Thesis
Reference
Spectroscopic studies of layered iridium oxides
DE LA TORRE, Alberto
Abstract
This thesis reports the doping evolution of the electronic structure of the single and bilayer idridates Sr2IrO4 and Sr3Ir2O7 from angle resolved photoemission (ARPES) experiments.
DE LA TORRE, Alberto. Spectroscopic studies of layered iridium oxides. Thèse de doctorat : Univ. Genève, 2015, no. Sc. 4866
URN : urn:nbn:ch:unige-790952
DOI : 10.13097/archive-ouverte/unige:79095
Available at:
http://archive-ouverte.unige.ch/unige:79095
Disclaimer: layout of this document may differ from the published version.
Department of Quantum Matter Physics
Spectroscopic Studies of Layered Iridium Oxides
THÈSE
présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de docteur ès Sciences, mention Physique
par
Alberto De la Torre
de
Córdoba (Espagne)
Thèse n◦4866
GENÈVE
Atelier d’impression ReproMail 2015
This thesis is the product of many hours of support, both scientific and emotional, from many different people.
First of all, I would like to thank Felix Baumberger for giving me the opportunity to join his group when I did not even know how to hold a screwdriver. It has been a pleasure to learn physics and the intricacies of ARPES from him.
This thesis would not exist without Robin Perry, who brought the iridates into my life. I also need to acknowledge Emily Hunter. Both welcomed me to their lab and opened my eyes to the fascinating world of crystal growth.
I would also like to thank Alaska Subedi, Christophe Berthod and Antoine Georges for their theoretical support, their patience and their capacity to explain difficult con- cepts.
It has been a real honor to work with Peter Baltzer. His vast knowledge and expertise never ceased to amaze me.
During the last two years I was fortunate enough to collaborate with many excellent scientists. Irene Battisti and Milan Allan put in many hours down in the bunker running the STM experiments with me. Ignacio Gutierrez, who, apart from being a transport wizard, was always there to cheer me up and play Thursday football. Without the support of the many synchrotron scientists, especially Timur Kim and Moritz Hoesch at Diamond, most of the data in this thesis would have been non-existent. And to everyone who helped in any way, big or small; a big thank you!
Special thanks goes to all the members of the ARPES group: Siobhan McKeown- Walker and Anna Tamai, who coped with me for so long, Phil King, Flavio Bruno, Sara Riccó, Zhiming Wang, Bram van Megen and Irene Cucchi. It was a pleasure to work with you all.
I would also like to thank all the people whom I can call friends, those back at home and those I have met over these last four years, both in St Andrews and Geneva.
Thank you for being there despite the little time I could spend with you. A special thank you goes to Fernando Aguado, whose guidance has been really important for me in the last 8 years.
I would also like to thank my parents, Mariano & Pilar, and Alejandro, who will always be "el Nano" despite now being as tall as I am. Gracias por estar siempre a una llamada de teléfono y por entender que no he podido pasar en estos últimos años tanto tiempo con vosotros como me habría gustado.
Finally, I have to thank Jennifer. Thank you for being patient when my English was nothing more than gibberish. Thank you for being there with me throughout this process. Without you none of this would have been possible.
La supraconductivité dans les oxydes de cuivre 2D ayant la physique de Mott apparaît lorsqu’on les dope positivement depuis un état avec des quasiparticules nodales et un pseudo-gap antinodal. La compréhension de ce phénomène est toujours un défi stimu- lant et ambitieux. Malgré des travaux expérimentaux concertés, le choix entre divers modèles théoriques est rendu difficile par un manque d’information sur les strucures électroniques de bande des isolants de Mott apparentés. Les films minces d’oxydes d’iridates à structure pérovskite de type Ruddlesden-Popper Srn+1IrnO3n+1présentent un comportement corrélé dû à la conjonction de l’interaction spin-orbite, de la levée de dégénérescence dûe au champ cristallin ainsi que des corrélations entre électrons écrantées,U. Partant de la forte similitude entre la structure cristalline, la dynamique de spin et la structure électronique supposée des iridates et des cuprates, plusieurs groupes ont prédit que les oxydes d’iridates présenteraient une supraconductivité similaire elle aussi à celle des cuprates. Cependant, jusqu’à ce jour, aucun matériau supraconduteur n’a été découvert dans les composés de cette famille.
Cette thèse fait état de différences marquées entre les états fondamentaux de Sr2IrO4 et de Sr3Ir2O7à partir d’études de l’évolution de leur structure électronique en fonction de leur dopage en ayant recours à la spectroscopie de photoémission résolue en angle (ARPES: Angle Resolved PhotoEmission Spectroscopy).
Nous présentons tout d’abord la croissance et la caractérisation des oxydes d’iridates à une couche dopés négativement (Sr1−xLax)2IrO4 avecx≤0,05. Les mesures de transport électronique ne révèlent aucune trace de supraconductivité dans ces échan- tillons. En utilisant la spectroscopie photoélectronique avec résolution angulaire et la microscopie à effet tunnel, nous contrôlons l’évolution de la structure électronique à basse énergie en fonction du dopage dans le composé (Sr1−xLax)2IrO4. Nous trou- vons un état dont la phénoménologie est étonnamment proche de celle des cuprates sous-dopés, caractérisée par une surface de Fermi large avec un poids spectral cohérent selon la direction nodale coexistant avec un pseudo-gap anti-nodal. Nous prouvons ensuite que cet état électronique inhabituel découle de l’effondrement du gap dû aux corrélations. Des études de la dépendance en température de ce gap avec ARPES mettent en évidence une transition vers une surface de Fermi sans gap.
Nous présentons ultérieurement une étude ARPES de l’évolution de la structure électronique à basse énergie de l’iridate á deux couches (Sr1−xLax)3Ir2O7 en fonction du dopage. Contrastant fortement avec (Sr1−xLax)2IrO4, nous constatons, au-delà de la transition métal-isolant pourx= 0,03, l’émergence d’une petite surface de Fermi ne contenant que les porteurs de charge additionnelsxainsi qu’une réduction progressive de l’interactionU avec le dopage. L’état métallique du composé (Sr1−xLax)3Ir2O7
présente plus de quasiparticules cohérentes et aucune trace d’un pseudo-gap dans toute la zone de Brillouin, indiquant un état métallique à faibles corrélations.
Nous en concluons ainsi que Sr2IrO4devrait être décrit par un isolant 2D de Mott suivant une dynamique de spin de type Heisenberg avec une unique bande pseudo-spin 2D. D’un autre côté, nous trouvons que la description de l’état fondamental de Sr3Ir2O7
se rapproche plus de celle d’un métal corrélé.
