Dendritic Growth in Si1−xGex Melts

We investigated the types of dendrites grown in Si1−xGex (0 < x < 1) melts, and also investigated the initiation of dendrite growth during unidirectional growth of Si1-xGex alloys. Si1−xGex (0 < x < 1) is a semiconductor alloy with a completely miscible-type binary phase diagram. Therefore, Si1−xGex alloys are promising for use as epitaxial substrates for electronic devices owing to the fact that their band gap and lattice constant can be tuned by selecting the proper composition, and also for thermoelectric applications at elevated temperatures. On the other hand, regarding the fundamentals of solidification, some phenomena during the solidification process have not been clarified completely. Dendrite growth is a well-known phenomenon, which appears during the solidification processes of various materials. However, the details of dendrite growth in Si1−xGex (0 < x < 1) melts have not yet been reported. We attempted to observe dendritic growth in Si1−xGex (0 < x < 1) melts over a wide range of composition by an in situ observation technique. It was found that twin-related dendrites appear in Si1−xGex (0 < x < 1) melts. It was also found that faceted dendrites can be grown in directional solidification before instability of the crystal/melt interface occurs, when a growing crystal contains parallel twin boundaries.


Introduction
Dendrite crystals appear in the solidification processes of various materials including metals [1][2][3], metallic alloys [4][5][6][7][8][9][10], semiconductors [11][12][13][14][15][16][17][18][19], organic materials [20][21][22][23][24][25], and so on. It has been shown that the growth shapes and growth mechanisms of dendrites are different between metallic alloys [26,27] and semiconductors [13,18]. Although the solute and thermal diffusion in the liquid are key in determining the growth shape and growth velocity of dendrites in metallic alloys, the kinetics of atomic attachment at the reentrant corner formed at the front of dendrites due to the existence of twin boundaries determines the growth shape and growth velocity of dendrites in semiconductors. The typical shape of alloy dendrites is schematically shown in Figure 1a. Generally, a dendrite grows in a preferential direction to form a primary arm, and secondary arms extend from the side of the primary arm. On the other hand, the growth shape of the dendrites of semiconductor materials, such as Si and Ge, are different from that of metal alloy dendrites, as shown schematically in Figure 1b. At least two parallel twin boundaries exist at the center of the dendrites, the surface of the dendrites is bounded by facet planes, and the preferential growth direction is <112> or <110>, which is related to the existence of parallel twin boundaries [13,18]. This type of dendrite is called a "faceted dendrite" or "twin-related faceted dendrite". Nagashio et al. showed that Si dendrites can grow preferentially in the <100> direction (<100> dendrite) in a highly undercooled melt when the extent of undercooling (∆T) is more than 100 K [17]. The growth shape of <100> dendrites of Si is similar to that of alloy dendrites, that is, the <100> dendrite does not contain any twin boundaries (twin-free dendrite). To realize such a highly undercooled melt, a container-less levitation method was used [17]. In a normal solidification process using a container (crucible), it is difficult to maintain the melt in a highly undercooled state. Therefore, only faceted dendrites appear in the lower undercooling region, such as ∆T 10 K [28], in the solidification process of pure Si in a crucible. Recently, we directly observed the growth shape of dendrites of GaSb, which is a compound semiconductor with the zinc blende structure [19]. The growth scheme of GaSb dendrites could be explained by the Si faceted dendrite model. Thus, it has been shown that the types of dendrites depend on the material. Si 1−x Ge x (0 < x < 1) is a semiconductor alloy with a completely miscible-type binary phase diagram. Si 1−x Ge x alloys are promising for use as epitaxial substrates of electronic devices owing to the fact that their band gap and lattice constant can be tuned by selecting the proper composition [29][30][31][32][33][34][35], and thus, bulk Si 1-x Ge x single crystals have been obtained from the melt by several methods [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. Si-rich Si 1-x Ge x alloys (0.2 x 0.3) are promising for thermoelectric applications at elevated temperatures [54][55][56][57][58][59][60], and have also been considered for use in solar cells [61,62]. On the other hand, knowledge of the fundamentals of solidification for Si 1−x Ge x alloys, including dendrite growth, is still limited. Miller et al. investigated cellular growth of Ge-rich Si 1−x Ge x single crystals by a phasefield simulation [63]. We reported on the instability of the crystal/melt interface during directional solidification [64][65][66] and re-melting phenomena during rapid solidification processes [67] in Si-rich Si 1−x Ge x . Herlach et al. reported on the crystallization process from highly undercooled Si-Ge melts using a container-less levitation method [68]. They focused on the transition of the growth mode as a function of undercooling and reported that the critical undercooling, where the interface shape changed from faceted to dendritic and dendritic to planar, occurred in a highly undercooled region (45K < ∆T < 315K). However, details on the dendrite growth of Si 1−x Ge x (0 < x < 1) alloys over a wide range of composition have not yet been reported. In particular, we are interested in the type of dendrites that are formed in Si 1−x Ge x (0 < x < 1) alloys. Si 1−x Ge x can form a solid solution over a wide range, and thus there is the possibility to grow metallic alloy-type dendrites (twin-free dendrites), which were observed in pure Si in highly undercooled melts [17]. On the other hand, Si 1−x Ge x is also a faceted material like pure Si, pure Ge, and GaSb, and thus faceted dendrites (twin-related dendrites) could be grown.
Based on this fundamental interest, in this study we attempted to observe dendritic growth in Si 1−x Ge x (0 < x < 1) melts over a wide range of composition using an in-situ observation technique. Also, it is crucial to maintain a planar crystal/melt interface during a directional solidification for the growth of bulk Si 1−x Ge x ingots. Therefore, we performed directional solidification experiments to investigate when dendrite growth appears at the crystal/melt interface during directional solidification.

