Three-dimensional SiGe/Si heterostructures: Switching the dislocation sign by substrate under-etching

We present a joint theoretical and experimental analysis of the dislocation distribution in graded epitaxial SiGe crystals grown on under-etched Si pillars by low-energy plasma-enhanced chemical vapor deposition. Dislocation dynamics simulations are used to investigate preferential positioning of ${60}^{\circ }$ dislocations introduced in the system to release the lattice misfit strain. Coupling to a finite-element solver is exploited to allow for the exact numerical treatment of the stress fields in the presence of a complex distribution of free surfaces. The results show that, by suitably under-etching the Si pillars, it is possible to reverse the sign of the Burgers vector of the dislocations. This helps explaining differences in the experimentally observed distribution of dislocations in SiGe crystals grown on vertical and under-etched pillars, leading to a strong reduction of defects in the latter case. The agreement between simulations and experiments is not simply qualitative: the predicted number of defects generated by multiplication processes in tall crystals is indeed fully consistent with the measured one.

Figure 1

Perspective SEM view of a Si pillars array with vertical $\left\{110\right\}$ sidewalls. (b) Cross-sectional SEM image of one Si pillar with vertical $\left\{110\right\}$ sidewalls, width $w=5\mu \mathrm{m}$, and patterned height $h=7\mu \mathrm{m}$. The scallops on the pillar sidewalls are typical for the Bosch etching process. (c), (d) Perspective view SEM images of under-etched Si pillars with $w=7$ and $15\mu \mathrm{m}$, respectively.

Figure 2

(a), (b) Cross-sectional and top-view SEM images of SiGe crystals deposited on vertical Si pillars with $w=2$ and $25\mu \mathrm{m}$, respectively. (c) The same of (a) but for under-etched Si pillars with $w=7\mu \mathrm{m}$. Here, the two light blue lines indicate the extreme values of the SiGe crystal width taken at the SiGe/Si interface ($L_{\mathrm{m}}$) and at the crystal top ($L_{\mathrm{M}}$). (d) Cross-sectional SEM image after selective defect etching of a SiGe/Si crystal. Emerging dislocations, both in the SiGe crystal and in the Si pillar, appear as whitish pits. The dashed light blue line marks the SiGe/Si heterointerface.

Figure 3

(a) Average dislocation density in SiGe crystals (${\mathrm{DD}}_{\mathrm{SiGe}}$) deposited on vertical (black spheres) and under-etched (red triangles) Si pillars with different widths. (b) Probability of having dislocation-free SiGe crystals (${\mathrm{DFP}}_{\mathrm{SiGe}}$) as a function of the vertical (black spheres) and under-etched (red triangles) Si pillars width. (c), (d) Analogous to (a) and (b), respectively, but for dislocations located in the Si pillars.

Figure 4

(a), (b) Cross-sectional SEM images of SiGe crystals deposited on vertical and under-etched Si pillars, respectively, after selective defect etching (width $w=5$ and $7\mu \mathrm{m}$), respectively. The dashed light blue line shows the SiGe/Si heterointerface. The white arrow indicates a dislocation in the SiGe crystal. The red dashed ellipses in (b) highlight dislocations piled up along $\left\{111\right\}$ in the Si pillars. (c) Probability distribution of having a certain number of dislocations per pileup along $\left\{111\right\}$ planes in under-etched Si pillars ($w=7\mu \mathrm{m}$).

Figure 5

Hydrostatic stress maps (${\sigma }_{xyz}$) and considered geometry for the vertical (a) and under-etched (b), (c) Si pillars. The SiGe/Si interface is marked with a black line. The two considered pillar bases $L_{\text{m}}$ (b) and $L_{\text{M}}$ (c) for the under-etched pillar are taken to mimic the extreme values of bases measured on the tapered geometry of the grown pillar as reported in Fig. 2. In (d) is reported the stress map for an under-etched pillar with the first dislocation introduced. The filled black circle reports its optimal position along the interface while the black arrow represents the result of the energy minimization by means of a dislocation dynamics simulation.

Figure 6

Energy gain for the introduction of the first dislocations in an under-etched pillar with base $L$ with respect to the pillar height $H$. The formation energy is negative only for dislocations with the opposite Burgers vector $b$.

Figure 7

Plot of the energy gain versus the number of dislocations per pileup in an under-etched pillar. The optimal number of dislocations can be deduced to be four by the minimum in this curve because the introduction of a further one will increase the total elastic energy.

Figure 8

Comparison between experimental SEM images and hydrostatic stress maps from the final stages of the dislocations dynamics simulations for an under-etched pillar [(a) and (b)] and a vertical one [(c) and (d)]. Experimental images in (a) and (c) are the same as in Fig. 4 and reported here for convenience. As we can see, the simulation can well reproduce the behavior shown in experiments with dislocations in vertical pillars floating in the SiGe region, while those in the under-etched ones are pushed into the silicon pillar underneath.