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Bragg-MOKE and Vector-MOKE Investigations: Magnetic Reversal of Patterned Microstripes DISSERTATION zur Erlangung des Grades eines Doktors der Naturwissenschaften der Fakultät für Physik und Astronomie an der Ruhr-Universität Bochum vorgelegt von Till Schmitte Bochum 2002 Mit Genehmigung des Dekanats vom 07.11.2002 wurde die Dissertation in englischer Sprache verfasst. Eine deutschsprachige Zusammenfassung befindet sich am Ende der Arbeit. Mit Genehmigung des Dekanats vom 11.11.2002 wurden Teile dieser Arbeit vorab veröffentlicht. Eine Zusammenstellung befindet sich am Ende der Dissertation. Dissertation eingereicht am 29.11.2002 Erstgutachter: Prof. Dr. H. Zabel, Bochum Zweitgutachter: Prof. Dr. W. Kleemann, Duisburg Disputation am 12.02.2003 Contents I. Introduction 5 1. Introduction 7 2. Magnetism of thin films and thin film elements 2.1. Free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Observation of domains and magnetic hysteresis in magnetic stripes 2.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Methods 11 11 14 16 17 19 3. Magneto-optical Kerr effect of thin films and thin film grating structures 21 3.1. Theory of the Kerr effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.1. Kerr effect - basics . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.2. Electro-magnetic theory of the Kerr effect . . . . . . . . . . . . . 23 3.1.3. Second order contributions to the longitudinal Kerr effect . . . . . 25 3.2. Vector-MOKE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3. Diffraction gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4. Bragg-MOKE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4.1. Review of Bragg-MOKE literature . . . . . . . . . . . . . . . . . 32 3.4.2. Some simulations of Bragg-MOKE effects . . . . . . . . . . . . . . 39 3.4.3. Interference between stripe and substrate . . . . . . . . . . . . . . 44 3.5. MOKE setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5.1. Standard setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5.2. Measurement method . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5.3. Extensions of the standard setup . . . . . . . . . . . . . . . . . . 54 4. Sample preparation 4.1. Thin film preparation . . . . . . . 4.1.1. Molecular beam epitaxy . 4.1.2. rf-Sputtering . . . . . . . 4.2. Lithography . . . . . . . . . . . . 4.2.1. Electron-beam lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 59 59 60 60 61 1 Contents 4.2.2. Other Lithography techniques . . 4.2.3. Image transfer . . . . . . . . . . . 4.3. Imaging . . . . . . . . . . . . . . . . . . 4.3.1. Scanning electron microscopy . . 4.3.2. AFM and MFM . . . . . . . . . . 4.3.3. Microscopy and Kerr microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 63 64 65 66 66 III. Results and discussion 69 5. Anisotropy of Fe(001) 71 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2. Measurements and discussion . . . . . . . . . . . . . . . . . . . . . . . . 71 6. Fe-nanowires 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Sample preparation and experimental setup . . . . . . . . . . . . . . . . 6.3. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1. Magnetic properties of the continuous Fe film . . . . . . . . . . . 6.3.2. Magnetic properties of the Fe nanowire array: longitudinal component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3. Magnetic properties of the Fe nanowire array: transverse component 6.4. Analysis and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 75 75 77 77 7. CoFe grating 7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Sample preparation . . . . . . . . . . . . . . . . . . . . . 7.3. Remagnetization process of the CoFe-grating . . . . . . . 7.3.1. Results from MOKE measurements . . . . . . . . 7.3.2. Results from Kerr-microscopy . . . . . . . . . . . 7.4. Bragg-MOKE measurements at the CoFe grating sample 7.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 87 87 88 88 92 93 95 . . . . . . 97 97 97 99 99 101 103 9. Fe-gratings 9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 105 105 108 8. Ni-gratings 8.1. Introduction . . . . . . . . . 8.2. Experimental setup . . . . . 8.3. Results and Discussion . . . 8.3.1. Bragg-MOKE . . . . 8.3.2. MFM measurements 8.4. Summary and Conclusion . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 80 82 85 Contents 9.3.1. Single crystal film, sample A . . . . . . . . . . . . . . . . . . . 9.3.2. Polycrystalline Fe-gratings . . . . . . . . . . . . . . . . . . . . 9.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1. Saturation Bragg-MOKE signal . . . . . . . . . . . . . . . . . 9.4.2. Shape of Bragg-MOKE curves of the single crystalline sample 9.4.3. Shape of Bragg-MOKE curves of the polycrystalline sample . 9.5. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 10.Co gratings on a Fe-film 10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 10.2. Experimental details . . . . . . . . . . . . . . . . . . 10.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1. Standard MOKE measurements . . . . . . . . 10.3.2. Bragg-MOKE measurements . . . . . . . . . . 10.3.3. Intensity measurements . . . . . . . . . . . . . 10.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1. Increasing Kerr effect in the spin valve region 10.4.2. Shape of Bragg-MOKE curves . . . . . . . . . 10.4.3. Bragg-MOKE amplitude . . . . . . . . . . . . 10.5. Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 117 124 124 126 127 129 . . . . . . . . . . . . . . . . . . . . . . 131 131 132 133 133 135 137 138 139 141 141 142 11.Further measurements 143 11.1. Diffuse Kerr effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 11.2. Fe grating with giant Kerr rotation . . . . . . . . . . . . . . . . . . . . . 144 12.Conclusions 147 Bibliography 153 Zusammenfassung 161 Publications 168 Acknowledgments 169 Lebenslauf 170 3 Contents 4 Part I. Introduction 5 1. Introduction Motivation The understanding of the magnetization reversal process of artificially structured magnetic islands and wires is important both from a fundamental point of view and also for potential magneto-electronic device applications [1] or mass storage devices. Of particular interest for the design of magnetic thin film devices such as read-heads and magnetic random access memories (MRAM) is the magnetic domain structure within these microor nano-structured elements, their remanent magnetization, and the shape of their magnetic hysteresis loop. On the one hand, these parameters primarily depend on both the shape and the aspect ratio of the magnetic elements, and on the other hand, they depend on the intrinsic magnetic anisotropy constants of the magnetic material used [2]. Particularly, if magnetic islands or wires are separated by only small distances, long-range magnetic dipole interaction between the elements also has to be taken into account [3]. Generally, the interest in this new materials raises new questions in the field of experimental techniques for the measurements of the micromagnetic properties. Magnetic domain structures as well as the magnetization reversal process of nanostructured magnetic elements may be investigated by a number of experimental methods. On the one hand, magnetic domains of single magnetic elements may be imaged in real space by various techniques such as Kerr-microscopy [4], Lorentz-microscopy [5], scanning electron microscopy with polarization analysis (SEMPA) [6], X-ray magnetic circular dicroism (XMCD) microscopy [7] or magnetic force microscopy (MFM) [8]. Hysteresis loops of magnetic elements are derived by evaluating the total size of magnetic domains having a particular direction of the magnetization vector with respect to the direction of the applied magnetic field. Also, the hysteresis loop of single domain magnetic elements may be measured with magnetic force microscopy, by using a calibrated MFM-tip [9]. On the other hand, hysteresis loops of magnetic elements may as well be measured via the magneto-optical Kerr effect (MOKE), superconducting quantum interference device (SQUID) magnetometry or vibrating sample magnetometry (VSM) for which the magnetization reversal process may as well be identified from the shape of the corresponding hysteresis loop. However, resolution and accuracy of the latter techniques ask for a large number of identical elements to be investigated in parallel, such that the obtained hysteresis loop yields information upon the average magnetization reversal process of all elements, and not just upon the magnetization reversal of a single element. Nevertheless, such techniques more easily allow for hysteresis loop measurements taken at various angles e.g. in orthogonal directions, from which the vector of the magnetization can be reconstructed. 7 1. Introduction In the present study two new techniques based on the magneto-optical Kerr effect (MOKE) are explored and used to investigate the remagnetization process of arrays of magnetic stripes or wires. First, the MOKE can easily be operated as a vector-magnetometer. In addition to the longitudinal MOKE geometry, also the perpendicular component of the magnetization of nanowires is measured by applying an external magnetic field in a direction normal to the plane of incidence. Both the longitudinal and the perpendicular field orientation allows to derive a vector model for the magnetization process, following previous work by Daboo et al. [10]. Second, a diffraction technique is introduced: Whereas the magneto-optical Kerr effect is a well-established method for the investigation of thin film magnetism, the application to samples with lateral structures of the order of the wavelength of the illuminating laser light is new and challenging, promising to be a powerful technique. Here the laterally structured sample acts as an optical grating leading to interference effects in the reflected laser light. Principally scattering-techniques on periodic arrays of stripes or dots can provide valuable information for the study of micromagnetism. When laser light is reflected from these samples, Kerr hysteresis loops cannot only be measured in specular reflection but also at diffraction spots of different order. This technique has been named Bragg-MOKE. For instance, for ferromagnetic line gratings, the combination of diffraction and the magneto-optical Kerr effect (MOKE) can yield information about the mean lateral magnetization distribution [11]. The technique can also be used to change the sign and the amplitude of the MOKE signal [12]. Whereas measurements of the Bragg-MOKE exist for the polar [12] and transverse [11, 13, 14, 15] MOKE configuration, Bragg-MOKE hysteresis measurements in the longitudinal geometry have not been published so far1 . Aim of this thesis This thesis has two main goals: One is to investigate the remagnetization process of micro- and nano-patterned grating arrays and systematic studies of the remagnetization process will be presented. Geometrical factors as aspect ratio and angle between stripes and magnetic field as well as material parameters are varied. Especially Fe is in the focus of this thesis. This material can be prepared in polycrystalline and single crystalline states each with different magnetic anisotropy and hence completely different remagnetization processes. A variety of stripe arrays will be analyzed mainly using the MOKE, and thus revealing integrated information of the magnetic properties. The second goal is to to explore the potential of the magneto-optical techniques BraggMOKE and vector-MOKE. In particular Bragg-MOKE needs further investigations as the observed effects are rather intriguing. Three main effects will be of interest: • The influence of interference effects, e.g. between light reflected by the grating structure and the surrounding substrate, may amplify the observed Kerr rotation. • In diffraction geometries the incident angle is not identical to the reflecting angle. 1 8 There is one exception: In [16] the authors report on Kerr-spectra in longitudinal geometry but in a different diffraction geometry (conical diffraction), no Bragg-MOKE hysteresis curves are discussed. Therefore the question arises for off-specular Fresnel coefficients and how this will influence the longitudinal Bragg-MOKE effect. • During the remagnetization process domains will occur inside the magnetic stripes. Correlated domain structures will influence the shape of the measured BraggMOKE curves. This thesis will demonstrate each of these manifestations of the Bragg-MOKE effect and qualitative explanations will be given. The two main subjects of this thesis, namely remagnetization processes of magnetic thin film elements and magneto-optics in diffraction geometry, are both discussed from the experimentalist point of view: Different parameters of the systems under investigation are varied systematically and the effects are recorded. The explanations given follow this phenomenological approach rather than an analytical or numerical description from first principles. Outline The thesis on hand is organized in three parts. The first part gives an introduction to the subject and a brief discussion of the domain structures observed in thin film elements. The second part deals with the methods used here to investigate ferromagnetic gratings in the nano- or micrometer scale. These are mainly the magneto-optical Kerr effect and sample preparation techniques. The MOKE and the particular detection techniques will be explained in detail and the actual state of research in the field of Bragg-MOKE is discussed. The theoretical section of the second part also consists of a section in which some basic models of the Bragg-MOKE effect are calculated analytically. The experimental results are reported in the third part of this thesis. The chapters in this part are organized following the different samples and series of samples prepared and analyzed. In addition, the chapters are partial extensions of previously published work. The third part ends with a conclusion and outlook. 9 1. Introduction 10 2. Magnetism of thin films and thin film elements The phenomenon of ferromagnetism in the 3d-metals (Fe, Co, Ni) is essentially due to a quantum mechanical exchange energy, resulting from the Pauli principle and the almost localized electrons of the 3d orbitals. The exchange energy leads to a spin asymmetry in the 3d sub-band and thus a permanent moment of the metal. In this chapter some basic facts about thin-film magnetism are summarized. The origin and the theory of magnetism itself is not discussed any further, as many textbooks on this subject are available [17, 18, 19] and a decent discussion would be beyond the scope of this thesis. In the first section ferromagnetic thin films are discussed in a thermodynamic context and qualitative arguments for the existence of domains are given. Subsequently, typical domain-structures of thin-film elements are reviewed and a summary of relevant literature on magnetic stripes and grating structures is given. 2.1. Free energy By definition, in thermodynamic equilibrium, any system will always be in a state of minimum total free energy. The magnetic fields acting on the magnetic moments of electrons create a local magnetization. In this field of micromagnetics the ferromagnetic film is described by a vector-field m(~ ~ r), where m is the reduced magnetization, m = M/Ms . In a phenomenological approach several energy-terms are contributing to the total free energy: Exchange stiffness This term expresses the preference of a ferromagnet for a uniform magnetization direction: Z Eex = A (grad m) ~ 2 dV, (2.1) where A is the exchange constant, a material parameter. This is the phenomenological description of the quantum mechanical exchange energy. From this equation follows that a infinite ferromagnet is in its energetic minimum if all magnetic moments are aligned parallel. For non equilibrium cases (non-uniform magnetization distribution) the free energy depends on the exchange constant A. Hard magnetic material (e.g. Co) has a higher exchange constant than soft magnetic material (e.g. permalloy). Crystalline anisotropy The energy of the ferromagnet depends on the relative orientation of the magnetization vector and the crystalline axes of the lattice. This is an 11 2. Magnetism of thin films and thin film elements effect of the spin-orbit coupling and the crystal symmetry. Three types of crystalline anisotropy can be distinguished: cubic, uniaxial and hexagonal. In this thesis mainly Fe with a fourfold, cubic anisotropy is considered. The energy density of a magnetic moment in polar coordinates is: EK = (K1 + K2 sin2 θ) cos4 θ sin2 φ cos2 φ + K1 sin2 θ cos2 θ, (2.2) where φ and θ are the in-plane angle and out-of-plane angle, respectively. K1 is the cubic anisotropy constant. In two dimensions and the case of a (001) oriented thin film this reduces to K1 EK = sin2 (2φ). (2.3) 4 The easy axes of Fe are aligned along the [100] directions of the crystal lattice. Surface anisotropy Several reasons can lead to a twofold, uniaxial anisotropy. A common example is the out-of-plane surface anisotropy leading to a strong easy axis perpendicular to the surface of a thin film. This is found in several thin film systems, such as Co/Pd. This effect is important for the technical implementation of magnetooptical storage devices, but will not be discussed in this thesis. Instead, many samples display an in-plane uniaxial anisotropy due to steps at the surface or due to the artificial structuring of the surface. In this cases a phenomenological expression is: EU = KU sin2 (φ − φU ), (2.4) where KU is the uniaxial anisotropy constant and φU is the angle between the coordinate axis and the easy axis of the uniaxial anisotropy. Stray field energy The magnetized specimen produces a magnetic field itself, the stray ~ d . The systems tries to minimize the energy density of this field. The stray field field H energy is given by: µ0 Z ~ d ∗ mdV. H ~ (2.5) Ed = − 2 sample The stray field (also called demagnetizing field, the corresponding anisotropy is also called shape anisotropy) depends on the shape of the specimen, a good approximation for many situations is to assume a general ellipsoidal shape for the sample. Than the demagnetizing field is ~ d = −N M ~S , H (2.6) with the symmetric demagnetizing tensor N . For the three axes of the ellipsoid, a, b, c, the x component of N is [2]: q abc Z ∞ 2 Na = [(a + η) (a2 + η)(b2 + η)(c2 + η)]−1 dη, 2 0 (2.7) analogous expression are valid for the other directions. This expression can be evaluated numerically to gain the demagnetizing tensor elements for an arbitrary ellipsoid. 12 2.1. Free energy For the case of an infinitely extended plate the magnetization depends only on the z-coordinate and Nc = 1. The stray field energy density is µ0 (2.8) Ed = Ms2 cos2 θ. 2 This model is a very good approximation for thin magnetic films. If no other anisotropy favors out-of-plane magnetization the magnetization will remain in the film plane. The stray field energy contribution takes the form of an uniaxial anisotropy with an anisotropy constant KU,d = µ20 Ms2 . From the measurement of the polar (out-of-plane) magnetic hysteresis of a thin film the anisotropy energy can be calculated by integrating the hysteresis curve. Another important geometry are small magnetic stripes with dimensions l, w, h, length, width and height, respectively. It can be shown that for l w h the demagnetizing factor Nw for a magnetization in the plane, but perpendicular to the stripe is h Nw = , (2.9) h+w and therefore the stray field energy is µ0 h Ed,w = M 2 sin2 φ, (2.10) 2 h+w S which is again of the form of a uniaxial (in-plane) anisotropy with the anisotropy constant h KU,w = µ20 Ms2 h+w . In this case the easy axis of the anisotropy is in-plane and along the stripes (φ = 0). Zeeman energy The energy of the magnetic moment in an external field is given by: EZ = −µ0 MS H cos(φ − φH ), (2.11) where H is the (homogenous) external field and φH the angle between the field and the coordinate axis. There are other contributions to the complete free energy function, like magnetoelastic and magnetostrictive contributions. These contributions are neglected throughout this thesis. Sum of energies The complete energy function is the sum of all terms discussed above. For the case of homogenously magnetized samples the exchange stiffness is always zero and for the case of in-plane magnetized samples one finds: K1 E(φ, H) = −µO MS H cos(φ − φH ) + sin2 (2φ) + KU sin2 (φ − φU ) (2.12) 4 where the uniaxial anisotropy may be due to the shape anisotropy of magnetic elements, to the (in-plane) surface anisotropy of a continuous film or a combination of both. In this model the magnetization rotates from one direction into the other during remagnetization, discontinuities are not described. However, several practical cases can be evaluated using the above formulas, as will be shown in the experimental sections. If the exchange constant is small, such that the magnetization is not always homogenous, domains are formed which is discussed in the next section. 13 2. Magnetism of thin films and thin film elements Figure 2.1.: Landau pattern of a square magnetic element. 2.2. Domains The most simple situation is found if only the exchange energy and the stray field energy is taken into account (these two contributions always exist). If the stray field energy is dominating, as for soft magnetic material, magnetization patterns are formed that prevent stray fields completely. An example is a magnetic disc. The magnetic moments will form a closed circular structure, however, paying an energy-penalty by increasing the exchange energy. These kind of structures have been observed in circular magnetic dots [20, 2]. Domains in the more common sense are established if additionally magnetic anisotropy is taken into account. Instead of smooth magnetization patterns domains with sharp boundaries (domain walls) occur. Inside a domain the magnetization is homogenous and parallel to an easy axis of the magnetic anisotropy. The occurrence and the shape of domains are thus depending on the relative strengths of the three energy terms: exchange energy, stray field energy and anisotropy. If an external field is applied it couples to the system via the Zeeman energy. Domains with discontinuous domain boundaries are also established in magnetic par- Figure 2.2.: Equilibrium domain states of different permalloy elements, together with the van-den-Berg construction of the domains, taken from [21] 14 2.2. Domains Figure 2.3.: Construction of domain states in rectangular elements, see main text. ticles like square or rectangular dots even without the existence of anisotropy axes. If the magnetic pattern has sharp corners the demand for a zero stray field can only be fulfilled by forming lines of discontinuous magnetization distribution. Therefore magnetic dots with square and rectangular shape display the typical Landau pattern, as depicted in Fig. 2.1. An extensive study of domains in thin film elements was performed by van Berg [22]. He invented a geometrical procedure to construct the equilibrium domain structure of magnetic thin film elements. The algorithm is as follows: • draw circles inscribed in the magnetic element which touch the edges at least at two points. • the centers of these circles form lines which correspond to the domain walls • the magnetization is oriented perpendicular to the radius which runs to the point of contact of the circle and the edge. • if one circle touches the edges in more than two points the center of this circles marks an intersection of domain walls. As an example for this construction some measurements and schematic domain patterns from [21] are reproduced in Fig. 2.2. In addition to the lowest energy pattern constructed as explained above other higher energy patterns are also observed. If for instance a magnetic rectangular element is divided into two virtual halves the domain pattern can be constructed for the two halves separately using the van-den-Berg algorithm, see Fig. 2.3. The resulting pattern is of higher energy but may be also stable depending on the demagnetization process. Comparable domain states have been observed in [23]. If additional anisotropy energy is taken into account, the domain pattern may get even more complex. An additional uniaxial anisotropy with the easy axis perpendicular to the long side of the rectangular element in Fig. 2.3 would result in a stabilization of the domains along the easy axis. The depicted state may than be the lowest energy state. The most important result of van den Berg is that in most cases domains along the edges (closure domains) will form in order to minimize the stray field. Closure domains were experimentally observed in [24] for stripes of permalloy with an external 15 2. Magnetism of thin films and thin film elements field perpendicular to the stripe axis: The internal region of the stripes is magnetized along the field but depending on width and thickness more and more edge domains are formed when the external field is reduced. Two cases can be distinguished: first the domain state is dominated by the shape of the specimen. This is the case for soft magnetic material. The domains can be constructed with the van-den-Berg method. In this case the angle of the magnetization between two domains is arbitrary reflecting the angle of the geometric shape of the element (e.g. 90◦ walls for square elements). Another case is a film or an element with anisotropy. In this case the domains will be magnetized along an easy axis of the anisotropy. This leads two 180◦ domain walls in uniaxial and 90◦ walls in fourfold anisotropy material. Wether a material is dominated by the anisotropy or by the stray field is given by the parameter Q = K/Kd , where K is a general anisotropy constant taking four- or two-fold crystal anisotropy into account. For soft magnetic material Q 1. 2.3. Observation of domains and magnetic hysteresis in magnetic stripes Several measurements of the domain structure and the hysteresis of magnetic stripes and wires can be found in the literature: • Ebels et al. [25] have investigated Fe stripes on GaAs with the rather large width of 15 µm. They found an induced uniaxial anisotropy due to edge effects and a two step magnetization process with two different domain types, due to the combination of four-fold anisotropy of Fe and the patterning. • Shearwood et al. [3] report upon magneto-resistance and magnetization loops of arrays of sub-micron sized Fe stripes. The stripes were held constant in shape (0.5 µm width) but arrays with different separations between the elements where produced. The result is that again an uniaxial anisotropy is induced and hints of dipolar interactions depending on the separation were found. • Hausmanns et al. [26, 27] show in combined work of experiment and simulation the remagnetization behavior of Co nanowires (width: 150 to 4000 nm). They show an increase of the coercive field with decreasing width of the wires proportional to 1/w. In addition, the coercive field for different in-plane angles of the external field was examined, showing a simple behavior consistent to a model where the magnetization first rotates into the wire direction and afterwards switches by 180◦ . • Because of the shape anisotropy 180◦ domain walls in small wires are expected to be of the head-to-head type. This was theoretically confirmed by McMichael et al. [28]. The head-to-head domain wall consist of additional intermediate domains with a magnetization perpendicular to the wire axis or forming vortex-like structures. • McCord et al [23] performed an intensive Kerr microscopy study on rectangular permalloy elements. Very different domain structures were detected depending on 16 2.4. Conclusion the magnetic history of the element. Typically closure domains with large internal domains having perpendicular direction were observed. • Mattheis et al. [24] also measured large edge domains aligned with the edge of the stripes using Kerr microscopy for an external field direction perpendicular to the wire axis. 2.4. Conclusion The measurement and interpretation of domains in thin film elements is a very important subject in the field of magneto-electronics and general research on magnetism. Therefore there exist a large amount of publications on the subject. There are several approaches to the problem: • Today’s computer-power enables to calculate the domain structure of thin film element. Several commercial and non-commercial programs are available. However, only in combination with the experiment the real domain structure can be concluded. For special cases, like the spin structure inside of domain walls, the numerical simulation is almost the only possibility to gain insight due to difficulties observing very small magnetic structures. • Measurements of integral physical properties like the hysteresis curves, transport phenomena or dynamic properties gain important information on the magnetic system and can be used together with numerical simulations of the domain structure. General features of the domain structure can be concluded. These kind of measurements provide important parameters of the complete system like remanence, saturation magnetization, time constants or the magnetization vector. • Direct observation of the domain structure using Kerr microscopy, MFM or other techniques has the obvious advantage of directly imaging the domains, no simulation or assumptions are needed. However, every method has its specific limitations such as resolution or problems with the contrast. In addition, it is often difficult to obtain integral quantities such as remanence or coercive fields. Only a small portion of a sample may be visualized and the overall behavior may not be detected. The most comprehensive review of the subject is found in [29], where many methods, the domain theory and a vast amount of examples are given. The present thesis contributes to this field. The combination of diffraction and MOKE will be shown to yield information about the domain structure and the hysteresis simultaneously. 17 2. Magnetism of thin films and thin film elements 18 Part II. Methods 19 3. Magneto-optical Kerr effect of thin films and thin film grating structures This chapter covers the theory and the experimental realization of Kerr effect measurements of thin films and ferromagnetic grating structures. The first sections explain the theory of the Kerr effect, the fundamentals of vector-MOKE and the theory of diffraction gratings. The next section provides a review of the literature of the Bragg-MOKE effect. This section closes with simulations of some of the Bragg-MOKE effects described in the literature. This simulations are very important for a comparison of the experimental results reported in the third part of this thesis. In the last section of this chapter the experimental setup is introduced which was used to measure the standard MOKE hysteresis curves, the vector-MOKE results and the Bragg-MOKE curves. 3.1. Theory of the Kerr effect 3.1.1. Kerr effect - basics In general the magneto-optical Kerr effect is the change of polarization and/or intensity of a light beam reflected by a ferromagnetic surface. The measured quantity, e.g. the rotation of the polarization, is a linear function of the magnetization of the ferromagnetic material. A corresponding effect which has a quadratic dependence on the magnetization is called Voigt or Cotton-Mouton effect. A special case of this will be discussed later in Sec. 3.1.3. Another magneto-optical effect is the Faraday effect: the polarization of light is rotated by transmitting light through dielectric material in the presence of a magnetic field. In this case the rotation is proportional to the applied field. This effect is used in the experimental setup (Sec. 3.5). The simplest model of MOKE is to consider a Lorentz-Drude model of a metallic film. The incident light wave causes the electrons in the metal to oscillate parallel to the plane of polarization. In the absence of any magnetization the reflected light is polarized in the same plane as the incident light, this is the regular component with an amplitude RN . If a magnetization is thought to be acting on the oscillating electrons like an internal magnetic field, the electrons exhibit a second motion due to the Lorentz force. This second component is perpendicular to the direction of the magnetization and 21 3. Magneto-optical Kerr effect of thin films and thin film grating structures Figure 3.1.: Geometry of the three magneto-optical Kerr effects, see main text. perpendicular to the primary motion. The second component, RK , generates a secondary amplitude of the reflected light which has to be superimposed onto the primary beam [4]. In this framework one can understand easily the three general geometries of the magneto-optical Kerr effect [4], which are displayed in Fig. 3.1: a) In the polar geometry the magnetization is perpendicular to the reflecting surface. A linear polarized wave generates a second component, which is strongest if the angle of incidence is zero (perpendicular incidence, αi = 0). In addition, the effect is independent of the direction of the polarization for αi = 0. b) In the longitudinal configuration the magnetization is parallel to the reflecting surface and parallel to the plane of incidence. The effect generates a polarization rotation of the reflected beam in both cases, perpendicular (s-) and parallel (p-) polarized light with respect to the plane of incidence. The sign of the Kerr rotation in the two cases is different. The special case of perpendicular incidence generates no Kerr rotation in either case, because RK points along the beam (s-polarization) or RK is zero (p-polarization). Thus the measured Kerr effect increases with αi . c) For the transverse configuration the magnetization is oriented perpendicular to the plane of incidence and parallel to the surface. For p-polarized light this configuration causes a change of the amplitude of the reflected beam, but no Kerr rotation. The three cases can be combined to yield a formula of the Kerr effect for an arbitrary magnetization and polarization of the electromagnetic wave. However in the present work only the longitudinal configuration is used. The samples under investigation usually exhibit no out-of-plane magnetization components, thus excluding the use of the polar Kerr effect. The longitudinal case is preferred over the transverse case because of the easier detection of polarization rotations than intensity shifts, as will be explained in Sec. 3.5. The longitudinal measurements are performed using s-polarized light. A polarization parallel to the plane of incidence would add an intensity modulation due to the transverse Kerr effect. The theory of the MOKE in the framework of a free electron gas as discussed above has several shortcomings and the complexity of band structures of the ferromagnetic materials demands a quantum-mechanical treatment. The magnetic moment of the 22 3.1. Theory of the Kerr effect Figure 3.2.: Definition of the coordinate system used in the discussion of the Kerr effect. ferromagnetic material is caused by the spin-asymmetry of the spin-up and spin-down sub-bands of the 3d-band structure of Fe, Co or Ni. An incoming polarized light wave interacts through its electric field with the electrons, and changes their orbital momentum. Because of the weak spin-orbit interaction this results in an interaction between the electric field of the light wave and the magnetization. The incoming electromagnetic wave can be split into left- and right-circular polarized eigenmodes, which have different quantum-mechanical probabilities to excite spin-up or spin-down electrons of the 3dband near the Fermi-level. The exited electrons will emit electromagnetic waves, with different circular polarization depending on their spin-state. Effectively this mechanism results in matrix elements of a 2 × 2 reflection matrix Rc relating the Jones vector of the incoming wave in a circular-polarization basis to the Jones vector of the emerging wave (for a discussion of Jones vectors see Sec. 3.5 and [30]). The calculation of magnetooptical effects from first principles is a complicated task (see [31, 32, 33, 34]), however, for the present investigations a theory in the framework of electro-magnetism is sufficient and will be discussed in the next section. For an introduction to the theory of MOKE see [35, 36]. 3.1.2. Electro-magnetic theory of the Kerr effect As discussed in the above section the reflection of a electromagnetic wave by a ferromagnetic surface can be split into the regular reflection, which is described by standard Fresnel formulas and Fresnel reflection coefficients, and into a second component which adds a small contribution polarized perpendicular to the regular component. The superposition of the two components leads to a rotation of the polarization axis. The goal of the electromagnetic theory is to generalize the Fresnel formulas in order to yield magneto-optical reflection coefficients. In the following the situation depicted in Fig. 3.2 is assumed: A linear polarized wave is incident on a ferromagnetic surface under the angle αi and reflected at αf = −αi . The magnetic medium has the refractive index n1 . For the reflection at a magnetic medium the use of a dielectric constant is not sufficient. In stead of this, in the dielectric law, 23 3. Magneto-optical Kerr effect of thin films and thin film grating structures ~ = E, ~ is a complex tensor, which can be written as [37, 38, 35]1 : D 1 −iQmz iQmy 1 −iQmx = xx iQmz , −Qmy iQmx 1 (3.1) ~ and the material where the mi are the components of the magnetization vector M constant Q is the magneto-optical constant, also called Voigts constant2 . In order to take the absorption of the electromagnetic wave into account, the regular dielectric constant xx is a complex number, the imaginary part corresponding to the absorption coefficient. Equivalently, the refractive index n is complex. With the above dielectric tensor the Maxwell equations have to be solved [38], which leads to a reflection matrix R= rpp rps rsp rss ! . (3.2) Eq. 3.2 relates the p- and s-polarized components of the incoming wave to the respective components of the reflected wave. The coefficients rij are the ratio of the incident j polarized electric field and reflected i polarized electric field. rss is the standard Fresnel reflection coefficient, rpp is the standard Fresnel coefficient plus a term depending on mx Q (transverse MOKE) and both the off-diagonal elements of R are functions of mz Q and my Q (polar and longitudinal MOKE: see Fig. 3.2 for the definition of the coordinate system). All coefficients are functions of the refractive index and the refraction angle inside the ferromagnetic material. Explicit formulas can be found e.g. in [38]. The complex Kerr angles are defined as following: rsp p ΘpK = θK + ipK = , (3.3) rpp rps s ΘsK = θK + isK = . (3.4) rss Here θK and K are the Kerr rotation and ellipticity, respectively, and the superscripts denote whether the incoming light wave is polarized in the p- or s-state. In Ref. [38] simplified formulas for different MOKE geometries are derived. As this thesis deals with the longitudinal MOKE with s-polarized light, only this case is discussed. In general, there are two situations which have to be distinguished. If the ferromagnetic film is thick compared to the wavelength in the material, the MOKE signal is independent of the thickness tF M . If the layer is thin, the MOKE signal is a function of tF M , for ultrathin layers this is a linear function. For the laser wavelength used in the present investigations (λ = 623.8 nm) bulk iron magneto-optical parameters are given in Tab 3.1 [39]. This leads to a wavelength inside of iron of ≈ 220 nm. The longitudinal Kerr effect for the case of thick ferromagnetic films (i.e. the Kerr effect is independent of the thickness) is derived from Eq. 3.3 and the formulas of rij for mz = mx = 0 [38]: cos αi tan α1 in0 n1 my Q s θK = , (3.5) cos(αi − α1 ) (n21 − n20 ) 1 Different sign conventions are used in literature, the discussion in this thesis follows the proposed scheme in [37] 2 ~ is also called the gyromagnetic vector The product QM 24 3.1. Theory of the Kerr effect n 2.89 + 3.07i Q 0.042 + 0.012i α β 1.00 0.13 Table 3.1.: Magneto-optical parameters of Fe. The refractive index and the Voigt constant were taken from [39]. The parameters for second order contributions, α, β, were measured in [40] for a small angle of incidence (13◦ ) and for single crystalline Fe(001)/GaAs films. where n0 is the refractive index of the medium above the ferromagnetic film, for air n0 = 1 is assumed. The angle of the refracted beam, α1 , has to be calculated using Snell’s law: n0 sin αi = n1 sin α1 . The interesting case of thin ferromagnetic films is difficult to solve as multiple reflections and interference have to be taken into account. However, a method described in [41, 38] implements a matrix formalism which allows to calculate the MOKE signal for complicated film structures and superlattices. A simplified formula for the case of a thin ferromagnetic film on a non-magnetic substrate with the refractive index nsub (angle of the refracted light in the substrate: αsub ) is given in [38] as s θK = 4πn0 n1 nsub QdF M cos αi sin α1 . λ(n0 cos αsub + nsub cos αi )(n0 cos αi − nsub cos αsub ) (3.6) In more realistic situations also a layer on top of the ferromagnetic film, e.g. an oxide layer, has to be taken into account. In the present study magnetic films are considered with a typical thickness of dF M = 20 ... 