Like other 2D materials, such as graphene and the transition-metal dichalcogenides (TMDCs), black phosphorus has dramatically different electronic and mechanical properties from its bulk, 3D, parent and so may find use in a host of novel device applications. And just like graphene (which is a sheet of carbon atoms arranged in a honeycomb lattice), black phosphorus is a layered material containing individual phosphorus atoms that are arranged hexagonally. Each atomic layer is held together by weak van der Waals forces. However, in phosphorene, the surface is puckered, and this seems to make all the difference when it comes to bandgap behaviour.

Bulk phosphorene is a semiconductor with a moderate bandgap of between 0.31 and 0.35 eV, but the monolayer material is predicted to be an insulator with a much larger bandgap that varies with the number of phosphorus layers. Although such predictions have already been confirmed in laboratory experiments, researchers are still unsure as to where this considerable bandgap broadening comes from as the material is scaled down to monolayers.

GW approximation model

Now, Alexander Rudenko and Mikhail Katsneslon have made the first detailed analysis of the electronic properties of monolayer, multilayer and bulk phosphorene using a tight-binding model. This model is based on very accurate ab-initio calculations - a so-called quasiparticle GW approximation - and is important in two respects, explains Rudenko. “First, it sheds light on the origin of the bandgap in phosphorene and how it evolves with the number of layers in the material. Second, it serves as a starting point for further theoretical studies of transport properties, disorder and many other effects in this structure.”

The semiconducting-to-insulating transition observed as bulk phosphorene scales down to monolayers is not seen in other known 2D sheets like the TMDCs, which go from being indirect bandgap semiconductors in the bulk to direct bandgap semiconductors when scaled down.

The computational approach we employed is different to more routinely used density-functional theory (DFT) calculations, said Rudenko. “Instead of dealing with effective density functionals, as in DFT, we solve the many-body problem of electron-electron interactions in a more systematic manner by explicitly calculating the material’s ‘self-energy’,” he told “To analyse the result for a single layer of black phosphorus, we construct a minimal four-band model that describes the most relevant region of the energy spectrum involved in the material.”

Completely different behaviour

The model shows that interlayer hoppings strongly affect the bandgap in multilayer phosphorene. Such behaviour is completely different to that seen in graphene and other 2D materials like the TMDCs, he explained. For example, in graphene just one nearest-neighbour electron hopping is enough to reproduce the main characteristics of this material’s energy spectrum, he said. “However, the model for single-layer black phosphorus involves two important parameters, describing the in-plane and out-of-plane nearest-neighbour hoppings. A second (repulsive) parameter, which comes about thanks to the puckered structure of phosphorene sheets, appears to be the main reason why a bandgap opens up in this multilayer material,” he confirmed.

The team detailed its calculations in Physical Review B DOI: