Before you can read a single formula on the parent page, you need a small toolbox of ideas. This page builds every one of them from nothing, in an order where each idea leans only on the ones before it.
Every symbol below describes some part of a transistor. So let us draw one first, with zero jargon.
Figure 1 — A transistor is an electric switch. The purple gate on top sits above a teal channel; orange source/drain blocks are the two ends; the black arrow shows current flowing through the channel when the switch is ON.
The three parts you must be able to name:
Gate
the control terminal; its voltage turns the switch ON or OFF.
Channel
the strip the current flows through when ON.
Source & drain
the two ends current flows between (in through source, out through drain).
The oldest meaning of "node" was a length, so we start with length.
Now the word that the node number used to equal:
Figure 2 — Pitch vs. half-pitch. Teal bars are repeated metal lines; the orange double-arrow spans one full wire plus one gap (the pitch p); the plum arrow marks half of it (p/2).
Because area is a length times a length (a 2-D thing), shrinking each length by k shrinks area by k×k=k2. That squared is the single most important idea for "why density doubles."
Power formulas need three electrical ideas. Build them in order.
Figure 3 — Energy to fill a capacitor is the shaded triangle. As charge grows from 0 to CV (x-axis), voltage rises linearly from 0 to V (y-axis); the orange triangle's area is 21CV2, the energy stored.
Now two more symbols, then we turn energy per switch into power:
Frequency f
how many times per second a switch could flip (in hertz).
Activity factor α
the fraction of switches actually flipping each cycle (between 0 and 1); not every transistor toggles every time.
A perfect switch leaks nothing when OFF. Real transistors do. These symbols describe it.
The parent writes Ileak∝e−qVth/(nkBT). Decode each new symbol:
e(…)
the exponential function — the "runaway" curve; a small drop in Vth multiplies leakage many-fold.
q
the charge of one electron (a fixed constant of nature).
kB
Boltzmann's constant (energy per degree of temperature). We write it kB — notk — so it is never confused with the shrink factor k of section 2.
T
absolute temperature (hotter chip = more leakage).
n
the subthreshold ideality factor. Physically it measures how well the gate voltage reaches the channel: n=1 is a perfect gate (all gate voltage controls the channel); n>1 means some voltage is "lost" to capacitance in the body, so the switch turns off more sluggishly. Real devices have n≈1.1–1.5; a flat (planar) short transistor has a larger n (worse control), which is exactly why leakage got so bad and why FinFET/GAA — which push n back toward 1 — were invented.
Everything above rolls up into the three goals the parent calls PPA (Power, Performance, Area).
Power — the two pieces we just built, added:
P=dynamic: cost of switchingαCV2f+static: cost of leakingVIleak
Performance — we need a symbol for how fast one switch flips. That time is the gate delay td: how long it takes to charge the bucket C up to voltage V using the transistor's drive (ON) current Ion.
Area / density — the metric that actually improves:
Every arrow means "you must understand the source before the target makes sense." The whole map funnels into the parent topic: the process-nodes topic.
