UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Analysis of coherent resonant x-ray scattering and reconstruction of magnetic domains Rahmim, Arman 2001

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2001-0501.pdf [ 13.02MB ]
Metadata
JSON: 831-1.0085183.json
JSON-LD: 831-1.0085183-ld.json
RDF/XML (Pretty): 831-1.0085183-rdf.xml
RDF/JSON: 831-1.0085183-rdf.json
Turtle: 831-1.0085183-turtle.txt
N-Triples: 831-1.0085183-rdf-ntriples.txt
Original Record: 831-1.0085183-source.json
Full Text
831-1.0085183-fulltext.txt
Citation
831-1.0085183.ris

Full Text

ANALYSIS  OF COHERENT RESONANT X-RAY SCATTERING AND RECONSTRUCTION  O F MAGNETIC DOMAINS by  ARMAN  B.Sc,  RAHMIM  T h e University of British Columbia, 1999  A THESIS S U B M I T T E D  IN P A R T I A L F U L F I L M E N T O F  THE REQUIREMENTS  FOR T H EDEGREE OF  MASTER O F SCIENCE in T H E F A C U L T Y O F G R A D U A T E STUDIES (Department of Physics and Astronomy)  We accept this thesis as conforming to the required standard  T H E UNIVERSITY  O F BRITISH C O L U M B I A  October 4, 2001 ©  A r m a n R a h m i m , 2001  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study.  I further agree that  permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives.  It is  understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Physics and Astronomy  T h e University O f British Columbia Vancouver, Canada  Date  Abstract  ii  Abstract  We have explored the use of coherent resonant x-ray scattering as a powerful technique to study, characterize and reconstruct magnetic domains for antiferromagnetic ( A F M ) and ferromagnetic ( F M ) thin films. T h i s method is capable of high-resolution imaging (as it is not limited by optical aberrations), is able to probe buried interfaces and is operational in the presence of other fields. Here we report the first experimental observation of x-ray speckle patterns generated by A F M domains. was performed on L a F e 0  3  Resonant x-ray scattering  thin films possessing two types of domains with  their A F M orientations perpendicular to each other. X-ray magnetic linear dichroism ( X M L D ) at the Fe L  3  absorption edge has been exploited in order  to give rise to modulations of the scattering amplitudes according to domain distributions, resulting in magnetic speckle. We also report resonant x-ray scatterng in the transmission geometry from F M domains of C o / P t multilayers. Magnetic x-ray circular dichroism ( M X C D ) has been utilized with the contrast arising from the dependence of scattering amplitude on magnetization direction of F M domains, which are oriented normal to the surface (i.e. parallel of antiparallel to photon helicity) due to the perpendicular interfacial anisotropy provided by the broken symmetry at the C o - P t interface. B y tuning the energy to the C o L  3  edge,  magnetic speckle is very clearly demonstrated. We have analytically shown that upon reversal of magnetic contrast (tuning of the scattering energy to either of the two crystal field split peaks of the Fe L  3  edge in the first experi-  ment, and changing the photon helicity in the second experiment) changes in  Hi  Abstract  speckle patterns will be observed solely arising from the interference between roughness a n d / o r pinhole scattering with magnetic scattering. We have developed a new reconstruction technique, upon extension of Fourier transform iterative algorithms previously utilized in other reconstruction tasks, capable of reconstructing A F M and F M magnetic structure from resonant x-ray scattering intensity measurements.  T h i s technique is shown  to be very successful upon application to noisy simulated data. Using this method, experimental speckle data from the F M domains of the C o / P t multilayer have been inverted resulting in magnetic domains showing a remarkable similarity to the worm-domain structure of the actual domain distribution imaged using magnetic force microscopy ( M F M ) . This, to our knowledge, has been the first reconstruction of magnetic domains from experimental data.  Moreover, direct (non-iterative)  reconstruction of F M domains has  been shown to be possible upon using small pinholes a n d / o r rough samples with roughness scale comparable to the size of domains.  Contents  iv  Contents  Abstract  ii  List of Figures  vii  Acknowledgements  xii  Part I  Coherent Resonant X-ray Scattering from Magnetically Ordered  Samples  1  1. Introduction  2  1.1  Study of Antiferromagnetic T h i n Films  4  1.2  Magnetic Speckle  7  1.3  Study of Ferromagnetic T h i n Films  9  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces . . . . 12 2.1  Coherent Soft X-ray Scattering Experiment  13  2.2  Experimental Results  16  2.3  Magnetic Intensity Correlation Function  19  2.4  Analytical Results and Comparison with Experiment  30  3. Resonant X-ray Scattering from Ferromagnetic Thin films  40  3.1  Contrast Mechanism  40  3.2  Coherent Soft X-ray Scattering Experiment  41  Contents  v  3.3  Experimental Results  45  3.4  Effect of Polarization on Observed Speckle  47  Part II Reconstruction of Magnetic Domains 4. The Phase Problem  52 53  4.1  Introduction  :  53  4.2  Iterative Algorithms  56  4.3  Uniqueness  58  4.4  D u a l Solutions  60  4.5  H y b r i d Input-Output (HIO) Algorithm  61  4.6  Oversampling  63  5. Reconstruction of Magnetic Domains in Reflection Geometry . . . . 65 5.1  Reconstruction of Roughness  66  5.2  Reconstruction of Magnetic Structure  69  5.3  Steepest-Descent Method  71  5.4  Application to Noisy D a t a  76  6. Reconstruction of Magnetic Domains in Transmission Geometry . . 83 6.1  Separation of Charge and Magnetic Scattering  83  6.2  Iterative Reconstruction of Magnetic Structure  86  6.3  Application to Noisy Simulated D a t a  89  6.4  Application to Experimental D a t a  92  6.5  Direct (non-iterative) Reconstruction  98  6.6  Future Experiments and Outlook  100  7. Conclusion  103  Bibliography  106  Contents  vi  Appendices A. Intensity Correlation in terms of Amplitude Correlation  113  B. Autocorrelation of Roughness Distribution for a Self-AfRne Surface 118  List of Figures  VII  List of Figures  1.1  A magnetic recording head (figure courtesy of J . Stohr, Stanford Synchrotron Radiation Laboratory)  1.2  3  A magnetic random access memory ( M R A M ) cell (figure courtesy of J . Stohr, Stanford Synchrotron Radiation Laboratory).  2.1  P E E M image of the LaFe03 sample used in the experiment  2.2  The figure shows calculated a) magnitude and b) phase of the re-  4 13  flected amplitude for both domain orientations. The magnetic dependence of reflected amplitude is obvious. Note that the magnetic contrast inverts when moving from the first split peak to the second.  However, phase patterns as a function of energy follow nearly  the same path 2.3  Grazing incidence geometry employed for scattering from the surface of L a F e 0  2.4  14  3  thin  films  15  (a) Overview of the 2D intensity distribution of the scattered x-rays, (b) Measured intensity at 710.0 eV. (c) Measured intensity at 708.2 eV. (d) A second measurement of intensity at 708.2 eV. Clear change of speckle by tuning the energy to either of the split peaks is observed  2.5  17  Average Intensity of pinhole scattering as a function of q . x  A  circular pinhole of radius R — 2.5 / i m illuminates a uniform surface at grazing incidence 9 = described in the text  7°.  T h e three curves are 23  viii  List of Figures 2.6  Sorensen's paradox: Exchange of the scattering amplitude of two magnetic domain orientations is equivalent to first inverting the magnetic amplitude of the magnetic domains and then adding a constant value  2.7  30  Comparison of 'y(q) calculated from simulated data according to E q . (2.1), as well as the theoretical 7(g) calculated according to E q . (2.47). The size of the q-box. over which averaging was performed is also shown  2.8  32  Plots of correlation vs. roughness (a) at q  = 2.3/i m  x  3.1 / / m , q — 4.2 umT - 1  1  x  and q = 6.2 ^ m  _ 1  x  _ 1  , q  =  x  ( q — 0 for all). The y  roughness at which our experiment was performed is indicated on the figure by the dotted line 2.9  34  Plots of correlation vs. magnetic scattering amplitude (a ) m  at q = x  l.8u m , q = 2.6 um~ , q = 3.1 ^ m " and q = 4.9 urn" l  - 1  x  x  (q = 0 for all). Value of a y  m  1  1  x  corresponding to the split resonances  of LaFeOs is indicated on the figure by the dotted line 2.10  36  The experimental intensity correlation values vs. q for different x  roughness values including the experimental value of a =  0.20.  The experimental calculations are also shown 3.1  Spectral absorption scans for C o .  37  Absorption contrast be-  tween the domains with their magnetization parallel and antiparallel to photon helicity is evident 3.2 3.3  Setup for the experiment performed at B E S S Y II  41 42  Magnetic force microscopy measurement of the C o / P t thin film studied in the experiment. Scanned dimension shown is  5 x 5 jum 3.4  Experimental scattering geometry  43 44  List of Figures 3.5  ix  (a) Measured intensity at off-resonant energy of 774.1 eV, upon scattering from C o / P t multilayer, pinhole used in the experiment,  (b) S E M image of  (c) Pinhole scattering calcu-  lated from image (b). Colorscales in images (a) and (c) span 3 orders of magnitude 3.6  46  Measured transmitted beam intensity at resonant energy of 779.5 eV, after transmission through the C o / P t multilayer.  4.1  T w o arbitrary images (a) and (b) are considered.  .  47  Image (c)  is obtained from the Fourier modulus of image (a) and the Fourier phase of image (b) as described in the text. Similarly, Image (d) is obtained from the Fourier modulus of image (b) and the Fourier phase of image (a). T h e Fourier phase clearly seems to be a more determining factor in image recovery. . . .  54  4.2  Block Diagram of the error-reduction algorithm  57  4.3  Block Diagram of the reconstruction algorithm for the inputoutput concept  5.1  . .  62  Input domain distribution (top), E R - o n l y reconstruction (middle), H I O - E R reconstruction as described in text  (bottom).  SNR=15 5.2  77  Errors in zero constraint (top), phase constraint (middle) and true error (bottom)  as defined by Eqs.  (5.42), (5.43) and  (5.44), respectively, vs. iteration number (SNR=15) 5.3  79  Errors in zero constraint (top), phase constraint (middle) and true error (bottom)  as defined by Eqs.  (5.42), (5.43) and  (5.44), respectively, vs. iteration number (SNR=5)  80  List of Figures 5.4  x  (a) P E E M image of the sample used in the simulation, (b) Reconstructed image obtained after three cycles of 250 iterations of the H I O algorithm and 50 iteration of the E R algorithm (SNR  of 10 has been included in the simulated intensity),  (c)  Result of passing image (a) through a \ow-q filter (0.4 / i m ) - 1  to remove high-g perturbations of the P E E M image not captured in reciprocal space (therefore in the reconstruction) due to size of detector  81  6.1.  Symmetric transmission scattering geometry utilized in Ref. [44].  85  6.2  (a) T h e object support is used as a priori knowledge for the reconstruction, (b) Reconstruction algorithm (inside the support) according to first line of E q . (6.9)  (c) Reconstruction  algorithm (outside the support) according to second line of E q . (6.9) 6.3  88  (a) Input surface on which simulated scattering has been performed,  (b) Resulting reconstruction if no oversampling is  considered,  (c) Resulting reconstruction upon oversampling  the image by 4 in both directions. . . . 6.4  89  Fourier domain error is monitored for 1000 iterations in the reconstruction of the image shown in F i g . (6.3a).  We have  considered 12 trials with different starting random inputs. T h e results exhibit high degree of dependence on the starting input. 90 6.5  Intensity scans at off-resonant energy of 774.1 e V and resonant energy of 779.5 eV. T h e effect of changing the polarization is also shown  6.6  92  Simulated resonant x-ray scattered intensity (a) with and (b) without charge scattering  94  List of Figures 6.7  xi  (a) A n unbalanced domain distribution is considered, (b) Resulting expected reconstruction upon removal of \ow-q counts from the measured intensity  6.8  . .  95  Reconstruction of F M domain distribution on the C o / P t multilayer on which transmission scattering experiment was performed. We have used three cycles of 300 iterations of the H I O algorithm and 100 iterations of the E R algorithm  6.9  96  C o / P d multilayer with various sizes of magnetic bit patterns ( f c / m m means flux changes per m m , i.e. 2000 f c / m m corresponds to a bit size of 500 nm).  T h e magnetic patterns are  written and imaged with a magnetic write/read head. SiN A.l  x  The  membrane is the large rectangle in the image  101  (a) Pinhole and (b) magnetic contributions from points on an 80-pixel I D surface, containing 20 randomly generated magnetic domains, at a point on the detector with qL = 5.6 > TT/2  (\rrik\ = 20%). Circular Gaussian nature of the magnetic contribution is observed  115  Acknowledgements  xu  Acknowledgements He who does not express gratitude to creatures, has not expressed gratitude to the Creator either. -Prophet Mohammad  Writing this thesis and doing the necessary research for it, like almost all other things we do as social beings, was a collective endeavor. I have been blessed all through my life with overwhelming kindness manifested in people with whom I have lived and worked. I am very grateful for everything that different individuals have done for me. Many people have contributed to my success directly or indirectly. I will mention a few in order to keep the tradition of gratitude alive. I would like to deeply thank my supervisor, Dr. Tom Tiedje, for his irreplaceable support and enthusiasm. Dr. Tiedje's knowledge of various subjects in the field of condensed matter physics as well as his availability to attend individuals' concerns despite his vast load of responsibilities have been truly inspirational. I would also like to thank Dr. Sawatzky for his invaluable comments upon reading my thesis. A very special thanks is due to Dr. Sebastian Tixier whose contributions to my work are many and diverse, without which my work would have been of much lesser quality. His devoted work habits as well as human integrity have been sources of admiration for me, and I wish him the very best life has to offer him. I would also like to thank Martin Adamcyk, Jens Schmitz, Ben Ruck, Eric Nodwell, Anders Ballestad, Eric Strohm, Jim MacKenzie and A l Shmalz for many helpful and vibrant discussions. In general, I cannot imagine a more lively and friendly atmosphere than the one prevalent in our group. I also thank Dr. Stefan Eisebitt for his continuous collaboration with our group in the course of completing my master's degree and his invitation of me to attend Berlin for x-ray scattering experiments performed at the synchrotron radiation source BESSY II. During my stay in Germany, I was truly overwhelmed by the hospitality of Marcus Lorgen and Mirko Frewald, and I wish both best of luck in the course of their lives and work. The contributions of my family members to my life and work are beyond my ability to recite. Without their love and labor I would not be where I am right now. My brother, Iman, has been of great understanding and kindness. My parents have sacrificed many things throughout their lives for my sake and I own them all that I have. I am humble before their dignity and I love them sincerely.  Part I Coherent Resonant X-ray Scattering from Magnetically Ordered Samples  2  1. Introduction  1. Introduction  T h e requirements of the magnetic storage industry coupled with the development of modern thin film growth techniques have led to the production of new artificially structured nanoscale magnetic materials with unique magnetic properties [1]. T h e preparation and ultimately the control of the properties of these systems depends critically on the development of a better understanding of the magnetism of surfaces, interfaces, and thin films. One phenomenon of great interest, from both the fundamental physics point of view as well as that of technological significance is that of interlayer magnetic coupling in multilayer structures.  Initial studies of coupling in  multilayers were motivated primarily by scientific curiosity (see  references  2-8 in Ref. [2]). U p o n the discovery of giant magnetoresistance ( G M R ) [3, 4, 5], the technological relevance of these scientific discoveries in the magnetic storage technology were made apparent. Consider for instance the example of magnetic recording heads. A n illustration is shown in F i g . (1.1). A magnetic recording head typically consists of:  (i) a magnetic write head consisting of a coil and a yoke guiding the  magnetic flux created by the coil to a pole tip. T h e large magnetic field from the pole tip is used to write the magnetic bits into a thin magnetic film on a rotating magnetic-recording disk, (ii) a magnetic read head used to retrieve the bit information written on the disk. T h e head senses the magnetic flux emerging from the transition regions between the bits on the disk that is being read. It consists of two ferromagnetic ( F M ) transition-element layers (e.g.  Fe, Co) separated by a noble-metal spacer layer (e.g.  Ag, Au). The  3  1. Introduction  Fig. 1.1: A magnetic recording head (figure courtesy of J. Stohr, Stanford Synchrotron Radiation Laboratory). flux in the disk is large enough to change the magnetization direction in the top F M layer of the head, whereas the magnetization direction in the second F M layer is pinned by interlayer exchange coupling to an antiferromagnet 1  (AFM) [7, 8]. The G M R effect, associated with spin-dependent scattering of conduction electrons from these antiferromagnetically aligned magnetic layers, may now be used to retrieve the orientation of the top F M layer. A sense current flowing through the system experiences a resistance that is higher by about 10% when the two ferromagnetic layers are magnetically aligned antiparallel rather than parallel [2]. A rather similar idea is used in spin-dependent transport in magnetic random access memory ( M R A M ) cells [9]. These cells, as illustrated in Fig. (1.2), Modeling studies have indicated that exchange coupling is caused by a small ferromagnetic moment of the antiferromagnetic surface. Imperfections such as domain walls, atomic steps, and grain boundaries are believed to be instrumental for the appearance and size of this moment, as they break the symmetry of the magnetic structure at the surface of the antiferromagnet [6]. 1  1. Introduction  4  Magnetic  Fig. 1.2: A magnetic random access memory ( M R A M ) cell (figure courtesy of J. Stohr, Stanford Synchrotron Radiation Laboratory).  consist of a tunnel junction in which two F M layers are separated by an insulator.  A s in the case of read heads, one of the F M layers is pinned by  exchange coupling to an A F M layer.  T h e magnetization direction in the  other F M layer can be rotated by the magnetic field of a current flowing in a nearby write line. T h e cell also consists of a read line, where the tunneling current flowing through the read line senses a resistance that depends on the relative orientation of the two F M layers.  1.1 Study of Antiferromagnetic Thin Films In general, coupling of structural and electronic degrees of freedom over many scales has resulted in intimate relationships between the structural, magnetic and optical properties of magnetic thin films, generating increasing applications in sectors ranging from communication to sensors to electro-optics to magnetic recording [2].  New applications and improvement of device per-  1. Introduction  5  formances rely on a microscopic understanding of the static and dynamic magnetic properties of these structures. However, difficulties arise due to the lack of appropriate experimental techniques when trying to image magnetic structures in buried layers under applied field or when trying to observe the dynamics of a magnetic structure on sub-micron length scales. A s a notable example, the detailed mechanism of exchange coupling between thin A F M and F M films, in spite of the numerous technological applications, is still poorly understood [10, 11]. In order to understand the exchange coupling process, it is critical to be able to study and image an A F M layer responsible for the 'pinning' of the magnetization of the F M layer it has been coupled to.  However, due  to the magnetically compensated nature of an antiferromagnet, no net magnetic moment can be detected when averaging over the unit cell of a solid or surface.  B u t recently, an antiferromagnetically ordered M n monolayer was  imaged with atomic spatial resolution using  spin-polarized scanning tunneling  microscopy ( S P - S T M ) [12]. Unlike conventional magnetic force microscopy ( M F M ) , which measures the magnetic dipole force between magnetic sample and magnetic tip, S P - S T M exploits the dependence of the tunneling current on the relative orientation of the magnetization of tip and sample. T h e magnetic tip acts as a source of spin-polarized electrons, probing the spin-split density of states of the magnetic sample. T h e ability to probe topography, crystallography, magnetism and surface chemistry at the same time renders S P - S T M a powerful tool for the investigation of magnetic surfaces and monolayers [11]. However, the extreme surface sensitivity of this method does not allow for the study of buried layers, a critical requirement for the investigation of interlayer coupling phenomena. Furthermore, the ability to study dynamics in magnetic structures over short periods of time is limited by S P - S T M , as this method is based on scanning the surface. Earlier, it was demonstrated that x-ray magnetic linear dichroism ( X M L D )  1. Introduction  6  spectromicroscopy can image A F M domains in epitaxial thin films [13], with a relatively long probing depth (3 to 5 nm) [11]. T h e technique combines a high-flux-density soft x-ray beam line with a photoelectron emission m i croscope ( P E E M ) . T h e method offers a high (though not comparable with S P - S T M ) spatial resolution (>10 nm) in conjunction with elemental and chemical specificity and surface sensitivity.  T h e contrast is due to the fact  that the scattering/absorption cross section under resonant excitation depends on the orientation of the electric field vector of the incident linearly polarized x-rays and the A F M axis.  T h i s dependence,  and the fact that  antiferromagnets can be probed, comes from a term in the absorption crosssection that is proportional to the expectation value of the local magnetic moment squared ( M ) , and the X M L D contrast arises due to the difference 2  in cross section for light polarized perpendicular or parallel to the magnetic moment [14, 15]. T h e X M L D contrast effect was first clearly demonstrated for antiferromagnetic N i O by Alders et al. (1995) [16]. While X M L D spectromicroscopy is extremely valuable as it provides a real space image of an antiferromagnetic surface, it has two drawbacks. First, the spatial resolution is limited by the aberrations of the secondary electron imaging system rather than by the wavelength of the incident radiation. Second, the study of domain dynamics in applied magnetic fields is impossible due to interaction between the secondary electrons and the magnetic field. These drawbacks can be improved by a method, which we will thoroughly explore in the course of this thesis, namely  coherent x-ray resonant scatter-  ing. In this method, one obtains speckle patterns induced by the magnetic structure of the sample of interest. T h e method, we will demonstrate, is capable of characterization and imaging (via  reconstruction techniques) of the  magnetic structure of antiferromagnetic as well as ferromagnetic thin films.  1. Introduction 1.2  Magnetic  7  Speckle  A speckle pattern is formed when a medium with a randomly varying height or refractive index is illuminated with coherent or partially coherent light. T h e resulting pattern is due to interference between waves which undergo various optical path differences or phase shifts after being scattered by different parts of the medium [17].  