X-ray Absorption and ResonantInelastic X-ray Scattering Calculationswith Ligand Field Single ClusterMethod on Praseodymium NickelOxidebyShadi BalandehB.Sc., Sharif University of Technology, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2013c? Shadi Balandeh 2013AbstractRNiO3 perovskites (R=rare earth) are one of the most interesting compounds in condensedmatter physics presenting various unusual physical properties. The detailed electronic structureof these materials are very controversial at the present time. The charge transfer energy andthe d-d Coulomb interaction are the two very important parameters which can explain theirelectronic behaviours nicely. However, predicting their values has been a challenge to the sciencesociety so far.X-ray Absorption Spectroscopy (XAS) and Resonant Inelastic Scattering (RIXS) are thetwo very useful techniques to probe the electronic structure of a solid state system in generaland predict these two energies in particular. In this thesis Multiplet Ligand Field ClusterCalculation (MLFCC) is used to calculate these two spectra, then the charge transfer energy(?), the covalent hopping integral(pds), and the d ? d Coulomb repulsion energy Udd areobtained by fitting the calculated spectra to the experiment.In this work, the calculated XAS results are compared with the experiment and the adjustedvalues are introduced as ? = 2.5 eV , pds=-1.9eV, Udd=7.5eV and 10Dq=0.5 eV.The low spin to high spin transition is also studied and the critical charge transfer energiesand covalent hopping integrals are calculated at which the abrupt transition happens. It isalso found that in almost all low spin cases the d8L9 configuration has the largest contributionto the ground state. Since the best fit of XAS is not satisfactory and displays considerabledifferences with the experiment, the study is followed with the RIXS calculations.Finally, the calculated RIXS results for different polarizations are compared with the exper-iment.It results in a smaller ? = 0.8eV and a smaller absolute value of pds=-1.4eV at whichthe double peak structure in XAS L3 vanishes. This could be an evidence to the fact that XASshould not be interpreted in the conventional way and the ? should not be fitted to keep thedouble peak which probably has another source than the multiplet structure.iiPrefaceThe program has been used for the calculations is the ?Ligand Field Theory Package?-version2013.2.4 written by Maurits W.Haverkort.The PrNiO3 sample for the X-ray absorption spectra was grown at Max Planck InstituteStuttgart by G. Logvenov and G. Christiani and the corresponding measurement was carriedout at the Canadian Light Source in Saskatoon by S. Macke, A. Radi, J. E. Hamann-Borrero,R. Sutarto and F. He and the first analysis including subtracting the background were done byS. Macke.The Resonant Inelastic X-ray Spectroscopy (RIXS) experimental data are provided byThorston Schmitt as the group leader and Valentina Bisogni both from Paul Scherrer Insti-tute (PSI) and Marta Gibert, Sara Catalano and Jean-Marc Triscone all from the university ofGeneva.One of the high spin cobalt oxide X-ray absorption measurements in chapter 3 was carriedout by Abdullah Radi in Canadian Light Source (CLS) with Ronny Sutarto and SebastianMacke involved. The other CoO spectrum is provided by Dian Rata and Alex Frano fromBerlin.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Background Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Transition Metal Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Covalency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 The Zaanen-Sawatzky-Allen Model . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Multiplets and Hund?s Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 2p-X-ray Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Resonant Inelastic X-ray Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 152 XAS calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Configuration Interaction Model in a Single Cluster Calculation . . . . . . . . . 172.2 Seeking the Best Agreement with Experiment . . . . . . . . . . . . . . . . . . . 203 Spin State Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Low Spin to High Spin Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 264 RIXS Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1 Seeking the Best Agreement with Experiment . . . . . . . . . . . . . . . . . . . 334.2 XAS Spectra with Parameters Driven from RIXS . . . . . . . . . . . . . . . . . 535 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55ivTable of ContentsBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59vList of Tables2.1 Ground state configurations and the corresponding energies. . . . . . . . . . . . . 192.2 Excited state configurations and the corresponding energies . . . . . . . . . . . . 194.1 The first 25 eigen-energies of the system with ? = 2.5eV . . . . . . . . . . . . . . 394.2 The first 25 eigen-energies of the system with ? = 0.5eV . . . . . . . . . . . . . . 434.3 The first relative eigen-energies of the system with the following values: tpp =0.8, pds = ?1.4,? = 0.8, 10Dq = 0.4, Udd = 7.5, Upd = 9 eV . . . . . . . . . . 45viList of Figures1.1 Insulator-Metal-Antiferromagnetic phase diagram for nickelates as a function ofthe tolerance factor or rare earth ionic radius and temperature.[19] . . . . . . . . 11.2 The atomic arrangement of a single unit cell of the perovskite crystal structureof RNO3. Green indicates the rare earth ion, red the oxygen and gray the nickelatoms.[3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The spherical coordinates used in the spherical harmonics.[2] . . . . . . . . . . . 31.4 The real part of the angular wavefunctions of d orbitals.[5] . . . . . . . . . . . . 41.5 Crystal field energy level diagram for the octahedral geometry . . . . . . . . . . . 51.6 Orbitals energy level diagram for a NiO6 cluster considering covalency andhybridizations.[8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 d(xy) and d(x2? y2) orbitals forming pi and ? bondings with oxygens? p orbitalsrespectively.[9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.8 The Zaanen-Sawatzky-Allen (ZSA) diagram[21] . . . . . . . . . . . . . . . . . . 81.9 Density of states versus energy of TM 3d and anion 2p states determining thetwo different types of insulator in ZSA scheme. EA and EI are electron affinityand ionization energies.[17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.10 Electron removal and addition in Mott-Hubbard insulator and charge transferinsulators.[17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.11 PNO 2p-XAS spectra from experiment [11] . . . . . . . . . . . . . . . . . . . . . 131.12 2p-XAS spectroscopy of different transition metal compounds. The L3 ? L2splitting is larger for the late transition metal compounds with the higher nuclearcharges. [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.13 Low energy excitations in a condensed matter system. The energy scales arerelevant for transition metal oxides.[1] . . . . . . . . . . . . . . . . . . . . . . . . 162.1 The relative on-site energies of the different configurations for the initial andfinal states in 2p-XAS in terms of ?, Udd and Upd [9] . . . . . . . . . . . . . . . . 192.2 The final calculated XAS spectra comparing the experiment . . . . . . . . . . . . 222.3 Comparing the L3 peaks in the calculations and the experiment . . . . . . . . . . 232.4 The same calculated L3 peak with the lower broadening of 0.2 eV compared withexperiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23viiList of Figures3.1 The energy levels of a d-transition metal oxide in an octahedral geometry con-sidering the covalency with surrounding oxygens. . . . . . . . . . . . . . . . . . . 243.2 It shows how by decreasing the charge transfer energy, splitting increases andsubsequently system changes from High spin State to Low spin State. . . . . . . 253.3 XAS spectra just before the spin state transition . . . . . . . . . . . . . . . . . . 273.4 XAS spectra just after the spin state transition . . . . . . . . . . . . . . . . . . . 273.5 XAS spectra just before the spin state transition . . . . . . . . . . . . . . . . . . 283.6 XAS spectra just after the spin state transition . . . . . . . . . . . . . . . . . . . 283.7 High spin CoO XAS spectra from experiment . . . . . . . . . . . . . . . . . . . . 293.8 Map of the spin values versus ? and pds showing the low spin to high spin transition 303.9 Energy level diagram versus pds . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.10 Energy level diagram versus charge transfer energy . . . . . . . . . . . . . . . . . 314.1 The experimental RIXS spectra form Geneva at T=300K in the left and atT=15K in the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 RIXS experimental geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 A schematic sketch of the d? d excitations in a d7 low spin system. . . . . . . . 354.4 Calculated RIXS spectra for the following values: tpp = 0.8, pds = ?1.9,? =2.5, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV and RIXS broadening of 0.3 eV . . . . . 374.5 The resonant energies taken for the RIXS calculations, at 852 ,852.5, 853, 853.2,853.5, 853.8, 854.1, 854.4 eV with the following values tpp = 0.8, pds = ?1.9,? =2.5, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV and RIXS broadening of 0.2 eV . . . . 384.6 Calculated RIXS spectra with the following parameters: ? = 2.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV and RIXS broadening of 0.2 eV. Thevertical axis is the intensity in an arbitrary unit and the horizontal axis is theenergy loss in eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.7 The resonant energies taken for the RIXS calculations, at 851.2 ,851.6, 852, 852.3,852.5, 852.7, 852.9, 853.4, 853.6 eV with the following values ? = 0.5, tpp =0.8, pds = ?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV . . . . . . . . . . . . . . . 414.8 Calculated RIXS spectra with the following parameters: ? = 0.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV . . . . . . . . . . . . . . . . . . . . . . 