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Colossal Magnetocapacitance and Scale-Invariant Dielectric Response in Mixed-Phase Manganites

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COLOSSAL MAGNETOCAPACITANCE AN D SCALE-INVARIANT DIELECTRIC RESPONSE IN MIXED-PHASE MANGANITES By RYAN PATRICK RAIRIGH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Ryan Patrick Rairigh

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To my wife, Nikki

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iv ACKNOWLEDGMENTS I am indebted to the many individuals w ho contributed to the successful completion of my graduate career, represented by this dissertation. First, and foremost, I extend my heartfelt thanks to Dr. Art Hebard for hi s guidance, motivation, and friendship. He provided an atmosphere that stimulated di scussion, collaboration, and enjoyment among his graduate students. Likewi se, Art was always ready to learn something. This made coming to work each day an exciting prospect that I will surely miss. These qualities and others made him an exceptional advisor and it was an honor to work for (and with) him, these past 5 years. I would also like to thank th e many members, both past and present, of the Hebard lab that I have worked with and learned from over the years: Jeremy Nesbitt, Partha Mitra (in particular, for teaching me what good Indi an food should taste like), Xu Du, Guneeta Singh, Sinan Selcuk, Kevin Mc Carthy, Nikoleta Theodoropoul ou, Steve Arnason, Rajiv Misra, Mitchell McCarthy, and Ritesh Das. I extend extra special thanks to Jeremy Nesbitt who has been my friend, lunch comp anion, and favorite skeptic for many years now. I also extend my thanks to the other members of my supervisory committee for taking time out of their incred ibly busy schedules to answ er questions and guide my education. Each of them has contributed to my academic career. David Tanner, taught me as an undergraduate the value of understa nding the importance of the error in a measurement. Selman Hershfield, was instru mental in shaping my early career. Cammy

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v Abernathy, taught me everything I know a bout crystal growth while simultaneously indulging my interest in talk ing about football. Amlan Bisw as, provided much of impetus for the experimental work carried out in my study and taught me everything I ever wanted to know about cricket. In additi on, I would like to thank Dr. Steve Detweiler whose teaching convinced me as an uncertain undergraduate that physics was indeed the right path for me to take. I would like to thank my family. First, I thank my parents, whose unwavering support and confidence in me has been a beacon, all of my life. They always went out of their way to value and solicit my opinion, and include it, even when I was too small to know what an opinion was. They taught me that you can learn a lesson from anyone, if you just pay attention and listen. This rec ognition of the value of knowledge and my worth, as part of our family, meant more to me than I can express in words. I also thank my extended family of friends (especially, Pa ul McDermott), aunts, uncles, cousins, and grandparents who have shaped who I am today. As the saying goes, last, but certainly not least, I thank my wife, Nikki. She is the reason that I have come to this point in my life with the le vel of satisfaction, completeness, and joy I feel. I thank her fo r being my partner, confidant, travel companion, motivator, best frie nd, and the love of my life.

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF FIGURES.........................................................................................................viii ABSTRACT....................................................................................................................... xi CHAPTER 1 INTRODUCTION........................................................................................................1 2 REVIEW OF MANGANITES.....................................................................................3 2.1 Introduction.............................................................................................................3 2.2 Structure.................................................................................................................. 4 2.3 Theoretical Models.................................................................................................7 2.3.1 Double Exchange Theory.............................................................................7 2.3.2 Beyond Double Exchange, Part 1: Jahn-Teller Polarons...........................16 2.3.3 Beyond Double Exchange, Part 2: Percolating Phase Separation..............21 3 DIELECTRIC RELAXATION PHENOMENA........................................................26 3.1 Time-Dependent Dielectric Resp onse: Polarization v. Conduction.....................26 3.2 Frequency-Dependent Dielectric Response..........................................................29 3.3 Dielectric Response Functions.............................................................................31 3.3.1 Impedance and Admittance........................................................................32 3.3.2 Dielectric Permittivity................................................................................34 3.3.3 A Dielectric Case-Study: Disordered Conductors......................................35 4 CAPACITANCE MEASUREMENTS AN D EXPERIMENTAL TECHNIQUES...38 4.1 Thin-Film Metal-Insulator-Metal Capacitors.......................................................38 4.2 Metal-InsulatorMeta l (MIM) Fabrication..........................................................39 4.3 Measuring Capacitance: Tools and Techniques...................................................40 4.3.1 Impedance Analyzer...................................................................................40 4.3.2 Lock-In Amplifier......................................................................................43 4.3.3 Capacitance Bridge.....................................................................................46 4.4 The Series Resistance, Co mplex Capacitance Problem.......................................48

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vii 5 COLOSSAL MAGNETOCAPACIT ANCE IN PHASE-SEPARATED MANGANITES..........................................................................................................54 5.1 Dielectric Measurements of Stro ngly Correlated Electron Materials...................54 5.2 (La1xPrx)5/8Ca3/8MnO3..........................................................................................55 5.3 The Colossal Magnetocapacitance E ffect: Measurement and Analysis...............61 6 SCALE INVARIANT DIELECTRIC RESPONSE IN PHASE SEPARATED MANGANITES..........................................................................................................76 6.1 Modeling the Competition of Phases in (La1xPrx)5/8Ca3/8MnO3 (LPCMO).........76 6.2 The Dielectric Response Due to Phase Separation in LPCMO: Universal Power-Law Phenomena..........................................................................................81 6.3 Crossover in the Dielectric Response: The Dynamics of Phase Separation.........93 7 SUMMARY AND FUTURE DIRECTIONS...........................................................101 7.1 Summary of Experimental Work........................................................................101 7.2 Future Work........................................................................................................102 LIST OF REFERENCES.................................................................................................104 BIOGRAPHICAL SKETCH...........................................................................................111

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viii LIST OF FIGURES Figure page 2-1 Three magnetoresistance versus temp erature curves for different La0.67Ca0.33MnOx samples ....................................................................................3 2-2 Schematic of the perovskite stru cture that encompasses the MnO6 octahedron that serves as the basis for all manganites..................................................................5 2-3 Crystal-field splitting of th e five-fold degenerate 3 d orbitals in the Mn3+ ion. A similar diagram for Mn4+ would have unoccupied eg states.......................................6 2-4 Schematic view of orbital and spin ordering on the Mn sites in LaMnO3. ..............11 2-5 Phase diagram for La1-xCaxMnO3 showi ng the effect of hole-doping on the magnetic ordering of the manganite.........................................................................12 2-6 Resistivity calculated from Eq. 2-8. ......................................................................... 14 2-7 Temperature dependent resistivity of La1-xCaxMnO3 ( x = 0.25) in various applied magnetic fields. ...........................................................................................15 2-8 The projection of the crys tal structure of LaMnO3 along the b axis. The unequal Mn-O bond lengths illustrate the Jahn-Teller distortion..........................................17 2-9 Jahn-Teller distortion and its effect on the energy level splitting of the Mn3+ orbitals…..................................................................................................................18 2-10 Magnetic field dependence of re sistivity calculated using the direct integration method … ..................................................................................................................21 2-11 Dark-field SEM images for La5/8yPryCa3/8MnO3 obtained by using a superlattice peak caused by charge ordering (CO)......................................................................22 2-12 Scans (6 m x 6 m) of LPCMO. On c ooling, FM regions (dark area) grow in size……....................................................................................................................23 2-13 Random resistor network resistivity for various metallic fractions, p in 2D and 3D (inset)..................................................................................................................24 2-14 Monte Carlo results for the RFIM model. ...............................................................25

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ix 3-1 Barrier-volume capacitor..........................................................................................32 3-2 The complex impedance diagram of the ba rrier-volume circuit. The large arc is due to the barrier region and the sma ll arc is due to the volume region..................33 3-3 Random resistor capacitor netw ork that describes the diel ectric behavior of the universal ac conductivity materials..........................................................................36 4-1 Schematic of MIM. A) the metallic counter electrode. B) the dielectric. C) the manganite base electrode. D) the substrate..............................................................39 4-2 Confidence intervals of the measured impedance as a function of the measurement frequency............................................................................................42 4-3 Schematic of a lock-in amplifier..............................................................................43 4-4 The magnitude of the impedance of a 400 LPCMO film as measured by the Solartron 1260A (black squares) a nd PAR 124A Lock-In (red circles)..................45 4-5 Schematic of the capacitance bridge circuit.............................................................47 4-6 Schematic of a complex capacitor............................................................................49 4-7 Equivalent circuit of MIM capac itor with a resistive electrode, Rs, and complex capacitance, C = C1 – iC2, where R2 = 1/C2..........................................................50 4-8 The temperature dependent impedance of the LPCMO film, that serves as the base electrode in our MIM.......................................................................................52 5-1 Temperature dependent resistivity of A ) La0.67Ca0.33MnO3 (LCMO) and B ) Pr0.7Ca0.3MnO3 (PCMO) .....................................................................................55 5-2 Temperature dependent re sistivity for an LPCMO x = 0.5 thin film with thickness d 500 ..................................................................................................56 5-3 Magnetoresistance data for LPCMO ( x = 0.5) and d 500 for various temperatures.............................................................................................................57 5-4 Field-cooled magnetization curves for La5/8xPrxCa3/8MnO3 for various x . .............59 5-5 Temperature dependence of the resistance (black circles) and capacitance (red squares) of LPCMO.................................................................................................62 5-6 Thickness dependence of the transition temperature...............................................64 5-7 Frequency-dependent dielectric resp onse of inset the model circuit. .....................66 5-8 Maxwell-Wagner relaxation.....................................................................................68

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x 5-9 Magnetic field and temperature dependen ce of the LPCMO capacitor measured at f = 0.5 kHz............................................................................................................69 5-10 Colossal changes in the magnetic fiel d dependent capacitance in the LPCMO MIM capacitor..........................................................................................................72 5-11 The field dependent capacitance at T = 10 K for the LPCMO MIM. The arrows indicate the direction the field is swept....................................................................73 5-12 The temperature dependent ac loss in applied magnetic fields................................74 6-1 General form of the equivalent ci rcuit for our LPCMO MIM capacitor..................76 6-2 The frequency dependence of the real, C, and imaginary, C, parts of the LPCMO MIM capacitor at temperatures, cooled in zero magnetic field.................79 6-3 The simplified equivalent circui t for our LPCMO MIM capacitor..........................80 6-4 Cole-Cole plots in zero magnetic fi eld on warming for 50, 100 and 200 K............82 6-5 Cole-Cole plot showing the effect of magne tic field on the dielectric behavior at T = 100 K.................................................................................................................83 6-6 Cole-Cole plot showing the effect of magne tic field on the dielectric behavior at T = 65 K, with same frequency steps as the T = 100 K data....................................84 6-7 Cole-Cole plot showing the effect of te mperature at H = 10 kOe on the dielectric behavior....................................................................................................................86 6-8 Universal dielectric behavior for the complex capacitance of the LPCMO MIM...87 6-9 Cole-Cole plot showing the comp lete dielectric relaxation of C............................88 6-10 Temperature dependent capacita nce in zero magnetic field....................................92 6-11 Cole-Cole plot for warming temperatur e from 10 to 300 K at fixed frequency and various magnetic fields......................................................................................95 6-12 T-H phase diagrams..................................................................................................96 6-13 The power-law scaling collapse (PLSC) region expands (con tracts) along the PLSC-NPLC line (open symbols) with increasing (decreasing) frequency.............98

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xi Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COLOSSAL MAGNETOCAPACITANCE AN D SCALE-INVARIANT DIELECTRIC RESPONSE IN MIXED-PHASE MANGANITES By Ryan Patrick Rairigh May 2006 Chair: Arthur Hebard Major Department: Physics We measure capacitance on thin-film metalinsulator-metal (MIM) capacitors using the mixed-phase manganite (La0.5Pr0.5)0.7Ca0.3MnO3 (LPCMO) as the base electrode. The LPCMO was grown by pulsed laser deposition (PLD), then an AlOx dielectric was deposited via rf magnetron spu ttering, and then Al was thermally evaporated as the counter-electrode. The LPCMO films exhibit colossal magnetoresistance (CMR) where resistance decreases by seve ral orders of magnitude w ith applied magnetic field. Correspondingly, capacitance shows a change of three orders of magnitude in the region of the resistance drop. These colossal magnetocapacitance (CMC) effects are related to magnetic-field-induced percolating changes in the relative propor tions of coexisting ferromagnetic metal (FMM) and charge ordere d insulating (COI) phases in the LPCMO. The widths of the temperature-dependent hys teresis loops (in capaci tance and resistance) are approximately the same, but the cen ter of the capacitanc e loop is shifted 20 K below the center of the resistance loop. When the el ectrode resistance is at a maximum (low

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xii capacitance) the electrode co mprises filamentary conductors threading a predominantly insulating medium. In this region, double l ogarithmic Cole-Cole plots revealed the intrinsic dielectric response of the LPCMO in which the data plotted either as a function of frequency or temperature for different fr equencies collapse onto si ngle straight lines, thus implying scale-independent phenome nology over a wide range of frequency, magnetic field and temperature. Phenomena witnessed in these experiments give a comprehensive picture of the percolative pha se separation inherent to LPCMO. These capacitance methods hold similar promise for th e study of other phase-separated systems.

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1 CHAPTER 1 INTRODUCTION Thin film metal-insulator-metal (MIM) capacitors are ubiquitous, with applications in cellular telephones, satel lite communications networks GPS receivers, test and measurement applications, filters, voltage controlled oscillators (VCOs) and RF amplifiers; they help control the modern electronic world. In addition to providing valuable aid in everyday applications, they also provide a system for investigating fundamental aspects of thin film physics. Tr aditionally, the bulk di electric properties of insulators such as their static or steady stat e response to a steady electric field and the dynamic response to time-varying electric fi elds have been studied by capacitance methods. However, it has been known for over 40 years1 that capacitance measurements on MIM capacitors reveal information about th e metal-insulator inte rface in addition to the bulk dielectric properties of the insula tor. These interface cap acitance effects are caused by the penetration length of the electric field into the metallic electrode and, in ultra thin insulating layers, they dominate the measured capacitance. In this way, it is also possible to study the material properties of th e electrode with a capacitance measurement. We studied the unique electronic and magne tic properties of the mixed-valence, phase-separated manganite (La1xPrx)5/8Ca3/8MnO3 (LPCMO) via a capacitance measurement. Our measurement used LP CMO as the base electrode in a MIM capacitance structure, whereby we can probe the dielectric response intrinsic to this complex strongly-correlated el ectron material. The experiment will study the effect of temperature, magnetic field and frequency on th e characteristic behavior associated with

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2 the fundamental mechanisms that drive the dynamic electronic, magnetic and structural properties of the LPCMO. The unique nature of our capacitance measurement will allow us a window into the phase separation that occurs due to percolation of ferromagnetic, metallic (FMM) regions in the presence of charge-ordered insulating (COI) regions. Traditional transport measurements such as the dc resistivity2-4 or dc magnetization5, capture some of the important properties of LPCMO like the magne tic field-dependence of the metal-insulator transition and the associ ated decrease in the resistivity by several orders of magnitude. However, our capaci tance measurements, performed at ac, will illuminate a wealth of additional informati on about the complicated competition between the FMM and COI phases that is not available via traditional dc methods.

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3 CHAPTER 2 REVIEW OF MANGANITES 2.1 Introduction In 1994, a new era in the study of doped oxides began with the discovery (by Jin, et al .6) of “colossal magnetoresistance” (CMR) effect in epitaxial thin films of La0.67Ca0.33MnO3 (LCMO), a mixed manganite compound. They observed a negative magnetoresistance three orders of magnitude larger than any magnetoresistance effect observed up to that time (Fig. 2-1). Figure 2-1. Three magnetoresistance versus temperature curves for different La0.67Ca0.33MnOx samples 1) as de posited 2) heated to 700 C in O2 for 0.5 hours and 3) heated to 900 C for 3 hours. Jin et al., Science 264, 413-415 (1994), Fig. 1, pg. 413. This magnitude of response of the electrical transport to an external magnetic field is not observed in bulk metallic systems wher e, in clean systems, at low temperatures,

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4 magnetoresistance is caused by magnetic fiel d dependence in the electronic mean free path. The CMR is also much larger in amp litude than the giant magnetoresistance (GMR) effect observed in Fe-Cr multilayer and other la yered structures where a large sensitivity to magnetic fields is due to the spin-valve effect between spin-polarized metals. Shortly after the discovery of the CMR effect, other researchers discovered that these manganites possessed magnetic-field-induced insulator-m etal and paramagnetic-ferromagnetic phase transitions, as well as, associated lattice-structural transitions3. Additionally, Park discovered7 that at temperatures below Tc, that manganites were half metals, meaning that they approached 100% spin polarization. These materials had been studied and so me of their electrical and thermal properties had been known for more than 50 years8, including a large magnetoresistance effect around the ferromagnetic Curie temperature, Tc. These magnetoresistive effects were understood in the context of the double exchange theory initially developed by Zener9 and further elaborated on by DeGennes10, and Anderson, et al.11. With the discovery of CMR and other associated effects, an explosion of new research began that continues unabated to this day. Specifica lly, experimentalists and theorists have endeavored to reexamine manganites for potential practical applications and to understand the underlying mechanisms responsible for the properties of these remarkable materials. 2.2 Structure La1xCaxMnO3 is the prototypical CMR material a nd belongs to a class of 3D cubic perovskite-based compounds commonly cal led mixed manganites. Depending on the hole-doping of the material, x LCMO crystallizes in vari ous distorted forms of the classic ABO3 perovskite structure. The Mn ions are placed in the center of an oxygen

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5 octahedron (Fig. 2-2), forming MnO6 octahedra with corner-shared oxygen atoms. The Mn and O ions are arranged in a MnO2 plane that is analogous to the CuO2 planes in the cuprate high-temperature superconductors. Figure 2-2. Schematic of the perovskite structure that encompasses the MnO6 octahedron that serves as the basis for all manganites. In LCMO, the MnO2 planes are then stacked in a variety of sequences with the MnO2 planes interlaced with (La, Ca)O planes. The complexes are named depending upon how many MnO2 planes are arranged between bilaye rs of (La,Ca)O planes. This is the Ruddlesden-Popper series, also known as the layered perovskite st ructure. The series has the formula unit (A, B)n+1MnnO3n+1 and the 3D cubic manganites are the end (n= ) constituent of it, which means that they have no (La,Ca)O bilayers. The 3D mixed manganites are generally expressed as A1-xBxMnO3, where A is a trivalent rare-earth ion and B is a divalent dopant. This mixing of cation valences produces an associated mixed valence in the Mn ions due to charge conservation, in that Mn3+ and Mn4+ coexist in the compound. Typically, these two ionic species are randomly distributed throughout the crystal lattice. The chemical substitution of the trivalent i ons (La, Pr, etc.) with the divalent ions (Ca, Sr, etc.) effectively dope s the manganite with holes that appear as

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6 vacancies in the electron conduction band (i.e. an increase in divalent ion substitution leads to an increase in the concentration of Mn4+). This mixed-valence chemistry and the octahedral atomic configuration become impor tant as we discuss the unique interactions and ordering phenomena that occur in the orbital, charge, and spin dependent properties of manganites. In mixed manganites, the local spins a nd conduction electrons are d-electron in nature. Specifically, they are of the 3d orbi tal levels associated with the previously mentioned octahedral-coordinated Mn ion. Normally, the 3d orbital electrons in Mn are five-fold degenerate, but the octahedral coordination partia lly lifts this degeneracy because of cubic crystal fiel d-splitting effects such as hybr idization and electrostatic interactions between neighboring ions (Fig. 2-3). This field splitting breaks the rotational invariance of the orbitals and resu lts in three degenerate, low-energy, t2g states egt2g 3 dorbitals Mn3+ egt2g 3 dorbitals Mn3+ Figure 2-3. Crystal-field splitting of the five-fold degenerate 3 d orbitals in the Mn3+ ion. A similar diagram for Mn4+ would have unoccupied eg states. separated by a few eV from two degenerate, high-energy, eg states. The three t2g orbitals are dxy, dyz and dzx. The two eg orbitals are 2 2y xdand2 23 r zd.

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7 The eg orbitals have a high degree of overlap with the 2p orbitals of the neighboring O ions and this results in a strong hybridi zation of the two. These hybridized orbitals comprise the pathways through which electron conduction occurs. The t2g orbitals have little to no hybridization with the O ions and thus behave more or less like localized states that are electrically inert, with a core spin of S = 3/2. Additionally, the electron-electron interactions among the t2g levels are such that it is energetically favorable for all electrons to have parallel spins. In turn, a strong intra-atomic exchange coupling, JH, occurs between the eg conduction electron spins a nd the core spins in the t2g orbitals. This strong coupling (a consequence of Hund’s rule and the Pauli Exclusion Pr inciple) sponsors an alignment of the spins between the two energy le vels at all accessible temperatures and is an important factor in unde rstanding the fundamental be havior of the manganites. 2.3 Theoretical Models 2.3.1 Double Exchange Theory Double exchange is the starting point for understanding the magnetoresistance and paramagnetic (PM)–ferromagnetic (FM) phase transitions inherent to manganites. Historically, the double exchange model for fe rromagnetism was proposed in its original form by Zener9 to explain the “empirical correlat ion between electr ical conduction and ferromagnetism in certain compounds of mangane se with perovskite structure” observed by Jonker and Van Santen8. This theoretical work was de voted mostly to understanding the decrease in resistivity with increasi ng magnetic field, and not to understanding the magnitude of the magnetoresistance effect s. In a system like the mixed-valence manganites “double exchange” refers to simulta neous transfer of an electron from a Mn3+ to an O2with the transfer of an electron from an O2to a Mn4+. In 1955, Anderson and Hasegawa11 proposed a variation on th is picture that involved an intermediate state

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8 between the two mentioned above, whereby a singly occupied O ion is flanked by two Mn3+ ions with parallel eg spins. This picture produ ces an effective hopping term, t, for the electron to transfer Mn3+ to Mn4+ that is proportional to the square of the hopping term for the 2p O orbital and the t2g Mn orbitals. For classi cal spins, this hopping is proportional to cos(/2), where is the angle between the spins of the adjacent t2g electrons. Equation 2-1 shows how this the electrical conduction is incumbent on the magnetic character of the manganite through t, and thus the tr ansfer of the eg hole from one site to another. The hopping term, t ~ cos(/2), indicates that FM alignment of the adjacent spins maximizes both t and the bandwidth, W, which is proportional4 to cos2 and is the distance between the maximum and minimum en ergy in the hole band. Uncorrelated spin alignment will result in a reduction of the hopping amplitude by cos( /2). This implies that an antiferromagnetic (AF) arrangement of the t2g spins would result in zero hopping amplitude and zero hopping conductivity. The phys ics of this important result of Hund’s rule coupling can be expressed in the double exchange Hamiltonian developed by Kubo and Ohata12 (Eq. 2-1) where iS is the spin of the core-like t2g electrons, ,' zz is the †† ,' ,,',, zz DEHiizijjz iziz izzijzHJScctcc (2-1) Pauli matrix,† izcand izcare the creation and annihilation operators, respectively, for an eg electron with spin z at site i in the conduction band, tij is the transfer matrix element governing electron hopping from site i to j and JH is the intra-atomic exchange coupling, with JH > 0 implying FM order.

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9 The physics of Eq. 2-1 states that for a sufficiently large JHS, when an electron hops from site i to j it must align its spin from being parallel with iS to being parallel withjS Therefore, the electron hopping amplitude is tuned by a parameter that shows a maximum when the core spins align in a pa rallel configuration a nd a minimum when the core spins are aligned anti-parallel. This s hows the basis for the claim that the electron conduction is intimately coupled with th e magnetic character of the system. We can write the Hamiltonian in Eq. 2-1 for a simple two state problem where an electron is confined to states i and j (Eq. 2-2). In the classical limit whereHijJS, the †††† ,',' izzjzz ijHizjzjziz izjzizjzHJSccScctcccc (2-2) core spins can be described in terms of unit vectors specified by polar angles ij and ij. After suitable manipulation of the coordinatesa, the transfer matrix generates the following expression for the hopping amplitude: ,cos(/2)cos(/2)sin(/2)sin(/2)exp()ijijijijtti (2-3) If iS is parallel to j S then i =j and ,ijt = t. If iS is anti-parallel to j S then i =j + and ,ijt= 0. Therefore, for FM alignment of the core spins, we see no reduction in the hopping amplitude and for AF alignment we see no electron hopping. If we make an appropriate choice of gauge we can remove the term exp()iji from the expression. a The goal is to take the electron axis in question (i.e. state i or j ) and rotate it parallel to the core spin axis, S, operating with the rotation matrix 'exp/2ijzz ijzzRin where ijn is the unit vector describing the classical spin stateiS and j S respectively.

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10 This complex quantity is a Berry phase, which is a quantum mechanical phase factor that mimics an internal magnetic field. The resulting real expression is as follows: ,cos/2ijijtt (2-4) Equation 2-4 illustrates that the eg band receives an kinetic energy gain-an increase in the ability of elec trons to move from site to si te in a crystal lattice-from FM alignment. The strong on site coupling between the conduction and core spins in this FM state also produces a unique state. At low temperatures, the spin-polarized conduction band is completely split, such that the mi nority spin band is empty. This corresponds to JH >> W, the eg bandwidth, which is in contrast to conventional ferromagnets which have J << W. This produces the experimentally obser ved half-metal nature of manganites mentioned previously7. Thus, we see the natural conn ection between magnetic order and electron hopping in double exchange. Specifical ly, that disorder in the electron spins causes randomness in tij, which decreases below Tc or with an applied magnetic field. FM alignment must in turn compete with other interactions such as superexchange13, which favors an AF exchange between two nearest neighbor non-degenerate t2g orbitals via their shared O2ion, with disorder influenced by thermal fluctuations in the crystal lattice and charge -exchange (CE) AF due to charge-ordering, which occurs when the Mn3+ and Mn4+ cations order coherently into two separate magnetic sublattices over long dist ances along the crystal lattice. It is important to note at this point that this competition between interactions and the overall magnetic ordering of manganites is heavily influenced by the hole doping of the host material. The magnetic phase diagram in Fig. 2-4 shows that in La1xCaxMnO3

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11 one can chemically change the inherent natu re of the spin ordering below the magnetic transition temperature. In this system, the host material, x = 0 (i.e. completely undoped), is LaMnO3 an insulating AF dominated by superexchange. This means that each Mn in the MnO6 octahedron has four FM bonds and two AF bonds (see Figure 2-4).When the hole doping is increased, the manganite moves to a mixed contribution of superexchange and double Figure 2-4. Schematic view of orbital a nd spin ordering on the Mn sites in LaMnO3. Tobe et al. Phys. Rev. B 64, 184421 (2001), Fig. 1a, pg. 2. exchange interactions for 0< x < 0.08 and 0.88< x < 1 which leads to a canted antiferromagnetic (CAF) insulating st ate, first predicted by DeGennes10. Theoretically, this CAF state is produced by a dominat e super-exchange in teraction in the ab-plane with a weak double exchange inter action occurring along the c-axis. However, more recent results2,14-16 show that the CAF state is a manifest ation of phase separation (PS) between coexisting FM and AF states. In this context, PS is a result of an inhomogeneous spatial density of eg electrons, resulting in regions that are hole doped (FM regions) and those that are undoped (AF regions). A charge-ordered (CO) insulating AF state first appears for 0.08 < x < 0.17 and a FM metallic state for 0.17 < x < 0.5. When x > 0.5, the Mn4+ becomes the dominant ion and the manganite re-e nters an AF state, w ith CO occurring at intermediate temperatures.

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12 At x = 0.5, the compound enters a very unique PS state. Initially, it undergoes a FM transition followed by a simultaneous AF and CO transition at T < Tc. This CO state can be easily “melted” into a FM metallic stat e by application of magnetic field, x-rays or external pressure4. It should also be noted that all the before mentioned antiferromagnetic states are insulating in nature and thus ag ree, in principle, with the double exchange picture that conduction and ferroma gnetism are inextricably linked. Figure 2-5. Phase diagram for La1-xCaxMnO3 showing the effect of hole-doping on the magnetic ordering of the manganite. CAF: canted antiferromagnet, FI: ferromagnetic insulator, CO: charge or dered. CAF and FI could have spatial inhomogeneity with both ferroand an tiferro-magnetic states present. Hwang et al. Phys. Rev. Lett. 75, 914-917 (1995), Fig. 4, pg. 917. Figure 2-5 also illustrates that, regardless of the hole-doping in La1xCaxMnO3, the high temperature magnetic phase is a parama gnetic insulator (PI). The double exchange model predicts this magnetic phase transition, in that for a properly doped material at low temperatures we have parallel core spins and carrier hopping amplitude of t x = 1/8 7/8 Blank space blank space blank 3/8 4/8 5/8 Paramagnetic insulator

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13 (ferromagnetic phase ), which above Tc, transforms to a state with non-collinear and/or orthogonal core spins and carrier hopping amplitude of ,ijt = tcos( /2) 0.7t (paramagnetic phase). This decrease in hopping amplitude by associated with the magnetic phase transition, should produce a co ncomitant response in the conduc tivity of the manganite. This is because the only ener gy scale in the double exchange theory is D, the spin wave stiffness17, which has a theoretical estimate of D ~ 2tn, where n is the electron density per unit cell per orbital and t is the hopping amplitude. Millis et al.17 show that the temperature de pendent resistivity for the double exchange model with large Hund coupling interaction between the t2g and eg electrons (Eq. 2-5) where a0 is the lattice constant and M (,T) is a “memory function”b. 2 0T0,/ aMTeD (2-5) Further evaluation of Eq. 25 to leading order in 1/ S and kFa0, where kF is the Fermi wave vector, reveals the double exchange resistivity (Eq. 2-6), where R labels the various sites 1224 1 2 ,,0/RTeSSSRSRSBR (2-8) b Since there is no rigorous expression for the dc resistivity in the double exchange model, due to an intractable integral equation, an approxima tion must be made. In this case, Millis et al. use a memory function method (which, in principle is valid at large frequency) was used to extract the dc resistivity in terms of the memory function. This memory functio n can be explicitly determined by a perturbation expansion in the dominant scattering mechanism; in this case, the scattering of the electrons off of the spin fluctuations or spin disorder. M(,T) takes the form: 0 01 ,,,,it t M TdteHjHj D (2-6) Where H is from Eq. 2-1 and j is a current operator: †† 221 2ii ab ijjbia ia jb ijabSS jitcccc S (2-7)

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14 on the crystal lattice, in the coordinate system that conn ect a site to it s nearest neighbor and B(R) is the electron current-current correlation functionc. The results of Eq. 2-8 are shown in Figure 2-6. They show that for all temperatures, including T < Tc, that the Figure 2-6. Resistivity calculated from Eq. 2-8. In this model, the resistivity increases below Tc and with increasing magnetic fiel d, in direct contradiction with experiment (see Figure 2-6). Millis et al. Phys. Rev. Lett. 74, 5144-5147 (1995), Fig. 1, pg. 5146. temperature moves through Tc, with a decrease in resistivity with increasing magnetic field. Also, at T > Tc, the resistivity exhibited by the CM R materials is much larger than interaction reflects insulating behavior with an increase in the resistivity of ~150% from Tc to 0.6Tc. Contrasting this experimental data in Figure 2-7 shows th e typical transport c In the free electron approximation B(R) is expressed as: 222 11 4222 0 11sinsin2sin 9 () 32()FFF F F FFkRkRkR BR ka kR kRkR (2-9)

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15 behavior for a CMR manganite18. The resistivity changes by orders of magnitude as anticipated by simple double exchange m odels. Nor does double exchange predict the rapid decrease in resistivity on either side of Tc. Other theoretical work has shown the wide gamut of responses that double exchange can produce in various limits and approximations. In contrast to Millis et al., Calderon et al. found19 via Monte Carlo simulations that in the classical spin limit, the double exchange model can produce metallic behavior for all temp eratures, which still contradicts all experi mental data. Yunoki et al.20,21 have produced the closest qualitative result to experiment utilizing a modified dou ble exchange model. Th ey used a one-orbital Figure 2-7. Temperature dependent resistivity of La1-xCaxMnO3 (x = 0.25) in various applied magnetic fields. Schiffer et al. Phys. Rev. Lett. 75, 3336-3339 (1995), Fig. 2, pg. 3337. or FM Kondo model with varying chemical potential, . In their work, Monte Carlo simulations of classical t2g spins produced low temperature instabilities at certain electron

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16 densities as was varied. When a Heise nberg coupling was introduced between the localized t2g electrons, these instabilities manifest themselves as two PS regions: one hole doped and FM, the other undoped and AF. Thus we have a double exchange interaction, albeit a sophisticated one, that at least hints at the metallic and insulating nature of CMR manganites. Double exchange theory may anticipate the ferromagnetic-paramagnetic phase transition, but it does not account for the experimentally observed metal-insulator transition (MIT). This is because the availabl e mechanism for describing resistivity in the double exchange model, the scattering of electro ns due to spin disorder, is insufficient in magnitude to cause this dynamic crossover from a metallic to very insulating state, as seen in experiments. It is this high temperat ure insulating state, either PI or CO, which seems key to understanding the CMR effect. In addition, the double exchange interaction does not predict the role of varying the hole-d oping, reflected in the complex structure of these materials at low temperatures as illustra ted in Figure 2-5. Understanding all of these phenomena will require augmenting the doubl e exchange mechanism with a universal theory that accounts for not only the orbital and spin ordering eff ects highlighted above, but also incorporates the various structural and lattice effects ignored thus far in this discussion. 2.3.2 Beyond Double Exchange, Part 1: Jahn-Teller Polarons In general, cation substitutions and ox ygen stoichiometry can both control the various physical properties of perovskite materials. This is especially true in distorted perovskites like the manganites. These dist orted structures experience electron-lattice coupling via two main mechanisms. One quan tity, the tolerance factor, involves the crystal structure’s effect on hopping conduc tion. Specifically, different choices for the

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17 trivalent and divalent cations in the mixed valence manganites change the internal stresses acting on Mn-O bonds due to the cati on’s varying size. These internal stresses have a strong effect on the hopping amplitude due to changes in the relative Mn-O-Mn bond angle22, via compression and tension of the bonds. The other mechanism involves electron-phonon coupling that links lattice de viations in the crystallographic structure with deviations in the electronic configur ation away from their average values. In LaMnO3, this electron-phonon coupling is an inhe rent electronic instability native to the MnO6 octahedra, the Jahn-Teller distortion23. La c a O2 Mn O1 1.92 2.15 La c a O2 Mn O1 1.92 2.15 Figure 2-8. The projection of the crystal structure of LaMnO3 along the b axis. The unequal Mn-O bond lengths illustrate the Jahn-Teller distortion. In the manganites, the MnO6 octahedra experience a distortion in the Mn-O bond lengths along different crysta llographic directions. This distortion arises from an electronic instability, the Jahn-Teller (JT) instability, intrinsic to a Mn3+ ion situated in an

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18 octahedral crystal fieldd (see Figure 2-8). Simply put, the JT distortion occurs when a given electronic level of an ion or cluster is degenerate in a structure of high symmetry, this structure is generally unstable, and the cluster will present a distortion toward a lower symmetry ionic arrangement23. JT electron-lattice interactions, which are an electron-phonon coupling mechanism, can be found in most orbitally degenerate d-electron ions, as in Ni3+ or Cu2+, located in octahedral systems4. The JT distortion has profound egt2g 3 dorbitals Mn3+ dxydxz dyz223 zrd22xyd Oxygen ion Mn3+ion egt2g 3 dorbitals Mn3+ dxydxz dyz223 zrd22xyd Oxygen ion Mn3+ion Figure 2-9. Jahn-Teller distorti on and its effect on the ener gy level splitting of the Mn3+ orbitals. The distortion stre tches the Mn-O bond in the z direction and compresses it in the x and y direction. d Two “breathing” modes can also occur around a given Mn which couples to changes in the eg density.

