Direct structure determination of systems with two-dimensional periodicity

We describe a new x-ray method for the direct measurement of structures which have two-dimensional (2D) periodicity, and are positionally correlated with an underlying substrate crystal. Examples include reconstructed crystal structures at interfaces, layered heterostructures, crystalline-amorphous interfaces, and self-assembled structures on crystalline substrates. The structure is obtained by determining the complex scattering factors along the Bragg rods and Fourier back-transforming them into real space. The method for determining the complex scattering factors has two variations. The first is generally applicable. It involves the measurement of the derivative of the diffraction phase along the Bragg rods and the subsequent determination of the diffraction phase using the known structure of the substrate. The second is applicable to 2D systems, with an unknown structure, that are buried within a crystal with a known structure. In this case the diffraction phase is determined without the need to measure its derivative first. We experimentally demonstrate both variations by determining the diffraction phase along one Bragg rod of a GaAs sample with four buried AlAs monolayers. Using simulated data along the Bragg rods within a volume in reciprocal space, we show that the method yields the three-dimensional structure of 2D systems with atomic resolution.