Insulators: Types, Properties and Uses : Types, Properties and Uses [1 ed.] 9781611223965, 9781617619960

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Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

ELECTRICAL ENGINEERING DEVELOPMENTS

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INSULATORS: TYPES, PROPERTIES AND USES

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ELECTRICAL ENGINEERING DEVELOPMENTS

INSULATORS: TYPES, PROPERTIES AND USES

KEVIN L. RICHARDSON

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EDITOR

Nova Science Publishers, Inc. New York Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Insulators : types, properties, and uses / editor, Kevin L. Richardson. p. cm. Includes index. ISBN: (eBook) 1. Electric insulators and insulation. I. Richardson, Kevin L. TK3421.I555 2010 621.319'37--dc22 2010036153

Published by Nova Science Publi

 New York

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CONTENTS Preface Chapter 1

Ferromagnetic Insulator in the Electron-Doped Manganites Jie Yang and Yuping Sun

1

Chapter 2

Insulator Inspection Technologies Joon-Young Park

Chapter 3

High-K Dielectric Insulators Used in Low-Voltage Organic Field-Effect Transistors Hu Yan, Toshihiko Jo and Hidenori Okuzaki

61

Quantum Non-Magnetic States near Metal-Insulator Transition: A New Candidate of Spin Liquid State Su-Peng Kou

75

Chapter 4

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vii

Chapter 5

Chapter 6

Chapter 7

Chapter 8

29

Theory of Normal State Transport in Cuprates in Magnetic Field Zhihao Geng

113

Electrodynamics of Mott Insulators and Insulator to Metal Transitions A. Perucchi, L. Baldassarre and S. Lupi

133

Leakage Current on High Voltage Contaminated Insulators P. T. Tsarabaris and C. G. Karagiannopoulos

173

Synthesis and Study of Nanoscale Magnetic Semiconductor and Magnetic Metal/Insulator Films: Role of Energetic Ions S. Ghosh

Index

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PREFACE An insulator, also called a dielectric, is a material that resists the flow of electric current. An insulating material has atoms with tightly bonded valence electrons. These materials are used in parts of electrical equipment, also called insulators or insulation, intended to support or separate electrical conductors without passing current through themselves. Some materials such as glass, paper or Teflon are very good electrical insulators. This book presents topical research data in the study of insulators, including design and development of a new type of ferromagnetic insulator; insulator inspection technologies; high-k dielectric insulators used in low-voltage organic field-effect transistors; the electrodynamics of Mott insulators and insulator-to-metal transitions; and the leakage current on high voltage contaminated insulators. Chapter 1 - In this chapter, the authors will focus on the investigation of the ferromagnetic insulator behavior for the electron-doped manganites La1-xTexMnO3 (x 0.15). The results show that the authors can obtain ferromagnetic insulators and tune their Curie temperatures TC in the electron-doped manganites La1-xTexMnO3 (x 0.15) by changing the oxygen stoichiometry (La0.9Te0.1MnOy), doping Pr at La-site (La0.9-xPrxTe0.1MnO3), and substituting Cu for Mn (La0.85Te0.15Mn1-xCuxO3). Moreover, the ferromagnetic insulating properties of LaMn1-xTixO3 (0 x 0.2) are also studied. The appearance of ferromagnetic insulator state in these compounds can be understood in terms of the combined effect of destruction of Mn2+-O-Mn3+ double exchange interaction network, the appearance of superexchange ferromagnetic interaction, and the introduction of random Coulomb potential caused by the introduction of oxygen vacancies and the doping effect giving rise to the localization of carriers. The present studies should provide a route to design and develop new type of ferromagnetic insulators. Chapter 2 - Insulators, which are used for insulation and support of power lines, may involve minute defects incurred during the manufacturing process, and thus require sufficient testing before they are installed on a steel tower. However, even so insulators deteriorate after installation due to high voltage stress, mechanical stress, thermal stress and environmental stress, which may lead to an outage. As power transmission plays a key role in supplying electricity to the industrialized world, such an insulator failure could have a drastic impact on the industries, economy, and even national security of a country. For this reason, the inspection operation has been carried out regularly for all the insulators. This chapter introduces various inspection technologies to prevent unforeseen power outages through the

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Kevin L. Richardson

detection of faulty insulators in advance. Chapter 3 - To date three kinds of approaches have been mainly proposed and studied in order to achieve low-voltage organic field-effect transistors (FETs). First one is simply using a high-k insulator to obtain certain electric capacitance at lower gate voltage, which is based on the high dielectric constant. Second one is using an ultrathin self-assembled monolayer of organic compound as the gate dielectric, which is based on the small thickness of the insulator layer. Third one is using polymer electrolyte containing ionic liquid, which is based on polarization by ion-moving at semiconductor/insulator interface. In the authors opinion the first one among the three approaches can provide robust platform for fabrication of the lowvoltage organic FETs and for fundamental studies of organic semiconductors. In this chapter the authors describe their recent efforts on a polymer-microfiber-based FET-channelpatterning, and air-stable ambipolar pentacen/C60 FET with a self-encapsulation, as well as fabrication of low-voltage organic FETs by using high-k insulators such as HfSiO or SrTiO3. At last, the authors also describe their researches on synthesis and colloidal dispersion of SrTiO3 nanoparticles, as well as preparation of SrTiO3 films by a wet-process. Chapter 4 – In this paper, the authors investigate the π Hubbard model and the Hubbard model on honeycomb lattice within an approach keeping spin rotation symmetry. Quantum non-magnetic insulators are explored near the Mott-Insulator transition. Such type of quantum non-magnetic insulator in bipartite lattices is driven by quantum spin fluctuations of relatively small effective spin-moments. By considering the fermionic nature of vortices (halfskyrmions), a new type of quantum state - nodal spin liquid is found to be the ground state of the non-magnetic insulator state. There exist three types of quasi-particles in nodal spin liquids: nodal fermionic spinons, gapped bosonic spinons and roton-like U 1 gauge field. Chapter 5 – Based on the t-J model, the theory of normal state transport in cuprates with the external magnetic field is presented. The influence of a strong external magnetic field on the normal state resistivity of the underdoped cuprates is discussed. There is general agreement that the parent compound of these materials is a Mott insulator with antiferromagnetic long range order. The authors begin with a discussion of the linear response approach and specular reflection model to establish the basic electromagnetic response scenario. The authors show the microscopic model in cuprate with external magnetic field and introduce the fermion-spin representation to the charge-spin separation formulation. The mean field theory and self-consistent equations approach in the case of external magnetic field is presented. The holon self-energy is obtained by employing the path integral representation of the Green function. Then the formalism of resistivity under an external magnetic field is given by the Kubo formula. Special emphasis is given to the analysis of the vector potential. The vector potential taken the coulomb gauge is discussed within the linear response method, which cannot be measured directly but plays a crucial role in quantum phenomenon. Theoretical ideas and analytical methods are introduced and presented in some detail. Some numerical result is given in the last part. It is shown that when the superconductivity in the underdoped cuprates is suppressed in the presence of a strong external magnetic field, the system reveals a low temperature normal state insulator-metal crossover. Chapter 6 - Here the authors will review the electrodynamic properties of Mott insulators and other strongly correlated systems across insulator to metal transitions. The authors will first introduce the theoretical framework within which strongly correlated systems are

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Preface

ix

presently understood. Next the authors will give an overview of the techniques discussed here, for temperature and pressure dependent measurements. The authors will then address the properties of several compounds as V2O3, VO2, V3O5, and NiS2. This will allow them to compare and contrast the optical properties across insulator to metal transitions, when the insulating state is of the Mott or of the charge-transfer type, or when electron-phonon mechanisms play some role. Chapter 7 - The present chapter will present a short review of research work regarding leakage current. In addition, measurement system of the leakage current is introduced using a high sampling frequency analogue/digital converter. The application of this system in 20 kV insulators contaminated from a compound of salt and kaolin are described. Using this system measurement in a time frame of one period (50 Hz) were done and an investigation of the observed phenomena is attempted with the assistance of i-u characteristic curves plotted for one cycle of voltage application. The fact that existence of partial discharges on the surface of the contaminated insulators, beyond threshold field intensity leads to radiation emission, is also be examined. An estimation of the free electrons energy has been done and the corresponding emitted radiation which seems to include acoustic waves, radio waves, microwaves and infrared waves is also examined. The classification of the leakage current values of a typical porcelain insulator of 20 kV, contaminated by salt and kaolin, is presented. The classification is based on the collaboration between the above high precision data acquisition system having high sampling rate and an unsupervised self-organized neural network. In addition a simulation model for contaminated insulators is presented. The proposed model will be provided, together with a mathematical function that simulates the behavior of the dry band resistance as a function of time, even in cases where arcs or partial discharges occur. The model‟s parameters of a typical porcelain insulator of 20 kV, contaminated by salt and kaolin, are presented. Chapter 8 - Nanoscale magnetic semiconductors and magnetic metal/insulator nanogranular films are under significant focus in materials science research because of their promising applications in spin-mediated devices. The upcoming technology where these materials are going to be used are, spintronics, opto-spintronics, data storage and sensors. Better performance of the devices depends on proper synthesis of these materials and engineering their properties. Ions with different energy ranges (eV – keV- MeV) play an important role in synthesis of these materials as well as modification of their properties for better performance. This review is divided in three sections. In the first section, a comprehensive review on diluted magnetic semiconductors (DMSs), transparent magnetic semiconductors (TMSs), their importance, mechanism behind ferromagnetism, ZnO based DMSs and importance of different magnetic metal-insulator nanocomposite is given. Role of energetic ions in each case is highlighted. The second section deals with experimental studies on synthesis and characterization of Ni implanted/doped ZnO films in the perspective of DMS/TMS. The results are discussed on the basis of carrier mediated exchange interaction and bound polaron model. In the third section, results on Ni-SiO2 nanocomposite films grown by atom beam sputtering technique are discussed. An attempt has been made to correlate microstructure, composition and magnetic properties of these nanocomposite films. Finally, the results obtained in these studies are summarized and future research scopes are highlighted.

Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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In: Insulators: Types, Properties and Uses Editor: Kevin L. Richardson, pp.1-27

ISBN: 978-1-61761-996-0 ©2011 Nova Science Publishers, Inc.

Chapter 1

FERROMAGNETIC INSULATOR IN THE ELECTRON-DOPED MANGANITES Jie Yang1,* and Yuping Sun1, 2 1

Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, People‟s Republic of China 2 High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, People‟s Republic of China

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ABSTRACT In this chapter, we will focus on the investigation of the ferromagnetic insulator behavior for the electron-doped manganites La1-xTexMnO3 (x 0.15). The results show that we can obtain ferromagnetic insulators and tune their Curie temperatures TC in the electron-doped manganites La1-xTexMnO3 (x 0.15) by changing the oxygen stoichiometry (La0.9Te0.1MnOy), doping Pr at La-site (La0.9-xPrxTe0.1MnO3), and substituting Cu for Mn (La0.85Te0.15Mn1-xCuxO3). Moreover, the ferromagnetic insulating properties of LaMn1-xTixO3 (0 x 0.2) are also studied. The appearance of ferromagnetic insulator state in these compounds can be understood in terms of the combined effect of destruction of Mn2+-O-Mn3+ double exchange interaction network, the appearance of superexchange ferromagnetic interaction, and the introduction of random Coulomb potential caused by the introduction of oxygen vacancies and the doping effect giving rise to the localization of carriers. The present studies should provide a route to design and develop new type of ferromagnetic insulators.

1. INTRODUCTION Ferromagnetic (FM) insulators have been attracted much attention since they combine ferromagnetism with desirable insulating properties, which enable FM insulators to have

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potential spintronic applications, e.g. spin filter. The concept of spin filter relies on the use of a FM or ferrimagnetic insulating tunnel barrier. In a FM insulator material, the conduction bands are spin split by exchange interaction, leading to two different barrier heights for spinup and spin-down electrons. The combination of a nonmagnetic electrode with a ferromagnetic barrier yields two very different currents for spin-up and spin-down electrons due to the exponential dependence of the transmission with the barrier height, resulting theoretically in a very large spin polarization. Strong insulation is a prerequisite for polarity, since conductors cannot support electric polarization. Actually, FM insulator states have been widely observed and discussed in EuO (TC = 79 K) [1], YTiO3 (TC = 29 K) [2], SeCuO3 (TC = 29 K) [3], BiMnO3 (TC = 105 K) [4]. Mixed-valence hole-doped manganites perovskites have attracted considerable attention in recent years because of the observation of colossal magnetoresistance (CMR) and more generally due to the unusually strong coupling between their lattices, spin, and charge degrees of freedom. The large variety and the tunability of the physical properties (ferroelectricity, ferromagnetism, antiferromagnetism, metallicity, superconductivity, optical properties, etc.) exhibited by transition-metal oxides with perovskite structure provide tremendous advantages for spintronics by bringing additional functionalities that do not exist in more conventionally used materials. Recently, FM insulators were also found in some lightly hole-doped manganites [5-6]. Then a salient question to ask is the following: are there any FM insulators in the electron-doped manganites? With this notion in mind, the focus of this chapter is on showing that we can also obtain FM insulators and tune their Curie temperatures TC in the electron-doped manganites La1-xTexMnO3 (x 0.15). The compounds La1-xTexMnO3 (x 0.15) with a mixed valence state of Mn2+/Mn3+ exhibit a FM metal state below TC [7]. It was found that FM insulator states begin to occur when we change the oxygen stoichiometry (La0.9Te0.1MnOy), dope Pr at La-site (La0.9-xPrxTe0.1MnO3), and substitute Cu for Mn (La0.85Te0.15Mn1-xCuxO3). Moreover, the FM insulating properties of LaMn1-xTixO3 are also studied. The following common features are found in these compounds: (1) TC decrease with the increase in the oxygen vacancies and the doping content of Pr, Cu, and Ti; (2) Paramagnetic (PM)-FM phase transition become broader implying a wider distribution of the magnetic exchange interactions in the Mn–O–Mn network, i.e., the increase in magnetic inhomogeneity; (3) The resistance increases and displays the insulating properties upon an oxygen content and a doping level of Pr, Cu, and Ti. The present studies provide a route to design and develop FMI. From the point of view of practical applications, however, it is still a challenging task to explore FM insulators with TC above room temperature due to the fact that none of the present FM insulators can be used at room temperature, since their TC are below 300 K.

2. FMI IN THE OXYGEN DEFICIENCY LA0.9TE0.1MNOY MANGANITES Polycrystalline La0.9Te0.1MnOy samples were prepared through the conventional solidstate reaction method in air. The preparation of the samples can be found elsewhere [8]. In order to change the oxygen content of the samples, we annealed the samples at 750 °C, 800 °C, and 850 °C, respectively, in N2 atmosphere under 2 MPa pressure for 4 h with graphite

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Ferromagnetic Insulator in the Electron-Doped Manganites

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powder placed near the samples. The oxygen content of the samples was determined by a redox (oxidation reduction) titration in which the powder samples taken in a quartz crucible were dissolved in (1+1) sulfuric acid containing an access of sodium oxalate, and the excess sodium oxalate was titrated with permanganate standard solution. The method is found to be effective and highly reproducible. The detailed method to determine the oxygen content of samples is reported in elsewhere [9]. The crystal structures were examined by x-ray diffractometer (XRD) using a CuK radiation at room temperature. The magnetic measurements were carried out with a Quantum Design superconducting quantum interference device (SQUID) MPMS system (2 T 400 K, 0 H 5 T). Both zero-field-cooled (ZFC) and field-cooled (FC) data were recorded. The resistance was measured by the standard four-probe method from 25 to 300 K. Figure 1 shows the x-ray diffraction (XRD) patterns of La0.9Te0.1MnOy samples of (A) asprepared and (B) 750 °C, (C) 800 °C, (D) 850 °C-annealed in N2 atmosphere. It shows that the XRD pattern of all the annealed samples is similar to that of the as-prepared sample, implying that no structure change occurs, although the oxygen content of the annealed samples is reduced remarkably as shown below. The oxygen content of samples is determined to be 3.01, 2.97, 2.86 and 2.83 corresponding to samples A, B, C, and D, respectively. It indicates that the oxygen stoichiometry decreases with increasing the annealed temperature. In addition, the powder XRD at room temperature shows that all samples are single phase with no detectable secondary phases and the samples can be indexed with the rhombohedral structure with the space group R 3C . The structural parameters of the samples are refined by the standard Rietveld technique. The obtained structural parameters are listed in Table 1. It shows that the lattice parameters of La0.9Te0.1MnOy samples vary monotonically with decreasing oxygen content. The Mn-O-Mn bond angle decreases and the Mn-O bond length increases with the reduction of oxygen content. It is mainly related to the local lattice distortion caused by introducing oxygen vacancies in the Mn-O-Mn network. Moreover, the unit cell volume of samples increases monotonically with reducing the oxygen content. With the reduction of oxygen content, the average manganese oxidation state decreases and thus the average manganese ionic size increases resulting in the increase of unit cell volume of the samples. Table 1. Refined structural parameters of La0.9Te0.1MnOy at room temperature. The space group is R 3C Sample A B C D

a (Å) 5.524 5.533 5.542 5.549

c (Å) 13.357 13.367 13.383 13.396

V (Å3) 353.010 354.407 355.928 356.859

d Mn-O (Å) 1.9644 1.9710 1.9783 1.9851

(º) 163.83 162.24 160.65 157.41

Mn-O-Mn

Rp (%) 8.03 7.59 8.72 9.47

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Figure 1. XRD of the as-prepared La0.9Te0.1MnOy sample (denoted as A) and samples annealed at 750 °C, 800 °C and 850 °C denoted as B, C and D, respectively

Figure 2. Magnetization as a function of temperature M (T) for samples A and B under the field-cooled (FC) and zero-field-cooled (ZFC) modes denoted as the filled and open symbols, respectively. The inset shows M (T) curves for samples C and D

Figure 2 shows the temperature dependence of magnetization of La9Te0.1MnOy under both ZFC and FC modes at H = 0.1 T for all samples. The Curie temperature TC (defined as the peak of dM dT in the M vs. T curve) is 239 K, 238 K, 213K and 165K for samples A, B, C and D, respectively. Obviously, the Curie temperature TC of sample B decreases slightly compared with that of sample A. However, the magnetization magnitude of sample B decreases obviously at low temperatures compared with that of sample A. With further reducing the oxygen content, TC of sample C and D drop rapidly compared with that of the as-

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Ferromagnetic Insulator in the Electron-Doped Manganites

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prepared sample. We suggest that the TC reduction should be attributed to the effect of the partial destruction of Mn2+-O-Mn3+ DE interaction network because of the introduction of the oxygen vacancies and the weakening of Mn2+-O-Mn3+ DE interaction arising from the decrease of the bandwidth of eg electrons due to the increase in the Mn-O bond length and the decrease in the Mn-O-Mn bond angle. In addition, Figure 2 shows that both samples A and B have a sharp PM to FM transition; however, PM-FM phase transitions of samples C and D obviously become broader, which implies the increase in the magnetic inhomogeneity with the decrease of oxygen content. It is worth noting that a bifurcation begins to occur between both FC and ZFC curves at low temperature regions for samples B, C and D implying the appearance of magnetic frustration arising from the competition between the FM and antiferromagnetic (AFM) exchange interaction. With the reduction of oxygen content, the difference between FC and ZFC curves becomes greater. For sample D, an obvious spin glass-like behavior appears as is evidenced by a cusp in the ZFC curve at 100 K and a distinctive separation of the FC and ZFC curves. So the reduction of oxygen content could result in the decreasing of

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TC and the broadening of PM-FM transition. The magnetization as a function of the applied magnetic field at 5K is shown in Figure 3. It shows that, for samples A, B and C, the magnetization reaches saturation at about 1T and keeps constant up to 5T, which is considered as a result of the rotation of the magnetic domain under the action of applied magnetic field. For sample D, the rapid increase of magnetization M (H) at low magnetic fields resembles ferromagnet with a long-range FM ordering corresponding to the rotation of magnetic domains, whereas the magnetization M increases continuously without saturation at higher fields, which reveals the existence of AFM phase corresponding to the linear variation of high-field region. It is well known that the competition between FM phase and AFM phase would lead to the appearance of the spin glass state. That is why the spin glass-like behavior exists in sample D. In addition, it is worth noting that the magnetization magnitude of sample C also increases slightly at low temperatures as H > 1 T compared with that of sample B. It is may be related to the excess reduction of oxygen content for sample C (y = 2.86) compared with that of sample B (y = 2.97).

Figure 3. Field dependence of the magnetization M (H) for samples A, B, C and D at 5 K Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Figure 4.The temperature dependence of the resistivity (T) for samples A, B, C and D at zero (solid lines) and 0.5T fields (dashed lines)

Figure 4 shows the temperature dependence of resistivity (T) for samples A, B, C and D at zero fields in the temperature range of 30-300K. For sample A, it shows that there exists an insulator-metal (I-M) transition at TP1 (= 246K) slightly higher than its TC (=239 K). In addition, there exists a shoulder at TP2 (=223 K) below TP1, which is similar to the double peak behavior observed usually in alkaline-earth-metal-doped samples of LaMnO3 [10, 11]. Double peaks (TP1=240K and TP2=205 K) shift to lower temperatures for sample B. Compared with sample A, I-M transition at TP1 becomes weak and the low-temperature resistivity peak at TP2 becomes more obvious. It shows that the oxygen deficiency can substantially enhance the low-temperature resistivity peak at TP2. Different from the origin of the TP1 peak, the resistivity peak at TP2 is believed to reflect the spin-dependent interfacial tunneling due to the difference in magnetic order between surface and core [11]. And this variation of double peak behavior is presumably related to an increase in the height and width of tunnel barriers with increasing oxygen deficiency. Moreover, the resistivity of sample B at low temperatures does not drop with decreasing temperature. Instead there is a slight upturn in (T) at low temperatures, implying the appearance of carrier localization caused by the oxygen vacancies because of the reduction of oxygen content. The experimental data measured at applied field of 0.5 T for samples A and B in the temperature range of 30-300K are also recorded. For samples A and B, it can be seen from Figure 4 that the resistivity of samples decreases under the applied magnetic field, TP1 peak position shifts to a higher temperature and TP2 peak position does not nearly change, which also means the origin of TP2 peak is different from that of TP1 peak. For samples C and D, (T) curves display the insulating behavior ( d

dT

0 ) in both high-temperature PM phase and low-

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Ferromagnetic Insulator in the Electron-Doped Manganites

7

temperature FM phase and the resistivity maximum increases by six orders of magnitude compared with that of the as-prepared sample implying the enhancement of the localization of carriers. This FM insulating behavior cannot be explained base only on the DE model because the FM and metallic nature must coexist within the framework of the DE model. Therefore the FM order at low temperatures for samples C and D does not mainly originate from DE interaction. As Goodenough predicts, a Mn-O-Mn 180 -superexchange interaction generally gives rise to AFM ordering while 90 -superexchange interaction will lead to FM ordering [12]. This superexchange ferromagnetism (SFM) has been suggested in explaining ferromagnetism of the Tl2Mn2O7 and CaCu3Mn4O12, which are of the character of no Mn mixed valence [13, 14]. The Mn-O-Mn bond angel for sample C and D are 160.65 and 157.10 , which deviate obviously from 180 . Therefore, for samples C and D, the observed FM transition may be related to the appearance of SFM caused by the local lattice distortion arising from the increase of oxygen vacancies in the Mn-O-Mn network. It should be mentioned that for La0.9Te0.1MnO2.97 (sample B), the ratio of 2+ Mn /(Mn2++Mn3+) is close to 16%, comparable to that of La0.84Te0.16MnO3. From the known experimental data [15], such a system should show higher I-M transition temperature and lower resistivity compared with that of sample A. For La0.9Te0.1MnO2.86 (sample C) and La0.9Te0.1MnO2.83 (sample D), the ratio of Mn2+/(Mn2++Mn3+) is close to 38% and 44%, comparable to that of La0.62Te0.38MnO3 and La0.56Te0.44MnO3, respectively. In our previous work [7], such two systems should show an I-M transition. However, the case is clearly not observed in samples C and D. Additionally, as we can see from Figure 3, the magnetization magnitude of sample C at low temperatures as H > 1 T is slightly higher than that of sample B, which is consistent with the experimental data due to the increase of the ratio of Mn2+/(Mn2++Mn3+). On the other hand, for sample C, the appearance of SFM caused by the local lattice distortion arising from the increase of oxygen vacancies in the Mn-O-Mn Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

network is also a possible reason. Moreover, the effect of oxygen content on TC of samples B, C and D is quite small comparable to that of La0.84Te0.16MnO3, La0.62Te0.38MnO3 and La0.56Te0.44MnO3, respectively. So oxygen content reduction in La0.9Te0.1MnOy is expected to cause three effects. The first effect is the increased Mn2+/(Mn2++Mn3+) ratio, giving rise to the carrier density increase and the resistivity decrease, and causing the enhancement of the FM coupling. The second effect is the partial destruction of double exchange (DE) interaction between Mn2+-O-Mn3+ and the localization of carriers because of the appearance of oxygen vacancies. The third effect is the occurrence of the local lattice distortion due to the introduction of oxygen vacancies in the Mn-O-Mn network, which is important for electrical conduction. The local lattice distortion caused by oxygen vacancies in samples is confirmed by the structural parameter fitting through the Reitveld technique. The local lattice distortion is also expected to increase the resistivity and destruct the DE interaction due to the reduced eg electron bandwidth [16]. As a result, the competitions of the three effects suggested above decide the complicated behavior of the electrical transport and magnetic properties of the samples La0.9Te0.1MnOy.

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3. FMI IN THE A-SITE DOPING LA0.9TE0.1MNO3 In this part, we examined a series of samples La0.9-xPrxTe0.1MnO3 (0 ≤ x ≤ 0.9) in which the average ionic radius of the A-site element < rA > is systematically varied while keeping the Mn2+/Mn3+ ratio fixed at 1/9 by doping Pr at La-sites. We find that the average ionic radius of the A-site elemen < rA > has strongly affected the structural, magnetic and transport properties in electron-doped manganites samples La0.9-xPrxTe0.1MnO3 (0 ≤ x ≤ 0.9). Moreover, we can obtain FM insulators and tune their Curie temperatures TC by changing the doping level of Pr in La0.9-xPrxTe0.1MnO3 (0 ≤ x ≤ 0.9). A series of ceramic samples of La0.9-xPrxTe0.1MnO3 (0 ≤ x ≤ 0.9) were synthesized by a conventional solid-state reaction method in air. The preparation of the sample can be found elsewhere [17]. The structure and lattice constant were determined by powder XRD with a CuKα radiation at room temperature. The resistance as a function of temperature was measured by the standard four-probe method from 25 to 300K. The magnetic measurements were performed on a Quantum Design SQUID MPMS system (2 ≤ T ≤ 400 K, 0 ≤ H ≤ 5 T). XRD at room temperature shows that all samples are single phase with no detectable secondary phases. XRD patterns of the samples with x = 0 and x = 0.18 can be indexed by

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rhombohedral lattice with the space group R3C . While XRD patterns of the samples with x = 0.36, 0.54, 0.72 and 0.9 can be indexed by orthorhombic lattice with space group Pbnm. The structural parameters are refined by the standard Rietveld technique and the fitting between the experimental spectra and the calculated values is relatively good based on the consideration of lower RP values as shown in Table 2. Figures. 5(a) and (b) show the experimental and calculated XRD patterns for the samples with x = 0 and 0.36, respectively. The structural parameters obtained are listed in Table 2. As we can see, for samples La0.9xPrxTe0.1MnO3 (0 ≤ x ≤ 0.9), the crystal structure at room temperature changes from rhombohedral phase ( R3C , Z = 2, x ≤ 0.18) to orthorhombic phase (Pbnm, Z = 4, x ≥ 0.36). The lattice distortion and the bend of Mn-O-Mn bond increase when the crystal structure varies from a lattice to an orthorhombic lattice. It is well known that one of the possible origins of the lattice distortion of perovskites structures is the deformation of the MnO6 octahedra originating from Jahn-Teller (JT) effect that is directly related to the concentration Mn3+ ions. But for the present study samples, the concentration of Mn3+ ions is fixed. And thus the observed lattice distortion should be only caused by the average ionic radius of the A-site element < rA >, which is governed by the tolerance factor t [ t = (rA + rO )

2 (rB + rO ) ], where ri (i=A, B, or O) represents the average ionic size of

each element. As t is close to 1, the cubic perovskite structure is expected to form. As < rA > decreases, so does t, the lattice structure transforms to rhombohedral ( R3C ), and then to orthorhombic (Pbnm) structure, in which the bending of the B-O-B bond increases and the bond angle deviates from 180°. For La0.9-xPrxTe0.1MnO3 samples, the structural transition at room temperature mainly originates from the variation of the tolerance factor t induced by the substitution of smaller Pr3+ for larger La3+ ions. Figure 6 shows the temperature dependence of magnetization of La0.9-xPrxTe0.1MnO3 (0 ≤ x ≤ 0.9) under both ZFC and FC modes at H = 0.1 T. The Curie temperature TC are 239 K, 207 K, 159 K, 120 K, 93 K and 75 K for x = 0, 0.18, 0.36, 0.54, 0.72 and 0.9, respectively,

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which are listed in Table 3. Obviously, the Curie temperature TC decreases monotonically with increasing Pr-doping level. We suggest that the TC reduction should be attributed to the reduction of Mn-O-Mn bond angle with decreasing the average ionic radius of the A-site element , and thereby reducing the matrix element b which described electron hopping between Mn sites. Thus the DE interaction between Mn2+-O-Mn3+ becomes weakening because of the narrowing of the bandwidth and the decrease of the mobility of eg electrons caused by the substitution of smaller Pr3+ ions for larger La3+ ions.

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Table 2. Refined structural parameters of La0.9-xPrxTe0.1MnO3 (0 x 0.9) at room temperature. O(1):apical oxygen; O(2): basal plane oxygen

In addition, from Figure 6, a sharp FM to PM transition is observed for the samples with x 0.36. However, as x 0.54, the temperature range of FM-PM phase transition become broader with increasing Pr-doping level implying a wider distribution of the magnetic exchange interactions in the Mn-O-Mn network, i.e., the increase in the magnetic inhomogeneity. Moreover, it is clear that the ZFC curve does not coincide with the FC curve below a freezing temperature Tf for the samples with x 0.36. With increasing Pr-doping content, the difference between M-T curves under FC and ZFC modes becomes greater because of the increase of the magnetic frustration arising from the bending Mn-O-Mn bond, which is in accordance with the structural refinement results. This discrepancy between ZFC and FC magnetization is a characteristic of cluster glass.

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Table 3. TC, TP and the fitting parameter of La0.9-xPrxTe0.1MnO3 (0

x

0.9) samples

Figure 5. XRD patterns of the compound La0.9-xPrxTe0.1MnO3, (a) x = 0 and (b) x = 0.36. Crosses indicate the experimental data and the calculated data is the continuous line overlapping them. The lowest curve shows the difference between experimental and calculated patterns. The vertical bars indicate the expected reflection positions

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Figure 6. Magnetization as a function of temperature for La0.9-xPrxTe0.1MnO3 (x = 0, 0.18, 0.36, 0.54, 0.72 and 0.9) measured at H = 0.1T under the field-cooled (FC) and zero-field-cooled (ZFC) modes that are denoted as the filled and open symbols, respectively

Figure 7. Field dependence of the magnetization in La0.9-xPrxTe0.1MnO3 (x = 0, 0.18, 0.36, 0.54, 0.72 and 0.9) at 5 K. The dashed lines represent the extrapolation lines and M 0 denotes a linear extrapolation M (H) to H = 0

The magnetization as a function of the applied magnetic field at 5K is shown in Figure 7. It shows that, for the samples with x 0.18, the magnetization reaches saturation at about 1T and keeps constant up to 5T. For the sample with x = 0.36, the magnetization slowly reaches saturation at about 4T, implying the appearance of a small amount of AFM phase at low temperatures. However, for the samples with x 0.54, Figure 7 exhibits that the rapid increase of magnetization M (H) at low magnetic fields, whereas the magnetization M increases continuously without saturation at higher fields, revealing a superposition of both FM and AFM components. The coexistence of and competition between ferromagnetic and antiferromagnetic interaction would favor the formation of a cluster glass state, as observed in

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La0.9-xPrxTe0.1MnO3 (x 0.54) samples. In fact, based on the temperature and magnetic field dependence of magnetization for these samples, the microscopic magnetic structure can be understood by presence of small sized FM clusters in the samples, as can be clearly observed from the broad magnetic transition range for the sample with x=0.9. Moreover, in order to determine the change in volume of the FM phase in respect to Pr doping level, a liner extrapolation of M (H) to H = 0 for the samples with x 0.54 is plotted in dashed line in Figure 7. At 5 K, the FM phase of the samples with x = 0.72 and 0.9 decreases by about 23% and 48%, respectively, in volume compared with that of the sample with x = 0.54. So it can be concluded that Pr-doping induces an increased AFM superexchange interaction. Figure 8 (a) shows the temperature dependence of resistivity (T) for the samples with x = 0, 0.18 and 0.36 at zero fields in the temperature range of 30-300K. For sample with x = 0, it shows that there exists an insulator-metal (I-M) transition at TP1 (= 246 K) which is close to its Curie temperature TC (= 239K). In addition, there exists a bump shoulder at TP2 (= 223 K) below TP1. More interesting phenomenon is that double I-M transitions show significant variation with changing the Pr-doping level. Double peaks (TP1= 210 and TP2= 186 K) shift to low temperatures for x = 0.18 sample. Compared with the x = 0 sample, I-M transition at TP1 becomes weak and I-M transition at TP2 becomes more obvious. It shows that the Pr-doping at La-site can substantially enhance the I-M transition at TP2. When Pr-doping level is increased to x = 0.36, I-M transition at TP1 (=153K) is almost invisible and displays an inflexion behavior. And I-M transition at TP2 (=105K) becomes more obvious. In other words, the I-M transition at TP1 has been almost suppressed. Moreover, there exists a low-temperature I-M transition at T*(= 66K) for the sample with x = 0.36, implying the presence of magnetic inhomogeniety due to the Pr-doping at La-site. The experimental data measured at applied field of 0.5 T for samples with x = 0, 0.18 and 0.36 in the temperature range of 30-300 K are also recorded. It can be seen from Figure 8(a), for the samples with x = 0 and 0.18, the applied field suppressed the resistivity peak at TP1 significantly and the resistivity peak shifts towards higher temperatures. Especially for the sample with 0.36, the resistivity peak at TP1 seems to be suppressed completely under the applied field. However, for the second I-M transition at TP2, it is worth noting that for the sample with x = 0 and 0.18, the applied field change the position of the resistivity peak at TP2 slightly, whereas for the sample with x = 0.36 the position of the resistivity peak at TP2 moves to higher temperature greatly under the applied magnetic field. The difference in the response of the resistivity peak at TP2 for the applied field in samples between x = 0 and x = 0.18, 0.36 indicate that they may have different origins. As to the origin of the low-temperature I-M transition at about 66 K for sample with x = 0.36, we consider it is mainly related to the opening of a new percolative channel. From Figure 8 (a), one can see that the (T) curve under zero fields exhibits an upturn from 70 K with further cooling, which is indeed the result of the competition between the AFM interaction and the ferromagnetic DE interaction in the sample. With further cooling, the amounts of small sized FM clusters increase and finally come into being a filament percolative channel. And thus the I-M transition at about 66K can be observed.

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Figure 8. (a)The temperature dependence of the resistivity of La0.9-xPrxTe0.1MnO3 (x = 0, 0.18, 0.36) samples at zero (solid lines) and 0.5T fields (dashed lines). (b) The temperature dependence of the resistivity of La0.9-xPrxTe0.1MnO3 (x = 0.54, 0.72, 0.9) samples at zero fields

For the samples with x = 0.54, 0.72 and 0.9, (T) curves display the semiconducting behavior (d /dT. The temperature phase diagram as a function of the tolerance factor t and the average ionic radius of the A-site element < rA > is plotted in Figure 10. As we can see, with the decrease of t and < rA >, the Curie temperature TC of the study samples decreases as well as. It is worth noting that TC shows a linear dependence upon the tolerance factor t . Similar relation between the average ionic radius of the A-site element < rA > and TC is also observed. As < rA > decreases,

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so does t, the lattice structure transforms to rhombohedral ( R3C ), and then to orthorhombic (Pbnm) structure. At the same time, the phase transition also occurs from PM-FMM to PMFMI. All these are ascribed to the increase of the bending of the Mn-O-Mn bond with decreasing the average ionic radius of the A-site element and the reduction of the tolerance factor t because of the substitution of smaller Pr3+ ions for larger La3+ ion.

4. FMI IN THE MN-SITE DOPING LA0.85TE0.15MNO3 To study the substitution of Mn sites by transitional elements for clarifying the mechanism of CMR is very important because of the crucial role of Mn ions in the CMR materials. The Mn-site doping is an effective way to modify the crucial Mn-O-Mn network and in turn remarkably affects their intrinsic physical properties. Among the investigation of all Mn-site doping, the Cu-doping at Mn-site is more interesting because of its special nature of Cu perovskite compounds, i.e., its high temperature superconductivity of cuprates. However, many controversial results have been reported concerning the influence of Cudoping on the properties of polycrystalline manganites. Ghosh et al. studied the effect of transitional element doping (TE = Fe, Co, Ni, Cr, Cu, Zn) in La0.7Ca0.3MnO3 and found that the doping content of 5% Cu and Zn both lead to remarkably drop in resistance in the system [22]. The slightly-doped Cu in polycrystalline La0.825Sr0.175Mn1-xCuxO3 (0 x 0.2) giving rise to the remarkable increase of the resistance has been found, which is attributed to the induced antiferromagnetic (AFM) superexchange interaction and the long range FM order becoming weaker [23]; however, in Cu heavily doped polycrystalline samples, two different kinds of conduction mechanism has been found in the doping range of (0.20 x 0.32) and

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(0.33 x 0.40), respectively [24]. In order to comprehensively understand the origin of the effect of Cu-doping at Mn-site of electron-doped manganites, we carefully investigate the effect of Cu-doping at Mn-site on structural, magnetic and electrical transport properties in the electron-doped manganites La0.85Te0.15Mn1-xCuxO3 (0 x 0.2). Polycrystalline La0.85Te0.15Mn1-xCuxO3 (0 x 0.2) samples were prepared through the conventional solid-state reaction method in air. The preparation of the samples can be found elsewhere [25]. The crystal structures were examined by XRD using a Cu-K radiation at room temperature. The magnetic measurements were carried out by a Quantum Design SQUID MPMS system (2 T 400 K, 0 H 5 T). The resistance was measured by the standard four-probe method from 25 to 300 K. Table 4. Refined structural parameters of La0.85Te0.15Mn1-xCuxO3 (0

x

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Parameter a (Å) c (Å) v (Å3) Mn-O (Å) Mn-O-Mn (º) Rp (%)

0.2) at room temperature. The space group is R 3C x=0 5.525 13.355 353.010 1.9644 163.83 8.03

x=0.05 5.530 13.348 353.273 1.9690 162.10 6.23

x=0.10 5.531 13.348 353.589 1.9721 161.28 9.29

x=0.15 5.535 13.346 354.049 1.9849 157.49 6.74

x=0.20 5.538 13.343 354.340 1.9911 155.92 6.73

Figure 11. X-ray diffraction patterns for La0.85Te0.15Mn1-xCuxO3 (x = 0, 0.05, 0.1, 0.15 and 0.2) samples

Figure 11 shows XRD patterns of La0.85Te0.15Mn1-xCuxO3 (0 x 0.2) samples. The powder x-ray diffraction at room temperature shows that all samples are single phase with no detectable secondary phases and the samples had a rhombohedral lattice with the space group

R3C . The structural parameters can be obtained by fitting the experimental spectra using the standard Rietveld technique. The obtained structural parameters are listed in Table 4. No structural transition has been observed and the lattice parameters for La0.85Te0.15Mn1-xCuxO3 (0 x 0.2) samples vary monotonically with increasing Cu content. The Mn-O-Mn bond

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angle decreases with increasing Cu-doping level, whereas the Mn-O bond length increases which displays the inverse correlation to the variation in the Mn-O-Mn bond angle. To describe the ion match between A and B site ions in perovskite structure compounds, a geometrical quantity, i.e., tolerance factor t is usually introduced and t is defined as t = (rA + rO ) 2 (rB + rO ) , where ri (=A, B, or O) represent the average ionic size of each

element. It is well known that some internal stress will be introduced into the structure when there is a size mismatch between atom A and B sites. Internal stress can be partially relieved by structure distortion. The cubic structure is distorted either by atom B moving off center in its octahedra or by the cage collapsing by rotation of the BO6 octahedra. For the present studied system, the distortion of MnO6 octahedra arises from partial replacement of Mn by Cu because of the fixed average ionic radius of A-site. It is well known that the departure from the average Mn radius R(Mnav) (defined as the average over the radii for Mn3+ and Mn4+ ions in the ratio (1-x): x in Ln1-xAxMnO3 compounds) at the dopant site would subject the neighboring Mn-O bonds to a centric push or pull [22]. When a bigger ion at the Mn site should compress the neighboring Mn-O bonds and result in decreasing the Mn-O-Mn bond angle. Similarly, we can calculate R(Mnav) (defined as the average radii for Mn3+ and Mn2+ ions in the ratio 85: 15 in La0.85Te0.15MnO3 compounds). Standard ionic radii [21] with values 0.83Å for Mn2+, and 0.65Å for Mn3+ in perovskite-structure manganites, respectively, are used to calculate R(Mnav). The average radius of Mn ion R(Mnav) is calculated to be 0.658 Å. Therefore, based on the monotonic varying of Mn-O bond length, lattice volume, and Mn-OMn bond angle with increasing Cu-doping level, we suggest that the doped Cu ions at Mn-site should be in the form of Cu2+ ions because the radius of the Cu2+ is 0.73 Å, i.e., R(Cu2+)>R(Mnav). The temperature dependence of magnetization M of La0.85Te0.15Mn1-xCuxO3 (0 ≤ x ≤ 0.2) under both ZFC and FC modes at H = 0.1 T are measured. Our measured samples may be considered as ellipsoid for which N=0.05 (SI units), and the applied field is parallel to the longest semi-axis of samples. So a uniform field can exist throughout the samples and the shape demagnetizing fields can be reduced as much as possible. The results are shown in Figure 12 for the samples with x 0.05, the temperature range of FM-PM phase transition becomes broader with increasing Cu-doping level, which implies the increase of magnetic inhomogeneity. The bifurcation begins to occur between the FC and ZFC curves at low temperature region for the samples with 0.10 ≤ x ≤ 0.20 implying the appearance of magnetic frustration. With increasing Cu2+ content, a bigger difference between M-T curves under FC and ZFC is observed. For samples with x = 0.15 and 0.20, an obvious spin-glasslike behavior appears, which is evidenced by a cusp in the ZFC curve at 40K and a distinctive

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separation of the FC and ZFC curves. Although the decrease of TC and broadening of PM-FM transition with increasing Cu-doping level, the magnetization magnitude of Cu-doping samples increases at low temperatures with increasing Cu-doping level as x 0.15. The magnetization as a function of the applied magnetic field at 5 K is shown in Figure 13, The magnetization reaches saturation at about 1 T and keeps constant up to 5 T for the samples with x = 0, 0.05 and 0.1, which is considered as a result of the rotation of the magnetic domain under external magnetic field, whereas for the samples with x=0.15 and 0.2, the M (H) dependences are unsaturated up to H = 5 T and they exhibit high slope. It is well known that the competition between FM phase and AFM phase would lead to the appearance of a spin-glass state. That is why a spin-glass-like behavior exists in the samples with x = 0.15 and 0.20. Moreover, in order to determine the change in volume of the FM phase in respect to Cu doping, a linear extrapolation of M (H) to H = 0 for the sample with x = 0.15 and 0.20 is plotted in solid line in Figure 13. At 5 K, the FM phase decreases by 27% in volume from x =0.15 to x=0.20, which is the result of the Mn2+-O-Mn3+ FM networks being destroyed gradually because of Cu-doping at Mn-site. Additionally, similar to the result shown in Figure 12, the magnetization magnitude of Cu-doping samples also increases at low temperatures as H > 1 T for x 0.15 compared with that of no Cu-doped sample. It may be related to the appearance of superexchange ferromagnetism (SFM) due to the obvious deviation of 180o of Mn-O-Mn bond angle. The SFM phase is in fact a FM phase with predominant superexchange ferromagnetic interaction. As Goodenough predicts, a Mn-O-Mn 180o superexchange interaction generally gives rise to AFM ordering while 90o superexchange interaction will lead to FM ordering [12]. This SFM has been suggested to explain ferromagnetism of Tl2Mn2O7, in which there is no Mn mixed valence [13]. On the other hand, the opening of the new DE channel between Mn3+-O-Mn4+ is also possible because the substitution of Cu2+ for Mn3+ introduces Mn4+ ion.

