The characterization of heteroepitaxial structures by high resolution X-ray diffractometer is a non-destructive technique that allows to reveal the structural properties of samples under analysis. In fact, application of this method requires an understanding of how the diffraction profile of the specimen relates to the crystal structure of the sample. From the rocking curve obtained through the HRXRD it is possible to determine the lattice constants (both parallel and perpendicular to the interface between layer and substrate) of the cubic cell of the heteroepitaxial layer, the density of dislocations present either in the layer or within the substrate and the thickness of the layer deposited. All these parameters are very important for the realization of virtual substrates and for their applications. Indeed, for multi-junction structures or for the integration of VSs within electronic devices, it is mandatory that the layer of Ge at the end of the deposition process features high crystalline quality and contains a low density of defects and that the obtained sample is not deformed neither structurally nor morphologically.
4.3.1. Strain
By considering Eqs. 1.45. and 1.47., in order to calculate out-of-plane and in-plane strains in a heteroepitaxial structure it is necessary to determine the lattice constant of the layer perpendicular and parallel to the interface between substrate and film. With this regard, it is usually appropriate to assume that the strain is constant as a function of depth. In this case, for a binary heteroepitaxial layer such as Ge (over Si substrate), the relaxed lattice constant is known so that there is only one independent unknown. Moreover, once the in-plane or out-of plane lattice constant is known, the other may be calculated. Another assumption is that the substrate, having Bragg angle , can be considered unstrained, i.e., it is thick with respect to the layer.
If crystallographic planes of the film are parallel to the surface, symmetric rocking curves are normally used to obtain a diffraction profile of the sample under investigation. In particular, for a Ge film grown over a (001) Si substrate, the (004) are thus the planes used for symmetric diffraction.
The out-of-plane lattice constant of the epitaxial layer can be determined from the Bragg angle for the layer in the rocking curve. For a diamond heteroepitaxial layer, e.g., Ge film, using the (00m) reflection it results
where
is the d_spacing of (00m) planes for the layer. From Bragg’s law one obtains
4.8.
where is the difference in 00m Bragg angles between the epitaxial film and the substrate, i.e., . It is worth noting that in real
heteroepitaxial structures a crystallographic tilt between the normals to the surfaces of the epitaxial layer and the substrate may occur. Hence, in order to find the strain in the epitaxial film, it is necessary to measure rocking curves at two or more azimuths φ. As an example, if RCs are measured at azimuths φ =0 and φ =90 , then the angular separation between the epitaxial layer and substrate diffraction peaks turns out to be
4.9.
Finally, the out-of-plane strain can be simply obtained from Eq. 1.47, which also gives the in- plane strain under assumptions of biaxial stress and tetragonal distorsion.
An alternative way to measure the strain, both perpendicular and parallel to the interface, is to carry out an asymmetric RC, i.e., by recording diffracted intensity from planes which are not parallel to the surface. For the case of Ge (001) layer, one can use (224) crystallographic planes as sketched in Fig. 4.3. As can be seen, the distance between planes is directly related to the in- plane lattice constant, thus allowing for an easy determination of all the strain parameters.
Figure 4.3: X-ray diffraction from planes which are not parallel to the surface, thus resulting in asymmetric RC.
4.3.2. Dislocation density
X-ray diffraction has been employed to determine the average dislocation density in the volume of heteroepitaxial binary SiGe samples. Indeed, as reported in [4.7], in single-crystal semiconductor specimens misfit and threading dislocations broaden the RC in two ways: (i) the dislocation introduces a rotation of the crystal lattice, thus directly broadening the FWHM of the RC (angular broadening); (ii) the dislocation is bounded by a strain field, in which the Bragg angle of the crystal is nonuniform (strain broadening).
The experimental X-ray rocking curve is assumed to be Gaussian in shape, with FWHM , and to represent the convolution of a number of Gaussian intensity distributions. Hence, results from the convolution of Gaussian intensity functions lead to
4.10.
where represents the Darwin width for the heteroepitaxial sample under analysis, is the instrumental broadening, is the broadening due to angular rotation at dislocations, being the width due to strain which surrounds dislocations, the broadening due to crystal thickness and the spread due to curvature of the specimen. According to Ref. [4.7], if the effects of the crystal size broadening and curvature can be considered negligible, then the broadening contribution due to dislocations can be found from the following formula
where , b being the length of the Burger vector [4.4] and D the dislocation density. The strain broadening due to dislocations has been modeled by Warren, Hordon and Averbach as , where represents the mean square strain in the direction of the normal to the diffracting planes.
Hence, in order to calculate dislocation density of SiGe heteroepitaxial samples, the FWHM of the RC has been measured for a number of hkl reflections and the extracted value
has been plotted as a function of . Then, the values of and represent the
intercept and slope of the obtained function, respectively. Finally, the dislocation density can be simply found by using the following equation
4.12.
It should be highlighted that the measurement of as few as three rocking curves allows accurate determination of dislocation density. For most (001) semiconductor crystals, e.g., SiGe heteroepitaxial samples, since the (004) RC width is mostly related to the angular broadening of dislocations while the (113) width is primarily determined by the strain broadening, then it is sufficient to measure the (004), (113) and (115) RCs for the application of this approach. Fig. 4.4 shows an example of the application of this method.
Figure 4.4: vs. for a 1.5-µm thick layer of Ge over Si (100), grown by LEPECVD technique. is the square of the dislocation broadening, extracted from measured rocking curve widths for various hkl reflections. is the Bragg angle. The filled circles represent the data extracted from measurements, and the line is the least squares fit.