Superconductivity
The 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)
Superconductivity – RJ Nicholas HT10
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Superconducting Elements
Critical Field
Superconductivity is
destroyed by magnetic
fields
Critical field depends on
temperature, typically
Bc = B0 (1 − (T / Tc ) 2 )
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Meissner Effect
It was discovered in 1933 that when cooled in a magnetic
field flux is expelled completely from a superconductor
Inside the superconductor
B = Ba + μ0M, giving M = -B/μ0 (χ = -1)
This is not the result of zero resistance
Flux expulsion
∫ E.ds
= ∂φ
∂t
Superconductor
Superconductor
with hole
T > Tc
Flux is expelled
from
superconductor,
T < Tc
B < Bc
Flux is trapped in
a zero resistance
metal
T < Tc
B=0
Superconductivity – RJ Nicholas HT10
Zero Resistance
Normal Metal
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Flux penetration
In order to cause the flux
∂j
ne 2τ
ne 2
j
=
E
E
=
,
so
expulsion it is necessary for
∂t
m
m
there to be a surface current to
∂j
ne 2
ne 2 ∂B
E
=
=
−
curl
curl
generate the internal flux
∂t
London & London assumed that:
m ∂t
m
ne 2
curl j = −
B
m
ne 2
ne 2
curl curl j = −
curl B = − μ 0
j = − ∇2 j
m
m
m
giving : j = j0 exp − λx, with λ =
μ 0 ne 2
λ is the London penetration depth (approx. 10 nm)
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
Bc
GS ( Bc , T ) = GS (0, T ) −
∫ M.dB ,
0
Bc
with M = −
B
μ0
Bc2
GS ( Bc , T ) = GS (0, T ) + ∫ dB = GS (0, T ) +
μ0
2μ 0
0
Superconductivity – RJ Nicholas HT10
B
4
At Bc the normal and superconducting phases are in
equilibrium, so their Gibbs functions are the same. Thus:
Bc2
GN (0, T ) − GS (0, T ) =
2μ0
We can deduce the Entropy difference from S = -∂G/∂T
ΔS = S N − S S
1 dBc2
= −
2 μ 0 dT
= −
Bc dBc
μ 0 dT
At Tc the value of Bc → 0 so SN = SS
dBc/dT is negative, so SN > SS for T < Tc
Entropy and Specific Heat
Entropy of two states is
the same at Tc
Specific heat is:
C = T
∂S
∂T
Discontinuity in C at Tc
Second order phase
transition
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Specific heat of superconductor
has a large discontinuity and
tends to zero at T = 0
Specific heat is activated with
⎛ Δ ⎞
⎟⎟
C S ∝ exp ⎜⎜ −
k
T
⎝ B ⎠
Infrared absorption
Infrared absorption when
hν > 2Δ
Value of energy gap 2Δ is
related to Tc
2Δ ≈ 3.5 k BTc
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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 for
k1 = -k2
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Electrons bind together in pairs with momenta kF and -kF.
Bonding pair have opposite spins in a spin singlet
wavefunction
1
φ = φ S (r1 , r2 )
(↑↓ − ↓↑)
2
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
Zero Resistance
Current 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.
Superconductivity – RJ Nicholas HT10
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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
Δ2 g ( E F )
Bc2
= gain in Gibbs free energy =
4
2μ0
Coherence Length ξ = vFτ
Estimate 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)
Type I and Type II Superconductors in B field
Magnetisation energy occurs over the
London penetration depth λ
Superconductivity is established over the
coherence Length ξ
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Type II superconductors
Have short coherence lengths and high Tc
Form vortices which make a flux lattice above Bc1
Typical materials:
One element, Nb, and many
alloys such as
NbTi, Nb3Sn, V3Ga….
High Tc Materials:
Ba0.75La4.25Cu5O5(3-y)
YBa2Cu3O7-x
High Tc Materials
Conduction takes
place in CuO planes
All properties are
highly anisotropic
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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
v =
1
(− =∇ − qA ) ,
m
∫ j.ds
Choose a
path inside
superconductor
0
j =
(
(
nq
(− =∇ − qA )
m
)
nq
= k.ds − ∫ qA.ds
m ∫
nq
2nπ= − q ∫ curlA.dS
=
m
=
)
∴ 2nπ= = q ∫ curlA.dS = q ∫ B.dS = qΦ
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Result is that flux is
quantised in units:
Φ0 =
h
h
=
= 2.07 × 10 −15 T m 2
q
2e
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
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