The voltage at which the slope is zero

A junction’s forward drop falls by about two millivolts for every kelvin it warms. Two millivolts a kelvin, and the wrong sign measured that coefficient and found where its sign comes from: the thermal voltage in the exponent rises with temperature, the saturation current it divides into rises far faster, and the drop is what the exponential needs to be to carry a fixed current against a saturation current that has grown by many orders of magnitude. Measured at a milliamp, the drop fell 1.828 millivolts per kelvin, and the slope itself drifted, so a line through the middle of the sweep missed the ends by about a millivolt and a half.

That essay closed by describing, in words, the one construction in which this drift is the signal rather than the error. Take a junction’s drop, which falls; add a quantity that rises; choose the weighting so the two cancel. The rising quantity is the difference between two junctions run at different current densities, and the construction is a band-gap reference — the circuit inside almost every converter, regulator and supervisor that needs a voltage the room cannot move.

Described in words, the construction has three loose ends. Is the difference really free of the device, as the textbook says, or only nearly? What does the sum come out at, and why that voltage? And how is a real circuit, built from parts that are each a few per cent from their nominal values, made to have zero slope — by measuring a slope, which needs two temperatures, or by something cheaper? This essay solves the junctions and measures each of those, and the third answer turns out to be the useful one.

A difference with no device left in it #

Put two junctions side by side and run the same current through both. One has an emitter area eight times the other’s, so eight times its saturation current; at any current, the larger one needs a smaller drop. The diode law says by how much. The drop is $V_T \ln(I/I_S + 1)$, and the difference between the two is

$$\Delta V = V_T \ln\frac{I/I_S + 1}{I/(8 I_S) + 1} \approx V_T \ln 8,$$

in which the current, the saturation current and the band gap have all cancelled. What is left is Boltzmann’s constant over the electron charge, times the temperature, times the logarithm of a ratio of areas that a layout fixes.

Solved at four currents spanning three decades, the drops themselves differ by 179 millivolts — three decades of current at 59.5 millivolts a decade — and their differences do not differ at all. Every one is 53.758 millivolts at 300 K, rising 179.19 microvolts per kelvin, and the straight line through them, extended to absolute zero, arrives at a thousandth of a microvolt. Across the 200 kelvin the sweep covers, each junction’s saturation current rises by fifteen orders of magnitude. None of that reaches the difference.

The one place the device does reach it is the “+1”. At ten microamps and 400 K, the larger junction’s saturation current has climbed to about a thousandth of the current, the approximation $\ln(x + 1) \approx \ln x$ is no longer exact, and the difference departs from the line by 0.04 per cent. At ten milliamps the departure is four parts in ten million. Two currents with one name says the real departure comes sooner, because the current a junction leaks in practice is generated in its depletion region and is far larger than the forward law’s saturation current. That is the first thing worth carrying out of the solve: the difference is proportional to absolute temperature until the current is small enough that leakage is a measurable fraction of it, and a low-power reference that runs its core at a few hundred nanoamps is exactly where that stops being a footnote.

The ratio of areas sets the slope and nothing else. At 2 to 1 the difference is 17.9 millivolts and rises 59.7 microvolts per kelvin; at 24 to 1 it is 82.2 millivolts and 274. The two currents are forced equal by an amplifier or by a mirror, and the copy and its two errors is a reminder that a mirror’s copy is itself a few parts in a hundred out — an error in the current ratio enters the difference exactly as an error in the area ratio does. A designer chooses a ratio for reasons of layout — eight is a common-centroid arrangement of nine units around one — and makes up the rest of the slope with a resistor ratio, which is the weight in the next section. What matching does about temperature met the same expression from the other side, as the offset a mismatched pair cannot help having: $V_T \ln m$ is a defect when $m$ is an accident of fabrication and a reference when it is chosen.

Two slopes made equal and opposite #

One drop at a hundred microamps falls 1,921 microvolts per kelvin at 300 K, with its current held proportional to absolute temperature — the way most reference cores are biased, for a reason the last section returns to. The difference rises 179.19. Multiply the difference by the ratio of the two, 10.7229, and add it to the drop.

The weight is found by differencing the solved drops: the slope of each at 300 K, from the two temperatures a hundredth of a kelvin either side, and the ratio that makes the sum’s slope vanish. Since the sum is linear in the weight, one ratio is exact. At that weight the two slopes agree to a part in a million and the sum is 1.171704 volts.

Over the automotive range, −40 to 125 °C, that sum moves 2.53 millivolts, against the 318 millivolts the drop alone moves. Both halves of the sum are large and change a great deal; their sum is a three-hundredth of either. The 2.53 millivolts that remain are not a failure of the weight — no weight removes them — and the bow a trim leaves is about them. This essay is about the 1.171704.

