What is the theoretical maximum efficiency for LEDs? Whatever efficiency you want to give in converting electricity to light. I'm guessing it would depend on the type of LED. I imagine limitations come from various reasons. It would be cool if there were an optoelectronics tag. 
 A: By efficiency, I will assume you are referring to power efficiency (the ratio of optical output power to electrical input power), although much of the loss mechanisms discussed below apply to other types of efficiencies as well.
There is no clear-cut "theoretical maximum" efficiency that I am aware of. There are several factors limiting the efficiency of an LED. The more important ones in most LEDs are as follows:


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*Optical (extraction) efficiency: $$\eta_{op} = \frac{\textrm{Number of photons extracted}}{\textrm{Number of photons generated}}$$ Not all of the photons generated within the LED can be extracted from the LED. Some of these are reabsorbed by the semiconductor or other materials, and some of them are reflected back from material interfaces. The device geometry, semiconductor refractive index and packaging play important roles in determining the extraction efficiency.

*Internal quantum efficiency:
$$\eta_i=\frac{\textrm{Number of photons generated}}{\textrm{Number of charge carriers passing through junction}}$$ Photons are generated as a result of the radiative (band-to-band) recombination of electrons and holes in the LED. All electron-hole pairs that recombine through non-radiative mechanisms (such as trap-assisted recombination and Auger recombination) produce heat instead of radiation, limiting the internal quantum efficiency. The requirement for high internal quantum efficiency is why LEDs are made of what are called direct band gap semiconductors. High efficiency LEDs typically suppress non-radiative recombination processes by confining electrons and holes in quantum well heterostructures, where they recombine radiatively.

*Power efficiency:
$$\eta_p=\frac{\textrm{Number of photons extracted}}{\textrm{Number of charge carriers passing through junction}} \frac{E_p}{qV_{bias}} = \eta_i \eta_{op} \frac{E_p}{qV_{bias}}$$
where $E_p$ is the energy of an emitted photon (approximately equal to the band gap of the LED active region material band gap energy), $q$ is the elementary charge and $V_{bias}$ is the LED bias voltage. During the normal operation of an LED $V_{bias}$ is typically close to $E_p$, it is often greater as a consequence of losses such as intraband thermalization.
