Confronting Vegard's rule in Ge1-x Sn (x) epilayers: from fundamentals to the effect of defects

Journal of Physics D: Applied Physics J. Phys. D: Appl. Phys. 55 (2022) 295301 (19pp) https://doi.org/10.1088/1361-6463/ac677a Confronting Vegard’s rule in Ge1−xSnx epilayers: from fundamentals to the effect of defects S Magalh˜ aes1,∗, M Dias1, B Nunes1, F Oliveira2, M F Cerqueira2,3and E Alves1 1IPFN, Instituto de Plasmas e Fus˜ ao Nuclear, Campus Tecnológico e Nuclear, Instituto Superior Técnico, Universidade de Lisboa, Estrada Nacional 10, 2695-066 Bobadela LRS, Portugal 2Centro de Física, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal 3International Iberian Nanotechnology Laboratory, Braga, Portugal E-mail: [email protected] Received 15 December 2021, revised 10 April 2022 Accepted for publication 14 April 2022 Published 3 May 2022 Abstract Comprehensive and systematic study challenging the application of Vegard’s rule to germanium tin solid solutions grown on germanium buffer layers and 100 silicon substrates is presented. The binary’s lattice parameters, composition and respective uncertainties are determined through x-ray diffraction via reciprocal space mapping technique employing newly developed software. The tin content is confirmed by Rutherford backscattering spectrometry and energy dispersive x-ray spectroscopy. The statistical agreement between the tin contents derived by the different structural characterization techniques suggests the binary to follow generically the Vegard’s rule in the range of low Sn molar fractions (<5%). Phase separation, Sn segregation, point defects, post-growing oxygen impurities, and deteriorated surface morphology are found to be relevant within the ∼200 nm germanium tin films. Although, complex mechanisms triggering composition/strain heterogeneities are found in the analysed Ge1−xSnxcompounds, the deviation from the perfect crystals is suggested to be not enough to distort the inand out-of-plane lattice parameters away from its empirical linear combination. Supplementary material for this article is available online Keywords: GeSn compound, x-ray diffraction, x-ray reflectivity, defects, ion channelling (Some figures may appear in colour only in the online journal) 1. Introduction Germanium tin (GeSn) and silicon germanium tin (SiGeSn) are examples of solid solutions based group-IV crystalline semiconductors. The binary and the ternary have been the objects of intense research in the last two decades mainly due to the possibility of making a silicon-compatible direct band gap material with controlled lattice parameters and band gap energy [1–10]. The possibility of epitaxial growth directly on ∗Author to whom any correspondence should be addressed. silicon substrates further motivates the research in the field despite the significantly different thermodynamic properties, ionic sizes and electronegativities of Si, Ge and Sn. In fact, the nature of the different atoms challenges the growth of single phase Ge1−xSnxwith excellent crystalline quality. The crystalline quality deteriorates with increasing tin content as phase separation and segregation tend to decrease the solubility for higher Sn content SiGeSn pure random compounds (<0.5%) [11]. Even for low Sn content GeSn (or SiGeSn), the relaxation of lattice strain in the epitaxial buffer layers acting as virtual substrates decreases the pseudomorphism degree while are often accompanied by layer tilting and development and 1361-6463/22/295301+19$33.00 Printed in the UK 1© 2022 The Author(s). Published by IOP Publishing Ltd This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 License. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. J. Phys. D: Appl. Phys. 55 (2022) 295301 S Magalh˜ aes et al propagation of dislocations [12]. The lack of solubility in the compound, on the one hand, affects the crystal size through a heterogeneous distribution of unit cells with slightly different lattice parameters. On the other hand, it limits the crystalline quality because of strain relaxation via defects’ formation. Composition and/or strain heterogeneities strongly affect the growing of quasi-perfect single crystalline GeSn (or SiGeSn) compound layered structures. Both crystalline quality limitations have similar effects in the x-ray diffraction (XRD) patterns. Specifically, in the case of ωor 2θ–ω(radial) scans, composition and strain shift the Bragg peaks. In fact, both set-ups are influenced by the lattice strain propagation due to deviations from the ideal pseudomorphic growth and because of defects in the semiconductor epitaxial layers. With respect to crystallographic defects, highly mosaic crystals foresee the broadening of the Bragg peaks. Therefore, it decreases the accuracy in the determination of the centroid peak. Executing reciprocal space mapping (RSM) of asymmetric reflections may overcome the above ambiguity by enabling the separation of the composition and strain related effects so that the Bragg peak positions can be precisely derived in the reciprocal space [13–18]. The determination with high accuracy of the lattice parameters and the (Si1–yGey)1−xSnxstoichiometry (xand y) is, thus, fundamental to understand the relaxation mechanism(s) responsible for the crystalline quality deterioration with increasing Sn content. Corrections to the Vegard’s rule are known for the Ge1−xSnxand the Si1−xGexcubic systems as well as for (Si1−yGey)1−xSnxwhere bowing parameters are incorporated into the linear relationship between the individual lattice parameters (band-gaps) [19,20]. Furthermore, calculations based on the density functional theory support the deviation observed in both Ge1−xSnxand Si1−yGeysystems [21,22]. To the best knowledge of the authors, it is the first time a systematic experimental analysis devoted to the validation of Vegard’s rule in the Ge1−xSnx compounds is put forward considering all relevant uncertainties. Impurities also distort the crystal lattice imposing a non-linearity of the relation between the lattice parameters and the derived composition [23–25]. In order to evaluate the composition, new software (method) to fit mathematically the RSMs employing 2D Gaussians is presented and made available to the scientific community. Besides the 2D fitting, the software provides the option of fitting the projections onto both orthogonal directions using Gaussians and pseudo-Voigt (PV) functions. From the RSM software, the lattice parameters of the Si substrate, Ge buffer layer and GeSn compound as well as its uncertainties are determined. The composition of the binary layer is then derived together with its uncertainty. Results obtained with prior versions of the software can be found elsewhere [14–17,26,27]. The software titled RSM can be found at www.mrox.eu and is addressed in supplementary information (S1) (available online at stacks.iop.org/JPhysD/55/295301/mmedia) where full description is presented. The crystal growing procedure of a set of GeSn compound epilayers by molecular beam epitaxy (MBE) on a Si substrate and the experimental characterization techniques are described in section 2. Section 3is devoted to the demonstration of the proposed analytical method. In section 4, the composition and strain dependency upon the Sn content is characterized by the x-ray RSM, x-ray 2θ–ω scans, x-ray reflectivity (XRR), ion and electron beam measurements. Reasonable discrepancies in the determination of the molar fraction of several systems using XRD and Rutherford backscattering spectrometry (RBS) (ex, the Al1−xInxN) have been reported [28,29]. The studies refer to high density of defects build-up hydrostatic strain into the layers, thus, motivating analogous analysis in the current manuscript. The main conclusions of the manuscript are addressed in section 5. 2. Samples and description of the experimental techniques A set of four samples (C1, C2, C3 and C4) were grown by MBE using an electron beam evaporator for germanium with a silicon crucible and a Knudsen cell for the tin with pyrolytic BN crucible and a base pressure lower than 10−9mbar [30,31]. Substrates of 100 silicon with 100 mm diameter were used. They were treated with a wet-chemical etching in a HF bath, to clean and create a hydrogenized layer at the surface, and when inside the deposition chamber, the substrate was heated to 700 ◦C to remove the hydrogenized layer by thermal desorption. This treatment was followed by the growth at 330 ◦C of an epitaxial buffer layer of Ge with a thickness of ∼100 nm and a growth rate of 0.43 Å s−1. The substrate temperature was reduced to 75 ◦C and 85 ◦C to grow a Ge1−xSnx layer with a thickness of ∼200 nm over the buffer layer. This layer was grown with the same Ge flux as the one used for the buffer layer whereas the Sn flux was increased from sample to sample (from C1 up to C4). A low temperature was used to grown the Ge1−xSnxlayer in order to mitigate tin aggregation and/or precipitation. The samples were finalized with a annealing step in-situ by ramping-up their temperature to ≈195 ◦C followed by an immediate ramp-down to room temperature (see figure 1). The relaxed in-plane (a||) and out-of-plane (a⊥) lattice parameters of the single elements materials (Si and Ge and Sn cubic crystals) as well as relevant stiffness coefficients (C13 and C33) are listed in table 1. The values depicted in table 1 reflect the average determined for the last two decades of the published parameters found in the literature. An exhaustive comparison between the inand out-of-plane Si, Ge and Sn lattice parameters as well as respective stiffness coefficients is addressed in S2. The samples were analysed by XRD/XRR, RBS/ion channelling (RBS/C), backscattered electron imaging (BSE) and secondary electron imaging (SE). Although the Sn contents derived from XRD, RBS and energy dispersive x-ray spectroscopy (EDS) were measured in the same region of the sample, the beam dimensions are different for each technique and lateral composition heterogeneities are not to be discarded. With respect to the XRD technique, RSM and MROX software’s were used. Related