Theory

Electrical resistance in metals arises because electrons propagating through the solid are scattered due to deviations from perfect translational symmetry. These are produced either by impurities (giving rise to a temperature independent contribution to the resistance) or the phonons (lattice vibrations in a solid - the temperature dependent occupancy of these boson states produces a temperature dependent resistivity p = p₀ + AT⁵ in a metal at low temperature.

In a superconductor below its transition temperature T_c, there is no resistance because these scattering mechanisms are unable to impede the motion of the current carriers. The current is carried in all known classes of superconductor by pairs of electrons known as Cooper pairs. The mechanism by which two negatively charged electrons are bound together is still controversial in "modern" superconducting systems such as the copper oxides or alkali metal fullerides, but well understood in conventional superconductors such as aluminium in terms of the mathematically complex BCS (Bardeen Cooper Schrieffer) theory.

The essential point is that below T_c the binding energy of a pair of electrons causes the opening of a gap in the energy spectrum at E_f (the Fermi energy - the highest occupied level in a solid), separating the pair states from the "normal" single electron states.

The size of a Cooper pair is given by the coherence length which is typically 1000Å (though it can be as small as 30Å in the copper oxides). The space occupied by one pair contains many other pairs, and there is thus a complex interdependence of the occupancy of the pair states. There is then insufficient thermal energy to scatter the pairs, as reversing the direction of travel of one electron in the pair requires the destruction of the pair and many other pairs due to the nature of the many-electron BCS wavefunction. The pairs thus carry current unimpeded.

BCS theory applies directly to superconductors such as Nb₃Ge (T_c = 23K) in which the electrons are bound together by their interaction with the vibrations of the underlying lattice: one electron in the pair polarises the lattice by attracting the nuclei towards it, leaving a region of excess positive charge (a potential well) into which a second electron is attracted - the positively charged nuclei thus mediate an attraction between the negatively charged electrons. Only electrons within the vibrational frequency of E_F can be paired by this interaction, and so only a small fraction of the electrons become superconducting.

The most obvious experimental signature of superconductivity is the observation of zero D.C. electrical resistance, and the possibility for low loss power transmission and large fast computers is one of the main reasons for technological interest in breakthroughs in superconductivity.

Zero resistance is hard to measure, and the most definitive evidence for superconductivity in fact arises from d.c. magnetic measurements. Persistent currents on the surface of the superconductor make it a perfect diamagnet below T_c i.e. it expels all magnetic flux. The difference between superconducting diamagnetism and that of benzene is both its larger size and more complex history dependence. This is illustrated by the effect of the cooling procedure on the measured superconducting diamagnetism:

• Cooling in zero applied magnetic field and measurement of the magnetisation on warming in a measuring field applied below T_c (zero field cooled (ZFC) measurement). The superconductor displays perfect diamagnetism (flux exclusion)
  
• Cooling in the measuring field through T_c followed by measurement on warming (field cooled or FC measurement). Magnetic flux present inside the sample above T_c is trapped inside by the shielding currents on the surface below T_c . The FC susceptibility is still diamagnetic (flux expulsion) but reduced in magnitude compared to the ZFC susceptibility.

This hysteresis of the diamagnetic susceptibllity below T_c is known as the Meissner effect and is definitive proof of superconductivity. The magnetic behaviour of a superconductor below T_c as a function of field is also more complex than that of a simple para- or diamagnet. The magnetisation initially varies linearly with applied field, but its behaviour above a critical field allows the division of superconductors into Type I or the more common Type II.

Type I superconductors lose their superconductivity above a critical field H_c as the field penetrates the material. In type II superconductors, the field penetrates the superconductor partially to form the Abrikosov flux lattice above the lower critical field H_c1. Above H_c1 the diamagnetism decreases with increasing applied field until superconductivity is quenched at the upper critical field H_c2, returning to the normal metallic state.

The value of HC2 is very important as it partially determines the current carrying capacity of the superconductor and its uses e.g. to produce high field superconducting magnets. In the copper oxides, H_c2 is of the order of 40T at liquid helium temperatures.

Copper oxide superconductors

The YBa₂Cu₃0₇ sample made in this experiment was the first material ever to have a superconducting transition temperature above the boiling point of liquid nitrogen (BCS based predictions had suggested a limit to T_c of about 30-40K). It is important to note that superconductivity only occurs for copper oxidation states either greater or less than, but not equal to, two e.g. La₂CuO₄ is an antiferromagnetic insulator whereas La_1.85Sr_0.15CuO₄ is a metal and superconductor. Question 6 asks you to account for the insulating nature of the Cu(II) compounds.

The binding mechanism leading to the formation of Cooper pairs in the copper oxides is thought to be related to the extremely strong superexchange interactions between the copper spins and not to result from the electron-vibration interaction which produces the lower T_c's.

The variation of electronic properties with copper oxidation state is well illustrated in the La_2-xSr_xCuO₄ series. For 0>x>0.03, the Neel temperature T_N decreases from 250K at x=0 to zero at 0.03, and a transition to a superconducting metal occurs at x = 0.06, with T_c rising to a maximum of 42K at 0.15. Beyond x=0.2, the superconductivity disappears and the compounds are non-superconducting metals. The importance of the copper oxidation state and the chemical environment of the copper cations in controlling the properties is shown by the very wide variation in T_c in the compounds listed in Table I. The influence of applied pressure on superconductivity is also remarkable, and suggests the likely influence of possible chemical substitutions. The highest superconducting transition temperature yet achieved reproducibly is 150K under 23.5 GPa of hydrostatic pressure in HgBa₂Ca₂Cu₃O_8+d.

Table I: Transition temperatures in inorganic superconductors