The roughness of the electronic interfaces of p-n GaAs multilayers has been investigated by cross-sectional scanning tunneling microscopy. Two physically different contributions to theroughness were found, both much larger than the underlying atomically sharp 'metallurgical' interface. The roughness arises from the individual electrostatic screening fields around each dopant atom near the interface and from a clustering of dopant atoms. The latter leads to charge carrier depleted zones extending locally through the entire nominally homogeneously doped layer for layer thicknesses close to the cluster dimension, hence limiting the precision of the spatial and energetic positioning of the Fermi energy in nanoscale semiconductor structures.