The electron states in the band gap are localised, whereas the states which contribute to the bands are not.
The electron states in a solid are not simple, there's a lot of non-trivial quantum mechanics going on. The electron states in a free atom are localised around the atom - the electrons in those states can't leave the atom without a lot of energy, so they can't conduct anything. When you pack lots of atoms together, the surrounding electron states overlap and mix. Which states mix with each other is dictated by the energy: similar energies means more mixing. You end up with a new set of states which extend over the whole block of material. If the material has a periodic lattice, these electron states group together into bands.
Every state in a band has some velocity (called a Fermi velocity) associated with it, and an electron in that state can be thought of as moving through the material with that velocity. The Fermi velocity of electrons in the conduction band is very large, but because the electrons are all going in different directions, there is no net current. An applied electric field moves some electrons from states which were going in one direction, to states going in the other. In a metal, one of the bands is part full, so there are plenty of nearby states to move electrons into. In a semiconductor, there is a gap between a full band and an empty one so it's much harder to push electrons into the higher band.
When dopants are added, they don't form a nice periodic lattice and they are much more spread out than the silicon atoms that host them. This means that the electron states around the dopant can't mix with states from other dopants to form a band. Since the energy levels of the dopant states are different from the silicon states, they don't mix (much) with them either. Instead, the electron states are localised around the dopant, much like the states around the free atom. An electron in that state can't conduct in the way one in a band can. It either has to jump up into the band, or jump to another nearby dopant. The former happens in semiconductors, the later is known as incoherent transport, and appears in some other materials.
I'm not sure how well I've explained this, but if you don't get a clear answer here, you could try the physics stack exchange. This definitely feels more like condensed matter physics than electrical engineering!
The diagrams appear correct - although as with all diagrams of this sort they don't quite tell the full story, but good enough for a basic understanding.
When you add the dopants in, they add extra energy levels within the gap between valence and conduction bands that electrons can sit at Reference. These energy levels represent the ionisation energy of the dopant.
When the device is very cold (e.g. if you put it in LN2), there is not enough energy to allow the electrons to move either up to acceptor energy levels, or up in to the conduction band from donor energy levels.
However, as soon as you warm the device up, there is enough energy to ionise the dopants. For p-type, electrons gain enough thermal energy to hop from the valence band into the energy states created by the p-type dopant ions. For n-type, the electrons gain enough energy to escape the donor ions and move in to the conduction band.
It is important to note though that the diagrams don't give a good idea of scale - the energy difference between the band edge and dopant levels are much smaller than the band gap of the semiconductor. The dopant levels might be only 0.05eV Reference from the band edge compared to a band gap of say 0.7eV Reference. As a result the excited electrons from the donors are far more likely to go in to the conduction band than the valence band, and the acceptors are far more likely to pull in electrons from the valence band.
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Yes, the sensor responds to the room light and it may be brighter than the signal from the remote control.
The signal from the remote control however is modulated with a signal that is typically at 38kHz. The TV is tuned to only respond to light modulation at that frequency. The 38kHz modulation is pulsed to transmit the actual control function.
The signals from room light will typically be at DC for steady light such as sunlight or at 100/120Hz from lights powered by the wall socket (the ripple is at twice the frequency usually because the lights illuminated on both positive and negative half-cycles). Some LED lights may modulate with other frequencies as well because of the way they control their power.
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