T h e speckle pattern manifests itself as a  modulation of the intensity of incoherent diffuse scattering provided that the incident radiation is sufficiently coherent. In the optical energy range speckle is an established technique [18]. It makes use of the high degree of coherence of laser light which is a consequence of the stimulated emission of radiation in the light amplification process. However, x-ray radiation must be used in order to extend the spatial resolution from optical length scales to atomic transition energies (soft x-rays) and to molecular and atomic length scales (hard x-rays). In the x-ray range, experimental speckle investigations became feasible by the development of undulators for electron storage rings which serve as bright x-ray sources. These devices consist of periodic magnet arrays in which the stored electron beam is deflected in a way that the radiation from different periods interferes coherently. T h e resulting high brilliance of these x-ray sources can directly be translated into high coherent flux which is required in speckle experiments [19]In the interaction of electromagnetic radiation with an atom, the magnetic moment associated with electronic spin is known to interact with the magnetic field of the radiation [20]. T h i s is true in general for light; however, soft x-rays are especially important in this regard as the magneto-optic effects are largest for resonant scattering near the L edges, which are in the soft x-ray region for the 3d elements [21]. This enhanced magnetic sensitivity is due to the spin polarization of electrons in the magnetic 3d levels. Resonant soft x-  8  1. Introduction  ray magnetic scattering results in the promotion of a spin-polarized electron from the 2p level into and out of the spin-polarized empty 3d intermediate states.  B u t the important property of these states is that the probability  for the electron to go into them is modulated by the polarization of the electron. In other words, one type of electron spin-polarization will respond more strongly to a particular excitation energy than another type. T h e modulation of the magnitude of scattering amplitude at an atomic site by the orientation of the electronic spin, or magnetization, would therefore enable performance . of soft x-ray resonant scattering experiments where speckle patterns arising from the magnetic contrast on the sample would be observed. T h e reader is referred to reference [22] for a thorough introduction to magnetic speckle. Theoretically, it has been shown that the resonant electrical dipole scattering amplitude can be written as [15, 23]:  / -e}-e F,( )-z(e}xeo)-M F( ) + (e}-M„)(eo-M„)Ff 0  n  1  0  where eo and e/  n  . (1.1)  are the polarization vectors of the incident and scattered  x-rays, respectively, M  n  is a unit vector in the magnetization direction at the  atomic site r„ in the lattice, and F^°\  and F^ \ 2  in the case of transition  elements, are defined in terms of different linear combinations of the spherical harmonics describing the 2p to 3d dipole transition matrix elements.  A  non-resonant charge contribution to the scattering is also included in the expression for F^°K  T h e measurable scattering signal is then obtained by  summing over all lattice sites with a phase factor containing the momentum transfer K :  I oc ^exp(i K • r ) / n  n  (1.2)  n  T h e first term in E q . (1.2) describes scattering from the charge distribution, while the remaining two terms are exclusively magnetic in character and are defined in terms of different linear combinations of the spherical harmonics describing the 2p to 3d dipole transition matrix elements. T h e term that  1. Introduction is linear in M  n  9  is responsible for the observation of x-ray magnetic circular  dichroism (XMCD), and vanishes for antiferromagnets due their magnetically compensated nature. The term that is quadratic in M  n  gives rise to  x-ray magnetic linear dichroism (XMLD), and is the effect that will allow for the study of antiferromagnetic thin films. For the A F M thin films studied in chapter 2, we have used the grazing incidence reflection geometry, in which the sample (i.e. LaFe0 ) domains, which are in-plane and have their respec3  tive A F M axis perpendicular to each other [33], should respond differently to the linearly polarized incoming beam. As a result, we have observed a speckle pattern in reciprocal space sensitive to the orientation of antiferromagnetic domains, as we shall demonstrate.  1.3  Study  of Ferromagnetic  Thin  Films  The method of coherent x-ray resonant scattering is also applicable to ferromagnetic thin films. Similar to the antiferromagnetic case, an increase in the sensitivity to the valence electron magnetization is obtained by tuning the photon energy to the L edge, where a 2p core electron is excited into an empty, magnetically aligned 3d state. In the case of F M thin films, we will in particular be studying Co/Pt multilayer materials. The magnetization tends to align normal to the interfaces due to perpendicular interfacial anisotropy provided by the broken symmetry at the Co-Pt interface, as first predicted by Neel [24]. Domains are formed on the multilayer, resulting from the competition between this perpendicular interfacial anisotropy and thin-film shape anisotropy. The latter effect favors the formation of regions with different magnetization directions such that the net magnetization of the sample, and therefore the magnetic stray field outside the sample, is canceled to a great extent. For more detail, consult Ref. [25], wherein the basic concepts associated with ferromagnetism in (ultra) thin films are nicely illustrated from an  10  1. Introduction experimentalist's viewpoint.  We will be interested in using x-ray magnetic circular dichroism ( X M C D ) as the contrast mechanism in scattering experiments performed on C o / P t multilayer thin films. Ferromagnetic spin-dependent absorption via X M C D was first successfully demonstrated by Schiitz et al. [26] on iron, and more dramatically by Chen et al. [27] on iron and cobalt.  T h i s technique has  since been established as a quantitative magnetometry tool, with capabilities not afforded by traditional magnetics techniques. Its foremost strengths are the element-specific, quantitative separation and determination of spin and orbital magnetic moments and their anisotropies [28], as well as its chemical sensitivity [29], its ability to identify moment orientations in ultrathin films and monolayer magnetic materials [30], which leads to its  element-specific  magnetic imaging capability [31] and its.sub-monolayer sensitivity [32]. We will be further interested in a transmission geometry where the intensity is measured in the near forward (z) direction. T h i s will maximize the sensitivity to the second term in E q . (1.2), as the domains have orientation M  n  also along  z (i.e. parallel or antiparallel to photon helicity). T h e third term of E q . (1.2) would then vanish by this geometry. T h e effective atomic scattering factors are then given by / ±  where (+) and f  m  =  / c ± / m  (1-3)  and (—) are used to denote the helicity of the photon, and  f  c  are the charge and magnetic components of scattering, respectively,  corresponding to the first and second terms of E q . (1.2). T h i s thesis contains seven chapters, where in chapter 2 we describe the first experimental observation of x-ray speckle patterns generated by resonant scattering from antiferromagnetic domains. We will also explore, experimentally and analytically, the dependence of speckle on reversal of magnetic contrast. In chapter 3 we report the successful observation of magnetic speckle arising from the magnetic structure of ferromagnetic multilayer thin films in  1. Introduction  11  the transmission geometry. T h e chapter will also investigate the dependence of speckle on reversal of photon helicity. In the second part of the thesis, we will be very interested in the question of reconstruction of magnetic structure from measured intensity in reciprocal space. to the general class of  T h i s problem is related  phase problems, as the phase of the scattererd ampli-  tude is lost at the detector and only the modulus is maintained. We have left the task of introducing this topic to chapter 4. We will next elaborate upon methods that we have developed for the purpose of reconstructing magnetic domains in antiferromagnetic (chapter 5) and ferromagnetic materials (chapter 6).  In these chapters, we will for the first time demonstrate iter-  ative algorithms capable of reconstructing magnetic structure, and we will, by means of simulation and experimentation, establish the developed reconstruction algorithms as valid and successful methods of imaging magnetic structure of thin magnetic films.  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 12  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces  In this chapter, we report the first experimental observation of x-ray speckle patterns from antiferromagnetic domains, as predicted by principles explained in sections (1.1) and (1.2) of the previous chapter.  A detailed theoretical  analysis of the speckle patterns, namely their dependence on the pinhole, magnetic and roughness components of scattering will also be presented. A n image of antiferromagnetic domains at the surface of a L a F e 0 3 thin film as measured using photoelectron emission microscopy ( P E E M ) is shown in F i g . (2.1). T w o types of domains are present with their respective A F M axis perpendicular to each other [33]. F i g . (2.2a) shows soft x-ray reflectivity spectra from a single domain taken at the Fe L  3  absorption edge for both  orientations of the x-ray polarization, that is perpendicular and parallel to the A F M axis. T h e y are found by first measuring the spectrum for the imaginary part of the refractive index by measurement of the absorption, and finding the real part via a Kramers-Kronig transformation of the absorption data. From the real and imaginary parts, and a knowledge of the angle of incidence, the reflectivity spectrum is then calculated. T h e splitting of the L a F e 0 spectrum at the Fe L edge into two resonance 3  3  peaks (708.2 e V and 710.0 eV) is clearly seen in the reflectivity calculation. Furthermore, a reversal of the magnetic contrast is observed when moving from one peak to another. T h e splitting of the two peaks occurs as a consequence of an interplay between crystal field and coulomb effects as well as spin-spin and spin-orbit couplings [34]. Especially important is the spin-orbit coupling of the 2p corr state and the multipole coulomb and exchange inte-  m 2. Resonant X-ray Scattering from Antiferromagnetic Surfaces  Fig. 2.1: P E E M image of the LaFeOa sample used in the experiment.  grals with the 3d states, in addition to 3d-3d coulomb interactions. We have shown, in F i g . (2.2b), a plot of the calculated phase of the reflected amplitude, and one readily observes that the phase patterns for both orientations follow nearly the same behavior as a function of energy.  This observation  shall be used later in this chapter (Sec. 2.3). We now describe the experiment.  2.1  Coherent  Soft X-ray  Scattering  Experiment  Linearly polarized light was generated by a U5 undulator and monochromatized at beamline 8.0 of the Advanced Light Source, Berkeley. T h e scattering geometry is shown in F i g . (2.3). Without refocusing optics, the monochromatized beam entered the home built end station and passed through a double pinhole coherence filter about 32 m downstream of the undulator. Low energy stray light was blocked before the pinholes by a C o transmission filter. In order to perform a coherent scattering experiment, two conditions need to be fulfilled: (a) the sample has to be illuminated by a laterally coherent beam and (b) the maximum path length difference has to be smaller than the longitudinal coherence length £ = A / 2 A A [19]. In a two pinhole coherence 2  filter with pinholes of diameter Pi and P  2  separated by L, condition (a) is  14 2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 0 0.1 perp. 0.08[ perp.  parallel -120 -140 0  710 715 705i Energy (eV)  -160  710 715 : 705i Energy (eV)  Fig. 2.2: Thefigureshows calculated a) magnitude and b) phase of the reflected amplitude for both domain orientations. The magnetic dependence of reflected amplitude is obvious. Note that the magnetic contrast inverts when moving from the first split peak to the second. However, phase patterns as a function of energy follow nearly the same path. fulfilled if  P\<\L/2P  2  [35]. In our setup, the arrangement P i = 4 0 / i m ,  P =5 2  /im, and L=230 m m was used, which fulfills this condition and produces a laterally coherent beam behind the second pinhole. The sample is located 30 m m downstream of P2 and the asymmetric pinhole arrangement produces a beam of 5 / i m diameter at the sample position, which at an incidence angle of 7° results in an elliptical footprint 5 /xm x 40 fj,m on the sample.  T o collect sufficient diffuse scatter on the detector,  one needs to perform x-ray reflection at grazing incidence. up,  W i t h this set-  the longitudinal coherence, according to condition (b), is sufficient to  observe coherent scattering at 17.5 A (710 eV) up to a scattering angle of 16° both parallel and perpendicular to the scattering plane. O u r soft x-ray  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 15  Sideview  s  Fig. 2.3: Grazing incidence geometry employed for scattering from the surface of LaFeOs thin films. detector records a 2D spatial image of the scattering within this solid angle. The detector is located 660 m m behind the sample and has an active area of 40 x 20 m m . The  LaFeOs film, on which scattering was performed, was grown in an  oxide molecular beam epitaxy system by means of a block-by-block growth method on a S r T i 0 3 (100) substrate [37]. T h e substrate surface was miscut by 2° which favors the growth of L a F e 0  3  crystallographic domains with the  c-axis parallel to the surface terraces over the second possible orientation perpendicular to the terraces, i.e. the LaFeC>3 film has a crystallographic inplane anisotropy. T h i s crystallographic anisotropy translates one-to-one in an antiferromagnetic anisotropy as shown for this particular film by X M L D P E E M in figure 2.1. The  sample was mounted such that its antiferromagnetic domains were  oriented parallel and perpendicular to the electric field vector of the incident  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 16 x-rays. T h e X M L D contrast occurs between the two split peaks of the Fe L  3  edge in the x-ray absorption spectrum. B y changing the excitation energy by 1.8 e V from one peak to the other, the relative absorption of one type of domain decreases while the relative absorption of the perpendicularly oriented domains increases. T h e same effect is observable in reflectivity. We performed the speckle experiment by tuning the incident x-ray energy to the two split peaks (Fig. 2.1) and comparing the resulting speckle patterns. T h e sample position is not changed during this procedure, so that the illuminated area on the sample remains the same.  T h e basic idea is  that at one of the split peaks, one set of domains will scatter more strongly (higher reflectivity) than the other set of domains. A t the other split peak the magnetic contrast is inverted. T h e sample has a unique spatial reflectivity pattern which, under coherent illumination, will translate into a unique speckle pattern. T h e anisotropy of the sample is not important for this effect.  2.2  Experimental  Results  In F i g . (2.4a), an overview image of the intensity distribution of'the scattered x-rays on the detector is presented. T h e largest intensity is found in the vicinity of the center, corresponding to the specular direction. For better visibility of the weaker structures, the color z scale in the image in the vicinity of the specular reflection has been saturated. A r o u n d the center, Fraunhofer diffraction rings are visible. T h e rings are not perfectly circular due to internal structure in the pinhole, which results in intensity streaks across the central peak. Furthermore, diffuse scattering from the surface" roughness and the A F M domains is observed in the regions outside the Fraunhofer rings and extends at a constant q\\ — ^Jq\ + q to form ellipses as shown in the fig2  ure. T h e elliptical shape of the constant q\\ contours is related to the grazing incidence geometry of the present experimental set-up.  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 17  Fig. 2.4: (a) Overview of the 2D intensity distribution of the scattered x-rays. (b) Measured intensity at 710.0 eV. (c) Measured intensity at 708.2 eV. (d) A second measurement of intensity at 708.2 eV. Clear change of speckle by tuning the energy to either of the split peaks is observed.  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 18 T h e black spots on the figure correspond to dead areas on the C s l coated multi-channel plate detector.  T h e horizontal line in the lower part of the  figure is due to the beam footprint intercepting the edge of the sample. Note that Fraunhofer rings are not present below this line while diffuse scattering is still observed. In addition, from the typical length scale of the surface roughness (0.1 //m) obtained by atomic force microscopy, and from the average size of the magnetic domains (1.5 /im) measured by P E E M , diffuse scattering is expected for momentum transfers up to cut-off values of 60 / i m 4 /zm  - 1  - 1  and  for the surface roughness and magnetic scattering, respectively. T h e  size of the speckles and the spacing between the Fraunhofer rings are solely determined by the wavelength of the radiation used, and by the size of the coherently illuminated sample area. Changes of the speckle patterns in a reduced (/-range are shown in Figs. (2.4b) and (2.4c), wherein the incident photon energy is changed by 2.8 e V from 710.0 e V to 708.2 e V . A repeated measurement of the speckle pattern at 708.2 e V is shown in F i g . (2.4d). One observes a clear change in the speckle pattern upon reversal of magnetic contrast, which occurs by tuning the energy from one split peak to another.  T h i s observed change cannot be at-  tributed to noise due to the reproducibility of repeated measurements at the same energy, as we have demonstrated. A 0.25% change in the photon wavelength does not introduce a significant change in the roughness scattering contribution to the speckle pattern, as verified by comparing off-resonance data for equal variations in photon energy. Consequently, the difference in the speckle pattern between F i g . (2.4b) and (2.4c) may only be a consequence of the reversal in magnetic contrast. Nevertheless, the very fact that the pattern of speckle is modified rather than merely its contrast reflects the presence of interference between magnetic and non-magnetic (i.e.  pinhole  a n d / o r roughness) components of the scattering, as we shall discuss in detail in the next section.  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 19 2.3  Magnetic Intensity Correlation  Function  In order to quantify the change in the speckle pattern with a change of energy, we have calculated the normalized correlation factor Y xp  scattering images represented by the data matrices M  hk  and N  hk  For two measured  at the incident photon energies E i and E2, this figure of merit is defined as:  -f (Ei  E) =  exv  ^h,k( hk - M )(N  - N)  M  2  hk  y/Z , (M h k  - M)  2  hk  hk  hk  Z (N hlk  ^  hk  hk  -  Ny hk  T h e correlation factor is 0 when the images being compared are uncorre-  cted,  1 when they are perfectly correlated, and -1 when they are perfectly We calculated j  exp  anticorrelated. Aq  y  m 5.2 / i m  - 1  for image areas of Aqx  «  0.8 ^ m  _ 1  x  in size.  It must be observed that roughness and magnetic contrast both lead to speckle. T h i s indicates that for any rough magnetic sample, if the coherence area of roughness and the size of magnetic domains are comparable, roughness and magnetic scattering terms will interfere with each other. Furthermore, for small areas of illumination, the diffraction rings from the pinhole will also interfere with these terms. In order to understand the interference of the pinhole, magnetic and roughness terms, we will derive a theoretical expression for the magnetic intensity correlation function upon reversal of magnetic contrast. T h i s occurs upon a 9 0 ° rotation of the sample or upon tuning of the energy to either of the two split peaks of the Fe L  3  edge.  T h e analysis shall reveal the necessity of interference between magnetic and non-magnetic components of the scattering for any change of speckle upon reversal of magnetic contrast. We now proceed to include surface parameters in our derivations.  We  shall denote the complex scattered amplitude and scattered intensity from the sample by A±(q)  and i ± ( q ) , respectively. Having the momentum transfer  20 2. Resonant X-ray Scattering from Antiferromagnetic Surfaces  defined as  K = k - k f  where k  0  (2.2)  0  and kf are the ingoing and outgoing wave vectors, respectively, it is  then split into its components perpendicular to the surface, q , and parallel z  T o take into account the contrast in the scattering from different magnetic domains, we introduce a scattering amplitude factor C(x)[l ± m(x)] where C(x) is the product of the aperture function and the magnetic scattering factor averaged over the two types of domains, and m(x) describes the variation in the dimensionless magnetic scattering amplitude at the sample due to the magnetic domains. We must note that for m{x) to be real, we are assuming the complex phase of the scattering amplitudes at both magnetic orientations to be the same. We have checked that this is true, as shown in F i g . (2.2b), where it is apparent that the phase patterns for both types of orientations follow nearly the same structure. In other words, we are correct in assuming that it is the amplitude of the atomic scattering factor that is modulated by the domain orientation and not the phase. One consequence of this observation is that the multiple-wavelength anomalous diffraction ( M A D ) method [36], in which the strong dependence of the relative phase near the scattering peaks is utilized to extract additional information about the magnetic structure, may not be used for the antiferromagnetic scattering under consideration. T h i s method shall be further explained in Sec. (6.5) in the context of reconstruction of magnetic domains from ferromagnetic thin films. We shall use /i(x) to denote the roughness of the sample.  In the B o r n  approximation and the Fraunhofer limit of scattering, following the above set of notations, the speckle amplitude is written as the 2D Fourier transform: (2.3) where  x = (x,y). T h e scattered intensity is given by i ± ( q ) = | ^ 4 ± ( q ) | - We 2  21 2. Resonant X-ray Scattering from Antiferromagnetic Surfaces  now introduce the following notation:  (q)  (2.4)  A/±(q) = J±(q) - (7±(q)>  (2.5)  ^o,±(q) = M±(q)>  (2.6)  A A ± ( q ) = A (cf) - A° ±  ±  and  where  with ( ) calculated over an ensemble of surface roughness and magnetic domain distributions. In other words, in order to derive an expression depending on the statistical properties of the particular surface (e.g.  roughness  exponent, average size of magnetic domains) but not the particular configuration of the surface, one must consider ensemble averages . We also make the following observation: finding ensemble averages ( ) will still result in rings arising from the deterministic contribution of the aperture in the overall scattering. However, in the experimental calculation of the intensity correlation given by E q . (2.1), aperture oscillations will be smoothed as the averaging is performed over q-space (in our case, the qbox over which averaging is performed covers about 4 rings). Therefore, to assure correspondence between theoretical and experimental calculations of the correlation, we choose to introduce a 'smoothing operator' [ J , which will smooth out aperture oscillations by performing averages in a q-box in reciprocal space. We then choose to define the normalized magnetic intensity correlation function in the following way:  L(A'/ (q)A'/-(q))J /L(A'7 (q)»)JL<A'/_(q)2)J  7(q)  +  N  (2.7)  +  where  A'/ (q) = / (q)-L(/±(q)>J ±  ±  (2.8)  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 22 and we recall that ( ) is an average over an ensemble of surface roughness and magnetic domain distributions. One may now recall that an ensemble average of intensity calculations for particular realizations of a sample yields no speckle (similar to the case of incoherent scattering). However, it must be noted that in the calculation of the numerator of Eq. (2.7), one first calculates the product A ' / ( q ) A ' / _ ( q ) +  for  one particular surface configuration and then averages over an ensemble  of surfaces.  