Bunches of misfit dislocations on the onset of relaxation of Si$_{0.4}$Ge$_{0.6}$/Si(001) epitaxial films revealed by high-resolution x-ray diffraction

TL;DR

This study uses Monte Carlo modeling of x-ray diffraction data to reveal that misfit dislocations in SiGe/Si films are bunched, causing diffraction peaks to underestimate the true relaxation level.

Contribution

It demonstrates that dislocation bunching significantly affects x-ray diffraction analysis of epitaxial film relaxation.

Findings

Dislocation bunching leads to underestimation of relaxation.

Monte Carlo modeling accurately reproduces diffraction patterns.

Inhomogeneous dislocation distribution is essential for understanding relaxation.

Abstract

The experimental x-ray diffraction patterns of a Si$_{0.4}$Ge$_{0.6}$/Si(001) epitaxial film with a low density of misfit dislocations are modeled by the Monte Carlo method. It is shown that an inhomogeneous distribution of 60$^\circ$ dislocations with dislocations arranged in bunches is needed to explain the experiment correctly. As a result of the dislocation bunching, the positions of the x-ray diffraction peaks do not correspond to the average dislocation density but reveal less than a half of the actual relaxation.

Full text

Si1-xGex films on Si substrate constitute a heteroepitaxial system that finds numerous applications in the whole compositional range Cressler (2006) and, at the same time, is a model system that demonstrates the whole spectrum of strain relaxation mechanisms. When either thickness or Ge content is small, the layers stay strained, by accepting the lateral lattice spacing of the substrate and expanding vertically due to the Poisson effect. Larger strain is relaxed by one of the two mechanisms, plastic relaxation in planar layers by introduction of misfit dislocations at the interface Fitzgerald (1991); Hull and Bean (1992); Jain, Harker, and Cowley (1997); Yu. B. Bolkhovityanov, Pchelyakov, and Chikichev (2001) or development of three-dimensional islands in the Stranski–Krastanov growth mode Brunner (2002). Planar films with controlled strain state are required for various applications. Particularly, the high compressive strain in the Si0.4Ge0.6 films, studied in the present work, ought to enhance a room-temperature two-dimensional hole gas mobility, which is important for application of such films in the field effect transistorsMyronov et al. (2014).

X-ray diffraction is a well established technique to characterize relaxed epitaxial films. Positions of the x-ray peaks provide the lattice parameters of a relaxed film and hence the density of misfit dislocations Heinke et al. (1994); Bowen and Tanner (2006); Benediktovich, Feranchuk, and Ulyanenkov (2014). An application of the same analysis at the onset of relaxation suffers from the peak broadening due to a small layer thickness Hartmann, Abbadie, and Favier (2011). Moreover, we show below that the inhomogeneity in the dislocation distribution plays an essential role in the x-ray diffraction analysis. The position of the coherent peak is given by the less strained regions of the film and hence underestimates the relaxation. The diffuse x-ray intensity occurs more sensitive to both the presence of misfit dislocations and their distribution.

We have chosen for a detailed x-ray diffraction study a 27 nm thick Si0.4Ge0.6 film on Si(001), demonstrating an early stage of the relaxation. Thinner unrelaxed and thicker relaxed films of the same series of samples were studied earlier Ulyanenkova et al. (2013). The samples were grown by reduced pressure chemical vapor deposition in an industrial ASM Epsilon 2000 system. Germane and disilane precursors were used to grow Si0.4Ge0.6 epilayers at the growth temperature of 450*∘* C. The critical thickness for plastic relaxation at a Ge content $x=0.6$ is 10 nm People and Bean (1985, 1986); Hartmann, Abbadie, and Favier (2011). The low growth temperature allows us to obtain 2.7 times thicker layer possessing a small relaxation.

High-resolution x-ray diffraction measurements were performed using a 9 kV SmartLab Rigaku diffractometer with a rotating anode. The diffraction setup included a two-crystal Ge monochromator in the 400 setting and a one-dimensional high-speed position-sensitive detector D/teX Ultra from Rigaku.

