Fermi Level Determination for Charged Systems via Recursive Density of States Integration

Determining the Fermi level position for a given material is important to understand many of its electronic and chemical properties. Ab initio methods are effective in computing Fermi levels when using charge-neutral supercells. However, in the case where charges are explicitly included, the compensating homogeneous background charge, which is necessary to maintain charge neutrality in periodic models, causes the vacuum potential to be ill-defined - which would otherwise have been a reliable reference potential. Here, we develop a method based on recursively integrating the density of states to determine shifts in the Fermi level upon charging. By introducing incremental charges, one can compute the density of states profile and determine the shift in the Fermi level that corresponds to adding or removing a given increment of charge δq, which allows the evaluation of the Fermi level for any arbitrary charge q. We test this method for a range of materials (graphene, h-BN, C₃N₄, Cu, and MoS₂) and demonstrate that this method can produce a reasonable agreement with models that rely on localized compensating background charges.