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2 4 Chapter 7 in the textbook

3 7.1 Introduction and Survey Current density:

4 Understanding of electrical conduction As Drude did, A free electron gas or plasma ; valence electrons of individual atoms in a crystal What is plasma? Without electric field: the electrons move randomly so that no net velocity results. Thermal velocity: 7.2 Conductivity-Classical Electron Theory For a monovalent metal, : # of atoms / cm 3 : Avogadro constant (#/mole)  : density (gram/cm 3 ) M : atomic mass of element (gram/mole) One calculates about to atoms per cubic centimeter, i.e., to free electrons per cm 3 for a monovalent metal.

5 If E is applied 1)Acceleration, 2)Constant velocity after the field is removed Free electron model should be modified. For a free electron (a)

6 In a crystal, collision Drift velocity Friction force (or damping force) Equation of Motion

7 For the steady state Where,  is a relaxation time: average time between two consecutive collisions Final drift velocity

8 Mean free path between two consecutive collisions N f : number of free electrons

9 Without fieldWith an electric field Fermi velocity At equilibrium, no net velocityWhat difference between Class. and QM 7.3 Conductivity-Quantum Mechanical Considerations The maximum velocity that the electrons can have is the Fermi velocity (i.e., the velocity of electrons at the Fermi energy) Only specific electrons participate in conduction and that these electrons drift with a high velocity (v F )

10 Fig.7.4 Population density We now calculate the conductivity by quantum mechanical means and apply, as before, Ohm’s law.

11 Population density at E F Fig.7.5

12 For a spherical Fermi surface, vsvs Fermi velocity, relaxation time, population density at Fermi energy Fig.7.6 Since,  = j/E Classical expression Contains the information that not all free electrons are responsible for conduction, i.e., the conductivity in metals depends to a large extent on the population density of the electrons near the Fermi surface. E M : monovalent metals E B : bivalent metals E I : Insulators

For Pure Metals Linear temperature coefficient of resistivity Ideal resistivity Matthiessen’s rule 7.4 Experimental Results and Their Interpretation Residual resistivity

14 Atoms of different size cause a variation in the lattice parameters local charge valence alter the position of the Fermi energy. Linde’s rule ∝ (  valence electrons) 1/2 For dilute single-phase alloys For alloys

Ordering vs Disordering (Nordheim’s rule) Two phase mixture:

Superconductivity High T c superconductors (T c >77K) 77K: boiling point of liquid nitrogen 20K: boiling point of liquid hydrogen 4K: boiling point of liquid helium

Experimental Results m a : atomic mass  material constant Type I superconductors Ceramic superconductors usually have a smaller H c than metallic super conductors, i.e., they are more vulnerable to lose superconductivity by a moderate magnetic field. Ellimination of the superconducting state also occurs by subjecting the material to a strong magnetic field.

18 Type II superconductors: Transition metals and alloys consisting of Nb, Al, Si, V Pb, Sn, Ti, Nb 3 Sn, Nb-Ti, ceramic superconductors Contains small circular regions called vortices or fluxoids, which are in the normal state with a mixed superconducting and normal conducting area. Flux quantum: The interval represents a state in which superconducting and normal conducting areas are mixed in the solid.

Theory Postulate superelectrons that experience no scattering, having zero entropy (perfect order), and have long coherence lengths. BCS theory: Cooper pair (pair of electrons that has a lower energy than two individual electrons) Electrons on the Fermi surface having opposite momentum and opposite spin forms cooper-pair These electrons form a cloud of Cooper pairs which drift cooperatively through the crystal. The superconducting state is an ordered state of the conduction electrons eV

Thermoelectric Phenomena Thermoelectric power, or Seebeck coefficient Peltier effect: a direct electric current that flows through the junctions made of different materials causes one junction to be cooled and the other to heat up. Contact potential: metals with different E F Explain the band diagram of metals to explain the contact potential (based on workfunction difference)