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Layered transition metal dichalcogenides do generally not exhibit characteristic electronic surface states localized perpendicular to the layers. Employing van der Waals epitaxy together with angle-resolved photoemission spectroscopy we show how surface-layer-derived electronic states can be generated on these materials. For a heterojunction consisting of one HfS 2 epilayer adsorbed on bulk WSe 2 , purely two-dimensional behavior as well as three-dimensional coupling of the epilayer to substrate bulk states is observed despite a large lattice mismatch between epilayer and substrate. The experimental results are discussed in the context of electronic structure calculations.The broken translational symmetry at surfaces of crystalline materials, in general, can lead to localized surface states and surface resonances. These features are twodimensionally periodic parallel to the surface. If the energy of these states lies within a band gap of the projected band structure of the bulk material, they are exponentially localized perpendicular to the surface. Surface resonances overlap in energy with bulk bands and usually extend several layers into the bulk because they can mix with bulk states.Two-dimensional layered materials of the transition metal dichalcogenide ͑TMDC͒ family, consisting of chalcogenmetal-chalcogen triple layers bonded by van der Waals interactions only, 1 generally do not exhibit surface derived states. This is due to the absence of broken bonds at their surface. Electronic states whose wave functions are mainly localized at the surface-which we label as surface-layer-derived states in the following-can relatively easily be generated, however, by heteroepitaxial growth of one kind of a chalcogen-metal-chalcogen triple layer on a different layered substrate material. Even in the case of a large lattice mismatch between epilayer and substrate such adlayers can be grown smoothly with their own lattice constant by van der Waals epitaxy ͑VDWE͒. 2 This way, it is possible to grow a large variety of insulating, semiconducting, or metallic TMDC heterojunctions 3,4 which are of technological importance in the field of photovoltaics 5 and in high-energydensity batteries. 6 WSe 2 , e.g., is a prototype for highefficiency solar cells. 7 More recently, Klein et al. 8 have shown that surface states on InSe layers prepared by van der Waals epitaxy on graphite do not couple to substrate states and exhibit a purely twodimensional character, therefore. In this case, the energy of the InSe surface layer states lies within a projected gap of the graphite substrate band structure so that the surface-layerderived states do not interact with the substrate states.On the contrary, here we are dealing with the opposite case in which the energy of all surface-layer-derived states coincides with the projected substrate bulk bands. In particular, we investigate which states exhibit merely twodimensional character and which display in addition elec-tronic coupling to the substrate, i.e., in the third dimension, through surface-de...

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ABSTRACT:We give a probabilistic analysis of the Multiple Depot Vehicle Routing Problem (MDVRP) where k depots and n customers are given by i.where f is the density of the absolutely continuous part of the law of the random variables giving the depots and customers and where the constant α depends on the number of depots. If k = o(n), α is the constant of the TSP problem. For k = λn, λ > 0, we prove lower and upper bounds on α, which decrease as fast as (1 + λ) −1/d .

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