Superconductors

1.1 Introduction
Superconductivity was discovered in 1911 by a Dutch physicist named Kamerlingh Onnes. In 1908 Onnes made helium a liquid by cooling it to 452 degrees below zero Farenheit (4 degrees Kelvin), allowing him to chill materials to nearly absolute zero. In his experiments he investigated the electrical properties of metals at extremely low temperatures. In one experiment he passed a current through a mercury wire, measuring the resistance as he chilled it see Fig 1. At 4.2 degrees K the resistance vanished, Onnes wrote "The experiment left no doubt about the dissapearance of the resistance of mercury. Mercury has passed into a new state, which because of its extraordinary electrical properties may be called the superconductive state."[1]. Onnes also found that this superconducting transition is reversible. When a superconducting sample was heated it recovered its normal resistivity at a critical temperature, Tc, confirming that he was observing a new state of matter, which depends on state variables, such as temperature, rather than the history of the sample. This transition into the superconducting state can be likened to other phase transitions, such as vapor-liquid at the vaporization point, or the ferromagnetic transition at the Curie point [2]. Onnes was aware of the large commercial potential and scientific importance of his discovery, as an electrical conductor with no resistance could carry current over extreme distances with no losses, or carry current in a loop indefinitely. In one experiment, Onnes had a current flowing in a loop of lead wire cooled below its Tc and one year later the current was still flowing undiminished [1]. Even though Onnes had made an immensely important contribution to the field of physics there were still aspects of the phenomena of superconductivity yet to be understood in order to make useful applications. Further contributions include but are not limited to, governing rules developed by B. Matthias, the two-fluid model introduced by Gorter and Casimir, the Meissner Effect and London equations, and the Bardeen, Cooper, and Schrieffer (BCS) theory, which will be briefly discussed in this page.

Fig 1. (Left) Kamerlingh Onnes. (Right) Plot of resistivity vs. Temperature for a liquid helium cooled mercury wire[3][4].

1.2 Empirical Rules from B. Matthias
Superconductivity does not appear in all substances and in the substances in which it does, Tc varies greatly. Therefore it was particularly useful to have a set of criteria that would indicate if there was the likelihood of observing superconductivity and if so, the expected value of Tc. The rules are listed below:

1. Superconductivity occurs only in substances in which the valence number per atom is between 2 and 8. In general, superconducting elements lie in the inner columns of the periodic table. The phenomenon has not yet been observed in noble metals.
2. The valance numbers 3, 4.7, 6.4 (nearly odd) are particularly favorable, i.e., they result in higher critical temperatures, while the numbers 2, 4 and 5.6 (nearly even) are particularly unfavorable.
3. A small atomic volume, accompanied by a small atomic mass, favors superconductivity.

These rules prescribed by B. Matthias allowed him to discover thousands of new superconductors [2].

1.3 The Two-Fluid Model
In 1934 physicists Gorter and Casimir developed the two-fluid model of superconductivity in order to help explain the thermodynamic properties of superconductors. The model suggests that in a superconducting material, a finite fraction of the electrons are condensed into a sort of superfluid that extends over the entire volume of the system. At zero temperature there is complete condensation and all electrons participate in forming the superfluid, though it should be noted that only the electrons near the Fermi surface have their motion significantly affected by the condensation[5]. The normal electrons behave as usual, as particles flowing in a viscous medium, whereas the electrons in the superfluid, experience no scattering, have zero entropy, and long coherence length, which give superconductors their unique properties[2]. As the temperature of the system is increased from zero, a fraction of the electrons in the superfluid evaporate from the condensate and form a weakly interacting normal fluid, which extends through the volume of the system, interpenetrating the superfluid[5]. As the temperature increases to Tc the fraction of electrons contained in the superfluid goes to zero and the system experiences a second-order phase transition from the superconducting to the normal state. Gorter and Casimir developed a formula relating the number of electrons in the superfluid to temperature that would agree with their experiments[2]:

(1)
\begin{align} n_s =n \left[ 1- ({T \over T_c})^4 \right] \end{align}

Where $n_s$ is the fraction of electrons in the superfluid.
Other experiments at low temperatures have shown that there is a peaking of the specific heat just below Tc, indicating an increase in entropy as the system transitions to the normal state, thus implying that the superconducting state has a greater degree of order than the normal state. The exponential decrease of the specific heat of the electrons is given by[2]:

(2)
\begin{align} C_v=ae^{-b({T \over T_c})} \end{align}

The exponential behavior suggests the presence of an energy gap in the energy spectrum of the electrons. The gap, lying just at the Fermi level is extremley small, ~10-4eV, but prevents electrons from being readily excitable, and leads to a small specific heat[2]. Figure 2 Shows a plot of the density of states, g(E), vs. energy and the resultant gap at the Fermi level which is greatly exaggerated.

