UFDC Home  myUFDC Home  Help 
DARK ITEM  



Full Text  
COLOSSAL MAGNETOCAPACITANCE AND SCALEINVARIANT DIELECTRIC RESPONSE IN MIXEDPHASE 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: JahnTeller Polarons .........................16 2.3.3 Beyond Double Exchange, Part 2: Percolating Phase Separation .............21 3 DIELECTRIC RELAXATION PHENOMENA................................... ..................26 3.1 TimeDependent Dielectric Response: Polarization v. Conduction...................26 3.2 FrequencyDependent 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 CaseStudy: Disordered Conductors................................35 4 CAPACITANCE MEASUREMENTS AND EXPERIMENTAL TECHNIQUES ...38 4.1 ThinFilm M etalInsulatorM etal Capacitors .............. ...................... ...........38 4.2 M etalInsulator 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 ockIn 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 PHASESEPARATED 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 (LalxPr,)5/sCa3/8MnO3 (LPCMO).........76 6.2 The Dielectric Response Due to Phase Separation in LPCMO: Universal P ow erL 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 21 Three magnetoresistance versus temperature curves for different La0.67Ca0.33M nOx sam ples. ............................................................................. 3 22 Schematic of the perovskite structure that encompasses the MnO6 octahedron that serves as the basis for all manganites .... ........... ........ .........................5 23 Crystalfield splitting of the fivefold degenerate 3d orbitals in the Mn3+ ion. A similar diagram for Mn4+ would have unoccupied eg states..............................6.. 24 Schematic view of orbital and spin ordering on the Mn sites in LaMnO3 .............. 11 25 Phase diagram for LalxCaxMnO3 showing the effect of holedoping on the magnetic ordering of the manganite.............................. ...............12 26 Resistivity calculated from Eq. 28.................. ........................ .................. 14 27 Temperature dependent resistivity of LaxCaxMnO3 (x = 0.25) in various applied m magnetic fields.. .......................................... .. .. ......... ........ .... 15 28 The projection of the crystal structure of LaMnO3 along the b axis. The unequal MnO bond lengths illustrate the JahnTeller distortion. .......................................17 29 JahnTeller distortion and its effect on the energy level splitting of the Mn3 orbitals ......................................................................................... 18 210 Magnetic field dependence of resistivity calculated using the direct integration m ethod .................................................................................. .......... 21 211 Darkfield SEM images for Las/sPryCa, _\MnO3 obtained by using a superlattice peak caused by charge ordering (CO)....................................................................22 212 Scans (6 [tm x 6 [tm) of LPCMO. On cooling, FM regions (dark area) grow in siz e ... ........................................................................ 2 3 213 Random resistor network resistivity for various metallic fractions, p in 2D and 3 D (in set).............................................................................................. 2 4 214 M onte Carlo results for the RFIM m odel...................................... ..................25 31 B arriervolum e capacitor............................................................ ............... 32 32 The complex impedance diagram of the barriervolume circuit. The large arc is due to the barrier region and the small arc is due to the volume region. ................33 33 Random resistor capacitor network that describes the dielectric behavior of the universal ac conductivity m materials .........................................................................36 41 Schematic of MIM. A) the metallic counter electrode. B) the dielectric. C) the manganite base electrode. D) the substrate. ................................... ............... 39 42 Confidence intervals of the measured impedance as a function of the m easurem ent frequ ency ......................................... .............................................42 43 Schem atic of a lockin am plifier ...................................... ............... ..... .......... .43 44 The magnitude of the impedance of a 400 A LPCMO film as measured by the Solartron 1260A (black squares) and PAR 124A LockIn (red circles). ................45 45 Schematic of the capacitance bridge circuit ........................................................47 46 Schem atic of a com plex capacitor................................................ ........ ....... 49 47 Equivalent circuit of MIM capacitor with a resistive electrode, Rs, and complex capacitance, C = C1 iC2, where R2 = 1/coC2....................................................... 50 48 The temperature dependent impedance of the LPCMO film, that serves as the base electrode in our M IM ........................................................... .....................52 51 Temperature dependent resistivity of A) La0.67Ca0.33MnO3 (LCMO) and B) Pr0.7Ca0.3M nO 3 (PCM O). ............................................. ............................ 55 52 Temperature dependent resistivity for an LPCMO x = 0.5 thin film with thickness d & 500 A ......... ......... ............... ...................................... ..................... 56 53 Magnetoresistance data for LPCMO (x = 0.5) and d & 500 A for various tem peratures....................................................... ........... ........... 57 54 Fieldcooled magnetization curves for La5/8xPrxCa, ,MnO3 for various x...............59 55 Temperature dependence of the resistance (black circles) and capacitance (red squares) of LPCM O ..................... ............ ... .............................62 56 Thickness dependence of the transition temperature. ........................ ...........64 57 Frequencydependent dielectric response of inset the model circuit.. ...................66 58 M axw ellW agner relaxation......................................................... ............... 68 59 Magnetic field and temperature dependence of the LPCMO capacitor measured atf 0.5 kH z ............................ ................ .............. ......... 69 510 Colossal changes in the magnetic field dependent capacitance in the LPCMO M IM capacitor .............. ...... .......... ........................... .. .. ......... 72 511 The field dependent capacitance at T = 10 K for the LPCMO MIM. The arrows indicate the direction the field is swept. ........................................ ............... 73 512 The temperature dependent ac loss in applied magnetic fields............................74 61 General form of the equivalent circuit for our LPCMO MIM capacitor.................76 62 The frequency dependence of the real, C' and imaginary, C parts of the LPCMO MIM capacitor at temperatures, cooled in zero magnetic field ................79 63 The simplified equivalent circuit for our LPCMO MIM capacitor.....................80 64 ColeCole plots in zero magnetic field on warming for 50, 100 and 200 K...........82 65 ColeCole plot showing the effect of magnetic field on the dielectric behavior at T = 100 K ...........................................................................83 66 ColeCole 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 67 ColeCole plot showing the effect of temperature at H = 10 kOe on the dielectric behavior ............................................... .. ................. .......................86 68 Universal dielectric behavior for the complex capacitance of the LPCMO MIM.. .87 69 ColeCole plot showing the complete dielectric relaxation of C' ............ .........88 610 Temperature dependent capacitance in zero magnetic field.. ..................................92 611 ColeCole plot for warming temperature from 10 to 300 K at fixed frequency and various m agnetic fields ...................... .... ................................ ............... 95 612 TH phase diagram s........... ..................................... ....... .. .......... 96 613 The powerlaw scaling collapse (PLSC) region expands (contracts) along the PLSCNPLC 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 SCALEINVARIANT DIELECTRIC RESPONSE IN MIXEDPHASE MANGANITES By Ryan Patrick Rairigh May 2006 Chair: Arthur Hebard Major Department: Physics We measure capacitance on thinfilm metalinsulatormetal (MIM) capacitors using the mixedphase 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 counterelectrode. 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 magneticfieldinduced percolating changes in the relative proportions of coexisting ferromagnetic metal (FMM) and charge ordered insulating (COI) phases in the LPCMO. The widths of the temperaturedependent 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 ColeCole 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 scaleindependent 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 phaseseparated systems. CHAPTER 1 INTRODUCTION Thin film metalinsulatormetal (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 timevarying 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 metalinsulator 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 mixedvalence, phaseseparated manganite (LalxPrx)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 stronglycorrelated 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 chargeordered 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 fielddependence of the metalinsulator 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. 21). 1500 Lao.6Cao.aMnOx 1000 2 500 0 0 100 200 300 T(K) Figure 21. 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, 413415 (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 FeCr multilayer and other layered structures where a large sensitivity to magnetic fields is due to the spinvalve effect between spinpolarized metals. Shortly after the discovery of the CMR effect, other researchers discovered that these manganites possessed magneticfieldinduced insulatormetal and paramagneticferromagnetic phase transitions, as well as, associated latticestructural 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 LalxCaxMnO3 is the prototypical CMR material and belongs to a class of 3D cubic perovskitebased compounds commonly called mixed manganites. Depending on the holedoping 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. 22), forming MnO6 octahedra with cornershared oxygen atoms. The Mn and O ions are arranged in a MnO2 plane that is analogous to the CuO2 planes in the cuprate hightemperature superconductors. B La Sr2 OMnTM n4f Figure 22. 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 RuddlesdenPopper 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 AlxBxMn03, where A is a trivalent rareearth 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 mixedvalence chemistry and the octahedral atomic configuration become important as we discuss the unique interactions and ordering phenomena that occur in the orbital, charge, and spindependent properties of manganites. In mixed manganites, the local spins and conduction electrons are delectron in nature. Specifically, they are of the 3d orbital levels associated with the previously mentioned octahedralcoordinated Mn ion. Normally, the 3d orbital electrons in Mn are fivefold degenerate, but the octahedral coordination partially lifts this degeneracy because of cubic crystal fieldsplitting effects such as hybridization and electrostatic interactions between neighboring ions (Fig. 23). This field splitting breaks the rotational invariance of the orbitals and results in three degenerate, lowenergy, t2g states .,= = eg Mn3+ 3d orbitals ,' t2g Figure 23. Crystalfield splitting of the fivefold 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, highenergy, 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 electronelectron interactions among the t2g levels are such that it is energetically favorable for all electrons to have parallel spins. In turn, a strong intraatomic 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 mixedvalence 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 21 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. 