The Bilayer Colossal MagnetoresistiveManganites

Wing Kiu SiuVan der Waals-Zeeman Institute

Faculty of Science, University of Amsterdamwing.siu@student.uva.nl

Supervised by:

Mark Golden

February 8, 2006

Contents

1 Introduction 3

2 Background 52.1 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Electronic level scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Doping phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 The colossal magnetoresistance effect . . . . . . . . . . . . . . . . . . . . . 82.5 Double-exchange theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Theoretical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Magnetic Phase diagram 12

4 Lattice structure and electrical conductivity 144.1 Magnetic and charge transport properties . . . . . . . . . . . . . . . . . . . 14

4.1.1 Temperature-dependence of the resistivity . . . . . . . . . . . . . . 154.2 Lattice structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.1 Lattice structure at room-temperature. . . . . . . . . . . . . . . . . 174.2.2 Lattice structure: effect of doping level and temperature . . . . . . 174.2.3 Lattice structure and electrical conductivity: effect of magnetic field 19

5 Polarons 215.1 Mn-O bond-lengths: temperature-dependence . . . . . . . . . . . . . . . . 215.2 Polarons in the paramagnetic insulating state (x = 0.40) . . . . . . . . . . 22

5.2.1 Polarons: short-range correlations . . . . . . . . . . . . . . . . . . . 245.2.2 Polarons: structure of the short-range polaron correlations . . . . . 265.2.3 Polarons dynamics: CE-type charge and orbital ordering in the x = 0.40

doping level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Charge and orbital ordering for LaSr2Mn2O7 316.1 Charge and orbital ordering . . . . . . . . . . . . . . . . . . . . . . . . . . 316.2 Competition between CE-type and A-type antiferromagnetic phases . . . . 34

7 Raman scattering 367.1 Charge-ordering in the x = 0.50 doping level . . . . . . . . . . . . . . . . . 367.2 Charge-order fluctuations in the x = 0.40 doping level . . . . . . . . . . . . 39

8 Optical studies 428.1 Optical spectra: doping dependence . . . . . . . . . . . . . . . . . . . . . . 428.2 Optical spectra: temperature-dependence for x = 0.30 and x = 0.40 with

metallic ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438.2.1 Theoretical models . . . . . . . . . . . . . . . . . . . . . . . . . . . 438.2.2 Incoherent charge dynamics in the low-energy region . . . . . . . . 46

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9 Phase separation 489.1 Theoretical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.2 Experimental evidence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

10 Conclusion 51

11 Bibliography 53

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1 Introduction

Transition-metal oxides have been an extensively studied family of compounds in the con-densed matter community for the last decades. Especially the discovery of High-Tc Super-conductivity in copper-oxide based materials in 1986 spurred the interest in this group ofmaterials.Recently the condensed matter community has taken interest in another family of thetransition-metal oxides, the manganese perovskites (manganites). The curiosity was arousedby the report of giant and colossal magnetoresistance in Mn-O based films by Helmolt et al.[1] and Jin et al. [2] respectivly. Magnetoresistance (MR) is the change in resistivity uponapplication of a magnetic field and is currently very important in magnetic data storagetechnology; the giant magnetoresistance-effect is currently used in every disc drive.The GMR-effect is exhibited by magnetic multilayered materials, where two magnetic lay-ers are closely separated by a thin nonmagnetic layer. The two ferromagnetic layers thenact like two polarizers in an optical experiment, where aligned polarizers allow light to passthrough and crossed polarizers do not. In the magnetic layer, the resistivity depends on thespin orientation of the conduction electrons relative to the magnetic moment of the layer,e.g electrons with parallel spin will undergo less scattering and have a lower resistivity.When the moments of the magnetic layers are antiparallel aligned with respect to eachother, electrons will have a high scattering rate in one of the layers and the resistance willbe high. When a magnetic field is applied, the moments of the layers will align parallel andelectrons with a spin parallel to these moments will have a low scattering rate, loweringthe resistance. An applied magnetic field will thus lower the resistance.

The magnetoresistance found in the Mn-O films however is not based on spin-dependentscattering of the magnetic multilayers. Helmolt et al. [1] found in 1993 a magnetoresis-tance of 60 % at room temperature in La-Ba-Mn-O films, which is significantly higher thanthe 40 % in the traditional GMR exhibiting Cu/Co multilayers. One year later, Jin et al.[2] observed a magnetoresistance of 127.000 % at 77K and 1300 % at room temperature inLa0.67Ca0.33MnOx and termed this thousandfold change in resistivity colossal magnetore-sistance. Shortly after the discovery of the CMR effect in the La-Ca-Mn-O 3D perovskitefilms, Moritomo et al. [3] investigated another approach to increase the magnetoresis-tance effect. They grew single-crystals of the two-dimensional bilayer La1.2Sr1.8Mn2O7

and found a magnetoresistance of 20.000% (H = 7T) at the transition temperature of 126K. They were the first to report the CMR-effect in a layered manganite. They foundthat the reduced dimensionality enhanced the magnetoresistance compared to analoguethree-dimensional systems. For example the (La, Sr)MnO3 system with an infinite numberof layers has a magnetoresistance of 2000% (H = 15T) at Tc ∼ 300K. The price for theenhanced magnetoresistance is however paid by a lower transition temperature. The neg-ative magnetoresistance effect at Tc is accompanied by a transition from a paramagneticinsulator to a ferromagnetic metal and the mechanism that produces the CMR-effect isthus inherently different from the GMR-effect, in which no magnetic phase transitions takeplace. Up to now it is not well understood what mechanism produces the colossal mag-netoresistance and elucidating its mechanism might be of great relevance for data storage

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technology. The interest in the colossal magnetoresistive properties of the manganites ishowever not only based on possible technological applications, but is also fundamental innature. In this family of compounds there is a complex and delicate balance between thecharge, spin, lattice and orbital degrees of freedom, leading to a variety of physical phe-nomena.The CMR-effect was initially understood in the framework of the double-exchange the-ory, which intuitively explains the coupling between the magnetic and resistive transitions.Later it was however recognized by Millis et al. [4] that double-exchange fails to explainthe quantitative change in resistivity and the low Tc and additional physics must be takeninto account. He argued that there is an unusually strong interaction between the electronsand lattice vibrations (phonons) [5] and that this coupling of the electrons to the latticemight explain the quantitative change in resistivity. The manganites therefore also providea playground to study the poorly understood physics of systems in which a high density ofelectrons strongly couple to phonons. It is thus not only relevant to the manganese oxides,but also for the coppper-oxide High-Temperature Superconductors, in which it is not clearwhat the dominant interaction is.

In addition to electron-phonon coupling, several mechanisms have been proposed toexplain the CMR-effect, such as the formation of polarons, charge/orbital ordering andelectronic phase separation. This literature study will review a wide range of experimentsdone on the bilayer manganites, to summarize the extensive literature available on thistopic. The first chapter will discuss some starting-points for understanding the manganites,including crystal structure, phase diagram and the double-exchange model. The secondchapter will discuss the magnetic phase diagram in more detail, being a starting point forunderstanding macroscopic properties such as electrical conductivity, magnetic propertiesand the lattice structure. The following two chapters will then describe the x-ray andneutron scattering experiments, which have been performed to study the lattice structurein detail. The formation of polarons and the existence of charge and orbital orderingmechanisms will be discussed in the light of these experiments. The structural chapterswill then supplemented by Raman scattering studies. The electronic structure will thenbe discussed in the light of optical measurements, which study the actual charge carrierdynamics. The last chapter will describe the essential ideas of the phase separation modeland describe a few experiments that might support or contradict this scenario.

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2 Background

2.1 Crystal structure

The schematic crystal structure of the manganites is shown in figure 1. The manganitesare built of MnO6 octahedra with the Mn atom in the center of the oxygen octahedron.The octahedra share their in-plane O atoms and thus form two-dimensional MnO2 planes,similar to the CuO2 planes in the High-Tc Superconducting (HTSC) cuprates. TheseMnO2 planes are stacked in various orders and are separated by rock salt layers (typicallycomposed of the trivalent rare-earth atom R and divalent alkaline atom A). The rock saltlayers bind the crystal together and act like charge reservoirs. By varying the ratio betweenthe trivalent and divalent atoms, one can tune the valency of the Mn atom. The manganitesare generally represented by the Ruddlesden-Popper (RP) notation (R, A)n+1MnnO3n+1

(R,A are often Lanthanum and Strontium in this review), where n stands for the numberof MnO2 planes in the unit cell. As one can see there are structurally two classes inthe CMR oxides: the layered samples with one (n = 1) and two (n = 2) layers and thethree dimensional perovskite samples (n =∞). The layered samples cleave nicely betweenthe rock salt layers as these are ionically bonded and are thus suited for surface sensitivetechniques such as photoemission spectroscopy. The perovskite samples do not exhibit anatural cleavage plane and may be fractured or scraped in vacuum. The layered samplesbelong to the tetragonal space group (I4/mmm, z = 2) and the n =∞ belongs to therhombohedral space group (R3c, z = 2) [3]. In this thesis the focus will be on the bilayeredLa2−2xSr1+2xMn2O7.

n=∞

Figure 1: Crystal structure of the manganites which are commonly denoted by the Ruddlesden-Popperseries (R,A)n+1MnnO3n+1. In this (a) La1−xSr1+xMnO4 (n = 1). (b) La2−2xSr1+2xMn2O7 (n = 2). (c)La1−xSrxMnO3 (n =∞). Bottom right: basic building block with the MnO6 octahedra and the R,A atomsat the corners. From Chuang et al. [6]

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2.2 Electronic level scheme

In the manganites the valency of the Mn atom will depend on the ratio between the trivalentand divalent atoms. The hole doping is denoted by x and is experimentally achieved byadding more of the divalent alkaline atom. To illustrate this, the compounds with x = 0and x = 1 form the end numbers of the family and correspond in the 3D perovskitesto LaMnO3 and SrMnO3. The Mn atom has in these compounds a formal valency ofrespectively 3+ and 4+. LaMnO3 has thus four 3d-orbital electrons and SrMnO3 three3d-orbital electrons. The 3d4 configuration has a special implication when the Mn atom isplaced in a crystal field. This is illustrated in figure 2, where a schematic plot of the Mn 3denergy levels is shown. In (a) the five 3d-orbital levels (illustrated in 2(d)) are degenerate.When the Mn atom is placed in a crystal field (e.g taking the oxygen atoms into account)the degeneracy is lifted. The orbitals pointing between the oxygen atoms (the t2g orbitals:xz, yz and xy) will be lower in energy and the orbitals pointing towards the 6 oxygen atomswill be higher in energy (the eg orbitals: z2 − r2 and x2 − y2). In the case of 3d4 therewill be one eg electron, as the strong Coulomb repulsion between the d-orbital electrons,as well as Hund’s rule and the exchange interaction J will favor a high spin state. Theorbitals will all first fill up with a single electron before double-occupancy happens. For the3d4 state, the system can gain energy by distorting its lattice. This is illustrated in 2(c),where the octahedron is elongated in the z-direction, so that the z2 − r2 orbital is furtheraway from the oxygen p-orbitals. The energy of the occupied z2 − r2 orbital will then be

Figure 2: Schematic energy levels for the Mn 3d energy levels for (a) the Mn ion alone with 5 degenerated-orbitals. (b) Mn atom in the oxygen octahedron and the crystal field will split the 5 d-orbitals in 3degenerate (xz, yz and xy) and 2 degenerate (z2 − r2 and x2 − y2) orbitals. The energy separation istermed 10 Dq. (c) Jahn-Teller distortion of the MnO6 octahedra lowers the energy of the system when theeg orbitals have an uneven number of electrons. (d) Schematic representation of the probability densitysurface of the five 3d-orbitals. From [7].

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lower, but the energy of the unoccupied x2 − y2 orbital will be higher as the octahedronis compressed in this direction. The electronic energy of the system will then be loweredwhen the eg orbitals have an uneven number of electrons. The 3d3 state will thus notexhibit such a distortion as there is no lowering of the energy. This distortion is called aJahn-Teller distortion and proves to be quite important in the manganites. Although theelectronic energy is lowered by this distortion, it competes with the gain in energy due tothe structural distortion. This is even more complicated by the effect the distortion has onits neighbors (formation of long-range strain fields). It must be noted that in this exampleit was chosen to occupy the z2 − r2 orbital, but in reality this is not so simple. This willbe discussed in more detail in chapter 5.

The end members (x = 0 and x = 1) of the manganites, are both antiferromagneticinsulators due to the super-exchange interaction. For intermediate doping levels howeverthe Mn atom will have a mixed valency between 3+ and 4+ and a non-integral numberof d-electrons. This mixed valency of the Mn atoms gives rise to a variety of phases asa function of the doping level. In the following section the doping phase diagram of thebilayered manganites will be discussed.

2.3 Doping phase diagram

The phase diagram for (0.30 6 x 6 1.0) is shown in figure 3. There is a wide range ofphases present as a function of doping level and temperature. The CMR-effect occurs for(0.30 6 x 6 0.48) and this literature study will thus mainly focus on this region of thephasediagram. Almost over the whole region of the phase diagram the high temperature

Figure 3: Magnetic and crystallographic phase diagram of La2−2xSr1+2xMn2O7 for (0.30 6 x 6 1.0). Thesolid data points represent the magnetic transitions as determined from neutron powder diffraction. Leftaxis: TN antiferromagnetic Neel temperature, TC ferromagnetic Curie temperature, TCO charge-orderingtemperature, TO orbital ordering temperature. Open circles represent structural phase transitions transi-tions. From Ling et al. [8].

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state is the paramagnetic insulator (PM-I). For low temperatures there is a wide variety ofphases, ranging from a ferromagnetic metal (FM-M) in the CMR-region to several typesof antiferromagnetic phases for higher doping levels. The CMR-effect is thus a phase tran-sition between the PM-I and FM-M state. The details of these magnetic structures forvarious doping levels will be discussed in chapter 3. While the magnetic phase diagram isquite complicated, the crystal structure remains quite simple over the whole doping regime.All the La2−2xSr1+2xMn2O7 crystals for (0.30 6 x 6 0.75) have a tetragonal structure withspace group I4/mmm(Z = 2) with lattice parameters (a = 3.874 A, c = 20.145 A) [9].

