Angle-resolved photoemission studies of the metal-insulator transition in lightly rare earth doped Ca2RuO4

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(1)Thesis. Angle-resolved photoemission studies of the metal-insulator transition in lightly rare earth doped Ca2RuO4. RICCO, Sara. Abstract Ca2RuO4 is a multi-band Mott transition system with a ground state that is known to be sensitive to external perturbations, such as chemical substitution, pressure, strain and electric fields. In this work I utilize such sensitivity to determine its electronic structure evolution by angle-resolved photoemission spectroscopy (ARPES) as the system is driven across the metal-insulator transition by doping, temperature and uniaxial strain.. Reference RICCO, Sara. Angle-resolved photoemission studies of the metal-insulator transition in lightly rare earth doped Ca2RuO4. Thèse de doctorat : Univ. Genève, 2018, no. Sc. 5270. DOI : 10.13097/archive-ouverte/unige:113119 URN : urn:nbn:ch:unige-1131199. Available at: http://archive-ouverte.unige.ch/unige:113119 Disclaimer: layout of this document may differ from the published version..

(2) UNIVERSITÉ DE GENÈVE Département de Physique de la Matière Quantique. FACULTÉ DES SCIENCES Professeur Felix Baumberger. Angle-resolved photoemission studies of the metal-insulator transition in lightly rare earth doped Ca2RuO4. THÈSE présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de docteur ès Sciences, mention Physique par. Sara Riccò de Parma (Italie). Thèse n◦ 5270. GENÈVE Atelier d’impression ReproMail 2018.

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(4) To my mother..

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(6) Résumé. Le Ca2 RuO4 est un système de transition de Mott multi-bandes avec un état fondamental connu pour être sensible aux perturbations externes, telles que la substitution chimique, la pression, la tension et les champs électriques. Dans cette thèse, j’utilise cette sensibilité pour déterminer l’évolution de sa structure électronique par spectroscopie photoélectronique résolue en angle (ARPES) lorsque le système est conduit à travers la transition métal-isolant en faisant varier son dopage, sa température et sa déformation uniaxiale. Pour cette étude, j’ai développé une série de monocristaux dopés aux terres rares Ca2−x Xx RuO4 et caractérisé leurs propriétés avec l’ARPES, avec l’analyse dispersive en énergie (EDX), avec la diffraction des neutrons sur monocristaux, avec la magnétisation, avec le transport et avec des mesures de chaleur spécifiques. Ces expériences révèlent que les électrons supplémentaires des ions trivalents de terres rares ne deviennent pas itinérants dans la structure cristalline à basse température de Ca2−x Xx RuO4 . Cependant, le dopage réduit les distorsions de la structure de l’état fondamental et supprime à la fois les transitions structurelles et les transitions métal-isolant qui coïncident pour tous les niveaux de dopage. Je montre que la transition métal-isolant est complètement supprimée pour une concentration de dopage de x = 0.11. L’état fondamental métallique émergeant dans Ca2 RuO4 substitué par des terres rares montre des excitations à basse énergie fortement renormalisées avec une surface de Fermi bien définie. Une analyse de sa structure électronique à basse énergie dans nos expériences ARPES et dans les calculs de théorie du champ moyen dynamique (DMFT) démontrent une forte augmentation du couplage spin-orbite (SOC) due aux corrélations électroniques. Enfin, je rapport des expériences ARPES sur des échantillons légèrement iii.

(7) dopés Pr tout en variant la température et la déformation uniaxiale quasicontinuellement pour amener le système à travers la transition métal-isolant. Ces études révèlent que la transition de Mott se caractérise par un effondrement soudain de la bande de Hubbard, que j’interprète comme une propriété générique de systèmes de Mott multi-bandes. En se déplaçant à travers la transition métal-isolant grâce à une variation de la tension uniaxiale, je ne trouve aucune preuve d’une divergence de masse à proximité de la transition. Au lieu de cela, ces données montrent des signatures claires de coexistence de phase. L’état fondamental métallique stabilisé par la tension est très similaire à l’état métallique induit par le dopage pour x = 0.11. En particulier, il ne montre aucun signe de splitting d’échange ferromagnétique, suggérant que l’état métallique à basse température induit par la contrainte ou le dopage diffère de l’état ferromagnétique itinérant rapporté précédemment pour Ca2 RuO4 non dopé sous stress uniaxial et tension épitaxiale.. iv.

(8) Abstract. Ca2 RuO4 is a multi-band Mott transition system with a ground state that is known to be sensitive to external perturbations, such as chemical substitution, pressure, strain and electric fields. In this work we utilize such sensitivity to determine its electronic structure evolution by angle-resolved photoemission spectroscopy (ARPES) as the system is driven across the metal-insulator transition by doping, temperature and uniaxial strain. For our studies we grew a series of rare earth doped single crystals Ca2−x Xx RuO4 and characterized their properties by ARPES, energydispersive X-ray analysis (EDX), single crystal neutron diffraction, magnetization, transport and specific heat measurements. These experiments reveal that the additional electrons of the trivalent rare earth ions do not become itinerant in the low-temperature crystal structure of Ca2−x Xx RuO4 . However, doping reduces the distortions of the ground state structure and suppresses both structural and metal-insulator transitions, which we find to coincide for all doping levels. We show that the metal-insulator transition is suppressed completely for a doping concentration of x = 0.11. The metallic ground state emerging in rare earth substituted Ca2 RuO4 shows strongly renormalized low-energy excitations with a well-defined Fermi surface. An analysis of its low-energy electronic structure in our ARPES experiments and in dynamical mean field theory (DMFT) calculations demonstrates a strong enhancement of spin-orbit coupling (SOC) due to electronic correlations. Finally, we report ARPES experiments on lightly Pr-doped samples performed while varying temperature and uniaxial strain quasi-continuously to tune the system across the metal-insulator transition. These studies reveal that the Mott transition is characterized by a sudden collapse of the lower v.

(9) Hubbard band, which we interpret as a generic property of multi-band Mott systems. Tuning through the metal-insulator transition by uniaxial strain, we find no evidence for a mass divergence near the transition. Instead, our strain-dependent data show clear signatures of phase coexistence. The metallic ground state stabilized by strain is very similar to the doping-induced metallic state for x = 0.11. In particular, it shows no signs of a ferromagnetic exchange splitting, suggesting that the low-temperature metallic state induced by strain or doping differs from the itinerant ferromagnetic state reported previously for undoped Ca2 RuO4 under uniaxial stress and epitaxial strain.. vi.

(10) Contents. 1 Introduction 1.1 Ca2 RuO4 : a multi-band Mott insulator 1.1.1 Crystal structure . . . . . . . . 1.1.2 Electronic properties . . . . . . 1.1.3 Magnetic order . . . . . . . . . 1.2 Ground state instability . . . . . . . . 1.2.1 Chemical substitution . . . . . 1.2.2 Pressure and strain . . . . . . . 1.2.3 Electric field and current . . . 1.3 Spin-orbit interaction . . . . . . . . . . 1.3.1 Excitonic magnetism . . . . . . 1.3.2 SOC in the metallic state . . . 1.4 Summary of thesis . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. 2 Methods 2.1 ARPES . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Phenomenology . . . . . . . . . . . . . . . . 2.1.2 Theoretical description . . . . . . . . . . . . 2.1.3 The ARPES laboratory at the University of 2.2 Theoretical methods . . . . . . . . . . . . . . . . . 2.2.1 Density functional theory . . . . . . . . . . 2.2.2 Dynamical mean field theory . . . . . . . . 2.3 Crystal growth . . . . . . . . . . . . . . . . . . . . 3 Results 3.1 Bulk characterization of doped Ca2 RuO4 vii. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . Geneva . . . . . . . . . . . . . . . . . . . .. 1 . 1 . 1 . 3 . 4 . 5 . 5 . 6 . 8 . 9 . 10 . 11 . 13 . . . . . . . .. 15 15 15 17 21 22 23 24 27. 31 . . . . . . . . . . . 31.

(11) CONTENTS 3.2 3.3 3.4 3.5 3.6. The electronic structure of doped Ca2 RuO4 . . . . . . . 3.2.1 Doping evolution . . . . . . . . . . . . . . . . . . 3.2.2 The metallic ground state . . . . . . . . . . . . . Single-band vs multi-band doped Mott insulators . . . . Temperature dependence of the MIT in doped Ca2 RuO4 In-situ strain tuning of the MIT . . . . . . . . . . . . . . Conclusions and perspectives . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 37 38 41 53 55 57 69. A Single crystal neutron diffraction. 71. B Tight binding model. 77. Acknowledgements. 81. Bibliography. 83. viii.

