  Heterostructures consisting of alternating GaN/AlN epitaxial layers represent
the building-blocks of state-of-the-art devices employed for active cooling and
energy-saving lightning. Insights into the heat conduction of these structures
are essential in the perspective of improving the heat management for
prospective applications. Here, the cross-plane (perpendicular to the sample's
surface) thermal conductivity of GaN/AlN superlattices as a function of the
layers' thickness is established by employing the $3\omega$-method. Moreover,
the role of interdiffusion at the interfaces on the phonon scattering is taken
into account in the modelling and data treatment. It is found, that the
cross-plane thermal conductivity of the epitaxial heterostructures can be
driven to values as low as 5.9 W/(m$\cdot$K) comparable with those reported for
amorphous films, thus opening wide perspectives for optimized heat management
in III-nitride-based epitaxial multilayers.