424.9 The resonant energies taken to calculate RIXS spectra with the following valuestpp = 0.8, pds = ?1.4,? = 0.8, 10Dq = 0.4, Udd = 7.5, Upd = 9 eV with energyshift of 865 eV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.10 Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9 eV . . . . . . . . . . . . . . . . . . . . . . 464.11 Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9 eV . . . . . . . . . . . . . . . . . . . . . . 474.12 Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9 . . . . . . . . . . . . . . . . . . . . . . . . 48viiiList of Figures4.13 Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9 . . . . . . . . . . . . . . . . . . . . . . . . 494.14 Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9 . . . . . . . . . . . . . . . . . . . . . . . . 504.15 Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9 . . . . . . . . . . . . . . . . . . . . . . . . 514.16 Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9 . . . . . . . . . . . . . . . . . . . . . . . . 524.17 Calculated XAS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9 eV . . . . . . . . . . . . . . . . . . . . . . 545.1 Calculated RIXS spectra with the following parameters: ? = 2.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV and RIXS broadening of 0.2 eV . . . . 595.2 Calculated RIXS spectra with the following parameters: ? = 2.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV and RIXS broadening of 0.2 eV . . . . 605.3 Calculated RIXS spectra with the following parameters: ? = 2.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV and RIXS broadening of 0.2 eV . . . . 615.4 Calculated RIXS spectra with the following parameters: ? = 0.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV . . . . . . . . . . . . . . . . . . . . . . 625.5 Calculated RIXS spectra with the following parameters: ? = 0.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV . . . . . . . . . . . . . . . . . . . . . . 635.6 Calculated RIXS spectra with the following parameters: ? = 0.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV . . . . . . . . . . . . . . . . . . . . . . 645.7 Calculated RIXS spectra with the following parameters: ? = 0.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV . . . . . . . . . . . . . . . . . . . . . . 65ixAcknowledgmentsFirst and foremost, I would like to express my sincere gratitude to my advisor Prof. GeorgeSawatzky for the continuous support of my M.Sc study and research whilst allowing me theroom to work in my own way, for his patience, contagious enthusiasm, and immense knowledge.I could not have imagined having a better advisor and mentor for my graduate study.Besides my advisor, I would like to thank Prof. Maurits Haverkort for his patience and deepknowledge in answering my questions regarding his program and its physics.I would also like to thank Dr. Sebastian Macke who was always willing to help and give hisbest suggestions. I also thank Abdullah Radi for training me on the X-ray diffraction machineand providing me with the TEY signal of CoO in the chapter three.My sincere thanks also goes to Prof. Andrea Damascelli for his insightful comments on mythesis.Last but not least, I would like to thank Dr. Ilya Elfimov for his support through thenetwork technical issues.xDedicationTo my parents, Zohreh and Mohammad and to my brother, Mehrdad for their endless love andencouragement.To the love of my life, my husband Ehsan for his continuous love and support.xiChapter 1Background Materials1.1 Transition Metal OxidesTransition Metal Oxides (TMOs) are believed to be one of the most fascinating systems insolids. They exhibit various unusual physical properties such as the high temperature superconductivity, huge variations in their magnetic properties and metal-insulator transitions. Theseries RNiO3 (R=Rare earth) are the transition metal oxides, which present a sharp insulator-metal transition strongly depending on temperature and tolerance factor.[19] Tolerance factor(t ? dR?O?2dNi?O) is used as a measure of the amount of distortion of a perovskite from an idealcubic unit cell. Therefore, for a given RNiO3, the closer the tolerance factor to one, the closerto cubic with Ni-O-Ni bond angle of ? = 180? .The detailed behaviour of these phase transitionsare shown in Fig. 1.1Figure 1.1: Insulator-Metal-Antiferromagnetic phase diagram for nickelates as a function of thetolerance factor or rare earth ionic radius and temperature.[19]The phase diagram illustrates how for different rare earth radius, RNO shows differentmagnetic and electronic properties. These transitions distinguish three different regimes: Metal,anti-ferromagnetic insulator and paramagnetic insulator. From the diagram, it can be observedthat at low temperatures, PrNiO3 shows anti-ferromagnetic behaviour with ordered anti-parallelspins, but when the temperature increases, it loses this ordering and experiences insulator-metal11.1. Transition Metal Oxidestransition with delocalized electrons.The sensitivity of these physical properties to the crystal chemistry and structural distortionsoffers many possibilities to manipulating their properties in order to engineer new applicationsin these materials, such as in the complex oxides heterostructure. [23] But for this aim, theelectronic structure of these materials itself has to be understood well.From the Zaanen-Sawatzky-Allen scheme[21], using only a few parameters (the charge trans-fer energy and the d ? d Coulomb interaction energy) is able to account for the electronic be-haviour of a great number of 3d TM oxides such as the phase transitions[13]. In this thesis wetry to find these two parameters by fitting the Multiplet Ligand Field Single Cluster Calcula-tions to the experiment. Another purpose of this work is to test this theoretical approach forthe specific case of PrNiO3 (And NdNiO3) and comment if in this case the physical propertiescan be described reasonably well by a single cluster calculation or not and if not what theprobable reasons could be.In this thesis we also seek the charge transfer energy and the covalent hopping integral atwhich in a given d7 valence configuration system a low spin to high spin transition can occur.Being in a low or high spin state is an important characteristic of a system which can controlits magnetic and electronic properties.The source of the unusual properties of d-transition metal oxides is the unique nature ofthe valence d electrons.Figure 1.2: The atomic arrangement of a single unit cell of the perovskite crystal structure ofRNO3. Green indicates the rare earth ion, red the oxygen and gray the nickel atoms.[3]Transition metal atoms with their incomplete d shells combine with oxygen to form thetransition metal oxides. These are relatively ionic compounds with O2? ions and a rare earthin a 3+ valence state leaving the nickel also in a 3+ valence state. Therefore, in PNO, nickelis 3+ resulting in 3d7 valence shell configuration. Therefore, the principle quantum number nand the orbital angular quantum number l of the valence electrons are 3 and 2 respectively. In21.1. Transition Metal Oxidesa spherical potential, the electrons? wave functions can be expressed as:?n,l,m(r, ?, ?) = Rn,l(r)Yml (?, ?) = Rn,l(r)pl(cos ?)eim?Rn,l(r) are the radial part of the wave functions depending on the n and the l and Y ml (?, ?)or the spherical harmonics are the angular part of the wave function depending on the magneticquantum number m and the l which can be express in terms of the Legendre polynomials orpl(cos ?) with:n = 1, 2, 3, ...l = 0, 1, 2, ..., n? 1m = ?l,?l + 1, ..., l ? 1, lSo for the d orbitals with l = 2, there are five different magnetic quantum numbers andtherefore, five different angular wave functions, with:?l ? m ? l? ?2 ? m ? 2? m = ?2,?1, 0, 1, 2By using the corresponding spherical harmonics and the spherical to Cartesian coordinateconversion, the real parts of the angular d orbitals take the following forms:Figure 1.3: The spherical coordinates used in the spherical harmonics.[2]x = r sin ? cos?y = r sin ? sin?z = r cos ?31.1. Transition Metal OxidesY m?l = (?1)mY ?mld(x2 ? y2) ? Re(Y ?22 ) ? sin2 ? sin 2? =x2 ? y2r2d(yz) ? Re(Y ?12 ) ? sin ? cos ? sin? =yzr2d(3z2 ? r2) ? Re(Y 02 ) ? (3 cos2 ? ? 1) =3z2 ? r2r2d(zx) ? Re(Y 12 ) ? sin ? cos ? cos? =xzr2d(xy) ? Re(Y 22 ) ? sin2 ? cos 2? =xyr2Figure 1.4: The real part of the angular wavefunctions of d orbitals.[5]For zero magnetic field and neglect of the spin orbit coupling, energy does not depend onthe magnetic quantum number m, therefore, in a spherical potential there are five different dwave functions with the same energy or in the other words, in a spherical potential, d orbitalsare five-fold degenerate. Upon filling these states with electrons however one does have to takeinto account the Coulomb interaction between the electrons which develops into a complicatedmultiplet structure which we will discuss later in a simplified version.However, to understand real materials, we are more interested in studying less symmetricalcases such as the octahedral potential in a single crystal of RNO, assuming a cubic crystalstructure and equivalent Ni-O bond lengths. The lattice is shown in Fig. 1.2. One way to treatthis group of potentials and geometries is using the Ligand Field Theory (LFT) where ligandusually refers to the oxygen 2p electron states forming bonds with the central Ni 3d states.Crystal Field Theory (CFT) is the simplest form of LFT, in which each of the ligandions is treated as a point charge. If consider each oxygen, as a negative point charge, thenif the d orbitals points right towards them that will be very repulsive with a huge Coulombrepulsion energy and if they points between them, less repulsive. So when d orbitals are off-axis41.2. Covalency(d(xy), d(xz), d(yz)) since oxygens are on the axes, it is less repulsive and in a lower energy.(Fig. 1.4 ). Therefore, The d(x2?y2) and d(3z2?r2) which are usually referred to as eg orbitals,have the same and a higher energy and d(xy), d(yz) and d(xz) or t2g orbitals are found to beat a lower energy. That is how the degeneracy is partly removed in an octahedral geometry.The splitting energy is usually referred as 10Dq. The eg orbitals go up by 6Dq and the t2gorbitals go down by 4Dq to keep the center of gravity the same as for the spherical potentialenergy levels. The crystal field energy level diagram is shown in Fig. 1.5Figure 1.5: Crystal field energy level diagram for the octahedral geometry1.2 CovalencyAlthough crystal field theory can explain some electronic features of the d orbitals, it completelyignores the nature of ligands by treating them as point charges.That is why it fails in, for example, explaining a large energy splitting caused by a neutralligand like CO or high spin to low spin transition by changing the charge transfer energy.LFT is the model, which considers all ionic, covalent and hybridization aspects of coordi-nation complexes.The basic idea behind both CFT and LFT is that when there is a metal atom, mostly a transi-tion metal ion, at the center of a coordination sphere, surrounded with donor ligand atoms, theenergy of its valence orbitals is going to be changed by the existence of those ligands. Figures1.5 and 1.6 show these two theories predictions in this regard in an octahedral geometry.Apparently, there is a huge contribution from covalency and electron hopping betweenligands and the metal ion. In the latter one, there are Ni-d orbitals at some energy and oxygenorbital at some lower energy. They interact with each other resulting in ? and pi bonds andanti-bonds.Fig. 1.7 shows how in cubic symmetry t2g and eg orbitals form pi and ? combinationsrespectively. Since eg orbitals point towards oxygens p orbitals, their hopping integrals arelarger and put ? bonding and anti-bonding energies at higher levels than pi ones. These hoppingintegrals are usually presented in the parameters pd? and pdpi.51.2. CovalencyFigure 1.6: Orbitals energy level diagram for a NiO6 cluster considering covalency andhybridizations.[8]Figure 1.7: d(xy) and d(x2 ? y2) orbitals forming pi and ? bondings with oxygens? p orbitalsrespectively.[9]tpp and pds are the other parameters related to the covalency which are defined as follows:[8]tpp = pp? ? pppiwhere pp? and pppi are the ? type and pi type hopping integrals between the oxygens p orbitals,defined relative to the O-O bond direction. In the other words, the bonding and anti-bondingorbitals are tpp below and above the Op onsite energy.Veg or pds and Vt2g are covalent hopping integrals are also defined as the follows:61.3. The Zaanen-Sawatzky-Allen ModelVeg = pds = ??deg |H|?Leg ? =?3pd?Vt2g = ??dt2g |H|?Lt2g ? = 2pdpiWherein pd? and pdpi are hopping energies between an electron in a Ni-d orbital and anelectron in ligand-p shell with ? and pi symmetries respectively. As an example consider thedx2?y2 orbital with four pi bonds with the oxygens p orbitals around in an xy plane as shownin Fig. 1.7 ThenVdx2?y2 = ??dx2?y2 |H|?Lt2g ? =1?4[??dx2?y2 |H|p1?+??dx2?y2 |H|p2?+??dx2?y2 |H|p3?+??dx2?y2 |H|p4?]= 4?4pdpi = 2pdpiWhere each p orbital has been considered normalized and 1?4is the normalization factor fortheir linear combination. The signs in the linear combinations are determined by the orbitalsphase and can be read off of the figure above. In defining the hopping integral the correctcoordinate system has to be selected to preserve the symmetry.1.3 The Zaanen-Sawatzky-Allen ModelThere are two main theories describing the outer electrons. Theories which mostly can describesystems in which the band width is large compared to the electron-electron Coulomb repulsionenergy. Alternatively, the systems in which there is a large overlap between the orbitals ofneighbouring atoms such as the systems with s or p valence shell. The other ones are the theo-ries which are more applicable to the systems with localized electrons and bandwidths smallerthan the Coulomb energy like systems with outer f shells. [15]Whereas the above systems, the TMOs with partially filled d shell, have an intermediate band-width and therefore show intermediate characters. That is why band theory or ligand fieldtheory alone fails in describing them perfectly. One solution can be the employment of boththeories together.Before Fujimori[6] and Sawatzky and Allen[16] introduced charge transfer energy as animportant parameter in the physics of transition metal oxides, it was thought for a long timethat nickel oxide is a Mott-Hubbard insulator where the band gap is determined by U , theon-site 3d ? 3d Coulomb repulsion energy. But it was not consistent with the experimentswhich presented huge differences in the band gap for the compounds with a same TM atomand different surrounding ions.[10] If it was a Mott-Hubbard insulator, the band gap should bedetermined by the Udd, which is not changed strongly with changing the ions. This controversywas settled by considering ?, the energy it costs to transfer an electron from a ligand tothe transition metal atom. Zaanen, Sawatzky and Allen presented a systematic scheme [21]interpreting the electronic structure of the 3d transition metal compounds in terms of the ?71.3. The Zaanen-Sawatzky-Allen Modeland the Udd.According to ZSA diagram,Fig. 1.8, there are two different types of gaps possible in tran-sition metal compounds. Namely, the charge transfer gap due to the ligand to metal chargetransfer energy and the Mott-Hubbard gap associated with Coulomb interaction energy. Thenassociated with these two gaps, the compounds can be categorized into two regimes: Mott-Hubbard regime where U is smaller than ? and the band gap is determined by the U, andcharge transfer regime where ? is smaller than U and the magnitude of band gap is given by?. Therefore, it can be concluded that when either the Mott-Hubbard gap or charge transfergap closes, there will be an insulator to metal transition. The diagram of their density of statesis shown in Fig. 1.9 which illustrates their differences in more details. Fig. 1.10 also showswhat kind of charge fluctuations happen in each of these two types of materials.Figure 1.8: The Zaanen-Sawatzky-Allen (ZSA) diagram[21]The on-site effective screened 3d?3d Coulomb energy U or U eff is here defined as the energyit costs to transfer an electron from a TM d orbital to another TM d orbital on a different site.So, basically we are left with a dn+1 and a dn?1 ions rather than the two initial dn ions. Inorder to find the U or the Hubbard conductivity gap, the Hund?s ground state energies 1.4 ofthe systems with n, n ? 1 (ionized state), and n + 1 electrons (electron-affinity state) have tobe calculated.[12]U = E(dn+1) + E(dn?1)? 2E(dn)The charge transfer energy (?) is an energy it costs to remove an electron from a ligand p81.3. The Zaanen-Sawatzky-Allen ModelFigure 1.9: Density of states versus energy of TM 3d and anion 2p states determining thetwo different types of insulator in ZSA scheme. EA and EI are electron affinity and ionizationenergies.[17]Figure 1.10: Electron removal and addition in Mott-Hubbard insulator and charge transferinsulators.[17]orbital and put it in a TM d orbital.? = E(dn+1L)? E(dn)91.4. Multiplets and Hund?s Ruleswhere L denotes one hole in a ligand atom.It is believed that high valence oxides like Ni3+ in RNOs belong to the charge transferregime where the smallest gap is associated with the ligand to metal charge transfer energy.Therefore, it should be strongly affected by the configurations of the transition metal and itssurrounding oxygen ions.That could be the reason makes RNiO3 electronic structure difficult to understand andchallenging. In this thesis we try to investigate this for Praseodymium Nickel Oxide (PNO)by doing Ligand Field Cluster Calculations (LFCC), employing M. W. Haverkorts codes[9] andexperimental data from CLS and PSI.1.4 Multiplets and Hund?s RulesThe Hund?s rules determine in a given system which state should be the ground state. Theysay that the atomic orbitals of a particular shell have to be filled according to the followingrules:Firstly, the total spin has to be maximized. It in fact implies that the electrons should haveparallel spins. Therefore, from the Pauli exclusion principle they must be in different spatialorbitals which results in a smaller Coulomb repulsion and lowers the energy.Secondly, the total orbital angular momentum has to be maximized. It leads to having lotsof angular lobes which again reduces the Coulomb repulsion.And the third one states that for the less than half filled shells the total angular momentumis J = L? S and for more than half filled shells, it is J = L+ SThen the Hund?s ground state energy can be determined by:[12]E(n,Hund) = ?I(n)I + ?F0(n)F0 + ?J(n)J + ?C(n)Cwhere: ?I = n, ?F 0 =n!2 and the ?J is the number of parallel spin pairs and I is the oneelectron potential. C describes the angular part of the multiplet splitting and J is the Hund?sexchange interaction energy which describes the exchange interactions between parallel spins:C(dd) =114(97F 2 ?57F 4)JH(dd) =114(F 2 + F 4)In Multiplet Ligand Field Theory (MLFT), there are several possible electronic configura-tions (multiplets) with different energies because of the different Coulomb interaction energiesfor each. These electron-electron Coulomb interactions can be written as:[18]??|N?i,j>i1rij|?? =(N2)??|1r12|??101.4. Multiplets and Hund?s RulesIn Hartree-Fock formalism the many body wavefunctions |?? are chosen to be in the formof the Slater determinant, so with P ?s as permutation operators, we will have:|?? = 1?N !?????????????1( ~x1) ... ?N ( ~x1)?1( ~x2) ... ?N ( ~x2)... ... ...... ... ...?1( ~xN ) ... ?N ( ~xN )????????????= 1?N !N !?n=1Pn{?1( ~x1)...?N ( ~xN )}??|1r12|?? =1N !N !?n,m=1?(?1)n?1Pn{??1( ~x1)...??N ( ~xN )}1r12(?1)m?1Pm{?1( ~x1)...?N ( ~xN )}d ~x1d ~xN=(N ? 2)!N !N?n,m=1[???n( ~x1)??m( ~x2)1r12?n( ~x1)?m( ~x2)d ~x1d ~x2????n( ~x1)??m( ~x2)1r12?n( ~x2)?m( ~x1)d ~x1d ~x2]? ??