50 nm, for which the approximation of a thick ferromagnetic film turned out to be satisfactory. In Fig. 3.3 the Kerr rotation as given by Eq. 3.5 is plotted for the bulk Fe parameters as a function of the incident angle. The function exhibits a maximum at αi = 55◦ . For most experimental situations an incident angle of 45◦ is realistic for which a Kerr rotation of θK = 0.068◦ can be expected. The ellipticity shows a maximum for the same angle of incident, which is K = 1.85 · 10−3 rad. 3.1.3. Second order contributions to the longitudinal Kerr effect The dielectric tensor in Eq. 3.1 is linear in the magnetization. As already mentioned, there are cases where a second order contribution to are important. Especially single crystalline Fe often exhibits strong second order effects. The second order effects are ~ and a second order term is added to the dielectric tensor in Eq. 3.1, quadratic in M which is given by [4]: B1 m2x B2 mx my B2 mx mz B2 my mz B1 m2y . B2 mx my 2 B2 mx mz B2 my mz B1 mz (3.7) From this tensor expressions for the MOKE can be derived, as is shown in [42]. Experimental examples for the case of Fe can be found in [43, 40]. In Ref. [40] fits to MOKE data in the longitudinal geometry show that the Kerr effect can be described effectively by θK ∝ my + αmy mx + βm2x . (3.8) 25 3. Magneto-optical Kerr effect of thin films and thin film grating structures 0.08 0.07 Re(θsK) [°] 0.06 0.05 0.04 0.03 0.02 0.01 0 0 15 30 45 αi [°] 60 75 90 Figure 3.3.: Plot of the Kerr rotation as a function of the incident angle αi , as described in Eq. 3.5 for bulk Fe and s-polarized light. The two phenomenological parameters α and β were estimated in [40] for the case of a single crystalline Fe(001) film on GaAs and are given in Tab. 3.1. As discussed in [42] the second order contributions depend on the angle of incidence. The values in Tab. 3.1 were taken at αi = 13◦ , for larger angles as were used in this thesis smaller second order effects are expected. For the longitudinal configuration the magnetization components mx and my in Eq. 3.8 can be identified with the two orthogonal magnetization components mt and ml along the transverse and longitudinal in-plane direction, respectively. Therefore the second order effects lead to a contribution in the longitudinal MOKE of transverse magnetization components. If the re-magnetization process of the sample under investigation is dominated by magnetization rotation processes, the orthogonal magnetization component increases around zero field and will lead to strong asymmetries in the measured hysteresis loop. Vice versa, if the re-magnetization process involves only 180◦ domain wall movements, the magnetization in any domain is always oriented parallel or antiparallel to the external field, thus no second order contributions are detected. For Fe, tending to 90◦ domain walls, a combination of the two limiting cases is expected. 3.2. Vector-MOKE It is often advantageous to measure not only the component of the magnetization along the applied field, but also the orthogonal magnetization component in order to reconstruct the magnetization vector from the measurement. The longitudinal MOKE can be used as a vector-magnetometer in the following manner: Magnetic hysteresis measurements were performed using a high resolution magneto- 26 3.2. Vector-MOKE Figure 3.4.: Definition of the sample rotation χ and the angle φ of the magnetization ~ for the case of the longitudinal setup (a) and the perpendicular vector M setup (b). In order to measure the transverse magnetization component mt the field and the sample are rotated by 90◦ , such that the angle χ is held constant, but the magnetization component mt is in the scattering plane. optical Kerr effect setup (MOKE) in the longitudinal configuration with s-polarized light, which is able to measure the exact Kerr angle as a function of the applied magnetic field. Details of the experimental setup can be found in Sec. 3.5. Here the magnetic field lies in the scattering plane and the resulting Kerr angle is proportional to the l component of the magnetization vector along the field direction, θK ∝ ml , where ml is ~ ~ the longitudinal component of M projected parallel to H. Additionally, the design of the setup enables one to rotate the sample around its surface normal (angle χ), in order to apply a magnetic field in different in-plane directions. This kind of measurement cannot distinguish between a magnetization reversal via domain rotation and/or via domain formation and wall motion. Therefore measurements were performed with the external magnetic field oriented perpendicular to the scattering plane and the sample rotated by 90◦ with respect to the scattering plane, keeping the rest of the setup constant. In this perpendicular configuration MOKE detects the magnetization component parallel t to the scattering plane and perpendicular to the magnetic field, θK ∝ mt , as has been shown by [10]. The geometry of the setup is sketched in Fig. 3.4. Both components, ml ~ sampled over the and mt , yield the vector sum for the average magnetization vector M region, which is illuminated by the laser spot. This area is ≈ 1mm2 . The magnetization vector can be written as ~ = M ml mt ! = |M | cos φ sin φ ! . (3.9) The proportionality constant between the Kerr angle θK and the two magnetization components is a priori unknown. For the samples under investigation it was found that in saturation the Kerr angle does not dependent on the sample rotation χ. Furthermore, the angle of incidence of about 40◦ was kept constant for both set-ups. Therefore the 27 3. Magneto-optical Kerr effect of thin films and thin film grating structures error - if at all - is tolerable by assuming the same proportionality constant for both configurations. A source of error may be a contribution from the polar MOKE effect, which would add a signal proportional to a magnetization component perpendicular to the sample surface. In addition second order magneto-optical effects [42] (see Sec. 3.1.3) could interfere with the following analysis. However, neglecting these potential problems, one can write: ml cos φ θl = = K , (3.10) t mt sin φ θK from which follows the rotation angle of the magnetization vector: θt φ = arctan K l θK ! . (3.11) Furthermore one can express |M |, normalized to the saturation magnetization: l |M | θK 1 = . l,sat sat |M | θK cos φ (3.12) Another possibility of yielding magnetic vector information from MOKE measurements is to use a combination of the transverse and longitudinal Kerr effect, as has been shown by [44]. In the case of the transverse Kerr effect the magnetic information is obtained from an intensity shift of the reflected light, which is proportional to the magnetization along the applied magnetic field perpendicular to the scattering plane. In this geometry obviously a rotation of the polarization can be attributed to the longitudinal Kerr effect which is then sensitive to the magnetization component perpendicular to the magnetic field. Thus, by measuring both, the rotation and the intensity one can extract information of two orthogonal magnetization components. Details of the procedure can be found in [44]. The advantage here is that the magnetic field and the sample stay in the same position and the two components can be measured simultaneously, as opposed to the geometry used in this work. The drawbacks are that the detection is more complicated and the two signals yielded are not directly comparable concerning their magnitude, because two different physical quantities are measured. The results of vector-MOKE measurements provide important information which allow to distinguish between different magnetization reversals. Two limiting cases can easily be separated (see Fig. 3.5): ~ |, is • Coherent rotation (Fig. 3.5(a)): If the length of the magnetization vector,|M constant during the reversal, the magnetization rotates from one direction into the other. • Domain formation (Fig. 3.5(b)): If only domains are formed, the angle of the magnetization stays always aligned with the external field but the magnitude changes. In this case the transverse component is zero. It is instructive to plot the transverse component and the angle φ as given by Eq. 3.11 and Eq. 3.12 as a function of the longitudinal magnetization component ml (the component parallel to the external field) For the case of coherent rotation the transverse component is increased if the longitudinal component is decreases and and vice versa. If no transverse component can be detected no rotation of the magnetization takes place and the reversal is governed by domain processes. 28 3.3. Diffraction gratings Figure 3.5.: Two limiting cases of magnetization reversal and the resulting vector-MOKE measurements. (a) shows the case of coherent rotation. The reversal is sketched and the transverse component, the angle and the magnitude of the magnetization are plotted as a function of the magnetization along the field. (b) depicts the case of domain formation and no rotation. The same quantities are plotted as in (a). 3.3. Diffraction gratings The most common diffraction experiments, and all experiments covered in this thesis, are performed in the so-called Frauenhofer diffraction geometry. This means that the source of light and the observer are at an infinite distance to the diffracting object. This condition is easily satisfied by the use of a laser as light source. Thus a plane wave is incident on the diffracting object, which is viewed as an object with a certain transfer function, f (y). The complex function f (y) describes the reflection or transmission of the amplitude of the electrical field vector and its absorption. In the following paragraphes the scalar diffraction theory from one-dimensional grating structures is outlined. In this context, scalar means that the diffraction is independent of the polarization of the incident light. Every point on the object acts as a new source of light, emitting a spherical wave, its amplitude and phase given by the transmission function. In one dimension the resulting diffraction pattern is the integral over the surface of the diffracting object multiplied with a phase factor: Z ψ(k) = f (y)exp[−i(ky)]dy, (3.13) where k is a reciprocal space vector defined in this case via k = k0 (sin αf − sin αi ). (3.14) The angles αi and αf define the directions of the incoming and diffracted beam, respectively. The wavenumber k0 is defined by k0 = 2π/λ, where λ is the wavelength of the electromagnetic wave. The simplest transfer function is that of a slit aperture of width a in one dimension: fslit (y) = { 1 if |y| ≤ a/2 . 0 if |y| ≥ a/2 (3.15) 29 3. Magneto-optical Kerr effect of thin films and thin film grating structures 7 18 x 10 16 14 Intensity 12 10 8 6 4 2 0 −40 −30 −20 −10 0 αf [°] 10 20 30 40 Figure 3.6.: Calculated diffraction pattern for perpendicular incidence, a = 2.3 µm, d = 5 µm, N = 8 and λ = 632 nm. The resulting diffraction pattern is given by: |ψ(k)|2 = a2 sin2 (ak/2) . (ak/2)2 (3.16) Another important case is the diffraction from a finite array of diffracting objects. If the transfer function is a regular spaced array of delta-functions the resulting intensity pattern is: sin2 (N dk/2) |ψ(k)|2 = , (3.17) sin2 (dk/2) where N is the number of delta-functions contributing to the diffraction pattern and d is the grating parameter. This function describes the well known intensity pattern from a diffraction grating with major and minor intensity maxima. The intensity in the major maxima is increasing with N , the number of minor intensity maxima between the major maxima is (N − 2). For perpendicular incidence the major intensity maximum of order n occurs if the Bragg-formula is satisfied: d sin αf = nλ. (3.18) Considering the more general case of non-zero αi leads to the grating-equation: d(sin αf − sin αi ) = nλ. (3.19) At this point it is important to note that the exact form of the grating equation depends on the sign convention chosen. For Eq. 3.19 the angles in the first and third quadrant are 30 3.4. Bragg-MOKE positive and angles in the second and fourth quadrant are negative (cartesian convention, see [45]). Another source of confusion often occurs in comparison with x-ray scattering techniques, where the angles are defined not relatively to the surface normal but with respect to the surface. Therefore every sin function in the above formulae would be converted to a cos for x-ray diffraction. On the other hand, the most common case for x-ray scattering is diffraction from the lattice perpendicular to the surface, which again leads to a rotation of the coordinate system of 90◦ . In total the form of the above equation is the same for diffraction-gratings and Bragg-diffraction from crystals. A more realistic case of a diffraction grating is to assume single slits with a transfer function as given in Eq. 3.15, which are convoluted with the regular grating of deltafunctions as discussed above. From Fourier-theory it is known that the Fourier-transform of a convolution of two functions is the product of the Fourier-transforms of the two single functions (convolution-theorem). Therefore the intensity function of this case is the product of Eq. 3.17 and Eq. 3.16, i.e. the intensity pattern shows maxima at the same positions as for the delta-function array (Eq. 3.19 still holds), but the intensity at the maxima display a modulation whose envelope is the intensity function of the single slit. This model can be used in the most cases considered in this thesis. An example of a diffraction pattern according to Eq. 3.17 assuming an envelope as given in Eq. 3.16 is plotted in Fig. 3.6. Of course, an even more realistic intensity function would be the Fourier-transform of the real transfer function, i.e. the modulation of the (complex) reflection coefficient when viewed perpendicular to the stripes, also taking the hight difference of the stripes and grooves into account. More details of the theory of diffraction gratings can be found in [45] and elementary textbooks on optics (e.g. [46]). It should be mentioned that the above theory of diffraction gratings is a scalar theory. That means that during diffraction the two orthogonal polarization directions are not coupled and the polarization state is conserved. Obviously this is inherently not the case for the combination of diffraction and Kerr effect. Even non-magnetic, metallic gratings couple the polarization directions, thus a vector-theory of diffraction is necessary [45]. A simple example is a grating consisting of thin metallic wires which has been used as a polarizer. In this case the E-vector of the transmitted beam is aligned parallel to the wires. The vector theory of diffraction is a broad subject in optics and several textbooks and articles cover the matter, e.g. [47, 48, 49], and references therein. However, in this thesis only the scalar theory is considered. Therefore results can not be fitted to models exactly, but it will be shown that main features of the measurements can be discussed in the framework of the scalar theory. 3.4. Bragg-MOKE The term Bragg-MOKE stems from the used combination of the usual Kerr effect measurement and diffraction from a lateral structure, e.g. a diffraction grating. Instead of analyzing the intensity or polarization rotation of the specular reflected beam, signals of the diffracted beams are measured. This section first reviews the literature on the matter and than simulations of several effects playing an important role for Bragg-MOKE are reported. 31 3. Magneto-optical Kerr effect of thin films and thin film grating structures 3.4.1. Review of Bragg-MOKE literature First experiments The first time the combination of Kerr effect and diffraction from grating structures was mentioned in literature was 1993 by Geoffroy et al. [11]. In this article measurements of the transverse Kerr effect from different diffracted intensities from a SmCo4 square arrays with a grating parameter of 4 µm are reported. The loops measured at the diffraction spots did not simply reproduce the standard MOKE curve but showed remarkable differences. The measurements are reproduced in Fig. 3.7. In addition, the magnitude of the measured Kerr effect in saturation changed with the order of diffraction. In [11] a simple explanation of the observed effects is offered in the framework of the scalar diffraction theory as it is outlined in Sec. 3.3. In this case two main contributions have to be taken into account: • The diffracted light originates not only from the ferromagnetic grating but also from the not ferromagnetic substrate. The phase shift, φh , between both contributions is depending on the height of the structure and the angle of diffraction and incidence: 2πh [1 + cos(αi + αf )], (3.20) φh = λ cos αi This contribution might change the amplitude of the measured signal. • The magnetization distribution in the magnetic elements (i.e. the domains) give rise to a weak modulation of the reflectivity in the elements. This magnetization modulation contributes via Fourier transformation to the signal in the BraggMOKE hysteresis loops, but does not change the Kerr amplitude in saturation. Especially the last point needs further explanation: As discussed in Sec. 3.1 the transverse Kerr effect measures the intensity shift as a function of the magnetization rather than the polarization rotation as it is the case for the longitudinal geometry which is discussed in this thesis. However, the magnetic contribution in the transverse Kerr signal is only a small contribution superimposed on the non-magnetic reflectivity signal. The authors of [11] assume the same to be true for the Bragg-MOKE effect in transverse geometry. The measured intensity as a function of the magnetization at different Bragg-spots is the product of the intensity due to the Fourier transform of the structural grating and the Fourier transform of the magnetization distribution. When measured as a function of the external field only the latter is changing, thus the Bragg-MOKE curve is related to the change of the Fourier component with wave number n/d of the spatial distribution of magnetization within the magnetic elements. The authors stress that the coherent Bragg diffraction peaks carry information on the mean magnetization contribution in the patches. Following this assumptions the authors of [11] develop a model assuming a simple two domain wall configuration in the square elements. The resulting model calculation could qualitatively reproduce the measurements but no quantitative correspondence was achieved. In particular the Kerr signal in saturation at different Bragg-spots could not be explained. Taking the article of Geoffroy et al. as a starting point several publications followed both experimentally and theoretically. Again the two lines of investigations are visible: 32 3.4. Bragg-MOKE Figure 3.7.: Transverse Bragg-MOKE measurements form a square dot array of hard magnetic material with a grating parameter of 4 µm. The figure is taken from Geoffroy et al. [11]. one is to understand the amplitude of the magneto-optical signal in saturation and the other is to understand the shape of the Bragg-MOKE curves as resulting from the domain states during the re-magnetization process. Measurements and simulations of the saturation Bragg-MOKE amplitude in transverse geometry In the article of van Labeke et al. [13] (also see [50]) an experimental situation is constructed, which is essentially easier than the experiments discussed in [11]. The sample under investigation was a one-dimensional grating structure, which was completely covered with a soft magnetic film, such that the grating and the grooves in between are covered with ferromagnetic material (relief grating). The resulting grating was expected to exhibit no domain structure related to the grating geometry. Therefore the measurements concentrated on the Bragg-MOKE amplitude in saturation as a function of the incident angle rather than the shape of the hysteresis loops. Measurements were carried out in transverse MOKE configuration and compared to calculations which used 33 3. Magneto-optical Kerr effect of thin films and thin film grating structures the Rayleigh expansion and a perturbation method. The complex matrix formalism calculations reproduced very well the measured Bragg-MOKE amplitudes without the need to introduce phenomenological parameters (the optical constants were measured on flat surfaces and used in the simulations). The variable used in this study was the sum of the incident and diffracted angle of the laser beam for a given order of diffraction, Θ = αi + αf,n . The result is that the transverse Bragg-MOKE effect increases for increasing Θ and for constant Θ the Bragg-MOKE amplitude increases with n. For small values of Θ the curves can be approximated by a linear function. Another important observation was that the curves always pass through the origin, i.e. for Θ = 0 the measured and calculated Kerr amplitude is zero. This means that if the diffracted beam is directed along the incident beam (Littrow geometry, see [45]) no Bragg-MOKE curve can be measured. Another important result from the calculation was, that in transverse geometry the p- and s-polarized eigenmodes are not coupled, there is neither depolarization nor rotation of the polarization and the transverse Bragg-MOKE effect can only be observed with p-polarized light, as already discussed for the standard transverse MOKE in Sec. 3.1 The experiments were extended to inhomogeneous gratings (i.e. the grooves were non magnetic) in [51, 52]. The same experiments were performed as before, but also the relation between the transverse Bragg-MOKE amplitude as a function of n and αi and the diffracted intensity without Kerr effect were investigated. It turned out that the Kerr effect can be increased dramatically for n 6= 0 and increasing αi . The increase was greatest for n = ±1, in this case a maximum combined with an abrupt change of sign is observed for a special αi at which the diffracted intensity reaches a minimum. The observed effects are explained with the interference of the light diffracted form the ferromagnetic grating and the light diffracted by the non-ferromagnetic substrate-grating formed by the grooves. Essentially the same effect takes place for the anti-reflection coatings commonly used to enhance the Kerr effect by interference of reflected beams at two surfaces. By varying the angle of incidence one can find a configuration in which the light diffracted by the non-magnetic sub-grating compensates the Fresnel component of the light diffracted by the ferromagnetic sub-grating. The total intensity is minimized and consequently the relative change of intensity due to the transverse Kerr effect reaches a maximum. The parameters for finding this maximum are the height of the stripes, the angle of incidence and the optical constants of the material. This interference effect of Bragg-MOKE amplitude amplification is superimposed to the effect due to the optical properties of the ferromagnetic grating itself as discussed in [13] without non-magnetic sub-grating. Bragg-MOKE hysteresis loop simulations in transverse geometry The article of Vial and Labeke [53] is an extension of the work of the same authors [11, 13] in two respects: first, the model for the simulation of the transverse BraggMOKE effect, as it was presented in [13] (previous paragraph of this section), is further extended to take into account inhomogeneous gratings and, second, the domain model discussed in [11] is used together with the vectorial diffraction theory in order to model Bragg-MOKE hysteresis loops. The general experimental and theoretical facts about the Bragg-MOKE amplitude at different orders of magnitude and varying angle of incidence 34 3.4. Bragg-MOKE from [13] are reproduced, but additionally Bragg-MOKE hysteresis loops are modelled, with the same magnetic model as was used in [11]. The magnetic model used is based on the assumption that the magnetization direction inside magnetic domains is directed only parallel or anti-parallel to the external field, which is in the transverse configuration perpendicular to the scattering plane and along the stripes of the assumed grating structure. Hence, the magnetization is always oriented along the easy axis of the stripe-induced uniaxial anisotropy. Two cases were distinguished: • one wall configuration: during re-magnetization only one 180◦ domain wall moves from one side of the stripe to the other, the domain wall is oriented parallel to the stripe. This configuration does not lead to a change of the shape of Bragg-MOKE hysteresis loops compared to the standard MOKE curve. • two wall configuration: in the middle of the stripe one domain with the opposite magnetization direction nucleates and the two domain walls propagate to the sides of the stripe. The case of two domains nucleating at the edges and travelling inwards is equivalent. The simulation of this case leads to Bragg-MOKE hysteresis curves similar to the one observed in [11], without the need of taking phenomenological parameters into account. As an example the simulations for a diffraction grating is reproduced in Fig. 3.8. Here the transverse Bragg-MOKE signal is plotted as a function of the magnetization (and not the external field), i.e. a Bragg-MOKE loop without change in shape as compared to the specular loop would display a straight line (see also [11]). In this representation the Bragg-MOKE loops only show the effect of the domains. Transverse Bragg-MOKE from anti-dot arrays Bragg-MOKE measurements performed at a different kind of sample were reported by Vavassori et al. [14] and Guedes [15]. Instead of arrays of dots or stripes, antidots, i.e. holes in a continuous thin film, are investigated. In the first article [14] a theory for the Bragg-MOKE amplitude in saturation for the case of antidot arrays is developed. The authors find simple expressions for the transverse MOKE signal at diffraction spots, valid for p-polarized light. The theory described here is also a scalar diffraction theory taking the form factor of magnetic discs into account. More interesting is the idea presented in [14] to split the hole array into two sub-films with opposite magnetization direction in order to model the hole array. In their description the hole array is represented by the sum of a continuous film and a dot array with opposite magnetization direction. The authors also report on small changes of the hysteresis loops taken at diffraction spots. They relate the small changes to blade-like domains forming around the holes in the Fe film. The small wiggles found around the coercive field of the Fe film are reported to be independent of the order of diffraction and the direction of the external field. From this it is concluded that the domains have the same periodicity as the hole array. The domain state is also described by the addition of several virtual films. In order to illustrate this a figure taken from [14] is reproduced in Fig. 3.9. These experiments are extended to hole arrays with elliptical holes in [15]. In this case the authors find a pronounced change of the slope of the Bragg-MOKE hysteresis 35 3. Magneto-optical Kerr effect of thin films and thin film grating structures Figure 3.8.: Simulation of transverse Bragg-MOKE measurements from a diffractiongrating. The Kerr intensity is plotted as a function of the magnetization for different order of diffraction p. The figure is taken from Vial and Labeke [53]. loops if the elliptical holes are aligned with their long axis perpendicular to the field. The shape of the Bragg-MOKE hysteresis loops also changes with order of diffraction. This effect is attributed to domains forming around the coercive field which connect holes in the 45◦ direction of the square lattice. The width of the domains is of the order of the grating parameter. In [15] also a domain image is presented confirming these assumptions. The Bragg-MOKE hysteresis loops are explained essentially with arguments already given by Geoffroy [11]. A quantitative method is described, which calculates the magnetic form factor of the domains. For standard MOKE the signal is proportional to one component of the magnetization, e.g. my . The case of BraggMOKE can be described by replacing my with a magnetic form factor, f , which is the Fourier-transform of the domain-structure in one unit cell, S, of the (anti)dot array: f= Z S my exp(ikr)dS, (3.21) where r is a position vector inside the area of the unit cell and k is the reciprocal space vector corresponding to the diffraction order to be analyzed, i.e. k = 2πn/d. The simulation presented in [15] for the assumed domains using Eq. 3.21 qualitatively reproduce the measurements. 36 3.4. Bragg-MOKE Figure 3.9.: Virtual addition of several domains, which was used by Vavassori et al. [14] in order to construct a model of the observed wiggles in the Bragg-MOKE curves. Bragg-MOKE measurements and calculations in polar geometry The article of Bardou et al. [54] reports measurements of the Bragg-MOKE effect in polar MOKE geometry from dot arrays with perpendicular magnetic anisotropy. The grating parameter ranges from 1.1 to 4 µm. The investigated set of samples consist of densely packed arrays, with the dot diameter close to the grating parameter, and a loosely packed array with the dot diameter equal to the half of the grating parameter. The shape of the hysteresis curves is reported not to change at the diffraction spots as compared to MOKE measurements at the specular reflected beam. However, interesting features were found with respect to the saturation Kerr rotation: • The Kerr rotation at the specular spot changes sign for the loosely packed dot array as compared to the continuous film and the densely packed arrays. • For a loosely packed array a change of sign of the Kerr rotation is reported for the specular MOKE curve, the Bragg-MOKE curves exhibit the same sign of the Kerr-rotation as the continuous film while the specular reflected beam changes sign. • The Bragg-MOKE Kerr rotation oscillates with a period of two as a function of the order for a dot array with the dot diameter half of the grating parameter. The qualitative explanation offered in [54] deals again with the phase difference between the light reflected from the background substrate and the magnetic dots. For different dephasing during reflection and subsequent interference, a change of the measured polarization rotation can be assumed. The theoretical work of Suzuki et al. [12] tries to build the theoretical basis for the observations in [54]. For the case of the polar Kerr effect, for large lattice parameters compared with the wavelength and the magnetic dots being large enough that edge effects can be ignored, the authors find very useful and concise relations for the BraggMOKE effect. The calculation first assumes a single unsupported dot of arbitrary shape, for which the retarded potential is evaluated. From this the authors find expressions for the polar Kerr-effect for diffracted light in non-specular directions, i.e. non-specular complex reflection coefficients. The most important result is that in the case of the single ferromagnetic island the polar Kerr effect is independent of the shape of the island. In a 37 3. Magneto-optical Kerr effect of thin films and thin film grating structures second step the calculation is extended to the case of an array of magnetic unsupported islands. In this case it is pointed out that nothing changes but the light obviously is only diffracted in certain angles for which the obtained relations for the unsupported islands remain valid. In the next step the substrate is taken into account and a phase factor taking the hight difference into account enters the calculation. For this case only the polar Kerr effect in specular reflection is explicitly discussed: For certain ratios of the reflection coefficient of the non-magnetic substrate and the magnetic islands a resonance-like amplification of the Kerr effect is expected as a function of the filling factor of the array. Also a change of sign is predicted. This corresponds well to the observations reported in [54]. What is left to do? - conclusion As a conclusion from the work in the literature one finds that three effects contribute to the subject: • optical constants: The relaxation of the constrain αi = αf leads to off-specular reflection coefficients, which have to be taken into account. • interference effects: Interference between substrate and grating may add a significant contribution comparable to the effect of anti-reflection coatings (like ZnS) commonly used in magneto-optics. • domain structure: As the diffraction process is mathematically a Fourier transformation of the reflection coefficient distribution, which is depending on the domain distribution, the Bragg-MOKE curves will depend on the actual magnetization distribution. As this will change during the remagnetization process, the shape of the Bragg-MOKE curve will be altered depending on the order of diffraction and on the particular domain structure. All Bragg-MOKE studies reported in the literature so far were carried out in the transverse and in the polar geometry. The longitudinal geometry was not considered although it is quite popular and has several advantages over the transverse Kerr effect: • the longitudinal effect measures the absolute Kerr rotation and not the relative intensity change. A more quantitative analysis may thus be possible. • the longitudinal effect is often easier to measure. • for two-fold symmetries (like diffraction gratings) the longitudinal effect enables to measure with the field perpendicular to the stripes if the diffraction spots are aligned in the plane of incidence. This geometry (hard axis) is magnetically more interesting than the easy axis configuration of the transverse Bragg-MOKE measurements. • the longitudinal Kerr effect can be used to construct a vector magnetometer. Obviously, there are also disadvantages of using the longitudinal Kerr effect, e.g. the additional second order effects may complicate the interpretation of the observed hysteresis loops and also a theory for the longitudinal Bragg-MOKE effect may be more 38 3.4. Bragg-MOKE complicated. This was proved by Bangert et al. [55], who showed that magnetic gratings already change the polarization state of the diffracted light without taking the Kerr effect into account. What is also missing in the literature is a systematic study of the influence of different geometries and materials of diffraction gratings on the observed Bragg-MOKE effects. Several studies are meaningful: • Change the stripe widths, but leave the grating parameter constant. In this situation the influence of different diffraction envelopes can be studies while the angle of diffraction and thus the optical constants are constant for a given order of diffraction. In addition, different stripe widths will change the influence of the demagnetizing effects of the edges. This will lead to different domains structures which can be analyzed. • Change the material and leave the grating structure unchanged. Two cases can be distinguished: a) only the material changes but magnetic anisotropies are constant. This may lead to a change of the optical parameters. b) change the magnetic anisotropy, e.g. isotropic soft magnetic material versus more hard magnetic material with intrinsic magneto-crystalline anisotropy. This will enable to compare different domain structures. • Change the grating parameters and leave the relative stripe width constant. This will change the angle of diffraction while the relative diffraction envelope of the stripes is constant. • Construct more complicated gratings, e.g. non-ferromagnetic gratings on ferromagnetic substrates, or ferromagnetic gratings on ferromagnetic substrates. This will further demonstrate the influence of the different parameters on the BraggMOKE effects. 3.4.2. Some simulations of Bragg-MOKE effects In this section some simulations of the effects which were described in the literature and which were reviewed in the previous section is presented. The aim of this section is to show the effect of simple cases, described with simple formulas, which will be used in the discussion of the measurements of this thesis. Fourier components of domains As pointed out especially in [11] and [15], the magnetic signal obtained from hysteresis measurements at diffraction spots represents the nth Fourier component of the magnetization distribution. Whereas Guedes [15] tries to model Bragg-MOKE hysteresis curves using numerical integration methods, Geoffroy analytically calculates the Fourier components for a simple, one-dimensional case. Both methods completely neglect the complicated optics of the interaction, which was pointed out in [53]. However, general and qualitative agreement between model and measurements were found. Several 39 3. Magneto-optical Kerr effect of thin films and thin film grating structures Figure 3.10.: Sketch of the re-magnetization models used for the simulation, see text. d denotes the grating parameter i.e. the period of the structure, w the width of the stripes and wd the width of the magnetic domain. archetypical domain structures can be calculated analytically. All calculations assume the magneto-optical signal to be given by the real part of the Fourier transformation : Z In = Re[ 0 d ikx m(x)e dx] = Z d m(x) cos(kx)dx, (3.22) 0 where d is the grating parameter (i.e. the volume of the unit cell in one dimension), the wave vector is given by k = 2π , and m(x) is the magnetization distribution. Note that d the magneto-optical signal as given in Eq. 3.22, is a function of the magnetization rather than the external field, thus the effects of anisotropy and pinnig of domain walls are not taken into account. These types of curves have been named diffraction hysteresis loops (DHL) in [11, 53]. Experimentally DHL’s are obtained by plotting the Bragg-MOKE curve of order n as a function of the normalized magnetization measured in specular geometry. (Therefore the DHL for n = 0 is a linear function through the origin with unit slope.) Two 180◦ domain-wall model This is the model proposed in [11]. In its original form it covers the case of magnetization reversal along the easy axis (external field parallel to the stripes) and magnetic sensitivity direction also along the stripes, but can be generalized to the case covered in this thesis, which is magnetization reversal along the hard axis with the magnetic sensitivity direction also perpendicular to the stripes. 40 3.4. Bragg-MOKE w=0.2*d 1 0.5 0 −0.5 −1 −1 n=1 n=2 n=3 0 In In 0.5 w=0.33*d 1 n=1 n=2 n=3 −0.5 −0.5 0 m 0.5 −1 −1 1 −0.5 w=0.52*d 1 1 w=0.75*d n=1 n=2 n=3 n=1 n=2 n=3 0.2 0 −0.5 −1 −1 0.5 0.4 0 In In 0.5 0 m −0.2 −0.5 0 m 0.5 1 −0.4 −1 −0.5 0 m 0.5 1 Figure 3.11.: Plots of Eq. 3.24 and Eq. 3.25: Model of Geoffroy [11] for the diffraction hysteresis loop (DHL) of a system with two 180◦ domain walls. The solid line corresponds to the ascending and the dashed line to the descending branch of magnetization. The magneto-optical signal at the diffracted spot of order n, In , is plotted as a function of the magnetization m of the sample. Different cases for w and n are considered as indicated in the figure. For saturation M = m(x)dx = ms ∗ w = 1, with the width of the stripe, w, and ms = 1/w, the saturation magnetization density (see Fig. 3.10(a)). For the descending branch (M is swept from positive to negative saturation), one domain with the reversed magnetization direction nucleates in the middle of the stripe, Fig. 3.10(b) (two 180◦ domain walls). Therefore M = ms ∗ w − 2 ∗ ms ∗ wd , where wd is the actual width of the )w reversed domain. From this one can easily find wd = (1−M . The domain pattern can 2 thus be viewed as a superposition of the saturated stripe with two times the domain of negative magnetization. Therefore the DHL at order n is: R In = ms "Z w cos(kx)dx − 2 0 # w/2+wd /2 Z cos(kx)dx , (3.23) w/2−wd /2 which can be evaluated to: d πwn cos In = πwn d " πwn πwn (1 − M ) sin − 2 sin d d 2 For the descending branch the reversed domain is given by wd = can be calculated to d πwn cos In = πwn d " !