21) where Si is the spin of the corelike t2g electrons, oz,z' is the HDE = H z (S',zz')CzC + I toc/cz (21) 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 intraatomic exchange coupling, with JH > 0 implying FM order. The physics of Eq. 21 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 antiparallel. 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. 21 for a simple two state problem where an electron is confined to states i andj (Eq. 22). In the classical limit where JHS. > oo, the Hi = JH (Si Czzz'C +Sj c ,z t +c jziz (22) 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 (23) If Si is parallel to Sj then 0, =0, and tj = t. If Si is antiparallel 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] (24) Equation 24 illustrates that the eg band receives an kinetic energy gainan increase in the ability of electrons to move from site to site in a crystal latticefrom 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 spinpolarized 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 halfmetal 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 nondegenerate t2g orbitals via their shared 02 ion, with disorder influenced by thermal fluctuations in the crystal lattice and chargeexchange (CE) AF due to chargeordering, 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. 24 shows that in LalxCaMnO3 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 24).When the hole doping is increased, the manganite moves to a mixed contribution of superexchange and double C Figure 24. 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 superexchange interaction in the abplane with a weak double exchange interaction occurring along the caxis. However, more recent results2'1416 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 chargeordered (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 reenters 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, xrays 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 25. Phase diagram for LalxCaxMnO3 showing the effect of holedoping 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 antiferromagnetic states present. Hwang et al. Phys. Rev. Lett. 75, 914917 (1995), Fig. 4, pg. 917. Figure 25 also illustrates that, regardless of the holedoping 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 noncollinear 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. 25) where ao is the lattice constant and M (o, T) is a "memory function"b p(T)= aoM (o) = O,T)/7re2D (25) Further evaluation of Eq. 25 to leading order in 1/S and kCao, where kF is the Fermi wave vector, reveals the double exchange resistivity (Eq. 26), where R labels the various sites p(T)= e2 [(SOS)S( IS (R)S(R+s, ) /S4]B (R) (28) 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 (26) 0 Where H is from Eq. 21 andj is a current operator: ijabi (bCajb CjbCia 1+ (27) on the crystal lattice, in the coordinate system that connect a site to its nearest neighbor and B(R) is the electron currentcurrent correlation functions. The results of Eq. 28 are shown in Figure 26. 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 26. Resistivity calculated from Eq. 28. In this model, the resistivity increases below To and with increasing magnetic field, in direct contradiction with experiment (see Figure 26). Millis et al. Phys. Rev. Lett. 74, 51445147 (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 27 shows the typical transport In the free electron approximation B(R) is expressed as: B(R) 9 sin2kF R+ ) sin2kF(R i) 2sin2(kFR) (29) 32(kB(R) Fa=)4 (kF R 1 ')2 (F R (29) 32(kF 4 +S,2 2k, k8F {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 oneorbital Sf M=H 0 . 100 1T S60 L 40 20 0 50 100 150 200 250 300 350 Temperature (K) Figure 27. Temperature dependent resistivity of LalxCaxMnO3 (x = 0.25) in various applied magnetic fields. Schiffer et al. Phys. Rev. Lett. 75, 33363339 (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 ferromagneticparamagnetic phase transition, but it does not account for the experimentally observed metalinsulator 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 holedoping, reflected in the complex structure of these materials at low temperatures as illustrated in Figure 25. 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: JahnTeller 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 electronlattice 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 MnO 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 MnOMn bond angle22, via compression and tension of the bonds. The other mechanism involves electronphonon coupling that links lattice deviations in the crystallographic structure with deviations in the electronic configuration away from their average values. In LaMnO3, this electronphonon coupling is an inherent electronic instability native to the MnO6 octahedra, the JahnTeller distortion23 01 1.92 La Mn 2.15 A 02 Figure 28. The projection of the crystal structure of LaMnO3 along the b axis. The unequal MnO bond lengths illustrate the JahnTeller distortion. In the manganites, the MnO6 octahedra experience a distortion in the MnO bond lengths along different crystallographic directions. This distortion arises from an electronic instability, the JahnTeller (JT) instability, intrinsic to a Mn3+ ion situated in an octahedral crystal field (see Figure 28). 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 electronlattice interactions, which are an electronphonon coupling mechanism, can be found in most orbitally degenerate delectron 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 29. JahnTeller distortion and its effect on the energy level splitting of the Mn3 orbitals. The distortion stretches the MnO 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 23 and lowers the electronic energy (see Figure 29). 