Calculations Of Exciton States And Stark Effect For Monolayer Transition Metal Dichalcogenides

Monolayer transition metal dichalcogenides is a new-type of two-dimensional?2D?atomic crystal discovered after graphene,the most remarkable of these is Monolayer MoS₂,which consists of a single layer of molybdenum atoms sandwiched between two layers of sulphur atoms in a trigonal prismatic structure.Distinct from multilayers and bulk of hexagonal MoS₂,monolayer MoS₂ is a direct-bandgap semiconductor with explicitly broken inversion symmetry.The conduction and valence band-edges are located in momentum space at two different sorts of points in the Brillouin zone,namely the K and K' valleys,thus corresponding to two valley degrees of freedom.Monolayer MoS₂ has novel optical properties and exciton properties that are interrelated.In terms of optical properties,optical excitation with circularly polarized light can create dynamic valley polarization due to the spin-valley coupling,causing absorption of left circularly polarized and right circularly polarized light at the K and K' valleys,respectively?called the optical selection rules?.Therefore control of valley polarization can be achieved by optical helicity.The excitons--namely electron-hole pairs bound by Coulomb attraction--created by photoexcitation,on the other hand,have a large binding energy,ranging from 0.5 to 1 eV,which are two orders of magnitude larger than in semiconductor quantum wells.In 2D materials such as monolayer?around 3 A thick?MoS₂,a complicated electron-hole interaction occurs together with nonlocal dielectric screening:at a long distance the effective interaction behaves like the Coulomb interaction with screening determined by its environment while at close proximity it diverges logarithmically with screening depending mainly on the 2D material,resulting in excitons with a large binding energy due to the strong electron-hole interaction.Exciton peaks and trion peaks observed are characteristic of measured optical absorption spectra as well as photoluminescence spectra,which arise from interband transitions associated with exciton absorption and radiative recombination.The main results of this thesis are summarized as follows.First,a plane-wave expansion method is used to solve the Wannier equation for 2D excitons.The exciton wave-functions are expanded in terms of the plane waves that form an orthonormal basis,and by imposing the periodic boundary condition,the Wannier equation is then transformed into a set of secular equations in momentum space?connected with a real symmetric matrix?,which are solved using standard numerical diagonalization technique such as the QR algorithm to yield a series of exciton states,including the ground state and excited states,and further the binding energies of these excitons.This is referred to as the 2D method in the thesis.For monolayer MoS₂,for instance,the calculated ground state?n = 1,m = 0?binding energy is 0.530 eV,while the first excited states?n = 2,m = ±1?are doubly degenerate and have a binding energy of 0.292 eV,and the binding energy of the second excited state?n = 2,m = 0?is 0.234 eV.Second,reducing the Wannier equation to a one-dimensional?1D?radial equation,then a fourth-order Runge-Kutta method is developed to solve the exciton radial equation.Appropriate boundary conditions are obtained by exploiting the asymptotic expressions of the electron-hole interaction,allowing for a fast and yet accurate solution of the exciton states as well as their binding energies.For the ground state and the first and second excited states,the binding energies calculated with the 1D method are 0.555,0.318 and 0.258 eV,respectively.Third,based on our numerical results and using the hydrogenic wave functions,we have constructed variation wave functions for the first three energy levels.The accuracy of these analytical wave functions has been proved by comparing the energy levels obtained using the variational method and the 1D method.The use of variational wave-functions will facilitate evaluations and simplify calculations of binding energy,exciton-phonon scattering and exciton relaxation.Fourth,the Stark effect of the 2D excitons is studied in the presence of an external electric field.As the effective exciton Hamiltonian in this case depends on the angle?the field direction?,the exciton equation cannot be simplified to a 1D radial equation any more,but it can be numerically solved using our 2D plane wave expansion method,specifically,by diagonalizing the resulting Hermitian matrix.The shifts of exciton energy levels are analyzed by comparison with previous studies.The ground state energy is redshifted,which increases quadratically with the field due to the second-order Stark effect;the exciton polarizability is ?1s = 5.6 × 10-18 eV?m/V?²,making a shift of about 1 millielectron volt for an electric field of 20 V/micro metre,in agreement with previous findings.The 2p and 2s excitons are also redshifted but with an order of magnitude larger polarizability than the ground state(?2p1 = 9.2× 10-17 eV?m/V?²,?^2p2 = 2.9 × 10-17 eV?m/V?²,?^2s = 5.0 ×10-17 eV?m/V?²).Our studies have provided a deeper understanding of the 2D excitons in monolayer dielectrics,and more importantly,new effective methods for studies on the exciton physics of 2D atomic crystals.