In this article we will discuss about:- 1. BCS Theory of Superconductors 2. High-Temperature Superconductors 3. Applications.
BCS Theory of Superconductors:
For energy gap to occur there must be a mechanism whereby electrons in superconductors can lower their energy. Since the gap is very small (≈ 10-4 eV), such a mechanism is a weak attractive interaction, which is the resultant of repulsive electron-electron interaction and the interaction between the electrons and the lattice ions.
The lattice interaction is attractive in nature as follows- In a lattice, an electron tends to pull towards itself the positive ions, so that it is surrounded by a region of positive charges. Another electron coming into the vicinity will be drawn towards this region of enhanced positive charge density of ions and as a result it will look as if it were attracted towards the first electron.
This phenomenon of attraction led Bardeen, Cooper and Schrieffer (BCS) to demonstrate that the ground state of an assembly of mutually attracting electrons is separated by an energy gap from the lowest excited levels of the energy spectrum. The attractive interaction forms two electron singlet bound states in momentum space called Cooper pairs (two electrons having equal and opposite momenta and spin).
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Because of limitations on the response time of the lattice to the motion of electron, only a proportion of the conduction electrons can participate in the pairing process. These conduction electrons lie within energy equal to KBθ0, where θD is the Debye temperature. This means that roughly a fraction 10-4 of the itinerant electrons form Cooper Pairs within a spherical energy shell of width KBθD centered at the Fermi surface.
The wave function of a Cooper pair extends over a fairly large volume and overlaps the wave functions of other Cooper pairs. This overlap gives rise to strong correlations among the motion of all the pairs (~106 pairs). Hence the superconducting state is a collective state in which all the conduction electrons act co-operatively.
According to Schrieffer, the conduction electrons in a superconductor are condensed into a single domain which extends over the entire volume of the superconducting system and is capable of motion as a whole.
Because of the existence of the energy gap, this superconducting domain in the ground state resists perturbation unless the energy of the perturbation exceeds the gap energy as this will break up Cooper pairs. This means that the wave function of the collective ground state has a stiffness or rigidity which explains the zero resistivity of the superconductor.
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An electric current along the superconductor involves an overall rigid translational motion of the single superconducting domain. Such a translational motion requires only a small amount of energy which proceeds without friction because the random scattering of an electron by an irregularity in the lattice would affect the correlations of the electrons and require a transition to an excited state, above the energy gap and thus the scattering is inhibited.
The rigidity of the wave function also explains the Meissner effect. When the superconductor is immersed in the magnetic field, the wave function of the collective state does not change, i.e., the orbital configuration of the electrons does not change. However, the electrons of cooper pair change their speed while remaining in the same orbital configuration.
This change of speed of the charge carrier’s amount to an induced current which expels the magnetic field forms the volume of the superconductor. Such a change of speed without a change of orbital configuration is analogous to what happens in the Bohr model if an election in a circular orbit is gradually immersed is a magnetic field. This change of speed gives rise to diamagnetism of atoms. The analogous change of speed of Cooper pairs in the collective state gives rise to perfect diamagnetism of superconductors.
The superconductivity is due to the mutual interaction and correlation of the behaviour of electrons which extends over a considerable distance. The maximum distance upto which the states of pair electrons are correlated to produce superconductivity is called coherence length . The paired electrons can be many thousands of atomic spacing apart, i.e., means free path is very small as compared to the normal state.
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This shows that the long range nature of the correlation. The properties of a superconductor depend on the correlation of electrons within a volume of 3 called the coherence volume. It is because, the large number of electrons in such a volume act together in superconductivity, that the transition is extremely sharp. For type-I superconductor ξ > Δ and for type-II superconductor ξ > Δ.
The BCS theory makes the following predictions which have been compared with experiments for a number of oxide superconductors:
(a) The transition temperature Tc and energy gap Eg are related to the Debye temperature θD, the electron-electron attractive potential V and density of electrons at Fermi level (N(EF) as:
(b) The energy gap Eg is proportional to the transition temperature through Eg/KbTc ≈ 3.5 except for gapless superconductors.
