A semiconductor assembly includes a semiconductor element having contacts on a first surface electrically connected with contacts of a carrier element by electrically conductive material. A second surface opposite the first surface has a convex curvature with a first radius or a concave curvature with a second radius. The second surface of the convex curvature or the second surface of the concave curvature is connected in a positive-fit manner to a cooling body surface of a concave cooling body curvature of the cooling body, and, during operation at a selected barrier layer temperature, the first radius of the convex curvature deviates by at most 10% from a third radius of the concave cooling body curvature, or the second radius of the concave curvature deviates by at most 10% from a fourth radius of the convex cooling body curvature.