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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 10 -9 10 -7 10 -5 10 -3 10 -1 10 3 Ag Cu Al Au Ni Pb Sb Bi Graphite Ge (doped) GeSi 10 5 10 7 10 9 10 11 10 13 10 15 10 17 Window glass Ionic conductiv ity Bakelite Porcelain Diamond Rubber Polyethyl ene Lucite Mica PVC SiO 2 (pure) Metallic materials Insulators Semi-conductors
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Classification based on Conductivity Semi-metals Semi-conductors Metals 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 → Planck’s constant Wave number vector (k) Non relativistic
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↑ → k ↓ → E ↓ E → k → Discrete energy levels (Pauli’s exclusion principle)
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If the length of the box is L n → integer (quantum number) Number of electrons moving from left to right equals the number in the opposite direction Electron in an 1D box L Quantization of Energy levels
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In 3D Each combination of the quantum numbers n x, n y, n z 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 (E F ) is called the Fermi level If E F 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 > 0 K P(E) → E → Increasing T 0K
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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 E → k → EFEF EFEF Electric Field
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Force experienced by an electron m → mass of an electron E → applied electric field
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Velocity → time → vdvd Collisions 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 v d → Drift velocity → Average collision time
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The flux due to flow of electrons → Current density (J e ) n → number of free electrons ~ Ohm’s law
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Mean free path (MFP) (l) of an electron l = v d 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 Sources of Electron Scattering Solute / impurity atoms Defects Thermal vibration → Phonons Grain boundaries Dislocations Etc.
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Thermal scattering At T > 0K → atomic vibration scatters electrons → Phonon scattering T ↑ → ↓ → ↑ Low T MFP 1 / T 3 1 / T 3 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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Resistivity ( ) [x 10 -8 Ohm m] → T (K) → Cu-Ni alloy 100 200 300 1 2 3 4 5 Cu-2%Ni Cu-3%Ni → 0 as T→ 0K With low density of imperfections Pure Cu Increased phonon scattering Impurity scattering ( r )
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Mattheissen rule = T + r Net resistivity = Thermal resistivity + Resistivity due to impurity scattering
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Conductors Power transmission lines → low I 2 R loss → large cross sectional area Al used for long distance distribution lines (Elastic Modulus Al 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 1300 o C Nichrome (80% Ni, 20% Cr), Kanthal (69% Fe, 23% Cr, 6% Al, 2% Co) ■ Upto 1700 o C: SiC & MoSi 2 ■ Upto 1800 o C: Graphite Mo and Ta need protective atmosphere at high T W (MP = 3410 o C) is used is used as filament in light bulbs → creep resistance above 1500 o C improved by dispersion hardening with ThO 2 Resistance thermometers: ► High temperature coefficient of resistivity ► Pure Pt
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SUPERCONDUCTIVITY
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Resistivity ( ) [x 10 -11 Ohm m] → T (K) → 10 20 5 10 Ag Sn Resistivity ( ) [x 10 -11 Ohm m] → T (K) → 5 10 20 0 0 TcTc Superconducting transition temperature Superconducting transition ?
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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 0 Hc [Wb / m 2 ] → T (K) → TcTc H c / J c Normal Superconducting J c [Amp / m 2 ] →
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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 Normal Superconducting
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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 < T c
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Type I and Type II superconductors
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M → H → HcHc Normal Superconducting Type I 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 H c
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M → H → HcHc Normal Type I Type II (Hard) superconductors Type II SC has three regions Vortex Region Gradual penetration of the magnetic flux lines Super conducting H c1 H c2
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As type II SC can carry high current densities (J c ) they are of great practical importance The penetration characteristics of the magnetic flux lines (between H c1 and H c2 ) 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) J c ↑ as H c2 ↑
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Nb – 40%Ti alloy, T = 4.2 K, Magnetic field strength = 0.9 H c2 MicrosctructureJ c (A / m 2 ) Recrystallized10 5 Cold worked and recovery annealed10 7 Cold worked and precipitation hardened10 8
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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 T c superconductivity CompoundTcTc Comments Nb 3 Ge23 KTill 1986 La-Ba-Cu-O34 KBednorz and Mueller (1986) YBa 2 Cu 3 O 7-x 90 K> Boiling point of Liquid N 2 Tl (Bi)-Ba(Sr)-Ca-Cu-O125 K
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Manufacture of YBa 2 Cu 3 O 7-x Please read from text book
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Crystal structure of YBa 2 Cu 3 O 7 x Y Ba Cu O
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