2.4 The colossal magnetoresistance effect

As discussed in the introduction, Moritomo et al. [3] grew single crystals of the bilayermanganite with a doping level of x = 0.40 and found a maximum magnetoresistance of20.000% when a magnetic field of 7 T was applied. In figure 4 the CMR effect is illus-trated in the resistivity vs temperature diagram. The resistivity decreases by two ordersof magnitude when a magnetic field is applied at the transition temperature of 126 K.The conductivity appears to be highly anisotropic, as ρc is ∼ 102 times as large as ρab.This indicates that the carriers are mainly confined in the bilayers. The CMR-effect isaccompanied by a phase transition between the paramagnetic insulator (PM-I) and theferromagnetic metal (FM-M). The easy axis seems to lie in-plane as the low-field (10-mT)magnetization is also rather anisotropic Mab/Mc ≈ 10.

b)a)

Figure 4: (a) Resistivity vs Temperature diagram. The CMR effect occurs near Tc = 126 K. (b) Magne-tization vs temperature diagram for different members of the Ruddlesden-Popper family. The CMR-effectis for the n = 2 and x = 0.40 accompanied by a transition from the paramagnetic to ferromagnetic state.From Moritomo et al. [3]

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2.5 Double-exchange theory

The CMR-effect was initially understood in the double-exchange theory (DE), the latterbeing proposed by C. Zener [10] in the 1950’s and later expanded and refined by P.-G deGennes [11], P.W Anderson and H. Hasegawa [12]. In the 1950’s it was already observedthat manganite perovskites with a doping level of x = 0, x = 1 were neither ferromagneticnor good conductors, while for (0.20 6 x 6 0.40) the compounds were strongly ferromag-netic and good conductors. Zener recognized that this behavior originates from the inter-action between incomplete d-shells and termed this specific interaction double-exchange.In compounds with intermediate doping levels, the Mn atoms will have a valency of 3+ or4+ and the electrical conductivity comes from the transfer of an electron from the Mn3+

to the Mn4+ ion. The coupling between the Mn atoms is indirect and is mediated by anoxygen atom with a full shell. The term double-exchange arises from the simultaneoustransfer of an electron from the Mn3+ to the O2− and from the O2− to the Mn4+. In thisapproach Zener assumed that the spins of the d-orbitals are all parallel aligned, obeyingHund’s Rule.Later P.-G de Gennes, P.W Anderson and H. Hasegawa expanded this theory more quanti-tatively by taking the angle between the spins into account. This is schematically illustratedin figure 5. In this picture the t2g electrons are aligned parallel due to a very strong on-site

Figure 5: Schematic illustration of the double-exchange theory. The hopping probability tij of the eg

electrons to hop from site i to j depend on the relative angle θ between the t2g spins. Figure from Chuanget al. [6]

Coulomb repulsion. The hopping probability of the eg electron between the Mn3+ to Mn4+

sites is related to the angle between the t2g spins on the two sites. This is explained bythe very strong Hund’s rule coupling JH , which connects the eg spins to the t2g spins andis approximated by JH → ∞. This strong coupling constant will favor a state in whichthe eg electron spin is aligned with the t2g electron spins, leading to ferromagnetism. Theessence of the double-exchange theory is that the hopping amplitude tij for the eg to hopfrom site i to site j depends on the relative angle θ between the t2g spins:

tij = t0 · cos(θ

2) (1)

where t0 is the bare hopping probability. For the ferromagnetic state, the t2g spins at thetwo sites are aligned parallel and the relative angle θ = 0o causes the hopping probabilitytij to be maximal. When the t2g spins are aligned antiparallel (antiferromagnetic state)then θ = 180o and tij will be 0, corresponding to the insulating state.

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In the bilayer manganite the CMR-effect is accompanied by a transition from the para-magnetic to the ferromagnetic state. The spins in the paramagnetic state are randomlyorientated and this is approximated by an average angle of θ = 90o. The hopping prob-ability in the paramagnetic state is thus approximated by tij = 2t0

3. When going from

the paramagnetic to the ferromagnetic state, the hopping probability is thus expected toincrease from 2t0

3to t0. Double-exchange thus explains the close connection between the

magnetic and resistive transitions. It also explains the decrease in resistivity when a mag-netic field is applied. The magnetic field will align the t2g spins, reducing θ and increasingthe hopping probability, consistent with the magnetoresistance effect.Although double-exchange gives an intuitive feeling about the coupling between chargeand spin, it fails to quantitatively explain the magnitude of the effect. Double-exchangepredicts a decrease of 33% in conductivity when going from the ferromagnetic to the para-magnetic state, this is however not consistent with the observed two orders of magnitudechange in resistivity. Also the rich magnetic phase diagram at low temperatures, whichexhibits charge and orbital ordering and several types of antiferromagnetism, is not pre-dicted by DE. Double-exchange is thus not sufficient to drive the system insulating at hightemperatures and additional physics must be incorporated to understand the CMR-effect.

2.6 Theoretical considerations

In 1995 Millis et al. [4] recognized that double-exchange fails to quantitatively explain theresistivity behavior and the low transition temperature of La1−xSrxMnO3. The calculatedresistivity above Tc, using double-exchange, was orders of magnitude too large and belowTc the resistivity increased whereas experimentally it was found to decrease. Millis et al.thus argued that additional physics must be taken into account. He suggested that polaronsmight play a significant role, due to the very strong electron-phonon coupling stemmingfrom the Jahn-Teller distortion of the MnO6 octahedra. The CMR-effect might then be across-over from a high T, polaron dominated magnetically disordered regime to a metallicmagnetically ordered regime at low T. This suggestion was supported by calculations byRoder et al. [13] who studied the effects of combining double-exchange with Jahn-Tellerlattice couplings to holes in La1−xAxMnO3. He found that the lattice effects were able todecrease the magnetic transition temperature and can explain the doping dependence ofthe transition temperature.Millis et al. [5],[14] performed calculations in which a model was used, whereby the elec-trons are ferromagnetically coupled to the t2g core-like spins and are coupled to phonons(Jahn-Teller lattice coupling). The strong electron-phonon coupling is then the crucialadditional physics to the double-exchange theory, which localizes the electrons as polaronsfor T > Tc and is weakened in the ferromagnetic state for T < Tc. The behavior of theelectron-phonon coupling is modeled by a dimensionless parameter λ = Elatt

teff, in which

Elatt is the energy lowering from the electron-lattice coupling and teff is the ’bare’ elec-tron kinetic energy. The electron kinetic energy teff can be changed over a wide rangeby varying temperature, magnetic field and doping level. The CMR-effect is explainedin the following way: in the high-temperature state, teff is sufficiently small and λ is

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then so large that the electron-phonon interaction localizes the electrons. As T is loweredthrough Tc, double-exchange aligns the spins ferromagnetically and teff increases and therelated λ decreases sufficiently, with metallic behavior as result. Millis’ theoretical resis-tivity calculations showed qualitative similarities with experimental data on manganiteperovskites, indicating that electron-phonon coupling is indeed important in the physics ofthe manganites.

Although it is clear that electron-phonon coupling is playing a significant role in theCMR-effect, the calculations performed by Millis and Roder are oversimplified and do nottake doping dependence, inter-site correlations and Coulomb interactions into account.More extensive calculations are thus needed to include these effects.

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3 Magnetic Phase diagram

In this chapter the magnetic phase diagram shown in section 2.3 will be described in moredetail. In figure 6 two phase diagrams are shown, (a) was presented by Ling et al. [8] and(b) was published by Kubota et al. [15]. The two phase diagrams are essentially identicalto each other, but differ slightly in the exact definition of the regions of the magneticphases. Although the differences are minor, they are relevant and will be pointed out.The magnetic structure is quite complicated below the ordering temperature, but is quitesimple above Tc. The bilayered manganite is for all doping levels in this region a param-agnetic insulator (PM-I). The complicated magnetic structures below Tc will be describedas a function of doping level, by illustrating the different magnetic structures.

Doping region (0.30 6 x 6 0.32) In Ling’s phase diagram, the spins are aligned parallelto the c-axis and are ferromagnetically coupled within the bilayers and antiferromagneti-cally coupled between the bilayers. This phase is denoted as AFM and is also illustratedbelow the phase diagrams.

Kubota’s phasediagram however is a little bit more complicated in this region. ForT 6 70K there is a ferromagnetic phase (FM-II) with the spins aligned parallel to the c-axis. For (70K 6 T 6 100K) the spins are aligned antiferromagnetic between the bilayersand ferromagnetic within the bilayers. In addition they start to rotate away from the c-axis, to align parallel to the ab-plane. This is markedly different from Ling’s phasediagram,who defines no ferromagnetism and an alignment of the spins in the ab-plane . Perringet al. [16] also found a rotation away from the c-axis from 60 K up to the 100 K to gaina substantial component in-plane, analogue to Kubota. They however do not report thepresence of ferromagnetic ordering. Argyriou et al. [17] however do mention the formationof a FM phase for x = 0.30 at low temperatures, but attributes this to a second layeredmanganite phase with (0.30 6 x 6 0.32) with FM alignment. The AFM ordering shouldthus probably be considered the dominant phase for this doping region.

Doping region (0.32 6 x 6∼ 0.40) For x = 0.32 the spins are aligned parallel to thec-axis and tilt away, for larger x, to lie in the ab-plane for x = 0.36. There is in-planeferromagnetism for (0.36 6 x 6 0.40). In Ling’s phasediagram there is a thus a tiltedferromagnetic state (FM) for (0.32 6 x 6 0.36).In Kubota’s diagram this tilted FM (FM-I) region is from (0.32 6 x 6 0.38).

Doping region (∼ 0.40 6 x 6 0.50) For x = 0.40 the CMR-effect is most pronouncedand the majority of the research is thus focussed around this doping level. Ling andKubota essentially define the same phases (C-AF and A-AFI) for larger x, but differ in thedefinition of the exact regions. There is for both phase diagrams, between (Tc 6 T 6 TN),a type-A AFI phase below the Neel temperature TN , in which the intrabilayer coupling isAFM and the interbilayer coupling is ferromagnetic. Ling defines this region to be between( 0.42 6 x 6 0.48), while Kubota finds this phase for ( 0.39 6 x 6 0.48). For T 6 Tc there

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Figure 6: Magnetic phase diagrams by Ling (a) and Kubota (b) of La2−2xSr1+2xMn2O7 for (0.30 6 x 60.6/0.5). (a)The solid data points represent the magnetic transition as determined from neutron powderdiffraction. CAF stands for canted-antiferromagnetic phase and A-AFI for the A-type antiferromagneticinsulator. From Ling et al. [8] (b) AFM-II (not shown) for x = 0.3 is the planar A-type AFM structurewith FM intrabilayer and AFM interbilayer coupling. FM-I and FM-II stand for ferromagnetic orderingwith the spins in-plane and along the c-axis, respectively. AFM-I is the A-type AFM. For x = 0.50 onlyAFM-I exists in phase I, while AFM-I and CE-type AFM coexist in phase II. From Kubota et al. [15].Below: corresponding magnetic structures for the different doping regimes. The thick arrows indicate themagnetic moment on the Mn atoms.

is a transition to a canted-AFM phase (Ling : ( 0.42 6 x 6 0.48)). Which is a cantingof the spins away from ferromagnetism towards antiferromagnetism. Kubota defines thisregion between ( 0.39 6 x 6 0.48), in which the canting angle between the spins within abilayer is finite at x = 0.39 and reaches 180o for x = 0.48. This is consistent with data fromHirota et al. [18], who found a canting angle of 6.3o for x = 0.40 to 180o for x = 0.48. It ispeculiar how Ling et al. define the canted-AF region, as Ling et al. also refer to Hirota’spaper.

Doping region (x = 0.50) For x = 0.50 a charge-ordered state appears below TCO

slightly above TN and disappears again to form the A-AFI phase (is called AFM-I by Kub-ota). Kubota further mentions the coexistence of CE-type and A-type antiferromagneticordering at intermediate temperatures. The details of this coexistence and structure ofthese magnetic ordered phases will be elaborated on in section 6.2.

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4 Lattice structure and electrical conductivity

After a detailed description of the magnetic phases, the magnetic properties and the elec-trical conductivity for several doping levels can be discussed. As the CMR-effect is essen-tially the response of electrical conductivity to an applied magnetic field, understandingthe magnetic properties and the charge transport is necessary in elucidating the physicsof the CMR-effect. The latter will be illustrated in the first part of this chapter. In thesecond part the lattice structure and electrical conductivity as a function of doping level,temperature and magnetic field (magnetoresistive properties) will be discussed. It wasalready argued in section 2.6, that electron-lattice coupling might play an important rolein the manganites. These lattice distortions can apart from scattering techniques, whichstudy the microscopic lattice structure, be studied by magnetostriction experiments andwill be presented in the last section.

4.1 Magnetic and charge transport properties

In this section the electrical conductivity and the magnetization for several doping levelswill be discussed, to provide a basic understanding of the charge transport and magneticproperties. In figure 7 the temperature-dependence of the resistivity along the ab-plane(ρab) and the c-axis (ρc) at zero field is shown for La2−2xSr1+2xMn2O7 single crystals to-gether with the magnetization at 0.5 T. It was already known from the magnetic phase

106

103

100

10-3

106

103

100

10-3

106

103

100

10-3

106

103

100

10-3

Figure 7: Temperature dependence of the in-plane ρab and interplane ρc resistivity at 0 field and themagnetization at H=0.5 T // ab-plane for La2−2xSr1+2xMn2O7 single crystals for (0.3 6 x 6 0.45). FromKimura et al. [19]

diagram that below Tc, a long-range ferromagnetic ordering is present. This is also appar-ent from the magnetization data, which show a steep rise in the magnetization at T ∼ 100K for (0.3 6 x 6 0.4). The decrease in magnetization for x = 0.30 at 60 K correspondsto the change of the easy-axis from in-plane to out-of-plane for lower temperatures. Thiswould actually correspond to Kubota’s version of the magnetic phase diagram in this re-gion. The saturated moment (Msat ≈ 3.5µB) of the crystals with ferromagnetic couplingwithin a bilayer (0.3 6 x 6 0.4) is near that expected for the full Mn moment.As one can see from the resistivity data for all doping levels, the anisotropy in the carrier

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motion in the ab-plane and along the c-axis is quite large, ρc/ρab ∼ 102. The anisotropyis due to the layered structure and suggests that the carrier motion is confined within theMnO2 bilayer. For (0.3 6 x 6 0.4) the onset of ferromagnetism is accompanied by a dropof several orders of magnitude in the resistivity (insulator-metal transition). The effect ismost pronounced for x = 0.40 and the majority of the research is thus focussed on thisdoping level.The temperature-dependence of the different crystals can give more information regardingthe nature of the charge transport. As one can see for (0.3 6 x 6 0.4), there is essentiallyno doping dependence of ρc above Tc and the temperature-dependence indicates semicon-ducting behavior. The in-plane resistance (ρab) above Tc does show doping dependence.For x = 0.40 the temperature dependence shows semiconducting behavior, similar to ρc,with an activation energy of 30-40 meV [3]. For smaller x the insulating behavior is sup-pressed and evolves towards a metallic-like temperature-dependence (dρab/dT > 0) in theregion (Tc 6 T 6 270K) for x = 0.30 [23]. It must be noted that the interplane resistivityof the x = 0.30 at low temperatures is almost an order of magnitude larger than the crys-tals with (0.3 6 x 6 0.4). This is probably due to the antiferromagnetic coupling betweenthe bilayers, which reduces the out-of-plane carrier motion.The in-plane resistivity for the x = 0.40 doping level is ρab ∼ 2×10−3Ω ·cm and indicates avery poor metallic state, e.g it is on the order of the inverse of Mott’s minimum of metallicconductivity.

