Real Space Periodic Electron Density Modulations and Vortex Core Spectroscopy in Heavily Overdoped Bi2Sr2CaCu2O8+δ
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(2) UNIVERSITÉ DE GENÈVE Département de Physique de la Matière Quantique. FACULTÉ DES SCIENCES Professeur Ch. Renner. Real Space Periodic Electron Density Modulations and Vortex Core Spectroscopy in Heavily Overdoped Bi2Sr2CaCu2O8+δ. THÈSE présentée aux Facultés des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention Physique. par. Tim Gazdić de Ljubljana (Slovènie). Thèse No 5494. GENÈVE Atelier d'Impression ReproMail 2020.
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(4) Résumé La supraconductivité, disparition de toute résistance électrique audessous d’un seuil de température (Tc ), est un phénomène connu depuis plus de 100 ans. Lorsque la théorie BCS a été en mesure de décrire les mécanismes microscopiques de cet état, les espoirs d’exploiter la supraconductivité dans une large gamme d’applications courantes ont été quelque peu refroidis, puisque les modèles prédisaient que la température critique ne pouvait guère excéder une vingtaine de Kelvins. La découverte en 1986 de composés devenant supraconducteurs à des températures bien supérieures à cette limite théorique est apparue comme une énorme surprise dans le monde scientifique. Ces nouveaux matériaux, dont la caractéristique commune est la présence de plans d’oxyde de cuivre dans la structure cristalline (cuprates), ont rapidement révélé des températures critiques supérieures à 100 K. De telles températures excédant largement les prédictions théoriques de la théorie BCS, un énorme effort a été fourni par la communauté dans le but de présenter des théories alternatives capables de décrire au mieux la supraconductivité dans ces matériaux. Le Bi2 Sr2 CaCu2 O8+δ (Bi2212) est l’un des membres de cette famille des cuprates. Son état stochiométrique de base est d’être un isolant électrique, mais il devient (supra-)conducteur lorsque l’on dope les plans d’oxyde de cuivre en y introduisant des lacunes électroniques (trous). Lorsque la densité de ces trous augmente, diverses phases électroniques apparaissent et viennent cohabiter avec la supraconductivité. La nature exacte de la relation entre ces phases et la supraconductivité fait actuellement l’objet d’intenses débats. La température critique maximale du Bi2212 est de l’ordre de 90 K au dopage optimal de 0.16 trous par cellule unité (p=0.16). Au-delà de ce seuil, dans le régime de surdopage, il est attendu que ces divers ordres électroniques s’estompent et laissent une certaine ”liberté” à la supraconductivité. En régime de très forts sur-dopages, les cuprates se comportent pratiquement comme des liquides de Fermi conventionnels, en opposition à leur comporte3.
(5) ment de métal étrange manifesté aux dopages plus modérés. Le défi majeur de cette étude est de tenter de comprendre ce régime de dopage extrême, et déterminer si la supraconductivité y revêt une forme plus conventionnelle que dans la phase sous-dopée. Nous utilisons ici la technique de microscopie à effet tunnel (STM) pour étudier le comportement de la supraconductivité et des phases électroniques alternatives. En offrant la possibilité de sonder la densité d’états électroniques locale (LDOS) avec une résolution spatiale de l’ordre des distances atomiques, le STM est l’outil idéal pour observer les interactions entre les ordres électroniques induits par le dopage et la supraconductivité. Cette étude se focalise sur le Bi2212 extrêmement sur-dopé (p>0.19), où les signaux électroniques provenant de la phase supraconductrice sont très clairement identifiables. A notre étonnement, nous avons détecté deux ordres de charges statiques (non-dispersifs), similaires à ceux qui proviendraient d’une onde de densité de charges, pourtant supposée absente en régime de fort sur-dopage. Une analyse détaillée de nos données révèle que l’un de ces ordres peut être expliqué de façon précise par un modèle d’interférences de quasi-particules (QPI) dans un milieu inhomogène, en accord complet avec la théorie BCS, et ceci malgré l’absence d’une claire dispersion des états électroniques. La seconde modulation statique n’est détectée que dans des zones localement confinées, et révèle des signatures spatiales et électroniques similaires à celles observées dans les régimes très sous-dopés. Certaines indications suggèrent que ces domaines se développent dans des régions où la supraconductivité reste celle d’une phase sur-dopée. L’origine de leur présence n’est pas expliquée, et mériterait une attention toute spécifique dans de futures études. Un autre résultat remarquable de cette thèse concerne les signatures électroniques des cœurs de vortex, une particularité des supraconducteurs de type II plongés dans un champ magnétique. L’une des prédictions les plus marquantes de la théorie BCS est l’existence d’un pic de conductance à énergie nulle (ZBCP) au centre d’un vortex. Ce pic se divise en deux structures dont la position en énergie augmente à mesure que l’on s’éloigne du centre du cœur. Une telle signature n’avait encore jamais été directement observée dans un vortex de Bi2212, où seuls des états à basse énergie (SGS) et des modulations de conductance avaient déjà été détectés. Nous démontrons que sous champ magnétique intense, les vortex mesurés dans les composés sur-dopés ne distinguent en rien de ceux étudiés aux dopages plus modérés. Les modulations de conductance montrent une dispersion énergétique en complet accord avec une interférence QPI, et infirme l’hypothèse d’une onde de densité 4.
(6) de charges dans les vortex. Les signatures à basses énergies (SGS) ne montrent quant à elles aucune dispersion. Des études théoriques ont montré que la signature électronique d’un cœur de vortex peut être considérablement affectée par la proximité immédiate des vortex voisins. Nous nous sommes efforcés de minimiser ces perturbations en appliquant des champs magnétiques aussi faibles que possible (160 mT), ce qui augmente la distance séparant les lignes de champ entre elles. Dans le même temps, les effets d’interférences QPI induits par les vortex eux-mêmes sont également réduits. Nos études montrent que : 1) le pic de conductance attendu à énergie nulle au centre du cœur est bel et bien présent ; 2) les pics à basses énergies se déplacent vers des énergies plus hautes en s’éloignant du centre du vortex ; 3) la décroissance de la hauteur du pic à énergie nulle est plus rapide le long de la direction cristallographique (11) que dans la direction (10). Toutes ces signatures confirment les prévisions de la théorie BCS et suggèrent que le phénomène de supraconductivité dans le B2212 à dopage extrême est effectivement plus conventionnel.. 5.
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(8) Abstract The phenomenon of vanishing resistance below a certain critical temperature (Tc ) has been known for over 100 years. When Bardeen-CooperSchrieffer (BCS) theory for the microscopic origins of superconductivity was developed, little hope remained for widespread application of superconductivity because it predicted that a critical temperature in excess of ∼20 K was unlikely. The observation of vanishing resistance far above this temperature in 1986 came as a huge surprise. Compounds whose common denominator are copper-oxide layers within the crystal structure (cuprates), reached critical temperatures around 100 K. Since this is far beyond BCS-expected temperatures, lots of effort has been made to provide alternative theories for the superconductivity in these compounds. Bi2 Sr2 CaCu2 O8+δ (Bi2212) is a member of the cuprate family which is insulating unless holes are introduced into the copper-oxide plane via doping. As the number of introduced holes is increased, many phases manifest themselves aside from superconductivity. Their interaction with superconductivity is still debated. The Bi2212 maximal critical temperature of about 90 K is reached at an optimal doping of 0.16 holes per unit cell (p=0.16). Beyond this doping, the overdoped side of the phase diagram is entered, in which alternative phases are believed to be suppressed but the superconductivity persists. Above the superconducting critical temperature, overdoped cuprates also behave more in accordance with conventional Fermi-liquids rather than the strange metal phase encountered at lower doping. The central challenge of this thesis is to asses whether the superconductivity at these high doping levels is also simplified. We use the scanning tunneling microscope (STM) to study the behavior of superconductivity and alternative phases. With the power of resolving the local density of states (LDOS) with atomic resolution, this local probe is well-suited for observing the mixing of doping induced charge density waves and superconductivity. 7.
