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Superconductors II-When a metal is cooled to the critical temperature, electrons in the metal form Cooper Pairs.-Cooper Pairs are electrons which exchange phonons and become bound together. -As long as kT < binding energy, then a current can flow without dissipation.-The BCS theory of Superconductivity states that bound photons have slightly lower energy, which prevents lattice collisions and thus eliminates resistance. -Bound electrons behave like bosons. Their wavefunctions dont obeyPauli exclusion rule and thus they can all occupy the same quantum state.

Types of SuperconductorsType ISudden loss of magnetisationExhibit Meissner EffectOne HC = 0.1 tesla No mixed stateSoft superconductorEg.s Pb, Sn, Hg

Type IIGradual loss of magnetisationDoes not exhibit complete Meissner EffectTwo HCs HC1 & HC2 (30 tesla)Mixed state presentHard superconductorEg.s Nb-Sn, Nb-Ti

-MHHCSuperconductingNormalSuperconducting-MNormalMixedHC1HCHC2HCooper Pairs

-Cooper pairs can tunnel together through the insulating layer of Josephson Junction.-This process is identical to that of quantum barrierpenetration in quantum mechanics.-Because of the superconducting nature (no resistance) and the fact that Cooper pairs can jointly tunnel through an insulator we canmaintain a quantum current through the Josephson Junction without an applied voltage.-Thus a Josephson Junction can be used as a very sensitive voltage, current or flux detector.-A changing magnetic field induces a current to flow in a ring of metal, this effect can be used to detect flux quanta. Radio Astronomy uses these devices frequently.Types I Superconductors There are 30 pure metals which exhibit zero resistivity at low temperature. They are called Type I superconductors (Soft Superconductors). The superconductivity exists only below their critical temperature and below a critical magnetic field strength. Types II SuperconductorsStarting in 1930 with lead-bismuth alloys, were found which exhibited superconductivity; they are called Type II superconductors (Hard Superconductors). They were found to have much higher critical fields and therefore could carry much higher current densities while remaining in the superconducting state. The Critical FieldAn important characteristic of all superconductors is that the superconductivity is "quenched" when the material is exposed to a sufficiently high magnetic field. This magnetic field, Bc, is called the critical field. Type II superconductors have two critical fields. The first is a low-intensity field, Bc1, which partially suppresses the superconductivity. The second is a much higher critical field, Bc2, which totally quenches the superconductivity. The Critical FieldResearcher stated that the upper critical field of yttrium-barium-copper-oxide is 14 Tesla at liquid nitrogen temperature (77 degrees Kelvin) and at least 60 Tesla at liquid helium temperature. The similar rare earth ceramic oxide, thulium-barium-copper-oxide, was reported to have a critical field of 36 Tesla at liquid nitrogen temperature and 100 Tesla or greater at liquid helium temperature. BCS Theory of Superconductivity The properties of type I superconductors were modeled by the efforts of John Bardeen, Leon Cooper, and Robert Schrieffer in what is commonly called the BCS theory. A key conceptual element in this theory is the pairing of electrons close to the Fermi level into Cooper pairs through interaction with the crystal lattice. This pairing results from a slight attraction between the electrons related to lattice vibrations; the coupling to the lattice is called a phonon interaction.BCS Theory of SuperconductivityThe electron pairs have a slightly lower energy and leave an energy gap above them on the order of .001 eV which inhibits the kind of collision interactions which lead to ordinary resistivity. For temperatures such that the thermal energy is less than the band gap, the material exhibits zero resistivity. Bardeen, Cooper, and Schrieffer received the Nobel Prize in 1972 for the development of the theory of superconductivity. JOSEPHSON EFFECTJOSEPHSON EFFECT, the flow of electric current, in the form of electron pairs (called Cooper pairs), between two superconducting materials that are separated by an extremely thin insulator. A steady flow of current through the insulator can be induced by a steady magnetic field. The current flow is termed Josephson current, and the penetration ("tunneling") of the insulator by the Cooper pairs is known as the Josephson effect. Named after the British physicist Brian D. Josephson, who predicted its existence in 1962.The Science.The superconducting state is defined by three very important factors: critical temperature (Tc), critical field (Hc), and critical current density (Jc). Each of these parameters is very dependant on the other two properties presentcritical temperature (T ) The highest temperature at which superconductivity occurs in a material. Below this transition temperature T the resistivity of the material is equal to zero.critical magnetic field (Hc ) Above this value of an externally applied magnetic field a superconductor becomes nonsuperconductingcritical current density (Jc) The maximum value of electrical current per unit of cross-sectional area that a superconductor can carry without resistance.

Flux-Pinning:The phenomenon where a magnet's lines of force (called flux) become trapped or "pinned" inside a superconducting material. This pinning binds the superconductor to the magnet at a fixed distance.

