# How close do electrons have to be to have 'collided' within a semiconductor?

In semiconductors like Zener diodes undergoing Avalanche effect, electrons are colliding all of the time and causing a cascade that leads to current flow.

So how close do the electrons have to get to call it a collision? Are we talking a direct hit on each other (<1 electron diameter), within the orbital diameter or just somewhere nearby? And if it's that latter case, what's considered nearby if you're a whizzing electron?

When people say "electrons collide" in the context of solid-state physics, they do not mean that they actually "come into contact" with one another. What is normally meant by "electron collision" is the act of two electrons exerting a force on each other due to their Coulombic interaction and thus causing both electrons to change the state in which they are, normally denoted by a Bloch wave vector $$\mathbf{k}$$ for each electron.
As to how close electrons need to be to collide, this is a generally complicated question to answer and depends greatly on the material. As I stated earlier, the key fact is that they interact through a Coulombic potential. However, this potential is "screened" by the other electrons and ions in the solid. This means that each electron does not see the full Coulomb potential generated by the other one, but an effective screened potential because all the other electrons and ions are also attracted by it and "get in the way". For long-wavelength interactions, the screened Coulomb potential $$U(r)$$ has the form $$U(r) \propto \exp(-r/\lambda)r^{-1},$$ where $$r$$ is the distance between both electrons and $$\lambda$$ is called the screening length. Hence, we can say that electrons at a distance in the order of magnitude of $$\lambda$$ or less interact effectively under the screened Coulomb potential. There is no general closed form of $$\lambda$$, however. In semiconductors obeying Boltzmann statistics, the screening length in the long-wavelength limit may be used to estimate its order of magnitude. In this limit, $$\lambda$$ given by the Debye-Hückel screening length $$\lambda_D$$ and it is equal to $$\lambda_D = \left(\frac{\varepsilon_L k_BT}{ne^2}\right)^{1/2}.$$
In this expression, $$\varepsilon_L$$ is the electric permittivity of the ion lattice, $$k_B$$ is Boltzmann's constant, $$T$$ is the temperature of the solid, $$n$$ is the electronic density, and $$e$$ is the electron's elementary charge.