Thesis
Reference
Scanning tunneling microscopy and spectroscopy on nickelate thin films
AMUNDSEN, Thomas
Abstract
We present a scanning tunnelling microscopy study of the metal insulator (MI) phase transition in NdNiO3 thin films grown on NdGaO3. Although topographic resolution is limited to grain structures and step edges, we did achieve high resolution and reproducible spectroscopic mapping of the thin film surface. The spectra we measure are either metallic or insulating, and mapping the local gap as a function of position shows the emergence of well-defined insulating domains in a conducting background upon cooling the film through the MI transition. When warming the film through the MI transition, the opposite is observed, with metallic domains appearing in the insulating background. There is no gradual shift in the measured insulating gap amplitude at the phase transition. The measured proportion of metallic surface area as a function of temperature follows the resistive hysteresis loop and allows a direct comparison with a percolation model for the MI phase transition.
AMUNDSEN, Thomas. Scanning tunneling microscopy and spectroscopy on nickelate thin films . Thèse de doctorat : Univ. Genève, 2016, no. Sc. 5049
DOI : 10.13097/archive-ouverte/unige:94798 URN : urn:nbn:ch:unige-947987
Available at:
http://archive-ouverte.unige.ch/unige:94798
Disclaimer: layout of this document may differ from the published version.
Département de la matière condensée
Scanning Tunneling Microscopy and Spectroscopy on Nickelate thin films
THÈSE
présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de docteur ès Sciences, mention Physique
par
Thomas Brecke Amundsen
Bergen (Norvège)de
Thèse n 5049
GENÈVE
Centre d’Impression de l’Université de Genève 2017
Ces dernières années, la physique de la matière condensée s’est fortement focalisée sur la recherche de matériaux exhibant des propriétés électroniques exceptionnelles.
Une des raisons de cet engouement est le fait que les technologies industrielles de l’électronique sont essentiellement basées sur le silicium, et qu’elles sont sur le point d’atteindre leurs limites aussi bien au niveau des performances que de leur miniaturi- sation. Parmi ces matériaux, on trouve tous ceux fabriqués à partir du bloc perovski- tique élémentaire ABO3, et dont l’énorme famille comprend entre autres les supra- conducteurs à haute température critique, les composés multiferroïques ou ceux à magnétorésistance colossale.
On y trouve également les nickelates, qui ont la forme RNiO3où R est une des terres rares à l’exception du Lanthane. Ces perovskites orthorhombiques présentent une remarquable transition métal-isolant lorsque l’on abaisse suffisamment la tem- pérature. La température de cette transition (TM I) est fortement dépendante de la nature de l’élément terre rare R : plus celui-ci est gros, plus la structure cristalline du composé se rapproche d’une perovskite cubique, et plus TM Iaugmente.
Les mécanismes qui se cachent derrière cette transition ne sont pas encore com- plètement établis, et l’intérêt pour une meilleure compréhension de ce phénomène est actuellement très important. Il est clairement établi qu’à la transition, le volume de la cellule-unité subit une brusque augmentation et que l’angle Ni-O-Ni augmente vers celui de la structure cubique. Dans le même temps, la structure cristalline passe d’orthorhombique à légèrement monoclinique, et les octaèdres NiO6 ne sont plus dégénérés. Le volume des octaèdres alterne entre petit et grand dans toutes les direc- tions.
Beaucoup d’études ont été faites sur la configuration électronique nécessaire pour avoir simultanément une transition structurelle et électronique. Il est reconnu que les électrons Ni 3d jouent un rôle prépondérant. L’hypothèse est qu’ils alternent entre
bienne et d’interactions de Hund, il est possible d’avoir une transition structurelle accompagnée d’une transition électronique vers un état isolant avec une bande inter- dite d’environ 200 mV d’énergie.
La connaissance de la bande interdite peut donner une information sur le mécan- isme à l’origine de la transition, en indiquant quels sont les états électroniques qui participent à cette transition. Des mesures indirectes comme des mesures de spec- troscopie optique ou de spectroscopie de photoélectrons (ARPES) montrent que la bande interdite se situe dans un intervalle entre 100 et 250 mV. L’outil utilisé dans cette thèse est le microscope à effet tunnel (STM). Cet outil peut directement mesurer la bande interdite de manière très locale avec une précision spatiale inférieure au nanomètre, et une résolution en énergie de l’ordre du mV.
Cette thèse s’est focalisée sur l’étude par microscopie/spectroscopie tunnel de couches minces de NdNiO3 déposées sur un substrat de NdGaO3. Nous avons mesuré les spectres tunnel en fonction de la position puis extrait de chacun la bande interdite à l’aide d’un modèle théorique. Cette étude nous a permis d’établir des cartes de la bande interdite en fonction de la température avec grande précision au voisinage de la transition métal isolant.
Il apparaît tout d’abord très clairement qu’au-dessus de TMI, les caractéristiques I(V) sont pratiquement linéaires, signature typique d’un régime purement métallique.
Au-dessous de TMI, les spectres montrent une déplétion d’états proche du niveau de Fermi, signe de l’ouverture d’une bande interdite dans la structure électronique.
Nous avons établi que la bande interdite se situe aux environs de 180 mV, en accord avec les valeurs qui ont été rapportées dans d’autres études par différentes tech- niques. Ce travail est tout-à-fait en accord avec le modèle de distortion de dispropor- tion de liens.
Grâce aux possibilités offertes par le STM, nous avons pu suivre l’évolution des caractéristiques électroniques locales en fonction de la température. A température ambiante toute la surface de l’échantillon est métallique, avec des caractéristiques I(V) linéaires. Au-dessous de la transition, toute la surface est isolante avec des spectres révélant une bande interdite. Aux températures proche de TMI , nous avons pu clairement démontrer la coexistence de deux phases, une phase métallique et une phase isolante. Tout au long de la transition, les spectres fournissent une unique valeur de bande interdite, et seule la proportion relative des deux phases varie. Les proportions relatives de ces phases sont en accord avec les mesures de résistivité qui ont été effectuées par transport lors de la caractérisation des échantillons. En particulier, le caractère hystérétique de la transition résistive (premier ordre) est aussi observé dans l’évolution en température de la proportion des domaines.
Nos mesures sont en accord avec un modèle percolatif comme mécanisme macro- scopique à l’origine de la transition résistive. Nous avons observé dans l’espace réel la séparation de phases, condition requise pour la percolation, et mesuré un seuil de
R Rare Earth.
XRD X-Ray Diffraction.
DOS Density Of States.
LDOS Local Density Of States.
AF Anti Ferromagnetic.
STM Scanning Tunneling Microscopy.
STS Scanning Tunneling Spectroscopy.
Energy gap in the DOS.
MIT Metal to Insulator Transition.
FT Fourier Transform.
XAS X-ray Absorption Spectroscopy.
PND Powder Neutron Diffraction.
TM I Metal to insulator transition temperature.
o Orthorhombic.
pc Pseudocubic.
ZBC Zero Bias Conductance.