Understanding how metallicity or even superconductivity emerges upon doping two- dimensional Mott insulators remains a daunting challenge. Despite concerted experi- mental efforts studying this problem in cuprates, discrimination between competing theoretical models remains challenging not least because of a lack of electronic struc- ture data from related doped Mott insulators. The Ruddlesden-Popper series of layered perovskite iridates Srn+1IrnO3n+1 shows correlated behavior due to the interplay between crystal-field splitting, spin-orbit interaction and moderate electron-electron correlationsU. Cuprate-like superconductivity has been predicted by several groups based on the strong similarities of the crystal structure, spin dynamics and putative electronic structure of that of cuprates. However, to date no superconductivity has been observed in the layered iridates.
This thesis reports the doping evolution of the electronic structure of the single and bilayer idridates Sr2IrO4and Sr3Ir2O7from angle resolved photoemission (ARPES) experiments.
Firstly, we present the growth and characterization of electron doped single layer iridates (Sr1−xLax)2IrO4 withx ≤ 0.05. Transport measurements show no signs of superconductivity in these samples. Using ARPES and scanning tunneling mi- croscopy we monitor the doping evolution of the low energy electronic structure of (Sr1−xLax)2IrO4. We find a state with intriguingly similar phenomenology to that of underdoped cuprates. In particular, our data shows a large Fermi surface with coherent spectral weight along the nodal direction coexisting with an antinodal pseudogap. We show that this unusual electronic state emerges from a rapid collapse of the correlated gap. Temperature-dependent ARPES studies indicate a transition towards an entirely ungapped Fermi surface with more isotropic interactions around the Néel temperature of the undoped parent compound.
Subsequently, we present an ARPES study of the doping evolution of the low energy electronic structure of the bilayer iridate (Sr1−xLax)3Ir2O7. In stark contrast to (Sr1−xLax)2IrO4we find that above the insulator to metal transition atx= 0.03a small Fermi surface containing only thexextra carriers emerges and that the correlated gap reduces gradually upon doping. The metallic state of (Sr1−xLax)3Ir2O7shows more coherent quasiparticles and no traces of a pseudogap throughout the entire Brillouin zone, which indicates a weakly correlated Fermi-liquid-like metallic state. Those results suggest that Sr3Ir2O7can be described as a correlated semi-metal.
Acknowledgments i
French Réssumé i
Abstract iii
1 Background 1
1.1 Mott Physics . . . 2
1.1.1 The Antiferromagnetic Insulator . . . 2
1.1.2 Mott Insulator . . . 3
1.1.3 Mott insulator above half-filling . . . 7
1.1.4 Mott Heisenberg physics and the t-J model . . . 7
1.2 Phenomenology of underdoped copper oxides . . . 9
1.3 5dTransition Metal Oxides . . . 14
1.3.1 Sr2IrO4 . . . 18
1.3.2 Sr3Ir2O7 . . . 20
2 Experimental Techniques 23 2.1 Angle resolved photoemission spectroscopy . . . 23
2.1.1 ARPES setup at the University of Geneva . . . 27
2.1.2 ARPES setup at Diamond Light Source . . . 28
2.1.3 Many Body description of the photoemission process . . . 29
2.2 Scanning tunneling microscopy . . . 34
2.2.1 SPECS JT-STM . . . 36
2.2.2 Many Body description of the tunneling process . . . 37
3 Results and discussion 39 3.1 Crystal Growth . . . 39
3.1.1 Introduction to flux growth . . . 39
3.1.2 (Sr1−xLax)2IrO4growth conditions . . . 40
3.1.3 Sample characterization . . . 44
3.2 Collapse of the Mott gap and emergence of a nodal liquid in Sr2IrO4 . 46 3.2.1 ARPES results on the undoped parent compound Sr2IrO4 . . 47
3.2.2 Doping evolution of (Sr1−xLax)2IrO4by ARPES . . . 48
3.2.3 Tight binding model . . . 49
3.2.4 Tight binding description of the experimental band dispersion 53 3.2.5 Large versus small Fermi surface . . . 54
3.2.6 Momentum dependent pseudogap . . . 57
3.2.7 Temperature dependence of the pseudogap . . . 59
3.3 Coherent Quasiparticles with a Small Fermi Surface in Lightly Doped Sr3Ir2O7 . . . 63
3.3.1 ARPES results on undoped Sr3Ir2O7. . . 63
3.3.2 Doping evolution of (Sr1−xLax)3Ir2O7 by ARPES . . . 64
3.3.3 Small Fermi surface and coherent quasiparticles . . . 65
3.4 The role of correlations in the low energy electronic structure of ligthly electron doped Sr2IrO4and Sr3Ir2O7 . . . 69
3.4.1 Reduction of the correlated gap of (Sr1−xLax)3Ir2O7as a func- tion of doping . . . 70
3.5 Preliminary scanning tunneling microscopy results . . . 74
4 Conclusions and Outlook 77 A Appendix1 79 A.1 Sample Growth conditions . . . 79
References 83
BACKGROUND
The Sommerfeld model of free electrons with quantum statistics developed in the 1930s was highly successful at explaining the electronic properties of the known materials at the time. In this picture a material is modeled as a combination of valence electrons and a periodic lattice of atoms. Bands and gaps emerge from the constructive and destructive interference of electron wavefunctions in~k-space. Thus, the metallic behavior is determined by the position of the chemical potential. If it lies within a half filled band, low energy excitations can create population imbalance in momentum space originating a charge current. On the other hand, insulating behavior will happen if the chemical potential lies in a band gap.
However, the discovery of some materials [1, 2] with insulating behavior not predicted by the known theories at the time posed a serious challenge for this theory and ultimately started the field of correlated physics. Correlated behavior occurs in systems where the interaction between electrons is not negligible, and, therefore, their ground state and their excitations are not explainable in the basis of the interaction of a single electron with the electrostatic field of the ions in the crystal. Strongly correlated materials are associated with low electron mobility. Hence, the ratio between the on-site Coulomb interaction (U) and the kinetic energy is large. As a result, electron delocalization is not energetically favorable and a many body description is more suitable.
A particular case, the so-called Mott Insulator [1], occurs in half-filled systems when the kinetic energy gained by tunneling of an electron, with amplitudet, from one atomic siteito an already occupied atomic sitejis smaller than the energy necessary to overcomeU. In other words, if the ratio betweenU and the bandwidth,W, is large (W ∝t). Due to the energy cost of double occupancy, a single half-filled band would then be split into an empty upper Hubbard band (UHB) and a completely filled lower Hubbard band (LHB) with one electron per site, resulting in insulating behavior.