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 AlxBxMnO3 (0 < x < 0.2), the JT distortion is a static effect that produces a cubictetragonal structural phase transition. This is a static or frozen effect because the bond length distortions are coherent and have longrange order throughout the crystal. The JT effect is very strong here with a bond length change of 10 % of the mean MnO distance24. As x is increased from zero doping, the structural transition temperature and the size of the longrange, 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 electronphonon 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 longrange order in favor of fluctuations from sitetosite, to the double exchange mechanism as follows: Hf = HDE + + g~ Qb (j)cjbz + Q2(j) (210) Sc jaz 2 Here we have used HDE from Eq. 21 and added a term which has external magnetic field h coupled to the core spins Sc. The lattice distortion terms have electronphonon 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 0h;=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, 175178 (1996), Fig. 2, pg. 177. calculated resistivity under this picture are shown in Figure 210. It is clear that the theoretical result correctly captures, qualitatively and somewhat quantitatively', the experimentally observed results seen in Figure 27. 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/syPryCa, MnO3 (LPCMO) (see Figure 211). 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 27, 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 211. Darkfield SEM images for La5/yPryCa3/8MnO3 obtained by using a superlattice peak caused by charge ordering (CO). a) shows the coexistence of chargeordered (insulating), the light area, and chargedisordered (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 darkfield 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 chargedisordered domains at T > T,. Uehara et al. Nature 399, 560563 (1999), Fig. 3, pg. 562. Zhang et al.27 showed direct observation of percolation in LPCMO using low temperature magnetic force microscopy (MFM) (Figure 212) 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 Moreo1416,20,28 showing the various mechanisms behind the intrinsic inhomgeneities 0 Hz 2 Hz Figure 212. 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, 805807 (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 213). 10 , pa, 1cm) (C sod , 10 o0 102 04 1 10 T .I 10 .0o 2D  0 100 200 300 T (K) Figure 213. 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, 135138 (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 214) reminiscent of the SEM images in Figure 211 and the MFM images in Figure 212. 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 214. 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, 135138 (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 JahnTeller polarons describe the FM metallic and insulating regions, respectively. In summary, CMR effects are best understood in terms of competing phase separation between double exchangegoverned, ferromagnetic regions and insulating regions formed by JahnTeller polarons. CHAPTER 3 DIELECTRIC RELAXATION PHENOMENA 3.1 TimeDependent Dielectric Response: Polarization v. Conduction Dielectric relaxation probes the interaction between a medium, with some polarization, P, and a timedependent 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 timescale, via the characteristic relaxation time or times, and amplitude of the chargedensity fluctuations inherent to the sample. The mechanism that governs these fluctuations varies from materialtomaterial. In most homogeneous materials, where the chargedensity 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 timescales associated with the reorientation of dipoles cover a broad range, with the associated frequencies spanning from 104 Hz, for SiTeAsGe (STAG) glass, to 1012 Hz, for lowviscosity liquids. It is important here to differentiate between the two main processes associated with the chargedensity 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 timedependence of the polarization response to the driving force. This timedependence is a delay in the polarization to follow the timedependence of the electric field. Below we show how the timedependence 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 timedependent 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) (31) where sEE is the induced free space charge of the electrode. The static, timeindependent, polarization, P, is related to the static electric field as follows: P = oZE (32) We introduce time dependence to the polarization in Eq. 32 by means of a dielectric response function,f(t), and assume the timedependent 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 (33) 0 where tris 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 (34) then the dielectric displacement in Eq. 31 takes on the following form: D(t) = oEo O(t) + f (r)dr (35) 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)] (36) 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 (34), 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) (37) 0 where (O) is the static susceptibility. As a consequence of Eq. 37, 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 timedependent dielectric permittivity, (t)29. 3.2 FrequencyDependent 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 timedomain, experimentally the most practical representation is usually in the frequencydomain. 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 frequencydomain utilizing Fourier transformations and numerical integration. Given a timedependent function, F(t), the Fourier transform of it into the frequencydomain has the following form: I +Do [F()] = F() 1 F(t)exp(i)t)dt (38) 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, timedependent function produces a complex, frequencydependent function that contains information about the amplitude and phase of the timedependent signal. Additionally, the Fourier transform allows us to write integrals of the form in Eq. 34 as the product of the Fourier transforms of the functions in the integrand30. In particular, it allows us to define a frequencydependent 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 (39) Therefore, we now have a way of simultaneously studying the dielectric response that is inphase, '(o), and outofphase, X"(o), with the ac electric field. The ability to study the inphase and outofphase 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. 