tunnel Junctions Scattering factor fA of sublattice A

Scattering factor fA of sublattice B

Scattering factor fA of sublattice C

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f Mn

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f Co

(1 − α ) f Mn + α f Si

f Co

(1 − α ) f Si + α f Mn

Way of disorder:

(4) Co(A)Mn(B)-Co(C) Interchange

(1 −

(5) Co(A)Si(D)-Co(C) Interchange

(1 −

(6) random disorder in all sublattices

α 2

) f Co +

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) f Co +

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2 α α α α 2 (1 − α ) f Co + ( f Mn + f Si ) (1 − α ) f Mn + (2 f Co + f Si ) (1 − α ) f Co + ( f Mn + f Si ) (1 − α ) f Si + (2 f Co + f Mn ) 3 3 3 3 3 3

Fig. 6.8. Scattering factors of the four sublattices A, B, C and D after [56] taken from [57]. fCo , fM n and fSi are the corresponding atomic scattering factors

Fig. 6.9. Summary of the evolution of the disorder parameters of the six most probable ways to disorder the L21 structure of Co2 MnSi taken from [57]. The intensities are normalized to the intensity value at αi = 0 the state of full atomic order

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the crystallographic texture of the Co2 MnSi layer imposes an additional constrain on these techniques. However, employing electron diﬀraction patterns taken from cross section samples in HRTEM would beneﬁt from an overall textured Co2 MnSi layer with (100) orientation in growth direction. This ensures a high probability to ﬁnd columnar Co2 MnSi grains close to a [58] type of zone axis which in turn contains strong superstructure reﬂection as is predicted by calculation and experimentally demonstrated in Fig. 6.10. Using X-ray diﬀraction patterns would also require textured Co2 MnSi layers with (100) orientation in growth direction so as to compare the intensity ratios of the (h00) family type of reﬂections. To extract information about the atomic ordering measured X-ray diﬀraction patterns could be ﬁtted to atomic ordering models taking into account all six possible ways of disorder discussed above by performing a Rietveld analysis. However, both techniques would reveal only qualitative results unless a Co2 MnSi single crystal can be used as a reference. Nevertheless, taking all these obstacles towards uncovering the ordering mechanism in these full Heusler alloys is worth while and will deﬁnitely contribute to a better understanding and hopefully tuning of the microstructural TMR-properties relationships in the near future. 6.4.3 Transport Properties of Heusler Alloy Based Magnetic Tunnel Junctions In this section the transport properties as well as chemical and magnetic interface properties of MTJs with a Heusler alloy electrode and Al-O barrier are discussed. A large TMR eﬀect is anticipated for many applications, whereas

Fig. 6.10. HRTEM micrograph of a Co2 MnSi grain in [58] zone axis orientation: (a) Calculated electron [58] diﬀraction pattern. (b) Live FFT of the HRTEM micrograph. Clearly visible are the superstructure reﬂection of [2] type family

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the TMR amplitude is connected with the Julli´ere spin polarization [3] Pa,b of electrode a and b: TMR = 2Pa Pb /[1 − Pa Pb ]. The ultimate magnetoelectronic material should have a gap in the minority (or majority) electron density of states at the Fermi energy EF and thus 100% spin polarization. These materials are called ferromagnetic half-metals. This property has been predicted theoretically for some half and full Heusler compounds [37, 41, 44, 59]. Starting in 1999 with Tanaka et al. [45] (NiMnSb), diﬀerent Heusler compounds have been implemented in magnetic tunnel junctions, whereas halfmetallicity has not been demonstrated so far in these structures. The largest reported spinpolarizations are found for Co2 Cr1−x Fex Al [60] and Co2 MnX (X=Si, Al or Ge) [61, 62, 63] single layer electrodes and very recently for [Co2 MnSi / Co2 FeSi]10x multilayer electrodes [64]. To achieve high eﬀective spin-polarization for Co2 MnSi based MTJs an insitu annealing procedure applied after barrier formation and before depositing the top electrode of the junction has been developed [50]. The in-situ annealing forces the atomic ordering of the Co2 MnSi thin ﬁlms and, hence, increases their spin-polarization. This preparation techniques has also been applied successfully to MTJs with Co2 FeSi single layer and [Co2 MnSi / Co2 FeSi]10x multilayer electrodes [64]. These MTJs will be addressed in more detail in the following: We will compare the temperature and bias voltage dependent TMR of our Heusler alloy based junctions with more conventional Co62 Fe26 B12 / AlOx / Co62 Fe26 B12 and Co70 Fe30 / AlOx / Ni80 Fe20 MTJs to show up the speciﬁc characteristics of the Heusler alloy based MTJs. The inﬂuence of the temperature dependent magnetic moments at the Heusler alloy / barrier interfaces as well as electronic band structure eﬀects will be discussed. The experimental results for the Co2 MnSi based MTJs will also be compared to bandstructure calculations for perfectly L21 -ordered Co2 MnSi bulk material (lattice constant 0.565 nm) obtained using the SPR-KKR program package [65]. The MTJs were deposited at room temperature by DC- and RF-magnetron sputteringfrom stoichiometric targets on thermally oxidized Si(100) wafers. On a 40 nm thick V buﬀer we deposited a magnetically soft electrode (Co2 MnSi100 nm and Co2 FeSi100 nm single layers or a [Co2 MnSi5 nm / Co2 FeSi5 nm ]x10 multilayer). The subsequently deposited thin Al ﬁlm with a typical thickness of 1.5 nm was plasma oxidized to form a tunnel barrier. To achieve high spin-polarization this bottom electrode including the barrier was in-situ annealed for 40–60 min. at 450◦C (Co2 MnSi) and 380◦ C (Co2 FeSi and [Co2 MnSi5 nm / Co2 FeSi5 nm ]x10 ), respectively. Afterwards, the barrier was shortly oxidized again to remove contamination of the barrier surface and covered with the top electrode (Co70 Fe30 5 nm ). The Co-Fe is pinned by Mn83 Ir17 10 nm and ﬁnally covered by the upper conduction leads (Ta / Cu / Ta / Au). Then, the layer stacks were ex-situ vacuum annealed at 275◦ C in a magnetic ﬁeld of 0.1 T to set the exchange bias of the pinned electrode. The junctions were patterned by optical lithography and ion beam etching. The transport properties of the junctions were measured as a function of magnetic ﬁeld, bias voltage and temperature by conventional two-probe DC technique.

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For probing the structural and magnetic properties of the Heusler alloy based electrodes and at the Heusler alloy / AlOx interfaces, half junctions without top electrode were also fabricated and investigated by an Alternating Gradient Field Magnetometer (AGM), x-ray absorption spectroscopy (XAS), x-ray magnetic circular dichroism (XMCD) and x-ray photoemission spectroscopy (XPS). Temperature dependent XAS and XMCD were performed at beamline 4.0.2 and 7.3.1.1 of the Advanced Light Source, Berkeley, USA. The Co-, Fe- and Mn-L edges and the Si-K edge were investigated. Surfacesensitive total electron yield (TEY) [66] as well as bulk-sensitive ﬂuorescence yield (FY) spectra [66] were recorded. Now, we give an overview on the characteristic transport properties of MTJs with Heusler alloy electrode. The low temperature TMR majorloops of Co2 FeSi single layer and [Co2 MnSi / Co2 FeSi]×10 multilayer based MTJs and their room temperature TMR amplitude as a function of Al thickness is shown in Fig. 6.11. An TMR of up to 114% has been observed corresponding to a spin polarization 0.74 [67]. The low temperature TMR majorloops of Co2 MnSi based MTJs prepared with in-situ at 450◦ C (‘CMS100’) and without in-situ annealing (‘CMS100ag’) are shown in Fig. 6.12a. Sample ‘CMS100’ shows up to 95% TMR at 20 K /1mV and a sharp magnetization reversal of the Co2 MnSi around zero magnetic ﬁeld. Without in-situ annealing (‘CMS100ag’) the TMR is strongly reduced to maximal 1.3% at 16 K / 20 mV (±2000Oe ﬁeld range). The Co2 MnSi electrode shows superparamagnetic behavior and is not saturated at ±2000Oe. The pinned Co-Fe top electrode of both samples shows an exchange bias of about –650 Oe. The TMR amplitude of 95% found for ‘CMS100’ corresponds to eﬀective spin polarizations of PCo2MnSi = 66% assuming PCo70F e30 = 49% [61]. The low temperature bias voltage dependence (±500 mV range) of the junctions is shown in Fig. 6.13a. ‘CMS100’ shows a considerably stronger bias voltage dependence than more conventional Co-FeB / Al-O / Co-Fe-B (‘CoFeB’) and Co-Fe / Al-O / Ni-Fe (‘CoFe-NiFe’) MTJs with a maximum TMR amplitude of 114% (‘CoFeB’, PCo62F e26B12 = 60%) [68] and 71% (‘CoFe-NiFe’) [68, 69], respectively. The same holds for the tem60 ML

90 60 30

SL

(a)

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0 -2 -1 0 1 2 magnetic field [kOe]

multi layer electrode

40 20 single layer electrode

0 0.5

1.0 1.5 2.0 2.5 Al thickness [nm]

Fig. 6.11. (a) Low temperature TMR majorloops of Co2 FeSi single layer (“SL”, black curve, 16 K) and [Co2 MnSi / Co2 FeSi]×10 multilayer (“ML”, red curve, 17 K) based MTJs measured with 10 mV bias for 1.5 nm thick Al. (b) Room temperature TMR amplitude (10 mV bias) of MTJs with Co2 FeSi single and [Co2 MnSi / Co2 FeSi]×10 multilayer electrode as a function of Al thickness. Taken from [67]

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100

TMR [%]

80

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60 40 "CMS100ag" (x30)

20 0 -2

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1

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Fig. 6.12. TMR majorloops for MTJ ‘CMS100’ (20 K / 1mV) and ‘CMS100ag’ (16 K / 20 mV). Taken from [68]

perature dependence of the TMR (Fig. 6.13b), which is identical for MTJs ‘Co-Fe-B’ and ‘MTJ-NiFe’, although their maximum TMR amplitudes are signiﬁcantly diﬀerent. Especially, their TMR(T)-curves are concave. In contrast, the Co2 MnSi based junction show a convex TMR(T)-dependence. A convex TMR(T) dependence is also found by Oogane et al. [62] for junctions with epitaxial Co2 MnSi electrodes. The Co2 MnSi based MTJs prepared without insitu annealing (‘CMS100ag’) show no TMR above 200 K. Especially, a linear bias voltage dependence of the TMR is observed for ‘CMS100’, ‘CoFeB’ and ‘CoFe-NiFe’ in the bias voltage (V) range of a few 10 mV. Remarkably, the in-situ annealed junctions ‘CMS100’ show an inversed TMR of up to –6.3% at room temperature for large negative bias voltage, i.e., when the electrons are tunneling from the Co-Fe into the Co2 MnSi electrode (see Fig. 6.13c). For positive bias voltage, when the electrons are tunneling from Co2 MnSi into CoFe, the TMR remains positive. This inversion of the TMR is not observed for the other three junction types ‘CMS100ag’, ‘CoFe-NiFe’ and ‘CoFeB’. Their monotonic decrease of the positive TMR with increasing bias voltage shown in Fig. 6.13 for the ±500 mV bias voltage range just goes on up to the dielectric breakdown of the junctions. E.g., the Co-Fe /AlOx / Ni-Fe junctions [69] ‘MTJ-NiFe’ still show a room temperature TMR of about +10% for ± 1200 mV bias voltage. For ‘CMS100ag’ junctions, which were not in-situ annealed during layer deposition, the TMR at 16 K is +0.2% for ± 1200 mV bias voltage. The inversion of the TMR turned out to be very characteristic feature for Heusler alloy based MTJs. It has been also observed for junctions with Co2 FeSi single layer and Co2 MnSi/Co2 FeSi multilayer electrode [64] and for Co2 MnGe / MgO / Co-Fe junctions [63]. Before these results can be interpreted, we have to focus on the chemical, magnetic and electronic properties at the Heusler alloy / Al-O interface. The case of a Co2 MnSi / AlOx interface will be discussed exemplarily: The element speciﬁc properties of Co, Mn and Si in ‘half’ junctions corresponding to the full MTJ stacks ‘CMS100’ and ‘CMS100ag’ were probed by x-ray absorption spectroscopy. Bulk sensitive FY spectra of Co and Mn are shown in Fig. 6.14. Pronounced XANES oscillations indicating the atomic ordering throughout the Co2 MnSi layers are found for the annealed sample ‘CMS100’

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bias voltage [V] Fig. 6.13. (a) Typical bias voltage dependence at low temperature for ‘CMS100’ (measured at 20 K), and ‘CoFeB’ (30 K) and for ‘MTJ-NiFe’ (10 K). All data is normalized to the maximum TMR at low bias. The inset shows the TMR amplitude for ‘CMS100ag’ (16 K) (b) Typical normalized temperature dependence of ‘CMS100’ (measured at 10 mV), ‘CoFeB’ (10 mV), ‘MTJ-NiFe’ (10 mV) and ‘CMS100ag’ (20 mV). (c) Typical bias voltage dependence in the ±1500 mV bias voltage range of a ‘CMS100’ junction, measured at room temperature. The insets show TMR minor loops measured at +10 mV and –1300 mV, when only the magnetization of the soft Co2 MnSi electrode is switched by the external magnetic ﬁeld. The magnetizations of Co2 MnSi and Co-Fe are aligned parallel (antiparallel) for magnetic ﬁelds of –60 Oe (+60 Oe). Taken from [68]

(see Fig. 6.14). Additionally, this sample shows shoulders at about 4 eV above the maximum intensities of the Mn- and Co-L2,3 resonances (marked by ‘A’ in Fig. 6.14) which are not present for ‘CMS100ag’. These shoulders reﬂect small peaks in the density of unoccupied d-like states [65] of the ordered Co2 MnSi. The TEY-spectra probing the Co2 MnSi / AlOx interface show these ﬁngerprints of the atomic ordering in the annealed sample ‘CMS100’ only for Co (see arrows in Fig. 6.15b). MnO identiﬁed by its characteristic XAS multiplett

