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Chapter: 2 General physics
    Section: 2.6 Electricity and magnetism
        SubSection: 2.6.4 Superconductivity

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2.6.4 Superconductivity

Superconducting properties of materials

Superconductivity is characterised by the complete loss of electrical resistance below some finite temperature, Tc, which is a characteristic of the material used. The phenomenon is rather common: one quarter of the elements become superconducting, as well as several thousand different alloys and compounds. Tc values vary from 0K to, at the time of writing, 160K. Above Tc superconductors exhibit ohmic conductivity, much as for a normal metal. At and below Tc, in the absence of a magnetic field, the low frequency resistivity falls to zero. Superconductivity can be destroyed by the application of a magnetic field which exceeds a critical value Hc(T), a decreasing function of temperature T, which reaches zero at T = Tc. The critical fields of superconductors at T = 0 vary widely, having rather low values for pure elemental superconductors (typically 10 mT) but attaining very high values, of order 100 T, for the new high temperature ceramic materials (see below).

(Click the Images below to view Larger Images)
Thumbnail image of 2.6.4 Fig A - Click to see fulsize image in new window

Superconductors fall into two classes: Type I and Type II. The principal difference between them is that the former possess a single well defined critical field whereas the latter have two critical fields, the lower of which (Hc1) defines the field below which magnetic flux is substantially excluded from the material (the Meissner effect). For fields above Hc1 but below the upper critical field (Hc2) the superconductor is said to be in the mixed state and magnetic flux penetrates it although a supercurrent can still flow without dissipation. In the mixed state any penetrating magnetic flux is confined within narrow flux vortex lines with each of which is associated a quantum of magnetic flux Φ0 ( = h/2e = 2 x 10−15 Wb). The two figures (left) show the variation of Hc1 with temperature for a number of elements (which are mainly Type I superconductors) and the temperature variation of Hc2 for a number of alloys. (Bc1 =  μ0Hc1 and Bc2 = μ0Hc2.)

Bc1 Tesla Vs Temperature (K)The critical current Ic of a Type I superconductor is that current which will produce the critical magnetic field at the conductor’s surface. The passage of a current down a Type II superconducting wire in the mixed state exerts a Lorentz force on the magnetic flux vortices, unless these lie exactly parallel to the current flow. If the vortices are caused to move by this force then dissipation will occur and appear as electrical resistance. To make conductors with high Ic values (essential for construction of high field solenoids etc.) Type II materials are designed so that the microscopic structure favours ‘pinning’ of flux vortices at fixed positions in the material, even in the presence of a large Lorentz force. This requires great attention to be paid to structural morphology but is readily achieved in a number of alloys based on niobium. The ability to produce critical current densities as high as 1010 A/m2 has been demonstrated routinely in helium cooled superconductors and in thin films of the high temperature cuprate materials (see below).

The resistance of superconductors to alternating currents is not strictly zero, due to the inertia of the paired charge carriers which are responsible for superconductivity in all presently known materials. The resistivity nevertheless can be extremely small at all temperatures below Tc. The real conductivity of normal metals means that the surface resistance Rs increases as f 0.5, where f is the frequency of the applied current. In contrast, the complex conductivity of superconductors leads to an f 2 dependence. For high purity conventional superconductors in single crystal form the surface resistance at temperatures well below Tc has the following dependence

   Rs (f2/T) exp(−Δ(T)/kT)

where Δ(T) is the temperature dependent energy gap between the superconductor ground state and the normal conducting state. For conventional superconductors the energy gap is of order 1 meV and its value at T = 0, Δ(0) scales with Tc. Δ(T) is a decreasing function of T reaching zero at T = Tc. The above equation applies for frequencies f << Δ/h and shows that for T/Tc < 0.2 the surface resistance of pure superconductors at microwave frequencies is many orders of magnitude lower than for conventional metals at the same temperature. Also the surface reactance Xs is to a good approximation proportional to f. This may be seen as a consequence of the frequency independent penetration depth λ(T) of the magnetic field.




Properties of superconducting elements

Element

Periodic table
group no.

Tc/K

Type

Hc(0)/mT

Hc1(0)/mT

Hc2(0)/mT

               

Beryllium

II A

 

  0.026

Lanthanum

III A

(α)

  4.88

I

80

 

 

(β)

  6.00

I

160  

Titanium

IVA

 

  0.4

I

    5.6

Zirconium

  0.61

I

    4.7

Hafnium

 

  0.128

I

      1.27

Thorium

  1.38

I

   16.0

Vanadium

V A

 

  5.4

II

  26

268

Niobium

 

  9.25

II

173

405

Tantalum

 

  4.47

II

  45

200

Protactinium

 

 

  1.4

Molybdenum

VI A

 

  0.92

     9.6

Tungsten

 

  0.015

I

        0.115

Technetium

VII A

 

  7.8

II

116

312

Rhenium

 

  1.7

I

20

Ruthenium

VIII A

 

