Electrical – XBee power supply design using a Li-Po 3.7V battery

Something like this could be very efficient (not much more than the LED draw) with a CMOS op-amp. For fairly constant visual brightness (+/-10% current) you could derive the 100mV with a voltage divider from the battery, or use a low current reference such as the TLV431 which would require another 100uA or so but would make the current very constant and accurate (of course the 1.24V output of the TLV431 would have to be divided down to the 100mV).

Q1 is a logic level MOSFET rated for 3V drive. U1 is a CMOS single-supply op-amp with R-R output. Careful layout is required around R2 so that trace resistance does not affect the current sensing.

^{simulate this circuit – Schematic created using CircuitLab}

In operation the op-amp drives the MOSFET gate to maintain a voltage of 100mV at the source so that the current through the LED is 100mV/5\$\Omega\$ or 20mA.

The resistor R1 and capacitor C1 are to prevent oscillation due to the MOSFET gate capacitance.

The minimum voltage across the MOSFET to maintain regulation is Rds(on)\$\times I_{LED}\$ + 0.1V- with a suitable MOSFET with low Rds(on) that won't be much more than 100mV, so it should work down to 3.3V with a 3.2V LED Vf.

The short answer is yes, but your voltage is going to track with the voltage of the battery. A slightly more elegant approach is the basic linear regulator shown below.

edit: just to be clear, this will still track with the battery voltage when the battery voltage is less than the breakdown voltage of the zener. If you can get away with a Vout of about 3V, you could select a zener with a breakdown voltage of 3.6V and get a steady output voltage over a relatively large section of your battery SOC.