to the former, it is first time the software is presented as final version to the scientific community. The latter, acronym for multiple reflection optimization package for XRD, in its single reflection mode has been often used 2 J. Phys. D: Appl. Phys. 55 (2022) 295301 S Magalh˜ aes et al Figure 1. Illustration of the samples’ growth steps. The numeration indicates the proceeding order. for the simulation (and fitting) of the 2θ–ωscans using the dynamical theory of XRD [32–38] while the multiple reflection mode has been recently presented [38]. The code employs the dynamical theory of XRD and considers the effects of the instrumental function and the presence of Cu Kα2. In particular, de-convoluting a layered structure with specific composition, thickness and crystalline quality determined via the static Debye–Waller (DW) factor into several layers allows deepening the knowledge on the ternary’s solubility. With respect to XRR, a developed code to be fully integrated into the MROX main software was developed. Several publications fall into prior versions of the software can be found elsewhere [39,40]. The XRD/XRR measurements were performed on a Bruker D8 AXS diffractometer. At the primary x-ray beam side, a Cu source was used together with a Göbel mirror and a germanium 220 monochromator in order to reduce the vertical divergence and to mitigate the Kα2radiation, respectively. A double-axis set-up was used, where a slit of 0.1 mm width faces the detector to perform the radial 2θ–ωscans and the reciprocal space maps. To probe the ω-scans, in particular to optimize the final measurements, no slit was placed before the detector. With respect to the XRR measurements, the 220 Ge monochromator was replaced by a Soller slit in order to maximize the reflected intensity. The diffracted/reflected intensity was collected with a point focus detector and recorded with the Bruker D8 AXS diffractometer acquisition software. RBS/C measurements were performed on a van de Graaff AN-2500 type-A accelerator using a 2 MeV 4He+ion beam of 1 mm diameter. A pin diode was placed at a backscattering angle of −165◦to collect the backscattered particles. Random RBS spectra were acquired by tilting the surface normal by 5◦away from the analysing beam and rotating the sample during the measurement to suppress channelling effects. Simulations of the random RBS spectra were performed using the Nuno’s data furnace (NDF) code [41]. The crystalline quality of the GeSn epilayers along the growing direction are determined from the axial <001> aligned scans by 4He+ion channelling measurements [42]. The XRD as well as the RBS/C measurements were performed at the Laboratório de Aceleradores e Tecnologias de Radiaç˜ ao, Campus Tecnológico e Nuclear, Lisbon, Portugal. The BSE and SE modes were performed using a JEOL JSM-7001F field emission gun scanning electron microscope equipped with an Oxford Instruments EDS Table 1. Si, Ge and Sn in- (a||) and out-of-plane (a⊥) relaxed lattice parameters, and C13 and C33 stiffness coefficients used to determine the Sn contents in the Ge1−xSnxbinary compounds. Element a|| (Å) a⊥(Å) C13 (GPa) C33 (GPa) Si 5.431 5.431 63.9 165.7 Ge 5.658 5.658 47.9 129.2 Sn 6.49 6.49 29.3 67.6 system. Quantitative analysis was performed by EDS using appropriate trademarked software. Each sample was analysed in more than ten randomly selected points. The EDS measurements were executed in the Electron Microscopy Laboratory (Microlab) at the Instituto Superior Técnico, Lisbon, Portugal. 3. Determination of the lattice parameters, composition and respective uncertainties to sample C4—application of the RSM software The fitting of the RSM around the 115 Si, 115 Ge and 115 Ge1−xSnxreciprocal lattice points were successfully accomplished through the developed RSM software. The software fits the experimental data employing a Marquardt–Levenberg algorithm [43]. Furthermore, it employs 2D-Gaussians taking into consideration the rotation of the diffraction spots in the reciprocal space. The software is applicable to asymmetric reflections as the case presented here but, furthermore, to symmetric reflections. The best solution found is refined using PV functions of the cuts along the measured domain. Cubic and hexagonal (wurtzite) Bravais lattices are implemented in the code. The software determines the (Qˆ x,Qˆz)centroids of the diffraction spots [44]. Thus, the crystal lattice inand outof-plane lattice parameters and chemical composition as well as the relevant uncertainties are determined using the Bragg’s rule and the Poisson law in order consider the biaxial strain [45–47]. Furthermore, the determination of δQˆ xand δQˆz, i.e, the widths of the diffraction spots along Qxand Qz, respectively, allows quantifying the crystalline mosaicity of a given crystal. The limitations of the RSM software are only restricted to an almost complete overlap diffraction spots where respective centres may not be accurately determined. The procedure to determine the lattice parameters of the tri-layer heteroepitaxial system