Figure 12. Magnetization as a function of temperature for La0.85Te0.15Mn1-xCuxO3 (x = 0, 0.05 and 0.1) measured at H = 0.1T under the field-cooled (FC) and zero-field-cooled (ZFC) modes that are denoted as the filled and open symbols, respectively. The inset show M (T) curves for the samples with x = 0.15 and 0.2

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Figure 13. Field dependence of the magnetization in La0.85Te0.15Mn1-xCuxO3 (x = 0, 0.05, 0.1, 0.15 and 0.2) at 5 K. M0 denotes a linear extrapolation M (H) to H = 0

The temperature dependence of resistivity for La0.85Te0.15Mn1-xCuxO3 (0 ≤ x ≤ 0.20) is shown in Figure 14. The experimental data are obtained at zero and applied field of 0.5 T for the samples with x = 0, 0.05 and 0.10 in the temperature range of 25-300 K. Figure 14 shows that ρ increase considerably with increasing Cu-doping level. For no Cu-doped sample, it shows that there exists an I-M transition peak at TP1 (=246 K) slightly higher than its TC (=239 K). In addition, there exists a shoulder at TP 2 (=223 K) below TC. Double peaks (TP1 = Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

196 and TP 2 = 176 K) shift to low temperatures for x = 0.05 sample. Compared with the x = 0 sample, I-M transition at TP1 becomes weak and I-M transition at TP2 becomes more obvious. It shows that the Cu-doping at Mn-site can substantially enhance the I-M transition at TP2. ρ (T) curve only behaves as single pronounced I-M transition at TP = 117 K for the sample with x = 0.1, which lies in well below its TC value of 169 K. It shows that the I-M transition at TP1 has been suppressed completely for x ≥ 0.1. This variation of double ρ peak behavior is presumably related to an increase both of the height and width of tunnel barriers with increasing Cu-doping [26]. Moreover, Figure 14 manifests that ρ increases several orders of magnitude as x > 0.05. For the samples with x = 0.15 and 0.20, the resistance at low temperature is so high that the data are collected merely in a limit temperature range in order to avoid exceeding our measuring limit and ρ (T) displays the semiconducting behavior (dρ/dT0.1 exhibit semiconducting behavior in both high-temperature PM phase and lowtemperature FM phase. We suggest that the remarkable increase of ρ for Cu-doping samples originates from the combined effect of destruction of Mn2+-O-Mn3+ DE interaction network, the appearance of SFM insulating phase and the introduction of random Coulomb potential caused by the substitution of Cu2+ for Mn3+. The temperature dependence of resistivity of the samples with x = 0, 0.05 and 0.1 at a magnetic field of 0.5 T is also plotted in Figure 14. It shows that the resistivity of samples decreases under the applied magnetic field, TP1 peak

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position shifts to a higher temperature and TP2 peak position does not nearly change, which also means the origin of TP2 peak is different from that of TP1 peak.

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5. FMI IN THE DIRECT MN-SITE DOPING OF LAMNO3 To explore CMR materials as well as obtain clues concerning the mechanism of CMR, it is worthwhile to investigate if the DE and CMR could be obtained directly by Mn-site element substitution. Previous study of the LaMn1-xMxO3 (M=Co, Ni) has thrown light on this topic. The substitution of magnetic cations for manganese in LaMnO3 leads to PM-FM transition but no I-M transition, i.e., FMI, as described mainly for Ni and Co doped LaMnO3 [27-31]. There is a lot of controversy about the mechanism which governs the FM properties for the FMI phase. Goodenough et al. [27] explained this behavior by ferromagnetic superechange interactions between Mn3+-O-M3+, where M3+=Mn3+, Ni3+, or Co3+. Asai et al. [28] proposed that Ni and Co are in the divalent state, and there are FM superexchange interactions between the Ni2+ (Co2+) and Mn4+. Park et al. [29] argued that the dopant Co was divalent, and explained the ferromagnetic state of this series by a DE mechanism between Mn3+-O-Mn4+. Although there is still controversy on the above experimental results, i.e., whether the Co is a real divalent in LaMn1-yCoyO3, the idea of inducing mixed valence by direct Mn-site doping persists. Hébert et al. [32] pointed out that the ferromagnetism can be induced by doping univalent and divalent cations, i.e., Li+ and Zn2+ due to the creation of Mn4+ ions which favor Mn3+/Mn4+ DE. With this notion in mind, we have performed investigations on the magnetic and magnetotransport properties of the LaMn1-xTixO3 (0 x 0.2) system because the Ti element is generally a stable tetravalent in transition-metal oxides. Moreover Ti4+ is a nonmagnetic cation and there are no interactions between Ti4+ and Mn3+, which avoids the complexity caused by the substitution of Co or Ni which are magnetic cations. We consider that the Mn2+ will be introduced for the sake of valence balance with doping the nonmagnetic cations Ti4+, and we suggest that the ferromagnetism in the rhombohedral samples LaMn1-xTixO3 (0.05 x 0.2) may arise mainly from the DE interactions between Mn2+-O-Mn3+ due to the substitution of Ti4+ for Mn3+. LaMn1-xTixO3 samples with nano-scale grains were prepared by a Pechini method [33], since this procedure can enhance the density and phase homogeneity of the samples at lower sintered temperatures. Stoichiometric amounts of high-purity La2O3 powders were dissolved in diluted nitric acid in which an excess of citric acid was added, followed by the addition of stoichiometric amounts of Mn(NO3)2 and Ti[OCH(CH3)2]4 dissolved in distilled water with continuous stirring, and then ethylene glycol was added to make a solution complex. The preparation of the samples can be found elsewhere [34]. The crystal structures were examined by x-ray diffractometer using a Cu-Kα radiation at room temperature. The magnetic measurements were carried out with a Quantum Design SQUID MPMS system (2 T 400 K, 0 H 5 T). Both ZFC and FC data were recorded. The resistance was measured by the standard four-probe method from 25 to 300 K. The oxygen content of the samples was determined by a redox (oxidation reduction) titration [9]. For no-Ti doped sample, the oxygen content can be determined as 3.11, whereas for the Tidoped samples, only slightly surplus oxygen has been observed and the oxygen content almost stabilizes around 3.01.

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Ferromagnetic Insulator in the Electron-Doped Manganites

Figure 14. Temperature dependence of the resistivity of La0.85Te0.15Mn1-xCuxO3 (x = 0, 0.05, 0.1, 0.15 and 0.2) samples at zero (solid lines) and 0.5T fields (dashed lines)

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Table 5. Room-temperature structural parameters for the rhombohedral ( R3C ) phase LaMn1-xTixO3. Rietveld refinements were carried out using the hexagonal setting of this space group. In this setting the La atoms reside at 6a: (0, 0, 3/4), the Mn/Ti atoms at 6b: (0, 0, 0), and the O atoms at 18e: (x, x, 1/4) Parameters a (Å) b (Å) c (Å) V (Å3) x (O) d Mn-O (Å) Mn-O-Mn (º) Rp (%)

x=0 5.5251(4) 5.5251(4) 13.3531(6) 353.0213(15) 0.5471(2) 1.9621(1) 164.77(1) 12.16

x = 0.05 5.5276(3) 5.5276(3) 13.3711(1) 353.8143(40) 0.5432(5) 1.9605(15) 165.99(74) 7.52

x = 0.1 5.5319(4) 5.5319(4) 13.4308(7) 355.9521(62) 0.5462(9) 1.9668(17) 165.04(76) 8.23

x = 0.15 5.5461(3) 5.5461(3) 13.4187(10) 357.4663(20) 0.5493(7) 1.9720(18) 164.04(75) 11.96

x = 0.2 5.5494(1) 5.5494(1) 13.4286(2) 358.1417(54) 0.5537(9) 1.9768(15) 162.63(59) 11.43

The powder x-ray diffraction (XRD) pattern of LaMn1-xTixO3 samples at room temperature shows that all samples are single phase with no detectable secondary phases and the samples have a rhomboheral structure with the space group R3C . The structural parameters of the samples are refined by the standard Rietveld technique. In the process of refinement, the substitution of Ti ions at Mn sites was considered. Detailed results of the structural refinements are listed in Table 5. It exhibits that the lattice parameters of LaMn1xTixO3 samples vary monotonously with increasing the Ti-doping level. The Mn-O-Mn bond angle ( Mn-O-Mn) increases with increasing x from 0 to 0.05, then decreases with further increasing x from 0.05 to 0.2, whereas, the Mn-O bond length (dMn-O) displays the inverse correlation to the variation in Mn-O-Mn.

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Figure 15. Magnetization as a function of temperature for LaMn1-xTixO3 (0 x 0.2) measured at H = 0.1T under the field-cooled (FC) and zero-field-cooled (ZFC) modes that are denoted as the filled and open symbols, respectively. The inset shows field dependence of the magnetization in LaMn1-xTixO3 (0 x 0.2) at 5 K. The arrows mark the direction of increasing and decreasing magnetic fields

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The main panel of Figure 15 shows the temperature dependence of magnetization M of LaMn1-xTixO3 (0 x 0.2) under both ZFC (MZFC) and FC (MFC) modes at H = 0.1 T. The Curie temperature TC is defined as the one corresponding to the peak of dM dT in the M vs. T curve. From Figure 15, we can see that LaMnO3+δ sample exhibits the wide PM-FM transition caused by the oxygen excess. Additionally, there exist distinct differences between MFC and MZFC curves at low temperatures, which are reminiscent of a characteristic of cluster glass. The Ti-doped samples undergo distinct PM-FM and the Curie temperature are 175 K, 146 K, 113 K, and 94 K for the samples with x=0.05, 0.10, 0.15, and 0.20, respectively. Obviously, the Curie temperature TC decreases monotonically with increasing Ti-doping level. We suggest that the TC reduction should be attributed to the reduction of Mn-O-Mn bond angle. The magnetization as a function of the applied magnetic field at 5 K is shown in the inset of Figure 15. For the sample with x=0, M (H) curve shows that the rapid increase of magnetization M (H) at low magnetic fields resembles of ferromagnet with FM ordering corresponding to the rotation of magnetic domains, whereas the magnetization M increases continuously without saturation at higher fields, revealing a superposition of both FM and AFM components. For the sample with x=0.05, the magnetization reaches saturation at about 1 T and keeps constant up to 4.5 T. For the sample with x=0.10, the magnetization slowly reaches saturation at about 4T, implying the appearance of a small amount of AFM phase at low temperatures. However, for the samples with x ≥ 0.15, the magnetization M increases slowly without saturation at higher fields. In fact, based on the temperature and magnetic field dependence of magnetization for the samples LaMn1-xTixO3 with x=0 and x 0.10, the microscopic magnetic structure at low temperatures can be understood by presence of FM clusters in the AFM matrix. Figure 16 shows the temperature dependence of resistivity (T) for the samples with x=0, 0.05, 0.10, 0.15, and 0.20 at zero fields in the temperature range of 30-300K. For the sample

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with x=0, it shows that there exists a bump shoulder at TP1 (=230 K), which is the same as its Curie temperature TC (=230 K). In addition, there exists a broader resistivity peak at TP2 (=138 K). For the samples with x=0.05, 0.10, 0.15, and 0.20, ρ(T) curves display the semiconducting behavior (dρ/dT 3.11), due to M 6= 0, the ground state becomes a nodal AF insulator with massive fermionic excitations. By contrast, there is only the insulating phase of the traditional Hubbard model (See Fig.5). MI transition for the Hubbard model on honeycomb lattice : From Fig.6, one could find that MI transition for the Hubbard model on honeycomb lattice occurs at a critical value about (U/t)c1 ' 2.23 at zero temperature[36, 38, 37, 39]. In the weak coupling limit (U/t < 2.23), the ground state is a semi-metal (SM) with nodal fermi-points; in the strong coupling region (U/t > 2.23), the ground state turns into NAI with M 6= 0.

4.

Effective Nonlinear σ Model in the Insulator State

In above section, we study the quantum phase transitions between semi-metal and nodal AF insulator of the π-flux Hubbard model and that in the Hubbard model on honeycomb lattice by HF approach. A question here is whether the nodal AF insulator with M 6= 0 has real long range AF order. The non-zero value of M by HF mean field method only means the existence of effective spin moments. It does not necessarily imply that the ground state of NAI is a long range AF order because the direction of the spins is chosen to be fixed along ˆ z-axis in the mean field theory. Thus one needs to examine stability of magnetic order against quantum fluctuations of effective spin moments based on a formulation by keeping spin rotation symmetry. In the following parts we will focus on the NAIs and try to deal with the spin fluctuations

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Figure 6. The staggered magnetization M of the Hubbard model on honeycomb lattice at T = 0. (U/t)c1 ' 2.23 is the critical point of the metal-insulator(MI) transition of the Hubbard model on honeycomb lattice. by using the path-integral formulation of electrons with spin rotation symmetry[1, 25, 26, 27, 28, 21, 22]. The interaction term in Eq.(1) can be handled by using the SU(2) invariant Hubbard-Stratonovich decomposition in the arbitrary on-site unit vector Ωi  n ˆ i↑ n ˆ i↓ =

cˆ†i cˆi 4

2

1 − [Ωi·ˆ c†i σˆ ci]2 . 4

(14)

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†

Here σ = (σx, σy , σz ) are the Pauli matrices. By replacing the electronic operators cˆi and cˆj by Grassmann variables c∗i and cj , the effective Lagrangian of the 2D generalized Hubbard model at half filling is obtained: X X X c∗i ∂τ ci − (ti,j c∗i cj + h.c.) − ∆ c∗i Ωi ·σci . (15) Leff = i

i

hiji

To describe the spin fluctuations, we use the Haldane’s mapping [26, 40, 41]: q i Ωi = (−1) ni 1 − L2i + Li

(16)

where ni = (nxi , nyi , nzi ) is the Neel vector that corresponds to the long-wavelength part of Ωi with a restriction n2i = 1. Li is the transverse canting field that corresponds to the z-axis for short-wavelength parts of Ωi with a restriction Li · ni = 0. We then rotate Ωi to ˆ the spin indices of the electrons at i-site:[1, 25, 26, 27, 28, 21, 22] ψi = Ui† ci Ui† ni · σUi = σz Ui† Li · σUi = li · σ where Ui ∈SU(2)/U(1).

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83

One then can derive the following effective Lagrangian after such spin transformation: Leff =

X

−

X

ψi∗∂τ ψi +

i

ψi∗a0 (i) ψi

i

(ti,j ψi∗eiaij ψj

−∆

X

X

+ h.c.)

  q ψi∗ (−1)iσz 1 − l2i + li · σ ψi

(18)

i

where the auxiliary gauge fields aij = aij,1 σx +aij,2 σy and a0 (i) = a0,1 (i) σx +a0,2 (i) σy are defined as eiaij = Ui†Uj , a0 (i) = Ui† ∂τ Ui . (19) In terms of the mean field result M = (−1)i hψi∗σz ψi i as well as the approximations, q

1 − l2i ' 1 −

l2i iaij , e ' 1 + iaij , 2

we obtain the effective Hamiltonian as: X X ψi∗∂τ ψi + ψi∗[a0 (i) − ∆σ · li ]ψi Leff ' i

i

X − [ti,j ψi∗(1 + iaij )ψj + h.c.] hiji

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−∆

X l2 X i (−1)iψi∗σz ψi + ∆M . 2 i

(20)

i

By integrating out the fermion fields ψi∗ and ψi , the effective action with the quadric terms of [a0 (i) − ∆σ · li ] and aij becomes Seff

1 = 2

Z

β

dτ 0

X 2∆2 2 [−4ς(a0 (i) − ∆σ · li )2 + 4ρs a2ij + l ] U i

(21)

i

where ρs and ς are two parameters as ζ=−

ρs = −

1 X ∆2 1 ∂ 2E0 (By ) | = B =0 y  3 , N ∂By2 N 2 k 4 |ξk |2 + ∆2

(22)

1 ∂ 2E0 (a) 1 X 2 | = a=0 1 . 2 2) 2 N ∂a2 N 4(|ξ | + ∆ k k

(23)

2 is a coefficient to be determined in different lattices. And one may see detailed calculations in Appendix. To learn the properties of the low energy physics, we study the continuum theory of the effective action in Eq.(21). In the continuum limit, we denote ni , li, iaij ' Ui†Uj − 1 and

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a0 (i) = Ui† ∂τ Ui by n(x, y), l(x, y), U †∂x U (or U † ∂y U ) and U † ∂τ U, respectively. From the relations between U †∂µ U and ∂µ n, 1 a2τ = a2τ,1 + a2τ,2 = − (∂τ n)2 , τ = 0, 4 1 2 2 2 aµ = aµ,1 + aµ,2 = (∂µ n)2, µ = x, y, 4 i a0 ·l=− (n × ∂τ n) · l, 2 the continuum formulation of the action in Eq.(21) turns into Z Z 1 β Seff = dτ d2 r[ς(∂τ n)2 + ρs (5n)2 2 0 2∆2 − 4i∆ς (n × ∂τ n) · l + ( − 4∆2ς)l2] U

(24) (25) (26)

(27)

where the vector a0 is defined as a0 = (a0,1, a0,2 , 0) . Finally we integrate the transverse canting field l and obtain the effective NLσM as Z β Z 1 1 dτ d2r[ (∂τ n)2 + c (5n)2 ] (28) Seff = 2g 0 c with a constraint n2 = 1. The coupling constant g and spin wave velocity c are defined as[1, 25, 26, 27, 28, 21, 22]: s ρs χ⊥ , c2 = ⊥ (29) g= ρs χ Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

where ρs is the spin stiffness ρs =

1 X 2 1 2 2 2 N k 2(|ξk | + ∆ )

(30)

and χ⊥ is the transverse spin susceptibility χ⊥ = [(

1 X ∆2 −1 − 2U ]−1. 3 ) 2 2 N 2 k 4(|ξk | + ∆ )

(31)

In addition, we need to determine another important parameter - the cutoff Λ. On the one hand, the effective NLσM is valid within the energy scale of Mott gap, 2∆ = U M. On the other hand, the lattice constant is a natural cutoff. Thus the cutoff is defined as the following equation [26] 2∆ ). (32) Λ = min(1, c For the π-flux Hubbard model, we get the corresponding coefficient 2 as
2 = t2 [cos (2kx) ∆2 + 8t2 + 4t2 cos (2ky ) + ∆2 + 3t2 + t2 cos (4kx )].

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85

Figure 7. The spin stiffness ρs and the spin wave velocity c of the π-flux Hubbard model at T = 0. The numerical results of ρs and c of the π-flux Hubbard model are illustrated in Fig.7. In the strong coupling limit(U = 25t) , the spin stiffness and the spin velocity are obtained as and c = 0.226278t = 1.41424J that match the earlier results ρs = 0.03936t √ = 0.2460J J 4t2 ρs = 4 , c = 2J (J = U ) obtained from the Heisenberg model [42, 43, 44, 45]. For the Hubbard model on honeycomb lattice, we obtain the coefficient 2 as √ 
1 2 t [6∆2 + 27t2 + 2∆2 + 27t2 cos 3ky 4 √  √  3ky /2 cos 3ky + 36t2 cos (3kx/2) cos   √
+ 2 5∆2 + 27t2 cos (3kx/2) cos 3ky /2  √  3ky ]. + 9t2 cos (3kx ) 1 + cos

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2 =

(34)

The numerical results of ρs and c are illustrated in Fig.8, where one can find that c = 0.168901985t = 1.055637J in the strongly coupling limit (U = 25t) also match the earlier 2 results, c = 1.06066J (J = 4tU ), obtained from Heisenberg model [46].

5.

Global Phase Diagram

In this section we will use the effective NLσM to study the magnetic properties of the insulator state. The Lagrangian of NLσM with a constraint (n2 = 1) by a Lagrange multiplier λ becomes i 1 h (∂τ n)2 + c2 (5n)2 + iλ(1 − n2 ) (35) Leff = 2cg where iλ = m2 and m is the mass gap of the spin fluctuations.

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Figure 8. The spin stiffness ρs and the spin wave velocity c of the Hubbard model on honeycomb lattice at T = 0.

5.1.

Large-N Approximation

√ Using the large-N approximation we rescale the field n → Nn and obtain the saddle-point equation of motion as [47, 48, 49] X Π(q, iωn) = 1. (36) (n0 )2 − kB T ωn ,q6=0

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In Eq.(36), n0 is the mean field value of n and Π(q, iωn) = −

gc ωn2

+

c2q2

+ m2

(37)

is the propagator of the spin fluctuations δn = n−n0 . Here ωn = 2πnkB T , n = integers. At finite temperature, the solution of n0 is always zero that is consistent to the MerminWigner theorem. From Eq.(36), we may get the solution of m as    2πc cΛ − gk −1 T m = 2kB T sinh e B sinh . (38) 2kB T In the limit T  Λ, Eq.(38) can be rewrite as:     2πc 1 1 −1 1 exp − − m = 2kB T sinh 2 kB T g gc

(39)

where

4π . (40) Λ Therefore, at zero temperature the solutions of n0 and m of Eq.(36) are determined by the dimensionless coupling constant α = gΛ. In particular, there exists a critical point αc = 4π (or gc = 4π Λ ): For the case of α < 4π, we get solutions of n0 and m: gc =

n0 = (1 −

g 1/2 ) ,m=0 gc

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87

QD

Figure 9. The dimensionless coupling constant α of the π-flux Hubbard model (cicle solid line). There are three regimes, semimetal(SM), quantum disordered(QD), antiferromagnetic(AF), separated by two critical points (U/t)c1 ' 3.11, (U/t)c2 ' 4.26, respectively. For the case of α > 4π, we get solutions of n0 and m: n0 = 0, m = 4πc(

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5.2.

1 1 − ) gc g

(42)

Global Phase Diagram of π-flux Hubbard Model

For π-flux Hubbard model, we calculate the dimensionless coupling constant α = gΛ and show results in Fig.9. The quantum critical point corresponding to αc = 4π turns into U/t ' 4.26 which divides the insulator state into two phases - quantum disordered state (QD) in the region of 3.11 < U/t < 4.26 and long range AF order in the regions of U/t > 4.26. In the regions of U/t > 4.26 (where α < αc ), at zero temperature, the mass gap vanishes (See Fig.10(a)) which means that long range AF order appears. To describe the long range AF order, we introduce a spin order parameter M0 =

M M g n0 = (1 − )1/2. 2 2 gc

(43)

As shown in Fig.10(b), the ground state of the long range AF ordered phase has a finite spin order parameter. One may see that in the strong coupling limit, U/t → ∞, the values naturally match the results derived from the Heisenberg model mapped from the π-flux Hubbard model. At finite temperature, because the energy scale of the mass gap m is always much smaller than the temperature, i.e., m  kB T (or ωn ), quantum fluctuations become negligible in a sufficiently long wavelength and low energy regime (m < |cq| < kB T ) . Thus in this region one may only consider the purely static (semiclassical) fluctuations.  Theeffec2 ρ˜s tive Lagrangian of the NLσ M then becomes L = 2 (5n) where ρ˜s = c 1g − g1c is the renomalized spin stiffness. So we call it renormalized classical (RC) region.

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Figure 10. The mass gap m of the spin fluctuations and the ordered spin moment M0 = n0 M of the π-flux Hubbard model at zero temperature. There are three regimes, semimetal (SM), quantum disordered(QD), antiferromagnetic (AF), separated by two critical points (U/t)c1 ' 3.11, (U/t)c2 ' 4.26, respectively. In the region of 3.11 < U/t < 4.26 (where α > αc ), there is a finite mass gap of spin fluctuations at zero temperature (See Fig.10): m = 4πc( g1c − 1g ). Therefore, the ground state of the insulator in this region is not a long range AF order. Instead, it is a quantum disordered state (or non-magnetic insulator state) with zero spin order parameter M0 = 0 (See Fig.10). The existence of a non-magnetic insulator state provides an alternative candidate for finding spin liquid state. Based on above results, we get the global phase diagram at finite temperature which is illustrated in Fig.11. One can see that at finite temperature, there are four crossover lines, THF , Tρ , Tν , that separate five regions. The highest crossover line is THF that is obtained from Eq.(13) and denotes the establish of the effective spin-moments. Above THF , it is metal phase without energy gap ∆ = 0. The crossover line Tρ ∼ ρs denotes the validity of the NLσM, where ρs is the energy scale of spin stiffness. In the region Tρ < T < THF , the free spin-moments are established (denoted by M 6= 0) that show a Curie-Weiss behavior. In this region one cannot use the effective NLσM. Below Tρ, short range spin-correlation exists and the effective NLσM is valid. The region below Tρ is dominated by the crossover lines Tν ∼ ρs |1 − g/gc| . In the region of Tν < T < Tρ, when g approaches gc , the spin-correlation length has a general scaling as ξ ∼ |g − gc |−1 .

(44)

So we call this region quantum critical (QC) region. Below Tν , it is RC region that has been discussed above. Obviously, such type of quantum non-magnetic insulator on bipartite lattices is induced neither by geometry frustrations regarded as the examples in varied spin models nor by the local charge fluctuations with finite energy gap. What is the physics origin of this quantum non-magnetic state? The key point is that, due to the special electron dispersion (the existence of nodal fermions for non-interacting case) the coupling constant g is almost

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Figure 11. Phase diagram of the π-flux Hubbard model at finite temperature. 1 near the MI transition (See Fig.12). Hence the non-magnetic state origproportional to M inates from quantum spin fluctuations of relatively small effective spin-moments, M → 0. The results show sharp contrast to those from the traditional Hubbard model, where the dimensionless coupling constant is always smaller than αc = 4π. Let us compare the properties of the insulator state in π-flux Hubbard model and those in the traditional Hubbard model. For the traditional Hubbard model on square lattice, due to the nesting effect, there is no MI transition at finite U and the insulator state here doesn’t 1 (See belong to NAI. In the U/t → 0 limit, the coupling constant g is not proportional to M 2 U 1/4 Fig.12). Instead, g is about g ∼ √π ( t ) that becomes smaller and turns into zero the weakly coupling limit(See more details in Ref.[26]). So the quantum fluctuations of the effective spin-moments are suppressed. Using the NL σM formulation, due to g < gc (See Fig.12), the ground state of the Hubbard model on square lattice always has a long range AF order.

5.3.

Global Phase Diagram of the Hubbard Model on Honeycomb Lattice

For the Hubbard model on honeycomb lattice, we calculate the dimensionless coupling constant α = gΛ and show results in Fig.13. The quantum critical points corresponding to αc = 4π turn into (U/t)c2 ' 2.88 and (U/t)c3 ' 2.93 which divide the insulator state into three phases - a quantum disordered state (QD) in the region of 2.88 < U/t < 2.93 and two long range AF order in the regions of 2.23 < U/t < 2.88 and U/t > 2.93. In the regions of 2.23 < U/t < 2.88 and U/t > 2.93 (where α < αc ), at zero temperature. the mass gap vanishes m = 0 while the spin order parameter is nonzero, M0 6= 0. In the region of 2.88 < U/t < 2.93 (where α > αc ), at zero temperature, there is a finite mass gap of spin fluctuations m 6= 0 while the spin order parameter is zero M0 = 0. Therefore, the ground state of the insulator in this region is non-magnetic insulator state in a narrow non-magnetic window. See the results in Fig.14 and Fig.15. In the regions of 2.23 < U/t < 2.88, the effective spin-moments are also small, M 2 ' 0.2, as becomes the

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Figure 12. Illustrations of the relations between the coupling constant g and the staggered magnetization M of the π-flux Hubbard model(circle solid line) and the traditional Hubbard model (square solid line in inset (b)).

Figure 13. The dimensionless coupling constant α of Hubbard model on honeycomb lattice at T = 0. Insert (b) shows non-magnetic region.

Figure 14. Spin order parameter M0 of the Hubbard model on honeycomb lattice at T = 0. Insert (b) shows non-magnetic region. Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Figure 15. The mass gap m of the spin fluctuations of the Hubbard model on honeycomb lattice at T = 0.

Figure 16. Phase diagram of the Hubbard model on honeycomb lattice at finite temperature. Insert (b) shows non-magnetic region.

origin of the non-magnetic state (It is noted that the cutoff in this regime is almost unit, Λ ∼ 1). Recently, the existence of the quantum non-magnetic state near Mott transition for the Hubbard model on honeycomb lattice is conformed by using the Schwinger boson technique followed by a mean field decoupling and exact diagonalization for small systems in Ref.[50]. At finite temperature, we plot the global phase diagram by identifying six regimes separated by five crossover lines (shown in Fig.16). Similarly, THF and Tρ are establishing temperatures of effective spin moments and short range spin-correlation, respectively. Below Tρ ∼ ρs , we may use O(3) NLSM to describe the physical properties of low energy degrees of freedom safely. Now the region below Tρ is dominated by two QCPs at (U/t)c2 = 2.88 and (U/t)c3 = 2.93 that control three crossover lines and four regimes : two RC regimes, a QD regime and a merged QC regime (see detail in the inset of Fig.16).

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6.

Su-Peng Kou

Half-skyrmion as Fermionic Excitation in Nodal AF Insulator

An interesting issue is the nature of the non-magnetic insulator. Is it a valence-band crystal [10], or algebra spin liquid state[1, 11], ...? To learn the nature of the non-magnetic insulator, one may exam its excitations. In the non-magnetic insulator, we get the effective model of massive spin-1 excitations Ln =

 Λ  (∂µ n)2 + m2 n2 . 2g

(45)

Using the CP(1) representation, we have Ln =

 2Λ  |(∂µ − iaµ )z| + m2z z2 g

(46)

zi σzi , ¯ zz = 1. Here aµ ≡ − 2i (¯ z∂µ z − where z is a bosonic spinon, z = (z1 , z2) , ni = ¯ zz) is introduced as an assistant gauge field. Because the bosonic spinons have mass gap, ∂µ ¯ we may integrate them and get

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Leff =

1 (∂µ aν )2 4e2a

(47)

with e2a ∼ mz = m 2 . Due to the instanton effect, the gauge field aµ obtains a mass gap and the bosonic spinons that couple the gauge field aµ are always confined. Now the lowest energy excitations are gapped spin wave. However, the answer is not quite right. Applying the Oshikawa’s commensurability condition and Hastings’ theorem to the present case, a uniform ground state with triplet excitation gap must be accompanied with other gapless excitations[51, 52]. Where is the gapless excitation? Our answer is the gapless excitations are just the topological vortices.

6.1.

Half-skyrmion

In the following parts, we focus on the topological vortices with half topological charge[53, 54, 55, 56, 57, 58, 21, 24, 59], Z 1 1 (48) Q = d2 r 0νλ n · ∂ ν n × ∂ λ n = ± . 4π 2 To stabilize topological vortices (half-skyrmions), a small easy-plane anisotropic term should be added to the original model. In the vector n representation, the solutions of the halfskyrmion of the continuum limit are λ(x − x0 ) λ(y − y0 ) , ±p , nhs = ( p 2 2 |r − r0| + λ |r − r0 |2 + λ2 λ ). ±p |r − r0|2 + λ2

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Here λ is the radius of the half-skyrmion at r0 = (x0 , y0). Inside the core |r − r0 |2 < λ, the spin is polarized; outside it |r − r0|2 > λ, one get a vortex-like spin configuration on XY plane. From the exact solutions, there exist two types of merons : one has a up-spin polarized core Q = 12 , λ(x − x0 ) λ(y − y0 ) , p , (nhs )1 = ( p 2 2 |r − r0| + λ |r − r0|2 + λ2 λ p ); |r − r0|2 + λ2

(50)

the other a down-spin polarized core Q = − 12 , λ(x − x0 ) λ(y − y0 ) (nhs )2 = ( p , p , 2 2 |r − r0| + λ |r − r0|2 + λ2 λ −p ). |r − r0|2 + λ2

(51)

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One can see Fig.18 that shows the two types of merons. So the topological charge of a half-skyrmion is determined by both the spin configuration and the polarized direction of AF order in the core. Also, there are two types of anti-merons : one has a up-spin polarized core Q = − 12 , λ(x − x0 ) λ(y − y0 ) , p , (nhs)3 = ( p 2 2 |r − r0| + λ |r − r0|2 + λ2 λ ); −p |r − r0 |2 + λ2

(52)

the other a down-spin polarized core Q = 12 , λ(x − x0 ) λ(y − y0 ) , −p , (nhs )4 = ( p 2 2 |r − r0| + λ |r − r0 |2 + λ2 λ ). −p |r − r0 |2 + λ2

(53)

Next, we calculate the mass gap of the half-skyrmions. In AF ordered state, the mass of half-skyrmion mhs is associated with the ordered staggered moment : mhs = 2π ρ˜s (g < gc )

(54)

where ρ˜s = ρs (1 − ggc ) is the renormalized spin stiffness. In the quantum non-magnetic insulator state (g > gc ), the mass of half-skyrmion vanishes[60], mhs = 0. When halfskyrmions become mobile, their quantum statistics becomes important. Is a half-skyrmion boson or fermion?

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Figure 17. The scheme of (nhs )1, the meron with topological charge −1/2. Down-spins locate at the center.

Figure 18. The scheme of (nhs )2 , the meron with topological charge 1/2. Up-spins locate at the center.

6.2.

Zero Modes on Half Skyrmions

To know the statistics of the half-skyrmion, in the first step, we solve the fermionic zero modes around them. Let us derive long wave-length effective Lagrangian of the hopping term in the extended Hubbard models. Near the two nodal fermi-points at k1 = ( π2 , π2 ), k2 = ( π2 , − π2 ) for the 2π 2π (1, √13 ) and k2 = √ (−1, − √13 ) for the π-flux Hubbard model and those at k1 = √ 3 3 Hubbard model on honeycomb lattice, the spectrums of fermions become linear. In the continuum limit, by replacing the operator cˆi and site number i by Grassmann number ψ(x) and continuum coordinates x, y, the Dirac-like effective Lagrangian describes the low energy fermionic modes for both cases Lf = iψ¯1γµ ∂µ ψ1 + iψ¯2γµ ∂µ ψ2

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95

where ψ¯1 = ψ1† γ0 = ( ψ¯↑1A, ψ¯↑1B , ψ¯↓1A, ψ¯↓1B ) and ψ¯2 = ψ2†γ0 = ( ψ¯↑2B , ψ¯↑2A, ψ¯↓2B , ψ¯↓2A )[33, 34, 35]. γµ is defined as γ0 = σ0 ⊗ τz ,   1 0 . τ x, τ y , τ z are Pauli matrices. We have set γ1 = σ0 ⊗ τy , γ2 = σ0 ⊗ τx , σ0 = 0 1 the Fermi velocity to be unit vF = 1. After considering the local spin moments, we get the long wave-length effective model of nodal AF insulator [62, 63, 64] X iψ¯αγµ ∂µ ψα + me (ψ¯1n · σψ1 − ψ¯2n · σψ2). (56) Leff = α

α = 1, 2 labels the two Fermi points. me denoted the mass gap of the electrons. Now we consider a half-skyrmion with a narrow core of lattice size Λ−1 = max(a, (2∆)−1) as ! x − x0 y − y0 nhs = p , ±p ,0 . (57) |r − r0|2 + Λ−1 |r − r0|2 + Λ−1

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See Fig.19, all spins of the meron are suppressed onto the XY plane[53, 54, 55, 61, 56, 57, 58, 59].

Figure 19. The scheme of the meron with a narrow core. All spins are suppressed onto the XY plane. Around a meron configuration, the fermionic operators are expanded as X ˆb‘αke−iEk t ψαk (r) ψˆα (r, t) =

(58)

k6=0

+

X

† dˆ†αk eiEk t ψαk (r) + a ˆ0α ψα0 (r),

k6=0

where ˆbαk and dˆ†αk are operators of k 6= 0 modes that are irrelevant to the soliton states † 0∗ 0∗ 0∗ 0∗ , ψ↑αBk , ψ↓αAk , ψ↓αBk (r) = ( ψ↑αAk ) are the functions of discussed below. ψαk 0 zero modes. ˆ aα are annihilation operators of zero modes. Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Su-Peng Kou To obtain the zero modes, we write down two Dirac equations from Eq.(56) i∂x γ1ψ10 + i∂y γ2ψ10 + me n · σψ10 = 0

(59)

and i∂xγ1 ψ20 + i∂y γ2ψ20 − me n · σψ20 = 0.





(60)

ξ1(˜ x)e−iθ  ξ2(˜  x)  , we Firstly we solve the Dirac equation for ψ10. With the ansatz ψ10 =   ξ3(˜  x) iθ ξ4(˜ x)e have ∂x˜ ξ2 = ξ3 , ∂x˜ ξ3 = ξ2, ξ1 ξ4 ∂x˜ ξ1 = − + ξ4, ∂x˜ ξ4 = − + ξ1 x ˜ x ˜ where r = | r | (cos θ, sin θ) and x ˜=

|r| me .

(61)

The solution has been obtained in Ref.[?] as 1

x) = ξ4 (˜ x) = 0, ξ2 (˜ x) = ξ3 (˜ x) = x ˜ 2 K 1 (˜ x) ξ1(˜ 2

x) is the modified Bessel function. So the solution of ψ10 becomes where K 1 (˜ 2   0  exp(− |r−r0 | )    me  |r−r0 |  .  exp(− m )  e 0

(62)

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To solve ψ20, we transform the equation i∂i γiψ20 − me n · σψ20 = 0 into U i∂i γi ψ˜20U −1 + me U n · σψ˜20U −1 = 0,

(63)

where U = eiπγ0 /2, U γi U −1 = −γi and U −1 ψ20U = ψ˜20. Then the solution of ψ20 is obtained as   0  − exp(− |r−r0 | )    me (64) .  |r−r0 |  exp(− m )  e 0 0 0 , ψ↓1B , It is noticeable that from above solutions of zero modes, the components ψ↑1A 0 0 ψ↓2A and ψ↑2B are all zero. We have set the Fermi velocity to be unit vF = 1. So the solution of zero modes is     0 0  exp(− |r−r0 | )   − exp(− |r−r0 | )      0 me me and ψ = ψ10 =    . 2 |r−r0 | 0|  exp(− |r−r    ) ) exp(− me me 0 0

In addition, the numerical results of the zero modes of a half-skyrmion with a narrow core on a 21 × 21 square lattice was shown in Fig.20. The exact charge is localized around the defect center within a lengthscale ∼ Λ−1. Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Figure 20. The fermionic zero mode of half skyrmion on a 21 × 21 lattice.

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6.3.

Induced Quantum Numbers on Half Skyrmion

In the second step, we calculate the induced quantum numbers on a half-skyrmion. For the solutions of zero modes, there are four zero-energy soliton states | soli around a half skyrmion which are denoted by | 1+ i⊗ | 2+ i, | 1− i⊗ | 2− i, | 1− i⊗ | 2+ i and | 1+ i⊗ | 2− i. | 1− i and | 2− i are empty states of the zero modes ψ10(r) and ψ20(r); | 1+ i ˆ0α and | soli and | 2+ i are occupied states of them. Thus we have the relationship between a as ˆ01 | 1+ i =| 1− i, a a ˆ01 | 1− i = 0,

(65)

ˆ02 | 2− i = 0, ˆ02 | 2+ i =| 2− i, a a ˆ P Firstly we define total induced fermion number operators of the soliton states, NF = ˆ Nα,F with α

ˆα,F ≡ N

Z

: ψˆα† ψˆα : d2r

ˆ0α + = (ˆ a0α)†a

(66)

X † 1 † (ˆbαkˆbαk − dˆαk dˆαk ) − . 2 k6=0

ˆ0α and | soli in : ψˆα† ψˆα : means normal product of ψˆα† ψˆα. From the relation between a Eq.(65), we find that | 1± i or | 2± i have eigenvalues of ± 12 of the total induced fermion Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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ˆF , number operator N ˆ1,F |1±i = ± 1 |1± i, N 2 ˆ2,F |2±i = ± 1 |2± i, N 2 1 2

ˆ1,F |2± i = 0, N

(67)

ˆ2,F |1± i = 0. N

z = Another important induced quantum number operator is staggered spin operator, Sˆ(π,π) R P † P † † † † † 1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ cˆi σz cˆi − 2 cˆi σz cˆi = 2 : [(ψ↑1Aψ↑1A + ψ↓1B ψ↓1B − ψ↓1A ψ↓1A −ψ↑1B ψ↑1B ) +

i∈A

i∈B

† † † † ψˆ↑2A + ψˆ↓2B ψˆ↓2B − ψˆ↓2A ψˆ↓2A − ψˆ↑2B ψˆ↑2B )] : d2r. For the four degenerate zero (ψˆ↑2A modes, it can be simplified into

1 ˆ z ˆ Sˆ(π,π) | soli = (N 2,F − N1,F ) | soli. 2

(68)

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0 0 0 Let us show the detailed calculations. From the zero solutions of ψ↑1A , ψ↓1B , ψ↓2A and 0 ψ↑2B , we obtain four equations Z † ψˆ↑1A : d2 r) | soli ≡ 0, (69) ( : ψˆ↑1A Z † ψˆ↓1B : d2r) | soli ≡ 0, ( : ψˆ↓1B Z † ( : ψˆ↓2A ψˆ↓2A : d2 r) | soli ≡ 0, Z † ( : ψˆ↑2B ψˆ↑2B : d2r) | soli ≡ 0.

