Superconductivity and Superfluidity
The effects of lattice vibrationsThe effects of lattice vibrations
The localised deformations of the lattice caused by the electrons are subject to the same “spring constants” that cause coherent lattice vibrations, so their characteristic frequencies will be similar to the phonon frequencies in the lattice
The Coulomb repulsion term, on the other hand, has a time scale defined by the plasma frequency and is therefore effectively instantaneous
If an electron is scattered from state k to k’ by a phonon, conservation of momentum requires that the phonon momentum must be Q=k-k’
The characteristic frequency of the phonon must then be the phonon frequency Q,
k-Q k´+Q
k´k
Q
The electrons can be seen as interacting by emitting and absorbing a “virtual phonon”, with a lifetime of =2/ determined by the uncertainty principle and conservation of energy
Lecture 12
Superconductivity and Superfluidity
The attractive potentialThe attractive potential
It can be shown that such electron-ion interactions modify the screened Coulomb repulsion, leading to a potential of the form
2Q
2
2Q
2s
2o
2
1)kQ(
e)Q(V
Clearly if <Q this (much simplified) potential is always negative.
1
11
)kQ(
e2Q
22s
2o
2
This shows that the phonon mediated interaction is of the same order of magnitude as the Coulomb interaction
The maximum phonon frequency is defined by the Debye energy ħD =kBD,where D is the Debye temperature (~100-500K)
The cut-off energy in Cooper’s attractive potential can therefore be identified with the phonon cut-off energy ħD
V)E(N
2exp2E2E
FDF
Lecture 12
Superconductivity and Superfluidity
The maximum (BCS) transition temperatureThe maximum (BCS) transition temperature
N(EF)V is known as the electron-phonon coupling constant:
2/V)E(N Fep
ep can be estimated from band structure calculations and from estimates of the frequency dependent fourier transform of the interaction potential, ie V(Q, ) evaluated at the Debye momentum.
Typically ep ~ 0.33For Al calculated ep ~ 0.23 measured ep ~ 0.175For Nb calculated ep ~ 0. 35 measured ep ~ 0.32
epDcB
1exp2Tk75.1
In terms of the gap energy we can write
which implies a maximum possible Tc of 25K !
Lecture 12
Superconductivity and Superfluidity
Bardeen Cooper Schreiffer TheoryBardeen Cooper Schreiffer Theory
In principle we should now proceed to a full treatment of BCS Theory
However, the extension of Cooper’s treatment of a single electron pair to an N-electron problem (involving second quantisation) is a little too detailed for this course
Physical Review, 108, 1175 (1957)
Lecture 12
Superconductivity and Superfluidity
Bardeen Cooper Schreiffer TheoryBardeen Cooper Schreiffer Theory
BCS theory requires:
(a) low temperatures - to minimise the number of random (thermal) phonons (ie those associated with electron-ion interactions must dominate)
(b) a large density of electron states just below EF (the electrons associated with these states are those that are energetically suited to form pairs)
(c) strong electron phonon coupling
BCS theory is an effective, all encompassing microscopic theory of superconductivity from which all of the experimentally observed results emerge naturally
Ginzburg-Landau theory can be derived from BCS theory, and the phenomenological coefficients introduced by Ginzburg and Landau are related to quantities introduced in the microscopic theory
Lecture 12
Superconductivity and Superfluidity
Superconducting MaterialsSuperconducting Materials
1910 1930 1950 1970 1990
20
40
60
80
100
120
140
160
Su
per
con
du
ctin
g t
ran
siti
on
tem
per
atu
re (
K)
Hg Pb NbNbCNbC NbNNbN
V3SiV3Si
Nb3SnNb3Sn Nb3GeNb3Ge
(LaBa)CuO(LaBa)CuO
YBa2Cu3O7YBa2Cu3O7
BiCaSrCuOBiCaSrCuO
TlBaCaCuOTlBaCaCuO
HgBa2Ca2Cu3O9HgBa2Ca2Cu3O9
HgBa2Ca2Cu3O9
(under pressure)
HgBa2Ca2Cu3O9
(under pressure)
Liquid Nitrogen temperature (77K)
Lecture 12
Superconductivity and Superfluidity
Superconducting compoundsSuperconducting compounds
Perhaps the most widely used class of superconducting compounds are the A3B family which crystallise in the A-15 structure.
