## research papers

## Characterization of dislocations in germanium layers grown on (011)- and (111)-oriented silicon by coplanar and noncoplanar X-ray diffraction^{1}

**Andrei Benediktovitch,**

^{a}^{*}Alexei Zhylik,^{a}Tatjana Ulyanenkova,^{b}Maksym Myronov^{c}and Alex Ulyanenkov^{b}^{a}Department of Theoretical Physics, Belarusian State University, Nezavisimosti Avenue 4, Minsk, Belarus, ^{b}Rigaku Europe SE, Am Hardtwald 11, Ettlingen, Germany, and ^{c}Department of Physics, The University of Warwick, Coventry, UK^{*}Correspondence e-mail: andrei.benediktovitch@atomicus.by

Strained germanium grown on silicon with nonstandard surface orientations like (011) or (111) is a promising material for various semiconductor applications, for example complementary metal-oxide semiconductor transistors. However, because of the large mismatch between the *Phys. Rev. B*, (1997), **55**, 1793–1810] has been generalized to an arbitrary surface orientation, arbitrary dislocation line direction and noncoplanar measurement scheme.

Keywords: strained germanium; silicon; complementary metal-oxide semiconductors; noncoplanar X-ray diffraction.

### 1. Introduction

With the downscaling of today's technology to the nanometre level, to create silicon-based complementary metal-oxide semiconductor (CMOS) transistors it becomes necessary to look for alternative channel materials, strain configurations and crystallographic orientations to realize the full potential of the semiconductor band structure, required to obtain the highest electron and hole mobility channels (Takagi *et al.*, 2008). Strained germanium appears to be one of the most promising alternative channel materials owing to both its intrinsically higher electron and hole mobility values and its compatibility with existing CMOS fabrication techniques (Dobbie *et al.*, 2012; Myronov *et al.*, 2014). In addition to strain, further improvements of the device performance can be made by using nonstandard surface orientations such as (011) and (111) to fully exploit the properties of the Ge band structure. High mobility electron and hole channel transistors have already been predicted and demonstrated by several groups using Ge substrates with different crystallographic orientations and strain (Ritenour *et al.*, 2007; Zimmerman *et al.*, 2006; Kuzum *et al.*, 2009; Shang *et al.*, 2003; Maikap *et al.*, 2007; Nishimura *et al.*, 2010; Chui *et al.*, 2002; Low *et al.*, 2004). However, because of the large mismatch between the constants of silicon and germanium, the growth of such systems is challenged by nucleation and propagation of threading and misfit dislocations that degrade the electrical properties.

To characterize thin films, a variety of techniques are used, such as *etc*. X-ray characterization techniques are favorable owing to their nondestructive nature and good matching of X-ray wavelength to the atomic scale of modern semiconductor devices. HRXRD is a suitable tool for nondestructive investigation of multilayer structures: the peak position delivers the parameters connected with composition and strain while the peak shape is conditioned by layer thickness and defects present in the sample, the dislocations playing a decisive role in peak profile formation (Benediktovitch, Feranchuk & Ulyanenkov, 2014).

A theoretical approach for calculation of the diffracted *et al.* (1997) and successfully applied to a number of systems (Kaganer *et al.*, 2006*b*; Benediktovich *et al.*, 2011; Kopp, Kaganer, Baidakova *et al.*, 2014). The peak shape is sensitive to the reflection used, the measurement geometry, the type of dislocations, their density and their degree of correlation. It was shown that from a set of peaks measured for a number of reflections one can get reliable information about the dislocation type (Kaganer *et al.*, 2006*b*). The underlaying idea is closely connected to the method of full-profile analysis in powder diffractometry: given a set of profiles for different reflections *hkl*, the fitting of the whole set enables the determination of the defect parameters (Ribarik & Ungar, 2010; Scardi & Leoni, 2002). The reason is that different defect types produce peak broadening that changes in a different way with *hkl* and, consequently, that enables one to disentangle the contribution from a distinct defect type. For example, the dependence of dislocation-induced broadening on *hkl* is governed by the dislocation contrast factor *C*_{hkl}, which is specific for each dislocation type, thus enabling one to find the dislocation type from the measured powder profile (Leoni *et al.*, 2007; Ungár *et al.*, 2001). A similar approach can be used for treatment of diffraction data measured in HRXRD mode from films containing dislocations. However, in this case one has to modify significantly the underlying theory. In particular, the dislocation contrast factor should be replaced with the dislocation contrast tensor (Benediktovitch, Feranchuk & Ulyanenkov, 2014), and special attention should be payed to the way that the diffraction signal is collected (Kaganer *et al.*, 2005). The quantities of interest can be calculated on the basis of the approach presented by Kaganer *et al.* (1997). However, for each sample normal orientation and dislocation line direction the expressions for intensity distribution should be derived again, accounting for specifics of the geometry. In the current paper we propose a general formalism to treat an arbitrary case of surface orientation and dislocation line direction; also, peculiarities of the application to noncoplanar measurements are discussed.

