Metal-insulator transitions in nickelate heterostructures

Thesis

Reference

Metal-insulator transitions in nickelate heterostructures

SCHERWITZL, Raoul

Abstract

Dans cette thèse nous avons étudié des hétérostructures basées sur les perovskites nickelates RNiO3, où R est une terre rare. Les nickelates présentent une transition spectaculaire d'un comportement métallique à un comportement isolant en fonction de la température. Le but de cette thèse consiste à identifier les mécanismes physiques responsables de la transition métal-isolant et d'explorer les moyens de les manipuler. Nous avons trouvé que la transition dans NdNiO3 peut être manipuler en utilisant des contraintes de croissance compressives, la technique de l'effet de champ et de la lumière. De plus, nous avons induit une transition métal-isolant et un ordre magnétique dans des hétérostructures de LaNiO3, le seul membre de la famille des nickelates, qui par lui-même, ne manifestent ni de transition ni d'autres phénomènes d'ordre.

SCHERWITZL, Raoul. Metal-insulator transitions in nickelate heterostructures. Thèse de doctorat : Univ. Genève, 2012, no. Sc. 4427

URN : urn:nbn:ch:unige-217406

DOI : 10.13097/archive-ouverte/unige:21740

Available at:

http://archive-ouverte.unige.ch/unige:21740

Disclaimer: layout of this document may differ from the published version.

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Metal-insulator transitions in nickelate heterostructures

TH` ESE

pr´ esent´ ee ` a la Facult´ e des sciences de l’Universit´ e de Gen` eve pour obtenir le grade de Docteur ` es sciences, mention physique

par

Raoul Scherwitzl de Vienne, Autriche

Th` ese N

4427

GEN`EVE

Atelier d’impression ReproMail 2012

Working towards my PhD thesis has been an exciting experience, involving many peo- ple, which were fundamental for its production. I would like to acknowledge everyone who contributed to it.

First, I would like to thank Prof. Triscone for giving me the opportunity of joining his group. I am glad I could learn from his vast knowledge of physics and his leadership skills. He showed great trust, optimism and enthusiasm throughout my thesis, which kept my motivation high. He always made time to discuss results and future projects.

In addition, I profited enormously from the creative and stimulating work environment he created, consisting not only of a large amount of in-house growth and measurement facilities, but also of strong interactions in a cordial ambience with highly competent, motivating and inspiring researchers at every level, including himself.

I am thankful to Prof. Keimer, Prof. van der Marel and Prof. Catalan for their key role in evaluating this thesis. I am in particular grateful to Prof. Catalan, whom I had many stimulating discussions about nickelates with.

A very special thanks goes to Pavlo Zubko, who co-designed this fascinating project and supervised me in my everyday work. Without his deep understanding of physics, his motivation, his creativity and his insatiable need to get to the bottom of things, this thesis would have not been possible. I learned a great deal from him. It was also a great pleasure to have had Marta Gibert on the nickelates team, whose rigorous and efficient working moral was vital to make this project successful. The nickelates team also benefited tremendously from Stefano Gariglio, especially from his know-how and his support.

I owe thanks to Andrea Caviglia for our interesting discussions during the time when he was a group member and for our excellent and productive collaboration since he joined the Max-Planck research group for structural dynamics in Hamburg.

I would like to take the opportunity to thank the remaining present and past group

v

members C´eline Lichtensteiger, Nicolas Stucki, Nicolas Reyren, Alessia Sambri, Clau- dia Cancellieri, Almudena Torres-Pardo, Alexandre Fˆete, Danfeng Li, Daniela Stor- naiuolo and St´ephanie Fernandez for their support and the jovial atmosphere.

During the past years, I have also had the pleasure to collaborate on exciting projects with Marc Gabay, Jorge I˜niguez, Andrea Cavalleri, Philip Aebi, Phil Willmott, An- drea Damascelli, Antoine Georges, Marisa Medarde, Manuel Bibes, Agn`es Barth´el´emy, Christian Ruegg and Urs Staub. In particular, I am grateful to Alberto Morpurgo and Ignacio Gutierrez Lezama for discussions and for introducing me to ionic liquids.

I also want to thank Nu˜no Couto, Niko Minder, Daniele Braga, Shimpei Ono, Benedikt Ziegler, Gijsbert Rispens, Jill Guyonnet, Iaroslav Gaporenko, Cedric Blaser, Yulia Lisunova, Alexandre Piriou, Ivan Maggio-Aprile, Carmine Senatore, Enrico Giannini, Julien Ruppen, Francois Bianco, Christophe Berthod, Louis Antognazza, Christoph Renner and Patrycja Paruch.

My research has benefited immensely from the technical assistance of Marco Lopes, S´ebastian Muller, G´eraldine Cravotto, and Spiros Zanos. I would also like to thank our group secretary Esther Schwarz.

I would like to express my warmest thanks to my father, my mother, my sisters Laura and Iris for their continuous support from day one. Ultimately, I thank my fianc´ee Elina for the wonderful life that we have been sharing in Geneva.

Les solides peuvent ˆetre class´es en deux cat´egories: les m´etaux et les isolants. Au z´ero absolu, un m´etal est capable de conduire un courant ´electrique, alors qu’un isolant ne l’est pas. Du point de vue de la physique du solide, les ´electrons, assujettis au potentiel p´eriodique cr´e´e par les ions du solide, n’ont la possibilit´e d’adopter que cer- taines plages de valeurs d’´energie, appel´ees les bandes d’´energie. Ces derni`eres sont s´epar´ees par un gap qui correspond `a une r´egion d’´energies interdites. Selon le rem- plissage des bandes par les ´electrons, le solide sera soit un m´etal (dans le cas d’une bande partiellement remplie) soit un isolant (lorsque la bande est vide ou enti`erement remplie). Cette th´eorie des bandes fournit une excellente description pour un grand nombre de mat´eriaux dans lesquels les ´electrons sont consid´er´es comme libres. Des exemples incluent le silicium semi-conducteur ou l’aluminium supraconducteur.

Cependant, il existe des mat´eriaux, notamment les oxydes de m´etaux de transitions, pour lesquels l’approche conventionnelle ´echoue. Par exemple, le NiO est un isolant alors que sa bande est `a moiti´e remplie. Dans ces mat´eriaux, les ´electrons ne sont plus libres, mais interagissent fortement entre eux, ce qui a pour cons´equence l’´emergence d’une grande diversit´e de ph´enom`enes physiques fascinants comme des transitions m´etal-isolant, de la supraconductivit´e `a temp´erature ´elev´ee ou de la magn´etor´esistance colossale. De plus, ces propri´et´es sont tr`es sensibles `a des perturbations comme la temp´erature, la pression, le champ magn´etique ou le dopage, donnant lieu `a des di- agrammes de phases complexes. Cette richesse en propri´et´es continue `a motiver un grand effort de recherche et a d´ej`a men´e `a de nombreuses applications pratiques.

