Is the Energy of an absorbed photon exactly the energy of the band gap? I was wondering, if the Energy of a Photon which is absorbed by an Electron, hast to be exactly the Energy of the bound gap.
So if i have two energy levels in an atom $E_2$ and $E_1$, does my Electron have to have exactly the Energy
$$h\nu = E_2 - E_1$$
or is it sufficient if the photon has a bigger Energy than that ?
I was wondering because if one assumes the spectrum to be continuous the chance of finding a photon with just the right energy of lets say $h\nu=10.2\text{eV}$ should be rather small.
 A: No, it is sufficient for the photon energy to exceed the band gap. Any excess energy is transformed into kinetic energy for the electron in the new band. You get exactly the same effect when ionizing an atom - the excess energy simply powers the electron into a faster continuum state.
You should also take into account that photon energies are never exactly defined except for monochromatic beams with infinite temporal duration. This is exactly because of the energy-time uncertainty relation: the only way to have a perfectly defined photon frequency, and hence energy, is to observe it for an infinitely long time. Thus, the photon energy is always spread out over a finite bandwidth.
A similar effect holds for atomic bound-bound transitions, which will always have a finite natural linewidth. This is caused by spontaneous emission, which means that if you leave the atom in an excited state for long enough then it will eventually return to the ground state. This then limits the amount of time in which you can coherently probe the frequency of the transition, and in turn limits the precision to which you can measure this frequency.