Therefore, the definition is compatible with the experimental  definition of 7 ( q ) given by Eq. (2.1). We also note that our definition of 7 ( q ) has the particular advantage of suppressing effects of partial coherence when studying magnetic speckle. T o see this, we note that partial coherence has the effect of decreasing the contrast in the scattered intensity, thus multiplying A ' 7 ( q ) by a factor a<l ±  in  order to account for the lowering of statistical variations. B u t the numerator and denominator in Eq. (2.7) both consist of A ' / ( q ) , and therefore the ±  factor a, due to cancellation, will not alter the values of 7 ( q ) . T o better understand the application of the operator [ J , consider the case of reflection from a uniform surface illuminated by a circular pinhole of radius RQ. W h e n incident on the sample at an angle 6 (with respect to the sample surface), the illumination forms an ellipse with the elongated radius in the scattering direction given by R = R /sm(9). T h e resulting diffraction 0  pattern (along the q direction) is [39] x  P  ( 5 ) =  P  o  [  ^  -  (.) 2 9  where PQ is the peak intensity and J\ is the Bessel function of the first kind, order one.  We shall refer to this as pinhole scattering.  In the asymptotic  form, one has [40] J (x) x  / 2 3 ~ W — cos (x - - 7 T ) V TTX 4  (2.10)  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 23  3 q(um  1  )  4  Fig. 2.5: Average Intensity of pinhole scattering as a function of q . A circular pinhole of radius R = 2.5 yum illuminates a uniform surface at grazing incidence 6 = 7°. The three curves are described in the text. x  from which we conclude  P(q)  r  ^w  cos  -r  {qR  4Pn  +  iqR)  cos[2(qR - |TT)]  '  3  ] (2.11)  (QR)  3  in the asymptotic limit. Note that averaging the second term over a few rings should yield a number close to zero. Thus, defining  R(q) =  4P  0  1  (2.12)  7r (gi?)  3  we conclude that R{q) should be a very good estimate of [P(q)\ away from the first few diffraction rings. T o see this, we have plotted in F i g . ( 2 . 5 ) the calculation of L^(9)J (P(Q)  I  S  averaged over a  q-box  extending about four  rings as also shown on the graph). It is clearly seen that L^*(9)J agrees very well with R(q) after the first few diffraction rings.  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 24 Using trigonometric manipulations, from E q . (2.11), one will find  16P  2  P { q )  2  ~  r  , 2 cos[2(qR - f TT)] , cos[A(qR - | T T ) ]  3  l%f  +  1  2(^)« J  +  ( 2  -  1 3 )  Note that upon averaging the above expression, the last two terms may again be ignored by the argument already given above. Thus we have  wi^Ivwf^"  '  2  (2 14)  Note that from this it follows that  LA'P(q) ] 2  = ~  [P(q) \-[P(q)\ 2  2  \R{qf  •  (2-15)  a result which will be useful in our future derivations. For an isotropic magnetic domain distribution, one has ( A ' J ( q ) ) 2  +  =  ( A ' J _ ( q ) ) , thus 2  L(A7 (q)A7_(q))j +  ^  ~  L<A'/ (q) >j 2  +  as both types of domains contribute equally.  (2  -  16)  T h e isotropic assumption is  used for simplicity in the derivation and the results can be trivially extended to the anisotropic case. Combining Eqs. (2.5) and (2.8), we have  A7 (q) = AJ (q)+0(q)  (2.17)  /%) = <J(q)> - L(/(q))J  (2-18)  ±  ±  where  Noting that /3(q) is statistically constant (already ensemble-aver aged), it is easy to see that from Eqs. (2.16) and (2.18), we have  l ( A 4 ( q ) A / - ( q ) ) J + \fi(<jf\  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 25 We have, in Appendix A , extended the theory developed by Pederson [41], derived in his case for the spectral speckle correlation in polychromatic speckle patterns, to the case of magnetic intensity correlation and have demonstrated that in our case, regardless of the sample roughness or magnetic contrast, in regions of reciprocal space outside the first diffraction ring, the following relation is valid: (A/ (q)A/ (q)) = |r +  ±  ( + i ± )  (q)|  2  - |F? , (q)| +  (2.20)  2  ± )  where we. have the following definitions  r  ( q ) = (A+(q)A* (q))  (2.21)  (q) = <(q)(A° (q))*  (2.22)  (+t±)  ±  and r^  ± )  ±  T h e advantage of E q . (2.20) is that it expresses intensity correlation functions in terms of amplitude correlation functions, which, as we shall see, are easy to calculate. It must also be noted that, according to the appendix, E q . (2.20) is correct for the case of  ensemble averaging, thus justifying why we had to  change the form of E q . (2.16) into E q . (2.19). Note also that  </(q)> = r  (+i+)  ( q ) = r _,_ (q) (  (2.23)  )  Using Eqs. (2.19) and (2.20), we finally arrive at  7 L Q  '  _ Lir,^-,(q)|'-|rt^,(q)|'J Wq)'J +  L ir , (q)l - R , ( q ) l J + l / W J 2  +  +  2  +  [  ' '  We now intend to find an explicit expression for the above equation. We start with E q . (2.21), which using E q . (2.3) can be expressed as  r ,±)(q) (+  =  (J J  x  [l + m ( x ) ] [ l ± m ( x ) ]  2  1  e  2  2  ^(M i)-M 2)) «i-(xi-x2)^ x  x  dxidx C{xi)C(x )  x  e  (2.25)  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 26 We will assume that roughness and magnetization vary independently, thus the ensemble averages may be taken separately for the roughness and magnetic terms. Under the assumption of Gaussian distributed fluctuations of surface height, we have [42] ^ igz(/i(xi)-/i(x ))^ _ e  (2.26)  -q a (l-p (X)) 2  2  e  z  h  where, p/i(X) is the normalized autocorrelation function of the surface height. It is demonstrated in Appendix B , following work by Sinha in Ref. [43], that upon assumption of a self-affine surface with a finite cut-off length for the roughness, the normalized autocorrelation function for a surface of fractal dimension D = 3 — h becomes s  r  p (X) = e-^  2hr  (2.27)  h  where £ is the effective cutoff length, and that such an assumption is consistent with the sample grown over a  finite period of time (governed by growth  models). For an isotropic magnetic domain distribution <[l + m (  X l  )][l±m(x )])  =  2  =  l±(m(  X l  )m(x )) 2  l±cr^p (X) m  .  (2.28)  where p ( X ) is the normalized magnetic scattering amplitude autocorrelam  tion function and am  = y '(m ) 1  2  is the  rms magnetic scattering amplitude at  the surface which is about 20% in the experiment. Substituting Eqs. (2.26) and  (2.28) into E q . (2.25), and defining X = x  x  — x , we obtain for the 2  amplitude correlation function:  r , ( q ) = e">> j  {X) [l ± a  2  (+  ±)  (X)]e^ ^e^ dX  2  Qc  2  x  mPm  (2.29)  where Q (X) = j C ( x ) C ( x + X ) d x C  (2.30)  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 27 It should be noted that letting R = | X | and assuming a circular illumination at the sample, one may change the variables into polar coordinates, thus expressing E q . (2.29) as the one dimensional integral  r , ± ) ( q ) = e~  qW  J Qc(R) [l ±  (+  q = \q + q \ ^ 2  where  2  1  2  r  o  (R)}e^ ^Jo(q R)RdR  2  2  mPm  (2.31)  r  and J is the Bessel function of zeroth kind. However, 0  this equation does not hold generally as the beam input is typically not circular at the surface due to grazing incidence, and therefore integration must be performed separately in both dimensions. Continuing on with E q . (2.29),  r  (+f±)  ( q ) = c-«2'V {&(X) q  [l ±  o  (X)}  2 mPm  C  ^ W }  (2.32)  where .F {} indicates a Fourier transform. Using the convolution theorem, q  we have  r  (+l±)  (q)  =  e-^XWX)}  * ^ {[l ± q  o  (X)}  2  c  mPm  ^P  f c  (X)  }  (  2  .  3  3  )  where the asterisk denotes a convolution. T h i s can be re-written in the form  r , (q) (+  ±)  =  e-^V { (X)}*[^ {l}  ±  <&FM*)e*^ ™}]  q  &  q  + ^  q  { e ^ (  h  x  ) - l } (2.34)  We note that the illumination function £ ( X ) is a very wide function comC  pared to P h ( X ) and p ( X ) . m  In this limit, J q { ^ ( X ) } behaves as a delta c  function when applied to the second and third terms in the bracket expansion of E q . (2.34). It thus follows that:  r , (q) (+  ±)  =  e - ^ ( P ( q ) + ff(q) ± M ( q ) ) 2  (2.35)  where P(q)=^ {^c(X)} q  (2.36)  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 28 H{q) = ^{e  ^  - 1}  q2(j2ph{  (2.37)  M(q) = ^ ^ { p ( X ) e ^ ^ ( ) } ]  (2.38)  x  q  m  Note that as £ ( X ) , p ( X ) and Ph(X) are symmetric, therefore P ( q ) , M ( q ) C  m  and i7(q) will be real valued. Note also that we can now derive an expression for | / ( q j recalling that the pinhole component of the scattering P ( q ) must be  |_-P(q)J  replaced by an averaged version  a  sw  e  explained previously in order to  remove the deterministic aperture rings. However, M ( q ) and H(q), which are ensemble averaged functions, do not exhibit oscillations.  It therefore  follows by combining Eqs. (2.23) and (2.35) that  L(/(q))J = e - ^ ( L P ( q ) J + H(q) + M ( q ) )  (2.39)  2  Recalling E q . (2.18), we get  /3(q) = P(q) - LP(q)J = A ' P ( q )  (2.40)  from which, using E q . (2.15), we have  L/%) J = ^(q) 2  2  (2-41)  where  *(<!) = rn(qR) TTT^  ( - ) 2  42  We further note that by Eqs. (2.3) and (2.6)  A° (q) ±  = j C(x)(l + -<72 "a 2  =  e  2 / 2  Q z  m{x))(e ^)e ^dx iq  ici  jC{x)e dx  (2.43)  i(iX  for a Gaussian distributed roughness and an isotropic magnetic domain distribution. F r o m this we conclude  rj ,_ (q) = r j +  )  + i + )  (q)  =  \e-^ ^^(C(x))\ 2  e  P(q)  2  (2.44)  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 29 where we have used the convolution theorem. Combining Eqs. (2.24), (2.35), (2.41) and (2.44), we have for the magnetic intensity correlation function  7  X + lfl(q)  W  2  '  1  +  where M  ±  =  [(P(q) +  =  L(2P(q) + ^ ( q ) ± M ( q ) ) ( H ( q ) ± M ( q ) ) j  ' =  (2P(q)  +  fY(q)±M(q)) -P(q) J 2  J  J  2  ff(q)±M(q))(^(q)±M(q))  (2.46)  where we have used the fact that the 'smoothing operator' [ J only considerably affects the pinhole term, and its effect on the magnetic and roughness terms is negligible. These terms are already ensemble-averaged and do not exhibit short-ranged oscillations over q-ranges covered by the  q-box, is neg-  ligible. Combining Eqs. (2.45) and (2.46), with a little algebra we get as our final result for the magnetic intensity correlation function  (fl(q) + H(q) + M(q)) where we recall that  4P  *  =T  ifl(qP  1  P  ( 2  '  4 8 )  F r o m this result, we make the following observation. It is clearly seen from E q . (2.47) that the correlation upon reversal of magnetic contrast would be unity (hence no intensity contrast) without the presence of the pinhole and roughness scatter terms, -P(q) and H(q).  In other words, beating of pinhole  a n d / o r roughness scattering with the magnetic scattering is the process responsible for any change in the speckle pattern. Next section is intended to further illustrate this point.  I.  30 2. Resonant X-ray Scattering from Antiferromagnetic Surfaces (a)  R,  (b)  •:....,in.,f(c)  ,.. ...in:..fz: ;  -[l+m(x)]  1-m(x)= -[l+m(x)]+2  Fig. 2.6: Sorensen's paradox: Exchange of the scattering amplitude of two magnetic domain orientations is equivalent to first inverting the magnetic amplitude of the magnetic domains and then adding a constant value.  2.4  Analytical  Results  and  Comparison  with  Experiment  Let's first make the following observation: Exchanging the scattering amplitude of the two magnetic domain orientations (i.e. reversal of magnetic contrast) can be mathematically thought of as first contrast of the magnetic domains and then  inverting the scattering  adding a constant value. T h i s is  shown in F i g . (2.6), where we have set the complex phase 4> = 0 in order to enable convenient plots on real axes. However, these observations are clearly applicable to the general case.  B y comparing Figs. (2.6a) and (2.6c), one  might then argue that since intensity is insensitive to sign change in the object, it must not change upon switching of the magnetic contrast (Sorensen's Paradox)! However, this is not the case due to the presence of pinhole and roughness interference fringes: rewriting the above statement upon the inclusion of these terms, and not just the magnetic contribution (as is done in the figure), we  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 31 have C(x)[l - m ( x ) ] e  i<?2/l(l)  = - C(x)[l + m(x)]e * i9  fc(x)  + 2C(x)e^  / l ( l )  (2.49)  Here, the Fourier modulus of the left hand side and of the first term on the right hand side of the equation are not equal. T h e second term on the right hand side can not always be neglected and results in a modification of the scattered intensity. In the particular case of a smooth 2 D surface with uniform circular i l lumination, the scattered amplitude upon switching of magnetic amplitude will be modified by a sign change and addition of the term 2J\{qR)/qR. Furthermore, if the area of illumination of this smooth surface is very large compared to the size of the magnetic domains, this latter term will only be considerable in the \om-q region of the reciprocal space, and as a result in regions where magnetic speckle is exhibited, no change upon switching magnetic amplitude will be observed as illustrated i n F i g . (2.6).  In the more  general case of a rough surface with roughness length scale of the same order of magnitude as the magnetic domain size a n d / o r a finite illumination area extending over several magnetic, the Fourier modulus of left hand side and of the first term on the right hand side of E q . (2.49) will not be equal and magnetic contrast will be observed.  This observation can be reformulated  in the following way: upon reversal of magnetic amplitude, while magnetic scattering inverts, pinhole and roughness scattering do not, which leads to the cross-beating of these terms and results in speckle pattern changes. We first perform a I D simulation of the correlation function j(q), wherein the roughness is not initially considered. T h e pinhole diameter is taken to be 5 /xm, illumination direction is taken to be normal to the surface (i.e. # = 0 ° ) , and a relative magnetic scattering amplitude o  m  = 0.2 is assumed. Magnetic  domains of average width 0.5 / i m are artificially created by the binning of sums of Gaussian functions located randomly on a I D grid. T h e correlation  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces  1  1  Sizeofq-box  S °-  8  32  1  \ !  ?>-  I  0.6  Simulation  c if  0.4  i  0.2  c g  I  Theory  1 1  11  h  1  8 o -0.2 10 q(um  15  20  )  Fig. 2.7: Comparison of 7(g) calculated from simulated data according to Eq. (2.1), as well as the theoretical 7 ( 9 ) calculated according to E q . (2.47). The size of the q-box over which averaging was performed is also shown.  function upon magnetic reversal of the simulated data is calculated according to the definition of E q . (2.1). It is compared with the theoretical calculation of 7(g)  according to E q . (2.47). Since the scattering is simulated in I D , we  use R(q) = P /2(qR) . 2  0  The  results are shown in F i g . (2.7).  averaging was performed is also shown. curves are in reasonable agreement. largely dominates and j(q)  T h e size of the q-box over which T h e simulated and the theoretical  A t small q values, the pinhole term  = 1. In the higher q limit, i.e. at q comparable  to 2 7 T over the average size of the domains, the ratio of pinhole to magnetic scattering decreases and beating results in the lowering of the correlation function. A t high q the magnetic term becomes the dominant term causing the correlation function to approach unity. The  quantitative differences exhibited between the two curves are due to  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 33 the fact that for the calculation of the simulated 7(g) only one particular sample has been considered, whereas in the calculation of the theoretical j{q),  the expression M(q)  is found by performing an ensemble average over  many magnetic domain distributions.  T h i s can be directly traced back to  definitions E q . (2.1) and E q . (2.7), where the latter involves taking  both an  average over a g-range and an ensemble average. A n ensemble average was included in the definition of E q . (2.7) in order to derive an analytic result that does not depend on a particular surface, but rather on the particular class of surfaces (e.g. roughness exponent). T h e differences between the two curves may also be attributed to the approximation, in the derivation of E q . (2.39), that the operator |_ J only has significant effect on the pinhole term due to its oscillatory nature. Doing so, the effect of averaging the magnetic and roughness terms over the q-box of interest is ignored. However, the curves clearly capture the important feature of the correlation function which is lowered in a region of reciprocal space in which the beating of the scattering from magnetic domains with the pinhole function is maximized. One should also make the observation that the correlation factor, despite the comparable size of the q-box over which averaging is performed and the width of the dip observed in the figure, drops to negative values at qregion corresponding to the size of domains.  T h i s is a property of very  smooth surfaces, wherein the intensity patterns arising from the magnetic effect, upon reversal of magnetic contrast, may be shifted such that the speckles in the two images being compared become rather anticorrelated (i.e. have negative correlation).  This effect, as we shall see, is diminished  with increasing roughness. Furthermore, we have noticed that as the size of the q-box is made smaller, the correlation factor (simulated) becomes lower in value.  T h i s is understandable since for a large  q-box, regions in which  the pinhole rings are maximized, put more emphasis on the observed speckle  34 2. Resonant X-ray Scattering from Antiferromagnetic Surfaces  1 0.8 o o0.6 c o S0.4 o V_  o  °0.2  0 0  0.05  0.1  0.15 , 0.2 o-(nm)  0.25  Fig. 2.8: Plots of correlation vs. roughness (cr) at q = 2.3/x m x  ^  = 4.2 ^ m  _ 1  and q  = 6.2 pm'  1  x  0.3  - 1  , q = 3.1 ^ m x  _ 1  ,  ( q = 0 for all). The roughness at y  which our experiment was performed is indicated on the figure by the dotted line. arising from the magnetic effect, and therefore render the anti-correlation effect more difficult as pinhole diffraction is deterministic and does not invert upon reversal of magnetic contrast. However, as the size of the q-box is made smaller until it reaches less than the width of an individual diffraction ring, the diffraction peak contained within the q-box is not going to be compared with nearby minima, and it becomes easier for speckle patterns to become anticorrelated within that limited g-region. In the remaining of this section, we apply the analytic expressions derived in the preceding section to the actual experimental conditions. We have taken our sample to be a smooth self-affine surface structure, i.e. h  r  = 1 (see E q .  (2.27)), as obtained by the P S D of the atomic force microscopy ( A F M ) image taken from our sample.  For the magnetic autocorrelation function p , m  we  have used the calculated autocorrelation of the P E E M data of the sample  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 35 (and averaged it over a few neighboring pixels). We have used q = 0.9 n m  1  z  as in the experiment, and the relative magnetic scattering amplitude o  m  is  taken to be 20%. In F i g . (2.8) the correlation function is plotted, as calculated by E q . (2.47), versus surface roughness o for four different regions in reciprocal space. is observed that at q  It  = 2.3 jum" , the roughness scattering is effectively 1  x  nonexistent, which results in the correlation factor to remain constant with increasing roughness. For q = 3.1 / / m " , the pinhole effect is less dominant; 1  x  this is why, when no roughness is present, the value of correlation factor at q  x  = 3.1 / x m  - 1  is considerably lower than the value at q = 2.3 / i m . In fact, - 1  x  the correlation values are negative for smooth samples at q = 3.1 / m i , but - 1  x  with increasing roughness the speckle patterns become more correlated. T h i s can be attributed to the independence of roughness contribution from reversal of magnetic contrast (i.e. as roughness scattering becomes more dominant, it favors the speckle pattern to remain correlated). A t q  = 4.2 / i m , the - 1  x  effect of pinhole scattering is still less present and a minimum (dip) is observed in the plot at which the interference of magnetic and roughness terms is maximized. A s the roughness increases beyond this point, the roughness component of scattering begins to dominate the magnetic contribution and 7 approaches unity. A t very low values of roughness the magnetic effect will not be able to interfere with a non-magnetic effect (as roughness and pinhole contributions are small) again causing the correlation value to approach unity. A similar effect is observed for q = 6.2 / i m , but the value of rough- 1  x  ness at which the correlation factor is minimized is lower as the magnetic term is less dominant in this region of reciprocal space, and the roughness term is able to interfere with it at smaller roughness values. T h e roughness at which our experiment was performed (0.25 nm) is indicated on the plot. Note that given the scattering geometry of the present experiment (q ~ 0.9 n m ) , one would estimate the scattering due to rough- 1  z  36 2. Resonant X-ray Scattering from Antiferromagnetic Surfaces  0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  Fig. 2.9: Plots of correlation vs. magnetic scattering amplitude  (cr ) at qx 1.8/i m , q = 2.6 p i n , q = 3.1 / i m and q = 4.9 / i m (q = 0 for all). Value of a corresponding to the split resonances of LaFe03 is indicated on the figure by the dotted line. - 1  - 1  x  - 1  TO  - 1  x  x  y  m  ness (q a ~ 0.22) to be comparable in magnitude to the magnetic scattering z  (<7 is 0.20), resulting in a reasonable degree of interference and sufficiently m  lowering the correlation factor from unity. T h i s is what is also observed in the figure. In F i g . (2.9) the correlation function is plotted, as calculated by E q . (2.47), versus the dimensionless magnetic scattering amplitude a  m  regions in reciprocal space.  It is clearly seen that a  m  for four different  at which change in  speckle pattern upon reversal of magnetic contrast is maximized (i.e. 7 is minimized) varies depending on region in g-space under consideration. For g-regions not corresponding to the size of magnetic domains (q = 1.8 / i m "  1  x  and q  x  = 4.9 / i m  - 1  in the plot where in the former, pinhole scattering is  dominant and in the latter, roughness scattering), a much larger magnetic contrast is required to bring about considerable change in speckle, than in q-  =  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 37  1  0.8 o  §0.6 H—  c o ^0.4 CD b. i—  o  °0.2 0 0  2  3  _ 4  q (i»m  )  5  1  x  6  Fig. 2.10: The experimental intensity correlation values vs. q for different roughness values including the experimental value of a = 0.20. The experimental calculations are also shown. x  regions corresponding to the size of the magnetic domains q  x  = 3.1 / m i  as a  m  - 1  (q = 2.6 jim." and 1  x  in the plot). Another observation that may be made is that  approaches very small or very large values, 7 approaches unity. T h i s  is easy to understand as at very small values of a , m  diminished, and at very large values of a , m  the magnetic contrast is  the magnetic effect becomes very  dominant, not able to interfere with non-magnetic terms and thereby rendering change in magnetic speckle negligible. Value of a  m  corresponding to the  split resonances of LaFeOs (i.e. 20%) is also indicated on the figure by the dotted line. It is therefore deduced that in our particular experimental setup, we should be seeing a noticeable change in speckle patterns at q ~ 3 / / m . - 1  T h e correlation factor 7 is plotted in F i g . (2.10) as a function of q for the x  roughness values a = 0, 0.25, 0.60 and 1.0 nm. Lowering of the correlation factor is observed in a region of intermediate q where the beating of the magnetic term with the pinhole and roughness terms is maximized. One also  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces 38 observes that in this intermediate g-range, the correlation factor drops to negative values for the smooth sample.  