Parmi ces mat´eriaux `a corr´elations fortes, nous nous concentrons, dans le cadre de cette th`ese, sur les perovskites nickelates qui poss`edent la formule chimique RNiO<sub>3</sub>, o`u R est une terre rare. Les nickelates pr´esentent une transition spectaculaire d’un comportement m´etallique `a un comportement isolant en fonction de la temp´erature manifest´ee par un changement de plusieurs ordres de grandeurs en r´esistivit´e. De

vii

plus, des ph´enom`enes comme la disproportionation des charges et un ordre antifer- romagn´etique apparaissent dans la phase isolante. Toutes ces propri´et´es d´ependent fortement du dopage ainsi que de la largeur de bande laquelle est modifiable par la substitution de terre rare ou par la pression hydrostatique. Le but de cette th`ese con- siste `a identifier les m´ecanismes physiques responsables de la transition m´etal-isolant et d’explorer les moyens de les manipuler.

Pour atteindre ce but, nous avons con¸cu et construit une chambre de pulv´erisation cathodique `a magn´etron `a radio-fr´equence en g´eom´etrie hors-axe, dedi´ee `a la crois- sance `a l’´echelle atomique de structures artificielles comme des h´et´erojonctions et des super-r´eseaux; a-priori absent dans la nature. Dans notre cas, il s’agit de couches minces de LaNiO₃ et de NdNiO₃ `a ´epaisseurs et contraintes variables ainsi que des super-r´esaux de LaNiO₃ combin´e avec du LaMnO₃ ferromagn´etique.

Trois d´emarches sont employ´ees pour manipuler la transition m´etal-isolant dans le NdNiO₃. Premi`erement, nous d´emontrons que la transition peut ˆetre maˆıtris´ee en utilisant la technique de l’effet de champ. En augmentant le nombre de porteurs, la temp´erature de transition est abaiss´ee de plus de 50 K et de giganteques change- ments de r´esistivit´e d’environ 60 000 % sont d´etect´es. Cette exp´erience prouve que la densit´e de porteurs joue un rˆole essentiel dans l’origine de la transition.

Deuxi`ement, des contraintes de croissance compressives sont utilis´ees pour non- seulement varier la temp´erature de transition, mais aussi pour la supprimer, stabilisant ainsi la phase m´etallique jusqu’aux plus basses temp´eratures. Un ´elargissement des bandes ou des effets orbitaux pourraient ˆetre responsable du ph´enom`ene observ´e.

Finalement, en irradiant l’h´et´erojonction avec de la lumi`ere dont la fr´equence corre- spond `a l’´energie d’absorption d’un mode de vibration du substrat, une phase m´etallique est induite instantan´ement.

Une grande partie de cette th`ese est aussi consacr´ee `a l’´etude de LaNiO3. Ce dernier est le seul membre de la famille des nickelates ne manifestant ni de transition m´etal- isolant ni d’autres ph´enom`enes d’ordre, restant un m´etal paramagn´etique `a toutes temp´eratures. Dans cette th`ese, nous montrons que le LaNiO₃ sous certaines condi- tions peut r´ev´eler des propri´et´es similaires aux autres nickelates.

Plus pr´ecis´emment, nous avons d´ecouvert qu’en r´eduisant l’´epaisseur des couches de LaNiO₃ `a quelques cellules unitaires, une transition m´etal-isolant ´emerge. La transition est associ´ee `a une r´eduction de dimensionalit´e de 3D `a 2D. De plus, la pr´esence d’importantes fluctuations de spins se r´ev`ele `a travers une magnetor´esistance isotropique. L’origine de ces fluctuations est probablement li´ee `a la proximit´e du

syst`eme de la phase isolante antiferromagn´etique des nickelates.

La r´ealisation de super-r´eseaux contenant des nanocouches de LaNiO₃ et LaMnO₃ semble renforcer ces observations. Effectivement, un champ d’anisotropie d’´echange est d´etect´e dans les multicouches, sugg´erant qu’un ordre magn´etique se d´eveloppe dans le LaNiO₃. Cet ordre magn´etique ressemblerait `a une onde de densit´e de spin.

Acknowledgments v

R´esum´e vii

1 Introduction 1

2 Theoretical Concepts 3

2.1 Electrical conduction . . . 3

2.1.1 Metals and insulators . . . 5

2.1.2 Mott insulators . . . 7

2.1.3 Anderson insulators . . . 10

2.2 Metal-insulator transitions . . . 12

2.2.1 Mott-Hubbard transitions . . . 13

2.2.2 Mott-Anderson transitions . . . 14

2.2.3 Other types of metal-insulator transitions . . . 16

2.3 Nickelates . . . 16

2.3.1 Pervoskites and related complex oxides . . . 16

2.3.2 RNiO₃ . . . 19

3 Experimental Setups 25 3.1 Thin-film deposition . . . 25

3.2 Structural Characterization . . . 28

3.2.1 Characterization procedure . . . 29 xi

3.2.2 Strained nickelate thin films . . . 31

3.3 Electrical measurement setups . . . 35

4 Metal-insulator transition in NdNiO₃ 37 4.1 Electrical transport in NdNiO₃ . . . 37

4.1.1 Metal-insulator transition in NdNiO₃ thin films . . . 37

4.1.2 Strain control of the metal-insulator transition . . . 39

4.1.3 Size effect . . . 42

4.2 Electric-field control of metal-insulator transition . . . 44

4.2.1 Field-effect . . . 44

4.2.2 Field-effect on NdNiO3 . . . 48

4.3 Light control of metal-insulator transition . . . 51

4.4 Conclusions . . . 55

5 Metal-insulator transition in LaNiO₃ 57 5.1 Physical properties of LaNiO₃ . . . 57

5.2 Metal-insulator transition in LaNiO₃ thin films . . . 62

5.2.1 Metal-insulator transition . . . 62

5.2.2 Weak localization . . . 64

5.2.3 Field-effect . . . 70

5.3 Conclusions . . . 71

6 LaNiO₃-based superlattices 73 6.1 LaNiO₃/LaMnO₃ superlattices . . . 73

6.1.1 Properties of LaMnO₃ films . . . 74

6.1.2 Structural Characterization of LaNiO₃/LaMnO₃ superlattices 76 6.2 Physical properties of LaNiO₃/LaMnO₃ superlattices . . . 80

6.2.1 Electrical properties . . . 80

6.2.2 Magnetic properties . . . 81

6.3 Conclusions . . . 87

7 Conclusions and Perspectives 89

Appendices 91

A Low temperature behavior of nickelate thin films 91

B List of publications 94

Bibliography I

Introduction

Fig. 1.1: Everything happens at the interface

Life happens at the interface between Earth and space. The human body interacts with its environment through interfaces, such as the skin, the lung tissue or cell membranes. To paraphrase Herbert Kroemer, it may often be said that everything happens at the interface. This certainly applies to condensed matter physics, where modern scientists have begun to create interfaces by combining different materials in so-called heterostructures in order to generate and control novel electronic proper- ties. A plethora of devices emerged from such heterostructures that revolutionized our everyday lives. The most remarkable example is the field-effect transistor, where a switch controlled by an electric field determines whether an electrical current can

1

flow trough the device. Its invention in 1947 marks the birth of the information technology revolution and these devices can be found today in nearly every electronic device, such as computers and smartphones. In 2007, it was estimated that 10 million transistors are produced per day per person on the planet. Conventional transistors or other devices are typically based on semiconductor materials, in which electrons travel throughout without interacting much with each other. Semiconductors by themselves lack exotic properties. All the functionalities of the devices we rely on today arise from interfaces.