A fan from one point #

Why 1.1717 volts? The drop of a junction carries its own answer. The saturation current goes as $T^{3}e^{-E_g/kT}$, and when the drop at a fixed current is written out with that in it, the logarithm of the exponential is a term $E_g/q$ that does not depend on temperature at all. As the temperature falls towards zero, the thermal voltage multiplying everything else falls with it, and the drop at any current tends to the band gap the model was given: 1.12 volts. The difference term is proportional to temperature, so it tends to zero. Every weighting of the sum, therefore, starts from the band gap at absolute zero.

Drawn from a quarter of a kelvin, five weightings of the sum fan out of one point. The heavier weights climb steeply and reach 300 K well above 1.17 volts; the lighter ones climb slowly or bend back. The one that is flat at 300 K is identified by where it is at 300 K as surely as by its slope, because a family of curves that all leave one point and do not cross is a family in which the height at any temperature orders the slopes there.

That is the reason the circuit is called what it is. Its output, in the flat condition, is the band gap extrapolated to absolute zero plus a small correction, and the correction can be written down. Differentiate the drop with the saturation current’s temperature law in it — exponent $x_{ti}$ on $T$, and the core current itself rising as $T^\alpha$ — and set the slope of the sum to zero:

$$V_\text{ref}(T_0) = \frac{E_g}{q} + (x_{ti} - \alpha)\,V_T(T_0).$$

With the model’s band gap of 1.12 volts, $x_{ti} = 3$ and a core current proportional to absolute temperature, $\alpha = 1$, that is 1.12 plus two thermal voltages, 1.171704 volts — the solved value to six figures, by a route that shares nothing with it but the diode law. One is a difference of solutions and the other a derivative taken on paper, which is the standard a number should meet before it is used.

The real silicon band gap, extrapolated linearly from room temperature, is nearer 1.21 volts than 1.12, because silicon’s band gap itself narrows as it warms; a model with a constant band gap puts all of that into the saturation current’s exponent. A silicon reference therefore sits near 1.25 volts rather than 1.17, and that number is the one on datasheets. The argument does not depend on which band gap is used: the fan leaves whichever value the junction has.

Slope read off a voltage #

The fan says something a designer can use. A reference whose slope at 300 K is wrong is a member of the family that reaches 300 K at the wrong height, and the error in height and the error in slope are the same error. From the expression above, the slope of any member at the trim temperature is

$$\left.\frac{dV_\text{ref}}{dT}\right|_{T_0} = \frac{V_\text{ref}(T_0) - V_\text{magic}}{T_0},$$

where $V_\text{magic}$ is the zero-slope voltage. A millivolt of output is 3.33 microvolts per kelvin of slope. The test of that is to build references wrong in several unrelated ways at once and ask whether the line still holds.

Two hundred references are drawn with three independent kinds of error: a saturation current scattered by a factor of 1.8, the kind of spread a process run produces; an area ratio half a per cent off; and a resistor weight one per cent off. Their outputs at 300 K spread from 1.126 to 1.217 volts and their slopes from −153 to +152 microvolts per kelvin. Every one of them lies on the line through the zero-slope voltage at a slope of one over 300 K, to 0.16 nanovolts per kelvin — the precision of the differencing that measured the slopes.

That is what makes a band-gap reference manufacturable. Measuring a slope needs two temperatures, an oven and the time for a die to settle in it, for every part. Measuring a voltage needs one temperature and a meter. Because each reference’s slope is its voltage’s distance from 1.1717 volts, a trim that adjusts the resistor weight until the output reads 1.1717 at room temperature has adjusted the slope to zero, whether the part was wrong because of its saturation current, its area ratio or its resistors. The trim that takes the drift with it found the same structure in a differential pair: a trim removes a drift only when it acts on the quantity whose temperature law the drift shares. Here the weight multiplies the term that is proportional to absolute temperature, so a single adjustment of it at one temperature is an adjustment of the slope at all of them.

A target that belongs to the bias #

The correction $(x_{ti} - \alpha)\,V_T$ has two numbers in it, and the second is a design choice. A core run at a constant current has $\alpha = 0$; a core whose current is itself made from a difference of junction drops across a resistor, the commonest arrangement, has $\alpha = 1$; some designs run the junction that supplies the falling term at a current rising faster still.

Each power of temperature in the bias lowers the zero-slope voltage by one thermal voltage, 25.85 millivolts, and the five solved references land at 1.1976, 1.1846, 1.1717, 1.1588 and 1.1459 volts to seven figures. The practical consequence is that the trim target is not a property of silicon. A designer who copies the number 1.1976 from a description of a constant-current core onto a core whose current is proportional to absolute temperature has trimmed it 25.85 millivolts high, and the part drifts 86.2 microvolts per kelvin — 74 parts per million per kelvin, worse than an untrimmed part from a good process.

The reason most cores are biased proportional to absolute temperature is partly this term and partly the circuit: a core that makes its own current from the difference voltage across a resistor produces exactly that law with no extra parts. It also changes the size of the 2.53 millivolts the sum leaves behind, which belongs to the next essay. Here the point is narrower: the voltage a reference is trimmed to encodes the law of its bias, and a production test that does not know that law is trimming to the wrong number.

