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Diffused Resistor

Module by: Bill Wilson

Summary: This module covers the diffused resistor and sheet resistance.

Sometimes, in a circuit design, we will need a resistor. This is usually made either with poly or with a diffusion. If we took our n-tank or similar n-type diffusion, we could make a long narrow strip of it, and use it as a resistor. As long as we keep the substrate at ground, and any voltages on the resistor greater than ground, the n-p junction will be reverse biased and the resistor will be isolated from the substrate. Now we all know

R=ρLA=LnqμtW R ρ L A L n q μ t W (1)
Figure 1
A Diffused Resistor
A Diffused Resistor (5.43.png)
The only trouble is, what is n for a diffused resistor? A quick look at the chart showing carrier concentration as a function of depth after a diffusion shows that when we do a diffusion, nn is not a constant, but varies as we go down into the wafer. We will have to do some kind of integral, assuming lots of parallel, thin resistors, each with a different carrier concentration! This is not very satisfactory.

In fact, it is so unsatisfactory that IC engineers have come up with a better description resistance than one involving n n and μμ. Note that we could write Equation 1 as

R=1nqμtLW R 1 n q μ t L W (2)
We define the first fraction (which contains the carrier concentration, thickness etc.) as the sheet resistance R s R s of the diffusion. While this can be more-or-less predicted, it is usually also a post-fabrication measured value.
R s ≡1nqμt R s 1 n q μ t (3)
R s R s has units of "ohms/square", and you are probably tempted to ask "per square what?". Well it can be any square at all, cm, μm, km, since all we really need to know is R s R s and the length to width ratio of the resistor structure to find the resistance of a resistor. We do not need to know what units are used to measure the length and the width, so long as they are the same for both. For instance if the resistor in Figure 1 has a sheet resistivity of 50 Ω/square, then by blocking the resistor off into squares WWxW W in dimension, we see that the resistor is 7 squares long (Figure 2) and so its resistance is given as:
R= 50 Ωsquare 7 squares = 350 Ω R 50 Ω square 7 squares 350 Ω (4)
Figure 2
Counting the Squares
Counting the Squares (5.44.png)

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