Two millivolts a kelvin, and the wrong sign

A bias point is a solution established the thing this field is built on: an operating point is not a choice but the root of a transcendental equation, found by Newton’s method on the whole netlist, and the diode “drop” that a schematic annotates with 0.7 V is whatever that root turns out to be. It measured 59.53 mV per decade of current at every supply and every series resistance tried, agreeing with $nV_T \ln 10$ to a part in 10⁵.

Every one of those numbers was computed at 300 K, and the model had no temperature in it at all.

That is a larger omission than it sounds, and the reason is arithmetic rather than physics. The effects this field spends whole essays on are millivolt-scale: distortion arrives at 1.03 mV of drive, gain error at 7.30 mV, a differential pair’s two boundaries at 10.4 and 18.2 mV. A diode’s drop moves by about two millivolts for every kelvin. A room that warms by ten degrees moves it further than every amplitude boundary in the field put together.

Two things depend on temperature and they pull opposite ways #

The diode’s law is

$$I = I_s\left(e^{V/nV_T} - 1\right)$$

and temperature appears in it twice.

The thermal voltage $V_T$ = kT/q is linear in absolute temperature and nothing else. It is 21.543 mV at 250 K and 34.469 mV at 400 K, and it is the same number for every semiconductor device ever made, because it contains two physical constants and a temperature and no material property whatever. Read the law with only this in mind and the conclusion is that the drop should rise with temperature: a larger $V_T$ needs a larger V to reach the same exponent.

The saturation current $I_s$ does not behave like that at all. It goes roughly as T³ times an exponential in the band-gap energy over kT, and the exponential dominates everything. Across the same 150 K it goes from 9.99 × 10⁻¹⁹ A to 1.20 × 10⁻⁹ A — nine orders of magnitude, for a temperature ratio of 1.6.

So the drop is what the exponential needs to be in order to carry a stated current against an $I_s$ that has grown by a factor of a billion, and the billion wins by a wide margin. The drop falls.

This is the site’s habit applied to a sign rather than to a boundary: an expression that is correct and, read in isolation, points the wrong way. The thermal voltage is the visible half of the model and the saturation current is the half that decides.

The two routes #

The slope is measured by least squares on thirty-one solved operating points. It is also evaluated from the closed form that the same law gives when differentiated at constant current,

$$\frac{dV}{dT} = \frac{V - V_g - 3V_T}{T}$$

where $V_g$ is the band-gap voltage extrapolated to absolute zero and the 3 is the T³ prefactor’s contribution. The two share the diode’s parameters and no arithmetic: one is thirty-one Newton solves and a regression, the other is one division.

They give −1.8284 mV/K and −1.8299 mV/K, which is 0.08% apart.

The agreement is the check, and what makes it a real check rather than a restatement is that the closed form contains $V_g$, which never enters the numerical route at all — it is buried inside the temperature dependence of $I_s$ that the solver only ever sees as a number. A sign error or a factor in either place would show immediately.

The rule of thumb is a point on a curve #

“Two millivolts per kelvin” is one of the most durable numbers in electronics, and the measurement says something more precise and more useful: it is 1.828 mV/K at one milliampere, and it moves.

The steps between those rows are 0.1985, 0.1984, 0.1984 and 0.1984 mV/K, which is not a coincidence and is the second exact result in this essay. Differentiating the closed form with respect to log₁₀ I at fixed temperature gives $V_T$ ln 10 / T, and at the midpoint of this range that is 64.48 mV / 325 K = 0.1984 mV/K per decade of current. Every decade of bias moves the temperature coefficient by the same amount, and the amount is the decade slope divided by the absolute temperature.

So the rule of thumb is a reading of a curve at a conventional point. A device biased at ten microamps drifts a quarter more than the rule says, and one at a hundred milliamps a fifth less. That is not a small correction in a circuit whose whole purpose is to hold a reference.

The other slope goes the other way #

The bias-point essay’s central measurement was 59.53 mV per decade of current, and an earlier explanation of why it was not constant had to be corrected — it is, and the figure now measures it rather than printing it.

What that essay could not say is that it is 59.53 mV at 300 K, because it had one temperature. It is $nV_T \ln 10$, so it is proportional to absolute temperature, and across this range it goes from 49.6 mV at 250 K to 79.4 mV at 400 K.

The two slopes are worth holding side by side, because they are the same device and they move in opposite directions:

• the drop at fixed current falls, at about 1.8 mV/K, driven by the saturation current;
• the change in drop per decade of current rises, at $(k/q)\ln 10$ = 0.198 mV/K, driven by the thermal voltage and by nothing else.