(9) We study the heavily overdoped Bi2212 in which we find clear superconducting signals. We also find two static (non-dispersive) orders akin to the charge density wave phase believed to be extinct on the heavily ovedoped side (p>0.19). One of the orders is found to be accurately described by interference of quasiparticles (QPI) in an inhomogeneous superconductor, in accordance with the BCS theory. This finding shows a flaw in the standard interpretations which sees QPI as a dispersive order. The second static modulation exists only locally and bears spatial and spectral signatures similar to the underdoped side of the phase diagram, but develops in regions of constant superconducting signatures. Its existence on the overdoped side deserves attention in future studies. Another remarkable result of this thesis concerns the spectral signatures of vortex cores, which manifest in Type II superconductors, in presence of a magnetic field. The clear prediction of the BCS theory of a zero bias conductance peak (ZBCP), splitting into two peaks which are at increasingly higher energies as a function of distance from the core, has not been observed in Bi2212. On the contrary, low energy states at constant energy and real space LDOS modulations have been found. We show that vortices in high applied fields on the overdoped side of the phase diagram are indistinguishable from optimally doped cores. Real space modulations disperse in accordance with vortex-enhanced QPI, cutting their link to density waves. Low energy (E< ∆SC ) spectral features remain at a constant energy. Theoretical works have shown that closely packed vortex cores can significantly affect the LDOS measured at their central positions. We reduce such effects, as well as the effects of vortex-enhanced quasiparticle interference by increasing the intervortex distance using very low fields, in the miliTesla range (160 mT). We observe: 1) ZBCP at the vortex core position, 2) a shift of low energy states to higher energy with distance from the core, and 3) a faster ZBCP decay along (11) crystallographic direction than the (10) direction. All of these signatures mirror the expectations of the BCS theory and suggest that the superconductivity in overdoped Bi2212 is indeed conventional.. 8.
(10) Contents Introduction. 14. 1 Superconductivity 1.1 Discovery & first descriptions . . . . . . 1.2 Theory . . . . . . . . . . . . . . . . . . . 1.2.1 Ginzburg-Landau . . . . . . . . . 1.2.2 Bardeen-Cooper-Schrieffer theory 1.3 High-temperature superconductivity . . 1.3.1 Cuprates . . . . . . . . . . . . . 1.3.2 Bi2 Sr2 CaCu2 O8+δ single crystals. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 15 15 18 18 19 23 23 27. 2 Scanning Tunneling Microscopy 2.1 History of the STM . . . . . . . . . . . . . 2.2 Theory of STM . . . . . . . . . . . . . . . 2.2.1 Tunneling and topographies . . . . 2.2.2 Local density of states (LDOS) . . 2.3 The hardware . . . . . . . . . . . . . . . . 2.3.1 Ultra high vacuum . . . . . . . . . 2.3.2 Low temperature . . . . . . . . . . 2.3.3 The STM head . . . . . . . . . . . 2.3.4 Noise reduction and magnetic field 2.4 The experiment . . . . . . . . . . . . . . . 2.5 Remarks regarding STM on Bi2212 . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 29 29 32 32 34 36 36 39 40 40 41 43. 3 Periodic Charge Patterns in STM Images 3.1 Fourier Transform Scanning Tunneling Spectroscopy (FT-STS) . . . . . . . . 3.1.1 Analysis tools . . . . . . . . . . . . . . . . 3.1.2 Towards the measurements . . . . . . . . 3.2 Quasiparticle interference (QPI) . . . . . . . . . 3.2.1 Introduction to QPI and the octet model 9. 47 . . . . .. . . . . .. . . . . .. . . . . .. 47 47 49 52 52.
(11) 3.2.2 Theoretical formulation of QPI . . . . . . . . . . Charge density waves and modulations . . . . . . . . . . 3.3.1 Mixing CDW and QPI . . . . . . . . . . . . . . .. 57 60 61. 4 Vortices 4.1 Ginzburg-Landau mixed state . . . . . . . . . . . . . . . 4.2 Density of states implications . . . . . . . . . . . . . . . 4.3 Vortices in cuprates . . . . . . . . . . . . . . . . . . . .. 65 66 69 70. 5 Ordered Electronic Phases 5.1 Spectroscopic comparison of underdoped and overdoped Bi2212 . . . . . . . . . . . . 5.2 Static periodic charge modulations in overdoped Bi2212 √ √ 2a0 × 2a0 periodic modulation - CDW or QPI? 5.2.1 5.2.2 Static periodic charge modulation at high energy . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .. 75. 3.3. 77 83 85 91 93. 6 Mixed State and Vortex Core Spectroscopy 95 6.1 Vortex cores in high magnetic field . . . . . . . . . . . . 96 6.2 Vortex cores in low magnetic field . . . . . . . . . . . . . 101 6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 110 7 Conclusions and Further Work. 111. A Density of States in an Inhomogeneous Superconductor 115 A.1 Chebyshev expansion . . . . . . . . . . . . . . . . . . . . 115 B Practical Challenges of Mixed State Measurements 119 B.1 Surface fluctuations . . . . . . . . . . . . . . . . . . . . 119 B.2 Same area imaging with and without a vortex . . . . . . 121 C Supplementary material 123 C.1 Double vortices . . . . . . . . . . . . . . . . . . . . . . . 123 C.2 Full Z-map dispersion . . . . . . . . . . . . . . . . . . . 126. 10.
(12) Acronyms ARPES angle-resolved photoemission spectroscopy a.u. arbitrary units BCS Bardeen-Cooper-Schrieffer Bi2212 the cuprate superconductor Bi2 Sr2 CaCu2 O8+δ CdGM Caroli-De Gennes-Matricon CDW charge density wave FOV field of view FT Fourier transform FT-STM/S Fourier transform croscopy/spectroscopy. -. scanning. G-L Ginzburg-Landau High-Tc ’s high temperature superconductors L/DOS local/density of states PID proportional, integral, derivative controller SGS sub gap state(s) Tc superconducting critical temperature UHV ultra-high vacuum YBCO the cuprate superconductor YBa2 Cu3 O7−δ. 11. tunneling. mi-.
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(14) Introduction Since the industrial revolution, scientists have embraced the challenge of understanding electricity and electronic conduction, which has now become an integral part of modern life. With incremental development, the entertainment, transport and information exchange we now take for granted have been achieved. On one hand, solving a scientific puzzle adds a piece to the mosaic of human knowledge, without immediate application in mind. On the other hand, contemporary research constantly envisions more complex and efficient devices at the core of our modern technological societies. The study of superconductivity was initially purely curiosity driven research, without specific applications in mind. In 1911, K. Onnes [1] found the resistance of mercury to vanish at 4.2 K. Research continued for more than two decades before another fundamental characteristic of superconductors was discovered - the expulsion of magnetic field from the bulk of a superconductor, unveiled by Ochsenfeld and Meissner, known today as the Ochsenfeld-Meissner effect [2]. The potential applications of superconductivity arguably reach far beyond current transport mentioned above. Lossless current transmission and energy storage could contribute to a much needed sustainable energy distribution, superconducting quantum interference devices could teach us more about the function of our brain, strong electromagnets could be applied in physics and medical imaging, vortices are envisaged as building blocks of quantum computers, and so on. With many obvious applications of superconductors, their necessary cooling has remained too expensive for many applications for years. Hence, the pursuit of the theoretical understanding of these fascinating phases was driven by curiosity and desire for a better understanding of their fundamental properties. A great hallmark for theoretical physics was achieved with the publication of the Bardeen-Schriefer-Cooper (BCS) microscopic theory of superconductivity in 1957 [3] in which the particles involved (electron/Cooper pairs or 3 He atoms) are described as a Bose-Einstein condensate. The 13.