Until the discovery of high temperature superconductivity Mott physics was mostly studied in three dimensional materials like VO2[2] or NiO [1] . The Ba - La - Cu - O system becomes superconducting from a single band half-filled spin-1/2 Mott insulator when doped with holes [3]. With the aim of understanding the phase diagram of the cuprate superconductors, a plethora of correlated materials found interest, together
with an intense drive to design new techniques by which to measure them [4]. De- spite this concerted effort, understanding how superconductivity emerges from the antiferromagnetic Mott insulator phase as a function of doping remains a fundamental challenge.
In recent years a new energy scale has been added to the picture of correlated elec- tron systems. Spin-orbit interaction (SOI) is the interaction between the orbital angular momentum and the spin of the electron. In many systems this interaction represents only a small correction to the band structure. However, in two dimensional systems or in the case of a large spin orbit coupling constant, SOI can be key to understanding the physical properties of a solid. One example are topological insulators, materials where SOI can invert the orbital character of the bands resulting in topologically different states at the surface [5].
New physical phenomena emerge and new insight into many body physics can be obtained when different degrees of freedom share similar energy scales. For example, unexpected correlated behavior has recently been found in iridium oxides resulting from the interplay between U and SOI opening a path to explore a region of the phase diagram of interactions not accessed before [6].
In the following we will discuss relevant concepts such as Mott physics, the phenomenology of cuprates and the basic properties of5dTransition Metal Oxides, to set a framework for the rest of this thesis.
1.1 Mott Physics
1.1.1 The Antiferromagnetic Insulator
To illustrate how an insulating ground state emerges out of a simple metal when electron-electron correlations,U are included, let us start from a non interacting band structure described by a nearest neighbor tight binding model (TB) for a half filled system in a cubic lattice of lattice constantawith all the hopping parameters equivalent:
H0 =−tX
hi,ji
c†icj =−2tX
j
cos(~k·~a) (1.1)
wheretis the hopping amplitude andc†i is the creation operator of an electron in site iin the free electron basis [7]. For an ordering vectorQ~ = (π/a, π/a, π/a)the low energy electronic structure fulfills the following relation:
−2tX
j
cos((~k+Q)~ ·~a) = (−1)3−2tX
j
cos(~k·~a) (1.2) where the(−1)3originates fromcos(kia+π) =−cos(kia). Therefore, it verifies the nesting condition:
ξ(Q~ +~k) =−ξ(~k) (1.3)
near the Fermi level. At sufficiently low temperature, this leads to a divergent spin susceptibility for any repulsive on-site interaction, causing an antiferromagnetic state with ordering vectorQ~ .
In the ordered state the Hamiltonian can be split into two different parts for spin up and spin down:
H↑ =H0+Hint =−tX
hi,ji
c†i↑cj↑−UX
l
(−1)lm0(nl↑−1/2)
H↓ =H0+Hint =−tX
hi,ji
c†i↓cj↓+UX
l
(−1)lm0(nl↓−1/2)
(1.4)
withU the on site electron-electron interaction andm0the staggered magnetization.
Therefore, each spin sees a self-consistent potential, ±U m0, with periodicity (2a,2a,2a)resulting from the interaction with the other spin species. The oscillating character ofHintfavors a periodic arrangement of the spins. For example, the inter- action promotes spin up electrons on even sites while reducing the presence of spin down electrons. In odd sites the behavior is inverted. This ground state results from the nesting of the Fermi surface and is independent of the value ofU. As the antiferro- magnetic instability is a "spin" ordering the average total density on each site remains unchanged, while the spin "density" varies with respect to a situation without nesting.
The spin ordering reduces the k-space periodicity opening a Peierls-Slater gap [8]
at the border of the new Brillouin zone, due to the destructive interference between states of momentum~kand~k+π/a, resulting in a insulating ground state. Therefore, a half-filled tight binding band on a cubic lattice is susceptible to a metal-insulator transition (MIT) in the presence of a very smallU.
1.1.2 Mott Insulator
Figure 1.1 –Metal-Insulator transition in VO2 as a function of doping and pressure, from [2].
The antiferromagnetic insulator discussed above is just a particular case of a half- filled cubic lattice. However, insulating behavior in paramagnetic materials, such as NiO [1], or with non cubic crystalline structures, for example VO2, [2], could not be explained in the same way. One of the first explanations to the unexpected electron localization was given by Sir Neville Mott [9]. Mott realized that in the presence of U, the energy cost for double occupancy in an atomic site, might be larger than the energy gained by electron delocalization. Hence, a ground state totally dominated by the charge properties of the system,with the conduction electrons "frozen" on respective atomic sites, might be a more favorable ground state than a delocalized Fermi sea.
The competition between kinetic energy and U can be described by the Hub- bard model [10]. While the kinetic energy favors carrier itinerancy, electron-electron correlations tend to confine the charge carriers into real space atomic sites.
H=H0+Hint=−X
hi,ji
ti,jc†icj+UX
i
ni,↑ni,↓ (1.5)
where the kinetic energy, is parametrized by a nearest neighbor hopping parameterti,j andni,σis the number operator for each spin class.
t U
2W > U Weakly correlated metal 2W < U Mott insulator 2 W = U
Simple metal
Correlated metal
Mott insulator K
U
Figure 1.2 –Top: Schematic of the phase diagram of the Hubbard Hamiltonian. Bottom:
In the weakly correlated limit electrons can hop to empty or partially occupied atomic states. WhenUis larger than the bandwith electrons a ground state with one electron per atomic site is more energetically favorable.
To understand the interplay between electron delocalization and on-site interaction we will discuss the energy balance of two extremal possible ground states: the Fermi sea (which corresponds to the extremal case ofU = 0) and a ground state characterized by one bound electron per atomic site (U → ∞) [7]. The energy of the Fermi sea|Fi
in the Hubbard model is given by:
EF =hF|H|Fi=hF|H0|Fi+UX
i
hF|ni↑ni↓|Fi (1.6) The kinetic energy term is strongly dependent on the particular band structure of each material. However, by assuming a constant density of states,N = 2Ln(0), whereLis the number of atomic sites, and a finite bandwidth2W, certain general properties can be obtained. Within these assumptions the kinetic energy is:
Ekin= 2 Z Ef
N(ξ)ξdξ=L(Ef2−W2)n(0) (1.7) where the factor of 2 accounts for the spin degeneracy andEf is the energy of the Fermi level. In the same approximation the density of particles for each spin species, Nσ/Lcan be easily calculated by integrating the constant density as a function of energy up to the Fermi cutoff.