1 Introduction 1
2 Nickelates 3
2.1 Discovery . . . 3
2.2 Crystal structure . . . 4
2.2.1 High temperature phase . . . 4
2.2.2 Low temperature phase . . . 6
2.3 Electronic structure . . . 11
2.3.1 Introduction . . . 11
2.3.2 Sommerfeld . . . 12
2.3.3 Band theory . . . 14
2.3.4 Mott insulators . . . 16
2.3.5 RNiO3 . . . 20
2.3.6 Insulating gap . . . 22
2.3.7 Epitaxial thin films . . . 25
2.3.8 Theoretical research . . . 25
3 Epitaxial thin films 29 3.1 Deposition techniques . . . 29
3.1.1 Chemical vapour deposition . . . 29
3.1.2 Physical vapour deposition . . . 30
3.2 Off-axis radio-frequency magnetron sputtering . . . 32
4 Scanning tunneling microscopy and spectroscopy 35 4.1 History . . . 35
4.2 Theoretical treatment of STM . . . 43
4.3 Experimental setup . . . 45
4.3.1 STM principles . . . 45
4.3.2 The in-house built Aurora STM . . . 48
4.3.3 Sample and tip preparation . . . 50
5 Thin film characterization 53 5.1 X-ray diffraction . . . 53
5.2 Transport resistance measurements . . . 55
5.3 Atomic force microscopy . . . 57
5.4 Surface morphology as measured by STM . . . 59
5.4.1 Lines on the surface of film B . . . 61
6 Tunneling spectroscopy of NdNiO3thin films 63 6.1 I(V) spectra for NdNiO3 . . . 64
6.2 Fitting . . . 66
6.3 Example . . . 67
7 Spatially resolved spectroscopy of NdNiO3thin films 69 7.1 Spectroscopy on film A . . . 70
7.2 Spectroscopy on film B . . . 71
7.2.1 Spectroscopy on film B, T = 290 K . . . 71
7.2.2 Spectroscopy on film B, thermal cycling . . . 73
7.3 Gap value distribution . . . 75
7.4 Hysteresis . . . 77
7.5 Effect of lines on the surface . . . 79
7.6 Spectroscopy on film C . . . 79
8 Percolation 83 8.1 History . . . 84
8.2 Results . . . 88
8.3 Discussion . . . 90
9 Conclusions 93
CHAPTER 1
Introduction
Nickelates with the chemical formula RNiO3, where R is a rare earth element, are an interesting group of materials in part because most exhibit a sharp metal to insulator transition as a function of temperature. With the personal computer and similar devices having grown to be ubiquitous, immense interest has developed in improving said devices. As of today the underpinnings of information technology are transistors that can be used to form logical circuits and memories. These transistors are for the most part built from silicon. The important features of the transistor are the possibility to amplify an electrical current and to open or close an electronic circuit.
The binary function of transistors has led to research into other materials that can fulfill the binary requirement. Nickelates are binary in the sense that as a function of temperature, they are either metallic or insulating. An understanding of the mecha- nism that causes this behaviour could make it possible to tune it as required. Great advances have been made in the research of nickelates and the transition temperature is tunable by changing the rare earth ion, by growing films on substrates that apply a strain in the film or by the application of external pressure. A motivation is often to be able to understand the material to a degree that allows its manipulation in a way that can be useful in a physical device.
In this thesis we present a scanning tunneling microscope (STM) study of the electronic propperties of nickelate thin films. The STM allows for spatially resolved spectroscopic imaging of the sample as the temperature is taken through the metal to insulator transition. This enables an unparalleled view of how the material responds to temperature changes and may give indications on the mechanism underlying the transition.
This work is structured as follows. Chapter 2 details the nickelate family. It includes the major steps in research, a description of the material as we know it and the status of research today. Chapter 3 details the fabrication and characteriza- tion of nickelate thin films. Chapter 4 describes the STM, the central experimental technique used in the study presented here. Its history, theoretical backing and con- struction are discussed. Chapter 5 shows the characterization of the thin films used in this work. Chapter 6 shows the morphology of the nickelate thin films as revealed by high resolution STM. Chapter 7 details the spectroscopy technique and the math- ematical models used to analyse our data. In chapter 8 we study the temperature de- pendence of the nickelate thin films electronic properties by spatially resolved STM spectroscopy. The STM is able to accurately and directly identify the local metallic or non metallic nature of a material. A statistical analysis of this data is presented.
Chapter 9 discusses percolation, a possible mechanism for the macroscopic resis- tance transition. The manuscript ends with concluding remarks and some outlook for future experiments.
In writing this work, I did greatly benefit from a range of very nice review articles by Medarde [1], Catalan [2], Fischer etal. [3], Imada etal. [4] and Landauer [5].
CHAPTER 2
Nickelates
2.1 Discovery
In the 1950s, perovskite-like compounds garnered a lot of interest because of a wealth of fascinating physical properties, among others semiconductivity. The solid state transistor had been discovered by Baardeen etal. [6] and it was based on ger- manium, a semiconductor. S. Geller determined the structure of Gadolinium Or- thoferrite GdFeO3to be a perovskite of space group Pbnm [7]. About a year later A. Wold etal. [8] synthesized Lanthanum Nickel Oxide LaNiO3and tried, but did not manage to form stable compounds involving samarium, gadolinium and yttrium.
The electron configuration of nickel is [Ar] 4s23d8. Ni(II) oxides are easily prepared at low pressures and high temperature, but Ni(III) oxidation state, necessary for the synthesis of RNiO3, where R is short for the Rare Earth element group, necessitates either high pressure or lower temperature [1]. Bismuth nickel oxide BiNiO3was synthesized by Tomashpol0skii etal. at 70 kbar and this served as inspiration for Demazeau etal. [9].
Their reasoning was that no RNiO3 compound except LaNiO3 was known to have been made and that their synthesis in high pressure and in oxygen might be a viable way forward to stabilize the trivalent nickel. Demazeau used a "Belt" device to achieve a pressure of 60 kbar. They mixed R2O3, NiO and KClO3in the proportion 1 : 2 : 1.5 and the thermal decomposition of potassium chlorate produces a very high local oxygen pressure. Using this procedure they managed to produce the rare earth nickelate compounds of yttrium, lanthanum, neodymium, samarium, europium, gan- dolinium, dysprosium, holmium, erbium, thulium, ytterbium and lutetium. They did
not succeed in synthesizing the trivalent nickelate versions of cesium, praseodymium or terbium. They were able to confirm with X-ray diffraction measurements Wolds results [8] that showed that LaNiO3is of a rhombohedral crystal structure in the space group R¯3c. In addition they were able to determine the high temperature crys- tal structure of the rest of the rare earth nickelates they syntesized as being of a orthorhombic perovskite structure in the space group Pbnm, an illustration of which they included and is shown in figure 2.1.
Figure 2.1: The crystal structure of the orthorhombic Pbnm perovskites. T indicates rare earth (terre rare in french). A pseudolattice is indicated with thick lines. From [9].
After being a dormant field of study for around two decades, the discovery of superconductivity with a high critical temperature in the cuprates [10] led to a re- newed interest in nickelate oxide systems. In 1990, Lacorre etal. [11] succeeded in synthesizing RNiO3versions of Sm, Nd and for the first time Pr. They discovered that while cooling the sample, the resistivity of the samples went through a metal to insulator transition (MIT) and that the transition was discontinuous for Nd and Pr, indicating a first-order transition, shown in figure 2.2.
