Origin of giant enhancement of phase contrast in electron holography of modulation-doped n-type GaN

The electron optical phase contrast probed by electron holography at $n$-$n^{+}$ GaN doping steps is found to exhibit a giant enhancement, in sharp contrast to the always smaller than expected phase contrast reported for $p$-$n$ junctions. We unravel the physical origin of the giant enhancement by combining off-axis electron holography data with self-consistent electrostatic potential calculations. The predominant contribution to the phase contrast is shown to arise from the doping dependent screening length of the surface Fermi-level pinning, which is induced by FIB-implanted carbon point defects below the outer amorphous shell. The contribution of the built-in potential is negligible for modulation doping and only relevant for large built-in potentials at e.g. $p$-$n$ junctions. This work provides a quantitative approach to so-called dead layers at TEM lamellas.