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This is a little bit puzzling.wenty said:

Why would there be a [tex]T^5[/tex] dependence of the resistivity at low temperature? (Note: it is resistivity or conductance data that would make sense and often quoted, since "resistance" depends on the geometry of the conductor). If you are applying Debye's law for molecular vibrations and specific heats, then you're missing something.

In metals, the transport properties are accurately described by the Fermi Liquid theory. In such a scenario, the resistivity depends directly on the electron-electron scattering rate of the charge carriers and depends on [tex]T^2[/tex]. However, you can only see such relationship at very low temperatures, below 10K. This is because at higher temperatures, the scattering with phonons or lattice vibrations will overwhelm the electron-electron scattering. So under most conditions, the [tex]T^2[/tex] dependence isn't observed.

Zz.

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Gokul43201

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Humm... can't say I do. The [tex]T^2[/tex] dependence of the scattering rate has been seen in Mo(110) surface state, which is a pseudo 2D system. So I can't recall other type of relationships, at least for the electron-electron scattering rate. I'm sure there are other type of temperature dependence in the current transmission once you go beyond the Fermi Liquid regime.Gokul43201 said:

Zz.

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But again, it depends on how low of a temperature. Remember that the phonons "freeze out" at some point. I threw out 10K as a number but I believe it is more likely below 5K.nbo10 said:

I need to go find some solid references on this...

Zz.

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Dr Transport

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The [tex] T^{5} [/tex] region is for [tex] \frac{T}{\Theta_{D}} \lessapprox \frac{1}{2} [/tex].

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Again, note that when electron-phonon interaction DOMINATES, the electron-electron scattering with its [tex]T^2[/tex] dependence is washed out. However, at very low temperatures, when the phonon modes freezes out, the electron-phonon scattering in metals grow very weak and the electron-electron scattering dominates. This is where you get the [tex]T^2[/tex] dependence. This is part of the Fermi Liquid model. See, for example

http://cscmr.snu.ac.kr/publish/prb_67_033103(2003).pdf [Broken]

http://www.physicstoday.org/pt/vol-54/iss-1/p42.html [Broken]

The scattering rate adds linearly, i.e. the total scattering rate is equal to the sum of the e-e scattering rate + e-ph scattering rate + e-impurity scattering rate.... The e-impurity scattering rate usually adds a constant term to the resistivity and often is the cause of the "residual" resistivity at T=0 extrapolation. Over most of the temperature range, the e-ph scattering dominates considerably (and note that the debye temperature for most metals is higher than room temperature and so the phonon spectrum doesn't reach saturation yet). However, at some point at very low temp., the [tex]T^2[/tex] dependence kicks in if this is a "standard" metal.

Zz.

http://cscmr.snu.ac.kr/publish/prb_67_033103(2003).pdf [Broken]

http://www.physicstoday.org/pt/vol-54/iss-1/p42.html [Broken]

The scattering rate adds linearly, i.e. the total scattering rate is equal to the sum of the e-e scattering rate + e-ph scattering rate + e-impurity scattering rate.... The e-impurity scattering rate usually adds a constant term to the resistivity and often is the cause of the "residual" resistivity at T=0 extrapolation. Over most of the temperature range, the e-ph scattering dominates considerably (and note that the debye temperature for most metals is higher than room temperature and so the phonon spectrum doesn't reach saturation yet). However, at some point at very low temp., the [tex]T^2[/tex] dependence kicks in if this is a "standard" metal.

Zz.

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Dr Transport

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wenty said:

Zapper,

The original question was about metals, not to get too specific but your references were to a paper about alloys and superconductors. The plot I refered to in Ziman was for pure metalic elements, Al, Au, Ag, Cu etc.....I agree, in alloys the resistivity in the Fermi-liquid model at low [tex] T [/tex] is [tex] T^{2} [/tex], but the classical theory does a fine job of describing pure elemental metals and will break down in the case of a complicated system like CeNiSi_2 which I would guess is more a semimetal than a metal not having read anything on the system before.

dt

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I gave those references because those are the only two I could find. I know for a fact that ordinary metals have been show to have the same property at extremely low temperature, but I can't find the references. Furthermore, The Fermi liquid model SHOULD work for a standard metal - this is where all of its assumptions and simplications are the most accurate. In fact, it tends to breakdown for more exotic metals and compound.Dr Transport said:Zapper,

The original question was about metals, not to get too specific but your references were to a paper about alloys and superconductors. The plot I refered to in Ziman was for pure metalic elements, Al, Au, Ag, Cu etc.....I agree, in alloys the resistivity in the Fermi-liquid model at low [tex] T [/tex] is [tex] T^{2} [/tex], but the classical theory does a fine job of describing pure elemental metals and will break down in the case of a complicated system like CeNiSi_2 which I would guess is more a semimetal than a metal not having read anything on the system before.

dt

The fundamental issue here is whether the scattering CAN be described as I have indicated - one can look those up easily. If it is, then below some temperature (the say way there is a cut-off temperature and energy for the phonon spectrum), if the phonons are no longer dominant (we know that can happen in superconductors where the virtual phonons take over from the real ones), then the electron-electron interactions take over. This e-e interaction NEVER goes away. It is just masked by other stronger interactions.

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ZapperZ said:I gave those references because those are the only two I could find. I know for a fact that ordinary metals have been show to have the same property at extremely low temperature, but I can't find the references. Furthermore, The Fermi liquid model SHOULD work for a standard metal - this is where all of its assumptions and simplications are the most accurate. In fact, it tends to breakdown for more exotic metals and compound.

The fundamental issue here is whether the scattering CAN be described as I have indicated - one can look those up easily. If it is, then below some temperature (the say way there is a cut-off temperature and energy for the phonon spectrum), if the phonons are no longer dominant (we know that can happen in superconductors where the virtual phonons take over from the real ones), then the electron-electron interactions take over. This e-e interaction NEVER goes away. It is just masked by other stronger interactions.

Zz.

This is why I want to find some experimental results of electrical resistivity.

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