Using above four equations, we obtain Z 1 † † † † z d2r : (−ψˆ↑1A | soli = ψˆ↑1A − ψˆ↓1B ψˆ↓1B − ψˆ↓1A ψˆ↓1A − ψˆ↑1B ψˆ↑1B Sˆ(π,π) 2 1 ˆ † † † † ˆ +ψˆ↑2A ψˆ↑2A + ψˆ↓2B ψˆ↓2B + ψˆ↓2A ψˆ↓2A + ψˆ↑2B ψˆ↑2B ) :| soli = − (N 1,F − N2,F ) | soli. 2 Then we calculate two induced quantum numbers defined above. Without doping, the soliton states of a half skyrmion are denoted by | 1− i⊗ | 2+ i and | 1+ i⊗ | 2− i. One can easily check that the total induced fermion number on the solitons is zero from the cancelation effect between two nodals ˆF | 1+ i⊗ | 2− i = 0. ˆF | 1− i⊗ | 2+ i = N N

(70)

It is consistent to the earlier results that forbid a Hopf term for the low energy theory of two dimensional Heisenberg model[65]. On the other hand, there exists an induced staggered spin moment on the soliton states | 1− i⊗ | 2+ i and | 1+ i⊗ | 2− i, 1 z | 1− i⊗ | 2+ i = | 1− i⊗ | 2+ i, Sˆ(π,π) 2 1 z | 1+ i⊗ | 2− i = − | 1+ i⊗ | 2− i. Sˆ(π,π) 2

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The induced staggered spin moment may be straightforwardly obtained by combining the z definition of Sˆ(π,π) and Eq.(67) together. In order to giving a clearly comparison, we show the induced quantum numbers on a half-skyrmion in Table.1:

Total fermion number staggered spin number Topological charge

| 1+ i⊗ | 2− i 0 − 12 1 2

| 1− i⊗ | 2+ i 0 1 2 − 12

From the table one can see that the topological charge of the half-skyrmions is spin-dependence : For the meron with 12 (− 12 ) spin number, the topological charge Q= − 12 (Q= 12 ) with upspin (down-spin) polarized in the core.

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6.4.

Fermionic Statistics of Half Skyrmion

In the third step, we examine the statistics of a half-skyrmion with an induced staggered spin moment. When half skyrmions become mobile, their quantum statistics becomes important. Let us examine the statistics of a half skyrmion with an induced staggered spin moment.  In z↑ CP(1) representation of n, a ”bosonic spinon” is introduced by n = ¯ zσz with z = z↓ and ¯ zz = 1. Since each ”bosonic spinon” z carries ± 12 staggered spin moment, an induced staggered spin moment corresponds to a trapped ”bosonic spinon” z. On the other hand, a half skyrmion can be regarded as a π−flux of the ”bosonic spinon”, Z Z 1 1 1 2 n · ∂x n × ∂y nd r = µν ∂µ aν d2 r = ± (72) 2π 2π 2 i z∂µ z 2 (¯

− ∂µ¯ zz). To be more explicit, moving a ”bosonic spinon” z around a   z↑eiφ 0 half skyrmion generates a Berry phase φ to z → z = where z↓eiφ Z (73) φ = µν ∂µ aν d2r = ±π.

with aµ ≡

As a result, a ”bosonic spinon” z and a half skyrmion (meron or antimeron ) share mutual semion statistics. Binding the trapped ”bosonic spinon”, a mobile half skyrmion becomes a fermionic particle. We may use the operator fσ to describe such neutral fermionic particle with half spin. Here σ is the spin index. The relation between the zero energy states and the fermionic states is given as (74) | 1+ i⊗ | 2− i = f↓† | 0if and †

| 1− i⊗ | 2+ i = f↑ | 0if .

(75)

The state |0if is defined through f↑ |0if = f↓ |0if = 0. We call such neutral object (fermion with ± 12 spin degree freedom) a ”fermionic spinon”. Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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6.5.

Effective Model of Half-skyrmions

In the fourth step, we study the dynamics of the half-skyrmion (fermionic spinon) and then obtain its effective model. Because z particle can be regarded as a π-flux of a half-skyrmion, with a single bosonic spinon on each site, the half-skyrmions show similar behavior of vortices in XY model : it can move on dual lattices (square lattice for the π-flux Hubbard model and triangle lattice for the honeycomb lattice) and feel an effective π-flux phase[55]. Then the leading order of the hopping term of the half-skyrmions is  X Hhs = − t˜I,J fI† fJ + h.c. (76) hI,Ji

where fI = (fI↑, fI↓)T are defined as the annihilation operators of fermionic spinons. I and J denote two nearest-neighbor dual sites. Here t˜I,J is the effective hopping of the fermionic spinons. Since there is also a π-flux phase for the half-skyrmions on both square lattice for the π-flux Hubbard model and triangle lattice for the honeycomb lattice, one needs to divide the dual-lattice into two sublattices, A and B. So there also exist two nodal fermi-points and the spectrum of fermions becomes linear in the vicinity of the two nodal points. In the continuum limit, we get a Dirac-like effective Lagrangian that describes the low energy fermionic spinons X X ¯ a γµ ∂µ Ψa + imhs iΨ ΨaΨa (77) Lhs = a

a

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where ¯ 1 = Ψ† γ0 = ( f¯↑1A , f¯↑1B , f¯↓1A , f¯↓1B ) Ψ 1 and ¯ 2 = Ψ† γ0 = ( f¯↑2B , f¯↑2A , f¯↓2B , f¯↓2A ). Ψ 2   1 0 . τ x , τ y , τ z are γµ is defined as γ0 = σ0 ⊗ τz , γ1 = σ0 ⊗ τy , γ2 = σ0 ⊗ τx , σ0 = 0 1 Pauli matrices. In the effective Lagrangian, the second term is added phenomenologically to describe the energy gap of the fermionic spinons. For simplicity, we set the Fermi velocity to be unit. Similar effective model of half-skyrmions has been obtained in Ref.[56, 55] from different points of view.

7.

Nodal Spin Liquid State

Whether the non-magnetic insulator is a VBC state? Because the VBC state is be characterized by the condensation of the half-skyrmions. If half-skyrmions are bosons, they will condense. Then one gets a VBC state (or quantum dimer state)[66]. However, in nodal AF order half-skyrmion obeys fermionic statistics by trapping a bosonic spinon z. Then the massless fermionic vortices lead to a new story - a new type quantum spin liquid state - nodal spin liquid (NSL). NSL is a spin liquid state with nodal fermion excitations and

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emergent U (1) gauge fields. We would like to say that the quantum non-magnetic state near Mott transition of the π-flux Hubbard model and the Hubbard model on honeycomb lattice are examples of NSL state.

7.1.

Mutual Chern-Simons Theory

From above results, there exist two types of fields, the bosonic spinon z and the fermionic spinon fσ† . Now the low energy effective model becomes X 1 (∂µ n)2 + i Ψaσ (γµ ∂µ + mhs ) Ψaσ . (78) Leff = Lhs + Ls = 2g aσ In particular, there exists non-trivial topological relationship between z and Ψaσ - the fields Ψaσ that carry ± 12 winding number of AF vector n, X σz 1 ¯ a γµ Ψa . µνλ n · ∂ν n × ∂λ n = −i Ψ (79) 8π 2 a z

Here the operator σ2 means that the topological charge of the fermionic spinons is spindependence : For the meron with 12 (− 12 ) spin number, the topological charge Q= − 12 (Q= 12 ) with up-spin (down-spin) polarized in the core. To ensure such constraint in Eq.(79), we add a new term in the effective Lagrangian, X σz 1 ¯ a γµΨa ). Ψ (80) Lconstraint = iAµ ( µνλ n · ∂ν n × ∂λ n + i 8π 2 a

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Finally the low energy effective theory is obtained as Leff = Lhs + Ls + Lconstraint X 1 = i Ψa (γµ ∂µ − iσ z Aµ + mhs ) Ψa + (∂µ n)2 + LM CS 2g a X 2 Ψa (γµ ∂µ + iσ z Aµ + mhs ) Ψa + |(∂µ − iaµ )z|2 + LM CS . = i g a

(81)

Here, there is a mutual-Chern-Simons term, 1 µνλ 1  Aµ ∂ν aλ = µνλ aµ ∂ν Aλ. π π To obtain above low energy effective theory, we have use the equation LM CS =

(82)

1 n · ∂ν n × ∂λn. (83) 2 Then the effective model becomes an U (1)×U (1) mutual-Chern-Simons (MCS) gauge theory. Fermions Ψaσ couple to an U (1) gauge field Aµ ; boson spinon z couples to an U (1) gauge field aµ . The results say that Ψaσ act as half topological vortices for aµ and z act as half topological vortices of Aµ . This effective Lagrangian proposed in here and earlier papers [59, 67] retains the full symmetries of translation, parity, time-reversal, and global spin rotation , in contrast to the conventional Chern-Simons theories where first two symmetries are usually broken. In the following parts of the paper, we use U (1) × U (1) MCS theory to learn the quantum non-magnetic state in π-flux state. ∂ν aλ − ∂λ aν =

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7.2.

Su-Peng Kou

Long Range AF Order

If g < gc , the ground state has a long range N´eel order, hzi = z0 6= 0 or hni 6= 0. The fermions acquire a mass gap mhs ∼ 2π ρ˜s and become irrelevant to the low energy physics. Hence the effective model turns into 2z02 Λ 1 | aµ |2 + µνλ aµ ∂ν Aλ geff π X +i Ψa (γµ ∂µ − iσ z Aµ + mhs ) Ψa

Leff '

(84)

a c where geff is renormalized coupling constant as geff = (gg·g . Here the mass term of the c −g) gauge field aµ is caused by z condensation. After integrating aµ we get a renormalized kinetic term for gauge field Aµ ,

Leff =

1 (∂µ Aν )2 + Lhs 4e2A

(85)

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g

with e12 = z2 πeff2 Λ . Due to the instanton effect, the gauge field Aµ also obtains a mass gap 0 A and a linear confinement appears for fermions[68]. The low energy physics is dominated only by spin waves and the effective Lagrangian becomes Leff = 2gΛ (∂µ n)2. eff At small T , the AF order is believed in the so-called renormalized classical (RC) region. In the RC region, the effective Lagrangian loses the Lorentz invariance and becomes Leff → 1 ˜s(∇n)2. Here ρ˜s is the renormalized spin stiffness as ρ˜s = ρs (1 − ggc ). Then at low 2ρ temperatures, fermionic vortices are paired by the logarithmic-attractive interaction V (r) = |r| 2π ρ˜s ln a0 . With the increase of temperature, neutral vortex-antivortex pairs like those in the XY model are thermally excited, leading to a conventional contribution to the screening effect. By the conventional Kosterlitz-Thouless (KT) theory, there exists a “deconfining” temperature, π ρ˜s . (86) Tde ' 2 Above Tde, free excited fermionic vortices exist. The spin-correlation becomes qualitatively different from the prediction from non-linear σ model.

7.3.

Nodal Spin Liquid

If g > gc , the ground state has no AF long range order. Now z has a mass gap mz , or one has the massive spin-1 quanta, i Λ h |(∂µ − iaµ )z|2 + m2z z2 . (87) Ln = 2g Then the effective model becomes X Ψa (γµ ∂µ + iσ z Aµ ) Ψa Leff = i a

+

i 1 Λ h |(∂µ − iaµ )z|2 + m2z z2 + µνλ Aµ ∂ν aλ . 2g π

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(88)

Quantum Non-magnetic States Near Metal-insulator Transition

103

After integrating the massive bosons z, there appears the kinetic term for gauge field aµ Leff = i

X

Ψa (γµ ∂µ − iσ z Aµ ) Ψa

a

1 1 + 2 (∂µ aν )2 + µνλ Aµ ∂ν aλ 4ea π

(89)

where e2a ' 3πmz . Furthermore, due to the MCS term, after integrating aµ we obtain a mass term for gauge field Aµ , X e2 Ψa (γµ ∂µ + iσ z Aµ ) Ψa + a2 A2µ . (90) Leff = i π a Due to exchanging the gauge field Aµ , a short range interaction is induced between fermions. It is obvious that the short range interaction is irrelevant. As a result, the massless Dirac particles that couple to Aµ become real low energy degrees of freedom. On the other hand, is bosonic spinon that couples to aµ real quasi-particle? To answer the question, we need to calculate the energy gap of the gauge field aµ . After integrating the massless fermions, the effective model turns into Leff = − where e2a ' 3πmz ,

1 4e2A

=

1 1 1 (∂ν aµ )2 − 2 (∂ν Aµ )2 + µνλ aµ ∂ν Aλ 2 4ea π 4eA 1 √ 16

p2

(91)

and p2 = p2µ is the momentum. After integrating Aµ we

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get the effective model of aµ as the Leff = −

1 e2A 2 2 (∂ a ) + a . ν µ 4e2a π2 µ

(92)

This Lagrangian has the schematic form − 2e12 p2 + π42 p. That means the momentum of a the gauge field aµ is not zero, p 6= 0. Then the gauge field aµ has a finite energy gap 2 a 2 a ) and shows roton-like behavior. As a result, the induced interaction by (Leff ∼ 8e π4 µ exchanging the gauge field aµ is irrelevant and the bosonic spinons are real quasi-particles. Finally, the low energy effective theory of nodal spin liquid becomes X

  ¯ a γµ ∂µ Ψa + Λ (∂µ z)2 + m2z z2 Ψ 2g a p 4 p2 2 1 a . − 2 (∂ν aµ )2 + 4ea π2 µ

Leff = i

(93)

There are three types of quasi-particles : two flavor gapless fermionic spinons Ψaσ , gapped bosonic spinons z and the roton-like gauge fields aµ . Fig.21 is a scheme of the dispersion of the nodal spin liquid : a gapless fermionic excitation and a roton-like excitation. Therefore, from the effective theory we conclude that NSL state is a new type of quantum spin liquid. Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Su-Peng Kou

Figure 21. Scheme of the dispersion of the nodal spin liquid : a gapless fermionic excitation and a roton-like excitation

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7.4.

Experimental Predictions

In addition, we show the experimental predictions of NSL state. Firstly, we calculate the spin-correlation in the NSL state. In NSL, the spin-correlation decay exponentially 



+ (94) S (x, y)S −(0) = eiQ·Ri n+ (x, y)n−(0) ∼ eiQ·Ri e−r/ξ p with Q = (π, π), r = x2 + y 2 and n± = nx ± iny . Here ξ is spin correlated length, ξ = Λg8πeff . In Fig.22, we give the spin-correlation length of the Hubbard model on honeyc at T = 0.02t, 0.1t, 0.2t, respectively. From the Fig.22, comb lattice defined as ξ = m taking T = 0.02t as an example, one can see that the spin-correlation length increases quickly with increasing interaction U/t. However, the spin-correlation length doesn’t increase monotonously with U/t - it will decrease and reach a minimum value near the MI transition U/t ∼ (U/t)c2 ∼ (U/t)c3 . The dip of the spin-correlation length will indicate the existence of the non-magnetic state near MI transition. When one increases the interaction further, the spin-correlation increases again and finally decreases in the strongly interacting limit due to J → 0. For other cases with higher temperature, T = 0.1t, 0.2t, there exist similar dip structure of the spin-correlation length via U/t which means that people may observe the anomalous spin dynamics more easily in experiments. In particular, one can see that the narrow non-magnetic insulator is sensitive to the choice of the cutoff Λ, however, the dip structure of the spin-correlation length via U/t is robust. Secondly, we calculate the special heat. Because the fermionic spinons have no energy gap, the special heat is dominated by fermions. So at low temperature, the the special heat is 12ζ (kB T )2 (95) CV = π R ∞ 2 dx = 34 Γ(3)ζ(3) ' 1.803. where ζ = 0 exx +1

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Quantum Non-magnetic States Near Metal-insulator Transition

105

Figure 22. Correlation lengh ξ of the Hubbard model on honeycomb lattice at T = 0.02, 0.1, 0.2. The unit of temperature is t. Thirdly, we calculate the spin susceptibility. The definition of the spin susceptibility is

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1 F = F (B = 0) − χB 2 2

(96)

where F is free energy and B is the external magnetic field. There are two contributions to the total spin susceptibility χ, one from bosonic spinons, the other is from the fermionic spinons as (97) χ = χb + χf . Here χb is given by[49, 69] χb =

2 2 (2µB M )2 ms β χ⊥ µ2B M 2 + [ − ln(emsβ − 1)] 3 πβ 1 − e−ms β

(98)

with β ≡ 1/kBT . The contribution from the fermionic spinons is almost linear temperature dependence 4 ln 2 (99) χf = πβ in units of (gµB)2 .

7.5.

Discussion

Finally, we give a comparison to different quantum orders with π-vortex (half-skyrmion or vison) and bosonic spinon. In general, for a system with π-vortex and bosonic spinon, three exist five types of quantum orders as[70] : 1. VBC state: If one has gapless bosonic π-vortices and massive bosonic spinons, then the ground state is always VBC state with spontaneous translation symmetry breaking;

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2. AF order: If one has massive bosonic π-vortices and gapless bosonic spinons, then the ground state is an AF order with spontaneous spin rotation symmetry breaking; 3. Z2 topological order : if one has massive (fermionic or bosonic) π-vortex and massive bosonic spinon, the ground state must be a Z2 topological (Z2T) order with topologically degenerate ground state; 4. Algebraic vortex liquid : if one has gapless fermionic π-vortices (the fermionic spinons in this paper) and massive bosonic spinons, then ground state may be an algebraic vortex liquid (AVL)[71]. In algebraic vortex liquid, fermionic excitations themselves couple to massless U (1) gauge fields, as gives an example to algebraic spin liquid. Thus there doesn’t exist any type of real quasi-particles; 5. Nodal spin liquid : if one has gapless fermionic π-vortices (the fermionic spinons in this paper) and massive bosonic spinons, then the ground state is a nodal spin liquid without any spontaneous symmetry breaking. In particular, there exist gapped rotonlike gauge modes. In order to giving a clearly comparison, we give Table.2 :

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VBC state AF order AVL state Z2T order NSL state

π-vortex massless Boson massive Boson massless fermion massive Boson (fermion) massless fermion

bosonic spinon massive Boson massless Boson massive Boson massive Boson massive Boson

We give a short remark on the relation between NSL state and the AVL in Ref. [11, 12, 72]. In AVL, fermionic excitations coupling to a massless U (1) gauge field and cannot be real quasi-particles. In NSL state, due to the protection from the mutual semion statistics between fermionic spinons and bosonic spinons, fermionic excitations are real excitations. There is neither bosonic spinon nor roton-like gauge mode in AVL state and the low energy effective theory of AVL state is also difference from that of NSL. In AVL, the spin-correlation shows critical behavior, while in NSL state, the spin-correlation decay exponentially.

8.

Conclusion

In the end of the paper, we draw a conclusion. In this paper, we investigate the NAIs of the π-flux Hubbard model and the Hubbard model on honeycomb lattice within an approach keeping spin rotation symmetry. We calculate the spin stiffness, the transverse spin susceptibility, the spin wave velocity and the coupling constant g. In the strong coupling Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

Quantum Non-magnetic States Near Metal-insulator Transition

107

limit (U/t → ∞), our results of spin velocity and spin order parameter agree with the results obtained from earlier calculations of the traditional Hubbard model. In particular, quantum non-magnetic insulators are explored near the MI transition that corresponds to the strong coupling region of the effective NLσM, g > gc . Such type of quantum nonmagnetic insulator in bipartite lattices is driven by quantum spin fluctuations of relatively small effective spin-moments. Furthermore, by considering the fermionic nature of vortices (half-skyrmions), a new type of quantum state - nodal spin liquid becomes the ground state of the non-magnetic insulator state. The low energy physics is basically determined by its U (1) × U (1) mutual Chern-Simons gauge theory. There exist three types of quasiparticles in nodal spin liquids: nodal fermionic spinons, gapped bosonic spinons and rotonlike U (1) gauge field. It is the mutual semion statistics between fermionic spinons and bosonic spinons that guarantee the stability of NSL state. In addition, because NSL state represents a new class of quantum state which may be applied to learn the nature of the spin liquid state in other systems, for example, 2D S = 1/2 Heisenberg model on Kag´ome lattice, 2D frustrated Hubbard models. NSL may be the possible candidate of the non-superconducting (insulator) nodal liquid explored in the under-doped high Tc cuprates by angle resolved photoemission spectroscopy in Ref.[73].

9.

Appendix: The Detailed Calculations of ρs and ς

To give ρs and ς for calculation, we choose Ui to be

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Ui =



∗ ∗ zi↑ zi↓ −zi↓ zi↑



,

(100)

zi σzi , zi = (zi↑ , zi↓)T , ¯ zi zi = 1. And the spin fluctuations around ni = ˆ zi is where ni = ¯ ni = ˆ zi +Re (φi ) x ˆ+Im (φi ) y ˆ   2
1 − |φi | /8 + O φ3i . zi = φi /2

(101) (102)

Then the quantities Ui† Uj and Ui† ∂τ Ui can be expanded in the power of φi − φj and ∂τ φi , Ui†Uj = ei  † Ui ∂τ Ui =

φi −φj 2

σy

0 1 2 ∂τ φi

− 12 ∂τ φi 0



(103) .

(104)

According to Eq.(19), the gauge field aij and a0 (i) are given as 1 (φi − φj ) σy 2 i a0 (i) = ∂τ φi σy . 2 aij =

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(105) (106)

108

Su-Peng Kou

Supposing aij and a0 (i) to be a constant in space and denoting ∂i φi = a and ∂τ φi = iBy , we have 1 aij = − a · (i − j)σy 2 1 a0 (i) = − By σy . 2

(107) (108)

The energy of Hamiltonian of Eq.(21) becomes E (By , a) =

1 2 1 ζB + ρs a2 . 2 y 2

(109)

Then one could get ζ and ρs from the following equations by calculating the partial derivative of the energy ζ=− ρs = −

1 ∂ 2 E0 (By ) |By =0 N ∂By2

(110)

1 ∂ 2 E0 (a) |a=0 . N ∂a2

(111)

Here E0 (By ) and E0 (a) are the energy of the lower Hubbard band  X ζ ζ E0 (By ) = E+,k + E−,k k

E0 (a) =

X

ρ ρ E+,k + E−,k



(112) (113)

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k ζ ζ ρ ρ , E−,k and E+,k , E−,k are the energies of the following Hamiltonian Hζ and where E+,k Hρ X X Hζ = − (ti,j ψi∗ψj + h.c.) − ∆ (−1)i ψi∗σz ψi

+

X

i

ψi∗a0 (i) ψi

(114)

i

Hρ = −

X

(ti,j ψi∗eaij ψj + h.c.) − ∆

X (−1)iψi∗σz ψi .

(115)

i

Using the Fourier transformations for Hζ , we have the spectrum of the lower band of Hζ s  By 2 ζ E±,k = − + ∆2 (116) |ξk | ± 2 where ξk has been obtained in Eq.(8). Using Eq.(110), ζ is obtained as ζ=

∆2 1 X  3 . N 2 2 2 k 4 |ξk | + ∆

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(117)

Quantum Non-magnetic States Near Metal-insulator Transition

109

Similarly, using the Fourier transformations for Hρ , we obtain the spectrum of the lower band of Hρ r h i1 2 ρ E±,k = − ∆2 + |ψ|2 + |ϕ|2 ± 4∆2 |ψ|2 − (ϕψ ∗ − ψϕ∗)2 (118) where ϕ and ψ are defined as  1 ϕ = −t e cos a·δ 2 δ   X 1 ik·δ e sin ψ = −t a·δ . 2 X

ik·δ



(119) (120)

δ

Here δ denotes the vectors of two nearest neighbors. For the π-flux Hubbard model, we have 1 = (a, √
0) , δ2 = (0, a);√for
the Hubbard model on honeycomb lattice, we have δ1 = a2 1, 3 , δ2 = a2 1, − 3 , δ3 = (−a, 0). Using Eq.(111), ρs is given as ρs =

1 X 2 1 . 2 2) 2 N 4(|ξ | + ∆ k k

(121)

2 is a coefficient to be determined in different lattices.

Acknowledgments

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This research is supported by NCET, NFSC Grant no. 10874017.

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[24] Kou S P and L. F. Liu, arXiv:0910.2070. [25] Dupuis N, Phys. Rev. B. 2002, 65, 245118-245122. [26] Borejsza K, Dupuis N, Europhys. Lett. 2003, 63, 722-728; Borejsza K and Dupuis N, Phys. Rev. B. 2004, 69, 085119-085125. [27] Schulz H J, Phys. Rev. Lett. 1990, 65, 2462-2465; Schulz H J, in The hubbard Model, 1995, Edited by D. Baeriswyl (Plenum, New York). [28] Weng Z Y, Ting C S, and Lee T K, Phys. Rev. B. 1991, 43, 3790-3793. [29] Hsu T C, Phys. Rev. B. 1990, 41, 11379-11387. [30] Jaksch D and Zoller P, New J. Phys. 2003, 5, 56-66. ¨ [31] Juzeli¯unas G and Ohberg P, Phys. Rev. Lett. 2004, 93, 033602-033605; Juzeli¯unas G, ¨ Ohberg P, Ruseckas J and Klein A, Phys. Rev. A. 2005, 71 , 053614-053622. [32] Lin Y J, et. al., Phys. Rev. Lett. 2009, 102, 130401-130404. [33] Hou C Y, Chamon C, and Mudry C, Phys. Rev. Lett. 2007, 98, 186809-186812. [34] Herbut I F, Phys. Rev. Lett. 2007, 99, 206404-206407. Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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[49] Sachdev S, Quantum Phase Transitions , 1999, Cambridge University Press. [50] D.C. Cabra, C.A. Lamas, and H.D. Rosales, cond-mat/10033226. [51] Oshikawa M, Phys. Rev. Lett. 2000, 84, 1535-1538. [52] Hastings M B, Phys. Rev. B. 2004, 69 104431-104443. [53] Belavin A A and Polyakov A M, JETP Lett. 1975, 22, 245-247. [54] Verg´es J A, Louis E, Lomdahl P S, Guinea F, and Bishop A R, Phys. Rev. B. 1991, 43, 6099–6108; John S, Berciu M and Golubentsev A, Europhys. Lett. 1998, 41, 31; Berciu M and John S, Phys. Rev. B. 1998, 57, 9521–9543; Berciu M and John S, Phys. Rev. B. 2000, 61, 16454-16469. [55] Morinari T, Phys. Rev. B. 2005, 72, 104502-104511. [56] Ng T K, Phys. Rev. B. 1995, 52, 9491–9499; Phys. Rev. Lett. 1999, 82, 3504-3507; Int. J. Mod. Phys. B. 2000, 14, 349-369. [57] Otsuka Y and Hatsugai Y, Phys. Rev. B. 2002, 65, 073101-073104. [58] Weng Z Y, Sheng D N, and Ting C S, Phys. Rev. Lett. 1998, 80, 5401-5404; Weng Z Y, Int. J. Mod. Phys. B. 2007, 21, 773-827. Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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[59] Kou S P and Weng Z Y, Phy. Rev. Lett. 2003, 90, 157003-157006; Kou S P, Qi X L, Weng Z Y, Phys. Rev. B. 2005, 71, 235102-235113. [60] Auerbach A, et al, Phys. Rev. B. 1991, 43, 11515-11518. [61] Balents L, Fisher M P A and Nayak C, Int. J. Mod. Phys. 1998, B 12, 1033-1068. [62] Tanaka A and Hu X, Phys. Rev. Lett. 2005, 95, 036402-036405. [63] Kim K S, Phys. Rev. B 72. 2005, 214401-214409. [64] Senthil T and Fisher M P A, Phys. Rev. B. 2006, 74, 064405-064415. [65] Wen X G and Zee A, Phys. Rev. Lett. 1988, 61, 1025-1028. [66] Senthil T, Balents L, Sachdev S, Vishwanath A, and Fisher M P A, Phys. Rev. B. 2004, 70, 144407-144439. [67] Kou S P, Levin M, Wen X G, Phys. Rev. B. 2008, 78, 155134-155144. [68] Polyakov A M, Nucl. Phys. B. 1977, 120, 429-458. [69] Kou S P, Li T, Weng Z Y, Europhys. Lett. 2009, 88, 17010-17015. [70] Xu C K and Sachdev S, Phys. Rev. B. 2009, 79, 064405-064420.

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[71] J. Alicea, O. I. Motrunich, M. Hermele, M. P. A. Fisher, Phys. Rev. B 72 (2005) 064407-064429; Alicea J, Motrunich O I, Fisher M P A, Phys. Rev. Lett. 2005, 95, 247203-247206; Ryu S, Motrunich O I, Alicea J, Fisher M P A, Phys. Rev. B. 2007, 75, 184406-184418. [72] Ghaemi P and Senthil T, Phys. Rev. B. 2006, 73, 054415-054431. [73] Chatterjee U, Shi M, Ai D, Zhao J, Kanigel A, Rosenkranz S, Raffy H, Li Z Z, Kadowaki K, Hinks D G, Xu Z J, Wen J S, Gu G, Lin C T, Claus H, Norman M R, Randeria M, Campuzano J C, cond-mat/0910.1648.

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In: Insulators: Types, Properties and Uses Editor: Kevin L. Richardson, pp.113-132

ISBN: 978-1-61761-996-0 c 2011 Nova Science Publishers, Inc.

Chapter 5

T HEORY OF N ORMAL S TATE T RANSPORT IN C UPRATES IN M AGNETIC F IELD Zhihao Geng ∗ Department of Physics, Beijing Normal University, Beijing, China

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Abstract Based on the t − J model, the theory of normal state transport in cuprates with the external magnetic field is presented. The influence of a strong external magnetic field on the normal state resistivity of the underdoped cuprates is discussed. There is general agreement that the parent compound of these materials is a Mott insulator with antiferromagnetic long range order [3]. We begin with a discussion of the linear response approach and specular reflection model to establish the basic electromagnetic response scenario. We show the microscopic model in cuprate with external magnetic field and introduce the fermion-spin representation to the charge-spin separation formulation. The mean field theory and self-consistent equations approach in the case of external magnetic field is presented. The holon self-energy is obtained by employing the path integral representation of the Green function. Then the formalism of resistivity under an external magnetic field is given by the Kubo formula. Special emphasis is given to the analysis of the vector potential. The vector potential taken the coulomb gauge is discussed within the linear response method, which cannot be measured directly but plays a crucial role in quantum phenomenon. Theoretical ideas and analytical methods are introduced and presented in some detail. Some numerical result is given in the last part. It is shown that when the superconductivity in the underdoped cuprates is suppressed in the presence of a strong external magnetic field, the system reveals a low temperature normal state insulator-metal crossover [10, 11].

1.

Introduction

The discovery of high-Tc cuprates by J. G. Bednorz and K. A. M¨uller (1986) was a major milestone in condensed matter physics [1]. Since then, a wealth of nonconventional and fascinating phenomena was encountered in studying these materials. Over the last two ∗ E-mail address:

[email protected]

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decades, a significant body of reliable and reproducible data has been accumulated through intense experimental investigations [2, 4]. By now it is clear that the parent compound of high-Tc cuprates was a Mott insulator with antiferromagnetic long range order [3], while the understanding of its ground state evolves with doping is a great challenge that has to be met. In the undoped case the cuprates show an antiferromagnetic (AF) long range order N´eel state, however as the carrier concentration is increased by ionic substitution or increasing the oxygen content, these compounds turn into strongly correlated metals leaving the AF short range order correlation still intact [2–4], and it also shows superconductivity as a striking part in the phase diagram [9]. As one can see from Fig. 1, the line drawn should

Figure 1. Schematic phase diagram of doped cuprates. be regarded as a crossover. It is widely believed that the strong electron correlation plays a crucial role to determine the fascinating behaviors in these systems, and much effort has been devoted to research in doped cuprates. Unfortunately, the microscopic mechanism governing these properties is still uncertain and more research is still required. Anderson proposed a RVB scenario soon after the discovery of high-Tc cuprates. The RVB scenario describes the state of these systems associated with notion of holons and spinons, and charge spin separations [12], where the degrees of freedom of the electron are decoupled as the charge and spin degrees of freedom, and the elementary excitations for the charge and spin degrees of freedom are the holon and spinon. This idea shed highlights in the theoretical research of cuprates, and based on this scenario many unusual properties of cuprates are successfully described qualitatively [8, 9]. Among the nonconventinal behaviors in cuprates, a hall mark is the charge transport [7], e. g., the conductivity shows a non-Drude behavior at low energies, and the resistivity exhibits a linear temperature behavior over a wide range of temperatures [6, 7]. Especially, with a pulsed high magnetic field to suppress conductivity, the resistivity in the underdoped samples shows an unusual insulator-metal crossover at low temperatures [5]. The discovery of the insulator-metal crossover under a pulsed magnetic field caused great interest and argument. Just as in conventional superconductors, the understanding of normal state trans-

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Theory of Normal State Transport in Cuprates in Magnetic Field

115

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port properties is regarded as a key step towards the elucidation of the pairing mechanism for high temperature superconductivity. It is believed that the charge transport measurement with a high magnetic field reveals the normal state of these systems extended to lower temperatures hidden by the existence of the superconducting phase, and provide an important clue to the pairing interaction. The insulator-metal crossover is argued to mainly provide an experimental evidence for the pseudogap and some argues it is closely related to the evolution of the spin excitation. However the origin of the insulator-metal crossover has not reach a common consensus and remains a mystery hard to reveal. In this article, we mainly focus on the discussion of the behavior of normal state resistivity with a pulsed magnetic field. The ultimate goal is to present a normal state transport theory of the underdoped cuprates with the external magnetic field and reveal the new physics in these systems, based on this we reproduce the insulator-metal crossover observed in the experimental investigation [10,11]. The paper is organized as follows. We begin with the discussion of electromagnetic response of the charge transport in the linear response approach and introduce the boundary condition in a specular reflection idea. The importance of the vector potential is emphasized. Then we discussed the strongly electron correlated microscopic model in cuprates with the external magnetic field. Based on the charge-spin separation to address the local constraint, we show the calculation of the Green function at the mean-field level and the self-consistent field theory, then we show the evaluation of holon self-energy. We present the idea and procession of how to couple the external magnetic field in the theoretical analysis in some detail. The formalism of the kernel of response function is given at the following and calculated with the unperturbed Hamiltonian, i.e. with A ≡ 0. Then the electron resistivity is calculated using the Kubo formula. Some numerical result is shown and discussed in the end.

2.

Linear Response and Specular Reflection Model

Generally the external magnetic field applied to the system represents a perturbation. For a static system without the charge transport, the electromagnetic effect is mainly governed by the coupling of spins and external magnetic field, however in the case of charge transport in the system, the induced field generated by currents can cancel the external field to some extent. Therefore for the analysis of the electromagnetic response of the dynamic system, we should treat the net field as a perturbation on the system as a whole, not the external magnetic field only. We begin with the discussion of the charge current. In the linear response approach, the average value J(r,t) = hj(r,t)i of the induced screening current j in the presence of the vector potential A is found as J(r,t) = hj(d)(r,t)i + hj(p)(r,t)i = hj(d)(r,t)i + i

Z t ∞

dt 0

Z

d 3 r0 h[j(p)(r,t), j(p)(r0 ,t 0) · A(r0 ,t 0)]i,

(1)

where j(d) is the part of the current operator explicitly dependent on (proportional to) the vector potential and j(p) is the current operator in the absence of external fields, i.e. for A ≡ 0. These parts are called diamagnetic and paramagnetic ones, respectively. The thermal average in Eq. (1) will be calculated with the unperturbed Hamiltonian, i.e. with A ≡ 0.

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116

Zhihao Geng Eq. (1) can be rewritten in the form of Fourier representation 3

Jµ (q, ω) = − ∑ Kµν (q, ω)Aν(q, ω).

(2)

ν=1

Here Kµν is the kernel of the response function and µ, ν = 1, 2, 3 label the x, y, z axes of the Cartesian coordinate system, respectively. We will show the kernel of the response function can be separated into two parts (d)

(p)

Kµν (q, ω) = Kµν (q, ω) + Kµν (q, ω), (d)

(3)

(p)

a diamagnetic Kµν and a paramagnetic Kµν one. The diamagnetic part can be obtained explicitly from the form of the diamagnetic current operator. The paramagnetic part is calculated approximately as it involves evaluation of the retarded current-current correlation function (p)

(p)

Rµν (q,t − t 0 ) = −ih[ jµ (q,t), jν (−q,t 0)]iΘ(t − t 0 ).

(4)

The retarded function is inconvenient for perturbation analysis, one usually proceeds with the corresponding imaginary-time-ordered Matsubara function (p)

(p)

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Pµν (q, τ) = −hT jµ (q, τ) jν (−q, 0)i,

(5)

where τ is the imaginary time. The retarded correlation function can be obtained from the imaginary time Fourier transform Pµν (q, iω) by analytic continuation to real frequencies. The kernel function derived here within the linear response theory describes the current reaction of an infinite system to an external magnetic field, and do not conclude the effect of the boundary condition. However the boundary condition should not be neglected, as one can see it involves the form of the vector potential. In order to take into account the confined geometry of physical systems, we introduce two surfaces being the boundary between the vacuum and the sample. The way to introduce the boundary condition is that the electrons can be specularly reflected at the surface. Since the electrons passing through the surface without scattering should have a past trajectory to a vector potential which is exactly the same as that of electrons specularly reflected at the surface in the actual case, so the condition A(x0 + 0+) = A(x0 − 0+) should be satisfied. One can do this by introducing an effect current sheet to stimulate the external magnetic field at the surface. This specular reflection idea is discussed in detail by Landau and Pitaevskii [22] as well as Tinkham [23]. As shown in Fig. 2 the current sheet at the edge x = 0 is introduced as 2 Jext1,y(x) = − B0 δ(x), µ

(6)

0

where µ0 is the magnetic permeability and B0 is the amplitude of the external magnetic field at the surface x = 0. It is clear that the current sheet introduces a discontinuity of 2 B0 in the z component of the local magnetic field and the condition A(0+) = A(−0+) is satisfied. Meanwhile the current sheet at the edge x = a is introduced following the same way as 2 Jext2,y (x) = − B0 δ(x − a). µ 0

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Theory of Normal State Transport in Cuprates in Magnetic Field

117

a

external field

B (0 ,0 , B0 ) total current J tot ! J int J ext

z local field

y

h (0 ,0 , hz (x))

a

0

x

Figure 2. specular reflection model The corresponding Fourier component of the external current reads 8π2 B0 δ(qy )δ(qz), Jext1,y(q) = − µ0 8π2 Jext2,y(q) = − B0 eiqx a δ(qy )δ(qz). µ0

(8) (9)

Then, following the Maxwell equation, the local magnetic field in this geometry reads

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rot h = µ0 Jtot = µ0 (Jint + Jext1 + Jext2 ) = µ0 Jint + (0, −2B0 [δ(x) + δ(x − a)], 0).

(10)

Since there is no current along the x and z axes, the only non-zero component of the curl of the local magnetic field is the y component. One can choose the vector potential as A(r) = (0, Ay(x), 0) and fix it at the Coulomb gauge divA = 0. Thus with the identity rot h = rot rotA = grad divA − ∇2 A,

(11)

one proceeds with Eq. (10) −∇2 A = µ0 Jtot = µ0 (Jint + Jext1 + Jext2 ).

(12)

As the vector potential has only the y component this simplifies to ∂2 Ay = −µ0 (Jint,y + Jext1,y + Jext 2,y), ∂x2

(13)

or in the Fourier transformed form q2x Ay (q) = µ0 [Jint,y (q) + Jext 1,y(q) + Jext 2,y(q)].

(14)

According to the linear response relation between the internal current and the vector potential as Jint,y (q) = −Kyy (q)Ay(q), and the expression of the introduced current sheet Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

118

Zhihao Geng

shown in Eq. (8) and (9), one obtains that in the dynamic system the vector potential satisfies the following equation (1 + eiqx a )δ(qy)δ(qz) . Kyy (q) + q2x

Ay(q) = −8π2 B0

(15)

This relation should be emphasized as one can find that the vector potential evolves with the kernel of response function and therefore evolves with doping and the temperature and thus can not be a constant. Once the kernel function is obtained, the vector potential can be presented. One can define a new quantity which will be used in the next sections AIy (qx, 0, 0) = −2B0

1 + eiqx a , Kyy (qx, 0, 0) + q2x

(16)

as in the calculation of Green function the δ(x) function in Eq. (15) will be canceled out through the integral of momentum space and only AIy (0) appears. Actually this equation should be solved simultaneously with the other self-consistent equations given in the next section. However the amount of calculation about this equation is great and therefore the full self-consistent approach with the vector potential AIy(0) is not accessible. Thus, the kernel of response function is only evaluated based on the unperturbed Hamiltonian, i.e. with A ≡ 0.

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3.

