Silicon-Germanium (SiGe) alloys are critical materials in modern semiconductor technology, particularly for high-speed electronics, optoelectronics, and strain-engineered devices. Precise characterization of SiGe composition, strain, and defects is essential for optimizing performance. X-ray diffraction (XRD) and transmission electron microscopy (TEM) are two of the most powerful techniques for this purpose, offering complementary insights into the structural and compositional properties of SiGe alloys.

XRD is a non-destructive technique widely used to determine the composition, strain state, and crystal quality of SiGe alloys. The lattice parameter of SiGe varies linearly with Ge content according to Vegard's law, allowing composition to be extracted from XRD peak positions. For a relaxed Si1-xGex alloy, the lattice parameter a(x) can be expressed as a(x) = aSi + x(aGe - aSi), where aSi is 5.431 Å and aGe is 5.658 Å. In pseudomorphic SiGe layers grown on Si substrates, the in-plane lattice parameter remains constrained to that of Si, inducing biaxial compressive strain. The out-of-plane lattice parameter expands, and the degree of relaxation can be quantified by comparing symmetric and asymmetric XRD reflections.

High-resolution XRD (HRXRD) using a multiple-crystal diffractometer provides exceptional angular resolution, enabling precise measurement of strain and composition. Reciprocal space mapping (RSM) around asymmetric reflections, such as the (004) and (224) peaks, is particularly valuable for distinguishing between strain and composition effects. The position of the SiGe peak relative to the Si substrate peak in the RSM directly reveals the strain state. Fully strained layers exhibit peaks aligned vertically with the substrate, while partially relaxed layers show a lateral shift. The relaxation percentage R can be calculated using the relationship R = (a|| - aSi)/(a0 - aSi), where a|| is the in-plane lattice parameter and a0 is the relaxed lattice parameter of the alloy.

XRD also detects defects such as dislocations and stacking faults through analysis of peak broadening and diffuse scattering. The Williamson-Hall plot, which plots the full width at half maximum (FWHM) of XRD peaks against the diffraction vector, helps distinguish between size broadening (due to finite crystallite dimensions) and strain broadening (due to dislocations). For SiGe alloys with high dislocation densities, the FWHM increases significantly, and the peaks may exhibit asymmetric tails.

TEM provides direct atomic-scale imaging of SiGe alloys, revealing defects, interfaces, and compositional variations that XRD cannot resolve. High-resolution TEM (HRTEM) images the crystal lattice, allowing visualization of individual atomic planes and defects such as threading dislocations, misfit dislocations, and antiphase boundaries. Strain fields around defects appear as distortions in the lattice fringes. Geometric phase analysis (GPA) of HRTEM images quantifies local strain with sub-nanometer resolution, mapping the displacement of lattice planes relative to a reference region.

Scanning TEM (STEM) combined with high-angle annular dark-field (HAADF) imaging offers Z-contrast, where the intensity scales with the atomic number squared. Since Ge has a higher atomic number than Si, HAADF-STEM can resolve compositional fluctuations at the atomic scale. Energy-dispersive X-ray spectroscopy (EDS) in STEM mode provides quantitative elemental mapping, with a spatial resolution of approximately 1 nm. Electron energy-loss spectroscopy (EELS) can also detect compositional variations by analyzing the core-loss edges of Si and Ge, though it requires thinner samples than EDS.

For defect analysis, TEM reveals the type, density, and distribution of dislocations in SiGe layers. Misfit dislocations at the SiGe/Si interface relieve strain when the critical thickness is exceeded. Their spacing can be measured directly from TEM images and correlated with the degree of relaxation. Threading dislocations propagating through the layer degrade device performance, and their density is a key metric for material quality. Weak-beam dark-field (WBDF) TEM enhances dislocation contrast, enabling accurate counting even at densities exceeding 10^8 cm^-2.

Convergent-beam electron diffraction (CBED) in TEM measures local strain with high precision by analyzing shifts in higher-order Laue zone (HOLZ) lines. The strain sensitivity of CBED is on the order of 10^-4, making it suitable for studying strain gradients near interfaces or defects. However, CBED requires careful sample preparation to avoid artifacts from thickness variations.

Both XRD and TEM face challenges in analyzing ultra-thin SiGe layers or nanostructures. For layers thinner than 10 nm, XRD signals become weak, and interference effects complicate data interpretation. TEM provides higher resolution but is limited by sample preparation artifacts and electron beam damage. Combining both techniques mitigates these limitations, with XRD offering statistical averaging over large areas and TEM providing localized details.

In summary, XRD and TEM are indispensable for characterizing SiGe alloys, each offering unique advantages. XRD excels at measuring average composition, strain, and relaxation over macroscopic regions, while TEM provides atomic-scale insights into defects and interfacial structure. Together, they enable comprehensive optimization of SiGe materials for advanced semiconductor applications. The choice of technique depends on the specific information required, with many studies benefiting from a combined approach to fully understand the material properties.