|N?i,j>i1rij|?? =12[N?m6=n???n( ~x1)?n( ~x1)1r12??m( ~x2)?m( ~x2)d ~x1d ~x2????n( ~x1)?m( ~x1)1r12??m( ~x2)?n( ~x2)d ~x1d ~x2] = ?nn|mm? ? ?nm|mn?Note that in the first integral the ??n( ~x1)?n( ~x1) and ??m( ~x2)?m( ~x2) terms are like the clas-sical charge densities of two electrons in orbital n and m so the whole integral acts like theclassical Coulomb repulsion. But the second term has no classical counterpart and comes fromthe ?Pauli exclusion principle?. It is subtracted from the first term and reduces the totalCoulomb repulsion energy compared to the classical case. They are usually referred as Slaterintegrals and exchange term respectively.These two terms can also be expressed in one term as:Vijkl =? ?1|r1 ? r2|??i (r1)??j (r2)?k(r2)?l(r1)dr1dr2It is convenient to express this integral in terms of the Legendre polynomials and the spher-ical harmonics:Rk(ij, kl) =? ?2rk<rk+1>P ?i (r1)P?j (r2)Pk(r1)Pl(r2)dr1dr2where r< = min(r1, r2), r> = max(r1, r2)However, the Coulomb interaction term is usually expressed in Slater integrals as: F k(i, j) =Rk(ij, ij) and Gk(i, j) = Rk(ij, ji) which are corresponding to the Coulomb term and exchangeterm respectively. It can be derived that for electron-electron interaction in d orbitals all terms111.5. 2p-X-ray Absorption Spectroscopyexcept F 0, F 2 and F 4 vanish, these are usually called monopole, dipole and quadrupole integralsrespectively. Because of the polarizations effect the monopole term (F 0) is strongly screenedand reduced. For example, in NiO it is about 7 eV[20], while its atomic value is about 18eV.[14] . In our calculations it is referred as Udd which is varied to get the best agreement withexperiment. The other Slater terms are hardly reduced because of the screening so we can trustthe abinitio results for them. Although experience shows that better is reduce them by about20% because of the atomic correlation effects and the hybridization with ligand orbitals in thecompounds.In transition metal chemistry these Slater integrals are usually expressed in terms of Racahparameters A,B and C with:A = F0 ? 49F4B = F2 ? 5F4C = 35F4With F0 ? F 0, F2 ? F249 and F4 ?F 4441In MLFT the energy of a state with n electrons in an open d shell is determined by:E(n,L, S, ?) = nI +12n(n? 1)Uav + U(n,L, S, ?)Wherein, L, S and ? are total orbit, spin and seniority quantum numbers. I is the oneelectron potential, Uav is the average multiplet Coulomb exchange interaction energy andU(n,L, S, ?) is the multiplet splitting.The average Coulomb repulsion energy between two electrons in a d orbital in terms ofSlater integrals is:Uav = F0 ?14441(F 2 + F 4)1.5 2p-X-ray Absorption SpectroscopyIn 2p-XAS, a hole is created in a 2p core level by transition of an electron from there to 3dvalence shell by shining x-ray photons to the sample. This excited state is very unstable sothe core hole decays by radiant x-ray emission or other radiation-less transitions like Augerdecay.[7]There are many different states in the valence shell but only transitions obeying dipoleselection rules can happen. Therefore for each initial state the set of available excited statesare different. That is why 2p-XAS can provide good information of the electronic structure ofthe ground state.The rate of transition from the initial state i to the final state f through the above process121.5. 2p-X-ray Absorption Spectroscopycan be determined by the Fermi?s Golden rule:Ri?f =?f2pi~|?f |A.p|i?|2?(Ef ? Ei)Where p is the momentum operator and A is the vector potential of the applied electromagneticfield. The delta term satisfies energy conservation and the sum is over all unoccupied final stateswith the energies of Ef . Then it can be seen that to have a finite transition probability we needto have specific conditions in the initial and final states. As an example, the transition from as orbital to py orbital with x polarized light is not allowed since the corresponding matrix termis zero which is an integral over an odd function over whole space. This also explains why XASspectra can be different for different light polarizations (but not in the cubic symmetry).In this report, 2p-XAS is used to study a nickelate single cluster in PNO.To understand XAS main features, let us take a look at Fig. 1.11 which is a PNO XASspectrum from experiment. There are two resonance structures near energies 853 and 870 eV.Figure 1.11: PNO 2p-XAS spectra from experiment [11]Therefore, there is a 17 eV energy difference between them. This cannot be due to the splittingenergy between t2g and eg orbitals. This splitting is only about 1eV or less. The splittingbetween these two peaks, which are called L3 and L2, with L3 at a lower energy,is resulted fromthe spin orbit coupling in the 2p-core level.The spin-orbit coupling is an interaction between electron?s spin magnetic moment and themagnetic field generated by the electron?s orbital motion around the nucleus. These interactionenergy is equal to:?H = ?s.Bl131.5. 2p-X-ray Absorption SpectroscopyThe electric field electrons travel through is radial so from the Biot-Savart law the magneticfield will be:B =Ze??4pir3[v ? r]l = r ?mev? B =Zeffe??4pir3me~l?s = ?gse2me~s? ?H =Zeffe2??8pim2e1r3(~s.~l)We also have:?1r3? ? Z3eff? ?H = ?~s.~lwith ? ? Z4eff where Zeff is the screened nuclear charge.The orbital angular momentum in 2p shell is 1 and spin of a single hole is 12 so the associatedtotal angular quantum number can take the following range of values:|l ? s| ? j ? |l + s| ?12? j ?32? j =12,32These two values give rise to the two different resonance peaks.J = L+ S ? J2 = L2 + S2 + 2L.S ? j(j + 1) = l(l + 1) + s(s+ 1) + 2l.s??l.s? = ?j(j + 1)? l(l + 1)? s(s+ 1)2=?2[j(j + 1)? 1(1 + 1)?12(12+ 1)] = ?2[j(j + 1)?114] =forj =12?= ??forj =32?=?2? splitting :3?2We saw earlier that the coefficient ? is proportional to Z4eff . Therefore this splitting in the2p-XAS spectra should increase strongly with increasing the effective nuclear charge.The Fig. 1.12 shows how for late transition metal compounds with higher nuclear chargethe L3 ? L2 splitting is larger compared to the early transition metal ones. There is a same141.6. Resonant Inelastic X-ray ScatteringFigure 1.12: 2p-XAS spectroscopy of different transition metal compounds. The L3 ? L2splitting is larger for the late transition metal compounds with the higher nuclear charges. [9]trend for transition metals with 3d, 4d, 5d and 6d configurations. The higher Zeff the largerspin-orbit coupling effect.L3 corresponding to j = 32 , has higher intensity because of the higher degeneracy of 2j+1 = 4comparing 2j + 1 = 2 in L2. In 1.11 the continuum steps in background intensity, which arefrom excitations to unbound continuum-like state, have been subtracted. In XAS, the spectralshape can also provide detailed information of the ligands? effect on the metal and the atom?selectron configuration. As discussed above the energy splitting is also an indication to thespin-orbit coupling effect.Another feature of a spectrum is its spectral line width. The spectral lines are not infinitelysharp. They are broadened and one of the most important reason of this broadening is called the??life-time? or the ??natural?? broadening. The reason is rooted in the ??uncertainty principle?which relates the excited state life time to the uncertainty of its energy. The shorter life time,the larger energy uncertainty or the wider spectral line. The experimental energy resolution isthe another source to the line width broadening.1.6 Resonant Inelastic X-ray ScatteringRIXS is a powerful tool to probe low energy excitations such as the charge transfer and d? dexcitations.In this process the photon makes an excitation from core to conduction band, then anelectron decays back from conduction band to the core and a photon comes out. The excitationsin which an electron decays back to the same d configuration are called d ? d excitations andthose falling to another configuration with different number of d electrons and ligand holes are151.6. Resonant Inelastic X-ray Scatteringknown as charge transfer excitations. [7]Therefore, This process should not be necessarily elastic. The second electron can decayfrom a lower level in the conduction band and photon can lose some energy. The amount of lostenergy depends on the occurred excitation. Fig. 1.13 shows how different kind of excitationscost different amount of energies,the energy scales are relevant for transition metal oxides. Theprocess is resonant because energy of the incident light has to be the same as the energy of thex-ray absorption edges to excite an electron from core to valence.RIXS usually is considered as a two-step process. One from initial state |i? to an intermediateone |m? and then from intermediate to final state |f? with the energies of Ei, Em and Efrespectively. The intermediate state is the state with a hole in the core level, which is the sameas the excited state in XAS.The corresponding intensity of this second order process is determined by the Kramers-Heisenberg (KH) formula:I ??f(?m?f |T ?|m??m|T |i?Ei + ~?? Em ? i?m2)2?(Ei ? Ef + ~?? ~?)? and ? are angular frequencies of the incident and emitted photons respectively. ?m is theintermediate state intrinsic line-width or the broadening due to the intermediate state lifetimeand T is the relevant transition operator. At resonant, when the intermediate state is the XASfinal state, the other terms in the denominator become zero (Em ? Ei = ~?) and leaves onlythe core-hole broadening.Figure 1.13: Low energy excitations in a condensed matter system. The energy scales arerelevant for transition metal oxides.[1]16Chapter 2XAS calculations2.1 Configuration Interaction Model in a Single ClusterCalculationIn a configuration interaction model, the wave functions for the single cluster are expressed asa linear combination of different possible configurations.[22] In PNO, where a nickelate singlecluster is going to be studied, nickel has d7 valence shell electron configuration and p6 core shell.NiO cluster has six oxygens around so there are 3 ? 6 = 18 O ? p orbitals. But, not all linearcombinations are taken into account, only those that couple and interact with Ni-d orbitals areconsidered. Based on the Goodenough-Kanamori hybridization rules the number of them isreduced to ten, because for each Ni-d orbital there is one such linear combination. In this caseeither one, two or three electrons can hop from ligand to nickel d orbitals.So the XAS ground state wavefunction can be expressed as:?g = ?1|p6d7L10?+ ?2|p6d8L9?+ ?3|p6d9L8?