CONDUCTIVITY

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CONDUCTIVITY

• Conductivity
• Superconductivity

Electronic Properties Robert M Rose, Lawrence A
Shepart, John Wulff Wiley Eastern Limited, New
Delhi (1987)
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Resistivity range in Ohm m ? 25 orders of
magnitude
Semi-conductors
Metallic materials
Insulators
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Metals
Semi-metals
Classificationbased on Conductivity
Semi-conductors
Insulators
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Free Electron Theory

• Outermost electrons of the atoms take part in
  conduction
• These electrons are assumed to be free to move
  through the whole solid ? Free electron cloud /
  gas, Fermi gas
• Potential field due to ion-cores is assumed
  constant ? potential energy of electrons is not
  a function of the position (constant negative
  potential)
• The kinetic energy of the electron is much lower
  than that of bound electrons in an isolated atom

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Wave particle duality of electrons

• ? ? de Broglie wavelength
• v ? velocity of the electrons
• h ? Plancks constant

Wave number vector (k)
Non relativistic
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? ? ? k ? ? E ?
E ?
Discrete energy levels (Paulis exclusion
principle)
k ?
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Electron in an 1D box
L
If the length of the box is L
n ? integer (quantum number)
Quantization of Energylevels
Number of electrons moving from left to right
equals the number in the opposite direction
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In 3D

• Each combination of the quantum numbers nx , ny
  , nz corresponds to to a distinct quantum state
• Many such quantum states have the same energy
  and said to be degenerate
• The probability of finding an electron at any
  point in box is proportional to the square of
  the amplitude ? there are peaks and valleys
  within L
• If the electron wave is considered as a
  travelling wave the amplitude will be constant

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Fermi level

• At zero K the highest filled energy level (EF)
  is called the Fermi level
• If EF is independent of temperature (valid for
  usual temperatures) ? Fermi level is that level
  which has 50 probability of occupation by an
  electron

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T gt 0 K
0K
P(E) ?
Increasing T
E ?
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Conduction by free electrons

• If there are empty energy states above the Fermi
  level then in the presence of an electric field
  there is a redistribution of the electron
  occupation of the energy levels

EF
ElectricField
EF
E ?
k ?
k ?
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Force experienced by an electron

• m ? mass of an electron
• E ? applied electric field

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• In the presence of the field the electron
  velocity increases by an amount (above its
  usual velocity) by an amount called the drift
  velocity
• The velocity is lost on collision with obstacles

• vd ? Drift velocity
• ? ? Average collision time

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The flux due to flow of electrons ? Current
density (Je)

• n ? number of free electrons

Ohms law
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Mean free path (MFP) (l) of an electron

• l vd ?
• The mean distance travelled by an electron
  between successive collisions
• For an ideal crystal with no imperfections (or
  impurities) the MFP at 0 K is ?
• Ideal crystal ? there are no collisions and the
  conductivity is ?
• Scattering centres ? MFP? , ?? ? ?? , ??

Scattering centres
Thermal vibration ? Phonons
Sources ofElectron Scattering
Solute / impurity atoms
Defects
Dislocations
Grain boundaries
Etc.
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Thermal scattering

• At T gt 0K ? atomic vibration scatters electrons
  ? Phonon scattering
• ? T ? ? ? ? ? ? ?
• Low T ? MFP ?? 1 / T3 ? ? ?? 1 / T3
• High T ? MFP ?? 1 / T ? ? ?? 1 / T

Impurity scattering

• Resistivity of the alloy is higher than that of
  the pure metal at all T
• The increase in resistivity is ? the amount of
  alloying element added !

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Cu-Ni alloy
Increased phonon scattering
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Cu-3Ni
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Cu-2Ni
Resistivity (?) x 10-8 Ohm m ?
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Impurity scattering (?r)
2
1
With low density ofimperfections
Pure Cu
100
200
300
T (K) ?
? 0 as T? 0K
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Mattheissen rule
? ?T ?r
Net resistivity Thermal resistivity
Resistivity due to impurity scattering
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Applications
Conductors

• Power transmission lines ? low I2R loss ? large
  cross sectional area
• Al used for long distance distribution
  lines (Elastic ModulusAl increased by steel
  reinforcement)
• OFHC (Oxygen Free High Conductivity) Cu (more
  expensive) is used for distribution lines and
  busbars. ? Fe, P, As in Cu degrade conductivity
  drastically

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Electrical contacts

• Electrical contacts in switches, brushes and
  relays
• Properties ? High electrical conductivity ?
  High thermal conductivity ? heat dissipation
  ?High melting point ? accidental overheating
  ? Good oxidation resistance
• Cu and Ag used
• Ag strengthened by dispersion strengthening by
  CdO CdO ? Strengthens Ag ? Improves wear
  resistance ? If arcing occurs ? decomposes (At
  MP of Ag) to absorb the heat