The second is the cleaner number of the two. It has no material property in it, no band gap, no prefactor — the assertion in the figure checks it against $V_T$ ln 10 at the coldest point measured and it agrees to a part in a thousand. It is the same quantity that makes a bandgap reference possible, and it is the subject of the next rung.

Where the curvature comes from #

The drop against temperature is not quite a straight line, and the figure draws a straight line through the middle precisely so that it is not quite. Over 150 K the departure is about 1.5 mV.

The reason is in the closed form. The slope is $(V - V_g - 3V_T)/T$, and $V$ is itself falling as the temperature rises — so the numerator grows more negative while the denominator grows, and the two do not cancel. The curvature is what is left over.

This matters for the same reason the linear coefficient does. A reference trimmed to be correct at two temperatures is not correct between them, and the residue is this curvature. In a bandgap reference it is the dominant error after the first-order term is cancelled, it is a few parts per million per kelvin squared, and it is why references are specified with a temperature box rather than a coefficient.

What it does to a bias network, which is where it is usually met #

The abstract statement is that the drop moves 1.8 mV/K. The concrete one is what that does to a circuit that was designed without it, and the bias-point essay’s own arrangement is the shortest demonstration.

A resistor from a supply to a diode fixes the current at roughly (supply − drop)/R. From a five-volt supply the drop is a seventh of the total, so a 274 mV change over 150 K moves the current by about 6% — a mild effect, and the reason a large supply is the standard defence.

From a one-volt supply the same 274 mV is more than a third of the headroom, and the current changes by nearly a factor of two. From a supply just above the drop, the arrangement stops working entirely somewhere in the range: at 250 K a 0.75 V supply delivers essentially nothing and at 400 K it delivers whatever the resistor allows. That is not a degradation but a qualitative change, and it is inside the temperature range a consumer product is specified over.

The version that bites hardest is the one where the diode is a transistor’s base-emitter junction and the current it sets is a collector current that heats the transistor. The junction warms, its drop falls, the bias network delivers more current, the transistor dissipates more, and the junction warms further. Whether that loop settles or runs away is decided by the sign and size of exactly the coefficient this essay measures against the thermal resistance from junction to ambient — and the loop is a feedback loop in the sense the feedback field means, with the collector current as the signal and 1.8 mV/K as part of the gain.

This site does not model self-heating, and that boundary is worth stating plainly rather than implying: every operating point in this field is solved at a stated ambient temperature with no path from dissipation back to temperature. The measurement above is what the loop’s gain would be built from; the loop itself is not here.

What the model still does not contain #

This site’s rule is that no model is drawn without the condition under which it stops applying, and the temperature-aware diode has several that this essay has not measured.

The ideality factor is held at 1. Real junctions run between 1 and 2 depending on which recombination mechanism dominates, and the mechanism that dominates changes with current — so n is weakly a function of bias as well as of temperature. Every slope above would move proportionally.

Series resistance is not in the device. At a hundred milliamps a few tenths of an ohm of bulk resistance contributes tens of millivolts, and it has a temperature coefficient of its own with the opposite sign. The figure’s highest current is where this omission first matters.

The band gap is a constant here and is not one. $V_g$ itself falls with temperature by about 0.3 mV/K. The closed form above uses its extrapolated value at absolute zero, which is the standard treatment and is exactly the kind of extrapolation this collection is elsewhere suspicious of.

Each of those is a correction to a number rather than a change of sign, which is why the essay stands without them. They are stated because a model whose limits are unstated is the thing this site is against.

What this does to the field’s other numbers #

Three essays in this field carry a number that is now known to have a temperature attached, and it is worth saying which and by how much rather than leaving the correction implied.

A bias point is a solution measures 59.53 mV per decade. That is $nV_T \ln 10$ at 300 K and it is the quantity that moves most simply of any here: multiply by T/300 and it is right anywhere.

The distortion a linear model cannot have measures the amplitude at which an exponential’s distortion reaches one per cent — 1.03 mV — and the amplitude at which gain error reaches one per cent, 7.30 mV. Both are amplitudes compared against $V_T$, so both scale with absolute temperature in exactly the same way, and their ratio of 7.06 is temperature-independent. That ratio was the essay’s actual claim, which is why it survives.

What a pair cancels measures a differential pair’s two boundaries at 10.4 and 18.2 mV and finds them 1.75 times apart because the pair’s leading distortion term is the third harmonic rather than the second. Same argument: both are amplitudes in units of $V_T$, the ratio is what was claimed, and the ratio does not move.