Nσ/L= (Ef+W)n(0) (1.8) we can define the density of each spin species asNσ/2L=n(0)(Ef+W). In the full band case,Ef =W, andNσ/L = 1for spin up and down. Then as our density of states is constant,n(0) = 1/2W.
Thus, we can rewrite the kinetic energy as:
Ekin/L= 2W n↑(n↑−1) (1.9) whereN↑/L = n↑. Thus, the kinetic energy is zero for an empty or completely filled band. Equation 1.9 recovers the expected behavior for the kinetic energy: it is maximum at half-filling but there is no energy gain when the states are localized.
The interaction energy for the Fermi sea is also easily computed as we are consid- ering the creation operator in the basis of the plane waves,c†|∅i ∝e−i~k·~r. Thus:
EU/N =UX
hi,ji
hF|ni↑|FihF|ni↓|Fi=U n2 (1.10) consistent with the high probability to have double occupancy due to the delocalized nature of plane waves.
Therefore, bringing both results together, the total energy of the Fermi sea at half-filling in the Hubbard Hamiltonian is:
EF S/N = 1
4(−2W +U) (1.11)
On the other hand, the ground state with one localized particles per atomic site,
|Mi=
N
Y
i=1
c†i,σ
i|∅i (1.12)
has no kinetic energy asH0 ∝c†jci. This term involves hopping of an electron from siteito sitej. However, by definition,|Mihas exactly one particle per site. H0|Mi
is therefore orthogonal to|Mi, as in this new state there would be at least one pair of empty/double-occupied sites. As a result, no energy gain is achieved by electron delocalization:
hF|H0|Fi= 0 (1.13)
Moreover, by definition, the lack of double occupancy implies no on-site interaction:
hF|Hint|Fi= 0 (1.14)
Therefore,
EM/N = 0 (1.15)
Hence, within this simple estimate, the Fermi sea is favorable forU <2W, but for large values of U electrons tend to localize. Thus, for a large kinetic energy Ekin∝t∝W electrons can overcome the cost of double occupancy. However, ifU is larger thanW, the electrons will localize into atomic sites and charge transport will not occur. The critical value ofU for which the localized ground state becomes more energetically favorable depends on the particular band structure, but remains valid for more realistic wavefunctions and it is usually of the order of U ≈ 2W. Thus, the Hubbard Model describes a metal-insulator transition as a function of the band width and of the strength of the interaction (Figure 1.2). These kind of MIT transitions have been thoroughly studied in recent years and will be discussed in more details in section 1.2.
Note, that in all the previous discussion of the Mott insulator we have ignored the magnetic structure. The insulating behavior does not require a magnetic interaction, and it is driven by charge fluctuations rather than spin-spin interactions. That does not exclude that magnetic order can coexist with Mott physics. Magnetic coupling arises, indeed, due to virtual hopping processes which often reduce the energy for anti-parallel configurations.
To understand how this insulating behavior emerges it is instructive to study the atomic limit for finite values ofU. Let us assumeLempty atomic sites. Filling up each atomic site with one single electron results in a narrow energy level (there is no hopping,t= 0, in the atomic limit) which defines the energy origin. However, when a second electron is added to a sitei, the on-site interaction has to be overcome by paying an energy costU. We can interpret this as the result of having a second atomic level of energyU and placing the second electron directly there. It is important to notice that these energy levels are fundamentally different from the usual single electron spectra:
the upper level at energyU is not there unless there is already one electron occupying each available state at the energy reference.
When we consider a finitet, the hopping part of the Hamiltonian broadens the atomic levels into a tight binding band of widthW ≈2zt, wherezis the number of nearest neighbors. This results in two tight binding bands separated by an energy gap U, known as upper and lower Hubbard bands. Note that this band structure is also very different form the usual semiconductor band structure picture. A tight binding band can host up to2Lelectrons, while each of the Hubbard subbands cannot contain more than 1 electron per site. In contrast to the antiferromagneic insulator, this phenomenon is not originated by a reduction of the~k-space periodicity. The Hubbard bands have the
same~k-vectors as the tight binding band in the limitU = 0. The splitting into Upper and Lower Hubbard band is a correlation effect, unexplained in the free electron model.
1.1.3 Mott insulator above half-filling
In the previous subsection we have described the existence of an insulator to metal transition as a function of the ratio between electron-electron correlations and band widthU/W. However, this discussion was restricted to a half-filled system. In the following, we will investigate doped Mott insulators, i. e., systems withn6= 1.
To start let us make the assumption that if the doped system is metallic atU/t→ ∞ then it will be metallic at any finiteU/t. This assumption is plausible considering the behavior atn <1. When the number of electrons is smaller than that of available sites, a largeU would forbid double occupation. Therefore, there would be only empty sites and singularly occupied sites. It is then, plausible to say that a strong HubbardU does not inhibit electron hopping to an empty neighboring site, despite the fact that by doing that the spin arrangement might change. Thus, in the frame of this assumption, electron delocalization does not cost any energy and the system is a metal. The argument would be the same for a more than half-filled system,n >1.
According to our simple discussion of the Hubbard model, a doped Mott insulator will behave as a metal. However, only a small number of free carriers will contribute to transport, resulting in a small Fermi surface. The metallic behavior is only carried out by the extra charges above/below half-filling and the Mott gap remains. However, it can happen that the addition of extra carriers increases the "effective" bandwidth, reducing theU/W ratio, destroying the Mott state. In this case, not contemplated in the previous discussion, the Mott gap will collapse and1 +xelectrons, wherexare the extra carriers, would contribute to the Fermi surface.
1.1.4 Mott Heisenberg physics and the t-J model
Despite describing the basics of the Mott insulator to metal transition, the Hubbard model is not ideal to study the low-energy excitations of a Mott insulator near half- filling. The Hubbard Hamiltonian allows for excited states with doubly occupied sites, making the study of the dynamics of a single carrier much more difficult. However, one can perform a canonic transformation of the Hilbert space resulting from the Hubbard model into a second one which excludes states with double occupancy. In that case, the low energy physics can be separated from the high energy excitations. The resulting Hamiltonian is known as the t-J model [11, 12] :
HtJ =−t X
hi,ji,σ
(˜c†i,σ˜cj,σ+H.C.) +JX
hi,ji
(S~i·S~j−ninj
4 ) (1.16)
with˜c†i,σ=c†i,σ(1−ni,¯σ)a term that avoids double occupancy,J = 4tU2 the exchange coupling constant,S~ithe spin operator at sitei.
In the following we will discuss the motivation for the t-J model, which has the following general properties:
• At half-filling it represents the antiferromagnetic Mott insulator.