exhaustively described in S3 is exemplified using the highest Sn content (Si1−yGey)1−xSnxlayer grown on a Si1−yGeybuffer layer and a 100 silicon substrate (sample C4). The choice of sample C4 is based upon the fact that the 115 Si1−yGeyand 115 (Si1−yGey)1−xSnxellipsoids (diffraction spots) are, as shown in figure 2, perfectly distinguishable. The experimental 115 RSM is depicted in figure 2(a)) while its simulation is plotted in figure 2(b)). The intensity scale on the right side of figure 2is calculated as log (I0+1), where I0 is the measured intensity. The logarithmic representation for the intensity emphasizes the differences between the diffraction patterns of all involved crystals while the increment of the unity in the logarithmic argument overcomes the log(I0→0) mathematical impossibility. 3 J. Phys. D: Appl. Phys. 55 (2022) 295301 S Magalh˜ aes et al Figure 2. Experimental RSM in the vicinities of the 115 Si reciprocal lattice point and simulated RSM using 2D-Gaussians of the highest tin content (Si1−yGey)1−xSnxsample. The indexation of the various diffraction spots is highlighted in figure 2(b). Peak #1 refers to the Cu Kα1115 Si diffraction spot while peak #2 is associated to the Cu Kα2115 Si. In fact, the Cu Kα2115 Si residuals were not completely mitigated by the optical system (Göbel mirror and 2-bounce 220 Ge monochromator). Placing an analyser at the secondary x-ray beam-path would further decrease the intensity associated to peak #2 but at the expenses of ∼1 magnitude order of intensity decrease to the Cu Kα1115 Si as well. Peaks #1 and #2 appear rotated while the 115 Si1−yGey(peak #3) one presents very low level of rotation. Peak #4 is attributed to the 115 diffraction of the (Si1−yGey)1−xSnxpseudo-ternary. The horizontal dash-dot black lines crossing figures 2(a) and (b) mark the Qzpositions of the 115 Si substrate, Si1−yGey1−x and (Si1−yGey)1−xSnx, from higher to lower Qz, respectively. The simulated Qzpositions using equation S3.1 agree perfectly with the experimental data. Moreover, the shape of all diffraction spots is well simulated with the 2D-Gaussians. The slight differences between the experimental data and the simulation are focussed on the intensity (colour distribution) and widths of the diffraction spots. With respect to the former, the most visible differences arise from the distribution of the intensities in the reciprocal space attributed to the 115 Si (peak #1) and centre of the 115 (Si1−yGey)1−xSnxdiffraction spot (peak #4). Nevertheless, the distribution of the levels of intensity through colours may induce ambiguities as simulated data is less clear under the 2D representation rather than 1D cut along a specific direction. What is more, typical problems originated by the big data analysis play an important role. In fact, individual RSM contains more than 20 000 (Qx, Qz, I) pairs adding extra computational efforts to the fitting process. In order to check the validity of the 2D-Gaussian in the angular region of interest (vicinities of the Bragg peaks), the proposed method refines the fitting of the RSM with PV functions of the individual cuts. Moreover, the advantage of employing the latter in the current work is related to the determination of the uncertainties through a fast fitting method rather than the time-consuming 2D-Gaussian fitting. In fact, a cut along a specific direction presents more than 50 times less experimental data points reducing dramatically the time for the convergence of the fitting. Figure 3compares the fittings performed with the Gaussian and PV functions along the red and green lines for the 115 Si Kα1(peak #1), (a), 115 Si Kα2(peak #2), (b), 115 Si1−yGey1−x(peak #3), (c) and 115 (Si1−yGey)1−xSnx(peak #4), (d) diffraction spots, respectively. The better fit quality obtained using PVs compared to the Gaussians, especially around the various peaks tails is evidenced in figure 3.ωas well as 2θ–ωscans constitute specific cuts along the asymmetric RSM [48]. In fact, according to [49], the mathematical peak-function which better describes both measurements is the PV, thus, by virtue of the reciprocity theorem, it applies to the RSM as well. To summarize, from the Qˆ x,Qˆzcentres derived from the 2DGaussian and PV fittings, the inand out-of-planes lattice parameters are determined (equations S3.2(a1) and (a2)). In the case of sample C4, (Qx,Qz)=(15.6791, 54.9857), thus, a|| =5.6673 ±0.0041 Å and a⊥=5.7135 ±0.0021 Å. Assuming a Ge relaxed lattice parameter of 5.658 Å (table 1) the measured quantities reveal higher lattice parameters evidencing the effect of the growing of the pseudo-ternary and the limited Ge buffer layer crystal size (∼100 nm). Composition is then determined using equations S3.3–S3.5. To determine the composition a combination of a bisection, secant and inverse quadratic interpolation methods are employed [50,51]. The procedure to derive the lattice parameters, composition and associated uncertainties using the RSM software is composed by IV steps exhaustively described in S3. In the case of sample