Electrical properties of materials

1. Electrical properties of materials

2. Free electron theory • Only kinetic energy considered • Independent electron approximation • Free electron approximation • Pauli principle • Fermi-Dirac statistics electron density, charge, mass, relaxation time Chemical potential electric field electrical current • Ohm’s law • Hall effect • Thermal conductivity (metals)

3. Energy Bands • Periodic potential due to nuclei in the solid • Bragg diffraction of electron waves • Forbidden energy gap opens up in the energy bands Electrons in a periodic potential Free electrons Energy gap

4. Electrical conductivity, • = number density, charge, mobility • of current carriers

5. Metals • Free charge carriers • High conductivity • Conductivity decreases with increasing temperature • mobility of charge carriers decreases • Resitivity of gold • Resitivity (10-8  m) • 0 2 4 6 8 • 0 200 400 600 800 1000 • Temperature (K) Courtsey: http://hypertextbook.com/facts/2004/JennelleBaptiste.shtml

6. Semiconductors • Activated charge carriers • Conductivity increases with increasing temperature • number of charge carriers increases • mobility of charge carriers decreases • Slope = Eg

7. Extrinsic (Impurity) semiconductors electrons donor level acceptor level holes n - type p - type thermistors, photoconductors p-n junction  diodes  transistors

8. Superconductors Kamerlingh Onnes, 26 October 1911

9. Critical temperature (Tc) • zero resistance, persistent current • perfect diamagnetism (Meissner effect) • critical field (Hc) • Electron-phonon coupling (BCS theory) • Examples: Hg [Tc ~ 4 K], Pb [Tc = 8 K], Nb3Sn [Tc ~ 23 K] • High-Tc superconductors (YBa2Cu3O7-x [Tc = 90 K] • High field magnets • SQUID • Magnetic levitation

10. Problem Set • At 0 K,  = F for free electrons in a metal. Demonstrate this using Fermi-Dirac statistics. • Heat capacity of free electron gas is about 1% of that expected on the basis of the law of equipartition of energy. Why ? • What are the basic assumptions of free electron theory ? What are the phenomena explained by it ? And what were its main failures ? • Show that the Hall voltage, Ey = -eBEx/mc in a rectangular bar sample when Bz = B, Bx = By = 0 and current is only in the x direction. • The Weidemann-Franz law is known to fail at low temperatures. Suggest a possible explanation. • Given the Bloch function, k(r) = uk(r).eikr, obtain the eigenvalue of the crystal translation operation T (ie. translation through a lattice vector, T). • For a simple cubic lattice, show that the kinetic energy of the free electron at a corner of the first Brillouin zone (ie. having wave vector at this point) is higher than that of an electron at the midpoint of a face of the zone. What bearing does this have on the conductivity of divalent metals ? • MnO is experimentally found to be a semiconductor. Draw the qualitative band diagram for MnO; is it expected to be a semiconductor on the basis of this band picture ? Suggest a possible explanation if there is a conflict. • Slater antiferromagnetic ordering which causes metal-insulator transition, induces the doubling of the unit cell. What experimental technique would be appropriate to detect such a transition ? • Calculate the number density of conduction electrons and holes in pure Ge at 300 K (assume, me = mh; Eg = 0.67 eV). • Qualitatively explain the origin of isotope effect in superconductors. • Give qualitative explanations for the entropy (S) and free energy (G) variations with temperature of the superconducting (S) and normal (N) states (see figure below). Suggest how the data for the normal states could be obtained below TC.