High-Tc Superconductivity

High-Tc Superconductivity

CHAPTER FOUR High-Tc Superconductivity Hans-Rudolf Ott Laboratorium für Festkörperphysik, ETH Zürich, 8093 Zürich, Switzerland Contents 1. Introduct...

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CHAPTER FOUR

High-Tc Superconductivity Hans-Rudolf Ott Laboratorium für Festkörperphysik, ETH Zürich, 8093 Zürich, Switzerland

Contents 1. Introduction 2. Prelude to High-TC Superconductors: Superconducting Ternary Oxides 2.1 SrTiO3 2.2 LiTi2O4 2.3 BaPb1–xBixO3 2.4 Summary 3. High-TC Superconductors: the Decisive Step with Cu Oxides 3.1 Crystallography and some basic physical properties 3.1.1 3.1.2 3.1.3 3.1.4

La2–xMxCuO4–y (M = Sr, Ba) MBa2Cu3O7–δ, MBa2Cu4O8 AmM2Rn–1CunO3n+m+1 Ca1–xSrxCuO2

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3.2 More on normal-state properties of doped Cu-oxide compounds 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5

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Electrical transport Magnetism Density of electronic states Pseudogap Bottom line

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4. Superconductivity of High-TC Cuprates 4.1 General aspects 4.2 The phase transition 4.3 Characteristics of the superconducting state

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4.3.1 4.3.2 4.3.3

Gap function and gap anisotropy Experimental methods for probing gap nodes Experimental methods for probing the gap symmetry

4.4 Summary 5. Superconducting Fullerides 5.1 Materials’ structure 5.2 Onset of superconductivity 5.3 Selected physical properties 6. Final Remarks References

Superconductivity in New Materials, Volume 04 ISSN 1572-0934, DOI 10.1016/S1572-0934(10)04004-7

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 2011 Elsevier B.V.

All rights reserved.

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1. INTRODUCTION The quest to find materials which are superconducting at tempera­ tures above the regime of liquid helium must have started shortly after the discovery of the phenomenon in 1911 [1]. Without a clear understanding of what exactly provokes the loss of electrical resistance at some well-defined critical temperature, this search was simply a trial-and-error enterprise, however. Before the discovery of the Meissner–Ochsenfeld effect in 1933 [2], it mainly involved measurements of the temperature dependence of the electrical resistivity of available materials. Later and until now, the observa­ tion of an abrupt diamagnetic signal, reflecting the expulsion of magnetic flux from the superconductor, is often easier to accomplish. This search first concentrated on metallic elements and later was extended to binary alloys and compounds. The success of the theoretical description of the phenom­ enon by Bardeen, Cooper, and Schrieffer (BCS) in 1957 altered the situa­ tion. In particular, the BCS equation for the critical temperature which in its simplest form reads Tc ¼ D e  1=g

½4:1

and relates Tc with other characteristic parameters of the material, opened the way for more directed searches. In the original BCS theory [3], the interaction between the itinerant electrons and the atomic crystal lattice and its vibrations (phonons) is shown to provoke the onset of superconductivity. The Debye temperature D is a measure for the range of frequencies or energies of the lattice vibrations and the parameter g captures the product of the strength of the interaction that leads to the pairing between the indivi­ dual electrons and the density of electronic states at the Fermi energy EF, the important parameter D(EF), which characterizes the ensemble of the con­ duction electrons of the considered material. Since for most materials, D is lower than room temperature and g << 1, it is clear why Tc must be low as observed. The BCS approach to the problem may, however, be regarded as a more general framework and Eq. [4.1] as a universal expression for Tc. In this sense, if pairing interactions among the itinerant electrons, other than the electron–phonon interaction exist, it is conceivable that related favorable parameters would lead to higher values of Tc. Among the first to discuss such scenarios and high-temperature super­ conductivity was V.L. Ginzburg. In the first half of the 1960s he argued

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that for direct electron–electron interactions in specially tailored sandwich structures, termed excitons, D in Eq. [4.1] should be replaced by the Fermi temperature TF [4]. Since for materials with average densities of itinerant charge carriers, TF is 1–2 orders of magnitude larger than D, a considerable enhancement of Tc might thus be expected, provided that at the same time, g is not much reduced. At approximately the same time, that is, even a bit earlier and using similar arguments, W. Little had suggested that suitable arrangements of molecules might result in materials which would be superconducting at temperatures as high as room temperature and even above [5]. Only shortly thereafter, the possibility of phononinduced room-temperature superconductivity in metallic solid hydrogen under very high pressure was predicted on the basis of band-structure calculations and discussing strong-coupling effects in connection with the electron–phonon interaction [6]. All these early predictions made use of Eq. [4.1] and discussed the possibility of inserting more favorable parameters than those of simple metals and alloys. Unfortunately, the calculated results were not corrobo­ rated by experiment. At that time, the situation was even worse in the sense that it turned out to be difficult to predict critical temperatures even of already known superconductors from this sort of calculations. A simple parametrization of Tc employing Eq. [4.1] seemed not to be an adequate approach to reach the goal. Experimental efforts to enhance Tc were still based on educated guesses, concentrated on materials, mostly binary com­ pounds and alloys containing elements of the d-transition regime of the periodic table. Only modest progress was achieved. The term high Tc was used arbitrarily for critical temperatures exceeding 10 K [7]. In the early 1970s, the highest Tc achieved in this way with a Nb-based alloy was approximately 23 K [8] which at least exceeded the boiling temperature of liquid hydrogen. The situation with respect to enhancing Tc did not really change until the unexpected and therefore amazing discovery of J.G. Bednorz and K.A. Müller in 1986, which suggested that certain copper-oxide com­ pounds enter a superconducting state at temperatures exceeding 30 K [9]. A quick verification of the conjecture [10, 11] and rapid progress in further enhancing Tc in other related materials, that is, other types of copper oxides, to temperatures above the boiling point of liquid nitrogen [12] pushed research in superconductivity into the limelight of science and public interest. The significance of this development is best captured in a diagram, such as that shown in Fig. 4.4 in the introductory chapter on history, where

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the discussed enhancement of Tc is plotted versus time. The highest critical temperature at ambient pressure that has been achieved until today is of the order of –140°C [13] and must, by all standards, still be regarded as a cryogenic, that is, low temperature. This is why in this context we use the term high-Tc rather than high-temperature superconductivity. From the materials’ point of view, we note that these compounds, as will be shown below, are composed of four or more chemical elements. At least in this sense, they are much more complicated than the substances men­ tioned above. Before describing and discussing the properties of these high-Tc cuprate superconductors, it is instructive to briefly review some properties of superconducting ternary oxide compounds that were known before 1986. With hindsight some of them might be regarded as a prelude to high-Tc superconductors although, considering their physical properties, none of them can be identified as direct indicator of the later development. Nevertheless, these materials gave some hints that superconductivity was not an exclusive property of simple metals and alloys but rather a phenom­ enon that also occurs in materials that seem, from a more conventional point of view, rather unfavorable for adopting a superconducting state at low temperatures. Although the main portion of this chapter will be devoted to a discus­ sion of the cuprate superconductors, other and more recent developments of high-Tc superconductivity involving materials that are neither oxides nor copper compounds will also be included.

2. PRELUDE TO HIGH-TC SUPERCONDUCTORS: SUPERCONDUCTING TERNARY OXIDES Between 1964 and 1975, in at least three cases superconductivity was found in materials which, at that time, did not a priori qualify as potential superconductors. This subsection is devoted to a short description of these cases, explicitly leaving out the superconducting ternary oxide compounds of the tungsten bronzes [14, 15] which are, no doubt, of interest in connection with superconductivity in systems with reduced dimensions D, that is, D < 3.

2.1 SrTiO3 Superconductivity of this compound, although reported already in 1964 [16] and occurring only at temperatures below 1 K, is of some interest here for various reasons. First, the pure compound is an electrical insulator and

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Ti Ca O

Figure 4.1 Schematic crystallographic unit cell of CaTiO3 (perovskite).

crystallizes with the cubic CaTiO3 (perovskite) structure (see Fig. 4.1) which is, as will be seen below, of some relevance in connection with other high-Tc materials. A metallic state can be achieved, however, by suitable doping of pure SrTiO3 via either replacing Ti by small amounts of an element of an adjacent column in the periodic table, such as Nb [17, 18], or by slightly depleting the oxygen sublattice [16, 17]. Later, electron or hole doping became well-known procedures for achieving electronic con­ duction in otherwise insulating Cu oxides. Many of these conducting cuprates prove to be superconductors at amazingly high temperatures. What is remarkable is the wide range of concentration nc of itinerant charge carriers, spread over 2 orders of magnitude, for which superconduc­ tivity of doped SrTiO3 is observed (see Fig. 4.2). For either doping type, the 0.5

0.4 0.3 TC 0.2 0.1 0

1019

1020

1021

n

Figure 4.2 Variation of Tc of doped SrTiO3 upon the concentration n of itinerant charge carriers. The circles in brackets are ficticious data used for calculations (see Ref. [17]).

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maximum Tc is found in the vicinity of nc = 1020 cm−3. With oxygen depletion, Tmax  0.3 K and upon Nb doping Tmax  0.7 K [18]. Although c c the reduction of the electrical resistivity from room to helium temperature covers at least 2 orders of magnitude, the residual resistivity at low temper­ tures is, with approximately 100 μΩcm, still very high, that is, 100 times larger than the corresponding value of pure Cu at room temperature. With respect to the Tc(nc) curve in Fig. 4.2, the reason for the decrease of Tc with increasing nc above 1020 cm−3 is of special interest because a similar decrease of Tc above an optimal doping concentration is also observed for cuprate superconductors. In the present case, a tendency to localize magnetic moments forming on the Ti sites with increasing doping might be the cause and, if true, would indicate a conventional superconductivity of this material. This scenario is considered as highly unlikely in the case of high-Tc superconductors, however. From the results of magnetization measure­ ments [19], it was concluded that doped SrTiO3 exhibits extreme type II superconductivity, very much like the later discovered cuprate superconductors.

2.2 LiTi2O4 The discovery of superconductivity of this compound in 1973 was rather surprising because its Tc of about 12 K was, at that time, considered as unexpectedly high [20]. This material crystallizes with a spinell-type struc­ ture (spinell = Al2MgO4) and the Ti atoms are again surrounded by oxygen octahedra. Contrary to SrTiO3, the stoichiometric compound is super­ conducting but loses this feature very rapidly by changing the chemical composition of the substance, for example, by varying the Li/Ti ratio only slightly [21]. For instance, the compound Li1.1Ti1.9O4 exhibits a semiconductor-type temperature dependence of the resistivity and no superconducting transition at low temperatures. It is remarkable that the disappearance of superconductivity is coupled to a metal–insulator transition which is not, as in other cases discussed below, related to a structural transition. The change from metallic to insulating behavior is reflected in corresponding variations of the low-temperature specific heat [22]. In metallic LiTi2O4, the electronic specific heat parameter is quite large and indicates a substantial mass enhancement of the conduction electrons. It is not clear whether the related large value of the electronic density of states N (EF), which influences the magnitude of the above-mentioned parameter g in Eq. [4.1], is responsible for the relatively high value of Tc.

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2.3 BaPb1–xBixO3 BaPbO3, in spite of in principle compensating valencies of its constituents, is metallic and crystallizes with a structure of orthorhombic symmetry. The electrical resistivity is almost temperature independent and very high, of the order of a few hundred μΩcm [23]. Gradually replacing Pb by Bi results in a structural transition to a crystal lattice of tetragonal symmetry at x  0.1 [24]. Further enhancing x leads to a superconducting phase which is restricted to a very narrow region around x = 0.25 and a maximum Tc of approximately 12 K [25]. This value was again unexpectedly high at the time of its discovery in 1975 [26], because none of the chemical components belongs to the class of d-transition elements, which was thought to be a must for achieving high critical temperatures. From measurements of the Hall con­ stant RH it was concluded that the concentration of conduction electrons increases with x, reaching a maximum at x = 0.25 [23]. The corresponding value and sign of RH are compatible with a count of one electron per Bi atom. In contrast to other oxide superconductors, in particular the high-Tc materials in the cuprate sector, the magnetic susceptibility  is negative in the normal state and largest in magnitude for x = 0.25. This indicates that the paramagnetic contribution is weakest for this composition. Considering that for this material Tc exceeds 10 K, the itinerant charge carrier density is anomalously low. Although some evidence for a strong coupling between electrons and phonons and hence a favorable condition for superconduc­ tivity in the traditional sense was claimed [27], theoretical arguments suggest that this interpretation is not compatible with band-structure calculations and their connection with the metal–insulator transition that is observed at x  0.35 [28, 29]. BaBiO3 is an insulator. Starting from this composition, a partial replace­ ment of Bi by K leads, via structural transition, to a cubic crystal lattice and to an onset of superconductivity around 29 K for Ba0.6K0.4BiO3 [30, 31]. This observation was made only after the discovery of high-Tc supercon­ ductivity in copper oxides, however.

