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 modelDetermine the valence of an atomDetermine the density of conduction electronsCalculate the electrical conductivity of a metalEstimate and interpret the relaxation time

Models of Atoms

Drude ModelZa = atomic number = # of electrons

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 ionValence electrons wander away from their parent atoms, called conduction electrons

isolated atom

in a metal

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)

Core electronsMetallic 11Na, 12Mg and 13Al are

assumed to have 1, 2 and 3 mobile electrons per atom respectively.

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.

Valence vs. Oxidization

Fe: 1s2 2s2 2p6 3s2 3p6 4s2 3d6

Number of outer electrons = 6The 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

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

How the mobile electrons become mobile

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.

+

+ + +

+ +

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.

Drude Model: A gas of electrons

X

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)

Classical (Drude)ModelDrude 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

TkEMB

Bef /~

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 = 6.022 x 1023 Z m /A

Group: Find n for Copper

n = N/V = 6.022 x 1023 Z m /A

Density of copper = 8.96 g/cm3

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

nN

Vrs 1

3

4 3

3/1

3/11

~4

3

nnrs

mean inter-electron spacing

in metals rs ~ 1 – 3 Å (1 Å= 10-8 cm) rs/a0 ~ 2 – 6

529.02

2

0 me

a

Å – Bohr radius

Effective Radius

Dr. Jie Zou

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)

v

Current: (Amps)dq

dti

q tidi

R

V

Macroscopic Microscopic

where resistivity

conductivity

EEJ

EEEEEEEEEEEEEE

“Quasi-Classical” Theory of Transport

Charge

Ohm’s Law

Current Density: J = I/A

= 1/

where resistivity

conductivity

EEJ

EEEEEEEEEEEEEE

Finding the Current Density

Current Density: J = I/A

Vd = drift velocity or average velocity of conduction electrons

Applying an Electric Field E to a Metal

F = ma = - e E, so acceleration a = - e E / m

Integrating gives v = - e E t / mSo 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

Needed from problem 1.2

J = E = - n e vavg = - n e (- e E / m)

Then conductivity = n e2 / m ( = 1/)

And = m / ( n e2) ~10-15 – 10-14 s

Calculating Conductivity

vavg = - e E / m

Group ExerciseCalculate the average time between collisions

for electrons in copper at 273 K.

Assuming that the average speed for free electrons in copper is 1.6 x 106 m/s calculate their mean free path.

= m / ( n e2)

where resistivity

conductivity

EEJ

EEEEEEEEEEEEEE

vavg = x/

Resistivity vs Temperature not accounted for in Drude model

Resistivity is temperature dependent mostly because of the temperature dependence of the scattering time .

• 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.

1020-

1010-

100 -

10-10-R

esis

tivi

ty (

Ωm

)100 200 3000

Temperature (K)

Diamond

Germanium

Copper

= m / ( n e2)

Metal

Semiconductor

Insulator

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)

From the opposite direction