Photoemission study of Bi-cuprate high-t c superconductors in the lightly-doped to underdoped regions

Transcription

1 Photoemission study of Bi-cuprate high-t c superconductors in the lightly-doped to underdoped regions Thesis Kiyohisa Tanaka Department of Physics, University of Tokyo December, 2004

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3 Contents 1 Introduction 1 2 Background Strongly correlated systems High-T c superconductors Crystal structure Phase diagram Electronic structure Doping evolution Transport properties Pseudogap behavior Photoemission Spectroscopy Energetics of photoemission spectroscopy Angle-resolved photoemission spectroscopy Spectral function and Self-energy Experiment Bi2212 samples ARPES experiment Evolution of the Electronic Structure Introduction Experimental Results and discussion Doping evolution of EDC s Rigid-band-like shift Spectral weight mapping in the momentum space Valence-band maximum in k-space Doping evolution and t Conclusion i

4 6 Doping and Temperature Dependence of ARPES Data Introduction Experimental Results and discussion Estimation of the electrical resistivity from ARPES spectra Temperature dependence of the momentum-distributioncurve width in underdoped Bi Momentum-distribution-curve analysis in lightly-doped Bi (Pseudo)gap behavior in spectra at low temperatures Conclusion Effects of next-nearest-neighbor hopping t Introduction Experimental Results and discussion Band dispersion and its width in parent insulators Effects of bilayer splitting in lightly-doped Bi Interpretation of the band structure along the t-t -t -U model Doping evolution of spectra at k=(π, 0) Possible relationship between t and T c,max Conclusion Temperature Dependence of the Chemical Potential Introduction Experiment Results and discussion Temperature dependence of the core-level spectra Temperature dependence of ARPES spectra Evaluation of the chemical potential shift Chemical potential shift compared with thermodynamic properties Comparison with the t-j model calculation Conclusion Summary and Concluding Remarks 85 ii

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6 Chapter 1 Introduction Since the discovery of the high-t c superconductors by Bednorz and Müller [1], enormous number of studies have been performed to understand the mechanism of high-t c superconductivity beyond the conventional BCS theory. The cuprate systems show not only high-t c superconductivity but also show various unusual behaviors in the vicinity of the filling-control Mott metal-insulator transition [2 4]. Unconventional behaviors of highly correlated electron system, in paticular transition-metal oxides, have been extensively investigated in recent years [5]. Understanding the nature of the correlated electron systems has been the most challenging issue in condensed matter physics due to the difficulties inherent in the many-body character of electron correlations. The cuprate systems drastically change their behaviors depending on the electron density in the two-dimensional CuO 2 planes, which is the stage of the high-t c superconductivity and related low-energy physics. The basic behaviors of the CuO 2 plane are common to all the cuprate superconductors. When the Cu d x 2 y 2 orbitals of the CuO 2 plane are half-filled, the system is an antiferromagnetic Mott insulator because of the strong on-site Coulomb repulsion between electrons [2 4]. Removing electrons from the insulating CuO 2 plane may be regarded as hole doping into the Mott insulator. Upon hole doping, the three-dimensional antiferromagnetic order is rapidly destroyed and a transition from the insulator to the superconductor takes place at the hole concentration per Cu of δ With further hole doping, the system becomes superconducting at low temperatures. The maximum T c (T c,max ) is achieved at δ 0.15, referred to as optimal doping, while lower and higher hole doping is referred to as underdoped and overdoped, respectively. While overdoped cuprates behave like a conventional metal above T c, underdoped cuprates in the normal state show behaviors strongly deviated from the standard Fermi-liquid behaviors and thus are commonly called anomalous metals. Angle-resolved photoemission spectroscopy (ARPES) is a powerful tool to 1

7 2 Chapter 1. Introduction probe the electronic structure of the band dispersion directly. Recent development of the ARPES method has enabled us to observe fine structures near the chemical potential (µ). Bi families of cuprates, especially Bi 2 Sr 2 CaCu 2 O 8+y (Bi2212) with its high critical temperature (T c 95 K) and an extremely good cleavage plane, are ideal systems for ARPES. It is also possible to measure the temperature dependence of photoemission spectra from low temperature to room temperature, which is essential to reveal the pseudogap properties, especially in the underdoped region. Therefore, extensive research has been performed and revealed a lot of important electronic properties of cuprates, for example, the d-wave symmetry of the superconducting gap [6], the normalstate gap [7 12] and so on. On the other hand, those measurements on Bi2212 were restricted only around the optimally doped composition (δ= ) because of the difficulty in sample synthesis. The electronic structure of the heavily-underdoped region (δ<0.10), therefore, has not been clarified. Recent progress in sample synthesis with a new approach of substituting La for Sr have systematically produced high-quality single crystals of lightly-doped Bi2212 [13] and it made it possible to study the electronic structure of lightly-doped Bi2212 by photoemission. In this thesis, using higher energy and momentum resolution and improved measurement geometry, we present details of the electronic structure of Bi2212 from the lightly-doped to underdoped regions observed by ARPES. The evolution of the electronic structure with hole doping from the Mott insulator to the superconductor has been a long-standing issue. In contrast to the conventional band picture, the carrier number n is given by the hole concentration δ rather than the band filling 1 δ in the underdoped region [14, 15]. Recent transport studies on lightly-doped cuprates have indicated metallic (dρ/dt > 0) behaviors at high temperatures even though they show localization behavior (dρ/dt < 0) at low temperatures [13, 16, 17]. In order to understand these unconventional behaviors, we have performed the direct observation of the electronic structure by ARPES. It is also a long-standing problem why the different families of cuprates show different T c,max s although they share the common two-dimensional CuO 2 planes and a similar phase diagram. Since the information of the lightly-doped Bi2212 gave us the total understanding of the doping evolution of the electronic structure in Bi2212 from the lightly-doped to overdoped regions, we have compared the electronic structure in Bi2212 with that in La 2 x Sr x CuO 4 (LSCO), whose T c,max is lower than Bi2212. The present thesis is organized as follows. The background for the present study on the high-t c system is described in Chapter 2. The techniques of photoemission spectroscopy is briefly reviewed in Chapter 3. In Chapter 4,