That is a pleasant result and it is not an accident. The field’s claims are ratios and orders — first order against second, this boundary against that one — and a quantity that enters every amplitude as a common scale factor cancels out of all of them. What does not cancel is the operating point itself, which is why the drop is where the temperature dependence lands and why this essay is about the drop.

Why a fixed current is the right thing to hold #

The measurement above holds the current constant and watches the voltage move. It could equally have held the voltage and watched the current, and that version is far more dramatic: at a fixed 650 mV the current changes by more than four orders of magnitude across the same 150 K.

Holding the current is the right choice, and the reason is that it is what a circuit does. A diode is almost never driven by a voltage source — a source of 650 mV into a junction is a source in series with nothing, and the first figure of the bias-point essay is about exactly why that arrangement has no useful operating point. What supplies a diode in practice is a resistor from a larger supply, or a current mirror, and both of those hold the current far more tightly than the voltage. So a fixed-current sweep is the sweep a circuit performs on its own when the room warms up, and the voltage change is what the rest of the circuit sees.

It is also what makes the comparison with the amplitude boundaries fair. The bias current sets $g_m$, $g_m$ sets the gain, and the amplitude boundaries are in units of $V_T$. Holding the current holds the first two fixed and lets the essay ask about the third.

The one place the drift is useful #

Everything above treats the temperature coefficient as an error, and there is one construction in which it is the signal.

Because the drop’s coefficient is negative and $V_T$’s is positive, a weighted sum of the two can have a coefficient of zero. That is the whole idea of a bandgap reference: take a junction’s drop, which falls at about 1.8 mV/K, add a term proportional to the difference between two junctions’ drops at different current densities — which is $V_T$ ln(ratio) and rises at $(k/q)\ln(\text{ratio})$ per kelvin — and choose the ratio so the two cancel. The sum lands near the band-gap voltage extrapolated to absolute zero, which is where the name comes from.

Both halves are measured in this ladder. The falling term is this essay’s 1.828 mV/K. The rising term is the 0.198 mV/K per decade above, which is exactly $(k/q)\ln 10$ and contains no material property at all — which is why a reference built this way depends on a resistor ratio and a current ratio rather than on anything about the silicon.

What is left after the cancellation is the curvature, and that is the paragraph above: a few parts per million per kelvin squared, irreducible by a first-order trim, and the reason a precision reference is specified over a temperature range rather than at a coefficient.

The rule this model stops being true under #

Every figure on this site carries the frequency, amplitude or size at which its model stops applying. For the field’s whole first three essays the missing condition was a temperature, and it was 300 K.

1.828 mV/K at a milliamp, 0.198 mV/K per decade of bias, and 49.6 to 79.4 mV per decade across 250 to 400 K. Those three numbers are what “at 300 K” was hiding, and the first of them is larger than every amplitude boundary this field has measured.

What the collection does with a coefficient this large #

Two millivolts per kelvin against a distortion edge at 1.03 millivolts and a small-signal edge at 7.3 is a coefficient that dominates its own subject, and the rest of the collection deals with it in exactly three ways.

Cancel it. What matching does about temperature is the standard answer and its result is exact: a pair cancels the 1.8 mV/K entirely, and what it leaves is the thermal voltage times the logarithm of a saturation-current mismatch — proportional to absolute temperature, so drifting at 3 333 parts per million per kelvin at every mismatch, because the ratio contains nothing about the device.

Use it. The sensor inside its own answer turns the coefficient into the instrument, and finds the price: a thermometer 4.26 kelvin warmer than what it is measuring, under-reporting every change in ambient by 9 584 parts per million because the sense current’s own dissipation falls as the reading rises, and out by 583 millikelvin at 85 degrees on a part calibrated at 25.

Or let it close a loop. The loss that depends on what it causes is where a temperature coefficient becomes a feedback path rather than an error — a ferrite whose saturation flux falls and whose permeability rises with temperature, making the temperature a fixed point with a stable root at 89 degrees and an ignition root at 191.

Which is the useful frame. A coefficient this size is not an error term to be budgeted; it is a mechanism, and the design question is which of the three things is being done with it.

What the coefficient reaches #

A temperature coefficient on a junction is not a correction to one number; it is a correction to every number in the field. A bias point is a solution, not a choice is the root that moves. The copy, and its two errors is where two junctions at the same temperature cancel most of it, and What a pair cancels, and what it only halves is where the cancellation is exact by symmetry rather than approximate. What matching does about temperature is the measurement of what is left after the cancellation, which is the offset divided by the absolute temperature. The distortion a linear model cannot have moves with it too, because the thermal voltage is the amplitude scale of the whole field, and Every model has an edge is where that scale becomes a boundary.