The existence of strongly bound excitons is one of the hallmarks of the newly discovered atomically thin semi-conductors. While it is understood that the large binding energy is mainly due to the weak dielectric screening in two dimensions (2D), a systematic investigation of the role of screening on 2D excitons is still lacking. Here we provide a critical assessment of a widely used 2D hydrogenic exciton model which assumes a dielectric function of the form (q) = 1 + 2παq, and we develop a quasi-2D model with a much broader applicability. Within the quasi-2D picture, electrons and holes are described as in-plane point charges with a finite extension in the perpendicular direction and their interaction is screened by a dielectric function with a non-linear q-dependence which is computed ab-initio. The screened interaction is used in a generalized Mott-Wannier model to calculate exciton binding energies in both isolated and supported 2D materials. For isolated 2D materials, the quasi-2D treatment yields results almost identical to those of the strict 2D model and both are in good agreement with ab-initio many-body calculations. On the other hand, for more complex structures such as supported layers or layers embedded in a van der Waals heterostructure, the size of the exciton in reciprocal space extends well beyond the linear regime of the dielectric function and a quasi-2D description has to replace the 2D one. Our methodology has the merit of providing a seamless connection between the strict 2D limit of isolated monolayer materials and the more bulk-like screening characteristics of supported 2D materials or van der Waals heterostructures.

Vertical stacking of two-dimensional (2D) crystals, such as graphene and hexagonal boron nitride, has recently lead to a new class of materials known as van der Waals heterostructures (vdWHs) with unique and highly tunable electronic properties. Ab-initio calculations should in principle provide a powerful tool for modeling and guiding the design of vdWHs, but in their traditional, form such calculations are only feasible for commensurable structures with a few layers. Here we show that the dielectric properties of realistic, incommensurable vdWHs comprising hundreds of layers can be calculated with ab-initio accuracy using a multi-scale approach where the dielectric functions of the individual layers (the dielectric building blocks) are coupled simply via their long-range Coulomb interaction. We use the method to illustrate the 2D-3D dielectric transition in multi-layer MoS2 crystals, the hybridization of quantum plasmons in large graphene/hBN heterostructures, and to demonstrate the intricate effect of substrate screening on the non-Rydberg exciton series in supported WS2.The class of 2D materials which started with graphene is rapidly expanding and now includes metallic and semiconducting transition metal dichalcogenides [1] in addition to group III-V semi-metals, semiconductors and insulators [2]. These atomically thin materials exhibit unique opto-electronic properties with high technological potential [3][4][5][6][7]. However, the 2D materials only form the basis of a new and much larger class of materials consisting of vertically stacked 2D crystals held together by weak van der Waals forces. In contrast to conventional heterostructures which require complex and expensive crystal-growth techniques to epitaxially grow the single-crystalline semiconductor layers, vdWHs can be stacked in ambient conditions with no requirements of lattice matching. The latter implies a weaker constraint, if any, on the choice of materials that can be combined into vdWHs.The weak inter-layer binding suggests that the individual layers of a vdWH largely preserve their original 2D properties modified only by the long range Coulomb interaction with the surrounding layers. Turning this argument around, it should be possible to predict the overall properties of a vdWH from the properties of the individual layers. In this Letter we show that this can indeed be achieved for the dielectric properties. Conceptually, this extends the Lego brick picture used by Geim and Grigorieva [8] for the atomic structure of a vdWH, to its dielectric properties. Specifically, we develop a semi-classical model which takes as input the dielectric functions of the individual isolated layers computed fully quantum mechanically and condensed into the simplest * kiran@fysik.dtu.dk † thygesen@fysik.dtu.dk possible representation, and couple them together via the Coulomb interaction, see Figure 1. Despite the complete neglect of interlayer hybridization, the model provides an excellent account of both the spatial and dynamical dielectric properties of vdWHs....