2.2 Crystal structure
2.2.1 High temperature phase
RNiO3compounds are of a ABX3type structure where X is an anion, B is a cation with preference for octahedral coordination and A is a cation with an oxidation state
Figure 2.2: The logarithm of electrical resistance for SmNiO3, NdNiO3, PrNiO3and LaNiO3. All shows a sharp MIT, except the La compound which remains metallic.
From [11].
so that the material is electrically neutral. In our case the formal oxydation states are R+3Ni+3O32, meaning that on average the oxygen atoms in the unit cell each get two more electrons and fill up their 2p shell, while R and Ni loose three electrons and leaves the Ni atom with seven electrons in its 3d shell and none in its 4s shell.
In the metallic regime, LaNiO3is rhombohedral, while the other RNiO3are or- thorhombic perovskites. They are distorted variants of the cubic perovskite (Pm¯3m) as exemplified by the mineral "perovskite" CaTiO3 at temperatures above 900˚C.
The structure which is drawn in fat stripes in figure 2.1 is the unit cell for cubic per- ovskites if one imagines the sides in all three directions being of equal length and the BX3being undistorted. The tilting of the oxygen octahedra is the main contributor to the orthorhombic distortion as Garcia-Munoz etal. showed in 1992. This tilting of the octahedra is accompanied with a slight shift in the R position in the a-b plane.
They managed to completely and accurately map the crystallography of RNiO3with R = La, Pr, Nd and Sm with a higher resolution than what had been achieved previ- ously using neutron diffraction [12]. The a and b axis of the orthorhombic unit cell are tilted by 45˚ compared to the cubic cell and the base vectors (001) and (010) are approximatelyp
2times longer while the (100) vector is approximately double the length.
The tilt of the NiO3octahedra occurs because the R atom does not fill up the entire space that is left in the center of the unit cell. This causes a buckling of the Ni - O - Ni atom chain. The smaller the central R atom is, the more severe will the tilt
be. The tilt can be parametrized by a tolerance factor t.
t= dN i O p2dR O
(2.1) In the ideal case for CaTiO3, t is equal to 1. In the case of La, the rare earth ion is slightly too small to fill up the central space and the resulting stable structure is rhombohedric with a tolerance factor t⇡0.986. For smaller rare earth ions, the structure is orthorhombic and t becomes gradually smaller, starting with t⇡0.975 for Pr to t⇡0.932 for Lu [11]. The orthorhombic structure can also be parametrized by the Ni-O-Ni angle ⇥. For the cubic perovskite, ⇥is equal to 180˚. ⇥for the rhombohedric LaNiO3is equal to 165.2˚. ⇥for PrNiO3, the biggest rare earth ion of the orthorhombic nickelates, is equal to 158.5˚. As the rare earth ion is reduced in size,⇥for LuNiO3is equal to 143.4˚ [13]. The angle reduction is significant. t and
⇥vary in a very similar way and their correlation follows
⇥⇡275 t˚ (2.2)
so that when the tolerance factor is reduced, the Ni-O-Ni angle is reduced.
2.2.2 Low temperature phase
As the temperature is lowered the crystal structure undergoes a first order transition.
Garcia-Munoz etal. showed that the unit cell volume changes discontinuously by 0.25% for PrNiO3, by 0.23% for NdNiO3and by 0.15% for SmNiO3(figure 2.3a).
The Ni-O bond length also exhibits a discontinuous jump (figure 2.3b). The magni- tude of the bond length change is of the same order as the relative volume change.
The Ni-O-Ni bond angle ⇥ shows a jump of about 0.5˚ at the transition. So in essence, the distortion of the unit cell becomes a bit less severe at the transition and the unit cell volume increases slightly.
More recent (2000) measurements by Alonso et al. [13–15] shows that there is a reduction in the crystal symmetry as the samples are cooled through the MIT.
Alonso etal. showed that RNiO3(with R = Y, Ho, Er, Tm, Yb, Lu) goes from being orthorhombic at high temperature to being slightly monoclinic (space group P21/n).
The monoclinic angle which is a measure of the tilt of the unit cell remains below 90.17˚, where = 90˚ corresponds to the orthorhombic structure. The monoclinic angle increases as the size of the rare earth ion decreases.
The monoclinic structure is accompanied by a breathing distortion of the NiO6
octahedra. As the TM Iis crossed, one octahedra is reduced in size while the other is increased with approximately inversely proportional reduction and increase of the bond lengths. The two types of octahedra alternate in all three directions.
After the discovery of structural change at the MIT for the smaller rare earth ions, much effort was done in studying the larger rare earths to see if the effect oc- curs for them as well. Electron diffraction and Raman scattering measurements done
Figure 2.3:Structural changes at the MIT. a) The change of the relative unit cell volume.
From reference [11]. b) dN i Oas a function of temperature. From reference [12].
by Zaghrioui etal. on NdNiO3seemed to indicate a symmetry breaking towards a monoclinic structure [16] as diffraction spots appear below the transition and lines in the Raman spectra splits below the transition. Staub etal. [17] performed resonant X-ray measurements on NdNiO3thin films and found again a lowering of the sym- metry of the material towards monoclinic. Piamonteze etal. [18] performed X-ray absorption measurements on RNiO3(R=Pr, Nd, Eu) and found splitting of the FT amplitude of the XAS signal similiar to the one occuring in the smaller rare earth compound YNiO3again indicating a transformation towards monoclinic structure.
These measurements were done from early 2000 until Piemonteze etal. in 2005.
In 2008 and 2009, the structure of the rare earth nickelates were unambiguously es- tablished. Medarde etal. [19,20] performed high resolution powder neutron diffrac- tion (PND) measurements on PrNiO3and x-ray absorption spectroscopy (XAS) on the whole family of RNiO3. Garcia-Munoz etal. [21] performed high-resolution synchrotron powder diffraction measurements on NdNiO3. These measurements and the ones cited so far in this thesis, with the exception of the study performed by Staub etal. were performed on polycrystalline samples, synthesised in much the same way as Lacorre etal. did in 1991. The largest single crystals reported to date do not exceed 100µm (figure 2.4).
Therefore, measured bulk samples are polycrystalline such as compressed pow- der. Thin films which offer new and exciting ways of preparing and controlling the material will be addressed later. In 2008 Medarde etal. performed a very nice high- resolution powder neutron diffraction (PND) study of PrNiO3. The authors observed a splitting in some of the high angle reflexions at temperatures below the TM I(figure 2.5d) and e)). The salient point is that a model based on the space group Pbnm for the neutron-diffraction pattern fits the data very well at high temperature above the TM I, but fails to reproduce the features in the pattern that appears below the TM I, notably the splitting of some high-2✓reflexions. In contrast, a model based on the P21/n space group accurately reproduces this splitting below TM I. This splitting
Figure 2.4: Example of single crystal samples of RNiO3. Shown is an as-grown NdNiO3sample measured by scanning electron microscopy at 1500x magnification.
From [22].
was not observed in previous neutron diffraction measurements probably due to lack of resolution. The splitting indicates a long range static monoclinic symmetry be- low TM I The inset of figure 2.5a) shows the evolution of the monoclinic angle.
