Reduction of threading dislocation density beyond the saturation limit by optimized reverse grading

The threading dislocation density (TDD) in plastically relaxed Ge/Si(001) heteroepitaxial films is commonly observed to decrease progressively with their thickness due to mutual annihilation. However, there exists a saturation limit, known as the geometrical limit, beyond which a further decrease of the TDD in the Ge film is hindered. Here, we show that such a limit can be overcome in SiGe/Ge/Si heterostructures thanks to the beneficial role of the second interface. Indeed, we show that ${\mathrm{Si}}_{0.06}{\mathrm{Ge}}_{0.94}/\mathrm{Ge}/\mathrm{Si}(001)$ films display a TDD remarkably lower than the saturation limit of Ge/Si(001). Such a result is interpreted with the help of dislocation dynamics simulations. The reduction of TDD is attributed to the enhanced mobility acquired by preexisting threading dislocations after bending at the new interface to release the strain in the upper layer. Importantly, we demonstrate that the low TDD achieved in ${\mathrm{Si}}_{0.06}{\mathrm{Ge}}_{0.94}/\mathrm{Ge}/\mathrm{Si}$ layers is preserved also when a second, relaxed Ge layer is subsequently deposited. This makes the present reverse-grading technique of direct interest also for achieving a low TDD in pure-Ge films.


I. INTRODUCTION
The epitaxial growth of high-germanium (Ge) content silicon-germanium (Si1-xGex) layers [x > 0.7] on silicon (Si) is still attracting a growing attention in the scientific community, owing to its potential impact on several technological applications in the field of photonic and electronic devices [1]. Only to name of few, we can cite the fabrication of pMOS fin field-effect transistors (FinFETs) based on highly strained Ge [2,3]; the use of GeSi/Ge/Si as material for near-infrared integrated light sources [4,5], photodetectors [6,7], and optical modulators in the integrated silicon photonic platform [8,9]; the role of high quality Ge/GeSi quantum wells in the realization of a semiconductor based qbits platform [10,11]; and finally, as active material for the fabrication of a Si-based THz quantum cascade laser (QCL), which might provide an electrically pumped compact source of coherent THz radiation at room temperature [12,13].
However, the heteroepitaxial growth of Ge-rich Si1-xGex layers directly on Si is challenging, due to the large lattice mismatch in the 3  4% range [5]. Because of this large mismatch, the critical thickness for plastic relaxation of the heteroepitaxial strain is limited to a few nm only.
After this threshold, 60 degree dislocation loops are nucleated, gliding on {111} planes and eventually dropping strain-relieving misfit segments at the heterointerface [14,15], which are then bounded by the screw-type arms of the threading dislocations (TDs) [16].
Two different approaches have been commonly proposed in literature to achieve Ge or Ge-rich Si1-xGex layers on Si. The first consists in relying on compositionally graded Si1-xGex thick buffers as virtual substrates (VSs), where the Ge content is gradually increased to accommodate the elastic energy by means of a progressive plastic relaxation, thus promoting less entangled dislocation networks [17]. Furthermore, the persistence of a strained upper region during the growth of the layer provides a steady driving force for the motion of TDs, favoring their interaction and annihilation [18]. This method, despite being widely reported for low [x < 0.5] Ge content [5], is unpractical to achieve with standard deposition techniques (such as reduced pressure chemical vapor deposition (RPCVD) or molecular beam epitaxy (MBE)) for Ge or Ge-rich Si1-xGex, due to the large buffer thickness (up to 20 µm [5,19]) needed to reach the final composition range and to the resulting high surface roughness, requiring a chemical mechanical polishing process step for any practical application [2,5].
