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

TL;DR

This paper introduces a new second quantization formalism for Frenkel excitons in cubic semiconductors, simplifying the representation of excitonic states and providing insights into their Coulomb interactions and mode splitting.

Contribution

It develops a fermionic operator-based approach to analyze Frenkel excitons, avoiding complex determinants and enabling systematic diagonalization of degeneracies.

Findings

Diagonalization of exciton states into wave, spin, and spatial modes.

Identification of longitudinal and transverse exciton modes.

Clarification of Coulomb scattering effects at small wave vectors.

Abstract

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 ($6N_s$ x $6N_s$) Slater determinants -- 2 for spin, 3 for spatial degeneracy and $N_s$ 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 $N_s$ lattice sites $\mathbf{R}_\ell$ into $N_s$ exciton waves $\mathbf{K}_n$, 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 $\mathbf{K}_n$ limit.