The exponent that is a square

Every active device in this collection has been an exponential. The diode, the bipolar transistor, the differential pair, the current mirror and the bias point all rest on one law — collector current proportional to the exponential of base-emitter voltage over the thermal voltage — and so does every number that has come out of them. Fifty-nine and a half millivolts of base-emitter voltage per decade of current. A one per cent distortion edge at 1.03 millivolts of drive. A matched pair whose offset drifts by exactly the offset divided by the absolute temperature. Each of those is the exponential seen from a different side, and none of them would survive a device that obeyed a different law.

A field-effect device obeys a different law. Its drain current in saturation is proportional to the square of the amount by which the gate voltage exceeds a threshold, and the quantity that appears everywhere is that excess — the overdrive — rather than a thermal voltage. Nothing about the temperature is in it. Nothing about a fundamental constant is in it. What is in it is a geometry and a mobility, which is to say a fabrication process, and the square law’s whole character follows from that.

The interesting part is that the device obeys the other law too. Below threshold it does not stop conducting; it conducts an exponential, with an ideality factor in front of the thermal voltage and a decade of current every seventy-seven millivolts. Above threshold it does not go on rising as a square either, because the carriers stop moving faster. So a real device passes through the square law rather than obeying it, and the question this essay is about is where — and how wide the passage is.

Two errors, and the coincidence between them #

The square law is wrong in two places for two reasons, and the reasons have nothing to do with each other.

Below, it is too small. It says the current is zero at zero overdrive and the device says it is 2.12 microamperes, because the exponential underneath has not finished. At fifty millivolts of overdrive the square law gives 45% of the current the device carries; at a hundred millivolts, 82%.

Above, it is too large. It says the current goes on rising as the square, and the device says it does not, because a carrier’s drift velocity saturates: past a field of a few megavolts per metre a carrier does not go faster, so the current stops being proportional to the field and becomes proportional to the charge alone. In this model that enters as a division by $1 + V_{ov}/V_c$ with $V_c = E_c L$ — a length appearing as a voltage. At a volt of overdrive on a half-micron channel the square law gives exactly 1.5 times the current the device carries, and the one-and-a-half is $1 + 1/2$ with nothing else in it.

Two errors with opposite signs is a different situation from one error. It means the square law is not right on one side and wrong on the other: it is exact at one overdrive, where the subthreshold lift and the velocity-saturation loss cancel, and that overdrive is a coincidence rather than a property. For a half-micron channel it is 156.5 millivolts, where the device carries 24.5 microamperes.

That single point is the whole of what a designer gets for free. Everything else is a band around it, and the band is what the slider on the figure above moves.

A model that becomes a point #

At a quarter of a micron the square law is inside one per cent from 128.7 to 134.2 millivolts of overdrive — a factor of 1.04. At half a micron, 152.0 to 161.7 millivolts, a factor of 1.06. At a micron and a quarter, 1.12. At twenty-five microns, 3.98.

Read the other way, that is a statement about when the square law was a good model. A channel of six microns is a device of about 1980; a channel of a quarter of a micron is a device of about 1998; and the expression that every account of the subject leads with went from covering a factor of four in overdrive to covering a factor of four per cent over that interval. It did not become wrong. It became a point.

Loosening the tolerance changes the size of the effect and not its shape. At ten per cent the same two channels give factors of 1.55 and 1.97, and at twenty-five microns, 35. The band always exists, it always shuts as the channel shortens, and the reason it shuts is that one of its two edges is fixed by a thermal voltage and the other by a length.

The lower edge deserves a sentence of its own, because it is the only quantity here that a process cannot move. Weak inversion is diffusion, its slope is $nV_T\ln 10$, and the ideality factor $n$ is between about 1.1 and 1.5 for anything made of silicon. So the overdrive at which the subthreshold lift falls below one per cent is set by the temperature and a number near unity — 263 millivolts for a long channel — and no amount of fabrication moves it. Both edges of the band are outside the designer’s control and only one of them is outside the fabricator’s.

What the square law is most confidently wrong about #

The expression a designer actually uses is not the current. It is the transconductance divided by the current — how much gain a device gives back for the current put through it — and it is the quantity on which the square law makes its largest and most attractive error.

