Ooo, a bandgap discussion. I must participate or people will think I've died
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I mostly agree with Kevin - the startup device is one of 5 gate-source voltages in series. Even the tiniest of currents will shut off M5. This is a really bad idea if you are using bipolars to generate the bias voltage across the resistor since they might not get enough current to operate correctly (low level injection, yada yada...). With MOSFETs... unless you made that bottom NMOS ratio every small you would have to have really bad mismatch between the PMOS mirror before it wouldn't start up. However, it could take a very long time to start up.
A better circuit (arguably better than the good one on Kevin's site) is to
1) replace M2-M4 with a single long device or large resistor (pull-down device)
2) connect source of M5 to Vdd
3) size the M1 device so that when the circuit is operating properly it provides 5 and 10 times the current necessary to pull up the gate of M5 and shut it off over all corners.
4) M5 should be sized so that the current it provides during startup is about the same as the normal current in the bias circuit.
Items 3&4 are my own rules of thumb. If the devices are too strong you can get into a large signal oscillation during startup. I feel this is the best circuit because it directly senses the current in the bias loop and won't turn off until it is well on its way to fully starting up.
All that said, with an entirely MOS circuit it would be unusual for it to not
eventually start up - basically the low-current gain around the loop would have to be less than one. Your mismatches and leakage would have to be horrible before that would happen. (again bipolar is a different story). Simply getting rid of M1-M4 and diode connecting M5 is becoming a popular circuit for this MOS bias circuit at low Vdd, and I'd say it is simpler and more robust than the one you show, but not as robust as the one I described above or on Kevin's great site.