Superconductivity and Superfluidity The effects of lattice vibrations The localised deformations of the lattice caused by the electrons are subject to

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


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


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


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


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


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


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


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


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


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


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


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


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


Organic superconductors under pressureOrganic superconductors under pressure

The systems are particularly interesting from a fundamental perspective

Is the superconductivity “conventional”?

Lecture 12


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


Organic superconductorsOrganic superconductors


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

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Superconductivity and Superfluidity The effects of lattice vibrations The localised deformations of the lattice caused by the electrons are subject to