(15) 14. theory gave little hope for superconductivity to ever approach more accessible temperatures useful for every day applications and experimental progress on increasing the critical temperature (Tc ) below which the resistance drops to zero was slow (see Fig. 1.1a). A great discovery shook the scientific community in 1986, when Berdnoz & Mueller revealed superconductivity at temperatures far exceeding other contemporary experiments as well as theoretical predictions [4]. Copper oxide superconductors were shown to exhibit superconductivity above the temperature of liquid Nitrogen! Nitrogen is abundant on the earth’s surface and easy to liquefy so these compounds instantly became a popular research topic. Furthermore, the sudden jump of the critical temperature greatly increased the scope of applications in advanced technologies. Room temperature superconductivity was no longer deemed impossible, and an intensive research effort was dedicated to this objective. However, perplexing aspects inexplicable by the BCS theory withstood the test of time and have remained unclear until present time. Researchers’ patience continues to be tested with new findings often raising more questions than they answer. In this thesis, I hope to shed a photon of light on this unsolved problem that has challenged physicists for decades. For our investigation we choose scanning tunnelling microscopy (STM), a technique developed in 1981 by Binning and Rohrer [5]. The power of this tool was soon demonstrated with atomic resolution imaging of crystal surfaces [6]. As our endeavour is contributing to the elusive microscopic theory, what better experiment can be imagined than one allowing imaging of individual atoms and furthermore resolution of electronic spectra at each position? In this thesis, we use STM to experimentally study copper-oxide superconductors with high critical temperature. The manuscript is organised as follows: in chapter 1 we introduce theoretical descriptions of superconductivity. We further give a brief overview of the complex high temperature cuprate phase diagram, whose ordered electronic phases are the prime objective of this thesis. In chapter 2, we introduce the principles and operating modes of the scanning tunnelling microscope. In chapter 3 we discuss standard STM data analysis methods and introduce quasiparticle interference used to gain insight into the band structure. Chapter 4 is a more detailed introduction into the cuprate mixed state and the corresponding literature. Original measurements showing static density modulations on heavily overdoped Bi2 Sr2 CaCu2 O8+δ (Bi2212) are presented in chapter 5. Our own studies of the vortex phase and vortex core spectroscopy is presented in chapter 6. Chapter 7 summarizes the important results and discusses possible directions of future experiments..
(16) Chapter 1. Superconductivity Perfectly lossless conductivity first appears like a poor assumption, a dream. Historically its astounding application potential was clear instantly after its experimental discovery. Furthermore, many elements exhibit superconductivity naturally, making it ever-present and seemingly easily available. What is more, theoretical descriptions of such exquisite behavior are always a welcome playing ground for physicists. These ingredients motivate us to pursue better explanations still, despite more complicated systems arising displaying an even wider variety of phenomena. Superconductivity has proven to be a tough nut to crack as more than 40 years passed between its first discovery [1] and a good microscopic theoretical description, the Bardeen-Cooper-Schrieffer (BCS) theory [3]. Among many predictions, the theory also indicated a limited temperature range where superconductivity can exist. Against all these predictions, the 1986 groundbreaking discovery of high temperature superconductors (high-Tc ’s) with critical temperatures in excess of 100 K [4] has skyrocketed the superconducting phase to unprecedented heights (see Fig 1.1a). This returned us to the previous path where the details of the microscopic mechanisms causing this phenomenon were unknown. We are approaching another 40 years since the discovery of high-Tc ’s and are still trying to marry new observations to modified theories. While some features are in agreement with theoretical predictions, many are yet to be understood.. 1.1. Discovery & first descriptions. The 1911 observation of mercury’s resistance disappearance below 4.2 K was an exception at first, but subsequent years revealed that it was 15.
(17) 16. Chapter 1. Superconductivity. far from the only material to exhibit this behavior. The critical temperature of various newly discovered compounds crept upwards in small increments for decades. The great leap forward in maximal critical temperature in 1986 was unexpected (see Fig. 1.1a). It took a relatively long time for the expulsion of applied magnetic field to be discovered by Meissner in 1933 [2] and it remains a popular demonstration of superconductivity today. Observation of the diamagnetic response led to a classification of superconductors according to their behaviour in the presence of an applied field. In Type I superconductors, the diamagnetic response of the bulk prevents penetration of magnetic fields beyond a thin layer at the surface of the material. At some critical field Hc , superconductivity is abruptly destroyed, and the material returns to its normal, resistive state (blue region in Fig. 1.1b). Type II superconductors behave similarly up to the first critical field Hc1 . However, superconductivity is not destroyed immediately above this field. Instead, it is energetically favorable for the superconductor to allow penetration of magnetic flux through the bulk. These penetrating flux lines, called vortices, can provide a glimpse at the normal state from which the superconducting pairing arises and host a variety of interesting phenomena within them. As the field is increased further, more and more flux lines penetrate until superconductivity is eventually supressed at Hc2 . Complete suppression of superconductivity is thus preceded by a mixed state in which the bulk of superconducting material allows flux lines to penetrate (red region in Fig. 1.1b). The mixed phase was first investigated theoretically by the London brothers in 1935 [7]. They considered free electrons affected by an external electric field and derived two well-known equations. The second one relates the circulating superconducting current density j to the magnetic field B inside the superconductor ns e2 B (1.1) m Where e and m are electron charge and mass, and ns is the density of superconducting carriers. Combining Eqn. 1.1 with Ampere’s circuital law ∇ × B = µ0 j, ∇×j=−.
(18) 1.1. Discovery & first descriptions. 17. (a) 140. (c) conventional Copper-Oxide Iron based Type I Type II. 120. Tc(K). 100 80. HgBa2Ca2Cu3O8+d Bi2Sr2Ca2Cu3O10 YBa2Cu3O7-d. Liquid Nitrogen T. Superconducting Normal B state state B0. SmFeAsO1-xFx. 60. MgB2. 40. BaxLa5-xCu5Oy. 20. Pb 0 Hg. Nb. LaOFeP. Nb3Sn. 𝜆. LaFeAsO1-xFx. B'=e-1B0 x. (B,x)=(0,0). 1911 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020. Year Type I. -M. (b). Superconducting state. Normal state. Superconducting state. Hc1. Mixed state. Hc. Type II. Normal state. Hc2. H(T). Figure 1.1: (a) Evolution of the superconducting transition temperature with time since the discovery of superconductivity for some important superconducting families. Conventional superconductors follow the BCS theory and phonons act as the attractive force between electrons. In copper-oxide and iron based superconductors pairing interaction remains unclear. (b) The main difference between Type I and Type II superconductors is that Type II experiences the mixed state in which some of the field may penetrate into the bulk of the sample. (c) Exponential suppression of the external field B0 inside the superconducting state, λ is the depth at which the external field is reduced by a factor of e−1 ..
(19) 18. Chapter 1. Superconductivity. one finds ∇2 B = where λ =. q. m µ0 ns e2 .. 1 B λ2. (1.2). From the Meissner effect, we know that deep. inside the superconductor B = 0. But some of the field can penetrate the superconducting state at its surface or interface to a normal phase as in Fig. 1.1c. Inside the superconductor, the solution to Eqn. 1.2 is −x B(x) = B0 e λs . In this solution, λ is the depth over which the magnetic field penetrating inside a superconductor is diminished by a factor of e−1 and is known as the London penetration depth (see Fig. 1.1c).. 1.2 1.2.1. Theory Ginzburg-Landau. The next notable progression from the London Eqn. 1.1 was the Ginzburg-Landau (G-L) theory of 1950. Its details are covered in many academic books (eg. [8]), and we outline some significant results here. The G-L theory describes superconductivity using an order parameter φ whose variations can describe phase transitions, similar to how the density order parameter ρ describes the liquid-to-gas transition of water, changing from high in the liquid to low in the gas1 . The G-L order parameter is complex φ(r) = |φ(r)|eiΘ(r) and n∗s = |φ|2 is the density of superconducting electrons which is zero above Tc and increases below Tc . The thermodynamic properties of the superconductor follow from the Gibbs free energy density G. The Gibbs free energy can be expanded in terms of the order parameter φ close to the phase transition temperature Tc where |φ|2 is small. Minimizing this free energy with respect to variations of the order parameter provides the first G-L equation 1 (i~∇ + e∗ A)2 φ + aφ + b|φ|2 φ = 0 (1.3) 2m∗ where m∗ , e∗ are the mass and charge of the superfluid electrons, A is the magnetic field vector potential and a and b are temperature dependent constants. Similarly, the free energy can be minimized with 1 There is a subtle difference between these two phase transitions as the density is abruptly changed upon boiling, whereas the number of superconducting electrons changes continuously..