For x = 0.45 both ρab and ρc show semiconducting behavior over the whole temperaturerange. There are two broad anomalies in the resistivity at 150 and 90 K, correspondingto the onset of respectively A-type antiferromagnetism and canting of the spins within abilayer. The latter is also evident from the magnetization data.

4.1.1 Temperature-dependence of the resistivity

A more detailed electrical transport study on La1.2Sr1.8Mn2O7 was carried out by Zhanget al. [20] and Chen et al. [21]. They also measured the temperature and magneticfield dependence of the resistivity, but in addition they fitted several temperature regionswith certain models. Chen et al. used the adiabatic small polaron hopping model to fitthe resistivity in the paramagnetic phase. In the adiabatic limit, the electron motion isassumed to be faster than the ionic motion of the lattice and all correlations except foron-site Coulomb repulsion are ignored. They found that for (T > 218K), all the maincharacteristics of the charge transport, e.g the H and T dependence, the resistivity cuspnear Tc and the decrease of resistivity with H, were well reproduced by their small-polaronhopping model. It will be shown in the following sections and chapter 5 that polaronsindeed play an important role in the paramagnetic state. The authors do not report on thetemperature region between (Tc > T > 218K). Zhang et al. found that the ferromagneticphase below Tc is characterized by two regions. A close-up of the in-plane and out-of-plane resistivity below Tc is shown in figure 8 (a) and (b). The first region between50K > T > 110K (Tc = 125K) is characterized by a T9/2 dependence of the resistivity.The fit to the data with a T9/2 dependence (solid line) is shown in figure 8 (c) and (d).

15

c)

d)

Figure 8: a,b Zoom of the in-plane ρab and out-of-plane ρc between (0K > Tc) for La1.2Sr1.8Mn2O7. c,dThe temperature-dependence between (50K > T > 110K) can be fit with a T9/2 dependence (solid line)and is a signature that the conductivity is dominated by two-magnon scattering processes. From Zhanget al. [20]

The authors contribute the temperature-dependence to a two-magnon scattering process,whereas electron-electron scattering would yield a T2 dependence and electron-phononscattering would give a T5 dependence.

Both the in-plane and out-of-plane resistivity show an upturn below 50 K and this weaknonmetallic behavior (metal-like behavior is characterized by (dρ/dT > 0)) could be fittedwith a σab ∼ T 1/2 dependence. This T1/2 dependence of the in-plane conductivity wasalso found in the x = 0.30 doping level between (30mK > T > 2K) by Okuda et al. [22]and was suggested to arise from weak localization effects in disordered 3D metals. As theout-of-plane conductivity σc shows the same T1/2 dependence, it confirms the 3D metallicnature of the bilayer manganite below Tc.

4.2 Lattice structure

It was already argued that electron-phonon coupling might play an important role in themanganites, due to the Mn3+ induced Jahn-Teller distortion of the lattice. As the CMR-effect is essentially the response of the resistivity upon application of a magnetic field, amagnetostriction study is very useful as it combines these two properties of the mangan-ites. Magnetostriction is defined as the mechanical response of a material when a magneticfield is applied. It is a property of ferromagnetic materials which change their physicaldimensions in response to changing their magnetization. The deformation is measured bya strain gauge as a function of magnetic field.As lattice distortions will become evident in anomalies in the lattice structure, a magne-tostriction study can give an intuitive feeling of the relation between the CMR-effect (themagnetoresistive properties are also shown) and lattice distortions which can self-trap the

16

electrons. Apart from the lattice degrees of freedom, one can also study the orbital degreeof freedom of the eg electron, as it is intimately connected to the lattice structure, via theJahn-Teller distortion of the Mn3+ ion.In the first part of this section the lattice parameters and Mn-O bond lengths at room-temperature will be illustrated, to give an intuitive feeling about the magnitude of thelattice distortion and the orbital occupancy. In the second part the lattice structure asa function of temperature will be discussed, to relate the lattice distortions to the metal-insulator transition. In the last part of this section, the magnetostrictive properties arerelated to the magnetoresistive ones, providing a direct connection between the latticedistortions and the CMR-effect.

4.2.1 Lattice structure at room-temperature.

In figure 9 the lattice parameters and the three different Mn-O bond lengths at room-temperature as a function of doping level are shown. It is seen that with increasing holedoping x, the a-axis increases only slightly while the c-axis decreases with 0.4 A (2% de-crease). This trend is also observed in the Mn-O bond lengths, in which the equatorialbond-length is almost independent of doping level, while the apical Mn-O bond (the un-shared oxygen atom in particular) changes significantly. As the Mn-O bond lengths aredirectly related to the orbital occupance of the eg electrons, via the Mn3+ mediated Jahn-Teller distortion, a change in the Mn-O bond lengths will reflect the preferential orbitaloccupancy of the eg electrons. Since the c-axis decreases with increasing hole concentration,it appears that the eg electrons prefer to occupy the z2 − r2 orbital. This is approximatelyconsistent with Magnetic Compton-Profile measurements 1 done by A. Koizumi et al. [25],who found that the eg state is dominated by the x2 − y2 type orbital with almost constantpopulation, while the population of the z2 − r2 type orbital decreases with increasing holeconcentration x.

4.2.2 Lattice structure: effect of doping level and temperature

It was already explained that magnetostriction measurements can observe the mechani-cal response of a system to changing its magnetization upon application of a magneticfield. In this section the lattice structure as a function of doping level and temperaturewill be discussed. Kimura et al. performed strain gauge measurements as a function oftemperature and found that the material changes its macroscopic dimensions around the

1In Compton scattering, an incident x-ray photon is scattered by an atom and is sometimes reflectedin a different direction. In this case it loses energy to an electron, which is then ejected from the atom.The angle between the scattered photon direction and the incident photon is measured as well as theangle between the scattered electron and the incident photon. A Compton profile is defined as the one-dimensional projection of the electron momentum distribution. Magnetic Compton scattering measures inaddition the spin magnetic moment. Experimentally it is achieved by substracting the two data sets takenwith reversed applied magnetic field, or by substracting the data sets obtained by using circular left andright polarization of the photons. Magnetic Compton scattering thus measures the spin-polarized electronmomentum distribution. For more detailed information see [24].

17

Figure 9: Lattice parameters and Mn-O bond lengths at room-temperature for La2−2xSr1+2xMn2O7 asa function of hole doping x. The different Mn-O bonds lengths are indicated in the octahedron. FromKimura et al. [19].

transition temperature. This physical change is termed striction by Kimura et al., insteadof magnetostriction, as no magnetic field is applied to induce a physical change. Strictionmeasurements can thus not only be used to study magnetic transitions, but also latticedistortions.It was already seen in the magnetic phase diagram that the transition temperature followsa parabolic path in the doping region (0.3 6 x 6 0.4), with the maximum Tc at x = 0.36.Kimura et al. [26] performed strain gauge measurements as a function of temperatureand doping level and found that lattice anomalies associated with transitions through Tc

reverse sign at x = 0.36. For example, for the x = 0.40 doping level the striction alongthe a- and b-axis (∆Lab/Lab), for decreasing temperatures, drops at Tc, while the stric-tion along the c-axis increases. For x < 0.36 the opposite behavior is observed, while noanomalies are seen for x = 0.36. Connection to the microscopic properties of the lattice isprovided by neutron powder diffraction studies by Medarde et al. [27], who confirm thatthe coherent lattice anomalies (lattice parameters and Mn-O bond lengths) reverse sign atx = 0.36, while the anomalies are completely suppressed at this doping level. In additionthe apical and equatorial Mn-O bond lengths were used to analyze the distortion of theMnO6 octahedra at the transition temperature. For compositions other than x = 0.36, theMn-O bonds showed pronounced lattice anomalies at the transition, whereas no expan-sions of the bonds were observed for this optimal Tc composition. It appears thus that themaximum Tc coincides with a minimum in the lattice rearrangement at the transition andthat the sign reversal in the macroscopic data by Kimura has its origin in the microscopicoctahedra distortion. Thus for x > 0.36 the octahedra elongate at Tc and for x < 0.36

18

opposite behavior is observed.

4.2.3 Lattice structure and electrical conductivity: effect of magnetic field

Kimura et al. [26] also studied the effect of a magnetic field on the lattice structure andcorrelated this to the magnetoresistive properties. The results for the magnetostrictionand the magnetotransport data along the ab-plane and c-axis for doping levels x = 0.30,0.40 and 0.45 are shown in figure 10. As one can see, all the crystals show giganticmagnetostriction at the transition temperature and this is accompanied by a steep dropin resistivity for the x = 0.30, 0.40 samples. For x = 0.30 there is an increase in ∆Lab/Lab

when passing through Tc and the lattice striction also increases when a magnetic field isapplied above Tc. Whereas ∆Lc/Lc decreases when H is applied. For x = 0.40 the oppositebehavior is observed. For both the doping levels it is seen that the magnetostriction andmagnetoresistance are pronounced around the transition temperature, but become smallwhen going away from Tc. In the x = 0.45 crystals there is semiconducting behavior

Figure 10: (a-c) Temperature dependence of the in-plane resistivity (ρab) and striction [∆Lab/Lab(300K)]and (d-f) Inter-plane resistivity (ρc) and striction [∆Lc/Lc(300K)] in various magnetic fields forLa2−2xSr1+2xMn2O7 for x = 0.30, 0.40 and 0.45. 1kOe = 0.1T. From Kimura et al. [26]

19

down to the lowest temperatures and only the application of a magnetic field induces theinsulator-metal transition. The drop in resistivity is accompanied by a decreasing strictionalong the ab-plane and an increasing striction along the c-axis, similar to the x = 0.40crystals.

It is remarkable that the striction and magnetotransport behavior becomes similar forthe x = 0.40 and x = 0.45 crystals at high fields, implying that the field-induced FM-spinarrangement is accompanied by a similar lattice striction in the higher doped regions. It wasalready seen in section 4.1 that the striction data could be related to the Mn-O bond lengthsand that the orbital occupancy as a function of doping level could be derived. Argyriouet al. [28] performed temperature-dependent neutron diffraction studies on samples withx = 0.40 and confirm that the Mn-O bond lengths follow the trends seen by Kimura inthe striction data. The magnetostriction data for the x > 0.40 samples thus indicates astabilization of the z2 − r2 orbital when going to the low-temperature state. This thusgives rise to a FM double-exchange interaction along the c-axis. The magnetostrictiondata for the x 6 0.36 samples seems to indicate a stabilization of the x2 − y2 orbital inthe cooling or field-induced state. The octahedra in this doping regime however appearto elongate along the c-axis and form a collective Jahn-Teller distorted stucture, as wasevident from figure 9. The lattice striction towards the low-temperature FM state thusseems to suppress a collective Jahn-Teller distortion. The FM ordered state thus exists ofa relatively isotropic mixture of the x2 − y2 and z2 − r2 orbital states.

In conclusion, magnetostriction measurements were correlated to the magnetoresistiveproperties of the bilayer manganite and significant lattice distortions were observed aroundthe PM-I to FM-M transition temperature. The lattice distortions could also be inducedby the application of a magnetic field, these measurement thus provide a direct relationbetween the charge transport properties, the CMR-effect itself and the importance of latticedistortions in the manganites. It was found that for x > 0.36 the octahedra tend to elongatein the ferromagnetic state, thus stabilizing the z2 − r2 orbital. Whereas opposite behaviorwas observed for the x < 0.36 samples. The lattice anomalies are completely suppressedat x = 0.36, coinciding with the maximum in Tc.

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5 Polarons

It was shown in the previous chapter that lattice distortions play an important role in themanganites and that anomalies in the lattice structure could be related to the magnetore-sistive properties. The importance of the coupling of the electrons to the lattice, via theMn3+ induced Jahn-Teller distortion was also pointed out at an early stage by Millis et al.(section 2.6). In this chapter these lattice distortions will be discussed in detail using thex-ray and neutron scattering studies performed on the bilayer manganite. The formationof polarons as a possible mechanism to explain the insulating behavior in the paramagneticstate will be discussed for the optimally doped x = 0.40 compound.

5.1 Mn-O bond-lengths: temperature-dependence

To connect the macroscopic striction data to the microscopic structure, the Mn-O bondlengths probed by powder neutron diffraction by Mitchell et al. [9], are shown in figure 11.These authors probed the Mn-O bond lengths as a function of temperature for x = 0.40.The Mn-O bond lengths have the same index as in figure 9 and are shown again here forclarity. As one can see, the equatorial Mn-O (3) contracts upon lowering the temperaturebelow Tc whereas the out-of-plane Mn-O (2) with the unshared oxygen atom expands.The apical Mn-O bond (1), with the shared oxygen atom, does not show any temperature-

a)

b)

c)

d)

Figure 11: (a,b) Mn-O bond lengths for La1.2Sr1.8Mn2O7 as probed by powder neutron diffraction. (c)Lattice parameters vs. temperature. The inset shows the unit cell volume as a function of temperature.No dramatic change is seen across Tc. (d) Octahedral distortion D =< Mn−Oapical > /Mn−Oequatorial

vs. temperature. Where the average of Mn-O(1) and Mn-O(2) is taken for dMn−Oapical. From Mitchell et

al. [9]

21

dependence. The trends in the apical and equatorial Mn-O bond lengths agree with thetrends seen in the striction data. The octahedral distortion, as a ratio of the equatorialto the apical Mn-O bond length is shown in figure 11 (d). The authors interpreted the’five long and one short’ geometry as an increased distortion of the octahedra in the fer-romagnetic state. The delocalization of the electrons thus results in an enhancement ofthe structural distortion in the metallic state. Such a local lattice distortion however isnot incompatible with the formation of lattice polarons and the authors referred to Ramanscattering data [29] on the x = 0.40 sample, which report on the formation of small polaronsin the paramagnetic state and a crossover to large polaronic behavior in the ferromagneticstate2. In a later paper [53] however the same authors interpret the Raman spectra as asignal of charge fluctuations in the ferromagnetic state, rather than the formation of largepolarons. These Raman scattering data will be discussed in chapter 7.