As is shown, remains at 90˚ at high temperature in agreement with an orthorhom- bic crystallographic structure. At TM I jumps in value, indicating a monoclinic crystallographic structure.
Figure 2.5:a) Neutron powder-diffraction pattern of PrNiO3(red dots) at 10K and best fit using a space group P21/n model (black line). Inset, evolution of the monoclinic angle in degrees. b) and c) Higher detail of two different angular intervals at 170K and using a space group Pbnm model for the fit. d) and e) Same angular interval, but at 10K and using a P21/n model for the fit.
Garcia-Munoz etal. observed the same features on a NdNiO3sample using syn- chrotron diffraction measurements in 2009. They concluded in the same way that a Pbnm model did not reproduce the features at low temperatures and that a P21/n model was appropriate. Medarde etal. performed K-edge high resolution x-ray ab- sorption spectroscopy in partial fluorescence yield detection (PFY-XAS) measure- ment on polycrystalline RNiO3(R= La, Pr, Nd, Sm, Eu, Gd, Ho, Er, Tm, Yb and Lu) in 2009 [20]. The authors scanned the energy of the incoming beam from 8330 to 8380 eV which is across the Ni K absorption edge. They then measured the inten- sity of the Ni K↵1fluorescence, meaning the intensity energy emitted by electrons jumping from the 2p shell to the 1s shell (figure 2.6a)).
Figure 2.6:a) First derivative of the PFY-XAS of the Ni K↵measured on RNiO3in the insulating state except for LaNiO3measured at room temperature in a metallic state. b) Evolution of the A and B peaks in energy (from a)) as a function of the radius of the rare earth ion. Also a comparison to compounds with a nominal valence of +2, +3 and +4
Medared etal. found a splitting in the first derivative of the Ni K↵PFY-XAS, strongly suggesting a differentiation between two distinct nickel sites. For the metal- lic LaNiO3compound the first derivative of the PFY-XAS signal is a single peak. For the smallest rare earth RNiO3compound, Lu, the splitting of the peak is the most pronounced. The severity of the splitting remains similar from R=Lu to R=Ho, but for bigger rare earth ions, the splitting of the peak becomes less pronounced. This is shown in figure 2.6b) where the energy of the peaks A and B is followed. Com- parisons with octahedrically coordinated Ni oxides with nominal valences +2, +3,
+4 are also shown. Therefore, RNiO3compounds with R smaller than Gd has two distinct NiO6octahedra with the smaller having a formal valence close to +4 and the bigger a formal valence close to +2. RNiO3compounds with a large rare earth ion have two distinct NiO6octahedra, but they are less different and both have a valence closer to +3.
Summarizing, RNiO3nickelates all undergo a MIT accompanied by a structural transition from orthorhombic to monoclinic at the TM Iwith the exception of LaNiO3
which remains metallic and rhombohedric at all temperatures. The severity of the monoclinic distortion grows bigger with smaller rare earth ions. The monoclinic distortion is accompanied by a lifting of the NiO6octahedra degeneracy, called the breathing mode or band disproportionation. This affects the electronic structure of the material.
2.3 Electronic structure
2.3.1 Introduction
A look at the history of research into condenced matter physics and nickelates is beneficial for a general knowledge of the material that is investigated. Here, I present a brief history of related research.
J.J. Thomson discovered the electron in 1897 [23]. Paul Drude formulated in 1900 [24] a relation between macroscopic measurable quantities and microscopic properties of the material:
j=nq2⌧
m E (2.3)
where j is the current density, n, m, q,⌧ is respectively the particle density, mass, charge and scattering time. This derivation of Ohm’s law was made before the in- vention of quantum mechanics. The classical theory saw many successes because electrons in simple metals essentially behave as quasipartics which obey the Drude model, but with a modified mass [25]. Drude assumed that the electrons in a metal had their momentum distributed according to Maxwell-Boltzmann distribution func- tions. The Drude model accurately describes electrical and thermal conductivity, but fails to predict the heat capacity or the magnetic susceptibility of metals.
Wolfgang Pauli introduced a new quantum number in 1925, later identified as the electron spin, to formulate the Pauli exclusion principle that states that no two electrons can have all their quantum numbers identical. This permitted an expla- nation for the way that electronic shells are filled in atoms. Later Fermi and Dirac expanded the Boltzmann distribution of systems that have reached thermodynamic equilibrium. Boltzmann showed that if you have a system with two possible energy states,✏and 0, and you connect it to a reservoir, the probability ratio of finding the
system at equilibrium with energy ✏to the probability of finding the system with energy 0 follows:
P(✏)
P(0) =exp( ✏/kbT) (2.4)
wherekbis Boltzmann’s constant.Taking into account that the condition P(✏) + P(0) = 1, we obtain:
P(✏) = 1
exp(✏/kbT) + 1 (2.5) Fermi and Dirac included the possibility of two systems exchanging particles as well as energy and found the famous Fermi-Dirac distribution function:
P(✏) = 1
exp((✏ µ)/kbT) + 1 (2.6) 2.3.2 Sommerfeld
Armed with the Fermi-Dirac distribution which takes into account the Pauli exclu- sion principle, Sommerfeld set out to investigate valence electrons in a metal [26].
Sommerfeld did not take into account electron-electron or electron-phonon interac- tions, but considered the case of a free electron gas in a solid. The corresponding Schrödinger equation is:
~ 2m( @2
@x2+ @2
@y2+ @2
@z2) k(r) =✏k k(r) (2.7) where k(r) is the wavefunction of the particle at r with wavevector k and✏kis the energy of the electron with wavevector k. The wavefunctions that satisfy this relation are of the form:
k(r) =exp(ik⇥r) (2.8)
Combining equations 2.7 and 2.8, one finds the energy of the electron with wavevector k:
✏k=~2k2
2m . (2.9)
According to the Pauli principle, in the ground state, the electrons occupy all the states up to some energy✏F, the Fermi energy. The states at the Fermi energy have a wavevector kF such that
✏F =~2kF2
2m . (2.10)
If one defines periodic boundary conditions for the wavefunction one finds that there is one wavevector per volume element (2⇡/L)3). Counting how many of these cubes fit in a sphere of radius kF, one gets the total number of states N, since kF is the maximum wavevector.
2⇥ 4⇡kF3/3 (2⇡/L)3 = V
3⇡2k3F =N (2.11)
We thus get the Fermi wave number:
kF = (3⇡2N
V )1/3 (2.12)
the Fermi energy:
✏F = ~2 2m
✓3⇡2N V
◆2/3
. (2.13)
and the total number of states:
N= V 3⇡2
✓2m✏
~2
◆3/2
(2.14) This allows us to evaluate the density of states D(✏)
D(✏)⌘dN d✏ = V
2⇡2⇥
✓2m
~2
◆3/2
⇥✏1/2=3N
2✏ (2.15)
Sommerfeld had understood how Fermi-Dirac statistics would affect the density of states (figure 2.7).