37 and writing the corresponding frequencydomain response as follows: I(Co)= cE(cw)+io)D(c) (310) where 3 [iD(t) / t] = iaD(m) Using Eq. 31, the current can be written in terms of the frequencydependent polarization, P(i)= coE(wo)X (o), such that, after some algebra: I() = [o + Z0o )"(o) + iog {1 + Z'())}]E(o) (311) f When we speak of the frequencydependent 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 inphase 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 outofphase 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 frequencydependent 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 frequencydependent 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 inphase and outofphase 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 31. Barriervolume capacitor. Cg and GB are the barrier capacitance and conductance, respectively. Cv and GB are the volume capacitance and conductance, respectively. barriervolume 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 31). This circuit is described by the MaxwellWagner 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 31 is written explicitly as follows: Z() /= + iGB (312) 1 +icr 1l+ir0IB where T = C Gv and B = C GB. A schematic representation of Eq. 313 (Figure 32) 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 32. The complex impedance diagram of the barriervolume circuit. The large arc is due to the barrier region and the small arc is due to the volume region. Figure 32 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. 313 we can write the admittance of the response (Eq. 314), where r = CB/ Cv. (1 + io) z,)(1 + io) z,) Y(o)) = G( (313) 1+ico, ,r A complex admittance plot of Eq. 314 (not shown) reveals that the volume and barrier regions are two distinct vertical lines with the xintercepts indicating Gv and GB, respectively. 3.3.2 Dielectric Permittivity The frequencydependent permittivity, e(o), is defined in terms of the complex susceptibility as follows: E(m) = co [1 + x'(m)ix"(Oc)] = E'(o)) iE"()) (314) 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. 315). E(o)) () ) (315) 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)] (316) 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 CaseStudy: Disordered Conductors The frequencydependent 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 powerlaw 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 33). 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 33. 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 highfrequency dielectric constant, Ec, the limit of a > oin Eq. 314. 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. 316). Dyre showed32 that V. imE + g(r)]V})= 0 (317) when Eq. 316 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) = LDI)2 (318) 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 ThinFilm MetalInsulatorMetal Capacitors Traditional MIM capacitors are easily approximated by the parallel plate model of capacitance. Where two largeg, twodimensional, 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. 41), where 0o= 8.85x1012 F m1, is the permittivity of free space. Cg EA (41) d The measured capacitance can be heavily modified from the geometrical capacitance (Eq. 41) when d becomes vanishingly small. Specifically, an interface capacitance due to an electron screening length dependent voltage drop at the metaldielectric 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 MetalInsulator Metal (MIM) Fabrication The capacitors used in our experiments have a consistent trilayer architecture comprised of a polycrystalline, thin film (LaliPrx)5/sCa3/8MnO3 (LPCMO) base electrode (C in Figure 41) deposited on a highly insulating NdGaO3 (NGO) substrate (D in Figure 41), an amorphous AlOx dielectric (B in Figure 41) and a polycrystalline, thin film Al counterelectrode (A in Figure 41). 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 41. 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 counterelectrode 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 impedancegain/phase analyzer, which is a self contained ac response unit that simultaneously measures the inphase and outofphase response of a system to a small ac perturbation (either current or voltage). Second, we discuss the lockin 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/GainPhase 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 lockin 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 inphase and outofphase 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. 42. 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 42. 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 frontend 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 LockIn Amplifier The Princeton Applied Research PAR 124A lockin amplifier allows for the measurement of a wide variety of ac responses with a high degree of precision. While a lockin amplifier is a powerful and extremely important tool it is also quite simple, especially when compared to the more complicated impedancephase/gain analyzers discussed below. A schematic diagram of a lockin amplifier is shown in Figure 43. 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 lowpass 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 Lowpass DC Sensitive Filter Amplifier Detector Reference In Reference Out Figure 43. Schematic of a lockin amplifier The major differences between an ac voltmeter, which will also report the voltage amplitude from an input signal, and a lockin amplifier are that: 1) a lockin uses a reference signal that is synchronized with the input signal being measured and 2) lockin J The reference signal is derived from a periodic voltage source and is usually on the order of 1Vpp. amplifiers will measure the inphase and outofphase 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)] (42) 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. 