2.4 Summary Upon varying the chemical composition the ternary oxides with perovskite­ type crystal structures exhibit a tendency to instabilities of the crystal lattice and concomitant changes in the electronic excitation spectrum. Although this might indicate a strong coupling between electronic and lattice excita­ tions and hence favorable conditions for superconductivity, no searches for

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higher critical temperatures following this line of arguments were successful prior to 1986.

3. HIGH-TC SUPERCONDUCTORS: THE DECISIVE STEP WITH CU OXIDES Although, as outlined in Section 1, some alternative suggestions of mechanisms for superconductivity based on electron pairing appeared in the literature [32–34] over the years, nobody really questioned the paramount importance of the electron–phonon interaction for the occurrence of the phenomenon. With this in mind, it seemed obvious that a strong interaction of this type would also be favorable for reaching high values of Tc. At the same time, a strong coupling between electronic and lattice degrees of freedom is also expected to lead to other instabilities, such as structural and/or electronic phase transitions. Examples of the latter are Fermi surface instabilities leading to spin or charge density wave states at low tempera­ tures. A well-known example for the former is the cooperative Jahn–Teller effect (see, e.g., [35]) which provokes distortions of the crystal lattice below a critical temperature. From the outset of his interest in the problem of raising Tc of super­ conductors, checking systems close to this latter type of instabilities was the working hypothesis of Alex Müller. Following these ideas, he and Georg Bednorz were finally led to investigate the temperature dependence of the electrical conductivity of a certain class of copper oxide compounds. Their truly amazing discovery, that is, the onset of the loss of electrical resistance upon decreasing temperature of a multiphased ceramic substance containing La, Ba, Cu, and O at temperatures exceeding 30 K, was published in the fall of 1986 [9]. As usual, the completely unexpected observation was first met with some scepticism by the specialists. The result was intriguing enough for others, however, and it took only 2 months to confirm the discovery [36] and not much later, the correct chemical composition of the compound was identified [37]. Frantic research activities around the globe followed and rapid progress in finding new compounds and in raising Tc above the boiling point of liquid nitrogen [12] was made. In the following sections the development and the principal aspects of superconductivity in copper oxide compounds are described and discussed. Because of the enormous wealth of available data obtained from probing a large number of different compounds, the material to be presented and discussed consists of typical examples and is not intended to provide a complete overview.

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As may be concluded from the preceding section, the general physical properties of multicomponent oxide compounds, especially the occurrence of their superconductivity, depend very much on the chemical composition of the investigated material. The same is true for the high-Tc copper compounds. Already soon after the discovery and based on the results of early experiments, it became clear that no acceptable explanations for this type of superconductivity could be achieved without the understanding of the normal-state features. For this reason it is imperative that also in this chapter, the variation of physical properties in both the normal state and the superconducting state are considered. As will be seen, for both regimes, new lines of thoughts in theoretical interpretations of the experimental data were followed.

3.1 Crystallography and some basic physical properties In this section, the most important and best-studied varieties of copper oxide compounds which have been found to be superconductors are introduced. They are briefly discussed not only with respect to their chemical composition, their crystal structure, and some of their basic phy­ sical properties in the normal state, but also with regard to the influence of the chemical composition on their critical temperatures Tc. 3.1.1 La2–xMxCuO4–y (M = Sr, Ba) A largely unknown early study [38] of substances of this type with M = Ca, Sr, Ba, and Pb revealed a metallic conductivity of some varieties of this series, simply in the form of a decreasing electrical resistivity ρ with decreas­ ing temperature, that is, ∂ρ/∂T > 0. Unfortunately for those workers and presumably because of a lack of the necessary cryogenic fluids, the investi­ gation was limited to temperatures T > 200 K. Nevertheless, the report contained important information concerning the crystal structure of this series of compounds. A few years later, a more detailed investigation [39] confirmed and extended these early results. Following their line of thoughts, these data were intriguing enough for Bednorz and Müller and lead them to extend the measurement of the resistivity of this type of material to lower temperatures. In this way they found the first evidence for the onset of superconductivity in these oxides between 30 and 40 K [9]. Ternary La2CuO4 (214-type compound) crystallizes in a tetragonal K2NiF4-type structure but at lower temperatures, at approximately 530 K, adopts an orthorhombically distorted version of this structure [40, 41]. Although a common count of the expected number of conduction

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electrons per formula unit would qualify this compound to be a metal, it is an insulator and, depending on the oxygen content, orders antiferromag­ netically with onsets in the range of 200–300 K [42]. The onset temperature Td of the orthorhombic distortion can be reduced by partly replacing trivalent La by a divalent alkaline-earth element. For x exceeding 0.2, the tetragonal structure is stable down to very low temperatures [43]. A sche­ matic view of the corresponding unit cell of the crystal lattice, adopting the so-called T structure, is shown in Fig. 4.3 [44]. The arrangement of atoms may be regarded as a stacking of different planes which are also components of the earlier mentioned perovskite structure. Between two CuO2 planes, the insertion of two LaO planes instead of only one weakens the threedimensional (3D) character of the structure and hence anisotropies in the physical properties may be expected. This expectation was later confirmed by numerous experiments on single-crystalline specimens. Typical for this structure is the octahedron formed by oxygen atoms with a Cu atom in its center. Already very shortly after the discovery of superconductivity of Cu oxides, it was realized that the occurrence of superconductivity in these materials is intimately related to the density of itinerant charge carriers [45] which, as it turned out, could be easily varied by simply changing the chemical compositions of the substances. Indeed the replacement of trivalent La by divalent Sr or Ba results in a rapid suppression of the antiferromagnetically ordered phase and eventually, the creation of itinerant holes leads to metallic behavior and superconductivity. The schematic La (Sr)

La, Gd (Sr)

Nd (Ce)

T

T*

T′

O Cu La/M Nd/Ce M = Ca Sr Ba La/Gd (Sr)

Figure 4.3 Schematic representation of crystallographic unit cells of (214)-type cuprates (see Ref. [44])

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low-temperature phase diagram of this type of compounds is shown in Fig. 4.4 [46]. It is often presented as the generic phase diagram of cuprate superconductors, although for other varieties of such compounds, the same type of diagram exhibits different features. In this case, beyond the onset of superconductivity at some nonzero value of x, the critical temperature Tc first increases with increasing doping with holes but Tc(x) reaches a maximum and beyond that concentration xmax, in the so-called overdoped region, ∂T/∂x < 0 and eventually, superconductivity is no longer adopted beyond a certain doping concentration. Later experimental evidence implied additional boundaries in the diagram shown in Fig. 4.4; they will be discussed at a later stage in this chapter. The metallic phase is obviously obtained by doping an intrinsic insulator; in this case, as mentioned above, by the introduction of holes. The forma­ tion of the insulating phase in the parent compound is, at first sight, unexpected. Its occurrence may be traced back to the interaction among the conduction electrons which, in the simplest models of solids, is usually neglected. The problem was recognized many years ago and it was mainly N.F. Mott and J. Hubbard who pioneered related work toward a solution [47, 48]. In their honor, materials which exhibit this type of insulating ground state due to correlation effects among the conduction electrons are termed Mott–Hubbard insulators. The increasing metallicity upon doping is very well reflected in the x-dependence of ρ(T) of this compound series,

600 La2–xSrxCuO4

400 T (K)

Tetragonal

200

Orthorhombic

AF

Spin-glass SC 0

0.1

0.2

0.3

x

Figure 4.4 Schematic low-temperature phase diagram of La2–xSrxCuO4 (see Ref. [46]).

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4 La2–xSrxCuO4

Resistivity (10–3 Ωcm)

ρab 3

x = 0.10 x = 0.12 x = 0.15

2

x = 0.20 x = 0.30

1

0

0

200

400

600

800

1000

T (K)

Figure 4.5 Temperature dependences of the in-plane electrical resistivities ρab(T) of La-214 compounds with different doping concentrations (see Ref. [49]). 7

La2–xSrxCuO4 Resistivity (10–1 Ωcm)

6

x = 0.12 x = 0.15

ρc

5

x = 0.10

x = 0.20 x =`0.30

4 3 2 1 0 0

200

400 T (K)

600

800

Figure 4.6 Temperature dependences of the electrical resistivities ρc(T), perpendicular to the Cu–O planes of La-214 compounds with different doping concentrations (see Ref. [49]).

which is shown in Figs. 4.5 and 4.6 for the in- and out-of-plane resistivity of single-crystalline samples [49]. The same results for ρab(T) and ρc(T) demonstrated the expected strong anisotropy of the electrical conduction. Particularly intriguing is the strong

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1000

La2–xSrxCuO4 x = 0.12

ρc /ρab

x = 0.15 x = 0.20 x = 0.25 x = 0.30

500

0

0

200

400 T (K)

600

800

Figure 4.7 Temperature dependence of the anisotropy ratio ρc/ρab of La-214 single crystals with different doping concentrations (see Ref. [49]).

temperature dependence of this anisotropy, captured in the ratio ρc/ρab(T) shown in Fig. 4.7. Especially in material with a low doping concentration x, that is, in the underdoped regime, it is reflected in the characteristic difference in the temperature dependences of the resistivities either along or perpendicular to the basal plane. While along the planes, the conduction is metallic and ∂ρ/∂T > 0, ∂ρ/∂T perpendicular to the planes is negative at low temperatures. Nevertheless, superconductivity prevails at low tempera­ tures. This anisotropy was found to be even more pronounced in subse­ quent work on other cuprates, reaching values of the order of 105 for the ratio ρc/ρab (see, e.g., [50]). It was mainly this observation which implied the view that the transport properties of the subsystem of the itinerant charge carriers could not be described in the same way as the conduction electrons in simple metals. We address this issue later in this chapter. The transition from a magnetically ordering insulator to a metal upon enhancing x is, of course, also reflected in the features of the normal-state magnetic susceptibility (T). The Curie-Weiss-type temperature depen­ dence (T–Q)−1 of  due to local moments is replaced by a rather low magnetic susceptibility, reflecting the magnetic behavior of the conduction electrons. Not surprisingly, the temperature dependence of  in the metallic regime also depends on x. This is shown in Fig. 4.8, where it may be seen that for superconducting material at low doping, (T) actually decreases with decreasing temperature, clearly not typical for a normal metallic

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5 x = 0.26

χ (10–7 emu/g)

4

La2–xSrxCuO4

0.22 0.20

3

0.18

2

0.14 0.10

1

0.08 0 0

200

400

600

T (K)

Figure 4.8 Normal-state magnetic susceptibilities (T) of La2–xSrxCuO4 (see Ref. [51]).

material [51]. This trend is increasingly compensated with increasing x and only far within the overdoped regime, ∂/∂T < 0 across the covered temperature region. This obviously anomalous behavior was first not under­ stood but, with hindsight, indicated a feature of the normal state of these materials that will be discussed in Section 3.2.4. In passing we note that other 214-type compounds (Ln2–xCexCuO4–) may be synthesized. Here, the La site is occupied by either Pr, Nd, or Sm, that is, light rare-earth or lanthanide (Ln) elements. They, however, crystal­ lize with a somewhat different structure, termed T’ structure [44]. This tetragonal structure, also shown in Fig. 4.3, is stable down to low tempera­ tures. The oxygen environment of each Cu atom is a planar square and thus distinctly different from the octahedron in the T structure. The ternary compounds are again insulating antiferromagnets. Both the Cu and loca­ lized Ln moments order spontaneously at respective Néel temperatures. Metallic conduction and superconductivity is obtained by partial replace­ ment of the trivalent Ln ions with tetravalent Ce and, in addition, a slight reduction of the oxygen content. Formally, the resulting itinerant charge carriers are now electrons [52]. Yet another type of Cu–O coordination is possible in the so-called T structure of the 214 variety of compounds (Ln2–x–yCexSryO4–) [53]. It is also shown in Fig. 4.3. Part of the Ln sites are occupied by Ce and/or Sr. Inspection of Fig. 4.3 reveals that the unit cell T lacks inversion symmetry. This remarkable feature is usually considered as not favorable for