However, as the surface roughness  increases, the correlation factor increases and becomes positive.  We have  already attributed this effect to the insensitivity of roughness scattering to reversal of magnetic contrast and thereby a tendency to keep the speckle patterns more correlated with increasing roughness contribution. T h e experimental intensity correlation 7  e i p  , as calculated by E q . (2.1) on  scattering measurements from the surface of LaFeOs, is also plotted in F i g . (2.10). T h e plot demonstrates the minimization of  j  exp  at  q ~ 3 pm~ as l  x  predicted by theoretical calculations. T h e data plotted was normalized by dividing the correlation factor of two images taken at different split resonances with the correlation factor of two images taken at different times but at the same resonant peak. T h i s procedure took the effect of experimental noise into account and resulted in a smaller magnetic effect (i.e. the lowering of the experimental correlation factor was significantly reduced).  Noise is  believed to come from small changes in the illuminated area, low statistics due to a small count rate in the diffuse scattering at high q and imperfections of the pinhole. T h e noise results in a lack of quantitative agreement between the experimental and theoretical calculations of the correlation factor, as shown on the figure. Following the analysis presented, future experimental results, upon diminishing the effect of noise, will demonstrate a higher degree of change in speckle patterns upon reversal of magnetic amplitude. In this chapter, we have demonstrated speckle patterns observed for the first time in soft x-ray resonant scattering from antiferromagnetic domains. Scattering contrast from the antiferromagnetic domains was obtained by exploiting the large x-ray magnetic linear dichroism ( X M L D ) at the Fe L edge. 3  B y tuning the energy to either of the two split peaks of the Fe L  3  edge, we  were able to invert the scattering contrast between the two domains. T h e study of antiferromagnetic thin films using soft x-ray resonant scattering  2. Resonant X-ray Scattering from Antiferromagnetic Surfaces will allow for the study of magnetic domains in buried layers and in the presence of magnetic fields, which is not possible with such electron based techniques as photoelectron emission microscopy ( P E E M ) . Furthermore, it can be used in combination with future 4th generation x-ray light sources to study dynamics of materials on a very short time scale.  In addition, x-ray  resonant scattering, not limited by the aberrations of optical imaging techniques, offers new opportunities to study nanoscale magnetic structure in antiferromagnetic materials. We will be interested later in this thesis to investigate the possibility of reconstruction of A F M domains from measurements of magnetic speckle. In this regard, we will have an overview of the phase problem in chapter 3, and then explore reconstruction methods applicable to imaging A F M domain in chapter 4.  39  3. Resonant X-ray Scattering from Ferromagnetic Thin films 40  3. Resonant X-ray Scattering from Ferromagnetic Thin films  In this chapter, we will demonstrate the ability of coherent resonant x-ray scattering, as already applied to the case of antiferromagnetic ( A F M ) thin films in reflection (chapter 2), to study ferromagnetic ( F M ) thin films in transmission. Dependence of magnetic speckle on polarization will also be investigated.  3.1  Contrast  Mechanism  We will in particular be investigating resonant scattering of x-rays from C o / P t thin films. A s explained in section (1.3), due the competition between perpendicular interfacial anisotropy and thin-film shape anisotropy, magnetic domains are formed in the multilayer, with magnetization direction normal to the plane (i.e. parallel or antiparallel to photon helicity). T o enhance sensitivity to magnetization of the particular domains, transmission geometry has been utilized. A s elaborated in section (1.3), the contrast mechanism used in the experiment is that of x-ray magnetic circular dichroism ( X M C D ) , where we will use (+) and (—) to denote the photon helicity. T h e response of the F M atoms was then shown to be  /±  where f  c  and f  m  scattering factor.  =  / c ± / m  (3-1)  are the charge and magnetic components of the atomic T h e term f  orientations of the domains.  m  changes sign depending on the magnetic  41 3. Resonant X-ray Scattering from Ferromagnetic Thin films  Energy (eV) Fig. 3.1: Spectral absorption scans for Co. Absorption contrast between the domains with their magnetization parallel and anti-parallel to photon helicity is evident.  Spectral x-ray absorption scans for Cobalt are shown in F i g . (3.1). Sensitivity of the absorption coefficient to domain magnetization is apparent. It follows from E q . (3.1), and is seen in the figure, that the relative magnetic contrast experiences reversal upon changing the polarization from right to left and vice versa. A s we will see in chapter 6, performing x-ray scattering measurements with  both photon helicities may yield additional information useful  for the reconstruction of F M domain distribution from measured speckle.  3.2  Coherent  Soft  X-ray  Scattering  Experiment  Experiments were performed at the U E 6 5 / 1 S G M beamline at B E S S Y II , 1  Berlin. T h i s beamline, which has become operational very recently, uses the synchrotron radiation generated in a helical undulator. T h e polarization of 1  http://www.bessy.de/BII/  3. Resonant X-ray Scattering from Ferromagnetic Thin films  Fig. 3.2: Setup for the experiment performed at BESSY II the emitted radiation can be varied from linear (in any direction perpendicular to the beam) to positive and negative circular. T h e degree of polarization is 95% for circular light. In our experiments, the third harmonic of the undulator output is further monochromatized in the beamline to provide soft x-rays tuned to the C o L  3  absorption edge. T h e monochromator resolution  is estimated to be on the order of E/ AE = 10 for the slit settings used in 3  our experiment. Details of the beamline are depicted in F i g . (3.2). Optical elements from the undulator going downstream consist of a horizontal deflection mirror (plane mirror) as well as a vertically focusing mirror which focuses on the entrance slit. T h e entrance slit of the monochromator is set to 70 / / m . T h e monochromator is a spherical grating which focuses the beam onto the exit slit, set to 150 p m .  T h e beam from the exit slit is in turn selected by a  pinhole, placed 4 m from the exit slit. T h e illumination reaching the sample to be studied is defined by a downstream pinhole located 2 m m in front of the sample. T h e two pinholes system is used for establishing a laterally coherent beam at the sample, with this condition being fulfilled if P ! < A L / 2 P , where 2  42  43 3. Resonant X-ray Scattering from Ferromagnetic Thin films  Fig. 3.3: Magnetic force microscopy measurement of the C o / P t thin film studied in the experiment. Scanned dimension shown is 5 x 5 am.  Pi and P  2  are the diameters of the downstream pinhole and the exit slit  (not the upstream pinhole as it is only used to reduce stray light from the monochromator), and L is their separation [19, 35]. In our setup, we have Pi = 5 pm, P  2  = 70 nm and L ~ 4 m , thereby satisfying the condition for  lateral coherence. T h e sample consists of a C o / P t multilayer with 50 periods. T h e thickness of the C o and P t individual layers are 3.1 nm and 0.7 nm, respectively. T h e sample was grown by magneton sputtering onto a smooth, low-stress, 160nm-thick S i N membrane. T h e substrate was heated to 2 5 0 ° C and the sample x  consists of a 20-nm-thick P t buffer layer at the bottom and a 3-nm-Pt cap on the top.  T h e magnetization is aligned normal to the interfaces due to  anisotropy provided by the broken symmetry at the C o - P t interface, as first predicted by Neel [24]. T h e sample has a magnetic order consisting of worm domains, which are magnetized parallel or anti-parallel to the sample normal. T h e magnetic domain distribution for the thin film studied was imaged using M F M (magnetic force microscopy) and is shown in F i g . (3.3). T h e average domain size is determined from the M F M image to be in the range 0.2-0.25  44 3. Resonant X-ray Scattering from Ferromagnetic Thin films  soft x-ray  Fig. 3.4:  Experimental scattering geometry  T h e transmission scattering geometry used in the experiment is shown in F i g . (3.4). T h e coherently scattered radiation was detected with a 40x20 m m position-sensitive detector for soft x-rays, consisting of a stack of multichannel plates ( M C P ) with a resistive anode readout. T h e top M C P plate was covered with C s l to induce photon-electron conversion. Due to a very low dark count rate (about 5-10 cps over entire area) as well as no readout noise, a high dynamic range is achieved in the measurement process, capable of measuring intensities in the central peak and the higher q-ranges. T h e distance from the sample to detector was 690 m m . T h e sample was glued to a C u holder, a 1-mm-thick sheet with a hole in it, coming down from a feedthrough, and the pinhole was then pressed onto the C u holder from the other side.  3. Resonant X-ray Scattering from Ferromagnetic Thin films 45 Having the momentum transfer defined as  K = k -k /  (3.2)  0  where ko and k / are the ingoing and outgoing wave vectors, we write q = (q , q ) x  as the parallel (to the surface) component of the total momentum  y  transfer and q as the perpendicular component. In this experiment, considz  ering the geometry, the parallel momentum transfer q = ^Jq + q% spans up 2  to a m a x i m u m ^ s i n ( 0 and  9  max  ~  m a : r  ) = 130 / m i  - 1  where A = 1.6 n m at the C o L  3  edge  y/20 + 10 /690 is determined by the size of the detector and 2  2  its separation from the sample. momentum transfer  This means that maximum perpendicular  q will span up to z  2-7r/A  \cos(9 ) — cos(0)| = 2.1 / j m " . 1  max  Clearly, we see that q is too small for probing individual cobalt layers each z  3.1 n m in thickness.  Since we will be later evaluating the possibility of re-  constructing the magnetic order in the multilayer, it is important to note in advance that any reconstruction algorithm developed will be that of the sum of magnetic distributions on the individual layers. Thus assuming that the magnetic distribution on the individual layers are highly correlated, the reconstruction will be meaningful.  3.3  Experimental  Results  T h e experiment was first performed at the off-resonant energy of 774.1 eV. In F i g . (3.5a), an overview image of the off-resonant intensity distribution of the scattered x-rays on the detector is presented. Fraunhofer diffraction rings due to the aperture are clearly visible. From the typical length scale of the surface roughness (0.1 /mi) obtained by atomic force microscopy, large diffuse scattering is expected, and is observed, for momentum transfers q up to the cut-off value of 60 / m i . - 1  A n image of the aperture used in the experiment  as scanned by S E M (scanning electron microscope) is shown in F i g . (3.5b).  46 3. Resonant X-ray Scattering from Ferromagnetic Thin films  Fig. 3.5: (a) Measured intensity at off-resonant energy of 774.1 eV, upon scattering from Co/Pt multilayer, (b) SEM image of pinhole used in the experiment, (c) Pinhole scattering calculated from image (b). Colorscales in images (a) and (c) span 3 orders of magnitude. The calculated pinhole scattering obtained from the S E M image is shown in Fig. (3.5c). Non-circularity of the rings due to the internal structure in the pinhole is clearly captured by this image. In the reconstruction process, we would make use of this S E M image in order to define our support constraint as a priori knowledge of the scattering. Upon tuning the experiment to the resonant energy at the Co L edge, 3  one would expect speckle, arising from the modulation of the transmitted amplitude across the sample, to occur in regions of q-sp&ce corresponding to the size of the magnetic domains (0.25-0.30 /mi). This corresponds to speckle in reciprocal space positioned at <7~20-25 / i m  -  1  .  This is exactly what  occurred as we tuned the energy to 779.5 eV. The result is shown in Fig. (3.6). The largest intensity is found in the vicinity of the center, corresponding to the specular (unscattered) beam direction. For better visibility of the weaker structures, the color z scale has been saturated for the central beam. Speckles positioned on the detector correspond clearly to the average size of domains as observed by M F M . The fact that the speckle are randomly distributed corresponds to the random positioning of the worm domains of the sample.  3. Resonant X-ray Scattering from Ferromagnetic Thin films  -40  -20  0 , q>™ )  20  40  Fig. 3.6: Measured transmitted beam intensity at resonant energy of after transmission through the Co/Pt multilayer.  779.5  eV,  From a reconstruction point of view, we will make further observations in chapter 6 with regards to the x-ray scattering measurements reported in this section. Phasing algorithms capable of successful recovery of magnetic structure from such measurements will be presented. 3.4  Effect  of Polarization  on Observed  Speckle  In this section, we would like to investigate whether/how speckle patterns from a F M thin film will change upon switching the helicity of the incoming photons from right to left, and vice versa. In this regards, we will derive an expression for the intensity correlation upon reversal of the photon helicity (+) -H- (—). The correlation function will allow one to quantitatively study the conditions under which magnetic speckle is polarization-dependent and thus reveal the extent to which additional information about the magnetic domain distribution can be derived from performing x-ray resonant scattering  47  3. Resonant X-ray Scattering from Ferromagnetic Thin Hlms 48 measurements at both right and left photon helicities. This will also be important from an image reconstruction point of view, as will be discussed in chapter 6, section. (6.5). We choose to define 7(q), the normalized intensity correlation function, exactly as we did in chapter 2, Eq. (2.7). As we previously discussed, the perpendicular momentum transfer q is z  negligible compared with the thickness of the multilayers. The scattered x-ray detector can therefore be written as a 2D Fourier transform of the scattering (3.3) where q = (q , q ),x x  y  = (x, y), (±) is used to denote the helicity, and  speci-  fies the area that is being illuminated. Note that /(x) actually represents the atomic scattering factor integrated normal to the sample (i.e. / f(x, y, z)dz). This is because we ought to be dealing with a three dimensional Fourier transform in Eq. (3.3), but since the phase e  iqzZ  is fairly constant for the  thin multilayer, the atomic scattering factor is simply integrated along the z direction. Instead of fl, we will include the pinhole illumination C(x) at the sample in our future derivations as we would like to include effects of pinhole scattering on the detected intensity. Decomposing the atomic scattering factor into its charge and magnetic components as given by Eq. (3.1), we then write for the scattered amplitude (3.4) where / (x) and / (x) describe the distribution of charge and magnetic scatc  m  tering amplitudes (integrated along the z direction) across the sample. Above expression may be written in the following form (3.5)  3. Resonant X-ray Scattering from Ferromagnetic Thin films 49 where / ° is the average multilayer charge scattering across the sample, m(x) is the modulus ratio of / ( x ) to / ° (with <j) the complex phase), and h(x) m  c  is the mean-zero dimensionless roughness profile, normalized with respect to the total width of the multilayer. The roughness is used to describe the variation of / (x) across the sample [44]. c  Assuming an isotropic magnetic profile as well as independence of the roughness and magnetic distributions, we note that the amplitude correlation factor r , ( q ) = <A (q)Ai(q)> ( +  ± )  (3.6)  +  where ( ) is over an ensemble of input roughness and magnetic domain distributions, can be written as r  (q) - |/ °| 1 1 d d x 2  ( + ) ± )  c  X l  2  c  *-(*i-*»>  C( )C(x ) . X l  2  x ( l + (/i( )/i(x )} ± (m( )m(x )>) Xl  2  Xl  2  .  (3.7)  Defining X = X i — x , and following the steps of the analysis presented in 2  Sec. (2.3), the above equation simplifies to r  ( q ) = | / ° | / dXe*-™Q (X)(l + olp {X)±o* p (X))  (3.8)  2  ( + l ± )  where o~h and o  c  m  c  h  m  m  are the dimensionless relative surface roughness and mag-  netic scattering amplitudes, respectively, and h{X), £> (X) and p ( X ) are P  C  m  the ensemble-averaged autocorrelation functions of the roughness, pinhole illumination and magnetic domains distributions, respectively, normalized to unity. Introducing the definitions  H(q) = a ^{p (X.)}} 2  h  M(q) =  h  o^ WX)}] q  (3.9) (3-10)  3. Resonant X-ray Scattering from Ferromagnetic Thin Rims 50 where T { q  } denotes a Fourier transform, it then follows that the magnetic  intensity correlation function, again following the exact lines of argument presented in Sec. (2.3), is given by  7(q)  (fl(q) + Hjcj) - M(q))  |i?(q)  2  -  (i?(q)+ f7(q) + M ( q ) ) - i R ( q ) 2  J  (3.11)  2  J  where we recall that -R(q) describes a uniformly illuminated 2D pinhole function averaged over its oscillations, given by E q . (2.48). transmission geometry M(q)  Note that in the  is purely magnetic, unlike the case of reflec-  t i o n geometry where it exhibited additional dependence on the roughness (Eq. 2.38).  T h e main conclusion is the same: lowering of the correlation  function from unity (thus observance of change in speckle) will occur in regions of reciprocal space corresponding to the size of the magnetic domains, given that the pinhole a n d / o r roughness scattering is present to interfere with magnetic scattering. T h i s would occur only if the size of illumination a n d / o r the length scale of the roughness is comparable to the size of the magnetic domains. We shall further study in chapter 6 the effects of charge-magnetic interference on the success of reconstruction algorithms applied to the transmission geometry. T h e analysis of this effect, however, is not important for demonstrating the success of the experiment presented in this chapter, since measurements on and off-resonant energies have exhibited a clear contrast. T h e speckle patterns arising at the resonant energy may only be attributed to the magnetic effect as they are not observed at off-resonant energies. T h i s is contrary to the case of reflection from surface of LaFeOa (chapter 2) where speckle patterns were observed for measurements both on and off-resonant energies. T h i s rendered the analysis and attribution of speckle patterns to magnetic scattering more challenging, as roughness scattering had an additional effect on the observed speckle patterns. In that case, we had to invert  3. Resonant X-ray Scattering from Ferromagnetic Thin films 51 the magnetic contrast (by tuning the scattering energy from one crystal field split resonance to another) in order to demonstrate presence of magnetic scattering, which through interference with non-magnetic scattering gave rise to changes in the speckle pattern.  T h e second part of the thesis will be de-  voted to reconstructing magnetic domains from speckle data obtained for the scattering geometries discussed in chapters 2 and 3.  Part II Reconstruction of Magnetic Domains  4. The Phase Problem  53  4. The Phase Problem 4.1  Introduction  We would now like to reconstruct the magnetic domain distribution from the speckle patterns discussed in Part I. T h i s involves a problem that is common to a number of fields, such as x-ray diffraction, electron diffraction, neutron diffraction, astronomy and remote sensing, in which only the magnitude of the Fourier transform can be measured and the phase of the Fourier transform is lost. T h e objective is then to retrieve the phase information lost in the measurement. T h i s is referred to as the  phase problem. Mathematically put,  given the modulus | A(q)\ of the Fourier transform  A(q) = \A(q)\e ^=T[f(x)} ia  (4.1)  of an object f(x), complex or real, one must reconstruct the object, or equivalently, reconstruct the Fourier phase a(q).  Because the autocorrelation of  the object is given by .F [|A(<7)| ], this is equivalent to reconstructing the -1  2  object from its autocorrelation. It must be noted that phase information lost in the act of measurement is a very important component of the image in the Fourier domain. Consider the two image  1  shown in Figs. (4.1a) and (4.1b). Referring to the Fourier  modulus and phase of image (a) as a and a , respectively, and similarly for a  1  Images belong to D r . Tom Tiedje and D r . Sebastian Tixier of the Department of  Physics and Astronomy, University of British Columbia.  4. The Phase Problem  54  Fig. 4.1: Two arbitrary images (a) and (b) are considered. Image (c) is obtained from the Fourier modulus of image (a) and the Fourier phase of image (b) as described in the text. Similarly, Image (d) is obtained from the Fourier modulus of image (b) and the Fourier phase of image (a). The Fourier phase clearly seems to be a more determining factor in image recovery. image (b), one may now wish to perform the following operation  \T\a]  (4.2)  (4.3) where we have performed an inverse Fourier transform on a matrix whose modulus is found from the Fourier modulus of one of the two images and whose phase is found from the Fourier phase of the other image.  T h e two  55  4. The Phase Problem  resulting images are shown in Figs. (4.1c) and (4.Id). It is surprising that the images recovered identify with the images whose Fourier phase (not modulus) were used!  In other words, having a device that would maintain only the  phase, as opposed to the modulus, would have been more convenient as phase seems to play a more important role in defining the image whose Fourier transform is taken. Depending on the physical process under consideration, it may prove possible to approach the phase problem in the following trivial manner: a certain contribution of the object to the Fourier domain may depend on an experimental parameter which may be varied in the measurement process. T h i s in effect will lead to a change in the observed pattern in the Fourier domain. U p o n performing multiple measurements, one will then be able to extract a set of linearly independent equations, which, upon being solved, reproduce the object under consideration. A n example of this is the method of  multiple-  wavelength anomalous diffraction ( M A D ) , in which the strong dependence of the object's phase on the scattering energy is used to extract additional information upon performance of multiple diffraction measurements on the material of interest [36, 45].  