42 is passed through the lowpass 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 lowpass filter by a gain of Gdc, resulting in an output of: Vout =GdcGaCAoV (43) 2 Figure 44, shows a comparison of the impedance of an LPCMO film as measured by the Solartron 1260A (black squares) and the PAR 124A Lockin 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. 43 will be Vo= GdcGcAoV cos (3+ (44) In this instance, the phase of the input reference would be tuned such that the signal in Eq. 44 is a maximum. 107 N ( RBallast 106 SamplOutput  LockIn 75 100 125 Temperature (K) Figure 44. The magnitude of the impedance of a 400 A LPCMO film as measured by the Solartron 1260A (black squares) and PAR 124A LockIn (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 twoterminal lockin technique (Figure 44 inset) over the impedance analyzer in terms of high impedance measurements. Thus lockin amplifiers are ideal for measuring ac responses where the signaltonoise 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 AndeenHagerling 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 1018 F and a loss' resolution of 3 x 1016 S. The bridge circuit is depicted in Figure 45. Like the lockin 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 (45a) N2 SLoss is in reference to the component of the impedance 900 outofphase 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 (45b) 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 45. Schematic of the capacitance bridge circuit. The standard impedance elements are fusedsilica capacitances for Co and variable pseudoresistors for Ro. The unknown sample is expressed by Cx, the inphase component, and Rx, the outofphase 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 threeterminal 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 45. 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 45). 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 45, 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 inphase and outofphase 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 outofphase, or imaginary terms: RDC that expresses the leaky behavior, and 1/o)C"(o) the lossy behavior" (Figure 46A). In the geometry of our capacitance measurements (Figure 41) 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 frequencydependent 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 46. 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 47, 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 outofphase with the capacitance is inphase with the series resistance. Therefore, the effective dielectric response will be compromised by the presence of the series resistor. Figure 46B. Therefore we must reconcile the seriesparallel model of Figure 47 with the pure parallel model of Figure 46. C1 Rs R2 Figure 47. 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 46 (with RDC ~ oc) is written as: Y, = +ioCef (46) Rff The impedance, Zs, of the series RC model of Figure 47 is written as: Zs = Rs + (47) 1 +io)R2C, Writing this in terms of the associated admittance, Ys, we arrive at: Ys = + (48) 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 (49) (R2 +Rs)2 + (CR2RC)2 (R, +Rs)2 +(CRC)2 Equations 46 and 49 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(410 a) (R2 + R) + 2R'1'RC2 Ceff = R )2+ ) 2 (410 b) e R2 +Rs + mR2R,Q2 In an ideal situation, Cff, would be independent of any of the outofphase components such that Ceff C1. We now apply the following constraints to Eq. 410b: Constraint 1: (mR2RsC1)2 (R2 +RS)2 (411 a) R2Rs 1 2R << (411 b) R2 +Rs oC, Constraint 2: Rs < 1 SRs << (412 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. 413). Rff Cff pR2C (413) R2 +Rs + 22RRsC2, Also, we can ignore the voltage drops along the highly resistive, LPCMO base electrode as long as the following condition (Eq. 414) is met: Rs The series resistance in our model, Rs, is equivalent to the twoterminal resistance of the LPCMO base electrode that is used in our MIM capacitor. Thus, it is an 52 experimentally measurable quantity0. Figure 48, shows the temperature dependence of Rs and the real and imaginary parts of the impedance of our MIM. Clearly, the conditions of Eq. 414 are met by our choice of frequency and the materials in our MIM capacitor. 108 N 106 100 200 300 Temperature (K) Figure 48. 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. 414 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 twoterminal measurements is necessary. Contact resistance between the lead of the instrument and the material is always present in a twoterminal 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 PHASESEPARATED 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 techniques3744. 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 some3739,43'44 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 (LaliPrx)5/8Ca3/8MnO3 As we discussed in Ch. 2, (LailPrx)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 2.1112). 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 51, 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 150k 0oe 0 670k Oe 70k OeF 0 50 100 150 200 250 300 350 70k C 101 Temperature (K) 0 5 0 150 200 250 300 Temperature [K] Figure 51. 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 metalinsulator transition, and at low temperature is very resistive, while LCMO has a metalinsulator 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: TH phase diagram indicating the onset of the MIT in PCMO and showing the hysteretic region. A) Schiffer et al. Phys. Rev. Lett. 75, 33363339 (1995), Fig. 2, pg. 3337. B) Tomioka, Y. et al. Phys. Rev. B 53, R1689R1692 (1996)45, Fig. 2, pg. R1690. metalinsulator transition (MIT) at all magnetic fields and shows colossal magnetoresistance (CMR), while PCMO has no zerofield 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 51B 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 zerofield MIT, a concomitant magnetic phase transition from chargeordered 101 100 0 4,J I) () a) Q^ 10 102 0 100 200 300 Temperature (K) Figure 52. 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 52. 