Figures 1(a-d) present experimental reciprocal space maps in the symmetric 004 and several asymmetric reflections, in a sequence of increasing asymmetry. The wave vectors are represented in the dimensionless units of the product of the components of the reciprocal space vector $(q_{x},q_{z})$ and the film thickness $d=27$ nm. Each map comprises a coherent scattering streak extended along the surface normal, and diffuse scattering. The presence of the coherent and the diffuse intensities is an indication of a weakly distorted film, possessing a low density of misfit dislocations. A closer inspection of the symmetric 004 map in Fig. 1(a) reveals that the positions of the coherent and the diffuse maxima do not coincide: with the origin $q_{z}=0$ chosen at the position of the coherent peak, the diffuse intensity is maximum at $q_{z}d\approx 5.5$.

Figure 2(a) shows line scans extracted from the maps perpendicular to the scattering vectors ($\omega$-scans) at the intensity maxima of the respective maps. For the 004 reflection, two scans are presented, one through the maximum of the coherent intensity (gray line) and the other through the maximum of the diffuse intensity (green line). Evidently, the former scan shows a larger peak intensity, while the latter has a higher diffuse scattering intensity. In asymmetric reflections, the scans through the coherent and the diffuse maxima reveal a less pronounced (albeit present) difference, and we present only the scans through the coherent maxima.

The reciprocal space maps in asymmetric reflections in Figs. 1(b-d) reveal a strong asymmetry of the diffuse intensity distributions. For each reflection, the coherent streak at $q_{x}=0$ separates the diffuse intensity in two lobes, the one at $q_{x}<0$ possessing notably higher intensity in comparison with the other at $q_{x}>0$. This asymmetry and a sharp border between the two lobes is clearly seen in the scans presented in Fig. 2(a).

Our aim now is to find a distribution of misfit dislocations that complies with all characteristic features of the experimental diffraction patterns in Figs. 1 and 2. The 60∘ dislocations with Burgers vectors $\nicefrac{{1}}{{2}}\left\langle 011\right\rangle$ that glide in the $\{111\}$ planes are the main type of dislocations in crystals with diamond or zinc blende structure Fitzgerald (1991); Hull and Bean (1992); Jain, Harker, and Cowley (1997); Yu. B. Bolkhovityanov, Pchelyakov, and Chikichev (2001). Calculated and experimental diffraction curves from uncorrelated 60∘ dislocations show characteristic side peaks (side lobes on the maps) P.Kidd, P.F.Fewster, and N.L.Andrew (1995); Kaganer et al. (1997); Alexe et al. (2002, 2005) that are absent in our experimental patterns. Since edge (Lomer type) misfit dislocations are also formed at the SiGe/Si interface as a result of a reaction between 60∘ dislocations Yu. B. Bolkhovityanov and Sokolov (2012); Yu. B. Bolkhovityanov et al. (2013); Marzegalli et al. (2013), we have tried various models for the arrangement of edge dislocations, using the Monte Carlo method Kaganer and Sabelfeld (2009). However, the calculated profiles are notably narrower than our experimental profiles.

A Monte Carlo method for the calculation of the x-ray diffraction intensity from relaxed epitaxial films has been formulated in Ref. Kaganer and Sabelfeld (2009). The method is applicable to any kind of misfit dislocations but used so far only for edge (Lomer type) dislocations in symmetric Bragg reflections. The positional correlations of dislocations were considered implying that the dislocations tend from random to more regular arrangements to reduce the elastic energy of the film. In the present work, we have included 60∘ dislocations and asymmetric reflections, and searched in a wider range of possible positional correlations.

Reciprocal space maps calculated by the Monte Carlo method in Figs. 1(e-h) and the diffraction profiles shown by black lines in Fig. 2 demonstrate a quantitative agreement with the experimental maps and curves. Now we describe the Monte Carlo model of the dislocation distribution that we have used. We assume two arrays of straight misfit dislocations with the dislocation lines in the two orthogonal $\left\langle 110\right\rangle$ directions. For dislocations with the lines normal to the scattering plane $(x,z)$, Burgers vectors $\mathbf{b}=\nicefrac{{1}}{{2}}\left\langle 011\right\rangle$ have the same component $b_{x}=-a/2\sqrt{2}$ releasing the misfit, while the signs of two other components, screw $b_{y}=\pm a/2\sqrt{2}$ and edge $b_{z}=\pm a/2$, are chosen on random and uncorrelated (here $a$ is the lattice parameter of the substrate). The position of the Bragg peak corresponding to the average relaxation is given by $q_{0x}=\rho Q_{x}b_{x}$ and $q_{0z}=-\frac{2\nu}{1-\nu}\rho Q_{z}b_{x}$, where $\rho$ is the linear density of misfit dislocations and $Q_{x},Q_{z}$ are the components of the reciprocal lattice vector Kaganer et al. (1997). The shift $q_{0z}$ is taken into account in Figs. 1(e-h), but the $q_{0x}$-shift is not made, as discussed below.