Experimental Procedures
An in situ observation system composed of a furnace and a digital optical microscope was used for the experiments [69]. Pure Si (99.9999%) and Ge (99.999%) were mixed with different compositions. Before the experiments, both of the raw materials were cleaned. First, they were ultrasonically cleaned in acetone to remove oil components from the surface. After rinsing with ultrapure water, the samples were dipped into aqueous HF solution to remove the surface oxide layer. Then, the cleaned Si and Ge were put into a silica crucible with a size of 22 × 12 × 11 mm, which was placed into a furnace. The furnace has a window to observe the sample surface by means of the digital optical microscope. The furnace temperature was controlled by two zone graphite heaters. After melting of the raw materials, the temperature was kept for 30 min at higher than the melting temperature to allow the mixing of Si and Ge. Then, the inside of the furnace was cooled to promote solidification.
We performed two experiments as follows. First, we simply cooled the whole melt at 20 • C/min just to observe the growth shape of the dendrites over a wide composition range to investigate the types of dendrites formed. Next, we carried out directional solidification experiments to investigate when dendrite growth appears at the crystal/melt interface during directional solidification. After the melting and mixing processes of the raw materials, we adjusted the temperature difference between the two zone heaters to 20 • C. Then, the temperature was gradually decreased while maintaining the temperature difference, and crystallization was started from one side of the crucible.
The solidification processes in each sample were monitored and recorded by PC. After the solidification, orientation analysis was performed by the SEM-EBSP (Scanning Electron Microscopy-Electron Back Scattering diffraction Pattern) method.

