# Dislocation-Free SiGe/Si Heterostructures

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

**2012**, 335, 1330–1334), immediately capturing attention due to the appealing possibility of growing micron-sized Ge crystals largely free of thermal stress and hosting dislocations only in a small fraction of their volume. Since then, considerable progress has been made in terms of extending the technique to several other systems, and of developing further strategies to lower the dislocation density. In this review, we shall mainly focus on the latter aspect, discussing in detail 100% dislocation-free, micron-sized vertical heterostructures obtained by exploiting compositional grading in the epitaxial crystals. Furthermore, we shall also analyze the role played by the shape of the pre-patterned substrate in directly influencing the dislocation distribution.

## 1. Introduction

^{7}/cm

^{2}. In order to lower the defect density by a further order of magnitude different approaches must be used. A key one was introduced by Fitzgerald and coworkers. They demonstrated [4,5] the possibility to lower typical TDDs in Ge/Si films down to only ≈10

^{6}/cm

^{2}by growing “graded layers”, i.e., by actually depositing Si

_{1−x}Ge

_{x}alloys with x gradually increasing during deposition. Grading allows for two main advantages with respect to constant-composition films: (a) threading arms are always subject to a nonzero gliding driving force [6] (b) the character of a dislocation can change during growth, threading segments bending and allowing for further strain relaxation without the need for nucleating new dislocations. While nowadays graded layers are still the main route to produce substrates with low dislocation densities, it is worth to emphasize that recent attempts have shown that 10

^{6}/cm

^{2}density values can be reached also by direct deposition at the desired final composition, provided that a suitable annealing and “etch back” procedure is exploited [7].

## 2. Vertical Heterostructures with Constant Composition

#### 2.1. Vertical Growth of Ge/Si by LEPECVD

^{2}Si pillars, separated by 2 and 4 μm trenches. Ge was subsequently deposited by Low-Energy Plasma-Enhanced Chemical Vapour Deposition (LEPECVD). LEPECVD [14] allows for the grow of crystalline Ge under strong out-of-equilibrium conditions, determined by the high deposition rate (≈4 nm/s for both structures in Figure 1) and by the low deposition temperature (440 °C in Figure 1a; 490 °C in Figure 1c). Under such conditions, typical diffusion lengths are much smaller than the micrometric pillar sizes, so that the Ge crystal has a tendency to grow vertically [11] from the very beginning. Some lateral enlargement also takes place, leading to a progressive shrinking of the lateral distance between crystals growing on adjacent pillars. This causes a strong self-shielding effect ultimately leading to (almost) perfect vertical growth. Once vertical growth is established, the crystals can be grown for several dozens of microns without ever touching, separated by a very small gap (Figure 1b). The vertical morphology offers two key advantages with respect to common 2D layers. On one hand, the free surface surrounding the crystals allows for very efficient relaxation of the thermal-stress field [15,16,17,18], therefore avoiding cracking. On the other hand, 60° dislocations forming at the Ge/Si interface and laying on (111) planes, are confined to the bottom of the crystal only (no 60° defect can reach the region located at a height h >1.4 b, where b is the Si pillar base width). While 60° dislocations are the dominant linear defects in Ge/Si systems grown under typical conditions, the LEPECVD out-of-equilibrium conditions lead to the formation of perfectly vertical defects [19,20] which can thread through the whole pillar, reaching the topmost surface as shown in Figure 2a,b. However, also these dislocations can be expelled laterally, provided that the top facet of the pillar is not parallel to the (001) substrate during growth. This can be easily achieved by raising the growth temperature [12], as shown in Figure 2c. Fortunately, indeed, linear defects tend to follow the growth front, so that the problem of vertical dislocations can be easily solved by properly tuning the growth conditions eliminating the top (001) facet.

_{x}Si

_{1−x}alloys of various Ge-content x. Actually, growth at low x is even more convenient in terms of lateral expulsion of defects, as the aforementioned vertical dislocations are not formed [20]. This is explicitly shown in Figure 3, where the etch-pit distribution clearly shows the lateral expulsion of defects.

#### 2.2. Other Deposition Techniques and Other Materials

^{2}Si pillar are reported [21]. As the typical growth conditions in RPCVD are very different from LEPECVD in both terms of flux (lower) and temperature (higher), longer diffusion lengths make vertical growth more difficult. To limit horizontal growth and material diffusing from the top regions to the pillar bottom, growth was therefore performed on pillars whose lateral walls were oxidized.

#### 2.3. Suspended Films

## 3. Graded, Vertical Heterostructures

#### 3.1. Elastic vs. Plastic Relaxation: Theory

_{1−x}Ge

_{x}/Si VHEs by means of a quasi-3D approach allowing for detailed estimation of thermodynamic plasticity onsets. This method exploits isotropic linear elasticity theory and Finite Element Method (FEM) calculations (see also [36]) while VHEs are modeled by simplified geometries as in Figure 7a. However, notice that it can be straightforwardly used to investigate realistic shapes as shown in Refs. [37,38,39].

_{1−x}Ge

_{x}/Si interface, respectively. The resulting elastic field is self-similar, i.e., its qualitative features depend on the height-to-base aspect-ratio $\mathrm{R}=\mathrm{h}/\mathrm{B}$ and it can be adapted to any specific size upon proper rescaling. Figure 7c–e shows the different distributions of ${\mathsf{\sigma}}_{xyz}$ in the central slice of a VHE for $\mathrm{R}=0.1$, $\mathrm{R}=0.5$, and $\mathrm{R}=1.0$, respectively.

_{1−x}Ge

_{x}/Si systems with respect to other dislocation types [40]. In particular, the stress field of the coherent system is evaluated in the 3D structure, as in Figure 7b, and it is extracted in the central slice to compute ${E}_{\mathrm{coh}}$. Then, the same elastic field is superimposed to the one of a straight (perpendicular to the 2D slice) 60° dislocation lying at the center of the Si

_{1−x}Ge

_{x}/Si interface, as in Figure 7f, to compute ${E}_{\mathrm{dislo}}$. When $\mathsf{\Delta}\mathrm{E}<0$, plastic relaxation is energetically favored, and dislocations are expected. The central position for the dislocation is the minimum energy configuration for relatively small and large values of $\mathrm{R}$. For intermediate aspect-ratios, the minimum energy configuration may be shifted towards the sidewalls. However, the central position gives a very good approximation of the global energy minimum. Notice that computing $\mathsf{\Delta}\mathrm{E}$ in the central slice corresponds to evaluate an energy per unit length of the dislocation misfit segment. Therefore, this approach describes the tendency of a dislocation to elongate, resembling the classical method used to evaluate the critical thickness for the insertion of dislocation, ${\mathrm{h}}_{\mathrm{c}}$, in planar structures [5,41]. Further details about the method, its assessment, and the simulation setup can be found in Refs. [35,36].

_{1−x}Ge

_{x}epilayer, $\mathrm{x}$, and the lateral size of the VHE, $\mathrm{B}$. Moreover, it is possible to determine the critical lateral size as a function of $\mathrm{x}$, namely ${\mathrm{B}}_{\mathrm{c}}(\mathrm{x})$. These quantities are shown in Figure 7: ${\mathrm{h}}_{\mathrm{c}}(\mathrm{B},\mathrm{x})$ is illustrated by means of dashed black isolines, while ${\mathrm{B}}_{\mathrm{c}}(\mathrm{x})$ corresponds to the solid red line. Above the red curve, $\mathsf{\Delta}\mathrm{E}>0$ for thicknesses smaller than ${\mathrm{h}}_{\mathrm{c}}$, at which plasticity is expected to set in. Below the red curve, $\mathsf{\Delta}\mathrm{E}>0$ for any thickness of the epilayer, i.e., the corresponding VHE is predicted to be always coherent.