Transition metal oxides are a class of materials in which electrons strongly inter- act with each other. Concomitantly, an incredible variety of functional properties, unattainable in conventional semiconductors, emerges such as metal-insulator transi- tions, superconductivity or colossal magnetoresitance. While displaying a remarkable diversity in physical properties, their underlying crystalline structure is similar, allowing heterostructures based on these materials to be fabricated. An interface can thus be formed between materials, which by themselves have fascinating properties. The potential for novel exciting functionalities and physics is high and a large amount of research groups in academia and industry are exploring such heterostructures.

In this thesis, heterostructures based on perovskite nickelates RNiO₃, where R is a rare earth, have been studied. By themselves, they display a bandwidth-controlled temperature-driven metal-insulator transition accompanied by charge and spin order- ing and large changes in resistance. Our goal is to control and understand the un- derlying mechanisms of the metal-insulator transition and to look for possibly new physical properties and functionalities.

In chapter 2, we will introduce the basic theoretical concepts relevant to metal- insulator transitions and the current status of the experimental and theoretical ef- forts focused on nickelates. Then, chapter 3 will be dedicated to the experimental techniques used during this thesis to fabricate, characterize and measure our nick- elate heterostructures. In chapter 4, we will demonstrate the ability to control the metal-insulator transition of NdNiO₃ heterostructures using three different ap- proached: strain, field-effect, and ultrafast optical pumping. We will then investigate inchapter 5, the nickelate compound LaNiO₃that does not display any metal-insulator transition nor any ordering phenomena. We will show that a metal-insulator transi- tion can be induced and we will discuss its origin. Finally, in chapter 6, LaNiO₃ and LaMnO₃ bilayers are repeatedly stacked to form so-called superlattices, containing many interfaces. A surprising functionality is unraveled with strong implications for the physical properties of LaNiO₃.

Theoretical Concepts

In this chapter, we will introduce the basic theoretical concepts related to Mott and Anderson insulators and their transition to the metallic phase. Special emphasis on the electrical properties will be given. These concepts will then be applied to describe the current consensus on experimental and theoretical aspects of the metal-insulator transition in perovskite nickelates.

2.1 Electrical conduction

At the heart of electrical conduction lies Ohm’s law [1],

V =R·I, (2.1)

which states that a current I flowing through a conductor is proportional to the applied voltageV. The proportionality factor is the resistanceR and its inverse is the conductance G = 1/R. A few decades later, in 1879, E. H. Hall found that when an external magnetic field is applied perpendicular to the current flowing through a conductor, a voltage V_H builds up between two electrodes located perpendicularly to both the magnetic fieldBand the current [2]. The following proportional relation was found:

V_H =R_H·B·I, (2.2)

where R_H is called the Hall coefficient.

These two empirical observations opened the field of electrical transport and magne- totransport. P. Drude was the first to propose a model for the underlying mechanism

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of conduction in a metal[3]. The so-called Drude model is able to relate the resis- tance to physical quantities such as the scattering rate τ, the carrier density n, and the dimensions of the sample. The resistivity ρ of a material can be calculated from the measured resistance R as follows:

ρ=R_sheetd =Rdw

L, (2.3)

wherew,Landd are the width, the length and the thickness of the sample respectively.

R_sheet is the sheet resistance, which is often used when dealing with two dimensional systems. The Drude conductivity can be written as

σ = ne²τ

m^? =neµ (2.4)

whereµ is the mobility andm^? the effective mass of the electron ¹. Within the same framework, the measured Hall resistance can be used to determine the sign and the number of carriers in the material

n = 1

R_Hed. (2.5)

The thickness d of the sample only enters in Eq. (2.5) when the three dimensional carrier density is required and can be omitted in the case of two dimensional systems.

With the advent of quantum mechanics, Drude’s model was revisited. Sommerfeld as- sumed that an electron gas obeys Fermi-Dirac statistics instead of Maxwell-Boltzmann statistics, which means that each orbital state can host at most two electrons, one for each spin direction. Therefore, electron states in a crystal are occupied up to a limiting energy, called the Fermi energyE_F [4]. In reciprocal k-space, occupied states are separated from unoccupied states by the Fermi surface k_F, which directly relates to the number of carriers n in the following way,

k_F = (3π²n)^1/3 (2.6)

The Fermi energy can be calculated using EF =~²k_F²/2m^?. Another important phys- ical quantity is the mean free path l_e, also called the elastic scattering length, which

1As will be shown later, electrons in solids lie in bands, which will renormalize their mass, called then the effective massm^?. Depending on the band structure of the material, the effective mass can range from 0 at the Dirac point in graphene to ∼10 for correlated materials, i.e. LaNiO₃, and even 1000 for some heavy fermion materials.

is the average distance travelled by the electron between two elastic collisions.

l_e =v_Fτ_e (2.7)

wherev_F = ~k_F/m^? is the Fermi velocity and τ_e is the elastic scattering time. Both can be calculated by measuring the resistivity and the carrier density at low tempera- tures (e.g. 4.2 K) using Eq. (2.4) and Eq. (2.5) (and assuming m^? is known).

Many different physical mechanisms can contribute to the scattering rate, hence to the dc resistivity: e.g. impurity scattering; electron-electron, electron-phonon, or spin- orbit interactions; or magnetic scattering. Their contribution to the overall resistivity will vary with temperature, but in a given temperature range and in a particular system, often only one or a few mechanisms that can be dealt with separately will be relevant.

A powerful method to tackle these problems is the quantum-many body formalism. It relates the dc resistivity to microscopic quantities like the self-energy by making use of Ohm’s law and linear response theory. Details of calculations can be found in Ref.

[5]. For example, in the presence of electron-electron interactions, the dc resistivity at low temperatures (below ∼50 K) is predicted to be

ρ(T) =ρ₀+AT². (2.8)

ρ₀ is the residual resistivity in the T →0 limit, arising from elastic scattering at im- purities, and A is a positive coefficient that is directly related to the square of the specific heat coefficientγ [6] and can be seen as an indicator of interaction strength.

More than a century after Drude published his theory, the first steps in the character- ization of new materials still often consist of measuring their resistance as a function of temperature and magnetic field and determining their various physical quantities (resistivity, carrier sign and density, mean free path, ... ).

2.1.1 Metals and insulators

Materials can be classified into two categories: metals and insulators. The rigorous definition of an insulator is that it displays zero conductivity at 0 K whereas a metal has a finite conductivity. Since experiments are generally preformed at finite temper- atures, it is common to define an insulator as a material that exhibits d ρ/d T < 0, i.e. an increase in resistance with decreasing temperature. Conversely, a metal shows a decrease in resistance with decreasing temperature, hence d ρ/d T >0.