Fig. 2 Density of states for a superconductor[2]

1.4 The Meissner Effect and London Equations
In 1933, Walther Meissner and R. Ochsenfeld found that along with unique electronic properties, superconductors had surprising magnetic properties as well. They found that a superconductor will not allow a magnetic field to penetrate all the way through it, because as a magnetic field approaches a superconductor, screening currents develop on the surface[1]. The screening currents set up an equal but opposite magnetic effect which cancels the magnetic field. "A magnet placed near a superconductor will literally see its mirror image and, since like poles repel, both superconductor and magnet will try to move away from each other"[1]. The Meissner Effect only occurs if the magnetic field is relatively small, but if the magnetic field becomes too large, it can penetrate through the superconductor causing it to lose it's superconductivity. Below are two videos[6][7] demonstrating the Meissner Effect. The first video demonstrates potential applications of superconductors in transportation systems while the second video just demonstrates the basic effect, where the black cube is a high temperature superconductor that can reach its Tc by being submerged in liquid nitrogen that has a boiling point of 77 K. A magnet is then able to levitate above the superconductor once the Tc of the superconductor is reached, and subsequently sets down once the superconductor begins to warm up.

Click play to see the Meissner Effect in action!

In 1935, two brothers F. and H. London developed two equations that govern the microscopic electric and magnetic fields of superconductors. Their phenomenological theory was based on the two-fluid model. The densities of the superfluid and normal fluid are $n_s$ and $n_n$ respectively and due to local charge neutrality the densities are restricted by $n_s + n_n = n$, where $n$ is the average number of electrons per unit volume. The first of the London equations is essentially F=ma applied to a set of free particles of charge $-e$ and density $n_s$. Since the electrons in the super fluid are unaffected by the usual scattering mechanisms which produce the conductivity, $\sigma_n$ in the normal fluid, the only force acting on an electron in the superfluid is the force due to the electric field[5]. The superfluid and normal fluid current densities are given by:

(3)
\begin{align} \mbox{\boldmath$J$}_s = -en_s \mbox{\boldmath$v$}_s \end{align}
(4)
\begin{align} \mbox{\boldmath$J$}_n = -en_n \mbox{\boldmath$v$}_n \end{align}

When we combine the density of the supercurrent $\mbox{\boldmath$ J$}_s$ with the equation of motion for an electron in the superfluid:

(5)
\begin{align} m {d\mbox{\boldmath$v$}_s \over dt} = -e \mbox{\boldmath$E$} \end{align}

to yield the first London equation:

(6)
\begin{align} {d\mbox{\boldmath$J$}_s \over dt}={n_s e^2 \over m} \mbox{\boldmath$E$} \end{align}

In the steady state, the current in a superconductor is constant, so ${d\mbox{\boldmath$ J$}_s \over dt}=0$, therefore, the electric field inside a superconductor vanishes, i.e, the voltage drop across it is zero[2]. To develop the second London equation we combine the result of the constant field with the Maxwell equation to obtain:

(7)
\begin{align} {d\mbox{\boldmath$B$} \over dt}=- \nabla \times \mbox{\boldmath$E$} \end{align}

From this we find that ${d\mbox{\boldmath$ B$} \over dt} =0$ showing that the magnetic field is constant in the steady state and independent of temperature. Though this contradicts the Meissner Effect, where when$T$ is raised to $T_c$ the flux penetrates the sample[2]. To account for the invalidity of ${d\mbox{\boldmath$ B$} \over dt} =0$, we substitute $\mbox{\boldmath$ E$}$ from equation(6) into equation(7) and eliminated the time differentiations to finally obtain the second London equation:

(8)
\begin{align} \mbox{\boldmath$B$}=-{m \over n_s e^2} \nabla \times \mbox{\boldmath$J$}_s \end{align}

Combining this equation with the Maxwell equation relating $\mbox{\boldmath$ B$}$and$\mbox{\boldmath$ J$}_s$ and applying the resulting field equation to a semi-infinite specimen with its surface lying in the $yz$ plane, and the field applied in the $y$-direction we get a differential equation with the solution:

(9)
\begin{align} B_y (x)=B_y (0)e^{-x \over \lambda} \end{align}
(10)
\begin{align} \lambda=(m/\mu_0 n_s e^2)^{1/2} \end{align}

