# How does an electron move through a metal or a semiconductor?

I understand that atoms in a metal or semiconductor are bonded and that they will have several eigenstates where electrons can reside. When a voltage is applied electrons in eigenstates will move to other eigenstates and generate a current. However, what is this called? What is the mechanism? Are there several mechanisms? I would like to know some textbooks that address this at a quantum mechanical level. Would I need to delve into quantum electrodynamics?

• I think you're looking for band theory of conduction, see here: en.m.wikipedia.org/wiki/Electronic_band_structure. In a nutshell, an (infinite) periodic potential allows continuous energy bands for electrons (dense states) which permit electrons to move across conductors. Mar 31, 2016 at 2:43
• Modern Physics By Bernstein quantum mechanical model of conduction of electrons in a metal or semiconductors Mar 31, 2016 at 5:01

The idea behind the Kubo formula is straightforward even though the math surrounding it can get ugly. You start with a Hamiltonian $H_0$ for the system in question. You can think of $H_0$ as describing a metal. You want to measure the current $j\sim\langle \hat{j}\rangle$ where $\hat{j}$ is the current operator and the brackets denote taking a thermal expectation value. If I do not attach a battery to my metal there will not be a current. In other words if I take the expectation value using the system described $H_0$ then there is no current.
To get a current I have to add a term $V$ to the Hamiltonian and take the expectation value with respect to the Hamiltonian $H=H_0+V$. This added term simply describes the coupling of the original system to a classical EM field. In some cases it is important to consider the EM field as a quantum object but in many cases classical EM suffices. Since it is difficult to diagonalize the combined Hamiltonian $H=H_0+V$ what people do is treat $V$ using first order perturbation theory, hence the name linear response. Since linear response is just first order perturbation theory the current that you calculate will depend on the eigenstates of $H_0$. Knowing these eigenstates gives you insight into the conductivity of a system.