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Fig. 6.14. XAS-FY spectra at (a) Mn- and (b) Co-L edge of ‘CMS100ag’ and ‘CMS100’. The spectra were measured at 15 K. The XANES oscillations found for the annealed ‘CMS100’ are marked by arrows and are also visible at 300 K. The intensities of the L2,3 -resonances are reduced because of saturation eﬀects [66]. Taken from [68]

structure [70] (see peaks ‘A1’–‘B2’ in Fig. 6.15c) is present at the interface for both samples and masks the ﬁngerprints of ordering for interfacial Mn in sample ‘CMS100’. Although the saturation eﬀects change the shape of the bulk sensitive Mn FY-spectra shown in Fig. 6.14a, it is obvious that the MnO multiplett structure is not present in the Mn FY-spectra. In addition to the

data fit

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665

670

Fig. 6.15. (a) Magnetization loop of the as grown sample ‘CMS100ag’ measured at 15 K (open circles). The black line is a ﬁt by the Langevin function (see text). (b) X-ray absorption spectra and magnetic circular dichroism in TEY-mode at the CoL edge of ‘CMS100’ measured at 15 K. (c) X-ray absorption spectra and magnetic circular dichroism in TEY-mode at the Mn-L edge of ‘CMS100’ measured at 15 K. The red curve δ(hν) is the diﬀerence of the Mn XMCD asymmetries of ‘CMS10’ measured at 15 K and 300 K, respectively. The inset shows diﬀerences of the Co XMCD asymmetry at 15 K between the ordered and disordered sample ‘CMS100’ and ‘CMS100ag’, the photon energy is deﬁned with respect to the energy position of maximum intensity of the Co-L3 resonance. Taken from [68]

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interfacial MnO we found some SiO2 at the interface in the Si-K edge XAS spectra. The temperature dependence of the interfacial magnetic moments of Co and Mn is adressed now. For these atoms the magnetic moments are governed by the 3d-electrons which can be probed by x-ray absorption at the L2,3 -edges. According to the sum rules analysis [71] the XMCD asymmetries measured at 15 K correspond to a ratio of the spin magnetic moments mMn for Mn and s Mn Co mCo for Co of m /m = 1.4–2.1. The large uncertainty of this value results s s s from the not precisely known factor to correct the jj-mixing for Mn [72]. Co However, this value is signiﬁcantly smaller than mMn s /ms =2.9 as expected from bandstructure calculations [44]. The reduced magnetic moment ratio is reasonable, because of the Mn-O formation at the interface. Compared to ‘CMS100’ the XMCD asymmetries of the as grown sample ‘CMS100ag’ are by a factor of 7 (Co) and 9 (Mn) smaller at 15 K (applied ﬁelds ±0.55 T), accordingly their magnetic moments are strongly reduced. Furthermore, the magnetization loop (Fig. 6.15a) calculated [73] from its Co and Mn XMCD asymmetries in TEY mode shows a typical superparamagnetic behavior. Compared to the 15 K values the XMCD asymmetries are reduced at 300 K by a factor of 14.0±1.3 for Co and 13.5±0.9 for Mn, respectively, which results from the intrinsic temperature dependence of the superparamagnetic clusters and from an additional reduction of mean magnetic moments of each cluster by a factor of 1.8. In contrast, the temperature dependence of the interfacial magnetic moments of the ordered sample ‘CMS100’ is much smaller than for ‘CMS100ag’, their magnetic moments of Co and Mn are only reduced to 94% and 90% of the low temperature values. Assuming a Bloch-like temperature dependence of the interfacial magnetic moments, m(T )/m(0) = (1−αT 3/2 ), this corresponds to spin wave parameters α of 1.17 × 10−5 K−3/2 for Co and 1.95 × 10−5 K−3/2 for Mn. These values are 4 (Co) and 6.7 (Mn) times larger than the spin wave parameter αbulk = 2.81 ± 0.21 × 10−6 K−3/2 of bulk Co2 MnSi measured by Ritchie et al. [74]. Whereas the majority of the interfacial Mn2+ ions remains paramagnetic or becomes antiferromagnetically ordered at low temperature, the generally larger temperature dependence of the Mn magnetic moment can be attributed to the metallic Mn in not perfectly ordered Co2 MnSi and might be a result of the antiferromagnetic Mn-Mn coupling in the alloy. Compared to ‘CMS100’, the temperature dependence of the interfacial magnetic moments of sample ‘CoFeB’ is weaker. Co and Fe magnetic moments at the Co-Fe-B / AlOx interface are only reduced to 98% and 97% of the low temperature moments, when the temperature increases from 25 K to 300 K. Therefore, thermal magnon excitation at the barrier–electrode interface is much more pronounced in the Co2 MnSi based junctions than in our MTJs with Co-Fe-B electrodes. Finally, we will discuss the observed transport properties on the base of the chemical and magnetic interface properties. The superparamagnetic behavior

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of the Co2 MnSi electrode found in the major loops of ‘CMS100ag’ junctions (Fig. 6.12) is consistent with the temperature dependent magnetic properties at the Co2 MnSi / AlOx interface. The superparamagnetism of the electrode can reduce its interfacial magnetization at ±2 kOe with raising temperature so strongly, that the TMR vanishes nearly at room temperature. In general, the temperature and bias voltage dependent transport properties of MTJs are rather complex because of the variety of diﬀerent contributions to the total conductance like direct tunneling including bandstructure eﬀects [75, 76] and the shape of the tunnel barrier [77], thermally induced changes of the interfacial magnetization [78], magnon [79, 80, 81] and phonon assisted tunneling [80], unpolarized conductance via defect states in the barrier [78, 82], spin scattering on paramagnetic ions in the barrier[83, 84] and possible spin ﬂip scattering on interfacial antiferromagnons [84] in the MnO. Because of the MnO formation at the lower barrier interface spin scattering on paramagnetic Mn2+ ions in the barrier [83, 84] has to be taken into account. For the small applied magnetic ﬁelds in our transport measurements (≤ 0.2 T) the Zeeman energy of paramagnetic Mn-ions is much smaller than the smallest thermal energy (kB T≈1 meV at 10 K) and bias voltage (1 mV). Accordingly, spin scattering on paramagnetic ions is a quasi-elastic process on our energy scale and should not inﬂuence the bias voltage or temperature dependence [84]. However, this process can reduce the eﬀective spin polarization of the Co2 MnSi, which is indeed PCo2MnSi = 66% for the in-situ annealed ‘CMS100’ junctions instead of theoretically expected 100%. The same holds for the unpolarized conductance via one defect state in the barrier [82], which can partly reduce the eﬀective polarization but does not have an inﬂuence on the bias voltage or temperature dependence. Please note, that the partial oxidation of Mn and Si at the barrier interface should disturb the ordering process of the interfacial Co2 MnSi during the in-situ annealing which should also contribute to the reduction of the effective spin-polarization of the Co2 MnSi. If MnO orders antiferromagnetically at low temperature, spin ﬂip scattering on interfacial antiferromagnons should result in a decrease of the TMR with the square of the bias voltage in ﬁrstorder approximation as shown by Guinea [84]. Phonon assisted tunneling [80] and unpolarized hopping conductance via two or more defect states [82] should also result in a non-linear bias voltage dependence of the TMR. In contrast, a linear bias voltage dependence as observed for ‘CMS100’ in the bias voltage (V) range of a few 10 mV has been predicted for magnon assisted tunneling [80, 81] assuming an energy independent band structure around the Fermi energy and surface magnon excitation. Compared with ‘CoFeB’ and ‘CoFeNiFe’ junctions TMR(V) is considerably larger, which would imply, that the magnon assisted tunneling is more pronounced in ‘CMS100’ junctions. That is consistent with the temperature dependent XAS and XMCD investigations which showed, that thermal magnon excitation is considerably larger at the Co2 MnSi / AlOx than at the Co-Fe-B / AlOx interface. Additionally to the linear bias voltage dependence, the convex TMR temperature dependence of ‘CMS100’ junctions is a characteristic feature for magnon-assisted tunnel-

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ing as proposed by Han et al. [81]. Taking only direct and magnon-assisted tunneling into account the resulting selfconsistent ﬁts of the bias voltage dependence of the TMR and the temperature dependence of the area resistance product for parallel (RP ) and antiparallel (RAP ) alignment are very good. Especially, for antiparallel alignment of the Co2 MnSi and Co-Fe electrodes, when magnon-assisted tunneling would be most important. Therefore, from the comparison of the transport properties and the temperature-dependent interfacial magnetic moments we suggest, that the TMR temperature and (low) bias voltage dependence of our in-situ annealed ‘CMS100’ junctions is stronger compared with conventional Co-Fe-B or Co-Fe / Ni-Fe based MTJs because of enhanced magnon-assisted tunneling. The inversion of the TMR for larger bias voltage is an astonishing feature for the Heusler alloy based MTJs which will be explained in the next paragraph. 6.4.4 Band Structure Calculations of Heusler Alloys Band structure calculations were performed using the SPR-KKR package [65] to investigate the theoretical behavior of Heusler based magnetic tunnel junctions and to get a better understanding of the properties that lead to high spin polarization and, thus, high magnetoresistance. A better understanding also results in in new possibilities of tailoring the properties of magnetic tunnel junctions. The calculations were carried out in the framework of local spin density approximation (LSDA), where the KKR-method for calculating the electronic structure was used. The Green‘s function is determined by treating the solid as a multi stray system, where the pertubating potentials of every atom are assumed to be spherically symmetric without overlapping (atomic sphere approach, ASA). The system was analysed taking relativistic eﬀect into account, so the used Dirac-Hamiltonian for the spin-polarized system is given by 2 cα · ∇ + βmc + Veﬀ (r) + βσ · Beﬀ (r) Ψi (r) = i Ψi (r) i δExc [n, m] , with Beﬀ (r) = Bext (r) + δm(r) where α,β are the Dirac matrices. Within this approach it can be shown, that Green’s function may be written as n × nn G(r, r , E) = ZΛn (r, E)τΛΛ (E)Z (r , E) Λ ΛΛ

−

(ZΛn (r, E)JΛn× (r , E)Θ(r − r)

Λ

+ JΛn (r, E)ZΛn× (r , E)Θ(r − r ))δnn , where Z and J are the regular and the irregular solution of the Dirac equation nn for a spin-polarized potential well, respectively and τΛΛ gives the scattering

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path operator. Green’s function provides all the information about the system, e.g. the charge density and magnetization are given by EF 1 n(r) = − tr dE G(r, r, E) π EF 1 dE βσz G(r, r, E). m(r) = − tr π Figure 6.16 shows the calculated bandstructure of a Co2 MnSi compound in L21 phase. The viewgraph illustrates the contributions of the diﬀerent Elements at diﬀerent energies. To put the calculation in relation to the expected tunnel magnetoresistance, the density of states is reduced to the DOS of the s-electrons around the Fermi-level. The s-electrons are believed to be the main contribution to the tunneling current in Alumina based tunnel barriers [86], although, the tunnel barrier material [87] and structure [58] can select diﬀerent electrons in diﬀerent tunneling structures. The left side of Fig. 6.17 shows only the s-electrons close to the Fermi-level to take this into account. The right side shows the Co-Fe counter electrode to complete the tunnel junction. One can see the gap just below the Fermi-level in the total DOS as well as in the s-electron DOS, followed by a sharp increase just above the Fermi-energy. The deﬁnition of the eﬀective spin polarization is applied: P =

N (E)↑ − N (E)↓ , N (E)↑ + N (E)↓

(6.5)

where N (E)↑ and N (E)↓ are the DOS of majority and minority s-electrons, respectively. The spin polarization of Co2 MnSi has positive values at and

Fig. 6.16. Bandstructure calculations using the SPR-KKR package [65] for a Co2 MnSi compound in L21 phase. Please note the Fermi-energy at the edge of the bandgap and the pronounced increase next to the edge

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Fig. 6.17. Bandstructure of the tunneling s-electrons in L2 1 Co2 MnSi and Co-Fe at bias-voltages of 500, 0 and –500 mV. Since mainly the density of states Figure taken from [85]

below the Fermi-energy and negative values in a region above it. To get the eﬀective spin polarization at a certain applied bias voltage in a tunnel junction, one has to integrate the DOS at that bias value (taking temperature Fermi smearing into account). Then it becomes obvious that the resulting eﬀective spin polarization is negative only for energies well above the minority peak. The TMR eﬀect is generally interpreted by Julli`ere‘s expression to calculate the TMR ratio out of a given spin polarization: ΔR RAP − RP 2P1 P2 = = , R RP 1 − P1 P2

(6.6)

where resistance in anti-parallel (RAP ) and parallel state (RP ) are related to the spin polarizations P1 and P2 of the two electrodes, PCo−Fe and PCo2 MnSi in our case. Therefore, a negative eﬀective spin polarization of only one electrode at a certain bias voltage leads to a negative resulting TMR-eﬀect. And exactly this behavior is seen in the bias voltage dependence of the TMR ratio at RT (Fig. 6.13c). A negative bias voltage applied to a Co2 MnSi / AlOx / Co70 Fe30 MTJ drives s-like electrons from occupied states at EF of Co-Fe through the Alumina tunnel barrier into unoccupied s-like states of the Co2 MnSi above its Fermi-level. Figure 6.13c shows a characteristic kink in the bias voltage dependence but no crossover at about 400 mV, because of the integration and Fermi-smearing the actual crossover is shifted to higher voltages. The (integrated) number of majority spins is higher at all energies for the Co-Fe counter electrode, in other words the spin polarization remains positive. Therefore, positive bias voltages lead to a positive TMR eﬀect amplitude which decreases with increasing bias voltage [88]. Consequently, we can select negative and positive magnetoresistance by applying diﬀerent bias

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voltages, although, the originally large TMR ratios have not been achieved at these voltages so far. However, in principle this opens up new ways for applications, e.g., in programmable magnetic logic devices [85].

