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PH 318- Introduction to superconductors 1

SUPERCONDUCTIVITY

property of complete disappearance of electricalresistance in solids when they are cooled below acharacteristic temperature. This temperature is calledtransition temperature or critical temperature.

Superconductive state of mercury (TC=4.15 K) wasdiscovered by the Dutch physicist Heike KamerlinghOnnes in 1911, several years after the discovery ofliquid helium.


Classical elemental superconductors

Element Transition temperature, KZinc 0.88

Aluminum 1.20Indium 3.41

Tin 3.72Mercury 4.15

Lead 7.19

Until 1983 record Tc=23.3 K was that of Nb3Ge alloy.


Progress in Tc of superconductor materials with time

High temperature superconductors discovered in1986: Tc=80-93 K, parent structure YBa2Cu3O7 .At present the record transition temperature (TBCCO)is now at TC= 134 K.


Effect of trapped magnetic flux

Consider a ring made out of superconductivematerial.

Perform the following thought experiment:1. At T>Tc the material is normal state. When the

external magnetic field is turned on, it penetratesthrough the ring.

2. Reduce the temperature so that T<Tc.

3. Remove the external magnetic field.


4. You discover that the magnetic field that waspenetrating through the opening of the ring magneticfield remains there. The magnetic flux remainstrapped in the ring opening.

This effect can be explained in terms of Faraday’s lawof induction

r rE d l

ddt

. = −z Φ


r rE d l

ddt

. = −z Φ

where E is the electric filed along the closed loop, Φis the magnetic flux through the opening of the ring.

Before the external magnetic field was turned offthere was a magnetic flux Φ=B.(area) through thering.

Below Tc the resistivity of superconductor becomesequal to zero and therefore at T<Tc the electric fieldinside the superconductor must be and is zero aswell. In view of this

r rE d l. =z 0

and therefore,the right side of Faraday’s equation

ddtΦ

= 0

which means thatΦ = =B area constb g

The magnetic flux Φ through the ring must remainconstant. For this reason the magnetic flux remainstrapped in the opening of the ring after the externalmagnetic field has been turned off.


There is no magic involved. The trapped magneticfield passing through the ring is due to the currentinduced in the ring when the external magnetic fieldwas turned off.

The induced current is called the persistent current.

The current persists, it does not decay because theresistance of the ring is zero. Actually no decrease ofcurrent was observed over the period of three years!Theoretically, the relaxation time of current carriers inthe superconductor is greater than the age ofuniverse.


Meissner effectexpulsion of magnetic field from the interior of thesuperconductorThought experimentConsider a sphere made out of superconductivematerial. At T>Tc the material is in normal state.When external magnetic field is turned on, theexternal magnetic field penetrates through thematerial.

On the basis of Faraday’s law,r rE d l

ddt

. = −z Φ

one would expect that at T<Tc the magnetic fieldwould remain trapped in the material after theexternal field has been turned off.


The trapping of magnetic field does not happen (theabsence of magnetic field inside the superconductoris the Meissner effect).This is what happens:

The magnetic field is expelled from the interior of thesuperconductor, inside the superconductor B=0.

Superconductor expels magnetic field from theinterior by setting up electric current at the surface.The surface current creates magnetic field thatexactly cancels the external magnetic field!

This electric current at the surface of thesuperconductor appears at T<Tc in order that B=0inside the superconductor.


Penetration of magnetic field below the surface ofsuperconductorsThe surface current is distributed in the surface layer,the layer carrying the electric current has a finitethickness, and because of this, the external magneticfield partially penetrates into the interior of thesuperconductor,

B x Bx

external( ) exp= −FHGIKJλ

λ = penetration distance at temperature T;


Temperature dependence of penetration distanceλ = penetration distance at temperature T;λ0 = penetration distance at temperature T=0.

λ λ=

− FHG

IKJ

04

1TTC

λ0 = 30 - 130 nm, depending on the superconductormaterial


The magnetic properties of superconductorsIn addition to the loss of resistance, superconductorsprevent external magnetic field from penetrating theinterior of the superconductor. This expulsion ofexternal magnetic fields takes place for magneticfields that are less than the critical field. Magneticfield greater than BC destroys the superconductivestate.


Critical magnetic fieldThe critical magnetic field depends upon thetemperature,

B T BTTC C

C

( ) = − FHG

IKJ

FHG

IKJ0

2

1

BC0 = critical magnetic field at T=0.


Relationship between resistivity (a), magnetic fieldinside the superconductive material (b) andmagnetization of superconductor as a function ofexternal magnetic field


Critical currentSuperconductive state is destroyed by magnetic field.

Consider a straight wire. Since electric current in thewire creates magnetic field

BIr

= µπ0

2

The wire can carry maximum superconductivecurrent, Ic, corresponding to the critical magnetic fieldBc at the surface of the wire, r=R,

IRB

CC= 2

0

πµ

µ0 = 4π 10-7 Tm/A is the magnetic permeability of freespace.