_{1−x}Ge

_{x}epilayer on Si with Ge content ${\mathrm{x}}_{\mathrm{epi}}$. With the configuration of VHEs as discussed so far, this can be achieved only up to a lateral size of $\sim {\mathrm{B}}_{\mathrm{c}}({\mathrm{x}}_{\mathrm{epi}}),$ that is less than $40$ nm for a pure-Ge epilayer. However, looking at the elastic field of the VHEs in Figure 7, one can easily notice that for a thickness $\mathrm{h}\sim \mathrm{B}$, i.e., for an aspect ratio $\mathrm{R}\sim 1$, the top of the epilayer is fully relaxed. Moreover, this is a purely geometric effect that is not dependent on $\mathrm{B}$ or $\mathrm{x}$ [34]. So that, even for basis larger than ${\mathrm{B}}_{\mathrm{c}}({\mathrm{x}}_{\mathrm{epi}})$, one can think about growing a first coherent layer with ${\mathrm{x}}_{1}\le {\mathrm{x}}_{\mathrm{c}}(\mathrm{B})$ up to $\mathrm{h}\sim \mathrm{B}$. Then, as far as the top of the structure is relaxed, a second layer can be considered, and the vertical structure is predicted to be coherent provided that the increase in the Ge content with respect to the first layer is lower than ${\mathrm{x}}_{\mathrm{c}}(\mathrm{B})$, i.e., the second layer can have a Ge content ${\mathrm{x}}_{2}$ such as ${\mathrm{x}}_{2}-{\mathrm{x}}_{1}\le {\mathrm{x}}_{\mathrm{c}}(\mathrm{B})$. So that a coherent structure with a Ge content up to $2{\mathrm{x}}_{\mathrm{c}}(\mathrm{B})$ can be grown. If this Ge content is smaller then ${\mathrm{x}}_{\mathrm{epi}}$, further layers can be considered exploiting the same idea up to the desired target. The discrete number of layers $\mathrm{n}(\mathrm{B})$ required to reach ${\mathrm{x}}_{\mathrm{epi}}$ for a given B is then given by the simple relation $\mathrm{n}(\mathrm{B})={\mathrm{x}}_{\mathrm{epi}}/{\mathrm{x}}_{\mathrm{c}}(\mathrm{B})$.

#### 3.2. Dislocation-Free Graded Heterostructures

## 4. Deposition on Under-Etched Pillars

#### 4.1. Graded Heterostructures on Under-Etched Pillars: Experimental Results

#### 4.2. Graded Heterostructures on Under-Etched Pillars: Theoretical Interpretation

_{m}) and largest (L

_{M}) dimensions. Figure 12 shows that necking changes the local misfit relaxation both in Si and in SiGe, introducing a strongly compressed region (blue color in the maps) within Si. This has a profound influence on the nucleation and distribution of dislocations.

**b**

_{1},

**b**

_{2},

**b**

_{3}, and

**b**

_{4}in Figure 12e. Two of them (

**b**

_{1}and

**b**

_{2}) provide expansion of the region above the core, the others have the opposite effect. These are the ones more often encountered in SiGe/Si planar films or in vertical pillars, as the tensile strain introduced by the dislocations reduces the lattice compression due to lattice-parameter misfit. The presence of the strong compressive stress in under-etched Si pillars (Figure 12b–d), instead, reverses the sign of the lowest-energy defects. This is shown in Figure 13: insertion of the same dislocations relaxing SiGe/Si films or SiGe on vertical pillars (“normal” case in Figure 13) raises the energy of the system for both explored sizes. On the contrary, the introduction of dislocations with opposite sign of the in-plane component of the Burgers vector (

**b**

_{3}, or

**b**

_{4}, helping to relax compression in the Si region) becomes energetically favored beyond a critical height. The difference in energy between the system with and without a dislocation located at its minimum-energy position (∆E

_{min}), indeed, becomes negative. In Ref. [46] it was shown by dislocation dynamics simulations that accounting for the change in Burgers vector orientation is fundamental in order to explain the typical dislocation distributions experimentally observed in SiGe crystals grown on underetched Si pillars. The typical pile-up in the Si region, well evident in Figure 11e, is indeed compatible only with dislocations removing half atomic plane in the Si region (Burgers vector

**b**

_{3}).

## 5. Conclusions and Perspectives

## 6. Patents

## Funding

## Conflicts of Interest

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**Figure 1.**Self-aligned vertical growth of Ge micro-crystals on 2 × 2 µm

^{2}Si pillars, 8 µm deep. (

**a**) Dark-field STEM view of 7 µm tall Ge crystals grown at 440 °C on pillars spaced by 2 µm. (

**b**) Plot of the distance between the adjacent crystals in panel a. (

**c**). SEM lateral view of 50 µm tall Ge crystals grown at 490 °C on pillars spaced by 4 µm. Reproduced with permission from [12].