6.5 Magnetic Tunnel Junctions with a Crystalline MgO(001) Tunneling Barrier As described in Sect. 6.1, magnetic tunnel junctions (MTJs) with an amorphous Al-O tunnel barrier have been extensively studied since the discovery of the room-temperature TMR eﬀect [4, 5]. In Al-O-based MTJs, magnetoresistance (MR) ratios of up to 70% have been achieved experimentally at room temperature (RT) [89] as shown in Fig. 6.18. However, these relatively small MR ratios are considered to severely limit the feasibility of applications of spintronic devices. For example, MR ratios signiﬁcantly higher than 70% at room temperature are indispensable for developing high-density magnetoresistive random-access-memory (MRAM, see Fig. 6.3). According to the Julliere’s model [3] the MR ratio of an MTJ can be expressed by spin polarization P of ferromagnetic electrodes as M R = 2P1 P2 /(1 − P1 P2 ). Here, P1 and P2 are spin polarizations of the two ferromagnetic electrodes deﬁned as P ≡ (N↑ (EF ) − N↓ (EF ))/(N↑ (EF ) + N↓ (EF )), where N↑ (EF ) and N↓ (EF ) are the density of states (DOS) of the electrode at Fermi energy (EF ) for majority-spin and minority-spin bands, respectively. Spin polarization of a ferromagnet (FM) at low temperature can be directly measured using FM/AlO/superconductor tunnel junctions [90]. According to this kind of measurement, spin polarizations of 3d-ferromagnetic metals and alloys based on iron (Fe), nickel (Ni) and cobalt (Co) are usually in the range 0 < P < 0.6 at low temperature [90, 91]. Julliere’s model with these spin polarizations yields a maximum MR ratio of about 100% at low temperature. MR ratio of about 70% at RT is therefore close to the Julliere’s limit for the 3d- ferromagnetic450 “6”: AIST

113

MR ratio (%) at RT

400

MgO(001) barrier

Tohoku 84

350

Amorhous Al-O barrier

300

AIST Tohoku

250

“5”: Anelva–AIST28 “4”: IBM77

200

“3”: AIST114

150 “2”: AIST14 NVE 104

100

Tohoku 68 MIT 70 50

0

1995

Nancy 52 ”1”: CNRS-Thales 61

2000

2005

Year

Fig. 6.18. History of improvement in MR ratio at room temperature (RT). Solid circles denote MTJs with an amorphous Al-O barrier; open circles represent MTJs with a crystalline MgO(001) barrier

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alloy electrodes if a reduction in P at ﬁnite temperature (due to thermal spin ﬂuctuations) is taken into account. One way to achieve a much higher MR ratio is the use of special kinds of ferromagnetic materials called “half metals”, which have a full spin polarization (| P |= 1) and, therefore, are theoretically expected to exhibit a huge MR ratio (up to inﬁnity, in principle) when used as the electrodes of MTJs. As candidates for such half metals, there are several types of materials: CrO2 , Heusler alloys such as Co2 MnSi, Fe3 O4 , and manganese-perovskite oxides such as La1−x Srx MnO3 . Very high MR ratios above several hundred percent have been achieved at low temperature in La1−x Srx MnO3 / SrTiO3 / La1−x Srx MnO3 MTJs [92] and Co2 MnSi / Al-O / Co2 MnSi MTJs [54]. However, such high MR ratios have never been observed at RT because of the large temperature dependence of MR ratio in these MTJs [93]. Another way to achieve a very high MR ratio is to use coherent spin-dependent tunneling in an epitaxial MTJ with a crystalline tunnel barrier such as MgO(001). This is the main subject of Sect. 6.5. Before going into the details of coherent tunneling, an incoherent tunneling process through the amorphous Al-O tunnel barrier is explained ﬁrst. Figure 6.19 a schematically illustrates a tunneling process in the MTJ with an amorphous Al-O barrier. Here, the top electrode layer is Fe(001) as an example of a 3d-ferromagnet. Various Bloch electronic states with diﬀerent symmetries of wave functions exist in the electrode. Because the tunnel barrier is amorphous, there is no crystallographic symmetry in the tunnel barrier. Because of this non-symmetric structure, Bloch states with various symmetries have ﬁnite tunneling probabilities. This tunneling process can be regarded as an incoherent tunneling. In 3d-ferromagnetic metals and alloys, Bloch states with Δ1 symmetry (mainly s-like states) usually have a large positive spin polarization at EF , while Bloch states with Δ2 symmetry (mainly d-like states) usually have a negative spin polarization at EF . Julliere’s model assumes that tunneling probabilities are equal for all Bloch states. This assumption corresponds to a completely incoherent tunneling, in which none of the momentum and coherency of Bloch states are conserved. However, even in the

Fe(001)

Δ2 Δ5

Δ2

k//

Δ1

Fe(001)

Δ5

Δ1

kZ

Δ1

MgO(001)

Al-O

Fe(001)

(a)

Δ1

(b)

Fig. 6.19. Schematic illustrations of electron tunneling through (a) an amorphous Al-O barrier and (b) a crystalline MgO(001) barrier

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case of an amorphous Al-O barrier, this assumption is not valid. Although spin polarization P deﬁned with band structure is negative for Co and Ni, the experimentally observed P is positive for these materials when combined with the Al-O barrier [90, 91]. This discrepancy indicates that the tunneling probability in actual MTJs depends on the symmetry of Bloch states. The actual tunneling process is explained in the following way. The Δ1 Bloch states with large P are considered to have higher tunneling probabilities than the other Bloch states [94, 95]. This results in a positive net spin polarization of the ferromagnetic electrode. Because the contribution of the other Bloch states, such as Δ2 states (P < 0), to the tunneling current is not negligible, the spin polarization of the electrode is reduced below 0.6 in the case of usual 3d-ferromagnetic metals and alloys. If only the highly spin-polarized Δ1 states coherently tunnel (Fig. 6.19b), a very high spin polarization and, thus, a very high MR ratio are expected to appear. Such an ideal coherent tunneling in an epitaxial MTJ with a crystalline MgO(001) tunnel barrier is theoretically predicted as explained below. It should be noted here that the actual tunneling through the amorphous Al-O barrier is considered to be an intermediate process between the completely incoherent tunneling represented by Julliere’s model and the coherent tunneling illustrated in Fig. 6.19 b. 6.5.1 Theory of Coherent Tunneling Through a MgO(001) Tunnel Barrier As shown in Fig. 6.20, a crystalline MgO(001) barrier layer can be epitaxially grown on a bcc Fe(001) layer with a relatively small lattice mismatch of about 3%. This amount of lattice mismatch can be reduced by lattice distortions in the Fe and MgO layers and/or absorbed by forming dislocations at the interface. A coherent tunneling transport in epitaxial

(a)

(b) : Mg 0.21 nm

:O aMgO aFe Fe[010]

MgO[010] MgO[100]

0.14 nm

: Fe

Fe[001] , MgO[001]

Fe[100] Fe[001]

MgO[001]

Fe[010] , MgO[110]

Fe[100] , MgO[110]

Fig. 6.20. Crystallographic relationship and interface structure of epitaxial bcc Fe(001) / NaCl-type MgO(001): (a) top view and (b) cross-sectional view. aF e and aM gO denote the lattice constants of bcc Fe and NaCl-type MgO unit cells, respectively

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Fe(001) / MgO(001) / Fe(001) MTJ is schematically illustrated in Fig. 6.19 b. In the case of ideal coherent tunneling, Fe-Δ1 states are theoretically expected to dominantly tunnel through the MgO(001) barrier by the following mechanism [96, 97]. For the k = 0 direction ([001] direction in this case), in which the tunneling probability is the highest, three kinds of evanescent states, Δ1 , Δ5 and Δ2 , exist in the band gap of MgO(001). When the symmetries of tunneling wave functions are conserved, Fe-Δ1 Bloch states couple with MgO-Δ1 evanescent states, as shown in Fig. 6.21 a. Figure 6.21b shows partial DOS (obtained by ﬁrst-principle calculations) of the decaying evanescent states in a MgO barrier layer in the case of parallel magnetic alignment obtained by ﬁrst-principle calculation [96]. Among these states, the Δ1 evanescent states have the slowest decay (and the longest decay length) of partial DOS in the MgO barrier. The dominant tunneling channel for parallel magnetic state is this Fe-Δ1 ↔ MgO-Δ1 ↔ Fe-Δ1 . Band dispersion of bcc Fe for the [001] (k = 0) direction is shown in Fig. 6.22a. The net spin polarization of Fe is small because both majority-spin and minority-spin bands have many states at EF . However, the Fe-Δ1 states are fully spin-polarized at EF (P = 1). A very large TMR eﬀect in the epitaxial Fe(001) / MgO(001) / Fe(001) MTJ is therefore expected. It should also be noted that even for anti-parallel magnetic states, a ﬁnite tunneling current ﬂows. Tunneling probability as a function of k wave vectors (kx and ky ) is shown in Fig. 6.23 [96]. For the majority-spin conductance in parallel magnetic state (P state) (Fig. 6.23a), tunneling takes place dominantly at k = 0 owing to the coherent tunneling of majority-spin Δ1 states. For the minority-spin conductance in P state (Fig. 6.23b) and the conductance in anti-parallel magnetic state (AP state) (Fig. 6.23c), spikes of tunneling probability appear at ﬁnite k points called “hot spots”. The origin of this “hot-spot tunneling” is resonant tunneling between interface resonant states [96]. Although a ﬁnite tunneling current ﬂows in AP state, the tunneling

Δ1

Δ1

Δ5

Δ5

Δ2

Δ2’ kZ k//

Δ1

Δ5

Δ2

MgO(001)

Fe(001)

Fe(001)

10 10

Fe(001)

Δ1 (spd)

1

Density of States

MgO(001)

Fe(001)

5

Δ5 (pd)

10

10

15

10

20

10

25

2

Δ2’ (d) 3

4

5

6

7

8

9 10 11 12 13 14 15

Layer Number

(a)

(b)

Fig. 6.21. (a) Coupling of wave functions between the Bloch states in Fe and the evanescent states in MgO for k = 0 direction. (b) Tunneling DOS of majority-spin states for k = 0 in Fe(001) / MgO(001)(8 ML) / Fe(001) with parallel magnetic state [96]

6 Magnetic Tunnel Junctions (a)

Δ1↓

Δ1↑

Energy ( eV )

2.0

(b)

Δ1↓

319

Δ1↑

1.0

EF Δ2’↑

EF

0

Δ2’↑

Δ5↑

Δ5↑ 1.0

Γ

Η Γ

kz

kz

Η

Fig. 6.22. (a) Band dispersion of bcc Fe in the [001] (Γ -H) direction. (b) Band dispersion of bcc Co in the [001] (Γ -H) direction redrawn from Bagayako et al. [98]. Solid and dotted lines represent majority-spin and minority-spin bands, respectively. Thick solid and dotted lines represent majority-spin and minority-spin Δ1 bands, respectively. EF denotes Fermi energy. To compare relative levels of EF in bcc Fe and bcc Co with respect to the majority-spin Δ1 band, bottom edges of the majority-spin Δ1 band in (a) and (b) are aligned at the same energy level

conductance in P state is much larger than that in AP state, resulting in a very high MR ratio. It should be noted that the Δ1 Bloch states are highly spin-polarized not only in bcc Fe(001) but also in many other bcc ferromagnetic metals and alloys based on Fe and Co. For example, band dispersion of bcc Co(001) (a metastable structure) is shown in Fig. 6.22b. The Δ1 states in (a)

Transmission

P state Majority spin

0.06 0.04 0.02 0 -0.6 -0.4 -0.2

0.6 0.4 0 0.2 -0.2 0 0.2 -0.4 y 0.4 0.6 -0.6

k

kx

(b)

Transmission

P state Minority spin

0.03 0.02 0.01 0 -0.6 -0.4 -0.2

0 -0.2 0 0.2 -0.4 0.4 0.6 -0.6

kx

(c)

0.6 0.4 0.2

ky

Transmission 0.004

AP state

0.003 0.002 0.001 0 -0.6 -0.4 -0.2

0 -0.2 0 0.2 -0.4 0.4 0.6 -0.6

kx

0.2

0.6 0.4

ky

Fig. 6.23. Tunneling probability in Fe(001) / MgO(001)(4 ML) / Fe(001) MTJ as a function of kx and ky wave vectors [96]. (a) Majority-spin conductance in parallel magnetic state (P state), (b) minority-spin conductance in P state, (c) conductance in anti-parallel magnetic state (AP state)

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bcc Co are fully spin-polarized at EF as in the case of bcc Fe. According to ﬁrst-principle calculations, a Co(001) / MgO(001) / Co(001) MTJ exhibits TMR eﬀect even larger than that of an Fe(001) / MgO(001) / Fe(001) MTJ [99]. Note also that a very large TMR eﬀect is theoretically expected not only for the MgO(001) barrier but also for other crystalline tunnel barriers such as ZnSe(001) [100] and SrTiO3(001) [101]. However, a large TMR eﬀect has never been experimentally observed in MTJs with these crystalline barriers, except for MgO(001), because of experimental diﬃculties in fabricating high-quality MTJs without pin-holes and inter-diﬀusion at the interfaces. 6.5.2 Giant TMR Eﬀect in Epitaxial Magnetic Tunnel Junctions with a Single-crystal MgO(001) Barrier Since the theoretical predictions of a very large TMR eﬀect in Fe / MgO / Fe MTJs [96, 97], there have been some experimental attempts to fabricate fully epitaxial Fe(001) / MgO(001) / Fe(001) MTJs [102, 103, 104]. Bowen et al. ﬁrst succeeded in observing a relatively high MR ratio of 30% at RT (“1” in Fig. 6.18) [103]. However, the MR ratios did not exceed the highest value for the Al-O-based MTJs (70% at RT) as shown in Fig. 6.18. The main diﬃculty at the early stage of experimental attempts was the fabrication of an ideal interface structure like that shown in Fig. 6.20b. It was experimentally observed that Fe atoms at the Fe(001) / MgO(001) interface were easily over-oxidized [105]. Results of ﬁrst-principle calculations on the ideal interface and the over-oxidized interface are shown in Fig. 6.24 [106]. In the case of the ideal interface (Fig. 6.24a), where there are no O atoms in the ﬁrst Fe monolayer at the interface, the Fe-Δ1 Bloch states eﬀectively couple with the MgO-Δ1 evanescent states in the k = 0 direction. In the case of the overoxidized interface (Fig. 6.24b), where excessive oxygen atoms are located in the interfacial Fe monolayer, the Fe-Δ1 states do not couple with the MgO-Δ1 :O

: Mg

: Fe

LDOS of Δ1↑ states Excessive O atom at interface

Fe[001], MgO[001]

(a) Fe[010], MgO[110]

Fe[100], MgO[110]

(b)

Fig. 6.24. Partial density of states at EF due to the majority-spin Δ1 states near the Fe(001) / MgO(001) interface, redrawn from Zhang et al. [106]. (a) Ideal interface and (b) over-oxidized interface

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states eﬀectively; this prevents coherent tunneling of Δ1 states and results in a signiﬁcant reduction in the MR ratio. In this way, coherent tunneling is very sensitive to the structure of barrier / electrode interfaces. Yuasa et al. fabricated high-quality fully epitaxial Fe(001) / MgO(001) / Fe(001) MTJs by using MBE growth [33, 107]. A cross-section transmission electron microscope (TEM) image of the MTJ is shown in Fig. 6.25. Single-crystal lattices of MgO(001) and Fe(001) are clearly identiﬁed in the TEM image. They succeeded in observing very high MR ratios of up to 180% at RT (“2” and “3” in Fig. 6.18), which exceeded the highest MR ratio for Al-O-based MTJs for the ﬁrst time. Magnetoresistance curves of the epitaxial Fe(001) / MgO(001) / Fe(001) MTJ are shown in Fig. 6.26a. The key for achieving such high MR ratios is considered to be clean Fe / MgO interfaces without over-oxidation. X-ray absorption spectroscopy (XAS) and x-ray magnetic circular dichroism (XMCD) studies revealed that interfacial Fe atoms adjacent to the MgO(001) layer are not oxidized at all and have a large magnetic moment, indicating that there are no oxygen atoms in the ﬁrst Fe monolayer at the interface [108]. Parkin et al. fabricated highly-oriented poly-crystalline (or textured) Fe-Co(001) / MgO / Fe-Co(001) MTJs using sputtering deposition on a SiO2 substrate with a TaN seed layer, which was used to orient the entire MTJ stack in the (001) plane, and they achieved high MR ratios of up to 220% at RT (“4” in Fig. 6.18) [32]. It should be noted that the fully epitaxial MTJs and the textured MTJs are basically the same from a microscopic point of view, if structural defects such as grain boundaries do not have a strong inﬂuence on transport properties. Fully epitaxial Co(001) / MgO(001) / Co(001) MTJs with metastable bcc Co(001) electrodes were also fabricated using MBE and were observed to show even higher MR ratios, i.e., above 400%, at RT (Fig. 6.26b, “6” in Fig. 6.18) [109] than those in the Fe(001) / MgO(001) / Fe(001) MTJs as theoretically predicted by Zhang et al. [99]. The very large TMR eﬀect in the MgO-based MTJs is called the “giant TMR eﬀect”.

