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The Materials Research Society Series
Koichi Hayashi Editor
Hyperordered Structures in Materials
Disorder in Order and Order within Disorder

The Materials Research Society Series

The Materials Research Society Series covers the multidisciplinary ﬁeld of materials research and technology, publishing across chemistry, physics, biology, and engineering. The Series focuses on premium textbooks, professional books, monographs, references, and other works that serve the broad materials science and engineering community worldwide. Connecting the principles of structure, properties, processing, and performance and employing tools of characterization, computation, and fabrication the Series addresses established, novel, and emerging topics.

Koichi Hayashi
Editor
Hyperordered Structures in Materials
Disorder in Order and Order within Disorder

Editor Koichi Hayashi Department of Physical Science and Engineering Nagoya Institute of Technology Nagoya, Aichi, Japan

ISSN 2730-7360

ISSN 2730-7379 (electronic)

The Materials Research Society Series

ISBN 978-981-99-5234-2

ISBN 978-981-99-5235-9 (eBook)

https://doi.org/10.1007/978-981-99-5235-9

© Materials Research Society, under exclusive license to Springer Nature Singapore Pte Ltd. 2024

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Preface
Solid matter can roughly be categorized into crystals, which have regular atomic arrangements over long distances and amorphous, which have random atomic arrangements. These materials possess various functional properties which we believe are governed by deviations from the perfect regularity or randomness of their atomic arrangements. Therefore, we decided to focus on these deviations, discuss its correlation with material functions, and summarize it in this technical book.
From the crystal side, the deviations from perfect crystallinity are represented by dopants. Dopants are aperiodic structures within regular matrix structures, and in semiconductors, play an important role in creating carriers. It can be said that most materials contain dopants. Although dopants are considered point defects, this book is written with more emphasis on defect complexes, which will play a leading role in materials design in the future. We call this structural feature “Disorder in Order.” On the other hand, there are no amorphous materials that have perfect randomness. A topological analysis of the atomic arrangement in glass has revealed that crystal-like topology can be extracted from the glass network structure, and that this topology correlates with the functional properties of the glass. We call this structural feature “Order within Disorder.” These imperfections, each from perfect regularity and perfect randomness, are integrated and called “Hyperordered Structure” in this book. These ideas and the structural features of “Hyperordered Structure” are described in detail in Part I, Hyperordered Structures: Disorder in Order/Order within Disorder.
Hyperordered structures exist in a wide range of material groups, such as dielectrics, functional glasses, zeolites, superconductors, and biological materials, and can be regarded as a treasure trove of material functionality. This book is organized from analytical, theoretical, and material perspectives to provide a comprehensive understanding of hyperordered structures. In Part II, Characterization of Hyperordered Structures, methods that can reveal hyperordered structures are introduced. In particular, ﬁve chapters introduce methods using quantum beams. Part III, Computational Approaches to Hyperordered Structures, introduces methods for unveiling electronic states and structural features to explore the function of hyperordered structures. Part IV, Physicochemical Properties of Hyperordered Materials,
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introduces hyperordered structures hidden in actual materials with examples of a wide range from protein and inorganic matter.
We hope that this book, which discusses materials science from a new perspective, will intrigue readers and bring new ideas to researchers.

Nagoya, Japan

Koichi Hayashi

Contents
Part I Hyperordered Structures: Disorder in Order/Order within Disorder
1 From Point Defects to Defect Complexes . . . . . . . . . . . . . . . . . . . . . . . . 3 Koichi Hayashi
2 Topological Order and Hyperorder in Oxide Glasses and Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Shinji Kohara
Part II Characterization of Hyperordered Structures 3 Atomic-Resolution Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Tomohiro Matsushita, Koji Kimura, and Kenji Ohoyama 4 X-Ray and Neutron Pair Distribution Function Analysis . . . . . . . . . . 93
Yohei Onodera, Tomoko Sato, and Shinji Kohara 5 Angstrom-Beam Electron Diffraction Technique
for Amorphous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Akihiko Hirata 6 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Ayano Chiba and Shinya Hosokawa 7 Property Measurements of Molten Oxides at High Temperature Using Containerless Methods . . . . . . . . . . . . . . . . . . . . . . 159 Takehiko Ishikawa, Paul-François Paradis, and Atsunobu Masuno Part III Computational Approaches to Hyperordered Structures 8 Density Functional Theory Calculations for Materials with Complex Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Ayako Nakata and Yoshitada Morikawa
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Contents

9 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Yu Takano, Takahiro Ohkubo, and Satoshi Watanabe
10 Reverse Monte Carlo Modeling of Non-crystalline and Crystalline Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Naoto Kitamura and Shinji Kohara
11 Structural-Order Analysis Based on Applied Mathematics . . . . . . . . 265 Motoki Shiga and Ippei Obayashi
12 Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Motoki Shiga and Satoshi Watanabe

Part IV Physicochemical Properties of Hyperordered Materials
13 Dielectric Materials with Hyperordered Structures . . . . . . . . . . . . . . . 313 Hiroki Taniguchi
14 Hyperordered Structures in Microporous Frameworks in Zeolites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Masanori Takemoto, Kenta Iyoki, and Toru Wakihara
15 Glasses with Hyperordered Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Atsunobu Masuno and Madoka Ono
16 Biological Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Hideaki Tanaka
17 Battery and Fuel Cell Materials with Hyperordered Structures . . . . 395 Naoto Kitamura
18 Superconductors with Hyperordered Structures . . . . . . . . . . . . . . . . . 411 Yoshihiro Kubozono and Jun Akimitsu
19 Ordered and Disordered Metal Oxide for Biomass Conversion . . . . 433 Daniele Padovan and Kiyotaka Nakajima

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

Abbreviations

2D 3DG AAL ABED ADL AFM AIMD ALD AMBER APD AT ATAT ATP BCC BCZT BLC BO BTE BVS C4mim CC CCTO CE CEF CHA CHARMM Chl CI CN CORs CV

Two-dimensional 3-Deoxy-glucosone Aero-Acoustic Levitation Angstrom-beam electron diffraciton Aerodynamic levitation Atomic force microscopy Ab initio molecular dynamics Anharmonic lattice dynamics Assisted model building with energy reﬁnment Avalanche photodiode PaseATP synthase Alloy Theoretic Automated Toolkit Adenosine triphosphate Body-centered cubic (Ba,Ca)(Zr,Ti)O3 Barrier layer capacitor Bridging oxygen Boltzmann transport equation Bond valence sum 1-butyl-3-methylimidazolium Coupled-cluster CaCu3Ti4O12 Cluster expansion Cyclic electron ﬂow Concentric hemispherical analyzer Chemistry at Harvard macromolecular mechanics Chlorophyll Conﬁguration interaction Coordination number Chemically ordered regions Cross-validation

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x
CVD Cyt b6f DAC DF DFT DI DMF DnC DOS DRIFT DRP DTA DW EDX/EDS EML EPDDs ESL EXAFS FAD FCC Fd FDCA FFP FFT FNR FSDP FTIR FWHM GAP GFL GGA GIC GL GP GROMACS GVL hcp HDNNP HF HIP HMF HOMO HREM HR-TEM HSMC

Abbreviations
Chemical vapor deposition Cytochrome b6f Diamond anvil cell Density Functional Density functional theory Distortion index Dimethylfuran Drop and Catch Density of states Diffuse reﬂectance infrared Fourier transform Dense random packing Differential Thermal Analysis Drop weight Energy dispersive X-ray spectroscopy Electromagnetic levitation Electron-pinned defect-dipoles Electrostatic levitation Extended X-ray absorption ﬁne structure Flavin adenine dinucleotide Face-centered cubic Ferredoxin Furan-2,5-dicarboxylic acid Forward focusing peak Fast Fourier Transform Ferredoxin–NADP+ reductase First sharp diffraction peak Fourier Transform Infrared Full Width at Half Maximum Gaussian approximation potential Gas Film Levitation Generalized gradient approximation Graphite intercalation compound Ginzburg Landau Gaussian process Groningen machine for chemical simulation γ-valerolactone Hexagonal close-packed High-dimensional neural network potential Hartree–Fock Hot isostatic pressure 5-hydroxymethylfurfural Highest occupied molecular orbital High-resolution electron microscopy High-resolution transmittance electron microscopy Hard sphere Monte Carlo

Abbreviations

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IMFP INS IPEH IR ISS-ELF IUVS IXS IZA-SC
IZO JAXAJ JG KMC KS LASSO LC-ωPBE LDA LET LHCI LHCII LIB LPSO LUMO LVP MAS MBP MC MCPB MD MO MP MRB MRO MTK++ MyPresto NADP+, NADPH NAMD NBO ND NH NH3-SCR NITO-x% NMR NN NPG

Inelastic mean free path Inelastic neutron scattering Inverse photoelectron holography Infrared absorption spectroscopy Electrostatic levitation furnace onboard the ISS Inelastic ultra violet scattering Inelastic X-ray scattering The Structure Commission of the International Zeolite Association In2O3-ZnO Japan Aerospace Exploration Agency Johari-Goldstein Kinetic Monte Carlo Kohn-Sham Least absolute shrinkage and selection operator Long-range corrected omega-Perdew–Burke–Ernzerhof Local density approximation Linear electron transfer Light-harvesting complex I Light-harvesting complex II Lithium ion battery Long-period stacking ordered Lowest unoccupied molecular orbital Large volume press Magic-angle spinning Maximum bubble pressure Monte Carlo Metal center parameter builder Molecular dynamics Molecular orbital Møller-Plesset Magnesium rechargeable battery Medium-range order Modeling toolkit plus plus Medically yielding protein engineering simulator Nicotinamide adenine dinucleotide phosphate Nanoscale molecular dynamics Non-bridging oxygen Neutron diffraction Neutron holography Selective catalytic reduction of NOx with NH3 x% Nb+In co-doped TiO2 Nuclear magnetic resonance Nearest neighbor Neopentylglycol

xii
NRIXS NRU NSE NV O(N) oct OEC OSDAs PALS PC PCA PCS PD PD PDB PDF PEF PEH PET PH PID PLA PMN PNRs PP PQ PQH2 PSD PSI PSII PVSVC PXRD PZT QENS RAA RESP RFA RMC RMSD RMSE RMSF SADs SAED SD SDD

Nuclear resonant inelastic X-ray scattering National Research Universal Neutron spin echo Nitrogen-vacancy Order-N Octahedral Oxygen-evolving complexes Organic structure-directing agents Positron annihilation lifetime spectroscopy Plastocyanin Principal component analysis Photon correlation spectroscopy Pendant drop Persistence diagram Protein Data Bank Pair distribution function Polyethylene furanoate Photoelectron holography Polyethylene terephthalate Persistent homology Proportional-Integral-Differential Polylactic acid Pb(Mg1/3Nb2/3)O3 Polar nano regions Principal Peak Plastquinone Plastoquinol Position sensitive detector Photosystem I Photosystem II PV siplit vacancy complex Powder X-ray diffraction Lead ziroconate titanate Quasi-elastic neutron scattering Reduced atomic arrangement Restrained electrostatic potential Retarding ﬁeld analyzer Reverse Monte Carlo Root-mean-square distance Root-mean-square error Root-mean-square ﬂuctuations Structure-directing agents Selected area electron diffraction Sessile drop Silicon drift detector

Abbreviations

Abbreviations

xiii

SEM SF Sn-Beta SOAP SOFC SPEA SRO STEM SX TDI TEM tet Ti/SBA-15 TM ToF TOF UV UV-Vis XAFS XFEL XFH XPCS XPS XRD

Scanning electron microscopy Symmetry function Sn-containing Beta zeolite Smooth overlap of atomic position Solid oxide fuel cell Scattering pattern extraction algorithm Short-range order Scanning transmission electron microscopy Soft X-ray Time-domain interferometry Transmission electron microscopy Tetrahedral TiO2-grafted mesoporous silica SBA-15 Trans membrane Time-of-ﬂight Turnover frequency Ultra-Violet Ultraviolet-visible X-ray absorption ﬁne structure X-ray free electron laser X-ray ﬂuorescence holography X-ray photon correlation spectroscopy X-ray photoelectron spectrum X-ray diffraction

Part I
Hyperordered Structures: Disorder in Order/Order within Disorder

Chapter 1
From Point Defects to Defect Complexes
Koichi Hayashi

Abstract Most of the materials have dopants to create functionalities. However, due to the high demands on material functionality in cutting-edge devices, the conventional concept of point defects is no longer sufﬁcient. Therefore, attempts to break through the limitations of point defects by doping different elements to create defect complexes have begun to progress. This section begins with examples of point defects and their role in a wide range of materials, including semiconductors, superconductors, catalysts, scintillators, metals, and glasses. Then, the limitations of the point defects and their solutions using defect complexes are presented with examples of semiconductor and glass. In the latter part of this section, examples of defect complexes are presented, divided into dopant-vacancy pairs and complex defects with sub-nanometer scales. These confer novel functionalities to materials. For example, nitrogen vacancy (NV ) centers in diamond have been considered as a promising for quantum bits, and Mn4CaO5 clusters in photosystem II protein play an important role in photosynthesis. Finally, I introduce characterization techniques and theoretical methods to correctly understand the structures and properties of the defect complexes.
Keywords Defect complex · Dopant–vacancy pair · Atomic resolution holography · Large scale DFT

1.1 Doping and Functionality

Most materials contain dopants that confer some functionalities. Si, which is widely used in the world, is one of the good examples. Pure Si is an insulator. However, by doping different elements such as boron or phosphorus to Si, it exhibits functions as a p- or n-type semiconductor, respectively. In this case, the functionality originates

K. Hayashi (B)
Department of Physical Science and Engineering, Nagoya Institute of Technology,
Nagoya 466-8555, Japan
e-mail: khayashi@nitech.ac.jp

© Materials Research Society, under exclusive license to Springer Nature Singapore Pte

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Ltd. 2024

K. Hayashi (ed.), Hyperordered Structures in Materials, The Materials Research

Society Series, https://doi.org/10.1007/978-981-99-5235-9_1

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from carriers, such as holes or electrons. pn junctions, which are interfaces between pand n-type semiconductors, are basic elements of semiconductor electronic devices, such as diodes, transistors, and integrated circuits. Similarly, the injection of carriers is necessary in many superconducting materials. To realize superconductivity, some dopants are added to host materials in many cases. For example, in the typical cuprate superconductor La2-xSrxCuO4, Sr is responsible for injecting holes at the LaO plane, which is the carrier doping layer [1]. The carriers generated at the LaO plane are transferred to the superconducting layer CuO2 plane. On the other hand, the electrondoped copper oxide superconductor Nd2-xCexCuO4 was developed later [2].
Perovskite materials exhibit a variety of functions, such as ferroelectricity or superconductivity. As a catalyst, perovskite oxides have long been known to exhibit gas-phase oxidation reactions. Therefore, they have been expected to have use in removing NOx and CO in automobile exhaust gas [3]. Moreover, their use in photocatalysis has also been studied [4]. Dopants have been used to generate carriers, and in 2000 a water-splitting reaction of more than 50% was demonstrated in La-doped NaTaO3. Many other such systems have been found, and the search for novel substances is underway. In particular, for Ca, Sr, and La-doped in KTaO3, the dopant sites have been determined by X-ray absorption ﬁne structure (XAFS) and X-ray ﬂuorescence holography (XFH). As a result, it was found that they characteristically occupy multiple sites, including substitutions in both A and B sites [5, 6]. It was reported that the electron-hole recombination can be controlled by dual-site doping [7].
Scintillators, which are implemented in radiation detectors, often require dopants because they need luminescent centers for visible light. For example, in a NaI singlecrystal scintillator, doped Tl+ becomes a luminescent center [8]. Here, Na and I atoms interact with X-rays or γ-rays, generating secondary electrons. The generated secondary electrons spread in a matrix, and Tl+ defects catch them and are subsequently excited. Tl+ luminescent centers emit light owing to the 6sp → 6s2 transition. Since the decay time of scintillation is as short as <1 ns, scintillation counters make single-photon detection possible. Many single crystals are also doped with rare-earths as luminescent centers, typically CaF2:Eu [9] and Lu2SiO5:Ce [10] which use the transitions of Eu3+ 5d → 4f and Ce3+ 5d → 4f, respectively. For neutron detectors, in addition to the luminescent center, element (nucleus) capturing neutrons are necessary. Li (6Li) is a good absorber of neutrons, and it emits α-rays, which excite luminescent centers. Ce-doped LiCaAlF6 is one of the typical neutron scintillation counters [11].
Among various magnetic compounds, rare-earth magnets are strong permanent ones made from alloys of rare-earth elements. They have a magnetic force 10 times greater than that of ferrite magnets. Samarium–cobalt (SmCo) magnets are typical rare-earth magnets, and then perform well as magnets. However, since Sm and Co are costly, an alternative rare-earth magnet had been sought in industries, and Nd–Fe–B magnets were developed eventually. In these magnets, the addition of B can provide signiﬁcant hard magnetic properties to ribbon alloys that are superquenched to a microcrystalline state [12, 13]. In addition, the Curie temperature can be increased by adding Dy [14]. Such magnets are used in motors for hybrid or electronic vehicles.

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Doping is also used to decrease thermal conductivity. For example, thermoelectric materials require a high Seebeck coefﬁcient and low thermal conductivity. Although the Heusler-type thermoelectric material Fe2VAl has a high Seebeck coefﬁcient, its low thermal conductivity has been problematic. For this reason, attempts have been made to reduce thermal conductivity by doping heavy elements [15]. In Ta-doped Fe2VAl, the local structure around Ta was analyzed in detail by XFH and XAFS. As a result, Ta was found to be substituted for the V site, and the XFH measurements showed that Ta was tightly bound to the surrounding Fe. In addition, the inelastic X-ray scattering was measured since the thermal conduction is essentially transferred by phonons. In this case, a low-energy optical mode identiﬁed on the basis of the presence of Ta arises and crosses with the acoustic modes of Fe2VAl, which contribute to heat transfer. The coherence of phonons should contribute to the enhancement of thermal insulation [16]. Inelastic X-ray scattering is introduced in Chap. 6.
Most magnets are composed of metals. The history of metallic materials is so old, dating back to the period BC. The most widely used structural material is steel, which is Fe with < 2% C added. Various other trace elements are also added to improve mechanical properties, workability, and corrosion resistance. On the other hand, Ti alloys are representative light alloys. Pure Ti has a hexagonal closed-pack (HCP) structure at room temperature; however, owing to its poor workability, β-Ti, which has a body-centered cubic (BCC) structure, is often used in industry. Here, elements such as Nb, Mo, and V, which are β-stabilizers, are added at about 10%. On the other hand, FeCo alloys [17] have attracted attention as excellent soft magnetic materials because they exhibit high saturation magnetization, low coercive force, high permeability, and high Curie temperature, which are useful for solenoid cores, injector cores, and sensors. Fe0.08Co0.92 is also used as a neutron polarizer. Its crystal structure is face-centered cubic (FCC). The structures of pure Fe and Co are BCC and HCP, respectively, at room temperature. However, by adding 8% Fe, FeCo alloy becomes FCC. In this case, Fe acts as an FCC stabilizer [18].
Like metallic materials, the history of glass dates back to the period BC. The main raw material, silica, has a high melting point (1713 °C) [19] and requires large amount of fuel such as wood, which has been a problem. Therefore, the technique of lowering the melting point (1000 °C) by mixing silica with soda (sodium carbonate) has been used since ancient times. This is known as “soda glass.” In addition, because of the history of glass making as a craft, dopants have been used to color glasses. For example, when iron (II) oxide is doped to glass, the glass becomes blue-green. Addition of chromium gives a darker green color. Dopants are also needed to produce optical ﬁbers, which are important in the infrastructure of modern information society. The primary material is SiO2, which exhibits the highest light transmittance. Regarding the structure of an optical ﬁber, the core is covered by a cladding, and light propagates accompanied by total reﬂection at the interfaces between the core and the cladding. This total reﬂection phenomenon is caused by the difference in the refractive indexes between the core and the cladding, and this is controlled by dopants such as Ge and F. However, these dopants increase transmission loss at the same time, and thus, improvements of optical ﬁbers are underway.

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Such improvements are important for submarine cables that require transmission over long distances.
In this section, we discussed the functionalization of various materials by doping. Similarly, the control of vacancies is an important issue for carrier generation in semiconductors and other materials. For example, the semiconductor compound CdTe, which is also used in solar cells, is usually of the p-type, because cadmium defects tend to occur [20]. Such vacancies are controlled by changing the synthesis conditions, such as the amount of Cd and Te prepared.

1.2 Beyond the Limitation of Point Defects
As mentioned in Sect. 1.1, doping is an essential process in imparting functionality to materials. However, the demand for materials used in cutting-edge devices is increasing yearly, and even higher functionality is required. For example, with innovations in communication technologies such as 5G and 6G, high-speed processing of large amounts of data and the accompanying increase in memory capacity are constant development goals in computers and smartphones. Moore’s law, which indicates the annual rate of increase in the number of transistors per integrated circuit, appears to still hold, although it is now slowing down [21]. The key to maintaining Moore’s law is circuit stacking and device miniaturization technologies. With wiring widths reaching several nm, hence, the next goal is to develop high-carrier-density semiconductors that can be driven by ever smaller transistors.
Here, it is necessary to introduce more carriers with high doping concentration and to reach the required current, However, these targets face the problem of the solid solution limit. For example, when As is highly doped into Si, not all As atoms are located at the substitution site. In this case, many As atoms form As4V or As2V clusters (V is Si vacancy) (Fig. 1.1), which are inactive clusters and do not contribute to carrier injection. Such clusters composed of As and vacancies have been observed by electron microscopy and photoelectron holography [22, 23]. However, these inactive clusters can be changed to active clusters. Tsutsui and Morikawa proposed that energetically stable As2B is formed by additionally doping B into the vacancy part of As2V [24]. Such a defect complex can be regarded as a new element that is not on the periodic table. This is just a “hyperordered structure,” which is the subject of this book.
In the previous section, soda glass is mentioned, where soda refers to sodium hydroxide NaOH. When sodium carbonate is added to silicon dioxide, the melting point drops from 1,723 °C to nearly 1,000 °C, which facilitates the processing. However, the ionic mobility of Na+ is rather high, and when the Na+ contacts with water, sodium silicate is formed, resulting in extremely poor corrosion protection. This is an example of the limitation of single-element doping, similarly to the case of As-doped Si.
Such poor chemical durability can be further improved by introducing another alkali ion, such as potassium. The improvement is due to the well-known “mixed

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Fig. 1.1 “Active” As2B cluster in Si
alkali effect,” and this method of improving is one of the most important methods for producing inexpensive and durable glass in practical processing. The reason for the improved chemical durability is the signiﬁcant decrease in the mobility of the doped alkali ions after mixing different alkali ions. Although the reason for this decrease is as yet not well understood, it is getting clearer with the use of X-ray and neutron diffraction measurements, as well as the reverse Monte Carlo methods [25] and molecular dynamic (MD) simulation, which can provide a model of atomic arrangements. For example, Onodera et al. studied the silica glass mixed with Na and K, and found that Na+ and K+ formed a tight pair (Fig. 1.2), which apparently inhibited the migration of the alkali ions. A topological analysis using persistent homology showed a signiﬁcant reduction in the number of large cavities as a result of the mixed alkali effect [26]. This pair of dissimilar alkali ions is another defect complex, a “hyperordered structure” that confers signiﬁcant stability in soda glasses. The mixed alkali glasses are also discussed in Chap. 15.
1.3 Dopant–Vacancy Pairs
Many defect complexes composed of dopants and vacancies have been studied for a long time. For example, the diffusion of dopants (impurity atoms) in a solid involves vacancies. Thermodynamically, there are no crystals without vacancies at ﬁnite temperatures, and the concentration of vacancies increases with temperature.

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Fig. 1.2 Na+–K+ pair in sodium potassium silica glass
Vacancies can form strain ﬁelds around them or trap electrons and produce Coulomb ﬁelds [27]. Therefore, vacancies are easily paired with dopants (impurity atoms). For example, dopants at substitution sites in silicon crystals have strong covalent bonds with four adjacent silicon atoms and cannot move to neighboring lattice points in principle. However, this situation changes markedly when a Si vacancy reaches an adjacent lattice site. Vacancies and dopants are swapped, and dopants can move from their original lattice positions. Thus, vacancies play a major role in the diffusion of dopants, and here the combination of vacancies and dopant defects is an essential process.
Historically, defect complexes have long been studied for semiconductors such as Si and Ge [28]. For example, when a silicon crystal is grown by the Czochralski method, oxygen is present at about 1018 atoms/cm−3. Oxygen contamination is inevitable because of the use of a quartz crucible and sometimes results in the formation of defect complexes. It has been shown by infrared spectroscopy measurements that defect complexes called A-centers, which are optically active, are promoted when samples are irradiated with electron beams or other radiations [29]. This would be a defect complex consisting of oxygen and vacancies. It is believed that oxygen, which normally locates at the interstitial sites, breaks the bonds between Si atoms upon irradiation and goes to Si sites, followed by the formation of adjacent Si vacancies. Electrons are trapped in these vacancies, yielding a spin resonance spectrum [30]. Electron spin resonance measurements revealed the existence of oxygen–vacancy defect complexes. Active defect complexes have been observed in other systems, such as

1 From Point Defects to Defect Complexes

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phosphorous-doped ﬂoating-zone Si, where the so-called E-centers (phosphorous– vacancy complexes) are induced by electron irradiation [31]. These defect complexes can form voids when they aggregate. For Si, cathodoluminescence has also been widely used to evaluate defects [32].
A well-known example of a dopant–vacancy defect complex is the N–vacancy pair in diamond [33]. Vacancies are formed in the proximity of the N–substituted site for C, where lone pairs of nitrogen are distributed. A lone pair prevents carbon from going to the vacancies, forming a defect complex in diamond called the NV center, which is a pair of N and a vacancy. This lone pair has an electron spin with the spin quantum number S = 1. The NV center has outstanding spin coherence in solids [34], and its ability to access a single spin by light is expected to lead to applications such as high-sensitivity quantum magnetic sensors and quantum information devices. While many types of quantum bits require cooling, the hardness and wide band gap of diamond allow the NV center to retain its quantum state even at room temperature. The NV center has excellent potential in that, in principle, its magnetic sensitivity comparable to that of a superconducting quantum interference device (SQUID) can be expected at room temperature.

1.4 Defect Complexes with Sub-Nanometer Scale
In this section, defect complexes with sub-nanometer scales are shown. The ﬁrst example is Co:TiO2, a ferromagnetic semiconductor [35, 36]. Ferromagnetic semiconductors are materials that possess the properties of both magnets and semiconductors. The Curie temperatures Tc of many ferromagnetic semiconductor materials are lower than room temperature. For example, Mn:GaAs, a prototypical ferromagnetic semiconductor, has a Tc of about 110–190 K [37]. If ferromagnetic semiconductors are to be implemented in devices, their Tc must be above room temperature, so there has been a race to improve their Tc of such ferromagnetic semiconductors.
In this context, Co:TiO2 is an excellent material that exhibits a Tc of about 600 K. The state of the doped cobalt was of interest. Hayashi et al. measured XFH hologram and XAFS of Co:TiO2 with a rutile structure containing 5% Co. Detailed analysis of the atomic images reconstructed from the hologram and the XAFS signals obtained revealed the presence of CoO2Ti4 clusters (Fig. 1.3). It has also been found that Co atoms occupy Ti sites when the concentration of Co is low (~1%). However, density functional theory (DFT) calculations have shown that the single CoO2Ti4 clusters cannot exist stably in rutile TiO2, and that they can exist when two or more clusters are linked together [38]. The CoO2Ti4-like structure is called a suboxide, which is an oxide with a small number of oxygen atoms, and is considered an intermediate in the reaction process that forms ordinary oxides. It is surprising that such nanostructures formed around magnetic elements may be associated with high-temperature ferromagnetism.
The popular light metals are Al and Ti alloys, which are widely used as a material for vehicle bodies, where lightness and rigidity are required. However, the density

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Fig. 1.3 CoO2Ti4 cluster in rutile TiO2. Reproduced from Ref. [38] with reprint permission
of Mg is smaller than that of Ti alloys, and has been expected to be applied to frames for automobiles and airplanes from the viewpoint of the energy conservation in transportation. However, Mg has problems such as its high ﬂammability and poor workability. Therefore, Kawamura et al. have been developing Mg alloys by doping transition metals and rare-earth metals to Mg to overcome its disadvantages. The developed metals are called long-period stacking ordered (LPSO) Mg alloys, which have novel structural features, and proposed a new concept of material reinforcement mechanism called “kink-zone reinforcement” [39].
In the LPSO phase, layers composed of added heavy metals are formed in a matrix of Mg, and they are arranged in periodic alley. In addition, in a hexagonal Mg matrix, stacking faults occur in the heavy-metal-enriched layer, which causes the appearance of cubic sites. In these enriched layers, peculiar clusters are formed by the codoped metal elements. For example, in Mg75Zn10Y15, where Zn and Y are co-doped, the principal FCC unit-like clusters such as Zn6Y8 are formed as shown in Fig. 1.4 [40, 41]. Similar clusters have been observed in other types of LPSO Mg alloy. Also, inelastic X-ray scattering study [42] revealed characteristic vibrational modes of ZnY clusters relating to microscopic elastic properties, which are described in Chap. 6. In the observations of microstructures of cast extrusions, the LPSO phase has kink deformation, which is the origin of excellent mechanical properties. For this reason, the Zn6Y8 cluster can be viewed as a “hyperordered structure” that improves its mechanical properties with respect to Mg, which is attracting attention as a light metal material.
Such metal clusters are also found in proteins. A typical example is photosystem II (PSII), which is involved in plant photosynthesis. Oxygen-evolving photosynthesis uses sunlight to split the water molecule to release oxygen and synthesize organic

1 From Point Defects to Defect Complexes

11

Fig. 1.4 Zn6Y8 cluster in the LPSO Mg alloy. Reproduced from Ref. [40] with reprint permission
matter from carbon dioxide, providing energy source necessary for the survival of life on earth. The reactions involved in light absorption, electron transfer, water splitting, and oxygen evolution are catalyzed by a series of membrane protein complexes. PSII is responsible for the production of O2, protons, and electrons from the oxidation of water by light energy. Here, electrons are derived from H2O through the catalysis of Mn4CaO5 clusters in PSII (Fig. 1.5), and a water splitting reaction occurs. The Mn4CaO5 cluster catalyzes the water splitting in a reaction cycle called the S-state transition. The structure of a Mn4CaO5 cluster was determined by synchrotron Xray diffraction to be a distorted chair-shaped one [43]. The reaction cycle called Kok cycle has ﬁve stages from S0 to S4, with oxidation progressing as the stages proceed [44]. In this cycle, a total of four electrons are withdrawn from two water molecules, resulting in the generation of O2. As can be seen from this unique photochemical reaction, Mn4CaO5 clusters can be regarded as “hyperordered structures” that play an important role in photosynthesis. No artiﬁcial catalyst has such a complex process, and we have much to learn from the diverse functions of living organisms. Chap. 16.
On the other hand, the most reduced Mn4CaO5 cluster is that in the S0 state, but the cluster in the S1 state is the most stable, where two Mn ions are trivalent and the other two are tetravalent. Since the valence of Mn in PSII is easily reduced by X-rays, there had been ambiguity in the determination of the Mn3+ and Mn4+ conﬁgurations. However, these conﬁgurations in the S1 state have been determined by structural evaluation by damage-free X-ray free electron laser (XFEL) diffraction and associated theoretical calculations [45]. The S2 and S3 states were then similarly evaluated by XFEL diffraction [46]. However, structural characterization by X-ray diffraction cannot directly evaluate the valence of Mn. Therefore, attempts have been recently made to determine the Mn3+ and Mn4+ conﬁgurations more directly using valence-selective X-ray ﬂuorescence holography [47].

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Fig. 1.5 Mn4CaO5 cluster in photosystem II. Green, purple, and red balls correspond to Ca, Mn, and O atoms
1.4.1 Toward Understanding Defect Complexes Based on Hyperordered Structures
In the ﬁrst half of this chapter, we described the roles of dopants in various materials by reviewing a number of papers on the basis as the origin of creating functions. One of the major roles of dopants is the generation of carriers such as electrons and holes in semiconductors and superconductors. In scintillators, dopants form impurity levels and are responsible for visible light emission. Dopants in metallic materials are effective in controlling the microstructure and crystal structure, which in turn leads to the workability and reinforcement of the metallic materials. Magnetic properties, such as magnetic force and Curie temperature can be improved by applying dopants. For thermoelectric materials, dopants have been used to reduce thermal conductivity and improve thermoelectric efﬁciency. As for glass, dopants were used to lower its melting point and control its refractive index.
Dopants are often paired with vacancies, as in the case of diffusion of dopants. In Si, oxygen and vacancies form optically active A-sites. The NV center in diamond can maintain its quantum state at room temperature and is therefore expected to be

1 From Point Defects to Defect Complexes

13

applied to quantum bits. On the other hand, there is a limit of the number of carriers in semiconductors owing to the solubility limit. In this case, dopants easily aggregate with vacancies, and form defect complexes such as As2V. However, it has been proposed that active defect complexes such as As2B can be created if B is inserted into vacancies. The Na–K pairing in alkali glass also achieves both melting point reduction and corrosion resistance. Finally, in this section, we present sub-nanoscale defect complex clusters that have actually been observed. The CoO2Ti4 cluster in Co:TiO2 may be the origin of ferromagnetism. The Zn6Y8 clusters in LPSO Mg alloys involve a new reinforcing mechanism: kink-band reinforcement. Metal clusters in proteins enable complex catalytic reaction, that is, they are the ideal “hyperordered structures.”
The point defects and defect complexes introduced here can hardly be measured by conventional X-ray and neutron diffraction methods. Therefore, element-selective local structure analysis techniques such as atomic resolution holography, anomalous X-ray scattering, and XAFS are required. Particularly for defect complexes with sub-nanometer scales, the application of atomic resolution holography and anomalous X-ray scattering, which can be used for middle-range local structure analysis, is effective. Dopants can be also observed using transmission electron microscopy, but the dopant elements must be heavier than the matrix elements. The angstrom beam electron diffraction method developed by Hirata et al. can be applied to amorphous materials and has been successfully applicable to discovering unique nanoscale structures [48]. These characterization methods are presented in Part 2 of this book.
The role of the observed hyperordered structure should be clariﬁed using ﬁrstprinciples calculations. Large-scale ﬁrst-principles calculations are required here because large numbers of atoms must be handled in a simulation box that spans several nm. The linear scaling density functional theory (DFT) CONQUST [49], which can handle about 100,000 atoms, is described in Chap. 3. The development of synthetic processes will also be an important endeavor in the creation of hyperordered structures. Although such structures can be developed by chance in some cases, it is essential to be able to control the system at will, for which the use of machine learning will become important in the future.

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S, Stellhorn JR, Hosokawa S, Matsushita T, Tajiri H, Ang AKR, Nishino Y (2020) Phys Rev 101:024302 17. Sundar RS, Deevi SC (2005) Int Mater Rev 50:157 18. Su X (2003) J Mater Sci 3:4581 19. Haynes WM (ed) (2014) Handbook of chemistry and physics, 95th ed. CRC Press, Boca Raton, FL, pp 4–88 20. Shepidchenko A, Sanyal B, Klintenberg M, Mirbt S (2015) Sci Rep 5:14509 21. https://en.wikipedia.org/wiki/Transistor_count 22. Oshima Y, Hashimoto Y, Tanishiro Y, Takayanagi Y, Sawada H, Kaneyama T, Kondo Y (2010) Phys Rev B 81:035317 23. Tsutsui K, Matsushita T, Natori K, Muro T, Morikawa Y, Hoshi T, Kakushima K, Wakabayashi H, Hayashi K, Matsui F, Kinoshita T (2017) Nano Lett 17:7533 24. Tsutsui K, Morikawa Y (2020) Jpn J Appl Phys 59:010503 25. McGreevy RL, Puztai L (1988) Mol Simul 1:359 26. Onodera Y, Takimoto Y, Hijiya H, Taniguchi T, Urata S, Inaba S, Fujita S, Obayashi I, Hiraoka Y, Kohara S (2019) NPG Asia Mater 11:75 27. Fahey PM, Grifﬁn PB, Plummer D (1989) Rev Mod Phys 61:289 28. Corbett JW, Watkins GD, Chrenko RM, Mc’Donald RS (1961) Phys Rev 121:1015 29. Wean RE (1976) Phys Rev 13:1653 30. Lee Y-H, Corbctt JW (1976) Phys Rev B 13:2653 31. Watkins GD, Corbett JW (1964) Phys Rev 134:A1359 32. Sugie R, Matsuda K, Ajioka T, Yoshikawa M, Mizukoshi T, Shibusawa K, Yo S (2006) J Appl Phys 100:064504 33. Loubser JHN, van Wyk JA (1978) Rep Prog Phys 41:201 34. Gruber A, Dräbenstedt A, Tietz C, Fleury L, Wrachtrup J, von Bor-czyskowski C (1997) Science 276:2012 35. Matsumoto Y, Murakami M, Shono T, Hasegawa T, Fukumura T, Kawasaki M, Mhmet P, Chikyow T, Koshihara S, Koinuma H (2001) Science 291:854 36. Yamada Y, Ueno K, Fukumura T, Yuan HT, Shimotani H, Iwasa Y, Gu L, Tsukimoto S, Ikuhara Y, Kawasaki M (2011) Science 332:1065 37. Ohno H, Shen A, Matsukura F, Oiwa A, Endo A, Katsumoto S, Iye Y (1996) Appl Phys Lett 69:363 38. Hu W, Hayashi K, Fukumura T, Akagi K, Tsukada M, Happo N, Hosokawa S, Ohwada K, Takahasi M, Suzuki M, Kawasaki M (2015) Appl Phys Lett 106:222403 39. Kawamura Y, Hayashi K, Inoue A, Masumoto T (2001) Mater Trans 42:1172 40. Egusa D, Abe E (2012) Acta Mater 60:166 41. Nishioka T, Yamamoto Y, Kimura K, Hagihara K, Izuno H, Happo N, Hosokawa S, Abe E, Suzuki M, Matsushita T, Hayashi K (2018) Materialia 3:256 42. Hosokawa S, Kimura K, Stellhorn JR, Yoshida K, Hagihara K, Izuno H, Yamasaki M, Kawamura Y, Mine Y, Takashima K, Uchiyama H, Tsutsui S, Koura A, Shimojo F (2018) Acta Mater 146:273 43. Umena Y, Kawakami K, Shen JR, Kamiya N (2011) Nature 473:55 44. Kok B, Forbush B, McGloin M (1970) Photochem Photobiol 11:457

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49. http://www.order-n.org/

Chapter 2
Topological Order and Hyperorder in Oxide Glasses and Liquids
Shinji Kohara

Abstract The advent of quantum beam sources, which can generate high-ﬂux highenergy neutrons and X-rays, and the development of advanced instruments make it feasible to probe atomic arrangement in disordered materials with high real space resolution [1–5]. A combination of quantum-beam (X-ray and neutron) diffraction (see Chap. 4), theoretical simulations such as density functional theory (DFT) (see Chap. 8) and molecular dynamics (MD) (see Chap. 9), and data-driven structural modeling such as reverse Monte Carlo (RMC, see Chap. 10) modeling [6–8] enables us to study topological order in disordered materials. In this chapter, recent research topics on probing the topological order in oxide glasses (see Chap. 15) and liquids are introduced. Moreover, the application of topological analyses (see Chap. 11) to uncover the hidden topological ordering in the pair correlation is addressed. Finally, we introduce hyperordered glasses and liquids that have been recently discovered to discuss the relationships among diffraction peaks, topological order, and hyperorder in disordered materials.
Keywords Topology · Ring size distribution · Cavity distribution · Homology

2.1 Topological Order−Disorder

The concept of topological order was proposed by Gupta in 1993 [9]. Figure 2.1 shows the schematics [9] and primitive ring size distributions in cristobalite (crystalline SiO2) (a) and silica (SiO2) glass obtained by RMC molecular dynamics (MD) modeling (b) [10, 11]. The schematics are two-dimensional analogs for the cornersharing tetrahedral network in both materials. The ring is deﬁned in Chap. 11. A representative crystalline SiO2, cristobalite, possesses only sixfold rings (i.e., it consists of six SiO4 tetrahedra), whereas silica glass has various ring sizes from threefold to

S. Kohara (B)
Research Center for Advanced Measurement and Characterization, National Institute for
Materials Science, Ibaraki 305-0047, Japan
e-mail: KOHARA.Shinji@nims.go.jp

© Materials Research Society, under exclusive license to Springer Nature Singapore Pte

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Ltd. 2024

K. Hayashi (ed.), Hyperordered Structures in Materials, The Materials Research

Society Series, https://doi.org/10.1007/978-981-99-5235-9_2

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(a)

(b)

1.0

0.3

Fraction Fraction

0.2
0.5
0.1

0.0 0 1 2 3 4 5 6 7 8 9 10 11 12 Ring size n

0.0 0 1 2 3 4 5 6 7 8 9 101112 Ring size n

Fig. 2.1 Schematics of a cristobalite and b silica glass together with primitive ring size distributions (obtained by RMC−MD modeling for the glass) [10]

tenfold rings, although sixfold rings are predominant. Indeed, Gupta characterized that the former is topologically ordered and the latter is topologically disordered [9].