**Figure 2.**Dislocations in Ge micro-crystals on Si pillars. (

**a**) Bright-field TEM cross-section in the Ge[220] Bragg condition showing both 60° and vertical dislocations. AFM view of the Ge crystal top after defect etching in iodine solution for (

**b**) a (001) flat-top morphology obtained by growing at 440 °C and (

**c**) a {113} pyramidal shape obtained at 560 °C. The schematics illustrate the different propagation of dislocation lines with respect to the faceting of the growth front. Reproduced with permission from [12].

**Figure 3.**Lateral expulsion of dislocations. SEM image showing confinement of etch-pits in the bottom pillar region for a Ge

_{0.2}Si

_{0.8}/Si(001) crystal. Reproduced with permission from [20]. American Institute of Physics.

**Figure 4.**Growth sequence of a Ge micro-crystal on a 2 × 2 µm

^{2}Si pillar by RPCVD. (

**a**) Lateral and perspective SEM views of samples after deposition of different duration. Multiple crystal seeds are observed at the early stages, coalescing into a single faceted structure later on. (

**b**) Profiles obtained by a phase-field simulation of crystal growth, matching the experimental behavior. Reproduced with permission from [21]. American Chemical Society.

**Figure 5.**Deposition of several different materials on Si pillars. (

**a**) SiC/Si, deposited by hot-wall Chemical Vapour Deposition. Reproduced with permission from [24]. Electrochemical Society (

**b**) GaN/Si, deposited by plasma-assisted molecular beam epitaxy. Reproduced with permission from [25]. American Chemical Society (

**c**) GaAs/Si, deposited by molecular beam epitaxy. Reproduced with permission from [26]. Copyright 2013, the American Institute of Physics (

**d**) GaAs/Ge/Si, where Ge was deposited by LEPECVD and GaAs by metal-organic vapour-phase epitaxy. Reproduced with permission from [28]. American Institute of Physics.

**Figure 6.**Coalescence of vertically-aligned Ge micro-crystals into a suspended film. (

**a**) Top and (

**b**) lateral SEM images of the as-grown crystals and of identical samples after annealing of different duration. (

**c**) Evolution sequence obtained by a phase-field simulation of surface diffusion. First, local rounding of the facets occurs; then, connection bridges form between neighboring crystals leaving holes, that are finally filled by material flow. Reproduced with permission from [30]. American Chemical Society.

**Figure 7.**Theoretical modeling of VHEs and stress fields. (

**a**) Simplified geometry of VHEs used to perform the theoretical analysis. (

**b**) Illustrative map of the hydrostatic stress ${\mathsf{\sigma}}_{xyz}$ in a VHE as in panel (

**a**) with $\mathrm{x}=1$. (

**c**–

**e**) ${\mathsf{\sigma}}_{xyz}$ in the central slice of a coherent 3D VHE as in panel (

**a**,

**b**) with an aspect-ratio $\mathrm{R}=\mathrm{h}/\mathrm{B}$ equal to 0.1, 0.5 and 1.0 respectively. (

**f**) ${\mathsf{\sigma}}_{xyz}$ as in panel (

**e**) superposed to the hydrostatic stress field induced by a dislocation lying at the interface ($B=25$ nm). Reproduced with permission from [35]. American Institute of Physics.

**Figure 8.**Model results. Critical pillar base ${\mathrm{B}}_{\mathrm{c}}$ as a function of the Ge content x (solid red curve) and critical thickness of Si

_{1−x}Ge

_{x}/Si VHEs as function of B and x (dashed black isolines, numbers correspond to ${\mathrm{h}}_{\mathrm{c}}$ expressed in nm). Reproduced with permission from [35]. American Institute of Physics.

