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I’m working on a lab report on the relationship between thickness and resistivity of thin metal films. I collected data that was approximated to the 2nd order exponential decay equation:

$$7\times 10^{-5}\exp\left(\dfrac{-x}{3.0453\times 10^{-7}}\right)+9.1\times 10^{-4}\exp\left(\dfrac{-x}{7.8344\times 10^{-9}}\right)+1.6\times 10^{-4}$$

(with resistivity on the y-axis and thickness on the x-axis). I’m supposed to find out at which point, according to my equation, the change in thickness stops having a great effect on the change in resistivity. The slope of my graph is pretty steep at small thicknesses but it becomes less steep the thicker the thin film. I figured it would be similar to finding the inflection point of my curve, but I don’t have any change in concavity, so that is not exactly how I’m supposed to go about it. Any help? What do you guys advice me to do?

I also need to find out to find the horizontal asymptote of my equation to determine an estimate for the eventual final resistivity of my metal sample, how do I go about doing that?

This is the graph of my data if it helps in any way:graph

Morg Man
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  • I have a hard time understanding how an equation of that form would come up physically, but alright :) Regrettably, I don't know the answer to your question (I will be deleting my earlier comment now) – Danu Sep 17 '14 at 20:41
  • Maybe there's a better exponential fit? I know I needed an exponential fit for the data, but a first order exponential fit didn't give me a good enough fit, so a tried using a second order exponential fit and got a line that was in better agreement with my data points. I'm not quite sure how to fit it correctly. – Morg Man Sep 17 '14 at 20:47
  • I don't know where you get the name second order exponential fit, but I really think something like the formula you're using should not be used without a physical justification why it should be like that. A normal exponential decay is much more logical in most cases. – Danu Sep 17 '14 at 20:49
  • That's how the program I used to find the fit called it, I apologize if I'm using the term incorrectly :/. The fit I got for a normal exponential decay didn't seem to be a good fit at all, so I decided to try something else. Would you advise me to stick with it anyway? – Morg Man Sep 17 '14 at 20:55
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    I think you need to consider what is physically going on. What type of a relation can you expect, based on the relevant physics? One can always find a perfect fit to data, but it's meaningless if you forced it to fit instead of having it fit naturally. – Danu Sep 17 '14 at 20:56
  • I see what you are saying, but I don't really know that to do :/. I was told that my data would possibly be an exponential decay, and technically, isn't my equation also an equation of exponential decay? Pardon my ignorance regarding the subject. – Morg Man Sep 17 '14 at 21:36
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    Can you put your data on log-log axes and show that? This data is begging to be freed from its linear prison :( – Kyle Oman Sep 17 '14 at 21:39
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    @MorgMan I refer back to my previous comment. Perform a theoretical analysis to try and form an expectation of what the expression should look like (in an ideal situation) – Danu Sep 17 '14 at 21:57
  • Should it be a straight line once I change the data to log-log axes? Bc that's not what I'm getting :( – Morg Man Sep 17 '14 at 22:03
  • You fit two differently decaying exponentials. This might be reasonable (see for instance some supernova light curves), but maybe not. There are other ways to add parameters to complicate a single exponential (like add a linear background, or have an $x^2$ in the exponent). As Danu says, though, extra parameters should be justified physically. Also, single exponentials will look straight on log-linear plots, power laws will look straight on log-log plots. –  Sep 18 '14 at 02:35
  • I'd expect resistivity to diverge as film thickness approaches some critical nonzero value, due to particle graining effects. Also be very careful when "fitting" heavily nonlinear data like this, you usually need to use nonuniform weights to emphasize the features that matter. – Nanite Sep 18 '14 at 05:34

1 Answers1

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I can't see your graph I supose it has been taken down. I have never seen the exponential fit to the experimental data of resistivity. Could you show me the reference? As I'm searching for experimental data on resistivity of ultrathin metallic layers I can show you few theoretical results.

I don't know how thick is your film but I can give you semiclassical approximation [1]. In this article you'll find formulas and few important references.

The quantum theory of electron transport is well described here [2]. Trivedi shows that gross resistivity is due to surface roughness which falls as the film thickness grows. This result is verry common in theoretical physics and has been proven experimentaly. Impurity scattering for low concentrations of impurities is neglible.

Yet neither of these two have exponential approx.

[1] H. Ishida, Physical Review B 60, 7 (1999)

[2] N. Trivedi, N.W. Ashcroft, Physical Review B 38, 17 (1988).

mayestre
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