Where the models stop

The constant that is a window

A diode's ideality factor is quoted as a number and defined as a derivative, which means it has a value at every current and no value anywhere. Eight one-decade fits to one curve return factors from 1.23 to 1.98, two of them straight to a few parts in a thousand — so the residual gives no warning at all. Asked for the forward voltage at a milliamp, the window containing it is right to a third of a millivolt and the worst is out by 186, which is a current a hundredth of the truth.

Assumes: The one current a constant is right at · How small is small signal · A bias point is a solution, not a choice

The one current a constant is right at took the 0.7-volt diode apart. The forward drop is the solution of a transcendental equation, it moves about sixty millivolts per decade of current, and quoting one number for it is quoting a point on a curve. The replacement offered there was the exponential law, and the exponential law has two constants in it: a saturation current and an ideality factor.

This rung is about those two constants, and the finding is the same shape one rung up. They are not constants either. They are the output of a straight-line fit over a window of current, they describe that window very well, and what they say about any other window is wrong by a factor.

The 0.7 volt constant, solved over eight decades of current. The forward voltage moves 59.5 mV for every factor of ten in current, so over the range drawn here it runs from 0.298 V to 0.774 V. The three marked points are solutions for 1 V, 5 V and 12 V through a kilohm, found by Newton's method; they span 88 mV. — Fig. 1 The law the constants belong to: current exponential in voltage, sixty millivolts a decade at an ideality factor of one. Everything below is about what happens when a real junction is asked to be this and the fit is taken somewhere in particular.

What the number is actually defined as

The ideality factor is the reciprocal slope of the logarithm of current against voltage, in units of the thermal voltage:

$n = (1/V_T) \cdot dv/d(\ln i)$ .

That is a derivative. It has a value at every current on the curve, and it is a constant only if the curve is a single exponential — which is to say, only if the device has exactly one conduction mechanism and no series resistance.

A junction has at least three things going on. Recombination in the depletion region carries the current at low bias and its factor is near two. Diffusion carries it above that and its factor is near one. And the bulk and contact resistance adds a voltage proportional to the current, which is not a junction mechanism at all but which a reciprocal slope cannot distinguish from one.

Fig. 2 The derivative, drawn. It starts at 0.67 at a nanoamp, where the recombination current is so near zero bias that the law’s own minus one pulls the reciprocal slope down; climbs to 1.99 near a microamp, where recombination runs as the exponential it is; falls as diffusion takes over, to 1.33 at a milliamp; and then climbs without stopping — 3.35 at a hundred milliamperes, which is the series resistance being read as ideality.

The device drawn here is built from those three: two exponentials in the junction voltage and a series resistance between the junction and the terminals, which makes the terminal law implicit and therefore solved rather than written down. It is not offered as a better diode. It is offered as something with a known answer to fit, so that what an extraction returns can be compared against what generated the curve instead of against another extraction.

Eight windows, eight answers

Take the curve over eight decades of current, from a nanoamp to a hundred milliamps, and fit ln(i) against v by least squares over each decade in turn. That is the extraction everybody does, and least squares in the logarithm is the right thing to do to a quantity whose errors are multiplicative — the same reasoning the digits the arithmetic did not have applies to relative error generally.

The eight answers run from 1.227 to 1.984, and they are not monotonic.

One diode curve, eight one-decade fits, and eight different ideality factors. computed by solving, not by drawing. A junction with two conduction mechanisms — recombination near n = 2 at low current, diffusion near n = 1 above it — and a series resistance, which is what a real diode is. Fitting ln(i) against v over each decade in turn returns an ideality factor for each, and they run from 1.227 to 1.984 without being monotonic: the factor rises through the recombination region, falls through the diffusion region, and rises again where the series resistance takes over. Two of the windows are straight to a few parts in a thousand, so the residual gives no warning. The bars are what each fit predicts for the forward voltage at 1 mA: the worst is out by -186 millivolts, which is a current 0.01 times the truth. — Fig. 3 One curve, eight fits. The lowest window straddles the junction’s approach to zero bias, the next ones sit in recombination, and the factor then falls through the diffusion region and rises again where the series resistance takes over — and beside each point is what that fit predicts for the forward voltage at a milliamp.

