RobG
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The emitter-base voltage is too large: on the order of 600mV at room temperature and 800mV at -40C biased at 1uA (the numbers are very rough and depend on size f transistor). You can get some improvement as you lower the current, but each time you half the current you only get a Veb reduction of 17mV at 30C, 13mV at -40C. You'd have to decrease current by 7 orders of magnitude (0.1pA) to get 500mV at -40C, and that wouldn't even give you enough headroom to bias the thing.
Also, at low currents you get low level injection which lowers the current gain if the transistor (beta or Hfe). This may not matter if you are just using it as a diode.
You can use Schotkey diodes for less base-emitter, but there is some issue or they would be used all the time.
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