Ashcroft and Mermin - Graz University of Technologyhadley/ss2/lectures16/may4.pdfAC Conductivity For constant voltage, conductors conduct and insulators don't. For low ac voltages

Ashcroft and Mermin

Dielectric function of BaTiO3

Dielectric function of silicon

absorption

extinction K

refractive index n

( ) ( ) ( )n iK

4 fKc

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 transport (low frequencies)

sc

dvF ma eE m

scd

e E vm

e dE v

scme

2nenem

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

http://lamp.tu-graz.ac.at/~hadley/ss2/linearresponse/dmetal.php

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