A trim is a model #

The other number in the correction is $x_{ti}$, the exponent on temperature in the saturation current. It comes from the physics of minority-carrier diffusion — the intrinsic carrier density squared, and the mobility and diffusion length’s own temperature dependence — and it is 3 in the textbook, between about 3 and 4 in fitted device models, and different between processes.

A reference trimmed to the voltage an exponent of 3 predicts, built from a junction whose exponent is actually 3.5, drifts −43.1 microvolts per kelvin; at 2.5 it drifts +43.1. The relation is exact and simple: Boltzmann’s constant over the electron charge, 86.2 microvolts per kelvin, for every unit the exponent is wrong, because the exponent enters the zero-slope voltage multiplied by one thermal voltage and the slope is that voltage error over 300 K.

So the single-temperature trim moves the problem rather than removing it. It measures a voltage to a microvolt, and the voltage it compares against is only as good as the exponent the designer believes. A tenth of a unit of uncertainty in $x_{ti}$ is 2.6 millivolts in the target and 8.6 microvolts per kelvin in every part trimmed to it — about 7 parts per million per kelvin, which is the order of the temperature coefficient a precision reference is sold at, and the order at which the floor a converter sets stops being a property of its bit count. That is why such references are characterised on a few parts across temperature before the target is fixed for the production line: the oven is still there, but it is paid for once per design rather than once per part.

Where this sits among the other temperature numbers #

Every operating point in the earlier essays was computed at 300 K until the junction’s own coefficient was measured, and the edges that move with the room then re-drew the site’s boundaries across −40 to 125 °C and found most of them proportional to the thermal voltage. The reference is the one circuit in which that proportionality is turned into a constant, and it is worth being exact about what it holds constant. The sum’s slope is zero at one temperature. Its value is not a physical constant: it is the band gap plus a term the designer chose, and its residual movement over the range is the bow the next essay measures.

The sensing junction in the sensor inside its own answer runs the same difference the other way: a thermometer reads the difference term and calibrates out everything else. The two circuits share their front end and differ only in what they add to it. And the bias point that is a solution is the reminder that every drop in this essay is the root of a transcendental equation rather than a constant; the reference is built from the one combination of such roots that does not move.

Why these numbers can be trusted, and where they stop #

Each drop is the diode law solved at its own temperature, with a saturation current that rises as $T^{3}$ times a Boltzmann factor in a constant band gap; the law is evaluated in logarithms, so that the same expression holds from a quarter of a kelvin, where the saturation current is below anything a double can store, to 400 K. At 300 K it agrees with the site’s ordinary junction model to a part in a trillion. Slopes are central differences a hundredth of a kelvin wide.

Three routes cross-check each other. The zero-slope weight is found by differencing and the zero-slope voltage it produces is held against $E_g + (x_{ti} - \alpha)V_T$ to seven figures; the scatter of two hundred references is held against the line those two predict; and the fan’s common point is held against the band gap at a quarter of a kelvin to a millivolt.

What the model leaves out sets the limits of every number above. The band gap is held constant, where real silicon’s narrows by about a quarter of a millivolt per kelvin and bends, which adds its own curvature to the bow and moves the target to the familiar 1.2 to 1.25 volts. The junctions are ideal: no series resistance, no base current, no ideality factor other than one — the logarithm is in the collector found a real diode holding its logarithm to a per cent over only 2.4 decades, which is why references are built from transistors rather than diodes — and no Early effect on the transistors a real core is built from. Every one of those adds a term, and each has a temperature dependence of its own. The resistors are perfect and do not drift. And the amplifier that forces the two branches to carry the same current has no offset — in a real reference, its offset is multiplied by the same weight of ten that multiplies the difference, which is the reason chopper-stabilised cores exist.

Still open: the leakage edge, the amplifier’s offset, and noise in the difference #

The current below which the difference bends. The “+1” put four parts in ten million into the difference at ten milliamps and four in ten thousand at ten microamps, both at 400 K. A nanopower reference runs its core at tens of nanoamps, where at high temperature the saturation current is a percent or more of the current; how far down the bias can go before the departure exceeds the bow, at each temperature, is a boundary this model can draw and has not.

The amplifier’s offset, multiplied by the weight. The weight that turns 179 microvolts per kelvin into 1,921 is about ten, and any offset in the amplifier that forces the two currents equal appears in the output at roughly that gain. An offset is itself a drifting quantity, and whether a millivolt of it is better handled by the voltage trim — which absorbs its value at one temperature — or worse, because its drift is not proportional to absolute temperature, is the same question the pair’s trim answered, asked of a different source.

The noise in a difference of ten. The difference term is 53.8 millivolts amplified to 576, and the junctions’ shot noise is amplified with it. The ratio of areas trades the amplification against the difference’s size, so there is a ratio at which the output noise is least for a given current, and it has nothing to do with the layout reasons that usually choose eight.