(20) 1.2. Theory. 19. respect to variations of the vector potential A, resulting in the second G-L equation − ∇2 A +. e∗2 i~e∗ ∗ ∗ (φ ∇φ − φ∇φ ) + A|φ|2 = 0 2m∗ m∗. (1.4). If the this equation is solved in the absence of a magnetic field (A=0) around a one-dimensional vacuum - superconductor interface, the order parameter can be shown to be [8] φ = φ∞ tanh. x √ 2ξ. (1.5). q ~2 where ξ = 2m∗ |a| . The length ξ is the coherence length and corresponds to the distance over which φ can vary significantly as seen in Eqn. 1.5. The superconducting order parameter is φ∞ deep inside the superconductor (x → ∞), but is vanishing linearly approaching the vacuum-superconductor boundary (x → 0). Such a boundary is also found around the vortices where the normal state in the vortex centres transitions to the superconducting state outside of the vortex.. 1.2.2. Bardeen-Cooper-Schrieffer theory. A tremendously successful theory which describes the origin of the superconducting state from microscopic principles is the Bardeen-CooperSchrieffer (BCS) theory [3]. It understands superconductivity as the coupling of electrons into pairs (the Cooper pairs) by means of a phonon exchange. Details of this theory are beyond the scope of this thesis but are covered in many textbooks (eg. Ref. [9]). Building blocks A first major building block of the BCS theory is the electrons’ ability to attract. This counter intuitive interaction results from the fact that the considered electrons are theoretically described as quasiparticles in a crystal, in which they interact via lattice deformations called phonons. An electron traveling through a lattice of positive ions deforms the lattice, leaving a local accumulated positive charge in its wake, which can attract a second electron passing at a later time, see Fig. 1.2. Prior to publication of the BCS theory, Cooper analyzed the problem of adding two electrons to a Fermi sea at 0 K [10] in which he showed that if the interaction between them is attractive they form a stable bound pair, no matter how weak the attraction between them is. The consequence of such electron coupling is that the whole Fermi surface is unstable to pair formation. A many-body wavefunction of.
(21) 20. (a). Chapter 1. Superconductivity. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. (c). (b). e1-. (d). +. +. +. + +. + +. e2-. +. Figure 1.2: A schematic of Cooper pairing. (a) A lattice of positively charged ions is arranged periodically. (b) Electron 1 propagates through the lattice of ions and deforms it. (c) An accumulation of positive charge remains in the wake of electron 1. (d) Electron 2 moving through the lattice is deviated by the charge accumulation left by electron 1, leading to a retarded attractive interaction..
(22) 1.2. Theory. 21. such a system is preferably written in k-space when assuming translation invariance as Y |ψBCS i = (uk + vk c†k↑ c†−k↓ )|0i (1.6) k. where uk and vk are parameters which can be adjusted to minimize the total energy, and |0i is the Fermi sea. The Fermi sea is filled to the Fermi surface to which an even number of electrons which form Cooper pairs are added. Knowing the inclination of the Fermi sea to form Cooper pairs in presence of phonon mediated pairing interaction, we can define the BCS Hamiltonian2 X X HBCS = ξk c†kσ ckσ + Vkk0 c†k↑ c†−k↓ c−k0 ↓ ck0 ↑ ≡ H0 + HI (1.7) k,σ. kk0. where Vkk0 = −V if |ξk | < ωD with ωD the Debye frequency, and Vkk0 = 0 otherwise. This mimics an attractive potential within ωD of the Fermi energy. The first term creates or annihilates electrons in the eigenstates ξ of the system, and the second term accounts for the two-body attractive interaction. The ground state of this Hamiltonian is not a Fermi liquid due to the Cooper pairs which are boson-like quasiparticles with zero momentum and spin. Since the above Hamiltonian is still not solvable, a mean field Hamiltonian in which some of the operators are replaced by their average values is used [9] X X X MF HBCS = ξk c†kσ ckσ − ∆k c†k↑ c†−k↓ − ∆∗k c†−k↓ c†k↑ (1.8) kσ. k. k. where the gap ∆k = −. X. Vkk0 hc†−k0 ↓ c†k0 ↑ i. k0. is introduced. This mean field Hamiltonian can be solved by using the Bogoliubov transformations [11], which diagonalize the Hamiltonian when ξk 1 ξk 1 2 2 1+ , vk = 1− (1.9) uk = 2 Ek 2 Ek p where ξk is the normal state dispersion and Ek = ξk2 + |∆k |2 is the Bogoliubov quasiparticle energy. This also shows the existence of a gap at the Fermi level (when ξk = 0, Ek = ∆k ). 2 This is the spin singlet Hamiltonian which couples electrons of opposite spin. Triplet superconductivity also exists..
(23) 22. Chapter 1. Superconductivity. Green’s function formalism All of the above can also be derived in terms of Green’s functions which correlate particle creation and annihilation operators [9]. This approach is often advantageous, for example it allows calculation of the density of states via the BCS Green’s function GBCS (k, iωn ) =. 1 iωn − ξk −. |∆k |2 iωn +ξ−k. (1.10). where ∆k and ξ have the same meaning as before. Eqn. 1.10 can also be recast in terms of the previously established Bogoliubov factors uk and vk as GBCS (k, iωn ) =. vk2 u2k + iωn − Ek iωn + Ek. (1.11). in which the first term diverges when iωn = +Ek , so uk is the electronlike excitation. Likewise, vk is the hole-like excitation due to the pole at −Ek . The Green’s function formalism also lets us analytically determine the relationship between the zero-temperature BCS order parameter ∆k and the critical temperature Tc , the ratio of which, for a superconductor with momentum independent pairing field, is 2∆(0) kB Tc = 3.53. As a rule of thumb then, the larger the gap, the higher the critical temperature of the material. All of the above equations are shown for the case of a homogeneous, translationally invariant superconductor. As we will see in the upcoming chapters, it is often desirable to model inhomogeneous materials in which much of the above is reformulated in terms of the Bogoliubov-De Gennes equations [12]. While an astounding number of experimentally confirmed predictions can be made using this formalism they are beyond the scope of this thesis (see Ref. [9]). G-L theory, useful for our understanding of the mixed state (see Ch.4), is also derivable from the BCS theory [13]. Density of states The density of states (DOS) of a superconductor can be calculated using the BCS Green’s function (Eqn. 1.10). To use it, we need to know the symmetry of the superconducting gap ∆k and the normal state dispersion ξk . In the simplest case of s-wave superconductivity (e.g. Al), the pairing field (superconducting gap and its phase) is momentum independent, so an electron travelling in any direction will see the same constant gap. The DOS of such a superconductor is shown in Fig. 1.3b. Due.
(24) 1.3. High-temperature superconductivity. 23. to the momentum independence of the gap, the position of coherence peaks at zero temperature identifies the superconducting gap size. However the gap of d-wave superconductors like Bi2 Sr2 CaCu2 O8+δ (Bi2212) is momentum dependent. As shown in Fig. 1.3c, the gap changes sign and magnitude as a function of momentum. Its magnitude evolves from zero along the nodal directions to a maximum along the antinodes. The spectrum is not fully gapped because the quasiparticles see no gap if travelling in the antinodal directions at 45o to the atomic lattice despite being a part of the phase coherent condensate. Note that the Fermi surface (red line) does not extend all the way to the antinodal points. Because the gap opens around the Fermi surface, the maximal gap realized in the system is smaller than the maximal gap allowed by the parametrization. A tunneling junction at a single point integrates over all momenta and will therefore see contributions of electrons encountering different gap sizes. The STM measured quasiparticle excitation spectrum will be similar to Fig. 1.3d and the position of its peaks accurately identifies the maximum gap realized in the system.. 1.3. High-temperature superconductivity. Most materials which were first found to superconduct like mercury and niobium are conventional superconductors whose behaviour is well described by the BCS theory with phonon-mediated pairing. Most intriguing superconductors though are ones exhibiting zero resistivity at temperatures far beyond what was envisioned by the BCS theory, the so-called high temperature superconductors (high-Tc ’s)3 . Cuprates, discussed in this thesis, belong to this class of materials. While the critical temperature clashes with BCS expectations and the question of the pairing mechanism is not settled, other features like the momentum integrated DOS measured in STS are in good agreement with BCS (Fig. 1.3d).. 1.3.1. Cuprates. As grown structure Cuprates are the class of superconductors at the forefront of High-Tc ’s since their discovery thanks to their extremely high Tc of 133 K in HgBa2 Ca Cu2 O1+x [14] at ambient pressure and above 160 K under applied pressure [15]. Their crystallographic structure is orthorhombic 4 with a copper oxide layer(s) located at its centre in which supercon3 The discovery of MgB also challenged the theory as its pairing is conventional2 phonon mediated- but it’s Tc is 39 K! 4 Some cuprates, are tetragonal or very close to it. e.g. for Bi2212 see Fig. 1.5a.
