Finally, we estimate the scaling of the exciton binding energy with substrate screening, using both the Bethe-Salpeter equation (BSE) and an effective-mass 2D Wannier model. In addition, we highlight the effects of semi-infinite substrates on the optical properties. In all cases, excitonic optical spectra in excellent agreement with experiments and ab initio theory are obtained. Hence, this system has many similarities with a two-band 2D semiconductor, and demonstrates the versatility of the current approach. Nonetheless, the surface states of Si(111) 2вЂ‰Г—вЂ‰1 are confined to a dвЂ‰в‰€вЂ‰4вЂ‰Г… surface region, with energies well-isolated from the bulk bands in most of the Brillouin zone 16, 17, 18.

The latter example represents a somewhat different system, for which the substrate cannot be eliminated for obvious reasons. However, most theoretical work are currently being carried out on free-standing monolayers, neglecting possibly important effects of substrate screening almost always present in experiments. The former two cases represent van der Waals bound 2D semiconductors - a family of materials whose optical response are currently being intensively studied 4, 5, 12, 13, 15. To test the applicability of the model presented here, we consider three 2D semiconductors for which rigorous ab initio results are available for comparison: Monolayer hBN, monolayer MoS 2 and the dangling ПЂ-electron bonds of a 2вЂ‰Г—вЂ‰1 buckling-chain reconstructed 14 Si(111) surface. This model is particularly useful in the context of excitons constructed on the basis of a minimal TB model, but may readily be generalized for ab initio methods. In the present work, we derive an analytical model dielectric screening function for 2D semiconductors, allowing for the description of substrate screening.

Thus, reliable model dielectric screening methodologies are highly attractive for the study of many-body optical properties of 2D systems. The latter approach, in particular, is questionable for free-standing 2D materials, where the dielectric screening is known to approach unity 3 for small q - hence a simple scaling of the effective screening by a constant factor leads to the wrong qualitative behaviour for small q. However, the lack of completeness in such restricted bases typically implies under-screening, necessitating simplified screening models 12 or ad hoc rescaling of RPA results 2. Furthermore, tight-binding (TB) states have recently been applied as a highly efficient basis for constructing exciton wave functions 2, 11, 12, 13. The study of many-body effects in bulk 3D semiconductors has been aided by the application of model dielectric functions 7, 8, 9, 10, where the essential q-dependence is extrapolated from simple analytical formulas, thus side-stepping numerically taxing RPA response functions. Hence, screening in 2D materials is known to be highly non-local, which in reciprocal space translates into a function depending strongly on momentum transfer q 3, 4, 5, 6. In particular, the question of translating the inherently 3D concept of a dielectric function, calculated ab initio in the random phase approximation (RPA) from super-cell geometries, to free-standing 2D materials has been scrutinized in great detail 3, 4, 5. Screening in 2D semiconductors, e.g., has recently attracted a tremendous amount of attention. The dielectric screening is an essential ingredient in such calculations 1, and especially so for low-dimensional systems 1, 2, 3, 4, 5.

In that case, replace Q by Q/Оµ, where Оµ is the relative permittivity due to these other contributions.It is by now recognized that correlated electron-hole pairs, termed excitons, are essential for reliably modelling the optical response of semiconducting materials. Note that there may be dielectric permittivity in addition to the screening discussed here for example due to the polarization of immobile core electrons. With k 0=0 (no screening), this becomes the familiar Coulomb's law. The Thomas-Fermi wavevector (in Gaussian-cgs units) is k 0 2 = 4 ПЂ e 2 в€‚ n в€‚ Ој ( cgs-Gaussian) It is a special case of the more general Lindhard theory in particular, ThomasвЂ“Fermi screening is the limit of the Lindhard formula when the wavevector (the reciprocal of the length-scale of interest) is much smaller than the fermi wavevector, i.e. ThomasвЂ“Fermi screening is a theoretical approach to calculating the effects of electric field screening by electrons in a solid.

Thomas fermi screening parameter