One possible approach for composition determination in strained layer heterostructures is strain state analysis. This method relies on measurement of local lattice parameters and displacements from high-resolution transmission electron microscopy (HRTEM) images. The figure shows, as an example, an InAs quantum dot, which was grown on a GaAs substrate, together with the displacement field evaluated with the DALI program.

The DALI program contains the following analysis steps:

- Noise reduction, if necessary
- Detection of lattice sites
- Gridding
- Calculation of a reference lattice
- Computation of dispalcements and lattice spacings
- Evaluation of the composition (using comparison with finite-element calculations, if necessary)

The lattice parameter of InAs is approximately 7% larger than that of GaAs. Strain state analysis of the elemental concentration x in a ternary semiconductor, for example InxGa1-xAs, is based on Vegard's law. This law predicts a linear dependence between the In-concentration and the bulk-material lattice parameter of the ternary semiconductor. Knowing the lattice parameters aGaAs and aInAs of the binary materials, the lattice parameter of the ternary material In xGa1-xAs is given by: aInGaAs = (1-x) a GaAs+x aInAs .

However, the lattice parameter in a strained layer significantly deviates from that in the bulk material. In a pseudomorphically grown layer, the lattice parameter parallel to the interface resembles that of the substrate, whereas the lattice parameter in the growth direction shows tetragonal distortion. This distortion is partially relieved in strained islands and layers. These effects can be taken into account by finite-element simulations.

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### Publications:

[1] A. Rosenauer, S. Kaiser, T. Reisinger, J. Zweck, W. Gebhardt, D. Gerthsen, *Digital analysis of high-resolution transmission electron microscopy lattice images*, Optik 102, 63 (1996)

[2] A. Rosenauer, T. Remmele, D. Gerthsen, K. Tillmann, A. Förster, *Atomic scale strain measurements by the digital analysis of high-resolution transmission electronmicroscopic lattice images*, Optik 105, 99 (1997)

[3] A. Rosenauer, D. Gerthsen, V. Potin, *Strain state analysis of InGaN/GaN – sources of error and optimised imaging conditions*, Phys. Stat. Sol. (a) 203, 176 (2006)