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 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

3. Classification based on Conductivity Semi-metals Semi-conductors Metals 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 → k → Discrete energy levels (Pauli’s exclusion principle)

7. 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

8. 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

9. 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

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

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 E → k → EFEF EFEF Electric Field

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

13. 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

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

15. 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.

16. 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 !

17. 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 )

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

19. 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

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  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

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 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

23. SUPERCONDUCTIVITY

24. 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 ?

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  0 Hc [Wb / m 2 ] → T (K) → TcTc H c / J c Normal Superconducting J c [Amp / m 2 ] →

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 Normal Superconducting

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 < T c

28. Type I and Type II superconductors

29.  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

30.  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

31.  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 ↑

32. 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

33. 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

34. 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

35. Manufacture of YBa 2 Cu 3 O 7-x Please read from text book

36. Crystal structure of YBa 2 Cu 3 O 7  x Y Ba Cu O