Roberto Escudero
Instituto de Investigaciones
en Materiales
Universidad Nacional Autónoma de México
Apartado Postal 70360, México, D.F. 04510. MEXICO
escu@physics.ucsd.edu
Abstract
Some of the physical phenomena of the superconducting materials are the main topics of discussion in this chapter, particularly in the superconducting ceramics. Before to start the discussion and analysis of those characteristics, a small background is introduced, and this is related with the phenomenological bases of the superconducting state. After this, an introduction to the most important theoretical models to describe the superconducting state will be given. We will start with a thermodynamic description, after with London theory, to follow with the Ginzburg Landau theory and finaly with a brief introdution to the BCS weak coupling theory. The former will be useful to define the two types of superconductors: type I and II. To define this, we introduced two important parameters or lengths, which helps to define the superconducting state, the London penetration depth and the Ginzburg Landau coherence length. Important aspects related to the superconducting state of the ceramics will be discussed. Special emphasis is given in two experimental techniques: electron tunneling and point contact spectroscopy which are two spectroscopics tools which probes with particular detail the superconducting state, particularly the energy gap. These two experimental techniques may give enough information to start to have a better understanding of the physical mechanisms that couple electrons in the high temperature superconductors. At the end we will give some of the phenomenological aspects of the behavior of the normal state of the ceramics.
Key words: Superconducting Ceramics, High Transition Temperature, Superconductivity, Energy Gap.
1 Physics of the Superconducting
State
1.1 Background
Before starting with a description about the superconducting state and its characteristics, let us cite some of the most important chronological events from the point of view of the new knowledge that was developing to understand more about the physical phenomena. This, since the discovery by Heike Kamerlingh Onnes, until recent days.
Chronological events:
1911  H. Kammerlingh Onnes discovered that the electric resistance goes to
zero when mercury is cooled at about 4.2 K. Kammerlingh Onnes call to
this state the supraconducting state (different to the superconducting
state, as we know today). He observed the phenomenon in different elements
of the periodic table, he found in general that the transition temperatures
(the temperature at which the resistance goes to zero) was very low and
of the order of 1 K to 10 K.
The width of the transition was very narrow: DT_{C} £ 10^{5} K And the resistivity change in the transition temperature of the order of: r £ 10^{17}r_{C}(T=0K). With r_{C} ~ 10^{9}Wcm. Which implies that all the electronic system is taking part in the phenomenon. 
1913  A magnetic field of a particular value destroys the zero resistance state, and the material goes back to the normal state. 
1916  The application of a electric current of enough intensity to a superconducting material turn this into the normal state. 
1924  Keesom and Onnes discover that there is not changes in the Xray spectra when the material pass from the normal to the superconducting state. 
1933  Discovery by Meissner and Ochsenfeld that the superconducting state is a diamagnetic state. The magnetic field is excluded from the superconductor, but also the field is expelled when the field is applied in the normal state and as is cooled through the superconducting state. So superconductivity now is defined as the physical state with two characteristics: R=0, and B=0. 
1934  H. and F. London explain the Meissner effect using Maxwell equation and constitutive equations of the electromagnetic theory, they find that Ñ^{2}B=l^{2}B and in the superconducting state the density of current is proportional to the vector A, thus J µ A. Ohm's law is not longer valid, and the magnetic field is exponentially screened to the interior of the sample. 
1934  Theory of two fluids by Gorter and Casimir to explain the thermodynamic behavior of the superconducting state. I_{T}=I_{N}+I_{S}, where I_{N}, and I_{S} are the current densities in the normal and in the superconducting states respectively. 
1935  Electrodynamic model by London to explain the electromagnetic behavior of a superconductor. 
1938  Keesom and Van Laer measured the electronic specific heat in Tin. 
1941  Discovery of high transition temperature in NbN (15 K). 
1950  Discovery of the Isotopic effect by Reynolds, Maxwell, Serin, Wright and Kesbitt. The transition temperature is proportional to the inverse of the square root of the atomic mass. 
1950  Phenomenological theoretical model by Ginzburg and Landau to explain superconductivity in terms of an order parameter which is equivalent to a macroscopic wave function. With this model it is possible to define type I and type II superconductors. 
1950  Fröhlich proposes the importance of the electronphonon interaction in superconductivity together with Bardeen to explain the isotopic effect. 
1954  Corak and coworkers find that the specific heat shows a behavior that could be explained as if exist a kind of activated process in the conduction band (energy gap in the conduction band?). Specific heat C ~ exp(E_{a}/K_{B}T_{C}). 
1955  With the neutrons experiments of Wilkinson et al, it was concluded that the superconducting transition is not in the real space (it is occurring in the momentum space). 
1956  Paper by L. Cooper where he proposes the instability of the Fermi sea by pairing two electrons with opposing momentum [k\vec] and spin up, S , and down, S¯ . 
1957  Paper in the Physical Review where is formulated the Bardeen Cooper and Schriffer theory (BCS theory)^{[A]} that explain from the microscopic point of view the electronic pairing mechanism that lead to the superconducting behavior of materials, based in the electronphonon interaction. 
1959  1960  Gor'kov shows the equivalence between the GinzburgLandau theory and the BCS theory in the limit close to the critical temperature. 
1961  First application of superconductivity: The construction of 1 Tesla magnet using A15 alloys. 
1961  Experimental demonstration of the quantization of the magnetic flux through a superconducting ring by Deaver and Fairbanks. 
1961  Giaever experiments using tunnel junctions that show the existence of the energy gap in a superconductor. 
1964  Theoretical paper by Josephson where he demonstrates the existence of a superconducting current (in absence of voltage) through a tunneling device made with two superconductors separated by a weak constriction. 
1965  Eliashberg theory. The exact general superconducting theory. All the information is content into the kernel a^{2}(w)F(w). 
1970  First commercial SQUID. 
1973  Intermetallic alloys with A15 crystalline structure with maximum critical temperatures of the order of 23 K. 
1970 's  Theoretical models to explain why the maximum transition temperature of superconductors must be about 30 K (of course all they were wrong !!). 
1986  Bednorz and Müller publish a paper which reports on the discovery of a new Copper based ceramic which has a maximum transition temperature of about 30 K. 
1987  Paul Chu's experiments that show the increasing of the transition temperature with external applied pressure in the Bednorz and Müller compounds. 
1987  Discovery of the Y_{1}Ba_{2}Cu_{3}O_{7d} superconducting compound by the group of P. Chu. This compound now is known as the [1:2:3], and has a maximum transition temperature of the order of 90 K. 
1988  1989  Discovery of a new family of superconducting cuprates with Bismuth and Thallium. Bismuth compounds with maximum transition temperatures of the order 110 K and Thallium based compounds with about 125 K. 
1991  Discovery of more Copper based ceramics, now type n with transition temperatures of the order of 30 K. 
1992  Discovery of a new organic superconductors based in Alkalines and C_{60}, with maximum transition temperatures of 35 K. 
1992  Discovery of the new family of cuprates but now with Mercury. The maximum transition temperature close to 135 K. 
1994  Increasing experimental evidences that the pairing mechanism in the high T_{C} cuprates have an order parameter in which the symmetry is primarily of type d_{x2y2}, but a mixing of symmetries of s and d is not ruled out. 
1996  Last experiments by IBM's group shows that the symmetry of the order parameter is of dwave character [2]. The implications of this fact is that antiferromagnetic fluctuations can generate this type of symmetry in the order parameter. Accordingly then, electronphonon coupling as the main mechanism for the electron pairing is almost discarded. 
It is clear from the above chronologic events that the long journey to understand the problem of superconductivity in terms of microscopic grounds, since the discovery by H. Kamerlingh Onnes in Leiden to our days was and actually is, a slowly and difficult task in many sense and sometimes caused by neglecting importants facts, let's cite only two. First at all, it is clear that the spectacular success of the BCS theory that required 46 years to be invented, was because the intrinsic difficulties of the problem. Before the development of this theory, any problem in physics ever solved, presented the strong correlation of so many bodies (10^{23} electrons) moving in a coherent way and given such amount of the new physical and distinct characteristics as occurs in the superconducting phenomena. This necessarily required the introduction of new mathematical methods in physics, as was the case of the introduction of the many body theoretical methods into solid state, that they were asked for from nuclear physics and transposed to the solid state problem to construct a macroscopic wave function that represents the interaction of many electrons moving in the material. In another hand, the elegant and important phenomenological model by Ginzburg and Landau formulated in the year of 1950 was not very much appreciated and passed unawared (at least in the western literature) by about nine years. Only until Gor'kov was able to demonstrate in the year of 1959 and 1960 that the Ginzburg Landau theory was a limiting form of the microscopic theory of BCS valid near the transition temperature, was when it started to be appreciated and used to understand more about the different types of superconducting materials, for instance type II superconductors.
Nevertheless, superconductivity is one of the most beautiful examples of the nature that shows with the most pristine clarity the quantum macroscopic behavior of the matter. The physical grounds of the microscopic theoretical models^{[B]} rest in the simple idea of a paring mechanism that occurs between electrons when the weak interaction between electrons and phonons is enough to surmount the electrostatic repulsion of the electrons and thermal fluctuations. Thanks to the BCS theory we are able to explain many of the experimental evidences about the behavior of many different superconducting materials, mainly pure elements and some alloys or compounds. In table 1 we show a list of many different superconducting materials; elements of the periodic table, compounds and alloys, that may be understood in base to BCS theory, those materials we called conventional ones. Also in the table, it is shown other materials catalogued as unconventionals, in the sense that discrepancies are found if we wanted to explain the behavior based in BCS theory. In these materials it is believed that the pairing mechanism is mediated by another kind of elementary excitation rather that phonons, and the pairing symmetry may be different. In this unconventional materials there are ceramics based in copper, heavy Fermions, Chevrel phases, fullerens, etc. However, the position in the table is not totally conclusive, so changes to different position from unconventional to conventional or viceversa may have place according to more precise experiments.
So accordingly to the last paragraphs, it is clear that the study of superconductivity is a very important topic, thanks to the superconducting high T_{C} cuprate ceramics that Bednorz and Müller discovered, but also to other materials (as for example heavy fermions, and organics). Also it is worth noting that very recently there are more evidences that the pairing mechanism in the cuprates is mediated by magnetic fluctuations, this last observation is according to Josephson tunneling experiments that show that the symmetry of the macroscopic wave function that defines the superconducting state is very probably of type dwave [2]. However, in the near future we will know with more detail what is this mechanism. It is important to emphasize that the knowledge of the pairing mechanism, will permit a major knowledge about these systems, and therefore in the future we will predict and find new superconducting materials with higher transition temperatures. To finish with this rapid overview of the history of superconductivity, I would like to recommend to those who want to appreciate another point of view of a scientist which contribution have been important in solid state physics and in particular superconductivity, to read the P.W. Anderson point of view[3].
SUPERCONDUCTORS  
conventional  
Elements  Alloys 
Hg(4.5)  V_{3}Si (15) 
Pb(7.2)  Nb_{3}Sn(18) 
Nb(9)  Nb_{3}Ge(23) 
Sn(3.7)  PbMo_{6}S_{8} 
Al(1.2)  HoMo_{6}S_{8} 
In(3.4)  LaMo_{6}S_{8} 
V(5.4)  ErRh_{4}B_{4} 
unconventional  
Heavy Fermions  Organic  Perovskites 
UPt_{3}  (SN)_{x}  Na_{x}WO_{3} 
UBe_{13}  [C] (TMTSF)_{2}X  SrTiO_{3d} 
URu_{2}Si_{2}  (ET)_{2}X  Ba(Bi_{1x}Pb_{x})O_{3} 
UNi_{2}Al_{3}  ETCuSn  La_{2x}Ba_{x}CuO_{4d} 
UPd_{2}Al_{3}  [D]A_{3}C_{60}  Y_{1}Ba_{2}Cu_{3}O_{7d} 
CeCu_{2}Si_{2}  TTFTCNQ  Bi_{2}SrCa_{2}Cu_{3}O_{10} 
Tl_{2}Ba_{2}Ca_{2}Cu_{3}O_{10}  
Ba_{1x}K_{x}BiO_{3}  
HgBa_{2}Ca_{2}Cu_{3}O_{8+d}  
Hg_{0.8}Pb_{0.2}Ba_{2}Ca_{2}Cu_{3}O_{x}  
Sr_{2}RuO_{4}  
1.2 Phenomenology of the Superconducting State
The various types of superconducting systems may be separated in the next list according to their different properties:
We can divide the four set of the above materials in conventional superconductors in the sense that follow the BCS theory, and other in which its behavior show a nonconventional form, far to be interpreted as BCS. Particularly, because it is thought that magnetic fluctuations are influencing the pairing process. Also, from the thermodynamic point of view we can divide the superconductors in two types: type I, and type II.
Before to describe those types of systems first we will describe the basic physical behavior of the superconducting state:
As we mentioned before in the introductory background, the superconducting state can be characterized by two important properties: R=0, and B=0. The first characteristic can be described in the next; when lowering the temperature of a material, like a metal, the electrical resistance tend to diminish in a form that commonly it is called metallic form. This is almost linear, but with a small inflection at about 30 to 50 Kelvins, which corresponds to approximately one tenth of the Debye temperature. In lowering the temperature for clean elements like Sn, Pb, Nb, etc. there is a temperature that is called the critical transition temperature T_{C} in which the resistance suddenly goes to zero in a very sharp transition. In some of those elements the sharpness is so abrupt that the interval of temperature where this occurs is of the order or less than 10^{5} K, as is the case of gallium. This abrupt change in some way may implies that almost all the electrons are participating in the process of this change of resistivity behavior.
The second characteristic that defines a superconducting material is the behavior under the influence of a magnetic field. Suppose that we are performing the experiment of cooling the material in the presence of a magnetic field; at the moment of reaching the transition temperature the electrical resistance goes to zero, as we mentioned before, but now the magnetic flux that was crossing the interior of the metal is suddenly expelled of the sample. This is called Meissner effect. We can see more clearly this effect if we consider the equation, that relates the Magnetization M, the magnetic induction B, and the external magnetic field H, we have, in cgs units:

(1) 
if the external magnetic field has a small value H, the material
is magnetized and above T_{C}, B ¹ 0. At the moment when the temperature reach T_{C}, B
is expelled of the interior of the sample, therefore:
0=H+4pM.

(2) 
From here we can obtain the Magnetic Susceptibility defined as:

(3) 
Accordingly, this implies that the material in the superconducting state is a perfect diamagnet, with a maximum value of diamagnetism. So we can conclude, that in a superconducting material below a certain temperature called T_{C}, the electrical resistance becomes zero, and is a perfect diamagnet. Those two phenomena are illustrated in figure 1.
Figure 1  Characteristics to define a superconducting material.
Experimentally ones find that a certain value of the applied magnetic field and depending of the temperature, the normal state is restored. This field is called the critical magnetic field H_{C}. So, the magnetic flux penetrates the sample and the resistance becomes different to zero. The equation that follows quite well this behavior of the magnetic critical field is given by:

(4) 
The materials which follow this behavior are called type I superconductors. Experimentally one finds two things about the behavior of superconductors in the presence of magnetic fields: the critical fields of the superconducting elements of the periodic table are in general small, so if one want to increase the density of current in a material that behaves in this way, as soon as we surpassed the critical field it will be back to the normal state. Accordingly then, those type of materials are not appropriated for applications, as for example to conduct high densities of electrical currents. The second aspect important to mention is that in fact the magnetic field penetrates a little bit inside of the superconductor, although this is rapidly attenuated in a short distance called the penetration length, or penetration depth, l. However, type II superconductors behaves in a quite different manner, as we can see in the next figure, where we have drawn the two types of systems.
In figure 2 we illustrated the two types of superconducting materials, in a) is the type I, and in b) the type II, and the penetration of the field in general.
Figure 2  Two type of superconducting materials, according to the behavior
in the presence of a mangnetic field.
The physics behind those figures is the following: As soon as the magnetic field reaches the value H_{C}, in figure a) the superconducting state is destroyed, the material is not longer superconducting. However for a type II superconductor as is illustrated in figure b), in the moment when the magnitude of the external field surpasses the critical field, now called H_{C1}, the magnetization start to decrease. Inside of the superconductor now there is a quantum of magnetic flux, which is localized in a very small region, the vortex region. In this region the material is in the normal state and the magnetic field penetrates. So Meissner effect does not exist^{[E]}, but still the materials has superconducting regions. If the magnetic field follows increasing more normal vortex regions appear inside the sample and those form a periodic hexagonal array. The increasing of the magnetic field creates more, and more vortices, and the periodic array follows diminishing the lattice periodicity. This implies that there are more and more normal regions. Therefore, the magnetization follows decreasing, until the moment that all the sample is complete compacted with normal regions. At this moment when the magnetization is zero the sample is now in the normal state. The magnetic field that destroys superconductivity in a type II superconductor is called the second critical field, denoted by H_{C2}. Many materials in nature are catalogued as type II superconductors, and in fact these present better characteristics appropriated for applications. With those materials it is possible to construct high intensity magnets, which particularly are one of the most important applications of the superconducting state. It is important to observe that after the second critical field has been reached the superconductivity is destroyed, but still remains some indications of superficial superconductivity, this last is also destroyed when the magnetic field applied to the sample reaches a magnitude known as H_{C3}.
1.3 Thermodynamic of the Superconducting State
The application of thermodynamic concepts to describe the superconducting state, was possible only until the important discovery by Meissner and Ochsenfeld in 1933 of the expulsion of the magnetic field by a superconductor. It is evident that in a physical system in which the superconducting behavior is due only to perfect conductivity, thermodynamics is not possible to apply. The reason of this easily can be seen in fig. 3a. In this figure we have the behavior of a perfect conductor in the presence of a magnetic field, we may note that the final state is very dependent on the way in which the magnetic field was applied, in other words; of the history of the sample. In comparison, in fig 3b we can observe that for a superconducting material in which the Meissner effect exists the final state is independent of the way the magnetic field was applied. In this form we conclude that with the Meissner effect it was possible formulate a state equation to describe the superconducting behavior of a sample. Gorter in 1934, concluded that H_{C}(T) is a phase diagram, and this was the principle to formulate a thermodynamic theory.
Figure 3  The behavior of a perfect conductor and a superconductor in presence of magnetic field.
It is useful to use the Gibbs free energy to describe the superconducting state in the following way:

(5) 
When T, V, and P are constants, then simply we have:

(6) 
In the above equation U is the internal energy, T is the temperature, S
is the entropy, P is the pressure, V the volume of the sample, M(H) is the
magnetization depending of the external Magnetic field H. If we do the calculation
for a superconducting cylinder, with the cylinder axis parallel to the increasing
magnetic field, and in addition we consider that the cylinder is type I,
then:

(7) 
since M=H/4p. This is the change of the energy due to the Meissner effect. In the normal
state B @ H, and in the superconducting state B=0. Then

(8) 

(9) 

(10) 

(11) 
Figure 4  The change of the entropy and the specific heat when a material transits from the normal to the superconducting state.
Remembering that dH_{C}/dT < 0 we see that S_{N}
> S_{S}; we may conclude according to this equation
that the order of the superconducting state is higher that the normal state.
Figure 4 shows some of those behaviors of the two thermodynamics quantities.
In addition from the definitions of entropy and the latent heat, Q, which
is defined as Q=T(S_{N}S_{S}) we see that this vanishes at the transition because
in that case S_{S}=S_{N}, which implies
that the thermodynamic transition is a second order in absence of magnetic
field, but first order in presence of magnetic field[^{F]}.
It is interesting to mention that in going from the superconducting to the
normal state in the presence of a magnetic field heat is absorbed, which
implies that Q > 0. Here Q is written as:

(12) 
We will mention in this space some experimental results that are important to understand the modification that normal material experiments when passing to the superconducting state.
First, is useful to remember that the specific heat in the normal state basically can be separated in two contributions: the contribution of phonons, the Debye contribution, which is important at high temperature, and the electronic contribution which is important at low temperatures. So, the specific heat can written as the sum of two parts:

(13) 
The important point in this discussion is that measurements of Keesom and Van Laer in 1938, indicated that the electronic contribution in Tin in the superconducting state may be written as; C_{Se} ~ 3gT^{3}/T_{C}, which is not very different to a normal electronic contribution in a metal. But in 1954, Corak, Satterthwaite, Goodman, and Wexter were able to adjust the electronic contribution for vanadium with similar precision as for the Keesom and Van Laer expression using a law

(14) 
It is worth mentioning, that the two expressions for the specific heat (Keesom and Corak) are compatibles with a two fluid model, as the proposed by Gorter and Casimir in 1934, but the important implication of the second form from Corak et al, is that this implies a forbidden energy gap in the electronic spectrum of the superconductor. This was the first time, that a modification in the electronic band of a normal metal was proposed to explain the characteristics of the superconducting state^{[G]}. The idea of the energy gap that separate two regions in a electronic system, one lower in energy, and filled with electrons, and other above and empty, but with available states to be occupied, arises from a model of semiconducting systems.
Before to go further in to the next section, it is important to remember that other studies in superconductors, using experimental techniques, as Xray diffraction, indicates that does not exist any structural change when passing in temperature from the normal to the superconducting states. In 1955 Wilkingson et al. investigate the diffraction of neutrons by electrons, with the result that the system is essentially the same above and below the transition. The distribution of electrons does not experiment any change at T_{C}, free electrons remain free, bound electrons remain bound: Then, from here again we can conclude that the ordering that we found in former sections is due to a condensation of the electrons in the momentum space.
Soon after the specific heat measurements by Corak et al. more experimental evidences pointed out to the idea that two fluids could describe the physical processes that happen in superconductors; one fluid that helps to describe the normal phase formed by normal electrons, and another fluid formed by superelectrons. With those physical evidences can be concluded that a two fluid model would be appropriated to describe the superconducting and normal state. Thermal conductivity measurements show that [4] the normal fluid is the only that transport the entropy. Figure 5, shows the diminishing of the thermal conductivity k, at the critical temperature T_{C}.
Figure 5  Thermal conductivity K_{s} for a superconducting sample in fuction of the temperature normalized to the thermal conductivity in the normal state K_{n}
Another experimental fact are added in this description of the superconducting state, but we are not going to go further in this topic, of course additional information can be found in the literature about superconductivity. However, the experimental bases for the formulation of the model of two fluids must deal with two fundamental assumptions:
This thermodynamic model, was important at that time, and was able to predict the behavior of some experimental quantities, as for example the form of the variation of the magnetic field as function of temperature, and the most important; the variation of the penetration depth l in function of the temperature, given by the equation: l = l_{0}[1(T/T_{C})^{4}]^{1/2}. In addition various of the concepts introduced in this theory were latter used in models of Ginzburg and Landau, and also in certain form in the BCS theory.
1.4 London Theory
The thermodynamic description and the two fluid models were important to describe much of the behavior of the specific heat, thermodynamic magnetic field, penetration length, etc. However, one of the main aspect related with transport properties or perhaps, better to call them, electronic properties; needed a physical interpretation as for example: the Meissner effect and how is transported the supercurrent in a superconductor. This two main issues were addressed by the London theory, and in the next paragraphs we will explain how those issues were solved.
We may consider that there are two fluids to explain the behavior of one superconductor; the total current can be written as the sum of the two fluids

(15) 

(16) 
and

(17) 
where for the normal electrons Ohms law is valid, of course, but for the superelectrons we have to obtain a different equation; we know that the force F acting in an electron is given in terms of the electric field, as

(18) 

(19) 
In this equation we can name 1/l^{2}_{L}=(4pe^{2}n_{S}/mc^{2}), then; the first London equation is:

(20) 
To deduce the second London equation related with the expulsion of magnetic field of the interior of a superconductor, we have to use the Maxwell equations:

(21) 
Therefore, with a little algebra and some manipulation, and assuming that B=H (without any lose of validity) we get that

(22) 
obtaining the derivative respect the time, and using the first London
equation, we have:

(23) 

(24) 
The solution of this equation is valid for a perfect superconductor, and is not totally valid for a superconductor; shows that the change respect to the time of a magnetic field decays exponentially with a characteristic length, l_{L}. whereas B is a constant in the interior of the superconductor, and of course that constant may be zero. To satisfies completely the fact that dB/dt decays exponentially inside the superconductor (as observed experimentally) London proposed that also B must decays exponentially inside the superconductor, so this implies that the last equation must be valid for dB/dt, and for B, and then it is necessary to propose the next relationship for the superconducting current:

(25) 
Then using this proposed equation in the first London equation, we find that

(26) 
which is the second London equation, and which solution shows a exponential
decays of B in the interior of the superconductor, until a distance
that is the London penetration depth:

(27) 
According to those equation we can explain most of the features in the behavior of the superconducting materials. Thus, we must remember that London theory replace the Ohm's law, valid in ordinary conductors, with a relation between the current and the magnetic field, equation 24^{[H]}. This theory permits a complete phenomenological description of the superconducting state which is in complete accordance with a thermodynamic description and electrodynamic behavior. However, the most important contribution from the London theory came from the physical interpretation about the properties of the macroscopic wave function or the order parameter. The important concept, which arises due to the demonstration that exists a proportionality between J and A implies the existence of a long range order in the momentum space of the superconducting electrons. London was able to demonstrate using quantum mechanics formalism, and the definition of p=mv+(e/c)A, that order in momentum space demands extension of the wave function over larger and larger dimensions. Therefore, this implies that the macroscopic quantum wave function of the condensate is rigid, without possibility of changes in the presence of magnetic fields. However, one of the problems with the London theory is the change of size of the penetration depth with the temperature, as experimentally was observed by Pippard. This anomaly in the London model can be explained in a qualitative form in the London framework in base that the wave function is not completely rigid, and in fact this is observed by the fact that n_{S} < n, where n is the total density of electrons in the system. Nevertheless, to correct this anomaly, Pippard introduces the concept of coherence length x_{0} using a nonlocal electrodynamic in the next way.
Pippard introduced the coherence length while proposing a nonlocal generalization of the London equation for the superconducting current. He uses the Chambers nonlocal generalization of the ohm law:

(28) 
where R=rr^{¢}, this formula takes into account the fact that the current at a point r depends on E(r^{¢}) throughout a volume of radius l about r. Pippard argued that the superconducting wave function should have a similar characteristic dimension x_{0} which can be estimated by an uncertainty principle.

(29) 
then;

(30) 
Where a is a numerical constant of the order of unit, and x_{0} may be taken as the smallest size of the wavepacket, and plays an analogous role to the mean free path in the nonlocal electrodynamic of normal metals.
Another contribution of the London theory for a better knowledge of superconductivity was the discovery that for superconductors (type I) must exist a positive energy that separates the normal state from the superconductor.
To explain this surface energy for the two type of superconductors, we must remember that in general for the superconducting state there exists a balance between the magnetic contribution of the free energy and the energy expended in the electron ordering in the system, and this balance is given by the size of the London penetration depth, that has to do with the magnetic contribution, and the Pippard coherence length that is related to the electron ordering. So if l is small than x, then the resulting balance will give a total free positive energy. This is exactly the case that occurs for type I superconductors. The contrary case is also true; for type II superconductors where l tend to be bigger than x the total surface energy will be negative. This behavior is illustrated in the graphic construction of Figure 6.
Figure 6  Contributions to the surface energy of a superconductor type I.
1.5 Basic Principles About the Ginzburg and Landau Theory
The Ginzburg and Landau theory was published in 1950 [5] and could explain with great precision the behavior of type II superconductors in terms of the ratio of two important parameters of the superconducting state: the coherence length, x, and the magnetic penetration depth, l: k = l/x. Whereas the coherence length describes the characteristic scale in which the order parameter varies in space, the magnetic penetration depth measures the spatial penetration of the magnetic field inside a superconductor. In type I superconductors, which in general are clean materials, are characterized by long electronic mean free path, and the coherence length is bigger than the penetration depth (x >> l ) whereas in type II superconductors; i.e. alloys, compounds, ceramics, and organic compounds, the electronic mean free path is small, and therefore the coherence length is small, or much more smaller than the penetration depth. Accordingly, in Ginzburg and Landau theory the parameter k defines the boundary between a superconductor type I or II, and this has the value: k = 1/Ö2. For Type I superconductors (mostly pure metals) k < 1/Ö2, with exception of Nb, V, and Tc which are type II materials. For the contrary for type II superconductors k exceeds the value of 1/Ö2, which means that the penetration length l is bigger than the coherence length x. It is important to remark that the superconducting ceramics of high transition temperature are extreme type II superconductors; in this case l >> x and of course k >> 1/Ö2.
Ginzburg and Landau theory shows that H_{C2} is given
by

