The window that closes from the top

The window a field-effect input wins in found the band of source resistance over which a field-effect input is quieter than a bipolar input biased for each source, and found both of its ends in closed form. The floor, 2.83 kilohms, is set by the two inputs’ voltage noise. The ceiling, five gigohms, is set by the field-effect input’s gate leakage: $2V_T/(\sqrt\beta\,I_G)$, the source resistance at which the leakage’s shot noise costs what the bipolar’s base current does. A nanoamp of leakage brought the ceiling down to five megohms.

It ended by naming what it had held fixed. The leakage is a reverse-biased junction’s current, and a reverse-biased junction’s current rises steeply with temperature — the semiconductor field’s essays measure exactly how. The ceiling is inversely proportional to the leakage, so it must fall as the input warms. This essay carries the junction’s law through the window and finds how fast the ceiling falls, the temperature at which a given sensor leaves the window, whether the window ever closes, and how wrong the familiar “doubles every ten degrees” is along the way.

The two inputs, and the leakage’s law #

The two inputs are the ones the earlier measurement compared. The bipolar input is biased at the collector current that minimises its noise figure for each source, with a current gain of a hundred and fifty ohms of base resistance; its best noise figure is $1 + r_b/R_s + 1/\sqrt\beta$. The field-effect input has a fixed milliamp of drain current, a channel voltage noise set by its transconductance, and a current noise that is the full shot noise of its gate leakage, $2qI_G$. The window’s two ends are found by bisecting the two noise figures against source resistance, not by the closed forms they are later compared with.

What is new is the leakage. A junction field-effect transistor’s gate is a reverse-biased junction, and its leakage is that junction’s reverse current at the transistor’s gate’s bias — five volts here. The semiconductor field’s model gives that current as two mechanisms: diffusion over the junction, scaling as $T^3 e^{-E_g/kT}$, and generation inside the depletion region, scaling as $T^{1.5} e^{-E_g/2kT}$ times the region’s width. The two are scaled together so that the total is one picoamp at 25 °C, the value the earlier essay used, and every temperature below takes its leakage from that law. Nothing assumes a doubling interval.

The ceiling falls with the leakage, exactly #

At every temperature the bisected ceiling times the leakage equals $2V_T/\sqrt\beta$ to within three per cent, so the ceiling is the leakage’s reciprocal with a slowly rising thermal voltage in front. From 0 to 150 °C it falls nearly four decades, from 39.5 gigohms to 5.79 megohms. Over the range in which instruments actually run — room temperature to the inside of a warm enclosure, 25 to 75 °C — it falls by a factor of 25, from 5.14 gigohms to 207 megohms.

That is a large movement for an end of a window whose other end does not move at all, and it changes what the window is. Read in noise figure rather than in resistance, the cost of being outside it grows quickly: a source ten times above the ceiling sees the leakage’s shot noise contribute ten times what the bipolar input’s base current would, and the field-effect input’s noise figure there is several decibels worse than the bipolar input’s best, not a fraction of one. At 25 °C it spans six decades of source resistance and the field-effect input is the obvious choice for anything above a few kilohms. At 75 °C it spans less than five, and a 1-gigohm source that was well inside the window at room temperature is outside it.

When a sensor leaves #

The question a designer asks is the reverse of the figure: for a given sensor, at what temperature does the bipolar input become the quieter choice?

A 100-megohm source keeps the field-effect advantage up to 88.6 °C, a 1-gigohm source to 48.6, and a 10-gigohm source only to 16.4 — below room temperature. A 100-gigohm source, a typical pH electrode or ionisation chamber, is outside the window at every temperature above −10 °C with this part’s leakage, and needs an input with far less leakage than a picoamp. Each decade of source resistance costs about 33 kelvin, which is the temperature rise over which the leakage grows tenfold.

This turns the advice “use a field-effect input for a high-impedance source” into a statement with a temperature in it. For a given part the advice is right below a temperature set by the source, and wrong above it, and the boundary for a gigohm source falls inside the range most instruments are specified for. The current the instrument draws found the same crossover for the offset a bias current causes — a field-effect input beating a bipolar one by a large factor at room temperature and losing to it somewhere above — and the noise window is the same physics read through the shot noise of the leakage rather than through its mean.

The floor does not move, and the window narrows from the top #

The window has two ends, and only one of them is a leakage.