5.2 Polarons in the paramagnetic insulating state (x = 0.40)

The importance of electron-lattice coupling and the formation of polarons in the hightemperature magnetically disordered regime, was already pointed out at an early stageby Millis et al. and Roder et al. and was discussed in section 2.6. They described thedoped manganites in terms of strong electron-lattice coupling via the Jahn-Teller distortion,surrounding the Mn3+ ions. The charge carriers then become self-trapped by the locallattice distortion and the system can be described by a polaronic transport model. BelowTc the lattice distortions relax due to the delocalization of the electrons via the double-exchange mechanism. They predicted that the Debye-Waller factors should change acrossthe insulator-metal transition in such a polaronic model. This change in D-W factors asT is lowered through Tc was indeed experimentally observed in the layered manganites(0.32 6 x 6 0.4) by Medarde et al. [27]. Several x-ray and neutron scattering experiments[30],[34] have been carried out to study the formation of polarons and their temperature-dependence. In this section the formation of polarons in the optimally doped x = 0.40samples will be discussed. A polaron is defined as a localized charge, with an associatedlattice distortion field. In the case of the manganites the electrons can be become self-trapped by a Jahn-Teller distortion, a polaron, and this lattice distortion will give rise tosome lattice strain. Polarons can be studied by x-ray scattering as the strain field (causedby the polarons) is observed as diffuse scattering around the expected Bragg peaks in x-rayscans. To be more specific, it is known as Huang scattering and is defined to be producedby a long-range strain field, associated with a localized point defect. Huang scatteringthus monitors the formation of individual polarons and does not observe potential polaroncorrelations. It must be noted that, apart from the Huang scattering contribution, diffusescattering is also produced by the vibrations (phonons) of the atoms in a lattice. The latteris termed the thermal diffuse scattering.

2Small polarons are bound to the lattice sites and can only move by jumps between nearest or next-nearest neighbors, whereas large polarons are free to move in the conductance or valence band of thecrystal and have far higher mobilities. Small polarons are thus also smaller in real space and have a highereffective mass and large polarons often span multiple unit cells [31].

22

a)b) c)

quasielastic peak

acoustic phonon

Figure 12: Diffuse x-ray scattering from La1.2Sr1.8Mn2O7 (a) Diffuse scattering in a butterfly pattern,around the (0,0,8), (0,0,10) and (0,0,12) Bragg reflections at T = 300K. The diffuse scattering partlyarises from the distortion field created by Jahn-Teller distortions (Huang scattering) and partly from thevibration of the atoms (TDS). (b) Temperature-dependence of two lobes of the diffuse scattering (Huangand TDS contributions) around the (2,0,0) peak. The straight line at low T is the estimated phononcontribution and is the thermal diffuse scattering. The abrupt increase in intensity at Tc is however due tothe formation of polarons and its associated distortion field. (c) Neutron energy scans at q = (2.05, 0, 0.25)which is on one of the lobes of the diffuse scattering around the (2,0,0) Bragg reflection. The excitation at2.4 meV is attributed to an acoustic phonon mode. The quasielastic contribution increases with decreasingtemperature and then collapses below Tc. A flat background (29 counts) and an elastic incoherent peak(89 counts) measured at 10 K have been subtracted from these data. From Vasiliu-Doloc et al. [30]

Vasiliu-Doloc et al. [30] indeed observed diffuse scattering in a butterfly pattern, closeto the expected Bragg peaks. The diffuse scattering in the (h0l)-plane as measured forLa1.2Sr1.8Mn2O7 is shown in figure 12 (a). This butterfly pattern thus partly arises fromthermal diffuse scattering (TDS, phonons) and partly from Huang scattering (individualpolarons). As the Huang scattering is the part of the diffuse scattering which contains theinteresting physics, it would be useful to separate it from the thermal diffuse scattering.As TDS arises from the vibration of atoms (i.e phonons), it has a stronger temperature-dependence than the Huang scattering, which is produced by point-defects. The TDScontribution to the diffuse scattering can thus be determined by tracking the temperature-dependence of the diffuse scattering. The latter is indeed done by the authors and is shownin figure 12 (b). As one can see, there is an almost linear temperature-dependence belowTc and this is characteristic of an acoustic phonon mode [34]. The authors confirmedthis observation by performing neutron scattering on one of the lobes of the butterfly, atq = (2.05, 0, 0.25). The neutron energy scan is shown in (figure 12 (c)) and exhibits aquasielastic (0 meV) and an inelastic peak at 2.4 meV. The latter was ascribed to acousticphonons3 and indeed confirms the former conclusion.

3A phonon mode is a collective vibration of the atoms around their equilibrium and is characterizedby a wavelength and an amplitude. In a longitudinal mode the atoms vibrate in the direction of thepropagation direction of the wave, whereas in the transverse mode the atoms move perpendicular to thepropagation direction. In materials one typically defines two phonon modes: an acoustic and an opticalmode. In the case of acoustic phonons, all the atoms move in the same direction and the motions ofadjacent atoms are positively correlated. For a mode with a very long wavelength the atom spacing is not

23

The intensity of the diffuse scattering however increases abruptly at Tc and this cannotbe ascribed to a conventional phonon mode. Vasiliu-Doloc et al. attribute this abruptincrease in the intensity to the formation of polarons. This observation is again verified bythe energy neutron scans 12 (c). As one can see, the inelastic phonon peak does not showa strong temperature-dependence, whereas the intensity of a quasielastic peak increases asthe temperature is lowered from 300 K to just above the transition temperature. As T islowered through Tc the intensity of the quasielastic peak collapses. The sudden decreasein diffuse scattering at the transition temperature can thus be solely associated with thecollapse of the quasielastic peak. The quasielastic peak can thus be associated with theformation of individual polarons. The timescale of these lattice distortions can then beestimated from the width of the quasielastic peak. Vasiliu-Doloc et al. estimate, usingτ ∼ ~/2∆E, that the lattice distortions are quasistatic on a time scale of τ ∼ 1ps (i.e theyare static on the time scale of typical phonon vibrations).

From the anisotropy of the butterfly pattern, one can estimate the geometry of the Jahn-Teller distortion. As Huang scattering typically displays a ~q−2 dependence, the anisotropic~q-dependence can be explained by an in-plane Jahn-teller distortion. The exact details ofthe structure will be explained in the next subsection.

Vasiliu-Doloc et al. thus prove the existence of quasistatic polarons above Tc and theircollapse in the ferromagnetic state. This is also consistent with neutron-diffraction data byArgyriou et al. [34], who also found an anomalous temperature-dependence of the diffusescattering (a sudden decrease at Tc). They compared the temperature-dependence of thediffuse scattering to the changes in Mn-O bond lengths and Debye-Waller factors and find agood correlation. From this observation they describe the high-temperature paramagneticstate with disordered Mn-O bond lengths and the low-temperature ferromagnetic state witha more uniform distribution. There is thus polaronic transport above Tc and a delocalizedstate below Tc, resulting in a uniform distribution of the Mn-O bond lengths.

5.2.1 Polarons: short-range correlations

Up to now the scattering data have only monitored the formation of individual polaronsas possible correlations between the polarons is not observable in the Huang scattering.More information regarding possible polaron correlations was provided by the presence ofbroad incommensurate superlattice peaks in the paramagnetic state. At high temperature(300 K) no superlattice peaks are present and the polarons and their distortion fields arenot correlated. As T is however lowered to T = 125K, close to the transition temperature,superlattice reflections start to appear. These superlattice peaks are shown in the contour

changed and there is no stretching of the bonds. In the case of a mode with a short wavelength, the atomicspacings do change and there is a small stretching of the bonds. In a dispersion relation (frequency vs.wavevector), the acoustic phonon mode is typically characterized by a frequency that goes to 0 for ~q = 0and a linear dependence of the frequency on the wavevector for ~q π/a. The speed of the propagation ofthe phonon, which is also the speed of sound in the lattice, is thus independent of the phonon frequency.The phonon modes for which ω(~q) 6= 0 at ~q = 0, are described by optical phonons. Optical phonons onlyarise in crystals which have more than one atom in the unit cell, as the optical branch is produced bynearest neighbor atoms that vibrate against each other [32][33].

24

a)b)

Figure 13: Polaron correlations in La1.2Sr1.8Mn2O7. (a)Diffuse superlattice reflections at ~q = (±0.3, 0,±1)and ~q = (0,±0.3,±1) at T = 125K, arising from correlated charge modulations. (b) Closed triangles:temperature-dependence of the Huang scattering. The latter was obtained by substracting the linearphonon contribution (TDS) from the diffuse scattering in (b). Closed circles: Temperature-dependenceof the x-ray intensity of the (2.3, 0, 1) incommensurate peak. open circles: the T-dependence of thequasielastic neutron peak around the (2,0,0) Bragg reflection. From Vasiliu-Doloc et al. [30]

plot of the hk-plane at l = 18 in figure 13(a). Three superlattice peaks (the fourth wasexperimentally not accessible) are observed at ~q = (±0.3, 0,±1) and ~q = (0,±0.3,±1)in units of 2π/a as measured from the nearest Bragg peak. Vasiliu-Doloc et al. arehowever not conclusive on the presence of two ~q-vectors. The second set of superlatticereflections with ~q = (0,±0.3,±1) is either attributed to the presence of 90o rotated (a,b)twin domains of a system which can be described by 1 ~q-vector (i.e stripes) or because it isan intrinsic feature and 2 ~q-vectors are needed to describe the system (i.e a checkerboard).Unfortunately a detailed explanation is missing and the authors only concentrate on the~q = (±0.3, 0,±1) peaks.

The presence of superlattice peaks thus indicate that the individual polarons are actu-ally correlated and form a modulated structure. From the magnitude of the ~q-vector theyfind that the modulation is confined to the ab-plane with an incommensurate period of 3.3lattice constants. From the width of the incommensurate superlattice peaks Vasiliu-Dolocet al. estimate that the correlations are short-ranged, e.g 26 A (6-7 lattice constants). Thisshort-ranged correlation is not unsurprising as the modulated structure is incommensuratewith the lattice. These correlations are once again quasistatic on a timescale of τ ∼ 1ps,as measured from energy neutron scans (not shown).

The temperature-dependence of the quasistatic correlated polarons can be very infor-mative regarding their importance for the CMR-effect, e.g a dramatic effect is expectedat Tc. The temperature-dependence is obtained by plotting the Huang scattering, thequasielastic neutron peak (around the (2,0,0) Bragg reflection) and the incommensuratesuperlattice peaks with ~q = (0,±0.3,±1) as a function of temperature in figure 13 (b). TheHuang scattering, monitoring the Jahn-teller distortion around the Mn3+ ion, is obtainedby subtracting the thermal diffuse scattering (phonon contribution) of the diffuse scatter-ing. The temperature-dependence of the quasielastic peak from the neutron energy scansalso monitors the formation of individual polarons. The temperature-dependence of the

25

superlattice peaks tracks the short-range correlations between the Jahn-Teller distortions.As one can see the temperature-dependence of the incommensurate peak intensity is

remarkably similar to the Huang scattering, indicating that these superlattice reflectionscan indeed be associated with the formation of polarons above Tc. Thus above Tc chargecarriers trap themselves and form polarons with an associated strain field. With decreasingtemperature the polarons become correlated and superlattice peaks start to appear. Thetemperature-dependence of the intensity of the superlattice peaks and Huang scattering in-creases with decreasing intensity and indicates that together with the formation of polaroncorrelations, the polarons themselves become more localized. Below Tc double-exchangefavors a delocalization of the charge carriers and the electrons move ’faster’ than that thelattice can trap them, the polarons and associated correlations melt.

A direct relation between the CMR-effect and the formation of polarons is also givenby the dependence on the magnetic-field. Upon increasing the magnetic field, the spins arealigned parallel and the intensity of the incommensurate peaks decreases, almost trackingthe decrease in resistivity [35].

5.2.2 Polarons: structure of the short-range polaron correlations

Additional information regarding the microscopic structure of the polaron correlations canbe obtained by analyzing the intensities of the superlattice peaks in more detail. Campbellet al. [36] performed a structural analysis based on the integrated intensities of the peakswith ~q = (h ± 0.3, 0, l ± 1) and determined the atomic displacements associated with theshort-range polaron correlations. The measurements were performed in the paramagneticstate at a temperature where the correlations are the strongest T = 125K [30]. They foundthat the ~q = (0.3, 0,±1) modulation structurally corresponds to a Jahn-Teller coupledcharge density wave (CDW). This modulation is also illustrated in figure 14. From thewidth of the superlattice peaks, the correlation length ξ is estimated to be 26 A (6-7lattice constants) in the ab-plane and ∼ 10A along the c-axis (is c/2). The presence of thesuperlattice peaks at only odd integers of l indicates that the modulation is 180o out ofphase in neighboring layers.

In the Jahn-Teller coupled CDW, the magnitude of the Jahn-Teller distortion dependson the Mn3+ (e.g eg orbital occupancy) character of the octahedron. They found that themagnitude of the distortion displays a sinusoidal character and can be associated with avarying eg electron density across the lattice within the coherence length of 26 A. As themagnitude of the octahedral distortion reflects the Mn3+ character, the eg electron densityvaries in a sinusoidal way with a period of 3.3 a and can be called a charge density wave.The octahedra thus exhibit a mixed Mn3+/Mn4+ character. Campbell et al. found that theoctahedra are elongated in the a-axis direction due to the occupation of the dx2−r2

4 orbitalby the Mn eg electrons. A few remarks remain regarding the structural features of thisJahn-Teller coupled CDW. It was found that the octahedra which were elongated alongthe a-axis, were compressed along the c-axis. The octahedra are rotated about the b-axis,

4The x2 − r2 orbital is treated as an in-plane rotation of the z2 − r2 orbital. A good mathematicaldescription is however missing.