  0.49

I

    6.9

Osmium

 

  0.66

I

    7.0

Iridium

 

  0.11

I

    1.6

Americium

(α)

  0.6

 

 

(β)

  1.0

 

 

 

 

Zinc

II B

 

  0.85

I

    5.4

Cadmium

 

  0.517

I

    2.8

Mercury

(α)

  4.154

I

  41.1

 

 

(β)

  3.949

I

  33.9

Aluminium

III B

 

  1.75

I

  10.5

Gallium

(α)

  1.083

I

    5.8

 

 

(β)

  5.9, 6.2

I

56 

 

 

(γ)

  7.62

II

>300

 

 

(δ)

  7.85

II

Indium

 

  3.41

I

 28.2

Thallium

 

  2.38

I

 17.8

Tin

IV B

 

  3.72

I

 30.5

Lead

 

  7.2

I

  80.3

Lutetium

VII B

 

  0.1

I

 35.0

               

    Many other elements become superconducting when deposited as thin films, when in the amorphous state, or under high pressure.




Recent developments in superconductivity

Since 1987 the number and variety of known superconductors has enlarged very significantly. The most important development has been the observation of superconductivity in a wide range of ternary or quaternary cuprate compounds. The original discovery of a Tc as high as 40 K in the LaSrCuO system by Bednorz and Muller almost doubled the record highest transition temperature. Further discoveries have resulted in a number of families of related compounds which are demonstrating potential for real applications and possessing Tc as high as 133 K at normal pressure. The cuprate superconductors are ternary, quaternary or even more complex oxides of copper in which planes of copper and oxygen are separated from one another by the other cations. An example of the structure of the unit cell of one of the most promising compounds for commercial applications YBa2Cu3O7 is shown in the figure below. Although the 2-dimensional planes are responsible for superconductivity the materials carry a supercurrent in three dimensions, adjacent copper-oxygen planes being coupled together sufficiently strongly to allow resistanceless current flow. The cuprate superconductors are all anisotropic to a greater or lesser degree however. This anisotropy is observed in penetration depth λ(0), coherence length, critical fields and critical current densities.

Crystal structure of YBa7Cu3O7In addition to the cuprates the transition temperature of organic superconductors has been raised to 11K and it has also been shown that doping of Buckminsterfullerene (a ‘soccer-ball’ shaped molecular form of carbon) with alkali metals can produce superconductivity at 40 K. Both of these latter developments have been somewhat eclipsed by the technological importance of the cuprates. A further class of metallic superconductors, called the heavy fermions, has also emerged recently. These superconduct mainly in the temperature region below 1 K. The interest in them has been mainly generated by the novel mechanism believed to be responsible for the charge carrier pairing and in the symmetry of the resulting macroscopic wave function. In all four cases of ‘exotic superconductivity’ mentioned above, the precise mechanism for superconductivity is as yet not clear though all involve paired charge carriers. It seems at the time of writing that the BCS mechanism responsible for superconductivity in the elemental metals, in which electron pairs couple through exchange of virtual phonons, does not provide the only or even the dominant mechanism in the high temperature cuprates superconductors.

The table below lists Tc, Hc2 (at a specified temperature) and λ(0) for a selection of alloy and compound superconductors. In general where a range of stoichiometries exhibit superconductivity the composition with the highest reported Tc has been given. Note that the list represents only a small selection of the total of known superconductors.




Properties of some superconducting alloys and compounds

Composition

Tc/K

μ0Hc2(T)/T, (/K)

λ(0)/nm

Nb0.67Zr0.33

11.0

>8.3 

(4.2)

100

Nb3Ge

23.6

37    

(4.2)

75

V3Si

17.1

23    

(4.2)

70

Nb3Al

19.1

29.5  

(4.2)

65

La1.85Sr0.15CuO4

37  

45     

(0)

220

YBa2Cu3O7

93  

140     

(0)

140

Bi1.6Pb0.4Sr2Ca2Cu3O10

110   

184     

(0)

250

Bi2Sr2CaCu2O8

92 

107     

(0)

250

Tl2Ca2Ba2Cu3O10

128  

75     

(0)

190

Tl2Ba2CaCu2O8

100  

99    

(0)

220

Hg1Ba2Ca2Cu3O10

133  

190    

(0)

180

Nd1.7Ce0.3CuO4

24

?

 

BaBi0.25Pb0.75O3

  13.0

7    

(0)

1000

Ba0.7K0.3BiO3

  35.0

17    

(0)

345

PbMo6S8

  14.4

51    

(4.2)

240

RbCs2C60

33 

78    

(0)

210

k-[BEDTTTF]2Cu[NCS]2

  10.5

10    

 

750

UPt3

      0.53

2.1 

 

700


Values of Tc, Hc2 and λ are to some extent dependent on composition and method of preparation. The above values represent only a guide to the optimal values to be expected. Where an entry is blank the parameter in question has not been measured or a number of measurements are discrepant.




J.C.Gallop

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