The recovery a Schottky does have

The heat a recovery leaves behind ends its list of remedies with the one that seems to end the subject: a Schottky diode conducts by majority carriers, has no minority charge to sweep out, and so, in that essay’s words, the entire analysis does not apply to one — no recovery, no switch loss from it, no thermal loop. The sentence is the received view and the datasheets agree with it; a silicon-carbide Schottky’s sheet often prints “zero reverse recovery” on its first page.

The sheet also prints, a few lines further down, a quantity called $Q_c$, total capacitive charge, in nanocoulombs, at a stated voltage. It is there because the received view leaves something out. A diode that blocks a voltage has a depletion layer, the depletion layer is a capacitor, and charging it to the rail takes charge that has to come from somewhere. In a converter it comes through the switch that is turning on, at the moment the switch is turning on, which is exactly the path the diode that conducts backwards traced for stored charge. This essay measures how much it is, against what a fast silicon diode returns in the same place, and finds the regime in which the diode with no recovery is the worse of the two.

Two charges that depend on different things #

The silicon diode’s charge is the one the earlier essays computed. A conducting junction holds a minority charge of $\tau I_F$; driving its current down at a rate $a$ removes it by the circuit and by recombination at once, and the junction cannot block until it is gone. The charge that flows backwards is the area under the reverse current, and it depends on the forward current, on the rate it is driven down, and through the lifetime on the temperature. It does not depend on the voltage the diode is about to block, because the stored charge is gone before that voltage appears.

The Schottky’s charge is the opposite in every respect. Its junction capacitance follows the abrupt junction law,

$$C(v) = \frac{C_0}{\sqrt{1 + v/\varphi}},$$

and the charge to bring it from zero to a reverse voltage $V$ is the integral of that,

$$Q_C(V) = 2C_0\varphi\left(\sqrt{1 + V/\varphi} - 1\right),$$

which rises as $\sqrt{V}$ once $V$ is well past the built-in voltage $\varphi$. It depends on the voltage and on nothing the junction was doing before: not the forward current, not the slope, not the temperature beyond the small drift of $\varphi$. The device here is stated rather than looked up, as a 650 V silicon-carbide Schottky with $C_0 = 603$ pF and $\varphi = 1.2$ V, chosen so that it holds 25.0 nC at 400 V, which is the size of the number printed on sheets for parts in that class. The silicon diode is a 600 V ultrafast with a 20 ns lifetime at 25 °C and 30 pF of junction capacitance of its own, which is added to its stored charge rather than ignored.

The closed form for $Q_C$ is checked on every figure against Simpson’s rule on $C(v)$ itself, to a part in ten million; the silicon diode’s stored charge comes from the recovery’s closed form, which is in turn checked against a march of the same charge equation.

The two waveforms make the difference in kind visible before any comparison of size. The silicon diode’s current is a triangle whose height is set by the ramp and whose width is set by the stored charge; it would be the same at 40 V as at 400. The Schottky’s is a displacement current, $C(v)\,dv/dt$, whose height is set by how fast the switch node moves and whose duration is the time it takes to reach the rail. Faster switching makes it taller and shorter; its area is the same at any slew rate, because the area is $Q_C(V)$ and nothing else.

A boundary in voltage, at a stated current #

At one ampere forward and 100 A/µs the silicon diode returns 18.1 nC of stored charge, and its own capacitance adds 1.59 nC at 400 V. The Schottky returns nothing at zero volts and 25.0 nC at 400 V. Somewhere between, they cross, and the figure at the top of this essay draws both curves against the reverse voltage at that current. The Schottky’s rises as the square root of the voltage, 11.8 nC at 100 V and 30.9 nC at 600 V; the silicon diode’s is nearly flat at its stored 18.1 nC plus a little capacitance. They are equal at 245 V.

At 245 V and one ampere the two diodes hand the switch the same charge. Above it, the Schottky hands over more. The slider on that figure moves the forward current, which moves only the silicon curve, and it shows how fast this boundary moves with load: at 300 mA it is 42 V, at 100 mA it is 9 V, and at 3 A the silicon diode returns more charge at every voltage up to 600, 40.1 nC against 25.0 at 400 V. The boundary is a statement about the product of two things, voltage and light load, and a converter running on a 400 V bus at a fraction of its rated current sits on the wrong side of it.

Read beside the figure at the top, this one makes the shape of the trade plain. Tripling the current roughly doubled the silicon diode’s charge and moved its curve wholly above the Schottky’s, while the Schottky’s curve is identical in both pictures. One device’s charge is a property of the operating point; the other’s is a property of the bus.