Fe(001)

MgO(001)

Fe(001) 2 nm

Fig. 6.25. Cross-section transmission electron microscope (TEM) image of epitaxial Fe(001) / MgO(001)(1.8 nm) / Fe(001) MTJ [33]. The vertical and horizontal directions correspond respectively to the MgO[001] (Fe[001]) axis and MgO[100] (Fe[110]) axis

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Fig. 6.26. Magnetoresistance curves at room temperature and 20 K at bias voltage of 10 mV for (a) epitaxial Fe(001) / MgO(001) / Fe(001) MTJ [33] and (b) epitaxial Co(001)/MgO(001)/Co(001) MTJ with metastable bcc Co(001) electrodes [109]. Arrows represent magnetization alignments. In these MTJs, the top ferromagnetic electrode layer is exchange-biased by an antiferromagnetic Ir-Mn layer

6.5.3 Other Phenomena Observed in Epitaxial Magnetic Tunnel Junctions with a MgO(001) Barrier The epitaxial MTJs with single-crystal MgO(001) barrier are a model system in studying the physics of coherent spin-dependent tunneling because of the well-deﬁned crystalline structure with atomically ﬂat interfaces. In addition to the giant TMR eﬀect, other interesting phenomena, as explained below, have been observed in epitaxial MTJs. The MR ratio of the epitaxial Fe(001) / MgO(001) / Fe(001) MTJs was observed to oscillate with respect to MgO barrier thickness, tMgO [33]. Figure 6.27a shows the tMgO -dependence of a resistance-area product, RA, namely, tunneling resistance for a unit junction area (in units of Ω · μm2 ). The tunneling resistance increases exponentially with respect to the barrier

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0 1.0

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Fig. 6.27. (a) MgO thickness (tM gO )-dependence of resistance-area product (RA) in parallel magnetic state (P state) and anti-parallel magnetic state (AP state) at 20 K for epitaxial Fe(001) / MgO(001) / Fe(001) MTJs. (b) tM gO -dependence of MR ratio at 20 K and 293 K for epitaxial Fe(001) / MgO(001) / Fe(001) MTJs

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thickness (tMgO ), which is a typical tunneling character. The tMgO -dependence of the MR ratio is shown in Fig. 6.27b. Surprisingly, the MR ratio was observed to oscillate as a function of tMgO with a single oscillation period of 0.30 nm at both low temperature and room temperature. It should be noted that such an oscillation of transport property has never been observed in regards to MTJs with an amorphous Al-O barrier. It might be thought that the TMR oscillation is a result of the layer-by-layer epitaxial growth of MgO(001), in which the growth of one monolayer is almost completed before the growth of a new monolayer begins. This growth could cause an oscillation in the MR ratio because the interface morphology (atomic step density) changes periodically, layer by layer. This cannot be the origin of the observed TMR oscillation, however, because the oscillation period (0.30 nm) is not the thickness of the monoatomic MgO(001) layer (0.21 nm). As an origin of the TMR oscillation, Butler et al. proposed an interference between tunneling states [96]. Regarding the evanescent states at EF in MgO, an interference between two states, which correspond to Δ1 and Δ5 at k = 0, could cause an oscillation of tunneling conductance as a function of tMgO . This oscillation is explained as follows. These states have complex wave vectors, the perpendicular components (z-components) of which are expressed as k1 = k1r + iκ1 and k2 = k2r + iκ2 . When k · Δz > 0.59 (Δz is an interlayer spacing of MgO(001)), k1r = k2r and κ1 = κ2 = κ [96], the tunneling transmittance T can be simply expressed as T = | exp(ik1 · tMgO ) + exp(ik2 · tMgO ) |2 = 2exp(−2κ · tMgO ) (1 + cos((k1r − k2r ) · tMgO )) . The tunneling transmittance thus oscillates as a function of tMgO with a period proportional to 2π/(k1r − k2r ). This transmittance oscillation could cause an oscillation in the MR ratio. It should be noted that (k1r − k2r ) is a function of k . The oscillation, therefore, in principle could not have a single oscillation period. However, it is also theoretically predicted that “hot spots” exist under both the P state and AP state (as shown in Fig. 6.23). The oscillation, therefore, could have a single period for k of the hot spots. Although the detailed mechanism is still not clear, the observed TMR oscillation could be evidence of coherent spin-dependent tunneling through MgO(001). The epitaxial Fe(001) / MgO(001) / Fe(001) MTJ is also a model system used in studying interlayer exchange coupling (IEC) between two ferromagnetic (FM) layers via an insulating non-magnetic (NM) spacer. IEC in a FM / NM / FM structure with a metallic NM spacer has been extensively studied and is well known to show oscillations as a function of spacer thickness [110]. IEC for a metallic spacer is induced by conduction electrons at EF . Although similar IEC is also theoretically expected in a FM / NM / FM structure with an insulating NM spacer (MTJ structure) [111], such intrinsic IEC has not been observed in MTJs with an amorphous Al-O barrier. Faure-Vincent et al. ﬁrst succeeded in observing intrinsic antiferromagnetic IEC in an epitaxial Fe(001) / MgO(001) / Fe(001) MTJ structure, in

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which an extrinsic ferromagnetic magnetostatic coupling was superposed [112]. Katayama et al. then obtained a more reﬁned experimental result, in which little extrinsic magnetostatic coupling was exhibited, as shown in Fig. 6.28 [113]. Antiferromagnetic coupling was observed for tMgO < 0.8 nm. With increasing tMgO , the sign of IEC reverses at tMgO = 0.8 nm and gradually approaches to zero. Because of the atomically ﬂat barrier / electrode interfaces (see Fig. 6.25), no extrinsic magnetostatic coupling was observed. The IEC for a MgO(001) spacer is, therefore, considered to be mediated by spin-polarized tunneling electrons. 6.5.4 Giant TMR Eﬀect in CoFeB/MgO/CoFeB Magnetic Tunnel Junctions As explained in Sect. 6.5.2, fully epitaxial MTJs with a single-crystal MgO(001) barrier and textured MTJs with a (001)-oriented poly-crystal MgO barrier exhibit the giant TMR eﬀect at room temperature, which is a desirable property for spintronic applications such as MRAM and the read head of a hard disk drive (HDD). However, the epitaxial and textured MTJ structures are not compatible with the manufacturing processes of these devices for the following reason. MTJs for practical applications need to have the following basic stacking structure: seed layer / AF / SyF / tunnel barrier / free FM. Here, AF denotes an antiferromagnetic layer for exchange biasing. Ir-Mn, Pt-Mn, or related alloys are used as the AF layer. Free FM denotes the top ferromagnetic electrode layer, which acts as a free layer of a spin-valve. SyF denotes a synthetic ferrimagnetic structure (i.e., an antiferromagnetically coupled FM / NM / FM trilayer such as Co-Fe / Ru / Co-Fe), which acts as a pinned layer of a spin-valve. This type of bottom structure is indispensable not only for a robust exchange-bias on the bottom pinned layer but also for reducing a stray magnetic ﬁeld acting on the top free layer. In practical MTJs, the bottom AF / SyF layers are based on a fcc structure with (111)-orientation. The problem concerning with this structure is that a NaCl-type MgO(001) 0.01

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Fig. 6.28. Interlayer exchange coupling (IEC) in epitaxial Fe(001) / MgO(001) / Fe(001) at room temperature [113]

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barrier and bcc(001)-oriented ferromagnetic electrode layers cannot be grown on the fcc(111)-oriented AF / SyF structure because of mismatch in structural symmetry. To solve this growth problem, Djayaprawira et al. developed a new MTJ structure, CoFeB / MgO / CoFeB, by using sputtering deposition [114]. Crosssection TEM images of the MTJ are shown in Fig. 6.29. As seen in the highresolution TEM image (Fig. 6.29a), the bottom and top CoFeB electrode layers have an amorphous structure in an as-grown state. Surprisingly, the MgO barrier layer grown on the amorphous CoFeB is (001)-oriented polycrystalline. Because the bottom CoFeB electrode is amorphous, the CoFeB / MgO / CoFeB MTJ can be grown on any kinds of underlayers by sputtering deposition at room temperature. As shown in Fig. 6.29b, for practical applications, the CoFeB / MgO / CoFeB MTJ can be grown on a standard AF / SyF bottom structure. After post-annealing at 360◦ C, this CoFeB / MgO / CoFeB MTJ exhibited a giant MR ratio of 230% at RT (“5” in Fig. 6.18). Because this CoFeB / MgO / CoFeB structure is highly compatible with manufacturing processes, recent research and development on spintronic devices such as MRAM and HDD read head is based on this MTJ structure. Up to now, MR ratios above 350% at RT have been achieved in CoFeB / MgO / CoFeB - MTJs [115]. The mechanism of the giant TMR eﬀect in CoFeB / MgO / CoFeB MTJs is explained below. As illustrated in Figs. 6.19b and 6.21a, four-fold symmetry of the crystalline structure in both the MgO barrier and the electrodes is essential for coherent tunneling under the Δ1 states. The giant TMR eﬀect is therefore not theoretically expected for amorphous CoFeB electrodes. (a)

(b) Free layer Pinned layer SyF structure AF layer for exchange bias

Fig. 6.29. Cross-section TEM images for CoFeB / MgO / CoFeB MTJ with a synthetic ferromagnetic (SyF) pinned layer and an antiferromagnetic (AF) layer for exchange-biasing underneath the MTJ part. (a) is a magniﬁcation of (b)

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Yuasa et al. experimentally observed that the amorphous CoFeB adjacent to the MgO(001) layer crystallizes in the bcc(001) structure by annealing above 250◦C, as illustrated in Fig. 6.30 [116]. This type of crystallization process is known as “solid phase epitaxy”, in which the MgO(001) layer acts as a template for crystallizing the amorphous CoFeB layers due to the good lattice matching between MgO(001) and bcc CoFeB(001). The observed giant TMR eﬀect in CoFeB / MgO / CoFeB can therefore be interpreted within the framework of the theory for epitaxial MTJs because the microscopic structure of bcc CoFeB(001) / MgO(001) / bcc CoFeB(001) MTJs is basically the same as that of epitaxial MTJs. 6.5.5 Applications of the Giant TMR Eﬀect As explained in the previous section, the CoFeB / MgO / CoFeB MTJs showing the giant TMR eﬀect are compatible with manufacturing process for spintronics devices because they can be fabricated on any kinds of underlayers by sputtering deposition at RT followed by ex-situ post-annealing. Besides giant MR ratios and manufacturing compatibility, industrial applications require the MTJs to satisfy many other factors such as small bias-voltage dependence of MR ratio, high break-down voltage, and appropriate resistance-area (RA) products. CoFeB / MgO / CoFeB MTJs basically satisfy all the major requirements for applications. As an example, the RA products of the MTJs are described below. Impedance matching in electronic circuits is indispensable for high-speed operations of devices. The RA product of MTJ should therefore be adjusted to an appropriate value to satisfy the impedance matching. MRAM applications require RA in the range from 50 Ω · μm2 to 10 kΩ · μm2 depending on the lateral MTJ size (i.e., areal density) of MRAM. In this RA range, giant MR ratios can be easily obtained, as shown in Fig. 6.27. On the other hand, a very low RA, below about 1 Ω · μm2 , is required for a read head of a high-density HDD with an areal recording density above 500 Gbit/inch2 (see Fig. 6.31). A high MR ratio above 50%, which is desirable for this application,

Amorphous CoFeB

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After annealing

(a)

(b)

Fig. 6.30. Schematic illustration of the structure of a CoFeB / MgO / CoFeB MTJ in (a) as-grown state and (b) after annealing above 250◦ C

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250 Recording densities above 500 Gbit / inch2

MR ratio (%) at RT

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Fig. 6.31. MR ratio at room temperature vs. resistance-area (RA) product. Open circles represent CoFeB / MgO / CoFeB MTJs [117]. Black areas represent Al-Obased MTJs and CPP-GMR devices. The Required zone for hard disk drives with recording density above 500 Gbit/inch2 is indicated by the gray area

has never been realized using conventional Al-O-based MTJs and CPP-GMR devices with ultra-low RA < 1 Ω · μm2 , as shown in Fig. 6.31. Using CoFeB / MgO / CoFeB MTJs, Nagamine et al. achieved both ultra-low RA, i.e., down to 0.4 Ω·μm2 , and high MR ratios, above 50%, as shown in Fig. 6.31 [117]. The ultra-low resistance MgO-based MTJ is thus considered a promising candidate for next-generation HDD read head. The giant TMR eﬀect is also useful in developing MRAM. In conventional MRAM, the writing process (i.e. magnetization reversal of a free layer) uses a magnetic ﬁeld generated by pulse currents, and the read-out process uses a resistance change between parallel and anti-parallel magnetic states (i.e., TMR eﬀect). The giant TMR eﬀect enables high-speed read-out because the high MR ratio yields a high output signal for read-out [118]. In the conventional MRAM, however, the writing pulse currents increase by shrinking the lateral size of MTJs, which makes it diﬃcult to develop gigabit-scale highdensity MRAM. In a new type of MRAM, the so-called spin-transfer MRAM, on the other hand, the writing process uses the current-induced magnetization switching (CIMS) [30] (see also Sect. 6.3). This phenomenon is especially important in developing high-density MRAM because the writing pulse current ﬂowing through the MTJ can be reduced by shrinking the lateral MTJ size. CIMS was ﬁrst experimentally demonstrated in CPP-GMR devices and later in Al-O-based MTJs [119]. CIMS in MgO-based MTJs, which is especially important for MRAM, has been successfully demonstrated in CoFeB / MgO / CoFeB MTJs [120, 121, 122]. An example of CIMS is shown in Fig. 6.32. Switching between parallel and anti-parallel magnetic states is induced not only by applying a magnetic ﬁeld (Fig. 6.32a) but also by sending a pulse current through an MTJ (Fig. 6.32b). Based on the giant TMR eﬀect and CIMS in MgO-based MTJs, a prototype spin-transfer MRAM was developed as shown in Fig. 6.33, and reliable read-out and writing operations were demonstrated [122]. At present, the intrinsic critical current density, Jc0 , or

