So let's walk through the design process:
Input voltage range
Something between 2·1.55 V = 3.10 V and what the discharge curve below tells us about what happens after drawing 50 mA for 4 h. That'd be 0.2 Ah, and we need to scale that from a 2.20 Ah battery down to the typical AAAA capacity of 0.5 Ah, so we need to look at the 0.2 Ah · 4.4 = 0.88 Ah
Discharge curve says about 1.2 V. That sounds realistic, so our overall input range, including the fact that anything has a non-100% efficiency, would be at least 2.3 V to 3.1 V.
That highlights all LDO/linear based approaches are doomed, because they can't step-up Voltage.
As shown, we need a switch-mode power supply. Assuming we don't first want to burn energy to always work below the target voltage of 2.75V, a switch-mode supply that is able to boost and buck is necessary.
To achieve the 40mm² space restriction, we must look into highly integrated circuitry – definitely ICs that include the switch, not only the controller, but if possible even the inductor, or use switched capacitance (because our current isn't that large).
We visit the websites of the "usual suspects", being
Let's assume TI is doing pretty well with their simple-switcher modules that integrate the inductor. Let's see if what we can find comes close to our space restrictions – if it doesn't, we can drop that approach (we probably won't do better than TI, will we?).
Also, let's look at ON, which are known for selling high-volume, highly integrated SMPS ICs – typically in packages that are chip-scale, and they also expert in things like mobile phone flash LED controllers.
Let's check TI's portfolio for that.
- the smallest simple-switcher module is 99mm²: http://www.ti.com/lsds/ti/power-management/buck-boost-negative-output-module-products.page Ouch. More than twice as large as allowable,
- On has a category for LED drivers: http://www.onsemi.com/PowerSolutions/taxonomy.do?id=16200&lctn=header . The CAT3224 seems to fit our bill very well, and has a 3 mm x 3 mm = 9mm² package – leaving plenty of room for the external components necessary for the switched capacitance design. You'd need something like (very rough guess) 25mF as supercap – but I think that should work. Needs more reading of datasheet. They'll need a lot of space, though – I'd really experiment with high-capacity MLCC and see whether that works if I don't need the "flash" mode.
- In the category of switched capacitance, an ONsemi device seems to be the most interesting for this application: the NCP1729 is an inverting driver that would fulfill your needs and is dead simple to use:
but I'm not 100% certain from skimming the datasheet it works with 50mA with as little input voltage as 2.4 V. Further charge pumps should be considered.