theory of superconductivity


BCS Theory of Superconductivity
The properties of Type I superconductors were modeled successfully by the efforts of John Bardeen, Leon Cooper, and Robert Schrieffer in what is commonly called the BCS theory. A key conceptual element in this theory is the pairing of electrons close to the Fermi level into Cooper pairs through interaction with the crystal lattice. This pairing results from a slight attraction between the electrons related to lattice vibrations; the coupling to the lattice is called a phonon interaction.

Pairs of electrons can behave very differently from single electrons which are fermions and must obey the Pauli exclusion principle. The pairs of electrons act more like bosons which can condense into the same energy level. The electron pairs have a slightly lower energy and leave an energy gap above them on the order of .001 eV which inhibits the kind of collision interactions which lead to ordinary resistivity. For temperatures such that the thermal energy is less than the band gap, the material exhibits zero resistivity.
Bardeen, Cooper, and Schrieffer received the Nobel Prize in 1972 for the development of the theory of superconductivity.


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.

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. 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.
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 is 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.


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.
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.
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.
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.
Experimental Support: BCS Theory
Electrons acting as pairs via lattice interaction? How did they come up with that idea for the BCS theory of superconductivity? The evidence for a small band gap at the Fermi level was a key piece in the puzzle. That evidence comes from the existence of a critical temperature, the existence of a critical magnetic field, and the exponential nature of the heat capacity variation in the Type I superconductors.

The evidence for interaction with the crystal lattice came first from the isotope effect on the critical temperature.

The band gap suggested a phase transition in which there was a kind of condensation, like a Bose-Einstein condensation, but electrons alone cannot condense into the same energy level (Pauli exclusion principle). Yet a drastic change in conductivity demanded a drastic change in electron behavior. Perhaps coupled pairs of electrons with antiparallel spins could act like bosons?

Measured Superconductor Bandgap
The measured bandgap in Type I superconductors is one of the pieces of experimental evidence which supports the BCS theory. The BCS theory predicts a bandgap of


where Tc is the critical temperature for the superconductor. The energy gap is related to the coherence length for the superconductor, one of the two characteristic lengths associated with superconductivity.

Energy Gap in Superconductors as a Function of Temperature
The effective energy gap in superconductors can be measured in microwave absorption experiments. The data at left offer general confirmation of the BCS theory of superconductivity. The data is attributed to Townsend and Sutton.

The reduction of the energy gap as you approach the critical temperature can be taken as an indication that the charge carriers have some sort of collective nature. That is, the charge carriers must consist of at least two things which are bound together, and the binding energy is weakening as you approach the critical temperature. Above the critical temperature, such collections do not exist, and normal resistivity prevails. This kind of evidence, along with the isotope effect which showed that the crystal lattice was involved, helped to suggest the picture of paired electrons bound together by phonon interactions with the lattice.
Vanadium Heat Capacity

The heat capacity of superconducting vanadium is very different from that of vanadium which is kept in the normal state by imposing a magnetic field on the sample. The exponential increase in heat capacity near the critical temperature suggests an energy bandgap for the superconducting material.

This evidence for a bandgap is one of the pieces of experimental evidence which supports the BCS theory of superconductivity.

Exponential Heat Capacity
As it is warmed toward its critical temperature, the heat capacity of vanadium increases 100-fold in just 4 K. This exponential increase suggests an energy gap which must be bridged by thermal energy. This energy gap evidence was part of the experimental motivation for the BCS theory of superconductivity.

From comparisons with other methods of determining the band gap, it is found that the constant "b" in the exponential heat capacity expression is one-half the band gap energy. If the slope of the line in the illustration is determined by scaling, it is about b= 7.4k, corresponding to an energy gap of about 1.3 meV. This is slightly lower than the value obtained by other methods. The value predicted for vanadium from its critical temperature of 5.38 K by the BCS theory is 1.6 meV, and the measured value is close to that


BCS Theory for Conventional Superconductors

In 1957 the underlying microscopic theory of superconductivity in metals was unveiled by J. Bardeen, L.N. Cooper and J.R. Schrieffer [24], in the now famous BCS theory. In normal metals, the situation is well described by free electron theory, where the electrons behave as free particles and the metallic ions play a limited role in conductivity. BCS theory outlines how in the presence of an attractive interaction between electrons (Cooper pairs), the normal state of an otherwise free electron gas becomes unstable to the formation of a coherent many-body ground state. The mechanism behind the weak attractive force binding the Cooper pairs was actually first suggested by Herbert Frölich [25]. He proposed that the same mechanism responsible for much of the electrical resistivity in metals (i.e. the interaction of conduction electrons with lattice vibrations) leads to a state of superconductivity. This hypothesis of an electron-phonon interaction was born out of experiments which found that the critical temperature Tcvaried with isotopic mass. In simple terms, an electron interacts with the lattice by virtue of the Coulomb attraction it feels for the metallic ions. The result is a deformation of the lattice (i.e. a phonon). A second electron in the vicinity of the deformed lattice correspondingly lowers its energy, resulting in an electron-electron attraction via a phonon. Viewed in this context, the superconducting order parameter

$\Psi (\vec{r})$ from GL-theory can be interpreted as a one-particle wave function describing the position of the center of mass of a Cooper pair [26].