Extended t − J Model and Green Function

Now we turn to discuss the microscopic model of doped cuprates with the external magnetic field. It is believed that the common features of cuprates are closely related to the twodimensional CuO 2 planes. The essential low energy physics of these system can be modeled by a single band Hubbard model, or its strong coupling limit, the t − J model on a square lattice [12, 13]. While in the presence of the external magnetic field the wave function of the electron hopping from the site j to the site i along a certain path acquires an additional phase factor eiΦi, j |ψi → eiΦi, j |ψi,

(17)

Z Ri

(18)

where Φi, j =

e ~

A(r)dr

Rj

is called the Peierls phase [14] with A(r) the vector potential. So for the unconventional charge transport properties of doped cuprates with the external magnetic field, the t − J model should be extended by including the exponential Peierls factors [16–20] as e R i+ηˆ A(r)·dr i

H = −t ∑ e−i ~ ˆ iησ

† † Ciσ Ci+ησ ˆ + µ ∑ CiσCiσ + J ∑ Si · Si+ηˆ , iσ

iηˆ

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Theory of Normal State Transport in Cuprates in Magnetic Field

119

† where ηˆ = ±x, ˆ ±y, ˆ Ciσ (Ciσ ) is the electron creation (annihilation) operator, µ is the chemical potential, J stands for the exchange integral, t represents the kinetic energy of the hopping, and the spin operator is

Si =

1 † ˆ αβCiβ , Ciα (σ) 2∑ α,β

(20)

with α, β =↑, ↓ label the spin indices and σˆ = (σx , σy , σz ) denotes a vector of Pauli matrices. Besides, the Hilbert space of the t − J model is restricted to one which excludes double occupation of any site due to the effect of the strong Coulomb repulsion [12, 13]. This strong electron interaction is usually written as a local constraint [12, 13, 21]

∑ Ciσ† Ciσ ≤ 1,

(21)

σ

which is hard to handle. Generally there are not too many analytic tools available to address these strongly correlated problems. The best tool available to us involved the constraint of no double occupation in this case of the t − J model is the slave-particle formulation. To incorporate this on-site single occupancy constraint in the formalism properly, here we employ the fermion-spin representation as the charge-spin separation approach proposed by S. Feng [27, 28]. Then the electron operator is decoupled as

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Ci↑ = h†i↑ S− i ,

Ci↓ = hi↓S+ i ,

(22)

where the spinful fermion operator hiσ = e−iΦiσ hi represents the charge degree of freedom together with some effects of the spin configuration rearrangement due to the presence of the doped hole itself (charge carrier) and is called dressed holon, the spin operator Si represents the spin degree of freedom. The spinless fermion hi obeys the anticommutation relation, so the spinful fermion hiσ obeys the same anticommutation relation as hi , and the − spin operators S+ i and Si obey Pauli spin algebra. † Ciσ = 1 − h†i hi ≤ 1 is exThen in this decoupling scheme the electron constraint ∑σ Ciσ − − + actly satisfied automatically with the relation S+ i Si + Si Si = 1, and the advantage of this scheme is that the local constraint can be treated properly in analytical calculations. Moreover the charge and spin degrees of freedom of the electron can be separated at the mean field level as we will show. As proposed in the charge-spin separation scenario [12], the elementary excitations are the collective modes for the charge and spin degrees of freedom called the holon and spinon, respectively. Within the fermion-spin representation, the extended t − J model is expressed as e R i+ηˆ A(r)·dr i

H = −t ∑ e−i ~ ˆ iησ

† + − − + (hi↑h†i+η↑ ˆ Si Si+ηˆ + hi↓ hi+η↓ ˆ Si Si+ηˆ )

† − − † + +µ ∑(hi↑S+ i hi↑ Si + hi↓ Si hi↓ Si ) i

+J(1 − h†i hi )Si · Si+ηˆ (1 − h†i+ηˆ hi+ηˆ ),

(23)

one defines the hole doping concentration as δ = hh†iσ hiσ i = hh†i hi i, and in cuprates one has the following relations − S+ i Si =

1 + Szi , 2

+ S− i Si =

1 − Szi . 2

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(24)

120

Zhihao Geng

Eventually, the Hamiltonian reads e R i+ηˆ A(r)·dr i

H = −t ∑ e−i ~ ˆ iησ

† + − − + (hi↑h†i+η↑ ˆ Si Si+ηˆ + hi↓ hi+η↓ ˆ Si Si+ηˆ )

+µ ∑ hiσ h†iσ + Je f f ∑ Si · Si+ηˆ ,

(25)

iηˆ

iσ

where Je f f = J(1 − δ)2 . The strong correlation effect manifests itself by this representation, and the above equation governs the behaviors of f the holon and spinon in the doped cuprates with the external magnetic field with a competition of the kinetic energy and the magnetic field. In order to discuss the charge dynamics, one defines the one-particle dressed holon and spinon Matsubara Green functions as, g(i − j, τ − τ0 ) = hhhiσ (τ); h†jσ(τ0 )ii = −hT hiσ (τ)h†jσ(τ0 )i = −θ(τ − τ0 )h[hiσ(τ), h†jσ(τ0 )]+i, D(i − j, τ − τ ) = 0

= Dz (i − j, τ − τ ) = 0

=

(26)

− 0 + − 0 hhS+ i (τ); S j (τ )ii = −hT Si (τ)S j (τ )i − 0 −θ(τ − τ0 )h[S+ i (τ), S j (τ )]− i, hhSzi (τ); Szj(τ0 )ii = −hT Szi (τ)Szj (τ0 )i −θ(τ − τ0 )h[Szi(τ), Szj (τ0 )]−i,

(27) (28)

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where τ, τ0 is the imaginary time. For the calculation of the Green functions, one starts with the standard mean field approach AB = AhBi + hAiB − hAihBi + (A − hAi)(B − hBi).

(29)

e R i+ηˆ

− − + −i ~ i A(r)·dr h† One defines the mean field parameter hS+ i Si+ηˆ i = hSi Si+ηˆ i = χ, he ˆ hiσ i = i+ησ φ, the last one is a key point which means one only couple the vector potential to the charge carriers h†iσ for the concreteness. In this case one will notice that the vector potential only appears in the holon Green function, not in the spinon Green function. Since here we do not consider the interaction of spins and the external magnetic field, thus this coupling scheme is reasonable and convenient. Actually one can couple A to h†iσ or S with the standard minimal coupling, but not to both and the result is independent of this choice. Then the Hamiltonian can be written as,

H = Ht + HJ + HI , Ht

(30)

R i+ηˆ −i ~e i A(r)·dr

= tχ ∑ e iηˆ

HJ =

† † (h†i+η↑ ˆ hi↑ + hi+η↓ ˆ hi↓ ) − µ ∑ hiσ hiσ ,

1 − − + z z Je f f ∑[ε(S+ i Si+ηˆ + Si Si+ηˆ ) + 2Si Si+ηˆ ], 2 iηˆ e R i+ηˆ A(r)·dr i

HI = t ∑[(e−i ~ iηˆ

e R i+ηˆ A(r)·dr i

+(e−i ~

(31)

iσ

(32)

+ − h†i+η↑ ˆ hi↑ − φ)(Si Si+ηˆ − χ)

− + h†i+η↓ ˆ hi↓ − φ)(Si Si+ηˆ − χ)],

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Theory of Normal State Transport in Cuprates in Magnetic Field

121

with ε = 1 + 2tφ/Je f f , and the mean field Hamiltonian is HMFA = Ht + HJ . At the mean field level, the dressed holon and spinon Matasubara Green functions can be obtained in the framework of the equation of motion. For the holon Green function the equation of motion reads d g(i − j, τ − τ0 ) = −δ(τ − τ0 )δi j + hh[HMFA , hiσ(τ)]−; h†jσ (τ0)ii. dτ

(34)

In the analysis of the holon Green function, it is justified to approximate the additional phase factor with an accuracy up to terms linear in the vector potential e R i+ηˆ A(r)·dr i

ei ~

= eiΦi,i+ηˆ == 1 + i

δΦi,i+ηˆ e ˆ Aβ (r) = 1 + i A(i) · η, δAβ (r) ~

(35)

and one assumes a static vector potential of the form A(i) = ∑q A(q)eiq·i. However the calculation of the spinon Green function is a tedious work, and it is essential to proceed to the hierarchy of the equation of motion at a further stage as d2 − 0 D(i − j, τ − τ0 ) = −δ(τ − τ0 )hT [[HMFA, S+ i (τ)]− , S j (τ )]− i dτ2 − 0 +hh[HMFA , [HMFA, S+ i (τ)]−]− ; S j (τ )ii.

(36)

It will turn out to be important to employ the Kondo-Yamaji strong decoupling theory [25] − z z + − hhSzk Szl S+ m ; S j ii → αhSk Sl ihhSm ; S j ii,

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− + − hhS+ k Sl Sm ; S j ii

→

− + − − + + − αhS+ k Sl ihhSm ; S j ii + αhSl Sm ihhSk ; S j ii,

(37) (38)

to cut off the spinon equation of motion. The decoupling parameter α here is introduced in − the calculation to violate the sum rule of the correlation function hS+ i Si i = 1/2 in the case without antiferromagnetic long range order, which can be regarded as the vertex correction. Define the Fourier transform of the Matsubara Green functions, i. e. 1 0 1 e−iωn (τ−τ ) ∑ eik·(i− j) G(k, iωn), β∑ N k n

(39)

d(τ − τ0 )eiωn (τ−τ ) ∑ e−ik·(i− j) G(i − j, τ − τ0 ),

(40)

G(i − j, τ − τ0 ) = and G(k, iωn ) = −

Z β 0

0

i− j

where β = 1/T , T is the temperature, and ωn = (2n + 1)π/β, ωn = 2nπ/β,

fermions bosons

(41) (42)

Moreover, one has the following relations δi, j =

1 eik·(i− j) , N∑ k

δ(τ − τ0 ) =

1 0 e−iωn (τ−τ ) . ∑ N n

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(43)

122

Zhihao Geng

Then the mean-field dressed holon and spinon Matsubara Green functions in the present case are obtained as

(0)

gσ (k, iωn ) = D(0)(k, iωn ) = (0)

Dz (k, iωn ) =

1 , iωn − ξk Bk , (iωn )2 − ω2k Bz (k) , (iωn )2 − ω2z (k)

(44) (45) (46)

respectively, where Bk = λ[2χz(εγk − 1) + χ(γk − ε)], Bz (k) = λχε(γk − 1), λ = 2ZJe f f , γk = (1/Z) ∑ηˆ eik·ηˆ , and Z is the number of the nearest neighbor sites, and the mean-field dressed holon and spion excitation spectra are given by

ξk = 2tχ(coskx + cos ky − AIy (0) sinky ) − µ, ω2k

(47)

= αελ (εχ + χ/2)(γk ) − ελ [α(χ + εχ/2) + (αC + 2

z

2

2

z

z

(1 − α)/(4Z) − αεχ/(2Z)) + (αC + (1 − α)/(2Z) −

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αχz /2)/2]γk + λ2 [αCz + (1 − α)/(4Z) − αεχ/(2Z) + ε2 (αC + (1 − α)/(2Z) − αχz /2)/2], 1 ω2z (k) = ελ2 (ε(αC + (1 − α)/(2Z)) − αχ − αχγk )(1 − γk ), Z

(48) (49)

with the mean field spinon correlation functions defined as χz = hSzi Szi+ηˆ i, C = + − z z z 2 (1/Z 2 ) ∑η, ˆ ηˆ 0 hSi+ηˆ Si+ηˆ 0 i, and C = (1/Z ) ∑η, ˆ ηˆ 0 hSi+ηˆ Si+ηˆ 0 i. In order to determine the mean field parameters, one uses the relation [24]

hBAi =

i 2π

Z ∞ hhA; Biiω+i0+ − hhA; Biiω−i0+ −∞

eβω ± 1

dω,

(50)

where + for fermions and − for bosons. The following identity is essential

1 1 = P ∓ iπδ(x). + x ± i0 x Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

(51)

Theory of Normal State Transport in Cuprates in Magnetic Field

123

Then one can get the following self-consistent equations about the parameters e R i+ηˆ A(r)·dr i

he−i ~

h†i+ησ ˆ hiσ i = φ = hh†iσ hiσ i = δ = − hS+ i Si i =

1 2

=

− hS+ i Si+ηˆ i = χ =

1 e AIy (0) 1 sin ky ][1 − tanh( βξk )], [γk − 2 ∑ 2N k ~ Z 2 1 1 [1 − tanh( βξk )], 2N ∑ 2 k 1 Bk 1 coth( βωk ), ∑ N k 2ωk 2 1 Bk 1 γk coth( βωk ), ∑ N k 2ωk 2

1 ∑0 hS+i+ηˆ S−i+ηˆ 0 i = C = Z 2 η, ˆ ηˆ

1 Bz (k) 1 γk coth( βωz (k)), ∑ N k 2ωz (k) 2 1 Bk 1 γ2k coth( βωk ), ∑ N k 2ωk 2

1 ∑0 hSzi+ηˆ Szi+ηˆ 0 i = Cz = Z 2 η, ˆ ηˆ

1 Bz (k) 1 γ2k coth( βωz (k)). ∑ N k 2ωz (k) 2

hSzi Szi+ηˆ i = χz =

(52) (53) (54) (55) (56) (57) (58)

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These equations reflect the dynamic correlations between the parameters, and the amount of equations is equal to the number of the mean field parameters together with the chemical potential µ and the decoupling parameter α. Then with the evolution of doping and temperature, the mean field parameters, the chemical potential µ and the decoupling parameter α can be determined by the solutions of the self-consistent equations automatically. Besides, the commonly used parameter t/J for the t − J model is usually chosen by the calculation from the local density functional theory [15].

4.

The Dressed Holon Self-Energies

The mean-field Green functions do not contain the scattering rate to the self-energy. In order to discuss the charge transport we now proceed to calculate the dressed holon self-energy with the help of holon and spinon Green function at the mean field-level. The full dress holon Green function can be expressed in the path integral representation as gσ (i − j, τ − τ0 ) = −hT hiσ (τ)h†jσ(τ0 )i = −

Z

Dh†σDhσ DS+ DS−hiσ (τ)e−

Rβ 0

dτL(τ) † h jσ (τ0 ),

(59)

with the partition function reads Z=

Z

Dh†σ DhσDS+ DS− e−

Rβ 0

dzL(τ)

,

(60)

where the Lagrangian L = LMFA + L0 , and L0 takes the form e R i+ηˆ A(r)·dr i

L0 = t ∑ e−i ~ iηˆ

† + − − + (h†i+η↑ ˆ hi↑ Si Si+ηˆ + hi+η↓ ˆ hi↓ Si Si+ηˆ ).

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(61)

124

Zhihao Geng Then in this formalism,

gσ (i − j, τ − τ ) = − 0

= −

Z Z

Dh†σ Dhσ DS+ DS−hiσ (τ)e−

(0) gσ (i − Z 1 β

−

2

0

dτL0 (τ)

Rβ

e

0

dτLMFA(τ) † h jσ (τ0 )

Dh†σ Dhσ DS+ DS−hiσ (τ)[1 + 1 2!

=

Rβ

0

Z β 0

dt1

Z β 0

dt2 L0 (t1 )L0 (t2 ) + · · ·]e−

Rβ 0

dτLMFA(τ) † h jσ (τ0 )

j, τ − τ ) 0

dt1

Z β 0

dt2 hT hiσ (τ)L0 (t10 )L0 (t20 )h†jσ(τ0 )i.

(62)

Since we are discussing the normal state properties, the combining terms with spin ↑ and spin ↓ in the expansion of the above equation is dropped, so one has

(0)

gσ (i − j, τ − τ0 ) = gσ (i − j, τ − τ0 ) Z β R l 0 +ηˆ 0 R l+ηˆ t2 A(r)·dr −i ~e l A(r)·dr −i ~e l 0 − ∑∑e e dt1 2 l ηˆ l 0 ηˆ 0 0 Z β

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0

+ − dt2 [hT hiσ (τ)h†l+η↑ ˆ (t1 )hl↑ (t1 )Sl (t1 )Sl+ηˆ (t1 ) † − 0 h†l 0 +ηˆ 0 ↑(t2 )hl 0↑ (t2 )S+ l 0 (t2 )Sl 0 +ηˆ 0 (t2 )h jσ (τ )i + + − hT hiσ (τ)h†l+η↓ ˆ (t1 )hl↓ (t1 )Sl (t1)Sl+ηˆ (t1 ) † − 0 h†l 0 +ηˆ 0 ↓(t2 )hl 0↓ (t2 )S+ l 0 (t2 )Sl 0 +ηˆ 0 (t2 )h jσ (τ )i].

(63)

Within the standard diagrammatic technique, one excludes the contribution of the disconnected bubble in the calculation, then the second-order dressed holon self-energy in the Fourier transform can be evaluated from the Dyson’s equation as

Σ(k, iωn ) =

t2 2 2 ∑ N 2 [Z γp+q+k − 4AIy (0) sin(py + qy + ky )Zγp+q+k ] p,q 1 ∑ ∑ g(0)(k + q, iωm )D(0)(p, iωl )D(0)(p + q, iωm + iωl − iωn ), (64) β2 iω m iωl

where the Peierls phase factor is approximated to terms linear in the vector potential. Completing the summation over Matasubara frequencies and the holon self-energy Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

Theory of Normal State Transport in Cuprates in Magnetic Field

125

reads, Σ(k, ω) =

t2 2 2 Bp Bp+q ∑ N 2 [Z γp+q+k − 4AIy (0) sin(py + qy + ky )Zγp+q+k ] 4ωpωp+q p,q  [nB(ωp ) − nB (ωp+q )]nF (ξk+q ) + nB (ωp+q )[1 + nB (ωp )] ω − ξk+q + ωp+q − ωp [nB (ωp+q ) + 1][nB (ωp ) + 1] − [nB (ωp+q ) + 1 + nB (ωp )]nF (ξk+q ) + ω − ξk+q − ωp+q − ωp [nB (ωp+q ) + nB (ωp ) + 1]nF (ξk+q ) + nB (ωp+q )nB (ωp ) + ω − ξk+q + ωp+q + ωp [nB (ωp+q + 1)]nB (ωp ) + [nB (ωp+q ) − nB (ωp )]nF (ξk+q ) + , (65) ω − ξk+q − ωp+q + ωp

where nB (ωp ) and nF (ξp ) are the boson and fermion distribution functions, respectively. One can see that the influence of the external magnetic field is mainly coupled to the holon self-energy here. The holon self-energy characterizes the scattering rate of the holons from the dressed holon-spinon interaction with the external magnetic field and dominates the properties of charge transport.

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5.

The Kernel of Response Function and the Vector Potential

One question is remained in the above analysis that even though the mean-field parameters, the chemical potential µ and the decoupling parameters α are determined by the selfconsistent theory, however the vector potential AIy(0) is not obtained through the same approach. Here it should be determined by the kernel of response function calculated from the unperturbed Hamiltonian, i.e. with A ≡ 0 as mentioned in the precious section. Now we turn to discuss the kernel of response function in the charge-spin separation scheme. Following the standard procedure, the charge current operator is j = e∂P/∂t. Using the mean-field-type approximation, the polarization operator P is obtained as † Ciσ = ∑ Ri − P = ∑ Ri ni = ∑ RiCiσ i

iσ

i

1 Ri h†iσ hiσ , 2∑ iσ

(66)

where Ri is the position operator. Thus, the current function is found from the Heisenberg equation of motion ∂P ie = [H, P] ∂t ~ e R i+ηˆ iet † + − − + ˆ †i+η↑ e−i ~ i A(r)·drη(h ∑ ˆ hi↑ Si Si+ηˆ + hi+η↓ ˆ hi↓ Si Si+ηˆ ). 2~ iηˆ

j(t) = e =

(67)

For the charge transport there is no direct contribution to the electron current-current correlation function from spins. In this case, we take the spin degree at the mean field level in the charge current operator, then j(t) =

e R i+ηˆ eχt ˆ †i+ησ e−i ~ i A(r)·drηh ∑ ˆ hiσ . 2~ iησ ˆ

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(68)

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Zhihao Geng

Expand the additional Peierls phase factor in the current operator to terms linear in the vector potential as before, j(t) =

e2 χt iχet † ˆ ηh ˆ i+ηˆ † hiσ ˆ h + [A(i) · η] ηh iσ ˆ i+ησ 2~ i∑ 2~2 i∑ ˆ ˆ ησ ησ

= j(p) + j(d) .

(69)

Introducing the Kr¨onecker symbol, one holds the following equality ηµˆ = δη,ˆ ˆ µ − δη,−ˆ ˆ µ,

(70)

ηµˆ ηνˆ = (δη,ˆ ˆ µ − δη,−ˆ ˆ µ )(δη,ˆ ˆ ν − δη,−ˆ ˆ ν) = δη,ˆ ˆ µ δη,ˆ ˆ ν + δη,−ˆ ˆ µ δη,−ˆ ˆ ν,

(71)

where µ, ν = x, y label the axes of the Cartesian coordinate system. Then, the total diamagnetic current along the µ axis is represented by the operator (d)

jµ (t) = =

e2 χt ∑ Aβ(i)ηβˆ ηµˆ h†i+ησ ˆ hiσ 2~2 i∑ ˆ β ησ e2 χt (h†i+ˆµσ hiσ + h†i−ˆµσ hiσ )Aµ (i), ∑ 2 2~ iσ

(72)

and the paramagnetic current operator along the µ axis is ieχt ηµˆ h†i+ησ ˆ hiσ 2~ i∑ ˆ ησ

(p)

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jµ (t) =

iχet (h†i+ˆµ hiσ − h†i−ˆµσ hiσ ). 2~ ∑ iσ

=

(73)

Going over to the Fourier representation 1 hmσ = √ ∑ eik·m hkσ , N k

1 h†mσ = √ ∑ e−ik·m h†kσ , N k

(74)

Thus, the average value of the Fourier component of the total diamagnetic current within the linear approximation reads Jµ (q) = h jµ (q)i = ∑( (d)

(d)

ν

e2 χt cos kµ hh†kσ hkσ iδµν )Aµ (q). ~2 ∑ kσ

(75)

Eventually, the diamagnetic part of the kernel of response function can be now identified as (d)

Kµν (q) = −

e2 χt coskµ hh†kσ hkσ iδµν , ~2 ∑ kσ

(76)

where the thermal average of the charge carrier number was evaluated by using the formula nk =

hh†kσ hkσ i =

1 2π

Z ∞ −∞

dωAσ (k, ω)nF (ω) =

1 gσ (k, iωn ), β∑ iωn

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(77)

Theory of Normal State Transport in Cuprates in Magnetic Field

127

and the dressed holon Green function can be expressed as frequency integrals in the spectral representation gσ(k, iωn ) =

1 2π

Z ∞ −∞

dω

Aσ (k, ω) , iωn − ω

(78)

with the holon spectral function Aσ(k, ω) = −2Imgσ (k, ω). The next step is to calculate the paramagnetic part of the response kernel. We may first proceed to the Matsubara current-current correlation function (p)

(p)

Pµν (q, τ) = −hT jµ (q, τ) jν (−q, 0)i,

(79)

(p)

(p)

where jµ (q, τ) is the Fourier transformation of the paramagnetic current operator jµ (τ), τ is the imaginary time. The evaluation of the current-current correlation function is more complicated and it is necessary to take the following approximation hT h†kσ (τ)hk+qσ(τ)h†k0 +q0 σ (0)hk0σ (0)i = −hT hk+qσ (τ)h†k0 +q0 σ (0)ihThk0 σ (0)h†kσ (τ)iδk,k0 .

(80)

Employing the standard many-body techniques the Matsubara correlation function in the imaginary frequency can be found as

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Pµν (q, iωn ) = −(

qµ χet 2 i (qµ −qν ) qν ) e2 sin(kµ + ) sin(kν + ) ∑ ~ 2 2 kσ

1 ∑ gσ (k + q, iωn + iωm )gσ(k, iωm ) β iω m qµ qν χet 2 i (qµ −qν ) ) e2 = −( ∑ sin(kµ + 2 ) sin(kν + 2 ) ~ kσ 1 4π2

Z ∞ ∞

dω dω0 Aσ (k, ω)Aσ(k + q, ω0 )

nF (ω) − nF (ω0 ) . iωn + ω − ω0

(81)

Thus, the paramagnetic part of the response kernel for the dc current reads (p)

Kµν (q, 0) = Pµν (q, ω → 0) = −

qµ qν (χet)2 i (qµ −qν ) e2 sin(kµ + ) sin(kν + ) ∑ 2 ~ 2 2 kσ

1 gσ (k + q, iωn )gσ(k, iωn ) β∑ iωn =

qµ qν (χet)2 i (qµ −qν) e2 sin(kµ + ) sin(kν + ) ∑ 2 ~ 2 2 kσ Z ∞ 1 1 dωdω0 Aσ (k, ω)Aσ(k + q, ω0 )(nF (ω) − nF (ω0 ))P . (82) 2 4π ∞ ω − ω0

Compiling Eqs. (76) and (82) ,the full kernel of the linear response function reads (d)

(p)

Kµν (q, 0) = Kµν (q, 0) + Kµν (q, 0). Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

(83)

128

Zhihao Geng

Then one obtains the vector potential AIy (0) as AIy(0) = −2B0

1 . Kyy (0)

(84)

As mentioned before, Kyy (0) is calculated by the Green function of the unperturbed Hamiltonian, i.e. with A ≡ 0. The numerical results shows a key that the vector potential AIy (0) is not a constant in the whole domain range of temperature, it behaves like a logarithm shape with a relatively greater absolute value at the low temperature than the high temperature. This striking result is closely related to the insulator-metal crossover of the resistivity under a pulsed magnetic field. By now, we have presented the theory for the charge dynamics in cuprates with the external magnetic field, the way to determine all the essential parameters is emphasized. More further discussions are based on the above calculation.

6.

Charge Transport

Now we are ready to discuss the charge dynamics in doped cuprates. We start from the charge current operator j(τ) of the imaginary time τ ∂P e = [P, H] ∂τ ~ e R i+ηˆ eχ1 t ˆ †i+ησ e−i ~ i A(r)·dr · ηh ∑ ˆ hiσ = ∑ ji (τ). 2~ iησ i ˆ

j(τ) = e

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=

(85)

According to the Kubo formula [26], one define the current-current Marasubara correlation function as Π(τ − τ0 ) = −hT j† (p, τ)j(p0, τ0 )i,

(86)

where j(p, τ) is the current density as j(p, τ) = ∑i eip·i ji (τ). For the dc conductivity, one should take the limit p → 0, then lim j(p, τ) = lim ∑ eip·i ji (τ) = ∑ ji (τ) = j(τ).

p→0

p→0 i

(87)

i

Thus, the current-current correlation function is evaluated as Π(iωn ) =

∑( kσ

eχt 2 2 ) [sin kx + sin2 ky + AIy (0) sin2ky ] ~

0 0 1 gσ (k, iωn + iωn )gσ (k, iωn ). ∑ β 0

(88)

iωn

where ωn is the Matasubara frequency. One can express the holon Green function as frequency integrals in the spectral representation by substituting Eq. (78) into Eq. (88) and Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

Theory of Normal State Transport in Cuprates in Magnetic Field

129

evaluate the frequency summation, then the conductivity is given by ImΠ(ω) ω (χt)2 (sin2 kx + sin2 ky + AIy (0) sin2ky ) = ∑ 4π kσ

σ(ω) = −

Z ∞

−∞

A(k, ω + ω0 )A(k, ω0)

nF (ω0 ) − nF (ω0 − ω) . ω

(89)

Then the resistivity under a pulsed magnetic field is evaluated from the dc conductivity σdc = limω→0 σ(ω) as ρ=

1 1 = . σdc limω→0 σ(ω)

(90)

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We have performed a numerical calculation and the results together with the normal state case are plotted in Fig. 3. We show the electron resistivity as a function of temperature for the external magnetic field B = 10.5T (solid circles) and B = 0 (open circles) in the doping δ = 0.08 with parameters t/J = 2.5 for a reasonably estimative value of J ≈ 100mev.

Figure 3. The electron resistivity as a function of temperature for the external magnetic field B = 10.5T (solid circles) and B = 0 (open circles) in the doping δ = 0.08 with parameters t/J = 2.5 for a reasonably estimative value of J ≈ 100mev. The solid lines are guides for the eyes. Our results obviously show that in zero magnetic field, the resistivity is characterized by a metallic-like behavior and exhibits a linear temperature dependence with deviations at low temperatures, which suggests a metallic ground state hidden by the present of superconductivity in the underdoped cuprates, not the commonly argued insulating ground state. However, when the strong external magnetic field B = 10.5T is applied to the system, the resistivity shows a crossover from the low temperature insulating-like to moderate temperature metallic-like behavior, which is also in qualitative agreement with experiments [10]. In

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Zhihao Geng

particular, this resistivity diverges as the logarithm of the temperature and suggests an unusual insulating ground state with a pulsed magnetic field. One indicates the insulator-metal crossover is not caused by the coupling of spins and external magnetic field, but possibly by the electromagnetic response of the charge current within the strong electron correlation, and it gives no evidence that the crossover reveals the behavior of the pseudogap. Finally, one notices that in the above analysis we do not consider the interaction of spins and external magnetic field. For the discussion of the coupling of spins and external magnetic field, there should be an additional Zeeman term to the t − J model, which reads † Hz = −εB ∑ σCiσ Ciσ ,

(91)

iσ

where εB = gµBB is the Zeeman magnetic energy, with the Lande factor g, Bohr magneton µB , and a uniform external magnetic field B. This term has a different origin of the Peierls Phase factor. Actually, we have already performed another calculation based on the Hamiltonian † † † Ci+ησ H = −t ∑ Ciσ ˆ + µ ∑ CiσCiσ + J ∑ Si · Si+ηˆ − εB ∑ σCiσCiσ ,

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ˆ iησ

iσ

iηˆ

(92)

iσ

where we only consider the coupling of spins and external magnetic field represented by the Zeeman term and exclude the contribution of the Peierls phase factor. The electron resistivity is obtained following the same procedure. However our results show that the interaction of spins and external magnetic field has negligible influence on the electron resistivity and there shows no phase crossover even with a high magnetic field in this case. Therefore for the discussion of the charge transport properties we only consider the contributions of the vector potential, and one can also involve both the Peierls phase facotr and the Zeeman term, while the results are almost the same as these shown here. It is also not surprising to notice the antiferromagnetic order induced by a pulsed magnetic field [29], since in the superconducting phase the external magnetic field is expelled by the superconductivity to some extent, when the superconductivity is destroyed with the existence of a pulsed magnetic field, there is nothing to prevent it.

Conclusion In summary, the theory of charge transport in doped cuprates with the existence of external magnetic field is established. We present the linear response approach and the specular reflection model. For the Green function we show the mean-field theory and the selfconsistent method, especially emphasis the analysis of the vector potential. We reveal that the vector potential has a striking behavior in the whole range of the temperature. Using the Kubo formula, we calculated the electron resistivity based on the charge-spin separation representation, and the insulator-metal crossover with a pulsed magnetic field is reproduced here. We also indicate the origin of the insulator-metal crossover induced by the magnetic field and discuss the contribution of the coupling of spins and external magnetic field. The author would like to express the gratitude to Ting Wang for the support and the help to complete the graphic. Many thanks also go to Professor M. Krzyzosiak and Professor S. Feng for the helpful and enlightening discussions.

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Theory of Normal State Transport in Cuprates in Magnetic Field

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References [1] J. G. Bednorz and K. A. M¨uller, Z. Phys. B 64, 189 (1986); Adv. Chem. 100, 757 (1988), Nobel Lecture. [2] See, e.g., M. A. Kastner, R. J. Birgeneau, G. Shirane, Y. Endoh, Rev. Mod. Phys. 70, 897 (1998) and references therein [3] P. A. Lee, N. Nagaosa, and X. G. Wen, Rev. Mod. Phys. 78, 17 (2006) [4] See, e.g., K. S. Bedell et al (ed) Proc. Los Alamos Symp. (Redwood city, CA: AddisonWesley) (1990) [5] G. S. Boebinger, Y. Ando, A. Passner, T. Kimura, M. Okuya, J. Shimoyama, K. Kishio, K. Tamasuku, N. Ichikawa, and S. Uchida, Phys. Rev. Lett. 77, 5417 (1996) [6] A. P. Mackenzie, S. R. Julian, D. C. Sinclair, and C. T. Lin, Phys. Rev. B 53, 5848 (1996) [7] A. I. Larkin and A. A. Varlamov, The Physics of Superconductors: Vol. I Conventional and High-Tc Superconductors , edited by K. H. Bennemann and J. B. Ketterson (Springer-Verlag, Tokyo, 2002) [8] See, e.g., P. W. Anderson, The Theory of Superconductivity in the High-Tc Cuprates (Princeton, NJ: Princeton University Press) and references therein [9] See, e.g., E. Dagotto, Rev. Mod. Phys. 66, 763 (1994) and references therein Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

[10] G. S. Boebinger et al., Phys. Rev. Lett. 77, 5417 (1996). [11] Y. Ando et al., Phys. Rev. Lett. 75 (1995) 4662; Yoichi Ando et al., Phys. Rev. B. 56, R8530 (1997) . [12] P. W. Anderson, in Frontiers and Borderlines in Many Particle Physics , edited by R. A. Broglia and J. R. Schrieffer (North-Holland, Amsterdam, 1987), p. 1; Science 235, 1196 (1987) [13] Zhang F C and Rice T M, Phys. Rev. B 37, 3759 (1988) [14] J. M. Luttinger, Phys. Rev 51, 814 (1951) [15] M. S. Hybertson, E. Stechel, M. Schuter, and D. Jennison, Phys. Rev. B. 41 11068 (1997). [16] S. Misawa, Phys. Rev. B 49, 6305 (1994) [17] J.E. Hirsch, F. Marsiglio, Phys. Rev. B 45, 4807 (1992) [18] D. J. Scalapino, S. R. White, S. C. Zhang, Phys. Rev. Lett. 68, 2830 (1992) [19] D. J. Scalapino, S. R. White, S. C. Zhang, Phys. Rev. Lett. 47, 7995 (1993) Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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[20] T. Kostyrko, R. Micnas, K. A. Chao, Phys. Rev. B 49, 6157 (1994) [21] C. Gros, R. Joynt, and T. M. Rice, Phys. Rev. B 36, 381 (1987) [22] L. D. Landau, L. P. Pitaevskii, Statistical Physics (Part 2) , Pregamon Press Ltd. (1980), sec. 52. [23] M. Tinkham, Introduction to Superconductivity , McGraw-Hill (1996), appendix 3. [24] D. N. Zubarev, Soviet Phys. Usp., 3, 320 (1960) [25] J. Kondo, K. Yamaji, Prog. Theor. Phys., 47 807 (1972) [26] G. D. Mahan, Many-Particle Physics, Kluwer Academic/Plenum Publishers. (2000), sec. 1.2. [27] S. Feng, Z. B. Su, L. Yu, Phys. Rev. B 49, 2368 (1994) [28] S.Feng, J. Qin, T. Ma, J. Phys. Condens. Matter 16, 343 (2004)

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[29] B. Lake, H. M. Rønnow, N. B. Christensen, G. Aeppli, K. Lefmann, D. F. McMorrow. P. Vorderwisch, P. Smeibidl, N. Mangkorntong, T. Sasagawa, M. Nohara, H. Takagi, and T. E. Mason, Nature. 415, 299 (2002)

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In: Insulators: Types, Properties and Uses Editor: Kevin L. Richardson, pp.133-172

ISBN: 978-1-61761-996-0 c 2011 Nova Science Publishers, Inc.

Chapter 6

E LECTRODYNAMICS OF M OTT I NSULATORS AND I NSULATOR TO M ETAL T RANSITIONS A. Perucchi, L. Baldassarre and S. Lupi Sincrotrone Trieste S.C.p.A., Basovizza, Trieste, Italy

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1.

Introduction

The impact of the band theory of solids, which has been developed mainly by Bloch and Bethe along the first half of the 20th century, goes well beyond condensed matter physics. The understanding of conducting and insulating states of matter has enabled indeed the possibility to tune material’s properties according to one’s wishes, thus leading to the fabrication of solid-state devices which have entered everyone’s life. The band theory of solids is directly linked to the quantization of the energy introduced by quantum mechanics. When many atoms are brought together to form a solid, the difference in their energy levels becomes so small that one can think about them as forming a continuum of levels, i.e. a band. Nevertheless there will always be some values of the energy which will not correspond to any available electronic state, thus forming a so-called band-gap. The concept of band-gap, its size, and its position relative to the chemical potential, are central in defining whether a solid is a metal, a semiconductor or an insulator. The electronic band theory has been proven to work for several simple materials, as typically semiconductors and noble metals. Nonetheless its application to more complex systems, involving 3d,4f , or 5f electron bands is much more troublesome, as pointed out by de Boer and Verwey, already in 1937. There are several mechanisms which can engender an insulating state at odds with band theory. The Peierls mechanism describes how some materials with half-filled bands are unstable towards the electron-phonon interaction and how an insulating state can result because of that. More in general, the interaction of electrons from poorly dispersing bands with the lattice can lead to enhanced masses and localization phenomena. Mott acknowledged the important role played by the intra-site electronic Coulomb repulsion energy (U ) in shaping the electronic properties of transition metal oxides. It is true indeed that if the bandwidth ( t) is comparable (or lower) than U , correlation may result in the splitting of the conduction band into so-called lower and upper Hubbard Bands, and the material is turned into a Mott insulator. This is the situation which is encountered

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A Perucchi, L. Baldassarre and S. Lupi

for instance in NiO, as well as in many other transition metal oxides (TMO). In this case one normally distinguishes between Mott and Charge-Transfer insulators, depending on the relative position of the split Hubbard bands with respect to the O2p level. Mott insulators, and more in general strongly correlated materials are not odd, and rare materials, we are trying to understand as a merely intellectual challenge. Instead, strong correlations, and Mott insulating properties are ubiquitous in carbon-based materials that are currently the subject of intense studies for technological applications in organic electronics. Moreover, it is known that exotic properties and gigantic electronic responses are sometimes found in close proximity to a Mott insulating phase. High-Temperature Superconductivity and Colossal Magnetoresistence are prominent examples of the extremely promising potentials of the (bad) metallic states arising from a Mott insulator. From an electrodynamic point of view, a Mott insulator presents a charge gap in the optical conductivity, similar to that of more common insulating or semiconducting materials. A fundamental tool for understanding the nature and the properties of insulating states is the description and the comprehension of the non-conventional metallic states near the Mott transition. In particular, by inducing a metal to insulator transition (MIT) in these compounds, it is possible to discriminate between the different mechanisms that stabilize the insulating state. As we will see in the following, the electrodynamics of the insulator to metal transition provides well-established signatures of strong correlation, thus enabling to determine the nature of the underlying insulator. Here we will review the electrodynamic properties of Mott insulators and other strongly correlated systems across insulator to metal transitions. We will first introduce the theoretical framework within which strongly correlated systems are presently understood. Next we will give an overview of the techniques discussed here, for temperature ( T ) and pressure (P ) dependent measurements. We will then address the properties of several compounds as V2 O3, VO2 , V3 O5, and NiS2 . This will allow us to compare and contrast the optical properties across insulator to metal transitions, when the insulating state is of Mott or of charge-transfer type, or when the electron-phonon interaction plays a role.

2. 2.1.

Theoretical Framework Electronic Correlation and Insulating Compounds

The band theory of solids, upon which relies the understanding of the properties of the materials presently employed in the semiconductor and electronics industry, is fundamentally an independent electron theory. Electrons (or holes) are treated as a gas of non-interacting particles, behaving according to the Hamiltonian determined by the lattice coordinates. While being successful in describing the transport properties of many materials of technological use, the assumption of non-interacting electrons is not realistic in general. Nevertheless, as recognized by Landau with his celebrated Fermi-liquid theory, in several cases the band theory framework can be generalized to materials in which sizable, though not exceedingly large, interactions between electrons do exist. In a Fermi-liquid, the net effect of the interaction between electrons is the renormalization of a few parameters, as e.g. the mass or the magnetic susceptibility. Still, within the Fermi liquid framework the concept of band does apply, and one can predict insulating or metallic behavior according to the

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Electrodynamics of Mott Insulators and Insulator to Metal Transitions

135

band-filling. Central to the Fermi liquid theory is the concept of quasi-particle, an excitation which maps one by one with the electron’s gas. Quasi-particles share with electrons the same spin, charge, and momentum, thus allowing to retain many of the concepts of band theory. Nonetheless, instabilities of the Fermi liquid are often observed in strongly correlated electron systems, thus leading to the disappearance of quasi-particles and to the occurrence of excitations of a new type. A thorough review of the theory of strongly correlated systems is well beyond the scope of this chapter. The interested reader will find excellent treatise in Refs. [1], [2] or [3]. Our focus will rather be on the electrodynamics of Mott and Charge-Transfer type insulators. We will also briefly outline topics as polarons and Wigner crystals, as well as the Peierls dimerization. These phenomena are sometimes found competing with the Mott mechanism in determining the type of insulating ground state.

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2.2.

Mott Insulators

Sir N. Mott was the first to explain how electron-electron (e-e) correlations could result in an insulating state where a metallic behavior is expected. Mott considered a lattice model with one single electronic orbital on each site. With no e-e correlation a single band is built up from the overlap of the atomic orbitals and is completely full when two electrons, with opposing spin, occupy each site. Mott argued that two electrons on the same ion feel a Coulomb repulsion (U ) that can be so large to determine a splitting of the the band in two: the lower band would be formed by a single electron placed on an empty site, while double occupancy on the same ion results in states forming the upper band. When the system has a band half-filled (e.g. one electron per site) the splitting induced by the Coulomb repulsion produces a completely full lower band and an empty upper one (see Fig. 1a). This insulating state is difficult to understand in a wave-like picture: it is not caused by the absence of available one-electron states, due to destructive interference in the k-space, as in the case of band insulators. Mott proposal, moreover, does not take into account any role of magnetic degrees of freedom: the system can be an insulator not depending whether it is magnetic or not. Slater [4], on the other hand, proposed the rise of an insulating gap due to the presence of long-range magnetic order. Since most Mott insulators present, at least at zero temperature, antiferromagnetic order, Slater put forward that the insulating state could appear due to a band gap generated by the superlattice structure of the magnetic periodicity. Materials presenting a significant electronic correlation are generally associated with moderate values of the bandwidth t (i.e. a small value of the hopping integral) with respect to the Coulomb repulsion U . This results in low kinetic energy, implying that delocalizing electrons over the whole solid may become less favorable. In a naive way, poor screening due to low-bandwidth, will make the electrons feel more the Coulomb repulsion. The balance between U and t can become so unfavorable that corresponding electrons will remain localized. If this happens to all electrons close to the Fermi level, the system will be insulating. It is therefore clear the the ratio U/t plays a key role in inducing the electrodynamic response of solids. Within a Fermi Liquid picture, the ground state and the low-energy excitations can be fully described by an adiabatic switching of the e-e interaction. Since the number of carriers can not change on approaching the MIT, the only way to reach an insulating state contin-

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uously is through the divergence of the single quasi-particle mass m∗, with the vanishing of the renormalization factor Z, defined by the relation 1/Z = m∗ /m where m is the band mass. When, on the other hand, a symmetry breaking occurs, the adiabatic continuity has not to be fulfilled any more and the MIT can be described by a vanishing of the mobile carrier number. Basing on this argument one can distinguish between two ways of approaching the MIT from the metallic side: the mass-diverging type and the carrier-number vanishing type [1].

2.3.

The Hubbard Model

The Hubbard model, first introduced in 1963 [5], is the prototype model to describe strongly correlated electron systems. This lattice model, which considers only electrons in a single band, adds to a tight-binding Hamiltonian a repulsive term between electrons at the same lattice site i: H = −t

X

(c†iσ cjσ + h.c.) + U

σ

X

ni↑ ni↓ .

(1)

i

Here the creation (annihilation) of the single-band electron at site i with spin state σ is denoted by c† iσ (ciσ ) and niσ = c†iσ ciσ is the number operator. t (the hopping integral within a tight-binding model) is supposed, for sake of clarity, isotropic and non-vanishing only for nearest-neighbor hopping: ¯ 2 ∇2 h φjσ (r)dr, (2) 2m m being the electron mass and φiσ the Wannier orbital. The repulsive Coulomb interaction felt by two electrons at the same atomic site is represented by U >0: tij =

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Z

U=

Z

φ∗ iσ (r)

φ∗iσ (r)φiσ (r)

e2 0 0 0 ∗ 0 φi,−σ (r )φi,−σ (r )drdr . |r − r |

(3)

This model does not take into account multi-band effect and holds only for s−orbitals. When dealing with d-orbitals it is implicitly assumed that orbital degeneracy is lifted out due to the crystal field, in order to describe the low-energy excitations in terms of one single band lying at the Fermi level. A further hypothesis is that either the ligand-band ( p) energy is far from the relevant d−band or it is so strongly hybridized that a single band can be considered. This Hamiltonian also neglects the inter-site Coulomb repulsion and often the electron hopping is restricted to the sum over pairs of nearest-neighbor sites < ij >. This can be justified by the screening effect that makes the long-range part of the Coulomb force to decrease exponentially with r. However, within those assumptions, the Hubbard model fails in describing important features such as charge ordering or magnetic effects that can arise from geometrical frustration. In spite of those simplifications, the Hubbard model well captures the features of the Mott insulating phase and the transition between Mott insulators and metals. The insulating state occurs at half-filling where the average electron number hniσ i is controlled at 1/2. At half-filling, under electron-hole symmetry, the Fermi level lies at µ = F = 0 and the

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appearance of the insulating phase is clearly due to the correlation effect, arising from the second term of Eq. 1. The Hubbard model can be solved exactly only in one dimension ( 1d) [6], while for d > 1 only the limits U/t → 0 and U/t → ∞ can be analytically discussed. In the former case the bands of non-interacting system are recovered, and a metallic phase occurs. Within the limit of strong correlation (large U/t), carrier hopping is unfavorable with respect to antiparallel alignment of two neighboring spins. The Hubbard model provides, therefore, an excellent description of the system in the insulating phase, as it becomes a Heisenberg 2 model (with J = 4tU > 0), where in the absence of magnetic frustration due to the lattice symmetry, an antiferromagnetic phase is attained.

