Commutation technique for an exciton photocreated close to a metal

Recently, we have derived the changes in the absorption spectrum of an exciton when this exciton is photocreated close to a metal. The resolution of this problem – which has similarities with Fermi edge singularities – has been made possible by the introduction of “exciton diagrams” in which the exciton metal-electron vertex has been essentially guessed. The validity of this procedure relied on a dreadful calculation based on standard free electron and free hole diagrams, with the semiconductor-metal interaction included at second order only, and its guessed extention to higher orders. Using a commutation technique similar to the one we recently introduced to deal with interacting excitons, we are now able to prove that this exciton diagram procedure we proposed is indeed valid for this problem at any order in the interaction. PACS. 71.10.Ca Electron gas, Fermi gas – 71.35.-y Excitons and related phenomena Interactions with excitons have always been a tricky problem to handle properly. The interactions being in fact interactions with free electrons and free holes, one a priori has to crack the excitons into electrons and holes, in order to really know their effects. This leads to see the exciton as the sum of ladder diagrams [1] between one electron and one hole, with possibly, once in a while, an interaction of this electron or this hole with something else. Although fully safe, this approach becomes very fast dreadfully complicated, as can be seen from the simplest problem on interacting excitons studied in reference [2], namely an exciton photocreated close to a metallic “mirror”. It is indeed the simplest problem on interacting excitons, in the sense that the photocreated electron and the metal electrons are discernable (being spatially separated) so that there is no Pauli exclusion between them. This Pauli exclusion, and the exchange processes associated to the indiscernability of the carriers, is an additional, but major, difficulty for interacting exciton problems. Rather recently, we have developed a “commutation technique” [3,4] which allows to cleanly identify contributions coming from Coulomb interaction between excitons and contributions coming from possible exchange between carriers. Using this commutation technique, we can derive the correlations between excitons at any order exactly. We have already been able to prove that the effective bosonic Hamiltonian for excitons quoted by everyone up to now cannot be correct: First, it is not even hermitian [3]; second, it misses purely a e-mail: [email protected] Pauli terms [3]; third, and worse, the concept of effective Hamiltonian itself has to be given up [5] because, whatever the exciton-exciton part is, it cannot reproduce the exciton correlations correctly, due to the complexity of the exchange processes. If such an effective Hamiltonian were correct, exciton diagrams, with boson-exciton propagators and interaction vertices deduced from the interacting part of the Hamiltonian, would of course be fully valid. However, as such an effective Hamiltonian is incorrect, the validity of the exciton diagram procedure is actually not established at all in spite of a widely spread belief. At the time we studied the problem of an exciton photocreated close to a metal and the changes in the exciton absorption spectrum induced by the semiconductormetal interaction, we had not yet developed this commutation technique. This is why we safely used standard diagrams [6] with free electrons and free holes and Coulomb interactions between them. We were able to put the electron-metal and hole-metal interactions at second order only. At this order, we proved that the sum of all the seven complicated diagrams corresponding to these second order processes ends up with the same result as the one derived in an extremely simple way, by using intuitive “exciton diagrams”: In these, the exciton propagator was taken to be Gx(ω; ν,Q) = 1 ω − Eν,Q + iη , (1) where Eν,Q = εν+EQ is the energy of the (ν,Q) exciton, ν being the relative motion state index and Q the center of 306 The European Physical Journal B mass momentum. The exciton-metal vertex was somehow cooked in a reasonable way from the bare electron-metal and hole-metal interactions. Since there were no hope to calculate standard electron-hole diagrams with more than two electron-metal and hole-metal interactions, we assumed that the exciton diagram procedure, which looked physically quite reasonable, should hold at any order. By studying this problem in the light of our commutation technique, we are now able to prove that this exciton diagram procedure is indeed fully correct in the studied case, at any order in the semiconductor-metal interaction. Let us reconsider this problem from the beginning: A highly doped 2D quantum well is set at a distance d from an empty quantum well in which one exciton is photocreated. The metal Fermi sea reacts to the sudden appearance of the photocreated electron-hole, and its change, in turn, modifies the photon absorption. Of course, similarities with Fermi edge singularities [7–9] follow from this Fermi sea reaction. Note that in this problem we have one exciton only, so that we do not have to take into account any kind of exciton-exciton interactions. The Hamiltonian of this semiconductor-metal coupled system reads H = Hsc +Hm + Wsc−m, where Hsc is the semiconductor Hamiltonian and Hm is the metal Hamiltonian. The semiconductor-metal potential Wsc−m reads Wsc−m = ∑