S-state pairs

we have have seen that g k  k  '  k ( E F  V kk 2  k ' g k ) '  ( E  1 2  k ) So returning to the spatial component of the wave function we have  o ( r )  k   k F cos 2 E K    .

r E ' where we measure the energies from the Fermi energy E k =  k -E F E’=2E F - E > 0 is the binding energy Therefore, since the weighting factor in the wave function depends only on E k this solution for the wavefunction has

spherical symmetry it is S-state and spin singlet

This is generally found to be the case - but many of the new exotic superconductors may have other symmetries High T c cuprates - d-wave Sr-Ru-O perhaps a p-wave spin triplet

Superconductivity and Superfluidity Lecture 11

Justification for Cooper’s potential

Note that the weighting factor (2E k +E’) 1 of 1/E’ when E k =0 has a maximum value , ie for electrons at the Fermi energy. It falls off with higher E k Therefore those electrons within a range E’ above E F are most strongly involved in forming bound pairs As E’<< ħ  c behaviour of for N(E F )V<1 V kk’ this shows that detailed out and around ħ  c will not have any great effect on the result This is justification for Cooper’s simplification.

Superconductivity and Superfluidity Lecture 11

Condensation

If the formation of one Cooper pair lowers the energy of the system, then formation of Cooper pairs from all electrons would lower the energy even further Such a state would resemble a

Bose-Einstein condensate

However if the Fermi sea collapses in this way then Cooper’s analysis is invalid as we no longer have a good Fermi surface Condensation into pairs therefore continues until

equilibrium is reached

ie until the state of the system has changed so much that the binding energy for one more pair has become zero At this stage all pairs of electrons are in the same state - therefore they are boson-like But the Cooper pairs a weakly bound and therefore constantly break apart and reform, usually with different partners so they preserve some fermion character

Superconductivity and Superfluidity Lecture 11

Why should the pairs have zero momentum

Imagine two shells (representing the Fermi momentum states) of radius p F thickness  p separated by a vector P , the momentum of a pair and (a) If P is arbitrary but finite, only those electrons in the shaded portions at the Fermi surface can contribute to pairs with a net momentum of P=p 1 + p 2 (b) The number of electrons that can contribute to pairs with a net momentum of P=p 1 + p 2 increases as P  0 and reaches a sharp peak when P=0 , ie when the shells coincide.

(c) When this happens

every electron

can find a partner with equal and opposite momentum on the Fermi sphere  p p 1 P p p 1 2 P (b) p 2 p 1 (a) P=0 p 2 (c)

This must be the most energetically favorable state - all electrons can participate Superconductivity and Superfluidity Lecture 11

Size of a Cooper pair - an approximation

The binding energy of a Cooper pair is approximately the gap energy  E This gap is centred at E F , where electrons have a wavevector k F The energy of one electron is So and If E~E F , k~k F ,  E~E g E  E   p 2  2 m 2 m  2 2 k  k 2 m  2 k 2  E  2  k E ~ k F E g E F  2 k  k .

2 m m  2 k 2  and, as E 2 g  k k /E F ~10 -4, At the top of a band, k F ~  /a, so  k~   10 -4 /2a so  k/k F ~0.5

 10 -4 Now, very approximately,  k~1/  x , where  x wavefunction, and as a~1.5Å , we have  x~10 4 Å is the

spatial extent

of the

Superconductivity and Superfluidity Lecture 11

Overlapping pairs

The density of conduction electrons in a metal is n~10 22 cm -3 The fraction that might be expected to form Cooper pairs in a superconductor is of the order of  k/k F ~0.5

 10 -4 So the number of electrons that will partcipate in pairing is approximately 0.5

 10 18 cm -3 so there are ~2.5  10 17 pairs cm -3 Note that the volume of one pair is approximately 4  (10 4 ) 3 /3Å 3 = 4  10 -12 cm 3

So in the volume of one pair there are ~10 6 pairs!!

Superconductivity and Superfluidity Lecture 11

Density of states

N(E) States available for singly excited electrons 2  ~2.6meV

Lecture 11

E F ~5eV E

The total number of states is unaltered by the presence of the gap Superconductivity and Superfluidity

Origin of the attractive interaction

We have introduced the interaction V kk '  1  s   V ( r ) e i ( k  k ' ).

r d r s in which V kk’ , the strength of potential to scatter a pair of electrons from (k’,-k’) (k,-k) , is the fourier transform of the real space potential V(r). to For convenience we can write V ( Q )  V ( k  k ' )  V kk ' We can start by looking at the form of V(Q) interaction V ( r )  4 e 2  o r when V(r)  1  s   s V ( r ) e i Q .

r d r is a bare Coulomb which, when fourier transformed using the above expression, gives V ( Q )  e 2  s  o Q 2  e 2  o Q 2 for unit normalisation volume

This V(Q) is always positive Superconductivity and Superfluidity Lecture 11

The screened Coulomb repulsion

If we now take the dielectric function of the medium  (Q,  ) into account V(Q) reduced by a factor of {  (Q,  )} -1 is The most obvious contribution to  (Q,  ) is the screening effect of the conduction electrons. This introduces a screening length of 1/k s ~1Å . In the Thomas-Fermi approximation  is given by   1  k s 2 Q 2 So the screened potential is V ( Q )  e 2  o ( Q 2  k 2 s ) The screening has removed the divergence at Q=0 but, however strong the screening, it still leaves a positive V(Q)

……..and no superconductivity Superconductivity and Superfluidity Lecture 11

The ion contribution

An attractive potential can arise if the

motion of the positive ion cores

is taken into account First

an electron polarises the medium

by attracting the positive ions (through a strong electron phonon coupling). The locally excess concentration of positive ions in turn

attract a second electron

, giving an effective positive interaction between electrons If this attraction is strong enough to override the screened Coulomb repulsion then the net interaction is attractive and superconductivity results

…...first suggested by Frolich in 1950 Superconductivity and Superfluidity Lecture 11