Figure 1. Schematic illustration of energy levels for (a) a Mott- Hubbard insulator and (b) a charge-transfer insulator generated by the d -site interaction effect. From Ref. [1].

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Although both the Mott insulating state and the MIT are reproduced in the Hubbard model, in order to make quantitative predictions more complex and realistic factors must be introduced. For instance, a single-band model is certainly inappropriate in the case of so-called Charge Transfer insulators. These are materials in which the energy separation between anion and cation bands is smaller than the energy separation U . As described in Fig. 1b, in this case the gap opens between anion and cation states, with a strongly hybridized character.

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2.4.

Approximate Solutions of the Hubbard Model

Since the early ideas of Mott, the metal to insulator transition has been the subject of numerous experimental and theoretical investigations. We shall briefly review the different theoretical scenarios that were put forward, corresponding to different approaches to the transition. The early work of Hubbard [5] described the transition in terms close to Mott’s original view. He attempted to give an effective band description of the correlated system. Starting from the insulating phase, upon reducing U there is a critical value Uc where the LHB and the UHB merge with each other and a metal is recovered. The so-called Hubbard approximations, however, fail in reproducing the discontinuous character of the MIT due to the gap closure. Moreover, they do not provide a description of the metal consistent with the Fermi-liquid properties, since in the DOS no quasi-particle peak shows up as the metallic phase is established. On the other hand Brinkman and Rice [7], developing a previous work of Gutzwiller [8], started from the metallic phase, described through a strongly renormalized Fermi liquid with a reduced low-energy scale (or effective Fermi energy) of order ZD. Z is the quasiparticle residue, as defined previously and D is the half-bandwidth (hereafter considered as equivalent to t in the U /t ratio). As the interaction strength increases, this energy scale vanishes at a critical value UBR, with Z ∝ (UBR −U ). The transition is therefore described in terms of localization of the Fermi-liquid quasiparticles ( m∗ /m ∝ 1/(UBR − U ) → ∞) which disappear in the insulator. This approach is a consistent low-energy description of the strongly correlated metal, but does not account for the high-energy excitations forming the Hubbard bands, which should be present already in the metallic state. Furthermore, it gives an oversimplified picture of the insulating state, which is described as a collection of local moments with no residual antiferromagnetic exchange. Classical mean field theories are not appropriate to describe simultaneously coherent low-energy excitations (as for the Gutzwiller approach) and incoherent higher-energy terms (Hubbard approximations). As we will see in the following section, the Dynamical Mean Field Theory (DMFT) developed for d = ∞, can instead describe, within the Hubbard model, both the insulating and the metallic phases.

2.5.

Dynamical Mean Field Theory: Definition of the Hubbard Phase Diagram

Mean-field theory approximates a lattice problem with many degrees of freedom by a single site effective problem. The idea is that the charge dynamics at a given site can be obtained Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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as the interaction of the local degrees of freedom with an external bath (created by all the other degrees of freedom on the other sites). This can be applied also to the Hubbard problem, allowing to substitute the lattice Hamiltonian of Eq.1 by a local one. In order to take into account the wide range of energy scales involved in a system with electronic correlations, the Dynamical Mean Field Theory has been introduced, which also considers dynamic quantum fluctuations. This leads to an energy-dependent Weiss effective field, at variance with the constant one in classical mean-field approximations and, in the case of the Hubbard model, DMFT allows one to describe both low-energy coherent and high-energy incoherent features.

Figure 2. Evolution of the density of states at T = 0 obtained with DMFT (G is the Green’s function, see text). The first four curves (from top to bottom, U/t = 1, 2, 2.5, 3) correspond to an increasingly correlated metal, while the bottom one ( U/t = 4) represents the insulating state. From Ref. [3].

At half filling, the evolution of the density of states (DOS, defined as πDρ(ω) = −Im[G][3], where G is the Green function and D the half-bandwidth) is shown for T = 0 in Fig. 2, for increasing U/D ratio. Therein it is possible to follow the progressive reduction of the central quasi-particle (QP) peak and the formation of the LHB and UHB with increasing U/t. At a critical value of U = Uc1 the Hubbard bands separate one from the other while the QP peak is still retained at the Fermi energy ω =  − F = 0. For higher values of U the peak increasingly narrows until it disappears for U = Uc2 and the two Hubbard bands only are left in the DOS. The vanishing of the QP peak is associated with a vanishing renormalization factor Z, and hence with a diverging value of m∗ as previously described by Brinkman and Rice (see Sec.2.4.).

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The insulating phase (U > Uc2 ) described by DMFT is in agreement with the results of the Hubbard approximations. The fact that DMFT provides correct results for both the cases of small and high interactions (i.e small or large values of U ) and that it becomes exact in the limit of infinite dimensions, suggests that also the results at intermediate values of U/t are probably correct. DMFT may be extended to include spatial correlations (cluster-DMFT) or to correctly consider the band structure of the actual system with the Local Density Approximation (LDA+DMFT). However, already single-band single-site DMFT can qualitatively capture important features within the Hubbard model. To fully describe the phase diagram of a strongly correlated system one needs to introduce as control parameters, in addition to the U/t ratio, both T and the degree of magnetic frustration (simulated by the ratio of the next-nearest-neighbor hopping amplitude to the nearest neighbor one: t2 /t1). As a function of these parameters, four phases are possible: a paramagnetic metallic phase (PM), a paramagnetic insulating (PI) state, an insulating phase with antiferromagnetic ordering (AFI) and, in the presence of magnetic frustration, an antiferromagnetic metallic state (AFM).

Figure 3. Approximate phase diagram predicted by the p model with nearest-neighbor ( t1 ) and next-to-nearest-neighbor hopping ( t2 ) (t2 /t1 = 1/3). The first order paramagnetic metal to insulator transition ends at the critical point Tc (square). From Ref. [3].

In the case of no magnetic frustration, for T < TN (where TN is the N´eel temperature) the AFI phase is obtained and the model maps onto the Heisenberg model in the case of large interaction values. The more realistic case of intermediate frustration leads to the phase diagram shown in Fig. 3. The presence of partial frustration reduces the value of TN with respect to the un-frustrated case. TN [9] turns out to be smaller than the critical temperature for the Mott transition in the case of small U . Therefore, a transition between the PM and the PI phase is possible.

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2.6.

141

Polarons and Wigner Crystal

Wigner suggested in 1934 that an electron gas propagating in an inert, uniform, positive background will crystallize in a lattice if the electron density is lower than a critical value [10]. The tendency towards the charge ordering is due to a subtle balance between the potential and the kinetic energy at low electronic densities. In this case the coulomb repulsion overcomes the kinetic energy and the detailed spatial arrangement of the electrons becomes important. While the formation of a Wigner crystal in 1-d or 2-d has been nowadays experimentally verified in nano-structured materials and on the liquid helium surface [11], the experimental realization of a 3-d Wigner crystal is more debated. Several transition metal oxides (TMO) show electronic ground states characterized by charge localization effects and, in some cases, localization may transform in a long range order resembling a Wigner crystallization. The formation of a Wigner crystal has been proposed as the explanation for the metal to insulator transition (Verwey transition) observed in Fe 3 O4 . Still, in this case the proposed mechanism is related to the crystallization of electrons displaying a sizably enhanced mass. The source of mass enhancement is identified with a polaronic instability. Indeed, the strong electron-lattice interaction often observed in TMO, may favor the charge localization and, in some cases, the Wigner crystallization [12]. In a polar lattice, the introduction of an electron (for example upon doping) results in a lattice distortion, because the electron will attract and repel the positive and negative ions respectively. Any electron transport mechanism should involve even the distortion of the polarized lattice and the composite quasi-particle which include the electron dressed with the phonon cloud is named polaron [14]. The problem of an electron moving in a polar lattice was first considered by Landau [13] by introducing an effective interaction between the charge and the distorted part of the lattice. He considered the distortion as a sphere of radius R centered on the electron assuming that the attractive potential had the simple form: V (r) =

(

−|V | if r ≤ R 0 if r > R

Landau showed that for V larger than a critical value V ∗ a discrete energy level becomes available for the electron. This corresponds to a bound state of the charge, that is said to be self-trapped in the lattice. This modeling describes naively the polaron formation, and does not consider that the electron-lattice interaction is the sum of at least two terms: a long-range component, due to the Coulomb interaction between the charge and the ionic dipoles, and a short-range component, due to the displacement of the ions around the charge from their equilibrium position. If we define the energy E(R) as that of the electronic ground state plus the distortion, in order to have a self-trapping, the attractive potential must balance the increase in elastic energy due to the distortion. When the long-range component, represented by a dipolar field ∝ R−1 , dominates the attraction, we can write the energy balance equation [14]: E(R) =

V t − β 2 2R 2R

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(4)

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where

1 1 2e − ); V ∼ , ∞ 0 a a being the inter-atomic spacing. Here the dipole lattice is represented as a continuous medium with a dielectric constant  that is given by: 1/ = 1/∞ − 1/0 . ∞ and 0 and are, respectively, the electronic and the static dielectric constant of the polar crystal. The difference 1/∞ − 1/0 arises because the ionic vibrations occur in the infrared. The parameter t is the electronic hopping term. We can then evaluate E(R) for realistic values of the parameters. As the attraction is mainly long-range, a minimum of E(R) is found for R equal to several lattice units. This means that the distortion involves many lattice sites and a large polaron is formed. On this regard, Fr¨olich [12] proposed a large polaron hamiltonian where the polarization, carried by the longitudinal optical (LO) phonons, is represented by a set of quantum oscillators with frequency ωq , and the interaction between the charge and the polarization field is linear in the field: β=(

H=

X p2 p

c† cp + 2m∗ p

X

ωq b†q bq +

X M0 eiq·r † √

q

p,q

V

q

cp+q cp(bq + b†q )

(5) h3 ω 3 ¯

1

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where m∗ is the effective electron mass, V is the volume, M02 = 4πα¯ h( 2m0 ) 2 and 1 2 α = ( e¯h )( 2¯hmωo ) 2 ( ε1∞ − ε10 ) is the Fr¨ohlic parameter which measures the electron-phonon interaction. In other terms in the large polaron concept the interaction strength depends on the static (ε 0 ) and dynamic (ε∞ ) dielectric functions as the spatially extended distorsion provides the possibility to use the classical electrodynamics of continuum media. If the short-range local distortion dominates the attraction, the shape of the potential depends strongly on the dimensionality d of the system, as in a confinement problem. We shall thus write WH t − d (6) 2R2 R with WH the hopping energy. We note that in one dimension a large polaron solution is still possible, while for d=2, 3 the minimum of E(R), if present, is found for R → 0, indicating that only the first neighboring ions are involved in the distortion. In this situation a small polaron is formed. E(R) =

At variance with that proposed by Fr¨olich, the prototype hamiltonian for small polarons was presented by Holstein [12] by introducing in the tight-binding hamiltonian an interaction term of the electron with a longitudinal optical phonon, and assuming an adiabatic regime. To achieve energy balance, an elastic lattice energy term is included: H − t

X

(c†i cj + h.c.) − g

X j,q

(b†q + bq )c†j cj +

X

hωq b†q bq ¯

(7)

q

In the case of the small polaron, the electron-phonon interaction is measured by λ = hω0 zt with z nearest neighbor number. g /¯ A polaron, although self-trapped in a bound state, can be thermally excited to a free charge continuum of states. After the thermal excitation in the continuum the charge selftraps in a neighboring site, giving rise to dc conduction by hopping. The large polaron in 2

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the Hamiltonian of Eq. 5 can indeed be described as a new quasi-particle of effective mass m∗ /m = 1 + α/6. In the case of small polarons with a high hopping energy with respect to kB T , the charge will instead hop to neighboring sites to be immediately re-trapped. The polaron mobility will then be low, showing a thermally activated transport. The mobility µ(T ) will indeed contain a term exp(WH /kB T ). Austin and Mott [15] studied the simple case of polaronic hopping transport in a molecular lattice. Therein it is possible to assess the relation between the activation energy for hopping WH and the polaron energy Wp . This relation is different in the adiabatic and the anti-adiabatic case resulting in WH = 1 1 2 Wp − D and WH = 2 Wp respectively, with D the half-bandwidth. In a polar lattice, details of hopping mechanism are clearly more complicated, we will only notice that a strong anisotropy may be found in m∗, as also reported by Austin and Mott [15]. Several authors have studied the optical properties of polarons [14] and both in the large and in the small polaron models σ1 (ω) presents an absorption peak at finite frequencies. This band can be naively explained in terms of a photo-ionization process of the polaronic charge which is extracted from its self trapping bound state by absorbing a resonant photon. Moreover, in the large polaron scenario, the ratio between the spectral weight of the incoherent part (which is linked to the localization dynamics) over that of the coherent component (which is instead related to the coherent motion of the polaron) may provide an estimate of the mass renormalization m∗/m due to the polaronic formation [14]. Finally, by applying the DMFT technique (see Sec. 2.5.) Fratini et al [12] have calculated the optical properties of a polaronic system in both weak, intermediate and strong coupling regime. In particular they showed that the Wigner crystallization results in a strong hardening of the polaronic photo-ionization band (see Fig. 5 in Ref. [12]). The difference between the energy of the photo-ionization band in the ordered polaronic state and that in the disordered phase is thus a quantitative parameter defining the energy gained from the polaronic system by forming the Wigner crystal. The mass renormalization and the crystallization energy of the prototype V 3 O5 polaronic material will be estimated directly from the experimental data in the section 6.

2.7.

The Peierls Mechanism

Besides the effect of electron correlation, one has also to keep in mind that other mechanisms not accounted for by conventional band theory can trigger an insulating ground state, especially in materials with poor bandwidth. This is indeed the case for charge and spin density wave instabilities (CDW and SDW) which are ground states of matter found typically in low-dimensional materials [16]. The formation of a CDW, as described by Peierls is a consequence of the singularities in the Lindhard response function, arising when sizable portions of the Fermi surface can be superimposed through a translation of wavevector → − q . Under these conditions the system finds favorable to undergo a distortion of the lattice which triggers a down-folding of the Brillouin zone because of the reduced symmetry. CDWs are normally found in low-dimensional materials, since in this case the nesting con− dition (i.e. the connection of portions of the Fermi surface through the wavevector → q ) is more favorable. As a consequence of the CDW instability, free charges condensate into a collective mode, which is generally pinned to the lattice. The electron condensate displays a charge

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− density modulation with periodicity and direction determined by → q . If the Fermi Surface is perfectly nested, the material is turned into an insulator across the Peierls transition. The microscopic force driving the CDW transition is the electron-phonon interaction, while correlation effects do not need to play any role in the Peierls mechanism. However many systems (between which several vanadium oxide compounds) experience a lattice distortion in correspondence with the MIT: in this case it becomes unclear whether the principal responsible for the MIT is the Peierls or the Mott mechanism. From an electrodynamic point of view the formation of a Charge-Density-Waves opens up a gap in the optical conductivity. A large absorption term also builds up at mid-infrared frequencies, corresponding to the excitations from the charge-density wave collective term to the electronic continuum. This feature is also called single-particle gap.

3. 3.1.

Methods Infrared Spectroscopy

Optical Reflectivity is a bulk, contact-less technique enabling to probe the low-energy electrodynamics of solids over a broad spectral range and with excellent energetic resolution. The reflectivity R(ω) is defined as the ratio between the electromagnetic intensity reflected from the sample (Is ) and that reflected from a reference mirror (I0 ):

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R(ω) =

Is (ω) . I0(ω)

(8)

Data are typically collected at near-normal incidence for ω between 30÷ 20000 cm−1 and the reference measurement is usually taken on a metallic film (gold, silver or aluminum, depending on the spectral range) evaporated in situ over the sample surface. R(ω) is related to the complex refractive index n ˜ = n + ik via the Fresnel equations[17, 18]: R(ω) = |˜ r|2 = |

(˜ n − nm ) 2 | (˜ n + nm )

(9)

where r is the reflection coefficient and nm is the refractive index of the medium that shares the front interface with the sample. Whether the reflectivity is measured at a vacuum-sample interface nW =1 and Eq. 9 reduces to the simple formula: R(ω) =

(n − 1)2 + k2 . (n + 1)2 + k2 p

(10)

Since R(ω) = r˜(ω)˜ r∗(ω) and r(ω) = R(ω)eiθ(ω) , R(ω) and θ(ω) are related by the Kramers-Kronig (KK) transformations that correlate the real and the imaginary part of a linear response function due to the causality principle. In order to perform KK transformations the reflectivity has to be suitably extrapolated towards low and high frequencies by using standard procedures (see Refs.[17, 18]). Once performed KK transformations, by employing either Eq.9 or Eq.10, depending on the experimental conditions, the complex refractive index can be calculated. Given the well

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known relations among the optical response functions we can also calculate the real part of the optical conductivity σ1 (ω). The optical conductivity is defined as the linear response function relating the electrical current J and the excitation due to an electric field E by the expression J(q, ω) = σ ˜(ω, q)E(q, ω).

(11)

We can derive an expression that relates the optical conductivity with microscopic observable that can be easily defined in a N-electron system. The fluctuation-dissipation theorem relates the fluctuations described by a correlation function to the dissipation described by the imaginary part of a susceptibility. In the case of response to the electromagnetic radiation, the interaction Hamiltonian, in the dipole approximation, reads Hint = − 1c J · A, where A is the perturbing vector potential. The following expression of the fluctuationdissipation theorem can be derived directly from the Fermi golden rule σ1 (ω) =

X n

1 hωV ¯

Z ∞

dthn|ˆ {J(0), ˆ J(t)}|nieiωt ,

(12)

0

called the Kubo formula [18]. The right hand side of Eq. 12 can be independently derived from the Fermi golden rule for the transition probability between two energy levels, once summed over all the possible initial and final states. This allows one to establish a proportionality relation between σ1 and the total transition rate W . One can get a microscopic interpretation of the conductivity, as the sum over all the possible ”jumps” of energy ω in a given distribution of states εn , weighted by their dipole matrix element, and whose ground state energy is εg :

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σ1(ω) =

πX 2 δ(ω − εn + εg ) ˆ |hn|J|gi| V n εn − εg

(13)

It can be demonstrated [18] that combining Kramers-Kronig relations with physical arguments about the behavior of the real and imaginary part of the response function it is possible to establish a set of so-called sum rules for various optical parameters. We will here define only the f-sum rule as: ωp 2 = 8

Z ∞ 0

σ1(ω, T )dω =

π X qj2 πN e2 = 2m 2 j Mj 

2

1/2

(14)

e ) and the charge and the where ωp is the plasma frequency (defined as ωp = 4πN m mass of the j charged objects in the solid unit cell have been generalized to q and M to include the case of phononic excitations. It is worth of noticing that N is the total number of electrons per unit volume, if the integral 14 is carried out to infinite frequencies. The sum rule evaluated up to high frequency expresses a constraint that σ1 must fulfill when some external parameters, such as temperature T or pressure P , vary. At low frequencies, the optical conductivity is normally dominated by the contribution from free electrons, if any. In normal metals this contribution is called the Drude term, and is described by the relation: 1 ωP2 τD , (15) σ ˜Drude = 4π 1 − iωτD

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where N is the number of free electrons, e the electron charge, m their band-mass, and τD is the electron scattering time. Besides the Drude term - normally associated to the coherent motion of charge carriers - other relevant electronic excitations are usually described phenomenologically in terms of Lorentzian Harmonic Oscillators which take into account electronic interband transitions. The contribution from the infrared active phonon modes to the optical conductivity is described through Lorentz Oscillators, as well. The measured optical conductivity is therefore expected to be described as the sum of the Drude term and of the Lorentzian Oscillators, taking into account both electronic and vibrational contributions. In its final form the Drude-Lorentz model is given by the formula

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σ ˜=

1 ωP2 τD 1 X ωl2 ω + 2 − ω 2) + ω τ . 4π 1 − iωτD 4π j j(ω0j j j

(16)

While this simple model nicely catches up the physics of most conventional materials, its application to strongly correlated electron systems is much more problematic. Strongly correlated electron systems in their (bad) metallic phase, are indeed characterized by the presence of a broad conductivity term spanning from zero frequency to the mid and near-infrared range. In order to describe this term, some author use the so-called extended Drude model approach, which makes use of a frequency dependent scattering rate [19]. Alternatively, one can make use of a two-component description, where the low frequency term is modeled in terms of one Drude plus one harmonic oscillator. In a simple picture, one can figure out these two terms as being associated to the coherent motion of quasiparticles and to the incoherent transitions from the lower lying electronic bands (Hubbard bands and charge transfer excitations) to the quasiparticle peak and from the quasiparticle peak to higher energy (Hubbard) bands. A second characteristic fingerprint of electron correlation is provided by the (apparent) violation of the so-called f-sum rule. The integral in Eq. 14 should remain constant for any variation of the external parameters (e.g. temperature, pressures, magnetic fields, etc.). For obvious reasons, in the real case one has to deal with partial sum-rules, extending up to the maximum frequency of the measurement. In conventional metals, the partial sum-rule is well satisfied within frequencies extending up to the plasma frequency [20]. On the other hand, a typical hallmark of strong correlation is the redistribution of the spectral weight within frequencies of several eV. This is seen in optical spectra through a violation of the partial sum-rule [20, 21], while ellipsometric techniques are normally used to precisely assess the frequency at which the spectral weight is effectively recovered [22].

3.2.

Diamond Anvil Cell

Hydrostatic pressure conditions on the GPa scale are achieved by making use of Diamond Anvil Cell’s (DAC’s). DAC’s consist of two opposing diamonds (see Fig. 4), with a sample compressed between the diamond’s culets. Since pressure scales with the inverse of the area, if one wants to achieve high pressures, the culet area has to be small, i.e. with a diameter of hundreds micron and below. The combination of the absorption from the diamonds (which, for IR spectroscopy, have to be type IIa, i.e. nitrogen free), and the small dimensions of the samples which can be placed in a DAC, make the optical throughput of the

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Figure 4. Pictures of the open (left panel) and closed (right panel) screw-driven DAC with a copper gasket. From Ref. [23].

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whole system rather low. Infrared measurements at high pressure are therefore a very challenging technique, which takes great advantage of making use of an infrared microscope coupled with a high brightness source, as a synchrotron infrared source [24, 25]. Between the two diamonds of the DAC one places a metallic gasket that can be of different materials (stainless steel, molybdenum, aluminum, etc.) depending on the pressure range to investigate. In the hole drilled at the center of the gasket is placed the sample together with a hydrostatic salt transparent to infrared (KBr, NaCl, CsI, etc.) and a ruby chip. The latter is used to monitor pressure in situ by measuring its pressure-dependent fluorescence. In the case of Reflectivity measurements, great care has to be taken in order to prepare a clean and flat sample-diamond interface. As a reference one can measure the reflectance from a metallic mirror (Au, Ag) placed inside the cell. ˜ is related to the reflectivity measured The complex refractive index of the sample n(ω) inside the DAC, through the Fresnel formula n ˜ (ω) − nd 2 , Rs−d (ω) = n ˜ (ω) + n

(17)

d

in which nd = 2.43 indicates the refractive index of diamond [26], assumed to be real and frequency independent over the whole considered spectral range. However, following Eq. 17, the pressure dependent optical properties are not univocally determined from Rs−d . In order to fully characterize optical properties of the sample and calculate the optical conductivity one can either fit Rs−d (ω) within the Drude-Lorentz framework, measure simultaneously reflectance and transmittance, or perform Kramers-Kronig Transformations.

4.

Vanadium Sesquioxide - V2 O3

V2 O3, which is considered as a paradigmatic example of a Mott-Hubbard material, shows a metal-to-insulator transition at T N ∼ 160 K [27] from a high-temperature stronglycorrelated paramagnetic metal (PM) to a low-T antiferromagnetic insulator (AFI). Through the substitution of a small amount (0.005 < x TN . While the PM-AFI transition is accompanied by a giant lattice rearrangement from a corundum to a monoclinic structure, in the PI-PM MIT only the a/c ratio (where a is the lattice

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parameter of the hexagonal corundum plane and c is the perpendicular axis), changes discontinuously. The systematic study of chemical substitutions in V 2 O3 with Cr and Ti ions led to the phase diagram reported in Ref. [28]. The phase diagram closely resembles that obtained with a single-band DMFT model, with a degree of intermediate magnetic frustration (see Fig. 3). For long time the properties of V 2O3 have been described in terms of a single-band model [30] and mainly neglecting the role of the lattice in determining its electrodynamic response. More recent studies [31, 32] have however pointed out that a single band picture is not enough to fully describe the properties of V 2 O3 and, more important, that some minor lattice modifications may play a role across the PI-PM transition. The picture seems therefore more complex and the role of electronic correlation needs to be better clarified. Studies at high temperature allow to asses the importance of correlation in a metallic system: In conventional metals well-defined long-lived quasiparticles exist up to a coherence temperature Tcoh of the order of the Fermi temperature. Typically, Tcoh has a value larger than the experimentally accessible temperatures, not allowing to observe the loss of coherence and of spectral weight expected for T > Tcoh . On the other hand, in strongly correlated electron systems the coherence scale is generally reduced. This is particularly true in the proximity of the Mott transition, where the disappearance of the QP across Tcoh [3] is expected. In the AFI state, σ1 (ω) - in good agreement with previous data [33, 34] - shows a well defined charge gap in the infrared. The absorption edge rises with increasing frequency, building up a broad feature in the visible. This is the Mott gap between the Hubbard bands of the strongly correlated AFI [3, 35]. On crossing TN the gap is abruptly filled and σ1 (ω) shows, below 1 eV, a metallic absorption due to the appearance of QP. Indeed the optical conductivity of V 2O3 shows at 200 K a well defined metallic contribution. In addition to the metallic term, a broad, incoherent band (MIR) is observed in the mid-infrared. As T is raised, σ1 (ω) presents a strong temperature dependence (see Fig. 5), with a huge transfer of SW from low to high frequency through an isosbestic point at about 6000 cm −1 . The ω → 0 limit of σ1 (ω) and the DC conductivity are in excellent agreement at almost all temperatures (see inset of Fig. 5). This agreement and the overall decreasing behavior of σ1 (ω) with increasing frequency indicate that a well-established metallic term is present. For T = 450 K, the low-frequency σ1(ω) starts to show a downturn: This change of sign is associated with the disappearance of QP. On increasing T from 450 to 600 K, the downturn eventually transforms into a pseudogap at low frequency, while the MIR broadens. The pseudogap formation in σ1 (ω) is in agreement with an anomalous enhancement of the DC resistivity of V 2 O3 when entering the crossover regime [37, 38, 39]. Such behaviour can be understood as being related to the disappearance of the QP’s because of their loss of coherence above Tcoh ' 425 K. As a first step one might discuss the optical results in a single-band half-filled Hubbard model, in particular if the loss of spectral weight (calculated with the restricted sum-rule) is correctly described: By using the same effective U values as in [35] (U = 2D for the PM and U = 4D for the AFI phase, D being the half-bandwidth), it is found that the Hubbard model provides correct estimates for the high temperature decrease, i.e. 40% to 70% for the different cutoffs (see Fig. 6). However the T scale is off by a factor of 2, thus confirming the need for a more complicated theoretical description.

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100 K 200 K 300 K 350 K 400 K 450 K 500 K 550 K 600 K

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T (K) SW(T)/SW(200K)

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Figure 5. Optical conductivity of V 2O3 plotted between 0 and 18000 cm −1 , for selected T . In the inset σ1 (ω) is shown on a reduced frequency scale together with the DC points from four-probe resistivity measurements. From Ref. [36].

1

0

0.05

0.10

b)

Ω=0.7D D 2D

0

Figure 6. Normalized SW for three different cutoffs Ω as a function of T in experiments (left) and in a single-band Hubbard model (right). The normalization of the theoretical data coincides with the experimental one if D = 1 eV. From Ref. [36]. Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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To go beyond a single-band theoretical discussion it is necessary to combine LDA+DMFT [40]. Tomczack and Biermann [32] discussed the optical properties of V 2 O3 by taking into account independently the role of the different orbitals ( eπg and a1g ) within DMFT. Following the theoretical framework described in [41], the authors assert that the eπg excitations are not coherent down to 390 K, while a1g carriers are. By raising temperature one therefore expects a larger impact on the a1g derived carriers and their consequent loss of coherence. Indeed, the optical response below 0.6 eV, which is dominated by the a1g − eπg transitions, will be more temperature dependent than that above 0.6 eV, which is mainly due to eπg − eπg transitions. Following [32], this can explain at least qualitatively, the opening of the pseudogap in the optical conductivity. Note that while the multiband model depicts a different scenario with respect to the single-band picture, the mid-infrared band observed in the optical spectra is explained in both cases as being due to transitions between a coherent peak at the Fermi level and an incoherent band (i.e. from the a1g to the lower lying eπg , in the multiband case). A further effect that should to be taken into account is the modification of the lattice parameters as T is increased [28]. The modification of the c/a ratio, that goes from 2.835 at room temperature, to below 2.800 at temperatures higher than 600 K, was considered in the LDA+DMFT calculation at 300 and 700 K. As a result of the distortion the LDA bandwidth shrinks, and the optical conductivity from DMFT (see Fig. 7) displays the presence of a well defined pseudogap, in agreement with the experimental data shown in Fig. 5. Noteworthy, if the lattice parameter values at 300 K were used in the high temperature DMFT calculation, the optical conductivity at 700 K would have displayed a strongly renormalized Drude-like peak, rather than the pseudogap. The results above discussed show that the loss of spectral weight is mainly due to a loss of coherence of the QP excitations. However a pseudogap in the spectra of excitation can be explained only by accounting for the T -dependence of lattice constants, which gives a small shrinking of the LDA bandwidth upon heating. Due to the proximity of the Mott instability, the small change in the lattice parameters effectively drives V 2 O3 into a quasi-insulating state.

5.

Vanadium Dioxide - VO2

Vanadium Dioxide is also a tremendously renowned material in the field of strong correlations. Its popularity is undoubtedly due to the presence of a MIT transition involving a resistivity jump of two orders of magnitude very close to room temperature (340 K, 67 ◦ C), thus making it extremely appealing from a technological point of view. Indeed Vanadium Dioxide has great potentials to be used for fabricating thermo-chromic windows but also for the realization of micro-thermometers or ultra-fast optical switches. Nevertheless the interest in Vanadium Dioxide has also been triggered by the controversial origin of its phase transition. It is true indeed that the electronic transition, is accompanied by a structural one, from monoclinic to rutile. For this reason it is still unclear whether the Mott-Hubbard or the Peierls mechanism dominates the transition. New advances in experimental techniques have recently led researchers to re-examine the nature of the insulating state of VO 2. Sorting out of many, we mention femtosecond time resolved studies,

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700 K 300 K

0.0

0.5

1.0

1.5

ω (eV)

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Figure 7. Optical conductivity for V 2 O3 at three different temperatures with LDA+DMFT, calculated by taking into account the experimental values of the lattice parameters [28]. From Ref. [36]. both at optical [42] and THz frequencies [43, 44], and Scanning Near-field Infrared Microscopy (SNIM) [45]. These recent research studies seem to indicate that the major role in the MIT of VO2 is played by the Mott-Hubbard mechanism. Here, we present recent optical spectroscopy data at high pressures [46, 47]. We will see in this paragraph how pressure is able to disentangle the electronic and structural transitions, thus providing a clear argument in favor of the Mott-Hubbard mechanism, as the originator of the phase transition. The pressure-dependent optical conductivity of VO 2 is shown in Fig. 8. At low pressure one can see the presence of a main gap feature slightly below 0.5 eV, which is followed by a smooth absorption tail extending down to 1500 cm −1 (0.2 eV). At first, spectra are only weakly dependent on pressure, approximately up to 10 GPa. Above this value an abrupt increase of σ1(ω) is observed, together with a partial filling of the insulating gap. By linearly extrapolating the conductivity curves towards zero frequency, the data collected above 10 GPa give positive, although small σ1 (ω = 0) values, compatible with a bad metallic behaviour. The frequency integrals of σ1 (ω) over 900-1600 cm −1 [SWL ∗ (P )] and 2600-5000 cm−1 [SWH ∗ (P )] at each pressure, and normalized to the lowest pressure values, are displayed in Fig. 9. The same pressure dependence is highlighted in the two frequency ranges, the main feature being an abrupt change of slope at 10 GPa. While data at far infrared frequencies would be desirable in order to provide a better estimate of σ1 (ω = 0), the data in Figs. 8 and 9 provide a strong indication for the occurrence of a pressure induced MIT above 10 GPa.

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σ1(ω) (Ω-1.cm-1)

VO2

0.2 GPa 5.9 10.1 11.9 13.9

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4000

5000

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Figure 8. Pressure dependent optical conductivity of VO 2 . From Ref. [46].

Infrared data can be compared with pressure-dependent Raman spectroscopy [46]. At ambient pressure, 15 narrow phonon lines of the 18 Raman-active modes predicted for the monoclinic M 1 phase, are identified. Applied pressure results in a phonon frequency hardening, with no significant change in the overall peak pattern. Since the Raman spectrum of VO2 in the rutile (R) phase is characterized by 4 broad peaks only, it is concluded that by applying pressure up to 19 GPa (i.e. the highest applied pressure value), no transition to the R phase (or to a more symmetric phase than M1) is achieved. A more careful analysis of the Raman data shows that all phonon peaks except ωV 1 (at 192 cm−1 ) and ωV 2 (at 224 cm−1 ), increase almost linearly with pressure. It is found indeed that both ωV 1 and ωV 2 experience a rather abrupt change in dω/dP at 10 GPa. While dω/dP is close to zero at low pressures, it increases to 1-1.5 cm −1 /GPa above 10 GPa. Following the absence of oxygen isotope effect in the Raman measurement in [47], these two modes are assigned to V-ions motion along the c-axis. It is therefore tempting to attribute the above-mentioned change of slope as a consequence of a rearrangement of the V dimers as the pressure is increased above 10 GPa. A further insight on the physics of VO 2 can be gained by addressing Cr-substituted compounds. Fig. 10 displays the V 1−xCrx O2 phase diagram as a function of the Cr substitution fraction x. The introduction of Cr has only a weak effect on TM IT , which moderately increases. On the other hand Cr induces (at room-temperature and below) structural transi-

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VO2

25 20 SWL* SWH*

15

2.0

10

SWH* (P)

SWL* (P)

2.5

1.5

5 0

4

8

12

1.0 16

Pressure (GPa)

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Figure 9. Normalized Spectral Weights SWL ∗ and SWH ∗, as defined by SWL ∗ (P ) = SWL(P )/SWL (0) and SWH ∗ (P ) = SWH (P )/SWH (0). From Ref. [46].

tions between three distinct monoclinic insulating phases. In the M 2 phase only one half of the V atoms dimerizes along the c-axis direction, while the remaining V ions form zigzag chains of equally spaced atoms. The M 3 phase is somehow intermediate between M 1 and M 2: The dimerized chain starts to tilt and the zigzag chains get partially dimerized. The M 3 phase, often also called the T phase, has been explained in terms of two V sublattices within a triclinic structure [48]. The M 2 and M 3 phases can also be regarded as metastable phases of pure VO 2, with a free energy which is only slightly larger than that of M 1 [49]. From an optical point of view, σ1 (ω) of the Cr-substituted sample (displayed in Fig. 11) is very much similar to that of the pristine sample. At low pressures, the conductivity is gapped, and a strong enhancement of σ1 (ω) is observed above 10 GPa, as an indication of a pressure-induced MIT. The absolute values of the normalized spectral weights (shown in Fig. 12) are almost the same for both samples, even though, in the case of the Cr-doped sample only, a slight increase is observed also at low pressures. While infrared measurements evidence a substantial similarity in the electronic properties of the two compounds, Raman spectra are of great help in order to underlie their structural differences. It is found indeed that at ambient pressure conditions the pure and 0.025% Cr doped compound display well distinct phonon peak patterns, as a consequence of their different monoclinic arrangements (i.e. M 1 and M 2 respectively). The same is true also for the x=0.007 (M 3) compound, displaying a Raman pattern intermediate between the two. By applying pressure, the Raman spectra of the three compounds progressively merge. For the x=0.007 compound one observes a continuous transition towards the M 1 phase, which is completed at about 3 GPa. Similarly, the x=0.025 sample completes its structural transition at 9 GPa. Such a difference in the transition pressure values reflects the larger structural changes between the phases M 2 and M 1, with respect to the intermediate

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V1-xCrxO2

R Temperature (K)

350

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M3

250

M1 200 0.000

0.005

0.010

0.015

0.020

0.025

x

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Figure 10. Phase Diagram of V1−x Crx O2 as a function of temperature and Cr concentration, following [50]. The abbreviations R, M1, M2 and M3 stand for the Rutile and Monoclinic (1,2,3) structural phases respectively, as described in the text. From Ref. [23].

phase M 3. An accurate inspection of the pressure dependence of the most relevant phonon modes, leads to conclude that pressure progressively compress and symmetrizes oxygen octahedra of the three samples, which then get the more and more similar between each other. On further increasing pressure above 9 GPa, the pressure dependence of the ωV 1 and ωV 2 modes changes slope, for all the considered samples. This suggests a slight rearrangement of the V chains, leading to a common metallic monoclinic phase ( Mx ), where the extent of the Peierls distortion is still unknown. However, the metallic state in the Mx phase is completely different to that observed in the Rutile symmetry. In this case a robust Drude term shows-up in the infrared conductivity surviving also at the highest measured temperature (T ' 450 K)[22] meanwhile this term is absent in our case. The combination of high pressure infrared and Raman results outline a scenario where the pressure-induced metallization process occurs within the monoclinic symmetry, which retains well above 15 GPa (at least up to 19 GPa for VO 2 )[51]. The structural differences observed between the various monoclinic phases have a very small impact on the electronic properties, as observed through infrared. As suggested by the Raman measurement, above 10 GPa the unit cell volume abruptly decreases for all the considered Cr concentrations, which suggests a common lattice instability within the monoclinic symmetry. Since the Monoclinic to Rutile structural change is considered as the signature for the Peierls tran-

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300

250

σ1(ω) (Ω-1.cm-1)

V0.975Cr0.025O2

0.5 GPa 6.4 9.2 10.7 13.1 14.4

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0 0

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3000

4000

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Figure 11. Pressure dependent optical conductivity of 0.025% Cr doped VO 2. From Ref. [47]

sition in VO 2, the outlined results support the major role of electron correlations against structural effects in driving the MIT. The important role of correlation is corroborated by recent Temperature dependent ellipsometry data [22] displaying a spectral weight redistribution across the MIT which extends to frequencies larger than 6 eV.

6.

V3 O5

V3 O5 is an intermediate member of the Magn´eli phase, undergoing a metal to insulator transition [52] for TM IT = 428 K at which the resistivity sharply decreases by more than two orders of magnitude. However in the conducting state dρ/dT remains negative, thus indicating an activated transport behavior [53]. The activation energy has been determined through resistivity measurements, resulting to 0.3 eV in the low-temperature insulating phase, and 0.13 eV in the high-temperature state. As in the case of VO 2 , the phase transition is concomitant to a modification of the lattice structure leading to the symmetry change P 2/c → I2/c, and to a reduction in the unit cell size of about 0.14 % [53, 54]. The monoclinic crystal structure is constituted by oxygen octahedra surrounding the vanadium atoms. These octahedra form two types of alternating one-dimensional chains [54, 55]: one chain (A) is composed by double octahedra sharing

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V0.975Cr0.025O2

3

SWL* SWH*

10

SWH* (P)

SWL* (P)

4 15

2

5

0

4

8

12

1 16

Pressure (GPa)

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Figure 12. Normalized Spectral Weights SWL and SWH for 0.025% Cr doped VO 2. From Ref. [47].

a face, where these double-octahedra are coupled together by sharing edges. The other chain (B) is made up of single corner-sharing octahedra. The chains A and B are mutually connected by sharing both octahedral edges or corners. In the conducting state at high temperature, two different kinds of octahedra (V(1) and V(2)) have been identified, which split into four below TM IT (V(11), V(12), V(21), V(22)) [54, 55]. Following [54], in the insulating phase, the V(11) octahedra host the V 4+ ions while in the three other kind of octahedra V3+ ions are found. The V(1) octahedra can host either V 3+ or V4+ vanadium ions, with a 3.5 mixed valence, while in V(2) octahedra there are only V 3+ ions. V(1) octahedra occupy the chains made of double octahedra, while V(2) compose the single octahedra chains. Besides hosting in the low- T phase only V 4+ ions, V(11) is by far the smallest and more distorted octahedron. The MIT seems therefore to be related to a charge disproportion mechanism involving also a spatial ordering of the V 4+ and V3+ ions [53]. It has been argued that [53] this mechanism may be similar to that of the so-called Verwey transition [56] observed in magnetite (Fe3 O4 ). A scenario suggesting the formation of a polaronic Wigner crystal (see subsection 2.6.) below TM IT , and its melting at high temperatures, have therefore been put forward [53]. A transition from an antiferromagnetic to paramagnetic phase [57] is also observed within the insulating phase at TN = 75 K. Since TM IT  TN , the MIT appears to be decoupled from the AF order. Therefore V3O5 is a suitable system to study the charge dynamics near a metal-to-insulator transition without any charge localization induced by magnetic effects. The optical reflectivity measured at nearly normal incidence is reported in Fig. 13a at

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selected T . For temperatures up to 373 K, R(ω) shows a nearly constant behavior for ω < 250 cm−1 suggesting an insulating character. Above TM IT , instead a conducting behavior can be deduced from the monotonous increase of R(ω) for ω → 0. 1.0

V3O5

Reflectivity

0.8

a)

0.6 10 K 300 373 423 433 473 573

0.4 0.2

Increasing T

0.0 Increasing T

b)

-1

-1

σ1(ω) (Ω cm )

1500

1000

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500

0 100

2

3

4 5 6

2

3

1000 -1 Frequency (cm )

4 5 6

2

10000

Figure 13. a) Optical Reflectivity of V 3 O5 between 100 and 20000 cm −1 at selected temperatures. b) Real part of the optical conductivity at the same temperatures and over the same spectral range, as extracted from KK transformations. From Ref. [58].