With an expression for the density of states it becomes possible to answer ques- tions that posed a great deal of difficulty to physicists at the time, namely the inex- plicable low value for the specific heat of metals. The specific heat of a material is a parameter that is easily measured and the specific heat of metals were measured to be around 1% of the value predicted by classical statistical mechanics. Without the knowledge of Pauli’s exclusion principle and the Fermi distribution function, it was assumed that all electrons in the material would contribute to the heat capacity of the material. With Pauli’s principle it was understood that not all the states were able to be thermally excited, but only those states within a range of kBT of the Fermi energy could be excited. Fermi’s distribution function specifically deals with which states are thermally excited. The specific heat is calculated by finding the increase in total energy as a system is heated from 0 K to T, U⌘U(T) U(0)and differentiating with respect to T. At low temperature, the result is:
CV =dU dT = ⇡2
2 kBT
✏F
nkB, (2.16)
a value, when compared to the classical value of⇠nkBis⇠100 times smaller.
Figure 2.7:The effect of temperature on the density of electronic states of a metal. At T = 0 K, the electrons fill the states up to the Fermi energy. At finite T, the states closest to the Fermi energy are exited according to the Fermi function. From [26]
2.3.3 Band theory
The free electron model of metals explains very well fundamental results for metals like measured resistance and specific heat, but it fails to explain why some materials are insulators or semiconductors. Felix Bloch considered the case where you add a periodic potential to the free electrons [27] and formulated the theorem that the solutions of the Schrödinger equation for a periodic potential must be of the form:
k(r) =uk(r)exp(ikr) (2.17) In 1931, Wilson was satisfied with how Blochs theory described conductivity in metals [28]. He also observed that the ideas of Bloch failed to explain the existance of insulators or semi-conductors. Wilson showed two ways of accounting for the ef- fect of an ionic cubic lattice on the electronic states. The first, nearly free electrons, is characterized by taking into account the periodicity of the lattice and treating the problem perturbatively. This led Wilson to find discontinuities in the energy distri- bution (figure 2.8).
The other way of looking at the problem was the case of tightly bound elec- trons. In this case he was able to distinguish between bands of energy and to identify forbidden ranges of energy. The method is to have a tight binding potential with a small overlap between wavefunctions which leads to a hopping integral t [29] and the Hamiltonian becomes
H=X
i
E0|iihi| tX
hi,ji
|iihj|. (2.18)
Figure 2.8:Discontinuities in the energy distribution of the electronic states. Forbidden zones of energy exists for the electrons. From [28].
Working in Fourier space and exploiting the periodicity of k to work in the first Brillouin zone we get the tight-binding Hamiltonian:
H= tX
k
2cos(ka)|kihk|+E0
X
k
|kihk|, (2.19)
which is diagonal in k. The Hamiltonian shows that it is favourable for the elec- trons to hybridize their orbits. Applying this Hamiltonian for a square lattice gives the energy dispersion:
✏(k) = 2X
l
tlcos(klal), (2.20) where l denotes the different coordinate axes, and a is the lattice spacing. The allowed energies for the electronic states constitutes what is called an electron band and it is formed because of the interaction between the electrons and the ions in the lattice, forming in this case the basis for the tight binding model. This understanding of band theory let physicists be able to predict if a material was an insulator or a metal based on where the Fermi energy is located with respect to the electron bands of the material and represented a great success of band theory. If the Fermi energy is located in the middle of an electron band, this would mean that there are states available at a higher temperature that the electrons are able to be excited into with a field or with temperature. If the Fermi energy is located in the gap between two electron bands, so that the lower band is filled and the upper one is empty, there are
no states close in energy to the states in the material with the highest energy so that they are difficult to excite. These materials are insulators.
If we look at the bottom of the band in equation 2.20, we can expand the cosine to get:
✏(k) =E0 2t+tk2. (2.21) By analogy with free electrons,✏(k) =~2k2/2mwe can define an effective mass m* = 1/(2t). The consequence of this is that the electrons move like free electrons, but with a renormalized mass caused by the formation of electron bands.
2.3.4 Mott insulators
Considerable surprise was expressed by physicists when it was reported by De Boer and Verwey in 1937 that NiO was an insulator [30]. This could not be explained by band theory. It was known that the d band of Ni was incomplete and one would expect the compound to be metallic since the d band is crossing the Fermi energy.
According to the Proceedings of the Physical Society (1937) [31], reported by N.F.
Mott with the help of some notes from R. Peierls, R. Peierls observed:
"The solution of the problem would probably be as follows: if the trans- parency of the potential barriers is low, it is quite possible that the elec- trostatic interaction between the electrons prevents them from moving at all. At low temperatures the majority of the electrons are in their proper places in the ions. The minority which have happened to cross the po- tential barrier find therefore all the other atoms occupied, and in order to get through the lattice have to spend a long time in ions already oc- cupied by other electrons. This needs a considerable addition of energy and so is extremely improbable at low temperatures."
This observation went to the crux of the matter and initiated the field of strongly correlated electrons. N.F. Mott made the first theoretical advances in describing how materials with a half filled band could be insulators [32]. These insulators are called Mott insulators. He considered a lattice model with one electronic orbital at each site. Without interactions between electrons, the electronic orbitals would overlap and form one single band. This band is full if each site has two electrons, one with spin up and one with spin down. However Mott considered interactions between electrons and found that two electrons on a single site experience strong Coulomb repulsive forces. These forces would split the band in two, one band consisting of electrons that occupy an empty site, the other consisting of electrons that occupy a site where there is already another electron.
Mott also described how the distinction between a metal and an insulator of this kind should be sharp, in other words a first order transition, with regards to the dis- tance between lattice ions. He looked at what happens when an electron leaves a site,
leaving behind a hole on that site. There will be a Coulomb attraction between this electron and the hole. The electron and the hole will form a bound state and not con- tribute to conductivity. However, if the electronic screening by the other electrons is strong enough, the electron and the hole will not feel the Coulomb attraction. There- fore they will not form a bound state and will be free to contribute to the conductivity of the metal.
The theoretical model by Hubbard [33] helped in the understanding of the tran- sition between a Mott insulator and a metal. The Hamiltonian is given by:
HH=Ht+HU µN, (2.22)
Ht= tX
hiji
(c†i cj +h.c), (2.23)
HU=UX
i
(ni,
1 2)(ni,
1
2), (2.24)
and
N⌘X
i
ni . (2.25)
The Hubbard model includes important simplifications, but in spite of this it is capable of reproducing the Mott insulating phase and the transition between metals and the Mott insulator. Regarding the splitting of the half filled band into a full lower band and an empty higher band, one speaks of the lower and upper Hubbard band and this specific type of insulator as a Mott-Hubbard insulator.
When one takes into account other electronic orbitals of the material, different MIT mechanisms can come into play. Zaanen, Sawatszky and Allen wrote a sem- inal paper in 1985 titled: "Band gaps and electronic structure of transition-metal compounds" [34]. They had noticed that the Mott-Hubbard (MH) theory works very well for compounds based on Ti and V in that the band gap that is measured is related to Coulomb repulsion from electrons in the 3d state. However, for Co, Ni and Cu based compounds, the gap seemed to be related to the electron affinity of the anion in the compound and therefore the gap origin was assumed to be of a charge-transfer type (CT).