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19 influence on the phenomena of the CMR mangan ites. This is because the distortion acts to couple the eg conduction electrons with a lattice instability, represented by an oxygen displacement. This distortion lifts th e orbital double degeneracy of the eg electrons seen in Figure 2-3 and lowers the electronic energy (see Figure 2-9). Note that manganites like SrMnO3, which have no eg electrons, do not have a JT distortion. Thus intermediate doping regimes will have a modified lattice from either pure Mn3+ ion materials or pure Mn4+ ion materials. In A1xBxMnO3 (0 x < 0.2), the JT distortion is a static effect that produces a cubic-tetragonal structural phase transition. This is a static or frozen effect because the bond length distortions are cohe rent and have long-range or der throughout the crystal. The JT effect is very strong here with a bond length change of ~10 % of the mean Mn-O distance24. As x is increased from zero doping, the st ructural transition temperature and the size of the long-range, coherent distor tion decrease rapidly. In the low doping regime ( x < 0.20), the interplay of this diminished static JT interaction and the large Hund coupling of the DE interaction dominate the cu rrent understanding of the physics of these manganites. A strong electron-phonon coupling can locali ze carriers, because an electron in a given Mn orbital causes a loca l lattice distortion (e.g. the eg electron in the Mn3+ ion’s 223 zrdorbital), which produces a pot ential energy minimum that can trap the electron in that orbital. If the coupling is strong enough, these tendencies lead to the formation of a trapped state called a polaron. Following this line of reasoning leads to a competition between electron localization, or trapping, this would result in insulating behavior, and electron delocalization through hopping conduction, which can lead to metallic behavior.

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20 This argument was first proposed by Millis et al.25 as an explanation for the observed CMR phenomena in manganites with intermediate doping, 0.2 x < 0.5. They used a model Hamiltonian that coupl ed the lattice distortion vi a dynamic JT coupling, which excludes long-range order in favor of fluctu ations from site-to-site, to the double exchange mechanism as follows: †2()() 2ab c effDEjazjbz jazj chk HHSgcQjcQj S (2-10) Here we have used HDE from Eq. 2-1 and added a term which has external magnetic field h coupled to the core spins Sc. The lattice distortion terms have electron-phonon coupling g and phonon stiffness k The electron is coupled to the distortion via a traceless symmetric matrix, Q = r [cos( ) z + sin( ) x], that parameterizes the cubic distortion in polar coordinates, r and Heff is solved in the dynamical mean field approximation26 with JH and treating core spins and phonons as classical. The calculation parameterizes the effective interaction in terms of a dimensionless quantity that is the ratio of the energy gained by an electron localizing and forming a polaron, EJT, and the effective hopping parameter, teff, which follows the DE mechanism. In the manganites, the DE mechanism and asso ciated magnetostructural effects can have strong dependence on temperature, ma gnetic field and doping. Likewise teff, and therefore should reflect these changes in the fundame ntal properties of the manganite, such as the resistivity. In this context when T > Tc the effective coupling should be relatively large due to a small effective hopping that leads to insulating behavior (electron localization), while for T < Tc the emergent FM increases teff so that decreases sufficiently to allow metallic behavior (e lectron delocalization). The results of the

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21 Figure 2-10. Magnetic field dependence of resi stivity calculated using the direct integration method for electron density n = 1 and = 1.12. h = 0.01t corresponds to 15 T if t = 0.6 eV and Sc = 3/2. Millis et al. Phys. Rev. Lett. 77, 175-178 (1996), Fig. 2, pg. 177. calculated resistivity under this picture are shown in Figure 2-10. It is clear that the theoretical result correctly captures, qualitatively and somewhat quantitativelye, the experimentally observed results seen in Figure 2-7. 2.3.3 Beyond Double Exchange, Part 2: Percolating Phase Separation The remaining feature of the experimental data not fully realized up to this point is the nature of the inherent phase separa tion (PS) that underlies the CMR phenomena. In 1999, Uehara et al.2 produced striking evidence for th e existence of pha se separation in La5/8yPryCa3/8MnO3 (LPCMO) (see Figure 2-11). In addi tion to highlighting its existence over a wide range of temperatures, they al so showed its percolating nature. Several experimentalists further studied the percolation picture. Kim et al.5 who used e The theoretical results underestimate the role of the eg electron density, n For n = 0.75, which would match the experimental results of Figure 2-7, th e theoretical result shows anomalous low temperature behavior25.

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22 magnetization correlated with c onductivity measurements to show that a classical 3D percolation threshold (~17 % metallic fraction ve rsus insulating fraction as predicted by the general effective medium (GEM) theory ) is reached at the MIT in LPCMO over a wide range of doping levels. Figure 2-11. Dark-field SEM images for La5/8yPryCa3/8MnO3 obtained by using a superlattice peak caused by charge ordering (CO). a) shows the coexistence of charge-ordered (insulating), the light area, and char ge-disordered (FM metallic), the dark area, domains at 20 K for y = 0.375. The curved dark lines present in CO regions are antip hase boundaries, frequently observed in dark-field images for CO states of La0.5Ca0.5MnO3. b) and c) obtained from the same area for y = 0.4 at 17 K and 120 K, respectively, show the development of nanoscale charge -disordered domains at T > Tc. Uehara et al. Nature 399, 560-563 (1999), Fig. 3, pg. 562. Zhang et al.27 showed direct observation of percolation in LPCMO using low temperature magnetic force microscopy (MFM) (Figure 2-12) which they correlated with resistivity measurements to show that the percolation of the FM metallic grains is responsible for the steep drop in resistivit y associated with all CMR manganites.

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23 Theoretical work on PS behavior of the manganites was spearheaded by Dagotto and Moreo14-16,20,28 showing the various mechanisms be hind the intrinsic inhomgeneities Figure 2-12. Scans (6 m x 6 m) of LPCMO. On cooling, FM regions (dark area) grow in size. Percolation is visible below 113 K. Zhang et al. Science 298, 805-807 (2002), Fig. 1a, pg. 806. that occur in the CMR manganites over a wi de range of hole doping, temperatures and magnetic fields. Specifically, they showed15 that the typical transport properties of the manganites, the room temperature insulator transforming to a low temperature “bad” metal, are directly correlated with the inherent inhomogene ity of the electronic phases. Initially they employed a random resist or network in both 2D (square) and 3D (cubic) clusters. They mimicked the mixed phase behavior of La5/8yPryCa3/8MnO3 with link resistances in parallel, randomly assigne d as metallic or insulating, with a fixed metallic fraction, p The metallic resistance values were taken from experimental data on LPCMO with y = 0 and the insulating values from y = 0.42, although the calculated results appear qualitatively independent of the material. Upon solving the Kirchhoff equations, they varied p for different temperature sw eeps and found the calculated resistivities showed pure insulating behavior for p = 0 and pure metallic behavior at p = 1 with intermediate values of p producing a MIT and percolatin g behavior (Figure 2-13).

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24 Figure 2-13. Random resistor network re sistivity for various metallic fractions, p in 2D and 3D (inset). A broad peak appear s at intermediate temperature and values of p in qualitative agreement with experimental evidence. Mayr et al. Phys. Rev. Lett. 86, 135-138 (2001), Fig. 1c, pg. 135. However, the broad peak found using this mode l does not accurately reflect the transport behavior of most manganites. To solve this, a temperature dependent percolation mechanism, separate from the variation of the metallic fraction must be introduced. The introduction of a temperature depende nt percolation mechanism is certainly reasonable in CMR materials, because of the resistivity’s extreme sensitivity to temperature changes, especially arou nd the MIT. A random field Ising model28 (RFIM) is used to ascribe a temperature dependent and disorder induced type of percolative PS, which could simultaneously capture the elect ronic and magnetic competing phases. This analysis produced beautiful phase behavior that showed resistivity very similar to experiment and magnetic phase separation (Fig ure 2-14) reminiscent of the SEM images in Figure 2-11 and the MFM images in Figure 2-12.

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25 Figure 2-14. Monte Carlo results for the RFIM model. A) the net resistivity with similar percolating behavior to the RRN model for intermediate metallic fractions and temperatures but with much sharper changes in resistivity. B) and C) the Monte Carlo results for the magnetic phase behavior in real space, with the regions in B) correlating with C). These results are in very good agreement with experiment. Mayr et al. Phys. Rev. Lett. 86, 135-138 (2001), Fig. 2, pg. 136. This PS work indicates that the observed CMR phenomena, including the complex electronic and magnetic phase behavi or, can be explained in terms of small changes in the metallic fraction with conc omitant changes in the conductivity of the insulating region. These last phenomena will be the driving force behind much of the new experimental work described in th e remainder of this dissertation. In essence, PS in the manganites provides the mechanism for the CMR properties, while the DE interaction and Jahn-Teller polarons describe the FM metallic and insulating regions, respectively. In summary, CMR effects are best understood in terms of competing phase separation betwee n double exchange-governed, ferromagnetic regions and insulating regions fo rmed by Jahn-Teller polarons. A B C

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26 CHAPTER 3 DIELECTRIC RELAXATION PHENOMENA 3.1 Time-Dependent Dielectric Resp onse: Polarization v. Conduction Dielectric relaxation probes the interaction between a medium, with some polarization, P and a time-dependent electric field, E(t) The relaxation consists of the recovery of the polarization on the sudden remo val or application of the electric field. The resulting polarization respons e of the medium (ionic, electronic, molecular, etc…), to the ac electric field, characterizes the time-s cale, via the characteristic relaxation time or times, and amplitude of the charge-density fluctuations inherent to the sample. The mechanism that governs these fluctuations va ries from material-to-material. In most homogeneous materials, where the charge-den sity can depend on molecular, ionic or electronic sources, the simplest polarization mechanism is a permanent or induced dipole moment that is reoriented by the ac electric field. The time-scales associated with the reorientation of dipoles cover a broad range with the associated frequencies spanning from 10-4 Hz, for Si-Te-As-Ge (STAG) glass, to 1012 Hz, for low-viscosity liquids. It is important here to differentiate betw een the two main processes associated with the charge-density of the medium in question: charge polarization, and charge current. In a static electric field, polarization occurs due to a net displacement of the charges in the medium by some finite amount, while charge conduction results from a finite average velocity in the motion of charges that depends on the dc conductivity, In this way, polarization mechanisms and the related charge dipoles are incapable of contributing to a dc charge current.

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27 At ac, we can draw an even sharper dis tinction between the charge current and the charge polarization because there is an a ssociated time-dependen ce of the polarization response to the driving force. This time-dep endence is a delay in the polarization to follow the time-dependence of the elect ric field. Below we show how the time-dependence of the polarizat ion leads to a polarization cu rrent that is fundamentally different than the dc charge current. A standard configuration for a capacitor ha s two parallel electr odes, separated by a dielectric with thickness, d On applying an ac voltage, V(t) to the capacitor, a spatially homogeneous electric field, E(t) = V(t)/d is induced perpendicula r to the plane of the dielectric. The free space charges in the elect rodes will respond instantaneously to this ac electric field, while the dielectric medium will exhibit a time-dependent delay. This response can be expressed in term s of the dielectric displacement, D(t), which gives the total charge density induced at th e electrodes of the capacitor due to E(t) as follows: 0()() DtEPt (3-1) where 0E is the induced free space charge of th e electrode. The static, time-independent, polarization, P, is related to the static electric field as follows: 0PE (3-2) We introduce time dependence to the polar ization in Eq. 3-2 by means of a dielectric response function, f(t) and assume the time-dependent electric field, E(t) is a continuous function of time such that the to tal amplitude can be written as the sum of discrete increments, E(t) d t The function, f(t), carries the response of the dielectric medium to the ac electric field. The time dependent polarization can be written as:

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28 0 0()()() PtfEtd (3-3) where t is the time before the measurement is made at time, t and we define the dielectric response function such that when t = 0 that f() = 0 If we now impose that E(t) take the following time dependence: 0 00 for 0 ()() for 0 t EtEt Et (3-4) then the dielectric displacement in Eq. 3-1 takes on the following form: 00 0()()()tDtEtfd (3-5) The total current, I(t) flowing in this capacitor can then be written as follows: 000000()() ()()() dtdPt I tEEEEtft dtdt (3-6) where E0 is the dc current flowing through the system. The polarization current arises from the inability of the charge species to fo llow the time dependence of the electric field that is driving the system. In the present form of the electric field (3-4), the polarization current must go to zero as t This implies that at infinite times that the polarization must approach a constant29, defined by: 0000 0()()(0) PEftdtE (3-7) where (0) is the static susceptibility. As a c onsequence of Eq. 3-7, no charge species may be transferred to or from the dielectric system. In contrast, the dc conduction current, E0, exists only in the presence of free charge species that move across the dielectric

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29 from one capacitor electrode to another. On ce the polarization of the system reaches saturation at t i.e. after a sufficiently long time, the electric field can be switched off and the resulting depolarization can be observed. This is an important aspect of dielectric relaxation, in that as the polarized char ge species begin to relax and ultimately depolarize, the dynamics of the charge species can be observed in th e absence of the dc charge conduction. In practice, this is how many experimenta lists ascertain the dielectric response function, and thus the time-dependent dielectric permittivity, (t)29. 3.2 Frequency-Dependent Dielectric Response While the most natural and readily acce ssible representation of the various dielectric and electrodynamic processes pr esent in insulating me dia may be in the time-domain, experimentally the most prac tical representation is usually in the frequency-domain. Dielectric response is us ually studied as a f unction of frequency, instead of time, because measurements can be made at a given frequency, or range of frequencies, with a high degree of accuracy and precision due to the present state of rf signal technology. It is po ssible to move back and forth from the timeto frequency-domain utilizing Fourier tran sformations and numerical integration. Given a time-dependent function, F(t) the Fourier transform of it into the frequency-domain has the following form: 1 ()()()exp() 2 FtFFtitdt (3-8) where is the frequency and therefore F() is the frequency spectrum associated with F(t) This is a powerful tool because the F ourier transform of a real, time-dependent function produces a complex, frequency-depe ndent function that contains information

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30 about the amplitude and phase of the time-de pendent signal. Additionally, the Fourier transform allows us to write integrals of the form in Eq. 3-4 as th e product of the Fourier transforms of the functions in the integrand30. In particular, it allows us to define a frequency-dependent susceptibilityf, which is just the Fourier transform of the dielectric response function, f(t) as follows: 0()()()()exp() iftitdt (3-9) Therefore, we now have a way of simultane ously studying the dielect ric response that is in-phase, () and out-of-phase, () with the ac electric fi eld. The ability to study the in-phase and out-of-phase components of a dielectric response is at the heart of dielectric relaxation spectroscopy. In partic ular, we would like to know what each of the components of the dielectric response, in this case the dielectric susceptibility, contribute to the relaxation process. We can answer this question by returning to the total current of Eq. 3-7 and writing the corresponding frequencydomain response as follows: ()()()IEiD (3-10) where ()/()DttiD Using Eq. 3-1, the current can be written in terms of the frequency-dependent polarization, P()=0E()(), such that, after some algebra: 00()()1()()IiE (3-11) f When we speak of the frequency-dependent susceptib ility is important to note that multiple unique and separate mechanisms may contribute to it. Therefore, () is comprised of the sum of the susceptibilities due to each of these independent mechanisms.

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31 Here we see that the imaginary component of the susceptibility, (), is in-phase with the dc conduction current and therefore contribut es to the total dissipa tion, or loss, of the charge polarization at ac. Therefore () is termed the dielectric loss. The real part of the susceptibility, (), is out-of-phase with the dc c onduction and thus acts to conserve charge polarization. The picture produced from this analysis is th at the total dielectric response and relaxation is driven by the competing mechanisms of charge polarization, or charge storage in terms of a capacitor, due to (), and charge dissipation, due to the ac loss, (), in the presence of the dc conduction current, E. 3.3 Dielectric Response Functions In the previous section we derived an e xpression for the total frequency-dependent current, I(), present in a capacitor comprised of a dielectric medium between two electrodes. This current was expressed in terms of the complex susceptibility, () and dc conductivity, (=0). When measuring the total dielectr ic response, at ac, it is usually not possible to look at the dc and ac contributions se parately. Specifically, the frequency-dependent current simp ly describes the movement of charge in response to the driving force provided by the ac electric field, E(). This movement of charge is due to both dc and ac processes and thus any dielectr ic response that is measured will contain contributions from both dc and ac conduction. Th erefore, we think in terms of measuring an effective dielectric respons e and using analysis to separa te the various contributions. This can be done in terms of several different, but equi valent, dielectric response functions that are consequences of the macr oscopic Maxwell’s equa tions. In particular, we will define the impedance/admittance and dielectric permittivity, from which all other dielectric responses can be eas ily generated or represented.

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32 3.3.1 Impedance and Admittance In a dielectric system, there are two ways to generate a dielectric response: the measurement can source a voltage, V(), that drives a current, I() or vice versa. There is a phase shift between I() and V(), such that the response can be resolved into in-phase and out-of-phase components. The response as sociated with this phase shift can be represented in terms of two dielectric functions, the admittance, Y()=I()/V(), and the impedance, Z()=V()/I(). In practice, the admittance is used when the experiment is sourcing voltage and measuring current, while impedance is used for sourcing current and measuring voltage29. In a system that is modeled as an ac dielectric process acting in parallel with a dc conduction pr ocess (evident in the analysis above in Section 3.2), the admittance is a natural representation because it involves a constant voltage, V(), across both parallel elements. However, the reciprocal nature of the admittance and the impedance allows for the easy transformation of one quantity to another when necessary as when a dc conduction path exists in series with the dielectric medium (Section 4.3.4). A system whose understanding benefits from impedance/admittance analysis is the CBGB CVGV CBGB CVGV Figure 3-1. Barrier-volume capacitor. CB and GB are the barrier capacitance and conductance, respectively. CV and GB are the volume capacitance and conductance, respectively.

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33 barrier-volume problem that generalizes two di stinct dielectric regi ons in series. Each dielectric region is m odeled as a capacitor, C, in parallel with a dc conductance, G (Figure 3-1). This circuit is desc ribed by the Maxwell-Wagner relaxation31 developed to describe the behavior of a c onducting volume or “bulk” dielectric material with a barrier or “contact” region that is highl y capacitive and less conducting (CV << CB and GB << GV). The impedance of the circuit in Figur e 3-1 is written explicitly as follows: 1/ 1/ 11V B VBG G Z ii (3-12) where V = CV/GV and B = CB/GB. A schematic representation of Eq. 3-13 (Figure 3-2) shows the two distinct dielectric regions: volume (small arc) and barrier (large arc). 0100200300 0 100 200 Z'' Z' V = 1B = 1 = GV/ CB0100200300 0 100 200 Z'' Z' V = 1B = 1 = GV/ CB Figure 3-2. The complex impedance diagram of the barrier-volume ci rcuit. The large arc is due to the barrier region and the sm all arc is due to the volume region.

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34 Figure 3-2 illustrates the utility of a complex impedance diagram. The diagram reveals information about the time constants of the volume and barrier regions, as well as indicating the crossover frequenc y that divides the two dielectric responses. If we invert Eq. 3-13 we can write the admittance of the response (Eq. 3-14), where r = CB / CV. r i i i G YV B V B 1 ) 1 )( 1 ( ) ( (3-13) A complex admittance plot of Eq. 3-14 (not s hown) reveals that the volume and barrier regions are two distinct vertical lines with the x -intercepts indicating GV and GB, respectively. 3.3.2 Dielectric Permittivity The frequency-dependent permittivity, () is defined in terms of the complex susceptibility as follows: 0()1()()()()ii (3-14) such that, D()=()E() In this way, the dielectric perm ittivity is a measure of the total response of the dielectric system, due to both the free space and material dependent contributions, to an ac electri c field. It does not however, include the dc conduction contributions which are intrinsic to any measurement, while being extrinsic to the true dielectric response. It is po ssible to express the dielectric permittivity in terms of the ac conductivity via 1) Ampere’s law, t E E H / ) ( ) ( and 2) the divergence theorem, 0 ) ( H (Eq. 3-15). i ) 0 ( ) ( ) ( (3-15)

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35 The main advantage in using the dielectric permittivity representation of the dielectric response is it is directly proportional to the cap acitance; the quantity most frequently measured to study dielectric rela xation. The complex capacitance is defined as follows: ()()()()() A CCiCi d (3-16) where A is the area of the electrode and d is the thickness of the dielectric. In this way, the complex capacitance is composed of a real part, C() which reflects the capacitance associated with charge storage and an imaginary part, C() which accounts for the ac loss. The study of the complex capacitance will dominate the remainder of this dissertation. 3.3.3 A Dielectric Case-Study: Disordered ConductorsThe frequency-dependent conductivity of disordered conductors exhibits striking universal behavior. In these systems, meas urements of the ac c onductivity as a function of frequency, at different temperatures, can be scaled onto a “master” curve. This “master” curve approximates a power-law dependence of frequency, with power 1. This behavior occurs independent of conductio n mechanism (electroni c or ionic) and the state of the system (amorphous or polycry stalline). The only commonality in these universal ac conductors is that they share strong disorder with conduction dominated by percolation. The ac conductivity of this broad class of mate rials is best understood in terms of a random resistor cap acitor network (RRCN) (Figure 3-3). The microstructure of these materials is a percolating network of conducting and capacitive regions. This model is confirmed theoretically by considering a macr oscopic model of disorder in a solid with

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36 Figure 3-3. Random resistor cap acitor network that describes the dielec tric behavior of the universal ac conductivity materials. heterogeneous conducting phases. This model assumes that the disordered system has free charge carriers with current density J(r,t) = g(r)E(r,t) where g(r) is the local conductivity at the corresponding position in the solid and displacement D(r,t )= E(r,t) with bound charges that are described by th e high-frequency dielectric constant, the limit of in Eq. 3-14. From Gauss’s law, D and the electrostatic potential, ) ( t r E we can write the continuity equation, 0 J in terms of the local conductivity and the bound char ge dielectric constant (Eq. 3-16). Dyre showed32 that 0 r g i (3-17) when Eq. 3-16 is discretized in the presence of a period ic potential across the RRCN circuit for a cubic lattice in D dimensions with spatial length L separating each node, that the free charge ac conductivity, () is as follows:

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37 i L YD 2) ( ) ( (3-18) The frequency dependence of the ac conductivity is governed by the capacitors influence on the local potential at each node, that in turn determines the resistor currents. In the high frequency limit the capacitor admittances (susceptance) dominate the response and produce a spatially homogeneous electric field.

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38 CHAPTER 4 CAPACITANCE MEASUREMENTS AN D EXPERIMENTAL TECHNIQUES 4.1 Thin-Film Metal-Insulator-Metal Capacitors Traditional MIM capacitors are easily a pproximated by the parallel plate model of capacitance. Where two largeg, two-dimensional, metallic electrodes with area A enclose an insulating medium, w ith dielectric constant and thickness d capable of storing charge as defined by the geometrical capacitance relation (Eq. 4-1), where 0 = 8.85x10-12 F m-1, is the permittivity of free space. 0 gA C d (4-1) The measured capacitance can be heavily modified from the geometrical capacitance (Eq. 4-1) when d becomes vanishingly small. Specifically, an interface capacitance due to an electron screening length dependent voltage drop at the metal-dielectric interface of the electrodes can come to dominate the measured capacitance33,34. These screening effects can be further enhanced in the presence of a magnetic field when the MIM has fe rroor paramagnetic electrodes35. In our experiments, additional complications to the traditional parallel plate model arise from the wide variability of the conductivity of our base electrode, with magnetic field and temperature. In the following sections we outline the methodology used and considerations made when meas uring our novel MIM capacitors. g Large in this case is defined as the area of the elect rode should be much larger than the thickness of the dielectric; A << d

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39 4.2 Metal-InsulatorMetal (MIM) Fabrication The capacitors used in our experiments ha ve a consistent trilayer architecture comprised of a polycrystalline, thin film (La1xPrx)5/8Ca3/8MnO3 (LPCMO) base electrode (C in Figure 4-1) deposited on a highly insulating NdGaO3 (NGO) substrate (D in Figure 4-1), an amorphous AlOx dielectric (B in Figure 4-1) and a polycrystalline, thin film Al counter-electrode (A in Figure 41). Contact is made to the baseh and counter electrodes using pressed indium and fine gold wire held in place with a dilute silver paint, respectively. A B C D A B C D Figure 4-1. Schematic of MIM. A) the metallic counter electrode. B) the dielectric. C) the manganite base electrode. D) the substrate. The LPCMO layer is grown using pulsed layer deposition (PLD) in a high vacuum chamber with a base pressure of ~ 1 x 10-6 Torr. A NGO substrate is loaded onto the high temperature sample holder a nd held at a temperature of ~800 C during the growth process. The LPCMO thin films used in our experiments ranged in thickness from 400 to 600 as determined by atomic force microscopy (AFM) profiling. The AlOx dielectric is deposited via rf magnetron sputtering of an alum ina target in an ultra high vacuum chamber with a base pressure of ~ 1 x 10-8 Torr. The dielectrics used were in a thickness range of 100 to 150 as measured by an Inficon in situ quartz crystal thickness h Silver paint was also used to make contact to the base electrode in some occasions. We found no qualitative distinction between the two contact methods.

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40 monitor. In the past33,34, this method has produced hi gh density, homogenous AlOx down to thicknesses of ~ 30 . The counter-electr ode is deposited via thermal evaporation of 99.999% pure, aluminum wire, through an alum inum shadow mask, onto the dielectric layer, in a high vacuum chamber with a base pressure of ~ 1 x 10-7 Torr. 4.3 Measuring Capacitance: Tools and Techniques Fundamentally, all capacita nce measurement techniques involve measuring a material’s ability to store and transfer electr ic charge in the presence of an ac electric potential. There are three major instruments that are used in our experiments. The first is the impedance-gain/phase analyzer, which is a self contained ac response unit that simultaneously measures the in-phase and out-of-phase response of a system to a small ac perturbation (either current or voltage). Second, we discuss th e lock-in amplifier; it is the most versatile in terms of its ability to m easure a wide variety of ac responses and its ability to be integrated into measurement schemes with other instruments. We then discuss the importance of the capacitance bri dge to the experimental work we have conducted. 4.3.1 Impedance Analyzer The impedance analyzer used in our e xperimental work is a Solartron 1260A Impedance/Gain-Phase Analyzer. It is desi gned to measure the complex impedance, Z*=Z1+ iZ2, of a sample in the frequency range of 10 Hzi to 32MHz and excitation voltage amplitude, Vout 3 Vrms. Whereas, the lock-in amplifie r usually uses an external reference signal that “locks” onto the input signal, the 1260A has a built in frequency i Some would argue that measurements at frequencies less than 1 Hz are no l onger “ac” measurements, but really dc in nature. The converse opinion is th at there is no true “d c” measurement because all measurements occur over some nonzer o time, and thus have a frequency inherently associated with them. This is left as a question for the ages.

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41 synthesizer that drives the signal for the te st sample, and the reference signal for the voltage measurement. The 1260A outputs a signal that drives the sample at any specified frequency and amplitude in the range stated above. In turn, a cu rrent amplifier input measures the associated current through th e sample while simultaneously, differential voltage amplifier inputs, VHI and VLO, measure the voltage across the sample using an internal amplifier. The applied sinusoidal, ac voltage and the resultant ac current are measured to arrive at the complex impedance, Z* = V / I The applied voltage, V(t)=V0cos(t) produces electronic polarization in the sample, and the resultant current, I(t)=I0cos(t+) will have the same frequency, but a different amplitude and phase, The 1260A will then report the complex impedance in a variet y of ways, the most general of which is the magnitude of the impedance, |Z| and the phase angle, such that the complex impedance is, Z*=|Z|exp( i ) This in turn can be written in terms of its real and imaginary parts to reveal the in-phase a nd out-of-phase components of the response. The major experimental disadvantage presented by the 1260 A is that the confidence interval of its impedance meas urements is severely reduced for high impedance samples, see Fig. 4-2. This is primarily because the input impedance at the VHI and VLO terminals is 1 M // 35 pF, according to spec ifications. This input impedance

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42 100 1k 10k 100k 1M 10MFrequency (Hz)100M 10M 1M 100k 10k 1k 100 10 1 100m 10mImpedance (Ohms)0.1% 0.1 0.2% 0.2 1% 1 10% 10 100 1k 10k 100k 1M 10MFrequency (Hz)100M 10M 1M 100k 10k 1k 100 10 1 100m 10mImpedance (Ohms)0.1% 0.1 0.2% 0.2 1% 1 10% 10 Figure 4-2. Confidence intervals of the measured impedance as a function of the measurement frequency. A sample with Z > 10 M will have a 10 % error in the measured impedance amplitude and 10 error in the measured phase angle at all frequencies. problem could have been circumvented by introducing a front-end for the 1260A that would include high input impedance buffers, such as operational amplifiers configured as voltage followers, between the electrodes and the differential amplifier that would increase the confidence levels at low frequenc y. However, this was not pursued in our lab because the bulk of our results were obtained with the capacitance bridge outlined in the following section.

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43 4.3.2 Lock-In Amplifier The Princeton Applied Research PAR 124A lock-in amplifier allows for the measurement of a wide variety of ac responses with a high degree of precision. While a lock-in amplifier is a powerful and extremely important tool it is also quite simple, especially when compared to the more complicated impedance-phase/gain analyzers discussed below. A schematic diagram of a lo ck-in amplifier is shown in Figure 4-3. It consists of five stages: 1) an ac signal amp lifier, which is the input for the signal to be measured; 2) a reference input; 3) a phase sensitive detector (PSD) that acts as a multiplier; 4) a low-pass filter; and 5) a dc am plifier. The output of the dc amplifier is a voltage that is proportional to the amplitude, V0, of the input signal, V(t) = V0cos(0 t+) where 0 is the frequency of the signal and is a phase shift. Signal Amplifier Phase Sensitive Detector Low-pass Filter DC Amplifier Signal Monitor Reference Out Signal In Reference In Output Signal Amplifier Phase Sensitive Detector Low-pass Filter DC Amplifier Signal Monitor Reference Out Signal In Reference In Output Figure 4-3. Schematic of a lock-in amplifier The major differences between an ac voltm eter, which will also report the voltage amplitude from an input signal, and a lock-i n amplifier are that: 1) a lock-in uses a reference signalj that is synchronized with the input signal being measured and 2) lock-in j The reference signal is derived from a periodic voltage source and is usually on the order of 1Vp-p.

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44 amplifiers will measure the in-phase and out -of-phase components of the input signal with precise frequency control. The signal amplifier receives th e input, amplifies the voltage by a gain of Gac and filters out most, but not all, extraneous signals at other frequencies. Then the amplified signal input, Vac(t) = GacV0cos(0 t) and the synchronized, or triggered, out put of the reference input, VRI(t) = A0cos(0 t) (we are assuming = 0)k are multiplied in the PSD that produces the following output: 0001 ()1cos2 2PSDacVtGAVt (4-2) The amplitudes of the dc term and the second harmonic term are both proportional to our input amplitude, and thus it is redundant to pursue both of them. Therefore, the signal in Eq. 4-2 is passed th rough the low-pass filte r that attenuates the second harmonic term and also integrates the signal greatly reducing random noise. This output is then fed into the dc amplifier which increases the signal from the low-pass filter by a gain of Gdc, resulting in an output of: 001 2outdcacVGGAV (4-3) Figure 4-4, shows a comparison of the impe dance of an LPCMO film as measured by the Solartron 1260A (black squares) and th e PAR 124A Lock-in with output read by a k Having 0 and 0 complicates the result because the input signal will have some unknown phase and the output in Eq. 4-3 will be 001 cos 2outdcacVGGAV (4-4) In this instance, the phase of the input reference would be tuned such that the signal in Eq. 4-4 is a maximum.

