This paper (and many others I've read) claim that searching for ways of producing THz radiation is a high-interest research topic.
However, something I've just never understood is why it's so hard compared to other frequency ranges: we seem to have no problem producing radio waves or visible light, so why is THz/IR so hard to produce? Basically, what's preventing us from using the same concepts that we use to produce other wavelengths to produce THz?
If I had to guess intuitively, it would be something having to do with the fact that it coincides with thermal excitations, so if you wanted to use the same concept as an LED but just use a material with a smaller band gap like HgCdTe, the fact that all the carriers are constantly excited at room temperature would be really inconvenient (which kind of meshes with my experience of having to use LN2 to cool FTIR machines).
Best Answer
First, for clarification: As Jon Custer already mentioned, THz radiation is not difficult to generate. It's simply part of the black body radiation. What is in fact difficult is to generate coherent or at least narrow-band THz radiation.
Regarding emission from semiconductor material: There is not so many materials, which would offer a bandgap in the THz range. You're right, HgCdTe is one of the candidates, InAsSb would be another option. And there are probably more. What they have in common is simply the fact that you can't get/fabricate them with a good crystal quality. There are a lot of engineering issues like missing substrate materials for functional layers, defects causing intrinsic carriers or mid-gap states, which could provide recombination paths, ... Furthermore, even if one could fabricate these materials, they would not be very robust, simply due to their material parameters. High bond energies go hand in hand with high bandgap and vice versa.
Therefore, people try to circumvent this by doing all kind of bandgap engineering. One prominent example are quantum cascade lasers, where the radiation is created from transitions between conduction band states, though these devices don't reach room temperature yet. Actually not even close, they are still limited to < 200 K. Another option is frequency doubling or tripling of resonant tunneling diodes (which to my knowledge can produce THz radiation but no coherence).
Finally, the THz range is not only characterized by missing sources, but it's hard to find good (meaning narrow-band, fast) detectors as well! To my knowledge, bolometric detection, where you heat a small chip and measure the conductivity change, is still the most common way. Also the detector side requires cryogenic cooling to eliminate thermal background.