1. Metals: Drude Model and Conductivity (Covering Pages 2-11) Objectives
By the end of this section you should be able to:
Understand the basics of the Drude model
Determine the valence of an atom
Determine the density of conduction electrons
Calculate the electrical conductivity of a metal
Estimate and interpret the relaxation time

2. Models of Atoms
What are some differences you notice between these bottom two atoms?
1s2 = 2
2s2 2p6 = 8
3s2 3p6 3d10 = 18

3. Drude Model Za = atomic number = # of electrons isolated atom
in a metal
Z valence electrons are weakly bound to the nucleus (participate in reactions)
Za – Z core electrons are tightly bound to the nucleus (less of a role)
In a metal, the core electrons remain bound to the nucleus to form the metallic ion
Valence electrons wander away from their parent atoms, called conduction electrons

4. Why mobile electrons appear in some solids and not others?
According to the free electron model, the valance electrons are responsible for the conduction of electricity, thus termed conduction electrons.
Na11 ￫ 1s2 2s2 2p6 3s1
This valance electron, which occupies the third atomic shell, is the electron which is responsible chemical properties of Na.
Valence = Maximum number of bonds formed by atom
Valance electron (loosely bound)
Metallic 11Na, 12Mg and 13Al are assumed to have 1, 2 and 3 mobile electrons per atom respectively.
Core electrons

5. Group: Find the Valence
What is the valency of iron, atomic Number 26? 1s2 2s2 2p6 3s2 3p6 4s2 3d6 Number of outer electrons = 6 Iron would like to suck up other electrons to fill the shell, so it has a valence of 4 when elemental.
Have them do

6. Valence vs. Oxidization
Fe: 1s2 2s2 2p6 3s2 3p6 4s2 3d6
Number of outer electrons = 6
The removal of a valance electron leaves a positively charged ion.
8O: 1s2 2s2 2p4 often steals two e-s -> O2-
Note: oxidization state (e.g. 2- in O) is not the valence, but can be used in combination with the original valence to find the new valence
Have them do
How many bonds in Mn4+?

7. How the mobile electrons become mobile
When we bring Na atoms together to form a Na metal, the orbitals overlap slightly and the
valance electrons become no longer attached to a particular ion, but belong to both.
A valance electron really belongs to the whole crystal, since it can move readily from one ion to its neighbor and so on.
Na metal

8. Drude Model: A gas of electrons
Drude (~1900) assumed that the charge density associated with the positive ion cores is spread uniformly throughout the metal so that the electrons move in a constant electrostatic potential.
All the details of the crystal structure is lost when this assumption is made. X + + + + + +
According to both the Drude and later the free electron model, this potential is taken as zero and the repulsive force between conduction electrons are also ignored.

9. Drude Assumptions
Electron gas free and independent, meaning no electron-electron or electron-ion interactions
If field applied, electron moves in straight line between collisions (no electron-electron collisions). Collision time constant. Velocity changes instantaneously.
Positives: Strength and density of metals
Negative: Derivations of electrical and thermal conductivity, predicts classical specific heat (100x less than observed at RT)

10. Classical (Drude)Model
Drude developed this theory by considering metals as a classical gas of electrons, governed by Maxwell-Boltzmann statistics
Applies kinetic theory of gas to conduction electrons even though the electron densities are ~1000 times greater
The expected number of particles
Took away: Drude introduced conduction electron density n=N/V, V=volume of metal, N = Avogadro’s #

11. Electron Density n = N/V
6.022 x 1023 atoms per mole
Multiply by number of valence electrons
Convert moles to cm3 (using mass density m and atomic mass A)
n = N/V = x 1023 Z m /A
Density = m/V, we want 1/V to go with the N from valence times Avogadro’s constant, so
1/V = density/mass

12. Group: Find n for Copper
Density of copper = 8.96 g/cm3
Gives very close to data value in book: 8.49x1022/cm3
n = N/V = x 1023 Z m /A

13. Effective Radius
density of conduction electrons in metals ~1022 – 1023 cm-3
rs – measure of electronic density = radius of a sphere whose volume is equal to the volume per electron
mean inter-electron spacing
Bohr radius = mean radius of the orbit of an electron around the nucleus of a hydrogen atom at its ground state
Draw box with lots of circles in it on the board
in metals rs ~ 1 – 3 Å (1 Å= 10-8 cm) rs/a0 ~ 2 – 6
Å – Bohr radius

14. A model for electrical conduction
Drude model in 1900
In the absence of an electric field, the conduction electrons move in random directions through the conductor with average speeds ~ 106 m/s. The drift velocity of the free electrons is zero. There is no current in the conductor since there is no net flow of charge.
When an electric field is applied, in addition to the random motion, the free electrons drift slowly (vd ~ 10-4 m/s) in a direction opposite that of the electric field.
(a)
(b)
Dr. Jie Zou

15. “Quasi-Classical” Theory of Transport
Microscopic
Macroscopic
Charge
Current  potential drop
Current Density: J = I/A
Ohm’s Law
 = 1/

16. Finding the Current Density
Vd = drift velocity or average velocity of conduction electrons
Current  potential drop
Current Density: J = I/A

17. Applying an Electric Field E to a Metal
Needed from problem 1.2
F = ma = - e E, so acceleration a = - e E / m
Integrating gives v = - e E t / m
So the average vavg = - e E  / m, where  is “relaxation time” or time between collisions
vavg = x/, we can find the xavg or mean free path 

18. Calculating Conductivity
vavg = - e E  / m
J =  E = - n e vavg = - n e (- e E  / m)
Then conductivity  = n e2  / m ( = 1/)
And  = m / ( n e2) ~10-15 – s

19. Group Exercise  = m / ( n e2)
vavg = x/
Calculate the average time between collisions  for electrons in copper at 273 K.
Time is 2.7 x 10^-14 s
Mean free path = 43 nm
Assuming that the average speed for free electrons in copper is 1.6 x 106 m/s calculate their mean free path.

20. Resistivity vs Temperature not accounted for in Drude model
1020-
1010-
100 -
10-10-
Resistivity (Ωm)
100
200
300
Temperature (K)
Diamond
Germanium
Copper
Insulator
Resistivity is temperature dependent mostly because of the temperature dependence of the scattering time .
Semiconductor
Metal
 = m / ( n e2)
Insulators have high resistance.
In Metals, the resistivity increases with increasing temperature.
The scattering time  decreases with increasing temperature T (due to more phonons and more defects), so as the temperature increases ρ increases (for the same number of conduction electrons n)
We will find the opposite trend occurs in semiconductors but we will need more physics to understand why.

21. In Terms of Momentum p(t)
Momentum p(t) = m v(t)
So j = n e p(t) / m
Probability of collision by dt is dt/
p(t+dt)=prob of not colliding*[p(t)+force]
(1-dt/)
= p(t)–p(t) dt/ + force +higher order terms
p(t+dt)-p(t) =–p(t) dt/ + force +higher order
Or dp/dt = -p(t)/ + force(time)
Relate to newton’s second law F=dp/dt when no damping force
Thus, collisions effectively just cause a damping force

22. From the opposite direction