MOSFET 1962 Verify Channel Inverts Capacitance Voltage Curve

# Verify the Channel Inverts: Measuring the MOS Capacitor's C-V Curve

## 1. Why a Capacitance Measurement Proves What a Simple Electrical Test Cannot

This step verifies Step 5's three electrostatic regimes directly, by sweeping the gate voltage on a simple test capacitor built from the same gate-oxide-silicon stack this series has built all along, while measuring the small-signal capacitance between gate and substrate at each point along the sweep — a technique this project has never needed before, because no earlier device in this project's history had an internal electrostatic state worth measuring as a capacitance rather than as a current or a voltage. In accumulation, majority carriers crowd the interface and the whole structure behaves as a simple parallel-plate capacitor set entirely by the oxide, so the measured capacitance equals $C_{\text{ox}}$ directly. Moving into depletion, a region stripped of free charge appears beneath the oxide and acts as a second capacitor in series with it, and since two capacitors in series always measure smaller than either one alone, the measured capacitance falls. What happens past the threshold voltage Step 5 identified is where this measurement becomes genuinely revealing, and genuinely dependent on how fast the measurement itself is made.

$$\frac{1}{C_{\text{meas}}} = \frac{1}{C_{\text{ox}}} + \frac{1}{C_{\text{dep}}(V_G)}$$

where $C_{\text{meas}}$ is the capacitance actually measured at a given gate voltage $V_G$, $C_{\text{ox}}$ the fixed oxide capacitance Step 2 set, and $C_{\text{dep}}(V_G)$ the depletion capacitance, which shrinks as the depletion region widens with increasing gate voltage until it saturates near the same strong-inversion condition Step 5's equation already identified — this series relation alone predicts a capacitance that falls toward a minimum and stays there, but it says nothing about what happens if the minority carriers forming the inversion layer can actually respond to the measurement's own AC signal, which is exactly the question this step's measurement is built to answer.

One Curve That Depends on How Fast You Measure the C-V curve of a MOS capacitor, low frequency against high frequency gate voltage, VG → Cmeas Cox low frequency — carriers keep up, C rises back toward Cox high frequency — carriers can't keep up, C stays at Cmin VT 1/Cmeas = 1/Cox + 1/Cdep(VG) — true until the curves split past VT the split itself is the proof that minority carriers are actually accumulating, not merely predicted to

## 2. Real Diagram: A Small AC Signal, Riding on a Slowly Swept Gate Bias

The apparatus below is deliberately simple — a DC bias slowly swept across the gate of a bare test capacitor, with a small AC signal superimposed to probe the capacitance at each bias point, and a meter reading the resulting small-signal capacitance directly.

DC Sweep, Small AC Probe, a Capacitance Meter Reading a bare gate-oxide-silicon test capacitor, no source or drain required DC BIAS SWEEP + AC PROBE GATE / OXIDE / SILICON TEST CAP CAPACITANCE METER RECORDED C-V TRACE the measurement reads the same structure's electrostatic state, not its current

## 3. This Project's First Measurement of an Internal State Rather Than a Terminal Behavior

Every verification step this project has performed before this one measured a terminal quantity — a breakdown voltage, a resistance, a switching time, a thickness read off an interference fringe — a number describing how a finished device behaves from the outside, under a specific operating condition. This step measures something different: not what current flows or what voltage is sustained, but the electrostatic state of the surface itself, inferred from how much charge a small additional voltage can add or remove at a given bias point. That inference only works at all because this device's threshold voltage, unlike any earlier series' turn-on condition, corresponds to a genuine change in what kind of charge sits at the interface — and the specific detail that reveals that change unambiguously, the frequency-dependent split between the two curves past threshold, has no equivalent in anything this project characterized before 1962, because no earlier device's internal state ever depended on whether minority carriers had enough time to respond to a probe signal.

Step 6 does not merely confirm that this device turns on near the voltage Step 5 predicted; it measures the microscopic electrostatic event — an interface switching from depleted to inverted — that Step 5 could only describe, turning a theoretical claim into a curve any later engineer can read directly off an instrument.

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