26

Figure 14: Jahn-Teller coupled Charge Density Wave associated with the diffuse ~q = (0.3, 0,±1) satellites.The modulation period is 1

0.3 = 3.3a and has a short coherence length of 25 A due to the incommensuratenature of the modulation. The atomic displacements in the z-direction (blue) and x-direction (red) areillustrated by the wave-like modulation. The peaks indicate +x or +z displacements and troughs indicate-x or -z displacements. Octahedra are compressed in the c-direction when they are elongated in the a-direction. The octahedra are in addition rotated about the b-axis and there is a 180o phase difference inthe modulation between the bilayers. From Campbell et al.. [36]

resulting in a buckled ab-plane [35], a distortion not present in the average structure. Allthis, together with the 180o phase difference between the adjacent layers, appear to worktogether to minimize the lattice strain caused by the dominant Mn3+ distortion.

A few critical remarks remain: as was mentioned earlier, Vasiliu-Doloc et al. reportedthe presence of superlattice reflections with two ~q-vectors. The origin of the presence oftwo ~q-vectors was however not entirely clear and was either due to twinning or because it isan intrinsic effect. Campbell et al. however does not mention the second ~q-vector and didnot show a cut of the hk-plane, which would show the presence of these peaks. They thusonly analyzed the intensity of the ~q = (h ± 0.3, 0, l ± 1) peaks. Li et al. [41] did find forthe x = 0.50 doping level a twinning of 90o rotated domains. The two sets of superlatticereflections might thus be due to twinning, but a definite answer cannot be given.

Now the structure of the Jahn-Teller distortion is illustrated, the temperature-dependenceof the polaron correlations can be explained. Campbell et al. [37] found from an analy-sis of the anisotropy of the Huang scattering, that at 300 K the eg electron occupies theout-of-plane z2 − r2 orbital, resulting in a Jahn-Teller elongation along the c-axis. Between300 K and Tc the ’orbital polarization’ shifts into the ab-plane (e.g the x2 − r2 and y2 − r2

orbitals). As these orbitals have a better overlap than the out-of-plane z2 − r2 orbitals,the interaction between the polarons will be stronger, facilitating the formation of polaroncorrelations.

27

5.2.3 Polarons dynamics: CE-type charge and orbital ordering in the x = 0.40doping level

Another type of superlattice reflection has been observed by Argyriou et al. [38]. Theyfound diffuse scattering at ~qL = (0.3, 0, 1), as discussed earlier, together with superlatticereflections at ~qCE = (1/4, 1/4, 0). The latter is usually observed in the x = 0.50 dopinglevel [46], As at this doping level there are equal amounts of Mn3+ and Mn4+ ions, a chargeordering in a checkerboard fashion takes place. The latter is then in addition accompaniedby an orbital ordering of the x2 − r2 and y2 − r2 orbitals, which gives rise to a CE-typeAFM ordering, to form a single long-range ordered phase. The details and structure of thischarge and orbital ordered phase for the x = 0.5 doping level will be discussed in the nextchapter. For doping levels where this ratio is not unity, the orbital ordering can becomefrustrated and double-exchange wins the competition at low temperatures.

Argyriou et al. indeed found no long-range charge and orbital ordering for a wide rangeof doping levels, but rather a coexistence of diffuse scattering at ~qL and ~qCE. The intensityof the former scales directly with the resistivity, increasing with decreasing temperatureand then collapses at the transition temperature, concomitant with the sudden decreasein resistivity. The intensity of the ~qCE reflections however was found to decrease withdecreasing temperature. To investigate the dynamics of these CE-correlations, Argyriouet al. performed temperature-dependent (T = 4− 460K) inelastic neutron scattering mea-surements for samples with doping levels of x = 0.38 and 0.40. The neutron energy scansat ~qCE = (1/4, 1/4, 0) for three temperatures are shown in figure 15.

The peak centered around 0 meV has an elastic and quasielastic contribution and werefitted with respectively a Gaussian and Lorentzian. The temperature-dependence of thesecontributions are respectively shown in figure 15 (a) and (b). From the width of thequasielastic peak at 360 K, they estimate the lifetime of the Jahn-Teller polarons to beon the order of femtoseconds (60 fs at 360 K) and are thus far more dynamic than the~qL correlations (1 ps). More information can be obtained from the constant energy cutsaround the (2,2,0) Bragg reflection along the [110], in figure 15 (c). The contour map of thequasielastic contribution was obtained by substracting the transverse acoustic phonon andHuang scattering contributions. There is thus a maximum of quasielastic scattering around(2.27, 1.73, 0)5 and shows that the Jahn-Teller polarons are still dynamically correlatedat high temperatures (360 K). The correlation length of these correlations is estimated tobe ∼ 12A. From the fits of the elastic and quasielastic contribution they basically definedthree regimes.

For T > T∗ = 310K the Jahn-Teller correlations with ~qCE are dynamic and fluctuatewithin a lifetime on the order of femtoseconds (60 fs at 360 K). These CE correlationsare determined to be short-ranged (12 A). The width Γ of the quasielastic peak decreaseslinearly between 460 K and 310 K and indicates that a continious freezing transition istaking place. In the neutron energy scans, a transverse acoustic phonon mode is present

5Argyriouet al. found that this peak has a weak dependence on the composition. For x = 0.38 themaximum was observed for (2.27, 1.73, 0). They argue that given the breadth of the scattering of theseshort-range correlations, the shift away from (1/4, 1/4, 0) is not significant.

28

q=(2.25, 1.75, 0)

b1)

b2)

b3)

a)

Figure 15: (a) Neutron energy scans at ~q = (2.75, 1.75, 0) for various temperatures (Tc = 114K). Thepeak centered around 0 meV was fitted with an Gaussian for the elastic and a lorentzian (solid line) forthe quasielastic contribution. The quasielastic contribution displays a strong temperature-dependence asseen from the width of the fit. The peak centered around 11 meV corresponds to the [110] transverseacoustic phonon. (b) Temperature-dependence of the elastic (b1) and quasielastic (b2) intensity. (b3)The quasielastic width Γ of the Lorentzian fit. (c) Color contour map of the quasielastic scattering at360 K. The map was obtained by substracting the transverse phonon mode and the Huang scatteringcontributions. From Argyriou et al.. [38]

at 11 meV.For Tc < T < T∗ = 310K the polaron correlations are static. This is indicated by an

increasing elastic component below 310 K and a concomitant decreasing quasielastic com-ponent. The short-range spatial correlations (the width Γ is constant) do not change andthis indicates a freezing of the polaron correlations.

For T < Tc the polarons melt and the diffuse scattering indicates a regime in whichtransverse optic phonons dominate. The latter was shown by temperature-dependent neu-tron energy scans (not shown) at ~q = (2.25, 1.75, 0). The transverse optical phonon wasvisible at 110 K and 7 K at 21.2 and 22.5 meV respectively and was found to disappearfor T > Tc.

There is thus a coexistence of the ~qL = (0.3, 0, 1) and ~qCE = (1/4, 1/4, 0) superlatticereflections in the paramagnetic phase, which exhibit different correlation lengths and times.The ~qL correlations are mainly elastic (τ ∼ 1ps) in nature, as determined from neutronenergy scans. Whereas the CE correlations were found to be dynamic above 310 K andfreeze below this temperature, akin to a glass transition. The latter observation suggeststhat the paramagnetic state arises from inherent orbital frustrations, which inhibit theformation of a long-range charge-ordered and orbital ordered state, like in the x = 0.50doping level.

In summary, x-ray and neutron scattering experiments on the x = 0.40 doping level havefound the existence polarons at higher temperatures, which trap the electrons above Tc and

29

drive the system insulating. For decreasing temperatures the polarons become correlatedwithin ξ = 26A and form an incommensurate Jahn-Teller coupled charge density. BelowTc double-exchange favors the delocalization of the charge carriers and the polarons andthe short-ranged correlations melt.

The magnetostriction data for this doping level find an increased striction along the c-axis and a decreased striction along the a- and b-axis. This corresponds to an elongationof the octahedra in the ferromagnetic state. In the light of the structure of the polaroncorrelations, these striction data can also be interpreted as a decreased distortion along thea-axis. This would inhibit the formation of the in-plane polaron correlations, consistentwith the metallic ferromagnetic state.

30

6 Charge and orbital ordering for LaSr2Mn2O7

In the x = 0.50 doping level, there are equal amounts of Mn3+ and Mn4+ ions and it wasalready recognized by Goodenough [40] in 1955 that in this situation, a charge ordering ofthe Mn3+ and Mn4+ might occur. In the model Goodenough proposed, the checkerboardordering of the charges is accompanied by a zigzag orbital ordering of the dx2−r2 and dy2−r2

orbitals. This charge ordering and staggered ordering of the orbitals is called a CE-typeordering. The staggered orbital ordering can in addition give rise to a CE-type magneticordering, in which there is ferromagnetic coupling between the filled Mn3+ and emptyMn4+ eg d-orbitals (double-exchange interaction) and antiferromagnetic coupling betweenthe empty eg orbital (90o rotated w.r.t the filled eg-orbital) of the Mn3+ ion and the emptyMn4+ eg orbital (super-exchange interaction). The charge and orbital ordering is thuscalled a CE-type ordering and the magnetic ordering associated with it, is termed theCE-type antiferromagnetic ordering.

6.1 Charge and orbital ordering

This proposed charge and orbital ordering was indeed confirmed by several groups, whichobserved superlattice reflections via electron diffraction [41][42]. The electron diffractionpatterns as found by Li et al. [41] are shown in figure 16. At room-temperature, no super-

Figure 16: [001] zone-axis electron diffraction of LaSr2Mn2O7 obtained at (a) 300 K and (b) 110 K. (c)Sometimes two sets of superlattice reflections are observed, indicated by ~q1 and ~q2, they arise from thepresence of 90o twin domains. (d) Intensity of the Bragg and superlattice reflections as a function oftemperature. From Li et al. [41]

lattice reflections are observed and the diffraction pattern reflects the tetragonal structure(a). Upon cooling, superlattice reflections with ~q = (1/4, 1/4, 0) become visible just below∼200 K (b) and with decreasing temperature the intensity of the superlattice reflectionsincreases (d). Occasionally two sets of superlattice reflections were observed (c), indicatedby ~q1 and ~q2. The second set of reflections was considered to originate from twin domains,

31

c)

d)

Figure 17: Temperature-dependence of (a) the in-plane resistivity ρab and out-of-plane resistivity ρc.(b) Magnetization with H=10 kOe // ab (c) Integrated intensity of the 7

4 ,− 14 , 0 and 7

4 , 14 , 0 superlattice

reflections as measured with x-ray diffraction. The closed triangles in (a) and (b) indicate the onset of thecharge ordered phase. (d) Proposed charge and orbital ordering in LaSr2Mn2O7. The dx2−r2 and dy2−r2

orbitals order in a zigzag fashion. The ordering wavevector ~q = (1/4, 1/4, 0) is indicated by the red arrow.From Kimura et al. [42]

where the superlattice vectors were rotated by 90o with respect to each other. Indeed twindomains were previously observed in the infinite [43] and one-layered manganites [44]. Sim-ilar reflections were observed by Kimura et al. and they found that the reflections alsodisappear again for lower temperatures. Their temperature-dependence of these reflections,measured with x-ray diffraction, are shown in figure 17 together with the resistivity andmagnetization data. As one can see, the sharp increase in the in-plane resistivity and ananomaly in the magnetization can be related to the onset of superlattice reflections.

Li et al. and Kimura et al. have interpreted the superlattice reflections with ~q =(1/4, 1/4, 0) as a charge and orbital ordering. In which the Mn3+ and Mn4+ ions orderon alternate sites and is accompanied by an zigzag orbital ordering of the dx2−r2 anddy2−r2 . This is more clearly illustrated in figure 17(d). Such a CE-type charge and orbitalordering was indeed originally proposed by Goodenough [40] for the half-filled cubic per-ovskite (n =∞). Similar superlattice reflections were also in fact earlier observed for thesingle-layered manganite La0.5Sr1.5MnO4 [44] and were interpreted as an orbital ordering.Murakami et al. [45] performed a direct x-ray observation of the orbital ordering andshowed that there is a concomitant charge and orbital ordering at TCO = 220 K.

A more detailed study of this doping level was performed by Argyriou et al. [46], usingneutron and x-ray diffraction. In the diffraction data they also observed the superlattice

32

reflections with ~q = (1/4, 1/4, 0), consistent with the data from Li et al. and Kimuraet al.. From the width of the superlattice reflections they estimate the coherence lengthto be hundreds to thousands of angstroms. To gain a better understanding of the crys-tallographic details of this charge-ordered phase, they performed single-crystal neutrondiffraction. From a detailed study of data they found that two distinct Mn sites could bedistinguished. The different sites arise from a structural distortion, in which the Mn(1)(purple) site has two short (1.923 A) and two long (1.962 A) in-plane Mn-O bonds andthe Mn(1’) (pink) site has two similar pairs of in-plane Mn-O bonds (1.918 and 1.903 A).The Mn(1) site thus corresponds to an in-plane distortion of the MnO6 octahedra dueto the occupation of the dx2−r2 and dy2−r2 orbitals. It can thus be viewed as an in-planeJahn-Teller distortion which traps the charge. The MnO6 octahedra are in addition co-operatively distorted, e.g the nearest Mn3+ sites are distorted in a different direction (a90o rotation). Such an in-plane elongation was in fact also found in the x = 0.40 dopinglevel by Campbell et al. [36] as discussed in the previous chapter. The crystal structureas found by Argyriou et al. is shown in figure 18. Due to the presence of two distinct Mnsites, a orthorombic supercell of a0 ∼

√2a, b0 ∼ 2

√2a, c0 = c rotated 45o with respect to

the tetragonal parent cell, can be identified.