Alternatively, a decade ago, it has been proposed to first deposit high quality Ge-rich Si1-xGex layers on Si, by using Ge/Si(001) VSs followed by the deposition of tensile strained layers featuring a decreasing Ge content [2,5,20]. In this method, known as reverse-graded (RG)-VS, first a relaxed Ge layer will be deposited on a Si substrate. For achieving low-defective Ge on Si layer growth, a large number of successful epitaxial approaches has been already reported in literature [21,22,25]. After this Ge/Si(001) VS creation, the process is followed by a Si1-xGex buffer growth with either an abrupt or reverse grading layer, being thick enough to relax the existing tensile strain. In this case, the excess strain built up in the RG layer during its deposition provides the additional mobility to the TDs, again leading to their eventual interaction and annihilation. Compared to the aforementioned forward graded thick VS approach, this RG method produces consistently smoother buffer layers with low root-mean-square (rms) roughness and low TD density (TDD). Shah et al., for instance, compared both methods for achieving a Si0.22Ge0.78 VS on Si(001). The results showed that, for obtaining a Si0.22Ge0.78 VS with a TDD in the 10 6 cm -2 range, the RG method required a significantly thinner heterostructure (2.8 µm instead of  8 µm), exhibiting also a six times smaller rms roughness (2.6 nm instead of  15 nm) [23].
Clearly, the goal is to reduce the TDD to the lowest possible value, since TDs are responsible for the degeneration of the structural and consequently electrical and optical properties of the material.
In this paper, by using an industry-grade RPCVD system hosted in a fabrication pilot line, we performed a systematic investigation of the growth of Si0.06Ge0.94/Ge/Si(001) RG-VSs. As a relevant case study, a fixed Ge content of x = 0.94 for the Si1-xGex part was chosen, since this is the typical RG-VS compositional range required to realize a strain-balanced Si-based THz QCL heterostructure [12,24]. The use of one Si0.06Ge0.94 layer on a Ge buffer as RG-VS is therefore in contrast to previous studies, which reported on RG-VS consisting of several Si1-xGex layers with decreasing Ge content [20,23]. All the samples were thoroughly analyzed in terms of their crystal defects (Misfit dislocations (MDs) and TDDs) and lattice properties (strain, composition profile, surface morphology). A close comparison between experimental results and theoretical Dislocation Dynamics (DD) simulations was carried out, allowing us to describe and understand the relaxation behavior as well as the dislocation propagation and annihilation during RG-VS growth. The main goals of this study were i) to understand the defect development and strainrelease in Si0.06Ge0.94/Ge/Si(001) RG-VS and ii) to optimize the growth processes leading to the minimum TDD in RG-VSs, as a key milestone for the potential subsequent deposition of highquality strain-compensated THz QCL structure on a Si platform.

II. EXPERIMENTAL AND SIMULATION METHODS
All the samples of this study were deposited on 200 mm Si(001) wafer in a commercial ASM Epsilon 2000 lamp-heated single wafer RPCVD reactor at a pressure of 80 Torr. Prior to the epitaxial growth, Si(001) substrates were wet chemically cleaned by a Radio Corporation of America solution, followed by a prebake in the chamber at 1000°C in hydrogen (H2) atmosphere. For all samples, a 100 nm Ge seed layer was grown at 350°C under reduced pressure condition using germane (GeH4) and nitrogen (N2) as carrier gas. After seed layer deposition, all samples have been annealed insitu at 800°C in H2. After annealing, variable thickness Ge bulk layers (targeting a total Ge buffer thicknesses of 0.5 -8 µm) were grown at 550°C under reduced pressure condition using H2 and GeH4 as carrier and reactant gas, respectively. For all Ge bulk layers, a cyclic growth and annealing procedure was used, consisting of two annealing steps at 800°C in H2, one interrupting the Ge growth at half of the targeted layer thickness and one at the end of the Ge bulk layer growth. It is worth noting that this procedure leads to a reduction of the TDD by two order of magnitude, as compared to the as-grown case [2,25]. As described in the following, we have also realized two sample series, in which Si0.06Ge0.94 layers with a thickness up to 1.2 µm were grown on top of Ge/Si(001) substrates at 550°C under reduced pressure condition, using silane (SiH4) and GeH4 as reactant gas. Except for two later described cases, no annealing steps were applied during or after the Si0.06Ge0.94 layer growth process.
