Floating gate transistor operation: Foundations — Hardware deep dive

Before you can follow Floating gate transistor operation, you must be fluent in a short list of pictures and symbols. We build each one from absolute zero, in an order where every idea leans only on the ones before it.

1. Electric charge, $Q$ , and the electron

The smallest free piece of charge that matters here is the electron. We keep the size of its charge and its sign clearly separate:

Let $q$ be the magnitude (always positive) of the electron's charge: $q = 1.6 \times 1 0^{- 19} C (q > 0, just the size) .$
The electron's signed charge is then $q_{e} = - q$ , because an electron is negative.

Sign convention for this page: $q$ is a positive magnitude; the minus sign that makes charge negative is written out explicitly every time. So a group of $N$ electrons carries a total signed charge

$Q = - N q (negative, because each electron contributes - q) .$

Why the topic needs this: the entire stored bit is a pile of electrons sitting on the floating gate. $Q_{F G}$ (the charge on the floating gate) is exactly $- N q$ for $N$ trapped electrons.

Look at figure s01: eight little violet dots each stamped with a " $-$ " are the trapped electrons; the orange arrow zooms in on a single one to show its magnitude $q = 1.6 \times 1 0^{- 19} C$ , and the caption reminds you the sign is negative. The whole cluster is the $Q = - N q$ on the left.

Figure — Floating gate transistor operation

2. Voltage, $V$ — the "electric height"

Voltage is always measured between two points, so we need a naming rule for the subscripts:

Choosing $V_{c h} = 0$ just fixes the bottom of the hill; every other height is counted upward from there.

Why the topic needs it: every terminal of the transistor sits at some voltage, and the formulas relate $V_{F G}$ to $V_{C G}$ . If you don't know the subscripts name a node's height above the channel, those formulas are meaningless.

3. The MOSFET — a voltage-controlled switch

You must already know a MOSFET as a switch. Here is the minimum picture.

The magic voltage is the threshold voltage:

Look at figure s02: the magenta curve is the drain current as the gate voltage $V_{C G}$ climbs. Left of the violet dashed line (the threshold $V_{T}$ ) the current is flat at zero — the shaded "OFF" region — and the instant $V_{C G}$ passes $V_{T}$ the current lifts off into the "ON" region. That kink is the threshold.

Why the topic needs it: a floating gate transistor is nothing but a MOSFET whose $V_{T}$ we can secretly change and lock in place. Storing a bit = choosing a high or low $V_{T}$ .

4. Insulator (oxide) — the glass wall

Two oxides matter:

Tunnel oxide — thin (~8–10 nm, where 1 nm $= 1 0^{- 9}$ m, about the width of a few atoms). Thin enough that, under a violent push, electrons can sneak through.
Control/inter-poly oxide — thicker, so charge can't escape upward.

Why the topic needs it: the floating gate is wrapped in oxide on all sides. That wall is the reason trapped charge can't leak — which is the reason the memory is non-volatile (keeps data with power off).

5. The floating gate — a sealed island

Two "shores" push on this island: the control gate above it and the channel below it.

6. Capacitance, $C$ — how strongly two plates feel each other

The chip is so small we use femtofarads: $1 fF = 1 \times 1 0^{- 15} F$ .

Two capacitors matter, because the floating-gate island has two shores:

$C_{C G}$ — capacitance between control gate and floating gate (the top plate pair).
$C_{F G}$ — capacitance between floating gate and channel (the bottom plate pair).

Look at figure s03: the orange bar on top is the control gate, the magenta bar on the bottom is the channel (held at $V_{c h} = 0$ ), and the violet bar in the middle is the floating-gate island. The double-headed arrow across the top gap is labelled $C_{C G}$ ; the one across the bottom gap is $C_{F G}$ . Notice the island touches no wire — it only feels the shores through these two capacitors.

Why the topic needs it: the floating gate is the middle plate of two stacked capacitors. Its voltage is a capacitive divider of the control-gate voltage — that's the whole reason the parent formula for $V_{F G}$ looks the way it does.

7. Capacitive division and the coupling ratio $α_{G}$

The floating gate is the middle plate of two capacitors in series: $C_{C G}$ up to the control gate (at $V_{C G}$ ) and $C_{F G}$ down to the channel (at $0$ ). Let us derive its voltage from scratch using charge conservation.

Step 1 — WHAT is conserved. The island is sealed, so with no trapped charge its total charge is zero. The charge the top capacitor deposits on the island plus the charge the bottom capacitor deposits must add to zero.

Step 2 — write each capacitor's charge. Using $Q = C \times (voltage across it)$ , and taking the island-side of each plate:

top capacitor holds $C_{C G} (V_{F G} - V_{C G})$ on the island (voltage of island relative to control gate),
bottom capacitor holds $C_{F G} (V_{F G} - 0)$ on the island.