The optical conductivity of V 3 O5 extracted through KK transformations is shown in Fig. 13b. At low temperatures, σ1 (ω) displays a clear insulating behaviour, with a chargegap superimposed to phononic peaks in the far-IR and an absorption band in the mid-IR. No major change is observed in the conductivity while crossing TN : the charge gap is indeed unchanged and this indicates the low sensitivity of the electronic properties of V 3O5 to the magnetic ordering. This behavior is at variance with V 2 O3 (see Section 4.) where the crossing of TN opens a large gap (nearly 1 eV) in the optical conductivity strongly redistributing the SW from low to high energy scales [22]. In order to better quantify

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-1 )

the effects of the MIT on the low-energy electrodynamics of V 3 O5 we have studied the evolution of the MIR band as a function of T . This behavior can be easily tracked by fitting the optical conductivity to a Drude-Lorentz (DL) model (see section 3.). While heating from 10 to 300 K the MIR (which can and well described by a single lorentzian curve) continuously narrows and shifts towards low frequencies (Fig. 14). This trend is much more pronounced between 300 K and TM IT as the characteristic MIR frequency (estimated from the DL fit) drops from 3000 cm −1 at 300 K to 1000 cm −1 at the highest temperature (573 K). In the following, these two energy scales (corresponding the former to the insulating state, the latter to the conducting one) are defined as HF-MIR and LF-MIR respectively. On the other hand, the width of the MIR band (ΓM IR , defined at half-width-half-maximum) abruptly increases above TM IT possibly indicating that a more disordered electronic state is generated in the high temperature conducting phase.

ω MIR Γ MIR

V3O5

6000 4000 2000 0

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0

100

200

300

400

500

600

Temperature (K) Figure 14. The MIR characteristic frequency (ωM IR) and its half-width-half-maximum (ΓM IR ), as a function of temperature are shown. The values have been obtained by a Drude-Lorenz fit of the optical conductivity (see text). From Ref. [58].

In the high temperature state the charge gap is nearly filled by a flat background as a consequence of the MIT. This background (causing the non-zero dc conductivity), that cannot be described by a simple Drude term, is superimposed to the LF-MIR absorption centered at about 1000 cm −1 and resembles the optical conductivity of a polaronic system (see section 2.6.). The spectral weight, reported in Fig. 15 at the various temperatures, shows a strong temperature dependence below 5000 cm −1 , whereas the f-sum rule is fully satisfied above 8000 cm−1 . This behaviour is very much reminding that of Fe 3 O4 [59]. In that case the charge-ordered ground state is attributed to polaron ordering. However, the energy scale of the SW recovery in these compounds (1 eV) is small if compared to V 2 O3 or VO2 where - as discussed previously - the spectral weight is not recovered yet at 6 eV [22]. Apart

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from giant structural transitions which affect, in some materials, the optical conductivity over a broad energy scale, only electron correlation, mixing low- to high -energy degree of freedoms, may induce in the optical spectral a strong T dependence until several eV. This suggests that in V 3 O5 the SW redistribution in a relatively narrow energy interval is related to a huge electron-lattice interaction. The pseudogap feature in the optical conductivity of WH

the conducting phase may be associated with a mobility of the form µ0 e kB T , where WH is the polaron hopping energy (see Sec. 2.6.). This behavior leads to an activated conductivity that is in agreement also with resistivity data. Therefore we can estimate the effective mass from the Mott equation [59, 60]: WH 1

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m∗ ≈ 5me 2 h¯ ω

(18)

where ¯ hω is an average phonon energy. Here WH can be estimated from the LF-MIR band peak position that corresponds to two times the polaronic binding energy Wp (see Sec. 2.6.). The formula (18) leads to m ∗ ≈ 10 m where m is the free-electron mass. The effective mass of the carriers can also be extracted from the integrated spectral weight (see Sec. 2.6.). From the Drude-Lorentz fit of the optical conductivity at 573 K (where a clear low-frequency contribution related to a finite dc conductivity appears) the polaronic coherent SW can be estimated to be 1.2 x10 6 cm−2 . The incoherent part that corresponds to the LF-MIR contribution is instead 5.5 x10 6 cm−2 . Therefore m∗ /m ≈ 5 in good agreement with the previous estimate. It is worth of noticing that different models (depending on different approximations) may be employed to estimate the effective polaronic mass renormalization, however the two estimates here presented are of the same order of magnitude, supporting in V 3 O5 a strong electron-lattice interaction scenario. Optical data therefore suggests an explanation of the MIT in V 3 O5 in terms of polaron ordering. Indeed vanadium octahedra, which host the holes carriers are strongly distorted. These distortions, both in the high and low temperature phases, arise from a strong latticecharge interaction and correlate to the formation of small polarons. In the conducting state, although strongly reduced, a difference in the lattice distortions among V(1) and V(2) octahedra is still present. This suggests that, at least for intermediate temperatures, the electrodynamics of V3 O5 can be described in terms of a disordered state of preformed polaronic charges. The infrared absorption at LF-MIR thus corresponds to photo-induced hopping between V(1) and V(2) octahedra. Moreover the fast exchange among V(1) lattice sites (which contain either V 3+ or V4+ vanadium ions) results in a bad metallic conductivity. This implies a polaronic diffusive motion in the conducting state which mirrors the absence of a Drude peak in the optical conductivity. In the insulating phase hole carriers instead are confined within the most distorted V(11) octahedra (with valence 4+). As indicate by diffraction, these octahedra spatially ordered at low temperature forming a Wigner crystal and the HF-MIR band observed in the optical spectra corresponds to photoinduced hopping processes of polaronic carriers from V(11) to the less distorted V 3+ octahedra. Moreover, the three V3+ octahedra display different distortions, thus determining a distribution of ordering energies which widens the bandwidth of HF-MIR. According with diffraction data, with increasing temperature the differences in the distortions between different octahedra reduces, thus explaining the decrease of ΓHF −M IR . Therefore the MIT transition may be explained in terms of a disorder to order transition of preformed polaronic charges. The crystallization energy (see Sec. 2.6.) can

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0.10

SW(Ω,T)

Increasing T

V3O5 0.05

10 K 300 373 423 573

0.00 0

2000

4000

6000

8000

10000

-1

Frequency (cm )

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Figure 15. Optical Spectral Weight of V 3O5 as a function of frequency at selected temperatures. From Ref. [58].

be estimated from the difference between the HF-MIR at 10 K and the LF-MIR energies at 573 K. This difference corresponds to an energy of about 2500 cm −1 (0.25 eV) showing that the polaronic Wigner crystal ground state energy is well beyond that of a disordered polaronic liquid state. The P − T phase diagram of V3O5 , as determined by Sidorov et al. [61], is reported in Fig. 16. TM IT reduces as pressure is increased, and vanishes at about 9 GPa suggesting the existence of a Quantum Critical Point for this material. At room temperature, a pressure-induced MIT occurs at about 6.3 GPa, where a discontinuity in the resisitivity was measured. The behaviour of TN has been determined as a function of pressure, as well [62]: only a slight increase of TN is found at a rate of 0.82 K/GPa, up to 5 GPa. Reflectivity measurements as a function of pressure were performed at room temperature. Pressure induces an enhancement of the reflectivity level over the whole measured range. The quantity R(P ) − R(P = 0.5GPa) ∆R = (19) R R(P = 0.5GPa) evaluated at 1600 cm −1 is reported in Fig. 18. A phenomenological sigmoid fit to the data permits to estimate directly from the reflectivity, the pressure PM IT = 6 GPa at which room temperature metallization occurs. This value is in very good agreement with the value determined by the pressure-dependent resistivity measurement [61]. ∆R/R has been evaluated also at other frequencies, with consistent results.

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500

Metal 400 300 200

PM Insulator

100

AF Insulator 0 0

2

4

6

8

10

Pressure (GPa)

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Figure 16. Pressure vs Temperature phase diagram of V 3 O5 as determined in [61]. The AF order is found at ambient pressure below TN = 75 K. The thick curve represents the P -dependence of TN [62]. The measurements performed in this work (vs. both T and P ) have been collected over the red markers shown in figure. From Ref. [23].

The optical conductivity, as depicted in Fig. 17b, has been obtained by Lorentz-Drude fitting (see Sec. 3.). The fits are consistent with two-absorption band scenario. Their characteristic energies at ambient P and room T well correspond to those of LF-MIR and HF-MIR measured in the T -dependent experiment showing the good quality of data in the anvil cell. At variance with the T -dependent data where the HF-MIR band gradually shifts towards low frequency, in the P -dependent data a coexistence of the two bands at any measured pressure can be observed. With increasing pressure the SW of HF-MIR decreases, while that of LF-MIR increases leaving the total SW almost constant. The optical high pressure data could be fitted also by using a Drude term instead of a LF-MIR. However, any reasonable fitting performed with the Drude term would lead to high values of σdc , not consistent with resistivity data [61]. Moreover, the activated behaviour of the resistivity at high pressures is a further indication for negligible coherent transport, at least at room temperature. The coexistence of the LF- and HF-MIR bands at high pressures suggests that the P induced MIT can be related to a different microscopic mechanism with respect to the T dependent one. X-ray pressure-dependent data indeed indicate the presence of an anisotropic compressibility. With pressure, the b-axis lattice parameter reduces more than a and c. As a consequence, the compression of the layers containing V 3+ octahedra only is larger in comparison to those containing both V 3+ and V4+ . The distortion differences between tri-

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V3O5

Rsd (ω)

0.8 0.6

Increasing P

P= 0 GPa 0.5 1.2 4.9 6.9 9.8

GPa GPa GPa GPa GPa

0.4 0.2 a) 0.0

Increasing P

b)

-1

σ1(ω) ( Ω−1cm )

1500

1000

500

0 Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

0

2000 4000 -1 6000 Frequency (cm )

8000

Figure 17. a) Rsd (ω) of V3O5 at the sample-diamond interface at room T for selected pressures and compared to the ambient pressure (black line) sample-diamond reflectivity. b) Pressure-dependent optical conductivity of V 3O5 , as extracted from the LD fit. Thin dashed lines are the HF- and the LF-MIR components of the fit for P = 9.8 GPa data. From Ref. [58].

and tetra-valent octahedra is reduced only along certain lattice directions, thus allowing the simultaneous presence of both LF-MIR and HF-MIR bands in the optical spectra. Therefore the conducting state of V 3O5 induced, at room-T , by pressure can be described in terms of a dynamical equilibrium among strongly localized small polarons (related to the HF-MIR) which do not contribute to the dc conductivity and less localized small polarons (related to the HF-MIR) which, due to the smaller localization energy, may thermally hop from different lattice site, determining a bad metallic conductivity as indicated by dc data. In conclusion the charge dynamics of V 3 O5 across both the MITs induced by temperature and pressure can be explained in terms of a localization-delocalization process of lattice

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polarons. This localization phenomenon transforms in a real ordering in the insulating state induced by temperature. This ordered phase is one of the rare example of polaronic Wigner crystal in solids. This suggests that the most important energy scale in V 3O5 , at variance with V 2 O3 and VO2, is due to charge-lattice interaction. 0.6

V3 O5

0.5

a)

6 GPa

∆R/R

0.4

0.3

0.2

0.1

0.0

b)

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SW

0.10

LF - MIR HF - MIR Total

0.05

0.00 0

2

4

6

8

Pressure (GPa) Figure 18. a) ∆R/R at 1600 cm−1 . The dashed line is a phenomenological sigmoid fit to data. b) Spectral weight of the LF-MIR, HF-MIR, and total as a function of pressure, from the LD fit. From Ref. [58].

7.

Nickel Disulfite - NiS 2

NiS2 has cubic pyrite structure, with Ni and S 2 pairs forming a three-dimensional rocksalt lattice. NiS 2 attracts particular interest as it easily forms a solid solution with NiSe 2 (NiS2−x Sex ), which, while being iso-electronic and iso-structural to NiS 2, is nevertheless Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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A. Perucchi, L. Baldassarre and S. Lupi 4000

σ1(ω) (Ω−1.cm-1)

NiS2-xSex 3000 x=1.2 2000 x=0.6 1000 x=0

x=0.55

∆σ1=σ1(x)-σ1(x=0)

3000 x=1.2 2000 x=0.6

Temperature (K)

a) 0

300 PI

200

PM

100 AFI

0 0.0

AFM

0.5 1.0 1.5 Se content

1000 x=0.55

b)

0

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0

2000

4000 Frequency (cm-1)

6000

8000

Figure 19. (Color online) a) Optical conductivities of NiS 2−x Sex for x = 0, 0.55, 0.6, 1.2 at ambient conditions. b) Difference ∆σ1 = σ1(x) − σ1(x = 0) spectra. Inset: Phase diagram of NiS2−x Sex [1]. Black dots correspond to the samples measured in this work. b) Optical conductivities from KK transformations. From Ref. [65]

a good metal. By increasing the Se content above x ≈ 0.6, one can induce at room temperature an insulator to metal transition. On the other hand a magnetic phase boundary from an antiferromagnetic to a paramagnetic metal is found at low T at about x = 1, as shown in the phase diagram reported in Fig. 19 [1]. Alternatively, one can induce a metallic state with pressure. For long time NiS 2 has been classified as a Charge-Transfer insulator, even though Zaanen, Sawatsky and Allen [63] suggested that, since holes were accommodated in the antibonding orbital of the sulfur pairs, NiS 2 had to be considered a band insulator. Recent XAS data, together with theoretical calculations, suggest this view [64]. The optical properties of the solid solution NiS 2−xSex upon varying x and external pressure have been studied in two recent papers: Ref. [65] by authors A.P and S.L. and Ref. [64] from L.B. . Both works address the substantial differences between pressure and alloying induced phase transitions, while different mechanisms underlying the MIT are singled out. In the following, data and interpretation from Ref. [65] only will be presented and discussed.

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The optical conductivities of NiS 2−x Sex compounds at different Se contents are strikingly different. The pristine NiS 2 material displays the clear presence of a gap, roughly located at 4000 cm −1 (taken at Full-Width-Half-Maximum), and associated here to the Charge Transfer gap. On increasing the Se-content x, a large amount of spectral weight (SW) is transferred from high to low frequency through an isosbestic point around 8000 cm −1 . Such a strong spectral weight redistribution suggests the important role played by electronic correlations [1] in driving the transition into the metallic state. It is possible to separate the low energy contribution to the σ1(ω) into two distinct and well-defined terms: one broad mid-IR band is peaked at about 2000 cm −1 and extends up to nearly 8000 cm −1 , while a much sharper contribution is present below 500 cm −1 only. This is very clearly seen by calculating the ∆σ1 = σ1(x) − σ1(x = 0) difference spectra, as shown in Fig. 19. Similarly to the case of V 2 O3 discussed before in Fig. 5, one can attribute the two terms to coherent and incoherent excitations. Indeed, the sharp term is related to the appearance of a quasi-particle peak at the Fermi level. On the other hand, the incoherent term which in the case of V 2O3 was due to optical transitions from the QP peak to the upper and lower Hubbard bands, will be now a sum of the transitions between the Fermi level and all the lowest energy excitations, i.e. the Hubbard split Ni d bands and the chalchogen 2p band. The two-component scenario can be confirmed by fitting the σ1 (ω) curves through a Drude-Lorentz (DL) model. Data can be described by a Drude term plus two Lorentzian oscillators. The Drude and the low energy oscillator (centered around 2000 cm −1 ) describe the coherent and the mid-IR excitations around EF , while the remaining oscillator at 10000 cm−1 mimics the CT and Hubbard transitions. The fitting components are reported as thick dashed lines in Fig. 19b for x = 0.6. As is often the case in correlated insulators pressure can also be used to induce an insulator to metal transition, and a MIT has been observed in pure NiS 2 for P > 4 GPa, in the transport properties [66, 67]. Optical Reflectivity curves obtained by applying pressure, instead of alloying, are reported in Fig. 20a. When pressure is increased, Rsd (ω) gets progressively enhanced at low frequencies. At the highest pressures, Rsd (ω) still displays an over-damped behavior which characterizes correlated bad metals. All Rsd (ω) curves are converging at about 10000 cm −1 . The Rsd (ω) data are fitted within the same DL framework used before for NiS2−x Sex at ambient-P . This allows to extract values for the quasi-particle spectral weight, defined here as the sum of the Drude and of the mid-IR contributions. As a next step we analyze the behavior of the quasi-particle SW as a function of the cubic lattice parameter a, as experimentally determined upon alloying and pressure application. The important difference is that the lattice is expanded by Se-alloying [68, 69] whereas it is compressed by pressure [70]. The quasi-particle SW, shown for pure NiS 2 at working P in Fig. 20b and for NiS 2−xSex at different Se-contents in Fig. 20c, thus reveals a striking non-monotonic behavior as a function of a. Its slow continuous increase ˚ (i.e. at the highest values of P ) reflects the progressive enhancement of the for a < 5.57 A kinetic energy due to the applied P and corresponds to a nearly-complete metallization of ˚ up to aeq ≈ 5.68 A ˚ (namely the lattice parameter corresponding NiS2. For a > 5.57 A to NiS2 at ambient conditions), correlation effects get larger and the SW drops rapidly to zero as a consequence of the Mott transition. On further increasing a above aeq due to the Se-alloying, the SW (Fig. 20c) restarts to increase, owing to the onset of the Se-induced

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MIT. A qualitative understanding of the two different MITs can be obtained through selfconsistent TB-LMTO LDA calculations [71]. At ambient P the Ni eg -states with a bandwidth Weg = 2.1eV and the antibonding ppσ ∗-S states are separated by a CT gap (∆LDA ) centered around 1.5 eV. Beside this gap, a second LDA-CT gap is present between occupied ppπ-S states below EF and the eg -Ni states (see e.g. Ref. [72]). Upon applying pressure, the lattice contracts and the entire band-structure around EF is renormalized; e.g. by a factor of 1.13 at P = 10 GPa (dashed curve in Fig.21a). Hence, the bandwidth-gap ratio Weg /∆LDA remains nearly constant. On the other hand, the interaction U can be assumed to be constant, so that Weg /U increases by the factor 1.13, triggering a bandwidthcontrolled MIT (BC-MIT). In the case of Se-substitution, the lattice expands (instead of shrinking) due to the larger atomic radius of Se ions. This leads to a very complementary scenario: While the changes of the eg -bandwidth Weg are negligible, the CT gaps shrink (see Fig. 21b). Assuming U again to be constant, the driving force for the MIT is now the reduction of the charge transfer gap (Weg /∆LDA increases, as Weg remains basically unaffected). If we now turn to the optical data of Fig. 20, we can expect - deep inside the metallic phase - the correlations to be weak. Thus LDA could provide proper results. The square plasma frequency calculated by LDA, as the average over the FS of the squared velocities, is compared in Fig. 20b-c to the experimental data. Because of the different changes in the band structure under pressure and upon Se alloying, LDA gives non-monotonic behavior and very different slopes in agreement with experiment, see Fig. 20b-c). For the insulating NiS2 compound and close to the phase transition, electronic correlations are not negligible and LDA is not enough. The results of a LDA+DMFT calculation is shown for NiS 2 at ambient pressure. For U > 3J (U being the intra-orbital Coulomb interaction between the Ni eg -states and J the Hund coupling), the two eg -orbitals split and a gap opens for NiS 2 . This insulating LDA+DMFT solution results in a very strong suppression of the square plasma frequency (as well as of the SW) as indicated by the vertical arrow in Fig. 20b-c. Besides the important similarities between the P - and Se-dependent phase diagrams [66, 68], the analysis of the optical SW results reveals that the two MITs rely on distinct microscopic mechanisms. These mechanisms can be understood theoretically in terms of the two fundamental parameters for the MIT in a CT insulator: Under pressure, Weg /∆LDA = const. and Weg /U increases, triggering the MIT; in contrast upon alloying Se, the increase of Weg /∆LDA is responsible for the MIT, whereas Weg /U even decreases.

8.

Conclusion

Scope of the present chapter was to provide the reader with an overview of insulating states of matter arising from electronic correlation and electron-lattice interaction. We have presented here the optical properties of various prototype materials showing an exotic insulating state which cannot be explained through the conventional band theory of Bloch and Bethe. We have illustrated how the analysis of the electrodynamics across the MIT, which can be induced in those systems by varying external parameters as temperature or pressure, provides fundamental information for the understanding of the nature of their insulating ground state.

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Se concentration (x) 0.0

a)

NiS2

SW (cm-2)

1.0

P=10.7 GPa P= 8.0 P= 4.9 P= 3.5 P= 2.1 P= 1.1 Fit (3.5, 10.7 GPa)

0.8

Rsd(ω)

1.0x10

0.5

1.0

1.5

9

0.8 0.6 0.4 Pressure Se

0.2 0.0

0.6

0

2

4

6

8

10

Pressure (GPa)

x=1.2 0.4

x=0.55 0.2

x=0 0.0 2

4

6

b)

P

9

1.0x10

8

12x10

c)

x (Se)

0.8

SW (cm-2)

3

10

Frequency (cm-1)

LDA 5.57 Å

0.6

LDA+ DMFT

P (GPa) 5

0.4

10

0.2

5.7 5.6

a (Å)

Data B-M LDA

5.5 Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

0.0 5.5

5.6

5.7

5.8

5.9

Lattice Constant (Å)

Figure 20. (Color online) a) Rsd (ω) measured for NiS2 at high-P (thick solid lines) (open symbols show DL fits) and calculated Rcal sd (ω) at sample-diamond interface for NiS 2−x Sex at selected x (dashed lines). The kink in Rsd (ω) at about 3700 cm −1 is instrumental. Inset: QP spectral weight vs P (bottom) and x (top); top and bottom scales are chosen consistently with a scaling factor f ≈ 0.14/GPa (see text). Lower panels: QP spectral weight (see text) vs lattice constant a for NiS2 (b) and NiS2−x Sex (c). Green circles are square plasma frequency values calculated with LDA (see text). A rescaling factor of 1.25 has been used for the comparison with the experimental data. The dashed-dotted vertical line marks the a value for NiS2 at ambient conditions. Inset: lattice parameter vs P. Experimental data from Ref.[70] (solid circles), calculated values using the B-M equation (open circles) and LDA (solid triangles). From Ref. [65].

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Total DOS NiS2 ambient-P Total DOS NiS2 10 GPa

15

10 ∆ LDA

LDA-DOS (a.u.)

5 ppσ∗ - S

eg - Ni 0 b)

Total DOS NiSe2 ambient-P 15

10

5

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0

-1

0

1 ω - EF (eV)

2

3

Figure 21. (Color online): a) Total LDA-DOS for NiS 2 . Solid line represent the DOS at ambient P , dashed lines at 10 GPa. b) Total LDA-DOS for the NiSe 2 compound. In the case of NiS2, the 10 GPa DOS can be rescaled by a factor 0.88 on that at ambient-P , resulting in an increase of Weg /U , keeping Weg /∆LDA fixed. In contrast, Se-substitution results in a decrease of Weg /U and an increase of Weg /∆LDA due to the shrinking of the charge-transfer gap ∆LDA . From Ref. [65]. As pointed out at the beginning of the chapter, the potentials of using correlated electron systems in future electronic devices triggers the large and long-standing effort in this field. Still much work remains to be done both towards the achievement of a full understanding of their peculiar behavior, and for overcoming technical challenges. Despite these difficulties, it has been possible to realize devices in which an external electric field modulates the electronic properties of strongly correlated materials, as hightemperature superconductors or colossal magnetoresistive manganites. Nonetheless, experiments are still rare, especially as for what concerns the optical characterization of working devices, a gap which needs to be addressed soon.

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Recent progresses in the growth of atomic-scale multilayers are now opening new possibilities in the design of material’s properties. An exciting development is the appearance of 2-dimensional metallic states at the interface between a Mott and a band insulator. Prominent examples are the SrTiO3 /LaTiO3 and SrMnO3 /LaMnO3 superlattices, which are now providing a new exciting playground for the study and exploitation of the unique properties of strongly correlated materials. Optical measurements are already providing a clean and contact-less probe of these new interfacial states.

Acknowledgments The authors wish to thank P. Postorino for the long-standing collaboration on pressure dependent optical studies, and P. Calvani, M. Marsi and S. Biermann for fruitful discussions. We also acknowledge all the coauthors of the works presented here: E. Arcangeletti, L. Boeri, M. Capone, D. Di Castro, K. Conder, P. Dore, P. Hansmann, K. Held, O. Jepsen, L. Malavasi, C. Marini, P. Metcalf, D. Nicoletti, M. Ortolani, E. Pomjakushina, G. Sangiovanni, D.D. Sarma, V.A. Sidorov, R. Sopracase, D. Topwal, A. Toschi, M. Valentini.

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[45] Qazilbash M M, Brehm M, Chae B G, Ho P C, Andreev G O, Kim B J, Yun S J, Balatsky A V, Maple M B, Keilmann F, Kim H T and Basov D N 2007 Science 318, 1750 [46] Arcangeletti E, Baldassarre L, Di Castro D, Lupi S, Malavasi L, Marini C, Perucchi A and Postorino P 2007 Phys. Rev. Lett. 98, 196406 [47] Marini C, Arcangeletti E, Di Castro D, Baldassarre L, Perucchi A, Lupi S, Malavasi L, Boeri L, Pomjakushina E, Conder K and Postorino P 2008 Phys. Rev. B 77, 235111 [48] Pouget J P, Launois H, Rice T M, Dernier P, Gossard A, Villeneuve G and Hagenmuller P 1974 Phys. Rev. B 10, 1801 [49] Rice T M, Launois H and Pouget J P 1994 Phys. Rev. Lett. 73, 3042 [50] Marezio M, Mc Whan D B, Remeika J P and Dernier P D 1972 Phys. Rev. B 5, 25411 [51] Preliminary high-pressure x-ray diffraction data extend the stability of the monoclinic phase up to 42 GPa [Baldini M, Malavasi L et al., unpublished]. [52] Terukov E I and Chudnovskii F A 1974 Fiz. Tekh. Poluprovodn. 8, 1266 [53] Chudnovskii F A, Terukov E I and Khomskii D I 1978 Solid State Commun. 25, 537 [54] Hong S H and Asbrink S 1982 Acta Crystallogr. B, 38, 713 Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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[65] Perucchi A., Marini C., Valentini M., Postorino P., Sopracase R., Dore P., Hansmann P., Jepsen O., Sangiovanni G., Toschi A., Held K., Topwal D., Sarma D. D. and Lupi S. 2009 Phys. Rev. B 80, 073101 [66] Miyasaka S., Takagi H., Sekine Y., Takahashi H., Mori N. and Cava R. J. 2000 J. Phys. Soc. Japan 69, 3166 [67] Niklowitz P. G., Steiner M. J., Lonzarich G. G., Braithwaite D., Knebel G., Flouquet J. and Wilson J. A. 2006 cond-mat/0610166v1. [68] Fujimori A. 2001 Phys. Stat. Sol. (b) 233, 47 [69] Kwizera P. Dresselhaus M. S. and Adler D. 1980 Phys. Rev. B 21, 2328 [70] Fujii T., Tanaka K., Marumo F. and Noda Y. 1987 Miner. Journ. 13, 448 [71] Andersen O. K. and Jepsen O. 1984 Phys. Rev. Lett. 53, 2571 [72] Matsuura A. Y., Shen Z. -X., Dessau D. S., Park C. -H., Thio T., Bennet J. W. and Jepsen O. 1996 Phys. Rev. B 53, R7584

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Chapter 7

LEAKAGE CURRENT ON HIGH VOLTAGE CONTAMINATED INSULATORS P. T. Tsarabaris and C. G. Karagiannopoulos† National Technical University of Athens, School of Electrical and Computer Engineering, Athens, Greece

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ABSTRACT The present chapter will present a short review of research work regarding leakage current. In addition, measurement system of the leakage current is introduced using a high sampling frequency analogue/digital converter. The application of this system in 20 kV insulators contaminated from a compound of salt and kaolin are described. Using this system measurement in a time frame of one period (50 Hz) were done and an investigation of the observed phenomena is attempted with the assistance of i-u characteristic curves plotted for one cycle of voltage application. The fact that existence of partial discharges on the surface of the contaminated insulators, beyond threshold field intensity leads to radiation emission, is also be examined. An estimation of the free electrons energy has been done and the corresponding emitted radiation which seems to include acoustic waves, radio waves, microwaves and infrared waves is also examined. The classification of the leakage current values of a typical porcelain insulator of 20 kV, contaminated by salt and kaolin, is presented. The classification is based on the collaboration between the above high precision data acquisition system having high sampling rate and an unsupervised self-organized neural network. In addition a simulation model for contaminated insulators is presented. The proposed model will be provided, together with a mathematical function that simulates the behavior of the dry band resistance as a function of time, even in cases where arcs or partial discharges occur. The model‟s parameters of a typical porcelain insulator of 20 kV, contaminated by salt and kaolin, are presented.

A version of this chapter was also published in Electric Power Research Trends, edited by Michael C. Schmidt, published by Nova Science Publishers. It was submitted for appropriate modifications in an effort to encourage wider dissemination of research. † Email: [email protected] Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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1. INTRODUCTION The flashover is probably the major visible undesirable effect on an insulator, which results in power loss. This effect usually appears on insulators that operate under adverse conditions, like industrial areas, near the sea, in a desert or in snow [1-10]. The probability of a flashover to occur on a contaminated insulator depends on the level and nature of the insulator surfaces‟ contamination, and also on the wetting conditions [11]. The flashover is the result of a series of other effects, which occur in the following order: insulator contamination, wetting of contaminants, generation of a thick electrolytic solution, increase of the leakage current, increase of the temperature, formation of a dry band, occurrence of partial discharges, occurrence of arcs, and finally the flashover [1]. Research on this phenomenon has been carried out for many years by scientists all over the world. Test methods have been developed and many papers published. Researchers have been studying the flashover on contaminated insulators, especially from salt contaminants, and most of them have worked on the correlation of the voltage level at which the flashover occurs either with the number of units on the insulator string or with the equivalent pollution density [2, 6, 7, 11-15]. Other researchers have been working on the correlation of the precipitation rate with the equivalent salt deposit density [14]. A large part of the bibliography [3, 8, 9, 14, 16-20] concerns the leakage current. There, has been studied the leakage current of the contaminated insulator in relation to the salinity [8], the leakage current for salt pollution in relation with time[17], as well as the exposure time in a foggy place [14]. There has also been recorded the leakage current in insulators polluted by a combination of salt and desert dust [9], and also in insulators exposed in a coastal area having snow deposited on their surface [16]. Furthermore, a measurement method of the leakage current has been proposed, for the calculation of the rms value of the leakage current [3]. The form of interference in the radio and TV sets in relation with the form of the leakage current under DC [20] and AC [17, 18] voltage has also been examined.

2. LEAKAGE CURRENT MEASUREMENT AND RECORDING SYSTEMS In order to record and study the behaviour of parameters what are related with the contaminated insulator behaviour, many researchers have used various recording and monitoring systems. The leakage current, speed and direction of wind, relative humidity, environment and insulator‟s surface temperature, the rainfall and finally the conductivity of polluted are the parameters commonly recorded because it is accepted that these parameters has influence on the leakage current behaviour. The presentation of sensors used for the measurement of speed and direction of wind, relative humidity, environment and insulator‟s surface temperature, rainfall and conductivity of polluted surface is believed that doesn‟t fit the scope of this chapter. For this reason only the common used leakage current sensors would be presented. A leakage current recording system composes by the sensor and the recording unit. The systems used, among other, for the leakage current measurements are given in the following paragraphs.

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Habib et al [21] and Khalifa et al [22] have proposes a monitoring system which gives an alarm signal when the number of leakage current pulses exceeding a threshold value, exceeds a predetermined limit. The system uses a toroidal coil, a current pulses comparator, a counter, a pulses number comparator and finally a circuit for the production of the alarm signal. The toroidal coil, which operation is based on Hall effect, is used for the current pulses measurement. The pulse comparator is used for the pulse width comparison concerning a predetermined threshold (which varies from 45 mA until 1 A or even more). The counter measures the number of current pulses (for a time interval 4-30 min) which have exceeded the threshold. Then the comparator compares if the number of current pulses has exceeded the in advance determined number of pulses and if yes it gives command for alarm. Vlastos et al [23] have proposed a system for the current pulses recording. The system uses leakage current measurement sensors, a voltmeter for the measurement of applied voltage, as well as measurement instruments for speed and direction of wind, atmospheric pressure as well as rainfall. The outputs of above instruments are connected in a sampling system of 100 channels. The sampling speed is 30 Hz and each input is sampled once each 10 seconds roughly. The leakage current measurement sensors give as output the biggest price of current for all the time interval of sampling. Marrone et al [24] have proposed a recording system for the quantity of insulator surface pollution. The system is constituted by a surface conductivity recording system. Richards et al [25] have proposed a system of pollution recording in the insulator surface. The system is constituted by a leakage current recording system, as well as sensors for the measurement of relative humidity, the environment temperature, the temperature of insulator surface (radiation sensor), the speed and the direction of wind, the level of condensation and the quantity of rain. The current recording measures the biggest value of leakage current for each half- period of applied alternative voltage. Gilbert et al [26] have proposed a system of current pulses recording. The system uses a very small resistance (5 Ω) for the leakage current measurement, and an integrating circuit which integrates the current values over a fixed period (10 seconds). The integrated current values (each 10 seconds) are stored in memory. The system checks also the current values in the case, where their value exceeds, for three sequentially time intervals ( of 10 seconds), a predetermined limit, then the system produces a alarm signal for the insulators washing procedure start up. Thalassinakis et al [27] have proposed a leakage current monitoring system. The system is constituted by a leakage current sensor connected with a recorder. The sensor is a high accuracy Hall effect current transducer. The recorder incorporates an analogue-to-digital (A/D) converter and a microprocessor data acquisition system, specially designed to record the leakage current activity on nine different insulators. All captured data are stored in the system and can be retrieved via an RS232 serial port or via an external dial-up modem. The sampling rate of the A/D converter of the instrument is 20 kHz and the whole system accuracy is 0.5% of full scale. The unit is also equipped with 3 voltage inputs in order to record the three phase to ground voltages through appropriate voltage transformers, 160/√3 kV to 120/√3 V. The recording system is additionally equipped with weather sensors for recording humidity, temperature, wind-speed and rainfall. Williams et al [28] and Sebo et al[29] have proposed a leakage current measurement systems constituted by a measuring resistance and a digital storage oscilloscope. The leakage current was observed on the oscilloscope as voltage drop across the measuring resistance.

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Sorquist et al [30] have used a digital multimeter connected with a PC for the leakage current measurement. The multimeter is directly connected to the nonenergized size of the insulators. Fernardo et al [31] have proposed a leakage current measurement systems constituted by a shunt (varied from 2Ω to 4.7 kΩ), a digital storage oscilloscope and a PC. The leakage current were observed and saved on the oscilloscope as voltage drop across the shunt. Tsarabaris et al [32–36] have proposed a leakage current recording system. The measurement circuit is provided schematically in figure 1. The voltage elevation transformer is supplied by a Variac, thus resulting in a controlled gradual voltage increase. The transformer‟s secondary coil is connected in parallel with a capacitive voltage divider composed by capacitors C1 and C2 respectively. In parallel with the voltage divider the series connection of the insulator and the measurement resistance Rm is coupled. The voltage across C2 and Rm is measured with the use of an A/D converter. The data transfer from the A/D converter is achieved via an RS 232 connection to a computer where it is recorded and visually displayed with the use of appropriate software. The technical characteristics of the capacitors C1 and C2 were 224 pF, 100kV / 50 Hz and 2 μF, 1000V / 50 Hz respectively. The values of the above-mentioned capacitors determine the appropriate voltage division ratio, as the maximum voltage across C2 must not exceed the permitted input voltage of the A/D converter. In this case, this maximum value was ±5 V. The resistance Rm, has zero selfinduction and its maximum permitted power is considerably higher than the one imposed by the measured leakage current. The thermal factor a of the resistance was practically zero. Applying high voltage techniques the electromagnetic interference from the high voltage components was eliminated and with a careful screening, low induced voltages caused by stray magnetic fields, were minimised. The whole sampling system was specifically designed and released for measuring fast rising voltage profiles and was capable of detecting voltages as low as ±1mV with a typical error 0,1% when the input measured voltage was 1V. The employed 12 bit A/D converters were of successive approximation type with fast conversion rate of 0,020μsec (2x20 MSPS). The above system has successfully used to measure leakage current pulses.

Figure 1. Measurement circuit 1: variac, 2: high voltage transformer, C 1–C2: voltage divider, Rm: measuring resistance

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It results that the recording systems proposed among others concern mainly the leakage current, speed and direction of wind, relative humidity, environment and insulator‟s surface temperature, the rainfall and finally the conductivity of polluted. From these recordings useful conclusions can be exported regarding the form and the value of leakage current and the effect of factors such as the speed and the direction of wind, the relative humidity, the environment temperature, the rainfall and the conductivity of surface in the leakage current. Also conclusions can be exported regarding the effect of speed and the direction of wind, relative humidity, temperature of environment, and rainfall in the conductivity of surface. These conclusions can help on one side in the planning and the manufacture of the high voltage insulators and on the other side in the better understanding of mechanisms of pollution deposition, the leakage current behaviour and flashover.

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3. LEAKAGE CURRENT BEHAVIOUR In the extensive bibliography for the insulator behaviour, a number of researchers [4-6] for the case of porcelain or glass insulators have studied in detail the form of leakage current. Woodson and McElroy [4] have used photographic camera for the study of arcs on surface of polluted porcelain insulator. Their results showed that in the surface arcs are presented having radial direction. The duration of these arcs depends considerably from the amount of humidity on the pollution layer. For humidity density in the surface equal to 3 mg / cm 2 the time duration of arcs is varied between 0.5 and 1 seconds while for density equal to 16 mg / cm 2 time duration of arcs is varied between 2 and 6 seconds. The intensity of above arcs is of the order of tens or hundreds mA. Lambeth et al[6], studying the current behaviour during salt fog tests, found that the current waveform is constituted by regions where the current has very low value (almost zero) and from parts with sinusoidal form, which are reported as current pulses, the duration and the width of which depend considerably from the density of salinity contained in the pollution layer. The time duration of sinusoidal form parts is varied from 5 ms to 7 ms for salinity density variation from 5 kg / m 3 to 40 kg / m 3 . The corresponding width variation is roughly between 112.5 mA and 300 mA . The current study at the phase of ice fusion on ice covered insulators, showed that the current presents pulses of width of the order of 10 mA and duration roughly 50 seconds as well as pulses having width of the order of 450 mA and duration of roughly one min. The vibrations that had width of the order of 10 mA were attributed corona streamers while the pulses with considerably bigger width in the white arc occurrence [5] between a number of disks. According to bibliography [8] the leakage current during one period of time (50 Hz) is initially practically zero and retains a prominent value only close to the maximum of the applied voltage. It is also documented that its waveform always consists of pulses appearing randomly, without discrimination among the pulses [19]. It has been reported [31] that the leakage current may have sinusoidal form as well as small high-f spikes and long duration discharges. At the application of the system in the leakage current measurement on porcelain insulators of 20 kV, Tsarabaris et al [32] have observed that the leakage current waveform is

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constituted of linear regions as well as of small and high amplitude pulses. By observing the waveforms of the leakage current in relation with the applied voltage (figures 2a and 3a) as well as the corresponding i-u characteristic curves (figures 2c and 3b), we can distinguish that the leakage current is zero for values of the applied voltage near zero (figure 2b point A). This is expected, as the potential difference at the edges of the insulator is too low and thus the corresponding field is also low. The current rises linearly with the increase of the applied voltage up to a point B. This linearity is obvious in the linear part of the i-u characteristic curves and is represented by symbol I in figure 2b. This implies that in the A-B area conductivity probably appears at the surface of the insulator [21]. The conductivity can be probably attributed to the low conductivity of the solid insulating material of the insulator (porcelain), and also to the conductivity of the deposits on the insulator‟s surface. The deposits‟ conductivity can be attributed to the conductivity of the dense electrolytic solution (such as NaCl), which is formed among the substances of the insulator‟s surface pollution and the water concentrated on its surface. The ions of the diluted substance obtain higher mobility under the influence of the potent electric field, causing the appearance of the conductivity[21].

Figure 2. a,b) Waveforms of leakage current and applied voltage in relation to time. c) i-u characteristic curves which have been obtained after the elimination of time from the previous measurements Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Figure 3. a) Waveforms of leakage current and applied voltage in relation to time. b) i-u characteristic curves which have been obtained after the elimination of time from the previous measurements

There is a point in the leakage current characteristic, where the current increases instantaneously, obtaining the shape of high amplitude - negligible width, pulses. The pulses vary in number and amplitude among different periods (50 Hz). They don‟t appear periodically. The leakage current pulses which appear randomly, are assumed to be caused by partial discharges on the insulator‟s surface, as well as by arcs on the insulator‟s surface among different disks or across dry bands. This is rational because it is documented [22-25] that the occurrence of partial discharges is followed by quick current fluctuations of the order of a few tens of μsecs. Therefore, a dense spectrum of current pulses may imply that on the insulator‟ surface there was a high activity of partial discharges and arcs. Looking at the waveforms of the leakage current on Figure 2a and 3a, we can see that the current pulses can be split into three categories. In the first category belong pulses with amplitude up to 10 mA and with negligible width of the order of few μsec. In the second category belong pulses with amplitude over 10 mA and with negligible width as before. In the third category we can find pulses with amplitude over 10 mA and with significant width usually of the order of a few thousands of μsec. First category pulses are probably caused by the partial discharges, because their amplitude is small, in contradiction with the pulses of the other two categories, whose amplitude is quite higher and which are probably caused by arcs. Referring to the second and third category pulses, those which have the less width are caused by momentary arcs, whereas those which have significant width are caused by long duration arcs. Regarding the pulses caused by continuous arcs, they start close to the maximum value of the applied voltage. Thereafter their amplitude rises linearly up to a maximum value of approximately 40 mA. After that their amplitude decreases linearly with the applied voltage and becomes zero when the applied voltage is almost zero. This implies that on the insulator‟s

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surface long duration arcs existed which die out when the applied voltage reaches a rather low value. The third part of the leakage current half period waveform (area III of figure 2b) is similar to the first, i.e. leakage current responds almost linearly to the variation of voltage, due to the existence of conductivity on the insulator‟s surface.

4. RADIATION EMISSION It has been shown that ageing and penetration of solid insulators during their electrical strain under high voltage are explained by methods that can be classified into two theories [37, 38]: macroscopic theory and quantum mechanical theory

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According to the macroscopic theory, the efforts of research have been aimed at an equivalent electrical circuit based on the combination of electrical measurements with optical observations or changes in the surface of the material caused by ageing. From measurements taken, it was determined that there are 4 basic factors which contribute to the reduction in the molecular stability of the insulating material and to the change of its molecular structure. These factors are [37]: Joule losses Partial discharges Coulomb forces Ambient temperature The most important factor of ageing is considered to be partial discharges. By the term partial discharge we mean each local discharge of restricted length in the solid insulator. The lack of homogeneity in the volume of the solid insulators constitutes the centre of partial discharges. Furthermore, lack of homogeneity and contaminants on the insulator surface also constitute centers for the initiation of partial discharges. The quantum mechanical theory principally concerns the energy of free electrical carriers (electrons) which are produced during the ageing of the material. The lack of homogeneity within the solid insulator induces the appearance of disturbance bands between conductivity and strength bands, which, like intermediate energy levels, facilitate the transmission of electrons to the conductivity band [37, 38]. A determining factor in the creation of free electrical carriers is the value of the imposed field [37]. Above a threshold value of the electric field the electrons acquire increased kinetic energy, which they emit as electromagnetic radiation when they are absorbed at the anode [38, 39]. The spectrum of the radiation during insulator ageing ranges from acoustic waves to ultra-violet radiation. [3739].