These equations imply that a magnetic field vanishes exponentially in the bulk of a superconductor[5]. The distance that is penetrated is given by $\lambda$, also known as the London penetration depth. It can now be concluded that the flux is not expelled completely from the superconductor but only from its surface[2]. The temperature dependence of $\lambda$ can be found by combining the solution to the two-fluid model, equation(1), with that for $\lambda$ to get:

(11)
\begin{align} \lambda=\lambda(0) {\left[ 1- {T^4 \over {T_c}^4} \right]} ^{-1/2} \end{align}

1.5 The BCS Theory
The BCS theory was developed by Bardeen, Cooper, and Schrieffer in 1957. This theory explains the properties of conventional superconductors well, however it does not adequately describe the properties of high temperature (ceramic) superconductors. One of the key concepts of the BCS theory is the Cooper pair. Cooper pairs are pairs of electrons that have lower energy than two individual electrons[7]. The Cooper pairs can be better understood by considering conduction electrons lying inside the Fermi sphere with two electrons inside the Fermi surface that repel each other, seeFig3[2]

Fig 3.[2]The Fermi sphere containing two electrons interacting at the Fermi surface.

To further understand the formation of Cooper pairs we can imagine and electron drifting through a crystal, causing a momentary perturbation of the ions in the lattice. After the electron passes an ion, displacing it, the ion reverts back to its original position, but does so in an oscillating manner as if held to its lattice position by springs. The oscillation creates a phonon, which interacts with a second electron causing it to lower its energy as well as emit a phonon itself, interacting with the first electron. The passing back and forth of phonons is what couples the two electrons, lowering their energy[8]. Figure 4 shows a schematic of a Cooper pair. The binding energy is strongest when the electrons have opposite moments and spins. The Cooper pairs formed are the same as the electrons of the superconducting fluid considered in the two-fluid model, and the binding energy of the pairs correspond to the energy gap shown in Fig.2[2]. Since the Cooper pairs have lower energy than unpaired electrons, the Fermi energy in the superconducting state can be considered lower than for the normal state
which suggests an energy gap. This gap straddles the Fermi energy level which can be seen on the plot of density of states for a super conductor, see Fig 5. The energy gap actually stabilizes the Cooper pairs against breaking apart. When considering all of the electrons on the Fermi surface, they form a cloud of Cooper pairs that drift cooperatively through the crystal. The superconducting state is an ordered state of conduction electrons where there is no scattering on the lattice atoms, resulting in zero resistance[8].

Fig 4.[8] Schematic of a Cooper pair. Fig 5[2] Density of states for a superconductor showing the resultant peaking of the density of states just below and above the gap.

1.6 Applications
In 1986 Bednorz and Muller found a new class of superconductors that involved copper oxide-based ceramics. These superconductors had critical temperatures that were twice as high than what was known at the time, triggering a great surge in research efforts on developing high Tc superconductors. Eventually compounds, named 1-2-3 superconductors because of their molar ratios of rare earth to alkaline earth to copper, were created. These new superconductors have critical temperatures above 77K, which is the boiling point of liquid nitrogen. Before this, the critical temperatures of superconductors only reached about 23 K, requiring liquid hydrogen or liquid helium for cooling, but now these high-Tc superconductors require liquid nitrogen cooling which is much less expensive and easier to handle. This greatly benefits technological applications of superconductors, not only because of the higher Tc but because of the higher critical magnetic field, and current density which allows their use in applications which require very large magnetic fields or the transfer of large currents[8].

Present Applications:[1]

Magnets

• Medical diagnostics research, MRI
• Ore refining
• R&D magnets
• Magnetic shielding
• Supercolliders, magnetic fusion machines

Electronics

• Defense systems
• Superconducting quantum interference devices, SQUIDs
• Electromagnetic shielding

Transportation

• High-speed trains, MAG-LEV
• Ship drive systems

Potential Applications:

Computers

• Semiconducting superconducting hybrid computers
• Active superconducting elements

Power Utility

• Energy prodution, magnetic fusion
• Large turbogenerators
• Energy storage
• Electrical power transmission

References
[1] Hunt, Daniel V., Superconductivity Sourcebook, (John Wiley & Sons, Inc, 1989), pp2-4;24
[2] Omar, Ali M., Elementary Solid State Physics, (Pearson Education, 1999), pp496-504
[3] (Online) http://www.nndb.com/people/521/000099224/
[4] (Online) http://hyperphysics.phy-astr.gsu.edu/Hbase/solids/scdis.html
[5] Schrieffer, J.R., Theory of Superconductivity, (Perseus Books, 1964), pp1-4;9-12