The phenomenon of superconductivity became accessible due to the achievement of low temper-atures. Based on the Joule-Thompson process [21], in 1895 Linde developed a method to liquify air in large amounts [22]. In 1908, Onnes reported liquified Helium [23]. Three years later, he showed that at 4.19 K the resistance of Mercury drops to zero [24]. This phenomena was not expected within the conduction theory of metals developed by Drude [25].
In 1933, Meissner and Ochsenfeld [26] showed that a superconductor cannot be explained by assuming only a ’perfect conductor’. When a superconductor is in an external weak magnetic field and cooled below the critical temperature, the magnetic flux is expelled from the bulk. To explain this effect, the London brothers assumed two kinds of conducting electrons: Normal-state and super-conducting electrons [27]. Within this theory, they defined the penetration depthλ, indicating the length that the magnetic field penetrates the bulk. Its value is proportional to the number of superconducting electrons.
In 1950, Ginsburg and Landau proposed a phenomenological theory based on Landau’s theory of second order phase transitions [28]. This theory introduces a complex pseudo-wave function ψ as an order parameter. Its square is the local density of superconducting electrons. The Ginzburg Landau parameter κ = λ/ξ was also defined. The ξ is called the coherence length and can be understood as the decaying length of the superconducting wave function. Abrikosov showed that the caseκ > 1/√
2 leads to a type II superconductor [29]. This was a theoretical explanation for an experimental observation; From early magnetization measurements, it was known that the superconductors can be ’hard’ (type I) or ’soft’ (type II) in their diamagnetic response (see [30] and references therein). In 1934, Shubnikov had already experimented with an alloyed system which could be transformed from type I to type II. The considerable difference between both types is that in a type I superconductor the increase of a magnetic field to a critical value abruptly destroys superconductivity. In a type II superconductor, there are two critical fields: Hc1 and Hc2. At Hc2 the superconductivity is destroyed; The lower Hc1 describes the boundary between the ’Meissner phase’ and the ’Shubnikov phase’. In the Shubnikov phase, the superconductor minimizes the effect of the penetrating magnetic field within the superconducting bulk by vortices carrying magnetic flux. These vortices are quantized by multiplies of Φ0 =h/2e.
In 1950, Maxwell [31] and Reynolds et al. [32] reported the so called ’isotope effect’ in Mercury.
They found that TC is inversely proportional to M1/2 where M is the isotope mass. Fröhlich predicted that the driving force for superconductivity is the electron-phonon interaction [33].
In 1957, Bardeen, Cooper and Schrieffer published their theory of superconductivity [4]. In this BCS theory, two Bloch electrons with opposite spin and momentum can balance out the Coulomb repulsion by interacting though a virtual phonon. The correlation of both these elec-trons represents a bosonic quasi-particle in its ground state, which is called a ’Cooper pair’.
Within this state, the boson can carry a current without dissipation. All this considered, the condition for a Cooper pair is therefore that the matrix element for electron-phonon interaction (Λ) is stronger than the electron-electron interaction (µ): −V =−Λ +µ < 0. The formation energy of the Cooper pair and its thermodynamics can be calculated within the weak coupling limit. This limit means that the matrix element V is for the Bloch electrons in an interval around the Fermi energy EF±~ωD isotropic (independent of the wavevector). This condition implicitly means thatTC ΘD. Here ΘD =~ωD/kB is the Debye temperature. Therefore, the superconducting transition temperature TC can be calculated as:
TC = 1.14 ΘDe−1/N(EF)V . (2.1)
Here,N(EF) is the electrons density of state at the Fermi energy. For the superconducting state it is also remarkable, that a gap opens below TC. For T=0 the gap is given as
∆(0) = 3.52kBTC/2 (2.2)
The gap has an s-wave symmetry for weak-coupling, which means that the gap opens isotropically at the Fermi surface.
The predicted gap was found [34, 35] and the BCS theory was able to explain all phenomena in the superconductors, e.g. the isotope effect and the Josephson tunneling [36]. Further research led to the BCS weak-coupling case being expanded towards a strong-coupling including better understanding for the electron-phonon interaction by Migdal [37], Eliashberg [38], and McMillan [39]. McMillan determined that, in most conventional cases, the maximum critical temperature TmaxC could be estimated. In Nb3Ge films, superconductivity1 could be leveled up to 22.3 K [40], with improved conditions to 23.2 K [41]. In 1973, Bardeen stated that in the common alloys, TmaxC for normal phonon mechanisms is almost achieved and other mechanisms must be suggested for obtaining higher transition temperatures [42]. Since that time, TmaxC could not be increased further.
