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AC Conductivity
For constant voltage, conductors conduct and insulators don't.
For low ac voltages in a conductor, electric field and the electron velocity are in-phase, electric field and electron position are out-of-phase.
For low ac voltages in an insulator, electric field and the electron position are in-phase, electric field and electron velocity are out-of-phase.
At high (optical) frequencies the in-phase and out-of-phase component of the response is described by the dielectric function.
Conductivity / Dielectric function
0 0
P nexE E
0
1 1i
j nev i nexE E E
0i
v i x
Below about 100 GHz the frequency dependent conductivity is normally used.Above about 100 GHz the dielectric function is used (optical experiments).
, , i t i t i tv v e x x e E E e Harmonic dependence
Diffusive metal
n nm mj E
The current is related to the electric field
( ) ( ) ( )dv t ev tm eE tdt
The differential equation that describes how the velocity changes in time is:
n nm mv E
The impulse response function :
1( ) exp 0etg t tm m
Inertial term
Steady state solution
Diffusive metal
( ) ( ) ( )dv t ev tm eE tdt
Assume a harmonic solution E()eit, v()eit
The differential equation is:
1 ( ) ( )i m v Ee
1
12 2
11 11
v ii m iE e
2 2
11
j v ine neE E
low ne 2
high inem
me Scattering time
Diffusive metal
2 2
0 0 0 0 0
11
P nex nev ne iE E i E i i
2 20
1 11
ne i
2
23 21 pi
Take the limit as goes to infinity
2
21 p
0 for 0 for 0
22
0p
nem
low frequency metal / high frequency insulator
Ibach & Lueth
Aluminum
Conducting transparent contacts for LEDs and Solar cells
Windows that reflect infrared
Reflection of radio waves from ionosphere
22
0p
nem
ITO
Intraband transitions
( ) ( )c vE k E k
When the bands are parallel, there is a peak in the absorption (")
Ibac
h &
Luet
h
Intraband (d-band) absorption
1 R
y =
13.6
ev