(25) 24. Chapter 1. Superconductivity. 2. (c) 0. - /2. -2. 2. -. (b) 1.2. 1. 0 Δ max. -1. -4 -3. -1. -2. - /2. 0. kx/a0. -Δ. 2. 1. -1. 0. /2. -. -. - /2. (d). Δ. 0. kx/a0 -Δmax. 2. 0. 2. -1. -. -3 0. -3. -1. antinodal. 0. 1. -1. 0. 1. 0. - /2. 0. -4. -1. -2. -2. -4. ky/a0. -3. 1Δ -1. 1. -3. -2. -1. 0. l. 2 0. -4. 2. -2. /2. -2. ky/a0. 1. -3. -1. -1. 1. 0. 1. 2. -2. /2. 1. -1. -1. da. 2. -1. no. (a). -1. /2. Δmax. DOS(a.u.). 1 0.8 0.6 0.4 0.2 0. -2. -1. 0 E/t. 1. 2. -2. -1. 0 E/t. 1. 2. Figure 1.3: (a) The constant gap of an s-wave superconductor is momentum independent, here overlayed over the 2D band structure. The superconducting gap opens around the Fermi level in red. (b) The DOS of an s-wave superconductor in which the position of superconducting peaks reflects the maximal superconducting gap size allowed by the parametrization at zero temperature. Unless the Fermi level extends exactly to the (π, 0) point, a Van Hove singularity will cause unequal coherence peak height or an additional peak. (c) The gap symmetry of a d-wave superconductor whose amplitude is maximal in the antinodal directions (white dashed line) but changes sign. Nodal directions marked with a dashed black line are momenta of electrons seeing no gap. (d) DOS of a d-wave superconductor, with the same parameters as in (a,b). Because the Fermi surface does not touch the antinodal regions, the maximal gap is not reached and the Van Hove singularity is visible. Quasiparticles with different momenta will see a different gap, including ungapped superconducting electrons..
(26) 1.3. High-temperature superconductivity. 25. ductivity resides. The in plane lattice constants a and b are typically similar (order of few Å) but much larger in the out of plane c direction (>10 Å). Likewise, resistivity anisotropy between in-plane and out-ofc = 104 [16] hinting at plane can be huge; for underdoped Bi2212, ρρab decoupled layers. What is more, superconductivity has been observed in monolayer compound with little change, illustrating that SC is really two dimensional in this material [17]. As grown crystals are typically highly correlated Mott insulators schematized in Fig. 1.4a,b in which electrons are localized to their atoms due to strong Coulomb repulsion of neighboring sites. The Pauli exclusion principle favours antiparallel ordering of neighbouring electrons [18], giving rise to long range antiferromagnetic order below room temperature. To dope the system, oxygen atoms are intercalated between crystal layers removing electrons from CuO2 planes and hole (p) doping the system. This induces a myraid of ordered phases, including superconductivity, across all cuprates. On the other hand, some elements can create superconductivity by substitution within the layers themselves, for instance Ce atoms which electron (n) dope Nd2−x Cex CuO4±δ . Superconductivity is present on both the electron and hole doped sides of the phase diagram (Fig. 1.4c), but past efforts focused on the hole doped side which appears more uniform across different compounds and displays the highest critical temperatures. Phase diagram of hole doped cuprates Cuprates at low hole doping outside the superconducting dome show a pseudogap (PG) and complex physical properties. Within the pseudogap phase a plethora of different orders manifest themselves. Stripes, nematic phases, spin density and charge density waves have been detected by a variety of different techniques including those sensitive to bulk response. Many of these can be studied by tunnelling spectroscopy experiments through characteristic spatial modulations of the local density of states. When the doping is increased sufficiently to induce the low temperature superconductivity, another characteristic of this littleunderstood phase appears. The tunneling spectroscopy measurements show a suppression of the density of states around the Fermi level prior to the entry into the superconducting phase (T>Tc ) [20], a possible explanation of which is that the electrons start pairing, but are not phase coherent yet. Along the same lines, angle resolved photoemission (ARPES) experiments see a momentum dependence of the spectra which appear to have clear quasiparticle coherence peaks below Tc around the nodal direction but lack them in the antinodal region. Above Tc , only the gaps in the antinodal region survive, where the.
(27) 26. Chapter 1. Superconductivity. (a). (c). e-. (b) e-. Figure 1.4: (a,b) Illustration of highly correlated parent state, an electron on a square lattice sees an available state on the neighboring atom. Its transition is felt by all others as they rearrange to fill the available states. (c) Phase diagram of cuprates from [19]. Though the extent of many phases is disputed, an overall correspondence of the main phases remains between electron and hole doped sides.. superconducting gap of the highest magnitude lies below Tc . Furthermore, regions of well developed quasiparticle coherence peaks below Tc extend towards the antinode with increasing doping. As the doping is increased further, the strange metal phase is entered. Its name derives from a failure of Fermi liquid description of electrons as dressed quasiparticles in a normal metal. Experiments show anomalies within this phase e.g. a linear dependence of the resistivity on temperature without saturation and a sign change of the Hall coefficient upon overdoping [21]. Interestingly, optimally doped cuprates evolve directly from a parent state of this strange metal when temperature is decreased. The elusive physics of this marginal Fermi liquid and how it relates to superconductivity is still unclear. Removing enough electrons from the CuO2 planes leads to overdoping, where reduced correlations and a more Fermi liquid -like behaviour of the normal state is observed, as evidenced by a surface of well defined quasiparticles [22]. Coherence peaks develop along the whole Fermi surface [23]. In light of these simplifications, the BCS theory may better describe the superconductivity on the overdoped side. Disentanglement of these phases has been an enormous challenge for physicists. A good theoretical description which can give rise to multiple intertwining phases could give clues about which of them we should try enhancing and which should be suppressed in order to enable.
(28) 1.3. High-temperature superconductivity. 27. even higher temperature superconductivity.. 1.3.2. Bi2 Sr2 CaCu2 O8+δ single crystals. In this thesis, we focus on Bi2 Sr2 CaCu2 O8+δ (Bi2212) single crystals. The crystallographic structure of this compound is shown in Fig. 1.5a. A top view of a CuO layer shows the dx2 −y2 and p electronic orbitals of Cu and O, respectively. Its layers are stacked on top of each other in the out of plane direction and the unit cells are bound together with weak van der Waals forces. The weak van der Waals binding allows exposing a fresh BiO surface by cleaving (breaking along a large surface). Our experiments are sensitive to surface contaminations, so we typically cleave the sample in extremely clean environments. A fresh cleave exposes an uncontaminated surface, ideal for surface sensitive techniques such as scanning tunneling microscopy. Adjacent Bi atoms on the surface are separated by a distance of a0 =3.8 Å. Intercalated oxygen induces the supermodulation [24] in Fig. 1.5b which is present in all of the layers and easily observed in STM topography5 . While the supermodulation has been shown to have a small effect on the size of the superconducting gap [25], its effect on other observed features such as charge density modulations is negligible. Single layer (Bi2 Sr2 CuO6+δ - Bi2201) and trilayer (Bi2 Sr2 Ca2 Cu3 O10+δ - Bi2223) parent compounds differ in the number of CaO & CuO planes, with a different c axis length and maximal Tc . Since superconductivity is induced in the CuO planes which are two atomic layers below the BiO surface (see Fig. 1.1a), tunneling into the superconducting DOS occurs through the intermediate BiO and SrO layers [26]. Its phase diagram on the hole doped side is in line with the general cuprate phase diagram of Fig. 1.4c and will be detailed in Ch. 5.. 5 The. supermodulation can be suppressed by Pb doping the compound..
