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12.a. Magnetism in Solids
12.b. Type I Superconductors
12.c. Type II superconductors
12.d. Josephson junction
12.e. High Tc superconductors
The origin of magnetism in matter is the spin and orbital angular momentum of atomic electrons.When an atom has an even number of electrons then the electrons spins and orbital angular momenta can pair up to result in no net magnetic moment. Atoms with an odd number of electrons have by necessity nonzero magnetic moments.
The best conducting metals do not become superconductors at any temperature. Of the ones that do have a critical magnetic field that depends on temperature as
Bc(T) = Bc(0) { 1 - (T/Tc)2 }
In other words, at B> Bc(0) there is no superconductivity at any temperature. Now Bc(0) =0.01-0.1 tesla. Type I superconductors are not the best for constructing superconducting magnets, because we cannot really reach high magnetic fields without destroying supeconductivity.
Superconductors must "arrest" magnetic fields accross them because Faraday's law states that the change of flux is equal to the line integral of the electric field in a closed loop, which is zero. Farady's law allows a constant magnetic field existing inside superconductors, but type I superconductors expel the magnetic field completely at the time of the transition. This is called the Meissner effect. What physically happens is that if at the time of going superconducting the sample is in magnetic field then a surface current is generated that develops its own magnetic field, cancelling exactly the external field inside the superconductor. This surface current changes with the external magnetic field.
The existence of the critical field imlies the existence of a critical current across the superconductor. The currrent induces a magnetic field, which if exceeds the critical value, destroys superconductivity.
The magnetic field does not vanish sharply at the surface of the superconductor. There is a certain penetration depth, inside which the surface current compensating the external field flows. The magnetic field drops exponentially as more and more of it screened out by the surface current. The exponential drop is characterized by the penetration depth, l. The penetration depth is also a function of temperature. The closer one gets to the critical temperature the deeper the penetration is
l(T) = l(0) / [1 - (T/Tc)2 ]1/2
The magnetic properties inside superconductors can be understood if we consider the relation inside matter:
B = B0 + m0M,
where M is the magnetization and m0 is the permeability of free space. Alternatively, we can define H (magnetic field strength) as
B= m0 (H + M),
i.e. B0 = m0 H
We also define the susceptibility, c, as M= c H. Then for superconductors B=0 so H= -M, or in other words, c=-1. A superconductor is a perfect diamagnet.
The Messner effect allowed a classical explanation for superconductivity. Suppose there are two states of the conductor, superconducting and normal. Since below the critical temperature the system is superconducting, that states must be at a lower energy. If the superconductor is in magneticc field then extra energy must be spent, B2 / 2 m0 per unit volume. Thus unless the energy difference is not larger then this
| Es - En | > B2 / 2 m0
no superconducting state can exist. In other words, the critical current is
En - Es = Bc2 / 2 m0
Certain alloys, when the magnetic field exceeds the critical value do not become normal conductors. The magnetic field penetrates the superconductor, concentrated in flux filaments named vortices. Inside vortices (having similar penetration length) there is a normal conductor. Further increasing the magnetic field increases the number of fluxlines and at a second critical field, Bc2 (also function of T) the superconducting state collapses. The key is that the critical field is usually very high 10s of Teslas, allowing to build powerful supercnducting magnets. These magnets do not require energy to maintain the magnetic field.
In all superconductors there are persistent currents: could last as long as 105years. Unlike in ordinary conductors: resistance is zero. Another important property is the existence of a coherence length. This is a quantum mechanical phenomenon,the range of interaction creating the superconducting state (details later). In type I superconductors thhe coherence length is large x > l. Impurities (like in alloys) cut down on the coherence length and x < l. that is required for type II superconductivity.
If the superconductor is not simply connected (topology of a torus) than magnetic field can and will be maintained inside the ring even when an external field is turned off. The flux of such a magnetic field (and the flux of vortices) is quantized. This essentially follows by the single valuedness of the wavefunction when one goes around inside the superconductor around an area of flux. The rule one obtains is
F = n h / q.
where n is an integer and q is the charge of the carriers of supercurrent. Experimental data shows that q=2e and not e !!! Magnetic flux quantum is h / q.
Specific heat: In normal metals or other solids the specific heat vanishes as a power of temperature near T=0. Think of the Debye formula, which results from the integration over all frequency modes of lattice oscillations. In contrast to the Debye formula the Einstein formula deals with a single frequency of oscillations and then the speciifc heat vanishes near T=0 as
C ~ e-hn/kT.