T h i s is conceptually similar to the method of  multiple isomorphous replacements, where one changes the structure factors by replacing one type of scatterer by another type, instead of changing the energy around a resonance [45]. We shall later consider the applicability of this approach, and limitations thereof, to our particular experiments described in chapters 2 and 3. In cases where the above approach is inapplicable or limited, we need to consider the more general case of having a single modulus Fourier transform measurement at hand from which the object is to be reconstructed. It is to be noted that in encountering this problem, in addition to  a priori knowledge  of the modulus Fourier transform of the object, which acts as a constraint in the  Fourier domain, it is critical to impose constraints in the object domain  56  4. The Phase Problem as well.  T h i s is because without partial knowledge of the object domain,  it is trivial to create an object satisfying the Fourier constraint by simply inverse Fourier transforming the product of the measured Fourier modulus  \A(q)\ and exp[ia(q)], with a(q) taken to be any arbitrary phase set. Note for instance that images (4.1a) and (4.1c) satisfy the same modulus Fourier transform but do not show any resemblance.  4.2  Iterative  Algorithms  Amongst the solutions proposed to the phase problem, the  iterative Fourier  transform algorithm appears to be most practical from the point of view of applicability under the most general assumptions, minimum computational complexity and minimum sensitivity to noise [46, 47, 48]. T h e first widely accepted phase algorithm was put forward in 1971 by Gerchberg and Saxton [49].  T h e idea is that if information about the magnitude of the object as  well as about the magnitude of the object's Fourier transform is supplied, the phase information may be recovered. We write f(x)  = \f(x)\e ^ 1  where the  modulus of the object |/(a;)| is known a priori, and the phase of the object  (j)(x) is to be reconstructed. T h e algorithm is summarized by the following pair of equations for the ktb step of the iterative method: (4.4)  (4.5) T h e process is initiated by a random Fourier phase set <J)Q(X) and bounces back and forth between the Fourier domain, where the resulting modulus Fourier transform \Gt{q)\ is replaced by the observed |^4(?)|, and the object domain, where the amplitude of the estimated image p(x) is discarded after every iteration and replaced by the known | / ( z ) | .  4. The Phase Problem  57  G^IG^IexpNq)]  A" V  -  Satisfy Object  Satisfy Fourier  Domain Constraint  Domain Constraint  gk ' OO  M  {G' (q)}<Z_ k  G\(x)=|A(q)|exp[ia(q)]  Fig. 4.2: Block Diagram of the error-reduction algorithm.  In 1978, Fienup [46] generalized the Gerchberg-Saxton phasing algorithm, as previously applied to the reconstruction of phase of objects, to problems in which partial constraints (in the form of measured data or information known a priori) exist in both object and Fourier domains. T h e iterative procedure may be summarized in the following four steps (for the kth iteration) [48]: (1) Fourier Transform gk{x), an estimate of the object (2) Replace the modulus  f(x).  \G(q)\ of the resulting Fourier transform with the  measured Fourier modulus  \A(q)\ = yl xp{q) to form G'(q), an estimate of e  A(q). (3) Inverse Fourier transform  G'(q).  (4) A p p l y changes on the resulting image g' [x) to allow it to satisfy k  the  object-domain constraints to form gk+i(x), a new estimate of the object. T h e algorithm is depicted in the diagram of F i g . (4.2). A n example of Fienup's application of the algorithm was to the retrieval of the phase of real and positive objects by the inclusion of finite supports  4. The Phase Problem and positivity constraints in the object domain.  58  For this case, the fourth  step may be written mathematically by  xE  C  (4.6)  where £ is the set of points at which  g' (x) is positive and does not exceed the  known diameter of the object (i.e.  satisfies the object-domain constraint).  k  T h i s generalized algorithm is often referred to as the error reduction ( E R ) algorithm, since as shown by Fienup [47], the error can only decrease (or stay the same) at each iteration.  4.3  Uniqueness  We must now turn our attention to the question of uniqueness. T h i s is very important since we are interested in establishing this method as an imaging technique capable of reconstructing objects correctly. In other words, one might worry that a solution formed after one performs the iterative Fourier algorithm, may be a correct solution in the sense that it satisfies all constraints in the object and Fourier domains, but a wrong solution in the sense that it does not agree with the actual object we are imaging. Until the late 1970's, there was much doubt that the phase-retrieval problem could be uniquely solved, because the one-dimensional theory of analytic functions available at the time indicated that there were ordinarily a huge number of ambiguous solution [50, 51, 52]. T h e first indications that a two (or more) dimensional case is usually unique, despite the lack of uniqueness in one dimension, came from empirical reconstruction results [46, 53]. T h i s promoted an extensive study of the uniqueness problem and the research has yielded fruitful results. For instance, the uniqueness of the phase of real and positive objects of dimension D > 1 with finite supports was shown theoretically in 1979 by Bruck and Sodin [54] and in 1982 by Hayes [55]. Following  4. The Phase Problem  59  some partial ideas proposed in 1947 by Boyes-Watson et al. [56] and in 1952 by Sayre [57], in 1982 Bates developed the oversampling method to retrieve the phase from the magnitude of the Fourier transform and demonstrated that solutions to Fourier phase problems in more than one dimension are effectively unique, almost always, for localized images.  In 1984, Barakat  and Newsam showed also that for general, complex valued objects the nonuniqueness of the phase is pathologically rare [58]. Thereupon, Bates and T a n in 1985 [59] and Lane in 1987 [60] argued that because of the loss of the positivity constraints, the phase retrieval from modulus Fourier transform of complex objects, though almost always unique theoretically, is much more difficult empirically than that from real and positive objects. However, in 1987 Fienup demonstrated empirically the possibility of reconstructing a complex-values object by having sufficiently strong support constraints including certain special shapes and separated supports. In 1998, Miao et al. [61] demonstrated phase retrieval from x-ray diffraction of a complex specimen with the use of the oversampling technique and positivity constraints on the real or imaginary parts of the object under reconstruction. In x-ray diffraction, when the incident x-rays are low-energy photons (soft x-rays), the electron density is complex. T h e real part of the complex density representing the effective number of electrons that diffract the x-rays in phase is usually positive, while the imaginary part represents the absorption of the x-rays by the specimen and thus is ordinary matter . 2  always positive for  Using the positivity constraint on the real or imaginary  parts of complex-valued objects was shown empirically to yield unique and successful phase retrieval. 2  The exception is where the material amplifies the incident x-ray beam, as with the  x-ray laser amplifier.  4. The Phase Problem 4.4  Dual  60  Solutions  T h e mentioned studies, it must be observed, address the uniqueness of solutions for a set of functions related in a  non-trivial manner. T h a t is, from  a general, complex function satisfying the Fourier constraint, one can always generate a trivial second solution also satisfying the same constraint. If a complex function  f(x) = \f(x)\e ^ is the first solution to the Fourier 1  constraint, then  f(x) = r(-x) is its dual. f(x)  = \f(-x)\e- «-4 i  (4.7)  gives the identical Fourier transform. In the reconstruction  of object phase, for instance, where the known object constraint is symmetric, it is impossible to choose between the dual solutions 4>(x) and — </>(—x), and in practice it is observed that the iterative Fourier algorithm converges to each of the dual solutions with an equal chance [62]. D u a l solution pose an additional obstacle in the reconstruction procedure. It was first observed by Fienup (1986) [48] that the image under construction sometimes stagnates at a certain point in the iteration containing features of both the true image f(x)  and its dual. T h e algorithm, unable to decide  upon a preference for the two sets, prevents the iterative reconstruction of the object from approaching features of one solution only and the result is stagnation. T o overcome this stagnation, the reduced-area support constraint method may be used, consisting of the following steps: (1) Replace the object support function, known a priori, with a temporary support function consisting only of a subset of the correct function which is not centrosymmetric. (2) Perform a few iterations with the temporary function. (3) Replace the temporary function with the correct one and continue with the iteration. Continuing with the example of reconstruction of object phase, the reduced-  61  4. The Phase Problem  area support constraint is seen in many cases to favor some of the features of either <j>(x) or —(f>(—x) over the other.  T h e n once the correct support  constraint is reinstated, one of the two twin images may have a sufficiently large advantage over the other that the algorithm then converges toward that image [48].  This method is not guaranteed to work; however, it may  be optimized with choosing the correct temporary mask. Alternatively, we have chosen to start the iterative algorithm with different random estimates for the object to be reconstructed, and we have noticed, as we shall show, that reconstruction is sensitive to random input and certain random inputs will not lead the reconstruction procedure into the dual image stagnation as discussed above. Thus, after performing a number of trial runs initiated with different random sets, the one yielding closest agreement with the object and Fourier-domain constraints is taken as the correct solution.  4.5  Hybrid  Input-Output  (HIO)  Algorithm  It is very commonly observed that the error reduction algorithm, discussed in Sec. (4.2), converges very slowly after the first few iterations [63, 47, 48, 64] as we have also observed in our calculations presented later. Often, although the image is forced to satisfy the constraints in real space at every iteration, subsequent application of the amplitude data in Fourier space followed by inverse Fourier transformation has the effect of negating the real space constraint modification. T h e algorithm seems to j u m p back and forth and stagnation is the result. Several modifications have been proposed with the most successful being that of the Hybrid Input-Output (HIO) algorithm [47, 48, 65]. In the application of the error reduction algorithm, recalling the diagram shown in F i g . (4.2), g +i(x), the new estimate for the object at every k  iteration, is formed from g' (x) by forcing it to satisfy the object-domain k  62  4. The Phase Problem  F{}  Input g  Satisfy Foiuior Domain Constraint.  Output g  F-'{)  1  Fig. 4.3: Block Diagram of the reconstruction algorithm for the input-output concept constraint.  T h e input-output concept, however, is based on having  g (x) k  thought of as the driving, or input, function for the next output g' (x), as k  depicted in F i g . of the input  (4.3). T h i s is motivated by the finding that a small change  gk(x) results in a larger change of the output g' (x) in the same k  general direction of the input [66]. Thus, for the next output to accomplish the correct change of value towards the desired constraint, the input value must be modified by a certain factor 3 times the desired change in the output. Thus, for instance, in the application of the support constraint according to the H I O algorithm, g i(x) is given by k+  ( \ — j 9k( )  xe S  x  9 h + l [ X )  ~ \ g (x) + 3(0 - g' (x)) x i S k  ,  ,  k  where S is the support area of the sample over which it is being illuminated, and ft is a constant typically between 0.5 and 1.0 used to drive points outside the support region towards zero.  63  4. The Phase Problem  T h e strength of the H I O algorithm is that it supplies more information about the true image by mixing the new image at every iteration with a constrained version of itself as to modify the input by every iteration to move towards the desired constraint.  For a detailed exploration of the behavior  of the H I O algorithm^ consult Refs. [67, 68].  We must now consider an  important requirement for a successful reconstruction: oversampling.  4.6  Oversampling  It has been argued by Miao et al. [61] that the Fourier domain must be sampled at twice the Bragg frequency or more in order to allow for successful phase retrieval. T h e argument may be presented in the following manner. In practice, we approximate the object and Fourier transform by arrays. Let us say that the Fourier transform is sampled at the Bragg peaks.  Writing  E q . (4.1) as a discrete Fourier transform  A(q )= E / ( ^ ) e ^ ' k  (4-9)  j=o  where Xi and & are the sampled points in the object and Fourier domains (both domains sampled N times). However, since only the magnitude of the Fourier transform can be experimentally measured, the data corresponds to  \Mq )\ = \Y f(x )e >\ iqkX  k  /  (4.10)  j  j=0  For a general one dimensional complex-valued object, E q . (4.10) should really be thought of as N sets of equations with 2N unknown, namely the real and imaginary components of f(xj)  where j = 0 , 1 , N  — 1. For the two  (three) dimensional case, the number of equations is i V ( i V ) , with the un2  knowns 2 i V  2  (27V ). Let us now consider f(xj) 3  3  to be real. This implies that  the Fourier transform is even and has central symmetry. Therefore, for a I D  4. The Phase Problem object, the number of equations drops to N/2,  64  with the number of unknowns  being N. Again, for the two (three) dimensional case, this would correspond to  N /2 (N /2) equations, with A^ (N ) unknowns. T h i s is an exact conse2  3  2  s  quence of the phase problem, where the loss of phase at each detected point in Fourier space, reduces the information obtained from the process, as only the amplitude may be detected.  Therefore, from the above argument, one  sees that if there is any hope of retrieving the phase, the Fourier domain must be oversampled such that twice as many equations will be produced. T h i s corresponds to oversampling the magnitude of the Fourier transform by > 2 for a I D object, > 2 / 1  > 2 / 1  3  2  in each dimension for a 2D square object, and  in each dimension for a 3D square object. Oversampling is a challenge  in crystallographic experiments, where the diffraction peaks (which occur at the Bragg frequency by definition) are very strong and therefore the intensity between the peaks may only be detected with the application of a very high dosage, resulting in radiation-damage problems [69, 70]. B u t when the sample, is non-crystalline, the diffraction pattern is weak and continuous, and can be easily oversampled.  In all the simulations and experiments we  have performed, as elaborated in this thesis, the degree of oversampling (i.e. number of pixels per aperture ring) has been at least four in each dimension, easily satisfying the oversampling condition. In the next two chapters, we will concentrate on the problem of reconstruction of magnetic domains in the following two cases: (i) Reflection scattering from rough, antiferromagnetically ordered thin films. (ii) Transmission scattering from ferromagnetic multilayer thin films.  5. Reconstruction of Magnetic Domains in Reflection Geometry 65  5. Reconstruction of Magnetic Domains in Reflection Geometry  In this chapter we will be interested in establishing phase retrieval from x-ray resonant scattering as a valid and powerful technique for the reconstruction (i.e. imaging) of magnetic domains. In particular, we will be considering resonant scattering in the reflection geometry, in which case the surface roughness has a phase effect on the scattered amplitude. Consider the problem of reflection from the antiferromagnetic thin films studied in chapter 2. We recall that the intensity measured is given by 2  (5.1) where C(x) is the function describing the illumination at the sample, m(x) the relative magnetic scattering amplitude at the surface, and  is  (f)(x) = q h(x) z  is the perpendicular momentum transfer multiplied by the surface roughness. It was noted in chapter 2 that the scattering phases for the two types of domain orientations in the antiferromagnetic sample exhibited similar patterns as functions of energy (see F i g . 2.2). Consequently, additional information could not be extracted via the multiple-wavelength anomalous diffraction ( M A D ) method [36], in which the strong dependence of the relative phase near the scattering peaks is utilized to arrive at linearly independent equations that allow for a direct solution of the magnetic distribution (see Sec. 6.5 for more details). We will be exploring a reconstruction algorithm we have developed for the purpose of reconstructing the magnetic domain distribution of A F M thin  5. Reconstruction of Magnetic Domains in Reflection Geometry films. T h e algorithm makes use of intensity measurements performed at offand on-resonant energies, as well as  a priori knowledge of the object support  (i.e. area of illumination).  5.1  Reconstruction  of  Roughness  We first note that the distributions of magnetic structure and surface roughness are  both initially unknown in a typical reconstruction task. T h e rough-  ness profile cannot be detected by real space imaging techniques (e.g. atomic force microscopy) as the exact position on the sample illuminated by the xray radiation is not typically known. T o tackle this issue, we note that upon making intensity measurements at an off-resonant energy where magnetic contrast diminishes, the scattering becomes insensitive to the domains, and the resulting speckle will exhibit pure dependence on the sample roughness. T h e roughness profile may then be first retrieved from the off-resonant data, with the resulting information being utilized later for the reconstruction of the magnetic domains. T h e reconstruction of roughness may be approached in different ways. First, we consider a closed-form relation which we have developed, in the perturbation limit  q h(x) <C 1, that would allow for reconstruction of roughness. z  We consider the one-dimensional case, with extension to the two-dimensional case being straightforward. We impose the condition that our object to be studied has non-zero values only on one side of the origin (say, x > 0). We will then need to re-define the pinhole function C(x), which we will assume is  even about the origin and has a width 2a, such that the above criterion is  satisfied. T h u s we write:  C'(x) = C(x - a)  (5.2)  and similarly for the roughness  h'(x) — h(x — a)  (5.3)  66  5. Reconstruction of Magnetic Domains in Reflection Geometry 67 in order to render the object non-zero only in the region x 6 [0,2a]. T h e diffraction amplitude is then given by: /•2a  A(q) = / C'(x)e * ^e dx Jo iq  h  (5.4)  iqx  In the perturbation limit, one may Taylor-expand  e  ^  tQzh  1+  iq h(x) in z  order to get /•2a  A(q) =  r2a  / C'(x)e dx + iq C'(x)h'(x)e dx Jo Jo iqx  iqx  z  r2a  a  C{x)e dx + iq / C'(x)h'(x)e dx / -a Jo e S(q ) + iq H(q) iqx  lQX  z  =  iqa  z  z  (5.5)  where  S(q) = f C(x)e dx a  (5.6)  iqx  J—a  is the diffracted amplitude from a smooth surface and is  real as C(x) is even  about the origin, and 2a  r  H(q) = H {q) + iH^q) = / R  JO  C'(x)h'(x)e dx  (5.7)  iqx  is the Fourier transform of the product of the surface height h'(x) and the illumination function  I{q) ~ =  C'(x). From E q . (5.5), we obtain for the intensity S(q) -2q S(q)lm[e- H(q) 2  iqa  z  S(q) -2q S(q)(H (q)cos(qa)-H (q)sm(qa)) 2  z  I  R  where the small quadratic term \H(q)\ has been neglected. 2  (5.8)  Consequently  one sees that in the roughness perturbation limit, information in reciprocal space also appears as a perturbation from the ideal Fraunhofer diffraction pattern S(q). Having had translated the entire object to the right of the origin, such that h'(x) = 0 for all x < 0, thus making it 'causal', then the real and imaginary  5. Reconstruction of Magnetic Domains in Reflection Geometry 68 components of H(q) satisfy the Hilbert transform relationship [71]  i  r *W)  7T 7 - o o  (5.9)  q' — q  where it is understood that the integral is taken in the the Cauchy principal value (p.v.) sense. Thus, combining Eqs. ,(5.8) and (5.9),  I(q)-S(q) + 2q S(q)(H (q)cos(qa)-r2  z  I  f °  7T yJoo  v  One may now wish to solve for H (q) r  ^^-dq q  sin(ga)) = 0 (5.10) J  — q  '  in the above equation, with the rest of  the parameters being known in the experiment. It is also noted that as the surface height h(x) is real, we have by E q . (5.7): /•2a (•2a  HAq) = / C'(x)h'(x)sm(qx)dx Jo 10  (5.11)  Thus, upon solving for #/(g), one may then perform an inverse sine transform to retrieve the roughness distribution. For a general function f(x) Fourier transform T(q),  the Fourier inversion formula is written 1  /(*)  and its  roo  = 7T  roo  -iqx  /  d  dq  (5.12)  x  U p o n defining the odd function  fix)  = f(x)  -  (5.13)  f(-x)  and plugging it into E q . (5.12), the following formula follows in a straightforward manner \  f( ) x  7T  In order for f(x)  roo  = ~ J-oo  roo  / f{x)sm(qx)dx sin(qx)dq  to coincide with the initial object f(x),  not an odd function, one must initially shift f(x) (say x > 0), rendering f(x)  (5.14)  J-oo  and f(x)  which in general is  to one side of the origin  identical in the region to which  f(x)  5. Reconstruction of Magnetic Domains in Reflection Geometry 69 was shifted. T h i s explains why we initially shifted C(x) and h(x) to the right of the origin . Considering this discussion, and letting 1  f(x) = C'(x)h'(x), we  have by Eqs. (5.11) and (5.14): 1  r°°  C'(x)h'(x) = - /  H (q)sm(qx)dq  x>0  I  (5.16)  7T J - o o  where Hj(q)  has been obtained by solving E q . (5.10). T h e above equation,  therefore, would yield a reconstruction of the surface morphology, as C'(x)  is  known. However, it must be noted that exact reconstruction, as seen in the above equation, would result from an infinite detector size (q —>• ± o o ) . In an experiment, therefore, the success of high resolution reconstruction will be determined by the size of detector. Therefore, we have constructed an alternative mathematical formulation for the reconstruction of roughness in the perturbation limit.  It must be  noted that E q . (5.10) is not in a form that would yield a direct solution. Alternatively, the roughness may also be reconstructed using the conventional Gerchberg-Saxton iterative algorithm as previously described by Eqs. (4.4) and.(4.5). T h i s is the method most commonly used in the literature. Further work is required in order to extract and compare particular computational advantages in the application of the the mathematical formulation we have presented versus the Gerchberg-Saxton algorithm.  5.2  Reconstruction  of Magnetic  Structure  Once the roughness is reconstructed from scattering data at an off-resonance energy, the scattering energy must be tuned back to one of the resonance 1  It turns our that if we do not have to worry about this issue (i.e. if h(x) (  reconstructed is odd about the origin), the factor e  iqa  to be  would not appear in E q . (5.5) and  we would obtain the much simpler formula  I(q) - S(q) + 2q S(q)H (q) 2  z  easily solvable for Hj(q),  I  =0  contrary to E q . (5.10), and invertible for roughness  (5.15) h(x).  