Handout 1
Drude Model for Metals
In this lecture you will learn:
• Metals, insulators, and semiconductors
• Drude model for electrons in metals
• Linear response functions of materials
Paul Drude (1863-1906)
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
Inorganic Crystalline Materials
Ionic solids
Covalent solids
Mostly insulators
Example: NaCl, KCl
Semiconductors
Si, C, GaAs, InP, GaN
PbSe, CdTe, ZnO
Insulators
SiO2, Si3N4
Metals
Au, Ag, Al,
Ga, In
Metals
1- Metals are usually very conductive
2- Metals have a large number of “free electrons” that can move in response to an
applied electric field and contribute to electrical current
3- Metals have a shiny reflective surface
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
3
Properties of Metals: Drude Model
Before ~1900 it was known that most conductive materials obeyed Ohm’s law
(i.e. I =V/R).
In 1897 J. J. Thompson discovers the electron as the smallest charge carrying
constituent of matter with a charge equal to “-e”
e
1.6 10
19
C
sea of
electrons
ions
In 1900 P. Drude formulated a theory for
conduction in metals using the electron
concept. The theory assumed:
1) Metals have a large density of “free
electrons” that can move about freely
from atom to atom (“sea of electrons”)
2) The electrons move according to
Newton’s laws until they scatter from
ions, defects, etc.
3) After a scattering event the momentum of
the electron is completely random (i.e.
has no relation to its momentum before
scattering)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
electron
path
+
+
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
Drude Model - I
+
Applied Electric Field:
electron
path
In the presence of an applied external electric
field E the electron motion, on average, can
be described as follows:
Let
be the scattering time and 1/
+
+
+
be the scattering rate
This means that the probability of scattering in small time interval time dt is:
dt
The probability of not scattering in time dt is then: 1 dt
Let p t
be the average electron momentum at time t , then we have:
pt
dt
1
dt
pt
e E t dt
If no scattering
happens then
Newton’s law
dp t
dt
eE t
dt
0
If scattering happens then average
momentum after scattering is zero
pt
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
4
Drude Model - II
Case I: No Electric Field
dp t
dt
pt
pt
Steady state solution:
0
Case II: Constant Uniform Electric Field
Steady state solution is:
pt
e
Electron path
Electron path
E
E
Electron “drift” velocity is defined as:
pt
m
v
e
E
m
= e /m = electron mobility
(units: cm2/V-sec)
E
Electron current density J (units: Amps/cm2) is:
J
n
e v
Where: n
ne
E
E
electron density (units : # /cm3 )
electron conductivity (units : Siemens/cm )
ne
ne 2
m
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
Drude Model - III
Case III: Time Dependent Sinusoidal Electric Field
dp t
dt
pt
eE t
There is no steady state solution in this case. Assume the E-field, average
momentum, and currents are all sinusoidal with phasors given as follows:
Et
Re E
dp t
dt
e
i
eE t
t
pt
e
1 i
p
pt
Re p
i p
E
e
eE
p
m
v
i
t
J t
Re J
e
i
t
p
e m
E
1 i
Electron current density:
J
n
e v
Where:
E
ne 2
m
1 i
0
Drude’s famous result !!
1 i
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5
Linear Response Functions - I
The relationship:
J
E
is an example of a relationship between an applied stimulus (the electric field in
this case) and the resulting system/material response (the current density in this
case). Other examples include:
P
o
electric polarization
density
E
e
electric
susceptibility
M
electric field
H
m
magnetic polarization magnetic
density
susceptibility
magnetic field
The response function (conductivity or susceptibility) must satisfy some
fundamental conditions …. (see next few pages)
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
Linear Response Functions - II
Case III: Time Dependent Non-Sinusoidal Electric Field
For general time-dependent (not necessarily sinusoidal) e-field one can
always use Fourier transforms:
d
2
Et
E
e
i
t
dt E t e i
E
t
(1)
Then employ the already obtained result in frequency domain:
J
E
And convert back to time domain:
d
2
J t
J
e
i
t
d
2
E
i
e
t
Now substitute from (1) into the above equation to get:
J t
J t
d
2
E
dt '
t
e
i
t
dt '
d
2
e
i
t t'
E t'
t' E t'
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6
Linear Response Functions - III
J t
dt '
t
t' E t'
d
2
t'
t
Where:
e
i
t t'
The current at time t is a convolution of the conductivity response function and the
applied time-dependent E-field
0
Drude Model:
t
d
2
t'
t
1 i
e
0
t'
i
d
2
t t'
0
e
1 i
i
t t'
t t'
e
t
t'
step function
t
t'
t
t'
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
Linear Response Functions - IV
The linear response functions in time and frequency domain must satisfy the
following two conditions:
1) Real inputs must yield real outputs:
Since we had:
J t
d
2
dt '
e
i
t t'
E t'
This condition can only hold if:
*
2) Output must be causal (i.e. output at any time cannot depend on future input):
Since we had:
J t
dt '
t
t' E t'
This condition can only hold if:
t
t'
0 for t
t'
Both these conditions are satisfied by the Drude model
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
7
Drude Model and Metal Reflectivity - I
When E&M waves are incident on a air-metal interface there is a reflected wave:
o
Ei
o
o
Et
Hi
Er
Hr
Ht
The reflection coefficient is:
Er
Ei
Question: what is
o
o
for metals?