2.2 Topological Disorder in Silica Polymorph
The primitive ring size distributions in crystalline SiO2 with the corner-sharing tetrahedra motif, α-cristobalite (d = 2.327 g cm−3), α-quartz (d = 2.655 g cm−3), coesite (d = 2.905 g cm−3), and silica glass (d = 2.2 g cm−3) by MD−RMC modeling [11] are shown in Fig. 2.2a−d. Intriguingly, high-density SiO2 crystal phases (α-quartz and coesite) are not topologically ordered, although the fraction of sixfold rings in α-quartz is indistinct. Onodera et al. have recently reported inelastic neutron scattering (INS) data of a series of silica polymorphs [12], with which we can discuss the relationship between dynamics and topological order–disorder in silica polymorphs. The details of inelastic neutron scattering are given in Chap. 6. Intermediate-range ordering of SiO2 glass has been discussed in terms of the low-frequency dynamics of a glass network manifested by boson peak [13−15]. Contour maps of the dynamical structure factor S(Q, E) of silica crystals and a series of densiﬁed glasses obtained by INS measurements are shown in Fig. 2.3. All samples show a distinct low-energy band. Note that the densiﬁed glasses were synthesized by hot compression (RT/7.7 GPa (d = 2.24 g cm−3), 400 °C/7.7 GPa (d = 2.54 g cm−3), and 1200 °C/7.7 GPa

2 Topological Order and Hyperorder in Oxide Glasses and Liquids

19

(d = 2.72 g cm−3)) [12]. It is found that peak position in energy shifts to higher energy side with increasing density as reported in ref. 15. Among the three spectra for the crystalline phases, only the S(Q, E) of coesite is very broad and the coesite spectrum appears similar to glass spectra, in which a distinct boson peak is observed at Q ~3 Å−1. This behavior suggests that the dynamics of coesite is rather similar to that of glass even though diffraction data shows prominent Bragg peak probably due to high density. Figure 2.4 shows the dynamical structure factors < S(Q, E) > of a series of silica crystals and densiﬁed silica glasses. RT/7.7 GPa glass shows a distinct boson peak at E of ~5 meV (more details about the boson peak are given in Chap. 6) and the peak shifts toward a large value with increasing density. Crystal phases show similar behavior, but both α-cristobalite and α-quartz show a sharp low-energy peak, whereas coesite shows very broad peaks that stretch toward higher energies. We consider that this behavior may be related to topological disorders in coesite. Moreover, the variation of ring size is a key feature in understanding the dynamics manifested by the boson peak in silica glass.

2.3 Cavity Distribution
Cavity distribution is a very important topological feature of oxide glasses and liquids. The details of the calculation of cavity distributions are described in Refs. 5 and 6. The cavity volume ratio of silica glass is approximately 32% [10, 11] calculated using the pyMolDyn code [16]. Onodera et al. reported the squeezing of a cavity associated with the densiﬁcation by hot compression determined using the MD-RMC models of densiﬁed silica glasses. Figure 2.5 shows the visualization of cavities (green) together with the histograms of cavity size distributions. It is found that a large cavity is transformed into small cavities by densiﬁcation [12].

2.4 Persistent Homology Analysis
Persistent homology analysis is a relatively new topological analysis method. The details of the analysis are given in Chap. 11. This method was developed to capture the shape of rings and cavities on the basis of a persistent diagram (PD); hence, it is very powerful to combine it with conventional ring size distribution and cavity distribution analyses.
Figure 2.2e−h show Si-centric PDs, which provide information on ring shape, of α-cristobalite (d = 2.327 g cm−3), α-quartz (d = 2.655 g cm−3), coesite (d = 2.905 g cm−3), and silica glass (d = 2.2 g cm−3) obtained by MD−RMC modeling [11]. The death values in intense death proﬁles indicated by arrows become small, suggesting that the rings are distorted with the density change from α-cristobalite to coesite (see Fig. 2.2e−g). On the other hand, the Si-centric PD of silica glass shows a distinct proﬁle along with the death axis, suggesting that silica glass includes the

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Fig. 2.2 a−d Primitive ring size distribution in (a) α-cristobalite, b α-quartz, c coesite, and d silica glass. e−h Si-centric PDs of e α-cristobalite, f α-quartz, g coesite, and h silica glass. Reproduced from [11]. CC BY 4.0

homology of a series of silica crystal polymorphs. Thus, the combination of ring size distribution analysis and persistent homology analysis provides us with crucial topological information that was hither to available.

2.5 Hyperordered Oxide Glasses and Liquids
Recently, extremely ordered (hyperordered) glasses and liquids have been discovered. We introduce hyperordered silica glass, alumina (Al2O3) glass, and erbia (Er2O3) liquid to discuss the relationships among diffraction peaks, topological order, and hyperorder.

2.5.1 Densiﬁed Silica Glass
Figure 2.6a shows the in situ neutron structure S(Q) of SiO2 glass under high pressure [18]. The details of the high-pressure technique are described in Chap. 4. The ﬁrst sharp diffraction peak (FSDP) and the principal peak (PP) are observed in the ambient pressure data (black curve) at Q ~1.5 Å–1 and ~3 Å–1, respectively [4]. The formation of FSDP is the result of atomic ordering along with cavities by corner-sharing SiO4 tetrahedra. The origin of the second PP seems to be some type of orientational

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Intensity (arb. unit)

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Fig. 2.3 Contour maps of the dynamical structure factor S(Q, E) of silica crystals and densiﬁed silica glasses. Reproduced from [12]. CC BY 4.0

correlation among oxygen atoms that occupy the corner of the tetrahedra, suggesting that PP reﬂects the packing of oxygen atoms [19]. More details are described in Chap. 4. Upon application of pressure at room temperature, the FSDP shifts to high Q and diminishes. In contrast, the PP becomes very sharp (see Fig. 2.6a) associated with the cavity volume reduction, as illustrated in Fig. 2.5. This extraordinarily sharp PP is a signature of hyperordered orientational correlations formed by oxygen atoms under high pressure.
Figure 2.6b shows the X-ray total structure factors S(Q) of densiﬁed silica glasses obtained by hot compressions (discussed in this chapter). These data are not in situ diffraction data, but the FSDP is the sharpest in the sample recovered at 1200 °C/

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<S(Q,E)>

0 1

0

0

5

10

15

20

25

E (meV)

Fig. 2.4 Dynamical structure factors <S(Q, E)> of a series of silica crystals and densiﬁed glasses. RT/7.7 GPa (black), 400 °C/7.7 GPa (red), 1200 °C/7.7 GPa (blue), α-cristobalite (green), α-quartz (magenta), and coesite (gray). Reproduced from [12]. CC BY 4.0

7.7 GPa. This behavior is very different from in situ neutron diffraction data shown
in Fig. 2.6a. We demonstrate that 1200 °C/7.7 GPa glass is the most ordered silica
glass in the world, the so-called hyperordered silica glass. This behavior is also captured by the Si-centric PDs obtained by the MD−RMC modeling. Figure 2.7a−c show the Si-centric PDs of densiﬁed silica glasses recovered at RT/7.7 GPa (d = 2.24 g cm−3), (400 °C/7.7 GPa (d = 2.54 g cm−3), and 1200 °C/7.7 GPa) (d = 2.72 g cm−3) together with three-dimensional representations of the PDs shown in Fig. 2.7e−g, respectively. The death values of the vertical proﬁle along with the death axis highlighted by rectangles in Fig. 2.2a−c become small after densiﬁcation, which is similar to the change in a series of silica crystal polymorphs shown in Fig. 2.2e−g. Moreover, the three-dimensional representation shows a marked sharpening in the
distribution of multiplicities with increased density. Therefore, we can conclude that
the three-dimensional representation of Si-centric PD is a good indicator of FSDP
of densiﬁed silica glasses.

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RT/7.7 GPa (30.7%)

400 °C /7.7 GPa (11.1%)

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Fig. 2.5 Visualization of cavities in a series of densiﬁed silica glasses (upper) together with the histograms of cavity volume (lower). Reproduced from [12]. CC BY 4.0

(a)
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Death bk (Å2)

Birth bk (Å2)

Fig. 2.7 a−c Si-centric PDs of silica glasses recovered at RT/7.7 GPa a, (400 °C /7.7 GPa b, and 1200 °C /7.7 GPa [12]. d−f three-dimensional representations of the PDs from the boxed regions (magenta) of silica glasses recovered at RT/7.7 GPa a, (400 °C /7.7 GPa b, and 1200 °C /7.7 GPa. Reproduced from [12]. CC BY 4.0

2.5.2 Intermediate Alumina Glass
Silica is a good glass former as mentioned above, whereas alumina is not a glass former and is classiﬁed as an intermediate according to Sun [20] (see Chap. 15) for more details). Indeed, it is impossible to form an aluminum glass by the conventional melt−quench technique. However, Hashimoto et al. have recently reported that the amorphous alumina synthesized by the anodization of aluminum metal exhibits glass transition [10]. Figure 2.8 shows neutron (a) and X-ray (b) total structure factors S(Q) of silica and alumina glasses. Silica glass shows a distinct FSDP because of its lower glass-forming ability. In contrast, alumina glass shows an extraordinarily sharp PP in the neutron S(Q) data similar to in situ high pressure data of silica glass shown in Fig. 2.6a, which suggests that the packing density of oxygen atoms is very high in alumina glass. We suggest that the alumina glass is a hyperordered glass. Figure 2.9 illustrates the atomic arrangements of alumina glass in stick representation (a) and with cavity visualization (b). We can see the lattice (crystal)-like atomic arrangement formed by the edge-sharing of AlOn polyhedra highlighted by black dotted lines. In addition, we can also recognize many more sparse regions formed by the tetrahedral corner-sharing motif in Fig. 9a. The cavity volume ratio of alumina glass is 4.5%, which is comparable to those of densiﬁed silica glasses recovered at 1200 °C/7.7 GPa. The average Al−O coordination number is 4.7, which is much higher than 4 in

2 Topological Order and Hyperorder in Oxide Glasses and Liquids

25

silica glass and the formation of AlO4, AlO5, and AlO6 is conﬁrmed. This variation of Al−O coordination is the reason for the formation of edge-sharing Al−O polyhedra,
which can disturb the evolution of intermediate-range ordering detected as an FSDP.

(a)
FSDP PP 5
4

(b)
FSDP PP 5
Silica glass 4

Silica glass

S (Q) S (Q)

3

3

Alumina glass

Alumina glass

2

2

1

1

0

5

10

15

20

0

5

10

15

20

Q (Å 1)

Q (Å 1)

Fig. 2.8 Neutron a and X-ray b total structure factors S(Q) of silica and alumina glasses. Reproduced from [10]. CC BY 4.0

(a)

(b)

Al

O

Fig. 2.9 Atomic arrangements of alumina glass in stick representation a and with cavity visualization (highlighted in green) b. Pink and red circles represent Si and O atoms, respectively. Reproduced from [10]. CC BY 4.0

26
2.5.3 Levitated Erbia Liquid

S. Kohara

Erbia (Er2O3) is a nonglass-forming material and its melting point is extremely high (T m = 2430 °C). To perform diffraction measurements of such a high-temperature liquid, we have developed several containerless techniques, which allow us to hold a liquid droplet without any container. The variation of the levitation technique is described in Chap. 7. The details of the combination of levitation techniques and diffraction measurements proposed by Price are reviewed in ref 21. The levitation technique has recently been in the investigation of structures of single-component oxide liquids, Al2O3 [22], Y2O3 [23], Ho2O3 [23], La2O3 [23], ZrO2 [23], UO2 [24], TiO2 [25], B2O3 [26], Lu2O3 [27], and Yb2O3 [27]. Here, we employed the aerodynamics levitation technique for X-ray diffraction measurement of liquid erubia, with which a sample is levitated using an inert gas from a conical nozzle [21].
Figure 2.10 shows X-ray structure factors S(Q) of erbia liquid (2650 °C) [28] (a) and zirconia (ZrO2) liquid (2800 °C) [29] (b) together with those obtained using several simulation techniques. We cannot observe any FSDP in both data sets because they are nonglass-forming materials. However, they exhibit a PP at Q ~2 Å−1. The FWHMs of PP for zirconia and erbia liquids are 0.7669 and 0.4299, respectively. In the case of zirconia liquid, the RMC−density functional (DF)/MD model of 501 particles (magenta curve) reproduces experimental data. However, we need the RMC−MD model of 5000 particles (red curve) to reproduce the extraordinarily sharp PP for erbia liquid. As a benchmark, we reduced the number of particles in the standard RMC approach and conﬁrmed that 500 particles (blue) are insufﬁcient to reproduce such an extraordinarily sharp PP. We consider that erbia liquid is an unusual hyperordered oxide liquid.

S(Q) S(Q)

(a)
3 2 1 0 -1
0

FWHM 0.4299

Experimental data
5000 particles RMC model
3000 particles 1000 particles 500 particles 250 particles

(b)
2
1

3

2

1

0

0

-1 1.0 2.0 3.0

5

10

15

0

Q (Å 1)

FWHM 0.7669

Experimental data 501 particles

5

10

15

Q (Å 1)

Fig. 2.10 X-ray structure factors S(Q) of erbia liquid (2650 °C) [28] a and zirconia (ZrO2) liquid (2800 °C) [29] b together with those obtained by RMC−MD, RMC, and RMC−DF/MD modeling.
Reproduced from [28]. CC BY 4.0

2 Topological Order and Hyperorder in Oxide Glasses and Liquids

27

Fig. 2.11 Atomic arrangement of erbia liquid in stick representation. Reproduced from [28]. CC BY 4.0
The cation−oxygen coordination numbers are approximately 6 for both liquids, which is extremely larger than 3.9 for SiO2 (2000 °C) liquid and 4.4 for Al2O3 liquid (2127 °C). Moreover, the oxygen−cation coordination number is 3.0 for zirconia liquid and 4.1 for erbia liquid, suggesting that a large fraction of an OEr4 tetracluster, which cannot be observed in other liquids, is observed.
Figure 2.11 illustrates the atomic arrangement of erbia liquid in stick representation. We can see extraordinarily densely packed atomic arrangement, which is found in alumina glass, but we can hardly see the sparse region, which is observed in alumina glass. This extraordinarily densely packed atomic arrangement highlighted by dotted lines is formed by the edge-sharing ErOn polyhedra associated with the formation of OEr4 tetraclusters and is the origin of extremely low glass-forming ability and the extraordinarily sharp PP.
2.6 Concluding Remarks
In this chapter, we introduced the concept of topological order and disorder proposed by Gupta [9]. He proposed on the basis of ring statistics. We applied this concept to silica polymorphs and also compared their data with inelastic neutron scattering data

28

S. Kohara

to understand the origin of topological disorder, which can be important information to understand the boson peak.
We have also introduced the analytical methods for cavity distributions and persistent homology because they are very powerful in combination with conventional ring size distribution analyses.
We have also reviewed several unusual structures of oxide glasses and liquids in terms of “hyperorder.” Their structures are very different from those of conventional oxide glasses and liquids. The characteristic features of these glasses and liquids are the variations of polyhedra in terms of coordination number, polyhedral connections (corner, edge, and face), ring size, and ring shape in terms of homology. These features should as a whole be called “topological disorder,” which is related to the formation of hyperorder in oxide glasses and liquids.

Acknowledgements This work was supported by a JSPS Grant-in-Aid for Transformative Research Areas (A) “Hyper-Ordered Structures Science”: Grants No. 20H05878 and No. 20H05881.

References
1. Kohara S, Salmon PS (2016) Adv Phys: X 1:640 2. Kohara S, Akola J (2021) In World scientiﬁc series in nanoscience and nanotechnology
advanced characterization of nanostructured materials, Ed by Sinha SK, Sanyal MK, Loong CK (World Scientiﬁc Co. Pte. Ltd., Singapore), pp 247−305 3. Ohara K, Onodera Y, Murakami M, Kohara S (2021) J Phys: Condens Matter 33:383001 4. Kohara S (2022) J Ceram Soc Jpn 130:531 5. Benmore CJ (2023) Comprehensive inorganic chemistry III (Third Edition)” ed by Reedijk J, Poeppelmeier KR (Elsevier, Amsterdam), pp 384−424 6. McGreevy RL, Pusztai L (1988) Molec Simul 1:359 7. McGreevy RL (2001) J Phys: Condens Matter 13:R877 8. Kohara S, Pusztai L (2022) Atomistic simulations of glasses: fundamentals and applications, ed by Du J, Cormack AN (Wiley-American Ceramic Society, Hoboken), pp 60−88 9. Gupta PK (1993) J Am Ceram Soc 76:1088 10. Hashimoto H, Onodera Y, Tahara S, Kohara S, Yazawa K, Segawa H, Murakami M, Ohara K (2022) Sci Rep 12:516 11. Onodera Y, Kohara S, Tahara S, Masuno A, Inoue H, Shiga M, Hirata A, Tsuchiya K, Hiraoka Y, Obayashi I, Ohara K, Mizuno A, Sakata O (2019) J Ceram Soc Jpn 127:853 12. Onodera Y, Kohara S, Salmon PS, Hirata A, Nishiyama N, Kitani S, Zeidler A, Shiga M, Masuno A, Inoue H, Tahara S, Polidori A, Fischer HE, Mori T, Kojima S, Kawaji H, Kolesnikov AI, Stone MB, Tucker MG, McDonnell MT, Hannon AC, Hiraoka Y, Obayashi I, Nakamura T, Akola J, Fujii Y, Ohara K, Taniguchi T, Sakata O (2020) NPG Asia Mater. 12:85 13. Sokolov AP, Kisliuk A, Soltwisch M, Quitmann D (1992) Phys Rev Lett 69:1540 14. Sugai S, Onodera A (1996) Phys Rev Lett 77:4210 15. Inamura Y, Arai M, Nakamura M, Otomo T, Kitamura N, Bennington SM, Hannon AC, Buchenau U, Non-Cryst J (2001) Solids 293–295:389–393 16. Heimbach I, Rhiem F, Beule F, Knodt D, Heinen J, Jones RO (2017) J Comput Chem 38:389 17. Wille TF, Rycroft CH, Kazi M, Meza JC, Haranczyk M (2012) Microporous Mesoporous Mater 149:134

2 Topological Order and Hyperorder in Oxide Glasses and Liquids

29

18. Zeidler A, Wezka K, Rowlands RF, Whittaker DAJ, Salmon PS, Polidori A, Drewitt JWE, Klotz S, Fischer HE, Wilding MC, Bull CL, Tucker MG, Wilson M (2014) Phys Rev Lett 113:135501
19. Salmon PS, Zeidler A (2019) J Stat Mech Theory E 2019:114006 20. Sun K-H (1947) J Am Ceram Soc 30:277 21. Price DL (2010) In High-temperature levitated materials (Cambridge University Press,
Cambridge), pp 2−19 22. Skinner LB, Barnes AC, Salmon PS, Hennet L, Fischer HE, Benmore CJ, Kohara S, Weber
JKR, Bytchkov A, Wilding MC, Parise JB, Farmer TO, Pozdnyakova I, Tumber SK, Ohara K (2013) Phys Rev B 87:024201 23. Skinner LB, Benmore CJ, Weber JKR, Du J, Neuefeind J, Tumber SK, Parise JB (2014) Phys Rev Lett 112:157801 24. Skinner LB, Benmore CJ, Weber JKR, Williamson MA, Tamalonis A, Hebden A, Wiencek T, Alderman OLG, Guthrie M, Leibowitz L, Parise JB (2014) Science 346:984987 25. Alderman OLG, Skinner LB, Benmore CJ, Tamalonis A, Weber JKR (2014) Phys Rev B 90:094204 26. Alderman OLG, Ferlat G, Baroni A, Salanne M, Micoulaut M, Benmore CJ, Lin A, Tamalonis A, Weber JKR (2015) J Phys: Condens Matter 27:455104 27. Pavlik A III, Ushakov SV, Navrotsky A, Benmore CJ, Weber JKR (2017) J Nucl Mater 495:385 28. Koyama C, Kohara S, Onodera Y, Småbråten DR, Selbach SM, Akola J, Ishikawa T, Masuno A, Mizuno A, Okada JT, Watanabe Y, Nakata Y, Ohara K, Tamaru H, Oda H, Obayashi I, Hiraoka Y, Sakata O (2020) NPG Asia Mater. 12:43 29. Kohara S, Akola J, Patrikeev L, Ropo M, Ohara K, Itou M, Fujiwara A, Yahiro J, Okada JT, Ishikawa T, Mizuno A, Masuno A, Watanabe Y, Usuki T (2014) Nat Commun 5:5892