+ ?4|p6d10L7?pndmLz denotes n electrons in the transition metal core p shell, m electrons in the transitionmetal ion d shell and z electrons in the ligands? p shell.The dimension of Hamiltonian matrix that has to be diagonalized can be obtained as follows:p6d7L10 : p shell is full so there is only one possible configuration here, there are seven elec-trons in the ten d orbitals so there are(107)= 120 ways to arrange them there. Ligand orbitalsare also full with ten electrons so there is only one possible configuration there. Therefore,there will be 1 ? 120 ? 1 = 120 possible states for this configuration.With the same counting, there will be 450, 450 and 120 states corresponding to p6d8L9, p6d9L8and p6d10L7 configurations respectively. Therefore, there are 1140 wavefunctions altogether anda 1140? 1140 Hamiltonian matrix.And for the XAS state since one electron is excited from Ni-p shell to the Ni-d shell the excitedwavefunction will be presented as:?XAS = ?1|p5d8L10?+ ?2|p5d9L9?+ ?3|p5d10L8?Again with the same counting there will be 270, 600, 270 states corresponding to p5d8L10, p5d9L9and p5d10L8 configurations so there are again 270 + 600 + 270 = 1140 wavefunctions in total.Due to the multiplet effect which is rooted in electron-electron interactions, these 1140172.1. Configuration Interaction Model in a Single Cluster Calculationfinal states do not all have the same energies. They spread out over an energy range. Eachconfiguration has its own multiplet and set of final states, which differ from one element toanother.Using ligand orbitals reduces the Hilbert space significantly and increases MLFT calculationsefficiency.[8]There are some important parameters that will be introduced in the following paragraphs.Many of them can be calculated abinitio like dipole and quadruple Slater integrals, and somewill be adjusted to experiment, which we mostly deal with in this thesis.Udd and Upd: The monopole Coulomb interactions which are strongly reduced because ofthe screening in a polarizable medium. They will be fitted to experiment to obtain the bestagreement. Udd is the one between electrons in Ni-d shell and Upd is between the Ni p-core holeand Ni-d electrons. From what has been discussed in the previous chapter, U3d?3d is given by:U3d?3d = E(d6Lm) + E(d8Lm)? 2E(d7Lm)Where E(dnLm) is the center of gravity of the dnLm multiplets.For multiple interactions or Slater integrals F 2 and F 4 ,the screening is not huge. Becausethey do not involve in changing the charge of ions. Therefore, they can be obtained abinitiofrom calculations and reduced by about 20%.The line width broadening, G: It is the energy broadening parameter as introduced in theprevious chapter.1.5Charge transfer energy ?: Another very important parameter in these calculations specif-ically, and in transition metal oxides generally, is the charge transfer energy or ?. As it wasintroduced in the last chapter,1.3 it is the energy cost to hop one electron from the ligand tothe metal d shell. In this case, it is the difference between on-site energy of d7L10 and d8L9 con-figurations. With this definition, the on-site energies for each configurations can be expressedin terms of ? and the monopole Coulomb repulsion energies, Udd and Upd and the bare on-siteenergies of electrons in each shell, p, d and L as follows:182.1. Configuration Interaction Model in a Single Cluster CalculationFigure 2.1: The relative on-site energies of the different configurations for the initial and finalstates in 2p-XAS in terms of ?, Udd and Upd [9]Ground state conf. On-site energy Relative on-site energyp6d7L10 6p + 7d + 10L +(72)Udd 0p6d8L9 6p + 8d + 9L +(82)Udd d ? L + 7Udd ? ?p6d9L8 6p + 9d + 8L +(92)Udd 2? + Uddp6d10L7 6p + 10d + 7L +(102)Udd 3? + 3UddTable 2.1: Ground state configurations and the corresponding energies.Excited state conf. On-site energy Relative on-site energyp5d8L10 5p + 8d + 10L +(82)Udd ?(81)Upd 0p5d9L9 5p + 9d + 9L +(92)Udd ?(91)Upd ? + Udd ? Updp5d10L8 5p + 10d + 8L +(102)Udd ?(101)Upd 2? + 3Udd ? 2UpdTable 2.2: Excited state configurations and the corresponding energiesAfter basis sets are defined, operators as matrices and wavefunctions as vectors also haveto be created in order to start the calculations. In this case they are created in cubic (Oh)symmetry.192.2. Seeking the Best Agreement with Experiment2.2 Seeking the Best Agreement with ExperimentNow we can set numerical values to the introduced parameters and generate the XAS spectra.We seek the best agreement with the experiment. To this aim, for each parameter, in a reason-able range of values the calculations are repeated in small steps. The best fitting is happenedat the following values: pds = ?1.9, tpp = 0.8, Udd = 7.5, Upd = 9.0, 10Dq = 0.5,? = 2.5, G =1eV . The final match corresponding to the above values are shown in Fig. 2.2 The criteria forgoodness of fit are mostly the intensity ratio between between the peaks in L3 (in the doublepeak structure), the line shape and the spin orbit coupling splitting energy.For each set of values, besides the XAS spectrum, the expectation values of some otherquantities are also calculated, which are presented at the top of each spectrum. Neg, Nt2g andNd are the average numbers of electrons in deg and dt2g orbitals and their sum in the d shellrespectively. The square of the coefficients ?1 ,?2,?3 and ?4 are also calculated. As they havebeen defined earlier, they are the indications of each configuration contribution to the groundstate.?g = ?1|p6d7L10?+?2|p6d8L9?+?3|p6d9L8?+?4|p6d10L7? = ?1|?7?+?2|?8?+?3|?9?+?4|?10?P7 = |??7|?g?2 = ?21P8 = |??8|?g?2 = ?22P9 = |??9|?g?2 = ?23P10 = |??10|?g?2 = ?244?i=1Pi = ?21 + ?22 + ?23 + ?24 = 1By comparing the calculated and experimental spectra, it can be seen that other than somevery general features like the spin-orbit coupling splitting, the other details do not agree. Thesplitting in calculated L3 peak is smaller than the experiment. The line shapes in both peaks donot match very well and the shoulder in the calculated L3 peak is not present in the experiment.The differences are shown more clearly in Fig. 2.3 of the L3 peaks alone. The differences areeven more considerable in Fig. 2.4 which the calculation is done with the smaller broadeningof 0.2 eV (compared the previous 1 eV broadening) which is in fact more realistic. There areat least three significant multiplet structure peaks which are not present in the experiment.While it is usually believed that the d7L10 configuration has the largest contribution to theground state[13] , it is found in our calculations that in the final result, d8L9 configuration hasa larger contribution number than d7L10 with ?22 larger than 50% .It is an important result as202.2. Seeking the Best Agreement with Experimentit can be the reason why the calculations do not work well here.That perhaps is an indication that the cluster is not big enough, because it seems that thestarting point should have been the starting point in which we start with nickel d8 configurationand a hole in the oxygens and then we can no longer use a single local cluster because all theother nickels are also contributing ligand holes. for example in Pr3+NiO3 the nickel would be2+ rather than the former 3+, and there will be one hole per each three oxygens or two holesfor the whole single octahedron with six oxygens and that is not included in our Hilbert space.These holes can propagate and form bands with other clusters which are not included in theHilbert space of a single cluster calculation either.It might sound strange that how with a positive ?, the ground state wavefunction has alarger d8L9 contribution than the d7L10, which basically indicates the d8L9 state has a lowerenergy than the other one. The explanation is hidden in the definition of the ?. As it hasbeen defined earlier, it is an average energy difference between the two states and does notinclude crystal field splitting. It is simply defined in terms of the single particle energies whichare kind of an average energy, d and L, and monopole Coulomb interaction energy, Udd anddoes not include the other Coulomb terms either. But the truth is, each of these states hasmany multiplets and therefore there are many energy differences between these two states.Apparently there must have been a particular d8L9 multiplet with a lower energy than a d7L10multiplet.212.2. Seeking the Best Agreement with Experimenttpp = 0.8, pds = ?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9, ? = 2.5, Distortion =0, Broadening = 1. eVNeg = 1.707, Nt2g = 5.966, Nd = 7.673?21 = 0.411, ?22 = 0.508, ?23 = 0.079, ?24 = 0.002S2 = 1.031Figure 2.2: The final calculated XAS spectra comparing the experiment222.2. Seeking the Best Agreement with ExperimentFigure 2.3: Comparing the L3 peaks in the calculations and the experimentFigure 2.4: The same calculated L3 peak with the lower broadening of 0.2 eV compared withexperiment23Chapter 3Spin State Transition3.1 IntroductionIt is generally accepted that if Ni is in a d7 state in these materials it is most likely in a low spinstate contrary to what it for example is in NiO where it is in a high spin state. The spin stateis of course of great importance in describing the magnetic properties but also the electronicstructure. So we should check if the XAS spectrum matches a high spin or a low spin state.To study this I present a brief introduction to the spin state transition that would occur if wechange the parameters for example to a very large charge transfer energy . I also then comparethe XAS spectra to demonstrate that we must be in a low spin state which provides limits tothe charge transfer energy and the covalent hopping integral pds.Fig. 3.1 is a simpler illustration of Fig. 1.6 which shows the energy levels of the d-transitionmetal oxide in an octahedral geometry considering the covalency.Figure 3.1: The energy levels of a d-transition metal oxide in an octahedral geometry consideringthe covalency with surrounding oxygens.in perturbation theory for large delta compared Veg the splitting is determined by:splitting energy =V 2eg ? V2t2g?The parameters are defined in the first chapter.Now Fig. 3.2 shows how by decreasing the ? or increasing the pds or Veg, which increasesthe splitting energy, the system can experience an abrupt transition from a high spin state toa low spin state ,as dictated by Hund?s rule.243.1. IntroductionFigure 3.2: It shows how by decreasing the charge transfer energy, splitting increases andsubsequently system changes from High spin State to Low spin State.Apparently, this transition can occur when the low spin state has a lower energy than thehigh spin one. These energies can be determined by the Hund?s energy expression as follows:In both cases the number of electrons or n is the same so only the Hund?s exchange energy andon-site energies are going to be considered:E(HS) = ?