Despite the success of the Drude model, it failed to explain several experimental observations in condensed matter. For instance, why some materials are insulators and some metals? Bloch, Peierls and Wilson established that the energies of electrons in a periodic lattice form bands, which are separated by regions with forbidden energies, called band gaps [7, 8, 9]. If the bands are either empty or full, the system will be an insulator, whereas, if the bands are partially filled, it will be a metal, as illustrated in Fig. 2.1. In other words, when the Fermi energy lies in the band gap, the material is an insulator and when the Fermi energy crosses a band, it is a metal. An insulator with a small gap is often called a semiconductor due to thermal excitations across the gap.

Fig. 2.1: Band structure for an insulator and a metal. DOS and E stand for density of states and energy respectively.

The band structure arises from Schroedinger’s equation

∇²ψ+ 2m

~²

(E −V(x , y , z))ψ= 0 (2.9) whereV is the periodic lattice potential andE and ψ the energies and eigenfunctions respectively. Solving Eq. (2.9) is not straightforward. It requires an appropriate choice of potential V (e.g. the Hartree-Fock potential, which takes into account the ionic potential and the potential of weakly interacting electrons) as well as appropriate techniques to calculate the wavefunctions and corresponding energies. A particularly suitable method for narrow bands, such as the ones formed byd-orbitals, is the tight- binding model, where bands are formed by electrons hopping from one site to an other.

Using the second quantization formalism, the Hamiltonian of the tight-binding model

is given by

H_t =−t X

<i ,j >,σ

(c_{i ,σ}^† c_j,σ+h.c .), (2.10) wherec_{i σ}^†(c_{i σ}) denotes the creation (annihilation) of an electron with spin σ at site i and t is the hopping integral between neighboring sites,

t = Z

drφ^?_{i σ}(r)[−~²

2m∇²+V]φj σ(r) (2.11) where φ_{i σ} denotes the wave function of an electron on site i with spin σ. The bandwidth B, or the kinetic energy, is then given by

B = 2z t (2.12)

withz being the coordination number.

In an undoped insulator or semiconductor, conduction is mediated by carriers generated by excitation of electrons from the valence band (top filled band) to the conduction band (lowest empty band). This requires an energy equal or higher to the band gap Eg. The conductivity can then be written as

σ ∼ ex p(−Eg/k_BT), (2.13) wherek_B is the Boltzmann constant. Other mechanisms can contribute to conduction (e.g. variable range hopping at low temperatures), and will be described later.

2.1.2 Mott insulators

In 1937, de Boer and Verwey reported that NiO is an insulator [10]. This cubic com- pound is made of Ni²⁺ and O²⁻ ions, where the Ni has the electronic configuration 3d⁸. There are five 3d states, which can host ten electrons in total. Due to the crystal field, the states split into three degeneratet2g orbitals fully occupied by 6 elec- trons and two degenerate e_g orbitals occupied by two electrons. Hence, NiO has the configuration t_2g⁶ e_g²: the t_2g band is filled and the e_g band is half-filled. According to conventional band theory, NiO should thus be metallic. In their report, de Boer and Verwey mentioned a series of other transition metal oxides with partially filledd-bands that were found to be insulators or poor conductors and thus failed to be described by Wilson’s model. At that time Peierls noted that”electrostatic interaction between the electrons prevents them from moving at all”. This marks the beginning of the field of strongly correlated systems.

Sir N.F. Mott was among the first to investigate how Coulomb repulsion, i.e. electron- electron interactions, can explain the insulating state in some materials [11]. His model consisted of a lattice with one electron occupying each site. In the absence of on-site electron-electron interactions, the overlap of the atomic orbitals forms a single band, which is half-filled. The band is considered full when two-electrons, one with spin-up and one with spin-down, occupy each site. However, in the case of a considerable Coulomb repulsion, the energy cost for having two electrons on the same site can be large enough to split the band into a lower band of singly-occupied sites and an upper band of double-occupied sites. For one electron per site, the lower band will be full and the upper band empty, hence the system will be insulating and called aMott insulator.

The theoretical framework around this idea was worked out by P.W. Anderson in 1959[12] and J. Hubbard 1963-1964[13, 14, 15], who introduced the so called Hubbard HamiltonianH_H

H_H =−t X

<i j >σ

c_{i σ}^†c_{i σ} +UX

i

n_i↑n_i↓ (2.14)

where n_{i σ} = c_{i σ}^†c_{i σ} is the number operator. U is the on-site Coulomb interaction, defined as

U =e² Z Z

drdr’φ^?_{i σ}(r)φi σ(r) 1

|r−r’|φ^?_i−σ(r’)φ_i−σ(r’) (2.15) Eq. (2.14) is therefore an extension of the tight-binding Hamiltonian of Eq. (2.10), which is recovered in the case where electron-electron interactions do not play a significant role, i.e. U ∼ 0. The additional energy provided by U > 0 leads to a splitting of the d-band of initial widthB into the so-called lower and upper Hubbard bands with widths ofB1 and B2 respectively, as illustrated in Fig. 2.2. As mentioned above, the lower Hubbard band is the band of singly-occupied sites and the upper Hubbard band is the band of double-occupied sites. From Eq. (2.14), it becomes clear that the ratioU/t orU/B is the determining factor for the physical properties of a system. IfU/B <<1, the tight-binding Hamiltonian applies, the band is partially full and the system is metallic. When U/B >> 1, the system is dominated by electron- electron interactions, which leads to insulating behavior with a energy gap equal to U− ¹₂(B1 +B2).

Strongly correlated or Mott materials are almost exclusively transition metal oxides with partially filledd bands. Other systems are 4f rare earth or 5f actinide elements.

The main reason stems from the competition of energy scales arising from the kinetic term and the Coulomb repulsion, i.e. between localized and itinerant aspects. d

Fig. 2.2: Band structure for a Mott-Hubbard insulator and a charge transfer insulator for 1/2-filling.

orbitals are particularly localized in space and the wave functions are narrow leading to bandwidths of only a few electron volts. TheU of Eq. (2.15) gives the unscreened value of the Coulomb interaction and is typically tens of electron volts. However screening effects ind bands (from the proximity of the lower lying 4s band for example) reduceU to only a few electron volts, which is comparable to the bandwidth.