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R(Ω)

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400

-1

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Ipulse ( mA )

H ( Oe )

Fig. 6.32. (a) Magnetoresistance curve (R − H loop) and (b) current-induced magnetization reversal (CIMS) curve (R − I loop) at room temperature in a CoFeB / MgO / CoFeB MTJ with a lateral size of 70 x 160 nm [120]

a pulse-current density with a pulse duration of 1 nsec necessary for CIMS (about 2 × 106 A/cm2 ) [31, 121, 122] is not small enough for developing highdensity MRAMs. If Jc0 is reduced to about 5 × 105A/cm2 ), it will be possible to develop gigabit-scale high-density MRAM. The MgO-based MTJs also have the potential for microwave-device applications. Tulapurkar et al. demonstrated that a DC voltage is generated between the two electrodes when an AC current with a microwave frequency is passed through a CoFeB / MgO / CoFeB MTJ [123]. This phenomenon named “spin-torque diode eﬀect” can be used for detecting microwaves. The spintorque diode eﬀect originates from the giant TMR eﬀect and a resonant precession of free-layer magnetic moment induced by spin-transfer torque [123].

MTJ

Capping layer

CoFeB (free)

Pt-Mn

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Bit Line

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Buffer layer

Ru 10 nm

1 nm

SyF

CoFe

Transistors

Fig. 6.33. Cross-section TEM images of 4 kbit spin-transfer MRAM using CoFeB / MgO / CoFeB MTJs (courtesy of Sony Corp.) [122]

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6.6 Summary and Conclusions Magnetic tunnel junctions are the basis for many magnetoelectronic applications. Their magnetoresistance eﬀect amplitude resulting from spin-dependent tunneling increased dramatically within a short time. MTJs based on amorphous Al-O barriers and 3d-transition metal alloy electrodes such as Ni-Fe, Co-Fe and Co-Fe-B reach a TMR amplitude of around 70% at room temperature. To increase the TMR further fully spin-polarized half-metallic electrode materials are very attractive. Half-metallicity is suggested for a large number of Heusler alloys (e.g., Co2 MnSi, Co2 FeSi and Co-Cr-Fe-Al). However, MTJs basing on these electrode materials and Al-O barriers generally show a large temperature and bias voltage dependence of the TMR. So far, the superior low temperature transport properties of the Heusler based MTJs with Alumina barrier compared to MTJs with conventional 3d-alloy electrodes can not be conserved up to room temperature. By utilizing a new barrier material, a crystalline MgO(001), giant TMR ratios above 400% at room temperature have been reached. It has been suggested, that coherent tunneling is responsible for this strongly enhanced magnetoresistance. Furthermore, very low area resistance values can be reached with MgO based MTJs, which enables to switch the magnetization of the electrodes not only by external magnetic ﬁelds but also by spin-polarized tunnel currents transfering a spin torque (CIMS). In applications, output performance is roughly proportional to MR ratio. The improvements of MTJs, especially the giant TMR eﬀect in MTJs with MgO(100) barrier is thus expected not only to extend the applications of existing devices but also to realize novel spintronic applications such as high-density MRAM using CIMS, ﬁeld programmable logic circuits, high-performance magnetic sensor for next-generation hard disk drives and biochips including magnetoresistive sensors and magnetic manipulation systems for sensing DNA or proteins.

Acknowledgment Some of the authors (G. R., A. H., A. T. and J. S.) acknowledge the ﬁnancial support by the Deutsche Forschugnsgemeinschaft (DFG) and the European Union (EU). J.S. gratefully acknowledges the opportunity to work at the Advanced Light Source, Berkeley, which is supported by the U.S. Department of Energy under Contract No. DE-AC03-76SF00098. S.Y. acknowledges the ﬁnancial support by the New Energy and Industrial Technology Development Organization (NEDO) of Japan.

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7 Ferromagnet/Semiconductor Heterostructures and Spininjection Martin R. Hofmann1 and Michael Oestreich2 1

2

AG Optoelektronische Bauelemente und Werkstoﬀe, Ruhr-Universit¨ at Bochum, D-44780 Bochum, Germany [email protected] Institut f¨ ur Festk¨ orperphysik, Universit¨ at Hannover, Appelstrasse 2, D-30167 Hannover, Germany [email protected]

Abstract. We review the recent progress for spin injection into semiconductors. After discussing the physical background on the basis of the optical selection rules we describe spin injection via magnetic semiconductors and ferromagnetic metal contacts. The concepts for optical detection of spin injection and the problem of spin relaxation in the semiconductor are analyzed before we ﬁnally address spin optoelectronic devices, namely the spin light-emitting diode (spin-LED) and the spin vertical-cavity surfaceemitting laser (spin-VCSEL).

7.1 Introduction The principle of semiconductor devices will drastically change in the next 20 years since quantum size eﬀects will dominate with further decrease of structure size. The semiconductor industry has recognized this upcoming revolution but so far nobody knows the optimal concept for future quantum devices. Classical semiconductor devices use the exact control of charge by moving electrons and holes in the conduction and valence band but the charge might not be the best adapted quantity for quantum devices since the coherence time of the electron space wave function is extremely short. In addition to the electrical charge, the electron also has a spin. This spin is from the principle point of view particularly suitable for quantum devices since the spin is a quantum mechanical entity that is under certain conditions extremely stable even at room temperature. At the same time, the required energy for changing the spin orientation is in principle very small while the Coulomb charging energy per electron increases dramatically with decreasing structure size. The utilization of the electron and hole spin for future semiconductor devices is a rapidly growing research ﬁeld known as semiconductor spintronics. In 1990, Datta and Das suggested the ﬁrst spintronic device which is nowadays known as spin transistor [1]. Figure 7.1 schematically depicts its prin-

M. R. Hofmann and M. Oestreich: Ferromagnet/Semiconductor Heterostructures and Spininjection, STMP 227, 335–360 (2007) c Springer-Verlag Berlin Heidelberg 2007 DOI 10.1007/978-3-540-73462-8 7

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Fig. 7.1. Schematics of a spin transistor [1]. Spin polarized electrons (red arrows) are injected from a ferromagnetic source contact into a semiconductor channel where the spin orientation is controlled by the electric ﬁeld of a non magnetic gate contact and analyzed by a ferromagnetic drain contact. The blue arrows schematically depict the eﬀective magnetic ﬁeld that results from the electric gate ﬁeld and the wave vector of the electrons (from Ref. [2])

ciple of operation. Spin polarized electrons are injected from a ferromagnetic source contact. The spin orientation of the electrons is controlled during their transport to the drain contact by the electrical ﬁeld of a gate contact which introduces a spin precession due to the Rashba Hamiltonian. The conductivity of the device is high/low if the spin orientation of the electrons at the drain contact is parallel/antiparallel to the spin orientation of the ferromagnetic drain contact at the Fermi energy. Thereby, the spin transistor switches between high and low output solely by the control of the spin orientation. This kind of spin transistor is at the moment not applicable as real device due to several reasons but it shows nicely the principles and the prerequisites of spintronic devices. One of the main prerequisites, which will be discussed in the following, is obviously the eﬃcient injection and especially the eﬃcient electrical injection of spin polarized carriers from a ferromagnetic injector into a semiconductor.

7.2 Spin Injection 7.2.1 Theory The optical injection of spin polarized electrons into direct semiconductors with ﬁnite spin orbit coupling is extensively studied in the literature and nearly 100% spin injection eﬃciency has been experimentally demonstrated [3]. Figure 7.2 shows schematically the optical selections rules for GaAs and optical excitation in growth direction [4]. In bulk material, heavy and light hole are degenerate and excitation with circularly polarized light yields in this 3−1 simpliﬁed picture 50% electron spin polarization P0 = ↑−↓ ↑+↓ = 3+1 since the transition matrix element of the heavy hole is three times stronger than the

7 Ferromagnet/Semiconductor Heterostructures and Spininjection + 1/2

T=3

+ 3/2

- 1/2

-

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CB +

T=1

-

+

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3D LH

+ 1/2

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- 3/2 VB

2D

- 1/2

Fig. 7.2. (left) Strongly simpliﬁed schematics of the optical selection rules in bulk GaAs (3D) and GaAs/AlGaAs quantum wells (2D) for right (σ− ) and left (σ+ ) circularly polarized excitation in growth direction. Circular polarization of light and spin polarization are inevitable connected in this conﬁguration. (right) Degree of spin polarization for a 20 nm GaAs quantum well calculated by k · p-theory. For high excitation energies or high conﬁnement energies in thin quantum wells, very fast spin relaxation due to the large spin splitting of the conduction band can strongly reduce the spin polarization in the ﬁrst picoseconds after excitation. (from [4])

transition matrix element of the light hole. In principle the holes are polarized, too but spin relaxation of free holes is usually extremely fast, i.e. on the time scale of the momentum relaxation time, so that free holes can be considered as unpolarized in most experiments. In quantum wells, the degree of spin polarization depends strongly on the excitation energy and accurate values must be calculated by an elaborate k · p-theory including excitonic effects [4]. The right part of Fig. 7.2 depicts the calculated energy dependence of the degree of spin polarization for a 20 nm GaAs quantum well and excitation in growth direction. The degree of polarization reaches for resonant excitation of the lowest heavy hole transition nearly 100%, changes sign for resonant excitation of the light hole, and has a well pronounced structure due to excitonic eﬀects even at high energies. The calculated spin polarization is for low excitation densities in excellent agreement with experiment. For high excitation densities, bleaching of the absorption reduces the degree of spin polarization [4]. The selection rules are valid for excitation and recombination, so that the degree of circular polarization of the photoluminescence from the heavy hole transition is a very good approximation to the degree of the electron spin polarization, if unpolarized holes are assumed. The selection rules depend strongly on the direction of excitation and recombination. The degree of circular polarization of the heavy hole transition in unstrained GaAs quantum wells is for example for in-plane excitation/recombination independent of the electron and hole spin polarization, i.e., the heavy hole quantum well side emission is not suitable for measuring electron or hole spin polarization. The electrical injection of spin polarized carriers from a ferromagnetic metal into a semiconductor is much more challenging than the optical spin injection, since the interface between metal and semiconductor plays a crucial

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role. Schmidt and coworkers pointed out that the basic obstacle for spin injection from a ferromagnetic metal into a semiconductor is the huge conductivity mismatch between these two materials [5]. They showed that the degree of spin polarization in the semiconductor is nearly negligible for Ohmic contacts and typical device geometries and concluded that contacts with almost 100% spin polarization are needed for eﬃcient spin injection. Only four months later, Rashba published a more detailed picture of electrical spin injection pointing out correctly that tunnel contacts can solve the problem of electrical spin injection from a ferromagnetic metal into a semiconductor [6]. 7.2.2 Experiment Two major concepts have been suggested to realize electrical injection of spin polarized carriers into semiconductors. The most obvious concept is to replace the commonly used nonmagnetic metal contacts by ferromagnetic metals such as Fe or Co. This concept with its potential and limitations will be discussed in the second paragraph. First, we address an alternative that is completely semiconductor based and uses dilute magnetic semiconductors for spin injection or spin alignment. Spin Injection by Dilute Magnetic Semiconductors As a consequence of the enormous progress in material science, the fabrication of new semiconductor materials with designed properties is a fast growing and successful research ﬁeld. In particular, the realization of semiconductor materials with magnetic properties has become feasible by introducing magnetic components as, for example, Mn. [7] So called dilute magnetic semiconductors are particularly attractive for spin injection because the conductivity mismatch problem discussed above does not appear. Two approaches have been successfully realized for spin injection with dilute magnetic semiconductors: spin injection out of ferromagnetic semiconductors [8] and spin alignment with paramagnetic semiconductors [9]. The idea to use ferromagnetic semiconductors for spin injection is straightforward but depends on the availability of ferromagnetic semiconductors of suﬃcient quality. It has been shown that the introduction of a few % of Mn into GaAs is indeed possible and that GaMnAs becomes ferromagnetic under certain conditions [10]. Ohno et al. have used a ferromagnetic GaMnAs layer on a GaInAs/GaAs light emitting diode (LED) structure to investigate spin injection. Figure 7.3a shows their sample structure. The GaMnAs is used as the p-contact since GaMnAs is usually p-doped [7]. Accordingly spin polarized holes are injected. Figure 7.3b shows the experimental results at a temperature of 6 K and an external magnetic ﬁeld of 1000 Oe (0.1 T) applied in in-plane direction. The degree of circular polarization of the LED emission out of the cleaved edge of the sample (i.e. perpendicular to the growth direction) is spectrally

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Fig. 7.3. (a) Schematic structure for spin injection with GaMnAs spin injector and (b) electroluminescence intensity and polarization degree as a function of energy (from Ref. [8])

analyzed in order to investigate the degree of spin injection. The degree of circular polarization exhibits a peak coincident with the maximum of the LED emission and at the quantum well ground state transition energy [8]. The authors report a background polarization which they attribute to a combination of sample strain and experimental geometry [8]. In order to eliminate these background eﬀects, the changes in polarization are also investigated as a function of magnetic ﬁeld. Figure 7.4 shows the relative change in polarization as a function of magnetic ﬁeld for diﬀerent temperatures. The maximum change in polarization at T=6 K is of the order of 1%. For low temperatures a hysteresis behavior

Fig. 7.4. Relative change in light polarization degree as a function of magnetic ﬁeld for diﬀerent temperatures. The inset shows the maximum polarization degree and the magnetization of the GaMnAs layer (determined from SQUID measurements) as a function of temperature (from Ref. [8])

