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I was recently caught up in a situation where I tried to explain to someone with only vary basic knowledge of physics (notion of atoms and electrons, etc.) what causes superconductivity. One thing I was unable to explain though was why superconductors work only at low temperatures.

Superconductivity is caused by electrons interacting with the phonons in such a way that the effective interaction between electrons becomes attractive (for electrons near the Fermi surface). The electrons with attractive interactions between them then form bosonic pairs and condense to the ground state which leaves a gap in the energy spectrum of the remaining electrons. The temperatures at which this no longer is possible is found by searching for a solution to the gap equation with gap parameter set to zero.

Without going through the mathematics and the gap equation, how would you explain to a layman or an undergrad or maybe even a physicist working on a different field why superconductivity ceases to exist at sufficiently high temperatures?

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I am very much confused about the answers presented so far and I believe they miss the main point. At any finite temperature it is not the normal (internal) energy E which is important but the free energy F = E - T S

At zero temperature the system is in a superconducting state because the pairing up the energy lowers the total energy of the system by the condensation energy (which is essentially the binding energy of the Cooper pairs). However the superconducting state is an ordered state: all the Cooper pairs are in a single macroscopic quantum state which can be described with a few degrees of freedom. Because of this the superconducting state has a much lower entropy than the corresponding normal state.

When going to any finite temperature it is not the state with the lowest energy which is realized but the one with the lowest free energy. Because of the difference in entropy it is clear that at some point the normal state will be favored.

Two points (which will maybe resolve some confusion):

1) for the superconducting state to exist and to be stable, it is not important that the single particle excitations are gapped. This is a red herring. In fact the single particle excitations in the high temperature superconductors (d-wave) are not gapped. The importance of gap stems from the fact that it is the order parameter which is a measure of the condensation energy.

2) the transition between a superconducting and a normal state driven by temperature is just the same as the transition between an (ordered) ferromagnet at low temperature (with low entropy) and the paramagnetic state at high temperature (with high entropy). In both cases, it is not the internal energy which drives the transition as at any temperature the ferromagnetic state enjoys the lower energy

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The simplest explanation is to consider a tray with a bunch of small depressions in it and a bunch of marbles on it. When the tray is stationary, the marbles fall in to the depressions and stay there. When you shake the tray very slowly, the marbles stay in their depressions. But if you shake the tray vigorously, the marbles pop out of the depressions and go all over the place. As long as you keep shaking vigourously, the marbles move around the tray almost as if the depressions weren't even there. And as soon as you slow your shaking enough, the marble fall back in to the depressions and cannot move beyond their local depression.

"Temperature" is a measure of how vigorously the elements of something are moving around randomly. A gas which is 4X as hot has its molecules moving, on average, 2 times as fast (square that to get 4 times the energy). The depressions represent the fact that many molecules have a slighly lower energy when they are at a "sweet spot" distance from each other which is actually pretty close together, but that energy is not very much lower. It is easy to see how the marbles in the depression is analogous to a solid, where the molecules are arrayed in fixed locations in a regular lattice.

But it is not much of a stretch to see how the marbles in the depression are also aanalagous to the liquid situation. In liquid, the molecules want to stay near each other, but they can "roll around over each other" pretty freely as long as they don't get too far apart from any other molecules.

And so we have described "phase transitions," how a weak attractive force (shallow depressions) only impose their order when the temperature (amount of random motion energy) is low enough.

Superconducting Phase Transition

In a metal which is potentially superconducting, at high temperatures its conduction electrons zip around pretty freely and energetically throughout the extent of the metal. It is so much like a gas of electrons that the very useful model is called the free electron model and at higher temperatures the electrons are referred to as a gas.

But it turns out that in some metals, (lead, tin, niobium, among others) there is a very weak attractive force between electrons. This is so week that at room temperature it is completely unnoticeable. Indeed, none of the gas electrons get "stuck" in the tiny "depressions" associated with such a weak force until the temperature is around one one-hundredth of normal room temperature, around 4 Kelvins for lead, whereas normal room temperature is about 290 Kelvins and liquid nitrogen is still 77 Kelvins.

The superconducting state is more like a liquid than a solid. So the electrons have some "correlation" with each other, which is broken when they are jostled out of their "depression" by thermal bumping from other electrons.

Now what this model tells you is a way to describe falling in to a more correlated state from a "gas" state, which applies to gases condensing to liquids, then solids, it also applies to gasses of electrons condensing into a superconducting fluid-like state. What it does NOT describe is WHY that state carries current with absolutely positively no voltage drop required. That is a different (and probably harder) post!

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Supercondivity happens because an electron energy gap appears that prevents scattering of the electrons. Echows is obviously aware of this because he specifically mentioned the gap. Your answer doesn't explain how the gap appears and why the gap falls to zero at the transition temperature. –  John Rennie Aug 2 '12 at 15:50
    
Actually if I think about it, my question could be rephrased as: why is there a temperature dependence in the gap? Yes, I can see from the gap equation that there is, but how would you interpret that information? –  Echows Aug 2 '12 at 18:04
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Since we haven't had any definitive answers I'll have a a go, but I expect comments from the big guns in the site! :-)

In the superconducting state you have electrons bound into pairs by an attractive force (due to lattice interactions). But there isn't just one electron pair, there are a huge number of them and they are all entangled. So you can't raise the energy of a single electron: you would have to raise the energy of the whole population of pairs and there are around Avagadro's number of them.

When you attempt to put energy in nothing happens until the energy breaks up a pair of electrons, and the isolated electrons now have a continuum of energies. So the smallest amount of energy you can put in is the binding energy of the Cooper pairs. This is why there is an energy gap.

It just remains to explain why the binding energy of the Cooper pairs falls with increasing temperature, and this is simply because the increased lattice vibrations interfere with and eventually destroy the long distance correlations required to form the pairs.

There is an interesting article on the history of superconductivity here, and it gives a popular science description of the processes inolved.

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The simplest explanation is that the thermal energy present in a system at low temperatures is not enough to break the paired state of the electrons.

The pairing of the electrons and the condensation of the pairs is what makes a superconductor. If you want a dead-simple lay persons explanation, consider two spheres glued together as the electron pair, and you trying to pull the spheres apart as the thermal energy. Only if you are strong enough will you be able to pull apart the spheres, and break down the pairing states. Now, if you have a million friends, all of whom have their own set of glued spheres and are as strong as you are, once you all have broken the glued sphere pairs, the spheres in your hands are free to move around willy-nilly, and bang into each other, giving rise to resistance.

This post gives me a reason to provide a link this video, which has its difficulties, but is visually appealing as a model : http://www.youtube.com/watch?v=O6sukIs0ozk

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