X-ray diffraction (XRD) measurements were performed using a Rigaku SmartLab diffractometer equipped with a 9 kW rotating Cu anode (Cu Kα = 0.1541 nm). In-plane and outof-plane lattice constants were determined analyzing the (004) and (224) Bragg reflections. The unstrained lattice constant was derived by assuming a tetragonal distortion of the lattice and a Poisson ratio corresponding to the nominal stoichiometry. Finally, the stoichiometry was determined by taking into account the deviation from Vegards law published by Dismukes et al. [26].
The surface morphology was studied by a Bruker Dimension Icon atomic force microscope (AFM), working in PeakForce Tapping mode while the structural quality was investigated by The theoretical investigation of plastic relaxation in RG-VS SiGe layers was performed by means of a DD approach [27]. All the numerical simulations reported in the following were carried out by using the microMEGAs (mM) [28] DD code. In the DD approach, the evolution of a given dislocation distribution is achieved by discretizing each dislocation line into small segments and evaluating the Peach-Kohler equation to obtain the driving force F acting on each of them: where is the Burgers vector of the considered segment, its line direction and is the local stress tensor acting at its position. The latter can be due to the self-interactions with other segments composing the same dislocation line, to the interactions with other dislocations and to external loads like the heteroepitaxial biaxial stress field. In mM, the stress field produced by the dislocation segments is evaluated by means of analytical expressions, as the ones reported by Hirth and Lothe in Ref. [29]. With this procedure, the evolution of the dislocation line can be modeled by the movement of all its composing segments following the driving force of equation (1) and eventually handling the possibility of local reactions between them. These topological reactions are evaluated whenever two segments during their motion cross each other and depending on the kind of the two interacting segments, the final results can be, for instance, the annihilation of the two segments or the formation of a junction between them.
The DD approach is applied here to study the relaxation of thin films and, thus, we exploited the capped-layer approach. As described by Schwarz [30], this approach permits to overcome the limitation of the classical DD approaches which rely on analytical expressions for the evaluation of the stress field produced by dislocation segments. Indeed, while the correct stress field of dislocations near free surfaces of a generic geometry can be obtained only by numerically tackling the partial differential equation of mechanical equilibrium [32,33], e.g. by means of the Finite Element Method, the capped layer approach can be conveniently exploited for the particular case of thin films with a flat free surface. As shown by several studies [33][34][35], the error introduced by this approximation, e.g. on the evaluation of the critical strain for TD movement, is less than 5%.
This level of uncertainty is comparable to that introduced by the uncertainty of other parameters used in the linear elasticity defect theory, e.g. the cut-off radius to include the core energy of dislocations in a continuum framework, and, thus, does not affect significantly the expected overall level of accuracy of a DD simulation.
In this work, we applied the DD approach to study the relaxation of the heterostructures presented in this section. This was achieved by setting the initial condition of all the simulations by introducing a random distribution of TDs with an areal density matching the TDD experimentally observed for all the considered thickness of the Ge buffer layer. No further dislocation nucleation was considered. Following the experimental results, the heteroepitaxial strain of the Ge buffer layer is considered to be, on average, fully relaxed by a network of misfit dislocation, not directly modeled in our approach. The TDs that are present in the buffer layer are propagated into the top SiGe layer and are then let to evolve under the influence of the misfit strain in the RG top layer.

III. RESULTS AND DISCUSSION
In order to understand the dislocation dynamics in our material system, we first investigate the evolution of the TDD as a function of the thickness of relaxed Ge layers.