• Forn <1it describes the so-calleddoped antiferromagnetwhere the itinerant particles are holes in the Mott-Hubbard insulator.
• Virtual hopping between nearest neighbors results in magnetic interaction be- tween the correlated itinerant particles.
• Away from half-filling it reproduces several properties of the copper oxide superconductors, including experimental band dispersions [13].
The t-J model originates from the necessity of describing the motion of electrons in the Hubbard subbands in the limitW/U <<1. In principle one could try to treat this perturbatively in terms of the hopping parametert. In stark contrast to theU/W <<1 limit many spin configurations are possible due to the electron being localized in atomic sites. This results in highly degenerate ground states. Therefore, the hopping parameter cannot be treated perturbatively: any small perturbation would not be adiabatic in the hopping parameter as it would break the spin degeneracy.
However, there is a way to overcome this problem. At first order, the eigenstates of the total Hubbard Hamiltonian are not eigenstates of the interaction Hamiltonian (HU) (the kinetic energy is not diagonal in theHU basis). However, at first order, there exists a basis in which the total Hamiltonian is diagonal. Of course, after this transformation one will find that at higher order the eigenstates are again mixed, so one would have to proceed iteratively.
To start the discussion one should note that the kinetic energy part of the Hubbard Hamiltonian can be split into three sub-terms:
Hkin=H++H−+H0 (1.17)
WhereH+(H−) is the part of the hopping term that increases(decreases) the number of doubly occupied sites, andH0 keeps it constant.H+andH−are highly complex terms. Double occupancy involves non-negligible changes in energy and may relate to states in the two Hubbard bands. Therefore,Hkin, acquires a many-body character which renders a perturbative treatment in the hopping parameter practically impossible.
The canonical transformation should be chosen such asH+andH−cancel out of the full Hamiltonian,H =H++H−+H0+HU. Thus,R, a canonical transformation, has to verify that:
Hef f =eiRHe−iR=H0+HU+O(t2) (1.18) The transformationR=−Ui(H+−H−)fulfills this condition. To ordert2/U it leads to:
Hef f =H0+HU+ 1
U[H+, H−] (1.19)
where the commutator term comes from the expansion of the exponential. It can be shown that this term relates to the kinetic antiferromagnetic exchange as:
1
U[H+, H−] = 2t2
U (S~i·S~j−ninj
4 ) (1.20)
and we therefore recover the t-J Hamiltonian:
Hef f =HtJ=−t X
hi,ji,σ
(˜c†i,σ˜cj,σ+H.C.) +JX
hi,ji
(S~i·S~j−ninj
4 ) (1.21) where we are neglecting terms of ordert3. The first term describes hopping in the subspace without double occupancy, essentially restricting the hopping to empty sites.
The second part of Equation 1.21 describes the nearest neighbor spin interaction with coupling constant J. At half-filling, Equation 1.21 reduces to the Heisenberg Hamiltonian:
H =JX
hi,ji
S~i·S~j (1.22)
This term can be understood in a toy model including two spin-1/2 particles in two atomic sites. If both particles have the same spin, electron hopping is forbidden by the Pauli principle. However, hopping is possible if the two spins are opposite.
The state with two electrons in the same atomic site is not energetically favorable and the system will favor a transition towards the initial state. In this transition the spin configuration can be changed with respect to the starting one, in a process known as virtual exchange. The virtual exchange has its origin in an interplay between the Pauli principle, kinetic energy and electron-electron interaction and acts as a magnetic exchange. IfJ is positive a configuration with opposite spins in neighboring sites will be favored. This explains the existence of antiferromagnetic ordering up to high temperature in Mott insulators despite negligible dipolar spin-spin interactions. The Mott antiferromagnet is indeed rather different to the classical Néel antiferromagnet.
While Néel antiferromagnteism requires static spin configurations, the Heisenberg Hamiltonian allows for dynamic spin configurations, known as magnons.
1.2 Phenomenology of underdoped copper oxides
EF
Cu 4d EF
LHB U UHB
O 2p O 2p
U
O 2p EF
Cu 4d EF
LHB U UHB
O 2p
U
Mott Insulator Charge transfer insulator
Figure 1.3 –Sketch of the electronic structure of a Mott insulator and a charge transfer insulator. In a charge transfer insulator the smallest gap is not the correlated gap and the LHB is embeded in a sequence of bands.
The physics of cuprates is dominated by the properties of the copper oxide planes.
Cu is in a2+oxidation state, therefore, it has 9 electrons in its outer3dshell. Thus,
the copper3dshell is partially filled as obtained within band theory. Experimentally though, correlated insulating behavior is found instead, which can be explained by the interplay between strong on-site Coulomb interaction,U ≈4eV, and crystal field splitting, which effectively results in a single half-filleddx2−y2 band.
The band structure of the cuprates is, in reality, slightly different from the simple idea of a Mott insulator. So far, when discussing Mott physics, we have considered an ideal isolated system with a single band that is split into an upper and lower Hubbard band due to charge correlations. However, in a real material, the Hubbard subbands might be embedded in a sequence of bands with larger bandwidth. In the so-called charge transfer insulator [14], the smallest of the gaps, and thus the most relevant excitation process, is not between Hubbard subbands. Cuprates follow this behavior;
their lowest energy excitation being the creation of a hole in the 2pO band and a quasiparticle in the3dupper Hubband band (Figure 1.3). A charge transfer insulator is, therefore, a Mott insulator with a smaller gap than the one between the Hubbard subbands, despite the fact that the insulating behavior results from the same process:
the Mott localization and the splitting of the half-filled band into an upper and a lower Hubbard band.
a) b)
Temperature
Hole concentration p Superconductivity
T* TN
TC
Incipent charge density Antiferromagnetism wave
Figure 1.4 –a) Schematic phase diagram of the copper oxides,from [15]. b) Evolution of the temperature dependence of the resisitivy as a function of hole doping in La2−xSrxCuO4. The upturn in resistivity is a characteristic signature of the pseudogap, from [16].
Cuprates crystallize in a perovskite-like structure. Superconductivity emerges as an instability of the normal state when extra carriers are introduced in the system.
Higher values of Tcare achieved in hole doped cuprates prompting a higher number of theoretical and experimental efforts. Hole doping is normally achieved by two different methods, either independently or in combination: (1)3+A-site ion substitution with a second ion of oxidation state2+, as in La2−xSrxCuO4, or (2) by introducing interstitial oxygen atoms, as in Bi2Sr2CaCu2O8+xor YBa2Cu3O7−xand the related compounds.
