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Superconductivity

Basic Properties

The Discovery ofSuperconductivity

• Using liquid helium,(b.p. 4.2 K), H.Kamerlingh Onnesfound that theresistivity of mercurysuddenly dropped tozero at 4.2 K.

H. K. Onnes, Commun. Phys. Lab.12,120, (1911)

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Making Liquid He

• Isothermal compression: compress He whilecooling it with LN2 (77 K).

• Expansion: Allow He to expand and cool.• The Hampson-Linde Cycle

Other Superconducting Materials

M. K. Wu, et al., Phys. Rev. 58, 908 (1987) B. T. Matthias, et al., Phys. Rev. 95, 1435, (1954)

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Elemental Superconductors

Types of Superconductor

• Type I: Very Sharp transition tosuperconducting state at the criticaltemperature TC.

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Types of Superconductors

• Type II: More gradual transition tosuperconducting state.

• Mostly alloys or compounds.• Generally higher TC than Type I.

Is it really zero resistance?

• If a current is generated in a superconducting ring,it will persist because of the zero resistivity.

• In a normal metal the current would dissipate dueto resistive effects.

• Experiments have demonstrated that currents insuperconducting coils can persist for years withoutany measurable degradation. Experimentalevidence points to a current lifetime of at least100,000 years.

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Superconductivity

Magnetic Properties

Perfect Diamagnetism

• According to Faraday’s Law aconductor will oppose any changein externally applied magneticfield. Circulating currents will beinduced to oppose the buildup ofmagnetic field in the conductor.

• A perfect conductor would be aperfect diamagnet. Inducedcurrents in it would meet noresistance, so they would persist inwhatever magnitude necessary toperfectly cancel the external fieldchange.

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Meisner Effect

• What happens if thematerial becomessuperconducting inan external field?

• For a perfectdiamagnet you wouldexpect the field to beunchanged.

Meisner Effect

• When a materialbecomessuperconducting itactively expelsthe magneticfield.

• This is theMeisner Effect.

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Type II

• In Type II superconductors the magneticfield is not excluded completely, but isconstrained in filaments within thematerial.

Type II

• These filaments are in the normal state,surrounded by supercurrents in what iscalled a vortex state.

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Critical Magnetic Fields

• Magnetic fields can destroy the superconductingstate.

• Above the critical magnetic field BC, the materialwill not remain superconducting even at absolutezero.

Critical Magnetic Fields

• The critical field depends on temperature:

• The existence of a critical magnetic field impliesthe existence of a maximum current in a wire ofthe superconducting material because the currentitself generates a magnetic field.

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Examples of Type I and II BC

Rohlf, James William, Modern Physics from a to Z0, Wiley 1994

Superconductivity

BCS Theory

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The Isotope Effect

• The criticaltemperatures ofdifferent isotopes ofthe same elementdepend on the massof the isotope.

• As mass increases thecritical temperaturedecreases.

The Isotope Effect

• The dependence is given by

• This implies that the lattice vibrations play arole in superconductivity.

• As M→∞ the TC → 0.

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BCS Theory

• A theory for the mechanism of Type Isuperconductivity was developed by JohnBardeen, Leon Cooper, and Robert Schrieffer.

• They received the Nobel Prize in 1972 for theBCS theory.

BCS Theory

• In the model the lattice mediates anattractive interaction between electrons.

An electron passing through the latticecauses the ions to move slightly together.The positive charge density propagates through the lattice.

Another electron passing by the increased positive charge density region experiencesan attractive Coulomb interaction.

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BCS Theory• The net effect of this is that the two electrons have

exchanged momentum via the lattice.• The interaction “between” the electrons is attractive as

each participates in an attractive Coulomb interaction (withthe lattice.)

• The BCS theory shows that in some instances the energyof this attractive interaction can exceed that of the(shielded) repulsive interaction that the electrons exert oneach other.

• In this case the electrons form a bound pair called aCooper Pair.

BCS Theory

• This pairing can be modeled interms of electron-phononinteractions.

• The 1st electron emits a phononand it is absorbed by the 2ndelectron. (Remember that aphonon is a quanta of latticevibration.)

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Forming Cooper Pairs

• The requirements for the formation of largenumbers of Cooper pairs are Low temperature so that random thermal phonons do

not interfere with the mediating phonons. Interaction between lattice and electrons be strong. The number of electrons lying just below the Fermi

energy must be large. The two electrons in each pair have opposite spin

orientations. The two electrons in a pair have equal and opposite

momentum (in the absence of an applied field.)

Cooper Pairs

• Because they are weakly bound Cooper pairsconstantly breaking up and reforming.

• The weak binding also causes them to be large.• In the region of the pair there are many electrons

that would like to bond into a pair.• If the total momentum of each Cooper pair is zero

then the number of pairs will be a maximum.

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The BCS Superconductor

• The state of the system is highly ordered with allof the pairs having the same center of massmotion.

• When an external field is applied the pairs behaveas particles with two electron charges movingthrough the lattice.

• The system moves together as a unit with all ofthe Cooper pairs locked together in motion.

• This is why there are no scatterings from latticeimperfections.

The Band Gap• According to BCS the binding energy of a Cooper pair is

about (7/2)kTC at 0 K.• It is energetically favorable for two electrons near the

Fermi level to to promote themselves above the Fermienergy so that they can participate in the Cooper pairinteraction.

• For each electron there is a nearly continuous distributionof single electron states

• For the system there is the superconducting ground stateand then an energy band gap, and above that normalconducting states.

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The Band Gap

• The incoming phonon (or photon) must have enough energy to promote theelectron to an unfilled level AND break the bond. This means that there is aminimum energy to go from a “superconducting” state to a “normal” state fora Cooper pair.

• The band gap between the superconducting ground state and the normal statesis related to the Cooper pair binding energy.

• The system (at 0 K) can only accept energies greater than the binding energyof a pair.

Fermi Energy

Evidence for the Band Gap

• A band gap is implied by the fact that the resistance isprecisely zero. If charge carriers can move through a latticewithout interacting at all, it must be because their energiesare quantized such that they do not have any availableenergy levels within reach of the interaction energy withthe lattice.

• Experiments measuring the absorption of microwavephotons by superconducting materials indicate an energygap.

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Band Gap and TC

• As predicted byBCS the band gap isdependent on thecritical temperature.

Superconductivity

Applications

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Superconducting Magnets• Type II superconductors such as niobium-tin and niobium-

titanium are used to make the coil windings forsuperconducting magnets. These two materials can befabricated into wires and can withstand high magneticfields.

• Typical construction of the coils is to embed a largenumber of fine filaments ( 20 micrometers diameter) in acopper matrix. The solid copper gives mechanical stabilityand provides a path for the large currents in case thesuperconducting state is lost. These superconductingmagnets must be cooled with liquid helium.

Superconducting Magnets• Used in MRI with highly uniform and stable fields of

about 1-7 Tesla.• Used in accelerator applications: bending magnets have

been designed to produce up to 20 Tesla.

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Power transmission• Reduction in power loss through cables when

transmitting electrical power can be achieved bythe use of Type II superconductors.

• In 2000 Los Alamos researchers were able toproduce meter lengths of YBCO superconductingtapes with critical current exceeding 100 amps andcurrent densities of one million amps per squarecentimeter at liquid nitrogen temperatures.