Atomic orbitals form bands because of the periodic potential of the lattice. In transition-metal oxides the band closest to the Fermi energy is the 3d band. The bandwidth of the d band is a result of the overlap of the d orbitals of two adjacent transition metal sites. The d orbitals have a small radius compared to the lattice constants so the bands are sharp. Between two Ni atoms there is an O atom in the perovskite structure of nickelates. From this follows that the bandwidth is modulated by the overlap of the d wavefunction at the Ni atom and the p wavefunction at the O atom. If the p wavefunction constitutes a bridge in this sense the wavefunction are hybridized. This has the effect of making the d bandwidth even more narrow.
Figure 2.9: The splitting of the 3d orbitals. As the crystal field has fewer and fewer symmetries, the 3d orbitals are split in energy. From [4].
In the transition-metal oxides the 3d orbital is fivefold degenerate for each spin.
The 3d band can accomodate five electrons for each spin and without the crystal structure these electrons all have the same energy. The crystal field splits this degen- eracy. Since the oxygen tends to have a strong negative valence, the crystal field in the direction of the oxygens will be stronger. In addition, deviation from the cubic structure will split the bands even further, see figure 2.9 for examples of crystal field splitting. In the nickelates, the orthorhombic structure doesn’t split the degeneracy and the eg band remains degenerate. In the transition-metal oxides it is the p or- bital of the oxygen ligand that will form the bridge between the transition metal 3d orbitals, illustrated in figure 2.10.
Figure 2.10: Illustration of the 3d orbitals and the p orbital that forms the bridge and hybridizes. From [4].
In the Zaanen-Sawatzky-Allen classification scheme of insulators, U indicates the Coulomb repulsion between two electrons on the same site and is the difference in energy between the lower and upper Hubbard bands. If U is big one would expect a large band gap insulator. Another possibility is that the anion electronic states comes into play by being close to the Fermi energy. In the case of transition-metal oxides, the anion states in question are the p shell of oxygen. If the p band of oxygen is close to the Fermi level one can have charge fluctuations of the formdni !dn+1i L.
The way this charge fluctuation is written means that the gap, notated as opposed to U, is the gap between states consisting of a singly occupied Ni d band with fully occupied O p band and doubly occupied Ni d band (two electrons in the eg band) and a hole in the oxygen ligand denoted byL, shown in figure 2.11.
Figure 2.11: Two different mechanisms for forming a gap in Mott insulators. a) Schematic of the formation of a Mott-Hubbard insulator. b) Schematic of the forma- tion of a charge transfer insulator. From [4].
Figure 2.11a) shows that without the interaction U, the d band is half filled and the p band is filled. With the interaction U, the d band is split and the charge gap is the energy with which one has to excite an electron to jump to a site and be doubly occupied of the formdnidnj !dni 1dn+1j . For figure 2.11b), the starting point is in principle the same as for the MH insulator, but when the U interaction is included the p band is closer to the Fermi energy than the lower Hubbard band either because of high U or because of the energy level of the p band. The charge gap is then the energy with which one has to excite an electron on the oxygen p band to become doubly occupied on the d band.
2.3.5 RNiO3
Demazeau measured the distance Ni-O to be 1.94 Å, subtracted the assumed ionic radius of oxygen 1.40 Å, and got the ionic radius of the Ni3+ ions to be 0.54 Å.
This suggested an electronic configuration of weak spin for the Ni3+meaning that there are six electrons in the t2gbands and one in the egbands. Neutron diffraction measurements [12] found the nickel ion to have a magnetic moment close to 1µB
which is consistent with a low spin (S=1/2) Ni3+configuration.
This low spin configuration could be expected to lead to orbital ordering like the cooperative Jahn-Teller effect. H.A. Jahn and E. Teller proved that for molecules that are not linear, stability and degeneracy is not possible at the same time [35]. To rid itself of this degeneracy the material undergoes a structural change and elongates the NiO6octahedra. This causes for example the 3d3z2 r2 to have a lower energy than the 3dx2 y2orbital so that the e1gelectron will be located in the orbital with the lowest energy. A Jahn-Teller effect should elongate the octahedra and lead to orbital ordering. This Jahn-Teller distortion could induce a gap to open like the Jahn-Teller distortion observed in the manganites [36]. The Jahn Teller effect was thought to have been seen in the RNiO3with the smallest rare earths, but not for compounds such as PrNiO3or NdNiO3[37]. Jahn-Teller has since ceased to be thought of as relevant in the nickelates. The direction from "Charge Ordering as Alternative to Jahn-Teller Distortion" by Mazin etal. [38] is accepted.
Even though the Jahn-Teller effect has been ruled out, coupling between the electronic orbitals and the lattice is important. Changing the isotope of oxygen from
16Oto18Ohas the effect of making the oxygen atoms heavier, but it has no effect on the crystal structure [39, 40]. The heavier oxygen slows down lattice vibration and makes the transport electrons more local which has the effect of increasing the TM I. Recently it has been shown that it is possible to influence the electronic properties of the material with optical excitation of the substrate lattice [41], again emphasizing the importance of the electron-phonon interactions.
In the earliest interpretations of the MIT in the RNiO3, a metal to charge-transfer insulator transition was found to be the most natural [42]. Based on the Zaanen- Sawatsky-Allen scheme of Mott insulators, RNiO3was found to be in the boundary
of CT insulators close to the insulator-metal boundary. As previously mentioned the tolerance factor describes how distorted the unit cell of the material is. As the rare earth ion becomes smaller, the unit cell becomes more distorted and the tolerance factor decreases. As the tolerance factor decreases, the orbital overlap between the Ni egorbital and the O p orbital decreases. This has the effect of increasing the TM I, represented in figure 2.12
Figure 2.12:A phase diagram of the RNiO3showing how the tolerance factor influences the TM Iand the Néel temperature. From [2].
Since the orbital overlap between the d and the p orbitals is reduced, the band- widths of the p and d overlap is reduced and a charge transfer gap is introduced. This idea from 1992 is represented in figure 2.13 where the examples are LaNiO3which is metallic at all temperatures because of the large rare earth and SmNiO3which has a high TM Iat around 400K because of the small rare earth ion. An important facet of the phase diagram of 2.12 is that for R = Nd and Pr, the Néel temperature is identical to TM I, while for compounds with a smaller rare earth, the TM Iis much higher than the TN.
To summarize, there are some problems with the traditional picture of Mott in- sulators as the driving force for the MIT in nickelates. Measurements in a SQUID magnetometer [37, 43] does not show a change in the magnetic susceptibility as a function of temperature as the material is taken through the MIT. This is incom- patible with a homogenous Mott-Hubbard transition. One would expect that as the electrons go from being itinerant to being localized, the magnetization would change.
Figure 2.13: A schematic showing the early idea of charge transfer mechanism for RNiO3compounds. From [42].
A second question is if the occupancy of the correlated orbitals is small enough to drive a MH or CT MIT. DFT + DMFT procedure was performed by Wang etal. [44]
and they found the occupancy of the d orbital to be around 2 compared to the value
⇡1.3 needed for the MH/CT MIT.
