Circuit Response
every model has an edge, and the figure carries it
Concept

Channel length — where it appears

The distance a field-effect transistor's carriers cross from source to drain under the gate. It sets how early velocity saturation bends the square law, and with it the transconductance, the output resistance and the bias at which temperature stops mattering.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The band closes over a stage's own bias at 7.93 microns. computed by solving, not by drawing. The overdrive over which the square law is within 1.0 per cent, against the channel length that sets it — the velocity-saturation voltage is Ec·L, so the axis is a size and not a bias. The shaded region is the band; the curve through it is the overdrive at which the square law is exact, which exists at every length because the subthreshold and velocity-saturation errors have opposite signs. Both edges move: the lower one from 79.6 mV to 252.9 mV and the upper from 81.2 mV to 1200.0 mV, so the band is a factor of 1.021 at 0.050 µm and 4.7 at 30 µm. The fourth curve is the overdrive a common-source stage with a fixed gate voltage and a fixed source resistor solves to, which barely moves at all; it leaves the band at 7.929 microns and is outside it for every shorter device. What the square law would have said about that stage is the last two rows: 220 per cent too much current on the 0.050 µm device and 53 per cent too much efficiency, against 0.15 and 0.66 per cent at 30 µm.

The length that is a voltage

The square law's band is closed from above by a parameter that is not a bias, a current or a temperature: it is the channel length, wearing a voltage's units. Swept, the band goes from a factor of 4.74 on a thirty-micron device to 1.021 at fifty nanometres — and the overdrive a stage actually biases itself to barely moves at all, so the two cross at 7.93 microns and every shorter device is biased outside the band. The band was also measured in the wrong quantity: the square law is exact in the current somewhere at every length, and its error in the transconductance is never below 42.5 per cent at fifty nanometres and reaches one per cent only above 6.502 microns.

semiconductors · Square law A half-micron field-effect device's current against its gate voltage at 250, 300, 350 and 400 K, and the gate voltage at which they agree. computed by solving, not by drawing: mobility falling as T^−1.5, threshold falling 2 mV a kelvin, velocity saturation at 2 V. Below the crossing a warmer device draws more, because its threshold has fallen; above it less, because its mobility has. The curves agree at 1.7000 V, 1.0000 V of overdrive — the closed form's 2κT·vc/(1.5·vc − κT) — where 300 K draws 0.667 mA and 250 and 400 K draw 0.669 and 0.673 mA.

The overdrive at which temperature cancels

A field-effect device's current has two temperature coefficients pulling opposite ways: its threshold falls about two millivolts a kelvin, which raises the current, and its mobility falls as the temperature to the −1.5, which lowers it. They cancel at one overdrive, (4/3)κT = 0.800 V for a long channel, and the device's curves at every temperature cross there. Velocity saturation moves the crossing up — to 1.000 V at half a micron and 2.400 V at 0.15 µm — and, if the saturation velocity does not itself change with temperature, removes it altogether below a hundred nanometres. Below the zero a warmer device takes more current, the bipolar runaway in a device said to be free of it: 26 per cent more for 10 K at 0.1 V of overdrive.

semiconductors · Square law

Named alongside it

The objects these essays reach for when they reach for this one.

Design tradeoffOverdrive voltageSquare lawVelocity-saturationModel rangeSubthreshold conductionTemperature coefficientThermal runawayTransconductanceTransconductance efficiency

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