C4, xSn =0.042 ±0.006. Finally, the uncertainties in the Si, Si1−xGexrelaxed lattice parameters, respective stiffness coefficients and measured lattice parameters are 4 J. Phys. D: Appl. Phys. 55 (2022) 295301 S Magalh˜ aes et al Figure 3. Representation of the cuts along specific directions determined using equation (1) of the experimental RSM highlighting the 115 Si Kα1, (a), 115 Si Kα2, (b), the Si1−yGeybinary, (c), and the (Si1−yGey)1−xSnxpseudo ternary, (d), respectively. Upper: cut of the RSM experimental data along the red line and fit using a Gaussian and PV functions. Right: cut of the RSM experimental data along the green line and fit using a Gaussian and PV functions. (e) Ge content in Si1−yGeyderived using the proposed method. (f) Sn content in (Si1−yGey)1−xSnxderived using the proposed method for sample C4. considered to derive the uncertainties in the Sn content (0.006). The uncertainties in yGe and xSn are calculated based on the individual uncertainties (inputs in the RSM software) for the relaxed inand out-of-plane lattice parameters (∆a||,⊥,0 (Si, Ge, Sn)) and respective C13 and C33 (∆C13, ∆C33 (Si, Ge, Sn)). Then, a set of 10 000 combinations referred as 5 J. Phys. D: Appl. Phys. 55 (2022) 295301 S Magalh˜ aes et al Figure 3. (Continued.) p-numbers containing a||,⊥,0(Si, Ge, Sn) are generated through a Monte Carlo algorithm. Finally, the average and standard deviation of the yGe and xSn are determined. As an example, considering the eight free-variables present in the determination of xSn (4 in a||,⊥,0(Ge, Sn) and 4 in C13 (Ge, Sn) and C33 (Ge, Sn)) the effect of employing a||,⊥,0 ±∆a||,⊥,0 and C13(C33) ±∆C13(C33) in the calculations of the Sn molar fractions results in 28permutations which is considerable less than the 10 000 p-numbers generated. Therefore, the 10 000 pnumbers generated foresee high accuracy in the determination of the xSn molar fraction and high precision in the determination of the respective uncertainty. Figures 3(e) and (f) show 6 J. Phys. D: Appl. Phys. 55 (2022) 295301 S Magalh˜ aes et al the derived yGe and xSn contents as function of the 10 000 p-numbers. In each figure, the upper and right side one-dimensional plots show the experimental data along the specific red and green lines cuts depicted in the 2D plot. The green and red lines were determined using equation (1): I(Qx,z) = a0×     (1−a3)×exp(−log(2)×((Qx,Qz)−Qˆ x,Qˆz δQx,z)2) +a3 1+((Qx,Qz)−Qˆ x,Qˆz δQx,z)2     .(1) For a pure Gaussian, a3=0, while a3=1 corresponds to a pure Lorentzian, i.e. a0and a3are the simulated intensity and Lorentzian fraction, respectively. Screenshots of the RSM software highlighting the multi-step procedure described above are addressed in S1 by exemplifying the required parameters used to input in the file with respect to sample C4. 4. Experimental results and discussion 4.1. Study of the validity of the Vegard’s rule of the Ge1−xSnx layers grown on Ge buffer layers and 100 Si substrates The procedure summarized in section 3is applied to the other three samples (C1 to C3). Representative reciprocal space maps around the 115 Si reciprocal lattice point evidencing the heteroepitaxial growth of Ge on the 100 Si substrate and the Ge1−xSnxbinary on the Ge template are shown in figure 4. The fits of the experimental data using 2D-Gaussians is shown on the right side of each measured RSM. The horizontal dashed lines in the figure are included for guiding purposes and to emphasize the similarities between the 115 Si and 115 Ge buffer layer Qˆzcentres among the three samples. Furthermore, it allows evidencing the effect of the increase of the tin content, xSn (in Ge1−xSnx), from samples C1 to C3. The fits together with the experimental data corresponding to the RSM cuts are highlighted in figures 5(a)–(j)) for the four samples. Figures 5(a)–(i) correspond to the fits of the Si 115 Cu Kα1, Si 115 Cu Kα2, Ge and Ge1−xSnxprojected against Qx[100] direction while the figures 5(b)–(j)), correspond to the same cuts projected against Qz[001] direction. In figure 5, the cuts for each measured RSM are fitted and translated vertically for clarity. In the case of figures 5(g) and (h), the cuts in the vicinities of peak #4 (Ge layer), corresponding to samples C2 to C4, are represented while in figures 5(i) and (j), for sample C3 only one cut is performed in order to evidence/satisfy the observed splitting of the diffraction spot associated to the Ge1−xSnxbinary (peaks #4 and #5). Moreover, the cuts around the 115 Ge1−xSnxof sample C1 are not included in the figures 5(g) and (h) due to the overlap between the Ge and the binary diffraction spots (peaks #3 and #4). The fits represented in red colour are executed using a PV function employing the optimized height, centre (Qj ˆ x,Qj ˆz), Gaussian width (by definition, twice of the full width at half maxima (FWHM)), and Lorentzian fraction outputs. The uncertainties of the derived PV coefficients are used to consider the lower and upper boundaries represented in green and blue colours, respectively. The good agreement between