Recent advances in nanofabrication reveal significant impact of nanostructuring on thermal transport in solids. This paves a way for novel thermoelectric devices with low thermal conductance. Nanostructured materials can be fabricated to have dimensions less than the mean free path of heat carrying phonons. Recently developed fabrication technology of air-gap heterostructures (AGH) enables to create GaAs nanopillars with controlled length of a few nanometers which can act as ballistic point contacts for phonons. The AGHs were fabricated using molecular beam epitaxy combined with in situ local droplet etching technique and ex situ selective chemical wet etching of a sacrificial layer. Homogeneity and strain distribution in AGHs is of high importance for the functionality of thermoelectric device. Despite a strain-free nature of nanopillars grown in nanoholes, their shape and strain distribution remain to be studied. Using coherently focused x-rays available at modern synchrotron sources it is possible to apply the Coherent Diffraction Imaging (CDI) for ab initio reconstruction of electron density and strain inside an individual nanostructure. At the same time, conventional reciprocal space mapping (RSM) can be used to evaluate the defect densities in thin surface layers. In this contribution we present the results of non-destructive investigation of AGHs by means of RSM and CDI experiments performed at the P10 Coherence Beamline of PETRA III synchrotron at DESY, Hamburg