(12) CHAPTER. 1. Introduction. 1.1. Ca2 RuO4 : a multi-band Mott insulator. Research interest in the prototypical multi-band Mott insulator Ca2 RuO4 was at first motivated by its vicinity to the unconventional p-wave superconductor Sr2 RuO4 [1]. Intriguingly though, the electronic properties of Ca2 RuO4 differ strongly from Sr2 RuO4 . While Sr2 RuO4 is a good Fermi liquid below TFL ∼ 25 K with a superconducting phase below Tc ∼ 1.5 K, the ground state of Ca2 RuO4 is a Mott insulator with antiferromagnetic order. This is generally attributed to the strong interplay between lattice and electronic degrees of freedom which also renders the ground state of Ca2 RuO4 particularly sensitive to perturbations and, therefore, a unique playground to explore a variety of different magnetic and electronic phases. Moreover, this strongly correlated 4d system is ideally suited to investigate the effects of spin-orbit interaction, a central thread in the search for novel quantum phenomena.. 1.1.1. Crystal structure. Ca2 RuO4 and Sr2 RuO4 are layered perovskite ruthenates of the RuddlesdenPopper series An+1 Run O3n+1 , where A is Ca or Sr. Sr2 RuO4 is a metal and has a I4/mmm tetragonal unit cell, which contains two Ru atoms. Conversely, because of the smaller ionic radius of Ca compared to Sr (rCa = 1.18 Å, rSr = 1.31 Å), the structure of Ca2 RuO4 is characterized by a rotation of the RuO6 octahedra in the ab plane, and a tilt about the c axis. Due to these 1.

(13) 1. Introduction a). b) 12.4 12.3. Ca. 12.2. c (Å). Ru O. 12.1 12.0 11.9 11.8. c. 11.7. b a. 0. 100 200 300 400 T (K). Figure 1.1: a) Orthorhombic unit cell of Ca2 RuO4 . Due to the structural distortions, the in-plane unit cell is rotated by 45◦ with respect to Sr2 RuO4 and contains two Ru atoms (see Fig. 1.2(a)-(b)), therefore the 3D unit cell contains four Ru atoms; b) Temperature dependence of the c lattice parameter, from [2]. The discontinuity at ∼ 360 K corresponds to a first order structural phase transition.. distortions the unit cell of Ca2 RuO4 , shown in Fig. 1.1(a), is orthorhombic and contains four Ru atoms. The a and b crystallographic axes are rotated by 45◦ with respect to the tetragonal in-plane basis vectors defined to be along the Ru-O bonds in Sr2 RuO4 (Fig. 1.2(a)-(b)). Correspondingly, in momentum space the in-plane Brillouin zone for the orthorhombic system is half the size and rotated by 45◦ compared to the tetragonal one (Fig. 1.2(c)). The space group is Pbca. a). b). c) M O. O. b. Y. X. Ru Γ. Ru a. b a. Figure 1.2: Sketches of the Ru-O plane for a) Sr2 RuO4 and b) Ca2 RuO4 . The red and blue marks indicate Ru and O atoms, respectively. The small black squares highlight the in-plane unit cell, which is tetragonal for Sr2 RuO4 and orthorhombic for Ca2 RuO4 . c) The large and small squares represent the tetragonal and orthorhombic in-plane Brillouin zone, respectively. For the small one, the Γ, X, Y and M high symmetry points are indicated.. 2.

(14) 1.1 Ca2 RuO4 : a multi-band Mott insulator Ca2 RuO4 is a Mott insulator [3, 4] with an insulating gap estimated to be about 0.2 eV [4–7] and it exhibits canted antiferromagnetic (CAF) order below the Néel temperature TN = 110 K. The metal-insulator transition (MIT) occurs at TM I = 360 K and is accompanied by a symmetry-preserving, first order structural phase transition (Fig. 1.1(b)). The ‘short-c axis’ insulating phase, also with a larger orthorhombicity, is called S-Pbca, while the quasitetragonal ‘long-c axis’ metallic phase is named L-Pbca.. 1.1.2. Electronic properties. The absence of magnetic order in the insulating state just below TM I (TN < TM I ) suggests that Ca2 RuO4 is a Mott-Hubbard-type insulator. Its low-energy electronic properties are determined by the four d electrons of the Ru4+ ions. The cubic environment splits the Ru d orbitals into a high-energy eg doublet and a low-energy t2g triplet. Taking into account spin degeneracy, the six t2g states are therefore 2/3-filled. Due to the layered structure of the system, the bandwidth W of the dxy orbital is approximately two times larger than for the dxz /dyz states. In the L-Pbca phase density functional theory (DFT) calculations find Wxy ∼ 2.8 eV and Wxz/yz ∼ 1.5 eV, while in the S-Pbca phase the bandwidths are slightly reduced by the structural distortion to Wxy ∼ 2.5 eV and Wxz/yz ∼ 1.3 eV [8]. Such a bandwidth ratio favors a larger occupation of the less extended out-of-plane orbitals and, therefore, a negative orbital polarization p = nxy − (nxz + nyz )/2, where ni denotes the orbital occupations. The distortion of the RuO6 octahedra, however, lifts the degeneracy of the t2g manifold by inducing a crystal field splitting ∆CF , lowering the energy of the dxy orbital with respect to the quasi-degenerate dxz /dyz states, as sketched in Fig. 1.3(a). This, conversely, a). b) xy xz yz sum. L-Pbca. eg S-Pbca. DOS. 4d4. S-Pbca. xz/yz. t2g ∆CF. -6. -4. xy -6. -4. -2 0 2 E-EF (eV) -2 0 2 E-EF (eV). 4 4. 6 6. Figure 1.3: a) Sketch of the low-energy states of Ca2 RuO4 . b) DMFT calculations (U = 2.3 eV, J = 0.4 eV) of the density of states (DOS) in the L-Pbca (upper panel) and S-Pbca (lower panel) phases.. 3.

(15) 1. Introduction favors the occupation of the in-plane orbital and thus a positive polarization. In the L-Pbca phase the local density approximation (LDA) estimates ∆CF ∼ 100 meV, sufficient to suppress the orbital polarization to p ∼ 0. Dynamical mean field theory (DMFT) calculations consistently find a metallic solution down to low temperature and no orbital polarization [8]. In the S-Pbca phase of Ca2 RuO4 the flattening of the RuO6 octahedra enhances the crystal field splitting to ∆CF ∼ 300 meV. Because the structural phase transition L-Pbca→S-Pbca removes the spatial degeneracy of the t2g orbitals, thus reducing the ground state energy, it has been attributed to the Jahn-Teller effect [2, 6, 9]. Band calculations alone, however, fail in predicting the opening of an insulating gap [10]. This is rectified by including correlations within DMFT [8], which finds a fully occupied dxy orbital and half-filled dxz ,dyz bands that then undergo a Mott transition and split into an empty upper Hubbard band (UHB) and a filled lower Hubbard band (LHB) as shown in Fig. 1.3(b). Despite the moderate values of the Coulomb repulsion U ∼ 2 eV and the broad bandwidths ∼ 3 eV arising from the extended nature of 4d orbitals, ruthenates in general are strongly correlated materials, with correlations that persist to low energies producing carrier masses highly enhanced over the LDA predictions (e.g. γ/γLDA ∼ 4 in Sr2 RuO4 ). Strong correlations in this multi-orbital materials have been attributed to Hund’s coupling, the intra-atomic energy scale that favors a configuration with two electrons in different orbitals with parallel spins, which reduces the energy cost of double occupation due to the Coulomb interaction [11].. 1.1.3. Magnetic order. The canted-antiferromagnetic (CAF) order of the ground state of Ca2 RuO4 originates from the oxygen-mediated superexchange coupling of the d orbitals [12], with spin canting attributed to the Dzyaloshinskii-Moriya interaction [4, 13]. Neutron diffraction studies reveal that two different kinds of magnetic modes exist in pure Ca2 RuO4 [14, 15]: an A-centered mode, which represents the major phase and stabilizes below ∼ 110 K, and a B-centered mode, which is a minor phase with ordering temperature ∼ 150 K. As sketched in Fig. 1.4(b), the in-plane alignment of the canted spins produces a magnetic moment which can be either parallel or antiparallel to the b crystallographic axis. The stacking along the c direction of the RuO planes determines the difference between A and B-centered modes. In the A-mode the magnetic moments of two consecutive planes have opposite in-plane directions, generating three dimensional AF order. In the B-mode, conversely, the planes are ordered to have all the in-plane moments pointing in the same direction, thus producing a ferromagnetic (FM) component along the c axis. The two 4.

(16) 1.2 Ground state instability a). b). 3.0. TN. TB a. 2.5. -3. M/H (10 emu/mol-Ru). 3.5. b 2.0 1.5 50. 100. 150 200 T (K). 250. 300 A - mode. B - mode. Figure 1.4: a) Magnetic susceptibility measured in a zero-field-cooled (ZFC) sequence. b) Spin configuration in the CAF phase, adapted from [15].. magnetic ordering temperatures are observed in the temperature dependent magnetic susceptibility (Fig. 1.4(a)) as a cusp and a kink, respectively.. 1.2. Ground state instability. 1.2.1. Chemical substitution. Lattice and electronic degrees of freedom are strongly entangled in Ca2 RuO4 [16], making the ground state particularly sensitive to perturbations. One way to tune the electronic ground state is to substitute divalent Ca by isovalent a). b). 400. 50. SC CAF. 0 I 0.0. M-M. 0.4 0.8 1.2 1.6 Sr concentration x. (I) (II) 100. 50. 0.5. μ0H = 0.1 T ∥ab. 0.1. 0.0. 0.55 0.7 2.0. 0. 0 0.0 Ca. 2.0. (III) M/H (emu/mol-Ru). -3. I4/mmm. se. 100. PM I. - pha. 150. χ(0) (10 emu/mol-Ru). 200. PM - M. d” “Tilte. 250. I41/acd. a S-Pbc. T (K). 300. 150 L-Pbca. 350. 0.5. 1.0 x. 10 T (K). 1.5. 20. 2.0 Sr. Figure 1.5: a) Phase diagram of Ca2−x Srx RuO4 , with abbreviations: PM for paramagnetic, CAF for canted antiferromagnetic, M for metal, M - M for magnetic metallic, I for insulator, SC for superconductor. b) Magnetic suceptibility at 2 K in the M - M (II) and in the PM - M (III) regions, from [9].. 5.