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(b)

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Figure 4. Typical waveform of discharge current and the corresponding voltage applied in relation to time and the corresponding i-u characteristic curves obtained after the elimination of time from the previous measurements. a. (temperature = 20 oC, pollution = 0.1 mg/cm2, and relative humidity = 80%). b. (temperature = 20 oC, completely clean, and relative humidity = 80%)

During measurements acquired for an applied voltage of 20 / 3 kV in porcelain post insulator samples, having a leakage distance of 540 mm, and contamination (with a compound of salt and kaolin) approximately 0.1 mg/cm2, Tsarabaris et al [32] have found that one region of the leakage current waveform (around the point where the applied voltage obtains its maximum value), where rapid current fluctuations appear. They presumed that these fluctuations are generated by non-self-supported discharges [40], where the required electrical field for the production of free electrical carriers is provided by the field that has evolved across a dry band. In figure 4a someone can also notice that the current jump (starting of instabilities) takes place at a distinct threshold value of the applied voltage. This phenomenon has been observed in solid dielectrics and has extensively analyzed in the bibliography [40]. Specifically, it is widely accepted that among the quantities of primary interest, concerning electronic transport in solids, are the carrier mobilities, the majority carrier sign, the carrier deep trapping lifetime, the efficiency of photogeneration (the number of free carriers produced per absorbed photon), the surface and bulk recombination lifetime and the bulk conductivity [41]. In disordered insulating materials, these properties are generally strong functions of variables such as the applied electric field, the temperature and in some case the overall geometry. The current-voltage characteristics of dielectrics stressed by high electric fields is often characterised by current instabilities which may appear beyond a threshold field intensity [42]. A widely used physical model describing charge transport in dielectrics is based upon the presence of deep electronic states in their energy gap. It is believed that electron traps are induced in the band-gap of these materials as a result of various kinds of defects, e.g. chain folding and branching, carbonyl groups, cavities and entanglements, unsaturated bonds and molecular oxygen [43]. The current flow (in the low field region) has mainly been associated with electron hopping conduction and quantum mechanical tunneling between adjacent electron sites. For higher applied fields, electrons can gain enough energy to be excited towards wide energy bands allowing kinetic energies of the

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order of l0 eV, in which case impact ionization phenomena may become of primary importance [37, 44]. Theoretical studies, based on quantum mechanics, prove that for an electron hopping conduction process in a narrow impurity band [45], or for a low-density electron gas in random potential [46], the electric field will induce composite traps leading to exponential conductivity decay and N-type negative differential conductance (NDC). For even higher field values, the free electron concentration becomes larger than the trap density and electron-electron interaction increases their escape probability, changing N-type NDC to superlinear or even S-type NDC. Although the above theories were initially developed to explain ionization phenomena in high field domains of III-V semiconductors [47], they also appear to be in remarkable agreement with the data obtained experimentally so far [40] of a large area mechanically contacted MIM structure, when excited by high voltages. It seems that the dry solid contamination on the insulator‟s surface (dry band), behaves as a solid dielectric between two electrodes (conducting wet areas on the insulator‟s surface). This assumption is verified by the fact that non-linear i-u characteristics are not presented in figure 4b, which represent completely clean insulators. The current-voltage loop observed in Figure 4b can be attributed to the dry band capacitance. Besides, the hysteresis effect is frequently noticed on the i-u characteristics obtained to electrode gaps bridged by dielectrics or vacuum and it has been attributed to the capacitive nature of the structure or even to the production of photoelectrons by the electrodes or the insulator's surface [48]. According to the previously mentioned model, during the free electrical carriers‟ incidence onto the positive electrode, their kinetic energy is given in the form of radiation. This energy is given by the following expression [40]:

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W

Δu 2

(eV)

(1)

where

Δu Δi R

(2)

By substituting the term Δu into equation (1) we obtain the following expression: W

R

Δi 2

(eV)

(3)

The emitted radiation energy, in the case of figure 4a‟s non-self-supporting discharges, is given in the histogram in figure 5. From the calculated emitted radiation energy values, it results that the range during the non-self-supported discharges mentioned above includes acoustic waves, radio waves, microwaves and infrared waves. This range is in accordance with the range of the radiation emitted in the case of partial discharges. Consequently, the fluctuations that appear in figure 4a‟s current waveform are probably due to partial discharges. They could probably take place on the insulator‟s surface and/or in the bulk of the insulator‟s material. But by observing figure 4b‟s current waveform, we can see that no fluctuations appear. This reasonably implies that partial discharges do not take place in the bulk of the insulator‟s material.

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Figure 5. Emitted radiation energy in the case of figure‟s 4a non-self-supporting discharges

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5. CLASSIFICATION OF LEAKAGE CURRENT USING NEURAL NETWORKS The application of neural networks in the field of the insulators has been reported a few years ago. Various systems have been developed for the forecasting of the flashover [49-51], for the analysis of the surface tracking on solid insulators [52], for the identification of faulty insulators using corona discharge analysis [53], for the modeling of partial discharges on epoxy resin post insulators [54], for the inspection of the insulators on a string based on the texture analysis of the images of the string‟s units [55],for the modeling of partial discharge inception and extinction voltages of sheet samples of solid insulating materials [56], for the ESDD modeling[57] and for the forecasting of number and the place of faults due to insulators contamination in a distribution network [58]. Fernardo et al [31] have used neural networks for the evaluation of the harmonic contents (3rd and 5th harmonics) of leakage current waveforms of contaminated polymeric surfaces. The neural networks which have been used are based on the following training algorithms: back - propagation [31, 50-51, 54-58] and learning vector quantization [31]. From these algorithms, back - propagation requires supervision while learning vector quantization doesn‟t [59]. Regarding leakage current Fernardo et al [31] have used neural networks for the classification of the leakage current half-period waveforms on contaminated polymeric surfaces. Tsarabaris et al [35] have used neural networks for the classification of the leakage current pulses on contaminated insulators. The proposed neural network (NN) is based on the known from the bibliography as “self-organized feature mapping” which has been developed by Kohonen[59, 60]. The neural network consists of the input layer and the output layer. The input layer includes a single neuron. The output layer is one-dimensional and consists of a variable number of neurons. The single neuron in the input layer corresponds to anyone of the input patterns. Each of the output layer neurons corresponds to each class. The output layer

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neurons‟ number is variable and is defined by the user each time the training procedure is executed. The Euclidean norm between the input vector and the weight vector of the corresponding neuron is used as the criterion for the selection of the winning neuron. The normalized values of momentary leakage current values have been considered to be the input patterns of the neural network. The normalization is carried out using the highest value among all samples of leakage current that are held for the training. A class is defined as a fixed region of normalized values of leakage current momentary values which has clearly defined limits and which doesn‟t overlap another class. Since a normalized value of a leakage current momentary value corresponds to a single momentary value of the leakage current, the input patterns and the classes in which they are classified come from and univocally correspond to momentary values of the leakage current. Initially, a large number of measurements of leakage current momentary values were acquired using the system previously described [32-34]. The contamination of the insulator‟s surface was in the range of 0.1–0.14 mg/cm2 [61]. Those measurements, approximately 1.000.000 leakage current momentary values, were stored in file. These measurements correspond to approximately 100 periods of the leakage current waveform. Then the normalization of all samples of that file took place, and the normalized samples were stored in a new file. The neural network was trained using as input patterns the normalized patterns of the momentary leakage current values. A summary of all the training parameters of the network is given in Table 1. It is required for the network training that the number of output neurons is defined. This number expresses the different categories of the leakage current momentary values. By observing the waveforms of the leakage current, drawn from measurements, someone can see that no more than five categories are needed, whereas less than three would be insufficient. This admission will be checked below. For that reason the NN was trained separately for three, four and five output neurons. The rest of the parameters remained unchanged, and are given in Table 1. After the completion of each training process, the convergence weights that resulted were stored in a file. The above procedure was conducted in order to choose the appropriate number of output neurons. It decided [35] to adopt the “three output neurons” approach. Table 1. The neural network’s training parameters Number of neurons in the output layer Initial neighborhood Minimum neighborhood‟s Radius Initial weights Initial learning rate Amount to change learning rate Minimum learning rate Δwlimit Maximum number of epochs

Variable All output layer‟s neurons 0 Between 0 and 1 0.9 0.005 0.005 0.001 2000

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The limits that determine each class, actually determine which current values belong to each class. In this way, a categorization of the levels of the leakage current values is achieved. The case where a large number of classes derived would imply significant fluctuations in the current values. On the other hand, the existence of only one class would imply absence of such fluctuations. The leakage current waveforms seem to present regions with negligible, as well as with intense current fluctuations. The objective is to make a clear distinction between the current values that are provocative or non-provocative of fluctuations and if they are, to make a distinction among the various levels of fluctuation. So, the right determination of the limits of the classes is particularly important. In order to achieve our objective, the contribution of the neural network is considered of great importance. The proposed neural network is able to determine the precise limits of the classes and to do this dynamically. This means that if in the future we wish to enlarge the training patterns base, then the new training of the network would result in the recalculation of the classes‟ limits. From the analysis that was achieved with the use of the NN, it seems that there are three classes in which the leakage current values can be categorized, which correspond to three regions of values. Comparing the results with optical observations and observing one typical waveform of the leakage current (like the waveform in Figure 3a), we can see the existence of regions where the current presents very small fluctuations, as well as regions with significantly large fluctuations. In order to investigate this subject, we plotted Figure 6, in which typical forms of pulses are presented, as they resulted from the measurements that were performed. In this figure we can distinguish pulses of small and large amplitudes, as well as parts of the waveform where no fluctuations are present. Based on the above, it is possible to correlate the classes (in which the leakage current is classified), with the pulses that are present in the leakage current waveform. Based on the current value when a pulse occurs, we see that it is possible to discriminate the pulses into those of small and those of large amplitude. The classification results that class 2 represents pulses of small amplitude. Similarly, class 1 represents pulses of large amplitude. Class 3, represents the linear region of leakage current waveform (Figure 3) where no fluctuations appear.

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Pulses having high amplitude can be separated into two subcategories, one with short width and one with long width pulses. This neural network classifies both pulses in the same class. That was expected, as the classification was based on the amplitude of the leakage current with the same range of variation. Short and significant width pulses, are probably caused by momentary and long duration arcs, respectively, due to the large current value. Therefore, in order to discriminate clearly the pulses according to their width, other additional parameters of the leakage current waveform must be sought for, which characterize time duration. This can be a topic for further investigation. This NN in cooperation with the measurement system previously described (figure 1) can be used in the approximate calculation of the number of arcs and partial discharges that occur on the insulator‟s surface for a specific time interval. This can be achieved by calculating the mean value of the number of measured samples that are acquired throughout the duration of an arc or partial discharge, and from that the mean time of arc and partial discharge duration, since we know the sampling rate. From this calculation one can make assumptions about the insulator‟s deterioration during the presence of arcs and partial discharges, since in the area, in which an arc is created, erosion on the insulator‟s surface is observed [1] (due to the heat release). Moreover the system can be used in the calculation of power loss on each insulator, since both the momentary value of the leakage current and the momentary value of the applied voltage is known. The power loss can be calculated both for cases of individual arcs and partial discharges, and also during a specific time period of leakage current. The system can also be used as alarm device for the washing of insulators, which are in service in substations. This can be done either by only checking the existence of arcs, or by counting the number of arcs (exceeding a limit) for a given time period. Finally, the system can be applied for the study of leakage current in the case of insulators manufactured from other material (e.g. composite), where the phenomena probably differ, as well as for insulators which are in service in different high voltage networks (66 kV, 150 kV, 400 kV) placed in different environments, where the level and the kind of pollution varies.

6. MONTELLING In the bibliography [62-65], regarding the study of pollution flashover phenomenon measurements have became and various models for the insulator surface behaviour have been proposed. A review of the main mathematical models for pollution flashover has been done by Rizk [62]. As results from this review the polluted surface can be simulating from an arc in line with a resistance which represents the resistance of it‟s not short circuit part of the pollution layer. For the study of longrod insulator behaviour a model of composed layer of pollution (two layers of pollution in line where one corresponds in the pollution of the cylindrical form shank‟s and the other in the pollution of disks) was used. During the study of the resistance of pollution layer it was considered that in a small region round the points of inception and extinguishment of the arc the resistance of pollution layer is different from that of rest of layer. It has been proposed a new relation for the calculation of mean resistance of pollution layer per unit of creep length which takes into consideration the occurrence of multiple arcs. Improvement factors of the pollution layer conductivity have also been proposed.

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Farzaneh et all [63] and Farzaneh et al [64] using models, based on that of Obenaus [62], for the simulation of surfaces covered by ice and for the arc that is propagated on them, they proposed a model for the propagation of discharges on the insulator surface covered by ice under DC voltage. In their study they used triangular samples of ice. This model allows among others the resistance determination of the non short circuit part of ice. Farzaneh et al [65], used a model, similar with that of Obenaus, for the simulation under AC voltage of surfaces that is ice covered. Vosloo and Holtzhausen [66] recently proposed a model of an equivalent circuit of a contaminated insulator, which is based on daily measurements of the leakage current. Specifically, the leakage current waveform at which the daily maximum value occurs is stored. A theoretical model for flashover voltage forecasting has also been proposed [67], which takes into consideration the re-ignition characteristics of the dry band arcs. In the above models, the equivalent circuit is mainly composed of the resistance of the conductive layer on the contaminated insulator‟s surface, which appears to be partly short-circuited along an area, implying an arc occurrence [66-68]. Tsarabaris et al [36] have proposed a simulation model for contaminated insulators. Experimentally investigating the behaviour of an insulator, they have seen that, when the insulator is clean and has no pollution on its surface, it behaves as a capacitor. As pollution is deposited on its surface, without humidity, it still behaves almost as a capacitor. When, the humidity level of the pollution on its surface rises because of, for example, hoarfrost, mist, drizzles, and frost, then an electrolytic solution is formed. This results in the development of a leakage current. Consequently, at that time the insulator behaves as a capacitor in parallel with a resistance (Figure 7a). The leakage current through the conductive layer on the insulator‟s surface results in an increase of temperature near the path of the leakage current. This causes evaporation and therefore the formation of dry bands. In this case, the equivalent circuit shown in Figure 7b simulates the insulator. In this circuit C Db , R Db are the insulator‟s dielectric capacity and surface resistance, respectively, in the dry band region, while C1, C2 and R1 , R 2 are the insulator‟s dielectric capacities and surface resistances, respectively, in the leakage regions on either side of the dry band region. Dry bands tend to retain the leakage current and at the same time undergo stress from the high voltage drop along them. It has been suggested that the voltage, which a dry band is stressed with, is stabilized at a slightly lower value than the threshold required for discharge development in air. If this unstable equilibrium is perturbed then a discharge will appear along the length of the dry band and the current pulse intensity will be reduced by the resistance of the residue polluted surface. The extension of arcs along the length of the dry bands depends on many factors, mainly the non-uniformity of the surface pollution, the deposit's resistivity and the peak leakage current value [3,7,69-71]. The development of an arc is simulated by a small value parallel resistance R(t), valid for the period of the arc‟s duration. This resistance practically short-circuits the dry band resistance. Considering that not all the dry band area is short-circuited, although a considerable part is, the resistance R(t) for the period of the arc‟s duration must be the resistance of the non-short-circuited section of the dry band. For the time when no arcs occur, R(t) must be large, so that its effect in the equivalent circuit will be practically negligible.

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Figure 7. Equivalent circuit of a polluted insulator: a) Before the formation of dry bands. b) After the formation of a dry band. c) The proposed model

From the above, the equivalent circuit of the insulator is as shown in Figure 7c, where Ceq and R eq are the capacitance and resistance of the insulator not involving the dry band

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and C Db and R Db are the capacitance and resistance of the dry band region, respectively. On the insulator’s surface and specifically on the dry band not subject to arcing, partial discharges also occur. The effect of partial discharges on the equivalent circuit of the insulator is also represented by R(t), which short-circuits for this occasion a rather small part of the dry band resistance. Because not all dry band is short-circuited, but only a part, R(t) at the time when partial discharges occur represents the non-short-circuited part of the dry band. Resistance R(t) that short-circuits a small or considerable part of the dry band, due to partial discharges or arcs respectively, is a non-linear time dependent resistance. For R(t) the following formula has been proposed [36]: 2 2 R (t )  K1 - [K1 - (rarc  r1)]  e a1*t - t1 -[K1 - (rpd  r 2)]  e a 2*t - t 2

(4)

where K1 is a constant, usually of the order of 1012 Ω, r1 and r2 are the resistances of the nonshort-circuited part of the dry band in the case of an arc or a partial discharge respectively, and rpd , rarc are the resistances of the partial discharges and arcs, respectively, when one of them is in progress. The constants a1 and a2 determine the time duration of the pulses, and have values of the order of 1014. Time parameters t1 and t2 correspond to the initial time when partial discharges and arcs occur, respectively. The model’s evaluation, using representative samples of 20 kV porcelain insulators having contamination on their surface in the range of 0.1–0.14 mg/cm2 , results that resistance

R eq was found to be of the order of 0.5 MΩ. Resistance R Db found to be of the order of 5 MΩ. Resistance r1 found to be of the order of 50 kΩ. Resistance r2 found to be of the order of 1.6 ΜΩ. The resistances rpd , rarc defined to be of the order of 10 Ω [72]. The total capacity

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of the insulator, C sum , derived from the bibliography [12] and from the measurements performed using a Schering Bridge, is of the order of 10 pF. The capacity C Db is of the order of 100 pF. and Ceq in the order of 11 pF. This model satisfactorily simulates 20 kV porcelain insulators subjected to partial discharges and arcs. To further improve the model, a study could be undertaken regarding the time partial discharges and arcs occur on the dry band. In addition, the behaviour of the proposed model could be studied for insulators installed in 150 kV and 450 kV networks.

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[6] [7]

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[14] Fujimura, T; Naito, K; Irie, T. “Performance of semiconducting glaze insulators under adverse conditions”, IEEE Trans. Power Apparatus and Systems, vol. PAS-97, 1978, 763-771. [15] Kimoto, I; Fujimura, T; Naito, K. “Performance of insulators for DC transmission line under polluted conditions”, IEEE T72, 556-9, 1972. [16] Takasu, K; Shindo, T; Arai, N. “Natural contamination test of insulators with DC voltage energization at inland areas”, IEEE Trans. Power Delivery, 1988, vol. 3, 18471853. [17] Macchiaroli, B; Turner, FJ. “A new contamination test method”, IEEE Trans. Power Apparatus and Systems, 1969, vol. PAS-88, 1400-1411. [18] Bernardelli, PD; Cortina, R; Sforzini, M. “Laboratory investigation on the radio interference performance of insulators in different ambient conditions”, IEEE T72, 1972, 193-6. [19] Sawada, Y; Fukushima, M; Yasui, M; Kimoto, I; Naito, K. “A laboratory studu on RI, TVI and AN of insulator strings under contaminated condition”, IEEE Trans. Power Apparatus and Systems, vol. PAS-93, No 2, 1974, 712-719. [20] Fukushima, M; Sanaga, Y; Sasano, T; Sawada, Y. “AN, RI and TVI from single unit flashover of HVDC suspension insulator strings”, IEEE Trans. Power Apparatus and Systems, 1977, vol. PAS-96, 1233-1241. [21] Habib, SED; Khalifa, M. "A New Monitor for Pollution on Power Line Insulators - Part 1: Design, Construction and Preliminary- Tests", IEE Proc. Gen. Trans, and Distrib, 1986, Vol. 133, Pt. C, No. 1, 105-108. [22] Khalifa, M; El-Morshedy, A; Gouda, OE; Habib, SED. "A New Monitor for Pollution on Power Line Insulators • Part 2; Simulated Field Tests", IEE Proc. Gen. Trans, and Distrib, 1988, Vol. 135, Pt. C, No.1, 24-34. [23] Vlastos, A; Orbeck, T. “Outdoor leakage current monitoring of silicone composite insulators in coastal service conditions”, IEEE Transactions on Power Delivery, 1996, Vol. 11, No.2, 1066 -1070. [24] Marrone, G; Marinoni, F. “New apparatus set at ENEL to monitor pollution deposit and pilot cleaning operations on outdoor insulators”, Cigre, Paper, 1996, 33-302. [25] Richards, CN; Renowden, JD. “Development of a remote insulator contamination monitoring system”, IEEE Transactions on Power Delivery, 1997, Vol. 12, No.1, 389 397. [26] Gilbert, R; Gillespie, A; Eklund, A; Eriksson, J; Hartings, R; Jacobson, B. "Pollution flashovers on wall bushings at a coastal site", Cigre, Paper, 1998, 15-102. [27] Thalassinakis, E; Karagiannopoulos, CG. “Measurements and interpretations concerning leakage currents on polluted high voltage insulators”, Meas.Sci.Technol, 2003, Vol.14, 421-426. [28] Williams, DL; Haddad, A; Rowlands, AR; Young, HM; Waters, RT. “Formation and Characterization of Dry Bands in Clean Fog on Polluted Insulators”, IEEE Trans.on DEI, 1999, Vol. 6, No. 5, 724-731. [29] Sebo, SA; Zhao, T. “Utilization of Fog Chambers for Non-ceramic Outdoor Insulator Evaluation”, IEEE Trans.on DEI, 1999, Vol .6 No. 5, 676-686. [30] Sοrqvist, T; Gubanski, SM. “Leakage Current and Flashover of Field-aged Polymeric Insulators”, IEEE Trans.on DEI, 1999, Vol. 6 No. 5, 744-753.

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[31] Fernando, MARM; Gubanski, SM. “Leakage Currents Patterns on Contaminated Polymeric Surfaces”, IEEE Trans.on DEI , 1999, Vol .6 No. 5, 688-694. [32] Tsarabaris, PT; Karagiannopoulos, CG; Bourkas, PD; Theodorou, NJ. “An experimental investigation of leakage current on high voltage contaminated insulators”, Iranian J. Electrical and Computer Engineering, 2003,, Vol. 2, No. 1, 30-34. [33] Tsarabaris, PT; Karagiannopoulos, CG; Polikrati, AD; Paisios, MP; Theodorou, NJ. “An investigation regarding leakage current on hight voltage non-ceramic insulators”, 3d Int. Conf. Power and Energy Systems (EuroPES 2003), Marbella, Spain, 3-5 September 2003, 717-720. [34] Tsarabaris, PT; Halaris, PG; Polykrati, AD; Karagiannopoulos, CG; Bourkas, PD. “Radiation Emission Phenomena in 20 kV High Voltage Porcelain Insulators”, 4th Int. Conf. Power and Energy Systems (EuroPES 2004), Rhodes, Greece, 28-30 June 2004, 512-515. [35] Tsarabaris, PT; Karagiannopoulos, CG; Bourkas, PD; Theodorou, NJ. “A classification of the leakage current pulses of high voltage contaminated porcelain insulators using a self-organised neural network”, Int. J. Engineering Intelligent Systems, 2006, Vol. 14, No. 1. [36] Tsarabaris, PT; Karagiannopoulos, CG; Theodorou, NJ. “A model for high voltage polluted insulators suffering arcs and partial discharges”, Journal of Simulation Modeling Practice and Theory, 2005, Vol.13, No.2, 157-167. [37] Zeller, HR. “Breakdown and pre-breakdown phenomena in solid dielectrics”, IEEE Trans. Electrical Insulation, 1987, vol. EI-22, 115-122. [38] Bourkas, PD; Kayafas, EA; Dervos, C; Stathopulos, IA.“Eine mogliche Erklarung der Glimmentladungen in festen Isolirstoffen bei Stosspannungsbeanspruchung“, Etz- Archiv Bd, 1989, 11, H.5, s. 163-165. [39] Bourkas, PD; Kayafas, EA; Dervos, C; Stathopulos, IA. “Enhanced partial discharges due to temperature increase in the combined system of a solid-liquid dielectric”, IEEE Trans. Electrical Insulation, 1989, vol. EI-25, 469-474. [40] Bourkas, PD. High voltage applications, NTUA Publication, Athens, 1996. [41] J; Mort, G. Pfister, Editors: “Electronic Properties of Polymers”, Wiley/Interscience, New York, 1982. [42] Dervos, C; Bourkas, PD; Kayafas, EA. “High-frequency current oscillations in solid dielectrics”, J. Phys. D: Appl. Phys., 1989, Vol. 22, No.2, 316-322. [43] Lowell, J; Rose-Innes, AC. “Contact electrification”, Advances in Phys., 1980, Vol. 29, No.6, 947-1023. [44] Sparks, M; Mills, DL; Warren, R; Holstein, T; Maradudin, AA; Sham, LJ; Loh, E; King, DF. “Theory of electron-avalanche breakdown in solids”, Phys. Rev. B., 1981, Vol.24, No.6, 3519- 3536. [45] Nguyen Van Lien and Shklovskii, BI. “Hopping conduction in strong electric fields and direct percolation”, Solid State Commun., 1981, Vol.38, 99-102. [46] ELL Levin and Shklovskii BL. “Negative differential conductivity of a low density electron gas in random potential”, Solid State Commun., 1988, Vol. 67, No. 3, 233-237. [47] Gelmont, BL; Shur, MS. “S-type current-voltage characteristic in Gunn diodes”, J. Phys.TfcAppL Phys., V, 1973, 842-850.

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[48] Jaitly, NC; Sudarshan, TS. “X-ray emission and pre-breakdown currents in plain and dielectric bridged vacuum gaps under DC excitation”, IEEE Trans. Electr. Insul, 1988, Vol.23, No.2, 231-242. [49] Cline, P; Lannes, W; Richards, G. “Use of pollution monitors with a neural network to predict insulator Flashover”, Electric Power Systems Research, 1997, Vol. 42, 27-33. [50] Ghosh, PS; Chakravorti, S; Chatterjee, N. “Estimation of Time to Flashover Characteristics of Contaminated Electrolytic Surfaces using a Neural Network”, IEEE Trans.on DEI, 1995, Vol.2, No.6, 1064-1074. [51] Lavalle, MM; Ortiz, GR. “Flashover Forecasting on HV Insulators with a Back Propagation Neural Net”, Can. J. Elec. And Comp. Eng., 1996, Vol.21 No.1, 29-32. [52] Ugur, M; Auckland, W; Varlow, BR; Emin, Z. “Neural Networks to Analyze Surface Tracking on Solid Insulators”, IEEE Trans.on DEI, 1997, Vol.4, No.6, 763-766. [53] Li, CR; Shi, Q; Cheng, YC; Yu, C; Yichao, Y. “Identification of Faulty Insulators by Using Corona Discharge Analysis Based on Artificial Neural Network”, Proc. Int. Symp. on EI, Arlington, USA, 1998, 382-385. [54] Ghosh, S; Kishore, NK. “Modeling PD Inception Voltage of Epoxy Resin Post Insulators using an Adaptive Neural Network”, IEEE Trans.on DEI, 1999, Vol.6, No.1, 131-134. [55] Hamed, M; El Desouky, A. “A computerized inspection for the high voltage insulating surfaces”, Electric Power Systems Research, 2000, Vol.53, 91-95. [56] Ghosh, S; Kishore, NK. “Modeling partial discharge inception and extinction voltages of sheet samples of solid insulating materials using an artificial Neural Network”, IEE Proc.-Sci. Meas. Technol., 2002, Vol.149, No.2, 73-78. [57] Arabani, MP; Shirani, AS. “An artificial neural network modeling of esdd”, CIGRE Paper, 1994, 33-101. [58] Tsanakas, AD; Papaefthimiou, GI; Agoris, DP. " Pollution flashover fault analysis and forecasting using neural networks", Cigre, 2002, 15-105. [59] Haykin, S. “Neural Networks- A Comprehensive Foundation”, Prentice-Hall, New Jersey, 1994. [60] Pandya, AS; Macy, RB. “Pattern Recognition with Neural Networks in C++”, CRC Press / IEEE Press, Boca Raton, 1996. [61] IEC, 60507: “Artificial pollution tests on high-voltage insulators to be used on a.c. systems”, Geneva, 1991. [62] Rizk, FAM. “ Mathematical models for pollution flashover”, Electra, 1981, No.78, 71103. [63] Farzaneh, M; Zhang, J; Chen, X. “Modeling of the AC arc discharge on ice surfaces”, IEEE Transactions on Power Delivery, 1997, Vol.12, No.1, 325-338. [64] Farzaneh, M; Zhang, J. “ Modelling of DC arc discharge on ice surfaces”, IEE Proc.Gener, Transm. Distrib., 2000, Vol. 147, No.2, 81-86. [65] Farzaneh, M; Fofana, I; Tavakoli, C; Chen, X. “Dynamic modeling of dc arc discharge on ice surfaces”, IEEE Trans. DEI, 2003, Vol. 10, No. 3. [66] Rizk, FAM; Nguyen, DH. “Ac source interaction in HV pollution tests”, IEEE Trans. on Power Apparatus and Systems, 1984, Vol. PAS-103, No. 4, 723-732. [67] Holtzhausen, JP. “Application of a reignition pollution flashover to Cap and Pin insulator strings”, Proc. 6th IASTED International Conference: Power and Energy Systems, Rhodes - Greece, 2001, 411-415.

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[68] Vosloo, WL; Holtzhausen, JP. “A model for electrical discharge and leakage current development on high voltage insulators”, Proc. 2nd IASTED International Conference: Power and Energy Systems, Crete-Greece, 2002, 640-643. [69] CIGRE WG 33-04 Taskforce 01: “A review of current knowledge: Polluted Insulators”, 1998. [70] Mizuno, Y; Nakamura, H; Naito, K. “Dynamic simulation of risk of flashover of contaminated ceramic insulators”, IEEE Trans. on Power Delivery, Vol.12, No. 3, 1997, 1292-1298. [71] Rahal, A; Huraux, C. “Flashover mechanism of high voltage insulators”, IEEE Trans. Power Apparatus and Systems, 1979, Vol. PAS- 98, No. 6, 2223-2231. [72] Psomopoulos, CS; Karagiannopoulos, CG. “Measurement of Fusible elements during current interruption and interpretation of related phenomena”, Measurement, 2002, Vol. 32, 15-22.

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Chapter 8

SYNTHESIS AND STUDY OF NANOSCALE MAGNETIC SEMICONDUCTOR AND MAGNETIC METAL/INSULATOR FILMS: ROLE OF ENERGETIC IONS S. Ghosh Nanostech Laboratory, Indian Institute of Technology, New Delhi, India

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ABSTRACT Nanoscale magnetic semiconductors and magnetic metal/insulator nanogranular films are under significant focus in materials science research because of their promising applications in spin-mediated devices. The upcoming technology where these materials are going to be used are, spintronics, opto-spintronics, data storage and sensors. Better performance of the devices depends on proper synthesis of these materials and engineering their properties. Ions with different energy ranges (eV – keV- MeV) play an important role in synthesis of these materials as well as modification of their properties for better performance. This review is divided in three sections. In the first section, a comprehensive review on diluted magnetic semiconductors (DMSs), transparent magnetic semiconductors (TMSs), their importance, mechanism behind ferromagnetism, ZnO based DMSs and importance of different magnetic metal-insulator nanocomposite is given. Role of energetic ions in each case is highlighted. The second section deals with experimental studies on synthesis and characterization of Ni implanted/doped ZnO films in the perspective of DMS/TMS. The results are discussed on the basis of carrier mediated exchange interaction and bound polaron model. In the third section, results on Ni-SiO2 nanocomposite films grown by atom beam sputtering technique are discussed. An attempt has been made to correlate microstructure, composition and magnetic properties of these nanocomposite films. Finally, the results obtained in these studies are summarized and future research scopes are highlighted. A version of this chapter was also published in Synthesis and Engineering of Nanostructures by Energetic Ions edited by Devesh Kumar Avasthi and Jean Claude Pivin published by Nova Science Publishers. It was submitted for appropriate modifications in an effort to encourage wider dissemination of research. Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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1. INTRODUCTION Transition metal (TM) doped nanoscale zinc oxide (ZnO) films as a diluted magnetic semiconductor (DMS) and TM-insulator nanocomposite films have been attracted great research attention because of their promising applications in future spintronics, optospintronics, data storage, high frequency etc. devices. Ions of different energy scales play a crucial role in synthesis as well as modification of these materials. Low energy ion beam sputtering and ion implantation have been routinely used for synthesis of DMS as well as nanocomposites. Ions with mega electron volt energy or swift heavy ions (SHI) have been used to engineer the properties of these materials down to nanoscale. This section of this review article deals with a brief survey of DMS materials, mechanism of ferromagnetism (FM) in these materials, ZnO based DMSs followed by important aspects of metal-insulator nanocomposite films and their importance. Role of energetic ions is also highlighted in each subsection.

1.1. Zinc Oxide Based Diluted Magnetic Semiconductor

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In this subsection, we initially discuss different DMS materials, mechanism behind ferromagnetism followed by different studies on ZnO based DMSs and role of energetic ions in this field.

1.1.1. Diluted magnetic semiconductor: Material aspects and importance The present day semiconductor electronic and optoelectronic devices utilize the charge degree of freedom in their wide functionality in signal processing or light emission. The future semiconductor spintronic and optospintronic devices exploit the spin degree of freedom of charge carrier in new generation of transistors, lasers and integrated magnetic sensors. The ability to control spin injection, transfer and detection with high efficiency demand their utilities in ultra-low power high speed memory, logic and photonic devices. For practical purpose, spintronic devices need to use semiconductors those maintain their magnetic properties at room temperature or above. This is a challenge, because most magnetic semiconductors lose their magnetic properties at temperatures well below room temperature and would require expensive and impractical refrigeration in order to use in practical applications. The major criteria for selection of semiconductor spintronics materials are (i) existence of ferromagnetism above 300 K, (ii) capability to tune the ferromagnetic properties, like saturation magnetisation (Ms), coercivity (Hc), remnant magnetisation (Mr) by changing dopants, their concentration, physical properties of the materials, micro/nanostructures etc. and (iii) integration in future device with much ease. Scientific venture on transition metal (TM) doped wide band gap semiconductors (like GaAs, ZnO, GaN) have attracted significant research attention due to their promising characteristics, which satisfy the criteria mentioned above. Among various wide band gap semiconducting materials, transition metal doped ZnO became the most extensively studied materials, since the prediction by Dietl at al. [1] based on mean field theory, indicate them promising candidate to realize a diluted magnetic semiconducting (DMS) material, with Curie temperature above room temperature. Another importance of these materials is that they can

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be readily incorporated in the existing semiconducting heterostructure system where a number of optical and electronic devices could be realized. The magnetic behaviour of such materials depends upon the concentration of TM ions in the crystal, the carrier density and the crystal quality. Generally, when 3d transition metal ions are substituted for the cations of the host, the resultant electronic structure is influenced by strong hybridazation of the 3d orbitals of the magnetic ion and mainly the p orbitals of the neighbouring host anions. There are two interacting subsystems of DMS materials, namely the delocalized conduction band electrons and valance band holes and the randomly localized magnetic moments associated with the magnetic atoms. For practical application in spintronic devices, a high Curie temperature (TC) is one of the most important issues of research and for proper design of these materials the mechanism behind ferromagnetism must be clearly understood.

1.1.2. Mechanism behind ferromagnetism Besides all experimental studies, various theoretical approaches have been taken under consideration to understand the ferromagnetism in these materials. The theory dealing with ferromagnetism driven by the exchange interaction between charge carriers and localized magnetic moments was first proposed by Zener [2]. The features of DMS are induced by the exchange interaction between d-shell electrons of the magnetic ions and the delocalized band carrier states (s or p origin). In recent days a large number of models, based on mean field theory, first principle calculation and bound magneton polaron etc. [3] have been proposed to explain the exchange interaction and the experimental results, although each has its own limitation. Broadly, ferromagnetism in ZnO based DMSs can be explained by two models. In case of materials having lower electrical resistance, charge carrier mediated exchange interaction fits well, whereas materials having higher resistance, model based on bound magnetic polaron (BMP) gives a proper explanation. BMP forms by the alignment of the spins of many transition metal ions with that of much lower number of weakly bound carriers such as excitons within a polaron radius. Formation of BMP is illustrated in Figure 1.1.

Figure 1.1. Formation of bound magnetic polaron and their ovelap leading to FM ordering. Circle represents the cation site, while square represents O-vacancy Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Due to the complexity of the DMS systems, based on ZnO and especially, the possible presence of secondary phase precipitates, it is still difficult to find an universal theory. Very recently, strong influence of microstructure on RT-FM of TM doped ZnO has been demonstrated [4], which adds another new aspect to this problem. The effect of crystallite size and overall morphology on RT-FM of Ni doped nanodimensional ZnO film is observed by our group recently [5]. This particular effect needs to be addressed in theory and therefore requires detailed investigations with more planned experiments. At this stage, it would be fair to state that the models on TM doped ZnO and GaN is still in the stage of its infancy and it is too early to give a definitive description of the exact mechanism, governing the experimental observations regarding the origin of reported magnetization behaviour.

1.1.3. Important experimental results of ZnO based DMS The research impact of ferromagnetic properties of transition metal doped ZnO has gained significant status with various promising results [6]. Values of TC above room temperature have been reported in Co doped ZnO films [7] Ando et al. [8] reported a large magneto optical effect in Zn1-xCoxO thin films as measured by magnetic circular dichroism (MCD) spectra, suggesting the suitability of this material as DMS. Ferromagnetic phase was obtained in Co implanted ZnO at lower temperature (~5K) by Norton et al. [9]. Recently, above room temperature ferromagnetism (RT-FM) is also observed in Co implanted ZnO films grown on sapphire substrate. Fukumura and co workers first reported [10] the ZnMnO grown by PLD upto 35% Mn in ZnO matrix, without degradation of crystal quality, a distinct contrast with respect to III-V semiconductor based DMS. Kundaliya et al. [11] reported that the observed ferromagnetism is due to a metastable phase (oxygen vacancy stabilized Mn2-xZnxO3- ) rather than by carrier induced interactions between separated Mn atoms in ZnO. Similarly, Ni and Fe doped ZnO prepared by chemical methods, physical vapour deposition and ion implantation also show promising RT-FM behaviour [6,12]. 1.1.4. Role of energetic ions Ion implantation in ZnO has been established as an important synthesis route to achieve ferromagnetism [5,12]. FM in ion implanted ZnO arises either by charge carriers mediated interaction generated due to ion induced defects or by phase segregated metals. In case of intrinsic FM in ion implanted ZnO films, oxygen vacancy plays an important role as it acts as a source of charge carriers. In case of nanodimensional film, micro/nano structure has significant influence over exchange interaction and can be utilized to tune ferromagnetic properties [4,5,13]. Apart from the questions related to the origin of FM in this material, combination of RT-FM with high optical transmission in visible region has also been a subject of important investigation as these materials demand potential applications in magneto optical devices [14]. SHI due to its large energy deposition capability in materials, has been used to dissolve clustered transition metals in ZnO matrix, leading to room temperature ferromagnetic property [15].

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1.2. Metal-Insulator Nanophase Composite Films

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In this subsection we discuss important materials aspects of different metal-insulator nanocomposite films and role of energetic ions in this research area.

1.2.1. Material aspects and Importance Magnetic granular films [16,17], consisting of nanoscale magnetic particles such as Fe, Co, Ni and of their alloys embedded in an insulating matrix, for example Al2O3 and SiO2 etc. have received much attention due to their unusual physical properties, and increasing possibilities for use in technological applications, such as reading heads, high frequency applications, sensors and high density magnetic recording [18]. These granular materials exhibit peculiar magnetic and magneto transport properties, like enhanced coercivity, superparamagnetism, giant tunnel magnetoresistance (GTMR) giant hall effect (GHE) and in many cases excellent soft magnetic properties [19]. All these properties lead to promising application of these materials as data-storage media and high frequency devices. GTMR originates from spin dependent tunnelling between magnetic metal granules through insulating intergranules. Besides, magnetic particles and clusters in nanometric size posses an increasing importance in quantum computing and as diagnostic and therapeutic in medical tools and other life sciences. The new phenomena in nanocomposites arise due to interplay between the intrinsic properties, size distribution of nanoparticles, finite size effects and the interparticle interactions. For an appreciable degree of coherence of inter particle interactions, long range domain structure is likely to form even much below the percolation threshold. All these properties can be controlled by varying the particle size and interoparticle distance. Overall, there are four major issues related in this area of research, which are (i) reduction of particle size for increasing their number density, (ii) uniform size distribution, (iii) better control over volume fraction of the magnetic particles and (iv) developing new techniques for synthesis and design. 1.2.2. Role of energetic ions Energetic ions so far have played a significant role to engineer the properties of these nanocomposite films. C. D. Orleans et al. [20] have shown formation of spherical Co nanoparticle in SiO2 film by 160 keV Co ion implantation, followed by an elongation of these particles under 200 MeV I ion irradiation. This induces an anisotropy in the magnetic properties of the nanoparticles as examined by SQUID magnetometry. In a work by J. C. Pivin et al. [21], It has been found that the shape of the precipitation kinetics of Fe particles in the silicon sub oxide as a function of the ion fluence, or of the average energy transferred per unit length, indicates that nucleation requires a threshold energy density per unit volume. The threshold value of specific energy loss indicated in this work is ~ 13 keV/nm and falls in the electronic energy loss regime [22]. Same authors have also reported „hammering‟ effect by swift heavy ions on silica matrix causing a tilt of easy magnetization axes when fraction of metal particles is only a few percent. Recently, Shirai et al. [23] have studied irradiation 210 MeV Xe ions in FePt nanogranular films, (FePt)47(Al2O3)53 using advanced TEM including tomography. Ion irradiation induced coarsening of FePt nanoparticles with elongation along beam direction. At very high fluence (~ 5 x 1014 ions/cm2) well coarsened FePt balls have been formed on the irradiated surface, whereas the particles in the film interior have been

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deformed into rods along the ion trajectory. Irradiation on similar system (FePt nanoparticles embedded in silica films ) by 120 MeV Au ion leads to elongation of the particles and perpendicular magnetization [24]. The enhancement of coercivity of the particles perpendicular to the films surface is favourable to the perpendicular magnetic recording at high density.

2. SYNTHESIS AND STUDY OF NI DOPED/IMPLANTED ZNO FILMS

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In this section, we discuss some of our results on room temperature ferromagnetism and transparent ferromagnetism in Ni doped/implanted ZnO films. The experimental tools used for these studies are following: Crystalline phases of the films were identified by Glancing angle X-Ray Diffraction (GAXRD) technique, with CuK x-ray radiation (λ = 1.54 Å). To identify the chemical states of the elements present in the surface of the films, X-ray photoelectron spectroscopy (XPS) measurements were carried out using Mg K radiation (h =1253.6 eV) after etching with argon ions (Energy = 5 keV) for 1 min. The C1s peak at 284.6 eV was used, as a reference for correction of the shifts, due to charging effect. The microstructure of the films was investigated by transmission electron microscopy (TEM) associated with selected area diffraction (SAED), using a 200 kV TECNAI G2 20 microscope. The magnetic characterization was performed using a MPMS SQUID magnetometer. Some implanted films (ZnO/Si by VPT) were also examined by alternating gradient magnetometry (AGM), at room temperature. The resistivity ( ) and the carrier concentration of the films were measured by van der Pauw method. Optical transmittance across UV-Vis region has been studied by HITACHI UV-Vis spectrophotometer. Results obtained from these studies are discussed in the following two subsections.

