We will then calculate and study interesting quantities of the superconducting system, and nally describe how the results predicted by the bcs theory fare against experimental evidence obtained about superconductors. From comparisons with other methods of determining the band gap, it is found that the constant b in the exponential heat capacity expression is onehalf the band gap energy. Mathematical analysis of the bcsbogoliubov theory 1. The theory yields a secondorder phase transition and a meissner effect in the form suggested by pippard. The stabilization energy thus obtained is called the gap energy k. The main point of the bcs theory is that the attractive electronelectron interaction mediated by. Students, researchers and academics interested in bcs theory and its origin.
The energy gap is most directly observed in tunneling experiments and in reflection of microwaves from superconductors. A central feature of the bcs theory is the presence of an energy gap in the electron densityofstates of superconductors. However, the temperature green function matsubara function is used from. The superconducting energy gap is a key aspect in the theoretical description of superconductivity and thus features prominently in bcs theory. For weak coupling superconductors, which aredescribedby the famous bcs theory 1957,thereis a relation between the gap energyand the superconducting transition temperature, 2. This promote the electron pair into the boson state. The ratio between the value of the energy gap at zero temperature and the value of the superconducting transition temperature expressed in energy units takes the universal. This pair condensation is the basis for the bcs theory of superconductivity. Bcs theory cornell laboratory of atomic and solid state physics. Abstract the bardeencooperschrieffer bcs theory is employed to study the energy gap in nuclear matter with various internucleonic potentials which fit singlet low energy scattering data and the swave phase shift at 310 mev. Originally this idea sounds like crazy, because electrons the same charge so they have repulsive interactions. This statement will be enhanced by another proof with all terms of interaction taken into account in the secular perturbation.
Bcs theory derivation of the gap equation we start with a general hamiltonian that describes manyfermion system interacting via a spinindependent interaction potential vx, h x. Pdf we consider a hypothetical substance, where interaction. Concepts of fermi liquid theory and instability of the normal state. At ttc, pairing is dissolved and the energy gap reduces to zero. Bardeen tried to explain the origin of the gap using the electronlattice interaction as suggested by the isotope effect, but did not have a complete theory. Phonon mediated attractive interaction and the cooper pairs. Bardeen, cooper, and schrieffer received the nobel. There is an energy gap for individualparticle excitations which decreases from about 3. The main point of the bcs theory is that the attractive electronelectron interaction mediated by the phonons gives rise to cooper pairs, i. The speci c heat of a superconductor c s and and normal metal c n.
Bcs theory predicts the dependence of the value of the energy gap. First working microscopic theory for superconductors. Deviations from the bcs theory were consistent with the physical arguments used by schrieffer and wada to explain the large ratio of energy gap to transition temperature for strong coupling superconductors and with the detailed numerical calculations of swihart, scalapino and wada. Even the ground state in the original bcs theory might be fictitious. Theory predicts at least 2 dominant superconducting gap values at low temperature and the twovalues at low temperature, and the two gap feature is robustgap feature is robust against pressure, doping and impurity scattering. If the slope of the line in the illustration is determined by. Hence it is doubtful whether there really exists energy gap in breaking a pair. So if you give some gap energy to superconductor, it generates excited phonons, which can be detected, they insist. If one tries to probe the system with a stm, for example, you will get no tunneling below this gap.
Theory of superconductivity carsten timm wintersemester 20112012 tu dresden institute of theoretical physics. The bcs theory has turned out to be of enormous predictive power and many of its predictions and implications like the temperature dependence of the en. A key feature of the bcs theory is the presence of an energy gap. Bcs theory thus predicts many electron ground states, as well as excited for the superconductor in the range 0 to tc and in these states, coopers pairs are supposed to be in the condensed state with a definite phase coherence. The green function has been effectively used by employing the nambu spinor.
