# Where to find the data of electrical resistance

## Main Question or Discussion Point

Does anyone know where to find the data of electrical resistance(conductance) of metals at different temperatures?I want to check the $$T^5$$ law at low temperature and the linear relation between conductance and temperature at high temperature.

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ZapperZ
Staff Emeritus
2018 Award
wenty said:
Does anyone know where to find the data of electrical resistance(conductance) of metals at different temperatures?I want to check the $$T^5$$ law at low temperature and the linear relation between conductance and temperature at high temperature.
This is a little bit puzzling.

Why would there be a $$T^5$$ 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 $$T^2$$. 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 $$T^2$$ dependence isn't observed.

Zz.

Gokul43201
Staff Emeritus
Gold Member
I recall something about $T^5, T^7$ type dependence in disordered systems (and perhaps in 2D, but I could be wrong, since I think there's some lnT dependence there). Ring any bells Zz ?

ZapperZ
Staff Emeritus
2018 Award
Gokul43201 said:
I recall something about $T^5, T^7$ type dependence in disordered systems (and perhaps in 2D, but I could be wrong, since I think there's some lnT dependence there). Ring any bells Zz ?
Humm... can't say I do. The $$T^2$$ 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.

Zz.

For metals at low temperatures the resistivity does go as $$T^5$$, if I remember correctly it has to do with the foward El-ph scattering processes dominating. Look up Gruneisen law for more info.

ZapperZ
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2018 Award
nbo10 said:
For metals at low temperatures the resistivity does go as $$T^5$$, if I remember correctly it has to do with the foward El-ph scattering processes dominating. Look up Gruneisen law for more info.
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.

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

Zz.

Dr Transport
Gold Member
From my well worn copy of Electrons and Phonons by JM Ziman, pg 365, the resistivity of a metal is proportional to $$T^{5}$$. He gives the details behind this in a well written arguement on that page. He also aludes to some experimental data. I'd start there, then look at either the Handbook of Physics or another reference to find the data. On the next page he plots a fit of the experimental data at the time to the Gruneisen-Bloch formula.

the $$T^5$$ region is between the phonon and impurity dominant scattering regions. It's somewhere below the Debye temperature for the conductor but the diagrams I checked didn't have a proper T-scaling so I have no idea where it occurs exactly.

Dr Transport
Gold Member
The $$T^{5}$$ region is for $$\frac{T}{\Theta_{D}} \lessapprox \frac{1}{2}$$.

ZapperZ
Staff Emeritus
2018 Award
Again, note that when electron-phonon interaction DOMINATES, the electron-electron scattering with its $$T^2$$ 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 $$T^2$$ 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 $$T^2$$ dependence kicks in if this is a "standard" metal.

Zz.

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Dr Transport
Gold Member
wenty said:
Does anyone know where to find the data of electrical resistance(conductance) of metals at different temperatures?I want to check the $$T^5$$ law at low temperature and the linear relation between conductance and temperature at high temperature.

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 $$T$$ is $$T^{2}$$, 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

ZapperZ
Staff Emeritus
2018 Award
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 $$T$$ is $$T^{2}$$, 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
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.

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.