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15Superconductors A Brief IntroductionProperties DC electrical resistivity goes to zero at a low temperature. (Called Tc: critical temperature)

A superconductor in a weak magnetic field will work like a perfect diamagnet.

Meissner effect: magnetic flux present in superconductor is ejected when superconductor cooled through Tc.

Superconductivity destroyed by strong enough magnetic field, Hc.

16Types of SuperconductorsType I Superconductors (soft superconductors) many types of pure materials have this behavior Zn, Cd, Hg, etc. Hc is too low for applications in superconducting magnets

Type II Superconductors tend to be alloys has a region that is a mixture of superconducting and non-superconducting states (vortex state) Hard superconductors (large magnetic hysteresis) MRIs

17TBCOTl2Ca2Ba2Cu3O10Ceramic cuprate superconductorTc=125 K (Beyers, R.) or 120K (Kittel)(2223)Three Cu perovskite-like units separated by Tl-O bi-layers.Tc seems to go up with number of CuO2 layers.18

TBCO nominal unit cellSource: R. Beyers, et. al.Crystallography and microstructure of Tl-Ca-Ba-Cu-O superconducting oxides19

Tl-O layerTl-O layerBa layerCuO2 layerCa layerCuO2 layerCa layerCuO2 layerBa layerTl-O layer20

Body centered (b.c.) tetragonal cellLattice Parameters:a = 3.822(4) angstromsc = 36.26(3) angstromsSource: R. Beyers, et. al.Crystallography and microstructure of Tl-Ca-Ba-Cu-O superconducting oxidesacaNote: c ~ 10 a21

Naming Scheme2223# of CuO2 planes within a conducting block.# of insulating layers between conducting blocks. (Tl-O layers)# of layers separating CuO2 planes.# of spacing layers between CuO2 blocks.Source: www.superconductors.org22Perovskite like structure

Source: J. Phillips. Physics of High-Tc SuperconductorsIn TBCO this structure is deformed to the tetragonal structure of the CuO2 units.OxygenCopperBa or Ca23

John Bardeen, Leon Neil Cooper, John Robert Schrieffer

Nobel Prize in Physics 1972 "for their jointly developed theory of superconductivity, called the BCS-theorye-e-PhononCoherence length Cooper pair modelA movement of the C-P when a supercurrent is flowing, is considered as a movement of a centre of the mass of two electrons creating C-P. Creation of a C-Pairs diminishes energy of electrons. Breaking a pair (e.g. through interaction with impurity site) means increase of the energy. All the C-P are in the same quantum state with the same energy. A scattering by a lattice imperfection (impurity) can not change quantum state of all C-P at the same time (collektive behaviour).e-e-Phonon2526Coherence length

Coherence length is the distance between the carriers creating a Cooper-Pair.GLSuperconductorConcentration C-PISLSLx< GLCoherence length is the largest insulating distance which can be tunneled by Cooper-Pairs.SCSC(Xi)26Xi

"for his theoretical predictions of the properties of a supercurrent through a tunnel barrier, in particular those phenomena which are generally known as the Josephson effects". Nobel Prize in Physics 1973 Brian David JosephsonJosephson discovered in 1963 tunnelling effect being 23-years old PhD studentThe superconducting tunnel Josephson) junction (superconductorinsulatorsuperconductor tunnel junction (SIS) is an electronic device consisting of two superconductors separated by a very thin layer of insulating material ISLSLx< GLSCSC28Penetration depth

depicts the distance where B(x) is e-time smaller than on the surface

TemperaturePenetration depth Superconductor2829Superconductor type II in a magnetic field

Vortex-lattice in superconductor type II. Magnetic flux of a vortex is quantized: 0=h/2e2.0710-15Tm2

Bi=Ba+0MOutside field BaOutside field BaMagnetization 0MAverage inside field BiMeissner phaseMixed phaseNormal condu-ctor2930Superconductor type II. B-T-Diagram

Mixed phaseMeissner phaseNormal stateTemperature TMagnetic induction B

STM (Scanning Tunneling Microscopy). Abrikosov-lattice in NbSe2

H. Hess, R.B. Robinson, and J.V. Waszczak, Physica B 169 (1991) 422The penetration depth

In the case where the dimension of the sample are much greater than , then inside the sample. Then is constant, if varied the gradient term in (1.10) would mean that the free energy increased. The constant value of is given from equation (1.13):

. (1.19)

Since the order parameter is constant, i.e. , equation (1.14) becomes

. (1.20)

Taking the curl of both sides of equation (1.20), and substituting for the vector potential yields to _1056746643.unknown

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. (1.21)

Equation (1.21) is identical to the second London equation (1.6) with a penetration depth given by

(1.22)

where is the penetration depth at zero temperature. The above equation, in contrast to the expression (1.8) of the London penetration depth, contains the temperature dependent parameter, , which is defined in terms of ._1056748914.unknown

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