In any case, both methods effectively hole dope the CuO2layers by pulling electrons out of it [17]. The added carriers are mobile, hopping via the neighboring oxygens, and drive the system towards metallicity, as shown in Figure 1.4-b).
Antiferromagnetic ordering is eventually destroyed atp≈1−2%. Increasing hole
concentration results in the emergence of superconductivity atp≈5%, with Tcfinding its maximum at aroundp≈15%. Strikingly, further increase in the number of holes results in a decrease of the critical temperature, with superconductivity vanishing at p≈30%, together with a transition to a normal metal for even higher doping.
1.0 1.0
0.5 0.5kx
yK
0.00.0
∆k I
II
1.0 1.0
0.5 0.5kx
K
y
0.00.0
∆k
I II
III
a) b)
Figure 1.5 – a) Fermi surface evolution of La2−xSrxCuO4 as a function of doping, from [18]. b) Schematic of the momentum space dependence of the superconducting gap (top) and pseudogap (bottom), from [19].
In stark contrast to the phonon-mediated Bardeen-Cooper-Schrieffer (BCS) super- conductors, photoemission experiments in cuprates observe a superconducting gap withd-wave symmetry (Figure 1.5-b)). Spin singlet pairing in BCS requiress-wave superconductivity, which suggests a different order parameter for the cuprate super- conducting phase, opening one of the major questions of today’s condensed matter physics [17]. Strikingly, the gap symmetry is not the only difference between the cuprate-like and BCS superconductors. The underdoped regime of the hole-doped cuprates is characterized by the pseudogap phase (PG), a suppression of low energy charge excitations at higher temperature than the superconducting phase. A simplified phase diagram of the cuprates is summarized in Figure 1.4-a).
The PG is characterized by the lack of states at the Fermi level for certain momen- tum values up to a temperature, T∗, higher than the superconducting critical temperature Tc. This effect, stronger in underdoped samples, persists to optimal doping and it might coexist with superconductivity in overdoped samples [20]. Initially observed as a reduction of the spin response of the cuprates (Knight shift experiments [21] and NMR measurements [22], for example), it was rapidly related to concomitant charge exci- tations. Many breakthroughs in understanding the pseudogap phenomenology come from angular resolved photoemission (ARPES) [4, 23] and scanning tunnel microscopy (STM) [24–26] experiments. This two techniques are, indeed, ideal and complementary
to study the momentum and space dependence of the pseudogap.
Intensity (a.u.)
−0.1 0.0 0.1
UD92 UD75 UD65 UD50 UD40 OD86 Bi2212
T << Tc Antinode ~ 0°
E − EF (eV) θ
Intensity (a.u.)
−0.1 0.0 0.1
30 K 51 K 71 K 81 K 92 K 100 K123 K 144 K163 K 180 K 190 K 200 K210 K T*
Bi2212 UD92Antinode
~ 0°
Tc
E − EF (eV) θ
50
40
30
20
10
0
Gap (meV)
40 30 20 10 0
Bi2212 UD928 K 102 K
(π,0) (0,π)
(°) θ θ Γ
a) b) c)
Figure 1.6 – a) Momentum dependence, b) doping and c) temperature dependence along the antinodal direction of the low energy electronic structure of Bi2Sr2CaCu2O8+x. Adapted from [27].
The first ARPES measurements within the superconducting dome of cuprates, found largely coherent quasiparticles with a large superconducting (SC) gap towards (π,0), or antinodal direction, that closes towards(π/2, π/2), or nodal direction, where no SC gap is found, consistent with thed-wave symmetry of the gap. Strikingly, at higher temperature a second energy gap is observed. This gap, which vanishes at a critical temperature T∗ higher than Tc[23], was consistent with values for the PG observed with other techniques. Moreover, ARPES data revealed a highly anisotropic momentum dependence of the PG: it reaches it maximum along the antinodal direction and vanishes towards the node, displaying similar symmetry to the SC gap. However, in stark contrast to the point-node of the coherent SC gap, segments of the Fermi surface, known as Fermi arcs, remain ungapped in the PG phase. Figure 1.5-b) sketches the momentum dependence of both gaps.
The pseudogap phenomenology varies between materials, however certain general properties are shared. Forpvalues in the pseudogap phase the Fermi surface consists of the characteristic Fermi arcs. As an example the doping evolution of the Fermi surface of La2−xSrxCuO4 from ARPES is shown in Figure 1.5-b). The gap size at the antinodal direction and the~k-space extension of the gapped area decrease with increasing doping, eventually vanishing for optimally doped samples, when the large underlying Fermi surface emerges. Figure 1.6 shows the evolution of the pseudogap as a function of doping and the transition into the superconducting phase along the antinodal direction, where the SC gap and PG have comparable magnitude. However, the pseudogap decreases as a function of hole concentration while the sample becomes more superconducting upon doping. Moreover, Energy Distribution Curves (EDCs)
0 0 1 0
5 0
0 1 -
1 234
5 6
Sample Bias (mV)
dI/dV (nS)
1.0 0.8 0.6 0.4 0.2
0.0 -50 0
20 meV 70 meV
∆1 = 55 meV N = 455
a) b)
Figure 1.7 – a)dI/dV map showing the spatial inhomogenity in the low energy electronic structure of Bi2Sr2CaCu2O8+x. b)dI/dV curves at selected points in the surface. From [19].
show coherent peaks (blue lines in Figure 1.6-c) which vanish above TC, indicating that coherence is related to the onset of superconductivity. The red lines correspond to the region below T∗, with the pseudogap value decreasing towards the onset temperature.
STM has also become a powerful technique to understand the origin of the pseu- dogap. The ability of STM to probe the density of states above the Fermi level has shown that the pseudogap phase has approximately particle-hole symmetry. Moreover, Scanning Tunneling Spectroscopy (STS) revealed strong spatial inhomogeneities in the underdoped regime, with superconducting regions (characterized by sharp coherent peaks on the density of states) coexisting with incoherent, pseudogapped patches on the sample. Figure 1.7, adapted from Ref. [19], shows typicaldI/dV curves and the PG spatial distribution on an underdoped sample of Bi2Sr2CaCu2O8+x. The darker areas correspond to completely pseudogaped regions.
With the improvement of these experimental techniques more information on the interplay and/or possible competition between these two phases has been obtained. In recent years, two gaps have been found to coexist in the superconducting dome [27–30], proving the coexistence of PG and SC gap in this region of the phase diagram. One, which near the nodal direction does not change size upon doping and closes above TC, has been associated with the SC gap and tracks the superconducting dome as a function of doping. The second one, which is zero near the node and increases when the doping is decreased, has been identified with the pseudogap above TC.