8 the physical properties of Bi2212 measured in this thesis and the experimental setup are described. In Chapter 5, we present the doping evolution of the electronic structure of Bi2212 from the lightly-doped to underdoped regions. Here, we confirm that the different families of cuprates show different doping evolution. In Chapter 6, we focus on the nodal k=(0, 0)-(π, π) direction and present how the observed ARPES spectra describe the unusual transport properties in the lightly-doped region. The antinodal region k (π, 0) will be focused in Chapter 7. In this Chapter, we discuss the origin of the different electronic structure between different families of cuprates, comparing the spectra of Bi2212 with those of LSCO. We introduce the importance of the next-nearest-neighbor hopping parameter t, which may also influence T c. The temperature dependence of the chemical potential is also discussed in Chapter 8. Taking into account those results, summary and concluding remarks are given in Chapter 9. 3

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10 Chapter 2 Background 2.1 Strongly correlated systems The high-temperature (high-t c ) superconductors belong to a large class of materials, that is, strongly correlated materials, which are characterized by strong interactions or correlations between electrons. The history of strongly correlated materials begins in the early days of modern solid state physics. In 1930 s, Bloch [18] and Wilson [19] developed band theory, which explained why some materials exhibit metallic behavior and others insulating. Boer and Verweij [20] soon pointed out that the Bloch and Wilson picture broke down for a large number of insulating 3d transition metal compounds, such as NiO and CoO, which were predicted to be metals. Peierls pointed out that the large local d-d Coulomb interaction between electrons overcame the energy gain by delocalization of the electrons. Mott [2] and Hubbard [3] attributed the insulating behavior to electron-electron correlation. Nowadays, the Hubbard model is frequently used to study strong correlated systems. In the mean time, Anderson [21] introduced super-exchange in a model with a large d-d Coulomb interaction of local 3d electrons caused by the energy lowering via hopping. This model is virtually identical to the Hubbard model. Local Coulomb interaction U plays an important role in another class of phenomena, for example, the Kondo [22] and heavy Fermion phenomena, where magnetic impurities in a metal or even a lattice of the magnetic impurities are concerned. Anderson [23] introduced the famous Anderson Hamiltonian to address these phenomena, where Coulomb interaction between the local d electrons or f electrons in the magnetic impurities plays an important role and determines whether local magnetic moments exist or not. In the large U limit, the Anderson Hamiltonian can be transformed to the Kondo Hamiltonian [24], where only the spin degrees of freedom of the impurities are retained. Coulomb interaction between two electrons, the basic ingredient of these 5

11 6 Chapter 2. Background models, is simple. However, when going from a two-body or three-body problem to a many body problem, correlation effects give rise to a vast number of complex and often surprising phenomena. The biggest challenge in condensed matter physics is to obtain a better understanding of the many body problems and their manifestation in a large variety of phenomena, which include metal insulator transition, insulator-superconductor transition, Kondo effects, heavy Fermion phenomena, high-t c superconducting, mixed valence, quantum Hall effect, colossal magneto-resistance, charge ordering, and so on. Some of them have been interpreted in the framework of the above theoretical models. Only in few cases could be exact solutions, e.g., the exact solution of onedimensional Hubbard model [25], and the single-impurity Kondo model. Yet both solutions are so complicated that even their physical consequences are a subject of intense research. Due to these difficulties, the theoretical paradigm is far from being built in the field of strongly correlated systems. On the other hand, unexpected new discoveries are made quite frequently in this field. 2.2 High-T c superconductors Crystal structure In the last twenty years, following the discovery of superconductivity in La 2 x Ba x CuO 4 by Bednorz and Müller [1], a large number of related compounds with high superconducting transition temperatures have been found. The most common feature of these high-temperature superconductors (HTSC s) is a layeredperovskite crystal structure containing CuO 2 planes separated by block layers. The block layers play a role of a charge reservoir. Electrons are withdrawn from the CuO 2 planes by replacing or adding ions. For example, in the bismuth (Bi) families of cuprates, doping can be controlled by adding extra oxygen in the Bi-O planes, by substituting rare-earth ions for Ca 2+ ions or by substituting La 3+ ions for Sr 2+ ions. Usually the block layers are insulating and have almost no contributions to the low-energy elctronic states. On the other hand, the CuO 2 planes have energy states around the Fermi level (E F ) and thus dominate the low energy physics. Therefore, the CuO 2 planes should be essential to HTSC s and are believed to be the stage of the superconductivity. Each HTSC has a different number of CuO 2 planes. For example, the Bi families of HTSC, Bi 2 Sr 2 CuO 6+δ (Bi2201), Bi 2 Sr 2 CaCu 2 O 8+δ (Bi2212), Bi 2 Sr 2 Ca 2 Cu 3 O 10+δ (Bi2223) have one, two and three CuO 2 layers, respectively as shown in Fig Other families of cuprates, for example, La 2 x Sr x CuO 4 (LSCO) and YBa 2 Cu 3 O 7 δ (YBCO) have one and two CuO 2 planes, respec-

12 2.2. High-T c superconductors 7 Figure 2.1: Crystal structures of bismuth families of high temperature superconductors. tively. In the Bi families of cuprates, the two Bi-O layers are weakly bonded to each other through the Van-der-Waals force, therefore, one can easily cleave samples and the Bi-O layer emerges at the top of the surface plane Phase diagram Figure 2.2 shows a schematic phase diagram of the cuprates. At half filling (hole concentration δ=0), the cuprates are antiferromagnetic insulators (AFI) with a Néel temperature of roughly 300 K. As the number of holes increases, the antiferromagnetic phase is quickly destroyed and the cuprates become strange metals characterized by a pseudogap. This phase with its many anomalous properties is poorly understood. For further hole doping, the metallic phase turns to more conventional behavior expected from Fermi-liquid theory. At low enough temperatures, for 0.07 δ 0.25, the system becomes superconducting. The maximum T c (T c,max ) is achieved at δ 0.15, referred to