There is a correction to the earlier essays hiding in the silicon number, and it needs stating before the comparison is trusted. The expression those essays use for the recovered charge is $\tfrac{1}{2}I_{RM}t_{rr}$, with $t_{rr}$ counted from the start of the turn-off ramp — that is, from the moment the current starts falling, while it is still forward. The charge that actually flows backwards is the area under the reverse current, counted from the instant it crosses zero, and the difference is $\tfrac{1}{2}I_{RM}\,I_F/a$. At one ampere, 100 A/µs and 20 ns that is 7.0 nC of a quoted 25.1, so the expression overstates the backward charge by 38 per cent at this operating point. Every number here uses the area, which is checked as the integral of the waveform drawn above to a part in ten thousand. Had it used the older expression, the silicon diode would look worse than it is and the boundary would sit lower — the comparison would favour the Schottky more than the physics does.

A boundary in current, at a stated voltage #

Turned round, the same comparison gives a current: at a fixed bus voltage, below what forward current does the Schottky hand over more?

The crossover current is located by bisection in the logarithm of the current and confirmed as a root: the two charges agree there to a part in a million. Its value, 1.42 A at 400 V, is the number worth having for a design, because the forward current is the variable a converter actually sweeps through as its load changes. A power-factor-correction stage on a 400 V bus with a ten-ampere rating runs below 1.42 A whenever it is at less than about a seventh of its load, which is much of the time for most equipment. In that region the silicon-carbide diode, bought precisely to remove the switch’s recovery loss, returns more charge to the switch than the silicon part it replaced.

The comparison is not therefore an argument for silicon. At full load the silicon diode returns 81.3 nC to the Schottky’s 25.0, more than three times as much, and at full load is where the heat is. What the boundary changes is the statement “zero recovery”. The Schottky’s charge is not zero; it is constant. Constant is a better property than small at full load and a worse one at light load, and which matters depends on where the converter spends its hours.

Not every coulomb costs the same #

The comparison so far is in charge, and the reason to care about charge is that the switch has to supply it at the full supply voltage. The heat a recovery leaves behind found that the silicon diode’s stored charge costs the switch $V Q_{rr}$: the charge passes through a transistor that is standing across nearly the whole supply, and none of that energy is stored anywhere. The Schottky’s charge is different, because it is not destroyed. It ends up on a capacitor, and a capacitor gives energy back.

The arithmetic is a familiar result with an unfamiliar exponent. The half that never arrives charges a linear capacitor from a step and finds that the resistor dissipates exactly half of what the source delivers, whatever the resistance. Here the source is the bus, the resistor is the switch’s channel, and the capacitor is the Schottky’s junction. The source delivers $VQ_C$; the junction stores

$$E_C = \int_0^V v\,C(v)\,dv,$$

and the switch dissipates the difference. A linear capacitor would make that difference one half. An abrupt junction makes it nearly two thirds, because its capacitance is largest at low voltage: most of its charge goes in while the voltage across it is small and the voltage across the switch is nearly the whole bus. At 400 V, far above the built-in 1.2 V, the ratio is 64.9 per cent, and in the limit it is exactly two thirds.

The energy is computed twice, by routes that share nothing: once as $VQ_C - E_C$ from the two closed forms, and once as the integral of the voltage across the switch times the charge it passes, $\int_0^V (V - v)\,C(v)\,dv$, by Simpson’s rule. They agree to a part in ten million.

The stored third is not free either, but it is not lost in the switch. At the next turn-off the switch node swings back down and the junction’s charge goes out into the inductor current — a lossless exchange in the ideal case. So in energy the Schottky’s charge is cheaper per coulomb than the silicon diode’s, and the crossover current falls from 1.42 A to 799 mA. The diode with no recovery loses the comparison over a smaller range of load than the charge alone suggests, but it still loses it.

Three capacitances, and which one the switch pays for #

A nonlinear capacitor has no single capacitance, and this one has three that matter, all at 400 V and all different. The slope, $dQ/dV$, is what a bridge or an impedance analyser reads and what a sheet plots as a curve: 33.0 pF. The charge-equivalent, $Q_C/V$, is the fixed capacitor that would take the same charge to the same voltage: 62.5 pF. The energy-equivalent, $2E_C/V^2$, is the fixed capacitor that would store the same energy: 43.9 pF. The capacitance that is not one number found the same three-way split on a ceramic capacitor, where a bias-dependent dielectric does what a depletion layer does here, and the lesson transfers whole: each answers a different question, and quoting one as though it answered the others is the error.

For this comparison the right one is the charge-equivalent, because the switch’s loss is set by the charge it passes and the voltage it passes it at. A designer who read the small-signal curve at the operating voltage and multiplied by 400 V would estimate 13.2 nC and halve the Schottky’s real charge — which, at the boundary, is the difference between the two diodes. The energy-equivalent is what matters for the third that comes back, and it is the one sheets now tabulate as an “energy-related” output capacitance on transistors for exactly that reason.

