Magnetic field effect on quaternion excitonic complexes in bilayer structures near metals

Two-dimensional monolayer structures of transition metal dichalcogenides (TMDs) have been shown to allow many higher order excitonic bound states such as trions (charged excitons), biexcitons (excitonic molecules) and charged biexcitons. Recently [1], an atom-like excitonic structure was reported experimentally in bilayer TMDs in accord with theory predictions – the quaternion – a complex of a free charge carrier in top layer bound to a like-charge trion in bottom layer placed close to a parallel metal layer to screen the excessive repulsive interaction in the system. Quaternions are doubly charged bosons that could be the basis for a new path to the robust Wigner crystallization effect or, at the other extreme, to superconductivity without Cooper pairing. Here, we study theoretically the effect of the perpendicular magnetic field on quaternions. We show that, contrary to the linear Zeeman shift known for excitons/trions in TMD monolayers [2], the quaternion ground state exhibits a quadratic magnetic field shift similar to that known for hydrogen-like atoms (muonium, positronium) but with unexpectedly different quadratic field dependences in the K(+) and K(-) valleys. [1] Z.Sun, et al., Nano Lett. 21, 7669 (2021); [2] D.MacNeill, et al., Phys. Rev. Lett. 114, 037401 (2015).