Now the part that makes this a trap rather than a caveat: two of those windows are fitted by a straight line to a few parts in a thousand. The one from a hundred nanoamps to a microamp has a root-mean-square residual of one point four thousandths in the logarithm; the one above it, one point seven. On a log plot they are straight lines with points on them.

So the residual — the one diagnostic an extraction naturally produces — reports that the model fits beautifully. It is telling the truth. A two-mechanism curve is very nearly a single exponential over one decade, because one mechanism dominates there. The fit is excellent and the numbers are about a decade.

What a good fit predicts somewhere else

Ask each of the eight fits for the forward voltage at one milliampere, and compare against what the curve itself says there, which is 0.6393 volts.

The window that contains a milliamp gets it right to three tenths of a millivolt. Of course it does: it was fitted there. The worst window is out by a hundred and eighty-six millivolts.

On an exponential a voltage error is a current error, and the exchange rate is brutal. A hundred and eighty-six millivolts at this ideality is a factor of about eighty in current — the fit that was straight to a part in a thousand predicts a current a hundredth of the truth. A window in the middle of the range is out by thirty-six millivolts, which is a current three times too large.

Fitting over the whole eight decades at once — which sounds more honest, and is what somebody who suspects the problem would try — returns 1.758 and misses the milliamp by eight millivolts. That is better than the worst window and worse than four of the eight, and it has a residual an order of magnitude larger than the good windows had, so at least it says something is wrong.

This is precisely the shape the exponent nobody put in found in a magnetic core: five windows on one measured loss curve giving Steinmetz exponents from 1.578 to 2.843, the tightest fit being the worst extrapolation, and a single fit across everything returning a sensible-looking number that misses by seventy-two per cent. The two devices have nothing in common and the failure is identical, because the failure is about fitting a power law to something that is not one.

Why the residual carries no warning

It is worth being precise about why the one diagnostic that comes free is useless here, because the same reasoning applies to every fitted coefficient in this collection.

A residual measures how well the model describes the data it was fitted to. It is a statement about interpolation. Extrapolation error is a different quantity and there is no general relationship between them — and here the relationship runs the wrong way. The windows with the smallest residuals are the ones deepest inside a single mechanism, which is exactly where the fitted pair is furthest from what any other region needs.

So the ranking is inverted: a residual of one point four thousandths goes with an error of a hundred and eighty-six millivolts, and a residual of nine hundredths — sixty times worse — goes with an error of forty-seven. The tightest fit is the worst prediction, on the same curve, in the same units.

The only diagnostic that would have caught it is the one nobody runs: fit twice, over two different windows, and see whether the two agree. Two fits that agree are evidence the law holds across the span between them. Two fits that disagree by half a unit in n are a statement that it does not, and it costs one extra least-squares solve.

The Steinmetz exponent is a local slope, and how far it moves is a property of the material. computed by solving, not by drawing. Loss per cycle against peak flux density over three decades, marched on a play-operator core, with the local exponent d ln W / d ln B drawn across the top of the same frame. It is not a constant anywhere: 2.797 at 5.5 millitesla, heading for the three that Rayleigh's law gives, and 1.462 near saturation where the material has run out of magnetisation to give — a range of 1.420. How wide that range is is itself a property of the material: over the same amplitudes a soft core's exponent moves by 1.73 and a hard one's by 0.21. A single power law fitted across the whole range returns β = 2.518 and misses by 72.2 per cent; the same law fitted over the quarter of it from 9.7 to 24 millitesla returns 2.743 and misses by 0.97. Below 0.58 millitesla this discretisation has no loss at all, which is the finite operator count showing and not the material; the sweep starts above it. — Fig. 5 The identical failure in a different field, drawn: windows on one magnetic loss curve, each fitted tightly, each giving a different exponent, and the tightest fit extrapolating worst. Two devices with nothing in common and one arithmetic mistake.

The saturation current is worse

The ideality factor at least stays inside a factor of two. The saturation current — the intercept — does not.

It is an extrapolation of a straight line back to zero volts, from a window that may be six decades away. A small error in the slope becomes an enormous error in the intercept, and the two are strongly correlated: a fit that returns a larger n also returns a larger $I_S$ , and the two errors partly cancel inside the window and compound outside it.

That correlation is the reason the pair is worth more than either half. Quoting a device’s saturation current alone is close to meaningless, and quoting it without the window it came from is entirely so.