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superconductivity but with the proper values of x, y, and , superconduc­ tivity is, nevertheless, achieved. The Sr content must be large enough to allow for a sequential ordering of (Sr,Ce) and (Nd,Ce) planes as shown in Fig. 4.3. 3.1.2 MBa2Cu3O7–δ, MBa2Cu4O8 Structurally, these two types of compounds, where M may be Yttrium or any rare-earth element except Ce and Tb, are related, but it turns out that the latter is chemically much more stable than the former. The positive pressure effect on Tc, that is, ∂Tc/∂p > 0 in the original 214 compounds [54] prompted early work aiming at preparing cuprates with a small cation such as Y. The result was a new type of compounds with another crystal structure but the really exciting feature was the climbing of Tc to temperatures above the boiling point of liquid nitrogen [12]. It was this observation which really boosted superconductivity in cuprates into the limelight and awakened enormous public interest. Meanwhile, compounds of this type, particularly for M = Y, have been studied in many different ways and their physical properties are known in great detail. Both structures may be regarded as stackings of different layers with different chemical compositions. For YBa2CueO7–∂ (YBCO-123) the stack­ ing sequence is Y–CuO2–BaO–CuOx–BaO–CuO2–Y, again containing elements of the perovskite structure mentioned earlier [55]. The symmetry of the structure is orthorhombic and the unit cell, assuming  = 0, is schematically depicted in Fig. 4.9. Two inequivalent Cu sites are most easily distinguished by the difference in their oxygen environment. The M atoms are captured between two identical blocks, each containing a CuO2 plane with the Cu(2) sites, a BaO plane, and a plane consisting of Cu–O chains along the b direction of the orthorhombic structure with the Cu(1) sites. Thus the oxygen coordination at the Cu(2) sites is pyramidal but linear at the Cu(1) sites. It is the missing oxygen between the chains, that is, along the a direction, which provokes the orthorhombic distortion of the crystal lattice. At the melting temperature, the compound crystallizes with a hightemperature tetragonal structure which, at approximately 750°C, transforms into the low-temperature orthorhombic structure via a rearrangement of atoms on the oxygen sublattice. By simply reducing the oxygen content, this rearrangement can be reversed at lower temperatures. For  = 0.6, the structure is again tetragonal also at low temperatures. Depletion of oxygen produces vacancies in the Cu–O chains which redistribute on that plane in such a fashion that on the average, oxygen atoms occupy sites along the a

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Ba/Y Cu O c

Cu(2) b

Y a

O(2) Ba

O(3) O(4)

O(1)

Cu(1)

δ=1 δ=0 Figure 4.9 Schematic representation of the crystallographic unit cells of YBa2Cu3O7– for (a)  = 0 and (b)  = 1, respectively.

and b directions with equal probability [56]. If  = 1, the Cu–O chains are fully depleted of oxygen; the resulting atomic arrangement in the unit cell is also schematically shown in Fig. 4.9. The variation of oxygen between  = 0 and  = 1 not only affects the crystal structure but also alters the ground state of the resulting materials. Before these aspects are addressed, some additional information with regard to the crystal structure in this type of compounds is in order. A special feature of this 123-compound series is the formation of superstructures in different regimes of oxygen content, leading to inhomo­ geneities and different orthorhombic structures, depending on the value of . A detailed discussion of these aspects can be found in Ref. [57]. A schematic representation of the unit cell of the chemically less fragile compound MBa2Cu4O8 (YBCO-124) [58, 59] is shown in Fig. 4.10 [60]. With respect to crystal growth, these compounds have the advantage to grow without twin boundaries which are a major obstacle in the growth of single crystals of the 123 variety where special measures are needed to avoid these for reliable transport measurements on single-crystalline samples. The addition of a second chain leads to a structural subunit consisting of Cu–O ribbons rather than chains. This structural alteration enhances the aniso­ tropy of the material, especially in the planes perpendicular to the c-axis. The quarternary system M–Ba–Cu–O allows for the formation of yet

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Cu1, O4 Ba, O1 Cu2, O2 Y

Figure 4.10 Schematic representation of the crystallographic unit cell of YBa2Cu4O8 (see Ref. [60]).

another stable compound with yet another chemical composition, namely Y2Ba4Cu7O14– or, in short notation, YBCO-123.5 [61, 62]. The unit cell results from the stacking of the unit cells of YBCO-123 and YBCO-124 on top of each other. This variety contains all previously identified Cu–O elements, that is, CuO2 planes, Cu–O chains, and Cu–O ribbons. If, as mentioned above, an element of the rare-earth series, except Ce or Tb, is chosen for M, the M site carries a local magnetic moment due to the incomplete filling of the 4f orbitals. Local magnetic moments in concentra­ tions of this order usually prohibit the onset of superconductivity in con­ ventional superconductors. This is clearly not the case here [63]. It appears that the magnetic ions and their electronic environment are well separated from the essential Cu–O subunits of the lattice and in this way do not significantly affect the formation of the superconducting condensate. One particular case is M = Pr. This compound is not superconducting even at low temperatures and it is still not quite clear why. 3.1.3 AmM2Rn–1CunO3n+m+1 Many of the now known cuprate superconductors may be classified under this heading. They were the result of intensified efforts to synthesize new layered copper oxide compounds and it so happened that some of them showed Tc’s exceeding 100 K [64–67]. So far compounds with A = Bi, Tl,

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Hans-Rudolf Ott

or Hg; M = Sr or Ba; and R = Ca or a heavy rare-earth element have been identified. They are important not only because some of them exhibit the highest critical temperatures Tc for superconductivity known today but also because some of them are believed to offer potential for technical applica­ tions. For A = Hg, m = 1; for Bi m = 2; and for A=Tl, m = 1 or 2 is possible. Bi-based compounds with m = 1 have been prepared but none of them exhibits superconductivity above 1.5 K [68]. Usually the parameter n may vary between 1 and 4 but sometimes (see below) even higher values seem possible. The parameter n indicates the number of CuO2 planes in the unit cell and often, in the context of superconducting cuprates, the term n-layer compound is used. These planes are separated by n – 1 layers formed by Ca ions and their number has been found to influence the magnitude of Tc significantly. The first such example was found in Bi-based cuprates where the Tc of the order of 20 K for Ca-free material [69] could be raised to temperatures above 80 K by intercalating Ca sheets between the Cu–O planes [64]. The structures of these compounds may be regarded as resulting from sequential stackings of elements characteristic for either the perovskite or the NaCl structure (see, e.g., [70]). The stacking sequences are often interrupted by crystallographic shears in the form of shifts between adjacent layers. Because of the relatively large number of chemical components and the variety of different structures, the preparation of these materials often results in multiphased samples. It has been noted that partial replacements of the above-mentioned constituents with other elements is possible. An impor­ tant example is Pb which may be used to substitute Bi. In particular it serves as a stabilizer for favorable phases, such as the Bi-2223 phase [71]. Since a comprehensive compilation of the atomic arrangements in the crystal lat­ tices of Bi- and Tl-based cuprates is given in Ref. [60], we refrain from displaying all these structures here. It turns out that most cuprate superconductors with a critical tempera­ ture exceeding the boiling point of nitrogen have at least two CuO2 planes in their crystallographic unit cell. It was therefore a surprise when the onset of superconductivity at a temperature of 94 K was reported for HgBa2CuO4 [72], a so-called single-layer compound. Subsequent work which aimed at enhancing the number of CuO2 planes per unit cell was soon successful and the expected increase of Tc significantly above 100 K was achieved [13]. The chemical composition of this compound series is HgBa2Can–1CunO3n+2. A schematic representation of the stacking of the individual layers with increasing n is shown in Fig. 4.11. As may have been expected, the idea that

99

High-Tc Superconductivity

Hg-1245 Hg-1234 Hg-1223 Hg-1212 Hg-1201 Hg Ba Ca Cu O

Figure 4.11 Schematic representation of the crystallographic unit cells of Hg-12(n–1)n cuprate compounds.

simply enhancing n would finally lead to a continuous increase of Tc was not corroborated. The highest Tc, of the order of 135 K, is achieved for n = 3. A more detailed sketch of the unit cell of this compound is shown in Fig. 4.12. Although in later work, thin layers of material with n exceeding 4 were synthesized, no further enhancement of Tc was achieved [73]. It turned out that with increasing n > 3, Tc continuously decreases. It may well be that a large number of Cu–O layers (n > 3) and the required interstitial Ca layers prevent an optimal distribution of itinerant electronic charges.

HgOδ BaO CuO2 Ca CuO2 Ca CuO2 BaO HgOδ

Figure 4.12 Schematic representation of the crystallographic unit cell of HgBa2Cu3O8+.

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Hans-Rudolf Ott

Ca/Sr Cu O c b a

Figure 4.13 Schematic representation of the crystallographic unit cell of Ca1–xSrxCuO2 (see Ref. [74]).

3.1.4 Ca1–xSrxCuO2 Recalling the above remarks concerning the number of CuO2 planes per unit cell, the structure of this type of compound is of importance because it may be viewed as the realization of a compound with infinitely many such planes. If the concept of n CuO2 layers separated by n – 1 Ca interlayers is taken to the limit of n >> 1, the stoichiometry of the compound will approach that of CaCuO2. With respect to the perovskite structure ABO3, the Ca(Sr) layers may be regarded as AO layers from which all oxygen atoms have been removed. CaCuO2, a hypothetical compound with tetragonal structure, has not yet been synthesized but it has been found that small amounts of Sr on Ca sites suffice to stabilize the structure which is shown in Fig. 4.13. It proved possible to grow single crystals [74] and later work employing high-pressure synthesis procedures demonstrated that in this way the compositional range of stability may be extended considerably [75].

3.2 More on normal-state properties of doped Cu-oxide compounds 3.2.1 Electrical transport An early recognized puzzle in understanding the physical properties of these cuprate materials was the temperature dependence of the electrical resistivity ρ(T) of the doped compounds. A representative example is shown in Fig. 4.5. The electrical resistivity of common metallic materials below room temperature is in the range of 100 μΩcm or less and decreasing with temperature. Because of intrinsic disorder, alloys often exhibit values of ρ

101

High-Tc Superconductivity

exceeding 200 μΩcm. In these cases, the mean free path of the conduction electrons is of the order of interatomic distances and thus ρ(T) is either constant or even increases slightly with decreasing temperature [76]. As may be seen in Fig. 4.5, the values of ρ above Tc are in the range of mΩcm and this is also true for all the other cuprate superconductors known today. In spite of this overall high electrical resistivity, ρ(T) varies significantly with temperature by decreasing approximately linearly with decreasing tempera­ ture. Aspects of the large anisotropy were already addressed above. Both these features are not easy to reconcile with the common view on electrical conduction in metals where the itinerant electrons are modeled, depending on the strength of the interaction between the electrons, as a Fermi gas or Fermi liquid and its interaction with the crystal lattice, providing the main source of scattering. Anomalous features were also observed in early experiments probing the Hall response of these compounds. Above Tc, the Hall coefficient RH was found to increase considerably with decreasing temperature [77]. For com­ mon metals and in the simplest approximation, R−1 H is a measure for the concentration of itinerant charge carriers and therefore is not expected to change with temperature. The description of thermally varying values of RH requires that details of the scattering in the electronic transport need to be taken into account. The Hall coefficient and the Hall resistivity ρxy are related via ρxy ¼RH ⋅ H

½4:2

where H is the externally applied magnetic field along the z direction. A number of subsequent measurements of the Hall effect confirmed this trend and results obtained from single-crystalline samples of YBa2CuO7– indi­ cated that, for currents parallel to the CuO2 planes and the magnetic field oriented along the c-axis of the crystal lattice, the Hall constant is positive and varies as T−1 [78]. Examples for this behavior, plotted in the form of R−1 H (T), are shown in Fig. 4.14 for single crystals of YBCO-123 and Bi-2212 [79]. The configuration with both current and magnetic fields oriented in the plane but perpendicular to each other, such that the Hall field EH is directed along the c-axis resulted in a negative and temperatureindependent Hall constant between Tc and 400 K [80]. Equally puzzling is the temperature variation of the in-plane Hall angle   ρxy QH ¼ arctg ½4:3 ρxx

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Hans-Rudolf Ott

RH–1 (kOe/μΩ cm)

30

25

20

Bi2Sr2CaCu2O8+δ YBa2Cu3O7–δ

15

100

200

150

250

T (K)

Figure 4.14 Temperature dependence of the in-plane Hall constant RH of single-crystalline YBa2Cu3O7 and Bi2Sr2CaCu2O8, plotted as R−1 H (T) (see Ref. [79]).

in the form of cotQH = ρxx/ρxy, where ρxx is the resistivity measured along the direction of the applied current. As may be seen in Fig. 4.15, this parameter varies very closely to T2 over a wide temperature range in the normal state of both YBCO-123 and Bi-2212 [79]. This type of Hall response is quite generally observed for these cuprates and hence taken as an additional evidence for an anomalous behavior of the subsystem carrying the electronic transport in these materials. In essence it is argued [81] that 350 300

cotθH

250 200 150 100

Bi2Sr2Cacu2O8+δ (H = 50 kOe) YBa2CU3O7–δ (H = 55 kOe)

50 1

2

3

4 5 T 2(104 K2)

6

7

8

Figure 4.15 Temperature dependence of the in-plane Hall angle H of single-crystalline YBa2Cu3O7 and Bi2Sr2CaCu2O8 in the form of cot H versus T2 (see Ref. [79]).