the experimental data and the PV fits along a given cut is evidenced in figure 5. In fact, as demonstrated in section 3, the fit using PVs satisfies better the experimental data than a pure Gaussian. In the executed fits, the Lorentzian fraction (a3, equation (1)) is derived to be above 0.5. Moreover, figures 5(g) and (i) suggest similar inside the experimental and instrumental errors (Qj ˆ x,)centres for all jdiffraction spots attributed to the Ge1−xSnxlayer. On the other hand, the splitting of the diffraction spots with respect to [001] direction is evident from figures 5(h) and (j)) where the xSn content is shown in the inset. The centres and the full widths at half maxima derived for both orthogonal directions of the diffraction spots, (Qˆ x,Qˆz) and (δQˆ x,δQˆz), the in-plane and out-of-plane lattice parameters, a|| and a⊥, and germanium, yGe, and tin, xSn, contents together with the respective uncertainties for all samples are depicted in table 2. The mentioned relevant outputs are determined directly by fitting the cuts through the diffraction spots with PV functions. According to table 2, for most of parameters derived from the Gaussian (not shown) and PV fittings, the absolute values determined for me former lie within the uncertainties derived for the latter. The inand out-of-plane lattice parameters were then determined using equations S3.2(a1)–(b1) and the deduced (Qb x,Qbz)co-ordinate pairs. The tin contents and respective uncertainties derived by the proposed method are: xSn,sample C1 =0.009 ±0.007, xSn,sample C2 =0.017 ±0.005, xSn,sample C3/Peak3 =0.032 ±0.006, xSn,sample C3/Peak4 =0.043 ± 0.004 and xSn,sample C4 =0.042 ±0.006, respectively. The uncertainties are mainly affected by the determination of the peak centroid. In fact, according to equations S3.2(a) and (b) and error propagation theory, the uncertainty in the composition will be strongly affected by the uncertainties in the determination of the lattice parameters, which, in turn, is calculated from the (Qb x,Qbz)centroid peaks. Consequently, the xSn =0.009 calculated for sample C1 presents the highest uncertainty (±0.007) due to the fact of high level of peaks overlapping (115 Ge0.991Sn0.009 and 115 Ge). The uncertainties in the determination of the Sn content for the other samples vary between 0.004 and 0.007. To validate the method, the Sn content was determined by RBS and depicted in table 2, as also is the Sn content obtained by EDS. The experimental data of the RBS random spectra and respective simulations using the NDF code [41] are shown in figure 6(a) for samples C1, C2 and C4. In figures 6(b)–(d), the energy interval between ∼1700 and ∼1950 keV of the RBS spectra is magnified in order to better visualize the differences between the random and aligned along the <001> axis spectra in the region close to the Sn-signal. Qualitatively, the sharp barriers depicted inset figure 6is attributed to the chemical elements present in the sample (Si, Ge and Sn). The continuous line arrows refer to 7 J. Phys. D: Appl. Phys. 55 (2022) 295301 S Magalh˜ aes et al Figure 4. Comparison between the experimental RSMs in the vicinities of the 115 Si and simulations employing 2D Gaussians for 3 Ge1−xSnxbinary layers grown on the Ge template and 100 Si substrate. The Si Cu Kα2centred on slightly higher Qˆ zcompared to the 115 Si Cu Kα1diffraction spots is also observed in all the three RSMs. a|| and a⊥are inversely proportional to Qˆ xand Qˆ z, respectively, showing increasing tin content from the left to the right side in the figure, i.e. from (a) to (c). Horizontal lines suggest the approximately the same a⊥ for the three Si wafer pieces and also for Ge buffer in the three samples. The Ge1−xSnxlayer is quasi-pseudomorphic with the respective Ge layer (both present approximately the same Qˆ x). the chemical elements present at the surface (Ge, Sn and O) while the dashed line arrow refers to the energy of the element (Si) targeted by the 4He+and backscattered at a given depth according to the random spectra simulations. On the one hand, the Ge buffer layer thickness is directly proportional to ∆EGe illustrated in the random spectra of figure 6. On the other hand, the Sn-signal is partially overlapped with the Ge-signal, thus, complicating the quantification of the binary layer thickness. The increase of the tin content is clear from the increasing backscattering yield observed in the Snyield in figures 6(b)–(d). Inset are depicted the RBS derived Sn contents. Although, as demonstrated in figures 6(a)–(d), very good qualities of the simulations are accomplished, several Ge1−xSnxlayers with different composition and thicknesses are required to satisfy the experimental data. The dispersion at xSn among the simulated layers is below 0.2% while its respective thicknesses vary by 10 nm, evidencing, thus, the low level of measured composition heterogeneities and thickness fluctuations. In order to compare with the stoichiometry derived by RSM, a method based on the weighted xSn averaged along the total thickness of the