Constant potential energetics of metallic and semiconducting electrodes: A benchmark study on 2D materials

Grand-canonical (GC) constant-potential methods within an implicit solvent environment provide a general approach to compute the potential-dependent energetics at electrified solid-liquid interfaces with first-principles density-functional theory. Here, we use a mindfully chosen set of 27 isostructural 2D metal halides MX2 to analyze the variation of this energetics when the electronic structure changes from metallic to semiconducting and insulating state. Apart from expectable changes due to the opening up of the electronic bandgap, the calculations also show an increasing sensitivity to the numerical Brillouin zone integration and electronic smearing, which imposes computational burdens in practice. We rationalize these findings within the picture of the total interfacial capacitance arising from a series connection of the electrochemical double-layer capacitance and the so-called quantum capacitance resulting from the filling of electronic states inside the electrode. For metals, the electrochemical double-layer capacitance dominates at all potentials, and the entire potential drop takes place in the electrolyte. For semiconductors, the potential drop occurs instead fully or partially inside the electrode at potentials within or just outside the bandgap. For 2D semiconductors, the increased sensitivity to numerical parameters then results from the concomitantly increased contribution of the quantum capacitance that is harder to converge. Fortunately, this understanding motivates a simple extension of the CHE + DL approximation for metals, which provides the approximate GC energetics of 2D semiconductors using only quantities that can be obtained from computationally undemanding calculations at the point of zero charge and a generic double-layer capacitance.