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High-Tc Superconductivity

these observations require that two different scattering mechanisms governing the in-plane electrical transport with different scattering rates τtr −1 and τH −1 have to be considered. As implied by the general feature of ρab(T), the longitudinal scattering rate τtr ~ T. Then, in order to explain the data in Fig. 4.15, the transverse scattering rate τH must vary as T2. As pointed out by Anderson [81], this all is compatible with postulating that the itinerant carriers can be modeled as quasiparticle-like excitations, the so-called spinons, of a two-dimensional (2D) Luttinger liquid as opposed to the Fermi-liquid­ type features of conduction electrons in common metals. Considering the inevitable presence of defects or impurities, the model predicts that cotQH = a T2 + C, with a as a constant parameter and C increasing with increasing impurity content. This latter behavior was actually observed by experiment. 3.2.2 Magnetism As mentioned above, the magnetic response of the normal state in the form of the magnetic susceptibility (T) is anomalous in the sense that it exhibits a considerable temperature dependence as opposed to the almost temperatureindependent Pauli-type susceptibility of common metals. Particularly in the underdoped regime, (T) decreases significantly with decreasing temperature. In metallic substances, the nuclear magnetic resonance (NMR) response is, to a large extent, influenced by the presence of itinerant charge carriers. First they contribute to the local magnetic field at the sites of the probed nuclei and thus provoke a shift of the resonance field, well known as the Knight shift K = ΔH/H0. Here, H0 denotes the resonance field of the nuclei in a non­ metallic environment and ΔH is the shift of the resonance field if the resonance is mapped at constant frequency. In the simplest case, K is given by [82] 8 hjuk ð0Þji2EF se ½4:4 3 and is thus related to the spin susceptibility of the conduction electrons K¼

se ¼ 2B2 DðEF Þ hjuk ð0Þji2EF

½4:5

is the average density of conduction electrons occupying where states at the Fermi energy, μB is the Bohr magneton, and D(EF) denotes the density of electronic states at the Fermi energy. In this approximation, se is obviously temperature independent and hence also K is expected not to vary with temperature. Conduction electrons are also the main actor in the NMR spin-lattice relaxation process in metals. Again, in the simplest approximation, due to Korringa, the relaxation rate T −1 1 is expected to

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Hans-Rudolf Ott

vary linearly with temperature, such that (T1T)−1 is approximately constant (see, e.g., [83]). This so-called Korringa constant is related to the Knight shift via ðT1 T Þ − 1 K 2

½4:6

Since, as shown above, se (T) of these cuprates does exhibit anomalous features, it is not surprising that the experimentally probed NMR response, usually from nuclei of Cu and O isotopes in these compounds, also deviates from that just quoted above. The interpretation of such data is not straight­ forward, however. Because these cuprates are structurally and electronically complicated materials, the total resonance shift, for instance, is the result of various different contributions. Next to the spin, orbital degrees of freedom of the itinerant carriers also have to be considered. Recounting the atomic arrangements of the structural unit cells shown above, it is obvious that Cu and O nuclei occupy different lattice sites with different local environment. Measurements of NMR are probing the response of the nuclei locally and hence also the influence of the local environments on K and T1 has to be taken into account. A clear case for a strongly temperature-dependent resonance shift was identified for the intrinsically underdoped compound YBCO-124 and the data for Cu and O nuclei, analyzed for different components, are displayed in Fig. 4.16 [84]. The reduction of K with decreasing temperature below 0.20

0.25 0.15 0.20

0.15

17

0.01 0.05

Kspin (%) ab

Kspin (%) oo

0.10

17

0.10

17

17

Kspin (%) iso

Kspin (%) ax

0.02 0.15

0.10

0.05 0.05 63

0.00

0.00

0.00

0.00

0

100

spin Kabspint 17Koo

200

17

spin Kiso

17

Kaxspin

300

T (K)

Figure 4.16 Temperature dependence of various contributions to the Knight shift of NMR signals from Cu and O nuclei in YBa2Cu4O8 (see Ref. [84]).

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High-Tc Superconductivity

Knight shift (%)

0.3

0.2

0.1

0.0

0

100

200

300

400

T (K)

Figure 4.17 Average temperature dependences of the magnetic shifts of the NMR signals from Cu nuclei on planar Cu sites of YBa2Cu3O7 (broken line) and YBa2Cu4O8 (solid line) (see Ref. [85]).

room temperature but above Tc is substantial. Similar observations were made for underdoped YBCO-123 and Bi-2212. The data is of quite different character if K is measured for optimally doped or overdoped material. A schematic representation of this difference is displayed in Fig. 4.17, where K(T) is shown for nuclei of Cu ions occupying sites in the CuO2 planes for optimally doped YBCO-123 and YBCO-124 [85]. Likewise, deviations from the quoted Korringa behavior of the spin­ lattice relaxation rate T −1 1(T) were observed. A selection of data, all obtained from 63Cu nuclei on planar Cu sites, is shown in Fig. 4.18 [86]. Included are some results that were obtained by probing cuprates that are not superconducting at low temperatures. Quite generally the relaxation is much faster in the compounds which are superconductors. At least at low temperatures the relaxation of the nonsuperconducting materials is of Kor­ ringa type, but clearly not so for the superconducting compounds. For the latter, T −1 1(T) = D + cT at elevated temperatures, indicating that at least two independent relaxation channels have to be considered. Although the second term may be ascribed to some sort of Korringa relaxation, the first, temperature-independent contribution, is obviously of different origin, possibly based on the spin dynamics of localized d-electrons on the Cu ions. As may be seen in Fig. 4.19, the anomalous character of T −1 1(T) is emphasized if the Korringa-type contribution is subtracted from the

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Hans-Rudolf Ott

10

1/T1 (103s–1)

La1.85Sr0.15CuO4

YBa2Cu3O6.52

5

YBa2Cu3O6.91

LaBa2Cu3O6.9 La1.7Sr0.3CuO4

0

0

100

200

La4BaCu5O13+δ

300

400

T (K)

Figure 4.18 Temperature dependence of 63Cu NMR spin-lattice relaxation rates for nuclei on planar Cu sites in different cuprate compounds. The solid lines are to guide the eye and the broken lines represent the linear-in-T Korringa-type variation (see Ref. [86]).

6

La1.85Sr0.15CuO4

5

(1/T1)Ks (103s–1)

4 YBa2Cu3O6.52

3 YBa2Cu3O6.91

2 LaBa2Cu3O6.9

1

0

La1.7Sr0.3CuO4

0

100

200

300

La4BaCu5O13+δ

400

500

T (K)

Figure 4.19 Same data as shown in Fig. 4.18, after subtraction of the Korringa-type background (Ks). The resulting data are interpreted as being due to Cu 3D electrons; the solid lines are to guide the eye (see Ref. [86]).

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High-Tc Superconductivity

experimental data. The strong reduction of T −1 1 well above the respective onset of superconductivity signals a loss of magnetic degrees of freedom in the electronic subsystem [86]. In retrospect, data of this type provided one of the first indications for the formation of a pseudogap in the normal-state quasiparticle excitation spectrum, a feature to be discussed in Section 3.2.4. 3.2.3 Density of electronic states The physical properties of a metal are, to a large extent, governed by the density of states D(E) that itinerant charge carriers may occupy, in particular those with energies E close to the Fermi energy EF. The value of this parameter depends not only on the density of conduction electrons but also on the strength of the interactions between them. The most direct way to access this parameter experimentally is offered by measuring the specific heat C at low temperatures, because the electronic part Cel of C is directly proportional to D(EF) and of the form 2 C el ¼ T ¼ 2 k2B DðEF Þ⋅T 3

½4:7

The measurements have to be done while the metal is in its normal state but at low temperatures, certainly much below the Debye temperature D, because only in this regime a separation of the electronic and the lattice contributions to the total specific heat is feasible. In addition, only for T << TF (EF = kBTF), the temperature dependence of Cel = gT is particularly simple. With this in mind it is immediately clear that it is difficult to access D(EF) for the copper oxide superconductors. In the range of their Tc values, of the same order of magnitude as D, the total specific heat Ctot is completely dominated by the specific heat due to lattice excitations. The problem of extracting reliable values for Cel(T) for these materials has been addressed in various ways and is described in the literature [87–89]. From those investiga­ tions,  values of the order of 25 and 10 mJ/moleK2 were obtained for optimally doped YBCO-123 and Bi-based material, respectively. This implies that in these cases, Cel in the normal state varies linearly with T even at temperatures of the order of 100 K. A relevant set of data obtained from measurements on overdoped YBCO-123 is displayed in Fig. 4.20a, where Cel/T is plotted versus T [90]. Here Cel(T) was obtained by subtraction of the lattice part from the original experimental data in an apparently controlled way [91]. Indeed, just above Tc, indicated by the anomaly in Cel(T), Cel/T is constant and does not change much with varying . A significantly different situation is met with Cel(T) data obtained from the same material but driven

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Hans-Rudolf Ott

5

0.25 0.20 0.29 0.19

(a)

δ = 0.04 0.080.13 0.08

4 3

γ (mJ/g–at K2)

2 1

Y0.8Ca0.2Ba2Cu3O7–δ

0

5

(b)

.32

.29

.37

4

.41

3

.44 .50

2

.60

1 0

.55

.67

0

20

40

60

80 T (K)

100

120

140

160

Figure 4.20 Temperature variation of the electronic-specific heat of Y0.8Ca0.2 Ba2Cu3O7– in the overdoped (a) and underdoped (b) regimes (see Ref. [90]).

into the underdoped regime by enhancing the value of  to above 0.3. Considerable deviations from Cel/T = constant above Tc are visible in Fig. 4.20b. As the NMR data quoted above, they again indicate an anom­ alous reduction of D(EF) in the normal state, distinctly different from the common features shown in Fig. 4.20a. 3.2.4 Pseudogap Various experimental results, among them the above-cited data on the temperature dependence of NMR relaxation and specific heat above Tc, provided indirect evidence for the reduction of electronic degrees of free­ dom with decreasing temperature in the normal state of superconducting cuprates. The onset of this reduction is in most cases hard to identify exactly and therefore the onset temperature T is ill defined. Nevertheless, its approximate dependence on the doping concentration, that is, T (x) with ∂T /∂x < 0, is well accepted and shown schematically in Fig. 4.21.

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High-Tc Superconductivity

T (K)

TN

T*

AF

Tc SC Hole doping

Figure 4.21 Schematic phase diagram of cuprate superconductors at low temperatures. TN(x) and Tc(x) represent the boundaries of the antiferromagnetically ordered and superconducting phases, respectively. The broken line T (x) indicates approximately the onset of the pseudogap formation upon decreasing temperature.

More direct evidence for this phenomenon was subsequently obtained from measurements employing spectroscopic methods and a first review of these observations was given in Ref. [92]. For our purposes, we briefly discuss a narrow selection of relevant experiments and concentrate on experimental techniques that were developed to full maturity after 1990. It may be regarded as a lucky coincidence that parallel with the develop­ ment of high-Tc superconductivity, major technical advances in the methods of photoelectron spectroscopy (PES) were made. Continuous progress in detector developments, leading to substantial improvements of the energy resolution in the recorded direction-integrated spectra, was complemented by the availability of intense photon sources in the form of electron synchrotrons together with insertion devices such as wigglers and undulators. This allowed for new opportunities in applying angle-resolved photoelectron spectroscopy (ARPES) with high momentum and energy resolution and adequate inten­ sities to keep the time span for recording a single spectrum of electrons emitted in a chosen direction at a feasible level. This new opportunity was used to probe the k-dependence of quasiparticle states at constant energy which resulted in 2D maps of Fermi edge EF(k), mainly of Bi-2212 single crystals. A review of corresponding efforts was published in Ref. [93]. Measurements of this type at different temperatures, T > Tc, revealed a gradual loss of the high-temperature Fermi arcs with decreasing temperature

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Hans-Rudolf Ott

M



— Y M

Γ

M Γ

— Y M



T ≤ Tc

Y





M Γ T < T*

M T > T*

Figure 4.22 Schematic representation of the growth of the Fermi surface boundary with increasing temperature, as obtained from ARPES experiments (see Ref. [94]).

[94]. This shrinking of the distinct Fermi boundary is schematically depicted in Fig. 4.22. It merges into the gap features of the superconducting state below Tc which will be discussed in Section 4.3.1. From this result alone, it appears that the pseudogap formation may be regarded as a precursor of the gap formation due to the onset of superconductivity. As often in the past, various kinds of tunneling experiments are well suited to probe gap structures in the electronic excitation spectrum, espe­ cially in connection with superconductivity [95]. With respect to the pre­ sent problem, we quote the results of measurements employing the technique of scanning tunneling microscopy (STM) at low temperatures, which allows for local tunneling experiments below room temperature. Measurements of the tunneling current I versus the applied voltage V and recording the differential conductivity ∂I/∂V as a function of V provide information on the energy dependence of the density of electronic states D(E) close to the Fermi energy EF. As shown in Fig. 4.23, a series of this type of spectra, recorded over an extended temperature range, gives clear evidence for the persistence of a gap feature to temperatures distinctly above Tc of the bulk material [96]. The data again imply an intimate relation between the two types of gaps and suggest a filling of the gap structure rather than the usually observed narrowing in energy with increasing temperature. Studies investigating the pseudogap phenomenon by STM techniques are reviewed in Ref. [97]. Other experiments, mainly probing optical properties and Raman spec­ troscopy, which are not mentioned here are covered in Ref. [92] and experiments probing Andreev reflection are reviewed in Ref. [98]. Explor­ ing the significance of this pseudogap formation in the normal state of cuprates, in particular also with respect to the occurrence of superconduc­ tivity, is still on the agenda of current experimental and theoretical investigations.