Ge1−xSnxlayer is applied. Specifically, xSn =PN j=1xj Sntj PN j=1tjwhere xj Sn and tjare the Sn contents and thicknesses of the individual (jth) layer, respectively. The Sn content derived via RBS for the 4 Sn-containing samples are 0.002, 0.017, 0.035 and 0.042. Uncertainties around 0.005 in xSn are suggested which agree with the ones found in literature, where the RBS manual analysis of Al1−xInxN/GaN layers is performed [52]. Therefore, the RBS derived xSn perfectly agrees with the ones derived via the XRD RSM counterpart. The estimation of the uncertainty in the determination of the tin content via the RBS is solely based upon graphical visualization of the obtained simulations using the NDF code [41]. Although the Sn content derived through both XRD and RBS techniques agree, it is of fundamental importance the procedure to calculate the uncertainties by the proposed method. Thus, no evidences of a deviation from Vegard’s rule are found for the Ge1−xSnxsystem, within the Sn content range studied. Indeed, in most of the studies such deviations appear only for considerably larger values of xSn, of the order of 0.2 (see [53]). Furthermore, Xu et al suggested the low levels of accuracy if studies focussing the deviation from Vegard’s rule are applied to highly strained Ge1−xSnx(and Si1−yGey) compounds (>1%) [54]. Therefore, it is fundamental to determine the level of strain in the different Ge1−xSnxlayers studied here. Figure 7shows the parallel (ε||) and the perpendicular (ε⊥) to the sample surface strains as function of the Sn content derived by XRD using either the two Sn contents derived for peaks #4 and #5 of sample C3 (figures 7(a) and (b)) and only the lower Sn content (peak #4) in figures 7(c) and (d). Inset figures 7(a)–(d)), the linear regression equation taking into account the uncertainties of the fitting parameters. It considers the ∆xSn, ∆ε||, and ∆ε⊥. The fitting procedure involves a Monte Carlo algorithm to obtain the hyperbolic boundary curves. The uncertainty in thefitting parameters is computed in 5000 iterations, assuming that errors are Gaussian and centred. The upper and lower hyperbolic curves for both orthogonal deformations cross the relaxed Sn content at around zero Sn molar fraction for both situations depicted in figure 7. The Sn molar fraction used in figure 7is the one derived through XRD. Nevertheless, the Sn contents derived by XRD and RBS are inside respective errors. The above behaviour suggests that the studied binary system does not require a correction for the 8 J. Phys. D: Appl. Phys. 55 (2022) 295301 S Magalh˜ aes et al Figure 5. Cuts along the Qx[100] direction, (a), (c), (e), (g) and (i), and Qz[001] direction, (b), (d), (f), (h), (j)), respectively, of C1–C4 samples. The fits are performed using PVs by employing the coefficients (red curve) and the derived upper and lower boundaries (in blue and green curves, respectively). The black open circles correspond to the experimental data. Inset is indicated the 115 Ge peak of samples C1 to C4 and xSn content (in Ge1−xSnx) of samples C2 to C4. Appropriate peak # assignment is illustrated. Vegard’s rule which is in agreement with the study reported in [54] for Sn contents up to 14%. The Si, Ge and Sn relaxed lattice parameters of 5.430 57, 5.656 92 and 6.489 31 Å used in [54] are inside the uncertainties employed in the calculations in this work (table 1). Although the above conclusion is supported considering the derived uncertainties, the 9 J. Phys. D: Appl. Phys. 55 (2022) 295301 S Magalh˜ aes et al Figure 10. (a) C1, C2 and C3 experimental and simulations of the XRR specular scans in the vicinity of the 000 reflection (surface) showing with the vertical dashed line that the density at the surface is relatively the same in the three samples. The fits were performed using the MROX code [39,40]. Inset is the magnification of the density (in g cm−3) as function of depth (in the region of interest of Ge1−xSnx+Ge). (b) Density as function of depth output from the simulations shown in (a). (c) Roughness (in Å) of the different layers used in the XRR simulations. vapour deposition (CVD) grown GeSn, the surface roughness is expected not to exceed 1 nm [63] while on the MBE layers studied here, the surface roughness easily reach twice of that value. GeSn compound grown by CVD usually shows better uniformity and crystalline quality than that by MBE [64]. Although, high quality MBE GeSn layers have been reported, the studies focus low tin content compounds [8,12,30,31]. It is interesting to note that depending on the gases and the growing techniques, particular kind of defects may be created which may drive segregation. For example, Ge2H6as well as well as SnCl4as being used as precursors in the CVD of high-quality GeSn with Sn compositions up to ∼13% [65]. The defects, on the other hand, are required to relax the lattice further motivated by the lack of appropriate substrate, i.e. with less lattice mismatch with respect to the tin-compound. The sizes of Ge and Sn atoms induces an elevated number of point defects which agrees with the high minimum yield derived from the ion channelling measurements and the low DW from the simulations of the 004 2θ–ωscans. Therefore, the deviation from pseumodorphic Ge1−xSnxon the Ge layers, observed compositional/strain heterogeneities along the growing direction, measurable phase separation in the binary, interstitial and/or point defects accumulation in the cubic surface bi-layer crystalline structure, interface roughness’s reaching the nanometre scale do not contribute to a deviation of the Vegard’s rule in the low tin germanium content (<5%) system. 