Despite being an extremely weak attraction, bound pairs form in part because of the presence of a Fermi sea of additional electrons. As a result of the Pauli exclusion principle, electrons that would prefer to be in a state of lower kinetic energy cannot populate these states because they are already occupied by other electrons. Thus a Fermi sea is required to ensure the formation of bound pairs of electrons; otherwise an isolated pair of electrons would just repel one another as a result of the Coulomb force between them. The Fermi sea itself is comprised of other distinct bound pairs of electrons. It follows that each electron is a member of both a Cooper pair and of the Fermi sea which is necessary for the formation of all Cooper pairs. The force of attraction between the electrons which comprise a Cooper pair has a range equivalent to the coherence length. It should be noted that the separation between electrons in a Cooper pair (and thus the correlation length $\xi$), for a type-I superconductor, is large enough that millions of other pairs have their centers of mass positioned between them. It is then assumed that the occupancy of a bound pair is instantaneous and uncorrelated with the occupancy of other bound pairs at an instant in time [27]. Armed with knowledge of the fundamental particles responsible for superconductivity (i.e. Cooper pairs), the substitutions q=2e and

$\vert \Psi_{\circ} \vert^{2} = n_{s}/2$ immediately transform the Ginzburg-Landau result for the penetration depth

$\lambda (T)$ [i.e. Eq. (2.6)] into the result predicted by London theory [i.e. Eq. (2.1)].

One of the most remarkable features emerging from BCS theory, is the existence of an energy gap

$\Delta (T)$ between the BCS ground state and the first excited state. It is the minimum energy required to create a single-electron (hole) excitation from the superconducting ground state. Thus the binding energy of a Cooper pair is two times the energy gap

$\Delta (T)$. BCS theory estimates the zero-temperature energy gap $\Delta$(0) as [26]:


 \begin{displaymath}\Delta (0) = 1.76k_{B}T_{c}<br />
            \end{displaymath} (17)



and near the critical temperature Tc,


 \begin{displaymath}\frac{\Delta (T)}{\Delta (0)} = 1.74\left( 1-\frac{T}{T_{c}}\right)^{1/2},<br />
            \ \ \ T \approx T_{c}<br />
            \end{displaymath} (18)



so that the energy gap approaches zero continuously as

$T \rightarrow T_{c}$. Superconductors which obey Eq. (2.17) are considered to be weakly-coupled, in reference to the weak interaction energy between electrons in a Cooper pair. Furthermore, the wave functions corresponding to electron pairs (Cooper pairs), are spatially symmetric like an atomic s-orbital with angular momentum L=0. That is to say, the wave function of a pair is unchanged if the positions of the electrons are exchanged. This immediately implies that the spin part of the wave function is antisymmetric in accordance with the Pauli exclusion principle. In particular, the electron pairs are in a spin-singlet state S=0with antiparallel spins. The pairing mechanism in a conventional superconductor is thus appropriately called, s-wave spin-singlet. The energy gap of an s-wave superconductor is finite over the entire Fermi surface. Under ideal circumstances, the magnitude of the gap is the same at all points on the Fermi surface. BCS theory assumes the Fermi surface is spherical (see Fig. 2.3). More realistically however, the energy gap reflects the symmetry of the crystal under consideration [28]. For a conventional s-wave superconductor with a weak-coupling ratio

$\Delta (0)/k_{B} T_{c} =1.76$, BCS theory predicts:


 \begin{displaymath}\frac{\lambda (T)}{\lambda (0)}-1 \approx 3.33 \left( \frac{T}{T_{c}}<br />
            \right)^{1/2} e^{-1.76T_{c}/T}<br />
            \end{displaymath} (19)



for small T (i.e.

T < 0.5Tc) [17,29]. At low temperatures, the energy gap is virtually independent of temperature and much larger than the thermal energy kBT. The probability of exciting a single electron with energy Ek is then proportional to the Boltzmann factor

e-Ek/kBT. The maximum value of this probability is proportional to

$e^{- \Delta (0)/ k_{B} T} = e^{-1.76 T_{c} /T}$, which is the exponential factor appearing in Eq. (2.19). Therefore in conventional superconductors,

$\lambda (T)$ shows an exponential decrease at low temperatures.

If the value of the energy gap is not constant over the entire Fermi surface, then the minimum value of the gap determines the density of quasiparticle excitations at these low temperatures. Hence the topology of the energy gap is crucial in deciding the low-temperature behaviour of

$\lambda (T)$. If the sample under investigation is riddled with impurities, then there will exist a broad range of transition temperatures

$\Delta T_{c}$ [30]. Equations (2.17) and (2.18) suggest that one should anticipate a corresponding distribution of gap energies in such materials.

 \begin{figure}% latex2html id marker 897<br />
\begin{center}\mbox{<br />
\epsfig{file=bcsga . . .<br />
. . . surface. The dotted lines outline the Fermi surfaces.<br />

Some systems (e.g. lead, mercury) produce experimental results which deviate substantially from the BCS results [26]. These materials are more appropriately described by strong-coupling theory where the coupling ratio

$\Delta (0)/k_{B}T_{c}$ is greater than the BCS prediction of 1.76. Under certain conditions superconductivity can occur without an energy gap in some materials. Tunneling experiments on superconductors with specific concentrations of paramagnetic impurities show this to be possible [31]. Theories exist which explain such anomalies, and the nature of the gap as we will soon see is a vital property to be considered in any theory describing superconductivity in the high-Tc compounds.

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Tue, 06/02/2009 - 11:00