I wonder if (actually, I suppose, and suspect that!) the soft knee is due to a finite number of crystal defects, impurity sites, or whatever.
The knee is very much softer than for a proper Si avalanche diode (which is pure enough to handle large surge currents, distributing that current over the whole die area fairly evenly).
It further stands to reason that, if the V(I) curve is made up of myriad (but finite!) defects with different current density (i.e., series resistance) and breakdown voltage, then the curve should be lumpy, at least if you can zoom in close enough to observe that lumpiness, and average it over enough time (since each breakdown site will bleed erratic, noisy current).
Another way to put it: if you vary the current into the junction, the voltage will vary in a noisier way than you would expect from an ideal junction (which would simply have a smooth exponential response). This reminds me of Barkhausen noise in ferromagnetic materials (where a "pop" is produced as magnetic domains realign across magnetically-hard "stuck" points; though that's driven by hysteresis, so not quite the same).
Tim