Roughly 10 6 or 10 7 can all be in the same state, importantly the cooper pairs provide the energy gap required to be brokenformed in order to transition in and out of the superconducting state. Superconductivity and bcs theory introduction electronphonon interaction, cooper pairs bcs wave function, energy gap and quasiparticle states predictions of the bcs theory limits of the bcs gap equation. The energy gap seen between the superconduct ing state top and. For this to occur the potential v needs to have at least some. Calculated values of specihc heats and penetration depths and their temperature variation are in good agreement with experiment. The effective net attraction between the normally repulsive electrons produces a pair binding energy on the order of millielectron volts, enough to keep them paired at extremely low temperatures. Theory of superconductivity university of illinois. For temperatures such that the thermal energy is less than the band gap, the material exhibits zero resistivity. The ratio between the value of the energy gap at zero temperature and the value of the superconducting.
Problems encountered during the creation of the bcs theory. Bcs theory, critical temperature, energy gap and phonons. For superconductors the energy gap is a region of suppressed density of states around the fermi energy, with the size of the energy gap much smaller than the energy scale of the band structure. The most important and famous relation, which bcs theory predicts is the relation between energy gap. Part i preliminaries the bcs theory is based on two important insights. The bcs theory of superconductivity 15, which has been used to interpret much of this experimental work 2,4,9, predicts in the limit of weak coupling that the reduced energy gap 2. Cooper problem ii 3 orbital part of the wave function spin singlet.
Nanophysics in superconductivity bcs theory flux vortices josephson junction coherence length knut tyse. Pdf bcs theory with the external pair potential researchgate. Cooper pairs and the bcs theory of superconductivity. Bcs theory of superconductivity hyperphysics concepts. Temperature dependence of the superconducting energy gap. But bcs theory relies on various rough approximations, as i explain later. Well, i would say that the bcs theory is valid for weak. Lecture 4 bcs ground state, variational calculation. Eremin, maxplanck institut fur physik komplexer systeme, dresden, germany.
Bcs from nuclei and neutron stars to quark matter and cold atoms g baym energy gap, mass gap, and spontaneous symmetry breaking y nambu bcs as foundation and inspiration. Bcs relation between the gap at zero temperature and the critical temperature, t c. Here is a simplified explanation of the bcs theory to help you, sing a clear explanation with animations, i help you understand the theory from a. The bcs hamiltonian in electronic systems the low temperature properties are determined by the longlived quasiparticles in an energy shell. Lecture notes on superconductivity a work in progress. A most direct evidence of the gap and of the electron structure of the superconductors is provided by the electron tunneling experiments, initiated by i. The superconducting state an electronic condensate. The temperature dependence of the gaps was not consistent with oneband bcs theory but was well described by a twoband hence, two gap bcs model with interband interactions. The energy gap is related to the coherence length for the superconductor, one of the two characteristic lengths associated with superconductivity. It can be verified that the energy is lower when this is nonzero for only one. The bcs theory predicts a bandgap of where t c is the critical temperature for the superconductor. So having laid the foundation of bcs theory, that is how, cooper established.
Below the transition, the superconductor speci c heat shows activated behavior, as if there is. Across the energy gap, there are many excited states for the superconducting cooper pairs. The quantumfield approach to superconductivity theory. The energy gap as a function of the temperature, evaluated from these.
This energy gap evidence was part of the experimental motivation for the bcs theory of superconductivity. Temperature dependence of the superconducting energy gaps. For comparison, the temperature dependence of the weak coupling bcs gap, scaled to 2d 4. Thesecooperpairsformthenacoherentmacroscopicgroundstate,whichdisplaysagapped spectrumandperfectdiamagnetism. Evidence of a band gap at the fermi level described as a key piece in the puzzle. The transmutation of symmetry f wilczek from bcs to the lhc s weinberg readership. Bardeen says that a gap in the density of states at the fermi energy, will probably explain the meissner effect and thus lead to an explanation of superconductivity. These fermion pairs are known as cooper pairs, and this theory of superconductivity is known as the bcs theory for bardeen, cooper and schrieffer. Notes on the ginzburglandau theory icmr summer school on novel superconductors university of california, santa barbara august 2 august 15, 2009. This is the key advance of bcs to construct a new quantum wavefunction for all the electrons fundamental change only for electrons within a energy range. The temperature dependence of the superconducting energy gap of pb was determined using pbpb tunnel junctions. The keystone of the core bcs results is the gap equation sec. The superconducting energy gap of rb 3 c 60 nature.
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