Superconductor thermodynamics

Example Superconductor Thermodynamics. Figure4.19 shows a phase diagram in the T-//-plane, i.e. in this system the magnetic //-field assumes the role of P. The two phases are named s and n. We easily can work out a version of the Clapeyron equation in this case. Our starting point is Eq. (4.44), i.e. 

Rankine cycle, 23 231—234 Thermodynamic stability, of MgB2 superconductors, 23 833 Thermodynamic state variables,... 

According to the relationship between the lattice volume and Tc as described, cubic CssCgo would be an ultimate candidate for a higher Tc superconductor, but the conventional vapor-solid reaction affords only the thermodynamically stable CsCso and CS4C60 phases. In 1995, noncubic CssCgo was obtained by a solution process in liquid ammonia, and the superconductivity was observed below 40 K under an applied hydrostatic pressure of 1.4 GPa [311]. 

Harsta A (1997) Thermodynamic modelling of CVD of high-T-c superconductors. Journal of Thermal Analysis 48(5), 1093-1104... 

Classical BCS theory dictated that superconductors with the highest Tc s would not be thermodynamically stable phases. Softening the phonons would raise Tc but would ultimately lead to structural instabilities. Increasing the density of states at the Fermi level would raise Tc but would eventually lead to an electronic instability. 

Trivial examples of metastability are solid solutions. Because these are inherently defect systems, they cannot be thermodynamically stable at low temperatures. Most of our high Tc superconductors need to be regarded as solid solutions which are then necessarily metastable phases. We could dismiss this as an irrelevant observation on the basis that solid solutions are merely required in order to adjust the carrier concentration to appropriate levels. However, we seem unable to generally make stable high Tc superconductors. One could even suggest that there is a correlation between Tc and metastability the higher the Tc, the more unstable. 

There are two possible exceptions to the rule that high Tc superconductors are metastable. Both YBa2Cu4Og and Y2Ba4Cu7015 appear to be stable under the conditions where they form, and they are superconductors without further oxidation. The remaining question to be answered is whether or not they are thermodynamically stable at room temperature and below. This is a difficult question to answer. Calorimetry can compare the heats of formation of these compounds with other compounds known in the Y/Ba/Cu/O system. However, it is always possible that some of the most stable phases in the Y/Ba/Cu/O system have not yet been prepared because they are kinetically inaccessible. 

Figure 13.16 Magnetization verses applied magnetic field for (a) a type I superconductor and (b) a type II superconductor. For the type I superconductor, the magnetic flux does not penetrate the sample below 9 Cc where the sample is a superconductor. Above rMc, the sample is a normal conductor. For the type II superconductor, the magnetic field starts to penetrate the sample at 3Cc, 1, a magnetic field less than rXc, the thermodynamic critical field. Superconductivity remains in the so-called vortex state between 9 c and Ci2 until WCt2 is attained. At this magnetic field, complete penetration occurs, and the sample becomes a normal conductor.

Not only do the thermodynamic properties follow similar power laws near the critical temperatures, but the exponents measured for a given property, such as heat capacity or the order parameter, are found to be the same within experimental error in a wide variety of substances. This can be seen in Table 13.3. It has been shown that the same set of exponents (a, (3, 7, v, etc.) are obtained for phase transitions that have the same spatial (d) and order parameter (n) dimensionalities. For example, (order + disorder) transitions, magnetic transitions with a single axis about which the magnetization orients, and the (liquid + gas) critical point have d= 3 and n — 1, and all have the same values for the critical exponents. Superconductors and the superfluid transition in 4He have d= 3 and n = 2, and they show different values for the set of exponents. Phase transitions are said to belong to different universality classes when their critical exponents belong to different sets. 

Chapters 13 and 14 use thermodynamics to describe and predict phase equilibria. Chapter 13 limits the discussion to pure substances. Distinctions are made between first-order and continuous phase transitions, and examples are given of different types of continuous transitions, including the (liquid + gas) critical phase transition, order-disorder transitions involving position disorder, rotational disorder, and magnetic effects the helium normal-superfluid transition and conductor-superconductor transitions. Modem theories of phase transitions are described that show the parallel properties of the different types of continuous transitions, and demonstrate how these properties can be described with a general set of critical exponents. This discussion is an attempt to present to chemists the exciting advances made in the area of theories of phase transitions that is often relegated to physics tests. 

The Josephson current, being an equilibrium supercurrent between two superconductors, can be calculated from the general thermodynamical relation... 

Ehrenfest s concept of the discontinuities at the transition point was that the discontinuities were finite, similar to the discontinuities in the entropy and volume for first-order transitions. Only one second-order transition, that of superconductors in zero magnetic field, has been found which is of this type. The others, such as the transition between liquid helium-I and liquid helium-II, the Curie point, the order-disorder transition in some alloys, and transition in certain crystals due to rotational phenomena all have discontinuities that are large and may be infinite. Such discontinuities are particularly evident in the behavior of the heat capacity at constant pressure in the region of the transition temperature. The curve of the heat capacity as a function of the temperature has the general form of the Greek letter lambda and, hence, the points are called lambda points. Except for liquid helium, the effect of pressure on the transition temperature is very small. The behavior of systems at these second-order transitions is not completely known, and further thermodynamic treatment must be based on molecular and statistical concepts. These concepts are beyond the scope of this book, and no further discussion of second-order transitions is given. 

From a thermodynamic standpoint the difference between type 1 and type 2 superconductors is that the surface energy of the interface between the superconducting and normal states is positive in the first case (a > 0), and negative (a < 0) in the second. 

Early was proposed to used the functional integral methods for calculation the thermodynamic properties of high-Tc superconductors including antiferromagnetic spin fluctuations [5],... 

For strong type-II superconductors a difference may exist between B 2 obtained from ac-susceptibility data and the thermodynamically relevant upper critical field, 5c2, for which (2.13) and (2.14) were derived originally. Prom extensive work on high-Tc cuprates it is known that B 2 is related to the so-called irreversibility line [196]. The basic idea of a semiquantitative flux-creep theory [197] is that pinned vortices can be activated thermally over an energy barrier Uq resulting in a reduced critical current of the form [196]... 

A thermodynamic quantity not very often measured for organic superconductors is the specific heat, C. Usually the crystal sizes are rather small and consequently a high sensitivity of the apparatus is needed. In most experiments, therefore, an assembly of many pieces of material is necessary to gain better resolution. In addition, the jump of C at Tc is expected to be rather small especially for compounds with higher transition temperatures because of the comparatively large lattice contribution to C owing to the low electron density and the low vibrational frequencies. 

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