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45 75100125 106107 Solartron 1260A PAR 124A Lock-in Amplifier |Z| ()Temperature (K) Sample Output Lock-In RBallast 75100125 106107 Solartron 1260A PAR 124A Lock-in Amplifier |Z| ()Temperature (K) Sample Output Lock-In RBallast Figure 4-4. The magnitude of the impeda nce of a 400 LPCMO film as measured by the Solartron 1260A (black squares) a nd PAR 124A Lock-In (red circles), at a frequency of 50 Hz. The arrow indicat es the where the i nput impedance of 1260A lies, and we note that is wher e the measurements collapse onto one another. Inset: circuit for ac resistance m easurement with PAR 124A. Keithley 182 nanovoltmeter (red circles) that has an input impedance of 10 G This clearly illustrates the advantag e of the two-terminal lock-in technique (Figure 4-4 inset) over the impedance analyzer in terms of high impedance measurements. Thus lock-in amplifiers are ideal for meas uring ac responses where the si gnal-to-noise ratio is very high, even when the signal of interest is a few nanovolts buried in a background signal orders of magnitude larger.

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46 4.3.3 Capacitance Bridge The capacitance bridge used in this work is the Andeen-Hagerling AH 2700A. Like the Solartron 1260A, it is a stand alone ac response unit that requires no external reference signals to perform its measuremen ts. Unlike the 1260A, it utilizes a small ac voltage source and a balancing ci rcuit to achieve its results, and measures with maximum accuracy and precision on samples that have medium to high impedances (105 < |Z| < 1015 ). The AH bridge operates in a frequenc y range of 50 to 20,000 Hz and ac voltage amplitude of 30 mVrms to 15 Vrms with a best capacitance resolution of 0.01 x 10-18 F and a lossl resolution of 3 x 10-16 S. The bridge circuit is depicted in Figur e 4-5. Like the lock-in amplifier, its operational components consist of five stages: 1) a sinusoidal, voltage generator that has precise amplitude and frequency control; 2) a high precision ratio tr ansformer; 3) Tap 1 which controls the voltage acr oss the standard impedance; 4) Tap 2 which controls the voltage across the sample, or unknown, impeda nce; and 5) the voltage detector. To balance the bridge, the voltage between Ta p 1 and ground and the voltage between Tap 2 and ground must be equal to the voltage acr oss Leg 3 and the voltage across Leg 4, respectively. This ensures that when the bri dge is balanced that the voltage across the detector is zero. When the detector reads zero voltage, or as close to zero as possible, the relationships between the standard an d sample impedances are as follows: 1 0 2 XN CC N (4-5a) l Loss is in reference to the component of the impedance 90 out-of-phase with the capacitive element. It is usually measured in terms of Siemens (S) with 1 S = 1 Ohm-1 = 1 mho, in older notation.

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47 2 0 1 XN R R N (4-5b) where N1 and N2 are the number of turns in leg 1 and leg 2, respectively, of the secondary windings of the transformer. This capacitance bri dge circuit illustrates the main benefit of GeneratorDetector Tap 1 Tap 2 Leg 3 Leg 1 Leg 2 Leg 4 Ratio Transformer Sample Impedance H L CXRXC0R0Standard Impedance BNC Connectors G GeneratorDetector Tap 1 Tap 2 Leg 3 Leg 1 Leg 2 Leg 4 Ratio Transformer Sample Impedance H L CXRXC0R0Standard Impedance BNC Connectors G Figure 4-5. Schematic of the capacitance bridge circuit. The standard impedance elements are fused-silica capacitances for C0 and variable pseudo-resistors for R0. The unknown sample is expressed by CX, the in-phase component, and RX, the out-of-phase component. all bridge technology, in th at the act of measuring th e unknown impedance requires measuring small deviations from zero. The success exhibited by the AH 2700A depends on one key external component: a guarded three-terminal meas urement. In an unguarded measurement there is a strong likelihood that additional current could be dr awn across the sample due to a net voltage drop between L and H in Figure 4-5. To a void this, a guard (G) is placed around the sample, in our case an electri cally isolated copper cylinder, which is connected to the ground of the bridge circuit (Figure 4-5). Therefore, no stray voltages, or cable capacitance can appear along the leads between the circuit and the guard. Additionally,

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48 the voltage between H and G w ill be across the secondary wi nding of the transformer and thus will be in parallel with the sample thus excluding it from any unwanted current. There is no voltage drop between L and G becau se it shunts the detector and is held at ground while the bridge is balanced, thus it can have no effect on the sample impedance. Another feature of the AH 2700A evident fr om Figure 4-5, is the bridge always assumes that the ac response of the sample is th at of a parallel RC mode l. In practice, this is a safe assumption because most “normal” MIM capacitors behave in this manner. However, many novel materials and confi gurations for MIM capacitors are not represented by this parallel RC model. In these instances caution must be used when interpreting the result s of the measurement. The AH 2700A does allow for reporting the in-phase and out-of-phase components based on a series RC model. This is done by a simple set of algebraic transformations36. A more complicated scenario that is applicable to our experiments is outlined in the following section. 4.4 The Series Resistance, Complex Capacitance Problem To interpret the measured complex capacitance, C( ) = C( ) – iC( ) the effective dielectric response, must be modeled in an equivalent circuit. Ideally, we would be able to express any complex dielectric re sponse, in terms of so me real capacitance, C() in parallel with the out-of-phase, or imaginary terms: RDC that expresses the leaky behavior, and 1/C() the lossy behaviorm (Figure 4-6A). In the geometry of our capacitance measurements (Figur e 4-1) the main concern is to ensure that the electric equipotential lines are parallel to the interface of the base electrode and the dielectric. m The dielectric of a “leaky” capacitor passes a small dc current in response to an applied dc voltage, reflected by a large dc resistance, RDC, in parallel with C() A “lossy” capacitor has frequency-dependent dissipation from the imaginary part C(), C() and acts in parallel with RDC.

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49 This is nontrivial because our base electrode is a highly resistive material over a wide range of temperature, magnetic field, etc…A highly resi stive electrode will introduce voltage drops along its length that will act as a resist or in series with the capacitor defined by base electrode/dielect ric interface. Of course in any MIM capacitor, the electrodes are not ideal and have some associated resistance. C ( ) RDC 1/ C ( ) CeffReff C ( ) RDC 1/ C ( ) C ( ) RDC 1/ C ( ) CeffReff CeffReff Figure 4-6. Schematic of a complex capacitor. A) C ( ) is the real part of the dielectric response and RDC and 1/ C ( ) are the impedances of the leakage and loss, respectively. B) effective circuit assumed by the capacitance bridge. Therefore, some criteria must be established that enables a high leve l of confidence that the series resistance can either be ignored, or at least quantified a nd subtracted from the dielectric responsen. Assume that we have a resistive el ectrode in our sta ndard MIM capacitance configuration in series with a lossy cap acitor that has no dc shunt resistance. The equivalent circuit for this is shown in Figure 4-7, with R2 = 1/C() As previously highlighted in Section 4.3.3, the capac itance bridge technique employed in our measurements, reports the measured respons e assuming the parallel RC model shown in n The real part of the capacitance will not be affected by the series resistor, but the imaginary term which is out-of-phase with the capacitance is in-phase with the series resistance. Therefor e, the effective dielectric response will be compromised by the presence of the series resistor. A B

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50 Figure 4-6B. Therefore we must reconcile the series-parallel model of Figure 4-7 with the pure parallel model of Figure 4-6. Figure 4-7. Equivalent circuit of MI M capacitor with a resistive electrode, Rs, and complex capacitance, C = C1 – iC2, where R2 = 1/C2. The admittance, YP, the inverse of the impedance, Z of the parallel RC model in Figure 4-6 (with RDC ~ ) is written as: 1 P eff effYiC R (4-6) The impedance, ZS, of the series RC* model of Figure 4-7 is written as: 2 211SSR ZR iRC (4-7) Writing this in terms of the associated admittance, YS, we arrive at: 21 2211 (1)S SiRC Y R RiRC (4-8) After some algebra we arrive at an expression for the r eal and imaginary parts of YS: 222 2 221 12 2222 221221 SS S SSSSRRRRC CR Yi R RRRCRRRRC (4-9) C1R2 Rs

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51 Equations 4-6 and 4-9 imply that the compone nts of the effective admittance can now be written in terms of YS as follows: 22 221 222 221 SS eff SS R RRRC R R RRRC (4-10 a) 2 12 22 221 eff SSCR C RRRRC (4-10 b) In an ideal situation, Ceff, would be independent of any of the out-of-phase components such that Ceff C1. We now apply the following constraints to Eq. 4-10b: Constraint 1: 22 212 SS R RCRR (4-11 a) 2 211S SRR R RC (4-11 b) Constraint 2: 2 S R R (4-12 a) 21SR C (4-12 b) The sum impact of these constraints is that the effective dielectric response of our MIM capacitor depends solely on the contributions of C1 and R2 (Eq. 4-13). 2 12 21 222 221 effeff SSCR R CRC RRRRC (4-13) Also, we can ignore the voltage drops along th e highly resistive, LPCMO base electrode as long as the following c ondition (Eq. 4-14) is met: 1211 min,SR CC (4-14) The series resistance in our model, RS, is equivalent to the two-terminal resistance of the LPCMO base electrode that is used in our MIM capacitor. Thus, it is an

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52 experimentally measurable quantityo. Figure 4-8, shows the te mperature dependence of RS and the real and imaginary parts of the im pedance of our MIM. Clearly, the conditions of Eq. 4-14 are met by our choice of frequenc y and the materials in our MIM capacitor. 0100200300 1041061081010 1/C' 1/C'' Rs LPCMO filmZ ()Temperature (K) Figure 4-8. The temperature dependent impedance of the LPCMO film, that serves as the base electrode in our MIM and th e real and imaginary parts of the impedance of our MIM, 1/C and 1/C, respectively. The experimental data tells us that we satisfy the condi tion laid out in Eq. 4-14 and thus the voltage drop along the LPCMO film is compared to the voltage drop across the capacitor negligible. This result assumes that the dc resistance of our thin film dielectric, AlOx, is effectively infinite. We separately confirmed this by attempting to measure the dc resistivity of the o A quick note about the perils of two-terminal meas urements is necessary. Contact resistance between the lead of the instrument and the materi al is always present in a two-term inal measurement. The best way to estimate its effects is to measure the resistance in both a two and four terminal configuration. A large discrepancy between the two measurem ents indicates a large contact re sistance. If the two numbers are approximately equal then the contact resistance is small. In our measurements on LPCMO, we see a small contact resistance, orders of magnitude smaller than the resistance of the LPCMO film itself.

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53 MIM across the electrodes (i.e. through the AlOx dielectric) and found it to be immeasurably large (infinite) by all technique s and tools available. For comparison, we used the same transport measurement pro cedures on a precision standard ceramic 109 resistor and found we were easily able to m easure its resistance. Therefore, we place the lower bound on the dc resistance of our MIM at RDC > 109 This further implies that our LPCMO MIM capacitor has very lit tle leakage curren t through the AlOx dielectric, and is thus best described as a “lossy” capacitor.

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54 CHAPTER 5 COLOSSAL MAGNETOCAPACITANCE IN PHASE-SEPARATED MANGANITES 5.1 Dielectric Measurements of Strong ly Correlated Electron Materials In recent years, a wide variety of stro ngly correlated electron materials (SCEMs) have been probed using standard bulk dielectr ic techniques37-44. These methods entail using the SCEM as the insulator in a MIM capacitance structurep and measuring the capacitance to determine the complex permittivity, *() dielectric constants, complex impedance, Z*() etc… of the SCEM. One of the main goals of these previous studies has been to identify a SCEM that possesses a lo w, or zero, frequency dielectric constant, (0) that is very large, or “colossal”, relative to that of standard di electric materials like Al2O3, which has (0) 9, or even SrTiO3, which has (0) 100. Colossal or giant magnetocapacitance has been observed in some37-39,43,44 of these SCEMs, with changes in the measured dielectric permittivity of up to 500 % with applied magnetic fields. In our experiments we observe a colossal magnetocapac itance effect that re sults in an increase in the measured capacitance of three orders of magnitude. However, we are not looking at colossal dielectric constants but rather a competition between metallic and insulating phases that produces a striki ng dielectric response as a function of magnetic field, temperature, and frequency. p Typically, these materials are grown as crystals, di ced and then either gold or some other metal are deposited on both sides of the insulator. Alternativ ely, some experiments have used pressed metallic electrodes due to the possib ility of damage at the interface due to sputtering.

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55 5.2 (La1xPrx)5/8Ca3/8MnO3 As we discussed in Ch. 2, (La1xPrx)5/8Ca3/8MnO3 (LPCMO) is a mixed valence manganite that exhibits a wide range of phe nomena, including the presence of competing electronic and magnetic phase separated (PS) regions pres ent over a wide range of temperatures and magnetic fields, (see Figures 2.11-12). LPCMO has become the prototypical manganite to study the effects pha se separation because its two host systems, LCMO and PCMO, exhibit wide ly different phase behavior with respect to similar temperature and magnetic field scales. As illustrated in Figure 5-1, LCMO exhibits a La1xCaxMnO3 x = 0.3 La1xCaxMnO3 x = 0.3 Figure 5-1. Temperature dependent resistivity of A) La0.67Ca0.33MnO3 (LCMO) and B) Pr0.7Ca0.3MnO3 (PCMO) in the applied magnetic field indicated in the legends. At zero field, th e PCMO has no metal-insulator transition, and at low temperature is very resistive, while LCMO has a metal-insulator transition, with low resistivity at low temperature. The PCMO exhibits pronounced hysteresis in the resistivit y between the cooling and warming curves indicated by the arrows in B. inset: T-H phase diag ram indicating the onset of the MIT in PCMO and s howing the hysteretic region. A ) Schiffer et al. Phys. Rev. Lett. 75, 3336-3339 (1995), Fig. 2, pg. 3337. B) Tomioka, Y. et al. Phys. Rev. B 53, R1689-R 1692 (1996)45, Fig. 2, pg. R1690. A B

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56 metal-insulator transition (MIT) at all magnetic fields and shows colossal magnetoresistance (CMR), while PCMO has no zero-field MIT, but does have a large CMR effect that is orders of magnitude la rger than the CMR effect in LCMO. Another feature evident in Figure 5-1B is that PCMO exhibits hysteresis in the resistivity as the temperature is cooled and warmed. LCMO does not show this hysteretic behavior. When LCMO and PCMO are combined to form LPCMO, the resulting material exhibits a zero-field MIT, a concomitant ma gnetic phase transition from charge-ordered 0100200300 10-310-210-1100101 R es i s ti v it y ( cm ) 0 kOe 20 kOeTemperature (K) (La4/8,Pr4/8)5/8Ca3/8MnO3 Figure 5-2. Temperature depende nt resistivity for an LPCMO x = 0.5 thin film with thickness d 500 showing the decrease in th e size of the hysteresis and size of the resistance transition with in creasing magnetic field, along with an increase in the TMI. The arrows indicate the direction the temperature is being swept.

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57 insulator (COI) to fe rromagnetic metal (FMM), a pronounced hysteresis in the resistivity and a very large CMR of approximately four or ders of magnitude at the MI temperature, TMI. Typical transport behavior for a pol ycrystalline thin film of LPCMO ( x = 0.5) with thickness d 500 is shown in Figure 5-2. As in PCMO, LPCMO shows a decrease in the size of the hysteresis and size of the re sistance transition with increasing magnetic field, along with an increase in TMI. In Figure 5-3, the magnetic field-depende nt resistivity is plotted for a LPCMO ( x = 0.5) film with thickness d 500 . The data illustrate that at temperatures 010203040506070 10-310-210-1100 300 K 200 K 175 K 150 K 125 K 100 K 75 K 5 K Resistivity ( cm)Magnetic Field (kOe) (La4/8, Pr4/8)5/8Ca3/8MnO3 Figure 5-3. Magnetoresistance data for LPCMO ( x = 0.5) and d 500 for various temperatures. The magnetoresistance is at a maximum at temperatures in the vicinity of the TMI where we also see a small degree of hysteresis in the effect.

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58 around TMI the LPCMO experiences a maximum in th e change of resistance with applied magnetic field. Also, we see a small degree of hysteresis in the magnetoresistance that is much smaller, as a percentage effect, than the hysteresis in the temperature dependent resistivity seen in Fi gure 5-2. In LPCMO, at all temp eratures, and thus for any ground state (COI, FMM, etc…), the magne toresistance is always negativeq with no indication of saturation at the fields shown here. To understand this phenomenon, we return to the PS picture introduced in Section 2.2.3 Phase separation is inherently coupled to the competition between the structural, electronic, and magnetic properties of the ma nganites, and thus is very sensitive to external perturbations such as mechanical pressure46, magnetic3 and electric47 fields and internal perturbations li ke structural pressure48 and chemical substitution2,5. This sensitivity has made these materials promising candidates for applications as magnetic sensors, strain gauges, etc. However, this same unique sensitivity to perturbations also makes it difficult to fabricate manganite-bas ed devices. In particular, the surfaces and interfaces of manganites are notor iously difficult to control. In fact, it has been suggested that the properties of manganites at the su rface can be entirely different from the bulk7,49. The current general picture for the phase behavior is as follows. The dominant ground states in these LPCMO manganite sy stems are a low temperature ferromagnetic metallic (FMM) phase with a fu lly spin-polarized conduction band7,50 and a high temperature orbital and charge ordered insu lating phase characterized by the presence of electron-phonon coupling due to Jahn-Teller distortions3,51. This high temperature phase exhibits paramagnetic behavior abov e the charge-ordering temperature, TCO 225 K, and q A negative magnetoresistance implies a decrease in resistance with increasing magnetic field; a positive magnetoresistance implies the opposite.

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59 antiferromagnetic charge-ordered in sulating (COI) behavior between TCO and TMI. Below TMI, LPCMO is dominated by the FMM phase whic h persists to low temperatures, with the specific onset of these various magnetic phases depending on factors such as the charge/spin/orbital stability of specific doped states3,52,53 (Figure 5-4). Figure 5-4. Field-cooled magnetization curves for La5/8xPrxCa3/8MnO3 for various x The magnetization data at x = 0.30 will correlate most closely with the resistivity data in Figure 52 and Figure 5-3 that has x = 0.313 in this notation. Collado et al. Chem. of Mat. 15, 167-174 (2003), Fig. 3, pg. 170. The main feature missed by the picture outlined above, which speaks in broad terms about the majority phase, is that unlike the superf luid-fluid transition in liquid He4, or the paramagnetic-ferromagnetic transition in an itinerant ferromagnet, like iron, the various electronic and magnetic phases in LPCMO coexist and compete with each other x = 0.30

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60 over a wide range of temperatur es, magnetic fields, etc…around TMI. This metastable phase separated behavior is the hallmark of first-order phase transitions. In the case of LPCMO, the competing phases have very sim ilar ground-state energies over a range of temperatures that stretch from below TCO at high temperatures to well below TMI. In particular, in the temperature range loos ely defined above, there is a dynamic and heterogeneous mixture of COI and FMM re gions (Figure 2-11 and Figure 2-12). In 1999, Uehara et al.2 showed that the MIT in LPCMO was governed by a percolation mechanism. They observe d that a critical concentration, xc = 0.41, existed for the MIT in La5/8xPrxCa3/8MnO3, with no MIT occurring for doping levels above xc. In LPCMO with x < xc, they plotted log(0), where 0 is the resistivity at T = 8 Kr as a function of log( xc-x ) at zero field. Percolation theory54 predicts that in a percolating metal-insulator system that the re sistivity should be of the form ~ ( xc-x), with -1.9 in three dimensions. Uehara et al. found that the log-log plot mentioned above produced a linear region over several orders of magnitude in resistivity for 0.275 x 0.40 with a slope of -6.9. This slope is much larger in magnitude than the one predicted for percolating systems. However, percolation theory pr edictions are for a nonmagnetic metal-insulator system. In LPCMO, the elec tron conductivity is inextricably linked with the ferromagnetic nature of the metallic regions as detailed in Ch. 2. This correlation between conductivity and ferromagnetism implies that if there is a high degree of misalignment between ferromagnetic regions, then the conductivity will be greatly reduced. This misalignment will be at a maximum for concentrations near xc, and for zero magnetic field because of an increasing presen ce of COI regions. In an applied magnetic r A temperature, and thus resistivity, deep in the metallic phase for all samples with x < xc.

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61 field of 4 kOe, a log-log plot produ ced a similar linear region with a slope of -2.6; much closer to the value pred icted by percolation theory a nd thus indicating percolation as the dominate mechanism for PS. This highl ights one of the unique aspects of LPCMO and manganites in general. Classical percol ation theory is inextricably linked with continuous or second-order phase transitions, yet LPCMO is a phase separated material with clear indications of first-order phase tr ansitions as previously mentioned. A solution to this apparent contradi ction is suggested by Li, et al.55 in that TMI and TC are separate quantities in manganites. Specifically the magnetic phase transition at TC is second-order in nature and the associated magnetic orde r enhances thermally activated conduction via the double exchange mechanism. This in turn leads to a first-order metal-insulator phase transition at TMI due to a non-classical percolatio n transition. Th is non-classical percolation transition takes into account inte ractions (analogous to surface tension) between the FMM and COI phases that classical percolation th eory does not consider. Although this result does not fully explain the percolation transition and other associated phenomena observed at TMI via conductivity and dc magnetization measurements5. The experiments detailed in the remainder of th is dissertation will fo cus on understanding the competition between phases, and the perc olation mechanisms that drive it. 5.3 The Colossal Magnetocapacitance Effect: Measurement and Analysis Each LPCMO MIM capacitor is inserted into a probe configured to provide true three terminal measurements as require d by the Andeen-Hagerling 2700A capacitance bridge (Section 3.3.3). All temperature and magnetic dependent measurements are performed in a commercial cryostat, th e Quantum Design QD6000, equipped with a 70 kOe magnet and housed in an electrically isolated, screen room or Faraday cage. All

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62 the capacitance measurements, unless explicitly stated otherwise, are carried out at 0.5 kHz. We performed our measurements ove r a range of temperatures from 5 K to 300 K and at magnetic fields from -70 kOe to 70 kOe. Shown in Figure 5-5 is a plot of the four-t erminal resistance (black circles) of an LPCMO x = 0.5 thin film with d = 600 , as measured by the PAR 124A lock-in 0100200300 103104105106107 100101102103 Resistance (Temperature (K) H = 0 kOe T T Capacitance (pF) T T A B C D Figure 5-5. Temperature depe ndence of the resistance (bl ack circles) and capacitance (red squares) of LPCMO. Right scale: Four terminal resistance of an LPCMO thin film used as the base electrode of the capacitor. All temperature sweeps were conducted at 2 K/min. The solid arrows indicate the direction the temperature was swept. Left scale: Three terminal capacitance of the LPCMO/Al2O3/Al capacitor measured at f = 0.5 kHz. Notice the offset in the maximum in the resistance and the minimum in the capacitance. Inset: Schematic of the MIM D: NdGaO3 substrate, C: LPCMO d = 400 , B: AlOx d = 100 , A: Al d = 1000 .

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63 amplifier and Hewlett Packard 182 nanovoltmeter, and the capacitance (red squares) of a MIM utilizing the same LPCMO film as its base electrodes. The LPCMO film shows the characteristic electronic tr ansport behavior outlined in Section 5.5.2 above, with TMI 95 K on cooling and TMI 106 K on warming, and a resistance transition spanning a range of approximately four orders of magnitude. The associated capacitance reaches plateaus at high and low temp erature where the LPCMO is in its PI and FMM states, respectively. In between these plateaus the cap acitance experiences a change in amplitude of approximately three orders of magnit ude, with minima occurring at 75 K and 93 K upon cooling and heating, respectively. This is a shift in transition temperature of approximately 20 K from the MIT in the associat ed resistance data. Th is is our first clue that the dielectric behavior of our MIM deviates from phenomena observed in LPCMO via electronic transport measurements, like the TMI determined by the longitudinal dc resistivity. The shift between the capacitance mini mum and the resistance maximum can be understood in terms of inherent strain present in LPCMO thin films. All thin films exhibit some strain introduced by the mismatch in th e lattice constants of the substrate and the film. In LPCMO, this strain field has a dire ct effect on the phase behavior of the film. When NGO is used as the substrate for LPCM O (as in our films) the lattice mismatch produces a tensile strain field concentrated at the substrate-thin film interface. NGO and the FMM phase in LPCMO have a cubic crysta l structure and thus the strained region stabilizes the FMM phase relative to the COI phase56,57. This implies that for two LPCMO thin films, the film with higher stra in will have more FMM phase present at a s For complete details on sample fabrication see Section 4.3.2.

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64 given temperature, magnetic field, etc. than the film with low strain. The strain-stabilized FMM phase will constitute a low resistance state at the substrate-film interface. In Figure 5-6A, the LPCMO capacitor (depicted as an infi nite parallel RC circuit) illustrates the potential drop of the capacitance measurement is sensitive to the resistance perpendicular to the surface of the film, R. A standard 4-terminal resistance measurement is dominated by the parallel resistance, R||, because the strain-stabilized FMM region shorts out the higher resistance state away from the NGO substrate. Our capacitance minimum is due to the MIT associated with R that occurs when the FMM percolates throughout the thickness of the LPCMO. This occurs at a lo wer temperature than the MIT seen in the dc resistivity. AlOXLPCMO RRR||R|| Strain-stabilized FMM NGO LPCMO AlOXLPCMO RRR||R|| AlOXLPCMO RRR||R|| Strain-stabilized FMM NGO LPCMO Strain-stabilized FMM NGO LPCMO Figure 5-6. Thickness dependence of the transition temperature. We now want to suggest a qualitative pi cture for the behavior of the capacitance seen in Figure 5-5. As the temperature d ecreases from 300 K, the LPCMO is in the PI phase. Concomitantly, the capacitance is at a plateau in this temperature range. This is due to the PI phase being a “bad” insulator, in the sense that it can still effectively screen

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65 the electric field and allow for a reasonable well-defined potential drop to occur across the AlOx dielectric. As the temperature is lowere d, the resistance increases and at ~225 K there is a first-order phase transition58 into the COI phase with FMM phase present as small volume fraction. This increase in resi stance reflects that as the temperature decreases, the COI phase becomes increasingly insulating at a faster rate than the FMM regions grow in size. Therefore, the associat ed capacitance decreases due to a decreasing ability of the LPCMO to screen with mob ile electric charge. When the electrode resistance is at a maximum (low capacita nce) the electrode comprises filamentary conductors of the FMM phase threading the pr imarily insulating COI. This resistance region corresponds to just before the 3D percolation threshold5. As the resistance decreases and the FMM regions grow in size, the capacitance slowly increases until the FMM part reaches a critical fraction and then the capacitance increases dramatically. Then the FMM domains are the dominant crys tal phase and the capacitance again reaches a plateau, due to the electrode becoming a “good” metal. One will note that the low temperature capacitance plateau is ~12 % sm aller in value than the high temperature plateau. This decrease is due to the temperature dependence of the AlOx dielectric, which shows an approximately equal li near decrease in temperature35. At this point it is necessary to revisit a topic introduced in Ch.4 concerning the perils of a two terminal measurement, and th e effect of the Maxwell-Wagner relaxation in our LPCMO MIM capacitors. In Chapter 3, we outlined the Maxwell-Wagner relaxation as a contact, or interface, effect where char ge polarization occurs across the interface of the contact and the electrode The Maxwell-Wagner model predicts the following: 0 01/1eff cccC C iRCiRC (5-1)

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66 where C0 is the bulk AlOx capacitance, Rc is the contact resistance and Cc is the contact capacitance. This equation predicts that in the limit of high frequency, the smaller capacitance will dominate. In the configuration of our MIM capac itor one would anticipate Maxwell-Wagner effects could exist between the contact, either pressed indium or silver paint, and the LPCMO. These effects would produce an extr insic, enhanced low frequency dielectric response that relaxed to the intrinsic dielectric properties of the insulator31,59 (Figure 5-7) and in fact many claims of “colossal” diel ectric effects have tu rned out to be Figure 5-7. Frequency-dependent dielectric response of inse t the model circuit. At low frequency the response is domina ted by the contact capacitance, CC. At frequencies above the characteristic re laxation, the dielect ric properties of the sample capacitance, C0, dominate the response. Lunkenheimer et al. Phys. Rev. B 66, 052105 (2002), Fig. 1, pg. 2. Maxwell-Wagner phenomena31,40,60. These extrinsic low frequency effects would mask any of the intrinsic properties of the insula tor, rendering the measurement uninformative of the bulk material at best. To insure th at Maxwell-Wagner effects do not dominate the measurement several checks can be made: 1) the contact material should be varied to check for consistency of the observed phenomen a; 2) the size of the contact should be

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67 varied to see if there is sc aling with area; 3) frequency should be swept to ascertain whether there is a characteristic relaxation pres ent as in Figure 5-7. Items 1) and 2) are employed because the Maxwell-Wagner effect is a contact effect, like a Schottky barrier in metal-semiconductor (MS) structures. MS co ntacts can generally behave in an Ohmic, or linear, fashion where there is no built-in electric potential acro ss the MS interface or they form a Schottky barrier. These barriers can occur if the electron work function of the metal is greater than the electron affinity of the semiconductor. This mismatch in energies results in an energy barrier equal in magnitude to the difference between the work function and the affinity. This barrier region has a lo wer electron concentration relative to the metal or semiconductor and thus becomes a depleti on region. The depletion region acts as a capacitance in series with the bulk semiconducti ng material as seen in the inset to Figure 5-7. Changing the contact material corres pondingly changes the el ectron work function that the Schottky barrier depends on and ther efore leads to very different dielectric behavior. Varying the size of the contact likewise changes the area of the depletion region and would result in a corresponding chan ge in the contributi on of its dielectric behavior. In Figure 5-8A, we plot the temperat ure dependent capacitance for the same LPCMO MIM capacitor with indium and then s ilver paint contacts. These results show that the capacitance measured in our experime nts is insensitive to the contact material. This implies that a barrier, or contact cap acitance response due to Schottky barrier or Maxwell-Wagner effects is absent, or at leas t they are not the dominant mechanism that governs the dielectric response of our syst em. Figure 5-8B and Figure 5-8C shows a

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68 100101102103 Indium contact Silver paint contact C (pF)0100200300 100101102103 C measured Maxwell-Wagner model C (pF)Temperature (K)H = 0 kOe 0100200300 101102103 C measured Maxwell-Wagner model C (pF)Temperature (K)H = 15 kOe 100101102103 Indium contact Silver paint contact C (pF)0100200300 100101102103 C measured Maxwell-Wagner model C (pF)Temperature (K)H = 0 kOe 0100200300 101102103 C measured Maxwell-Wagner model C (pF)Temperature (K)H = 15 kOe Figure 5-8. Maxwell-Wagner relaxation. A) temperature dependent capacitance, on cooling, for the same LPCMO MIM capaci tor with indium (black squares) and silver paint (red circles) contacts. The behavior is almost identical for the two contact materials. This implies th at Schottky barrier-like effects are not present in our measurement. B) comparison of the measured real part of the complex capacitance (black) and that predicted by the Maxwell-Wagner model (pink), see Eq. 5-1. The model fails to capture the transition temperature in the measured capacitance. C) the model fails to capture the magnetic field dependence as well. comparison of the measured capacitance and the relaxation model prediction shown in Eq. 5-1 in zero and nonzero field, respectiv ely. The shift in th e capacitance minimums between the measured capacitance (black s quares) and the Maxwell-Wagner model (pink line) is due to the use of the longitudinal resistivity, governed by R|| in Figure 5-6A, in Eq. 5-1. The capacitance is sensitive to the MIT associated with R in Figure 5-6A which A B C

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69 occurs at lower temperatures (higher F MM concentrations) than the MIT for the longitudinal resistivity. As we mentioned, the small capacita nce dominates at high frequency and therefore the response can be parameteri zed by the ratio of the two capacitances, c = Cc/C0. Additionally, the value of RM = Rtwo-terminal, the two terminal resistance of the LPCMO film and C0=C(300 K) were measured experimentally. In our modeling, we found that c = 10-4 produced results that most closely ma tched our data in zero field. The 0100200300 100101102103 405060708090100101102103 0.0 0.4 0.8 1.2 0 kOe 10 kOe 20 kOe 50 kOeC (pF)Temperature (K)H to plane C C (pF)H=0 kOe Loss Loss (S)TT Figure 5-9. Magnetic fiel d and temperature dependence of the LPCMO capacitor measured at f = 0.5 kHz. Large magnetic field causes an increase in the transition temperature, and a decrea se in the size of the capacitance transition around Cmin. Inset, Right scale: Loss as a function of temperature. Sharp loss peaks indicate an inflection in the associated capacitance. Left scale: Associated zero fi eld capacitance behavior, cl early showing hysteresis and the sharp transition.