We say that such a system has an energy gap: the spectrum does not start at n=0 but at a higher value. Superconductors clearly show such a dependence of the specific heat on temperature.
C ~ e-D/kT
where D is the energy gap. The physical meaning is the same as of the gap in semiconductors: The system (charge carriers) are in a state such that they cannot absorb an arbitrary small energy. One needs to provide at least D to excite it.
BCS theory:
Clearly the classical theory cannot explain superconductivity: collisions always create resistance. The quantum mechanical theory cannot understand either why no scattering on impurities or phonons takes place. Amazingly, it turned out that lattice vibrations, phonons, that normally cause resistance also make superconductivity possible. The role the lattice plays in superconductivity became clear when the isotope effect was discovered. This is the dependence of critical temperature on the isotope of the superconducting material. Such a dependence is possible only if lattice vibrations are relevant for superconductivity.
It turns out that due to interactions with the lattice (phonons) electrons form S=0 pairs in s-bound state. The binding energy of these pairs is very small (10-3eV) but nonzero. So if one tries to excite these pairs, one needs a certain energy to break them up. The electron gas is substituted by a gas of Cooper pairs, which are, in a sense, similar to a He atom. Cooper pairs are bosons, so they all can be in the same quentum state. At low temperature every gas of bosons condenses: a macroscopic fraction of them occupies the quantum mechanical ground state. The superconducting state is a superfluid state of the Cooper pairs. The energy gap is nothing else but the energy needed to break up Cooper pairs. The role of the magnetic field in superconductivity can be understood in the following way: If the magnetic field is strong it will be energetically favorable to break up the Cooper pairs and align electron spins parallel to each other. This will of course destroy th esuperconducting state.
One important quantum mechanical property of condensed Cooper-pairs: They have a single wavefunction all across the superconducting material, Y. The phase of this wavefunction will become very important in the dicussion of the Josephson effect.
Tunnelling junction: A junction of a superconductor and a normal metal with thin insulator. Single electrons can tunnel through only, because cooper pairs do not exist on the normal side. Cooper pairs must be broken up before they tunnel through so a certain minimal energy (Voltage) is needed so that the tunneling current would start. The voltage required is Vt = Eg/2e. Eg is the gap energy needed to break a pair.
Electromagnetic Radiation Superconductors can absorb electromagnetic radiation of frequency higher than n > Eg/ h. The reason is similar to the non-conduction of semiconductors (a very small) energy is needed to raise electrons from the lower superconducting state to the normal state.
Brian Josephson proposed to form junctions from two superconductors. His idea was that Cooper pairs will also be tunnel through narrow junctions, not only single electrons. The tuneling rate is dependent, of course, on the area of the junction and the size of the junction (decreases exponentially with distance, as in any other tunneling case). At zero phase difference and no applied field there is no reason for tunneling. Quantum mechanically the rate can only be a function of the phase difference. This can be calculated from straightforward quantum mechanical considerations and it is ( Quantum mechanical currents are proportional to (i/2m)[y* d y /dx - y d y* /dx ], i.e vanish for vanishing phase, real wavefunction)
Is = I0 sin ( f1 - f2 ) = I0 sin ( d ).
I.e. a persistent current is detected accross the junction even if no external voltage is applied.
Even more interesting phenomena happen when an external dc voltage is applied. The junction generates an ac current.
Is = I0 sin ( d + 2 p f t),
where f is the quantum mechanical (deBroglie) frequency: f = 2e.V/h, corresponds to the exact Coulomb energy of a Cooper at potential V.
Observation of ac Josephson effect:
1) Apply dc voltage and observe the radiation of frequency f.
2) Apply external radiation of frquency f'. If at the same time the gap is changed, by changing the voltage at certain values of V the external frequency is exactly the right value to raise a Cooper pair to the other side of the barrier. This happens when nhf'= hf= 2e V. At these values of the voltage the current jumps. Measuring the voltage, frquency exactly allows one to calculate e/h to extremely high precision.
Quantum interference devices: (SQUIDS). If the current is split trough two junctions then the maximal current will be a function of magnetic flux quanta enclosed. The maximal current is a periodic function of the number of flux quanta across trhe junction. Very small magnetic fields can be measured. (10-14T)