In some transition metal-oxide systems, the direct overlap between d-orbitals is so small that d-electrons can hybridize with ligand oxygen atoms. If, in addition, the oxygen p-band lies closer to the Fermi energy, the energy gap is no longer formed between two d-bands, but between the upper Hubbard d-band and the O 2p band, as shown in Fig. 2.2. This type of insulator, discussed by Zaanen, Sawatzky and Allen in Ref. [16], is called a charge-transfer insulator. The gap is then dominated by the charge-transfer energy ∆ = _d −_p, which represents the energy difference between the average position of the oxygen _p and transition metal bands _d. While the electron transfer in a Mott insulator occurs betweend −d orbitals of neighboring unit cells, i.e. d_iⁿd_jⁿ ↔d_iⁿ⁻¹d_jⁿ⁺¹ (d_i, d_j denote transition-metal sites), electrons in a charge-transfer insulator are transferred between p−d orbitals within a unit cell, i.e.

d_iⁿ ↔ d_iⁿ⁺¹L, where L denotes a hole in the anion valence band. The hybridization strength becomes larger as one moves from lighter transition metal elements (Ti, V) to heavier ones (Cu, Ni). According to this scheme, cuprates and nickelates would be charge transfer insulators whereas titanates and vanadates should be Mott insu- lators. Differences between these two types of insulators can indeed be observed in high-energy spectroscopy.

Up to this stage, long-range interactions, such as antiferromagnetism, have been ne- glected. Indeed, antiferromagnetic long range order can lead to insulating behavior as was worked out by Slater in 1951 [17]. Electrons with spin up and spin down have different potentials, therefore, the antiferromagnetic superlattice doubles its period- icity with respect to the non-magnetic lattice, which splits each energy band in half and opens a gap. While this argument is certainly valid at low temperatures, it fails to explain why many antiferromagnetic insulators remain insulating above their N´eel temperature, as is the case, for example, in NiO.

The precise use of the term Mott insulator can vary throughout the literature. It was mentioned above that a Mott insulator arises from on-site electron-electron in- teractions, which split an half-filled 3d-band into two bands: a full lower band where each site is occupied by one electron, and an empty upper band where each site is occupied by two electrons. The electrons on each site have a magnetic moment, but no long-range order such as antiferromagnetism for example is required for the insu- lating state. Electrons are excited across thed−d gap. In practice, the picture is less simple. Mott insulators are not always half-filled and can have several electrons per site, where the spins are coupled according to Hund’s rule. Additionally, various other phenomena (spin-orbit interaction, Jahn-Teller distortions, orbital or charge ordering) can take place, such that the simple Hubbard Hamiltonian, which considers only a sin- gle band, needs to be extended in order to take them into account. As was mentioned above, depending on theU and the ∆, some materials will form an energy gap between p−d orbitals, instead of d−d orbitals. A Mott insulator therefore generally refers to a system that is insulating due to strong electron-electron interactions, which cannot be neglected in the description of its physical properties.

A striking feature of Mott insulators is the transition to a metallic phase, which exhibits strong correlations characteristics. This will be discussed in more detail in the following section. Extensive reviews on the topic of Mott insulators and metal- insulator transitions can be found in N.F. Mott’s textbook [18] and in the review by M. Imada, A. Fujimori and Y. Tokura [19].

2.1.3 Anderson insulators

Both band and Mott insulators were described assuming a perfect crystal. However in reality, crystals are never free of disorder, especially the more complex compounds such as transition metal oxides. Disorder typically appears in these materials either as vacancies, impurities or dislocations, hence disrupting locally the periodicity of the

lattice. This leads to scattering. As disorder increases, the material can become amorphous, loosing its translational symmetry, i.e. its lattice periodicity. This means that above a certain amount of disorder, the mathematical description of the previous sections has to break down and that, as a consequence, Wilson’s description of an insulator is not accurate anymore. The question arises then how the insulating state can be described? To formulate the question more precisely: what is the wave func- tion of an electron in a random potential?

Anderson was the first to point out that the wave function can change dramatically in the presence of strong disorder, i.e. in a random potential [20]. He argued that the wavefunction becomes localized and its envelope decays exponentially from some point in space r₀.

|ψ(r0)| ∼ex p(|r−r₀|/ξ) (2.16)

Fig. 2.3: Wave function of an extended state a) and of a localized state b) with localization lengthξ. Adapted from [21].

This is illustrated in Fig. 2.3. In the presence of weak disorder (Fig. 2.3 a)), a Bloch wave will loose phase coherence over the length scale of the inelastic scattering length, but overall remain extended. Disorder will induce some randomness in the potential, but nearby localized orbitals will be nearly degenerate in energy and therefore overlap with each other and form an extended wave. However if disorder is strong, the random potential fluctuates strongly over the characteristic distance of an orbital exponential decay, and therefore nearby states are very unlikely to have similar energies. States with comparable energies might be very far apart in space, such that no overlap and hence no extended state can be formed.

In this strongly localized regime, conductivity at low temperatures (i.e. in the absence of thermal activation from the conduction band) arises from electrons hopping between localized states and is called variable range hopping. The hopping probability depends

on two parameters: the spatial separation between the two sites and their separation in energy. Conductivity then follows the relation

σ =Cex p(−(T0/T)^α) (2.17) where C depends on electron-phonon interactions and T₀ on the density of localized states at the Fermi level and the exponential decay rate of the envelope of the localized wave. The exponent α will depend on the dimensionality of the system: in three dimensions α = 1/4 [22] and in two dimensions α = 1/3 [23]. In the case where interaction effects are present, α= 1/2 [24].

2.2 Metal-insulator transitions

The transition from an insulator to a metal or vice versa is called the metal-insulator transition and continues to be one of the most central and debated topics in condensed matter physics.

It was again Mott who took the first important steps towards understanding the metal- insulator transition [11, 25]. He considered that each positive charge, which attracts electrons to it, will be screened by the available free carriers. The potential around the positive charge will decay exponentially as

V(r) =−e²

r e^{−r /λT F} (2.18)

withλ_{T F} being the Thomas-Fermi screening length given byλ_{T F} =p

(2πE_F/3e²n).

is the dielectric constant. In a normal metal, the screening length is typically ∼1

˚A. The same positive ion will also have an intrinsic effective Bohr radius aH, which is the distance of the outermost bound electron experiencing the Coulomb potential unscreened by free carriers.

If the effective Bohr radius is bigger than the Thomas-Fermi screening length, the oth- erwise bound electron is no longer bound because the attractive potential is screened by the free charges. Conversely, if the Thomas-Fermi screening length becomes too large, the free electrons are insufficient to screen the attractive Coulomb potential and become bound, hence the system becomes insulating.

Mott thus predicted that a transition between the metallic and insulating state occurs when λ_{T F} =a_H. Sinceλ_{T F} ∼0.22n^−1/3, Mott’s criterion can be rewritten as

a_Hn_c^1/3≈0.22. (2.19)

If aHnc^1/3 > 0.22, the system is metallic and if aHn^1/3c < 0.22 it is insulating. The precise value of the constant in Eq. (2.19) can vary depending on interactions. Nev- ertheless, this criterion has been successfully applied to a large number of systems en- compassing several orders of magnitude in carrier densities and Bohr radii [26]. Mott initially illustrated the transition from an insulating to a metallic state by bringing the atoms closer together, as can experimentally be realized in pressure experiments.

Introducing carriers into an insulating matrix is another approach, which will be dis- cussed later.

Among the various types of metal-insulator transitions, two have garnered a partic- ular amount of interest in the field of condensed matter, namely the Mott-Hubbard transition and the Mott-Anderson transition.