(29) 28. Chapter 1. Superconductivity. (a). (b). Cleaving plane Bi Sr Ca Cu O. 5. 42. A. 15.4 A. a0=3.8 A. Bi Sr CuO Ca CuO Sr Bi. 5.41 A. Figure 1.5: (a) Bi2212 crystal structure with the cleaving plane and a top view of the CuO lattice orbitals. (b) The supermodulation seen in STM topographies is a real structural property spanning throughout the bulk of a crystal as found from bulk sensitive studies [24]..
(30) Chapter 2. Scanning Tunneling Microscopy Superconductors have been under scrutiny from many different angles. In this thesis we tackle them using scanning tunneling microscopy. Developed in 1981 by Binning and Rohrer [5], this surface sensitive technique’s power lies in its ability to resolve the energetic distribution of electrons (density of states - LDOS) with unparalleled resolution. Because of the STM’s surface sensitivity, clean surfaces and an inert environment are imperative for optimal performance. In materials like Bi2212, separate unit cells are weakly bonded by van der Waals forces, which are overcome when the unit cells are separated by cleaving the crystal. This exposes clean and atomically flat planes which lend themselves naturally to STM. Among materials examined by this technique, High-Tc ’s and particularly Bi2212 discussed in this thesis have a long history of studies. With pioneering experiments starting more than 20 years ago [27, 28], many phenomena are still not well understood and new questions emerge each year. In this section, a brief history of the technique is followed by a short description of its fundamental working principles. Finally the real world of experiments is entered, starting with instrumental details of the system, a practical walkthrough of a typical experimental setup and a final description of common challenges.. 2.1. History of the STM. Tunneling spectroscopy was instrumental in the early history of superconductivity, many years prior to the development of the STM. In Giaever’s groundbreaking experiment in 1960, a device of aluminium 29.
(31) 30. Chapter 2. Scanning Tunneling Microscopy. (Tc =1.2 K)/alumina (insulator,15 Å)/lead (Tc =7.2 K) was immersed in liquid helium (4.2 K) and a potential difference was applied across it [29]. This junction could be tuned between normal metal-insulatornormal metal (NIN) junction to normal metal-insulator-superconductor (NIS) by tuning either an applied field, or by modifying the temperature. The current-voltage characteristics were recorded and found to change following the NIS-NIN transition. The shape of its derivative (a measure of the electronic density of states described in detail below) matched the BCS prediction of the quasiparticle density of states in lead. The obvious next step in approaching the theoretical model from an experimental point of view was to replace the insulator by vacuum. For the tunneling effect to take place, the middle insulating layer must be kept tremendously thin and replacing a solid, thin insulator barrier with vacuum requires angstrøm precision for the mechanical approach of the two metals, clean environment and above all a vibration-free setup. Binnig and Rohrer achieved the required conditions in 1981 [30], demonstrating the exponential dependence of the tunneling current on barrier width. They demonstrated the power of this tool when used as a probe a short while after. Imaging atomic steps of single atom depth on the surface of CaIrSn4 [5] demonstrated unprecedented depth resolution and single atom resolution was achieved on clean Si surfaces [6]. The doors were burst open and experimental imagination led the way not only with surface imaging but also molecular and atomic scale surface modification [31], local superconducting gap spectroscopy [32] and alternative phase imaging of superconducting vortex cores [33]. High-temperature superconductivity was discovered not long after the STM’s inception. While imaging remained a challenge, promising successes were achieved on Bi2212 in the early 1990s [28, 37, 27]. As years progressed, noise levels and temperatures were lowered, vacuum systems were improved, and cryogenic hold times were increased to enable the acquisition of large scale spectroscopic maps which can provide information on the band structure of High-Tc ’s [35]. Nowadays cutting-edge STMs are capable of millikelvin temperatures, magnetic fields in excess of 10 T and keeping cryogenic for days, sometimes weeks, which are the most important factors in acquiring good quality data [38]. More often than not, the challenge is identifying the precise measurement adding a piece to the puzzle of High-Tc ’s, rather than finding a suitable instrument to do so..
(32) 2.1. History of the STM. 31. H. 1975. 1980. 1985. s Tc. 1970. h. 1965. ig. 1960. 1990. 1995. 2000. 2005. Year. Figure 2.1: Some hallmarks in tunneling spectroscopy, STM, STS and its application to high-Tc ’s. Chronologically from left to right: 1960, superconducting tunneling spectrum from Giaever [29], 1981-83 development and application of the STM by Binnig and Rohrer [30, 5, 6]. High temperature superconducting data came later; Bi2212 spectra [27], vortex lattice and detection of low energy states (YBCO) [34] followed by large field of view mapping which enabled detection of periodic DOS modulations in vortex cores (Bi2212) [35], as well as in zero field [36], the latter was related to band structure (Bi2212)..
(33) 32. 2.2. Chapter 2. Scanning Tunneling Microscopy. Theory of STM. The origin of the STM’s fantastic real space resolution can be understood through rudimentary quantum mechanics. Though the real tunneling process is anything but trivial, reasonable assumptions can make the theoretical modelling of the STM intuitively understandable while remaining a good approximation to reality.. 2.2.1. Tunneling and topographies. We consider the simplest case of an STM junction consisting of a tip and a sample of the same metallic material separated by vacuum, depicted in Fig. 2.2a. The separation is reduced to a few Å and an electron moving from tip to sample is considered with vacuum in its way. Such a problem can be described as a single quantum mechanical system of a particle with energy E moving from left to right with a potential barrier of height V0 and thickness d in its way as shown in Fig. 2.2b. Classically, a potential barrier of height V0 is insurmountable to a particle with energy E<V0 . The Schrødinger equation describes the state function of a particle in such a quantum system. The general time-dependent Schrødinger equation may be separated into two parts whose product gives the solution as Ψ(r, t) = φ(t)ψ(t). Focusing on the position dependent part, we want to solve Hψ(r) = Eψ(r), where H is the Hamiltonian, sum of the kinetic and potential energies of the involved particle and E are the eigenenergies of the system. 2 −~ 2 (2.1) ∇ + V (r) ψ(r) = Eψ(r). 2m The potential landscape of the tunnel junction consists of three regions. The sample, the potential barrier (vacuum) and the tip. The Schrødinger equation (Eqn. 2.1) can be solved in each of them separately. The height of the potential barrier, V(r), depends on the position in the system, being equal to V0 inside the barrier and 0 otherwise. In all three positions, the Schrødinger equation can be solved. Outside the barrier, solutions are superpositions of left and right moving waves of the general form ψ(r) = Aeikr + Be−ikr .. (2.2). In the barrier region, where V=V0 , the Schrødinger equation can be rewritten as d2 ψ(r) − κψ(r) = 0, (2.3) dr2.
(34) 2.2. Theory of STM. 33. (a). Piezo fee db. ac k. Bias e-. (b) E. V0. 0. d. Figure 2.2: (a) An illustration of an STM tip in the process of acquiring the height of all the points along the red line on the surface of Bi2212. (b) shows an electron tunneling through a potential barrier of height V0 ..
(35) 34. Chapter 2. Scanning Tunneling Microscopy. q. where κ = 2m(V~02−E) . The solution of this equation inside the barrier is not a standing wave, but an exponentially decaying function ψ(r) ∝ e−κr for a particle traveling from left to right. The probability of the particle to tunnel through the barrier, so-called transmission probability | ψ(r) |2 , can be shown to be | ψ(r) |2 ∝ e−2κd ,. (2.4). for a wide and tall barrier(κ 1). The finite probability a particle with energy E has to pass through a potential barrier V despite it being classically forbidden (V > E) is what we call quantum tunneling. The tunneling probability decreases exponentially with the barrier thickness d, which is precisely the reason STM is so sensitive to changes of separation between the tip and the sample. Depending on the distance, a majority of electrons might not tunnel through the junction but be reflected as in the classical case. Nevertheless, it is enough that a tiny fraction is transmitted in order to sustain a typical STM setpoint current of picoamperes (10−12 A). This simple model description allows to understand constant current STM topography. A mechanism approaches the tip in Fig. 2.2a close to the surface of a sample, until a desired current Isetpoint is detected for a given setpoint bias. As it moves along the surface the tip encounters corrugations and the current varies exponentially. A feedback system continuously adjusts the height Z of the tip to maintain a constant Isetpoint . This way topographic information about a sample can be recorded, one of the fundamental operating modes of STM. Looking at the problem in more details (see next section), we find that the tunnelling current depends not only on the tip’s proximity to the surface but also on the number of states below the tip allowed to tunnel, which can be distributed inhomogeneously. A real constant current topography therefore provides a convolution of structural and electronic information.. 2.2.2. Local density of states (LDOS). STM is not the only technique with the ability of resolving atoms in real space, it is also achievable in atomic force microscopy (AFM) or transmission electron microscopy (TEM). The advantage of STM is its ability to accurately measure the density of states as a function of energy with high spatial resolution. To understand how this works, we need to delve a little further into theory. As before, we consider two independent systems left (L) and right (R), separated by vacuum. Each has its own number of particles (Nl , Nr ), Hamiltonians (Hl , Hr ), and one-particle wavefunctions (ψλ (l),.
(36) 2.2. Theory of STM. 35. ψρ (r)). The exchange of particles between the two systems is introduced via a tunneling Hamiltonian HT X Tλρ c†ρ cλ + Tρλ c†λ cρ = X + X † . (2.5) HT = λρ. The Tλρ term and its Hermitian conjugate are the tunneling matrix elements, i.e. the amplitudes for an electron to tunnel from a state |ψλ i on the left to |ψρ i on the right and vice versa. The current due to the particle exchange between the two systems is measured by the rate of change of particles on one side (multiplied by electron charge e). Let us consider the right system: I = −eh. d Nr i. dt. (2.6). d Assuming a linear response of h dt Nr i due to a perturbation HT , the current can be shown to be R 2ieV t R I = −2eIm{CXX CXX (eV )}, † (eV ) + e. (2.7). where the two parts correspond to the single particle and to the Josephson tunneling currents respectively. Even though this thesis concerns superconductors, the Josephson current remains zero for a non-superconducting tip, and we therefore focus on the single particle current, Is . The theoretical construct of imaginary time allows us R to evaluate the correlation function CXX † (eV ) in imaginary time with analytic continuation. Furthermore, the assumptions of independent subsystems and tunneling only between a single point on the tip (l0 ) to a single point directly below the tip (r0 , equivalent to setting the tunneling matrix element to T (l, r) = tδ(l − l0 )δ(r − r0 ), where l0 and r0 ) reduces the single particle current to 2π|e| I= | t |2 ~. Z dω[f (ω − |e|V ) − f (ω)]A(l0 , l0 , ω − |e|V )A(r0 , r0 , ω),. (2.8) where f are the Fermi functions and A the spectral functions. The diagonal part of the spectral function is the density of states. With the left-hand system as a probe, we can choose a material with energy independent DOS such that A(l0 , l0 , ω) = Nl (l0 , ω) ≈ Nl (l0 , 0) in which case the single particle current becomes. I=. 2π|e| | t |2 Nl (l0 , 0) ~. Z dω[f (ω − |e|V ) − f (ω)]Nr (r0 , ω).. (2.9).
(37) 36. Chapter 2. Scanning Tunneling Microscopy. Its derivative with respect to the voltage is called the conductance (σ). Since the only voltage-dependent part is the Fermi function, we find Z dIs ∝ dω[−f 0 (ω − |e|V )]Nr (r0 , ω). (2.10) σ(V ) = dV Since the derivative of the Fermi function at 0 K is the delta function, the conductance is just proportional to NR (r0 , ω), the DOS at a point below the tip! At a finite temperature, the conductance is thermally broadened as the Fermi function’s derivative is no longer a delta function; so the quantity measured in experiments is a thermally broadened density of states. In practice, the conductance measures the sample DOS at the position of the centre of the tip apex. A spectroscopic experiment is performed by placing a tip over a specific point, disabling the feedback, sweeping the bias and recording the response of the current. The acquired I(V) curve is then differentiated with respect to the voltage providing the LDOS at the point of dI measurement also called the conductance ( dV = σ).. 2.3. The hardware. Since its invention in 1981, incremental development of STM has brought us to the state of the art systems nowadays used routinely. These are worthy of a thesis on their own. Here we discuss some of the main components, without any one of which the experiments of this thesis would not have been possible • Ultrahigh vacuum • Low temperature • STM head • Noise reduction and magnetic field Most of the measurements described in this thesis were done on a commercial Joule-Thompson-Tyto STM (JT-Tyto).. 2.3.1. Ultra high vacuum. Since the tunnelling Hamiltonian description of Ch. 2.2.2 does not account for scattering events within the tunnelling junction, we want to limit the impurities within it. Therefore, an important component of STM experiments is a clean working environment. To reduce reactivity with any materials present in the chambers, STM chambers are usually made of non-magnetic, unreactive stainless steel. To ensure optimal experimental conditions, the tunneling junction must be pure. This is.
(38) 2.3. The hardware. 37. partly ensured by preparation of a stable tip (see Ch. 2.4) and a clean surface, but is also dependent on the purity of the vacuum barrier. Its quality is measured in terms of the pressure P, which is a measure of the energy dissipated by the molecules in a container by collision with its walls. These molecules can interact with the sample or the tip and can deteriorate the scanning conditions. It is in our interest to reduce the number of collisions and reduce the energy dissipated by the them. The ideal gas law P V = nRT describes a relationship between pressure, number of molecules (n) and temperature (T) in a container of volume V. From it we see that if the volume is kept constant, pressure can be reduced by decreasing the number of molecules present (pumping the chamber) or decreasing the temperature (see next section). Pressure is decreased in four stages. A mechanical roughing pump decreases the pressure from atmospheric pressure to rough vacuum (1 Bar-10−3 mBar). Thereafter, a turbomolecular pump joins in, decreasing the pressure from rough to high vacuum (10−3 mBar-10−8 mBar). Finally, ion and titanium sublimation pumps remove the final molecules, bringing the system to ultra-high vacuum (10−8 mBar-10−10 mBar). Ultra high vacuum (UHV) is the cleanest environment achievable on earth. To put the junctions cleanliness into perspective, 1 Bar corresponds to ≈1018 molecules/mm3 , at UHV this drops to ≈105 molecules/mm3 . Upon sealing the vacuum chambers off from ambient atmosphere, a cleaning procedure is carried out. This usually consists of decreasing the pressure using only the turbomolecular pump on all the chambers, achieving a pressure of 10−5 mBar after a few days. Next, the temperature of the entire system is raised for several days (bakeout), during which wall adsorbates such as water which evaporates or oxygen are desorbed from the walls and aspired by the turbomolecular pump. Once high vacuum is reached, ion and titanium sublimation pumps are activated and degassed by heating them above their usual operating temperature. The temperature is then decreased back to room temperature with all of the pumps running simultaneously on different chambers. Ultimately the system is cooled to cryogenic temperatures and the cooling itself acts as a cryopump by condensing any gaseous remnants in the system, further decreasing the pressure. In order to decrease the contamination of the STM head, any tools and samples used are cleaned and inserted into the STM head in 3 stages that can be seen in Fig. 2.3, monitoring their degassing..
(39) 38. Chapter 2. Scanning Tunneling Microscopy. Cryostat 10-10mBar. Preparation chamber LN2 reservoir. sam. LHe reservoir. ple p ath. 10-7mBar. Loadlock. 10-10mBar. STM chamber JT cooler. Figure 2.3: Tyto-JT-STM system used during this thesis. Anything inserted into the system follows the dashed sample path to minimize contamination of the STM chamber..
(40) 2.3. The hardware. 2.3.2. 39. Low temperature. Low temperature is not only important for decreasing the pressure and achieving superconductivity, it also improves tip stability, decreases thermal drift, and improves energy resolution. The JT-Tyto is capable of achieving temperatures down to 1.2 K. The three component cryostat capable of achieving this temperature is shown on the right side of Fig. 2.3. The outer shield is filled with liquid nitrogen (77 K) which acts as a first protection layer from outside heat radiation. The inner shield is filled with liquid helium (4.2 K) which must be thermally decoupled from the nitrogen shields to reduce helium loss. Helium shields can be thermally anchored to the STM head if we choose to operate at 4.2 K. The JT-cooler is connected to the helium bath via a small tube, thermally decoupled from the helium reservoir but coupled to the STM head. Another tube of high impedance is connected to an external pump. By pumping liquid helium through the impedance, it undergoes Joule-Thompson expansion, decreasing its temperature down to 1.2 K. If the STM head is connected to the helium bath it is kept at a constant temperature of 4.2 K. Upon removing this thermal connection the temperature can be either decreased to 1.2 K as just described, or increased above by using a heater installed close to the sample. In both cases it is essential to limit the thermal anchoring between the sample and the helium vessel..