(17) 1. Introduction Sr, thus realizing a quasi-continuous connection between the multi-band Mott insulator Ca2 RuO4 and Sr2 RuO4 , the only known superconductor isostructural to cuprates. The phase diagram of Ca2−x Srx RuO4 , sketched in Fig. 1.5(a), shows a variety of structural, magnetic and electronic phases [2, 9, 17] stabilized by the distortions of the crystal structure, which generally increase with increasing Ca concentration. For Sr concentration 1.5 < x < 2 the crystal structure is tetragonal, space group I4/mmm. For x = 1.5 a rotational distortion appears (space group I41 /acd), while for x ≤ 0.5 a tilt of the RuO6 octahedra is found. In the range x ≤ 0.2 the symmetry is lowered to Pbca. The ground state undergoes the MIT at x = 0.2. Next to this point a correlated metallic region with AF short-range order and a non-Fermi liquid-like resistivity is found. At xc = 0.5 the low-temperature magnetic susceptibility shows a critical enhancement (Fig. 1.5(b)) suggesting that the system is at the verge of FM order at 0 K [18] and marking the crossover to a nearly ferromagnetic state which then evolves into a Fermi liquid (FL) phase with enhanced paramagnetism and a superconducting ground state. The emergence of different magnetic phases supports the idea, which stems from band structure calculations [19], that FM order in layered ruthenates is favored by rotational distortions, while tilting of the RuO6 octahedra stabilizes antiferromagnetism.. 1.2.2. Pressure and strain. Pressure allows to control the bandwidth of the t2g orbitals and thus to tune the MIT in Ca2 RuO4 . In contrast to chemical substitution, it does not introduce disorder into the system and is therefore, at least in principle, a cleaner tuning parameter. At room temperature 0.5 GPa hydrostatic pressure is sufficient to induce a MIT in Ca2 RuO4 [20], which is accompanied by a first order structural phase transition. Coexistence of S-Pbca and L-Pbca phases at low temperature has been observed up to ∼ 1 GPa [21]. In the metallic phase the in-plane resistivity ρab (T ) decreases for decreasing temperature, while the out-of-plane ρc (T ) increases (Fig. 1.6(a)), showing a quasi-2D nature comparable to cuprates and to Sr2 RuO4 [20]. Below ∼ 10 ÷ 25 K ρc (T ) starts decreasing and ρab (T ) drops sharply to its residual value. This temperature range corresponds to the onset of FM order, as also observed in the hysteretic behavior of the magnetization curve shown in Fig. 1.6(b). The small saturated magnetic moment µ ∼ 0.4 µB suggests that this low-temperature magnetic phase is itinerant [22]. The tilting of the oxygen octahedra is suppressed by pressure and disappears around 5.5 GPa, which corresponds to the maximum of the Curie temperature. Interestingly, coexistence of tilt and ferromagnetism is found, as opposed to Ca2−x Srx RuO4 where tilt appears to drive the system 6.

(18) 1.2 Ground state instability a). b) 3. 10. Ca2RuO4. 0. ρab. 8.0 GPa 3.0 GPa 2.0 GPa. 10. -1. 10. 0.10 ρc. M (µB/Ru ion). 1. 10. ρ (Ω·cm). ρc. 0.1 MPa. 2. 10. -2. 10. -3. 10. 2.0 GPa. -4. 10. ρab. -5. 10. 3.0 GPa. 0.05. Ca2RuO4 H // ab at 2 K. 0.00 -0.05 0.1 Mpa 0.7 GPa. -0.10. -6. 10. 0. 100. 200. -0.4. 300. T (K). -0.2. 0.0 0.2 µBH (T). 0.4. Figure 1.6: a) In-plane (ρab ) and out-of-plane (ρc ) resistivity at fixed pressure, from [20]. b) Magnetization curve at 2 K. For P = 0.7 GPa ρab is metallic and the hysteresis in the susceptibility indicates FM ordering, from [20].. away from the FM instability. Superconductivity has been reported close to the border of ferromagnetism, above 9 GPa, in stark contrast to cuprates in which the superconducting state appears in proximity of the AFM phase. According to ac-susceptibility measurements, Tc reaches 0.4 K at 14 GPa, the highest pressure achieved in the experiment [23]. Uniaxial strain applied along the ab plane is even more effective than hydrostatic pressure to tune the properties of Ca2 RuO4 by coupling to the orbital degrees of freedom. The critical pressure for the transition to the FM-metallic phase is ∼ 0.2 GPa, more than an order of magnitude smaller than under hydrostatic pressure, where ∼ 5.5 GPa are needed to induce a similar transition [24]. Epitaxial strain in Ca2 RuO4 thin films grown by pulsed laser deposition (PLD) provides an alternative method of suppressing the Mott insulating ground state [25, 26]. For sufficiently thin films the in-plane lattice parameters are strained to match the substrate, while the out-of-plane constant accomodates depending on the applied strain and on the material’s Poisson’s ratio [26]. The behavior of these systems, therefore, depends on the substrate used. Ca2 RuO4 thin films grown on LaSrAlO4 (001) (LSAO) [25], for instance, can be considered to be 2%÷3% compressively strained in the ab plane compared to the bulk structure at 295 K. They display a metallic behavior in the temperature range 2 K - 400 K, as shown in Fig. 1.7. Similar to the behavior of Ca2 RuO4 under hydrostatic pressure, epitaxial films on 7.

(19) 1. Introduction. Figure 1.7: In-plane resitivity of Ca2 RuO4 on a LSAO film, from [25].. LSAO substrates show a clear downturn of the resistivity at ∼ 20 K, which coincides with the onset of FM order (see inset of Fig. 1.7).. 1.2.3. Electric field and current. Remarkably, a modest electric field of ∼ 40 V/cm applied at room temperature induces a MIT in bulk Ca2 RuO4 , accompanied by a S-Pbca→L-Pbca structural phase transition (Fig. 1.8(a)). This induced metallic state, moreover, can be stabilized down to 4 K by maintaining a weak constant current [27]. Fig. 1.8(b) shows the temperature dependence of the in-plane resistance with a flowing current of 420 mA. The sudden drop of the resistance at ∼ 15 K marks the transition to a FM phase, in analogy to the pressureinduced metallic state. The mechanism that drives the E-field induced MIT is not yet understood. Unlike pressure, electric fields do not couple directly to the structural distortion. Moreover, the observed critical field of ∼ 40 V/cm is much smaller than the value of ∼ 4 MV/cm predicted by the conventional Zener breakdown model, suggesting that an unconventional breakdown mechanism based on many-body Zener effect might be essential [28]. Lower dc currents were recently shown to drive the system towards a non-equilibrium semimetallic phase [29], which differs from the usual metallic L-Pbca state. This phase, remarkably, shows extraordinarily strong diamagnetism exceeding the values found in most of the non superconducting materials [30]. The transition between the two current-induced phases is continuous, with a 8.

(20) 1.3 Spin-orbit interaction a). b) 7.5 12.3 7.0. Resistance (Ω). c (Å). 12.2. 12.1. 6.5 6.0 5.5. Ca2RuO4 E // ab. E // c 290 K. 12.0. 5.0 11.9. 0. 20. 40 60 E (V/cm). 80. 0. 100. 200 T (K). Figure 1.8: a) E-field dependence of the c lattice parameter, from [27]. L-Pbca and S-Pbca coexist in the shaded region. b) Temperature-dependent variation of the resistance in the current-maintained metallic phase (I = 420 mA), from [27].. gradually decreasing resistivity for increasing current.. 1.3. Spin-orbit interaction. Spin-orbit coupling (SOC) is a relativistic effect that couples spin and orbital degrees of freedom of electrons in atoms. While SOC is small for light elements and can be treated perturbatively, it is now understood to produce novel quantum phenomena, such as topological insulators [31] and the anomalous Hall effect in certain compounds with heavy elements [32]. The interplay of SOC with strong electronic correlations in 4d and 5d transitionmetal oxides (TMOs) is a subject of intense study [33]. A prominent class of materials in this respect are the perovskite iridates Sr2 IrO4 and Sr3 Ir2 O7 , whose insulating ground state is shaped by the combination of SOC and Coulomb repulsion [34]. Strong SOC in Ca2 RuO4 was originally suggested based on O 1s X-ray absorption spectroscopy (XAS) studies [35] that report a temperature-dependent mixing of the dxy , dxz , dyz orbitals in the two unoccupied states of insulating Ca2 RuO4 . In particular, they estimate 0.2 ÷ 0.5 holes in the dxy state. These observations were explained as an effect of SOC and a spin-orbit coupling constant λ ∼ 200 meV was obtained using a simple model which 9.