In Fig. 1, we display the results of the TDD count obtained on samples featuring a Ge thickness up to 7.8 µm. Remarkably, all the investigated samples were crack-free, despite the large bowing, due to the thermal strain accumulated during the growth owing to the different coefficient of thermal expansion of the Ge epi-layer and the Si substrate [36,37]. Furthermore, we would like to mention here that, even for Ge thickness up to 7.8 µm, all grown Ge/Si(001) VSs exhibit a rms roughness value < 1 nm (  0.8 -0.3 nm ). As an exemplary case, we show in the inset of Fig. 1   By analyzing in detail Fig. 1, we note that, initially, the TDD decreases fast with increasing Ge thickness, showing an almost two-decade reduction for a Ge layer thickness of 2.5  3 µm. A further increase of the epi-layer thickness leads nearly to a saturation of the TDD which reaches a saturation value of 7×10 6 cm -2 .
The reduction of TDD for increasing thickness of a fully relaxed epi-layer is commonly reported in literature [2,20,25] and it is interpreted within the "geometrical model" [40]. As the system is fully relaxed, the extension of existing misfit segments is hindered and the only driving force for TD motion is the interaction with other TDs, possibly leading to their mutual annihilation when TDs featuring opposite Burger's vector are close enough. By increasing Ge layer thickness, since the dislocations glide on {111} planes, the probability of two TDs approaching each other is increased and, thus, their annihilation enhanced. However, the probability of an annihilation event is progressively reduced for decreasing TDD and the geometrical effect is gradually less important, thus explaining the observed saturation behavior.
As a subsequent step, we investigated how the plastic relaxation occurs and the TDD evolves This corresponds to a total thickness of the heterostructures up to 2.4 µm, i.e. close to the thickness at which we observed the onset of the TDD saturation in the case of Ge/Si(001) layers (see. Fig.1).
As described in Section II, the TDDs of all samples were measured using the etch pit count method based on the Secco etch. As an exemplary image, we show in Fig. 2(a) the SEM picture of the surface of the thickest sample of this series (1.2 µm Si0.06Ge0.94/1.2 µm Ge/Si(001) sample). To demonstrate that the rms roughness is not worsened with respect to a Ge/Si(001) VS of the same total thickness, we show, in Fig. 2(b), a (10 x 10) µm² AFM image of the 1.2 µm Si0.06Ge0.94/1.2 µm Ge/Si(001) sample. The corresponding rms roughness value measured on the image is 0.41 nm, matching well that reported in Fig. 1 for a 2.3 µm Ge/Si(001) sample (0.44 nm). We can then see that the introduction of a Si0.06Ge0.94 layer has very limited impact on the surface roughness.
In Fig. 2(c), we show the corresponding XRD reciprocal space mapping (RSM) around the (2 ̅ 2 ̅ 4) reflections of SiGe and Ge. We can clearly see that the Ge layer features a slight overrelaxation, with a residual tensile strain   2×10 -3 , corresponding to a relaxation value respect the Si substrate of R  105 %.  Fig. 2(e). It is worth noting that the post-annealing process did not affect, within the experimental error, the relaxation nor the TDD. Upon comparing these data with the Ge/Si sample having the same total thickness, we note that the Si0.06Ge0.94 layer relaxation reaches a saturation value R = 106 %, slightly larger than what was observed for the Ge/Si layers (R = 105%). While, in principle, the same R value should be expected for the two cases, since the coefficients of thermal expansion of Ge and Si0.06Ge0.94 are practically equal and the thermal budget used in the growth is also identical, the slightly larger value of R observed for the Si0.06Ge0.94/Ge/Si(001) sample can be attributed to the hardening effect, hindering the complete relaxation of the second epilayer [37]. Thus, we can conclude that a 1.2 µm Si0.06Ge0.94 layer added on top of a 1.2 µm Ge/Si(001) heterostructure results in a fully relaxed RG-VS, with no remaining driving force for TD motion. The resulting TDD is a factor 2× larger to that measured in the case of Ge/Si. Based on these findings, two questions arise: Why is this happening and, more generally, what is the impact of the TDs already developed in the Ge/Si layer for the relaxation process of the tensile strained Si0.06Ge0.94 /Ge layer?
