The overdrive at which temperature cancels

The exponent that is a square set a field-effect device beside the junction this subject had used for everything before it, and found it obeying two laws — an exponential below threshold, a square above — with a narrow band where the square is right. The length that is a voltage found the band’s upper edge set by the channel’s length, wearing a voltage’s units as the velocity-saturation voltage, and the band collapsing at short channels. Both measured the device at one temperature.

A junction’s temperature behaviour was a single number with a sign: two millivolts a kelvin, and the wrong sign found a diode’s drop falling with temperature at a rate set by its current. A field-effect device has two numbers and they have opposite signs. Its threshold falls with temperature, so at a fixed gate voltage a warmer device is further above threshold and draws more. Its carriers’ mobility falls with temperature, so a warmer device conducts less for the same overdrive. Which wins depends on the overdrive, and at one overdrive neither does. This essay finds where, what the channel’s length does to it, and what it means for two devices sharing a current.

The device, warmed #

The device is the earlier essays’: a β of two milliamperes per volt squared, a threshold of 0.7 volts, a subthreshold exponential joined to the square by a smooth interpolation, and velocity saturation as a voltage $v_c$ in the denominator, $I = \tfrac12\beta V_{ov}^2/(1 + V_{ov}/v_c)$. What is new is that each parameter moves with temperature, by the standard first-order laws. The mobility, and with it $\beta$, falls as $T^{-1.5}$. The threshold falls by $\kappa = 2$ millivolts a kelvin. The thermal voltage that sets the subthreshold slope is $kT/q$. And the velocity-saturation voltage is the saturation velocity times the length over the mobility, so as the mobility falls it rises, as $T^{1.5}$ — with the saturation velocity itself held constant, which is the one assumption the short-channel results depend on and the one place a real device’s numbers would differ.

The current at four temperatures, plotted against the gate voltage, fans out on both sides of one gate voltage and passes through it nearly together. For the half-micron device that is 1.700 volts, an overdrive of one volt, where the current at 300 kelvin is 0.667 milliamps and at 250 and 400 kelvin is 0.669 and 0.673 — within one per cent across a hundred and fifty kelvin.

Where the two coefficients cancel #

The temperature coefficient of the current, $d\ln I/dT$, is the sum of the two laws’ contributions. Writing the current as $\ln I = \ln\beta + 2\ln V_{ov} - \ln(1 + V_{ov}/v_c)$ and differentiating at a fixed gate voltage, the mobility gives $-1.5/T$, the threshold gives $+2\kappa/V_{ov}$, and the velocity-saturation term gives a correction that depends on how $v_c$ moves. For a long channel the last term vanishes and the coefficient is zero where $2\kappa/V_{ov} = 1.5/T$:

$$V_{ov}^* = \tfrac43\kappa T,$$

0.800 volts at 300 kelvin. With velocity saturation the same algebra gives

$$V_{ov}^* = \frac{2\kappa T\,v_c}{1.5\,v_c - \kappa T}.$$

The device’s own coefficient, from a difference of a kelvin either side of 300, crosses zero exactly where the closed form says, to two parts in a hundred thousand at every length: 0.800 volts for a long channel, 1.000 for half a micron, 2.400 for 0.15 microns. Near threshold the coefficient is large and positive — about two and a half per cent a kelvin at a tenth of a volt of overdrive — because a small overdrive is a large fraction of itself to lose to the threshold’s two millivolts. Deep in strong inversion it is negative and small. The zero between them is where a designer can put a device whose current does not care about its temperature.

A zero that leaves at a length #

The closed form has a denominator that can vanish. As the channel shortens, $v_c$ falls, and once $1.5\,v_c$ is no larger than $\kappa T$ there is no overdrive at which the coefficient is zero.

The zero climbs as the channel shortens — 0.811 volts at seven and a half microns, a volt at half a micron, 2.4 at 0.15, 7.2 at 0.11 — and it leaves for good at $v_c = \kappa T/1.5$, 0.40 volts, a channel of a hundred nanometres. The reason is the velocity-saturated current’s own temperature law. Deep in velocity saturation the current is nearly $\tfrac12\beta v_c V_{ov}$, and $\beta v_c$ is the mobility times the saturation velocity over the mobility — the mobility cancels — so the only temperature dependence left is the threshold’s, which raises the current. With nothing to oppose it, the coefficient is positive at every overdrive.

