Electron wavefunctions in semiconductors

[erst]

I have a very poor understanding of how an electron "actually exists" in a crystal -- how it can be visualized.

So conduction band electrons are supposed to be plane waves modulated by a periodic wavefunction (my understanding of Bloch theorem). This means they're basically everywhere in the crystal simultaneously. And yet we often talk about the real space location of electrons and holes. For example:

1. How electron-hole pairs are pulled apart by the electric field in the depletion region of a solar cell, which gives rise to photocurrent (or else they find each other and recombine).
2. Electrons/holes scattering from ionized dopants, which have specific real space locations.
3. Electrons/holes diffusing here and there because of concentration gradients from varied doping.
4. Boltzmann Transport equation or Monte Carlo simulations which treat them as particles.

Etc.

What gives? Is an individual electron everywhere or is it localized? I just don't see how to reconcile what I've seen in EE device courses with the "electron as a plane wave" notion from solid state physics. Just how physically extensive is an electron's wavefunction in, say, silicon?

 

[zhanghe]

Erst,
In serious speaking, the wave picture in solid physics is true. And the concept to treat an electron as a ball (as in your EE device class) is only a kind of simplification for people to think and manage the operating of device, who are so familiar with the classical picture in the world. Further, of course this simplification is correct by serious reasoning based on statistics and quantumm physics ... which make sure that the free carrier (electron and hole) you talked about in EE class is indeed a common behavior of all electrons(in every state of cond.band and vallence band, respectively)

 

[DrDu]

Electrons are not supposed to be modulated plane waves. Only effective one electron energy eigenstates of some effective one electron hamiltonian are supposed to be modulated plane waves.
Hence, even in those substances and that energy region where the quasi-particle picture is approximately true, it is possible to consider localized particles as a superposition of plane wave states. After all, we can't expect a localized creation event of an electron hole pair to produce completely delocalized energy eigenstates.
The book by Ashcroft and Mermin discusses quite some detail when and where a semi-classical treatment of electrons is appropriate.

 

[chrisbaird]

Even in the full quantum picture (avoiding classical "ball" models), electrons behave as waves and as particles at the same time. I am sorry if this is hard to visualize, but such is the current theory that best matches experiment. The wavelike picture is appropriate when discussing wave-like phenomena such as electron diffraction, wave dispersion relations in solids, etc. and the particle-like picture is correct when discussing particle-like phenomena such as collisions, electron-hole pair creation.

A helpful picture is to envision the electron as spread out in a wave most the time (actually a sum of its quantum wavefunction eigenstates), such as the atomic orbitals in a solid, or the quantum well states in a quantum-well laser, and then envision it collapsing to a particle when it scatters off a phonon, photon, another electron, etc., and then eventually spreading out back into a wavefunction when it settles down into a new state.