SUPERCONDUCTORS
INDEX
INTRODUCTION
REVIEW OF LITERATURE
IDEAS LEADING TO THE BCS THEORY
BCS THEORY OF SUPERCONDUCTIVITY
MATHEMATICAL NOTATION OF BCS THEORY
SCIENTISTS WHO WORKED FOR BCS THEORY
APPLICATIONS OF BCS THEORY
FUTURE SCOPE
BIBLIOGRAPHY
INTRODUCTION
I Ravneet kaur of B.tech IT(hons) department working on the topic called “ BCS Theory ” The topic of my term paper include mainly the approach towards the BCS theory of superconductors ie what is BCS theory, how it could be explained, Review of literature for BCS Theory and contribution of five scientists in working of BCS theory with applications and future scope of BCS Theory.
REVIEW OF LITERATURE
The BCS Theory of Superconductivity:
"a complete theoretical explanation of the phemonenon",
“The phenomenon of superconductivity was discovered by the Dutch physicist Kamerling Onnes in 1911. Already his first measurements indicated that one had found a fundamentally new state of matter... Many remarkable properties were discovered in the following decades. However, the central problem, the question about the underlying mechanism for superconductivity, remained a mystery up to the late 50's... A significant step forward was taken around 1950 when it was found theoretically and experimentally that the mechanism for superconductivity had to do with the coupling of electrons to the vibrations of the crystal lattice. Starting from this mechanism, Bardeen, Cooper and Schrieffer developed in 1957 a theory of superconductivity, which gave a complete theoretical explanation of the phenomenon.
[The BCS Theory] was indeed very successful in explaining in considerable detail the properties of superconductors. The theory also predicted new effects and it stimulated an intensive activity in theoretical and experimental research, which opened up new areas for research. One may as examples mention the use of the quantum mechanical tunnel phenomena to study superconductors, the discovery of magnetic flux quantization and the remarkable Josephson effects. These more recent developments are intimately connected with the fundamental theory of superconductivity and have confirmed in a striking way the validity of the theoretical concepts and ideas developed by Bardeen, Cooper and Schrieffer."
Ideas Leading to the BCS Theory
The BCS theory of superconductivity has successfully described the measured properties of Type I superconductors. It envisions resistance-free conduction of coupled pairs of electrons called Cooper pairs. This theory is remarkable enough that it is interesting to look at the chain of ideas which led to it.
1. One of the first steps toward a theory of superconductivity was the realization that there must be a band gap separating the charge carriers from the state of normal conduction.
o A band gap was implied by the very fact that the resistance is precisely zero. If charge carriers can move through a crystal lattice without interacting at all, it must be because their energies are quantized such that they do not have any available energy levels within reach of the energies of interaction with the lattice.
o A band gap is suggested by specific heats of materials like vanadium. The fact that there is an exponentially increasing specific heat as the temperature approaches the critical temperature from below implies that thermal
energy being used to bridge some kind of gap in energy. As the temperature increases, there is an exponential increase in the number of particles which would have enough energy to cross the gap.
2. The critical temperature for superconductivity must be a measure of the band gap, since the material could lose superconductivity if thermal energy could get charge carriers across the gap.
3. The critical temperature was found to depend upon isotopic mass. It certainly would not if the conduction was by free electrons alone. This made it evident that the superconducting transition involved some kind of interaction with the crystal lattice.
4. Single electrons could be eliminated as the charge carriers in superconductivity since with a system of fermions you don't get energy gaps. All available levels up to the Fermi energy fill up.
5. The needed boson behavior was consistent with having coupled pairs of electrons with opposite spins. The isotope effect described above suggested that the coupling mechanism involved the crystal lattice, so this gave rise to the phonon model of coupling envisioned with Cooper pairs.
BCS Theory of Superconductivity
An intuitive description of superconductivity is sufficient for public and non-technical uses. However, ultimately, a more rigorous mathematically- based explanation must be formulated. Superconductivity was not sufficiently explained until 1957 when John Bardeen and his graduate assistants Leon Cooper and John Schreiffer proposed a microscopic explanation that would later be their namesake: the BCS Theory. This theoretical explanation later earned them the Nobel prize, making John Bardeen the only man in history to be awarded this honor twice.
The BCS Theory is, in its simplest form, actually contradictory to our crude macroscopic view expressed earlier. As discussed earlier, superconductivity arises because electrons do not interact destructively with atoms in the crystal lattice of the material. The BCS Theory says that electrons do actually interact with the atoms, but constructively.