The situation changed in 1986 when Bednorz and Müller found the high-temperature supercon-ductor La2−xBaxCuO4(LBCO) with a TC of about 30K [2]. This was the first high-temperature superconductor of the hole-doped cuprate family to be found. At this point the ’goldrush’ for cuprate high-temperature superconductivity began: In January 1987, La2−xSrxCuO4 (LSCO) with TC=36K was reported by Cava et al. [43], in February 1987,YBa2Cu3O7−x (YBCO) with TC=93K by Wu et al. [44] - the first at temperatures warmer than liquid Nitrogen. In De-cember 1987, Bi2Sr2CuO6+δ (Bi2201) was reported by Michel et al. [14]: A high-temperature superconductor without rare earths. The events in January of 1988 may illustrate the excite-ment of the time: Maeda et al. found a 105 K phase in the Bi-Sr-Ca-Cu-O system [45]. The article was received by the Japanese Journal of Applied Physics on January 22 and accepted a day later. On January 26 of 1988, Chu et al. sent their report about superconductivity up to
1Historically, the superconductivity of this materials was already called ’high-temperature superconductivity’
2.1From conventional to unconventional superconductivity
Figure 2.1: Composed from [49], [50] and [51]: Collection of some important superconductors.
114 K in the Bi-Al-Ca-Sr-Cu-O system to Physical Review Letters [46] and the phase identifi-cation followed later [47]. The material which was found by both was Bi2Sr2CaCu2O8+δ, also called Bi2212 or BSCCO2. The discovery of other high-temperature superconducting cuprates followed. The highest transition temperature of all cuprates is observed for the Hg-family with the triple-layered Hg2223 having the maximum critical temperature of 134 K [48] under nor-mal conditions. Under pressure of about 30 GPa, it can even reach 164 K [3]. Compared to the BCS-superconductors the remarkable difference is that the cuprates are bad metals in the normal-state and close to a metal-insulator transition. They are layered, quasi two-dimensional structures with a central CuO2-plane. The superconducting order parameter (gap) has a d-wave symmetry. Most conventional, low-TC superconductors have an s-wave symmetry and are normal-state metals.
Electron-doped cuprates also exist, e.g. Pr2−xCexCuO4 (PCCO), Nd2−xCexCuO4 (NCCO)
2More exactly it seemed to be a mixture of the two-layer Bi2212 and the three-layer Bi2223, with a contami-nation caused by the AlO2-crucible, in the case of the latter.
[52, 53], and Sr1−yNdyCuO2 [54]. These materials possess the essential building block of the cuprate family: The CuO2-planes. An important question is whether its superconductivity is the same as in the hole-doped cuprates. Further investigations that reach this conclusion could be very interesting for testing the mechanism.
The mechanism of cuprate high-temperature superconductivity is strongly debated. The most prominent theories will not be reviewed here, but will be addressed in a later section. In the following I would like to summarize briefly other known superconducting materials. The subjectively ordered collection of these materials is depicted in Fig. 2.1.
There are also organic superconductors. For a more detailed review, please see [55, 50]. They consist of a packed repetition of a building block, which is a donor molecule plus an acceptor complex. Certain packing produces a quasi two-dimensionality. The first superconductivity in this class of materials was reported in 1980 by Jerome et al. in (TMTSF)2PF6 [56]. Here, TMTSF is the donor and is fully written as tetramethyltetraselenafulvalene. The organic super-conductors show some similarities to the cuprates; They are typically quasi two-dimensional and show indications of a d-wave superconductivity. They can be considered correlated materials, because they are Mott-insulating and antiferromagnetic. Under ambient pressure, the record is TC=11.2 K for κ-(ET)2Cu[N(CN)2]Br [57]. Here, ET is the donor and the short form of BEDT-TTF (bisethylenedithio-tetrathiafulvalene) and κ is one of three possible phases for the packing of the ET molecules.
The alkaloid-doped C60 is the most prominent member of the so-called ’molecular superconduc-tors’ or ’fullerene superconducsuperconduc-tors’. The first report of superconductivity at 18K was in K3C60as reported by Hebard et al. [58] in 1991. Until now, the highest TC found is 38 K in Cs3C60[59].
It is widely believed that A3C60 are s-wave, BCS-like superconductors, driven by the coupling to phonons and probably with some strong-coupling effects. But there is no conclusive evidence that this picture is correct or that another electronic mechanism is excluded [60].
Powdered MgB2 has been used since the 1950s, but in 2001 Nagamatsu et al. [61] discovered that it is a high-temperature superconductor with a TC of 39 K. This ignited a new ’gold rush’.
Similar to high-TC superconductors, MgB2 is a layered material. However, while cuprates are bad metals or even insulators in their normal-state, MgB2 is always a metal. The material is mainly considered to be a multigap superconductor. In this system two conduction bands are involved in a BCS-like phonon-mediated pairing and two superconducting gaps also evolve (see e.g [62, 63, 64]).
In 2006 it was not widely recognized that the Iron-based layered superconductor LaOFeP showed superconductivity up to 4 K [65]. In February 2008, Kamihara et al. reported that La[O1−x
Fx]FeAs has a TC up to 26K [66], then it was found to be 43 K [67]. Again a ’gold rush’
ensued. The record is now (June 2008) TC=56K for Gd1−xThxOFeAs [68]. I suggest that the superconducting series can be described by the formula (RE)[O1−x Fx]Fe(P), where RE is a rare earth and P a pentel. As the papers are published before the ink is dry, a discussion of a possible mechanism would only be premature.