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Is it possible to make a photovoltaic cell that would only absorb the invisible part of the electromagnetic spectrum, while letting visible light pass through or bounce off its surface?

I guess that if the n-type Si and p-type are added special dopants so the energy gap needed for the electron to escape would be the same as the one provided by the photons of such radiations, it would work. But this is when it gets tricky as visible light stands in the middle of IR and UV light (what you would normally look for in the non-visible spectrum, ignoring the other waves), so if the cell would have a specific band gap that would be satisfied from UV light, it wouldn't be from IR as it has lower energy (provided visible light is let intact).

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Yes, you can have photovoltaic tuned to different areas of the EM Spectrum.

No, you can't cover the whole EM spectrum with one detector.[1]

In fact, limiting yourself to photovoltaics is throwing out most of the spectrum- you are only looking at materials where EM radiation will cause an electron to jump bands.

Is there a particular bandwidth you are looking for? Would multiple sensors, covering different areas of the spectrum, work?

You may be able to use a filter to remove the areas you aren't interested in. Photographers, for example, do this with "IR filters" that block visible light.

[1]Well, "can't" is strong. I am failing to imagine a detector with that bandwidth.

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Yes you can make transparent solar cells. Obviously their efficiency won't be nearly as high as their opaque cousins.

http://web.mit.edu/newsoffice/2011/transparent-solar-windows-0415.html

There is nothing impossible about a single material that absorbs UV and IR but not visible. Organic semiconductors can do that. Inorganic materials like silicon almost certainly cannot.

There are not too many UV photons from the sun, so if you are making a single-junction IR-absorbing transparent solar cell, it doesn't much matter whether or not it also absorbs UV. The UV only helps a little bit. (The UV would help significantly more in a transparent two-junction tandem solar cell.)

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