(21) 1. Introduction includes only SOC and crystal field splitting [36]. DMFT, however, predicts nxy ∼ 2 and nxz ∼ nyz ∼ 1 orbital occupations in the S-Pbca Mott phase and does not reproduce such fractional occupations. Moreover, it is difficult to reconcile fractional orbital occupations with the insulating ground state of Ca2 RuO4 . This suggests a shortcoming of the single-ion interpretation of XAS used in Ref. [36] possibly related to the structural distortions in Ca2 RuO4 . These in fact lead to a deviation of the local coordinates used to define the orbital character from the global crystalline axes, with respect to which the transition matrix elements are defined.. 1.3.1. Excitonic magnetism. The energy scale determined by SOC competes with the crystal field splitting to determine the ground state of the system. Neglecting SOC, the crystal field lowers the energy of dxy , thus stabilizing the crystal field eigenstates and suppressing the orbital degrees of freedom. In this case the ground state magnetism is described by the Heisenberg Hamiltonian for S = 1 states. If SOC is strong enough, on the other hand, the orbital angular momentum is unquenched and given by Leff = 1. This is coupled to the spin S = 1 by SOC and the eigenstates are described by the quantum number J = L + S, which can take values 0, 1, 2. The ground state singlet J = 0 is non-magnetic, in a). b) Tz 0.5. δ. 0.4. Tx/y. M // c. 0.3. I/λ. M // ab. 0.2. λ. 0.1 0.0 0.0. s. PM 0.1. 0.2 0.3 δ/λ. 0.4. 0.5. Figure 1.9: a) Sketch of the low-energy structure in strong SOC regime. Singlet and triplet states are split by the spin-orbit interaction, the triplet is further split by the crystal field that favors in-plane magnetization. b) Phase diagram resulting from the interplay of the different energy scales: SOC (λ), crystal field (δ) and exchange coupling (I ), from [37].. 10.

(22) 1.3 Spin-orbit interaction apparent contrast with the AF ground state of Ca2 RuO4 . Van-Vleck type magnetism, however, can emerge due to gapped singlet-triplet excitations, represented by hard-core bosons called triplons, deriving from the superexchange interaction [37, 38]. The low-energy structure of such a system is sketched in Fig. 1.9(a). The magnetic properties of the ground state depend on the competition among SOC strength λ, crystal fields and superexchange energy scales, as shown in Fig. 1.9(b). Magnetic order can be obtained from a Bose-Einstein condensate (BEC) of triplons [39], which is separated from the J = 0 vacuum state by a quantum critical point. In general, the long-range order of condensates is described by a complex wavefunction Ψ(r,t), whose phase oscillations represent the Goldstone modes of the broken U(1) continuous symmetry, while amplitude modes are associated to the Higgs mode of particle physics. The same analogy can be extended to magnetic systems, where the condensate amplitude is the magnetization of the system and the phase defines its specific direction in space. In a system of interacting spins described by the XY model Goldstone and Higgs excitations are spin-waves corresponding to fluctuations in the direction and in the magnitude of the magnetization, respectively [37]. The exotic scenario of excitonic magnetism in Ca2 RuO4 , despite claims of experimental evidence [40–42], seems to be contradicted by first-principles theoretical [43] and experimental [44] studies.. 1.3.2. SOC in the metallic state. The importance of the inclusion of SOC in the theoretical description of the electronic properties of Sr2 RuO4 and Sr2 RhO4 has been investigated within the LDA [45–49]. These studies, however, neglect the renormalization of SOC and do not quantify its effective strength in the presence of correlations. A recent theoretical study of Sr2 RuO4 [50] explores the consequences of the inclusion of SOC in DMFT calculations unveiling its effects at all frequencies. Ref. [50] shows that SOC does not affect the frequency dependent orbital diagonal components of the self energy, because the energy scale below which the orbital degrees of freedom are frozen is higher than the one characteristic of SOC, as a consequence of the Hund’s metal nature of ruthenates [11]. In contrast, the orbital off-diagonal terms of the self energy, which vanish by symmetry when SOC is not included, acquire an energy independent finite value, which can be interpreted as a correlation-induced enhancement of the bare SOC coupling. SOC, moreover, affects significantly the band structure at points where bands cross in its absence. Degenerate points are eff split in momentum by δk ∼ λv and in energy by ∆ ∼ Zλeff , where Z is the quasiparticle weight obtained from the linearization of the real part of the self energy, as shown in Fig. 1.10(a) and (b). The effect of correlations on SOC is therefore two-fold and strongly frequency dependent: on one hand 11.

(23) 1. Introduction a). b) 6.4 H. Zxz. 6.0. Re Σ(ω) (eV). Δε ~ Z λeff EF δk ~ λeff / v. 5.6. L. Zxz L. Zxy. 5.2 4.8. xz (w/oSOC) xz (SOC) xy (w/oSOC) xy (SOC). 4.4 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 ω (eV). Figure 1.10: a) Effect of the inclusion of SOC at band crossings. b) DMFT energy dependence of the real part of the self energy with and without SOC. Quasiparticle renormalization factors Z are highlighted, from [50].. correlations enhance the coupling in an energy independent way, on the other hand they suppress it by energy dependent quasiparticle renormalizations.. 12.

(24) 1.4 Summary of thesis. 1.4. Summary of thesis. This work is structured as follows. In Chapter 2 we will present the main experimental and theoretical techniques we used. The floating zone method allowed us to grow high-quality single crystals of lightly rare earth doped Ca2 RuO4 , which we probed by angle-resolved photoemission spectroscopy (ARPES), a unique technique to investigate the microscopic electronic structure of solids. Our experimental studies are discussed and compared to the theoretical descriptions provided by density functional theory (DFT) and dynamical mean field theory (DMFT). Inclusion of SOC in the latter description proved to be essential in reproducing our results. In Chapter 3 we introduce our materials of study by presenting their bulk properties, characterized through neutron scattering, magnetization, resistivity and specific heat measurements. We explore the metallic phase of lightly doped Ca2 RuO4 by ARPES and compare it to the theoretical predictions, finding that SOC has an important effect on the electronic structure of the system and that, remarkably, its bare value is enhanced by correlations. We investigate the nature of the metal-insulator transition of Ca2 RuO4 and compare it to the case of lightly doped single-band Mott insulators, namely cuprates and iridates. Finally, we demonstrate in-situ strain-tuning of the metal-insulator transition in ARPES experiments and highlight the capabilities of this new technique.. 13.

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(26) CHAPTER. 2. Methods. 2.1. ARPES. Angle-resolved photoemission spectroscopy (ARPES) is based on the photoelectric effect, i.e. the emission of electrons when light shines on a material, first observed by Heinrich Hertz [51] in 1887 and explained in 1905 as a manifestation of the quantum nature of light by Albert Einstein [52], who was awarded the Nobel prize in 1921 for his breakthrough. By probing both the kinetic energy and the angular distribution of the photoelectrons emitted from a sample, ARPES is a unique experimental tool to investigate the momentum-dependent electronic band structure of solids.. 2.1.1. Phenomenology. A schematic of an ARPES spectrometer is shown in Fig. 2.1. The photoelectrons emitted from the sample are mapped onto an angle-energy plane by the hemispherical analyzer, which comprises a multi-element electron lens, a hemispherical deflector to disperse the energies and a detector [53]. The lens collects the electrons emitted within an acceptance angle centered at the direction of its axis, defined with respect to the sample plane by the angles (θ, φ) (see Fig. 2.1) and produces a Fourier-space image at the entrance slit of the hemisphere. Photoelectrons are accelerated or decelerated in the lens in order to match their energy to the working energy of the deflector, the pass energy. The deflector consists of two concentric hemispheres of radius R1 and 15.

(27) 2. Methods. Hemispherical Analyzer. θ, k UV source Lens Lens. z. 2D detector Ekin. θ ϕ x. Sample. y. Sample. Figure 2.1: Schematic of an ARPES experiment.. R2 kept at a potential difference ∆V . Only the electrons whose energy is within a window centered at the pass energy Epass = e∆V /(R1 /R2 − R2 /R1 ) can travel through the capacitor and their trajectories are dispersed depending on their kinetic energy. In the detector the photoelectrons thus dispersed in angle and in energy are amplified through an avalanche effect by a multi-channel plate (MCP) and accelerated to a phosphor screen which is imaged by a CCD camera. The energy resolution of the apparatus is α2 ∆E = Epass ( w+dx 4R0 + 4 ), where w is is the width of the entrance slit, dx is the size of one hit on the CCD camera, R0 is the mean of the radii and α is the acceptance angle, namely the maximal angle that the photoelectrons can have to enter the analyzer. The kinetic energy Ekin of photoelectrons propagating in vacuum as free √ particles fully determines the modulus of their momentum k = 2mEkin /~, where m is the electron mass. The kinetic energy, moreover, is related to the binding energy EB of the electrons in the material by Ekin = hν − Φ − EB ,. (2.1). where hν is the incident photon energy and Φ is the work function of the material, as shown in Fig. 2.2. The momentum of the electrons inside the solid, considered here in the extended zone scheme, can be written as the sum of two components k = kk + k⊥ , parallel and orthogonal to the surface, respectively. Neglecting the photon momentum, the first one is conserved in the emission process as the surface is invariant under translations, and its modulus is p kk = sin θ 2mEkin /~. (2.2) The orthogonal component, in contrast, is not conserved because the presence of a surface breaks translation symmetry along the perpendicular direction 16.