The floor is where the field-effect input’s channel voltage noise costs what the bipolar input’s base current does, and both of those are set by the inputs’ own bias currents and by the thermal voltage, which rises only in proportion to the absolute temperature. So the floor sits at 2.83 kilohms from freezing to 150 °C and has moved only to three kilohms by 250. The window narrows entirely from the top.

It closes — the two ends meet — at 285 °C, where the input junction leaks enough that a bipolar input wins at every source. That is far above any temperature a silicon instrument survives, so in practice the window never closes; it narrows until the sensors a designer cares about fall outside it. The useful number is therefore not the closing temperature but the ceiling at the operating temperature, read against the source.

The bipolar input is held at room-temperature parameters throughout, and that is a simplification worth naming: a bipolar input’s current gain rises with temperature, which lowers its base-current noise for a given collector current and moves its best noise figure slightly in its favour. That makes the crossover temperatures above slightly optimistic for the field-effect input, not pessimistic.

Ten degrees is one temperature’s rule #

The familiar statement is that a junction’s leakage doubles every ten degrees. Carried through the junction’s law, it is true at one temperature only.

The doubling interval is $\ln 2$ divided by the logarithmic derivative of the current with temperature, and for a thermally activated current that derivative is dominated by the activation energy over $kT^2$. As the temperature rises the activation term falls as $1/T^2$, so each doubling takes longer than the last: 7.5 kelvin at freezing, ten near 50 °C, fifteen by 150 °C. The transistor’s gate’s leakage here is mostly generation — the mechanism activated by half the band gap — whose own interval at 27 °C is 9.0 kelvin; diffusion, activated by the whole gap, would double in 4.5. A part whose leakage is dominated by diffusion, which happens in wider-gap or cleaner junctions at high temperature, would double faster than the rule says.

For the window this cuts both ways. Below room temperature the leakage falls faster than the rule predicts, so cooling an input buys more ceiling than “ten degrees a doubling” would suggest; above about 50 °C it rises more slowly, so a hot instrument loses its ceiling a little less quickly than the rule threatens. The halving intervals of the ceiling — 8.5 to 15.7 kelvin across the range — are the leakage’s doubling intervals read the other way. The junction that is a resistor at zero volts used the same saturation current to set a photodiode’s noise at zero bias, and that noise inherits exactly this temperature law.

Why the leakage is shot noise and not Johnson noise #

The ceiling is set by the transistor’s gate leakage’s shot noise, $2qI_G$, and it is worth checking that this is the right noise for the current to carry, because a reverse-biased junction also has a resistance and that resistance might seem to carry Johnson noise as well. The floor a current sets found that a current’s noise and a resistor’s are equal when the voltage across the element is $2kT/q$, fifty millivolts at room temperature, and that a junction’s own noise sits between the two. An input junction reverse-biased by five volts is a hundred times past that point: its current is fully developed, its noise is pure shot noise, and its incremental resistance is enormous and contributes nothing. The junction that is a resistor at zero volts measured the opposite limit, where the same junction at zero bias is Johnson noise of its shunt resistance and nothing else.

So the temperature law that matters is the current’s, not a resistance’s, and the ceiling falls with the current directly. If an input were run with its gate near zero bias — some electrometer stages are — its gate noise would follow the shunt resistance instead, which falls with temperature by the same activation law but enters the window through Johnson noise, $4kT/R$ rather than $2qI$, and the ceiling’s dependence would change form. The window measured here is the reverse-biased case, which is the normal operating point of a junction field-effect input.

The count picture behind the density matters at the ceiling for a different reason. The count under the density found that a picoamp in a ten-kilohertz noise bandwidth is only 312 carriers a window, and that the tails of such a count are far heavier than a Gaussian of the same variance. At room temperature a picoamp of leakage sits exactly in that regime; warming the input junction raises the count and brings the tails back toward Gaussian, so a cold high-impedance input has the lower noise density and also the less Gaussian noise. For a threshold detector that is a second, opposite effect of temperature on the same current.

Cooling, guarding, and the leakage the window cannot see #

The ceiling’s form names the two remedies, and the measured curves price them. Cooling the input from 75 °C to 25 °C raises the ceiling from 207 megohms to 5.14 gigohms, a factor of 25 for fifty kelvin; that is often the cheapest way to keep a gigohm sensor inside the window, by moving the input stage away from whatever is heating it. Cooling below room temperature pays even better per kelvin, because the doubling interval shortens there, which is why the most sensitive charge and current measurements are made with cooled inputs.