We have measured a series of

maps (RSMs) and profiles for a number of reflections in coplanar as well as noncoplanar measurement geometries to get a consistent data set for analysis of dislocation microstructure. The noncoplanar measurement geometry was achieved by rotations of the detector arm around two orthogonal axes (Rigaku SmartLab diffractometer) without tilting the sample. The developed formalism is applied to the measured data set to obtain information about the dislocation ensemble.### 2. Sample growth and measurement

#### 2.1. Sample growth

The epitaxial Ge layers investigated here were grown on 100 mm (111)- and (110)-oriented Si substrates by reduced pressure chemical vapor deposition in an ASM Epsilon 2000 using a GeH_{4} standard as a gaseous precursor diluted in H_{2} All used wafers were initially baked at 1423 K for 2 min in H_{2} in order to desorb any native oxide on the Si substrates prior to epitaxial deposition. To grow the Ge layer, a fixed GeH_{4} precursor flow rate and chamber pressure of 100 Torr were used (1 Torr = 133.3 Pa), in such a way that the GeH_{4} was held constant at 75 mTorr for both low-temperature (LT) and high-temperature (HT) stages. It was shown earlier by Shah *et al.* (2011) that a similar approach is capable of producing high-quality relaxed Ge buffers on Si(001). The growth temperatures for the LT and HT layers were kept constant at 673 and 943 K, respectively, with no Ge growth occurring during the ramp between these temperatures, and with H_{2} flowing inside the chamber. The temperature ramping rate was fixed at 4.5 counts per second. Post-growth *in situ* annealing was carried out on some wafers at 1103 K for 10 min in H_{2}. Different thicknesses for the LT and HT layers were achieved by varying the deposition times for each layer for growth on (111) and (110). The Ge growth rates at 673 K were determined to be approximately 0.05 nm s^{-1} on (111) and 0.1 nm s^{-1} on (110); at 943 K these were 0.6 and 0.5 nm s^{-1}. These growth rates are lower than the corresponding values on (100) of 0.3 and 1.5 nm s^{-1} (Shah *et al.*, 2011). In the case of the (110) samples the thickness of the as-grown Ge epilayers was ∼420 nm, and in the (111) samples it was ∼500 nm. The thickness in each sample was controlled by cross-sectional (TEM).

#### 2.2. X-ray diffraction

Room-temperature measurements were performed using an in-plane-arm-equipped 9 kW SmartLab Rigaku diffractometer with a rotating anode providing Cu *K*α radiation (see Fig. 1). X-ray diffraction measurements were carried out in a parallel-beam geometry. A patented cross-beam optics unit was used for this purpose, which provides the parallel beam collimated vertically. A high-resolution setup with the combination of a four-crystal Ge monochromator in the 220 setting, a two-crystal Ge analyzer in the same setting and a was used to achieve sufficient resolution for the measurement of a set of samples.

In the case of the used diffractometer, the positions of the source and the detector can be described by the following instrumental angles (see Fig. 1):

(*a*) angle , which is the angle between the line connecting the sample and X-ray source and the plane of the sample holder;

(*b*) angle , which is the angle between the axis of in-plane arm rotation and the plane of the sample holder; in the case of no in-plane arm rotation is the angle between the line connecting the sample and detector and the plane of the sample holder;

(*c*) angle , which is the angle of the in-plane arm rotation.

The angle is specific to the model of in-plane diffractometer, and the measurement mode involving this additional detector movement degree of freedom will be considered below.

Sets of maps (in coplanar geometry, ) and scans (in noncoplanar geometry, ) were measured for the two types of samples. The ω positions; the point detector was used. For the diffractometer used, the sample was not moved, the angle being and the ω angle being . In the case of a (111)-oriented substrate, the sample was aligned in such a way that the 513 reflection was in the diffraction plane and the 513 and 153 RSMs were measured. Then the sample was aligned in such a way that the 242 reflection was in the diffraction plane and the 242, 333 and 404 RSMs were measured. The 333 RSM also was measured after rotation of the sample by 90°. In the case of an Si substrate with (011) orientation the sample was aligned in such a way that the 133 reflection was in the diffraction plane and the 33 and 133 RSMs were measured; then the sample was aligned to the 026, and reflections and the 062, 242 and 153 RSMs were measured correspondingly.

mapping was performed by a series of – scans at variousThe in-plane movement of the diffractometer arm (the angle ) provides an additional degree of freedom in exploring the *L*_{x}*L*_{z} (see Fig. 1*b*) and explore the without sample tilting. For the (111) substrate orientation, the sample was aligned in such a way that the 224 reflection was in the diffraction plane *L*_{x}*L*_{z}. In this case . The – and scans around the 040, 044, 133 and 242 reflections were measured. A sketch of the incoming and the outgoing beam arrangements for the noncoplanar profile measurement of reflection 044 is presented in Fig. 2. For the (011) substrate orientation, the sample was aligned in such a way that ; the scans around the 02, 24, 13, 33 and 026 reflections were measured.

### 3. Diffracted distribution

#### 3.1. General expressions for distribution in reciprocal space

The distribution of the diffracted (diffuse scattering) intensity from a crystal with defects in the *G* (Krivoglaz, 1996; Kaganer *et al.*, 2006*b*; Benediktovitch, Feranchuk & Ulyanenkov, 2014):

where is the displacement at the site **r** due to randomly distributed dislocations, is the scattering vector, is the deviation of the scattering wavevector from reciprocal-lattice point , and the average is performed over the dislocation positions. In the case of an epitaxial film corresponds to the pseudomorphic strained film on the substrate.

We will focus below on almost completely relaxed films. In this case the film's crystalline *d* is the film thickness. Also, we will analyze the vicinity of the Bragg peak where most of the scattered intensity is concentrated; quantitatively, we will consider the case of . Under these assumptions the correlation between atomic positions drops off quickly and the main contribution to the scattered signal comes from closely spaced points centered far from dislocation lines, *i.e.* in the crystal areas that are most weakly distorted. In this case we can assume

which considerably simplifies the calculations.

To calculate the correlation function one has to know the displacement fields from a single defect. Below we will consider two types of defects: misfit dislocations and threading dislocations. In the case of misfit dislocations the displacement field is given in a coordinate system associated with the direction of the dislocation line and sample normal . Let us denote this system Dm and define the direction of its axis as

In this coordinate system the displacement field at the point (*x*, *z*) due to the dislocation line passing through the point is expressed as

where is the displacement field in the infinite medium of a dislocation at the origin, the first two terms on the right-hand side of equation (4) correspond to the dislocation itself and the image with respect to the surface, and the third term is the remaining surface relaxation. The explicit expressions for all these terms and for all Burgers vector orientations are given by, for example, Kaganer *et al.* (1997).

Performing the averaging over dislocation positions following the method outlined by Krivoglaz (1996) and Kaganer *et al.* (1997), and in the frame of approximation (2), the correlation function of the displacement fields of defects results in

where is the tensor of mean strain due to misfit dislocations, which is given by (in the Dm coordinate system)

Here *b* is the Burgers vector of the dislocation and ν is the Poisson ratio. The tensor *E* in equation (5) is a fourth-rank tensor describing strain fluctuation. In analogy to the approach used in powder X-ray diffraction, this tensor is the elastic component of the dislocation contrast factor (Klimanek *et al.*, 1988; Martinez-Garcia *et al.*, 2009). In the case of epitaxial layers, this tensor becomes *z* dependent (which is not the case for powder X-ray diffraction), its components in the Dm coordinate system being equal to

The integral over d*x* can be calculated analytically; the explicit expressions are given in Appendix *A*. At large dislocation densities the elastic interaction between dislocations leads to spatial correlation between dislocation positions (Freund & Suresh, 2004). This positional correlation within the validity of approximation (2) leads to factor *g* in equation (5), which has the meaning of the ratio of the dispersion of distances between dislocation lines divided by the square of the average distance [see detailed discussion by Kaganer & Sabelfeld (2011)].

The second type of defect that will be important for us is threading dislocations running through the layer. Below, the threading dislocations are considered to follow the direction of the Burgers vector to maximize their screw nature (Bolkhovityanov & Sokolov, 2012). The expressions below are derived for screw threading dislocations; however, the tensor-based formalism presented here is general. Let us introduce the coordinate system Dt associated with the direction of the threading dislocation line : the *z* axis of the Dt system is directed along ; the directions of the *x* and *y* axes can be chosen arbitrarily in the plane normal to :

The correlation function becomes

Here ρ_{s} is the threading dislocation density. To calculate the dislocation contrast elastic tensor *E* one has to know the displacement from a threading dislocation in the half-space and to calculate the two-dimensional integral over dislocation positions. The elastic displacement fields from an inclined dislocation in isotropic half-space were found by Yoffe (1961); the explicit expressions with corrected misprints are given by Shaibani & Hazzledine (1981). For this displacement field one can find that the displacement field derivative depends on the coordinates like 1/*z* *f*(*x*/*z*,*y*/*z*). Hence from equation (9) it follows that the tensor *E* does not depend on *z*, *i.e.* it is constant within the layer. Besides the actual form of the displacement field the value of tensor *E* depends on how the two-dimensional integral is calculated. Since the displacement field derivative has an asymptotic behavior like 1/*r*, the integral is logarithmically divergent at both lower and upper limits. The truncation at the lower limit is done at a length scale corresponding to the termination of the validity range of the assumption of equation (2), while the truncation at the upper limit is done at a length scale corresponding to the dislocation correlation length (Kaganer & Sabelfeld, 2010; Benediktovitch, Feranchuk & Ulyanenkov, 2014). Both length scales are quite ill defined, but since the actual value of the tensor *E* depends on them logarithmicaly their influence is weak. Because of this fact we will neglect fine effects due to modification of the displacement fields related to the presence of the boundary and will use the solution for an infinite medium. In papers by Kopp and co-workers (Kopp, Kaganer, Baidakova *et al.*, 2014; Kopp, Kaganer, Jenichen & Brandt, 2014) the effect of boundary terms on the X-ray diffraction profile was accounted for using the direct assumption-free Monte Carlo approach and it was shown to be subtle. For the mentioned reasons we will use the displacement fields for an infinite medium,

which results in the following nonzero components:

Here the logarithmic term that appeared as a result of normalization is estimated according to Kaganer & Sabelfeld (2014). *R* is the correlation length of dislocation positions.

#### 3.2. Measured distribution

The expressions in the previous section provide the intensity distribution in three dimensions in

To experimentally access this distribution one would need a resolution function confined in three dimensions in which is not the case in most of the used measurement modes. The recorded intensity and the intensity distribution in are connected by the resolution function :In commonly used measurement modes the resolution function does not provide the resolution in one or two directions. Accounting for this fact is crucial for processing the diffraction data. Several cases will be considered below.

With the help of the instrumental angles , , it is convenient to get the components of the wavevectors in the coordinate system associated with the sample stage and X-ray source. Let us denote this laboratory coordinate system as L and define the directions of its axes as

where *L*_{x} is parallel to the projection of the incoming vector on the sample surface. As one can see from equations (5) and (6), we will have need to perform convolutions of the vector defined in the L system with the tensor *E* defined in the Dm or Dt system. To do this one has to transform both quantities to the same coordinate system. This can be easily done for the particular cases described by Kaganer *et al.* (1997) and Kopp, Kaganer, Baidakova *et al.* (2014), but for a general case one would need a universal recipe. In order to provide it, let us consider the crystallographic coordinate system C:

Here a cubic system has been considered for simplicity. The vectors and **τ** are needed to define the Dm/Dt system. The direction of the incoming beam is known in the C system and is usually expressed in Miller indexes. The matrix *T*^{CD}_{ij} that transforms the components from the C system to the D system is easily obtained: its rows are the components of vectors , , written in the C system. In a similar way one can obtain the elements of *T*^{CL}_{ij}. The needed matrix *T*^{DL}_{ij} is obtained as

##### 3.2.1. mapping

Let us consider a triple crystal arrangement with a monochromator and an analyzer and the coplanar arrangement with . We assume that the resolution function in the diffraction plane is much narrower than the peak width due to the dislocation broadening, but in the direction normal to the diffraction the beam is not well conditioned and the resolution function is much wider than the peak. In this case equation (12) transforms to integration of in the *q*_{y}^{(L)} direction. Performing this integration in the expression (1) one gets the δ function in *y*^{(L)}, and for the measured intensity distribution one obtains

Here ρ stands for for the misfit dislocations and for threading dislocations. The term *T*_{2} can be written in the form

Here a 2 × 2 reflection-dependent matrix determines the shape of the peak. It has two contributions: the elastic due to the tensor *E* and the geometric due to the tensor *G*, which describes the dependence on the used reflection. Comparing to the similar expressions used in the powder diffraction case (Klimanek *et al.*, 1988; Martinez-Garcia *et al.*, 2009; Ungár *et al.*, 2001; Scardi & Leoni, 2002) one can see that is the analog of the dislocation contrast factor, but it has two significant distinctions: (i) it is not a factor but a 2 × 2 matrix and (ii) in the case of misfit dislocations it is *z* dependent owing to the effect of the boundary.

##### 3.2.2. Noncoplanar out-of-plane scans

The in-plane degree of freedom of a detector described by the angle is favorable for the analysis of thin films (Ofuji *et al.*, 2002; Yoshida *et al.*, 2007), for texture analysis (Nagao & Kagami, 2011), for residual stress gradient investigation (Benediktovitch, Ulyanenkova, Keckes & Ulyanenkov, 2014) and for other applications. In the case of the epitaxial films studied here, the in-plane arm movement enables us to explore without sample rotation. Now we consider a measurement performed with a monochromator and an analyzer at . We will assume that compared to the peak width there is a large divergence of the source in the horizontal direction and a large acceptance of the detector in the direction. In this case the measured signal is given by the integral of intensity in over the plane normal to the vector:

Here the components in the L coordinate system were calculated on the basis of the expressions for the scattering vector given by Benediktovitch, Ulyanenkova & Ulyanenkov (2014) and Benediktovitch, Ulyanenkova, Keckes & Ulyanenkov (2014). φ_{s} is the angle between the vector **k**_{in} and plane *L _{x}L_{z}* (see Fig. 1); it describes the divergence of the source in the horizontal direction. The integration over this plane results in

Here the quantity *C*(*z*) is again an analog of the dislocation contrast factor. As one can see from equations (17) and (19), the influence of the measurement mode on the measured intensity distribution is encapsulated by the geometric tensor *G*. Just for the illustration of this method of description let us consider several more measurement modes in terms of the geometric tensor *G*.

##### 3.2.3. Coplanar double-crystal scans

In the case of the absence of an analyzer the measured intensity is integrated over the *et al.*, 2006*b*, 2005). In this case for the intensity distribution one can use equation (19) with the vector replaced by

and the geometrical tensor

##### 3.2.4. Powder diffraction

In the powder case, the intensity distribution provided by equation (1) corresponds to the intensity from a single grain, and the measured signal comes from grains with the different orientations. Because of this one has to integrate the intensity in over the plane normal to the transferred wavevector (Kaganer *et al.*, 2005; Benediktovitch, Feranchuk & Ulyanenkov, 2014). Hence for the geometrical tensor one obtains (Klimanek *et al.*, 1988)

The elastic part *E* of the dislocation contrast factor depends on the dislocation type. By measuring various reflections and/or changing the measurement mode one will get the different geometric tensor *G* and hence the values of different combinations of the elements of the elastic tensor *E*. This way of obtaining information about the dislocation type was successfully demonstrated for a number of polycrystalline samples (Ungar, 2004; Ungár, 2001; Leoni *et al.*, 2007). Below we will try to adopt this approach to the characterization of the dislocation structure in Ge epitaxial layers.

### 4. Data processing and analysis

With the help of equations (16)–(19) one can simulate the measured intensity distribution. We will consider the area close to the peak, where the approximation (2) holds. The results of calculations after equations (16)–(19) and experimental data show that the shape of the RSM is close to a two-dimensional Gaussian, and the shape of the profile is close to a Gaussian curve as long as the deviations from the peak center are of the order of (see Figs. 3 and 4 for examples of measured data; other data look similar). At larger deviations from the peak center the shape transforms to a power law (Kaganer *et al.*, 2006*a*). However, we will not use this low-intensity region in further analysis. Some asymmetry in the measured profile not predicted by the current formalism may be due to other defect types causing the peak broadening.

Within the considered approach the shape of all RSMs and profiles is determined by a small number of parameters, namely and . To process the data we propose to use a simple and illustrative approach close to a modification of the Williamson–Hall plot, which is intensively used in microstructure analysis of powder and polycrystalline samples (Ungár, 2001) [Shalimov *et al.* (2007) take a similar approach for application to heteroepitaxial film analysis]. The basis of the approach is to analyze the peak width dependence on the reflection. To assign a peak width to the data we will describe the profile by a single parameter *w*, which is the half-width at the 1/*e* level, and in the case of the RSM three parameters *a*, *b*, ψ were used to describe the elliptic isointensity contour at the 1/*e* level (see Fig. 3).

#### 4.1. Ge/Si(111)

For the Ge layer on the (111)-oriented Si substrate we have considered the dislocation configuration shown in Fig. 5. Three {111} slip planes are considered to be equally populated with, in total, six types of Burgers vector orientation (Nguyen, 2012). Assuming that all six misfit dislocation types have an equal dislocation density and are independent of each other, with the help of equation (6) transformed from the Dm to the L coordinate system we obtain for the average deformation

From the peak positions on the RSMs one can find that the Ge layer is completely relaxed. In order to compensate the mismatch between the Ge layer and Si substrate from equation (23), the necessary density of misfit dislocations is found as nm^{-1}. At such values of the dislocation density the coherent scattering is almost completely suppressed, and hence the observed diffraction signal is of diffuse origin, corresponding to the formalism presented above. The peak broadening due to dislocations is much higher than that due to the finite film thickness, and for this reason no thickness fringes are observed. The broadening due to the instrumental function effect was estimated by the substrate peak broadening, which transpired to be much narrower than the observed peaks and was omitted in the calculations.

The expression for the quantity in the case of the considered dislocation system becomes

Here *g*_{60} is a parameter for the positional correlation of misfit 60° dislocations, the index α denotes the dislocation type, is a combination of four transformation matrices calculated for each misfit dislocation type α,

and is the corresponding quantity for threading screw dislocations. The parameters *a*, *b*, , *w* for the measured RSMs and profiles were calculated for the parameter in the range 0–1, which is enough to find the parameters *a*, *b*, , *w* for any dislocation densities by simple scaling following from equation (24). The fitting of *a*, *b*, *w* found from the measured data resulted in 1/*g*_{60} = 44, cm^{-2}. Since equation (11) and subsequent equations only contain the product one cannot find without the knowledge of *R*. However, the dependence on *R* is logarithmic and hence has only a weak influence on the result. We will further assume a typical value of *R* according to Kaganer & Sabelfeld (2014) and take . The misfit dislocations are strongly positionally correlated, which is expected for such a thick layer with a high mismatch. The density of threading dislocations qualitatively agrees with the value 6 ×10^{8} cm^{-2} obtained by Nguyen (2012) by TEM for a similar sample.

Fig. 6 plots the experimentally measured peak width parameters (*a*_{exp}, *b*_{exp}, *w*_{exp}) *versus* calculated ones (*a*_{th},*b*_{th},*w*_{th}) corresponding to the value of *f*_{m/s} found from the fit and the value of . This plot can be considered as a modified Williamson–Hall plot (Ungár, 2001) adapted for thin-film analysis. In the ideal case all points should fall on the same line; at the given *f*_{m/s} the slope of this line gives the absolute value of or . The intersection of this line with the ordinate axis gives the broadening due to the crystallite size and instrumental effects. In our case this contribution to the broadening is negligible owing to the large film thickness, which supports the assumptions made above.

#### 4.2. Ge/Si(011)

A similar approach was applied for characterization of the Ge layer on the (011)-oriented Si substrate. The peak positions showed that this layer was also completely relaxed. However, the 60° dislocations are able to provide the relaxation only in one direction (Elfving *et al.*, 2006). For relaxation in the orthogonal direction we considered 90° dislocations. The resulting dislocation configuration is shown in Fig. 7. For the considered set of the four 60° misfit dislocations with the Burgers vector on {111} slip planes, a 90° dislocation and four threading dislocations that are considered to follow the direction of the Burgers vector, we obtain for an average deformation

Here the setting is considered. In order to compensate the mismatch between the Ge layer and Si substrate in all directions, the necessary density of the misfit dislocations is found from equation (26) as nm^{-1} and nm^{-1}.

The expression for the quantity in the case of the considered dislocation system becomes

where *g*_{60} is a parameter for the positional correlation for the misfit 60° dislocations and *g*_{90} the equivalent for the 90° ones. The parameters *a*, *b*, , *w* for the measured RSMs and the profiles were calculated for the parameters *f*_{m 60/s} = , in the range defined by inequality , which is enough to find the parameters *a*, *b*, ψ, *w* for any dislocation densities by simple scaling following from equation (27). The fitting of *a*, *b*, *w* found from the measured data resulted in 1/*g*_{60} = 28, 1/*g*_{90} = 20, cm^{-2}. One can see that in this case the misfit dislocations are also strongly positionally correlated, which is expected for such a thick layer with a high mismatch. The density of the threading dislocations is underestimated. One of the reasons may be that we did not include the broadening due to the stacking faults (Huy Nguyen *et al.*, 2013). The incorporation of stacking faults into the current formalism will be the topic of future investigations.

The analog of the modified Williamson–Hall plot demonstrated in Fig. 8 shows that, similar to the Ge/Si(111) case, the contribution to the broadening due to finite film thickness is negligible.

### 5. Conclusions

An approach to calculate the intensity distribution in *E* and *G*, the first being determined by the strain fields produced by the defects and the second being dependent on the measurement mode only. Several examples of measurement modes are discussed in terms of the geometrical tensor *G*; the corresponding values of *G* are given in equations (17), (19), (21) and (22). The approach was applied for processing sets of RSMs and profiles measured in noncoplanar geometry for Ge/Si(111) and Ge/Si(011) layers. The measured intensity distributions were well described by Gaussians, which enabled us to use a small number of parameters associated with the shape and treat them in a manner similar to the modified Williamson–Hall plot. The misfit dislocations were found to be strongly positionally correlated, and the density of threading dislocations for the Ge/Si(111) layers were in qualitative agreement with TEM observations (Huy Nguyen *et al.*, 2013).

### APPENDIX A

The tensor *E*_{ijkl} is a symmetric over permutation of the pair of indexes: *E*_{ijkl} = *E*_{klij}. The nonzero components are given below:

Here ν is the Poisson ration, *b*_{x}, *b*_{y}, *b*_{z} are the components of the Burgers vector, and all quantities are given in the Dm coordinate system. Its origin is taken at the free surface. The value of *z* in the above expressions is dimensionless and equal to the ratio *z*^{(Dm)}/*d*, where *d* is the film thickness; hence *z* = 1 corresponds to the interface where misfit dislocations are lying.

### Footnotes

^{1}This article will form part of a virtual special issue of the journal, presenting some highlights of the 12th Biennial Conference on High-Resolution X-ray Diffraction and Imaging (XTOP2014).

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