13 8 Chapter 2. Background Figure 2.2: A schematic phase diagram of the cuprates. At half filling, the cuprates are antiferromagnetic insulators (AFI). As holes are doped into the CuO 2 plane with increasing δ, the system becomes a strange metal, which has been characterized by many unusual properties. At very large hole doping, more typical metallic properties associated with a Fermi liquid appear. At low temperature, the system shows superconductivity (SC). as optimal doping, while lower and higher hole doping is referred to as underdoped and overdoped, respectively. While the T c,max of different families of cuprates are different, the T c roughly scales with T c,max and follows an universal curve as a function of hole concentration. As shown in Fig. 2.2, the cuprates have two kinds of pseudogaps, namely, the large and small pseudogaps. We will cover this topic in Section Electronic structure In the CuO 2 plane, the d x 2 y 2 band originated from Cu has the highest energy among the five d bands as shown in Fig. 2.3, because the Cu d x 2 y 2 orbital well hybridizes with oxygen p orbitals through the in-plane Cu-O bonds which

14 2.2. High-T c superconductors 9 have shorter length than the out-of-plane Cu-O bonds. Therefore, in the parent compound with nine d electrons (or one d hole), the d x 2 y 2 band is half-filled and each Cu atom has spin 1/2. Since the on-site Coulomb repulsion U between d x 2 y 2 electrons is stronger than the band width W (the kinetic energy of the electrons) in the cuprates, the electrons tend to avoid the double occupancy of the d x 2 y 2 orbital and, at half-filling, the electrons are localized as a Mott insulator. Previous photoemission experiments have shown that the Cu d 8 state is 8 ev below the Cu d 9 L state, where L refers to a hole in the ligand orbital (which in this case are the oxygen orbitals) [26]. U is large compared to the charge-transfer energy, which classifies the parent compounds of the cuprates as charge-transfer type insulators ( <U) rather than Mott-Hubbard type insulators ( >U). According to optical conductivity studies, the chargetransfer gap ( CT ) is about 1.5 ev for undoped parent insulators [15,27]. Thus one cannot neglect the oxygen sites when constructing an effective Hamiltonian to study the low energy physics. A three band Hubbard model was first proposed by Emery [28]. H = ɛ d n d iσ + ɛ p n p jσ + t pd (p jσ d iσ + H.c.) + t pp (p jσ p j σ + H.c.) iσ jσ σ<ij> σ<ij > + U d n d i n d i + U p n p j np j + U pd n d iσn p j σ, (2.1) i j σ<ij> where p and d refer to oxygen p and copper d orbitals, respectively. The three bands in the CuO 2 plane, d x 2 y 2, p x and p y, and their energies relative to the chemical potential µ are illustrated in Fig. 2.4 (a) and (c). Zhang and Rice reduced this Hamiltonian to a more simplified form [29]. They consider a single CuO 4 plaquett containing one or two holes. They found that one hole located at the Cu site is hybridized most strongly with a hole in a molecular orbital of a linear combination of the four surrounding oxygens, forming a singlet, a nonbonding, and a triplet state. The large energy separation between the singlet and the triplet, which has been calculated to be 3.5 ev by Eskes and Sawatzky [30], implied that the triplet state could be projected out from the low energy physics. Thus the p hole in the Zhang-Rice singlet (ZRS) carries another spin 1/2 in the opposite direction to the spin of the d hole of Cu. An effective Hamiltonian can then be constructed to allow the ZRS to hop between the CuO 4 plaquettes over the entire CuO 2 plane. One may regard the ZRS band as the effective lower-hubbard band (LHB) having the d x 2 y 2 symmetry at the Cu site with the effective Coulomb repulsion U eff = CT as illustrated in Fig. 2.4 (b) and (d). If one ignores the upper-hubbard band (UHB), the effective Hamiltonian becomes the t-j model, which was also an effective Hamiltonian for the single Hubbard model. Hereafter, most of discus-

15 10 Chapter 2. Background sion will be based on the one-band model, namely, the single-band Hubbard model or the t-j model. Figure 2.3: Schematic diagram for the energies of Cu 3d and O 2p orbitals in the cuprates. In undoped La 2 CuO 4, for example, the Cu atom is surrounded by an oxygen octahedron elongated along the z-direction [Fig. 2.1]. (a) (b) p x d x 2-y2 d x 2-y2 Zhang-Rice singlet p y (c) (d) Cu 3d x 2-y2 Cu 3d x 2-y2 UHB W UHB µ CT U eff µ ZRS O 2p U (LHB) Cu 3d x 2-y2 LHB Figure 2.4: Schematic pictures for the electronic orbitals (a)(b) and the electronic structure (c)(d) of the CuO 2 plane. Three bands, Cu 3 d x 2 y 2,O2p x and 2p y, are considered in (a) and (c). Only the d x 2 y 2 band at the Cu site is considered in (b) and (d).