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Resistor

• Properties ? Uniform resistivity ?
  homogenous alloy ? Stable resistance ? Avoid
  aging / stress relaxation / phase change ?
  Small T coefficient of resistance (??R) ?
  minimizes error in measurement ? Low
  thermoelectric potential wrt Cu ? Good
  corrosion resistance
• Manganin (87 Cu, 13 Mn, ?R 20 x 10?6 / K)
  and Constantan (60 Cu, 40 Ni) are good as
  resistor materials ?R (Cu) 4000 x 10?6 / K
• Low thermoelectric potential wrt to contact
  material (usually Cu) reduces error due to
  temperature difference between junctions. For
  high precision dissimilar junctions should be
  maintained at same temperature
• Ballast resistors are used in maintaining
  constant current ? I ? ? T ? ? R ? ? I ?
  Requriement high ?R (71 Fe, 29 Ni ? ?R 4500
  x 10?6 / K)

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Heating elements

• Properties ? High melting point ? High
  resistivity ? Good oxidation resistance ?
  Good creep strength ? Resistance to thermal
  fatigue ? low elastic modulus ? low
  coefficient of thermal expansion
• Upto 1300oC Nichrome (80 Ni, 20
  Cr), Kanthal (69 Fe, 23 Cr, 6 Al, 2 Co)
  Upto 1700oC SiC MoSi2 Upto 1800oC
  Graphite
• Mo and Ta need protective atmosphere at high T
• W (MP 3410oC) is used is used as filament in
  light bulbs ? creep resistance above 1500oC
  improved by dispersion hardening with ThO2
• Resistance thermometers ? High temperature
  coefficient of resistivity ? Pure Pt

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SUPERCONDUCTIVITY
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Superconducting transition
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10
Sn
Ag
Resistivity (?) x 10-11 Ohm m ?
Resistivity (?) x 10-11 Ohm m ?
10
5
?
5
10
0
10
0
Tc
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T (K) ?
T (K) ?
Superconducting transition temperature
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Current carrying capacity

• The maximum current a superconductor can carry
  is limited by the magnetic field that it
  produces at the surface of the superconductor

Hc / Jc
Normal
Jc Amp / m2 ?
?0 Hc Wb / m2 ?
Superconducting
T (K) ?
Tc
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Meissner effect

• A superconductor is a perfect diamagnet
  (magnetic suceptibility ? ?1)
• Flux lines of the magnetic field are excluded
  out of the superconductor ? Meissner effect

Superconducting
Normal
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Theory of low temperature superconductivity-
Bardeen-Cooper-Schreiffer (BCS) theory

• Three way interaction between an tow electron
  and a phonon
• Phonon scattering due to lattice vibrations felt
  by one electron in the Cooper pair is nullified
  by the other electron in the pair ? the
  electron pair moves through the lattice without
  getting scattered by the lattice vibrations
• The force of attraction between the electrons in
  the Cooper pair is stronger than the repulsive
  force between the electrons when T lt Tc

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Type I and Type II superconductors
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Type I (Ideal) superconductors

• Type I SC placed in a magnetic field totally
  repels the flux lines till the magnetic field
  attains the critical value Hc

Type I
?M ?
Normal
Superconducting
H ?
Hc
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Type II (Hard) superconductors

• Type II SC has three regions

Vortex Region Gradual penetration of the
magnetic flux lines
Type I
?M ?
Superconducting
Vortex
Normal
H ?
Hc1
Hc
Hc2
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• As type II SC can carry high current densities
  (Jc) they are of great practical importance
• The penetration characteristics of the magnetic
  flux lines (between Hc1 and Hc2) is a function
  of the microstructure of the material ?
  presence of pinning centres in the material
• Pinning centres ? Cell walls of high
  dislocation density (cold worked/recovery
  annealed) ? Grain boundaries (Fine grained
  material) ? Precipitates (Dispersion of very
  fine precipitates with interparticle spacing
  300 Å)
• Jc ? as Hc2 ?

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Potential Applications

• Strong magnetic fields ? 50 Tesla (without
  heating, without large power input)
• Logic and storage functions in
  computers Josephson junction ? fast switching
  times ( 10 ps)
• Magnetic levitation (arising from Meissner
  effect)
• Power transmission

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High Tc superconductivity
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Manufacture of YBa2Cu3O7-x
Please read from text book
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Crystal structure of YBa2Cu3O7?x
Y
Cu
O
Ba