What layout does about it

Random differences and gradients

Two devices drawn alike differ in two ways. Random mismatch (E7.4, E7.5) differs on every die and shrinks as the square root of the area. A gradient is a slow change across the chip in oxide thickness, doping, temperature or stress. It shifts each device by an amount set by its position, the same way on every die of a wafer.

Layout cannot remove a gradient, but it can make both devices see the same average of it. The trick is to cut each device into fingers, smaller transistors in parallel, and to choose which finger goes where.

The simulator models it with one source per finger. A threshold shift ΔV_th acts like a voltage taken off the gate, as E5.4's mismatch sources did. So each finger gets a gate source worth its position (0 to 7 along the row) times the gradient.

Placement against a gradient

Three ways to place four A fingers and four B fingers on eight positions. The centre of each device is the average of its finger positions.

Under a linear gradient the offset is g times the distance between the two centres. That is 4g side by side, g interdigitated and 0 for the common centroid, whose devices share one centre.

Real gradients are not perfectly straight. The next run bends the gradient: the shift grows as the square of the position, 0.1 mV × p² at the end of the sweep. Now equal centres are not enough, because a square weighs far positions more.

The common centroid cancels the linear part of any gradient and shrinks the rest. In two dimensions the cross-coupled quad does the same in both directions. It places A B in one row and B A below, so both centres sit in the middle of the square.

Row ends and dummy fingers

The common centroid has a weakness of its own: ABBAABBA puts both ends of the row on A. The fingers share one strip of diffusion, and the trench oxide around it squeezes the silicon most near its edges. So an end finger has a different threshold and mobility from an inner one.

The SKY130 model includes this length-of-diffusion effect through SA and SB, the distances from a gate to the two ends of its strip; the run sets them per finger. A dummy finger is an extra transistor at each end that carries no current: it only pushes the edge away from the active fingers.

With no dummies the end effect adds −3.24 mV to the common centroid. The first dummy at each end cuts it more than five times. AAAABBBB (like ABABABAB) is a mirror image end to end, so it cancels the end effect but fails against a gradient. A careful layout uses both: common centroid in the middle, dummies at the ends.

What else layout does, and what this lesson models

1. Same orientation

   Implants arrive at a small angle and stress differs with direction, so matched devices point the same way.

2. Close together

   A gradient grows with distance, so matched devices sit side by side with the same neighbours.

3. Symmetric wiring

   Equal wire lengths and contacts make series resistance and parasitic capacitance match too.

4. Guard rings

   A guard ring is a ring of substrate or well contacts around a sensitive block. Current a noisy neighbour pushes into the substrate leaves through it instead of moving the analog devices' body voltage.

Try it

Run it: systematic is the follower's offset with no gradient, gradient_offset the offset that 1 mV per finger pitch adds to the AAAABBBB placement.

Now edit the .param line to a1=0 a2=3 a3=4 a4=7 b1=1 b2=2 b3=5 b4=6 (ABBAABBA) and predict the result before you run. Then change .func grad(p) {p} to {p*p/10} to bend the gradient, and try a1=0 a2=2 a3=4 a4=6 b1=1 b2=3 b3=5 b4=7 (ABABABAB) under both shapes.

Check yourself

* AAAABBBB under a gradient of 1 mV per finger: the offset it adds, in mV, rounded to a whole number
.lib sky130 tt
.func grad(p) {p}
.subckt otag inp out vdd gr a1=0 a2=1 a3=2 a4=3 b1=4 b2=5 b3=6 b4=7
IB vdd nb 20u
XMB nb nb 0 0 sky130_fd_pr__nfet_01v8 W=8 L=1
XM5 tail nb 0 0 sky130_fd_pr__nfet_01v8 W=8 L=1
XM3 x x vdd vdd sky130_fd_pr__pfet_01v8 W=8 L=1
XM4 out x vdd vdd sky130_fd_pr__pfet_01v8 W=8 L=1
* M1 = fingers A1-A4 (gate = +), M2 = fingers B1-B4 (gate = out)
EA1 ga1 inp gr 0 {-grad(a1)}
EA2 ga2 inp gr 0 {-grad(a2)}
EA3 ga3 inp gr 0 {-grad(a3)}
EA4 ga4 inp gr 0 {-grad(a4)}
EB1 gb1 out gr 0 {-grad(b1)}
EB2 gb2 out gr 0 {-grad(b2)}
EB3 gb3 out gr 0 {-grad(b3)}
EB4 gb4 out gr 0 {-grad(b4)}
XA1 x ga1 tail 0 sky130_fd_pr__nfet_01v8 W=2.5 L=1
XA2 x ga2 tail 0 sky130_fd_pr__nfet_01v8 W=2.5 L=1
XA3 x ga3 tail 0 sky130_fd_pr__nfet_01v8 W=2.5 L=1
XA4 x ga4 tail 0 sky130_fd_pr__nfet_01v8 W=2.5 L=1
XB1 out gb1 tail 0 sky130_fd_pr__nfet_01v8 W=2.5 L=1
XB2 out gb2 tail 0 sky130_fd_pr__nfet_01v8 W=2.5 L=1
XB3 out gb3 tail 0 sky130_fd_pr__nfet_01v8 W=2.5 L=1
XB4 out gb4 tail 0 sky130_fd_pr__nfet_01v8 W=2.5 L=1
.ends
VDD vdd 0 1.8
VIN in 0 0.9
VGR gr 0 0
X1 in out vdd gr otag a1=0 a2=1 a3=2 a4=3 b1=4 b2=5 b3=6 b4=7
.control
dc VGR 0 1m 1m
let mv = floor(1000*(v(out)-v(out)[0])+0.5)
meas dc offset_mv FIND mv AT=1m
.endc
.end