# Electrical conductivity

1. Jan 13, 2005

### JohanL

How do you with a simple model explain the temperature dependence of the
electrical conductivity.
If you use the Drude model you get for the electrical conductivity

sigma = ne^2t / m

where n is the density of mobile electrons and t is the relaxation time.
t is the time between collisions and must be the only variabel here that depends on temperature. How can you estimate t(T).

Maybe there is a better model that describes the temperature dependence of the electrical conductivity.

2. Jan 13, 2005

### Crosson

I never liked the way the drude model gets bogus at the end:

"The only relevant quantity with dimensions of time is the time between collisions".

Alright, thermal collisions are much more frequent than conduction drift collisions. Calculate the distance between electrons N (number of conductivity electrons per cubic meter) arranged in a 1m^3 sphere (fun!). Then calculate the speed of the electrons from temperature using:

KE = 3/2 *k*T

where KE is kinetic energy, k is boltzmans and T is temperature.

Use the mean free path and velocity to compute time between collision.

After all that, throw away the drude model and study quantum mechanics.

3. Jan 19, 2005

### JohanL

I have a question about mean free path p.
I calculate it from the density of electrons n (electrons/m^3)
then p = 1/(third root of n),

and then it should be indepent of temperature.
But I know that it should be different for different temperatures.
How can you estimate p for different temperatures?

_____________________

Then I used your model to estimate the temperature dependence of the thermal conductivity of the free electrons in a metal.

K = C*T*t = D * sqrt(T)

Where C and C are constants and t again is the time between collisions.
But the experimental curve of K doesnt have this form. Only for low temperatures it has. Then it reaches a maximum and goes down.
Why?

4. Jan 21, 2005

### Gokul43201

Staff Emeritus
Briefly :

The effective relaxation time comes from two contributions : scattering off of the lattice/phonons (not other electrons - the Drude model does not include electron-electron interactions), and scattering off of impurities and lattice imperfections.

$$\frac{1}{\tau} = \frac{1}{\tau _{lat}} + \frac{1}{\tau _{imp}}$$

Speaking of resistivities instead of conductivities, you have

$$\rho = \rho _ {lat} + \rho _{imp}$$

For most elemental metals, $\tau _ {imp}$ is fairly independent of the temperature. The lattice interactions are largely result of the fact that the lattice is vibrating rapidly, providing a large scattering cross section, so much so, that as $T \rightarrow 0$, $\rho _ {lat} << \rho _{imp}$

So, at 0 K : $\rho \approx \rho _ {imp}$

INCOMPLETE...

Just realized you are now suddenly talking about thermal conductivity $\kappa$, rather than electrical conductivity $\sigma$. Which one is it ? Drude does NOT try to explain thermal conductivity, and can not, because this is a largely phonon process. Seminal work on thermal conductivity was done by Debye and Pierls.

Last edited: Jan 21, 2005