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70 model does capture some degree of the capacita nce response but clearl y illustrates that the dielectric phenomena we observe is not du e to a contact effect, nor is it solely dominated by the transition in the longitudinal resistance. In Figure 5-9, we plot the temperature dependent capacitance in applied magnetic fields. Like the resistivity, the magnitude of the transition decreases by orders of magnitude as the magnetic field increases. As well, the size of the hysteresis decreases and the transition temperature increases for both cooling and warming curves. The capacitance shows almost no tr ansition for H =50 kOe, with the temperature dependence approaching that of a standard Al-AlOx-Al capacitor and at H = 70 kOe (not pictured) the capacitance is essentially linear in temp erature with no visible transition. The inset in Figure 5-9 shows a magnificat ion of the transition region. The real part of the capacitance, C and the ac losst are shown together to illustrate the sharp loss peaks that indicate the inflection point in the temperature dependence of the capacitance transition. The magnetic field dependence of th e dielectric response il lustrates that there are two fundamental contributions present in our measurement: 1) at high and low temperatures the capacitance is dominated by the AlOx; 2) in a broad intermediate range of temperatures the capacitance is dominated by the LPCMO via some heretofore unknown mechanism. Unlike the work mentioned in Section 5. 5.1 where the authors posited that the SCEMs under study possessed colossal dielec tric constants that exhibited large temperature and/or magnetic field depende nce, we believe the dominant LPCMO dielectric response is refl ecting the competing FMM and COI percolating phases. The t Loss C

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71 minimum in the capacitance occurs below the TMI, as determined by resistivity, but we know from the work of Uehara et al.2, Zhang et al.27 and others52,61-63 that the phase separated region extends to well below this temperature. We can extend the phase competition explanation by using an effective area argumentu. In Figure 5-5, we see, on cooling, that the capacitance starts to decrease dramatically at a temperature approximately equal to TCO. The capacitance continues to decrease by orders of magnitude as the underlying phase behavior of the LPCMO is mostly COI with small, filamentary FMM regions randomly distribut ed throughout resulting in a large, and increasing resistivity. As th e resistivity passes through TMI, the LPCMO reaches the 3D percolation threshold5,62, which corresponds to a metallic fraction of ~17 %, and the resistivity drops by orders of magnitude b ecause a metallic conducting pathway exists between any two points in the LPCMO. Concomit antly, the capacitan ce is still decreasing and continues to decrease until we ll past the previously mentioned TMI. At 20 K below TMI a minimum in the capacitance is reached a nd then there is sudden, sharp transition to a high capacitance state. In this temperatur e range, the LPCMO has a growing metallic fraction and a decreasing resistivity. This increase in capacitance corresponds to the LPCMO reaching a critical meta llic fraction that screens ou t the polarization contribution of the competing phases in the LPCMO electrode and reintroduces the AlOx as the dominant dielectric contribut ion to the capacitance. Then, at low temperatures, the FMM domains are the dominant crystal phase and th e capacitance again reaches a plateau, due to the LPCMO electrode becoming a “good” metal. u Recall that 0 g A C d and thus the capacitance measures, and is governed by, the effective area of the electrode.

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72 -60-40-200204060 100101102103 150 K 125 K 100 K 80 K 70 K C' (pF)H (kOe) warming H to plane Figure 5-10. Colossal change s in the magnetic field dependent capacitance in the LPCMO MIM capacitor. In Figure 5-10, the magnetic field dependence of the capacitance isotherms at temperatures above and below Cmin at H = 0 kOe are plotted. The magnetic field sensitivity increases as the temperature a pproaches the sharp capacitance transition and thus the capacitance saturates at lower magnetic fields than it does at higher temperatures. Also, with increasing proximity to the capacita nce transition there is the development of hysteresis (see T = 70 K) indi cating the increasing presence of the FMM phase in the LPCMO. This hysteresis continues to grow as a percentage of the total magnetocapacitance at low temperatures (Figure 5-10).

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73 -60-40-200204060 704 706 708 710 C (pF)H (kOe) T = 10 K Figure 5-11. The field dependent capacitan ce at T = 10 K for the LPCMO MIM. The arrows indicate the direction the field is swept. The large hysteresis is due to the large FMM fraction that occurs at low temperatures in the LPCMO. The fact that there is any magnetocapacitance reflects that the COI phase is still present but is a small fraction of the total film. Up to this point we have largely ignored the behavior of the imaginary part of the dielectric response. The ac loss provides valuable information about the various dissipation mechanisms present in the dielect ric response. In Figur e 5-12 we plot the temperature dependent ac loss in applied ma gnetic fields, as measured by the AH 2700A. Unlike the real part of the complex capacitance, the ac loss is almost two orders of magnitude larger at high temperatures than at low temperatures. This confirms that the high temperature PI phase of the LPCMO is a “bad”, somewhat lossy, insulator and that

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74 the low temperature phase is a relatively “good” metal which shows little dissipation in terms of the measured response. We also not e that from Figure 512 that at H = 50 kOe there are no longer any sharp loss peak s and thus no capaci tance transitions. 0100200300 0.01 0.1 1 H to plane 0 kOe 20 kOe 50 kOeLoss (S)Temperature (K) Figure 5-12. The temperature dependent ac loss in applied magnetic fields. The sharp peaks indicate the capacitance transition as mentioned for the inset to Figure 5-9. Notice that unlike the real part of the complex capacitance shown in Figures 5.6 and 5.9, the high temperatur e loss is orders of magnitude larger than the low temperature value. In the following chapter we will address the underlying dielectric response of the LPCMO film that is hinted at by the data shown above. We will propose an equivalent circuit that captures the relevant physics of our dielectric measurements. Additionally, we will study the interplay of charge conservation and charge dissipation in these LPCMO

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75 MIM capacitors over a wide range of frequencies, temperatur es and magnetic fields and see striking and univers al scaling behavior.

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76 CHAPTER 6 SCALE INVARIANT DIELECTRIC RE SPONSE IN PHASE SEPARATED MANGANITES 6.1 Modeling the Competition of Phases in (La1xPrx)5/8Ca3/8MnO3 (LPCMO) In the previous chapter we outlined the temperature and magnetic field dependent capacitance associated with the colossal magnetocapacitance (CMC effect observed in LPCMO MIM capacitors. Moreover, we proposed that the underlying mechanism that drove this effect was the competition betw een the ferromagnetic metallic (FMM) and C* AlOx( ) RAlOx Rseries RMC* M( ) C* AlOx( ) RAlOx Rseries RMC* M( ) Figure 6-1. General form of the equiva lent circuit for our LPCMO MIM capacitor. A) Rseries is the two terminal LPCMO dc resi stance between the contact and the boundary defined by the AlOx insulator, (see Figure 4-1). In Section 4.3.4, we showed (Fig. 4-8) that the Rseries contribution to the dielectric response could be ignored due to Eq. 4-14. The measured complex capacitance, C*() = C() iC() is composed of two sepa rate dielectric components representing the contribution of the LPCMO capacitance, CM and RM and the AlOx capacitance, CAlOx and RAlOx. B) the LPCMO dielectric response is modeled as an infinite parallel RC ci rcuit where all capacitors are complex with value C* M() and each resistor represents the dc resistance, RM. This is a macroscopic model which captures the response of an inhomogeneous conductor to an ac electric potential. C) the capacitance due to AlOx is a simple parallel RC circuit with RAlOx effectively infinite (Section 4.3.4). charge-ordered insulating (COI) phases. To support and expand on this idea, we model the effective dielectric response in terms of an equivalent circuit, seen in Figure 6-1. A B C

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77 Figure 6-1 shows the most general repr esentation possible for our LPCMO MIM capacitor. However, it can be simplified by inco rporating some of the assertions made in previous chapters with new data shown in the present chapter. In Section 4.3.4, we outlined how a highly resistive electrode could contribute to the total dielectric response and established a criterion, Rseries << min { 1/C (), 1/C () }, under which it could be excluded. In Figure 4-8, we plot temperat ure dependent impedance data revealing that this criterion is met by our LP CMO MIM capacitor and therefore Rseries in Figure 6-1A is also excluded from the dielectric response. In Figure 6-1C we note that the dc resistance of the AlOx, RAlOx, has been measured to be effectively infinite in experiments detailed in Section 4.3.4. This infinite resistance implie s that there is no dc leakage current through the AlOx and therefore RAlOx can be ignored because it is not incorporated into the measured dielectric behaviorv. In addition, the frequenc y dispersion due to the AlOX is negligible. Finally, below we will demonstrate how the infinite parallel resistor-capacitor network in Figure 6-1B further excludes the potential for voltage drops along the LPCMO film. In Chapter 3, we showed that an inhomoge neous solid in an ac electric field can be accurately modeled by an infinite parallel resistor-capacitor ne twork. All capacitors were equal in value and proportional to the bound charge dielectric constant,, and the resistors were proportional to the free charge current at a specific positi on in the solid. We now extend this picture to model the LP CMO base electrode of our MIM capacitor (Figure 6-1B). LPCMO can certainly be thought of as an “inhomogeneous solid” v This is not to say that RAlOx does not affect the dielectric response at a ll. If the resistance of the AlOx was comparable to that of the LPCMO, then we would not be able to confidently extr act the dielectric behavior of the LPCMO MIM.

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78 (Chapter 3). The phase separation between the FMM and COI phases is analogous to the conducting, free charge and insulating, bound charge model, respectively. An important condition imposed by the ge ometry of the capacitance measurement and the geometry of our MIM is that the LP CMO must not have elec tric potential drops along its width; that is, there must always be a well-defined equipot ential parallel to the planar interface of the LPCMO and AlOx. If this were not the case then, even at ac, the voltage drops could produce a current and an associated resistan ce in the LPCMO film that would be indistinguisha ble from the measured dielec tric response, preventing the capacitance due to the AlOx from revealing the underlying dielectric phenomena in the LPCMO. The necessary conditions for an e quipotential surface are met, in part, by having an infinite AlOx dc resistance, which we reported in Section 4.3.4, and also by the infinite parallel resistor-capacitor network. In the high-frequency limit, M >> 1, where M =RMCM, the capacitance contribution to the admittance of the infin ite circuit in Figure 6-1B dominates and produces a spatially uniform electric field32 and thus the required equipotential surface. This high-frequency limit must be ba lanced by the condition that if M is too large then the impedance constraints (Eq. 4-14) on our m easurement will not be met. Therefore we must restrict ourselves to frequencies that ensure we are in the high-frequency limit, but not outside of our impedance constraints. In Figure 6-2, we plot th e frequency dependence of C and C in zero magnetic field at various temperatures, on cooling. Th e figure shows that for temperatures within the colossal magnetocapacitance (CMC) region de fined in the previous chapter to be between 55 and 175 K for cooling in zero magnetic field, the dielectr ic response is in

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79 10210310410-310-210-1100101102 250 K 200 K 150 K 125 K 115 K 100 K 80K 75 K 50 K C '' (pF)f (Hz)H = 0 kOe 10-1100101102103 250 K 200 K 150 K 125 K 115 K 100 K 80K 75 K 50 K C (pF)H = 0 kOe 10210310410-310-210-1100101102 250 K 200 K 150 K 125 K 115 K 100 K 80K 75 K 50 K C '' (pF)f (Hz)H = 0 kOe 10-1100101102103 250 K 200 K 150 K 125 K 115 K 100 K 80K 75 K 50 K C (pF)H = 0 kOe Figure 6-2. The frequenc y dependence of the real, C, and imaginary, C, parts of the LPCMO MIM capacitor at temperatures cooled in zero magnetic field. the high frequency limit because the peak in C, where M = 1, occurs at frequencies well below the range of our measurements. Th is ensures that we do have an equipotential

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80 surface in the LPCMO base electrode. In Figur e 6-3, we present the equivalent circuit that takes into consideration the limits imposed by the preceding analysis. C* AlOx( ) RMC*M( ) C* AlOx( ) RMC*M( ) Figure 6-3. The simplified equivalent circuit for our LPCMO MIM capacitor. The LPCMO is still modeled as an infinite parallel resistor-capacitor network, with dissipation due to RM, the dc resistance, and the ac loss originating from the imaginary part of C* M(), CM() We now have a framework within which we can present our analysis of the dielectric behavior of our LP CMO MIM capacitor. Specifically, we should note that this equivalent circuit is analogous to the Maxwe ll-Wagner relaxation model. In that model, the dielectric response is comp rised of two complex capacitors in series that represent a contact capacitance, Cc, in parallel with a contact resistance, Rc, and a bulk capacitance, C0, respectively. Associated with those two capacitors are tw o time constants, c = RcCc and 0 = RcC0, with 0 << c. These time constants divide the total dielectric response into two separate regions such that for < 1/c, the low-frequency limit, the larger contact capacitance is dominant and for 1/c < < 1/0 the smaller bulk capacitance dominates and is largely frequency independent. For > 1/0, the dielectric response approaches the “high frequency permittivity”, or bound charge dielectric constant, In our MIM capacitor, the dielectric response is also modeled by two complex capacitors in series, with the exception that one (the LPCM O) is an infinite ne twork of equivalent

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81 complex capacitors shunted by resistance RM. Nevertheless, we will see that the complex capacitance of our LPCMO MIM is likewise divided into re gions due to two separate time constants, M = RMCM and 0 = RMCAlOx, with the major exception that in the LPCMO MIM, M << 0. This implies that in the hi gh-frequency limit, the dielectric response will be governed by the LPCMO sin ce the “contact” capacitance which is the LPCMO capacitance is small compared to the AlOx. 6.2 The Dielectric Response Due to Phas e Separation in LPCMO: Universal Power-Law Phenomena Figure 6-2 shows that the dispersion in the comple x capacitance of our MIM reflects strong temperature dependence. Speci fically, at temperatures above and below the CMC region, there is very little frequency dispersion in C, and C shows the onset of a small peak. The presence of a loss peak indicat es that the dielectric response is in the vicinity of a crossover from a low to high-frequency limit. Analogously, this implies that for these high and low temperature regions that the complex capacitance is dominated by the AlOx. This supports our supposition that the complex capacitance is reflecting the phase competition occurring within the LPCMO, in that at high and low temperatures the phase of the manganite is in a relatively homogeneous and conducting state that screens out any charge polarization or dipole formation in the LPCMO. To further explore these dielectric ph enomena, we now look at our complex capacitance data in terms of Cole-Cole plots64. In a Cole-Cole plot, C is presented as a function of C while some external parameter, like fr equency, temperature, etc., is varied. This is a popular way of presenting dielectric data because it is compact and allows the physical properties the system, such as the ch aracteristic frequencies and time constants,

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82 101001000 100 200 300 C '' (pF)C (pF) H = 0 kOe T = 200 K increasing 610620630 10 100 C '' (pF)C (pF) H = 0 kOe T = 50 K increasing 0.1110 1 10 100 C '' (pF)C (pF) H = 0 kOe T = 100 K increasing 101001000 100 200 300 C '' (pF)C (pF) H = 0 kOe T = 200 K increasing 610620630 10 100 C '' (pF)C (pF) H = 0 kOe T = 50 K increasing 0.1110 1 10 100 C '' (pF)C (pF) H = 0 kOe T = 100 K increasing Figure 6-4. Cole-Cole plot s in zero magnetic field on warming for 50, 100 and 200 K. A) at 50 K the LPCMO is mostly FMM and the plot shows that as the frequency increases the dielectric response passes through a relaxation region, located around the cusp specif ied by the dotted arrow, indicating a crossover in the dominan t capacitance elements. B) at 100 K the LPCMO is deep in the phase separated region and the Cole-Cole plot shows very different behavior than the data at 50 or 200 K. As freque ncy increases, the response decreases in an apparent powe r law fashion, eventually rolling off as C approaches C ~ C) at 200 K, the dielectric behavior shows a relaxation region like the 50 K data, with a cusp signifying the AlOx capacitance. Also note that the high fr equency portion of the curve (the circled region) is decreasing with in creasing frequency in a manner very similar to the 100 K data. to be shown in terms of curve shapes a nd points of discontinuity. In Figure 6-4, C and C data from Figure 6-2, are converted to Cole-Col e plots at three differe nt temperatures in A B C

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83 zero field. These three data sets come from temperatures that ar e above (200 K), below (50K), and inside (100 K) the CMC region de fined in the temperat ure dependent data taken at zero magnetic field a nd at a frequency of 0.5 kHz (Figure 6-4). The behavior observed at 100 K, in Figure 6-4B, over the entire measured frequency range hints at unique intrinsic LPCMO dielectric response due to phase separation. 0.1110100 0.1 1 10 100 C'' (pF)C' (pF) 1 kOe 5 kOe 10 kOe 20 kOe T= 100 K warming increasing Figure 6-5. Cole-Cole plot showing the effect of ma gnetic field on the dielectric behavior at T = 100 K. Th e raw data reveal that increasing magnetic field causes a shift in the diel ectric response along a master curve, such that increasing magnetic field behaves like decreasing frequency and vice versa. Now that the dielectric response of the CM C region has been singled out, we will explore the nature of competition of phases in this region. Specifically, we will look at the magnetic field dependence of the comple x capacitance. Figure 6-5 shows a Cole-Cole

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84 plot of the magnetic field dependence of the LPCMO MIM at T = 100 K, with the response measured at /2 = {0.1, 0.2, 0.5, 1, 2, 5, 10, 12, 16, 20 kHz}. The dielectric behavior shown in Figure 6-5 illustrates that in the LPCM O base electrode there is a unique interplay occurring betw een magnetic field and freque ncy that produces a master curve along which all the data fall. For comparison, Figure 6-6 shows a 101001000 10 100 C'' (pF)C' (pF) 1 kOe 5 kOe 10 kOe 20 kOe T= 65 K warmingincreasing Figure 6-6. Cole-Cole plot showing the effect of ma gnetic field on the dielectric behavior at T = 65 K, w ith same frequency steps as the T = 100 K data. The data collapses onto a single curve as it did at T = 100 K but at high magnetic field, or low frequency, a cr ossover from the LPCMO to AlOx dielectric response is clearly visible. magnetic field dependent Cole-C ole plot at T = 65 K, whic h is outside the CMC region defined above. We again see that the data collapse onto a single master curve with a similar relationship between frequency and magne tic field. However, at low frequency, or

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85 high magnetic field (see H = 20 kOe (blue down tr iangles)), there is the signature of a dielectric crossover from the high freque ncy, LPCMO to the low frequency, AlOx dielectric response. This further supports the claim that the comp lex capacitance depends on the competition of phases in LPCMO becau se an increasing magnetic field always causes an increase in the FMM fraction of the film. An interesting consequence of this inverse relationship between frequency and ma gnetic field is that the high frequency complex capacitance at 65 K closely matches the response seen at T =100 K. At 0.5 kHz and T = 65 K, the LPCMO base electrode is well outside of the CMC region but, as the frequency increases above ~ 10 kHz, the CM C region seems to expand in temperature. To pursue this further we need to observe if the dielectric behavior at T = 100 K (Figure 6-5) is indeed a signature of the entire CMC region and in turn the underlying phase separation. Figure 6-7 shows a Cole-Cole plot at H = 10 kOe over a range of temperatures that lay within the CMC region at 0.5 kHz. Th e figure illustrates that at each temperature, the dielectric behavior collapses onto the sa me master curve that described the magnetic field dependent data at T = 100 K. In addition, at /2 = 0.5 kHz, the measurement frequency used for the data in the previous chapter, the dielectric behavior falls on the same line. This collapse implies that the be havior seen in Figure 6-7 and Figure 6-5 are signatures of the CMC region and correspond ingly the competition between the FMM and COI phase separated areas. To explore th e physical significance of the behavior revealed in the Cole-Cole plots of Figures 6.5-6.7 we will determine the functional form of the master curve that governs the dielectr ic response in the CMC region. Also we will

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86 incorporate the unique frequency, temperatur e and magnetic field dependence revealed by the data. 0.1110100 1 10 100 increasing 90 K 100 K 125 K 150 KC''(pF)C'(pF) H = 10 kOe Figure 6-7. Cole-Cole plot showing the effect of temp erature at H = 10 kOe on the dielectric behavior. The raw data reveals that increasing temperature causes a shift in the dielectric response along a master curve, such that increasing temperature behaves like decreasing frequency and vice versa. In each of the Cole-Cole plot s that reveal a master curv e, the behavior appears to have power-law dependence for sufficiently hi gh frequencies or low magnetic fields and temperatures. In addition, it appears as if th is power-law is ubiqu itous and regardless of which parameter, frequency, magnetic field or temperature, is varied the dielectric response is governed by it. Figure 6-8 presen ts a Cole-Cole plot that includes the

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87 dielectric behavior with a pair -wise variation of all parameters. In this plot, we witness the collective character of the dielectric re sponse to these external perturbations. 0.1110100 0.1 1 10 100 f (0.1 20 kHz) 100 K, 1 kOe f (0.1 20 kHz) 90 K, 10 kOe T(90 148 K) 0.5 kHz, 1 kOe T(89 140 K) 0.5 kHz, 5 kOe H(0 17 kOe) 0.5 kHz, 100 K H(0 23 kOe) 0.5 kHz, 125 KincreasingC''(pF)C'(pF) T, H increasing Figure 6-8. Universal dielec tric behavior for the complex capacitance of the LPCMO MIM. The legend indicates the paramete r being swept and its range, and the two parameters being held constant respectively. The temperature and magnetic field sweeps are taken from th e data presented in Figures 5.9 and 5.10, respectively. In the region below ( C, C) (5, 0.5) in Figure 6-8, there is a pronounced roll-off in the dielectric response and as the dissipation due to the ac loss vanishes. This is because at high, or infinite, freque ncy the permittivity, and the corresponding capacitance, is dominated by the free space contribution, 0, and other instantaneous polarization mechanisms29. This limit produces the infinite frequency capacitance

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88 contribution, C, that is an intrinsic dielectric pr operty of a given MIM system, just as is the bare dielectric response in a material To see the underlying relaxation behavior we will look at C as a function of C C. 0.010.1110100 0.1 1 10 100 [C' C](pF)increasingC''(pF) T, H increasing = 0.7 Figure 6-9. Cole-Cole pl ot showing the complete dielectric relaxation of C. This plot shares the same legend as Figure 6-7. C is determined by extrapolating the high frequency roll-off to C = 0, at which C = C. The data are then fit to the power-law expression in Eq. 6-1, and the fit is plotted as the orange line. The exponent for the response is = 0.7. Figure 6-9 shows the pow er-law response of C as a function of C C over four orders of magnitude. The data are fit to a general power-law expression of the form: ,,,, CTHCTHC (6-1)

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89 and plotted in Figure 6-9 (orange line). C is determined by extrapolating the high frequency roll-off to C = 0, at which C = C. We employed a non-linear curve fitting routine to perform the additional analysis. We used two fitting parameters, and and obtained = 9.26 (0.08) and = 0.701 (0.004) with a shared variance of r2 = 0.99997 between the data and our pow er-law model. Alternativel y, we performed a three parameter fit, leaving C as a free parameter and obtained quantitatively similar results with comparable levels of confidence. Power-law responses are frequently s een in physical systems that exhibit scale-invariance. Thermodynamically speaking, scale-invariance implies that the physical properties of a system governed by the equati on of state will remain unchanged for an appropriate change, either an increase or decr ease, in the scale of the quantities contained in that equation. The features of the system do not change if the space is magnified, i.e. length scales (or energies) are multiplied by a common factor. This phenomenon is a general property of all physical systems in the vi cinity of a second-order phase transition65. For example, consider the equation of state for a ferromagnetic system, where is the order parameter: 01 1 log 21c BTH TkT (6-2) If we make the following substitutions, cTT T and 0BH x kT then expand Eq. 6-2 in a power series we arri ve at the following: 33 x (6-3)

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90 This equation of state can be scaled with respect to and x using the following scaling relations: (6-4 a) 2 (6-4 b) 3 x x (6-4 c) Upon substituting these relations into Eq. 6-3 we find that the equation of state retains the same form, but with scaled parameters: 33 x (6-5) Second-order phase transitions are char acterized by these “scaling laws” and by power-law relationships that depend on critical exponents. These critical exponents are universal in all systems that exhibit second-or der phase transitions, w ith the value of the exponents depending on only the dimensionality of the system and the symmetry of the order-parameter66. The question now becomes, does th e power-law relationship in our data imply scale-invariance, and if so, what is its source and what el se can this tell us about the underlying phase separation? In addition to second-order phase transitio ns, another set of phase transitions exist that exhibit critical phenomena: percol ation transitions. In 1992, Isichenko67 wrote that, “The percolation problem describes the simple st possible phase tran sition with nontrivial critical behavior.” We previously discussed (Section 2.2.3 and Secti on 5.5.2) the key role that percolation plays in th e phase separation mechanism of the manganites. In 1972, the work of Fortuin and Kasteleyn68 identified the correspondence between second-order phase transitions and the percolation tran sition. They showed that the critical

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91 temperature, Tc, of a second-order phase transition was analogous to the percolation fraction, pc, in a percolation transition. The sca ling laws outlined a bove were connected to the percolation transition later by Straley69 and Kirkpatrick70. In the scaling theory of pe rcolation transitions, the dc conductivity and dielectric permittivity, should show power-law behavior in | ppc| where p is the metallic fraction of the two-phase system. There should be two different power-l aws in percolating metal-insulator systems: one for p < pc, where the dielectric permittivity of the metal-insulator system dominates and one for p > pc, where the conductivity of the system dominates. In add ition, predictions were made71-73 that for p < pc in the vicinity of p=pc, there should be a divergence in the dielectric constant, the real part of the dielectric permittivity. This divergence was confirmed in 1981 by Grannan et al.74, who found that the dielectric c onstant showed the following power-law behavior: s c cpp c p (6-6) with the mean value of s = 0.73 (0.07) and pc 0.21. This experimentally determined relationship was explained theoretically75, in part, in terms of a random resistor network, composed of two, separate types of resistor s, one with small, finite resistance and one with infinite resistance. Th is theoretical work found a power-law for the effective conductivity, of the two-phase system. These nume rical simulations produced a similar power-law to Eq. 6-6 for p < pc, ~ (pcp)-s, with s = 0.70 (0.05). In addition, it had been previously suggested69,72,73 that in a two conductor system, like the one used in the above numerical work, that the dielectric constant and the conductivity w ould exhibit the same power-law dependence on pc for p
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92 Based on the previous anal ysis, we would expect that if our power-law dependence is due to the ensuing percolation transition we would see divergent behavior in C as the LPCMO approached and passed through the percolat ion threshold. Therefore, we return to the temperature-dependent data originally shown in Figure 5-5, (reproduced below in Figure 6-10). 50100150200 10-1100101102103 C (pF)Temperature (K) H = 0 kOe T T 50100150200 10-1100101102103 C (pF)Temperature (K) H = 0 kOe T T Figure 6-10. Temperature dependent capacitan ce in zero magnetic field. On cooling, as the temperature moves through Cmin, indicated by the two arrows for cooling and warming, the capacitance undergoes a dramatic increase, and it begins to diverge as the temperature continues to decrease. The divergence is abruptly cut off when the capacitance of the AlOx becomes dominant. In Figure 6-10, the data show that, on cooling and warming, the capacitance reaches a minimum, Cmin, at Tmin, indicated by the arrows. On cooling, as the temperature decreases past Tmin, the capacitance experiences a rapi d increase. This divergence of the

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93 capacitance is similar in nature to the behavi or of the dielectric constant just below a classical percolation transition. However, th e divergence of the m easured capacitance in Figure 6-10 reflects a macroscopic change in the phase sepa ration of the LPCMO, not a classical percolation transition. In the LPCM O, this divergence occurs because as the temperature decreases, the metallic frac tion increases and many conducting channels connect the entire length and thickness of th e film while being separated by thin barriers of insulating material. The measured capaci tance is sensitive to small changes in temperature because, in the region around Tmin, these temperature changes produce large-scale (macroscopic) changes in the FMM concentration. Eventually, this diverging capacitance is effectively cut off when th e FMM phase shorts out the polarization mechanisms of the COI phase in the LPCM O. At this point the LPCMO becomes a relatively homogeneous metal and the meas ured capacitance is dominated by the AlOX dielectric. The picture that emerges from this analysis is that in the CMC, or phase-separated, region the mixed phases are pe rcolating in nature and their dielectric response exhibits a scale-inva riant, power-law response. As the metallic fraction reaches a critical value, or equivalently when the te mperature or magnetic field reaches a critical value, the dielectric response of the LPCMO is screened out, resu lting in a different dielectric response. 6.3 Crossover in the Dielectric Response: The Dynamics of Phase Separation The question of the percolati on threshold raises some questions. In the resistivity data of Figure 5-5, the perc olation threshold has been obs erved to coincide with the resistivity peak. Clearly, the temperatures at which we see this universal power-law dielectric response are well above and below th is threshold. To further explore the full

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94 spectrum of dielectric response, we sweep temperature in a Cole-Cole plot at fixed frequency, /2 =0.5 kHz, and for various magnetic fields in Figure 6-11. Sweeping temperature gives us the broadest probe to access each of three dominant ground state phases, FMM, COI+FMM and PMI at high te mperatures. In this way we hope to distinguish the percolation gove rned scale invariant, power-l aw dielectric response from other responses. The Cole-Cole plot shows that the dielectr ic response is divided into two branches by a very sharp and pronounced crossover. Th is crossover point o ccurs at a magnetic field dependent temperature, indicated in the inset of Figure 6-11. As well, this crossover point corresponds exactly to Tmin where the capacitance reached a minimum as indicated in Figure 6-11. This provides a precise definition for the shift in transition temperature from that at the resistivity peak to that of the capacitance minimum, which could reflect the difference between ac and dc. Othe r work has attributed a shift in Tc to different magnetic properties at the surface of manganites49,50,76. In our experiments, the shift in transition temperature reflects a pronounced cros sover in dielectric response rather than a depressed Tc associated with surface physics. In the temperature range just above these crossover points, the dielectric response s hows behavior with a universal power-law scaling collapse (PLSC). For temperatures belo w this crossover, the dielectric behavior

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95 0300600900 0 300 600 900 036912 0 20 40 T T 1 kOe 5 kOe 10 kOe 15 kOe 22.5 kOe C''(pF)C'(pF) 0.5 kHz warming 90 K 89.6 K 95.2 K 100 K Figure 6-11. Cole-Cole plot for warmi ng temperature from 10 to 300 K at fixed frequency and various magnetic fields. The plot reveals a sharp and very pronounced crossover from a region with power-law scaling collapse (PLSC at temperatures above the crossover point and a region with no universal power-law behavior or s caling collapse (NC). At high temperatures, the response shows a scaling collapse, but without a universal power law, i.e. non power law collapse (NPLC). The cr ossover from PLSC to NPLC is indicated by the deviatio n of the response from th e power-law fit (orange line) with exponent = 0.7. This is the same fit obtained in Figure 6-8. Inset: A magnified look at the crossove r region. The orange 0.7 line is clearly visible as it passes through al l the data on the PLSC side of the crossover. Each crossover point oc curs at a magnetic field dependent temperature, indicated by the temperature tags. shows neither universal powe r-law nor scaling collapse (NC). These sharp, magnetic field dependent crossover temperatures determine a T-H boundary for PLSC to NC behavior at /2 = 0.5 kHz. The NC region corresponds to the low temperature plateau in

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96 01020304050 75 100 125 150 cooling warming Temperature (K)H (kOe) COI FMM 0102030 75 100 125 150 C''(H,T) ~ [C'(H,T) C]0.7warming PLSC NPLC PLSC NC cooling PLSC NPLC PLSC NC Temperature (K)H (kOe) 0.5 kHz 01020304050 75 100 125 150 cooling warming Temperature (K)H (kOe) COI FMM 0102030 75 100 125 150 C''(H,T) ~ [C'(H,T) C]0.7warming PLSC NPLC PLSC NC cooling PLSC NPLC PLSC NC Temperature (K)H (kOe) 0.5 kHz Figure 6-12. T-H phase diagrams. A) resistivity and B) dielectric response. Lower boundary-crossover from PLSC to NC (inset of Figure 6-11), Upper boundary-crossover from PLSC to NPLC (deviation from solid orange line of Figure 6-11). Inset: The equation for power-law behavior governing the dielectric response in the shaded regi on. The boundary of this upper region expands to higher temperatures a nd magnetic fields with increasing frequency. A B

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97 Figure 6-10 where the AlOx capacitance dominates. The magnetic field dependence in this region is due to the small remnant COI phase that persists at low temperatures. At high temperatures, the re sponse shows a scaling collaps e, but without a universal power law (NPLC). The crossover from PLSC to NPLC is indicated by the deviation of the response from the power-law fit (orange line) with exponent = 0.7. This is the same fit obtained in Figure 6-8. In addition, we can define an upper T-H boundary that divides the PLSC from the NPLC regions. Figure 6-12A and B shows a T-H phase diagram for the resistivity and the resulting phase diagram for th e dielectric behavior at /2 = 0.5 kHz as defined above, respectively. The phase diagram for the resi stivity shows the magnetic field dependent metal-insulator transition temperature, TMI, for warming and cooling. Phase diagrams of this kind are found throughout the manganite literature, with th e caveat that some experimentalists use the inflection point in the resistance56,62,77 and others use the resistivity peak3,5. Our plot is generated from the peak in the resistivity. The width between the cooling and warming curves defines the region where the mixed phase exhibits hysteretic behavior. The lines dete rmine a crossover (I-M on cooling, M-I on warming) between mixed phase regions that are predominantly COI in the upper left portion and FMM in the lower right portion. We emphasize that this is a qualitative determination of the boundaries of the CO I and FMM regions without a clear-cut determination of the extent of the mi xed phase regions. In contrast to the T-H phase diagram for resistivity in Figure 6-12A, the phase diagram of Figur e 6-12B incorporates well-defined boundaries to the power-law scaling behavior discussed above.

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98 Figure 6-12B indicates that in the shaded region where the PLSC behavior occurs, the response “walks” around in HT -space and always stays on the same master curve. This two dimensional slice at /2 = 0.5 kHz tells only a fraction of the story. Previously, in Figure 6-4 C and Figure 6-6, we saw that the dielectric response shown in the Cole-Cole plots seemed to fall into a power-law behavior at sufficiently high frequencies. When these data are incorporated into the master curve plot of Figure 6-8 we find they obey the same universal power-law scaling (Figure 6-13). Th is implies that the PLSC region broadens as fre quency increases. Below 0.5 kHz, the PLSC region shrinks 0102030405060 80 100 120 140 160 180 0.1 kHz 0.5 kHz Temperature (K)H(kOe)5 kHz cooling Figure 6-13. The power-law s caling collapse (PLSC) region expands (contr acts) along the PLSC-NPLC line (open symbols) with increasing (decreasing) frequency. The PLSC-NC line (solid sy mbols) is frequency independent and represents the crossover from dynami c (PLSC) to static (NC) phase separation.

