Download Superconductivity and Superfluidity PHYS3430 Professor Bob

The 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
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
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’
k-Q
k´+Q
Q
k
k´
The characteristic frequency of the
phonon must then be the phonon
frequency Q,
Lecture 12
Superconductivity and Superfluidity
The attractive potential
It can be shown that such electron-ion interactions modify the screened
Coulomb repulsion, leading to a potential of the form
2


Q
e2
V (Q ) 
1


2 
o (Q2  k 2s )  2  Q



e2
1

1



2
o (Q2  k 2s )  2 Q
 1
This shows that the phonon mediated interaction is of the same order of
magnitude as the Coulomb interaction
Clearly if <Q this (much simplified) potential is always negative.
The maximum phonon frequency is defined by the Debye energy ħD =kBD,
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 ħD
 2 

E  2EF  2D exp 
N
(
E
)
V


F
Lecture 12
Superconductivity and Superfluidity
The maximum (BCS) transition temperature
N(EF)V is known as the electron-phonon coupling constant:
ep  N(EF )V / 2
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 ~ 0.33
For Al calculated ep ~ 0.23 measured ep ~ 0.175
For Nb calculated ep ~ 0. 35 measured ep ~ 0.32
In terms of the gap energy we can write
 1 

  1.75kBTc  2D exp  

  ep 
which implies a maximum possible Tc of 25K !
Lecture 12
Superconductivity and Superfluidity
Bardeen 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 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 transition temperature (K)
Superconducting Materials
160
HgBa2Ca2Cu3O9
(under pressure)
140
HgBa2Ca2Cu3O9
120
TlBaCaCuO
BiCaSrCuO
100
YBa2Cu3O7
Liquid Nitrogen
temperature (77K)
80
60
(LaBa)CuO
40
20
Hg Pb Nb
1910
Lecture 12
NbC
1930
NbN
Nb3Sn
Nb3Ge
V3Si
1950
1970
1990
Superconductivity and Superfluidity
Superconducting compounds
Perhaps the most widely used class of
superconducting compounds are the A3B
family which crystallise in the A-15
structure.
B
A
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
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 compounds
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
V3Si
Nb3Sn
Nb3Al
Nb3Ga
Nb3Sn
15.4K
17.1K
18.3K
18.9K
20.3K
23.0K
23T
23T
24T
33T
34T
38T
B
A
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 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
Nb
Nb3Sn
Cu
Cu
Sn
swaging
Lecture 12
annealing
Superconductivity and Superfluidity
The 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
The M atoms form a nearly cubic lattice in
which the Mo6X8 uinits are inserted
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.
Lecture 12
Superconductivity and Superfluidity
The Chevrel phase compounds
Some Chevrel compounds have relatively high
transition temperatures, and very high critical
fields
Compound
Tc
B*
SnMo6S8
PbMo6S8
LaMo6S8
PbMo6Se8
12K
15K
7K
3.6K
34T
60T
45T
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 borocarbides
The rare earth nickel borocarbides, discovered in
1994 have relatively high transition temperatures
but also order magnetically at temperatures
comparable to Tc
Y
Yb
Lu
Tm
Er
Ho
Dy
Tb
Gd
TN(K)
Tc(K)
0
0
0
1.5
6.5
6
10
15
19.5
15
0
16
10.8
10.5
8.5
6.2
0
0
(g-1)2J(J+1)
0
(HF?)
0
1.17
2.55
4.5
7.08
10.5
15.5
…an ideal system for probing the interplay of
superconductivity and magnetism
Y, Lu, Tm, Er, Ho, Dy
(Tb, Gd, Nd, Pr, Ce, Yb)
Ni
B
C
Superconductivity and Superfluidity
Organic Superconductors
The Bechgaard salts are nearly
one dimensional conductors
with very low carrier densities
The electronic properties are
extremely anisotropic
Most of the class of compounds
(TTMTSF)2-X, where X is an anion are
only superconducting under pressure
X
ClO4
PF6
ReO4
Lecture 12
pc/kbar
0
9
9.5
CH3
Se
Se
CH3
CH3
Se
Se
CH3
TMTSF
tetramethyltetraselenafulvane
Tc
1.2K
1.2K
1.4K
Superconductivity and Superfluidity
Organic superconductors under pressure
The systems are particularly interesting from a fundamental perspective
Is the superconductivity “conventional”?
Lecture 12
Superconductivity and Superfluidity
Organic Superconductors
The b-(BEDT-TTF)2X salts, where X
is an anion such as I3, IBr2 or AuI2 are
largely 2d organic superconductors
X
I3
bL
I3
bH
IBr2
Cu(NCS)2
Tc
1.2K
8.1K
2.5K
10K
There is recent evidence
that superconductivity in
some of the BEDT
compounds can only exist
in high magnetic fields
H
H
H
H
In this state the electron pairs may
have finite momentum!
Lecture 12
S
S
S
S
S
S
S
S
BEDT-TTF
Bis-ethelenedithio-tetrathiafulvane
Superconductivity and Superfluidity
H
H
H
H
Organic superconductors
Superconductivity and Superfluidity
The 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
K2 RbC60
Rb2KC60
Rb3C60
Cs3C60
19K
22K
25K
29K
47K
Although the isotope effect is BCS-like in C60 there is some evidence that
superconductivity might not be “conventional”
Lecture 12
Superconductivity and Superfluidity