16 2.2. High-T c superconductors 11 Mott-Hubbard type insulator Charge-transfer type insulator Figure 2.5: Effect of carrier doping for the three kinds of insulators: A band insulator, a Mott-Hubbard type insulator and a charge-transfer type insulator. The electron removal spectra (photoemission) are indicated by the shaded regions. The top shows the undoped insulator while the middle and bottom spectra are for hole doping and electron doping, respectively. Spectral weight for single hole or electron doping is also shown Doping evolution Although the Hubbard models correctly predict the antiferromagnetic insulating state in undoped parent insulators, it is unclear how the Mott insulator evolves into the metallic state. Figure 2.5 shows the doping evolution of the electronic structure for a band insulator, a Mott-Hubbard type insulator and a charge-transfer type insulator. In the band insulator case with either hole or electron doping, the chemical potential is simply shifted according to the number of added holes or electrons. The Mott-Hubbard type insulator is similar, however, there is a significant difference. When a hole is doped into the system, it removes one doubly occupied state, which means the removal of the state both from the UHB and LHB. Hence, the spectral weight is transferred from both the UHB and LHB as the chemical potential is shifted to the LHB. The situation is analogous when electron is doped into the system. For a charge-transfer type insulator, the electron doping evolution remains the same, although it should be noted that the spectral weight still comes from the LHB, not from the oxygen p band. In the case of hole doping of the p- d hybridization is negligibly weak, an occupied state in the oxygen p band simply becomes unoccupied, which is what happened in the case of the band

17 12 Chapter 2. Background Figure 2.6: Two possible evolutions of the chemical potential upon hole doping. (a): Undoped Mott insulator. (b): Chemical potential is pinned. (c): Chemical potential is shifted upon doping. In either case, spectral weight is transferred from the UHB and ZRS to the states near the chemical potential. insulator. As the hybridization is increased, the ZRS band is formed above the oxygen p band and the spectral weight transfer occurs in a similar way as the p-d hybridization is negligibly weak, leading to a behavior similar to one-band Hubbard model. The above description, however, fails to capture another possibility which appears to have been realized in cuprates. Namely, in the case of LSCO, it is believed that the chemical potential remains fixed inside the gap upon doping [31]. These states may be related to stripe formation. The additional scenario along with the case described avobe is shown in Fig. 2.6 (b). Here, upon doping states are created which fix the chemical potential inside the gap. This is a strongly differing view from the case where the chemical potential shifts upon doping (Fig. 2.6 (c)) Transport properties The transport properties of the cuprates are unusual from the viewpoint of standard Fermi-liquid theory. One of the most unusual properties is the temperature dependence of the electrical resistivity. While Fermi-liquid theory predicts that the temperature dependence of the resistivity ρ is given by ρ = ρ 0 +AT 2, an optimally doped δ 0.15 sample shows a robust T -linear dependence, ρ ρ 0 + AT, in a wide temperature range as shown in Fig. 2.7 [32].

18 2.2. High-T c superconductors 13 Figure 2.7: Temperature dependence of the in-plane resistivity for various doping concentrations in LSCO. T -linear resistivity is seen in a wide temperature region around optimally doped (x 0.15) sample [32]. This T -linear behavior is widely observed in other high-t c cuprates around optimum doping. In the underdoped region, ρ at low temperatures eventually shows insulating behavior with a logarithmic T dependence, particularly when the superconductivity is suppressed by a high magnetic field [33]. Recently, the electronic properties of lightly-doped cuprates have attracted much interest, since LSCO and YBCO surprisingly show metallic behavior (dρ/dt > 0) in the in-plane resistivity near room temperature in the lightly-doped region even down to the extremely light doping limit inside the antiferromagnetic insulating phase, even though the in-plane resistivity shows localization behavior at low temperature. It should also be noted that the magnitude of the resistivity at the metallic region is far above the Ioffe-Regel limit (as shown in Fig. 2.8) [16, 17] Pseudogap behavior As shown in Fig. 2.9, the underdoped cuprates show characteristic temperatures well above T c : T χ below which the magnetic susceptibility χ(t ) decreases from the maximum value [34,35], T H below which the inverse of the Hall coefficient 1/R H (T ) is reduced from the temperature independent value deviates downwards from the linear behavior extrapolated from the high-temperature data [36]. These characteristic temperatures have a similar value, which is much higher than T c.asδdecreases towards the AFI, the characteristic temperatures increase, namely, the strange metallic region grows wider. This

19 14 Chapter 2. Background Figure 2.8: Temperature dependence of the in-plane resistivity for lightly-doped LSCO and YBCO [16]. energy scale is associated with the opening of a large pseudo-gap PG, which has been observed in angle-integrated photoemission spectra as shown in Fig. 2.9 (d) [37]. On the other hand, at lower temperatures near T c, remarkable anomalies are observed in the underdoped region. One is the spin-gap behavior observed by nuclear magnetic resonance (NMR) for underdoped YBCO [38, 39]. Another obvious anomaly is the normal-state gap behavior observed in the ARPES spectra of underdoped Bi2212 (Fig (b)), which is in other word called small pseudogap [7 12]. While the superconducting gap on the Fermi surface near (π, 0) disappears above T c for an overdoped sample, the energy gap survives as a pseudogap even at a temperature in the normal state for the underdoped sample. A similar behavior of the density of state also has been reported in tunneling spectra [40]. It has also been reported that, below a temperature somewhat above T c, the electrical resistivity ρ(t ) and the uniform magnetic susceptibility χ(t ) slightly deviate downwards from the T -liner extrapolation from higher temperatures as shown in Fig (a)(c)(d) [34,35]. All these anomalies have similar crossover temperatures, which decrease as the hole concentration increases and, in some systems, appear to merge into T c in the overdoped region (x >0.2). This small pseudogap seems to scale with T c,max (Fig (e)) in Bi2212 and LSCO [41]. It is often said that the underdoped region is the key to reveal the high-t c mechanism because of its unfamiliar behavior and that the existence of the pseudogap is strongly related to the high-t c mechanism.

20 2.2. High-T c superconductors 15 (a) (b) (c) (d) Figure 2.9: Temperature dependence of the magnetic susceptibility in LSCO [34] (a), and in Bi2212 [35] (b), and of the Hall cofficient in LSCO [36] (c), showing characteristic temperatures much higher than T c. Doping dependence of angleintegrated photoemission spectra [37] is shown in (d). The large pseudogap energy PG is determined by the binding energy from which the spectra is suppressed toward µ.