Part II
Characterization of Hyperordered Structures

Chapter 3
Atomic-Resolution Holography
Tomohiro Matsushita, Koji Kimura, and Kenji Ohoyama

Abstract This chapter describes details of atomic-resolution holography, which gives a three-dimensional atomic image around a target atomic site. It can visualize the atomic structures of an impurity, a dopant, an adsorbate on a crystal, and a thin-ﬁlm interface as well as the positional ﬂuctuation of a target atom. These atomic structures cannot be measured by conventional measurement methods, such as X-ray diffraction (XRD), X-ray absorption ﬁne structure (XAFS), and electron microscopy. There are several types of atomic-resolution holography: X-ray ﬂuorescence holography, neutron holography, photoelectron holography, and inverse photoelectron holography. The principle and apparatus of these techniques are introduced together with their features. Some excellent examples of hyperordered structures embedded in crystals are presented, such as the Co suboxidic structure in TiO.2, the Eu dopant structure in CaF.2, the P dopant structure in the diamond, the defect structure between an insulating layer and a diamond semiconductor, and so forth. Section 3.1 gives a brief introduction to atomic-resolution holography. Section 3.2 describes its principle in detail, especially, the recording processes of holograms and the reconstruction algorithms. Individual holographic techniques are introduced in Sects. 3.4, 3.5, 3.6, and 3.7. Finally, the features of these techniques are summarized in Sect. 3.8.
Keywords Atomic-resolution holography · X-ray ﬂuorescence holography · Neutron holography · Photoelectron holography · Inverse photoelectron holography · Three-dimensional atomic image

T. Matsushita (B)
Graduate School of Science and Technology, Nara Institute of Science and Technology, Ikoma, Nara 630-0192, Japan e-mail: t-matusita@ms.naist.jp
K. Kimura Department of Physical Science and Engineering, Nagoya Institute of Technology, Nagoya 466-8555, Japan
K. Ohoyama Graduate School of Science and Engineering, Ibaraki University, Hitachi 316-8511, Japan

© Materials Research Society, under exclusive license to Springer Nature Singapore Pte

33

Ltd. 2024

K. Hayashi (ed.), Hyperordered Structures in Materials, The Materials Research

Society Series, https://doi.org/10.1007/978-981-99-5235-9_3

34
3.1 Introduction

T. Matsushita et al.

Holography is a method for generating a three-dimensional (3D) image from an interference pattern of waves recorded on a physical medium. Hungarian physicist Gabor invented holography in 1948, for which he received the Nobel Prize in Physics in 1971 [1]. Although his study was intended to improve the resolution of an electron microscope by using the interference of electron waves, his proposed principle of holography is applicable to any type of wave. In particular, optical holography is used worldwide in applications such as the anti-counterfeiting of banknotes. The principle of holography has been applied to the 3D reconstruction of atomic-scale images; this is called atomic-resolution holography. In 1986, Szöke presented the basic principle of atomic-resolution holography to record atomic arrangements [2]. In this study, it was proposed that a 3D atomic arrangement can be reconstructed from a holographic oscillation contained in electron waves or electro-magnetic waves emitted from atoms (e.g., photoelectrons and ﬂuorescent X-rays). In 1988, Barton presented a Fourier transform method for reconstructing an atomic image from an atomic-resolution hologram [3, 4].
Figure 3.1 shows a schematic image of the atomic-resolution holography (as noted below, this is the normal-mode measurement method). When a material is irradiated with X-rays, photoelectrons (or ﬂuorescent X-rays) are emitted from the atoms in the material. These particles have wave properties. These waves are scattered by the surrounding atoms, resulting in scattered waves. Naturally, the scattered and direct waves interfere with each other, producing an interference pattern in the emission angular distribution. This interference pattern is an atomic-resolution hologram, which contains information on the 3D arrangement of atoms around the emitter of the particle. By analyzing this hologram, a 3D atomic arrangement around the emitter can be obtained. This method is especially powerful for elucidating local structures around a dopant in materials.
Recent advances in experimental and analysis methods for atomic-resolution holography have revealed that doped elements can induce not only a simple point defect but also novel structural features such as dopant clusters, defect complexes, and strongly off-centered atomic arrangements. The following sections describe how these hyperordered structures in crystals (i.e., disorder in order) can be characterized by atomic-resolution holography.

3 Atomic-Resolution Holography

35

a) Excitation

d) Angular distribution = Hologram

b) direct wave

c) scattered wave

Emitter atom
Reconstruction calculation

3D atomic image around the emitter atom
Fig. 3.1 Schematic diagram of atomic-resolution holography. a Irradiation with excitation particles (X-rays, etc.). b Emission of secondary particles. c Scattering of waves of secondary particles, producing scattered waves. d Interference pattern between the direct and scattered waves in the emission angular distribution. This interference pattern is a hologram
3.2 Principle of Atomic-Resolution Holography
3.2.1 Recording Process
The recording process of atomic-resolution holography can be of two modes: normal mode and inverse mode. Figure 3.2 shows the principle of the recording process. The normal mode utilizes the scattering and interference of spherical waves emitted from a speciﬁc element. The inverse mode accesses the scattering and interference of

36

T. Matsushita et al.

(a) Normal mode

(b) Inverse mode

Hologram

Hologram L

L

θ,ϕ

θ,ϕ

F

F

E

S

a

E

S

a

Fig. 3.2 Recording process of atomic-resolution holography: a normal mode and b inverse mode

incident plane waves. Considering the time-reversal symmetry of the laws of physics, these two methods result in the same hologram being obtained.

3.2.2 Normal Mode

Figure 3.2a shows the recording process for a normal-mode hologram. A beam (photons or electrons; labeled as L) excites the secondary particles (electrons or photons) emitted from the atom (labeled as E). Here, the atom that emits the secondary particle is the emitter. The secondary particle has the property of a wave; therefore, a spherical wave spreads around the emitter. The wave is scattered by surrounding atoms to form the scattered wave. One scatterer atom is labeled as S. The scattered and unscattered waves interfere with each other, and an interference pattern is formed by the angular distribution of the emitted secondary electron (labeled as F). Because the scattered and unscattered waves can be regarded as the object waves and reference wave, respectively, the interference pattern can be regarded as the hologram. This hologram contains information on the 3D atomic structure around the emitter. Figure 3.3 shows a schematic view of the formula for interference in the normal mode. The emitted wave (.s-wave: isotropic spherical wave) is described as

.ϕ(k, r)

≡

exp(i k |r |) |r|

,

(3.1)

where.k and.r are the wave number and position vector, respectively. There are many scatterer atoms, and their index is written as .h. The wave reaches a scatterer atom

3 Atomic-Resolution Holography

37

Fig. 3.3 Schematic view of formula for interference in normal mode

located at .ah. The wave function at the scatterer is given by

.ϕ(k,

ah)

=

exp(i k|ah |ah|

|)

.

(3.2)

Then, the scattered wave is formed as

.ψ(k, r, ah)

=

exp(i k |ah |) |ah|

·

f

(θah

k

)

exp(i k |r |r −

− ah

ah |

|)

,

(3.3)

where . f (θ ) is the atomic scattering factor (also called the atomic form factor, or atomic structure factor) and.θah k is the angle between vectors.ah and.k. As described below, this scattering factor varies depending on the measurement method. For exam-
ple, in the case of ﬂuorescent X-rays, the atomic scattering factor has the following
relationship with the atomic scattering factor of X-ray diffraction:

. f (θah k) = − fxray(θah k).

(3.4)

The interference intensity is expressed by

.I (k, r) = |||||ϕ(k, r) + E ψ(k, r, ah)|||||2 .

(3.5)

h

This interference pattern is observed at a distant detector. (The atomic scale is nm,

and the detector is nearly 1 m away, which is distant in terms of the atomic scale.)

Therefore,

.I (k) = |||||ϕ(k) + E ψ(k, ah)|||||2 .

(3.6)

h

38

T. Matsushita et al.

Here, the following relation is used:

.ϕ(k) = lim r ϕ(k, r).
r →∞

(3.7)

When .r approaches inﬁnity, the vector .k becomes parallel to the vector .r; consequently, .(k, r) → k. Then, the hologram function is deﬁned by subtracting the reference wave intensity:

.χ (k) = |||||ϕ(k) + E ψ(k, ah)|||||2 − |ϕ(k)|2 .
h

.χ

(k)

=

E
h

2R

|

f

(θah

k)

exp(i

(kah − ah

k

·

ah ))

|

.

(3.8) (3.9)

This equation contains each atomic position vector, and the hologram is the simple sum of each atom’s hologram. Therefore, a 3D atomic image can be reconstructed from the hologram without requiring additional information such as a model of the initial atomic arrangement. This normal mode is used in photoelectron holography (PEH) and normal-mode X-ray ﬂuorescence holography (XFH), which are described later.

3.2.3 Inverse Mode

Figure 3.4 shows a schematic view of the formula for the inverse mode. The incident

plane wave is given by

.ϕ(k, r) = exp(i k · r).

(3.10)

.

.

.

Fig. 3.4 Schematic view of formula for interference in inverse mode

3 Atomic-Resolution Holography

39

Here, the origin is the atom that emits the secondary particle (X-ray, .γ -ray, or electron). This plane wave is scattered by a scatterer atom located at .ah, where .h is the index for the atoms. The wave function at the scatterer is given by

.ϕ(k, ah) = exp(i k · ah).

(3.11)

Then, the scattered wave is formed as

.ψ

(k,

r,

ah

)

=

exp(i

k

·

ah)

f

(θah

k)

exp(i k |r |r −

− ah|) ah|

.

(3.12)

Here, many atoms are present. The interference intensity is expressed by .I (k, r) = |||||ϕ(k, r) + E ψ(k, r, ah)|||||2 .
h

(3.13)

This interference pattern is observed at the emitter atom (.r = 0). Therefore,

.I (k) = |||||ϕ(k) + E ψ(k, ah)|||||2 .
h

(3.14)

Then, the hologram function is deﬁned by subtracting the reference wave intensity:

.χ (k) = |||||ϕ(k) + E ψ(k, ah)|||||2 − |ϕ(k)|2 .
h

.χ

(k)

=

E
h

2R

|

f

(θah

k)

exp(i

(kah + ah

k

·

ah ))

|

.

(3.15) (3.16)

The obtained equation is similar to that for the normal mode. When .k/ = −k is adopted, the two equations are identical.

3.2.4 Effect of Thermal Vibration and Fluctuation
The positions of atoms ﬂuctuate due to thermal vibration and other factors. If the atoms are vibrating isotropically and the standard deviation of their vibration amplitude of the .hth atom is deﬁned as .σh, the hologram is given by Matsushita et al. [5]

40

T. Matsushita et al.

. χ

(k)

=

E
h

2R

|

f

(θah

k)

exp[i(kah − |ah|

k

·

ah

)]

exp[−σh2|k|2(1

−

cos

θah k

| )]

.

(3.17)

From this equation, the larger the angle .θahk, the smaller the term .exp[−σh2|k|2(1 − cos θahk)] becomes. This means that the amplitude of backscattering is suppressed

by the ﬂuctuation of the atomic positions.

3.3 Overview of Atomic Image Reconstruction
Atomic image reconstruction methods are important in practical uses. Barton [3, 4] proposed a Fourier transform method because the hologram has the term.exp(i(kah + k · ah)). The amount of information is preserved by the Fourier transformation, and a 3D volume hologram is required. When using a single-energy hologram, the twinimage formation, where a false image is generated at the point-symmetrical position, is a serious problem as described later. The normal mode of XFH suffers from this problem because it is difﬁcult to obtain a multi-energy hologram. In contrast, when a multi-energy hologram is available, Barton’s method is effective. Inverse XFH and neutron holography (NH) satisfy this condition.
Here, Barton’s method and its application to XFH are described. In addition, an advanced reconstruction method called the scattering pattern extraction algorithm (SPEA) [6, 7] is brieﬂy presented, which is applicable to single-energy X-ray ﬂuorescence holograms to obtain reasonable atomic images. This advanced method was originally developed for the atomic image reconstruction in PEH to deal with complicated scattering processes of electrons (i.e., the complicated form of. f (θah k)), and a detailed explanation will be given in Sect. 3.6.

3.3.1 Barton’s Algorithm

Atomic images, .U (r), can be obtained from the Fourier-transform-based formula

[3]

{

.U (r) = w(θkr )χ (k)R[exp(i(kr − k · r)]d k,

(3.18)

k=|k|

where .w(θkr ) is a weight function. This weight function is related to the scattering factor and therefore depends on the measurement method. For example, in XFH, it is .w(θkr ) = −1. However, the atomic image reconstructed from Eq. (3.18) suffers from the twin-image problem, shown in Fig. 3.5. Suppose a dimer model contains Fe as an emitter atom at the origin and Cu as a scattering atom at.(x, y, z) = (3, 0, 0)Å (Fig. 3.5a). The calculated hologram at a photon energy of 6.4 keV is shown in Fig. 3.5a. Figure 3.5b shows the reconstructed image obtained by substituting this

3 Atomic-Resolution Holography

41

holographic oscillation into .χ (k) in Eq. (3.18). We can observe a twin image at .(x, y) = (−3, 0) Å, as indicated by the arrow, in addition to the real atomic image observed at.(x, y) = (3, 0)Å. As shown in Fig. 3.5c, the oscillation of the hologram signal is almost symmetric about.θ , resulting in the twin image at the centrosymmetric position of the real Cu atom, as shown in Fig. 3.5b. Such symmetric oscillation appears when.kah ∼ π N (N = 1, 2, ...). Because twin images are inherent to holography, it is impossible to avoid this problem by enhancing the statistics or increasing the measurement angular range.
One of the most effective methods to suppress twin images is multi-wavelength measurements. As shown in Fig. 3.6a, hologram oscillations are dependent on the energy (wavelength), which is included in .k in Eqs. (3.9) and (3.16). In Fig. 3.6a, the oscillations are in-phase for various energies in the forward-scattering region at around .θ = 0◦ and not in-phase in the back-scattering region at around .θ = 180◦. Therefore, by superimposing the images with different energies by using [4]

{{ .U (r) = dk w(θkr )χ (k)R {exp[i(kr − k · r)]} d k,

(3.19)

k |k|=k

twin images can be effectively suppressed. Figure 3.6b shows the image reconstructed by multi-wavelength holograms from 7.5 to 11.0 keV in steps of 0.5 keV using Eq. (3.19). We can conﬁrm that the twin image almost completely disappears.
Note that Eq. (3.19) includes the integral with respect to .k = |k| = 2π/λ, where .λ is the wavelength of ﬂuorescent or incident X-rays for the normal or inverse mode, respectively. Because the wavelength of ﬂuorescent X-rays cannot be tuned, Eq. (3.19) is not applicable to the holograms recorded by the normal-mode XFH. Therefore, almost all XFH experiments have been performed using the inverse mode [8].
Notably, the twin-image problem can be ignored for PEH because the holographic oscillations of photoelectrons attenuate much more strongly with increasing .θ , and thus, the photoelectron holograms are far from symmetric with respect to .θ . This is why the normal mode works well for PEH. In contrast, the attenuation of the holographic oscillation with increasing .θ is even weaker for NH than that for XFH, because the scattering length.bh of neutrons is constant for.θ . Therefore, twin images in NH have a greater inﬂuence on the reconstructed image than in XFH. As described in Sect. 3.5, this difﬁculty has been successfully overcome by using the time-ofﬂight (ToF) method, which enables us to record numerous holograms with different wavelengths at the same time.

3.3.2 Fitting-Based Atomic Image Reconstruction
By using the atomic distribution function, .g(r), the atomic-resolution hologram, .χ (k), can be written as

42

T. Matsushita et al.

( ) ( )

(a) 6.4 keV
Fe Cu
3
(c) 6.4 keV
peak peak peak dip dip
150 100 50 (deg.)

(b)

4

2 Fe
0

-2 Twin Image
-4

-4 -2 0 2 4 ( )

0.1

1.0

peak

peak

peak

180°
0 dip dip

= 0°

Fig. 3.5 Origin of twin images. a Calculated hologram of a dimer model of Fe and Cu for.hν = 6.4 keV. b Atomic image reconstructed from the hologram shown in (a). c Line proﬁle of the hologram along the dashed line indicated in the hologram to the right

(a) 10.0 keV
9.0 keV 8.0 keV
6.4 keV

150 100 50

0

(deg.)

(b)

4

2 Fe
0

-2

-4

-4 -2 0 2 4

( )

0.1

1.0

Fig. 3.6 Effect of multi-wavelength holograms on suppressing twin images. a Calculated hologram oscillations of Fe-Cu dimer model (Fig. 3.5a) with different incident energies. b Atomic images using eight energies (7.5 to 11.0 keV in steps of 0.5 keV) obtained from Eq. (3.19)

3 Atomic-Resolution Holography
{ .χ (k) = 2 g(r) f (θak)R[exp[i(kr − k · r)]]d r.

43
(3.20)

As will be described in Sect. 3.6 (see Eq.e(3.48)), the atomic distribution function .g(r) is conveniently expressed by.g(r) = h δ(r − ah)/r . With this deﬁnition, Eq. (3.20) is rewritten as

E .χ (k) = 2 g(rh) f (θak)R[exp[i (kr − k · r)]]d r.
h

(3.21)

By minimizing the evaluation function.E

=

e
i

|χi(exp)

−

χi(sim)|2

+

λ

e
j

gj

(see Eq.

(3.51) in Sect. 3.6), the atomic image can be obtained. The second term is the so-

called penalty term in L1 regularization, thus, SPEA with this evaluation function is

called SPEA-L1 [10, 11]. The iterative calculation

.g (n+1) ( r )

=

g(n)

−

α

∂

E (n) ∂r

(3.22)

is used in the minimization process. Here, the parameter .α is optimized using the gradient method.
Figure 3.7 shows a comparison of the atomic images around Zn in ZnTe obtained using Barton’s algorithm and SPEA-L1 [10]. The holograms obtained from the XFH experiment were used for the reconstruction. Figure 3.7a shows the hologram obtained at the incident X-ray energy of 12 keV. As schematically shown in Fig. 3.7b, the Te-plane around the Zn atom was reconstructed as shown in Fig. 3.7c–f. Here, atomic images were reconstructed using Barton’s algorithm and SPEA-L1 by using nine holograms from 11 to 15 keV in steps of 0.5 keV (Fig. 3.7c and d) and a single hologram at 12 keV (Fig. 3.7e and f). Using nine holograms, both Barton’s method and SPEA-L1 well reconstruct the Te atomic images, although the reconstructed image obtained using Barton’s algorithm contains sidelobes of the Fourier transform, as indicated by circles in Fig. 3.7c. By contrast, the quality of the reconstructed image from the single-energy hologram obtained by SPEA-L1 (Fig. 3.7f) is markedly different from that obtained by Barton’s method (Fig. 3.7e). In other words, distinct Te atomic images can be observed in Fig. 3.7f, whereas clear atomic images are hardly observed in Fig. 3.7e. In recent years, SPEA-L1 has been successfully used in some studies [12–14], especially in valence-speciﬁc XFH studies [13, 14].
This result demonstrates the potential of XFH to obtain reliable atomic images even with a single-energy hologram, and thus, it provides a good opportunity to revisit normal-mode XFH. As described in the next section, normal-mode XFH has some advantages over inverse mode XFH, such as a much shorter measurement time with a 2D detector and the ability to measure small samples with focused incident X-rays. In this context, experimental and analytical techniques based on normal-mode XFH are being investigated further.

44

T. Matsushita et al.

Fig. 3.7 Comparison of Barton’s method and SPEA-L1. a X-ray ﬂuorescence hologram of ZnTe obtained using the incident X-ray energy of 12 keV. The data were taken from Ref. [9]. b Structure of ZnTe. c Cross section of the image reconstructed by Barton’s method. Nine holograms from 11 to 15 keV were used. The distance of the xy-plane from the emitter is 1.5 Å. d Image reconstructed from nine holograms using SPEA-L1. e Image reconstructed from a single hologram obtained at 12 keV using Barton’s method. f Image reconstructed from a single hologram using SPEA-L1. Adapted from Ref. [10] with reprint permission

3 Atomic-Resolution Holography

45

3.4 X-ray Fluorescence Holography (XFH)

X-ray ﬂuorescence holography was ﬁrst demonstrated by Tegze and Faigel in 1996 [15]. Because the hologram amplitude is only approximately some 0.1% of the ﬂuorescent X-ray intensity, XFH experiments using laboratory X-ray sources are very time-consuming. However, with the advent of third-generation synchrotron radiation facilities together with signiﬁcant developments in analytical methods to obtain improved 3D atomic images [16–25] and experimental techniques to efﬁciently collect ﬂuorescent X-rays [26–30], XFH has been practically applied to various structural [8, 31], functional [8, 12, 31–38], and biological materials [31, 39, 40]. Here, the apparatus of XFH and several hyperordered structures in materials found by XFH will be described. Furthermore, recent developments in experimental and analytical techniques will be presented.