(28) 2.1 ARPES. Figure 2.2: Energetics of the photoemission process, from [54].. and further assumptions or experimental data are needed to determine k⊥ . A simple approach which assumes a nearly-free-electron dispersion for the final evanescent states gives p k⊥ = 2m(Ekin cos2 θ + V0 )/~, (2.3) where V0 is the inner potential, namely the energy of the bottom of the valence band with respect to the vacuum reference Ev , as illustrated in Fig. 2.2. The inner potential can be determined experimentally from the periodicity of the E(k⊥ ) dispersion observed for fixed kk while varying the photon energy.. 2.1.2. Theoretical description. To date, due to the high degree of complexity involved, a comprehensive formal description of the photoemission process has not been achieved. A phenomenological approach combines Fermi’s golden rule to obtain the transition probability between two quantum states with a three-step model 17.

(29) 2. Methods which describes the process as a sequence of three independent events: the optical excitation of a bulk electron into a final state, its transport to the surface and its transmission through the surface into vacuum. A more rigorous one-step approach, conversely, treats photoemission as a single quantum-mechanical coherent process. Both models rely on the sudden approximation, which is in principle applicable only to high-energy electrons, and assumes that the particle removal is instantaneous. Here we will discuss a treatment within response theory, which envisions the photocurrent as a steady response of the system to an applied electromagnetic field [55]. Let us introduce the vector potential A(r, t) to describe the light in a photoemission experiment, and let us fix a gauge where the scalar potential vanishes. The coupling between the electromagnetic field and the electron current j(r) is described as Z Vt = −e dr j(r) · A(r, t), (2.4) where e is the elementary charge. We are interested in the average current collected at a point R outside the sample hj(R)i. The contribution at zeroth order in the electromagnetic field describes thermionic emission, i.e. the emission of current in absence of an external field due to thermally excited electrons, while the first order term vanishes outside the sample. Photoemission is a second order process in the electromagnetic field and the photocurrent at second order depends on the three-current correlation function Cjµ (R)jν1 (r1 )jν2 (r2 ) according to hjµ (R, t)i(2) =. e2 2. Z. XZ dω1 dω2 dr1 dr2 exp(−i(ω1 + ω1 )t) 2π 2π ν ν. (2.5). 1 2. Aν1 (r1 , ω1 )Aν2 (r2 , ω2 )Cjµ (R)jν1 (r1 )jν2 (r2 ) (iΩ1 → ω1 + i0+ , iΩ2 → ω2 + i0+ ). In imaginary time τ , the three-current correlation function reads Cjµ (R)jν1 (r1 )jν2 (r2 ) (τ, τ 0 ) = hTτ jµ (R, τ )jν1 (r1 , 0)jν2 (r2 , τ − τ 0 )i,. (2.6). where Tτ is the time-ordering operator and h. . . i denotes the thermal average evaluated in absence of the external perturbation. By expressing the current using the creation and annihilation operators, the correlation function we need to compute is Cc†. † † Rσ cRσ cr1 σ1 cr1 σ1 cr2 σ2 cr2 σ2. =. (τ, τ 0 ). hTτ c†Rσ (τ )cRσ (τ )c†r1 σ1 (0)cr1 σ1 (0)c†r2 σ2 (τ. (2.7) − τ 0 )cr2 σ2 (τ − τ 0 )i,. which is represented by the generic Feynman diagram shown in Fig.2.3(a). The lower vertex (r1 , σ1 , 0) represents the excitation of a photoelectron by 18.

(30) 2.1 ARPES. (r2 σ2, τ-τ') G. (R σ, τ). Gbulk(r1-r2, τ'-τ). fre e (r 2 -R. (R ee. , -τ. r 1,. '). τ). G fr. (r1 σ1, 0). Figure 2.3: a) Generic diagram describing the photoemission experiment, and b) diagram corresponding to the sudden approximation, from [55].. light, the electron is indicated by the line leaving to the right. The shaded box includes all the interactions (impurities, phonons, other electrons, surface etc.) experienced by the electron during its travel to the detector at point R, as well as by the photo-hole which propagates ‘back in time’ from point r1 to r2 . The line connecting R and r2 describes the neutralization of the sample, achieved by putting sample and detector in electrical contact. The sudden approximation allows us to simplify this problem and find an expression that relates the photocurrent measured in ARPES to the oneparticle excitations of the bulk material. Specifically, we make the following approximations: (i) the photoelectrons are excited directly into vacuum and, therefore, behave as free particles as soon as they are excited (which is reasonable since photoemission only probes the first few layers below the surface), (ii) the propagation of the photo-hole from r1 to r2 takes place in the bulk and is unaffected by the surface, and (iii) the electromagnetic field in the solid is the same as in the vacuum (which is not so sensible for metals, where the penetration length is a few 10 nanometers). In the sudden approximation the original correlation function (Fig. 2.3(a)) simplifies to the product of three single-particle Green’s functions (Fig. 2.3(b)) describing the free propagation of the photoelectron and the neutralization process, and the propagation in the bulk material of the photo-hole left behind. Evaluating the correlation function within the sudden approximation and assuming linearly polarized light, i.e. A(r, t) = A(r) cos(ω0 t), we can finally express the angle and energy-resolved ARPES intensity measured through a solid angle dΩ along the direction n as d2 I(n, ω) = M (n, hν − Φ) A(k, ω) f (ω), dΩdω. 19. (2.8).

(31) 2. Methods where Φ is the work function of the material, f (ω) is the Fermi function, A(k, ω) is the spectral function of the bulk Green’s function and M (n, hν −Φ) is the matrix element which depends on the light intensity, polarization and frequency, as well as on the geometry of the experiment according to M (n, hν − Φ) =. e2 V κ 3 |n · A|2 , 8π 2 m~. (2.9). where κ is the wavevector of the free photoelectron, m is the electron mass and V is the volume of the system. The spectral function, in turn, describes the one-particle excitations of a system and it is related to the retarded Green’s function according to 1 A(k, ω) = − Im G(k, ω). π. (2.10). For a free electron gas the excitations above the ground state consist in the creation of either a particle above the Fermi level or a hole in the Fermi sea. The excitations are therefore single particles described by the quantum numbers of electrons or holes. The Green’s function reads G0 (k, ω) =. 1 , ω − ξk + iη. (2.11). where ξk is the particle dispersion and η is an infinitesimal quantity. The spectral function is thus a Dirac delta A(k, ω) = δ(ω − ξk ) peaked at the particle energy ξk , measured with respect to the chemical potential. For interacting systems, the Green’s function is given by G(k, ω) =. 1 , ω − ξk − Σ(k, ω) + iη. (2.12). where the self energy Σ(k, ω) = Σ0 (k, ω) + i Σ00 (k, ω) includes the effect of interactions. The spectral function therefore becomes A(k, ω) = −. Σ00 (k, ω) 1 . 0 π [ω − ξk − Σ (k, ω)]2 + [Σ00 (k, ω)]2. (2.13). If the self energy depends weakly on energy and momentum, the spectral function can be approximated as a Lorentzian. The pole represents the energy of the excitation and has an offset with respect to the bare dispersion given by the real part of the self energy, while the width, measured by the imaginary part of the self energy, is related to the inverse lifetime of the excitation. A system which has well defined electron-like one-particle excitations is a Fermi liquid and its excitations are called quasiparticles. ARPES thus represents a unique tool for investigating the band structure of materials and the one-particle excitations of Fermi liquids. 20.

(32) 2.1 ARPES Let us consider the low-energy excitations close to the Fermi level, neglecting the momentum dependence of the self energy, which is in many cases a reasonable approximation. By expanding the real part of the self energy according to Σ0 (k, ω) = Σ0 (0) − λω + O(ω 2 ), (2.14) 0. with λ = − ∂Σ∂ω(ω) |ω=0 , the spectral function takes the form A(k, ω) ≈. 1/2τ (ω) Z . π (ω − Ek )2 + (1/2τ (ω))2. (2.15). Such low-energy quasiparticles thus have a reduced spectral weight Z = 1/(1+ λ), a finite lifetime τ (ω) = (2Z|Σ00 (ω)|)−1 and a renormalized dispersion Ek , yielding a reduced velocity vk∗ = vk /(1 + λ) and an enhanced mass m∗ = (1 + λ)m.. 2.1.3. The ARPES laboratory at the University of Geneva. Fig. 2.4 shows a photograph of the ARPES laboratory at the University of Geneva. The system is in ultra-high vacuum (UHV), at pressures lower than ∼ 10−10 mbar. The samples are transferred from air to the load lock (LL), which can be pumped down to a pressure p ∼ 10−8 mbar. The load lock is. He lamp. AC. PC. analyzer. sputtering chamber. LL. 11 eV 6 eV. Figure 2.4: Photograph of the ARPES setup at the University of Geneva. Highlighted are the three light sources (He discharge lamp, 11 eV and 6 eV lasers), the hemispherical analyzer, the analysis chamber (AC), the preparation chamber (PC), the load lock (LL) and the sputtering chamber.. 21.

