Larger transistors (Ic >= 1A?) typically fail from a single avalanche discharge event, even with no external impedance connected, more or less because their entire junction capacitance gets discharged through a tiny (100s nm?) channel; but again, this only happens when all conditions are present to trigger the effect. Other than that, for most if not all transistors, the current in question here should be too low for that breakdown mode.
Most things can handle at least a little avalanche, before either breaking down catastrophically at higher currents (but without overheating) (or being biased into such a condition as above), but I haven't seen any GaN parts yet that tolerate any avalanche at all; blue/white LEDs for example typically fail suddenly at 20-30V, no matter how slowly you approach it.
IGBTs are another example; they're rarely if ever rated for any avalanche energy at all (and with good reason, being four-layer devices..!), but are regularly rated in terms of breakdown -- Vces (C-E sustain voltage), meaning, biased under, say, a fraction of a mA. They're likely to latch up or fail outright if driven harder, but for small currents, yeah, you absolutely can measure it.
BJTs are mostly the same way, as Vcbo transitions to Vceo depending on base current or wiring, and Ic. With base open, high avalanche currents are not a good idea, but with it grounded or reverse-biased, there can be some energy handling. (If nothing else, a switching type BJT must handle dynamic recovery, i.e. the C-B junction turning off, which acts like a rising avalanche voltage during the turn-off slope. RBSOA plots are often given for these types, which show what V and I are safe for what time scale.)
BJTs have been used as zeners or TVS from time to time; a decidedly off-label use, but maybe not uncommon in the 70s before TVS proper came along. Not sure if there were specific types for this or it was more of a shoot-from-the-hip idea? Although the only example that comes to mind is for input protection / clamping in several Heathkit oscilloscopes.
The main downside of BJT breakdown is the noise, either from poorly controlled breakdown compared to a proper regulator zener, or the jaggy avalanche I keep mentioning.
Note that single events aren't repetitive events: even if something like a BJT or rectifier happens to survive a single avalanche event of nearly any current, it's no guarantee it always will. This is the main concern, at least as a design matter, distinguishing TVS from zeners: zeners are simply never rated for surge, or not the same way TVSs are anyway, while TVS obviously are. They might be identical dies, but without the datasheet saying so, you never really know for sure.
Repetitive avalanche is a known hazard for many types; modern MOSFETs rarely if ever handle it, more or less a consequence of the gate oxide being too close to the avalanche site (along with other implementation details). Over time, charges get trapped in the gate oxide, until gate oxide failure occurs. Older types had repetitive as well as single-pulse avalanche ratings (e.g. most or all? HEXFETs); modern (trench and SJ) types usually have single-pulse only. Older generation also used comparable I_AS to I_D(max), whereas modern types it's usually a fraction thereof. (But to be fair, you rarely if ever use modern MOSFETs anywhere close to full ratings, so as to keep conduction losses down; which means, even if you do make use of avalanche (and, you probably shouldn't, but for those rare times when you do?), it's probably at comparable to the load current, so, this is fine. There are definitely some parts that use a suspiciously low I_AS (lower than the load current you would use) though.
Speaking of old types: possibly, modern "linear" type MOSFETs are just the old planar design, rebranded. Though there's really just IXYS/Littelfuse making them, I think, and they only give single-pulse in their datasheets.
I also suspect TI's NEXFETs might handle repetitive, thanks to a unique wrap-around design that puts more distance between the channel and avalanche site; though it might just be that the pitch is fine enough that it's affected anyway. In any case, they aren't testing repetitive avalanche, so the safe assumption is they do not, in fact, handle it.
Tim
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