2.2.1 Mott-Hubbard transitions

A Mott-Hubbard transition or just Mott transition describes the transition from a Mott insulator to a metal. As was discussed in the previous section, the relevant parameters in the Hubbard model are the U/t ratio and the band filling n. The phase diagram of the metal-insulator transition is shown in Fig. 2.4. If each atomic site has 0 or 2 electrons per site, the material is a band insulator. If the filling is 1, the material will be a metal or an insulator depending onU/t. Mott and Hubbard argued that the transition occurs whenU = 1/2(B₁+B₂), i.e., when the gap closes and the upper and lower Hubbard bands cross each other. Increasing the strength ofU/t will bring the system deeper into the insulating state, while decreasing it will render the system metallic. Changing the U/t ratio can be achieved experimentally by applying hydrostatic pressure or internal/chemical pressure (by substituting atoms). Bringing atoms closer together will enhance the overlap between orbitals, therefore increase the bandwidth and weaken U/t, eventually turning an insulator into a metal. This is called a bandwidth-controlled metal-insulator transition. A canonical class of materials showing this behavior are the nickelates (RNiO₃, R being a rare earth), which will be the subject of this thesis. Another approach to trigger a transition into a metallic state is to change the filling fromn = 1 to n = 1±x (x <1), either electrostatically using the field-effect technique (see chapter 4) or by introducing dopants/vacancies into the material. This is called a filling-controlled metal-insulator transition. Famous examples are the high-T_c superconductors based on cuprates (i.e. La₂CuO₄) and the colossal magnetoresistive manganites (i.e. LaMnO₃).

The metallic phase emerging from such transitions is often referred to as anoma- lous and highly correlated. Staying within the Fermi-liquid description, such highly

Fig. 2.4: Phase diagram of the Mott metal-insulator transition with the ratio U/t shown as a function of filling. FC-MIT and BC-MIT stands for filling controlled and bandwidth controlled metal-insulator transition respectively. The shaded area close to the Mott insulating state is strictly speaking metallic, but may also be insulating due to the competition of ordered phases.

From [19].

correlated metals exhibit a large enhancement in effective mass and in magnetic sus- ceptibilities, as well as in other parameters, such as theT²-coefficient from Eq. (2.8).

Metals that are close to their Mott insulating state are subject to significant fluctua- tions and ordering of spin, charge and orbital degrees of freedom. In some cases, the Fermi-liquid description breaks down all together and other theories are required to describe the behavior (e.g. high-T_c superconductivity).

Finally, it is worth mentioning that despite being electronically driven, the metal- insulator transitions are generally accompanied by a structural change, such as a change in volume or even crystal structure, which however is a consequence of the transition rather than its origin.

2.2.2 Mott-Anderson transitions

A Mott-Anderson transition or just Anderson transition describes the transition from an Anderson insulator to a metal, and therefore deals with disordered electronic sys- tems, which can be interacting or non interacting. In the previous sections, we have seen that strong disorder leads to an Anderson insulator. The question is what hap- pens for intermediate disorder. Instead of changing the amount disorder, we can also consider varying the Fermi energy. There is a continuous range of states from the

bottom of the band until the middle of the band. The lowest states at the bottom of the band are generally localized and trapped whereas the states in the middle of the band are extended. Hence, increasing the Fermi energy will eventually lead to a transition from localized to extended states. This critical energy Ec is called the mobility edge and separates insulators (with E_c > E_F) and metals (with E_c < E_F).

As opposed to band and Mott insulators, disordered insulating systems do not have a band gap, but have a finite density of states at the Fermi level. Only in the presence of interactions, a gap can develop.

In the insulating regime, conduction occurs when carriers are excited from the Fermi energy across the mobility edge, analogous to semiconductors. While the conductiv- ity at low temperature proceeds via variable range hopping, it follows the thermally activated behavior of Eq. (2.13) at higher temperatures, except that the band gap is replaced byE_c −E_F, the difference between the Fermi energy and the mobility edge (see Fig. 2.5).

Fig. 2.5: Density of states in the conduction band of non-crystalline material with the mobility edge separated by ∆E from the band edge. From [27].

Ioffe and Regel pointed out that the kinetic theory of conductivity has to break down when the particle wavelength becomes longer than the mean free path [28]. Hence, when the product of the Fermi wavenumber and the mean free path, k_Fl, becomes smaller than unity, a metal to insulator transition is expected. This translates into a minimum conductivity required to sustain a metallic state. In three dimensions, the value depends on geometrical factors [29], but in two dimensions it was shown to be universal [30]

k_Fl = h/e²

R_sheet ≈ 25 kΩ

R_sheet. (2.20)

Eq. (2.20) demonstrates thatkFl in two dimensions solely depends on the sheet resis- tance and that a metal-insulator transition should occur at a sheet resistance of 25 kΩ.

From an experimental point of view, k_Fl can be varied continuously by changing the thickness, which directly affects the sheet resistance. Hence, in a disordered system one expects a metal-insulator transition as the thickness is reduced. These predictions were confirmed in a series of experiments on amorphous metals and superconductors [31]. Another phenomenon characteristic of such systems is weak localization which will be discussed in detail in Chapter 5.

2.2.3 Other types of metal-insulator transitions

Besides Mott and Anderson metal-insulator transitions, there exists other types of transitions, which are outside the scope of this thesis. However, it is worth mentioning the Peierls transition. Peierls considered a one-dimensional lattice with one electron per site. He stated that in the absence of electron-electron interactions, the atoms would be separated by their distance a (see Fig. 2.6 a)). However, in the presence of interactions, this configuration would be inherently unstable and prefer re-arranging itself as sketched in Fig. 2.6 b), hence doubling the lattice periodicity and opening a gap. The resulting ground states are called charge or spin density waves and extensive reviews on their properties were written by G. Gruner [32, 33]. Density waves typically develop in highly anisotropic, mostly one-dimensional, crystals, but they can occur in two or three dimensional band structures as well.

While Mott transitions are predominantly electronically-driven, Peierls transitions are considered to be structurally-driven, but in both cases, interactions are required to describe the insulating state.

2.3 Nickelates

2.3.1 Pervoskites and related complex oxides

Perovskites or perovskite-like compounds, which fall into the broader category of complex oxides, are a fascinating class of materials in which the interactions between charge, spin, orbital and lattice degrees of freedom strongly compete with each other.

Their strong interplay leads to a wealth of physical properties such as ferroelectricity, multiferroicity, metal-insulator transitions, colossal magnetoresistance and high-Tc

superconductivity [34, 35, 36, 19, 37, 38] and continues to fascinate the scientific community as well as to drive new technological developments.

Fig. 2.6: a) Metal without interactions. b) Peierls insulator in the presence of interactions.

From [32].

A perovskite compound has the chemical formula ABO₃, where A and B are cations, B being generally a 3d transition metal. At elevated temperatures, the structure is often cubic of the space group P m¯3m as shown in Fig. 2.7 a). However, their ground state seldom remains cubic as they can undergo structural transformations as the temperature is lowered, hence reducing their symmetry. Examples of different symmetries and their structural properties can be visualized in Fig. 2.7 b).