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is observed which disappears for temperatures above 52 K. This disappearance is consistent with the Curie temperature of the GaMnAs layer which was determined to be 52 K from SQUID measurements. A comparison with a reference structure without magnetic GaMnAs layer supports the claim of the authors to observe spin injection. However, these results have been controversially discussed for several reasons. First, the sample geometry does not allow a clear interpretation of the results via the optical selection rules. In quantum well structures, the selection rules as shown in Fig. 7.2 only hold for vertical direction (i.e. with light emission and spin orientation parallel or antiparallel to the growth direction) and in the vicinity of k = 0. In addition, the GaMnAs contact injects holes instead of electrons but the hole spin relaxation is extremely fast. Besides the serious problems concerning the interpretation of these results, an additional problem is the low Curie temperature of the GaMnAs layer of only 52 K. The need of cryogenic cooling is not attractive for future applications and therefore, large material development eﬀort has to be invested to develop ferromagnetic semiconductors with a Curie temperature TC considerably above room temperature. Dietl and Ohno have analyzed the potential candidates for ferromagnetic semiconductors in detail [11]. Though TC can exceed 100 K in GaMnAs, a TC above room temperature can most probably not be reached with this material [11]. Among the various candidates for ferromagnetic semiconductors currently the Nitrides like GaMnN [11, 12] and MnAs clusters in GaAs environment [13] seem to have the highest application potential due to their high Curie temperatures and compatibility to existing optoelectronic semiconductor technology. The latter might be a critical issue for other ferromagnetic semiconductors with TC above room temperature as, for example, ZnCrTe [14], Cr-doped In2 O3 [15], CdMnGeP2 [16], or ZnMnO [11]. Instead of using ferromagnetic semiconductors for spin injection, one might also use paramagnetic semiconductors with large Zeeman splitting for spin alignment [17]. Fiederling et al. have successfully followed this approach [9]. The left part of Fig. 7.5 shows their sample geometry. They use an GaAs/AlGaAs LED structure with a paramagnetic BeMnZnSe layer in the n region. The huge g-factor due to super exchange interaction leads in BeMnZnSe to a large Zeeman splitting in an external ﬁeld at low temperatures [9]. Accordingly, the occupation of states with one particular spin orientation (depending on the direction of the magnetic ﬁeld) is strongly preferred and the electron spins are aligned when they pass this layer. The right part of Fig. 7.5 shows the measured degree of circular polarization of the light emitted in growth direction as a function of magnetic ﬁeld in the same direction [9]. A maximum polarization degree of 43% is reported with a thick (300 nm) BeMnZnSe layer. The value is lower for a thin (3 nm) layer and disappears for a non-polarizing BeMgZnSe layer. Though this successful proof for spin alignment with considerable polarization degrees of the optical emission, this concept also suﬀers from two

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Fig. 7.5. (left) Device geometry of a BeMnZnSe spin aligner LED. (right) The degree of circular polarization of the electroluminescence for a 300 nm thick BeMnZnSe spin aligner (squares), a 3 nm thick BeMnZnSe spin aligner (circles) and with a non-polarizing BeMnZnSe layer (triangles). The crosses show the intrinsic polarization degree of the GaAs layer (from Ref. [9])

drawbacks. First, the g-factor in BeMnZnSe dramatically decreases with increasing temperature such that room-temperature operation is unrealistic [9]. Second, and more severe, this spin aligner concept requires large magnetic ﬁelds of the order of 1–5 T to ensure considerable Zeeman splitting. Such ﬁelds usually require external superconducting magnets which are a major barrier for practical applications. Spin Injection by Ferromagnetic Metals Spin injection from ferromagnetic metals is an alternative to using dilute magnetic semiconductors and has other advantages and drawbacks. In contrast to the magnetic semiconductors, most of the ferromagnetic metal injectors have Curie temperatures above room temperature and thus spin injection at room temperature is feasible. However, the injection eﬃciencies with ferromagnetic metal contacts are usually considerably smaller than those of magnetic semiconductor based injectors. The ﬁrst successful spin injection with ferromagnetic contacts at room temperature was published by Zhu et al. [18]. They used an 20 nm thick Fe ﬁlm epitaxially grown onto a GaAs/GaInAs LED structure in one MBE machine with a separate chamber for the Fe growth. The device structure is shown schematically in the left part of Fig. 7.6 [18]. The spin injection eﬃciency was investigated by analyzing the polarization degree and orientation of the LED emission. However, the magnetization of the thin Fe layer is in plane without external ﬁeld due to the shape anisotropy. As mentioned above, the selection rules require a vertical orientation of the spins in the active layer in order to achieve an unambiguous connection between spin polarization and circular polarization of the light emission (which is also in direction perpendicular to the layer plane). That means that the magnetization of the ferromagnetic metal contact also has to be perpendicular to the layer plane. Accordingly, an external magnetic ﬁeld has to be applied to tilt the magnetization in the

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Fig. 7.6. (left) Spin LED with Fe injector. (right) Circular polarization degree as a function of external magnetic ﬁeld with (squares) and without (triangles) Fe contact at 25 K (from Ref. [18])

injector into vertical orientation. For Fe, a ﬁeld of about 2 T is required to achieve complete vertical magnetization of the injector. To investigate the spin injection, one usually analyzes the degree of circular polarization of the LED emission. The right part of Fig. 7.6 shows the degree of circular polarization of the spin LED at 25 K [18] as a typical example for such experiments. The eﬀects due to spin injection are small and superimposed by other contributions. To unambiguously isolate the contribution of spin injection, the authors additionally analyzed a reference sample without ferromagnetic Fe spin injector. This reference sample also exhibits circularly polarized light contributions to the LED emission for nonzero magnetic ﬁeld. These contributions are due to Zeeman splitting in the semiconductor and don’t have anything to do with spin injection. Due to the Zeeman splitting of the spin up and spin down states there is a diﬀerence in occupation of states with diﬀerent spin orientation, and thereby a slight preference for the emission of light with one circular polarization occurs. Accordingly, even without ferromagnetic contacts circularly polarized light is emitted in the presence of a magnetic ﬁeld. In the regime of interest, the degree of circular polarization due to Zeeman splitting varies linearly with magnetic ﬁeld [18]. However, this Zeeman contribution is also present in structures with ferromagnetic spin injectors and will be superimposed to the spin injection component. This can clearly be seen in the right part of Fig. 7.6. Even without Fe contacts, the circular polarization increases linearly with magnetic ﬁeld and reaches a maximum value of below 1%. With Fe contacts, an additional contribution due to spin injection is superimposed. It increases strongly in the regime between –2 T and 2 T and saturates at magnetic ﬁelds above ±2 T. At zero external ﬁeld the circular polarization degree is zero showing that indeed a high external

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ﬁeld is needed to achieve detectable spin injection in this geometry. Fields of the order of 2 T, however, require a superconducting magnet which introduces considerable complexity, in particular with regard to future applications. The maximum circular polarization degree due to the spin injection component is in this experiment 2%. It should be noted, however, that this value does not directly correspond to the spin injection eﬃciency. The spin injection efﬁciency can be supposed to be higher because a considerable fraction of the spin oriented carriers injected into the semiconductor will loose their spin orientation due to spin relaxation eﬀects. We will address this issue which is of particular relevance at higher temperatures approaching room temperature in more detail below. However, Zhu et al. observe spin injection with circular polarization degrees of about 2% (in saturation) up to room temperature. The circular polarization degree at room temperature measured as a function of magnetic ﬁeld is shown in Fig. 7.7. In particular for such small circular polarization degrees, it is essential to unambiguously prove that the observed signatures really arise from spin injection and not from artefacts as, e.g. magnetoptic eﬀects [19, 20] at the contacts. Zhu et al. therefore analyze the LED emission at diﬀerent spectral positions corresponding to the electron to heavy hole and electron to light hole transitions, respectively. Considering the optical selection rules shown in Fig. 7.2 and assuming that, for example, the +1/2 electron state is preferentially occupied due to spin injection, the heavy and light hole transitions from this state necessarily have opposite circular polarization: the transition into the +3/2 heavy hole is σ+ polarized and the transition into the –1/2 light hole is σ− polarized. Both polarizations change sign when the direction of the magnetic ﬁeld is inverted and the electron –1/2 state is preferentially occupied. Accordingly, the heavy

Fig. 7.7. Circular polarization degree as a function of external magnetic ﬁeld at room temperature. The open sqares correspond to the light hole transition and the ﬁlled squares to the heavy hole transition, respectively. (from Ref. [18])

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and light hole transitions have opposite ﬁeld dependence. This behavior is exactly seen in Fig. 7.7 and ﬁnally proves the spin injection. Though this was the ﬁrst successful realization of spin injection from a ferromagnetic metal into a semiconductor, the concept still suﬀers from a few drawbacks. The ﬁrst is the need for high external magnetic ﬁelds. This issue will be addressed at the end of this section. The second is the Fe/GaAs interface. The growth conditions for the iron layer are diﬃcult because at growth temperatures considerably above room temperature interfacial components between Fe and GaAs might form which destroy the magnetic functionality of the injector. Any heating of the device when it is operated might also lead to the formation of such interfacial components and to a degradation of the device. Most of the groups working with Fe/GaAs or Fe/AlGaAs contacts have taken great care on the preparation of the interface by growing the whole structure including the Fe layer in one single MBE machine [18, 21]. Though it could be shown that successful spin injection can also be realized growing the semiconductor and the ferromagnetic metal layer in separate machines with a well controlled transfer procedure,[22] large eﬀort has been invested to ﬁnd alternatives for the spin injectors which are more stable than the Fe/GaAs or Fe/AlGaAs interfaces. A ﬁrst attempt was to use MnAs as the ferromagnetic metal for spin injection [23]. Similar injection eﬃciencies as with the Fe injectors could be achieved. The problem of this approach is that MnAs only has a Curie temperature of 40◦ C. The use of Fe3 Si provides injection eﬃciencies in the same range as MnAs and Fe but seems to be more promising because of its favorable thermal stability [24]. These ﬁrst proofs for spin injection from ferromagnetic metals into a semiconductor rebut the prediction by Schmidt et al. [5] who thought this concept for spin injection to be impossible. According to the analysis of Rashba [6], it is now commonly assumed that the spin polarized carriers tunnel through the Schottky barrier from the ferromagnetic Fe layer into GaAs. Detailed analysis of the transport process over the Fe/GaAs interface by Hanbicki et al. supports the assumption that the carriers predominantly tunnel through the Schottky barrier [25]. Consequently, the quality and shape of the Schottky barrier is crucial for the spin injection [26]. Figure 7.8 shows schematically the ﬂat band conduction band structure for an Fe ﬁlm on a GaAs/AlGaAs structure [21]. The Schottky barrier has a triangular shape and its thickness is determined by the doping proﬁle in the semiconductor. Hanbicki et al. have optimized the Schottky barrier using an adapted doping proﬁle in the semiconductor and succeeded in considerably higher injection eﬃciencies [21]. They reported a polarization degree of the emission of more than 10% at low temperatures but observed a strong decrease with rising temperature. Their analysis yields spin injection eﬃciencies of up to about 30% in their sample though this analysis and the exact values have been controversially discussed [27]. However, it was obvious from these ﬁrst publications on spin injection with ferromagnetic metal contacts that the interface between the ferromagnet and

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Fig. 7.8. Schematic ﬂat band conduction band structure of Fe on a GaAs/AlGaAs LED structure

the semiconductor and, in particular, the shape of the tunnel barrier is crucial for the spin injection. Instead of using Schottky contacts as the tunnel barriers, tunneling through insulator based tunnel contacts has also been investigated [19]. A comparison between Fe Schottky contacts and Fe/Al2 O3 tunnel barrier contacts provided slightly better results for the latter concept [28]. This ﬁnding is strongly supported by the work of Jiang et al. [29] who report a room temperature spin injection eﬃciency of 32% using a CoFe/MgO (001) tunnel injector into GaAs/AlGaAs LED structures. The data of Jiang et al. are shown in Fig. 7.9. At low temperatures, the polarization degree of the LED emission is above 50% and even at room temperature 32% polarization was reported. In addition to spin injection eﬀects, the authors observe linear contributions to the polarization as a function of magnetic ﬁeld which they attribute to Zeeman splitting and to a magnetic ﬁeld dependent spin relaxation process [29]. The data by Jiang et al. currently represent the published record value for spin injection at room temperature. It should be noted that postgrowth thermal annealing of the structures provided a substantial improvement of the spin injection eﬃciency [30]. This again conﬁrms the enormous importance of the interfaces for spin injectors. Though considerable spin injection eﬃciencies have been achieved by optimization of the tunnel injectors, one major drawback for applications still remains. In all the work mentioned in this section so far, spin injection could only be achieved in the presence of signiﬁcant external magnetic ﬁelds of the order of 1 T to 2 T. This problem is due to the fact that the magnetization of all injectors discussed so far is in plane whereas the selection rules require perpendicular magnetization. With these common ferromagnetic injectors, this can only be achieved with external magnetic ﬁelds that turn the magnetization out of plane. For Fe contacts, for example, about 2 T are required to achieve maximum perpendicular orientation. An alternative which is particularly attractive for applications is to use injector materials that exhibit a magnetization with a strong out of plane component even for zero external ﬁeld. Such injectors could enable spin injection in remanence which would be a

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Fig. 7.9. Magnetic ﬁeld dependence of the electroluminescence of the GaAs /AlGaAs LED with the CoFe/MgO (001) tunnel injector for 100 K (a) and 290 K (b). Sample I contains 8% Al, sample II 16%. The crosses in (a) show the linear Zeeman contribution. In (c) and (d) the linear contribution is subtracted from the data of (a) and (b), respectively. The solid lines show the scaled SQUID magnetometer results for the contacts (from [29])

remarkable progress because no superconducting magnet is necessary for spin injection in that case. Gerhardt et al. have recently suggested to use Fe/Tb multilayers to turn the magnetization of the injecting Fe layer out of plane [22]. Figure 7.10 shows the device structure they investigated. The LED structure used is a GaInAs/GaAs quantum well structure [22] similar to that used by Zhu et al. [18]. The spin injection is via a Schottky barrier contact between a thin epitaxial Fe layer and GaAs. In contrast to the work by Zhu et al. [18] the semiconductor structure and the Fe layer were grown in separate MBE machines with a well controlled transfer procedure between the machines [31]. On top of the Fe layer, an Fe/Tb multilayer sequence is grown which causes a partially vertical orientation of the magnetization in the injecting Fe layer [31]. Figure 7.11 shows the polarization degree of the LED emission at 90 K as measured with a Stokes polarimeter [22] in comparison to the SQUID measurement of the contact. The data on the polarization degree are corrected for the linear Zeeman contribution [31]. The polarization degree reaches a maximum value of 0.8% for magnetic ﬁelds above 0.5 T. This value is lower

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Fig. 7.10. Schematic structure of a Fe/Tb spin injector on a GaInAs/GaAs LED [22]

than that reported by Zhu et al. who showed polarizations of 2% [18]. However, one has to consider that the structures by Zhu et al. most probably exhibit a more perfect interface structure between Fe and GaAs due to the growth in only one MBE machine. The most important results of the work of Gerhardt et al. is that even for zero ﬁeld, i.e. in remanence, a considerable polarization is observed [22]. This conﬁrms the ﬁrst spin injection from a ferromagnetic metal contact into a semiconductor in remanence. The dependence of the polarization on magnetic ﬁeld reproduces well the hysteresis loop of the magnetic layers as measured by a SQUID magnetometer. The switching from one saturated magnetization state to the opposite takes place in less than 0.1 T, indicating an almost single domain state at low temperatures. To exclude artifacts, Gerhardt et al. also investigated the spectral dependence of the polarization. Like Zhu et al., they proved spin injection with the inversion of the polarization orientation when detecting at the light hole transition instead of the heavy hole transition [31]. Even at higher temperatures up to room temperature they report hysteresis behavior of the emitted polarization degree conﬁrming spin injection in remanence up to room temperature. Recently, van‘t Erve et al. have also reported spin injection in remanence but in a diﬀerent geometry [32]. They use an edge emitting AlGaAs/GaAs LED with an Fe contact for spin injection and demonstrate circular polarization degrees of about 5% in remanence but at a low temperature of 20 K only. Though they observe a clear hysteresis behavior the interpretation of these data is rather diﬃcult because the selection rules do not provide a clear connection between circular polarization degree and spin polarization for quantum wells in that geometry.