The BCS Theory makes a crucial assumption at the beginning: that an attractive force exists between electrons. In typical Type I superconductors, this force is due to Coulomb attraction between the electron and the crystal lattice. An electron in the lattice will cause a slight increase in positive charges around it. This increase in positive charge will, in turn, attract another electron. These two electrons are known as a Cooper pair. If the energy required to bind these electrons together is less than the energy from the thermal vibrations of the lattice attempting to break them apart, the pair will remain bound. This explains (roughly) why superconductivity requires low temperatures- the thermal vibration of the lattice must be small enough to allow the forming of Cooper pairs. In a superconductor, the current is made up of these Cooper pairs, rather than individual electrons.
So, Cooper pairs are formed by Coulomb interactions with the crystal lattice. This is also what overcomes resistance. Remember, an electron inside the lattice causes a slight increase of positive charge due to Coulomb attraction. As the Cooper pair flows, the leading electron causes this increase of charge, and the trailing electron is attracted by it. This is illustrated below.
This BCS theory prediction of Cooper pair interaction with the crystal lattice has been verified experimentally by the isotope effect. That is, the critical temperature of a material depends on the mass of the nucleus of the atoms. If an isotope is used (neutrons are added to make it more massive), the critical temperature decreases. This effect is most evident in Type I, and appears only weakly in Type II.
"...recall that early researchers made the somewhat paradoxical observation that the best conducting materials could not be made to exhibit superconductivity. A good conductor is, by definition, a material that will allow electrons to carry current with a minimum resistance. Therefore, since the primary cause of resistance is the electrons collisions with the lattice, a good conductor must have a minimal interaction between the electrons and the lattice. Consequently, the lattice is unable to mediate an attractive force between the electrons and the superconducting phase transition cannot occur. The converse of this observation also holds: metals exhibiting poor conductivity make excellent superconductors with relatively higher critical temperatures because the electrons greatly interact with the lattice." (Orlando 527)
This superconductivity of Cooper pairs is somewhat related to Bose-Einstein Condensation. The Cooper pairs act somewhat like bosons, which condense into their lowest energy level below the critical temperature, and lose electrical resistance.
The BCS Theory did exactly what a physical theory should do: it explained properties already witnessed in experiment, and it predicted experimentally verifiable phenomena. Though its specific quantitative elements were quite limited in their application (it only explained Type I s-wave superconductivity), its essence was quite broad and has been modified applied to various other superconductors, such as Type II perovskites.
Applications of BCS Theory
1. An induced superconducting state caused by charge transfer between intrinsically superconducting (α) and intrinsically normal (β) subsystems is studied. A most interesting case is a layered system with some layers being normal. An analysis of the general Hamiltonian describing the phenomenon allows us to evaluate Tc and the spectrum, which displays a two-gap structure. A superconducting state can be induced through different charge transfer channels (intrinsic proximity effect; inelastic two-band channel). A very important contribution comes from the ‘‘mixed’’ channel. Systems with various strengths of the coupling are described. The presence of magnetic impurities leads to an induced gapless superconductivity. This model is applied to high-Tc cuprates (in particular, to Y-Ba-Cu-O), as well as to conventional systems. The spectroscopy of Y-Ba-Cu-O appears to be very sensitive to the oxygen content whereas Tc changes relatively slowly. The model is directly related to such phenomena as residual microwave losses, zero-bias anomalies, the ‘‘plateau’’ effect, etc.
2. An induced superconducting state caused by charge transfer between intrinsically superconducting (α) and intrinsically normal (β) subsystems is studied. A most interesting case is a layered system with some layers being normal. An analysis of the general Hamiltonian describing the phenomenon allows us to evaluate Tc and the spectrum, which displays a two-gap structure. A superconducting state can be induced through different charge transfer channels (intrinsic proximity effect; inelastic two-band channel). A very important contribution comes from the ‘‘mixed’’ channel. Systems with various strengths of the coupling are described. The presence of magnetic impurities leads to an induced gapless superconductivity. This model is applied to high-Tc cuprates (in particular, to Y-Ba-Cu-O), as well as to conventional systems. The spectroscopy of Y-Ba-Cu-O appears to be very sensitive to the oxygen content whereas Tc changes relatively slowly. The model is directly related to such phenomena as residual microwave losses, zero-bias anomalies, the ‘‘plateau’’ effect, etc.