We present a generalized hydrogen model for the binding energies (EB) of excitons in twodimensional (2D) materials that sheds light on the fundamental differences between excitons in two and three dimensions. In contrast to the well-known hydrogen model of three-dimensional (3D) excitons, the description of 2D excitons is complicated by the fact that the screening cannot be assumed to be local. We show that one can consistently define an effective 2D dielectric constant by averaging the screening over the extend of the exciton. For an ideal 2D semiconductor this leads to a simple expression for EB that only depends on the excitonic mass and the 2D polarizability α. The model is shown to produce accurate results for 51 transition metal dichalcogenides. Remarkably, over a wide range of polarizabilities the expression becomes independent of the mass and we obtain E 2D B ≈ 3/(4πα), which explains the recently observed linear scaling of exciton binding energies with band gap. It is also shown that the model accurately reproduces the non-hydrogenic Rydberg series in WS2 and can account for screening from the environment.A striking property of two-dimensional semiconductors is the ability to form strongly bound excitons. This was initially predicted theoretically for hBN [1], graphane [2] and various transition metal dichalcogenides [3][4][5] and has subsequently been confirmed experimentally [6][7][8].The quantum confinement of excitons in 2D comprises a tempting and intuitively appealing explanation for the large binding energies in these materials [9]. However, it is now well understood that the confinement of the local electronic environment in 2D plays a crucial role in the formation of strongly bound excitons [3,10]. The 2D electronic system is rather poor at screening interactions and the effective Coulomb interaction between an electron and a hole is simply much stronger in 2D than in 3D.From a first principles point of view, the treatment of excitons requires advanced computational methodology such as the Bethe-Salpeter equation [11,12]. This approach has been applied to obtain absorption spectra for numerous insulators and usually yields very good agreement with experiments [13]. However, only systems of modest size can be treated this way and simplified models of excitons will be an inevitable ingredient in calculations of realistic systems. For example, if the effect of substrates or dielectric environment is to be included in the calculation of excitons in 2D systems [14], the computations become intractable with a standard BetheSalpeter approach. For three-dimensional materials the Mott-Wannier model comprises a strong conceptual and intuitive picture that provides a simple framework for calculating exciton binding energies [15]. In the center of mass frame, an excited electron-hole pair can be shown to satisfy a hydrogenic Schrödinger equation, where band structure effects are included through an excitonic effective mass µ and the dielectric screening from the environment is included through the static d...

Two-dimensional (2D) semiconducting materials are promising building blocks for optoelectronic applications, many of which require efficient dissociation of excitons into free electrons and holes. However, the strongly bound excitons arising from the enhanced Coulomb interaction in these monolayers suppresses the creation of free carriers. Here, we identify the main exciton dissociation mechanism through time and spectrally resolved photocurrent measurements in a monolayer WSe2 p–n junction. We find that under static in-plane electric field, excitons dissociate at a rate corresponding to the one predicted for tunnel ionization of 2D Wannier–Mott excitons. This study is essential for understanding the photoresponse of 2D semiconductors and offers design rules for the realization of efficient photodetectors, valley dependent optoelectronics, and novel quantum coherent phases.

van der Waals heterostructures (vdWH) are ideal systems for exploring light-matter interactions at the atomic scale. In particular, structures with a type-II band alignment can yield detailed insight into carrier-photon conversion processes, which are central to, for example, solar cells and light-emitting diodes. An important first step in describing such processes is to obtain the energies of the interlayer exciton states existing at the interface. Here we present a general first-principles method to compute the electronic quasi-particle (QP) band structure and excitonic binding energies of incommensurate vdWHs. The method combines our quantum electrostatic heterostructure (QEH) model for obtaining the dielectric function with the many-body GW approximation and a generalized 2D Mott-Wannier exciton model. We calculate the level alignment together with intra- and interlayer exciton binding energies of bilayer MoS/WSe with and without intercalated hBN layers, finding excellent agreement with experimental photoluminescence spectra. A comparison to density functional theory calculations demonstrates the crucial role of self-energy and electron-hole interaction effects.