  Experiments at synchrotron radiation sources and X-ray Free-Electron Lasers
in the soft X-ray energy range ($250$eV--$2$keV) stand to benefit from the
adaptation of the hybrid silicon detector technology for low energy photons.
Inverse Low Gain Avalanche Diode (iLGAD) sensors provide an internal gain,
enhancing the signal-to-noise ratio and allowing single photon detection below
$1$keV using hybrid detectors. In addition, an optimization of the entrance
window of these sensors enhances their quantum efficiency (QE). In this work,
the QE and the gain of a batch of different iLGAD diodes with optimized
entrance windows were characterized using soft X-rays at the
Surface/Interface:Microscopy beamline of the Swiss Light Source synchrotron.
Above $250$eV, the QE is larger than $55\%$ for all sensor variations, while
the charge collection efficiency is close to $100\%$. The average gain depends
on the gain layer design of the iLGADs and increases with photon energy. A
fitting procedure is introduced to extract the multiplication factor as a
function of the absorption depth of X-ray photons inside the sensors. In
particular, the multiplication factors for electron- and hole-triggered
avalanches are estimated, corresponding to photon absorption beyond or before
the gain layer, respectively.