The deviation from the cubic structure can be quantified using the tolerance factor t, which is given by

t = r_A−O

√2r_B−O. (2.21)

The tolerance factor compares the ionic bond distancesr_A−O andr_B−Oand is equal to 1 in the ideal cubic case when all the anions just touch the cations. In general however, the tolerance factor will deviate from unity. Ift >1, the material will have a tendency towards a polar distortion, where a displacement of the cation with respect to the oxygen anion generates an electrical dipole, which can ultimately lead to ferroelectric- ity. In the case wheret <1, non-polar instabilities are more common and the oxygen octahedra surrounding the B cation will rotate or buckle, hence directly affecting and reducing the so-called superexchange angle B-O-B. This angle is particularly relevant

Fig. 2.7: a) A cubic perovskite of the type ABO3. The red atom corresponds to A, the blue to B and the green to the oxygen O. b) Examples of different symmetries which are relevant for the perovskite nickelates.

in materials where conduction or magnetism originates from states that overlap be- tween the 3d orbitals of the B atom and the 2p orbitals of the O cation. In this case, the transfer integral or bandwidth is approximated by t ∼ t₀c os(π−θ), where t₀ is the transfer integral for the cubic perovskite andθ the B-O-B angle. This shows that the bandwidth, hence the interaction strength of U/B, is very sensitive to changes in the tolerance factor. Excellent examples illustrating how structural distortions affect physical properties are rare-earth nickelates.

2.3.2 RNiO

The perovskite rare earth nickelates have the generic formula RNiO₃, where R is a rare earth and Ni is in its 3+ low spin state with fully occupied t_2g bands and one electron in thee_g band. Conduction states are believed to be formed by the hybridiza- tion of the Ni 3d and O 2p orbitals. The nickelates are mainly known for their sharp temperature-driven metal-insulator transition with orders of magnitude changes in the resistance. Excellent reviews on this topic were written by M. Medarde and G. Catalan [39, 40]. Nickelates were first synthesized in 1971, however it was only from 1991 onwards that they were systematically studied [41, 42]. The main motivation at that time was to find compounds other than cuprates that would display high temperature superconductivity. Nevertheless, nickelates remained largely overlooked in comparison to other perovskite families such as titanates, manganites or cuprates. One of the main reasons for this was that nickelates are extremely difficult to synthesize because the stabilization of the Ni³⁺ state requires very high temperatures and high pressures.

Up to now, bulk single crystals are practically nonexistent (largest crystals are smaller than 100 µm), hence most of the initial research was performed on powdered poly- crystalline samples. Single crystals of nickelates with rare earths ranging from La to Sm have so far only been achieved via thin film growth [43, 44, 45]. Thin films with smaller rare earth radii have not yet been grown. Epitaxial thin films therefore remain the closest system available to study the intrinsic properties of single crystalline nick- elates.

Ground state: charge and spin ordering

Except for LaNiO₃, which is rhombohedral (R¯3c) and a paramagnetic metal at all temperatures in the bulk, perovskite nickelates exhibit a metal-insulator transition from a high temperature orthorhombic (P bnm) metallic phase to a low temperature monoclinic (P2₁/n) insulating phase. The monoclinic distortion is very small, but has

been detected by Xray diffraction and Raman scattering [46, 47]. The ground state of these materials are characterized by two phenomena.

The first is a charge ordering phenomena, also called charge disproportionation of the Ni site in the form: 2Ni³⁺ → Ni^3+δ + Ni^3−δ. The Ni sites splits their unique 3+

valence in the metallic phase, into two adjacent Ni sites in the insulating phase with respective valences of 3 +δ and 3−δ (δ is below 1 and depends on the distortion (rare earth) and the temperature). The charge disproportionation has been observed for the entire family in the bulk and for NdNiO₃ in thin films [46, 48, 49].

The second phenomenon is a peculiar antiferromagnetic ordering that develops in all nickelates with a wave vector of (1/2, 0, 1/2)or tho in the orthorhombic notation or (1/4, 1/4, 1/4)_pc in the pseudocubic notation [50, 51]. The spins are ferromagnet- ically coupled in the (111)_pc planes, which themselves are stacked in a ↑↑↓↓ fashion along the [111]_pc.

The distances between the ferromagnetically- and antiferromagnetically-coupled planes should be different due to differences in magnetostriction of ferromagnetic and anti- ferromagnetic bonds. Together with the alternating charges from the charge ordering, this should lead to a net ferroelectric polarization along the [111]_pc direction. Hence the nickelates should be magnetoelectric multiferroics. However this has yet to be proven experimentally [52, 53]. A sketch of the charge and spin ordering in the nick- elates is illustrated in Fig. 2.8.

Fig. 2.8: Charge and spin ordering in RNiO3. The empty and filled circles correspond to the Ni sites with valences of 3±δ in δ → 1 limit. The red and blue arrows show the spins on each Ni site. The black arrow indicates the ionic displacements due to magnetostriction and the resulting polarization P is also drawn. Adapted from [52].

Phase diagram

The next remarkable feature of the nickelates is the dependence of their physical properties on the chosen rare earth, which makes the nickelates a canonical example for bandwidth controlled metal-insulator transitions. This is illustrated by the phase diagram in Fig. 2.9. By choosing a smaller rare earth, the tolerance factor and subse- quently the Ni-O-Ni superexchange angle is decreased and the transition temperature TMI between the metallic and the charge ordered phase increases continuously from

∼130 K for PrNiO3 to ∼600 K for LuNiO3. In fact, by choosing a combination of different rare earths, one can tune T_MI to lie anywhere on this line, e.g. at room temperature for the mixed compound Nd0.5Sm0.5NiO3. LaNiO3, which has the largest tolerance factor, does not undergo a metal-insulator transition. Note that LaNiO₃ also has a different crystal structure; it is rhombohedral as opposed to orthorhombic.

Reducing the Ni-O-Ni angle leads to a reduction of the bandwidth, hence to an in- crease of the U/B strength (assuming U does not change much), which drives the system deeper into the insulating state. The rare earth A site cation is not directly relevant to the electronic properties in the Ni-O network, but it enables the bandwidth to be tuned by 30-40 % [19].

In the antiferromagnetic phase, two regimes can be observed. In the first part of the diagram, the N´eel temperature TN of the antiferromagnetic ordering coincides with T_MI and also increases with increasing U/B strength. In this regime, the transition is first-order and the resistance displays hysteresis upon thermal cycling. This is the case for PrNiO₃and NdNiO₃. For Sm and smaller rare earths,T_N is decoupled fromT_MI and decreases with decreasing (increasing) tolerance factor (U/B). The system therefore undergoes two transitions: from a paramagnetic metallic to a paramagnetic insulating at high temperatures, and eventually to an antiferromagnetic insulating phase at lower temperatures. Photoemission spectroscopy and neutron measurements confirm these two different regimes [54]. In the resistance versus temperature curves, no hysteresis is observed. The two antiferromagnetic regimes, where T_N first increases and then decreases, can be ascribed in a more general way to a passage from a weak to a strong Mott insulator [55, 56].