# (1+M )w 2 πwn πwn (1 + M ) − sin + 2 sin d d 2 . (3.24) and the DHL !# . (3.25) 41 3. Magneto-optical Kerr effect of thin films and thin film grating structures Some examples are plotted in Fig. 3.11. From Eq. 3.24 and Eq. 3.24 and Fig. 3.11 some important conclusions can be drawn: • Specular case: limn→0 In (M ) = M , as expected • Saturation: For both the ascending and descending branch one finds: In (M ± 1) = d sin( 2πwn ) 2πwn d • For small πnw the ascending and descending branch fall together, and for saturation d In = ±1. Thus, for small stripe widths relative to the lattice parameter and small order of diffraction the DHL is identical to the specular case. • An interesting case is w = d2 , for which one finds In (M = ±1) = 0 ∀n, see also Fig. 3.11. • Because of the symmetry of Eqs. 3.24 and 3.25, the two values of the ascending and descending branch at M = 0 are equal with opposite sign and for most cases (see Fig. 3.11) non-vanishing. As m = 0 corresponds to the coercive field, Hc , in the specular MOKE curve, it is clear that in general Hc of the Bragg-MOKE curves is not identical to Hc of the specular measurement. • All curves display a point-symmetry to the origin. • The same results, only with exchanged signs, are obtained assuming not a central reversed domain but two reversed domains at the edges of the stripes. 90◦ edge domains Another straightforward model is to assume domains formed at the edges of the stripes with the magnetization direction along the stripe (two 90◦ domain walls). For the present geometry this leads to effective ”domains” at the edges of the stripes with zero magnetization, as the magnetic sensitivity direction is oriented perpendicular to the stripes, see Fig. 3.10(c). During the magnetization reversal the central domain changes its width wd . Thus the reduced magnetization is M = wd /w. In this model the ascending and descending magnetization branch is identical, no splitting in the DHL occurs3 . The magneto-optical signal is given by: In (M ) = 1 Z w/2+wd /2 cos(kx)dx, w w/2−wd /2 (3.26) which leads to d πnw πnw In (M ) = cos sin M . (3.27) πnw d d Some examples are plotted in Fig. 3.12. The two extremal cases n → 0 and M = ±1 are identical to those of the model with two 180◦ domain walls as discussed in the previous paragraph. The DHL of this model always passes through the origin, In (0) = 0, and all curves show point symmetry around the origin. 3 Therefore the term diffraction hysteresis loop is not exact and, in fact, in [11] the term DHL is only used if the two branches of magnetization are distinct. However, for clarity in this thesis DHL denotes the representation of the Bragg-MOKE curve as a function of the magnetization regardless on wether splitting occurs or not. 42 3.4. Bragg-MOKE Figure 3.12.: Plots of Eq. 3.27: Model with edge domains, see Fig. 3.10(c). The magneto-optical signal at the diffracted spot of order n, In , is plotted as a function of the magnetization m of the sample. Different cases for w and n are considered as indicated in the figure. w=0.2*d 2 0 −1 −2 −1 n=1 n=2 n=3 0.5 In In 1 w=0.33*d 1 n=1 n=2 n=3 0 −0.5 −0.5 0 m 0.5 −1 −1 1 −0.5 w=0.52*d 0.5 1 w=0.75*d 0.1 0.5 n=1 n=2 n=3 0.05 0 In In 0 m n=1 n=2 n=3 0 −0.05 −0.1 −1 −0.5 0 m 0.5 1 −0.5 −1 −0.5 0 m 0.5 1 Figure 3.13.: Plots of Eq. 3.28: Remagnetization via coherent rotation or irregular domain formation. The magneto-optical signal at the diffracted spot of order n, In , is plotted as a function of the magnetization m of the sample. Different cases for w and n are considered as indicated in the figure. 43 3. Magneto-optical Kerr effect of thin films and thin film grating structures Coherent rotation For magnetic measurements along the hard axis a case without the formation of domains is possible by the coherent rotation of the magnetization from the hard axis in saturation into the easy axis in remanence (along the stripe axis). Because only the magnetization component perpendicular to the stripes is recorded this leads essentially to a change of the reduced magnetization without changing the domain width, wd = w, as depicted in Fig. 3.10(d). The same model also covers the case of irregular domains formed inside the stripe which leads upon averaging to a reduction of the magnetization. Because the argument of the cos function in Eq. 3.22 is independent on M , the resulting DHL’s are linear functions of M with zero axis intercept, given by: d 2πnw sin M, 2πn d In (M ) = (3.28) some examples are plotted in Fig. 3.13. Note that for this case the shape of the BraggMOKE curve is not altered with respect to the specular MOKE curve. Only the Kerr effect amplitude and sign may change. One-domain wall configurations In addition to the above models, configurations with only one domain wall can be considered. The first case is one 180◦ wall moving perpendicular to the wire axis, where the wall can start from both sides of the stripe. Assuming equal probability of the reversed domain nucleating on either side leads to an average magnetization profile with zero magnetization on each side of the stripe. Therefore this model is identical to the model of two 90◦ walls nucleating on both sides and travelling inwards as described in the above paragraphs. The same considerations are true for one 90◦ wall. The averaging, however, does not lead to zero magnetization at the edges. Therefore this case can be described by a superposition of the model for two 90◦ walls and the coherent rotation model. Both cases with one domain wall do not introduce new features to the observed DHL. This situation changes if the assumption is lifted that the domains nucleate with equal probability on each side. If one assumes one reversed domain starting always from the same side of the stripe, the calculation leads to solutions in general similar to the ones depicted in Fig. 3.11, i.e. the two branches are distinct and point symmetry is fulfilled. However, the actual shape and oscillation periods are different. For the case of one 90◦ wall always starting at the same side, similar results are expected. For both cases it is assumed that the single domain always starts on one side for the ascending branch and on the other side for the descending branch. If this degeneracy is also lifted and the domain always starts to nucleate at on given position, the calculation shows that the two branches are distinct and additionally the point symmetry is lifted. This leads to solutions where the end point of one complete magnetization cycle is not identical with the starting point. 3.4.3. Interference between stripe and substrate The ferromagnetic stripes do not form a grating in vacuum, but of course they are supported by the substrate and buffer layers. In this cases of reflection gratings the substrate also forms a diffraction grating with the same lattice parameter as the ferromagnetic 44 3.4. Bragg-MOKE 1 6 (b) E =E =1; θ =0.01° intensity 1 θB [°] K 0.5 0 (a) E1=E2=1; θK=0.01° 0.5 (d) E1=1; φ=8/10 π; θK=0.01° intensity θB [°] K −3.1416 0 3.1416 φ [rad] 6 (c) E1=1; φ=8/10 π; θK=0.01° 0 −0.5 K 2 0 −3.1416 0 3.1416 φ [rad] 2 4 1 2 3 4 2 0 1 E2 2 3 E2 Figure 3.14.: Superposition of two polarized plane waves, as calculated in Eq. 3.29. In B the top row (a,b) the observed Kerr rotation, θK , and the intensity is plotted as a function of the phase shift , E1 = E2 = 2 and θK = 0.001◦ are held constant. In the bottom row (c,d) the observed Kerr rotation and 8 intensity is plotted as a function of E2 , E1 = 1, θK = 0.001◦ and φ = 10 π are held constant. stripes, but with a width of wsub = d − wF M . The substrate grating alone produces diffraction spots at the same positions as the ferromagnetic grating, because the grating parameter is equal. However, the substrate grating produces a pattern with altered intensities and with an additional phase shift with respect to the ferromagnetic grating. The resulting diffraction pattern is a coherent superposition of the two subgratings. It is instructive to consider first two plain waves with arbitrary intensity, a phase shift and, in addition, one of the waves exhibits a small rotation of the polarization direction. Assuming that the main polarization direction is s-polarization the two waves are given by: ! ! Φ1,s exp(iφ) Φ1 = = E1 , Φ1,p 0 ! ! Φ2,s cos θK Φ2 = = E2 . Φ2,p sin θK (3.29) (3.30) The coherent superposition leads to the following observed Kerr-rotation and intensity: <(sin θK ) , <(exp(iφ) + cos θK ) I = |sin θK + exp(iφ) + cos θK |2 , B θK = (3.31) (3.32) 45 3. Magneto-optical Kerr effect of thin films and thin film grating structures d=10; a=5.1 d=10; a=2.5 15 E1*0.5 E2 10 Intensity Intensity 15 5 10 0 0 5 3 3 2.5 2.5 2 2 θB / θK K B 5 0 −5 θK / θK E1*0.5 E2 1.5 −5 0 5 −5 0 n 5 1.5 1 1 0.5 0.5 −5 0 n 5 B Figure 3.15.: Plots of θK (Eq. 3.34) and intensity for d = 10 and a = 5.1 (left) and a = 2.5 (right). The amplitude of the substrate-grating is diminished (factor 0.5) and a phase shift is neglected. which is plotted for some values in Fig. 3.14. It can be seen that for a phase shift B of φ = ±π the observed Kerr rotation, θK , has a discontinuity (a). For the same situation the intensity displays a minimum (c). For a given phase shift the observed Kerr rotation also exhibits a discontinuity when varying the intensities. It can be shown B that a discontinuity of θK is observed if cos φ = − E2 cos θK E1 (3.33) is fulfilled. More explicit, one finds for the observed Kerr rotation: B θK = E2 sin θK . E1 cos φ + E2 cos θK (3.34) B A simple case is E1 = E2 = 1, if in addition θK and φ is assumed to be small θK → θ2K ; π B if θK is small and φ = 2 one finds θK → θK . The phase shift between the substrate and the grating is due to the path difference and thus to the height of the stripes h. The phase difference is given in Eq. 3.20 [11]. It can be seen that the phase difference increases with increasing αi and exhibits a maximum for the case αi = −αf (Littrow-mounting, i.e. the diffracted beam is reflected back into the incident beam). As a rule of thumb one finds that h must be significantly larger than a tenth of the wavelength in order to find large influence. This condition, however, is rarely fulfilled as technical problems often impede very high aspect ratios. If the phase shift is smaller, B B 2 θK depends only on the relative amplitudes: θK ≈ θK ( E1E+E ). 2 46 3.4. Bragg-MOKE If the diffraction pattern of the substrate and the stripes are calculated separately, the diffracted amplitudes E1 from the substrate and E2 from the ferromagnetic layer may differ significantly for a given order of diffraction n. This leads to modulations in the observed Kerr effect as a function of n. Using Eqs. 3.16, 3.14, 3.18, the diffraction pattern for the stripes and the substrate-grating can be calculated. The result can be inserted in Eq. 3.34. As example two situations are plotted in Fig. 3.15. For both cases the phase shift is neglected and the amplitude of the substrate grating is multiplied by 1 in account for different reflection coefficients. The grating parameter is assumed to be 2 10. The left column displays the case for the stripe-width of 5.1 and the right column for the stripe-width of 2.5. In the top panels the intensity for the two subgratings is B plotted separately and in the bottom panels the obtained values of θK are plotted. It is clearly seen that the observed Kerr rotation changes significantly and oscillates with a period of 2 for a ≈ d/2 and with a period of 4 for a = d/4. 47 3. Magneto-optical Kerr effect of thin films and thin film grating structures 3.5. MOKE setup All measurements in this thesis have been performed in the longitudinal MOKE configuration. As pointed out in Sec. 3.1 this means that the Kerr rotation angle θK has to be detected in order to measure a magnetic hysteresis curve. The signal of the longitudinal MOKE is up to first order proportional to the component of the magnetization lying in the intersection of the scattering and the sample plane. In the following sections the experimental realization of this concept will be discussed and an example will be given. Furthermore extensions of the standard setup will be introduced which allow to measure two orthogonal magnetization components (vector MOKE) and the setup for measuring the Bragg-MOKE effect will be explained. 3.5.1. Standard setup The most common application of the longitudinal MOKE is to measure hysteresis curves of a thin magnetic film as a function of the relative in-plane orientation of the sample to the external field. In this case the sample can be rotated around its surface normal (angle χ) while the external field, H, is fixed in the scattering plane defined by the incoming and outgoing light beams. From a sequence of hysteresis curves as a function of χ one can deduce the in-plane anisotropy of the ferromagnetic film or the coupling of ferromagnetic multilayers [56, 57, 58, 59]. This kind of anisotropy measurements provide the basis for all subsequent investigations of the Bragg-MOKE effect and other magnetic measurements. Therefore the measurement technique of the standard setup will be discussed in detail in the following sections. The setup and the measuring algorithm is based on the work of Th. Zeidler [60, 61]. More technical details can be found in this references. Mechanical alignment Fig. 3.16 displays the geometry of the setup. A HeNe laser is used to shine polarized light at an angle αi onto the sample, the sample environment with the magnet is depicted in Fig. 3.17. The light is reflected under an angle αf = −αi and passes into the polarization detection system, which is pictured in Fig. 3.18. The incident beam and the axis of the detector system define the scattering plane. The sample is mounted between the poles of an electromagnet (see Fig. 3.17), such that the field is in the scattering and the sample plane. The sample can be rotated around its surface normal using a stepper motor. It is important that during rotation the sample stays aligned perpendicular to the scattering plane, because otherwise the reflected beam would not pass through the detector system for any angle χ. Therefore the sample holder attached to the stepper motor can be adjusted with a three-screw mechanism. The magnet provides a field of ≈ ±2600Oe for a gap between the poles of 2.5 cm. As can be seen from Fig. 3.16 the accessible range of αi is reduced for smaller gaps between the magnetic poles, because the magnetic coils would disturb the optical path from the laser to the sample. For the gap of 2.5 cm the maximum αi is 45◦ . The Kerr effect depends on αi , reaching a maximum of the longitudinal MOKE effect typically around 55◦ (for Fe, see Sec. 3.1). From this follows that higher external fields are only obtained at the cost of smaller Kerr signals. 48 3.5. MOKE setup magnet coils sample rotation c sample rotator af ai s-polarizer modulator p-analyzer HeNe laser photodiode Figure 3.16.: Schematic drawing of the standard setup to measure the longitudinal Kerr effect. Optical path The light emitted from the HeNe laser is polarized perpendicular to the scattering plane (s-polarized) in a Glan-Thompson prisma before it is reflected at the sample. The Figure 3.17.: Photography of the sample environment of the MOKE apparatus. 49 3. Magneto-optical Kerr effect of thin films and thin film grating structures Figure 3.18.: Photography of the detector setup. reflected light passes through two Faraday setups before the polarization state is determined with an analyzer (crossed with respect to the polarizer: p-state) and a photodiode acting as a detector. In the two Faraday setups (see Fig. 3.18) the light passes through a glass-rod upon which an axial magnetic field is applied. The Faraday effect causes the linear polarization of the light to rotate an angle θF , which is given by θF = νlHa , (3.35) where ν is the Verdet constant, l is the length of the glass-rod and Ha is the axial magnetic field. Because the magnetic field is provided by a solenoid which is operated with a current IF , the Faraday-rotation is essentially proportional to the current IF : θF = kF IF , (3.36) with kF being an experimental quantity which was determined to be kF = 0.7165 ◦ A . One of the two Faraday setups is operated with an AC current at a frequency a little smaller than 1kHz (the modulator) and the other Faraday-setup is operated with a DC current (the rotator). The rotator is used to compensate the Kerr-rotation of the sample and the modulator is used to modulate the polarization in order to use lock-in techniques to detect the polarization state. This will be explained in the next section. Technical details of the Faraday setup can be found in [60]. The high power (5 mW) of the HeNe laser is not necessarily needed for the standard measurements, but it is helpful when the light diffracted from gratings or dot-arrays is analyzed. The detector is a photodiode. A photomultiplier turned out to be less useful as it is sensitive to the magnetic stray fields of the magnetic coils in the setup. Measurements of very small Kerr rotations were improved by using the photodiode. 50 3.5. MOKE setup 3.5.2. Measurement method The optical arrangement can be further analyzed by using the Jones-matrix method [30]. The electrical field vector of an electromagnetic plane wave can be written as: ~ t) = E(z, |Ex | exp i(kz − ωt + δx ) |Ey | exp i(kz − ωt + δy ) ! Ex Ey = ! , (3.37) where z is the propagation direction of the light wave and ω, k, δ are defined as usual. As only the state of the polarization is of importance, the wave can be normalized and the time and position dependent parts can be separated. The polarization of the wave can thus be represented by a vector of two complex numbers: |Ex | exp(iδx ) |Ey | exp(iδy ) ~ = E ! . (3.38) This is the Jones vector of a plane wave. Linearly polarized light with an angle α to the scattering plane can be expressed as ~ = E cos α sin α ! . (3.39) The general case of an elliptically polarized wave is given by ~ = E cos α cos − i sin α sin sin α cos + i cos α sin ! , (3.40) with being the ellipticity and α the tilting angle of the major axis of the polarization ellipse. Every optical element can be represented by a 2x2 matrix, T , expressing the change of the polarization by passing light through this element: ~f = T E ~ i. E (3.41) The easiest case is the Faraday rotator, which rotates the polarization for an angle α and is thus represented by a simple rotation matrix: TF araday = cos α sin α − sin α cos α ! . (3.42) For the case of the Faraday modulator the rotation angle α is an oscillating function: α = α0 sin ωt. The Jones matrix for every kind of optical element, like polarizers, λ/4plates and so forth, are tabulated in [30]. A sequence of optical elements is expressed as the product of the representing Jones matrices. In this manner the polarization state can be calculated for a complex setup. For the actual MOKE setup the product of Jones matrices of the polarizer, the sample, the Faraday rotator, the Faraday modulator and the analyzer has to be calculated. In general the sample is represented by the Jones matrix: ! r̃pp r̃ps S= , (3.43) r̃sp r̃ss 51 3. Magneto-optical Kerr effect of thin films and thin film grating structures which is the magneto-optical Fresnel reflection matrix, where the complex numbers rij express the ratio of the incident j polarized electric field and reflected i polarized electric field. The measured Kerr rotation is given by: ΘK = θK + iK = r̃ps r̃ss (3.44) From this considerations the intensity function of the setup can be calculated4 : 2 2 rps rps 1 2 I = rss 1 + 2 − 1 − 2 cos(2δ0 sin ωt)+ 2 rss rss rps 2 cos(φss − φps ) sin(2δ0 sin ωt)) , rss " ! (3.45) where δ = δ0 cos ωt is the modulation amplitude of the Faraday modulator and the rotation of the Faraday rotator is assumed to be zero. In this equation the Kerr-rotation cos(φss − φps ). The intensity function Eq. 3.45 is plotted can be identified: θK = − rrps ss for a certain set of parameters in Fig. 3.19(a). Obviously two frequency components are detected: the double fundamental frequency of the modulation and a smaller component corresponding to the fundamental frequency. For zero Kerr rotation the fundamental frequency vanishes completely. The panels (b) and (c) of Fig. 3.19 show the power spectrum and the phase of the Fourier transform of the intensity function in (a), calculated using the FFT method. In (b) the two peaks corresponding to the fundamental and the double frequency can be seen. The phase information depicted in (c) is important to detect the sign of the Kerr rotation. The difference of the two extremal points around the ground frequency of the phase signal as a function of the Kerr rotation is plotted in Fig. 3.20. This kind of signal can be obtained with a phase sensitive lock-in amplifier. The figure shows that the signal S proportional to the Kerr-rotation. The same result can be obtained by an expansion of Eq. 3.45 into a series of Bessel functions [62]: " 1 rps S= 1+ 2 rss 2 ! rps 2 − J0 (2δ0 ) 1 − − 4J1 (2δ0 )θK sin ωt− rss # ! rps 2 2J2 (2δ0 ) 1 − cos 2ωt + O(sin 3ωt) . rss (3.46) This also proves: S ∝ θK . It follows that two possible measuring techniques can be used: One can measure directly the signal S. The advantage is that this is fast, the drawback is that S in Eq. 3.46 also depends on the intensity of the incident laser light. As a result the hysteresis has to be measured several times and an average has to be taken in order to yield a sufficient resolution, which will slow down the measurement, or, alternatively, one has to use intensity stabilized lasers, which are more expensive. In this case the DC Faraday coil can be omitted. For the direct measurement of S it is more difficult to extract the exact value of the Kerr rotation, because the signal also depends on the other parameters 4 The calculation has been performed by W. Kleemann, Universität Duisburg, Germany [62]. 52 3.5. MOKE setup Intensity 0.0115 0.011 0.0105 0.01 0 0.002 0.004 0.006 0.008 0.01 2000 f [Hz] 4000 t[s] 0.2 (b) phase(FFT) power(FFT) 0.08 0.06 0.04 0.02 0 0 2000 f [Hz] 4000 (c) 0.1 0 −0.1 0 Figure 3.19.: (a) the intensity function Eq. 3.45 for f = ω/2π = 963Hz, δ0 = 2◦ , θK = −0.2◦ , θF = 0. (b) the power spectrum of the signal calculated with FFT, and (c) the phaser of the FFT 0.15 0.1 phase−signal 0.05 0 −0.05 −0.1 −0.15 −0.2 −40 −20 0 Kerr−angle θK [°] 20 40 Figure 3.20.: Phase signal as it can be extracted from the Fourier-spectrum in Fig. 3.19 as a function of the Kerr-rotation. 53 3. Magneto-optical Kerr effect of thin films and thin film grating structures (intensity, modulation amplitude) which may vary from experiment to experiment and depend on material parameters of the sample (ellipticity, absorption). The second modus is to compensate the Kerr rotation for each external field using the DC Faraday rotator. In an ideal case S in Eq. 3.46 is zero if θK = −θF . Therefore the signal is independent of the incident intensity and no stabilized lasers have to be used. Also it is advantageous that the actual Kerr rotation is directly proportional to the current through the Faraday coil and only has to be calibrated once. The drawback here is the longer time needed for the compensation at each measuring point. However, this is the technique used for all measurements in this thesis, a resolution of typically 10−4 ◦ has been achieved. From Eq. 3.46 also follows, that the signal can be increased by increasing the amplitude of the Faraday modulation, which will also result in a higher resolution of the measurement. For technical reasons the modulation amplitude is only ≈ 2◦ . Furthermore a high ellipticity of the light to be analyzed will result in a reduced signal and thus reduced resolution. Up to here only completely polarized light and ideal polarizers have been assumed. If the polarizers are non ideal or the sample depolarizes a portion of the light beam, a constant term has to be added to Eq. 3.46. In this case the compensation can not be done to zero but only to a minimum of the lock-in signal. Of course this will also reduce the accuracy of the measurement. Data acquisition The complete MOKE setup is computer controlled5 . The program controls the feedbackloop responsible for the compensation measurement, discussed in the previous section, and controls the external field at the sample position. For each measuring point the current IF and the external field H, measured in-situ using a calibrated Hall probe, are written into a file. After one hysteresis measurement is performed, the Kerr rotation is calculated from eq. 3.36 using the calibrated proportionality constant kF . The programm also controls the movement of the motor responsible for rotating the sample (angle χ). The software provides an easy to use graphical user-interface as well as a simple programming interface for specific extensions of the software, like measurements of the intensity, angle dependent measurements and so forth. 3.5.3. Extensions of the standard setup The standard MOKE setup described above has been modified in several ways in order to meet the needs of the different measuring techniques applied. Bragg-MOKE The complete setup was mounted onto a goniometer in order to be able to chose the angle of the incoming beam and the exit beam freely. A double rotary stage with two independent axes was equipped with computer controlled step motors. The first stage rotates the laser together with the polarizer and the second stage rotates the sample 5 The software was rewritten using the graphical programming language LabView 54 3.5. MOKE setup Figure 3.21.: Geometry of the Bragg-MOKE setup. The s-polarized light in perpendicular incidence (αi = 0) is diffracted from the grating. The sample rotation χ is selected, such that the magnetic field is perpendicular to the stripes. The Kerr detector can be rotated perpendicular to the plane of incidence by an angle αf . environment consisting of the magnet and the sample holder. This setup is necessary for the Bragg-MOKE measurements in order to measure at different diffracted beams. The setup is schematically depicted in Fig. 3.21 For most of the Bragg-MOKE measurements the angle of incidence αi was chosen 0◦ , because this is the most symmetric case. Only in this case the exit angles of the diffracted beams obey the relation αfn = −αf−n . Furthermore, for general symmetry reasons hysteresis curves measured at an order n must be identical to the Bragg-MOKE curve at order −n with only the sign changed, if αi = 0◦ . This can be seen with the following argument: 1. For the geometry described above the incoming laser beam and the sample normal form a common axis. The setup consisting of the sample, the magnet and the detector shall be viewed as rotatable around this axis. 2. If the complete setup is rotated around this axis by 180◦ the detected MOKE signal must be identical (including the sign). This rotation consists of the rotation of the magnet and the detector. 3. If only the magnet is rotated 180◦ the sign of the measured MOKE signal is 55 3. Magneto-optical Kerr effect of thin films and thin film grating structures reversed, because the field is reversed. 4. Therefore a rotation of the detector only (which is identical to a transformation n → −n) must also lead to a change of the sign of the MOKE signal. However, this highly symmetric configuration results in a smaller phase difference of the waves diffracted by the grating structure and the waves diffracted from the substrate as compared to a situation with non-zero incident angle and constant exit angle. Vector-MOKE A second magnet was constructed such that the direction of the magnetic field is perpendicular to the scattering plane. This geometry is used to measure the perpendicular magnetization component as described in Sec. 3.2. In the experiment first the sample is placed in the longitudinal configuration and the hysteresis curves are measured for different in-plane rotation angles with increments of ∆χ. Then the setup is changed to the perpendicular configuration and the sample rotation is repeated. A general source of error in the setup is the definition of the angle χ: For both the measurement of the longitudinal and transverse magnetization component the angle has to be redefined, which adds an uncertainty of ±2◦ to the measurement of χ. Vector-Bragg-MOKE At this point it is important to note that combinations of Bragg-MOKE measurements and Vector-MOKE measurements are drastically limited in the case of grating structures (more generally: diffracting arrays with a two-fold symmetry): 1. The diffraction pattern of a grating structure lies in the plane of incidence only if the stripes are oriented perpendicular to the plane of incidence (χ = 90◦ , hard axis configuration). 2. For other χ the conical diffraction geometry [16] is established, in which the detector has to be tilted out of the plane of incidence. This is technically difficult for the present setup, as the Faraday-rods and the analyzer are attached to the detector. 3. Therefore, in the longitudinal geometry only the hard axis orientation is accessible for Bragg-MOKE measurements. For the perpendicular configuration only the easy axis orientation (field along the stripes) is accessible probing the perpendicular magnetization component. 4. Therefore no Vector-Bragg-MOKE can be established with the present setup. 5. If the detector could be moved out of the plane of incidence, it remains to further investigations if the results of Vector-Bragg-MOKE can be used for calculating a vector-model of the remagnetization process. As in this case the scattering plane is not identical to the plane of incidence, the constrains for the longitudinal Kerr effect are not obeyed and a mixture of longitudinal and transverse Kerr effects can be anticipated complicating the analysis of the results. 56 3.5. MOKE setup Figure 3.22.: Example of the diffraction pattern of a magnetic grating. The figure displays a photography of the pattern as is can be observed on a screen. However, these considerations are not true for square arrays of magnetic dots or comparable systems with a four-fold symmetry of the diffracting structure. In this case equivalent diffraction spots occur in the plane of incidence as well as perpendicular to it, therefore no rotation of the sample is necessary when changing from the standard geometry to the perpendicular geometry. Intensity-measurements In addition to these modifications a simple photodiode can replace the complete polarization detection unit. This can be used to measure the integrated intensities of specular or diffracted beams as a function of the involved angles. This information was useful for clarifying the Bragg-MOKE effect as will be shown in the subsequent sections. As an example the diffraction pattern of a diffraction grating is depicted in Fig. 3.22. The photodiode is directed to the diffracted beams so that the complete visible intensity around a given spot covers the photo-sensitive area of the detector. 57 3. Magneto-optical Kerr effect of thin films and thin film grating structures 58 4. Sample preparation In this chapter the sample preparation including thin film preparation techniques and lithography methods are explained. The resulting grating structures were analyzed using some microscopy techniques which are also introduced. 4.1. Thin film preparation The samples discussed in this thesis are laterally structured ferromagnetic thin films. The samples were prepared by different physical vapor deposition techniques such as molecular beam epitaxy or re-sputtering. The typical thickness of the ferromagnetic films discussed here is in the range from 10 to 50 nm, thus no thin film effects like perpendicular anisotropy or reduced critical temperatures have to be taken into account. Apart from single films also more complex structures as magnetic multilayers, i.e. stacks of different materials, are discussed. While the sequence of the layers deposited onto the substrate is determined during the sample growth, the lateral structure of the sample is achieved by means of electron beam lithography or other lithographic techniques. In the following sections the different growth techniques are briefly presented and the growth procedure of a single Fe layer is used to illustrate the two techniques. 4.1.1. Molecular beam epitaxy Molecular beam epitaxy (MBE) uses directed beams of atoms or molecules which condense on a substrate. Ultra-high vacuum conditions ( 10−10 mbar) prevent contamination of the growing film as well as corrosion of the heated source material. The material is heated with a resistive heater (Knudsen cell) or the material is heated locally with a high energy electron beam. The system has a variety of materials build in, and a shutter system enables to grow complicated multilayers. During the growth the substrate temperature is typically enhanced in order to yield a single crystal film growth. The growth can be monitored using the Reflection high energy electron diffraction (RHEED) technique, where high energy electrons under grazing incidence are diffracted from the surface of the sample. The resulting interference pattern gives information about the quality of the samples [63]. A general review of MBE and related methods can be found in [64, 56]. Using the MBE method single crystal Fe films were grown as follows: A sapphire (Al2 O3 ) substrate in the (101̄2) (r-plane) orientation, previously annealed at 1050◦ C, is first covered with a Nb buffer layer of ≈ 20 nm thickness. The growth temperature is chosen to be 900◦ C. Afterwards the Nb buffer is annealed at 950◦ C. Subsequently a 59 4. Sample preparation Cr buffer of about the same thickness is deposited at a substrate temperature of 450◦ C and is annealed at 750◦ C. The resulting high quality Cr buffer is (001) orientated and has about the same lattice parameter as the subsequent Fe film. The 20 nm Fe film is deposited at 300◦ C and is not further annealed to prevent the intermixing of Fe and Cr at the surface. The Fe film is covered with a 2 nm thin Cr cap layer. In this configuration the Fe film grows along the (001) direction, has smooth surfaces and is single crystalline in a single structural domain [65, 63] and exhibits a fourfold magnetocrystalline anisotropy [59]. For a detailed description of the growth of Fe(001) on this system see [66, 58, 63, 65, 67]; a detailed description of the actual MBE machine used can be found in [65]. 4.1.2. rf-Sputtering Another technique of fabricating thin films is the sputter technique, where a plasma between the target material and the substrate is generated. The plasma ions, typically Ar ions, ballistically extract target atoms from the cathode. Due to the special geometry of the target, atoms are deposited onto the substrate. A special case is the sputter-process for insulating material, where an rf driven plasma is used. A general introduction to sputter deposition techniques can be found in [64]. The sputter process can be reversed, i.e. it can be used to dry-etch material of the sample. In order to do this, the sample to be etched is placed at a target position and material is removed from the sample due to the Ar-ion bombardment. This process is important for the patterning of the lateral structures as will be explained further below. More detailed information about the actual sputter machine used can be found in [68, 69] Some of the samples discussed in this work have been fabricated by sputtering Fe onto an Al2 O3 (112̄0) (a-plane) substrate at room-temperature. This results in polycristalline Fe films, basically without magnetic in-plane anisotropy. 4.2. Lithography In order to produce any kind of lateral structures in the micrometer or even nanometer scale a lithographic process is used. The main common feature of a great variety of different lithographic techniques is the production of a mask made from organic material and subsequently transferring this mask into the desired material. Essentially it is a three step process: • Production of the mask: an organic material is exposed to certain radiation, which can be by visible light, UV-light, x-rays or electrons, changing the chemical properties of the material. In the developing process the exposed portions of the mask are washed away and the not exposed parts of the film remain. • thin film preparation: use MBE or sputter techniques to produce a thin film. • image transfer: the mask has to be transferred into the thin film. There are two possible ways to do so: 60 4.2. Lithography Figure 4.1.: Two possible image transfer technologies: A) lift off technique; B) etching. For further explanation see main text. – lift-off: The mask is prepared on the substrate. Afterwards the thin film is prepared onto the mask. In the lift-off process a reactive bath (e.g. aceton) is used to wash out the remaining mask. The parts of the sample where material was deposited on the mask are removed. The deposited material remains in the ditches of the mask. This is illustrated in Fig. 4.1 A). It can be seen that the lift-off technique essentially forms the negative image of the mask. – etching: The thin film is prepared first on the substrate. Afterwards the mask is fabricated on top. Different etching techniques can be used to transfer the mask into the film. The illustration in Fig. 4.1 B) shows that this procedure results in a positive image of the mask in the film. For this thesis several techniques have been used to produce the grating structures. In the following the mask preparation used is explained and afterwards the particular image transfer is described. 4.2.1. Electron-beam lithography Most of the grating samples were prepared using the electron-beam lithography technique. First the samples are coated with a double layer of resist (PMMA = Polymethylmethacrylate) of different molecular weight. The coating is done with a spin-coater: the sample is rotated at high speed and a drop of PMMA is deposited on the sample (3000 61 4. Sample preparation electron energy beam size beam current dwell time step size write field magnification 20 kV 100 nm 0.6 nA 0.006 s 0.05 µm 819.22 µm2 75x Table 4.1.: Typical parameters of the electron-beam lithography setup which were used to produce a PMMA grating mask for stripes of 5 µm grating parameter and stripe widths from 1 to 4 µm rpm for all samples in this thesis for 30 s). Afterwards the layer is tempered at 180◦ C for 60 mins. This procedure is done twice to obtain two PMMA layers on top of each other. The bottom layer has a lower molecular weight (200kg/mol) than the top layer (950kg/mol), so that the bottom layer is more sensitive to the electron irradiation. The electron irradiation cracks the long organic molecules of the resist and after exposure the cracked, exposed, parts of the resist film are removed with a developing agent 1 . The double layer technique results in a structure with a mushroom-like profile as depicted in Fig. 4.2. This is especially useful for the lift-off techniques as the remover can more easily solve the remaining PMMA than with a single layer technique. A general review of electron beam lithography methods can be found in [70] A commercial scanning electron microscope2 was equipped with a software3 that scans the focused electron beam over the surface of a sample to form a previously designed image. The desired image or pattern is designed with a integrated software of the lithography system. The program is useful to design arrays of elements, like stripes, due to the ability of the program to automatically multiply one element to form an array. The software also controls the exposure time of the image. More information on the particular implementation of electron beam lithography can be found in [71]. As an example the lithography parameters of one grating structure are given in Tab. 4.1. The smallest pattern produced with this system were stripes with a width of 0.5 µm. 4.2.2. Other Lithography techniques Aside from the electron beam lithography technique other techniques exist, which are used to pattern thin film samples. Particularly optical lithography is the standard tool in the semiconductor industry. In this case an image of a previously prepared mask is projected onto the resist layer using an optical setup. This technique is fast and the mask can be used very often to produce the same pattern in the resist. For scientific use this is a disadvantage, because the fast and easy procedure is payed by inflexibility in the mask design. Principally the optical method is limited to the wavelength of the light 1 all chemical agents, the PMMA, the developer, and the remover are supplied by All Resist GmbH, Berlin, Germany with the trade names AR-P-679-04, AR-600-56 and AR-600-70, respectively. 