(33) 2. Methods connected through a valve to the preparation chamber (PC), p ∼ 10−10 mbar, which hosts a heating stage allowing for sputtering and annealing of the samples. From the PC through a second valve one can access the analysis chamber (AC) (p ∼ 5 × 10−11 mbar), where samples are cleaved in situ. In the AC surface deposition and surface structure analysis through low energy electron diffraction (LEED) can be performed. The samples are held on a motorized 6-axis manipulator, and can be cooled down to ∼ 4 K by a He-flow cryostat. To perform photoemission experiments there are three light sources available: a He discharge lamp (MBS) with rotatable linear polarization providing monochromatized 21.2 eV photon energy, a continuous wave laser (LEOS Solutions) producing 6.01 eV photon energy (206 nm) by passing a 824 nm laser through two second harmonic generation (SHG) stages in series and a 11 eV (113 nm) laser (Lumeras) which generates the 9th harmonic of a 1024 nm pumped laser. The linear polarization state (vertical or horizontal) of both lasers can be controlled. The hemispherical analyzer is a MBS-A1 model. This has one one deflector inside the lens and by tilting the lens axis one can acquire Fermi surface maps without moving the sample to vary the (θ, φ) angles. The ARPES data showing the low-energy electronic structure of doped Ca-ruthenates investigated in this thesis were acquired at the I05 beamline of Diamond Light Source. We chose to use synchrotron light because the 6 eV did not allow us to access a wide enough region in momentum space, the matrix elements were not sufficiently good with the 11 eV laser, and the 0.5 mm spot of the He lamp was too large. The I05 station offers a high photon flux, with spot size 50 × 50 µm2 , produced by a variable polarization Apple II undulator. Photon energies range from 18 eV to 240 eV and four polarization states are available (linear vertical and horizontal, circular left and right). The system is in UHV, p ∼ 5 × 10−10 mbar, the samples are cleaved in situ and held on a 6-axis motorized manipulator. As in the home laboratory, cooling down to ∼ 6 K is achieved through a He-flow cryostat. The hemispherical analyzer is a Scienta R4000 and the highest energy resolution of the system is 3 meV [56]. The energy resolution in our experiments is 10 meV.. 2.2. Theoretical methods. Having shown that angle-resolved photoemission can provide us experimental insight into the electronic structure of solids, we now discuss density functional theory and dynamical mean-field theory, two well-established techniques that allow for theoretical predictions. Confronting experimental data with theoretical models is of fundamental importance to understand and interpret the data, to test the limits of the model, and to ultimately 22.

(34) 2.2 Theoretical methods advance our knowledge of the underlying physics.. 2.2.1. Density functional theory. Density functional theory (DFT) is an ab initio approach to solve the many-body problem of an inhomogeneous gas of interacting electrons by transforming it into a single-electron one. The only parameters needed are the atomic number Z, the number of electrons and the positions of atoms in the crystal. At the foundation of DFT is the proof, given by Hohenberg and Kohn in 1964 [57], that the many-body ground state energy is a unique functional of the density ρ(r). Kohn and Sham in 1965 [58] carried that approach further to derive a set of self-consistent equations. Hohenberg and Kohn showed that the ground state energy of an inhomogeneous electron gas in a static external potential Vext (r) (usually the attractive nuclear potential) can be written as Z Z Z ρ(r)ρ(r0 ) e2 dr dr0 + G[ρ], (2.16) E[ρ] = dr Vext (r)ρ(r) + 2 |r − r0 | where the second term is the Hartree energy due to the Coulomb interaction and G[ρ] is a universal functional of the density, valid for any number of particles and any external potential. Expression 2.16 has a minimum for the correct density. Kohn and Sham made the key assumption that there exists a system of non-interacting electrons with the same density as the interacting one, represented by a Slater determinant ΨS [ρ], and wrote the functional G[ρ] as G[ρ] = TS [ρ] + Exc [ρ], (2.17) where TS [ρ] is the kinetic energy of the non-interacting gas and Exc [ρ] is the exchange-correlation energy, which does not have a simple exact expression. Introducing a basis of one-electron wave functions {ψi }, we obtain the energy functional Z N ~2 X Ω[{ψi }] = − hψi |∇2 |ψi i + dr Vext (r)ρ(r) (2.18) 2m i=1 Z Z ρ(r)ρ(r0 ) e2 + dr dr0 + Exc [ρ], 2 |r − r0 | PN where N is the total number of electrons and ρ = i=1 |ψi |2 . To find a stationary set of one-particle orbitals {ψ1 , ..., ψN } we need to solve the Euler-Lagrange equations N. δΩ[{ψ1 , ..., ψN }] X − ij ψj = 0, δψi j=1 23. (2.19).

(35) 2. Methods where the Lagrange multipliers ij guarantee the orthonormality of the oneelectron wave functions. Equations 2.19 lead to the Kohn-Sham equations h. −. i ~2 2 ∇ + Veff (r) ψi = i ψi , 2m. where we have introduced the effective potential Veff defined as Z δExc [ρ] ρ(r0 ) 2 + . Veff (r) = Vext (r) + e dr0 |r − r0 | δρ(r). (2.20). (2.21). The Kohn-Sham equations are a set of one-particle Schrödinger equations describing non-interacting electrons moving in an effective potential. They need to be solved iteratively until self-consistency is achieved. DFT applications are mostly based on the local density approximation (LDA), where the exchange-correlation energy, assuming a slowly-varying density, is defined according to Z Exc [ρ] = dr ρ(r)xc (ρ(r)), (2.22) where xc (ρ) is the exchange-correlation per electron of a homogeneous gas of density ρ. The DFT approach has been very successful and it is widely applied in many branches of chemistry and material science. This technique, however, can provide qualitatively wrong results when applied to the study of strongly interacting systems, where the effect of correlations cannot be described in a single-electron approach. Despite the fact that DFT allows the rigorous calculation of the ground state energy and its derivatives only, one of the main applications of LDA is the calculation of electronic band structures. This is achieved by solving the Kohn-Sham equations obtained from the Hamiltonian h i XZ ~2 2 HLDA = dr Ψ† (r, σ) − ∇ + Veff (r) Ψ(r, σ), (2.23) 2m σ expanding the field operator Ψ(r, σ) on a properly chosen basis set (e.g. the d orbitals) and interpreting the Lagrange multipliers i as the one-particle energies of the system.. 2.2.2. Dynamical mean field theory. The intuitive idea at the heart of mean-field theory is that the dynamics of a single site in a lattice can be thought of as resulting from the interaction of the site with an external bath produced by all the remaining sites. Weiss mean-field theory provides an approximate solution (yet correct in the limit 24.

(36) 2.2 Theoretical methods of infinite lattice coordination) to the Ising model describing a lattice of spins in a magnetic field and with nearest neighbor ferromagnetic interaction. Let us see how this can be generalized to quantum many-body physics. We consider a system described by the Hubbard Hamiltonian X X H=− tij (c†iσ cjσ + c†jσ ciσ ) + U ni↑ ni↓ , (2.24) i. hi,ji,σ. where tij represents the hopping between nearest neighbor sites and U is the on-site Coulomb interaction. Pioneering work carried out by Metzner and Vollhardt [59] proves that for infinite dimensions, hence coordination z → ∞, the only relevant quantum-mechanical processes are those described by the local interacting Green’s function Gii (ω), which implies that the self energy itself is local, i.e. k-independent, Σ(k, ω) = Σ(ω). Based on this key observation, dynamical mean-field theory (DMFT) [60–63] reduces the lattice many-body problem to a single-site one assuming that the self energy obtained from the latter is a local approximation of the full self energy of the lattice. The approach is called dynamical because the local quantum fluctuations in time, i.e. the transitions among the four possible states |0i, | ↑i, | ↓i, | ↑↓i through the exchange of electrons with the rest of the lattice, are reproduced by the hybridization of the single site with an effective bath. As first noted by A. Georges and G. Kotliar [60], the single site embedded into an effective medium can be mapped to an Anderson impurity model, described by the Hamiltonian X X HAM = i a†iσ aiσ + Vi (a†iσ coσ + c†oσ aiσ ) (2.25) iσ. iσ. + U no↑ no↓ − µ. X. c†oσ coσ ,. σ. where a†iσ (aiσ ) are creation (annihilation) operators for non-interacting conduction electrons playing the role of the bath and c†oσ (coσ ) represent the localized interacting impurity orbital. Integrating out the conduction electrons, we obtain an effective problem for the interacting electrons, with an imaginary-time action given by β. Z Seff = −. Z dτ. 0. Z +U. 0. β. dτ 0. X. c†oσ (τ ) G0−1 (τ − τ 0 ) coσ (τ 0 ). (2.26). σ. β. dτ no↑ (τ ) no↓ (τ ). 0. Here G0 (τ − τ 0 ) is the non-interacting Green’s function for the impurity model describing the amplitude for the creation of a fermion on the site at 25.