**Figure 11.**Experimental characterization of dislocations in SiGe crystals. (

**a**) Average dislocation density in SiGe crystals (DDSiGe) deposited on vertical (black spheres) and under-etched (red triangles) Si pillars with different widths. (

**b**) Probability of having dislocation-free SiGe crystals (DFPSiGe) as a function of the vertical (black spheres) and under-etched (red triangles) Si pillars width. (

**c**,

**d**) Analogous to (

**a**,

**b**), respectively, but for dislocations located in the Si pillars. The strong tendency towards dislocations piling-up in the Si region is well evident in the SEM image reported in panel (

**e**). Reproduced with permission from [46]. American Physical Society.

**Figure 12.**Model results. Hydrostatic stress maps (σxyz) and considered geometry for the vertical (

**a**) and under-etched (

**b**), (

**c**) Si pillars. The SiGe/Si interface is marked with a black line. Two considered pillar bases L

_{m}(

**b**) and L

_{M}(

**c**) are taken to mimic the extreme values of bases measured on the tapered geometry of the grown pillar (Figure 9c). In (

**d**) is reported the stress map for an under-etched pillar once the first dislocation is introduced. In panel (

**e**) a schematic representation of the four possible Burgers vectors is reported along with their in-plane projection. Reproduced with permission from [46]. American Physical Society.

**Figure 13.**Model results. Energy gain for the introduction of the first ‘normal’ (red-dashed curve) or ‘opposite’ (purple solid curve) dislocations in under-etched pillars with base L with respect to the pillar height H. The formation energy is negative only for dislocations with the ‘opposite’ Burgers vector

**b**. Reproduced with permission from [46]. American Physical Society.

**Table 1.**Critical parameters and design of fully-coherent VHEs as function of $\mathrm{B}$. The values of critical Ge content ${\mathrm{x}}_{\mathrm{c}}(\mathrm{B})$, grading rate $\mathrm{r}(\mathrm{B})$, number of layers $\mathrm{n}(\mathrm{B})$ to achieve ${\mathrm{x}}_{\mathrm{epi}}=1$ and corresponding total thickness $\mathrm{t}(\mathrm{B})$ under the assumption of $\mathrm{h}\sim \mathrm{B}$ are reported.

$\mathit{B}$ | ${\mathit{x}}_{\mathit{c}}(\mathit{B})$ | $\mathit{r}(\mathit{B})$ | $\mathit{n}(\mathit{B})$ | $\mathit{t}(\mathit{B})$ |
---|---|---|---|---|

150 nm | 0.360 | 0.23%/nm | 2 | 300 nm |

200 nm | 0.291 | 0.45%/nm | 3 | 600 nm |

300 nm | 0.210 | 73.37%/μm | 4 | 1.2 μm |

500 nm | 0.161 | 32.27%/μm | 6 | 3.0 μm |

1.0 μm | 0.116 | 11.62%/μm | 8 | 8.0 μm |

1.5 μm | 0.101 | 6.73%/μm | 9 | 13.5 μm |

2.0 μm | 0.093 | 4.66%/μm | 10 | 20.0 μm |

3.0 μm | 0.085 | 2.85%/μm | 11 | 33.0 μm |

5.0 μm | 0.079 | 1.59%/μm | 12 | 60.0 μm |

7.5 μm | 0.076 | 1.01%/μm | 13 | 97.5 μm |

10.0 μm | 0.075 | 0.75%/μm | 13 | 130.0 μm |

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## Share and Cite

**MDPI and ACS Style**

Montalenti, F.; Rovaris, F.; Bergamaschini, R.; Miglio, L.; Salvalaglio, M.; Isella, G.; Isa, F.; Von Känel, H. Dislocation-Free SiGe/Si Heterostructures. *Crystals* **2018**, *8*, 257.
https://doi.org/10.3390/cryst8060257

**AMA Style**

Montalenti F, Rovaris F, Bergamaschini R, Miglio L, Salvalaglio M, Isella G, Isa F, Von Känel H. Dislocation-Free SiGe/Si Heterostructures. *Crystals*. 2018; 8(6):257.
https://doi.org/10.3390/cryst8060257

**Chicago/Turabian Style**

Montalenti, Francesco, Fabrizio Rovaris, Roberto Bergamaschini, Leo Miglio, Marco Salvalaglio, Giovanni Isella, Fabio Isa, and Hans Von Känel. 2018. "Dislocation-Free SiGe/Si Heterostructures" *Crystals* 8, no. 6: 257.
https://doi.org/10.3390/cryst8060257