A fresh view on Frenkel excitons: Electron–hole pair exchange and many-body formalism

Abstract

Excitons are coherent excitations that travel over the semiconductor sample. Two types are commonly distinguished: Wannier excitons that are formed in inorganic materials, and Frenkel excitons that are formed in organic materials and rare-gas crystals.(i) Wannier excitons are made from delocalized conduction electrons and delocalized valence holes. The Coulomb interaction acts in two different ways. The intraband processes bind a free electron and a free hole into a Wannier exciton wave, while the interband processes separate the optically bright excitons made of spin–singlet electron–hole pairs from dark excitons made of spin–triplet pairs, with an additional transverse–longitudinal splitting that results from a nonanalytical Coulomb scattering which depends on the exciton wave vector direction with respect to the crystal axes.(ii) Frenkel excitons are made from highly localized excitations. The Coulomb interaction then acts in one way only, the intralevel Coulomb processes between sites, analogous to intraband processes, being negligible for on-site excitations in the tight-binding limit. The interlevel Coulomb processes, analogous to interband processes, produce both, the Frenkel exciton wave and its singular splitting, through electron–hole pair exchange between sites: the excitonic wave is produced by delocalizing the on-site excitations through the recombination of an electron–hole pair on one site and its creation on another site. These interlevel processes also split the exciton level in exactly the same singular way as for Wannier excitons. Importantly, since the interlevel Coulomb interaction only acts on spin–singlet pairs, just like the electron–photon interaction, optically dark pairs do not form excitonic waves; as a result, Frenkel excitons are optically bright in the tight-binding limit.We here present a fresh approach to Frenkel excitons in cubic semiconductor crystals, with a special focus on the spin and spatial degeneracies of the electronic states. This approach uses a second quantization formulation of the problem in terms of creation operators for electronic states on all lattice sites — their creation operators being true fermion operators in the tight-binding limit valid for semiconductors hosting Frenkel excitons. This operator formalism avoids using cumbersome (6Ns×6Ns) Slater determinants – 2 for spin, 3 for spatial degeneracy and Ns for the number of lattice sites – to represent state wave functions out of which the Frenkel exciton eigenstates are derived. A deep understanding of the tricky Coulomb physics that takes place in the Frenkel exciton problem, is a prerequisite for possibly diagonalizing this very large matrix analytically. This is done in three steps:(i) the first diagonalization, with respect to lattice sites, follows from transforming excitations on the Ns lattice sites Rℓ into Ns exciton waves Kn, by using appropriate phase prefactors;(ii) the second diagonalization, with respect to spin, follows from the introduction of spin–singlet and spin–triplet electron–hole pair states, through the commonly missed sign change when transforming electron-absence operators into hole operators;(iii) the third diagonalization, with respect to threefold spatial degeneracy, leads to the splitting of the exciton level into one longitudinal and two transverse modes, that result from the singular interlevel Coulomb scattering in the small Kn limit.To highlight the advantage of the second quantization approach to Frenkel exciton we here propose, over the standard first quantization procedure used for example in the seminal book by R. Knox, we also present detailed calculations of some key results on Frenkel excitons when formulated in terms of Slater determinants.Finally, as a way forward, we show how many-body effects between Frenkel excitons can be handled through a composite boson formalism appropriate to excitons made from electron–hole pairs with zero spatial extension. Interestingly, this Frenkel exciton study led us to reformulate the dimensionless parameter that controls exciton many-body effects — first understood in terms of the Wannier exciton Bohr radius driven by Coulomb interaction – as a parameter entirely driven by the Pauli exclusion principle between the exciton fermionic components. This formulation is not only valid for Wannier excitons but also for composite bosons like Cooper pairs.