(52)J + 5(?4Dq) + 2(6Dq) = ?10J ? 8DqE(LS) = ?2(32)J + 6(?4Dq) + 6Dq = ?6J ? 18DqTo HS to LS transition:?6J ? 18Dq < ?10J ? 8Dq4J < 10DqTherefore, when the energy 4J is smaller than the energy 10Dq the low spin state has alower energy and the spin transition can occur.253.2. Low Spin to High Spin Transition3.2 Low Spin to High Spin TransitionIn this section the critical values of the hopping integrals and the charge transfer energy whereinPNO experiences a transition from low spin to high spin states, are obtained and discussed.The expectation value of S2 operator in the ground state is also calculated, which shouldbasically be s(s + 12). As it has been discussed in the previous section the total spin squaredvalue in a nickelate single cluster is either 3.75 in the high spin state or 0.75 in the low spinstate while the calculated numbers here are either about 3 in HS or about 1 in LS. The reason ishidden in the program?s spin operator definition and calculation. Here the total spin operatoronly acts on the nickel atom not on the whole cluster including the ligands and that is why thenumbers differ. But they can still be an indication for the spin state .In changing the parameters most changes in the shape of the spectra are continues andgradual except at some points in changing the ? and pds. At some certain values even a 0.05eV change in them alters the shape drastically. It is also true that at those points the spinvalue abruptly changes from about 1 to about 3. So it seems that at those specific values of? or pds, the material changes from low spin to high spin state and that is why it also showsa very different behaviour in its XAS. Fig.3.3 and 3.4 are the spectra just before and after thetransition with increasing ? only by 0.05 eV from 3.75 to 3.80 eV and Fig. 3.5 and 3.6 are theones with changing the pds by 0.05 eV from -1.65 to -1.60 eV.Remember, at this point, the fitting does not matter any more, here we only care about thespin state for any d7 system with the d ? d Coulomb energy of 7.5 eV, 10Dq of 0.5 eV and azero distortion which also gives us limits to the charge transfer energy and the covalent hoppingintegral pds in our low spin system.It is worth looking at another high spin d7 system to compare the high spin XAS spectrawith. In Fig. 3.7 you may find the XAS spectra for high spin cobaltate which is a d7 high spinmaterial. [4] It can be observed that the both high spin systems show some similar behavioursin their XAS line shapes.As it explained above by increasing ? or decreasing the absolute value of pds the splittingenergy decreases and system experiences a low spin to high spin transition ( 3.2) So for eachspecific value of ? there is a specific critical value of pds which can produce the needed splittingenergy to make the spin transition and vice versa. To find these set of values, the map of the?spin squared? values in terms of pds and ? is obtained. It is shown in Fig.3.8 which has twodistinct spin areas corresponding to low spin and high spin states. Each point in the red orhigh spin area gives the corresponding high spin pds and ? values. Therefore, for any givenvalue of pds and ? the spin state can be predicted by the map.263.2. Low Spin to High Spin Transitiontpp = 0.8, pds = ?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9, ? = 3.75 Dist = 0 eVNeg = 1.618, Nt2g = 5.958, Nd = 7.575?21 = 0.485, ?22 = 0.455, ?23 = 0.058, ?24 = 0.001S2 = 0.993Figure 3.3: XAS spectra just before the spin state transitiontpp = 0.8, pds = ?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9, ? = 3.80, Dist = 0 eVNeg = 2.386, Nt2g = 4.993, Nd = 7.379?21 = 0.646, ?22 = 0.329, ?23 = 0.025, ?24 = 0.000S2 = 3.099Figure 3.4: XAS spectra just after the spin state transition273.2. Low Spin to High Spin Transitiontpp = 0.8 pds = ?1.65 10Dq = 0.5, Udd = 7.5, Upd = 9, ? = 2.5, Dist = 0 eVNeg = 1.673, Nt2g = 5.960, Nd = 7.633?21 = 0.434, ?22 = 0.499, ?23 = 0.065, ?24 = 0.001S2 = 1.046Figure 3.5: XAS spectra just before the spin state transitiontpp = 0.8, pds = ?1.60, 10Dq = 0.5, Udd = 7.5, Upd = 9, ? = 2.5, Dist = 0 eVNeg = 1.673, Nt2g = 5.960, Nd = 7.633?21 = 0.434, ?22 = 0.499, ?23 = 0.065, ?24 = 0.001S2 = 3.049Figure 3.6: XAS spectra just after the spin state transition283.2. Low Spin to High Spin TransitionFigure 3.7: High spin CoO XAS spectra from experiment293.2. Low Spin to High Spin TransitionFigure 3.8: Map of the spin values versus ? and pds showing the low spin to high spin transition303.2. Low Spin to High Spin TransitionAnother interesting feature regarding spin state transition is illustrated in the ground stateenergy level diagrams. The first fifty ground state eigenvalues versus pds and ? are shownrespectively in Fig.3.9 and 3.10 It can be seen that in both cases exactly at the spin statetransition critical values, the first two energy levels are crossed. It basically says that whilebefore the low to high spin transition, the lowest ground state energy is corresponding to thespin half state and the first excited one is corresponding to the spin 32 , after the transition thesituation goes the other way around making the spin 32 state the new ground state and the firstexcited state the spin half state.Figure 3.9: Energy level diagram versus pdsFigure 3.10: Energy level diagram versus charge transfer energyHowever by doing the MLFT cluster calculation in a nickelate single cluster, some interesting313.2. Low Spin to High Spin Transitionfeatures such as the spin transition have been obtained, the final XAS result does not match tothe experiment very well unlike the result for NiO, MnOor SrT iO3. [8] Therefore we can notreally rely on the values have obtained by fitting the above calculations to the XAS experiment.In the next chapter the same method is used to obtain RIXS spectra in order to understandthe material and the employed theory more deeply.32Chapter 4RIXS Calculations4.1 Seeking the Best Agreement with ExperimentHere again we need an experimental reference which is RIXS results from Geneva on NdNO3.In NNO, nickel is again 3+ and since we are treating a single cluster of nickelate, it should notbe very different than the Nickel 3+ in PNO [19]. The spectra is shown in Fig.4.1As explained in the previous chapter, here photon excites an electron from the 2p to the3d shell, then an electron decays into the core hole and a photon comes out. The first peakcorresponds to the elastic process where no energy is lost and the second electron decays fromthe same level of energy as the excited one.Fig. 4.1 shows the experimental RIXS spectra at different resonance energies about theXAS L3 peak for two different temperatures. RIXS can be conducted by employing light withtwo different polarizations as here in Fig.4.1 LV and LH denote vertical and horizontal lightpolarizations.The geometry of this experiment is illustrated in Fig.4.2. It clarifies what the different lightpolarizations imply here.From Fig. 4.2 the following vectors? orientations are concluded. The calculations have alsodone with respect to the experiment geometry.Kin = {cos 15?, 0, sin 15?}Kout = {cos 65?, 0, sin?}?in = LVin = {0, 1, 0}?out = LVout = {0, 1, 0}piin = LHin = {? sin 15?, 0, cos 15?}piout = {? sin 65?, 0, cos 65?}334.1. Seeking the Best Agreement with ExperimentFigure 4.1: The experimental RIXS spectra form Geneva at T=300K in the left and at T=15Kin the right.To calculate these kind of spectra in principle three basis sets and three Hamiltonians areneeded, corresponding to the initial, intermediate and final states. But in this case initial andfinal basis sets are the same and the intermediate state has the same basis set as the XAS state.A very simple sketch of this second order process for a d7 system is illustrated in Fig.4.3 Itonly shows one of the d? d excitation possibilities, which actually should be the first one.For each incoming polarization, the RIXS spectra are obtained by adding the both possibleoutgoing light polarizations. Now by choosing the resonance energy from XAS spectra, thecorresponding RIXS spectrum can be calculated.To be able to compare the results with experiment, the same resonance energies near theL3 peak, on the left shoulder, are chosen. The energies are relative, they are shifted to matchthe experiment as much as possible for convenience. The other parameters such as hopping344.1. Seeking the Best Agreement with ExperimentFigure 4.2: RIXS experimental geometryFigure 4.3: A schematic sketch of the d? d excitations in a d7 low spin system.integrals, charge transfer energy and 10Dq are set as the final values in the second chapterobtained by matching to the XAS from experiment . The intermediate state(XAS) broadeningand RIXS spectra broadening are set to be 0.2 and 0.3 eV respectively. The resonant energiesare shown on the XAS spectra in Fig.4.5 and the corresponding RIXS results are presented inFig. 