Figure 18: The crystal structure of the charge ordered state of LaSr2Mn2O7. The black dotted linesindicate the tetragonal parent unit cell, while the thin green lines show the

√2a× 2

√2a× c orthorombic

supercell. The arrows show the displacements along the a0 direction of the atoms away from the parenttetragonal unit cell. The various Mn-O bond lengths are also shown in the figure and make the two distinctMn(1) (purple) and Mn(1’) (pink) sites. From Argyriou et al. [46]

Such a localization of the charge and an associated orbital ordering, as seen from theMn-O lengths, of the in-plane dx2−r2 and dy2−r2 orbitals is indeed consistent with the CE-type charge and orbital ordering as proposed by Li et al. and Kimura et al.. In their modelhowever there is a discrete ordering of the charges, consistent with the discrete chargeordering as proposed by Goodenough. According to Argyriou et al. however such an ionicmodel of charge order is oversimplified and a degree of covalency between the Mn d-orbitalsand O p-orbitals is not unlikely. This conclusion was based on the bond valence sums of

33

the two distinct Mn sites, which were found to be more similar than compared to the cubicperovskites. They thus argue that the charge ordering in the layered manganites is lesspronounced compared to the cubic perovskites La0.5Ca0.5MnO3[47]. This partial chargeordering is also reflected in the partial obital ordering. From the Mn-O bond lengths theyestimate that the charge is unequally shared between the dz2−r2 and dx2−y2 and that in anionic charge ordering picture the former orbital would be underoccupied and the latter isnot completely empty.

6.2 Competition between CE-type and A-type antiferromagneticphases

Argyriou et al. have thus found a CE-type charge and orbital ordering in the LaSr2Mn2O7,analogue to the ordering in the half-filled 3D perovskites. The staggered orbital orderingis however associated with a CE-type magnetic ordering, where there is staggered ferro-magnetic interaction between filled Mn3+ and empty Mn4+ (double-exchange) and antifer-romagnetic interaction (superexchange) between empty Mn3+ and Mn4+ d-orbitals. Thisspin, charge and orbital ordering pattern of the CE-antiferromagnetism type is shown infigure 19.

Figure 19: Spin, charge and orbital ordering, exhibiting the CE-type antiferromagnetic ordering arisingfrom the in-plane zigzag ordering of the dx2−r2 and dy2−r2 orbitals on the Mn3+ ions. The Mn4+ areindicated by the closed circles. Figure from Salomon et al. [48].

Up to now, only experimental evidence of the charge and orbital ordered phase havebeen provided. Argyriou et al. expected based on the orbital ordering patterns to find a CE-type antiferromagnetism in this charge and orbital ordered phase. To verify this assumptionthey performed low-temperature neutron diffraction measurements. They however foundmagnetic reflections indicative of a type-A antiferromagnetic ordering (FM MnO2 sheetsand AFM within and between bilayers see chapter 3). Such a A-AFM is usually associatedwith a ferromagnetic ordering of the dx2−y2-orbital and is inconsistent with the CE-typemagnetic ordering (ordering of the dx2−r2 and dy2−r2 orbitals). Argyriou et al. solve

34

this apparent contradiction by reporting the coexistence rather than the coincidence ofthe CE-type charge-ordered phase and the type-A antiferromagnetic phase. They foundthat the A-type AFM ordered phase starts to develop below TN = 170K and forms atthe expense of the charge-ordered phase. At T = 125K there are nearly equal amountsof the phases and at T = 50K the antiferromagnetic phase has entirely consumed thecharge-ordered phase, forming the type-A AF consistent with the magnetic phasediagramin chapter 3. Argyriou et al. thus do not find magnetic reflections representative of theCE-type AFM, but assume this magnetic ordering from the orbital ordering. Their resultsand temperature-dependence are also consistent with the data from Chatterji et al. [49],who found an increasing A-AFM component below 170 K and a melting charge-orbitalordering at 100 K. They also did not find the associated CE-type magnetic ordering.The data from Argyriou et al. are also consistent with the conclusions of Kimura et al.who found a disappearance of the superlattice reflections below 100 K and argued thatthe charge-ordered phase is altered to the layered antiferromagnetic state with loweringtemperature.

There is thus a static charge-ordered phase between (210K > T > 170K) and a regionbetween (170K > T > 50K) in which there is a coexistence between the CE-type chargeand orbital ordering and type-A AF phases. It is thus not a real electronic phase separation,as the charge is essentially localized in both the CO and A-AF phases. Below 50 K thereis only the type-A antiferromagnetic phase.

In summary, electron and single-crystal neutron diffraction have observed the presenceof charge and orbital ordering in the x = 0.50 bilayer manganites. Such a charge and orbitalordering was proposed by Goodenough to be accompanied by a CE-type antiferromagneticordering of the spins. Argyriou et al. however found a charge-ordered phase between(210K > T > 170K) and a region (170K > T > 50K) in which there is a coexistencebetween the CO and type-A AF phases. They have also determined the crystal struc-ture associated with the charge-ordered phase and found the presence of two distinct Mnatoms. The different Mn sites arise from a lattice distortion, in which the Mn4+-like is lessdistorted than the Mn3+- like site. The structural distortion is produced by a Jahn-Tellerdistortion, which is induced by the occupation of the in-plane dx2−r2 and dy2−r2 orbitals. Itis however not entirely clear what drives the orbital ordering in the half-filled manganites.Calculations on the antiferromagnetic perovskite LaMnO3 with a 3d4 however have shownthat a cooperative Jahn-Teller distortion is the dominant source of the orbital ordering,rather than the superexchange interaction between orbitals on different sites. Althoughthe 3d4 is different from the half-doped 3d3.5 manganites (no charge ordering and no CE-type AFM), it is not unlikely that the cooperative Jahn-Teller distortion drives the orbitalordering in LaSr2Mn2O7 as the electronic energy effect of the superexchange interaction isrelatively small compared to the effect of the structural distortion [50][51].

35

7 Raman scattering

Raman scattering is a sensitive probe to structural changes and is thus a useful techniqueto study the complex interaction between the charge, orbital and lattice degrees of freedomin the manganites. In Raman scattering, light is scattered by the system and the main partof the photons will be elastically scattered (Rayleigh scattering), but a small fraction of thephotons (typically a factor of 106) will have absorbed or lost some energy to the system.These inelastically scattered photons will thus respectively have a frequency higher (anti-Stokes line) or lower (Stokes-line) than the incident photons. The difference in energy6

between the incident photon and the inelastically scattered photon is equal to the energyof the vibration of the system, (e.g the lattice vibrations of the system) [32]. For the anti-Stokes line to be visible in the spectrum, the elementary excitation for example phonons,has to be already excited in the solid. Whereas the former can induce the elementaryexcitation with the loss in energy. Raman scattering thus involves the inelastic scatteringof light and is a very sensitive probe to the lattice vibrations of the system. In this chapterthe few Raman scattering experiments performed on the x = 0.50 and x = 0.40 dopinglevels will be discussed. The first section will discuss the former doping level and thecharge and orbital ordering as probed by Raman scattering. The latter doping level willbe discussed in the second section.

7.1 Charge-ordering in the x = 0.50 doping level

It was already described in the previous chapter that in the x = 0.50 doping level a chargeand orbital ordering takes place between (170K 6 T 6 210K). Whereas for lower tem-peratures the type-A AFM phase starts to develop at the expense of the latter and hasformed a single phase below 100 K. The formation of this charge and orbital ordered phaseand the A-type antiferromagnetic phase has stimulated Yanamoto et al. [52] and Romeroet al. [53] to study the possible novel interplay of the charge and orbital ordering withthe lattice dynamics. The structural changes associated with this CO were also studied bysingle-crystal neutron diffraction and were discussed in chapter 6.1. As Raman scattering isa very sensitive probe to local or dynamical structural changes, it is a good complementarystudy to the neutron diffraction study.

In figure 20 (a-c) the Raman spectra as measured by Romero et al. [53] are shown.For the tetragonal crystal structure [14/mmm D17

4h] one expects 4A1g + B1g + 5Eg phononsto be seen. They are respectively observed in the following scattering geometries: x′x′ ⇒A1g + B2g, x′y′ ⇒ B1g + A2g and zz ⇒ A1g. Where z is parallel to the c-axis and x′ andy′ are along the Mn-O bonds in the MnO2 plane. For T > TCO in figure 20(a) the spectraonly exhibit the 4A1g + B1g + 5Eg phonons as expected from the tetragonal symmetry andthere are thus no structural distortions.

As T is lowered below TCO however, new peaks (indicated by the arrows) are seen inthe x′x′ and x′y′ geometries, but not in the zz geometry. As these new peaks are not

6The difference in energy between the incident and scattered photon is expressed in wavenumber cm−1

and is equal to 0.1239 meV

36

a)

b)

c)

d)

e)

f)

g)

in-plane out-of-plane

Figure 20: (a-c) From Romero et al. [53] (a) For T > TCO only the expected phonons of the tetragonallattice structure of LaSr2Mn2O7 are observed. (b) For T < TCO new phonon peaks in the (x’x’) and (x’y’)spectra are observed. They are activated by the charge and orbital ordering within the MnO2 plane. (c)These new modes are absent in NdSr2Mn2O7. (d-g) From Yamamoto et al. [52]. (d,f) In-plane andout-of-plane spectra of La0.5Sr1.5MnO4 in the paramagnetic insulating (290 K) and charge-orbital-orderedphase (10 K). (e,g) In-plane and out-of-plane spectra of LaSr2Mn2O7 in the paramagnetic insulating (290K), charge-orbital-ordered (120 K) and type-A AFM phase (10 K).

expected from a symmetry analysis of the tetragonal unit cell, they are attributed byRomero et al. to arise from a charge and orbital ordering, which breaks the tetragonalsymmetry. The notion of the charge and orbital ordering was also based on the spectrafrom NdSr2Mn2O7, in which the charge-ordering is quenched by the substitution of La byNd. The suppression of the charge-ordering was attributed to the relaxation of the in-plane Jahn-Teller distortion [54]. Also the similarity between the spectra of LaSr2Mn2O7

and La0.5Sr1.5MnO4 [52], in which the latter was known to exhibit charge-ordering fromdiffraction measurements [44] [45], was supportive of this conclusion.

The Raman spectra for the LaSr2Mn2O7 and La0.5Sr1.5MnO4 are actually measured byYanamoto et al. [52] and are shown in figure 20 (d-g). As one can see in the Raman spectraof La0.5Sr1.5MnO4, two peaks (532 and 637 cm−1) in the x′x′ and two peaks (532 and 693cm−1) in the x′y′ spectra arise in the low-temperature charge-ordered state. As they arenot expected in the tetragonal I4/mmm symmetry, the authors attributed these peaks tophonon modes that are activated by the lattice distortion upon charge-ordering. From asymmetry analysis of these phonon modes, the authors estimated that charge-ordered phaseexhibits a cooperative Jahn-Teller distortion, similar to the CE-type charge and orbitalordering in La0.5Sr0.5MnO3 and LaSr2Mn2O7 and discussed in the previous chapter. Usingthis CE-type ordering unit cell, which is orthorombic, they determined these phonon modes

37

to correspond to breathing modes and in-plane Jahn-Teller distortions. These modes areillustrated in figure 21. The peaks at 532 and 637 cm−1 in the (x′x′) spectrum correspondrespectively to the JT mode (2) and breathing mode (1). Whereas the peaks at 532 and693 cm−1 in the (x′y′) spectrum correspond to the JT mode (1) and breathing mode (2).The breathing mode (1) and JT mode (1) were considered by the authors to be activatedby the charge-ordering alone and the breathing mode (2) and JT mode (2) to arise fromthe concomitant charge and orbital-ordering.

As one can see, the in-plane spectra of LaSr2Mn2O7 in the charge-ordered state at 120K show a great resemblance to that of La0.5Sr1.5MnO4. The charge and orbital orderingpattern of the bilayer is thus expected to be the same as for the single-layer manganite.Using the same symmetry arguments as for single-layered manganite, they assigned the502 and 623 cm−1 peaks in the (x’x’) spectrum to the JT mode (2) and breathing mode(1). The peaks at 505 and 671 cm−1 in the (x’y’) spectrum are assigned to the JT mode(1) and breathing mode (2). Two major modes at 466 and 582 cm−1 are observed in the zzspectra (allowed within the original I4/mmm tetragonal structure) and correspond to thestretching mode (1) and (2), shown in figure 21 (b) As the intensities of the Jahn-Tellermodes are always larger than the breathing modes, the authors argued that the majorlattice distortion is the Jahn-Teller distortion.

Yamamoto et al. have thus shown the presence of charge and orbital ordering transitionsin the LaSr2Mn2O7 and La0.5Sr1.5MnO4 compounds via Raman scattering. They observedfour phonon modes below TCO, which were not expected in the I4/mmm tetragonal unitcell and were not seen in the high temperature state. The presence of these phonon modescould then be explained when an orthorombic unit cell, consistent with a CE-type chargeand orbital ordering, was assumed. From the symmetry of the phonon modes, they could

Mn3+

Mn4+

Breathing mode (1)Ag

SL: 637 cm-1BL: 623 cm-1

J-T mode (1)B1g

SL: 532 cm-1BL: 505 cm-1

J-T mode (2)Ag

SL: 532 cm-1BL: 502 cm-1

Breathing mode (2)B1g

SL: 693 cm-1BL: 671 cm-1

MnO

Stretching mode (1)Ag

BL: 466 cm-1

Stretching mode (2)Ag

BL: 582 cm-1

Figure 21: (a) In-plane Raman active modes in the charge-orbital-ordered phase. The observed energiesof the Raman peaks are indicated under the corresponding modes for the single-layer (SL) La0.5Sr1.5MnO4

and bilayer (BL) LaSr2Mn2O7. (b) Out-of-plane Raman modes for LaSr2Mn2O7. From Yamamoto et al.[52]

38

assign the peaks to correspond to breathing and in-plane Jahn-Teller modes, where thelatter, arguing from the major intensity, is probably the dominant lattice distortion. Theythus correlated the concomitant charge and orbital ordering with breathing mode andin-plane Jahn-Teller distortions.

Up to now no remarks have been made regarding the absence of the four phonon modesat low temperatures. This would according to Yamamoto et al. correspond to the onset oftype-A antiferromagnetism and the disappearance of the charge-ordered phase. Which wasalso found from the temperature-dependence of the superlattice reflections, as discussedin the previous chapter. In the Raman spectra measured by Romero et al. however, thesame four phonon peaks of Yamamoto et al. are observed in the (x’x’) and (x’y’) spectraat 5 K. This would thus be inconsistent with Yamamoto’s data. Romero et al. attributethis discrepancy in the data to a difference in doping level, in which Yamamoto’s sampleswere considered to have a lower doping than x = 0.50.

The latter conclusion was also based on measurements by Romero et al. on the x = 0.40doping level, in which they observed a similar quenching of the phonon peaks at lowertemperatures.

