Superconductivity – RJ Nicholas HT10 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)
Superconductivity – RJ Nicholas HT10 2
Superconducting Elements
Critical Field
Superconductivity is destroyed by magnetic fields
Critical field depends on temperature, typically
))/(1( 20 cc TTBB −=
Superconductivity – RJ Nicholas HT10 3
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.
Superconductivity – RJ Nicholas HT10 4
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)
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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
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B
ScS
B
ScS
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c
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∫
∫BMBM
Superconductivity – RJ Nicholas HT10 5
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
Superconductivity – RJ Nicholas HT10 6
Specific heat of superconductor has a large discontinuity and tends to zero at T = 0
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ−∝Tk
CB
S 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
Superconductivity – RJ Nicholas HT10 7
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
Superconductivity – RJ Nicholas HT10 8
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.
Superconductivity – RJ Nicholas HT10 9
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
Superconductivity – RJ Nicholas HT10 10
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
Superconductivity – RJ Nicholas HT10 11
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
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πChoose apath insidesuperconductor
Superconductivity – RJ Nicholas HT10 12
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