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This is the real trick with FET specs: the numbers on the first page of the datasheet are basically useless in most applications. They assume very optimistic operating conditions, such as continuous conduction at Tj=25C, or ideal switching scenarios. As soon as you operate in between fully-on and fully-off for any significant period of time, the real numbers get a lot lower, and you have to look through the rest of the datasheet to figure out what they are. Putting three of those in parallel might be fine, but the IRLz24 is a pretty old part. The graphs scanned in from an old datasheet and the Rds(on) of 0.1R give that away immediately. You mentioned wanting to run a 6V diode from a 7V source, so you would need at least three of those in parallel just to get the Vds low enough for that to work.
Spreading the thermal dissipation of the driver across three transistors isn't a bad idea, but might not be necessary with the right part, depending on the operating point. There are more modern parts that will allow you to operate with lower Vds with a single transistor, which will significantly improve the efficiency of the driver and make driving the FET(s) easier too. And a lot of those parts will have SOA charts that include DC, so you won't have to guess about how they might perform off the chart. For a single transistor operating at 13A, you probably want to start looking at parts rated for at least 20A continuous.
One thing that makes this a bit trickier is that the Vf of the laser diode is not fixed, so some amount of power dissipation shifts between the driver FET, the shunt, and the diode itself depending on the VI curve (and temperature, and conversion efficiency) of the diode. Your worst case operating point for the FET might not be at full output current, but it's probably somewhat close.
On the subject of driving the FET, gate charge can be an issue. The more charge has to be driven into/out of the gate to change the operating point, the longer the FET will take to respond to the driving op amp, and this directly affects the stability of the driver as well as overshoot. The driver circuit will need to be compensated to deal with this, and it's a good idea to include provisions for different compensation strategies when you build it so you can make adjustments.
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Many thanks- I think the IRLZ24 is out. I have my eye on some TI models (CDS18537) that have clear DC ratings, ~20mR Rds, and low input charge at ~20 nC (although not as low as the IRLZ24).
If I have three in parallel, then I need to move 60 nC around for full on/off (which is overkill). To be safe, if this happens in 1 MHz (targeting 70 khz cleanly), then I need about 60 mA gate driving current, right? The ultra-stable op amp I have in mind can't do this, so I'll need to make a gate driving circuit of some sort...