A radiation-emitting semiconductor component has a high p-type conductivity. The semiconductor body of the component includes a substrate, preferably an SiC-based substrate, on which a plurality of GaN-based layers have been formed. The active region of these layers is arranged between at least one n-conducting layer and a p-conducting layer. The p-conducting layer is grown in tensile-stressed form. The p-doping that is used is preferably Mg.