Theoretically you can make integrated circuit with any semiconductor, but I don't know if anyone has been brave enough to try it.
The smallest integrated circuit is a diode, it has 1/2 transistors :). OK, a diode is not actually an integrated circuit, but if you can use a material to make a diode, you probably can use the same material to make an integrated circuit with enough money, time and ingenuity.
Some of the diodes are made of germanium. It has a very low band gap, that is useful for "crystal radios" ( http://en.wikipedia.org/wiki/Crystal_radio#Crystal_detector ). Perhaps this can be useful to make very low voltage CI, to reduce the power and heat. But perhaps there is a technical problem that I don't know.
The diodes in the LED have many semiconductors. The band gap of the material is related to the color ( http://en.wikipedia.org/wiki/Light-emitting_diode#Colors_and... ). With this you could make high voltage CI (like 10V?). Perhaps it may reduce the noise ratio??? But perhaps there is a technical problem that I don't know.
[kens: If you are reading this, I'd love to see a technical post about this subject.]
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Edit: According to Wikipedia, one of the first CI was made of Germanium: http://en.wikipedia.org/wiki/Integrated_circuit#Invention
> Half a year after Kilby, Robert Noyce at Fairchild Semiconductor developed his own idea of an integrated circuit that solved many practical problems Kilby's had not. Noyce's design was made of silicon, whereas Kilby's chip was made of germanium. Noyce credited Kurt Lehovec of Sprague Electric for the principle of p–n junction isolation caused by the action of a biased p–n junction (the diode) as a key concept behind the IC.