Guarding addresses a different leakage. The transistor’s gate’s own junction leakage is inside the part and is what the window measures; but at the board, leakage from neighbouring conductors to the input node adds a current with its own shot noise and its own temperature and humidity law. The current that does not reach the input measured a guard ring held at the input’s own potential reducing that external leakage to almost nothing. A guard cannot touch the transistor’s gate’s internal leakage, so it does not move the ceiling drawn here; what it prevents is a second, external ceiling arriving below this one on a humid, warm board.

The thermal side has one more twist. The input’s own dissipation warms its gate, so the operating point chosen for low voltage noise — a higher drain current — warms the die and raises the leakage, lowering the ceiling. The boundary that is a starting point measured a loss that depends on the temperature it causes; the window has the same structure, and its ceiling is properly read at the die temperature that the chosen bias produces.

What the window is worth, warm #

The practical reading is a design procedure in three steps. Find the source resistance and the highest temperature the input will see — the die’s, not the room’s, since a field-effect input’s own dissipation and its neighbours’ warm it. Read the window’s ceiling at that temperature from the leakage law, not from a room-temperature data sheet line. If the source is above the ceiling, the bipolar input is the quieter choice there, and the field-effect input’s advantage at room temperature is not the relevant one.

Two remedies follow from the ceiling’s form. The ceiling is $2V_T/(\sqrt\beta\,I_G)$, so the remedy is less leakage — a part with a smaller gate junction, a MOS input whose gate is insulated rather than a junction, or an input guarded so that its leakage path is held at zero volts — and keeping the input cool, which buys a factor of two every eight to fifteen kelvin depending on where on the curve the input sits. The two generators that are one current found that a bipolar input’s two noise generators are one current divided two ways; the field-effect input’s are two unrelated things, and temperature acts on only one of them, which is why the window moves at one end and not the other.

What the measurement leaves out #

The leakage model is the junction’s own, scaled to a picoamp at 25 °C. A real gate’s leakage includes surface and package paths that do not follow the junction law, which at low temperatures can dominate and set a floor under the ceiling’s rise; below room temperature the figures here may promise more than a real part delivers. A MOS-input amplifier has no gate junction and its leakage is set by its protection diodes, which obey the same law but at a different scale, typically far smaller.

The channel’s voltage noise is taken as white. At low frequencies it has a flicker component that raises the window’s floor, and at high source impedance the transistor’s gate’s capacitance turns the channel’s voltage noise into a current noise rising with frequency, which brings the ceiling down at high frequency independently of the leakage. Both are named as open below and neither is in the figures.

And the noise figure is a spot figure at one frequency. A real measurement integrates over a band, and if the band includes the flicker region the comparison between the two inputs changes in the bipolar input’s favour at the floor, as well as in its favour at the ceiling when warm.

The number to keep #

A field-effect input’s window over a bipolar one closes from the top as it warms: the ceiling is $2V_T/(\sqrt\beta\,I_G)$ and falls as the gate leaks, from 5.14 gigohms at 25 °C to 913 megohms at 50 and 207 at 75, while the floor stays at 2.83 kilohms. A 1-gigohm sensor leaves the window at 48.6 °C and a 10-gigohm one at 16.4 — about 33 kelvin a decade of source resistance. The leakage’s doubling interval is 7.5 kelvin at 0 °C and 15.0 at 150, lengthening as the activation term falls with $1/T^2$; “ten degrees” is the rule at about 50 °C.

Still open: the flicker floor, the capacitance ceiling, and the die that heats itself #

The flicker corner as a second floor. With flicker noise in both inputs’ voltage generators the floor becomes a function of frequency, and the window at low frequency is narrower from the bottom. Solving it at each frequency for stated corners would give the frequency below which the bipolar input wins at every source.

The transistor’s gate capacitance as a ceiling of its own. The channel’s voltage noise across the input capacitance is a current noise rising with frequency. Where that ceiling crosses the leakage’s, as a function of temperature, would say which limit a warm high-impedance front end meets first.

Self-heating. A field-effect input’s drain current heats its own input junction. The leakage then depends on the bias chosen for low voltage noise, so lowering the floor raises the leakage and lowers the ceiling: a fixed point between the input’s dissipation and its window, of the kind the thermal-feedback essays measure.