LECTURE 1 DRUDE LORENTZ MODEL, Lecture notes of Solid State Physics

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TOPIC 1

FREE ELECTRON THEORY OF METAL

DRUDE-LORENTZ MODEL

Introduction: Drude Lorentz model

• Electrons free to move – behave like gas molecules
• Treat like ideal gases Ion core Free electron gas Free e model
• Valence electrons are free - > form a ‘gas’ of ‘non-interacting electrons
• Behave as independent electron
• Not affected by electron-ion interaction
• but subjected to Pauli principle
• But scattered by defect in lattice→ this limit its conductivity (relates with relaxation time approximation)

Drude-Lorentz model

• Metals have free electrons – thus show good electrical and thermal conductors
• Free e in metals obey Maxwell-Boltzmann statistic which is first used to explain the behavior of gas molecules
• 2 principles from the theory of gas:
  1. e moves at random and the mean square velocity is given by U^2 = Ux^2 + Uy^2 + Uz^2 = 3 Ux^2 = 3 Uy^2 = 3 Uz^2 Because of Ux^2 = Uy^2 = Uz^2
  2. Random kinetic energy of the e is given by the kinetic energy of the gas molecule K= 1 2 mv^2 = 3 2 mvx^2 = ⋯ = 3 2 kT Where T is the temperature of electron gas k is the Boltzman constant m is the electron mass

• Conducting materials contain finite density, N of mobile and free charge carriers at the microscopic level
• these free carriers could be electrons, positive or negative ions, or positive “holes”
• The free carrier can be model as m d𝐯 dt = q𝐄 − m 𝐯 τ Equation ( 1 ) Macroscopic friction force proportional to - v Note : Friction is a consequence of “collisions” of charge carriers with the neutral background at a frequency of ν = 1 τ collisions per unit time, and causes the decay of v as v (t)= v (0)exp(−t/τ) in the absence of field E q- carrier’s charge m- carrier’s mass qE- electrical force  - is the average time between collision

Assuming that mean velocity or the average drift velocity, < Vx( 0 ) > at t= 0 under influence of Ex When Ex is suddenly disconnect (Ex =0) , < Vx > → 0 exponentially < Vx(t) >= < Vx( 0 ) > exp(−t/τ) , τ is known as relaxation time Vx t Vx 0 t τ small τ large

• The relaxation time is the free time between collision with condition that the velocity after collision is random.
• The distance travel between collision is different, but on the average the electron travels a distance known as mean free path, l.
• Thus τ is the average time taken by the electron to travel distance l.
• τ depends on the velocity, u due to thermal kinetic energy, not on the drift velocity < Vx > τ = l u

l 3kT/m 1 /^2

K=

1 2 mv 2 = 3 2 kT Hence σ = ne^2 l 3kTm 1 /^2 If l is assumed to be constant then σ is inversely proportional to T 1 / 2 which is not agreement with the experiments

Thermal conductivity, K=

𝐶𝑣

v

Electron conductivity, σ = nq 2 τ m Apply classical ideal gas to evaluate electronic specific heat and mean square velocity : Cv =

nkB and average translational kinetic energy ,

mv

=

kBT Hence K σ = 3 2 kB e

T In 1850 Wiedemann and Franz have investigated experimentally metal and found that K σT =constant = 2.4 × 10

Js

K

Drude model match with the experimental finding by Wiedemann and Franz Wiedemann & Franz - > Themal conductivity electrical conductivity = constant

Success of Drude model:

1. Verify ohm’s law
2. Explain electrical and thermal conductivities
3. Derive the Wiedemann- Franz law Failure of Drude model:
4. At low temperature ,  and K are varied in different ways and

K

σT is not a constant at all T. Drude gives

K

σT constant at all T.

2. Cannot predict the electronic specific heat (experimental date is much smaller than predicted)
3. Electrical conductivity of semiconductor cannot be explained −due to the conclusion that ρα T, provided that number of electron per unit volume , n is independent on T