3.4.1 Hologram Oscillations in Fluorescent X-Rays

Suppose a material containing a dopant is irradiated with X-rays. If the energy of the X-rays is higher than the absorption edge of the dopant, the dopant emits ﬂuorescent X-rays. Then, part of the ﬂuorescent X-rays are scattered by the surrounding atoms. The interference between the unscattered and scattered waves slightly modulates the intensity of the ﬂuorescent X-rays, which corresponds to the holographic oscillation. The unscattered and scattered ﬂuorescent X-rays correspond to “Direct wave” and “Scattered wave” in Fig. 3.3, respectively, which illustrates the formation process of the interference pattern in the normal mode. Then, the intensity of the ﬂuorescent X-rays .I for the normal mode can be written as [8, 32, 41]

. I

(k)

=1 +

−|||||Eh2RrE ehfxra|ayhr(eθafxkr)aayhe(xθpa[ki)(kexaph[−i (kka·h

−k· ah )]|||||2

ah ,

| )]

(3.23)

where.k is the wave vector of the ﬂuorescent X-rays,.re ≈ 2.8 × 10−5 Å is the classi-
cal electron radius,.ah is the position of the.hth atom,. f is the atomic scattering factor, and .θak is the angle between .k and .ah. The second term represents the holographic oscillation, .χ (k):

.χ (k)

=

−2R

E
h

| re

f xray (θa k ) ah

exp[i (kah

−

k

·

| ah )]

.

(3.24)

46

T. Matsushita et al.

Here, the amplitude of the holographic oscillation,.χ (k), is only approximately some

0.1% of the intensity of the ﬂuorescent X-rays. Note that Eq. (3.24) can be obtained

by substituting . f (θak) in Eq. (3.9) into the negative value of the atomic scattering factor,.− fxray(θak). The negative sign originates from the phase shift of X-rays by.π

through Thomson scattering [42].

Equation (3.24) can be converted into the holographic oscillation for the inverse

mode by deﬁning .k as the wave vector of incident X-rays and replacing .k with .−k,

that is,

.χ (k)

=

−2R

E
h

| re

f xray (θa k ) ah

exp[i (kah

+

k

·

| ah )]

.

(3.25)

3.4.2 Apparatus
Nowadays, XFH experiments are actively being carried out in synchrotron facilities such as SPring-8 and KEK-PF in Japan [32]. Because the holographic oscillation is only approximately some 0.1% of the intensity of the ﬂuorescent X-rays, the highintensity X-rays available at a synchrotron facility are necessary. Furthermore, the synchrotron X-ray source can offer energy tunability to record multi-wavelength holograms by inverse mode XFH.
Figure 3.8 illustrates the experimental setup for inverse mode XFH. A sample on a two-axis (.θ, φ) goniometer is irradiated with X-rays. The ﬂuorescent X-rays from a target element in the sample are selectively extracted by adjusting the position of the analyzer crystal. Owing to the round shape of the analyzer crystal, the ﬂuorescent X-rays are focused on the detector, thereby enhancing the ﬂuorescent signals

Sample
Incident X-rays ϕ θ

Analyzer crystal
Pb shield Detector

Fluorescent X-rays

Fig. 3.8 Apparatus for inverse mode XFH

3 Atomic-Resolution Holography

47

Fluorescent X-rays

45° pin-hole mirror
Incident X-rays

sample

SDD

Camera

Fig. 3.9 Apparatus for normal-mode XFH

2D detector

signiﬁcantly. An avalanche photodiode or silicon drift detector (SDD) is typically used as a detector. The ﬂuorescent intensities are recorded as a function of .θ and .φ in Fig. 3.8, and thus, 2D data can be obtained, which is referred to as a hologram.
Figure 3.9 schematically shows the experimental setup for normal-mode XFH [43, 44]. In normal-mode XFH, ﬂuorescent X-rays from the ﬁxed sample are recorded as a function of the emission direction. Because the sample is ﬁxed during the measurement, holograms from small samples can be readily measured using a focused X-ray beam, which is available in synchrotron facilities. The typical size of the focused X-rays is.∼1 × 1.µm.2, and thus, a sample with a size of .∼10 × 10 .µm.2 or less can be a target of normal-mode XFH. A telescopic camera and a 45.◦ pinhole mirror are installed to observe the sample from the direction of the incident X-rays, thereby enabling the sample position to be precisely adjusted with an accuracy of about 1 .µm. Such high accuracy is required to measure small samples such as protein crystals [44]. In a recent setup, a 2D detector was used to efﬁciently measure ﬂuorescent X-rays emitted in a certain range of solid angles. In addition, the energy spectra are observed using an SDD.

3.4.3 Applications
Recently, various hyperordered structures have been found in materials such as thermoelectrics, lightweight structural materials, and superconductors [31]. Here, some examples of hyperordered structures revealed by XFH are presented.

48

T. Matsushita et al.

Co-doped TiO.2: Dilute Magnetic Semiconductor

As already described in Sect. 1.1, Co-doped TiO.2 is a promising ferromagnetic semiconductor with a high Curie temperature of about 600 K. The suboxidic structure shown in Fig. 1.1.3 embedded in this material is a typical hyperordered structure. Figure 3.10a and b present atomic images around Co in a Co-doped TiO.2 thin ﬁlm [45]. It is conﬁrmed that the positions of atomic images around Co in paramagnetic Ti.0.99Co.0.01O.2 (Fig. 3.10a) agree well with those predicted from the rutile-type TiO.2 structure (Fig. 3.10c). This observation indicates that the Co atoms in Ti.0.99Co.0.01O.2 are located at the substitutional sites of rutile-type TiO.2. By contrast, the atomic images around Co in ferromagnetic Ti.0.95Co.0.05O.2 (Fig. 3.10b) cannot be explained by the simple substitutional model. Figure 3.10d shows the corresponding atomic conﬁguration around Co, where fewer oxygen atoms are present around Co than in the simple substitutional model shown in Fig. 3.10c; in other words, suboxidic coordination is formed around Co in ferromagnetic Ti.0.95Co.0.05O.2. Although this suboxidic structure is energetically unstable, a ﬁrst-principles calculation showed that the dimerization of the Co atom stabilizes this suboxidic state, as shown in Fig. 1.3 in Chap. 1.
This study on Co-doped TiO.2 demonstrates the capability of XFH to reveal not only simple point defects but also more complex local structures, namely, hyperordered structures.

Zn, Y-doped Mg Alloy: Lightweight Structural Material
Mg–Zn-Y alloys [46] are promising candidates for next-generation lightweight structural materials because of their excellent mechanical properties, such as a very high yield strength of more than 600 MPa and reasonable ductility. Scanning transmission electron microscopy (STEM) revealed the Zn and Y atoms from a Zn.6Y.8 cluster in the .hcp Mg matrix structure as shown in Fig. 1.1.4 [47] in Chap. 1. As already mentioned in Chap. 1, this Zn.6Y.8 cluster can be regarded as a hyperordered structure.
While STEM can visualize the atomic arrangement in a selected area of a sample, XFH gives a statistically averaged structure around the doped elements in the whole sample, which is useful for understanding the effect of doping on the macroscopic properties of materials. For this purpose, XFH was applied to Mg.75Zn.10Y.15 alloy [48].
An in-plane atomic image around Zn in Mg.75Zn.10Y.15 is shown in Fig. 3.11a. Here, the atomic images obtained from the experimental and calculated holograms are displayed using Barton’s algorithm. The calculated atomic image was derived from the structural model proposed in the STEM study [47]. The upper part of Fig. 3.11a illustrates the three different in-plane atomic conﬁgurations of Zn. Here, Zn atoms form a regular triangle along the .ab plane, and we labeled the vertices of this triangle as A, B, and C (Fig. 3.11a and b). The atomic image obtained by XFH corresponds to the superposition of the local structures around these three Zn atoms, as shown in Fig. 3.11b.

3 Atomic-Resolution Holography

49

Fig. 3.10 Reconstructed real-space images around Co in a Ti.0.09Co.0.01O.2 and b Ti.0.95Co.0.05O.2 thin ﬁlms. c and d Structure models obtained from the images in (a) and (b), respectively. Adapted from Ref. [45] with reprint permission
In both the experimental and calculated results in Fig. 3.11, sixfold atomic images can be clearly observed at the positions marked by the solid circles, corresponding to the positions of the surrounding Zn atoms inside the Zn.6Y.8 cluster. However, atomic images cannot be conﬁrmed at the positions marked by the dashed circles in the experimental results, which is in contrast to the calculation. These positions correspond to the surrounding Zn atoms included in the neighboring clusters. Therefore, weak intercluster and strong intracluster correlations are indicated, as schematically shown in Fig. 3.11c. According to the calculation of the atomic images, where the intercluster positional ﬂuctuations were expressed as a Gaussian distribution, the magnitude of ﬂuctuation (half width at half maximum of the Gaussian) was evaluated to be at least 0.33 Å.
A study of the deformation behavior of Mg-Zn-Y alloys [49] demonstrated that in-plane ordering is closely related to the ductility. Namely, the ductility of this alloy deteriorates as the in-plane ordering of the Zn.6Y.8 cluster develops. Thus, the intercluster positional ﬂuctuations revealed in the present XFH study can contribute to the enhancement of ductility.
Recently, Hagihara et al. [50, 51] reported that the addition of a small amount of Zn and Y (.<1 at%) can induce outstanding mechanical properties comparable

50
Fig. 3.11 a In-plane atomic images around Zn. Experimental and calculated results are shown. The correspondence to the in-plane positions of Zn atoms in the structural model [47] is also shown. b Schematic illustration of the superposition of the three different atomic conﬁgurations around Zn, corresponding to the atomic image obtained by XFH. c Schematic drawing of robust Zn.6Y.8 clusters and weak intercluster correlations. Adapted from Ref. [48] with reprint permission

T. Matsushita et al.
(a) : Emitter Zn
: Surrounding Zn Intracluster
B A
C

(b) Zn plane (c)

A

B

C

to those of Mg–Zn-Y alloy with higher Zn/Y concentrations. This dilute Mg–Zn-Y alloy is noteworthy for reducing the cost of rare earths as well as for having an even lower weight than that with higher Zn/Y concentrations. Therefore, it is a future target for XFH studies.
Ca-doped BaTiO.3: Pb-free Piezoelectric Material
Piezoelectric materials can interconvert mechanical and electrical energy, and they are used in various applications such as sensors and inkjet printers. Currently, lead zirconate titanate (PZT) is widely used; however, it contains harmful Pb atoms. Thus, Pb-free piezoelectric materials are being explored. Recently, BaTiO.3-based Pbfree materials with a perovskite structure have been actively studied. In particular,

3 Atomic-Resolution Holography

51

(a)

(b)

(c)

10

10

010 ( )

5

5

010 ( )

0

Ca

0

Ba

-5

-10-10

-5 0 5 100 ( )

-5

10 -10-10

-5 0 5 100 ( )

Central Ion 10 (Ca or Ba) Reconstructed plane

Ca/Ba

Ti

O

010 ( ) Intensity (arb. units)

(d) 1.0
0

(e) 1.0
0

(f) 8 6 4

Around Ca Around Ba

-1.0 3.0 4.0 5.0 100 ( )
2.5

-1.0 3.0 4.0 5.0 100 ( )
5.0

2
0 3.2 3.4 3.6 3.8 4.0 4.2
100 ( )

010 ( )

Fig. 3.12 Atomic images around a Ca and b Ba in (Ba.0.9Ca.0.1)TiO.3. c Corresponding reconstructed plane. d and e Close-up views of the atomic images indicated by arrows in (a) and (b), respectively. f Line proﬁles of atomic images along the solid lines in (d) and (e). Adapted from Ref. [53] with reprint permission

(Ba,Ca)(Zr,Ti)O.3 (BCZT) is known to exhibit piezoelectric properties superior to those of PZT at temperatures below approximately 373 K [52]. However, the origin of the high piezoelectricity of BCZT is not yet sufﬁciently understood.
Because the radius of a Ca.2+ ion (1.34 Å) is approximately 17% smaller than that of a Ba.2+ ion (1.61 Å), the local structure around the Ca.2+ ions is expected to be signiﬁcantly different from that around the Ba.2+ ions in the base material of BaTiO.3. Therefore, XFH was applied to (Ba.0.9Ca.0.1)TiO.3, and the local structures around Ca and Ba, occupying the same A site, were separately analyzed [35].
Figure 3.12a and b respectively show atomic images around Ca and Ba obtained by XFH measurement using Barton’s algorithm. As shown in Fig. 3.12c, the .<001.> plane containing Ca or Ba is reconstructed in Fig. 3.12a and b, respectively. The circles in Fig. 3.12a and b correspond to the ideal positions of the A site cations predicted from the crystal structure of BaTiO.3. Although the atomic images around Ba are observed at the center of the circles, those around Ca are shifted toward the center. This observation indicates that the lattice locally contracts around Ca, which is consistent with the smaller ionic radius of Ca ions than that of Ba ions.
Figure 3.12d and e show enlarged views of the atomic images indicated by the arrows in Fig. 3.12a and b, respectively. A signiﬁcant difference is seen not only in the position but also in the shape of the atomic images around Ca and Ba. Speciﬁcally, the atomic image around Ca (Fig. 3.12d) is elongated in the radial direction, while

52 (a)

T. Matsushita et al. (b)
Short Ca-Ba distance

Long Ca-Ba distance

Ca

Ba Ti

O

Fig. 3.13 a Schematic illustration of the atomic displacements of Ca and surrounding Ba ions evaluated by XFH measurements. b Ca and NN Ba ions. The displayed displacements are magniﬁed threefold to make them easy to recognize. Adapted from Ref. [53] with reprint permission

no such elongation is observed around Ba (Fig. 3.12e). This feature indicates that the central element of Ca itself is displaced. Figure 3.12f shows the line proﬁle of the atomic images plotted along the solid lines in Fig. 3.12d and e. Here, a shoulder is observed in addition to the main peak around Ca; this is different from the proﬁle around Ba, which shows a single peak. From the Gaussian ﬁtting to the line proﬁle around Ca, the peak positions are evaluated to be 3.52 and 3.95 Å.
Figure 3.13a shows that a structural model is used to reproduce these peak positions. Here, the Ca ion is displaced by 0.36 Å in the .<111.> direction and the neighboring Ba ion is displaced by 0.22 Å in the.<100.> direction. The displacement of the Ba ion corresponds to a local lattice contraction around Ca. The displacement of the Ca ion deforms the shape of the atomic image, as shown in Fig. 3.12d. Figure 3.13b shows that the Ca and nearest neighbor (NN) Ba ions are extracted from the structural model. The displacement of Ca in the .<111.> direction is understood to result in the appearance of long and short Ca-Ba distances. These two distances correspond to the peaks at 3.52 and 3.95 Å, respectively, shown in Fig. 3.12f.
The large displacement of Ca ions revealed in this study is similar to that of Pb ions in PZT [54, 55], which also occupy the A site. This result suggests the possibility that the Ca ions, which are biocompatible, can play a similar role to Pb ions. The observed displacement of the Ca and surrounding Ba ions is an excellent example of disorder in order that can contribute to enhancing material functionality.

3.4.4 Recent Developments
Valence-Speciﬁc XFH
An element with multiple valence states often plays an important role in inducing material functionalities, such as magnetic, catalytic, and electrochemical proper-

3 Atomic-Resolution Holography

53

Fig. 3.14 Yb.LIII XANES spectra of YbInCu.4 at 7 and 300 K. Adapted from Ref.
[14] with reprint permission

ties. To understand the structural origins of these properties, it is crucial to perform
valence-speciﬁc local structural analysis around the element. Recently, it has been
demonstrated that XFH can reveal valence-speciﬁc local structures around an ele-
ment with different valence states [13, 14]. Here, a valence-speciﬁc XFH study of Yb.2+ and Yb.3+ coexisting in YbInCu.4 [14] is presented.
YbInCu.4 is a well-known valence-transition material, which shows a signiﬁcant change in the valence from 2.94 to about 2.8 upon cooling to 17 K, as observed by
a combination of multiple X-ray spectroscopic techniques [56]. Figure 3.14 shows
Yb.LIII XANES spectra of YbInCu.4 at 7 and 300 K reported in the XFH study [14]. It is clearly observed that a shoulder appears at around 8.939 keV upon cooling
from room temperature to 7 K as indicated by the downward arrow. This feature is attributable to Yb.2+ ions, and thus it is possible to selectively excite Yb.2+ ions by
setting the incident X-ray energy to 8.939 keV.
In Ref. [14], the XFH measurements were performed at 300 and 7 K with incident
X-ray energies of 8.947 and 8.939 keV. Figure 3.15a shows the 3D atomic image
measured at 300 K with the incident X-ray energy of 8.947 keV. This image corresponds to the local structure around Yb.3+ ions, because the contribution of Yb.2+
ions is negligible at this temperature. As indicated by arrows, the 12 ﬁrst-neighbor
Yb ions are observed in good agreement with the crystal structure of YbInCu.4 shown in Fig. 3.15b. When the temperature is decreased to 7 K, the 3D atomic images are
markedly changed as presented in Fig. 3.15c and d, which were obtained at 7 K with
incident X-ray energies of 8.497 and 8.939 keV, respectively. At 8.497 keV, both Yb.3+ and Yb.2+ ions are excited, and thus Fig. 3.15c is the superposition of the local environments around Yb.3+ and Yb.2+ ions. At 8.939 keV, on the other hand, only Yb.2+ ions are excited, and consequently, Fig. 3.15d corresponds to the 3D atomic arrangement around Yb.2+. We can observe a marked difference between Fig. 3.15c
and d. The shape of the atomic images in Fig. 3.15d exhibits tails along the.a,.b, and
.c axes, in contrast to Fig. 3.15c, which can be more clearly seen in Fig. 3.15e and f. This feature can be attributed to the larger ionic radius of Yb.2+ than that of Yb.3+,

54

T. Matsushita et al.

Fig. 3.15 a 3D atomic image of YbInCu.4 reconstructed from an Yb-.Lα hologram measured at 300 K with an incident X-ray energy of 8.947 keV. b Crystal structure of YbInCu.4. c and d 3D atomic images reconstructed from Yb-.Lα holograms measured at 7 K with incident energies of 8.947 and 8.939 keV, respectively. e and f Close-up views of 2nd and 3rd nearest neighbor atomic images, respectively. Adapted from Ref. [14] with reprint permission
which can cause the positional ﬂuctuation of Yb.2+ ions. The “cross” shape of the atomic images indicates that the direction of the ﬂuctuation is .<100>.
Note that advances in analytical and experimental techniques played important roles in this valence-speciﬁc XFH study on YbInCu.4. In valence-speciﬁc XFH measurements, only a single-energy hologram can be obtained, and thus the ﬁtting-based algorithm described in Subsect. 3.3.2 is necessary to reconstruct reasonable atomic images. In fact, the atomic images in Fig. 3.15 were reconstructed by SPEA-L1 [10, 11]. Furthermore, the XFH experiment was successfully performed at a very low temperature below the boiling point of liquid nitrogen (.∼80 K) in this study. The details of the experimental technique for low-temperature XFH are described in the following.
XFH at Low Temperature and On-Going In Situ Measurements
Various intriguing phenomena are induced in materials at low temperatures, such as superconductivity, magnetic ordering, ferroelectricity, and valence transition. Structural information obtained by XFH will provide a valuable insight into the origin of

3 Atomic-Resolution Holography

55

Fig. 3.16 a Schematic drawing and b photograph of the cryostat head specially designed for lowtemperature XFH. Adapted from Ref. [57] with reprint permission
such curious low-temperature properties of materials. Many XFH works have been carried out at low temperatures using a liquid nitrogen open-ﬂow cooler. However, the lowest temperature achievable with this equipment is only about 80 K, corresponding to the boiling point of liquid nitrogen. In this context, a cryostat specially designed for low-temperature XFH down to 4 K has been developed.
Figure 3.16a and b show schematic drawing and photograph of the cryostat head designed for low-temperature XFH, which is sealed in a stainless steel tube having a Be X-ray window. The cryostat head is cooled by a helium cryogenic compressor under vacuum. To precisely control.φ (see Fig. 3.8) at a low temperature, a coolable piezoelectric rotator is employed. The value of .φ is recorded as the resistivity of a built-in variable resistance with a resolution of 0.006.◦. The rotator and the cryostat head are connected by a Ag wire to ensure good heat transfer. The angle .θ (see Fig. 3.8) is controlled by rotating the whole cryostat installed on a goniometer. Using this specially designed cryostat, the atomic images shown in Fig. 3.15 were obtained.
One of the advantages of XFH over other atomic-resolution holographies is a ﬂexible sample environment, which makes it feasible to perform in situ XFH measurements under various experimental conditions. Low-temperature XFH is a good example of adopting this advantage. High-pressure XFH using diamond anvil cell (DAC) installed in the normal-mode XFH setup shown in Fig. 3.9 is currently under development. Also, in situ XFH experiments on piezoelectric materials under an electric ﬁeld are now in progress. These developments will draw out the potential of

56

T. Matsushita et al.

XFH to provide detailed and systematic atomic-scale information on external-ﬁeldinduced phenomena such as structural phase transition and dielectric response.