(31) 
Before to end with this brief description of the Ginzburg and Landau theory, let us give a very small introduction about the quantitative physical behavior of the superconducting state according to this model.
The theory can describe with great precision the behavior of type II superconductors, although one complete theory to describe the type II superconductors was given by also Abrikosov and Gor'kov. In this model one writes the behavior of the superconducting order parameter (The macroscopic wave function) in terms of the free energy in the following way:

(32) 
Where F_{N} is the free energy in the normal state, in presence of zero magnetic field, e^{*} is the effective charge of the electron, which is taken as 2e, H is the magnetic field, and a, b, and m^{*} are constants, depending on the material. To minimize the free energy we used a variational method which leads to the two Ginzburg and Landau equations:

(33) 

(34) 
In absence of magnetic field this last equation looks more simple;

(35) 
which in turn looks like the classical solution for the probability of current of the Schrödinger equation for one electron. This expression is gauge invariant provided that indeed Y has a charge e^{*}, which then under a gauge transformation we can write

(36) 
where

(37) 
Furthermore, if Y is independent of r, then the expression for the current, equation 34,
reduces to the London equation:

(38) 
The Ginzburg and Landau free energy functional represent a plausible extension
of the London theory and the two fluid model for situations where the superconducting
order parameter is position dependent through the introduction of
the complex order parameter which is reminiscent of a Schrödinger wave
function. The order parameter may also be related to the critical field:

(39) 

(40) 

(41) 
This equation simply tell us that the order parameter is the most important quantity that defines a superconducting material, which inclusive in many cases can be gapless, but always existing the order parameter.
So accordingly, the description of the superconducting state in the Ginzburg
Landau theory is characterized by two lengths: the coherence length, given
by

(42) 
and the London penetration length

(43) 
From here we can obtain the Ginzburg Landau parameter k, which may be obtained from the ratio between the two lengths.

(44) 
where the two types of superconductors can be defined as:
k < 1/Ö2, for Type I superconductors, and k > 1/Ö2 for type II materials. The boundary between the two is at k = 1/Ö2.
With these conditions one can construct a phase diagram of the critical fields H_{C}(T), H_{C1}(T), H_{C2}(T), and H_{C3}(T). Accordingly then, Meissner effect will exist until the superior limit of the H_{C1}(T). Between H_{C1}(T) and H_{C2}(T) is the region known as the Shubnikov phase, or also known as the vortex state, and from H_{C2}(T) to above until a superior limit H_{C3}(T) will exists surface superconductivity. This behavior is drawn in figure 7. It is worth mentioning that Saint James and de Gennes in 1963 showed that surface superconductivity occurs when the parameter k ³ 0.42. They also showed that the magnitude of H_{C3} has values of the order H_{C2} £ H_{C3} £ 1.7H_{C2}, depending on the angle between the surface and the applied magnetic field. In the case of type I superconductors evidently also will exist surface superconductivity if k has the appropriate value.
Figure 7  The magnetic phase diagram of Type II superconductors as function of the temperature for the low (intermetallic) and high transition (ceramic0 superconductors.
In View of the great success of the Ginzburg and Landau theory to describe many characteristics of the type II superconductors, it was convenient to extend it to situations where the superconductor has imposed a transport current. Here, the superconductor develops a resistance which is associated with the driven motion of the order parameter due to the vortex structure; in this case it is convenient to describe the situation using a time dependent Ginzburg Landau equations. This derivation is out of the context of this small training course in superconductivity, but we wanted to mention because give us a good idea about other applications and expands of this useful model. Last to mention, is that in some way the Time Dependent Ginzburg Landau theory is the equivalent to the Boltzmann transport equation in normal metals.
1.6 Basic Principles About BCS Theory
The basic idea in the BCS theory is the formation of electronic pairs when an effective attractive interaction occurs between electrons, resulting from the electronelectron phonon mediated interaction. The interaction is attractive when the difference between electronic states is less than the phonon energy (^{h}/_{2p}) w_{D}. The strength of the electronphonon interaction is peaked when the electrons are in states of equal and opposite momentum and in opposing spins. Only electrons close to the Fermi surface will tend to sense the attractive interaction and therefore they will tend to form electronic pairs. This point was discovered by Leon Cooper and gave rise to the name of Cooper pairs. However, to include all the electrons that are participating in the formation of the superconducting state we have to describe it by constructing a ground state wave function. This wave function is such as the probability of occupation of one pair is independent of that of the other. This then implies a Hartreelike wave function. The Hamiltonian that takes in to account the electron pair occupation, and the hole pair occupation above and below the Fermi energy, and the attractive pair interaction potential, which is assumed constant, gives us a beautiful physical description of the problem of superconductivity and most of the experimental results are explained with this model.
BCS theory has three important parameters, in which are based the properties of all superconducting system. The most important and crucial one is the electronphonon interaction V, in which is based the electron pairing of all BCS superconductors. So accordingly, an electronic system with a very weak electronphonon interaction will be superconducting only at very low temperature (speaking in a simplistic way). In BCS this interaction potential is taken as a constant, and is useful to determine the transition temperature. However, in BCS theory we can not calculate the T_{C} from first principles. The other two parameters are related to the normal state properties: the density of states at the Fermi surfaces N(e_{F}), and the speed of the electrons at the Fermi surfaces, v_{F}. According to the BCS theory, in a normal metal the density of states N(e_{F}) is radically changed when this pass to the superconducting state.
Around the Fermi surface it is opened a gap, whose size is only about a few Kelvins, a feature very characteristic of all strongly correlated electronic systems. In fact the size of the gap is between 10 to 30K (about 0.5 to 3 meV). This gap evolves with temperature, growing very fast close to the transition temperature, and stabilizing at almost a constant value at low temperature. The size is indicative of the strength of the interaction, and of the transition temperature. In fact in BCS theory the value of the gap 2D is given by the ratio 2D/K_{B}T_{C}=3.53, which gives the characteristic of a weak coupling system and is intrinsic to the theory. From the gap equation and at temperatures close to the transition, the value of the gap D® 0, and then T_{C} is determined as

(45) 

(46) 
To find the evolution of the energy gap in all ranges of temperatures, has to be done, solving numerically the gap equation:

(47) 

(48) 

(49) 
Figure 8 display the form of this modification, and shows the evolution of the energy gap respect to the temperature.
Figure 8  The modification of the density of states of a normal metal and the evolution of the energy gap in a superconductor as a function of the temperature.
At T_{C} the specific heat shows a discontinuity, the
theory gives the change of this discontinuity as

(50) 
It is worth mentioning that the ratios of 2D/K_{B}T_{C}=3.53, and the changes of the specific heats at the transition, are followed rather well by many superconducting materials. For most of the superconductors the ratio of the energy gap over the transition temperature is very close to 3.5, the only differences are found in Pb and Hg which are strong coupled superconductors.
In this theory the size of the coherence length x_{0} is given as

(51) 