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99 in scope until it effectively occupies only th e PLSC to NC boundary line of Figure 6-12 B. This occurs in the range of 0.05 < /2 <0.07 kHz. This unique connection between frequency and the PLSC response, which is li nked to the percolation of the mixed phase regions, indicates an analogous relationshi p between the spatial extent of the scale-invariance and the time-scales of our measurement. If we think of the measurement fr equency as our “window” into the scale-invariant mixed phase regions that govern the dielectric response then as the frequency increases from low (long time scales ) to high (short time scales) our “window” decreases in size and becomes more sensitive to local fluctuations in the phase separation. This claim is bolstered by the fact that ther e are two kinds of phase separation present in LPCMO: static and dynamic phase separation78-80. Large energy barriers exist in LPCMO that enable the system to reach low temperatures, well below the percolation threshold, in a blocked, metastable state composed of coexisting FMM and COI phase s with relative fraction changing as a function of time. Ghivelder and Parisi79 performed temperature dependent magnetization measurements on LPCMO ( x = 0.6) and studied the magnetic viscosity and time dependence of the system. They found that at low temperatures there exist multiple blocked states in the phase-separated region. At very low temperatures, T<
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100 fluctuations. In this way, LPCM O behaves like a magnetic glass79 in this region. Therefore, we propose that the PLSC dielectric behavior and its expansion in size with increasing frequency is a signature of the in trinsic dynamic phase separation between the FMM and COI phases. Furthermore, the PLSC to NC boundary is a “hard” line in the sense that even as the PLSC region will not expa nd into this low temperature region because the phase separation is fr ozen and does not provide the temporal fluctuations necessary to produce the PLSC response. Conversely, the PLSC region will expand and overtake much of the NPLC region because this region exhibits fluctuations in the phase separation at very short time interv als. In short, the scale-invariant power law behavior witnessed in our LPCMO dielectric response is a consequence of the competition of FMM and COI phases, in both space and time.

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101 CHAPTER 7 SUMMARY AND FUTURE DIRECTIONS 7.1 Summary of Experimental Work The dielectric response of LPCMO MIM capacitors has be en studied as a function of temperature, magnetic field and frequenc y and has produced a comprehensive picture of the percolative phase sepa ration inherent to LPCMO. We have demonstrated that a two-terminal measurement can be quite revealing of the intrinsic nature of the material under study providing that strict impedance constraints are met. The use of a highly insulating dielectric, like our AlOx, in series with a phase-separated electrode allows for the dielectric nature of the phase separa tion to be studied. In this way, the AlOx capacitance “decloaks” the underlying dielec tric phenomena of the LPCMO. Unlike dc measurements or standard bulk capacitance tech niques, this method allows for the direct observation of the dielectric re sponse of a metal-insulator system both above and below the metal-insulator temperature and the associated percolation transition. We have observed a colossal magneto capacitance (CMC) effect in our LPCMO/AlOx/Al capacitance structures that is du e to the magnetic field dependence of the competition between ferromagnetic metallic (FMM) and charge-ordered insulating (COI) phases: with increasing magnetic field the LPCMO has an increasing concentration of FMM and thus a greater ability to sc reen electric charge. This dielectric characterization has revealed the following st riking features that ar e inherent to phase separated manganites:1) the percolation of the phase-separated regions leads to scale-invariance in the dielectric response in both space and time; 2) this scale-invariance

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102 in the phase-separated regions of LPCMO o ccurs over a broad range of temperatures, magnetic fields and frequencies which we attr ibute to the dynamic nature of the phase separation; and 3) at ac, the percolation mechanism that drives the competition of phases shows unique behavior unseen in measurements taken at dc. 7.2 Future Work The LPCMO MIM capacitance structure reve aled unique and intriguing dielectric behavior that revealed bulk properties of the LPCMO which were inaccessible via dc measurements. Specifically, the full dielectric behavior of the phase-separated LPCMO was “decloaked” by the presence of the bulk AlOx insulator. Additional capacitance measurements should be carried on LPCMO, specifically; the thickness of the AlOx dielectric should be decreased; to probe the interface capacitance regime. The interface capacitance could provid e important information about th e effect phase separation on the screening length of the LPCMO, a quantit y that is of this writing, unknown. The full potential of capacitance measuremen ts as a tool for studying the nature and mechanisms that underlie phase-separati on has not been realized. The two-terminal capacitance measurement and sample geometry employed in this work could lend itself to the study of the intrin sic behavior of other phase-sep arated systems, beyond LPCMO. Most notably, two-terminal capacitance meas urements should be carried out on other manganites that exhibit phase separation like, La1xSrxMnO3 and La1xCaxMnO3, to see if the various parameters, such as the power-law exponent, are universal. Beyond manganites, the technique could be employed to measure the dielectric response of other phase separated, strongly-corr elated electron materials like the electron-doped cuprate,

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103 Pr2xCexCuO4 or hole-doped cuprate, La2CuO4+ which are both materials that exhibit superconducting and insulating phases at va rious doping levels, temperatures, etc. In this dissertation, we have shown that capacitance measurements can reveal a wealth of information about materials that pos sess complex behavior. The future uses of these techniques will undoubtedly aid in the discovery of new and exciting dielectric phenomena.

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109 65. Kadanoff, L. P., Gotze, W., Hamblen, D., Hecht, R., Lewis, E. A. S., Palciaus,V., Rayl, M., Swift, J., Aspnes, D. and Kane, J. Static Phenomena near critical pointstheory and experiment. Rev. Mod. Phys. 39, 395-406 (1967). 66. Dyre, J. C. & Schroder, T. B. Universa lity of ac conduction in disordered solids. Rev. Mod. Phys. 72, 873-892 (2000). 67. Isichenko, M. B. Percolation, statisti cal topography, and tr ansport in randommedia. Rev. Mod. Phys. 64, 961-1043 (1992). 68. Fortuin, C. M. & Kasteley, P. Random-c luster model .1. intr oduction and relation to other models. Physica 57, 536-540 (1972). 69. Straley, J. P. in Electrical Transport and Optical Properties of Inhomogeneous Media (eds. Garland, J. C. & Tanner, D. B.) 118 (AIP Conference Proceedings, New York, 1978). 70. Kirkpatrick, S. in Electrical Transport and Optical Properties of Inhomogeneous Conductors (eds. Garland, J. C. & Tanne r, D. B.) 99 (AIP Conference Proceedings, New York, 1978). 71. Efros, A. L. & Shklovskii, B. I. Crit ical Behavior of C onductivity and DielectricConstant near Metal-Non-Me tal Transition Threshold. Physica Stat. Sol. B 76, 475-485 (1976). 72. Bergman, D. J. & Imry, Y. Critical behavior of complex dielectric constant near percolation threshold of a heterogeneous material. Phys. Rev. Lett. 39, 1222-1225 (1977). 73. Bergman, D. J. Exactly solvable micr oscopic geometries and rigorous bounds for the complex dielectric constant of a 2-component composite material. Phys. Rev. Lett. 44, 1285-1287 (1980). 74. Grannan, D. M., Garland, J. C. & Tanner, D. B. Critical behavi or of the dielectric constant of a random composite ne ar the percolation threshold. Phys. Rev. Lett. 46, 375-378 (1981). 75. Straley, J. P. Critical exponents fo r conductivity of random resistor lattices. Phys. Rev. B 15, 5733-5737 (1977). 76. Garcia, V., Bibes, M., Barthelemy, A., Bowen, M., Jacquet, E., Contour, J.-P. & Fert, A. Temperature dependence of th e interfacial spin polarization of La2/3Sr1/3MnO3. Phys. Rev. B 69, 052403 (2004). 77. Kuwahara, H., Tomioka, Y., Asamitsu, A., Moritomo, Y. & Tokura, Y. A firstorder phase-transition in duced by a magnetic-field. Science 270, 961-963 (1995).

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PAGE 123

111 BIOGRAPHICAL SKETCH Ryan Rairigh was born on March 22, 1979, in Tampa, Florida. He is a second-generation Florida native, a rarity to be sure. He was a curious (some might say nosy) child who enjoyed the company of adults because they usually knew things that he did not. When he was a toddler, he was keen on being read to which his mother indulged him in regularly. At age 8, Ryan enrolled in th e gifted program at Forest Hills Elementary and fell in love with learning in all its shapes and forms. He especially loved reading and writing. He also had a strong attraction to ma th and science, motivated partly by a love for science fiction. However, his true love for these subjects did not show up until his junior year at Vivian Gait her High School when two influential teachers helped change his life. Terry Adams and Susan Hamme r taught him physics and pre-calculus, respectively, and separately they showed him how to think about scientific and mathematical problems in the abstract and the practical. In Fall 1997, Ryan began his undergraduate career at the University of Florida studying engineering science, wi th the idea of getting a degr ee in biomedical engineering and going to medical school. Thes e goals faded, by his sophomo re year, when he realized that he had to walk through the door that Mr. Adams had opened for him a few years before. He changed his major to physics and happily embarked on a terrific ride, with several ups and downs, that he is still on today. During his senior year at UF, he met two people who again changed his life: Dr. Arthur Hebard, who became his research advisor, and Nikki Rairigh n Whitney, his wife. In Art, Ryan found a mentor and friend whose

PAGE 124

112 infectious curiosity and persistence was an inspiration. In Nikki, Ryan found a friend, a confidant, a kindred spirit, and a special l ove that made him the luckiest man (within experimental error) on Earth. This dissertation represents the culmina tion of his research efforts over the past 5 years.


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COLOSSAL MAGNETOCAPACITANCE AND SCALE-INVARIANT DIELECTRIC
RESPONSE IN MIXED-PHASE MANGANITES















By

RYAN PATRICK RAIRIGH


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Ryan Patrick Rairigh

































To my wife, Nikki















ACKNOWLEDGMENTS

I am indebted to the many individuals who contributed to the successful completion

of my graduate career, represented by this dissertation. First, and foremost, I extend my

heartfelt thanks to Dr. Art Hebard for his guidance, motivation, and friendship. He

provided an atmosphere that stimulated discussion, collaboration, and enjoyment among

his graduate students. Likewise, Art was always ready to learn something. This made

coming to work each day an exciting prospect that I will surely miss. These qualities and

others made him an exceptional advisor and it was an honor to work for (and with) him,

these past 5 years.

I would also like to thank the many members, both past and present, of the Hebard

lab that I have worked with and learned from over the years: Jeremy Nesbitt, Partha Mitra

(in particular, for teaching me what good Indian food should taste like), Xu Du, Guneeta

Singh, Sinan Selcuk, Kevin McCarthy, Nikoleta Theodoropoulou, Steve Arnason, Rajiv

Misra, Mitchell McCarthy, and Ritesh Das. I extend extra special thanks to Jeremy

Nesbitt who has been my friend, lunch companion, and favorite skeptic for many years

now.

I also extend my thanks to the other members of my supervisory committee for

taking time out of their incredibly busy schedules to answer questions and guide my

education. Each of them has contributed to my academic career. David Tanner, taught me

as an undergraduate the value of understanding the importance of the error in a

measurement. Selman Hershfield, was instrumental in shaping my early career. Cammy









Abernathy, taught me everything I know about crystal growth while simultaneously

indulging my interest in talking about football. Amlan Biswas, provided much of impetus

for the experimental work carried out in my study and taught me everything I ever

wanted to know about cricket. In addition, I would like to thank Dr. Steve Detweiler

whose teaching convinced me as an uncertain undergraduate that physics was indeed the

right path for me to take.

I would like to thank my family. First, I thank my parents, whose unwavering

support and confidence in me has been a beacon, all of my life. They always went out of

their way to value and solicit my opinion, and include it, even when I was too small to

know what an opinion was. They taught me that you can learn a lesson from anyone, if

you just pay attention and listen. This recognition of the value of knowledge and my

worth, as part of our family, meant more to me than I can express in words. I also thank

my extended family of friends (especially, Paul McDermott), aunts, uncles, cousins, and

grandparents who have shaped who I am today.

As the saying goes, last, but certainly not least, I thank my wife, Nikki. She is the

reason that I have come to this point in my life with the level of satisfaction,

completeness, and joy I feel. I thank her for being my partner, confidant, travel

companion, motivator, best friend, and the love of my life.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST O F FIG U R E S .......... .......................... ........ .. ...... .......................... .. viii

ABSTRACT .............. .................. .......... .............. xi

CHAPTER

1 INTRODUCTION ............... ................. ........... ................. ... ..... 1

2 REVIEW OF MANGANITES .......................................................................3

2.1 Introduction..................................................... ................... .. ....... ...... 3
2.2 Structure ..................................... .................. .............. .......... 4
2.3 Theoretical M models ....................................... ..... ............. .. ........ ..
2.3.1 D ouble Exchange Theory ........................................................ ............... 7
2.3.2 Beyond Double Exchange, Part 1: Jahn-Teller Polarons .........................16
2.3.3 Beyond Double Exchange, Part 2: Percolating Phase Separation .............21

3 DIELECTRIC RELAXATION PHENOMENA................................... ..................26

3.1 Time-Dependent Dielectric Response: Polarization v. Conduction...................26
3.2 Frequency-Dependent Dielectric Response............................... ............... 29
3.3 D electric R response Functions ........................................ ......... ............... 31
3.3.1 Im pedance and A dm ittance ............................................. ............... 32
3.3.2 D electric P erm ittivity .................... ......................... ............... .... 34
3.3.3 A Dielectric Case-Study: Disordered Conductors................................35

4 CAPACITANCE MEASUREMENTS AND EXPERIMENTAL TECHNIQUES ...38

4.1 Thin-Film M etal-Insulator-M etal Capacitors .............. ...................... ...........38
4.2 M etal-Insulator- M etal (M IM ) Fabrication .................................. ............... 39
4.3 Measuring Capacitance: Tools and Techniques ....................................... 40
4.3.1 Im pedance A nalyzer........................................................ ............... 40
4 .3 .2 L ock-In A m plifier ........................................................... .....................43
4.3.3 C apacitance B ridge................. ........................ ................... 46
4.4 The Series Resistance, Complex Capacitance Problem ......................................48









5 COLOSSAL MAGNETOCAPACITANCE IN PHASE-SEPARATED
M A N G A N IT E S ............................................................................... ................ .. 54

5.1 Dielectric Measurements of Strongly Correlated Electron Materials...................54
5.2 (L al-,P rx)5/sC a3/8M nO 3 ......................... .... ... ....... ................................ 55
5.3 The Colossal Magnetocapacitance Effect: Measurement and Analysis ..............61

6 SCALE INVARIANT DIELECTRIC RESPONSE IN PHASE SEPARATED
M AN GAN ITES ............... ................. ............... ........... ............76

6.1 Modeling the Competition of Phases in (Lal-xPr,)5/sCa3/8MnO3 (LPCMO).........76
6.2 The Dielectric Response Due to Phase Separation in LPCMO: Universal
P ow er-L aw P henom ena ........................................................................................81
6.3 Crossover in the Dielectric Response: The Dynamics of Phase Separation.........93

7 SUMMARY AND FUTURE DIRECTIONS..........................................................101

7.1 Sum m ary of Experim ental W ork.................................... ........................ 101
7 .2 F u tu re W ork ...................................... ......................................... 102

LIST O F R EFEREN CE S ......................................... .... ....................... ............... 104

BIOGRAPH ICAL SKETCH ..................................................................................11
















LIST OF FIGURES


Figure pge

2-1 Three magnetoresistance versus temperature curves for different
La0.67Ca0.33M nOx sam ples. ............................................................................. 3

2-2 Schematic of the perovskite structure that encompasses the MnO6 octahedron
that serves as the basis for all manganites .... ........... ........ .........................5

2-3 Crystal-field splitting of the five-fold degenerate 3d orbitals in the Mn3+ ion. A
similar diagram for Mn4+ would have unoccupied eg states..............................6..

2-4 Schematic view of orbital and spin ordering on the Mn sites in LaMnO3 .............. 11

2-5 Phase diagram for Lal-xCaxMnO3 showing the effect of hole-doping on the
magnetic ordering of the manganite.............................. ...............12

2-6 Resistivity calculated from Eq. 2-8.................. ........................ .................. 14

2-7 Temperature dependent resistivity of La-xCaxMnO3 (x = 0.25) in various
applied m magnetic fields.. .......................................... .. .. ......... ........ .... 15

2-8 The projection of the crystal structure of LaMnO3 along the b axis. The unequal
Mn-O bond lengths illustrate the Jahn-Teller distortion. .......................................17

2-9 Jahn-Teller distortion and its effect on the energy level splitting of the Mn3
orbitals ......................................................................................... 18

2-10 Magnetic field dependence of resistivity calculated using the direct integration
m ethod .................................................................................. .......... 21

2-11 Dark-field SEM images for Las/sPryCa-, _\MnO3 obtained by using a superlattice
peak caused by charge ordering (CO)....................................................................22

2-12 Scans (6 [tm x 6 [tm) of LPCMO. On cooling, FM regions (dark area) grow in
siz e ... ........................................................................ 2 3

2-13 Random resistor network resistivity for various metallic fractions, p in 2D and
3 D (in set).............................................................................................. 2 4

2-14 M onte Carlo results for the RFIM m odel...................................... ..................25









3-1 B arrier-volum e capacitor............................................................ ............... 32

3-2 The complex impedance diagram of the barrier-volume circuit. The large arc is
due to the barrier region and the small arc is due to the volume region. ................33

3-3 Random resistor capacitor network that describes the dielectric behavior of the
universal ac conductivity m materials .........................................................................36

4-1 Schematic of MIM. A) the metallic counter electrode. B) the dielectric. C) the
manganite base electrode. D) the substrate. ................................... ............... 39

4-2 Confidence intervals of the measured impedance as a function of the
m easurem ent frequ ency ......................................... .............................................42

4-3 Schem atic of a lock-in am plifier ...................................... ............... ..... .......... .43

4-4 The magnitude of the impedance of a 400 A LPCMO film as measured by the
Solartron 1260A (black squares) and PAR 124A Lock-In (red circles). ................45

4-5 Schematic of the capacitance bridge circuit ........................................................47

4-6 Schem atic of a com plex capacitor................................................ ........ ....... 49

4-7 Equivalent circuit of MIM capacitor with a resistive electrode, Rs, and complex
capacitance, C = C1 iC2, where R2 = 1/coC2....................................................... 50

4-8 The temperature dependent impedance of the LPCMO film, that serves as the
base electrode in our M IM ........................................................... .....................52

5-1 Temperature dependent resistivity of A) La0.67Ca0.33MnO3 (LCMO) and B)
Pr0.7Ca0.3M nO 3 (PCM O). ............................................. ............................ 55

5-2 Temperature dependent resistivity for an LPCMO x = 0.5 thin film with
thickness d & 500 A ......... ......... ............... ...................................... ..................... 56

5-3 Magnetoresistance data for LPCMO (x = 0.5) and d & 500 A for various
tem peratures....................................................... ........... ........... 57

5-4 Field-cooled magnetization curves for La5/8-xPrxCa-, ,MnO3 for various x...............59

5-5 Temperature dependence of the resistance (black circles) and capacitance (red
squares) of LPCM O ..................... ............ ... .............................62

5-6 Thickness dependence of the transition temperature. ........................ ...........64

5-7 Frequency-dependent dielectric response of inset the model circuit.. ...................66

5-8 M axw ell-W agner relaxation......................................................... ............... 68









5-9 Magnetic field and temperature dependence of the LPCMO capacitor measured
atf 0.5 kH z ............................ ................ .............. ......... 69

5-10 Colossal changes in the magnetic field dependent capacitance in the LPCMO
M IM capacitor .............. ...... .......... ........................... .. .. ......... 72

5-11 The field dependent capacitance at T = 10 K for the LPCMO MIM. The arrows
indicate the direction the field is swept. ........................................ ............... 73

5-12 The temperature dependent ac loss in applied magnetic fields............................74

6-1 General form of the equivalent circuit for our LPCMO MIM capacitor.................76

6-2 The frequency dependence of the real, C' and imaginary, C parts of the
LPCMO MIM capacitor at temperatures, cooled in zero magnetic field ................79

6-3 The simplified equivalent circuit for our LPCMO MIM capacitor.....................80

6-4 Cole-Cole plots in zero magnetic field on warming for 50, 100 and 200 K...........82

6-5 Cole-Cole plot showing the effect of magnetic field on the dielectric behavior at
T = 100 K ...........................................................................83

6-6 Cole-Cole plot showing the effect of magnetic field on the dielectric behavior at
T = 65 K, with same frequency steps as the T = 100 K data...................................84

6-7 Cole-Cole plot showing the effect of temperature at H = 10 kOe on the dielectric
behavior ............................................... .. ................. .......................86

6-8 Universal dielectric behavior for the complex capacitance of the LPCMO MIM.. .87

6-9 Cole-Cole plot showing the complete dielectric relaxation of C' ............ .........88

6-10 Temperature dependent capacitance in zero magnetic field.. ..................................92

6-11 Cole-Cole plot for warming temperature from 10 to 300 K at fixed frequency
and various m agnetic fields ...................... .... ................................ ............... 95

6-12 T-H phase diagram s........... ..................................... ....... .. .......... 96

6-13 The power-law scaling collapse (PLSC) region expands (contracts) along the
PLSC-NPLC line (open symbols) with increasing (decreasing) frequency.............98















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

COLOSSAL MAGNETOCAPACITANCE AND SCALE-INVARIANT DIELECTRIC
RESPONSE IN MIXED-PHASE MANGANITES

By

Ryan Patrick Rairigh

May 2006

Chair: Arthur Hebard
Major Department: Physics

We measure capacitance on thin-film metal-insulator-metal (MIM) capacitors using

the mixed-phase manganite (Lao.sPro.5)o.7Cao.3MnO3 (LPCMO) as the base electrode. The

LPCMO was grown by pulsed laser deposition (PLD), then an AlOx dielectric was

deposited via rf magnetron sputtering, and then Al was thermally evaporated as the

counter-electrode. The LPCMO films exhibit colossal magnetoresistance (CMR) where

resistance decreases by several orders of magnitude with applied magnetic field.

Correspondingly, capacitance shows a change of three orders of magnitude in the region

of the resistance drop. These colossal magnetocapacitance (CMC) effects are related to

magnetic-field-induced percolating changes in the relative proportions of coexisting

ferromagnetic metal (FMM) and charge ordered insulating (COI) phases in the LPCMO.

The widths of the temperature-dependent hysteresis loops (in capacitance and resistance)

are approximately the same, but the center of the capacitance loop is shifted -20 K below

the center of the resistance loop. When the electrode resistance is at a maximum (low









capacitance) the electrode comprises filamentary conductors threading a predominantly

insulating medium. In this region, double logarithmic Cole-Cole plots revealed the

intrinsic dielectric response of the LPCMO in which the data plotted either as a function

of frequency or temperature for different frequencies collapse onto single straight lines,

thus implying scale-independent phenomenology over a wide range of frequency,

magnetic field and temperature. Phenomena witnessed in these experiments give a

comprehensive picture of the percolative phase separation inherent to LPCMO. These

capacitance methods hold similar promise for the study of other phase-separated systems.














CHAPTER 1
INTRODUCTION

Thin film metal-insulator-metal (MIM) capacitors are ubiquitous, with applications

in cellular telephones, satellite communications networks, GPS receivers, test and

measurement applications, filters, voltage controlled oscillators (VCOs) and RF

amplifiers; they help control the modem electronic world. In addition to providing

valuable aid in everyday applications, they also provide a system for investigating

fundamental aspects of thin film physics. Traditionally, the bulk dielectric properties of

insulators such as their static or steady state response to a steady electric field and the

dynamic response to time-varying electric fields have been studied by capacitance

methods. However, it has been known for over 40 years1 that capacitance measurements

on MIM capacitors reveal information about the metal-insulator interface in addition to

the bulk dielectric properties of the insulator. These interface capacitance effects are

caused by the penetration length of the electric field into the metallic electrode and, in

ultra thin insulating layers, they dominate the measured capacitance. In this way, it is also

possible to study the material properties of the electrode with a capacitance measurement.

We studied the unique electronic and magnetic properties of the mixed-valence,

phase-separated manganite (Lal-xPrx)5/8Ca -, MnO3 (LPCMO) via a capacitance

measurement. Our measurement used LPCMO as the base electrode in a MIM

capacitance structure, whereby we can probe the dielectric response intrinsic to this

complex strongly-correlated electron material. The experiment will study the effect of

temperature, magnetic field and frequency on the characteristic behavior associated with









the fundamental mechanisms that drive the dynamic electronic, magnetic and structural

properties of the LPCMO. The unique nature of our capacitance measurement will allow

us a window into the phase separation that occurs due to percolation of ferromagnetic,

metallic (FMM) regions in the presence of charge-ordered insulating (COI) regions.

Traditional transport measurements, such as the dc resistivity2'4 or dc magnetization5,

capture some of the important properties of LPCMO like the magnetic field-dependence

of the metal-insulator transition and the associated decrease in the resistivity by several

orders of magnitude. However, our capacitance measurements, performed at ac, will

illuminate a wealth of additional information about the complicated competition between

the FMM and COI phases that is not available via traditional dc methods.















CHAPTER 2
REVIEW OF MANGANITES

2.1 Introduction

In 1994, a new era in the study of doped oxides began with the discovery (by

Jin, etal.6) of "colossal magnetoresistance" (CMR) effect in epitaxial thin films of

Lao.67Cao.33MnO3 (LCMO), a mixed manganite compound. They observed a negative

magnetoresistance three orders of magnitude larger than any magnetoresistance effect

observed up to that time (Fig. 2-1).


-1500
Lao.6Cao.aMnOx




-1000-
2




-500




0
0 100 200 300
T(K)
Figure 2-1. Three magnetoresistance versus temperature curves for different
La0.67Ca0.33MnOx samples 1) as deposited 2) heated to 700 OC in 02 for
0.5 hours and 3) heated to 900 OC for 3 hours. Jin et al., Science 264,
413-415 (1994), Fig. 1, pg. 413.

This magnitude of response of the electrical transport to an external magnetic field

is not observed in bulk metallic systems where, in clean systems, at low temperatures,









magnetoresistance is caused by magnetic field dependence in the electronic mean free

path. The CMR is also much larger in amplitude than the giant magnetoresistance (GMR)

effect observed in Fe-Cr multilayer and other layered structures where a large sensitivity

to magnetic fields is due to the spin-valve effect between spin-polarized metals. Shortly

after the discovery of the CMR effect, other researchers discovered that these manganites

possessed magnetic-field-induced insulator-metal and paramagnetic-ferromagnetic phase

transitions, as well as, associated lattice-structural transitions3. Additionally, Park

discovered7 that at temperatures below Tc, that manganites were half metals, meaning

that they approached 100% spin polarization.

These materials had been studied and some of their electrical and thermal

properties had been known for more than 50 years8, including a large magnetoresistance

effect around the ferromagnetic Curie temperature, T,. These magnetoresistive effects

were understood in the context of the double exchange theory initially developed by

Zener9 and further elaborated on by DeGenneso0, and Anderson, et al.1. With the

discovery of CMR and other associated effects, an explosion of new research began that

continues unabated to this day. Specifically, experimentalists and theorists have

endeavored to reexamine manganites for potential practical applications and to

understand the underlying mechanisms responsible for the properties of these remarkable

materials.

2.2 Structure

Lal-xCaxMnO3 is the prototypical CMR material and belongs to a class of 3D cubic

perovskite-based compounds commonly called mixed manganites. Depending on the

hole-doping of the material, x, LCMO crystallizes in various distorted forms of the

classic ABO3 perovskite structure. The Mn ions are placed in the center of an oxygen








octahedron (Fig. 2-2), forming MnO6 octahedra with corner-shared oxygen atoms. The

Mn and O ions are arranged in a MnO2 plane that is analogous to the CuO2 planes in the

cuprate high-temperature superconductors.


B La Sr2




OMnTM n4f





Figure 2-2. Schematic of the perovskite structure that encompasses the MnO6
octahedron that serves as the basis for all manganites.

In LCMO, the MnO2 planes are then stacked in a variety of sequences with the

MnO2 planes interlaced with (La, Ca)O planes. The complexes are named depending

upon how many MnO2 planes are arranged between bilayers of (La,Ca)O planes. This is

the Ruddlesden-Popper series, also known as the layered perovskite structure. The series

has the formula unit (A, B)n+1MnnO3n+1 and the 3D cubic manganites are the end (n=o0)

constituent of it, which means that they have no (La,Ca)O bilayers. The 3D mixed

manganites are generally expressed as Al-xBxMn03, where A is a trivalent rare-earth ion

and B is a divalent dopant. This mixing of cation valences produces an associated mixed

valence in the Mn ions due to charge conservation, in that Mn3+ and Mn4+ coexist in the

compound. Typically, these two ionic species are randomly distributed throughout the

crystal lattice. The chemical substitution of the trivalent ions (La, Pr, etc.) with the

divalent ions (Ca, Sr, etc.) effectively dopes the manganite with holes that appear as








vacancies in the electron conduction band (i.e. an increase in divalent ion substitution

leads to an increase in the concentration of Mn4+). This mixed-valence chemistry and the

octahedral atomic configuration become important as we discuss the unique interactions

and ordering phenomena that occur in the orbital, charge, and spin-dependent properties

of manganites.

In mixed manganites, the local spins and conduction electrons are d-electron in

nature. Specifically, they are of the 3d orbital levels associated with the previously

mentioned octahedral-coordinated Mn ion. Normally, the 3d orbital electrons in Mn are

five-fold degenerate, but the octahedral coordination partially lifts this degeneracy

because of cubic crystal field-splitting effects such as hybridization and electrostatic

interactions between neighboring ions (Fig. 2-3). This field splitting breaks the rotational

invariance of the orbitals and results in three degenerate, low-energy, t2g states


.,= = eg
Mn3+

3d orbitals ,'






t2g
Figure 2-3. Crystal-field splitting of the five-fold degenerate 3d orbitals in the Mn3+ ion.
A similar diagram for Mn4+ would have unoccupied eg states.
separated by a few eV from two degenerate, high-energy, eg states. The three t2g orbitals

are dxy, d, and dzx. The two eg orbitals are dX2_Y2 and d3z2_ 2 .









The eg orbitals have a high degree of overlap with the 2p orbitals of the neighboring

O ions and this results in a strong hybridization of the two. These hybridized orbitals

comprise the pathways through which electron conduction occurs. The t2g orbitals have

little to no hybridization with the O ions and thus behave more or less like localized states

that are electrically inert, with a core spin of S= 3/2. Additionally, the electron-electron

interactions among the t2g levels are such that it is energetically favorable for all electrons

to have parallel spins. In turn, a strong intra-atomic exchange coupling, JH, occurs

between the eg conduction electron spins and the core spins in the t2g orbitals. This strong

coupling (a consequence of Hund's rule and the Pauli Exclusion Principle) sponsors an

alignment of the spins between the two energy levels at all accessible temperatures and is

an important factor in understanding the fundamental behavior of the manganites.

2.3 Theoretical Models

2.3.1 Double Exchange Theory

Double exchange is the starting point for understanding the magnetoresistance and

paramagnetic (PM)-ferromagnetic (FM) phase transitions inherent to manganites.

Historically, the double exchange model for ferromagnetism was proposed in its original

form by Zener9 to explain the "empirical correlation between electrical conduction and

ferromagnetism in certain compounds of manganese with perovskite structure" observed

by Jonker and Van Santen8. This theoretical work was devoted mostly to understanding

the decrease in resistivity with increasing magnetic field, and not to understanding the

magnitude of the magnetoresistance effects. In a system like the mixed-valence

manganites "double exchange" refers to simultaneous transfer of an electron from a Mn3

to an 02- with the transfer of an electron from an 02- to a Mn4+. In 1955, Anderson and

Hasegawa11 proposed a variation on this picture that involved an intermediate state









between the two mentioned above, whereby a singly occupied O ion is flanked by two

Mn3 ions with parallel eg spins. This picture produces an effective hopping term, t, for

the electron to transfer Mn3+ to Mn4+ that is proportional to the square of the hopping

term for the 2p O orbital and the t2g Mn orbitals. For classical spins, this hopping is

proportional to cos(0/2), where Ois the angle between the spins of the adjacent t2g

electrons. Equation 2-1 shows how this the electrical conduction is incumbent on the

magnetic character of the manganite through t, and thus the transfer of the eg hole from

one site to another.

The hopping term, t cos(8/2), indicates that FM alignment of the adjacent spins

maximizes both t and the bandwidth, W, which is proportional4 to cos20 and is the

distance between the maximum and minimum energy in the hole band. Uncorrelated spin

alignment will result in a reduction of the hopping amplitude by cos(0/2). This implies

that an antiferromagnetic (AF) arrangement of the t2g spins would result in zero hopping

amplitude and zero hopping conductivity. The physics of this important result of Hund's

rule coupling can be expressed in the double exchange Hamiltonian developed by Kubo

and Ohata12 (Eq. 2-1) where Si is the spin of the core-like t2g electrons, oz,z' is the


HDE = -H z (S',zz')CzC + I toc/cz (2-1)
i,z,z' i,j,z

Pauli matrix, cz and ciz are the creation and annihilation operators, respectively, for an eg

electron with spin z at site i in the conduction band, t, is the transfer matrix element

governing electron hopping from site i toj and JH is the intra-atomic exchange coupling,

with JH > 0 implying FM order.









The physics of Eq. 2-1 states that for a sufficiently large JHS, when an electron

hops from site i toj it must align its spin from being parallel with Si to being parallel

with Sj Therefore, the electron hopping amplitude is tuned by a parameter that shows a

maximum when the core spins align in a parallel configuration and a minimum when the

core spins are aligned anti-parallel. This shows the basis for the claim that the electron

conduction is intimately coupled with the magnetic character of the system.

We can write the Hamiltonian in Eq. 2-1 for a simple two state problem where an

electron is confined to states i andj (Eq. 2-2). In the classical limit where JHS. -> oo, the


Hi = JH (Si Czzz'C +Sj c ,z t +c jziz (2-2)

core spins can be described in terms of unit vectors specified by polar angles 0, and ,.