3.5 Neutron Holography (NH)
3.5.1 Roles of Neutrons in Studies of Hyperordered Structures
In materials science, light elements such as B, H, and Li currently play important roles in applications such as Li-ion batteries, hydrogen storage materials, and fuel cells. A typical example is semiconducting Si. By doping 10.20/cm.3 of B, the n-type semiconductor performance is drastically enhanced. The main role of dopants is to provide charge carriers; on the other hand, from the viewpoint of structural physics, clarifying the positions of B in the lattice and the effects of B doping on the lattice (e.g., generation of hyperordered structures) is indispensable to understand the origins of the properties. However, local structures around light elements, like B, cannot be observed even by X-ray ﬂuorescence holography (XFH) because the ﬂuorescent X-ray is strongly absorbed by the air. Moreover, because X-rays interact with electrons in atoms, the interaction with light elements is weaker than that with heavy elements. Speciﬁcally, scattering intensity of X-rays from an atom is proportional to the square of the scattering amplitude (atomic scattering factor for X-rays) from the atom. As a result, the signals from light elements are obscured by those from heavy elements. Thus, lighter elements are not suitable targets for XFH as emitters or scattering atoms despite their importance. This is a problem in studies of local atomic structures or hyperordered structures.
Figure 3.17 shows the atomic number dependence of the scattering amplitude (i.e., atomic scattering factor for X-rays (. fxray) and scattering length for neutrons (.b)). Although neutrons do not interact with electrons in atoms, except in magnetic scattering, because they are electrically neutral, they are scattered by the nucleus; thus,.b has no direct relation with the atomic number, whereas. f is proportional to the atomic number. Note that the signs of .b can be both positive and negative, whereas that of . f is positive for all elements. An important difference in the observation of atoms by neutrons and X-rays is the difference in scattering amplitude between light and heavy elements; for instance, . f of U is 98 times larger than that of H, whereas the difference in .b is only 2.3 times. Consequently, neutrons have higher sensitivity than that of X-rays to observe light elements in materials containing heavy elements. For example, Fig. 3.18 shows atomic images around .10B in 0.26 at% (atomic percent) .10B (center) doped Si [58] as visualized by inverse mode NH, which is discussed below. Atomic images of Si around.10B are successfully observed; however, as mentioned above, B cannot be a good emitter of XFH because of the strong absorption of ﬂuorescent X-rays by the air. Thus, Fig. 3.18 shows an advantage of NH for light elements as emitters as explained above. Because of this advantage

Neutron Scattering Length (fm)

X-ray Atomic Scattering Factor

3 Atomic-Resolution Holography
20

15

10

Fe

D O
5

Neutron X-ray
La

57
100
75
50 U
25

0 Li
H -5
0 1

Ti Mn 20

40

60

Atomic No.

0

-25

80

100

Fig. 3.17 Atomic number dependence of scattering amplitude of X-rays (atomic scattering factor . f :dashed line) at momentum transfer.Q = 0, and neutrons (scattering length.b: circles)..b of some elements, such as H, Li, and Mn, is negative

Fig. 3.18 Atomic image around.10B in 0.26 at%.10B (center) doped Si [58]. Reproduced with permission from Ref. [58]

and the importance of light elements, atomic-resolution holography using neutrons will be important for novel materials science, particularly studies of hyperordered structures.
In comparison with photoelectron holography (PEH) (see Sect. 3.6), which can also be used to accurately observe the atomic arrangements of light elements, NH still has some advantages. NH data are not affected by the conditions of the surface and shape of the samples because of the much larger penetration depth of neutrons (of the order of 1 cm); therefore, NH can be used to observe the bulk properties of samples. Moreover, NH can be applied to insulators, whereas PEH samples must be metals or semiconductors.

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NH, like XFH, has two modes: normal and inverse modes. Normal-mode NH was ﬁrst proposed by Cser et al. [59] and demonstrated using a single crystalline sample of Al.4Ta.3O.13(OH) by Sur et al. later in 2001 at the National Research Universal (NRU) reactor, Chalk River Laboratories (Canada) [60]. The principle of normal mode is explained in the references in detail [59, 61]. Inverse mode NH was also ﬁrst performed by Cser et al. on a single crystal of 0.26 at% Cd doped Pd at the highﬂux research reactor of the Institute Laue-Langevin in Grenoble, France in 2002 [62]. Several experiments of NH have been reported since then [61, 63–66].
However, experiments were initially performed using neutrons with only one wavelength in a practical beam because of the weakness of the monochromatic beam in reactor facilities. Because observations of many holograms with different wavelengths are indispensable to accurately visualize atomic images by reducing twin images (a false image generated at the point-symmetric position of the real atom. See in Sect. 3.3.1) as shown in Fig. 3.5, investigations of local structures/hyperordered structures by NH did not progress in these days. To overcome this problem, Hayashi et al. utilized white neutrons in an accelerator-type neutron facility, J-PARC in Tokai, Japan, and succeeded in drastically enhancing the accuracy of inverse mode NH in 2017 [67].

3.5.2 Inverse Mode NH Using White Neutrons
The principle of inverse mode NH is the same as that of XFH, except for the difference in the measurand: ﬂuorescent X-rays for XHF and prompt.γ -rays for NH. Therefore, the detailed explanation of the principle is omitted here to avoid repetition; only some important differences from XFH are discussed.
When neutrons are absorbed by emitter atoms, a prompt .γ -ray with a particular energy is instantly generated isotropically from the nucleus of the emitter, and its intensity is proportional to the neutron intensity at the emitter. Because the energy of the prompt .γ -ray is speciﬁc to each element (e.g., 477 keV for .10B (.7Li)), the energy analysis of prompt .γ -ray from the samples assures element selectivity in a manner similar to that of ﬂuorescent X-ray for XFH. Thus, by measuring the intensity of the prompt .γ -ray with the speciﬁc energy of emitters, one can measure the amplitude of interference wave at the emitter, which is generated by interference between a spherical wave scattered by each atom around the emitter (scattered wave) and incident plane wave (direct wave), as shown in Fig. 3.4. Based on the element selectivity by .γ -ray spectroscopy, holograms that provide information on the local structures around the emitters (dopants) can be obtained. Notably, inverse mode NH is currently a unique method that affords element selectivity among various neutron scattering techniques for structural physics.
A neutron hologram, .χ (.k), in the inverse mode can basically be represented by Eq. (3.24) for XFH as follows:

3 Atomic-Resolution Holography
.χ (k) = 2R E bh ei(kah+k·ah) h ah

59
(3.26)

where .bh and .ah are the neutron scattering length and positional vector of the .h-th nucleus, respectively. The origin of .ah is deﬁned as the emitter. .k is the incident neutron wavenumber vector as deﬁned in Fig. 3.4; the amplitude of .k is .k = |k.| = 2.π /.λ, and it corresponds to the radius of a hologram in the.k space. The sum includes all atoms that contribute to the hologram. Equation (3.26) is slightly different from Eq. (3.25), because .bh is generally a complex number. Moreover, note that .bh is independent of scattering angle, .θakh , in Fig. 3.4. The sign of Eq. (3.26) is opposite to that of Eq. (3.25) for XFH because of the difference in the scattering process of X-rays and neutrons by an atom. For X-rays, the phase shifts by .π in the scattering process for all atoms, though. f is deﬁned as positive; this explains the negative sign in Eq. (3.25). In contrast, for neutrons, the phase does not shift in the scattering process for atoms with positive .b whereas it shifts by .π for limited atoms with negative .b, such as H, Li, and Mn (see Fig. 3.17). This difference in the treatment of the phase changes explains the opposite sign in Eqs. (3.25) and (3.26).
At present, NH experiments can be performed at beamline 10 (BL10) of the Materials and Life Science Experimental Facility (MLF) at the Japan Proton Accelerator Complex, J-PARC, in Tokai, Japan [68, 69]. The most important reason to perform experiments at J-PARC is that white neutrons can be used; white neutrons are neutron beams that have a wide and continuous wavelength range. For NH experiments in JPARC, the typical wavelength range is 0.3 –6.6 Å. Moreover, in MLF, white neutrons are generated as a pulse every 40 ms. The time width of the neutron pulse is approximately 10.µs at a neutron energy of 100 meV, which is negligible in comparison with the duration between neutron generation and detection (.∼10 ms). Thus, neutrons with wide wavelengths range can be assumed to be generated simultaneously at the neutron source of MLF.
For such pulsed neutron facilities including MLF, data are collected using Time-of-Flight (ToF) method. Figure 3.19 shows the principle of the ToF method. Because all neutrons with different velocities (i.e., different wavelengths) start to ﬂy from the source simultaneously, neutrons as particles reach the sample position in descending order of their velocity, .t1 < t2 < t3 for .v1 > v2 > v3. The neutron wavelength is inversely proportional to its velocity or proportional to ToF as .λ(Å) = 3956. × ToF(s)/L1 (m), where L1 is the distance from the neutron source to the sample position. Based on this relation, the wavelength dependence of the intensity of neutrons or .γ -rays from the NH samples can be obtained by measuring the ToF without any mechanical motions. By using white neutrons with the ToF method, one can obtain a large number of holograms with different wavelengths through one measurement: 130 holograms in the range of 0.3–6.6 Å with the typical operating conditions at BL10. This is important to reduce the twin images in reconstructed atomic images from the holograms; the effect of multi-wavelength measurement is shown in Fig. 3.6 using Barton’s algorithm. The total number of holograms of NH can be changed easily by changing the binning of the wavelength or ToF.

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Fig. 3.19 Principle of the ToF method in J-PARC

Atomic images .U (r) can be visualized from the obtained holograms, .χ (k), by using Barton’s method as in XFH explained in Sect. 3.3.1:

{

{

.U (r) = eikr dk χ (k)e−ikrdk

(3.27)

|k|=k

The second integral in Eq. (3.27) is performed on the surface of a hologram with a ﬁxed radius.k, and the ﬁrst integral is over an amplitude of.k. For a practical analysis of the experimental data, the ﬁrst integral over.k replaces to the sum about the discrete .k selected from the 130 holograms obtained by the ToF method. In comparison with XFH experiments in Synchrotron radiation facilities, in which 5–10 wavelengths can be obtained for one XFH experiment, holograms with up to 130 different wavelengths are an excellent advantage of NH to improve the reliability of atomic images despite the weaker beam ﬂux of neutrons than those of photons in synchrotron radiation facilities.
In 2017, Hayashi et al. succeeded in improving the accuracy of atomic images drastically by using white neutrons in MLF and in visualizing local atomic structures around Eu in a typical scintillation crystal 2 at% Eu-doped CaF.2 [67]. From the detailed analysis of the visualized atomic images, some characteristic ﬂuctuation of Ca by Eu doping was found [67]. This means that inverse mode NH with white neutrons can be used for practical investigations in materials science. Thus far, many results have already been obtained in J-PARC, such as semiconductor 0.26 at% Bdoped Si [58], the cage-type rare earth compound 2 at% Sm-doped RB.6 (R: Yb, La) [70], thermoelectric materials, 0.75 at% B-doped Mg.2Si [71], 0.25 at% B-doped Mg.2Sn [72], and power semiconductor 0.06 at% B-doped SiC [73]. Some results are introduced in a later section.

3 Atomic-Resolution Holography

61

3.5.3 Apparatus

Figure 3.20 shows the main part of the instruments for NH installed at BL10 of MLF in J-PARC. The instruments are simple; they consist of a.φ-.ω dual-axes goniometer, .γ -ray detectors, and position-sensitive neutron detectors (PSD) for the neutron Laue
camera method (not shown in Fig. 3.20). A single crystalline sample is set at the tip of the horizontal aluminum rod on the.φ goniometer set on the.ω goniometer; the.ω (corresponding to .θ in Fig. 3.8) and .φ axes are vertical and horizontal, respectively, so that the sample rotates about the two intersecting axes. The .ω axis for neutron experiments corresponds to the.θ axis for XFH in Figs. 3.3 and 3.8. The.γ -ray intensity is plotted on a sphere in the k-space with a radius of.k = |k.| = 2.π /.λ as a function of .φ-.ω, which is referred as a NH hologram.
The angular range of the.ω axis for NH is 0.◦ .≤ ω ≤ 170.◦, which is larger than the typical range of 0.◦ .≤ θ ≤ 70.◦ for XFH because of the difference in the penetration depth of neutrons and X-rays. The larger .ω range is an advantage of NH for obser-
vations of low-symmetry materials, such as materials without inversion symmetry.
The directions of the crystal axes can be determined accurately by the neutron Laue camera method using PSD with an accuracy of approximately 0.3.◦ at the beginning
of each experiment, which avoids ambiguity in analysis of holograms. Currently, the maximum neutron beam size of BL10 is 25.× 25 mm.2 at the sample position. It is
larger than the size of the samples, because the samples have to be fully in the beam during the.φ-.ω rotations; this is a different requirement from those in XFH and PEH.
The element selectivity of inverse mode NH is directly indicated by the energy resolution of the.γ -ray spectra. Two types of.γ -ray detectors with different resolutions
are used for NH on BL10 at present: a Bi.4Ge.3O.12 (BGO) detector for high efﬁciency and low energy resolution and poor element selectivity (./E/E = 30% at.∼400 keV), and a CeBr.3 detector for higher energy resolution (4% at .∼400 keV). To enhance the counting efﬁciency in experiments using a CeBr.3-type detector, a multidetector system is used as shown Fig. 3.20, because prompt.γ -rays emit isotropically. Though

Fig. 3.20 Main part of instruments for NH installed at BL10 of MLF in J-PARC, which consists of a dual-axis goniometer,.γ -ray detectors and Pb shields

Sample γ-ray

ϕ

ω

γ-ray detectors

Incident Neutrons

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Ge detectors have excellent energy resolution (0.3% at.∼400 keV) and provide ideal element selectivity, they are not effective for NH at J-PARC because of insufﬁcient counting speed.

3.5.4 Emitter Elements for NH
For XFH, elements heavier than K can be used as emitters; therefore, a wide variety of materials can be used as targets of XFH. In contrast, only a few emitter elements are available for inverse mode NH. Basically, elements that generate a stronger prompt .γ -ray are suitable for use as the emitters in NH. Figure 3.21 shows the intensity of the prompt .γ -ray from each element. The horizontal and vertical axes indicate the neutron absorption cross section, .σa, and prompt .γ -ray intensity, respectively. Figure 3.21 shows the ease/difﬁculty of NH experiments by using these elements as emitters.
However, the elements in the upper right have a larger .σa, implying that when their concentrations are excessively high, only the surface regions will be observed. Thus far, only elements in the dashed circles in Fig. 3.21 are observable as dopants:

10-15 10-16

10B

Gd

Strong

Cd

B

Sm

Hg

Highest Analytical Sensitivity (cps /atom)

10-17

Ho Er Dy

Eu

10-18 10-19

Nd Au Ta

W Mn

In Sc
Tm

Weak

V

Yb Co Ag

Ir

Pd

Tb Cl Cr Ti

Na Ge Cu

10-20 P H CeZnY

Fe

Al Sn

Si

10-21

10-1

100

101

102

103

104

105

Absoption Crosssection (barn)

Fig. 3.21 Intensity of.γ -ray from each element [74]. Horizontal axis indicates neutron absorption cross section [75]. Dashed circle indicates elements that have already been used as emitters

3 Atomic-Resolution Holography

63

.10B/natural B, Sm, Eu, and Cd with a very small concentration of 1 at% or less. At the same time, the visualization of atomic images of Cu and Si, whose.γ -ray intensity is 4–5 orders weaker than that of.10B, has been successful as standard samples, implying that for practical experiments, the optimization of the emitter concentration under the balance of the.γ -ray intensity and absorption effects is important. Materials that can serve as targets for NH remain limited; therefore, complementary use with other atomic-resolution holography techniques (e.g., XFH, PEF, IPEH) is indispensable and effective for investigations of local structures/hyperordered structures.

3.5.5 Applications
Scintillation Crystal Eu-doped CaF.2
This material is important in the history of atomic-resolution holography because it was the ﬁrst one in which accurate atomic images around a dopant (Eu) were successfully visualized by NH using white neutrons [67]. Subsequently, studies have actively investigated local structures/hyperordered structures of materials containing light elements.
Eu-doped CaF.2 is a typical scintillation crystal that is used for radiation measurements because of its high stability and humidity resistance [76]. Because divalent and trivalent ions are stable for Eu, doped Eu ions are naturally expected to be located at Ca.+2 positions in the form of Eu.+2. However, X-ray absorption ﬁne structure (XAFS) measurements conﬁrmed that Eu ions are trivalent in Eu-doped CaF.2 [67]. Thus, the position of Eu.+3 in the CaF.2 lattice and the origin of electric neutrality should be conﬁrmed.
Figure 3.22 shows the multi-wavelength hologram obtained by superimposing holograms for 34 different wavelengths along the radial direction because the radius of a hologram in.k-space is deﬁned as.|k| = 2.π /.λ. From the 3D holograms in Fig. 3.22, atomic images around doped Eu were successfully visualized as shown in Fig. 3.23a.

Fig. 3.22 Holograms of prompt.γ -rays from Eu in 2% Eu-doped CaF.2 [67]. a 3D volume hologram, and b hologram at.k = 4.05Å.−1. Reproduced with permission from Ref. [67]

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Fig. 3.23 3D atomic images reconstructed a from the obtained holograms and b from simulated holograms using a simple undistorted model of CaF.2 [67]. Reproduced with permission from Ref. [67]
First, the experimentally obtained atomic image shown in Fig. 3.23a is basically consistent with that obtained from simulated holograms by using a model in which Eu is located at the Ca position in the undistorted CaF.2 lattice shown in Fig. 3.23b. This indicates that the doped Eu.3+ is deﬁnitely located at the Ca.2+ position as expected, despite the difference in valency. Thus, the origin of the electric neutrality remains unclear. Previous studies claimed that the modiﬁcation of a local structure is stabilized by the capture of excess ﬂuoride ions that might be located at a neighboring interstitial position. Careful analysis of the atomic images shown in Fig. 3.23a revealed excess F around doped Eu. Figure 3.24 shows the (100) Ca plane including doped Eu (green). The red image in the rectangle in which the maximum scale expanded by a factor of 4 can be found at 2.3 Å from doped Eu. The distance between Eu-excess F is consistent with that obtained by XAFS. Although Eu.3+-interstitial F.− coupling has been proposed in previous studies, Fig. 3.24 shows the ﬁrst evidence obtained directly through experiments.
From a comparison between the experimental and simulated atomic images, another important ﬁnding is that the nearest Ca (dashed circles) in Fig. 3.23a is not obvious and is split into two parts, implying that the positions of Ca ﬂuctuate/distribute around the ideal position of Ca owing to Eu doping, while the nearest neighbor (NN) F is not affected. Figure 3.24b shows the cross sections of the 3D experimental images in the Ca (100) plane. The Ca images split with distance of 1.2 Å. In contrast, the NN F around doped Eu is clearly visualized in Fig. 3.23a, and it indicates that Eu doping does not affect the NN F site despite the shorter distance from Eu than that from the split nearest Ca.

