Dark current and thermal excitation in CCDs In charge-coupled devices (CCDs), doped semiconductors are used to produce an electronic signal from incoming photons - the underlying principle being the photoelectric effect. This simple law tells us that the maximum kinetic energy of an ejected electron is the photon's energy minus the work function of the material, i.e.
$$K_{\rm max} = \frac{hc}{\lambda} - \phi,$$
where in this case $\phi$ is the work function of silicon. The photons' energy allows electrons to overcome the band gap energy of the material, which, as I understand, is intrinsic to the material (only modified in doping).
One problem with CCDs is that of dark current. Thermal excitation can cause electrons to overcome the band gap energy without the need for photoelectric ejection. This is largely resolved by cooling the instrumentation/CCD to help prevent the electrons from becoming thermally excited. 
The question I have is about the application of this in optical/visible versus infrared applications, particularly for astronomical observing. In instruments where CCDs are now being used for infrared observing, the CCD's need to be cooled down more than in optical instruments. I know that infrared photons are higher wavelength and thus lower energy than visible photons. As a result, it seems to me that $K_{\rm max}$ should be lower in total for infrared photons, making it harder for them to overcome the band gap. 
So why isn't it that infrared instruments are used at a higher operating temperature than optical instruments?
 A: The band gap of the light sensitive materials used in infrared CCD's is typically smaller than the band gap of the materials used in CCD's optimized for visible light, so that electrons making up photocurrent could be excited by low energy infrared photons. 
Because of that difference, at any given temperature, infrared CCD's should have greater dark current - the current due to the temperature of the sensor rather than due to the received infrared radiation. Therefore, in order to maintain a similar level of the dark current or similar signal to noise ratio, infrared CCD's require more cooling.
The table below (copied from this presentation) shows band gaps for various materials used for IR detectors. It also shows corresponding maximum IR wavelengths that can be detected by sensors made out of these materials (cut-off column) and their typical operating temperatures 

We can see here that for detecting infrared radiation with wavelengths beyond $1\mu m$, materials with a smaller band gap than silicon are required. 
A: Strictly speaking none of the other materials are used as CCDs. Astronomers tried for years to make CCDs out of these materials but the amount of engineering expertise in Silicon is so much greater we couldn't catch up for other materials.
Generally  the longer wavelength detectors are low bandgap materials grown on a separate substrate and each pixel connected to a silicon CMOS imager for readout.
They are, as VF, said cooled further to reduce dark noise.

So why isn't it that infrared instruments are used at a higher
  operating temperature than optical instruments?

Silicon CCDs do experience reduced long wavelength sensitivity at very low temperatures. In silicon most of the dark current is generated by surface states and so it is possible to greatly reduce this with special treatments and get good dark noise without extreme cooling.
