The Drude Theory of Metals

Alexander Su

June 17, 2011

Drude Model Overview

• Motivation• Basic Assumptions of the Model• Collision and Relaxation Time• DC, AC Electrical Conductivity• Hall Effect and Magnetoresistance• Why are metals shiny?

Motivation

• Metals are excellent conductors of heat and electricity, are ductile and malleable

• Physicists have tried to construct simple models of the metallic state that account the characteristic metallic properties

• Theory of Paul Drude in 1900 came only a few years after J.J. Thompson discovered the electrons

Metals – Valence Assumption

• Number of protons is Za

• Number of valence electrons is Z• Each metallic atom has:

– A nucleus of charge +Zae– Z valence/conduction electrons free to leave the atom to form electron gas– Za-Z core electrons of charge –(Za-Z)e tightly bound to the nucleus

Classical Thoery: Drude ModelDrude proposed valence electrons constantly bounce between heavier, stationary crystal ions

Drude Model Assumptions• Ions are arranged in fixed positions and cannot move• Collisions between e’s and ions are instantaneous, uncorrelated

events• Ignore all other interactions

– Independent electron approximation– Free electron approximation

• Electron experiences a collision with a probability per unit time 1/– is the mean time between collisions– independent on the electron position or momentum

Equations of MotionElectrons collisions

No collisions

EoM averaged over electrons

Taking a derivative

Average momentum

DC Electric Field

In uniform DC electric field

For steady state Drift velocity

vdtNumber of electrons passing an Area A in time dt

Total charge passing through A in time dt

Current density

DC Electrical Conductivity

With

Ohm’s Law

and

Resistivity

Conductivity

Current is caused by an electric fieldCurrent depends linearly on the strength of the electric fieldCurrent density depends on the conductivity of the material

or

The Hall Effect (1879)

Charge carriers in a current deflected to one surface of a conductor

Hall Effect (1879)

Positive Charge Carriers Negative Charge Carriers

Charge carriers deflected toward the top surfaceExcess surface charges creates an electric field Steady state reached

Hall’s Observation

Magnetoresistance

Magnetoresistance independent of magnetic field

Drude Model: Magnetoresistance

Force acting on each electron

Momentum per electron

In steady state where

In Drude model, ρ(H) is independent of magnetic field.

Magnetoresistance

with

AC Conductivity and the Shiny

Oscillating electric field

Ohm’s Law where

Maxwell’s equations with no net charge

Wave equation where

At high frequencies where

AC Conductivity and the Shiny

ε real and negative, no wave propagationmetal is reflective

ε real and positive, propagating wavesmetal is transparent

For metals the plasma frequency is in the ultraviolet.

Wave equation

Bibliography

Ashcroft, N., & Mermin, N. (1976). Solid State Physics. New York: Harcourt, Inc.

Tompkins, H., & Irene, E. (2005). Handbook of Ellipsometry. New York: William Andrew, Inc.

Knight, R. (2004). Physics for Scientists and Engineers. San Francisco: Addison Wesley