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Magn. moment (10-4emu) Circ. pol. degree (%)

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0.8

(a)

0.4 0.0 -0.4 -0.8 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5

(b)

-2

1 2 -1 0 Magnetic field (T)

3

Fig. 7.11. Polarization of the LED emission as a function of magnetic ﬁeld (top) and SQUID measurement of the contacts (bottom) at 90 K. The circles are for the ﬁrst measurement series with the ﬁeld varied from zero to –2 T, the inverted triangles for the ﬁeld varied from –2 T to 2 T, and the triangles for the ﬁeld varied from 2 T to –2 T. The linear Zeeman contribution was determined with a reference sample and subtracted [31]

7.3 Spin Relaxation Optical detection is the most common tool for the analysis of spin injection like in the examples discussed in the last paragraph. However, one has to be aware of that the detection introduces a conceptional problem to the interpretation of the spin injection data. After spin injection, the spin polarized carriers have to be transported over distances of a few hundred nanometers to the active region of the LED. This is crucial because the spin relaxation in the semiconductor is so strong that, in particular at room temperature, all holes and also a considerable fraction of the spin polarized electrons have lost their orientation before they recombine. Accordingly, the measured polarization degree is usually lower than the real spin injection eﬃciency. A quantitative analysis to determine the real injection eﬃciency is not straightforward and has introduced controversial discussions [21, 27]. Moreover, when spintronic devices are considered, electron spin relaxation processes always have to be taken into account. Therefore, we discuss this issue in more detail in this section.

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Electron spin relaxation in semiconductors is governed by ﬁve most relevant spin relaxation mechanisms named D’yakonov-Perel’ (DP), intersubband spin relaxation (ISR), Bir-Aronov-Pikus (BAP), Elliott-Yafet, and the hyperﬁne-interaction mechanism. The D’yakonov-Perel’ spin relaxation mechanism emerges in systems lacking inversion symmetry due to spin-orbit coupling and arises therefore for example in GaAs and ZnSe but not in Si [33]. The starting point for the theoretical description of the DP spin relaxation is the Dresselhaus-Hamiltonian for binary semiconductors 2 2 Hspin = Γ σi ki (ki+1 − ki+2 ), (7.1) i

where i = x, y, z are the principal crystal axes with i + 3 → i, Γ is the spin-orbit coeﬃcient for the bulk semiconductor, σi are the Pauli spin matrices, and k is the wave vector of the electron [34]. The comparison of the Dresselhaus-Hamiltonian with the Hamiltonian for a free electron in a mag netic ﬁeld, H = 12 i μB σi Bi , directly shows that the k-dependent spin splitting in (7.1) can be interpreted as a k-dependent magnetic ﬁeld. The wave vector of the electron scatters randomly in time, leading to an eﬀective magnetic ﬁeld that changes randomly in amplitude and direction. This random magnetic ﬁeld destroys the average of the spin orientation of an ensemble of diﬀusive electrons irrecoverable since individual spins precess around diﬀerent directions and the momentum scattering annihilates the memory about the eﬀective magnetic ﬁeld. The DP increases in bulk with increasing temperature and in quantum wells with increasing conﬁnement energy due to the occupation of higher k states with larger spin splitting. The faster spin relaxation at higher temperatures is partially reduced due to motional narrowing eﬀects, i.e. the spin lifetime is inversely proportional to the momentum scattering time due to the faster change of the precession direction. As a consequence the DP mechanism is less eﬃcient in low mobility samples. At room temperature the DP mechanism is usually the most eﬃcient spin relaxation mechanism in semiconductors without inversion symmetry. The direction of the eﬀective magnetic ﬁeld from the DP-mechanism is special for (110)-oriented quantum wells. The Hamiltonian from 7.1 reads in (110)-quantum wells 1 1 HDP = −Γ σz kx [ kz2 − (kx2 − ky2 )] , 2 2

(7.2)

2 ¯ where z2 [110] is the conﬁnement direction, x [110], y [001], kz = |∇Ψz | dz, and Ψz denotes the z part of the electron wave function. The resulting magnetic ﬁeld ﬂuctuates randomly in amplitude but points for all k in growth direction. Therefore, a spin in z or −z direction does not precess in the random magnetic ﬁeld, does not relax by the DP mechanism, and exhibits very long spin relaxation times at room temperature. Ohno et al. measured spin relaxation times of several nanoseconds in 7.5 nm (110)-GaAs quantum

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Fig. 7.12. (left) Temperature dependence of the electron spin lifetime τs for B = 0 T (closed circles, spin along z) and B = 0.6 T (open circles, spin precesses around x) in a 20 nm (110)-GaAs quantum well. (from [36]). (right) Comparison of the spin relaxation rates due to the DP and the ISR mechanism measured in the same 20 nm n-doped (110)-GaAs quantum well. The open circles show the spin relaxation rates for in-plane spins where the DP mechanism is dominant. The ﬁlled circles show the spin relaxation rates of for spins in z-direction, where the DP mechanism is absent and ISR is dominant. The solid lines are calculations with a single momentum relaxation time as ﬁt parameter (from [37]). The ISR spin relaxation rate drops by one order of magnitude for a quantum well width decreased by a factor of two

wells at room temperature [35]. Figure 7.12 shows the temperature dependence of the spin relaxation in an n-doped 20 nm (110)-GaAs quantum well measured by time-resolved photoluminescence [36]. The spin relaxation time τs increases between 5 K and 120 K with increasing temperature since the coupling of electrons and photo-created holes decreases. This spin relaxation due to electron hole coupling is known as BAP spin relaxation and is discussed in the next paragraph. Above 120 K the spin relaxation time decreases again with increasing temperature due to the intersubband spin relaxation (ISR) mechanism. The ISR is based on the Dresselhaus spin splitting and resulting spin ﬂip transitions at intersubband transitions. The ISR spin relaxation rate decreases drastically with decreasing quantum well width since the intersubband transition probability decreases. The decrease of the electron spin relaxation rate with decreasing QW width is only observed in n-doped (110)-GaAs QWs where the ISR mechanism is the dominant spin relaxation mechanism. In photo-excited and especially in p-doped semiconductors, scattering in combination with exchange interaction between electrons and holes yields an eﬃcient spin relaxation mechanism for conduction band electrons, as ﬁrst pointed out by Bir, Aronov, and Pikus [38]. Figure 7.13(a) depicts the room-temperature τs versus the QW conﬁnement energy. In contrast to ISR, τs strongly decreases with decreasing quantum well width since the exchange interaction between electrons and holes, which is governed by the Hamiltonian H = AS · Jδ(r) ,

(7.3)

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Fig. 7.13. (a) Room-temperature spin relaxation time τs of undoped (110)-GaAs quantum wells versus the conﬁnement energy of the conduction electron Ee1 for low excitation density. (b) Room-temperature τs vs excitation density for a 7 nm and an 8.5 nm wide (110)-GaAs QW. (from [40])

increases with decreasing QW width. Here, A is proportional to the exchange integral between electron and hole, J is the angular momentum operator for the holes, S is the electron-spin operator, and r is the relative position of electrons and holes [39]. Figure 7.13(b) depicts τs versus excitation density. The spin relaxation time increases with increasing excitation density since the electron hole exchange interaction decreases with increasing density, i.e. the excitonic electron hole coupling is less pronounced in a high density electron hole plasma. Another important electron spin relaxation mechanism in semiconductors and metals was pointed out by Elliott [41] and studied in detail by Yafet. The single electron Bloch wavefunctions in a semiconductor are in the presence of spin-orbit coupling no eigenstates of the Pauli matrix σz but mixtures of Pauli spin-up | ↑> and spin-down | ↓> states. These spin-up and spindown states couple in the case of momentum scattering which leads to spin relaxation. The EY spin relaxation time increases with momentum relaxation time and can be approximated for conduction electrons with energy Ek in III-V semiconductors by 1 =A τs (Ek )

δso Eg + δso

2

Ek Eg

2

1 , τp (Ek )

(7.4)

where Eg is the band gap energy and δso is the spin-orbit splitting. The numerical factor A depends on the dominant electron scattering mechanism (phonon, electron-electron, electron-hole, impurity). Equation (7.4) shows that the EY

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mechanism is most important in small band gap semiconductors with large spin orbit coupling. In most semiconductors also the nuclei have a ﬁnite spin and the magnetic moments of the electrons interact with the magnetic moments of the nuclei. This hyperﬁne interaction is extremely weak for free electrons since the electron wave function averages over many nuclei. Electrons localized on donors or in quantum dots average on the other hand only over a ﬁnite number of typically 104 − 106 nuclei and hyperﬁne spin relaxation becomes important. Typical Larmor precession periods of these localized electrons due to the variance of the nuclear magnetic ﬁeld Bn are in GaAs at thermal equilibrium and ﬁnite temperatures around 1 ns.[42]. These precessions can be reversed by spin-echo experiments. Temporal ﬂuctuations of Bn lead to irreversible spin dephasing, i.e. spin relaxation. Spin relaxation in semiconductors is a multi-layered problem and depends on the semiconductor material, temperature, internal and external electric and magnetic ﬁelds, electron and hole density, conﬁnement energy, disorder, scattering, the nuclear polarization, etc.. The spin relaxation times range from femtoseconds for free holes in semiconductors with large spin orbit coupling over a few ps in InGaAs quantum wells [43] or several nanoseconds in (110) GaAs quantum wells for electron spins at room temperature [35] up to nearly seconds for electron spins in 28 Si at low temperatures [44]. Knowledge of the spin relaxation mechanisms is therefore important for the development of spin optoelectronic devices and yields the opportunity to design speciﬁc spin relaxation times for speciﬁc spintronic devices.

7.4 Spin Optoelectronic Devices The spin relaxation in the semiconductor is a major problem and makes the transport of spin information over distances longer than a few micrometer in most semiconductors diﬃcult or even impossible. The orientation of the (optical) polarization of a light ﬁeld, in contrast, is much more stable than the spin orientation in a semiconductor. Accordingly, spin information could potentially be transported optically over long distances. Therefore, it might be attractive to use optoelectronics not only as a tool for detecting spin injection into semiconductors but to consider spin optoelectronic devices as an alternative to pure electrical spintronic devices. In this section we will discuss the potential of two speciﬁc spin optoelectronic devices: the spin LED and the spin laser. 7.4.1 Spin LED Figure 7.5 shows a typical spin LED device design. A spin LED is the most straightforward concept for a spin optoelectronic device. The achievements discussed above make room temperature operation of a spin LED feasible

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when ferromagnetic metal contacts are used. Spin injectors based on dilute magnetic semiconductors do not yet operate at room temperature and are therefore not attractive for applications. The state of the art for room temperature spin injection concerning eﬃciency is deﬁned by the work of Jiang et al. [29]. They achieved an injection eﬃciency of 32% but required high external magnetic ﬁelds due to the in plane magnetization of their contacts. Combining their MgO (001) tunnel injector with the Fe/Tb multilayer ferromagnetic contact by Gerhardt et al. [22] would potentially provide a spin LED with considerable injection eﬃciency and without need for high external magnetic ﬁelds. Polarization degrees of the order of 30% in remanence combined with switching of the polarization orientation with less than 0.1 T should be available without major further development eﬀort. However, much higher polarization degrees cannot be expected since the magnetic polarization degree of the ferromagnetic metal injectors is considerably below 100% and the conversion from spin polarized carriers to polarized light in an LED does not involve an intrinsic ampliﬁcation process. But polarization degrees of the order of 30% will most probably be too low for practical applications, e.g. in communication technology. Such applications require eﬀects close to 100% which, based on the current state of the art, cannot be reached with spin LEDs. Moreover, LEDs in general exhibit only very moderate modulation speed and are therefore not attractive for high end information technology applications. 7.4.2 Spin Laser Spin controlled semiconductor lasers have higher application potential than spin LEDs but their realization still faces severe technological problems. We will address these problems at the end of this section and ﬁrst discuss the concept of a spin laser in principle. The selection rules require a vertical geometry for spin controlled lasers. Therefore, the concepts for spin controlled lasers are all based on vertical cavity surface emitting lasers (VCSELs). Hallstein et al. showed that spin precession in an external magnetic ﬁeld can modulate the emission of an optically pumped VCSEL structure, even with extremely high modulation speed [45]. The control of the emission of an optically pumped VCSEL by the photon spin was also reported by Ando et al. [46]. These reports highlight the general potential of spin VCSELs and accordingly, a device concept for an electrically pumped spin VCSEL was suggested by Oestreich et al. [47]. This concept is shown in Fig. 7.14 The idea is to combine a VCSEL with the spin injectors discussed above. If two injectors with opposite vertical magnetization are used, one might be able to control the polarization of the VCSEL emission simply by switching between the two contacts. Before discussing details and possible complications of this concept we analyze in more detail the general advantages of a spin VCSEL over a spin LED and over a conventional VCSEL.

