Download PHY583 - Note 1e - Free Electron Theory of Metal

Lecture 1e - Free Electron Theory of Metal
FREE ELECTRON THEORY OF METAL
Classical Free Electron Model of Electrical Conduction in Metals
First proposed by Paul Drude in 1900.
Leads to Ohm’s law & show that resistivity can be related to the motion of electrons in metals.
The conduction electrons (free electrons) although bound to their respective atoms when the atoms
are not part of a solid, becomes free when the atoms condensed into a solid.
In the absence of an electric field, the conduction electrons move in random directions through the
conductor with average speeds on the order of 106 m/s.
The situation is similar to the motion of gas molecules confined in a vessel.
Some scientists refer to conduction electrons in a metal as an electron gas.
When an electric field is applied, the free electrons drift slowly in a direction opposite to the electric
field, with an average drift velocity (vd) that is much smaller (typically 104 m/s than their average
speed between collisions (typically 106 m/s).
Fig. 12.11 (a) Random successive displacements of an electron in a metal without E field. (b)
Combination of random displacement & displacement produced by an external E field.
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Lecture 1e - Free Electron Theory of Metal
Fig. 12.12 The connection between current density J & drift velocity vd. The charge that passes
through A in time dt is the charge contained in the small parallelepiped, neAvddt.
The force (F) due to the electric field (E) acting on an electron of charge e:
F = eE .....(i)
By applying Newton’s 2nd law to the motion of electrons with constant acceleration (because the
electric field is uniform):
F = me a .....(ii) ;
......(iii)
average time interval between successive collision also referred to as mean free time.
Random initial velocities ( = 0) of the electrons is ignored since this average to zero.
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Lecture 1e - Free Electron Theory of Metal
Using eqn. (i), (ii), and (iii) gives:
..............12.10
Current density (J):
..................12.12
n is the electrons per unit volume of the conductor.
The proportionality of J to E given by equation 12.12 shows that the classical free electron model
predicts the observed Ohm’s law dependence of J on E.
Ohm’s law: J =
Comparing eqn. 12.12 to Ohm’s law: conductivity
...................12.13
Therefore resistivity () in terms of microscopic quantities:
By using
, L= mean free path i.e. the average free path between successive collision:
Eqn. 12.13 becomes
.............12.14
The relationship between vrms & temperature T can be determined from the classical equipartition
theorem:
.................12.5
Substituting the Maxwell-Boltzmann rms thermal speed given by 12.5 into Eqn. 12.14 we have:
The conductivity
The resistivity
.................12.15
.............12.16
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Lecture 1e - Free Electron Theory of Metal
Eqn. 12.15 & 12.26 represent the classical expressions for conductivity & resistivity.
Problems with the Classical Model
1. Although the classical electron gas model does predict Ohm’s law, Eg. 12.1(p.418) shows
that the value of conductivity calculated by using this model differ from the measured value
by an order of magnitude.
(
less than
by a factor of 10)
2. The measured resistivity of most metals is found to be proportional to the absolute
temperature
(as shown in Fig. 12.13) yet the classical gas model incorrectly predicts a
much weaker dependence
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Lecture 1e - Free Electron Theory of Metal
Quantum Theory of Metals
The deficiencies of the classical electron gas model -- in the numerical values of  & the temperature
dependence of  -- can be rectified by:
1. Replacing the Maxwell-Boltzmann distribution with the Fermi-Dirac distribution for the
conduction electrons in the metal.
2. Calculating the electron mean free path while explicitly taking into account the wave nature
of the electron.
Since the quantum mechanical calculations of electron mean free path is complicated, we shall rely
on qualitative arguments & inferences from measured quantities to give physical insight into the
surprising transparency of metals to conduction electrons.
Replacement of vrms with vF
Fig. 12.14 shows the velocity distribution in 3 dimensions of conduction electrons in the metals.
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Lecture 1e - Free Electron Theory of Metal
Fig. 12.14 (a) Allowed velocity vectors, or positions in velocity space, of conduction electrons in a
metal without an applied electric field. (b) The net effect of an applied electric field is a small
displacement of the Fermi sphere, the displace electrons having the speed of  vF.
Essentially all the electrons have velocities within the radius
in velocity space
The net effect of the E field is:
1. to leave an intact core of states for electrons with E < EF and
2. to produce a displacement of those electrons near the Fermi surface having v  vF.
Therefore only electrons with v  vF are free to move & participate in the electrical conduction:
Replacing the Maxwell-Boltzmann rms speed with the Fermi speed:
...................12.22
Quantum Mean Free Path of Electrons
Changes from the classical values of electron mean free path arise from the wave properties of the
electrons.
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Lecture 1e - Free Electron Theory of Metal
This discrepancy implies that we are using the wrong value for L & that the scattering sites for
electrons are not adjacent ion cores but more widely separated scattering centres.
We can account for the unexpectedly long electron mean free path by taking the wave nature of the
electron into account.
Quantum mechanical calculations shows that electron waves with a broad range of energies can
pass through a perfect lattice of ion cores
1. unscattered,
2. without resistance, and
3. with an infinite mean free path.
The actual resistance of a metal is due to
1. the random thermal displacements (thermal vibrations) of ions about lattice points
2. other deviations from a perfect lattice such as impurity atoms & defects that scatter electron
waves.
The lack of electron scattering by a perfect lattice can be understood by noting that the electron
wave generally travels through unattenuated, just as does light through a transparent crystal.
Strong reflections of electron waves are set up only for specific electron energies, and when this
occurs, the electron waves cannot travel freely through the crystal.
These strong reflections occur when the lattice spacing is equal to an integral number of electronic
wavelengths, resulting in a discrete set of forbidden energy bands for electrons.
From a classical point of view, the observed proportionality of resistivity to absolute temperature at
high temperature is the result of the scattering of electrons by lattice ions vibrating with larger
amplitude at the higher temperature.
From a quantum viewpoint, lattice vibrations have a quantized energy
where  is the angular
frequency of vibration of the lattice ions. These quantized lattice vibrations are called phonons, &
for purpose of calculation, the vibrating lattice ions are replaced by phonons.
The number of phonons with energy
that are available at temperature T is denoted as nP &
proportional to Bose-Einstein distribution function:
At higher temperature kBT >> than
& the above eqn. Becomes:
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Lecture 1e - Free Electron Theory of Metal
Hence, the number of phonons available to scatter electrons is directly proportional to T.
Since
In addition to temp – dependent part of resistivity, there is also a temp independent contribution to
the resistivity of metal which manifest clearly at T  10 K.
This residual resistivity
, which stays as T
, is produced by electron waves scattering from
impurities & structural defects (imperfections) in a given samples as illustrated in Fig. 12.25.
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Lecture 1e - Free Electron Theory of Metal
The parallel nature of the curves in Fig. 12.15 implies:
1. the resistivity caused by thermal motion of the lattice,
, is independent of the impurity
concentration,
2. resistivity is independent of temperature.
This result is formalized in Matthiessen’s Rule which state that:
The resistivity of a metal may be written in the form
...................(12.28)
Where
depends only the concentration of crystal imperfections and
depends only on temperature.
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