2 Philips SEM 515 located at the institute in Bochum 3 Elphy Quantum 1.3 of Raith GmbH, Dortmund, Germany 62 4.2. Lithography used for the image. However, industrial solutions nowadays use UV light and are able to produce structures smaller than 100 nm. In contrast, state of the art electron beam facilities can provide structure sizes in the regime of 10 nm. The setups used for this thesis, however, only provide a resolution of about 500 nm in the case of the electron beam lithography and 1 µm in the case of optical lithography, which is sufficient as the goal of this thesis is to perform diffraction experiments with visible light. The samples prepared by optical lithography were provided by other teams, as will be indicated in the chapters dealing with the experimental results. An additional optical lithography method is the interference lithography technique. In this case the resist is exposed by a laser interference pattern created by the interference of a split laser beam (wavelength of 457.8 nm ) using a Michelson type interferometer setup [72]. Samples patterned with this technique have stripes or dots with a lattice parameter down to 150 nm. The advantage of this technique are the large areas (up to several square centimeters), which can by patterned and the extremely well defined period of the grating. The disadvantages are the inflexibility of the pattern and the typically rounded profile of the developed patches. Samples prepared using this technique also have been provided by a different team as will by indicated in respective subsequent sections. 4.2.3. Image transfer As already discussed above there are principally two different techniques: the lift-off and the etching technique. Both procedures have been used for some of the samples under investigation, but it turned out that a combination of both was especially suitable. The procedure is shown in Fig. 4.2. First the thin film of Fe or other material is produced by means of MBE or sputter techniques. Afterwards the grating mask is produced using the double layer PMMA technique as described above. (step (b) in Fig. 4.2) In the next step a Al2 O3 layer of about the same thickness as the Fe layer is sputtered on the mask. Subsequently, the Al2 O3 film is patterned by means of the lift-off process. This results in a Al2 O3 mask on top of the Fe film, which can be used as an etching mask. (step (d) - (e) in Fig. 4.2). The etching process is done using the reversed sputter process. The sputter-rate of Fe is approximately the same as for Al2 O3 , such that the sapphire mask is completely removed when the grooves reach the bottom of the Fe layer. If the PMMA mask is directly used as the etching mask one might run into problems, because the thickness of the PMMA layer is much less homogeneous than the thickness of sputtered Al2 O3 . In addition, the sputter rate of PMMA is approximately ten times higher than the sputter rate of the metals used as thin films. Therefore it is difficult to produce elements with a large aspect ratio, i.e. the ratio between height and width of the elements. A disadvantage of the described combination of etching and lift-off is the additional image transfer process compared to a simple lift-off or etching procedure. For each image transfer, the image looses resolution and the edges become more rounded. However, as will be shown further down, the resulting gratings are of excellent quality with respect to edge sharpness and uniformity. In practice, it turned out that the reversed sputter process used to etch the structure into the thin film was difficult to control with respect to the etching rate. Therefore the sample was dry-etched only for a short time and afterwards the thickness of the mag- 63 4. Sample preparation Figure 4.2.: Patterning process used for fabricating array of stripes. (a) the (epitaxial) Fe-film on the substrate is covered with two different photo-resists, (b) the resist is exposed and developed, (c) a sapphire mask is deposited on the resist, (d) via lift-off the resist is removed and the sapphire mask is transferred in the continuous film, (e) the resulting pattern is a negative image of the exposed portions of the sample. netic film was checked at an uncovered part of the sample. As it is difficult to measure thicknesses from small portions of a film in the thickness-range of some nanometers, the magneto-optical Kerr effect (see section 3.1) was used to monitor the relative thickness as a function of the etching time. The Kerr signal for thin films is approximately a linear function of the thickness [73] and, even more important, the goal of the grating fabrication is to produce gratings with alternating magnetic properties, which are directly monitored with the use of the Kerr effect. As an example Fig. 4.3 shows the dependence of the Kerr signal as a function of the etching time. The main disadvantage of this procedure is that after each etching step the sample has to be transferred out of the sputter machine and has to be adjusted in the MOKE setup. In addition, due to the difficulties in controlling the etching rates, the procedure has to be repeated several times. 4.3. Imaging Imaging of the produced structures is important for the production process as well as for the interpretation of the results of the different MOKE techniques used. The lateral dimensions of the stripes are an important input to calculations e.g. the stray field energy. In addition, the Bragg-MOKE technique will be shown to produce a Fourier transformation of the domain state of the samples, which needs to be confronted with the real space domain structure of the gratings. Therefore some domain imaging tools (magnetic force microscopy and Kerr microscopy) were used. Information of the lateral structure is mainly obtained from scanning electron microscopy and atomic force microscopy. 64 4.3. Imaging 0.07 0.06 Kerr signal θK [°] 0.05 0.04 0.03 0.02 0.01 0 5 10 15 20 25 30 35 40 total etching time [min] 45 50 55 Figure 4.3.: Kerr signal as a function of the etching time in the sputter machine. The sample is a 20 nm thick single crystalline Fe film prepared by means of MBE techniques on a Nb/Cr buffer. 4.3.1. Scanning electron microscopy Scanning electron microscopy (SEM) is by far the most important and convenient instrument in the field of nano- and micro-lithography. It is used to structure the PMMA mask and can than be used to analyze the mask and the resulting structure after developing, lift-of or etching. There is a large amount of literature about this standard analysis tool, e.g. [74]. The actual SEM used is described in [71]. The instrument is especially useful to study the resist (PMMA) masks, because there is usually a large contrast between the organic material and the underlying metallic layer due to different secondary electron yield. SEM was widely used to improve the imaging and developing parameters of the lithography steps. There are, however, several shortcomings when thin metallic structures on metallic underlayers should be imaged. The contrast is poor and the height differences cannot be imaged. In addition, no absolute information of the height of a structure can be obtained. Therefore, the atomic force microscopy technique described below was used to study the completed grating samples. As the SEM was used for the lithography as well as the imaging of the mask it defines the lateral length scale of all structure discussed in this thesis. Measurements of calibration samples reveal that this definition is accurate to better than 10% [75]. The SEM used was additionally equipped with a material analysis tool, the energy dispersive x-ray analysis (EDX). The energy of the x-ray spectrum caused by the absorbed high energy electrons is analyzed. From this information the material composition of a 65 4. Sample preparation sample can be gained. Interesting here was the possibility to analyze the sample locally resolved, e.g. between or on the structures (with a resolution of ≈ 1 µm). The etching processes were optimized using this information. 4.3.2. AFM and MFM Atomic force microscopy (AFM) is a very versatile tool for analyzing nano- and microstructures. A tip attached to a cantilever is scanned using piezo-electric manipulators over the sample and the tip to sample force is measured by the deflection of the cantilever. General information about AFM and the used instrument4 can be obtained from [76]. The instrument is ideal for imaging single stripes or small portions of a grating, the lateral resolution is ≈ 10 nm. AFM is the only possibility to gain absolute height information of patterned media, the height resolution is ≈ 0.1 nm. The AFM is very easy to use and fast to set-up, however, there are several shortcomings: • one must be very careful with image abberations and artefacts • the maximal field of sight is 30 × 30 µm2 , which makes it sometimes difficult to find the interesting structures of the sample. If the AFM is equipped with a magnetic tip and operated in the non-contact mode, it can be used to image magnetic domains (magnetic force microscopy, MFM). The stray field of the tip interacts with the stray field of the domains and domain walls. The MFM mainly images the perpendicular components of the magnetic stray field in the vicinity of the sample surface. Therefore it is especially suited to image films with perpendicular magnetic anisotropy. The interpretation of domain images with in-plane magnetization is difficult. Several setups at different laboratories were used, as will be indicated in the following chapters. 4.3.3. Microscopy and Kerr microscopy The most traditional way to image and analyze the lateral structures and gratings is optical microscopy. It is by far the fastest and easiest method and important information during the fabrication-process were gained. In addition, it is the best way to yield overview images of large areas. The contrast for all systems under investigation is very good, and special techniques like dark field images and transmitting illumination can be used to further improve the contrast. Obviously, the resolution is limited and therefore is is only a complementary method together with AFM and SEM. An optical microscope can be modified that the sample is illuminated with polarized light and the polarization of the reflected light is analyzed. In this way a Kerr-microscope can be constructed, as explained in [4]. For several magnetic grids the domain structure was imaged by Kerr microscopy in the longitudinal mode5 (see Sec. 3.1 and [4]). The weak magneto-optical contrast was digitally enhanced by means of a background subtraction technique [77]. The experimental setup has the option to apply in-plane magnetic fields in any direction independently of the magneto-optical sensitivity direction. To visualize 4 5 Park Scientific Instruments, AutoProbe CP the measurements were performed by J. McCord, IFW Dresden, Germany 66 4.3. Imaging the magnetic domains within the narrow stripes, the highest possible optical resolution, which is on the order of 0.3 µm for the given visible light illumination, was chosen. For examples of Kerr microscopy studies at ferromagnetic elements see e.g. [23, 24]. 67 4. Sample preparation 68 Part III. Results and discussion 69 5. Anisotropy of Fe(001) 5.1. Introduction Measurements of the anisotropy of a single crystal Fe/GaAs(001) film are presented1 as an example for the standard longitudinal MOKE geometry. This is a model system for ferromagnetic/semiconducting heterostructures with promising potential technical applications in the field of spin-electronics [1, 79] and has attracted much work in recent years [80, 81, 82, 83]. In this particular case a Fe film was prepared on a twodimensional electron-gas structure 2 [83, 78] which was intended to use for subsequent magneto-electronic experiments. Here the results from the magneto-optical measurements are highlighted. The measurements and the interpretation can serve as an example for the investigation of Fe(001) on other substrate materials, which are more important for the microstructures discussed in the following chapters. The sample discussed here is a sample out of a large set with different preparation conditions and compositions. Further details can be found in [83, 78, 84]. However, all samples showed rather similar magnetic behavior. 5.2. Measurements and discussion A thin Fe film was prepared on GaAs(001) substrate with a buried 2-dimensional electron gas as described in [78]. The thickness of the Fe layer is tF e = 7 nm. Ex-situ magnetic measurements at room temperature were carried out using the highresolution longitudinal magneto-optic Kerr effect (MOKE) set up as described in Sec. 3.5. Magnetic hysteresis curves were measured with different in-plane rotational angles φH between the in-plane applied field H and the in-plane crystallographic axes of the substrate (Fig. 5.1, inserts). A small coercive field, Hc , of 15 ± 5 Oe is measured for all directions φH . The φH -dependence of the magnetic remanence (Fig. 5.1, full squares) indicates the superposition of an in-plane 2-fold (uniaxial) magnetic anisotropy and an in-plane 4-fold anisotropy, in agreement with earlier reports [80, 82, 83] about Fe films on bulk GaAs(001). The uniaxial anisotropy has hard axes at φH = 90◦ and 270◦ , i.e. along the [11̄0] direction of the substrate, and easy axes along [110]. The 4-fold hard axes are observed at φH = 0◦ , 90◦ , 180◦ and 270◦ , i.e. along [1̄1̄0], [11̄0] etc., and easy 1 This section is based on a part of the article Magnetism and Interface Properties of Epitaxial Fe Films on High-Mobility GaAs/Al0.35 Ga0.65 As(001) Two-Dimensional Electron Gas Heterostructures [78]. 2 The GaAs substrate with a two-dimensional electron gas was prepared in the group of Wieck et al., Ruhr-Universität Bochum, Germany. The Fe layer was prepared by means of a MBE process in the group of Keune et al., Gerhard-Mercator-Universität Duisburg, Germany. 71 5. Anisotropy of Fe(001) φ Figure 5.1.: MOKE results of the Fe/GaAs sample: The large figure displays the remanence as a function of the sample rotation with respect to the magnetic field and the insets show typical hysteresis curves as example. The straight line is the result of a simulation as is described in the main text. directions along [01̄0], [100] etc., as expected for bulk bcc Fe. The origin of the 4-fold anisotropy is the crystalline cubic anisotropy of bcc Fe, while the uniaxial anisotropy is due to interface anisotropy [82]. In order to describe the behavior of the measured remanence (Fig. 5.1) a simple model for the in-plan magnetization of a single-crystal thin film was used assuming a coherent in-plane rotation of the magnetization vector. The total magnetic energy is given by (see Sec. 2) E(φ, H) = −µO MS H cos(φ − φH ) + K1 sin2 (2φ) + KU sin2 (φ − φU ) 4 (5.1) The first, second and third term in Eq. 5.1 describe the Zeeman energy, the cubic anisotropy energy, and the uniaxial anisotropy energy, respectively. φ, φH and φU are the angles between the coordinate axis and the magnetization vector MS , the applied field H, and the uniaxial easy axis, respectively, all oriented in the film plane. The magnetizationversus-field curve is a trajectory on the energy surface E(φ, H) starting at the maximum 72 5.2. Measurements and discussion MS K1 6 1.67 · 10 A/m 3.3 · 104 J/m3 KU 1.8 · 104 J/m3 φU 45◦ Table 5.1.: Magnetic parameters extracted from FMR and SQUID measurements [85] of the Fe/GaAs film, which were used for the simulation of the MOKE results (Fig. 5.1). The parameters are defined in the text, φU is the angle between the hard axis of KU ([11̄0]) and the easy axis of K1 ([100]). applied field (with MS and H aligned), and φ travelling through a local minimum on the energy surface upon decreasing H. From the values φ(H = 0) on this trajectory the remanence of the hysteresis loop can be calculated. (In the actual simulation a small field value of H = 10 Oe prior to field reversal was chosen rather than H = 0). The magnetization curves for different in-plane angles φH were simulated using this model. From each simulation the angle φ at a small field (H = 10Oe) is recorded, and the lowfield magnetization, Ml , is calculated according to Ml (10Oe) = MS cos[φ(10Oe) − φH ]. The resulting function Ml = Ml (φH ), normalized to MS , is compared to the experimental data. The full-drawn line in Fig. 5.1 is the result of the simulation, where the magnetic parameters as given in Tab. 5.1 were used (extracted from SQUID magnetometry and ferromagnetic resonance (FMR) measurements on epitaxial Fe(7.7 nm)/GaAs(001) [86, 85]). The experimental and simulated data are in good agreement (Fig. 5.1). Note, that the full-drawn line in Fig. 5.1 is not a least-squares fit to the experimental data. However, the FMR parameters K1 and KU used here are in fair agreement with the corresponding parameters of epitaxial Fe on bulk GaAs(001) obtained from Fig. 3 in Ref. [82] for tF e = 53.7 ML (or 7.7 nm Fe): K1 = 3.7 · 104 J/m3 and KU = 1.6 · 104 J/m3 . Obviously, the magnetic properties of the present epitaxial Fe(001) films on GaAs/Al0.35 Ga0.65 As(001) heterostructures and of epitaxial Fe(001) films on bulk GaAs(001) (grown under similar conditions [82]) are of similar quality. At this point it should be stressed again that the Fe(001) films prepared by MBE techniques used in this thesis for preparing microstructures show very similar behavior with respect to the coexistence of two- and fourfold in-plane anisotropy. Therefore the present study can be regarded as a model for the system Fe on Cr(001)/Nb(001)/Al2 O3 (11̄02). However, the observed two-fold anisotropy for the films used in this thesis is generally smaller, because the two-fold anisotropy decreases with increasing film-thickness. The latter is a clear sign that the origin of the two-fold anisotropy in Fe on Cr(001)/Nb(001)/Al2 O3 (11̄02) is a surface or interface effect. Furthermore the uniaxial anisotropy decreases with decreasing miscut of the substrate. The miscut of the sapphire leades to steps at the surface which results in the observed uniaxial anisotropy [59]. The magnetism and magnetic anisotropy of Fe on several substrates is the subject of many publications, however, there is not much literature on Fe(001) on the particular system Fe(001)/Cr(001)/Nb(001)/Al2 O3 (11̄02). 73 5. Anisotropy of Fe(001) 74 6. Fe-nanowires 6.1. Introduction In this chapter1 the results of magneto-optical Kerr-effect (MOKE) measurements are discussed, which were performed on a thin Fe film of 13 nm thickness, which has been patterned into a periodic arrangement of nanowires by means of optical interference lithography. The resulting array of nanowires consist of stripes having a width of 150 nm and a periodicity of 300 nm. MOKE hysteresis loops are measured within magnetic fields which are aligned in different directions, both parallel and perpendicular with respect to the direction of the nanowires as well as for various angles in between. A particular arrangement of the longitudinal Kerr effect measurement allows to identify both the longitudinal and the transverse component of the magnetization of Fe nanowires (vectorMOKE, see Sec. 3.2). From this both the angle and the magnitude of the magnetization ~ is derived. For a non-parallel alignment of the nanowires with respect to the vector M direction of the external magnetic field, the hysteresis loops consist of a plateau region with two coercive fields Hc1 and Hc2 , which is discussed as resulting from an anisotropic pinning behavior of magnetic domains in a direction along and perpendicular to the nanowires. 6.2. Sample preparation and experimental setup The sample under investigation is a thin Fe film of 13 nm thickness which has been transformed into a periodic nanowire array by an anisotropic plasma etching process after film deposition. The Fe film was grown within a UHV-MBE system on a 1.7 cm × 1.7 cm Al2 O3 (11̄02) (r-plane) substrate onto a 150 nm thick Nb buffer layer. In this case the Nb layer has a (001) orientation as can be derived from the 3-dimensional epitaxial relationship between niobium and sapphire [88] and as was revealed by x-ray scattering. The Fe growth temperature was 120 ◦ C. Unlike the formation of epitaxial Fe(110) on Nb(110) under the same growth conditions [89], Fe forms a polycrystalline layer on Nb(001). The polycrystallinity of the Fe film is advantageous as it suppresses the the intrinsic magnetic in-plane anisotropy, which is important for the present study as will be discussed below. The Fe film was additionally covered with a 4 nm thick Nb film in order to protect it against oxidation. Both the thickness and the roughness of the film were analyzed by x-ray reflectometry. After film preparation, the sample was spin coated with 1 This chapter is based upon the article Magneto-optical study of the magnetization reversal process of Fe nanowires, see Ref. [87] 75 z [Å] 6. Fe-nanowires 222 111 2 0 1 2 y [mm 1 ] 0 ] x m [m 0 Figure 6.1.: Surface morphology of a periodic array of Fe nanowires on a Nb/sapphire substrate imaged with atomic force microscopy. The 3-dimensional surface graph covers a region of 2.5 × 2.5 µm2 . a positive-photoresist. The resist was then exposed by a periodic line pattern created by interference of a split laser beam (wavelength of 457.8 nm) by using a Michelson type interferometer setup2 [90]. Subsequently, the resist was developed, resulting in a periodically modulated resist mask on top of the Fe film with a cosine-squared thickness modulation of 300 nm periodicity. The sample was then ion etched in a conventional rfsputtering system, using the modulated resist as an etching mask. During the sputtering process, the etching rate of the resist is ≈ 10 times higher than that of the Fe film. Therefore, the thickness of the resist mask was chosen such that it was initially about 10 times thicker than the metal film. As a result of the etching process, the periodically structured resist mask is transferred into the underlying Fe film so that finally an array of Fe wires on top of a Nb buffer was obtained. After lift-off of remains of the resist the quality of the sample was checked by imaging its surface morphology with an atomic force microscope (AFM). Fig. 6.1 shows a corresponding AFM image of 2.5 × 2.5 µm2 size. The measurement confirms (i) the regularity of the Fe nanowires having a width of ≈ 150 nm and a periodicity of 300 nm and (ii) that the wires are completely separated from each other. Note, that the stripes have a sinusoidal shape, which is a consequence of the lithography technique used. The sample was measured by means of the standard longitudinal Kerr effect as described in Sec. 3.5, including rotation measurements in order to investigate the samples magnetic anisotropy. In addition vector-MOKE measurements as described in Sec. 3.2 were carried out for different in-plane rotation angles of the sample. The measurement geometry is sketched in Fig. 3.4. 2 The resist mask was prepared by S. Kirsch, Institut für Tieftemperaturphysik, Universität Duisburg, Germany 76 6.3. Experimental results 6.3. Experimental results 6.3.1. Magnetic properties of the continuous Fe film After film deposition and before plasma etching of the stripes reference MOKE measurements were carried out on the continuous Fe film. The upper panel of Fig. 6.2 presents a typical MOKE hysteresis loop. The lower panel shows the Kerr rotation measured at remanence normalized to the Kerr rotation in saturation as a function of the angle of rotation χ about the surface normal of the Fe film. As can be readily seen from Fig. 6.2, the remanent magnetization amounts to about 88% of the saturation magnetization and it is almost independent of the angle of rotation. Thus, the overall in-plane magnetic anisotropy is indeed negligible, as was purposely intended by choosing the above mentioned growth conditions during film deposition. The coercive field of the continuous film is Hc = 250 Oe, as is typical for a poly-crystalline Fe film containing many grain boundaries. Therefore, the Fe film is a good candidate for the investigation of an induced magnetic anisotropy caused by a lithographic patterning process. 6.3.2. Magnetic properties of the Fe nanowire array: longitudinal component Fig. 6.3 shows three typical MOKE hysteresis loops taken for the array of Fe nanowires with different in-plane angles χ as measured in the longitudinal configuration. The hysteresis loop in (a) has been recorded with the external magnetic field oriented parallel to the wires. In this case, an almost squared hysteresis loop with a relatively large coercive field Hc is found, which represents the typical behavior of a sample when magnetically saturated along an easy axis of the magnetization. The coercive field Hc = 250 Oe indeed matches the value determined for the continuous film. However, the absolute values of the Kerr rotation in saturation are reduced when compared to the unpatterned film. This simply results from the fact that less material contributes to the measured signal in the case of the patterned film. Note also, that the slope of the hysteresis loop at Hc is less steep for the structured sample in comparison to the unpatterned film. This indicates that the magnetization reversal is dominated by a different switching mechanism. Moreover, the patterning process obviously leads to an increase of the absolute value of the remanence in the easy direction (parallel to the wires) suggesting a suppression of domain formation. Fig. 6.3(b) shows the corresponding hysteresis loop for the Fe nanowire sample after the sample was rotated by 90◦ , i.e. when the Fe wires were oriented perpendicular to both the external field and the plane of incidence. Here, a typical hard axis hysteresis loop is obtained. The coercive field reduces to Hc = 50 Oe and the saturation field increases to values beyond 1000 Oe, which is the maximum magnetic field value of the present experimental setup. Fig. 6.3(c) shows a MOKE hysteresis loop taken for an intermediate angle of rotation χ = 45◦ . The hysteresis loop clearly exhibits a step-like behavior, when the direction of the external field is reversed after saturation. Some characteristic features of the hysteresis loops shown in Fig. 6.3 are presented in Fig. 6.4 as a function of the angle of rotation χ. Fig. 6.4(a) shows the remanent 77 6. Fe-nanowires 0.04 θ [°] 0.02 K 0 −0.02 −0.04 −1500 −1000 −500 0 H [Oe] 500 1000 1500 0.8 0.7 θK Rem / θK Sat 0.9 0.6 0.5 0 45 90 χ [°] 135 180 Figure 6.2.: MOKE hysteresis loop of the unpatterned thin Fe film (upper panel). The lower panel depicts the results of MOKE hysteresis loop measurements as rem a function of the angle of rotation of the unpatterned sample, where θK sat as measured at remanence is normalized to θK as measured at saturation, and plotted as a function of the angle of rotation χ, which is a measure of the magnetic anisotropy. rem sat Kerr signal θK normalized to the Kerr signal at saturation θK as a function of χ, which yields information about the squareness of the hysteresis loops. According to Fig. 6.4(a) the remanent Kerr signal is reduced significantly at certain angles χ without reaching zero-values signifying the hard axis orientations (around 90◦ and 270◦ ). For the corresponding angles χ along the easy axis orientations (0◦ and 180◦ ) the ratio rem sat θK /θK measures almost unity. Similar behavior is obtained when the coercive field Hc is plotted as a function of the angle of rotation χ in Fig. 6.4(b), from which two pronounced minima of χ at around 90◦ and 270◦ are found. The small local maxima within the minima at 90◦ and 270◦ reflect small changes of the anisotropy which are also found in the continuous film at these angles (compare the lower panel of Fig. 6.2). The hysteresis loops measured for intermediate angles of rotation χ exhibit steps, which occur at characteristic magnetic fields Hc2 . Fig. 6.4(c) shows Hc2 as a function of the angle of rotation χ. Hc2 exhibits maxima at the hard axes directions at 90◦ and 270◦ . As mentioned above, hysteresis loops which are measured at intermediate angles of rotation show certain steps at magnetic fields Hc2 . Some of those hysteresis loops are 78 6.3. Experimental results Figure 6.3.: MOKE hysteresis loops as measured in the longitudinal configuration. (a) the external magnetic field is oriented parallel to the nanowires, χ = 0◦ ; (b) the magnetic field is oriented perpendicular to the nanowires, χ = 90◦ ; (c) the nanowires are rotated by an angle of 45◦ with respect to the direction of the external field. The insets depict correspondingly the orientation of the stripes with respect to the directions of both the external magnetic field H and the reflected laser beam. displayed again in Fig. 6.5 on an enlarged scale, where only half of the hysteresis loops as measured in longitudinal configuration are plotted. Obviously, Hc2 almost coincides with Hc1 when measured along the easy axis (χ = 0◦ see Fig. 6.3(a)) and starts to increase when the sample is rotated away from the easy axis (see Fig. 6.5). Finally, Hc2 l fades out into the region of saturation when χ = 90◦ . Note that θK values in between Hc1 and Hc2 increase from almost zero to the saturation Kerr rotation which is measured in the direction of the hard axis. Note also that even the hysteresis loop measured at χ = 0◦ (Fig.6.3(a)) reveals a small step in the vicinity of the coercive field. From the results of longitudinal MOKE measurements with the magnetic field direction within the plane of incidence one may indeed conclude that patterning of the thin Fe film into an array of nanowires induces a strong uniaxial anisotropy, which results 79 6. Fe-nanowires 1 θrem / θsat K K 0.8 0.6 0.4 0.2 0 (a) 45 90 135 180 225 270 315 360 90 135 180 225 270 315 360 90 135 180 χ [°] 225 270 315 360 300 (b) Hc1 [Oe] 200 100 0 0 45 600 (c) Hc2 [Oe] 500 400 300 0 45 Figure 6.4.: Results from hysteresis loop measurements at different angles of rotation χ: (a) Kerr rotation as measured at remanence normalized to the Kerr rotation as measured at saturation, (b) coercive field Hc1 and (c) the position of steps in the hysteresis loops measured at intermediate angles of rotation between the hard axis and the easy axis (Hc2 ). from the shape anisotropy of Fe nanowires. The saturation field measured with MOKE along the hard axis is rather high, exceeding 1000 Oe. 6.3.3. Magnetic properties of the Fe nanowire array: transverse component As discussed above, also the transverse component of the magnetization vector was determined by Kerr effect measurements. Corresponding hysteresis loops for three directions of the external magnetic field relative to the direction of the Fe nanowires are shown in Fig. 6.6. The hysteresis loop reproduced in Fig. 6.6(a) was measured within a magnetic field direction parallel to the Fe nanowires, i.e. along the easy axis of the magnetization. Ideally, within this configuration the measured Kerr rotation should remain zero unless 80 6.3. Experimental results 0.03 χ=99° χ=117° χ=154° 0.025 ↑ Hc2(99°) θl [°] 0.02 K ↑ Hc2(117°) 0.015 0.01 ↑ Hc2(154°) 0.005 0 −200 0 200 400 H [Oe] 600 800 1000 Figure 6.5.: Evolution of the characteristic magnetic field Hc2 as a function of the angle of rotation χ. Different line shapes represent typical hysteresis loops as taken in the longitudinal configuration for intermediate angles of rotation. χ = 99◦ is close to the hard axis direction, i.e. when the external field is applied perpendicular to the nanowires. certain components of the magnetization are directed along the plane of incidence during the magnetization reversal process. As can be seen from Fig. 6.6(a) the measured Kerr rotation is indeed almost zero. A small signal is seen with a maximum value at zero field. The origin of this behavior will be discussed below. The coercive field is the same as measured for the easy direction in the longitudinal configuration. Fig. 6.6(b) shows the hysteresis loop for a magnetic field pointing along the hard axis direction, i.e. in a direction perpendicular to the wires and the plane of incidence. In this configuration the measured Kerr rotation is larger than in the previous case and non-zero up to the highest field values. From this behavior one can infer that the sample could not be magnetically saturated in the hard axis direction, in agreement with the result shown in Fig. 6.3 (b). Both measurements therefore represent minor loop measurements. Fig. 6.6 (c) shows the transverse component of the Kerr rotation measured for an angle of rotation of χ = 45◦ . Also here, the hysteresis loop exhibits steps similar to the ones shown in Fig. 6.3(c). Note, that the coercive field, denoted here as Hc2 , is almost identical with the switching field Hc2 as defined in the previous section for the longitudinal magnetization component. According to the MOKE hysteresis shown in Fig. 6.6(a), the transverse magnetization component never reaches zero even at high magnetic fields, indicating that the sample is never fully saturated in the present setup. This also corresponds to the hysteresis loop measurements in the longitudinal configuration, for which always a small final slope in the high-field region was found, even in the easy axis configuration (Fig. 6.3(a)). Reason for this may be the presence of canted spins, which may result from the rough surface of the sample. In addition, this effect may be pronounced within our measurement through the high surface sensitivity of MOKE. 81 6. Fe-nanowires Figure 6.6.: MOKE hysteresis loops with the magnetic field oriented perpendicular to the plane of incidence, which probes the transverse component of the magnetization (a) the array of Fe nanowires is aligned parallel to the direction of the external magnetic field (easy axis configuration); (b) the magnetic field is applied perpendicular to the array of nanowires (hard axis configuration); (c) the field is oriented at 45◦ to the Fe nanowires (intermediate configuration). The insets depict the orientation of the stripes with respect to both the external magnetic field H and the reflected laser beam. Note that the θK -scale is divided by two in comparison with what is shown in Fig. 6.3 for the longitudinal configuration. 6.4. Analysis and discussion The results of MOKE hysteresis loop measurements as a function of the sample rotation and expressed in terms of the longitudinal and transverse component of the magnetization can be combined in order to calculate the angle and magnitude of the magnetization vector by using equations 3.11 and 3.12. The corresponding results of such an analysis are displayed in Fig. 6.7, which provide an overview of the magnetization reversal process of the Fe nanowires. Here, two ideal cases can easily be distinguished. If the 82 6.4. Analysis and discussion → ← 360 φ [°] |M| / |M|sat 270 180 90 1 (a) χ=0° 0.8 0.6 (b) 0.4 0.2 0 −1000 0 → ← 360 φ [°] 180 (c) χ=90° 0 1000 0 1000 0 H [Oe] 1000 1 |M| / |M|sat 270 90 0 −1000 1000 0.8 0.6 (d) 0.4 0.2 0 −1000 0 → ← 360 1 φ [°] |M| / |M|sat 270 180 90 0 −1000 1000 (e) χ=45° 0.8 0.6 (f) 0.4 0.2 0 −1000 0 H [Oe] 1000 0 −1000 Figure 6.7.: Conversion of the results of MOKE measurements from Fig. 6.3and 6.6 by using equations 3.11 and 3.12 into the angle of the magnetization vector (left column) and the magnitude of the magnetization (right column). (a) and (b) display the results for the external magnetic field directed along the ~ Fe nanowires (the easy axis); (c) and (d) show the results as obtained for H perpendicular to the direction of the Fe nanowires (the hard axis); in (e) and (f) the magnetic field is tilted by 45◦ with respect to the direction of the Fe nanowires. The solid lines represent the measurements with the magnetic field increased from negative to positive fields, the dashed lines correspond to measurements with the magnetic field decreased from positive to negative field values. angle of the magnetization vector changes without changing the magnitude |M |, then the magnetization reversal occurs via coherent rotation. On the other hand, if the angle of the magnetization vector remains constant within a certain magnetic field range but the magnitude |M | changes, then magnetic domains are formed. Firstly, the magnetization reversal with the magnetic field oriented parallel to the Fe nanowires (Fig. 6.7(a,b)) is discussed, i.e. when the magnetic field is directed along the easy axis of the nanowires having a two-fold magnetic anisotropy. When the magnetic field is swept from negative to positive values, the magnetization vector within the Fe nanowires remains parallel to the direction of the external field (φ = 0◦ in Fig. 6.7(a)), 83 6. Fe-nanowires until the magnetic field reaches a value which matches the coercive field for the easy e.a. ~ suddenly flips from 0◦ to 180◦ and further on remains axis Hc1 ≈ +250 Oe. Then, M parallel to the external magnetic field direction up to the highest magnetic field values. Upon reduction of the magnetic field value and subsequent reversal of the magnetic field direction the magnetization vector again rotates but in the opposite sense and reaches e.a. φ = 360◦ after passing the coercive field at Hc1 ≈ −250 Oe. Both the starting point ~ (φ = 0◦ ) and the end point (φ = 360◦ ) are degenerate. The plot of the magnitude of M sat ~ normalized to the saturation value M in Fig. 6.7(b) shows a decrease of |M | over a e.a. relatively wide magnetic field range, reaching almost zero around ±Hc1 . This indicates e.a. that the magnetization reversal process around Hc1 is dominated by domain formation within a wide magnetic field range. The shape of |M |(H) suggests that a number of domains nucleate along the nanowires, rather than just one single domain wall travels along each wire during the magnetization reversal process. A single domain would result in a much more narrow range over which |M | switches. The situation changes when the magnetic field is directed perpendicular to the nanowires, i.e. along the hard axis direction, as shown in Fig. 6.7 (c and d). Here, the magnitude of the magnetization |M | starts to decrease as soon as the magnetic field is reduced from its maximum value. At the same time the magnetization starts to change its h.a. direction. When approaching the coercive field for the hard axis orientation Hc1 = ±75 Oe, the magnitude of the magnetization drops in an even more rapid fashion, albeit it also does not reach zero as in the previous case. Thus, for this case we assume that a number of domains are formed along the wires immediately after leaving the saturation state, most likely as a result of large demagnetizing fields. Coherent rotation can clearly be excluded to be the relevant magnetization reversal mechanism, however, some rotational processes still remain to be active close to the coercivity. Around Hc1 the angle of the magnetization changes rapidly (Fig. 6.7(c)). This indicates the switching of domains which are oriented parallel to the field but perpendicular to the wire axis. The hysteresis loops as measured in both the hard and easy directions also differ significantly from each other in terms of the coercivity. Whereas Hc1 for the easy axis loop more or less reproduces the value of the unpatterned film, its value is strongly reduced for the hard axis loop. Obviously, patterning of the continuous Fe film does not change the pinning behavior of domains along the wire axis. However, for the hard axis orientation the domain walls parallel to the external field only have to move from one side of the wires to the other side, which is probably aided by the rounded shape of the wires, thus, leading to much smaller coercivities. When the magnetic field is oriented 45◦ with respect to the Fe nanowires, we expect to obtain a combination of the easy and hard axis behavior. Both the angle and the magnitude of the magnetization vector are plotted for this case in Fig. 6.7 (e and f). Starting at saturation within negative magnetic field values towards zero magnetic ~ changes from 0◦ to almost 45◦ , while the magnitude of M ~ only field, the direction of M slightly decreases. Within this first part of the hysteresis loop the situation is comparable to what is found for the hard axis loop. Magnetic domains are nucleated along the nanowires, which are oriented in only one direction, because the external magnetic field has a component parallel to the wires. Close to Hc1 the remaining domains, which are still oriented parallel to the external field, switch by 180◦ resulting in a decrease of the 84 6.5. Conclusions ~ . A further increase of the magnetic field leads to a meta-stable situation magnitude of M within the plateau region between Hc1 and Hc2 . Here, some of the domains appear to be oriented parallel to the field and some domains are oriented parallel to the stripes. ~ is almost perpendicular to This can be verified from the fact that the orientation of M 1 ~ the stripes, and the magnitude of M is reduced to ≈ 2 of its saturation value. Only after application of a magnetic field larger than Hc2 does the magnetization switch into a direction parallel to the magnetic field direction and the magnitude regains its saturation value. The domains which are oriented parallel to the nanowires switch their direction, comparable to the case of the easy axis loop. The above interpretation for the three main orientations of the Fe nanowires with respect to the direction of the magnetic field (0◦ , 90◦ and 45◦ ) is consistent with measurements taken in various other in-plane directions. Fig. 6.4(c) displays Hc2 as a function of the direction of the external field. Hc2 increases when approaching the hard axes directions (90◦ and 270◦ ), and it is small otherwise. This can be explained by the corresponding component of the external magnetic field that drives the switching process. For instance, close to the hard axis orientation the external magnetic field component along the stripes becomes small, therefore, a higher external field is required to switch the domains which are oriented parallel to the nanowires. This can be expressed approximately through the relation: Hc2 = Hce.a. / cos χ. The true domain structure of the nanowires is likely to be affected by the morphology of the stripes. In particular, it might by affected by their waviness and by the fluctuations of the wire-width and wire-height. Future magnetic force microscopy investigations or spin-polarized scanning tunneling microscopy [91] will help to identify the domain structure of the Fe nanowires. It should be mentioned that in contrast to the data presented in [3], here no evidence for a dipolar interaction between the wires could be discerned. This, however, most likely results from the fact that the separation of the nanowires is of the same magnitude as compared to the wire-width. 