Monte Carlo on an amplifier's offset

Thirty dies in one netlist

Two transistors drawn alike differ by a small random threshold shift, the mismatch of E7.4. On a real wafer every die draws its own shifts, so every die has its own offset. A Monte Carlo run imitates that. It builds many copies of the circuit, gives each transistor its own random draw from the model's statistics, and measures every copy.

The lab defines OTA-A as a follower in a subcircuit, with its own 20 µA bias, and calls it 30 times: one call per die. The library keeps its mismatch switch MC_MM_SWITCH at 0; .param MC_MM_SWITCH={mm} inside the subcircuit turns it on for that copy. .options seed=5 fixes the random draws, so every run gives the same 30 dies.

Here are the 30 offsets as a histogram: how many dies fall in each 1 mV slice.

Two numbers describe the spread. The mean is the average offset, −2.62 mV. The standard deviation σ measures how far dies scatter around it: 1.83 mV. The copy with its switch off (mm=0) has no mismatch at all. Its offset, −2.17 mV, is the systematic offset of E5.4, built into the design. The mean of the dies estimates it.

The estimate has an error of its own. The mean of N dies scatters by σ/√N, here 0.33 mV, and this sample's mean lies 0.45 mV below the systematic value. σ from 30 dies is known to about ±13 % (σ/√(2N)). Sign-off uses hundreds of dies; the lab keeps 30 to run in a second.

A designer reports both numbers and signs off at the mean ± 3σ. For a normal spread, 99.73 % of dies land inside that window. Here it runs from −8.11 mV to +2.88 mV: a ±6 mV spec fails, even though every one of the 30 dies passes it.

Where the spread comes from

Each die's offset is a sum of parts, one per mismatched pair of devices. The run reads every transistor's threshold shift from the model (delvto) and builds two of them. The pair part is M2's shift minus M1's. The load part is M4's shift minus M3's, times that die's g_m3/g_m1.

The two parts add up to the whole: what is left over spreads by 24 µV. Their own spreads are 1.39 mV for the pair and 1.55 mV for the load. The two halves of OTA-A matter about equally.

Pelgrom's law (E7.4) predicts each part by hand. For a pair of devices, σ(ΔV_th) = A_VT/√(WL). The SKY130 file gives 3.356 mV·µm per NMOS and 5.856 mV·µm per PMOS, and a pair's difference is √2 times that: A_VT = 4.75 mV·µm for the NMOS pair and 8.28 mV·µm for the PMOS pair. The NMOS model also multiplies small devices by a size factor, 1.05 at 10/1.

1. Pair

   4.75 mV·µm × 1.05 / √(10 × 1) = 1.58 mV.

2. Load, at its own gates

   8.28 mV·µm / √(8 × 1) = 2.93 mV.

3. Load, referred to the input

   × g_m3/g_m1 = 0.567 from the operating point: 1.66 mV.

4. Total

   Independent parts add as powers: σ = √(1.58² + 1.66²) = 2.29 mV.

The hand budget says 2.29 mV; the 30 dies say 1.83 mV. This sample is low: its pair part spreads 1.39 mV (law: 1.58) and its load part 1.55 (law: 1.66). The parts are independent in the model, yet in 30 dies they lean against each other, so the total is below √(1.39² + 1.55²) = 2.08 mV.

Grow the device that dominates

The team's 40/4 pair cut the pair part by more than four, as Pelgrom promised. The load part did not move, and it was already the larger half. Here are all four sizings on the same 30 dies.

Growing both halves four times in area (20/2 and 16/2) halves σ, 1.83 → 0.88 mV, with less than half the area of the 40/4 pair alone. Growing the load to 32/4 next to the 40/4 pair brings σ to 0.44 mV, under the team's 0.5 mV.

Area is the expensive lever: halving σ costs four times the area and the gate capacitance. A smaller g_m3/g_m1 (a load at lower g_m/I_D) shrinks the load's share instead. And offset can be removed after the fact: trimming adjusts each part once at test, chopping and auto-zeroing cancel it continuously.

Try it

Run it and read the five meas lines: the mean, σ and the systematic offset, then the lowest and highest die with its number after at=. Open Plots to see off, the offset of each die against its number.

Now change .param wp=10 lp=1 to wp=40 lp=4 and predict σ before you run, using the budget above. Then grow the load too (wl=32 ll=4). Last, change the seed: the sample changes, and σ moves by roughly the ±13 % that 30 dies allow.

Check yourself

* five OTA-A followers with mm=0 on the .subckt line: the spread in microvolts, rounded
.lib sky130 tt
.options seed=5
.subckt ota inp out vdd mm=0
.param MC_MM_SWITCH={mm}
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
XM1 x inp tail 0 sky130_fd_pr__nfet_01v8 W=10 L=1
XM2 out out tail 0 sky130_fd_pr__nfet_01v8 W=10 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
.ends
VDD vdd 0 1.8
VIN in 0 0.9
X1 in o1 vdd ota
X2 in o2 vdd ota
X3 in o3 vdd ota
X4 in o4 vdd ota
X5 in o5 vdd ota
VK k 0 0
.control
op
compose os values v(o1)-v(in) v(o2)-v(in) v(o3)-v(in) v(o4)-v(in) v(o5)-v(in)
dc VK 1 2 1
let uv = floor(1e6*stddev(op1.os)+0.5)+0*v(k)
meas dc sigma_uv FIND uv AT=1
.endc
.end