Free Electron Model

So far: –How atoms stick together (bonding) –Arrangement of atoms (crystal lattice) – Reciprocal lattice –Motion of atomic cores (phonons) Now: –Effects arising from electrons (especially bonding electrons)

Drude Model Start with simplest of models Only 3 years after discovery of electron (1897), Paul Drude built model that explained several properties of metals Assumed valence e - are free to move, applied principles from kinetic theory of gases

Drude model can address the following metallic properties: –dc electrical conductivity –Thermal conductivity –Hall effect (behavior in uniform magnetic field) –ac electrical conductivity Primary assumptions of the model: 1)Electrons in metals behave like particles in gas the background of the ions keep the gas neutral

2) Thermalisation is due to collisions, Collisions are instantaneous 3) Probability of collision is dt/  -  is the relaxation time or collision time or mean free time - Assume  ≠  (r e, p e ) 4) Collisions cause e - to emerge in thermal equilibrium at that particular point - velocity after is unrelated to velocity before - velocity after is random in direction, with speed a random statistical function of local T

Note: We are not attempting to understand mechanism determining  (explanations of which are most troublesome and can be considered weakest link of theory) Drude model is strongest for effects that are independent of the value of 

Ohm’s Law-DC conductivity V: voltageL: length I: currentA: area

DC Electrical Conductivity j is flux of e - across section In a field E, the free e - are pulled in opposite direction and pick up speed n: electron density

F: forceE: electrical field e: electron charge  : conductivity

Ohm’s Law V: voltageL: length I: currentA: area

From estimates of n, measurements of , we find typical value of  = sec Relaxation time, , determined by collisions –Can be with phonons or lattice imperfections (e.g. impurities or defects)

Matthiessen Rule 1/  is probability of scattering Probabilities add So  = (m/ne 2 )(1/  ) means residual, ideal specimen indep. of T  ph =  ph (T)

Known as the Matthiessen rule At low T, certain deviations from strict Matthiessen rule are known (e.g. Kondo effect – scattering from magnetic moments of impurities)

Thermal Conductivity Weidmann Franz law  is the thermal conductivity,  – the electrical conductivity

Thermal Conductivity Weidmann Franz law

Phonon thermal conductivity can be comparable to e - contribution in metals –In pure metal  el is dominant at all T –In impure metals or disordered alloys, l is reduced and  ph may be comparable Ratio of Thermal to Electrical Conductivity in Metals

Values of  for various metals 0 C100 C Ag x W  /K 2 Au Cd

Hall Effect Uniform Magnetic Field Current in x direction Uniform H field in z  Lorentz force in negative y direction H j e-e-

Deflected e - build up on wall, creating field in y direction that opposes further build up We expect the field to be proportional to current j x and field H z Call constant of proportionality R H (known as Hall coefficient) Insert Lorentz force into momentum equation for f(t)

In steady state, dp/dt = 0 so p x, p y independently satisfy:

Multiply by (  ne/m)

After charge build up, j y = 0 Therefore Hall coefficient is Depends only on n, carrier density of metal

What does experiment say? In fact, contrary to simplistic theory, R H can depend on H,T and even sample preparation Also, some R H are positive However, at low T, high H and good sample, R H values do approach a limiting value

R H for some metals: MetalValence-1/R H n (at) e Li Na K Cu Ag Mg Al In To understand positive values of R H, need quantum theory of solids Metal

AC Electrical Conductivity For periodic (oscillating) E field In equation of motion We look for solutions E , p  are complex Real and imaginary parts must hold

1)As   0 (dc), we recover dc result 2)To explore effects of propagation of EM radiation in metals, use in Maxwell’s equations

For  <  p,   is real and negative Solutions decay exponentially in space, i.e. no propagation of radiation For  >  p, solutions are oscillatory For metals, radiation is absorbed, material opaque up until  p and then material becomes transparent

In fact, alkali metals become transparent in the UV Again, only free material parameter in  p is n, so we can link resistivity to onset of transparency. For alkali metals, agreement is not bad: of onset of transparency TheoryExperiment Li150 nm200 nm Na K Rb Cs For other metals, agreement is poorer.

Despite some impressive successes, the Drude model is inaccurate in several other cases For example, e - contribution to specific heat is 100X less at RT than predicted classical value of C v = 3/2nk B Must turn to quantum statistics: Sommerfeld Theory