The trion: two electrons plus one hole versus one electron plus one exciton

Abstract

We first show that, for problems dealing with trions, it is totally hopeless to use the standard many-body description in terms of electrons and holes and its associated Feynman diagrams. We then show how, by using the description of a trion as an electron interacting with an exciton, we can obtain the trion absorption through far simpler diagrams, written with electrons and excitons. These diagrams are quite novel because, for excitons being not exact bosons, we cannot use standard procedures designed to deal with interacting true fermions or true bosons. A new many-body formalism is necessary to establish the validity of these electron-exciton diagrams and to derive their specific rules. It relies on the "commutation technique"we recently developed to treat interacting close-to-bosons. This technique generates a scattering associated to direct Coulomb processes between electrons and excitons and a dimensionless "scattering" associated to electron exchange inside the electronexciton pairs -this "scattering" being the original part of our many-body theory.It turns out that, although exchange is crucial to differentiate singlet from triplet trions, this "scattering" enters the absorption explicitly when the photocreated electron and the initial electron have the same spini. e., when triplet trions are the only ones createdbut not when the two spins are different, although triplet trions are also created in this case. The physical reason for this rather surprising result will be given.

Trion ground state, excited states, and absorption spectrum using electron-exciton basis

“…In Ref. 37, we have shown (see Eq. (3.16) in this reference) that the wave function of a trion with center-of-mass momentum K, relative motion index η and spin S = (0, 1) splits as r e , r e ′ , r h |K, η, S = R T |K r, u|η, S , where R T = (m e r e + m e r e ′ + m h r h )/(2m e + m h ) is the center-of-mass coordinate, r = r e − r h is the distance between one electron and the hole, while u = r e ′ − (m e r e + m h r h )/(m e + m h ) is the distance between the other electron and the center-of-mass coordinate of the electron-hole pair.…”

Influence of free carriers on exciton ground states in quantum wells

“…In addition, starting from very low concentrations of free electrons in the quantum well, a bound trion (negatively charged exciton) state [11][12][13][14] is observed in experiments [6][7][8][9][10] . The present article will not be concerned with properties related to the trion state, but its existence provides another channel for exciton-electron interactions.…”

Effect of fermionic components on trion–electron scattering

“…By comparing the two above equations, we note that the trion center-of-mass momentum k j splits between electron and exciton according to their masses, just as the exciton does. In the same way, the prefactor ν, p|η, S in the trion expansion (9) is the ''Fourier transform in the exciton sense'' of the trion relative motion wave function, as shown in previous works [17,18]. Let us briefly recall a few important points for the trion physics we have obtained in these works.…”

“…Within these trion variables, the orbital parity reads r, u|η, S = (−1) S r , u |η, S . This condition leads, for the Fourier transform in the exciton sense, to [18] ν, p|η, S = drdu ν|r p|u r, u|η, S = (−1) S ν ,p ν|p + α e p p + α e p |ν ν , p |η, S . (14) It is then possible to show that expression (9) for trion creation operator also reads (14) are generated by crossed scalar products when using Eq.…”