For all nickelates, the transition is also associated with a small structural change that leads to a volume increase in the insulating phase of∼0.2% and an additional buckling of the octahedra with a reduction of the Ni-O-Ni angle by less than one degree. These changes are relatively small compared to the textbook Mott insulator V2O3, where a 3.5% change in volume, as well as a reduction of crystal symmetry is observed.

Classic experiments to perform on systems that exhibit metal-insulator transitions are

0.86 0.88 0.9 0.92 0.94

Tolerance factor (t)

0 100 200 300 400 500 600

T [K]

T_MI

Pr Nd Sm

Eu Gd Dy

ErHo Lu Y

Sm_1/2Nd_1/2

T_Neel

La (R3c) (Pbnm)

(P2₁/n)

(P2₁/n) CO-PM insulator

CO-AFM insulator

PM metal

Fig. 2.9: Phase Diagram of RNiO₃. CO, PM and AFM stands for charge ordered, paramag- netic and antiferromagnetic respectively.

the application of hydrostatic pressure and the introduction of dopants into the insu- lating matrix. Both experiments were performed on nickelates.

First, it was shown that hydrostatic pressure can suppress the metal-insulator transi- tion and stabilize the metallic phase, due to the fact that atoms are brought closer together and enhancing the hybridization and facilitate conduction [57, 58, 59].

A similar behavior is observed upon doping. The transition temperature decreases con- tinuously with carrier doping, whether these dopants are holes or electrons [60, 61].

Even with a careful structural analysis, however, it is difficult to disentangle the ef- fects of intrinsic doping and extrinsic structural deformations due to the sensitivity of nickelates to changes in the lattice.

Theory of nickelates

The nature of the insulating ground state and the origin of the metal-insulator transi- tion in the nickelates are still being debated. As mentioned above, the nickelates have a 3d⁷ configuration with a filledt_2g band and one electron in the degeneratee_g band.

In principle, the Ni³⁺should be Jahn-Teller active, like Mn³⁺in LaMnO₃, which would lift the degeneracy of the e_g orbitals (x²−y² and 3z²−r²). However, no changes of the NiO₆ octahedra across the transition have been observed, nor have any other signs suggesting the presence of orbital ordering in the nickelates [62, 51].

Electronic correlations are most likely to play a key role in the nickelates, which of- ten places them into the broad category of Mott insulators and Mott metal-insulator transitions. Due to their high valence, the nickelates were initially located in the Zaanen-Sawatzky-Allen scheme at the boundary between the charge transfer insu- lators and low-∆ metals making them ”bad” metals (rather low conductivity for a metal) and ”bad” insulators (rather low resistance for an insulator). This is in agree- ment with the observation that the bandwidth can be continuously tuned by changing the orthorhombic distortion, i.e. the Ni-O-Ni angle. Assuming that U remains more or less constant among the different RNiO₃, the interaction strength U/B therefore increases with the distortion and the nickelates move from a metallic (LaNiO₃) to an insulating state with continuously increasing transition temperature.

However, M. Medardeet al. have found in their oxygen isotope experiment (replacing O¹⁶ by O¹⁸) that the transition temperature is greatly affected despite the bandwidth remaining constant [63]. This shows that the bandwidth is not solely responsible for the transition and requires strong electron-lattice interactions to be taken into account [64]. In addition, the discovery of charge disproportionation demonstrated that nick- elates cannot be charge-transfer insulators. The nickelates are somehow the opposite of it since it is energetically more favorable to put two electrons on the same site than keeping them apart. Instead, it was argued that nickelates can be placed in a regime

where ∆ is small and even negative, which is sometimes called negative charge transfer insulator or self-doped Mott insulators [19, 65]. With small or negative ∆, the highest part of the oxygen 2p bands can overlap with the lowest part of the upper Hubbard band, such that some holes are transferred from the 3d orbitals to the 2p orbitals, or conversely some electrons from the oxygen to the transition metal. Indeed, using high-energy spectroscopy, it was found that in PrNiO₃, ∆ is smaller than 1 eV and that the ground state is a mixture ofd⁷(Ni³⁺) andd⁸L(Ni²⁺with one hole on the oxygen) configurations [66, 67]. Mizokawa et al. demonstrated that, in this regime, charge and spin order can emerge naturally in the form of charge and spin-density waves, which are able to explain most of the experimental observations especially the unusual magnetism in nickelates [68, 69, 70]. According to this scenario, the nickelates would undergo the following three phase transitions: metal to spin-density wave, metal to charge-density wave and charge-density wave to spin-density wave. Note that the unit cell in the insulating state associated with the wave vector (1/4,1/4,1/4)_pc is four times larger than in the metallic state and that thee_g band is filled to a quarter. In this case, thee_g band would split and open a gap between the filled and empty states in a similar way as in a Peierls transition. Another system that displays this charge ordering is CaFeO₃ with a t_2g³ e_g¹ configuration [71]. Both are Jahn-Teller active, but prefer to disproportionate instead, which is counter intuitive since U/B is generally large in 3d transition metal oxides. It was shown that for systems that are close to the itinerant limit (such as RNiO₃), charge disproportionation is indeed energetically cheaper and presents an alternative route to the Jahn-Teller distortion to lift the or- bital degeneracy [67, 72].

The field of nickelates is relatively young and rapidly evolving. Above is a summary of the current consensus on the mechanisms behind the gap opening, which is far from being mature and might evolve with future experiments.

Experimental Setups

In this chapter, an overview of the experimental techniques used for thin film de- position, structural analysis, device processing and electrical characterization will be given.

3.1 Thin-film deposition

In order to grow high-quality epitaxial single crystalline thin films of perovskite nicke- lates, one must be able to stoichiometrically transfer the three constituting elements (R, Ni and O) from the source material to a substrate. The techniques available for thin film deposition fall into two classes: chemical vapor deposition and physical vapor deposition. A more detailed description of the process of thin film growth can be found in several textbooks [73, 74]. Perovskite nickelate thin films have been grown both by chemical vapor deposition [75] and by physical vapor deposition techniques such as molecular beam epitaxy [76], pulsed laser deposition [44] and sputtering [77].

For this thesis, we chose radio-frequency off-axis reactive magnetron sputtering. Sput- tering is the process of removing atoms from the surface of the target by bombardment by positive ions of inert gas, such as Argon. The target is constituted of the desired elements in stoichiometric proportions: e.g. in the case of LaNiO3, the target contains 68.6% of La₂O₃ and 31.4% of NiO. An electric field of sufficient strength is applied between the target (cathode) and the grounded chamber (anode) in order to ionize Argon. The substrate is typically fixed on a heater, which is grounded to the chamber.

These ions are accelerated by the electric field towards the target and transfer their kinetic energy to the target (∼1 keV), which causes the atoms close to the surface

25