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EEE 315 - Electrical
Properties of Materials
Lecture 3
The Drude Model
 The Drude model of electrical
conduction was proposed in 1900 by
Paul Drude to explain the transport
properties of electrons in materials.
 The model assumes that the
microscopic behavior of electrons in a
solid may be treated classically and
looks much like a pinball machine,
with a sea of constantly jittering
electrons bouncing and re-bouncing
off heavier, relatively immobile
positive ions.
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Drude Model electrons (shown here in
blue) constantly bounce between
heavier, stationary crystal ions (shown
in red).
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The Drude Model:
Assumptions
 The Drude model considers the metal to be formed of a mass of
positively-charged ions from which a number of "free electrons" were
detached. These may be thought to have become delocalized when
the valence levels of the atom came in contact with the potential of
the other atoms.
 The Drude model neglects any long-range interaction between the
electron and the ions or between the electrons. The only possible
interaction of a free electron with its environment is via
instantaneous collisions. The average time between subsequent
collisions of such an electron is 𝝉, and the nature of the collision
partner of the electron does not matter for the calculations and
conclusions of the Drude model.
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The Drude Model
 The two most significant results of the Drude model are
an electronic equation of motion
 And a linear relationship between current density, J and
electric field, E
 Where t is the time and p, q, n, m, and 𝝉 are respectively
an electron's momentum, charge, number density, mass,
and mean free time between ionic collisions.
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Drude Conductivity
Ohm’s “Law”: V = IR
The Resistance R is a property of the conductor (e.g. a wire) which
depends on its dimensions, V is a voltage drop & I is a current. In
microscopic physics, it is more common to express Ohm’s “Law” in
terms of a dimension-independent conductivity (or resistivity) which
is intrinsic to the material the wire is made from.
In this notation, Ohm’s “Law” is written
E = rj or j = sE (1)
Here, E = the electric field, j = the current density, r = the resistivity
& s = the conductivity of the material.
Consider n electrons per unit volume, all moving in the direction of
the current with velocity v.
The number of electrons crossing area
A in time dt is nAvdt
A
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The charge crossing A in dt is -nevAdt, so
j = -nev. (2)
In the real material, we expect the electrons to be moving randomly
even in zero electric field due to thermal energy. However, they will
have an average, or drift velocity along the field direction.
(3)
vdrift = -eEt / m
This comes from integrating Newton’s 2nd Law over time t.
This is the velocity that must be related to j.
Combining (2) & (3) gives
j = (ne2t / m)E.
Comparison of this with (1) gives the Drude conductivity:
ne 2t
s=
m
(3)
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Electron Scattering
 Electron scattering is the
process whereby an electron is
deflected from its original
trajectory.
 Electrons can be scattered by
other
charged
particles
through
the
electrostatic
Coulomb forces.
 Furthermore, if a magnetic
field is present, a traveling
electron will be deflected by
the Lorentz force.
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What causes scattering?
Phonon Scattering
Ionized Impurity Scattering
Neutral Atom/Defect Scattering
Carrier-Carrier Scattering
Piezoelectric Scattering
Mobility
The electron mobility characterizes how
quickly an electron can move through a
metal or semiconductor, when pulled by
an electric field.
In semiconductors, there is an analogous
quantity for holes, called hole mobility.
The term carrier mobility refers in general
to both electron and hole mobility in
semiconductors.
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Mobility
 Electrical mobility is the ability of charged particles (such as
electrons or protons) to move through a medium in response to an
electric field that is pulling them.
 When an electric field is applied across a piece of material, the
electrons respond by moving with an average velocity called the
drift velocity. Then the electron mobility μ is defined as
where: E is the magnitude of the electric field applied to a material, vd
is the magnitude of the electron drift velocity (in other words, the
electron drift speed) caused by the electric field, and µ is the electron
mobility.
 SI unit of mobility is (m/s)/(V/m) = m2/(V·s).
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Drift Velocity in an Electric
Field
 Without any applied electric field, in a solid, electrons (or, in the
case of semiconductors, both electrons and holes) move around
randomly. Therefore, on average there will be no overall motion of
charge carriers in any particular direction over time. However, when
an electric field is applied, each electron is accelerated by the
electric field. If the electron were in a vacuum, it would be
accelerated to faster and faster velocities (called ballistic transport).
However, in a solid, the electron repeatedly scatters off crystal
defects, phonons, impurities, etc. Therefore, it does not accelerate
faster and faster; instead it moves with a finite average velocity,
called the drift velocity. This net electron motion is usually much
slower than the normally occurring random motion.
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Resistivity and Conductivity
Electrical resistivity (also known as resistivity, specific electrical
resistance, or volume resistivity) quantifies how strongly a given
material opposes the flow of electric current.
Electrical conductivity or specific conductance is the reciprocal of
electrical resistivity, and measures a material's ability to conduct an
electric current.
A conductor such as a metal has high conductivity and a low resistivity.
An insulator like glass has low conductivity and a high resistivity. The
conductivity of a semiconductor is generally intermediate, but varies
widely under different conditions, such as exposure of the material to
electric fields or specific frequencies of light, and, most important, with
temperature and composition of the semiconductor material.
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Temperature Dependence
of Metal Resistivity
 In general, electrical resistivity of metals increases with temperature.
Mathematically the temperature dependence of the resistivity ρ of a metal
is given by the Bloch–Grüneisen formula:
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Change of resistivity with temperature
for a metal
r
r
= T
where  = temperature coefficient of
electrical resistivity
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Matthiessen’s Rule
The resistivity of a metallic material is given by
the addition of a base resistivity that accounts
for the effect of temperature, and a
temperature independent term that reflects the
effect of atomic level defects, including
impurities forming solid solutions.
Matthiessen's Rule:
Mobility
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