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I want to create a set of silicon based materials that have been doped with different materials and/or different amounts of dopants. The purpose of this is to see how the spring constant of silicon changes with the different electrical properties, specifically, with the change in resistivity (because we are all set up to measure resistivity to a highly accurate degree).

Now, I know that the physical deposition of the interstitial dopants will, by themselves, change the spring constant. What I am looking for is effects beyond that, effects that are due to changes in the electron mobility as quantified via resistivity.

As such, the particular material (silicon) and the various dopants are actually irrelevant to this question. They were included for context only. My true question is as follows:

Are there any formulas or concepts that will explain or predict how does the spring constant of a given material change as its resistivity changes?

Any papers cited and/or books (Kittel, Mermin, Dieter, etc) would be appreciated.

If the only way to understand this is through the densities of the dopants vs substrate, or any other of the usual suspects, that's fine. But what I would really like is a more "magical" argument I guess:

I have a material with a certain resistivity. I wave a magic wand and change its resistivity. What is the new spring constant?

Hmm, I hope that's clear. Let me try one more:

I am looking for an Ohm's "law" like equation that relates spring constant to resistivity with a few physical constants thrown into a constant somewhere in the equation. Does such a thing exist?

Okay, well, any help is appreciated!

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My gut feeling would be that the equation you're looking for doesn't exist, or at least that it involves a lot of other variables. But you never know, physics is surprising sometimes. –  David Z Jun 7 '11 at 18:12
    
@David That's my fear. I certainly haven't been able to find it in the books in my office. I'm hoping that someone with more knowledge can at least point me in the right direction. –  pballjew Jun 7 '11 at 18:21
    
I hope so too. You came to the right place, at least ;-) (well, we'd like it to be the right place) –  David Z Jun 7 '11 at 18:51
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I think the same as David. Such a relation would be a surprise. So, look for it experimentally, and let others sweat to find an explanation :=) –  Georg Jun 7 '11 at 20:09

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Re: I am looking for an Ohm's "law" like equation that relates spring constant to resistivity with a few physical constants thrown into a constant somewhere in the equation. Does such a thing exist?

No.

For a specific material with a specific dopant, you can get a relationship between the two, but there is no general relationship.

While a component of the spring constant is related to the electron "pressure" -- and hence related to the Fermi energy -- the exact nature of the relationship (and how big a role it would play in the overall spring constant) will depend on the specific material. The resistivity is also related to the Fermi energy, but how much and in what manner will depend on the specific material.

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Can you possibly provide an example - for any given material and dopant - relating spring constants to Fermi energy and the same for a relationship between resistivity and Fermi energy? That may provide a helpful starting point to relate the two for my particular set up. Alternatively, can you tell me where to look (book w/ chapter, article, paper, etc) where such relations are derived? Thanks, Sam –  pballjew Jun 10 '11 at 2:15
    
For the elasticity, it's very minor, but look up "electron degeneracy pressure". For the other, more important parts, see Ch 22 of Ashcroft and Mermin. For the conductivity, it depends very strongly on what kind of material you're talking about. Look up any book on semiconductors. –  Anonymous Coward Jun 10 '11 at 17:24

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