Effect of bonding and static atomic displacements on composition quantification in ${\text{In}}_x{\text{Ga}}_{1-x}{\text{N}}_y{\text{As}}_{1-y}$

This study addresses the influence of static atomic displacements (SAD) and chemical bonding in (InGa)(NAs) alloys on structure factors for electron scattering and stresses their importance for compositional analysis. First, SAD are derived using valence force field (VFF) methods and their reliability is demonstrated by calculating residual atomic forces using density-functional theory (DFT). A systematic study of structure factors for low indium and nitrogen contents is given by means of full DFT calculations on the one hand and atomistic models on the other. We show that the consideration of SAD via VFF together with the inclusion of bonding via modified atomic scattering amplitudes is in best agreement with a full DFT calculation, providing the possibility to include bonding effects also if large cells containing ${10}^6$ atoms are considered. Second, a technique is presented, which allows the extraction of atomically resolved indium and nitrogen composition maps by evaluating strain and contrast in transmission electron microscopic two-beam images simultaneously. As the fringe contrast is compared with Bloch wave simulations, which in turn require accurate structure factors, we exemplify the influence of SAD and bonding on composition profiles regarding an ${\text{In}}_{0.08}{\text{Ga}}_{0.92}{\text{N}}_{0.03}{\text{As}}_{0.97}$ quantum well structure. In this respect, imaging conditions may be chosen for which SAD are of minor influence for conventional transmission electron microscopy, whereas relative errors larger than 25% can arise for the compositions if bonding is neglected.

Figure 1

(Color online) Residual forces resulting from DFT for a supercell containing 12 In and 4 N atoms before (blue) and after (black) VFF optimization. The trihedra top and bottom left refer to the scaling of the respective force vectors.

Figure 2

(Color online) Static atomic displacements (blue, enlarged by a factor of 5) obtained by VFF and difference vectors (black, enlarged by a factor of 50) which point from the VFF atomic positions to those found by DFT. Because of the computational demanding DFT relaxation, we used a smaller $2\times 2\times 2$ supercell in this case containing 3 In atoms and 1 N atom. The nomenclature of the atoms is the same as in Fig. 1. See the blue or black trihedron top and bottom left for an absolute scaling of the SAD and difference vectors, respectively.

Figure 3

$V_{200}$ (real part) in volts as a function of In and N content obtained from first principles for the relaxed $3\times 3\times 3$ (InGa)(NAs) supercells. Concentrations correspond to 0, 1, 2, 3, 4 N and 0, 3, 6, 9, 12 In atoms, respectively.

Figure 4

(Color online) Histogram to show the reliability of different atomistic models (see legend) for the 200 structure factor. Differences between the 25 structure factors in Fig. 3 and those obtained from the respective model have been calculated and the absolute frequency is shown for 29 difference intervals. Data sets (a) and (c) contain bonding effects via MASA, (b) and (d) correspond to isolated atom scattering data due to Weickenmeier and Kohl (WK[14]). Furthermore, (a) and (b) contain SAD, (c) and (d) do not.

Figure 5

(Color online) Diagram to show the largest difference between structure factors obtained from several atomistic models (see legend) and those calculated by DFT [see Fig. 3 for $\text{Re}\left(V_{200}\right)$]. Real and imaginary parts are treated separately, and only moduli of differences are given. Data sets (a) and (c) contain bonding effects via MASA, (b) and (d) correspond to isolated atom scattering data due to Weickenmeier and Kohl (WK[14]). Furthermore, (a) and (b) contain SAD, (c) and (d) not.

Figure 6

Evaluation of contrast and strain in an experimental 200 lattice fringe image. The central part shows the recorded fringe image (part) which has been noise filtered for better visibility. The map on the left depicts the fringe amplitude normalized to GaAs as defined in Eq. (4). On the right, the local strain is shown, which was measured from local cosine fit in the fringe image.

Figure 7

Reference data for $a_{200}\left(x,y,50\text{nm}\right)$ (greyscale) and strain (dashed white contour lines). In the case shown here, atomic scattering factors contained bonding and SAD via MASA and SAD correction factors, respectively. Simulations have been carried out for a thickness range of 0–100 nm, of which the 50 nm simulation is shown. Both data sets correspond to a relaxation of $r=0.5$.

Figure 8

Local distributions of indium and nitrogen in a quantum well region derived from reference amplitudes which include MASA and SAD according to Fig. 7.

Figure 9

Indium composition profiles calculated from maps analogous to the right composition map in Fig. 8 by averaging horizontally. The four curves differ in the reference data sets used, which are explained in the legend. The result for the mean indium concentration in a ternary (InGa)As well grown on the same sample during the same session is indicated by the gray arrow.

Figure 10

Nitrogen composition profiles calculated from maps analogous to the left composition map in Fig. 8 by averaging horizontally. The four curves differ in the reference data sets used, which are explained in the legend.

Figure 11

Graphs to estimate systematical errors due to inaccurately known specimen thickness (black) and chemical shifts by concentration gradients (gray). Calculations were performed using the MASA model and SAD. The black curve depicts the standard deviation ${\sigma }_t$ of $a_{200}$ according to Eq. (4) calculated from a thickness range $t∊\left[10\dots 100\text{nm}\right]$. The phase ${\phi }_{12}$ occurring in Eq. (4) can cause invalid strain measurements if strongly depending on composition. As the gray curve exhibits, this is the case mainly near the phase jump region at $x\approx 0.15$. Only the result for $y=0.03$ is shown because the behavior of ${\sigma }_t$ and ${\phi }_{12}$ is dominated by the dependence on $x$.