7.2 Charge-order fluctuations in the x = 0.40 doping level

Romero et al. [53] performed magneto-Raman scattering spectroscopy (application of amagnetic field) to study the competition between charge, orbital and ferromagnetic inter-actions. It was observed in the x = 0.50 doping level that the charge-orbital-ordered peaksare unstable against ferromagnetic correlations as demonstrated by the disappearance ofthe peaks upon application of a magnetic field in figure 20 (b). Yamamoto et al. [55]alsoperformed Raman scattering on the x = 0.40 doping level and found charge and orbitalordering peaks that were similar to the peaks in the spectra of the x = 0.50 doping level.As the temperature was lowered below Tc, they also found that the peaks would disappear.

To gain a better understanding of the dynamics of the melting of these phonon peaks,Romero et al. also measured samples with doping levels of x = 0.50 and x = 0.40. Theyindeed found similar phonon peaks in the x = 0.40 and x = 0.50 doping levels, where thespectra of the latter were already shown in figure 20. The two phonon peaks at 514 and623 cm−1 in the (xx) spectra of La1.2Sr1.8Mn2O7 are shown in figure 22. These new modeswere termed Oab phonons by Romero et al. and were considered to arise from an orbitalordering. The authors also found a new peak at 240 cm−1, which was attributed to acharge ordered activated Mn phonon and was also observed in the charge-ordered phaseof the x = 0.50 samples. The Mn phonon is induced by the Mn3+ −Oab −Mn4+ ordering,which breaks the symmetry and leads to the observed Raman intensity. The Oab phononpeaks are similar to the peaks observed in the spectra of the x = 0.50 doping level fromYamamoto et al. and were previously assigned to breathing modes and JT modes, eitheractivated by charge ordering or concomitant charge and orbital ordering.

To study the competition between charge, orbital and ferromagnetic interactions, Romeroet al. performed temperature-and field-dependent Raman scattering measurements. It wasobserved that the phonon peaks are quenched by either lowering the temperature below

39

a) b) c)

d)

e)

Figure 22: Raman scattering of La1.2Sr1.8Mn2O7 (a) as a function of temperature and (b) and (c) in thepresence of a magnetic field. The charge (dashed arrow) and orbital (solid arrow) correlations are seen tomelt by lowering T or by increasing H. The Raman response function Imχ(ω, T ) is the measured Ramanintensity I(ω, T ) upon correction by the Bose-Einstein thermal factor. (d,e) Temperature and magneticfield dependence of the normalized intensity of the (d) CO activated Mn and (e) orbital ordering inducedOab phonons. From Romero et al. [53]

Tc figure 22(a) or by applying a magnetic field figure 22(b) and (c). The authors thusargue that the melting of the phonon modes is driven by the ferromagnetic alignment ofthe spins. The temperature-dependence of the intensity of the charge-ordered activatedMn phonon peak and the orbital ordered induced Oab phonon are displayed in figure 22(b) and (c). As one can see, the charge order sets in near 270 K and grows with decreasingT and has a maximum just below Tc at T ≈ 100K, whereas the orbital ordering existsin the entire paramagnetic insulating state and has a maximum just above Tc. It mustbe noted that the temperature-dependence of the orbital ordering is remarkably similarto that of the short-range polaron correlations discussed in chapter 5. Where the latterwas shown to exhibit an orbital ordering of the in-plane dx2−r2 and dy2−r2 orbitals, due tothe incommensurate Jahn-Teller coupled charge density wave. The charge-order howeverseems to persist even in the low-temperature ferromagnetic state and the authors arguethat these correlations develop into a collective charge density wave excitation well belowTc. The latter would be consistent with angle-resolved photoemission data by Dessau etal. [56], in which the presence of a pseudogap was reported and was considered to arisefrom a Jahn-Teller coupled charge-density wave. These polaron correlations were howeverreported to melt below Tc [30]. The indications of these charge order fluctuations andthe development into a collective CDW in the ferromagnetic state would thus support the

40

charge-density wave pseudogap opening scenario.Romero et al. have thus shown the existence of orbital ordering in the paramagnetic

insulating state and their similar temperature-dependence with the short-range polaroncorrelations. They have also observed the persistence of charge order correlations in theferromagnetic metallic state and their possible relation to the formation of a pseudogap.

In conclusion, Raman scattering is a good complementary study to the x-ray and neutronscattering experiments and have confirmed the presence of charge and orbital ordering inthe x = 0.50 doping level. The nature of these phonon peaks, activated by a break ofsymmetry due to charge ordering, was ascribed to Jahn-Teller and breathing modes, ei-ther activated by charge order alone or concomitant charge and orbital ordering. Basedon the similarity between the phonon peaks of the x = 0.50 and X = 0.40 samples,Romero et al. conclude that there is charge and orbital ordering in the latter doping level.The temperature-dependence of the intensity of the orbital ordering peaks is similar tothe T-dependence of the short-range polaron correlations. In which in the latter, also acharge and orbital ordering takes place via the Jahn-Teller coupled charge density wave.The CO fluctuations were however found to persist in the ferromagnetic state and mightthus be related to the formation of a pseudogap, as found by angle-resolved photoemissionmeasurements.

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8 Optical studies

Up to now the literature has mainly focussed on the coupling of the lattice to the charge andorbital degrees of freedom. In this chapter the focus will be on the charge carriers, whichin the end determine the transport properties of these CMR manganites. The electronicstructure as studied by optical measurements for various doping levels will be discussed.The angle-resolved photoemission studies performed on the CMR-exhibiting compoundswill be discussed in detail elsewhere [57]. In the first section the optical spectra at room-temperature for various doping level dependence will be described. In the second sectionthe optical spectra for the CMR-exhibiting doping levels will be discussed in detail.

8.1 Optical spectra: doping dependence

Ishikawa et al. [58] measured the optical conductivity spectra7 of several doping levels atroom-temperature and these in-plane and out-of-plane spectra are shown in figure 23. Atthis temperature all compounds are a paramagnetic insulator and differences in magneticstructures are thus not taken into account. Both the in-plane and out-of-plane spectraexhibit a broad peak at 4 eV and was assigned to the charge-transfer (CT) type transitionbetween the O 2p and Mn t2g-like (down-spin) states. Whereas the spectrum below 3 eVis dominated by intra- and interband transitions of the O 2p and Mn eg hybridized states.In the in-plane spectra, a broad peak centered around 1 eV is observed and the mechanism

Figure 23: Doping dependence of the room-temperature polarized optical conductivity spectra forLa2−2xSr1+2xMn2O7 (x=0.30, x=0.40 and x=0.50). (a) In-plane (E⊥c) and (b) out-of-plane (E//c)spectra. Inset in (a): doping dependence of the effective number of electrons Neff at 3.0 eV for the in-planespectra (closed circles) and out-of-plane spectra(open circles). Double circles is the total number of Neff

and is the sum of the in-plane and out-of-plane Neff . From Ishikawa et al.[58]

7The optical conductivity is obtained by a Kramers-Kronig analysis of the polarized reflectivity data.

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which produces the high peak energy is up to now not entirely clear. It will however bediscussed in more detail in the following paragraphs. The spectral weight of this peakincreases and the peak energy decreases with increasing doping level x. Whereas in theout-of-plane spectra, a broad peak around 1.8 eV is observed and decreases with increasingx. As the out-of-plane spectral weight is produced by the occupance of the dz2−r2 orbital,the decreasing spectral weight with decreasing electron concentration indicates a changeof orbital occupancy. The latter is also observed in the effective number of electrons Neff

as a function of x, displayed in the inset of figure 23 (a). Neff increases for the in-planeand decreases for the out-of-plane spectra with increasing x, whereas the sum of the twois nearly x-independent. There is thus a change in eg orbital occupancy from dz2−r2 todx2−y2 as the doping increases. This is also consistent with the magnetostriction data andmagnetic Compton scattering results, as discussed in chapter 4.2.

8.2 Optical spectra: temperature-dependence for x = 0.30 andx = 0.40 with metallic ground state

In figure 24 (a) and (b), the temperature-dependence of the optical conductivity of thex = 0.30 and x = 0.40 doping levels is shown [58][59]. There is almost no temperature-dependence for the out-of-plane spectra, whereas for T < Tc the in-plane spectra show alarge spectral change for the energy range of 0-3 eV. In the paramagnetic state a broadpeak centered around 1 eV is observed and forms a pseudogap structure (≈ 0.2eV). As thetemperature is lowered close to Tc the peak height increases and the gap feature becomesclearer and is thus consistent with the insulating state above Tc. When the temperatureis lowered below Tc, large changes in the spectral weight in the energy region of 0-3 eVare observed. The most important observation is that the transferred spectral weight doesnot form the Drude-like peak centered around ω = 0, as expected from the simple double-exchange model, but instead forms a peak at 0.4 eV. The absence of the expected Drudepeak for metallic systems is a strong indication that the charge dynamics in the metallicstate is not conventional. This was also signalled by the high resistivity of the metallicstate (ρab ∼ 2 × 10−3Ω · cm), which was found to be of the inverse of Mott’s minimumfor metallic conductivity. Similar spectral weight change was observed for the analogousthree-dimensional perovskites. Although in the latter compounds, the spectral weight wastransferred to a broad mid-infrared absorption band and a finite Drude-like peak at 10 K.These peaks were respectively interpreted as the incoherent and coherent absorption bandof a large lattice polaron [61] [62].

8.2.1 Theoretical models

As the T-dependent spectral weight change of the layered manganites is very similar tothat of the 3D perovskites, Ishikawa et al. explains the low-energy conductivity spectrumby using the 3D analog.

In the latter compound, the spectra were interpreted with a simple double-exchangemodel, in which the Hund’s rule coupling energy SJH between the eg conduction electrons

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c)

Figure 24: (a,b) Temperature-dependence of the in-plane (E⊥c) and out-of-plane (E//c) optical conduc-tivity spectra for La2−2xSr1+2xMn2O7 of (a) x = 0.30 and (b) x = 0.40. Solid lines represent the spectraat 10, 90 K (x = 0.30) or 10, 130 K (x = 0.40). From Ishikawa et al. [58]. (c) Schematic illustration of thevariation of the spin-polarized eg conduction band in the framework of the double-exchange mechanism.From Okimoto et al. [61]

and t2g core-like spins exceeds the one-electron band width of the eg electrons. WhereS = 3

2is the magnitude of the t2g local spin. The t2g electrons are treated like core-like

electrons and the bandwidth is thus not taken into account in this model. When the SJH

exceeds the eg bandwidth, the authors presume that in this case the eg bands should splitup into two bands, which are then separated by SJH . These two bands, are then termedthe exchange split bands. This schematic model is illustrated in figure 24 (c). The lowerexchange split band is in the high-temperature paramagnetic state equally composed of aspin-down (left) and a spin-up (right) subband. Although it is not clear from the articleof from the figure, the upper exchange spit band should thus be composed of a spin-up(left) and spin-down (right) subband, as SJH is the penalty which has to be paid when thespins are aligned parallel. As SJH exceeds the eg electron bandwidth, the subbands withantiparallel spin will spit up in two separate bands and as mentioned earlier, termed theexchange split bands.

There is thus a gap-like feature in the conductivity spectra due to the inter-band tran-sitions between the exchange split bands. As T is lowered below Tc however, the double-exchange mechanism aligns the t2g spins parallel and the spin sub-bands are no longerequally occupied. The density of states (DOS) for the lower (upper) up-spin band willincrease (decrease), whereas the DOS of the lower (upper) down-spin band will decrease(increase). At T = 0 the exchange split bands will be completely spin-polarized, betweenwhich optical transitions are no longer allowed. The spectral weight of the inter-band tran-

44

sitions will thus decrease and the spectral weight of the intra-band transition within thelower up-spin band will increase, explaining the spectral weight change of the peak fromhigher energy to lower energy.

It must be noted that in this model a SJH ≈ 2eV was assumed for the layered com-pounds, consistent with the bandstructure calculations by Hamada, who used a local spindensity approximation with an on-site coulomb repulsion of U= 2 eV [63].

The temperature-dependent spectral weight change can thus partly be explained bya change in the character of the conduction-electron related transitions, in which in theparamagnetic state inter-band transitions between the exchange spit bands and in theferromagnetic state intra-band transitions dominate.

Although the overall temperature-dependence can thus be explained by this model,some features deserve a little more attention. It was already observed that the low-energy< 1eV spectral weight increases with decreasing temperature and the 1 eV peak shiftstowards lower energies to form a broad peak around 0.4 eV. This broad mid-infrared peakwas interpreted by Ishikawa et al. as the incoherent part of the spectrum. Whereas thecoherent part, the Drude peak, is suppressed. The formation of a broad peak in the mid-infrared region and the shift with temperature was further interpreted by the authors asa signature of small-polaronic conduction in the system. When they fit this peak withina small-polaron model however, the shape of the low-temperature 0.4 eV peak could notbe reproduced. Whereas they used a model in which the electrons couple to a dynamicalJahn-Teller lattice distortion, with a temperature-dependent coupling strength, they couldqualitatively reproduce the spectra down to the transition temperature. This model wasin fact proposed by Millis et al. [14] and was discussed in section 2.6. When a couplingstrength of λ ≈ 1.08 was used, the observed gradual shift of the 1 eV peak around and aboveTc to a lower energy (down to 0.4 eV) with increasing spectral weight, could qualitativelybe reproduced. The peak shift is then explained by a decrease of the effective electron-phonon coupling due to the increase of kinetic energy with decreasing temperature. Themodel is however not quantitative about the shape of the peak in the metallic state andother considerations, such as the collective nature of the Jahn-Teller distortion and/orshort-range orbital ordering, must be taken into account.

Lee et al. [60] have measured a similar temperature-dependence of the spectral weightchange and used a different approach than Ishikawa et al. to determine the nature of themid-infrared band. Lee et al. used a small polaron model at all temperatures to analyzethe optical conductivity spectra. They based this method on the presence of polaronsin the paramagnetic state and significant lattice distortions in the ferromagnetic state asobserved by by x-ray and neutron scattering studies [9]. They fitted the incoherent bandwith two Gaussian functions, in which the first corresponds to small polaron absorptionrelated to a nearest neighbor hopping from the Mn3+ to the Mn4+ ion. The second peak wasassigned to correspond to on-site d − d transitions. From the temperature-dependence ofthe fitted results, the authors conclude that the polaron and charge correlations grow withdecreasing temperature down to Tc and then disappear below Tc due to the ferromagneticordering. This would thus also be consistent with the x-ray and neutron scattering resultsfrom Vasiliu-Doloc et al. [30]. These short-range correlations could then explain the

45

spectral weight change above Tc and the formation of the small polaron peak in the mid-infrared region below Tc according to Lee et al. The nature of the small-polaron peak washowever not specified and a small polaron hopping mechanism is often associated with alow conductivity. The origin of the peak at 0.4 eV thus remains controversial and additionalexperiments are thus needed to gain a better understanding of its nature.