The A-atoms are typically the transition metals V or Nb, whilst the B atoms are non-transition metals such as Sn, Al, Ga, Si, Ge
BA
Six A15 compounds have transition temperatures over 17K
Nb3Ge thin films held the record for the highest known Tc of 23K for a number of years up to 1986
This was thought to be close to the limit imposed by BCS theory
Lecture 12
Superconductivity and Superfluidity
The A15 compoundsThe A15 compounds
BA
A structural instability associated with soft phonon modes and a lattice distortion are believed to be responsible for the high transition temperatures
Compound Tc B*
V3Ga 15.4K 23T V3Si 17.1K 23T Nb3Sn 18.3K 24T Nb3Al 18.9K 33T Nb3Ga 20.3K 34T Nb3Sn 23.0K 38T
Nb3Sn is the most widely exploited material for the construction of high field superconducting magnets for NMR, MRI etc
Lecture 12
Superconductivity and Superfluidity
The A15 compoundsThe A15 compounds
The materials properties that give the A15 compounds their relatively high Tcs give the compounds brittleness, which makes cable construction difficult:
The so called Rutherford method is generally used
Cu
Nb
Sn
swaging annealing
Nb3Sn
Cu
Lecture 12
Superconductivity and Superfluidity
The Chevrel phase compoundsThe Chevrel phase compounds
The Chevrel phases were discovered in 1971
They are ternary molybdenum chalcogenides of the type MxMo6X8
M could be any one of a number of metals at rare earth (4f) elements and X is S, Se or Te
Interestingly, these were the first class of superconductors in which magnetic order and superconductivity were found to coexist
With M=Gd, Tb, Dy, Er the superconducting transition temperatures are between 1.5 and 2K, while the Neel temperatures are between 0.5 and 1K.
The M atoms form a nearly cubic lattice in which the Mo6X8 uinits are inserted
Lecture 12
Superconductivity and Superfluidity
The Chevrel phase compoundsThe Chevrel phase compounds
Some Chevrel compounds have relatively high transition temperatures, and very high critical fields
Compound Tc B*
SnMo6S8 12K 34TPbMo6S8 15K 60TLaMo6S8 7K 45TPbMo6Se8 3.6K 3.8T
Critical current densities as high as 3x105A.cm-2 have been observed at 4.2K
Unfortunately the material is extremely brittle and making wires is problematic
Lecture 12
Superconductivity and Superfluidity
The nickel borocarbidesThe nickel borocarbides
Y, Lu, Tm, Er, Ho, DyY, Lu, Tm, Er, Ho, Dy(Tb, Gd, Nd, Pr, Ce, YbTb, Gd, Nd, Pr, Ce, Yb)
NiNi CCBB
TN(K) Tc(K) (g-1)2J(J+1)
Y 0 15 0Yb 0 0 (HF?)Lu 0 16 0Tm 1.5 10.8 1.17Er 6.5 10.5 2.55Ho 6 8.5 4.5Dy 10 6.2 7.08Tb 15 0 10.5Gd 19.5 0 15.5
The rare earth nickel borocarbides, discovered in 1994 have relatively high transition temperatures but also order magnetically at temperatures comparable to Tc
…an ideal system for probing the interplay of superconductivity and magnetism
Superconductivity and Superfluidity
Organic SuperconductorsOrganic Superconductors
The Bechgaard salts are nearly one dimensional conductors with very low carrier densities
Se
Se Se
Se CH3
CH3
CH3
CH3
TMTSF
tetramethyltetraselenafulvaneMost of the class of compounds(TTMTSF)2-X, where X is an anion are only superconducting under pressure
The electronic properties are extremely anisotropic
X pc/kbar Tc
ClO4 0 1.2KPF6 9 1.2KReO4 9.5 1.4K
Lecture 12
Superconductivity and Superfluidity
Organic superconductors under pressureOrganic superconductors under pressure
The systems are particularly interesting from a fundamental perspective
Is the superconductivity “conventional”?
Lecture 12
Superconductivity and Superfluidity
Organic SuperconductorsOrganic Superconductors
The -(BEDT-TTF)2X salts, where X is an anion such as I3, IBr2 or AuI2 are largely 2d organic superconductors
S
S
S
S
S
S
S
S
HHH
H
HHH
H
BEDT-TTF
Bis-ethelenedithio-tetrathiafulvane
X Tc
I3 L 1.2K I3 H 8.1K IBr2 2.5KCu(NCS)2 10K
There is recent evidence that superconductivity in some of the BEDT compounds can only exist in high magnetic fields
In this state the electron pairs may have finite momentum!
Lecture 12
Superconductivity and Superfluidity
Organic superconductorsOrganic superconductors
Superconductivity and Superfluidity
The Bucky ballsThe Bucky balls
Buckminsterfullerene contains 60 carbon atoms at the apices of a triacontaduohedron 7.1Å in diameter
C60 itself is not a superconductor, but it can be doped with alkali metals (which form an fcc lattice with a lattice parameter of 10Å) giving A3C60
Compound Tc
K3C60 19KK2 RbC60 22KRb2KC60 25KRb3C60 29KCs3C60 47K
Although the isotope effect is BCS-like in C60 there is some evidence that superconductivity might not be “conventional”
Lecture 12