21 16 Chapter 2. Background (a) (b) (c) (e) (d) Figure 2.10: Reduction in the electrical resistivity ρ(t ) in Bi2212 (a) and in LSCO (c), and the uniform magnetic susceptibility χ(t ) in LSCO (d) starting at somewhat above T c [34, 35]. Panel (b) shows the normal-state gap behavior observed in the ARPES spectra for underdoped Bi2212 [7]. The small pseudogap in Bi2212 and LSCO determined from various kinds of experiments are summarized in panel (e) [41].

22 Chapter 3 Photoemission Spectroscopy The technique of photoemission spectroscopy has been widely used as one of the most suitable methods by which the electronic structure of solids can be revealed. In this section, we will summarize the principles of this method [42 44]. 3.1 Energetics of photoemission spectroscopy Photoemission spectroscopy (PES) is a powerful tool to directly investigate the electronic structure of solids. Photoemission is the phenomenon that a solid irradiated by light emits electrons. Knowing the kinetic energy E v kin of the emitted electron in the vacuum, one can deduce how strong the electron was bound in the solids. Owing to the energy conservation law, E v kin = hν E B Φ, (3.1) where hν is the energy of the incident photons, E B is the binding energy relative to the Fermi level E F (chemical potential µ) and Φ is a work function of the material under study. The work function Φ is the energy required for the electron to escape from the solid through the surface and to reach the vacuum level E vac, that is, Φ = E vac E F. In practical PES experiments, since both the sample and the electron energy analyzer are grounded, the measured kinetic energy E kin of the emitted electron is referred to E F. Then, we obtain E kin = hν E B. (3.2) In the mean-field (Hartree-Fock) picture, the energy E B is approximately regarded as the energy ε of the electron inside the solid before it is emitted. Therefore, the energy distribution of the electrons inside the solid can be directly mapped by the distribution of the kinetic energies of photoelectrons 17

23 18 Chapter 3. Photoemission Spectroscopy E E kin e- e- Electron Energy Analyzer Kinetic Energy v E kin Photoemission Spectrum E kin E vac E F, µ Energy hν Valence Band hν Intensity Φ E B hν Core Levels Density of States (DOS) Figure 3.1: Schematic diagram of the principle of photoemission spectroscopy [45]. emitted with monochromatic incident photons. Figure. 3.1 diagrammatically shows how the electronic density of state (DOS) is mapped by the electronic distribution curve (EDC), namely, the photoemission spectrum. In this thesis, photoemission spectra will be displayed so that the energy relative to E F, E E B, (3.3) is taken for the horizontal axis. In real many-electron systems, the photoemission spectra are affected by the relaxation of the whole electron system in the solid, in addition to the one-electron energy ε in the simple view (frozen-orbital approximation). In response to the hole produced by the photoemission process, the surrounding

24 3.2. Angle-resolved photoemission spectroscopy 19 electrons tend to screen the hole to lower the total energy of the system. Therefore, considering the whole electron system, the binding energy E B is given by the energy difference between the N-electron initial state Ei N and the (N 1)- electron final state E N 1 f as E B µ = E N 1 f E N i (3.4) In other words, E B is the energy of the hole produced by the photoemission process, including the relaxation energy of the total electron system. Hence, much information about the electron correlations can be derived by analyzing the photoemission spectra. 3.2 Angle-resolved photoemission spectroscopy In angle-resolved photoemission spectroscopy (ARPES), not only the energies but also the momenta of electrons in the materials are probed. First, upon the photoexitation process, the crystal momentum should be conserved. Since the momentum of the ultraviolet light hν/c is negligible compared to the size of the Brillouin zone, we obtain K = k + G, (3.5) where k and K are the crystal momenta of the electron in the initial and final states, respectively, and G =(2n x π/a, 2n y π/b, 2n z π/c) is an arbitrary reciprocal lattice vector. Next, when the photoelectron escapes from the solid to the vacuum, part of the momentum perpendicular to the surface is lost due to the finite work function Φ, whereas the crystal momentum parallel to the surface is conserved owing to the translational symmetry along the surface. Therefore, if the momentum of the electron outside the vacuum is denoted by p, p / = K + G. (3.6) In the ARPES experiments, the kinetic energy E kin and the direction of the momentum, θ and φ, of photoelectrons are measured using an angle-resolved electron energy analyzer. Therefore, we obtain the parallel momentum of the emitted electron p =(p x,p y )as p x = 2m e Ekin v cos φ sin θ, (3.7) p y = 2m e Ekin v sin φ. (3.8) Summarizing Eqs. (3.2), (3.5), (3.6), (3.7) and (3.8), we can directly observe both the energy E E B and the parallel momentum k of the hole produced

25 20 Chapter 3. Photoemission Spectroscopy in the sample by the photoemission process as E = E kin hν, (3.9) 2me Ekin V k x = cos φ sin θ + 2n xπ a, (3.10) 2me Ekin V k y = sin φ + 2n yπ, (3.11) b where n x and n y are integers. Figure 3.2 schematically illustrates the procedure for the band mapping by ARPES. If the material under study is a two-dimensional system such as a high-t c cuprate, E and k yield enough information to map the energy-momentum dispersion of the band structure. Kinetic Energy, E kin ARPES spectra k 6 k 5 k 4 k 3 k 2 k 1 E v kin = -E B + hν E kin = E + hν E vac E F Energy hν hν hν hν k 1 k 2 k 3 k 4 k 5 k 6 Momentum Figure 3.2: Schematic diagram showing the principle of angle-resolved photoemission spectroscopy (ARPES). The band dispersions in the material are directly mapped by the ARPES spectra [45].

