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What does current look like on a quantum level? : askscience

Conduction band electrons aren’t treated discretely, and aren’t part of any particular orbital in general. Rather they are said to be ‘delocalized,’ with probability distributions that are spread out over the entire lattice instead of atomic or molecular orbitals. This is usually modeled as a ‘sea’ of electrons around cations. One model if this you can look up is the Drude model.
A more quantum phenomenon is the promotion of electrons from the valence band into the conduction band. Electrons can tunnel across this potential barrier.
The picture of conduction you’re describing is actually that used in the Hubbard model – used to study graphene nanoribbons for example. In these materials electrons are localised (or located) on individual atoms and this is very much the way to think about it.
But metals are very different. Some of the electrons in metals are actually spread over the whole material. This is why metals conduct electricity so well, as these electrons are free to move through the whole of the metal.
These two types of “orbitals” come up all over quantum mechanics. If you want to read more about this, the localised states are called bound states, where as the delocalised states in metals behave like free electrons.
The heart of your question lies in solid state physics. This is a subset of quantum mechanics aimed at understanding why solids behave the way they do. On a quantum level, current is still just the amount of charge that moves through some space in a given time. The only difference is that the charge is now packaged up into discrete units (electrons, protons and other charged fundamental particles). To understand how current flows in a material you first have to understand electrons behave in a material. The key feature of solid state physics is that many materials are crystals. This means that the atoms are spaced periodically. As you mention, band structures are the way that we summarize the effect of this periodic potential. Basically, a band structure just relates an electrons momentum (p=mv=hbar k) to its energy. The momentum can be positive or negative, the sign only denotes direction. In free space this is very boring, Energy=(m v2 )/2 = p2 /2m=(hbar k)2 /2m. When you throw in a periodic potential, this becomes modified and results in bands. Actually calculating band structures is quite difficult. The key idea is that there are ranges of energy where the electron can live and ranges of energy where the electron cannot live.
The electrons in a crystal live in the band structure. Each atom of the crystal brings a certain number of electrons with it. They fill the states in the bands starting from the lowest energy. Each of these states has a specific momentum associated with it. When a band is filled, the next electron has to be placed in a state in the next highest band. Applying a voltage to a material is the same as applying an electric field to the material (E=V/l where l is the length of the material). In the semiclassical picture, electrons with charge -e, feel a force F=-eE in the applied electric field. This force accelerates the electrons from lower voltage to higher voltage (they are negatively charged so lower voltage is actually higher energy for them as Energy=V*q where q is the charge, including the sign). These moving electrons constitute your current. A caveat to this is that electrons really live in quantum states and no two electrons can live in the same state(Pauli exclusion principle as electrons are Fermions). The electric field really moves electrons from states with one momentum to states with a momentum that is in the direction of the electric field. If the band is full, all the states are full and the electric field cannot change the electron’s state so no current flows. This is an insulator. When a band is partway filled, there are states that the electric field can move the electrons to. This allows a current to flow. Transistors are a little more complicated. The main thing you have to understand is p doping and n doping semiconductors. Imagine you have a crystal of silicon. If you take out a silicon atom and put a phosphorus atom in its place, you suddenly have an extra electron. A single phosphorus atom won’t change your band structure as you still have 1023 silicon atoms so it’s like you just added an extra electron to your system. Semiconductors have a filled band with another band with only slightly more energy (.5ish eV). This extra electron from the phosphorus can’t live in our filled band, called the valence band, because there are no more states. It must live in the next band, the conduction band. If you apply an electric field, this electron in the conduction band can flow because pretty much all the states in its band are empty. This is called n doping because we added an extra negative charge, the extra electron. If instead of a phosphorus atom we add an aluminum atom, we have one less electron. If the aluminum steals an electron from a neighbor, this neighbor now is missing an electron. Instead, of thinking of the aluminum as stealing an electron, you can think of the aluminum as giving the neighboring atom an empty state. This empty state is called a hole in solid state physics. A hole is basically a missing electron and it behaves like a particle with charge +e. If you apply an electric field to it, it can move around by trading places with an electron. Again, you get a current. We call this p doping a material as it is now missing an electron or you can think of it as having positively charged particles, holes. Transistors are semiconductors with a p doped region surrounded on both sides by an n doped region or vice versa. Honestly, I study physics and not material science or electrical engineering so I’m not super familiar with the details of how a transistor works.
I hope this helps.

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