For the exponential device the answer is famous and flat. $g_m = I_C/V_T$, so $g_m/I_C = 1/V_T$: 38.68 per volt, at every current whatever, from a nanoampere to an ampere, decided by the temperature alone. There is nothing to design.

The square law gives $g_m/I_D = 2/V_{ov}$, which is a different kind of answer entirely — it depends on the bias, and it grows without limit as the overdrive falls. Read as it stands, it says a device biased gently enough gives arbitrarily much transconductance for its current, and at twenty millivolts of overdrive it promises a hundred per volt: nearly three times what a bipolar transistor can do.

It stops at $1/nV_T$. Not approximately — that is what a weak-inversion exponential’s efficiency is, by the same one line of algebra that gives the bipolar’s, and the only difference between the two devices is the ideality factor. So the field-effect device’s ceiling is the bipolar’s ceiling divided by $n$: 29.76 per volt against 38.68, and the ratio is 1.3000 exactly, carrying no dimension, no current and no temperature.

That is the sharpest statement this essay has. A designer who wants transconductance for the least current has a hard limit, it is the same limit for both device families up to a factor of $n$, and the square law does not contain it.

The cost, in a number a designer feels #

Take a stage carrying 121 microamperes. A bipolar transistor at that current has a transconductance of 4,678 microsiemens. A field-effect device biased through a source resistor to the same current has 584.8 — exactly eight times less — and the reason is entirely in the overdrive it needs to get there: 379 millivolts against a thermal voltage’s 25.9.

The comparison is not a criticism of the device. It is the trade the device offers, and the other half of it is that no current flows into the gate at all, so the input impedance is a capacitance rather than a resistance and there is no $\beta$ anywhere in the answer. The mirror essay’s 1.32 per cent base-current error, which is a whole transistor’s worth of circuitry to repair, does not exist here.

The bias point is a solution here too #

Nothing above chose an overdrive. The overdrive is an answer, and getting it is the same procedure this field has used since its first essay: Newton’s method on the whole netlist, with the current law rebuilt from the device’s own expression afterwards to check it.

A common-source stage with a gate at 1.2 volts, a kilohm in the source and twenty in the drain settles in eleven iterations to a residual of $6.7\times10^{-16}$ — a part in $10^{15}$ of the largest current in the circuit. The source resistor and the gate voltage decide the current; the current decides the overdrive through a law with no closed inverse; and the answer is 379 millivolts of overdrive at 121 microamperes.

At that bias the square law’s $2/V_{ov}$ says the efficiency is 5.276 per volt and the device gives 4.835 — nine per cent optimistic, at an overdrive well outside the one per cent band, which is exactly where a real stage is biased. A designer who uses the square law for the current and the square law for the transconductance makes two errors in the same direction, and the second is the one that shows up in the gain.

Where this leaves the small-signal boundary #

The limits field computes an amplitude at which a small-signal model of an exponential device is one per cent wrong: 7.30 millivolts for gain error and 1.03 millivolts for distortion, both of them a fraction of the thermal voltage and neither of them a fraction of a supply rail.

The square-law device’s version of that boundary is a different number and, more importantly, a different kind of number. Linearising a square is exact to second order in a way linearising an exponential is not: a square has no third derivative at all, so a perfectly square-law device driven by a sinusoid makes a second harmonic and nothing else, at an amplitude that is a fraction of the overdrive rather than of a thermal voltage. With 379 millivolts of overdrive that fraction is large, which is the useful half of the trade — the same device that gives eight times less transconductance takes about fifteen times more drive before it distorts.

That comparison is worth making carefully rather than quoting, and it is what the second rung of this argument is for. What matters here is only that the boundary moves from a constant of nature to a bias condition, which is the same shift the whole essay has been about.

What the model is for #

None of this makes the square law a bad model. It makes it a model with a stated range, which is what every model in this collection is asked to be, and the range turns out to be narrow and to depend on a fabrication detail rather than on anything a circuit designer controls.