(52) 
Many experimental verifications has been made to check the validity of the theory. One, is to check if the mechanism of generation is due to the electronphonon interaction; for example checking the isotope effect. This can be done by using one of the prediction of the theory that shows that T_{C}=M^{a}, where M is the ionic mass and a = 1/2. However, care must be taken when interpreting experimental results. For example, when coulombic interactions and renormalizing processes are included in the theory, as is made in the Eliashberg theory, related with strong coupled effects, then the exponent a changes. We must note that a in BCS theory has a value of 1/2. However, according to strong coupling theory the exponent can be written as a = 1/2{1[m^{*}/(l^{*}m^{*})]^{2}}, with m^{*} being the renormalized Coulomb interaction, given by the expression m^{*}=m/[1+mln(E_{F}/E_{D})] and l^{*}=l/(1+l). We are not going to elaborate more about this problem but only we advice to experimentalist when making these type of measurements. But nevertheless an experimental detection of an isotope effect will be a clear indication of at least certain participation of the phonon mechanism in the formation of the pairing condensate. A very good reference for students interested in more details about this kind of effects can be found in the excellent two books of Superconductivity edited by Parks [6].
2 Some Experimental Techniques to Study the Superconducting State
Among the several experimental methods to study or characterizing the superconducting state, two are the most basic and common, and more easy to perform. One is the measurement of the electrical resistance as a function of the temperature, the other is to study the magnetic susceptibility, or simply the magnetization, also as a function of the temperature. We will describe this two experimental methods. After that we will describe electron tunneling (ET) and point contact spectroscopies (PCS). From these two experimental techniques the former, is the most direct method to probe the superconducting state. The second technique is also very appropriate to study the behavior of the superconducting state, as is also appropriate to study elementary excitations in normal electronic systems as metal, alloys, etc.
Various methods may be found in the literature to make measurement of the electrical resistivity or electrical resistance of a metal or a superconductor. These methods will be simple or complicated, depending on the precision in the measurements, and of the anisotropy of the specimen to be measured. For instance, if we want to measure the tensor of resistivity in anisotropic materials then the measurements will have to be made in two or tree different directions, depending of the symmetry of the crystal structure of the sample. For example, in the superconducting system [1:2:3] due to the crystalline orthorhombic symmetry we have to determinate the three components; i.e. r_{a}, r_{b}, and r_{c}. However, due to the similarity of the a and b directions and mainly to the twinned characteristics of the single crystals, in general only will be necessary to measure two directions. In that case one component will be r_{ab}, and the other r_{c}. In this case the most appropriate method will be the Montgomery method [7]. The Montgomery method requires the elaboration of point contacts in the four corners of the crystalline specimen in which will be determined the components of the resistivity tensor. If for example we numbered the corners as 1, 2, 3, and 4, then in this method one measured the tension V_{12}, when the current is injected in the points 3 and 4, thus to obtain the resistance R_{1}. Then, the current is injected between the position 2 and 3, and the tension V_{14} is determined. In this way the component R_{2} is determined, and so on. Nonetheless, this is only one of the cautions it will be necessary to have in mind. Other important points are concerned with the size of the current injected into the material to be measured, this applied current must be as small as possible in order to avoid heating of the sample and associated effects. Also at this point it is convenient to remember the form to glue the contacts to the specimen for injecting the current and measurement the voltage. In general in specimens with metallic conductivities, which is the simplest case, can be used indium or other alloy or metal with low melting point. In the case of ceramic or organic compounds silver paint is the faster and easier method. The four glued or soldered contacts have to be small preferentially and will be necessary to take care that the current contacts be separated from the other two voltage contacts. This configuration is shown in schematic form in figure 9. Here we show the form that the connections are glued to the specimen, and also a simple electronic circuit with enough sensitivity to measure with low noise the superconducting transition temperature of different type of specimens.
Figure 9  An electronic circuit to perform measurements of the electric resistance at low noise, and schematic of the method to glue the electric wires to the specimen.
2.2 Magnetic Susceptibility vs Temperature
With the measurements of the resistance versus the temperature, we have characterized one of the properties of the superconducting state. The second property of the superconducting state to be determined, is the diamagnetic behavior below the transition temperature. This second technique is without doubt the most clear indication when a superconducting transition exists. According to the size of the change of the magnetic susceptibility we can obtain the amount of superconducting material in the measured specimen in term of the percent of sample. In a measure, of this property the change in susceptibility is so strong that the transition temperature may be clearly defined, we must to remember that the expulsion of the magnetic field in the Meissner region is total. So accordingly if the sample is a pure superconductor then the amount of the diamagnetic contribution will be a factor very close to the maximum value of 1/4p, providing than the applied magnetic field is smaller than H_{C1}. In the magnetic measurements care will be necessary to make the measurement using powder samples, in case that we want to determine the amount of superconducting material. The measurement will be made; first, cooling the sample at low temperature and below the superconducting transition, and once the temperature is stable, the magnetic field is applied. In this stage the measure is made rising the temperature at a slowly rate, until the maximum desired temperature, say room temperature or as in many cases only about 10 or 20 degrees above T_{C}. After this, the measure will be repeated now cooling the sample, under the same magnetic field. This cycle provides, in the first case the measured of the shielding effects, and when cooling, the Meissner fraction. Of course in these measurements the contribution of demagnetization factors will need to be taken in to account. In the case of metals or intermetallic alloys, care also will be necessary as mentioned before, but in those simple cases the material is cut in cylindrical shape, and putting the sample oriented in the magnetic field with the cylindrical axis parallel to the magnetic field. In this form one can avoid to subtract demagnetization factors. The measure can be made with a variated number of apparatus, among them one can find Faraday balances, vibrating sample magnetometers, arranges of coils where one can fix different electronic setups in such a way that with simple parts it is possible measure with good precision the transition temperature, and also to determine the amount of superconducting material. However, in general the most precise apparatus to perform these kind of measurements is a magnetometer based in a Superconducting Quantum Device, called SQUID. With this apparatus one can performs measurements in the range of sensitivities of the order of 10^{9} emu.
2.3 Electron Tunneling and Point Contact Spectroscopy
2.3.1 Gap Spectroscopies
In a superconductor the energy gap D, is one of the most characteristic features of the electronic condensate; it arises as a consequence of the many body interaction between quasiparticles that are scattered coherently in a region of the k space which is centered around the Fermi surface within a width K_{B}T_{c}. Due to its nature, the evolution with temperature, size, and value of the ratio 2D/K_{B}T_{c} might give enough information that can be directly related to the microscopic processes that form the pairing condensate.
Several methods are currently used to measure and study the energy gap of a superconductor, e.g. infrared reflectivity, Raman scattering, ultraviolet angleresolved photoemission spectroscopy, and electron tunneling. Each one is important by itself and may provide information related to the superconducting state as well as information concerning the normal state. For details related to those techniques, The reader is addressed to the vast literature in the field (see for example references park,wolf,hasegawa,duif at the end of this chapter). In order to establish the historical background on this subject it is worth mentioning, that the first experimental confirmation and measurement of the energy gap in a superconductor came from infrared spectroscopy. The experiment was realized by Glover and Tinkham in the late 50's. More recently, Raman and photoemission spectroscopies have demonstrated that both are very powerful techniques to probe the nature of the interaction between electrons and phonons, or any other kind of elementary excitation in solids, as well as to study the basic mechanisms of the superconducting and the normal state.
On the other hand, electron tunneling is the only direct technique to study the superconducting energy gap, and it is perhaps the most sensitive probe for studying and analyzing the superconducting state. This spectroscopic technique, senses the microscopic processes which form the superconducting condensate, and can give information to completely characterize many of the microscopic processes that occur in the formation of a superconductor. The information that can be extracted from tunneling experiments is the temperature dependence of the energy gap, the phonon density of states, the coupling function a^{2}(w)F(w), where a^{2}(w) and F(w) are respectively the functions that give the strength of the electronelectron interaction, and the density of phonons in the material under study. It is worthwhile noticing that the coupling function or the weighted distribution function of phonons, a^{2}(w)F(w), plays a central role on the strong coupling superconductivity theory. Every piece of information related with the electrons, phonons, and the interaction electronphonon is important for superconductivity is included here. A material differs from another according to the value of a^{2}(w)F(w). It should be also important to remark that the distribution function F(w), may have different physical meaning; in Eliashberg theory the only basic concept that needs to be introduced is that the distribution function must have to be involved with a bosonic kind of object distribution, e.g. plasmons, polaron, bipolaron, exitons, holons, demons, etc.
Once has been collected the above information, it can be used to solve the two coupled Eliashberg equations. From there, can be extracted all the thermodynamic information that gives the physical behavior of one particular material, e.g. the critical magnetic field H_{c}(T), the deviation function D(t), the specific heat jump, the low temperature energy gap, the zero temperature energy gap, etc.
In tunneling studies many of the effects observed in the current vs voltage characteristics are so small, that it is usually convenient to study the dynamic conductance s(V)=dI/dV, the differential resistance s^{1}(V) as well as its derivative ds/dV. The wealth of information on the tunneling process itself, and on microscopic excitation spectrum that can be obtained from those tunneling measurements, was in the past one of the most reliable pieces of information in the study of conventional superconductors. With the discovery of new superconducting material, workers in the field immediately tried to use this powerful tool to study the superconducting state.
However, it was soon realized that reliable tunneling data in these new ceramic would be a very difficult task for various reasons; mainly, the problem of making reproducible tunnel junctions. The particular difficulty attributable to the nature of the parameters involved in the high T_{c} superconductors, is the small coherence length, x_{o}, which converts the technique from a bulk technique to a surface technique. The consequence of this change is that the experimental data depends on the behavior and characteristics of the surface. For example, changes on the surface due to possible variation of the oxygen stoichiometry, may have effect on the superconducting characteristics of the surface that consequently will be reflected on the tunneling data. Degradation processes, defects, contamination, granularity, etc., will also bring changes on the superconducting properties, that again will modify the tunneling results. Nevertheless, the second and most serious problem concerns to the physical interpretation of the tunneling data, assuming of course that it is correct, due to the fact that actually there is no theoretical model to obtain or guide the interpretation of the experimental information, and many experimentalists believe that this information must be understood in terms of the BCS model, without thinking that perhaps, nature is trying to indicate us a different kind of physical behavior for these exotic new materials. So here at this point, a warning should be given related to the interpretation that must be taken very seriously.
The tunneling techniques used in the past, to fabricate the tunneling devices with conventional materials, were in general made using evaporated thin films on a glass substrate. This consists on to deposit a metallic first film, or electrode, typically Aluminum metal of a thickness of the order of 1000Å to 3000Å , the surface of this film is oxidized in a well controlled manner, in such a way that the thickness of the oxide is maintained in the range of 10Å to 50 Å . The second step in the formation of the device, consists on evaporating the second electrode on top of the first film, trying to maintain the junction area as small as possible. The purpose of the small area is to have the minimum capacitance, and to maintain the tunnel current distribution as homogeneous as possible.
Figure 10  Current versus Voltage and Differential conductance versus Voltage characteristics of tunnel junctions. At very low temperatures the normalized differential conductance approaches the superconducting density of states (see also fig. 8)
This method gives the normal procedure to fabricate tunnel junctions using
metals, e.g. Lead, Tin, Indium, etc, and was used for the first time, many
years ago by I. Giaever. In fig. 10, it is shown the current vs voltage
characteristics and the differential conductance normalized to the normal
state, of junctions with one electrode normal and the other superconducting,
and both electrodes superconducting, these are denoted as NS, and SS. It
is important to mention that the reason to take the quotient between the
differential conductances in the superconducting state, and in the normal
state is the extraordinary fact that at low enough temperatures this is
strictly identical to the superconducting electronic density of states (see
also equation 49 that shows how the density of states is modified when the
material becomes superconducting.):

