Day 7: Body effect & channel-length modulation — sweep the 2N7000
Beyond the ideal square-law
The clean square-law from Day 6 is a first approximation. Two second-order effects matter enough to be interview staples: the body effect (threshold isn't fixed) and channel-length modulation (saturation current isn't perfectly flat). Today you'll see both in a real 2N7000 MOSFET's measured curves.
Body effect
V_T is only constant if the source and body are at the same potential. When the source sits above the body (Vsb > 0), the threshold *rises*: V_T = V_T0 + γ·(√(Vsb + 2φF) − √(2φF)), where γ is the body-bias coefficient. This bites in stacked transistors — the upper NMOS in a 3-input NAND pull-down has its source lifted above ground, so it's harder to turn on. It's a real reason series stacks are slower.
Channel-length modulation
In saturation the channel is pinched off near the drain, but raising Vds pushes the pinch-off point slightly back toward the source, *shortening* the effective channel and nudging Id up. We model it as Id = (k/2)·Vov²·(1 + λ·Vds). The small slope this adds in saturation is a finite output resistance `ro ≈ 1/(λ·Id)` — the MOSFET analog of the BJT's Early effect, and the thing that limits analog gain.
Illustrative 2N7000 output characteristics: steep triode rise, then a gently-sloped saturation plateau (that slope is λ).
Measuring the 2N7000
Set a fixed Vgs, sweep Vds, and measure Id across a small sense resistor with the Arduino ADC; repeat for a few gate voltages to build the family of curves above. Then match what you see to theory: the steep initial rise is triode, the plateau is saturation, and the plateau's slight upward tilt is channel-length modulation — its slope gives you λ.
Don't expect textbook numbers
The 2N7000 is a discrete power-ish part with V_T ≈ 2.1 V — nothing like a 0.4 V logic transistor, and it can sink enough current to need a sensible sense resistor and current limit. You are matching shapes and regions, not absolute values. Getting a real device to trace out the curves you drew on paper is the win.
Key terms
- Body effect
- The rise in V_T when the source is biased above the body (Vsb > 0); worst for stacked transistors.
- Body-bias coefficient γ
- The parameter setting how strongly Vsb raises the threshold voltage.
- Channel-length modulation
- The gentle rise of Id with Vds in saturation, from the pinch-off point moving toward the source.
- λ (lambda)
- The channel-length-modulation parameter; the (1 + λ·Vds) factor and the saturation slope.
- Output resistance ro
- ro ≈ 1/(λ·Id): the finite slope of the saturation curve; sets the ceiling on analog gain.
- Early effect (analogy)
- The BJT counterpart of channel-length modulation — same idea, different device.
Ship for Day 7
In a 3-input NMOS NAND pull-down stack, why is the transistor nearest the output node harder to turn on than the one at ground?