2.3.6 Insulating gap
Knowledge about the magnitude of the insulating gap is invaluable for the compre- hension of the nature of the metal to insulator transition in the nickelates. A precise determination of the gap can give information on which states are involved in the transition. The gap value may also give an indication of which interactions are in- volved. How the gap is formed and how it varies with parameters like temperature, strain, pressure and doping may further inform on the nature of the states and interac- tions involved. Decreasing the tolerance factor for example makes the material more insulating because it is assumed the hybridization of the d and p orbitals become less pronounced. This theory can be checked with a precise measurement of the gap. As the hybridization decreases, the bandwidth of the orbitals involved decreases and the insulating gap should increase, all else equal.
La2NiO4is a charge transfer Mott insulator with a band gap of about 4eV, much the same as NiO [45, 46]. It is interesting to compare these values with gap value estimations for the RNiO3 compounds since NiO for example is reasonably well
characterized. One can look at the eventual differences and identify the mechanisms that differentiate them.
Without direct measurements it is difficult to accurately determine the insulating gap in RNiO3compounds. Transport measurements of PrNiO3and NdNiO3have been performed and the insulating part of the resistive curve have been fitted to a semiconducting activation model [47,48]. Granados etal. found an activation energy of respectively 22meV and 25-28 meV. Catalan etal. used a model based on the activation energy of a semiconductor in addition to the effect of low temperature variable-range hopping:
1
R=A exp
✓ T0 T
◆1/4
+B exp
✓ T1 T
◆
(2.26) The effect of including variable-range hopping is that the model is a good fit for the entire insulating part of the resistive curve. See figure 2.14 for a side by side comparison of the effect of including the variable range hopping.
Figure 2.14:Side by side comparison of the fit by a) a model based on the semiconductor activation energy and b) also including the variable-range hopping. a) from [48] and b) from [49].
Catalan etal. finds the activation energy to be 17-20 meV.
Optical measurements performed on NdNiO3[50] report a larger gap than the one predicted from activation energy of the resistive curve. In the optical mea- surements the gap seems to be in the range 100 - 250 meV. Ultraviolet photoemis- sion spectroscopy(UPS) performed on PrNiO3[51] shows a loss of intensity going through the MIT indicating a gap size of around 200 meV (figure 2.15).
Ultraviolet photoemission spectroscopy performed on NdNiO3shows close to identical results [52] as for PrNiO3. Nd1 xSmxNiO3shows the same results for x
0.4 [53]. UPS measurements performed on NdNiO3thin films 100 nm thick are the same as for previous results on polycristalline samples [54].
Figure 2.15:Ultraviolet photoemission spectroscopy on PrNiO3and LaNiO3at temper- atures above, at and below the MIT for PrNiO3and high, medium and low temperatures for LaNiO3which remains metallic throughout. From [51].
2.3.7 Epitaxial thin films
Epitaxial NdNiO3thin films with good resistive switching characteristics were pre- pared by high pressure sputtering for the first time in 1995 on the (100) surface of single crystal LaAlO3substrates by J. F. DeNatale and P. H. Kobrin [55]. The authors discovered a sharp MIT with a hysterisis loop of about 10 K. The TM I= 116 K was found to be substantially smaller than the one reported in bulk polycrystalline sam- ples at about 200 K. The difference in TM Iis thought to be due to the induced strain from the substrate. Laffez etal. later prepared thin films of NdNiO3on SrTiO3, NdGaO3and Si (100) [56, 57].
Preparations of thin films without the need for high pressure were eventually done by pulsed laser deposition [49] and by chemical vapour deposition [58]. The success of synthesizing RNiO3thin films means that it is possible to control the character of the material to a level not possible with polycrystalline bulk samples.
RNiO3thin films can be grown on a variety of different substrates with different lattice constants. These substrates will induce a different strain in the thin film and changes the TM Iaccordingly. The samples that have been investigated for this thesis are thin films and the subject will be explored in more detail in the following chapter.
2.3.8 Theoretical research
A key question for the RNiO3compounds is what is the electronic orbital structure.
From synchrotron diffraction measurements and x-ray absorption measurements, it is clear that there is alternating sizes of the Ni-O bond length. There are uncertainties as to what causes these bond disproportionations. There are even questions about whether or not these bond disproportionations are pre-formed in the metallic state [20, 59].
The discovery of a monoclinic crystallographic structure in the small rare earth compounds [13–15] and the larger rare earth compounds [19, 21] was an indication of a possible charge-ordered ground state since the monoclinic symmetry leads to a difference in the volume of the NiO6octahedra. X-ray scattering measurements were performed by Staub etal. [17] and Scagnoli etal[60]. The authors observed directly x-ray scattering intensity at forbidden reflexion (033) assigned to the phenomenon of charge ordering below TM I. The charge disproportionation was found to be 2 ⇡ 0.45. A search for intensity at the wave vectors predicted for orbital ordering yielded no evidence for orbital ordering. Bodenthin etal. [61] observed in 2011 with the help of high-resolution x-ray powder diffraction that the members of RNiO3with the smallest rare earths show a charge ordered ground state of the type 2Ni3+ ! Ni3+ + Ni3 .
Mizokawa etal. [62, 63] in 1995 and 1996 performed photoemission and x-ray absorption spectroscopy and analysed their data using a cluster model and unre- stricted Hartree-Fock calculations. The authors found that in the ground state of PrNiO3, the d7orbital configuration is strongly hybridized with the d8Lconfigura-
tion. This result is not in conflict with the magnetic moment measurements showing close to ionic value because the local magnetic moment is calculated to be 0.9µB. The result means that an important portions of the Ni-O bonds are formed with a hole on the O p orbital. The formation of holes on the oxygen sites is referred to as
"negative charge-transfer" insulators [64]. The negative charge transfer energy refers to cases of Mott insulating charge transfer insulators where O p energy lies so high that the charge transfer gap becomes very small or even negative.
In 2000 Mizokawa, Khomskii and Sawatzky [65] argued that RNiO3compounds are self-doped Mott insulators, meaning that there is a small or negative charge trans- fer gap so that holes in the upper Hubbard band are transfered to the O p band because of an overlap between the two bands in energy. In this picture the holes are on the ligand O atoms and for NdNiO3and PrNiO3the material is an oxygen-site charge ordered state. The authors model the ground state as 3d8Linstead of 3d7, giving for the gap the excitation 3d8L+ 3d8L!3d8+ 3d8L2, instead of 3d7+ 3d7!3d7+
+ 3d7 . For smaller rare earths the charge order can be on the transition metal ion and be accompanied a shift in the O ions which gives a breathing distortion.
Mazin etal. [38] showed in 2007 that the Mott transition in orbitally degenerate systems can happen through charge order rather than orbital order as an alternative.
The authors showed that the system can rid itself of degeneracy by making one or- bital empty and one doubly occupied as an alternative to orbital ordering, e1g+ e1g! e0g+ e2g, if U - 3J is small. U is the local Coulomb repulsion and J is the Hund rule interaction. It may be energetically advantageous for the system to order according to the Hund rule rather than Jahn-Teller if the value of U-3J is small.