In the following, we aim at addressing these puzzling questions. To this purpose, we realized a sample series by depositing identical 1.2 µm Si0.06Ge0.94 layers on Ge/Si VS featuring five different thicknesses and consequently, as shown in Fig. 1 and previously discussed, different TDDs. All the Si0.06Ge0.94 show the same value of the strain relaxation R= 106 % (not shown) and a decreasing TDD value, as depicted in Fig. 3 (blue dots).
Taking as a reference the preexisting TD population in Ge/Si(001) samples (semi-transparent red squares), we can observe that the drop of TDD with the total heterostructures thickness ttot = t(Si0.06Ge0.94) + t(Ge) is more rapid for the Si0.06Ge0.94/Ge/Si(001) than for the Ge/Si(001) VS. As such, for ttot < 3 µm, the Si0.06Ge0.94/Ge/Si(001) samples have a larger TDD than Ge/Si samples. The opposite holds for the region ttot >3 µm, where we observed the saturation by the geometrical effect. We point out here that the lowest TDD achieved reaches the remarkable value of 1.7×10 6 cm -2 , for ttot 7 µm, being a factor 4× lower than the corresponding value for a Ge/Si VS of similar thickness. Importantly, by growing an additional sample featuring another 1.2 µm 100% Ge film on top of such heterostructure and inserting in this way another heterointerface, the TDD is found to be 2.5×10 6 cm -2 . This value is in-line with the one obtained in the Si0.06Ge0.94/Ge/Si(001) heterostructures and thus still well below the saturation limit for the geometrical effect in Ge/Si(001) (green diamond in Fig. 3). Therefore, this approach represents a viable path to obtain fully relaxed Ge layers featuring a lower TDD at equal total thickness.
Having previously shown that a 1.2 µm-thick SiGe/Ge/Si layer is stable against any reduction of the TDD during annealing cycles (see Fig. 2(d)), this result clearly points to a pivotal role of the Si0.06Ge0.94/Ge heterointerface on the TDD evolution.
To shed light on this behavior, we performed some dedicated DD simulations and an additional cross-section TEM characterization.
The effect produced by the introduction of a RG layer at high heterostructure thickness can be appreciated by looking at the simulation results reported in Fig. 4. Here, the simulation parameters were set in order to match the samples analyzed in Fig. 3. In particular, a 4.5 µm Ge buffer layer was considered with a 1.2 µm top layer consisting of pure Ge [ Fig. 4(a)] or Si0.06Ge0.94 [ Fig. 4(b)].
As explained in Section II, our simulation approach consists in setting the initial areal density of TDs in order to match the values reported in Fig. 3 for the corresponding Ge buffer thickness (TDD of about 8×10 6 cm -2 for the case under consideration). In Fig. 4(a), one can observe how these TDs evolve only marginally over time, with no annihilation events taking place due to the low probability of TDs encountering at this low TDD value. This is precisely what is expected to occur once the saturation limit, clearly visible in Fig.1, is reached and the geometric effect ceases to be effective [40]. On the contrary, the addition of a strained SiGe RG layer as in Fig. 4(b) provides additional mobility to the TDs, bending them at the new interface in order to release the excess strain in the top layer and enhancing the probability of annihilation. We actually verified that, during the evolution depicted in Fig. 4(b), bending of pre-existing threading arms results in nearly full strain relaxation of the upper layer. With this respect, there is no need to invoke further nucleation of dislocations to explain the experimentally observed relaxation of the Si0.06Ge0.94 layer. The resulting TDD evolution for the two simulated cases is reported in Fig. 4(c). Based on the insights on TDs behavior we obtained, we can propose the following scenario.