Substantial changes in the magnetic structure of the cuprate superconductors in the undoped regime have been observed. The expected neutron scattering spectrum for a 2D Mott insulator with Heisenberg dynamic is characterized by a spin-wave with ordering vectorQ~ = (π, π), as seen in the right panel of Figure 1.8. However, upon doping the characteristic "hour glass" dispersion is found in neutron scattering on La-based [32,33]
and Bi-based [34] cuprates suggesting a general property of hole doped cuprates [35]. The spectrum can be split in two components, the upper part resembles that of
(3/4,1/4) (1/2,1/2)0 (1/2,0) 50
100 150 200 250 300 350
En ergy (meV )
Wave vector (h,k)
250
0 50 100 150
En ergy (meV )
200
-0.3 0.3
(0.5 + h, 0.5) h(rlu) La2CuO4 La2-xBaxCuO4
0
x = 1/8 x = 0.16
Figure 1.8 –The spin excitations of La2CuO4are consistent with that of a Heisenberg antiferromanget [31] (right pannel). Substantial changes appear upon doping in the magnetic spectrum: an universal pattern has been identified, the "hour glass" dispersion.
Left panel shows the evolution of the magnetic excitations of La2−xBaxCuO4for two different doping levelsx= 0.125andx= 0.16. [32, 33].
antiferromagnetic spin fluctuations with a gap, while the origin of the incommensurate downward dispersion is more controversial and can be related to "stripe" charge ordering [17, 32, 35]. If "stripe" order is present the antiferrofangetism regions on the sample are interrupted andQ~ becomes incommensurate, shifting proportionally to the spacing and direction of the stripes.
In recent years the phase diagram of cuprates has become even more complicated (Figure 1.4) with the observation of charge order (CO) in the PG phase by different spectroscopic techniques like STM and resonant x-ray scattering [36, 37]. The relation between CO, PG and stripe order is still unknown.
Despite not having yet a full picture of the pseudogap phenomenology, it is clear that a complete understanding of the interplay between ordered phases and supercon- ductivity is exceptionally difficult. Further investigations with momentum and real space probes in new materials displaying similar properties to that of the cuprates could provide new insight into this problem.
1.3 5d Transition Metal Oxides
As we discussed in section 1.1.2 Mott correlated physics is strongly dependent on the bandwidth of the near Fermi level bands. In3dmaterials, for example, the electron-
electron correlations play a defining role in their physical properties. Cu-perovskites display Mott physics due to the interplay between electron-electron correlations and crystal field splitting. On the other hand, as we move down in the periodic table metallic uncorrelated behavior is expected: electron localization is more favorable as the4dand 5dorbitals are more spatially extended than their3dcounterparts.
However, there is a second energy scale that should be taken in account. Spin-orbit interaction (SOI) is a relativist effect which couples the orbital momentum and electron spins in atoms. Responsible for the Zeeman effect [38] in atomic physics, it is usually ignored when talking about the electronic structure of solids. Nonetheless, SOI scales with the forth power of the atomic number,Z4, and, for heavy atoms, it can not be treated perturbatively. In uncorrelated materials with heavy atoms, like Bi1−xSbx[39], Bi2Se3[40] and other families of topological insulators, SOI is key for the emergence of topologically protected surface states with properties that are right now under intense study [5].
Spin liquid Quadrupolar
Axion insulator
Topological insulatorMott insulatorMott Spin-orbit coupled
Mott insulator
Simple metal or band insulator
Topological insulator or semimetal
λ/t U/t
semimetalWeyl
Figure 1.9 –Sketch of the phase diagram resulting from Eq. 1.23, from [41]. Iridates are described by the top-left region of this diagram where correlated ground states emerge from the interplay betweenU and SOI.
In the5dtransition metal oxides electron correlations and SOI have comparable energy scales allowing for exciting new phenomena. While the large spatial extent of the5dorbitals favors electron delocalization, SOI can break the degeneracy between the dorbitals, effectively reducing the bandwidth and, thus, enhancing the effect of electron-electron correlations. In a recent review, Witczak-Krempa et al. [41]
introduced a generic Hamiltonian, based on the Hubbard model, useful to describe the interplay between these two phenomena:
H = X
i,j;α,β
ti,j;α,βc†jαcjβ+HC+λX
i
L~i·S~i+UX
i,α
niα(niα−1) (1.23)
withc†jαthe creation operator for an electron in orbitalαat sitei,niαthe occupation number,tthe hopping amplitude andλthe atomic SOI coupling constant. Figure 1.9 describes the schematic phase diagram resulting from this model as a function of the relative values of U andλ.
We are interested in the top-left region of the phase diagram where the narrow bands created by SOI can host Mott insulator physics. The main actor in the correlated SOI regime is Ir4+. Different iridates families containing this ion have attracted much attention recently. A2IrO3, with A an alkali metal, displays hexagonalJeff = 1/2 antiferromagnetism [42], while pyrocholore and spinel iridates display exotic magnetic states [43–45]. However, the most intensely studied compounds are the Ruddlesden- Popper series of layerd iridate Srn+1IrnO3n+1 which display an insulator to metal transition as a function of dimensionality. Sr2IrO4is a Mott insulator isostructural to the parent compound of the cuprates. Sr3Ir2O7also displays similar antiferromagnetic insulating behavior, but lies closer to the insulator to metal transition. Then=∞limit is a a correlated metal [46].
Figure 1.10 – a) Resistivity as a function of temperature and, b), magnetization as a funtion of temperature along the a and c axes for Sr2IrO4. From [47].
In the following, we will try to explain this insulating behavior, discuss the ex- perimental evidence and the implications of this newly discovered insulating state in the first two members of the Ruddlesden-Popper series of layered iridate perovskites Srn+1IrnO3n+1.
Crystal field splitting is key to explain the insulating ground state of Sr2IrO4and the other Ir4+compounds. The oxygen octahedral environment splits the5dorbitals into partially-filledt2gstates, while the emptyeg orbitals are pushed above the Fermi level.
Similar behavior is observed in Sr2RuO4 the4dlayered perovskite counterpart.
However, as we discussed previously SOI needs to be considered. In Sr2RuO4the SOI constant is small (λ ≈ 0.15 eV) compared to the other energy scales in the system and can be treated perturbatively. On the other hand, in the Ir4+compounds the SOI constant is three times larger (λ≈0.5eV) and comparable to the bandwidth and the Coulomb interaction. Therefore, it has to be included in the discussion of the electronic structure in a non-perturbative way.