Considerable theoretical work has been devoted to studying the charge ordering of the Ni 3d and O p orbitals and in particular the self doped Mott insulator alterna- tive. Park, Millis and Marianetti [66–68] take the concept of holes on the O ligands further by recognizing a novel correlation effect. They recognize that the d elec- trons on the Ni1site in the large NiO6octahedra become largely decoupled from the surrounding lattice while the d electrons on the Ni2site in the small NiO6octahe- dra bind with the holes on the O sites to form a singlet state. Ni1is the sites that have long Ni-O bonds (LB) and Ni2are the sites that have short Ni-O bonds (SB).
The authors term this state a site-selective Mott insulator. The electronic structure is site-dependent.
Figure 2.16 shows the comparison between DFT calculation and DFT + DMFT of the d electron spectral function for LuNiO3in the insulating state with crystallo- graphic structure P21/n. The density functional calculation predicts the ground state to be metallic, while the DFT + DMFT captures more of the correlation effects and predicts an insulating gap of about 200 meV at the Fermi level in the monoclinic phase. The model predicts a gap both for the small bonds and the long bonds.
From figure 2.16 it can be seen that the Ni1sites have more spectral weight below the Fermi level while the Ni2sites have more spectral weight above the Fermi level.
Mazin etal. [38] interpreted this as evidence for charge order. Park etal. notes that
Figure 2.16:Momentum integrated spectral function A(!computed for P21/n LuNiO3. a) DFT spectral function. b) DFT + DMFT spectral funciton obtained at U=5 eV, J=1 eV, Nd=8.0 and T=116 K. From [66].
the charge density of the Ni sites is the integral of the spectral function over the entire frequency range and that the integral difference between the two Ni sites is minute, concluding that the charge difference is not the cause of the insulating behaviour.
The authors further conclude from calculations that the Ni1sites show a Curie behaviour for the local magnetic susceptibility like the one expected of a paramgnetic Mott insulator, while the N2sites have a largely temperature independent magnetic susceptibility. Giving rise to the site-selective Mott insulator appelation. The expla- nation for this behaviour is that the Ni sites have a ground state valence of d8and that the Ni1site is largely decoupled from the lattice giving a spin S=1 from Hund’s coupling. On the short bond Ni site the two electrons in the eg orbital are strongly coupled to the two holes on the O sites, giving a total spin of 0 so that the local susceptibility is largely temperature independent. Using this model the calculation reproduces the insulating gap in the paramagnetic state (for small rare earth ions) as well as the structural (Pbnm to P21/n) and electrical (metal to insulator) dependence on the rare earth ion and external pressure.
Subedi, Peil and Georges proposed a low energy description of the MIT showing that for a system subjected to U and J, the charge disproportioned state is a param- agnetic insulator for a large range of parameters and that when U-3J is small, the system spontaneously favours bond disproportionation for large J [69]. There is an
effective attractive interaction between electrons with parallel spins on different or- bitals, stabilizing a bond density wave. The density of states will be as shown in figure 2.17a) where below the Fermi energy the lower Hubbard band is mainly of long bond character. For the states above the Fermi energy, two sets of states can be observed separated by a Peierls LB/SB site modulated pseudogap. The optical transitions associated with the two peaks in 2.17b), by Ruppen etal. [70] correspond to a transition across the insulating gap (A) and across the insulating gap and the Peierls pseudogap (B). This experimental result gives support to the bond dispropor- tion model because evidence of the Peierls gap is seen as the temperature is lowered below the MIT.
Figure 2.17: Possible transitions in the RNiO3when U-3J is small and J is large. a) Momentum-resolved spectral function for U=1.0 eV, J=0.85 eV. The colour represents the character of the site, red means more long bond character while blue means pre- dominantly short bond states. The circles indicate the moment-energy locations for the Peierls gap. Side panel: Density of states for LB (red) and SB (blue) sites. The arrows indicate the possible optical transitions. b) Real part of the optical conductivity, with peaks A and B at energies that correspond well with the density of states. From [70].
CHAPTER 3
Epitaxial thin films
The thin films used in our study were synthesized by the Triscone group, DQMP, University of Geneva. The films are of excellent quality [71–76] with sharp switch- ing characteristics. The group grows RNiO3on a variety of substrates. Although not involved, we briefly present the film growth for completeness.
3.1 Deposition techniques
There are two main ways of depositing thin films: chemical vapour deposition and physical vapour deposition. For both of these techniques the goal is to deposit a thin film of a desired compound on a substrate that controls the crystallographic parameters. If the film grows epitaxially, the in-plane crystallographic parameters will be taken from the substrate while the out-of-plane parameter has more freedom.
If the substrate is of the same composition as the film, homoepitaxy, there is in principle no stress induced in the deposited film. Strain can be induced in the film if the growth is heteroepitaxial. If the structure is orthorhombic, the c axis of the thin film will be oriented in-plane or out-of-plane depending on the lattice mismatch.
3.1.1 Chemical vapour deposition
Chemical vapour deposition consists of having the desired atoms in a volatile so- lution (precursor) that will deposit on the substrate given favourable conditions in the chamber where the substrate is located. Gorbenko and Bosak performed a metal organic chemical vapour deposition (MOCVD) to produce a thin film of LaNiO3on
a single crystalline (001) MgO substrate [77]. Their experimental setup serves as an example for this technique (figure 3.1).
Figure 3.1: MOCVD setup: 1) vibrational feeder, 2) powder mixture of precursors, 3) evaporator, 4) heated transport lines, 5) quartz reactor, 6) furnace, 7) substrate, 8) substrate holder with rotation gear, 9) pressure gauge, 10) pumping line. From [77].
The precursors are evaporated into the substrate chamber where different envi- ronmental condictions lead to a deposition on the substrate. MgO is a substrate that would induce a tensile stress in the film of five to ten percent. The film did not grow epitaxially due to this lattice mismatch. Novojilov etal. prepared an epitaxial thin film of RNiO3where R=Pr, Nd, Sm, and Gd) on LaAlO3substrate [58]. He succeeded because the lattice mismatch was less severe.
3.1.2 Physical vapour deposition
While CVD uses a chemical solution to properly vapourize the desired compound, PVD uses a physical process to vapourize the compound. Common PVD techniques are molecular beam epitaxy (MBE), pulsed laser deposition (PLD) and sputtering.
As briefly described in chapter two, NdNiO3thin films with good switching char- acteristics were first made by DeNatale and Kobrin in 1995 by RF magnetron sput- tering on (100) oriented LaAlO3substrates. Sputtering involves the ionization of an inert gas in the chamber. The positive ions produced are accelerated towards a target which is stochiometrically suited for the thin film to be grown. When the ions strike the target, their kinetic energy is enough to eject atoms close to the surface. The ejected atoms are transferred to the substrate where, if the conditions are right, a thin film will grow.
LaNiO3 on LaAlO3were grown by King etal. using MBE [78]. La and Ni samples are heated to produce a flux that enters a chamber with a partial ozone pressure which ensures the oxidization of the film on the substrate.
Pulsed laser deposition were used by Catalan, Bowman and Gregg in 2000 to grow NdNiO3on different substrates with NdGaO3yielding the best switching char- acteristics [79]. PLD involves a high power laser that is directed and pulsed at a target. The target is stochiometrically suited to the thin film to be created and the material is vapourized from the surface of the target and deposited on the substrate.
The properties of the film are independent of the growth techniques employed if the growth has succeeded.