At high heterostructure thickness (> 3 µm), when the "geometric" process has reached its saturation, so that increasing the thickness of the Ge buffer layer produces a negligible probability of interaction between the remaining TDs [40], the deposition of a top layer of different composition and lattice parameter (like Si0.06Ge0.94) can provide an alternative mechanism to TDD reduction. Indeed, the additional strained layer needs to be plastically relaxed. To do so, instead of nucleating new dislocations surmounting the associated kinetic barrier, it is easier to exploit preexisting TDs. This requires "bending" them at the additional heterointerface to develop a new network of misfit segments [41]. This re-established dislocation mobility can favor again the annihilation of TDs and, consequently, be responsible of the observed decrease of TDD. With this respect, the RG layer effectively acts like a "filtering layer" [42]. On the other thickness limit (< 3 µm), the introduction of the Si0.06Ge0.94/Ge RG interface is found to be detrimental for the action of the geometric effect, because bending the TDs effectively lowers the layer thickness where they can encounter each other and annihilate.
The above described bending mechanism can be better appreciated by looking at Fig. 5, where we display an additional TEM analysis based on lamellas taken out of two selected heterostructure samples, consisting of 0.5 µm Ge (Fig. 5(a)) and 0.5 µm Si0.06Ge0.94 (Fig. 5(b)) on 1.2 µm Ge/Si(001) VS. From the cross-section STEM images, it can be seen that, for the same total thickness of the layers, the pure-Ge sample exhibits TDs running through the whole heterostructure, reaching also the upper layer ( Fig. 5(a)). On the contrary, for the sample featuring the Si0.06Ge0.94/Ge RG heterointerface displayed in Fig. 5(b), an increased contrast of the heterointerface can be observed, due to the higher MD density required for accommodating the lattice mismatch during the plastic relaxation of the Si0.06Ge0.94 layer. Two exemplary STEM images of the heterointerface are seen in Figs. 5(c) and 5(d). In Fig. 5(c), a TD arm along the heterointerface is bending into the Ge buffer layer (red arrow mark) after encountering a MD. In Fig. 5(d), a TD arm coming from the Ge layer is "captured" at the heterointerface, where we observe the presence of the MDs needed to plastically relax the SiGe top layer. We point out that that some of these MDs may originate from the bending of other TDs coming from the Ge layer.
In Figs. 5(e) and 5 (f), we show simulated TEM images obtained by properly sectioning the simulation cells at the final stage of DD simulations. These were performed in order to reproduce the samples analyzed in the TEM images of Fig 5(a) and (b) with, respectively, 0.5 µm-thick Ge and Si0.06Ge0.94 on top of 1.2 µm-thick Ge/Si(001). In the pure-Ge case (Fig. 5(f)), the simulation shows TDs running unhindered through the whole heterostructure, also reaching the upper Ge layer. On the contrary, Figure 5(e) shows a totally different scenario, with the formation of a second dislocation network at the new Ge/SiGe interface due to the bending of TDs, relaxing the strain of the top layer. It is safe to state that the DD simulated TEM images match well the experimental TEM images.

IV. CONCLUSIONS
In this paper, we have proposed and quantitatively analyzed a strategy to lower the TDD in both tensile films grown on Ge/Si(001) and Ge films grown on Si(001). After having established a saturation limit for the TDD in Ge/Si(001) films grown by our RPCVD reactor ( 7×10 6 cm -2 for our growth procedure), we explored the possibility to achieve a further reduction of TDD by Si0.06Ge0.94/Ge/Si(001) RG-VS. This proved to be possible only when the TDD in the Ge/Si(001) film underneath was sufficiently low. Under these conditions, dislocation annihilation by the simple geometrical effect ceases to take place, while bending and gliding of threading arms at the Si0.06Ge0.94/Ge interface promote further interaction and annihilation, as directly shown by DD simulations. By proper tuning of the growth parameters, we were able to reach a very low TDD value of  1.7×10 6 cm -2 . Very interestingly, when a low TDD is reached in Si0.06Ge0.94/Ge/Si(001) RG-VS, an overgrown Ge film displays a TDD being much lower than the Ge/Si(001) saturation limit. We therefore believe that the present methodology can be extremely useful in achieving low TDDs also in 100% Ge films, without the need to use neither continuous grading nor etching-back procedures.