(53) 
The implementation of this technique for alloys, or A15 superconductors needs a little more care on the evaporating procedures, but however, gives in general good results. For materials such as single crystals or compounds which are difficult to evaporate, or where the stoichiometry and the characteristics of the compound can be changed with the evaporation, and therefore the properties, other type of tunneling junctions have been attempted, one type frequently used, is the well known as point contact tunnel junction. It consists of a metallic wire with a very fine tip, in close proximity with the sample to be studied, the idea is to obtain enough tunneling current by regulating the distance of the tip and the sample. Worth mentioning that this procedure is the basic idea of the tunneling microscope, where one can control with exquisite precision the distance between the tip and the sample.
To end with this description about the construction of tunneling devices, only rest to say that today, many type of tunneling techniques have been implemented to study the new ceramic superconductors with relative success, An interesting description of the state of the art on tunneling studies is the review by Hasegawa et al. [9].
Point contact spectroscopy is today a very well established technique to study diverse electronic mechanisms of interaction between electrons and elementary excitations in metals, alloys, and superconductors. Basically, the technique consist of injecting electrons in to the specimen to be studied by using a metallic contact with very small dimensions in the area of contact with the specimen. This constriction in principle, need to be as small as possible in comparison to the electronic mean freee path, in order to observe with detail the scattering that the electrons suffer inside of the specimen when are injected from the constriction. This limit is named the Sharvin's limit in honor to V. Sharvin, who first proposed the method in 1964 to study Fermi surfaces. In order to consider when a constriction is small and appropriate to observe excitations (the Sharvin limit) the ratio k, known as the Knudsen ratio must be; k = l/2a >> 1. Here l is the electronic mean free path, and 2a is the diameter of the constriction. In this limit, the injected electrons enter ballistically and are scattered deeply inside the specimen, and can sense many of the interactions that are dressing the conduction electronic gas. Yanson in 1984, discovered that in place to take only IV characteristics (current vs voltage) in his experimental contacts, takes the second derivative d^{2}V/dI^{2} of the voltage with respect to the current, the observed structure can be related almost directly with the Eliashberg function a^{2}(w)F(w). This double function, as was mentioned before is related to the electronphonon interaction. However, more recently, other experiments have also shown that other kind of scattering mechanisms may be observed with this simple technique.
There are three regimes of behavior in point contact experiments, where one can observe different features in the sample under study. The first one, was the ballistic regime, already explained, and perhaps the most interesting to observe all kind of scattering mechanisms that occurs in a material. For instance, processes related with Andreev reflections in superconductors can be observed with great clarity, and is interesting to remember that this process is directly related to the existence of an electronic energy gap. According to this, material with electronic condensates such as charge density waves, spin density waves and other may be studied with great detail using this technique. Fig. 11 shows how the features related with Andreev reflection are seen in point contact spectroscopy, and in comparison with tunneling spectroscopy. The z parameter is related with the transmittance of the barrier; if the contact is entirely direct or ''metallic'' then z=0. In the pure tunneling regime z® ¥.
Figure 11  Current versus Voltage and Differential Conductance versus Voltage characteristics of a point contact.
In the second regime; the diffusive regime, the Knudsen ratio is k ~ 1, still it is possible to see features related with electronic scattering processes, but the features in the d^{2}V/dI^{2} may now look smeared, the reason of this is that the effective volume of the scattering process is greatly reduced, and the chance of the electrons to be back scattered to the region where were injected, is also reduced at great scale. This reduction of the volume may be seen in terms of the elastic mean free path, l_{e}. So if l_{e} is reduced the Knudsen ratio is also reduced and the effective volume will be W_{eff}=pa^{2}l_{e}/4. Therefore, inelastic processes ( that were contributing in the ballistic regime) outside of this volume do not contribute to the back scattering current given rise to that the features will be smeared or diffuses.
The third regime, known as the Maxwell regime is that in which the Knudsen ratio is very small, the electronic mean free path now is small in comparison to the radius of the injecting contact: l << 2a. Here in this regime, all the injected energy is dissipated in the area of the junction, rising therefore, the temperature in the region of the junction or constriction.
To end with this short section, only rest to recommend the review article by Duif et al. [10] to whom be interested in this simple point contact spectroscopy, but not trivial to implement, and difficult to interpret as compared with tunneling spectroscopy.
3 Superconducting Materials
After 86 years from the discovery of superconductivity, we have today a great diversity of many different materials, with variated physical characteristic that present superconducting behavior at different transition temperatures. This enormous number of materials are known mainly, for the long time has passed since the discovery of the superconducting phenomena, but more important however, for the long standing interest to understand this wonderful state of the matter. Among all these sort of different materials with different physical characteristics and behaviors we might mention: pure elements, intermetallic alloys, ceramics, compounds without any transition metal as Ba_{1x}Pb_{x}BiO_{3} and Ba_{1x}K_{x}BiO_{3}, compounds known as blue bronzes, intercalated compounds with bidimensional structure, low dimensional systems, heavy Fermions, Chevrel phases, organic compounds, and superconducting ceramics of high transition temperature based in copper, etc. Among all these sets of diverse compounds, in many of them it is thought that the superconducting phenomena might be due to other mechanisms than the classical electron phonon interaction (eph). In some of them, it is presumed that the responsible interaction to form the Cooper pairs must be necessarily related with magnetic processes, excluding subsequently the eph interaction, as the only process involved in the pair formation. This unconventional type of interactions has been maturing in the last few years, and now with more precise experiments, and more sophisticated tools further evidences are clearly indicating that in heavy Fermions, at least, the Cooper pairing formation is of magnetic nature. This kind of interaction, of magnetic character, clearly is in contradiction with the canonical form of the BCS theory, in which both the magnetic and superconducting phenomena always compete. Nonetheless, the discovery of those new processes have increases our knowledge and new physics is emerging which brings new and important phenomena and for sure more new phenomena will be discovered in the future.
According to the type of pairing interaction, different to the eph interaction, many of the materials that present superconducting behavior might be catalogued as unconventional superconductors. Among these, for example, are the heavy Fermions superconductors, which present strong correlation between electrons of very large effective masses. In these heavy Fermions, it is possible to talk about the interplay between two kind of electronic gases; one formed by the normal conducting electrons of type d, and other formed by magnetic electrons or type f electrons, which present strong localization. Another type of unconventional systems are those as the reentrant superconductors, among them we can found the Chevrel phase HoMo_{6}S_{8}, the metallic alloy ErRh_{4}B_{4}, the recently discovered HoNi_{2}B_{2} C, and some compounds based in Tin and rare earths. These alloys or compounds, present at different temperatures, first superconductivity, and at lower temperature; ferromagnetism. The competition between the two phenomena clearly can be seen in the reentrance to the normal state at a temperature T_{C1} once the compound is in the superconducting state at T_{C2}, here T_{C2} is a higher temperature than T_{C1}. The explanation for this kind of behavior is that the magnetic process, settled at lower temperature (below the superconducting transition temperature) is stronger than the pairing coupling, and at the end, is stronger and destroy superconductivity. In all these reentrant superconductors always the magnetic process is of ferromagnetic nature; why, antiferromagnetism never produces this effect is still not very well understood. Some other unconventional materials are the organic compounds, in which recently new phenomena have been discovered and the type of interactions are indicating unconventional behavior in both the normal state and superconducting state. Of course other materials clearly catalogued as unconventional are the superconducting ceramics of high transition temperature, which as is very well known present a series of many different behaviors in comparison to normal metals, and are far to be totally understood. So in this case also is difficult to assign to the eph coupling mechanism the great variety of electronic processes that are observed in these ceramics.
In the rest of this chapter we will describe briefly some of these characteristics of superconducting cuprates, the idea mainly is to draw a panoramic scheme and situate at the interested reader in a modern context about the study of some of these materials.
3.1 Superconducting Cuprates of High T_{C}.
Material  T_{C} (K)  Year 
Hg  4.1  1911 
Pb  7.2  1913 
Nb  9.2  1930 
NbN_{0.96}  15.2  1950 
Nb_{3}Sn  18.1  1954 
Nb_{3}(Al_{3/4}Ge_{1/4})  2021  1966 
Nb_{3}Ga  20.3  1971 
Nb_{3}Ge  23.2  1973 
Ba_{x}La_{5x}Cu_{5}O_{y}  3035  1986 
(La_{0.9}Ba_{0.1})_{2}CuO_{4d} at 1 GPa  52  1986 
YBa_{2}Cu_{3}O_{7d}  95  1987 
Bi_{2}Sr_{2}Ca_{2}Cu_{3}O_{10}  110  1988 
Ta_{2}Sr_{2}Ca_{2}Cu_{3}O_{10}  125  1988 
Ta_{2}Sr_{2}Ca_{2}Cu_{3}O_{10} al 7 GPa  131  1993 
HgBa_{2}Ca_{2}Cu_{3}O_{8+d}  133  1993 
HgBa_{2}Ca_{2}Cu_{3}O_{8+d} at 25 GPa  155  1993 
Hg_{0.8}Pb_{0.2}Ba_{2}Ca_{2}Cu_{3}O_{x}  133  1994 
HgBa_{2}Ca_{2}Cu_{3}O_{8+d} at 30 GPa  164  1994 