6.5. Conclusions With an analysis of magneto-optical hysteresis loop measurements taken for both longitudinal and transverse configurations the complex magnetization reversal behavior of an array of Fe nanowires has been demonstrated. The Fe nanowires were fabricated by patterning a continuous polycrystalline Fe film by using optical interference lithography. Before patterning, essentially no in-plane magnetic anisotropy could be detected for the continuous Fe film. The pattering process induces a uniaxial magnetic anisotropy with the easy axis of magnetization directed parallel to the nanowires. Due to the imposed shape anisotropy of the Fe nanowire array it is expected that the magnetization reversal behavior is quite different for a magnetic field orientation parallel and perpendicular to the nanowires. The magnetization reversal properties of the nanowires were investigated by analyzing both the transverse and the longitudinal components of the magneto-optical Kerr rotation. For a magnetic field orientation parallel to the nanowires an easy axis behavior of the magnetization reversal is observed which is dominated by domain nucleation. For the magnetic field oriented perpendicular to the nanowires the magnetization reversal is found to be also dominated by domain 85 6. Fe-nanowires nucleation but with different pinning potentials for magnetic domains magnetized either parallel or perpendicular to the wire-axis, leading to smaller coercivities for the latter ones. For intermediate orientations of the magnetic field with respect to the direction of nanowires (between hard and easy axis) a superposition of both magnetization reversal processes is found, resulting in a plateau-like region in the hysteresis loop due to a metastable domain configuration. In summary, the analysis shows the potential usage of a vector magnetometer in order to unveil the complex magnetization reversal processes of nanoscaled magnetic wire-arrays. Because of the small grating parameter no Bragg-MOKE experiments could be carried out on this sample. The laser wavelength is too large. This problem does not exist for radiation with smaller wavelengths like thermal neutrons or soft x-rays, both being also very powerful magnetic probes. Measurements using polarized neutron reflectivity were performed at this particular grating structure [92]. In this experiments hysteresis loops were measured at the specular reflection and at a first order Bragg diffraction spot of the lateral periodicity. The results of the neutron hysteresis loops agree well with the presented MOKE measurements. However, for some orientations of the in-plane rotation (angle χ) deviations of first order neutron hysteresis loop are detected compared with zeroth order MOKE curve. Deviations between specular measurements and magnetic measurements at Bragg-spots are generally expected for the case of Bragg-MOKE as will be discussed in the following sections. Equivalent effects for neutron-scattering are thus also probable. The results will be published [92]. 86 7. CoFe grating 7.1. Introduction In this chapter1 studies of the magnetization reversal of a laterally structured CoFe film is reported using the magneto-optical Kerr effect in specular and diffraction geometry. The lateral structure consists of 90 nm thick and 1.2 µm wide Co0.7 Fe0.3 stripes with a grating period of 3 µm. The magnetization vector is measured by means of the vector-MOKE technique for different orientations of the sample with respect to the field directions. In addition, Kerr-microscopy was used for visualizing the domain state. Due to the high aspect ratio of the individual stripes, the remagnetization process of the stripe array is dominated by a single domain state over most of the field range. For the easy axis direction a nucleation and domain wall movement is observed at the coercive field. However, for all other orientations of the stripe array the magnetization reversal is dominated by a coherent magnetization rotation up to the coercive field. For the hard axis orientation the coherent rotation is complete. In addition, the measurements in diffraction geometry (Bragg-MOKE) reveal small contributions from closure domains, which were not detected using the standard methods. 7.2. Sample preparation The basis of sample preparation for the present study are Co0.7 Fe0.3 thin polycrystalline films grown by DC magnetron sputtering. A polycrystalline film is preferred to average the intrinsic magneto-crystalline anisotropy. The films have no further intentionally induced anisotropy, such that their anisotropy is dominated by the shape anisotropy. CoFe films with the quoted composition have been shown to exhibit the largest tunneling magnetoresistance [94] and have been introduced as electrodes in magnetic tunnel junctions for magneto-electronic devices [95]. Here2 a 20 x 10 mm2 Al2 O3 (11̄02) substrate is used. The sample was spin coated with Novolak photoresist, which was exposed by 442 nm light in a scanning laser lithography setup and developed afterwards. A layer stack consisting of 5 nm Ta, 90 nm Co0.7 Fe0.3 and a 5 nm Ta protection layer was deposited onto the patterned photoresist. Finally, the photoresist was removed via lift-off. The described procedure resulted in Co0.7 Fe0.3 stripes of 1.2 µm width and a grating parameter of d = 3 µm as can be seen from 1 This chapter is partially based on an excerpt of the article CoFe-stripes: magnetization reversal study by polarized neutron scattering and magneto-optical Kerr effect [93]. 2 The sample studied in this chapter was supplied by K. Rott and H. Brückel, Universität Bielefeld, Germany 87 7. CoFe grating Figure 7.1.: Surface topography of the array of Co0.7 Fe0.3 stripes obtained with an atomic force microscope shown in a 3-dimensional surface view. The displayed area is 20 x 20 µm2 . the AFM picture in Fig. 7.1. The grating structure covers the complete area of the substrate. 7.3. Remagnetization process of the CoFe-grating 7.3.1. Results from MOKE measurements The left row of Fig. 7.2 shows four typical longitudinal MOKE hysteresis loops taken from the CoFe stripes with different in-plane angles χ. The hysteresis loop in (a) corresponds to an external magnetic field oriented parallel to the stripes. In this case, an almost square hysteresis loop is found, which represents the typical behavior of a sample when magnetically saturated along an easy axis of the magnetization. The coercive field is Hc = 140 Oe. The coercive field increases to Hc = 200 Oe and Hc = 320 Oe for intermediate angles of rotation of 45◦ (c) and 63◦ (e), respectively. Fig. 7.2 (g) shows the corresponding hysteresis loop for the CoFe stripe array oriented perpendicular to both the external field and the plane of incidence. Here, a typical hard axis hysteresis loop is obtained. From the results of longitudinal MOKE measurements with the magnetic field direction within the plane of incidence follows that patterning of the thin CoFe film into an array of stripes induces a strong uniaxial anisotropy, which results from the shape anisotropy of CoFe stripes. The saturation field measured with MOKE along the hard axis is rather high, exceeding 1000 Oe. As discussed above, also the transverse component of the magnetization vector was determined by Kerr effect measurements. Corresponding hysteresis loops for four directions of the external magnetic field relative to the direction of the CoFe stripes are 88 7.3. Remagnetization process of the CoFe-grating θLK (long.) 0.02 θTK (trans.) (a) χ=0° (b) (c) χ=45° (d) (e) χ=63° (f) (g) χ=90° (h) 0 −0.02 0.02 0 θK [°] −0.02 0.02 0 −0.02 0.02 0 −0.02 −3 −2 −1 0 1 2 3 −3 −2 H [kOe] −1 0 1 2 3 Figure 7.2.: MOKE hysteresis loops measured in the longitudinal configuration (left figures) and in the transverse configuration (right figures). For both configurations in (a) and (b) the external magnetic field is oriented parallel to the stripes (easy axis configuration); in (c) and (d) the stripes are rotated by 45◦ and in (e) and (f) by 63◦ with respect to the direction of the external field; in (g) and (h) the magnetic field is oriented perpendicular to the stripes, (hard axis configuration). shown in the right column of Fig. 7.2. The hysteresis loop reproduced in Fig. 7.2(b) was measured with a magnetic field parallel to the CoFe stripes, i.e. parallel to the easy axis. Ideally, within this configuration the measured Kerr rotation should remain zero unless components of the magnetization lie in the plane of incidence during the magnetization reversal process. As can be seen from Fig. 7.2(b), the measured Kerr rotation is indeed almost zero, thus no rotation of the magnetization occurs in this configuration. Fig. 7.2 (d) and (f) show the transverse component of the Kerr rotation measured for angles of rotation of χ = 45◦ and χ = 63◦ , respectively. Fig. 7.2 (h) shows the hysteresis loop for a magnetic field pointing along the hard axis direction, i.e. in a direction perpendicular to the stripes and the plane of incidence. In these directions the measurements of the transverse magnetization component always proves a rotation of the magnetization away from the magnetic field. From these measurements important conclusion concerning the remagnetization pro- 89 7. CoFe grating 1 |θK Rem Sat / θK | 0.8 0.6 0.4 0.2 0 −90 −45 0 45 90 χ [°] 135 180 Rem Figure 7.3.: Kerr rotation in remanence, θK , normalized to the saturation Kerr roSat tation, θK , as a function of the sample rotation. The closed circles are the measurements in the longitudinal configuration and the open squares denote the measurements in the perpendicular configuration. The solid line is a plot of the model according to Eq. 7.1 and the dashed line is a plot of Eq. 7.2 cess of the CoFe stripes can be drawn. In general, two ideal cases can easily be distinguished: If the angle of the magnetization vector changes without changing the magnitude |M |, then the magnetization reversal occurs via coherent rotation. On the other hand, if the angle of the magnetization vector remains constant within a certain magnetic field range but the magnitude |M | changes, then magnetic domains are formed. The hysteresis curves in Fig. 7.2 exhibit no discontinuity besides the jump at Hc , therefore the magnetization process must be smooth, either dominated by rotation or by domain nucleation. If only coherent rotation takes place, the magnetization in remanence must be directed along the easy axis of the twofold anisotropy, i.e. along the stripes, without Rem forming any domains. In this case the longitudinal Kerr rotation in remanence, θK must follow: Rem,L θK = | cos χ|, (7.1) Sat θK and, correspondingly, the Kerr rotation angle in perpendicular configuration must follow: Rem,T θK = | sin χ|. Sat θK (7.2) The measurements perfectly obey these rules as is depicted in Fig. 7.3. Therefore it is 90 7.3. Remagnetization process of the CoFe-grating 3000 2500 Htc [Oe] 2000 1500 1000 500 0 0 45 90 χ [°] 135 180 Figure 7.4.: Coercive field taken from the measurements in the perpendicular geometry as a function of the sample rotation χ. The circles denote the measurement and the solid line is a plot of Eq. 7.3. clear that the magnetization process is governed by coherent rotation processes. Furthermore, the coercive field should also show such a simple behavior: Along the easy axis the magnetization switches by 180◦ if the external field is equal to the pinning field of the domains. If the sample is rotated away from the easy axis, the component of the external field driving the switching process is reduced. Therefore one finds: Hc (χ) = Hce.a. , cos χ (7.3) which is plotted in Fig. 7.4. Clearly the plot confirms this notion. A similar behavior was found in [26] for the remagnetization process of Co wires. In summary: For fields parallel to the stripes the remagnetization process proceeds by nucleation and domain wall motion within a very narrow region around the coercive field, Hc . For field directions 0 < χ < 90◦ the remagnetization process is dominated by a coherent rotation up to the coercive field, where some domains are formed. For χ = 90◦ the remagnetization process appears to proceed entirely via a coherent rotation of the magnetization vector and no switching takes place. In the hard axis direction the magnetization vector describes a complete 360◦ rotation during the full magnetization cycle without any discontinuity. For other directions a switching of the magnetization of 180◦ is observed at Hc which can be viewed as a head-to-head domain wall movement [28] through the stripe. Similar dependencies were found in [24], where the switching process was found to be dominated by edge curling walls. 91 7. CoFe grating Reason for this clear and straightforward remagnetization process is the high aspect ratio of the stripes ( height width = 0.075), leading to a strong shape anisotropy. The shape anisotropy results in a two-fold anisotropy with an anisotropy energy density ES . From a numerical integration of the magnetization curve in the hard axis direction follows ES ≈ 730 Oe · Ms , where Ms is the saturation magnetization of the CoFe alloy. The energy density of the shape anisotropy can be calculated according to Ref. [2], using the relation Es = 2πN · Ms2 , where N is the tensor element of the demagnetizing tensor in the appropriate direction. N can be calculated as an approximation for an ellipsoid inscribed in the wire by using a formula given in [2]. Assuming Ms =1775 G for Co0.7 Fe0.3 , the theoretical shape anisotropy constant is 780 Oe · Ms , which is in good agreement with our measurements. The value of the saturation magnetization Ms for Co0.7 Fe0.3 is higher than the value for pure Fe and is given in Ref. [96] and references therein. More recent experiments confirm this value [97]. 7.3.2. Results from Kerr-microscopy Figure 7.5.: Kerr microscopy pictures taken around Hc for different magnetic field to sample alignment of χ = 0◦ (left, easy axis alignment) and χ = 73◦ (right). The plane of incidence results in a top-down magneto-optical sensitivity axis parallel to the stripes. The magnetization directions are indicated by arrows in the χ = 0◦ map. Kerr microscopy studies support the conclusions from the MOKE measurements. Images at different magnetic fields show a single domain state for a field below and above Hc (not shown). Around Hc , domain nucleation and domain wall movement sets in. The proposed mechanism of head to head domain walls through the stripes can be confirmed. From our observations we assume independent domain nucleation and domain 92 7.4. Bragg-MOKE measurements at the CoFe grating sample wall movement for each single stripe. Possible existence of dipole-dipole interactions between domains of adjacent stripes cannot be extracted from the Kerr microscopy observations in Fig. 7.5 which shows Kerr-microscopy pictures taken at Hc for two different field angles χ of the sample rotation. In Fig. 7.5 (left) the easy axis case with no magnetization rotation for H 6= Hc is shown, and Fig. 7.5 (right) displays a picture taken from an intermediate angle χ = 73◦ close to the hard axis case. In agreement with the MOKE results, at Hc , similar Kerr microscopy pictures were found for different angles χ of the sample rotation between χ = 0◦ and χ = 88◦ . 7.4. Bragg-MOKE measurements at the CoFe grating sample At the CoFe grating as described above Bragg-MOKE measurements were carried out in the geometry discussed in Sec. 3.5. In particular, the angle of incidence was chosen αi = 0◦ and the grating is oriented with the stripes perpendicular to the external field (hard axis). In this geometry the first four orders of diffraction are accessible in the given setup (see Sec. 3.5.3 and Fig. 3.21). The results of the Bragg-MOKE measurements are reproduced in Fig. 7.6. For symmetry reasons (see Sec. 3.5.3) it is sufficient to discuss only the measurements of positive order of diffraction which are shown in the figure. The Bragg-MOKE curve at n = 2 is identical to the measurement of n = 1, therefore it is omitted. The three Bragg-MOKE curves in Fig. 7.6 are normalized to the measured Kerr-rotation in magnetic saturation. In order to compare the Bragg-MOKE amplitude sat (n), this value is plotted in Fig. 7.6(d) as a function of the order of in saturation, θK diffraction. Sat (n) is an increasing function of n. This effect is As can be seen in Fig. 7.6(d), θK a common feature observed in other measurements of the Bragg-MOKE effect as well and will be discussed later in more detail (Sec. 9.4). At this point the discussion will concentrate on the shape of the Bragg-MOKE curves independent on their amplitude. At a first glance the three plotted Bragg-MOKE curves in Fig. 7.6(a-c) reproduce well the hard axis MOKE loop depicted in Fig. 7.2(g). The loops are completely closed and have a rather high saturation field. However, upon closer inspection, it is seen that the saturation field differs for the different order of diffraction. In order to demonstrate this, Fig. 7.7 shows the diffraction hysteresis loops (DHL) for the four orders of diffraction. In this representation the Bragg-MOKE measurement is plotted as a function of the MOKE curve in specular geometry which is assumed to be proportional to the averaged magnetization, also see Sec. 3.4.1. Every deviation of the DHL from a straight line indicates a difference in the Bragg-MOKE curve in the corresponding magnetization regime. It is clearly seen that close to saturation magnetization (m/ms ≈ 1) the DHL of order n = 1 and n = 2 display a higher Bragg-MOKE signal than expected (saturation is approached at smaller fields than for the specular geometry). For n = 3 the opposite is true. In a regime where the sample is already saturated on average the Bragg-MOKE curve is still increasing. Only the DHL for n = 4 reproduces the ideal straight line with unit slope. This result is not expected as the anticipated remagnetization process from the vector- 93 7. CoFe grating (b) n=3 1 1 0.5 0.5 θK / θSat K θK / θSat K (a) n=1 0 −0.5 −0.5 −1 −1 −4 −2 0 H [kOe] (c) n=4 2 4 −4 −2 0 2 H [kOe] B−MOKE amplitude 4 0.4 1 0.3 θsat [°] K 0.5 θK / θSat K 0 0 0.2 −0.5 0.1 −1 −4 −2 0 H [kOe] 2 4 0 1 2 3 4 n Figure 7.6.: Results of Bragg-MOKE measurements from the CoFe stripe array. (a), (b) and (c) display the normalized Bragg-MOKE curves of order n = 1, 3 and 4, respectively. (d) shows the dependence of the saturation Bragg-MOKE amplitude as a function of n. MOKE and Kerr-microscopy studies above, together with the simulation of DHL’s in Sec. 3.4.2 would lead to DHL which are linear functions for all n. For a straightforward coherent rotation remagnetization process only the saturation Bragg-MOKE rotation would be altered. DHL’s as measured from the present CoFe sample must be related to a reversible formation of domains up to the highest field values measured. This was demonstrated with the simulations in Sec. 3.4.2. However, compared to Fig. 3.12 the deviation of the observed DHL’s from the ideal linear behavior is small. The conclusion drawn from the Bragg-MOKE measurements therefore is that edge-domains are established which exist up to very high fields. These edge-domains effectively reduce the magnetic width of the stripes. The measured DHL’s are therefore a superposition of two model cases as discussed in Sec. 3.4.2, namely the edge-domain and the coherent rotation models. Qualitatively the measured DHL agree with the simulations of these two case in Sec. 3.4.2. These domains probably reside at the edges of the stripes and are necessary for partly compensating the magnetic flux at the sharp edges of the stripes. Examples of such closure domains have been observed e.g. by McCord et al. [23]. As the magnetization direction in these domains is alternating up and down the average magnetization of the edge domains is zero. 94 7.5. Summary (a) n=1 (b) n=2 1 B−MOKE signal B−MOKE signal 1 0.5 0 −0.5 −1 −1 −0.5 0 0.5 m / ms (c) n=3 −0.5 −0.5 0 0.5 m / ms (d) n=4 1 −0.5 0 m / ms 1 1 B−MOKE signal B−MOKE signal 0 −1 −1 1 1 0.5 0 −0.5 −1 −1 0.5 −0.5 0 m / ms 0.5 1 0.5 0 −0.5 −1 −1 0.5 Figure 7.7.: CoFe-stripe array: Conversion of the Bragg-MOKE curves to DHL’s. 7.5. Summary In conclusion, we have studied the magnetization reversal behavior of CoFe stripes using Vector-MOKE, Bragg-MOKE and Kerr-microscopy. As was found from the VectorMOKE and Kerr-microscopy measurements, the remagnetization process of the present sample presents a very simple case consisting from purely coherent rotation and a 180◦ domain wall at the coercive field. In addition, no interaction of the stripes where found. However, the Bragg-MOKE measurement proofed that flux closure domains exist, which were not detected using Kerr-microscopy at high magnetic fields. (Note that the Kerrmicroscopy images depicted in Fig. 7.5 were produced in small fields around Hc ). The Bragg-MOKE techniques can give additional information which are difficult to obtain with other methods. Certain orders of diffractions probe particular Fourier components of the magnetization distribution and may be very sensitive to edge domains. Qualitatively the simulated archetypes of magnetization processes in Sec. 3.4.2 could be Sat reproduced, however, no qualitative agreement for the dependence of θK (n) was found using this model. From this sample an extensive study with polarized neutron reflectometry (PNR) was performed [93]. PNR at patterned magnetic films is a new and challenging task and the straightforward remagnetization process of the CoFe stripes is an ideal sample for a comparative study in order to evaluate the potential of PNR. The magnetization reversal process was observed at the first order Bragg peak as function of the orientation of the stripes with respect to the applied magnetic field and the scattering plane (the field was 95 7. CoFe grating applied perpendicular to the scattering plane). From the different cross-sections, spin asymmetry curves were calculated and compared to standard MOKE hysteresis loops. The curves show a very good agreement with the observed MOKE hysteresis loops, from which was concluded that PNR is a suitable method for studying patterned magnetic samples. Note, that here the standard specular MOKE curve is compared to the first order diffracted PNR magnetization loop (Bragg-PNR). From the discussion above and the simulations in Sec. 3.4.2 it is clear that also Bragg-PNR probes a certain Fourier component of the magnetization distribution. Therefore one would expect changes of the Bragg-PNR curves qualitatively similar to those observed with Bragg-MOKE (Fig. 7.6), which is not the case in [93]. However, due to the special physics of diffraction with neutron beams [93, 98] it is possible to observe Bragg-PNR curves for in-plane sample rotations from χ near to the easy axis up to χ near to the hard axis, but not exactly in the hard axis configuration, which is the only possible configuration for Bragg-MOKE (Sec. 3.5.3). Therefore Bragg-MOKE and Bragg-PNR display a certain lack of comparativeness. In addition Bragg-PNR curves of higher order than n = 1 could not be observed. 96 8. Ni-gratings 8.1. Introduction In this chapter1 measurements are presented of the Bragg-MOKE effect from Ni gratings with rather large lattice parameter of 20 µm. Because of the large grating parameter many interference spots are observable from which hysteresis loops can be taken. The second idea of this system was to separate effects due to domains in the stripes from pure optical phenomena. In order to do so two samples were prepared, the first one consists of an Al-grating on top of a Ni film, and the second one of a simple Ni grating. For the first case no correlated domains with the period of the grating are expected, thus no influence of domains to the Bragg-MOKE curves should occur. 8.2. Experimental setup The first optical grating (sample No. 1) has been prepared by depositing a Ni-film of 20 nm thickness on Si(111) and then preparing an array of Al-stripes with a lattice parameter d = 20 µm, a thickness of 20 nm and a width of a single Al-stripe of wAl = 6 µm on top of the Ni-film. The Al-stripes are thick enough to prohibit the laser beam to penetrate the Ni film below the stripes. The second sample (sample No. 2) is an optical grating of Ni-stripes on Si(111) with a lattice parameter of d = 20 µm, a width of a single Ni-stripe of wN i = 4 µm, and a thickness of 20 nm. Both gratings were prepared by optical lithography and a lift-off process2 . MOKE measurements with a HeNe laser as light source were carried out using a high resolution Kerr angle setup in the longitudinal configuration as described in Sec. 3.5, also see Fig. 8.1. The entire setup as depicted in Fig. 8.1 is mounted on a goniometer such that the angle of incidence αi can be varied between 0◦ and 45◦ and the detector can be moved from −45◦ to 45◦ . This is necessary in order to investigate the MOKE effect at different orders of the diffracted light. The grating is oriented with the normal of the surface in the scattering plane and the magnetic field perpendicular to the direction of the stripes. The diffraction spots are numbered by the order of diffraction n, n = 0 denoting specular reflection. Positive (negative) n denotes an increasing (decreasing) detector angle with respect to the surface normal. In addition to the measurements of the Kerr angle we also measured the integrated intensity at the interference spots. 1 This chapter is based upon the article Magneto-optical Kerr Effects of Ferromagnetic Ni-gratings see Ref. [99] 2 The samples were provided by K. Schädler and U. Kunze from Lehrstuhl für Werkstoffe der Elektrotechnik, Ruhr-Universität Bochum 97 8. Ni-gratings Figure 8.1.: Schematic set-up of the Bragg-MOKE experiment with the Ni-grating on Si between the pole faces of an electromagnet. The angle of incidence αi is fixed, the grating can be rotated by the angle χ. The Kerr rotation is measured at the different diffracted spot positions n. Figure 8.2.: Representative examples of Bragg-MOKE hysteresis loops of sample No. 1 (Al stripes on Ni) measured at different orders of diffraction n. 98 8.3. Results and Discussion 8.3. Results and Discussion 8.3.1. Bragg-MOKE First, the results obtained from the Al diffraction grating on Ni (sample No. 1)are discussed. In Fig. 8.2 selected examples of hysteresis loops measured at different orders of diffraction n are shown. The shape of the hysteresis loops is identical for all n, but the amplitude of the Kerr signal changes strongly. In Fig. 8.3 the amplitude of the Kerr signal at remanence and the intensity measured at different orders of diffraction n is plotted. One clearly observes a periodic behavior with an oscillation period of ∆n = 3. The Kerr intensity exhibits the same periodicity with a phase shift: the maximum of the Kerr angle matches exactly the minimum in the intensity. The intensity distribution in Fig. 8.3 results from a superposition of the diffraction pattern of the periodic lattice with a lattice parameter of d = 20 µm and the diffraction pattern of a single stripe with a width wAl ≈ 1/3d. The apparent periodicity in Fig. 8.3 is due to the Al-stripes with a width of about 6 µm. The modulation from the diffraction due to the Ni-spacing between the Al-stripes cannot be resolved, since its stripe width is approximately a factor of 2 larger. The close correlation between the maxima of the Kerr angle and the minima of the intensity can be attributed to the dominant contribution of the Ni stripes rem at these positions. In addition, the oscillation of θK (n) is superimposed to a linear rem decrease of θK (n) from negative to positive n. The linear contribution changes sign for n ≈ −18. As the measurements were carried out with αi ≈ 40◦ , the angle αf (n = −18) approximately matches the position where the diffracted beam is directed back along the surface normal. Vial and van Labeke [53] predicted a Kerr signal (in the transverse Kerr geometry) of zero and a change of sign for the Littrow -mounting, in which the diffracted beam is directed back into the incoming beam (αf = −αi ). This condition would be fulfilled for the present case for n ≈ 36. Therefore the result in Fig. 8.3 is in contradiction with the results in [53]. The linear contribution and the change in sign will be subject to further investigations reported in the following chapters. In Fig. 8.4 examples of hysteresis loops measured from the Ni grating on Si (sample No. 2) are shown. One first should note that for this grating a strong enhancement of the amplitude of the Kerr signal by a factor of up to 30 at certain diffraction spots is observed. Contrary to sample No. 1 the shape of the hysteresis loops strongly depends on the order of diffraction. Whereas at certain orders of diffraction (n = 1 shown in Fig. 8.4) the hysteresis loop has a shape identical to that observed at specular reflection n = 0, one observes anomalous hysteresis loops at other orders of diffraction (n = −3 and n = 9 in Fig. 8.4). Apparently these anomalous hysteresis loops contain a second component of the Kerr signal with a high saturation field and a sign change when going from +n to −n. The occurrence of the anomalous contribution in the hysteresis loops is correlated with a rather low value of the Kerr signal for the normal hysteresis loop. It is supposed that the anomalous contribution is caused by the magnetic microstructure on the edges of the stripes, e.g. by closure domains with a magnetization component perpendicular to the film plane. Similar anomalous contributions to the Bragg-MOKE hysteresis loops in transverse geometry have been observed in [11, 53]. The authors show by model calculations that the anomalous component can be attributed to the domain pattern of the ferromagnetic stripes. In Fig. 8.5 the intensity and the Kerr amplitude 99 8. Ni-gratings Figure 8.3.: Intensity of the diffracted light (upper panel) and Kerr signal amplitude at remanence (lower panel) versus the order of diffraction for sample No. 1. Figure 8.4.: Representative examples of Bragg-MOKE hysteresis loops of sample No 2 (Ni stripes on Si) measured at different orders of diffraction n. 100 8.3. Results and Discussion Figure 8.5.: Intensity of the diffracted light (upper panel) and Kerr signal amplitude at remanence (lower panel) versus the order of diffraction for sample No. 2. for sample No. 2 is plotted for different orders of diffraction n. The periodicity in the intensity distribution and the Kerr signal amplitude is ∆n = 5. This fits perfectly to the width of the Ni-stripes wN i = 4 µm, which is 1/5 of the lattice parameter d = 20 µm. Compared to the intensity pattern in Fig. 8.3, the intensity pattern in Fig. 8.5 appears strongly smeared out. This may be caused by a superimposed damped modulation from the diffraction of a single Si-stripe with a width of 16 µm. Contrary to the result in Fig. 8.2 there is no phase shift between the intensity and the Kerr angle, i.e. the maxima in the Kerr amplitude correlate with the maxima in the diffracted intensity. This is plausible, since in this case the Ni-stripes are responsible for the intensity modulation. 8.3.2. MFM measurements In order to further clarify the role of domains in this system force microscopy measurements have been carried out. Fig. 8.6(a) and (b)3 depict AFM and MFM measurements, taken simultaneously from the identical portion of a Ni stripe of sample 2. From the AFM measurement (a) the important information can be extracted, that the edges of the Ni stripes show large ridges which stem from the lift-of process (white lines along the stripe). These ridges are as high as 100 nm measured from the top of the stripes, exceeding the stripe height by a factor of 10. Therefore it can be expected that a local 3 The measurements (a) and (b) were carried out from A. Carl and co-workers at the Institut für Tieftemperaturphysik, Universität Duisburg, Germany 101 8. Ni-gratings Figure 8.6.: (a) AFM image of a portion of one Ni stripe and (b) MFM image of the same area taken from sample 2. (c) MFM image of a polycrystalline Ni/Si film. shape anisotropy will lead to strong magnetization components perpendicular to the sample plane. The MFM image of the same stripe was taken in zero field. It displays a magnetic contrast mainly at the edges of the stripes with alternating dark and light regions. The center of the stripes show only small magnetic contrast. As the MFM tip is mainly sensitive to perpendicular, out-of-plane, magnetization components it can be concluded that edge domains with perpendicular magnetization triggered by the ridges are formed. The MOKE signal in longitudinal configuration is proportional to the inplane and out-of-plane magnetization. Therefore the conclusion is reasonable that the additional components in the diffraction MOKE curves in Fig. 8.4 are connected to the out-of-plane magnetization of the edge domains. Fig. 8.6(c) displays a MFM measurement of a small portion of a Ni film, which was equivalently prepared as sample 2, but not structured. This measurement clearly shows stripe domains which stem from the very complicated domain structure of Ni films. Al- 102 8.4. Summary and Conclusion though the shape anisotropy leads to a overall in-plane magnetization of this film, the fine domain structure shows out-of-plane magnetized stripe domains which are favored because of an additional magnetostrictive anisotropy found in these films [100]. Hysteresis curves measured from this sample also shows a reduced remanence which can be attributed to portions of the sample magnetized alternating perpendicular to the film. More details of this measurements can be found in [76]. However, the size of the domains is so small that many of the domains can be placed in the width of one stripe. Therefore no diffraction signals at the position of the interference spots can be obtained which are correlated to this small magnetic patterns. The stripe domains were not resolved in (b), because the edge domains and the large structural hight differences cover the smaller domains. 8.4. Summary and Conclusion In the present chapter Bragg-MOKE measurements in longitudinal configuration were demonstrated. Gratings with a relatively large lattice parameter of 20 µm were used which enables one to observe the Bragg-MOKE effect up to higher orders of diffraction than could be done before. It has been shown that it is possible to increase the amplitude of the Kerr rotation by making use of diffraction from a ferromagnetic grating. In principle, this effect is similar to the amplification of the Kerr signal of a ferromagnetic homogeneous film using interference effects by coating the film with a dielectric cap layer. The Bragg-MOKE effect may be separated into optical effects only changing the overall Kerr amplitude and effects of the magnetic substructure and it seems possible to gain information about the magnetic domain pattern of the ferromagnetic stripes. Close to the angle of extinction of the diffracted light from a single ferromagnetic stripe, the Bragg-MOKE effect appears to be sensitive to the specific domain pattern at the edges of the stripes. The observed effects will be further analyzed and clarified in the subsequent chapters, where measurements of gratings with constant grating parameter but changing stripe width will proof the assumptions made above. 103 8. Ni-gratings 104 9. Fe-gratings 9.1. Introduction The magneto-optical Kerr effect in longitudinal configuration is used to study hysteresis loops of several Fe-gratings patterned by electron beam lithography. The Kerr effect is not only detected in specular reflection but also off-specular at the intensity maxima of the grating at different orders of diffraction n (Bragg-MOKE). The Kerr rotation in saturation increases linearly with the order of diffraction n if the incoming beam is at normal incidence, independent of the grating geometry. The shape of the hysteresis curves changes for special orders of diffraction, indicating an enhanced sensitivity for the formation of magnetic domains close to remanence: the nth order of diffraction is sensitive to the nth order Fourier component of the magnetization distribution. In this chapter1 two sets of gratings are under investigation. The first is made from polycrystalline and the second set is made from single crystalline, (001) oriented Fe-film. For both sets the grating parameter is constant at 5 µm and the stripe width is varied. The magnetism and domain structure of the single crystalline gratings are analyzed in different states of the patterning process showing the influence of connecting Fe between the stripes. 