제 4 장 Metals I: The Free Electron Model Outline 4.1 Introduction 4.2 Conduction electrons 4.3 the free-electron gas 4.4 Electrical conductivity 4.5 Electrical resistivity (T) 4.6 heat capacity of conduction electrons 4.7 Fermi surface 4.8 Electrical conductivity: effects of the Fermi surface 4.9 Thermal conductivity in metals 4.10 Motion in a magnetic field; cyclotron resonance and the Hall effect 4.11 AC conductivity and optical properties 4.12 Thermionic emission 4.13 failure of the free-electron model

4.1 Introduction Metals in life from ancient to future. Example: Duralumin (Cu, Mn, and Mg) in Car, Cu in electrical wire, Ag and Au as jewelry Common properties: physical strength, density, electrical and thermal conductivity Optical (visible) reflectivity Explanation of many physical properties by assuming cloud of free-electrons all over sample Most of optical properties can be explained but with a limit.

4.2 Conduction electrons Na gas; a simple collection of free atoms. Each atom has 11 electrons orbiting around the nucleus. 11 Na: 1s 2 2s 2 2p 6 3s 1 1 valence el. loosely bound to the rest of the system(=Na 1+ ). can explain ordinary chemical reaction. Na atoms to form a metal bcc structure (section 1.7) d = 3.7  Å Core electrons Valence electron d

4.2 Conduction electrons (p138) Na atoms to form a metal. bcc structure (section 1.7) two atoms overlap slightly  valence el. can hop from one to the neighboring atom  move around freely all over the crystal.  conduction electrons in a crystal !! Naming since they carriers electrical current in an electric field. Cf) core electrons do not give electrical current. N=concentration M’=atomic weight, Z = atomic valence (1+, 2+,..),  m =density of the substance Fig. 4.1 overlap of the 3s in solid sodium

4.3 Free electron gas (p140) e cond. is completely free, except for a potential at the surface. Free motion except occasional reflection from the surface which confines e cond.. Like gas!! So called free electron gas! How about interaction between 1) e cond. and e core.? a)Strong coulomb attraction between e cond. and core ion is compensated by repulsive potential due to array of core ion by quantum effect. b)e cond. spend tiny time near core ion. c)Short-range screened coulomb potential rather than a long-range pure coulomb potential.

interaction between 2) conduction electrons themselves? Pauli exclusion principle:  =  1 (a)+  2 (b) 1)electrons of parallel  or  spins tend to stay away. 2)electron pair having opposite spin  tends be stay way from each other to minimize the energy of the system. Each electron is surrounded by a hole” The hole with a radius of about 1 Å. The hole move with each electron. Interaction between two specific electron is always screened by the other electrons.  very low interaction between the two electrons. Free electron gas vs ordinary gas 1)charged, i.e. like plasma 2)N is very large N~10 23 cm -3 while ordinary gas ~ cm -3 Jellium model : ions form a uniform jelly into which e cond. move around

4.4 electrical conductivity (p142) Law of electrical conduction in metal : Ohm’s law J: current density per area, E=electrical field Substitute into (4.2) Substitute J and E (4.3) and (4.4) Electric field accelerates e cond. (not ions in the crystal) thus electric current.

Newton’ equation with friction force with  = collision time. Collision and friction tends to reduce the velocity to zero. Final (or steady) state solution. Terminal velocity (drift velocity) vs. random velocity 4.4 electrical conductivity (p142)

Newton’ equation with friction force with  = collision time. Collision and friction tends to reduce the (average vector)velocity to zero. Terminal velocity (drift velocity) vs. random velocity J: current density per area is proportional to E field,

dt A

 = collision time, mean free lifetime?  ~ sec

metal semiconductor

 = relaxation time?  ~ sec  vs. mean free path and random velocity

Origin of collision time l ≈ m ≈ 10 2 Å Quantum mechanics, deBroglie relation: A light wave traveling in a crystal is not scattered at all. The ions form a perfect lattice, no collision at all, that is l =∞ and hence τ≈∞, leads to infinite conductivity. l ≈ 10 2 Å. d≈ 4Å 25x unit cell