When in 1986 George Bednorz and K. Alex Müller discovered the superconducting cuprates with composition LaBaCuO, with critical temperatures of the order of 30 K [11], they awaken great interest amongst the scientific and technological communities. Since this discovery many new superconducting compounds have been found, in fact in a few months after the discover of the new superconductor, rapidly were found more compounds with much higher critical temperatures, and in fact very soon the utopical range of 90 K was reached. Now, after 10 years from the Bednorz an Müller discovery, the world of superconductivity has been completely changed and today with the mercury based compounds with similar compositions as other cuprates and with maximum transition temperatures in the order of 133 K^{[I}] apparently have reached the maximum limit in T_{C}. So now perhaps it will be necessary to look in another directions to try to find new compositions which will produce new superconducting compounds. A branch worths to explore with the idea to find new superconducting materials is the low dimensional organic (and inorganic) materials, as for example the systems formed with fullerenes, or the interesting ladders compounds. One approach might be to try to construct crystalline structures in which the internal chemical pressure can reach the external hydrostatic pressure that already we know increases the transition temperature in the cuprates. Other approach will be using different atoms in place of Copper to construct different materials and crystalline structures. At this point it is important to ask about the importance of the copper atoms in the superconducting compounds, or if we can produce other high T_{C} compounds without Copper, or if that really there are something very special in the combination of Copper and Oxygen atoms that still we do not know or understand. A concept that is emerging shows the importance in the superconducting ceramics, of the magnetic correlations caused by Copper atoms which are playing a very important role.
Table 2 shows the increasing of the transition temperature with different materials, and the particular composition in the superconducting ceramics that produces the maximum critical temperature in the compound.
All groups of those superconducting cuprates share many similarities. First, the crystalline structure is of the perovskite type, and basically in all cuprates always have the same building blocks to form the different compounds. For example, in the lower transition temperature compound, La_{2x}Sr_{x}CuO_{4d}, the crystallographic structure is of the K_{2} NiF_{4} type, but can be seen as a stacking of three perovskites. Figure 12 shows the crystalline structure of this compound.
Figure 12  Crystalline structure of La_{2x}Sr_{x}CuO_{4d} compound and the twin compound Sr_{2}RuO_{4}. This Ru based compound shows clearly the importance of copper atoms to obtain high transition temperature of about 37K, while the Ru based is less than 1K.
In other ceramic cuprates, as for example the Y_{1}Ba_{2}Cu_{3}O_{7d} (the [1:2:3] compound). The basic blocks are also stacked of three perovskites. These building blocks are formed as shown in figure 13. The most important parts, of course are all atoms, but important and more related to the electronic properties are the CopperOxygen chains, and the CopperOxygen planes. The planes are directly related with the transport properties. Yttrium or other of the rare earths that take part in the formation of crystalline structure of the [1:2:3] are used to form the structural frame that are needed to conform the main body of the compound. The contribution of these rare earth atoms to the electronic properties, is assumed to be small. However, this last assumption have to be taken carefully, because we have to remember the case of the Pr [1:2:3] compound, which is not superconductor, and however has all the crystallographic features of all other superconducting [1:2:3] compounds. In all superconducting cuprates, an interesting experimental evidence is that the transition temperature is very dependent of the number of copper planes in the compound: with three planes the maximum critical temperature is always obtained. More than three planes show that the maximum critical temperature is out of the optimum, and a saturation occurs, and the critical temperature start to decrease.
Figure 13  Perovskite building blocks that conform the [1:2:3] compound.
3.2 Physical Characteristics
In this section, we will describe some of the physical characteristics universally present in the superconducting cuprates. These always can be found when the compounds are in one of the three different regimes depending of the oxygen content (in general), or the alkaline earth atoms, when these are substituting lanthanum or other rare earth. The three regimes that we will describe are; the underdoped, optimum doped, and overdoped regimes. To analyze the influence of doping in the compounds we can start the discussion using three typical ceramic compounds as examples. We can use as typical compounds the following: La_{2x}Ca_{x}CuO_{4±d}, REBa_{2}Cu_{3}O_{7d}, and Nd_{2x}(Pr, Ce)_{x}CuO_{4±d}.
In the situation called the zero doping limit the compounds present the following stoichiometries or compositions: La_{2}CuO_{4}, REBa_{2}Cu_{3}O_{6}, and Nd_{2}CuO_{4}. In this limit, the compound is insulating and antiferromagnetic, with the magnetic ordering due to the copper atoms. The strength of the magnetic ordering can be measured according to the value of the Néel temperature. As we start to increase the level of doping it is found that the compound starts to change the behavior, presenting a semiconducting behavior. This kind of semiconductor is known in the literature as a Mott insulator. If we plot the resistance versus the temperature, we find that at low temperature the material tends to have an insulating or semiconducting character which is very sensitive to the degree of doping. Of course the level of doping is related directly to the number of holes or electrons injected at the Fermi level. What is found experimentally is the two characteristics: the insulating or semiconducting behavior disappear together with the antiferromagnetic Néel temperature, so while the more the insulating the material, higher the antiferromagnetic temperature. If we continue increasing the doping, more holes or electrons are added, the Néel temperature follows decreasing, until the moment that this disappear, and the material appears with the characteristics of a good conductor. But here we have to be very careful, because this apparently good conductor has now a very particular properties which are easily seen in the R versus T characteristics. For instances it shows a kind of ``metallic type behavior'', never ever observed in other materials or compounds until the discovery of the ceramic cuprates. This new extraordinary characteristic, clearly can be observed if we plotted the resistance versus temperature characteristic. What it is observed, is a perfect straight line which has a zero intercept at the origin, if this is extrapolated at low temperatures, below the superconducting temperature. With this level of doping, the compounds show at high temperatures the ``metallic'' behavior, and at low temperatures (between 30 K to 134 K, depending of the compound) the superconducting transition. This kind of metallic state persists with the zero intercept at the origin (and the high transition temperature) only in a very narrow optimal concentration of holes or electrons. In this optimum concentration the superconducting transition temperature is maximum and the Neel temperature is zero. With additional increase of the doping, the superconducting transition temperature start to decreases and reach a point of saturation, this mainly due to the stoichiometry or chemical nature of the compound which does not permit more increase in the dopant. At this point the ``metallic state'' becomes similar to a real metallic state.
It is worth to mention that this extraordinary interplay between this antagonist phenomena, magnetism and superconductivity, only was observed in the past in heavy Fermion systems and in some organic materials. This point of view makes us to think about a pairing mechanism which might to be related with interactions different to the electronphonon, and related in some way with magnetic interactions, as occurs in heavy Fermion systems. In figure 14 we show a diagram that presents all the features above mentioned with all the occurrence of the physical phenomena described in this section.
Figure 14  Phase diagram of ceramic superconductors, in the inset it is shown the resistence versus temperature characteristic in the three regimes described in the text. The main portion of the figure shows the variation of the number of holes in the CuO planes, and the three level of doping.
As we mentioned before, the level of doping changes the conducting properties of the copper ceramics. Why this change of behavior?, it is the normal question one can formulate. To respond to this question, one has to remember some of characteristics of the crystalline structure of the compounds. First at all, the ceramics cuprates in general are formed by two types of coppers according to the formal valence state presented; chains of CuO, and CuO_{2} planes. There exist a direct relationship to the insulating or conducting properties with the copper planes, because the electron orbitals of copper, 3d_{x2y2}, and 2p_{x2,y2} of the oxygen overlaps covalently, and depending of this degree of covalency the electronic density of states will be empty or half full, providing then that the transport properties can change between the different conducting states. From the chemical point of view the substitution of atoms like alkaline earth which have a strong tendency to free the two external electrons, in place of a rare earth atom, which in general have tendencies to work with a formal valence of 3^{+} produces in the copper atoms a state of mixed valence. Although it seems a fact that the valence changes is directly related with the transport properties in the superconducting ceramics, the detailed relationship remains to be understood. The experimental evidence indicates that the copper mixedvalence state and the low dimensionality are the characteristics necessarily present in these superconducting materials but there is no microscopic theory to explain the relationship between these evidences and the mechanism for their superconducting properties.