A diode's drop from 250 to 400 K, at 1.00 mA. computed by solving, not by drawing. Thirty-one operating points, each Newton's method on the exponential at its own temperature. The drop falls at 1.828 mV/K measured against 1.830 mV/K from the closed form — falls, although the thermal voltage in the exponent rises, because the saturation current rises faster. Over the same range the slope per decade of current goes the other way, from 49.6 mV to 79.4 mV, because that one is Vₜ ln 10 and nothing else. — Fig. 6 The other reason the pair travels badly: the saturation current is the quantity with the temperature dependence in it, and it rises 4.46 times for every ten kelvin, doubling every 4.49. The doubling every ten kelvin usually quoted belongs to a different current, the reverse leakage. An $I_S$ extracted at one bench temperature and one window carries two conditions and usually neither.

The window a data sheet supplies

A signal diode’s data sheet does not print an ideality factor at all. It prints a forward voltage at two or three currents — say 0.72 volts at ten milliamps and 0.62 at one — and that is a strictly better specification than a fitted pair, because it is a measurement rather than an extrapolation from one.

Two points determine a slope, and the slope those two determine is the local ideality factor somewhere between them. Sixty millivolts per decade of current corresponds to $n V_T \ln 10$ , so a hundred millivolts across a decade is n = 1.68 — an average over the decade, honestly bounded by the two currents it was taken between, and considerably more useful than a number to three figures with no range beside it.

The complaint this rung makes is therefore not about diode data sheets. It is about what happens when somebody needs a model, reaches for the two-constant exponential because that is what a simulator wants, and extracts it from whichever part of the curve was convenient — most often the leakage measurement, because it is the one an instrument takes without heating anything.

The 0.7 volt constant, solved over eight decades of current. The forward voltage moves 80.4 mV for every factor of ten in current, so over the range drawn here it runs from 0.402 V to 1.045 V. The three marked points are solutions for 1 V, 5 V and 12 V through a kilohm, found by Newton's method; they span 144 mV. — Fig. 7 The law with a middling ideality factor in it, which is what such an extraction typically produces. It is right along one stretch of the real curve and it is a straight line everywhere, which the real curve is not.

Where the model actually is right

None of this says the exponential law is wrong. It says it is a local law, and the range over which it holds is the range over which one mechanism dominates.

That range is narrower than it looks, and not where it looks. Held to one per cent of a logarithm, this device’s terminal current is a single exponential over 2.40 decades, from 50 nanoamps to 12.6 microamps, with an ideality of 1.98 — in the recombination region, not the diffusion one. Recombination still carries 97.6 per cent of the current at ten microamps and 82.8 per cent at a hundred, and the two mechanisms carry equal shares only at 0.80 milliamps, so the span from ten microamps to ten milliamps that looks like one diffusion law is a crossover between two. That is exactly the kind of range this collection asks every model to state, and the logarithm is in the collector measures it and finds where a far longer one is kept.

What is not legitimate is carrying a pair extracted at a microamp to a milliamp, which is what happens whenever a low-leakage measurement is used to characterise a part that will conduct.

What the thermal voltage is doing in the definition

There is a quantity in n’s definition that is not a property of the device, and it deserves a paragraph because it is where a second window sneaks in.

n is defined with $V_T = kT/q$ in it — about 25.85 millivolts at three hundred kelvin — so the ideality factor extracted from a measurement depends on the temperature the measurement was taken at, through a factor nobody usually records. A ten-kelvin error in the assumed junction temperature is a three per cent error in $V_T$ and therefore a three per cent error in n, which is 0.04 on a factor of 1.3.

That is small compared with the half-unit spread above, and it is not negligible compared with the precision people quote n to. A part specified as n = 1.02 has been measured to a precision that requires the junction temperature to be known to a few kelvin, and a junction dissipating a milliwatt in a plastic package is not at ambient.

The junction temperature is itself a fixed point of the kind the loss that depends on what it causes solves — the dissipation is i·v, which depends on the temperature, which depends on the dissipation. At a nanoamp that loop is irrelevant and at a hundred milliamps it is not, which means the top of the extraction range has a self-heating error in it that pushes n in the same direction the series resistance does.

The consequence in a circuit

A forward drop wrong by tens of millivolts is not an abstract complaint.