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
Drude Model and Metal Reflectivity - II
From Maxwell’s equation:
Ampere’s law:
H r ,t
Phasor form:
H r
J r ,t
J r
i
E r
i
eff
E r ,t
t
o
E r
o
i
o
E r
Effective dielectric
constant of metals
eff
o
1 i
o
E r
Metal reflection coefficient becomes:
Er
Ei
o
eff
o
eff
Using the Drude expression:
0
1 i
the frequency dependence of the reflection coefficient of metals can be
explained adequately all the way from RF frequencies to optical frequencies
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
8
Drude Model and Plasma Frequency of Metals
For metals:
eff
1 i
o
ne 2 m
1 i
and
o
0
1 i
1 :
For small frequencies
ne 2
m
0
eff
0
1 i
o
o
1 (collision-less plasma regime):
For large frequencies
0
ne 2
m
i
i
eff
where the plasma frequency is:
o
ne 2
om
p
2
p
2
1
For most good metals
this frequency is in the
UV to visible range
Electrons behave like a collision-less plasma
Note that for
1
p
the dielectric constant is real and negative
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
Plasma Oscillations in Metals
Consider a metal with electron density n
Now assume that all the electrons in a certain region got displaced by distance u
-ve charge
+ve charge left
accumulated
behind
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
E
The electric field generated
neu
o
Force on the electrons
F
eE
+
+
+
+
+
+
+
+
+
+
+
E
u
u
2
ne u
o
As a results of this force electron displacement u will obey Newton’s second law:
m
d 2u t
dt 2
F
Solution is: u t
eE
n e2 u t
d 2u t
A cos
pt
2
pu
dt 2
o
B sin
pt
t
second order
system
Plasma oscillations are charge
density oscillations
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
9
Plasma Oscillations in Metals – with Scattering
From Drude model, we know that in the presence of scattering we have:
dp t
dt
pt
eE t
m
d 2u t
As before, the electric field generated
m du t
dt
eE t
dt 2
E t
(1)
neu t
(2)
o
Combining (2) with (1) we get the differential equation:
d 2u t
2
p
dt 2
1 du t
dt
ut
Or:
d 2u t
dt
1 du t
dt
2
2
p
ne 2
om
p
ut
0
+
+
+
+
+
+
+
second order system with damping
+
+
+
+
+
+
+
+
+
+
+
E
u
u
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
Plasma Oscillations in Metals – with Scattering
Case I (underdamped case):
p
1
2
Solution is:
ut
e
t
A cos
pt
B sin
Damped plasma
oscillations
pt
Where:
1
2
2
p
p
Case II (overdamped case):
p
2
1
2
Solution is:
ut
Ae
1t
Be
2t
No oscillations
Where:
1
1
2
1
4
2
2
p
1
1
2
1
4
2
2
p
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10
Appendix: Fourier Transforms in Time OR Space
Fourier transform in time:
dt f t e i
f
t
Inverse Fourier transform:
d
2
f t
f
e
i
t
Fourier transform in space:
gk
dx g x e
ik x
Inverse Fourier transform:
dk
g k ei k
2
g x
x
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
Appendix: Fourier Transforms in Time AND Space
Fourier transform in time and space:
h k,
dx dt
h x,t e
ikx
ei
t
Inverse Fourier transform:
h x,t
dk
2
d
2
h k,
ei k x e
i
t
ECE 4070 – Spring 2010 – Farhan Rana – Cornell University
11
Appendix: Fourier Transforms in Multiple Space Dimensions
Fourier transform in space:
h k x , k y , kz
dx dy dz
h x, y , z e
i kx x
e
i ky y
e
i kz z
Need a better notation!
Let:
k
k x xˆ
k y yˆ
r
x xˆ
y yˆ
k z zˆ
d 3r
dx
dy
dz
z zˆ
hk
d 3r h r
e
ik .r
Inverse Fourier transform:
hr
d 3k
2
3
h k ei k . r
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12