That conclusion carries the essay’s one stated assumption. A saturation velocity that falls a little with temperature, as a real one does, puts a small negative term back and a zero back with it, at a very large overdrive; the figure’s edge is then where the zero runs off the useful range rather than off the arithmetic. The direction of the effect — a zero that climbs as the channel shortens until it is out of reach — does not depend on the assumption, and the length that is a voltage found the square law’s own band leaving at a length in the same way.

How far the zero is a point #

The zero is a statement about a derivative at one temperature, and over a wide range the curves do not all cross at one point. Pairwise, they cross at gate voltages that move up as the pair warms: 1.676 volts for 250 and 300 kelvin, 1.722 for 300 and 350, 1.767 for 350 and 400. Biased exactly at the 300-kelvin zero, the current rises 0.35 per cent at 250 kelvin and 0.95 per cent at 400: both the same sign, the curvature a first-order zero leaves behind. A bias that must hold the current over a stated range is best placed at the crossing of the range’s two ends — 1.716 volts for 250 to 400 kelvin — where the error is shared between the middle and the ends.

That is the same structure as the band-gap reference’s: the voltage at which the slope is zero found a reference’s output flat to first order at one temperature and bowed either side, and the bow a trim leaves measured what remains of that bow. A field-effect device biased at its zero is a reference of the same kind, with a bow of under one per cent over a hundred and fifty kelvin.

A current source, biased three ways #

The practical meaning of the zero is clearest in the simplest circuit that uses it: a device with its gate held at a fixed voltage, used as a current source. Biased at three-tenths of a volt of overdrive, the half-micron device draws 78.6 microamps at 300 kelvin, and its coefficient there is about 8,000 parts per million a kelvin: at 400 kelvin it draws 78 per cent more and at 250 kelvin 40 per cent less. At half a volt the coefficient has fallen to 3,200 parts per million, and the same range moves the current by +30 and −17 per cent. At the zero, one volt of overdrive, the range moves it by under one per cent.

So a current source biased at a fixed gate voltage is a poor current source almost everywhere and an excellent one at a single bias. The single bias is fixed by the device rather than by the designer — one volt of overdrive, two-thirds of a milliamp for this geometry — so to use it at another current the designer changes the device’s width rather than its gate voltage, which scales the current and leaves the zero where it is. That is the field-effect version of a choice the bipolar transistor never offers, and the next section says why.

Why a junction has no such zero #

A bipolar transistor’s collector current at a fixed base-emitter voltage rises with temperature at every bias, by roughly seven per cent a kelvin, because its saturation current grows as the cube of temperature times a Boltzmann factor in the band gap, and nothing in the device opposes that. The table that has to be refitted found the junction voltage’s coefficient at fixed current to be $(v_j - E_g/q - 3kT/q)/T$, negative at every current a junction can carry; turned round, the current at a fixed voltage always rises. There is no second mechanism with the opposite sign, so there is no zero, and every bipolar current source needs its bias to be held by feedback — an emitter resistor, a mirror — rather than by a choice of voltage.

The field-effect device has the second mechanism because its current depends on the carriers’ mobility as well as on how many there are, and mobility falls with temperature. Below threshold the device is a junction in all but name, its current exponential in the gate voltage, and its coefficient is the junction’s: about 4.6 per cent a kelvin just at threshold and rising further below it. Above threshold the exponential has become a square, the count of carriers grows only as the square of the overdrive, and the mobility’s fall can catch it. The zero is the place where the field-effect device stops behaving like a junction in this one respect.

Two devices sharing a current #

The sign of the coefficient decides something more serious than a drift.

Two devices in parallel, driven from one gate voltage, share the current according to their temperatures. Above the zero a warmer device carries less — 0.990 of the cooler’s at two volts of overdrive for a ten-kelvin difference — so it cools, and the pair shares by itself. That is the property field-effect devices are known for, and the reason they are paralleled without the ballast resistors bipolar transistors need. Below the zero the reverse holds: the warmer device carries 1.032 times the cooler’s at half a volt of overdrive and 1.264 times at a tenth, so it heats further and takes more again. That is the bipolar transistor’s thermal runaway, in a device usually said to be immune to it.

