I understand that digital circuits that can be run at high frequency can be called "high speed," but what is the meaning of "Schottky diode having a high switching speed?" Does this describe the maximum slew rate of the output?
2 Answers
As Schottky diodes are unipolar devices, they have zero reverse recovery time.
To expand on what that means:
Semiconductors have what's called a carrier lifetime, which is the average length of time minority carriers, injected (via heat, photons, or injection from other parts of the circuit) into an otherwise-in-equilibrium semiconductor, remain free before recombining with their opposites. While enough minority carriers are present, the semiconductor can conduct.
A forward biased pn diode constantly injects minority carriers across the junction, and when that injection stops it takes a finite length of time for the diode to stop conducting--even if the applied voltage reverses; reverse current can flow for a brief time (tens of nanoseconds for fast-recovery diodes, up to tens or even hundreds of microseconds for standard diodes). This is also true, incidentally, of bipolar transistors (including IGBTs); they can't come out of saturation very fast.
A Schottky diode, on the other hand, doesn't function by the same mechanism. It's a unipolar device, meaning there's only ever one type of carrier involved in conduction, and minority carriers are basically irrelevant. This means there's no increase in the minority carrier count when the diode is conducting, and consequently no delay between the applied voltage switching polarity and the conduction stopping. (This is an idealized explanation; in reality, there will be some reverse conduction as the diode's parasitic capacitance charges. But this is a very small capacitance and charges quickly.) This makes Schottky diodes ideal for high-speed switching, when their low breakdown voltage and high reverse leakage aren't a problem.

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There are probably two meanings here, which you need to be careful of.
First: marketing.
Manufacturers will claim something is "high speed" just because they can. And "can" can be very weakly justified indeed. Is a million times a second "fast"? Pretty damn fast to a human, great, let's roll with that.
Fast in relation to what? If it's the first of its kind, it must be fastest at it!
Fastest in its class? There have been numerous op-amps claiming "high speed", that are basically remakes of the µA741, or LM358, or etc. That is, having around 3MHz GBW, and very unimpressive DC specs.
There is still an argument to be made here, along a more subtle line (which might not be apparent if you aren't well versed in op-amp specs): supply current consumption may be reduced. In fact, there is a very clear and persistent tradeoff, between speed and current consumption, in a given technology. Op-amp designs get more and more complicated over time, eking out a tiny bit of performance while consuming less supply current; tricks like gain boosting (the amp goes faster (and draws more current!) when driven harder, but still settles at the same slow rate), feed-forward tricks to reduce phase margin (which otherwise forces GBW to decrease, for the unity-gain-stable case), and other proprietary methods.
Such amplifiers can be said to be "high performance" when they place below the Iq(GBW) curve of their predecessors. This leads to the peculiar situation that "nanopower" amps might claim "high performance" with the impressive GBW of... all of 3kHz. (To be fair, that they're able to produce an IC that moves so slowly at all, while still being an analog circuit, is an achievement by itself.)
Applying this to diodes, we have the situation that traditional PN junction diodes have gradually improved over time.
Some diodes may achieve speed by sacrificing Vf, sometimes quite severely; I recall some early ~30ns types in the 2000s, which claimed Vf max ~3V at rated current. (This is typically a case of carrier lifetime reduction with Au or Pt doping, which generally reduces trr, while increasing n (emission coefficient) and thus Vf, and Ir.)
There is also a softness tradeoff, where reverse recovery can be intentionally slowed to reduce EMI, at the expense of switching losses (effectively as the diode turns off, its breakdown voltage gradually rises, up to the rating; by carefully controlling junction doping, turn-off dV/dt can be controlled). Diodes claiming "low loss" might take the opposite approach.
Second: on schottkys.
Schottky diodes do not suffer recovery losses at all. To see the significance of this, first understand the PN diode behavior.