After suitable manipulation of the coordinates, the transfer matrix generates the

following expression for the hopping amplitude:

t = t [cos(O, /2)cos(O / 2)+ sin(O, / 2) sin(08 / 2) exp i(q% j)}1 (2-3)


If Si is parallel to Sj then 0, =0, and tj = t. If Si is anti-parallel to Sj then 0, =0j + i7

and t', = 0. Therefore, for FM alignment of the core spins, we see no reduction in the

hopping amplitude and for AF alignment we see no electron hopping. If we make an

appropriate choice of gauge we can remove the term exp i(4 Oj)} from the expression.




a The goal is to take the electron axis in question (i.e. state i orj) and rotate it parallel to the core spin axis,
S, operating with the rotation matrix Rzz, = exp iniy uz' / 2 where ny is the unit vector describing
the classical spin state Si and S respectively.









This complex quantity is a Berry phase, which is a quantum mechanical phase factor that

mimics an internal magnetic field. The resulting real expression is as follows:

t = tcos [(, -O)/2] (2-4)

Equation 2-4 illustrates that the eg band receives an kinetic energy gain-an

increase in the ability of electrons to move from site to site in a crystal lattice-from FM

alignment. The strong on site coupling between the conduction and core spins in this FM

state also produces a unique state. At low temperatures, the spin-polarized conduction

band is completely split, such that the minority spin band is empty. This corresponds to

J >> W, the eg bandwidth, which is in contrast to conventional ferromagnets which have

J<< W. This produces the experimentally observed half-metal nature of manganites

mentioned previously7. Thus, we see the natural connection between magnetic order and

electron hopping in double exchange. Specifically, that disorder in the electron spins

causes randomness in t,, which decreases below To or with an applied magnetic field.

FM alignment must in turn compete with other interactions such as

superexchange13, which favors an AF exchange between two nearest neighbor

non-degenerate t2g orbitals via their shared 02- ion, with disorder influenced by thermal

fluctuations in the crystal lattice and charge-exchange (CE) AF due to charge-ordering,

which occurs when the Mn3+ and Mn4+ cations order coherently into two separate

magnetic sublattices over long distances along the crystal lattice.

It is important to note at this point that this competition between interactions and

the overall magnetic ordering of manganites is heavily influenced by the hole doping of

the host material. The magnetic phase diagram in Fig. 2-4 shows that in Lal-xCaMnO3









one can chemically change the inherent nature of the spin ordering below the magnetic

transition temperature.

In this system, the host material, x = 0 (i.e. completely undoped), is LaMnO3 an

insulating AF dominated by superexchange. This means that each Mn in the MnO6

octahedron has four FM bonds and two AF bonds (see Figure 2-4).When the hole doping

is increased, the manganite moves to a mixed contribution of superexchange and double






C






Figure 2-4. Schematic view of orbital and spin ordering on the Mn sites in LaMnO3.
Tobe et al. Phys. Rev. B 64, 184421 (2001), Fig. la, pg. 2.

exchange interactions for 0< x < 0.08 and 0.88< x < 1 which leads to a canted

antiferromagnetic (CAF) insulating state, first predicted by DeGenneso1. Theoretically,

this CAF state is produced by a dominate super-exchange interaction in the ab-plane with

a weak double exchange interaction occurring along the c-axis. However, more recent

results2'14-16 show that the CAF state is a manifestation of phase separation (PS) between

coexisting FM and AF states. In this context, PS is a result of an inhomogeneous spatial

density of eg electrons, resulting in regions that are hole doped (FM regions) and those

that are undoped (AF regions). A charge-ordered (CO) insulating AF state first appears

for 0.08 < x < 0.17 and a FM metallic state for 0.17 < x < 0.5. When x > 0.5, the Mn4

becomes the dominant ion and the manganite re-enters an AF state, with CO occurring at

intermediate temperatures.








At x = 0.5, the compound enters a very unique PS state. Initially, it undergoes a FM

transition followed by a simultaneous AF and CO transition at T < T,. This CO state can

be easily "melted" into a FM metallic state by application of magnetic field, x-rays or

external pressure4. It should also be noted that all the before mentioned antiferromagnetic

states are insulating in nature and thus agree, in principle, with the double exchange

picture that conduction and ferromagnetism are inextricably linked.


350-
Paramagnetic insulator
300 3/8 4/8 5/8

250
20 x=1/8
X 1/8 Charge
S 200 ordered
insulator 7/
S 150 Ferromagnetic
100 I metal
S 100 0
1-
50 Antiferromagnetism

0 I
0.0 0.2 0.4 0.6 0.8 1.0
Cax
Figure 2-5. Phase diagram for Lal-xCaxMnO3 showing the effect of hole-doping on the
magnetic ordering of the manganite. CAF: canted antiferromagnet, FI:
ferromagnetic insulator, CO: charge ordered. CAF and FI could have spatial
inhomogeneity with both ferro- and antiferro-magnetic states present.
Hwang et al. Phys. Rev. Lett. 75, 914-917 (1995), Fig. 4, pg. 917.

Figure 2-5 also illustrates that, regardless of the hole-doping in Lal_-CaxMnO3, the

high temperature magnetic phase is a paramagnetic insulator (PI). The double exchange

model predicts this magnetic phase transition, in that for a properly doped material at low

temperatures we have parallel core spins and carrier hopping amplitude of t









(ferromagnetic phase ), which above Tc, transforms to a state with non-collinear and/or

orthogonal core spins and carrier hopping amplitude of tj = tcos(7t/2) = 0.7t

paramagneticc phase). This decrease in hopping amplitude by associated with the

magnetic phase transition, should produce a concomitant response in the conductivity of

the manganite. This is because the only energy scale in the double exchange theory is D,

the spin wave stiffness1, which has a theoretical estimate ofD 2tn, where n is the

electron density per unit cell per orbital and t is the hopping amplitude.

Millis et al.17 show that the temperature dependent resistivity for the double

exchange model with large Hund coupling interaction between the t2g and eg electrons

(Eq. 2-5) where ao is the lattice constant and M (o, T) is a "memory function"b

p(T)= aoM (o) = O,T)/7re2D (2-5)

Further evaluation of Eq. 2-5 to leading order in 1/S and kCao, where kF is the Fermi wave

vector, reveals the double exchange resistivity (Eq. 2-6), where R labels the various sites

p(T)= e2 [(SOS)-S( IS (R)-S(R+s, ) /S4]B (R) (2-8)
R,6,,62

b Since there is no rigorous expression for the dc resistivity in the double exchange model, due to an
intractable integral equation, an approximation must be made. In this case, Millis et al. use a memory
function method (which, in principle is valid at large frequency) was used to extract the dc resistivity in
terms of the memory function. This memory function can be explicitly determined by a perturbation
expansion in the dominant scattering mechanism; in this case, the scattering of the electrons off of the spin
fluctuations or spin disorder. M(ra, T) takes the form:


M (o, T) = dtelt [H, j]t ,[H, j]0 (2-6)
0

Where H is from Eq. 2-1 andj is a current operator:


ijabi (bCajb -CjbCia 1+ (2-7)









on the crystal lattice, in the coordinate system that connect a site to its nearest neighbor

and B(R) is the electron current-current correlation functions. The results of Eq. 2-8 are

shown in Figure 2-6. They show that for all temperatures, including T < Tc, that the

2




1.5-


E

E

h = 0.15 Tc
0.5 -
h = 0---



0.2 1 2 3
T/Tc
Figure 2-6. Resistivity calculated from Eq. 2-8. In this model, the resistivity increases
below To and with increasing magnetic field, in direct contradiction with
experiment (see Figure 2-6). Millis et al. Phys. Rev. Lett. 74, 5144-5147
(1995), Fig. 1, pg. 5146.

temperature moves through Tc, with a decrease in resistivity with increasing magnetic

field. Also, at T > Tc, the resistivity exhibited by the CMR materials is much larger than

interaction reflects insulating behavior with an increase in the resistivity of -150% from

To to 0.6T,. Contrasting this experimental data in Figure 2-7 shows the typical transport


In the free electron approximation B(R) is expressed as:


B(R) 9 sin2kF R+ ) sin2kF(R- i) 2sin2(kFR) (2-9)
32(kB(R) Fa=)4 (kF R 1 ')2 (F R- (2-9)
32(kF 4 +S,2 2k, k-8F {R)2








behavior for a CMR manganite18. The resistivity changes by orders of magnitude as

anticipated by simple double exchange models. Nor does double exchange predict the

rapid decrease in resistivity on either side of Tc.

Other theoretical work has shown the wide gamut of responses that double

exchange can produce in various limits and approximations. In contrast to Millis et al.,

Calderon et al. found19 via Monte Carlo simulations that in the classical spin limit, the

double exchange model can produce metallic behavior for all temperatures, which still

contradicts all experimental data. Yunoki et al.20,21 have produced the closest qualitative

result to experiment utilizing a modified double exchange model. They used a one-orbital

Sf M=H 0 .
100
1T


S60

L 40

20



0 50 100 150 200 250 300 350

Temperature (K)
Figure 2-7. Temperature dependent resistivity of Lal-xCaxMnO3 (x = 0.25) in various
applied magnetic fields. Schiffer et al. Phys. Rev. Lett. 75, 3336-3339
(1995), Fig. 2, pg. 3337.

or FM Kondo model with varying chemical potential, i. In their work, Monte Carlo

simulations of classical t,2 spins produced low temperature instabilities at certain electron









densities as [i was varied. When a Heisenberg coupling was introduced between the

localized t2g electrons, these instabilities manifest themselves as two PS regions: one hole

doped and FM, the other undoped and AF. Thus we have a double exchange interaction,

albeit a sophisticated one, that at least hints at the metallic and insulating nature of CMR

manganites.

Double exchange theory may anticipate the ferromagnetic-paramagnetic phase

transition, but it does not account for the experimentally observed metal-insulator

transition (MIT). This is because the available mechanism for describing resistivity in the

double exchange model, the scattering of electrons due to spin disorder, is insufficient in

magnitude to cause this dynamic crossover from a metallic to very insulating state, as

seen in experiments. It is this high temperature insulating state, either PI or CO, which

seems key to understanding the CMR effect. In addition, the double exchange interaction

does not predict the role of varying the hole-doping, reflected in the complex structure of

these materials at low temperatures as illustrated in Figure 2-5. Understanding all of these

phenomena will require augmenting the double exchange mechanism with a universal

theory that accounts for not only the orbital and spin ordering effects highlighted above,

but also incorporates the various structural and lattice effects ignored thus far in this

discussion.

2.3.2 Beyond Double Exchange, Part 1: Jahn-Teller Polarons

In general, cation substitutions and oxygen stoichiometry can both control the

various physical properties of perovskite materials. This is especially true in distorted

perovskites like the manganites. These distorted structures experience electron-lattice

coupling via two main mechanisms. One quantity, the tolerance factor, involves the

crystal structure's effect on hopping conduction. Specifically, different choices for the









trivalent and divalent cations in the mixed valence manganites change the internal

stresses acting on Mn-O bonds due to the cation's varying size. These internal stresses

have a strong effect on the hopping amplitude due to changes in the relative Mn-O-Mn

bond angle22, via compression and tension of the bonds. The other mechanism involves

electron-phonon coupling that links lattice deviations in the crystallographic structure

with deviations in the electronic configuration away from their average values. In

LaMnO3, this electron-phonon coupling is an inherent electronic instability native to the

MnO6 octahedra, the Jahn-Teller distortion23








01

1.92

La Mn



2.15 A 02







Figure 2-8. The projection of the crystal structure of LaMnO3 along the b axis. The
unequal Mn-O bond lengths illustrate the Jahn-Teller distortion.

In the manganites, the MnO6 octahedra experience a distortion in the Mn-O bond

lengths along different crystallographic directions. This distortion arises from an

electronic instability, the Jahn-Teller (JT) instability, intrinsic to a Mn3+ ion situated in an









octahedral crystal field (see Figure 2-8). Simply put, the JT distortion occurs when a

given electronic level of an ion or cluster is degenerate in a structure of high symmetry,

this structure is generally unstable, and the cluster will present a distortion toward a lower

symmetry ionic arrangement23. JT electron-lattice interactions, which are an

electron-phonon coupling mechanism, can be found in most orbitally degenerate

d-electron ions, as in Ni3+ or Cu2+, located in octahedral systems4. The JT distortion has

profound


x2 _y2


Mn3+ -
3z2 2
3d orbitals /





t2g xy
"--xz d yz


q Oxygen ion

0 Mn3+ ion


Figure 2-9. Jahn-Teller distortion and its effect on the energy level splitting of the Mn3
orbitals. The distortion stretches the Mn-O bond in the z direction and
compresses it in the x and y direction.


d Two "breathing" modes can also occur around a given Mn which couples to changes in the eg density.









influence on the phenomena of the CMR manganites. This is because the distortion acts

to couple the eg conduction electrons with a lattice instability, represented by an oxygen

displacement. This distortion lifts the orbital double degeneracy of the eg electrons seen

in Figure 2-3 and lowers the electronic energy (see Figure 2-9). Note that manganites like

SrMnO3, which have no eg electrons, do not have a JT distortion. Thus intermediate

doping regimes will have a modified lattice from either pure Mn3+ ion materials or pure

Mn4+ ion materials.

In Al-xBxMnO3 (0 < x < 0.2), the JT distortion is a static effect that produces a

cubic-tetragonal structural phase transition. This is a static or frozen effect because the

bond length distortions are coherent and have long-range order throughout the crystal.

The JT effect is very strong here with a bond length change of -10 % of the mean Mn-O

distance24. As x is increased from zero doping, the structural transition temperature and

the size of the long-range, coherent distortion decrease rapidly. In the low doping regime

(x < 0.20), the interplay of this diminished static JT interaction and the large Hund

coupling of the DE interaction dominate the current understanding of the physics of these

manganites.

A strong electron-phonon coupling can localize carriers, because an electron in a

given Mn orbital causes a local lattice distortion (e.g. the eg electron in the Mn3+ ion's

d 2 2 orbital), which produces a potential energy minimum that can trap the electron in

that orbital. If the coupling is strong enough, these tendencies lead to the formation of a

trapped state called a polaron. Following this line of reasoning leads to a competition

between electron localization, or trapping, this would result in insulating behavior, and

electron delocalization through hopping conduction, which can lead to metallic behavior.









This argument was first proposed by Millis et al.25 as an explanation for the observed

CMR phenomena in manganites with intermediate doping, 0.2 < x < 0.5. They used a

model Hamiltonian that coupled the lattice distortion via dynamic JT coupling, which

excludes long-range order in favor of fluctuations from site-to-site, to the double

exchange mechanism as follows:


Hf = HDE + + g~ Qb (j)cjbz + Q2(j) (2-10)
Sc jaz 2

Here we have used HDE from Eq. 2-1 and added a term which has external magnetic field

h coupled to the core spins Sc. The lattice distortion terms have electron-phonon coupling

g and phonon stiffness k. The electron is coupled to the distortion via a traceless

symmetric matrix, Q = r[cos(P)Tz + sin(P)Tx], that parameterizes the cubic distortion in

polar coordinates, r and p. Heffis solved in the dynamical mean field approximation26

with JH cO, and treating core spins and phonons as classical.

The calculation parameterizes the effective interaction in terms of a dimensionless

quantity X that is the ratio of the energy gained by an electron localizing and forming a

polaron, EjT, and the effective hopping parameter, teff, which follows the DE mechanism.

In the manganites, the DE mechanism and associated magnetostructural effects can have

strong dependence on temperature, magnetic field and doping. Likewise teff, and therefore

k, should reflect these changes in the fundamental properties of the manganite, such as

the resistivity. In this context when T > To the effective coupling X should be relatively

large due to a small effective hopping that leads to insulating behavior (electron

localization), while for T < To the emergent FM increases teff so that k decreases

sufficiently to allow metallic behavior (electron delocalization). The results of the













....... h 01





60.0 II0
40 0-h;=O.D035










0.00 005 0.10 015 0.20

T

integration method for electron density n = 1 and k = 1.12. h = 0.01t
corresponds to 15 T ift = 0.6 eV and Sc = 3/2. Millis et al. Phys. Rev. Lett.
77, 175-178 (1996), Fig. 2, pg. 177.

calculated resistivity under this picture are shown in Figure 2-10. It is clear that the

theoretical result correctly captures, qualitatively and somewhat quantitatively', the

experimentally observed results seen in Figure 2-7.

2.3.3 Beyond Double Exchange, Part 2: Percolating Phase Separation

The remaining feature of the experimental data not fully realized up to this point

is the nature of the inherent phase separation (PS) that underlies the CMR phenomena. In

1999, Uehara et al.2 produced striking evidence for the existence of phase separation in

Las5/s-yPryCa-, MnO3 (LPCMO) (see Figure 2-11). In addition to highlighting its existence

over a wide range of temperatures, they also showed its percolating nature. Several

experimentalists further studied the percolation picture. Kim et al. 5 who used



e The theoretical results underestimate the role of the eg electron density, n. For n = 0.75, which would
match the experimental results of Figure 2-7, the theoretical result shows anomalous low temperature
behavior25.









magnetization correlated with conductivity measurements to show that a classical 3D

percolation threshold (-17 % metallic fraction versus insulating fraction as predicted by

the general effective medium (GEM) theory) is reached at the MIT in LPCMO over a

wide range of doping levels.
























Figure 2-11. Dark-field SEM images for La5/-yPryCa3/8MnO3 obtained by using a
superlattice peak caused by charge ordering (CO). a) shows the coexistence
of charge-ordered (insulating), the light area, and charge-disordered (FM
metallic), the dark area, domains at 20 K fory = 0.375. The curved dark
lines present in CO regions are antiphase boundaries, frequently observed in
dark-field images for CO states of Lao.5Cao.5MnO3. b) and c) obtained from
the same area fory = 0.4 at 17 K and 120 K, respectively, show the
development of nanoscale charge-disordered domains at T > T,. Uehara et
al. Nature 399, 560-563 (1999), Fig. 3, pg. 562.

Zhang et al.27 showed direct observation of percolation in LPCMO using low

temperature magnetic force microscopy (MFM) (Figure 2-12) which they correlated with

resistivity measurements to show that the percolation of the FM metallic grains is

responsible for the steep drop in resistivity associated with all CMR manganites.









Theoretical work on PS behavior of the manganites was spearheaded by Dagotto and

Moreo14-16,20,28 showing the various mechanisms behind the intrinsic inhomgeneities










0 Hz 2 Hz

Figure 2-12. Scans (6 tm x 6 itm) of LPCMO. On cooling, FM regions (dark area) grow
in size. Percolation is visible below 113 K. Zhang et al. Science 298,
805-807 (2002), Fig. la, pg. 806.

that occur in the CMR manganites over a wide range of hole doping, temperatures and

magnetic fields. Specifically, they showed15 that the typical transport properties of the

manganites, the room temperature insulator transforming to a low temperature "bad"

metal, are directly correlated with the inherent inhomogeneity of the electronic phases.

Initially they employed a random resistor network in both 2D (square) and 3D

(cubic) clusters. They mimicked the mixed phase behavior of Las/s,8PryCa \MnO3 with

link resistances in parallel, randomly assigned as metallic or insulating, with a fixed

metallic fraction, p. The metallic resistance values were taken from experimental data on

LPCMO with y = 0 and the insulating values from y = 0.42, although the calculated

results appear qualitatively independent of the material. Upon solving the Kirchhoff

equations, they varied p for different temperature sweeps and found the calculated

resistivities showed pure insulating behavior forp = 0 and pure metallic behavior atp = 1

with intermediate values ofp producing a MIT and percolating behavior (Figure 2-13).









10 ,
pa, 1cm) (C sod ,





10 -o0
102 04 1

10 T .I





10 .0o 2D -
0 100 200 300
T (K)
Figure 2-13. Random resistor network resistivity for various metallic fractions, p in 2D
and 3D (inset). A broad peak appears at intermediate temperature and
values ofp in qualitative agreement with experimental evidence. Mayr et
al. Phys. Rev. Lett. 86, 135-138 (2001), Fig. c, pg. 135.

However, the broad peak found using this model does not accurately reflect the transport

behavior of most manganites. To solve this, a temperature dependent percolation

mechanism, separate from the variation of the metallic fraction must be introduced.

The introduction of a temperature dependent percolation mechanism is certainly

reasonable in CMR materials, because of the resistivity's extreme sensitivity to

temperature changes, especially around the MIT. A random field Ising model28 (RFIM) is

used to ascribe a temperature dependent and disorder induced type of percolative PS,

which could simultaneously capture the electronic and magnetic competing phases. This

analysis produced beautiful phase behavior that showed resistivity very similar to

experiment and magnetic phase separation (Figure 2-14) reminiscent of the SEM images

in Figure 2-11 and the MFM images in Figure 2-12.









S104
A
103

E 102

10


10
10-


loo
80
60
40
20
0 20 40 60 80 100

L w, i


0 ,6 --_- -
iL 2I -
0 100 200 300 o L- ...
T(K) 0 2 4 6 8
Figure 2-14. Monte Carlo results for the RFIM model. A) the net resistivity with similar
percolating behavior to the RRN model for intermediate metallic fractions
and temperatures but with much sharper changes in resistivity. B) and C) the
Monte Carlo results for the magnetic phase behavior in real space, with the
regions in B) correlating with C). These results are in very good agreement
with experiment. Mayr et al. Phys. Rev. Lett. 86, 135-138 (2001), Fig. 2, pg.
136.

This PS work indicates that the observed CMR phenomena, including the

complex electronic and magnetic phase behavior, can be explained in terms of small

changes in the metallic fraction with concomitant changes in the conductivity of the

insulating region. These last phenomena will be the driving force behind much of the new

experimental work described in the remainder of this dissertation.

In essence, PS in the manganites provides the mechanism for the CMR properties,

while the DE interaction and Jahn-Teller polarons describe the FM metallic and

insulating regions, respectively. In summary, CMR effects are best understood in terms

of competing phase separation between double exchange-governed, ferromagnetic

regions and insulating regions formed by Jahn-Teller polarons.














CHAPTER 3
DIELECTRIC RELAXATION PHENOMENA

3.1 Time-Dependent Dielectric Response: Polarization v. Conduction

Dielectric relaxation probes the interaction between a medium, with some

polarization, P, and a time-dependent electric field, E(t). The relaxation consists of the

recovery of the polarization on the sudden removal or application of the electric field.

The resulting polarization response of the medium (ionic, electronic, molecular, etc...), to

the ac electric field, characterizes the time-scale, via the characteristic relaxation time or

times, and amplitude of the charge-density fluctuations inherent to the sample. The

mechanism that governs these fluctuations varies from material-to-material. In most

homogeneous materials, where the charge-density can depend on molecular, ionic or

electronic sources, the simplest polarization mechanism is a permanent or induced dipole

moment that is reoriented by the ac electric field. The time-scales associated with the

reorientation of dipoles cover a broad range, with the associated frequencies spanning

from 10-4 Hz, for Si-Te-As-Ge (STAG) glass, to 1012 Hz, for low-viscosity liquids.

It is important here to differentiate between the two main processes associated with

the charge-density of the medium in question: charge polarization, and charge current. In

a static electric field, polarization occurs due to a net displacement of the charges in the

medium by some finite amount, while charge conduction results from a finite average

velocity in the motion of charges that depends on the dc conductivity, r. In this way,

polarization mechanisms and the related charge dipoles are incapable of contributing to a

dc charge current.









At ac, we can draw an even sharper distinction between the charge current and the

charge polarization because there is an associated time-dependence of the polarization

response to the driving force. This time-dependence is a delay in the polarization to

follow the time-dependence of the electric field. Below we show how the

time-dependence of the polarization leads to a polarization current that is fundamentally

different than the dc charge current.

A standard configuration for a capacitor has two parallel electrodes, separated by a

dielectric with thickness, d. On applying an ac voltage, V(t), to the capacitor, a spatially

homogeneous electric field, E(t) = V(t)/d, is induced perpendicular to the plane of the

dielectric. The free space charges in the electrodes will respond instantaneously to this ac

electric field, while the dielectric medium will exhibit a time-dependent delay. This

response can be expressed in terms of the dielectric displacement, D(t), which gives the

total charge density induced at the electrodes of the capacitor due to E(t), as follows:

D(t) = oE+P(t) (3-1)

where sEE is the induced free space charge of the electrode. The static, time-independent,

polarization, P, is related to the static electric field as follows:

P = oZE (3-2)

We introduce time dependence to the polarization in Eq. 3-2 by means of a

dielectric response function,f(t), and assume the time-dependent electric field, E(t), is a

continuous function of time such that the total amplitude can be written as the sum of

discrete increments, E(t)dt. The function,f(t), carries the response of the dielectric

medium to the ac electric field. The time dependent polarization can be written as:









0O
P(t) = o f f ()E(t- r)dr (3-3)
0

where t-ris the time before the measurement is made at time, t, and we define the

dielectric response function such that when t r= 0 thatf(r) = 0. If we now impose that

E(t) take the following time dependence:

0f for t < 0
E(t) = EoO(t) = Eo for t > 0 (3-4)


then the dielectric displacement in Eq. 3-1 takes on the following form:


D(t) = oEo O(t) + f (r)dr (3-5)


The total current, I(t), flowing in this capacitor can then be written as follows:

dO(t) dP(t)
I(t) = oEo + oEo + =Eo + = + EOE [3(t) + f(t)] (3-6)
dt dt

where oEo is the dc current flowing through the system. The polarization current arises

from the inability of the charge species to follow the time dependence of the electric field

that is driving the system. In the present form of the electric field (3-4), the polarization

current must go to zero as t ->o. This implies that at infinite times that the polarization

must approach a constant29, defined by:


P(oo) = soEo f (t)dt = EoZ(O) (3-7)
0

where -(O) is the static susceptibility. As a consequence of Eq. 3-7, no charge species

may be transferred to or from the dielectric system. In contrast, the dc conduction current,

oEo, exists only in the presence of free charge species that move across the dielectric









from one capacitor electrode to another. Once the polarization of the system reaches

saturation at t oo, i.e. after a sufficiently long time, the electric field can be switched off

and the resulting depolarization can be observed. This is an important aspect of dielectric

relaxation, in that as the polarized charge species begin to relax and ultimately

depolarize, the dynamics of the charge species can be observed in the absence of the dc

charge conduction. In practice, this is how many experimentalists ascertain the dielectric

response function, and thus the time-dependent dielectric permittivity, (t)29.

3.2 Frequency-Dependent Dielectric Response

While the most natural and readily accessible representation of the various

dielectric and electrodynamic processes present in insulating media may be in the

time-domain, experimentally the most practical representation is usually in the

frequency-domain. Dielectric response is usually studied as a function of frequency,

instead of time, because measurements can be made at a given frequency, or range of

frequencies, with a high degree of accuracy and precision due to the present state of rf

signal technology. It is possible to move back and forth from the time- to

frequency-domain utilizing Fourier transformations and numerical integration.

Given a time-dependent function, F(t), the Fourier transform of it into the

frequency-domain has the following form:

I +Do
[F()] = F() 1 F(t)exp(-i)t)dt (3-8)


where o) is the frequency and therefore F(o)) is the frequency spectrum associated with

F(t). This is a powerful tool because the Fourier transform of a real, time-dependent

function produces a complex, frequency-dependent function that contains information









about the amplitude and phase of the time-dependent signal. Additionally, the Fourier

transform allows us to write integrals of the form in Eq. 3-4 as the product of the Fourier

transforms of the functions in the integrand30. In particular, it allows us to define a

frequency-dependent susceptibilityf, which is just the Fourier transform of the dielectric

response function,f(t), as follows:

0O
Z(0) = '()- i"(m)= f (t)exp(-iot)dt (3-9)


Therefore, we now have a way of simultaneously studying the dielectric response that is

in-phase, -'(o), and out-of-phase, X"(o), with the ac electric field. The ability to study

the in-phase and out-of-phase components of a dielectric response is at the heart of

dielectric relaxation spectroscopy. In particular, we would like to know what each of the

components of the dielectric response, in this case the dielectric susceptibility, contribute

to the relaxation process.

We can answer this question by returning to the total current of Eq. 3-7 and writing

the corresponding frequency-domain response as follows:

I(Co)= cE(cw)+io)D(c) (3-10)

where 3 [iD(t) / t] = iaD(m) Using Eq. 3-1, the current can be written in terms of the

frequency-dependent polarization, P(i)= coE(wo)X (o), such that, after some algebra:

I() = [o + Z0o )"(o) + iog {1 + Z'())}]E(o) (3-11)






f When we speak of the frequency-dependent susceptibility is important to note that multiple unique and
separate mechanisms may contribute to it. Therefore, -(a)) is comprised of the sum of the susceptibilities
due to each of these independent mechanisms.









Here we see that the imaginary component of the susceptibility, X"(co), is in-phase with

the dc conduction current and therefore contributes to the total dissipation, or loss, of the

charge polarization at ac. Therefore X"(o) is termed the dielectric loss. The real part of

the susceptibility, '(ao), is out-of-phase with the dc conduction and thus acts to conserve

charge polarization. The picture produced from this analysis is that the total dielectric

response and relaxation is driven by the competing mechanisms of charge polarization, or

charge storage in terms of a capacitor, due to '(ao), and charge dissipation, due to the ac

loss, X"(a), in the presence of the dc conduction current, oE.

3.3 Dielectric Response Functions

In the previous section we derived an expression for the total frequency-dependent

current, I(o), present in a capacitor comprised of a dielectric medium between two

electrodes. This current was expressed in terms of the complex susceptibility, X(0) and

dc conductivity, u(o)=0). When measuring the total dielectric response, at ac, it is usually

not possible to look at the dc and ac contributions separately. Specifically, the

frequency-dependent current simply describes the movement of charge in response to the

driving force provided by the ac electric field, E(co). This movement of charge is due to

both dc and ac processes and thus any dielectric response that is measured will contain

contributions from both dc and ac conduction. Therefore, we think in terms of measuring

an effective dielectric response and using analysis to separate the various contributions.

This can be done in terms of several different, but equivalent, dielectric response

functions that are consequences of the macroscopic Maxwell's equations. In particular,

we will define the impedance/admittance and dielectric permittivity, from which all other

dielectric responses can be easily generated or represented.









3.3.1 Impedance and Admittance

In a dielectric system, there are two ways to generate a dielectric response: the

measurement can source a voltage, V(c), that drives a current, I(o) or vice versa. There is

a phase shift between I(o) and V(o), such that the response can be resolved into in-phase

and out-of-phase components. The response associated with this phase shift can be

represented in terms of two dielectric functions, the admittance, Y(i) =I(i)/V(O), and the

impedance, Z(io) V(o) /I(o). In practice, the admittance is used when the experiment is

sourcing voltage and measuring current, while impedance is used for sourcing current

and measuring voltage29. In a system that is modeled as an ac dielectric process acting in

parallel with a dc conduction process (evident in the analysis above in Section 3.2), the

admittance is a natural representation because it involves a constant voltage, V(o), across

both parallel elements. However, the reciprocal nature of the admittance and the

impedance allows for the easy transformation of one quantity to another when necessary

as when a dc conduction path exists in series with the dielectric medium (Section 4.3.4).

A system whose understanding benefits from impedance/admittance analysis is the


CB CV









GB GV
Figure 3-1. Barrier-volume capacitor. Cg and GB are the barrier capacitance and
conductance, respectively. Cv and GB are the volume capacitance and
conductance, respectively.








barrier-volume problem that generalizes two distinct dielectric regions in series. Each

dielectric region is modeled as a capacitor, C, in parallel with a dc conductance, G

(Figure 3-1). This circuit is described by the Maxwell-Wagner relaxation31 developed to

describe the behavior of a conducting volume or "bulk" dielectric material with a barrier

or "contact" region that is highly capacitive and less conducting (C << CB and GB <

Gv). The impedance of the circuit in Figure 3-1 is written explicitly as follows:

Z() /= + iGB (3-12)
1 +icr 1l+ir0IB

where T = C Gv and B = C GB. A schematic representation of Eq. 3-13 (Figure 3-2)

shows the two distinct dielectric regions: volume (small arc) and barrier (large arc).

I I
G)TB= 1

200- orv= 1






N 100- -






0 ) = GV/CB \
0 100 200 300

Z'
Figure 3-2. The complex impedance diagram of the barrier-volume circuit. The large arc
is due to the barrier region and the small arc is due to the volume region.









Figure 3-2 illustrates the utility of a complex impedance diagram. The diagram

reveals information about the time constants of the volume and barrier regions, as well as

indicating the crossover frequency that divides the two dielectric responses. If we invert

Eq. 3-13 we can write the admittance of the response (Eq. 3-14), where r = CB/ Cv.

(1 + io) z-,)(1 + io) z-,)
Y(o)) = G( (3-13)
1+ico, ,r

A complex admittance plot of Eq. 3-14 (not shown) reveals that the volume and barrier

regions are two distinct vertical lines with the x-intercepts indicating Gv and GB,

respectively.

3.3.2 Dielectric Permittivity

The frequency-dependent permittivity, e(o), is defined in terms of the complex

susceptibility as follows:

E(m) = co [1 + x'(m)-ix"(Oc)] = E'(o)) iE"()) (3-14)

such that, D(o) =(o))E(o). In this way, the dielectric permittivity is a measure of the total

response of the dielectric system, due to both the free space and material dependent

contributions, to an ac electric field. It does not however, include the dc conduction

contributions which are intrinsic to any measurement, while being extrinsic to the true

dielectric response. It is possible to express the dielectric permittivity in terms of the ac

conductivity via 1) Ampere's law, V x H = a(o)E + e(o))EI / /t and 2) the divergence

theorem, V (V x H) = 0 (Eq. 3-15).


E(o)) () ) (3-15)
1CO









The main advantage in using the dielectric permittivity representation of the

dielectric response is it is directly proportional to the capacitance; the quantity most

frequently measured to study dielectric relaxation. The complex capacitance is defined as

follows:


C() = C'(c) iC"(co)= A ['(o) iE"(o)] (3-16)
d

where A is the area of the electrode and d is the thickness of the dielectric. In this way,

the complex capacitance is composed of a real part, C'(o), which reflects the capacitance

associated with charge storage and an imaginary part, C"(w), which accounts for the ac

loss. The study of the complex capacitance will dominate the remainder of this

dissertation.

