Two-dimensional excitons in monolayer transition metal dichalcogenides from radial equation and variational calculations

Abstract

Exciton spectra of monolayer transition metal dichalcogenides (TMDs) in various dielectric environments are studied using an effective mass model incorporating a screened two-dimensional (2D) electron-hole interaction described by the Keldysh potential. Exciton states are calculated by solving a radial equation (RE) with a shooting method including Runge-Kutta integration. Particular attention is paid to the simple models for 2D exciton calculation. The 2D hydrogen model yields much lower exciton energies than the Rydberg series from the RE solution. The screened hydrogen model (SHM) [Phys. Rev. Lett. 116, 056401 (2016)] is examined by comparing its exciton spectra with the RE solutions. While the SHM is found to describe the nonhydrogenic exciton Rydberg series (i.e., the energy's dependence on main quantum number n) reasonably well, it fails to account for the linear decrease of the exciton energy with the orbital quantum number m. The exciton Bohr orbit shrinks as |m| becomes larger resulting in increased strength of the electron-hole interaction and a decrease of the exciton energy. The exciton effective radius expression of the SHM can characterize the exciton radius's dependence on n, but it cannot properly describe the exciton radius's dependence on m, which is the cause of the SHM's poor description of the exciton energy's m-dependence. For monolayer WS 2 on the SiO 2 substrate, our calculated s exciton Rydberg series agrees closely with that measured by optical reflection spectroscopy [Phys. Rev. Lett. 113, 076802 (2014)], while the calculated p excitons offer an explanation for the two broad features of a two-photon absorption spectrum [Nature 513, 214 (2014)]. Our calculated exciton energies for monolayer TMDs in various dielectric environments compare favourably with experimental data. Variational wave functions are obtained for a number of strongly bound exciton states and further used to study the Stark effects in 2D TMDs, an analytical expression being deduced which yields a redshift of the ground state energy to a good accuracy. The numerical solution of the RE combined with the variational method provides a simple and effective approach for the study of 2D excitons in monolayer TMDs. * Electronic address: phyjzzhang@jlu.edu.cn 2

Ultrafast Optoelectronic Processes in 1D Radial van der Waals Heterostructures: Carbon, Boron Nitride, and MoS2 Nanotubes with Coexisting Excitons and Highly Mobile Charges

“…[1][2][3][4][5] Significant exciton binding energies E B arise from the weak, non-local dielectric screening of the Coulomb interaction in 2D atomically-thin TMDs, combined with the relatively high effective masses of carriers: 5 for instance E B = 0.24 − 0.27 eV for the A exciton in a MoS 2 monolayer. 6,7 Heterostructures of atomically-thin 2D materials show unique emergent properties associated with interlayer coupling and charge transfer, opening up new possibilities for the development of nanoelectronic devices. 8,9 In such van der Waals heterostructures, excitons can remain tightly bound even across lateral and vertical heterojunctions.…”

Electrostatic Interaction of Point Charges in Three-Layer Structures: The Classical Model

“…However, both authors analyzed the interaction between two charges located at different sides of the slab, so that the z-coordinates of the charges were fixed (z = − 1 2 and z = 1 2 in our notation), and the interaction energy w was z-independent. The concrete results obtained in those articles [48,113] are often applied to a number of charge configurations without specifying them and applying the RKA formula mostly to exciton physics in heterostructures [68,72,73,[76][77][78]98,101,[114][115][116][117][118][119].…”

“…Even this approach turned out to be quite fruitful to uncover salient features of electrostatic interaction in three-layer systems. The results can be applied to metal-oxide-semiconductor structures [60,61], heterostructures [62][63][64][65][66] (often with special emphasis on excitons [48,[67][68][69][70][71][72][73][74][75][76][77][78] and possible electron-hole superfluidity [46,47,[79][80][81][82][83][84][85][86]), levitating electrons [29,30,50,87,88], trapped ions [29,87,89,90], and biological systems [91][92][93], where electrostatic interactions play an important role.…”

Crossover from weakly indirect to direct excitons in atomically thin films of InSe

“…2). The non-hydrogen-like energy sequence [48,49] is due to the 2D screening of the e-h interaction in the film leading to a Keldysh-like potential for L = 1 and 2.…”

“…Comparison of binding energies in meV as obtained from the harmonic oscillator basis against analytical and calculated results[49] for suspended MoS 2 . Basis size used in the comparison with MoS 2 monolayer corresponded to N max = 12 (basis size=91 states) and λ was optimized.…”