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High-Tc Superconductivity

δI/δVT/δI/δV82k

1.0

Tcbulk

0.5

T*

0 –100

–50

0

50

0.275 K 1.6 K 2.5 K 3.6 K 4.8 K 6.3 K 8.8 K 9.7 K 11.9 K 17.2 K 19.7 K 24.7 K 29.6 K 40.3 K 50.2 K 61.9 K 67.4 K 74.1 K 82.0 K

100

V (mV)

Figure 4.23 Temperature dependence of the normalized conductance versus voltage as measured by STM on a Bi-2201 single crystal. The broken line indicates the background signal measured at T = 82 K, to which all other curves are normalized. The gap feature persists to well above the critical temperature Tc of the bulk (see Ref. [96]).

3.2.5 Bottom line Based on this brief compilation of some unusual features of the normal state of cuprate superconductors, it may be concluded, as did many workers in the field, that the previously existing views on the metallic state derived from studies of common metals are insufficient for capturing the essence of the electronic properties of these cuprate materials. Among the numerous theore­ tical attempts to cope with this situation, some started from the notion that in these cases the hitherto successful Fermi-liquid model for the general descrip­ tion of the metallic state must be abandoned and other quasiparticle-like excitations have to be considered. Even if this view is accepted, the experi­ mental data seem to indicate that the character of these excitations changes with the level of doping, a feature that is particularly hard to understand. More recently, magneto-oscillatory effects have been observed in cor­ responding experiments on single crystals of underdoped YBCO at very low temperatures and very high magnetic fields [99, 100]. The observation of these effects is notorious for common metals and therefore, in our case, suggests that also for the electronic subsystem of these cuprates the

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Fermi-liquid model is applicable for describing the normal state close to T = 0 K. Some support for Fermi-liquid-type features of the quasiparticles was also found from experiments probing the thermal conductivity in the superconducting state at very low temperatures, to be discussed in one of the following sections. At temperatures of the order of Tc, however, a convincing modeling of the electronic properties is still missing. An outstanding feature of underdoped cuprates, not observed in studies of common metals, is the above-cited evidence for the reduction of the densitiy of states for quasiparticles at EF with decreasing temperature, but above Tc. As noted above, this feature is ascribed to the formation of a pseudogap in the quasiparticle excitation spectrum, indicating a loss or shift of electronic degrees of freedom. As may be seen in Fig. 4.20b, the reduction of the quasiparticle density of states is matched by a sizeable reduction of the specific-heat anomaly at and just below Tc. The latter indicates that the entropy loss due to the onset of superconductivity is also reduced as should be expected. While the anomalies are clearly documented for underdoped material, this is not really the case for overdoped compounds and different varieties of diagrams as displayed in Fig. 4.21 are found in the literature. Because the above-mentioned recently obtained evidence for Fermi-liquid-type beha­ vior of material in the underdoped regime, it seems questionable that the occurrence of the pseudogap is due to some unknown exotic quasiparticles as suggested in earlier work. Nevertheless, some of the anomalous properties of the normal state of cuprates of which a selection has been discussed above need a convincing explanation. If the pseudogap is simply due to the formation of pairs without the coherent properties of a true superconduct­ ing state, some theoretical explanation for the fact that not all superconduc­ tors exhibit this type of behavior in some form needs to be found. As long as these problems are without undisputed solutions, it seems difficult to achieve a convincing description of the superconducting state of these materials and to identify the reasons why Tc of these superconductors is distinctly higher than those of common metals.

4. SUPERCONDUCTIVITY OF HIGH-TC CUPRATES 4.1 General aspects Not only was the occurrence of superconductivity in these cuprate materials completely unexpected but what was even more intriguing was the unpre­ cedented high temperature at which the first onset was observed. The

High-Tc Superconductivity

113

subsequent rapid progress in raising Tc into the range of 100 K gave the final push to the obvious questions: is this the same type of superconduc­ tivity as that of common metals and alloys, and what type of mechanism is causing critical temperatures of this magnitude? At least concerning the first question, some undisputed statements can be made. Early experiments probing the flux trapping in superconducting rings of cuprates made it clear that the objects providing the supercurrent in these materials carry twice the charge of a single electron, that is, 2|e|, as do the Cooper pairs in common superconductors [101]. Next, attempts were made to check whether the magnetic flux expulsion from the interior of a singly connected superconducting sample in a static external magnetic field H or, more specifically, the decay of H over a characteristic length, the London penetration depth   2 1=2 mc λL ¼ ; ½4:8 4ns e2 with the relevant parameters m , the effective mass of the condensed quasiparticles; c, the speed of light; ns, the number of condensed quasipar­ ticles; and e, the electronic unit charge, is also observed for these materials. It did not take long before experiments using muon spin rotation (μSR) techniques established convincingly that the magnetic flux expulsion, that is, the Meissner–Ochsenfeld effect, is also present in cuprate superconduc­ tors [102]. Another early finding, the rather universal relation between Tc and the density of the itinerant charge carriers nc = 2ns near T = 0 K has been mentioned in Section 3.1.1. Corresponding data, derived from measure­ ments of the low-temperature μSR relaxation rate, 1 σ/ 2 /ns ½4:9 λ are shown in Fig. 4.24 (see, e.g., [103]); no similarly distinct trend was previously established for common superconductors, such as metallic ele­ ments. Recalling the layered structures of these cuprate materials, it should be noted that the magnitude of Tc is dictated by the number n of CuO2 planes per unit cell, at least for low values of n. An instructive example is the series of Hg-based cuprates for which, according to Fig. 4.25, Tc(n) exhibits a parabolic curve with ∂Tc/∂n < 0 for n > 3. As already mentioned in Section 3.1.3., the reason for the decrease of Tc above n = 3 is probably due to difficulties in optimizing the doping across the entire unit cell if the number of planes in it exceeds 3.

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150

2223

TC (K)

100 123 & 2212 50

214 0

0

1

2

3

4

5

σ (T → 0) (μs–1) Figure 4.24 Variation of the critical temperature Tc of cuprate superconductors with respect to the low-temperature μSR relaxation rate σ ∝ ns. The closed triangles represent La-214 compounds, the closed and open circles indicate Y-123 material, the crosses are data from Bi-2212 compounds, and the closed diamonds are for Bi-2223 material. The solid lines are to guide the eye (see Ref. [103]).

140

130

TC (K)

120

110

100

90

80 0

1

2

5 3 4 6 Number of Cu–O Layers (n)

7

8

Figure 4.25 Variation of Tc of Hg-based compounds of the form HgBa2Can–1CunO2n+2+ with respect to the number n of Cu–O layers in the unit cell (see Ref. [73]).

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High-Tc Superconductivity

As almost all multicomponent superconductors, the cuprates are type II superconductors. Simply from measurements of magnetization curves it may be concluded that the ratio between the upper and the lower critical field, Hc2/Hc1, is very large. Considering the relations [104]   ½4:10 Hc1 ¼ Φ0 =4λ2L ln ! and Hc2 ¼

Φ0 2 2

½4:11

with Φ0 denoting the flux quantum, it is clear that under these circum­ stances, the Ginzburg–Landau (GL) parameter  = λL / , the ratio between the penetration depth λL and the GL coherence length , must be very large or, in other words, λL >> . As a side remark we note that the same is true for all technically relevant superconducting materials. Analogous to the above-mentioned anisotropies of physical parameters in the normal state, considerable anisotropies have also been found for the characteristic lengths of the superconducting state, λL and . The most extreme anisotropies are again found for the Bi-based cuprates. Absolute values for λL were first calculated, as mentioned above, from results of relaxation rate measurements employing the μSR technique. Values between 150 and a few hundred nanometers were found for the penetra­ tion depths in the CuO2 planes and much larger values for those along the crystallographic c direction. From these values, which were later confirmed using other experimental methods, it may be concluded that the corre­ sponding coherence lengths are of the order of a few nanometers or less, that is, very small in comparison with the values of a few hundred nanometers or more, which are observed for common superconductors. This was confirmed by various types of experiments, indicating that , which is a measure for the spatial extension of the Cooper pairs, is very small, even less than the spacing between different atomic layers in the crystallographic lattice for along the c direction. Because in common superconductors, the size of the Cooper pairs is rather large, much larger than the interatomic distances, it is usually emphasized that the transition to the superconducting state should not be regarded as a condensation of bosons, that is, a Bose–Einstein condensation. An analogous argument is obviously less well justified for the superconducting state of these cuprates. Other characteristics of the superconducting state adopted by these cuprates are addressed later in this chapter.

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Concerning the second question, related to the mechanism triggering superconductivity in these materials, it is probably fair to say that still no consensus on how to answer it has been reached among the specialists. A major difficulty is, as we pointed out in Section 3.2.5, that the objects which undergo pairing are not yet really well defined. Without this insight, it seems difficult to establish a microscopic description of the pairing, the reason for it, and its consequences. Still unsettled is, for instance, the controversy concerning the issue whether lattice degrees of freedom are of importance in the formation of the superconducting state in these cuprates or whether the phenomenon is entirely due to interactions and instabilities affecting exclusively the ensemble of the electronic quasiparti­ cles, which exhibits anomalous properties already in the normal state. The most straightforward experiments for probing the role of the lattice excitations in common superconductors are measurements of the variation of Tc, either under external pressure or by replacing part of the atoms by isotopes of the same element. It turns out that external pressure mainly affects the Debye temperature D. To a good approximation, the electron– phonon interaction parameter is given by  λep ¼

A Mi 2D

 ½4:12

where A is a constant for a given crystal structure and Mi is the ionic mass. External pressure usually stiffens the lattice and the corresponding enhance­ ment of D accounts rather well for the usually observed pressure-induced reduction of Tc. For optimally doped cuprates, the pressure effect on Tc is usually quite small. An important exception is Hg-1223. A particularly large pressureinduced enhancement of Tc from 80 K at ambient pressure to 108 K at 10 GPa was reported for stoichiometric YBCO-124 (see, e.g., [105]), which is thought to be an intrinsically underdoped compound. A sizeable pressureinduced shortening of the Cu2-O1 bond most likely leads to a charge transfer into the CuO2 planes. The stoichiometric compound Hg-1223 appears to be optimally doped at ambient conditions but a significant enhancement of Tc from 134 K to above 150 K may be achieved by external pressures between 15 and 30 GPa [106–108]. Since pressure most likely mainly alters the degree of doping in the CuO2 planes, the observed pressure effects cannot be claimed to demonstrate the importance of the electron–lattice interaction.

117

High-Tc Superconductivity

However, measurements of the isotope effect, probing the ionic-mass dependence of the critical temperature which, because of the direct influ­ ence of M on D in Eq. [4.1], may be approximated by Tc /Miα

½4:13

are still considered as the most direct way to investigate the influence of the lattice excitations on the formation of the superconducting state. In interpreting the results of such measurements, it should be kept in mind that the simple form of Eq. [4.13] is only strictly valid for elements and that even under these circum­ stances, the classical value of the exponent α = –1/2 is verified in only a few cases [109]. Deviations have been observed (see, e.g., [110]) and were claimed to be due to large effective Coulomb interactions between the itinerant charge carriers and their variation upon isotope exchange on the ionic lattice [111, 112]. For this reason, an observation of α = 0 does not necessarily indicate that the electron– phonon interaction is negligible or not important for pairing. Severe complica­ tions are to be expected for multicomponent compounds because the isotope effect may be different for each of the j elements in the chemical composition and X j d ln Tc ¼ αj d ln Mi ½4:14 j

where αj are isotope-specific exponents, has to be introduced to capture the effect. At any rate, a variation of Tc upon an isotope exchange of an atomic species has to be considered as an indication for the influence of lattice excitations on the onset of superconductivity. Although traditionally the observation of an isotope effect has been taken as evidence for an electron– phonon-based mechanism of superconductivity, this need not necessarily be the case. Considering the remarks made above, the ionic-mass dependence of the Coulomb interaction between electronic quasiparticles, notoriously strong in these cuprate superconductors, may be the dominant reason for observing an isotope effect on Tc and hence this phenomenon may also be observed for purely electronically induced superconductivity [112]. Because of the chemistry of these cuprate superconductors, it is not easy to obtain reliable results on the isotope effect on Tc of these compounds. The most obvious choice for isotope exchange in these copper oxides is oxygen, from 16O to 18O and vice versa. Because of the high mobility of oxygen at elevated temperatures, removing and replenishing oxygen in these compounds is not a problem. However, because the oxygen content is known to affect the doping level quite critically, it is often argued that this oxygen-exchange procedure may lead to uncontrolled changes of doping and hence Tc. From

118

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a careful study [113] of the impact of exchanging 16O with 18O and back on Tc of La2–xSrxCuO4 with 0.1 £ x £ 0.15, it was concluded that the compounds with 18O had distinctly lower Tc’s than if the oxygenation was made with 16 O. Results of control experiments were interpreted as indicating that the isotope exchange process did not affect the doping level. However, significant shifts of the effective mass of the itinerant carriers, interpreted as polarons, were identified. Similar results were obtained in analogous investigations of other cuprates and therefore, for some of these high-Tc superconductors, the isotope effect, although small, has to be accepted as a reality. More recently, evidences for an isotope effect on the London penetration depth λL, and hence the ratio ns/m , were presented and discussed [114, 115]. It was argued that these results and the observation that the oxygen isotope effect increases with decreasing doping level support the view that the relevant quasiparticles involved in the pairing are Jahn–Teller polarons and hence the superconductivity in these cuprates is largely due to a favorable electron–phonon interaction [116]. In view of a possible interdependence between the formation of a pseudogap at T in the normal state and the onset of superconductivity at Tc, isotope effect experiments involving neutron scattering techniques prob­ ing possible variations of T upon exchange of the oxygen isotopes men­ tioned above were made [117]. It turned out that the invoked variations of T are much larger than those of Tc and of opposite sign. T of the 18O variety of a given compound is distinctly higher than that of its 16O counterpart and hence the parameter α in an equation equivalent to Eq. [4.14] and relating T to Mi is positive. In this sense, the dynamics of the lattice are to some extent also involved in the formation of the pseudogap in the normal state. Under­ standing the significance of the different sign of α in connection with T and Tc may well help to clarify the relation between the two phenomena. Theory-based arguments [118] suggested that these lattice dynamics may lead to structural and thus also electronic inhomogeneities, usually denoted as phase separation and stripe formation. Some support for this view was obtained from a number of experiments probing the structure of some of these materials in detail (see, e.g., [119]). Nevertheless, no consensus about the significance of these observations on the onset of superconductivity in these materials has been reached yet.

4.2 The phase transition In common elemental superconductors, the parameter  < 1 and hence the correlation length or, in other words, the spatial extension of the Cooper

119

High-Tc Superconductivity

pairs is rather large. The phase transition to the superconducting state is therefore, to a very good approximation, of mean-field type, that is, very sharp and not affected by fluctuations. It was pointed out in the previous section that the superconductivity of these cuprates is of extreme type II character with  >> 1. As a consequence, it has to be considered that although Tc is rather high, the coherence lengths are very short. Under these circumstances, it is expected that the phase transition to the super­ conducting state is significantly affected by fluctuations. The relevant para­ meter that measures the strength of fluctuations is the Ginzburg number [120]  2 Tc Gi ¼ =2 ½4:15 2 ð0Þ 2 Hc2 ab c where ab and c are the zero temperature coherence lengths parallel and perpendicular to the Cu–O planes, respectively. The temperature regime in which fluctuations are expected to influence experimental observations is ∣T – Tc∣ £ Tc · Gi. For common superconductors, Gi is of the order of 10−6 or less and, as mentioned above, the transition observed in various experi­ ments is very abrupt. This is clearly not the case for cuprates where the inherent large anisotropy of the materials even enforces fluctuation effects. In case of very large anisotropies, the relevant Ginzburg number is Gi2D ¼

Tc pffiffiffi εd 2

½4:16

 where ε ¼ Φ0 =4λL Þ2 and d is the thickness of an individual 2D Cu–O layer. An example for a resistive transition, here for the strongly anisotropic Bi-2212, is shown in Fig. 4.26 [121]. Transitions of this type monitored in

1.0 Bi-2212

ρ (arb.units)

0.8 0.6 0.4 0.2 0 0

50

100 150 T (K)

200

250

Figure 4.26 Temperature dependence of the in-plane electrical resistivity of Bi-2212, demonstrating the growing slope ∂ρ/∂T far above Tc (see Ref. [121]).

120

Hans-Rudolf Ott

1200

4 Bi-2212

1000

2

600 400

ρc (Ω cm)

ρab (μΩ cm)

3 800

1 200 0 0

0 100 200 300 400 500 T (K)

Figure 4.27 Anisotropy of the temperature dependence of the electrical resistivities parallel (ρab) and perpendicular (ρc) to the Cu–O planes of single-crystalline Bi-2212. Note the different scales for ρab and ρc, respectively (see Ref. [122]).

ρ(T) or the temperature dependence of the magnetization M(T) are annoy­ ing in the sense that they usually impede an exact evaluation of the critical temperature. The enhanced in-plane conductivity or, in other words, the reduction of ρab(T) already above Tc is often accompanied by an increase of the out-of-plane resistivity ρc with decreasing temperature. This intriguing coincidence is demonstrated in Fig. 4.27 [122]. Nevertheless, in spite of ∂ρc/ ∂T < 0 at Tc, the transition to the superconducting state is obviously still a 3D phenomenon. A further consequence of the material’s inherent anisotropies is the broadening of the transition in an external magnetic field. This was first observed in measurements of the electrical resistivities along and perpendi­ cular to the Cu–O planes in nonzero magnetic fields. The effect is particu­ larly prominent in data of ρ(T) for Bi-2212, as is demonstrated in Figs. 4.28 and 4.29 [122, 123]. First considering ρab(T), it may be seen that an increasing external magnetic field H along the c-axis does not simply shift the transition to lower temperatures but extends the drop in resistivity over a range in temperature of increasing width. Analogous features are seen in ρc(T) where, in addition, the shift of the transition reveals a further enhance­ ment of ρc, confirming the characteristically different in- and out-of­ plane electronic transport in these materials. The phenomenon of relatively high resistances in the nominally superconducting state is mainly due to vortex motion in the mixed state of these type II superconductors and, of course, a serious obstacle in technical applications in nonzero external magnetic fields.

121

High-Tc Superconductivity

60

Bi2.2Sr2Ca0.8Cu2O8+δ 50 40 30 H⊥ a,b

20

12 5 2T

ρab (μΩ cm)

10

12

52

0T

H || a,b

0 0.5

0.4

12

5

2

H|| a,b

0T

0.3 H⊥ a,b

12

5

2T

0.2

0.1

0.0 0

20

40

60

80

100

T (K) Figure 4.28 Influence of magnetic fields of different orientations on the in-plane resistivity ρab(T) at the onset of superconductivity of single-crystalline Bi-2212. The lower part of the figure emphasizes the low-resistance regime (see Ref. [123]).

Clear deviations from mean-field behavior are also seen in experiments probing the specific-heat anomaly at and below Tc. This is demonstrated in Fig. 4.30, where experimental data of the specific heat C(T) of a single crystal of YBCO-123 in the vicinity of Tc and zero external magnetic field [124] is compared with the result of a model calculation employing the socalled 3D XY model [125]. The justification for the relevance of this model, distinctly different from the mean field approximation, was given in Ref.

122

Hans-Rudolf Ott

30

Bi2Sr2CaCu2Oy 25

H||c

14T 12T

ρc (Ω cm)

20

10T 8T

15

6T 4T 3T 2T 1T 0.8T 0.6T 0.4T 0.2T 0.1T 0.01T

10 5 0 0

0.0T

20

40

60

80

100

120

T (K)

Figure 4.29 Influence of external magnetic fields on the out-of-plane resistivity ρc(T) of single-crystalline Bi-2212 (see Ref. [122]).

1490

Cp/T (mJ/mol–1 K–2)

1480

1470

1460

1450 85

90

95

100

T (K)

Figure 4.30 Specific heat anomaly at the superconducting transition of single-crystalline YBa2Cu3O7– in zero magnetic field. The solid line is a fit employing the 3D XY model mentioned in the text (see Ref. [124]).

123

High-Tc Superconductivity

1490 0 Tesla 0.5 Tesla

1480 Cp/T (mJ/mol–1 K–2)

1 Tesla 2 Tesla

1470

3 Tesla 4 Tesla

5 Tesla 1460

6 Tesla 7 Tesla

1450

1440

1430 82

86

90

94

98

T (K)

Figure 4.31 Influence of external magnetic fields on the anomaly of the specific heat anomaly at the transition to the superconducting state of single-crystalline YBa2Cu3O7–d (see Ref. [124]).

[126] for this case. Deviations from mean-field behavior are increasingly accentuated by external magnetic fields of increasing strength. The result of pertinent measurements, demonstrating the broadening and reduction of the anomaly, is shown in Fig. 4.31. The influence of the materials’ aniso­ tropy on the character of the phase transition is very clearly seen in the specific heat anomaly at the superconducting transition of Bi-2212, a compound exhibiting anisotropies in transport and other physical proper­ ties. As may be seen in Fig. 4.32, the anomaly is already very broad in zero magnetic field and is barely visible in magnetic fields that shift Tc by a few kelvins only. An intriguing aspect is the crossing point of all the curves at approximately 87 K which may be the result of some scaling phenomena related to phase transitions. Scaling phenomena at phase transitions is a broad topic in both experimental and theoretical physics (see, e.g., [127]) and in connection with cuprate superconductors, scaling typical for 2D systems has attracted a lot of attention (see, e.g., [128]).

4.3 Characteristics of the superconducting state 4.3.1 Gap function and gap anisotropy It was already mentioned above that superconductivity in these materials involves the pairing of quasiparticles which in the normal state exhibit an

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Hans-Rudolf Ott

1775

Cp/T (mJ/mol–1 K–2)

Bi2Sr2CaCu2O8

1770

1765 1760 1755 1750

80

85

90 T (K)

95

100

Figure 4.32 Specific heat of Bi-2212 across the transition from the normal to the superconducting state in different external magnetic fields. In zero magnetic field, Tc = 91 K. From top to bottom, the plots are for μH = 0, 0.1, 0.5, 1, 2, 3, 4, and 7 Tesla, respectively. Intriguing is the crossing point of all curves at approximately 87 K.

excitation spectrum which is, to some extent, anomalous but still contin­ uous in energy. In simple metals, the stability of the superconducting state carrying a persistent current is stabilized by the formation of a gap in the excitation spectrum of the quasiparticles below Tc. At this point, we do not consider the special case of gapless superconductivity (see, e.g., [129]) which is observed if by some measures the mean free path of the quasiparticles is made shorter than the GL coherence length. In common metals and alloys, the gap Δ(k) which opens at the Fermi energy EF of the quasiparticles is, in the simplest case, isotropic in k space or, at least, nonzero across the entire Fermi surface. Interaction anisotropies may lead to some small deviations from isotropy but, in general, they are difficult to identify experimentally. The conventional gap function of s-wave symmetry varies with temperature and Δ(T = 0) is a measure for the energy difference between the normal and the superconducting state. In the BCS theory, this condensation energy is

DðEF Þ⋅Δ2 ð0Þ Econd ¼ ½4:17 2 with Δ(0) as the gap value at T = 0 K. The flux expulsion phenomenon, intimately related to the superconducting state, is another manifestation of a nonzero condensation energy and the above-mentioned observation of the Meissner–Ochsenfeld effect thus requires the formation of a gap also for these cuprate superconductors. The most direct way to identify such a gap is to probe the quasiparticle excitation spectrum directly. Historically, the most

125

δI/δV (arb.units)

High-Tc Superconductivity

80

0

I (μA)

40

–40 0 –200 –100

0

100

200

V (mV)

Figure 4.33 Current I and differential conductivity ∂I/∂V versus voltage V of a break junction of a Bi-based cuprate superconductor. The latter clearly reflects the loss of available states at low energies (see Ref. [130]).

suitable and also successful experimental method employs tunneling of qua­ siparticles from a suitable electrode through a barrier into the probed super­ conductor [95]. Fig. 4.33 displays an example of early results of tunneling experiments using break junctions of a Bi-based cuprate superconductor [130]. The voltage dependence of the differential conductivity ∂I/∂V clearly reflects the loss of occupied states near EF, indicating the presence of a gap also in this case. Theoretically, the zero temperature value of the gap Δ(0) is found to be related to the value of Tc and the original BCS theory [3] gives   2Δð0Þ ¼ 3:52 ½4:18 kB Tc If this relation is, even approximately, valid in our case as well, the high values of Tc naturally imply equally enhanced values of Δ(0). This and the above-mentioned timely achievements in enhancing the energy resolution in measurements of energy spectra of photoemitted electrons via photo­ electron spectroscopy (PES) provided other experimental approaches to observe the opening of a gap in the superconducting state. Respective first PES experiments at temperatures above and well below Tc identified the expected distinct differences in the spectra at EF [131]. It seems fair to say that conventional wisdom would not have predicted the occurrence of superconductivity in these cuprates in the first place and certainly not at the relatively high values of Tc. Hence the discovery also provoked early speculations and discussions concerning the characteristics of

126

Hans-Rudolf Ott

this superconducting state. The characterization of a superconducting state usually involves a discussion of the details of the pair configuration and, intimately linked to that, the symmetry of the gap function Δ(k). The concept of symmetry breaking at phase transitions was introduced by Landau (see, e.g., [132]) and later, in more general terms, by Anderson [133]. In the wake of the BCS theory, some varieties of unconventional pairing and gap configurations were considered theoretically [134, 135]. The notion of unconventional superconductivity is chosen, if the symmetries or anisotropies of the Fermi surface and the gap function are different. This usually happens if, in addition to the inevitable breaking of the global gauge symmetry U(1) at the transition to the superconducting state, other symmetries, such as the time-reversal symmetry T or the spin-rotation symmetry R, are broken. The pairing configurations may adopt an even or odd symmetry, depending on the spin arrangement of the participating quasiparticles and either symmetry allows for numerous varieties of gap configurations which also depend on the symmetry of the crystal of the considered material (see, e.g., [136]). If the pairing interaction is, other than in the original BCS solution of the problem, k dependent, these gap configurations are distinctly different from the men­ tioned spherically symmetric solution of the original BCS theory. In these cases, gap nodes appear on points or lines on the Fermi surface, allowing for zero-energy excitations at these specific sites in k space. Gap nodes in the excitation spectrum have a drastic influence on the energy dependence of the quasiparticle density of states D(E) and hence the temperature dependence of physical properties in the superconducting state that are dominated by quasi­ particle excitations. Here E is measured from EF. An isotropic nonzero gap implies that D(E) = 0 for |E| < Δ, leading to an exponential reduction of excitations at temperatures T << Tc. The nodes, however, provoke that D(E) vanishes in power laws toward E = 0 and therefore, the respective properties exhibit power-law-type temperature dependences, which can be verified experimentally. This approach was chosen in first attempts to verify the unconventional nature of superconductivity in heavy-electron compounds, shortly before the advent of cuprate superconductivity (see, e.g., [137]) and it is also discussed in chapter 3 of this book. Nevertheless, a selection of experi­ mental evidence which indicates that the gap function Δ(k) of cuprate super­ conductors is not spherically symmetric and exhibits nodes is presented and discussed below. Early model calculations investigating the possible character­ istics of the superconducting states concluded that the formation of a gap with d-wave symmetry and hence with line nodes was a likely scenario for these cuprates [138].

127

High-Tc Superconductivity

4.3.2 Experimental methods for probing gap nodes A direct measure for the occupation of thermally excited quasiparticle states is the electronic component of the specific heat Cel(T) because it only depends on D(E). For conventional superconductors and T << Tc, it varies as Cel(T) ~ exp(–Δ/T) but in the presence of gap nodes, Cel(T) ~ Tn and the exponent n depends on the node configuration. Usually, gap nodes on points or lines are reflected in n = 3 and n = 2, respectively. The main obstacle to test the character of the superconducting state of cuprates in this manner is the difficulty to separate the electronic quasiparticle compo­ nent of Cel(T) from other, clearly dominating contributions to the specific heat, mainly due to thermal excitations of the crystal lattice. Nevertheless, attempts of this sort have been made. Results of measurements of C(T) of close to optimally doped YBCO-123 [139] and single-crystalline La1.85Sr0.15CuO4 [140], both in zero and nonzero magnetic field H, were analyzed and power-law-type variations of Cel(T) at T << Tc and H = 0 were claimed to have been observed. The analysis of Cel(T,H) data obtained for H ≠ 0 is even more involved, needs to consider excitations in the normal cores of vortices in the mixed state, and employs theoretical predictions for scaling laws to be obeyed [141, 142]. An example for the latter is shown in Fig. 4.34. In this sense, the interpretation of specific heat data, although simple in principle, is not straightforward but rather complicated in these

3.0

Cel = 0.91 H1/2 T 3T 5T 7T 9T

Cel/T 2 (mJ/K–3 mol–1)

2.5 2.0 1.5 1.0 0.5 0.0

0.0

0 .5

1.0

1.5

2.0

2.5

3.0

3.5

Z –1 = H 1/2 T –1 (T 1/2 K–1)

Figure 4.34 Scaling behavior of the specific heat of YBa2Cu3O7– at low temperatures and in different external magnetic fields (see Ref. [142]).

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Hans-Rudolf Ott

cases. The conclusion is the same from both types of analyses for the two materials and claims the presence of line nodes in the gap function Δ(k). Early indications for gap nodes were obtained from measurements of the penetration depth λL(T). As may be seen from Eq. [4.8], it depends on the ratio between the effective mass of the quasiparticles and the density ns of the pairs. The latter reflects the condensed pairs and is therefore temperature dependent and maximum at T = 0 K. As mentioned above, measurements of the relaxation rate in μSR were used to map the temperature dependence of λL. Also here, particularly important is the information at T << Tc and in order to achieve a high resolution in λL(T) in this regime, other types of experiments seem more appropriate. A favorable method employs measure­ ments of the microwave absorption of the superconducting specimen. It is well suited to measure the temperature-induced variation of λL rather than its absolute value. In this way ΔλL ¼λL ðT Þ−λL ðT0 Þ

½4:19

where T0 is an arbitrarily chosen temperature well below Tc, can be obtained. Respective data for the field penetration along the CuO2 planes of high-quality single crystals of YBCO-123 that was obtained from this type of measurements at temperatures above T0 = 1.3 K are shown in Fig. 4.35 [143]. It demonstrates that Δλab L increases linearly with temperature

80

YBa2Cu3O7

Δ λ (Å)

60

40

20

0

0

5

10

15

20

T (K)

Figure 4.35 Incremental increase of the in-plane penetration depth of single crystalline Y-123 upon increasing temperature. The different symbols represent data of different samples (see Ref. [143]).

129

High-Tc Superconductivity

between 1.3 and 20 K, that is, well below Tc of approximately 90 K. This result is compatible with line nodes in the gap function. The method may be considered as reliable because analogous experiments in the same work on a sample of a PbSn alloy, a conventional type II superconductor, resulted in an exponential increase of ΔλL at temperatures well below Tc = 7.2 K. From later experiments on cuprates, it was concluded that the reduction of the mean free path of the quasiparticles due to a controlled introduction of impurities results in ΔλL ~ T2 (see, e.g. [144]). At low temperatures, the transport of heat or energy in solids is usually based on itinerant quasiparticles and lattice excitations. It turns out that in the superconducting state of cuprate superconductors the thermal conductivity th in the Cu–O planes is mainly due to excited electronic quasiparticles and measurements of th ab(T) for T << Tc are expected to provide results that are dictated by the gap configuration Δ(k). For the case of a gap function with d-wave symmetry and line nodes in two dimensions, a general theoretical result [145] predicts that the heat transport carried by quasiparticles is inde­ pendent of the scattering rate for T approaching 0 K. The impurity-driven enhancement of the amount of quasiparticles in the node region just com­ pensates the impurity-imposed reduction of the mean free path, leading to a universal limit of the heat transport coefficient. Experimental evidence [146] for this type of behavior is shown in Fig. 4.36 for th(T) measured along the YBa2(Cu1–xZnx)3O6.9

λ / T (mW/K 2 cm)

2 x=0 x = 0.006 x = 0.02 x = 0.03

1

0

0

0.2

0.4 T (K)

Figure 4.36 Changes of the thermal conductivity along the a-axis of a singlecrystalline specimen of Y-123 at very low temperatures upon replacing small amounts of Cu by Zn (see Ref. [146]).

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Hans-Rudolf Ott

a-axis of single crystals of YBCO-123 with different small amounts of Zn replacing Cu. The data converge to a zero temperature limit of th/T which is independent of the impurity content and hence the scattering rate, as predicted. In connection with the problems of characterizing the normal state encountered above in Section 3.3, it is interesting to note that a more detailed analysis of such data [147] seems to indicate that the involved quasiparticles act as those of a Fermi liquid, at least at low energies, anticipat­ ing the much later identified quantum-oscillatory behavior in transport and magnetization which were mentioned in Section 3.2.5. Results of measurements of NMR and related relaxation processes in the superconducting state of metals were considered as early supporters of the validity of the BCS theory [148, 149]. The spin-lattice channel of the relaxation of nuclear spins via excited electronic quasiparticles is expected to be influenced by the absence or presence of nodes in the gap function. If gap nodes exist, the corresponding NMR relaxation rate T−1 1 (T), already briefly discussed in Section 3.3, is expected to decrease with decreasing temperature in a power-law-type fashion at T << Tc and not exponentially as for the case of a gap without nodes [136]. The data shown in Fig. 4.37 demonstrate quite clearly that the reduction of T−1 1 below Tc is not expo­ nential in 1/T [150]. The same data are significant in view of another NMR 105 63

Cu NQR H=0

TC = 38 K

104

TC = 92 K TC = 109 K

1/T1(s–1)

103 102

101

La0.925Sr0.075CuO4 YBa2Cu3O7

1

Bi⋅Pb⋅Sr⋅Ca⋅Cu⋅O

10–1 1

101

102 T (K)

103

Figure 4.37 Temperature dependences of the 63Cu NMR spin-lattice relaxation rates of three different cuprate superconductors (see Ref. [150]).

131

High-Tc Superconductivity

feature at the onset of superconductivity. The pairing configuration in the original BCS solution implies particular coherence effects, an exclusive feature of this s-wave type of paring, which influence the temperature dependence of properties of superconductors that are dictated by the absorption of energy via excited quasiparticles [104]. In the case of the NMR spin-lattice relaxation, one expects an abrupt increase of T−1 1 at Tc and, with decreasing temperature, a maximum and a subsequent reduction exponential in 1/T well below Tc [148]. It is obvious that this is not observed in the data shown in Fig. 4.37 and therefore it may be concluded that the pairing configuration of quasiparticles in the cuprates is different from that in common metals and alloys. The presence of nodes in the gap function is also indicated from results of NMR Knight-shift experiments [151, 152]. The above-mentioned experimental progress in ARPES was also instru­ mental in measurements of the gap configuration in k-space. The result of such an experiment probing Δ(k) is shown in Fig. 4.38 [153]. It confirms the vanishing of the gap at nodes but in addition it reflects very well a fourfold symmetry in the k variation of the gap and in this sense supports the view of a gap with d-wave symmetry and line nodes.

40

|Δ| (meV)

30

20

10

0

0

20

60 40 Angle (degrees)

80

Figure 4.38 The leading-edge shift of ARPES spectra of Bi-2212 relative to the position of the Fermi edge of Pt as a function of the average emission angle. The solid line is calculated assuming a d-wave-type gap function (see Ref. [153]).

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4.3.3 Experimental methods for probing the gap symmetry The results of all these experiments cited in the preceding section confirm that the gap function Δ(k) exhibits nodes. Some support is obtained for nodes along lines, compatible with an even or d-wave symmetry of Δ(k) and hence singlettype pairing of the quasiparticles. Taking into account the symmetry of the essential electronic orbitals in the Cu–O planes and considering various argu­ ments based on phenomenological and computational approaches lead to the conclusion that the pairing configuration and hence also Δ(k) adopt the dx2 − y2 symmetry. This implies that the gap function changes sign along the line nodes, synonymous with a discontinuous phase shift π of the order parameter and a feature that is not probed with the above-mentioned experimental methods. An unequivocal proof for the dx2 − y2 symmetry of the order parameter there­ fore requires another type of experiments, probing the relative phase of the gap function on different parts of the Fermi surface, separated by the gap nodes. Experiments of this type are based on the macroscopic quantum coherence of the superconducting state which is manifested in the flux quantization inside singly connected superconducting loops [154, 155] or in the Josephson effects [156] that form the basis of superconducting quantum interference devices (SQUIDs) [157]. The first suggestion for investigating unconventional superconductors in this way, at that time heavy-fermion superconductors, is due to Geshkenbein and Larkin [158]. The test involves the probing of phase coherence in closed superconducting loops, consisting of material of both the unconventional superconductor and some simple conventional superconduc­ tor. The concept was then adapted to the case of cuprate superconductors with the special aim to find evidence for the dx2 − y2 symmetry of the order parameter [159]. A schematic diagram of the set up is shown in Fig. 4.39, where it is assumed that the four lobes of the d-wave function are each oriented toward one of the edges of the sample. The arrangement asserts that one of the two Josephson contacts (1 or 2) is between two superconduc­ tors with opposite sign of the order parameter amplitude. Because of the phase change of π at this contact, the notion of π junction has been introduced [160]. Singly connected loops with none or an even number of π junctions exhibit a conventional flux quantization with an integer number ν of trapped flux quanta Φ0 = hc/2e, as observed in measurements of the critical current in loops of common superconductors. Loops with an odd number of π junctions, however, are expected to be different. The most stable situation, as indicated by maxima of the critical current in the loop, is established if the trapped flux Φ = (ν + ½)Φ0 is a half-integer multiple of flux quanta. Different varieties of experiments, pioneered by Wollman and coworkers [161], based on these

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IB

1 d-wave

Φ 2

s-wave

IB

Figure 4.39 Schematic SQUID-type configuration of a superconducting loop for phasesensitive tests of the order parameter symmetry of cuprate superconductors (see Ref. [159]).

facts were designed. All of them set out to measure the flux dependence of the critical current in the loop and the analyses of the results supported the assumption of an order parameter dx2 − y2 in cuprate superconductors. Details are described in the relevant literature which was reviewed in Ref. [162]. Instead of probing the trapped flux configuration with direct measure­ ments of the critical current in the loop, the trapped flux may also be measured indirectly, that is, inductively, with a scanning SQUID probe. The method makes use of the fact that a ring with an odd number of π junctions will always stabilize itself by generating a spontaneous current, even if the external flux is zero. The current-induced flux can be monitored with the scanning SQUID probe. In an experimental tour de force, many different experiments based on this approach were made and then reviewed in detail by Tsuei and Kirtley [163]. Thin small ring-shaped films of cuprate superconductors with diameters of the order of 50–100 μm were epitaxially deposited on single-crystalline SrTiO3 substrates with the Cu–O planes parallel to the substrate surface. Differently oriented substrates were mended in such a way that allowed for the preparation of different rings consisting of differently oriented arcs with contacts between them at the mending lines of the substrate. An example is shown in Fig. 4.40. In this way, different ring configurations with an even, including zero, or odd number of π junctions were fabricated. The trapped flux, induced in different ways, was then monitored with a scanning micro-SQUID probe. The results of the

134

0) (01

0)

(10

0)

60°

(010)

30°

(01

(10 0)

Hans-Rudolf Ott

(100)

Figure 4.40 Example of superconducting loops consisting of ring-shaped thin films of Y-123 deposited at chosen sites on a tricrystal substrate. Note the different crystallographic orientations of the components. The central ring is expected to have one π-junction (see Ref. [163]).

experiments were all compatible with a dx2 − y2 symmetry of the order parameter, both for hole- and electron-doped cuprate superconductors. While all these experiments confirmed the dx2 − y2 symmetry of the order parameter in the Cu–O planes, other tunneling experiments with the tunneling current along the c-axis gave results whose interpretations were less straightforward [164–166]. Some of them suggested a different symmetry for the order parameter and hence the gap function in this direction. It was postulated that a scenario of a mixed d- and s-wave symmetry of the overall gap function has to be considered [167]. The results of more recent experiments probing the temperature dependence of flux penetration into cuprate superconductors with μSR techniques were inter­ preted as to strongly support this conjecture [168].

4.4 Summary While the oxide superconductors cited in Section 2 are most likely con­ ventional superconductors, many different experiments gave evidence for nodes in the gap function of cuprate superconductors and therefore

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confirmed the unconventional nature of this superconductivity in the sense stated in Section 4.3. Much less clear is the situation concerning the mechanism which is responsible for the onset of superconductivity in these materials and at this time no consensus has been found to answer this question. Even without being completely understood, cuprate super­ conductors are, in principle, of high interest for technical applications. This is because at this time cuprates are the only class of materials for which superconductivity sets in at temperatures above the boiling point of nitro­ gen, a relatively cheap cryogenic liquid. This temperature range may also be accessed by techniques employing thermoelectric effects, another applica­ tion friendly cooling method. Naturally the lack of fundamental insight hampers a directed search for new materials with even higher Tc’s and therefore, a successful search still demands experience in materials science, some intuition, and, last but not least, some luck.

5. SUPERCONDUCTING FULLERIDES 5.1 Materials’ structure Shortly before the discovery of the cuprate superconductors, another class of new materials made the headlines not only in the scientific literature but also in daily journals and weekly magazines. These articles reported and dis­ cussed the discovery of a new type of stable molecules formed by carbon atoms and consisting of closed carbon networks with chemical compositions C60, C70, and others [169]. Their shape reminded of earlier architectural structures introduced by Mr. R. Buckminster Fuller and soon they were known under the name of buckyballs. For our purposes, only the C60 variety of molecules is of interest because up to present, only solids formed by them exhibit superconductivity. Under normal circumstances, a pure solid formed by a regular arrangement of C60 molecules is an electrical insulator (see, e.g. [170]). These solids, termed fullerenes, crystallize with an fcc structure, schematically shown in Fig. 4.41. Upon lowering the tem­ perature, they adopt a simple cubic structure at a transition temperature of 260 K which is accompanied by an orientational order of the individual C60 molecules [171]. Most likely inspired by the developments in the field of cuprate superconductors, metallic varieties of C60-based solids, called full­ erides, were obtained by doping [172] which is achieved, for example, by inserting atoms of alkali elements into the structure, such that compounds of the form A3C60 with A = K, Rb, Cs, or combination of these are obtained.

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Figure 4.41 Schematic view of the unit cell of fcc C60.

Another possibility is Na2AC60 where A is again one of the other three alkali elements [170]. From the chemical composition, it is clear that in these cases, the itinerant charge carriers are of electronic character. The crystal structure of these materials is of fcc type, with varying space group symmetries, how­ ever. Noncubic structures of superconducting fullerides were identified for Cs3C60 and NH3K3C60 under pressure. Simply adding alkali atoms does not always lead to metallic conductivity. Compounds with a composition of the type A4C60 usually crystallize with a bct-type structure and are insulators. More details on compositions and structures of C60-based superconductors can be found in Ref. [173].

5.2 Onset of superconductivity Immediately after having achieved metallic conduction in C60-based com­ pounds, the onset of superconductivity at, on conventional grounds, remark­ ably high temperatures was discovered [174, 175]. From numerous subsequent investigations, it is concluded that the magnitude of Tc is strongly

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40

Tc(K)

30

Na2RbxCs1–x C60 MxM´3–xC60(ambient pressure) Rb3C60 (p var.) K3C60 (p var.)

20

10

0 13.7

13.9

14.1

14.3

14.5

14.7

a(Å)

Figure 4.42 Onset temperatures of superconductivity of C60 compounds as a function of the lattice parameter. The variation of a is achieved by either doping or applying external pressure (see Ref. [170]).

linked to the lattice parameter in a rather universal, material-independent way, as exemplified in Fig. 4.42. The exceptions are limited to materials with rather complicated compositions. It turns out that an increasing separation a of the C60 molecules along the cube edge by changing the chemical composition results in a monotonous enhancement of the critical temperature [176]. Likewise, a compression of the lattice upon increasing external pressure leads to a reduction of Tc [177]. This equivalence of the influence of chemical and mechanical volume change on Tc is more an exception than a rule, as many such studies on other classes of super­ conductors have shown. Although the plot in Fig. 4.42 suggests that an enhancement of Tc is achieved by simply expanding the lattice, deviations from a universal Tc(a) and even bonafide exceptions with pressureinduced enhancements of Tc were indeed reported [178, 179]. Extended sets of Tc(n) data for a chosen alloy series, with n as the concentration of conduction electrons per unit cell, were obtained. From plots of the form of Tc(n)/T max where T cmax is the maximum critical temperature of the c series, it was claimed that the maxima of all considered Tc’s were reached if n = 3 and that no superconductivity was established if n < 2.5 or n > 4 [180]. For bulk material, the highest Tc was of the order of 40 K and was measured for the noncubic compound compound Cs3C60 [181].

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5.3 Selected physical properties As briefly mentioned above, metallic conductivity in C60-based materials is obtained via electron doping upon insertion of alkali-metal elements, resulting in compounds of the type A3C60. Some such combinations turn out to be electrical insulators, however. Results of calculations of the widths W of the essential electronic energy bands and of the on-site Coulomb correlation energies U indicate that the ratios U/W exceed the value of 1. It is thus argued that the metallic varieties of these compounds reside on the metallic side of a Mott–Hubbard-type metal insulator transition [182, 183]. Theoretical attempts to describe the electronic properties of these materials would therefore require that strong correlation effects be taken into account. With respect to superconductivity, another difficulty arises. It turns out that the energies of the essential phonon excitation branches are not very different from the calculated band widths W. This is clearly in conflict with the situation for common superconductors where this differ­ ence is considerable and, captured in Migdal’s theorem [184], is used as an argument for explaining the electron pairing as a consequence of a retarded electron–phonon interaction. Invoking this view in relation to supercon­ ductivity of these materials is thus rather questionable. Not surprisingly, the absolute values of the electrical resistivities ρ of these materials are relatively high and the corresponding electronic mean free paths are short, of the order of the separation between the molecular C60 clusters or even less. Examples of the temperature dependence ρ(T) of A3C60 films are shown in Fig. 4.43 [185]. Results of ρ(T) measurements on single crystals reveal values of the order of 1 mΩcm even at low temperatures and the observed temperature variations are sample depen­ dent. Therefore, it is difficult to identify the scattering mechanisms dominating this transport property. The electronic mean free path at room temperature must be very short but no trend to saturation of the resistivity is observed. According to Fig. 4.5, the same feature has been observed for ρ(T) of the cuprates. Nevertheless, it is quite unlikely that for the two types of materials the same scattering processes cause this phenomenon. In both cases, no convincing theoretical explanation for this behavior has yet been given. The magnetic susceptibility in the normal state is, quite generally, positive and almost temperature independent [186]. The latter is also true for the resonance shift in NMR experiments [187]. The electronic density of states D(EF), as calculated using Eq. [4.5], is considerably larger than the bare density of states Db(EF) resulting from band structure calculations, but

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6 5 Rb3C60

ρ (mΩ cm)

4 3 2

K3C60 1 0

0

100

200

300

400

500

600

T(K)

Figure 4.43 Electrical resistivities ρ(T) of thin films of K3C60 and Rb3C60, respectively (see Ref. [185]).

is consistent with estimates of D(EF) from spin-lattice relaxation rate data. As expected for metals, the relaxation rate T −1 1 (T) varies approximately linearly with temperature. The transition to the superconducting state is, as usual, most clearly reflected in ρ(T) and (T) but also manifest in a number of other experi­ mentally accessible properties which are dominated by electronic excita­ tions. One such case is the specific heat. Because these C60-based materials are fragile and very sensitive to air exposure, measurements of C(T) are technically rather difficult [186]. In the temperature range of the transition, the electronic part of C(T) is matched by the specific heat due to lattice excitations and, as shown in Fig. 4.44, the expected anomaly at Tc is almost masked by a large background of nonelectronic origin. Estimates of the electronic-specific heat parameter , which is proportional to ΔC(Tc)/Tc, and hence D(EF) indicate a rather low value of the effective electronic density of states at EF. Combining  and the average value of (T), estimates for the electronic mass enhancement (1 + λph) due to the electron–phonon interaction can be made. The resulting values between 0.5 and 0.7 are quite large. Theoretically, it was concluded that electron–phonon-induced super­ conductivity could not be achieved by a coupling to low-energy phonons but that instead the coupling to high-energy phonons was essential [186]. Measurements of the isotope effect were made by replacing 12C by 13C. The scatter in the data is substantial but claims for isotope-induced shifts of

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0.2

ΔC/T (J/mole K2)

K3C60

18.9K

0.1

0 10

15

20

25

T(K)

Figure 4.44 Specific heat anomaly indicating the transition to the superconducting state of K3C60. The solid line is to guide the eye and the broken line reflects the estimated background (see Ref. [186]).

Tc were made [188, 189]. Since no such changes were observed upon exchanging isotopes of the alkaline-metal elements, the importance of lattice modes invoking the C60 network is evident. Apart from delivering the itinerant charge carriers to the system, the influence of the alkaline atoms on Tc, other than via their size, is rather weak. The onset of superconductivity is also reflected in the decrease of the NMR spin-lattice relaxation rate T −1 1 upon decreasing temperature below Tc. The data shown in Fig. 4.45 [190] reveal a weak but distinct enhance­ ment of (T1T)−1 just below Tc, reflecting the earlier-mentioned coherence effect which is typical for conventional superconductors with an s-wave­ type gap function. This type of feature was confirmed by results of μSR experiments [191] and also the measured temperature dependence of the Knight shift K(T) was found to be consistent with this interpretation. It seems thus reasonable to qualify these alkaline-metal-doped C60-based superconductors as conventional s-wave-type superconductors. From the results of various types of experiments [190, 192, 193], the ratio 2Δ0/kBTc was extracted. Most of the reported values are between 3.0 and 4.2, indicating a moderate coupling strength between electrons and phonons. The characteristic lengths related to the superconducting state of these C60 materials were deduced from results of μSR [194, 195], NMR [187], and optical [196] experiments, supplemented by data of measurements of

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(T1T )–1(s–1K–1)

10–1

10–2

133Cs

10–3

87RB 13C

10–4 10

30

100

300

T(K)

Figure 4.45 Temperature dependences of NMR spin-lattice relaxation rates plotted as (T1T)−1 versus T as obtained from probing different nuclei in Rb2CsC60. Note the features just below the transition at approximately 30 K, providing some evidence for classical coherence effects mentioned in the text (see Ref. [190]).

the upper critical field Hc2(T) [197]. The spread of the values found for λL and is considerable. Values for λL(0) between 400 and 800 nm were reported and 0 of the order of 3 nm was deduced. The ratio of these two parameters, the earlier-mentioned GL parameter , is large and qualifies these materials as extreme type II superconductors. The large average value for λL(0) is compatible with the previously cited indications for a low concentration of itinerant charge carriers and the considerable enhancement of their effective mass. This brief overview of experimental results characterizing the super­ conducting state of these materials leads to the conclusion that C60-based superconductors are of conventional variety based on a at most moderately strong electron–phonon interaction [170, 198].

6. FINAL REMARKS The unexpected development, starting in 1986, which demonstrated that superconductivity above 30 K is possible not only triggered a frantic search for new materials but has also induced new ideas on electronic properties of solids in general. Topics that were addressed with particular vigor included the understanding of the metallic and the insulating state, respectively, and transitions from one to the other. The suitability of the previously very successful Fermi-liquid model for describing any metallic

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state was questioned. New ways to model the transition from a normal to a superconducting state, as well as to establish the character of the latter, have been considered.

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