  We show evidence that tunnel junctions (TJs) in GaN grown by metal-organic
vapor phase epitaxy are dominated by trap-assisted (Poole-Frenkel) tunneling.
This stems from observations of the careful optimized doping for the TJs.
Especially the p$^{++}$ and the n$^{++}$ layers are far from ideal. The
n$^{++}$ layer induces 3D growth, which can be seen by a rising oxygen signal
in Secondary Ions Mass Spectroscopy (SIMS). Furthermore, Mg segregation
observed by SIMS indicates a depletion region of more than 10 nm. Still, we
could realize TJ based LEDs with a low penalty voltage of 1.1 V and a specific
differential resistance of about 10$^{-2}$ $\Omega$.cm$^2$ at 20 mA without
using an InGaN interlayer.