Forward-bias of a PN junction injects charge carriers of both types (electrons and holes) into the junction, which need to be cleared before it becomes non-conductive again. This takes some time, and during that time the junction remains highly conductive (voltage drop ~ Vf). During that time, current can increase quite substantially -- that is increased magnitude, even in the negative-going direction, and Irr (peak reverse recovery current) can be several times the If that proceeded it before it finally turns off.
Here is an example waveform, measured on my own test jig (source: my website, https://www.seventransistorlabs.com/Images/Diode_Recovery_DSEP29-06A.png )
Current scale is -5A/div; voltage, 20V/div. From left, the diode is forward-biased at about 12A, then hard-switched to turn it off. Current reverses, reaching a peak of about 13A, then decays. The whole process takes less than 60ns, pretty fast generally speaking.
Some inductance and other speed limitations are evident from the construction of my jig; Vf appears to rise stepwise, but really it's just dI/dt creating a drop across the diode's inductance. Similarly, general bounciness is due to transistor and current shunt inductances.
Compared with datasheet specs, about 750 A/µs is applied, which should give Irm closer to 30A; but the graph is done at 100°C, while my measurement was at 25°C. (The Kf graph gives the adjustment to 25°C however, which looks reasonable given the quirks of my test jig.)
The same measurement on a schottky diode, however, produces a very different result. It's noteworthy that schottky capacitance is generally higher, so there is still some impact at turn-off -- but it does not scale with If (because it's not an injected charge phenomenon), and scales proportionally with dI/dt (because more dI/dt means more I = C dV/dt sooner).
Physically, a schottky diode is a metal-semiconductor junction. Only electrons are injected from the metal, making it a majority-carrier device. These charge carriers do not require time to recombine, therefore as soon as the junction voltage reverses, current flow ceases. This is a very fast effect: as fast as the carriers can move, limited basically by device strays (inductance and capacitance).
It is in this sense -- comparing schottky diodes to PN junction diodes -- that they can be said to be "high speed".
(Zener and avalanche effects are also very fast; you occasionally see TVS diodes or MOVs claiming fast response time, sometimes ~ns (consistent with their device strays), sometimes ~ps (consistent with physics but utterly imperceptible at the device level).)
It's worth noting that schottky undergo the same tradeoffs as PN diodes, though with fewer degrees of freedom -- so there has been less improvement over time. (Compare the ancient 1N5819 to a modern PMEG4010: Vf and Ir (typ) are almost identical, with Cj reduced slightly.) The primary tradeoff is high vs. low Ir, giving a slight edge in Vf.
The main thrust of development, then, has been extending schottky to higher voltage ratings, as the basic planar PtSi-Si junction (or other materials) breaks down around 40V. Through various means, ratings have been extended up to 100V or so in (I think?) planar Si, and 300V or more with SuperBarrier / trench MOS / other exotica (many of which indeed incur reverse recovery once again(!), though it's quite modest, and ~independent of temperature). And the widespread introduction of SiC has extended their range beyond 1200V.
[Note to editors: I use "schottky" as the genericized noun referring to the component type, which is named in honor of Schottky the proper noun. This seems consistent, to me, with SI units for example, which uses the general noun tesla for the unit, named in honor of the man Tesla. This has come up before (see also: "schmitt trigger", etc.?); I'm open to style suggestions. Or perhaps take it to Meta.]

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1Meta: genericized nouns emerging from proper nouns by losing their capital letter is a legitimate and well-documented language phenomenon. Inventors and discoverers are proud (also documented) when they see their name without the capital letter. – fraxinus Jun 18 '23 at 12:37
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1Side note on the mention of SiC schottkys: While they give you reverse-recovery-free diodes up to kV (highest I've seen is 3300 V, incidentally), and their leakage is much lower than other schottkys, they *do* have a very high forward voltage of typically around 1 V, even higher than typical silicon pn diodes. – Hearth Jun 18 '23 at 15:14
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@Hearth Indeed, 1V is pretty much where they start, and the internal resistance is much higher than Si (a downside of the material), which isn't much of a problem within ratings, but does mean the surge rating is quite low. Tradeoffs all the way up! – Tim Williams Jun 18 '23 at 17:15