superconductivity - · superconductivity – rj nicholas ht08 3 meissner effect it was discovered...

Superconductivity – RJ Nicholas HT08 1

SuperconductivityThe basic facts:

• Resistivity goes to zero below the critical temperature Tc (the most sensitive measurements imply R < 10-25 Ω)

• Many different materials show superconductivity

• Tc values range from a few mK up to 160K

• Superconductors expel flux (the Meissner effect) and act as perfect diamagnets.

• Superconductivity is destroyed by a critical magnetic field Bc

• Specific heat, infrared absorption, tunnelling, .. all imply that there is an energy gap associated with superconductivity

Resistivity

Transition is very sharp in pure materials (as narrow as 10-3 K), broader when impurities are present.

Very good conductors (simple free electron materials) do not superconduct.

Superconductivity is destroyed by high currents (critical current Jc)


Superconducting Elements

Critical Field

Superconductivity is destroyed by magnetic fields

Critical field depends on temperature, typically

))/(1( 20 cc TTBB −=


Meissner EffectIt was discovered in 1933 that when cooled in a magnetic

field flux is expelled completely from a superconductorInside the superconductorB = Ba + μ0M, giving M = -B/μ0 (χ = -1)This is not the result of zero resistance

Superconductor Superconductorwith hole

Zero ResistanceNormal Metal

T > Tc

T < TB < B

c

c

T < TB = 0

c

Flux expulsion

Flux is expelledfrom superconductor,

Flux is trapped ina zero resistancemetal

∫ ∂∂= td φsE.


Flux penetrationIn order to cause the flux expulsion it is necessary for there to be a surface current to generate the internal flux

London & London assumed that:

λ is the London penetration depth (approx. 10 nm)

tmnecurl

mne

tcurl

mne

tso

mne

∂∂

−==∂∂

=∂∂

=

BEj

EjEj

22

22

,τ

Bjm

necurl2

−=

20

0

22

0

2

with,exp:givingnemxjj

mnecurl

mnecurlcurl

μλλ

μ

=−=

∇−=−=−= jjBj

Thermodynamics of the Superconducting phase transition

In magnetic field we define a Gibbs free energy as:

G = E - TS -M.B, where the M.B term includes the energy of interaction of the specimen with the external field. Thus:

dG = (dE - TdS - B.dM) - SdT - M.dB = - SdT - M.dB

dE = dQ + dW

0

2

0 0

00

2),0(),0(),(

with,.),0(),(

μμ

μ

cS

B

ScS

B

ScS

BTGdBBTGTBG

dTGTBG

c

c

+=+=

−=−=

∫

∫BMBM


At Bc the normal and superconducting phases are in equilibrium, so their Gibbs functions are the same. Thus:

We can deduce the Entropy difference from S = -∂G/∂T

At Tc the value of Bc → 0 so SN = SS

dBc/dT is negative, so SN > SS for T < Tc

0

2

2),0(),0(

μc

SNBTGTG =−

dTdBB

dTdBSSS ccc

SN0

2

021

μμ−=−=−=Δ

Entropy of two states is the same at Tc

Specific heat is:

Discontinuity in C at Tc

Second order phase transition

TSTC∂∂

=

Entropy and Specific Heat


Specific heat of superconductor has a large discontinuity and tends to zero at T = 0

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−∝

TkC

BS exp

Specific heat is activated with

Infrared absorption

Infrared absorption when hν > 2Δ

Value of energy gap 2Δ is related to Tc

cBTk5.32 ≈Δ


BCS Theory• A field theory developed by Bardeen, Cooper and Schrieffer

• Explanation for the formation of an energy gap

• based on the formation of ‘Cooper pairs’ of electrons

• electrons experience an attraction caused by interaction with crystal lattice leading to binding in pairs

Evidence for phonon interactions:

• Isotope effect. For different isotopes Tc ∝ M-1/2

• Good conductors at high temp. (Cu, Na, Au etc) do not superconduct, poor conductors do (Hg, Pb, Sn…)

Cooper pairs

Two in a bed. Exchange of virtual phonons

Strongest interaction fork1 = -k2


Electrons bind together in pairs with momenta kF and -kF. Bonding pair have opposite spins in a spin singlet wavefunction

Pair has charge 2e and mass 2m

Pairs gain a binding energy of Δ per electron

Energy gap of 2Δ occurs at the Fermi energy EF

)(2

1),( 21 ↓↑−↑↓= rrSφφ

Zero ResistanceCurrent flows by displacement of entire Fermi surface. Because of the energy gap no scattering can occur until pairs can be excited across gap. Causes a Critical current Jc once electrons gain enough energy.

Energy gap is temperature dependent, leading to temperature dependence of Bc, Jc.


Energy and Coherence

Average energy gain per electron is approx. Δ/2 (actually Δ/4 with full theory) so as Δ × g(EF) electrons are shifted down

Coherence Length ξ = vFτEstimate τ from energy gap: /τ = 2Δ

so ξ = vF /2Δ (accurate result: vF /πΔ)

typical values are 1000 - 1 nm

(much shorter in exotic and high Tc materials)

0

22

2energyfreeGibbsingain

4)(

μcF BEg

==Δ

Type I and Type II Superconductors in B field

Superconductivity is established over the coherence Length ξ

Magnetisation energy occurs over the London penetration depth λ


Type II superconductors

Typical materials:One element, Nb, and many

alloys such asNbTi, Nb3Sn, V3Ga….

High Tc Materials:Ba0.75La4.25Cu5O5(3-y)

YBa2Cu3O7-x

Have short coherence lengths and high Tc

Form vortices which make a flux lattice above Bc1

High Tc Materials

Conduction takes place in CuO planes

All properties are highly anisotropic


Superconducting Tunnelling

Tunnelling between two superconductors with a very thin (few nm) barrier

Tunnel current shows features due to alignment of energy levels either side of barrier. Measures energy gaps

Flux Quantisation

Resistivity = 0 means no scattering. Therefore there is macroscopic phase coherence of the supercurrent over the entire length of a superconductor

( ) ( )AjAv qmnqq

m−∇−=−∇−= ,1

( )( )

Φ===∴

−=

−=

∫∫∫

∫∫∫

qdqdcurlqn

dcurlqnmnq

dqdmnqd

SBSA

SA

sAsksj

..2

.20

...

π

πChoose apath insidesuperconductor


Result is that flux is quantised in units:

Proof of existence of Cooper pairs.

Leads to many more sophisticated quantum interference effects (Josephson effect), and applications such as very sensitive measurement of small magnetic fields (and fluxes) e.g. SQUIDs

2150 1007.2

2mT

eh

qh −×===Φ

superconductivity - · superconductivity – rj nicholas ht08 3 meissner effect it was discovered...

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