(37) 2. Methods time τ and the annihilation at time τ 0 via exchange with the bath. In order to obtain a closed set of mean-field equations, expression 2.26 has to be supplemented by a self-consistency condition that relates G0 to some local quantity that is derived from Seff itself. In our case the local quantity is the impurity Green’s function, while the self-consistency condition reads G0−1 (iωn ) = G−1 (iωn ) + Σ(iωn ),. (2.27). where ωn = (2n + 1)/β are the Matsubara frequencies. Let us outline the DMFT self-consistent scheme. Starting with a trial self energy Σ(iωn ), we can compute the local Green’s function of the original lattice according to X G(iωn ) = G(k, iωn ) (2.28) k. =. X k. 1 . iωn + µ − ξk − Σ(iωn ). Since the local Green’s function and self energy have to be the same for the lattice and for the impurity, we obtain the non-interacting bath’s Green’s function G0 by equation 2.27. This in turn allows us to calculate the impurity’s Green’s function from the action 2.26 and to finally obtain a new estimate for the self energy Σnew = G0−1 −G−1 . The process is iterated with Σ = Σnew until convergence is achieved. Several numerical methods are available to solve the Anderson impurity model, yielding different results. The main approaches are the Quantum Monte Carlo (QMC), the Numerical Renormalization Group (NRG) and the Iterative Perturbation Theory (IPT) [64]. In the LDA+DMFT [61, 65] approach the LDA Hamiltonian 2.23 is treated in the framework of DMFT to take into account local correlations induced by the local Coulomb interaction, e.g. between d and f electrons. Considering t2g orbitals, the complete starting point Hamiltonian reads X X X nmσ nlσ H = HLDA + U nm↑ nm↓ + U 0 nm↑ nl↓ + (U 0 − J) m. m6=l. m<l,σ. (2.29) −J. X m6=l. c†m↑ cm↓ c†l↓ cl↑. +J. X. c†m↑ c†m↓ cl↓ cl↑. m6=l. −. X. ∆ nlσ ,. l,σ. where the indices m and l indicate the different orbitals, U and U 0 correspond to the intra-orbital and inter-orbital repulsion, respectively, while the spinexchange and the pair-hopping integrals, which coincide for real-valued wave functions, are represented by J. Rotational invariance requires that U 0 = U − 2J. The last term corresponds to the contribution to the Coulomb interaction already included into LDA and has to be subtracted to avoid double counting [61]. To perform self-consistent LDA+DMFT the first step is 26.

(38) 2.3 Crystal growth to compute the LDA band structure LDA (k) and to obtain U , J and ∆ via the constrained LDA method [66, 67] to define the many-body problem 2.29. This is solved by DMFT and the resulting self energy is used to compute a new density, which allows to construct a new HLDA . The process is iterated until convergence is achieved. By taking into account local correlations DMFT has been successful at capturing the Mott transition [65, 68] as well as at describing many other physical phenomena. DMFT, however, cannot reproduce all the effects due to the presence of non-local correlations, such as the critical behavior in the proximity of phase transitions. Cluster extensions of DMFT [69, 70], which consider a cluster of sites instead of a single site, can be used to include non-local correlations.. 2.3. Crystal growth. All of the single crystals of Ca2−x Xx RuO4 , with X = La, Pr, Nd, studied in this work were grown by me in collaboration with R. Perry through the floating zone (FZ) technique [71–74] using a commercial four-mirror image furnace (Crystal Systems Corporation, model FZ-T-10000-H-VII-VPO-PC), shown in Fig. 2.5. The initial step of the crystal growth process is grinding together CaCO3 , RuO2 and X2 O3 in the ratio obtained considering the chemical reaction x (2 − x) CaCO3 + X2 O3 + y RuO2 → Ca2−x Xx Ruy O4 + (2 − x) CO2 , 2 where an excess (y − 1) RuO2 is added to compensate for the volatility of the oxide during the growth. Since CaCO3 is hygroscopic, it is first dried at 600 ◦C for 18 hours. The resulting powder is pressed into cylindrical pellets with a diameter of approximately 2 cm and a height of about 5 mm. The pellets are placed on a Pt plate, piled up on top of a thinner bedding pellet to prevent contamination, and baked in air at 1100 ◦C for 18 hours, to allow the solid-state reaction to take place. The pellets are then ground to a powder, which is put inside a narrow cylindrical balloon, to form a rod of about 5 mm diameter and 10 cm length. The rod is inserted into a straw to keep it straight, and pressed in water using an isostatic press. After removing the straw and cutting the balloon, the rod is placed inside an alumina boat on top of a layer of bedding powder and sintered in air at 1200 ◦C for 2 hours. The so obtained feed rod is then suspended from a metallic hook at the center of the mirror stage of the image furnace (see Fig. 2.5) and vertically aligned to a seed rod held from below. At the first growth a small part of the feed rod was broken to make a seed rod, while later the remaining from the previous growth was used. The FZ method consists of melting the bottom end of the feed rod and connecting the molten zone to the feed 27.

(39) 2. Methods. Figure 2.5: The four-mirror image furnace used for the growth of Ca-ruthenate samples.. rod, as shown in Fig. 2.6(a). The hot spot is created at the center of the four-mirror stage, which focuses the light of four halogen lamps, reaching a maximum temperature of 2200 ◦C. As seed and feed elements are lowered at different speeds, new material containing single crystals or, in the best case, a single crystalline rod is grown from the molten zone on top of the seed rod. The growth rate is defined by the speed of the seed rod and is characteristic of the desired final crystal, the higher the symmetry, the higher the speed. During the growth the rods rotate in opposite directions at 400 rpm, to help stabilize the melt. A quartz tube, moreover, is used to keep the growth in partial O2 pressure, to reduce the evaporation of RuO2 . 28.

(40) 2.3 Crystal growth a). b). Figure 2.6: a) The molten zone. b) A typical single crystal obtained from FZ growth.. The FZ technique allowed me to grow high-quality single crystals, shown in Fig. 2.6(b), with minimal contamination thanks to the lack of contact with a crucible during the growth. Phase purity has been confirmed by powder X-ray diffraction measurements.. 29.

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(42) CHAPTER. 3. Results. 3.1. Bulk characterization of doped Ca2 RuO4. Single crystals of Ca2−x Xx RuO4 , with rare earth dopant X = La, Pr, Nd, were grown for nominal doping concentrations x = 0, 0.03 (Pr only), 0.05 (La, Pr, Nd), 0.10 (La, Pr, Nd) and 0.15 (La and Nd only). Unlike Ca and Sr these ions are trivalent, hence donate one electron to the system. We determined the actual doping concentrations by energy dispersive X-ray (EDX) analysis, finding values (x = 0, 0.03, 0.04, 0.07 and 0.11) which are systematically slightly lower than the nominal ones. In this work we will always refer to actual doping concentrations. The bulk transport and magnetic properties of the La-doped system were previously explored by Fukazawa and Maeno [15]. In agreement with their study, we find that light rare earth doping suppresses TM I and stabilizes a metallic paramagnetic ground state for x = 0.11, which is illustrated by the phase diagram shown in Fig. 3.1. As in the parent compound Ca2 RuO4 , the MIT is accompanied by a first order structural phase transition from metallic L-Pbca to insulating S-Pbca phase. Even though the rare earth dopants considered have a smaller ionic radius than Sr (rLa = 1.22 Å, rNd = 1.16 Å, rPr = rCa = 1.18 Å and rSr = 1.31 Å) and therefore are less effective at relaxing the structure of pure Ca2 RuO4 , we find that the doping threshold for the ground state metallicity is lower than in the Sr-doped system (x = 0.2), suggesting that the additional electron plays a role in precipitating the MIT. Magnetic order in the insulating S-Pbca ground state 31.