Results and Discussion
We directly observed the growth shape of dendrites in Si 1−x Ge x (0 < x < 1) melts cooled at 20 • C/min. Figure 2a shows a dendrite grown from a Si 0.7 Ge 0.3 melt. For comparison, a dendrite grown from a pure Si melt is also shown in Figure 2b. The growth shapes of the dendrites are similar; the surfaces of the dendrites are surrounded by well-developed facets, and there are no branches on the side of the dendrites like those seen for alloy dendrites (Figure 1a). It was also observed that twin boundaries existed at the center part of the dendrite, as indicated by the arrow in the middle picture in Figure 2a. To confirm this, orientation analysis was performed using the SEM-EBSP method after solidification. Figure 3 shows an SEM image of the dendrite (left figure) and the results of SEM-EBSP analysis performed at the center of this dendrite (right figures). The analyzed area is indicated by a dotted red square in the SEM image. The colored figure shows an orientation map and the gray image below the orientation map shows the grain boundary types as colored lines. It was found that two Σ3 twin boundaries, indicated by red lines, existed in the dendrite. Therefore, it was confirmed that the dendrite observed in Figure 2a was similar to Si faceted dendrites. Similar experiments were conducted over a wide range of composition for Si 1−x Ge x (0 < x < 1) melts. Figure 4 shows dendrites grown from Si 1−x Ge x (0 < x < 1) melts with different compositions. Dendrites could be observed in all samples. In all cases, the growth shapes of dendrites were similar to those of Si faceted dendrites, and not similar to the shape of alloy dendrites. We also confirmed the existence of parallel twin boundaries in the dendrites by SEM-EBSP analysis after the solidification as in Figure 3. Nagashio et al. observed alloy-type dendrites in the solidification of pure Si in a highly undercooled melt at ∆T > 100K. They realized such a highly undercooled melt using a container-less levitation method [17]. Unfortunately, our experimental system is not equipped with a measurement system able to measure the melt temperature directly. Therefore, the degree of undercooling at the initiation of dendrite growth could not be determined. However, one can understand that because we used a crucible, it was difficult to maintain a melt in a highly undercooled state.   As above, it was shown that the dendrites grown in the Si 1−x Ge x (0 < x < 1) melts in a solidification process (in a crucible) were twin-related faceted dendrites, similar to those in pure Si and Ge. In order to grow a faceted dendrite, a crystal must contain twin boundaries. It is known that the stacking fault energies of Si (around 30~50 mJm −2 [70,71]) and Ge (around 50~60 mJm −2 [70,71]) are quite low, and that of Si 1−x Ge x (0 < x < 1) is between these values [71]. Therefore, it appears that twin boundaries are easily generated in the crystal during the solidification, which leads to the appearance of faceted dendrite growth.
Next, we performed directional growth experiments to observe the crystal/melt interface of Si 1−x Ge x alloys for the investigation of the effect of twin boundaries on the shape of the crystal/melt interface. According to the theory of Mullins-Sekerka interface instability [72], the interface shape is determined by the temperature gradient in the melt at the interface, as schematically explained in Figure 5. Figure 5a shows the case where the planar interface is maintained in a positive temperature gradient in the melt at the interface. In this case, when a fluctuation is generated at the planar interface, as in the middle figure of Figure 5a, the growth velocity at the tip of the fluctuation is lower than that at the bottom part due to the lower degree of undercooling, and thus, the bottom part will catch up with the tip and a planar interface is maintained. On the other hand, Figure 5b shows the case where a zigzag faceted interface is formed under a negative temperature gradient. In this case, the growth velocity at the tip of the fluctuation at the interface becomes higher than that at the bottom part, as shown in the middle of Figure 5b, and thus the fluctuation is amplified to form a zigzag faceted interface. In case of non-faceted materials, the interface shape transforms from planar to cellular (or dendritic) under the negative temperature gradient, as also schematically shown in Figure 5b. This transformation of the interface shape from planar to wavy to cellular or zigzag faceted is well known as Mullins-Sekerka instability (instability of the crystal/melt interface) [72]. It was experimentally confirmed in pure Si single crystals [73]. It was also reported that the instability of a crystal/melt interface like Figure 5b is initiated at a portion of the grain boundary in multi-crystalline Si due to the difference in thermal conductivity between the interior of the crystal grain and the grain boundary [74]. We also observed interface instability for Si-rich Si 1−x Ge x (0 < x < 0.1) alloys [64][65][66]. In that case, it was shown that the critical growth velocity for interface instability to occur becomes lower with increasing Ge composition due to constitutional undercooling, and the transformation of the interface shape followed, as shown in Figure 5b. A question remained: if the growing crystal contains parallel twin boundaries and dendrite growth appears during directional solidification, does the dendrite growth appear before the interface instability or after the instability? Since it is difficult to artificially create parallel twin boundaries, we relied on chance and conducted the experiment many times, and were able to observe an interface with parallel twin boundaries. Figure 6a,b show snapshots of directional solidification from Si 1−x Ge x melts. The two melt compositions are almost the same; (a) Si 0.92 Ge 0.08 and (b) Si 0.9 Ge 0.1 . As shown in Figure 6a, the planar interface changed to a zigzag faceted interface due to the interface instability. Similar results for the transformation of the interface shape have been reported in pure Si [73] and Si-rich Si 1−x Ge x [65,66]. In those cases, the growing crystal did not contain parallel twin boundaries. On the other hand, when the growing crystal contained parallel twin boundaries (Figure 6b), dendrite growth appeared at a portion of the parallel twin boundaries, while a planar interface was maintained for the other part. This shows that the dendrite growth appears before interface instability occurs. According to the Mullins-Sekerka theory, as explained in Figure 5, interface instability appears under a negative temperature gradient in the melt at the crystal/melt interface, while a planar interface is maintained under a positive temperature gradient [63], which was experimentally confirmed in single crystal Si [73]. The result observed in Figure 6b shows that the planar interface is maintained in the portion without dendrite growth; thus, the temperature gradient appeared to be positive. This result agrees with our previous report for pure Si crystals containing twin boundaries [75]. Therefore, it was indicated that dendrite growth appears even in a positive temperature gradient when a crystal contains parallel twin boundaries, as shown in Figure 7. In this case, the amount of undercooling at the tip of the dendrite is smaller than that at the planar interface; however, the atomic attachment at the tip of the dendrite is known to be much faster than that at the planar interface due to the formation of a reentrant corner at the dendrite tip [18], and thus a dendrite can be grown.

Conclusions
We investigated dendrite growth in Si 1−x Ge x (0 < x < 1) alloys by in situ observation. Twin-related faceted dendrites appear in melt growth of Si 1−x Ge x (0 < x < 1), which is similar to the case with pure Si and pure Ge. Dendrite growth appears to occur from a planar crystal/melt interface during directional solidification when a Si 1−x Ge x crystal contains a parallel twin boundary. This indicates that a planar interface is not maintained even under a positive temperature gradient when the growing crystal contains parallel twin boundaries.