In a floating gate transistor, the gate is electrically isolated. I have a few questions regarding the quantum nature of this device:
It is sometimes mentioned that, to store or deduct charges from the isolated gate, quantum tunneling is utilized. Is invoking quantum mechanics really necessary here? can the effect also be explained in a classical framework (e.g. electrical breakdown) ?
It is also known that overtime, there could be a leakage of charges from the gate (even without applying any voltage). Does this too require quantum tunneling to explain?
Some quantum phenomena can only be observed in a coherent setting – where the system is sufficiently isolated from its environment and therefore decoherence does not occur (or is small). Does the effect of quantum tunneling in the current context also require coherence?
The electrons which are stored in the transistor’s gate – how can we estimate the typical coherence time after they are transferred to the gate? (by "coherence time" I mean the time before they get strongly entangled with their immediate environment). (I understand that this question might not make sense. If so, I would appreciate if you can explain why it does not make sense.)
The electrons in the gate – can we say that they have a discrete energy spectrum?
More precisely, how can we estimate the energy gaps between energy eigenvalues of the electrons?
Classical mechanisms can be used. In fact, the wikipedia article you linked to says so in the first paragraph. Hot-carrier injection is a classical phenomena. (You use carriers with enough energy to go over the barrier.) If you want to get past a barrier, you have two options: go thru it (tunneling) or over it (classical). I believe that some devices depend exclusively on quantum tunneling, in which case you really can't escape quantum tunneling.
Whatever mechanism can be used to get charge onto the gate can also be used to get charge off of the gate. However, unless you're heating up your device a lot, I'm guessing that tunneling is the dominant leakage mechanism.
No, tunneling does not require coherence. It seems like the unspoken premise of these questions is that tunneling is some exotic phenomenon, so can it really manifest itself in "normal" semiconductor devices? The answer is a resounding "yes"! Many semiconductor devices depend exclusively on quantum tunneling to work. Zener diodes are the best example. So are resonant tunneling diodes. Schottky diodes also rely heavily on quantum tunneling.
Many coherence times in MOS semiconductor devices at room temperatures are typically very short. I would effectively estimate them at zero. That said, you can make quantum computers with standard semiconductor fabrication techniques, but the coherence times generally aren't stellar, and it requires very low temperatures. Coherence times depend a lot on the details of your system --- especially what impurities are present. Going into details is beyond the scope of this answer --- it's an active area of research --- but here's an article if you're interested. (All that said, there are many coherence times. At room temperature, maybe the spin coherence time is non-trivial even if the phase coherence time is basically zero.)
Yes, the energy levels are discrete. Unless the gate is really tiny, to a good approximation, you can calculate the energy levels classically. The spacing is basically the charge of an electron times the capacitance between the gate and the rest of the system. I expect this to be like the energy levels of a single-electron transistor. If the gate is really small, the electrons interact with each other more and things get more complicated.