8.2.2 Incoherent charge dynamics in the low-energy region

In this paragraph the low-energy region will be discussed in more detail. In figure 25the low-energy optical conductivity spectra is shown for x = 0.40. The four peaks corre-spond to optical phonon modes and are almost temperature-independent, indicating thatthe tetragonal lattice symmetry is indeed preserved in the low-temperature states. Thisin contradiction to the low-temperature Raman spectra in which the Mn phonon peak,indicative of charge ordering, has remnant intensity in the ferromagnetic state. When the

Figure 25: Temperature-dependence of the in-plane optical conductivity in the low-energy region (0-0.1eV). Open circles represent the value of the dc conductivity at the respective temperatures. The dashedlines are guides to the eyes. From Ishikawa et al. [59].

optical conductivity spectra are extrapolated to (ω → 0), the zero-frequency conductivityσ0 should in theory coincide with the dc conductivity value, from resistivity measurements.The zero-frequency extrapolation appears however to be higher than the σdc values. It isinteresting to note that at low temperatures (e.g 50 and 10 K) the Drude-like increase ofthe optical conductivity σ(ω) is barely observed below 0.05 eV and seems to fall againbelow 0.02 eV towards the σdc value. The low-energy spectra are thus not consistent withthe Drude model at low temperatures. Several transport experiments on the x = 0.30 [22]and x = 0.40 [20] have reported the upturn of the resistivity and a T1/2-dependence of theresistivity below 50 K. It was suggested by Okuda et al. [22] that this T1/2-dependenceis characteristic of a weak-localization effect in a three-dimensional system. Such a three-dimensionality in this quasi-2D system might then come from some carrier hopping be-tween the bilayers, although the c-axis resistivity is two orders of magnitude larger thanthe in-plane resistivity.

46

Ishikawa et al. thus conclude that the carrier motion in the metallic state is governedby two energy scales: the first is due to the 0.4 eV incoherent band, which is ascribedto the dynamical lattice distortion and will relatively reduce the Drude peak. The secondenergy scale is the suppression of the Drude peak below 0.02 eV and might arise from somelocalization effect, of which the nature is not yet clear.

In conclusion, optical conductivity measurements on samples with different doping levelshave confirmed the decrease of the eg z2 − r2 orbital occupancy with decreasing electronconcentration, as discussed in chapter 4.2. In the temperature-dependent spectra of thex = 0.40 doping level, a broad band at high temperatures is observed at 1 eV and forms apseudogap feature of 0.2 eV. As T is lowered below Tc this peak shifts to lower energies andforms a broad incoherent band at 0.4 eV. The spectral weight change is thus not transferredto the expected coherent Drude-like peak for metallic systems and is thus suppressed by theincoherent band. The absence of a Drude peak thus signals that the low temperature chargecarrier dynamics are unconventional. The nature of the 0.4 eV peaks remains controver-sial and several mechanisms which include lattice distortions were proposed to explain it.The unconventional charge dynamics was also evident from angle-resolved photoemissionmeasurements [56], which found that the spectral weight was severely suppressed in themetallic state, consistent with the absence of a Drude peak. Recently however ARPESmeasurements [64][65] have observed the presence of quasiparticle-like peaks in the fer-romagnetic state and thus opens the possibility to study the electronic dispersion, thescattering rates and how electrons couple to collective excitations. The details of theseARPES measurements are discussed elsewhere [57].

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9 Phase separation

9.1 Theoretical models

Another prominent candidate for the explanation of the CMR-effect is the phase sepa-ration scenario [66]. Several computational simulations using a one-orbital model8 havedemonstrated that certain electron densities are not stable and the resulting ground stateis separated into two regions with different electron densities, which then are stable. Tobe more specific, it was found that there is a separation between hole-undoped antiferro-magnetic (AFM) and hole-rich ferromagnetic (FM) regimes [67]. This phenomenon hasbeen termed phase separation. It must be noted that in experimental techniques such asneutron scattering, a canted spin state is difficult to distinguish from a a mixed AFM-FMstate.

A more quantitative calculation is however the two-orbital model, in which two activeinstead of one active eg orbital per Mn ion is used. The orbital-degree of freedom isthus also taken into account to reproduce orbital ordering effects, which are known tooccur in the manganites [45]. As was pointed out in the previous sections, Millis et al.and Roder et al argued that the dynamical Jahn-Teller distortions and the electron-phononcoupling strength λ cannot be neglected. This electron-lattice couplings was thus also takeninto account. The phase diagram, that was calculated by using different electron-phononcoupling strengths λ, exhibited a wide range of phases, such as metallic and insulatingregimes with orbital ordering. Also the two-orbital model predicted the presence of phaseseparated doping regions in the electron-rich region for an intermediate to strong couplingstrength (λ = 1− 2) and in the electron poor region for λ = 0− 2 [68].

The macroscopic phase separation of two phases with different electron densities andthus different charges, should actually be prevented by long-range Coulomb interactions,which were not taken into account thus far. When Coulomb interactions are taken intoaccount a stable state is formed, in which clusters of one phase are embedded in the other.The competition between the attractive double-exchange interactions between the carriersand the repulsive Coulomb forces will then determine the size and shape of the clusters.A ’cluster’ which spans only one charge carrier is then termed a polaron and when severalcharge carriers are inside a common large distortion it is termed a cluster. Depending onthe size of the interactions either polarons or sizable droplets may arise as the most likelyconfiguration. Such a situation is then termed a charge inhomogeneous state (CI), and themicroscopic phase segregation is thus the net resulting effect of DE-Coulomb competition.

The phase separation scenario can then explain the CMR-effect by effectively increasingthe resistivity in the high-temperature phase, from the competing interactions betweenthe charge inhomogeneities (with AFM/FM mixed character, where carriers cannot hopbetween antiparallel spins) and ferromagnetism. The insulating state, which could notbe explained by double-exchange alone, then arises from the cluster formation, which is

8In the one-orbital model, the t2g spins are treated to be mainly localized, whereas the eg electronis mobile and uses the oxygen p orbitals as a bridge between the Mn atoms. In addition the eg orbitaldegeneracy is in the undoped limit split by Jahn-Teller distortions and a one-orbital treatment is used.

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dynamical at high temperatures. The metallic state is then obtained if the clusters growin size as the temperature is lowered and eventually reaches a limit where the percolationbetween the clusters is possible. At this stage the carriers can move over long distancesand a metallic state is formed.

9.2 Experimental evidence?

It would be interesting however to see whether these theoretical predicted tendency towardsphase separation are also observed in experimental studies. Magnetic neutron scatteringexperiments by Perring et al. [69] indeed report the coexistence of long-lived antiferromag-netic clusters within ferromagnetic fluctuations in La1.2Sr1.8Mn2O7. The antiferromagneticclusters are of the C-type AFM ordering, in which the spins are parallel to the c-axisand are AFM coupled within the MnO2 plane and ferromagnetically coupled within thebilayer. The AFM clusters have a correlation length of 9.3± 1.3A (approximately 2.4 a0)and have a relaxation time of 0.4× 10−12 s at 142 K. When the temperature is increasedto room temperature it was found that the spatial coherence was lost and the fluctuationrate increases. Whereas below Tc the AFM fluctuations rapidly decrease as the metallicand ferromagnetic state is entered. The charge carriers in the paramagnetic phase arethus hopping in a medium with mixed ferromagnetic and antiferromagnetic bonds. Theresistivity of the paramagnetic state is then explained by the traditional double-exchangemodel, in which hopping across FM bonds is allowed and across AFM is suppressed. Thisis thus consistent with the phase separation model described by Moreo et al.

Recently Rønnow et al. [70] performed atomic scale microscopy and temperature-dependent spectroscopy measurements on the x = 0.30 doping level and found evidencefor the rejection -for this two-dimensional metal- of the electronic phase separation sce-nario, in which different electronic states coexist in adjacent regions (e.g AF insulator andFM metal). This conclusion was based on the thousands of temperature-dependent spec-troscopy measurements, which were performed at different locations over 10 to 106 nm2.When they plot the zero-bias conductance (slope of the tunneling current at V = 0) as afunction of temperature, they found that all the slopes fall on a normal distribution, whichgradually shifts away from zero at from 44 K to 298 K. This normal distribution of thezero-bias conductance is shown in figure 26(a). It must be noted that the zero conduc-tance at low-temperatures is inconsistent with the bulk resistivity data, which as shown infigure 7, is metallic at low temperatures. The discrepancy between the two methods wasattributed to the fact that the out-of-plane conductance mimics the in-plane conductancein the bulk resistivity measurements. In this doping level there is antiferromagnetic bi-layer coupling below Tc and tunneling is the main channel for c-axis conductivity. Whenthe domain walls in the adjacent layers however do not coincide exactly (figure 26 (b)),there will be regions where the interbilayer magnetization is parallel, in which the tunnel-ing probability is enhanced compared to the surrounding AF correlated regions. In thelow-temperature state there is three-dimensional antiferromagnetism and the FM coupledregions will form a well-connected network. In this case the c-axis conductivity will mimicthe in-plane conductivity. As the in-plane hopping is inoperative in the STM tunneling

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a)

Figure 26: (a) Temperature-dependence of the zero-bias conductance. The open symbols where obtainedfrom the averages of thousands of I(V) spectra measured over large areas at fixed temperatures and thesolid symbols were obtained from a single I(V) curve on slowly warming. The inset shows the distributionof the conductance obtained from thousands of I(V) curves measured at individual points on layer areas atfixed temperatures. The uniform distribution thus indicates any absence of a metal-insulator separation.(b) Schematic illustration of the c-axis hopping mechanism (red line). In which the c-axis conductivitywill mimic the in-plane conductivity due to the overlapping ferromagnetic domains. From Rønnow et al.[70].

process, an insulating character is observed.It is thus observed that all the zero-bias conductance values fall on a normal distri-

bution. The absence of a bimodal splitting of this normal distribution thus demonstratesthe absence of electronic phase separation into metallic and insulating regions at any tem-perature. Such an electronic phase separation would be observed as a bimodal splittingof the normal distribution as the insulating and metallic regions give rise to different con-ductivities. Rønnow et al. thus conclude that there is no electric phase separation atany temperature and that the phase separation model is thus not suited for this bilayermanganite. The latter conclusion is also supported by the x-ray and neutron scattering ex-periments, in which no experimental evidence was found to support such a phase separationtendency.

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10 Conclusion

In this literature study an overview was given of a wide range of experiments done on thebilayer manganites. From this overview it became clear that this family of the transition-metal oxides provides a rich playground to study the complex interplay between the charge,spin, orbital and lattice degrees of freedom. In this system the energies of the competinginteractions are very close to each other and a small perturbation, such as temperature,magnetic field or doping level can thus tip the balance and a different groundstate isobserved. In this review the emphasis was around the x = 0.40 doping level, as thiscompound exhibits the most pronounced CMR-effect.

From the wide range of experiments it is clear that there is a strong coupling of theelectron and lattice degrees of freedom, due to the Jahn-Teller splitting of the Mn3+ ion.The magnetostriction and x-ray and neutron scattering measurements have pointed outthe importance of this electron-lattice coupling in the physics of the CMR-effect, as largelattice anomalies were observed as a function of temperature and magnetic field. Indeed,based on a detailed analysis of the x-ray and neutron scattering data, a model was pro-posed in which the charge carriers are trapped by self-induced lattice distortions, so-calledpolarons. These polarons formed short-range correlations, which strikingly tracked thetemperature-dependence of the conductivity. The insulating state, which could not be ex-plained by double-exchange alone, then arises from the self-trapping of the charge carriersby polarons and the associated short-range polaron correlations. These correlations canin fact be viewed as an incommensurate charge and orbital ordering via the Jahn-Tellercoupled charge density wave, providing a direct correlation between the charge, orbital andlattice degrees of freedom. Below Tc however, double-exchange favors the delocalizationof the electrons and the polarons and associated correlations melt, forming the conductingferromagnetic state. The CMR-effect can thus be intuitively explained by this polaronicmodel. The presence of an electron-lattice coupling was indeed also found by Ramanscattering experiments, which found phonon peaks associated with a charge and orbitalordering, in which the latter indeed tracked the short-range polaron correlations.

The tendency towards charge and orbital ordering is even more pronounced in the half-doped manganites, in which there are equal amounts of the Mn3+ and Mn4+ ions. Indeed aCE-type charge and orbital ordering, predicted by Goodenough in the 1950’s, was observedfor LaSr2Mn2O7 in an intermediate temperature-range. The proximity of the energy scalesof the competing interactions became evident from the coexistence of CE-type CO and OOand a type-A antiferromagnetism at lower temperatures. At even lower temperatures, thetype-A antiferromagnetic phase wins the competition over CE-type ordering and forms asingle long-range ordered phase. The notion of charge and orbital ordering was once againsupported by the Raman scattering experiments, which observed charge order activatedphonon modes.

Although it is clear for the x = 0.40 doping level that above Tc, a coupling of thecharge, orbital and lattice degrees of freedom is crucial in understanding the CMR-effect,the issue is not yet settled for the low-temperature ferromagnetic state. It was alreadyevident from transport measurements that the metallic state has an unusual high resistivity

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and is in fact on the border of being a metal. The latter observation was confirmed byoptical studies in which a Drude-like peak, characteristic for metallic conductivity, wasabsent. Instead a broad incoherent band was observed at 0.4 eV, the nature of whichis still controversial. The charge carrier dynamics in the ferromagnetic state are thusunconventional and also here a coupling of the electrons to lattice, orbital or spin degreesof freedom might play an important role. Raman scattering indeed observed charge orderfluctuations in the ferromagnetic state and were argued to develop into collective chargedensity wave excitation. The latter was then related to the formation a pseudogap, whichsuppresses the spectral weight at the Fermi level. Recently ARPES measurements haveobserved the presence of quasiparticles in the ferromagnetic state, providing a possibility tostudy the unconventional carrier dynamics directly. Additional measurements using opticalprobes, x-ray, neutron and Raman scattering might thus elucidate the unconventionalferromagnetic ’metallic’ state, which is still poorly understood up to this point.

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