A bias point is a solution makes the general point: a diode-connected reference or a bias string is a solve, and the solve’s answer moves with the device’s law. Thirty-six millivolts of error at room temperature is about one and a half thermal voltages, and a current source biased through such a string is out by the exponential of that.

What a 0.7 V constant costs, in the quantity it is used to predict. computed by solving, not by drawing by Newton's method on the exponential at 94 supplies through four resistors. The model is exact at 5.748 mA — the current at which the true drop is 0.7 V — and every curve crosses zero there, at four different supplies. Below it the model is low and above it high, and how much depends on the headroom rather than on the diode. Through the 87 Ω curve the drop is 49 mV out at 0.725 V and 147 mV out at 150.7 V — a factor of 3.0 — while the error in the current falls from -66% to 0.10%, a factor of 674, because the headroom underneath it has grown by 2017. On that curve the model is inside one per cent only above 1.12 V. — Fig. 8 The error the constant-drop model makes, from the rung below. The point of this rung is that replacing that constant with a fitted exponential does not remove the error — it moves it, from a number that is obviously an approximation to a pair of numbers that look like measurements.

What a measurement would have to say

The honest form of the specification has three parts rather than two: the ideality factor, the saturation current, and the current range the pair was fitted over. All three, or none of them means anything.

That is the same statement the exponent nobody put in reached about a Steinmetz exponent and how small is small signal reaches about a transconductance: a coefficient extracted by fitting is a description of a window, and the window is part of the coefficient.

And there is one thing this rung deliberately does not do, which is propose a better model. A three-parameter fit — two exponentials and a resistance — would return the numbers that generated the curve, because the curve was generated by exactly that. That is a demonstration of arithmetic and not of anything about diodes. The honest position is the one every model has an edge argues for: any model has a range, more parameters buy a wider range at the cost of an extraction nobody will perform, and the useful output of this page is not a recommendation but a condition to attach to the numbers that are already in use.

The number worth carrying is the pair 1.4 × 10⁻³ and 186 millivolts: a residual that says the fit is excellent, next to an error that says it is useless. Neither is wrong. They are answers to different questions, and a residual only ever answers the first one.

Three other constants with the same defect

A number extracted under a condition and used without it is a shape this collection keeps finding, and the other three instances are worth naming together because the remedy is the same in all of them and is never “measure more carefully”.

The coefficient that is about one reading is a ceramic capacitor’s temperature coefficient: a coefficient of the one capacitance a bridge reports at zero bias, on a part whose model has two parameters — and one number cannot determine two, so every way of dividing a ±15 per cent envelope between them honours the envelope exactly while putting the working capacitance anywhere across twenty-nine points. Two parts a bridge cannot tell apart differ by 1.80 at the voltage they are used at.

The exponent nobody put in is a core loss law’s flux exponent, and it is this essay’s defect exactly: β is a local slope rather than a material constant, running from 2.94 at half a millitesla to 1.46 near saturation, so five windows on one measured curve give β from 1.58 to 2.84 and predictions three times apart at a hundred and fifty millitesla.

A floor, or a line is the same complaint about a clock’s jitter specification, where two clocks with identical picoseconds put their error twenty-eight decibels apart because a total is a projection of a spectrum.

The common remedy in all four is to print the condition beside the number — a current, a bias, a flux range, a spectrum — and in all four the condition is already known to whoever made the measurement. The edges that move with the room is the collection’s own version of the same failure committed by this site rather than by a manufacturer, which is the reason it is stated as a habit rather than as a complaint.

There is one more thing the four have in common and it is the reason none of them is caught in practice. In every case the number is extracted by a fit whose residual is excellent — 1.4 parts in a thousand here, and a Steinmetz fit that reproduces its own measured curve to a per cent there — so the usual signal that something is wrong is not merely absent but pointing the other way. A large residual would send somebody looking; a small one confirms the extraction and says nothing at all about whether the extracted quantity means what its name suggests.

Which is why the diagnostic that works is not a better fit but a second window. Fit the same curve over a different decade and compare: if the parameter is a material constant the two agree, and if it is a local slope they do not. That costs one more fit, needs no extra data, and is the whole difference between the 1.23 and the 1.98 in this essay’s own table.

That test is available to a reader as well as to whoever made the measurement, which is the useful part. A data sheet that quotes one ideality factor has already thrown the second window away; one that quotes a forward-voltage curve has not, and two points a decade apart on it recover the pair.