4.4The room temperature RIXS spectra in Fig. 4.1 shows a continuum behaviour in its d? dlike excitations. By looking at the phase transition diagram for NdNo in chapter one,1.1 it can354.1. Seeking the Best Agreement with Experimentbe seen that at 300?, the system is in its metallic phase. So there is no gap to jump and makean excitation. That is why the spectra basically says that for any given energy an excitationcan occur. Our calculations does not include metallic phase. It models the insulting behaviouras discussed in the previous chapters.In the low temperature RIXS results, the lowest excitations happen at about 0.8 eV. Fornow, we assume it is corresponding to the lowest d ? d excitation and no excitation peak islost in the elastic peak broadening. On the other hand, in the calculated spectra the firstobservable excitation peak happens at about 1.4 eV as shown in Fig. 4.4 To understand whichexcitation this peak corresponds to, the first eigen-energies of the system are calculated. Theyare presented in table 4.1 The peaks? energies in the RIXS spectra are basically the relativeeigen-energies of the system. The magnon and phonon excitations are not considered here.table 4.1 suggests that in fact the first d ? d excitation peak in our calculation happens at atoo low energy (0.25 eV) and can not be observed in our spectra. In order not to loose it, theRIXS broadening is decreased to 0.2 eV. The results at the resonant energies of 853 and 853.2eV are presented in Fig.4.6 and the relative eigen-energies are also shown on them with thevertical lines. The other spectra at the other incident energies are presented in the appendix.364.1. Seeking the Best Agreement with ExperimentFigure 4.4: Calculated RIXS spectra for the following values: tpp = 0.8, pds = ?1.9,? =2.5, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV and RIXS broadening of 0.3 eV374.1. Seeking the Best Agreement with ExperimentFigure 4.5: The resonant energies taken for the RIXS calculations, at 852 ,852.5, 853, 853.2,853.5, 853.8, 854.1, 854.4 eV with the following values tpp = 0.8, pds = ?1.9,? = 2.5, 10Dq =0.5, Udd = 7.5, Upd = 9 eV and RIXS broadening of 0.2 eV384.1. Seeking the Best Agreement with ExperimentTable 4.1: The first 25 eigen-energies of the system with ? = 2.5eV .394.1. Seeking the Best Agreement with ExperimentFigure 4.6: Calculated RIXS spectra with the following parameters: ? = 2.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV and RIXS broadening of 0.2 eV. The vertical axis isthe intensity in an arbitrary unit and the horizontal axis is the energy loss in eV.404.1. Seeking the Best Agreement with ExperimentThis inconsistency between the experiment and the calculation can be addressed by tuningthe charge transfer energy (?) and hopping integral (pds). The splitting enegy which determinesthe energy at which the first d ? d excitation happens, is inversely proportional to the chargetransfer energy. Therefore, to increase this energy from about 0.25 to about 0.8 eV we haveto decrease ? dramatically. By calculating the eigen-energies of the system with the differentsmaller ?s, for ? = 0.5eV , the splitting of 0.67 eV is obtained which is almost equal to thesplitting energy in the experiment.The XAS spectra with this ? and the given resonant energies are shown in Fig.4.7 Thecorresponding RIXS spectra at the two of these energies are also presented in Fig. 4.8, theother ones are shown in the appendix.Figure 4.7: The resonant energies taken for the RIXS calculations, at 851.2 ,851.6, 852, 852.3,852.5, 852.7, 852.9, 853.4, 853.6 eV with the following values ? = 0.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV414.1. Seeking the Best Agreement with ExperimentFigure 4.8: Calculated RIXS spectra with the following parameters: ? = 0.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV424.1. Seeking the Best Agreement with ExperimentTable 4.2: The first 25 eigen-energies of the system with ? = 0.5eV .What we have done so far is decreasing the charge transfer energy from 2.5 to 0.5 eV toincrease the splitting or the first d ? d excitation energy to 0.67 eV in order to match thecalculated RIXS to the experimental ones. Now the problem with the new RIXS spectra is theintensity ratios and the polarization dependence. To adjust the splitting energy by changing?, we had assumed that the first peak in the experiment is the one associated with the firstto the second eigen-states. It might not be the case. The first inelastic peak in experimenthas an intensity comparable to the elastic peak or even larger. It also has a double peak whichshows polarization dependence at some resonant energies. They all are the characteristics of oursecond inelastic calculated peak. Therefore, it can be concluded that the first excitation in theexperiment is also happening in a very low energy smaller than the experiment resolution (less434.1. Seeking the Best Agreement with Experimentthan 0.1 eV) and then the second inelastic excitation happened at bout one eV. By consideringthe relation between the charge transfer energy and the hopping integral and the splittingenergy, and calculating the first eigen-energies for several cases, it has been obtained that forthe values of ? = 0.8 and pds = ?1.4, the best agreement can be obtained. Remember wehave to consider the spin transition and not let the system make the transition to the high spinstate. The allowed low spin ? and pds values can be taken from the spin state transition mapin the previous chapter. 3.8The final RIXS results are also shown in figures 4.10 to 4.16.Figure 4.9: The resonant energies taken to calculate RIXS spectra with the following valuestpp = 0.8, pds = ?1.4,? = 0.8, 10Dq = 0.4, Udd = 7.5, Upd = 9 eV with energy shift of 865eV444.1. Seeking the Best Agreement with ExperimentTable 4.3: The first relative eigen-energies of the system with the following values: tpp =0.8, pds = ?1.4,? = 0.8, 10Dq = 0.4, Udd = 7.5, Upd = 9 eV454.1. Seeking the Best Agreement with ExperimentFigure 4.10: Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9 eV464.1. Seeking the Best Agreement with ExperimentFigure 4.11: Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9 eV474.1. Seeking the Best Agreement with ExperimentFigure 4.12: Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9484.1. Seeking the Best Agreement with ExperimentFigure 4.13: Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9494.1. Seeking the Best Agreement with ExperimentFigure 4.14: Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9504.1. Seeking the Best Agreement with ExperimentFigure 4.15: Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9514.1. Seeking the Best Agreement with ExperimentFigure 4.16: Calculated RIXS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9524.2. XAS Spectra with Parameters Driven from RIXSTherefore, in order to match the elementary excitation energies in the experimental RIXSexperiment with the calculation, we had to decrease charge transfer energy significantly andadjust the hopping integral and 10Dq a little bit. In terms of the polarization dependencein the calculated final RIXS, it has been seen that while there is a considerable polarizationdependance at left side of the L3 peak, there is no such a dependence at the right side, wherethe double peak should have existed. Considering the above points and the fact that even withthe big ? of 2.5 eV, the calculated and experimental XAS spectra presented huge differences,it can be concluded that the double peak in the L3 may not be a result of multiplet structurebut could have an origin beyond what can be included in a isolated cluster approach.We should mention that the resonant inelastic x ray scattering spectra extended out to anenergy region including the second peak of the L3 XAS demonstrate a linear dispersion withincident energy after having passed through the first peak. This clearly demonstrates that thesecond peak has a very different origin and must involve a continuum state in which the excitedelectron in a state decoupled from the core hole and so does not participate in the decay to thecore hole state. The resulting spectrum then is that of x ray fluorescence rather than resonantx ray inelastic scattering.4.2 XAS Spectra with Parameters Driven from RIXSThe new fitted parameters from the previous section are ? = 0.8, tpp = 0.8, pds = ?1.4, 10Dq =0.4, Udd = 7.5, Upd = 9, these new values will alter the calculated XAS spectra a lot. Withthis smaller ?, the double peak in the L3 peak will disappear. From the XAS calculation,we already know that in these type of calculations changing the charge transfer energy mostlychanges the L3 double peak intensity ratio, but from the RIXS results we concluded that maybeit is not the right approach to get the ? from. This new calculated XAS is presented in Fig.4.17 and the configuration contribution numbers and the number of electrons in the d shell arealso calculated and presented. It can be seen that with these new values for the charge transferenergy and pds, d8L9 configuration still has the largest contribution to the ground state andsystem is still in the low spin state.534.2. XAS Spectra with Parameters Driven from RIXStpp = 0.8, pds = ?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9, ? = 0.8, Distortion = 0. eVNeg = 1.79, Nt2g = 5.97, Nd = 7.76?21 = 0.34, ?22 = 0.56, ?23 = 0.09, ?24 = 0.00S2 = 1.10Figure 4.17: Calculated XAS spectra with the following parameters: ? = 0.8, tpp = 0.8, pds =?1.4, 10Dq = 0.4, Udd = 7.5, Upd = 9 eV54Chapter 5ConclusionFrom the Zaanen-Sawatzky-Allen scheme, using only a few parameters (the charge trans- ferenergy and the d d Coulomb interaction energy) is able to account for the electronic be-haviour of a great number of 3d TM oxides such as the phase transitions. In order to find theempirical values for these energies for PNO, 2p-core X-ray absorption spectra were calculatedwithin multiplet ligand field theory for the PNO single cluster. Then by adjusting the calculatedspectra with the experiment, the best agreement happened at ? = 2.5 eV and pds=-1.9 eV.Changing the charge transfer energy and the covalent hopping integral mostly changes theL3 double peak intensity ratio until it reaches the spin transition and changes the line shapedrastically. Therefore, they mostly are chosen to satisfy this ratio to the most. Even though thebest match was not satisfactory and differed with experiment in this ratio and the presence ofa shoulder there. The differences are even more considerable for a smaller life-time broadeningof 0.2 eV with multiple peaks present.Then, the low spin to high spin state transition was studied and the critical values of ? andpds at which the system experiences an abrupt spin transition