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Electrical conductivity

  1. Jan 13, 2005 #1
    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. jcsd
  3. Jan 13, 2005 #2
    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.
  4. Jan 19, 2005 #3
    thx for your answer. :smile:

    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.
  5. Jan 21, 2005 #4


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    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.

    [tex] \frac{1}{\tau} = \frac{1}{\tau _{lat}} + \frac{1}{\tau _{imp}} [/tex]

    Speaking of resistivities instead of conductivities, you have

    [tex] \rho = \rho _ {lat} + \rho _{imp} [/tex]

    For most elemental metals, [itex] \tau _ {imp} [/itex] 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 [itex]T \rightarrow 0 [/itex], [itex] \rho _ {lat} << \rho _{imp} [/itex]

    So, at 0 K : [itex] \rho \approx \rho _ {imp} [/itex]


    Just realized you are now suddenly talking about thermal conductivity [itex]\kappa[/itex], rather than electrical conductivity [itex]\sigma [/itex]. 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
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