5. Reconstruction of Magnetic Domains in Reflection Geometry 70 energies at which magnetic contrast is exhibited (e.g. split peaks of the Fe Z/3 peak as described in chapter 2), and the reconstructed roughness will then itself turn into an object-domain constraint that must be satisfied upon successful phase retrieval. It must be emphasized in advance that we are assuming that it is the amplitude (and not the phase) of the scattering amplitude at any particular domain that is modulated by the magnetization of that domain. In other words, the phase <j>{x) is only modulated by the surface roughness and not the magnetic domains, as we have discussed in Sec. (2.3). We then propose the scheme for the reconstruction of the magnetic distribution to consist of iterations repeating the following four steps: (1) Fourier Transform an estimate of the object. In practice, one deals with sampled data in the computer. T h e n one uses the discrete Fourier Transform ( F F T algorithm):  G (q) = \G (q)\e ^  = ^ ^ ( i J e ^ W e ^  ia  k  k  (5.17)  x=0  where initially t {x) 0  is set to be a random input and (f) {x) is initiated by k  q h(x) where h(x) is the reconstructed roughness. z  (2) Replace the modulus of the resulting Fourier transform with the measured Fourier modulus (|^4(g)| =  ^I (q)) to form an estimate of the Fourier exp  transform. We note here that typical measurements do not cover the entire reciprocal space, thus one only replaces the modulus in the reciprocal region covered by the detector, and one does not alter Fourier values outside this range. (3) Inverse Fourier transform the estimate of the Fourier transform: N-1  4(x)eW = AT  2  Y,  \A(q)\e "'Me- - / io  iq  x N  (5.18)  (4) A p p l y finite support as well as phase constraints on the resulting image.  5. Reconstruction of Magnetic Domains in Reflection Geometry 71 T h e support constraint would read  w*)={<  i  w  :^  (5.19)  where S is the support area of the object over which the illumination is nonzero. For the phase constraint, the object i.e.  <f>' (x) is replaced by the known phase of k  q h(x) at every iteration. Thus a priori knowledge of the size z  of illumination as well as the phase of the object are used as constraints in the object domain. T h e fourth step of the E R algorithm can be easily modified to yield the H I O algorithm. For the finite support constraint, we will use tk+l{x)  =  {l U + P[0-t' (x)] lis k  k  k  Similarly, as we also know the phase of the object  .  "  (5  20)  <j)(x), the output phase  4>' (x) would undergo phase constraint in the following manner k  A where  _l<Pk(x)  if W {x) -4>{x)\ <e  M  k  ,  .  <j)(x) = q h(x) is known, and 3' again is a constant between 0.5 and z  1.0 used to derive the phase towards the known phase. Since the constraints usually do not need to be satisfied exactly, following work by Millane and Stroud [63], we have allowed for a small tolerance denoted by e.  Similar  to previous observation in reconstruction of real objects [47, 48], complex objects with strong support constraints [64] or with special positivity conditions [61], and reconstruction of antiphase domains [65], we also observe that the H I O algorithm improves image quality significantly faster than the E R method, as we will show.  5.3  Steepest-Descent  Method  A n alternative approach to the phase problem is to employ one of the gradient search methods. One such method, the steepest-descent method, was shown  5. Reconstruction of Magnetic Domains in Reflection Geometry by Fienup [47] to be closely related to the error reduction algorithm for the reconstruction of real and positive objects.  T h e connection was less  close for the case of reconstruction of phase of complex object (e.g. roughness).  surface  We will in this section derive the identity of the double-length  step steepest-descent method with the E R algorithm in the reconstruction of magnetic domains, as shown in the previous section, and thus, show that E R algorithm is a special case of the more general class of gradient search methods.  Some of these algorithms are seen to converge much faster than  the E R algorithm [47]. In the Fourier domain, the squared error is the sum of the squares of the amounts by which  G (q), the Fourier transform of the estimate of the object k  g (x), violates the Fourier-domain constraint. T h e squared error can then k  be expressed as  B = N-*Y[\Gk(q)\-\A(q)\]  2  (5.22)  k  One would like to reconstruct f(x) Fourier modulus  =  \f(x)\e ^ 1  whose phase 4>(x) and  \A(q)\ are known. A t each iteration, one first performs a  Fourier transform on  t {x)e ^ \ according to E q . (5.17), where t (x) is an l  x  k  k  estimate of the object magnitude | / ( x ) | . Next, the steepest descent method seeks to minimize the error matrix, B , k  the two dimensional case, the N  2  by varying a set of parameters. For  values of t (x) k  pendent parameters. Starting at a given point  are treated as N  2  inde-  t (x), in the N dimensional 2  k  parameter space, one would reduce the error by computing the partial derivatives of B  k  with respect to each of the points t (x)  in a direction  k  and then move from  t (x) k  opposite that of the gradient to form a new point t"(x) that  minimizes the error. T h e new estimate t i(x) for the reconstructed image k+  is then formed from  t' (x) by forcing the object-domain constraints to be k  satisfied. T h i s is iteratively performed until a minimum (hopefully a global minimum) is found. T h a t is, one minimizes the error as a function of the  72  5. Reconstruction of Magnetic Domains in Reflection Geometry N  2  parameters t(x) and then subjects the t(x) values to the object-domain  constraint. Ordinarily the computation of N  2  partial derivatives would be  a very lengthy task, but the line of argument presented in Ref. [47], when extended to our case greatly reduces the computation, as we describe below. Using E q . (5.22), the partial derivative of B with respect to a value at a given point, t(x), is  ' =m ~™ - W  d B  =2N  9)l ]Am?  -  (5 23)  We then have  d\G(q)\ _ d[\G(q)\^ 8t(x) dt(x)  _  1 d(G(q)G*(q)) 2\G(q)\ dt(x)  (5.24)  Note that by E q . (5.17) dG  d  ^  ^ ) - ^ ) ?  s  %  )  i<p(y)Jq.y/N  e  6  ( 5  '  2 5 )  which combined with the observation that  as the i V parameters t(x) are treated independently, will result in 2  a  U  r  _ i(t>(x) iq.x/N e  (5-27)  e  dt(x) F r o m Eqs. (5.24) and (5.27), it then follows  d\G(l)\ dt(x)  G(g)e-^e-^ + c.c. 2\G(q)\ N  =  =  1 w , ) ^  x  ,  )  5  2  g  )  2  Where c.c. means the complex conjugate. Inserting E q . (5.28) into E q . (5.23), and defining  t'(x)e '' i4  {x)  = AT  2  \MQ)\e e- iaiq)  iq  x/N  (5.29)  73  5. Reconstruction of Magnetic Domains in Reflection Geometry 74 we get  d B = [t(x)-t'(x)e ^'^-' ' ]-rc.c. i  = 2[t{x)-t'{x)cos{(j)'{x)-(t){x))] (5.30)  t (x))  t  where we have used the fact that t(x) and t'(x) are real valued by definition. Therefore, the computation of the entire gradient is done very simply by: (1) Fourier transforming t (x), (2) Replacing the amplitude of the Fourier transform by | A ( x ) | , (3) A p p l y i n g E q . (5.29), which happens to be an inverse Fourier transform to arrive at f(x) and c/>'(x), and (4) Evaluating E q . (5.30). Note that steps (2) and (3) of iterations in this procedure, which have been arrived at by gradient calculation arguments,  independently reproduce steps (2) and (3) of the error-reduction algorithm as discussed in Sec. (5.2). The optimum step size to take along the gradient, in order to minimize the error, can be determined by forming a first-order Taylor series expansion of B as a function of t(x) about the point tk(x),  + J2B [t(x)-t (x)}  B~B  k  k  (5.31)  k  X  One would now argue that  t (x), the output estimate for the object using k  the gradient search technique, should be the root of the above equation such that the Fourier-domain error B is set to zero. One would then get  «  l  H  t  '  - O T  w  (5  '  32)  which one easily verifies by inserting the above expression into t(x) in E q . (5.31). T h i s procedure can be applied throughout the iteration. Note now that by Parseval's theorem [71]  B = N- Y,\\G (q)\e ^ 2  ia  k  k  q  - \A(q)\e ^\ ia  2  = £\t (x)e^ k  -  t' {xY^\  2  k  x  (5.33)  5. Reconstruction of Magnetic Domains in Reflection Geometry 75 that is,  B  k  =  ^V" E [\G„{q)\ ~ \A( )\f = £\t (x) - t' (x)e^  =  B +B  2  q  c  k  (x)-<t> (x)) k  k  (5.34)  s  k  k  where  B = E [**(*) - t' (x)cos[<t>' (x) - <t> (x)\] c  k  k  k  (5.35)  k  X  B = Y [t' (x)sm[<f>' (x) - <j> {x)]] s  2  k  k  k  k  (5.36)  X  thus recalling Eqs. (5.30), (5.32) and (5.34), we get:  •^)-tik(aO = - ± ( l+  |  ^  (5.37)  Now if we consider the iteration process to have been repeated already for a number of times such that  (/>' (x) has been made sufficiently close to 4> (x) k  k  (i.e. <p' {x) — (f) {x) <§C 1), then it follows by Eqs. (5.35) and (5.36) that k  B  k  <§C B . k  k  T h i s also means that cos((/4(x) — (f)k{x)) may be replaced by unity  in E q . (5.37), resulting in the following simple relation  tl{x)-t (x)= -[t' {x)-t {x) l  k  k  k  (5.38)  T h e above equation, therefore, predicts the optimum step size to take in order, to minimize the error, arrived at by first-order Taylor series expansion of B ( E q . (5.31)). However, as B is quadratic in t(x), it may be argued that the linear approximation can be expected to yield a step size half as large as the optimum. Using the double-length step, E q . (5.38) becomes  %{x)-t (x) = \t' (x)-t (x)  (5.39)  tl(x)=t' (x)  (5.40)  k  k  k  or k  5. Reconstruction of Magnetic Domains in Reflection Geometry A s a final step in one iteration, the new estimate object domain constraints.  t' (x) is made to satisfy the k  Comparing this with the error reduction algo-  rithm, it is seen that they are identical. T h a t is, the error-reduction iterative Fourier transform algorithm can be thought of as a rapid method of performing a double-length step steepest-descent method.  O f course, if  ff\(x)  already satisfies all the conditions in real space, it is the desired reconstruction and the error is zero. T h e advantage of the previous derivations has been to demonstrate the possibility of performing gradient search calculations in the reconstruction of magnetic domains in a  non-expensive manner. For the  straightforward extension of the steepest-descent method to the conjugatesearch method, which is superior, the readers are referred to Ref. [47], with the results applicable to our case according to the above derivations leading to E q , (5.40).  5.4  Application  to Noisy  Data  A s a test of the E R and H I O algorithms, we have simulated a I D magnetic domain distribution with roughness included. T o examine the applicability of this approach to experimental data, we have studied the sensitivity of the algorithm to noise. A random noise noise = s i g n a l / S N R x random  (5.41)  is added to the diffraction pattern, where 'signal' refers to the intensity of the diffraction patterns, ' S N R ' is the desired signal-to-noise ratio, and 'random' is a generated array with its dimension the same as that of the diffraction pattern and each pixel value randomly selected between -0.5 and 0.5. First, the roughness is reconstructed according to the Gerchberg-Saxton algorithm from simulated data with no magnetic contrast (corresponding to making experimental measurements at off-resonant energies), with the  76  5. Reconstruction of Magnetic Domains in Reflection Geometry 2  0  350  400  450 500 550 position pixel  600  650  700  Fig. 5.1: Input domain distribution (top), ER-only reconstruction (middle), HIOE R reconstruction as described in text (bottom). SNR=15.  knowledge of the support (illumination function) used as the object-domain constraint. Next, finite support and phase constraints are utilized to reconstruct the magnetic domain distribution from simulated scattering at the resonant energy such that magnetic contrast is existent. T w o algorithms are considered, i) 200 iterations of the E R - o n l y algorithm; and ii) 2 cycles of 70 iterations of H I O and 30 iterations of E R . T h e original domain distribution and the two reconstructions are shown in F i g . (5.1). It is clearly observed that the utilization of the H I O algorithm results in a perfect reconstruction of data, whereas E R - o n l y algorithm does not recover the exact domain distribution. In order to monitor the degree by which constraints imposed on the outputs are satisfied at every iteration and to identify convergence of a solution, we have calculated the two following two error metrics,for the support and phase constraints, as follows:  (5.42)  77  5. Reconstruction of Magnetic Domains in Reflection Geometry which is also commonly used in scattering experiments (e.g.  reconstruction  of electron density [61, 69, 70]), and  1 1/2 E* —  xes  I  (5.43)  xes  which we. have considered for the first time as it only corresponds with our case in which  a priori knowledge of phase is used to reconstruct magnetic  domain distribution. However, it is commonly observed that error metrics do not always correlate well with image quality during the performance of the H I O algorithm, whereas the correspondence of object-domain error metrics to image quality is significantly improved in the application of the E R algorithm [47, 48, 64, 65]. T h i s has been attributed to the fact that while performing the H I O algorithm, the solution is escaping local minima, and while improving the image, the error actually increases. T h i s is the motivation for performing a number of cycles of iterations, in which one cycle consists of a number of iterations of the H I O algorithm followed by few iterations of the E R algorithm. T h e performance of E R results in the error falling rapidly to a level consistent with the improved image quality accomplished by the H I O algorithm. T o quantitatively observe the improvement of image while object-domain error might actually be increasing during H I O algorithm, we have introduced a true error metric:  Etrue —  £I^)-/(*)I7£I*)I  (5.44)  I xes  xes where  -11/2  f(x) = C(x)[l-\-m(x)]e ^ is the true object which one is supposed to l<t>  reconstruct, in accordance with E q . (5.1). However, in a typical experiment, we would not be able to measure this error as the true domain distribution would not be known, but we shall use this concept in our simulations to  78  5. Reconstruction of Magnetic Domains in Reflection Geometry  200  200  200.  iteration number  Fig. 5.2: Errors in zero constraint (top), phase constraint (middle) and true error (bottom) as defined by Eqs. (5.42), (5.43) and (5.44), respectively, vs. iteration number (SNR=15) better understand improvement of image quality while performing different algorithms. Figure (5.2) shows plots of EQ, E$ and E  TRUE  as the iteration progresses  for the case of S N R = 1 5 . It is seen that while the E R - o n l y algorithm clearly stagnates during the first few iterations, the application of the H I O algorithm significantly lowers the error. One also observes that upon the performance of the H I O algorithm, both E  0  and E^ increase while the true error  E  TRUE  actually decreases and image quality is improved. Performance of the extra E R iterations at the end of each cycle brings E  0  with the true error.  and E^ to better correlation  It is also observed that the largest drops in the true  error and thus improvement in image quality tend to occur during the H I O iterations. Same qualitative effects are observed when more noise is added. Plots of EQ, E^ and E  TRUE  for the case of S N R = 5 are shown in F i g . (5.3).  79  5. Reconstruction of Magnetic Domains in Reflection Geometry 80  10  1  0  ' 50  1  100 iteration number  ' 150  1  200  Fig. 5.3: Errors in zero constraint (top), phase constraint (middle) and true error (bottom) as defined by Eqs. (5.42), (5.43) and (5.44), respectively, vs. iteration number (SNR=5)  It is observed that for the first few tens of iterations, both curves appear to be stagnated.  T h i s corresponds mathematically to the algorithms being  trapped in an error local minimum. Nevertheless, after about 140 iterations, the H I O solution dips out of the local minimum and its value significantly lowers. We do clearly observe the trend that as S N R decreases, the time it takes for the iterative solution to converge to the true solution increases, and it becomes increasingly difficult for the solution to escape local minima. We  next consider to apply the reconstruction algorithm we have devel-  oped to noisy simulated data with experimentally relevant parameters. We have considered the rough, antiferromagnetically order L a F e 0 chapter 2. A F M and P E E M  3  studied in  images of the roughness and magnetic domain  distributions are used as the sample on which scattering has been simulated. The  PEEM  image is shown in F i g . (5.4a).  We have, included a signal-to-  5. Reconstruction of Magnetic Domains in Reflection Geometry 1.1  1.05  1  0.95  0.9 Fig-. 5.4: (a) P E E M image of the sample used in the simulation, (b) Reconstructed image obtained after three cycles of 250 iterations of the HIO algorithm and 50 iteration of the E R algorithm (SNR of 10 has been included in the simulated intensity), (c) Result of passing image (a) through a low-<? filter (0.4 ^ m ) to remove high-g perturbations of the P E E M image not captured in reciprocal space (therefore in the reconstruction) due to size of detector. - 1  noise (SNR) ratio of 10, and allowed for an oversampling factor of four in both directions. The resulting reconstruction is shown in F i g . (5.4b). It is obtained after three cycles of 250 iterations of the H I O algorithm and 50 iteration of the E R algorithm. T h e reconstructed image seems to be in good agreement with the actual P E E M image. However, it is observed that the reconstructed image does not capture high resolution perturbations of the P E E M image. T h i s is understandable as the maximum q probed by the detector is 15 / m i , cor- 1  x  responding to a resolution of 27r/15 = 0.4 /xm in real space. Consequently, we use a \ow-q filter on the P E E M image with a critical size of 0.4 /mi, the result being shown in F i g . (5.4c). It is seen that the filtered P E E M image and the reconstructed image are in excellent agreement.  Nevertheless, we have not  81_  5. Reconstruction of Magnetic Domains in Reflection Geometry 82 attempted to invert the actual experimental data shown in chapter 2 due to the overlap with the diffraction from the pinhole. T h e streaks arising from the pinhole scattering adversely interfere with the magnetic and surface roughness scattering. T h e pinhole scattering streaks could be reduced with a more uniform circular pinhole, possibly fabricated by ion beam milling. However, the success of the reconstruction from noisy simulated data clearly demonstrates the validity of our reconstruction method and is certainly promising for future experiments. T h i s has been the motivation for performing another x-ray resonant experiment, this time on ferromagnetic materials, which, as we shall show in the next chapter, has resulted in successful reconstruction of magnetic domains from experimental data.  6. Reconstruction of Magnetic Domains in Transmission Geometry  6. Reconstruction of Magnetic Domains in Transmission Geometry  In chapter 3, observation of magnetic speckle in coherent resonant x-ray scattering from ferromagnetic ( F M ) thin films was reported. In this chapter we will present algorithms capable of reconstructing F M domain distributions, similar to the algorithms applied to A F M structures in chapter 5. T h e algorithms will be subsequently applied to simulated noisy data and experimental resonant x-ray scattering results from C o / P t multilayers.  6.1  Separation  of Charge  and Magnetic  Scattering  We begin this section by analytically expressing the measured intensity in resonant x-ray scattering in terms of its charge and magnetic scattering components. T h e geometry is taken to be that of transmission as we discussed in chapter 3. It was shown in Sec. (3.4) that the scattered x-ray detector for F M thin films could be written as the following 2D Fourier transform (6.1) where q =  (q ,q ), x = (x,y), / ( x ) represents the atomic scattering factor x  y  integrated normal to the sample (i.e. / helicity, and  f(x, y, z)dz), ( ± ) is used to denote the  specifies the area that is being illuminated. We also recall that  the atomic scattering factor consists of charge and magnetic contributions (namely f± = f ± f ) c  m  as was discussed in Sec. (1.3). Therefore, defining (6.2)  83  6. Reconstruction of Magnetic Domains in Transmission Geometry  •Mq)  = /  (6.3)  / (x)e - dx i q  x  m  it follows from E q . (6.1) that the intensity is given by (6.4)  = |^c(q) ± ^ m ( q )  4(q)  We see that E q . (6.4) may be expanded in the form = |^c(q)f + | ^ ( q ) |  4(q)  ± 2 Re{^ (q)J^(q)}  2  c  (6.5)  leading to sums of purely magnetic and charge scattering terms as well as a cross term where the charge and magnetic contributions interfere. A s evident from E q . (6.5), performing the experiment with right and left circularly polarized light, and taking the average of the two intensities would result in the cancellation of the cross term. i.e. / (q) + / ( q ) c  m  (6.6)  where ^ (q)| =  // (x)e^ dx  (6.7)  T  // (x)e^ dx  (6.8)  2  c  x  c  and >(q)f  =  x  m  One consequence of using right and left circular polarizations is the separation of intensities in E q . (6.6), which means that the magnetic profile may be reconstructed without an actual reconstruction of charge distribution, as 7 (q) may be measured separately at an off-resonance energy where there c  is no magnetic contrast. T h i s is in contrast to the case of reflection, as we considered in chapter 5, where roughness h(x) was expressed in the form of a phase exp[z<7 7i(x)] z  multiplied by the magnetic distribution, as shown in  E q . (5.1). T h e roughness therefore had to be reconstructed first before the reconstruction of magnetic domains. In the present case of transmission, the  84  6. Reconstruction of Magnetic Domains in Transmission Geometry 85  Fig. 6.1: Symmetric transmission scattering geometry utilized in Ref. [44]. separation of the charge and magnetic scattering in g-space will result in a higher probability of converging to the correct solution, as the results will not be dependent on another reconstruction (i.e. that of charge distribution). However, an intensity measurement taken at an off-resonant energy must be correctly scaled, before being used in combination with on-resonance measurements, as the charge scattering amplitude f  c  is energy dependent [21].  M a n y possibilities exist to model the charge amplitude. From atomic force microscopy ( A F M ) data, a reasonable model is to assume that surface and related interlayer roughness of the multilayer are the source of charge scattering [44].  In our particular case of C o / P t multilayers, even though the  experiment is performed at the C o L  3  edge, it is not sufficient to merely  consider the measurement of spectral charge amplitude for C o , which we can find by averaging the two curves shown in F i g . (3.1). T h i s is because the P t capping and buffer layers would still contribute to the  charge scattering.  A n interesting method to tackle this issue has been proposed by K o r tright et al. [44], where the symmetric transmission scattering geometry, as  6. Reconstruction of Magnetic Domains in Transmission Geometry shown in F i g . (6.1), is used to constrain the scattering vector K = k / — k  0  in the film plane thereby optimizing the coupling to magnetic structure . 1  Spectral intensity measurements are then performed at the position in reciprocal space corresponding to the length scale of the roughness. T h e surface roughness as measured by A F M is then iteratively modeled assuming that scattering contrast is given by the difference between some linear combination of C o and P t scattering factors and vacuum. Magnetic scattering effects are also included in the calculations as the corresponding widths of the domains and the roughness may be comparable in which case interference between charge and magnetic contributions will be present. Iterative modeling, as elaborated in Ref. [44], leads to the determination of the ratio of C o to Pt contribution in the scattering, which combined with knowledge of the charge amplitudes for C o and P t individually, would determine the spectral dependence of the charge scattering for the entire sample. T h e reader is referred to the aforementioned study which has revealed an i m pressive sensitivity of resonant soft x-ray small angle scattering to magnetic and charge heterogeneity from 230 nm down to 20 n m , while also capable of distinguishing magnetic from charge scattering.  U p o n knowledge of the  spectral dependence of the charge scattering, the off-resonant results can be scaled compared to the on-resonance measurements in a straightforward manner.  6.2  Iterative  Reconstruction  of Magnetic  Structure  O u r next task is to reconstruct the magnetic order in the sample from the knowledge of J ( q ) , which we have found according to E q . (6.6) after perTO  forming measurements of i+(q) and i _ ( q ) at the C o L edge, and of I (q) at 3  c  This geometry also has the advantage of avoiding varying depth sensitivity with q [21] and distorted wave effects often associated with diffuse scattering near the specular beam and the critical angle for total reflection [21, 72]. 1  86  6. Reconstruction of Magnetic Domains in Transmission Geometry as we have described. We note that while the magnetic scattering factor  f  m  is complex in general, it simply changes sign from one domain to another. Therefore, since 7 (q) is not sensitive to an overall phase rotation of f m  m  the complex plane, f  m  in  may simply be thought to lie on the real line. T h a t  is, in the reconstruction of magnetic structure from  7 (q), the F M domains m  may be that of as belonging to a larger class of domains, referred to as an-  tiphase domains, which alter the phase (and not the amplitude) of the input beam by 0 or 7r depending on the particular d o m a i n . Ref. [65] has discussed 2  phase retrieval in coherent diffraction from the antiphase domains of the binary alloy C U 3 A U , which exhibit themselves below the critical temperature (668 K ) of the alloy with a length scale determined by growth conditions and annealing history. We have similarly applied an implementation of the Error-Reduction ( E R ) and H y b r i d Input-Output (HIO) algorithms to recover the phase information lost by the detector. Recall, according to diagram (4.2) of our discussion on the phase problem in chapter 4, that at every iteration of the phase retrieval algorithm, the output  g' (x) is encountered with an object-domain constraint, k  e.g. finite support constraint, which it must exactly satisfy in the case of the E R algorithm, or partially satisfy in the case of the H I O algorithm, to produce the new object estimate g +i(x) used as an input for the next k  iteration. Following our discussion on antiphase domains, we may impose an  antiphase constraint c (x) inside the area of illumination on the sample upon k  our knowledge of the fact that the reconstructed magnetic order must have phase 0 or 7r. In the application of the E R algorithm, this constraint as well as the finite support area constraint are then used in combination to give  xeS x£S  (6.9)  Of course, this statement is not true in the overall scattering as / = f + f exhibits variation of both amplitude and phase as f changes sign from one F M domain to another. But upon separation of charge and magnetic scattering, the statement is true. 2  c  m  m  87  6. Reconstruction of Magnetic Domains in Transmission Geometry 88  Fig. 6.2: (a) The object support is used as a priori knowledge for the reconstruction, (b) Reconstruction algorithm (inside the support) according to first line of Eq. (6.9) (c) Reconstruction algorithm (outside the support) according to second line of E q . (6.9)  where S is the support area of the illumination, and the antiphase constraint c (x) k  is given by  I/max I i s  \ - I  r  (t  c  ^  x  min(|/ -min(|/  m a x  > - \  | , |^(x)|) |,|^(x)|)  m a x  if if  Re{g' (x)} > 0 Re{g' (x)} < 0 k  k  . ^  . i  U  an upper limit for the magnitude of the scattering factor when  being reconstructed, and it has been included for the purpose of damping the magnetic scattering factor at any domain orientation to the m a x / m i n allowed value.  T h e amplitude of the wavefront at each point is allowed  to be less than \f  \ to allow for the smooth crossing of the points from  max  representing one type of domain to the other. T h e algorithm is depicted in F i g . (6.2). Similarly, the E R algorithm can be easily modified to the H I O algorithm  j  6. Reconstruction of Magnetic Domains in Transmission Geometry  0.5  -0.5  Fig. 6.3: (a) Input surface on which simulated scattering has been performed, (b) Resulting reconstruction if no oversampling is considered, (c) Resulting reconstruction upon oversampling the image by 4 in both directions.  as shown below  ' 9k(x) < g (x) + 6(0 - g' (x)) . 9k(x) + B'(c (x) - g' {x)) k  k  k  k  if x E S, \g' (x) - c (x)\ < e if x if x G S, \g' (x) - c (x)\ > s k  k  k  k  (6.11) where fi and fi are between 0.5 and 1.0. Also, since the constraints usually 1  do not need to be satisfied exactly, we have allowed for a small tolerance denoted by e [63].  6.3  Application  to Noisy  Simulated  Data  T o verify the applicability of the procedure we have developed to the reconstruction of magnetic domains, we have chosen to first study the problem from a simulation perspective and to include the effect of noise.  Thus we  have simulated scattering from a magnetic sample where the domain distri-  89  6. Reconstruction of Magnetic Domains in Transmission Geometry 90  HIO  200  400 600 Iteration  800  1000  Fig. 6.4: Fourier domain error is monitored for 1000 iterations in the reconstruction of the image shown in Fig. (6.3a).  We have considered 12 trials  with different starting random inputs. The results exhibit high degree of dependence on the starting input. bution, as shown in F i g . (6.3a), has been generated by the binning of sums of Gaussian functions located randomly on grids created on a 2D surface. In the previous chapter, the error metrics were defined in terms of objectdomain constraints. It is equally valid to define the error metric in terms of the Fourier-domain constraint (i.e. the measured intensity). defined the Fourier-domain error E  F  L  9  where  Thus we have  as  J  /  I I  \G (q)\, recalling from the previous chapter, is the modulus Fourier k  transform of  g (x), the estimate for the object at every iteration. k  T h e reconstruction algorithm was initiated by different random inputs, and the one yielding minimum error in the Fourier domain was chosen as the best reconstruction. We have included a signal-to-noise ratio (SNR) of  6. Reconstruction of Magnetic Domains in Transmission Geometry 5.  T h e iteration undergoes three cycles of 200 iteration of the H I O algo-  rithm and 100 iterations of the E R algorithm. F i g . (6.3b) demonstrated the reconstructed image when no oversampling is considered (i.e.  intensity is  detected at the Bragg frequency corresponding to the peaks of the aperture rings).  T h e reconstructed image is not in good agreement with the input  image, though the size and positioning of the domains has been captured in a qualitative manner. However, in the geometry the experiment on C o / P t multilayer was performed, the oversampling is about 4-5 in one direction and twice as much in the other. Therefore we choose to oversample our simulated sample by a factor of four in each direction. T h e resulting reconstruction, although rather low S N R is considered, agrees perfectly with the input domain distribution, as shown in F i g . (6.3c). It is clearly observed that oversampling is a very determining factor of the reconstruction. T h e readers are referred to Ref. [73] for an investigation of success of phase retrieval with varying degrees of oversampling, which, as mentioned in the previous chapter, only poses difficulties for scattering experiments from crystallographic material, where it is difficult to detect the intensity in-between the sharp Bragg peaks. We have found that the reconstructed image is highly dependent on the initial random estimate for the domain distribution. F i g . (6.8) shows how E  F  varies for different inputs as the reconstruction algorithm progresses.  Similar to many other works [47, 48, 64, 65], we have observed that the error actually increases upon the performance of the H I O algorithm, whereas performance of the E R algorithm results in the error falling rapidly to a level consistent with the improved image quality accomplished by the H I O algorithm. We clearly see that after the performance of the second cycle, one of the iterations converges, with the error rapidly falling to 2.5 x 1 0 , whereas - 2  upon reaching the third cycle, more solutions begin to converge.  However,  some of the solutions are clearly stagnated, demonstrating the necessity of performing the reconstruction iterations with many different random inputs  91  6. Reconstruction of Magnetic Domains in Transmission Geometry 5  -50  0 _., Q (\i m v  50  )  Fig. 6.5: Intensity scans at off-resonant energy of 774.1 eV and resonant energy of 779.5 eV. The effect of changing the polarization is also shown.  until convergence occurs. We have also observed that increasing the noise added to the scattered intensity does not noticeably affect the quality of the best reconstruction, rather it reduces the probability of convergence, in the sense that fewer trials being initiated with different random input converge to a solution satisfying all of the object and Fourier-domain constraints.  6.4  Application  to Experimental  Data  F i g . (6.5) shows a horizontal cut through the measured intensity at the resonance energy (right and left helicities) of 779.5 e V as well as the off-resonance energy of 774.1 e V . T h e intensities have been calibrated with respect to the duration of the measurement and the brightness of the source. We have made the following important observation: the off-resonance intensity appears to be unexpectedly low in the central peak (by about an order), whereas one would expect the resonant and non-resonant measurements to yield close  92  6. Reconstruction of Magnetic Domains in Transmission Geometry intensities in this region of reciprocal space, since introduction of magnetic contrast should only considerably affect measurements in regions of reciprocal space corresponding to the size of the domains. We have attributed this observation to the  detector dead time: since the geometry is that of transmis-  sion, the direct beam illuminating the sample is not considerably absorbed at off-resonant energies and would directly reach the detector unaffected by the sample. This, we believe, has resulted in the saturation of the detected intensity at the central pixels of the detector due to the number of photon counts reaching these pixels per second exceeding the maximum allowed value before the counts are saturated. T o encounter this issue, we make the following observation. In our sample, the size of the magnetic domains (~ 0.2 — 0.25 //m) is very small compared to the size of illumination on the sample (~  5.0 /zm).  the low-g regions on the detector contain information  T h i s implies that  only corresponding  to the shape of the pinhole illumination on the sample and that pinhole and magnetic scatterings would have very small interference. T h i s is shown by the following simulation. We have used the S E M image of the pinhole and the M F M image of the magnetic domains in the sample used in our experiment and simulated the scattered intensity.  T h e ratio of charge to  magnetic scattering has been roughly deduced from the absorption spectrum of C o shown in F i g . (3.1).  We have considered a smooth sample (though  inclusion of roughness is trivial).  F i g . (6.6a) shows the resulting intensity  calculation, while in F i g . (6.6b), the charge contribution has been ignored and only the diffusely scattered component has been taken into account (thus no aperture rings are observed). It is clearly seen that the speckle pattern in both figures are similar, demonstrating negligible interference between charge and magnetic scattering. T h i s suggests that it is permissible to simply neglect the low-g regions on the detector and to reconstruct the magnetic distribution from the magnetic  93  6. Reconstruction of Magnetic Domains in Transmission Geometry  20 40 . 60 q^um" ) 1  20 40 . 60 q (Mm") x  Fig. 6.6: Simulated resonant x-ray scattered intensity (a) with and (b) without charge scattering  speckle. It must, however, be noticed that if there are more domains of one type than the other (i.e. unbalanced domain distribution), then there would be a net D C component in the magnetic scattering which would affect the measured counts in low-g regions in reciprocal space, similar to the case of pinhole scattering.  A s observed in F i g . (6.5), changing the photon helicity  (thus reversal of the magnetic contrast), has resulted in a change in the intensity at the central peak (q~0). This may be due to an asymmetry of the magnetic domain distribution. In the reconstruction process, nevertheless, if the low-g values are ignored, the reconstructed domain  distribution should  still be the same, but the reconstructed magnetic scattering amplitude for the less dominant domain type will be increased relative to the other domain type in order to render the net magnetic contribution zero , as depicted in 3  Note that this implies that the upper limit constraint \fmax\ in E q . (6.10) must be set rather loosely in the case of an unbalanced domain distribution in order for this relative 3  94  6. Reconstruction of Magnetic Domains in Transmission Geometry  00  R f +f  X  1  1 X  Fig. 6.7: (a) A n unbalanced domain distribution is considered, (b) Resulting expected reconstruction upon removal of low-q counts from the measured intensity. Fig.  (6.7). For future experiments, nevertheless, it would be desirable to  take the measurements over a longer period of time with less photon counts per second  (cps), or to perform the experiments with a detector of higher  cps capacity, in order to ensure problems with detector dead time are not encountered. T h e following observation must also be made: since resonant x-ray scattering in the transmission geometry is performed on thin films, a non-negligible percentage of the direct beam entering the sample passes through without being diffracted. One consequence of performing 4  transmission scattering ex-  periments therefore is that the central pixels on the detector will contain, in addition to photon counts from the diffracted component of the beam, counts from the direct beam which has passed through the sample undiffracted. In this case, one will need to ignore the central few pixels of the measured intenincrease to be taken into account. This percentage varies exponentially with thickness, vanishing as the sample thickness 4  becomes large.  95  6. Reconstruction of Magnetic Domains in Transmission Geometry  Fig. 6.8: Reconstruction of F M domain distribution on the C o / P t multilayer on which transmission scattering experiment was performed. We have used three cycles of 300 iterations of the HIO algorithm and 100 iterations of the E R algorithm.  sity in the application of the reconstruction algorithms. Simulated analysis has revealed that ignoring only the few central pixels of the central Fraunhofer peak in transmission experiments will not have a particularly adverse affect on the success of phase retrieval [73]. Following the reconstruction algorithm we have proposed for the sample under consideration, we have used a circular inverted step-function to suppress the measured intensity in regions of reciprocal space with q < 1 5 / r n i , -1  wherein aperture rings (and not magnetic speckle) have been observed. This, as we have discussed, and as we shall observe, has the effect of removing the charge scattering component of intensity, as the charge and magnetic scattering components have negligible overlap in g-space. We have used as S E M image of the pinhole, shown in F i g . (3.5b), to enforce the finite support constraint in the iterative process.  T h e pinhole has been oriented so as to  96  6. Reconstruction of Magnetic Domains in Transmission Geometry maximize agreement between measured intensity at an off-resonant energy and calculated scattering from the S E M image (see F i g . (3.5)). In F i g . (6.8), we have shown the resulting reconstruction upon the application of three cycles of 300 iterations of the H I O algorithm and 100 iterations of the E R algorithm. We have not been able to compare the reconstructed image with the sample, as our pinhole was not attached to the sample and therefore the exact area of illumination could not be determined. However, from a qualitative point of view, it is observed that the resulting reconstruction captures the worm-like domain structure of the sample as imaged by M F M (see F i g . 3.3). T h e worm-domain image presented, to our knowledge, is the first reconstruction of magnetic domains from soft x-ray resonant scattering. We have observed that the separate application of the H l O - o n l y and E R only algorithms yield the same reconstructed structure.  However, in the  application of the H I O algorithm, the average size of reconstructed domain walls, which are regions where one type of domain orientation changes into the other type (i.e. magnetization becomes in-plane), were observed to be larger. Nevertheless, one must note that in our particular experiment, the reconstructed patterns obtained via any algorithm, may not be considered as true reflections of the domain wall width. T h i s is because the m a x i m u m parallel momentum transfer q = yjq% + q , probed by the detector (130 / m i ) 2  is not sufficient to image the domains walls (width<50 nm).  - 1  F r o m this it  follows, that discrepancies in the imaging of domain walls by H I O and E R algorithms, are artifacts of the methods, and the differences are expected to vanish upon extension of g-space probed by the detector. We speculate that upon improvement of image resolution (i.e. larger detector size and higher photon count), we will be able to image the domain walls successfully. Extension to higher spatial resolution will allow for dynamic studies of domain wall formation and evolution. T h e pinning of domain walls to substrate imperfections visible in the film tomography have been reported for  97  6. Reconstruction of Magnetic Domains in Transmission Geometry ultrathin fcc-Fe/Cu(100) films grown at different temperatures (see Ref. [25]). T h e study has revealed that minute changes in the growth mode of ultrathin films play a key role in their macroscopic behavior, possibly through pinning of domain walls at defects and dislocations. Therefore, detailed study of mechanisms of domain wall formation will serve to shed light on intimate relations of topography and magnetism that have been observed qualitatively on a microscopic scale.  6.5  Direct  (non-iterative)  Reconstruction  In this section, we will describe the possibility of using a direct approach in the reconstruction of magnetic strucutre from resonant scattering measurements.  In cases where the interference of charge and magnetic ampli-  tudes is considerable, the problem under consideration may be approached from a holographic point of view.  T h e basic idea is that the phase infor-  mation required in the reconstruction algorithm is derived from the interference between the charge and magnetic scattering processes.  One particular  method of implementing this idea is explored in Ref. [36], wherein multiplewavelength anamalous diffraction ( M A D ) is utilized.  T h e general idea is  based on the strong wavelength dependence of the phase of the scattering amplitude around a resonance, from which linearly independent equations may be obtained by performing multiple measurements.  In other words,  M A D utilizes the fact that the phase information for the structure factors are in the interference terms between the charge and magnetic scattering terms, which can be retrieved upon solving the obtained linearly independent equations. We choose to further illustrate this idea in a different context  (though  conceptually similar) applicable to X M C D experiments: performing double measurements upon reversal of photon helicity. We write E q . (6.5) in terms of  98  6. Reconstruction of Magnetic Domains in Transmission Geometry sums and differences of right and left photon-helicity intensity measurements  i+(q) and I_(q): 7+(q) + /_(q) 2 J (q)-/_(q) +  , _ . .12 JT(q)| + R e { ^ ( q ) } + I m { ^ ( q ) } 2  m  (6.13)  2  ro  Re{J- (q)}Re{^ (q)} + Im{^ (q)}Im{^ (q)} c  m  c  ro  (6.14) Assuming an exact knowledge of the charge scattering term .F c (q), these two equations contain the real and imaginary parts of .F m (q) as the only unknowns, which can therefore be solved for. T h e domain distribution can then be non-iteratively retrieved by a simple inverse Fourier transformation of ^" m (q). T h i s method is applicable to cases where the roughness of the area of illumination is exactly known, from which ^ ( q )  may be calculated. T h i s  would require the aperture to be attached to the sample, such that exact area under illumination can be measured. If the charge amplitude is not known a priori, one must then perform an experiment at an off-resonant energy, to remove effects of magnetic scattering, and reconstruct the charge scattering from the intensity. T h i s latter method involves an iterative reconstruction. Nevertheless, this might prove to be useful in dynamical studies where one would have to perform the charge reconstruction only once, and obtain the dynamic magnetic distributions directly from subsequent resonant scattering measurements. It must be noted that this approach will be limited to cases where charge and magnetic scattering terms exhibit considerable interference. T h a t is, if the last term of E q . (6.5) (i.e. the interference term) is negligible, J+(q) and i _ (q) will not demonstrate a noticeable difference, and therefore will not lead to two linearly independent equations. It was shown in Sec. (3.4) that changes in magnetic speckle upon reversal of photon helicity (left versus right) oc-. cur only in cases where the size of the illumination and/or the roughness is  99  6. Reconstruction of Magnetic Domains in Transmission Geometry comparable to the size of the magnetic domains. T h e additional information useful for direct reconstruction will therefore be limited, for instance, in the case of a thin film illuminated over many magnetic domains (typically the case) with no interfacial or surface roughness [36]. Successful reconstruction in this case would require a very large dynamic range (i.e. very long exposure time and low dark noise in the measurement). However, by using smaller pinholes a n d / o r rough samples with roughness length scale comparable to the size of domains, we speculate that this method might yield fruitful results. W i t h regards to the application of direct reconstruction methods to dynamical studies, we expect the rapidly evolving magnetic structure to pose time-related concerns for multiple measurements of the as required by these methods.  same distribution  Iterative methods on the other hand would  require only one resonant measurement per magnetic distribution.  How-  ever, in situations where direct reconstruction is applicable, one can avoid the lengthy task of performing iterative reconstruction algorithms on every frame obtained, and also have a greater chance of retrieving correct reconstruction, assuming that the reference charge distribution has been correctly obtained.  6.6  Future  Experiments  and  Outlook  In the future, we intend to perform similar scattering experiments on magnetic samples were the magnetic domains have been pre-patterned, such that their structure upon correct reconstruction would be recognizable. ample is shown in F i g . (6.9).  