9.2. Sample Preparation The stripe arrays were prepared by means of electron beam lithography. In the first step continuous thin film samples were prepared by molecular beam epitaxy (MBE) and by rf-sputtering (see Sec. 4.1. The MBE sample of the present study (sample A) is a 20 nm thick single crystalline Fe(001) film grown on a Cr(20 nm)/Nb(20 nm) buffer system on Al2 O3 (11̄02) [58, 63, 65] (Fig. 9.1(a)). The base pressure during the MBE process was 10−10 mbar. The crystalline quality was checked in situ using RHEED and ex-situ by x-ray diffraction techniques. The sputtered sample (B) is a polycrystalline Fe film produced by rf sputtering at a base pressure of 5 · 10−8 mbar. The Fe film was grown on Al2 O3 (112̄0) at room temperature (see Fig.9.1(b)). Again the structural properties were checked by ex-situ x-ray diffraction and small angle x-ray reflectivity. Before further processing the film, MOKE measurements of the samples were carried out, in order to have hysteresis loops of the unpatterned film as a reference. The standard MOKE measurements were carried out in longitudinal geometry at an angle of incidence 1 This section is an extension of the article Magneto-optical Kerr effect in the diffracted light of Fe gratings[101] 105 9. Fe-gratings Figure 9.1.: Sample design of the Fe-gratings. (a) shows sample A2 (single crystal Fe(001)) in which four gratings with the stripes parallel to the in-plane Fe(100) direction and one grating parallel to the Fe(110) direction were etched. The situation after the completed etching procedure is depicted. (b) displays the design of sample B (polycrystalline Fe) with four gratings. wF e , t and d denote the stripe width, the thickness of the stripes and the lattice parameter, respectively. Figure 9.2.: Atomic force microscopy image of one grating of sample B with wF e = 2.1 µm 106 9.2. Sample Preparation αi ≈ 45◦ with s-polarized light, i.e. with a polarization direction perpendicular to the plane of incidence. The magnetic anisotropy was measured by rotating the sample around its surface normal (angle χ in Fig. 3.21) and taking hysteresis curves for different angles. Details of the MOKE setup can be found in Sec. 3.5. From each of the thin films A and B several grating structures with different grating geometries were produced. Therefore the film thickness and crystal quality is the same for one set of gratings originating from one thin film. In a first step, the samples were spin-coated with a double layer of PMMA. Using electron-beam lithography, the grating structures were written onto the polymer film, which was subsequently patterned by developing. Finally the grating structure was transferred into the metallic film using ion-etching. The resulting grating structures were analyzed using scanning electron microscopy and atomic force microscopy techniques. An example of the gratings thus prepared is depicted in Fig. 9.2. The grating geometries used for the present study are shown schematically in Fig. 9.1. In more detail, the sputtered sample (B) was patterned using the double imagetransfer technique described in Sec. 4.1, i.e. after the mask fabrication via e-beam lithography a Al2 O3 film of the same thickness as the Fe film was deposited and structured using the lift-of. Afterwards, the (negative) Al2 O3 image of the desired pattern is etched into the Fe film. The Fe stripes therefore form a positive image of the stripes, i.e. the surrounding, unstructured area of the sample is completely etched away. In contrast, the single crystalline Fe sample were etched directly using the resist mask for the etching process. Therefore the negative image is formed, leaving the partes of the sample without grating untouched. Furthermore, the etching was performed in a two step process, the first step results in grating structures where the gap between the stripes is still covered with Fe connecting the adjacent stripes (this stage will be called sample A1 ). The second etching step results in gratings which are completed in the sense that no Fe connects the stripes (sample A2 )2 . After the patterning process, Bragg-MOKE hysteresis curves were taken. The geometry of the Bragg-MOKE measurements for αi = 0 is sketched in Fig. 3.21, more details of the Bragg-MOKE setup can be found in Sec. 3.5.3. The integrated intensities of the diffraction spots were measured using a photo-diode instead of the Kerr-detector. In addition, Kerr microscopy measurements were performed as described in Sec. 4.3.3. 2 Here an error in the publication [101] is corrected: in contrast to what is reported in [101], the measurements in the article on the single crystalline Fe gratings are performed in the not completely etched state (sample A1). Therefore some conclusions about the domain structure of this sample are also wrong in [101]. This, however, does not change the results and conclusions of the article with respect to the Bragg-MOKE effect, which is the main objective of [101] 107 9. Fe-gratings 0.05 K θ [°] (a) 0 −0.05 −400 −200 0 200 H [Oe] 400 0 200 H [Oe] 400 0.05 θK [°] (b) 0 −0.05 −400 −200 Figure 9.3.: MOKE measurements of the single crystalline Fe film (sample A) in the as prepared state. The two curves show MOKE measurements performed along an easy axis (a) and a hard axis (b) of the fourfold magneto-crystalline anisotropy. For the standard MOKE measurements αi = 45◦ . 9.3. Results 9.3.1. Single crystal film, sample A Unstructured film Longitudinal MOKE hysteresis curves of the unstructured Fe film are depicted in Fig. 9.3 for two orientations of the magnetic field along the magnetic easy (100) and hard (110) axis. As expected, hysteresis curves along the easy axis are of square type, whereas the hard axis hysteresis curve shows a reduced remanence. Measurements with the external field applied in different in-plane directions, i.e. at different sample rotation angles χ, reveal a fourfold symmetry, as expected for the Fe(001) film. Additionally, small overshoots, i.e. asymmetries, of the MOKE hysteresis curve can be seen in Fig. 9.3, which 108 9.3. Results Figure 9.4.: Kerr microscopy image of one of the single crystalline Fe-gratings before they were completely etched through (sample A1). The contrast in this image is mainly due to the magnetization direction of the domains. can be attributed to second order magneto-optical effects [40, 43, 42], see Sec. 3.1.3. In the case of the reversal process along the (110) axis the magnetization process combines ~ away from the hard axis (0◦ , along the applied a rotation of the magnetization vector M field) into the direction of the easy axis, at 45◦ from the applied field and by domain wall ~ has a nonzero transverse movements. During the rotation the magnetization vector M component, which results in the observed second order MOKE components. Both curves show very small coercive fields, Hc ≈ 10 Oe, the observed behavior is consistent with the coherent rotation model of the magnetization process [44]. Connected single crystalline Fe gratings, sample A1 In this section the measurements on the single crystalline Fe stripes (sample A1, see Fig. 9.1(a)) are reported. As the Fe strips are connected to each other by the residual Fe film in the gap, the magnetic domains in this situation are mostly independent on the actual stripe structure, i.e. no edge domains or induced anisotropy are expected. The situation was demonstrated by Kerr microscopy: In Fig. 9.4 domains which are larger than the stripes can clearly be observed. First measurements on the gratings from sample A1 with the direction of the stripes parallel to the in-plane (100) axis are discussed. As the magneto-crystalline anisotropy is fourfold, one easy axis lies parallel to the stripes, the other one perpendicular to the stripes. The patterning of the gratings did not add a measurable twofold shape 109 9. Fe-gratings 0.1 n=1 n=2 n=3 K θ [°] 0 −0.1 0.1 0.1 n=−2 0 n=−1 0 −0.1 −1 0 1 n=−3 −0.1 −1 0 1 −1 H [kOe] 0 1 Figure 9.5.: Bragg-MOKE hysteresis curves for one grating of sample A1, Fe-stripe width is 3.9 µm. 0.1 wFe=3.9µm w =2.4µm Fe w =2.0µm Fe w =1.3µm Fe sat θK [°] 0.05 0 −0.05 −0.1 −3 −2 −1 0 n 1 2 3 Figure 9.6.: Bragg-MOKE angle in saturation as a function of the diffraction order for sample A1 110 9.3. Results θK [°] 0.1 0 (a) n=1 −0.1 −2 −1 0 1 H [kOe] 2 0 1 H [kOe] 2 θK [°] 0.1 0 (b) n=2 −0.1 −2 −1 Figure 9.7.: Bragg MOKE hysteresis loops with the grating parallel to the hard axis of the magnetization for n = −1 (a) and n = −2 (b); wF e = 1.4 µm and d = 5 µm. anisotropy to the gratings of sample A1, which has been verified by rotating the gratings around the surface normal and measuring sequences of MOKE hysteresis curves for different angles χ. Thus in the geometry for the Bragg-MOKE measurements described above the gratings are oriented perpendicular to the external field and with an easy axis along the field. In this geometry, Bragg-MOKE measurements have been performed for the first three positive and negative orders of diffraction of all four gratings. In Fig. 9.5 Bragg-MOKE hysteresis curves for the grating with a Fe line width of wF e = 3.9 µm are presented as an example. It is clearly seen that the shape of the curves is independent of the diffraction order n and that the shape of the hysteresis loops basically reflects the easy axis curve from the unstructured film in Fig. 9.3. Only the coercive force is increased sat to Hc ≈ 45 Oe. In contrast, the absolute value of the Kerr rotation in saturation, θK , depends on n and changes sign when moving from positive to negative n. Fig. 9.6 111 9. Fe-gratings log(I) [a.u.] w =1.3µm Fe w =2.0µm Fe wFe=2.4µm wFe=3.9µm −5 −4 −3 −2 −1 0 n 1 2 3 4 5 Figure 9.8.: Integrated intensities for the gratings of sample A with the stripes parallel to (100) sat sat displays θK (n) for all four gratings. The θK (n)-curve is linear and independent of the width of the Fe stripes wF e . sat No dependency of θK on the diffracted intensities was found. Fig. 9.8 shows the diffracted intensities as a function of n for perpendicular incidence. Although the intensity varies differently for different stripe widths, representing the diffraction envelope of the single stripes, the measured Bragg-MOKE signal is not affected. The situation changes if the stripes of the grating are along the in-plane (110) axis of the Fe film, i.e. along the hard axis of the magnetization. Fig. 9.7 shows the Bragg-MOKE hysteresis curves for n = 1, 2 and for wF e = 1.4 µm. The Bragg-MOKE sat rotation in saturation θK (n) shows the same behavior as for the stripes along (100), but the shape of the curves changes significantly. The overshoots are also seen in the as prepared state of the sample A (see Fig. 9.3). However, the features are enhanced, indicating that the contributions of second order magneto-optical effects have increased. The contribution of second order effects decreases with increasing positive or negative n (see Fig. 9.7). Single crystalline Fe-gratings after etching, sample A2 After the remaining Fe connecting the stripes has been removed in an additional etching procedure, the magnetic properties of the single crystal Fe stripes were drastically altered. Fig. 9.9 shows hysteresis curves for different stripe widths and different orientations of the field to the stripes. Only the gratings with the stripes parallel to the 112 9.3. Results e.a. h.a. 0.015 45° 2.4µm 0 Hc2 −0.015 θK [°] 0.015 2.0µm 0 H −0.015 0.015 c2 1.7µm 0 H −0.015 −0.5 c2 0 0.5−0.5 0 H [kOe] 0.5−0.5 0 0.5 Figure 9.9.: Standard MOKE measurements from the single crystalline Fe grating A2 in three different directions with respect to the external field. In the left column measurements with the field perpendicular to the stripe axis are depicted. The middle column represents the measurements with the field along the stripes, and the right column with the field aligned 45◦ to the stripes. The measurements were carried out at the arrays with wF e = 2.4, 2.0 and 1.7 µm (top to bottom row) as indicated in the figure. The definition of Hc2 is also indicated in the hard axis plots. in-plane easy axis of the magneto-crystalline anisotropy are under investigation, i.e. the stripes are parallel to the Fe(100) direction. It is clearly seen that in this situation the patterning induces an additional twofold anisotropy, which is strongest for the smallest stripes and surmounts the four-fold crystalline anisotropy. The superposition of the two-fold anisotropy (with the hard axis perpendicular to the stripes) and the fourfold crystalline anisotropy of Fe (with the easy axes along and perpendicular to the stripes) has very interesting consequences. Below a certain stripe width (the transition is between wF e = 3.9 µm and wF e = 2.4 µm) the hysteresis exhibits a plateau region around zero field with almost zero magnetization in the hard axis (with respect to the stripes). The system enters the plateau region at a switching field Hc2 before the external field is reversed. At the field H = Hc2 the condition: Ek + Ez = Ek + EU (9.1) must be satisfied, with Ek , EU and Ez being the crystalline anisotropy energy density, 113 9. Fe-gratings −3 4.5 x 10 Hc/(Ms/2−Hc) 4 3.5 3 2.5 2 4 4.5 5 5.5 6 1/w [1/m] 6.5 7 7.5 8 5 x 10 c2 ( 1 ) for the gratings of sample A2 (see Eq. 9.2). From the Figure 9.10.: Plot of 1 MHs −H c2 w 2 linear fit to the data the height of the stripes can be determined, see text. the anisotropy energy density and the Zeeman-energy, respectively. The Zeeman energy at Hc2 is Ez = µ0 Ms Hc2 , and the uniaxial anisotropy is defined as EU = 12 µ0 N Ms2 . Here h the demagnetizing factor, N , is given by N = h+w , where h is the thickness of the Fe stripes and wF e the width (see Sec. 7 and [2, 56]). This leads to (in SI units): 1 h Ms 2 h + wF e Hc2 1 ⇐⇒ 1 =h . w Ms − Hc2 2 (9.2) Hc2 = Fig. 9.10 displays the data of Hc2 , plotted as 6 Hc2 1 Ms −Hc2 2 (9.3) as a function of 1 . w Here the bulk saturation magnetization of Fe (1.7 · 10 A/m) is chosen. From the linear fit to the data follows h = 4.4 nm. Obviously, the error is quite large. However, AFM measurements at the gratings confirm a height between the top of the stripes and the bottom of the groove of ≈ 9 nm. This discrepancy may be explained by the presumably deeper grooves and longer etching times than intended, i.e. during the etching the Fe-stripes are already ablated and the grooves are etched into the underlying Cr buffer, thus the thickness of Fe is smaller than the height of the stripes. Bragg-MOKE measurements at the gratings from sample A2 have been performed, some examples are plotted in Fig. 9.11. The measurements at the grating with wF e = 2.0 µm are representative for the measurements at the gratings for wF e = 2.4 µm and 114 9.3. Results wFe = 2.0 µm 0.15 wFe = 3.7 µm n=1 n=1 n=2 n=2 n=3 n=3 0 −0.15 K θ [°] 0.15 0 −0.15 0.15 0 −0.15 −0.4 0 0.4 −0.4 H [kOe] 0 0.4 Figure 9.11.: Example Bragg-MOKE hysteresis curves of the single crystalline Fe grating, sample A2. The Fe stripe width and order of diffraction is given in the figure. wF e = 1.3 µm. However, the very interesting Bragg-spot n = 2 and wF e = 2.4 µm could not be measured, as the diffracted intensity was too low. All these Bragg-MOKE hysteresis curves show generally the same shape as the specular MOKE curves depicted in Fig. 9.9. The only difference is that the plateau region around zero field is even more pronounced and exhibits a smaller remanence compared to the specular MOKE curves. In this case the diffraction at the gratings acts like a filter which excludes all light reflected at the ferromagnetic surroundings of the gratings. Therefore the BraggMOKE curves show only the magnetic behavior of the gratings, whereas the signal at the specular spot may be contaminated with signals from the neighborhood of the stripes. The Bragg-MOKE curves at the stripes with wF e = 3.7 µm show for n = 1 and n = 2 no change of the shape with respect to the specular curve. However, the curve for n = 3 sat displays an increase in the saturation field. The θK (n) curve displays, as was observed earlier, a monotonous behavior with increasing Kerr rotation for increasing n. The domain pattern of the gratings on sample A2 was measured using Kerr microscpy3 . In all cases a demagnetized state is imaged, which was achieve by a decreasing, oscillating in-plane field. The results are shown in Fig. 9.12 for the gratings with wF e = 2.4, 2.0 and 1.3 µm. For all images the magneto-optical sensitivity direction was chosen to be along the stripes, i.e. black domains are magnetized parallel and white domains antiparallel to the stripes, different shades of gray correspond to perpendicular 3 The measurements were performed together with J. McCord at the IFW Dresden, Germany 115 9. Fe-gratings Figure 9.12.: Kerr microscopy images of the single crystal Fe-grating. All measurements were performed in a demagnetized state with the magnetic sensitivity direction along the stripes. In the top row the field was oriented along the strips during demagnetization (easy axis), in the bottom row the field was oriented perpendicular to the stripes (hard axis). The Fe stripe width is indicated in the figure. For this particular sample the gratings are surrounded by the unstructured Fe film. Therefore the domains of the Fe film can be observed in some of the pictures, e.g. (a) and (c) in the bottom part of the image. magnetized or not-magnetized regions. In the first row the situation for a demagnetizing field along the easy axis and in the bottom row along the hard axis is presented. In the images (a) and (c) of Fig. 9.12 unpatterned surroundings of the gratings are also visible. In these regions clearly the fourfold anisotropy can be detected, which usually leads to 90◦ domain walls, i.e. in most cases neighboring domains exhibit a transition from black or white into gray, but almost never from black to white (this would correspond to a 180◦ domain wall). The direction of the walls is thus along the hard axis, i.e. 45◦ to the stripes. In these two images it can be observed how the stripes act as centers for the nucleation of domains in the unpatterned region. Inside the stripes in the images (a), (c), (e) and (f) basically only black and white contrasts are observed. The additional uniaxial anisotropy in this regions leads to locally homogenous magnetized stripes with only 180◦ domain walls which are perpendicular to the stripes. As 180◦ domain walls are unfavorable in the Fe system, the systems tries to extend the domain wall width over a larger stripe-length, which can be observed 116 9.3. Results particularly in image (a). In this case the uniaxial anisotropy is smallest and the regions of opposite magnetization collinear to the stipes are separated by several small domains displaying many 90◦ walls. For the cases of the smaller stripe widths the 180◦ wall does not extend over a larger area because the uniaxial anisotropy favoring a 180◦ domain wall is increased. The situation changes slightly if the external field is directed perpendicular to the stripes, as depicted in image (b) and (d). Because the external field essentially lessens the uniaxial anisotropy, domains directed perpendicular to the strips are more favorable. This can be observed in image (b). The regions of the stripes magnetized collinear with its axis are separated by large areas with domains magnetized perpendicular to the stripes of rhombic shape. These rhombic central domains (gray) are separated by triangular edge-domains magnetized along the stripes (black or white). This pattern fades out by approaching the homogenous magnetized regions giving the impression of a helix-like structure. For the case of the smallest stripes (images (e) and (f) in Fig. 9.12) the uniaxial anisotropy is always big enough to force all domains in a collinear direction with respect to the stripes. 9.3.2. Polycrystalline Fe-gratings Next MOKE results obtained from the polycrystalline Fe-film in which five different gratings have been etched (sample B, see Fig. 9.1(b)) are presented. In this case the polycrystalline state and the larger thickness of the Fe-film leads to a different magnetization behavior. Standard MOKE Fig. 9.13 shows specular MOKE hysteresis curves obtained on sample B with different gratings and with the magnetic field oriented along and perpendicular to the stripes. The shape of the hysteresis loops in this case is dominated by the shape anisotropy induced by the geometry of the gratings. The shape anisotropy is strongest for the case of the narrowest stripes. Bragg-MOKE For these gratings measurements of the Bragg-MOKE effect in the same geometry as for sat sample A1 have been performed. The θK (n) curve shows the same linear dependence as seen for sample A1. If the interference spots from the gratings of sample B are in the plane of incidence, which is the case for all measurements performed, the stripes are perpendicular to the external field, i.e. the Bragg-MOKE hysteresis curves are obtained in the hard axis configuration. Selected examples of the measured Bragg-MOKE hysteresis curves are depicted in Fig. 9.14. Because the measurement was performed in the hard axis direction, the hysteresis curves have to be compared with the hard axis measurements in Fig. 9.13. The rows of Fig. 9.14 display the Bragg-MOKE hysteresis curves for one stripe width at different orders of diffraction n. The columns represent the same order of diffraction 117 9. Fe-gratings Kerr signal [a.u.] 1 (a) 0.5 µm (b) 2.1 µm (c) 2.5 µm (d) 3.7 µm 0 −1 1 0 −1 −2 −1 0 1 2 −2 −1 H [kOe] 0 1 2 Figure 9.13.: MOKE hysteresis loops measured along and perpendicular to the stripes for the gratings with (a)wF e = 0.5 µm , (b) wF e = 2.1 µm, (c) wF e = 2.5 µm,(d) wF e = 3.3 µm. The hard axis (dashed line) is measured with the external field perpendicular to the stripes and the easy axis (straight line) with the field along the grating. for different Fe-stripe widths wF e . Obviously the shape of the hysteresis curves do not always represent the normal MOKE hysteresis curves as displayed in Fig. 9.13. This is in strong contrast to the previous measurements of sample A. In detail, for the case of wF e = 0.5 µm (first row in Fig. 9.14) essentially no change of the shape of the Bragg-MOKE hysteresis curves as a function of n is observed. In the case of wF e = 2.1 µm (second row in Fig. 9.14) no change for n = 1, positive overshoots for n = 2 and a negative contribution for n = 3 are observed, which leads to an apparent increase of the saturation field. In the third row the case of wF e = 2.5 µm is displayed, where we find no change of the Bragg-MOKE hysteresis curves of n = 1 and n = 3, but a strong negative contribution, which leads to sharp negative overshoots for n = 2. For wF e = 3.3 µm (bottom row) almost no change for n = 1 and n = 2 is detected, and again a negative contribution, which leads to an apparent increase of the saturation field for n = 3. For some of the Bragg-MOKE curves the DHL’s (see Sec. 3.4.2) were calculated. The results are depicted in Fig. 9.15. In this representation the Bragg-MOKE signal is plotted as a function of the normalized magnetization as obtained from the specular measurements. This representation is independent on M as a function of H. The normalized curves in Fig. 9.15 can be viewed as a superposition of a linear and a sine function. The splitting between the two magnetization paths is small only for n = 2 and wF e = 2.5 µm (c) a significant splitting occurs. These results can be interpreted 118 9.3. Results n=1 0.05 n=2 n=3 wFe= 0.5µ 0 −0.05 0.05 2.1µ 0 θK [°] −0.05 0.05 2.5µ 0 −0.05 0.05 3.7µ 0 −0.05 −2 0 2 −2 0 2 −2 H [kOe] 0 2 Figure 9.14.: Bragg-MOKE hysteresis curves from the polycrystalline gratings (sample B). Only the positive first three orders of diffraction are shown. Each row in the figure represents the measurements of one grating with constant wF e , as indicated in the figure. wF e is increasing from the top to the bottom. Each column displays measurements of a given constant order of diffraction (n = 1...3 as indicated in the figure). using the simple models developed in Sec. 3.4.2. The linear part thus stems from a remagnetization process due to irregular domain formation or coherent rotation (i.e. the magnetization component is reduced for decreasing fields, but no correlated effects 119 9. Fe-gratings (a) w =2.1 n=1 (b) w =2.1 n=2 BMOKE−signal Fe Fe 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1 −0.5 0 0.5 1 −1 (c) w =2.5 n=2 BMOKE−signal 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 0 0.5 m / ms 0.5 1 Fe 1 −0.5 0 (d) w =3.7 n=3 Fe −1 −0.5 1 −1 −0.5 0 0.5 m / ms 1 Figure 9.15.: Diffraction hysteresis loops, i.e. the Bragg-MOKE signal as a function of the specular MOKE measurement in the h.a. axis orientation, of the Fegrating. The order of diffraction and Fe stripe width is indicated in the figure. occur) and the sine-like contribution can be attributed to edge-domains. If the domain formation for the two magnetization paths is identical no splitting is expected. This results will be discussed further in the subsequent sections. Intensity Now these results are compared with the measurements of the integrated intensity of the diffraction spots depicted in Fig. 9.16. Here again oscillating intensities were found which are due to the diffraction envelope of the single stripes. The intensity minima are observed near the expected minima of the diffraction envelope of a single Fe-stripe. From Figs. 9.14 and 9.16 it can be concluded that Bragg-MOKE hysteresis curves with an anomalous shape coincide with a minimum in the intensity. For wF e = 0.5 µm no intensity modulation is detected and all curves preserve the shape of the standard curve measured in specular configuration. For wF e = 2.1 µm the anomalous Bragg-MOKE curves are close to the intensity minimum at n = 2. In the case of wF e = 2.5 µm a sharp intensity minimum exists at n = 2, which coincides with the Bragg-MOKE curve with the strongest anomaly. The intensity pattern for wF e = 3.7 µm exhibits a 120 9.3. Results w =0.5 µm log(I) [a.u.] Fe wFe=2.1 µm w =2.5 µm Fe wFe=3.7 µm −6 −4 −2 0 n 2 4 6 Figure 9.16.: Intensity pattern of sample B. The specular reflected intensity is omitted. minimum at n = 4, which is out of the range of the Bragg-MOKE setup, but already the Bragg-MOKE curve for n = 3 is slightly anomalous. Vector-MOKE at the polycrystalline Fe-grating samples In order to further clarify the origin of the additional components measured in the Bragg-MOKE curves of the polycrystalline sample (B) two different methods have been used. First, a virgin curve of the grating with wF e = 2.1 µm has been measured in BraggMOKE geometry (n = 2) identical to the result in Fig. 9.14 but after the sample was saturated perpendicular to the stripes. In this case the stripes are first saturated in the easy axis, than the sample is rotated into the hard axis direction and the Bragg-MOKE curve is measured starting at zero field. The result is plotted in Fig. 9.17. Obviously the additional contribution to the Bragg-MOKE signal is increased with this procedure indicating that the additional component is somehow connected to magnetic domains magnetized along the stripes, i.e. perpendicular to the field during the Bragg-MOKE measurements. Second, from the same sample and grating (B, 2.1 µm) a vector-MOKE measurement was performed in specular geometry, which is presented in Fig. 9.18. The figure shows the standard MOKE hysteresis curve in the hard axis as was already shown in Fig. 9.13. In addition, a measurement is plotted after rotating the sample and the magnet, which is proportional to the perpendicular magnetization component, MT , for a hysteresis along the hard axis. It can be seen that this component is non-zero but small compared to the 121 9. Fe-gratings 0.06 θK [°] 0.03 0 −0.03 −0.06 −2000 −1500 −1000 −500 0 H [Oe] 500 1000 1500 2000 Figure 9.17.: Virgin curve of n = 2, wF e = 2.1 µm of the polycrystalline Fe stripe array after saturation along the stripe axis. longitudinal measurement. From this one can conclude that the remagnetization process is mainly governed by domain processes, however, a small transverse magnetization component exists and is not completely averaged out. Kerr-microscopy Kerr microscopy was used to image the domains at the coercive field of all gratings from sample B. Fig. 9.19 displays some representative examples for hard axis magnetization. In the top row the magneto-optical sensitivity direction is oriented parallel to the field and perpendicular to the stripes. In the bottom row the the sensitivity direction is along the stripes. In Fig. 9.19 measurements from the grating with wF e = 3.7µm (left column) and wF e = 2.1µm (right column) are presented. The measurements of the grating with the larger stripe-width shows that only few and small edge domains with a magnetization along the stripes are formed (dark and light contrast in (b)). Consistent with this observation, it is found in (a) that the magnetic domains are mainly oriented perpendicular to the stripes. Moreover the domain pattern in (a) seems to proof a certain correlation of the domains in adjacent stripes, i.e. the dark or light domains are extended over more than one stripe. This result can be explained by the combination of three effects: • A small uniaxial anisotropy perpendicular to the strips. This was found with Kerr microscopy at an unpatterned portion of the sample B. The anisotropy is very small, so that the usual MOKE hysteresis measurements did not detect it. 122 9.3. Results 0.02 0.015 0.01 K θ [°] 0.005 0 −0.005 −0.01 −0.015 −0.02 −2500 −2000 −1500 −1000 −500 0 500 H [Oe] 1000 1500 2000 2500 Figure 9.18.: Vector-MOKE of wF e = 2.1 µm of the polycrystalline Fe stripe array in the h.a. axis orientation. The large curve is the longitudinal MOKE curve, the smaller, gray curve is a plot of the measurement in the perpendicular configuration, i.e. is proportional to the transverse magnetization component during reversal. • Dipolar coupling between neighboring stripes. The coupling increases with decreasing distance. This effect is only observed for the broadest stripes. • Vanishing shape anisotropy. The shape induced anisotropy with an easy axis along the stripes is reduced with increasing stripe width. Consistent to this the Kerr microscopy images of the grating with wF e = 2.1 µm (Fig. 9.19(c,d)) do not show this effect. The image (c) displays no correlation and in (d) strong edge domains are observed. In this case the dipolar coupling is reduced and the shape anisotropy is dominant. The small uniaxial anisotropy perpendicular to the stripes play no role for this stripe-width and separation. In order to explore the domain pattern any further in Fig. 9.20(a) a magnification of the Kerr image of the grid with wF e = 2.5 µm (sample B) in the demagnetized state is displayed. Here, a very regular domain structure with closure domains at the stripe edges is observed, as depicted schematically in Fig. 9.20(b). In the remanent state (not shown) one essentially observes similar domains with one magnetization direction in the interior of the stripes. 123 9. Fe-gratings Figure 9.19.: Kerr microscopy images of two of the Fe gratings as indicated in the figure. All depicted measurements were performed at the coercive field, when magnetized along the hard axis. The magneto-optical sensitivity direction is parallel to the field in the top row and parallel to the stripes in the bottom row. 9.4. Discussion 9.4.1. Saturation Bragg-MOKE signal For an interpretation of the results first the dependence of the saturation Bragg-MOKE signal as a function of the order of diffraction n is discussed, as shown in Fig. 9.6 (sample sat A1). This almost linear dependence of θK (n) was observed not only for the samples discussed in this chapter, but also for other ferromagnetic interference gratings with different sample design and different materials. If the angle of incidence, αi , is chosen sat to be non-zero, the zero-point of θK (n) shifts to the point at which the angle of the diffracted spot crosses the surface normal (see Sec. 8 and [99]). This is a clear indication that the sign of θK depends on the scattering geometry chosen. sat In order to explain the θK (n) dependence in Fig. 9.6 qualitatively, a Lorentz-Drude type model for the magneto-optical Kerr effect is assumed and the symmetry of the 124 9.4. Discussion Figure 9.20.: (a) Kerr microscopic image in the demagnetized state of a grating from sample B with the lattice parameter d = 5 µm and the stripe width wF e = 2.5 µm. The field direction during demagnetization was perpendicular to the stripes. (b) Orientation of the magnetization within the domains schematically ~ setup is considered: The incoming electromagnetic wave with the E-vector parallel to the stripes (see Fig. 3.21) induces a current oscillating perpendicular to the applied field. ~ of the ferromagnet cause a tilting The Lorentz forces from the magnetic induction B of this current out of the film-plane. The projection of the tilt angle in the direction of the observer increases with the angle proportional to sin(αf ), thus explaining the linear dependence at low angles in Fig. 9.6. This geometrical consideration is schematically depicted in Fig. 9.21. In other words, the observed dependence results from the off-specular magneto-optical constants, which can in principle be measured with the presented geometry, and which have been calculated in [12] for the case of the polar sat Kerr-effect. In addition, it was found that θK (n) does not dependent on the stripe width, wF e , i. e. is independent of the diffracted intensity. This result has also been obtained from theoretical considerations in Ref. [12]. In contrast to this, in Sec. 8 an sat (n) was found, which was attributed to the diffraction envelope of a oscillation of θK single stripe. The main difference between the present diffraction gratings and those studied in Sec. 8 is the smaller grating parameter of d = 5 µm compared to d = 20 µm in Sec. 8. In Ref. [12] the authors expect an important contribution from the interference between the metallic stripes and intermediate stripes on the substrate. The light diffracted from the metallic stripes and from the substrate has a phase difference, which should lead to interference phenomena. This effect has not been observed for our gratings, 125 9. Fe-gratings Figure 9.21.: Illustration of the effect that leads to the increasing Kerr rotation with increasing diffraction angle. The incoming beam is s-polarized at perpen~ out of the dicular incidence. The Lorentz-force results in a tilting of E sample plane. The projection into the direction of the observer increases at higher order of diffraction. because the height h = 20 . . . 50 nm of the stripes is small compared to the wavelength λ = 632 nm of the illuminating laser. As has been discussed in Sec. 3.4.3, the contribution from interference phenomena depends on two factors: first, the phase shift between the diffracted beams of grating and substrate, and, second, on the relative amplitudes. The present result can be interpreted by assuming that the amplitude of the diffracted waves from the substrate is always small compared to the amplitude diffracted from the ferromagnetic grating. In this case the observed Kerr rotation (according to Eq. 3.34) is mainly proportional to the Kerr effect of the ferromagnetic grating and essentially no interference effects can be observed. The detected intensity pattern in Figs. 9.16 and 9.8 can easily be modelled by assuming a simple transmission grating as given in Eq. 3.16. The substrate-grating is made of sapphire in the case of sample B which, as an insulator, has a smaller reflectivity constant than the metallic stripes. The situation is different in the case of sample A, where also the substrate-grating is of metallic material. One possibility is that the etching procedures have created more roughness in the grooves than on top of the stripes, and therefore have reduced the reflected intensity from in the grooves. 9.4.2. Shape of Bragg-MOKE curves of the single crystalline sample To start with, the Bragg-MOKE curves for sample A1 for the case of the Fe-stripes oriented along the in-plane Fe(110) direction (Fig. 9.7) are discussed. The standard MOKE curves in this direction (Fig. 9.3(b)) clearly exhibit contributions from second order MOKE effects. As discussed in [42, 40, 43], the Kerr angle in the longitudinal 126 9.4. Discussion configuration can be expanded as (the polar Kerr effect is neglected) θK = αML + βML MT + γMT2 , (9.4) where ML and MT are the longitudinal and transverse magnetization components of the ~ . In the longitudinal MOKE geometry ML is the component magnetization vector M ~ in in the film plane along the external field and MT denotes the component of M the film perpendicular to H. The coefficients β and γ depend on the ferromagnetic material and the optical geometry. The occurrence of second order MOKE effects proves that the magnetization rotates into the direction of the magnetic field. Details of the magnetization process are very sensitive to the exact in-plane orientation (angle χ) of the anisotropy axes relative to the external field [10]. This is the reason why the situation of the virgin film (Fig. 9.3) is not exactly reproduced after patterning (Fig. 9.7). The second term in Eq. 9.4 produces an asymmetric contribution to the MOKE hysteresis curves, which obviously increases in the Bragg-MOKE effect as compared to the standard MOKE measurement. The increasing second order effects can be explained with the small angles involved in the Bragg-MOKE geometry at perpendicular incidence. Second order effects increase in specular MOKE if αi = αf is small, for the case of perpendicular incidence the linear longitudinal component completely vanishes and only second order terms remain [42]. This geometry is comparable to the present case with αi = 0. Unfortunately, a theory for second order effects which relaxes the constrain αi = αf is not available at present. Aside from the increasing second order effects, the Bragg-MOKE curves of sample A1 and A2 do change shape significantly as compared to the specular curves. As discussed in Sec. 3.4.2 this must be connected to the fact that on average the magnetization distribution does not form domains inside the stripes. This can be seen from the different Kerr microscopy images taken. The image of sample A1 (connected Fe stripes, Fig. 9.4) shows domains which are larger than the Fe stripes, the magnetic structure is not correlated to the diffracting system. Therefore averaging over the stripes will just result in a homogenous magnetization with changing amplitude as given by the specular hysteresis curve. This case is equivalent to the coherent rotation model (of Sec. 3.4.2) and therefore a Fourier decomposition can only result in changing amplitude but not changing shape of the hysteresis. The same argument holds for the Bragg-MOKE curves of sample A2. As was measured by Kerr microscopy the magnetization in the stripes are not coupled and always directed up or down parallel to the stripes. The regions with domains form to some extent broad 180◦ walls which, on average, have zero magnetization. The Fourier transform of this case will again change only the amplitude but not the shape of the curves. 9.4.3. Shape of Bragg-MOKE curves of the polycrystalline sample The situation is different for the Bragg-MOKE hysteresis curves of the polycrystalline Fe-film (sample B). For this sample no macroscopic magnetocrystalline anisotropy exist and the domain structure is essentially defined by the geometry of the stripes. The stripes induce an uniaxial magnetic anisotropy, causing the formation of closure domains at the edges of the stripes for a magnetization direction perpendicular to the stripes. 