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)
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
WWx
W
W in dimension, we see that the resistor is 7 squares
long (
Figure 2) and so its resistance is given as:
R=50Ωsquare7squares=350Ω
R
50
Ω
square
7
squares
350
Ω
(4)
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