Extension from Excitons to Trions

Question

I am curious about the fundamentals of how a trion state is defined. An exciton state is considered a bound electron-hole pair. This can be formed by is we have a ground state which consists of a full valence band and a photon comes along with momentum $\vec{Q}$ and excites an electron to the conduction band, from which we leave a hole (absence of an electron) in itself place which is bound together through an interaction term (i.e Coulomb). Therefore such a state could be written as: $$c^\dagger(\vec{k}+\vec{Q})b(\vec{k})|GS\rangle$$

If we then look at a trion which can be considered as a charged exciton for this case. We have a three-body particle that either contains 2 holes and 1 electron (positive trion) or 2 electrons and 1 hole (negative trion).

Therefore we can consider a trion to be formed when a photo-excited exciton (as mentioned above binds with either an extra electron or a hole. My question here is how does one write this state? Does it have an impact on the extra momentum $\vec{Q}$ mentioned for the exciton? Because in theory how in this set-up would one have an extra electron in the conduction band to bind with the exciton system to form a trion? Hence, for a negative trion would this be:

$$c^\dagger(\vec{k}_1)c^\dagger(\vec{k}_2+\vec{Q})b(\vec{k}_1+\vec{k}_2)|GS\rangle$$ or would both of the electrons now in the system have equal extra momentum i.e $$c^\dagger(\vec{k}_1+\vec{Q}/2)c^\dagger(\vec{k}_2+\vec{Q}/2)b(\vec{k}_1+\vec{k}_2)|GS\rangle$$ In turn, both of these states can give the same results when going to calculation trion energies, but I am wondering what the correct formalism is

Answer 1

Trions are usually localized states. Although ions of different chemical elements are ubiquitous, they are non-existent for Hydrogen, since a complex of a proton and two electrons cannot have negative energy. This is different in semiconductors structures, where, in addition to the Coulomb-like attraction between a hole and electrons, there is also a confining potential and different shape of the dispersion relation.

The simplest case to visualize is a trion in a quantum dot, where the motion of all the quasiparticles is restricted, and they occupy discrete quantum states (image source):

The problem is whether this could be really described as an exciton-like state, since the particles are not held together by the Coulomb potential alone.

One can relax somewhat the confinement by allowing motion in one or two dimensions - e.g., in quantum wires/carbon nanotubes or quantum wells. Perhaps, trions are possible even in bulk materials with peculiar dispersion relations (i.e., the kinetic energy very different from that of protons and electrosn in free space).