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Electronic Devices

The MOSFET: threshold, channel equation and operating regions

Structure, the MOS capacitor and inversion, the triode equation with proof, saturation and the square law, operating regions, second-order effects, pMOS and power MOSFETs.

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The MOSFET (metal-oxide-semiconductor field-effect transistor) is the most manufactured device in human history: a modern processor contains tens of billions of them. It works very differently from the BJT: current flows in a thin channel near the surface, and the gate voltage controls it through an insulator, drawing no DC current. In this lesson we derive its equations.

The structure

In a p-type substrate (body, B) there are two heavily doped n⁺ regions, the source (S) and the drain (D). Between them, on top of the surface, lies a very thin insulating layer (silicon oxide, or higher-permittivity materials) and above that the gate electrode (G). The distance between source and drain is the channel length L, the dimension perpendicular to the drawing is the width W.

Cross-section of an n-channel MOSFET with source, gate, drain and body
In the p substrate there are the two n+ regions of source and drain. Above the region between them, of length L, lies the thin oxide layer of thickness t_ox and above it the gate. With a large enough gate voltage a thin electron channel forms under the oxide, connecting source and drain.

With no gate voltage, between source and drain there are two back-to-back pn junctions: whatever the sign of v_{DS}, one of them is reverse biased and no current flows.

The MOS capacitor and the threshold

Gate, oxide and substrate form a capacitor with capacitance per unit area C_{ox} = \varepsilon_{ox}/t_{ox}. With t_{ox} = 2 nm, C_{ox} \approx 1.7 µF/cm².

Applying a positive voltage to the gate:

  1. the substrate holes are pushed away from the surface, leaving a depletion region with the negative acceptor ions;
  2. as the voltage increases, the surface also attracts electrons (minority carriers in p-type), which gather in a very thin layer under the oxide;
  3. beyond the threshold voltage V_t the surface electrons outnumber the substrate holes: the surface has inverted to n-type, and this layer connects source and drain. It is the channel.

Beyond threshold, every extra volt on the gate adds a charge per unit area C_{ox}(v_{GS} - V_t) to the channel: gate and channel are the two plates of the capacitor. In current processes V_t is between about 0.2 and 0.5 V.

The MOSFET equation

Theorem. For v_{GS} \gt V_t and v_{DS} \lt v_{GS} - V_t (the triode region), the drain current is

    \[ i_D = \mu_n C_{ox}\frac{W}{L}\left[(v_{GS} - V_t)\,v_{DS} - \frac{v_{DS}^2}{2}\right] \]

Proof (gradual channel approximation). Let v(y) be the channel potential at point y, from 0 at the source to v_{DS} at the drain. At that point the effective gate-to-channel voltage is v_{GS} - v(y), and the electron charge per unit area is

    \[ Q_n(y) = -C_{ox}\left(v_{GS} - V_t - v(y)\right) \]

The current is drift along the channel and is the same through every cross-section: i_D = -W Q_n(y)\,\mu_n\,\frac{dv}{dy} (charge times width times velocity, with velocity \mu_n\,dv/dy). Separating variables and integrating from source to drain:

    \[ i_D\int_0^L dy = \mu_n C_{ox}W\int_0^{v_{DS}}\left(v_{GS} - V_t - v\right)dv \]

    \[ i_D L = \mu_n C_{ox}W\left[(v_{GS} - V_t)\,v_{DS} - \frac{v_{DS}^2}{2}\right] \]

which is the formula. ∎

Saturation

The triode formula is a parabola in v_{DS} with its maximum at v_{DS} = v_{GS} - V_t. At that point the channel charge at the drain vanishes: the channel is pinched off. Increasing v_{DS} further, the pinch-off point moves only slightly towards the source and the extra voltage drops across the short depleted stretch near the drain, which electrons cross pushed by the field. The current stays at the maximum value:

    \[ i_D = \frac12\,\mu_n C_{ox}\frac{W}{L}\,(v_{GS} - V_t)^2 \qquad (v_{DS} \ge v_{GS} - V_t) \]

This is the square law. The quantity v_{OV} = v_{GS} - V_t is called the overdrive voltage, and k_n' = \mu_n C_{ox} is the process parameter.

Output characteristics of a MOSFET with triode and saturation regions
Drain current as a function of drain-source voltage for several gate voltages. The dashed curve, where the drain-source voltage equals the overdrive, separates the triode region on the left from the saturation region on the right, where the curves are almost flat.

Example. k_n'W/L = 2 mA/V², V_t = 0.5 V, v_{GS} = 1 V: in saturation i_D = \frac12\cdot 2\cdot 0.5^2 = 0.25 mA. With a very small v_{DS}, instead, the quadratic term is negligible and the transistor is a gate-controlled resistance: R_{on} = 1/(k_n'(W/L)v_{OV}) = 1 kΩ.

Operating regions

RegionConditionCurrentUse
cut-offv_{GS} \lt V_talmost zeroopen switch
triodev_{DS} \lt v_{GS} - V_ttriode formulaclosed switch, variable resistance
saturationv_{DS} \ge v_{GS} - V_tsquare lawamplifier, current source

Mind the name: MOSFET “saturation” corresponds to the BJT active region, not to BJT saturation.

Second-order effects

  • Channel length modulation. In saturation the pinch-off point moves and the effective channel gets a little shorter: i_D grows slightly with v_{DS}, as in the Early effect, written i_D(1 + \lambda v_{DS}). The output resistance is r_o = 1/(\lambda I_D).
  • Body effect. If the source is not at the substrate potential, the threshold increases.
  • Subthreshold conduction. Below V_t the current is not zero but drops exponentially, at best by a factor of 10 every 60 mV. This is why the threshold cannot be lowered as much as one would like: billions of “off” transistors would leak too much current.
  • Short channel. In modern transistors, with L of a few tens of nanometres, the velocity saturates and the square law becomes almost linear in v_{OV}.

Variants

  • The pMOS has an n substrate, p⁺ source and drain and a hole channel: voltages and currents have opposite sign and, for the same size, it conducts about half as much because of the lower hole mobility.
  • Depletion-mode MOSFETs have a channel already present at v_{GS} = 0 and a negative threshold.
  • Power MOSFETs have vertical structures with thousands of cells in parallel and an on-resistance R_{DS(on)} of a few milliohms: they switch tens of amperes in power supplies and motor drives.

The insulated gate is the MOSFET’s great advantage: at DC it draws no current, so driving it costs only the charge of the gate capacitance. It is also its weak point: the oxide is extremely thin, and an electrostatic discharge punches through it. That is why integrated circuits have the protection diodes seen in the lesson on diode circuits.

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