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 53, the magnetic fielddependent 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 0101 ". 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 53. 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. 100 103 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 52. 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 manganitebased 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 spinpolarized conduction band7'50 and a high temperature orbital and charge ordered insulating phase characterized by the presence of electronphonon coupling due to JahnTeller distortions3'51. This high temperature phase exhibits paramagnetic behavior above the chargeordering 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 chargeordered 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 54). 2' I .I_ O x= 3 SX O. ~= 0.2 ". 5= U.25 .30 wl crrae mt c w r0375 resistivity data in Figure 52 and Figure 53 that has x 0.40.313 in this Sx=0412 notation. Coado et a ofMat. 15, 167174(2003), Fig. 3, pg. 170. H x 50 A x= 0625 0 IM M 300 T (K) Figure 54. Fieldcooled magnetization curves for Las/8PrCa, ,\11nO3 for various x. The magnetization data at x = 0.30 will correlate most closely with the resistivity data in Figure 52 and Figure 53 that has x = 0.313 in this notation. Collado et al. Chem. ofMat. 15, 167174 (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 superfluidfluid transition in liquid He4, or the paramagneticferromagnetic 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 firstorder phase transitions. In the case of LPCMO, the competing phases have very similar groundstate 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 211 and Figure 212). 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 La5/8sPrxCa, 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 metalinsulator 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 loglog 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 metalinsulator 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 loglog 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 secondorder phase transitions, yet LPCMO is a phase separated material with clear indications of firstorder 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 secondorder in nature and the associated magnetic order enhances thermally activated conduction via the double exchange mechanism. This in turn leads to a firstorder metalinsulator phase transition at TMI due to a nonclassical percolation transition. This nonclassical 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 AndeenHagerling 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 55 is a plot of the fourterminal resistance (black circles) of an LPCMO x = 0.5 thin film with d= 600 A, as measured by the PAR 124A lockin 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 55. 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 substratethin 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 strainstabilized FMM phase will constitute a low resistance state at the substratefilm interface. In Figure 56A, 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 4terminal resistance measurement is dominated by the parallel resistance, R because the strainstabilized 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 I9 Figure 56. Thickness dependence of the transition temperature. We now want to suggest a qualitative picture for the behavior of the capacitance seen in Figure 55. 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 welldefined potential drop to occur across the AlOx dielectric. As the temperature is lowered, the resistance increases and at 225 K there is a firstorder 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 MaxwellWagner relaxation in our LPCMO MIM capacitors. In Chapter 3, we outlined the MaxwellWagner relaxation as a contact, or interface, effect where charge polarization occurs across the interface of the contact and the electrode. The MaxwellWagner model predicts the following: C() ff = C (51) 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 MaxwellWagner 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 57) 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 57. Frequencydependent 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. MaxwellWagner 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 MaxwellWagner 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 57. Items 1) and 2) are employed because the MaxwellWagner effect is a contact effect, like a Schottky barrier in metalsemiconductor (MS) structures. MS contacts can generally behave in an Ohmic, or linear, fashion where there is no builtin 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 57. 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 58A, 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 MaxwellWagner effects is absent, or at least they are not the dominant mechanism that governs the dielectric response of our system. Figure 58B and Figure 58C shows a 68 103 A 102 C measured 101 MaxwellWagner model 0 103 0 Indium contact 100 .* Silver paint contact . Q 102 10 .u H = 15 kOe C' measured I B MaxwellWagner 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 58. MaxwellWagner 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 barrierlike effects are not present in our measurement. B) comparison of the measured real part of the complex capacitance (black) and that predicted by the MaxwellWagner model (pink), see Eq. 51. 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. 51 in zero and nonzero field, respectively. The shift in the capacitance minimums between the measured capacitance (black squares) and the MaxwellWagner model (pink line) is due to the use of the longitudinal resistivity, governed by R\ in Figure 56A, in Eq. 51. The capacitance is sensitive to the MIT associated with R1 in Figure 56A 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 = Rtwoterminal, the two terminal resistance of the LPCMO film and Co=C(300 K), were measured experimentally. In our modeling, we found that c = 104 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 59. 