26 3.3. Spectral function and Self-energy Spectral function and Self-energy In this section, the relationship between the photoemission spectrum and the retarded Green function and the details of the momentum distribution curve (MDC) are reviewed [44,46]. The retarded Green function at zero temperature is given by, G s (r, r,t t ) = { i 0 {ϕs (r,t),ϕ s(r,t )} 0 for t t 0 for t<t, (3.12) where s stands for the spin variable, 0 represents the ground states and {A, B} = AB + BA. The Fourier transform of the above equation gives G s (k,t t ) = 1 drdr G s (r, r,t t )e ik (r r ) V = i 0 a k,s (t),a k,s (t ) 0. (3.13) Here, we define n as an excited state of the system and E n as the energy of the state. From the completeness of the eigenfunctions n n n =1,we obtain for t t G s (k,t t ) = i n = i n 0 a k,s (t) n n a k,s (t ) 0 i 0 a k,s (t ) n n a k,s (t) 0 n n a k,s 0 2 e i (En E 0 )(t t ) i n a k,s 0 2 e i (E 0 En)(t t ). n (3.14) Fourier-transforming the above equation with respect to the single-particle energy ω, we have G s (k,ω) = 1 ωτ i dτg s (k,τ)e 0 = i n a k,s 0 2 dτe i (ω+en E 0 )τ n 0 i n a k,s 0 2 dτe i (ω+e 0 En)τ. (3.15) Then, the imaginary part becomes n 0 1 π ImG s(k,ω) = n n a k,s 0 2 δ(ω E n + E 0 ) + n 0 a k,s n 2 δ(ω E 0 + E n ) = A ω E F s (k,ω)+a ω E F s (k,ω). (3.16)

27 22 Chapter 3. Photoemission Spectroscopy The first term expresses the spectrum above the Fermi energy, which is the (angle-resolved) inverse-photoemission spectrum (IPES) and the second term is the (angle-resolved) PES. G s (k,ω) can be given using the self-energy Σ(k,ω) which renormalizes all interaction between particles, 1 G s (k,ω)= (3.17) ω ɛ 0 k Σ(k,ω). Thus the photoemission spectrum is given by A(k,ω) = 1 π ImG(k,ω) = 1 π ImΣ(k,ω) (ω ɛ 0 k ReΣ(k,ω))2 + (ImΣ(k,ω)) 2. (3.18) The real part of the pole of G(k,ω), ω = Ek is determined by the equation and the residue of the pole Z k is given by ( Z k = 1 ReΣ(k,ω) ω E k ɛ 0 k ReΣ(k,ω)=0, (3.19) ω=e k Near ω = Ek, we can expand Eq.(3.19) as follows, Therefore, Eq.(3.18) is written as ) 1. (3.20) ω ɛ 0 k ReΣ(k,ω) 1 Z k (ω E k). (3.21) A(k,ω) = Z k π Z k ImΣ(k,ω) (ω E k )2 +(Z k ImΣ(k,ω)) 2. (3.22) In the vicinity of E F, Ek can be written as E k = v k (k k F ), where vk ( E k ) is the renormalized Fermi velocity and k is taken perpendicular to the Fermi surface. Then, the momentum distribution curve (MDC) at the Fermi level (ω = 0) is given by A(k, 0) = Z k/vk Z k ImΣ(k, 0)/vk. (3.23) π (k k F ) 2 +(Z k ImΣ(k, 0)/vk )2 Thus, the MDC is given as a Lorentzian with a full width at half maximum (FWHM) of k =2 Z k ImΣ/vk, if the k-dependence of Z k,imσ(k, 0) and vk can be neglected. Since the inverse life time of the quasi-particle is given by 1/τ k = 2Z k ImΣ, 1/ k represents the mean free path l k : l k = v kτ k = 1 k. (3.24)

28 3.3. Spectral function and Self-energy 23 (0,π) (π,π) (c) MDC A(k, ω=0) k (0,0) (π,0) Energy relative to E F (ev) (b) EDC A(k=k F, ω) δ = (a) (d) Intensity (arb. units) k x (=k y ) (π/a) k Figure 3.3: Data taken in the angle mode of a Scienta analyzer. (a): Intensity plot. (b): Energy distribution curve (EDC) at k=k F. (c): Momentum distribution curve (MDC) at E F. (d): Energy dependence of the MDC width derived by a fitting of MDC s to a Lorentzian. Figure 3. 3 summarizes the relationship between the obtained ARPES spectra and momentum distribution curve (MDC) and energy distribution curve (EDC). This relationship shall be used for the analysis of ARPES spectra in Chapter 6.

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30 Chapter 4 Experiment 4.1 Bi2212 samples As mentioned in Chapter 2, different families of cuprates behave in different ways. While they all share the common feature of the CuO 2 planes, each shows its unique physical properties. Bi families of cuprates, especially Bi 2 Sr 2 CaCu 2 O 8 (Bi2212) with its high critical temperature (T c 95 K) and an extremely good cleavage plane, are ideal systems for angle-resolved photoemission spectroscopy (ARPES). In fact, intense research has been performed so far and revealed a lot of important electronic properties of the cuprates, for example, the d-wave symmetry of the superconducting gap [6], the normal-state gap [7 12] and so on. It is also possible to measure the temperature dependence of photoemission spectra from low temperatures to room temperature, which is useful to study the pseudogap properties, especially in the underdoped region. However, those measurement were restricted only near optimally doping because of the difficulty in sample synthesis. Recent progress in sample synthesis by Fujii et al. [13], with a new approach of substituting La for Sr, produced high-quality single crystals, which show metallic electrical resistivity around room temperature and localization behavior at low temperature, like La 2 x Sr x CuO 4 (LSCO) and YBa 2 Cu 3 O 7 δ (YBCO) [16,17]. Here, the physical properties of Bi2212 from lightly-doped to slightly overdoped measured in this thesis are summarized. As mentioned above, highquality single crystals of Bi 2 Sr 2 x La x CaCu 2 O 8+y have been systematically grown by the traveling-solvent floating-zone (TSFZ) method and carrier concentration was controlled by x, the concentration ratio of Sr 2+ and La 3+ [13]. The doping dependence of the in-plane thermopower is shown in Fig. 4.1 (a). The absolute value of the thermopower decreases with decreasing x, which confirms that the carriers are actually introduced into the CuO 2 planes. The hole concentration (δ), which was estimated using the empirical relationship between 25