However, spin is not a good quantum number in the basis given by (dxy↑,dyz↓,
dzx↓) and its time reversal partner. Thus, it is necessary to find a basis, where spin-orbit coupling can be included in the description. As we discussed in equation 1.23, spin enters the Hamiltonian proportional to the angular momentum:
HSOI ∝~λ~L (1.24)
In the absence of crystal field the degenerated-orbitals have angular momentum,
|L|= 2. However, when the isotropic environment is broken due to the presence of a crystal field the angular momentum gets "quenched" and a canonical transformation into a basis that diagonalizes the spin-orbit Hamiltonian is possible. As en example of the reduction of the angular momentum the expectation value ofLzin the basis of the crystal field splitd-orbitals,|ψα,βi= |dxzi,|dyzi,|dxyi,|d3r2−z2i,|dx2−y2i
, is shown in the following:
hψα|Lz|ψαi=
0 i 0 0 0
−i 0 0 0 0 0 0 0 0 2i 0 0 0 0 0 0 0 2i 0 0
(1.25)
The bottom right corner corresponds to the matrix elements of theeg subspace, which are totally quenched. On the other hand the expectation value ofLzin thet2g
basis is that of thep-orbitals,|ψα,βi= (|pxi,|pyi,|pzi), up to a minus sign:
hψα|Lz|ψαi=
0 −i 0 i 0 0 0 0 0
(1.26)
The previous argument can be extended to the other components of the angular momentum. As such, the angular momentum of thet2gorbitals is equal to the angular momentum of thep-orbitals up to a minus sign,L~0=−L. Thus, the angular momentum~ of thed-orbitals, which have|L|= 2in the absence of crystal field, is now|L|= 1in for thet2gstates and|L|= 0for theegorbitals. Therefore, the "effective" total angular momentum for thet2g orbitals takes the following values:
J~eff =L~0+S~ =−~L+S~
|J~eff|= 1 2,3
2
(1.27)
Where theJ~effstates are a linear combination of thet2g given by the following e basis transformation:
|1/2,1/2i
|3/2,1/2i
|3/2,−3/2i
=
−1/√
3 1/√
3 −i/√ 3
−2/√
6 −1/√
6 i/√ 6 0 1/√
2 i/√ 2
|dxy↑i
|dyz↓i
|dzx↓i
(1.28)
Note, that the new states witheffectiveangular momentum are eigenstates of the SOI Hamiltonian. Therefore,J~eff plays the role of pseudospin in the5dmaterials.
With this canonical transformation thet2g states lose their degeneracy due to SOI and they are split into a fully occupiedJeff = 3/2band and a half-filledJeff = 1/2. Let us recall our condition to have a Mott insulator (U >2W). Thus, as the bandwidth has been reduced, a correlated insulating ground state can appear in the Ir4+compounds.
Thus, SOI in these materials effectively reduces the bandwith allowing for a new ground state, the so-called spin-orbit driven Mott insulator, with an relatively smallU (Figure 1.11).
SPIN ORBIT
CRYSTAL FIELD
5d
eg
t2g
Jeff=1/2
Jeff=3/2
U µ
UHB LHB
Figure 1.11 –Schematic of the effect of crystal field splitting, SOI andU on the atomic levels of Ir4+.
However, for the layered iridates, particularly for Sr3Ir2O7, the validity of this model has been subject to debate [48]. The problem emerges in the difficulty to determine the crystalline structure in these materials. Complete determination of the atomic positions of the apical oxygen is hindered by the presence of Ir in the structure.
On one hand, the oxygen signal in x-ray scattering measurements is hidden behind the relatively strong iridium peaks. On the other hand, the high cross section of the Ir atoms, together with the small size of the available samples, makes it virtually impossible to refine the structure by neutron scattering. Different values for the octahedral rotation or a crystalline structure with a slightly different symmetry [49] might affect the crystal field splitting causing a larger overlap between states of different total angular momentum effectively increasing the ratio W/U above the metal insulator transition.
Recent theoretical and experimental efforts have been made to tackle this prob- lem [48, 49, 51–54]. For a model where the spin degree of freedom has not been considered, calculations of x-ray scattering matrix elements show equal resonant scat- tering intensities at the L3 (corresponding to the(2p3/2 → 5d) transition) and L2 (2p1/2 →5d). However, when spin is considered, i.e. in theJeff = 1/2case, a ratio L3/L2>>1is expected [50] (Figure 1.12). Recent resonant inelastic x-ray scattering (RIXS) experiments in Sr2IrO4[50] and Ba2IrO4[54] have found a enhancement of theL3edge, confirming theJeff = 1/2character of their ground state, in agreement with Dynamical Mean Field Theory (DMFT) results [48]. Interestingly, this behavior has also been observed in Sr3Ir2O7 [52, 53] despite the larger bandwidth of the bilayer compound.
1.3.1 Sr2IrO4
Sr2IrO4 is isostructural to the parent compound of the cuprates, La2CuO4 up to a 11◦ in-plane rotation of the oxygen octahedra which causes a √
2×√
2 structural
Figure 1.12 – a) RIXS data illustrating the difference in intensity for the L3and L2edge of Sr2IrO4. b) Right (left) panel shows a schematic of the allowed transitions when SOI is (not) included. From [50].
reconstruction [55]. Sr2IrO4was initially though to be a ferromagnetic insulator with a very small magnetic moment below 240K [47]. Later on it was found that the Jeff moments lock to the oxygen octahedra in an alternating manner. As the oxygen octahedra are canted, the magnetic moment is non-zero within a layer. The next IrO2
plane has the same net-magnetic moment in the opposite direction, resulting in an antiferromagnetic ordering of the total angular momentum. The red arrows in Figure 1.13 represent the "effective" total angular momentumJeff = 1/2moments [50]. When a magnetic field greater than the critical field is applied the net moments in each of the planes align ferromagnetically to produce a macroscopic field, explaining the magnetic signal seen in figure 1.10.
Intriguingly, not only the electronic and crystalline structure of Sr2IrO4is similar to that of the cuprates. RIXS experiments on the single layer iridate found magnetic excitations consistent with a Heisenberg antiferromagnet where the role of spin is played by the pseudospinJeff = 1/2. Similar to the magnetic excitations of La2CuO4
[31] (Figure 1.8) a gapless magnon with ordering vectorQ~ = (π, π)joining the center of the zone with the corner of the Brillouin zone is found in Sr2IrO4 [56] (Figure 1.14-a)). Note that the in-plane magnetic unit cell coincides with the one resulting from the octahedral distortion.
Thus, Sr2IrO4is a single band half-filled pseudospin-1/2 Mott-Heisenberg insulator.
Therefore, its low energy electronic structure can be described in a Hubbard model