2.1. Room Temperature Transparent Ferromagnetism in Ni Doped ZnO Films This subsection deals with synthesis of pure ZnO and Ni doped ZnO films grown on quartz substrates by fast atom beam (FAB) sputtering technique, and study their magnetic and optical properties. Fast atom beam sputtering technique has been demonstrated as a potential synthesis route to grow metal-insulator composite and Ni doped ZnO films down to nanoscale [25,26]. Details of film deposition processes are described in our works [26, 27]. Figure 2.1 (a) and (b) shows M versus H and optical transmittance across UV-VIS range in sputtered Ni doped ZnO film. It is clear from the figures that the film is ferromagnetic at room temperature with high optical transmittance (~ 80%). M-H studies at various temperature shows that saturation magnetization (Ms) value [Table 2.1] decreases between 15 and 5 K. This is attributed to the thermal agitation effect at higher temperature. However, Hc is the highest at 15 K [Table 2.1] and is attributed to the thermally activated magnetization reversal, involving domain wall pinning [27].

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Figure 2.1. (a) M-H curves of Ni doped ZnO film grown by atom beam sputtering technique indicating RT-FM property 100

Transmittance (%)

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80 Pure ZnO ZnO:Ni

60

40

20

0 300

400

500

600

700

800

Wavelength (nm)

Figure 2.1. (b) Optical transmittance versus wavelength of the undoped ZnO and Ni doped ZnO films indicating ~ 80% transmission

Table 2.1. Saturation magnetization (Ms) and coercivity (Hc) values at 5, 15 and 300 K of Ni doped ZnO film grown by atom beam sputtering technique Temperature 5K 15 K 300 K

Ms (emu/gm) 6.22 5.32 4.73

Hc (Oe) 192 310 100

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2.2. Studies on DMS Properties of 200 Kev Ni+2 Ion Implanted ZnO Films Zinc oxide films deposited by vapor phase transport (VPT) and pulsed laser deposition (PLD) were used for implantation with 200 keV Ni2+ ions using LEIBF facility of IUAC, New Delhi. Incident ion fluence (incident ions/cm2) were varied to obtain 2-7 at.% of Ni in ZnO. In case of Ni implanted ZnO/Si films grown by vapor phase transport method clear ferromagnetic phase is obtained, at room temperature. A typical M-H plot of the sample irradiated with fluence 8 x 1015 ions/cm2 is shown in Figure 2.2. It has also been observed that the FM strength (in terms of Ms) is maximum at this fluence (an intermediate fluence), which corresponds to 3 at. % of Ni in the film. The Ms value decreases with the increase in fluence. Similar result is also obtained in case of Ni implanted ZnO/sapphire films, grown by pulsed laser deposition. Maximum Ms value with high optical transmission (~ 80%) is observed in the film having 3 at.% Ni films. The room temperature FM and transmittance versus wavelength corresponding to this film are shown in Figure 2.3 (a) and (b) respectively. According to Zener model, direct superexchange between the magnetic ions is not possible but is mediated by carriers mediation is. It is supposed that FM is induced by the exchange interaction between localized d shell electrons of the magnetic ions and the delocalized band carrier states (s or p origin). In the Ni implanted ZnO films, we have seen that FM strength increases with decrease in conductivity and increase in charge carrier density, which is a clear indication that such exchange interaction plays an important role in RT-FM properties. The enhancement of charge carrier density is correlated with generation of defects, like O-vacancies, which has been quantified by X-ray photoelectron spectroscopy. A typical O1s spectrum of Ni implanted is shown in Figure 2.4. It is fitted by three Gaussian. The region marked as Ob represents O-vacancy mediated peak [28] and the area under this peak gives the total O-vacancies.

Figure 2.2. M-H plot of the ZnO/Si film irradiated with 200 keV Ni2+ ion with fluence 8 x 1015 ions/cm2 (3% Ni ZnO film)

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Figure 2.3. (a) M-H curve of 200 keV Ni (~3%) implanted ZnO film grown by PLD technique at room temperature, indicating RT-FM property

Figure2.3. (b) Optical transmittance versus wavelength of the same film indicating ~ 80% transmittance

Figure 2.4. A typical O1s spectrum of Ni implanted ZnO film. The peak marked as Ob represents Ovacancy Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Figure 2.5. (a), (b) and (c) AFM micrographs of the Ni (2%, 3% and 7%) implanted ZnO films, indicating enhancement of grain size from 2 to 3% followed by decrement of grain size in the film with highest fluence

An influence of film microstructure and overall morphology over exchange interaction has also been shown in case of VPT grown films. In case of a film, having smaller grains (for highest Ni concentration), trapping of charge carriers at grain boundaries can reduce the exchange interaction probability and hence overall FM strength [4,5]. The surface morphology of the films are shown in the following AFM micrographs (Figures 2.5 (a), (b) and (c)): However, in case of ZnO:Ni/sapphire films, no systematic variation of FM with conductivity of the films are seen. The resistivity values as observed by four probe techniques are of the order of few tens of Ohm-cm and therefore, the charge carrier mediated exchange interaction may not be the probable cause of RT-FM properties of these films. Earlier, Kaminski and Sharma [29] theoretically studied the development of spontaneous magnetization in magnetic semiconductors, arising from percolation of bound magnetic polaron (BMP) and derived analytic agreement with the experimental results. Recently, BMP model based on the presence of defects, has been used to explain the intrinsic and extrinsic behavior of ZnO based diluted magnetic semiconductors [30]. It has been shown that ferromagnetic exchange is mediated through localized donor electrons in the impurity band derived from defect states. An electron, associated with a particular defect, is confined in a hydrogenic orbit. All the Ni ions within the polaronic radius interact ferromagnetically, shaping the BMPs, which try to spread out to overlap and interact with the adjacent BMPs to realize magnetic ordering, resulting in RT-FM. Ion implantation is a tool to create defects in solids [31,32]. Oxygen vacancies and defect complexes are observed in different materials and its role in oxides and their role in the modification of the physical properties is well emphasized [33]. Defect production is chiefly caused by nuclear stopping, i.e. elastic collisions between a recoiling ion and the atoms in the medium. When an energetic ion collides with an atom in a crystal lattice and departs enough energy to it, the lattice atom will collide with other lattice atoms, resulting in a large number of successive collisions. All the atomic collisions, initiated by a single ion are called a collision cascade. A collision cascade can be divided into three phases. The initial stage, during which atoms collide strongly, is called the collisional phase, and typically lasts about 0.1 - 1 ps. As a result of the collisions, one can assume that all atoms near the initial ion path are in thermal motion at a high temperature. The high temperature will spread and be reduced in the crystal by heat conduction. This phase is called the thermal spike, and lasts roughly ~1 ns. When the thermal spike has cooled down, there will usually be left a large quantity of defects in the crystal. The defects can be of several different shapes, ranging from vacancies and interstitial atoms to complex interstitial-dislocation loops and volume defects.

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Generation of O-vacancies due to Ni implanted ZnO films from XPS analysis have already shown [5]. O-vacancy acts as an electron donor in case of ZnO and hence plays a vital role in exchange interaction mediated RT-FM of Ni implanted ZnO films. The same defect under certain condition can play role to create and shape BMPs in ZnO:Ni films, causing FM ordering. Interaction between BMPs is higher in case of a film having higher defect density. For Ni implanted films, it has been observed that O-vacancies is the highest at an intermediate concentration (3%). Overall, it has been established that Ni doped/implanted ZnO films show room temperature ferromagnetism and high optical transmission. Carrier mediated exchange and formation of BMP are the possible reasons for ferromagnetism. However, further investigations are required to completely rule out the possibility of formation of metal clusters and effects like superparamagnetism.

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3. SYNTHESIS AND STUDY OF NI: SILICA NANOGRANULAR FILM In this section we discuss some of the results on Ni:SiO2 nanocomposite films grown by fast atom beam sputtering. The experimental tools used for these studies are following: Ni:SiO2 films were deposited for two different Ni concentration by varying the number of Ni foils on SiO2 target by FAB technique. In another run, Ni:SiO2 film having same Ni conc. was annealed for 30 minutes in a mixture of Ar-H2 (5%) atmosphere at 400 oC. In this section firstly the (i) determination of composition of Ni-SiO2 film by theoretical calculation and experimental studies and secondly (ii) the correlation between composition, microstrature and other properties of these films are discussed. The characterization tools used for the above mentioned studies are Glancing angle XRay Diffraction (GAXRD) technique for structural phase identification, X-ray photoelectron spectroscopy (XPS) to identify the chemical states of the elements present in the surface of the films, transmission electron microscopy (TEM) associated with selected area diffraction (SAED) to examine microstructure and SQUID for magnetic characterization. The composition and the thickness of the films were determined by Rutherford backscattering spectrometry (RBS) using 2.4 MeV He2+ ions provided by ARAMIS accelerator of CSNSM, Orsay (France). Ni atomic fraction in the films was estimated by fitting backscattering spectrum using RUMP simulation code [34]. The microstructure of the films was investigated by a 200 kV TECNAI G2 20 microscope. Results obtained from these studies are discussed in the following two subsections.

3.1. Composition Analysis of Ni-Sio2 Films In nanophase metal insulator composite films, accurate estimation of metal fraction is an important job. The atomic fraction of an element of a nanocomposite film can be calculated by dividing the number of atoms of corresponding element by sum of metal silicon and O atom by the following equation [35]:

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(i) where i stands for metals, Si and O, M stands for metal only (here Ni) A, the area fraction and Y represents the sputtering yield. Sputtering yield can be estimated from Sigmund‟s theory [36] or a modified form of the same [37] or from SRIM simulation code [22]. Another way of determining the same is based on RBS analysis. A typical RBS spectrum, associated with Rump plots of Ni:SiO2 nanogranular film is shown in Figure 3.1. The Ni atomic fraction in this film can be determined by standard RBS formulation [34]. Now the theoretical estimate of at.% of Ni is compared with that of RBS analysis for a film having area ratio of Ni and SiO2 ~ 1:1 (i.e. 50 % each) and given in table 3.1. It has clearly evident that the two results are in good agreement with each other. Energy (MeV) 1.0

30

1.5

2.0

Normalized Yield

25

20

15

10

5 O

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0 100

Si

150

200

Ni

250

300

350

400

Channel

Figure 3.1. A typical RBS spectrum associated with Rump plots of Ni:SiO2 nanogranular film

Table 3.1. Area fraction of Ni and SiO2, sputtering yield of Si, O and Ni, theoretical and experimental atomic composition of a typical Ni: SiO2 film AM (%) 50

ASiO2 (%) 50

YO, YSi

YM

YO = 3.41 YSi = 0.902

YNi = 5.43

Atomic Composition (theory) Ni55.7Si9.3O35

Atomic Composition (experiment) Ni58.4Si12O29.6

Table 3.2. Ni (at.%) and coercivity values of all Ni:SiO2 granular films grown by atom beam sputtering. Film A to C shows the effect of annealing and B to D shows metal content effect Sample A C B D

Ni at. % 55 % 55 % 40 % 90%

Hc (Oe) 90 170 30 150

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3.2. Correlation between Properties of Ni:Sio2 Films with Composition and Microstructure

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In this study four films are considered, 55% (A), 40% (B), 55% and annealed (C) and 90% (D) as per their at.% of Ni determined by RBS analysis. Transmission electron microscopy shows that the particle size enhances with the increase in Ni content (B to D) as well as with increase in annealing temperature (A to C). Typical TEM micrograph, and SAED pattern are shown in Figure 3.2 (a) and (b). Overall TEM analysis shows increase in Ni content (B to D), during deposition or annealing at higher temperature (A to C), after deposition leads to better crystallinity of the films and increase in crystallite size. The thickness of the film estimated from RBS analysis is ~ 238 nm. In case of other samples under present study, RBS analysis based atomic composition will be considered. A typical hysterisis (M-H) curve corresponding to the film D, is shown in Figure 3.3. Magnetometry result shows enhancement of saturation magnetization (Ms) and coercivity (Hc) values with increase in Ni content as well as with annealing temperature. The Hc values of all the films are given in Table 3.2. This can be easily understood on the basis of size dependent magnetic properties of such granular films [38]. However, dipolar interaction and magnetic percolation are also expected to play important role behind the properties of these films. This needs further investigations.

(a)

(b) Figure 3.2. Typical (a) TEM micrograph, and (b) SAED pattern corresponding to film C Insulators: Types, Properties and Uses : Types, Properties and Uses, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Figure 3.3. A typical M-H curve of Ni:SiO2 film at room temperature

A detailed analysis microstructure and volume fraction dependent magnetic properties of Ni:SiO2 granular films are in progress and will be reported elsewhere.

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CONCLUSION Room temperature ferromagnetism is observed Ni implanted ZnO films grown by VPT and PLD techniques. The strength of RT-FM is maximum at an intermediated Ni concentration (~ 3%). Theoretical models, based on Zener exchange and bound magnetic polaron have been used to explain RT-FM properties, in the two cases. Ni implanted ZnO/sapphire films grown by PLD technique also exhibit very high optical transmittance (~ 80%), and hence and is a candidate of transparent magnetic semiconductor. ZnO/quartz films grown atom beam sputtering, shows higher conductivity alongwith transparent magnetic properties. A comparative analysis between theoretical and RBS based experimental estimation of Ni content in Ni:SiO2 granular films, grown by atom beam sputtering technique is made. Improvement in crystallinity, enhancement in saturation magnetization and corecivity have been observed by increasing metal volume fraction and annealing temperature.

ACKNOWLEDGMENTS Author gratefully acknowledges the scientific contribution and inspiration of Dr. D. K. Avasthi, Dr. D. Kanjilal, Dr. D. Kabiraj and Dr. Pravin Kumar from IUAC, New Delhi. This contribution would not have been materialized without constant effort and sincere suggestion of Mrs. Bhawana Pandey, Mr. Hardeep Kumar and Dr. P. Srivastava of IIT Delhi. Author is also grateful to Dr. J. C. Pivin, CSNSM, France, Dr. S. Zhou and Dr. H. Schmidt of FZD, Dreseden, Germany for many important suggestions and fruitful discussion. The experimental facilities like TEM, AFM provided by IIT Delhi, ion implantation and FAB by IUAC, New Delhi, RBS by CSNSM, France and SQUID by FZD, Germany are thankfully acknowledged.

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[12] [13]

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Dietl, T; Ohno, H; Matsukura, M; Cibert, J; Ferrand, D. Science, 2000, 287, 1019-1022. Zener, C. Phys. Rev., 1951, 81, 440-4. Coey, JMD; Venkatesan, M; Fitzgerald, CB. Nat. Mater, 2005, 4, 173-179. Yu, W; Yang, LH; Teng, XY; Zhang, JC; Zhang, L; Fu, G. S. J. Appl. Phys., 2008, 103, 093901-4. Pandey, B; Ghosh, S; Srivastava, P; Kumar P; Kanjilal, D. J. Appl. Phys., 2009, 105, 033909-5. Pearton, SJ; Norton, DP; Ivill, MP; Hebard, AF; Zavada, JM; Chen, WM; Buyanova, IA. IEEE Trans. Electron. Dev., 2007, 54, 1040-1048, and references therein. Lee, HJ; Jeong, SY; Cho, CR; Park, CH. Appl. Phys. Lett, 2002, 81, 4020-4022. Ando, K; Saito, H; Zin, J; Fukumura, T; Kawasaki, M; Matsumoto, Y; Koinuma, H. Appl. Phys. Lett, 2001, 78, 2700-2702. Norton, DP; Pearton, SJ; Hebard, AF; Theodoropoulo, N; Boatnar, LA; Wilson, RG; Appl. Phys. Lett, 2003, 82, 239-241. Fukumura, T; Toyosaki, H; Yamada, Y. Semicond. Sci. Technol., 2005, 20, S103-S111. Kundaliya, DC; Ogale, SB; Lofland, SE; Dhar, S; Metting, CJ; Shinde, SR; Ma, Z; Varughese, B; Ramanujachary, KV; Salamanca-Riba L; Venkatesan, T. Nat. Mater, 2004, 3,709-714. Zhou, S; Potzger, K; Talut, G; Grenzer, von Borany, J; Skorupa, W; Helm, M; Fassbender, J. Appl. Phys., 2008, 103, 07D530-3. Ronning, C; Gao, PX; Ding, Y; Wang, ZL; Schwen, D. Appl. Phys. Lett, 2004, 84, 783785. Ando, K. Solid State Sciences: Magneto-Optics, vol. 128, Springer, New York, 2000, 211. Angadi, B; Jung, YS; Choi, WK; Kumar, R; Jeong, K; Shin, SW; Lee, JH; Song, JH; Khan, MW; Srivastava, JP. Appl. Phys. Lett, 2006, 88, 142502-3. Asakura, S; Ishio, S; Okada, A; Saito, H. J. Magn. Magn. Mater, 2002, 240 485-489. Denardin, JC; Pakhomov, AB; Knobel, M; Liu, H; Zhang, XX. J. Magn. Magn. Mater, 2001, 226, 680-682. Wang, C; Xiao, X; Rong, Y; Hsu, TY. J. Mater. Sci., 2006, 41, 3873-.3879. Franco, V; Batlle, X; Labarta, A. J. Magn. Magn. Mater, 2001, 210, 295-301. D‟Orle´ans, C; Stoquert, JP; Estourne`s, C; Cerruti, C; Grob, JJ; Guille, JL; Haas, F; Muller, D. Phys. Rev. B 2003, 220101-4. Pivin, JC; Esnouf, S; Singh, F; Avasthi, DK. J. Appl. Phys., 2005, 98, 023908-1 to 023908-6. Ziegler, JF; Biersack, JP; Littmark, U. The Stopping Power of Ions in Solids, (Pergamon), Oxford, 1980. Shirai, M; Tsumori, K; Kutsuwada, M; Yasuda, K; Matsumura, S. Nucl. Instr. and Meth. Phys. Res. B, 2009, 267, 1787-1791. Pivin, JC; Singh, F; Angelov, O; Vincent, L. J. Phys. D, Appl. Phys., 2009, 42, 0250056. Avasthi, DK; Mishra, YK; Kabiraj, D; Lalla, NP; Pivin, JC. Nanotechnology, 2007, 18, 125604 -125604 - 4.

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[26] Ghosh, S; Srivastava, P; Pandey, B; Saurav, M; Bharadwaj, P; Avasthi, D.K; Kabiraj, D; Shivaprasad, SM. Appl. Phys. A, 2008, 90, 765-769. [27] Pandey, B; Ghosh, S; Sriastava, P; Avasthi, DK; Kabiraj, D; Pivin, JC. J. Mag. Mag. Mater, 2008, 320, 3347-51. [28] Chen, M; Wang, X; Yu, YH; Pei, ZL; Bai, XD; Sun, C; Huang, RF; Wen, LS. Appl. Surf. Sci., 2000, 158, 134-140. [29] Kaminski, A; Sharma, DS. Phys. Rev. Lett, 2002, 88, 247202-4. [30] Liu, XJ; Zhu, XY; Song, C; Zeng, F; Pan, F; J. Phys. D: Appl. Phys., 2009, 42, 0350047. [31] Saarinen; Hautojärvi, P; Keinonen, J; Rauhala, E; Räisänen, J. Phys. Rev. B, 1991, 43, 4249-4262. [32] de la Rubia, TD; Guinan, MW; Phys. Rev. Lett, 1991, 66, 2766-.2769. [33] Kucheyev, SO; Williams, JS; Jagadish, C; Zou, J; Evans, C; Nelson, AJ; Hamza, AV. Phys. Rev. B, 2003, 67, 094115-11. [34] Doolittle, LR. Nucl. Instr. and Meth. B, 1985, 9, 344-351. [35] Kumar, H; Mishra, YK; Mohapatra, S; Kabiraj, D; Pivin, JC; Ghosh, S; Avasthi, DK. Nucl. Instr. and Meth. B, 2008, 266, 1511-1516. [36] Sigmund, P. Phys. Rev. B, 1969, 184, 383-416. [37] Anderson, HH; Bay, HL. J. Appl. Phys., 1975, 46, 2416-2422. [38] Löffler, JF; Meier, JP; Doudin, B; Ansermet, J; Wagner, W. Phys. Rev. B, 1998, 57, 2915-2924.

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INDEX

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A absorption, 143, 144, 146, 148, 151, 157, 158, 159, 161 accelerator, 205 acetone, 62 acid, 20 activation energy, 143, 155 AFM, 5, 7, 11, 12, 13, 15, 18, 22, 25, 70, 140, 164, 204, 208 Africa, 59 ageing, 180 alternative, 175 amorphous silicon, 62 amplitude, 116, 140, 178, 179, 185, 186 anisotropy, 143, 199 annealing, 70, 206, 207, 208 annihilation, 77, 95, 100, 119, 136 argon, 200 assumptions, 186 Athens, 191 atmospheric pressure, 175 atomic orbitals, 135 atoms, vii, 21, 24, 78, 133, 153, 155, 197, 198, 204, 205

B backscattering, 205 band gap, 61, 135, 196 bandwidth, 5, 7, 9, 13, 17, 25, 133, 135, 138, 139, 143, 148, 150, 166 barium, 66 barriers, 6, 19, 23 batteries, 55 behaviors, 114, 120 Beijing, 75, 113

bending, 8, 9, 15 bias, 66 binding energy, 159 Bluetooth, 57 bonds, 13, 17, 181 bosons, 91, 93, 100, 103, 121, 122 branching, 181 breakdown, 191, 192

C capacitance, 182, 188 capsule, 63 carbon, 65, 134 carbonyl groups, 181 carrier, 181 categorization, 185 category b, 179 cation, 20, 25, 133, 138, 165, 197 causality, 144 ceramic, 8, 43, 190, 191, 193 channels, 175 China, 1, 26, 38, 40, 48, 75, 113 clarity, 136 class, 76, 107, 183, 184, 185, 186 classes, 184, 185 classical electrodynamics, 142 classification, vii, 173, 183, 185, 186, 191 cleaning, 190 closure, 138 clusters, 12, 13, 22, 25, 199, 205 coherence, 148, 150, 199 collaboration, vii, 173 collisions, 204 combined effect, vii, 1, 14, 17, 19, 25 competition, 5, 11, 12, 18, 120 complexity, 20, 198 components, 176

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Index

composition, vii, 24, 68, 70, 195, 205, 206, 207 compounds, vii, 1, 2, 13, 15, 17, 114, 134, 144, 152, 153, 158, 165 comprehension, 134 compressibility, 161 compression, 161 concentration, 182 concreteness, 120 condensation, 100, 102, 175 conductance, 182 conduction, 2, 7, 15, 62, 133, 142, 181, 191, 197, 204 conductivity, 114, 128, 129, 134, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 157, 158, 159, 161, 162, 174, 175, 177, 178, 180, 181, 186, 191, 202, 204, 208 conductors, vii, 2, 29, 48, 62 configuration, 62, 63, 66, 93, 95, 119 confinement, 102, 142 consensus, 115 consumption, 46 contaminants, 174, 180 contamination, 35, 38, 40, 41, 51, 174, 181, 182, 183, 184, 188, 190 contradiction, 179 convergence, 184 conversion, 176 conversion rate, 176 cooling, 12 copper, 147 corona discharge, 47, 48, 49, 183 correlation, 17, 21, 72, 88, 91, 102, 104, 106, 114, 116, 120, 121, 122, 125, 127, 128, 130, 133, 134, 135, 137, 143, 144, 145, 146, 148, 155, 159, 165, 166, 174, 205 correlation function, 116, 121, 122, 125, 127, 128, 145 Coulomb gauge, 117 Coulomb interaction, 136, 141, 166 counter measures, 175 Crete, 193 critical value, 138, 139, 141 crystal structure, 3, 8, 16, 20, 155 crystalline, 70 crystallinity, 71, 207, 208 crystallization, 66, 70, 141, 143, 159 crystals, 135 cuprates, vii, 15, 76, 107, 113, 114, 115, 118, 119, 120, 128, 129, 130 current ratio, 62, 63, 68, 70

D data processing, 49 data transfer, 176 decay, 25, 104, 106, 182 decomposition, 82 decoupling, 91, 119, 121, 123, 125 defects, vii, 29, 31, 38, 66, 69, 181, 198, 202, 204 deficiency, 6 deformation, 8 degenerate, 98, 106 degradation, 198 density, 174, 177, 182, 191 density functional theory, 123 deposition, 65, 69, 72, 177, 198, 200, 202, 207 deposits, 178 destruction, vii, 1, 5, 7, 15, 17, 19 detection, vii, 29, 31, 37, 40, 42, 45, 48, 49, 51, 52, 53, 54, 55, 196 detection devices, 52, 53, 54 detection system, 45 deviation, 18 diagnosis, 56 diamonds, 146, 147 dielectric constant, vii, 61, 63, 66, 69, 70, 142 dielectrics, 62, 66, 70, 181, 182, 191 diffraction, 3, 16, 21, 71, 159, 171, 200, 205 dimensionality, 142 dimerization, 135 diodes, 62, 191 Dirac equation, 96 discharges, vii, 173, 174, 177, 179, 180, 181, 182, 183, 186, 187, 188, 189, 191 discontinuity, 116, 160 discrimination, 177 dislocation, 204 disorder, 13, 25, 159 dispersion, vii, 61, 62, 72, 73, 78, 80, 88, 103, 104 displacement, 141 distilled water, 20 distortion, 3, 7, 8, 13, 17, 141, 142, 143, 144, 150, 154, 161 distortions, 159 distribution, 183 distribution function, 125 divergence, 136 division, 176 domain structure, 199 dopants, 196 doping, vii, 1, 2, 8, 9, 12, 13, 15, 17, 18, 19, 20, 21, 22, 25, 98, 114, 118, 119, 123, 129, 141 duration, 177, 179, 180, 186, 187, 188

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Index

E

F

eigenvalues, 97 electric current, vii electric field, 32, 36, 38, 40, 49, 68, 70, 145, 168, 178, 180, 181, 191 electrical resistance, 197 electricity, vii, 29 electrodes, 63, 68, 182 electrolyte, vii, 61, 62 electromagnetic, vii, 32, 42, 113, 115, 130, 144, 145, 176, 180 electromagnetic waves, 32 electron, vii, 1, 2, 7, 8, 9, 13, 15, 16, 24, 25, 61, 63, 64, 69, 75, 76, 77, 79, 88, 114, 115, 118, 119, 125, 129, 130, 133, 134, 135, 136, 141, 142, 143, 144, 145, 146, 148, 155, 159, 166, 168, 181, 191, 196, 204, 205, 207 electron microscopy, 207 electron state, 135 electronic structure, 197 electrons, vii, 2, 5, 9, 13, 17, 77, 79, 80, 81, 82, 95, 116, 133, 134, 135, 136, 141, 145, 146, 173, 180, 181, 197, 202, 204 elongation, 199 elucidation, 115 emission, vii, 47, 173, 192, 196 encapsulation, vii, 61 energy, vii, 173, 180, 181, 182, 183 energy density, 199 energy transfer, 199 engineering, vii, 195 entanglements, 181 environment, 174, 175, 177 environmental conditions, 41 epoxy, 183 equality, 126 equilibrium, 141, 162, 187 erosion, 186 etching, 200 ethylene, 20 ethylene glycol, 20 evaporation, 66, 70, 187 excitation, 46, 92, 103, 104, 115, 122, 142, 145, 150, 192 experimental condition, 144 exploitation, 169 exposure, 174 extinction, 183, 192

fabrication, vii, 61, 62, 63, 69, 70 fault analysis, 192 fault detection, 46 faults, 183 Fermi level, 135, 136, 150, 165 Fermi surface, 143 fermions, 76, 77, 78, 79, 88, 94, 100, 102, 103, 104, 121, 122 ferromagnetism, vii, 1, 2, 7, 18, 20, 24, 195, 196, 197, 198, 200, 205, 208 field theory, vii, 80, 81, 113, 115, 130, 138, 196, 197 filament, 12 films, vii, 61, 62, 69, 73, 195, 196, 198, 199, 200, 201, 202, 204, 205, 206, 207, 208 filters, 46, 49 finite size effects, 199 fluctuations, vii, 76, 77, 81, 82, 85, 86, 87, 88, 89, 91, 107, 145, 179, 181, 182, 185 fluorescence, 147 foils, 205 forecasting, 183, 187, 192 formula, vii, 113, 115, 126, 128, 130, 144, 145, 146, 147, 159, 188 France, 205, 208 free energy, 80, 105, 153 freedom, 2, 91, 99, 103, 114, 119, 135, 138, 139, 196 freezing, 9 frequencies, 49, 116, 124, 144, 145, 146, 151, 155, 158, 160, 165 frequency distribution, 46, 49 friction, 56 frost, 187 fusion, 177

G gauge theory, 107 Geneva, 192 Germany, 208 glass, 177 grain boundaries, 23, 66, 70, 204 granules, 199 graph, 36, 37, 38 graphite, 2 Greece, 173, 191, 192, 193 Guinea, 111

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Index

H

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hafnium, 66 Hamiltonian, 77, 79, 80, 83, 108, 115, 118, 120, 121, 125, 130, 134, 136, 139, 143, 145 heat release, 186 height, 2, 6, 19, 23 helium, 141 highways, 30 Hilbert space, 119 histogram, 72, 182 homogeneity, 20, 180 Hong Kong, 60 host, 156, 159, 197 Hubbard model, vii, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 94, 100, 101, 104, 105, 106, 107, 109, 118, 136, 137, 139, 140, 149 humidity, 174, 175, 177, 181, 187 hysteresis, 68, 70, 182

ionization, 143, 182 ions, vii, 8, 9, 13, 15, 17, 20, 21, 24, 25, 141, 142, 148, 152, 153, 156, 159, 166, 178, 195, 196, 197, 198, 199, 200, 202, 204, 205 IR spectroscopy, 146 irradiation, 199 isotope, 152 Italy, 32, 133

J Japan, 32, 34, 41, 43, 51, 52, 55, 59, 61, 172 joints, 48

K kinetics, 199 Korea, 29, 42, 60

I

L

ice, 177, 187, 189, 192 illumination, 31 images, 63, 65, 66, 71, 73, 183 impacts, 35 in transition, 20 incidence, 144, 156, 182 India, 195 induction, 32, 176 infancy, 198 inhomogeneity, 2, 5, 9, 17 initiation, 180 insects, 48 insertion, 48 instruments, 175 insulation, vii, 2, 29, 31, 34, 36, 38, 40, 41, 42, 48, 49, 52, 56, 68, 70 insulators, vii, 1, 2, 8, 29, 30, 31, 32, 34, 35, 36, 38, 40, 41, 42, 43, 46, 48, 49, 51, 52, 56, 61, 62, 66, 70, 73, 75, 76, 77, 107, 134, 135, 136, 138, 165, 173, 174, 175, 176, 177, 180, 182, 183, 186, 187, 188, 189, 190, 191, 192, 193 integration, 196 intensity, vii, 173, 177, 181, 187 interaction, 182, 192 interaction effect, 137 interface, vii, 61, 63, 66, 69, 144, 147, 162, 167, 169 interference, 3, 135, 174, 176, 190 interpretation, 193 interval, 175, 186 ion implantation, 196, 198, 199, 208

lasers, 196 lattice parameters, 3, 16, 21, 150, 151 lattice size, 95 lattices, vii, 2, 75, 76, 80, 83, 88, 100, 107, 109 leakage, vii, 42, 46, 66, 68, 70, 173, 174, 175, 176, 177, 178, 179, 180, 181, 183, 184, 185, 186, 187, 190, 191, 193 learning, 183, 184 lens, 45 life sciences, 199 lifetime, 181 ligand, 136 light emitting diode, 62 linear dependence, 15 linearity, 178 liquids, vii, 75, 76, 107 localization, vii, 1, 6, 7, 13, 15, 23, 25, 133, 138, 141, 143, 156, 162, 163 low temperatures, 4, 5, 6, 7, 11, 12, 13, 18, 19, 22, 23, 25, 114, 129, 157

M magnetic effect, 136, 156 magnetic field, vii, 5, 6, 11, 12, 18, 19, 22, 23, 78, 105, 113, 114, 115, 116, 117, 118, 120, 125, 128, 129, 130, 146, 176 magnetic moment, 25, 197 magnetic particles, 199

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Index magnetic properties, vii, 7, 77, 85, 195, 196, 199, 207, 208 magnetic sensor, 196 magnetic structure, 12, 22 magnetization, 4, 5, 7, 8, 9, 11, 12, 17, 18, 19, 22, 80, 81, 82, 90, 198, 199, 200, 201, 204, 207, 208 magnetoresistance, 2, 23, 24, 199 manganese, 3, 20, 25 manganites, vii, 1, 2, 8, 15, 17, 24, 25, 168 manpower, 54 manufacture, 177 manufacturing, vii, 29 mapping, 82, 183 markers, 161 materials science, vii, 195 matrix, 9, 22, 80, 145, 198, 199 measurement, vii, 173, 174, 175, 176, 177, 186 measures, 175 mechanical stress, vii, 29 media, 142 mediation, 202 melting, 156 memory, 69, 175, 196 metal oxides, 2 metal-oxide-semiconductor, 63 metals, 75, 114, 136, 145, 146, 148, 165, 198, 206 meter, 32, 42 micrometer, 62, 73 microscope, 69, 147, 200, 205 microstructure, vii, 195, 198, 200, 204, 205, 208 microwaves, vii, 173, 182 mixing, 159 mobility, 178 modeling, 141, 183, 189, 192 models, 186, 187, 192 modification, vii, 150, 155, 195, 196, 204 moisture, 40 molecular oxygen, 181 molecular structure, 180 molybdenum, 147 momentum, 77, 79, 103, 118, 135 monitoring, 60, 174, 175, 190 monolayer, vii, 61, 62 morphology, 71, 198, 204 motivation, 72 multiplier, 85

N NaCl, 147, 178 nanocomposites, 196, 199 nanometers, 69 nanoparticles, vii, 61, 62, 69, 70, 71, 72, 73, 199

nanostructures, 196 National Science Foundation, 26 national security, vii, 29 neon, 31 network, 183, 184, 185 neural network, vii, 173, 183, 184, 185, 186, 191, 192 neural networks, 183, 192 neurons, 183, 184 New Jersey, 192 next generation, 66 nitrogen, 68, 146 noble metals, 133 noise, 29, 30, 48 nonlinear systems, 60 nucleation, 199

O observations, 180, 185 one dimension, 137, 142 optical lattice, 78 optical parameters, 145 optical properties, vii, 2, 143, 147, 150, 164, 166, 200 optimization, 70 orbit, 204 oscillations, 191 overlap, 47, 135, 184, 204 oxalate, 3 oxidation, 3, 20, 24 oxygen, vii, 1, 2, 3, 4, 5, 6, 7, 9, 20, 22, 24, 25, 70, 114, 152, 154, 155, 181, 198

P pairing, 115 parallel, 17, 176, 187 parity, 101 particle mass, 136 particle size distribution, 72 patents, 52 pattern recognition, 49 percolation, 191, 199, 204, 207 performance, vii, 36, 38, 57, 58, 66, 69, 70, 189, 190, 195 periodicity, 135, 144 permeability, 116 phase diagram, 15, 77, 88, 91, 114, 140, 148, 152, 160, 161, 164, 166 phase transitions, 5, 81, 164 phonons, 142

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Index

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photoelectron spectroscopy, 202 photoemission, 107 physical properties, 2, 15, 91, 196, 199, 204 physics, 75, 78, 83, 88, 102, 107, 113, 115, 118, 133, 146, 152 planning, 177 platform, vii, 61, 62 platinum, 72 Poland, 59 polarity, 2 polarization, vii, 2, 61, 62, 125, 142 polarization operator, 125 pollution, 174, 175, 177, 178, 181, 186, 187, 189, 190, 192 polymer, vii, 36, 40, 61, 62 power, 174, 176, 186 power lines, vii, 29 precipitation, 174, 199 probability, 145, 174, 182, 204 probe, 3, 8, 16, 20, 144, 149, 169, 204 production, 175, 181, 182 project, 60, 189 propagation, 183, 187 proportionality, 145 prototype, 56, 136, 142, 143, 166 pulses, 175, 176, 177, 178, 179, 183, 185, 186, 188, 191 pure water, 62 purity, 20 pyrite, 163

Q quanta, 102 quantization, 133, 183 quantum computing, 199 quantum fluctuations, 81, 87, 89, 139 quantum mechanics, 133, 182 Quantum Phase Transition, 111 quantum state, vii, 75, 107 quartz, 3, 65, 200, 208 quasiparticles, 138, 146, 148

R radiation, vii, 3, 8, 16, 20, 47, 145, 173, 175, 180, 182, 183, 200 Radiation, 180, 191 radio, vii, 173, 174, 182, 190 radio waves, vii, 173, 182 radius, 8, 9, 14, 15, 17, 38, 93, 141, 166, 197, 204 rain, 175

rainfall, 174, 175, 177 Raman spectra, 153 Raman spectroscopy, 152 range, 182, 184, 186, 188 reading, 32, 199 reality, 49 recombination, 181 redistribution, 146, 159, 165 reduction, 180 reflectivity, 144, 147, 156, 160, 162 refractive index, 144, 147 reliability, 38 remote sensing, 43 renormalization, 134, 136, 139, 143, 159 replacement, 17 representative samples, 188 resistance, vii, 2, 3, 8, 15, 16, 19, 20, 34, 35, 40, 41, 42, 46, 49, 51, 56, 72, 173, 175, 176, 186, 187, 188, 197 resolution, 36, 39, 49, 144 rings, 38 rods, 32, 56, 200 room temperature, 2, 3, 8, 9, 16, 20, 21, 25, 150, 160, 196, 198, 200, 202, 203, 205, 208 roughness, 69 rutile, 150, 152

S salinity, 174, 177 salt, vii, 173, 174, 177, 181, 189 sampling, vii, 173, 175, 176, 186 sapphire, 198, 202, 204, 208 saturation, 5, 11, 18, 22, 63, 64, 66, 68, 70, 72, 196, 200, 207, 208 scaling, 88, 167 scattering, 116, 123, 125, 146 screening, 102, 115, 135, 136, 176 semiconductor, vii, 61, 133, 134, 196, 198, 208 semiconductors, vii, 61, 62, 63, 66, 73, 133, 182, 195, 196, 204 sensitivity, 32, 40, 157 sensors, vii, 174, 175, 195, 199 series, 174, 176 shape, 17, 38, 70, 128, 142, 179, 199, 205 shortage, 54 sign, 181 signal processing, 46, 196 signals, 45, 46, 49 silica, 199 silicon, 62, 63, 66, 68, 69, 71, 72, 199, 205 silver, 144 simulation, vii, 173, 187, 189, 193, 205, 206

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Index Singapore, 38 sintering, 69, 72 SiO2, vii, 62, 63, 66, 68, 70, 73, 195, 199, 205, 206, 208 SiO2 films, 205 skyrmions, vii, 75, 77, 92, 93, 99, 100, 107 sodium, 3 solitons, 77, 98 South Korea, 29 Spain, 60, 191 specifications, 43, 45 spectroscopy, 107, 151 spectrum, 179, 180 speed, 174, 175, 177 spin, vii, 2, 5, 6, 17, 18, 23, 75, 76, 77, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 95, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 113, 114, 115, 119, 124, 125, 130, 135, 136, 143, 195, 196, 199 spin dynamics, 76, 104 spintronic devices, 196, 197 square lattice, 76, 78, 89, 96, 100 stability, 180 stabilization, 72 standard deviation, 72, 73 statistics, 93, 94, 99, 100, 106, 107 steel, vii, 29, 38, 49, 147 stoichiometry, vii, 1, 2, 3 storage, vii, 175, 176, 195, 196, 199 storage media, 199 strain, 180 strength, 180 stress, 187 strong interaction, 76 strontium, 69, 70 structural changes, 153 structural transitions, 151, 159 substitution, 8, 9, 13, 15, 18, 19, 20, 21, 25, 114, 147, 166, 168 substitutions, 148 substrates, 69, 200 suffering, 191 sulfur, 164 sulfuric acid, 3 superconductivity, vii, 2, 15, 113, 114, 115, 130 superlattice, 135 suppression, 166 surplus, 20 susceptibility, 84, 105, 106, 134, 145 symmetry, vii, 75, 76, 77, 81, 105, 106, 136, 137, 143, 154, 155 synthesis, vii, 61, 62, 69, 70, 195, 196, 198, 199, 200 systems, 174, 175, 176, 177, 183, 192

T TEM, 63, 65, 199, 200, 205, 207, 208 temperature dependence, 4, 6, 8, 12, 13, 17, 19, 22, 23, 24, 25, 129, 148, 158 tension, 43, 45, 55, 56 testing, vii, 29, 30, 46, 48 texture, 183 theory, 180 thermal treatment, 73 thin films, 198 threshold, vii, 173, 175, 180, 181, 187 time, vii, 173, 174, 175, 177, 178, 179, 181, 184, 186, 187, 188, 189 time frame, vii, 173 titanate, 66, 69 tracking, 183 trajectory, 116, 200 transducer, 175 transformations, 77, 83, 108, 109, 127, 144, 157, 164 transition metal, 133, 134, 141, 196, 197, 198 transition metal ions, 197 transition rate, 145 transition temperature, 7 translation, 101, 105, 143 transmission, vii, 2, 29, 31, 35, 40, 43, 46, 49, 50, 54, 69, 180, 189, 190, 198, 200, 201, 202, 205 transmission electron microscopy, 200, 205 transport, vii, 7, 8, 13, 15, 16, 23, 25, 113, 114, 115, 118, 123, 125, 130, 134, 141, 143, 155, 161, 165, 181, 199, 202 triggers, 143, 168 triplet excitation, 92 tunneling, 6, 23, 78, 181

U UK, 32 ultraviolet range, 47 UV, 62, 200

V vacancies, vii, 1, 2, 3, 5, 6, 7, 25, 202, 204, 205 vacuum, 62, 63, 66, 69, 70, 72, 116, 144, 182, 192 valence, vii, 2, 7, 18, 20, 24, 76, 92, 156, 159 values, vii, 173, 175, 176, 178, 182, 184, 185, 188 vanadium, 144, 155, 156, 159 vapor, 202 variables, 181, 183 variation, 177, 180, 186

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Index

vector, vii, 40, 82, 84, 92, 101, 113, 115, 116, 117, 118, 119, 120, 121, 124, 125, 126, 128, 130, 145, 183, 184 velocity, 84, 85, 86, 95, 96, 100, 106, 107 vibration, 45 video, 45 vision, 47 VPT, 200, 202, 204, 208

W

X-ray, 16, 65, 69, 192, 200, 202, 205 X-ray diffraction (XRD), 3, 4, 8, 10, 16, 21, 66, 69, 71 X-ray photoelectron spectroscopy (XPS), 68, 69, 70, 200, 205

Z zinc, 196 zinc oxide, 196 zirconium, 66 ZnO, vii, 195, 196, 197, 198, 200, 201, 202, 203, 204, 205, 208

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weak coupling limit, 81 weak interaction, 76 wetting, 174 wide band gap, 196 wind, 174, 175, 177

X

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