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(c) The London equation is a consequence of the BCS theory and hence one expects the state of perfect diamagnetism to exist below Tc with x = -1.
(d) The transition temperature depends upon the average isotopic mass m through the relation
Tc ≈ m-1/2
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which gives a clear guide to the theory that electron-electron interactions exist via lattice ions.
(e) There is discontinuity in the electronic contribution to the specific heat at the transition temperature given by-
Cs – Cn/Cn = 1.43
where subscripts s and n denote the superconducting and normal states respectively.
High-Temperature Superconductors:
High Tc or HTS denotes superconductivity in materials, chiefly ceramic oxides, with high transition temperatures accompanied by high critical currents and magnetic fields. The first group of such superconductors discovered was La2-x, Mx CuO4 (M = Ba, Sr, Ca) with Tc ranging from 25 to 40 K and is usually referred to as 214 system. This discovery was followed by the discovery of another important system Ln Ba2 Cu3 O7-x (Ln = Y, Nd, Sm, Eu, Gd, Dy, Ho, Er, Tm, Yb) with x ≈ 0.2. This is called 123 systems and has orthorhombic structure.
The transition temperature for some of the important HTS is given below:
La1.85Ba0.15CuO4; Tc = 36 K
YBa2Cu3O7; Tc = 90 K
Tl2Ba2Ca2Cu3O10; Tc = 120 K
HgBa2Ca2Cu3O8 +δ ; Tc = 135 K
The BCS theory predicts the transition temperature Tc very low (maximum ≈ 40 K), so important issue is to understand as to why the Tc of Cu superconductors is so high. In weak coupling (electron-lattice interaction) limit, Tc can be expressed in this theory as;
where hWD is a cut off energy comparable to the maximum energy of excitation which mediate the pairing. Within the BCS framework there are the ways to increase Tc: (a) increase N (EF) (b) increase interaction potential V and (c) increase Debye frequency of lattice.
It is proposed that an attractive interaction between carriers can be mediated by the exchange of virtual electron-hole pairs the so called exciton. In order for this mechanism to work, two distinct types of carriers are required- (i) conducting carriers in a metallic region which pair to form the superconducting condensate and (ii) polarizable electrons in adjacent region which interact with the metallic carriers to mediate the pairing process.
The above mechanism is predicted to yield a significantly higher Tc than the electron-lattice interaction mechanism because the cut-off energy hWD is expected to be comparable to electronic excitation energies which are generally much higher than typical Debye energies. The occurrence of a strong electronic band in the infrared spectrum and its correlation with superconductivity provides evidence that the superconductivity of 123 materials is mediated by an excitonic mechanism.
One of the leading theories to high temperature superconductors of the class is the Resonating Valence Band (RVB) model proposed by Anderson and his co-workers. According to this model, the pairing mechanism is magnetic in origin and not of conventional BCS type.
The conductivity in (Sr, Ba, Y, La)2 CuO4 superconductors is increased by the resonance along the O-Cu…O-Cu lines of atoms, and superconductivity is probably achieved through interaction with lattice vibration.
Many of the properties of these conventional high – Tc superconductors are identical to those of conventional low – Tc metallic superconductors. These include the existence of energy gap over the entire Fermi surface below Tc and the Josephson tunneling. These superconductors possess certain properties which cannot match with those of conventional ones.
These are small isotope effect, small coherence length and unconventional temperature dependence of normal state response functions. Also, the pressure is found to increase the Tc in high conventional superconductors. The identification of the possible conduction mechanism in the high – Tc superconductors is perhaps the most challenging problem in condensed matter physics these days.
Applications of Superconductors:
(a) Superconductors are used for producing very strong magnetic field of about 20-30 Tesla, which is much larger than the field obtained from an electromagnet and such high magnetic fields are required in power generators.
(b) Magnetic energy can be stored in large superconductors and drawn as required to counter the voltage fluctuations during peak loading.
(c) These superconductors can be used to perform logic and storage functions in computers.
(d) A superconductor’s material can be suspended in air against repulsive force from permanent magnet. The levitation effect can be used in transportation.
(e) As there is no heat losses in a superconductor (i.e., I2R = 0), so power can be transmitted through the superconducting cables without any losses.