3. We propose a simple theory for the I-V curves of normal-superconducting microconstriction contacts which describes the crossover from metallic to tunnel junction behavior. The detailed calculations are performed within a generalized semiconductor model, with the use of the Bogoliubov equations to treat the transmission and reflection of particles at the N-S interface. By including a barrier of arbitrary strength at the interface, we have computed a family of I-V curves ranging from the tunnel junction to the metallic limit. Excess current, generated by Andreev reflection, is found to vary smoothly from 4Δ / 3eRN in the metallic case to zero for the tunnel junction. Charge-imbalance generation, previously calculated only for tunnel barriers, has been recalculated for an arbitrary barrier strength, and detailed insight into the conversion of normal current to supercurrent at the interface is obtained. We emphasize that the calculated differential conductance offers a particularly direct experimental test of the predictions of the model.
4. A new scheme to study exotic phase transitions is formulated by introducing the concept of a super-effective field. A general mechanism of phase transitions is elucidated and a general criterion of order parameters is proposed on the basis of the newly formulated super-effective-field theory. An alternative formulation based on a decoupled density matrix is also given. It is easily shown using these formulations that a quantum chiral order appears in the antiferromagnetic XY model on the triangular lattice. A super-effective-field theory of spin glasses is also presented. ©1988 The Physical Society of Japan
SCIENTISTS WHO WORKED FOR BCS THEORY
Dr. Bardeen's main fields of research since 1945 have been electrical conduction in semiconductors and metals, surface properties of semiconductors, theory of superconductivity, and diffusion of atoms in solids. The Nobel Prize in Physics was awarded in 1956 to John Bardeen, Walter H. Brattain, and William Shockley for "investigations on semiconductors and the discovery of the transistor effect," carried on at the Bell Telephone Laboratories. In 1957, Bardeen and two colleagues, L.N. Cooper and J.R. Schrieffer, proposed the first successful explanation of superconductivity, which has been a puzzle since its discovery in 1908. Much of his research effort since that time has been devoted to further extensions and applications of the theory.
Professor Cooper has received many forms of recognition for his work in 1972, he received the Nobel Prize in Physics (with J. Bardeen and J.R. Schrieffer) for his studies on the theory of superconductivity completed while still in his 20s. In 1968, he was awarded the Comstock Prize (with J.R. Schrieffer) of the National Academy of Sciences. The Award of Excellence, Graduate Faculties Alumni of Columbia University and Descartes Medal, Academie de Paris, Université Rene Descartes were conferred on Professor Cooper in the mid 1970s. In 1985, Professor Cooper received the John Jay Award of Columbia College. He holds seven honorary doctorates.
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He served as Director of the Institute for Theoretical Physics in Santa Barbara from 1984-89. In 1992 he was appointed University Professor at Florida State University and Chief Scientist of the National High Magnetic Field Laboratory.He holds honorary degrees from the Technische Hochschule, Munich and the Universities of Geneva, Pennsylvania, Illinois, Cincinnati, Tel-Aviv, Alabama. In 1969 he was appointed by Cornell to a six-year term as a Andrew D. White Professor-at-Large.He is a member of the American Academy of Arts and Sciences, the National Academy of Sciences of which he is a member of their council, the American Philosophical Society, the Royal Danish Academy of Sciences and Letters and the Academy of Sciences of the USSR.The main thrust of his recent work has been in the area of high-temperature superconductivity, strongly correlated electrons, and the dynamics of electrons in strong magnetic fields.
FUTURE SCOPE
The ideas from which this area has quite recently emerged can be traced back at least some 40 years. Then, and quite independently, Larkin and Ovchinnikov (LO) and Fulde and Ferrell (FF) proposed on purely theoretical grounds what amounted to a new type of superconductivity, now often referred to as the LOFF phase. This phase of inhomogeneous superconductivity can be usefully viewed as a proposed generalization of the Bardeen-Cooper-Schrieffer (BCS) state which is appropriate to describe many properties
of elemental metallic superconductors. Whereas the basic building block of the BCS theory is the Cooper pair, where the two electrons have momenta equal in magnitude and opposite in direction,
in the so-called LOFF phase a salient feature is that momenta do not add to zero. Then an almost immediate consequence of the LOFF proposal
is that the energy gap, or order parameter, has a spatial variation.The LOFF proposal, to our knowledge, has not yet been confirmed beyond reasonable doubt in condensed matter, but expectations are high that such a phase will come up in real materials in the fore see able future. But what seems remarkable, and worthy of much fuller exploration, is that the same basic ideas of LOFF may also prove to play an important role in the future in nuclear physics and in the theory of some aspects of the properties of pulsars which are commonly identified with neutron stars.
BIBLIOGRAPHY
www.manhattanrarebooks-science.com
www.hyperphysics.com
www.ffden.com
www.wikipedia.com
www.brittanica.com
www.quench_analysis.com
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