The exchange of that stored third is also less tidy than “returned to the load” suggests. The energy two capacitors cannot share measured how much is lost whenever charge moves between capacitances without an inductor to carry it, and the swing a diode keeps found what an inductor and a diode do to that loss. In a hard-switched converter the junction’s charge leaves through the inductor current at turn-off, which is the lossless path; in a converter that turns the switch on before the inductor has discharged the node, part of the stored third goes into the channel as well, and the energy crossover moves back toward the charge crossover.

The boundary moves with temperature, and in one direction #

The silicon diode’s stored charge grows with its carrier lifetime, and the lifetime grows with temperature; two loops, and one heatsink built a runaway out of exactly that dependence, and the pulse the heatsink does not feel showed that at converter frequencies the junction integrates its heat, so the temperature that sets the lifetime is the steady one rather than anything that happens within a cycle. The Schottky’s capacitance is set by doping rather than by any carrier’s lifetime and is taken here as independent of temperature, which is a fair approximation for silicon carbide over this range. So heat moves only one side of the comparison.

Every one of the twenty-one crossovers is located by bisection and every row falls monotonically, which is checked rather than assumed. The direction is the one that favours the Schottky: a hot silicon diode recovers worse, so the region in which the Schottky hands over more charge shrinks as the junction heats. A designer comparing the two at room temperature, where many measurements are made, sees the silicon diode at its best and sets the boundary more than twice as high as it will be in operation.

The voltage dependence goes the other way and is stronger: the boundary at 600 V is three and a half times the one at 100 V at room temperature, because the Schottky’s charge grows as the square root of the bus and the silicon diode’s barely moves with it. The light-load penalty is therefore a high-voltage phenomenon, which is why silicon-carbide Schottkys meet it — they exist to run at hundreds of volts — and low-voltage silicon Schottkys on a twelve-volt rail mostly do not.

Where the light-load charge comes from, when the load is lighter still #

The slider on the figure at the top of this essay goes down to 100 mA, where the boundary falls to 9 V and the Schottky hands over more charge at almost every voltage that matters.

This is the regime of a converter idling: a few hundred milliamperes of magnetising or standby current through a diode rated for ten amperes. The silicon diode’s stored charge shrinks with its current, roughly in proportion to it once the ramp is fast compared with the lifetime, and the Schottky’s does not shrink at all. At light enough load, a silicon diode’s turn-off is almost entirely capacitive too — its own 1.59 nC at 400 V — and a silicon-carbide Schottky is a diode with sixteen times as much capacitive charge at that voltage.

That last ratio is the whole story stated in device terms. A silicon-carbide Schottky trades a stored charge that scales with current for a capacitance that scales with area, and to carry ten amperes at a low forward drop it needs a large area. The capacitance is the price of the area, and the area is the price of a majority-carrier device carrying the current.

What the comparison does not include #

Three effects are left out on purpose and each would move the numbers.

The first is the switch’s own output capacitance, which is charged and discharged on every cycle whatever diode is chosen; it adds the same loss to both sides and so moves neither boundary, but it does make both diodes’ share of the total smaller.

The second is leakage. A silicon-carbide Schottky’s reverse leakage is low, but a silicon Schottky’s rises steeply with temperature and closes a thermal loop of its own; that is a separate argument and belongs with the conduction losses rather than the switching ones.

The third is the shape of the silicon diode’s recovery. The triangle used here, with its softness set to half the conducting time, is the model the earlier essays established and the area is what matters for charge; a snappier recovery puts the same charge through the switch faster and rings harder, which the diode that conducts backwards and its successors treat as a separate problem.

The number worth carrying #

At 400 V the silicon-carbide Schottky hands the switch 25.0 nC at every current, and below 1.42 A forward that is more than a 20 ns silicon diode does — below 632 mA at 125 °C, and below 799 mA in energy.

The habit that goes with it is to read “zero” on a specification as a statement about mechanism rather than about magnitude. A Schottky has no stored charge; that is a true sentence about how it conducts. It does not follow that it has no charge to return, and the quantity a converter pays for is the charge, whatever mechanism it is held by.

Still open: what the switch’s own capacitance does to the boundary #

The comparison here is between two diodes with the switch treated as ideal apart from its channel, and the next question is what happens when the switch’s own output capacitance joins in. A superjunction transistor’s output capacitance is strongly nonlinear, like the Schottky’s but much more so, and at turn-on it discharges into the channel at the same instant the diode’s charge arrives. Whether the two capacitive charges add, or whether the transistor’s large low-voltage capacitance absorbs part of the Schottky’s displacement current before it reaches the channel, decides whether the light-load boundary measured here survives in a real half-bridge.

The second open question is the ring. The heat a recovery leaves behind and its neighbours found the silicon diode’s snap exciting the loop inductance. A displacement current of twelve amperes at twenty volts a nanosecond excites it too, with a different spectrum — set by the slew rate rather than the recovery’s softness — and the comparison of the two rings is a separate measurement.