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.

Apr 10, 2023 2

Drude Model electrons (shown here in blue) constantly bounce between heavier, stationary crystal ions (shown in red).

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.

Apr 10, 2023 3

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 are 𝝉respectively an electron's momentum, charge, number density, mass, and mean free time between ionic collisions.

Apr 10, 2023 4

5

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 = j or j = E (1)

Here, E = the electric field, j = the current density, the resistivity & 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

6

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.

Combining (2) & (3) givesj = (ne2m)E.

Comparison of this with (1) gives the Drude conductivity:

vdrift = -eEm

This comes from integrating Newton’s 2nd Law over time . This is the velocity that must be related to j.

(3)

m

ne 2

(3)

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.

Apr 10, 2023 7

What causes scattering?

Phonon ScatteringIonized Impurity ScatteringNeutral Atom/Defect ScatteringCarrier-Carrier ScatteringPiezoelectric 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.

Apr 10, 2023 9

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).Apr 10, 2023 10

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.

Apr 10, 2023 11

Resistivity and Conductivity

Apr 10, 2023 12

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.

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:

Apr 10, 2023 13

T

where = temperature coefficient of electrical resistivity

Change of resistivity with temperature for a metal

©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

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

Apr 10, 2023 20