What survives outside the range is more useful than the expression itself. The ordering survives: more overdrive is more current and more transconductance, always. The efficiency ceiling survives and is exact. And the shape of the trade survives — transconductance against drive headroom, bought with overdrive — which is why designers who left the square law behind decades ago still think in overdrive.

What does not survive is any number computed by squaring, at any bias a real circuit uses.

What changes when the law is a polynomial #

Every other device in this field is an exponential, and a square law changes three of its results and leaves one. The distortion a linear model cannot have is the harmonic content, which for a pure square law terminates at the second harmonic instead of continuing. What a resistor in the emitter buys is the degeneration, which now changes an exponent rather than a scale. The copy, and its two errors loses its first error entirely, because there is no base current. What does not change is A bias point is a solution, not a choice — the operating point is still a root — and How small is small signal still has a boundary, at a different amplitude and for a different reason.

What is checked #

Four assertions, and the second is the one that would catch a wrong device model rather than a wrong number.

That the two errors have opposite signs, so the square law is exact at one overdrive rather than wrong on one side — asserted as a sign change located on the continuous error curve, with the band around it widening when the tolerance is loosened.

That far below threshold the device is an exponential with the diode’s own decade slope, times the ideality factor. That is a check on the interpolation itself: a model that merely bent smoothly between two regimes would pass every other assertion here and fail this one, because the slope it reaches is a number the interpolation was never told.

That the ratio of the two efficiency ceilings is the ideality factor exactly, to the last bits of a double, and that the device approaches its own ceiling from below at every overdrive drawn without ever exceeding it. A ceiling that a curve crosses is a fit.

And that the stage’s operating point closes Kirchhoff’s current law to a part in $10^{15}$, with the currents rebuilt from the device’s own law rather than from anything the iteration produced.

Which of this field’s results are about exponentials and which are about devices #

A field built on one device law has an obvious question hanging over it, and this essay is the place to answer it: how much of what the semiconductors field has measured is a fact about transistors and how much is a fact about $e^x$?

The arithmetic that comes straight from the exponential does not survive. How small is small signal’s 7.3 millivolts, the distortion a linear model cannot have’s 1.03 millivolts and the 59.5 millivolts a decade are all statements about the thermal voltage in an exponent, and a square-law device has none of them — its distortion is a second harmonic that stops at second order, and its transconductance goes as the square root of the current rather than in proportion to it.

What does survive is everything about symmetry and everything about feedback. What a pair cancels, and what it only halves turns on the transfer characteristic being odd, which is a property of the arrangement rather than of the law, so a square-law pair cancels its even harmonics at the arithmetic’s floor in the same way. And what a resistor in the emitter buys is about local feedback linearising a curve, which works on any monotone curve and differs only in the exponents.

Which is the useful reading of a device that obeys both laws in different regions. Below threshold it is the field’s exponential and every number transfers; above it the numbers change and the structure does not. That is a stronger claim than either half alone, and it is available only because the same solver can be handed either law and asked the same questions.

A range measured in a ratio rather than a value #

The square law being within one per cent over a factor of 1.06 in overdrive is the narrowest range in this field, and the way it is stated is worth noticing. It is a ratio — a factor of 1.06 — rather than a voltage, which makes it a statement about the shape of the transition between two laws rather than about where either of them sits.

That is the same construction a band rather than an edge uses for a switch, where the band is between 49.5 ohms and 1.01 megohms and both ends belong to the same part. A range with two ends and a stated ratio between them is what a model has when two mechanisms bound it, and the ratio is the durable number: it survives a change of device size, a change of bias and a change of process, in a way neither endpoint does.

The two errors bounding this one point in opposite directions, which the essay’s own gate asserts, and that is what makes the ratio meaningful rather than an artefact of where the tolerance was set. A range bounded on one side is a threshold with a tolerance attached; a range bounded on both is a property of the object.

At half a micron the factor of 1.06 is narrow enough that the square law is a convenience rather than a description, which is the finding worth carrying out of this essay. What survives is not the law but the pair of laws and the transition between them — a device that is exponential below threshold, approximately square above it, and neither over a band whose width is a property of the process rather than of the design.