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Fig. 7.14. Schematic depiction of a spin VCSEL [47]. The circular polarization of the stimulated laser emission is switched between σ+ and σ− by electrical injection of either spin-up or spin-down electrons

The fundamental advantage of a spin VCSEL over a spin LED is due to the fact that a spin VCSEL is a highly nonlinear device, in particular at threshold. Accordingly, a small spin polarization in the active region due to spin injection, i.e. a slight diﬀerence in carriers with opposite spin orientations, has diﬀerent consequences than in an LED structure. In particular, when inversion is reached, this diﬀerence leads to diﬀerences in the optical gain spectra for σ+ and σ− polarization as shown schematically in Fig. 7.15. In the particular situation of Fig. 7.15, the gain for the σ+ polarization reaches

Fig. 7.15. Gain spectra for σ+ and σ− polarization under spin injection. The dashed line shows the loss level

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the loss level, which is equivalent to the laser threshold, while the gain for the σ− polarization is below threshold. Accordingly, the laser starts operating in σ+ polarization and the opposite σ− polarization is suppressed. Note that this, in principle, leads to a 100% polarization of the laser output with only a small spin polarization in the active region. Various all optical test experiments have been performed to analyze this concept in more detail. For that purpose, a VCSEL is excited optically with circularly polarized light. According to the optical selection rules, one can achieve up to 100% spin polarization in the active region in a quantum well system if only the heavy hole to conduction band transition is excited. However, because of the stopband of the Bragg reﬂectors, one usually excites with considerable excess energy such that one heavy hole and one light hole transition are excited with circular polarized excitation (see Fig. 7.2). In this case, one can achieve a theoretical maximum of about 50% spin polarization in the active region because the heavy hole transition is three times stronger than the light hole transition. But one has to be aware that spin relaxation will reduce the maximum spin polarization considerably at room temperature. However, by variation of the degree and orientation of the polarization of the excitation, one can practically control the spin polarization in the active region between zero and an upper limit of 50%. H¨ ovel et al. and Gerhardt et al. have analyzed the conditions for spin control of an optically pumped VCSEL in detail [48, 49]. They varied the polarization of the excitation with a pulsed Ti:sapphire laser and analyzed the polarization degree and orientation of the VCSEL emission. A typical room temperature result of this study is shown in Fig. 7.16.

Fig. 7.16. Circular polarization degree (squares) of an optically excited VCSEL versus input polarization orientation (stars). The solid line shows the theoretical maximum for the spin polarization in the active region. The experiment was done at room temperature

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The VCSEL polarization is obviously controlled by the polarization of the excitation. Moreover, the VCSEL polarization is in most cases even higher than the polarization of the excitation conﬁrming that the nonlinearity of the laser leads to an eﬀective ampliﬁcation of spin information. In detail, less than 30% spin polarization in the active region is required to achieve a polarization degree of the VCSEL emission of 100% and only 8% spin polarization is suﬃcient to achieve an output polarization degree of 50% [48]. Note that 30% spin polarization is already achievable by electrical room temperature spin injection [29]. Gerhardt et al. have recently shown that spin control of a VCSEL also works with continuous wave excitation at room temperature, i.e. under realistic device conditions [49]. Before considering the technical aspects of an electrically pumped spin VCSEL it is worthwhile to think about further advantages and application potential of the spin VCSEL concept in addition to the pure spin controlled emission discussed so far. First, a spin VCSEL can operate more eﬃciently than a conventional VCSEL. For circularly polarized emission, only electrons of one spin orientation are required in the active region. For linear or unpolarized emission, in contrast, electrons with both orientations are required. With other words, circularly polarized, spin controlled emission only requires half of the electrons as compared to unpolarized or linearly polarized emission. Accordingly, the threshold of a spin VCSEL should be considerably reduced. Indeed, Rudolph et al. have demonstrated this eﬀect [50, 51, 52]. The threshold reduction is signiﬁcant at low temperatures. At room temperature, there is still a threshold reduction for a spin laser as shown in Fig. 7.17 but the reduction is less pronounced so far due to the fast room temperature spin relaxation in (100) GaAs QWs. A further important aspect is that spin controlled VCSELs should enable extremely high modulation speed. This holds in particular when modulation between the opposite circular polarizations in envisioned. Hallstein et al. demonstrated that modulation frequencies of up to 120 GHz might

Fig. 7.17. Integrated VCSEL emission as a function of average pump power for random spins (triangles) and aligned spins (circles) [50]

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be accessible with spin control. In their particular experiment, this was achieved at low temperatures by fast spin precession in an optically pumped VCSEL [45]. To make use of all these advantages for real applications, optical excitation has to be replaced by electrical spin injection. Though, in principle, suﬃcient spin injectors are available as discussed in earlier sections, several technical complications arise in the details of the realization of an electrically pumped spin VCSEL. A ﬁrst major problem for the realization of electrically pumped spin VCSELs is that the output polarization of a VCSEL is mostly determined by geometrical factors. Though electrically pumped VCSEL structures usually have cylindrical shape, and thus no particular polarization orientation is preferred, the output polarization of most devices is linear and exhibits complicated dynamics including, for example, polarization switching. This complicated behavior is due to inhomogeneities, strain, or birefringence caused by the internal electrical ﬁelds [53, 54]. For spin controlled VCSELs a ﬁrst but major step is to remove or balance these sources for polarization imprint to enable polarization control only by the spin orientation of the carriers in the active region. The second and even more severe technological problem is the long transport path of the carriers (electrons) from the spin injector into the active region. In electrically pumped VCSELs, the injection is usually either through the Bragg mirror layers or through lateral contacts circumventing the mirrors. The ﬁrst concept will most probably fail for spin VCSELs because the transport path is a few micrometers long and passes several heterointerfaces. Accordingly, spin relaxation will be so strong that no usable spin orientation reaches the active region. Lateral contacts, in contrast, remove the need for passing many interfaces but still, the transport paths are in the few μm range. Clever device concepts will therefore have to be developed to really access the high application potential of electrically pumped spin VCSELs. A possible concept is to use semiconductor spin aligners which can be introduced into the semiconductor with small enough separation from the active region. But the state of the art spin aligners do not operate at room temperature. That reduces the attractiveness of this particular approach considerably though a ﬁrst attempt to realize a spin VCSEL was based on this concept [55]. However, this device operated at low temperature with high external magnetic ﬁeld and low eﬃciency only. Moreover, it has been controversially discussed whether the observed polarization is due to injection of spin polarized holes or due to reabsorption eﬀects in the GaMnAs spin aligner [56].

7.5 Summary and Outlook Electrical spin injection into semiconductors is an intricate but most important problem in semiconductor spintronics. While optical spin injection is well understood and yields spin injection eﬃciencies of up to 100%, electrical spin

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injection is still a challenge. Paramagnetic semiconductors yield electrical spin injection eﬃciencies which are comparable to optical eﬃciencies but only at low temperatures and high magnetic ﬁelds. Spin injection by ferromagnetic dilute magnetic semiconductors is from the device point of view probably the best approach, but high degrees of electron spin polarization have not been demonstrated so far. Several dilute magnetic semiconductors or ferromagnetic metal clusters in semiconductors exhibit apparently high Currie temperatures but either the electrons responsible for transport are not spin polarized so far or holes mediate the ferromagnetism and the spin relaxation times of free holes is much too fast for most devices. Metallic ferromagnet/semiconductor heterostructures yield at the moment the highest degrees of electron spin polarization at room temperature and are very promising for several spintronic devices. However, spin relaxation constrains the distance between spin injector and the active spintronic region and metallic spin injection contacts are in ﬁrst approximation limited to the semiconductor surface. Thereby metallic spin injectors are less ﬂexible than their potential counterpart, the ferromagnetic semiconductor spin injector, and are therefore for several spintronic device geometries not of practical interest.

Acknowledgement The authors thank N.C. Gerhardt and S. H¨ ovel for stimulating discussions. M.R.H. thanks the German Science Foundation for support within the SFB 491. M.O. thanks the German Science Foundation and the BMBF for ﬁnancial support.

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Index

Anisotropic magnetoresistance, 294 Anisotropy parameters, eﬀective, 190 Area resistance, 295 Biquadratic exchange coupling, 206–208 Blocking temperature, 151–152 Brillouin light scattering, 188, 193–198 CIMS, 298–300 CMS100, 306–312 Co/Cu superlattices, 31–33 CoFeB, 310–311 CoFeB/MgO/CoFeB magnetic tunnel junctions, 324–326 Coherent tunneling, 317 Cooper pairs, 252 Co:TiO2 , 26 Co:ZnO, 27 Critical spanning vector, 202 CrO2 , 24–25 Cr:TiO2 , 27 Cryptoferromagnetism, 265, 283–285 Current induced magnetization switching, 298–300 DAFF, 125 Deposition techniques, 4–5 Diﬀusion coeﬃcient, 252, 256 Dilute magnetic semiconductor, 338 Dipol-dipole interaction, 50 Domain state model, 125–130 Dresselhaus Hamiltonian, 349 D’yakonov-Perel spin relaxation, 348–349

Eﬀective spin polarization, 313, 336 Electroluminescence, magnetic ﬁeld dependence, 346 ERDA, 12 Exchange bias eﬀect block temperature, 151–152 coercive ﬁeld, 105, 109, 145–148 discovery, 103–106 domain state model, 125–130 ﬁnite size systems, 168–172 granular systems, 169–170 lateral size dependence, 171–172 magnitude, 110–112 nanoparticles, 168–169 partial domain wall model, 137–139 phase diagram, 145–146 phenomenology, 106–107 sign of, 110 spin glass model, 139–152 temperature dependence, 163–164 thickness dependence, 112, 148–151 Exchange bias ﬁeld, 107, 117, 131–132 azimuthal dependence, 118–119, 135–137, 146–148 Exchange coupling Ag interlayer, 215–217 Au interlayer, 215–217 bilinear, 186 biquadratic, 186, 206–208 Cu interlayer, 210–215 dependence on magnetic layer thickness, 229–230 eﬀect of hydrogen, 218

362

Index

eﬀect of interface alloying, 223–228 eﬀect of lattice strain, 236 Fe/Cr/Fe, 221, 223 Fe/Mn/Fe, 233 loose spin model, 234–235 magnetic, 185–244 neutron studies, 230–232 role of multiple scattering, 228–229 temperature dependence, 217–219 time dependence, 236–243 Fe/Cr superlattices, 33 Fe/GaAs, 82–87 Fe/MgO, 79–82 Fe3 O4 , 21–24 Fermi spanning vector, 202 Fermi surface, Cr, 222 Ferromagnetic resonance, 70–73 technique, 193–198 Fe/V superlattices, 29 Field cooling procedure, 106 Field programmable logic gate arrays, 296 Giant tunneling magnetoresistance, 320–322 applications, 326–328 Growth modes, 2 Heusler alloys, 300–315 band structure calculations, 312–315 order parameter, 302 structure factor, 302 transport properties, 304–312 High spin polarized materials, 300–301 Hysteresis, vertical shift, 164–168 Incoherent tunneling, 316–317 Interface(s) compensated and uncompensated, 123 transparency, 256 Interlayer exchange coupling, 198–203 Inverse proximity eﬀect, 254, 266–267, 285

Landau-Lifshitz equation, 66–70 Landau-Lifshtz-Gilbert equation, 186 Long range triplet component, 263 Magnetic anisotropy, 46–50 free energy density, 51–54 temperature dependence, 73–78 Magnetic biochip, 297 Magnetic oxides, 19–27 Magnetic random access memory, 294 Magnetic semiconductors, 25–27 Magnetic tunnel junction, 291–329 Magnetite, 21 Magnetization reversal, 120 Magnetoelastic anisotropy, 65–66 Malozemoﬀ random ﬁeld model, 122–125 Mauri model, 130–137 Mean free path, 252 Meikeljohn-Bean model ideal, 108–112 realistic, 112–117 Metal oxide superlattices, 35–37 Molecular beam epitaxy, 8–9 Mo/V superlattices, 29 MRAM, 294 N´eel domain wall coupling, 121–122 N´eel interface anisotropy, 51 Ni/Cu, 88–92 Odd triplet superconductivity, 259, 281–283 Optical selection rules, 336–337 Orange peel coupling, 205–206 Oscillation of exchange coupling strength, 203, 208, 210, 223 Oscillation of superconducting transition temperature, 268 Oxide superlattices, 36–37 Partial domain wall model, 137–139 Pinhole coupling, 206 Proximity eﬀect conventional, 251 with helical ferromagnets, 263–264

Julliere’s model, 315, 316–317 Kim-Stamps model, 137–139

Quantum interference, 200–203 Quantum well states, 200, 211, 212

Index Rare earth metals, 17–19 Rare earth superlattices, 34 Resonance equation, 70 Rotational hysteresis, 121 Schottky barriers for spin injection, 344 Shape anisotropy, 62–63 Single crystalline tunneling barriers, 320–322 Single ion anisotropy, 50–51 Singlet pairing, 253 Spin density waves, 219–232 Spin glass model, 139–152 Spin injection, 336–348 by dilute magnetic semiconductors, 338–341 by ferromagnetic metals, 341–348 Spin laser, 353–357 Spin LED structure, 346, 352–353 Spin optoelectronic devices, 352–357 Spin orbit interaction, 48–49 Spin relaxation mechanisms, 348–352 Spin torque diode eﬀect, 328 Spin transistor, 335–336 Spin valve, superconducting, 274–281 Spin VCSEL, 353 Sputtering, 8–9

363

Stoner-Wohlfarth asteroid curve, 102, 103 Stoner-Wohlfarth model, 99–103 Substrates, 6–7 Superconducting coherence length, 261, 262, 272 Superconducting spin valve, 274–281 Superconducting transition temperature, 268–274 Surface anisotropy, 63–65 Surface diﬀusion, 3 Ti:Fe2 O3 , 27 Torque magnetometer, 104, 121 Training eﬀect, 152–158 Triplet pairing, 253 Tunneling magneto-resistance(TMR), 292–293 barrier thickness dependence, 322 oscillations, 323 Tunneling through amorphous barriers, 315–328 Uniaxial symmetry, 63–65 Unidirectional anisotropy, 105 Vertical cavity surface emitting laser, 353

Magnetic Heterostructures: Advances and Perspectives in Spinstructures and Spintransport (Springer Tracts in Modern Physics, 227)