To reach an agreement between the experimental and the calculated curves in Figs. 1 and 2, we varied the dislocation density and the distribution of dislocations. The dislocation density $\rho d=0.5$ used in the Monte Carlo calculations corresponds to a relaxation degree $R=0.05$. The positions of the dislocations are modeled as a Markov chain, with the probability $P$ to have a distance $\rho^{-1}P$ between two subsequent dislocations possessing a lognormal distribution. The probability density is generated as $P=\exp[\mu+\sigma N(0,1)]$, where $N(0,1)$ is the standard normal distribution with the mean 0 and dispersion 1. The standard deviation of the lognormal distribution is taken to be $s=10$ times larger than its mean value. Explicitly, the parameters of the lognormal distribution are $\sigma=\sqrt{\ln(1+s^{2})}$=2.15 and $\mu=-\sigma^{2}/2=-2.31$.

Figure 3(a) shows an example of the dislocations distributed according to our model, and the surface displacement caused by these dislocations. The positions of the dislocations are marked by vertical bars. Blue and green bars correspond to dislocations with opposite signs of the tilt component of the Burgers vector $b_{z}$. The large standard deviation of the distribution gives rise to bunches of dislocations separated by dislocation-free regions. The black curve in Fig. 3(a) is the surface displacement calculated for this dislocation array. On a mesoscopic scale, the surface relief exhibits rather sharp peaks caused by dislocation bundles separated by relatively flat regions. In calculating surface displacements from dislocations, we assume that the slip steps are eliminated by surface diffusion, as proposed by Andrews et al. Andrews et al. (2000, 2002, 2004), so that the surface displacement due to each dislocation is a continuous function of the coordinate $x$. The x-ray diffraction profile, calculated for this distribution of the dislocation positions, is shown in Fig. 3(d) by the black line. It is calculated at $q_{z}=q_{0z}$, i.e., at the $q_{z}$ position in between the ones presented in Fig. 2(a), for the same model of the dislocation distribution.

Figures 3(b,c) present, for a comparison, more homogeneous dislocation distributions modeled in the literature Andrews et al. (2000, 2002, 2004). The dislocation positions in Fig. 3(b) are chosen on random independently from each other, and the signs of the tilt components of their Burgers vectors are also not correlated. The dislocation density is the same, $\rho d=0.5$. The surface displacements (red line) are qualitatively similar to those in Fig. 3(a) but have less pronounced peaks and smaller flat areas. However, the x-ray diffraction profile calculated for this model by the same Monte Carlo method [red curve in Fig. 3(d)] looks qualitatively different. It possesses the side maxima described theoretically for uncorrelated random misfit dislocations Kaganer et al. (1997) and observed experimentally for In0.1Ga0.9As/GaAs(001) P.Kidd, P.F.Fewster, and N.L.Andrew (1995), Si0.75Ge0.25/Si(001) Kaganer et al. (1997), and ZnSe/GaAs(001) Alexe et al. (2002, 2005) epitaxial films.

Figure 3(c) shows a model with random dislocation positions but correlated Burgers vectors. Alternating groups containing a given number of dislocations with the same tilt component of the Burgers vector $b_{z}$ were considered in Ref. Andrews et al. (2004). In our model, the number of dislocations in a group is taken on random: the sign of $b_{z}$ is changed with the probability $p=0.1$, so that there are on average 10 dislocations in a group. The surface profile (blue line) varies over a larger lateral scale and a larger height. The diffraction profile calculated for this model [blue line in Fig. 3(e)] exhibits the same side maxima as the one for the case of uncorrelated positions and Burgers vectors above. Thus, the bunching of dislocations is required to explain our experimental diffraction profiles.