were obtained. A map also waspresented in which for any values of the charge transfer energy and covalent hopping integralthe spin state can be predicted.Then, another approach was employed to adjust these crucial parameters. The RIXS spec-tra were calculated in the same theory, basis set and ground state. In experimental RIXS,the first excitation peaks appeared at less than 1 eV. They were double peaked and showedpolarization dependence at some resonant energies on the left side of the L3. By calculatingseveral RIXS spectra and corresponding the relative eigen-energies to the d?d excitation peaksand considering the above features of the first observed inelastic peaks in the experiment, itwas concluded that these first peaks in fact correspond to the second d? d excitation in about1 eV and the first excitation should have been in a very low energy lower than the experimentenergy resolution (less than 0.1 eV).The values which satisfies these energies (the second relative eigen-energy at less than 0.1eV and the third one at about 1 eV) were found to be ? = 0.8, tpp = 0.8, pds = ?1.4, 10Dq =0.4, Udd = 7.5, Upd = 9 eVTherefore, to get the best agreement with RIXS in terms of the d ? d excitation energies,the charge transfer energy and covalent hopping integral change a lot. These new values do notgive the best agreement with XAS any more and the double peak on L3 will be lost. However,the fact that even the best XAS match was not satisfactory at all and the fact that no RIXS55Chapter 5. Conclusionpolarization dependence was found on the right side of the L3 unlike the left side, and the factthat the resonant inelastic x ray scattering spectra extended out to an energy region includingthe second peak of the L3 XAS demonstrate a linear dispersion with incident energy after havingpassed through the first peak demonstrates that the second peak has a very different origin andmust involve a continuum state in which the excited electron in a state decoupled from the corehole and so does not participate in the decay to the core hole state and the true values for ?and pds should not be obtained by keeping this second peak in L3 in MLFT calculations.There are some other reasons which might have caused the differences between the calcula-tions and the experiment. In our calculations it has been shown that for almost all the low spincases, the d8L9 configuration has the largest contribution in the ground state. It can suggestthat the true starting point for this problem might be the starting point in which we start withnickel d8 configuration and a hole in the oxygens and then we can no longer use a single localcluster because all the other nickels are also contributing ligand holes.Here also for simplicity, the cubic symmetry and zero distortion have been assumed whichis not exactly the case in the real material and might have caused the disagreements.56Bibliography[1] Resonant inelastic x-ray scattering. http://en.wikipedia.org/wiki/Resonant_inelastic_X-ray_scattering, October 2013.[2] Spherical coordinate system. http://en.wikipedia.org/wiki/Spherical_coordinate_system, October 2013.[3] Martin Buschke. Orbital reconstruction at the surface of lanthanum nickelate thin films.B.Sc thesis, 2012.[4] C. F. Chang, Z. Hu, Hua Wu, T. Burnus, N. Hollmann, M. Benomar, T. Lorenz, A. Tanaka,H.J. Lin, and H.H. Hsieh. Spin blockade, orbital occupation, and charge ordering inLa1.5Sr0.5CoO4. Physical Review Letters, 102(116401), 2009.[5] Jew-Tin Chen. Coordination chemistry. http://www.ch.ntu.edu.tw/~jtchen/course/inorganic/coord%20chem.html, October 2013.[6] A. Fujimori and F. Minami. Physical Review B, 30(957), 1984.[7] Frank de Groot and Akio Kotani. Core Level Spectroscopy of Solids. CRC press, 2008.[8] M.W. Haverkort, M. Zwirzycki, and O.K. Andersen. Multiplet ligand-field theory usingwannier orbitals. Physical Review B, 85(165113), 2012.[9] W. Maurits Haverkort. Spin and orbital degrees of freedom in transition metal oxides andoxide thin films studied by soft x-ray absorption spectroscopy. PhD thesis, 2005.[10] S. Hufner. Solid state Commun, 49(1177), 1985.[11] S Macke, A Radi, J. E. Hamann-Borrero, R. Sutarto, F. He, G. Logvenov, and G. Chris-tiani. X-ray absorption spectrum. Canadian Light Source and Stuttgart Max Planck In-stitute.[12] Dick van der Marel. The electronic structure of embedded transition metal atoms. PhDthesis, 1985.[13] Maria Luisa Medarde. Structural, magnetic and electronic properties of rnio3 per-ovskites(r=rare earth). J. Phys.: Condens. Matter 9, (16791707), 1997.[14] C.E. Moore. Atomic energy levels. NBS Circular, (467), 1949.57Bibliography[15] C.N.R Rao. Transition metal oxides. Annu. rev. phys. chem., 40(291-326), 1989.[16] G.A Sawatzky and J.W. Allen. Physical Review letter, 53(2339), 1984.[17] George Sawatzky. Electronic structure of correlated electron systems. http://www.phas.ubc.ca/~berciu/TEACHING/PHYS555/FILES/, September 2013.[18] Attila Szabu and Neil S. Ostlund. Modern Quantum Chemistry. 1996.[19] J.B. Torrance, P. Locorre, A.I. Nazzal, E.J. Ansaldo, and Ch. Niedermayer. Systematicstudy of insulator-metal transitions in perovskites RNiO3(R=Pr,Nd,Sm,Eu) due to closingof charge-transfer gap. Physical Review B, 45(14), 1991.[20] J. Zaanen. PhD thesis, 1986.[21] J. Zaanen, G. A. Sawatzky, and J.W. Allen. Band gaps and electronic structure oftransition-metal compounds. Physical Review letters, 55(4), 1985.[22] J Zaanen, C. Westra, and G. A. Sawatzky. Determination of the electronic structure oftransition-metal compounds: 2p x-ray photoemission spectroscopy of the nickel dihalides.Physical Review B, 33(12), 1986.[23] Pavlo Zubko, Stefano Gariglio, Marc Gabay, Philippe Ghosez, and Jean-Marc Triscone.Interface physics in complex oxides heterostructures. Annu. Rev. Condens. Matter Phys.,65(141), 2011.58AppendixHere the resonant inelastic scattering spectra at various incident energies and with the chargetransfer energy of 2.5 and 0.5 eV are presented. The photon energies are about the X-rayabsorption L3 peak. The vertical axis is the intensity in an arbitrary unit and the horizontalaxis is the energy loss in eV. The descriptions can be found in the chapter 4 of this thesis.Figure 5.1: Calculated RIXS spectra with the following parameters: ? = 2.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV and RIXS broadening of 0.2 eV59AppendixFigure 5.2: Calculated RIXS spectra with the following parameters: ? = 2.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV and RIXS broadening of 0.2 eV60AppendixFigure 5.3: Calculated RIXS spectra with the following parameters: ? = 2.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV and RIXS broadening of 0.2 eV61AppendixFigure 5.4: Calculated RIXS spectra with the following parameters: ? = 0.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV62AppendixFigure 5.5: Calculated RIXS spectra with the following parameters: ? = 0.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV63AppendixFigure 5.6: Calculated RIXS spectra with the following parameters: ? = 0.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV64AppendixFigure 5.7: Calculated RIXS spectra with the following parameters: ? = 0.5, tpp = 0.8, pds =?1.9, 10Dq = 0.5, Udd = 7.5, Upd = 9 eV65
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X-ray absorption and resonant inelastic X-ray scattering calculations with ligand field single cluster… Balandeh, Shadi 2013
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Title | X-ray absorption and resonant inelastic X-ray scattering calculations with ligand field single cluster method on praseodymium nickel oxide |
Creator |
Balandeh, Shadi |
Publisher | University of British Columbia |
Date Issued | 2013 |
Description | RNiO3 perovskites (R=rare earth) are one of the most interesting compounds in condensed matter physics presenting various unusual physical properties. The detailed electronic structure of these materials are very controversial at the present time. The charge transfer energy and the d-d Coulomb interaction are the two very important parameters which can explain their electronic behaviours nicely. However, predicting their values has been a challenge to the science society so far. X-ray Absorption Spectroscopy (XAS) and Resonant Inelastic Scattering (RIXS) are the two very useful techniques to probe the electronic structure of a solid state system in general and predict these two energies in particular. In this thesis Multiplet Ligand Field Cluster Calculation (MLFCC) is used to calculate these two spectra, then the charge transfer energy (Δ), the covalent hopping integral(pds), and the d-d Coulomb repulsion energy Udd are obtained by fitting the calculated spectra to the experiment. In this work, the calculated XAS results are compared with the experiment and the adjusted values are introduced as Δ=2.5eV , pds=-1.9eV, Udd=7.5eV and 10Dq=0.5 eV. The low spin to high spin transition is also studied and the critical charge transfer energies and covalent hopping integrals are calculated at which the abrupt transition happens. It is also found that in almost all low spin cases the d8L9 configuration has the largest contribution to the ground state. Since the best fit of XAS is not satisfactory and displays considerable differences with the experiment, the study is followed with the RIXS calculations. Finally, the calculated RIXS results for different polarizations are compared with the experiment.It results in a smaller Δ=0.8eV and a smaller absolute value of pds=-1.4eV at which the double peak structure in XAS L3 vanishes. This could be an evidence to the fact that XAS should not be interpreted in the conventional way and the Δ should not be fitted to keep the double peak which probably has another source than the multiplet structure. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2013-10-25 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-ShareAlike 2.5 Canada |
IsShownAt | 10.14288/1.0085560 |
URI | http://hdl.handle.net/2429/45392 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2013-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
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