A 4 probe conductivity in discs (Smits)

[Nony]

See ref (Smits, 1958, Bell Technical Journal: http://onlinelibrary.wiley.com/doi/10.1002/j.1538-7305.1958.tb03883.x/abstract (also Google will show a pdf copy that is not pay-walled)

The paper describes correction factors for doing 4 probe conductivity measurements on cylinders (semiconductor wafers, sintered ceramic specimens, etc.). Intuitively, it makes sense to me that there would be these sorts of correction factors (paths of current flow are in parallel, but going deeper into the sample, would cover a longer distance, thus higher resistance). I have used the paper routinely, but IANAP and I never verified the paper itself--just plug and chugged the factors from the table.

1. Was wondering, for those who are physicists how "hard" is this as a problem (to do the method of images and create the table of correction factors for geometry)? Is it a routine HW problem in undergrad or graduate E&M? If so what text (edition and page #)? If not, I would think it would be nice HW problem because it actually is a practical result and kind of an intuitive geometry. If too easy for grad E&M than junior year E&M? It definitely feels harder than intro physics, which is the only course I have taken.

2. Also, I don't know if this has been done yet, but some extension of the paper would be useful:

A. Slight deviations from centerline (in and perpendicular to the 4 points line) of pellet. How much does the true answer change, based on using the assumption that we are centered? Note, that there are no correction factors for this since you sort of "by eye" center the sample. [I guess if you had an industrial, repeating process, you could build a jig, but this is not the normal instrument.]

B. How much will warping of the pellet affect the true answer versus measured? [Uniaxially pressed ceramic pellets usually have a slight but visible warping from strains created in the specimen during compaction prior to heating.]

[It might be more "easy" classical physics calculation, but would still be very nice for people who just use the instrument and want more corrections! (The people who make new material samples are not the type to derive E&M correction factors.) Even if you think the result is trivial, could be more HW problems for the courses.]

 

[Nony]

Is my question too hard or too easy? Help, IANAP. I don't even want to know an answer, I just want to know how hard a question is and how/where it would fit into classical E&M? Undergrad? Grad? Neglected topic? [I hope not the last, since this is actually a practical result and not just a math drill.]

 

[marcusl]

I glanced at the first two pages. For an infinite planar sample, this problem can be solved using "potential theory." Solutions can be obtained directly from solving the Laplace equations in elliptic cylinder coordinates, or possibly by conformal mapping. The former involves Mathieu functions, so would be conventionally classified as graduate level, while the latter approach would probably be senior level except that conformal mapping is an "obsolete" technique that is often no longer taught in modern curricula. The presence of rectangular or circular boundaries is a complication that, I would guess, makes it insoluble without numerical calculation.

BTW, grinding the sample so that its boundary has the shape of a confocal ellipse (one whose foci are exactly the positions of the two excitation electrodes) would restore this to an analytically soluble problem, using the methods mentioned.

 

[Nony]

Thanks so much! Exactly the type of answer I wanted. The math comments very helpful.

So I guess Smits used numerical methods in 1958? Do you all assign HW problems that require numerical methods?

The grinding the boundary idea is neat, thanks. I would think the non-negligible thickness would still require numerical solution though? It's not a thin film. Typical instrument has the 4 points in 1 mm spacing. So a Si wafer or a ceramic HtSc pellet has thickness on the order of the spacing or larger.

P.s. If you can't tell, I love this paper! Even though I never knew the E&M involved in it. Those correction factors were sooooo nifty.