127 9. Fe-gratings Kerr microscopy images of domains of some of the Fe grids (sample B) are reproduced in Figs. 9.19 and 9.20. For the magnetization reversal of this polycrystalline film one would not expect to observe contributions in the specular MOKE signal from the second order term in Eq. 9.4, since the transverse component MT for a randomly oriented polycrystalline grain structure has as many positive as negative components. This is consistent with the fact that no anomalous hysteresis loops in the specular MOKE measurements are found for any of the polycrystalline Fe-gratings (see Fig. 9.13). Furthermore, the observed anomalous hysteresis loops in Fig. 9.14 display a symmetry with respect to the origin, which is not consistent with second order effects [14]. As has been discussed in Sec. 3.4.1 and 3.4.2 the anomalous shape of the BraggMOKE curves has been proven to be caused by the magnetic domain structure of the patterned films. Following the derivation in Ref. [11, 15], the Bragg-MOKE hysteresis curve for a diffraction spot of order n corresponds to the nth Fourier component of the mean magnetization distribution within each stripe, i.e. is given by the magnetic form factor of a single stripe, see Eq. 3.21. Knowing the magnetic domain structure as a function of the applied field one can calculate the Bragg-MOKE hysteresis curve of order n by numerical integration. In special cases the domain state can be described by simple models which can be solved analytically, as has been shown in Sec. 3.4.2. The domain structure of the Fe gratings B, as have been imaged in 9.19 and 9.20 can be described by the edge-domain model of Sec. 3.4.2, i.e. a central domain aligned with the field and edge domains of effectively zero magnetization. The simulations of this model in Fig. 3.12 show that the diffraction hysteresis loops (DHL, see Sec. 3.4.1 for the definition) are basically sinusoidal functions of the magnetization and are identical for the ascending and descending branch. The strongest effect can be found for the case of the width of the stripes being half the grating parameter. In this case the simulation is qualitatively identical with the measurement, as depicted in Fig. 9.15. A pronounced anomalous shape of the hysteresis curve is expected if the form factor fm (n) vanishes in saturation and only the formation of domains gives a finite value for fm . This is the case if g wF e = 2π n wF e = ±2π, ±4π, . . . . d (9.5) For the series of Bragg-MOKE curves in Fig. 9.14 the reflection n = 2 and wF e = 2.5 µm fulfills this condition and actually shows the most anomalous shape of the hysteresis loop. The shape can also be explained regarding the calculations presented in Sec. 3.4.2. The Bragg-MOKE curves were converted into DHL’s in Fig. 9.15. In this representation they can be compared to the calculated curves in Sec. 3.4.2. Most of the curves can be explained with a superposition of the edge-domain and the coherent rotation model. The first of this models covers the case that the effective width of the stripes changes during the remagnetization process and the latter that the magnetization changes homogenously its magnitude (coherent rotation or irregular domain formation). The calculations showed that in this cases sine like curves superimposed to linear curves are expected, the sine like component being maximal at the situations given by Eq. 9.5. The average domain distribution during remagnetization is therefore a combination of a general decrease of the magnetization due to irregular domain formation combined with 128 9.5. Summary and Conclusions the correlated formation of edge domains. For this cases a closed DHL is expected, i.e. the domain formation is reversible, as is found in Fig. 9.15(a,b,d). This seems not to be the case for the measurement of n = 2 and wF e = 2.5 µm, as depicted in Fig. 9.15(c). In this case also a mechanism where the two magnetization paths are not identical have to be taken into account. This was demonstrated in Sec. 3.4.2 with the two 180◦ domainwall model. However, an exact fit of the measured DHL to the calculated curve was not possible, the model obviously gives only qualitative results. This explanation is consistent with the measurements of the virgin curve after transverse saturation as was shown in Fig. 9.17. The area of the edge domains obviously increases when magnetizing along the strips. Therefore the virgin curve displays an enhanced anomalous contribution. Furthermore, the Vector-MOKE measurement in Fig. 9.18 proves that for the hard axis magnetization process the transverse component of the magnetization is small, i.e. the direction of the edge domains cancel each other out. From the Bragg-MOKE measurements no sign of dipolar coupling between the magnetic stripes was found. The coupling proved by Kerr-microscopy in the case of the broadest stripes presumably only exist for very small fields. The observed coupling in Fig. 9.19 leads to domains magnetized perpendicular to the stripes, the absence of edge domains and the domains being extended over more than one stripe. In the onedimensional models discussed in Sec. 3.4.2 this would only lead to an averaged decrease of the magnetization for small fields, as it is described by the coherent rotation model. Dipolar coupling that would lead to any alternating magnetization in adjacent stripes (i.e. the one-dimensional equivalent of a checker-board pattern) is not covered by the models in Sec. 3.4.2, however, is also not observed by Kerr-microscopy for the present gratings. 9.5. Summary and Conclusions In this work the Bragg-MOKE effect in longitudinal geometry on three different types of Fe-gratings (single crystalline connected, unconnected and polycrystalline) , keeping the grating parameter constant, was investigated. The results can be separated into the effects concerning the Bragg-MOKE amplitude in saturation and changes of the shape of the hysteresis loop. Regarding the amplitude in saturation, a monotonous dependence on the diffraction order is found. A qualitative explanation for this effect based on geometrical considerations is given. Concerning the shape of the hysteresis loops there are two different cases for the observation of anomalous loops. If rotation processes are involved, as e.g. for the single crystalline Fe-film with the field parallel to the hard magnetic axis, second order terms contribute asymmetric components to the MOKE signal for specular reflection as well as in higher order diffraction. For n 6= 0 the second order terms sometimes appear to be enhanced compared to the specular case. A different mechanism producing anomalous hysteresis loops which can be observed only in higher order Bragg-MOKE is correlated with the domain patter formed during the magnetization process. For certain orders of diffraction the domain formation can drastically alter the shape of the hysteresis curve. Contrary to the case when second 129 9. Fe-gratings order effects contribute to the hysteresis curve, the hysteresis loops in this case keep inversion symmetry [14]. The anomalies in the hysteresis loops in diffraction of order n represent the nth order Fourier component of the domain pattern and can get very pronounced if the symmetry of the domain pattern matches with the wave vector of the Fourier component. It has been shown that although the domain pattern is very similar for comparable stripe width, the anomalies in the hysteresis loops depend very sensitively on the ratio of the stripe width and the grating parameter. 130 10. Co gratings on a Fe-film 10.1. Introduction In this chapter the magneto-optical Kerr effect in the diffracted light from Co gratings with a grating parameter ranging from 5 to 15 µm on top of an Fe thin film is examined. The Co gratings are decoupled from the Fe film with a Cr spacer layer. The system forms a spin-valve structure as the coercive fields from Fe and Co are different. The hysteresis loops measured at the diffracted spots reveal interesting amplification of the Kerr signal in the field regime where the magnetization of Fe and Co is antiparallel. In the field of magneto-electronics spin-valve structures play an important role. These structures consist of two magnetic layers with different switching behavior so that in a certain field region the magnetization inside the two layers is antiparallel leading to an enhanced magneto-resistance. These systems can be realized with two methods: either one layer is coupled to an antiferromagnet which provides a unidirectional pinnig and the other layer is free. The coupled layer displays the exchange bias effect and has therefore a modified switching field. The other possibility is to use two layers with different coercive fields because of different materials used or different thicknesses prepared. For the present study the second approach was used. An epitaxial Fe layer with a small coercive field was deposited. On top a Co layer with a larger Hc was grown. The two layers where completely decoupled by a thick Cr interlayer. Due to the surface sensitivity of MOKE, magneto-optical measurements only probe the top layer of this stack. Therefore the top Co and Cr layers were etched into different stripe arrays in order to enable one to measure hysteresis curves of both layers simultaneously. Furthermore, this setup enables to use the Bragg-MOKE effect as an additional tool for the investigation of the system. The grating structures investigated in this chapter always show a ratio of width to lattice parameter of 12 . In this situation the intensity of the even order diffraction spots is strongly reduced and the signal is very sensitive to edge domains as has been shown in Sec. 9.4. The prepared gratings have grating parameters ranging from 5 to 15 µm. Increasing the width and grating parameter of the structures should decrease potential dipolar coupling and should decrease the uniaxial anisotropy induced in the Co stripes. As was shown in the previous sections, the amplitude of the Kerr effect as a function of the diffraction order is a linear function. With the present sample design it will be tested if this effect is indeed a function of the order of diffraction or a function of the diffraction angle. Furthermore, in Sec. 8 interesting oscillations of sat θK (n) were found for rather large grating parameter of 20 µm, an effect not reproduced for the case of smaller Fe gratings in Sec. 9.4. However, in order to separate the signals from the underlaying Fe layer and the Co stripes it would also be a good idea to use asymmetric grating designs (width not half 131 10. Co gratings on a Fe-film Figure 10.1.: Atomic force microscopy image of one of the Co on Fe gratings. The imaged grating has a grating parameter of d = 5µm. The width of the Co stripes is wCo = 2.8 µm and the hight of the structures is t ≈ 410 nm. of grating parameter), because the diffraction envelope would lead to different weighting of the two signals depending on the diffraction order under investigation, as has been shown in Sec. 8. 10.2. Experimental details Using the standard UHV-MBE process as explained in Sec. 4.1 a thin film sample on Al2 O3 (11̄02) was prepared. On a Nb(001) (6 nm) / Cr(001) (27 nm) buffer system a 20 nm thin Fe(001) film was evaporated. The growth quality was checked in-situ using RHEED. On top a 14 nm Cr spacer and a 27 nm Co layer was prepared, which is capped with a thin 2 nm Cr film for oxidation protection. The sample quality was checked with standard analysis techniques. The thin film stacking sequence is sketched in Fig. 10.2. From the thin film sample several grating structures with different grating geometries were fabricated in order to keep the structure of the gratings constant with respect to film thickness and crystal quality. The gratings were produced with electron beam lithography and ion etching, as described in Sec. 4.1. The resulting grating morphology was visualized using scanning electron microscopy and atomic force microscopy techniques, a AFM image is presented in Fig. 10.1. The geometries of the gratings are depicted in Fig. 10.2. Because of the inhomogeneity of the etching process, the difficulties in fitting XRD data of such complex films and the errors of the AFM and EDX methods used, the total inaccuracy of all thicknesses given is ≈ 10%. The lateral dimensions are better defined, the error for wCo and d is < 1%. The width of the stripe, wCo , is approximately half of the grating parameter, d, for all gratings. For technical reasons the deviation from this ideal case is higher for the gratings with smaller grating parameters. Before the etching process standard MOKE measurements of the samples were carried out, in order to have a reference result of the unstructured film. After the patterning 132 10.3. Results Figure 10.2.: Geometry of the five gratings prepared from the Fe/Cr/Co trilayer. The Co strips and the Fe film in the grooves is still capped with a thin (≈2 nm) Cr film, which is not shown in the drawing. The thicknesses t of the film system is constant for all gratings. The grating parameters d and the stripe widths wCo for the five gratings are given in the figure. process Bragg-MOKE hysteresis curves were measured. The MOKE and Bragg-MOKE technique was described in Sec. 3.5.3. For the present study the angle of incident was zero (perpendicular incidence). As the grating parameter is different for the five gratings under investigation the diffraction angle varies with the simple relation d sin αf = nλ, where αf is the diffraction angle (see Fig. 3.21), n the order of diffraction, and λ = 632.2 nm is the Laser wavelength. This holds only if the angle of incidence αi = 0◦ . Therefore an increasing number of diffraction spots was observed in the accessible range of αf = ±40◦ of the setup for an increasing grating parameter. 10.3. Results 10.3.1. Standard MOKE measurements The data obtained from the unstructured film is presented in Fig. 10.3(a). The MOKE measurement is only sensitive to the Co film on top of the sample, because the Co film and the Cr spacer are too thick to detect signals from the underlying Fe film. The in-plane rotation angle, χ, for the measurement in Fig. 10.3(a) is chosen identical to the Bragg-MOKE measurements, i.e. the hysteresis loop is directly comparable to the Bragg-MOKE measurements of the structured sample. The orientation of χ is almost parallel to an easy axis of the Co film as can be seen from the hysteresis loop with 100 % remanence and a coercive field of HcCo = 340 Oe. Fig. 10.3(b) shows a hysteresis curve of a Fe film on the same substrate buffer system, at identical growth conditions as the sample described above, but without the Cr spacer and the Co layer. Therefore this hysteresis represents the single Fe magnetization curve in the unstructured sample. The coercive field is HcF e = 15 Oe< HcCo and the nucleation field, HNF e , of the Fe hysteresis 133 10. Co gratings on a Fe-film 0.05 (a) 0 −0.05 θK [°] 0.05 (b) 0 −0.05 0.05 (c) 0 −0.05 −1000 −500 0 H [Oe] 500 1000 Figure 10.3.: Standard (specular) MOKE measurements of (a) the unstructured film, only the Co film contributed to the MOKE signal; (b) an Fe film prepared identically as the Fe/Cr/Co sample, but without the Cr/Co layers on top; (c) the Co grating with d = 7.5 µm. loop has the same sign as the coercive field and is only slightly smaller. The measured Fe hysteresis curve is not perfectly square, but after the magnetization switched, i.e. at a little higher field than HcF e the hysteresis curve exhibits a small ”kink”. This behavior is characteristic for a magnetic film with fourfold crystalline anisotropy and a superimposed uniaxial anisotropy as is often found for Fe films on this particular substrate / buffer system. The particular epitaxial relation of Nb(001) on Al2 03 (11̄02) leads to a tilting of the (001) axis. The resulting steps of the Fe surface induce the uniaxial anisotropy. The magnetization reversal does not consist of a simple 180◦ domain wall, but after switching into a direction of a local minimum of the anisotropy energy the magnetization rotates into the direction of the external field. Because of the different coercive fields a spin valve behavior is expected in the combined sample. The standard MOKE hysteresis loop obtained from the reflected light of 134 10.3. Results one of the Co gratings (d = 7.5 µm) is shown in Fig. 10.3(c). This combined hysteresis loop displays steps which are characteristic for spin-valve systems. The coercive field of the Fe film, HcF e , is preserved, the etching has not affected this property, in addition the small ”kinks”of the pure Fe loop (b) are still visible in the Fe/Cr/Co grating measurement (c). In contrast, the coercive field of the Co stripes, HcCo,stripes , increased to ≈ 700 Oe after the patterning process (compare (a) and (c)). In addition, the Co stripes width, wCo , has no effect on HcCo,stripes . 10.3.2. Bragg-MOKE measurements Shape of Bragg-MOKE curves Bragg-MOKE measurements on all observable diffraction spots from all gratings were performed. The most obvious results is, that only two principally different shapes of Bragg-MOKE hysteresis curves were detected. As an example normalized Bragg-MOKE curves for the grating with d = 7.5 µm are presented in Fig. 10.4. Displayed are the first six orders of diffraction for positive αf (positive n). It can be clearly seen that the shape only differs for even and odd n. In order to point out the differences between even and odd n, the case for n = 1 and 2 is displayed in an extra plot in Fig. 10.5. For the odd orders of diffraction (Fig. 10.3(a,c,e)) the measured Kerr rotation remains constant for decreasing field from saturation towards zero. At a small field just before the external field is reversed, the Kerr rotation decreases into a steep dip around zero. Note that this decrease cannot be identified with the nucleation field of Fe, because in Fig. 10.3 HNF e was found to be positive, whereas in the Bragg-MOKE measurements the first decrease of the Kerr rotation at negative external fields. After the dip the Kerr rotation increases again and in the spin-valve region (between HcF e and HcCo ) a Kerr rotation is observed which is essentially governed by the Co loop and not by the Fe loop in contrast to the standard MOKE curve. In the standard MOKE measurement (Fig. 10.3(c)) the Kerr rotation in the spin valve region has the same sign as the Kerr rotation in saturation, whereas in the Bragg-MOKE measurements the Kerr rotation in this region has the same sign as in remanence. Therefore the coercive field in the standard MOKE measurement reproduces HcF e and in the Bragg-MOKE measurement it reproduces HcCo of the patterned film. The Bragg-MOKE hysteresis curves for even order of diffraction are reproduced in Fig. 10.4(b,d,f). For these curves the coercive field is also that of the Co stripes and the influence of the Fe film is smaller than in the standard MOKE curve. However, the shape is different to the shape of the odd order diffraction loops. The dip around zero is not observed and the Kerr rotation in the spin-valve regime exhibits a constant decrease when the field is increased. In addition for n = 2 in Fig. 10.4(b) small kinks around HcCo are detected. This feature exists only for n = 2, the higher even order of diffraction exhibit only small steps at this position, see Fig. 10.4(d,f). For even order of diffraction the apparent change in sign of TNF e is also observed as for the odd order loops, but due to the absence of the ”dip” it is less pronounced. 135 10. Co gratings on a Fe-film 1 (a) n=1 (b) n=2 (c) n=3 (d) n=4 (e) n=5 (f) n=6 0 −1 θK / θK sat 1 0 −1 1 0 −1 −1 −0.5 0 0.5 1 −1 −0.5 H [kOe] 0 0.5 1 Figure 10.4.: Normalized Bragg-MOKE hysteresis curves of the first six positive orders of diffraction of the grating with wCo = 7.5 µm. Bragg-MOKE amplitude The amplitude of the Bragg-MOKE curves as a function of the diffraction angle are displayed in Fig. 10.6(a-e) for all five gratings under investigation. An increase of the measured Kerr rotation for increasing αf and a change of sign for αf = αi = 0 is observed. The dependency is almost linear with a small superimposed oscillation, which has a period of ∆n = 2. The superimposed oscillation is more pronounced for larger grating parameters. For d = 5 (Fig. 10.6(a)) almost no oscillation is found, whereas the oscillation is very strong for the largest grating parameters observed. In Fig. 10.6(f) the result of a standard angle-dependent MOKE measurement of the 136 10.3. Results 1 n=1 n=2 θK / θsat K 0.5 0 −0.5 −1 −2000 −1000 0 H [Oe] 1000 2000 Figure 10.5.: Normalized Bragg-MOKE hysteresis curves for n = 1, 2 of the grating with wCo = 7.5 µm. grating with d = 7.5 µm is shown. In this case the condition αi = αf is fulfilled (n = 0). It is clearly seen that the measured Kerr rotation for the specular geometry is smaller than in the Bragg-MOKE geometry for αi = 0 and n 6= 0. Assuming a linear function ∆θsat sat for θK (αf ) one can extract a slope of ∆αKf ≈ 4 × 10−3 . For the specular case in Fig. 10.6(f) the slope is only ≈ 8 × 10−4 . Therefore the measured Kerr signal is increased effectively by a factor of 5 by using the Bragg-MOKE geometry. 10.3.3. Intensity measurements The result of measurements of the integrated intensity of the diffraction spots is shown in Fig. 10.7 for the five gratings as a function of the diffraction angle. As the ratio between the Co stripe width and the grating parameter is nearly 12 for all gratings, the oscillation of ∆n = 2 is expected. However, as mentioned above, the gratings deviate slightly from this ideal case. wCo is up to 12% larger than d/2 for the smallest grating parameters. Therefore the oscillations in Fig. 10.7 are more and more pronounced when increasing the grating parameter. Note that the oscillation of the integrated intensities correspond to the oscillations of θK sat(n) in Fig. 10.6. The even orders of diffraction show a strongly reduced intensity, which correspond to the minima of the oscillation in sat the θK (n) functions. Furthermore the shape of the hysteresis curve changes for odd and even n (see Fig. 10.4), i.e. changes with constructive or destructive interference conditions. 137 10. Co gratings on a Fe-film 0.1 (a) d=5µm (b) d=7.5µm (c) d=10µm (d) d=12µm (e) d=15µm (f) d=7.5µm,specular 0 −0.1 K θsat [°] 0.1 0 −0.1 0.1 0 −0.1 −40 −20 0 20 40 −40 −20 αf [°] 0 20 40 Figure 10.6.: (a)-(e): saturation Bragg-MOKE rotation as a function of the diffraction angle αf for the gratings as indicated in the figure. For all Bragg-MOKE measurements the angle of incidence αi = 0. (f): saturation Kerr-rotation for the grating with d = 7.5 µm as a function of the reflection angle αf = αi . The scale in all subfigures is identical. 10.4. Discussion There are basically three features which can be separated: first, the increased Kerr effect in the spin valve region, second, the change of shape of the Fe curve (dip around zero) and the change of shape of the Co curve (different slope around Hc for even order loops) sat and, third, the behavior of θK (αf ). 138 10.4. Discussion 16 14 d=15 µm w 12 =7.5µm Co 10 d=12 µm wCo=6.2µm log(I) [a.u.] 8 6 d=10 µm wCo=5.2µm 4 2 d=8 µm wCo=4.2µm 0 −2 d=5 µm wCo=2.8µm −4 −60 −40 −20 0 α [°] 20 40 60 f Figure 10.7.: Integrated intensities of the diffraction spot of the five gratings. The grating parameter is indicated in the figure. The specular intensity (n = 0 is omitted. 10.4.1. Increasing Kerr effect in the spin valve region The change of Kerr amplitude in the spin-valve region can be explained by the superFe Co position of the signal of the two subgratings. Assume θK (H) and θK (H) to the Kerr loops of the Fe and the Co layer, respectively. Measurements of these curves were given in Fig. 10.3(a) and (b). The resulting signal of the combined sample can viewed as the m mean value, θK of the two signals: m θK (H) = Fe Co θK (H) + θK (H) . 2 (10.1) The result of this is plotted in Fig. 10.8(a). A better choice is to add the two curves up coherently taking phase and amplitude differences into account. The reflected beam from the Fe grating is assumed to be given by: EsF e EpF e ! = Fe E0F e eiφ Fe cos θK , Fe sin θK ! (10.2) where φ is a relative phase shift and E0 is the reflected amplitude. An equivalent expression shall be valid for E Co . The combined signal is than: EpF e + EpCo θK = arctan . EsF e + EsCo ! (10.3) 139 10. Co gratings on a Fe-film 0.05 (a) 0 −0.05 −1000 −500 0 500 1000 −500 0 500 1000 −500 0 H [Oe] 500 1000 0.05 θK [°] (b) 0 −0.05 −1000 0.05 (c) 0 −0.05 −1000 Figure 10.8.: (a) median of the Fe and Co hysteresis as measured in Fig. 10.3(a) and (b); (b) and (c) display the addition of the two curves according to Eq. 10.3 for the case φF e = φCo = 0, E0F e = 1 and E0Co = 5 in (b) and for the case E0F e = E0Co = 1, φCo = 0 and φF e = 0.45π in (c). This is plotted for the case φF e = φCo = 0, E0F e = 1 and E0Co = 5 in Fig. 10.8(b) and for the case of E0F e = E0Co = 1, φCo = 0 and φF e = 0.45π in Fig. 10.8(c). It is clearly seen that an enhanced Co amplitude or a large phase shift can result in the observed effect. The large phase shift of nearly π2 cannot be due to the height difference of the two gratings as was discussed in Sec. 3.4.3. Therefore it can be concluded that the enhanced Kerr rotation in the spin valve regime is additionally due to an increasing amplitude of the light diffracted by the Co grating compared to the Fe grating. This cannot be attributed to the diffraction envelope of the two subgratings as in the present study the width of the Fe and Co stripes are nearly equal. The increase in relative amplitude of the Co grating can be explained: • If the etching process is stopped too early the Fe film remains covered with a Cr film, whereas the Co film may be already uncovered. 140 10.4. Discussion • If the etching process continued too long the Fe film may be thinned out more than the Co film. • the etching process may roughen the surface of the Fe layer more than the Co layer. 10.4.2. Shape of Bragg-MOKE curves As already mentioned, the shape of the Bragg-MOKE curves is altered with respect to the specular curves in two respects. First the dip of the loops around zero will be discussed: The dip around zero external field, mainly observed in the odd order loops in Fig. 10.4, can be attributed to the Fe film because it is far from the Co coercive field in a region where the Co stripes show full remanence. In Sec. 9.4 (Fig. 9.9 hysteresis loops of single crystalline Fe stripes where shown. In that situation the additional uniaxial anisotropy resulted in hysteresis curves with a sharp step just before reaching zero field, i.e. a negative nucleation field HN . Assuming that the Fe film of the present study also exhibits an induced uniaxial anisotropy it can be concluded that equivalent hysteresis loops would be measured for the Fe film separately. However, originally it was intended not to change the properties of the underlying Fe film. This obviously was not achieved. An additional uniaxial anisotropy in the Fe film can be induced in two ways: • Either, the Co stripes on top couple to the Fe film via their dipolar fields • or, the etching process already modified the Fe film. The uniaxial anisotropy is only observed at the diffracted intensities because in this case only the parts of the sample having the grating period are filtered out. Contributions of the surroundings or of not correlated domain formation are reduced. The difference of shape of the curves of even and odd order of diffraction is mainly observed around the coercive field of the Co stripes. In the odd order loops the behavior is identical to the specular loops when taking the effects discussed above into account. For the even order loops basically the slope of the Co hysteresis loops seems to be altered. An additional component with a Kerr rotation opposite to the normal loop around Hc would explain the observation. Such an additional component was found in Sec. 9.4 for n = 2 and w = d/2 and was attributed to the existence of edge domains. As discussed in Sec. 9.4 and Sec. 3.4.2 the Bragg-MOKE curve at diffraction order n is proportional to the nth order Fourier component of the magnetization distribution of the stripes. Edge domains lead to an effective reduction of the stripe width and can cause the observed effects. In addition, the formation of closure domains at the edges is expected because of the strong demagnetizing fields for an external field perpendicular to the edges, see Sec. 2. 10.4.3. Bragg-MOKE amplitude sat (αf ) clearly consists of two contributions. An almost linear increase of The curve of θK sat θK (αf ) was observed earlier, e.g. in Sec. 9.4. In was explained there using a straightforward picture and in the frame work of a Lorentz-Drude model of the Kerr effect, 141 10. Co gratings on a Fe-film see Fig. 9.21. Superimposed to this an oscillating contribution is found in Fig. 10.6. This can be explained by the interference effects of the underlaying Fe layer and the Co grating, as it was discussed in Sec. 3.4.3. An example of this effect was plotted in Fig. 3.15. It is clearly seen that a width to grating parameter ratio of 12 can lead to an oscillation period of ∆n = 2. However, in the situation plotted in Fig. 3.15 the intensity minima correspond to Kerr amplitude maxima, just opposite to the case observed in the present study, see Fig. 10.6. This discrepancy may be solved if the actual situation of two magnetic subgratings is taken into account in the simulations. The oscillating contribution increases with increasing grating parameter. This is due to the fact that the ideal case of w/d = 1/2 is approached for increasing grating parameters. In addition, an oscillation is better observable if the points are close to each other and the number of points is much larger than the oscillation period, which also leads to an apparent increase of the oscillating contribution with the grating parameter. 10.5. Summary and Conclusion A spin valve like structure was under investigation. The first (Fe) layer was separated by the second (Co) layer by a thick 20 nm Cr interlayer. The top Co layer and the Cr interlayer were patterned into different grating structures. In this study it has been shown that Bragg-MOKE is a valuable tool for the investigation of such complex ferromagnetic microstructures. • The existence of edge domains was observed in the Co stripes. • Several hints indicate that a uniaxial anisotropy is induced in the underlying Fe layer. In addition, the Bragg-MOKE amplitude is increased significantly which may help technically to analyze these kind of structures. However, in order to study further the Bragg-MOKE effect it will be instructive to correlate the present measurements to Kerr microscopy images. Furthermore the hysteresis curves of the two subsystems could be measured separately using soft x-ray scattering techniques. This will assure the above conclusions and also help in the understanding of the observed effects. The soft x-ray techniques can also be used in a diffractional setup which may open up a new and challenging area of research. Moreover, a decent theoretical treatment of the observed optical effects is desirable. 142 11. Further measurements This chapter groups two different experiments which are too short for discussing them in separate chapters, but give additional important insight in the nature of the BraggMOKE effect that they make them worth mentioning. 11.1. Diffuse Kerr effect In order to further clarify the dependence of the saturation Kerr rotation as a function of the diffraction angle an experiment was designed which also allows to measure in the off-specular condition but without actually fabricating a grating structure. The sample prepared is rather simple: the unpolished backside of a standard sapphire substrate was covered with a 30 nm thick Fe film using the sputter process as explained in Sec. 4.1. The resulting film is extremely rough and no specular reflection can be observed when the sample is mounted in the MOKE setup and subjected to the laser 0.07 0.06 0.05 K θsat [°] 0.04 0.03 0.02 0.05 θK [°] 0.01 0 0 −0.05 −0.01 −0.02 −2000 −1000 −0.03 0 10 20 α [°] 30 0 H [Oe] 1000 40 2000 50 f Figure 11.1.: Saturation Kerr rotation as a function of αf for a diffuse scattering Fe film and perpendicular incidence. In the inset an example of the measured hysteresis curves is plotted. 143 11. Further measurements beam. Instead, a diffuse intensity distribution is observed in the complete solid angle of 2π above the film plane. The angle of incidence of the laser was again chosen to be perpendicular to the film plane. In this situation the diffuse scattered intensity in a certain solid angle around three reflection angles was collected using a two lens setup and focussed onto the Kerr-detector. The solid angle was comparatively small, the aperture was 4 cm at a distance of 30 cm from the sample. The angle of reflection chosen were 32, 40 and 50◦ . sat The result is displayed in Fig. 11.1. The main panel shows the results of θK (αf ). The saturation Kerr rotation is the maximum value measured up to the field range of 2 kOe. sat The straight line is a linear fit to the data and the origin showing that θK (αf ) indeed has the same dependence as found from the grating experiments, thus fully justifying the simple explanation of the experiment in terms of the Lorentz-Drude picture. The inset in Fig. 11.1 shows the measured hysteresis at 50◦ . The measurement is of course somewhat noisy, however, it proves that the MOKE hysteresis curves can be measured also in the diffuse scattered light of rough surfaces. 11.2. Fe grating with giant Kerr rotation In this section a set of samples produced from a 30 nm thick polycrystalline, sputtered Fe film on sapphire (a-plane) is under investigation1 . The grating parameter is d = 4, 6, 8 µm and the Fe stripe width is wF e = 1.9, 2.7, 3.7 µm for the three gratings, respectively. The detailed sample preparation technique is discussed in Sec. 4.1. Thus wF e is ≈ 10% smaller than d/2, however, a strong ∆n = 2 oscillation in the intensity spectrum is found. In this respect the three gratings are comparable to the set of samples measured in Sec. 10 and the motivation for this study also follows the lines discussed there. The standard longitudinal MOKE curves show an increasing uniaxial anisotropy with decreasing stripe width, but as wF e is generally larger than the Fe stripes discussed in Sec. 9.4 the effect is less pronounced. Some examples of Bragg-MOKE hysteresis loops (in the hard axis orientation) are depicted in Fig. 11.2. It is clearly seen that for n = 1 no anomalous hysteresis is detected. The same is true for other odd order of diffraction loops. The loops for n = 2 show a different behavior. For d = 6 µm an additional component is measured, which is identical to the measurements reported in Sec. 9.4. However, this was only observed for d = 6 µ. For larger (d = 8) and smaller (d = 4) Fe stripe widths this component was not observed. The reason for the anomalous shape of the hysteresis cures for n = 2 is, as discussed in the previous chapters, the existence of edge domains which lead to a variation of the effective width of the stripes for the magneto-optical detection for the setup. The Fourier transformation (Sec. 3.4.2) of this remagnetization behavior leads to sine-like additional components in the diffraction hysteresis loop (DHL, see Sec. 3.4.2) which result in the observed modification of the Bragg-MOKE hysteresis loop. For the broader Fe stripes (d = 8 µm and wF e = 3.7 µm) the absence of this component means that edge domains play a minor role for this grating. This is not surprising because the overall uniaxial shape anisotropy is reduced and the relative region influenced by edge effects is reduced. 1 This section is based on parts of the article Magnetooptical Kerr effect of Fe-gratings [102] 144 11.2. Fe grating with giant Kerr rotation n=−1 0.2 n=−2 8µm 0 −0.2 θK [°] 0.2 6µm 0 −0.2 0.2 4µm 0 −0.2 −0.5 0 0.5 −0.5 0 0.5 H [kOe] Figure 11.2.: Bragg-MOKE hysteresis loops for n = −1 and n = −2 and the grating parameter d = 4, 6 and 8 µm of the polycrystalline Fe grating. However, the absence of the additional component for the smallest stripe width of this set of gratings is surprising, but can be explained by the generally strongly reduced Kerr signal for this particular Bragg-MOKE loop. This brings the focus of the discussion to the Kerr signal as a function of the diffraction order. The measurement is depicted in Fig. 11.3. Different to the measurements in previous chapters the angle of incidence was non-zero, αi ≈ 40◦ . The Kerr signal in Fig. 11.3 is plotted as a function of the diffraction angle, αf , for a better comparison of the results of the three gratings with different d. The curve in Fig. 11.3 shows the up to now expected behavior. The Kerr signal θK (αf ) consists of two components: first, a linear dependence is observed which increases with increasing diffraction angle and displays a change of sign for αf = 0. This behavior fits perfectly to the qualitative 145 11. Further measurements d=8µm d=6µm d=4µm 0.5 0.4 θsat [°] K 0.3 0.2 0.1 0 −0.1 −60 −40 −20 0 20 α [°] Figure 11.3.: Bragg-MOKE amplitude in saturation of the polycrystalline Fe gratings as a function of the diffraction angle. model of the Kerr amplitude in the framework of a Lorentz-Drude model as it was developed in Sec. 9.4. In that case only perpendicular incidence was discussed but it is clear that similar arguments for non-zero angle of incidence must also lead to a change of sign of the Bragg-MOKE amplitude at αf = 0 (see Fig. 9.21). A similar dependence was already depicted in Sec. 8 (Fig. 8.3). Second, an oscillation with the period ∆n = 2 is superimposed. The minima of the oscillating contribution coincide with the intensity minima at the even order diffraction spots. The same behavior was already reported from the Co on Fe gratings in Sec. 10. There are, however, two rather remarkable exceptions: The Kerr loops for n = 2, d = 6 and 8 µm display a giant Kerr-signal amplification. This giant Kerr effect cannot be explained with the interference of light diffracted by the substrate and the grating. The phase difference between the two components can be calculated using Eq. 3.20. The result for the present geometry is φ ≈ π4 for n = −2 and even smaller for n = 2. Following the calculation in Sec. 3.4.3, which correlate the phase and intensity difference of the two components of the diffracted light to the observed Kerr rotation, no such giant Kerr effect can be anticipated. Only for larger phase difference > π2 an influence is predicted. All other attempts for the understanding of the observed Bragg-MOKE effects discussed up to here also seem to fail to explain the observed giant Kerr signal. Clearly, a decent theory of the longitudinal Bragg-MOKE effect, actually calculating the vectorial amplitudes of the electromagnetic field, is necessary. However, an amplification of the observed Kerr rotation of > 10 (compared the specular hysteresis and the curve for n = −2) might be very useful for technical applications. 