3 Atomic-Resolution Holography

65

a

b

Fig. 3.24 2D images at typical atomic (100) planes. a Ca plane at z = 0.0 Å. Eu (green) was located at the origin. b Ca plane at z = 2.70 Å from Eu [67]. Reproduced with permission from Ref. [67]
Cage-Type Rare Earth Compound: Sm-doped RB.6 (R: Yb, La)
Rare earth hexaborides (RB.6, where R is a rare earth) have attracted interest because of their unique electric, magnetic, and physical properties, including the competition between their magnetic and multipolar orderings, Kondo effects, and valence ﬂuctuations. Figure 3.25 shows the crystal structure of RB.6 (space group: Pm.3¯m), in which a rare earth atom (yellow) is surrounded by the cage of 24 B.
Inelastic X-ray scattering revealed anomalous phonon softening in GdB.6, TbB.6, and DyB.6 [77–79]. Such softening was also observed by Raman scattering, where the energy and intensity of the R motions can be scaled by the free space in the B-cage [80, 81]. Thus, the free space for the R motion in the B-cage in Fig. 3.25 must be an important factor for the unusual properties of this system. In contrast, the dispersion relation of YbB.6 along the [100] direction does not show particular softening. The difference in phonon behavior must be due to the divalency of Yb in YbB.6, whereas R in RB.6 (R = Gd, Tb, Dy) is trivalent because of the difference in ionic radius.
Fig. 3.25 Crystal structure of RB.6 (R: rare earth)(Space group: Pm.3¯m)

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Fig. 3.26 Hologram of Sm dope Yb.11B.6 at Sm at .λ = 2.37Å [70]. Reproduced with permission from Ref. [70]

Fig. 3.27 Atomic images (red) of Yb (a) and B (b) in the (100) plane of 2% Sm dope Yb.11B.6 [70]. Circles and dashed square are expected positions of Yb and unit cell. Reproduced with permission from Ref. [70]
Figure 3.26 shows a hologram of 2% Sm-doped Yb.11B.6 at .λ = 2.37 Å [70]; it is consistent with the simulated one, implying that the hologram patterns were observed successfully. Note that because natural B has a strong neutron absorption cross section, .σa, the sample was prepared using .11B as it has a negligible .σa. Figure 3.27 shows atomic images (red) of (a) Yb and (b) B in the (100) planes of 2% Sm-doped Yb.11B.6 [70]. Circles and dashed squares indicate the expected positions of Yb/B and unit cell, respectively. Yb and B are clearly observed around Sm up to approximately 15 Å from the central Sm. From the comparison with atomic images of the undoped Si, the ﬂuctuation of Sm in the B-cage could be estimated; the isotropic mean-square displacement .σ = 0.25(4) Å for YbB.6.
Figure 3.28 shows the distance dependence of the intensity of atomic images of (a) Yb and La and (b) B. The horizontal axes indicate the distance from Sm. The dashed

3 Atomic-Resolution Holography

67

Fig. 3.28 Distance dependence of intensity of a Yb and b B. Horizontal axis indicates the distance from Sm. Dashed lines indicate the simulation using the undistorted RB.6 lattice. Reproduced with permission from Ref. [70]
lines indicate the simulation using the undistorted RB.6 lattice. The intensity of Yb and La (Fig. 3.28a) is consistent with the simulation, indicating that the ﬂuctuation of Yb/La is negligible. For B (Fig. 3.28a), the intensity of the NN B shows an obvious reduction of 28%, and those of the second NN and further B are consistent with the simulation (dashed line). Because the intensity is reduced if the atoms are ﬂuctuating, this indicates that the NN B is ﬂuctuating owing to Sm doping, and the second NN and further B show no obvious ﬂuctuations. By quantitative analysis based on comparisons with simulations of the effects of ﬂuctuations, the isotropic mean-square displacement of B is estimated as .σ = 0.28 Å, which cannot be interpreted just by the thermal vibration in undoped RB.6 as determined by the conventional diffraction technique. Thus, the ﬂuctuation of B must be caused by Sm doping. The doping effect on Yb/La and the second NN and further B is suppressed because of strong valence bonding in the B-cage. This means that only the NN B cage has the distortion caused by Sm doping, implying that Sm doping realizes disorder in ordered structure (lattice) for hyperordered structures.
3.6 Photoelectron Holography (PEH)
When a sample is irradiated by photons, electrons in the core level are excited, and photoelectrons or Auger electrons are emitted from the atoms in the sample. Photoelectron spectroscopy is used to analyze the kinetic energy of photoelectrons. Because the energy level of a core electron depends on the element and valence, the chemical composition of a sample can be determined. PEH is a method based on core-level photoelectron spectroscopy, from which information on the atomic arrangement around a photoelectron emitter can be obtained.

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The recording process of PEH was mentioned in the previous section. When a monochromatic excitation light is incident on a sample surface, photoelectrons are excited. A spherical wave of photoelectrons centered on the emitter atom is generated. The spherical wave is scattered by the surrounding atoms to produce a scattered wave. The spherical wave and the scattered wave interfere with each other. If the initial spherical wave and the scattered wave are considered to be the reference wave and the object wave, respectively, this interference pattern can be regarded as a hologram. This interference pattern is the photoelectron hologram. In some cases, it is also called photoelectron diffraction, which is referred to as a photoelectron hologram hereafter. This hologram reﬂects the 3D atomic structure around the atom that emits the photoelectrons. Through computations, a 3D atomic arrangement can be reconstructed without a structural model.
In 1990, Harp performed PEH and attempted to reconstruct a 3D atomic image. A photoelectron hologram is easy to observe because the hologram amplitude is more than 10% of the photoelectron intensity [82]. However, owing to the complexity of the electron scattering process, a good atomic image could not be obtained despite the efforts of many researchers until a new method was developed. In 2004, Matsushita demonstrated that an atomic image could be obtained by using a ﬁtting-based calculation method [6, 83] that differs from the Fourier-transform-based method. Subsequently, this method has been improved, and sparse-modeling-based theories have been proposed in recent years.
PEH has the excellent feature of being able to reproduce the local structure around the atom that emits the photoelectrons. The binding energy of the core level is element-speciﬁc. Furthermore, chemical shifts can be used to distinguish atomic sites with different bonding patterns and chemical environments. Owing to the atomic site selectivity of photoemission spectroscopy, impurities in crystals, interfaces, surface adsorbates, and thin ﬁlms can be observed if the local structure is oriented even without any long-range order. The energy of excited photoelectrons is attenuated by inelastic scattering in materials. The inelastic mean free path (IMFP) is short. Therefore, information can be obtained from the surface to a depth of about the length of the IMFP. For example, photoelectrons excited with soft X-rays have a kinetic energy of a few hundred eV. In this case, the measurement can be limited to a depth of approximately 1 nm from the surface. When hard X-rays are used, the kinetic energy of the photoelectrons is several keV, which results in a longer IMFP and allows bulk-sensitive measurements.
The authors have developed an atomic image reconstruction theory that incorporates the scattering process of photoelectrons [6, 7, 83–87]. This method has a feature that 3D atomic images can be reconstructed from a single-energy photoelectron hologram without using the multiple-energy (wavelength) method.

3 Atomic-Resolution Holography

69

3.6.1 Apparatus

Some apparatus can be used to measure photoelectron holograms. There are two types of excitation light sources: X-ray tubes in a laboratory and synchrotron radiation at a synchrotron radiation facility. In the case of X-ray tubes, emissions from magnesium (K.α 1253.61 eV) and aluminum (K.α 1486.6 eV) are used. In a synchrotron radiation facility, micrometer-sized, bright, monochromatic X-rays with variable energy are available; this enables the measurement of small samples. To measure a photoelectron hologram, the angular distribution of the photoelectrons emitted from a sample should be detected. Therefore, an electron analyzer with good angular resolution is required. Two types of electron analyzers can be used to measure the two-dimensional (2D) angular distribution. One is a commercial electron energy analyzer, namely, a concentric hemispherical analyzer (CHA), which measures the 2D angular distribution by rotating both the polar and azimuthal angles of the sample as shown in Fig. 3.29a. This energy analyzer provides good energy resolution; however, it requires a long measurement time because of the scanning over the angles. The another is the 2D display-type spherical mirror analyzer (DIANA) [87], whose structure is shown in Fig. 3.29b. Photoelectrons emitted from the sample are bent by the electric ﬁeld such that they converge on the aperture; after passing through the aperture, the 2D angular distribution of photoelectrons is projected onto the screen. This analyzer can measure 2D angular distributions instantaneously. Recently, a high-resolution retarding ﬁeld analyzer (RFA) has been developed [88, 89], as shown in Fig. 3.29c. By applying a negative voltage to the spherical grid in the RFA, photoelectrons with a higher kinetic energy are projected onto the screen. The RFA is a photoelectron high-pass ﬁlter. Lock-in measurement by varying the negative voltage makes it a bandpass ﬁlter. By developing the spherical electrode, both high energy resolution and high angular resolution were achieved. The RFA has a simple structure and is expected to be widely used in future PEH measurements.

3.6.2 Analysis Method

To analyze photoelectron holograms, it is important to understand the recording process. Energy is conserved before and after photoelectron excitation (Fig. 3.30). The binding level of the initial state is denoted as .EB. Usually, the origin of .EB is taken at the Fermi energy (.EF). Similarly, the energy .Ef of the ﬁnal state is simply expressed as follows when the origin is taken from the Fermi surface:

.Ef = hν − EB,

(3.28)

where .hν is the photon energy. When a photoelectron is excited into vacuum, the
vacuum level becomes the origin. The difference between the vacuum level (.EV) and Fermi energy (.EF), that is, the work function .φ, is expressed as

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~600mm
(a) CHA

Screen (b) DIANA

(c) RFA

Sample Rotations
Screen

Screen

Sample

~40mm
SX

SX
electron orbits

Fig. 3.29 Apparatus for PEH. a Concentric hemispherical analyzer (CHA). The angular distribution of photoelectrons excited by soft X-rays (SX) is measured by scanning the sample angles. b Displaytype spherical mirror analyzer (DIANA). Emitted photoelectrons are focused on a pinhole and projected on the screen. c Retarding ﬁeld analyzer (RFA). The photoelectron angular distribution is directly projected on the screen

.Ekvac = Ef − φ = hν − EB − φ.

(3.29)

The energy from the Fermi level to the bottom of the valence band is called the internal potential .V0. Therefore, the kinetic energy of electrons in a solid can be written as follows, which is used for the formation of the hologram:

.

E

holo k

=

hν

−

EB

+

V0.

(3.30)

The emitted photoelectron or Auger electron can be deﬁned as
.ϕL (k, r, θ, φ) = k E ALlmil+1hl(1)(kr )Ylm (θ, φ),
lm

(3.31)

3 Atomic-Resolution Holography

Fig. 3.30 Energy conservation in PEH

Energy

Photoelectron E
f
E vac k
E V ϕ
E F V 0
hν
E B

71
E holo k Valence band

Surface

Core level
Vacuum Solid Depth

√ where.k is the wavenumber given by.k = 2m Ek/h,. ALlm is a coefﬁcient, and.L is an index for multiple excited states. .hl(1)(kr ) is a Hankel function of the ﬁrst kind. .Ylm(θ, φ) are spherical harmonics. The coefﬁcient . ALlm depends on the transition
process. For example, when the emitted wave function is an .s-wave, the coefﬁcient
is set to . As00 = 1, and the wave function is given as

.ϕs(k, r, θ, π )

=

√1 4π

eikr r

.

(3.32)

Here, the angles .θ and .φ can be regarded as the motion vector:

.ϕL (k, r, θ, φ) = ϕL (k, r) = ϕL (k, r ).

(3.33)

Then, the emitted wave is scattered by the surrounding atoms. The scattered wave function caused by the scatterer atom located at the position vector .a is deﬁned as .ψL (k, r, a). The whole wave function is given by

E .ϕL (k, r) = ψL (k, r, ah).

(3.34)

h

Here, .h is the index of the scatterer atom. The wave function at a distant location is

given as

.ψL (k,

a)

≡

lim
r →∞

r e−ikr ψL (k,

r,

a)

(3.35)

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.ϕL (k)

≡

lim
r →∞

r e−ikr

ϕL (k,

r ).

(3.36)

The observed intensity is given as

.I (k) = E |||||ϕL (k) + E ψL (k, ah)|||||2 .

L

h

(3.37)

The hologram function .χ (k) is deﬁned by removing the reference wave intensity

.I0(K ). Therefore,

EE

.χ (k) =

2R[ϕL∗ (k)ψL (k, ah)].

(3.38)

hL

Photoelectron Emitter

According to the perturbation theory, the photoelectron wave function is given by

E . ALlm = −2πi <olm(k, r)|e · r|φL (k)>,
lm

(3.39)

where .φL (k) is the initial state of the core-level electron and .L is an index used

to distinguish the initial states. For example, in the case of . p-initial states, .L =

px , py, pz should be used..olm(k, r) is a basis function of the excited photoelectron

given by

(

)

.olm (k, r) =

Rl (k, r )Ylm(θ, φ) r < b eiδˆl il+1khl(1)(kr )Ylm (θ , φ) r ≥ b

.

(3.40)

.δˆl is the phase shift of the excitation process, which is determined by the smooth connection to the radial function of.Rl (k, r ) at the border of the mufﬁn-tin sphere (.b is the radius of the mufﬁn-tin sphere)..e is the polarization vector of the light, which

is given by

.e

·

r

=

/ 4π 3

E1 em// Y1 m// .
m // =−1

(3.41)

Therefore, the transition probability is given by

E

E1

.<olm (k, r)|e · r|φL (k)> = RLl/m/

em// c1(l, m, l/, m/)δ(/m − m/m//).

l/m/

m // =−1

(3.42)

For example, the photoelectron wave function excited from the .s-state is shown in Fig. 3.31. When linearly polarized light is utilized, the amplitude of the photoelectron is strong along the vector of polarization. When circularly polarized light is utilized, a

3 Atomic-Resolution Holography

73

Initial state: s-state Emitted photoelectron wave: p-wave

∝h(1)(kr)[-Y (θ,φ)-Y (θ,φ)]

11

1-1

∝h(1)(kr)Y11(θ,φ)

Linearly polarized light

Circularly polarized light

Fig. 3.31 Excited electron waves when the initial state is the.s-state. The wave shape depends on the light polarization

spiral wave is formed. This means that in PEH, the wave function of the photoelectron strongly depends on the initial state and polarization of the light. To simulate the hologram pattern or to reconstruct the atomic image from the hologram, this effect should be taken into account.

Auger Electron Emitter
When an electron is removed from a core level and a vacancy is formed, an electron from a higher energy level falls into the vacancy. This energy excites another electron. The second emitted electron is called an Auger electron. The energy of Auger electrons is determined by the level of the core states, and it is independent of the energy of the excitation light. The Auger electron and X-ray ﬂuorescence emission processes are competitive transitions. As the core vacancy deepens below 1 keV, X-ray ﬂuorescence gradually becomes dominant.
The wavefunction of Auger electrons is also important for analysis. If the orbital quantum number of the core hole is.l, the probability that an electron in orbital.l + 1 will fall into the core hole increases. Then, Auger electrons with the .l + 2 orbital quantum number are emitted. For example, Cu .L V V is close to the . f -wave.

Inelastic Scattering Process
Some photoelectrons lose energy by inelastic scattering; simultaneously, the probability of elastic scattering is high. The average distance a particle travels in a material

74

Fig. 3.32 Energy

10

dependence of IMFP

T. Matsushita et al.

IMFP (nm)

1

102

103

104

Electron energy (eV)

before an inelastic collision is expressed as the inelastic mean free path (IMFP). Figure 3.32 shows the energy dependence of the IMFP. The IMFP exhibits roughly the same energy dependence regardless of the material composition. The solid line in the ﬁgure is represented by

143

√

.IMFP = E2 + 0.054 E.

(3.43)

The IMFP varies slightly with the substance; this variation is reprsented by the

gray area in Fig. 3.32. This IMFP is related to the surface sensitivity of PEH. The

surface depth sensitivity of PEH is almost the same as that of the IMFP. When

using photoelectrons with higher kinetic energy, the surface sensitivity decreases and

the bulk sensitivity increases. This IMFP changes the hologram pattern. The wave

function of the emitted photoelectrons is attenuated at the scatterer atom located at

.ah. When the IMFP length is deﬁned as .l, the hologram is given by

.χ (k)

=

E

E

2R[ϕL∗ (k)ψL (k,

ah )]

exp

|

−|ah | 2l

|

.

hL

(3.44)

Finally, when the ﬂuctuation effect is taken into account, the hologram is given by

.χ (k)

=

E

E

2R[ϕL∗ (k)ψL (k,

ah )]

exp

|

|

−|ah |

2l

exp[−σh2|k|2(1

−

cos

θahk )].

hL

(3.45)

3 Atomic-Resolution Holography

75

Elastic Scattering Process and Hologram Formation

Photoelectrons interact with the electrostatic potential of atoms and are scattered. The interference between the scattered and unscattered waves forms a photoelectron hologram. When photoelectrons are scattered by atoms, the atoms act as both a scatterer and a convex lens. Here, consider an inward.s-wave progressing toward the origin as shown in Fig. 3.33a. If there is no potential at the origin, the wave will pass through the origin and become an outward .s-wave with the same amplitude. Then, consider an inward . px -wave progressing toward the origin as shown in Fig. 3.33b. This . px -wave also passes through the origin and becomes an outward . px -wave.
Next, consider the scattering potential located at the origin as shown in Fig. 3.33c. The inward wave propagates to the origin. Around the origin, the radial frequency of the wave function increases because of the scattering potential. Then, this wave becomes an outward wave, with the same amplitude as that before scattering, although

t = -Δ

t=0

t = Δ

(a) s-wave

(b) px-wave

(c)
Before sattering

Interaction Higher frequency

Phase shift Scattered wave Original phase Incident wave

Wave Potential
Fig. 3.33 Motion of spherical wave for scattering calculation

76

(a)

E k

=

1000eV

T. Matsushita et al.

y (nm)

y (nm)

Emitter Cu atom
(b)
FFP
Emitter Cu atom
x (nm)
Fig. 3.34 a Wave function scattered by a Cu atom. b Electron density of (a)
the phase of the wave is changed. Scattering calculations involve the determination of this phase shift.
Figure 3.34 shows the scattering results calculated using these equations. The wave function around the scatterer Cu atom is modiﬁed, and the wave has a strong amplitude behind the Cu atom as shown in Fig. 3.34a. A strong electron density, called the forward focusing peak (FFP), is formed behind the Cu atom as shown in Fig. 3.34b.
Figure 3.35 shows the photoelectron angular distribution of a single Cu atom scatterer. An FFP is formed behind the Cu scatterer. Interference rings appear around the FFP. When the atomic distance is increased, the number of interference rings increases. This means that the center of the FFP and interference rings indicates the direction of the scatterer atom, and the number of interference rings indicates the atomic distance between the emitter atom and the scatterer atom.

3 Atomic-Resolution Holography

77

(a)

(b)

Cu Cu

Emitter

Emitter

Fig. 3.35 Photoelectron angular distribution of a single Cu atom scatterer

Consider a case with many atoms, such as a crystal. From Eq. (3.45), we deﬁne

. t

(k,

a)

=

|a|

E

2R[ϕL (k)ψL

(k,

a)]

exp

|

−|a| 2l

|

exp[−σ

2|k|2(1

−

cos

θak )].

L

(3.46)

The photoelectron angular distribution is given by

.I

(k)

=

E
h

t

(k, ah) |ah|

.

(3.47)

This means that the hologram generated by many atoms is described as the sum of the holograms generated by each scatterer atom. PEH features the effect of multiple scattering. The effect of multiple scattering is large in the direction of the FFP. When multiple scattering is taken into account, the intensity of the FFP may decrease under some conditions. Multiple scattering is essential for accurate simulation. However, for simplicity, some explanation is possible without considering multiple scattering effects. Figure 3.36 shows a simulated hologram using multiple scattering method and a measured hologram of Cu Auger. FFPs of [001], [101], and [111] were observed, and the theoretical calculations reproduced the experimental pattern well.

Analysis by Using Scattering Pattern Function
As discussed in the previous section, the pattern of the measured hologram is the sum of the holograms made by single atoms. Hereafter, the pattern created by a single atom (Eq. 3.46) is referred to as the scattering pattern function. We consider whether it is possible to obtain the atomic position .ah from the hologram using this equation..ah is described by a nonlinear equation. Therefore, it is difﬁcult to solve in its original form. Thus, we introduce a 3D function that represents the distribution of

78
(a) [101]

[001]

(b) [111]

[001]

T. Matsushita et al.

[010]

[010]

[100]

[100]

Simulation

Experiment

Fig. 3.36 a Simulated.LVV Auger hologram of Cu crystal. b Measured.LVV Auger hologram [84]. The kinetic energy of an Auger electron is 914 eV

atoms. When this function is composed of .δ-functions at the locations where atoms

exist, as given by

.g(r)

=

E
h

δ(r − ah) |ah|

,

(3.48)

it is immediately apparent that the hologram can be calculated by integrating the

following equation:

{

.χ (k) = g(r)t (k, r)d r.

