Abstracts: The In composition was investigated in InGaN/GaN MOCVD quantum wells by measurement of the local lattice distortion in high-resolution electron microscopy images. Experimental results were compared with the strain field measured on the simulated images of indium rich clusters. The atom positions in the supercell used for image simulations were generated by 3D finite elements modelling. The proposed model of Indium distribution agrees well with experimental data.
The last ten years have brought about a large development of GaN-based light-emitting diodes (LEDs) and laser diodes (LDs). The active areas of these devices are made of GaN/InGaN quantum wells (QWs) and many reports suggest that their high-efficiency luminescence may be due to indium clustering. One of the most powerful techniques for chemical composition determination is the measurement of the tetragonal distortion from cross-sectional high-resolution electron microscopy (HREM) images. The presence of In reach clusters leads to local lattice distortion, non-homogeneous relaxation at the thin film surface, and strong atomic column curvature. Recently, Ruterana et al. used 2D finite elements (FE) modelling to analyse the composition in InGaN QWs. In the following, we combine 3D FE modelling and image simulation to determine the In composition in InxGa1-xN (QW) grown by MOCVD.
2. Experimental results
The nominal thickness of InxGa1-xN multi QWs was 2.5 nm for a nominal composition of 15-17 at. % In. For this analysis, the HREM observations were carried out along the  GaN zone axis and the 0002, 0004, , beams were included in the aperture. The lattice distortion was measured by the peak–finding procedure on Wiener filtered images. Fig 1a shows the whole zone used for processing taken at first inverse contrast defocus. No visible differences in average contrast are visible between the QW and the GaN matrix. The foil thickness increases slowly from the border and is estimated to be around 10 nm in the centre. Fig 1b shows the area of interest and figure 1c exhibits the contour plot of the measured zz distortion component.
3. Finite element modelling and image simulations
The initial model of the spherical shape in rich cluster was constructed on the basis of the experimentally measured zz and previously determined scaling curves for composition calculation (Ruterana et al 2002). The size and composition of the clusters was adjusted in a recursive way by looking at the best match. Only the final models are discussed here (fig 2b).
To determine the stress state and the displacement field in GaN/InGaN structure, we used a FE mesh with 8-node elements (fig 2b). The dimension of the elements was chosen to ensure compatibility with the GaN lattice. The total dimension of the sample (regardless of the indium concentration model) was about 8nm x 14nm x 14nm (XYZ). The X direction was taken parallel to electron beam and Z direction was taken as the layer growth direction. In our modelling, we assumed an indium gradient in cluster as well as in the QW matrix. This gradient was approximated by using 8 InGaN areas with different In concentrations as shown on fig 2a, 2b. Anisotropic elastic coefficients (Robert et al) and lattice parameters of the individual layers were calculated using the Vegard Law. Due to symmetry of the cluster and its assumed position in the middle of well, the calculation was made for 1/8th of sample. Total number of elements, which depends on the used model for the In composition, was 19x33x36. This number of elements gives good accuracy and reasonable time of computation. Taking into account the symmetry of the sample, we blocked displacements normal to the OXY, OXZ, OYZ planes, respectively, and by introducing Multi Point Constraints (MPC) on XZ back plane, we simulate periodic boundary condition in the Y direction (what means that all nodes out of plane have the same displacement in Y direction). This 3D Finite Elements calculation uses a Taylor’s FEAP program (Zienkiewicz and Taylor 1989), which was modified to take into account the finite deformation (Crisfield 1997).
The FE model was transformed to atom positions in a supercell by applying the displacement of mesh to nearest atoms of a perfect GaN crystal. 3D Interpolation between nodes was performed to determine the ux, uy, uz components on the atoms in a GaN lattice. Figure 3c shows the ux superimposed on deformed mesh (deformation enlarged 10x). We have translated indium composition to the probability of appearance of In or Ga atoms using a random number generator instead of occupation parameter. The indium atoms in the model are shown in fig 3d. This approach gives quite noisy images corresponding better to reality and helps to study the influence of the statistical distribution of the random shift of the maximum contrast position in HREM images. At initial stage of refinement of the model we use the “projection of supercell “. This means that each atom was represented as small Gaussian function and finally the “atomic resolution like” HR image was obtained by superposition of such shapes. The image was next processed by the peak finding procedure and the “projected” strain field was extracted, figures 3a shows the measured distortion.
Fig.2 Model used for indium distribution. (a) 3D material distribution and FE mesh for In rich cluster 1/8 of supercell with applied boundaryr conditions (b) model of indium distribution used in FE calculations (1-GaN, and InxGa1-xN with respectively x= 0.02,0.06,0.10, 0.105,0.11, 0.4,0.6,0.8). (c) the mesh deformation enlarged 10x with grey levels corresponding to ux, scale in meters, d) The indium atoms in a supercell containing 135000 of atoms used for image simulation.
For image simulation, the model supercell was sliced and the high-resolution images were calculated using the EMS package (Stadelmann 1987). The imaging parameters for the used Topcon 002B microscope operating at 200 kV are: spherical aberration coefficient 0.5 mm, 0.8 mrad beam convergence, 8 nm spread of focus, and 12 nm-1 objective aperture. The absorption constants were for Ga (0.052), In (0.07), N (0.028) and the Debye–Waller factors were 0.004 nm2 for all atoms. The images useful for the peak finding procedure (relatively simple pattern) were obtained in the defocus range: 5-15 nm and for inverse contrast: 55-70 nm. At the theoretical Scherzer defocus of 43 nm, the image pattern along the GaN  zone axis is too complex for the peak finding procedure. An image calculated at the inverse contrast condition is shown on fig 3b. The zz component extracted from this image is shown on fig 3c. A Comparison of fig 3a and fig 3c shows that the zz peaks are very similar. However the shape of the peak is much more complicated due to imaging conditions. The model gives a slightly lower deformation in the cluster center and higher in inter cluster area than what is measured experimentally. This result was obtained after 3 iterations of model adjustment and improvement is still possible. These results show that most of the indium is inside small regions of about 2 nm diameter where the concentration is higher than 60% , they are surrounded by a matrix with In concentration below 10%.
4. Discussion and Conclusion
This example shows that the simple averaging of the 3D distortion calculated be FEM gives a good approximation of the distortion, which is obtained from the simulated images. This kind of relatively simple approach can give information about indium distribution for very thin areas compared to the cluster size with higher accuracy than spectroscopic methods or Z contrast which need thicker samples. At this point, it is not possible to conclude on the unicity of the determined Indium 3D distribution. The sample thickness has a large influence on the averaging of distortion. The precision will be better when accurate estimation of the local thickness is possible. However, for smallest In fluctuations, it will not be possible to distinguish between the real and statistical fluctuations, this sets a limit of cluster diameters of abound 1 nm for detectability.
This work was partially supported by the EU under contract no. HPRN-CT-1999-00040 and the CELIS excellence center ICA1-CT-2000-70018 as well as by the Committee for Scientific Research in Poland under grant 4 T11F 008-25.
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