Figure 3(e) presents the surface relief of the investigated Si0.4Ge0.6/Si(001) film as measured by atomic force microscopy (AFM). The cross-hatch pattern observed in this micrograph is a well-known manifestation of plastic relaxation Olsen (1975); Lutz et al. (1995); Gallas et al. (1999); Andrews et al. (2000, 2002, 2004); Saha, Cassidy, and Thompson (2013). With the dislocation density $\rho d=0.5$ as determined from the x-ray data, the number of dislocations on a 10 µm interval is 185, while only about 35 randomly spaced parallel lines are seen in Fig. 3(e). Hence, a single line in the AFM image corresponds to a group of dislocations, rather than a single dislocation. This is in a good agreement with our analysis and the calculated profile in Fig. 3(a). The formation of the cross-hatch pattern is a result of a complicated interplay between surface diffusion and dislocation generation. It remains debatable, if the dislocations cause the surface undulations or, oppositely, the undulations due to surface diffusion are sources of dislocations Saha, Cassidy, and Thompson (2013). Hence, we do not make a quantitative comparison of the measured and the modeled surface profiles.

The expected shift $q_{0z}$ of the coherent 004 peak due to an average strain calculated by the expression given above for the dislocation density of our sample $(\rho d=0.5)$ is equal to $q_{0z}d\approx 4.4$. This shift is taken into account in Fig. 2(b), so that the intensity maximum is expected to be at the origin, $q_{z}-q_{0z}=0$. We have verified this prediction by additional Monte Carlo calculations (not shown) of similar diffraction profiles for uncorrelated or more ordered dislocations, which give the intensity maxima at the expected position. However, the peak of the calculated curve in Fig. 2(b) is at $(q_{z}-q_{0z})d\approx-2.65$. Hence, the coherent peak position corresponds to less than half of the actual film relaxation.

The difference between expected and calculated peak positions can be explained by the dislocation bunching, which gives rise to regions with large and small strains, as it is reflected in the surface profile in Fig. 3(a). The dislocation-rich regions possess large strain and large strain inhomogeneity, so that they contribute mostly to the diffuse scattering. In contrast, the dislocation-depleted regions possessing small strain and small strain gradients contribute to the coherent intensity. As a result, the positions of the coherent and the diffuse intensity maxima on the calculated reciprocal space maps in Figs. 1(e-h) do not coincide: the coherent peak reflects the areas in the sample which are less strained than the average, while the diffuse peak represents the more strained ones.

The coherent peaks in Figs. 1(f-h) remain at the same lateral position $q_{x}=0$ as they are in an elastically relaxed dislocation-free film. This peak position has been analyzed in Ref. Kaganer et al. (1997) [see discussion after Eq. (27)] and in Ref. Benediktovitch et al. (2011). One can also see from the experimental maps in the insets in Figs. 1(a-d), that the $q_{x}$-positions of the substrate and the film peaks coincide. The intensity maxima move to $q_{x}=q_{0x}$ when the dislocation density is increased, the coherent peak weakens, and the diffuse peak dominates.

Summarizing, diffuse x-ray intensity from misfit dislocations can be revealed at the very early stages of relaxation of the epitaxial films, when the shift of diffraction peaks due to these dislocations is not yet visible. The diffuse intensity distribution is sensitive to the spatial arrangement of misfit dislocations. We model the dislocation distribution by the Monte Carlo method and find that the diffraction pattern from a Si0.4Ge0.6/Si(001) epitaxial film on the onset of relaxation is due to a very inhomogeneous dislocation distribution. Distances between dislocations vary very broadly, so that the standard deviation of the dislocation spacings is 10 times larger than the mean distance between dislocations. In other words, dislocations form bunches, as a result of the action of small number of dislocation sources. These bunches are seen as cross-hatch patterns in the AFM images of the film.

The inhomogeneous dislocation distribution results in peculiar features of the diffraction patterns. The positions of the coherent and the diffuse peaks do not coincide, since the former is mostly due to undisturbed regions between dislocation bunches while the latter is due to the inhomogeneous strain at the bunches. Moreover, since the coherent peak represents the undisturbed regions, rather than the strain averaged over the whole film, its position at the onset of relaxation does not correspond to the actual density of misfit dislocations and underestimates relaxation by more than a factor of 2.

The authors thank Bernd Jenichen and Oliver Brandt for a critical reading of the manuscript.