  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.
