Volume 46 Received 13 December 2012  Modified statistical dynamical diffraction theory: analysis of model SiGe heterostructures^{a}College of Nanoscale Science and Engineering, University at AlbanySUNY, 257 Fuller Road, Albany, NY 12203, USA A modified version of the statistical dynamical diffraction theory (mSDDT) permits fullpattern fitting of highresolution Xray diffraction scans from thinfilm systems across the entire range from fully dynamic to fully kinematic scattering. The mSDDT analysis has been applied to a set of model SiGe/Si thinfilm samples in order to define the capabilities of this approach. For defectfree materials that diffract at the dynamic limit, mSDDT analyses return structural information that is consistent with commercial dynamical diffraction simulation software. As defect levels increase and the diffraction characteristics shift towards the kinematic limit, the mSDDT provides new insights into the structural characteristics of these materials. 
The ability to effectively characterize strainrelaxed structures such as silicongermanium (SiGe) material commonly seen in modern thinfilm technology is one of the ongoing challenges in the semiconductor industry. Highresolution Xray diffraction (HRXRD) is commonly used for such analyses, since this method offers a highly sensitive nondestructive approach that can be easily used to collect data from these materials. The analysis of HRXRD data is often based on the TakagiTaupin equations (Takagi, 1962, 1969; Taupin, 1964) or their derivatives. The TakagiTaupin (TT) equations are in turn based on the dynamical theory of Xray diffraction, which assumes that the structures are crystallographically perfect (or nearly so), with very small fluctuations of lattice displacements.
The use of this approach for the analysis of highly defective layers is not recommended, however. For example, fully relaxed silicongermanium epitaxial layers grown with high Ge concentrations on silicon substrates are very poorly described using the dynamical diffraction theory (Shreeman & Matyi, 2010), as is structurally defective ionimplanted SiGe with a range of germanium concentrations (Shreeman & Matyi, 2011). A purely dynamical analysis strategy is not appropriate for the analysis of these cases of fully or partially relaxed samples.
An approach that has consistently shown promise for the analysis of structurally defective materials is the statistical dynamical diffraction theory (SDDT). It was first devised by Kato (1980a,b) and further developed by others, including Bushuev (1989a,b), Punegov (1990a,b, 1991, 1993), Punegov & Kharchenko (1998) and Pavlov & Punegov (2000). The typical approach in the SDDT is to integrate incoherent scattering (due to defects) and coherent scattering (dynamicalbased from crystallographically perfect structures) using two parameters: a static DebyeWaller factor (E) and a complex correlation length (). On the basis of the Bushuev treatment, we have a set of dynamical TT equations given by
where E_{o}^{c} and E_{h}^{c} are the coherent (dynamic) amplitudes, is the simplified deviation parameter, and the coefficients a_{ij} are Bushuev's (1989a,b) definitions of susceptibility given by
Here k is the wavevector, _{h} is the direction cosine of the diffracted beam, C gives the polarization of the incident Xrays, and _{o} and _{h} are the usual electrical susceptibilities. The specific procedures of statistical averaging are explained in detail by Bushuev (1989a,b) and are also discussed elsewhere (Li et al., 1995; An et al., 1995; Punegov, 1990a,b, 1991, 1993; Pavlov et al., 1995; Punegov & Kharchenko, 1998; Pavlov & Punegov, 2000; Authier, 2001).
The two new parameters and E were introduced by Kato (1980a). The longrangeorder parameter (E) is called the static DebyeWaller factor; E ranges from a value of unity for a fully strained (dynamical) structure to zero for the fully relaxed (kinematically limited) case. The second factor is the shortrange complex correlation length , which characterizes the scale over which definite spatial relationships between lattice sites can be considered applicable in a defective crystal (Authier, 2001). The magnitude of this quantity can be complex, since it is affected by phase relationships in the lattice as well as distance and thus presents a number of mathematical difficulties. Fortunately, a mosaic block model based on the method of Bushuev (1989b) has been found to provide a useful simplified approach for incorporating this parameter. As discussed by Shreeman & Matyi (2010) we can use the Bushuev mosaic block model and consider only the real part of by using
where = and = ( with ). In these expressions, _{o} is the width of the reflection of the individual blocks, _{M} is the width of the angular distribution of the mosaic blocks (assumed to be Gaussian), and s represents the convolution of the individual mosaic block diffraction and the angular distribution of the blocks.
The modified TT equations are part of the coherent scattering calculation within the SDDT framework. Typically, the SDDT model assumes that the substrate diffraction profile remains perfectly dynamical. We have found, however, that broadening effects due to the presence of defects in the substrate will have an impact on the diffracted amplitude and need to be incorporated into the SDDT model. This essential modification of the SDDT is detailed in recent work (Shreeman & Matyi, 2010, 2011; Shreeman, 2012). In this approach the basic definition E_{h}^{c} = R_{m  1}E_{o}^{c} is used, where the reflection coefficient from a given layer is defined in terms of the amplitude emerging from the material beneath. We can revise this term by instead incorporating a broadening effect (B_{e}) by defining E_{h}^{c} = (B_{e}R_{m  1} )E_{o}^{c}, in which
Here A is a normalization factor with respect to the substrate peak. As discussed in detail by Shreeman (2012), this modification explicitly addresses the impact of structurally defective layers on the scattered coherent amplitude and the resultant observed intensity distribution from an otherwise dynamically diffracting substrate in a layeronsubstrate materials system.
The preceding discussion has considered only the coherent (dynamic) component of the diffracted intensity. As discussed elsewhere (Shreeman & Matyi, 2010), the incoherent or kinematic component is described by the energy transfer equations
where the superscripts i and c refer to the incoherent and coherent contributions in the incident (o) and diffracted (h) beam directions, is the photoelectric absorption coefficient (divided by the direction cosine _{i}), and is the diffuse scattering kinematic cross section (Kato, 1976a,b; Bushuev, 1989a,b). The total intensity is given by the sum , where .
We refer to the incorporation of equation (7) into (1) and (2) as the modified statistical dynamical diffraction theory (mSDDT). It offers a method of incorporating broadening of the substrate peak using the same parameters (E and ) employed for layer peak broadening. In the present study, we have examined a set of well characterized model samples; the analysis of these materials should illustrate the utility of mSDDT for providing characterizations of partially and fully strain relaxed thinlayer materials.
Samples of SiGe were grown at approximately 823 K by molecular beam epitaxy, with Table 1 illustrating the experimental design. The first set (denoted SiGe2, 3 and 4) had a nominal consistent thickness (40 nm) with varying targeted Ge compositions ranging from 25 to 75%. In contrast, the second set (SiGea, b and c) had nominally constant composition (approximately 50% Ge) with thickness ranging from 20 to 70 nm. With these parameters, a full range of pseudomorphic strain behavior  from fully strained to fully relaxed  was sought.

Highresolution Xray analyses were performed using a Bruker D8 diffractometer equipped with an Eulerian quarter circle, a graded parabolic mirror and a twocrystal fourreflection [symmetric Ge(220)] monochromator. Supporting characterizations were performed using secondaryion mass spectrometry (SIMS) and transmission electron microscopy (TEM). TEM specimens were prepared in an FEI Nova Nanolab 600 focused ion beam/scanning electron microscope equipped with a Pt gas injection system for the deposition of surface protection layers and an OmniProbe nanomanipulator used for in situ liftout. Diffraction contrast and lattice images were recorded in a Jeol 2010F field emission transmission electron microscope operated at 200 kV. SIMS analyses were performed with an IonTof V300 spectrometer with Cs^{+} bombardment. Analyses were performed with pointtopoint normalizing to Si^{2} ions, a procedure that has been shown to minimize socalled matrix effects. Standard SiGe layers that were close in composition to the current materials and which had been previous calibrated by Auger electron spectroscopy and Rutherford back scattering spectrometry were used to determine layer composition.
For clarity, we will consider the experimental results based on the nominal structural characteristics of the sample sets, namely constantthickness versus constantcomposition samples.
Fig. 1 illustrates symmetric 004 scans obtained from the samples with a nominally fixed thickness but a variable Ge concentration. Sample SiGe2 appears to be fully strained and exhibits the typical signs of a dynamically diffracting thinfilm epitaxial material system: a sharp substrate reflection, a relatively narrow layer peak and well defined interference fringes. In contrast, the diffraction scan from sample SiGe3 showed a strong nondynamical layer diffraction peak while exhibiting no subsidiary dynamical fringes. The layer peak seen in the diffraction scan from sample SiGe4 was consistent with a highly relaxed layer with an even broader and lowerintensity layer peak. SIMS analysis from samples SiGe2 and SiGe3 showed that the nominal layer thickness of 40 nm and the nominal concentrations of 25% Ge and 50% Ge were reasonable estimates for these samples.
 Figure 1 Experimental 004 diffraction scans from the constantthickness sample set. 
All of the Xray data from the constantthickness sample set were then fitted by the mSDDT method. In our current implementation, the fitting process was performed by calculation of a trial curve followed by a visual comparison against an experimental diffraction profile and subsequent manual adjustment of the input structural model. For mathematical simplicity, the background was incorporated from a scan of a perfect singlecrystal silicon standard sample. We recognize that these approaches are inferior to a more sophisticated automated process for optimization of a structural model via minimization of the error between the experimental and calculated profiles (using, for instance, a LevenbergMarquardt fit optimization algorithm). However, for our current purposes of exploring the utility of the mSDDT approach and benchmarking its performance with technologically relevant materials, a manual fitting procedure was found to be acceptable.
Fig. 2(a) illustrates the 004 scan from sample SiGe2 along with the fit obtained with the mSDDT approach. In calculating the fit the composition was fixed at 25% Ge. Pseudomorphic strain () was incorporated into the fit by calculating a relaxed lattice parameter at the assumed composition and then calculating the strain based on the final fitted lattice parameter. Additionally, the fit was achieved using a longrangeorder parameter E equal to unity and a value of _{M} equal to zero, corresponding to a perfectly dynamically diffracting sample.
 Figure 2 Fit results for SiGe2: (a) mSDDT fit, (b) LEPTOS fit. 
While the fit generated for sample SiGe2 by the mSDDT approach appears to be reasonable, it was desirable to verify this result through comparison with a commercial dynamical fitting software package. Hence Fig. 2(b) illustrates the fit that was attained using a well known commercially available dynamical diffraction fitting package (LEPTOS, version 7.03; Bruker AXS GmbH, Karlsruhe, Germany). Although the LEPTOS package uses a recursive matrix formalism rather than the TT equations, its proven capabilities for fully describing dynamically diffracting thinfilm structures make it a useful tool for assessing the basic characteristics of the mSDDT approach developed here.
A qualitative comparison shows that the two fits (mSDDT and LEPTOS) give very similar results. Quantitatively, the thickness and composition values generated by the LEPTOS fit (40.4 nm and 28.3% Ge, respectively) are similar to the values (40 nm and 25% Ge) used for the mSDDT. The difference in the two composition values comes from the fact that the LEPTOS value was generated by an automatic fitting procedure based on a genetic algorithm, while the mSDDT analysis employed the nominal layer composition as input to a direct calculation with no fit minimization. The c/c strain from LEPTOS ( = 23 × 10^{3}) was similar to the value ( = 24 × 10^{3}) returned by the mSDDT analysis.
Fig. 3 shows the experimental 004 Xray data and the subsequent mSDDT fits for samples SiGe3 and SiGe4. As mentioned above, these samples are no longer purely dynamically diffracting systems, so they are not accessible by conventional HRXRD analytical approaches. However, the mSDDT is not subject to this limitation. For SiGe3, the composition was set at the nominal value of 50% Ge for the mSDDT fitting, and the nominal thickness of 40 nm was used. With these values in place the optimum fit returned with values of E = 0.25 and _{M} = 5.5''. As expected, these values indicate a diffraction characteristic that is intermediate between the purely dynamic and purely kinematic limits. Somewhat surprisingly, the strain that was returned by the mSDDT fit was = 25 × 10^{3}, which is almost identical to the value seen in the dynamically diffracting sample SiGe2.
 Figure 3 mSDDT fits from (a) SiGe3 and (b) SiGe4. 
The experimental 004 Xray scan and the resultant mSDDT fit from SiGe4 where the composition and layer thickness were set at 75% Ge and 40 nm are also shown in Fig. 3. The fit of the experimental data is apparently quite reasonable, especially considering that the structure is defined as fully relaxed, and that it is fitted with only a single layer. The mSDDT analysis showed that this sample was essentially at the kinematical limit with the value of the static DebyeWaller factor E = 0.1. Further evidence of relaxation in this sample with high germanium content is found in the reduced value of strain ( = 11 × 10^{3}) and the increased value of _{M} = 9''.
The samples constituting the second set (SiGea through SiGec) are characterized by a nominally constant germanium composition of 50% Ge but thicknesses ranging from 20 to 70 nm. SIMS results from these three samples confirmed that the nominal compositions and thicknesses are reasonable assessments of the structural characteristics of these samples.
Fig. 4 shows the symmetric 004 diffraction scans that were generated by this second sample set. The experimental HRXRD scan from the thinnest (and presumably the most highly perfect) sample in this set shows that sample SiGea appears to diffract dynamically. The mSDDT fitting shown in Fig. 5(a) confirmed this by returning values of unity and zero for E and _{M}, respectively. Again, the apparent dynamically diffracting nature of this sample invites comparison and validation, so the corresponding fit to the data achieved with the commercially available LEPTOS software is shown in Fig. 5(b). Both fits are consistent with a layer thickness of 18.5 nm and a germanium content of 50% Ge and show similar values of strain (32 × 10^{3} and 38 × 10^{3} for mSDDT and LEPTOS, respectively).
 Figure 4 Experimental 004 diffraction scans from the constantcomposition sample set. 
 Figure 5 Fit results for SiGea: (a) mSDDT fit, (b) LEPTOS fit. 
Sample SiGec was observed to be similar to SiGe3 in that it shows a single broad peak with no evidence whatsoever of dynamical fringes. Sample SiGeb showed an intermediate behavior of exhibiting some dynamical fringes along with a broadened diffraction profile from the layer. Fig. 6 presents the mSDDT fits obtained for the thicker samples (nominally 50 and 70 nm) in the constantcomposition set. Sample SiGeb was notable because it shows what is apparently a superposition of dynamic and kinematic diffraction effects, with dynamic fringes visible on a broadened (presumably partially kinematic) layer peak. The mSDDT fit was performed using a single layer in the structural model, and examination of Fig. 6(a) reveals that this fit is relatively poor. From this mSDDT analysis, the strain was found to be relatively high (32 × 10^{3}) while the longrangeorder parameter (E) had a neardynamic value of 0.9 and the bestfit value of _{M} was 5.7''. The experimental scan from the thickest sample of this set (Fig. 6b) displayed little evidence of dynamical behavior, and not surprisingly, the static DebyeWaller factor was very small (E = 0.01). However, the fit also returned a strain that was surprisingly large (22 × 10^{3}) for a thick layer that one might presume would be significantly relaxed; the bestfit value of _{M} was found to be 5.5''.
 Figure 6 mSDDT fits from (a) SiGeb and (b) SiGec. 
The above results suggest that the mSDDT may be a useful tool for characterizing defective layered structures that are inaccessible to conventional methods based solely on the physics of perfect crystal dynamical diffraction. As mentioned earlier, the experimental plan was to use a set of model samples with defined characteristics in order to benchmark the performance of the mSDDT approach. Such an approach is commonly used in the development of metrology tools, where the analysis of well defined materials is used to validate a new measurement technology. Additionally, it is desirable to use complementary analytical approaches to further confirm the results gained via an unproven approach; we have chosen TEM for this purpose.
Fig. 7 shows a series of crosssection TEM images that were recorded from the constantcomposition sample set (SiGea,b and c). The samples were prepared for imaging using a focused ion beam process; consequently the micrographs show the platinum layer that was used as a protection layer during the sample preparation, in addition to the SiGe layer and the silicon substrate.
 Figure 7 Crosssection TEM images from the constantcomposition samples. (a) SiGea; (b) SiGeb; (c) SiGec. 
The micrograph of the thinnest layer (SiGea) is unremarkable in that the SiGe appears defectfree. The thickness was determined to be 18.0 (5) nm, a value that agrees well with the dynamical fits (both mSDDT and LEPTOS) of the Xray data. More intriguing is the TEM micrograph of sample SiGeb, where the measured thickness was 47.3 (15) nm, in close correspondence with the mSDDT analysis. This sample appears defectfree and does not show any threading dislocations, although this observation is based solely on crosssectional (rather than planview) imaging and thus may not be sensitive to the presence of nonthreading misfit dislocations. The micrograph of the thickest sample (SiGec) produced a layer thickness measurement of 66.2 (7) nm, a value that again correlates reasonably well with the presumed 70 nm layer thickness used for the mSDDT model structure.
In addition, the micrograph from SiGec shows visible threading dislocations; estimates of dislocation density range from 7.3 × 10^{9} cm^{2} (top surface area calculation) to 1.06 × 10^{12} cm^{2} (length/volume calculation). Ultimately, of course, it would be desirable to relate the dislocation density directly to either the static DebyeWaller factor (E) or the mosaic spread (_{M}). In the case of fully kinematic scattering, expressions for the correlation function for an array of misfit dislocations are available (Pietsch et al., 2004), although it is not clear how effectively a kinematic treatment could be integrated into a semi to fully dynamical treatment such as the mSDDT.
The micrographs of the thickest samples (SiGeb and SiGec) showed that, in addition to the SiGe layer itself, there is evidence for strain in the vicinity of the Si/SiGe interface as indicated by a nonuniform intensity distribution. This observation suggests that the use of a singlelayer structural model may not be the optimum choice for the mSDDT fitting process. The use of a lamellar model with multiple layers is a common practice in the fitting of HRXRD data via dynamical simulation, particularly when a single layer cannot generate an acceptable fit to the data. Under these conditions it is typically assumed that the true sample structure is more complex than a simple single layer, so the use of a more complicated multilayer model structure is justified.
Because of the potential impact of an interfacial strain region shown in the TEM data, we returned to sample SiGeb to examine the impact of using a more complex structural model. Fig. 8 shows two results. In the first (Fig. 8a), a thin, highly strained (33 × 10^{3}) and defective (E = 0.6) Si layer was presumed to be the sole thin film on the silicon substrate. The mSDDT fit to the data is obviously still poor, but this trial does indeed suggest that the kinematic contributions to the layer peak in the experimental data can be rationalized by a defective interfacial layer. Next, a thick (48 nm) relatively perfect SiGe layer (E = 0.95) was combined with the interfacial layer and the resultant calculated mSDDT curve is shown in Fig. 8(b). Inspection shows that the fit has improved considerably, particularly in the vicinity of the layer peak.
 Figure 8 mSDDT fits of SiGeb: (a) interfacial layer only, (b) twolayer fit. 
These results suggest that the mSDDT may be useful for the modelbased fitting of defective thinfilm heterostructures. There are, however, issues that need to be resolved before the statistical theory or its variants could be used as a routine tool for applications such as semiconductor metrology. For instance, mention was made earlier of the fact that, in this study, the relationships between parameters such as the static DebyeWaller factor and the mosaic spread (E and _{M}) are currently not well linked in detail to the physical nature of defects in real materials. If such a correspondence could be achieved, the scope of materials problems that could be addressed using techniques such as the mSDDT would probably be enlarged considerably.
A separate issue involves the sensitivity of the fitting process to parameters such as E and _{M}: currently we have relatively little knowledge of the ranges that these parameters can attain while still generating an acceptable fit. Closely tied to this is the question of the uniqueness of a fit generated by the statistical diffraction theory, as well as the likelihood that the mSDDT fit variables may be correlated with more common parameters such as layer thickness and composition. Although it is likely to be a difficult task, the successful resolution of these issues should help broaden the applicability of methods based on the statistical dynamical theory to a wider range of materials characterization problems.
Finally, it is worth noting that the thicknesses of the samples investigated in this study are far less than the extinction distance (typically several micrometres) that defines the dynamical coupling between the incident and diffracted wavefields. However, it is well known (Bartels et al., 1986) that the transition from dynamical to kinematical diffraction is indicated by a continuous decrease in the influence dynamical interactions, such as a reduction in multiple reflections. Hence an analysis of the diffraction characteristics that accompany a transition from high to low structural perfection in epitaxial materials, versus the changes in dynamic coupling that come with changes in layer thickness, may represent a fruitful area of investigation.
The authors are grateful to SEMATECH for supplying the SiGe samples used in this work. We also thank the manuscript reviewers for many useful comments and suggestions.
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