The case where it bites is the one where the overdrive is small and the dissipation large: a power device used in its linear region, as an electronic load or the pass element of a linear regulator, driven at low overdrive with a large drop across it. Paralleled devices there do not share, and the one that is slightly warmer takes the current until it fails. The bias point that heats itself found a bias point’s own dissipation moving it, and the zero is where that feedback changes sign. What a resistor in the emitter buys is the bipolar cure — a resistance whose drop rises with current — and below the zero a field-effect device needs it as well.

Two routes to the zero #

The zero has been found two ways that share only the device law. One is a bisection on the device’s own temperature coefficient, measured as a difference of currents a kelvin apart; the other is the closed form from differentiating the law by hand. They agree to two parts in a hundred thousand at every length, which is the central difference’s own error at a kelvin’s step. A model whose temperature laws were misapplied — a sign on the threshold, a wrong power on the mobility — would still give plausible curves and would move the bisected zero off the closed form at once. The long-channel limit, $(4/3)\kappa T$, is the point both meet at.

A design rule, and its numbers #

For a long-channel device the rule is short: bias at four-thirds of $\kappa T$ above threshold, about 0.8 volts at room temperature, and the current is independent of temperature to first order. At half a micron, one volt. Below a few tenths of a micron the zero climbs past any overdrive a low-voltage process allows, and a short-channel device’s current rises with temperature at every useful bias — a fact about modern devices that the long-channel rule of thumb hides. A designer who relies on field-effect devices sharing current by themselves should check which side of the zero the bias sits on, because the answer depends on the channel length as much as on the bias.

How much the zero depends on the process #

The long-channel zero, $(4/3)\kappa T$, has only the threshold’s coefficient in it, and that coefficient is a property of the process — of the channel’s doping and the oxide’s thickness — that varies from about one and a half to three millivolts a kelvin between processes. The zero moves in proportion: 0.6 volts of overdrive at one and a half millivolts, 1.2 at three. Within one process the coefficient is held far more tightly than that, so a zero measured on one device is a good guide to its neighbours, but a zero quoted from one process is no guide to another.

The mobility’s exponent enters the same way, in place of the 1.5: a process whose mobility falls as $T^{-2}$ puts the long-channel zero at $\kappa T$, 0.6 volts, and one whose mobility falls as $T^{-1.2}$ at $(5/3)\kappa T$, a volt. The closed form is a recipe for re-deriving the number from two measured coefficients rather than a number to carry from one part to the next, which is the same lesson two millivolts a kelvin, and the wrong sign drew for a junction’s coefficient: a temperature coefficient belongs to a bias point and a device, and is not a constant of the material.

What the model leaves out #

The temperature laws are first-order: a mobility exponent of 1.5 that in real silicon ranges from about 1.2 to 2, a threshold coefficient of two millivolts that depends on doping and oxide, and a saturation velocity held constant. Each moves the zero’s position, and the closed form shows how — $\kappa$ in the numerator, the mobility exponent in place of 1.5 — without changing its existence for a long channel. The device is also isothermal: a device that heats itself moves its own temperature with its current, which turns the parallel-sharing figure into a feedback loop whose stability is the question still open after this one.

The number to keep #

A field-effect device’s current stops depending on temperature at an overdrive of $(4/3)\kappa T$ for a long channel, 0.800 V at 300 K, and at $2\kappa T v_c/(1.5v_c - \kappa T)$ with velocity saturation — 1.000 V at half a micron, 2.400 at 0.15 — until, with the saturation velocity constant, the zero leaves at a hundred nanometres. Below the zero a warmer device takes more current: 26 per cent more for 10 K at 0.1 V of overdrive.

Still open: the device that heats itself, a mobility that is not a power law, and the subthreshold reference #

A device that heats itself. Below the zero a warmer device takes more current and so heats further; above it, less. With the device’s own thermal resistance in the loop, the question becomes whether the feedback below the zero is strong enough to run away, and at what dissipation — the linear-mode safe operating area that power device data sheets draw and rarely explain.

A mobility that is not a power law. The $T^{-1.5}$ law is a fit, and at high fields the mobility’s temperature dependence weakens. A device whose effective exponent falls with overdrive has a zero that moves differently, and measuring the zero against an exponent that varies would say how much of the long-channel rule survives in a real process.

A reference made of the subthreshold region. Below threshold the current’s coefficient is large and positive and very predictable, set by the thermal voltage. Combined with the strong-inversion region’s negative coefficient, two devices at different biases could cancel each other’s drift, the field-effect version of a band-gap reference, and its curvature against the bipolar one is the comparison that would follow.