31 26 Chapter 4. Experiment 160 (a) 1000 (b) Thermo power (µv/k) δ = 0.02 δ = 0.03 δ = 0.05 δ = Temperature (K) Resistivity (Ωcm) δ = 0.02 δ = 0.03 δ = Temperature (K) (a): In-plane ther- Figure 4.1: Transport properties of Bi 2 Sr 2 x La x CaCu 2 O 8+y. mopower. (b): In-plane electrical resistivity. Figure 4.2: Temperature dependence of the Hall coefficient for Bi 2 (Sr,Ca,La) 3 Cu 2 O 8+y (δ=0.02) and Bi 2 (Sr,La) 2 CaCu 2 O 8+y (δ=0.04) samples [13]. δ and the room-temperature thermopower [47], varies from δ=0.02 (x=0.8) to 0.06 (x=0.6). The carrier numbers estimated from the Hall coefficient (shown in Fig. 4.2) also indicate similar values. Figure 4.1 (b) shows the temperature dependence of the in-plane electrical resistivity for lightly-doped Bi2212. The δ 0.03 samples shows metallic transport behavior (dρ/dt > 0) at room temperature like lightly-doped LSCO and YBCO [16, 17]. This metallic transport can be achieved by removing the disorder effect from the Ca site. On the other hand the δ=0.02 sample shows insulating transport behavior (dρ/dt < 0) at

32 4.1. Bi2212 samples 27 (a) (b) (c) Figure 4.3: Transport properties of Bi 2 Sr 2 Ca 1 x R x Cu 2 O 8. (a): In-plane resistivity. (b): In-plane thermopower. (c): T c and the Néel temperature [48]. room temperature, which is caused by the disorder effect from excess La substituted for Ca site. All the lightly-doped Bi2212 samples show localization behavior at low temperatures. In this thesis, we also have measured single crystals of Bi 2 Sr 2 Ca 1 x R x Cu 2 O 8+y (R = Pr, Er) and Bi 1.2 Pb 0.8 Sr 2 ErCu 2 O 8, which were grown by the self-flux method. Details of the growth conditions and characterization are described in [48, 49]. In these systems, substitution of trivalent rare-earth ions such as

33 28 Chapter 4. Experiment (a) (b) Figure 4.4: Transport properties of Bi 2 x Pb x Sr 2 ErCu 2 O 8. (a): In-plane resistivity. (b): In-plane thermopower [49]. Pr, Er and Y for the divalent Ca ion changes the Cu valence, and the hole concentration can be controlled from the lightly-doped to the slightly overdoped region by changing the concentration of the rare-earth ions. In Fig. 4.3, the in-plane electrical resistivity, thermopower and T c s in Bi 2 Sr 2 Ca 1 x R x Cu 2 O 8+y (R = Pr, Er) are shown. The monotonic change in thermopower indicates that the doping level is systematically controlled. Figure 4.4 shows the in-plane electrical resistivity and thermopower in Bi 2 x Pb x Sr 2 ErCu 2 O 8. It is clear that the in-plane resistivity always shows insulating behavior (dρ/dt < 0) even at the vicinity of insulator-superconductor transition in Bi 2 Sr 2 Ca 1 x R x Cu 2 O 8+y (R = Pr, Er) and Bi 2 x Pb x Sr 2 ErCu 2 O 8 in the non-superconducting samples. This can be caused by the strong disorder effect from rare-earth ions substituted for Ca site. 4.2 ARPES experiment The angle-resolved photoemission spectroscopy (ARPES) measurements in this thesis were performed at beamline 5-4 of Stanford Synchrotron Radiation Laboratory (SSRL). Here, we briefly describe the beamline and the measurement system. Beamline 5-4 is an undulator beamline with a normal incidence monochromator (NIM) as shown in Fig The NIM has advantage over incidence monochromators for photons in the vacuum ultraviolet (VUV) range light. NIM monochromators have much higher throughput for low energy pho-

34 4.2. ARPES experiment 29 Undulator Horizontal focusing mirror Vertical condensing mirror Entrance slit Monochrometer e- Exit slit Sample Vertical deflecting mirror Horizontal refocusing mirror Figure 4.5: Schematic layout of beamline 5-4 at SSRL. Manipulator Characterization chamber Scienta analyzer Beam in ARPES chamber Figure 4.6: Overview of the measurement system at beamline 5-4. tons. This gain is countered by the low reflectivity for higher photon energies for normal incidence. For this reason, the NIM monochromator at beamline 5-4 is also limited to the photon energy range ev. This is advantageous for most of high-t c materials as they have large photo-ionization cross-sections in this energy range. In typical operating configurations, the energy resolution

