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The modified Becke-Johnson meta-GGA potential of density functional theory has been shown to be the best exchange-correlation potential to determine band gaps of crystalline solids. However, it cannot be consistently used for the electronic structure of nonperiodic or nanostructured systems. We propose an extension of this potential that enables its use to study heterogeneous, finite, and low-dimensional systems. This is achieved by using a coordinate-dependent expression for the parameter c that weights the Becke-Russel exchange, in contrast to the original global formulation, where c is just a fitted number. Our potential takes advantage of the excellent description of band gaps provided by the modified Becke-Johnson potential and preserves its modest computational effort. Furthermore, it yields with one single calculation band diagrams and band offsets of heterostructures and surfaces. We exemplify the usefulness and efficiency of our local meta-GGA potential by testing it for a series of interfaces (Si/SiO2, AlAs/GaAs, AlP/GaP, and GaP/Si), a Si surface, and boron nitride monolayer.