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For the past two and a half decades, anomalous heating of trapped ions from nearby electrode surfaces has continued to demonstrate unexpected results. Caused by electric-field noise, this heating of the ions’ motional modes remains an obstacle for scalable quantum computation with trapped ions. One of the anomalous features of this electric-field noise is the reported nonmonotonic behavior in the heating rate when a trap is incrementally cleaned by ion bombardment. Motivated by this result, the present work reports on a surface analysis of a sample ion-trap electrode treated similarly with incremental doses of Ar ^+ ion bombardment. Kelvin probe force microscopy and x-ray photoelectron spectroscopy were used to investigate how the work functions on the electrode surface vary depending on the residual contaminant coverage between each treatment. It is shown that the as-fabricated Au electrode is covered with a hydrocarbon film that is modified after the first treatment, resulting in work functions and core-level binding energies that resemble that of atomic-like carbon on Au. Changes in the spatial distribution of work functions with each treatment, combined with a suggested phenomenological coverage and surface-potential roughness dependence to the heating, appear to be related to the nonmonotonic behavior previously reported.