(3.49)

To obtain the function .g(r) representing the distribution of atoms from the photo-

electron hologram using this formula, we use voxels that divide the real space into

a 3D lattice, rather than the sum of the .δ-functions. The following equation is then

obtained:

E .χi = g j t (ki , r j )/V ,

(3.50)

j

where.i and. j are indices for the hologram pixels and real space-voxels, respectively, and ./V is the volume of the voxel. This is a linear equation that transforms a 3D real-space voxel to a 2D hologram. This voxel can be created independently of the
crystal lattice structure. However, this system of equations cannot be solved by a sim-
ple gradient method because the total number of unknowns is much larger than the
number of hologram data. For example, if a single-energy hologram is measured at a solid angle of 1.◦, the total number of measurement points is approximately 20,000. Further, if a .±1 nm real-space region is divided into 0.01 nm voxels, the number of voxels is.2003. The number of voxels (i.e., unknowns in real space) is 400 times the
number of hologram pixels (i.e., equations). To solve this equation, we use the prop-

3 Atomic-Resolution Holography

79

Emitter 4

2

0

4 -2
0

-4

-2

0

2

-4 4

Fig. 3.37 Atomic image reconstructed from Cu hologram in Fig. 3.36

erty of the atomic image that it is positive and zero in most regions. Therefore, sparse

modeling with L1 regularization, which is used in machine learning, is effective. The

evaluation function using sparse modeling is given as

.E

=

E |χi(exp)

−

χi(sim)|2

+

E λ

gj,

i

j

(3.51)

where .λ is a parameter used to control the sparseness. The computation with this equation is called SPEA-L1 (scattering pattern matrix extraction algorithm using L1 regularization). A 3D atomic image of copper atoms reconstructed from the ﬁgure using SPEA-L1 is shown in Fig. 3.37. A clear fcc atomic arrangement is seen to be obtained.
In this light, some key points should be noted when measuring photoelectron holograms. The types of samples suitable for PEH include single crystals, doped single crystals, and surfaces of single crystals. Amorphous materials are not suitable. In addition, the same sample conditions as those for photoelectron spectroscopy are required. The sample must be able to be placed in a vacuum and should be electrically conductive. In addition, when using photoelectrons with low kinetic energy, surface cleaning is required. Because the amplitude of a photoelectron hologram is as high as 10% of its intensity, measurement is relatively easy. In contrast, when measuring impurities, it is important to ensure a sufﬁcient signal-to-noise ratio. The kinetic energy of photoelectrons should be 400 eV or higher for the following reasons. First,

80

T. Matsushita et al.

the higher the kinetic energy of the photoelectrons (i.e., the shorter the wavelength of the electrons), the better the spatial resolution. Second, the shape of the scattering pattern function is better and easier to analyze. Third, a higher kinetic energy weakens the phenomenon of photoelectron refraction at the surface, thereby distorting the hologram by the inner potential. Finally, the sensitivity is affected by the kinetic energy of the surface. For example, PEH using hard X-rays has been studied. Because the kinetic energy of the photoelectrons is very large, surface cleaning is no longer required. (Its practical application would greatly expand the range of its use.) When measuring surface structures, a lower kinetic energy should be used. Although atomic image reproduction is possible with a single energy, the 2D angular distribution of photoelectrons must be measured over a wide range of solid angles. An angular accuracy of at least 1.◦ is required. In addition, the orbital quantum number of the core level must be taken into account. If the orbital quantum number of the ﬁnal state is large, the interference fringes are weakened because many nodal directions appear in them. The.s-state tends to be the most accurate for hologram reproduction, whereas the . f -state tends to be less accurate.

3.6.3 Applications
As-doped Silicon
Silicon is widely used in the production of semiconductors, which are key components in electronic devices. When silicon is doped with small amounts of other elements (dopants), such as boron or phosphorus (arsenic), it can become either a p-type or n-type semiconductor, respectively. This means that it can be used to make diodes, transistors, and other electronic components that are essential to modern technology. Direct observation of the atomic arrangement is important because the state of dopants changes the state of the charge carrier. High doping is necessary to increase the number of charge carriers, but doping has its limitations. Even if the dopant concentration is increased, the number of charge carriers will not increase beyond a certain value. Photoelectron holograms have been measured for highly Asdoped Si [90]. As shown in Fig. 3.38a, the X-ray photoelectron spectrum (XPS) of As in Si are composed of three elements. This indicates that As atoms take on three different valence states in Si, labeled as BEH, BEM, and BEL. The XPSs at various emission angles were observed. Then, peak ﬁttings were carried out to obtain the three holograms shown in Fig. 3.38c–e. The hologram of Si 2p is also shown in Fig. 3.38b. No hologram pattern was observed for BEL. This indicates that the atoms around the As are random, and the As-Si compound is presumably formed. In contrast, a clear pattern similar to the Si 2. p hologram was observed for BEH. The Si atomic arrangement was reconstructed from the atomic image using the SPEA-L1 method. This indicates that BEH is a substitution site. However, since the nearest Si atom was not clearly observed, the position of the nearest atom was found to be ﬂuctuating, as shown in Fig. 3.38f. Finally, the reconstructed atomic image of BEM is close to

3 Atomic-Resolution Holography

81

that of the substitution site but is unclear. Using the information obtained from the molecular dynamics calculations, it was found that BEM is an As.2V hyperordered structure consisting of two As atoms and a vacancy (V) as shown in Fig. 3.38g. Thus, PEH is a powerful technique for measuring the chemical states and atomic structure of dopants.

Phosphorus-doped Diamond
Diamond is a semiconductor material with several unique properties that make it an attractive option for various applications. Diamond is known for its exceptional hardness, high thermal conductivity, and wide bandgap, which allows it to withstand high temperatures and high voltages. To make a diamond semiconductor of a speciﬁc type (either p-type or n-type), impurities are added to the diamond lattice during its growth. These impurities, known as dopants, introduce additional electrons or holes into the diamond lattice, which can be used to conduct electrical current. In the case of p-type diamond, boron is often used as a dopant, which introduces holes into the diamond lattice. Doping diamond with phosphorus is expected to make it an n-type semiconductor. However, it is difﬁcult to produce an n-type semiconductor, the reason for which was discovered via PEH [91].
Diamond whose (111) surface was doped with phosphorus was used as a sample. Figure 3.39a shows its core-level XPS. The spectrum shows that phosphorus has two chemical states in the doped diamond. Then, we measured the photoelectron holograms of the .α and .β peaks, which are respectively shown in Fig. 3.39b and d. First, a 3D atomic image of the .α peak was obtained, as shown in Fig. 3.39f, where the circles in the ﬁgure indicate the expected atom positions in the diamond structure. The ﬁgure indicates that the diamond structure is surrounded by phosphorus atoms. The phosphorus atoms were found to be substituted with carbon atoms. There are two atomic sites, A and B, in the diamond unit cell. In Fig. 3.39f, the atomic images of the A site are clear while those of the B site are unclear. This ﬁgure indicates the occupation ratio of the A and B sites. Additional analysis revealed that the occupation of the A site was approximately 80%; this result was unexpected because if substitution occurs randomly, the occupation ratio should be 50%. This biased occupancy was inferred to occur during the crystal growth. In the crystal growth of the (111) surface, after the surfaces of the A site grow, those of the B site grow. Then, the surfaces of the A site grow again. This process occurs repeatedly. When the A site grows, three bonds to carbon atoms appear. In contrast, when the B site grows, only one bond appears. Therefore, when A sites grow, phosphorus atoms can easily bond at the surface owing to the large number of bonds. This explains why phosphorus atoms selectively occupy the A sites that appear in the hologram.
Next, the atomic arrangement of the .β peak was estimated. The hyperordered structure with two missing carbon atoms, where a phosphorus atom was located at the center of the vacancies, was found with a PV split vacancy complex (PVSVC). The electronic structure was calculated from this atomic arrangement, and it was found to trap electron carriers. The XPS peak intensity of the .β site is larger than

82
(a) As 3d

T. Matsushita et al.

Intensity (arb. unit)

(b) Si 2p

BEH

BEM

BEL

3d 3d

3/2

5/2

Kinetic energy (eV)
(c) BEH

(d)

(e)

BEM

BEL

(f) BEH

(g) BEM

V As

As

Si

As

Si (fluctuated)

Si

Fig. 3.38 a XPS of As dopants in Si. three components, BEH, BEM, and BEL, were observed. b Photoelectron hologram of Si 2p. c–e Photoelectron holograms of BEH, BEM, and BEL. f Atomic arrangement deduced from the photoelectron hologram of BEH; As atoms were substituted for Si atoms. The surrounding Si atoms were ﬂuctuated. g Atomic arrangement deduced from the photoelectron hologram of BEM, which was presumed to be a hyperordered structure consisting of two As atoms and a vacancy.V

3 Atomic-Resolution Holography

83

(a) P 2p

2p1/2 2p3/2

α
2p 2p

β

1/2 3/2

Intensity (a.u.)

560 562 564 566 568 570 572 Kinetic energy (eV)

(b)

(c) (d)

(e)

Exp.

Sim. Exp.

Sim.

a = 3.57

(f)

(001) = 0.9

(010)

(g)
A site

(100) B site

View from A site View from B site
(h) PV Split Vacancy

Fig. 3.39 a XPS of P.2 p in diamond. Two components,.α and.β, were observed. b and d Observed photoelectron holograms of the.α and.β peaks, respectively. f Cross-sectional image of the reconstructed 3D atomic image of the.α peak. The plane located at (001) = 0.9 Å is displayed. Large and small circles indicate the expected positions viewed from the A and B sites in the diamond structure, respectively. g and h Atomic arrangements obtained from the photoelectron holograms of.α and.β, respectively. c and e Simulated holograms calculated using the obtained atomic arrangements
that of the.α site, indicating that electron carriers generated at the.α sites are trapped by the structure of the .β sites. Therefore, all the electron carriers generated at the .α sites are trapped, and this diamond does not become an n-type semiconductor.
The structure also has four different bond orientations. Further analysis revealed an occupancy bias. The vertical structure on the surface is produced more than the

84

T. Matsushita et al.

structures in other directions. This was also presumed to be due to crystal growth. Phosphorus was adsorbed at the A site, as mentioned above. Because phosphorus has a lone pair, it was difﬁcult to adsorb a carbon atom above the phosphorus. Therefore, vacancies often formed above phosphorus atoms. After this process, the phosphorus atoms moved and formed a PVSVC. Thus, PEH can be used to visualize the movement of dopants during crystal growth from their structure. The atomic arrangement of dopants is very important structural information for the development of new materials.

Defect at Interface of H-Terminated Diamond and Amorphous Al.2O.3
The study of insulating layers is important for the formation of diamond devices. To fabricate diamond devices, it is necessary to form a thin insulating layer, such as Al.2O.3 or SiO.2, on the diamond surface. Defects at the interface of insulating layers on diamond semiconductors can affect the electrical properties and lead to reduced device performance or failure. To minimize the number of defects, it is important not only to use high-quality deposition techniques but also to know the atomic structure of the defects to understand their physical features.
The atomic arrangement of defects formed between Al.2O.3 amorphous insulator and hydrogen-terminated diamond (001) was observed by PEH [92]. Figure 3.40a shows the XPS for carbon 1.s. The C–C bonding originates in the diamond bulk. In addition, C–H and C–O bonding components were observed. Ideally, there should be only C–C and C–H components; the C–O component is considered to be a defect. The observed C–C hologram (Exp.) and simulated C–C hologram (Sim.) are shown in Fig. 3.40b. Two holograms are in good agreement. Figure 3.40c shows the observed and simulated holograms of the C–O defect. The atomic arrangement was reconstructed from this hologram and analyzed, resulting in the structure shown in Fig. 3.40d. It is known that the hydrogen termination of diamond forms a structure of dimer rows of surface carbons. It was found that a C–O–Al–O–C bridge was formed that bridged the two dimer rows. Knowledge of the atomic arrangement of the defects will provide important insights for the evaluation of physical properties and the search for better deposition conditions in the future. Thus, PEH is a powerful tool for exploring the atomic arrangement of buried interfaces.

3.7 Inverse Photoelectron Holography (IPEH)
PEH and XFH are often performed at synchrotron radiation facilities, whereas NH requires a pulsed white neutron source such as J-PARC (Tokai, Japan). In other words, experiments need to be performed at a large-scale facility. To perform atomicresolution holography in the laboratory, Hayashi and Matsushita developed an inverse photoelectron holography (IPEH) [93]. This method applies energy-dispersive X-ray spectroscopy (EDX/EDS) in scanning electron microscope (SEM). EDX is used for

Intensity (a.u.)

3 Atomic-Resolution Holography

85

(a) C 1s

C-C
C-H C-O

698

699

700

701

702

Kinetic energy (eV)

(b) C-C

[111] (c) C-O
[101]

Exp.

Exp.

Sim. Exp.

Sim.

(d)
Amorphous Al O
23
C-O-Al-O-C
C-H Diamond bulk

[001] Al

H

OO

C Emitter (C-O)

[010] [100]

Fig. 3.40 a XPS of carbon .1s of H-terminated diamond with Al.2O.3 amorphous layer. Three components, C–C, C–H, and C–O, were observed. b Observed photoelectron hologram of C–C (Exp.) and the simulated photoelectron hologram of C-C (Sim.) using the diamond structure. c Observed photoelectron hologram of C–O (Exp.) and simulated holograms (Sim.) using the atomic structure of (d). The expected forward focusing peak positions are marked by circles. d Obtained atomic structure of a defect located at the interface of the H-terminated diamond and the Al.2O.3 amorphous insulating layer. A C–O–Al–O–C bridge is formed between the carbon dimer rows of the H-terminated diamond

performing elemental analysis and composition analysis by detecting the characteristic X-rays generated by electron irradiation. Because the energy of the characteristic X-rays is element-speciﬁc, it provides information on the element composition of the samples. Further, because the incident electrons have the nature of plane waves, the

86

T. Matsushita et al.

conditions are the same as those of the inverse mode of holography. Therefore, by measuring the intensity of characteristic X-rays while scanning the incident angles on the sample, a hologram similar to a photoelectron hologram can be obtained. As with PEH, the 3D atomic arrangement around a particular element can be measured. In addition, the energy of the electron beam can be easily changed. Thus, multipleenergy holograms can be easily measured. A multiple-energy hologram is more informative and thus provides a more accurate image of the atomic arrangement than an ordinary single-energy hologram. After the ﬁrst experiment, some subsequent experiments were performed [94, 95].
However, the disadvantages of the current commercially available SEM-EDX are that the intensity of the characteristic X-rays is weak and the measurement takes long time, because the electron beam current is low for reasons such as the consideration of sample damage. In addition, the size of the X-ray detector is small. This problem can be solved by developing a special apparatus for IPEH that can use high-current electron beams and a large-sized X-ray detectors.

3.7.1 Apparatus
A SEM with additional EDX is available as an instrument for IPEH. In addition, a sample rotation stage is required to scan the incident angles of the sample. The measurement apparatus is shown in Fig. 3.41. Electrons emitted from the electron gun are injected into the sample. X-rays emitted from the sample are detected by an X-ray detector capable of spectroscopy to detect the characteristic X-rays of the element of interest. Holograms are obtained by measuring the intensity variation of the characteristic X-rays while manipulating the sample orientation, .θ and .φ.

3.7.2 Applications
An example of the ﬁrst observation of IPEH is shown in Fig. 3.42 [93]. A single crystal of SrTiO.3(001) is used. IPEH is performed inside a SEM (Hitachi High-Technologies Corporation S-3400N). A Ge solid-state detector (CANBERRA GUL0055P) with a detection area of 50 mm.2 and a fast digital signal processor (XIA LLC DXP Saturn) were used to measure the Ti .K line spectra. Two angles of the sample stage were scanned to record the hologram. The solid angle of the detector was 0.564 sr, and the emission current of the electron gun was 0.00375 mA. Characteristic Ti .K X-rays were excited by using 6.00–6.30 keV electron beams. These electron acceleration voltages were selected because they are close to the Ti.K absorption threshold (4.965 keV). This effectively suppresses the Ti .K lines caused by inelastically scattered electrons, which are incoherent and thus do not contribute to the formation of the hologram. The use of an electron beam energy slightly higher than the absorption threshold is crucial.

3 Atomic-Resolution Holography

87

Electron gun Fluorescent x-rays

Electron beam
ϕ Sample

X-ray detector
θ

Fig. 3.41 Apparatus for IPEH (a) Experiment

(b) Simulation

Fig. 3.42 Inverse photoelectron holograms of SrTiO.3. a Experimental and b simulated holograms in k space. The kinetic energy of the electron beam is 6.0 keV. In the hologram simulation, a spherical 5457 atom cluster with a radius of 25 Åwas used
Figure 3.43 shows reconstructed 3D real-space images around the Ti atom by using the measured holograms. The atoms of all elements are clearly shown at the theoretical coordinates. The light element O is normally difﬁcult to reconstruct owing to its low-electron-scattering cross section. Generally, with increasing atomic number, the image becomes more distinct. The light O atoms are also successfully reconstructed in this experiment.

88

T. Matsushita et al.

[001]
Ti

O Sr

[001]
Ti: Emitter of characteristic X-ray
Fig. 3.43 3D images of SrTiO.3 reconstructed from multiple-energy holograms taken at 6.00, 6.08, 6.15, 6.22, and 6.30 keV
3.8 Summary and Outlook
This chapter has described the principle, experimental setup, and application of atomic-resolution holography. We demonstrated that atomic-resolution holography is a powerful tool to characterize hyperordered structures, especially disorder in order in crystalline materials.
We have discussed four types of atomic-resolution holographies: PEH, XFH, NH, and IPEH. Each technique has advantages and disadvantages, and thus, it is important to choose or combine these techniques depending on the aim of experiments. Tables 3.1 and 3.2 summarize the characteristics, features, and requirements of these techniques for practical applications.
Note that the information presented in these tables is based on the present developmental status of these techniques. Further improvements of experimental and analytical methods are ongoing, and thus this information may change in the future. Such upgrades will signiﬁcantly extend the applicable range of atomic-resolution holography, and the role of this technique will become increasingly important in understanding the origin of the functionalities of various materials including hyperordered structures.

3 Atomic-Resolution Holography

89

Table 3.1 Characteristics of PEH, XFH, NH, and IPEH.

PEH

XFH

NH

IPEH

In/Out Light element

X-

X-ray/Fluo.

ray/Photoelectron X-ray

√

.

.×

Neutron/.γ -ray √
.

Electron/Fluo. X-ray √
.

Bulk or surface Surface

Bulk

Bulk

Surface

Insulators Sample size

1–10 nm .× .>0.01 mm.2

0.01–1 mm √
.
.>1 mm.2*

.∼1 cm √
.
0.5–1 cm.3

1–10 nm
. × .>1 mm.2**

Vacuum is

Yes

No

No

Yes

necessary

.*For normal-mode XFH, .10 × 10 .µm.2 or less can be a target. Further, such a size is possible for inverse mode XFH, if a technique to keep the incident X-ray from irradiating a speciﬁc area during
sample rotation can be developed .**Ideally, 1.µm.2 or less is possible. A technique needs to be developed to keep the electron beam from irradiating a speciﬁc area

Table 3.2 Features and requirements of PEH, XFH, NH, and IPEH

Features

Requirements

PEH – Chemical-state-speciﬁc structural analysis is possible – Surface cleaning is necessary

XFH – High reliability of atomic images

– Lightest measurable element is potassium

– Sample environment is ﬂexible

NH

– High reliability of atomic images

– Observable emitters are limited

– Multi-wavelength holograms can be efﬁciently collected

IPEH – Laboratory experiment is possible

– Surface cleaning is necessary

– Measurement time is long

Acknowledgements The authors gratefully acknowledge the Ministry of Education, Culture, Sports, Science and Technology, MEXT, of Japan for the ﬁnancial supports through JSPS Grantsin-Aid for Transformative Research Areas (A) “Hyper-Ordered Structures Sciences” via Grant Nos. 20H05878, 20H05881, 20H05884, and 21H05547, Innovative Areas “Hydrogenomics” via Grant Nos. 19H05045 and 21H00013,"Materials Science on Mille-Feuille Structure" via Grant Nos. 20H05878 and 20H05881, “3D Active-Site Science” via Grant Nos. 26105001, 26105006, 26105013, 26105014, and 26105010, “Materials Science on Synchronized LPSO Structure” via Grant No. 23109001, Scientiﬁc Research (A) through Grant Nos. 19H00655 and 20H00303, Scientiﬁc Research (B) through Grant Nos. 21H01027 and 22H01774, and Young Scientists via Grant No. 20K15024. This work was also supported by the Nippon Sheet Glass Foundation for Materials Science and Engineering and Japan Association for Chemical Innovation. The synchrotron experiments were carried out with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) and Photon Factory Program Advisory Committee. The neutron experiments were performed under the user program of the Materials and Life Science Experimental Facility of J-PARC.

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