146 12. Conclusions Understanding the remagnetization processes of artificially produced lateral magnetic patterns, such as stripe or dot arrays, is of fundamental as well as of technical importance. Technical innovations such as magneto-electronics, MRAM’s, magnetic readheads and patterned magnetic media are not possible without fundamental research in this subject. There are two possibilities of analyzing the remagnetization process of magnetic thin film elements. First, one can use lateral resolving, microscopic methods to image the domains in single elements, and second, integrating methods such as MOKE or SQUID can be used to gain information about remanence, coercive field and hysteresis. Regular arrays may also be investigated using scattering methods, measuring magnetic properties in reciprocal space. The subject of this thesis was to explore the potential of new magneto-optical techniques and determine as example the magnetization reversal of patterned magnetic systems. The vector-MOKE technique gains additional integrated information about the average magnetization vector. The Bragg-MOKE technique uses diffraction to gain further information about the lateral magnetization distribution in reciprocal space. The systems under investigation were arrays of ferromagnetic stripes or wires with grating parameters ranging from 300 nm to 20 µm and stripe widths from 10% to 75% of the grating parameter. The grating structures were prepared from 10 - 20 nm thick Fe, Ni or CoFe alloy films of single crystal or polycrystalline quality. The samples were mainly prepared using electron beam lithography in combination with standard thin film preparation techniques like MBE or sputtering. The main tool of this thesis was the magneto-optical Kerr effect in the longitudinal configuration. If grating structures are subjected to the laser beam of a MOKE setup additional Bragg-reflections are observed which were used to gain further information of the domain distribution and are used to enhance the Kerr-signal (Bragg-MOKE). Furthermore, the MOKE technique can be used to gain vectorial information of the magnetization process. In the following the main results are briefly summarized and proposals for further experiments are given. Remagnetization process of ferromagnetic stripes The measurements in the preceding chapters dealt with the remagnetization process and the domain structure of ferromagnetic stripes and wires on the nano- and micrometer scale. As the measurements were mainly done using integrating methods no detailed analysis was possible, thus the main aim of this thesis was the exploration of the new 147 12. Conclusions magneto-optical techniques rather than micromagnetic studies. However, several interesting magnetization processes were measured and could be described combining vectorMOKE, Bragg-MOKE and microscopy techniques such as MFM and Kerr-microscopy: • The remagnetization pattern of Fe nanowires (Chap. 6) with a rounded shape and a grating parameter of 300 nm consists of a complex domain nucleation process. A plateau region in the hysteresis curve is most probable caused by different pinning potentials of domains magnetized along or perpendicular to the wire axis. • The remagnetization process of Ni stripes (width: 4 µm, Chap. 8)is almost not affected by edge effects, due to the fine domain structure, one order of magnitude smaller than the stripe width. Anomalies observed with the Bragg-MOKE technique are probably due to ridges at the edge of the stripes caused by the unperfect production of the sample. • The remagnetization process of CoFe stripes (width: 3 µm, see Chap. 7) is dominated by coherent rotation of the magnetization into the easy axis (along the stripes) and a 180◦ domain wall switching. This behavior is due to the high shape anisotropy of the CoFe stripes. In addition, the Bragg-MOKE technique provided evidence for the existence of small edge domains magnetized along the wire edge. • The remagnetization process of Fe stripes with different widths (ranging from 0.5 to 3.7 µm, see Chap. 9.4) showed very interesting features. The domain pattern depends on the relative strength of the fourfold crystalline anisotropy and the stripeinduced twofold anisotropy. For small separations of the stripes a coupling between the magnetization of the individual stripes was observed with Kerr-microscopy. If the fourfold anisotropy can be neglected the domains form a regular pattern with strong edge domains observable with Bragg-MOKE. Stripes with crystalline anisotropy showed a plateau region with zero magnetization. Vector-MOKE The vector-MOKE technique is an extension of the standard longitudinal Kerr effect. An additional field applied perpendicular to the scattering plane allows for a measurement of the two orthogonal in-plane components of the magnetization and leads thus to a measurement of the length and the angle of the magnetization vector. Therefore, domain processes can be distinguished from rotational processes during the remagnetization. The technique was described in detail in this thesis and several examples of measurements were given. In particular it was proven that the remagnetization process of an array of Fe nanowires (Chap. 6) consists of a complex domain structure and that an array of CoFe (Chap. 7) stripes changes its magnetization mainly by rotation of the magnetization. In other cases the vector-MOKE techniques gave important information for the interpretation of Bragg-MOKE measurements (Chap. 9.4). Bragg-MOKE In this thesis it was shown that the Bragg-MOKE technique is a valuable addition to the MOKE technique for the examination of laterally patterned magnetic films. Mainly 148 Figure 12.1.: Illustration of the linear dependence of the Bragg-MOKE signal in rotation as a function of the diffraction angle (all figures are taken from previous sections of this thesis). From left to right: Because of the diffraction geometry the exit angle is not simply the negative angle of incidence; this off-specular geometry causes increasing Kerr rotations for increasing exit angles; the effect was clearly observed for several grating structures. qualitative explanations for the observed effects were given, which may help experimentalist to interpret their results. However, a more detailed theoretical analysis of the longitudinal Bragg-MOKE effect, as it exists for other geometries [53] is certainly necessary. The deeper understanding of these effects also seems of technical interest since it allows for a fast and effective characterization of the quality of patterned magnetic media, which will play an important role in future magnetic data recording techniques. In more detail, the Bragg-MOKE hysteresis curves show three effects: • For measurements in diffraction geometry the constrain αi = αf is lifted, therefore the usual Fresnel formulas describing the polarization cannot be valid. Qualitative explanations in the framework of the Lorentz-Drude model of metals were developed in the preceding chapters. The saturation Kerr rotation of Bragg-MOKE curves consists of an almost linear part which increases with increasing |αf | (increasing n) and is zero for αf = 0 (parallel to the surface normal). The effect is depicted schematically in Fig. 12.1. This may be in contradiction with [53] were a vanishing Kerr signal is found if the diffracted beam is antiparallel to the incident beam (Littrow-mounting). • The light diffracted by the spacing between the stripes will also contribute to the total signal. The interference between light diffracted by the not ferromagnetic substrate and the ferromagnetic stripes may lead to a modification of the observed Kerr rotation much in the same way as anti-reflection coatings on thin films help to increase the magneto-optical contrast in Kerr-microscopy. However, the thickness of the stripes is small compared to the wavelength and thus the phase difference is also small – no amplification of this kind was observed. Another related effect is that the diffraction envelope of stripe and spacing will exhibit maxima or minima 149 12. Conclusions Figure 12.2.: Illustration of the oscillating Bragg-MOKE signal in saturation as a function of the diffraction angle (all figures are taken from previous sections of this thesis). From left to right: the sample morphology leads to phase and amplitude differences in the light diffracted from the stripes and the grooves; the diffraction envelope can be clearly observed by measuring the intensity of the diffraction spots; the oscillating Kerr amplitude superimposed on a linear increase (see Fig. 12.1) reflects the oscillation period of the intensity oscillations. at different order of diffraction, thus the ferromagnetic signal may be enhanced at certain order of diffraction. The effect of oscillating saturation Kerr signal is schematically depicted in Fig 12.2. The understanding of the oscillating Kerr signal in detail is a complicated task, in particular non-scalar diffraction theories have to be used. • The domain structure results in a non-uniform distribution of the magneto-optical reflectivity constants at the surface of the stripes. Therefore, the nth order diffraction spots carries information of the nth order Fourier component of the magnetization distribution of the grating. If the individual stripes of the grating behave more or less identical a magnetic form-factor can be defined which is the Fourier transformation of the average magnetization distribution inside the stripes. In this thesis several limiting cases in one dimension were calculated analytically (Sec. 3.4.2) and compared with measured data. In Fig. 12.3 this effect is depicted schematically. Although no exact fitting was possible, qualitative correspondence was reached. In particular, if edge domains are formed, the effective width of the stripes is reduced, which leads to a sinusoidal behavior of the Kerr signal as a function of the magnetization. If only irregular domains are formed or coherent rotation takes place, the effective width is constant but the magnitude of the magnetization inside the stripe is reduced. This case leads to linear functions of the Kerr signal. The use of diffraction for the magneto-optical measurements has also some technical advantages: • The Kerr signal in saturation can be increased by choosing high order of diffraction. In addition, for special cases a resonance may occur, an enhancement of the Kerr 150 Figure 12.3.: Illustration of the effect of edge domains leading to anomalous BraggMOKE hysteresis shapes (all measurements were previously presented in this thesis). From left to right: the Kerr-microscopy image reveals the existence of edge domains; a one dimensional model of one stripe is developed; the Fourier-transformation of this model can be calculated analytically; the calculation qualitatively corresponds to the Bragg-MOKE measurement in a representation of the Kerr-rotation as a function of the magnetization. rotation of a factor > 10 was reported (Sec. 11). However, the magneto-optic figure of merit (product of Kerr rotation and intensity) may not be enhanced, because the gain in Kerr rotation is payed by a strong decrease of the intensity at high order of diffraction. This problem does not affect the measurements for the present thesis, as the detection technique applied is only sensitive to the rotation and is to a large extent independent of the intensity. • Bragg-MOKE acts like a filter. Only signals from the grating can pass. Therefore, gratings of only small total size, surrounded by ferromagnetic material can be measured. Other integrating methods (like SQUID) have difficulties if the desired signal is small and is accompanied by other ferromagnetic signals. • It has been shown that also in the diffuse light of rough surfaces a MOKE signal can be isolated (Sec. 11). • The sensitivity to the magnetization at the edges of the stripes can be increased. If measurements are carried out at reflexes with strongly reduced intensity (e.g. n = 2 and w = d/2), the signal is very sensitive to changes of this ideal situation (Chap. 9.4). 151 12. Conclusions Outlook The research field of nano- and microstructured magnetic elements is still growing triggered by the potential technical applications and the general interest in magnetic domains and patterns. The combination with the fascinating subject of diffraction makes it even more challenging. This thesis proofed that the combination of diffraction and magneto-optics is a valuable tool in this research field and that in itself it is not completely understood. This thesis helped to unravel some of the complicated effects which were observed by Bragg-MOKE. However, many things remain to do, therefore this thesis closes with a list of proposals for new experiments which should be done in the future. Two main directions of future work can be presented: first, just use the gathered knowledge and investigate more complicated samples and magnetic structures and, second, try to further understand the Bragg-MOKE effect and develop methods to make the Bragg-MOKE hysteresis curves predictable: • Bragg-MOKE studies of the remagnetization process of dots or squares will provide interesting results. The subject of coupling between adjacent elements has also only been touched. Square lattices open up the possibility for further analysis using diffraction techniques. In addition, a combination of vector-MOKE and Bragg-MOKE is possible for arrays with a fourfold symmetry, i.e. the BraggMOKE signal can be detected with the applied field perpendicular and parallel to the scattering plane with out the need to rotate the sample (see Sec. 3.5.3). • The studies can be extended to different materials and a variety of other geometric factors (e.g. height of the elements). • Relief gratings and gratings of non-magnetic material on magnetic films have already been demonstrated, but may be further studied in order to measure offspecular magneto-optical constants independent of the domain structure. • Innovative magnetic structures utilizing exchange bias or interlayer exchange coupling in order to modify the domain structure are of growing interest. BraggMOKE measurements are very sensitive to changes of the domain structure and may thus help in this research subject. Other MOKE methods like MOKE with a rotating field (ROTMOKE, see [103]) can be combined with the Bragg-MOKE technique to gain additional information [104] and to separate non-linear magneto-optical effects [103]. Last but not least, the Bragg-MOKE studies presented here can be viewed as a preliminary study for measurements of inplane Bragg-spots of artificial magnetic structures using neutrons and soft x-rays. These two probes have some advantages over the visible photons of the laser light, as larger penetration depth (for neutrons), chemical sensitivity (for soft x-rays) and a generally smaller wavelength enabling the investigation of even smaller patterns. Some general physical ideas like the diffraction from the magnetization distribution are the same and MOKE is some orders of magnitude cheaper. Furthermore, methods have to be developed to model the Bragg-MOKE curves taking only the optic constants and the magnetization distribution into account. This has 152 already been done for the transverse Bragg-MOKE effect [53] and needs the expertise of scientist in the field of diffractional optics. Another viewpoint is that the Fourier spectrum of a given magnetization distribution can be calculated numerically, which should at least predict the shape of measured Bragg-MOKE curve. Such an experiment has recently been reported in [104]. The authors were able to model certain features of a Bragg-MOKE measurement by numerically calculating the two-dimensional Fourier transformation of a domain pattern obtained by using standard micromagnetic simulation tools. This is a very promising path for future investigations. In addition, one could directly try to calculate the Fourier compounds using experimental data obtained with Kerr microscopy. Finally, it should be noted, that besides all technical and fundamental interest in micro- and nanostructured magnetic media, it is also the general fascinating combination of classical Kerr-effect and diffraction which makes the subject of Bragg-MOKE attractive to physicist of several disciplines. Because of this combination unexpected data is produced and the shape of measured hysteresis loops astonished long-established experts in magnetism. Only by careful analysis it is possible to separate the optics from the magnetics. This thesis hopefully helped to develop physical transparent pictures for some of the peculiarities of Bragg-MOKE. 153 12. Conclusions 154 Bibliography [1] G. Prinz, K. Hathaway. Physics Today, 4, 24 (1995). [2] A. Hubert, R. Schäfer. Magnetic Domains, chapter 3. Springer-Verlag Berlin Heidelberg (1998). [3] C. Shearwood, S. J. Blundell, M. J. Daird, J. A. C. Bland, M. Gester, H. Ahmed, H. P. Hughes. J. Appl. Phys., 75, 5249 (1994). [4] A. Hubert, R. Schäfer. 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Fischer (2002). Private communication. 159 Bibliography [101] T. Schmitte, K. Westerholt, H. Zabel. J. Appl. Phys., 92, 4527 (2002). [102] T. Schmitte, O. Schwöbken, S. Gök, K. Westerholt, H.Zabel. J. Magn. Mag. Mat., 240, 24 (2002). [103] R. Mattheis, G. Quedau. J. Magn. Mag. Mat., 205, 143 (1999). [104] P. Garcia-Mochales, J. L. Costa-Krämer, G. Armelles, F. Briones, D. J. J. I. Martin, J. L. Vicent. Appl. Phys. Let., 81, 3206 (2002). 160 Zusammenfassung Zusammenfassung Einführung Das Verständnis von Ummagnetisierungsvorgängen von künstlich strukturierten magnetischen Inseln, Partikeln oder Streifen, ist sowohl von fundamentalem Interesse, als auch für die Anwendung von herausragender Bedeutung. Insbesondere die aktuelle Erforschung und Entwicklung der Magneto-Elektronik, wie z. B. MRAMs (magnetischen Speicher mit wahlfreiem Zugriff), Leseköpfe für Festplatten und strukturierte magnetische Speichermedien, beruhen zu großen Teilen auf dem Verständnis von Ummagnetisierungsvorgängen und Domänenstrukturen. Um die Ummagnetisierung experimentell zu bestimmen, sind zwei Zugänge möglich: Zum einen kann man die Domänenstruktur ortsaufgelöst mit Methoden wie der KerrMikroskopie oder der magnetischen Kraft-Mikroskopie (MFM) bestimmen. Zum anderen lassen sich magnetische Hysteresen mit integrierenden Methoden, wie dem magnetooptischen Kerr-Effekt (MOKE) oder SQUID (Supraleitende Quanteninterferenz Messung zur Bestimmung der magnetischen Streufelder), messen. Hysteresen, Koerzitivfelder und die remanente Magnetisierung können so sehr genau gemessen werden. Außerdem lassen sich viele gleichartige magnetische Elemente gleichzeitig untersuchen. In der vorliegenden Arbeit werden zwei neue magnetooptische Verfahren eingeführt und auf Probensysteme aus magnetischen Streifen in der Größenordnung von Mikrometern angewandt. • Die Vektor-MOKE Technik benutzt den longitudinalen Kerr-Effekt, um bei externen magnetischen Feldern in paralleler und senkrechter Orientierung zwei orthogonale Komponenten des Magnetisierungsvektors zu bestimmen. Aus diesen Informationen lassen sich schließlich, mittels der Länge und Orientierung des Magnetisierungsvektors, Aussagen treffen, ob der Ummagnetisierungsvorgang aus einer reinen Rotation der Magnetisierung, reinen Domänenprozessen oder aus einer Kombination von beiden besteht. • Die Bragg-MOKE Technik nutzt Interferenzphänomene an regelmäßigen lateralen Strukturen, um Aussagen über die gemittelte Domänenstruktur der magnetischen Inseln zu erhalten. Dabei wird der longitudinale Kerr-Effekt an gebeugten Laserstrahlen der als Interferenzgitter wirkenden Probe detektiert. Insbesondere läßt sich sehr einfach feststellen, inwiefern sich korrelierte Domänen, wie zum Beispiel Randdomänen, ausbilden. Die dabei untersuchten Probensysteme sind künstlich hergestellte magnetische Streifengitter. Die Breite der Streifen liegt zwischen 150 nm und 5 µm, die Schichtdicke ist typisch 20 nm und der Gitterparameter wird von 300 nm bis zu 20 µm variiert. Die Proben wurden mittels verschiedener lithographischer Methoden hergestellt. Ziel dieser Arbeit ist insbesondere die Untersuchung des Bragg-MOKE Effekts selbst. Dabei sind drei Punkte von Bedeutung: • Die Beugungsgeometrie ist nicht mehr mit dem eigentlichen longitudinalen KerrEffekt identisch, da nun der Einfallswinkel nicht mehr gleich dem Ausfallswinkel 161 Zusammenfassung ist. Daher spielen nicht-spekuläre Fresnel-Koeffizienten eine Rolle. Es stellt sich die Frage, inwiefern sich ein Einfluss auf die gemessenen Kurven ergibt. • Der Einfluss von Interferenzeffekten zwischen den magnetischen Streifen und der dazwischen liegenden Substrat-Region kann zu zusätzlichen Phasenverschiebungen führen, wie es beispielsweise bei der Benutzung von Anti-ReflexionsBeschichtungen beobachtet wird. • Aus grundsätzlichen Überlegungen folgt, dass die gemessene Bragg-MOKE Kurve der Ordnung n die nte Komponente der Fourier-Transformation der Magnetisierungsverteilung des Gitters darstellt (magnetischer Formfaktor). Kann man dies nutzen, um qualitative Informationen über die Domänenstruktur der Probe zu erhalten, ohne ortsauflösende Methoden zu benutzen? Die zentralen Aspekte dieser Arbeit liegen in der Untersuchung des Ummagnetisierungsvorgangs von magnetischen Gittern und der Untersuchung von optischen Effekten in der Bragg-MOKE Geometrie. Beide Aspekte werden vom Standpunkt der experimentellen Physik angegangen, verschiedene Parameter werden systematisch variiert und daraus phänomenologische Beschreibungen hergeleitet. Analytische oder numerische Herleitungen aus ersten Prinzipien stehen nicht im Vordergrund. Magnetismus von magnetischen Nano- und Mikrostrukturen Für den Magnetismus der untersuchten Streifengitter spielen zwei physikalische Phänomene eine herausragende Bedeutung: Einerseits ist dies die magnetische Anisotropie, d.h. das Bestreben der Magnetisierungsverteilung innerhalb eines magnetischen Elements nicht nur gleichförmig sondern auch entlang einer bestimmten Richtung ausgerichtet zu sein. Anisotropien werden entweder durch die Kristallstruktur oder durch die äußere Form vorgegeben. Beispielsweise sind dünne Filme in der Regel in der Filmebene und magnetische Streifenstrukturen entlang der Streifenrichtung magnetisiert (diese Richtungen bezeichnet man dann als leichte Richtungen). Die Kristallstruktur von Fe führt dagegen bei einkristallinen Proben mit einer (001) Wachstumsrichtung zu einer vierzähligen Anisotropie mit leichten Achsen entlang der (010) und (100) Richtung. Andererseits bilden sich in vielen Fällen magnetische Domänen aus, d. h. die Magnetisierungsverteilung ist nicht mehr gleichförmig, sondern es existieren Bereiche mit unterschiedlichen Magnetisierungsrichtungen und diskontinuierlichen Grenzen dazwischen. Domänen entstehen, um das Streufeld der Probe bzw. des magnetischen Elements zu reduzieren. Die Domänenverteilung ist entweder durch die äußere Form (Richtung der Magnetisierung entlang von Kanten) oder durch die oben erwähnten Anisotropien (Magnetisierung in einer Domäne entlang einer leichten Richtung) bestimmt. Methoden Probenpräparation Die Herstellung von lateral strukturierten Proben geschieht mit Hilfe von lithographischen Methoden. Im interessanten Bereich von Strukturgrößen von einigen hundert Nanometern bis hin zu einigen Mikrometern stehen mehrere Präparationsmethoden zur Verfügung. Allen Methoden gemein ist das Vorgehen in zwei Schritten: 162 Zusammenfassung Erst wird ein organischer Film strukturiert (Maske), dann die resultierende Maske in einen dünnen Film übertragen. Zur Herstellung der Maske wird der organische Film mit Licht (optische Lithographie, Raster-Laser-Lithographie) oder mit Elektronen (RasterElektronen-Lithographie) belichtet und dann entwickelt. Die Struktur wird entweder durch einen Ätzprozess, durch einen sog. lift-off -Prozess oder durch eine Kombination beider in einen dünnen magnetischen Film übertragen. Der dünne Film (hier handelt es sich um Schichtdicken von typischerweise 20 nm) wird mit Standartpräparationsmethoden wie Sputtern oder der Molekularstrahl-Epitaxie (MBE) hergestellt. Auf diese Weise wurden regelmäßige Matrizen von magnetischen Streifen produziert, die mit typischen Untersuchungsmethoden wie Raster-Elektronen-Mikroskopie, RasterKraft-Mikroskopie oder optischer Mikroskopie auf ihre Struktur untersucht werden konnten. Magnetische Untersuchungsmethoden Zur magnetischen Charakterisierung der Mikrostrukturen wurde der longitudinale Kerr-Effekt benutzt. In seiner normalen Konfiguration wird linear polarisiertes Licht von der Probe reflektiert. Die Magnetisierung der Probe führt daraufhin zu einer Änderung des Polarisationszustands des Laserlichts. Diese Drehung der Polarisation wird mittels einer speziell entwickelten, hochauflösenden Analysetechnik als Funktion des äußeren Feldes detektiert. Daraus erhält man eine Kurve, die proportional zur magnetischen Hysterese der Probe ist. Bei dieser Methode wird die Magnetisierungskomponente entlang des Feldes bestimmt, wobei das Feld dazu in der Einfallsebene des Lichts und parallel zur Probenoberfläche orientiert ist. Die Probe kann nun in verschiedene Richtungen (Streifenrichtung relativ zum Feld) gedreht werden. Aus den jeweiligen Hysteresen lassen sich sehr detaillierte Aussagen über die magnetische Anisotropie treffen. Wie bereits oben erwähnt, wird diese Methode im Rahmen dieser Arbeit erweitert, um sowohl Vektor-Informationen über die Magnetisierung, als auch qualitative Aussagen über die Domänenstruktur zu erhalten. Zur Messung von Bragg-MOKE Hysteresen wird die Probe so gedreht, dass die magnetischen Streifen senkrecht zur Einfallsebene des Lasers und senkrecht zum äußeren Feld stehen. Dies entspricht im Allgemeinen der schweren Richtung der durch die Form der Streifen induzierten Anisotropie. Die Streifen wirken wie ein Beugungsgitter und erzeugen in der Einfallsebene ein Beugungsmuster. Der Detektor wird nun so verstellt, dass das Licht eines Beugungsmaximums detektiert wird. Ansonsten wird die Hysterese normal gemessen. Ein anderer Weg wird mit ortsaufgelösten Messungen beschritten. Mit der KerrMikroskopie wird die Domänenstruktur eines kleinen Teils der Probe direkt sichtbar gemacht. Dabei wird die oben erwähnte Polarisationsänderung im Mikroskop als Kontrast sichtbar. Eine zweite wichtige Methode ist die der Raster-Kraftmikroskopie (AFM) verwandte Raster-Magnetokraft-Mikropskopie (MFM). Dabei wird eine nur Nanometer große magnetische Spitze über die Probe bewegt und die Wechselwirkung mit den magnetischen Streufeldern der Domänen detektiert. Diese beiden abbildenden Methoden wurden mit den aus MOKE Messungen erhaltenen Daten verglichen und erwiesen sich bei der Interpretation der Bragg-MOKE Ergebnisse als sehr nützlich. 163 Zusammenfassung Ergebnisse und Diskussion Ummagnetisierung von ferromagnetischen Streifen Obwohl das Hauptaugenmerk auf der Entwicklung der neuen magnetooptischen Methoden lag, wurden ferner interessante Ummagnetisierungsvorgänge an verschiedenen magnetischen Streifenstrukturen gemessen: • Der Ummagnetisierungsvorgang von Fe-Nanostreifen (Kap. 6) mit einem abgerundeten Querschnitt und einem Gitterparameter von 300 nm besteht aus einem komplexen Domänen-Nukleationsprozess. Ein Plateau-Bereich in der Hysterese deutet auf unterschiedliche Pinnig-Potentiale für Domänen hin, welche entlang oder senkrecht zur Streifenrichtung magnetisiert sind. • Der Ummagnetisierungsprozess von Ni-Streifen (Kap. 8) hingegen wird kaum von Randeffekten beeinflusst, da eine sehr feine Domänenstruktur vorliegt, die eine Größenordnung kleiner als die Streifenbreite ist. Anormale Bragg-MOKE Hysteresen konnten auf die Präparationsmethode, die zu starken Graten an den Rändern führte, zurückgeführt werden. • Reine Rotationsprozesse dominieren das magnetische Verhalten einer CoFeStreifen Probe (Kap. 7). Die Magnetisierung rotiert erst in die leichte Magnetisierungsrichtung (entlang der Streifen), bevor eine 180◦ -Domänenwand am Koerzitivfeld durch den Streifen läuft. Hier konnte mit der Bragg-MOKE Methode nachgewiesen werden, dass sich zusätzlich sehr kleine Randdomänen ausbilden. • Sehr intensiv wurden mehrer Streifengitter aus Fe untersucht (Kap. 9.4). Die Domänenstruktur hängt stark von der relativen Größe der vierzähligen Kristallanisotropie und der, durch die Form der Streifen gegebenen, zweizähligen Anisotropie ab. Für kleine Abstände wurde zusätzlich mit Hilfe der Kerr-Mikroskopie eine Kopplung zwischen den Streifen beobachtet. Falls die vierzählige Kristallanisotropie vernachlässigt werden kann, bilden die Domänen ein regelmäßiges Muster mit starken Randdomänen. Dies wurde auch mit Bragg-MOKE nachgewiesen. Bei einer Dominanz der Kristallanisotropie wurde in den Hysteresen eine Plateau-Region beobachtet. Vektor-MOKE Die Vektor-MOKE Technik ist eine Weiterentwicklung des herkömmlichen longitudinalen Kerr-Effekts. Dabei wird ein äußeres Feld senkrecht zur Einfallsebene des Lasers angelegt und die Probe um 90◦ gedreht. Daraus kann dann der Winkel und die Länge des Magnetisierungs-Vektors abgeleitet werden. Die Methode wurde im Detail erläutert (Kap. 3.2). Die Domänenstruktur insbesondere der CoFe-Streifen (Kap. 7) und der Fe-Nanostreifen (Kap. 6) wurde mit Hilfe dieser Technik untersucht. Auch in anderen Fällen lieferte Vektor-MOKE wertvolle Hinweise zur Interpretation, z.B. der Bragg-MOKE Messungen (Kap. 9.4). Bragg-MOKE In dieser Arbeit wurde gezeigt, dass die Bragg-MOKE Technik ein wertvolles zusätzliches Werkzeug zur normalen Hysterese-Messung lateral strukturierter magnetischer Filme darstellt. Es wurden hauptsächlich qualitative Erklärungen für die 164 Zusammenfassung beobachteten Effekte diskutiert, theoretische Erklärungen stehen noch aus. Allerdings sind bereits die qualitativen Erklärungen beachtenswert. Das tiefere Verständnis der beobachteten Bragg-MOKE Effekte ist auch von technischem Interesse, da Bragg-MOKE zur schnellen Analyse der Qualität strukturierter magnetischer Speichermedien geeignet scheint. Im Detail wurden drei Effekte beobachtet: • Für den Fall der Messung in Beugungs-Geometrie ist die Bedingung Einfallswinkel=Ausfallswinkel aufgehoben. Das führt dazu, dass die gewöhnlichen FresnelKoeffizienten nicht benutzt werden können. Für diesen Fall wurde ein Modell im Rahmen der Lorentz-Drude Theorie beschrieben (Kap. 9.4). Die Sättigungs-KerrRotation der Bragg-MOKE Hysteresen als Funktion des Beugungswinkels besteht aus einem fast linearem Anteil, der mit zunehmendem Beugungswinkel zunimmt. Dabei kommt es zu einem Vorzeichenwechsel, falls der gebeugte Strahl parallel zur Proben-Normalen verläuft (0te Ordnung für senkrechten Einfall). Dieser Effekt ist schematisch in Abb. 12.1 dargestellt. • Das Licht, das von den Zwischenräumen gebeugt wird, trägt auch zum Gesamtsignal bei. Dabei kann es zu einer Verstärkung des Kerr-Signals kommen, wenn der Phasenunterschied der vom Substrat und der von den Streifen gebeugten Wellen annähernd π beträgt. Dieser Effekt ist der Verstärkung des Kerr-Effekts durch Anti-Reflexions-Schichten äquivalent. Allerdings wurde dieser Effekt in Praxis nicht beobachtet, da die benutzten Schichtdicken relativ zur Laserwellenlänge zu klein sind. Ein ähnlicher Effekt tritt auch aufgrund der unterschiedlichen Amplituden der beiden gebeugten Wellen auf. An bestimmten Beugungsordnungen kann es dann zu einer Verstärkung oder Abschwächung des detektierten Bragg-MOKE Signals kommen. Dieser Effekt ist mit der Intensität des gebeugten Lichtes korreliert und führt zu einer Oszillation des Kerr-Signals als Funktion der Beugungsordnung (siehe Kap. 8 und 10). Diese Oszillation ist der oben erwähnten, linearen Abhängigkeit überlagert. Schematisch ist dieser Effekt in Abb. 12.2 dargestellt. • Die Domänenstruktur führt zu einer nicht homogenen Verteilung der magnetooptischen Reflektionskoeffizienten innerhalb der Streifen. Daher enthält die BraggMOKE Hysterese nter Ordnung Informationen über die nte Fourier-Komponente dieser Magnetisierungsverteilung. Falls sich die einzelnen Streifen alle ähnlich verhalten, ist es möglich einen magnetischen Formfaktor zu definieren, welcher die Fourier-Transformierte der gemittelten Magnetisierungsverteilung darstellt. Verschiedene Spezialfälle für diesen magnetischen Formfaktor wurden in Kap. 3.4.2 analytisch berechnet. Die Ergebnisse lassen sich qualitativ mit den Messungen von Bragg-MOKE Hysteresen vergleichen. Das Verfahren ist schematisch in Abb. 12.3 zusammengefasst. Insbesondere für den Fall von Randdomänen ergibt sich ein anschauliches Bild: Durch die Magnetisierung der Randdomänen entlang der Streifen tragen diese bei der Hysteresemessung in schwerer Richtung nicht mehr zum magnetooptischen Signal bei. Effektiv wird also die Breite der Stege verändert. Die Fourier-Transformation eines solchen Streifens variabler Breite ist eine Sinusfunktion. Falls nur unregelmäßige Domänen entstehen, verringert sich durch die Mittelung nur die Größe der Magnetisierung, die effektive Stegbreite bleibt jedoch konstant. In diesem Fall ist die Fouriertransformation eine lineare Funktion. Diese 165 Zusammenfassung einfachen Fälle konnten in Kap. 7 und 9.4 an experimentellen Daten nachgewiesen werden. In der Realität werden beide Fälle gleichzeitig vorkommen oder noch weit kompliziertere Domänenmuster entstehen. Die Kombination aus Interferenz, Beugung und dem magnetooptischen Kerr-Effekt hat einige technische Vorteile: • Mit Hilfe der Bragg-MOKE Technik kann es möglich sein den Kerr-Effekt erheblich zu verstärken, indem man z.B. eine hohe Beugungsordnung zur Messung ausnutzt. Eine Verstärkung von einem Faktor > 10 wurde in Kap. 11 nachgewiesen. Allerdings wird nicht unbedingt auch das Produkt von Intensität und Kerr-Rotation vergrößert, da die Intensität bei hohen Ordnungen stark abnimmt. Messungen mit der in Kap. 3.5 beschriebenen Detektionstechnik, sind davon weitgehend unabhängig. • Bragg-MOKE wirkt wie ein Filter, so dass nur Signale der regelmäßigen Gitterstruktur detektiert werden. Daher ist es möglich Gitter von nur geringer Gesamtgröße magnetisch zu vermessen. Andere Methoden, wie SQUID, stossen hier u.U. aufgrund der geringen Menge magnetischen Materials auf ihre Grenzen. Hinzukommt, dass bei Messungen der gesamten Probe auch nicht-magnetisches Material zum Gesamtsignal beitragen kann. • In der vorliegenden Arbeit wurde demonstriert, dass es möglich ist, auch von diffus streuenden Oberflächen eine magnetischen Hysterese mit MOKE zu messen (Kap. 11). • Die Sensitivität auf die Magnetisierung an den Rändern der Streifen kann erhöht werden. Für bestimmte Beugungsordnungen mit stark reduzierter Intensität (z.B. n = 2 und wF e = d/2) reagieren die Ergebnisse der Bragg-MOKE Untersuchung sehr empfindlich auf leichte Veränderungen dieser idealen Situation. Kleine Veränderungen der optischen Eigenschaften können schon durch eine Änderung der Magnetisierung an den Kanten hervorgerufen werden. Ausblick Die Forschung an magnetischen Nano- und Mikrostrukturen ist wegen der hohen technischen Relevanz nach wie vor ein sehr interessantes Gebiet. Die in dieser Arbeit demonstrierten faszinierenden Effekte der Interferenz und Beugung stellen eine weitere Herausforderung dar. Es wurde gezeigt, dass die Kombination aus Magnetooptik und Beugung ein wertvolles zusätzliches experimentelles Werkzeug bereitstellt, allerdings sind die beobachteten Effekte noch nicht alle vollständig verstanden. Obwohl die vorliegende Arbeit einen wesentlichen Beitrag zum Verständis geliefert hat, bleiben wichtige Experimente durchzuführen. Im Wesentlichen erscheinen zwei Richtungen für die zukünftige Forschung mit BraggMOKE sinnvoll: Zum einen kann man die bekannten Effekte ausnutzen und weitere, auch kompliziertere, Probensysteme untersuchen. Zum anderen sollten Experimente zum tieferen Verständnis des Bragg-MOKE Effekts selbst durchgeführt werden: • Bragg-MOKE Untersuchungen des Ummagnetisierungsprozess an Gittern mit kreisförmigen, rechteckigen oder quadratischen Elementen können interessante Ergebnisse liefern. Zum Beispiel wurden Effekte der magnetischen Kopplung zwischen 166 Zusammenfassung den magnetischen Elementen bisher kaum behandelt. Auch bieten 2-dimensional Gitter neue Möglichkeiten Interferenz und Beugungserscheinungen auszunutzen und eine Kombination von Bragg-MOKE mit Vektor-MOKE bietet sich an. • Die bisherigen Untersuchungen können auf weitere Materialklassen ausgedehnt werden. • Reliefartige Strukturen wurden bereits ansatzweise untersucht und bieten die Möglichkeit die magnetooptischen Konstanten unabhängig von der Domänenstruktur zu bestimmen. • Innovative magnetische Strukturen, die auf dem sog. exchange-bias Effekt oder der Zwischenlagenaustauschkopplung basieren, zeigen neue und interessante Domänenstrukturen. Da Bragg-MOKE sehr sensitiv auf die Änderung der Domänenstruktur reagiert, können neue Erkenntnisse erwartet werden. Nicht zuletzt sollte erwähnt werden, dass die vorgestellten Untersuchungen des BraggMOKE Effekts eine Vorstudie zu Messungen von magnetischen Phänomenen mit anderen Strahlungsarten angesehen werden können. So können vergleichbare Experimente mit weichen Röntgen-Photonen oder Neutronen durchgeführt werden. Diese beiden Sonden haben verschiedene Vorteile gegenüber dem benutzten Laserlicht: so weisen Neutronen eine größere Eindringtiefe auf und weiche Röntgenstrahlung kann benutzt werden, um chemisch selektive Informationen zu erhalten. Außerdem haben beide Strahlungen erheblich kleinere Wellenlängen, so dass kleinere Strukturen untersucht werden können. Desweiteren sollten Methoden entwickelt werden, wie die Bragg-MOKE Hysteresen allein aus den optischen Konstanten und der Magnetisierungsverteilung hergeleitet werden können. Dazu wird man die Expertise von Wissenschaftlern aus dem Bereich der Optik benötigen. Ein anderer Zugang ist über numerische Berechnungen der Fourierzerlegung von Magnetisierungsverteilungen möglich, welche die Form der Bragg-MOKE Hysteresen richtig voraussagen sollten. Dazu können entweder mikromagnetische Modellrechnungen oder ortsaufgelöste Messungen des kompletten Ummagnetisierungsvorgangs (z.B. mit einem Kerr-Mikroskop) als Ausgangspunkt gewählt werden. Abschliessend sollte bemerkt werden, dass, neben dem technischem und fundamentalem Interesse an mikro- und nanostrukturierten magnetischen Materialien, es gerade die Kombination aus klassischem Kerr-Effekt, Interferenz und Beugung ist, die die Faszination von Bragg-MOKE ausmacht. Deswegen kommt es zu ungewöhnlichen und unerwarteten Messergebnissen, die auch erfahrene Magnetismus-Experten staunen lassen. Nur durch die sorgfältige Analyse der Daten kann eine Separation der optischen von den magnetischen Effekten erreicht werden. Die vorliegende Arbeit konnte hoffentliche einen Beitrag zur Entwicklung physikalisch transparenter Vorstellungen zur Interpratation von Bragg-MOKE Messungen leisten. 167 Publications List of publications which have resulted from this work • [78]: B. Roldan-Cuenya, M. Doi, W. Keune, S. Hoch, D. Reuter, A. Wieck, T. Schmitte, H. Zabel Magnetism and Interface Properties of Epitaxial Fe Films on High-Mobility GaAs/Al0.35Ga0.65As(001) Two-Dimensional Electron Gas Heterostructures in: Appl. Phys. Lett., accepted (2003). • [87]: T. Schmitte, K. Theis-Bröhl, V. Leiner, H. Zabel, S. Kirsch, A. Carl Magnetooptical study of the magnetization reversal process of Fe nanowires in: J. Phys.: Cond. Mat. 14, 7527 (2002). • [93]: K. Theis-Bröhl, T. Schmitte, V. Leiner, H. Zabel, K. Rott, H. Brückl, J. McCord CoFe-stripes: magnetization reversal study by polarized neutron scattering and magneto-optical Kerr effect in: Phys. Rev. B, accepted (2003). • [99]: T. Schmitte, T. Schemberg, K. Westerholt, H. Zabel, K. Schädler, U. Kunze Magneto-optical Kerr effects of ferromagnetic Ni-gratings in: J. Appl. Phys. 87, 5630 (2000). • [101]: T. Schmitte, K. Westerholt, H. Zabel Magneto-optical Kerr effect in the diffracted light of Fe gratings in: J. Appl. Phys. 92, 4527 (2002). • [102]: T. Schmitte, O. Schwöbken, S. Goek, K. Westerholt, H.Zabel Magnetooptical Kerr effect of Fe-gratings in: J. Magn. Mag. Mat. 240, 24 (2002). 168 Acknowledgments Acknowledgments First, I would like to thank my advisor Prof. Dr. H. Zabel, who gave me the opportunity to work at his institute, for the valuable, open-minded discussions, his motivation and constant support in every aspect of this thesis. His abilities to raise money, to put science into intelligible (English) words and to teach all aspects of the theory of diffraction and scattering laid the basis for this thesis and also my personal progress in the last years. I owe many thanks to Prof. Dr. K Westerholt, for the discussions of new experimental results and his valuable advice. In addition, I would like to thank him for the critical reading of the manuscripts for the publications which have resulted from this work. I would like to address my special thanks to all co-workers of external institutions who have contributed to this successful work about magnetic microstructures. In particular, I like to mention: Prof. Dr. U. Kunze, Dr. S. F. Fischer, Th. Last and K. Schädler from the Institut für Werkstoffe der Elektrotechnik, Ruhr-Universität Bochum for valuable discussions and the preparation of several lithographic samples; Prof. W. Keune, Dr. M. Doi and B. Roldan-Cuenya from the Laboratorium für Angewandte Physik, Universität Duisburg for the preparation of Fe samples; Dr. H. Brückel and K. Rott from the Institut für experimentelle Festkörperphysik, Universität Bielefeld for the preparation of the CoFe sample; J. McCord from IFW Dresden for valuable discussions and the Kerr-microscopy measurements; S. Kirsch and A. Carl from Laboratorium für Tieftemperaturphysik, Universität Duisburg for the preparation of the Fe nanowires and MFM measurements and R. Meckenstock and D. Spoddig from Institut für Festkörperspektroskopie, Universität Bochum for valuable discussions. For the nice working atmosphere in the MOKE laboratory I would like to thank M. Etzkorn, T. Schemberg, S. Gök, F. Radu and A. Westphalen. The sample preparation is the basis of all this kind of research, for help with MBE, sputtering, lithography and many other kinds of technical problems I would like to thank O. Schwöbken, W. Oswald, P. Stauche, S. Erdt-Böhm and J. Podschwadek. Many things would not have been possible without the aid of our mechanic and electronic workshops. For help with any kind of computer problems I would like to thank M. Kneppe, M. Hübener, S. Hachmann and H. Glowatzki. For the proof-reading of this thesis I have to thank Dr. A. Remhof. I would like to thank the complete group in Bochum for the nice and constructive working-atmosphere, in particular I would like to mention Dr. R. Nötzel, Dr. J. Pflaum, Dr. Ch. Sutter. Dr. D. Labergerie, Dr. G. Piaszinski, A. Bergmann, J. Grabis, Dr. A. Schreyer, Dr. K. Theis-Bröhl for co-working and sometimes more sometimes less important discussions during lunch, coffee or beer. Especially I would like to thank my long-year friend and colleague Vincent Leiner and my room-mate Murat Ay. Last but not least I have to thank my wife Stephanie and my child Jonas for their support and, of course, my parents, relatives and friends are not forgotten! This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the Sonderforschungsbereich SFB 491. 169 Lebenslauf Lebenslauf Name Geburtsdatum Geburtsort Familienstand Till Schmitte 30. Mai 1971 Herne verheiratet, 1 Kind 1977 - 1981 1981 - 1990 Mai 1990 1990 - 1991 Grundschule in Bochum Gymnasium Freiherr-vom-Stein Schule in Bochum Abitur Zivildienst 1991 - 1997 1993 1994 - 1995 Studium der Physik an der Ruhr-Universität in Bochum Vordiplom Austauschstudent an der University of Sussex, Brighton, UK experimentelle Studienarbeit: ”Scanning electron microscopy with polarization analysis”, 1996 - 1997 Diplomarbeit am Lehrstuhl für Experimentalphysik / Festkörperphyik , Ruhr-Universität Bochum (Prof. Dr. H. Zabel), Thema: Magneto-optische Untersuchungen der Austauschkopplung an nanostrukturierten epitaktischen Schichten Dezember 1997 Diplom 170 1998 - 2002 Wissenschaftlicher Mitarbeiter am Lehrstuhl für Experimentalphysik / Festkörperphysik, Ruhr-Universität Bochum (Prof. Dr. H. Zabel) seit 2003 Wissenschaftlicher Mitarbeiter am Mannesmann Forschungsinstitut Duisburg