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9.5.10: Diffused Resistor

  • Page ID
    89990
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    Sometimes, in a circuit design, we will need a resistor. This is usually made either with poly or with a diffusion (shown in Figure \(\PageIndex{1}\)). 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 that \[\begin{array}{l} R &= \dfrac{\rho L}{A} \\ &= \dfrac{L}{nq \mu tW} \end{array}\]

    A diffused resistor, shown as a long, thin rectangular prism of width W, length L, and height t, that connects two rectangles.Figure \(\PageIndex{1}\): A diffused resistor

    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, \(n\) 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 \(n\) and \(\mu\). Note that we could write Equation \(\PageIndex{1}\) as \[R = \frac{1}{nq \mu t} \frac{L}{W}\]

    We define the first fraction (which contains the carrier concentration, thickness, etc.) as the sheet resistance \(R_{s}\) of the diffusion. While this can be more-or-less predicted, it is usually also a post-fabrication measured value. \[R_{s} \equiv \frac{1}{nq \mu t}\]

    \(R_{s}\) has units of \(\mathrm{"Ohms}/\mathrm{square"}\), and you are probably tempted to ask "per square what?". Well, it can be any square at all, measured in centimeters, micrometers, kilometers, etc., since all we really need to know is \(R_{s}\) 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 \(\PageIndex{1}\) has a sheet resistivity of \(50 \ \Omega / \mathrm{square}\), then by blocking the resistor off into squares that are \(W\) in both dimensions, we see that the resistor is 7 squares long (Figure \(\PageIndex{2}\)) and so its resistance is given as: \[\begin{array}{l} R &= 50 \ \left(\frac{\Omega}{\mathrm{square}}\right) 7 \ \ (\mathrm{squares}) \\ &= 350 (\Omega) \end{array}\]

    The top face of the resistor from Figure 1 above is divided into 7 squares, each having dimension W in both width and length.Figure \(\PageIndex{2}\): Counting the squares

    This page titled 9.5.10: Diffused Resistor is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Bill Wilson via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.