35 30 Chapter 4. Experiment of photons is on the order of few mev, which can be measured by a gas-phase cell. The endstation of beamline 5-4 is composed of a characterization chamber and an ARPES measurement chamber as shown in Fig The samples were cleaved in the characterization chamber. A low energy electron diffraction (LEED) system is installed in the characterization chamber for checking the qualities and atomic structures of the sample surfaces. The cleaved sample can be easily transferred to the chamber for angle-resolved photoemission measurements. The manipulator has five degrees of freedom for sample motion: three translational and two rotational ones. In the ARPES chamber a hemispherical analyzer Gamma-data Scienta SES200 with a two dimensional multichannel plate detection system is attached. The advantage of this analyzer is an ability to run in the so-called angle mode. In this mode, the voltages of the electron lens are configured such that instead of focusing the real-space image of the sample on the entrance slit of the analyzer, the lens focuses image from the infinity distance on the entrance slit. In this way, all electrons with parallel trajectories are collimated to a specific point on the entrance slit and thereby a specific point on the multichannel detector. Then we can get the angle-resolved spectra. This mode has the advantage of collecting the maximum number of photoemitted electrons in a certain angular range and increasing the energy resolution significantly. In this mode, the angular range is usually set to ±7, and the energy resolution of analyzer is set to 10 mev, as measured by gold Fermi edge. Figure 4.7 shows the experimental geometry used in the present measurements of the sample surface and the electron energy analyzer. One can change the emission angle by rotating the sample with two axes (θ and φ), while the analyzer position is fixed. It should be noted that the Bi2212 system has some drawbacks for photoemission measurements, namely, the bilayer splitting (shown in Fig. 4.8) [50 53] caused by the hopping between two CuO 2 planes and the superstructure caused by the structural modulation of the Bi-O layers [54]. As we will discuss in Chapter 7 and Appendix A, we found that ARPES spectra at k (π, 0), which are expected to show the strongest effect of bilayer splitting as well as strong photon energy dependence of the relative intensities of the split peaks, showed almost no photon energy dependence. Therefore, we concluded that the problem of the bilayer splitting in lightly-doped Bi2212 was not so serious in studying the electronic structure. This smaller effect of bilayer splitting in the lightly-doped region than the superconducting region may be caused by suppressed interlayer hopping due to the smaller number of carriers. As for the superstructure, we have avoided its effect by choosing a suitable

36 4.2. ARPES experiment 31 z xyz axis for chamber xyz axis for sample surface φ sample surface x θ φ θ y Electron energy analyzer Syncrotron radiation E hν Figure 4.7: Experimental geometry of ARPES measurement between sample surface, the analyzer and the polarization vector. Figure 4.8: ARPES studies of overdoped Bi2212 [50]. (a): Fermi surface mapping. (b): EDC s along the cut indicated by the arrow in (a).

37 32 Chapter 4. Experiment experimental setup. In the previous ARPES studies of Bi2212, the ghost images of the Fermi surface were reported [51, 55 57]. It occurs because of the diffraction of the outgoing photoelectrons by the modulation of k=±(0.21π, 0.21π) in the Bi-O layers. Figure 4.9 shows the Fermi surface of Bi2212 plotted in black lines and the diffraction replica caused by the superstructure plotted in blue lines in the momentum space. The crystal momentum k=(k x,k y )is shown in units of 1/a, where a is twice the Cu-O bond length within the CuO 2 plane. Here, the extended zone scheme is adopted. The first Brillouin zone (BZ) is surrounded by the bold black lines. In this thesis, in order to enhance the matrix elements as shown in Appendix A, we have mainly studied ARPES spectra along the Fermi surface in the second BZ plotted in a bold red line. We have measured the shaded region where the diffraction replica does not overlap the main band, and have successfully measured ARPES spectra along the Fermi surface. It should be noted that since we have measured ARPES spectra in the second BZ, the (k x,k y )=(2π, 0)-(π, π) direction in the second BZ is the nodal direction. It is known that the structural modulation of the Bi-O layers can be suppressed by substituting Pb for Bi. Figure 4.10(a) and (b) shows the Laue patterns of the Bi 2 Sr 2 ErCu 2 O 8 (Pb-free) and Bi 1.2 Pb 0.8 Sr 2 ErCu 2 O 8 (Pb-doped) samples. In Bi 2 Sr 2 ErCu 2 O 8, one can see 4-fold like symmetry, but there are many satellite spots and 4-fold symmetry is blurred. This observation can be Figure 4.9: Fermi surface of Bi2212 plotted in black lines and the diffraction replica caused by the superstructure plotted in blue lines in the momentum space. ARPES spectra were collected in the shaded region.

38 4.2. ARPES experiment 33 Figure 4.10: Laue patterns of Bi 2 Sr 2 ErCu 2 O 8 (a) and Bi 1.2 Pb 0.8 Sr 2 ErCu 2 O 8 (b). regarded as the effect of the modulated structure. (The vertical direction of Fig. 4.10(a) indicates the ΓY direction.) In Bi 1.2 Pb 0.8 Sr 2 ErCu 2 O 8, the Laue pattern shows a clear 4-fold symmetry. The disappearance of the satellite spots implies that the superstructure modulation has vanished. The rotations of all samples were defined by Laue measurements before ARPES measurements.

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40 Chapter 5 Evolution of the Electronic Structure from the Insulator to the Superconductor in Bi2212 Cuprates 5.1 Introduction The question of how the electronic structure of high-t c cuprates evolves from the Mott insulator to the superconductor with hole doping is one of the most fundamental, important issues in condensed matter physics. This subject has been investigated experimentally and theoretically, but still remains highly controversial. Previous angle-resolved photoemission spectroscopy (ARPES) studies have revealed two different cases. In underdoped La 2 x Sr x CuO 4 (LSCO), a two-component electronic structure has been observed, that is, upon hole doping in-gap states appears primarily well ( 0.4 ev) above the lower Hubbard band (LHB), the chemical potential stays in the in-gap-state region and spectral weight is transferred from the LHB to the in-gap states for further hole doping. [31, 58,59]. Already in the lightly-doped region, a weak quasiparticle (QP) peak crosses the Fermi level (E F ) in the (0,0)-(π, π) nodal direction and is responsible for the metallic transport behavior [59]. ARPES spectra of underdoped Ca 2 x Na x CuO 2 Cl 2 (Na-CCOC), on the other hand, have shown that upon hole doping the chemical potential moves to the top of the LHB and continues to shift downward for further hole doping [60]. As for the Bi 2 Sr 2 CaCu 2 O 8 (Bi2212) family of compounds, the apparently smooth evolution of ARPES spectra in the (π, 0) region in a combined plot of undoped Ca 2 CuO 2 Cl 2 (CCOC) and underdoped Bi2212 implies that a behaviour similar to Na-CCOC is expected for Bi2212 [61]. A core-level pho- 35