Those above mentioned facts are always present in all the ceramics cuprates. This will be the necessary feature that need to be present to obtain high temperature superconductivity if other element than copper are used? Recently, however it was demonstrated that indeed copper has a very special role in the size of the value of the critical temperature of the high Tc superconductors. A Ruthenium oxide, that presents exactly the same crystalline structure that the based ceramic superconducting cuprates of 30 K was synthetized by Maeno et al. ( see figure 13 and compare the two compounds, cuprates and rutheniutes) they found a superconducting transition temperature of only about 1 K, in this ceramic compound the crystalline structure is exactly the same like in the cuprate, the atomic positions are exactly the same, the only difference is that in place of copper there are ruthenium. However, going back to our initial discussion, we may also ask about the relative importance of the other atoms that form the crystalline structure of the ceramics?, also what are the different role of each atom that forms the structure. What so important is the oxygen, the alkaline earths, the different lantanides, like; Pr, Nd, Sm, Gd, etc. All these questions still remains a mystery in terms of microscopic grounds.
Other interesting characteristic, different to conventional or low T_{C} superconductors is related to the strong depression of the transition temperature when non magnetic atoms are substituted in place of copper atoms. This behavior is totally different to other systems, but similar to occurs in heavy Fermions. One example of this behavior is observed with the substitution of magnetic atoms in the [1:2:3] compound. If for instances Fe is substituting copper sites, the depression of the transition temperature is only small, in comparison as happen when Zn is substituted [12]. One can observe that for the case of conventional superconductors magnetic pair breaking is the only mechanism that depresses superconductivity, and well explained by the theoretical model of Abrikosov and Gor'kov. This interesting effect of Zn may implying the possibility of a magnetic interaction that is pairing the system, and also might imply a different symmetry in the order parameter, as for example dwave pairing. In this case if it were the case, would implies a strong sensitivity to non magnetic impurities. In spite of this, and according to this experiment of substitution of Zn and Fe atoms in place of Cu and to other more recently related with Josephson tunneling, and with flux quantization, there are more evidences that the symmetry of the order parameter is of the type dwave [2]. Particularly, it is worth mentioning that a beautiful experiment by Tsuei et al. had almost concluded this controversy and those experiments show that the order parameter is may be type d.
3.3 The Compound Ba_{1x}K_{x}BiO_{3}
The last topic that we would like to mention, also in a very compress form, is related to the superconductivity of another high temperature superconductor, owing to the perovskite family [13,14,15].
The discovery of superconductivity in Ba_{1x}K_{x}BiO_{3} provides the possibility of comparing its characteristics to conventional phonon mediated superconductors and to the new Cubased, highT_{c} materials [11]. This compound has received much attention because it is the first oxide superconductor without Cu in which the transition temperature, T_{c}, is of the order of 30 K, well above the best A15 strong coupled superconductors. The crystal structure of this compound is a simple cubic perovskite formed from cornershared BiO_{6} octahedra with Ba or K on the cell origin.
This three dimensional structure is isotropic with no magnetic effects, implying that the pairing mechanism might be of a different nature to that of the CuO based compounds, where low dimensional behavior and/or magnetism could play a fundamental role which is responsible for their abnormal high transition temperatures and for the anomalous normal state properties. The absence of magnetic order in both BaBiO_{3} and Ba_{1x}(K,Rb)_{x}BiO_{3} compounds is in contrast to the competition between antiferromagnetism and superconductivity in the cuprates.
From another point of view, Uemura [16] has found that it is possible to classify superconductors according to the value of the ratio T_{c}/T_{F} where T_{F} is the Fermi temperature, and T_{c} is the critical temperature. Uemura found that T_{c}/T_{F}=1/10 to ~ 1/100 is high for materials such as cuprates, BKBO, organic superconductors including C_{60}'s, Chevrel phases and heavyfermions, whereas ordinary BCS superconductors have ratios of the order of T_{c}/T_{F} £ 1/1000. The point that Uemura suggests is the possibility that these systems may be classified as a kind of superconductors with exotic properties, different to the BCS systems. These kind of superconductors have features that differ from the BCS's systems, e.g. high H_{c2}, small x, proximity to metalinsulator transition, spin and/or charge instabilities, highly correlated electronic behavior, and electrical resistivity close to the Mott limit. Ba_{1x}K_{x}BiO_{3} compound together with Ba_{1x}Rb_{x}BiO_{3}, belongs to the family of compounds BaBiO_{3}, BaPbO_{3}, BaPb_{1x}Sb_{x}O_{3}, and BaPb_{1x}Bi_{x}O_{3} all of these holding a perovskite crystalline structure. All the above ceramics are superconductors, except for BaBiO_{3}, showing transition temperatures ranging from 0.5 K to 30 K, for BaPbO_{3} to Ba_{1x}K_{x}BiO_{3} respectively. An interesting feature of all these BaBiO_{3} based superconductors, is that they have some peculiar characteristics which do not exactly fit the BCS simple theory to explain the transition temperatures, i.e. their low electron density, Coulomb repulsion seems nearly absent, and the highest transition temperature at doping levels in the limit of the metalinsulator transition. Also the insulating state is intriguing and difficult to explain. It is improbable that charge density wave induces the insulating state due to the nesting of the Fermi surface, because over a wide range of composition the crystal structure is nearly cubic, and implying a total nesting of the three dimensional Fermi surface to obtain the insulating behavior. In the case of the formation of charge or spin instability both will compete to oppose the formation of the superconducting state. Another interesting feature of this compound is that the isotope effect has a considerable value close to the BCS prediction that again remarks the importance of the electronphonon interaction. On the other hand the compound is diamagnetic over the entire range of compositions; no magnetism exist, nor Motttype insulator behavior; then the following questions may arise, what is the origin of the insulating state? Why is the transition temperatures are so high despite of the low electronic density? Why is the superconducting state close to the boundary with the insulator state?, Do both bismuthates and cuprates present a new state of the matter? and lastly, are these ceramics in a way similar to the cuprates? All these kind of physical properties, deserve further investigation; in particular the study of the normal properties will bring light about the anomalous superconducting behavior in these interesting materials.
To end with this chapter on superconductivity I would like to recommend to the interested in these topics four important books given at the last of this chapter: references [6,17,18,19].
4 Acknowledgments
I would like to thank F. Morales and D. Mendoza for valuable discussion, and for the help on the writing of this chapter. Also I would like to thanks G. Guevara, and D. Gutierrez which help me with the use of the computers and digitalization of the figures. This work was supported by the Programa Universitario de Superconductores Cerámicos de Alta Temperatura Crítica, by the Dirección General de Apoyos al Personal Académico, of the Universidad Nacional Autónoma de México, by the Consejo Nacional de Ciencia y Tecnología and by the Organization of American States.
References
[3] P.W. Anderson in Superconductivity in the past and the future chapter 23 of SUPERCONDUCTIVITY edited by R.D. Parks. page 1343, 1969. Marcel Dekker, inc New York.
[4] Mendelssohn K. in Progress in Low Temperature Physics, Vol. 1, North Holland 1955.
[5] V. L. Ginzburg, and L. D. Landau., Exp. J. Theor. Phys. 20, 1064 (1950).
[6] the most important reference about superconductivity is the book SUPERCONDUCTIVITY edited by R.D. Parks with 23 chapters. Marcel Dekker inc. New York 1969.
[7] H. C. Montgomery, J. Appl. Phys. 42, 2971 (1971).
[8] E. L. Wolf Principles of Electron Tunneling Spectroscopy Oxford University Press, New York, 1985.
[9] T. Hasegawa, H. Ikuta, and K. Kitazawa in Physical Properties of High Temperature Superconductors III. (ed.) D. M. Ginsberg, World Scientific Publishing, 1992.
[10] A. M. Duif, A. G. M. Jansen, and P. Wyder, j. Phys.: Condens Matter. 1, 3157 (1989).
[11] J. G. Bednorz, and K. A. Müller.,Z. Phys. B 64, 189 (1986).
[12] J. Axnas, et al., Phys. Rev. B53, R3003 (1996).
[13] A. W. Sleight, J. L. Gillson, and P. E. Bierstedt, Solid State Commun. 17, 27 (1975).
[14] L. F. Mattheiss, E. M. Gyorgy and D. M. Johnson, Jr., Phys. Rev. B37, 3745 (1988).
[15] R. J. Cava, B. Batlogg, J. J. Krajewski, R. Farrow, L. W. Rupp Jr, A. E. White, K. Short, W. F. Peck, and T. Kometani, Nature 332, 814 (1988).
[16] Y.J.Uemura,Physica C 185189, 733 (1991).
[17] A general reference related with type II superconductors is "Superconductivity of Metals and Alloys by P. G. DeGennes., Benjamin, New York., 1966.
[18] A important book about superconductivity that explain with great clarity the difficult mathematical part of the theory of BCS is "Theory of Superconductivity" by J. R. Schrieffer. Benjamin, New York., 1964.
[19] M. Tinkham Introduction to superconductivity. Robert & Krieger Publishers Company, Inc., 1980.
Footnotes:
[A] BCS comes from the initial letters of the first names of John Bardeen, Leon Cooper and Robert Schrieffer [1].
[B] Actually there are many models to explain high T_{C} superconductivity in the cuprates, superconductivity in organic materials, and in heavy Fermions. All they are necessarily based in a pairing mechanism in which the elementary excitation to pair the system is not completely understood
[C] X might be PF6, CIO4, AsF6, etc.
[D] A is one Alkaline atom or a combination of 2
[E] Meissner state is defined as a state where perfect diamagnetism exist, this mean that there is a total expulsion of the magnetic field inside of the superconducting material
[F] More about of these thermodynamic quantities and the formulae can be found in currently elementary books about superconductivity
[G] As we mention in the above section, this modification in the electronic spectrum is taking place in the phase space and not in the real space
[H] It is important to remark again the differences between ordinary conductors, where the current is proportional to the electric field, and in superconductors, where the current is proportional to the vector potential.
[I] It is worth noting, that in all the
cuprates the T_{C} suffers a dramatic increases when
an external pressure is applied. In the Hgbased compounds the maximum T_{C}
under applied pressure is about 164 K
Obs.: This paper was presented as an invited talk at the Simpósio Matéria 2000, Rio de Janeiro, Oct. 23  27, 2000.