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ECE 535 Notes for Lecture # 2

Class Outline: • Classical Conductivity – Part 2 • Sommerfeld Model– Part 1

M.J. Gilbert ECE 535 – Lecture 2

Classical Conductivity – 10

We will continue our examination of the Drude Model for metals and refresh our memories about quantum mechanics needed for the Sommerfeld model.

I want to cover: 1.  Drude Model for Metals

•  Time-Dependent Dynamics •  Response Functions

2.  Sommerfeld Model

•  Problems with Drude •  Schrodinger equation

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M.J. Gilbert ECE 535 – Lecture 2

Classical Conductivity – 9 Let’s look at a few cases…

Case I: No Electric Field

This gives the steady-state solution:

Case II: Constant and Uniform Electric Field

M.J. Gilbert ECE 535 – Lecture 2

Classical Conductivity – 11 Let’s examine the time-dependent response of metals.. Case III: Time Dependent Sinusoidal Electric Field

But now there is no steady state solution.

•  So we assume that the E-field, average momentum, currents are all correspondingly sinusoidal

•  We represent them with phasors.

We may now obtain the current density:

where

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M.J. Gilbert ECE 535 – Lecture 2

Classical Conductivity – 12 Let’s examine the relationship between current density and the electric field…

This is a relationship between an applied stimulus (the electric field) and the material response (the current density). But we have seen this many times before in different contexts…

Let’s keep these linear response functions in mind as we progress.

M.J. Gilbert ECE 535 – Lecture 2

Classical Conductivity – 13 Let’s now examine another case… Case IV: Time Dependent Non-Sinusoidal Electric Field

For some general time-dependent e-field excitation, we can always simplify our lives by using Fourier transforms.

(*)

Then employ the result we obtained for the frequency domain response:

And move back to the time domain:

Substitute (*) into the above result:

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M.J. Gilbert ECE 535 – Lecture 2

Classical Conductivity – 14 Let’s examine the last result a bit more…

where

•  This brings us to an interesting conclusion. The current at some time, t, is a convolution of the conductivity response function and the applied time-dependent electric field.

Consider the Drude model again:

M.J. Gilbert ECE 535 – Lecture 2

Classical Conductivity – 15 There are two important conditions that linear response functions must satisfy in both the time and the frequency domain:

Both of these conditions are satisfied by the Drude model..so that’s a good thing.

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M.J. Gilbert ECE 535 – Lecture 2

Classical Conductivity – 16 Can the Drude model tell us anything about waves incident on a metal?

For instance, we already know that EM waves incident from air onto a metal will have a reflected component…

Basic undergrad EM tells us the reflection coefficient:

What is the relative permittivity for a metal as a function of frequency?

M.J. Gilbert ECE 535 – Lecture 2

Classical Conductivity – 17 Start from Maxwell’s equations:

Write it in phasor form:

The metal reflection coefficient becomes:

Now use the Drude expression:

Using this, we can explain the frequency dependence of the reflection coefficient of metals from RF to optical frequencies.

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M.J. Gilbert ECE 535 – Lecture 2

Classical Conductivity – 18

For metals we have the following two results:

and

Let’s examine the frequency response for small frequencies:

For large frequencies, we see something interesting:

Plasma Frequency The electrons behave like a plasma free of collisions.

M.J. Gilbert ECE 535 – Lecture 2

Classical Conductivity – 19 Consider a metal with electron density, n, and assume all the electrons in a certain region are displaced by, u.

The electric field generated:

The force on the electrons:

Displacement obeys Newton’s second law:

Charge density oscillations..

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M.J. Gilbert ECE 535 – Lecture 2

Classical Conductivity – 20 One obvious problem is that we have, thus far, ignored scattering. From the Drude model, we know that:

We also know the electric field:

Combine the two equations to get a new differential equation:

M.J. Gilbert ECE 535 – Lecture 2

Classical Conductivity – 21 Let’s examine two cases of plasma oscillations in the presence of scattering:

Case I: Underdamped oscillations

Where:

Case II: Overdamped oscillations

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M.J. Gilbert ECE 535 – Lecture 2

Sommerfeld Model – 1 But…the Drude theory has some problems…

He was nominated 84 times for the Nobel Prize but never won…most ever.