How can I simulate a model electronic hole?

Question

Suppose I can solve time-dependent Schrödinger equation for several 1D particles (currently 3). I'd like to see, what an electronic hole is and how it behaves — in a series of numerical experiments.

I know that a hole is lack of electron in a sea of occupied electron states, but I think I don't quite understand, what it really is: do I need to have electron repulsion to see any useful effects? Do I need much more electrons for such a simulation than 3? What initial conditions should I use to see the hole? I'm thinking of using a model "crystal" potential with 3 cells — would this be enough?

As an example of what I'd like to actually see is exciton, namely how its wave packet wanders across the crystal. For an electron in conductivity band, taken as a wavepacket, I'd expect it to look like increase in probability density in the place where the wavepacket is located, and for hole it might be a similar decrease in charge density, so I guess the exciton should look like an increase of charge density in a wave packet and decrease around this wave packet. Is it right? Or is it actually invisible as a charge density?

Another example would be seeing excitonic states in band gap, but for this I don't really understand, what to compare it with — should I use some single-particle approximation to compute band structure without excitonic states, and then after computation of true energy states search for extra states in band gap, or how should I go about finding them?

Answer 1

I think you're overcomplicating things here. The whole point of the hole-model is to simplify the picture... instead of trying to model an entire sea of electrons, you model the hole itself.

For example, consider a bubble rising in a glass of cola... yes, you could create a complicated model of all the liquid moving within the glass, but it's much easier to just focus on the way the bubble moves (i.e. the tiny area with no liquid). In the same way, you can just pretend that the "hole" in the valence band is a single positively-charged particle rather than a gap in the middle of a complicated sea of electrons. The simplest model then just assigns an effective mass to the hole $m_h^*$, and its motion in the solid is described using a dispersion relation: \begin{equation} E_k = \frac{\hbar^2k^2}{2m_h^*} \end{equation}

The simplest model of an exciton (in a bulk semiconductor) could then treat the hole and an electron as an orbital pair with a reduced mass of: \begin{equation} \frac{1}{\mu} = \frac{1}{m_e^*} + \frac{1}{m_h^*} \end{equation}

The Bohr radius and exciton binding energy for the lowest (1s) exciton state are then given by an analogy with a hydrogen atom: \begin{equation} \lambda = \frac{4\pi\epsilon_r\epsilon_0\hbar^2}{\mu e^2} \end{equation} and \begin{equation} E_b = \frac{\mu e^4}{32\pi^2\hbar^2\epsilon_r^2\epsilon_0^2} \end{equation}

Note, though that there are several energy bands that a hole can occupy, and the simple effective-mass model breaks down as the kinetic energy becomes larger. In those cases, you'd need a more sophisticated model, such as $k\cdot p$ theory. Also, the bulk exciton model only really works if you are considering a single electron-hole pair in a material. You start to need complicated many-body models if you want to go any further than that. It is possible, though, to consider single excitons in heterostructures (quantum wells etc) through a variational approach to minimising the binding energy. See, for example, Chapter 6 in "Quantum Wells, Wires and Dots", P. Harrison, 3rd Ed., Wiley, Chichester (2009).