The influence of atomic size effect on the quantitative compositional analyses by means of local lattice parameter measurements
Authors: Schulz Tobias , Markurt Toni , Remmele Thilo , Korytov Maxim , Albrecht Martin 
Published: December 2016

A quantitative analysis of the chemical alloy composition down to the near atomic scale is a key issue for understanding physical properties of semiconductor nanostructures. One method that is commonly applied to this problem is ‘strain mapping’ in a high‐resolution transmission electron microscopy (HRTEM) image. The idea is to measure the out‐of‐plane lattice parameter locally in the HRTEM image and to calculate a local chemical composition within a specific probe volume using elastic continuum theory (see e.g. Refs. [i,ii]. Prerequisites for this approach are that (i) the HRTEM image contrast yields local information about the mean atomic position along the projection direction and (ii) the probe volume possesses a tetragonal distortion, i.e. the local in‐plane lattice parameter is fixed to that of the substrate. We show that the latter assumption only holds for comparatively large probe volumes, while it is a poor approximation as the probe volume approaches the near atomic scale. Such small probe volumes, however, are mandatory for studying small composition variations to distinguish e.g. statistical alloy fluctuations from clustering effects. For demonstration, we have performed HRTEM multislice image simulations using relaxed supercells that correspond to realistic sample thicknesses of around 10 nm. The supercells were relaxed using an interatomic potential that has been fitted to DFT calculations[iii]. The supercells contain typical (In,Ga)N quantum structures, such as wells or dots, coherent to GaN. Fig. 1 shows a random Indium distribution within a quantum well, as determined by counting the ratio of Ga and In atoms within a probe volume of (0.519 x 0.317 x 10) nm³ (corresponding to the GaN unit cell times the sample thickness). Fig. 2 displays the c‐lattice parameter map, measured in the corresponding HRTEM image simulation. As can be seen by naked eye, the out‐of‐plane lattice parameter map does only poorly reflect the true In distribution within the supercell. Even a non‐existing ordering phenomena might be mistakenly interpreted from the c‐lattice parameter map. As we will show, this effect stems from local strain fields caused by In fluctuations in the alloy, which do not only extent in out‐of‐plane but also in in‐plane directions, making the Indium quantification error prone. We have developed a novel approach, which measures next to the out‐of‐plane, also the local in‐plane lattice parameter to extract the local chemical composition. This method allows a much more reliable calculation of the chemical composition from these complex strain fields. Fig. 3 shows the resulting ‘strain revised' c*‐lattice parameter of the (In,Ga)N quantum well, which reveals a much better agreement with the actual In distribution. In addition to the theoretical works, we will also demonstrate first experimental images, where the feasibility of our approach is demonstrated. We propose that our method is versatile, working also for other compounds and is not limited to 2D objects but also applies for quantum wires or dots.