Type I and Type II Superconductorsexhibit different magnetic response to externalmagnetic field.

In Type I superconductor the magnetic field iscompletely expelled from the interior for B<BC.


Type II superconductors have two values of criticalmagnetic field, for B<BC1 the magnetic field iscompletely expelled (Type-I behavior), whereas forBC1<B<BC2 the magnetic field partially penetratesthrough the material.


The bulk of superconductor material breaks downinto two regions: superconductive from which theexternal field is completely expelled, and normalthrough which the external field penetrates.

The normal regions are distributed as filaments filledwith the external magnetic field. The flux of magneticfield through the filaments is quantized. Electriccurrent is induced at the interface between thenormal and the superconductive regions, the“surface” of filaments is “wrapped” in current whichcancels the magnetic field in the superconductiveregions.

The electric current is carried by the superconductiveregions of Type-II material.


Superconductive magnetsThe main advantage of the superconductive magnet,in contrast to the electromagnet, is that it does notneed to use (dissipate) energy to maintain themagnetic field.However,

IRB

CC= 2

0

πµ

In order to achieve high critical currents insuperconductive magnets we need materials withhigh Bc. Type-I superconductors are not suitablebecause of low Bc. Type-II materials are used forsuperconductive magnets.

Superconductive magnets achieving magnetic field ofabout 20 Tesla use wire from niobium alloys, andoperate at temperature of 4 K (cooled by liquidhelium).

Quantization of magnetic flux

Magnetic flux is quantized, the quantum of flux is

Φ015

22 07 10= = −h

ex weber.

(Wb=Tesla.m2)In general, the magnetic flux is

Φ Φ= n 0

where n is an integer.


Mechanism of superconductivity

Isotope effect, Tc depends on the mass of atoms

Tmass of atoms constituting the crystal latticec ∝ 1

Interaction between electrons and lattice atoms iscritical for the existence of superconductive state.Good conductors (weak scattering from the lattice)are poor superconductors (low TC).

Electrons on their flight through the lattice causelattice deformation (electrons attract the positivelycharged lattice atoms and slightly displace them)which results in a trail of positively charged region.This positively charged region of lattice atoms attractsanother electron and provides for electron-electroncoupling.


Electron pairs, and not single electrons, are chargecarriers in superconductors

The electron-electron coupling is weak and can bedestroyed by thermal motion of the lattice. For thisreason superconductivity exists only at lowtemperatures.

The electron-electron coupling results in electronpairing - formation of Cooper pairs. The Cooper pairsdo not have spin 1/2 and therefore do not followPauli’s principle (1 electron per state). Large numberof Cooper pairs can populate one collective state.This state is stable and requires some additionalenergy input (thermal energy) to be destroyed. Thebinding energy of Cooper pairs in the collective stateis several meV.


Formation of Cooper pairs is a spontaneous processresulting in lower energy state of electrons in thesuperconductor. In superconductors, the filled stateare occupied by Coopers pairs, and the empty band,above Eg, is occupied by “broken” Cooper pairs.The band gap Eg is a measure of binding energy ofCooper pairs, the greater binding energy, the greaterTc.

E k Tg B c= 3 53. .

Eg confirmed from absorption spectra. For hc/λ>Eg

electromagnetic radiation absorbed.


“No scattering, no resistance”

The formation of collective state of Cooper pairs takeplace at T<TC. In the collective bound state theCooper pairs do not scatter from the lattice and theconductivity of superconductor is infinitely large.

Scattering of electrons from the lattice atoms requirea change of state of electron.

In the superconductive state the current carryingspecies is the electron pair. For the Cooper pair toscatter it would have to change its state (like anelectron in normal metal). However, the Cooper pairis coupled to a large number of other Cooper pairsand so the whole collective of Cooper pairs wouldhave to be involved in scattering at once. This doesnot happen, and therefore there is no scattering ofCooper pairs and therefore the conductivity is infinite.


Current-voltage characteristics of metal-insulator-superconductor junction


Josephson effect

Consider two superconductors separated by a thininsulating layer, few nm thick.

Brian Josephson noted (1962) that

1. Electron pairs in the two superconductors can forma single collective state and the electron pairs cantunnel through the insulating layer.

DC Josepson effect = electron tunneling curentacross the junction in the absence of applied voltage.


2. If a DC voltage bias is applied across the junction,there is an AC current through the junction thatoscillates with frequency

fe

hV=

2

The existence of ac current through the biasedjunction = AC Josephson effect.

The AC Josephson effect provides a method for themost accurate measurement of the electric potentialdifference because f can be determined accurately by“frequency counters”.The value of 2e/h=483.6 MHz/µV.


SQUID=Superconductive QUantum Interference Deviceconsist of two Josephson junctions forming a ring.

SQUIDs are used to measure extremely weakmagnetic fields (for example, magnetic fields createdby currents in the brain in response to various stimulior thinking).

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