3.3.3 A Dielectric Case-Study: Disordered Conductors

The frequency-dependent conductivity of disordered conductors exhibits striking

universal behavior. In these systems, measurements of the ac conductivity as a function

of frequency, at different temperatures, can be scaled onto a "master" curve. This

"master" curve approximates a power-law dependence of frequency, with power y< 1.

This behavior occurs independent of conduction mechanism (electronic or ionic) and the

state of the system (amorphous or polycrystalline). The only commonality in these

universal ac conductors is that they share strong disorder with conduction dominated by

percolation. The ac conductivity of this broad class of materials is best understood in

terms of a random resistor capacitor network (RRCN) (Figure 3-3). The microstructure of

these materials is a percolating network of conducting and capacitive regions. This model

is confirmed theoretically by considering a macroscopic model of disorder in a solid with




























Figure 3-3. Random resistor capacitor network that describes the dielectric behavior of
the universal ac conductivity materials.

heterogeneous conducting phases. This model assumes that the disordered system has

free charge carriers with current density J(r,t) = g(r)E(r,t) where g(r) is the local

conductivity at the corresponding position in the solid and displacement D(r, t)= cE(r,t)

with bound charges that are described by the high-frequency dielectric constant, Ec, the

limit of a -> oin Eq. 3-14. From Gauss's law, V D = p, and the electrostatic potential,

E(r, t) = -Vp, we can write the continuity equation, p + V J = 0, in terms of the local

conductivity and the bound charge dielectric constant (Eq. 3-16). Dyre showed32 that

V. imE + g(r)]V})= 0 (3-17)

when Eq. 3-16 is discretized in the presence of a periodic potential across the RRCN

circuit for a cubic lattice in D dimensions with spatial length L separating each node, that

the free charge ac conductivity, o(o), is as follows:






37


C(w) = LD-I)2 (3-18)
LD-

The frequency dependence of the ac conductivity is governed by the capacitors influence

on the local potential at each node, that in turn determines the resistor currents. In the

high frequency limit the capacitor admittances (susceptance) dominate the response and

produce a spatially homogeneous electric field.














CHAPTER 4
CAPACITANCE MEASUREMENTS AND EXPERIMENTAL TECHNIQUES

4.1 Thin-Film Metal-Insulator-Metal Capacitors

Traditional MIM capacitors are easily approximated by the parallel plate model of

capacitance. Where two largeg, two-dimensional, metallic electrodes with area A, enclose

an insulating medium, with dielectric constant e, and thickness d, capable of storing

charge as defined by the geometrical capacitance relation (Eq. 4-1), where 0o= 8.85x10-12

F m-1, is the permittivity of free space.


Cg EA (4-1)
d

The measured capacitance can be heavily modified from the geometrical

capacitance (Eq. 4-1) when d becomes vanishingly small. Specifically, an interface

capacitance due to an electron screening length dependent voltage drop at the

metal-dielectric interface of the electrodes can come to dominate the measured

capacitance33,34. These screening effects can be further enhanced in the presence of a

magnetic field when the MIM has ferro- or paramagnetic electrodes35. In our

experiments, additional complications to the traditional parallel plate model arise from

the wide variability of the conductivity of our base electrode, with magnetic field and

temperature. In the following sections we outline the methodology used and

considerations made when measuring our novel MIM capacitors.


g Large in this case is defined as the area of the electrode should be much larger than the thickness of the
dielectric; A << d.









4.2 Metal-Insulator- Metal (MIM) Fabrication

The capacitors used in our experiments have a consistent trilayer architecture

comprised of a polycrystalline, thin film (Lali-Prx)5/sCa3/8MnO3 (LPCMO) base electrode

(C in Figure 4-1) deposited on a highly insulating NdGaO3 (NGO) substrate (D in Figure

4-1), an amorphous AlOx dielectric (B in Figure 4-1) and a polycrystalline, thin film Al

counter-electrode (A in Figure 4-1). Contact is made to the base and counter electrodes

using pressed indium and fine gold wire held in place with a dilute silver paint,

respectively.





D



Figure 4-1. Schematic of MIM. A) the metallic counter electrode. B) the dielectric. C) the
manganite base electrode. D) the substrate.

The LPCMO layer is grown using pulsed layer deposition (PLD) in a high

vacuum chamber with a base pressure of 1 x 106 Torr. A NGO substrate is loaded onto

the high temperature sample holder and held at a temperature of -800 C during the

growth process. The LPCMO thin films used in our experiments ranged in thickness from

400 to 600 A as determined by atomic force microscopy (AFM) profiling. The AlOx

dielectric is deposited via rf magnetron sputtering of an alumina target in an ultra high

vacuum chamber with a base pressure of 1 x 10- Torr. The dielectrics used were in a

thickness range of 100 to 150 A as measured by an Inficon in situ quartz crystal thickness


h Silver paint was also used to make contact to the base electrode in some occasions. We found no
qualitative distinction between the two contact methods.









monitor. In the past33'34, this method has produced high density, homogenous AlOx down

to thicknesses of- 30 A. The counter-electrode is deposited via thermal evaporation of

99.999% pure, aluminum wire, through an aluminum shadow mask, onto the dielectric

layer, in a high vacuum chamber with a base pressure of- 1 x 107 Torr.

4.3 Measuring Capacitance: Tools and Techniques

Fundamentally, all capacitance measurement techniques involve measuring a

material's ability to store and transfer electric charge in the presence of an ac electric

potential. There are three major instruments that are used in our experiments. The first is

the impedance-gain/phase analyzer, which is a self contained ac response unit that

simultaneously measures the in-phase and out-of-phase response of a system to a small ac

perturbation (either current or voltage). Second, we discuss the lock-in amplifier; it is the

most versatile in terms of its ability to measure a wide variety of ac responses and its

ability to be integrated into measurement schemes with other instruments. We then

discuss the importance of the capacitance bridge to the experimental work we have

conducted.

4.3.1 Impedance Analyzer

The impedance analyzer used in our experimental work is a Solartron 1260A

Impedance/Gain-Phase Analyzer. It is designed to measure the complex impedance,

Z*=Zi+ iZ2, of a sample in the frequency range of 10[tHzi to 32MHz and excitation

voltage amplitude, Vot < 3 Vms. Whereas, the lock-in amplifier usually uses an external

reference signal that "locks" onto the input signal, the 1260A has a built in frequency


1 Some would argue that measurements at frequencies less than 1 Hz are no longer "ac" measurements, but
really dc in nature. The converse opinion is that there is no true "dc" measurement because all
measurements occur over some nonzero time, and thus have a frequency inherently associated with them.
This is left as a question for the ages.









synthesizer that drives the signal for the test sample, and the reference signal for the

voltage measurement. The 1260A outputs a signal that drives the sample at any specified

frequency and amplitude in the range stated above. In turn, a current amplifier input

measures the associated current through the sample while simultaneously, differential

voltage amplifier inputs, VHI and VLO, measure the voltage across the sample using an

internal amplifier.

The applied sinusoidal, ac voltage and the resultant ac current are measured to

arrive at the complex impedance, Z = V/I. The applied voltage, V(t)=Vocos(ct),

produces electronic polarization in the sample, and the resultant current, I(t) =ocos(ct+ 3)

will have the same frequency, but a different amplitude and phase, 3. The 1260A will

then report the complex impedance in a variety of ways, the most general of which is the

magnitude of the impedance, |Z|, and the phase angle, 0, such that the complex

impedance is, Z*= Z exp(iO). This in turn can be written in terms of its real and

imaginary parts to reveal the in-phase and out-of-phase components of the response.

The major experimental disadvantage presented by the 1260 A is that the

confidence interval of its impedance measurements is severely reduced for high

impedance samples, see Fig. 4-2. This is primarily because the input impedance at the

VHI and VLO terminals is 1 MQ // 35 pF, according to specifications. This input

impedance









100M

10M

1M

100k

10k

1k

100

10

1

100m
10m
l10M


100 1k 10k 100k 1M 10M

Frequency (Hz)
Figure 4-2. Confidence intervals of the measured impedance as a function of the
measurement frequency. A sample with Z > 10 MQ will have a 10 % error
in the measured impedance amplitude and 100 error in the measured phase
angle at all frequencies.

problem could have been circumvented by introducing a front-end for the 1260A that

would include high input impedance buffers, such as operational amplifiers configured as

voltage followers, between the electrodes and the differential amplifier that would

increase the confidence levels at low frequency. However, this was not pursued in our lab

because the bulk of our results were obtained with the capacitance bridge outlined in the

following section.


V
__A _oi /







x^v^
__X__X









4.3.2 Lock-In Amplifier

The Princeton Applied Research PAR 124A lock-in amplifier allows for the

measurement of a wide variety of ac responses with a high degree of precision. While a

lock-in amplifier is a powerful and extremely important tool it is also quite simple,

especially when compared to the more complicated impedance-phase/gain analyzers

discussed below. A schematic diagram of a lock-in amplifier is shown in Figure 4-3. It

consists of five stages: 1) an ac signal amplifier, which is the input for the signal to be

measured; 2) a reference input; 3) a phase sensitive detector (PSD) that acts as a

multiplier; 4) a low-pass filter; and 5) a dc amplifier. The output of the dc amplifier is a

voltage that is proportional to the amplitude, Vo, of the input signal, V(t) = Vocos(coo t+ ),

where coo is the frequency of the signal and 5is a phase shift.

Signal Monitor
Signal In
Signal
Amplifier

/ni \Output
Phase Low-pass DC
Sensitive Filter Amplifier
Detector




Reference In Reference Out
Figure 4-3. Schematic of a lock-in amplifier

The major differences between an ac voltmeter, which will also report the voltage

amplitude from an input signal, and a lock-in amplifier are that: 1) a lock-in uses a

reference signal that is synchronized with the input signal being measured and 2) lock-in


J The reference signal is derived from a periodic voltage source and is usually on the order of 1Vp-p.









amplifiers will measure the in-phase and out-of-phase components of the input signal

with precise frequency control. The signal amplifier receives the input, amplifies the

voltage by a gain of Gac and filters out most, but not all, extraneous signals at other

frequencies. Then the amplified signal input, Vac(t) = GacVocos(coo t), and the

synchronized, or triggered, output of the reference input, VI(t) = Aocos(oo t), (we are

assuming 3,q= 0)k are multiplied in the PSD that produces the following output:


VPsD(t) = GacAoVo [1+ cos (2owt)] (4-2)

The amplitudes of the dc term and the second harmonic term are both

proportional to our input amplitude, and thus it is redundant to pursue both of them.

Therefore, the signal in Eq. 4-2 is passed through the low-pass filter that attenuates the

second harmonic term and also integrates the signal greatly reducing random noise. This

output is then fed into the dc amplifier which increases the signal from the low-pass filter

by a gain of Gdc, resulting in an output of:


Vout =GdcGaCAoV (4-3)
2

Figure 4-4, shows a comparison of the impedance of an LPCMO film as measured

by the Solartron 1260A (black squares) and the PAR 124A Lock-in with output read by a





k Having # 0 and 6# 0 complicates the result because the input signal will have some unknown phase
and the output in Eq. 4-3 will be


Vo= GdcGcAoV cos (3+ (4-4)

In this instance, the phase of the input reference would be tuned such that the signal in Eq. 4-4 is a
maximum.
















107



N- ( RBallast


106 SamplOutput

-- Lock-In

75 100 125

Temperature (K)

Figure 4-4. The magnitude of the impedance of a 400 A LPCMO film as measured by
the Solartron 1260A (black squares) and PAR 124A Lock-In (red circles), at
a frequency of 50 Hz. The arrow indicates the where the input impedance of
1260A lies, and we note that is where the measurements collapse onto one
another. Inset: circuit for ac resistance measurement with PAR 124A.

Keithley 182 nanovoltmeter (red circles) that has an input impedance of 10 GO. This

clearly illustrates the advantage of the two-terminal lock-in technique (Figure 4-4 inset)

over the impedance analyzer in terms of high impedance measurements. Thus lock-in

amplifiers are ideal for measuring ac responses where the signal-to-noise ratio is very

high, even when the signal of interest is a few nanovolts buried in a background signal

orders of magnitude larger.









4.3.3 Capacitance Bridge

The capacitance bridge used in this work is the Andeen-Hagerling AH 2700A.

Like the Solartron 1260A, it is a stand alone ac response unit that requires no external

reference signals to perform its measurements. Unlike the 1260A, it utilizes a small ac

voltage source and a balancing circuit to achieve its results, and measures with maximum

accuracy and precision on samples that have medium to high impedances (105 < Z <

1015 Q). The AH bridge operates in a frequency range of 50 to 20,000 Hz and ac voltage

amplitude of 30 mVrms to 15 Vrms with a best capacitance resolution of 0.01 x 10-18 F and

a loss' resolution of 3 x 10-16 S.

The bridge circuit is depicted in Figure 4-5. Like the lock-in amplifier, its

operational components consist of five stages: 1) a sinusoidal, voltage generator that has

precise amplitude and frequency control; 2) a high precision ratio transformer; 3) Tap 1

which controls the voltage across the standard impedance; 4) Tap 2 which controls the

voltage across the sample, or unknown, impedance; and 5) the voltage detector. To

balance the bridge, the voltage between Tap 1 and ground and the voltage between Tap 2

and ground must be equal to the voltage across Leg 3 and the voltage across Leg 4,

respectively. This ensures that when the bridge is balanced that the voltage across the

detector is zero. When the detector reads zero voltage, or as close to zero as possible, the

relationships between the standard and sample impedances are as follows:


C, = N C (4-5a)
N2




SLoss is in reference to the component of the impedance 900 out-of-phase with the capacitive element. It is
usually measured in terms of Siemens (S) with 1 S = 1 Ohm1 = 1 mho, in older notation.









R=2 (4-5b)
N,

where N1 and N2 are the number of turns in leg 1 and leg 2, respectively, of the secondary

windings of the transformer. This capacitance bridge circuit illustrates the main benefit of

Tap 1 Standard Impedance

Leg 3
S Leg 1



Generato Detector
Sample Impedance


Leg2 L Sm e
BNC Connectors -O
C--^ Rx G
Tap2 RCx G
Leg4 H
Ratio Transformer

Figure 4-5. Schematic of the capacitance bridge circuit. The standard impedance
elements are fused-silica capacitances for Co and variable pseudo-resistors
for Ro. The unknown sample is expressed by Cx, the in-phase component,
and Rx, the out-of-phase component.

all bridge technology, in that the act of measuring the unknown impedance requires

measuring small deviations from zero.

The success exhibited by the AH 2700A depends on one key external component:

a guarded three-terminal measurement. In an unguarded measurement there is a strong

likelihood that additional current could be drawn across the sample due to a net voltage

drop between L and H in Figure 4-5. To avoid this, a guard (G) is placed around the

sample, in our case an electrically isolated copper cylinder, which is connected to the

ground of the bridge circuit (Figure 4-5). Therefore, no stray voltages, or cable

capacitance can appear along the leads between the circuit and the guard. Additionally,









the voltage between H and G will be across the secondary winding of the transformer and

thus will be in parallel with the sample thus excluding it from any unwanted current.

There is no voltage drop between L and G because it shunts the detector and is held at

ground while the bridge is balanced, thus it can have no effect on the sample impedance.

Another feature of the AH 2700A evident from Figure 4-5, is the bridge always

assumes that the ac response of the sample is that of a parallel RC model. In practice, this

is a safe assumption because most "normal" MIM capacitors behave in this manner.

However, many novel materials and configurations for MIM capacitors are not

represented by this parallel RC model. In these instances caution must be used when

interpreting the results of the measurement. The AH 2700A does allow for reporting the

in-phase and out-of-phase components based on a series RC model. This is done by a

simple set of algebraic transformations36. A more complicated scenario that is applicable

to our experiments is outlined in the following section.

4.4 The Series Resistance, Complex Capacitance Problem

To interpret the measured complex capacitance, C(o) = C'(c) iC"(o), the

effective dielectric response, must be modeled in an equivalent circuit. Ideally, we would

be able to express any complex dielectric response, in terms of some real capacitance,

C'(ow), in parallel with the out-of-phase, or imaginary terms: RDC that expresses the leaky

behavior, and 1/o)C"(o) the lossy behavior" (Figure 4-6A). In the geometry of our

capacitance measurements (Figure 4-1) the main concern is to ensure that the electric

equipotential lines are parallel to the interface of the base electrode and the dielectric.


m The dielectric of a "leaky" capacitor passes a small dc current in response to an applied dc voltage,
reflected by a large dc resistance, RDC, in parallel with C(a). A "lossy" capacitor has frequency-dependent
dissipation from the imaginary part C(c), C "(a) and acts in parallel with RDc.








This is nontrivial because our base electrode is a highly resistive material over a wide

range of temperature, magnetic field, etc... A highly resistive electrode will introduce

voltage drops along its length that will act as a resistor in series with the capacitor defined

by base electrode/dielectric interface. Of course in any MIM capacitor, the electrodes are

not ideal and have some associated resistance.

A C'() B Ceff




o -----A M^-- _

I/mC"( )

R"R
RDC Reff
RDC
Figure 4-6. Schematic of a complex capacitor. A) C'(o) is the real part of the dielectric
response and RDC and 1/coC"(o) are the impedances of the leakage and loss,
respectively. B) effective circuit assumed by the capacitance bridge.
Therefore, some criteria must be established that enables a high level of confidence that

the series resistance can either be ignored, or at least quantified and subtracted from the

dielectric response".

Assume that we have a resistive electrode in our standard MIM capacitance

configuration in series with a lossy capacitor that has no dc shunt resistance. The

equivalent circuit for this is shown in Figure 4-7, with R2 = 1/oC"(c). As previously

highlighted in Section 4.3.3, the capacitance bridge technique employed in our

measurements, reports the measured response assuming the parallel RC model shown in

n The real part of the capacitance will not be affected by the series resistor, but the imaginary term which is
out-of-phase with the capacitance is in-phase with the series resistance. Therefore, the effective dielectric
response will be compromised by the presence of the series resistor.








Figure 4-6B. Therefore we must reconcile the series-parallel model of Figure 4-7 with the

pure parallel model of Figure 4-6.


C1


Rs







R2
Figure 4-7. Equivalent circuit of MIM capacitor with a resistive electrode, Rs, and
complex capacitance, C = C1 iC2, where R2 = 1/oC2.

The admittance, Yp, the inverse of the impedance, Z, of the parallel RC model in

Figure 4-6 (with RDC ~ oc) is written as:

Y, = +ioCef (4-6)
Rff

The impedance, Zs, of the series RC model of Figure 4-7 is written as:

Zs = Rs + (4-7)
1 +io)R2C,

Writing this in terms of the associated admittance, Ys, we arrive at:

Ys = + (4-8)
R2 + Rs(I+ ioR2C,)

After some algebra we arrive at an expression for the real and imaginary parts of Ys:

Y = (R2R) 2 C2 i C (4-9)
(R2 +Rs)2 + (CR2RC)2 (R, +Rs)2 +(CRC)2









Equations 4-6 and 4-9 imply that the components of the effective admittance can now be

written in terms of Ys as follows:

(Rp +Rs)2 +((OR2RsC) 2
Ri = 2 1(4-10 a)
(R2 + R) + 2R'1'RC2


Ceff = R )2+ ) 2 (4-10 b)
e R2 +Rs + mR2R,Q2

In an ideal situation, Cff, would be independent of any of the out-of-phase components

such that Ceff C1. We now apply the following constraints to Eq. 4-10b:

Constraint 1: (mR2RsC1)2 (R2 +RS)2 (4-11 a)

R2Rs 1
-2R <<- (4-11 b)
R2 +Rs oC,

Constraint 2: Rs <
1
SRs << (4-12 b)
o>C2

The sum impact of these constraints is that the effective dielectric response of our MIM

capacitor depends solely on the contributions of C1 and R2 (Eq. 4-13).


Rff Cff pR2C (4-13)
R2 +Rs + 22RRsC2,

Also, we can ignore the voltage drops along the highly resistive, LPCMO base electrode

as long as the following condition (Eq. 4-14) is met:


Rs )C1 mC2

The series resistance in our model, Rs, is equivalent to the two-terminal resistance

of the LPCMO base electrode that is used in our MIM capacitor. Thus, it is an






52


experimentally measurable quantity0. Figure 4-8, shows the temperature dependence of

Rs and the real and imaginary parts of the impedance of our MIM. Clearly, the conditions

of Eq. 4-14 are met by our choice of frequency and the materials in our MIM capacitor.


108


N


106


100


200


300


Temperature (K)


Figure 4-8.


The temperature dependent impedance of the LPCMO film, that serves as
the base electrode in our MIM and the real and imaginary parts of the
impedance of our MIM, 1 coC'and 1/coC respectively. The experimental
data tells us that we satisfy the condition laid out in Eq. 4-14 and thus the
voltage drop along the LPCMO film is compared to the voltage drop across
the capacitor negligible.


This result assumes that the dc resistance of our thin film dielectric, AlOx, is effectively

infinite. We separately confirmed this by attempting to measure the dc resistivity of the

o A quick note about the perils of two-terminal measurements is necessary. Contact resistance between the
lead of the instrument and the material is always present in a two-terminal measurement. The best way to
estimate its effects is to measure the resistance in both a two and four terminal configuration. A large
discrepancy between the two measurements indicates a large contact resistance. If the two numbers are
approximately equal then the contact resistance is small. In our measurements on LPCMO, we see a small
contact resistance, orders of magnitude smaller than the resistance of the LPCMO film itself.


," 1/0C'
-I --; ,, --1/-C"

S' ---- R LPCMO film
It ,






i i
I -


I

^-S
I




I ,I I
_I *'
\\~






53


MIM across the electrodes (i.e. through the AlOx dielectric) and found it to be

immeasurably large (infinite) by all techniques and tools available. For comparison, we

used the same transport measurement procedures on a precision standard ceramic 109 Q

resistor and found we were easily able to measure its resistance. Therefore, we place the

lower bound on the dc resistance of our MIM at RDc > 109 Q. This further implies that

our LPCMO MIM capacitor has very little leakage current through the AlOx dielectric,

and is thus best described as a "lossy" capacitor.















CHAPTER 5
COLOSSAL MAGNETOCAPACITANCE IN PHASE-SEPARATED MANGANITES

5.1 Dielectric Measurements of Strongly Correlated Electron Materials

In recent years, a wide variety of strongly correlated electron materials (SCEMs)

have been probed using standard bulk dielectric techniques37-44. These methods entail

using the SCEM as the insulator in a MIM capacitance structurep and measuring the

capacitance to determine the complex permittivity, e*(o), dielectric constants, complex

impedance, Z*(co), etc... of the SCEM. One of the main goals of these previous studies

has been to identify a SCEM that possesses a low, or zero, frequency dielectric constant,

E(0), that is very large, or "colossal", relative to that of standard dielectric materials like

A1203, which has E(0) = 9, or even SrTiO3, which has E(0) = 100. Colossal or giant

magnetocapacitance has been observed in some37-39,43'44 of these SCEMs, with changes in

the measured dielectric permittivity of up to 500 % with applied magnetic fields. In our

experiments we observe a colossal magnetocapacitance effect that results in an increase

in the measured capacitance of three orders of magnitude. However, we are not looking at

colossal dielectric constants but rather a competition between metallic and insulating

phases that produces a striking dielectric response as a function of magnetic field,

temperature, and frequency.





p Typically, these materials are grown as crystals, diced and then either gold or some other metal are
deposited on both sides of the insulator. Alternatively, some experiments have used pressed metallic
electrodes due to the possibility of damage at the interface due to sputtering.









5.2 (Lali-Prx)5/8Ca3/8MnO3

As we discussed in Ch. 2, (Lail-Prx)5/Ca-, ,\MnO3 (LPCMO) is a mixed valence

manganite that exhibits a wide range of phenomena, including the presence of competing

electronic and magnetic phase separated (PS) regions present over a wide range of

temperatures and magnetic fields, (see Figures 2.11-12). LPCMO has become the

prototypical manganite to study the effects phase separation because its two host systems,

LCMO and PCMO, exhibit widely different phase behavior with respect to similar

temperature and magnetic field scales. As illustrated in Figure 5-1, LCMO exhibits a



A *o Hr 0 I- ulator -

0 50


J0 10
B0 Oe



40
4T 100

200 2T
---E-- 50k Oe
S4T 40k Oc
04 0 2 2100 1-50k 0oe
-0 670k Oe
70k OeF
0 50 100 150 200 250 300 350 70k C
10-1
Temperature (K) 0 5 0 150 200 250 300
Temperature [K]
Figure 5-1. Temperature dependent resistivity of A) La0.67Ca0.33MnO3 (LCMO) and
B) Pr0.7Ca0.3MnO3 (PCMO) in the applied magnetic field indicated in the
legends. At zero field, the PCMO has no metal-insulator transition, and at
low temperature is very resistive, while LCMO has a metal-insulator
transition, with low resistivity at low temperature. The PCMO exhibits
pronounced hysteresis in the resistivity between the cooling and warming
curves indicated by the arrows in B. inset: T-H phase diagram indicating the
onset of the MIT in PCMO and showing the hysteretic region. A) Schiffer et
al. Phys. Rev. Lett. 75, 3336-3339 (1995), Fig. 2, pg. 3337. B) Tomioka, Y.
et al. Phys. Rev. B 53, R1689-R1692 (1996)45, Fig. 2, pg. R1690.








metal-insulator transition (MIT) at all magnetic fields and shows colossal

magnetoresistance (CMR), while PCMO has no zero-field MIT, but does have a large

CMR effect that is orders of magnitude larger than the CMR effect in LCMO. Another

feature evident in Figure 5-1B is that PCMO exhibits hysteresis in the resistivity as the

temperature is cooled and warmed. LCMO does not show this hysteretic behavior.

When LCMO and PCMO are combined to form LPCMO, the resulting material

exhibits a zero-field MIT, a concomitant magnetic phase transition from charge-ordered


101


100


0

4,J


I)
()
a)
Q^


10-


10-2


0 100 200 300

Temperature (K)
Figure 5-2. Temperature dependent resistivity for an LPCMO x = 0.5 thin film with
thickness d 500 A showing the decrease in the size of the hysteresis and
size of the resistance transition with increasing magnetic field, along with an
increase in the TMI. The arrows indicate the direction the temperature is
being swept.


w 0 kOe
o 20 kOe


(La 4/8' Pr4/85/8 Ca3/8M 3






57

insulator (COI) to ferromagnetic metal (FMM), a pronounced hysteresis in the resistivity

and a very large CMR of approximately four orders of magnitude at the MI temperature,

TMI. Typical transport behavior for a polycrystalline thin film of LPCMO (x = 0.5) with

thickness d& 500 A is shown in Figure 5-2. As in PCMO, LPCMO shows a decrease in

the size of the hysteresis and size of the resistance transition with increasing magnetic

field, along with an increase in TMI.

In Figure 5-3, the magnetic field-dependent resistivity is plotted for a LPCMO

(x = 0.5) film with thickness d 500 A. The data illustrate that at temperatures


(La PrI48)5C I Mn
(La4/8, Pr4/8)5/8Ca3/8M nO3


E
010-1




". 10-

(/)
(I
Q^


-1,^-----


1.,.IEFMVtIi.I.E


-- 300 K
*- 200 K
175 K
- 150 K
125 K
- 100 K
,-75 K
*-5K
"4`m


0 10 20 30 40 50 60 70

Magnetic Field (kOe)

Figure 5-3. Magnetoresistance data for LPCMO (x = 0.5) and d& 500 A for various
temperatures. The magnetoresistance is at a maximum at temperatures in the
vicinity of the TMI where we also see a small degree of hysteresis in the
effect.


100


10-3









around TMI the LPCMO experiences a maximum in the change of resistance with applied

magnetic field. Also, we see a small degree of hysteresis in the magnetoresistance that is

much smaller, as a percentage effect, than the hysteresis in the temperature dependent

resistivity seen in Figure 5-2. In LPCMO, at all temperatures, and thus for any ground

state (COI, FMM, etc...), the magnetoresistance is always negativeq with no indication of

saturation at the fields shown here. To understand this phenomenon, we return to the PS

picture introduced in Section 2.2.3

Phase separation is inherently coupled to the competition between the structural,

electronic, and magnetic properties of the manganites, and thus is very sensitive to

external perturbations such as mechanical pressure46, magnetic3 and electric47 fields and

internal perturbations like structural pressure48 and chemical substitution2'5. This

sensitivity has made these materials promising candidates for applications as magnetic

sensors, strain gauges, etc. However, this same unique sensitivity to perturbations also

makes it difficult to fabricate manganite-based devices. In particular, the surfaces and

interfaces of manganites are notoriously difficult to control. In fact, it has been suggested

that the properties of manganites at the surface can be entirely different from the bulk7'49

The current general picture for the phase behavior is as follows. The dominant

ground states in these LPCMO manganite systems are a low temperature ferromagnetic

metallic (FMM) phase with a fully spin-polarized conduction band7'50 and a high

temperature orbital and charge ordered insulating phase characterized by the presence of

electron-phonon coupling due to Jahn-Teller distortions3'51. This high temperature phase

exhibits paramagnetic behavior above the charge-ordering temperature, Tco = 225 K, and

q A negative magnetoresistance implies a decrease in resistance with increasing magnetic field; a positive
magnetoresistance implies the opposite.










antiferromagnetic charge-ordered insulating (COI) behavior between Tco and TMI. Below

TMI, LPCMO is dominated by the FMM phase which persists to low temperatures, with

the specific onset of these various magnetic phases depending on factors such as the

charge/spin/orbital stability of specific doped states352,53 (Figure 5-4).


2-' I .I_
----O x= 3
SX O.
~= 0.2
". 5= U.25
.30 wl crrae mt c w
-r0375














resistivity data in Figure 5-2 and Figure 5-3 that has x 0.40.313 in this
Sx=0-412














notation. Coado et a ofMat. 15, 167-174(2003), Fig. 3, pg. 170.
H x 50
--A x= 0625

















0 IM M 300
T (K)
Figure 5-4. Field-cooled magnetization curves for Las/8-PrCa-, ,\11nO3 for various x.
The magnetization data at x = 0.30 will correlate most closely with the
resistivity data in Figure 5-2 and Figure 5-3 that has x = 0.313 in this
notation. Collado et al. Chem. ofMat. 15, 167-174 (2003), Fig. 3, pg. 170.

The main feature missed by the picture outlined above, which speaks in broad

terms about the majority phase, is that unlike the superfluid-fluid transition in liquid He4,

or the paramagnetic-ferromagnetic transition in an itinerant ferromagnet, like iron, the

various electronic and magnetic phases in LPCMO coexist and compete with each other









over a wide range of temperatures, magnetic fields, etc... around TMI. This metastable

phase separated behavior is the hallmark of first-order phase transitions. In the case of

LPCMO, the competing phases have very similar ground-state energies over a range of

temperatures that stretch from below Tco at high temperatures to well below TMI. In

particular, in the temperature range loosely defined above, there is a dynamic and

heterogeneous mixture of COI and FMM regions (Figure 2-11 and Figure 2-12).

In 1999, Uehara et al.2 showed that the MIT in LPCMO was governed by a

percolation mechanism. They observed that a critical concentration, xc = 0.41, existed for

the MIT in La5/8s-PrxCa-, M1nO3, with no MIT occurring for doping levels above xc. In

LPCMO with x < x, they plotted log(po), where pois the resistivity at T = 8 Kr as a

function of log(x,-x) at zero field. Percolation theory54 predicts that in a percolating

metal-insulator system that the resistivity should be of the form p (x,-x), with y & -1.9

in three dimensions. Uehara et al. found that the log-log plot mentioned above produced a

linear region over several orders of magnitude in resistivity for 0.275 < x < 0.40 with a

slope of y = -6.9. This slope is much larger in magnitude than the one predicted for

percolating systems. However, percolation theory predictions are for a nonmagnetic

metal-insulator system. In LPCMO, the electron conductivity is inextricably linked with

the ferromagnetic nature of the metallic regions as detailed in Ch. 2. This correlation

between conductivity and ferromagnetism implies that if there is a high degree of

misalignment between ferromagnetic regions, then the conductivity will be greatly

reduced. This misalignment will be at a maximum for concentrations near xc, and for zero

magnetic field because of an increasing presence of COI regions. In an applied magnetic

r A temperature, and thus resistivity, deep in the metallic phase for all samples with x < xc.









field of 4 kOe, a log-log plot produced a similar linear region with a slope of y = -2.6;

much closer to the value predicted by percolation theory and thus indicating percolation

as the dominate mechanism for PS. This highlights one of the unique aspects of LPCMO

and manganites in general. Classical percolation theory is inextricably linked with

continuous or second-order phase transitions, yet LPCMO is a phase separated material

with clear indications of first-order phase transitions as previously mentioned. A solution

to this apparent contradiction is suggested by Li, et al.55 in that TMI and Tc are separate

quantities in manganites. Specifically, the magnetic phase transition at Tc is second-order

in nature and the associated magnetic order enhances thermally activated conduction via

the double exchange mechanism. This in turn leads to a first-order metal-insulator phase

transition at TMI due to a non-classical percolation transition. This non-classical

percolation transition takes into account interactions (analogous to surface tension)

between the FMM and COI phases that classical percolation theory does not consider.

Although this result does not fully explain the percolation transition and other associated

phenomena observed at TM via conductivity and dc magnetization measurements5. The

experiments detailed in the remainder of this dissertation will focus on understanding the

competition between phases, and the percolation mechanisms that drive it.

5.3 The Colossal Magnetocapacitance Effect: Measurement and Analysis

Each LPCMO MIM capacitor is inserted into a probe configured to provide true

three terminal measurements as required by the Andeen-Hagerling 2700A capacitance

bridge (Section 3.3.3). All temperature and magnetic dependent measurements are

performed in a commercial cryostat, the Quantum Design QD6000, equipped with a

70 kOe magnet and housed in an electrically isolated, screen room or Faraday cage. All








the capacitance measurements, unless explicitly stated otherwise, are carried out at

0.5 kHz. We performed our measurements over a range of temperatures from 5 K to 300

K and at magnetic fields from -70 kOe to 70 kOe.

Shown in Figure 5-5 is a plot of the four-terminal resistance (black circles) of an

LPCMO x = 0.5 thin film with d= 600 A, as measured by the PAR 124A lock-in



107 1103
T

S106 T
0 102

S10 10






LPCMO thin film used as the base electrode of the capacitor. All








the direction the temperature was swept. Left scale: Three terminal
capacitance. Inset: Schematic of the MIM D: NdGa substrate, C:


10PC
103

0 100 200 300

Temperature (K)
Figure 5-5. Temperature dependence of the resistance (black circles) and capacitance
(red squares) of LPCMO. Right scale: Four terminal resistance of an
LPCMO thin film used as the base electrode of the capacitor. All
temperature sweeps were conducted at 2 K/min. The solid arrows indicate
the direction the temperature was swept. Left scale: Three terminal
capacitance of the LPCMO/A1203/Al capacitor measured atf= 0.5 kHz.
Notice the offset in the maximum in the resistance and the minimum in the
capacitance. Inset: Schematic of the MIM D: NdGaO3 substrate, C:
LPCMO d= 400 A, B: AlOx d= 100 A, A: Al d= 1000 A.









amplifier and Hewlett Packard 182 nanovoltmeter, and the capacitance (red squares) of a

MIM utilizing the same LPCMO film as its base electrodes. The LPCMO film shows the

characteristic electronic transport behavior outlined in Section 5.5.2 above, with

TMIh 95 K on cooling and TMI = 106 K on warming, and a resistance transition spanning

a range of approximately four orders of magnitude. The associated capacitance reaches

plateaus at high and low temperature where the LPCMO is in its PI and FMM states,

respectively. In between these plateaus the capacitance experiences a change in amplitude

of approximately three orders of magnitude, with minima occurring at 75 K and 93 K

upon cooling and heating, respectively. This is a shift in transition temperature of

approximately 20 K from the MIT in the associated resistance data. This is our first clue

that the dielectric behavior of our MIM deviates from phenomena observed in LPCMO

via electronic transport measurements, like the TMI determined by the longitudinal dc

resistivity.

The shift between the capacitance minimum and the resistance maximum can be

understood in terms of inherent strain present in LPCMO thin films. All thin films exhibit

some strain introduced by the mismatch in the lattice constants of the substrate and the

film. In LPCMO, this strain field has a direct effect on the phase behavior of the film.

When NGO is used as the substrate for LPCMO (as in our films) the lattice mismatch

produces a tensile strain field concentrated at the substrate-thin film interface. NGO and

the FMM phase in LPCMO have a cubic crystal structure and thus the strained region

stabilizes the FMM phase relative to the COI phase56'57. This implies that for two

LPCMO thin films, the film with higher strain will have more FMM phase present at a


" For complete details on sample fabrication see Section 4.3.2.






64


given temperature, magnetic field, etc. than the film with low strain. The strain-stabilized

FMM phase will constitute a low resistance state at the substrate-film interface. In Figure

5-6A, the LPCMO capacitor (depicted as an infinite parallel RC circuit) illustrates the

potential drop of the capacitance measurement is sensitive to the resistance perpendicular

to the surface of the film, R1. A standard 4-terminal resistance measurement is dominated

by the parallel resistance, R because the strain-stabilized FMM region shorts out the

higher resistance state away from the NGO substrate. Our capacitance minimum is due to

the MIT associated with R1 that occurs when the FMM percolates throughout the

thickness of the LPCMO. This occurs at a lower temperature than the MIT seen in the dc

resistivity.


AlOx7


LPCMO
I

----------- -----



----- ------- ---- -- -------
A R RR LPCMO


NGO
I-----9------------



Figure 5-6. Thickness dependence of the transition temperature.

We now want to suggest a qualitative picture for the behavior of the capacitance

seen in Figure 5-5. As the temperature decreases from 300 K, the LPCMO is in the PI

phase. Concomitantly, the capacitance is at a plateau in this temperature range. This is

due to the PI phase being a "bad" insulator, in the sense that it can still effectively screen









the electric field and allow for a reasonable well-defined potential drop to occur across

the AlOx dielectric. As the temperature is lowered, the resistance increases and at -225 K

there is a first-order phase transition58 into the COI phase with FMM phase present as

small volume fraction. This increase in resistance reflects that as the temperature

decreases, the COI phase becomes increasingly insulating at a faster rate than the FMM

regions grow in size. Therefore, the associated capacitance decreases due to a decreasing

ability of the LPCMO to screen with mobile electric charge. When the electrode

resistance is at a maximum (low capacitance) the electrode comprises filamentary

conductors of the FMM phase threading the primarily insulating COI. This resistance

region corresponds to just before the 3D percolation threshold5. As the resistance

decreases and the FMM regions grow in size, the capacitance slowly increases until the

FMM part reaches a critical fraction and then the capacitance increases dramatically.

Then the FMM domains are the dominant crystal phase and the capacitance again reaches

a plateau, due to the electrode becoming a "good" metal. One will note that the low

temperature capacitance plateau is -12 % smaller in value than the high temperature

plateau. This decrease is due to the temperature dependence of the AlOx dielectric, which

shows an approximately equal linear decrease in temperature35

At this point it is necessary to revisit a topic introduced in Ch.4 concerning the

perils of a two terminal measurement, and the effect of the Maxwell-Wagner relaxation in

our LPCMO MIM capacitors. In Chapter 3, we outlined the Maxwell-Wagner relaxation

as a contact, or interface, effect where charge polarization occurs across the interface of

the contact and the electrode. The Maxwell-Wagner model predicts the following:


C() ff = C (5-1)
e 1+ ic RcCo/(1 + ioRcc)








where Co is the bulk AlOx capacitance, R& is the contact resistance and Cc is the contact

capacitance. This equation predicts that in the limit of high frequency, the smaller

capacitance will dominate.

In the configuration of our MIM capacitor one would anticipate Maxwell-Wagner

effects could exist between the contact, either pressed indium or silver paint, and the

LPCMO. These effects would produce an extrinsic, enhanced low frequency dielectric

response that relaxed to the intrinsic dielectric properties of the insulator31'59 (Figure 5-7)

and in fact many claims of "colossal" dielectric effects have turned out to be

10-
io4 ,g g B







10 I 1 I I I
10 10o 104 10 10 101a

v (Hz)
Figure 5-7. Frequency-dependent dielectric response of inset the model circuit. At low
frequency the response is dominated by the contact capacitance, Cc. At
frequencies above the characteristic relaxation, the dielectric properties of
the sample capacitance, Co, dominate the response. Lunkenheimer et al.
Phys. Rev. B 66, 052105 (2002), Fig. 1, pg. 2.

Maxwell-Wagner phenomena31,40,60. These extrinsic low frequency effects would mask

any of the intrinsic properties of the insulator, rendering the measurement uninformative

of the bulk material at best. To insure that Maxwell-Wagner effects do not dominate the

measurement several checks can be made: 1) the contact material should be varied to

check for consistency of the observed phenomena; 2) the size of the contact should be









varied to see if there is scaling with area; 3) frequency should be swept to ascertain

whether there is a characteristic relaxation present as in Figure 5-7. Items 1) and 2) are

employed because the Maxwell-Wagner effect is a contact effect, like a Schottky barrier

in metal-semiconductor (MS) structures. MS contacts can generally behave in an Ohmic,

or linear, fashion where there is no built-in electric potential across the MS interface or

they form a Schottky barrier.

These barriers can occur if the electron work function of the metal is greater than

the electron affinity of the semiconductor. This mismatch in energies results in an energy

barrier equal in magnitude to the difference between the work function and the affinity.

This barrier region has a lower electron concentration relative to the metal or

semiconductor and thus becomes a depletion region. The depletion region acts as a

capacitance in series with the bulk semiconducting material as seen in the inset to Figure

5-7. Changing the contact material correspondingly changes the electron work function

that the Schottky barrier depends on and therefore leads to very different dielectric

behavior. Varying the size of the contact likewise changes the area of the depletion

region and would result in a corresponding change in the contribution of its dielectric

behavior.

In Figure 5-8A, we plot the temperature dependent capacitance for the same

LPCMO MIM capacitor with indium and then silver paint contacts. These results show

that the capacitance measured in our experiments is insensitive to the contact material.

This implies that a barrier, or contact capacitance response due to Schottky barrier or

Maxwell-Wagner effects is absent, or at least they are not the dominant mechanism that

governs the dielectric response of our system. Figure 5-8B and Figure 5-8C shows a







68



103
A

102

C measured
101 Maxwell-Wagner model
0 103
0 Indium contact
100 .* Silver paint contact .
Q 102 10
.u H = 15 kOe
C' measured I
B Maxwell-Wagner model 0
103 -
101
102
LI- H=OkOe 0 100 200 300

u lo 10 Temperature (K)


100

0 100 200 300
Temperature (K)
Figure 5-8. Maxwell-Wagner relaxation. A) temperature dependent capacitance, on
cooling, for the same LPCMO MIM capacitor with indium (black squares)
and silver paint (red circles) contacts. The behavior is almost identical for the
two contact materials. This implies that Schottky barrier-like effects are not
present in our measurement. B) comparison of the measured real part of the
complex capacitance (black) and that predicted by the Maxwell-Wagner
model (pink), see Eq. 5-1. The model fails to capture the transition
temperature in the measured capacitance. C) the model fails to capture the
magnetic field dependence as well.

comparison of the measured capacitance and the relaxation model prediction shown in

Eq. 5-1 in zero and nonzero field, respectively. The shift in the capacitance minimums

between the measured capacitance (black squares) and the Maxwell-Wagner model (pink

line) is due to the use of the longitudinal resistivity, governed by R\ in Figure 5-6A, in

Eq. 5-1. The capacitance is sensitive to the MIT associated with R1 in Figure 5-6A which









occurs at lower temperatures (higher FMM concentrations) than the MIT for the

longitudinal resistivity.

As we mentioned, the small capacitance dominates at high frequency and

therefore the response can be parameterized by the ratio of the two capacitances, c =

Cc/Co. Additionally, the value of R = Rtwo-terminal, the two terminal resistance of the

LPCMO film and Co=C(300 K), were measured experimentally. In our modeling, we

found that c = 10-4 produced results that most closely matched our data in zero field. The


H L to plane

--- 0 kOe
-*- 10 kOe
20 kOe
V50 kOe
103 1.2
C
H=O kOe Loss
102- 08
10 -0.8

Q-"
l10o.


0 0.0
40 50 60 6 70 80 9
100 200 300
1 100 200 300


Temperature (K)

Figure 5-9. Magnetic field and temperature dependence of the LPCMO capacitor
measured atf= 0.5 kHz. Large magnetic field causes an increase in the
transition temperature, and a decrease in the size of the capacitance
transition around Cmin. Inset, Right scale: Loss as a function of temperature.
Sharp loss peaks indicate an inflection in the associated capacitance. Left
scale: Associated zero field capacitance behavior, clearly showing hysteresis
and the sharp transition.


102


101


100









model does capture some degree of the capacitance response but clearly illustrates that

the dielectric phenomena we observe is not due to a contact effect, nor is it solely

dominated by the transition in the longitudinal resistance.

In Figure 5-9, we plot the temperature dependent capacitance in applied magnetic

fields. Like the resistivity, the magnitude of the transition decreases by orders of

magnitude as the magnetic field increases. As well, the size of the hysteresis decreases

and the transition temperature increases for both cooling and warming curves. The

capacitance shows almost no transition for H =50 kOe, with the temperature dependence

approaching that of a standard Al-AlOx-Al capacitor and at H = 70 kOe (not pictured) the

capacitance is essentially linear in temperature with no visible transition.

The inset in Figure 5-9 shows a magnification of the transition region. The real

part of the capacitance, C', and the ac losst are shown together to illustrate the sharp loss

peaks that indicate the inflection point in the temperature dependence of the capacitance

transition. The magnetic field dependence of the dielectric response illustrates that there

are two fundamental contributions present in our measurement: 1) at high and low

temperatures the capacitance is dominated by the AlOx; 2) in a broad intermediate range

of temperatures the capacitance is dominated by the LPCMO via some heretofore

unknown mechanism.

Unlike the work mentioned in Section 5.5.1 where the authors posited that the

SCEMs under study possessed colossal dielectric constants that exhibited large

temperature and/or magnetic field dependence, we believe the dominant LPCMO

dielectric response is reflecting the competing FMM and COI percolating phases. The


t LOSS oC"









minimum in the capacitance occurs below the T]i, as determined by resistivity, but we

know from the work of Uehara et al. 2, Zhang et al.27 and others52,61-63 that the phase

separated region extends to well below this temperature. We can extend the phase

competition explanation by using an effective area argument". In Figure 5-5, we see, on

cooling, that the capacitance starts to decrease dramatically at a temperature

approximately equal to Tco. The capacitance continues to decrease by orders of

magnitude as the underlying phase behavior of the LPCMO is mostly COI with small,

filamentary FMM regions randomly distributed throughout resulting in a large, and

increasing resistivity. As the resistivity passes through TMI, the LPCMO reaches the 3D

percolation threshold5'62, which corresponds to a metallic fraction of 17 %, and the

resistivity drops by orders of magnitude because a metallic conducting pathway exists

between any two points in the LPCMO. Concomitantly, the capacitance is still decreasing

and continues to decrease until well past the previously mentioned TMI. At 20 K below

TMI a minimum in the capacitance is reached and then there is sudden, sharp transition to

a high capacitance state. In this temperature range, the LPCMO has a growing metallic

fraction and a decreasing resistivity. This increase in capacitance corresponds to the

LPCMO reaching a critical metallic fraction that screens out the polarization contribution

of the competing phases in the LPCMO electrode and reintroduces the AlOx as the

dominant dielectric contribution to the capacitance. Then, at low temperatures, the FMM

domains are the dominant crystal phase and the capacitance again reaches a plateau, due

to the LPCMO electrode becoming a "good" metal.



" Recall that C = A and thus the capacitance measures, and is governed by, the effective area of the
S del
electrode.








103




102

LL -- 150 K
125 K
0 10 1 K warming




100
l i l lH to plane:I




-60 -40 -20 0 20 40 60
H (kOe)
Figure 5-10. Colossal changes in the magnetic field dependent capacitance in the
LPCMO MIM capacitor.
In Figure 5-10, the magnetic field dependence of the capacitance isotherms at
temperatures above and below C,n, at H = 0 kOe are plotted. The magnetic field
sensitivity increases as the temperature approaches the sharp capacitance transition and
thus the capacitance saturates at lower magnetic fields than it does at higher temperatures.
Also, with increasing proximity to the capacitance transition there is the development of
hysteresis (see T = 70 K) indicating the increasing presence of the FMM phase in the
LPCMO. This hysteresis continues to grow as a percentage of the total
magnetocapacitance at low temperatures (Figure 5-10).









' I I 'I I I I I '


710



LL708 -

T=IOK

706



704

-60 -40 -20 0 20 40 60

H (kOe)

Figure 5-11. The field dependent capacitance at T = 10 K for the LPCMO MIM. The
arrows indicate the direction the field is swept. The large hysteresis is due to
the large FMM fraction that occurs at low temperatures in the LPCMO. The
fact that there is any magnetocapacitance reflects that the COI phase is still
present but is a small fraction of the total film.
Up to this point we have largely ignored the behavior of the imaginary part of the

dielectric response. The ac loss provides valuable information about the various

dissipation mechanisms present in the dielectric response. In Figure 5-12 we plot the

temperature dependent ac loss in applied magnetic fields, as measured by the AH 2700A.

Unlike the real part of the complex capacitance, the ac loss is almost two orders of

magnitude larger at high temperatures than at low temperatures. This confirms that the

high temperature PI phase of the LPCMO is a "bad", somewhat lossy, insulator and that








the low temperature phase is a relatively "good" metal which shows little dissipation in

terms of the measured response. We also note that from Figure 5-12 that at H = 50 kOe

there are no longer any sharp loss peaks and thus no capacitance transitions.










---0 kOe
c 0.1 20 kOe
0q .--- 50 kOe


H to plane


0.01
0 100 200 300

Temperature (K)

Figure 5-12. The temperature dependent ac loss in applied magnetic fields. The sharp
peaks indicate the capacitance transition as mentioned for the inset to Figure
5-9. Notice that unlike the real part of the complex capacitance shown in
Figures 5.6 and 5.9, the high temperature loss is orders of magnitude larger
than the low temperature value.
In the following chapter we will address the underlying dielectric response of the

LPCMO film that is hinted at by the data shown above. We will propose an equivalent

circuit that captures the relevant physics of our dielectric measurements. Additionally, we

will study the interplay of charge conservation and charge dissipation in these LPCMO






75


MIM capacitors over a wide range of frequencies, temperatures and magnetic fields and

see striking and universal scaling behavior.














CHAPTER 6
SCALE INVARIANT DIELECTRIC RESPONSE IN PHASE SEPARATED
MANGANITES

6.1 Modeling the Competition of Phases in (Lai-xPrx)5/8Ca3/8MnO3 (LPCMO)

In the previous chapter we outlined the temperature and magnetic field dependent

capacitance associated with the colossal magnetocapacitance (CMC effect observed in

LPCMO MIM capacitors. Moreover, we proposed that the underlying mechanism that

drove this effect was the competition between the ferromagnetic metallic (FMM) and


C C*AlOx()


series


Figure 6-1.


General form of the equivalent circuit for our LPCMO MIM capacitor. A)
Rseries is the two terminal LPCMO dc resistance between the contact and the
boundary defined by the AlOx insulator, (see Figure 4-1). In Section 4.3.4,
we showed (Fig. 4-8) that the Rseries contribution to the dielectric response
could be ignored due to Eq. 4-14. The measured complex capacitance, C*(o)
= C'(o) -iC"(o), is composed of two separate dielectric components
representing the contribution of the LPCMO capacitance, CM and RM and the
AlOx capacitance, CAlOx and RAlOx. B) the LPCMO dielectric response is
modeled as an infinite parallel RC circuit where all capacitors are complex
with value C*M(co) and each resistor represents the dc resistance, RM. This is
a macroscopic model which captures the response of an inhomogeneous
conductor to an ac electric potential. C) the capacitance due to AlOx is a
simple parallel RC circuit with RAlOx effectively infinite (Section 4.3.4).


charge-ordered insulating (COI) phases. To support and expand on this idea, we model

the effective dielectric response in terms of an equivalent circuit, seen in Figure 6-1.









Figure 6-1 shows the most general representation possible for our LPCMO MIM

capacitor. However, it can be simplified by incorporating some of the assertions made in

previous chapters with new data shown in the present chapter. In Section 4.3.4, we

outlined how a highly resistive electrode could contribute to the total dielectric response

and established a criterion, Rseres << min {/ioC '(o), 1/coC "(p)}, under which it could

be excluded. In Figure 4-8, we plot temperature dependent impedance data revealing that

this criterion is met by our LPCMO MIM capacitor and therefore Rseries in Figure 6-1A is

also excluded from the dielectric response. In Figure 6-1C we note that the dc resistance

of the AlOx, RAlox, has been measured to be effectively infinite in experiments detailed in

Section 4.3.4. This infinite resistance implies that there is no dc leakage current through

the AlOx and therefore RAlox can be ignored because it is not incorporated into the

measured dielectric behavior. In addition, the frequency dispersion due to the AlOx is

negligible. Finally, below we will demonstrate how the infinite parallel resistor-capacitor

network in Figure 6-1B further excludes the potential for voltage drops along the

LPCMO film.

In Chapter 3, we showed that an inhomogeneous solid in an ac electric field can

be accurately modeled by an infinite parallel resistor-capacitor network. All capacitors

were equal in value and proportional to the bound charge dielectric constant, and the

resistors were proportional to the free charge current at a specific position in the solid.

We now extend this picture to model the LPCMO base electrode of our MIM capacitor

(Figure 6-1B). LPCMO can certainly be thought of as an inhomogeneouss solid"


v This is not to say that RAlOX does not ti,.... t the dielectric response at all. If the resistance of the A1Ox was
comparable to that of the LPCMO, then we would not be able to confidently extract the dielectric behavior
of the LPCMO MIM.









(Chapter 3). The phase separation between the FMM and COI phases is analogous to the

conducting, free charge and insulating, bound charge model, respectively.

An important condition imposed by the geometry of the capacitance measurement

and the geometry of our MIM is that the LPCMO must not have electric potential drops

along its width; that is, there must always be a well-defined equipotential parallel to the

planar interface of the LPCMO and AlOx. If this were not the case then, even at ac, the

voltage drops could produce a current and an associated resistance in the LPCMO film

that would be indistinguishable from the measured dielectric response, preventing the

capacitance due to the AlOx from revealing the underlying dielectric phenomena in the

LPCMO. The necessary conditions for an equipotential surface are met, in part, by

having an infinite AlOx dc resistance, which we reported in Section 4.3.4, and also by the

infinite parallel resistor-capacitor network.

In the high-frequency limit, o) m >> 1, where im =RMCM, the capacitance

contribution to the admittance of the infinite circuit in Figure 6-1B dominates and

produces a spatially uniform electric field32 and thus the required equipotential surface.

This high-frequency limit must be balanced by the condition that if C)T m is too large then

the impedance constraints (Eq. 4-14) on our measurement will not be met. Therefore we

must restrict ourselves to frequencies that ensure we are in the high-frequency limit, but

not outside of our impedance constraints.

In Figure 6-2, we plot the frequency dependence of C'and C"in zero magnetic

field at various temperatures, on cooling. The figure shows that for temperatures within

the colossal magnetocapacitance (CMC) region defined in the previous chapter to be

between 55 and 175 K for cooling in zero magnetic field, the dielectric response is in










I ,
8:nfI


H = 0 kOe


102


103


8 -oo
13



-v-
V VV




104


103



102


f (Hz)
The frequency dependence of the real, C' and imaginary, C"' parts of the
LPCMO MIM capacitor at temperatures, cooled in zero magnetic field.


the high frequency limit because the peak in C where Co)T = 1, occurs at frequencies

well below the range of our measurements. This ensures that we do have an equipotential


-W X.


LL


C


* 250 K
* 200 K
150 K
v 125 K
115K
S100 K
80K
S75 K
-- 50 K








o 250 K
o 200 K
150 K
v 125K
115K
<- 100 K
- 80K
o 75 K
+ 50 K


O


100



10-1
102


101


1 100
LL-

10-1
o


10-3


Figure 6-2.


D-


>-I>1>
I+'+


07
fl


H = 0 kOe


m m,,, I .ini, ..,


' '''1


''I
a-g;









surface in the LPCMO base electrode. In Figure 6-3, we present the equivalent circuit

that takes into consideration the limits imposed by the preceding analysis.
i i i
0-r --- -------- -)



...... -....O x
C*((O)



R,
Figure 6-3. The simplified equivalent circuit for our LPCMO MIM capacitor. The
LPCMO is still modeled as an infinite parallel resistor-capacitor network,
with dissipation due to RM, the dc resistance, and the ac loss originating
from the imaginary part of C*M(oa), C"M'(o).

We now have a framework within which we can present our analysis of the

dielectric behavior of our LPCMO MIM capacitor. Specifically, we should note that this

equivalent circuit is analogous to the Maxwell-Wagner relaxation model. In that model,

the dielectric response is comprised of two complex capacitors in series that represent a

contact capacitance, Cc, in parallel with a contact resistance, Rc, and a bulk capacitance,

Co, respectively. Associated with those two capacitors are two time constants, r = RcCc

and o = RcCo, with o << rc. These time constants divide the total dielectric response

into two separate regions such that for o) < 1/r, the low-frequency limit, the larger

contact capacitance is dominant and for 1/rt < c < 1/To the smaller bulk capacitance

dominates and is largely frequency independent. For c1 > I/zo, the dielectric response

approaches the "high frequency permittivity", or bound charge dielectric constant, sE. In

our MIM capacitor, the dielectric response is also modeled by two complex capacitors in

series, with the exception that one (the LPCMO) is an infinite network of equivalent









complex capacitors shunted by resistance RM. Nevertheless, we will see that the complex

capacitance of our LPCMO MIM is likewise divided into regions due to two separate

time constants, rm = RMCM and to = RMCAlOx, with the major exception that in the

LPCMO MIM, TM << To. This implies that in the high-frequency limit, the dielectric

response will be governed by the LPCMO since the "contact" capacitance which is the

LPCMO capacitance is small compared to the AlOx.

6.2 The Dielectric Response Due to Phase Separation in LPCMO: Universal
Power-Law Phenomena

Figure 6-2 shows that the dispersion in the complex capacitance of our MIM

reflects strong temperature dependence. Specifically, at temperatures above and below

the CMC region, there is very little frequency dispersion in C' and C" shows the onset of

a small peak. The presence of a loss peak indicates that the dielectric response is in the

vicinity of a crossover from a low to high-frequency limit. Analogously, this implies that

for these high and low temperature regions that the complex capacitance is dominated by

the AlOx. This supports our supposition that the complex capacitance is reflecting the

phase competition occurring within the LPCMO, in that at high and low temperatures the

phase of the manganite is in a relatively homogeneous and conducting state that screens

out any charge polarization or dipole formation in the LPCMO.

To further explore these dielectric phenomena, we now look at our complex

capacitance data in terms of Cole-Cole plots64. In a Cole-Cole plot, C"is presented as a

function of C'while some external parameter, like frequency, temperature, etc., is varied.

This is a popular way of presenting dielectric data because it is compact and allows the

physical properties the system, such as the characteristic frequencies and time constants,














increasing
\\\ /

H = 0kOe /
T = 50 K K


610 620 630
C' (pF)


ip




Increasing

H = 0 kOe
T = 100 K

. i . .


300


200
LL
U-

0
100


100
C' (pF)


1000


C' (pF)
Cole-Cole plots in zero magnetic field on warming for 50, 100 and 200 K.
A) at 50 K the LPCMO is mostly FMM and the plot shows that as the
frequency increases the dielectric response passes through a relaxation
region, located around the cusp specified by the dotted arrow, indicating a
crossover in the dominant capacitance elements. B) at 100 K the LPCMO is
deep in the phase separated region and the Cole-Cole plot shows very
different behavior than the data at 50 or 200 K. As frequency increases, the
response decreases in an apparent power law fashion, eventually rolling off
as C'approaches C, E,. C) at 200 K, the dielectric behavior shows a
relaxation region like the 50 K data, with a cusp signifying the AlOx
capacitance. Also note that the high frequency portion of the curve (the
circled region) is decreasing with increasing frequency in a manner very
similar to the 100 K data.


to be shown in terms of curve shapes and points of discontinuity. In Figure 6-4, C'and C"

data from Figure 6-2, are converted to Cole-Cole plots at three different temperatures in


100


100

B


Figure 6-4.


// 0 increasing


S* H = 0 kOe
-T = 200 K

* '. | .. ,. *'"""
..0. .





83

zero field. These three data sets come from temperatures that are above (200 K), below
(50K), and inside (100 K) the CMC region defined in the temperature dependent data
taken at zero magnetic field and at a frequency of 0.5 kHz (Figure 6-4). The behavior
observed at 100 K, in Figure 6-4B, over the entire measured frequency range hints at
unique intrinsic LPCMO dielectric response due to phase separation.


V
100 T= 100 K
warming


10 -
H o increasing

1 1 kOe
1 -- 5 kOe

1 10 kOe
v-- 20 kOe
0 1 I I .
0.1 1 10 100
C' (pF)
Figure 6-5. Cole-Cole plot showing the effect of magnetic field on the dielectric
behavior at T = 100 K. The raw data reveal that increasing magnetic field
causes a shift in the dielectric response along a master curve, such that
increasing magnetic field behaves like decreasing frequency and vice versa.
Now that the dielectric response of the CMC region has been singled out, we will
explore the nature of competition of phases in this region. Specifically, we will look at
the magnetic field dependence of the complex capacitance. Figure 6-5 shows a Cole-Cole







plot of the magnetic field dependence of the LPCMO MIM at T = 100 K, with the
response measured at o/27i = {0.1, 0.2, 0.5, 1, 2, 5, 10, 12, 16, 20 kHz}. The dielectric
behavior shown in Figure 6-5 illustrates that in the LPCMO base electrode there is a
unique interplay occurring between magnetic field and frequency that produces a master
curve along which all the data fall. For comparison, Figure 6-6 shows a



ST= 65 K -
warming /

100-
w increasing l
L.
--- 1 kOe
S -- 5 kOe
10 kOe
-- 20 kOe

1 0 -. . .
10 100 1000
C' (pF)
Figure 6-6. Cole-Cole plot showing the effect of magnetic field on the dielectric
behavior at T = 65 K, with same frequency steps as the T = 100 K data. The
data collapses onto a single curve as it did at T = 100 K but at high magnetic
field, or low frequency, a crossover from the LPCMO to AlOx dielectric
response is clearly visible.
magnetic field dependent Cole-Cole plot at T = 65 K, which is outside the CMC region
defined above. We again see that the data collapse onto a single master curve with a
similar relationship between frequency and magnetic field. However, at low frequency, or









high magnetic field (see H = 20 kOe (blue down triangles)), there is the signature of a

dielectric crossover from the high frequency, LPCMO to the low frequency, AlOx

dielectric response. This further supports the claim that the complex capacitance depends

on the competition of phases in LPCMO because an increasing magnetic field always

causes an increase in the FMM fraction of the film. An interesting consequence of this

inverse relationship between frequency and magnetic field is that the high frequency

complex capacitance at 65 K closely matches the response seen at T =100 K. At 0.5 kHz

and T = 65 K, the LPCMO base electrode is well outside of the CMC region but, as the

frequency increases above 10 kHz, the CMC region seems to expand in temperature.

To pursue this further we need to observe if the dielectric behavior at T = 100 K (Figure

6-5) is indeed a signature of the entire CMC region and in turn the underlying phase

separation.

Figure 6-7 shows a Cole-Cole plot at H = 10 kOe over a range of temperatures

that lay within the CMC region at 0.5 kHz. The figure illustrates that at each temperature,

the dielectric behavior collapses onto the same master curve that described the magnetic

field dependent data at T = 100 K. In addition, at 0o/27: = 0.5 kHz, the measurement

frequency used for the data in the previous chapter, the dielectric behavior falls on the

same line. This collapse implies that the behavior seen in Figure 6-7 and Figure 6-5 are

signatures of the CMC region and correspondingly the competition between the FMM

and COI phase separated areas. To explore the physical significance of the behavior

revealed in the Cole-Cole plots of Figures 6.5-6.7 we will determine the functional form

of the master curve that governs the dielectric response in the CMC region. Also we will







incorporate the unique frequency, temperature and magnetic field dependence revealed
by the data.


100


LL
000 10
10


UI


I ' '1


' ' "11111


H = 10 kOe


Increasing


90 K
-100 K
125 K
150 K
I I I .III


Y

/ ,,,,,,,


, , I


V-


0.1 1 10 100

C'(pF)
Figure 6-7. Cole-Cole plot showing the effect of temperature at H = 10 kOe on the
dielectric behavior. The raw data reveals that increasing temperature causes
a shift in the dielectric response along a master curve, such that increasing
temperature behaves like decreasing frequency and vice versa.
In each of the Cole-Cole plots that reveal a master curve, the behavior appears to
have power-law dependence for sufficiently high frequencies or low magnetic fields and
temperatures. In addition, it appears as if this power-law is ubiquitous and regardless of
which parameter, frequency, magnetic field or temperature, is varied the dielectric
response is governed by it. Figure 6-8 presents a Cole-Cole plot that includes the


' ' '''1


I I
P
-t7/








dielectric behavior with a pair-wise variation of all parameters. In this plot, we witness
the collective character of the dielectric response to these external perturbations.


'I 1 ' 1 ' '1111 '

T, H increasing





So increasing

S--m-f (0.1 ~20 kHz) 10
-*- f (0.1 20 kHz) 90
T(90 148 K) 0.5
: o T(89 140 K) 0.5
SA H(0 17 kOe) 0.5
v H(0 23 kOe) 0.5
.l ... .....I. .. .. .. .... ,


^ooO


0 K, 1 kOe
K, 10 kOe
kHz, 1 kOe
kHz, 5 kOe
kHz, 100 K
kHz, 125 K
. ...i


0.1 1 10 100

C'(pF)
Figure 6-8. Universal dielectric behavior for the complex capacitance of the LPCMO
MIM. The legend indicates the parameter being swept and its range, and the
two parameters being held constant, respectively. The temperature and
magnetic field sweeps are taken from the data presented in Figures 5.9 and
5.10, respectively.
In the region below (C'" C) = (5, 0.5) in Figure 6-8, there is a pronounced roll-off

in the dielectric response and as co -> o, the dissipation due to the ac loss vanishes. This

is because at high, or infinite, frequency the permittivity, and the corresponding

capacitance, is dominated by the free space contribution, so, and other instantaneous

polarization mechanisms29. This limit produces the infinite frequency capacitance


100


LL
U-
Ii


10



1



0.1








contribution, Cm, that is an intrinsic dielectric property of a given MIM system, just as c.

is the bare dielectric response in a material. To see the underlying relaxation behavior we

will look at C"as a function of C'- C.

'' I" 'I "1 "1 I 1 "1 I 1 I'1


100
T7 H increasing



LL 10- y=0



0 1 o increasing




0 .1 ....., ..... ...... ...... ......
0.01 0.1 1 10 100

[C' C ](pF)

Figure 6-9. Cole-Cole plot showing the complete dielectric relaxation of C' This plot
shares the same legend as Figure 6-7. C, is determined by extrapolating the
high frequency roll-off to C"= 0, at which C' Co. The data are then fit to
the power-law expression in Eq. 6-1, and the fit is plotted as the orange line.
The exponent for the response is 7= 0.7.
Figure 6-9 shows the power-law response of C "as a function of C" C, over

four orders of magnitude. The data are fit to a general power-law expression of the form:


C"(o,,T,H)= A[C'(o,T,H)- CJ, '


(6-1)