(43) 3. Results. 400 TMI. T (K). 300 PM - metal L-Pbca. 200. PM - insulator S-Pbca. 100. TN AF - insulator S-Pbca. 0 0.00. 0.02. 0.04. 0.06 0.08 doping x. 0.10. 0.12. Figure 3.1: Phase diagram obtained from the bulk characterization of lighly rare earth X (La, Nd and Pr) doped Ca2−x Xx RuO4 , consistent with earlier studies on Ca2−x Lax RuO4 [15].. is preserved and the Néel temperature does not vary much over the whole doping range. The crystal structure of our samples was measured at 10 K and 300 K by single crystal neutron diffraction using the time-of-flight Laue technique at the SXD diffractometer of the ISIS Neutron and Muon source [75]. Data acquisition and refinement were carried out by our collaborator M. J. Gutmann. The measurements, reported in Appendix A and summarized in Fig. 3.2, show that doping gradually drives the insulating S-Pbca phase towards the metallic L-Pbca structure, while it has little effect on the latter. At 10 K the system is in the S-Pbca phase for all doping concentrations except x = 0.11; at 300 K, conversely, it is in L-Pbca for non-zero doping. Reference values for the L-Pbca structure of pure Ca2 RuO4 at 400 K are taken from the literature [2]. At 10 K the b lattice parameter, displayed in Fig. 3.2(a), decreases for increasing doping and since a is almost unchanged the orthorhombicity 1 − a/b (Fig. 3.2(c)) is also reduced, approaching zero in the quasi-tetragonal L-Pbca phase. The c parameter (Fig. 3.2(b)), in contrast, increases with x, driving the distortion of the RuO6 octahedra 1 − (rx + ry )/2rz , where rx,y and rz indicate the in-plane and apical bond lengths, respectively, from negative to positive values, i.e. from compressive to tensile (Fig. 3.2(d)). Since the octahedral distortion determines the local crystal field at the Ru ion, the reduced compression with increasing doping should suppress the crystal field splitting between the dxy and dxz /dyz orbitals and one may thus expect the orbital polarization found in the S-Pbca phase to be continuously reduced too. We will further discuss this issue later in this chapter. The Ru-O-Ru in-plane bond angle and the 32.

(44) 3.1 Bulk characterization of doped Ca2 RuO4 b). a) La Pr Nd. 10 K. 12.2 c (Å). b (Å). 5.60. 5.50 300 K 5.40 400 K 0.00. 12.0 11.8. 0.04 0.08 doping x. 0.12. 400 K. 300 K 10 K. 0.00. c). 0.04 0.08 doping x. 0.12. 0.04 0.08 doping x. 0.12. 0.04 0.08 doping x. 0.12. d) 0.05 0.03 0.02 0.01. 0.04 10 K. 1-(rx+ry)/2rz. 1-a/b. 0.04. 300 K. 0.00 400 K 0.00. 0.04 0.08 doping x. 0.00 300 K 10 K -0.02. 0.12. 0.00. e). f) 153.0. octahedral tilt (deg). Ru-O-Ru angle (deg). 0.02. 152.0 300 K 151.0 150.0. 10 K. 0.00. 0.04 0.08 doping x. 0.12. 12 10 K 10. 300 K. 8 6 0.00. Figure 3.2: Doping dependence of the structural parameters at 10 K, 300 K (red and black symbols, respectively, from neutron scattering experiments) and 400 K (yellow symbols, from [2]). Shaded blue and white regions represent S-Pbca and L-Pbca phases. Circles, triangles and squares indicate La, Pr and Nd doping, respectively. a), b) Evolution of b and c lattice constants, c) Orthorhombicity 1 − a/b, d) Octahedral distortion 1 − (rx + ry )/2rz , where rx , ry and rz indicate the in-plane and apical Ru-O bond lengths, respectively, e) In-plane Ru-O-Ru angle, d) Tilt angle of the RuO6 octahedra with respect to the c axis.. octahedral tilt, displayed in Fig. 3.2(e) and (f), respectively, quantify the distortion with respect to a perfect tetragonal perovskite structure. In an undistorted system the RuO6 octahedra are not rotated and the Ru-O-Ru angle is 180° while the tilt is 0. Deviations from these values reduce the in-plane overlap of the Ru t2g orbitals and, therefore, the bandwidth. In addition, they promote hybridization between in-plane and out-of-plane orbitals. In summary, our neutron diffraction data show that the doping evolution 33.

(45) 3. Results of the crystal structure is similar for the three dopants considered and is dominated by the doping concentration. The L-Pbca phase found at 300 K in the doped system, furthermore, has a negligible doping dependence. The S-Pbca structure, in contrast, evolves continuously for increasing doping towards the isosymmetric yet less distorted L-Pbca configuration. The transport properties of our crystals were probed by a four-point method in a Physical Properties Measurement System (PPMS). Since the samples tend to shatter as they undergo the first-order structural phase transition, loss of electrical contacts during the measurement was an issue. This was overcome by the use of highly ductile In contacts. Fig. 3.3(a) shows resistivity measurements for three doping concentrations. For the highest doping x = 0.11 the system is metallic in the whole temperature range and the resistivity displays an almost linear temperature dependence (see inset), indicating a deviation from the simple ρ(T ) ∝ T 2 Fermi liquid picture. For x = 0.04 and 0.07 the system undergoes a MIT below ∼ 250 K and ∼ 90 K, respectively. The hysteretic behavior of the MIT shown for x = 0.04 upon warming up (solid line) and cooling down (dashed and dotted line) proves a). b) 1500 10. ρab (μΩ·cm). 10. 9. 10 10 10 10. 10. 10. 500. 8. 10 0. 10. 10. La 0.11. 1000. 200 T (K). 7. ρab (μΩ·cm). 10. ρab (μΩ·cm). 10. 6. Pr 0.07. Pr 0.04. 5. 10 10 10. 4. 10. 3. La 0.11 0. 10 100. 200. 300. T (K). La2-xSrxCuO4 9. Sr2-xLaxIrO4 8. 0. 7. 6. 5. 4. 3. 0.06 0.05 0.1 0.075 0 100. 200. 300. T (K). Figure 3.3: a) Temperature dependence of the resistivity for La x = 0.11 and Pr x = 0.04, 0.07, plotted on a logarithmic scale. Continuous and discontinuous lines indicate acquisition upon warming up and cooling down the sample, respectively. The inset highlights the resistivity for La x = 0.11 on a linear scale, showing a deviation from the ρ ∝ T 2 behavior characteristic of Fermi liquids. b) Resistivity for the lightly doped cuprate La2−x Srx CuO4 (red line), from [76], and iridate Sr2−x Lax IrO4 (green line), from [77].. 34.

(46) 3.1 Bulk characterization of doped Ca2 RuO4 that the transition is first order. For comparison we show in Fig. 3.3(b) the in-plane resistivity for similar doping levels in the cuprate La2−x Srx CuO4 and in the single layer iridate Sr2−x Lax IrO4 reported in Refs. [76, 77]. In the latter systems, carriers rapidly delocalize away from the stoichiometric Mott phase, driving the system towards metallicity. In contrast, doped Ca2 RuO4 shows a pronounced MIT and remains highly insulating at low temperature, with resistivities that are more than five orders of magnitude higher than in cuprates and iridates. This provides compelling evidence for the complete localization of the additional rare earth electrons in the S-Pbca phase of Ca2−x Xx RuO4 . The specific heat of our samples was probed by a relaxation method in a PPMS. Owing to the structural phase transition, the crystals tend to detach from the grease holding them as they are cooled down, making these measurements challenging. Due to this issue we could obtain clean measurements for the x = 0.11 crystals only, as their transition is suppressed by doping. For a metallic solid, the specific heat can be written as the sum of electron and phonon contributions, CP = γT + βT 3 , which are linear and cubic in temperature, respectively. The electronic specific heat coefficient γ in turn is proportional to the density of states at the Fermi level, which gives an indication of the strength of correlations in the system. By plotting the ratio CP /T as a function of T 2 in the range 0 ÷ 25 K (Fig. 3.4) and extracting the offset of the linear fit at T = 0 K we estimate γ ∼ 89 mJ/mol Ru·K2 , more than two times larger than the value of γSr214 ∼ 37.5 mJ/mol Ru·K2 300. 2. CP/T (mJ/mol-Ru·K ). 250 La 0.11 200 150 100 γ ≈ 89 mJ/mol-Ru·K 50 0. 100. 200. 2. 300 2 2 T (K ). 400. 500. 600. Figure 3.4: Specific heat CP divided by T displayed as a function of T 2 for La x = 0.11. The linear fit allows to estimate the electronic specific heat coefficient γ.. 35.

(47) 3. Results reported for Sr2 RuO4 , [78]. This indicates that the metallic ground state for La doping x = 0.11 is strongly correlated, which implies strong renormalization of the quasiparticle mass and velocity induced by electron-electron interactions. The magnetic properties of our crystals were studied by a Superconducting Quantum Interference Device (SQUID) using a Magnetic Properties Measurement System (MPMS). The measurements in zero-field-cooled (ZFC) and field-cooled (FC) sequences obtained upon warming up with a field H = 1000 Oe applied parallel to the ab plane are displayed in Fig. 3.5. For x = 0 doping (Fig. 3.5(a)) we find a magnetic transition at TN = 112 K, india). 1.1 ZFC FC. M (emu/mol-Ru). 1.0. x=0 0.9 0.8 0.7. b). 250. M (emu/mol-Ru). 200. x = 0.04 x = 0.07. 150 100 50 0. M (emu/mol-Ru). c). 6 5 x = 0.11. 4 3 2 1 50. 100. 150 T(K). 200. 250. 300. Figure 3.5: Temperature dependence of the magnetic moment measured in ZFC (dashed and dotted line) and FC (solid line) sequences for a) x = 0, b) Pr x = 0.04 (black curve) and x = 0.07 (red curve), c) La x = 0.11.. 36.