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A battery in a circuit normally sources electrons at the negative terminal and sinks electrons at the positive terminal. The chemical reaction at the positive terminal consumes them. The total number of electrons in the battery does not change. Similarly, a diode accepts electrons at the negative terminal and sources electrons at the positive terminal, ...

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Anyway, the conjecture is that, possibly if the intensity of the light is strong enough, we can still get photoelectric effect with $\omega < \omega_c$? The frequency treshold for the photoelectric effect is observed with light of common intensities; even very low intensity light will work. There are ways to ionize atoms without such high-frequency ...

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By absorbing more than one photon an electron can still be detached. However, the probability will be small. On the other hand, using a focussed intense laser beam, the situation improves. In fact, isolated atoms can be ionised in this way. The process is called multi-photon ionisation and has been verfied experimentally.

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Yes. Whether the result is a good conductor depends on how localized electrons and holes in the respective bands are (or, saying the opposite phenomenologically: on their mobility). Their recombination time limits for how long you will have even just two charge carriers available for conduction. This tends to make such conduction very energy-inefficient ...

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