Step 3 — set the sum to zero (charge conservation, $Q_{F G} = 0$ case): $C_{C G} (V_{F G} - V_{C G}) + C_{F G} V_{F G} = 0.$

Step 4 — solve for $V_{F G}$ . Expand and collect $V_{F G}$ :

\;\Rightarrow\; V_{FG} = \underbrace{\frac{C_{CG}}{C_{CG}+C_{FG}}}_{\text{coupling ratio } \alpha_G}\, V_{CG}.$$ *Why this is a divider:* the two capacitors "share" the voltage in proportion to their sizes, exactly like two resistors in a resistor divider — the island lands at a fixed fraction of $V_{CG}$. > [!definition] Coupling ratio $\alpha_G$ > **$\alpha_G$** (a plain number between 0 and 1) says *what fraction of the control-gate voltage actually reaches the floating gate*. The picture: two people pulling a rope; $\alpha_G$ is how far your side wins. Bigger $C_{CG}$ → the control gate "wins" more → $\alpha_G$ closer to 1. **The intrinsic threshold $V_{T0}$.** The floating gate transistor still turns on when the *floating-gate* voltage reaches the MOSFET's own turn-on level. > [!definition] Intrinsic threshold $V_{T0}$ > **$V_{T0}$** is the threshold voltage the transistor would have *measured at the floating gate itself* — the built-in turn-on level of the underlying MOSFET, before any coupling or trapped charge is accounted for. It is the "$V_T$ of section 3" but named $V_{T0}$ to stress it is the *bare* value seen by the floating gate. Because we drive the *control* gate, not the floating gate, we must push $V_{CG}$ high enough that $V_{FG}$ reaches $V_{T0}$. Putting $V_{FG}=V_{T0}$ into the divider (now allowing trapped charge $Q_{FG}$) gives the **observed** threshold at the control gate: $$V_{T,\text{observed}} = \frac{V_{T0}}{\alpha_G} - \frac{Q_{FG}}{C_{CG}}.$$ **Why the topic needs it:** $\alpha_G$ and $V_{T0}$ are exactly the two symbols in that observed-threshold formula. Without the divider derivation and the definition of $V_{T0}$, the first term is a mystery. --- ## 8. Putting sign and shift together Now the payoff. Add negative charge $Q_{FG} = -Nq$ to the island. Being negative, it drags the island's voltage *down*, so the control gate must push *harder* to reach the turn-on point. The extra push you need is the **threshold shift**: $$\Delta V_T = -\frac{Q_{FG}}{C_{CG}}.$$ Read the sign carefully: $Q_{FG}<0 \Rightarrow \Delta V_T > 0$ → higher threshold → **programmed** → stores a **0**. No trapped charge → low threshold → **erased** → stores a **1**. > [!recall]- Why does adding *negative* charge *raise* the threshold? > Negative charge on the gate repels the electrons the control gate is trying to summon into the channel, so a larger $V_{CG}$ is needed to overcome it — a higher $V_T$. ::: Everything about writing that charge across the oxide is the job of [[Fowler-Nordheim tunneling]] and [[Channel hot-electron injection]] — those are the *mechanisms*, covered separately. Here you only needed the *symbols*. --- ## Prerequisite map ```mermaid graph TD Q["Charge Q and electron q"] --> V["Voltage V as electric height"] V --> MOS["MOSFET switch and threshold VT"] OX["Oxide insulator wall"] --> FG["Floating gate sealed island"] MOS --> FG Q --> FG FG --> CAP["Capacitance CCG and CFG"] V --> CAP CAP --> DIV["Capacitive divider and coupling alphaG"] DIV --> SHIFT["Threshold shift delta VT"] Q --> SHIFT SHIFT --> TOPIC["Floating gate transistor operation"] ``` --- ## Equipment checklist Test yourself — you are ready for the parent note only if you can answer each without peeking. What is the magnitude $q$ of an electron's charge, and what sign does the electron carry? ::: $q = 1.6\times10^{-19}\ \text{C}$ (a positive magnitude); the electron's signed charge is $-q$ (negative). What does the subscript in $V_{CG}$ mean? ::: The voltage of the control gate measured relative to the channel/ground reference ($V_{AB}$ = potential of $A$ relative to $B$). What does the threshold voltage $V_T$ of a MOSFET mean physically? ::: The gate voltage at which the switch just turns on; below it, no channel current. Why can trapped charge on the floating gate not leak away? ::: It is fully wrapped in insulating oxide, connected to no wire. What does capacitance $C$ measure, and what is its one-capacitor law? ::: How strongly two plates couple; $Q = CV$ (charge equals capacitance times the voltage across the plates). What is the coupling ratio $\alpha_G$, and how is it derived? ::: The fraction of control-gate voltage reaching the floating gate, $C_{CG}/(C_{CG}+C_{FG})$, from charge conservation on the middle plate (a series-capacitor divider). What is $V_{T0}$? ::: The intrinsic turn-on threshold of the underlying MOSFET, measured at the floating gate itself, before coupling and trapped charge. Adding negative charge to the floating gate does what to $V_T$? ::: Raises it, because negative charge repels the field forming the channel. Convert $1\ \text{fF}$ to farads. ::: $1\times10^{-15}\ \text{F}$.

1. Electric charge, Q, and the electron

2. Voltage, V — the "electric height"