A Common Man’s Derivation to the van der Pauw Method of Extracting Sheet Resistance for Conductive Films

by Nabil El-Hinnawy 

Almost every electrical engineer that does fabrication or electrical measurements of thin films (or cares about them) has used or run across Leo J. van der Pauw’s formula for sheet resistance:

But how many know how this magical formula came to be?  Or why it works?  Or what dude had enough free time to figure this out?  Hopefully what lies below answers these questions, as I too was perplexed by this as a young process integrator and vowed to understand where and why the “swastika-looking” shapes came from before showing up on my mask layouts.  I also had very little trust of any information that was listed as “trivial” in technical journals from a time period before the civil rights movement, so I made sure to do every integral and prove it was true.

Leo J. van der Pauw worked for Philips Research Laboratory in Eindhoven, Netherlands back in the 1950’s, and he is the dude who came up with all this craziness that is an absolute standard in the semiconductor/RF/electronics industry.  He wrote a paper for Philips Research Reports back in a time when private companies published their own research journals.  He published a paper called “A method of Measuring Specific Resistivity and Hall Effect of Discs of Arbitrary Shape” in Philips Research Reports back in February of 1958.  You’ll have hard a time trying to find the actual paper online for reasons I’m not aware of, but back when I originally did this derivation (like 2009 or 2010 when people weren’t trying to save every pdf on the internet so they could sell it back to you later) I was able to find a copy of this thanks to a Wikipedia link.  You should donate to them and you should support people and websites that link to free PDFs of amazing historical papers, like Microwaves101.

The Derivation:

Most van der Pauw (vdP) derivations start out with a picture that looks like some theoretical nonsense, which I’ve redrawn below for people with shorter attention spans that need some color:

While the original is technically correct, it also has that “theory” feel to it that makes my brain just sort of nod-off every millisecond I’m staring at it, so I redrew it with color and I’ll tell you in 5 bullet points why you care about this drawing:

• Current is injected at point A, leaves at point B.
• Voltage difference is measured between points C and D.
• No, this example as drawn right now makes no sense in a real world setting because no one would set something up like this if you were getting a paycheck and take any pride in your work.
• It is important (and make sense) in a theoretical setting because the point of the derivation is that the 4 points can be anywhere, even somewhere as stupid as the theoretical example.
• All of the contacts are on the EDGE of the sample, and that contact area at each of these points is assumed to be much smaller than the entire 2D area of the sample being measured by like an order of magnitude (this is what the Ph.D’s apparently call the semi-infinite plane assumption)
• What you’re measuring (the blue material in the figure) has to behave like the cartoon metal you draw in all your power points:

1. Should be the same thickness across the entire area
2. Should be 1 piece (no holes)
3. Should be identical in composition in all directions (no mobility gradients)

Assuming you followed me so far, which you should have, because all I’m saying is that we have a piece of uniform metal and 4 randomly placed contacts on the edges, all that happens now is some Gauss’ Law and mild calculus:

Current density going into point A:

Which is also equal to:

Where I is the total current we injected at A, r is the radius that current has expanded to, and t is the thickness of the film.  There’s no 2 in front of the π because we are only looking at half the radius of a circle because the contact is on the edge

Now if we want to know what the voltage drop is between points C and D because of the current I injected at point A, we take advantage of the following basic formulas I say are basic because I remember I used to know them as a 22 year old:

And

We get:

The limits of integration are where some of the magic happens, because the distance “D” and “C” is the distance from the point in question to point A, where the current is injected.  The integral actually now looks like this:

And if you’ll believe me that this is the generic solution to the above integral (check your favorite math book or search engine if you don’t believe me):

then

Where ln is the natural logarithm.  This is the voltage between points D and C due to current I at point A.  We also have to account for the current –I at point B.  By doing the same math we just did but using the different distances, we get:

If we define R_DC,AB as:

Then:

Also known as:

Because of some mathematical insight I don’t have, nor do I ever care to have, we want to get rid of the natural log in favor of an exponential.  Again, I don’t have the skills to know why, but it turns out it’s pretty important.

By getting rid of the exponential:

Why do they swap the numerator and the denominator to get a negative in the exponential?  Beats me, because as you’ll see you get the same equations anyway.  But if you followed this derivation so far, then you literally have followed the hardest part of this van der Pauw (vdP) derivation. 

If you go ahead and do literally the exact same derivation but inject the current at B and C and calculate the voltage at A and D (also known as rotating the sample 90 degrees in the real world)

You get:

And:

And now:

So that we can make this expression:

Now if you add expressions (1) and (2) together, which math says you are allowed to do but again I have no pre-existing intuition as to why you would ever do it:

The left hand side of this equation actually simplifies to 1.  You can trust me, or you can put it into wolframalpha, but I promise you it’s true:

So now we are at a point where:

Now if by some miracle of fabrication and luck you managed to make a structure where:

Then the previous equation reduces to this:

And there you have it.  All you needed was one integral of electrostatics, two simple equations of electrodynamics, and one generic integral formula, and a bunch of theory that some dude way smarter than me came up with, and you get a pretty fundamental relation for measuring thin films.

Now it turns out that you don’t actually need a miracle of fabrication and luck to make this structure.  If you follow the assumptions listed in the bullet points at the beginning, and you make a symmetrical structure with symmetrical contacts that follow the bullet points listed above, this structure is shockingly simple to make.  So simple, in fact, that grad students can fabricate it on their $3/hr salary, unless you are a terminal Masters student, in which case your salary is of course -$90/hr for every hour you are breathing.

Now that you know exactly where the formula comes from, you can also now properly design vdP structures such that you get the most accurate measurements.  You can also test out how accurate your vdP structures are because if you rotate the measurement 90 degrees, you should get nearly the same answer.  If you don’t, then you need to curl into a ball and cry or set your wafer on fire.

It should also go without saying that the measurements you do for vdP structures should be 4 point measurements, meaning you force a current on adjacent probes and measure the voltage on adjacent probes….just like in the derivation.

If you want to know more about what proper vdP structures should look like, find your oldest process integrator, product engineer, or layout jockey, and they can probably bore you to death about different kinds of process control monitors (PCMs).  Many old-timey GaAs foundries used or still use standard vdP cells that were developed on the DARPA MIMIC program over 30 years ago!  For people of a younger generation, they will look awfully similar to swastikas, but if you look closely at the original ones, the connections to the “Greek Cross” (what it more closely resembles actually go in the opposite directions than the emblem on the 1935-1945 German flag, which  is is prohibited from public display  in Germany today by The German Strafgesetzbuch (Criminal Code) section 86a.