MOSFET Channel Charge Profile: A 2D Gradual-Channel Model

Input & Supply

Input voltage V_in 0.60 V

Supply V_DD 1.20 V

Transistor Sizing

Width ratio W_p/W_n 2.20

Body / Junction Doping

Relative body doping N_a 1.00×

Live State

Output V_out

1.20 V

Channel current

0.0 µA

Transistor regions

NMOS: cutoffPMOS: sat

NMOS pinch-off x_p/L

—

PMOS pinch-off x_p/L

—

NMOS drain depletion W

—

PMOS drain depletion W

—

How this 2D model differs from the 3D scene

The 3D scene solves a single lumped 0D current per transistor (I vs. V_out) by bisection and animates carriers riding a fixed sinusoidal path — it never resolves anything along the channel. This 2D simulator instead spatially resolves each device with the gradual-channel approximation (GCA): the channel potential V(x) between source (x=0) and drain (x=L) obeys

I·dx = k·(V_ov − V(x))·dV(x)   [charge continuity along the channel]
⇒ V(x) = V_ov − √( V_ov² − 2·I·x / k )

Integrating this ODE (verified independently by Runge–Kutta stepping against the closed form below) gives the local inversion-charge density Q_i(x) = k·(V_ov − V(x)) at every point, not just the terminal current — that is what the tapering channel band actually plots. In saturation the profile reaches V_ov exactly at the drain end (x_p/L = 1, the idealized long-channel pinch-off), after which this model (like the 3D one) does not add channel-length modulation, so I stays a function of V_ov alone.

A second, entirely new mechanism has no counterpart in the 3D scene at all: the drain–body pn junction is modeled as a one-sided step junction whose depletion width grows with reverse bias and shrinks with doping,

W(V_r) = W₀ / √(N_a/N_a0) · √(1 + V_r / V_bi),   V_bi = 0.7 V

drawn as the dashed boundary eating into the drain diffusion. Because V_out sets the NMOS drain's reverse bias directly and V_DD−V_out sets the PMOS drain's, this depletion width visibly breathes in and out as the inverter switches — a real electrostatic effect the 3D model's carrier animation cannot show because it has no notion of space at all.

Width ratio W_p/W_n — same role as in the 3D scene: re-centers V_M by scaling k_p, which changes where each channel's charge profile pinches.
Relative body doping N_a — new here: higher doping compresses the depletion region (W ∝ 1/√N_a), the same trade-off real process engineers make against latch-up and junction breakdown.

NMOS inversion charge Q_i(x) PMOS inversion charge Q_i(x) channel potential V(x) depletion boundary

⚠ Couldn't start the canvas2D canvas context failed to initialize. Reload the page.