Superconductors take the diamagnetic effect to the extreme, since in a superconductor the field B is zero — the field is completely screened from the interior of the material. Thus the relative permeability of a superconductor is zero. An important characteristic of a superconductor is that its normal resistance is restored if a sufficiently large magnetic field is applied. The nature of this transition to the normal state depends on the shape of the superconductor and the orientation of the magnetic field, and it is also different for pure elements and for alloys.
In this subsection we describe the behaviour in the simplest situation; we shall discuss other more complex behaviour in Section 4. If an increasing magnetic field is applied parallel to a long thin cylinder of tin at a constant temperature below the critical temperature, then the cylinder will make a transition from the superconducting state to the normal state when the field reaches a well-defined strength.
Experiments indicate that the critical magnetic field strength depends on temperature, and the form of this temperature dependence is shown in Figure 11 for several elements. At very low temperatures, the critical field strength is essentially independent of temperature, but as the temperature increases, the critical field strength drops, and becomes zero at the critical temperature.
At temperatures just below the critical temperature it requires only a very small magnetic field to destroy the superconductivity. Estimate the magnetic field strength necessary to destroy superconductivity in a sample of lead at 4. From Equation 1,. It is interesting to compare the magnetic behaviour of a superconducting element with typical curves for diamagnetic, paramagnetic and ferromagnetic materials.
The magnetic behaviour of magnetic materials can be represented by B versus H graphs. Figure 12a shows the behaviour of typical diamagnetic and paramagnetic materials. The linear graphs in Figure 12a are similar to those for a superconductor above the critical field strength. The current density for a steady current flowing along a wire in its normal state is essentially uniform over its cross-section.
Within a superconductor, however, the magnetic field B is zero. The current must therefore flow in the surface of the wire. How does the magnetic field just outside the surface of a superconducting wire, radius a , carrying current I , compare with the field just outside the surface of a normal wire with the same radius, carrying the same current? The fields just outside the surface are identical.
This critical current is proportional to the radius of the wire. In the previous subsection you saw that the critical field strength is dependent on temperature, decreasing to zero as the temperature is increased to the critical temperature. This means that the superconducting current that a wire can carry will also decrease as the temperature gets closer to the critical temperature. Now, the current carried by a superconducting wire actually flows in a thin layer at the surface; it cannot be restricted to an infinitesimal layer, because that would lead to an infinite current density.
As you will see in Section 3, this means that the magnetic field penetrates into this thin layer, and we derive there relationships between the field and the current density. But in the present context, the point to note is that the transition to the normal state takes place when the magnetic field strength at the surface corresponds to the critical field strength, and this occurs when the current density at the surface reaches a critical current density. The magnetic field at the surface of a superconductor may have a contribution from an external source of magnetic field, as well as from the field produced by the current in the wire.
This external field will set up screening currents in the surface layer of the material. The transition to the normal state then occurs when the vector sum of the current densities at the surface due to the current in the wire and due to the screening current exceeds the critical current density, or, equivalently, when the magnitude of the vector sum of the magnetic fields that are present at the surface of the wire exceeds the critical field strength. From Equation 1, we also have that.
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As was mentioned earlier, a substantial dose of quantum mechanics would be required to provide a full explanation of the properties of superconductors. This would take us too far away from electromagnetism, and we shall therefore restrict our discussion to aspects that can be discussed using classical concepts of electromagnetism. We shall model the free electrons within a superconductor as two fluids.
They are accelerated by an electric field E , but are frequently scattered by impurities and defects in the ion lattice and by thermal vibrations of the lattice. The current density J n due to flow of these electrons is. Interspersed with the normal electrons are what we shall call the superconducting electrons, or superelectrons, which form a fluid with number density n s.
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The superconducting electrons are not scattered by impurities, defects or thermal vibrations, so they are freely accelerated by an electric field. If the velocity of a superconducting electron is v s , then its equation of motion is. Compare this with Equation 2, which relates current density and electric field in a normal conductor. Scattering of the normal electrons leads to a constant current in a constant electric field, whereas the absence of scattering of the electrons in a superconductor means that the current density would increase steadily in a constant electric field.
Therefore the normal current density must be zero — all of the steady current in a superconductor is carried by the superconducting electrons. Of course, with no electric field within the superconductor, there will be no potential difference across it, and so it has zero resistance. When discussing the Meissner effect in Subsection 2.
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We shall now show how that conclusion follows from an application of Maxwell's equations to a perfect conductor. We can then see what additional assumptions are needed to account for the Meissner effect in a superconductor. We assume that the electrons in a perfect conductor or a proportion of them are not scattered, and therefore the current density is governed by Equation 3.
