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side,the momentum ka = ka - q. For another side , kb = kb +
q
ka + kb = ka + kb= K. In addition ,changing phonon will make
the
sum of net attractive energy and net repulsive energy smaller
than zero.
This may make the two electron be bounded. Theres another fact
an
electron will change the electron which pair with it during
the
propagation in the lattice. This helps keep the stability of the
entire
system.
But how the two electrons being close enough to make the
exchange of phonon? When the electron passes by the lattice of
positive charged ions, they
tend to crowed in on the electron. If theres another electron in
nearby area, it will be attracted
to the high density of positive charge. Now we see the electron
as the matter wave, if two
matter waves are coherence, it will overlap considerably with
each other. Then we can see the
propagation as the matter wave of cooper pair.The most important
fact is that the cooper pair is boson. The feature of bosons is
that they
tend to stay at the same state. The behavior is totally
different with normal electrons ,which are
fermions, tend to stay in different state and obey the Pauli
exclusion principle. Since they are
in the same state, we can see the superconductor as a giant
molecule. Therefore, theres no
obstruction for electron flow, implies theres no resistance.
Since cooper pair is boson, all of the cooper pair is at the
same state. In order to make the
system equilibrium ,the total momentum of all of the cooper pair
is zero. From this we can get
another fact is that two electron in the cooper pair should have
different spin because by themomentum conservation
ka + kb = K=0 ka=-kb .
Now if theres a current flow in the superconductor,
what will happen?We can use a parabolic to show how
electron interact when theres a current flow. For the free
electron the electron energy is proportional to the
square of the wave function, we can use a wave vector
k to represent it. In the figure of the left.It is in
theequilibrium state.The right shows the case when theres
a current flow.In the normal metal ,the electron in the
right side,which is in the higher energy level, tend to
scatter to the left side of the parabolic to maintain the
system equilibrium. This will cause the resistance. But in the
superconductor ,to jump to higher
energy level the cooper pair should be separated and excited
together. In a superconductor
theres no enough energy to do this. So the scattering of
electrons is suppressed, this cause
the zero resistance. It gives the most well-known property of
superconductivity ,the PersistentCurrent.
Reference:The rise of the superconductors by P.J Ford and G.A.
Saunders
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