5. Conclusions The ∼200 nm thick Ge1−xSnxepilayers stacked on ∼100 nm Ge buffer layers and Si 100 substrates were successively grown by MBE. The tin contents were derived by new XRD RSM software. The software fits the asymmetric 115 reciprocal lattice point using 2D-Gaussians and refines the solution by means of PVs functions for the cuts along orthogonal directions calculated for the rotation of the different diffraction spots. Sn contents of 0.009 ±0.007, 0.017 ±0.005, double layer containing 0.032 ±0.006 and 0.040 ±0.004 and 0.042 ±0.006 for samples C1, C2, C3 and C4 are derived. The high uncertainty derived for the Sn content in sample C1 is a consequence of the partial 115 diffraction spot overlapping between the binary and the Ge buffer layer. Vegard’s rule was tested for the Ge1−xSnxcubic system by comparing the Sn contents derived through XRD and RBS. With respect to the latter, Sn contents of 0.002, 0.017, 0.035 and 0.042 were obtained with uncertainties of 0.005 estimated by direct observation of its effect in the spectra. Within the derived uncertainties, a linear relation between the single chemical elements representing the binary is sufficient. Thus, within the compositional range studied no evidences of soliciting a correction are found. The complex heteroepitaxial growth is ascertained from the dispersion of the relaxation degrees found for the individual samples. While the highest Sn content found for sample C4, i.e. Ge0.968Sn0.042, and the lowest Sn content derived for one of the layers of sample C3 (Ge0.958Sn0.032) present degrees of relaxation of 41.17% and 51.38%, the highest Sn content layer of sample C3 grew almost pseudomorphic with the Ge buffer layer. Moreover, the binary layer with the second highest Sn content (C2) is 30.35% relaxed. Complex mechanisms triggering composition/strain heterogeneities are found. In particular, phase separation reaching 0.8% is observed for sample C3 (0.032), while variations up to 0.6% and 0.2% of Sn content with depth are simulated by XRD and RBS, respectively. Furthermore, point defects, 16 J. Phys. D: Appl. Phys. 55 (2022) 295301 S Magalh˜ aes et al oxygen-rich regions, Sn segregation and surface morphology are found to constitute limitations to the Sn solubility within the ∼200 nm germanium tin films. With respect to the former, Debye Waller factors of 0.3 and minimum yields between 0.3 and 0.4 are found. Ion channelling complements point/ interstitial defects analysis showing Sn to be slightly more deviated from the regular lattices sites if compared to Ge. Therefore, the Ge layers are slightly crystalline quality enhanced if compared to the Ge1−xSnxsurface layers. The samples are not oxygen-free. Sample C3 revealed the lowest amount of oxygen compared to the other three samples. What is more, the most strained layer is observed for the highest Sn content layer of sample C3 suggesting strain relaxation of the lattice due to the oxygen impurities in samples C1, C2 and C4. Densities close to pure Sn (white tin 7.26 g cm−3and grey tin 5.75 g cm−3) at the surface decreasing its magnitude towards the border of the ∼200 nm tin film are calculated through the simulation of the XRR measurements. The simulations also suggest the presence of metallic Sn clusters at the surface. The Sn clusters and oxygen rich regions may induce the high average roughness of above 4 nm as obtained for the surface. The presented work demonstrates the single crystalline properties of the Ge1−xSnxcompounds revealing that the incorporation of Sn, even for low Sn contents, is a tough technological task. Breaking the limited solubility Sn in Ge is, thus, fundamental to develop Si-compatible direct band gap material with controlled lattice parameters and band gap energy crucial for optoelectronic and microelectronic devices in the near infrared region. Data availability statement The data that support the findings of this study are available upon reasonable request from the authors. Acknowledgments This work was supported by the Portuguese Foundation for Science and Technology (FCT) in the framework of the Plurianual Strategic Funding UID/FIS/50010/2019. F O acknowledges the FCT PhD Grant and thanks the Institut für Halbleitertechnik, Universität Stuttgart for hospitality. The authors acknowledge Professor J Schulze for providingthe MBE facilities and the growing of the growing of the germanium tin films. 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