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CONDUCTIVITY

CONDUCTIVITY. Conductivity Superconductivity. Electronic Properties Robert M Rose, Lawrence A Shepart, John Wulff Wiley Eastern Limited, New Delhi (1987). Resistivity range in Ohm m  25 orders of magnitude. Semi-conductors. Metallic materials. Insulators. Metals. Semi-metals.

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CONDUCTIVITY

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  1. CONDUCTIVITY • Conductivity • Superconductivity Electronic Properties Robert M Rose, Lawrence A Shepart, John Wulff Wiley Eastern Limited, New Delhi (1987)

  2. Resistivity range in Ohm m  25 orders of magnitude Semi-conductors Metallic materials Insulators

  3. Metals Semi-metals Classificationbased on Conductivity Semi-conductors Insulators

  4. 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

  5. Wave particle duality of electrons •  → de Broglie wavelength • v → velocity of the electrons • h → Planck’s constant Wave number vector (k) Non relativistic

  6.  ↑ → k ↓ → E ↓ E → Discrete energy levels (Pauli’s exclusion principle) k→

  7. 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

  8. 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

  9. 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

  10. T > 0 K 0K P(E) → Increasing T E →

  11. 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 ElectricField EF EF E → k→ k→

  12. Force experienced by an electron • m → mass of an electron • E → applied electric field

  13. Collisions vd Velocity →  time → • 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

  14. The flux due to flow of electrons → Current density (Je) • n → number of free electrons ~ Ohm’s law

  15. 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.

  16. Thermal scattering • At T > 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 !

  17. Cu-Ni alloy Increased phonon scattering 5 Cu-3%Ni 4 Cu-2%Ni Resistivity () [x 10-8 Ohm m] → 3 Impurity scattering (r) 2 1 Pure Cu With low density ofimperfections 100 200 300 T (K) → → 0 as T→ 0K

  18. Mattheissen rule  = T + r Net resistivity = Thermal resistivity + Resistivity due to impurity scattering

  19. 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

  20. 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 • Cuand 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

  21. 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)

  22. 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

  23. SUPERCONDUCTIVITY

  24. Superconducting transition 20 10 Sn Ag Resistivity () [x 10-11 Ohm m] → Resistivity () [x 10-11 Ohm m] → 10 5 ? 5 0 10 0 Tc 10 20 T (K) → T (K) → Superconducting transition temperature

  25. 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

  26. 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

  27. Theory of low temperature superconductivity- Bardeen-Cooper-Schreiffer (BCS) theory • Three way interaction between an two 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 < Tc

  28. Type I and Type II superconductors

  29. 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

  30. 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

  31. 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↑

  32. 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

  33. High Tc superconductivity

  34. Manufacture of YBa2Cu3O7-x Please read from text book

  35. Crystal structure of YBa2Cu3O7x Y Cu O Ba

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