A n ex-  T h e image size is 45 pm and 30 /xm in the  horizontal and vertical directions, respectively. It is obtained by a magnetic read head. T h e various sizes of the domains are indicated on the figure. T h e SiN  x  membrane is the large rectangle in the image. T h e S i N  fabricated by growing S i N  x  x  membrane is  on a Si wafer and then etching a hole in the Si  100  6. Reconstruction of Magnetic Domains in Transmission Geometry 101  Fig. 6.9: Co/Pd multilayer with various sizes of magnetic bit patterns (fc/mm means flux changes per mm, i.e. 2000 fc/mm corresponds to a bit size of 500 nm). The magnetic patterns are written and imaged with a magnetic write/read head. The SiN membrane is the large rectangle in the image. x  from the back (anisotropic etching), which in this case, has turned out to be rectangular. A magnetic layer is then evaporated on the S i N  x  side. T h e  magnetic layer is a C o / P d multilayer with 12 repeats with a total thickness of 14.4 n m . Markers are writen on this side with a focussed ion beam (FIB) in an S E M , as using S E M one can see the membrane when looking at the magnetic layer side. W i t h the aid of the markers, the membrane area can be located from the magnetic film side, and following that the domains are written using a magnetic write head. T h e magnetic tracks are purposefully put non-parallel to the membrane border, in order to distinguish charge/magnetic components more easily in q-space.  5  The pattern has been made at the IBM Almaden Research Center. It was designed by Andreas Moser and Stefan Eisebitt. The deposition of Co/Pd multilayers, marking the membrane using FIB, and writing/imaging the sample with the magnetic read head were performed by Kentaro Takano, Charlie Rettner and Andreas Moser, respectively. 5  6. Reconstruction of Magnetic Domains in Transmission Geometry T h e advantage of having a patterned sample is that once a certain number of domain types are illuminated and the resulting transmitted intensity is inverted, we would be able to fully evaluate the success of the reconstruction as the magnetic patterns may be easily recognized. We expect, considering successful inversion of noisy simulated data and the previously inverted experimental data, that future application of the methods we have discussed in this chapter to patterned samples would result in direct agreement of the reconstructed image and the true domain distribution, which in turn would establish the reconstruction technique as experimentally valid.  102  103  7. Conclusion  7.  Conclusion  In this work, we have described experiments that were performed in order to demonstrate coherent resonant x-ray scattering as a powerful method for the analysis and imaging of magnetic domains. Speckle patterns were observed for the first time in coherent soft x-ray scattering from antiferromagnetic ( A F M ) domains on the surface of LaFeOs thin  films.  Linearly polarized  coherent soft x-rays reflected at grazing angle from the surface. Scattering contrast from the A F M domains was obtained by exploiting the large xray magnetic linear dichroism ( X M L D ) at the Fe L  3  edge.  B y tuning the  energy to either of the two crystal field split peaks of the Fe Z / edge, we 3  were able to invert the scattering contrast between the two domains, and to demonstrate dependence of speckle on the magnetic structure. A detailed analytical treatment of the contributions of pinhole, magnetic and roughness components of scattering was presented.  It was shown that at g-regions  of reciprocal space corresponding to the size of the domains, the pinhole a n d / o r roughness scattering must be present to interfere with the magnetic scattering in order to produce observable change in the speckle upon reversal of magnetic contrast. In order to demonstrate the applicability of this method to ferromagnetic ( F M ) samples, we performed resonant x-ray scatterng on a C o / P t multilayer. T h e geometry was that of transmission, and magnetic x-ray circular dichroism ( M X C D ) was utilized with the contrast arising from the dependence of the atomic scattering amplitude on the magnetization direction of the domains which were oriented either parallel or antiparallel to the photon  7. Conclusion helicity. B y tuning the energy to the C o L  3  104  edge, magnetic speckle was very  clearly demonstrated. These experiments clearly demonstrated the ability of coherent resonant x-ray scattering to produce speckle patterns arising from magnetic contrast on A F M or F M thin films. We next concentrated on the reconstruction of magnetic domains as a new method of imaging magnetic structure for both A F M and F M thin films. T h e x-ray scattering experiment in the reflection geometry was first considered. It was noted that both the surface roughness and the magnetic domain distributions would be unknown in a typical reconstruction. T o tackle this issue, we noted that upon making intensity measurements at an off-resonant energy where magnetic contrast would diminish, the resulting speckle patterns would exhibit pure dependence on the sample roughness. T h e roughness profile could then be retrieved from the off-resonant data. We then developed an iterative reconstruction technique in which  a priori knowledge of the fi-  nite support and the reconstructed roughness would be used in combination with measurements performed at a resonant energy to retrieve the correct magnetic distribution. T h e technique was shown to be very successful upon application to noisy simulated data. Similarly, algorithms were developed for the reconstruction of F M domains studied in the transmission geometry. It was shown that right and left helicity measurements would allow for the separation of charge and magnetic scattering, and therefore allow for the reconstruction of magnetic structure without a need to reconstruct the charge distribution. Using this method, reconstructed magnetic domains from experimental data showed a remarkable similarity to the worm-domain structure of the actual domain distribution imaged using magnetic force microscopy ( M F M ) . This, to our knowledge, has been the first successful reconstruction of magnetic domains from experimental data. We furthermore discussed the possibility of using right and left circularly polarized photons to enable direct (i.e. non-iterative) reconstruc-  7. Conclusion tion of magnetic structure.  105  It was shown that the success of this method  is dependent on the presence of charge-magnetic interference. It was subsequently demonstrated that by using pinhole illumination comparable to the size of domains a n d / o r rough samples with comparable roughness scale, noniterative reconstruction of magnetic distribution would be possible without requiring a very large dynamic range (i.e. long exposure time and low dark noise in the measurement). One issue of great scientific and technological interest is the question of how a domain evolves as its magnetization is being reversed by the flux of a nearby domain. For instance, in the process of magnetic storage in M R A M cells, the study of magnetic reversal is of great interest. However, problems with resolution, buried interfaces and the presence of other fields have limited research in this area. While S P - S T M is uniquely capable of atomic resolution imaging, it does not work in the presence of other fields and cannot image buried layers.  Similarly, x-ray and electron real space imaging techniques  such as P E E M are limited by the aberrations of the optics involved (not the wavelength of the radiation). In addition, imaging using P E E M is virtually impossible in the presence of other fields due to the interaction of secondary electrons with the interacting magnetic fields. X - r a y resonant scattering is able in principle to overcome these shortcomings, and in this regard, we are very optimistic that the continued application of the techniques exhibited in the course of this thesis, coupled with the arrival of 4th generation x-ray light sources, will provide more information about domain evolution during reversal in films exhibiting perpendicular anisotropy. In addition, these resonant scattering techniques offer new opportunities to study nanoscale magnetic structure and phase transition phenomena in a broad range of samples.  Bibliography  106  Bibliography  [1] L . M . Falicov, D . T . Pierce, S. D . Bader, R. Gronsky, K . B . Hathaway, H . J . Hopster, D . N . Lambeth, S. S. P. Perkin, G . Prinz, M . Salamon, I. K . Schuller, R. H . Victoria, J . Mater. Res. 5, 1299 (1990). [2] J . B . Kortright, D . D . Awschalom, j . Stohr, S. D . Bader, Y . U . Idzerda, S. S. P. Parkin, I. K . Schuller, H . C . Siegmann, J . Magn. Magn. Mater. 207, 7 (1999). [3] M . N . Baibich, J . M . Broto, A . Fert, F . N . V a n D a u , F . Petroff, P. Etienne, G . Creuzet, A . Friederich, J . Chazelas, Phys. Rev. Lett. 61, 2472 (1988). [4] G . Binasch, P. Griinberg, F . Saurenbach, W . Zinn, Phys. Rev. B 39, 4828 (1989). [5] S. S. P. Parkin, Z . G . L i , D . J . Smith, A p p l . Phys. Lett. 58, 2710 (1991). [6] A . E . Berkowitz, K . Takano, J . Magn. Magn. Mater. 200, 552 (1999). [7] B . T . Jonker, ed.,  Magnetic Ultrathin Films, M R S Symposia Proceedings  No. 313 (Materials Research Society, Pittsburgh, 1993). [8] S. S. P. Parkin, N . More, and K . P . Roche, Phys. Rev. Lett. 64, 2304 (1990). [9] G . Prinz, K . Hathaway, Phys. Today 48, 24 (1995).  Bibliography  107  [10] L . Thomas et. al. Phys. Rev. Lett. 84, 3462 [11] A . Scholl, Science 288 (5472), 1762 (2000). [12] S. Heinze, M . Bode, A . Kubetska, O . Pietzsch, X . Nie, S. Bliigel, R . Wiesendanger, Science 288 (5427), 1805 (2000). [13] J . Stohr, A . Scholl, T . J . Regan, S. Anders, J . Liming, M . R . Scheinfein, H . A . Padmore, and R. L . White, Phys. Rev. Lett 83, 1862 (1999). [14] B . T . Thole, G . van der Laan, G . A . Sawatzky, Phys. Rev. Lett. 55, 2086 (1985). [15] J . P. Hannon, G . T . Trammell, M . Blumel, D . Gibbs, Phys. Rev. Lett. 61, 1245 (1988). [16] D . Alders, J . Vogel, C . Levelut, S. D . Peacor, T . H i b m a , M . Sacchi, L . H . Tjeng, C . T . Chen, G . van der Laan, B . T . Thole, G . A . Sawatzky, Europhys. Lett. 32, 259 (1995). [17] Z . H . C a i , B . L a i , W . B . Y u n , I. McNully, K . G . Huang, and T . P . Russel, Phys.  Rev. Lett. 73, 82 (1994).  [18] J . C . Dainty, ed.,  Laser Speckle and Related Phenomena, (Springer-  Verlag, Berlin, 1984). [19] D . Atwood,  Soft X-rays and Extended Ultraviolet Radiation: Principles  and Applications, p. 73 (Cambridge University Press Cambridge,2000). [20] F . D e Bergevin, M . Brunei, A c t a Cryst. A 37, 314 (1981): [21] J . B . Kortright, S. K . K i m , Phys. Rev. B 62, 12216 (2000). [22] B . H u , P. Geissbuhler, L . Sorensen, S. D . Kevan, J . B . Kortright, and E . E . Fullerton, Synchrotron Radiation News 14, 11 (2001).  Bibliography  i08  [23] H . A . Diirr et al., Science 284, 2166 (1999) . [24] M . J . Neel, J . Physique R a d . 15, 225 (1954). [25] R . Allenspach, J . Magn. Magn. Mater. 129, 160 (1994). [26] G . Shiitz, W . Wagner, W . Wilhelm, P. Kienle, R. Zeller, R . Frahm, G . Materlik, Phys. Rev. Lett. 58, 737 (1987). [27] C . T . Chen, Y . U . Idzerda, H . - J . L i n , N . V . Smith, G . Meigs, E . C h a b a n , G . . H . H o , E . Pellegrin, F . Sette, Phys. Rev. Lett. 75, 152 (1995). [28] J . Stohr, J . Magn. Magn. Mater. 200, 470 (1999). [29] K . Baberschke, A p p l . Phys. A 62, 417 (1996). [30] Y . U . Idzerda, L . H . Tjeng, H . - J . L i n , C . J . Gutierrez, G . Meigs, C . T . Chen, Phys. Rev. B 48, 4144 (1993). [31] J . Stohr, Y . W u , B . D . Hermsmeier, M . G . Samant, G . R . Harp, S. Koranda, D . D u n h a m , B . P. Tonner, Science 259, 658 (1993). [32] W . L . O'Brien, B . P. Tonner, Phys. Rev. B 52, 1 (1995). [33] A . Scholl, J . Stohr, J . Liming, J . A . Seo, J . Fompeyrine, H . Siegwart, J . P. Locquet, F . Nolting, S. Anders, E . E . Fullerton, M . R. Scheinfein, H . A . Padmore, Science 287, 1014 (2000). [34] D . Alders, P h . D . thesis, University of Groningen, T h e Netherlands (1997). [35] A . C . Price, L . B . Sorensen, S.D. Kevan, J . Toner, A . Poniewierski, R . Holyst, Phys. Rev. Lett. 82, 755 (1999).  Bibliography  109  [36] T . 0 . Mentes, C . Sanchez-Hanke, C . C . K a o , "Reconstruction of M a g netization Density in 2-Dimensional Samples From Soft X - r a y Speckle Patterns Using Multiple-Wavelength Anomalous Diffraction Method", in preparation. [37] J . P. Locquet, J . Perret, J . Fompeyrine, E . Machler, J . W . Seo, G , V a n Tendeloo, Nature 394, 453 (1998). [38] S. Eisebitt, M . Lorgen, R . Scherer, S. Tixier, A . R a h m i m , T . Tiedje, J . Liming, J . Stohr, A . Scholl, to be published (2001). [39] Matthews,  J , R. L . Walker,  Mathematical Methods of Physics, (New  York, W . A . Benjamin, 1964). [40] M . Abramowitz, I. A . Stegun,  Handbook of Mathematical Functions  (Dover Publications, New York, 1964). [41] H . M . Pederson, Optica A c t a , 22, 15 (1975). [42] J . W . G o o d m a n ,  Laser speckle and related phenomena, edited by J . C .  Dainty, p. 65 (New York:Springer-Verlag) (1975). [43] S. K . Sinha, A c t . Phys. P o l . A . 89, 219 (1996). [44] J . B . Kortright, S. K . K i m , G . P. Denbeaux, G . Zelter, K . Takano, E . E . Fullerton, submitted for publication L B N L - 4 7 2 7 3 (2001). [45] W . A . Hendrickson, Science 254, 51 (1991). [46] J . R . Fienup, O p t . Lett. 3, 27 (1978). [47] J . R . Fienup, J . O p t . Soc. A m . A 4AppI. O p t . 21, 2758 (1982). [48] J . R . Fienup, C . C . Wackerman, J . O p t . Soc. A m . A 3,1897 (1986)  Bibliography  110  [49] R . W . Gerchberg, W . O . Saxton, Optik 35, 237 (1972). [50] W . Wolf, Proc. Phys. Soc. (London) 80, 1269 (1962). [51] A . Walther, O p t . A c t a 10, 41-49 (1963). [52] E . M . Hofstetter, I E E E Trans. Inf. Theory IT-10, 119 (1964). [53] P. J . Napier, R . H . T . Bates, Astron. Astrophys. Suppl. 15, 427 (1974). [54] Y u . M . Bruck, L . G . Sodin, O p t . C o m m u n . 30, 304 (1979). [55] M . H . Hayes, I E E E Trans, on Acoustics, Speech, Sig. Proc. A S S P - 3 0 , 140 (1982). [56] J . Boyes-Watson, K . Davidson, and M . F . Perutz, Proc. R . Soc. London, Ser. A 191, 83 (1947). [57] D . Sayre, A c t a Crystallogr. 5, 843 (1952). [58] R . Barakat, G . Newsam, J . M a t h . Phys. 25, 3190 (1984). [59] R . H . Bates, D . G . H . T a n , in Inverse Optics II, A . J . Devaney and R. H . Bates, eds., Proc. S P I E 558, 54 (1985). [60] R . G . Lane, J . O p t . Soc. A m . A 4, 118 (1987). [61] J . Miao, D . Sayre, H . N . Chapman, J . O p t . Soc. A m . A 15, 1662 (1998). [62] I. A . Vartanyants, J . A . Pitney, J . L . Libbert, I. K . Robinson, Rev. B . 55, 13193 (1997). [63] R . P. Millane, W . J . Stroud, J . O p t . Soc. A m . A . 14, 568 (1977) [64] J . R . Fienup, J . O p t . Soc. A m . A 4, 118 (1987).  Phys.  111  Bibliography [65] J . A . Pitney, I. A . Vartaniants,  I. K . Robinson,  S P I E Conference on  Digital Image Recovery and Synthesis I V , S P I E Vol. 3815, 199 (1999). [66] O p t . E n g . 19, 297 (1980). [67] H . Takajo, T . Takahashi, R . Ueda, M . Taninaka, J . O p t . Soc. A m . A . 15, 2849 (1998). [68] H . Takajo, T . Takahashi, T . Shizuma, J . O p t . Soc. A m . A . 16, 2163 (1999). [69] J . Miao, J . K i r z , D . Sayre, A c t a Cryst. D56, 1312 (2000). [70] J . Miao, D.Sayre, A c t a Cryst. A56, 596 (2000). [71] R . N . Bracewell,  The Fourier Transform and Its Applications (McGraw-  H i l l , New York, 1965). [72] S. K . Sinha, E . B . Sirota, S. Garoff, and H . B . Stanley, Phys. Rev. B 38, 2297 (1988). [73] J . Miao et al., "Coherent diffraction imaging by using the oversampling method", in preparation. [74] P. Beckmann, A , Spizzichino,  The Scattering of Electromagnetic Waves  from Rough Surfaces (New York, Pergamon Press, 1963). [75] L . Mandel. L . , Progress  in Optics, Vol. 2, edited by E . Wolf (Amsterdam,  North-Holland Publishing C o . , 1963), p. 181. [76] B . B . Mandelbrot,  The Fractal Geometry of Nature (Freeman, New York,  1982). [77] M . Kardar, G . Parisi, Y . Zhang, Phys. Rev. Lett, 56, 889 (1986).  Bibliography  112  [78] A . Ballestad, M . Sc. Thesis, The University of British Columbia, Canada (1998). [79] A . - L . Barabasi, H . E . Stanley. Fractal Concepts in Surface Growth. (Cambridge University Press, Cambridge, 1995). [80] F . Family, T. Viscek, J . Phys. A 18: L75 (1985).  A. Intensity Correlation in terms of Amplitude Correlation113  A . Intensity Correlation in terms of Amplitude Correlation  Pederson [41] has shown for the case of spectral speckle correlation in polychromatic speckle patterns that the intensity correlation may be conveniently expressed in terms of the amplitude correlation. A n identical relation is derived here in a more elaborate manner and is shown to be applicable to the case of magnetic intensity correlation in the region where qL S> 7r/2 (L is the size of the sample), regardless of the sample roughness or magnetic contrast. The scattered intensity is given by  7 (q) = |A (q)| ±  where A±(q),  (A.l)  2  ±  the complex scattered amplitude, consists of two parts: A (q)  = A° (ci)  ±  ±  + AA (d)  (A.2)  ±  where A±(q) = (A±(q)) is the mean scattered amplitude with () denoting an ensemble-average over surface roughness and magnetic domain distributions, while AA± is a fluctuating amplitude component with zero mean. Taking I± = 7±(q), the intensity may now be written  I±  = \A \  = \Al +  2  ±  =  AA \  2  ±  /£ + li + 2 Re{(A° yAA } ±  where we have defined 7± = | A ^ | and 7^ 2  (I )=I° ±  = |AA | . 2  ±  + (I ) d  ±  ±  ±  (A.3) The mean value is  (A.4)  A. Intensity Correlation in terms of Amplitude Correlation114 and the fluctuating part A / ± = Ali + 2 Re{(A° )*AA } ±  (A.5)  ±  The intensity correlation becomes (A/+A/ ) = ±  (AI*Ali)  +  (4 Re{(A° )*AA }  +  2Re{(AI*(Al)*AA  +  x Re{(A° )*AA±}  +  )  ±  + Ali(A )*AA )}  (A.6)  0  ±  +  +  We are interested in investigating whether AA±, over ensemble of input surface roughness and magnetic distributions, exhibits independent Gaussian distribution of its real and imaginary components with common variance and zero mean (i.e. circular Gaussian statistics). Note that in the literature, it is typically desired for A± (not AA±) to obey circular Gaussian statistics [18] which requires much stronger constraints than in the present case. For ease of demonstration, the roughness is omitted in the rest of the derivations, however results are easily extendable to the case of rough samples. Assuming that the detector at each point receives contributions from N points on a I D surface (equally illuminated by the pinhole), the scattered amplitude can be expressed as a I D discrete Fourier transform: A (q) = J2( n)e 1±m  (A.7)  iqHn/N)  ±  n=l  where L is the size of the sample, and m is as defined in Eq. (2.3). It follows n  that AA± (q) = ± £  me ^ iqL  (A.8)  N)  n  71=1  Fig. (A.la) shows the result of adding the Fourier phase e ( l } iqL  n N  contri-  butions as n ranges from 1 to N. It is clear that as qL reaches and exceeds a quarter of a full rotation (i.e. 7r/2), the pinhole contribution begins to be in  A. Intensity Correlation in terms of Amplitude Correlation115  real  real  0.5  Fig. A.l: (a) Pinhole and (b) magnetic contributions from points on an 80-pixel ID surface, containing 20 randomly generated magnetic domains, at a point on the detector with qL = 5.6 > 7r/2 (|mfc| = 20%). Circular Gaussian nature of the magnetic contribution is observed. both the imaginary and real directions. Fig. (A.lb) shows the corresponding magnetic contribution to the diffracted amplitude, given by Eq. (A.8). In the simulations, domains are generated randomly and they are such that as one moves from one to another, the contribution m changes sign (\m \ = 20%). n  n  The figure captures the random walk nature of AA±(q) in both the real and imaginary directions. It must be kept in mind that we are assuming that the number of magnetic domains in the area of illumination is large, enabling us to make use of In regions qL 3> 7r/2, therefore, AA±(q)  the central limit theorem.  will  be circularly Gaussian distributed over ensemble of surfaces even though the pinhole contribution J2  e  ^l  %qL  k=l  k  NS>  may still be completely dominant. On the  contrary, this statement would certainly be untrue for A±(q) when pinhole scattering, which certainly does not follow Gaussian statistics, is dominant. Note also that the first diffraction minimum occurs at qL = 2n; therefore, we are correct in assuming AA± to have independent Gaussian distributed  A.  Intensity Correlation in terms of Amplitude Correlation116  real and imaginary parts with common variance and zero mean beyond the central diffraction ring. Under the assumption of circular Gaussian statistics for two general random variables B and C , we have the following two properties [75] (Re{B}Re{C}) = ( I m { 5 } I m { C } )  (A.9)  and  (\B\ \C\ ) = (\B\ )(\C\ ) + \(BCr)\ i  2  a  2  (A.10)  2  We now make the observation that the last term in E q . (A.6) contributes zero because it contains only odd powers of AA±.  It also follows from E q . (A.9)  that Re{{AA (AA±)*)}  = 2 (Re{A,4+}Re{A.4 })  +  ±  (A.ll)  Therefore, E q . (A.6) may be simplified to the form (A7 A/ ) = ( A / J A l l ) +  +  ±  (2Re{(A° YA (AA (AA y)}  (A.12)  0  +  ±  +  ±  Furthermore, it follows from E q . (A. 10) that (A/4 A/4)  = |rf  + ; ± )  |  (A.13)  2  where we have introduced the correlation function  ' rf  +i±)  =  (AA AA* ) +  (A.14)  ±  of the fluctuating component of the scattered amplitude. U p o n the introduction of two other correlation functions rj r  ( + i ± )  = A° {A%r  = (A A* ) +  (A.15)  +  + f ± )  ±  = r?  + ! ± )  + rf  + > ± )  (A.ie)  A. Intensity Correlation in terms of Amplitude Correlation117 we arrive at an equation for the intensity correlation:  (A/ A/ ) = |rf , | + 2Re{(r° )Tf } 2  +  ±  + ±)  = |r as quoted in E q . (2.20).  +)±)  | -|r? , | 2  (+i±)  +  ±)  2  +)±)  (A.i7)  B. Autocorrelation of Roughness Distribution for a Self-Affine Surface 118  B. Autocorrelation of Roughness Distribution for a Self-Affine Surface  Let W(X)  denote the mean-square height-deviation function  W(X) = ([h(x)-h(x + X)} ) 2  (B.l)  where ( ) represent a statistical average over position x on the surface. surface which is  A  self-affine has the property that [76, 72] W(X) oc R  2a  (B.2)  .where a has a value between 0 and 1, and determines how smooth or jagged the surface is. T h e surface fractal dimension is given by [76]  D = 3- a  (B.3)  a  However, E q . (B.2) represents an idealization, as the function W(X)  diverges  at infinity. In practice the mean-square roughness may saturate at a value a  2  for many reasons, including finite growth time as we will describe later.  We thus write  W{X,t) = 2a [l-exp[-(j) }} 2  2a  (B.4)  which reduces to form E q . (B.2) for X <C £. Thus, £ is an effective cut-off. length for the roughness of the form given by E q . (B.2). T h e justification of E q . (B.4) may to some extent be found in the equations which govern the growth of deposited films [43]. Consider for instance the Kardar-Parisi-Zhang ( K P Z ) equation [77]: the height function  h(x,t) as  a function of position and time is governed by an equation of the form  B. Autocorrelation of Roughness Distribution for a Self-Affine Surface 119  ^ where v,\  = ^h+^(Vh)  + ,  2  V  (B.5)  are constants and n = r)(x,t) represents the random noise asso-  ciated with the deposition and incorporation of atoms on the surface. A n analysis of the solutions of this equation, either numerically or using renormalization group methods, reveals that the height scales according to the so called  Family-Viscek scaling relation [78, 79, 80]: W(X,t) = X f(t/X ^) 2a  where fix)  a  (B.6)  a and 0 are referred to as the roughness and growth exponents, and  is a scaling function which must satisfy the properties that  f{x < 1) ~ x  20  f{x >  1) -> constant  (B.7) (B.8)  A function which would satisfy these conditions is  fix) = C x ^  (B.9)  2  x  where C  x  is a constant.  One will then recover E q . (B.4) by substituting  E q . (B.9) into E q . (B.6), and letting  o = {\cl)t» 3  (B.10)  and  e=t  1/z  where z is called the  (B.II)  dynamic exponent and satisfies relation z = a/B  (B.12)  Thus, we conclude that a self-affine surface with a finite cut-off length for the roughness is consistent with growth models if we assume that it corresponds  B. Autocorrelation of Roughness Distribution for a Self-AfRne Surface 120 to the surface which results after finite growth time. Combining Eqs. ( B . l ) and (B.4), it follows easily that <7i(x)/i(x + X)> = a e x p [ 2  (^)  2 Q  ]  with the normalized autocorrelation function as quoted in E q . (2.27).  (B.13)  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085183/manifest

Comment

Related Items