We are interested in the magnetic field in a perfect conductor, so we shall apply Maxwell's equations to this situation. Faraday's law is valid in all situations,.
We can use a standard vector identity from inside the back cover to rewrite the left-hand side of this equation:. The uniform external field in the x -direction means that the field inside the conductor will also be in the x -direction, and its strength will depend only on z. So Equation 7 reduces to the one-dimensional form. This indicates that any changes in the external magnetic field are attenuated exponentially with distance below the surface of the perfect conductor. A simple but useful description of the electrodynamics of superconductivity was put forward by the brothers Fritz and Heinz London in , shortly after the discovery that magnetic fields are expelled from superconductors.
Their proposed equations are consistent with the Meissner effect and can be used with Maxwell's equations to predict how the magnetic field and surface current vary with distance from the surface of a superconductor.
What is Superconductivity?
In order to account for the Meissner effect, the London brothers proposed that in a superconductor, Equation 4 is replaced by the more restrictive relationship. This equation, and Equation 3 which relates the rate of change of current to the electric field, are now known as the London equations. It is important to note that these equations are not an explanation of superconductivity. They were introduced as a restriction on Maxwell's equations so that the behaviour of superconductors deduced from the equations was consistent with experimental observations, and in particular with the Meissner effect.
Their status is somewhat similar to Ohm's law, which is a useful description of the behaviour of many normal metals, but which does not provide any explanation for the conduction process at the microscopic level. To demonstrate how the London equations lead to the Meissner effect, we proceed in the same way as for the perfect conductor. Thus, a uniform magnetic field like that shown in Figure 10b cannot exist in a superconductor. If we consider again the simple one-dimensional geometry shown in Figure 14, then we obtain the solution to Equation 11 by simply replacing the partial time derivatives of the fields in the solution for the perfect conductor Equation 8 by the fields themselves, that is,.
Therefore, the London equations lead to the prediction of an exponential decay of the magnetic field within the superconductor, as shown in Figure Many experiments have been done with samples that have a large surface to volume ratio to make the penetration effect of the field appreciable. Thin films, thin wires and colloidal particles of superconductors have all been used for this purpose. But it is also possible to use large specimens if the measurement is sensitive to the amount of magnetic flux passing through the superconductor's surface, and not to the ratio of flux excluded by the superconductor to flux through the normal material, which is close to unity.
In a classic experiment performed in the s, Schawlow and Devlin measured the self-inductance of a solenoid within which they inserted a long single-crystal cylinder of superconducting tin, 7. The inductance of the solenoid was therefore determined mainly by the magnitude of the penetration depth. The number density of superconducting electrons depends on temperature, so the penetration depth is temperature dependent.
Figure 16 shows this temperature dependence for tin, which is well represented by the expression. The London equations relate the magnetic field in a superconductor to the superconducting current density, and we derived the dependence of field on position by eliminating the current density. However, if we eliminate the magnetic field instead, we can derive the following equation for the current density:. Assume that the currents are steady. Equation 13 has exactly the same form as Equation The number density of free electrons in tin is 1. Calculate the penetration depth predicted by the London model, assuming that all of the free electrons are superconducting, and compare the result with the value measured by Schawlow and Devlin.
The numerical discrepancy between the London model prediction for the penetration depth of tin and the experimentally measured value indicates that this model has limitations. One limitation is that the model is essentially a local model, relating current density and magnetic field at each point. Superconductivity, though, is a non-local phenomenon, involving coherent behaviour of the superconducting electrons that are condensed into a macroscopic quantum state. This distance represents the distance over which the number density of the superconducting electrons changes, and is a measure of the intrinsic non-local nature of the superconducting state.
Since the penetration depth increases sharply as the temperature approaches the critical temperature Figure 16 , the London model becomes a good approximation in this limit. More importantly, the coherence length of superconductors decreases as the critical temperature increases and as the scattering time for normal electrons decreases.
Both of these effects mean that the coherence length is short compared with the penetration depth in alloy and ceramic superconductors, so the London local model is a good approximation in these cases too, and predicted and experimental results for the penetration depth are in good agreement. For pure elements, well below their critical temperatures, the penetration depth is generally much shorter than the coherence length, so a local model is not appropriate.
The ratio of the penetration depth to the coherence length is an important parameter for a superconductor, and we shall return to this subject in Subsection 4. The two main types of superconducting materials are known as type-I and type-II superconductors , and their properties will be discussed in the remainder of this course. All of the pure elemental superconductors are type-I, with the exception of niobium, vanadium and technetium. The discussion of the effects of magnetic fields and currents on superconductors earlier in this course has been confined to thin cylinders of type-I materials like lead or tin in a parallel magnetic field.
In Subsection 4.