Semiconductor detectors in general have a dead layer at their surfaces that is either a result of natural or induced passivation, or is formed during the process of making a contact. Charged particles passing through this region produce ionization that is incompletely collected and recorded, which leads to departures from the ideal in both energy deposition and resolution. The silicon p-i-n diode used in the KATRIN neutrino-mass experiment has such a dead layer. We have constructed a detailed Monte Carlo model for the passage of electrons from vacuum into a silicon detector, and compared the measured energy spectra to the predicted ones for a range of energies from 12 to 20 keV. The comparison provides experimental evidence that a substantial fraction of the ionization produced in the "dead" layer evidently escapes by diffusion, with 46% being collected in the depletion zone and the balance being neutralized at the contact or by bulk recombination. The most elementary model of a thinner dead layer from which no charge is collected is strongly disfavored.