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 59, 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 AlAlOxAl capacitor and at H = 70 kOe (not pictured) the capacitance is essentially linear in temperature with no visible transition. The inset in Figure 59 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 others52,6163 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 55, 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 510. Colossal changes in the magnetic field dependent capacitance in the LPCMO MIM capacitor. In Figure 510, 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 510). ' I I 'I I I I I ' 710 LL708  T=IOK 706 704 60 40 20 0 20 40 60 H (kOe) Figure 511. 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 512 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 512 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 512. The temperature dependent ac loss in applied magnetic fields. The sharp peaks indicate the capacitance transition as mentioned for the inset to Figure 59. 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 (LaixPrx)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 61. 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 41). In Section 4.3.4, we showed (Fig. 48) that the Rseries contribution to the dielectric response could be ignored due to Eq. 414. 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). chargeordered 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 61. Figure 61 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 48, we plot temperature dependent impedance data revealing that this criterion is met by our LPCMO MIM capacitor and therefore Rseries in Figure 61A is also excluded from the dielectric response. In Figure 61C 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 resistorcapacitor network in Figure 61B 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 resistorcapacitor 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 61B). 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 welldefined 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 resistorcapacitor network. In the highfrequency limit, o) m >> 1, where im =RMCM, the capacitance contribution to the admittance of the infinite circuit in Figure 61B dominates and produces a spatially uniform electric field32 and thus the required equipotential surface. This highfrequency limit must be balanced by the condition that if C)T m is too large then the impedance constraints (Eq. 414) on our measurement will not be met. Therefore we must restrict ourselves to frequencies that ensure we are in the highfrequency limit, but not outside of our impedance constraints. In Figure 62, 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 100 101 102 101 1 100 LL 101 o 103 Figure 62. D >I>1> I+'+ 07 fl H = 0 kOe m m,,, I .ini, .., ' '''1 ''I ag; surface in the LPCMO base electrode. In Figure 63, we present the equivalent circuit that takes into consideration the limits imposed by the preceding analysis. i i i 0r   ) ...... ....O x C*((O) R, Figure 63. The simplified equivalent circuit for our LPCMO MIM capacitor. The LPCMO is still modeled as an infinite parallel resistorcapacitor 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 MaxwellWagner 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 lowfrequency 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 highfrequency 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 PowerLaw Phenomena Figure 62 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 highfrequency 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 ColeCole plots64. In a ColeCole 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) ColeCole 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 ColeCole 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 64, C'and C" data from Figure 62, are converted to ColeCole plots at three different temperatures in 100 100 B Figure 64. // 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 64). The behavior observed at 100 K, in Figure 64B, 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 65. ColeCole 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 65 shows a ColeCole 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 65 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 66 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 66. ColeCole 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 ColeCole 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 65) is indeed a signature of the entire CMC region and in turn the underlying phase separation. Figure 67 shows a ColeCole 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 67 and Figure 65 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 ColeCole plots of Figures 6.56.7 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 67. ColeCole 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 ColeCole plots that reveal a master curve, the behavior appears to have powerlaw dependence for sufficiently high frequencies or low magnetic fields and temperatures. In addition, it appears as if this powerlaw is ubiquitous and regardless of which parameter, frequency, magnetic field or temperature, is varied the dielectric response is governed by it. Figure 68 presents a ColeCole plot that includes the ' ' '''1 I I P t7/ dielectric behavior with a pairwise 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 Smf (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 68. 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 68, there is a pronounced rolloff 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 69. ColeCole plot showing the complete dielectric relaxation of C' This plot shares the same legend as Figure 67. C, is determined by extrapolating the high frequency rolloff to C"= 0, at which C' Co. The data are then fit to the powerlaw expression in Eq. 61, and the fit is plotted as the orange line. The exponent for the response is 7= 0.7. Figure 69 shows the powerlaw response of C "as a function of C" C, over four orders of magnitude. The data are fit to a general powerlaw expression of the form: C"(o,,T,H)= A[C'(o,T,H) CJ, ' (61) 