(41) 40. 2.3.3. Chapter 2. Scanning Tunneling Microscopy. The STM head. The STM head is the central component enabling high resolution imaging and spectroscopy. It consists of coarse positioning motors to select specific locations on the sample and a fine mechanism to scan the tip with picometer sensitivity along the sample surface. Coarse motors move the sample stage in the XY plane (in plane) with a precision better than 1 µm per step. The tip can move in the out of plane (Z) direction using a similar motor. It’s mechanism relies on the piezoelectric effect which causes certain materials to deform when a voltage is applied across them. The tip sits on a rod which is in contact with the piezoelectric crystal with no applied voltage. A slow increase in the applied voltage continuously deforms the crystal which remains stuck to the rod, moving the tip towards the sample. Once the maximum allowed deformation of the crystal is reached, the voltage is abruptly set to zero, quickly deforming the crystal, which makes it slip along the surface of the rod. The cycle can begin again to perform the next step of the slip-stick motion. Retracting the tip works in an identical way but the slowly applied voltage is of the opposite polarity. The fine movement actuator must move precisely and reversibly on the picometer scale. It is also made of a piezoelectric material which is a continuous ceramic hollow cylinder. Five contact electrodes allow selecting specific regions where an electric field is applied to deform the piezo along X, Y, or Z. Depending on the voltages applied, the piezo will deform and move the tip in a particular direction. It can scan a maximum XY area of 10×10 µm2 at room temperature and its response is temperature dependent and must be re-calibrated at each temperature using some known material. The components of the STM head and the parts isolating it from the outside temperature radiation and vibrations are shown in Fig. 2.4.. 2.3.4. Noise reduction and magnetic field. A fundamental component of high quality measurements is a low noise environment. The entire STM system is placed in a Faraday cage, eliminating outside electromagnetic radiation, and is supported by air legs while in operation and only put back on the ground during cryogenic refills. The head itself is mechanically suspended from springs to further isolate it from external vibrations. Care must be taken when selecting the electronic components as well as in the design of all mechanical parts, to avoid components with low resonant frequencies. Another source of noise can be the resonant frequency of the building in which the STM operates. This type of noise is typically of low frequency which is difficult to damp. In some labs this is mitigated by construct-.
(42) 2.4. The experiment. 41. Figure 2.4: A sketch of the Tyto-JT-STM head and photographs of the head with radiation shields removed (right) and with everything is back in place (left).. ing special rooms, basements or entire buildings to reduce it. Due to its noise sensitivity, the STM can be used to measure the working hours of nearby roadworks or as an excellent seismograph. The instrument we used features a dry niobium-titanium coil magnet enabling us to apply a magnetic field up to three tesla along the c direction, perpendicular to the sample surface.. 2.4. The experiment. A successful STM experiment can take weeks and requires many subsequent steps to go right. In the following, we outline the execution of an experiment from tip preparation to data acquisition.. Sample As seen in Ch. 1, Bi2212 is a van der Waals material meaning its unit cell layers are weakly bound together. A sample is prepared by gluing it on a clean sample plate with a standard conductive EPO-TEK H20E glue [39]. Once the glue dries, the top surface is peeled off with scotch tape until the exposed surface is clean and flat. On the best samples this means flat regions of up to 1x1 mm2 where no flakes are seen under the microscope. Once this stage is reached, a screw is glued upside-down onto the sample. The sample is now prepared for transfer to the head of the system (see Fig. 2.5a), going through the steps outlined in Section 2.3.1 finally being deposited in a sample storing position just outside the STM head..
(43) 42. Chapter 2. Scanning Tunneling Microscopy. It then cools to the temperature of this slot. When the tip is in good condition as verified by a calibration process on a well-known metallic sample, the sample can be cleaved by breaking off the screw glued on top of the sample and inserting it on the STM head as quickly as possible.. Tips To make the tip, a solution of CaCl2 + H2 O + HCl is prepared. An iridium wire with a diameter of 0.25 mm is immersed into the solution with a lead electrode inside, applying potential difference V=25 V until a current of ∼250 mA flows. A tip etched in this way will have a clear, sharp apex. In STM experiments the flat area on which we must approach the tip is usually a surface smaller than 1 mm2 . We judge the tip’s landing spot by the reflection of the tip we see on this surface, which can be assessed with higher precision if the tip is sharp. The etched tip must then be cleaned of any oxide, water or other contaminants. Tip sputtering can be performed for initial cleaning of the apex. In this process, positively charged argon atoms are accelerated towards a tip which also has a slight positive charge [40]. This creates concentrated field lines at the sharp end due to the Poynting effect. As a consequence the apex is sputtered less than the rest of the tip, thus preserving its sharpness. The hitting argon ions peel off contaminants on the tip’s surface. Further cleaning and calibration is done on a clean single crystal Au(111). In anticipation of the experiment, the Au is also cleaned by repeated sputtering - annealing cycles in UHV. The tip is scanned repeatedly along the surface and trained. Training consists of controlled tip immersions and bias pulses. A trained tip is sharp, stable and has good energy resolution. To verify this is the case, we perform the following tests (see Fig. 2.5b-e): • Continuous scanning of a particular area. Repeated scans should be identical with no additional matter appearing on the surface, no notable tip changes (surface remains clean but the tip seems to have jumped to a different area) and no identical feature repetition (multiple tip effect). • Scanning over an atomic step tells us if our tip is multiple. If so, a copy of the step is seen. A further test for stability is increasing the setpoint current and scanning speed, abrupt changes in the scan are indicative of a poor tip condition. • Controlled atomic-scale crashes of less than 1 nm tell us about the sharpness of the apex. If a consequence of a crashed.
(44) 2.5. Remarks regarding STM on Bi2212. 43. tip is a hole (or a peak) at the location of the crash, and the tip continues scanning without a major change, the tip is stable. • Rapid low bias scanning or low bias scanning over an atomic step is another excellent way to verify tip stability. Scanning at low bias integrates over a smaller number of electrons so the tip is approached closer to the surface. • Tunneling spectroscopy of surface states. Au(111) is a metal with constant density of states except a sharp change at around -450 mV which should be detected. If the above tests are passed, the tip is in good condition and the experiment on the desired sample can proceed. When necessary, final steps of the tip training can be done on the sample itself. If the imaging appears slightly blurred despite passing all of the above tests and we are measuring on Bi2212, small but controlled crashes on its surface (no more than ∼400 pm in depth, Fig. 2.5e) may favorably rearrange the final atoms of the tip - but there is a risk involved as this can also lead to tip or sample destruction.. Approach and experiment With the tip and sample ready, we begin the approach. Light shining onto the sample surface from outside the STM head must be aligned so that we can see the reflection of the tip on the sample surface. We identify a flat part on the surface and move the tip around until its reflection is seen. We coarsely approach the tip to a distance of a few mm using the slip - stick piezo motors. The STM head is then isolated from the environment using external shields to limit ambient heat radiation and suspended onto springs to isolate the head from external vibrations. When the head temperature stabilizes, automatic approach is started. When the approach finishes, a small region on the surface is scanned (5×5 nm2 ) to verify surface condition. If atoms are resolved without tip changes, the field of view is gradually increased up to the desired size. The experiment can then begin. Spectroscopic maps measured for this thesis took from 2 hrs to 60 hrs.. 2.5. Remarks regarding STM on Bi2212. The exponential dependence of the tunneling current on the distance between the tip and the sample, the effect of tunnelling matrix elements, and the Bi2212 crystallographic structure have been introduced in Chs. 1.3.2 and 2.2.2. The cleaving plane is between the BiO planes, but.
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